RTMNU M.Sc Sem 4 New Syllabus 2020 | RTMNU M.Sc Revised Sem 4 Syllabus 2020 | Nagpur University M.Sc Syllabus Part 2| RTMNU Second Year M.Sc Syllabus

RTM Nagpur University M.Sc Semester New Revised Syllabus is given below for Downloading. The students can Download the respective Syllabus from following given details. Just go through the given links & read the given syllabus carefully. Nagpur University Second Year New Semester Online Detail syllabus given below.

**M. Sc. Mathematics**

**Semester-IV**

**Paper – XVI (Code: 4T1)**

**Dynamical Systems**

Unit 1:

Dynamical systems and vector fields. The fundamental theorem. Existence and uniqueness. Continuity of solutions in initial conditions. On extending solutions. Global solutions. The flow of a differential equation.

Unit 2:

Nonlinear sinks. Stability. Liapunov function. Gradient systems. Gradients and inner products.

Unit 3:

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 4:

Asymptotic stability of closed orbits, discrete dynamical systems. Stability and closed orbits. Non Autonomous equations and differentiability of flows. 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.

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**M. Sc Mathematics**

**Semester-IV**

**Paper – XVII (Code: 4T2)**

**Partial Differential Equations**

Unit 1:

First order partial differential equations in two independent variables and the Cauchy problem. Semilinear and quasi linear equations in two independent variables. First order non linear equations in two independent variables. Complete integral.

Unit 2:

Classification of second order partial differential equations. Potential theory and elliptic differential equations (sections 2.1-2.5).

Unit 3:

The diffusion equation and parabolic differential equations (sections 3.1-3.4).

Unit 4:

The Wave equation (sections 4.1, 4.2, 4.4, 4.8, 4.9)

Text Book:

Partial Differential Equations: Phoolan Prasad and Renuka Ravindran; New Age International (P) Limited.

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**M. Sc Mathematics**

**Semester-IV**

**Paper – XVIII (Code: 4T3)**

**Advance Numerical Methods**

Unit 1:

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 2:

Polynomial interpolation theory, Newton’s divided differences, finite difference and table oriented interpolation formulas. Forward-differences. Hermite interpolation.

Unit 3: 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 4:

The trapezoidal rule and Simpson’s rule, Newton- Cotes integration formulas.

Text book:

An Introduction to Numerical Analysis by K. E. Atkinson, Johan Wiley and sons, Inc.

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**M. Sc Mathematics**

**Semester-IV**

**Core Elective***

**Paper – XIX (Code: 4T4)**

**Fluid Dynamics-II**

Unit 1:

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.

Unit 2:

Nature of magneto-hydrodynamics, 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 3:

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.

Unit 4:

Turbulence: Definition of turbulence and introductory concepts. Equations of motion for turbulent flow. Reynolds Stresses Cylindrical coordinates. Equation for the conservation of a transferable scalar quantity in a turbulent flow. Double correlations between turbulence-velocity components. Change in double velocity correlation with time. Introduction to triple velocity correlations. Features of the double longitudinal and lateral correlations in a homogeneous turbulence. Integral scale of turbulence.

Text Books:

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

2. Fluid Mechanics: Joseph Spurk; Springer.

3. Turbulence by J.O. Hinze, 2nd edition, Mc Graw-Hill, chapter 1 sections 1.1 to 1.7

4. Fluid Mechanics by M.D. Raisinghania, S. Chand and Company, Delhi.

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.

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**M. Sc Mathematics**

**Semester-IV**

**Core Elective**

**Paper – XIX (Code: 4T4)**

**Cosmology**

Unit 1:

Static cosmological models of Einstein and de Sitter and their derivation and its Properties: (i) The geometry of the Universe (ii) Density and pressure (iii) Motion of test particle (iv) Doppler shift (v) comparison with actual universe, Comparison between Einstein and de-Sitter models.

Unit 2:

Cosmological principle, Hubble law, Weyl’s postulate,Derivation of Robertson Walker Metric and its properties, Motion of a particle and light rays in FRW model, Red shift, Deceleration parameter and Hubble’s constant, Matter Dominated era.

Unit 3:

Friedman Model, Fundamental equation of dynamical cosmology, density and pressure of the present universe, Matter dominated era of the universe, critical density, flat, closed and open universe, age of the universe.

Unit 4:

Steady state cosmology, Distance measure in cosmology, Comoving distance, Apparent luminosity and luminosity distance, Angular diameter and Lookback time, Galaxy count

Text Books:

1. Relativity, Thermodynamics and Cosmology: Richard C. Tolman, Oxford Press

2. Gravitation and Cosmology : Principles and Applications of the General Theory of Relativity by Steven Weinberg.

References Books:

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

2. Lecture on General Relativity , Sonu Nilu Publication (2004) by T M Karade, G S Khadekar and Maya S Bendre

3. The Theory of Relativity Moller C, Pub. Oxford University Press (1982).

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

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

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

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**M. Sc Mathematics**

**Semester-IV**

**Core Elective**

**Paper – XIX (Code: 4T4)**

**Algebraic Topology- II**

Unit 1:

Simplicial mappings. Chain mappings. Barycentric Subdivision. The Brouwer Degree. The fundamental theorem of algebra.

Unit 2:

No retraction theorem and Brouwer fixed point theorem. Mappings into spheres. Relative homology groups. The exact homology sequence. Homomorphisms of exact sequences

Unit 3:

The excision theorem. The Mayer-Vietoris sequence. Eilenberg-Steenrod axioms for homology theory. Relative homotopy theory. Cohomology groups. Relations between chain and cochain groups.

Unit 4:

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.

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**M. Sc Mathematics**

**Semester-IV**

**Core Elective**

**Paper – XIX (Code: 4T4)**

**Non-linear Programming-II**

Unit 1 :

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 2 :

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 3 :

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.

Unit 4 :

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.

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**M. Sc Mathematics**

**Semester-IV**

**Core Elective**

**Paper – XIX (Code: 4T4)**

**Advanced Algebra**

Unit 1:

Tensor product of modules. Exact sequences – Projective, Injective, and Flat Modules.

Unit 2:

Noetherian Rings and Affine Algebraic sets. Radicals and Affine varieties. Integral Extensions and Hilbert’s Nullstellensatz. Localisation. The prime spectrum of a ring.

Unit 3:

Artinian Rings . Discrete valuation rings. Dedekind domains. 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 4:

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

Text Book :

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

NOTE*: Candidates can choose any one paper from Core elective

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**M. Sc Mathematics**

**Semester-IV**

**PAPER XX : FOUNDATION (For Students other than Mathematics )**

**Paper – XX (Code: 4T5)**

**MATHEMATICS-II**

**Elementary Discrete Mathematics**

Unit 1:

Mathematical Logic: Introduction, Proposition, compound Proposition, Proposition and truth tables, logical equivalence, algebra of Proposition, conditional Proposition, converse, contra positive & inverse, bi conditional statement, negation of compound statements, tautologies & contradictions, normal forms, logic in proof.

Unit 2:

Lattice: Lattice as partially ordered sets, their properties, lattices as algebraic system, sub lattices, and some special lattices eg. Complete, complemented and distributive lattices.

Unit 3:

Boolean algebra and Logic Circuits: Boolean algebra, basic operations, Boolean functions, De-Morgan’s theorem, logic gate, sum of products and product of sum forms, normal form, expression of Boolean function as a canonical form, simplification of Boolean expression by algebraic method, Boolean expression form logic & switching network.

Unit 4:

Graph Theory: Basic terminology, simple graph, multigraph, degree of a vertex, types of a graph, sub graphs of isomorphic graphs, matrix representation of graphs, Euler’s theorem on the existence of Eulerian path & circuits, directed graph, weighted graphs, strong connectivity, chromatic number.

Text Book:

Discrete Mathematical structures with applications to computer science by J.P. Tremblay and R. Manohar, McGraw-Hill book company,1997.

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**M. Sc Mathematics**

**Semester-IV**

**CORE SUBJECT CENTRIC (Only Students of Mathematics )**

**Paper – XX (Code: 4T5)**

**Operations Research–II**

Unit 1:

Integer programming.

Unit 2:

Queuing theory and sequencing.

Unit 3:

Non- Linear programming- one and multi- Variable unconstrained optimization, Kuhn-Tucker conditions for constrained optimization.

Unit 4:

Quadratic programming, fraction programming and goal programming.

Text book:

Kanti-Swarup P.K. Gupta and Man Mohan: Operations Research, Sultan Chand and Sons New Delhi.

Reference books :

1. G. Hadley: Linear programming, Narosa Publishing House 1995.

2. 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