MSc mathematics syllabus: Science college Jabalpur

                                         Govt. Model Science College (Auto), Jabalpur 

Department of Mathematics & Computer Science

M.Sc. I Semester (Mathematics)

Unit Paper I : Advanced Abstract Algebra MM : 35

I. Another counting Principle, Conjugency relation Normalizer, Class Equation Cauchy's theorems, Sylow's theorems, Double cosets, Second and third part of Sylow's theorems, Application of Sylow's theorems in Finite Groups. 

II. Series of Group : Normal and subnormal series, Composition series, Zassenhaus, Schreir refinement, Jordan-Holder theorem. 

III. Solvable groups and its properties Commutator subgroup, Nilpotent groups and its properties. 

IV. Fields: Extension field, Finite extension, Algebraic element, Algebraic and transcendental extension, Roots of polynomials, Splitting field. 

V. More about roots: Derivative of a polynomial, Simple extension, Primitive element, Separable and inseparable extension, Perfect field, Finite field. 

 

Unit Paper II : Real Analysis MM : 35

I. Definition and existence of Riemann-Stieltjes integral and its Properties, Integration and differentiation, The fundamental theorem of Calculus. 

II. Integration of vector-valued functions, Rectifiable curves. Rearrangements of terms of a series. Riemann's theorem. 

III. Sequences and series of functions, pointwise and uniform convergence, Cauchy criterion for uniform convergence, Weierstrass M-test, Abel's and Dirichlet's tests for uniform convergence, uniform convergence and continuity, uniform convergence and Riemann-Stieltjes integration, uniform convergence and differentiation, Weierstrass approximation theorem, Power series, uniqueness theorem for power series, Abel's and Tauber's theorems. 

IV. Functions of several variables, linear transformations, Derivatives in an open subset of Rn, Chain rule, Partial derivatives, interchange of the order of differentiation, Derivatives of higher orders, Taylor's theorem, Inverse function theorem. 

V. Implicit function theorem, Jacobians, extremum problems with constraints, Lagrange's multiplier method, Differentiation of integrals, Partitions of unity, Differential forms, Stokes theorem. 

Unit Paper III : Topology MM : 35

I. Definition and examples of topological spaces, Basis for a topology, Standard topology, lower limit topology, The order topology, The product topology on X ×Y. Projections, The Subspace topology, Closed sets and limit points, Closure and interior of a set. 

II. Continuous functions, Equivalence with ε-δ condition, Examples of continuous functions, Homeomorphisms, topological property, topological Imbedding, Examples of Homeomorphisms, Rules for Constructing continuous functions, The Pasting lemma, Maps into products, The product topology, Box topology, Projection mapping, comparison of the product topology and the box topology. 

III. The Metric topology, Metrizable space, Standard bounded metric, The spaces $R^n$ and $R^w,$ Euclidean metric, square metric, Metrizability of $R^n$ and $R^w,$ Uniform metric, The sequence lemma, Uniform limit theorem. 

IV. Connected space, Separation, Definition and examples, Cartesian product of connected spaces, Connected sets in the real line, Linear continuum, Intermediate value theorem, Path connectedness. 

V. Compact spaces, Finite product of compact spaces, The Tube Lemma, Finite intersection condition, compact sets in the real line, Maximum and minimum value theorem, Limit point compactness, The Lebesgue number Lemma, Second countable and first countable spaces, Separation Axioms ($T_1,$ $T_2,$ $T_3$ spaces). 

Unit Paper IV : Complex Analysis -I MM : 35

I. Complex integration, Cauchy Goursat theorem, Cauchy integral formula, Higher order derivatives 

II. Morera's theorem, Cauchy’s inequality, Liouville's theorem, The fundamental theorem of algebra, Taylor's theorem.  

III. The maximum modulus principle, Schwartz lemma, Laurent series, Isolated singularities, Meromorphic functions, The argument principle, Rouche's throrem, Inverse function theorem. 

IV. Residues, Cauchy's residue theorem, Evaluation of integrals, Branches of many valued functions with special reference to $arg z,$ $logz$, $z^a$. 

V. Bilinear transformations, their properties and classification, Definitions and examples of conformal mappings. 

Unit Paper V : FUNCTIONAL ANALYSIS MM : 35

I. Convergence, Completeness and Baire's Theorem, Complete metric spaces. Limit & Limit point. Cantors intersection Theorem, Continuous mappings, Uniformly continuous mapping, examples. 

II. Spaces of continuous functions, Euclidean and Unitary spaces, Cauchy, Minkowski and Holders inequalities, Normed linear spaces, Examples and elementary properties, quotient spaces. Equivalence of normsm, 2-norm, supnorm, p-norm, 1≤ p < ∞, Minkowski and Holders inequalities for p-norm, examples. 

III. Banach space and examples, Continuous linear transformations, norm of an operator, Banach spaces B(N, N’). Functionals and their extensions, related Lemma, Hahn- Banach Theorem for normed linear spaces. Conjugates of normed linear spaces. 

IV. The natural embedding of normed linear space in its second conjugate space, Reflexive Banach spaces, open mapping theorem, Closed graph theorem. Conjugate of an operator, Uniform boundedness principle, examples. 

V. Inner product spaces, examples, elementary properties of Inner Product, Parallelogram law, Schwartz inequality and polarization identity, Hilbert Space and examples. orthogonal complements in Hilbert spaces. Orthonormal sets, Bessel's inequality, Gram Schmidt orthonormalization process. Conjugate space $H^*$, Riesz Representation Theorem.

 Govt. Model Science College (Auto), Jabalpur

Department of Mathematics & Computer Science

M.Sc. II Semester (Mathematics)

Unit Paper I : Advanced Abstract Algebra MM : 35

I. The elements of Galois theory: Automorphism of a field, Group of automorphisms of a field, Fixed field, Normal extension, Galois group of a polynomial, Fundamental theorem of Galois theory, Solution of polynomial equations by radicals, Insolvability of the general equation of degree 5 by radicals.

II. Introduction to Modules, Examples, Submodules and Direct sum of submodules, R-homomorphisms and Quotient modules, Finitely generated modules, Cyclic module.

III. Simple modules, Semi-simple modules, Schur's lemma, Free modules, Rank of a module.

IV. Noetherian and Artinian modules, Ascending and Descending chain condition (ace & dcc), Noetherian and Artinian rings, Examples, Hilbert basis theorem.

V. Fundamental Structure theorem of finitely generated modules over a Principal Ideal Domain and its applications to finitely generated abelian groups

Unit Paper II : LEBESGUE MEASURE AND INTEGRATION MM : 35

I. Lebesgue outer measure. Measurable sets. Regularity. Measurable functions. Borel and Lebesgue measurability. Non-measurable sets.

II. Integration of ,Non-negative functions. The General integral. Integration of Series, Reimann and Lebesgue Integrals.

III. The Four derivatives. Functions of Bounded variation. Lebesgue Differentiation Theorem, Differentiation and Integration.

IV. The LP-spaces, Convex functions, Jensen's inequality. Holder and Minkowski inequalities. Completeness of LP.

V. Dual of space when 1 ≤ P ∞, Convergence in Measure, Uniform Convergence and almost Uniform Convergence.

Unit Paper III(A) : Topology MM : 35

I. The Tychonoff Theorem, Completely regular spaces, The Stone-Cech compactification.

II. Metrization Theorems and Paracompactness, Local finiteness, The Nagata Smirnov Metrization Theorem (Nec & suff), Paracompactness

III. Complete Metric spaces & function spaces, Complete metric spaces, A spacefilling curve, Compactness in metric spaces Ascoli's theorem, Pointwise and Compact convergence.

IV. The Fundamental Group and covering spaces, Homotopy of paths, the Fundamental Group, Homomorphism, covering spaces, The Fundamental Group of the circle

V. Nets & filters : Topology and Convergence of Nets, Hausdorffness & nets, Compactness & nets, Filters and their convergence, Ultra Filters and compactness

Unit Paper IV : Complex Analysis -II MM : 35

I. Weierstrass factorization theorem, Gamma function and its properties, Riemann zeta function, Riemann’s functional equation

II. Rung’s theorem, Mittage-Leffler’s theorem, Schwartz Reflection principle, Analytic Continuation, Analytic Continuation along a path..

III. Monodromy theorem and its consequences, Harmonic function, Harmonic functions on a disk, Harnack’s inequality and theorem.

IV. Dirichlet’s problem, Green’s function, Jenson’s formula, Poisson-Jenson Formula.

V. Order of an entire function, Hadamard’s three circle theorems, Hadamard’s factorization theorem, Bloch’s theorem, The Little Picard theorem.

Unit Paper V: ADVANCED DISCRETE MATHEMATICS MM : 35

I. Algebraic Structures : Introduction, Algebraic Systems : Examples and General Properties : Definition and examples, Some Simple Algebraic Systems and General properties, Homomorphism and isomorphism, congruence relation, Semigroups and Monoids : Definitions and Examples, Homomorphism of Semigroups and Monoids.

II. Lattices: Lattices as Partially Ordered Sets : Definition and Examples, Principle of duality, Some Properties of Lattices, Lattices as Algebraic Systems, Sublattices, Direct product, and Homomorphism.

III. Some special Lattices, e.g. Complete, Complemented and Distributive Lattices, Boolean Algebra: Definition and Examples, Subalgebra, Direct product and Homomorphism, join irreducible, atoms and antiatoms

IV. Graph Theory: Definition of a graph, applications, Incidence and degree, Isolated and pendant vertices, Null graph, Path and Circuits: Isomorphism, Subgraphs, Walks, Paths and circuits, Connected graphs, disconnected graphs, and components, Euler graph.

V. Trees: Trees and its properties, minimally connected graph, Pendant vertices in a tree, distance and centers in a tree, rooted and binary tree. Levels in binary tree, height of a tree, Spanning trees, rank and nullity.

 Govt. Model Science College (Auto), Jabalpur

Department of Mathematics & Computer Science

M.Sc. III Semester (Mathematics)

Unit Paper I : Fuzzy Sets and Their Application-I MM : 35

I. Idea of fuzzy set and membership function, Definition of a fuzzy set, membership function, representation of membership function, General definitions and properties of fuzzy sets, Support, height, equality of two fuzzy sets, containment, examples.

II. Union and Intersection of two fuzzy sets, Complement of a fuzzy set, normal fuzzy set, α-cut set of a fuzzy set, strong α-cut, convex fuzzy set, a necessary and sufficient condition for convexity of a fuzzy set (Theorem 1), Decomposition of fuzzy sets, Degree of sub sethood, Level set of a fuzzy set, Cardinality, fuzzy cardinality, examples.

III. Other important operations on fuzzy sets, Product of two fuzzy sets, Product of a fuzzy set with a crisp number, Power of a fuzzy set, Difference of two fuzzy sets, Disjunctive sum of two fuzzy sets, example.

IV. General properties of operations on fuzzy sets, Commutativity, associativity, distributivity, Idempotent law, identities for operations, Transitivity, involution, Demorgans laws, proofs and examples, Some important theorems on fuzzy sets, set inclusion of fuzzy sets and corresponding α-cuts and strong α-cuts (Theorem 1).

V. Comparison of α-cut and strong α-cut, Order relation of scalars α is inversely preserved by set inclusion of corresponding α-cuts and strong α-cuts, α-cut of union and intersection of two fuzzy sets, α-cut of complement of a fuzzy set (Theorem 2), Examples, α-cuts and strong α-cuts of union and intersection of arbitrary collection of fuzzy sets.

Unit Paper II : Advanced Special Function -I MM : 35

I. Gamma and Beta Functions : The Euler or Mascheroni Constant γ, Gamma Function, A series for Г' (z) / Г (z) , Difference equation Г(z+1) = zГ(z).

II. Beta function, value of Г(z) Г(1-z), Factorial Function, Legendre's duplication formula, Gauss multiplication theorem.

III. Hypergoemetric and Generalized Hypergeometric functions: Function 2F1 (a,b;c;z) A simple integral form evaluation of 2F1 (a,b;c;z) , Values of F (a,b;c;1) and F (-n, b;c;1) etc.

IV. Contiguous function relations, Hyper geometrical differential equation and its solutions, F (a,b;c;z) as function of its parameters.

V. Elementary series manipulations, Simple transformation, Relations between functions of z and 1-z

Unit Paper III : Advance Numerical Analysis -I MM : 35

I. Piece wise and spline interpolation: Piecewise Linear Interpolation, Piecewise Quadratic Interpolation, Piecewise cubic Interpolation, Piecewise cubic Interpolation using Hermitetype data, Quadratic and cubic spline Interpolation, Bivariate interpolation

II. Approximation : Least squares Approximation, Gram-schmitt orthogonalization process, chebsyshev polynomials ,legendre polynomials .

III. Uniform approximation : Uniform norm , uniform polynomial approximation, best Approximation, best Uniform approximation condition for uniform best approximation .

IV. Rational approximation, choice of method, Runge`s example. 

V. Numerical differentiation: Methods based on interpolation Method, Methods based on finite difference operators , methods based on undetermined coefficients, optimum choice of step length.

Unit Paper IV : Operation Research –I MM : 35

I. Operations Research and its scope, Origin and Development of Operations Research, Characteristics of Operations Research.

II. Model in Operations Research, Phase of Operations Research, Uses and Limitations of Operation Research, Linear Programming Problems.

III. Graphical procedure, Graphical solution of property behaved L.P problems. Graphical solution in some exceptional cases.

IV. General Linear Programming Problem : Simplex Method exceptional cases, artificial variable techniques ; Big M method, two phase Method and problem of degeneracy.

V. Concept of Duality : Definition of Primal-dual problems ,Symmetric Primal-dual problems, Unsymmetric Primal-dual problems, General rules for converting any primal into its dual. Fundamental theorem of duality.

Unit Paper V : INTEGRAL TRANSFORM-1 MM : 35

I. Problem related to Laplace transform Initial and bounding value problems, simultaneous ordinary differential equations. Problem related to solution of partial differential equations. Application of Laplace Transformed in Differential Equations

II. Two dimensional Laplace`s Equation (Cartesian and Polar form). Three dimensional Laplace`s Equation to related problems.

III. Notion of wave Equation. General solution of wave Equations. Solution by separation of variables. Solution of two dimensional wave equation , three dimensional wave equation.

IV. Definition: Integral Equations, problems related to Integral Equations of convolution type. Integral differential equation . Abel`s differential equation

V. Notion of Heat Equations. One and two dimensional heat conduction equation. Solution by separation of variables and problems based on it.

 Govt. Model Science College (Auto), Jabalpur

Department of Mathematics & Computer Science

M.Sc. IV Semester (Mathematics)

Unit Paper I : Fuzzy Sets and their Applications - II MM : 35

I. Fuzzy sets: Basic Definitions, α-level sets, Convex fuzzy set, Basic operations on fuzzy sets, types of fuzzy sets, Extensions: Types of fuzzy sets, Further operations on fuzzy sets, Cartesian product, Algebraic products, Bounded sum and Difference, t-norm & t-conorm.

II. Extension principle and applications, Zadeh extension principle, image and inverse image of fuzzy sets, fuzzy numbers, algebraic operations with fuzzy numbers, extended operation and its properties, Special extended operation, addition, subtraction, product and division of fuzzy numbers.

III. Fuzzy relations on fuzzy sets, The union & intersection of fuzzy relations, Composition of fuzzy relations, max-* and max-product compositions, minmax composition and its properties, reflexivity, symmetry, transitivity, and their examples, special fuzzy relations, similarity relation.

IV. Fuzzy graphs: Definition and Examples, Fuzzy sub-graph, Spanning subgraph, path in a fuzzy graph, strength and length of a path,μ -length and μ- distances, connected nodes, fuzzy forest, fuzzy tree, Examples, Fuzzy Analysis: Fuzzy functions on fuzzy sets, classical function, fuzzy function, Examples.

V. Fuzzy Logic; An overview of classical logic, Its connectives, Tautologies, Contradiction, Fuzzy logic, logical connectives for fuzzy logic, Examples, Approximate reasoning, its rules, examples, other of implication operations, Linguistic hedges, Fuzzy quantifiers, Examples.

Unit Paper II - ADVANCED SPECIAL FUNCTION MM : 35

I. Legendre polynomials : Definition of Pn(x), Generating functions, recurrence relations, Beltrami`s result Christoffels summation formula , Murphy formula, Rodrigues formula Bateman's generating relations and other generating relations.

II. Legendre differential equation and its solutions . Laplace first and second integral for Pn(x) . Orthogonal properties of Legendre polynomials. Expansion involving Legendre polynomials . Fourier - Legendre Expansion

III. Bessel functions : Definition of Jn (z), Generating functions Bessel's differential equation, recurrence relations , Bessel's integral with index half and an odd integer, Orthogonality of Bessel functions

IV. Hermite polynomial : Definition of Hermite polynomials Hn(x), Pure recurrence relations, Differential recurrence relations, Rodrigue's formula, Other generating functions, Orthogonality, Expansion of polynomials, more generating functions

V. Laguerre Polynomials : The Laguerre Polynomials Ln(X), Generating functions, Pure recurrence relations, Differential recurrence relation, Rodrigues formula, Orthogonality, Expansion of polynomials, Special properties, Other generating functions.

Unit Paper - III : Advanced Numerical Analysis II MM : 35

I. Extrapolation Methods for Numerical Differentiation. Multistep methods for Numerical Solution of initial value problems. Explicit and implicit Multi step methods.

II. General Multistep methods : r (x) ans s (x) for linear multiple step methods . Convergence of Multi step methods. Predictor correctior methods.

III. Stability analysis of Multistep methods: First order differential Equations, Stability of Predictor- Corrector Methods. Stability of PMp CMc methods , second Ordinary Differential.

IV. Ordinary differential Equations: Three kind of Boundary conditions . Finite Difference methods, Linear second order differential Equations, Non linear second order differential Equations

V. Finite element method : Finite element Ritz Finite element method methods, Linear Boundary Value Problems, mixed boundary conditions

Unit Paper - IV : Operations Research MM : 35

I. Replacement Problems : Replacement of Items that deteriorate , Replacement policy for items whose maintenance cost increase with time and money value is constant. Money value . present worth factor (PWF) and discount rate . Replacement policy for items whose maintenance cost increase with time and money value with constant rate. Individual Replacement policy, Mortality theorem, Group replacement policy.

II. Assignment problems : Mathematical formulation, Fundamental theorems, Hungarian method for assignment problem. Unbalanced assignment problem , The Travelling Salesman (Routing) problem, Job sequencing Processing n Jobs through 2 machines , Processing n Jobs through 3 machines, a graphical method.

III. Transportation problems : North - West Corner Method Least - Cost Method. Vogel's Approximation Method, MODI Method, Exceptional cases and problem of degeneracy.

IV. Network analysis, constraints in Network, Construction of network, Critical Path Method (CPM) PERT, PERT Calculation, Resource Leveling by Network Techniques and advances of network (PERT/CPM)

V. Game theory - Two persons, Zero - Sum Games, Maximix - Minimax principle, games without saddle points -Mixed strategies, Graphical solution of 2xm and mx2 games, Solution by Linear Programming, Non-Linear programming Techniques - Kuhn - Tucker Conditions, Non - negative Constraints

Unit Optional Paper - V : Integral Transforms-II MM : 35

I Application of Laplace Transform to boundary value problems. 

II Electric Circuits problems, related to application of Electric Circuits. Application to dynamics, application to heat conduction equation, application to wave equations, Application to Beams.

III The complex Fourier Transform, Inversion Formula, Fourier cosine and sine transform.

IV Properties of Fourier Transforms, Convolution & Parseval's identity. 

V Fourier Transform of the derivatives, Finite Fourier Sine & Cosine Transform, Inversion Operational and combined properties Fourier transform