# Mathematics - Research Groups

### Special Functions:

The Special Functions is a branch of classical Analysis. In the last five decades the subject has witnessed a tremendous growth because of its wide applications in many branches of physical and engineering sciences. The most important object in the theory of Special Functions is Hypergeometric Function and its q-extension as it houses most of the functions of mathematical physics. Furthermore, the q-extension or simply q-hypergeometric function has been closely related to many of the works of Ramanujan. At present, people in the Department are working on q-hypergeometric functions, mock theta functions and topics related to Ramanujan's mathematics that has some latest developments resulting in rapid growth and a speedup research in many areas of Science such as String Theory.

**Faculty: Dr. S. Ahmad Ali, Dr. S. Nadeem Hasan Rizvi**

### Theory of summability and approximation theory:

The old hazy notion of convergence of infinite series was placed on sound foundation with the appearance of Cauchy's monumental work " Cours d' Analyse algebrique " in 1821 and Able's researches on the binomial series in 1826. However it was observed that there where certain non convergent series which particularly in Dynamical Astronomy furnish nearly correct results. A theory of divergent series was formulated explicitly for the first time in 1890 when Cesaro published a paper on multiplication of series. Since then the theory of series whose sequence of partial sums oscillates has been the centre of attraction and facination for a number of pioneering mathematical analysts. After persistent efforts in which a number of celebrated mathematicians took part, it was only in the closing decade of 18th century and in the beginning of 19th century a satisfactory method was discovered which is best on the Cauchy concept of convergence, which is known as " Summability Theory." Some of most familiar methods of summability are cesaro summability, Norlund summability, ordinary summability, ordinary and absolute summability, Euler summability, triangular matrix summability and degree of approximation. At the present Fourier series, Conjugate series of Fourier series, derived series of Fourier and degree of approximation of Fourier series, Conjugate series of Fourier series, derived series of Fourier has become centre of attraction for mathematical analysts.

**Faculty: Dr. U.S. Yadav**

### Differential geometry:

The Differential geometry is a mathematical discipline that uses the techniques of differential calculus and integral calculus as well as linear algebra and multilinear algebra to study problems in geometry. The study of calculus on differentiable manifolds is differential geometry. The study of manifold combines many important areas of mathematics; it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology. Certain special classes of manifolds also have additional algebraic structure; they may behave like groups, for instance. In that case, they are called Lie Groups. Alternatively, they may be described by polynomial equations, in which case they are called algebraic varites, and if they additionally carry a group structure, they are called algebraic groups. The geometry of manifold was introduced by Bernhard Riemann his famous habilitation lecture in 1867. The name manifold comes from Riemann's original German term, Mannigfaltigkeit. A differentiable manifold is a topological manifold equipped with an equivalence class of atlases whose transition maps are all differentiable. Therefore the differential geometry can also be described as 'Study of calculus on differential manifolds'. Differential manifold are widely used in the field of Mathematical Physics , Classical Mechanics, general relativity and Yang-Mill's Theory etc. Riemannian Geometry, Finsler geometry, Symplectic geometry, Contact geometry, Complex and Kahler geometry are different Branches of Differential Geometry. Finsler Geometry Is just Riemannian Geometry without the Quadratic Restriction. The name ``Finsler geometry'' came from Finsler's thesis of 1918. It is actually the geometry of a simple integral and is as old as the calculus of variations.

**Faculty: Dr. R. K. Pandey, Dr. Abhishek Singh, Dr. Gautam Lodhi**

### Hydro-dynamic Dispersion:

With respect to the importance of hydro-dynamic dispersion process studies in water quality management and pollution control particularly in aquifers, the dispersion has been referred to as a hydraulic mixing process by which the waste concentrations are attenuated while the waste pollutants are being transported downstream. Pollution can be classified in many ways. On the basis of the medium of the environment where it occurs most, it can be classified as air pollution, soil pollution, surface water pollution and ground water pollution. Solute particles released from different sources in these media degrade their quality for variety of uses. Its source may be natural or anthropogenic. One type of the source of these pollutions is a point source. Stationary point sources include volcanoes, factories, electric power plants, mineral smelters, petroleum refineries and different small scale industries; while non-point sources include all sorts of transport vehicles moving by road, rail or air. Groundwater pollution occurs due to infiltrations of wastes through rain water, from garbage disposal sites, septic tanks, mines, discharge from surface water bodies polluted due to industrial and municipal influents. Immiscible solute or tracer particles of pollutants are major cause of degradation of the hydro-environment in the surface water bodies and aquifers. The sources of such pollutants originate from human activities on the earth. Solute particles reach a surface water body with waste water drainage and reach an aquifer due to infiltrations from wastes disposal sites, underground septic tanks, mines and polluted water bodies that recharge the aquifers. Solutes are transported down the stream along the flow and disperse due to combined effects of diffusion and advection.

**Faculty : Dr. Atul Kumar**

### Variational Inequalities, Fixed Point Theory and Nonlinear Analysis

The theory of variational inequalities and equilibrium problems has emerged as an interesting branch of applicable mathematics and become a rich source of inspiration and motivation to study a large number of problems arising in optimization, operation research, and economics etc. The theory of variational inequalities was initiated by Stampacchia and Fishera independently in early sixties during the study of potential problems. But the first general principle for the existence and uniqueness of the solution of variational inequality problem was proved by Lion's and Stampacchia in 1967. Since then this theory has a great importance in theoretical and practical point of view.

**Faculty: Dr. Shuja Haider Rizvi**

### Faculty Research Publications

**S. Ahmad Ali and S. Nadeem Hasan Rizvi:**On Certain Transformations of Bilateral Basic Hypergeometric Series, Journal of Advanced Mathematics and Applications, Vol. 5, 1–5, 2016**S. Ahmad Ali and S. Nadeem Hasan Rizvi**, Transformations from Bilateral Basic Hypergeometric Series to Unilateral Basic Hypergeometric Series, International Bulletin of Mathematical Research,Vol 02, (4), 1-4, 2015.**S. Ahmad Ali And S. Nadeem Hasan Rizvi**, Certain Transformations of Basic and Poly- basic Hypergeometric series, Italian Journal of Pure and Applied Mathematics, 35, 617- 624, 2015**S. Ahmad Ali**and A. Agnihotri: Parameter Augmentation for Basic Hypergeometric Series using Cauchy Operator,*Palestine J. Math*., Vol. 5(1), 2016, pp. 192-197.**S. Ahmad Ali:**On Certain Basic Hypergeometric Series & Continued Fractions,*Gen. Math. Notes*, 2016. (Accepted)**S. Ahmad Ali**and A. Agnihotri: On Applications of Ramanujan Sum,*J. Math. Comput. Sci*., 2016. (Accepted).**Abhishek Singh**, R. K. Pandey, S. Khare: On horizontal and complete lifts of (1, 1) tensor fields F satisfying the structure equation - F(2K + S, S) = 0,*International Journal of Mathematics and Soft Computing*, Vol.6(1), 2016, pp. 143-152. ISSN No. 2249 - 3328 (Print), ISSN No. 2319 - 5215 (Online).**S. Ahmad Ali**and S N H Rizvi: Certain Transformations and Summations of Basic and Poly-Basic Hypergeometric Series,*Italian J. Pure Appli. Math*., Vol. 35, 2015, pp. 617-624.**S. Ahmad Ali:**On Generalization of a Partial Theta Function Identity of Ramanujan,*Journal Inequalities & Special Functions*, Vol. 6(2), 2015, pp. 1-4.**S. Ahmad Ali:**A Continued Fraction for Second Order Mock Theta functions,*Int. J. of Math. Anal*., Vol. 9(24), 2015, pp. 1187-1189.**S. Ahmad Ali**and S N H Rizvi: Certain Transformations and Summations of Basic Hypergeometric Series,*J. Math. Comput. Sci*., Vol. 5(1), 2015, pp. 25-33.**S. Ahmad Ali**and S N H Rizvi: On Certain Transformations of Bilateral Basic Hypergeometric Series,*J. Adv. Math. Appl*., Vol. 4(2), 2015.**Abhishek Singh**, R. K. Pandey, A. Prakash and S. Khare: On a pseudo projective recurrent sasakian manifolds,*Journal of mathematics and computer Science*, Vol. 14, 2015, 309-314.**S. Ahmad Ali and S. Nadeem Hasan Rizvi:**Certain transformations and summations of basic hypergeometric series,*J. Math. Comput. Sci*., Vol. 5(1), 2015, pp. 25-33. ISSN No. 1927-5307.- K. R. Kazmi,
**S. H. Rizvi**and Rehan Ali: A hybrid iterative method without extrapolating step for solving mixed equilibrium problem,*Creative Mathematics and Informatics*, Vol. 24(2) 2015, pp. 165-172. [ISSN 1843-441X; Refereed & Indexed Journal] (Sinus Association Publication). - K. R. Kazmi,
**S. H. Rizvi**and M. Farid: A viscosity Cesaro mean approximation method for split generalized vector equilibrium problem and fixed point problem,*Journal of the Egyptian Mathematical Society*, Vol. 23, 2015, pp. 362-370 [ISSN 1110-256X; Refereed & Indexed Journal] (Elsevier). **S. Ahmad Ali**and A. Agnihotri: Certain Basic Hypergeometric Identities through q-exponential operator technique,*International Bulletin of Mathematical Research*, Vol. 1(1), 2014, pp. 49-53.- K. R. Kazmi and
**S. H. Rizvi**: An iterative algorithm for generalized mixed equilibrium problems,*Afrika Mathematika*, Vol. 25(4), 2014. pp. 857-867, [ISSN: 1012-9405 (Print version), ISSN: 2190-7668 (Electronic version); Refereed & Indexed Journal] (Springer). - K. R. Kazmi and
**S. H. Rizvi**: An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping,*Optimization Letters*, Vol. 8, 2014, pp.1113-1124, [ISSN 1862-4480; Refereed & Indexed Journal; Impact Factor: 1.654] (Springer) (SCI Listed). - K. R. Kazmi and
**S. H. Rizvi**: Implicit iterative method for approximating a common solution of split equilibrium problem and fixed point problem for a nonexpansive semigroup,*Arab Journal of Mathematical Sciences*, Vol. 20(1), 2014, pp. 57-75. [ISSN 1319-5166; Refereed & Indexed Journal] (Elsevier). **Abhishek Singh**, R. K. Pandey and Sachin Khare: Parallelism of Distributions and Geodesics on F(2K+S, S)- Structure Lagrangian Manifolds,*International Journal of Contemporary Mathematical Sciences*, Vol. 9, 2014, pp. 515-522. ISSN 132-7586.**A. Kumar**and R. R. Yadav: One-dimensional solute transport for uniform and varying pulse type input point source through inhomogeneous medium,*Environmental Technolog*, Vol. 36(4), 2014, pp. 487-495. ISSN No. 0959-3330, 1479-487X (Online).**A. Kumar**and R. R. Yadav: Analytical solution of one - dimensional solute transports for unsteady flow through inhomogeneous semi-infinite porous domain: Dispersion being proportional to square of velocity,*Discrete and Continuous Dynamical Systems*, Vol. 34, 2013, pp. 457-466. ISSN 1078-0947(print), 1553-5231(online).- B. D. Rouhani, K.R. Kazmi and
**S. H. Rizvi**: A hybrid-extragradient-convex approximation method for a system of unrelated mixed equilibrium problems,*Transactions on Mathematical Programming and Applications*, Vol. 1(8), 2013, pp. 82-95. [ISSN 2325-405X; Refereed & Indexed Journal] (International Publication U.S.A.) - K. R. Kazmi and
**S. H. Rizvi**: Iterative algorithms for generalized mixed equilibrium problems,*Journal of the Egyptian Mathematical Society*, Vol. 21(3), 2013 pp. 340-345. [ISSN 1110-256X; Refereed & Indexed Journal] (Elsevier). - K. R. Kazmi and
**S. H. Rizvi**: Iterative approximation of a common solution of a split generalized equilibrium problem and a fixed point problem for nonexpansive semigroup,*Mathematical Sciences*, Vol. 7, 2013, pp. 10 (doi:10.1186/2251-7456-7-1) [ISSN 2251-7456; Refereed & Indexed Journal] (Springer). - K. R. Kazmi,
**S. H. Rizvi**: Iterative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem,*Journal of the Egyptian Mathematical Society*, Vol. 21(1), 2013, pp. 44-51. [ISSN 1110-256X; Refereed & Indexed Journal] (Elsevier). - K. R. Kazmi, N. Ahmad and
**S. H. Rizvi**: System of implicit nonconvex variationl inequality problems: A projection method approach,*The Journal of Nonlinear Sciences and its Application*, Vol. 6 2013, pp. 170-180. [ISSN 2008-1898; Refereed & Indexed Journal; Impact Factor: 1.0756] International Scientific Research Publications (ISRP) (SCI Listed). - K.R. Kazmi and
**S. H. Rizvi**: A hybrid extragradient method for approximating the common solutions of a variational inequality, a system of variational inequalities, a mixed equilibrium problem and a fixed point problem,*Applied Mathematics and Computation*, Vol. 218(9), 2012, pp. 5439-5452. [ISSN 0096-3003; Refereed & Indexed Journal; Impact Factor: 1.672] (Elsevier) (SCI Listed). - D. K. Jaiswal,
**A. Kumar**and R. R. Yadav: Unsteady dispersion in two-dimensional homogeneous porous media,*International Conference on Modeling & Simulation of Diffusion Processes and Applications-12*, BHU, Varanasi, 2012, pp 75-79. **A. Kumar**, Jaiswal, D. K. and R. R. Yadav: One-dimensional nutrient transport through soil,*International Conference on Modeling & Simulation of Diffusion Processes and Applications-12*, BHU, Varanasi, 2012,pp. 132-137.**A. Kumar**, D. K. Jaiswal and R. R. Yadav: Analytical Solutions of One-Dimensional Temporally Dependent Advection-Diffusion Equation along Longitudinal Semi-Infinite in Homogeneous Porous Domain for Uniform Flow,*Journal of Mathematics*, Vol. 2(1), 2012, pp. 01-11. ISSN No. 2278-5728 (Online), 2319-765X (Print).**A. Kumar**, D. K. Jaiswal and N. Kumar: One-dimensional Solute Dispersion along Unsteady Flow through heterogeneous Medium: Dispersion being proportional to square of velocity,*Hydrological Science Journal*, Vol. 57(6), 2012, pp. 1223-1230. ISSN No. 0262-6667 (Print), 2150- 3435 (Online)..- S. K. Yadav,
**A. Kumar**, and N. Kumar: Horizontal solute transport from a pulse type source along temporally and spatially dependent flow: Analytical solution,*Journal of Hydrology*, Vol. 412-413, 2012, pp. 193-199. ISSN: 0022-1694. - C. K. Mishra and
**Gautam Lodhi**: On curvature inheritance symmetry in Finsler space,*Acta Universitatis Apulensis*, Vol. 30, 2012, pp. 39-48. ISSN 1582-5329. - C. K. Mishra,
**Gautam Lodhi**and Meenakshy Thakur: On Decomposability of the curvature tensor in second order recurrent conformal Finsler spaces,*International Journal of Mathematics and Architecture*, Vol. 30, 2012, pp. 39-48. ISSN 1582-5329. **S. Ahmad Ali**and S M H Rizvi: On Continued Fraction Representation of Certain Function of Hypergeometric Type,*Jour. Ramanujan Math. Soc. & Math Sci.*, Vol. 1(2), 2012.**S. Ahmad Ali**:Some new identities of eight order Mock theta Functions,*Italian J. Pure Appl. Math*., Vol. 27 (2), 2011.**A. Kumar**, D. K. Jaiswal, and R. R. Yadav: One-Dimensional Solute Transport for Uniform and Varying Pulse Type Input Point Source with Temporally Dependent Coefficients in Longitudinal Semi- Infinite Homogeneous Porous Domain: Analytical Solutions,*International Journal of Mathematics and Scientific Computing*, Vol. 1(2), 2011, pp. 56-66. ISSN No. 2231-5330.- S. K. Yadav,
**A. Kumar**, D. K. Jaiswal and N. Kumar: One-dimensional unsteady solute transport along unsteady flow through inhomogeneous medium,*Journal of Earth System Science*, Vol. 120(2), 2011, pp. 205-213. ISSN: 0253-4126 (print), 0973-774X (online). - D. K. Jaiswal, and
**A. Kumar**: Analytical solutions of advection-dispersion equation for varying pulse type input point source in one-dimension,*International Journal of Engineering*, Science and Technology, Vol. 3(1), 2011, pp. 22-29. ISSN No. 2141-2839 (Online), 2141-2820 (Print). - D. K. Jaiswal, and
**A. Kumar**: Analytical solutions of time and spatially dependent one-dimensional advection-diffusion equation,*Earth and Environmental Science Pollution*, Vol. 32, 2011, pp. 2078-2083. ISSN No. 2229-712X (online). - D. K. Jaiswal,
**A. Kumar**, and R. R. Yadav: Analytical solution to the one-dimensional advection-diffusion equation with temporally dependent coefficients,*Journal of Water Resource and Protection*, Vol. 3, 2011, pp. 76-84. ISSN No. 1945-3094 (print), 1945-3108 (online). . - D. K. Jaiswal,
**A. Kumar**, N. Kumar, and M. K. Singh: Solute transport along temporally and spatially dependent flows through horizontal semi-infinite media: dispersion being proportional to square of velocity,*Journal of Hydrologic Engineering*, Vol. 16(3), 2011, pp. 228-238. ISSN No. 1084-0699 (print), 1943-5584 (online). - C. K. Mishra and
**Gautam Lodhi**: Torse-forming curvature inheritance in Finsler Spaces,*International J. of Math. Sci. & Engg. Appls*. (IJMSEA), Vol. 5 VI, pp. 347-353, 2011. - C. K. Mishra, D. D. S. Yadav and
**Gautam Lodhi**: Decompostion of Weyl Tensor Field in a Recurrent Finsler space,*ActaCiencia*, Vol. XXXVII M. No. 1, pp. 193-200, 2011. (India) **U. S. Yadav**: On (N,p,q)(C,1) summability of a sequence of Fourier coefficients,*Ganit Sandesh*, Vol. 16(1), 2011, pp. 61-66.**A. Kumar**, D. K. Jaiswal, and N. Kumar: Analytical solutions to one-dimensional advection-diffusion equation with variable coefficients in semi-infinite media,*Journal of Hydrology*, Vol. 380, 2010, pp. 330-337. ISSNNo. 0022-1694 (online).- C. K. Mishra and
**Gautam Lodhi**, Torse-forming Projective N-Curvature Collineation in NP-Fn,*ADJM (A.M.S., U.S.A.)*, Volume 10, Number 1, pp. 39-48, 2010. **Abhishek Singh**: On CR-structures and F-structure satisfying F8 + F7 + F6+ F5 + F4 + F3 + F2 +F=0,*J. Nat. Acad. Math*., Vol. 24, 2010.