Curriculum Vitae of Dr. S. Ahmad Ali
EDUCATIONAL PROFILE
M. Sc.: Mathematics (Statistics); Avadh University, (India) ;1987.
Ph. D.: Kanpur University, Kanpur (India) ; 1995
Topic of Ph. D. Thesis: A Study of Distribution Theory and an Application of Fractional Integration to Certain Integral Transformations and Integral Equations
RESEARCH PROFILE
Areas of Interest
Special Functions, q-Series, Distribution Theory, Integral Transforms, Fractional Calculus.
Research Contribution
In my earlier research work, I have studied the Distributional aspect of the various integral transforms and the fractional integral operators. This such a study of these distributional integral transforms and fractional integral operators helps us in solving the dual integral equations which are of frequent occurrence in mathematical physics. To further generalized these aspects and make them more useful, currently I am involved in making a study of generalized hypergeometric functions theory which is expected to yield more useful results.
I have introduced the two new forms of the extended Hankel transforms given by A. L. Schwartz and G. Ahuja by the use of 'cut' Bassel function of H. Kober. I also have introduced their variants. I have studied the mapping properties of these operators on Zemanian and Koh spaces of test functions and corresponding spaces of generalized functions. I have obtained the relations of one of my operators with some of the elementary differential operators.
The Erdelyi-kober operators of fractional integration were extended by L.G.Makarenko. I have extended the definition of Riemann-Liouville and Weyl operators of fractional integration for two-dimensional cases. I have studied the behavior of Makarenko's and aforesaid operators for generalized functions. For this purpose I also have constructed a testing function space and its dual. I have asserted that these operators can be used to solve the pair of dual integral equations.
I have used the definition of two distributions to prove an improved form of the Leibnitz theorem for the fractional derivative of the product of two distributions of arbitrary order. I further have generalized the above result and have given another form of the theorem by differentiating fractionally the product with respect to a continuously differentiable function.
I have introduced certain modified forms of Ramanujan's mock theta functions. By the use of q-fractional integral operators of R.P.Agarwal and W.Al Salam, I have been able to produce some very interesting integral relations between the mock theta functions of different order, particularly between the fifth order functions belonging to the two different groups. I have defined the partial series of the mock theta functions, recently found in Ramanujan’s ‘Lost Notebook’. I have given some alternative forms to these functions and have established certain relations between them which definitely through light on their structure and also they may have number theoretic interpretations. I have produced some hypergeometric representations of the mock theta functions found in the ‘Lost’ Notebook through some known hypergeometric series transformations. These representations suggest a new approach of study of these functions. Very recently, I have introduced four generalized q-functions which yield the mock theta functions of order six and ten as their special cases. Interestingly, the said q-functions have certain differential properties and satisfy the certain difference equations. Further, I have given certain alternative expansions to these functions and have also related them with their partial forms through q-series. The generalized partial function relations are expected to yield interesting results in the theory of restricted partitions as in the case of partial theta functions.
In a recent work I have produced certain continued fraction representations for the ratios of basic hypergeometric series and for certain hypergeometric series of special type. These representations cover a number of known continued fraction as its special cases including some of those given by Ramanujan.
The Extended Schwartz's Hankel Transformation for Generalized Functions; Jour. M.A.C.T., Vol. 25(1992).
The n - Dimensional Extended Hankel Transformation for Generalized Functions (with G. Ahuja); Ganita, Vol. 45, No.1(1994).
Leibnitz Theorem for Fractional Derivative of Generalized Functions. Ranchi Univ. Math. Jour., Vol. 25 (1994).
Riemann - Liouville & Weyl Fractional Integral Operators for generalized Functions; Ranchi Univ. Math. Jour., Vol. 47(1999).
On the Generalized Operators of Fractional Integration. Ganita Sandesh, Vol.14, No.2 (2000)
An Application of Generalized Operators of Fractional Integration. Math. Stud., Vol.69. No.1-4 (2001).
A Unified Mock Theta Function. South East Asian Jour. Math., Vol.1(2), 2002
An Application of Fractional Integral Operators to the pair of Dual Integral Equations for Generalized Functions. To appear in Math Stud.
Fifth order Mock Theta Functions and their relations. Alig. Bull. Math. Vol. 22, No. 1, 2003.
Properties of Certain Fractional q-Integral Operators and 'Cut' q-Hankel Transform. Proc. Nat. Acad. Sci.(India), 74 (A), I, 2004.
Certain Mock Theta Functions found in the ‘Lost’ Notebook. To appear in J. Ind. Math. Soc..
Transformations of Mock Theta Functions found in the ‘Lost’ Notebook. (with R.Y. Denis & S.N. Singh). Jour. Ind. Acad. Math., Vol. 26 (1), 2004.
A basic Hypergeometric approach to Mock Theta Functions found in the ‘Lost Notebook. Proc. Nat. Acad. Sci.(India), 74 (A), I, 2004.
On Certain q-Functions associated with Mock Theta Functions. To appear in South East Asian Jour. Math.
An infinite family of mock theta function. Proc. 4th Annual Conference of SSFA, University of Rajsthan, Jaipur (India), 2004.
Leibnitz Theorem for Fractional Derivative of Generalized Functions -II. Under publication
The Boundedness of Generalized Lowndes Operators. Proc. Nat. Acad. Sci.(India),
Some new identities of eight order Mock theta Functions. To appear in J. Ind. Acad. Math.
On Certain Basic Hypergeometric Series Transformations. To appear in Ind J. Math.
On Continued Fraction Representation of Certain Basic Hypergeometric Series. Under Publication
On Continued Fraction Representation of Certain Basic Hypergeometric Series of Special Type. Under Publication
A Bilateral Extension of Second Order Mock Theta Functions. Under Publication
Certain Transformations of Basic Bilateral Hypergeometric Series. Under Publication
On q-Series and Continued Functions. Under Publication
Research Project Undertaken
Title of Project: A Study of q-identities associated with partial Mock Theta Functions
Position in Project: Principal Investigator
Project Duration: Two year.
Sponsor: Council of Science & Technology, U.P.
TEACHING EXPERIENCE
I have more than eighteen years experience of teaching Undergraduate Mathematics course of the University of Lucknow. During this period I have taught Algebra (Linear and Modern), Matrix theory, Calculus, Coordinate Geometry, Vector Calculus, Differential Equations and Analysis to the students of B.A. and B.Sc. courses in the following institutions:
1.
Worked as a Lecturer (in temporary position) in Mathematics in Shia P.G. College (an associated college of the University of Lucknow), Lucknow from Nov. 21, 1988 to May 30, 1997.2. Presently working as a Lecturer and Head, Department of Mathematics, Amiruddaula Islamia Degree College (an associated college of the University of Lucknow), Lucknow since June 9, 1997
EXPERIENCES OTHER THAN TEACHING
Besides having a teaching experience of about eighteen years, I hold/ have held the following administrative positions:
Head, Department of Mathematics.
Proctor, Faculty of Science.
Maintenance Incharge.
Editor (English/ Science Section), College Magazine
Incharge, College Science Association.
AWARD
I have been awarded V. M. Shah Prize of the year 2000 of Indian Mathematical Society at its 66th Annual Conference held at Aurangabad from Dec 20-22 , 2000.
FELLOWSHIP
I am an elected Fellow of the Institute of Mathematics & its Applications, U. K.
OTHER PUBLICATIONS
Books
Theory of the Functions of a Complex Variables (Class Room Notes), 1989
Matrix Theory. Publication Center, Lucknow (India), 1995
Differential Calculus & Vector Calculus*, Unitech House, Lucknow, 2004
Advanced Calculus**, Unitech House, Lucknow, 2004
Complex Analysis, Anamaya Publications, New Delhi, (Under Preparation).
*Joint author (Ist author), **Joint author (IInd author)
COMPUTER SOFTWARE/ PACKAGES KNOWN
Windows, MS Office, Mathematica 3.1, Hyp, qHyp, Hyper, qHyper, fastZeil, Zeilberger, MultiSum, qZeil, qMultiSum
PAPERS/ INVITED TALKS IN CONFERENCES
Papers
Extended Swartz's Hankel Transformations for generalized functions. Annual conference of Bharat Ganita Parishad, Lucknow University, Dec. 1992.
Leibnitz Theorem for Fractional Derivative of Generalized Functions. 64th Conference of Indian Mathematical Society, Gurukul Kangri University, Dec. 1998
Properties of Certain Fractional q-Integral Operators and 'Cut' q-Hankel Transform. National Conference of the Society for Special Functions and their Applications held at J N Vyas University, Jodhdpur, March 2 – 4, 2000
Fifth order Mock Theta Functions and their relations, International Conference on the Works of Ramanujan, University of Mysore, July 1-3, 2000
Certain Mock Theta Functions found in the ‘Lost’ Notebook, . 66th conference of Indian Mathematical Society held at Aurangabad, Dec, 2000
A Unified Mock Theta Function, International Conference on Special Functions and their Applications of S.S.F.A., India held at the Institute of Management & Development, Lucknow, Feb 2-4, 2001.
A basic Hypergeometric approach to Mock Theta Functions found in the ‘Lost Notebook, Conference on Mathematics, 67th conference of Indian Mathematical Society held at Aligarh Muslim University, Aligarh, Jan 27-30, 2002.
On Functions associated with Mock Theta Functions, International Conference on Special Function and their Applications, Institute of Mathematical Sciences, Chennai. Sept. 23-27, 2002.
Certain q-functions associated with Mock Theta Functions found in the 'Lost' Notebook, 68th Annual Conference of Indian Mathematical Society, Shivaji University, Kolhapur, Dec. 20-23, 2002.
An infinite family of Mock Theta Functions, 4th Annual Conference of SSFA, University of Rajsthan, Jiapur. Feb. 20-22, 2003.
On an infinite family of Mock Theta Functions, 19th Annual Conference of RMS, BR Ambedkar Agra University, Agra, July 21-24, 2004.
Invited Talks
Symposium on Ramanujan, 87th Session of Indian Science Congress Association held at the University of Lucknow, Jan. 3-7, 2002.
Annual conference of the Society of Special Functions & their Applications held at Banaras Hindu University, Varanasi, March 4-6, 2002
Conference on Mathematics, University of Lucknow, Lucknow, Nov. 24-25, 2001.
69th Annual Conference of Indian Mathematical Society, Dec. 26-29, 2003. University of Lucknow, Lucknow.
Conference on Special Functions and their Applications - the Fifth Annual Conference of SSFA, February 8 - 10, 2004, Lucknow.
National Conference on Advances in Mathematics and its Applications, ISMMS, February 13 - 15, 2004, Gorakhpur
70
th Annual Conference of Indian Mathematical Society, Dec. 26-29, 2004. J. N. Vyas University, Jodhpur.92nd Session of Indian Science Congress Association held at the Nirma University of Science & Technology, Ahmadabad, Jan. 3-7, 2005.
Sixth Annual Conference of SSFA, Jaunpur, June 7-9, 2005
72nd Annual Conference of Indian Mathematical Society to be held at Rani Durgawati University, Jabalpur during Dec. 26-29, 2006.
National Symposium on Special Functions & their Applications, 12th Annual Conference of Vigyana Parishad of India, JN Vyas University, Jodhpur, October 25-27, 227
WORKSHOP/ SUMMER SCHOOL
I have participated in the following workshop & Summer School
Workshop on Hyp & qHyp Package, Institute of Mathematical Sciences, Chennai, September 15-19, 2001.
Arithmetic & Geometry Around Hypergeometric Functions, UNESCO-CIMPA Summer School, Galatasaray University, Istanbul (Turkey), June 13-25, 2005.
OTHER ACADEMIC ACTIVITIES
Assistant Secretary, Society for Special Functions and their Applications, India.
Regional Coordinator, National Mathematical Olympiad (Delhi), India.
Member, Sectional Committee of Mathematical Sciences Section,
Indian Science Congress Association (2005-06).Member, Editorial Board of the Proceedings of the Fifth Annual Conference of the Society for Special Functions and their Applications, India.
Member, Editorial Board, Bulletin of Pure & Applied Mathematics.
Reviewer, Mathematical Reviews, American Mathematical Society, USA.
Reviewer, Zentralblatt Math, European Mathematical Society, Germany.
I have organized the following Seminar/ Conferences:
·
Seminar on Role of Mathematics in Real Life Problems, January 4, 2000, Lucknow.·
International Conference on Special Functions & their Applications, February 2-4, 2001, Lucknow.·
Conference on Special Functions & their Applications (Int. Conf.), February 8 - 10, 2004, Lucknow.MEMBERSHIPS
Indian Mathematical Society
Indian Mathematical Academy
Society for Special Functions and their Applications
Bharat Ganita Parishad
Rajasthan Ganita Parishad
Indian Society for History of Mathematics
UP Association for Advancement of Science and Technology
Indian Science Congress Association
Calcutta Mathematical Society
Indian Society of Mathematics & Mathematical Sciences
Ramnujan Mathematical Society
PERSONAL PROFILE
Father’s Name : Mr. Syed Ali
Date of Birth : July 1, 1967
Marital Status : Married
Present Position: Head, Department of Mathematics
Amiruddaula Islamia Degree College
Lalbagh, Lucknow 226 001, India
Address for communication: 292, Chandralok, Aliganj, Lucknow 226 024, India
Tel: Res. +91-522-2323905, Off. 2272139, Ext. 16, Mobile 9415583389
References
1. Prof. R. P. Agarwal
Former Head, Department of Mathematics & Astronomy, University of Lucknow, Lucknow.
Ex Vice Chancellor, University of Lucknow (Lucknow) & University of Rajsthan (Jaipur).
Address: B – 1/201, Nirala Nager,
Lucknow 226 020. India.
Tel. +91 522 2789116
2. Prof. M. A. Pathan
Ex. Chairman, Department of Mathematics,
Aligarh Muslim University, Aligarh.
Tel.: +91 571 2701282
mapathan@gmail.com
3. Prof Rajiv K. Srivastava
Department of Mathematics,
Institute of Basic Sciences,
Dr. BR Ambedkar Agra University,
Khandari, Agra 282 002.
rajivksri@rediffmail.com
4. Prof. K. Srinivasa Rao
Senior Professor (Retd.)
Institute of Mathematical Sciences, Chennai.
ksrao18@yahoo.co.in
5. Prof. R. Y. Denis
Department of Mathematics,
Gorakhpur University,
Gorakhpur
ddry@sancharnet.in
6. Prof. P. K. Banerji
Departmnet of Mathematics,
JN Vyas University,
Jodhpur.
banerjipk@yahoo.com