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ThesisItem Open Access MAGNETOHYDRODYNAMIC FLOW OF FLUIDS THROUGH A POROUS CHANNEL(DEPARTMENT OF MATHEMATICS AND STATISTICS SAM HIGGINBOTTOM UNIVERSITY OF AGRICULTURE, TECHNOLOGY AND SCIENCES PRAYAGRAJ-211007, INDIA 2019, 2019) DWIVEDI, KARUNA; Paul, Prof (Dr.) AjitIn this chapter, effect of inclined magnetic field on the flow of visco-elastic fluid flowing through a porous channel has been analyzed and the conclusion of the study is given as fallows- In all the cases, the role of magnetic field is very important as it controls the velocity of the fluid. As the fluid is considered to visco-elastic, therefore, magnetic sensitive particles are attracted when magnetic field comes into play. The velocity profile with respect to magnetic field has a steady increase in a linear way for different values of Grashoff number The velocity of fluid with respect to exponentially increases as the visco-elastic parameter increases taking a resonance character. The increasing value of inclination parameter increases the velocity of the fluid with respect to magnetic field. The velocity of fluid with respect to layer distance increases for Hartman number. The temperature profile increases when frequency of oscillation increases and also increases for decreasing Peclet’s number. The concentration profile with respect to layer distance increases for increasing Schmidt number. MHD flow through vertical channel having porous medium in the presence of magnetic field have been observed and the following conclusions have been derived. The velocity of the fluid decreases with increasing angle of inclination with respect to the magnetic field. As the boundary layer distance increases, the velocity of the fluid increases for different Prandlt number and value of velocity of fluid for all Prandlt number start with almost same values after that increases gradually. The velocity decreases for increasing different radiation parameters with respect to the layer distance. The Graph concludes that the velocity decreases for increasing different Soret number with respect to the layer distance. The temperature of fluid increases and when reach at the maximum value of temperature then decreases gradually and end at the same value of temperature for increasing Suction parameter ( ). The concentration of fluid decreases for increasing Schmidt number and for Schmidt number (2.62) line of graph become almost straight. In this chapter, MHD flow through a horizontal channel containing porous medium placed under an inclined magnetic field have been observed as follows- Velocity profile decreases for increasing value of angle of inclination with respect to magnetic field. The graph shows the decreasing velocity pattern foe angle (30, 20, 10, 40) and then increases for angle (50). Velocity profile decreases for increasing depth (h) with respect to the magnetic field. Velocity of fluid start with different values and decreases sharply then meet at the same value of velocity. Velocity profile consistently decreases and all lines of graph meet for same velocity at same point for increasing value of permeability with respect to the magnetic field. Velocity profile increases sharply with respect to layer distance for different magnetic field Temperature profile first decreases then coincide at same value then again increases shows resonant character for different inclination of magnetic field. Temperature profile shows resonant character for different Brinkman number with respect to layer distance.ThesisItem Open Access ON CERTAIN TRANSFORMATION FORMULAE FOR BASIC HYPERGEOMETRIC FUNCTIONS AND BASIC INFINITE PRODUCTS”(DEPARTMENT OF MATHEMATICS AND STATISTICS SAM HIGGINBOTTOM UNIVERSITY OF AGRUCULTURE, TECHNOLOGY AND SCIENCES, ALLAHABAD-211007 INDIA 2019) SHRIVASTAV, AKASH KUMAR; HERBERT, Dr. NEERA A.Abstract: In the present work, Certaion transformation formulae for Basic Hypergeometric Functions and Bibasic series are established by using Bailey‟s Transform. Certaion transformation formulae for Basic Hypergeometric Functions are established by using some WP Bailey‟s Pairs in well known theorems. The results involving the Multi summation Identity in terms of ratio of infinite products are obtained by using m-Dissection. The results involving two Basic Bilateral Hypergeometric Functions and ratio of infinite products are derived by using some known results , similarly the results involving three Basic Bilateral Hypergeometric Functions and ratio of infinite products are derived by using some known results.ThesisItem Open Access Magnatohydrodynamic flow of fluids different channels.(DEPARTMENT OF MATHEMATICS AND STATISTICS SAM HIGGINBOTTOM UNIVERSITY OF AGRICULTURE, TECHNOLOGY AND SCIENCES ALLAHABAD-211007, INDIA 2018, 2018) Srivastava, Monika; Khare, Dr. RajeevThesisItem Open Access Magnatohydrodynamic flow of fluids different channels.(Department of Mathematics and Statistics Sam Higginbottom University of Agriculture, Technology and Sciences Allahabad-211007 (U. P.) India, 2018) SRIVASTAVA, Monika; Khare, Dr. RajeevThesisItem Open Access FLOW OF VISCOUS FLUIDS IN VARIOUS CHANNELS PLACED IN A MAGNETIC FIELD(DEPARTMENT OF MATHEMATICS AND STATISTICS SAM HIGGINBOTTOM UNIVERSITY OF AGRICULTURE, TECHNOLOGY AND SCIENCES ALLAHABAD-211007, INDIA 2018, 2018) HANVEY, RISHAB RICHARD; Paul, Prof. (Dr.) AjitThesisItem Open Access A STUDY OF FUZZY TRANSPORTATION PROBLEM (FTP) THROUGH FUZZY MINIMUM DEMAND SUPPLY METHOD(DEPARTMENT OF MATHEMATICS & STATISTICS SAM HIGGINBOTTOM INSTITUTE OF AGRICULTURE, TECHNOLOGY & SCIENCES ALLAHABAD SAM HIGGINBOTTOM INSTITUTE OF AGRICULTURE, TECHNOLOGY & SCIENCES ALLAHABAD-211007 INDIA 2016, 2016) MATHUR, NIRBHAY; PAUL, Dr. AJITThesisItem Open Access A Study on p-Maps through Right Transversals(Sam Higginbottom Institute of Agriculture, Technology & Sciences (SHIATS), 2015) Kumar, Punish; Paul, AjitThe thesis has been divided into five chapters. The first chapter is the introduction of the whole thesis. The second chapter is devoted to review of the literature in which we have tried to show the researches in group theory. In the third chapter namely, methodology, we have defined concepts which are necessary tools to make the thesis self contained. This is divided into four sections. First section contain information about loops, c-groupoid and c-homomorphism. In second section, we have talked about some general topology. Third section is devoted to categories and functors. In fourth section we have introduced homotopy. Fourth chapter namely, result and discussion, is the main chapter of the thesis. This chapter is divided into four sections. First section of it is devoted to the study of p-maps and results based on them. The following are some significant results of this part: Proposition (4.1.2.6): LetG be a group with identity e and p :GG be a pmap . The subset p(G) p(g) : gG of G is a subgroup of G . Proposition (4.1.2.9): Let G be a group with identity e and p :GG be a pmap . Let S be a subset defined in proposition (4.1.2.7). Define a binary operation on S by -1 x y p(xy) xy for all x, yS . Then (S, ) is a right loop. Proposition (4.1.2.13): Let G be a group with identity e and p :GG be a pmap . Let S be a right transversal to p(G) in G . Let us define : S p(G)S by 1 x h p(xh) xh . Then is a right action of p(G) on S . Proposition (4.1.3.1): Let G be a group with identity e and p :GG be a pmap . Let S be a right transversal with identity to p(G) in G . If pmap satisfies the condition 1 2 1 2 p(g g ) p(g p(g )) , then the right loop (S, ) is a group. ii Proposition (4.1.3.11): Let G be a group with identity e . Then the total number of distinct pmaps in G is the total number of distinct factorizations of G as HS where H is a subgroup of G and S is a right transversal (with identity e ) of H in G . Theorem (4.1.3.15): Let G be a group with identity e and p be a pmap . Then G be an extension of the subgroup p(G) with a right transversal S to p(G) in G . In section second of it, we have described another map, namely p-tilda map and then proved some results: Proposition (4.2.2.6): Let G be a group with identity e and p :GG be a p -map . Then the set H {gG: p(g) e} is a subgroup of G . Proposition (4.2.2.9) [67]: Let G be a group with identity e and p :GG be a p -map . Let 1 2 g , g G . Let S be a subset defined in proposition (4.2.2.7) . Define a binary operation on S by 1 2 1 2 p(g ) p(g ) p(p(g )p(g )) for all 1 2 p(g ),p(g )S . Then (S, ) is a right loop. Proposition (4.2.3.1): Let G be a group with identity e and p :GG be a p -map . Let S be a right transversal with identity to H in G . If p -map satisfies the condition 1 2 1 2 p(g p(g )) p(g )p(g ) for all 1 2 g , g G , then the right loop (S, ) is a group. Theorem (4.2.4.3): Let G be a group with identity e and p :GG be a p -map . Let H and S be as defined in proposition (4.2.2.6) and (4.2.2.7) respectively. Then G be an extension of the subgroup H with a right transversal S to H in G . In section third of it, we have introduced the concept of H-group and H-transversal and proved the following important results: Proposition (4.3.2.14): Let 0 (X, x ) and 0 (Y, y ) be two pointed topological spaces. Then 0 X { : : I X is a loop based at x } is an H-group with continuous multiplication . Similarly 0 Y { : : I Y is a loop based at y } is an Hiii group with continuous multiplication . Let f : (Y, y0 )(X, x0 ) is a continuous map. Then (Y,) is an H-subgroup together with an H-map ( , ) ( , ) f Y X . Theorem (4.3.3.2): Let (G,) be an H-group with base point identity element e of the group G . Let p be an H-transversal in an H-group (G,) . Then there is a canonical H-group structure on p(G) with respect to which the inclusion ( ) i p G G is an H-subgroup of (G,) . In section four of it, we have introduced another H-transversal for an H-group and proved the following important result: Theorem (4.4.2.2): Let (G,) be an H-group with base point identity element e of the group G . Let p be an H-transversal in an H-group (G,) . Then p(G) is an Hgroup with respect to the operation defined as follows 1 2 1 2 ( p(g ), p(g )) ( p )( p(g ), p(g )) for all 1 2 g , g G . In the chapter fifth, namely summary and conclusion, we have summaries the whole thesis, by mentioning that the following four are the main results of the thesis. Theorem (4.1.3.15): Let G be a group with identity e and p be a pmap . Then G be an extension of the subgroup p(G) with a right transversal S to p(G) in G . Theorem (4.2.4.3): Let G be a group with identity e and p :GG be a p -map . Let H and S be as defined in proposition (4.2.2.6) and (4.2.2.7) respectively. Then G be an extension of the subgroup H with a right transversal S to H in G . Theorem (4.3.3.2): Let (G,) be an H-group with base point identity element e of the group G . Let p be an H-transversal in an H-group (G,) . Then there is a canonical H-group structure on p(G) with respect to which the inclusion ( ) i p G G is an H-subgroup of (G,) . Theorem (4.4.2.2): Let (G,) be an H-group with base point identity element e of the group G . Let p be an H-transversal in an H-group (G,) . Then p(G) is an Hiv group with respect to the operation defined as follows 1 ( p(g ), p(g2)) ( p )( p(g1), p(g2)) for all 1 2 g , g G . In the end of chapter five, we have shown some importance and application of group theory in different areas. Hence, we have worked upto some extent in the direction of the existing problem of classifying all finite groups by determining all groups G (up to isomorphism) with a fixed subgroup H as a normal subgroup such that the quotient group G/ H is also a given group K . By making p and p to be continuous, we have also worked on Hgroup and H-transversal.ThesisItem Open Access Heat and Mass Transfer Studies on Unsteady Convective Flow over a Vertical Plate(Sam Higginbottom Institute of Agriculture, Technology & Sciences (SHIATS), 2016) NARENDRA BABU, N. V.; Paul, Ajit; Murali, G.Objectives: 1. To study Heat and Mass Transfer effects on MHD flow porous medium. 2. To study the Chemical reaction effects on unsteady MHD flow porous medium. 3. To evolution of a new mathematical model governing these flows based on the results obtained in the study. 4. The present problems have lot of Industrial and Engineering applications. 5. Discussing various Magnetohydrodynamics flow problems and corrspondind methodology.ThesisItem Open Access Some Homogeneous and Inhomogeneous Models of Universe with Reference to Alternative Theories of Gravitation(Sam Higginbottom Institute of Agriculture, Technology & Sciences (SHIATS), 2015) DUBEY, ANAND SHANKAR; KHARE, Dr. RAJEEV KUMAR; PRADHAN, Dr. ANIRUDHObjectives of the Research Work: 1. To discuss the following Bianchi cosmological models – Type–I cosmological model with respect to scalar tensor theory of gravitation. Type–II string cosmological model in Saez–Ballester theory of gravitation. Type–V cosmological model with perfect fluid and heat flow in Saez – Ballester theory of gravitation with deceleration parameter. 2. To evolve a model for accelerating bulk viscous Friedman Robertson Walker universe with respect to scale covariant theory of gravitation.