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
\nRTM 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.<\/p>\n
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Unit 1:
\nDynamical 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.<\/p>\n
Unit 2:
\nNonlinear sinks. Stability. Liapunov function. Gradient systems. Gradients and inner products.<\/p>\n
Unit 3:
\nLimit 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.<\/p>\n
\u00a0Unit 4:
\nAsymptotic 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.<\/p>\n
Text Book:
\nDifferential equations, dynamical systems & linear algebra: M.W. Hirsch & S. Smale, Academic Press, 1975.<\/p>\n
Reference Book:
\nDynamical systems: V.I. Arnold, Springer Verlag, 1992.<\/p>\n
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Unit 1:
\nFirst 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.<\/p>\n
Unit 2:
\nClassification of second order partial differential equations. Potential theory and elliptic differential equations (sections 2.1-2.5).<\/p>\n
Unit 3:
\nThe diffusion equation and parabolic differential equations (sections 3.1-3.4).<\/p>\n
Unit 4:
\nThe Wave equation (sections 4.1, 4.2, 4.4, 4.8, 4.9)<\/p>\n
Text Book:
\nPartial Differential Equations: Phoolan Prasad and Renuka Ravindran; New Age International (P) Limited.<\/p>\n
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Unit 1:
\nSimple enclosure methods, Secant method, Newton\u2019s 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\u2019s method for non- linear systems.<\/p>\n
Unit 2:
\nPolynomial interpolation theory, Newton\u2019s divided differences, finite difference and table oriented interpolation formulas. Forward-differences. Hermite interpolation.<\/p>\n
Unit 3: The Weierstrass theorem and Taylor\u2019s theorem. The minimax approximation problem, the least square approximation problem, orthogonal polynomial, economisation of Taylor series, minimax approximation.<\/p>\n
Unit 4:
\nThe trapezoidal rule and Simpson\u2019s rule, Newton- Cotes integration formulas.<\/p>\n
Text book:
\nAn Introduction to Numerical Analysis by K. E. Atkinson, Johan Wiley and sons, Inc.<\/p>\n
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Unit 1:
\nStress 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.<\/p>\n
Unit 2:
\nNature 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\u2019s theorem, The magnetic body force. Ferraro\u2019s Law of Isorotation.<\/p>\n
Unit 3:
\nDynamical similarity, Buckingham Theorem. Renold number. Prandtl\u2019s boundary layer, Boundary layer equation in two dimensions, Blasius solutions, Boundary layer thickness, Displacement thickness. Karman integral conditions, Separation of boundary layer flow.<\/p>\n
Unit 4:
\nTurbulence: 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.<\/p>\n
Text Books:
\n1. Text book of Fluid Dynamics: F. Chorlton; CBS Publishers, Delhi 1985.
\n2. Fluid Mechanics: Joseph Spurk; Springer.
\n3. Turbulence by J.O. Hinze, 2nd edition, Mc Graw-Hill, chapter 1 sections 1.1 to 1.7
\n4. Fluid Mechanics by M.D. Raisinghania, S. Chand and Company, Delhi.<\/p>\n
Reference Books:
\n1. An Introduction to fluid Mechanics: G.K. Batchelor; Foundation Books, New Delhi, 1994.
\n2. Boundary Layer Theory: H. Schichting; Mc Graw Hill Book Company, New York 1971.<\/p>\n
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Unit 1:
\nStatic 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.<\/p>\n
Unit 2:
\nCosmological principle, Hubble law, Weyl\u2019s 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\u2019s constant, Matter Dominated era.<\/p>\n
Unit 3:
\nFriedman 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.<\/p>\n
Unit 4:
\nSteady state cosmology, Distance measure in cosmology, Comoving distance, Apparent luminosity and luminosity distance, Angular diameter and Lookback time, Galaxy count<\/p>\n
Text Books:
\n1. Relativity, Thermodynamics and Cosmology: Richard C. Tolman, Oxford Press
\n2. Gravitation and Cosmology : Principles and Applications of the General Theory of Relativity by Steven Weinberg.<\/p>\n
References Books:
\n1. The Classical Theory of Fields, By Landau I.D. and Lifshitz E.M., Pub. Pergamon Press (1978).
\n2. Lecture on General Relativity , Sonu Nilu Publication (2004) by T M Karade, G S Khadekar and Maya S Bendre
\n3. The Theory of Relativity Moller C, Pub. Oxford University Press (1982).
\n4. Introduction to theory of relativity, Rosser W.G.V., ELBS (1972).
\n5. Relativity Special, General and Cosmology, Rindler W., Pub. Oxford University Press (2003).
\n6. Relativity: The General Theory, Synge J.L., North Holland Pub. Comp. (1971).<\/p>\n
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Unit 1:
\nSimplicial mappings. Chain mappings. Barycentric Subdivision. The Brouwer Degree. The fundamental theorem of algebra.<\/p>\n
Unit 2:
\nNo retraction theorem and Brouwer fixed point theorem. Mappings into spheres. Relative homology groups. The exact homology sequence. Homomorphisms of exact sequences<\/p>\n
Unit 3:
\nThe excision theorem. The Mayer-Vietoris sequence. Eilenberg-Steenrod axioms for homology theory. Relative homotopy theory. Cohomology groups. Relations between chain and cochain groups.<\/p>\n
Unit 4:
\nSimplicial and chain mappings. The cohomology product. The cap product. Exact sequences in cohomology theory. Relations between homology and cohomology groups.<\/p>\n
Text Book:
\nTopology : J.G. Hocking and G.S. Young : Addison Wesley, 1961<\/p>\n
Reference Books :
\n1. Topology : J.R.Munkres, Prentice Hall, Second Edition, 2000
\n2. Basic Concepts of Algebraic Topology : Fred H.Croom , Springer Verlag 1978.<\/p>\n
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Unit 1 :
\nDuality in non-linear programming \u2013 Weak duality theorem. Wolfe\u2019s duality theorem. Strict converse duality theorem. The Hanson \u2013 Huard strict converse duality theorem. Unbounded dual theorem. Duality in quadratic and linear programming.<\/p>\n
Unit 2 :
\nQuasi 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.<\/p>\n
Unit 3 :
\nOptimality and duality for generalized convex and concave functions \u2013 Sufficient optimality theorem. Generalized Kuhn \u2013 Tucker sufficient optimality theorem. Generalized Fritz John stationary point necessary optimality theorem, Kuhn-Tucker necessary optimality conditions under the weak constraint qualifications.<\/p>\n
Unit 4 :
\nDuality. Optimality and duality in the presence of nonlinear equality constraints \u2013 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.<\/p>\n
Text Book :
\nO.L. Mangasarian, Non- linear programming. Mc Graw Hill, New York.<\/p>\n
Reference Book :
\nMokhtar S. Bazaraa and C.M.Shetty, Non- linear programming, Theory and Algorithms, Wiley, New York.<\/p>\n
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Unit 1:
\nTensor product of modules. Exact sequences \u2013 Projective, Injective, and Flat Modules.<\/p>\n
Unit 2:
\nNoetherian Rings and Affine Algebraic sets. Radicals and Affine varieties. Integral Extensions and Hilbert\u2019s Nullstellensatz. Localisation. The prime spectrum of a ring.<\/p>\n
Unit 3:
\nArtinian 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).<\/p>\n
Unit 4:
\nLinear Actions and Modules over Group Rings. Wedderburn\u2019s theorem and some consequences. Character Theory and orthogonality Relations.<\/p>\n
Text Book :
\nAbstract Algebra: David S. Dummit & Richard M. Foote (Second Edition) John Wiley & Sons Inc.<\/p>\n
NOTE*: Candidates can choose any one paper from Core elective<\/p>\n
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Unit 1:
\nMathematical 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.<\/p>\n
Unit 2:
\nLattice: Lattice as partially ordered sets, their properties, lattices as algebraic system, sub lattices, and some special lattices eg. Complete, complemented and distributive lattices.<\/p>\n
Unit 3:
\nBoolean algebra and Logic Circuits: Boolean algebra, basic operations, Boolean functions, De-Morgan\u2019s 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.<\/p>\n
Unit 4:
\nGraph Theory: Basic terminology, simple graph, multigraph, degree of a vertex, types of a graph, sub graphs of isomorphic graphs, matrix representation of graphs, Euler\u2019s theorem on the existence of Eulerian path & circuits, directed graph, weighted graphs, strong connectivity, chromatic number.<\/p>\n
Text Book:
\nDiscrete Mathematical structures with applications to computer science by J.P. Tremblay and R. Manohar, McGraw-Hill book company,1997.<\/p>\n
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Unit 1:
\nInteger programming.<\/p>\n
Unit 2:
\nQueuing theory and sequencing.<\/p>\n
Unit 3:
\nNon- Linear programming- one and multi- Variable unconstrained optimization, Kuhn-Tucker conditions for constrained optimization.<\/p>\n
Unit 4:
\nQuadratic programming, fraction programming and goal programming.<\/p>\n
Text book:
\nKanti-Swarup P.K. Gupta and Man Mohan: Operations Research, Sultan Chand and Sons New Delhi.<\/p>\n
Reference books :
\n1. G. Hadley: Linear programming, Narosa Publishing House 1995.
\n2. G. J. Lieberman: Introduction to operations Research (Sixth Edition) Mc Graw Hill International Edition 1995.
\n3. H.A Taha: Operations Research \u2013 In Introduction, Macmillan publishing company inc, New York<\/p>\n","protected":false},"excerpt":{"rendered":"
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