Research



RESEARCH

Areas of Research : General Topology, Fuzzy Mathematics
Details of Ph. D. Thesis:
Faculty : Science
Subject: Mathematics
Area: General Topology
Title:  Some Problems in Set Topology Relating Group of Homeomorphisms and Order
Guide:  Prof. T. Thrivikraman F.N.A.Sc.,Dept. of Mathematics& Statistics, CUSAT, Cochin -
682022
University: Cochin University of Science& Technology

Brief synopsis of the thesis:

In this thesis we investigate some problems related to the concepts of the group of homeomorphisms and order.  Most of the problems considered are related to various levels of homogeneity  of topological structures. The representation of a subgroup of the group of all permutations S(X) of a set X as the group of homeomorphisms of a topological space (X,T) for some topology T is discussed first. The equivalence of anti- homogeneity and hereditarily rigidity is established. Cech closure spaces and the lattice of Cech closure operators are investigated with special reference to the complementation problem. The use of order theoretic and group theoretic methods in the study of topological structures is the main feature of the work.

Thesis available at   shodhganga.inflibnet.ac.in
Date of Submission: 18.11.1985
Year of Award: 1986 w.e.f. 7.6.1986
Date of Registration for Ph.D. : 01.02.1982
Full Time from 1.2.1982 to 31.8.1984
Part Time from 1.9.1984 to
Fellowship:   C.S.I.R   J.R.F. from 1.1.1984 to 31.8.1984
University Fellowship from 1.2.1982 to 31.12.1983

Some of the Research Papers Published

  • Ramachandran P.T. :  Anti-homogeneity is Equivalent to Hereditary Rigidity Ganita Vol.35.No.1(1984) pp 1-9
  • Ramachandran P.T. : Groups of Homeomorphisms and Normal Subgroups of the Group of permutations International Journal of Mathematics and Mathematical Sciences, Vol.14,No.3(1991) pp:475-480.
  • Ramachandran P.T. : Complementation in the Lattice of Cech closure operators, Indian Journal of Pure and Applied Mathematics, 18(2)(1987)pp.152-158.
  • P.M.Mathew, C.Sheela and P.T. Ramachandran , Associated Completely Regular Spaces , TAJOPAM, Vol.1, No. 1, Dec 2002, pp.101-106 .
  • P.T. Ramachandran , A Note on the Group of L-Fuzzy Homeomorphisms , Proceedings of the International Conference on Recent Trends in Mathematical Analysis held in St.Joseph's College, Irinjalakuda, Kerala, Allied Publishers Private Limited 2003 pp.166-175.
  • Ramachandran P.T.: â€œThe group of Homomorphisms and the Group of L-Fuzzy Homeomorphisms”, Proc.National conference of Mathematical Modelling Organised by Baselius College Kottayam and Kerala Mathematical Association, 14-16 March 2002, pp. 117- 122.
  • Ramachandran P.T. : Graph Theoretical Approach to Digital Topology, Proceedings of the national Seminar on Digital Topology and Morphology organised by Mar Thoma College for women, Perumbavoor during November 27-28 (2008) 31-36
  • Kannan V. and Ramachandran P. T .:”Hereditarily Homogeneous Spaces”, Proceedings, International Seminar on Recent Trends in Topology and its Applications, St.Joseph ‘s College,Irijalakuda,2009,48-51
  • Ramachandran P.T.: Lattices of Topologies and Fuzzy Topologies, Proceedings, International Seminar on Recent Trends in Topology & its Applications, St. Joseph's College, Irinjalakuda (2009) 210-218
  • Ramachandran P.T.: Semigroup of Continuous Functions, Proceedings of the International Conference on Groupoids, Semigroups and Automata, University of Kerala, Thiruvananthapuram, (2010),pp.89-94
  • Sini P. and Ramachandran P.T.: “On the Group of Homeomorphisms”, Bulletin of Kerala Mathematical Association Vol. 9, No. 1, (2012, June)  55-61 ISSN 0973-2721
  • Ramachandran P.T., Complemented Elements in the Lattice of Cech Closure Operators, Journal of Computer and Mathematical Sciences Vol. 3, Issue 5, 31 October (2012) 498-501 ISSN 0976-5727
  • Ramachandran P.T., Measurable Cardinals in Analysis, Proceedings of the National Seminar on Analysis, Algebra and Discrete Mathematics, University of Kerala, Thiruvananthapuram (2012) 121-125
  • Madhavan Namboothiri N.M., Anilkumar V.& Ramachandran P.T.:Cayley Fuzzy Graphs, Far East Journal of Mathematical Sciences,  Vol-73, No.1, 2013,  15-39. ISSN: 0972-0871
  • Ramachandran P.T., Pythagorean Triples, Proceedings of the National Level Instructional Seminar on Recent Advances in Mathematics Organised by St. Mary's College Thrissur & Kerala Mathematical Association (2014) pp.13-17
  • Ramachandran P.T., An Introduction to Digital Topology,Proceedings of the U.G.C. Sponsored National Seminar on Mathematical Morphology and Digital Topology Organised by Department of Mathematics, Newman College, Thodupuzha (2014) pp.31-38
  • Ramachandran P.T., Pi and the Kerala School of Mathematics , Kerala School of Mathematics: Trajectories and Impact, N.K. Sundaresan (Ed.), Ganga Books, Calicut (2014) 112-125 ISBN 978-81-931168-0-7
  • Dhanya P.M. &  Ramachandran P.T.: On the Simple Expansion of Generalized Topologies , International Journal of Pure and Applied Mathematics,Vol.104, No.4(2015) 621-630. ISSN: 1311-8080.  doi: http://dx.doi.org/10.12732/ijpam.v104i4.12
  • Kunheenkutty M., Kavitha T. and Ramachandran P.T.: Adjacency in the Lattice of Cech Closure Operators, International Journal of Pure and Applied Mathematics Vol.105 No.1 (2015) 73-86  ISSN  1311-8080  doi: http://dx.doi.org/10.12732/ijpam.v105i1.
  • Sini P. and Ramachandran P. T.: On t-representability  of the Cyclic Subgroups of  the Symmetric Group- I , International Journal of Pure and Applied Mathematics Vol. 106, No.3 (2016), 851-857 ISBN: 1311-8080 (Printed Version) ISBN: 1314-3395 ( online Version) doi: 10.12732/ijpam.v106i3.11
  •  Kavitha T., Sini P. and Ramachandran P. T. : On the Group of Closure Isomorphisms, Journal of Advanced Studies in Topology, 7(3)132-136 (2016), eISSN 2090- 388X ,  pISSN 2090-8288
  •  Sini P., Dhanya P. M. and Ramachandran P. T. :  On Completely Homogeneous L- topological Spaces,  Annals of fuzzy mathematics and informatics, 12(5), 703- 718(2016) pISBN- 2093-9310, eISSN-2287-6235

Papers Presented at International/National Level Conferences and Seminars

  • Ramachandran P.T. : Cech closure operators and Reflexive Relations,  51th Conference of Indian Mathematical Society at University of Cochin, Cochin682022, Dec 28-30, 1985.
  • Ramachandran P.T. : Complemented Elements in the Lattice of Cech Closure Operators Presented at National Level Conference on the Recent Trends in Topology March 1 &2 1997, Department Mathematics, N.G.M. College , Pollachi,Tamilnadu
  • Ramachandran P.T. : Fixed Points of the Lattice of Cech Closure Operators Presented at U.G.C. Sponsored National Seminar on Discrete Analysis at Sri Ramakrishna Mission Vidyalaya,College of Arts and Science,(Autonomous) Coimbatore 641020 from 22 to 24 th March 1999.
  • Ramchandran P.T. : A Note on the Group of Closure Isomorphisms of a Cech closure space in the UGC sponsored National Seminar on Topology and fractal Geometry organised by the Department of   Mathematics,Govt. College, Chitoor, November  19-21, 2010
  • Sini P. and Ramachandran P. T. :On t-representability  of the Cyclic Subgroups of  the Symmetric Group- I , International  Conference on  Mathematics 2015,  Nov:26-28, University of Kerala.
  • Kavitha T., Sini P.  and Ramachandran  P. T.: On Group of Closure Isomorphisms, Proceedings IC- AMMN-2016,FISAT, Angamaly, January 7-9, 2016 .

Malayalam Articles on Mathematics Published

Karal Marxum Ganita Sastravum, Marxian Darsanam, Vol. 4. Issue 1, November 2006, 15-18

Ariyapedaatha Ganithasastra Parambaryam, 47 th Sate Conference Souvenir, Kerala Sastra Sahithya Parishad,  2010, 38-39

Ganitam Malaylathil, Kavana Koumudi, Vol.16. Issue. 2 November- January 2014, 53-62



Research Guidance:

Order No. Pl.D.G1/6058/87/K Dis dated 24.7.87

Research Scholars  Under My Supervision:

Degree Awarded:

1. Dr. Joymon Joseph P., Government Engineering College, Kozhikode. (Awarded w.e.f. 15.05.2004)

Title: Neighbours in the Lattice of Topologies

2.Dr. Baby Chacko, St. Joseph's College, Devagiri, Kozhikode. (Joined on 29.03.2005, Thesis submitted  in September 2008, Awarded w.e.f. 03.03.2009)

Title: Some Lattice Theoretic Problems Related to General and Fuzzy  Topology

3.Dr. Madhavan Namboothiri N.M., Government College, Kasargod.(Joined on 23.03.2009, Submitted in March 2012, Awarded w.e.f.  30.11.2012)

Title: Some Problems Relating Order and Fuzzy Topologies

4.Dr. Kunheenkutty M., Principal, Government College, Kondotty, Malappuram Dt. (Joined on 24.1.2008, Part- Time, Thesis Submitted in January 2013, Awarded w.e.f. 17.03.2014)

Title: Some Problems on Local Properties of the Lattice of Topologies

5. Dr.Sreeja M. ( Joined on 19.04.2010, Synopsis Submitted in August 2013,Thesis Submitted in November 2013, Open Defence on 14.07.2014, Awarded w.e.f  14.07.2014)

Title: Some Problems Related to Homogeneity in Topological Spaces

6. Dr. Santhosh P.K., Dept. of Mathematics, Brennan College, Thalassery, (  U.G.C. F.D.P.), ( Joined on 12.04.2010, Joined on 12.04.2010 as Part Time, Synopsis Submitted on 20.12.2014. Thesis submitted on 5.1.2015. Open Defence and Viva- Voce on 10.12.2015. Awarded w.e.f 10.12.2015 )

Title: Some Problems on c- Spaces

7. Dr. Dhanya P.M.  Asst. Professor, Department of Mathematics, Government Victoria College, Palakkad, C.S.I.R. S.R.F. (Joined on 1-11-2012, Awarded w.e.f 17.8.17 )

Title: Some Problems on Generalized Topologies and Fuzzy Generalized Topologies

8. Dr. Sini P.,  Asst. Professor, Dept. of Mathematics, University of Calicut (Joined on 1.11.2012, Joined as FDP on 1-8-15, Submitted on 31.01.2017, Awarded w.e.f. 31.8.17)

Title: Studies on the Group of Homeomorphisms and the Group of L-fuzzy Homeomorphisms

9. Dr. Kavitha T.   U.G.C. S.R.F. (Joined on 26-03-2012)( Full Time) Submitted on 30.8.2017



Ongoing:

  1. Ms. Sruthi A.K.  U.G.C. J.R.F. (Full Time) (Joined on 19.8.2016)
  2. Ms. Naheeda Farhath ( Full Time) ( Joined on 03.01.2018)
  3. Ms. Saji N.R. ( Part Time) 

Papers of  research scholars under my supervision:

Baby Chacko

Baby Chacko: Cech Closure Operators on a Fixed Set: International Journal of Mathematical Analysis vol.6,2012.no.52,pp 2597-2603  ISSN 1312-8876

Baby Chacko: The Group Of Automorphisms of the Lattice of Fuzzy Topologies: Far East Journal of Mathematical Sciences vol.72,no.1,2013,pp 79-92 ISSN: 0972-0871

Baby Chacko: Some Results Related To the Lattice of Fuzzy Topologies on a fixed set: IOSR Journal of Mathematics vol.2,issue 4,2012,pp 29-32

Baby Chacko: On the Sublattice[I,Co] of the Lattice of Cech Closure Operators: IOSR Journal of Mathematics vol.3,issue 1,2012,pp 37-39

Baby Chacko: Fuzzy Topologies and t-irreducible Subsets: Journal of Advances in Theoretical and Applied Mathematics,vol 7,no.4,2012,pp 373-378

Baby Chacko: Fuzzy Closure Operator Induced by a Fuzzy Pseudo Metric: International Journal of Fuzzy Mathematics and Systems,vol.2,no.2,2012,pp 345-350

Baby Chacko: On Quasi-discrete Closure Operators: Global Journal of Mathematics and Mathematical Sciences,vol.2,no.1,2012,pp 31-35

Baby Chacko: A Problem Related to the Lattice of Topologies

Proceedings, International Seminar on Recent Trends in Topology & its Applications, St. Joseph's College, Irinjalakuda (2009) 210-218

Madhavan Namboothiri N.M.

N.M. Madhavan Namboothiri: On Order Properties and Separation Axioms in L- Fuzzy Spaces Proceedings, International Seminar on Recent Trends in Topology & its Applications, St. Joseph's College, Irinjalakuda (2009) 210-218

Madhavan Namboothiri N.M. : On Order Properties in L-Fuzzy Topological Spaces, Proceedings of the National Seminar on Topology and Its Applications to Computer Science and Related Areas, Kannur University, Kerala, 2009, 1-4.

N.M.Madhavan Namboothiri: Order Induced L-Fuzzy Topological Properties

Advances in Fuzzy Sets and Systems 9(2) (2011), 167-197

N.M.Madhavan Namboothiri: A Categorical Study between Preorder Structure and Fuzzy Topological Structure

Far East Journal of Mathematical Sciences Vol. 67 Number 1, 2012, Pages 121-140 ISSN  0972-0871 Madhavan Namboothiri N.M.

On Fuzzy Cech Closure Spaces and C-Reflexive Relations
Bulletin of Kerala mathematical Association
Vol. 9, No.1(2012, June) 63-77 ISSN 0973-2721

Madhavan Namboothiri N.M., Anilkumar V.& Ramachandran P.T.:Cayley Fuzzy Graphs, Far East Journal of Mathematical Sciences,  Vol-73, No.1, 2013,  15-39. ISSN: 0972-0871

Ratheesh K.P. and Madhavan Namboothiri N.M.:On C-spaces and Connective Spaces, South Asian Journal of Mathematics, 4(2013), 1-13 ISSN 2251-1512

Madhavan Namboothiri N.M. : Cayley Graphs and Graphical Topology, Bulletin of Kerala Mathematics Association, Vol. 11, No.1, (2014 June),  57-65 ISSN 0973-2721


Sreeja M.

M. Sreeja: On Anti Rigid Spaces
Bulletin of Kerala mathematics Association
Vol.8,No.1,(2011,June) 67-74 ISSN 0973-2721

Sreeja M.: On the Total Negation of Rigidity International Journal of Pure and Applied Mathematics, Vol.82 No.3 (2013) 391-398. ISSN 1311-8080 Sreeja M.:  Bankston Iteration Sequence of Rigidity, Journal of Advanced Studies in Topology 5(1), 34- 38  eISSN 2090-388X | pISSN 2090-8288


Joymon Joseph P.
Joymon Joseph P.: Upper Neighbours in the Lattice of Topologies
Proceedings, International Seminar on Recent Trends in Topology & its Applications, St. Joseph's
College, Irinjalakuda (2009) 69-77

Kunheenkutty M., 
Kavitha T. and Ramachandran P.T.: Adjacency in the Lattice of Cech Closure Operators, International Journal of Pure and Applied Mathematics Vol.105 No.1 (2015) 73-86  ISSN 1311-8080  doi: http://dx.doi.org/10.12732/ijpam.v105i1.7

Sini P.

Sini P. and Ramachandran P.T.: “On the Group of Homeomorphisms”, Bulletin of Kerala Mathematical Association Vol. 9, No. 1, (2012, June)  55-61 ISSN 0973-2721

Sini P. : On L-f representability of the Permutation Groups, Proceedings ,IC- AMMN-2016,FISAT, Angamally (2016) 93-96.

Dhanya P. M. and Sini P. : On Homogeneous Generalized Topological Space, Proceedings IC- AMMN-2016,FISAT, Angamaly (2016) 87-92.

Sini P and Ramachandran P. T.: On t-representability  of the Cyclic Subgroups of  the Symmetric Group- I , International Journal of Pure and Applied Mathematics Vol. 106, No.3 (2016), 851-857 ISBN: 1311-8080 (Printed Version) ISBN: 1314-3395 ( online Version)                                   doi: 10.12732/ijpam.v106i3.11

Kavitha T., Sini P. and Ramachandran P. T. : On the Group of Closure Isomorphisms, Journal of Advanced Studies in Topology, 7(3)132-136 (2016), eISSN 2090-388X ,  pISSN 2090-8288

Sini P., Dhanya P. M. and Ramachandran P. T. :  On Completely Homogeneous L-topological Spaces, Annals of fuzzy mathematics and informatics, 12(5), 703-718(2016) pISBN- 2093-9310, eISSN-2287- 6235

Sini P. : t-representability of Maximal Subgroups of  the Symmetric Groups, Journal of Advanced Studies in Topology,9(1)27-33(2018), eISSN 2090-388X | pISSN 2090-8288, https://doi.org/10.20454/jast.2018.1294.

Sini P. : The Group of Homeomorphisms and the Cyclic group of Permutation, Palestine Journal of Mathematics, 8(2), 53–60 (2019), ISSN 2219-5688.

Sini P. Hereditary Homogeneous Generalized Topological Space. Iranian Journal of Mathematics and informatics, 14(2),  9-18 (2019), e-ISSN: 2008-9473 , ISSN: 1735-4463,DOI: 10.7508/ijmsi.2019.02.002.


Papers Presented at  Conferences and Seminars

Sini P. and Ramachandran P. T. :On t-representability  of the Cyclic Subgroups of  the Symmetric Group- I , International Conference on “Mathematics” on 26-11-2015 to 28-11- 2015 at University of Kerala in collaboration with International Multidisciplinary Research Foundation.

Kavitha T., Sini P.  and Ramachandran  P. T.: On Group of Closure Isomorphisms, Proceedings IC- AMMN-2016,FISAT, Angamaly January 7-9, (2016).

“On f-representability of Dihedral groups” , International Conference on “ Advances in Applied Mathematics, Material Science and Nanotechnology for Engineering and Industrial Applications ” on 07-01-2016 to 09-01-2016 at Federal Institute of Science and Technology.

“Homogeneity in Generalized Topological Spaces”, MESMAC International Conference on Existence and Expression: Science, Philosophy and Art on 14-02-2017 to 16-02-2017 at MES Mampad College, Malappuram.

“Group of Homeomorphisms and Maximal Subgroups of Symmetric Group”, 29 th  Kerala Science Congress on 28-01-2017 to 30-01-2017 at Marthoma College Thiruvalla.

Lf representability of Cyclic Permutation Groups, International conference on Advances in Applicable Mathematics on 07th and 08th December 2017 at Bharathiyar University.

 Completely Homogeneous C-spaces, International Workshop and Conference on Topology and Applications from December 5th to 11th ,2018 at Ragagiri School of Engineering and Technology.

Ratheesh K. P.

Ratheesh K.P. and Madhavan Namboothiri N.M.:On C-spaces and Connective Spaces, South Asian Journal of Mathematics, 4(2013), 1-13 ISSN 2251-1512

Santhosh P.K.

Santhosh P.K. : On Products and Embeddings of C-spaces,Journel of Advanced Studies in Topology, Vol. 4, No. 4, 2013, pp. 18-24, ISSN 2090-8288|EISSN 2090-388X Santhosh P.K.: On the Quotients of C-Spaces,  accepted for publication in Boletim da Sociedade Paranaense de Matematica (Bulletin of Parana´s Mathematical Society, Brazil)  ISSN 0037-8712 (print) and ISSN 2175-1188 (on-line).

Dhanya P.M.

P. M. Dhanya, On the Lattice of Generalized Topologies, Int. J. Pure. Appl. Math. 102, No. 1, (2015) 85-95, doi:  http://dx.doi.org/10.12732/ijpam.v102i1.8

Dhanya P.M. & Ramachandran P.T.: On the Simple Expansion of Generalized Topologies , International Journal of Pure and Applied Mathematics,Vol.104, No.4(2015) 621-630. ISSN: 1311-8080.  doi: http://dx.doi.org/10.12732/ijpam.v104i4.12

Dhanya P. M. :On Completely Homogeneous Generalized Topological Spaces , International Conference on  Mathematics 2015,  University of Kerala,  Nov:26-28, 2015. Sini P., Dhanya P. M. and Ramachandran P. T. :  On Completely Homogeneous L-topological Spaces,  Annals of fuzzy mathematics and informatics, 12(5), 703-718(2016) pISBN- 2093-9310, eISSN-2287-6235

Papers Presented at International Level  Conferences and Seminars

Dhanya P. M. and Sini P. : On Homogeneous Generalized Topological Space, Proceedings IC- AMMN-2016,FISAT, Angamaly (2016) 87-92.

Kavitha T.

Kunheenkutty M., Kavitha T. and Ramachandran P.T.: Adjacency in the Lattice of Cech Closure Operators, International Journal of Pure and Applied Mathematics Vol.105 No.1 (2015) 73-86  ISSN 1311-8080  doi: http://dx.doi.org/10.12732/ijpam.v105i1.7

Papers Presented at International Level  Conferences and Seminars

Kavitha T., Sini P.  and Ramachandran  P. T.: On Group of Closure Isomorphisms, Proceedings IC- AMMN-2016, FISAT, Angamaly, January 7-9 (2016) .

M.Phil. Dissertations Guided:

1. Baby Chacko On the Lattice of Topologies (1989)
2. Sheela C.            Reflective and Coreflective Subcategories in Topology (1990)
3. Mary O.T.            On Rings of Continuous Functions (1990)
4. Rajendran P.S.        Semigroup of Continuous Functions  (1992)
5. Nisha Oommen           Applications of Boolean algebras in Topology
6. Annie Kurien             Digital Topology (2003)
7. Kunheenkutty M.          Local Properties of the Lattice of Topologies (2008)
8. Libeeshkumar K.B.        Group of Automorphisms of Graphs (2009) (Awarded in May,2011)
9. Ms. Sruthi A.K.          c-spaces (2016)
10.Mr. Rashid A.            A Study on Homogeneity and Rigidity in Topological Spaces (2017)
11.Ms. Shifna M.K.

 Nonstandard Analysis and Toplogy
12.Ms. Seira Susan Prasad A Study on Soft Set Relations and Soft Topology