Department of Mathematics

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About Us

Department of Mathematics was established in the year 1963 under the headship of Dr B.S. Fadnis. The post

 graduate students in Mathematics are encouraged to carry out research work leading to the award of Ph.d. of Rashtrasant Tukadoji Maharaj, Nagpur University. The department has presently nine sanctioned teaching staff comprising of two Professors, three Readers and Four Lecturers. However, due to the superannuation of some faculty members, the present strength  is reduced to seven. Research projects have been awarded to the faculty carrying out research work in various fields in Mathematics. Since, then the department has been active updating the curriculum and encouraging research. Various fields of Mathematics in which research is being carried out which includes Relativity, Cosmology, Fluid Mechanics, Thermoelasticity, Operational research, Magneto-hydrodynamics, Topology, Algebra. Recently the department also conducted M.Phil. Course. Faculty of the department actively involved in various academic activities Nationally and Internationally.

 

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Departmental Profile:

Name of the Head of the Department

Dr. Kishor C. Deshmukh

 

Professor & Head

Post Graduate Teaching Department of

Mathematics, R.T.M.Nagpur University Campus,

Nagpur – 440 033.

Ph. 0712-2500575,  Mob. 09665062708

Email:      kcdeshmukh2000@rediffmail.com 

Courses:  M.Sc. Mathematics and Ph.D. by Research

 

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Revised Syllabus For M. Sc. ( 2 year ) Course in Mathematics
 

Total Maximum Marks :1000
Maximum Marks per Paper : 100
Hourly Periods Allotted per week per paper:05

 

 

Courses:     

M.Sc. I:

(Compulsory Papers)

 

 

1. Paper-I  Algebra
2. Paper-II Real Analysis
3. Paper- III  Topology
4 Paper- IV Complex Analysis
(Optional Papers)
Opt any one
5. Paper- V Integral Equations and Integral Transforms
6. Paper- VI Fuzzy Mathematics
7. Paper-–VII  Advanced Discrete mathematics
8. Paper- VIII  Numerical Analysis
9 Paper- IX  Classical Mechanics and Differential geometry

 

M.Sc. II:

No. of Papers

(Compulsory Papers)

 

 

1. Paper-I  Dynamical Systems
2. Paper-II Functional Analysis
(Optional Papers)
Opt any three
3. Paper- III  Fluid Dynamics
4. Paper- IV General Relativity and Cosmology
5. Paper- V Algebraic Topology
6. Paper- VI Operations Research
7 Paper-–VII  Advanced Algebra
8. Paper- VIII  Differentiable Manifolds and Riemannian Geometry
9. Paper- IX  Coding Theory
10. Paper-X Mechanics of Solids
11. Paper- XI Harmonic Analysis
12. Paper- XII  Non-Linear Programming
13. Paper- XIII Banach Algebras
14. Paper- XIV  Fundamentals of Applied Functional Analysis

         

 

 

Paperwise Syllabus

 

M.Sc. Part I

 

Paper I

Algebra

 

Unit I:

Group Theory: Normal Series. Solvable groups. Nilpotent groups. Simplicity of An.

Structure Theorems of Groups: Direct Products. Finitely generated abelian groups. Invariants of a finite abelian group. Sylow theorems. Groups of order  and pq.

Module Theory : Completely reducible modules. Free modules. Uniform modules.

 

Unit II:

Field Extensions: Irreducible  polynomials and Eisenstein criterion. Adjunction of roots.  Algebraic extensions. Algebraically closed fields. Splitting fields. Normal extensions. Multiple roots. Finite fields. Separable extensions.

 

Unit III: Galois Theory and applications: Automorphism groups and fixed fields. Fundamental theorem of Galois theory. Fundamental theorem of algebra. Roots of unity and cyclotomic polynomials. Cyclic extensions. Polynomials solvable by radicals. Ruler and compass constructions.

 

Unit IV: Noetherian rings and affine algebraic sets. Radicals and Affine varieties. Integral extensions and Hilbert’s nullstellensatz.

 

Unit V: Artinian Rings. Discrete Valuation Rings. Dedekind domains

 

Text Book:

 

1) Basic Abstract Algebra : Bhattacharya, Jain and  Nagpal , (Second edition),

  Cambridge University Press.

 

2) Abstract Algebra : David S. Dummit and Richard M. Foote, (Second edition), John Wiley & Sons.

 

Reference Books :

1.Topic in Algebra: I. N. Herstein, Second Edition, John Wiley.

2. Basic Algebraic Geometry: Shafaravich Vol I

3. Algebraic Geometry: Robin Hartshone, GTM Springer

4. Algebraic Curves: William Fulton; W.A. Benjamin, 1969

 

 

 

Paper-II

Real Analysis

 

Unit I :  Sequences and series of Functions: Uniform convergence. Uniform convergence and continuity. Uniform convergence and integration. Uniform convergence and differentiation. Equi-continuous  families of functions. The Stone- Weierstrass theorem.

 

Unit II : Functions of Several Variables: Differentiation. The Contraction Principle. The   Inverse  Function Theorem. The Implicit Function Theorem . The Rank Theorem. Partition of Unity.

 

Unit III  :  Differentiable manifolds and submanifolds : The space of tangent vectors at a point of Rn. Another definition of Ta( Rn). Vector fields on open subsets of Rn.

Topological manifolds. Differentiable manifolds. Real projective space. Grassman manifolds. Differentiable functions and mappings. Rank of a mapping. Immersions. Sub manifolds.

 

Unit IV :

Outer measure. Measurable sets and Lebesgue measure. A non-measurable set. Measurable functions. Littlewood’s three principles.

The Riemann integral. The Lebesgue integral of a bounded function over a set of finite measure. The integral of a non-negative function. The general Lebesgue integral. Convergence in measure;

 

Unit V :

Differentiation of monotone functions. Functions of bounded variation. Differentiation of an integral. Absolute continuity. Convex functions. Lp-spaces. Holder and Minkowski inequality. Riesz-Fischer theorem.

 

Text books :

 

1. Principles of Mathematical Analysis (Third Edition): Walter Rudin

    Mc GRAW – HILL Book Company.

 

2. An Introduction to Differentiable Manifolds and Riemannian Geometry : W. Boothby,    

    Academic Press, 1975.

 

3. Real Analysis ( Third edition ) : H.L.Royden  Macmilan Publishing Co.

 

Reference books :

1.Methods of Real Analysis: R.R. Goldberg ,  John Wiley and sons.

2.Calculus of Several Variables: C Goffman ,  Harper and Row and John Weather Hill.

 

 

Paper-III

 

Topology

 

 

Unit I: Equipotent sets, cardinal numbers. Open sets and limit points. Closed sets and closure operators. Neighbourhoods, bases and relative topologies.

 

Unit II: Connected sets and components. Compact and countably compact spaces. Continuous functions, homeomorphisms. To- and T1- spaces, T2- spaces & sequences. Axioms of countability. Separability .

 

Unit III: Regular and normal spaces. Completely regular spaces. Metric spaces as topological spaces. Topological properties.

 

Unit IV: Finite products. Product invariant properties. Metric products. Nets and filters. Tichonov topology. Tichonov theorem.

 

Unit V: Quotient topology. Urysohn’s lemma. Urysohn’s metrization theorem. Paracompact spaces.

 

Text books:

 

(1) W.J. Pervin, Foundations of General Topology, Academic press, 1964.

 

(2) K.D. Joshi, Introduction to general Topology, Wiley Eastern Ltd. (for Nets and filters) 

     1983.

 

Reference books:

 

(1) Topology :  J.R. Munkres, (second edition), Prentice Hall of India, 2002.

(2) Introduction to topology & modern analysis:  G.F. Simmons,  Mc Graw Hill 1963.

(3) General Topology : J.L. Kelley, Van Nostrand, 1995.

 

 

  

Paper-IV

 

 Complex Analysis

 

Unit I : Power series representation of analytic functions. Zeros of an analytic function. The index of a closed curve. Cauchy’s theorem and integral formula. Counting zeros.The open mapping theorem. Goursat’s theorem.  

 

Unit II : Classification of singularities. Residues. The argument principle. The maximum principle,  Schwarz’s lemma. Convex function and Hadamards three circles theorem, Phragmen-Lindelof theorem.

 

Unit III : Spaces of analytic functions. The Riemann mapping theorem. The Weierstrass factorization theorem. Factorisation of Sine function.Gamma function, Riemann zeta function.

 

Unit IV : Runge’s theorem. Mittag – Leffler’s theorem, Schwarz reflection principle, Analytic continuation along a path, Monodromy theorem.

Basic properties of Harmonic functions. Harmonic functions on a disk Green’s function.

 

Unit V: Entire Functions: Jensen’s formula. Genus and order of an entire function. Hadamard factorization theorem.

Range of an Analytic function: Bloch’s theorem. The little Picard theorem. Schottky’s theorem. The Great Picard theorem.

 

Text Book :

 

1) Functions of one complex variable: John B. Conway. Second edition, Springer

      international Student Edition.

 

Reference Book:

 

1) Complex Analysis: L.V. Ahlfors. Mc-Graw Hill, 1966.

 

 

 

Paper-V

 

Integral Equations and Integral Transforms

 

(Optional)

 

 

Unit I : Preliminary concepts of integral equations. Some problems which give rise to integral equations. Conversion of ordinary differential equations into integral equations. Classification of linear integral equations. Integro- differential equations. Fredholm equations. Degenerate kernels. Hermitian and symmetric kernels. The Hilbert- Schmidt theorem. Hermitization and symmetrization of kernels. Solutions of integral equations with Green’s function type kernels.

 

Unit II :Types of Voltera equations. Resolvent kernel of Voltera equations, Convolution type kernels. Some miscellaneous types of Voltera equations. Non-linear Voltera equations.

 

Unit III : Fourier integral equations. Laplace integral equations. Hilbert transform. Finite Hilbert transforms. Miscellaneous integral transforms. Approximate methods of solutions for linear integral equations. Approximate evaluation of Eigen values and Eigen functions.

 

Unit IV : Fourier integral theorem. Fourier transform. Fourier cosine and sine transform. The convolution integral. Multiple Fourier transform. Solution of partial differential equation by means of Fourier transform. Calculations of the Laplace transform of some elementary functions. Laplace transform of derivatives. The convolution of two functions. Inverse formula for the Laplace transform. Solutions of ordinary differential equations by Laplace transform.

 

Unit V : Finite Fourier transform. Finite Sturm – Liouville transforms. Generalised finite Fourier transform. Finite Hankel transform. Finite Legendre transform. Finite Mellin transform.

 

Text Books :

 

1.  Integral Equations: A short course: LI. G Chambers International text book 

    company Ltd, 1976.

2. The use of integral transforms: I N. Sneddon, Tata Mc Graw Hill Publishing

   Company Ltd.

 

References Books :

1. Modern Mathematics For Engineers: Edwin F Beckenbach, Second series, 

   Mc Graw  Hill Book Company.

2. Operational Mathematics: R.V. Churchill,  3rd Edition , Mc Graw Hill Student  

   Edition

 

 

Paper VI

 

Fuzzy Mathematics

 

( Optional )

 

Unit I : Crisp Sets. Fuzzy Sets. Properties of a- cuts. Representation of Fuzzy sets. Extension principle for fuzzy sets. Operations on Fuzzy sets. Fuzzy Complements. Fuzzy intersections : t-Norms. Fuzzy unions : t-conorms. Combinations of operations. Aggregation operations.

 

Unit II : Fuzzy Numbers. Linguistic variables. Arithmetic Operations on intervals. Arithmetic operations on Fuzzy Numbers. Fuzzy equations. Crisp versus Fuzzy relations. Projections and Cylindric extensions. Binary Fuzzy relations. Binary relations on a single set. Fuzzy equivalence relations. Fuzzy compatibility relaltions. Fuzzy ordering relations. Fuzzy morphisms. Sup-i-compositions of Fuzzy relations. Inf-wi compositions  of Fuzzy relations.

 

Unit III : Fuzzy relation equations: General discussion. Problem partitioning. Solution method. Fuzzy Relation Equations based on sup-I-compositions. Fuzzy Relation Equations based on Inf-wi compositions. Approximate solutions. Use of Neural networks. Possibility theory : Fuzzy measures. Evidence theory. Possibility theory. Fuzzy sets and Possibility theory. Possibility theory versus Probability theory.

 

Unit IV : Fuzzy Logic: Classical logic. Multivalued logics. Fuzzy Propositions. Fuzzy quantifiers. Linguistic Hedges. Inference from conditional Fuzzy  propositions. Inference from conditional and qualified propositions. Inference from quantified propositions. Uncertainty based Information. Information and Uncertainty. Nonspecificity of Crisp sets. Nonspecificity of Fuzzy sets. Fuzziness of Fuzzy sets. Uncertainty in Evidence theory. Uncertainty measures. Principles of uncertainty.

 

Unit V : Constructing Fuzzy sets and operations on Fuzzy sets. Approximate reasoning. Fuzzy decision making : Methods of construction. Direct methods with one expert. Direct methods with multiple experts. Indirect methods with one expert. Indirect methods with multiple experts. Constructions from sample data. Fuzzy expert systems. Fuzzy implications. Selection of Fuzzy implications. Multiconditional Approximate reasoning. Role of Fuzzy relation Equations. Interval valued Approximate reasoning. Individual decision making. Multiperson decision making. Multicriterion decision making. Multistage decision making. Fuzzy ranking methods. Fuzzy linear programming.

 

Text Book:

 

Fuzzy Sets and Fuzzy Logic, theory and applications. George J. Klir and Bo Yuan , Prentice Hall, India.

 

 

Paper- VII

Advanced Discrete Mathematics

(Optional )

 

Unit I : Formal logic – statements. Symbolic representation and Tautologies. Quantifier predicates and validity. Propositional logic. Semigroups and monoids – Definition and examples of semigroups and monoids, homomorphism of semigroups and monoids, congruence relation and quotient semigroups, subsemigroups and submonoids, direct products, basic homomorphism theorem.

 

Unit II : Lattices – Lattices as partially ordered sets, their properties, Lattices as algebraic systems. Sublattices. Direct products and homomorphisms. Some special lattices, e.g. complete, complemented and distributive lattices. Boolean algebras -  Boolean algebras as lattices, various Boolean identities. The switching algebra example. Subalgebras, direct products and homomorphisms. Join-irreducible elements. Atoms and minterms, Boolean forms and their equivalence, Minterm Boolean forms, sum and products, canonical forms, minimization of Boolean functions, Applications of Boolean Algebra to switching theory (using AND, OR and NOT gates). The Karnaugh map method.

 

Unit III : Graph theory -  Definition of undirected graphs, Paths, circuits, cycles and subgraphs. Induced subgraphs. Degree of a vertex. Connectivity. Planar  graphs and their properties. Trees.   Euler’s formula  for connected planar graphs. Complete graphs. Kuratowski’s theorem  and its use. Spanning  trees. Cut – sets, fundamental cut-sets and cycles. Minimal  spanning trees and Kruskal’s algorithm. Matrix  representation of graphs. Euler’s  theorem on the existence of Eulerian paths and circuits. Directed  graphs. Indegree  and outdegree of a vertex. Weighted undirected graphs. Dijkstra’s algorithm. Strong  connectivity and Warshall’s algorithm. Directed trees. Search trees. Tree traversals.

 

Unit IV : Introductory Computability Theory – Finite state machines and their transition table diagrams. Equivalence of finite state machines. Reduced machines. Homomorphism. Finite automata. Acceptors. Non-deterministic finite automata and equivalence of its power to that of deterministic finite automata. Moore and Mealy machines. Turing machine and partial recursive functions.

 

Unit V : Grammars and Languages :  Phrase-Structure Grammars, Rewriting rules, Derivations. Sentential forms. Language generated by a grammar. Regular, context- free and context-sensitive grammars and languages. Regular sets, regular expressions and the pumping lemma. Kleene’s theorem. Notions of syntax analysis. Polish notations. Conversion of infix expressions to Polish notations. The reverse Polish notations.

 

Text Book :

 

Discrete Mathematical structures with Applications to Computer Science: J.P. Tremblay and R.Manohar,  McGraw Hill Book Co. 1997. 

 

 

 

Paper-VIII

 

Numerical Analysis

 

(Optional)

 

Unit I : Root finding for Non- Linear Equations : Simple enclosure methods, Secant method, Newton’s method, general theory for one point iteration methods. Aitken extrapolation for linearly convergent sequences, Error tests, Numerical evaluation of multiple roots, roots of polynomials, Mullers method, Non-linear systems of equations, Newton’s method for non- linear systems.

 

Unit II : Interpolation Theory : Polynomial interpolation theory, Newton’s divided differences, finite difference and table oriented interpolation formulas. Forward-differences. Hermite interpolation. Approximation of Functions: The Weierstrass theorem and Taylor’s theorem. The minimax approximation problem, the least square approximation problem, orthogonal polynomial, economisation of Taylor series, minimax approximation.

 

Unit III : Numerical Integration :The trapezoidal rule and Simpson’s rule, Newton- Cotes integration formulas. Gaussian quadrature. Asymptotic Error formulas and their applications. Automatic Numerical integration. Singular integrals. Numerical differentiation.

 

Unit IV: Numerical Methods for ordinary differential equations. Existence, uniqueness and stability theory. Euler’s method. Multistep methods. The mid point method. The trapezoidal method. A low order predictor corrector algorithm.

 

Unit V: Numerical solution of systems of linear equations. Gaussian elimination method. Pivoting in Gaussian elimination method. Variants of Gaussian elimination. Error analysis. The residual correction method. Iteration method. Error prediction and acceleration. Numerical solution of Poisson equation.

 

Text  book :

 

An Introduction to Numerical Analysis : Kendal E Atkinson, Johan Wiley and  sons, Inc.   

 

 

 

Paper-IX

 

Classical Mechanics and Differential Geometry

(Optional)

 

Unit I : Variational Principle and Lagrange’s equations; Hamilton’s Principle, some techniques of calculus of variations, Derivation of Lagrange equations from Hamilton’s principle, Extension of principle to nonholonomic systems. Legendre transformations and the Hamilton equations of motion. Cyclic coordinates and conservation theorems. Routh’s procedure and oscillations about steady motion, The Hamiltonian formulation of relativistic mechanics, The Principle of least action.

 

Unit II : Canonical Transformations: The equations of canonical transformation. Examples of canonical transformation. The symplectic approach to canonical transformations. Poisson brackets and other canonical invariants. Equations of motion. Infinitesimal canonical transformations and conservation theorems in the Poisson bracket formulation, the angular momentum, Poisson bracket relations, symmetry groups of mechanical systems.

 

     Unit III : Definition of surface. Curves on a surface. Surfaces of revolution.  

     Helicoids. Metric. Direction coefficients. Families of curves. Isometric  

     correspondence. Intrinsic properties. Geodesics. Canonical geodesic equations.    

     Normal property of geodesics. Existence theorems. Geodesic parallels.

 

     Unit IV : Geodesic curvature. Gauss Bonnet theorem. Gaussian curvature. Surfaces of  

      constant curvature. Conformal mapping. Geodesic mapping. Second fundamental

      form. Principal curvatures. Lines of curvature. Developables. Developables

      associated with space curves. Developables associated with curves on surfaces.

      Minimal surfaces and ruled surfaces.

  

     Unit V : Differential geometry of surfaces in the large: Introduction. Compact

     surfaces whose points are umbilics. Hilbets lemma. Compact surfaces of constant

     Gaussian or mean curvature. Complete surfaces. Characterisation of complete

     surfaces. Hilbert’s theorem. Conjugate points on geodesics. Intrinsically defined

     surfaces. Triangulation. Two dimensional Riemannian manifolds. The problem of

     metrization. The problem of continuation. Problems of embedding and rigidity.  

 

Text Books:

 

1. Classical mechanics:  H. Goldstein, Narosa publishing house, New Delhi

 

 

 

     2. An introduction to Differential Geometry: T.J. Wilmore; Oxford University Press

 

                                 

M.Sc. Part II

 

        Paper I

     

          Dynamical Systems

 

Unit I : Review of topology in Rn. New norms for old. Exponentials of operators. Homogeneous linear systems. A non-homogeneous equation. Higher order systems. The primary decomposition. The S+N decomposition.

 

Unit II : Nilpotent canonical forms. Jordan and real canonical forms. Canonical forms and differential equations. Higher order linear equations on function spaces. Sinks and sources. Hyperbolic flows. Generic properties of operators. Significance of genericity.  Dynamical system and vector fields. The flow of a differential equation.

 

Unit III : Nonlinear sinks. Stability. Liapunov function. Gradient systems. Gradients and inner products. RLC circuits. Analysis of the circuit equations. Van der Pol’s equation. Hopf bifurcation.

 

Unit IV : Limit sets, local sections and flow boxes, monotone sequences in planar dynamical systems. The Poincare- Bendixson theorem, Applications of Poincare-Bendixson theorem; one species, predator and prey, competing species.

 

Unit V :   Asymptotic stability of closed orbits , discrete dynamical systems  Stability of closed orbits, existence, uniqueness & continuity for non autonomous differential equations, differentiability of the flow of autonomous equations, persistence of equilibria, persistence of  closed orbits, structural stability .

 

Text Book:

 

Differential equations, dynamical systems & linear algebra: M.W. Hirsch & S. Smale, Academic Press, 1975.

 

     Reference Book :

Dynamical systems : V.I. Arnold, Springer Verlag, 1992.

 

 

 

 

Paper- II

 

Functional Analysis

 

Unit I : Normed spaces, Banach spaces, Further properties of normed spaces. Finite dimensional normed spaces and subspaces. Compactness and finite dimension. Bounded and continuous linear operators. Linear functionals. Normed spaces of operators. Dual spaces.

 

Unit II : Inner product space. Hilbert   space. Further  properties of inner product spaces. Orthogonal  complements and direct sums. Orthonormal sets and sequences. Total orthonormal sets and sequences. Representation  of functionals on Hilbert spaces. Hilbert  adjoint operators, self adjoint, unitary and normal operators.

 

Unit III : Hahn-Banach Theorem, Hahn-Banach Theorem for complex vector spaces and normed spaces. Adjoint operator on a normed space, Reflexive spaces, Category theorem, Uniform Boundedness Theorem. Strong  and weak convergence, convergence of sequences of operators and functionals, Open Mapping Theorem, Closed linear operators, Closed Graph Theorem.

 

Unit IV : Spectral theory of bounded linear operators: Basic concepts, further                  properties of resolvent and spectrum, use of complex analysis in spectral theory ; Compact linear operators on normed spaces, further properties of compact linear operators, spectral properties of compact linear operators on normed spaces.

 

Unit V : Spectral properties of bounded self-adjoint linear operators, further spectral properties of bonded self-adjoint linear operators, positive operators, square roots of positive operators, projection operators, Definition of ‘spectral family’ and statement of ‘Spectral Representation Theorem.’

 

Text Book:

 

Introductory Functional analysis with applications by E. Kreyszig, John Wiley &     

      Sons.

  

     Reference book:

  

     Introduction to Functional Analysis, by A.E. Taylor & D.C. Lay, John Wiley & Sons.

 

  

Paper III

 

Fluid Dynamics

 

( Optional )

 

Unit I : Kinematics of fluid in motion: Real fluids and ideal fluids. Velocity of a fluid at a point. Stream lines and path lines. Steady and unsteady flows. Velocity potential. Velocity vector. Local and particle rate of change. Equation of continuity. Acceleration of  a fluid. Condition  at a rigid boundary. General  analysis of fluid motion. Euler’s equation of motion. Bernoulli’s equation. Worked examples. Discussion of the case of steady motion  under conservative body forces. Some further aspects of vortex motion.

 

Unit II :  Some three dimensional flows: Sources, sinks and doublets. Images in a rigid  infinite plane. Images  in solid spheres. Axi-symmetric flows. Stokes’ stream function. The complex  potential for two-dimensional irrotational, incompressible flow. Complex velocity potential for standard two dimensional flow. Uniform stream. Line source and line sink. Line  doublets. Line  vortices. Two dimensional  image systems. The Milne-Thomson Circle Theorem. Some  applications of circle theorem. Extension of circle theorem. The theorem of Blasius.

 

Unit III : Elements of Thermodynamics: The equations of state of a substance, the first law of thermodynamics, internal energy of a gas, functions of state, entropy, Maxwell’s thermodynamic relation, Isothermal Adiabatic and Isentropic processes.

Gas Dynamics : Compressibility effects in real fluids, the elements of wave motion. One dimensional wave equation, wave  equation in two and three dimensions, spherical waves, progressive and stationary waves. The speed of sound in a gas, equation of  motion of a gas. Sonic, subsonic,  supersonic flows; isentropic gas flow. Reservoir  discharge through a channel of varying section, investigation of maximum mass flow through a nozzle, shock waves, formation of shock waves, elementary analysis of normal shock waves.

 

Unit IV : Viscous Flows : Stress components in a real fluid, relation between  Cartesian components of stress translation motion of fluid elements, the rate of strain quadric and principal stresses,  some further properties of the rate of the strain quadric, stress analysis in fluid motion, relation between stress and rate of strain, the coefficient of viscosity and laminar flow, the Navier-Stokes equations of motion of a viscous fluid, some solvable problems in viscous flow, diffusion of vorticity, energy dissipation due to viscosity, steady flow past a fixed sphere.

 

Magnetohydrodynamics: Nature of magnetohydrodynamics, Maxwell electromagnetic field equations; Motion at rest, Motion in medium , Equation of motion of conducting fluid, Rate of flow of charge, Simplification of electromagnetic

      field equation. Magnetic Reynold number; Alfven’s theorem,  The magnetic body   

      force. Ferraro’s Law of Isorotation.

 

Unit V : Dynamical similarity, Buckingham  Theorem. Renold number. Prandtl’s boundary layer, Boundary layer  equation in two dimensions, Blasius solutions, Boundary layer thickness, Displacement thickness. Karman integral conditions, Separation of boundary layer flow.

Stability and onset of turbulence. Reynold’s equations, turbulence shear flow near a wall. 

   

    Text Books :

 

1. Text book of   Fluid Dynamics: F. Chorlton,CBS Publishers, Delhi 1985.

      2. Fluid Mechanics: Joseph Spurk,  Springer.

 

Reference Books :

 

1 An Introduction to fluid Mechanics,:G.K. Batchelor, Foundation Books, New Delhi,   1994.

2. Boundary Layer Theory: H. Schichting, Mc Graw  Hill Book Company, New York

    1971.

3   Fluid Mechanics: M.D. Raisinghania, S. Chand and Company , Delhi.

4. Boundary layers: A. D. Young,  AIAA Edvcation series, Washington DC, 1981.

5. Foundation of Fluid Mechanics: S. W Yuan, Prentice Hall of India Private Limited,   Delhi, 1976.

 

 

 

Paper-IV

 

General Relativity and Cosmology

(Optional)

 

Unit I : Review of tensor Calculus, Bianchi identity and Einstein Tensor, Principle of equivalence, and covariance, geodesic principle, Newtonian approximation of relativistic equations of motion, Einstein field equation and its Newtonian approximation, Energy momentum tensor of perfect fluid .

 

Unit II: Schwarzchild exterior solution and its isotropic form, planetary orbits, Experimental tests of general relativity : (i) Advance of  perihelion of planet (ii) Bending of light rays (iii) Gravitational red shift in spectral lines. Schwarzchild’s interior solution.

 

Unit III: Einstein field equations with cosmological term, static cosmological Model of Einstein and de-Sitter and their derivation, properties and comparison with actual universe, Cosmological principle, Hubble laws, Weyl’s postulate, Robertson Walker Metric, further properties, Hubble and deceleration parameters, Red shift.

 

Unit IV: Matter dominated era of the universe, Angular size, Apparent luminosity, Galaxy count, Friedmann Model, Fundamental equation of  dynamical cosmology, critical density, flat, closed and open universe, age of the universe.

 

Unit V:  The Physics of Early Universes: Cosmic background radiation, The creation of matter and photons, A brief history of the early Universe, problems with standard big bang model: (i) flatness problem (ii) horizon problem, inflationary scenario.

 

 

Text Books :

 

1.Introduction to General Relativity : Ronald Adler, Maurice Bazin and Menamem     

    Schiffer.(Second Edition) Pub. McGraw-Hill, KOGAKUSHA Ltd. Articles 9.1,   

    9.2,9.3,9.4,14.1.

 

2.Relativity and Thermodynamics and Cosmology:  Richard C. Tolman, Pub.  

   Oxford at the clarendon Press . Articles : 133 to 145.

 

3. Gravitation and Cosmology : Principles and Applications of the General Theory of     

    Relativity. By Steven Weinberg.  Articles : Part Five: Cosmology: 1,2,8. Cosmology :   

    The Standard Model : 3, page No. 409 to 414, 415to 417, 459 to 468, 481 to 490.

 

 

4. General Relativity and Cosmology: A First Course,  By Tai L Chow, Wuerz   

    Publishing Ltd. Winnipeg, Canada

    Article:  Elementary Cosmology: page no. 205 to 210, Cosmic Dynamics: page no. 211    

    to 223, The Physics of Early Universes: page no. 233 to 252 and 263 to 267.

 

 

 References Books :

  1. Lectures on General Relativity By T. M. Karade, G.S. Khadekar & Maya S. Bendre., Pub. Sonu Nilu.

  2. Relativity and Thermodynamics and Cosmology, By Richard C. Tolman, Pub. Oxford at the clarendon Press.

  3. The Classical Theory of Fields, By Landau I.D. and Lifshitz E.M., Pub. Pergamon Press(1978).

  4. The Theory of Relativity Moller C, Plub. Oxford University Press(1982).

  5. Introduction to theory of relativity , Rosser W.G.V., ELBS(1972).

  6. Relativity Special, General and Cosmology, Rindler W., Pub. Oxford University Press(2003).

  7. Relativity: The General Theory, Synge J.L., North Holland Pub. Comp.(1971).

  8. Tensor Calculus, Synge J.L. and Schild A., University of Toronto Press(1949).

  9. Riemann Geometry and Tensor Calculus, Weber J.,Interscience Publishers, (1961).

 

Paper V

 

Algebraic Topology

 

(Optional)

 

Unit I: The Elements of Homotopy theory: Introduction. Homotopic mappings. Essential and inessential mappings. Homotopically equivalent spaces. Fundamental group. Knots and related embedding problems. Higher homotopy groups. Covering spaces.

 

Unit II: Polytopes and triangulated spaces: En as a vector space over E1.Barycentric coordinates. Geometrical complexes and polytopes. Barycentric subdivision. Simplicial mappings and simplicial approximation theorem. Abstract simplicial complexes. Embedding theorem for polytopes.

 

Unit III: Simplicial homology theory: Introduction. Oriented complexes. Incidence numbers. Chains, cycles and groups. Decomposition theorem for abelian groups. Betti numbers and torsion coefficients. Zero dimensional homology groups. Universal coefficients. Euler Poincare formula. Universal coefficients.

 

Unit IV:  Simplicial mappings. Chain mappings. Barycentric Subdivision. The Brouwer Degree. The fundamental theorem of algebra. No retraction theorem and Brouwer fixed point theorem. Mappings into spheres.

 

Unit V: Relative homology groups. The exact homology sequence. Homomorphisms of exact sequences. The excision theorem. The Mayer-Vietoris sequence. Eilenberg-Steenrod axioms for homology theory. Relative homotopy theory. Cohomology groups. Relations between chain and cochain groups. Simplicial and chain mappings. The cohomology product. The cap product. Exact sequences in cohomology theory. Relations between homology and cohomology groups.

 

Text Book : 

 

Topology : J.G. Hocking and G.S. Young : Addison Wesley, 1961

 

Reference Books :

 

 1. Topology : J.R.Munkres, Prentice Hall, Second Edition, 2000

 2. Basic Concepts of Algebraic Topology : Fred H.Croom , Springer Verlag 1978.

                                  

 

 

Paper-VI

 

Operations Research

 

(Optional)

 

Unit I :   Linear programming: Simplex method. Duality in linear programming.

Unit II :  Transportation problem. Assignment problem.

Unit III : Games and strategies. Network scheduling by PERT/CPM.

Unit IV : Integer programming. Sequencing problem. Queueing theory.

Unit V :  Non-Linear programming. Non-Linear programming methods.

 

Text book:

 

Operations Research : Kanti-Swarup P.K. Gupta and Man Mohan: Sultan Chand and   

Sons New Delhi. 12th edition.

 

Reference  books :

 

1.G. Hadley: Linear programming, Narosa Publishing House1995.

 

2.F.S. Hillier and G.J.Lieberman: Introduction to operations Research (Sixth Edition)      Mc Graw Hill International Edition 1995.

 

3.H.A Taha: Operations Research – In Introduction, Macmillan publishing company inc, New York

 

Paper- VII

 

Advanced Algebra

 

(Optional)

 

 

Unit I : Localisation. The prime spectrum of a ring

 

Unit II: Tensor Product of Modules. Exact sequences-Projective, injective, and flat modules.

 

Unit III :Introduction to Homological Algebra and Group Cohomology : Introduction to Homological Algebra- Ext and Tor. The Cohomology of Groups. Crossed homomorphisms and H1(G,A). Group Extensions, Factor Sets and H2(G,A).

 

Unit IV: Representation Theory and Character Theory : Linear Actions and Modules over Group Rings. Wedderburn’s theorem and some consequences. Character Theory and orthogonality Relations.

 

Unit V: Examples and Applications of Character Theory: Characters of Groups of Small Order. Theorems of Burnside and Hall. Introduction to the theory of induced characters.

 

Text Book :

 

Abstract Algebra: David S. Dummit & Richard M. Foote (Second Edition) John Wiley & Sons Inc.

Reference Book

Commutative Algebra: N.S. Gopalkrishnan;Oxonian Press

 

 

Paper- VIII

 

Differentiable Manifolds and Riemannian Geometry

 

(Optional )

 

 

Unit I :  Topological groups, Lie groups, Examples of Lie groups, product of two Lie groups,    Lie subgroups, The action of a Lie group on a manifold, transformation groups,  The action of a discrete group on a manifold. Covering manifolds.

 

Unit II :   The tangent space at a point of a manifold, Vector fields, One- parameter and local one-parameter groups acting on a manifold, Existence theorem for ordinary differential equations.

 

Unit III : Some examples of one-parameter groups acting on a manifold, One- parameter subgroups of Lie groups, The Lie algebra of vector fields on a manifold, Homogeneous spaces.

 

Unit IV : Covectors on manifolds, covector fields and mappings, Bilinear forms. The Riemannian metric, Riemannian manifolds as metric spaces, Some applications of the Partition of unity, Tensor fields, Multiplication of tensors on a vector space, Exterior multiplication of alternating tensors, The exterior algebra on manifolds, Exterior differentiation.

 

Unit V :  Differentiation of vector fields along curves in Rn, the geometry of space curves, curvature of plane curves, Differentiation of vector fields on submanifolds of Rn,  Formulas for covariant derivatives, Differentiation of vector fields, Differentiation on Riemannian manifolds, Constant vector fields and parallel displacement, The curvature tensor Geodesic curves on Riemannian manifolds, Symmetric Riemannian manifolds,

 

 

Text book :

An Introduction to Differentiable Manifolds and Riemannian Geometry : W. Boothby,   

Academic Press, 1975.

 

 

Paper-IX

 

Coding Theory

 

(Optional)

 

Unit I : Algebra, Krawtchouk polynomials, combinatorial theory, probability theory.

 Shannon’s theorem,  coding gain. Block codes, linear codes, Hamming codes.

 

Unit II :  Majority logic decoding, weight innumerators, the Lee metric. Hadamard codes and generalization, the binary Golay code, the ternary Golay code, constructing  codes from other codes. Reed-Muller codes, Kerdock codes. The Gilbert bound, Upper bounds, the linear programming bounds.

 

Unit III :  Definition of cyclic codes, generator matrix and check polynomial, zeros of a cyclic code, Idempotent of a cyclic code, other representations of cyclic codes, BCH codes, decoding BCH codes, Reed-Solomon codes, quadratic residue codes, binary cyclic codes of length 2n (n odd), generalized Reed-Muler codes.

 

Unit IV : Lloyd’s theorem, the characteristic polynomial of a code, uniformly packed codes, examples of uniformly packed codes, non existence theorems.

Quaternary codes, binary codes derived from codes over  Z4. Galois rings over Z4. Cyclic codes over Z4.

 

Unit V : Goppa codes, the minimum distance of Goppa codes, Asymptotic behavior of Goppa codes, decoding Goppa codes, generalized BCH codes.

 algebraic curves, divisors,  differentials on a curve, Riemann Roch theorem, codes from Algebraic curves, some geometric codes, improvement of the Gilbert-Varshamov bound.

 

Text Book:

 

Introduction to coding theory: J.H. van Lint, 3rd edition, GTM Vol 86, Springer Verlag 1999   

 

Paper- X

Mechanics of solids

(Optional)

 

Unit I : Analysis of Strain-Affine transformations. Infinite simal affine deformation. Geometrical interpretation of the components of strain. Strain quadric of Cauchy. Principal strains and invariants. General infinite simal deformation. Saint-venant’s equations of compatibility. Finite deformations. Analysis of stress-Stress tensor. Equations of equilibrium. Transformation of coordinates. Stress quadric of Cauchy. Principal stress and invariants. Maximum normal and shear stresses.

 

Unit II : Equations of elasticity-Generalized Hooke’s law. Homogeneous isotropic media, elasticity moduli for isotropic media. Equilibrium and dynamic equations for an isotropic elastic solid. Strain energy function and its connection with Hook’s law. Uniqueness of solution. Beltrami-Michell compatibility equations. Saint-venant’s principle.

 

Unit III : Torsion-Torsion of cylindrical bars. Tortional rigidity. Torsion and stress functions. Lines of shearing stress. Simple problems related to circle, ellipse and equilateral triangle.

 

Unit IV : Two dimensional problems: Plane stress. Generalized plane stress. Airy stress function. General solution of Biharmonic equation. Stresses and displacements in terms of complex potentials, Simple problems. Stress function appropriate to problems of plane stress. Problems of semi-infinite solids with displacements of stresses prescribed on the plane boundary. Waves-Propagation of waves in an isotropic elastic solid medium . Waves of dilatation and distortion. Plane waves. Elastic surface waves such as Rayleigh and Love waves.

 

Unit V : Variational methods-Theorems of minimum potential energy. Theorems of minimum  complementary energy. Reciprocal  theorem of Betti and Rayleigh. Deflection of elastic string. Central line of a beam and  elastic membrane. Torsion of cylinders. Variational problem related to biharmonic equation. Solution  of Euler’s equation by Ritz, Galerkin and Kontorovich methods.

 

Text Book :

 

Mathematical theory of elasticity: I. S. Sokolnikoff, Tata McGraw-Hill  Publishing Company Ltd, New Delhi, 1977.

 

Reference book: 

 

1. A Treatise on the mathematical theory of elasticity: A.E. Love, Cambridge University Press, London, 1963.

2.Foundations of Solid Mechanics: Y. C. Fund, Prentice Hall, New Delhi, 1965.

 

 

Paper- XI

 Harmonic Analysis

(Optional)

 

Unit I : Fourier series and Integrals : Definition and easy results, The Fourier transform, Convolution, Approximate Identities, Fejer’s theorem,Unicity theorem, Parseval relation, Fourier – Stieltjes coefficients, The classical kernels, Summability : metric theorems, Point wise summability.

 

Unit II : The Fourier Integral : Introduction, Kernels on R, The Plancherel theorem, Another convergence theorem, the Poisson summation formula, Bochner’s theorem, The continuity theorem.

 

Unit III : Discrete and Compact Groups : Characters of discrete groups, characters of compact groups, Bochner’s theorem, Examples, Minkowski theorem, Measures on infinite product spaces, Continuity of semi norms.

 

Unit IV : Hardy Spaces : Hp( T ), Invariant subspaces, factoring, proof of the theorem of F. Riesz and M. Riesz, Theorems of Szego and Beurling, Structure of inner functions,  Theorem of  Hardy and Littlewood , Hilbert’s inequality, Hardy spaces on the line.

 

Unit V : Conjugate functions :  Conjugate series and functions, Theorems of Kolmogorov and Zygmund, Theorems of Riesz and Zygmund, The conjugate function as a singular integral, The Hilbert transform, Maximal functions, Rademacher functions, absolute Fourier multipliers.

 

Text Book

 

Harmonic Analysis ( Second Edition ) : Henry Helson, Hindustan Book Agency, New Delhi, 1995

 

 

 

Paper- XII

Non-linear Programming

(Optional)

 

Unit I : The non-linear programming problem  and its fundamental ingredients. Linear inequalities  and the   theorem of the alternative. The optimality   criteria of linear programming. Tucker’s  lemma and existence theorems. Theorems of the alternative Convex sets – Separation theorems. Convex and concave functions -  basic properties and some fundamental theorems for convex functions. Generalised Gordan theorem. Bohnenblust – Karlin – Shapley theorem.

Unit II : Saddle point optimality criteria without differentiability – The minimization and the local minimization problems and some basic results. Sufficient  optimality theorem. Fritz John Saddle point necessary optimality theorem. Slater’s and Karlin’s constraint qualifications  and their equivalence. The strict  constraint qualification. Kuhn – Tucker saddle point optimality theorems. Differentiable concave and convex functions -  Some basic properties. Twice differentiable  convex and concave functions. Theorems in cases of strict convexity  and concavity of functions.

Unit III : Optimality criteria with differentiability- Optimality theorems, Fritz John stationary point necessary optimality theorem. The Arrow – Hurwicz – Uzawa  constraint qualification. Kuhn – Tucker  stationary – point necessary optimality theorem.

Duality in non-linear programming –  Weak duality theorem. Wolfe’s  duality theorem. Strict  converse duality theorem. The Hanson – Huard  strict converse duality theorem. Unbounded  dual theorem. Duality  in quadratic and linear programming.

Unit IV : Quasi convex, strictly quasi convex and  strictly quasi concave functions. Karamardian theorem. Global minimum ( maximum ). Pseudo convex and pseudo concave functions. Relationship between pseudo convex functions and strictly quasi convex functions. Differentiable  convex functions and pseudo convex functions.

Unit V :   Optimality and duality for generalized convex and concave functions – Sufficient optimality theorem. Generalized Kuhn – Tucker sufficient optimality theorem. Generalized Fritz John stationary point necessary optimality theorem, Kuhn-Tucker necessary optimality conditions under the weak constraint qualifications. Duality. Optimality and duality in the presence of  nonlinear equality constraints – Sufficient optimality criteria. Minimum principle necessary optimality criteria. Minimum principle necessary optimality theorem. Fritz John and Kuhn-Tucker stationary point necessary optimality criteria. Duality with nonlinear equality constraints.

 

Text Book

 

O.L. Mangasarian, Non- linear programming. Mc Graw Hill, New York.

 

Reference Book

 

Mokhtar S. Bazaraa and C.M.Shetty, Non- linear programming, Theory and Algorithms, Wiley, New York.  

 

 

Paper- XIII

 

Banach Algebras

 

(Optional)

 

Unit I :  Fundamental algebraic concepts, Topological algebras, Normed algebras

 

Unit II : Symmetric algebras, Realisation of a commutative normed algebra in the form of an algebra of functions.

 

Unit III : Homomorphism and isomorphism of commutative algebras, algebra ( or Shilov boundary ), Completely symmetric commutative algebras, Regular algebras, Completely regular commutative algebras.

 

Unit IV: Fundamental concepts and propositions in the theory of representations, Embedding of a symmetric algebra in an algebra of operators, In decomposable functionals and irreducible representations.

 

Unit V : Application to commutative symmetric algebras, Generalised Shur lemma, Some representations of the algebra B ( H ).

 

Text Book :

 

 M.A.Naimark, Normed Algebras, Noordhoff, Groningen, Netherlands, 1972.

 

Reference Book :

 

1) General Theory of Banach  Algebras  : C.E.Rickart, Von Nostrand,  1960,

2) Banach Algebras , Vol. I: T.W.Palmer ; Cambridge University Press, 1994

 

  

 

Paper- XIV

 

Fundamentals of Applied Functional Analysis

 

(Optional)

 

Unit I : Weak convergence, non-linear functionals and generalized curves, Hahn-Banach theorem, support functional of a convex set, Minkowski functional, support mapping, Separation theorem.

 

Unit II : Application to convex programming, generalization to infinite dimensional inequalities, a fundamental result of game theory, minimax theorem, theorem of Farkas, Linear operators and their adjoints, spectral theory of operators.

 

Unit III : Spectral theory of compact operators, operators on separable Hilbert spaces, L2 spaces over Hilbert spaces.

 

Unit IV : Definition and general properties of semi-groups, generation of semi-groups, semi-groups over Hilbert spaces, dissipative semi-groups, compact semi-groups, analytic

( holomorphic ) semi-groups, extensions.

 

Unit V : Differential equations, Cauchy problem, Controllability, state reduction observability, boundary input, evolution equations. Linear  quadratic regulator problem. The same  problem with infinite time interval. Hard  constraints. Final  value control. Time optimal  control problems.

 

 

Text Book :

 

Applied Functional Analysis : A.V.Balakrishnan, Springer Verlag.

 

Reference Books :

 

1. Linear Operators, Vol. I and II : N.Dunford and J.T. Schwartz, Interscience, 1958,

    1963.

2. Functional Analysis : K.Yosida, Springer Verlag, 1974.

 

 

 

 

M.Phil. Course:

No. of Papers (Any two)

  1. Differentiable Manifolds and Lie Groups

  2. Boundary Value Problems

  3. Algebraic Topology

 

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Admissions:

(Eligibility, General Procedure)

For admission to M.Sc. I course in Mathematics the candidate must have passed B.Sc. examination with Mathematics is one of the subject. Candidates are admitted on merit basis i.e. aggregate in B.Sc. exam and class wise (B.Sc. I, II, III) marks obtained in the subject in which admission being sought.

 

Teaching Faculty:

 

  Name Designation
1. Dr. K. C. Deshmukh    Professor
2 Dr. R. V. Saraykar   Professor (CAS)
3 Dr. N. W. Khobragade  Professor
4 Dr. G. S. Khadekar Professor (CAS)
5 Dr. M. S. Borkar  Reader
6 Dr. S. H. Ghate Reader (CAS)
7 Dr. (Mrs) S. A. Meshram Reader (CAS)

 

                                                          

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Faculty Profile:

 

(i) Dr. K. C. Deshmukh   

Dr. K. C. Deshmukh is working as Professor and Head, Department of Mathematics and is having 26 years of

 teaching experience in undergraduate level in Engg. College and postgraduate level in University.

Dr.  Deshmukh published 72 research papers in National and International journal of having good impact factor. Presently 10 students are working under his guidance for their Ph.d. work. He guided 12 students for Ph.d. work. Six students have also completed M.Phil. Dissertation under his supervision. He is carrying out his research work in the field of thermoelasticity having the applications in Mechanical and Aerospace Engg..  Dr. Deshmukh have completed two major research project successfully sanctioned by U.G.C., New Delhi.

Contact no. 9665062708

Email: kcdeshmukh2000@rediffmail.com

 

(ii) Dr. R. V. Saraykar      

Dr. R. V. Saraykar is working as Professor in Department of Mathematics and is having 35 years of teaching experience in postgraduate level in University Dr.  Saraykar published 29 research papers in National and International journal of having good impact factor. Presently 06 students are working under his guidance for their Ph.d. work. He guided 04 students for Ph.d. work. He is carrying out his research work in the field of Theory of Relativity, Fluid Dynamics. He visited the University of Natal at Durban, South Africa.

Contact no. 9923111526

Email: r_saraykar@rediffmail.com

 

(iii)  Dr. G. S. Khadekar

Dr. G. S. Khadekar is working as Professor in Department of Mathematics and is having 22 years of teaching experience in undergraduate level in Engg. College as well as postgraduate level in University. Dr.  Khadekar published 70 research papers in National and International journal of having good impact factor. Presently 10 students are working under his guidance for their Ph.d. work. He guided 10 students for Ph.d. work. Six students have also completed M.Phil. Dissertation under his supervision. He is carrying out his research work in the field of Theory of Relativity and Cosmology, Higher Dimensional space-time.  Dr Khadekar has completed one minor research project successfully sanctioned by U.G.C. New Delhi. He visited Turkey, France, Germany, Italy, U.S.A. and Iran.

Contact no. 9011323123

Email: gkkhadekar@hotmail.com

 

(iv) Dr. N. W. Khobragade

Dr. N. W. Khobragade is working as Professor in the Department of Mathematics and is having 28 years of teaching experience in postgraduate level in University. Dr.  Khobragade published 140 research papers in National and International journal. Presently 06 students are working under his guidance for their Ph.d. work. He guided 09 students for Ph.d. work. Thirteen students have also completed M.Phil. Dissertation under his supervision. He is carrying out his research work in the field of Thermoelasticity and Operational Research.  Dr. Khobragade has completed one major research project successfully sanctioned by U.G.C., New Delhi.

Contact no. 9730776567

Email: khobragade_nw@rediffmail.com

 

(v)  Dr. M. S. Borkar

Dr. M. S. Borkar is working as a Reader in Mathematics and is having 25 years of teaching experience in undergraduate level in Institute of Science College and postgraduate level in University.

Dr.  Borkar published 19 research papers in National and International journal of having good impact factor. Presently 03 students are working under his guidance for their Ph.d. work. He guided 01 student for Ph.d. work. Four  students have also completed M.Phil. Dissertation under his supervision. He is carrying out his research work in the field of Relativity, Operational Research.  Dr. Borkar has completed one minor research project successfully sanctioned by U.G.C. New Delhi.

Contact no. 7798155871

Email: borkar.mukund@rediffmail.com

 

(vi) Dr. S. H. Ghate

Dr. S. H. Ghate is working as a Reader in the Department of Mathematics and is having 31 years of teaching experience in postgraduate level in University. Dr. Ghate published 07 research papers in National and International journal of having good impact factor. Presently 02 students are working under his guidance for their Ph.D. work. He guided 01 student for Ph.D. work. He is carrying out his research work in the field of Algebra, Relativity, Economics and Fluid Mechanics. He visited the country U.S.A. and Thailand.

Contact no. 0712-2570690 , 8806664849

Email: sureshghate@gmail.com

 

 

(vii) Dr. (Mrs) S. A. Meshram

 

Dr. (Mrs) S. A. Meshram is working as a Reader in Department of Mathematics and is having 25 years of teaching experience postgraduate level in University.

Dr.  Meshram published 12 research papers in National and International journal. Presently 04 students are working under her guidance for their Ph.D. work. She is carrying out her research work in the field of Thermoelasticity, Solid Mechanics.  Dr  Meshram has completed one minor research project successfully sanctioned by U.G.C. New Delhi.

 

Contact no. 9890009224

 

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Project Sanctioned by UGC:

 

Name of Teacher

Title of the project

 

Duration

Funding

Agency

Amount

Dr. N. W. Khobragade

Study of direct and inverse thermoelastic problems

 

01.04.2007 to 31.03.2010

U.G.C.

Rs.5,62,100=00

Dr. K. C. Deshmukh

Critical study of some Thermoelastic problems

01.04.2007 to  31.03.2010

 

U.G.C.

Rs.4,31,600=00       

Dr. M. S. Borkar

Study of some cosmological  Models

01.04.2007 to 31.03.2009

 

U.G.C.

Rs.1,00,000=00      

Dr .Mrs. S. A. Meshram

Study of heat conduction Problem in some solids and                                        

01.04.2007 to 31.03.2009

U.G.C.

Rs.1,00,000=00       

 

Projects completed:

 

  1. “Some Aspects of Thermoelastic Problems” sanctioned by University Grant Commission, New Delhi, India, for three years duration i.e. 1st July 2003 to 30th June 2006. of Rs. 4.5 Lakhs ; The Principal Investigator Dr K. C. Deshmukh 

  2. Minor Research Project on: “Wesson Theory of Gravitation” sanctioned by University   Grant Commission, New Delhi, India.

    The above Minor research project of Rs. 30,000/- (Thirty thousand only) was successfully completed in the year 2004 by Dr. G.S.Khadekar 

  3. Minor Research Project on “Structure Consideration of Top Space” International Grant from Mahan Mathematical research center, Kerman, Iran.

    This minor research project in collaborating with Department of Mathematics,  Kerman University, Kerman, IRAN is successfully completed. By Dr. G. S. Khadekar Duration one year (2003-2004) Amount: $ 500. 

     

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Ongoing Projects

University Grants Commission New Delhi sanctioned THREE Major Research Project of Rs 20,88, 600 /- for three years duration i.e. 1-4-2011 to 31 03-2013. The details are as under :

Name of Principal Investigator Title of the Project Sanction Amount

  1. Dr K.C.Deshmukh , “ Study of some direct and indirect heat conduction problems in solids and their thermal behavior’’ Rs 5,28,000 /-
  2. Dr M.S.Borkar “ The Study of cosmological modelsin Einetein and in Rosen theory of gravitation ’’ Rs 8,08,800 /-
  3. Dr N.W.Khobragade “ Critical study of thermoelastic problems on different solids ’’ Rs 7,51,800 /-.

SPECIAL ASSISTANCE PROGRAMME

University Grants Commission , New Delhi has sanctioned the Assistance to the Department of Mathematics, R.T.M. Nagpur University, Nagpur at the level of DRS- I for Five years i.e. 01-04-2011 to 31-03-2016 under the Special Assistance Programme ( SAP ).
The proposal of the Department of Mathematics was examined by the Expert Committee on 15th Feb , 2011 at UGC office , New Delhi and after a very careful and critical in-depth examination of the academic achievement of the Department, the Expert Committee recommended the Department for consideration by the Commission to support the Department at the level of DRS-I.
This Special Assistance Programme is intended through constants efforts to raise the quality of teaching and research in different disciplines in Sciences in general and in Mathematics particular. The essence and primary aim of the scheme is combination of teaching and research to encourage group research efforts in pursuit of excellence.
Special Assistance Programme have identified for major two thrust areas :
1. Boundary value problems in Thermoelasticity
2. General theory of Relativity and Cosmology.
Dr K.C.Deshmukh and Dr R.V.Saraykar are nominated as the Co-ordinator and Deputy Co-ordinator ,respectively by UGC for functioning of the SAP programme.
The financial assistance approved for implementing the present phase at the level of DRS-I for a period of Five years i.e. 01-04-2011 to 31-03-2016 is of Rs 23.50 Lakh plus one Project Fellow @ Rs 8000 /- per month. The sanction SAP programme is very prestigious to the Department in particular and University in general. Department of Mathematics is the Fourth Department in the University which has been sanctioned SAP – assistance.

Dr R.D.Giri , Retired Professor of Mathematics , Mumbai University , Mumbai joined as a Emeritus Professor in Department of Mathematics , R.T.M. Nagpur University , Nagpur on 18-06-2011 for Two years.
 

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