INDIAN MATHEMATICAL SOCIETY

INDIAN MATHEMATICAL SOCIETY (Founded in 1907; Reg. No. S-550, Delhi) Registered Office: Department of Mathematics, Savitribai Phule Pune University, Pune-411007

http://www.indianmathsociety.org.in NEWSLETTER

NO. 45

MARCH-APRIL  ∑ 2021 π α Ώ

Facsimile of the Commemorative Postage Stamp on the 'Indian Mathematical Society' issued by the Department of Posts (Philately Division, Government of India, to mark the completion of hundred years of the Society. Released on the Inaugural day of the Platinum Jubilee 75th Annual th Conference of the Society on 27 December 2009. CONTENTS

1. A Brief Report of the 86th Annual Conference of the 1 Indian Mathematical Society

2. Minutes of the 86th Annual General Body meeting of the 4 Society

3. Call for Applications for the IMS Awards 11

4. Call for Applications for the IMS Prizes 13

5. Periodicals published by the Society 14

6. Membership of the Society 15

7. Guidelines for acceptance of Donations by the Society 17

8. An Appeal to all members of the IMS 18

9. Abstracts of the Plenary Talks, IMS Memorial Award 19 Lectures and Invited lectures, Talks in Symposia, delivered at the 86th Annual conference of IMS held at The Vellore Institute of Technology (VIT), Vellore, Tamil Nadu, India during December 17-20, 2020

10. Abstracts of the papers received for prizes and for 42 presentation at the 86th Annual conference of IMS held at The Vellore Institute of Technology (VIT), Vellore- 632 014. Tamil Nadu, India during December 17-20, 2020

11. Council for the session 2021-22 110 1

A BRIEF REPORT OF THE 86th ANNUAL CONFERENCE OF THE INDIAN MATHEMATICAL SOCIETY

The 86th Annual Conference of the Indian Mathematical Society- An international Meet was held online under the auspices of The Vellore Institute of Technology (VIT), Vellore (Tamil Nadu) during December 17-20, 2020 under the presidentship of Prof. B. Sury, I.S.I. Bangalore. The Conference was attended by more than 200 delegates, Two presi- dential addresses (General and Technical). Two plenary talks, one by Prof. M. S. Raghu- nathan, Chairman, National Centre of Mathematics, IIT, Bombay, and another by Harald Upmeie, University of Marburg, Germany, Four Memorial Award lectures, Four IMS Award lectures and Seven invited lectures were delivered in the conference. Also, six symposia were organized during the conference and thirty one invited speakers gave talks in the symposia. Moreover, in all 224 research papers were accepted for presentation at the Conference including 19 research papers for the paper presentation competition for various prizes.

The Conference was inaugurated by Prof. M.S. Raghunathan, FRS, Chairman, National Centre of Mathematics, IIT, Bombay. The function was presided over by Prof. B. Sury, ISI Bangalore, the President of the IMS. Dr. A. Mary Saral Dean, SAS, VIT, welcomed the delegates. The General Secretary of IMS, Prof. Satya Deo spoke about the Indian Mathematical Society and on behalf of the Society expressed his sincere and profuse thanks to the host for organizing the Conference. Announcements of IMS prizes and awards were also made by the General Secretary. Prof. Peeyush Chandra, the Academic Secretary of the IMS, reported the academic program of the Conference. The function ended with a vote of thanks by Dr. B. Rushi Kumar, Organizing Secretary, IMS- 2020.

The first Plenary Talk was given by Prof. M. S. Raghunathan, FRS, Chairman, National Centre of Mathematics, IIT Bombay, on “Some Major Indian Contributions to Mathematics in the 20th Century”. Second Plenary Talk was given by Harald Upmeier, University of Marburg, Germany, on “The Toeplitz Operators and Hilbert Modules on Bounded Symmetric Domains”

The 34th P. L. Bhatnagar Memorial Award Lecture was delivered by Prof. Neela Nataraj, Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, on “Lower-order Nonstandard Finite Element Methods for Biharmonic Plates”.

The 31st V. Ramaswami Aiyar Memorial Award Lecture was delivered by Prof. Sanoli Gun, Institute of Mathematical Sciences, Chennai, on “On bounds of Fourier- coefficients of Half-integer Weight Cusp Forms”.

The 31st Srinivasa Ramanujan Memorial Award Lecture was delivered by Prof. Anish Ghosh, School of Mathematics, TIFR Bombay, Mumbai on “The Unreasonable Effectiveness of Ergodic Theory in Number Theory ”.

The 31st Hansraj Gupta Memorial Award Lecture was delivered by Prof. Dinesh Khurana, Department of Mathematics, Panjab University, Chandigarh on “Some Glimpses into Noncommutative Ring Theory”.

Satish Bhatnagar Award Lecture was delivered by Prof. M. D. Srinivas, Centre for Policy Studies, Chennai on “Pandiagonal Magic Squares: From N¯ag¯arjunaTo N¯ar¯ayan. a Pan. d. ita To Vijayaraghavan”.

A. K. Agarwal Award Lecture was given by Jagmohan Tanti, Central University of Jharkhand, Ranchi, Jharkhand, India on “Euler’s Criterion for lth Power non residues 2 with l a Prime”.

A. M. Mathai Award Lecture was given by P. Prakash, Dept. of Mathematics, Amrita Vishwa Vidyapeetham, Coimbatore, India, on “Invariant Subspaces and Exact Solutions of Nonlinear PDEs”.

A. Narasinga Rao Prize Lecture was given by R. B. Yadav, Dept. of Mathematics, Sikkim University entitled “On Some Categories of Riemannian Manifolds”.

Various prizes for the Paper Presentation Competition:

For the IMS prizes 19 papers were received : two in Group 1, six in Group 4, two in group 5, two in group 6, two for AMU Prize and five for V. M. Shah Prize. No paper was received for Group 2 and 3. The papers were presented in the competition section. The following is the result for the award of these prizes. IMS Prize - Group-1: Two paper was presented in this group. The prize was awarded to Ranganatha Dasappa, Department of Mathematics, Central University of Karnataka, Kalaburagi-585367, Karnataka, India. IMS Prize - Group-2: No paper was received in this group IMS Prize - Group-3: No paper was received in this group. IMS Prize - Group-4: Six papers was received and presented. The prize was awarded to Manisha Chowdhury, Indian Institute of Technology, Kanpur, Uttar Pradesh, India. IMS Prize - Group-5: Two papers were received in this group. The prize was awarded to P. K. Sahu Department of Mathematics, Government Shyama Prasad Mukharjee College, Sitapur -497 111, Chhattisgarh, India IMS Prize - Group-6: Two papers were received in this group. However, the prize was not awarded to anyone. A. M. U. Prize: Two papers were received in this group and the prize was awarded to Pranav Sharma, Department of Mathematics, Lovely Professional University, Punjab. V. M. Shah Prize: Five papers were received and presented. The prize was awarded to Rishikesh Yadav, Applied Mathematics and Humanities Department, Sardar Vallabhbhai National Institute of Technology, Surat, Gujarat.

Invited Lectures delivered (1) Prof. S. P. Tiwari, IIT Dhanbad: Automata Theory Based on Residuated Lattices

(2) Prof. Shakir Ali, Aligarh Muslim University, Aligarh: Jordan ∗-derivations and Related Maps in Rings

(3) Prof. K. S. Charak, Jammu University, Jammu: Normal Families of Holomorphic Functions of Several Complex Variables

(4) Dr. Pradip Majhi, Calcutta University, Kolkata: Cotton Solitons within the Framework of Almost Kenmotsu 3-h-Manifolds

(5) Prof. Samares Pal, University of Kalyani, Kalyani: Catastrophic Changes in Coral Reef Dynamics under Macroalgal Toxicity, Overfishing and Invasion of Predators

(6) Prof. S. Saravanan, Bharathiar University, Coimbatore: Sharp Nonlinear Stabil- ity Limits for Centrifugal Convection in Porous Media 3

Symposia organized

Six symposia were organized and the details are as follows.

(1) Chandrasekharan centenary symposium in Number theory Convener : Prof. K. Srinivas, IMSc, Chennai.

(2) History of Indian Mathematics Conveners :Prof. M. S. Sriram, Prof. K. V. Sarma Research Foundation.

(3) Industrial Mathematics: Modeling, Optimization, Simulation Convener : Prof. S. Sundar, IIT Madras.

(4) Graph Theory and Combinatorics (in honour of S S Shrikhande) Convener : Prof. S Sane, CMI, Chennai.

(5) Topology and Geometry Convener : Prof. Parameswaran Sankaran, CMI, Chennai.

(6) Biomechanics Convener : Prof. B. V. Rathish Kumar, IIT Kanpur. 4

MINUTES OF THE 86th ANNUAL GENERAL BODY MEETING OF THE INDIAN MATHEMATICAL SOCIETY -2020

The Annual General Body Meeting of the Indian Mathematical Society-2020 was held on Sunday, the 20th December, 2020 at 12 Noon in Online mode from VIT, Vellore using zoom meeting app. The meeting was presided by the President of the IMS Prof. B. Sury.

At the beginning of the meeting the members of the General Body offered condolences for the following members of the IMS who passed away very recently: (a) Prof. A. M. Vaidya from Gujarat who served the Society as Editor of Math Student, (b) Prof. P. V. Arunachalam, who was the President of IMS during the year 2001-2002.

Then the President of the IMS welcomed all the members present and the following business was transacted:

Item No. 1: To confirm the Minutes of the General Body meeting held on November 25, 2019 at IIT Kharagpur, Kharagpur.

No comments or queries were received and so the Minutes of the General Body meeting held on November 25, 2019 at IIT Kharagpur, Kharagpur were confirmed.

Item No. 2: To Receive the Report of the General Secretary.

The Report of the General Secretary. 1. The IMS newsletters No. 43 and No. 44 were published in April 2020 and in August 2020, respectively. These were also uploaded on the website of the Indian Mathematical Society. The soft copies of these newsletters have been sent by e-mails to all the life members of the Society. Letters to the newly elected IMS president and three council members whose terms start w.e.f. April 1, 2020 were sent to them and their acceptances were received.

2. The meeting of the Academic Planning Committee (APC) for the IMS Conference 2020 to be held at VIT, Vellore was held in online mode due to Covid-19 pandemic on Sunday, the 10th June 2020 from 11.00 A.M. The meeting was presided over by the President of the IMS Prof. B. Sury of ISI, Bangalore. The names of the four memorial award lecturers, plenary speakers, invited speakers, list of symposia and their conveners were finalized. The Academic Secretary, in consultation with General Secretary, has completed the job of contacting all the speakers, inviting them for all the talks and finalizing the full academic programme of the conference.

3. A. Narasinga Rao Memorial Prize for the year 2020: Dr. R. B. Yadav of ISI, Tezpur has been awarded the Narasinga Rao memorial prize for his paper enti- tled On Some Categories of Riemannian Manifolds published in the JIMS 86(1-2) (2019), 199-210.

4. A. K. Agarwal Award for the year 2020: Dr. Jagmohan Tanti, Central University of Jharkhand, Ranchi has been awarded this prize for his paper Eulers criterion for prime orders in PID case published in Acta Arithmetica, Nov 2019.

5. A. M. Mathai Award for the year 2020: Dr. P. Prakash, Amrita Vishwa Vidyapeetham, Coimbatore, has been given this award on his paper New ex- act solutions of Generalized convection-reaction-diffusion equation published in European Physical Journal Plus (2019), 131-261.

6. Satish Bhatnagar Award for the year 2020: Prof. M. D. Srinivas, Formerly Professor, Department of Theoretical Physics, University of Madras, Chennai, has 5

been awarded this prize for his paper “The VAsanAbhASyas of BhAskarAcArya” pp. 249 -290 published in the book Bhaskara Prabha, Edited by K. Ramasubra- manian, Takao Hayashi and Clemency Montelle, and published by the Hindustan Book Agency, New Delhi, 2019 and Springer, 2019.

7. P. L. Bhatnagar Memorial Prize for the year 2020: This year the IMO compe- tition was held online but India did not participate in the competition because of the various conditions imposed by the IMO organizers due to the Covid-19 pandemic and so the prize has not been awarded to anybody.

8. Digitization of the back volumes of JIMS published from the year 1976 to the year 2005 has been completed by the Informatics Publishing Ltd., Bangalore. The digitized volumes are available online from the platform of Informatics India Pvt Ltd. These volumes were retrieved by Prof. Shikare from the IMS Library housed at Ramanujan Institute for Advanced Study in Mathematics, Madras Uni- versity, Chennai and the library of Pune University, Pune. The usual guidance and help has been provided to Prof. Rushi Kumar, Local Organizing Secretary of the 2020 IMS Conference-An International Meet, per- taining to various works like local arrangements, invitations, website, inaugural function of the conference etc.

10. The complete catalogue of the back volumes of the periodicals published by the Society as well as those received in exchange by the Society and available in the IMS Library, Chennai, is now available on the IMS website.

11. Registration of the IMS plot of land in Pune (about 7 km East of the Pune Air- port): The preliminary work for this important job was done by Prof. B. N. Waphare, Prof. M. M. Shikare and the treasurer Prof. S. K. Nimbhorkar. The registry of the land was completed in the presence of myself, Prof. Waphare, Prof. Shikare, Prof. Nimbhorkar on Feb 10, 2020 in Pune.

12. Preparing a Master Plan for the IMS campus on the IMS plot of land: Prof. Satya Deo invited views from all the Council Members about the possible layout of the campus and finalized the Master Plan to be developed. Summarizing all the views, a consolidated plan has been given to Prof. N. K. Thakare, who has very kindly agreed to get the Master Plan prepared by an eminent architect in Pune. The work is in progress. Acknowledgements

The General Secretary thanks Prof. Waphare and Prof. Shikare for drafting the IMS newsletters 43 and 44 and sending them to me for finalization. The Pune team of the IMS office bearers deserve thanks for their generous help in procuring the IMS plot which we have now purchased. Prof. Waphare is specially thanked for providing a loan of about 40 lacs for the purchase of the IMS plot. Prof. J. R. Patadia is thanked for maintaining and updating the website of the IMS from time to time

Item No. 3: To Receive the Report of the Academic Secretary

The Report of the Academic Secretary.

The 86th Annual Conference of Indian Mathematical Society an International Meet was held in virtual mode on Zoom platform under the auspices of VIT Vellore during Dec 17 - 20, 2020. During the conference two plenary talks, four Memorial award lectures and seven invited talks were delivered. The details are as follows:

Plenary Talk: (i) Prof. M. S. Raghunathan, CEBS, Mumbai 6

(ii) Prof. Harald Upmeier, University of Marburg, Germany

34th P. L. Bhatnagar Memorial Award lecture: Prof. Neela Nataraj, IIT Bombay, Mumbai 31st Srinivasa Ramanujan Memorial Award Lecture: Prof. Anish Ghosh,TIFR Mumbai 31st V. Ramaswami Aiyer Memorial Award Lecture: Prof. Sanoli Gun, IMSc Chennai 31st Hansraj Gupta Memorial Award lecture: Prof. Dinesh Khurana, Panjab Uni- versity Chandigarh.

One hour Invited Talks: Prof. Ravi P. Agarwal , Texas A & M University, USA

Half-hour invited talks: (i) Prof. S. P. Tiwari, IIT Dhanbad (ii) Prof. Shakir Ali, Aligarh Muslim University, Aligarh (iii) Prof. K. S. Charak Jammu University, Jammu (iv) Dr. Pradip Majhi, Calcutta University, Kolkata (v) Prof. Samares Pal, University of Kalyani, Kalyani (vi) Prof. S. Saravanan, Bharathiar University, Coimbatore

Six Symposia were conducted of one was in the memory of Prof. S. S. Shrikhande and another was to celebrate the birth centenary of renowned number theorist Prof. K. S. Chandrasekharan. The convenors of symposia identified speakers from India and abroad. All the symposia were conducted well. The detail of symposia are as follows: Symposia: (i) Chandrasekharan centenary symposium in Number theory - Prof. K. Srinivas, IMSc (ii) History of Indian Mathematics Prof. M. S. Sriram, Prof. K. V. Sarma Research Foundation (iii) Industrial Mathematics: Modeling, Optimization, Simulation - Prof. S. Sundar, IIT Madras (iv) Graph Theory and Combinatorics (in honour of S. S. Shrikhande) - Prof. S. Sane, CMI (v) Topology and Geometry - Prof. Parameswaran Sankaran, CMI (vi) Biomechanics - Prof. B. V. Rathish Kumar, IIT Kanpur

Apart from it the awardees of Narasinga Rao Prize, A. K. Agrawal Award, A. M. Maathai award and Satish Bhatnagar Award also delivered their talks. There was a great response for Competition Section as well as for Contributory papers. In all 200 papers were presented in the conference. A session on Funding Opportunities for Mathematics was also conducted.

All the sessions were conducted well. I wish to place on record my sincere thanks to all the speakers, the chairpersons of various sessions and the Local Organizing Secretary, Prof. Rushi Kumar for the success of the conference. 7

Item No. 4: To consider the Audited Statement of Accounts for the year 2019 - 2020 and budget for the Financial year 2021 - 2022.

The Audited Statement of Accounts for the year 2019 - 2020 and the budget for the Financial year 2021 - 2022 presented by the Treasurer, Prof. S. K. Nimbhorkar were ap- proved.

Item No. 5: To Receive the Report of the Editor, The Journal of the Indian Mathemat- ical Society for 2020.

Report of the Editor, the Journal of the Indian Mathematical Society for 2020.

Manuscript status:

(a) Number of manuscripts pending with the referees/ under process at the 15 end of 2018 : (b) Number of manuscripts received during 2019: 74 —— Total: 89

(a) Number of manuscripts accepted in 2019: 16 (b) Number of manuscripts not accepted in 2019 36 (c) Number of manuscripts with the referees/ under process: 33 (d)Number of manuscripts with authors for final revision: 04 ——– Total: 89

Publication Status (online by the IPL) (a) Volume 87 (1-2) 2020 of the JIMS published in March 2020. (b) Volume 87 (3-4) 2020 of the JIMS published on July 2020. (c) Volume 88 (1-2) 2021 of the JIMS is expected to be published online in Jan 2021. Publication Status (Print by the IMS) (a) Volume 87 (1-2) 2020 of the JIMS was published in August 2020 and has been sent to the subscribers. (b) Volume 87 (3-4) 2020 of the JIMS was also published in August 2020 and was sent to the subscribers then. There was a shortfall of some copies and so a reprinting was done. The copies were received a few days ago and in December 2020 and are being sent to the remaining subscribers by the Administrative Secretary, IMS. (c) Volume 88 (1-2) 2021 of the JIMS is expected to be published in February 2021.

Acknowledgements:

The Chief Editor, JIMS expresses his sincere thanks to numerous referees and the members of the Editorial Board, JIMS, especially Prof. Peeyush Chandra and Prof. Ravi Agarwal, for their help in refereeing and/or processing the manuscripts received for JIMS. He also records his sincere thanks to Prof. B. N. Waphare for extending all possible help in printing and dispatching copies of the JIMS to the subscribers of the Journal. He is also very grateful to Prof. Satya Deo for able guidance and to Prof. Shriram Nimbhorkar for his help in managing various matters related to the JIMS. Thanks are also due to Ms. Poornima of IPL for kind cooperation. 8

Item No. 6: To Receive the Report of the Editor, The Mathematics Student for 2020.

Report of the Editor of the Mathematics Student for 2020.

Publication Status of the Journal

The Volume 89 (Nos. 1-2) January-June, 2020 of The Mathematics Student was pub- lished in June 2020. The soft copy of the issue is sent to all the life members of the IMS by e-mail. The Volume 89 (Nos. 3-4) July-December, 2020 of The Mathematics has also been published in October 2020. The soft copy of this issue has been sent to the life members of the IMS. The soft copies of both the issues have been uploaded on the website of the Indian Math- ematical Society. Both the volumes have been printed out by Parshuram Process, Pune and the hard copies have been preserved in the Library of the Department of Mathematics, Savitribai Phule Pune University, Pune.

Status of the Manuscripts

In all 131 research papers were received by the Chief Editor for publication in the Mathematics Student till December 10, 2020. Out of these 131 papers, 18 papers have been accepted for publication in the journal, 36 papers were not accepted, 32 papers are pending with the referees, 5 papers were withdrawn by the authors and 40 papers are under process.

Status of the Journal

The University Grants Commission (UGC) has now included The Mathematics Student in the list of the UGC approved journals. The journal has been also included in the list of Scopus Indexed Journals. Scopus is Elseviers abstract and citation database launched in 2004. These facts will certainly help the journal to boost its popularity and reputation.

Digitization of Back Volumes

We propose to digitize the back volumes of the Mathematics Student with the help of Informatics Publishing Limited, Bangalore. The back volumes of the journal can be made available from the IMS library, Chennai and the Library of the Savitribai Phule Pune University, Pune.

Acknowledgments

We take this opportunity to put on record our sincere thanks and profuse gratefulness to the Members of the Editorial Board and the learned referees for their continuous sup- port and assistance in our sustained efforts for timely publication of The Mathematics Student. We would like to thank Prof. George E. Andrews, Prof. M. Ram Murty and Prof. B. Sury for proposing Problems for the Problem section, verifying solutions received from researchers and providing solutions to unsolved problems.

We are grateful to Prof. J. R. Patadia for extending his help to upload the soft copies of the volumes on the website of The Indian Mathematical Society. We thank the General Secretary Prof. Satya Deo and the Treasurer Prof. S. K. Nimbhorkar for reading the camera ready copies of the issues carefully and providing their inputs. We express our gratitude to the Administrative Secretary Prof. B. N. Waphare for getting the issues printed from Parshuram Process, Pune, sending the hard copies of the journal 9 to the subscribers and arranging to preserve the hard copies of the journal in the library of the Mathematics Department of S. P. Pune University, Pune.

Item 7: To Consider the Venue of the 87th Annual Conference of the Society to be held in December 2021.

The General Secretary Prof. Satya Deo received the proposal from the Vice Chancellor of the Mahatma Gandhi Mission University, Aurangabad (Maharashtra) for organizing the Annual conference of the IMS during December 2021. The Council, after discussion, accepted the proposal. So the venue for the Annual Conference to be organized in De- cember 2021 is Mahatma Gandhi Mission University, Aurangabad (Maharashtra).

Item 8: Announcement of the results of the following elections. (i) President for 2021-2022; (ii) Three members of the Council for a period of three years w. e. f. April 01, 2022.

(i)The Office of the President :

The Council nominated Prof. Dipendra Prasad, IIT Bombay, Mumbai for election to the Office of the President of the Society with effect from April 01, 2021. No other nominations were received from the members of the IMS for election to the office of the President. Therefore no election was held and Prof. Dipendra Prasad is declared elected unopposed to the office of the President of the IMS for one year with effect from April 01, 2021.

(ii) Election of members to the Council of the Society :

The Council made the following nominations for the election of the Council members of the Society for three years with effect from April 01, 2021. 1. Prof. Veeramani, IIT Madras, Chennai. 2. Prof. Pankaj Jain, South Asian University, Delhi 3. Prof. Jitendra Kumar, IIT Kharagpur, Kharagpur. No other nominations were received from the members of the IMS for election of the Council members. Therefore no elections were held and the above members are declared elected unopposed as Council members of the IMS with effect from April 01, 2021

Item 9. Any other item with the permission of the chair.

No other item was submitted for consideration.

The efforts put in by the Local organizing Secretary Prof. Rushi Kumar, his colleagues and the Vellore Institute of Technology administration for successful organization of the conference were appreciated by the office bearers and the members of the IMS. The Meeting ended with a vote of thanks to the chair, members of the IMS who were present, Prof. Rushi Kumar, his colleagues and the Vellore Institute of Technology ad- ministration.

Satya Deo General Secretary Indian Mathematical Society 10

IMS Sponsored Lectures

To popularize mathematics and to create awareness regarding the Society and its ac- tivities in the Country, the Society has a Scheme of Sponsored Lectures. It provides a token support of Rs. 1000/- to a number of Departments / Institutions for organizing popular and semi technical lectures.

Prof. Ravi Kulkarni has donated Rs. 1,25,000/- to organize Meenakshi Sundaram- Patoudi lectures.

Members arranging such lectures are required to send the report of the arranged lec- tures to The Treasurer, IMS, with a copy to The Editor, The Mathematics Student.

Society intends to enhance this activity of organizing such lectures at more and more centers. Members desirous to organize such lectures at their centers may write to the General Secretary Prof. Satya Deo through their respective Head of the Department. 11

Call for Applications for Various Awards to be given by the IMS for the Year 2021

Applications are requested from researchers in mathematics for the following Awards to be given by the Indian Mathematical Society for the year 2021. The last date for re- ceiving the applications is June 30, 2021. The applications should be sent to Prof. Satya Deo, the General Secretary of the IMS, along with the copy of the published paper and the proof of the age on his e-mail address : [email protected]. Papers published in paid journals will not be considered.

(1) A. K. Agarwal Award

Terms and Conditions for the Award: (a) The paper should be in the area of Number theory, Combinatorics, Discrete mathe- matics, Analysis and Algebra. (b) The paper should be in single authorship. In exceptional cases, the paper by two au- thors could be considered. In this event the prize amount will be equally divided (between the two authors). (c) The upper age limit is 45 years as on 31st December 2021. In case of two authors both must be below 45 years. (d) The papers considered must have been published either online or in print version dur- ing the year 2020. The paper must have been published in an internationally refereed journal. More weight age could be given to well established journals. (e) The author(s) should be Indian citizen and must have carried out the said research work in India. (f) The author(s) should not submit more than one publication for this award. (g) The prize carries a certificate and a cash amount up to Rs. 10,000/- depending on the rate of interest on the accrued amount.

(2) A. M. Mathai Award

Terms and Conditions for the Award : (a) The paper should contain significant original contribution in any branch of mathemat- ics which has some applications in other fields such as Physical Sciences, Biological and Medical Sciences, Social Sciences, Probability and Statistics. (b) The paper should be a single-author paper. The paper sent for this award should not have been submitted or rejected for any other award. (c) The upper age limit is 35 years as on December 31, 2021. (d) The author should not submit more than one publication for this award. (e) The papers must have been published either online or in print version during the year 2021. The paper must have been published in an internationally refereed journal. More weightage could be given to well established journals. The papers on routine generaliza- tions, computations of formulae without proper analysis should not be considered for this award. (f) The author should be associated with any university/college/ institution in India where the work was done and the paper must have a mention of the name of that institution as affiliation (the person need not be an Indian citizen). (g) The award carries a certificate and cash amount up to Rs 25,000/- depending on the rate of interest on the accrued amount. 12

(3) Prof. Satish C. Bhatnagar Award

Terms and Conditions for the Award: (a) The paper must be in the area of History of Mathematics focusing on a person, prob- lems, region, system of education or government. (b) The paper should be in single authorship. In exceptional cases, the paper by two authors is considered. In this event, the prize amount will be equally divided. (c) The minimum age limit for this award is 35 years as on 31st December 2021. There are no limits on the citizenship of the applicant. (d) In the case of joint authorship of the awarded paper, the prize amount will be equally divided between the two authors. (e) The author(s) need not be Indian citizen(s) and must have a Ph D degree in any subject. (f) The author(s) can submit only one publication for this award and the paper should not have been submitted for any award anywhere. (g) The award consists of a citation and a cash prize of Rs. 10,000/-.

(4) Subhash Bhatt Award

Terms and Conditions for the Award: (a) The paper should be in the area of Functional Analysis/ Harmonic Analysis/ Operator Theory and related areas. (b) The paper should have been published either online or in print in an internationally reputed journal during the previous year to the year of Annual Conference in which award is to be conferred e.g. for the award to be given in the Annual Conference of the year 2021, the papers published in 2020 will be considered. (c) The paper should be under the authorship of at most two authors and at least one of them should be below the age of 45 years as on 31st December of the year of the award. (d) In the case of joint authorship of the awarded paper, the prize amount will be equally divided between the two authors. (e) The author(s) should be Indian citizen and must have carried out the said research work in India. (f) The author(s) can submit only one publication for this award and the paper should not have been submitted for any other award anywhere. (g)The award carries a citation and a cash prize of Rs 25,000.

(5) P. K. Jain Award

Terms and Conditions for the Award: (a) The paper should be in the area of Functional Analysis/ Harmonic Analysis/ Function Theory and related areas. (b) The paper should have been published either online or in print in an internationally reputed journal during the previous year to the year of Annual Conference in which award is to be conferred e.g. for the award to be given in the Annual Conference of the year 2021, the papers published in 2020 will be considered. (c) The paper should be under the authorship of at most two authors and at least one of them should be below the age of 45 years as on 31st December of the year of the award. (d) In the case of joint authorship of the awarded paper, the prize amount will be equally divided between the two authors. (e) The author(s) should be Indian citizen and must have carried out the said research work in India. (f) The author(s) can submit only one publication for this award and the paper should not have been submitted for any other award anywhere. (g)The award carries a citation and a cash prize of Rs 25,000. 13

Call for Research Papers for Various Prizes to be given by the IMS during the Annual Conference of the IMS in 2021

In order to encourage and inspire the young and budding researchers in mathematics, the IMS organizes a Special Session of Paper Presentation Competition during its Annual Conferences for various Prizes to be awarded to the best research paper presented in dif- ferent categories. This Special Session is organized as a part of the Academic Programme with no other parallel session. Each of the eight prizes listed below carries a Certificate and a Cash Amount of Rs. 1000/-

Interested researchers should submit their research paper (in pdf format), Abstract (not exceeding 250 words, in tex and pdf format), proof of age and CV along with the covering letter to The Academic Secretary, IMS via e-mail on [email protected].

The last date of receiving applications is August 15, 2021

The details of the prizes, groups and areas are as follows: (1) A. M. U. Prize: Algebra, Differential Geometry and Functional Analysis. (2) V. M. Shah Prize: Real Analysis, Complex Analysis, Fourier Analysis, Har- monic Analysis, Approximation Theory and related areas. (3) IMS Prize-group-1: Discrete Mathematics (Combinatorics, Graph Theory, Posets), Lattice Theory, Set Theory, Logic, Number Theory and related areas. (4) IMS Prize-group-2: Geometry, Algebraic Geometry, Topology, Algebraic Topol- ogy, and related areas. (5) IMS Prize-group-3: Measure Theory, Probability Theory, Stochastic Pro- cesses, and related areas. (6) IMS Prize-group-4: Differential / Integral / Functional equations and inequal- ities, Special Functions, Numerical Analysis and related areas. (7) IMS Prize-group-5: Solid Mechanics, Fluid Mechanics, Electromagnetic The- ory, Magneto- Hydrodynamics, Astronomy, Astrophysics, Relativity and related areas. (8) IMS Prize-group-6: Operations Research, Optimization, Computational Math- ematics, Information Technology, Bio mathematics, History of Mathematics and related areas.

Terms and Conditions for the applicants to participate in the Competetion:

1. Only the Members of the Society are eligible for participation in the Competition.

2. The upper age limit of a candidate is 40 years as on December 31, 2021.

3. (i) The paper to be presented for the competition has to be under single authorship. (ii) The author should give a declaration that the work is unpublished and has not been submitted for competition anywhere else. In case of research scholars, the supervisor should verify that the work has been carried out independently. (iii) The work must have been carried out in India. 14

Periodicals published by the Society

The Society publishes two periodicals: The Journal of the Indian Mathematical Society (JIMS; the Journal; Print ISSN 0019-5839, Online ISSN 2455- 6475 ) and The Mathematics Student (Math Student; the Mathematics Student; Print ISSN 0025-5742), both of which are quarterly. The details can be found on the website: www.indianmathsociety.org.in

Subscriptions Annual subscription for the Journal / the Mathematics Student : For each periodical • Rs. 2500/- for Libraries of Educational Institutions in India - provided the sub- scription is direct. If an agent subscribes for an educational Institution in India, the subscription is Rs. 3000/- • Rs. 12000/- for others for personal use or to the agents who do not supply the name and address of the end user. • $200/- for personal use or for Libraries outside India. The agents are entitled to 15 % discount on their orders.

From the 2012 issue of The Mathematics Student onwards, the life Members are given online access to The Mathematics Student / are sent the soft copy of The Mathematics Student, instead of supplying the hard copy, for their personal use (not for circulation) at their E-mail address registered with the Society.

Those Members who have not registered their e-mail address are requested to register it online on [email protected]

It may please be noted that the contents of The Mathematics Student will continue to be available on the Society’s website www.indianmathsociety.org and a physical copy of The Mathematics Student will continue to be available at the IMS Library (Ramanujan Institute of Advanced Study in Mathematics, Madras University, Chennai) as well as at the Registered Office of the Society (Department of Mathematics, Savitribai Phule Pune University, Pune 411 007) for reference during office hours. 15

IMS MEMBERSHIP DETAILS

1. Membership terms:

1. Applicant should be a graduate and should have interest in the Objectives of the IMS. 2. All such persons as the Council of the IMS may admit from time to time to membership shall be the members of the Society. 3. Applications for membership should be made on the form available on the IMS website. 4. The Council of the IMS may refuse to admit to membership any person without as- signing any reason for the refusal.

2. There are three types of members of the Society:

1. Life Members: Any eligible person can be enrolled as a life member by applying on the prescribed form and by paying the membership fees as prescribed from time to time by the IMS. 2. Annual members: Any eligible person can be enrolled as an annual member by applying on the prescribed form and by paying the annual membership fees as prescribed from time to time by the IMS. This membership will come to end on March 31, irrespective of the date of paying the membership fees. 3. Sessional Members: Any person desirous of participating in the Annual Conference of the IMS will be enrolled as a sessional member by paying the membership fees as pre- scribed from time to time by the IMS. This membership is only for participating in one conference after paying the fees.

3. Rights and Obligations of the Members:

1. All Annual and Life members shall be entitled to receive communications about the activities of the Society, to participate in its conferences. 2. A member with good standing shall attend the General Body Meeting and will be eligible to vote if necessary. 3. Only life members will be eligible to be elected as a member of the IMS Council At present anybody has free access to the Mathematics Student on IMS website.

4. Membership Fees:

With effect from January 1, 2021, the membership fees is as follows i) Life Membership fee - For Indian citizens Rs. 3000/- For others US $150/- For members of the Societies having Reciprocity arrangement with the IMS,US $ 100/- For members form SAARC countries - US $ 50/- ii) Annual Membership - Rs. 500/- (US $25/- for foreigners). iii) Sessional Membership- Rs. 300/- (US $15/- for foreigners).

The membership fees can be paid either online or through DD/ payable at par Cheque. We prefer to have Online transfer of membership fee for which bank details are given as follows:

Name of the Account Holder : Indian Mathematical Society. Account No. : 0981000100312287 Name of the Bank : Punjab National Bank. IFSC Code : PUNB 0375900 Branch Name & Address : Adalat Road Branch, Aurangabad-431001.

For Payment through DD or Cheque payable at par it should be in the name of Indian Mathematical Society. In case, the demand draft is purchased from the State Bank of 16

India, please use the branch code for Aurangabad, Maharashtra, as 1716.

5. Application for Membership: Application for membership should be made on the Mem- bership form available on the IMS website.

The completed membership form can be sent to Prof. S. K. Nimbhorkar, Treasurer, IMS, C/o Ankur Hospital, Tilaknagar, Aurangabad 431001.

Alternatively, one can send scanned copy of completed form to Prof. S. K. Nimbhorkar by email at [email protected] (or [email protected])

Business Correspondence and Payments:

All business correspondence be addressed to Prof. S. K. Nimbhorkar, Treasurer, IMS; c/o Dr. Mrs. Prachi Kulkarni, Ankur Hospital, Tilaknagar, Aurangabad 431001. All payments should be sent to Prof. S. K. Nimbhorkar, Treasurer, IMS by DD / payable at par cheque drawn in favor of “The Indian Mathematical Society” payable at Aurangabad (Maharashtra), India at the address mentioned in the above.

IMS Library:

The information pertaining to the IMS library is available on the website www.indianmathsociety.org.in of the society. 17

Guidelines for acceptance of Donations to the Society

There will not be any further institution of Memorial Award Lectures. (This point was discussed in the earlier meetings of the Council and such was the consensus). The donation amount will not be less than Rupees Five Lacs. (There could be an upward revision of this amount from time to time). The donor may be an individual or a trust or a group of individuals. The Indian Mathematical Society will solely and independently own the amount donated to it. A prospective donor should approach the General Secretary of the Indian Mathematical Society with an Offer. Keeping with the spirit of this Policy Guidelines and if so felt nec- essary, referring to the Council whether the proposal be negotiated or not, in his wisdom, the General Secretary will negotiate the terms and conditions for each donation proposal and will put it before the Council for its consideration and approval. The Council will de- liberate on the proposal, and after modifications, if any, may accept the proposal through a special resolution with specific details mentioning the terms and conditions. This will be published in the IMS Newsletter after the Donor agrees to the resolution of the Council. Ordinarily during every Annual Conference of the Society there are several Invited Lec- tures and Symposia running in parallel sessions. One of these academic programmes may be permanently marked / identified as so and so sponsored programme in the (fond) memory of or so and so sponsored programme in the honor of as per the wish of each donor by the Council. This programme may be arranged in a parallel session during the Conference. The Council through its Academic Planning Committee (APC) will be the final authority in this regard to finalize the name of a speaker of an invited talk or the names of the Symposia speakers for this sponsored programme. The modus operandi for identifying the speaker(s) may be decided by the Council. The invited speaker(s) will be the guest of the host institution. In case of an honorarium, if any, to the invited speaker, the amount of the honorarium will not exceed the honorar- ium amount for the existing Memorial Award Lectures. Ordinarily train travel to the extent of AC-2 Tier be reimbursed. However, in special cases the domestic air travel may be considered. Not withstanding the above, (A) An offer of a donation with a stipulated purpose (not as part of the corpus), may be accepted by the Council on its merit. (B) An offer of a donation of any amount in general, without any stipulated conditions, may be accepted by the Council on its merit as a part of the General Purpose Corpus. The Council reserves its right whether or not a particular donation be accepted. 18

An Appeal to all members of the IMS

As a part of the “Green Initiative”taken by the Society (for further details, refer So- ciety’s website www.indianmathsociety.org.in), the Council of the Society has decided to send online the soft copy of the Mathematics Student / give online access to the Math- ematics Student to all the life members instead of supplying the hard copy. For this purpose, all the members of the Society are requested to register their e-mail address on- line, along with Name and the Unique Membership Number therein, to Prof. J. R. Patadia on e-mail [email protected] or [email protected] so that further necessary action can be taken. Important Change: Beginning from the 84th annual conference of the IMS the abstract of the papers accepted for presentation and invited talks etc are being published in our newsletter and this policy will continue in future also. Such abstracts will no longer be published in the Mathematics Student. 19 Abstracts of the papers presented at the 86th IMS Conference, VIT, Vellore, Tamil Nadu Plenary Talks Some Major Indian Contributions to Mathematics in the 20th Century by M.S. Raghunathan, Centre for Excellence in Basic Sciences, Mumbai

During the 20th century, Indian mathematicians working in India have made highly sig- nificant contributions to diverse mathematical fields. The first notable contribution ante- dates Srinivasa Ramanujan. There were quite a few works of note in the pre-independence period, but the mid fifties ushered in a new era when Indian mathematicians’ contribu- tions became a considerable influence in the very evolution of certain fields. In this talk I will describe some high points of Indian mathematics during the last century. The choice of material I will speak on is necessarily mostly limited to areas of mathematics I am familiar with; so will be far from comprehensive in its coverage. Even in the areas I have some acquaintance with, my emphasis will naturally be dictated by my taste. Also I will not elaborate much on work by Indian mathematicians working outside India nor on work done in the present century. I do not have the same kind of familiarity with work done in the last two decades that I have with the earlier work.

The Toeplitz Operators and Hilbert Modules on Bounded Symmetric Domains by Harald Upmeier, University of Marburg, Germany [email protected]

Bounded symmetric domains D are domains of holomorphy in Cn which can be realized as homogeneous spaces D = G/K, where G is a semi-simple Lie group and K is a maximal compact subgroup. They play a fundamental role in representation theory of G and the theory of automorphic functions for discrete subgroups of G. The (scalar) holomorphic se- ries of representations discovered by Harish-Chandra can be realized as weighted Bergman spaces of holomorphic functions on D. These Bergman spaces Hν , depending on a scalar parameter ν, carry an important class of operators, the so-called Toeplitz operators, which are related to the geometric quantization program, viewing D as a non-compact K¨ahler manifold.

In the talk we present two recent results concerning Toeplitz operators which are in- variant under the maximal compact subgroup K. By the Peter-Weyl decomposition of the Bergman spaces, these operators are characterized by their eigenvalues, labeled by all integer partitions m1 ≥ m2 ≥ · · · ≥ mr of length r = rank(D). Using deep results in the combinatorics of partitions (Young diagrams) we determine these eigenvalues for a certain class of “fundamental” partitions, revealing an interesting interplay between the covariant and contravariant quantization method.

In the second part of the talk we consider Hilbert submodules of the weighted Bergman spaces and construct the associated eigenbundle, which classically is spanned by the re- producing kernel functions, but in our general situation is not a line bundle any more but a complex linear fibre space in the sense of coherent analytic sheaves. The fibres of the bundle are identified with the space of sections on a certain Peirce manifold, generalizing projective space and the Grassmannians.

This work originates from the author’s stay at the Indian Institute of Science, Ben- galuru. 20 Memorial Award Lectures

31st Srinivasa Ramanujan Memorial Award Lecture

The Unreasonable Effectiveness of Ergodic Theory in Number Theory by Anish Ghosh, School of Mathematics, TIFR Bombay, Mumbai

Ergodic theory can be described as the mathematical study of the long term behaviour of dynamical systems. Recent decades have seen several spectacular applications of er- godic theory to a seemingly completely different branch of mathematics, namely number theory. I will describe some instances of this fruitful mathematical interaction. 34th P. L. Bhatnagar Memorial Award Lecture Lower-order Nonstandard Finite Element Methods for Biharmonic Plates by Neela Nataraj, Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076

The few popular piecewise quadratic schemes for the biharmonic equation based on tri- angles are the nonconforming Morley finite element method, the discontinuous Galerkin finite element method, the C0 interior penalty scheme and the WOPSIP scheme. All the schemes are modified in their right-hand side F replaced by F ◦ (JIM ) and then are quasi- −2 optimal in their respective discrete norms even for data F ∈ H (Ω). The smoother JIM is defined for piecewise smooth input function by a (generalized) Morley interpolation IM followed by a companion operator J . Energy norm and piecewise lower-order error estimates without data oscillations for the modified schemes are discussed. 31st Hansraj Gupta Memorial Award Lecture Some Glimpses into Noncommutative Ring Theory by Dinesh Khurana, Department of Mathematics, Panjab University, Chandigarh [email protected]

Besides listing some basic differences between commutative and non-commutative ring theory, we will present our work on some of the typical problems which are trivial for commutative rings. 31st V. Ramaswami Aiyer Memorial Award Lecture On bounds of Fourier-coefficients of Half-integer Weight Cusp Forms by Sanoli Gun, Institute of Mathematical Sciences, Chennai

In this talk, we will discuss about omega results of Fourier-coefficients of half-integer weight cusp forms which are not necessarily eigenforms. (This is a joint work with Kohnen and Soudararajan) 21 IMS Award Lectures Satish Bhatnagar Award Lecture

Pandiagonal Magic Squares: From N¯ag¯arjunaTo N¯ar¯ayan. a Pan. d. ita To Vijayaraghavan by M. D. Srinivas, Centre for Policy Studies, Chennai [email protected]

A pandiagonal magic square is a magic square where the entries along all the broken diagonals also add up to the magic sum. In India, the mathematics of magic squares was known as Bhadragan. ita and the Indian scholars paid special attention to the study of pandiagonal magic squares from ancient times. The ancient text Kacchaput.a ascribed to N¯ag¯arjuna(1st century CE) describes a method for constructing 4 × 4 pandiagonal magic squares. Var¯ahamihira(c.550 CE) describes a 4 × 4 pandiagonal magic square in the chapter on perfumery in his Br.hatsam. hit¯a,and the properties of such a sarvatobhadra have been discussed by his commentator Bhat.t.otpala (c.10th century). Pandiagonal magic squares have been found in the inscriptions at the Jaina temples in Dudhai and Khajuraho (c.10-11 centuries). The seminal text Gan. itakaumud¯i (c. 1350) of N¯ar¯ayan. a Pan. d. ita devotes an entire chap- ter to Bhadragan. ita, which contains many significant results on the construction of magic squares in general, and pandiagonal squares in particular. N¯ar¯ayan. a formulates and solves a linear indeterminate equation in order to characterise all the arithmetic sequences which can be employed to construct magic squares of a given order and specified sum. He de- scribes a method for constructing 4 × 4 pandiagonal magic squares using a succession of turagagati or horse movements and explicitly demonstrates that there are 384 pandiag- onal squares of order four. N¯ar¯ayan. a also describes a folding method of construction of doubly-even and odd order magic squares, which can be used to construct pandiagonal magic squares of all odd orders not divisible by 3, and all even orders divisible by 4.

Further properties of 4 × 4 pandiagonal magic squares have been discovered recently by B. Rosser and R. J. Walker (1938), and by the renowned Indian mathematician T. Vijayaraghavan (1941), who were all unaware of N¯ar¯ayan. a’s work and came up with new demonstrations of his result that there are 384 pandiagonal squares of order 4. Vija- yaraghavan has also presented a much simpler algorithm for the construction of these magic squares.

A. M. Mathai Award Lecture Invariant Subspaces and Exact Solutions of Nonlinear PDEs by P. Prakash, Department of Mathematics, Amrita Vishwa Vidyapeetham, Coimbatore [email protected], p [email protected]

In this talk, a systematic study for constructing the invariant subspaces of scalar and coupled nonlinear PDEs using the invariant subspace method. Also, we explicitly present that the generalized convection-reaction-diffusion equation admits more than one invariant subspaces in different dimensions which in turn helps to derive more than one different type of exact solution. The obtained exact solutions can be expressed in terms of polynomial, exponential, and trigonometric functions. In addition, we show how the invariant subspace method can be extended to fractional nonlinear PDEs. Finally, we also extend this method to time-fractional nonlinear PDEs with time-delay. 22

A. K. Agarwal Award Lecture

Euler’s Criterion for lth Power non residues with l a Prime by Jagmohan Tanti, Central University of Jharkhand, Ranchi, Jharkhand, India [email protected]

Let l ≥ 2 be a prime, p a prime ≡ 1 (mod l) and γ a primitive root (mod p). If an integer D with (p, D) = 1, is an lth power nonresidue (mod p) then D(p−1)/l is an lth root of unity α(6≡ 1) (mod p). Euler’s criterion of order l (mod p) studies the explicit (p−1)/l (p−1)/l conditions when D ≡ γ (mod p), i.e., when Indγ D ≡ 1 (mod l). In this talk we discuss the Euler’s criterion of order l when the ring of integers in the cyclotomic extension of Q of order l is a PID. Conditions are obtained in terms of Jacobi sums of order l.

A. Narasinga Rao Prize Lecture On Some Categories of Riemannian Manifolds by R. B. Yadav, Department of Mathematics, Sikkim University [email protected]

In mathematics we study mathematical structures and relations between them. For example, continuous functions between topological spaces in topology, homomorphisms between groups in group theory, linear transformations between linear spaces in linear algebra, smooth functions between manifolds in the theory of smooth manifolds and func- tors between categories in category theory are such relations. These relations are defined in such way that they relate the structures of the objects between which they are defined. Such relations are called morphisms in category theory. Only well known morphisms be- tween two Riemannian manifolds are isometries and conformal maps. These morphisms are isomorphisms in their respective categories of Riemannian manifolds. But these mor- phisms are too restrictive. In this talk we discuss two categories of Riemannian manifolds different from the previously known categories. We also discuss some properties of such categories and compare them with the previously known ones.

Invited Talks

1. Are We Prepared To Accept The Reality? by Ravi P. Agarwal, Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd., Kingsville, TX, USA [email protected]

In this lecture an attempt is being made to convince the world, especially we Indians that in reality Bharat, the Greater India, is the origin of mathematics. This will correct what most of the historians of Mathematics have claimed in their books, and we have accepted it religiously. We will also try to reassure that Mathematics as well as Proof cannot be defined. Further, we shall dis- cuss features of Upapattis (word in Pali, Sanskrit, and Marathi languages) in Indian Mathematics. 2. Sharp Nonlinear Stability Limits for Centrifugal Convection in Porous Media by S. Saravanan, UGC DRS Centre for Differential Equations and Fluid Dynamics, Department of Mathematics, Bharathiar University, Coimbatore 641 046, Tamil Nadu [email protected]

This talk is about determining nonlinear stability limits for a thermal convective flow in a ro- tating porous medium subjected to an alternating direction of centrifugal force field. The medium is homogeneous and exhibit rotationally invariant hydrodynamic and thermal properties. The 23

Darcy and the Brinkman models of flow through porous media are used to describe the momen- tum balance and the Boussinesq approximation is invoked to represent buoyancy. In order to understand possible instabilities a linear theory based on normal mode approach is applied first. By introducing a suitable energy functional a nonlinear analysis is then carried out. The uncon- ditional nonlinear stability limits are found exploiting the variational principles. The compound matrix method is then employed to solve the eigenvalue problems arising from the nonlinear and linear theories. Effects of various control parameters on the stability characteristics are predicted. The region of subcritical bifurcation is demarcated and failure of the linear theory is established. 3. Cotton Solitons within the Framework of Almost Kenmotsu 3-h- Manifolds by Pradip Majhi, Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata -700019, West bengal, India. [email protected]

In this paper, we consider the notion of Cotton soliton within the framework of almost Ken- motsu 3-h-manifolds. First we consider that the potential vector field is pointwise collinear with the Reeb vector field and prove a non-existence of such Cotton soliton. Next we assume that the potential vector field is orthogonal to the Reeb vector field. It is proved that such a Cot- ton soliton on a non-Kenmotsu almost Kenmotsu 3-h-manifold such that the Reeb vector field 2 is an eigen vector of the Ricci operator is steady and the manifold is locally isometric to H (−4)×R. 4. Automata Theory Based on Residuated Lattices by S.P. Tiwari, Department of Mathematics & Computing, Indian Institute of Technology (ISM), Dhanbad-826004, India. [email protected]

The concept of category theory introduced by Eilenberg and Mac Lane has shown to be useful in the development of many aspects of theoretical computer science. The minimal realization problem (which states that given a language, one can design a machine that realizes it) was stud- ied in a fairly general category theory setting by Goguen, is well known. However, an analogous minimal realization could not be provided for stochastic automata. In the case of automata based on multi-valued logic, we recently studied minimal realization problem under certain conditions. In this talk, we briefly explain the use of mathematical concepts involved in residuated lattices to study automata based on such lattices; and the construction of a minimal automaton based on residuated lattices. 5. Normal Families of Holomorphic Functions of Several Complex Vari- ables by K.S. Charak, Department of Mathematics, University of Jammu, Jammu-180 006, India kscharak7@rediffmail.com

Theory of normal families of meromorphic functions initiated by Paul Montel in 1907 now forms an integral part of function theory. In fact, this theory is responsible for many exciting advances in the area of complex dynamics, but there has also been many far reaching internal developments in the theory during the last over hundred years. In my talk, I shall present some recent developments in normal families of holomorphic functions of several complex variables for which a parallel theory is under process. 6. Catastrophic Changes in Coral Reef Dynamics under Macroalgal Toxicity, Overfishing and Invasion of Predators by Samares Pal, Department of Mathematics, University of Kalyani, Kalyani 741235, India [email protected]

Coral reefs can undergo relatively rapid changes in the dominant biota, a phenomenon re- ferred to as phase shift. Degradation of coral reefs is often associated with changes in community structure towards macroalgal dominated reef ecosystem due to the reduction in herbivory caused by overfishing. We investigate coral-macroalgal phase shift due to the effects of harvesting of herbivorous reef-fish by means of a continuous time model in a food chain. It is shown that the system is capable of exhibiting the existence of two stable configurations of the community under the same environmental conditions by allowing saddle-node bifurcations that involves in creation and destruction of fixed points and associated hysteresis effect. Moreover, it is observed that in 24 presence of low coral recruitment rate on algal turf and reduction in herbivory, the system exhibits hysteresis through a saddle-node bifurcation and a transcritical bifurcation. 7. Jordan ∗-derivations and Related Maps in Rings by Shakir Ali, Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India [email protected]

Let R be a ring with involution 0∗0. An additive map D : R −→ R is called a ∗-derivation (resp. Jordan ∗-derivation) if D(xy) = D(x)y∗ + xD(y)(resp. D(x2) = D(x)x∗ + xD(x)) holds for all x, y ∈ R. A Jordan ∗-derivation D of R is called inner if there exists a ∈ R such that ∗ ∗ Da(x) = xa − ax for all x ∈ R. The study of Jordan -derivations was motivated by the problem of representability of quadratic forms by bilinear forms (see [Proc. Amer. Math. Soc. 100(1987), 133-136] for details). It turns out that the question of whether each quadratic forms can be rep- resented by some bilinear form is intimated connected with the structure of Jordan ∗-derivations (viz., [Stud. Math. 97(1991), 157-165] and [Proc. Amer. Math. Soc. 119(1993), 1105-1113], where further reference can be found). In this talk, we will discuss the recent progress made on the topic and its related areas. Moreover, some examples and counter examples will be scrutinize for questions raised naturally.

Symposia Talks

1. Graph Theory and Combinatorics 1. The List Chromatic Number of a Graph by R. Balakrishnan, Department of Mathematics, Bharathidasan University, Tiruchirappalli - 620 024. [email protected]

A list assignment of a graph G is a mapping L which assigns to each vertex v of G a se L(v) of permissible colours. An L− colouring of G is a coloring c of G such that c(v) ∈ L(v) for each vertex v of G. We say that G is L−colourable if there exists a proper L−colouring of G. A k−list assignment of G is a list assignment L of G with |L(v)| = k for each vertex v of G. A graph G is k−choosable if G is L− colourable for each k−list assignment L. The choice number ch(G) of G (also called the list chromatic number χ`(G)) of G is the least k for which G is k−choosable. Clearly χ(G) ≤ ch(G), for each graph G. However, the difference ch(G) − χ(G) can be arbitrarily large. In fact, for each positive integer k ≥ 3, there exists a bipartite graph G with χ`(G) = k.

We present the result χ`(Kn,n) ≤ blog2 nc+2 : A celebrated result of Carsten Thomssen states that every planar graph is 5-choosable. However, there are planar graphs which are not 4-choosable. We present one such planar graph due to Choi and Cowen. In addition, we also discuss chromatically choosable graphs, that is, graphs for which the chromatic number and the choice number are equal. 2. p-adic Invariants for Quasi-symmetric 2-designs by Bhaskar Bagchi, Indian Statistical Institute, Bengaluru Centre, Bengaluru

We try to push to its logical limit the technique introduced by Professor Shrikhande in 1953 to prove analogues of the Shrikhande-Ryser-Chowla theorem for affine resolvable de- signs. The result is the introduction of certain invariants for strongly regular graphs : one invariant called the discriminant and a p-adic invariant for each prime p. Given a strongly regular graph G, we prove analogues of Schutzenberger theorem and Shrikhande-Ryser- Chowla theorem : necessary conditions for existence of a quasi-symmetric 2-design with block graph G and a given ‘defect’ (absolute difference of the two intersection numbers). The conditions are solely in terms of the graph invariants introduced, and the defect, so the results are effective when we can calculate the graph invariants explicitly. We do this in two cases : complete multi-partite graphs and co-triangular graphs, leading to para- metric restrictions on strongly resolvable 2-designs and triangular 2-designs, respectively. 25

3. Subspaces, Subsets, and Motzkin paths by Murali K. Srinivasan, Department of Mathematics, IIT Bombay, Mumbai [email protected]

In a beautiful paper Vogt and Voigt solved the problem of constructing an explicit sym- metric chain decomposition of the subspace lattice. We use their idea to code subspaces by Motzkin paths and give an explicit symmetric Boolean decomposition of the subspace lattice. 4. Contributions of S. S. Shrikhande towards λ-design Conjecture by Rajendra M. Pawale, Department of Mathematics, University of Mumbai, Kalina, Mumbai 400098 [email protected]

Let v, k and λ be integers with 0 < λ < k < v.A(v, k, λ) symmetric design is a pair (X, β), where X is a finite set with v elements, called points and β is a finite family of subsets of X, called blocks with |β| = |X|, such that each block contains k points and each pair of points occurs in λ blocks. A λ-design is a pair (X, β), where X is a finite set with v elements called points and β is a family of subsets of X called blocks, with |β| = |X| such that

(1) for all Bi,Bj ∈ β, i 6= j, |Bi ∩ Bj | = λ; (2) for all Bj ∈ β, |Bj | = kj > λ, and not all kj are equal. The only known examples of λ-designs are obtained by the following procedure: Let k 6= 2λ, and (X, ζ) be a symmetric (v, k, k −λ) design with block set ζ = {C1,C2,...,Cv}. Then

β = {B ⊂ X : B = C1 or B = (Ci \ C1) ∪ (C1 \ Ci) for some 2 ≤ i ≤ v}, is the block system of a λ-design. This construction is called complementation with respect to the block C1 and λ-designs obtained by this procedure are called type-1 λ-designs. The λ-design conjecture, also known as Ryser-Woodall conjecture due to Ryser and Woodall, states that all λ-designs are type-1. Shrikhande and Singhi had made significant contributions towards establishing truthful- ness of this conjecture, which is still open. We discuss the present status of the λ-design conjecture. 5. An Introduction to the Life and Work of S. S. Shrikhande by Sharad S. Sane, Chennai Mathematical Institute, Chennai-603103 [email protected], [email protected]

Since the symposium is in the memory of Professor S.S. Shrikahnde, who passed away this year, the talk is an attempt to present some of the milestones of mathematical achieve- ments of Shrikhande along with some personal reminiscences about his towering person- ality. Report on the symposium on Graph Theory and Combinatorics (in honour of Prof. S. S. Shrikhande)

The convener (Sharad Sane) began the Symposium with remembrance and obituary to Professor S. S. Shrikhande. Four eminent speakers gave talks besides the convener. Professor R. Balakrishnan of Trichy spoke on the difficult colour choosability problem for graphs. Professor Bhaskar Bagchi at the IISc, Bangalore spoke on the number theory connections of Shrikhande’s body of earlier work. Professor M. K. Srinivasan of the IIT Bombay spoke on Boolean decomposition of the subspace lattice and the Motzkin paths. Professor Rajendra Pawale of Mumbai University will speak on Shrikhande’s favourite λ-design conjecture. The convener finally concluded with a few well known and some unknown things about Shrikhande. 26

2. Biomechanics

1. Bio-Fluid Dynamics: Fluid Mechanical Aspects of Microcirculation by G. Radhakrishnamacharya,Indian Institute of Information Technology Design and Manufacturing, Kurnool

The fundamental principles of bio-fluid dynamics and, in particular, cardiovascular system are explained. The fluid mechanical aspects of pulse and blood pressure are il- lustrated. The fluid mechanical and physiological reasons for studying the flows in larger blood vessels and microcirculation separately are presented.

With a view to better understand the flow situation in microcirculation (flow in smaller vessels like arterioles, capillaries and venules), a two-fluid model has been proposed to de- scribe fluid flow in small diameter tubes consisting of a non-Newtonian fluid (Jeffrey fluid model) in the core region and Newtonian fluid in the peripheral region. Analytical expres- sions for effective viscosity, core hematocrit and mean hematocrit are obtained. The effects of various parameters, namely, Jeffrey parameter (λ1), tube hematocrit (H0) and tube ra- dius (a) on effective viscosity, core hematocrit and mean hematocrit have been studied. It is found that the effective viscosity decreases as the Jeffrey parameter increases but in- creases with tube hematocrit and tube radius. The mean hematocrit decreases as Jeffrey parameter increases but increases with tube hematocrit and tube radius. It is also noticed that the flow exhibits the anomalous Fahraeus-Lindquist effect.

2. Magnetic Drug Targeting in a Microvessel with Multifunctional Nanopar- ticle based Carrier Particle by P.V.S.N. Murthya, Sachin Shawb, Abhijit Sutradharc aDepartment of Mathematics, Indian Institute of Technology Kharagpur, West Bengal 721302, India bDepartment of Mathematics and Statistical Sciences, Botswana International University of Science and Technology (BIUST), Private Mail Bag 16, Palapye cDepartment of Basic Sciences, Indian Institute of Technology Bhubaneswar, Odisha 752050, India.

A review of non-invasive magnetic drug targeting of a multifunctional carrier particle in blood stream flowing in impermeable and permeable microvessels is done. Mathematical modelling of magnetic drug targeting in these microvessels is discussed using most popu- lar Casson and Herschel-Bulkley models to describe the non-Newtonian nature of blood flowing in the microvessels of diameter 100 micrometers. A dilute solution suspended with the carrier particle is injected into the microvessel upstream from the tumor located inside the body. A rare-earth cylindrical magnet is positioned in the vicinity of body surface to capture the carrier particle near the tumor zone.

A particular model considering the significant buoyancy force, fluidic force, drag force and inertia effect in the equation of motion of the carrier particle and the external mag- netic force experienced by the carrier particle is analyzed. Along with the therapeutics, biocompatible F e3O4 nanoparticles of spherical shape are assumed to be present in the carrier particle. Consequently, an effective density for the carrier particle is introduced. The numerical solution of the resulting coupled nonlinear equations of motion is obtained. The trajectories of the carrier particle is obtained under varying parametric conditions. It is evident that the non-Newtonian nature of blood significantly changes the particle trajectories. These results also indicated that lesser magnetization is required to attract the carrier particle more effectively near the tumor location due to the consideration of buoyant force along with the acceleration of the carrier particle. An optimal placement of the external magnet from the axis of the microvessel makes an efficient capture of the carrier particle near the targeted site. 27

3. Mathematical models of physiological fluid flows by P. Muthu, Department of Mathematics, National Institute of Technology (NIT), Warangal [email protected]

In this talk, we will discuss about mathematical models of certain physiological fluid flows such as (i) flow in renal tubule (ii) solute transfer in glomerular capillaries and (iii) synovial joint lubrication. Relevant literature from the past and the present is described along with an explanation of importance of studying these flow situations. An overview of various computational methods which are used to solve these problems is presented. The simulated results are interpreted with available experimental data to emphasize the need of modelling of physiological systems for better understanding of their functioning. 4. Cardiac Electrical Activity in a Human Cardiac Tissue: Theory, Computation and Application by Meena Pargaeia, B.V. Rathish Kumarb aG.P.G.C. Champawat, Affiliated to Kumaun University, U.K. ([email protected]) bIndian Institute of Technology, Kanpur ([email protected])

The heart electrical conduction system propagates the electrical impulse originated from the sinoatrial node (SA) called cardiac pacemaker, situated in the left atrium, due to which heart muscle starts to contract, and then this signal travels to the whole heart, and the contraction of entire heart muscle takes place. The basis of the electrical activity is the action potential, which is facilitated by ionic channels and the ionic cell transporters that enable the movement of ions across the cardiac cell membrane. Myocardial ischemia takes place when the blood flow to the heart and oxygen supply to the heart is abnormal. It is one of the leading causes of sudden death. Due to this myocardial ischemia metabolism and electrophysiological changes appear which results in the alteration of cardiac electrical activity. In this talk, a modified human ventricular TT06 cell level model (ODEs) coupled with the tissue level Monodomain model (PDE) is considered to analyze the influence of my- ocardial ischemia on the cardiac electrical activity of human cardiac tissue (HCT). The apriori finite element error estimate of this PDE-ODE system for the numerical scheme is found to be as o(h2 + k). A HCT (domain) is modeled in such a way that it consists multiple ischemic subregion (subdomain). The coupled system of partial and ordinary differential equations in this modeled domain is solved numerically using the Q1 finite ele- ment in space and backward Euler finite difference (BEFD) method in time. We evaluate the impact of the increasing size of the ischemic region and the presence of the multiple ischemic regions having equal or different intensities on the neighboring healthy part of the cardiac tissue. It is observed that with the increase in the ischemic region size by a factor five times, there is an additional almost 10 % drop in the action potential duration (APD) in the neighboring healthy regions. Increasing the number of ischemic regions from 1 to 4 leads to a 39 % drop in APD. 5. An Overview of Cardiovascular Flow Studies : Mathematical Theory, Numerical Analysis and Computational Simulation by B.V. Rathish Kumar, IIT Kanpur [email protected]

Virtual Cardiovascular flow analysis has now become a widely accepted viable approach to assist in clinical diagnosis and surgical decision making across the world. In this talk, to begin with we will touch upon the essential mathematical and numerical analytical aspects that play a crucial role prior to carrying out computational simulations to unravel the physics behind the cardiovascular flow dynamics, which by far continues to remain a challenge especially in the clinically declared pathological conditions. Then we will move forward to get glimpses of diseased condition of arterial vessels and the ways to model the same including patient specific modelling based on CT/MRI data. Subsequently we will 28 discuss in a nutshell the outcome of computational simulations on HPC platforms of few pathological cases based on both academic, which are essential in developmental phase, and clinical data, including few of the essential parallel computing methods to realize such solution in real time. Report on the Symposium on Biomechanics

The symposium on Biomechanics was convened by Prof. B. V. R. Kumar of IIT Kan- pur. In this symposium on the theme of BIOMECHANICS, we had five speakers. Each speaker delivered a 25 minute talk followed by 5 min for Q & A. The topics covered in order by five experts are: Introduction of Mathematical Modelling of Blood Flow, Modeling of Drug Delivery in stenosed artery, Modeling and analysis of renal flow dynamics, modeling and simulation of cardiac electrical activity and theoretical and parallel computational aspects of cardiovascular flow simulations.

3. Chandrasekharan Centenary Symposium in Number Theory

1. Mod p modular forms and simple congruences by Jaban Meher, NISER, Bhubaneswar [email protected]

We first give a description of the algebra of modular forms on the congruence subgroup Γ0(2) modulo a prime p. This result parallels results of Swinnerton-Dyer in the SL2(Z) case, Katz on the subgroup Γ(N) for N ≥ 3, Gross on the subgroup Γ1(N) for N ≥ 4. We next apply the theory of mod p modular forms on Γ0(2) to prove the non-existence of simple congruences for Fourier coefficients of quotients of certain Eisenstein series on Γ0(2). This is a joint work with Sujeet Kumar Singh. 2. Lucas sequences and its arithmetic by Shanta Laishram, Indian Statistical Institute, New Delhi [email protected]

The Fibonacci sequence, which is one of the most well known integer sequences, is a special example of a general family of Lucas sequences. In this talk, I will some arithmetic of the Lucas sequences, and in particular, I will discuss about the following result proved recently. Let {Un}n≥0 be a Lucas sequence. We show that if Un is a product of factori- als, then n ∈ {1, 2, 3, 4, 6, 8, 12} which is independent of the Lucas sequence. Further we show that if Un is a product of Catalan numbers or the middle binomial coefficients, then 2m
n ∈ {1, 2, 3, 4, 6, 8, 12, 16}. Here the m−th middle binomial coefficient is Bm = m and 1 2m
the m−th Catalan number is Cm = m+1 m .

3. The Discrete Logarithm Problem Over Prime Fields: Non-canonical Lifts and Logarithmic Derivatives by R. Padma, Department of Mathematics School of Advanced Sciences, VIT Vellore [email protected]

Many public key cryptographic protocols currently in use are based on the computa- tional hardness of the discrete logarithm problem. A polynomial time algorithm for the discrete logarithm problem in the case of anomalous curves was invented by N. P. Smart, I. A. Semaev, T. Satoh and K. Araki independently. In this talk we will modify their method using ideas from elementary number theory to attack the discrete logarithm problem over prime fields. This is joint work with H. Gopalakrishna Gadiyar. 29

4. Hardy’s theorem for general L-functions by K. Srinivas, The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai-600113 [email protected]

The Riemann Hypothesis (RH) asserts that all the non-trivial zeros of the Riemann 1 zeta-function ζ(s), s = σ + it lie on the critical line σ = 2 . As a first step towards RH, G. H. Hardy, in 1914 showed that infinitely many of them lie on the critical line. In this talk, we shall discuss the extension of Hardy’s approach to more general L-functions. 5. Prime powers dividing products of consecutive integer values of x2n + 1 by Stephan Baier, RKMVERI [email protected]

This is joint work with Pallab Kanti Dey (RKMVERI). Let n be a positive inte- n ger and f(x) := x2 + 1. We study orders of primes dividing products of the form 12 n+1 Pm,n := f(1)f(2) ··· f(m). We prove that if m > max{10 , 4 }, then there exists a n−1 prime divisor p of Pm,n such that ordp(Pm,n) ≤ n · 2 . For n = 2, we establish that for every positive integer m, there exists a prime divisor p of Pm,2 such that ordp(Pm,2) ≤ 4. Consequently, Pm,2 is never a fifth or higher power. This extends work of Cilleruelo who studied the case n = 1. Report on the Symposium on Chandrasekharan Centenary Symposium on Number Theory

A symposium entitled “Chandrasekharan Centenary Symposium on Number Theory” was organized on December 19, 2020 during the 86th Annual Conference of the IMS. It was convened by Prof. K. Srinivas of The Institute of Mathematical Sciences, Chennai. The event started with recounting the numerous contributions of the legendary mathematician K. Chandrasekharan in shaping the number theory culture in India by means of his books and fundamental papers. The following people delivered talks: 1. Prof. Shanta Laishram, ISI Delhi : Lucas sequences and its arithmetic 2. Prof. Stephan Baier, RKMVERI, Belur Math : Some powers dividing products of consecutive n integer values of x2 + 1 3. Prof. Jaban Meher, NISER, Bhubaneswar : Mod p modular forms and simple congruences 4. Prof. R. Padma, VIT, Vellore : The discrete logarithm problem over prime fields: non-canonical lifts and logarithmic derivatives 5. Prof. K. Srinivas, IMSc, Chennai : Hardy’s theorem for general L-functions

4. Topology and Geometry 1. On the volume of Fano manifolds by Harish Seshadri, IISc Bengaluru

A recent result of K. Zhang which gives an optimal volume bound for Khler manifolds with positive Ricci curvature was discussed. 2. Doodles on surfaces and associated groups by Mahender Singh, Indian Institute of Science Education and Research (IISER) Mohali [email protected]

Study of certain equivalence classes of a finite collection of immersed circles without triple or higher intersections on closed oriented surfaces can be thought of as a planar analogue of virtual knot theory where the genus zero case corresponds to classical knot theory. It is intriguing to know which class of groups serves the purpose that braid and virtual braid groups serve in classical and virtual knot theory, respectively. Khovanov 30 proved that twin groups, a class of right angled Coxeter groups with only far commuta- tivity relations do the job for genus zero case. We showed in some recent works that an appropriate class of groups called virtual twin groups fits into the theory for higher genus case. The talk would give an overview of some recent developments on the subject.

3. The Topology of Real Bott Manifolds by Raisa Dsouza, St Josephs College, Bengaluru

In this talk we introduce small covers and a particular class of examples of small covers known as Real Bott Manifolds. We discuss a few topological properties of these objects and also see an interesting relation between these objects and acyclic directed graphs.

m 4. Smooth Structures on CP for 5 ≤ m ≤ 8 by Ramesh Kasilingam, Department of Mathematics, IIT Madras, Chennai [email protected]

We classify up to diffeomorphism all smooth manifolds homeomorphic to the complex m projective m−space CP for m = 5, 6, 7 and 8. As an application, for m = 7 and 8, we obtain a bound on the number of smooth homotopy complex projective m−spaces with given Pontryagin classes up to orientation-preserving diffeomorphism. We also show that there exists a smooth manifold which is tangentially homotopy equivalent but not home- 8 omorphic to CP . 5. On the zero-divisor cup-length of real oriented Grassmann manifolds by Vimala Ramani, Anna University [email protected]

For a path-connected topological space X, the K-zero-divisor cup-length, where K is any field, is a cohomological lower bound for the topological complexity of X. For real oriented Grassmann manifolds G˜n,k, 3 ≤ k ≤ [n/2], we compute the rational zero-divisor cup-length in terms of n and k. We also prove that, for certain infinite families of real ˜ oriented Grassmannians Gn,3, the Z2-zero-divisor cup-length is a better lower bound for the topological complexity than the Q-zero-divisor cup-length. Report on the Symposium on Geometry and Topology.

There were five talks in the symposium, covering a wide range of topics in the broad areas of geometry and topology. Dr. Vimala Ramani, Anna University, Chennai, spoke on the zero-divisor cup length of oriented Grassmann manifold, which has application to lower bound on the topological complexity. Dr. Raisa Dsouza, St Joseph College, Bengaluru, spoke on the topology of real Bott towers. Dr. Ramesh Kasilingam, IIT Madras, gave a talk on the differential structures on the complex projective spaces. Dr. Mahender Singh, IISER Mohali, gave a talk on doodles on surfaces. Dr. Harish Seshadri gave a talk on the volume of Fano manifolds. The talks were well attended by students and faculty mem- bers from many parts of the country. The symposium was organized by Parameswaran Sankaran, Chennai Mathematical Institute, Kelambakkam. 31

5. History of Indian Mathematics

1. Brahmagupta’s Bh¯avan¯aand reading mathematics from Sanskrit texts by C S Aravinda, TIFR Centre for Applicable Mathematics, Bangalore [email protected]

As an example of deciphering mathematical results from old Sanskrit texts, we will show how to read Brahmagupta’s celebrated composition law, on solutions of the Brahmagupta- Pell equation, from his original Sanskrit verses.

2. On Zero-Divided Numbers in Ancient Indian Mathematics by Amartya Kumar Dutta, Indian Statical Institute, Kolkata [email protected]

It was in ancient India that zero received its first clear acceptance as an integer in its own right. In 628 CE, Brahmagupta describes in detail rules of operations with numbers — positive, negative and zero — and thus, in effect, imparts a ring structure on integers with zero as the additive identity. The various Sanskrit names for zero include kha, ´s¯unya,p¯urn.a. There was an awareness about the perils of zero and yet ancient Indian mathematicians not only embraced zero as a number but allowed it to participate in all four arithmetic operations, including as a divisor in a division. But division by zero is strictly forbidden in the present edifice of mathematics. Consequently, verses from ancient stalwarts like Brahmagupta and Bh¯askar¯ach¯arya referring to numbers with “zero in the denominator” shock the modern reader. Certain examples in the B¯ijagan. ita (1150 CE) of Bh¯askar¯ach¯arya appear as absurd nonsense. But then there was a time when square roots of negative numbers were considered non- existent and forbidden; even the validity of subtracting a bigger number from a smaller number (i.e., the existence of negative numbers) took a long time to gain universal acceptance. Is it not possible that we too have confined ourselves to a certain safe convention regarding the zero and that there could be other approaches where the ideas of Brahmagupta and Bh¯askar¯ach¯arya, and even the examples of Bh¯askar¯ach¯arya, will appear not only valid but even natural? Enterprising modern mathematicians have created elaborate legal (or technical) ma- chinery to overcome the limitations imposed by the prohibition against use of zero in the denominator. The most familiar are the methods of calculus with its concept of limit, results like l’Hˆopital’srules, and a language which enables one to express intuitive ideas 1 like 0 = ∞ through legally permitted euphemisms. Less well-known are the devices of commutative algebra, algebraic geometry and algebraic number theory like “localisation” which describes a legal structure for directly writing fractions with zero in the denom- inator without any subterfuge, and the more sophisticated ideas of “valuation theory” which admit multiple levels of infinities and thereby provide higher-dimensional algebraic analogues of l’Hˆopital’srules. In this talk we shall highlight an algebraic model proposed by Prof. Avinash Sathaye for understanding Bh¯askar¯ach¯arya’s treatment of khahara, (numbers with) zero in the de- nominator, including the apparently erroneous examples in the algebra treatise B¯ijagan. ita. A crucial ingredient of this model is the ubiquitous concept of “idempotent” in modern 2 algebra (elements e satisfying e = e). The commentary by Kr.s.n. adaivaj˜na(c.1548)indi- cates that idempotence was indeed envisaged as a natural property of numbers like zero and its reciprocal, the khahara. While historians of mathematics have tried to analyse Bh¯askar¯ach¯arya’s khahara in the framework of calculus, the difficulties with his examples disappear in the algebraic interpretation based on idempotents. Prof. Sathaye’s interpretation of Bh¯askar¯ach¯arya’s khahara also gives a new meaning to certain mysterious utterances of Ramanujan recorded by P.C. Mahalanobis.

3. Continued Fraction Technique in the Kerala School of Astronomy by Venketeswara Pai R, Department of Humanities and Social Sciences, Indian Institute of Science Education and Research (IISER) Pune 411008 32 [email protected]

A brilliant school of astronomers and mathematicians founded by M¯adhava (c. 1340 - 1420) flourished in Kerala between 14th - 17th century CE. One among them was Pu- tumana Somay¯aji,the author of Karan. apaddhati (c. 1532 - 1560), which explains the mathematical basis of the v¯akya system of computing the planetary positions, in which the true positions of planets are found directly from mnemonics along with some sim- ple arithmetical operations. Now, the calculation of the mean positions of the planets would be the first step in the computation of the true positions.These would be propor- tional to the rates of motion of the planets which would involve ratios of large numbers. Some other variables needed for obtaining the true positions also involve ratios of large mulitipliers and divisor. Karan. apaddhati describes a mathematical technique known as vallyupasam. h¯ara which is a variant of kut..taka method for solving linear indeterminate equations. In Karan. apaddhati, this method is used for obtaining smaller multipliers and divisors for the aforesaid ratios. vallyupasam. h¯ara method of transforming the vall¯i (a row of numbers) is essentially the recursive process of calculating the successive convergents of the continued fraction associated with the ratio. In Karan. apaddhati, it is the simple continued fraction expansions that is used. In a later text called Dr.kkaran. a (1606 CE), probably authored by Jyes.t.hadeva, we have a variant of this method in which semi-regular continued fraction expansions are prescribed, with modified recursion relations. Consider the problem of computing the true position of the Moon on some specific date, using the V¯akya method. This would involve finding a date which is close to the specific date, on which the mean and true longitudes are equal at Sunrise. This would imply that the anomaly of the Moon(an auxiliary variable which appears in the expression for the true longitude) is zero at that time. This known as the khan. d. adina. Finding the khan. d. adina is indeed an interesting mathematical problem, involving the kut..taka method to solve a linear indeterminate equation.

4. Generalized Brahmagupta-Jayadeva-Bh¯askara Problem by Avinash Sathaye, Department of Mathematics, University of Kentucky, Lexingon KY, USA [email protected]

One of the famous problems in the history Indian Mathematics is the one stated by Brahmagupta in the seventh century and completely me common fixedlved by Bh¯askara in twelfth century. The problem is to solve the equation Dx2 + 1 = y2 where D is a given nonsquare positive integer and the desired solutions x, y are integers. In seventeenth century, the same equation was contemplated by Fermat without the knowledge of Indian history and he posed it as a challenge to the mathematicians. Its solution led to many techniques in Quadratic forms in number theory and the high point was the work of Gauss in eighteenth century. We discuss a generalized version of the original problem which asks for all possible integers c such that Dx2 + c = y2 for some integers x, y. The generalized question naturally arises from the method of the Indian solution which consists of solving the generalized problem for some convenient c and then cleverly modify it to lead to the desired solution. We present a solution to the generalized problem by the same method as Bh¯askara by using the Theory of Bh¯askara forms in Ayyangar’s paper “New light on Bh¯askara’s chakravala ...” in JIMS(1929-1930) vol. 18. We also compare and contrast the solution with the techniques of using the Gauss’ theory of Quadratic forms in Number Theory.

5. Second Order Taylor Series for the Sine and Cosine Functions in the Kerala School of Astronomy and Mathematics by M.S. Sriram, Prof. K.V. Sarma Research Foundation, Adyar, Chennai [email protected] 33

The sine function is a necessary ingredient for most calculations in the mathemati- cal astronomy of the yore. The Indian texts on siddh¯antic astronomy beginning with Aryabhat¯ .¯iya (499 CE) give the table of sines, mostly corresponding to a 24-fold divi- sion of the quadrant: the values of sin(iα) are specified, where i = 1, 2,..., 24, and 90◦ 0 α = 24 = 225 . The value of sin θ for a general value of θ = iα + βα, (β < 1) is [sin((i+1)α)−sin(iα)] obtained from interpolation: sin(iα + βα) = sin(iα) + βα × α . In his Khan. d. akh¯adyaka (665 CE), Brahmagupta gives a second-order interpolation formula for the sine function (actually valid for any function). The sine of the anomaly appears in the expression for the true longitude of a planet. So, the rate of change of the true longitude, or the true daily motion would involve the rate of change of the sine function. The correct expression for this, employing the cosine as the rate of change of the sine, was given by Munj¯ala(932 CE) in his Laghum¯anasa and also by Aryabhat¯ .aII (950 CE) in his Mah¯asiddh¯anta. In his Siddh¯anta´siroman. i (1150 CE), Bh¯askara-II recognises the true daily motion as the instantaneous velocity, and explains how the cosine function appears in the expression for it. Actually, the true longitude involves the inverse-sine function, and in his Tantrasa ˙ngraha (1500 CE), N¯ilakan. t.ha So- may¯aj¯i gives the correct expression for the instantaneous velocity, involving the derivative of the inverse-sine. In Yuktibh¯as.¯a(1530 CE) of Jyes.t.hadeva (well known for the proofs of the infinite series expansions of π and sine and cosine functions), the rate of change of the sine and cosine functions is discussed in more detail. It gives the second order Taylor series for the sine and cosine functions, explicitly. In later Kerala texts like Sphut.anirn. ayatantra (1593 CE) of Acyuta Pis.¯arat.i and also Dr.kkaran. a (1606 CE), possibly a work of Jyes.t.hadeva, the second order Taylor series for the sine function is explicitly used. This is in the context of the procedure for finding the true longitudes of planets. [Note : In Indian mathematics and astronomy, the sine is actually the length of the opposite side in a right triangle with a given hypotenuse . Moreover, the argument of the sine is the relevant arc, not the angle.]

6. Industrial Mathematics: Modelling, Optimization, Simulation 1. Covid-19-Simulations for Germany by Thomas Goetz, Mathematical Institute, University Koblenz, Germany [email protected]

In the talk we will report about the different modeling and simulation approaches in Koblenz. As part of the MOCOS consortium we are conducting stochastic microstruc- ture simulations together with the universities of Kaiserslautern, Trier and Wroclaw. In Koblenz we are also doing parameter estimations based on SIR-type models. These mod- els allow to simulate effects of schools openings, the influence of household structures onto the disease dynamics and to estimate the number of undetected cases. Since all this is ongoing and very active current research work, the special focus of the talk is not yet fixed.

2. A Point Source Model to Represent Heat Distribution Without Calculat- ing the Joule Heat during Radiofrequency Ablation by Panchatcharam Mariappan, Department of Mathematics and Statistics, IIT Tirupati, India [email protected]

Numerous liver cancer oncologists suggest bridging therapies to limit cancer growth un- til donors are available. Interventional radiology including radiofrequency ablation (RFA) is one such bridging therapy. This locoregional therapy aims to produce an optimal amount of heat to kill cancer cells, where the heat is produced by a radiofrequency (RF) needle. Less experienced Interventional Radiologists (IRs) require a software-assisted smart solu- tion to predict the optimal heat distribution as both overkilling and untreated cancer cells 34 are problematic treatment. Therefore, two of the big three partial differential equations, (1) heat equation to predict the heat distribution and (2) Laplace equation for electric potential along with different cell death models are widely used in the last three decades. However, solving two differential equations and a cell death model is computationally expensive when the number of finite compact coverings of a liver topological structure increases in millions. Since the heat source from the Joule losses q = σ|∇V |2 is obtained from Laplace equation σ∆V = 0, it is named as Joule heat model. In this lecture, we represent the heat distribution from the RF needle by a point source model instead of the traditional Joule heat model. The idea behind this model is to solve σ∆V = δ0 where 2 √1 −x /4 δ0 is a Dirac-delta distribution which can also be represented by lim → 0 2 π e . Therefore, using the fundamental solution of Laplace equation we represent the solution of Joule heat model using an alternative model called Point Source model which is given by Gaussian distribution

2 1 |x−xi| X X − 2 q(x) = Cj e 2σ Ki xi∈Ω j where Ki and Cj are obtained by using needle parameters. This model is employed in one of the software solution called RFA Guardian which predicted the treatment outcome very well for more than 100 patients.

3. Fast direct solver for high frequency EM scattering by Sivaram Ambikasaran, IIT Madras [email protected]

The talk will focus on constructing a fast direct solver that scales as O(n log n) instead of O(n3), where ’n’ is the number of unknowns in the discretized linear system, for high frequency electromagnetic scattering problems.

4. A meshfree particle method for simulations of fluid flows and interacting particle systems by Sudarshan Tiwari, Department of Mathematics, University of Kaiserslautern, Germany [email protected]

The talk will be divided into two parts. The first part consists of the introduction of a mesh free particle method, called the Finite Pointset Method (FPM) and its application to simulate complex flow problems. This is a fully Lagrangian method, where particle move with their velocities. The spatial differential operators at an arbitrary particle position is approximated from its surrounding cloud of points based on the weighted least squares method. This is a generalized finite difference approximation. We simulate flows by solving the incompressible Navier-Stokes equations. We use the Chorin’s projection scheme in the meshfree framework. We present different simulation results for single- and multiphase flow problems. . In the second part, we present models for interaction particle system. We present the microscopic model first and then show its hydrodynamic equations. By modeling different forces we present the simulations of material flows, pedestrian flow and the swarming of birds. The hydrodynamic limit equations are solved by the FPM.

5. Pedestrian Crowd Dynamics Models by S. Sundar, DAAD Research Ambassador, Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036

Pedestrian crowd dynamics is a complex and intriguing topic which is being researched widely. Social scientists opine that crowd behaviour is not completely irrational and indi- viduals are mostly guided by reason and rules of behaviour. Mathematical modeling of the pedestrian crowd is a valuable tool for understanding the dynamics of the problem. Varied discrete and continuous modelling approaches ranging from macroscopic to microscopic length scales have been explored by scientists and mathematicians. This talk presents a 35 simple and elegant model The Social Force Model proposed by D. Helbing and P. Molnar in 1995. In this microscopic model, the motion of the pedestrians is assumed to be driven by non-physical social or behavioural forces and governed by kinematic equations. The model reproduces many of the self-organization phenomena, observed in reality, through com- puter simulations. This modeling approach is well-founded and the governing equations give a framework to understand and predict qualitatively and sometimes quantitatively certain aspects of pedestrian and crowd movement.

6. Selection of the Informative Frequency Band in a Bearing Fault Diagnosis in the Presence of Non-Gaussian Noise-the stochastic-based approach by Agnieszka Wyloma´nska, Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, Poland. E-mail: [email protected]

The vibration signals acquired on machines usually have complex spectral structure. As the signal of interest (SOI) is weak (especially at an early stage of damage) and covers some frequency range (around structural resonance), it requires its extraction from a raw observation. Until now, most of the techniques assumed the presence of Gaussian noise. Unfortunately, there are cases when the non- informative part of the signal (considered as the noise) is non-Gaussian due to the random disturbances or nature of the process executed by the machine. Thus, the problem can be formulated as the extraction of the SOI from the non-Gaussian noise. Recently this problem has been recognized by several authors and some new ideas have been developed. In this presentation, we would like to compare these techniques for benchmark signals (Gaussian noise, cyclic impulsive signals, non-cyclic impulsive signals with random amplitudes and locations of impulses and a mix- ture of all of them). Our analysis will cover classical methods and recently introduced algorithms based on the stochastic analysis of the vibration signals represented in time- frequency maps. A discussion on the efficiency of each method will be provided. Keywords: local damage; vibration; time-frequency analysis; band selectors; stochastic analysis. Report on the Symposium on Industrial Mathematics: Modeling, Optimization, Simulation

The IMS symposium on “Industrial Mathematics: Modeling, Optimization, Simulatio” turned out to be a very successful meet and this was due to the topics covered by our Invited Speakers. All the credit goes to them. Prof. Agnieszka Wylomanska of Wrocklaw University of Science and Technology, Poland started the presentation with her recent findings on Time Series Modeling applied to real life problems, Prof. Thomas Goetz of Uni-Koblenz, Germany gave a detailed description of Biological Modeling of Covid19 spread applicable to Germany, Dr.Sudarshan Tiwari of TU Kaiserslautern, Germany pre- sented his current research on Pedestrian Crowd Dynamics models (combining macro and micro scales) and demonstrated his models with a powerful meshfree method, Dr.Sivaram Ambikasaran of IIT Madras gave a detailed account his recent research and demonstrated with powerful examples and finally Dr.Panchatcharam Mariappan of IIT Tirupathi pre- sented his work on Medical Imaging which is a problem taken along with an Industry and highlighted the challenges. All the Invited Speakers, including the Coordinator, thanked the IMS for this opportunity 36

Papers for Competition Session

A. M. U. Prize 1. Modules Invariant under Clean Endomorphisms of their Injective Hulls by Manoj Kumar Patel, Department of Mathematics, National Institute of Technology, Nagaland, Dimapur -797103, Nagaland, India [email protected]

A module is quasi-injective if and only it is invariant under endomorphisms of its in- jective hull. In this paper we study the class of modules which are invariant under all clean endomorphisms of their injective hulls and show that this class of modules coin- cide with the class of quasi-injective modules. Some facts and results of this class of modules are obtained. We also establish some relations of clean-invariant modules with automorphism-invariant modules, idempotent-invariant modules, pseudo-continuous mod- ules and Utumi modules. Apart from this we have given several sufficient conditions under which automorphism-invariant modules to be clean-invariant.

2. Duality of Locally Quasi-Convex Convergence Groups by Pranav Sharma, Department of Mathematics, Lovely Professional University, Punjab-144411. [email protected]

In the realm of the convergence spaces, the generalisation of topological groups is the convergence groups, and the corresponding extension of the Pontryagin duality is the continuous duality. We prove that local quasi-convexity is a necessary condition for a convergence group to be c-reflexive. Further, we prove that every character group of a convergence group is locally quasi-convex.

V. M. Shah Prize 1. A Note on the Value Distribution of a Differential Monomial and some Normality Criteria by Bikash Chakraborty, Department of Mathematics, Ramakrishna Mission Vivekananda Centenary College, Rahara, West Bengal 700 118, India [email protected], [email protected]

In this paper, we prove some value distribution results which lead to some normality criteria for a family of analytic functions. These results improve some recent results.

2. A New Two-dimensional Quaternion Fractional Fourier Transform by Khinal Parmar, SVKM’s NMIMS MPSTME, V. L. Mehta Road, Vile-Parle (W), Mumbai-400056 [email protected]

In this paper, a new two-dimensional quaternion fractional Fourier transform is de- veloped. The properties such as linearity, shifting and derivatives of quaternion valued function are studied. The convolution theorem, Plancherel type theorem and inversion formula are also established.

3. Inequalities and Applications of Quaternion Windowed Linear Canonical Transform by Manab Kundu Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad 826004, Jharkhand, India. [email protected] 37

In this paper, we study the logarithmic uncertainty principle, Lieb uncertainty princi- ple and local uncertainty principle for the quaternion windowed linear canonical transform (QWLCT). Further we briefly introduce some applications of QWLCT of a linear time varying system.

4. Quantitative Approximation on a New Class of Sz´asz-Mirakjan Operators having Preserving Property by Rishikesh Yadav, Applied Mathematics and Humanities Department, Sardar Vallab- hbhai National Institute of Technology, Surat, Gujarat 395 007, India [email protected]

This article deals with the approximation properties of generalized version of Sz´asz- Mirakjan operators which preserve ax, a > 1 (fixed) and x ≥ 0. We study uniform convergence of the operators by using some auxiliary results and also error estimations are determined by considering the function from different spaces. The convergence of said operators is shown and analyzed by graphics, also, in the same direction, we compare the proposed operators with Sz´asz-Mirakjanoperators for the rate of convergence. A Voronovskaya-type theorem is studied and a comparison is shown under a sense of con- vexity with Sz´asz-Mirakjanoperators. To describe the quantitative means of an asymp- totic formula, we quantitatively approach to the Voronovskaya-type theorem, moreover, a Gr¨ussVoronovskaya type theorem is proved. In the last section, a modified sequence is constructed in the space of integrable function.

5. Binomial Distribution and its Geometric Properties Associated with Univalent Functions by S. Santhiya, School of Advanced Sciences, Vellore Institute of Technology, Vellore, [email protected]

The applications of hypergeometric function, confluent hypergeometric functions, Wrights function, generalized Bessel functions are interesting topics of research in Geomet- ric Function Theory. In 2014, Porwal (J. Comput. Anal., ID 984135) introduced Poisson distribution series and give a nice application on analytic univalent functions and co- relates probability density function with univalent function. After the appearance of this paper several researchers introduced hypergeometric distribution series, hypergeometric distribution type series, conuent hypergeometric distribution series, Binomial distribution series, and obtain sufficient conditions and inclusion relations for various classes of univa- lent functions. Based on their work, in this paper we introduced new functions belonging to the class L (A, B, θ) of analytic functions involving a power series whose coefficients are probabilities of Binomial distribution series and studied its geometric properties associ- ated with univalent functions. Motivated by the work given by Sheeza et al. (Kyungpook Math. J., vol. 59, pp. 301-314) and Nazeer et al. (J. Comput. Anal. Appl., Vol. 26, pp. j−n j−n j−n j−n 11-17), we define the following new functions Qp (z),Np,λ (z),Mp,λ,γ (z),Kp (z) and further find some analogous conditions for the functions defined by binomial distribution series belongs to the class L (A, B, θ).

IMS Prize Group 1 1. Some Properties of k-tuple t-core Partitions by Ranganatha Dasappa, Department of Mathematics, Central University of Karnataka, Kalaburagi-585367, Karnataka, India [email protected]

In this paper, we generalize some results due to Saikia and Boruah on congruence properties for A3,9(n), A9,3(n) and A4,8(n), where At,k(n) denote the number of k-tuple partitions of n where each partition is t-core. 38

2. Direct Summands of Goldie Extending Elements in Modular Lattices by Rupal C. Shroff, School of Mathematics and Statistics, MIT World Peace University, Pune 411038 rupal.shroff@mitwpu.edu.in

In this paper we study some results on direct summands of Goldie extending elements in modular lattice. An element a of a lattice L with 0 is said to be a Goldie extending element if and only if for every b ≤ a there exists a direct summand c of a such that b∧c is essential in both b and c. Some characterizations of a decomposition of a Goldie extending elements in a modular lattices are given.

IMS Prize Group 4 1. On Some Series Representation between R-function and Fractional Calculus Operators by Ankit Pal, Department of Applied Mathematics and Humanities, Sardar Vallabhbhai National Institute of Technology, Surat, Gujarat [email protected]

In this paper, we propose some fractional integral identities between R-function and Riemann-Liouville fractional integral operators. Further we consider R integral operator to derive series expansions in terms of classical Riemann-Liouville fractional operators We also investigate an analytical solution of fractional free electron laser equation involving R-function.

2. Approximation of Solutions for Nonlinear Functional Integral Equations using Homotopy Perturbation by Lakshmi Narayan Mishra, Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology (VIT) University, Vellore [email protected], [email protected]

This manuscript deals with the solutions of some nonlinear functional integral equa- tions using the concept of measure of noncompactness. Firstly, we verify the existence of solutions to the equation using the generalised Darbo fixed point theorem and then come up with an efficient iterative algorithm to find an approximate solution by applying ho- motopy perturbation method and Adomian decompostion. Also, we justify the efficiency of the algorithm with the help of an example. Finally an error analysis with the upper bound of errors is presented.

3. Aposteriori Error Estimation of Subgrid Multiscale Stabilized Finite Element Method for Transient Stokes Model by Manisha Chowdhury, Indian Institute of Technology, Kanpur [email protected]

In this study, we present a novel stabilized finite element analysis for transient Stokes model. The algebraic subgrid multiscale approach has been employed to arrive at the stabilized coupled variational formulation. Derivation of the stabilized form as well as sta- bility analysis of it’s fully discrete formulation are presented elaborately. Discrete inf-sup condition for pressure stabilization has been proven. For the time discretization the fully implicit schemes have been used. A detailed derivation of the aposteriori error estimate for the stabilized subgrid multiscale finite element scheme has been presented. Numeri- cal experiment has been carried out to verify theoretically established order of convergence.

4. Latest Inversion Free Iterative Scheme for Solving a Pair of Non-linear Matrix Equations by Sourav Shil, Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 39 [email protected]

In this work, the following system of non-linear matrix equations are considered, ∗ −1 ∗ −1 ∗ −1 ∗ −1 X1 + A X1 A + B X2 B = I, X2 + C X2 C + D X1 D = I, where A, B, C, D are arbi- trary n × n matrices, I is identity matrix of order n. Some conditions for existence of positive definite solution are discussed here and also convergence analysis of newly devel- oped algorithm for finding the maximal positive definite solution and its convergence rate as well. Two examples are also provided here in support of our results.

δ,ξ 5. The Polynomial Ln (x) and Fractional Calculus Vinod Kumar Jatav, Department of Applied Mathematics and Humanities, Sardar Vallabhbhai National Institute of Technology, Surat [email protected]

In this paper, we give composition formulas of generalized fractional integral involving δ,ξ the polynomials Ln (x) and also obtain Riemann-Liouville fractional derivative and some δ,ξ integral transforms of polynomials Ln (x). 6. Finite Difference Heat Transfer Analysis in Square Lattice when Pivotal Points Exist near Curved Boundaries by P. Reddaiah, Department of Mathematics, Kadapa, Andhra Pradesh [email protected]

In this paper I did Finite Difference Heat Transfer Analysis in Square Lattice When Pivotal Points Exist Near Curved Boundaries. To determine Pivotal Points at other than curved boundaries we use Central Difference Partial Operator Molecule. To determine Pivotal Points at curved boundaries we use ∇2 operator for unevenly spaced points. Af- ter finding pivotal point values I did heat transfer analysis after plotting in Graphical Representation. How temperature is distributed in Square Lattice is plotted as Contour Diagram. I did heat transfer analysis in x traverse direction. Using Mathematica 9 Ver- sion Software I did Mathematical Computations.

IMS Prize Group 5 1. The Influence of Magnetic and Gravitational Fields in a Non-ideal Dusty Gas with Heat Conduction and Radiation Heat Flux by P. K. Sahu, Department of Mathematics, Government Shyama Prasad Mukharjee College, Sitapur-497111, Chhattisgarh [email protected]

Similarity solutions for a spherical shock wave in a mixture of small solid particles of micro size and a non-ideal gas are discussed under the influence of the gravitational field and magnetic field with conductive as well as radiative heat fluxes. The solid particles are uniformly distributed in the mixture, and the shock wave is assumed to be driven by a piston. It is assumed that the equilibrium flow-conditions are maintained and the moving piston continuously supplies the variable energy input. Due to the central mass at the origin (Roche model), the medium is considered to be under the influence of the gravitational field. The density of the undisturbed medium is assumed to be constant in order to obtain the self-similar solutions. Distribution of gas-dynamical quantities are discussed through figures. It is obtained that the magnetic field has a decaying effect on shock strength. It is shown that due to an increase in the gravitational parameter the compressibility of the medium at any point in the flow field behind the shock front de- creases. The non-idealness of the gas causes a decrease in the shock strength and widens the disturbed region between the piston and the shock. It is interesting to note that in the presence of azimuthal magnetic field the pressure and density vanish at the piston and 40 hence a vacuum is formed at the center of symmetry, which is in excellent agreement with the laboratory condition to produce the shock wave.

2. Mathematical Study of Reflection of qP and qSV Waves from the Stress- free/rigid Surface of a Micro-mechanically Modeled Piezoelectric Fiber- Reinforced Composite Half-space by Sayantan Guha, Department of Mathematics and Computing, Indian Institute of Tech- nology (Indian School of Mines) Dhanbad [email protected]

The present article has two primary objectives: Firstly, to present the micro-mechanics model of Piezoelectric Fiber-Reinforced Composite (PFRC) and demonstrate some of its advantages over monolithic piezoelectric materials. Secondly, to analytically study wave reflection phenomenon at the stress-free/rigid surface of a PFRC. The PFRC struc- ture comprises of PZT-5A fiber-epoxy matrix combination and is modeled employing the Strength of Materials technique with Rule of Mixtures. Some advantages of the PFRC in contrast to monolithic piezoelectric materials are graphically demonstrated. Due to the incidence of a quasi-longitudinal or quasi-transverse wave, three reflected waves viz. quasi- longitudinal (qP), quasi-transverse (qSV), and electro-acoustic (EA) waves are generated in the PFRC. The closed-form expressions of amplitude ratios of all reflected waves are derived utilizing appropriate electro-mechanical boundary conditions at both the stress- free and the rigid surface. However, the amplitude ratios of reflected waves cannot be used exclusively to validate the numerical results. Hence, the expressions of energy ratios of all reflected waves and interaction energy are derived using the amplitude ratios, which exhibit the influence of existing parameters, and the Law of Conservation of Energy is established. Despite the endless advantages of PFRC, no mathematical studies have been performed yet on wave reflection phenomenon at the stress-free/rigid surface of a PFRC half-space. The present work is framed to explore the same for contemplating the phe- nomenon in constructed smart structures. Therefore, this work presents a novel effort to develop a connection between derivation of the composite’s micro-mechanics model and analyze wave reflection phenomenon in it.

IMS Prize Group 6 1. Cross Diffusion Induced Spatiotemporal Pattern in Diffusive Nutrient- Phytoplankton Model with Nutrient Recycling by Sarita Kumari, Department of Mathematics & Computing, Indian Institute of Technology (ISM),Dhanbad [email protected]

In this paper we have proposed a mathematical model of spatiotemporal interaction between the nutrient and phytoplankton. The interaction between nutrient and phy- toplankton has been considered with Holling type-III functional response and nutrient recycling. We have also consider the effect of cross diffusion in the model system. The stability analysis for non-spatial and spatial model system has been carried out. Spe- cial focus has been given to investigate the selection of spatiotemporal patterns in the neighbourhood of a critical parameter using the amplitude equation. Choosing apprope- riate control parameter from the Turing space, existence conditions for stable patterns are derived using the amplitude equations. We have performed the numerical simula- tion and observed the effect of time evolution, cross diffusion and rate of toxin release by phytoplankton on the density distribution of species by spatial patterns. Cross diffusion plays an important role for Turing instability and formation of spots to stripe like pattern.

2. An Answer to the Challenges in a CT-Data based Realistic Complex Artery Network Flow Study by Sumit Kumar, School of Biomedical Engineering, Indian Institute of Technology (BHU), Varanasi 41

[email protected]

Study of realistic human arterial network flow dynamics is well known to be faced with the challenges like a) apt geometric modelling, b) bifurcation zone meshing and c) selec- tion of right rheological model for the capturing the hemodynamics correctly in the entire network under study. In this study, we provide an answer to correctly trace the physics behind the hemodynamics in a complex and realistic 3D human arterial network consist- ing of abdominal artery (AA) branching into left and right iliac arteries and their further branching. For this, we accurately reconstruct the 3D geometry using subject specific CT-Scan DICOM imaging data and devise a correct strategy for mesh generation with MIMICS-18 in complex bifurcation zones and also establish the right rheological model choice for unraveling the physics behind the blood flow dynamics in a complex network. Localized grid generation strategy is used for discretization of the domain. Detailed large scale numerical simulations are carried out by FLUENT solver under finite volume para- digm and results are validated with the experimental studies from the literature. While the results based on Newtonian model and Power-law based Non-Newtonian model look nearly the same in large large arterial segments like AA the Non-Newtonian rheological model is found to give better results in iliac bifurcation zone, where the size of arterial vessel gets reduced to less than one-fourth of that of AA. This work present new tools for the three-dimensional model reconstruction, geometric analysis, mesh generation and the CFD investigation applied to blood flow in complex arterial network. 42

Contributory Papers

Section A: Combinatorics, Graph Theory, Logic, Discrete Mathematics

1. Generalized Fuzzy P and Q-Continuous Maps by Sujeet Kumar Chaturvedi, J.K. Maitra Department of Mathematics and Computer Sciences, R.D. University Jabalpur M.P. [email protected], jkmrdvv@rediffmail.com

In this chapter we have introduced the concept of fuzzy P-sets and fuzzy Q-sets in the category of generalized fuzzy topological spaces. Further we establish several results regarding P-Continuity and Q-Continuity in generalized fuzzy topological spaces.

2. Some Properties of Bipolar Doubt Intuitionistic Fuzzy K-ideals in BCK/ BCI-algebras by R. Angelin Subaa, K.R. Sobhab aAssistant Professor, Department of Mathematics, Women’s Christian College, Nagercoil- 629001, Tamilnadu, India. bAssistant Professor, Sree Ayyappa College for Women, Chunkankadai, Nagercoil. (Affil- iated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli) [email protected], [email protected]

In this research paper, we investigate bipolar doubt intuitionistic fuzzy K-ideals in BCK/ BCI-algebras. The purpose of the study is to introduce the concept of bipolar doubt intuitionistic fuzzy K-ideals are connected with bipolar doubt intuitionistic fuzzy subalgebra and bipolar doubt intuitionistic fuzzy ideals. Here we implement the charac- terization of Bipolar doubt intuitionistic fuzzy K-ideals using doubt intuitionistic positive α-level cut and doubt intuitionistic β-level cut set and some of its properties are analyzed.

3. A Real Life Problem in Decision Making using Hendecagonal Fuzzy Num- bers by Javaid Ahmad Shah, Rakesh Kumar Tripathi Department of mathematics, Dr. APJ Abdul Kalam University Indore (M.P) [email protected]

Most issues in life involve decision processes of one or another form, as trivial as we might consider them. To have ability to make consistent and correct choices is the essence of any decision process pervade with uncertainty. There are number of techniques to solve decision making problems, one such technique is to use Hendecagonal fuzzy numbers. Such fuzzy numbers are used to represent the uncertainty or vagueness of eleven linguistic variables. This paper presents an evaluation regarding the selection of teaching staff for an institute, among several candidates so that appropriate persons are selected which are suitable for the required job and also which will improve the performance of the institute and that of students as well.

4. A Note on the Relationship between Julia Set and Independence Polyno- mial of a Graph by K.U. Sreejaa, P.B. Vinodkumarb, P.B. Ramkumarb aDepartment of Mathematics,K.K.T.M. Government College, Pullut, Thrissur bDepartment of Mathematics, Rajagiri School of Engineering and Technology,Kochi [email protected]

Graph polynomials are widely studied and have found many applications in different fields of science. There are number of graph polynomials.We use independence polyno- mial to construct and study some Julia sets.The various relations between independence polynomial, energy, Julia set and Hausdorff dimension of graphs are closely examined.As a special graph Petersen graph’s connectivity is examined and it is found that its Julia 43 set is disconnected.

5. A Real Life Decision Making Problem as an Application Of Fuzzy Logic by Rakesh Kumar Tripathi, Showkat Ahmad Bhat Dr. A.P.J. Abdul Kalam University, Indore [email protected], [email protected]

Fuzzy sets are very useful tool to tackle with the concept of vagueness and uncertainty in decision making process. In this paper, we apply the theory of fuzzy sets to solve a real world decision making problem. The aim of this paper is to present a fuzzy logic framework for employee selection process.

6. Edge Chromatic Polynomials of S-valued Graphs by A. Arul Devia, V. Thiruvenib aDepartment of Mathematics, Sri Ramanas College of Arts and Science for Women Aruppukottai bDepartment of Mathematics, S.B.K. College Aruppukottai [email protected], [email protected]

In his monograph “Semirings and their applications” Jonathan Golan has introduced the notion of R valued graphs where R is a semiring. In the year 2015 Chandramouleeswaran and his scholars introduced the notion of semiring valued graphs (briefly called S-valued graphs). Many research has been carried out in S-valued graphs such as vertex domina- tion, edge domination, vertex edge domination, colouring of S-valued graphs and so on. This paper discusses the concept of edge S-chromatic polynomial associated with a given S-valued graph. Also we introduce the notion of vertex-edge colouring of a S-valued graph and the corresponding S-chromatic polynomial.

7. Blocks in a S−valued Semigraph by S. Nivetha, M. Chandramouleeswaran Department of Mathematics, Sri Ramanas College of Arts and Science for Women Aruppukottai [email protected], [email protected]

In his monograph on “Semigraphs and their applications”, Sampath Kumar introduced the notion of semigraphs in the year 2000. Since then many researchers worked on the theory of semigraphs. Motivated by introduction of the term R− valued graphs, where R is a semiring, in the year 2015, Chandramouleeswaran and others studied the concept of semiring valued graphs (S−valued graphs). The above two works motivated us to study semiring valued semigraphs. This paper studies the connected sets in a S−valued sem- igraph. Also we discuss the graphs associated with the given S−valued semigraph. We prove some simple characterisations of a block in a S−valued semigraph.

8. Characterization of Semi Splitting Block Graph by Nivedha Baskar, Tabitha Agnes Mangam, Mukti Acharya Department of Mathematics, Christ (Deemed to be University), Bengaluru [email protected]

For any graph G, the semi splitting block graph SB (G) of a graph is obtained by taking a copy of G, for each vertex in G a new vertex is added which is made adjacent to all the vertices of G adjacent to that vertex and for each block in G a new vertex is added which is made adjacent to all the vertices of that block. We study the distance parameters like diameter, radius of semi splitting block graph of cyclic and acyclic connected graph. A graph G is said be self centered if all the vertices of G are of same eccentricity. We charac- terized graphs whose SB (G) are self centered. A graph G is said to be semi splitting block graph if there exist a graph H such that SB (H) =∼ G. We characterized graphs which are 44 semi spitting block graph.

9. Local Vertex Antimagic Labeling for Disconnected Graphs by R. Shankar, M. Nalliah Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore [email protected], [email protected]

Let G = (V,E) be a graph of order p and size q having no isolated vertices. A bijec- tion f : E → {1, 2, 3, ..., q} is called a local antimagic labeling if for all uv ∈ E we have P w(u) 6= w(v), the weight w(u) = e∈E(u) f(e), where E(u) is the set of edges incident to u. A graph G is local antimagic if G has a local antimagic labeling. The local antimagic chromatic number χla(G) is defined to be the minimum number of colors taken over all colorings of G induced by local antimagic labelings of G. In this paper, we determine the local antimagic chromatic number for disconnected graph of path graph, cycle and friendship graph.

10. Local Distance Antimagic Vertex Coloring of Graphs by T. Divya, S. Devi Yamini Department of Mathematics, Vellore Institute of Technology, Chennai [email protected]

Let G = (V,E) be a graph of order n without isolated vertices. The function f : V → P {1, 2, 3, ..., n} be a bijection. The weight w(v) of a vertex v is w(v) = z∈N(v) f(z), where N(v) is the open neighbourhood of v. If w(u) 6= w(v) for any two adjacent vertices u and v, then f is said to be a local distance antimagic labeling. A graph G is local distance antimagic if G admits a local distance antimagic labeling. This induces a proper coloring where the vertex v is assigned the color w(v). The minimum number of colors required for proper coloring of G induced by local distance antimagic labeling of G is called the local distance antimagic chromatic number denoted by χld(G). In this paper, we introduced this new parameter and determined the local distance antimagic chromatic number of some graphs.

11. Topological Indices of the Clustered Graphs by T. Yogalakshmi, S. Gurunathan Department of Mathematics, SAS, VIT, Vellore [email protected]

Clustering deals with different kinds of attributes to identify communities within large group. For the past few decays telecommunication, radar detection, image processing plays a major role in networking. Networking strategy was first used in telecommuni- cation to reduce the time required for a call to go through. Interconnection networks is more efficient in parallel system performance and helps to realize the transportation of data between processors and memory modules. In these recent days, Optoelectronic system architecture have the capability of multiprocessors which are effectively empha- sized by networking. Topological indices transform the molecular graphs into a numeric quantity which characterize the topology of that graph. Topological descriptors find ap- plications in quantitative structural properties, quantum chemistry, crystalline materials and pharmaceutical fields. Now-a-days researchers scrutinize the concepts on Topological indices such as, Randic index, Zagreb index, atom-bond connectivity index which are ex- ploited to estimate the bioactivity of chemical compounds and the premises of topology in networking. In this paper, clustered graphs have been orginated from an algorithm. De- gree based indices such as first and second Zagreb, first and second Multiplicative Zagreb, Hyper Zagreb, Atom-Bond Connectivity (ABC) and Geometric indices of the clustered graphs have been computed. Moreover, the regression analysis on randic, SCI, ABC, ISI and GA indices of the clustered graphs have been established which intreprets that those indices are highly correlated with R = 0.99. Besides the neighbourhood based indices such 45

as ABC4 and GA5 have been found.

12. A Note on Reversibility Related to Idempotents by P. Jaish, Ramanujan School of Mathematical Sciences, Department of Mathematics, Pondicherry University, Puducherry - 605 014, India

Let R be a ring and I(R) denote the set of all idempotent elements in R . In this article, we show that the ring of 2 × 2 matrices over an arbitrary reversible ring R with I(R) = {0, 1} is quasi-reversible, which is an answer to the question in [Bull. Korean Math. Soc., 56(4) (2019) 993-1006] given by Da Woon Jung et al. Further, we show that the converse part is also true.

13. Solving Transportation Problem using Graph algorithms by Kanchana M, Kavitha K Department of Mathematics, Vellore Institute of Technology, Vellore [email protected]

Graphs are the tool for modeling and description of real life network systems. Many graph algorithms plays huge role in finding the optimal path for the transportation prob- lems with many aspects on it. The transportation, social networks, pipelines are all considered as a graph or a network model and solved using the shortest path algorithms. In this we obtain the optimal path by Dijsktra, Kruskal, Generalized Dijsktra, Extension of Kruskal and Path labeling algorithm. The complexity of the algorithms are listed.

14. Results on Generalized Divisor Sum Function ση of Certain Standard Graphs by R. Vignesha, Kalyani Desikana, A. Elamparithib aDivision of Mathematics,SAS, VIT, Chennai bDepartment of Mathematics, Dr. G S Kalyanasundaram Memorial School, Kumbakonam [email protected]

In general, for any non-negative number n ∈ R, the generalized divisor sum function th ση(n) is defined as the sum of the non-negative factors of n that are raised to η pow- P η ers, where η ∈ N.It can be represented as ση(n) = f This paper is concerned with f/n calculating the general form of ση(n) which we denote as Sη and we arrived at the exact expressions for the generalized divisor function for few standard graphs. Some standard graphs that we considered are Crown Graph, Gear Graph, Helm Graph, Flower Graph, Web Graph, Generalized Peterson Graph, CnCm, Cocktail Party Graph, Tadpole Graph, Windmill Graphs, Zero-Divisor Graphs, and certain classes of Divisor Function Graphs.

15. Local Distance Antimagic Graphs by V. Priyadharshini, M. Nalliah Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore - 632014 [email protected], [email protected]

Let G be a graph with n vertices and m edges with no isolated vertices. A bijection f : V → {1, 2, 3, . . . , n} is called local distance antimagic labeling, if any two adjacent ver- tices u and v with its vertex-weights w(u) 6= w(v), where vertex-weight w(u) is defined as P w(u) = wN(u) f(w). The local distace antimagic chromatic number χda(G) is defined to be the minimum number of colors taken over all colorings of G induced by local distace antimagic labelings of G. In this paper, we introduced a new parameter χda(G) and we obtained some basic results. We determied the local distance antimagic labeling for star graph, sub-divisional star graph and double star graph. 46

16. Local Edge Antimagic Labeling of Cycle Related Graphs by S. Rajkumar, M. Nalliah Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore - 632014 [email protected], [email protected]

Let G = (V,E) be a graph of order p and size q having no isolated vertices. A bi- jection f : V → {1, 2, 3, ..., p} is called a local edge antimagic labeling if any two ad- 0 0 jacent edges e = uv and e = vw of G, with w(e) 6= w(e ), where the edge weight is 0 w(e = uv) = f(u) + f(v) and w(e ) = f(v) + f(w). A graph G is local edge antimagic if G 0 has a local edge antimagic labeling. The local edge antimagic chromatic number χlea(G) is defined to be the minimum number of colors taken over all colorings of G induced by local edge antimagic labelings of G. In this paper, we obtain the local edge antimagic chromatic number is maximum degree of G, where G is a Generalized friendship graph of cycle graph, Generalized friendship graph of fan graph and Disconnected graph of Cycles kCn.

17. Genetic Algorithm to the Biobjective Multiple Travelling Salesman Problem by Shayathri Linganathan, Purusotham Singamsetty Department of Mathematics, Vellore Institute Technology, Vellore [email protected], [email protected]

The travelling salesman problem (TSP) and its variants have been studied extensively due to its wide range of real world applications, yet there are challenges in providing the efficient algorithms to deal some of its variants. The multiple travelling salesman problem (MTSP) is the generalization of TSP, aims to determine m-shortest routes for msalesmen all of which start and end at a depot city such that they cover all the n-cities without intervening. This leads to disproportionate distribution of number of cities to be covered by each salesman. This paper presents, a biobjectve MTSP (BMTSP), where the first objective is to minimize the total travel distance and the other minimizes the total cost along with the load balancing constraint. The optimal solution of this problem may not be possible at one point as it involves tradeoff between two objectives. A metaheuristic al- gorithm called Genetic Algorithm (GA) is proposed to obtain the Pareto efficient routes of BMTSP. Firstly, the n-cities are proportionately distributed to msalesmen next the initial solution is generated with the help of greedy search algorithm feed into GA. Further, GA is designed by inducting different operators such as crossover, mutation and reverse oper- ators to provide the Pareto routes. The experiments are carried out on different datasets from TSPLIB. The simulation results show the efficiency of the proposed algorithm.

18. Signed Roman Domination in an Interval Graph with Adjacent Cliques of Size 3 by M. Reddappaa, C. Jaya Subba Reddyb, B. Maheswaric aDepartment of Mathematics, S. V. University, Tirupati-517502. bDepartment of Mathematics, S. V. University, Tirupati-517502. cProfessor (Retd.), Department of Applied Mathematics, S. P. MahilaVisvavidyalayam, Tirupati-517502. [email protected], [email protected], [email protected]

The theory of Graphs is an important branch of Mathematics that was developed exponentially. The theory of domination in graphs is rapidly growing area of research in graph theory today. It has been studied extensively and finds applications to various branches of Science & Technology. Interval graphs have drawn the attention of many researchers for over 40 years. They form a special class of graphs with many interesting properties and revealed their practical relevance for modeling problems arising in the real world. The theory of domination in 47 graphs introduced by Ore and Berge is a fast growing area of research in graph theory today. An introduction and an extensive overview on domination in graphs and related topics is surveyed and detailed in the two books by Haynes et.al. In this paper a study of signed Roman domination in an interval graph with adjacent cliques of size 3 is carried out.

19. Claw-decomposition of Kneser Graphs by C. Sankari, Department of Mathematics, A.V.V.M. Sri Pushpam College (Affiliated to Bharathidasan University), Poondi, Thanjavur, Tamil Nadu [email protected]

A claw is a star with three edges. The Kneser graph KGn,2 is the graph whose vertices are the 2-element subsets of n-elements, in which two vertices are adjacent if and only if their intersection is empty. In this paper, we prove that KGn,2 is claw-decomposable, for all n ≥ 6. Further, we discuss about claw-decomposition of KGn,2(λ).

20. On the Zero Forcing Number of Complementary Prism graphs by M.R. Raksha, C. Dominic Department of Mathematics, Christ(Deemed to be university), Karnataka [email protected]

Zero forcing number of a graph G denoted by Z(G) is the minimum cardinality among all the zero forcing sets of the graph G. In this paper, we find few bounds on zero forcing number of the complementary prism graph GG based on the maximum degree of the graph G. Also we determine the zero forcing number of few complementary prism graph of some basic graphs. Finally, we find Zero forcing number of some special type of complementary prism graph. We establish the result, if G or G is disconnected graph of order n, then Z(GG) = n − 1, as one of the main result.

21. Edge Chromatic number of Zero-divisor graphs of some Semi-local rings by Subhash Mallinath Gadeda, Nithya Sai Narayanab aR.K. Talreja College of Arts, Science & Commerce, Ulhasanagar−03, District: Thane, Mumbai bN.E.S Ratnam College of Arts, Science & Commerce, Bhandup(W), Mumbai [email protected], [email protected]

A simple graph G is said to be of Class-1 if the Edge Chromatic number χ1(G) = ∆(G). In this paper we prove that the zero-divisor graphs of the Semi-local ring R = F1 ×F2 ×F3, where F1,F2,F3 are finite fields, belongs to Class-1.

22. S-cordial and Total S-Cordial Labeling in Signed Graphs by Divya An- toney, Tabitha Agnes Mangam, Mukti Acharya Department of Mathematics CHRIST (Deemed to be University), Bengaluru [email protected]

Let S = (G, σ) be a signed graph, where G is the underlying graph. The line signed graph L(S) of a signed graph S is defined by V (L(S)) = E(S) and the vertices are joined by an negative edge in L(S) if and only if the corresponding edges are negative and adja- cent to each other in S and positive in all other cases. Let L(S) be a line signed graph of S whose vertices have signs of the corresponding edges in S. The L(S) is said to be S-cordial if the difference between the positive and negative signs of the vertices (edges) differ by atmost one in L(S). The L(S) is said to be total S-cordial if the difference between the positive and negative signs in L(S) differ by atmost one in L(S). In this paper, we char- acterized signed graphs such as signed paths, signed cycles, signed star and signed bistars whose line signed graph admit S-cordial and total S-cordial labeling. We also find the characterization of some classes of signed graphs whose line signed graph admit S-cordial 48 and total S-cordial labeling with respect to negation and switching in them.

23. A Note on Edge-fault Tolerance in Augmented Cubes by Amruta Shinde, Y.M. Borse Department of Mathematics, Savitribai Phule Pune University Pune, 411 007 [email protected]

The augmented cube AQn is a variation of the hypercube network. In this paper, we obtain an upper bound on conditional h-edge connectivity of AQn, for odd integer h. Fur- thermore, we prove that in augmented cube AQn (n ≥ 3) if h = 3 then the upper bound is same as the lower bound on conditional 3-edge connectivity which is equal to 8n − 16.

Section B: Algebra, Number Theory, Lattice Theory and History of Mathematics

1. On Cechˇ Fuzzy Interior Spaces and Fuzzy Pretopological Spaces deter- mined by Implicators by Abha Tripathi, S.P. Tiwari Department of Mathematics & Computing, IIT (ISM), Dhanbad [email protected]

This paper is towards the study of Cechˇ fuzzy interior spaces and fuzzy pretopolog- ical spaces determined by general implicators. Further, we discuss some results based on Cechˇ fuzzy interior spaces and fuzzy pretopological spaces. Also, we show that the results regarding one-to-one correspondence between fuzzy reflexive approximation spaces and Cechˇ fuzzy interior space as well as fuzzy reflexive approximation spaces and fuzzy pretopological spaces.

2. Demonstration and Proof of a Unique Property of Mersenne Primes by Pranav Narayan Sharma Delhi Public School, Ahmedabad [email protected]

Till date, as many as more than 50 Mersenne primes, have been discovered using GIMPS. A Mersenne prime holds unique properties which can be used in computational processing. There is only one Mersenne prime that ends with unit digit three and rest of them end with either one or seven. This paper presents the unique property exhibited by Mersenne Primes using cyclicity which can be extended to double Mersenne Primes.

3. Data Encryption to Decryption by Using Laplace Transform by B. Ramu Naidua, K.P.R. Sastryb, D.M.K. Kiranc aFaculty of Mathematics, AU PG Campus, Vizianagaram, Andhra Pradesh bGVP College of Engineering (Autonomous), Visakhapatnam, Andhra Pradesh cDepartment of Mathematics, Vizag Institute of Technology, Visakhapatnam [email protected], [email protected],[email protected]

In this paper, we introduce an encryption and decryption procedure with high security by mathematical model, using Laplace transformation and Inverse Laplace Transforma- tion for the given transforming data from one end to other end. We also give an example. Here we convert plain text to ASCII code.We take two primes as a primary key for en- cryption and decrypt in of the original data.

4. Principal Ideal in Regular Rings by Anupam Rachna, B.N. Mandal University Madhepura [email protected] 49

In this paper we have study the impact of Von-Neumann regularity of a ring on some of the other concept in a ring. The impact is most pronounced in case of ideals. For rings in general, the collection of principal right ideals need not form a lattice, let alone a complemented lattice and so Von-Neumann was led to the class of regular rings, which may be described as those rings in which the collection of principal right ideals form a lattice under the obvious lattice operations. Von-Neumann showed that in a regular ring, the sum and the intersection of finitely many principal right ideals are also principal right ideals. Thus in any regular ring, the collection of principal right ideals forms a comple- mented modular lattice.

5. Notion on Rough Fuzzy Ideals in γ−rings and its Properties by P. Durgadevi, Ezhilmaran Devarasan Department of Mathematics School of Advanced Sciences Vellore Institute of Technology, Vellore [email protected], [email protected]

Rough set theory plot a new mathematical approach to inadequate understanding. In this approach ambiguity is expressed by a boundary region of the set. Rough set concept can be defined by means of algebraic operator union and intersection called approxima- tions. In this article, we define the notion of rough fuzzy ideals in γ−rings and proved some of its properties.

6. Cubic Magnified Translation on β−Subalgebras by P. Muralikrishnaa, R. Vinodkumarb, G. Palanic aPG and Research Department of Mathematics, Muthurangam Government Arts College (Autonomus),Vellore-632002 bDepartment of Mathematics, Prathyusha Engineering College, Thiruvallur cDepartment of Mathematics, Dr. Ambedkar Government Arts College, Chennai [email protected]

The theory of fuzzy sets introduced by Zadeh which has immense range of applications in various branches of mathematics. Jun et al. proposed a new concept called cubic sets which is applied into several algebraic structures like BF,KU & β-algebras and so on. In this article, the notion of cubic magnified translation on β−subalgebra is introduced and verified some results of cubic β-subalgebra using the idea of cubic magnified transla- tion(CMT). Also, the characteristics of cubic magnified translation on β−subalgebra is studied based on the fundamental operations like union and intersection.

7. Structure of MBJ - Neutrosophic Set applied on β-Filter by Prakasam Muralikrishna, Surya Manokaran PG & Research Departement of Mathematics, Muthurangam Govt. Arts College, Vellore [email protected]

In 1965, Zadeh introduced Fuzzy Set and researchers developed the fuzzy sets to many kinds of fuzzy sets like intuitionistic, interval valued fuzzy set and many. And Smaran- dache came up with the concept of Neutrosophic set is welcomed by the current researchers and many research work are explored using Neutrosophic Set. In recent days the study of MBJ-Neutrosophic set is investigated and is merged with some algebraic structures like BCK/BCI and β-algebra. The notion of filters was introduced by Henri Cartan in 1937. In 1991, C. S. Hoo introduced the concept of the filters in BCI-algebras. Also in 2013, A. Rezaei and A. Bourmand introduced the notion of geberalized fuzzy filters of BE-algebras. In this paper, the structure of MBJ-Neutrosophic β-subalgebra is considered and is cul- tivated to MBJ-Neutrosophic β-filter. Several results such as image, preimage and levels on filters are discussed.

8. A Fast Prime-Factorization of Large Integers and Its Applications to Affine Ciphers 50 by Blankson Henrya,b, Chattamvalli Rajana aDepartment of Mathematics, Vellore Institute of Technology, Vellore, Tamil Nadu, India bStatistics, Mathematics and Computer Studies Department, Cape Coast Technical Uni- versity, Cape Coast, Ghana [email protected]

This paper seeks to provide an alternative and fast prime-factorization algorithm for large integers. We also considered the decryption operation in a given Affine Cipher with a modulus (m). We also assumed that the gcd (A, m) = 1. To decrypt any cipher text, we need to solve the congruence Y ≡ AX +B(mod m) for x. The encryption function is of the form Y = Ek(X) = (AX + B)(mod m), where A, B ∈ Zm. Consequently, the decryption −1 function will also be of the form X = Dk(Y ) = D (YB)(mod m). Since gcd (A, m) = 1, A has a multiplicative inverse modulo m. It was established that the congruence will have a unique solution but it does not give us an efficient and fast method of finding the solution. We will therefore require an efficient algorithm to do this. Furthermore, a modular arithmetic and a fast prime-factorization of large integers provided us with the efficient decryption algorithm that we seek. This paper considered the application of fast prime-factorization of large integers and how to apply them to Affine Ciphers.

9. The G-vetex Colour Partition Algebra as a Centralizer Algebra of An ×G by A. Joseph Kennedy, P. Sundaresan Department of Mathematics, Ramanujan School of Mathematical Sciences, Pondicherry University, Puducherry - 605 014, India

We are going to restrict the generalized Jones result in [PK] which says that the G- vertex colored partition Pk(n, G) is the centralizer algebra of an action of the direct product of symmetric group and G Sn × G on tensor products of its permutation representation to the action of the direct product of alternating group and G An × G. Herein, we determine a basis for the centralizer algebra and exhibit at the moment the centralizer is isomorphic to the G-Colored partition algebra. Also we do the same for Extended G-vertex colored partition algebras Pˆk(n, G). 10. Commutativity of Prime Rings with Symmetric Biderivations Satisfying Certain Relations by C. Jaya Subba Reddya, Ramoorthy Reddyb aDepartment of Mathematics, S V University, Tirupati bS V Engineering College, Tirupati [email protected]

For any prime ring R,U be a nonzero ideal of R and B1(., .): R × R → R be a sym- metric biderivation of R. In the current paper, it was showed that R is commutative if and only if it satisfies one of the following properties (i) B1(uv, w) − uv ∈ Z(R), (ii) B1(uv, w) + uv ∈ Z(R), (iii) B1(uv, w) − vu ∈ Z(R), (iv) B1(uv, w) + vu ∈ Z(R), (v) B1(u, v)D(v, w) − uw ∈ Z(R), and (vi) B1(u, v)D(v, w) + uw ∈ Z(R), for any u, v, w ∈ U.

11. Some Results on the Density of Integral Sets with Missing Differences by Neha Rai, Ram Krishna Pandey Department of Mathematics, Indian Institute of Technology, Roorkee [email protected]

Let M be a set of positive integers. Motzkin posed the problem of finding the maximal density µ(M) of the sets S of nonnegative integers in which no two elements of S are allowed to differ by an element of M. In 1973, Cantor and Gordon find µ(M) for |M| ≤ 2. Until now, there is no known general formula for µ(M) in the case |M| ≥ 3. Several partial results are known in the case |M| ≥ 3 including some results in the case when M is an infinite set. Motivated by the earlier families M = {a, b, a + b} and M = {a, b, a + b, b − a} taken by Liu and Zhu (J. Graph Theory 47(2) (2004), 129-146), we study the maximal 51 density problem for the families M = {a, b, b − a, n(a + b)} and M = {a, b, a + b, n(b − a)}. For both these families, we find some exact values and some bounds on µ(M). We also see the connection of parameter κ(M) with µ(M). This number theory problem is also related to various chromatic number problems of the distance graphs generated by M in graph theory.

12. Matrices over Non-commutative Rings as Sums of Fourth Powers by Deepa Krishnamurthi, St Mira’s College for Girl’s, Pune [email protected]

Let R be a noncommutative ring with unity and n ≥ 2 . In this paper we prove that an n × n matrix A over R is a sum of fourth powers if and only if trace(A) is a sum of fourth powers and 2(sum of squares) and commutators modulo 4R. This extends the results of S. A. Katre, Anuradha Garge in the case of commutative rings.

13. Connecting Monomiality Questions with the Structure of Rational Group Algebras by Gurmeet K. Bakshia, Gurleen Kaurb aCentre for Advanced Study in Mathematics,Panjab University, Chandigarh bDepartment of Mathematics, Sri Guru Gobind Singh College, Chandigarh [email protected]

In recent times, there has been a lot of active research on monomial groups in two differ- ent directions. While group theorists are interested in the study of their normal subgroups and Hall subgroups, the interest of group ring theorists lie in the structure of their ra- tional group algebras due to varied applications. Revisiting Dade’s celebrated embedding theorem which states that a finite solvable group can be embedded inside some monomial group, it is proved here that the embedding is indeed done inside some generalized strongly monomial group. Still unresolved monomiality questions have been correlated by proving that all the classes of monomial groups where they have been answered are generalized strongly monomial. The study also raises some intriguing questions weaker than those asked by Dornhoff and Isaacs in their investigations.

14. Quasi Factorable Incidence Functions by L. Madhavia, Y. Rajasekhara Gowdb aDepartment of Applied Mathematics Yogi Vemana University, Kadapa -516005. bDepartment of H & B S G. Pulla Reddy Engineering College, Kurnool - 518007 [email protected]

The study of multiplicative and completely multiplicative arithmetic functions consti- tutes an important branch of Number Theory. Weisner and Ward have studied arith- metical functions from Lattice Theoretic point of view by generalizing the notion of an arithmetical function to that of an incidence function of a locally finite partially ordered set. D.A. Smith and H. Schield introduced the notion of factorable incidence function cor- responding to multiplicative arithmetic function and studied the properties of factorable functions analogous to the well-known properties of multiplicative functions in Number Theory. Recently the authors have introduced the notion of completely factorable inci- dence function analogous to the notion of completely multiplicative arithmetic function and obtained certain conditions under which a factorable incidence function is completely factorable. D. B. Lahiri introduced the notion of quasi-multiplicative arithmetical function and studied the conditions under which a multiplicative arithmetic function is quasi-multiplicative. Haukkanen also studied quasi-multiplicative arithmetical functions of n-variables as well as quasi A-multiplicative functions of one variable. One can also find a discussion of quasi-multiplicative functions in Chapter XI of “Classical Theory of Arithmetical Functions”. 52

In this paper we introduce the concept of quasi-factorable incidence function of a lo- cally finite partially ordered set analogous to quasi-multiplicative arithmetical function and obtain conditions under which a factorable incidence function is quasi-factorable.

15. Commutarias and Commutator Subgroup of finite p-groups by Rahul Kaushik, Manoj K. Yadhav Harish-Chandra Research Institute, Prayagraj [email protected]

Let G be a finite group with γ2(G) its commutator subgroup and K(G) := {[x, y]|x, y ∈ G}. The problem whether γ2(G) is equal to K(G) or not for a group G has been inves- tigated for various classes of finite groups. It was proved that if γ2(G) is an Abelian p-group, p > 3, generated by atmost 3 elements, then K(G) = γ2(G). Recently this result has been proved for finite p-groups, by relaxing the condition of commutativity of γ2(G), i.e., if G is a finite p-group, p > 3, and γ2(G) is generated by at most 3 elements, then K(G) = γ2(G). But such a result is not true in general for all finite p-groups G such that γ2(G) is minimally generated by more that 3 elements. In this paper I will present 4 a classification of finite p-groups G with γ2(G) of order p and exponent p such that each element of γ2(G) is a commutator.

Section C: Real and Complex Analysis (including Special Functions, Summability and Transforms etc) and Teaching of Mathematics 1. On Subclasses of Univalent Functions Defined By Opoola Differential Operator by A.N. Metkaria, N.D. Sangleb, S.P. Handec aDepartment of Mathematics, Visvesvaraya Technological University, Belagavi, Karnataka- 590018, India bDepartment of Mathematics, Annasaheb Dange College of Engineering and Technology, Ashta, Maharashtra-416301, India cDepartment of Mathematics, Vishwanathrao Deshpande Institute of Technology, Haliyal, Karnataka-581329, India [email protected]

The subclass AR (n, ρ, σ, δ, γ, µ, η) of the univalent functions with negative coefficients determined by the differential operator Opoola Dn has been introduced in this paper. Sharp results for coefficient estimates have been obtained, Hadamard product, Bounds of closure, Bounds of distortion and some other results.

2. On Multiplication and Division Theorems of Entire Algebroidal Func- tions of their Relative Growth Indicators of Higher Index in the Light of p-adic Analysis by Sanjib Kumar Dattaa, Aditi Biswasb aProfessor, Department of Mathematics, University of Kalyani P.O. Kalyani, Nadia - 741235, West Bengal, India bAssistant Professor,Department of Mathematics, Fakir Chand College, Diamond Harbour, Diamond Harbour, South 24 Parganas-743331, [email protected] ,[email protected]

Several ways of estimating of comparative growth analysis of entire algebroidal func- tions of their different kind of higher order relative growth indicators in the light of p-adic analysis, where p being a prime integer have been elaborately studied in this paper. Some examples are provided in order to justify the results obtained here. 53

3. Bounds for Probability of the Genaralized Distribution for Certain q-starlike and q-convex Error Functions Related to Shall-Shaped Region by K. Saritha, K. Thilagavathi School of Advanced Sciences, Vellore Institute of Technology, Vellore [email protected],[email protected]

The error function occurs widely in multiple areas of mathematics, mathematical physics and natural sciences. In this paper , using the concept of subordination and notion of q-operators, we defined a new subclass of q-difference operator for certain classes of the Spirallike Starlike and Convex error function . Also we estimate, The Fekete-Szego and Hankel determinant results associated with Shell-Shaped function for the new function.

4. Boas Transform of Wavelets and their Applications by Leena Kathuriaa, Nikhil Khannab aDepartment of Mathematics, Amity Institute of Applied Sciences, Amity University, Sector 125, Noida - 201313 (U.P.) bDepartment of Mathematics, Motilal Nehru College, University of Delhi, Delhi [email protected]

In 1936, Boas introduced an integral transform associated to the Hilbert transform which emerged due to the study of the class of functions having Fourier transform which vanishes on a finite interval. Later, in 1960, Goldberg studied this transform in detail and gave some significant results and properties. This transform was known by Boas transform. In this talk, we introduce the notion of Boas transform of wavelets and give its applications in the form of Boas transform wavelet convolution and cross-correlation theorems to analyze Boas transform of convolved (cross-correlated) signals. Analogously to Bedrosian theorem, Boas transform product theorem is also given.

5. On Configurations of Five Periodic Herman Rings by Gorachand Chakraborty, Department of Mathematics, Sidho-Kanho-Birsha University Purulia, West Bengal, Pin-723104 [email protected]

In this paper, we have studied the configurations of Herman rings for a special class M0 of meromorphic functions having at least one omitted value. We have shown that possible number of configurations of a 5-periodic Herman rings of function in M0 is six. Also we have investigated that the number of 5-cycles of Herman rings of the function is at most one. We have given a result about the non-existence of a 3-periodic Herman rings and a 5-periodic Herman rings simultaneously. Few examples of transcendental meromorphic functions which do not have any Herman rings are discussed. Finally, we end up with the conclusion section which may spark new problems for future research interest.

6. Convolution Conditions for New Subclass of Negative Analytic Functions Associated with Polylogarithm Functions Defined by Linear Differential Operator by M. Thirucheran, C. Selvi Post Graduate and Research Department of Mathematics, L N Government College, Ponneri, Chennai - 624 302, University of Madras, Tamil Nadu, INDIA. [email protected]

n In this paper, we introduce and study some properties for the new subclass Tβ,γ,δ,b (a) of polylogarithm function which is associated with the convolution of differential operator n P∞ n 1+c P k Lλ,δf(ξ) = ξ − k=2[1 + (k − 1)δ] k+c λ akξ Also, we obtain coefficient inequalities, growth and distortion, partial sums and integral operator. 54

7. Obtain Subclass of Multivalant Function Connected with Convalution of Polylogarithm Functions by M. Thirucheran, A. Anand Post Graduate and Research Department of Mathematics, L N Government College, Ponneri, Chennai - 624 302, University of Madras, Tamil Nadu, INDIA. [email protected]

n,p In this present work, we investigate some properties for the subclass Pβ,λ,δ,b(φ(z)) of n,p analytic function related with the linear differential operator Rλ,δ f(z) defined by poly- logarithm functions . And also, we obtain coefficient inequalities, extreme points, radii n,p of convexity and starlikeness, growth and distortion bounds for the subclass Pβ,λ,δ,b(φ(z)). 8. On Extension of Mittag-Leffler Function by Sunil Joshi, Department of Mathematics & Statistics, Manipal University Jaipur, Jaipur, India drsuneeljoshi@rediffmail.com, [email protected]

In this paper, we study the extended Mittag-Leffler function by using generalized beta function and obtain various differential properties, integral representations. Further we discuss Mellin transform of these functions in terms of generalized Wright hypergeometric function and evaluate Laplace transform, Whittaker transform in terms of extended beta function. Finally, several interesting special cases of extended Mittag-Leffler functions have also be given.

9. A Note on the Convergence of Wavelet Fourier Series by Varsha Karanjgaokara, Namrata Shrivastavb aDepartment of Mathematics, Govt. N.P.G.College of Science, Raipur bDepartment of Mathematics, Govt. Kavyopadhyay Hiralal College, Abhanpur, Raipur [email protected], [email protected] In this paper we discuss the rate of convergence of Wavelet Fourier series of periodic functions. Our result generalizes the results of Skopina,M.(Localization Principle for wavelet expansion, Self seminar systems, Dubna,(1999), 125-133) and Karanjgaokar,V.et al(communicated).

10. A New Subclass of Negative Multivalent Functions Involving Polyloga- rithm Functions by M. Thirucheran, M. Vinoth Kumar Post Graduate and Research Department of Mathematics, L N Government College, Ponneri, Chennai - 624 302, University of Madras, Tamil Nadu [email protected]

In this current work, we introduce and study some properties for the new subclass n,p n,p Nβ,γ,δ,b(φ(ξ)) of polylogarithms functions associated with the differential operator Dλ,δ f(ξ). Also, we obtained coefficient inequalities, integralmeans of inequalities, extreme points and distortion of the class.

11. On the Study of Deficiencies of Differential Equation under the Flavour of p-adic Co-prime Polynomial by Sanjib Kumar Dattaa, Ashima Bandyopadhyayb aDepartment of Mathematics, University of Kalyani P.O.: Kalyani, Dist: Nadia, Pin: 741235, West Bengal, India bRanaghat Brojobala Girls High School (H.S.) P.O.: Ranaghat, Dist.:Nadia, Pin: 741201, West Bengal, India [email protected] 55

Let K be an algebrically closed field of charecteristic 0, complete with respect to a p-adic absolute value and A (K) represents the K-algebra of analytic functions in K i.e, the set of power series with an infinite radius of convergence. In this paper,we will give a brief outline on co-prime polynomial of p-adic meromorphic functions. Also we wish to investigate some properties of deficiencies of p-adic differential equation. We also discuss here about holomorphic curve in the same line. A few examples are given here to justify the result obtained.

12. Few Results on Relative (k, n) Valiron Defects from the View Point of Integrated Moduli of Logarithmic Derivative of Entire and Meromorphic Functions by Sanjib Kumar Dattaa, Sukalyan Sarkarb, Ashima Bandyopadhyayc, Lakshmi Biswasd aDepartment of Mathematics, University of Kalyani P.O.: Kalyani, Dist.: Nadia, PIN: 741235, West Bengal, India bDepartment of Mathematics, Dukhulal Nibaran Chandra College P.O.: Aurangabad, Dist.: Murshidabad, Pin: 742201, West Bengal, India cRanaghat Brojobala Girls High School (H.S) P.O.: Ranaghat, Dist.: Nadia, Pin: 741201, West Bengal, India dKalinarayanpur Adarsha Vidyalaya P.O.: Kalinarayanpur, Dist.: Nadia, Pin: 741254, West Bengal, India [email protected]

Xiong (1967) has shown various relations between the usual defects and the rela- tive defects. Singh (1984) introduced the term relative defect for distinct zeros and poles and established various relations between it. The relative (k, n) Nevanlinna de- (k) (k) fect Rδ(n)(α; f) and the relative (k, n) Valiron defect R∆(n)(α; f) of ‘α’ with respect to f (k) for k = 1, 2, 3, ...... and n = 0, 1, 2, 3, ..... are respectively defined as limit inferior of the       ratio N r, α; f (k) and T r, f (n) as r → ∞ and limit superior of the ratio N r, α; f (k)   and T r, f (n) as r → ∞ . The prime target of this paper is to compare some relative (k, n) Nevanlinna defects with relative (k, n) Valiron defects from the view point of inte- grated moduli of logarithmic derivative of entire and meromorphic functions where k and n are any two non-negative integers.Some related examples are provided here in order to validate the result obtained.

13. Common Fixed Point Theorems for a Pair of Mappings in Bicomplex Valued Metric Spaces by Sanjib Kumar Dattaa, Dipankar Palb, Rakesh Sarkarc, Arghyatanu Mannad aDepartment of Mathematics, University of Kalyani, P.O.: Kalyani, Dist: Nadia, PIN- 741235, West Bengal, India. bDepartment of Mathematics, Prof. Syed Nurul Hasan College, P.O.: Farakka Barrage, Dist: Murshidabad, PIN-742212, West Bengal, India. cDepartment of Mathematics, Gour Mahavidyalaya, P.O.: Mangalbari, Dist: Malda, PIN- 732142, West Bengal, India dMousini Co-operative High School(H.S.), Bagdanga, Fraserganj Coastral, Kakdwip, South 24 Parganas, PIN-743357, West Bengal [email protected]

The theory of bicomplex numbers is acknowledged as an area of active research for quite a long period of time in search of special algebra. The concept of bicomplex num- bers is widely used in the literature as it becomes a viable commutative alternative to the non-skew field of quaternions, both are four dimensional and generalizations of complex numbers. In this paper we define the ‘max’ function for the partial order -i2 on a set of bicomplex numbers and study some common fixed point theorems for a pair of mappings satisfying a quasi-contraction condition in a bicomplex valued metric space. Some relevant examples are provided in support of our theorems. Our result is the generalization of the 56 result obtained by Verma & Pathak (2013).

14. On g-Mellin Transform: Construction, Convexity and Applications by Chandrani Basu, Pankaj Jain, Vivek Panwar Department of Mathematics, South Asian University, New Delhi, India [email protected]

Integral transforms play an important part in solving many differential and integral equations. Historically, in 1876, Riemann, first recognized the Mellin transform. In 1894, Mellin gave an elaborate discussion of the Mellin transform and its inversion formula. In the framework of g-calculus, the Mellin transform has been defined and studied. For the new g-Mellin transform, the appropriate convolution is defined and its connection with the Hausdorff operator is pointed out. The notion of pseudo-logarithmic convexity (concavity) has been introduced and it is proved that the g-Mellin transform is pseudo-logarithmically convex (concave) for a suitable pseudo-exponential function. This leads to defining the g- gamma function. Finally, certain applications of g-Mellin transform are provided, namely, solving integral equations and a Titchmarsh type theorem.

15. Common Fixed Point Theorems for Three Self Mapping in Bicomplex Valued Metrix Spaces by Sanjib Kumar Dattaa, Rakesh Sarkarb, Nityagopal Biswasc, Jayanta Sahaa aDepartment of Mathematics, University of Kalyani, P.O.: Kalyani, Dist.: Nadia, Pin: 741235, West Bengal, India bDepartment of Mathematics, Gour Mahavidyalaya, Mangalbari, Malda, West Bengal, India cDepartment of Mathematics, Chakdaha College, Chakdaha, Nadia, West Bengal, India [email protected]

During the last fifty years, fixed point theories in complex valued metric spaces are emerging areas of works in the field of the complex as well as functional analysis. Banach’s fixed point theorem plays a major role in the fixed point theory. It has applications in many branches of mathematics. The famous Banach’s theorem states that if (X, d) be a metric space and T be a mapping of X into itself satisfying d(T x, T y) ≤ kd(x, y), ∀x, y ∈ X, where k is a constant in (0, 1), then T has a unique fixed point x∗ ∈ X. In the paper we prove some common fixed point theorems for three self mappings in a bicomplex valued metric space. Our results generalize the literature due to Azam (2011) & S. K. Mahanta (2012) by using both the ideas of two weakly compatible mappings and rational contrac- tions for a pair of mappings in bicomplex valued metric space.

16. Some Common Fixed Point Theorems in Bicomplex Valued Metric Spaces under both Rational type Contraction and Coupled Fixed Point Map- pings by Sanjib Kumar Dattaa, Rakesh Sarkarb, Nityagopal Biswasc, Ashima Bandyopadhyayd aDepartment of Mathematics, University of Kalyani, P.O.: Kalyani, Dist.:Nadia, Pin:741235, West Bengal, India bDepartment of Mathematics, Gour Mahavidyalaya, P.O.: Mangalbari, Malda, Pin: 732142, West Bengal, India cDepartment of Mathematics, Chakdaha College, P.O.: Chakdaha, Nadia, Pin:741222, West Bengal, India dRanaghat Brojobala Girls High School(H.S), Rabindra Saranii, P.O.: Ranaghati, Dist.: Nadia, Pin: 741201, West Bengal, India [email protected]

During the past decades, enormous works by different researchers have been carried out in fixed point theory on metric spaces. The theory of bicomplex numbers is also a matter of active research for quite a long period of time in search of special algebra. The algebra 57 of bicomplex numbers are widely used in the literature as it becomes viable commutative alternative to the non-skew field of quaternions, both are four dimensional and general- izations of complex numbers. The concept of the coupled fixed point was first introduced by Bhaskar & Laxikantham (2006). The aim of this paper is to investigate some common fixed point theorems for a pair of mappings satisfying certain rational type contraction condition and having a unique common coupled fixed point in the framework of bicomplex valued metric spaces. Our results are the generalizations of existing literature of coupled fixed point theorems of Bhatt et. al (2011) and Savitri & Hooda (2015). A few examples are provided to justify the results obtained and the course of future prospect of works as carried out is sketched in the paper.

17. A Study of Extended Beta function with its Applications by Ekta Mittal Departement of Mathematics, IIS (Deemed to be University), Jaipur [email protected]

Present study is to provide a systematic analysis of new type of extended beta func- tion and hypergeometric function using a confluent hypergeometric function to investigate different properties and formulas of this function such as integral representations, deriv- ative formula, transformation formula, summation formula and much more. In addition, we also explore extended Riemann-Liouville fractional integral operator with associated properties.

18. On the Location of Zeros of Transcendental Entire Functions by Sanjib Kumar Dattaa, Tanchar Mollab, Mukul SKa, Jayanta Sahaa aDepartment of Mathematics, University of Kalyani P.O.: Kalyani, Dist: Nadia, Pin: 741235, West Bengal, India bDepartment of Mathematics, Dumkal College, P.O.: Basantapur,P.S: Dumkal, Dist.: Murshidabad, Pin: 742406, West Bengal, India. [email protected]

A function of one complex variable analytic in the finite complex plane C is called an entire function and whenever it has an essential singularity at point at infinity it will be transcendental. If a function f(z) is entire then it can be represented by an everywhere convergent power series like

n f (z) = a0 + a1z + ... + anz + ...

Thus the entire functions form natural generalization of polynomials.

The prime purpose of this paper is to derive zero free region for some transcendental entire functions of finite order under various conditions using the coefficients an’s. A few examples with related figures are given here to justify the results obtained.

19. On the Generalization of Enstr¨om-Kakeya Theorem for Entire Functions by Sanjib Kumar Dattaa, Tanchar Mollab, Mukul SKa, Tandra Sarkara aDepartment of Mathematics, University of Kalyani, P.O.: Kalyani, Dist: Nadia, Pin: 741235, West Bengal, India bDepartment of Mathematics, Dumkal College, P.O.: Basantapur, P.S: Dumkal, Dist.:Murshidabad, Pin: 742406, West Bengal, India [email protected]

n P j The classical Enstr¨om-Kakeya theorem states that if P (z) = aj z is a polynomial j=0 of degree n with real coefficients satisfying 0 ≤ a0 ≤ a1 ≤ ... ≤ an, then all the zeros of P (z) lie in the unit disk |z| ≤ 1 in the complex plane C. Many results on generalization of Enstr¨om-Kakeya theorem by putting various conditions on the coefficients of the poly- nomials exist. The prime concern of this paper is to extend the classical Enstr¨om-Kakeya 58 theorem for entire functions of non zero finite order having lacunary type power series ex- pansion. A few examples with related figures are given here to justify the results obtained.

20. Inclution Relation between Subclass of Pascu Type Harmonic Functions Based on Mittag-Lefflar Functions by K. Vijaya, V. Malathi Department of Mathematics School of Advanced Sciences, Vellore Institute of Technology, Vellore. India [email protected], [email protected]

In this paper, we investigate an association between certain subclasses of harmonic uni- valent functions by applying certain convolution operator concerning generalized Mittag- Leffler functions. To be more precise, we discuss such connections with PASCU-type harmonic univalent functions in the open unit disc D.

21. Certain Subclass of Meromorphic Functions Associated with Bessel Function by Santosh M. Popadea, Rajkumar N. Ingleb, P. Thirupathi Reddyc aDepartment of Mathematics, Sant Tukaram College of Arts & Science, Parbhani - 431 401, Maharashtra, India bDepartment of Mathematics, Bahirji Smarak Mahavidyalay, Basmathnagar - 431 512, Maharashtra, India cDepartment of Mathematics, Kakatiya University, Warangal- 506 009, Telangana, India [email protected]

In this paper, we introduce and study a new subclass of meromorphic univalent func- tions defined by Bessel function. We obtain coefficient inequalities, extreme points, radius of starlikeness and convexity. Finally, we obtain partial sums and neighborhood properties ∗ for the class σp (η, κ, λ, ν, α, β) 22. On Relative Defects of Special type of Differential Polynomial in Connec- tion with their Integrated Moduli of Logarithmic Derivative by Sanjib Kumar Dattaa, Sukalyan Sarkarb, Lakshmi Biswasc, Ashima Bandyopadhyayd aDepartment of Mathyematics, University of Kalyani P.O.: Kalyani, Dist: Nadia, Pin: 741235, West Bengal, India bDepartment of Mathyematics, Dukhulal Nibaran Chandra College P.O.: Aurangabad, Dist: Murshidabad, Pin: 742201, West Bengal, India cKalinarayanpur Adarsha Vidyalaya P.O.: Kalinarayanpur, Dist: Nadia, Pin: 741254, West Bengal, India dRanaghat Brojobala Girls High School (H.S.) P.O.: Ranaghat, Dist: Nadia, Pin: 741201, West Bengal, India [email protected]

The paper aims at the comparison between the relative Valiron defect and relative Nevanlinna defect of special type differential polynomials from the view point of their integrated moduli of logarithmic derivative. Let f be a meromorphic function in the com- plex plane. We consider F = f nQ [f], Q[f] being a differential polynomial in f with n = 1, 2, 3, ... and compare with the relative Valiron defect and relative Nevanlinna defect of F under the flavour of integrated moduli of logarithmic derivative. A few examples are provided here to validate the conclusion of the results obtained.

23. Bicomplexial Approch of Some Well Known Result In Complex Analysis by Debasmita Duttaa, Satavisha Deyb, Sukalyan Sarkarc, Sanjib Kumar Dattad aDepartment of Mathematics, Lady Brabourne College, P-1/2 Suhrawardy Avenue, Beniapukur, Dist. : Kolkata, PIN : 700017, West Bengal, India bDepartment of Mathematics, Bijoy Krishna Girls’ College, M.G. Road,Dist. : Howrah, PIN : 711101,West Bengal, India 59 cDepartment of Mathematics, Dukhulal Nibaran Chandra College, P.O.: Aurangabad, Dist.: Murshidabad, PIN : 742201, West Bengal, India dDepartment of Mathematics, University of Kalyani, P.O.: Kalyani, Dist.: Nadia, PIN :741235, West Bengal, India [email protected]

In this paper, we explore the fundamental results of zeros and poles of a bicomplex valued function and relation between them. Moreover some well known results in complex analysis like Jenson’s Inequality as well as some results on univalent functions are proved here in the bicomplex context.

24. Zalcman Conjecture and Hankel Determinant of Order Three for Re- ciprocal of Bounded Turning Functions and α-Convex Functions Associated with Exponential Function by V. Suman Kumara, R. B. Sharmab aDepartment of Mathematics, TSMS Chigurumamidi, Karimnagar, Telangana, India bDepartment of Mathematics, Kakatiya University, Warangal, Telangana, 506009, India. [email protected]

α In this work, we make an attempt to introduce two new subclasses denoted by Me and RTde which are analytic. The purpose of this investigation is to determine an upper bound of H3(1) | for the functions f in these classes associated with Exponential Function. Similar outputs are obtained for the rational function and the inverse function of f.

25. On a Common Fixed Point Theorem in Bicomplex Valued b-metric Space by Sanjib Kumar Dattaa, Dipankar Palb, Rakesh Sarkarc, Arghyatanu Mannad aDepartment of Mathematics, University of Kalyani, P.O.: Kalyani, Dist: Nadia, PIN- 741235, West Bengal, India bDepartment of Mathematics, Prof. Syed Nurul Hasan College, P.O.: Farakka Barrage, Dist: Murshidabad, PIN-742212, West Bengal, India cDepartment of Mathematics, Gour Mahavidyalaya, P.O.: Mangalbari, Dist: Malda, PIN- 732142, West Bengal, India dMousini Co-operative High School(H.S.), Bagdanga, Fraserganj Coastral, Kakdwip, South 24 Parganas, PIN-743357 [email protected]

Segre’s introduction of bicomplex numbers has stimulated the conceptualization of the bicomplex valued metric space by the subsequent mathematicians. The study of fixed point theorems within the framework of bicomplex valued metric spaces can unfold an area of research endowed with infinite scope of growth. This paper can be regarded as a humble effort towards the exploration of that scope. Rao et al. (2013) defined the complex valued b-metric space. This provided the opportunity to formulate the concept of bicomplex valued b-metric space by the researchers like Datta et al. (2020). The main purpose of this paper is to investigate a common fixed point theorem in bicomplex valued b -metric space satisfying some rational inequalities for two pairs of weakly compatible self contracting mappings. Our result is the generalisation of the findings of Mitra (2015).

26. Some Exceptional Value Theorems of Entire Functions Under the Treat- ment of Bicomplex Analysis by Satavisha Deya, Debasmita Duttab, Surajit Hazrac, Sanjib Kumar Dattad aDepartment of Mathematics, Bijoy Krishna Girls’ College, M.G. Road, Howrah-711101, West Bengal, India bDepartment of Mathematics Lady Brabourne College, Kolkata-700017, West Bengal, In- dia cDepartment of Mathematics Ananda Mohan College, 102/1, Raja Rammohan Sarani, Kolkata-700009, West Bengal, India 60 dDepartment of Mathematics University of Kalyani, P.O.: Kalyani, Dist: Nadia, PIN- 741235, West Bengal, India [email protected]

The study of exceptional values of entire functions was started with the famous theo- rem of Picard, one of the most important theorems of complex analysis. The exceptional values may be defined in many sense. Picard’s theorem admits the possibility of having an exceptional value in case of an entire function like as 0 for exp z. The value with this property is called Exceptional-P. There is another sense in which the value may be exceptional. An entire function may take the value a only at the points which have expo- nent of convergence less than the order of the function. The value with this property is called Exceptional-B, i.e., in the sense of Borel. In this paper our aim is to derive a few exceptional value theorems of entire functions such as Schottky’s theorem, Little Picard’s theorem, Landau’s theorem etc. under the flavour of bicomplex analysis. Some examples are given here to validate the results obtained.

27. Fractional Fourier transforms with the Flip Operator by Syed Papia Nawaz, V.R. Lakshmi Gorty SVKM’s NMIMS University, MPSTME Vile Parle (W), Mumbai [email protected]

In this paper, a modified Fourier transform known as fractional Fourier transform is introduced. Different properties of the transform including shifting properties and deriva- tives are studied in this work. Fractional Fourier transform is demonstrated with some examples. Relations with flip operators of Fractional Fourier transform have also been studied in this context.

28. Starlike Functions Associated with the Parabolic Region in the Right Half Plane by Sushil Kumar, Bharati Vidyapeeth’s College of Engineering, Delhi [email protected]

In this note, we consider the class of parabolic starlike functions defined on the open unit disk, introduced by F. Rønning. These functions are closely associated with the par- abolic region in the right plane that is given by Ω = {w ∈ C : |w − 1| < <(w)}. We determine sharp estimate on Hermitian–Toeplitz determinant of third order and a bound on third order Hankel determinants for parabolic starlike functions. In addition, parabolic starlikenees radius estimate and subordination result for various functions with positive real parts are also examined.

29. Certain Subclass of Analytic Function with Negative Coefficients De- fined by Catas Operator by G.M. Birajdara, N.D. Sangleb aDepartment of Mathematics, Shivaji University, Kolhapur (M.S) India bDepartment of Mathematics, Annasaheb Dange College of Engineering & Technology, Sangli, (M.S.) India [email protected]

In this paper, we investigate subclass of analytic function with negative coefficient defined by Catas operator in the unit disc U = {z ∈ C : |z| < 1}. The results included coefficient estimates, closure theorem and distortion theorems of several functions belong- ing to this subclass. Also, we presented detailed study of uniformly convex and uniformly starlike functions.

30. Analytic Functions of Complex Order Involving Hadamard Product by Lateef Ahmad WANI, Anbhu Swaminathan Department of Mathematics, Indian Institute of Technology, Roorkee, Uttarakhand, India 61 [email protected]; [email protected]

Let k ≥ 0, α ∈ [0, 1), γ ∈ [0, 1] and b ∈ C\{0}. We use Hadamard product to introduce a novel function class TUMγ (g, b, k, α) consisting of normalized analytic functions f(z) with negative coefficients satisfying the inequality

  0 2 00  0 2 00 1 zΦ (z) + γz Φ (z) k zΦ (z) + γz Φ (z) Re (1 − α) + − 1 > − 1 , b (1 − γ)Φ(z) + γzΦ0(z) |b| (1 − γ)Φ(z) + γzΦ0(z) where Φ(z) = (f ∗ g)(z) for some analytic function g(z). In this paper, we solve certain coefficient and radii problems, and the Silverman’s conjecture related to TUMγ (g, b, k, α). It is also shown that the class is invariant under certain well-known integral operators. Furthermore, some previously known results are obtained as special cases.

31. Growth Properties of Solutions of Linear Difference Equations with Co- efficients Having Finite Logarithmic Order by Nityagopal Biswas Assistant Professor, Department of Mathematics, Chakdaha College Chakdaha, Nadia, Pin: 741222, West Bengal [email protected]

In this paper, we investigate the relations between the growth of entire coefficients and that of solutions of complex homogeneous and non-homogeneous linear difference equa- tions with entire coefficients of finite logarithmic order by using a slow growth scale, the logarithmic order. We extend some precedent results due to Zheng and Tu (2011) (X.M. Zheng and J. Tu, Growth of meromorphic solutions of linear difference equations, J. Math. Anal. Appl. 384(2011), pp. 349 − 356.) and others.

32. Implications of Baker Omitted Value by Subhasis Ghora, Tarakanta Nayak School of Basic Sciences, IIT Bhubaneswar, Bhubaneswar, India. [email protected], [email protected]

For a transcendental meromorphic map f : C → Cb with only one essential singularity at n ∞ ∞, the set of points z ∈ Cb for which {f (z)}n=0 is defined and normal is called the Fatou set of f. The Julia set is the complement of the Fatou set. A maximal connected subset of the Fatou set is called a Fatou component. A Fatou component U is called completely −1 invariant if f(U), f (U) ⊆ U. A value z0 ∈ Cb is called an omitted value of f if f(z) 6= z0 for any z ∈ C. An omitted value is called a Baker omitted value (bov) if each boundary component of the pre-image of every open ball containing the omitted value is bounded. It is called stable if it is in the Fatou set. It is known that the bov (if exists) is the only asymptotic value also for a function with the stable bov, there is only one unbounded Fa- tou component which is infinitely connected and all other Fatou components are bounded. Certain dynamical issues of meromorphic maps with the stable bov are investigated. It is shown that the bov is always a limit point of the critical values and if the bov of a function is contained in an invariant Fatou component, then the Fatou component must be com- pletely invariant. The non-existence of Baker domain of a function with the stable bov and non-existence of invariant Baker domain of a function with non-stable bov are evinced.

33. Existance of Entire Solutions of Difference Equations by Renukadevi Sangappa Dyavanal Department of Mathematics, Karnatak University - Dharwad, India [email protected]

The main objective of this paper is to investigate the problem of existence of transcen- dental entire solution of a difference equation generated by general difference polynomial of a transcendental entire function of finite order. 62

n 34. Fractional Wavelet Transform in R by Bivek Gupta, Amit K. Verma Department of Mathematics, Indian Institute of Technology Patna [email protected]

n In this paper, we study continuous fractional wavelet transform (CFrWT) in R with n scaling parameter a = (a1, a2, . . . , an) ∈ R . We derive some of its basic properties like inner product relation and inversion formula. We also characterize the range of the trans- form. Moreover, we also study the boundedness and the approximation property of the transform in Morrey space.

35. A note on the bicomplex version of Enstr¨om-Kakeya theorem by Sanjib Kumar Dattaa, Tanchar Mollab, Mukul Ska, Jayanta Sahaa Department of Mathematics, University of Kalyani, P.O.:Kalyani, Dist.:Nadia, Pin:741235 West Bengal, India Department of Mathematics, Dumkal College, P.O: Basantapur, P.S: Dumkal, Dist.: Mur- shidabad Pin: 742406, West Bengal, India. [email protected]

Fundamental theorem of algebra only gives information about the number of zeros of a polynomial but not location of the zeros. All zeros of a quadratic polynomial can be derived algebraically for all possible values of its coefficients. But difficulty arises when degree of polynomial increases. So, it is desirable to know a region where the zeros of a polynomial lie. Classical Enstr¨om-Kakeya theorem is a result in this direction which says Pn j that if P (z) = j=0 aj z is a polynomial of degree n with non negative real coefficients satisfying non decreasing relation, then all the zeros of P (z) lie in the unit disc contained in the finite complex plane. Bicomplex algebra is the generalization of the field of complex numbers. Like complex entire function, a bicomplex entire function f(z) is also repre- P∞ j sented by an everywhere convergent power series as f(z) = j=0 αj z , where αj ’s and z are bicomplex numbers. Thus, bicomplex entire functions are the natural generalization of bicomplex polynomials. The prime concern of this paper is to revisit the Enstr¨om-Kakeya theorem with some of its consequences under the flavor of bicomplex analysis. Some ex- amples with related figures are given here to validate the results obtained.

36. Inequalities for the Maclaurin’s coefficients of spiralike functions involv- ing q-differential operator by K. Amarender Reddy, G. Murugusundaramoorthy, K.R. Karthikeyan Department of Mathematics, VIT University, Vellore [email protected]

The main purpose of this paper is to unify, extend and discretize several results related to spiralike and strongly spiralike functions. We achieve this purpose by making use of q-analogue of the well-known differential operator. We provide a formal extension of a bi-univalent spiralike and bi-univalent strongly spiralike functions. We obtain the inequal- ities for the Maclaurins coefficients of the functions belonging to the defined subclasses.

Section D: Functional Analysis, Measure Theory, Probability Theory and Stochastic Processes, and Information Theory 1. A generalization of the density zero ideal by Sumit Som, Department of Mathematics, National Institute of Technology Durgapur [email protected]

Let

F = (Fn) 63

be a sequence of nonempty finite subsets of ω such that limn |Fn| = ∞ and define the ideal

I(F ) := {A ⊆ ω : |A ∩ Fn|/|Fn| → 0 as n → ∞} .

The case Fn = {1, . . . , n} corresponds to the classical case of density zero ideal. We show that I(F ) is an analytic P-ideal but not Fσ. As a consequence, we show that the set of real bounded sequences which are I(F )-convergent to 0 is not complemented in `∞. 2. A note on Toeplitz, Hankel and Composition operators on the Bergman space by Pabitra Kumar Jena, P G Department of Mathematics, Berhampur University, Bhanja Bihar, Berhampur-760007, Ganjam, Odisha [email protected]

In this article, we characterize the sufficient conditions for Toeplitz and Hankel op- erators defined on the Bergman space to be unitary and mean of unitaries. Further, unitariness of Composition operators on the Bergman space are also studied.

3. Analysis of Retrial Queueing System with Two Way Communication, Working Breakdown and Collision by G. Ayyappana, B. Somasundaramb, G. Archanaa, S. Sankeethac aDepartment of Mathematics, Pondicherry Engineering College, Puducherry, India bDepartment of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R & D Institution of Science and Technology, Tamilnadu, India cDepartment of Mathematics, Saradha Gangadharan College, Puducherry, India [email protected]

This paper consider two way communication retrial queueing system with working breakdown and collision. Arriving primary incoming calls send to orbit while the server is busy. A retrial incoming calls on arrival, enter for service, if the server is found to be idle and may collide the service, if the server is busy. The outgoing calls made by the server when it is idle. The incoming calls is consider as a high priority call and an outgoing call is consider as a low priority call. The system may become defective when it is in regular service. After getting breakdown, instead of stopping service, the server will continue the service at a slower rate. The incoming calls arrive to the system according to Markovian arrival process(MAP), outgoing call and service time follow phase type (PH) distribution. The resulting QBD process is investigated in the steady state by using matrix-analytic method. Some of the performance measures are computed. Finally, numerical and graph- ical results are presented.

4. Simulation of Markov Chain Monte Carlo Method for Analysis of Sunspot Cycles by Shikhar Chandra, School of Electronics (SENSE), VIT University, Vellore, India [email protected]

Monte Carlo methods have been in application in various fields such as finance mod- elling, weather forecasting, chemical kinetics and nanotechnology. The method uses ran- dom sampling on a probability distribution after constructing a random process for a problem. This paper covers a specific technique of Monte Carlo method called the Markov Chain Monte Carlo (MCMC) technique. We use particularly, the Metropolis-Hastings al- gorithm for the implementation of this technique. The technique is further used to simulate probability distribution of time series solar activity in order to find recurring patterns and correlations in sunspot cycles. The data used for simulation is the “Monthly mean total sunspot number”, for each month from January 1749 to November 2018, publicly available by ‘World Data Center for the production, preservation and dissemination of the inter- national sunspot number’. Through simulations performed using Python programming, we find that the Markov Chain Monte Carlo technique provides a statistically accurate 64 probability distribution model for the sunspot cycles data.

5. Iterative approximation of common fixed points with simulation results in Banach spaces by Ashis Beraa, Ankush Chandab, Lakshmi Kanta Deya aDepartment of Mathematics, National Institute of Technology Durgapur, India bDepartment of Mathematics, Vellore Institute of Technology, Vellore, India [email protected]

In this article, we propose the Abbas-Nazir three step iteration scheme and employ the algorithm to study the common fixed points of a pair of generalized α-Reich-Suzuki non-expansive mappings defined on a Banach space. Moreover, we explore a few weak and strong convergence results concerning such mappings. Our findings are aptly validated by non-trivial and constructive numerical examples and finally, we compare our results with that of the other noteworthy iterative schemes utilizing MATLAB 2017a software. However, we perceive that for different set of parameters and initial points, the newly proposed iterative scheme converges faster than the other well-known algorithms. To be specific, we give an analytic proof of the claim that the novel iteration scheme is also faster than that of Liu et al.

6. The Mean Deviation Generating functions and a New Measure of Dis- persion by Rajan Chattamvelli, Department of Mathematics, VIT University, Vellore, Tamil Nadu 632014 [email protected]

The mean deviation (MD), also called average absolute deviation (AAD), is a popular measure of dispersion used in several applied science fields. The population MD is not easy to find for some distributions, as it requires meticulous arithmetical work. Most of the graduate courses in biostatistics, mathematical statistics, econometrics and manage- ment sciences devote minimal discussion on MD. The purpose of this paper is to derive new expressions for the mean absolute deviation from the mean and median. This for- mulation is not only of interest to students and teachers in various fields, but also useful to professionals and practitioners when explicit expressions for MD are unknown, but the tail sums are easy to find. These results are used to form new combinatorial identities and a new measure of dispersion that has potential for generalizing to higher dimensions.

7. Inequality for Maximum Modulus of Rational Functions by D. Tripathi, Deepa Arora Department of Mathematics, Manav Rachna University, Faridabad-121001 [email protected]

In this paper, we establish some inequalities for rational functions with prescribed poles and restricted zeros in the sup-norm on the unit circle in the complex plane. Some gener- alizations and refinements of rational function inequalities of Xin Li et.al [Some Rational Inequality Inspired by Rahman’s Research;Progress in Approximation Theory and Appli- cable Complex Analysis, Springer 2017 ] are obtained.

8. Associate space of grand Bochner Lebesgue spaces without Radon-Nikod´ym property by Monika Singh, Lady Shri Ram College for Women, (University of Delhi), Lajpat Nagar, New Delhi [email protected]

The Bochner Lebesgue space, denoted by Lp(Ω, µ; X), is the collection of all X-valued µ-measurable finite almost everywhere (a.e.) functions f i.e., k.kX value is finite a.e., such 65 that 1 Z  p 1 p kfkLp(Ω,µ;X) := kf(t)kX dµ(t) < ∞, µ(Ω) Ω where µ(Ω) < ∞. In, Kokilashvili, Meskhi and Rafeiro introduced grand Bochner Lebesgue spaces (GBLS) on finite measure space and studied its associate space assuming Radon- Nikod´ymproperty on X. In this paper presentation our aim is to talk about the associate space of GBLS without Radon-Nikod´ymproperty on X.

9. Rubio de Francia Extrapolation Theorem in Variable Lebesgue Spaces by Arun Pal Singh, Dyal Singh College, (University of Delhi), Lodhi Road, New Delhi [email protected]

In this paper presentation, we shall talk about the Rubio de Francia extrapolation re- sults for pair of non-increasing functions with Bp(·)-weights. And, mention an application to obtain the extrapolation result in the framework of variable exponent Lebesgue space p(·) Lw with Luxemburg norm.

10. Analysis of MMAP/P H1,PH2/1 Pre-emptive Priority Retrial Queue- ing System under Constant Retrial Policy with Orbital Search, Standby Server, Vacation, Impatient Behavior of Customers, Breakdown and Repair by G. Ayyappan, K. Thilagavathy Department of Mathematics, Pondicherry Engineering College, Pillaichavady, Puducherry, India [email protected], [email protected]

In this article, we consider a single server queue in which two types of heterogeneous customers arrive according to the marked Markovian arrival process and their correspond- ing service based on phase-type(PH) distribution. While the main server is offering service to the high/low priority customer who may be struck with breakdown immediately go for repair meanwhile, standby server would interrupt and take over service up to the main server rejuvenated from repair and return to the service station. Whenever the main server becomes idle due to completion of the repair, vacation and service times who may do an orbital search of low priority customers but in the case of standby server who do an orbital search when become idle due to completion of service to high/low priority cus- tomers. Using the Matrix-Analytic method, we investigated the expected number of high priority customers in the system as well as number of low priority customers in the orbit with the aid of steady-state probability vector. We examined the stability condition, a busy period of the system, cost analysis and characteristics of some performance measures of the system are discussed. Numerical results are tabulated and graphical representations are provided for a clear view of our model.

11. Fractals in Controlled Hausdorff Metric Space by C. Thangaraj and D. Easwaramoorthy Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632 014, Tamil Nadu [email protected]

This paper explores a new fractal space called Controlled Hausdorff Metric Space by using the controlled metric, which is extended from b-metric. Then the HutchinsonBarns- ley Operator(HB Operator) is defined by using the Iterated Function System (IFS) of contractions on a controlled complete metric space. It is proved that, the HB Operator of IFS is contraction on a controlled Hausdorff metric space and also assured that it has a unique fixed point in a controlled Hausdorff metric space, called Controlled Fractal.

12. New Generalized ‘Useful’ Entropies using Weighted Quasi-linear Mean with Utility 66 by Aakanksha Singhal, D.K. Sharma Jaypee University of Engineering and Technology, Raghogarh, Dist. Guna [email protected]

R´enyi entropy was the first generalized entropy derived using the concept of quasi- linear mean. Various other generalized entropies were later analyzed and expressed as the quasi-linear mean of elementary or generalized information. On similar lines, was introduced a generalized entropy: supra-extensive entropy. It was initially perceived that supra-extensive entropy may not be expressed as quasi-linear mean of information. In this paper, supra-extensive entropy is demonstrated as quasi-linear mean of generalized information for the first time. A new concept of weighted quasi-linear mean with utility is introduced and used to derive few existing generalized ‘useful’ information measures. Using the introduced concept of weighted quasi-linear mean with utility, new generalized ‘useful’ information measures based on generalized entropies namely: Tsallis, Sharma- Mittal and supra-extensive are defined. The concept defined can be used to find the generalized ‘useful’ information measure corresponding to any generalized information measures which are quasi-linear means of elementary or generalized information.

13. An Application of Intuitionistic Fuzzy Multisets in an Investment Deci- sion Making Problem by V. Inthumathi, A. Gnanasoundari Department of Mathematics, Nallamuthu Gounder Mahalingam College, Pollachi 642001 [email protected].

Making decisions on investment is certainly the most important task of an investor. The aim of this paper is to show how the concept of intuitionistic fuzzy multisets plays an important role in making good decisions using various distance measures.

14. ∆m-Statistical Convergence of Order α of generalized difference se- quences in Probabilistic Normed Spaces by Gursimran Kaur, Meenakshi Department of Mathematics, Chandigarh University, Mohali, Punjab, India. [email protected]; [email protected]

In the captioned paper, we define ∆m-statistical convergence of order α of generalized difference sequences in Probabilistic Normed Spaces and give their characterization. We also define the notion of ∆m-statistical Cauchy of order α for these types of sequences in Probabilistic Normed Spaces. We have also given few examples which demonstrates that this notion is more generalized in the Probabilistic Normed Spaces.

15. Distance functions (π, β) and a fixed point result in ordered metric spaces by Amit Sharmaa, Reeta Bhardwaja, Kamal Kumara, Naveen Manib aDepartment of Mathematics, Amity University, Haryana, India bDepartment of Mathematics, Sandip University, Nashik, India [email protected]

In present work, we derived a common fixed point result in an ordered complete metric space by using distance functions (π, β). Our main result, improves and unify the results of Yan et al. [Fixed Point Theory and Applications. 2012 (2012)] and Gupta and Mani [J. Fixed Point Theory Appl. 19 (2017), 1251 − 1267]. An example has been given in support of our findings.

16. Bulk service queueing system with multiple vacation and remaining ser- vice by standby server during the breakdown period by S. Karpagam, Department of Mathematics, Vel Tech Rangarajan Dr.Sagunthala R&D Institute of Science and Technology, 67

Chennai, India [email protected]

The purpose of this article is to bring out certain silent features of an M [X]/G(a, b)/1 queueing model with multiple vacation and remaining service by standby server for the service interrupted batch due to breakdown. For this model, some important performance measures are obtained and the stability conditions are derived. Finally, numerical results of the proposed model are presented.

I O I O 17. Analysis of MAP1 , MAP2 /P H1 ,PH2 /1 Retrial Queue with Single Vacation, Closedown, Setup, Optional Service, Balking and Two Way Com- munication by G. Ayyappana, G. Archanaa, S. Sankeethab aDepartment of Mathematics, Pondicherry Engineering College, Puducherry, India bDepartment of Mathematics, Saradha Gangadharan College, Puducherry, India [email protected]

In this paper we consider a single server constant retrial queueing system with close down, vacation, setup, optional service and balking. There are two types of arrivals namely incoming calls which are made by the customers and outgoing calls which are made by the server during the idle period. If server is idle, the arriving incoming call will be served immediately. If server is busy, the arriving incoming call will send the orbit. After the service completion the server become idle. During the idle time, the server will make the outgoing calls for their popularizing various schemes. The server goes vacation only if the system is empty. Before takes the vacation server will be close down the system and after the completion of vacation the server will set up the system for ready to give the service. The server will give optional service for incoming calls if its need additional ser- vice. The arriving incoming call may leave without entering the system, if server is busy. The incoming calls and outgoing calls follows the Markovian Arrival Process (MAP ) and service times of incoming and outgoing calls follow the Phase-type distribution and the rest of the random variables are exponentially distributed. The resulting QBD process is investigated in the steady state by employing matrix-analytic method. We have also done the busy period analysis of our model and discussed about the waiting time distribution of our system. Some of the performance measures of the system are derived and illustrated graphically/numerically.

18. Best approximations, distance formulas and orthogonality in C∗-algebras by Priyanka Grover, Sushil Singla Department of Mathematics, Shiv Nadar University, India [email protected]

The length of the perpendicular in a right-angled triangle is always less than or equal to the length of the hypotenuse. In an inner product space H, if v ∈ H and W is a subspace of H, then v is orthogonal to W if and only if kvk ≤ kv − wk for all w in W . This characterization of orthogonality to a subspace in an inner product space is taken as the definition of orthogonality of an element to a subspace in a normed space, called Birkhoff-James orthogonality. We characterize Birkhoff-James orthogonality of an ele- ment a ∈ A to a subspace B of A. We prove that that a is Birkhoff-James orthogonal to B if and only if there exists a state φ on A such that φ(a∗b) = 0 for all b ∈ B and φ(a∗a) = kak2. And we will see many known results which follow as corollary to this result.

19. A study on MAP/P H1,PH2/2 queue with unreliable servers and vaca- tion by G. Ayyappan, R. Gowthami Department of Mathematics, Pondicherry Engineering College, Puducherry [email protected] 68

In this article, we discuss about a two heterogeneous classical queueing model with working breakdown and Bernoulli vacation for server-1 and starting failure with single vacation for server-2. Customers arrive to the system by following Markovian Arrival Process(MAP) and service times follow Phase-type(PH) distribution. We have studied our model by using matrix analytic method. We have provided practical application for our model. We have done the busy period analysis and also derived the waiting time distribution for our system. To the end, some numerical and graphical exemplifications are provided.

20. MAP/P H(1),PH(2)/2 with interaction, multiple vacation and repair by G. Ayyappana, S. Sankeethab, Archana Gurulakshmia aDepartment of Mathematics, Pondicherry Engineering College bDepartment of Mathematics, Saradha Gangadharan College, India [email protected]

In this paper we consider two types of servers namely main and regular, the main server will be interrupted when he is providing service to the regular server customers. Inter arrival time of the system follows Markovian Arrival Process (MAP), the service time follows phase type distribution under the rest of the random variables are exponen- tially distributed. This system has been modeled into a QBD process investigating steady state using matrix analytic technique where the block elements of the generated matrix have finite dimensions. We have also discussed the busy period, waiting time distribution for our model. Performance measures of the system are derived and illustrated numeri- cally/graphically.

21. On Existence Results of Generalized Evolution Equation with Non- Instantaneous Impulses over the Banach space by Haribhai R. Katariaa, Prakashkumar H. Patela, Vishant Shahb aDepartment of Mathematics, Faculty of Science, The M. S. University of Baroda, Vado- dara - 390 002, India bDepartment of Applied Mathematics, Faculty of Technology and Engineering, The M. S. University of Baroda, Vadodara - 390 001, India [email protected], [email protected], [email protected]

This article established sufficient conditions for the existence of mild solution for the generalized impulsive evolution equation with non-instantaneous impulses with local and non-local conditions. Conditions for existence of mild solution of evolution equations with local conditions are established through Banach fixed point theorem while, with non-local condition is established through Krasnoselkii’s fixed point theorem. Finally, an illustra- tion is added to validate derived results.

Section E: Differential, Integral and Functional Equations

1. First-Order Nonlinear Dynamic Initial Value Problems by Martin Bohnera, Sanket Tikareb, Iguer Luis Domini dos Santosc aDepartment of Mathematics and Statistics, Missouri University of Science and Technol- ogy, Rolla, MO 65409-0020, USA. bDepartment of Mathematics, Ramniranjan Jhunjhunwala College, Ghatkopar, Mumbai, India cDepartamento de Matem´atica,Faculdade de Engenharia de Ilha Solteria, UNESP-Univ Estadual Paulista, Rua Rio de Janeiro, 266, Ilha Solteria, S˜aoPaulo CEP 15385-000, Brazil [email protected] 69

We prove three existence theorems for solutions of first-order dynamic initial value problems, including corresponding continuous and discrete cases. The main tools are fixed point theorems and dynamic inequalities. Two more results are given that discuss depen- dence of solutions on the initial conditions as well as convergence of sequences of solutions.

2. Solutions of the iterative root problem for a class of continuous functions by Veerapazham Murugan, Rajendran Palanivel Department of Mathematical and Computational Sciences, National Institute of Technol- ogy Karnataka, Surathkal, Mangaluru- 575 025, India [email protected]

In this paper, we introduce the concept of iteratively closed set of the set of all continu- ous self-maps on a compact interval. We obtain solutions of the iterative root problem for a class of continuous functions with non-monotonicity height equal to one by extending solutions from the characteristic interval.

3. Stability and Boundedness Criteria for Impulsive Fractional Differential Equations in Caputo Sense with Initial Time Difference by Pallvi Mahajan, Beant College of Engineering & Technology, Gurdaspur, Punjab [email protected]

In the past few decades, fractional differential equations have gained considerable im- portance and recently researchers are getting more interest in impulsive fractional dif- ferential equations due to its widespread applications in various fields of science and en- gineering. In the present work, an impulsive fractional differential equation in Caputo sense is investigated with initial time difference for the first time and some novel criteria has been derived to investigate their stability and boundedness behaviour. The stability with respect to initial time difference is the generalization of the basic stability concept in the literature. The investigations are carried out by perturbing Lyapunov function and by using comparison results. A generalized piecewise Lyapunov function has been used to obtain the desired results. The results that are obtained to investigate the stability significantly depend on the moment of impulses.

4. Classification of Delay Differential Equations with Constant Coefficients to Solvable Lie Algebras by Jervin Zen Lobo, Department of Mathematics, St. Xavier’s College, Mapusa, Goa - 403507 [email protected]

In this paper, we shall apply symmetry analysis to first and second order delay dif- ferential equations with constant coefficients. The determining equations of the admitted Lie group are constructed in a manner different from that of the existing literature for delay differential equations. We define the standard Lie bracket and make a complete classification of linear delay differential equations with constant coefficients, to solvable Lie algebras. We also classify some non-linear delay differential equations with constant coefficients, to solvable Lie algebras.

5. A Study of the Reproducing Kernel Hilbert Space Method for Poor Nu- trition in the Life Cycle by Gautem Patel, Kaushal Patel Department of Mathematics, Veer Narmad South Gujarat University, Gujarat, India [email protected]

In this work, the reproducing kernel Hilbert space method is applied for solving the mathematical model of the poor nutrition in the life cycle in the form of non linear system 70 of ordinary differential equations. The exact solution of the system has been obtained in terms of a convergent series. The results in figure show that the proposed method is effective for solving such non linear system of ordinary differential equations as compare to numerical methods.

6. Existence Assertion of Solution in the Space `p, p > 1 for Fractional Infi- nite System of Integral Equations of Nonlinear Type by Vijai Kumar Pathak, Lakshmi Narayan Mishra Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technol- ogy, Vellore 632 014, Tamil Nadu, India. [email protected], [email protected]

This article deals with existence assertion of solution in the space `p, p > 1 of fractional infinite system of integral equations of nonlinear type via measure of noncompactness (MNC) together with generalized Meir-Keeler fixed point theorem. The obtain result of existence assertion of solution is proved under rather general hypotheses. Some examples are given to illustrate the natural appreciations of our results presented in this work. We analyze the existence assertion of solution of the system of nonlinear fractional integral equations. Finally, we will bring up an example to show the advantage of our results. We use the homotopy perturbation method together with Adomian decomposition technique to obtain our existence assertion.

7. Periodic Boundary Value Problem for System of Caputo Sequential Dif- ferential Equations of Fractional Order by Jagdish A. Nanware, Department of Mathematics, Shrikishna Mahavidyalaya, Gunjoti, Dt.Osmanabad (M.S) India. jag skmg91@rediffmail.com

Monotone method coupled with lower and upper solutions is developed for periodic boundary value problem for system of Caputo sequential differential equations of frac- tional order. Method is successfully applied to obtain existence and uniqueness results for periodic boundary value problem for system of Caputo sequential differential equations of fractional order.

8. Solutions of Rossby Waves with Dissipation through Symmetries by Amlan K. Haldera, C.T. Dubab, P.G.L. Leachc,d aDepartment of Mathematics, Pondicherry University, Kalapet - 605014, Puducherry, In- dia bSchool of Computer Science and Applied Mathematics, University of Witwatersrand, Jo- hannesburg, South Africa cInstitute For Systems Science, Durban University of Technology, Durban, South Africa dSchool of Mathematics, Statistics and Computer Science, University of KwaZulu - Natal, Durban, South Africa [email protected]

The Lie symmetry analysis method is applied to (1+1) and (1+2)−dimensional Rossby wave with a dissipative term in the f and β−plane. We employ the canonical coordinates to obtain certain new reductions and equations which are generally devoid of any point symmetries. Some symmetry reductions leads to an Abel’s equation of the first kind and maximally symmetric second-order ordinary differential equation. The dispersion relation points that the (1 + 1)−dimensional Rossby wave propagation and energy transporta- tion is in both the directions and the presence of dissipation coefficient reduces the speed whereas the energy transportation increases for the viscous case in the β- plane. From the dispersion relation for the (1 + 2)−dimensional Rossby equation, it can be deduced that there are two waves propagating at the same time one in the eastward and the other westward, as evident from the values of the phase and group velocities. Moreover, the energy propagation is towards the east and it remains consistent throughout the flow. For 71 a particular symmetry, the graphical representations of the solutions for various values of the parameter β, presents a pictorial observations of Rossby wave on the Sun. We also present a comparative study with the inviscid case to prove our findings.

9. Optical Solitons with Generalized Third-order Nonlinear Schr¨odinger Equation via Lie Symmetry Analysis by Sachin Kumar, Sandeep Malik Department of Mathematics and Statistics, Central University of PunjabBathinda–151001, Punjab, India [email protected]

In this article, we investigate the generalized third order nonlinear Schr¨odingerequa- tion (NLSE), which is utilized in optical fibers to model pulses of ultra-short. By the implement of Lie symmetry analysis, we derive optical solutions of this model. These optical solutions are received in the form of dark, bright, and singular soliton solutions.

10. A Novel Technique for Solving the Higher-dimensional System of Non- linear Coupled Partial Differential Equation by Kumbinarasaiah S, Department of Mathematics, Bangalore University, Bengaluru-560056, India [email protected]

In this study, we propose an effective numerical algorithm to find numerical solutions to the system of partial differential equations. This algorithm includes the collocation method and the truncated Laguerre wavelet series. Here, we reduce the system of (2+1) dimensional partial differential equations into a set of algebraic equations that have un- known Laguerre wavelet coefficients. Some numerical examples are solved to validate the proposed technique’s efficiency and, also, discussed the comparison between the present method and other methods of solutions with the exact answer. The obtained results reveal that the current algorithm provides a better result than other methods.

11. A Result on the Approximate Controllability of Fractional Differential Equations of Order 1 < r < 2 M. Mohan Raja, V. Vijayakumar, R. Udhayakumar Department of Mathematics, School of Advanced Science, Vellore Institute of Technology, Vellore 632 014, Tamil Nadu [email protected]

This manuscript is mainly focusing on approximate controllability for fractional differ- ential evolution equations of order 1 < r < 2 in Hilbert spaces. We consider a class of control systems governed by the fractional differential evolution equations. By using the results on fractional calculus, cosine and sine functions of operators, and fixed-point ap- proach, a new set of sufficient conditions are formulated which guarantees the approximate controllability of fractional differential evolution systems. The results are established un- der the assumption that the associated linear system is approximately controllable. Lastly, we present the applications to support the validity of the study.

12. Existence and Uniqueness of Classical and Mild Solutions of Impulsive Fractional Evolution Equations by Jaita Sharmaa, Vishant Shaha, Raju K. Georgeb aDepartment of Applied Mathematics, Faculty of Technology and Engineering, The M. S. University of Baroda, Vadodara, India bDepartment of Mathematics, Indian Institute of Space Science and Technology, Triven- drum, Kerala, India [email protected]; [email protected]; [email protected] 72

In this article, we are deriving a set of sufficient conditions for existence and unique- ness of classical and mild solutions of fractional semi-linear evolution equation with and without impulses on the Banach spaces by generalizing the concept of semigroup in terms of generators and using generalized fixed point theorem. The conditions for the existence and uniqueness of classical and mild solutions obtained using this concept of generators are weaker than previously derived conditions. We also derived conditions under which mild solution becomes a unique classical solution. Illustrations are provided to validate our results.

13. On Approximate Controllability of a Class of Neutral Hilfer Fractional Stochastic Differential Systems by using Wright Function by C. Dineshkumar, R. Udhayakumar, V. Vijayakumar Department of Mathematics, Vel- lore Institute of Technology, Vellore - 632 014, [email protected]

This article mainly focusing approximate controllability of a class of neutral Hilfer fractional stochastic differential systems. By applying some ideas about the semi-group theory and fractional calculus, the main results of this article are proved. At first, the investigation about the existence of mild solution and then we study about the approx- imate controllability of the considered equation. Next, we extend our study to system with non-local conditions. Finally, a theoretical application is presented for proving the validity of controllability results obtained.

14. An Effective Iterative Method and Existence Result for a Class of Second-order Four-point Nonlinear BVPs by Nazia Urus, Amit Kumar Verma Department of Mathematics, Indian Institute of Technology Patna, India [email protected]

In this article, we develop a monotone iterative technique (MI-technique) with upper and lower solutions for a class of four-point nonlinear boundary value problems (NLBVPs). The nonlinear source term is dependent on the derivative of the solution. To study the ex- istence of a solution, we construct iterative sequences for the corresponding linear problem. We use quasilinearization to construct these iterative sequences. We prove maximum and anti-maximum principle to establish monotonicity of sequences of lower solutions (ln(x))n and upper solution (un(x))n such that ln(x) ≤ un(x) as well as ln(x) ≥ un(x), ∀n ∈ N. Then under certain assumptions, we prove that these sequences converge uniformly to the solution in the specific region. The motivation for this work came from many recent in- vestigations on MI-technique for four-point NLBVPs. To demonstrate that the proposed technique is impressive, we compute the solution of the NLBVPs which may not be com- puted effortlessly. We have also plotted upper and lower solutions for the test examples and have shown that under some sufficient conditions, the derived upper and lower solu- tions are monotonic in nature.

Section F: Geometry and Topology

1. Some ξ-Pre-Continuous Maps by Nazir Ahmad Ahengar, J.K. Maitra Department of Mathematics and Computer Sciences, R.D. University, Jabalpur [email protected], kmrdvv@rediffmail.com

In this paper the concept of ξ-pre-continuous and ξ-regular continuous maps in ξ- topological spaces are introduced and all the possible relationships of these maps have been discussed and established. Further we introduce and study ξ-pre-generalized closed 73 sets and ξ-pre-generalized continuity in ξ-topological spaces and investigate various rela- tionship by making the use of some counter examples.

2. p∗-Continuous Maps and its Generalization via Ideal by Rajesh Kumar Tiwari, J.K. Maitra Department of Mathematics and Computer Science, Rani Durgawati University, Jabalpur, India [email protected]; jkmrdvv@rediffmail.com

A collection of sets which are derived from topological space with respect to general- ized topology is said to be the p∗-open sets. In this paper the define p∗-continuity on topological space with respect to generalized topology in sense of p∗-open set. Also us- ∗ ing the ideal in p -open set, we presented Ip∗ -open sets. On the basis of Ip∗ -open sets, we propose to define Ip∗ -continuity and weakly Ip∗ -continuity on topological space with respect to generalized topology via ideal. Further, classical properties of P ∗-continuity, ∗ P -continuity and weakly Ip∗ -continuity are presented on topological space with respect to generalized topology.

3. Intuitionistic Fuzzy Almost Generalized e-continuous Mappings by Chandra Prakash Rathor Department of Mathematics and Computer Science, Rani Durgawati University, Jabalpur, India [email protected]

In this paper, we introduce and study the concept of Intuitionistic fuzzy almost gen- eralized e-continuous mappings. Several interesting properties of these mappings are also given. Examples and counter examples are also given to illustrate the concepts introduced in this paper. We also introduce the concept of Intuitionistic fuzzy T 1 e-space. It is seen 2 that a Intuitionistic fuzzy almost generalized e-continuous mapping from a Intuitionistic fuzzy T 1 1 e-space to another Intuitionistic fuzzy topological space becomes Intuitionistic 2 fuzzy almost continuous mapping.

4. Kaehlerian Spaces Admitting in H-Projective Vector Field with Constant Scalar Curvature by U.S. Negi H.N.B. Garhwal (A Central) University, S.R.T. Campus Badshahithaul, Tehri Garhwal - 249199, (U.K.), India. [email protected]

Ishihara (1959) has studied holomorphically projective changes and their groups in an almost complex manifold and also proved on holomorphic planes. Obata (1965) has defined and studied Riemannian manifolds admitting a solution of a certain system of differential equations. In this paper, we have defined and studied Kaehlerian spaces ad- mitting in H-projective vector field with constant scalar curvature and several theorems have been proved. Also, then find necessary and sufficient conditions for such a Kaehlerian space to be isometric to a complex projective space with Fubini-Study metric.

5. Lower Bounds for Regular Genus and Gem-complexity of PL 4-manifolds with Boundary by Biplab Basak, Manisha Binjola Department of Mathematics, Indian Institute of Technology Delhi [email protected]

Let M be a connected compact PL 4-manifold with boundary. In this article, we have given several lower bounds for regular genus and gem-complexity of the manifold M. In particular, we have proved that if M is a connected compact 4-manifold with h boundary 74 components then its gem-complexity k(M) satisfies the following inequalities: k(M) ≥ 3χ(M) + 7m + 7h − 10 and k(M) ≥ k(∂M) + 3χ(M) + 4m + 6h − 9, and its regular genus G(M) satisfies the following inequalities: G(M) ≥ 2χ(M) + 3m + 2h − 4 and G(M) ≥ G(∂M) + 2χ(M) + 2m + 2h − 4, where m is the rank of the fundamental group of the manifold M. These lower bounds en- able to strictly improve previously known estimations for regular genus and gem-complexity of a PL 4-manifold with boundary. Further, the sharpness of these bounds has also been shown for a large class of PL 4-manifolds with boundary.

6. On Some Intuitionistic p−Sets and Intuitionistic q−Sets by Poonam Agrawal, J.K. Maitra Department of Mathematics and Computer Science, Rani Durgawati University, Jabalpur (M.P.) India poo [email protected]

In 2002, Thangavelu and Chandrasekhara Rao introduced a new class of sets, namely p−sets in topological space. Also they have introduced a new class of set, namely q−sets in topological space. They have studied some basic properties of p−sets and q−sets in topological space. In this paper we have introduced intuitionistic p−sets and intuitionistic q−sets in intuitionistic toplogical spaces and obtained its significant properties. We have constructed some examples which are quite useful in theory of intuitionistic p−sets and intuitionistic q−sets.

7. Comprehensive Quasi-Einstein Spacetime with Application to General Relativity by Punam Gupta, Department of Mathematics & Statistics, School of Mathematical & Physical Sciences, Dr.Harisingh Gour University, Sagar- 470 003 [email protected]

The aim of this work is to introduce and investigate geometric and physical properties of the comprehensive quasi Einstein spacetime C(QE)n under certain conditions. Firstly, we prove the existence of C(QE)n by constructing non trivial examples. Finally, we study conformally flat comprehensive quasi Einstein spacetimes and homogeneous, isotropic, rel- ativistic two-fluid comprehensive quasi Einstein spacetimes.

8. On W2-Curvature Tensor of the Projective Semi-Symmetric Connection by T. Raghuwanshia, S.K. Pandeyb, M.K. Pandeya, A. Goyala aDepartment of Mathematics, U.I.T. R.G.P.V. Bhopal, M.P., India bDepartment of Mathematical Sciences, A. P. S. University, Rewa [email protected]

We study the curvature conditions of semi-symmetry type on an SP -Sasakian mani- fold admitting a projective semi-symmetric connection. It is shown that an SP -Sasakian manifold satisfying the conditions R˜ · W˜2 = 0 and W˜2 · R˜ = 0 is a quasi Einstein manifold.

9. On Selection of Generalized Continuous Multifunctions by Pankaj Jain, Chandrani Basu, Vivek Panwar Department of Mathematics, South Asian University, Akbar Bhawan, Chanakya Puri, New Delhi-110021, India [email protected]

Continuity of functions is an important property. However, often, we encounter func- tions with discontinuities of various types. In the past many decades, people have classified discontinuous functions in weaker classes of continuous functions, e.g., quasi continuity, semi continuity, B-continuity, B∗-continuity, and many more. 75

Very recently the authors have obtained and studied new classes of weak continuity, namely, weak B∗-continuity, contra B∗-continuity and slight B∗-continuity. The notion of weaker (or generalized) continuity also exist in the case of multifunctions or multivalued mappings. The question arises that can we obtain a continuous selection for these multifunctions? In 1956, Michael showed that a closed convex valued continuous multifunction F : X → Y from a paracompact space X to a Banach space Y admits a continuous selection. After that several authors obtained continuous selections for non-convex valued multifunctions in Banach spaces. Carbone showed that not always continuous selection exists. In this di- rection, many authors have studied the problems of existence of various types of selections for generalized continuous multifunctions. In this presentation, we discuss about the existence of quasicontinuous selection for slightly B∗-continuous multifunctions.

10. On Somewhat Pairwise Fuzzy β Continuous Map by Lalita Verma, J.K. Maitra Department of Mathematics and Computer Science, Rani Durgawati University, Jabalpur, M.P. 482001 [email protected], jkmrdvv@rediffmail.com

In this paper we have defined and characterized the concept of somewhat pairwise fuzzy β-continuous map and somewhat pairwise fuzzy β-open map in fuzzy bitopological spaces. We have obtained significant properties of it and constructed basic examples.

11. On Nonempty Intersection Properties in Metric Spaces by Ajit Kumar Gupta, Saikat Mukherjee Department of Mathematics, National Institute of Technology Meghalaya [email protected]

The classical Cantor’s intersection theorem states that in a complete metric space X, the intersection of every decreasing sequence of nonempty closed bounded subsets, with diameter approaches zero, has exactly one point. In this article, we deal with decreasing sequences {Kn} of nonempty closed bounded subsets of a metric space X, for which the Hausdorff distance H(Kn,Kn+1) tends to 0, as well as for which the excess of Kn over X \ Kn tends to 0. We achieve nonempty intersection properties in metric spaces. The obtained results also provide partial generalizations of Cantor’s theorem.

12. On Interval Type-2 Fuzzy Rough Sets and their Topological Structures by Shambhu Sharan, Department of Mathematics & P.G Center, College of Commerce, Arts & Science, Patna-800020 Patliputra University, Patna [email protected]

The present paper studies the relationship between interval type-2 (IT2) fuzzy rough sets and interval type-2 fuzzy topologies (IT2F-topologies) induced by IT2 fuzzy relations. Specifically, we establish some notable results: (i) any serial IT2 fuzzy relation induces an IT2F-topology (ii) any reflexive IT2 fuzzy relation and its transitive closure induce the same IT2F-topology. Subsequently, we obtain the interior and closure IT2 fuzzy operators of IT2F-topology induced by reflexive IT2 fuzzy relation and investigate their connection with IT2 fuzzy relation. Finally, the corresponding results are obtained when the relation is a similarity IT2 fuzzy relation.

13. Space-time Admitting Generalized Conharmonic Curvature Tensor by S.P. Maurya, S.K. Pandey, R.N. Singh Department of Mathematical Sciences, A.P.S. University, Rewa486003, India 76 [email protected]

The object of the present paper is to study Space-time Admitting Generalized Con- harmonic Curvature Tensor. In this paper we have studied the basic algebraic properties of generalized conharmonic curvature tensor. Next, it is proved that a 4-dimensional rel- ativistic generalized conharmonic flat space-time is an Einstein space-time and it is of constant curvature. Moreover, it is of O-type. It is also observed that in a 4-dimensional relativistic perfect fluid generalized conharmonically flat space-time following Einsteins field equation in the absence of cosmological constant, energy momentum tensor is covari- ant constant. Finally, it is proved that a 4-dimensional relativistic conservative generalized conharmonic space-time M with constant scalar function ψ is a GRW space-time.

14. Some Geometric Estimates on Warped Product Lightlike Submanifolds of Indefinite Kaehler Manifolds by Sangeet Kumar, Department of Mathematics, SGTB Khalsa College, Sri Anandpur Sahib - 140118, Rupnagar, India [email protected]

The purpose of present paper is to investigate SCR-lightlike warped product subman- ifolds of indefinite Kaehler manifolds and to find geometrical estimates arising in such warped products. We derive several geometric characterizations for squared norm of the second fundamental form of SCR-lightlike warped product submanifolds in an indefinite Kaehler manifold. Then we find a sharp estimate for the squared norm of the second fundamental form in terms of Hessian of the warping function, for SCR-lightlike warped product submanifolds in an indefinite complex space form. Finally, we present a geometric inequality for the existence of SCR-lightlike warped product submanifolds in indefinite complex space forms.

15. Screen Generic Lightlike Submanifolds of Indefinite Nearly Kaehler Man- ifolds by Megha, Sangeet Kumar Department of Mathematics, SGTB Khalsa College, Sri Anandpur Sahib - 140118, Rup- nagar, India [email protected]

In the process of generalization from Riemannian to semi-Riemannian manifolds, there is a natural existence of lightlike submanifolds and for which the local and global geometry is completely different than non-degenerate case. The main difference between the light- like submanifolds and non-degenerate submanifolds is that the tangent bundle intersects with the normal bundle. The concept of lightlike submanifolds has perceived several im- portant contributions in complex and contact semi-Riemannian geometries and has been successfully applied in differential geometry and mathematical physics, particularly, in theory of general relativity. Considering the growing importance of lightlike submanifolds and interesting geometrical and topological properties indefinite nearly Kaehler manifolds, we study screen generic lightlike submanifolds of indefinite nearly Kaehler manifolds. We prove the existence of screen generic lightlike submanifolds in indefinite nearly Kaehler manifolds of constant holomorphic sectional curvature c and of constant type α. Fur- ther, we derive several characterizations for the integrability of distributions associated with such submanifolds. Finally, we discuss totally umbilical screen generic lightlike sub- manifolds and minimal screen generic lightlike submanifolds of indefinite nearly Kaehler manifolds. 77

Section G: Numerical Analysis, Approximation Theory and Computer Science

1. A Fitted Galerkin Finite Element Method for Singularly Perturbed Dif- ferential Equations with a Small Negative Shift by N. Sathya Kumar, R. Nageshwar Rao Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore – 632 014, Tamil Nadu [email protected]

In this paper a fitted Galerkin finite element method is presented for the boundary value problem of singularly perturbed differential equations with a small negative shift in the convection term. A fitting parameter is introduced in the Galerkin finite element scheme and is obtained from the theory of singular perturbations. The resultant 3-term recurrance relation is solved by Thomas algorithm. Numerical results are obtained for test problems, which demonstrate the efficiency of the method.

2. Numerical Solution for Fractional Variable-order Differential Equation with Delay by Sarita Nandala,b, Dwijendra N. Pandeya aTechnology Studies Department, Woosong University, South Korea bDepartment of Mathematics, Indian Institute of Technology Roorkee, India [email protected]

In this paper, we propose a new efficient numerical approach for fractional variable- order differential equation with delay. The topic of variable-order fractional differential equations has not been explored extensively using delay, and to augment the order of convergence. Noting that O(h4) and O(τ 1−α) are the best convergence orders achieved in the spatial and temporal dimensions respectively, our aims are to improve the spatial con- vergence order to 4.5 and temporal order to 2, and to provide rigorous proof of solvability, convergence and stability of the proposed method. In the existing literature, a few work is available to construct higher-order numerical methods for the variable-order fractional differential equations with delay. Construction of a mathematical approach for such equa- tions involves the more numerical analysis. Here, we propose to use parametric quintic spline in the spatial dimension and L2 − 1σ formula for time dimension. The stability, convergence, and solvability will be proved using discrete energy method. The proposed numerical approach will improve the convergence in both aspects (spatial-dimension and time-dimension). Numerical simulation will be carried out using the MATLAB software to demonstrate the effectiveness of numerical scheme.

3. A Fixed Point Approach to the Existence-Uniqueness of Coupled-Elliptic Nonlinear Partial Differential for Convection in Porous Media by Saginta Dey, B.V.R. Kumar Department of Mathematics and Statistics,IIT-Kanpur

The problem of convection in a porous media, which is topic of great interest for researchers all over the world, is governed a system of coupled elliptic nonlinear par- tial differential equations. While the model, based on experimental validation, has been widely used the fundamental question of existence-uniqueness has not been addressed so far. In this study, we prove the existence-uniqueness of a coupled non-linear elliptic par- tial differential equation system using Feado-GalerkinMethod, compactness theorems and Brouwer’s fixed point theory.

4. On Fuzzy Contra g∗β-Continuous Functions by Madhulika Shuklaa, J.K. Maitrab aDepartment of Applied Mathematics, Gayan Ganga Insitude of Technology and Sciences Jabalpur (M.P.) 482011 78 bDepartment of Mathematics and Computer Science, R.D. University, Jabalpur 482001 [email protected], jkmrdvv@rediffmail.com

The notion of fuzzy sets was introduced by L. A. Zadeh in 1965. C. L. Chang has extended the concept of topology by taking a collection of fuzzy sets instead of crisp sets and developed the theory of Fuzzy Topological spaces. N. Levine introduced the concepts of generalized closed sets in general topology in the year 1970. In 2006, Eradal and Etienne introduced the notation of fuzzy contra continuous mapping. S. S. Benchalli and G. P. Siddapur introduced the notation of generalized pre-closed sets in fuzzy topological space in 2011. Recently M. Shukla introduced the concept of fuzzy contra g∗p-continuous, fuzzy contra g∗s-continuous and fuzzy contra g∗α-continuous in fuzzy topological space. In this paper we introduce and study the new class of mappings called fuzzy contra g∗β- continuous and fuzzy almost contra g∗β-continuous functions in fuzzy topological spaces. Also we define the relation between of fuzzy contra g∗β-continuous and fuzzy almost con- tra g∗β-continuous spaces and study some of their properties.

5. Generalized Differential Quadrature Method for Vibration Analysis of Non-homogeneous Orthotropic Thin Rectangular Plates by Renu Saini, Department of Mathematics, Maharaja Agrasen College, University of Delhi, India [email protected]

In present paper generalized differential quadrature method (GDQM) is used to ana- lyze the vibration characteristics of non-homogeneous orthotropic rectangular plates. It is assumed that the non-homogeneity arises due to the exponential variation in youngs moduli, shear modulus and density of the plate material along the direction of orthotropy. In GDQM the derivative of a function with respect to space variable at a given grid point is approximated as the weighted linear sum of function values at all the grid points in the computational domain of that variable. Based on Kirchhoffs plate theory, the gov- erning differential equation for such a plate model is derived. The solution procedure by means of GDQM has been implemented in a MATLAB code. The Numerical results are computed for CCCC boundary condition for first three modes of vibration. The effect of various plate parameters on natural frequencies is analyzed and presented graphically. A comparison of results with other methods available in the literature is presented. A close agreement of results shows the versatility of the present method.

6. Power Series Convergence Method for an Operator Based on Multivariate q-Lagrange Polynomials by Rahul Shukla, Purshottam Narain Agrawa Department of Mathematics, Indian Institute of Technology Roorkee, India [email protected]

Let C(I), k.k
be the Banach space of all continuous functions on I = [0, 1] with the sup-norm. Motivated by the work of Erkus et al.(Appl. Math. Comput. 182(2006), 213–222), for f ∈ C(I), Behar et al.(J. Math. Anal. Appl., 491(2) (2020), 1–24) defined the following class of operators

( r ) ∞ n  β(1),...,β(r) Y  (k) n X X n n n Bn,q (f(s); x) = 1 − xβn q (q ; q)l1 (q ; q)l2 ...(q ; q)lr k=1 p=0 l1+l2+···+lr =p (β(1))l1 (β(2))l2 ...(β(r))lr  [l ]   n n n f r q xp, (q; q)l1 (q; q)l2 ...(q; q)lr [n + lr − 1]q and studied the approximation properties of the bi-variate and GBS(Generalized Boolean Sum) operators associated to the above operators in the terms of the partial moduli of continuity and the mixed modulus of smoothness, respectively. 79

In the present work, using multivariate q-Lagrange polynomials, we construct an inte- gral type generalization of the above linear positive operator. We prove some interesting inequalities concerning to the moments and the central moments. Further, we study the convergence of our operator by means of Power series convergence method. In this quest, we prove a non-trivial Korovkin type theorem to show the convergence of the proposed operator. It is worth to mention that the power series convergence method is more general form of classical convergence.

7. A Novel Two Steps Numerical Method to Solve Non-linear Equations by Jogendra Kumar, Department of Mathematics, School of Physical Sciences, DIT University, Dehradun, Uttarakhand-248009 [email protected]

Present work proposes a numerical method of order three based on fundamental theo- rem of calculus and composite Simpson rule for solving nonlinear equations. The accuracy of method is shown with the help of numerical examples and a comparative study is done with some well-known existing numerical schemes.

8. A Numerical Scheme based on Haar Wavelet Nonstandard Finite Differ- ence Method for the Solution of a Class of Generalized Burgers’ Equation by Mukesh Kumar Rawani, Amit Kumar Verma Department of Mathematics, Indian Institute of Technology Patna, India [email protected] Solving Burgers’ equation always posses challenges for small values of viscosity. Here we propose a method to compute the numerical solution for a class of generalised Burgers’ equation based on the Haar wavelet (HW) coupled with non- standard finite difference (NSFD) method. In the solution process, the time derivative is discretised by nonstandard finite forward difference and spatial derivatives are approxi- mated by Haar wavelet. The quasilinearization process is used to tackle the non-linearity in the equation. Accuracy and efficiency of the method are assessed by computing L∞ and L2-error norms. It is observed that the proposed method produces accurate results and quite easy to implement.

9. An Effective Numerical Technique to Solve Lane-Emden Equations based on the Galerkin Finite Element Method by Biswajit Pandit, Amit Kumar Verma Department of Mathematics, Indian Institute of Technology Patna, Patna–801106, Bihar [email protected]

Lane-Emden type equations arise various physical phenomena in mathematical and as- trophysics like stellar structure, thermionic currents, thermal explosions, radiative cooling, CTC, etc. In this work, we consider a model by considering the equation

( 0 xβ y0(x)
+ xβ f(x, y) = 0, 0 < x < 1, & 0 0 y (0) = 0, b1y(1) + a1y (1) = c1.&

Here, we apply continuous Galerkin finite element method (CGFEM) by choosing the Lagrange linear and quadratic polynomials as a test and trial functions. We approximate 1 the integral value of nonlinear function by using trapezoidal and Simpson’s 3 rule. We compare the numerical results with other numerical results computed by various methods. Residue table shows that our proposed techniques are efficient, convenient and promising in the existing literature.

10. Application of Haar Wavelet on a Class of System of Coupled Lane- Emden Equation by Narendra Kumar, Amit K. Verma Department of Mathematics, Indian Institute of Technology Patna [email protected] 80

Coupled Lane-Emden equation appears in several branches of science and engineering such as catalytic diffusion reactions, dusty fluid models, and in the study of concentra- tions of Carbon Dioxide and Phenyl Glycidyl Ether. In this work, we consider a system of Coupled Lane-Emden equations with boundary conditions. We propose an efficient numerical technique based on the Haar wavelets collocation method together with the Newton-Raphson approach to solve the above differential equation. In this technique, we use the Haar wavelets collocation method and get the system of nonlinear equations. Then, we solve the system of nonlinear equations using the Newton Raphson method to get the solution of the system of Lane-Emden equations. We discuss some test problems based on it. We compare our results with the other existing methods such as the Ado- mian decomposition method, Taylor series solution, Successive iteration technique etc. and check the accuracy and efficiency of the proposed method.

11. Contour based Analysis for Image Classification by Ezilmaran D, Vinoth Indira D Vellore Institute of Technology, Vellore [email protected], [email protected]

Image processing has trending research in current decades. It contains lots of research challenges to increase results of the images. Classification behave major role for image pro- cessing to solve the challenges. Contour segmentation helps us to find the detecting edges of the images. The contrario approach, the starting point is defining the conditions where contours should not be detected soft gradient regions contaminated by noise. Ion mobil- ity spectrometry (IMS) increase the peal clarity of different measurement and it signifies signal to noise ratio. So in this article, we will approach the contour technique and IMS classsification to cluster the image through that this article make an analysis of the image.

12. Exact and Nonstandard Finite Difference Schemes for the Generalized Form of Burgers Fisher Equation by Sheerin Kayenat, Amit K. Verma Department of Mathematics, Indian Institute of Technology Patna [email protected]

We consider a generalized form of Burgers Fisher (BF) equation subject to certain ini- tial and boundary conditions. We propose an explicit exact finite difference (EFD) scheme for the BF equation using its solitary wave solution. Furthermore, a non-standard finite difference (NSFD) scheme is also proposed. The properties like positivity and bound- edness is proved to be preserved by the proposed NSFD scheme. The method is shown to be stable under certain conditions and the local truncation error is calculated. The principal part of the local truncation error of NSFD scheme is O (∆x + ∆t) . Approximate solutions of the BF equation is obtained using the proposed NSFD scheme. We have com- pared our result with two methods. First is compact finite difference (CFD) method in which a combination of a sixth-order CFD scheme in space and a low-storage third-order total variation diminishing Runge-Kutta scheme in time have been used and second is Adomian decomposition method. Comparisons indicate the supremacy of our proposed NSFD method. It gives encouraging result for various different parameters. Also the proposed NSFD scheme is very easy to handle and the computation work takes very less time. Moreover it gives accurate result for relatively bigger values of step size. 81

Section H: Solid Mechanics, Fluid Mechanics, Astrophysics and Relativity, and related areas

1. A Study of Unsteady Magnetohydrodynamic Flow of An Incompressible, Viscous, Electrically Conducting Fluid Bounded By Two Non-Conducting Vertical Plates in Presence of Inclined Magnetic Field by Mrinmoy Goswami, Krishna Gopal Singha Kaziranga University, Assam, India, Pragya Academy Junior College, Assam [email protected]

The present article addresses the effect of unsteady MHD flow of an incompressible, viscous fluid bounded by two no-conducting parallel plates placed vertically in presence of uniform inclined magnetic field. One of the plates is considered to be in motion with con- stant velocity whereas the other plate is adiabatic. Using transformation associated with decay factor, we have deduced a set of ordinary differential equations which are solved analytically for the flow field, temperature field and induced magnetic field for different values of MHD flow parameters. The results obtained for velocity, temperature and in- duced magnetic field are discussed and analyzed graphically.

2. Nonlinear evolution of weak discontinuity waves in Darcy-type porous media by Mithilesh Singh, Department of Applied Science, Rajkiya Engineering College, Sonbhadra-231206, INDIA. [email protected]

The propagation of nonlinear waves in one dimensional space, unsteady and compress- ible flow in Darcy-type porous media is analyzed. It is assumed that the weak discontinuity propagates long the characteristic path using the characteristics of the governing quasi- linear system as the reference coordinate system. Evolution equation in the characteristic plane is derived. As an application of the theory the breaking point at the wave front is determined. It is assessed as to how the porosity of the medium affects the process of steepening and flattering of acceleration waves with planar, cylindrical, and spherical symmetry. The critical amplitude of the initial disturbance has been determined such that any compressive disturbance with initial amplitude greater than the critical one always grows into a shock wave, while the initial amplitude less than the critical one always decays.

3. On Swirling Flows Near Rotating Disks by Bikash Sahoo, Department of Mathematics, National Institute of Technology Rourkela, India. [email protected]

In this paper we have discussed about different kind of revolving flows near rotating disks. These flow problems have immense technical and industrial applications. Though most of the flow problems admit self-similar solutions, it is difficult to find closed form analytic solutions even for the simplest of these problems. The mathematical and com- putational challenges arising due to such flow problems will be discussed. The challenges become worse if one considers non-Newtonian fluids, specifically viscoelastic fluids. The higher order derivatives present in the constitutive equations give rise to momentum equa- tions, whose order exceeds the number of available boundary conditions. Finally, we will focus on a specific flow problem, namely the problem arising due to the flow of a viscous Newtonian fluid near an infinite rotating disk with rough surface. The no-slip boundary conditions are replaced by partial slip boundary conditions. Simple mathematical analy- sis will be used to prove the existence of the solution before proceeding for the numerical computation of the self-similar equations.

4. Heat and Mass Transfer Effects on Linearly Accelerated Isothermal In- clined Plate 82 by J.L. Ramaprasada, K.S. Balamuruganb, B. Rushi Kumarc aDepartment of Mathematics, PB Siddhartha College of Arts and Science, Vijayawada, Andhra Pradesh, India bDepartment of Mathematics, RVR & JC College of Engineering, Guntur, Andhra Pradesh, India cDepartment of Mathematics, School of Advanced Sciences, VIT University, Vellore, Tamilnadu, India [email protected]

Current work reports heat and mass transfer effects on unsteady free convection flow past a linearly accelerated isothermal inclined plate with variable temperature and mass diffusion with thermal radiation. The fluid is gray and non-scattering medium. When time t > 0, the plate is accelerated with velocity u0t, the plate temperature is raised linearly with respect to time and the mass is diffused from the plate linearly with time. The Mathematical equations represent the present flow problem are solved by Laplace transform method. The influences of significant involved parameters on velocity, temper- ature and concentration are tested. The rate of heat transfer in terms of Nusselt number and the rate of mass transfer in terms of Sherwood number have also been computed and their impacts for various parameters are discussed through the graphs.

5. MHD Three-dimensional Flow of Powell Eyring Fluid over a Bidirectional Non-linear Stretching Surface with Temperature Dependent Conductivity, Heat Absorption/generation by R. Meenakumari, P. Lakshminarayana Department of Mathematics, SAS, VIT, Vellore-632014, India [email protected]

The Present work addresses the MHD three-dimensional boundary layer flow of Powell Eyring fluid over a bidirectional non-linear stretched surface in the presence of thermal ra- diation and heat generation/absorption. We also considered the temperature dependent thermal conductivity. The governing flow partial differential equations are transmuted into ordinary differential equations with the help of suitable similarity transformations. The resultant non-linear coupled system is solved numerically by shooting technique. The effects of various pertinent parameters on the present flow are presented graphically and explained in detail. The numerical values of heat transfer coefficient on various parameters are tabulated and analyzed.

6. Finite Difference Computation of Free Magneto-Convective Powell-Eyring Nanofluid Flow over a Permeable Cylinder with Variable Thermal Conduc- tivity by G. Kumaran, R. Sivaraj, Department of Mathematics, SAS, VIT, Vellore-632014 [email protected]

In this paper, a theoretical examination is implemented to analyze the impact of ther- mal conductivity variation and thermal radiation on chemically reacting, free convective Powell-Eyring nanofluid flow over a cylinder. The nanoscale effects are accounted by employing the Buongiorno model. The transformed governing equations are numerically solved by using Keller box method under suitable boundary conditions. The comparison results reveal that the obtained results find an excellent match with the results in the literature. The graphs and tables elucidate the impacts of various pertinent parameters on thermo-solutal transport characteristics. It is to be noted that amplifying thermal conductivity variation rises fluid velocity and temperature. Magnifying the radiation cor- responds to weak radiative flux and stronger thermal conduction which decrease the heat transfer whereas the mass transfer is increased.

7. Coupled Radiative and Convective Heat Transfer in Enclosures by N. Rajaa, S. Saravananb 83 aDepartment of Mathematics, KPR Institute of Engineering and Technology, Coimbatore bDepartment of Mathematics, Bharathiar University, Coimbatore [email protected]

The combined effect of surface radiation and buoyancy induced convection in a closed enclosure is investigated numerically. A vertical walls of the enclosure are cooled at a con- stant temperature whereas horizontal ones are perfectly insulated. Two discrete heaters with higher constant temperature are placed either side by side or one above the other inside the enclosure. The surfaces of the enclosure walls and the heaters are assumed to be opaque, gray and diffuse emitters and reflectors of thermal radiation. Air is considered as a working fluid which is radiatively non-participating under moderate temperatures conditions. The nonlinear partial differential equations for the resulting flow were solved by the finite volume method on a uniform staggered grid system. The fluid flow and energy distribution inside the enclosure is studied for different possible values of Rayleigh number, Ra and the emissivity, , of the surfaces concerned. It was found that, when the heaters remains side by side, the surface radiation plays a prominent role in altering the flow pattern. On the other hand the effect of surface radiation is minimal when the heaters are placed one above the other.

8. Impact of Inclined Magnetic Field on the Peristaltic Flow of a Couple Stress Fluid with Heat Transfer by R. VijayaKumar, Nirmala P. Ratchagar R. VijayaKumara,b, Nirmala P. Ratchagarc aDepartment of Mathematics, Annamalai University, Tamilnadu, India. bDepartment of Mathematics, Periyar Government Arts College, Cuddalore, Tamilnadu, India cDepartment of Mathematics, Annamalai University, Tamilnadu, India [email protected]

The aim of the present paper is to investigate the mathematical study of peristaltic transport of an incompressible couple stress fluid in an asymmetric channel under the influence of an inclined magnetic field and heat transfer. The channel walls contains inner and outer tube are rigid and sinusoidal wave and it is defined in cylindrical coordinate. The non-dimensional governing differential equations have been tackled under the assumption of long wave approximation and low Reynolds number. The velocity equation is solved an- alytically by utilizing the perturbation technique and exact solutions are computed from temperature equation. Impact of various parameters on flow characteristic have been analysed by plotting graphs and discussed numerically in details. The study exposes that the flow is appreciably influenced by the presence of a inclined magnetic field and it is contributed in biomedical engineering such as transport phenomenon in peristaltic micro pumps.

9. On the Azimuthal Shear Instability of Inviscid Incompressible Swirling Flows by S. Prakash, M. Subbiah Department of Mathematics, Pondicherry University, Kalapet, Puducherry-605014, India [email protected]

Bounds on Complex Eigenvalues corresponding to unstable azimuthal normal mode disturbances of inviscid incompressible variable density swirling flows are obtained.

10. Numerical Study of an Electrically Conducting three-dimensional Cas- son Fluid Flow over Porous Elastic Sheet with Non-Uniform Heat Source/Sink and Soret Effect by L. Padmavathia, S. Venkateswarlub, M. Suryanaryana Reddy c aDepartment of Mathematics, Jawaharlal Nehru Technological University Anantapur, 84

Ananthapuramu 515002, A.P, India bDepartment of Mathematics , RGM college of Engg &Tech, Nandyal 518501, A.P, India cDepartment of Mathematics , JNTUA college of Engineering, Pulivendula 516390, A.P, India [email protected]

The three-dimensional flow of Casson fluid over a porous Elastic sheet in the presence thermal radiation , non-uniform heat source/sink, soret effect and first ordered chemical reaction with the diffusion slip condition. The governing set of partial differential equa- tions is converted into the set of nonlinear ordinary differential equations using suitable similarity transformations and they are solved numerically by using bvp4c shooting tech- nique. For an analysis of the problem, velocity, temperature and concentration fields are demonstrated graphically and also skin friction, Nusselt number, and Sherwood number are represented through tables. The current results are compared to the previous results in an excellent agreement.

11. Effect of Viscoelasticity and Internal Current on Wave Attenuation by Shyam Sunder Iyer, M.J. Vedan Department of Computer Applications, CUSAT, Cochin [email protected]

Wave attenuation on a viscoelastic sea bed in the presence of internal current is studied. The sea bed characteristic is seen to have a significant influence on wave attenuation. This and the influence of internal current are investigated both analytically and numerically. This study is related to the modelling of the phenomenon of mud bank formation observed in the south-west coast of India

12. Mathematical Model of MHD Flow and Heat Transfer between a Solid Rotating and Stationary Permeable Disk by Maraika Alexander, Sreedhara Rao Gunakala, Victor M. Job Department of Mathematics and Statistics, The University of the West Indies, St. Augustine, Trinidad and Tobago. [email protected]

This study examines some of the velocity and temperature characteristics of the steady MHD flow of a viscous incompressible fluid between a rotating disk and a stationary per- meable disk, the depth of which is equal to that of the free fluid. The flow is conceptually divided into two layers, including: a free fluid region; and the porous layer (where the flow is naturally restricted). The Navier-Stokes and Brinkman equations are used to model the flow in the respective layers. The solution strategy involves the use of a series expansion method to approximate the velocity distributions and temperature effects. The velocity profiles are sketched for variations in the Reynolds number Darcy parameter, and Hart- mann number; while temperature profiles are sketched for variations in Reynolds number, Darcy parameter and thermal conductivity ratio. The influence of the above mentioned parameters on streamlines is also discussed.

13. Radiative Newtonian Carreau Nanofluid through Stretching Cylinder Considering First Order Chemical Reaction by Nasru Sivakumara, B. Rushi Kumarb, P. Durgaprasadc aDepartment of Mathematics, SRM IST,Kattankuluthur, India bDepartment of Mathematics, SAS, VIT, Vellore,India cDivision of Mathematics, VIT, Chennai, India [email protected] 85

A Mathematical model of MHD radiative Carreau nanofluid is investigated for lami- nar, steady and incompressible flow. The power law under the influence of heat genera- tion/absorption and radiation is taken into consideration. The geometrical model com- prises the effects of thermophoresis and Brownian motion. The governing conservation flow equations are converted into non-linear ordinary differential equations by suitable similarity transformations. The emerging nonlinear governing equations are tackled by numerical technique Runge-Kutta method of fourth order. The insight of the physical significances of the problem is presented graphically using MATLAB Software BVP4C. The interference impacts of Carreau nanofluid flow characteristics are presented in the formation of velocity field, temperature field and concentration field distributions. The non-dimensional physical parameters, magnetic parameter, radiation parameter, Ther- mophoresis parameter, Brownian motion parameter, Weissenberg parameter and chemical reaction parameter are analyzed. This investigation outlined that the ability of Magnetic field slow down the momentum of Carreau nanofluid to reduce the governing flow.

14. Bioconvective Flow of Eyring-Powell Fluid Suspended with Microorgan- isms in the Presence of Non-linear Thermal Radiation, Activation Energy and Variable Thermal Conductivity by A. Sumithra, R. Sivaraj Department of Mathematics, SAS, VIT, Vellore [email protected]

The dynamics of Eyring-Powell nanofluid suspended with microorganisms on a plate, wedge and stagnation point is explored. The flow field is subject to non-linear thermal radiation, activation energy, variable thermal conductivity, chemical reaction, and bio- convection. The governing equations are modified into a system of ordinary differential equations via similarity transformation which are solved numerically through Runge-Kutta (R-K) shooting technique.This study addresses the influence of numerous pertinent param- eters on the fluid flow, mass and heat transfer characteristics and the solutions obtained are elucidated through graphs and tables. It is witnessed that the Eyring-Powell fluid ma- terial parameters (λ1) and (λ2) exhibit a contrary nature on the velocity profile. Improved values of Prandtl number aids in heat transfer. Larger values of Schmidt number weaken the concentration boundary layer. In addition, the density of micro-organisms depreciates for growing values of Peclet number and bioconvective Schmitt number.

15. Numerical Simulation of Blood Nanofluid Flow over Three Different Ge- ometries by Means of Gyrotactic Microorganisms: Applications to the Flow of a Circulatory System by H. Thameem Basha, R. Sivaraj Department of Mathematics, Vellore Institute of Technology, Vellore [email protected]

This work is performed to express the significance of the induced magnetic field and gyrotactic microorganisms on the flow of tangent hyperbolic nanofluid over a plate, wedge and stagnation point of plate. Suitable self-similarity variables are employed to convert the fluid transport equations into ordinary differential equations which have been computed with the use of the Runge-Kutta-Fehlberg (RKF) approach. The impacts of active param- eters on flow field are illustrated with graphs and tables. The growing magnetic parameter lessens the blood nanofluid velocity over three different geometries. Blood nanofluid has a higher heat transfer rate over a stagnation point than a wedge and plate. Blood nanofluid temperature augments by uplifting the thermophoresis parameter. Peclet number shows a high impact on microorganisms density in a blood nanofluid. This exploration can provide a clear view regarding the heat and mass transfer behavior of blood flow in a circulatory system and various hyperthermia treatments like the treatment of cancer.

16. Cross diffusion and heat source effects on a three dimensional MHD flow of Maxwell nanofluid over a stretching surface with chemical reaction by M. Vinodkumar Reddy, P. Lakshminarayana 86

Department of Mathematics, Vellore Institute of Technology, Vellore [email protected]

This paper investigates the three-dimensional magnetohydrodynamic (MHD) flow of an upper convected Maxwell (UCM) nanofluid with the thermal radiation, cross-diffusion and heat source effects along a stretching sheet. The effects of chemical reaction, ther- mophoresis and Brownian motion are also studied. We have applied proper similarity variables and transformed the governing boundary layer equations into a system of non- linear ordinary differential equations. The present problem is solved numerically by R-K based shooting technique. The variations of temperature and concentration profiles are shown graphically and discussed in detail. The numerical results of Nusselt and Sherwood numbers are presented in tabular form for different physical parameters. We noticed that the Dufour and thermal radiation parameters decrease the temperature field and increase the concentration field. Heat source and chemical reaction parameters decrease the Nus- selt number and increase the Sherwood number. Also, noticed that the Dufour and Soret numbers raise the Nusselt number, but they decline the Sherwood number.

17. Similarity Solutions of One-dimensional MHD Shock Wave in a Non- Ideal Gas with the Effect of Viscosity by Narsimhulu Dunnaa, Ravilisetty Revathib aDepartment of Statistics and Applied Mathematics, School of Mathematics and Com- puter Sciences, Central University of Tamil Nadu, Neelakudi, Thiruvarur - 610 005, Tamil Nadu bDepartment of Mathematics, Birla Institute of Technology and Science Pilani, Hyder- abad - 500078, Telangana [email protected]

The interaction of shock waves with viscosity into a different medium is one of the most important problems in the regime of compressible gas flow. The effect of viscosity and non- idealness parameter on unsteady cylindrical strong magnetohydrodynamic (MHD) shock n wave driven out by a piston moving with the time according to a power law [vp ∝ (t/t0) ] in a non-ideal gas for both adiabatic and isothermal flows are investigated. The governing equations considered are reduced into a set of ordinary differential equations (ODEs) by using similarity transformations. To obtain distinct features of shock propagation, it is assumed that the dusty gas flow is the mixture of real gas and small solid particles which are uniformly distributed in the medium and the equilibrium flow condition is maintained. Numerical calculations have been performed to obtain the profiles of flow variables be- tween the piston (η = ηp) and shock front (η = 1) using Runge-Kutta method of 4th order. It is found that the flow variables have distinct effects in perfect gas, dust-free gas, and a mixture of perfect gas and small solid particles by increasing values of non-idealness parameters in the presence of viscosity. These effects are more significant in the case of isothermal flow when compared with adiabatic flow. The findings confirmed that the viscosity and non-idealness parameters have major effect on the flow variables and shock strength.

18. Gravity Effects on the Onset of Transient Convection in a Porous Medium by S. Vigneshwaran, S. Saravanan Department of Mathematics, Bharathiar University, Coimbatore - 641 046, Tamil Nadu [email protected]

This study is concerned with the onset of transient convection in a fluid saturated horizontal porous layer heated uniformly from below. The layer is subjected to a grav- ity gradient along its height. Bottom heating is imposed suddenly that can introduce temperature gradients within the layer, in particular adjacent to the bottom surface. A linear stability analysis is carried out employing the propagation theory. The resulting eigenvalue problem is solved using the shooting technique. The conditions representing 87 the onset of transient convection are obtained in terms of critical Rayleigh-Darcy number and critical wave number.

19. A Study with Magnetic field on Stenosed Artery of Blood flow by Sarfraz Ahmed, Biju Kumar Dutta Department of Mathematics, School of Basic Sciences, Assam Kaziranga University, Jorhat- 785006

The present study and its mathematical modelling was done to decide the impact of the magnetic field on blood moving through a pivotally asymmetric but radially sym- metric atherosclerotic conduit. Herschel-Bulkley fluid model condition has been taken to non-Newtonian character of blood flow in the presence of applied magnetic effect. The mathematical model is analyzed graphically and numerically. It was revealed that within the sight of applied magnetic field, blood didn’t definitely change the stream designs, yet caused an apparent decline in the shear stresses and a marginally lower protection from stream. This hypothetical demonstration to cardiovascular infections is considered in our study.

20. Soret and Heat Generation Effects on an Unsteady Free Convective Flow Past an Exponentially Accelerated Plate with Constant Mass Flux by J. Prakasha, R. Swethab, S. Vijaya Kumar Varmac aDepartment of Mechanical Engineering Science Faculty of Engineering and The Built Environment University of Johannesburg, Auckland Park Kingsway Campus Johannes- burg,South Africa bDepartment of Mathematics, Gudalvalleru Engineering College, Gudlavalleru,521356, A.P. India. cDepartment of Mathematics, S. V. University, Tirupati (A.P.), India. [email protected]

Closed form analytical analyzation has been accomplished to examine the consequences of Soret and heat absorption on a free unsteady convective flow with mass and heat trans- fer of a viscous, electrically conducting and incompressible fluid over a vertical plate which is exponentially accelerated. The plate is accelerated exponentially in its plane with a ve- at locity u = u0e at time t > 0 and at the same time, the level of temperature near the plate raises to TW with constant mass flux. The Boussinesqs dimensionless equations are figured out by the method of Laplace transforms in closed form. The results of these flow parameters on profiles of temperature,concentration, velocity are presented in graphs h ∂u i h ∂θ i and the effects of velocity gradient ∂y ,surface heat transfer rate tables. ∂y and y=0 y=0 h ∂c i surface mass transfer rate ∂y are discussed through tables. y=0

21. Dufour and Soret Effects on MHD Flow of Cu−water and Al2O3−water Nanofluid Flow over a Permeable Rotating Cone by Padmaja K, B. Rushi Kumar Department of Mathematics,School of Advanced Sciences, VIT, Vellore-632014, Tamil Nadu [email protected]

In this paper, we investigate numerically the nanofluid flow about a permeable, ver- tical rotating cone with Dufour and Soret effects in the presence of thermal radiation, magnetic field and chemical reaction. The heat and mass transfer of a MHD nanofluid about a porous vertical rotating cone is analysed. The fluid flow considered is steady, laminar and incompressible. A uniform suction/injection of the fluid is present on the surface of the cone. The cone is symmetric about the axis of rotation and is rotating with an angular rotating velocity. The governing equations pertinent to the fluid flow and the thermophysical properties of the nanofluid are nonlinear partial differential equations (PDEs). Using similarity transformation variables, these partial differential equations are 88 converted into ordinary differential equations (ODEs). MATLABs bvp4c solver is used to solve the converted system of ODEs. To achieve a clear understanding about the physical insights of the problem, the two nanofluids- copper in water and alumina in water are analysed. The graphical representations of tangential, normal, circumferential velocity profiles, temperature profiles and concentration profiles with respect to various fluid flow parameters are investigated. The Dufour and Soret numbers and the thermal radiation parameter have significant impact on the rates of heat and mass transfer.

22Entropy Generation on EMHD Stagnation Point Flow of Hybrid Nanofluid over a Stretching Sheet: Homotopy Perturbation Solution by Shaik Jakeer, P. Bala Anki Reddy Department of Mathematics,School of Advanced Sciences, VIT, Vellore-632014, Tamil Nadu [email protected]

The intention of this article is to explore the entropy generation in EMHD hybrid nanofluid on a stagnation point in the presence of slips, heat generation and viscous dissipation. The fluid in the enclosure is water containing hybrid nanoparticles Ag-Cu. Suitable self-similarity variables are employed to transform the nonlinear differential sys- tems into an ordinary differential system, which computed via homotopy perturbation method (HPM). The comparison with the homotopy perturbation method (HPM) gives an accurate and reliable result than the numerical method (Runge-Kutta method). The graphical results are expressed for velocity, temperature, entropy generation, Bejan, skin friction and Nusselt number with an impact of active parameters. The higher values of electric field enhancing the velocity whereas the opposite nature for a magnetic field pa- rameter. The entropy generation rises for higher values of a magnetic parameter, Eckert number and α1. In the magnetic field and electromagnetic field plays an important role in biomedical applications especially radiofrequency ablation (RFA), magnetic resonance imaging (MRI), cancer therapy, tumor therapy, malaria infection. This theoretical inves- tigation may be profitable in biomedical engineering, especially cardiology, cure of skin disorders and treat tumors in Uterus.

23. Natural Convection in a Nanofluid Saturated Porous Medium under Time-Periodic Gravity Modulation by M. Kousalya, S. Saravanan UGC DRS Centre for Differential Equations and Fluid Dynamics, Department of Mathematics, Bharathiar University, Coimbatore 641 046, Tamil Nadu [email protected]

Time-periodic gravitational field is of immense importance in space laboratory experi- ments involving crystal growth and other related fields. Hence the effect of timeperiodic gravity modulation on the onset of natural convection in a nanofluid saturated porous medium is investigated. In particular, water based nanofluids containing conventional metallic and metal oxide particles are considered. The Khanafer-Vafai-Lightstone model with more realistic empirical correlations for the physical properties will be used. The medium is assumed to be heated uniformly from below and a stability analysis is carried out on this configuration with the help of the Floquet theory. The emerging instabilities of synchronous and subharmonic types and the transition between them will be examined. The applications of the results of the present study in physical problems will be addressed.

24. Soret and Dufore Effects on MHD Flow through the Porous Medium about a Rotating Vertical Cone in Presence of Thermal Radiation by M. Chitra, S. Jeevitha Department of Mathematics, Thiruvalluvar University, Vellore [email protected] 89

The objective of this paper is to analysis the Soret and Dufour effect with heat and mass transfer on MHD flow through a porous medium of a binary fluid mixture about a vertical rotating cone along with thermal radiation. The governing equations are non- linear partial differential equations and so by using similarity transformation. They are converted into ordinary differential equations. Matlabs built in solver bvp4c is employed to solve numerically the ordinary differentials equations. The effect of various parame- ter on the Velocity, Temperature and Concentration profiles are computed and discussed through graphs.

25. Effective Thermal Conductivity and MHD Convection Flows of Non Newtonian Nanofluid from Horizontal Circular Cylinder by P. Vijayalakshmi a, R. Sivarajb aDepartment of Mathematics, Muthurangam Government arts college, Vellore, Tamilnadu, India bVIT University, Vellore, India [email protected] , [email protected]

This paper illustrates the combined magneto hydrodynamic (MHD) flows of casson viscoplastic nanofluid from a horizontal isothermal circular cylinder in non-Darcy porous medium. The effect of Brownian motion and thermophoresis are studied and validated. The overall heat and mass transfer shape are enhanced for improving the thermal and solutal dispersion effect. The governing partial differential equations are turned into non- linear ordinary differential equations using appropriate non similarity transformation and are solved numerically using Keller-Box finite difference technique. Numerical results for velocity, temperature, concentration along with skin friction coefficient, heat and mass transfer rate computed for different values of physical parameters. It is identified that velocity, heat and mass transfer rate are increased with increasing casson fluid parame- ter whereas temperature, concentration and skin friction are effectively decreased. Also observed that velocity is decreased with increasing Forchheimer parameter whereas tem- perature and nano-particle concentration are enhanced. The temperature profiles are enhanced for thermal dispersion coefficient and concentration profiles are enhanced for increasing solutal dispersion coefficient.

26. Cattaneo-Christov Heat Flux Model for MHD Sakiadis Flow of a Car- rearu Fluid Subject to Quartic Autocatalysis Chemical Reaction by V. Nagendramma, Deparment Of Mathematics, Presidency University, Banglore-560064 [email protected]

This paper explored the heat and mass transfer characteristics in magnetohydrody- namic Sakiadis flow of a Carreau fluid in the presence of autocatalysis chemical reaction. Cattaneo-Christov heat flux model is considered to develop the energy equations. The governing partial differential equations of motion were reduced to a system of ordinary differential equations with the aid of local similarity variables. These ordinary differential equations were further solved by using the Runge-Kutta Fourth order method with BVP5C technique. The thickness of the thermal boundary layer increases with increasing values of magnetic field parameter and Weissenberg number while a reduction is noticed with power law index and thermal relaxation parameters and Prandtl number. Concentration distribution decreases for higher values of strength of homogeneous reaction parameter K. Strength of heterogeneous reaction parameter ks results in the enhancement of concentra- tion.

27. Significance of Nanoparticle Aggregation of Nanofluid flow in an Irreg- ular Channel by Sandra Jestine, B Mahanthesh Department of Mathematics, CHRIST (Deemed to be University), Bangalore 560029, Karnataka, India. 90 [email protected]

The study of incompressible unsteady laminar flow and free convective heat transfer of nanofluid between a vertical long wavy wall and a parallel flat wall is carried out. For the investigation, Ethylene glycol-based nanofluid with titania nanoparticles were used. The study examines the role of nanoparticles aggregation effect under the influence of applied magnetic field, thermal radiation and internal heat absorption. Effective proper- ties of the nanofluid are measured by mixture theory and effective medium theory. The semi-analytical solution of the problem is obtained by regular perturbation method. Ef- fect of different physical parameters on the velocity profile and temperature profile have been studied. In addition, the skin friction and Nusselt number are also examined and presented graphically.

28. Effects of MHD and Electro-Magnetic Fields in Nanofluid over a Stretch- ing Sheet by G. Lakshmi Devi, H. Niranjan Department of Mathematics, School of Advanced Sciences Vellore Institute of Technology, Vellore-632014 [email protected]

The present paper targets to explode the effects of Electro-magnetic fields and heat transfer in a nanofluid over a stretching sheet near a stagnation point. In this mathemat- ical model hires the electric field, thermophoresis, Brownian motion and magnetic field of the system is explode. The governing partial differential equations are non-dimensionalized via related similarity transformation and the results are solved numerically. The impres- sion of some governing flow constants on the temperature, velocity, and concentration of nanoparticles are described through graphs. The variation of engineering quantities such as the Nusselt number and Sherwood number are calculate.

29. The Study of Rayleigh-B´enardConvection in Vertically Oscillating Hy- brid Nanoliquids by Gayathri, S. Pranesh Department of Mathematics, CHRIST (Deemed to be University), Bangalore, India. [email protected]

Rayleigh-B´enard convection in vertically oscillating hybrid nanoliquids is studied in this paper. The gravity modulation effect is studied on fifteen different hybrid nanoliquids by carrying out linear and non-linear analysis. Venezian approach is used to obtain the ex- pression for the correction Rayleigh number and wave number. Fourier series method is used in the non-linear analysis to obtain the expression for Nusselt number. It is observed that with the increase in amplitude of modulation, average Nusselt number decreases and with the increase in frequency of modulation, average Nusselt number increases.

30. Analysis of Fluid Flow in Triangular Cavity using FEM by Mariya Helen Mercy JK, V. Prabhakar Department of Mathematics, Vellore Institute of Technology, Chennai [email protected]

In the present work the temperature distribution and distortion of fluid flow inside a triangular cavity is validated using ANSYS. Three cases are dealt here. Case 1: The vertical wall is insulated, bottom wall heated up and the inclined wall is kept cold. Case 2: The vertical wall is hot, the bottom wall insulated and the inclined wall is kept cold. Case 3: The vertical wall is cold, the bottom wall insulated and the inclined wall is heated. The penalty finite element method is applied to solve the residuals of the non-dimensional form of the governing differential equation. Different types of fluids from air to engine oil is studied. The solid within which the fluid flows also was varied and analyzed. A no slip boundary condition is applied. The laminar nature of the fluid is desired with a different 91 combination of fluids and solids. The same model can be extended for a field problem like plastic injection mould flow with objective function with the results and a mathematical model can be developed.

31. The Effect of The Viscosity of The Porous Solid on The Parallal Plate Channal Flow of Ree-Eyring Liquid when the Dividers are Provided with Non-Erodible Porous Lining by R.L.V. Renuka Devi, Department of Mathematics, Sri Venkateswara University, Tirupati, Andhra Pradesh [email protected]

In this analysis the effect of the viscosity of the porous solid on the parallel plate channel flow of Ree-Eyring liquid when the dividers are provided with non-erodible porous lining is contemplated. The governing partial differential equations are changed to ordinary differential equation by utilizing non-dimensional quantities and solved it analytically. The effects of governing parameters on the liquid velocity are showed graphically. We researched the stream in the free stream area and permeable stream districts by utilizing Darcy law and Ree-Eyring liquid model respectively.

32. Wall Slip Effects on Nanofluid Flow in a Porous Channel by A. Subramanyam Reddya, S. Srinivasb aDepartment of Mathematics, VIT, Vellore -632 014, Tamil Nadu, India. bDepartment of Mathematics, VIT-AP University, Amaravati- 522237, Andhra Pradesh, India. [email protected]

The present analysis deals with the nanofluid flow in a porous channel with slip ef- fects. In this work, blood is considered as a Newtonian fluid and gold (Au)/copper (Cu) as nanoparticles. System of nonlinear ordinary differential equations is derived from the governing flow equations and is solved with the aid of homotopy analysis method. Conver- gence of series solutions is analysed. The influence of nanoparticle volume fraction, wall expansion ratio and slip parameters on the various flow variables have been discussed in detail.

33. Effect of Heat Transfer in a Micropolar Fluid on the Onset of Rayleigh- B´enard-ChandrasekharConvection with Porous Medium under Time Peri- odic Boundary Temperature and Internal Heat Source by Maria Anncy , Joseph T V , Pranesh S Department of Mathematics, CHRIST (Deemed to be University), Bangalore, India. [email protected]

The impact of heat transfer rate over a surface containing voids whose temperature at the boundaries are modulated in a micropolar fluid is investigated to understand the thermal instability of the system exposed to magnetic field and internal heat source. The heat transfer rate within the system with respect to time is found using Lorentz model. The outcome of the study conveys that as the internal heat of the system is increased the heat transfer rate within the system also increases indicating that the thermal instability is destabilized. Moreover, increase in Chandrasekhar number shows a decrease in heat transfer rate this is because, the impact of strong magnetic field is to reduce the rate of heat transport.

34. MHD Combined Convection Flow over a Moving Non-Isothermal Verti- cal Plate with Soret and Dufour Effects and Viscous Dissipation by A. Neerajaa, R.L.V. Renuka Devib, N. Bhaskar Reddyb aAditya College of Engineering, Surampalem, East Godavari Dt. Andhra Pradesh, India bSri Venkateswara University, Tirupati, Andhra Pradesh, India 92 [email protected]

This paper focuses on the numerical solution of a steady MHD combined convection flow of a viscous incompressible electrically conducting fluid along a moving, non-isothermal vertical plate in the presence of mass transfer, Soret and Dufour effects and viscous dis- sipation. The governing boundary layer equations have been transformed to a two-point boundary value problem in similarity variables and the resultant problem is solved numer- ically using the fourth order Runge-Kutta method along with shooting technique. The influence of various governing parameters on the fluid velocity, temperature, concentra- tion, skin-friction coefficient, Nusselt number and Sherwood number are computed and discussed in detail.

35. Triple Diffusive Convection in Temperature and Electric Field Depen- dent Variable Viscosity in a Newtonian Dielectric Liquid with Internal Heat Source by S. Pranesh, P.G. Siddheshwar, Ansa Mathew Department of Mathematics, CHRIST (Deemed to be University), Bangalore [email protected]

The linear and non-linear analysis of the triple diffusive convection in a temperature and electric field variable viscosity dielectric Newtonian liquid with internal heat source (or sink) are studied analytically. The strength of heat source and electric field are char- acterised by internal Rayleigh number and electric Rayleigh number respectively. The linear stability analysis shows that the increase in variable viscosity and internal heat source is to stabilize the system. The effect of internal Rayleigh number, electric Rayleigh number and variable viscosity on heat and mass transfer is investigated by deriving the generalized Lorenz model and solving it numerically. It is found the effect of increase of all these parameters increases the heat and mass transfer.

36. Linear and Non-Linear Analysis of Internal Heat Modulation on Rayleigh- B´enardConvection in Ferromagnetic Liquids with Couple Stress Meghana J, Pranesh S Department of Mathematics, CHRIST (Deemed to be University), Bangalore, India [email protected]

RayleighB´enard convection in ferromagnetic liquids with couple stress in the pres- ence of internal heat modulation is studied using linear and non-linear analysis. A non- autonomous Lorenz model for the problem is derived and both linear and nonlinear anal- yses are performed using this Lorenz model. The expression for the critical Rayleigh number and the correction Rayleigh number is found from the linearized Lorenz model. The Lorenz system of equations is solved for the amplitude to arrive at the Nusselt number which quantifies the heat transport. The influence of various non-dimensional parameters on the onset of convection and heat transfer are analysed. The study reveals that Couple stress parameter stabilizes the system and decreases the heat transfer. It is also found that square wave type of internal heat modulation is more stable compared to other wave types of internal heat modulation.

37. Linear and Nonlinear Analysis of Two-frequency Time-periodic Bound- ary Temperature on Rayleigh-B´enardConvection by Ansa Mathew, S. Pranesh, P.G. Siddheshwar Department of Mathematics, CHRIST (Deemed to be University), Bangalore. [email protected]

The paper analyses the effect of two-frequency temperature modulation at the onset of convection and heat transfer in a Newtonian fluid by carrying out a linear and non-linear analysis. The Venezian approach is assented encompassing the correction Rayleigh num- ber and wave numbers for meagre amplitude two-frequency temperature modulation The 93

Lorenz model is derived and is solved numerically to quantify the heat transport through Nusselt number. The effects of various combinations of sinusoidal and non-sinusoidal waveforms have been studied on the onset of convection and on heat transfer. On com- parison with no-modulation and single-frequency temperature modulation, it is seen that two-frequency temperature modulation delays the onset of convection and decreases the heat transport with increase in mixing angle. It is observed that temperature modulation results in sub-critical motion however out-of-phase temperature modulation is more stable compare to others.

38. Nonlinear Rayleigh-B´enardConvection in Variable Viscosity Ferromag- netic Liquids by Prakash Ra, Jayalatha Ga, Siddheshwar P.Gb,Sekhar G.Nc aDepartment of Mathematics, RV College of Engineering, Bengaluru 560059, Karnataka, India bDepartment of Mathematics, CHRIST (Deemed to be University), Bengaluru 560029, Karnataka, India cDepartment of Mathematics, BMS College of Engineering, Bengaluru 560019, Karnataka, India [email protected]

The paper deals with linear and nonlinear stability subjected to thermal convection of a variable viscosity Newtonian ferromagnetic liquid in the existence of uniform vertical magnetic field. The truncated Galerkin expansion is employed to study the perturbations in the system due to external constraints. The nonlinear stability is based on truncated Fourier series. The modified Lorenz model together with variable viscosity parameter is first derived and then it is used to describe both linear and nonlinear analyses of the sys- tem. The expressions for the critical Rayleigh number, R, Nusselt number (Nu) is found by solving the Lorenz system of equations. The effects of different parameters on the heat transport, thermal convection have been discussed. The results obtained agree truly with those of limiting cases.

39. Analytic Solution of Bloch Equation for a Time Varying Magnetic Field in the Transverse Direction by Parameswaran Ra,b, M.J. Vedanb aDepartment of Mathematics, Amrita School of Arts & Sciences, Kochi.Amrita Vishwa Vidyapeetham bDepartment of Computer Applications, Cochin University of Science & Technology [email protected]

A solution for Bloch Equation in the case of time varying magnetic field in the trans- verse direction is obtained. For this, purely analytical techniques with fundamental matrix generally used in solving a system of differential equations with non constant coefficients is employed. When the magnetic field increases exponentially, magnetization vector satisfies a second order differential equation which has the form of a Sturm Liouville equation. The solution in this case is also obtained.

40. Stability of Microscopic Body Cosmological Model in Barber Self- Creation Theory of Gravitation by J.S. Wath, A.S. Nimkar Shri Dr. R.G. Rathod Arts and Science College, Murtizapur, Dist. Akola (M.S.) India. [email protected], [email protected]

In this paper, we have studied stability of cosmological model in Rubans background in the context of Barber Self-Creation theory of gravitation in the presence of macroscopic 94 body. Exact solutions are obtained by using relation between metric coefficients and ra- diation. Also, we discuss the features of the obtained model.

41. Bianchi Type-III Cosmological Model Barber Self-Creation Theory of Gravitation by A.M. Pund , P.M. Lambat Department of Mathematics, Shri Shivaji Education Society Amravatis Science college, Congress Nagar, Nagpur (M.S.) India. ashokpund64@rediffmail.com

In this paper, we have investigated the self-creation theory of cosmology proposed by Barber with wet dark fluid as a source of matter in Bianchi type-III space time. The solution of field equation and cosmological model is obtained by using relation between metric coefficients and radiation universe. Also, we have discussed some Physical and kinematical properties of the obtained model.

42. Exact solution for Cosmological constant problem, Variable Gravita- tional Constant Problem and other Cosmological Problems and the Con- tinuity between Anisotropic and Isotropic Cosmology with Single type of Scalar Field and Scale Factor by Shouvik Sadhukhana, Alokananda Karb aIndian Institute of Technology, Kharagpur, West Bengal, India. bUniversity of Calcutta, Kolkata, West Bengal, India. [email protected], [email protected]

We have derived exact solutions for several cosmological problems in case of both Isotropic and Anisotropic cosmology. We have established the direct mapping between the anisotropic Bianchi Type I model and Isotropic FRW model. We have assumed cos- a 2 a mological constant as Λ = 3α a β a which helps us to solve some cosmological problems in generalized way. We have established all the cosmological phases with our derived general- ized scale factor. In our last few papers we had given some predictions about anisotropy as well as viscosity and here we have derived those predictions true. At last we have defined Quintessence scalar field dependent potential and its time evolution using a special type of canonical transformation technique. We have also predicted a direct transition between dark energy and baryonic matters. One of the most important areas of this paper is the derivation of Hyperinflation in Friedmann model. We get time evolution of G, Λ, viscosity η and anisotropic shearing constant σ which are controlling the phases and its transformations with time. We derived that both shear- ing scaler and cosmological constant are decreasing function of time whereas coefficient of viscosity and gravitational constant are partly decreasing and partly increasing and this fact caused the graceful exit as well as reheating phase in cosmic evolution. The most interesting and new part of this research. So the Hyperinflationary model is as a = a0 exp((H0 exp(bt))t). We also derived the inflation which matched completely with well known inflation solution but we proved that inflation is not a phenomena at w = −1. It happened just after a while of this point. Finally we got Scalar field vs Field potential relation that helped us to plot the following graphical representation.

43. Bianchi Type VI0 String Cosmological model in Lyra’s Manifold by S.R. Hadole, A.S. Nimkar Shri Dr. R.G. Rathod Arts and Science College, Murtizapur, Dist. Akola (M.S.) India. [email protected], [email protected]

A solution of field equations has been obtained for a Bianchi Type VI0 space-time with cosmic string in Lyras Manifold by using relation between metric coefficients and Reddy string. Certain physical and kinematic properties of the model have been examined. 95

44. Wet Dark Fluid Cosmological Model In Barber Self-Creation Theory Of Gravitation by S.C. Wankhade, A.S. Nimkar Shri Dr. R.G. Rathod Arts and Science College, Murtizapur, Dist. Akola (M.S.) India. [email protected], [email protected]

In this paper, we investigate a Bianchi type VIII cosmological model with wet dark fluid in Barber Self-Creation theory of gravitation. To get the determinate model of the universe, we have assumed the relation between metric coefficients R and S i.e S = Rn Also, the behavior of the model in radiation universe and physical implications of the model are discussed in detail.

45. Stability of Bianchi Type-IX Cosmological Model in Brans -Dicke The- ory of Gravitation by A.S. Nimkar, Shri Dr. R.G. Rathod Arts and Science College, Murtizapur, Dist. Akola (M.S.) India. [email protected]

In this paper ,we have investigated stability of Bianchi Type-IX cosmological model in the presence of energy momentum tensor for matter and the holographic dark energy in the framework of scalar tensor theories of gravitation proposed by Brans-Dicke(1961) . To obtain the exact solution we have used variation law for Hubble parameter. Also, we discuss the physical and kinematical properties of the model.

46. Hamiltonian Formalism of Bianchi Type 1 Model for Different types Cos- mic Fluid and Effect of Bulk Viscosity on Quintessence Model and Scalar Field Potential by Alokananda Kara, ShouvikSadhukhanb aDepartment of Physics, University Of Calcutta, Kolkata, India. bDepartment of Physics, Indian Institute of Technology, Kharagpur, India. [email protected],[email protected]

We have proposed the Hamiltonian formalism of Bianchi type-1 cosmological model for different types of cosmic fluids including both viscous and non-viscous cases. We have used generalized equation of state (EOS) parameter ω and the cosmological constant Λ. We have proposed a Lagrangian for the anisotropic Bianchi type-1 model in view of a variable mass-system moving in a variable potential. We have considered a canonical transforma- tion from expanding scale factor to scalar field ϕ, which gives a proper transition from classical theory to field theory of cosmology. At the end we have derived Wheeler-DeWitt equation from the Hamiltonians in operator form and predicted the coupling of Coeffi- cient of viscosity and anisotropy. We have also predicted the time variation of scalar field potential with coefficient of viscosity. We have also defined the effect of bulk viscosity on Quintessence model and scalar field potential as well as on classical field. In the derivation we have also shown the possibility of time dependent evolution of gravitational constant G. The evolution profile for anisotropy has also been shown. It is observed that viscosity doesn’t modify the relationship between scalar field potential V (φ) and Scalar field φ in case of viscous fluid of universe.

47. Propagation Of Stoneley Waves In Non-Local Elastic Medium by Ajmeera Chandulal, Department of Mathematics, National Sanskrit University, Tirupati 517 507 [email protected]

In particular, the nonlocal continuum is formulated and developed by Kunin, Edelen and Eringen. This theory is physically valid because we can compare this theory with 96 classical and molecular theories. In this paper, the propagation of stoneley waves in non- local elastic medium is considered. The period equation is derived and it is used to draw the stoneley boundary curves, taking two limiting cases. The nature of stoneley wave in nonlocal elastic medium is found to be dispersive unalike the nondispersive nature of the stoneley waves in classical elasticity. The two limiting cases of the period equation are obtained. These two equations are solved numerically for ratios of densities and rigidity moduli of two elastic halfspaces. Using these results, the stoneley boundary curves are drawn.

48. Remarks On The Dynamic Responce Of Irregular Orthotropic Viscoelas- tic Half-Sace Sunjected To a Moving Line Load by M.K. Pal, A.K. Singh Department of Mathematics & Computing, IIT(ISM) Dhanbad [email protected]

In this paper, we analyse that effect of induced stresses (compressive, shear and tensile) due to moving line load on irregular functionally graded orthotropic viscoelastic half-space. The expressions for said induced stresses are deduced in closed-form by using analytical approach. The effect of various physical parameters viz. maximum depth of irregularity, functionally gradedness, irregularity factor, and frictional coefficient on induced stresses for the considered model has been investigated. For numerical computation, the half-space are comprised with Carbon-fiber and Prepreg materials. Moreover, some notable charac- teristics have been outlined and delineated through graphs by using Matlab software.

49. Investigation of thermal excitation induced by laser pulses and thermal shock in the half space medium with variable thermal conductivity by Rakhi Tiwari Department of Mathematics, Nitishwar Mahavidyalaya, A Constituent Unit of Babasaheb Bhimrao Ambedkar Bihar University, Muzaffarpur, India [email protected]

The present study is concerned with the investigation of the transient responses of a half-space medium with the variable material properties in the context of dual phase lag thermo -elasticity. Boundary of the medium is subjected to a thermal shock. Further, the bounding surface is considered to be heated by a non-Gaussian laser beam. Kirchhoff transformation and Laplace transform technique have been adopted to obtain the ana- lytical solutions. Graphical results of the field-components – non-dimensional conductive temperature, non-dimensional displacement as well as non-dimensional stress have been achieved and illustrated for the different values of time and variable thermal conductiv- ity. Significant results are obtained and it is believed that these results may support in designing the structures heated by a non-Gaussian laser beam in engineering.

50. Investigation on SH-wave propagation in a porous piezoelectric compos- ite with mechanically and electrically perfect interfacial boundaries by Sharmistha Rakshita, Anirban Lakshmana, Kshitish Ch. Mistrib aDepartment of Mathematics, IIIT Kalyani, Kalyani,741235. bDepartment of Mathematics, Chandrapur College, Purba Burdwan 713145. sharmistha [email protected]

The propagation of shear horizontal(SH) waves in tri-layered structure consisting of a perfectly bonded porous piezoelectric layer sandwiched by two isotropic layer is studied analytically in this paper. With the aid of constitution equation for a porous piezoelec- tric structure, the solution of mechanical displacement and electric potential is explored. Dispersion relation is derived for electrical-mechanical boundary conditions at the free surface and interface, solved numerically and illustrated by means of graphs. The effect of porosity, width ratio on the dispersion curve are studied extensively.PZT-5H materials is considered for validation of numerical procedure and further investigation of results .The 97

findings of this mathematical schema can be used for designing and developing of under- water acoustic devices for sensing, non-destructive testing and health monitoring devices.

51. Study of torsional problem in Micro-isotropic, Micro-elastic solid by E. Rama, Department of Mathematics Osmania University, Hyderabad, Telangana [email protected]

In the present paper torsional problem in a Micro-isotropic, Micro-elastic solid is stud- ied and obtained the components of displacement, microrotation, stress and couple stress and shown them graphically.

Section I: Mathematical Modelling, Bio-Mathematics, Operations Research

1. Bifurcation and Chaos in a Discrete Predator-Prey Model with Holling Type-III Functional Response and Harvesting Effect by Anuraj Singh, Preeti Deolia ABV-Indian Institute of Information Technology and Management Gwalior, M.P. [email protected], [email protected]

Nowadays, due to the indiscrete and unconventional harvesting of biotic assets, over- exploitation of biological resources is becoming a topic of much concern among researchers. In this study, a discrete resource-consumer model with Holling type III functional response and harvesting effort is investigated. Bifurcation theory and center manifold theorem are used to establish the conditions for the existence of various bifurcations of codimension 1 such that Neimark-Sacker bifurcation, transcritical bifurcation and flip bifurcation. Nu- merical simulation is performed to demonstrate the analytical findings.

2. Nonlinear Dynamical Behaviour of an SEIR Mathematical Model: Ef- fect of Information, Saturated Treatment and Time Delay by Tanuja Dasa, PK Srivastavaa, Anuj Kumarb aDepartment of Mathematics, Indian Institute of Technology Patna, Patna 801103, India bSchool of Mathematics, Thapar Institute of Engineering and Technology, Patiala 147004, India [email protected]

When the disease spreads in a population, individuals tend to change their behaviour due to presence of information about disease prevalence. Therefore, the infection rate is affected and the incidence term in the disease model should be appropriately modified. In addition, it will be also interesting to see how limitation of medical resources have their impact on the dynamics of the disease. In this work, we propose and analyse an SEIR epidemic model which accounts for the information induced non-monotonic inci- dence function and saturated treatment function. The model analysis is carried out and it is found that when the basic reproduction number is below one, the disease may or may not die out due to the saturated treatment (i.e a backward bifurcations may exist and cause multi-stability). Further, we note that in this case disease extinction is possible if the medical resources are available for all. When basic reproduction number exceeds one, there is possibility of existence of multiple endemic equilibrium points. These multiple equilibrium points give rise to rich and complex dynamics by showing various bifurcation and oscillations (via Hopf-bifurcation). A global asymptotic stability of unique endemic equilibrium (when it exists) is established under certain parametric conditions. An impact of information is shown and also sensitivity analysis of model parameters is performed. Further, a corresponding delay mathematical model is proposed and analysed incorporat- ing delay in incubation. The delay model is analysed and existence of periodic oscillations 98 in population is established via Hopf-bifurcation. Various cases are considered numer- ically to provide the insight of model behaviour mathematically and epidemiologically. Our study underlines that saturation in treatment i.e. limitation of medical resources may cause bi (multi)-stability in the model system. Also, information plays significant role and gives rise a rich and complex dynamical behaviour of the model. Delay in incu- bation, may cause oscillations in the populations.

3. Painleve Property Analysis of Self Interacting Four Species Food Chain Mathematical Model and its Generalization to N-Species by B.V. Baby 3/88, Jadkal Post,Udupi District, Karnataka [email protected]

Long time scale dynamics of self interacting two types of Four Species Food Chain Mathematical Models and their generalization to N-species model are studied. It is found that, more than two species models are always of non Painleve‘ types and so they have erratic dynamics and that may lead to chaos in long time scale. Models are extended to N-species, where N is any finite integer greater than two.

4. The Role of Media on the Dynamics of Zika Outbreak: A Modeling Ap- proach by Naba Kumar Goswamia, B. Shanmukhab aDepartment of Mathematics, University of Mysore, Mysuru, India bDepartment of Mathematics, PES College of Engineering, Mandya, India [email protected]

A non-linear mathematical model has been proposed and analysed for the impact and role of media on the spread of Zika virus disease during the epidemic. First, we investi- gated the epidemiological feasible equilibrium points and the threshold parameter, basic reproduction number R0 is computed using the next-generation matrix method. The in- fected mosquito biting rate and the rate of human to human sexual transmission are the main parameters of the basic reproduction number. The stability of different equilibria of the model is studied and backward bifurcation is discussed, which suggests that merely re- ducing R0 less than one is not enough to make disease-free equilibrium globally stable. We present the sensitivity analysis based on the parameters involved in the basic reproduction number and identify some of the key parameters which can be regulated to control the transmission dynamics of the Zika virus. Secondly, we integrate time dependent control measures into the model and then examine the conditions that are requisite for the disease to be controlled optimally by employing Pontryagin’s Maximum Principle. Finally, effects of the deterministic and optimal control model witnessed by using numerical simulations.

5. Mathematical Model of Corona Virus (COVID-2019) with Limited Re- sources: A Case Study of India Akhil Kumar Srivastav, School of Advances Sciences, Vellore Institute of Technology, Chennai Campus, India [email protected]

The COVID-19 is now one of the deadliest pandemic in human history and has had tragic consequences affecting millions of people worldwide. In this paper, we propose a mathematical model where we classify the infective into two subcategories: asymptomatic and symptomatic. In the developing country like India, medical resources is very limited for this kind of pandemic situation, So in this model we have incorporated treatment fac- tor with limited resource. We analyse the model and also exploit the available data for assessing the pattern and future prediction. We calibrate the proposed model to fit the four data sets, viz. data for the states of Maharashtra, Tamil Nadu, Delhi and overall India, and estimate the transmission rate of symptomatic individuals and recovery rate of quarantined individuals. We also estimate basic reproduction number for the regions 99 under study. Our simulations predict that the infective population will be on increasing curve for Maharashtra and India whereas we can see the settling of active cases for Tamil Nadu and Delhi. Sophisticated techniques of sensitivity analysis are employed to deter- mine the impacts of model parameters on basic reproduction number and symptomatic infected individuals. Further, we perform stability analysis and established local stability of both disease free and endemic equilibrium.

6. Effect of Catheter on Unsteady Fluid Flow through an Inclined Stenosed Artery by Chhama Awasthi, Department of Mathematics, Harcourt Butler Technical University Kanpur-208002, Uttar Pradesh, India [email protected]

In the epidemic situation of COVID-19, patients suffering from cardiovascular diseases are at high risk of this disease. Therefore, it’s necessary and significant to understand the hemodynamic properties of the rheology of blood by developing a mathematical model on it. The non-linear differential equations along the suitable boundary conditions governing the fluid flow of the above mathematical model have been solved by the Perturbation method. In this study, the catheterization process is used that uses a long flexible tube called a catheter. This process determines the location and the percentage of blockage of the stenosis in an artery. Stenosis develops due to the invasion and deposition of fats and cholesterol and it obstructs partially or completely the blood flow in arteries. The combined effect of catheterization, body acceleration, slip, pressure gradient, yield stress, stenosis height, and inclination has been optimized and analyzed with the help of MATLAB. The graph shows that the axial velocity and flow rate increases with the increase in body acceleration, inclination angle, and slip velocity while axial velocity di- minishes on increasing the catheter radius. Wall shear stress increases with the increase in catheter radius and body acceleration but the presence of slip velocity reduces the wall shear stress. Effective viscosity diminishes on increasing body acceleration and inclination angle, whereas slightly augmented in the non-inclined stenosed artery.

7. ICU domain adaptation on survival prediction models built with neural networks by Lintu M.K.a, Asha Kamathb, Sudarsan N.S. Acharyaa aManipal School of Information Sciences, Manipal Academy of Higher Education, Mani- pal, Karnataka, India bDepartment of Data Science, Manipal Academy of Higher Education, Manipal, Kar- nataka, India [email protected]

Understanding the relationship between different disease factors and survival time in the presence of censoring is the core concept of survival analysis. The conventional sur- vival approaches include the Kaplan-Meier method, the Cox proportional hazards model, and extended Cox models. With the availability of large medical datasets, deep learning approaches are emerging to compete with conventional survival techniques, especially in clinical healthcare applications. PhysioNet challenge 2012 provided one such dataset with adult patients from ICU where the survival methodology is applicable. Rather than pre- dicting the usual mortality, an attempt to analyze the survival aspect of the PhysioNet data is made in this paper. The dataset consists of the heterogeneous population from different ICU domains that form different sub-populations. The concept of domain adap- tation was suggested recently to overcome this problem and make the predictions more accurate with minimal mismatch in training and testing sets that leads to improved pa- tient care. In this study, different readily available deep survival methods are implemented to the ICU data to address the benefits of the domain adaptation.

8. Analysis of a Modified Fractional Predator-Prey Model with Disease In- fection 100 by Chandrali Baishya, Department of Studies and Research in Mathematic, s Tumkur University, Karnataka, India [email protected]

In order to depict a situation of possible spread of infection from prey to predator a fractional-order model is developed and its dynamics is surveyed in terms of boundedness, uniqueness and existence of the solutions. We introduce several threshold parameters to analyze various points of equilibrium of the projected model and in terms of these threshold parameters we have derived some conditions for the stability of these equilib- rium points. Novelty of this model is that fractional derivative is incorporated in a system where susceptible predators get the infection from prey while predating as well as from infected predators and both infected preys and predators do not reproduce. The occur- rences of transcritical bifurcation for the proposed model are investigated. By finding the basic reproduction number, we have investigated whether the disease will become preva- lent in the environment. We have shown that the predation of more number of diseased prey allows to eliminate the disease from the environment, otherwise the disease would have remained endemic within the prey population. We notice that the fractional-order derivative has a balancing impact and it assists in administering the co-existence among susceptible prey, infected prey, susceptible predator and infected predator populations. Numerical computations are conducted to strengthen the theoretical findings.

9. Analysis of effect of Social Status on Depression by using Logistic regres- sion by Ishika Ahuja a,K Karthikeyanb aSchool of Computer Science and Engineering, VIT Vellore, Tamil Nadu, Vellore bDepartment of Mathematics, School of Advanced Sciences, VIT Vellore, Tamil Nadu, Vellore [email protected]

Depression has recently gained the attention of researchers and people are finally pay- ing attention to mental health. Numerous studies have shown that the effect of various things on different age groups is different. This paper uses binomial logistic regression to analyze the impact of age and social status on depression in the case of villagers. In this paper, we concentrate on farmers. Most of the studies regarding psychological disorders are performed on educated people who are aware of these mental health issues. On the other hand, farmers from small villages who are not aware of such problems, naturally stand at a different stance compared to these people, are not considered for such analysis yet. Thus I have used regression analysis to find the influence of social status and age on the mental health of farmers. Upon inspecting the data, it indicated that a binomial logistic regression model would be a good fit in all cases. In examining this relationship, we included variables such as age, education level, saved asset, and living expenses. The results showed evidence of a significant effect of age, education level, saved assets, and living expenses on the probability of a farmer suffering from depression.

10. Mathematical Modeling of COVID-19 Pandemic Dynamics with Non- pharmaceutical Interventions as Control Strategy Amit Sharmaa, Bhanu Guptaa, Sanjay K. Srivastavab aDepartment of Mathematics, JC DAV College Dasuya, Punjab, India bDepartment of Applied Sciences, B.C.E.T., Gurdaspur, Punjab, India [email protected]

Despite the major advances in the medical sciences, infectious diseases continue to cause significant morbidity and mortality in human populations worldwide. The new virus is normally responsible for an annual epidemic. Currently, we are facing COVID-19 (corona virus) outbreak, already accounted 4.5 crore infected and 11.8 lakh deaths worldwide. In this paper, we tries to build various mathematical models for studding dynamics of COVID-19 pandemic with quarantine, hospitalized and non-pharmaceutical interventions 101 as control strategies. As, mathematical models and computer simulations are useful in determining important biological thresholds because experiments with infectious disease spread in human populations are often expensive, unethical and sometimes impossible.

11. Stabilty Analysis of a Predator-Prey System with Square Root Func- tional Response Md Golam Mortujaa, Mithilesh Kumar Chaube,a, Santosh Kumarb aDiscipline of Mathematics, International Institute of Information Technology, Naya Raipur, India bDepartment of CSE, International Institute of Information Technology, Naya Raipur, India [email protected]

In this paper, we discuss the stability of predator-prey system with square root func- tional response in the presence non-linear harvesting in prey and group defence. We analyze the boundedness of the solutions, existence and stability conditions of the equi- librium points of the system. The results provide that, when hunting the prey or predator for economic interest, the harvesting rate must be chosen at a suitable value not smaller than the maximum sustainable yield to maintain the coexistence of both populations to maintain ecological balance. To verify our analytical results, several numerical simulations are carried out.

12. A Mathematical Study on Corona Virus Model with two Infectious States by Naga Soundarya Lakshmi V.S.V, A. Sabarmathi Department of Mathematics, Auxilium College, Vellore Department of Mathematics, Auxilium College, Vellore

A SIR model is formulated for COVID-19 with initial and secondary states. Existence and uniqueness of solutions, stability of the model and basic reproduction number were derived. Here the vulnerability of COVID-19 in Tirupathur district, Tamilnadu, India is discussed to exhibit the flow of variables of the model using numerical simulations. Also analysis of recovered is explored for Siddha and allopathy treatments.

13. Dynamical Behaviors of Fuzzy Prey Predator in SIR Epidemic Model P. Vinothini, K. Kavitha Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore [email protected], [email protected]

The aim of this paper is to study and analyze a fuzzy prey predator in SIR epidemic model. We have formulated a simple SIR type epidemic model in the presence of virus and we have discussed the dynamical behavior of the system. Further we analyze the fuzzy system and interpretation of SIR fuzzy model. Finally we find the existence and stability analysis of the fuzzy prey predator in SIR model system.

14. Dynamics of Fractional Illicit Drug Consumption Model with Holling Type-III Functional Response by Sindhu J Achar, Chandrali Baishya Department of Studies and Research in Mathematics, Tumkur University, Tumkur-572103, Karnataka [email protected]

In this paper we suggest a fractional differential equation describing the intake of illegal drugs in a population made up of drug consumers and non-users. The model describes the dynamics of non-users, experimental users, recreational users and addicts within a 102 population. This is an effort by analogy to the traditional multi-species predator-prey models to suggest a model that considers non-users as prey, experimental users and recre- ational users as predators as well as preys and addicts as predators. Growth of non-user population is represented by logistic law and pattern of influence of various categories of drug users on non-users as well as users are represented by Holling type-III functional response. The proposed model is analysed in terms of boundedness, existence and unique- ness, positivity of solutions. Sufficient conditions are derived for existence and stability of points of equilibrium. We have used the data from Hanley to forecast the marijuana drug consumption in the states of Colorado and Washington and analysed how incorporation of fractional derivatives affect the outcome of the model. The theoretical results are then validated by numerical simulation. Our results suggest that fractional can be a fantastic tool for a deeper understanding of illegal drug use.

15. A Robust Technique for Brain Tumor Detection Using Type-II Fuzzy Logic by Ananya Das, Subhashis Chatterjee Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad [email protected]

Detection of brain tumor in an automated way paves a significant advancement in the space of medical image processing. Classification is one of the significant and crucial step in the detection of brain tumor to help precise treatment. Nonetheless, manual recogni- tion with the assistance of human translation is time taking and furthermore subject to erroneous conclusion. Owing to these limitations, an automated brain tumor classification algorithm is proposed in this article. The current work is categorized into the following stages, viz. pre-preparing, clustering or segmentation, extraction of features, selection of key features, ranking of the chose features and lastly classification of the segmented tumor. Three major categories of features, viz. the Gray Level Co-occurrence Matrix, Law’s Texture and Mass Effect features, are extracted from the tumor and selection of features is implemented from each type followed by the ranking of the individual feature types. The concluding step comprises of the classification algorithm where a three phase classifier utilizing Type-II Fuzzy Inference System is developed to classify the segmented tumor into benign or malignant class. Finally, the work is tested and validated using BRATS dataset and performance comparison is showcased between the proposed work and Type-I Fuzzy Inference System.

16. Dynamics in a Prey-Predator Model with Susceptible-Infected-Recovered (SIR) Epidemic Disease in the Prey by Divya B, Kavitha K Department of Mathematics,School of Advanced Sciences, Vellore Institute of Technology Vellore [email protected], [email protected]

This paper proposes a non-linear mathematical model to study the dynamics of disease transmission among the prey population. We have formulated the SIR model with prey and used the Holling type II functional response and the equilibrium points are deter- mined.The stability analysis of the system is analysed using the Jacobian matrix and the next generation matrix method.

17. The Difference Equation Based Mathematical Model for Life-Cycle of Host-Parasitoid Systems by Suresh Rasappan, Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R & D Institution of Science and Technology, Tamilnadu, India. [email protected]

The Host-Parasitoiod life cycle is modeled under difference equation concept. A two- species model in which both species have a number of lifecycle stages that include eggs, 103 larvae, pupae, and adults is considered for this analysis. An adult female parasitoid finds a host on which to deposit its eggs. The larval parasitoids consume and eventually kill their host. A novel mathematical model is constructed under the difference equation concept. The dynamical properties and stability of the Host-Parasitoid cycle is analyzed.

18. Dynamics of IGP System with Provision of Additional Food to both Prey and Predator M.S. Bhuvaneswari, B.S.R.V. Prasad Department of Mathematics, Vellore Institute of Technology, Vellore [email protected]

Intraguild predation (IGP) explores the predatory interaction between heterospecific species that use similar, often limiting, resources and thus are potential competitors. In- traguild predation is claimed to be ubiquitous in nature. Intraguild predation has impor- tant implications for diversity maintenance in the fields of biological control, community ecology, wildlife management programs, and spatial ecology. Most of the theoretical stud- ies pertaining to biological control consider natural enemies as a specialist in nature and ignores the competitive interactions faced by these predators. However, experimental works carried out in biological control reveals that the natural enemies are not specialists always, and hence IGP is inherent in systems. In this work, we investigate the intraguild predation system, when the predators and prey are provided with supplementary food, and investigate the role of IGP in suppressing pests population. The predatory interactions of the proposed system are modeled by using Holling-II functional form with supplementary food. Here, we assume that the system has indirect competition between prey and preda- tor due to the presence of alternative food. We further assume that the prey population is affected by the intraspecific competition. A detailed mathematical analysis is carried out to study the stability, permanence, and various bifurcation of the considered system. We examined the conditions for the coexisting state and the prest-free state. The global dy- namics of the system are studied through the local and global bifurcation curves plotted in three-dimensional control parametric region consisting of additional food quality towards predator(α1), prey(α2) and the quantity of additional food (ξ). Numerical simulations are performed to validate the theoretical findings. The findings of current studies caution the eco-manager on the choice of supplementary food quality and quantity to achieve success- ful biological control.

19. Describing Tumour Growth through Mathematical Modelling by Keshav Kumar K, Department of Mathematics, Jawaharlal Nehru Technological University, Hyderabad [email protected]

Cancers are a large family of diseases that involve abnormal cell growth with the po- tential to invade or spread to other parts of the body. Indian Council of Medical research (ICMR) report estimates that there will be 15.7 lakh cases of cancer in India by 2025 and nearly 27.1 percent of Indias all cancer cases will be tobacco related. Cancer biology is incredibly complicated, as illustrated by the difficulties surrounding the diagnosis and treatment of cancer. However, Mathematical modeling has the potential to mediate this complexity by abstracting the system using simplifying assumptions into a mathemati- cal framework that can be analyzed and/or solved numerically to gain biological insight. The growth and development of solid tumours occurs in two distinct stages-the avascular growth phase and the vascular growth phase. This paper will present several mathemati- cal models which deal with the various stages of growth and development of solid tumours.

20. Ring Construction for Error Correcting Codes using Jacobson Radical: A Coding Theoretic Model for Genetic Sequence Analysis by Rajrupa Singh, Selvakumar R, Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, Tamil Nadu, India 104 [email protected]

A novel coding model for the genetic sequence analysis is proposed in our work. One of the basic structures of the theory of rings is the ideal, in which we define the Jacobson rad- ical. In the classic polynomial ring, it coincides with the nilradical. In this presentation we study the Jacobson radical of the polynomial rings with endomorphism, R[x, σ]. We are interested to study some of the questions on the coded region of a DNA sequence that are answered using the classical polynomial rings but specifically in the context of Jacobson Radical of polynomial rings. In particular, here we study the conditions required for the nilradical of such rings that coincides with the Jacobson radical in order to identify the coding region in a DNA sequence. These results strongly suggest that some deterministic rules must be involved in the genetic code origin. In the cellular level, the information in DNA is transformed into proteins. The transmission of genetic information is done by a sequence of bases like Adenine(A), Thymine(T), Guanine(G) and Cytosine(C) in any DNA structure which can be considered as digital codes. Further, the working of our coding model is demonstrated through an example.

21. Compartment Modelling and eigenvalue Expansion to Study the Drug Concentration in Capillary and Tissue Regions Surrounding the Malignant Tumour by M.A. Khanday, Department of Mathematics, University of Kashmir, Srinagar, 190006 Jammu and Kashmir, India [email protected]

The drug transport mechanism in the biological tissue can be modelled for its effective and efficient performance. Mathematics is playing a key role in almost all biomedical research problems including drug kinetics. A mathematical model based on reaction- diffusion equation has been formulated to understand the drug transport and its diffusion in a cancerous tissue. The eigenvalue expansion has been used to obtain the solution of the ordinary differential equations concerning the rate of change of drug concentration in different compartments including capillary and tissue regions surrounding the malignant tumour cells. The graphs were plotted to illustrate the variation of drug concentration with respect to time using MATLAB software. It has been observed from the graphs that the drug concentration decreases in the first compartment and gradually increases in the second compartment to some value and then decreases again in association with the concentration of the drug. Moreover, the behaviour of the tumour cells with changing drug concentration is simulated with respect to time and the results were compared and verified with the empirical data of Unni and Seshaiyer.

22. Forecasting Electric Energy Consumption in India using Univariate Time-Series Analysis by D. Karthika, K. Karthikeyan Department of Mathematics, SAS, VIT Vellore, Tamil Nadu, India [email protected]

Modelling electric energy consumption is useful in planning generation and distribution by power utilities. Univariate time-series analysis has been used for modeling and fore- casting domestic electric energy consumption in India. Autoregressive integrated moving average (ARIMA) models were developed using data from 1990 to 2017 and evaluated on forecasting new data for ten years. Compared to Holts model and Simple Exponential Smoothing model developed on the same data, ARIMA models require less data, have fewer coefficients, and are more accurate. The optimum ARIMA model forecasts yearly data for ten years with Mean Absolute Percentage Error (MAPE) of 1.4260%. The pro- posed ARIMA models are used to provide a ten-year forecast of the electricity demands in India. 105

23. Analysis of Lap Times in Formula-1 Motorsport due to Regulation Changes using Polynomial Regression by Thejineaswar Guhana, K. Karthikeyanb aSchool of Information Technology and Engineering, Vellore Institute of Technology Vellore bDepartment of Mathematics, School of Advanced Sciences, Vellore Institute of Technology Vellore [email protected], [email protected]

In the motorsport of Formula 1 engine regulation changes are quite common and when a change occurs the sport evolves by adapting to the new regulations. However, this transition from one regulation to another are not always uniform. So, we study how an engine regulation change affects the performance of the car by comparing the lap times and predict future performance due to new regulations passed using polynomial regres- sion. For this study, we compare data spanning over 3 engine eras-V10, V8 and V6-during a 25 year-time period. As mentioned, the performance of these engines is analyzed and predicted using the lap times: it is the time taken in seconds to cover the entire track once. A lap time which is relatively smaller in a specific track, indicates better performance of the car in that track.

24. An EPQ Model for Delayed Deteriorating Items with Time Dependent Cubic Demand and Shortages by R. Pavithra, K. Karthikeyan Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology Vellore [email protected], [email protected]

An economic production quantity model plays a dominant role in production and manu- facturing components. In this article, an EPQ model is proposed for delayed deteriorating items with time dependent cubic demand in before deterioration and after deterioration the demand is constant. In this inventory model shortages are allowed and it is partially backordered. A mathematical model is developed and the best cycle length which opti- mizes the total inventory cost and an economic production quantity. Finally, numerical example and sensitivity analysis of the developed model are presented to see the effect of changes in some parameter in the inventory system.

25. Hexadecagonal Fuzzy Transportation Problem by R. Saravanana, M. Valliathalb aNIFT-TEA College of Knitwear Fashion, Tirupur bChikkaiahNaicker College, Erode

Transportation problem is one of the sub classes of linear programming problem. The objective of the transportation problem is to minimize the transportation cost or maximize the profit. Fuzzy set theory has been applied in many fields of Science, Engineering and Management. In this article, a Hexadecagonal fuzzy number is ranked by using Robust ranking method. Fuzzy transportation problem is transformed into crisp transportation problem and solved by MODI method. A numerical example is presented and the optimal solution obtained by using proposed method.

26. Tandem Fluid Model Driven by an MX/M/1 Queue Subject to Balking and Vacations by M. Deepa, K. Julia Rose Mary Nirmala College For Women, Red Fields, Coimbatore, Tamilnadu - 641018 . [email protected], [email protected]

Generally, balking is an impatient customers behavior who refuses to enter the queue on arrival. This paper studies the bulk arrival of a tandem fluid queue with multiple 106 exponential vacations subject to balking. To provide greater flexibility on control of net input rate when flow is high, we introduce a bulk arrival fluid model with multiple ex- ponential vacations and balking. We derive explicit expression for the stationary buffer content by using Laplace transform. Then the performance measures such as mean of the buffer content is obtained and it is found to be independent of the vacation parameter θ. Finally, the effect of vacation and balking parameters on the mean buffer content is also illustrated by sensitivity analysis.

27. An Economic Production Quantity Model for Three Levels of Produc- tion with Weibull Distribution Deterioration and Shortage under Inflation by G. Viji, K. Karthikeyan Department of Mathematics, SAS, VIT Vellore. Tamil Nadu [email protected], [email protected]

An EPQ model has a wide range of scope in production and manufacturing sectors. This article presents three levels of economic production inventory model for deteriora- tion items with three different levels of production. Inflation and the rate of deterioration are also considered along with this inventory model which follows two parameter Weibull distributions. This article has a great influence in achieving minimum quantum stock of manufacturing items at the initial stage through which holding cost will be reduced at great extent. This article facilitates production organization in achieving reduced total cost, desired productivity and earning potential profit along with customer satisfaction. The objective of this article is to find the optimal solution for reducing total production cycle time so that total cost of the whole production cycle will be minimized. Eventually numerical example and sensitivity analysis on parameters are made to validate the results of the proposed inventory system.

28. Recent Trends of Applications of Business Analytics using Operations Research by Gunda Srinivasa Rao, Department of Mathematics, CMR university, Bangalore

Supply chain has long been an area where advanced analytics such as statistics and optimization were used for forecasting, planning, network design, inventory optimization, routing, warehouse slotting etc. Data science and data analytics are at the core of every modern globalized industry. Working in todays technology-centric workforce not only re- quires superior leadership skills, but the ability to translate data problems into the bigger picture for the organization. Business Analytics is the science of data-driven decision mak- ing. The use of analytics across industries for decision-making, automation of business processes, products, and solutions driven by analytics makes it an essential skill for every student graduating from management and engineering disciplines. Many organizations generate solutions to their problems using analytics and innovation in many companies is driven by analytics. Analytics is used as a competitive strategy by many successful companies. Organizations such as Amazon, Apple, Facebook, Google, IBM and Wall mart have created solutions using analytics. Amazon has created solutions such as recom- mender systems and Amazon go that are driven by analytics. Apples predictive keyboard is another example of solutions driven by analytics. Analytics is relevant not only to profit- making companies but also for government and nongovernment organizations(NGOs).The Akshaya patra foundation, an NGO based out of Bangalore, has used several analytics models for effective management of its free meal program which provides free meals to about 1.5 million school children in India. Analytics is not just about number crunching. It has evolved into a competitivestrategy that drives innovation across several organiza- tions. Business analytics is a set of statistical and operations research techniques, Artificial intelligence, Information technology and management strategies used for framing a busi- ness problem, collecting data, and analyzing the data to create value to organizations. Business Analytics can be broken into 3 components: 1)Business context, 2)Technology 107 and 3)Data Science.IT is used for data capture, data storage, data preparation, data anal- ysis and data share. An important output of analytics is automation of actionable items derived from analytical models; automation of actionable items is usually achieved using IT. Data science is the most important component of analytics, it consists of statistical and Operations research techniques, machine learning and deep learning algorithms.

29. Solving Open Travelling Salesman Subset-Tour Problem through a Hy- brid Genetic Algorithm by Jayanth Kumar Thenepallea, Purusotham Singamsettyb aDepartment of Science and Humanities, Sreenivasa Institute of Technology and Manage- ment Studies, Chittoor-517127, Andhra Pradesh, India bDepartment of Mathematics, School of Advanced Sciences, VIT, Vellore-632014, Tamil Nadu, India [email protected], [email protected]

In open travelling salesman subset-tour problem (OTSSP), the salesman needs to tra- verse a set of out of cities and after visiting the last city, the salesman does not necessarily return to the central depot. The objective is to minimize the overall traversal distance of covering cities. The OTSSP model comprises two kinds of problems such as subset selec- tion and permutation of the cities. Since a salesmans tour, do not contain all the cities, the problem of selection takes place. Another problem is to find the optimal sequence of the cities from the selected subset of cities. To solve this problem efficiently, a hybrid nearest neighbour technique based crossover-free Genetic algorithm (GA) with complex mutation strategies is proposed. To best of the authors knowledge, this is the first ever hybrid GA for the OTSSP. As there are no existing benchmark instances for OTSSP, a set of test instances is generated by using TSPLIB. The extensive computational experiments show that the proposed algorithm is having great potential in achieving better results for the OTSSP. Our proposed GA being the first evolutionary-based algorithm for OTSSP, which will help as the baseline for future research on OTSSP.

30. Analysis of Intuitionistic Fuzzy Transportation Problem by K.R. Sobha, Sree Ayyappa College for Women, Chunkankadai [email protected]

The objective of this paper is find out the optimum cost (maximum profit) of the In- tuitionistic fuzzy transportation Problem. Using Yager’s ranking method fuzzy quantities are transformed in to crisp quantities. Finally a numeric illustration is given to check the validity of the proposal.

31. An Economic Order Quantity model with Reverse Logistics inventory model in circular economy by S. Vennila, K. Karthikeyan Department of Mathematics, SAS, VIT Vellore, Tamil Nadu, India [email protected], [email protected]

The Economic Order Quantity (EOQ) model has developed enormously over long time on the quality of incorporating realistic factors. Reverse Logistics is the strategy for man- aging gathering or assembling back the pre-owned items from a definitive customer. It can take any structures with differing expenses and advantages to the business. The return products in an effort to recycling, reusing, repairing is measured by circular economy to recover assets. In this article we developed a reverse logistics economic order quantity inventory model in circular economy. The inventory issues is formed by the product was return to company when to request, how much amount to arrange where there is reverse flow of products into the system is manipulated into the form of optimal profit maximi- sation. To exist the result to optimal solution of the model by using non-linear Karush Kuhn-Trucker conditions for the objective functions are introduced. 108

Review Article on Queueing Inventory Models P. Indumathi, K. Karthikeyan Department of Mathematics, SAS, VIT Vellore, Tamil Nadu, India [email protected]

A Queueing inventory system has been extensively studied because of their widespread applications in the real world. The goal of this paper is to provide sufficient information about the queueing inventory theory. In this paper, we discuss different components of queueing systems along with the inventory models such as queueing systems with produc- tion inventory, queueing inventory systems with stochastic, queueing inventory systems with perishable products, queueing inventory systems with different types of demands, general queueing inventory systems with different service times, and deteriorating inven- tory systems.

33. Bargaining of a Wholesale Price for an Optimal Manufacturer with a Retailer in a One-Channel Supply Chain by K. Valli, P. Rajendran Department of Mathematics, SAS, VIT Vellore, Tamil Nadu, India [email protected], [email protected]

In this paper we investigate the ideal planning when a manufacturer bargains with a dealer price with a retailer in one channel supply chain that comprises of the manufac- turer and the retailer. To identify the problem, we consider the decision-making method in which the manufacturer can sell the products directly to the retailer andfinally to the consumers. We assume that the manufacturer decides the direct retail price by the one channel through the retailer. The manufacturer gets the highest profit by bargaining the wholesale price with the retailer. If manufacturer shows the one channel supply chain for practical decision making by using the multi-criterion decision making method [MCDMM], then the profit increases to a greater extent. To show the effectiveness of the MCDMM, we provide numerical examples. This paper presents the methodology, findings and con- clusions with the scope for further research.

34. Modelling of Deteriorating Systems Using Fuzzy Warranty Cost with Preventive Repairs by M. Mubashir Unnissa, D. Kalpanapriya Department of Mathematics, SAS, VIT Vellore, Tamil Nadu [email protected], [email protected]

This paper studies a repairable deteriorating. In order to minimize the operating cost of the system and to improve system availability, a preventive repair, not as good as new, is adopted during the warranty period. A cost model has been developed to show the importance of fuzzy preventive repairs during warranty period. Also the paper focuses on the various warranty expenses for the above model. Numerical illustrations have also been provided to show the importance of fuzzy repair times during warranty

35. Rough Hesitant Bipolar Neutrosophic Linear Programming Problem by E.R. Meena Kumaria, M. Thirucheranb aDepartment of Mathematics, Bharathi Womens College, Chennai 108. bDepartment of Mathematics, L.N. Govenment College , Ponneri - 601 204 [email protected], [email protected]

In this paper, we have proposed a bipolar rough hesitant neutrosophic number to solve the neutrosophic linear programming problem which is a most powerful technique in decision making. A new ranking function is proposed to convert the neutrosopic lin- ear programming problem to crisp linear programming problem. Solve this crisp linear programming problem using a standard method to get the optimal solution. We have 109 compared this proposed method with the existing methods to prove the optimality of the solution.

36. Lexi-Search approach for the Three Index Assignment Problem by Sumathi Pa, Viswanatha Reddy Gb,Purushotham Sc aDepartment of Mathematics, College of Agricultural Engineering (ANGRAU), Madakasira, A.P, India bDepartment of Mathematics, Sri Venkateswara University,Tirupati, A.P, India cDepartment of Mathematics, School of Advanced Sciences,VIT, Vellore, Tamilanadu, In- dia [email protected]

The classical assignment problem is to find a one-one correspondence between a set of persons and a set of machines with the least weight. The 3-index assignment problem is an NP-complete problem which is a generalization of classical assignment problem. Let there be three n−disjoint sets which denote the set of persons, machines and facilities. Let C(i, j, k) be the cost associated to each triplet (i, j, k) i.e. the weight is carried on working ith person on jth machine by utilizing kth facility. The 3-index assignment problem is to find a combination of n−triplets which covers the union of the three sets with least weight in such a way that no i, j, k is repeated. The 3-index assignment problem is expressed as a zero-one programming problem. In order to obtain the optimal combination of triplets among n3, a deterministic lexi-search algorithm is developed. The algorithm searches feasible solutions systematically and then moves towards the optimal solution with the help of effective backtracking and bounding strategies. The problem is explained with a suitable numerical example. The algorithm is coded in PYTHON programming language and computational results are tabulated. The computational experiments over a large set of random data sets exhibited that the proposed LSA requires fairly less CPU runtime to find the efficient solutions. Based on this experience, we strongly feel that the proposed algorithm is fairly efficient in resolving higher dimensional problems also.

37. Bipolar Vague ELECTRE 1 method for MCDM problems by Venkata Kalyani U, T. Eswarlal Department of Mathematics, Koneru Lakshmaiah Education Foundation, Vaddeswaram, AP,India [email protected], [email protected]

In numerous aspects dealing with bipolar information is essential. Elimination and choice translating reality (ELECTRE 1) is a widely used method to solve multi-criteria decision making problems. We established and proposed the Elimination and choice trans- lating reality (ELECTRE 1) with bipolar Vague sets to solve such problems.We verified the proposed method with a numerical example which shows the effectiveness of the method with a decision graph. 110

INDIAN MATHEMATICAL SOCIETY Council : 1st April, 2021 to 31st March, 2022

1) Prof. Dipendra Prasad 2) Prof. B. Sury President, Immediate Past President, Department of Mathematics, Theoretical Stat. and Math. Unit, IIT Bombay, Indian Statistical Institute Mumbai - 400 076, Bangalore - 560 059. [email protected] [email protected] 3) Prof. Satya Deo, 4) Prof. B. N. Waphare, General Secretary, Administrative Secretary HRI, Chhatnag Road, Jhusi, Department of Mathematics, Allahabad - 211 019. Savitribai Phule Pune University, [email protected] Pune - 411 007. [email protected] 5) Prof. Peeyush Chandra 6) Prof. S. K. Nimbhorkar Academic Secretary, Treasurer Professor (Retired) C/o Dr. Mrs. Prachi Kulkarni, Department of Mathematics Ankur Hospital, Tilaknagar, IIT, Kanpur - 208 016. Aurangabad - 431 001. [email protected] [email protected] 7) Prof. Sudhir Ghorpade 8) Prof. M. M. Shikare Editor-in-Chief, J. Indian Math. Soc. Editor-in-Chief, The Math. Student Department of Mathematics Department of Mathematics, IIT, Mumbai - 400 076. Savitribai Phule Pune University [email protected] Pune - 411 007. [email protected] 9) Prof. M. Pitchaimani, Librarian, 10) Prof. A. K. Das Ramanujan Institute for Dept. of Mathematics SMVDU, Advanced Study in Mathematics, Katra - 182 301, J & K. Chennai - 600 005. [email protected] [email protected] 11) Prof. G. P. Youvaraj 12) Prof. N. D. Baruah Ramanujan Institute, Dept. of Mathematical Sciences, University of Madras, Tezpur University, Chennai - 600 005. Assam, Pin 784 028. [email protected] [email protected] 13) Prof. Jitendra Kumar 14) Prof. S. Sreenadh Department of Pure Mathematics, Department of Mathematics, IIT Kharagpur, Kharagpur- 721 302. Sri Venkateswara University, [email protected] Tirupati - 517 502. [email protected] 15) Prof. Veeramani P.V. 16) Prof. Pankaj Jain Department of Mathematics Department of Mathematics IIT Madras, Chennai - 600 036. South Asian University, Akbar Bhawan, [email protected] Chanakya Puri, New Delhi - 110021. [email protected] 17) Prof. Nita Shah 18) Prof. Ahmed Ali Department of Mathematics Dept. of Math. & Comp. Science Gujarat University, Dean, School of Basic Sciences, Navarangpura, Ahmedabad - 380 009. Babu Banarasi Das University, [email protected] Lucknow - 226 028. [email protected]