Choice Based Credit System (CBCS) For M. Sc. Mathematics-I MATA GUJRI COLLEGE FATEHGARH SAHIB DEPARTMENT OF MATHEMATICS

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1 Choice Based Credit System (CBCS) For M. Sc. Mathematics-I MATA GUJRI COLLEGE FATEHGARH SAHIB DEPARTMENT OF MATHEMATICS POST-GRADUATE PROGRAMME (Courses effective from Academic Year )

2 SYLLABUS M.Sc. Mathematics (Part-I) Session Semester I Paper Code Paper Name Credits L T P Core course MM 101 Core Course MM 102 Core Course MM 103 Core Course MM 104 Lebesgue Theory of Integration Maximum Marks Internal Marks External Marks 51 0 (6) Algebra I 51 0 (6) Differential 51 0 (6) Geometry Linear Algebra 51 0 (6) CHOOSE ANY ONE OF THE FOLLOWING ELECTIVE COURSES Elective Course MM 105 Elective Course MM 106 Optimization Technique Number Theory 51 0 (6) (6) Seminar (1) Total

3 CORE COURSE MM 101: LEBESGUE THEORY OF INTEGRATION Internal Assessment: 30 INSTRUCTIONS FOR THE PAPER-SETTER Candidates are required to attempt three questions in all selecting two question from each SECTION-A Properties of Monotonic Functions, Functions of Bounded Variations, Total Variation, Additive Property of Total Variation, Total Variation on [a, x] as a Function of x, Function of Bounded Variation as difference of increasing function, Continuous function of Bounded Variation. [Scope as in Chapter -6( ) of RR-2] Algebras, σ- algebra and their properties, General measurable spaces, measure spaces, properties of measure, Complete measure, Lebesgue outer measure and its properties, measurable sets and Lebesque measure, A non measurable set. [Scope as in Chapter-3 of RR-1( )] SECTION-B Measurable function w.r.t. general measure, Borel and Lebesgue measurability. Integration of non-negative measurable functions, Fatou s lemma, Monotone convergence theorem, Lebesgue convergence theorem, The general integral, Integration of series, Riemann and lebesgue integrals. Differentiation; Vitalis Lemma, The Dini derivatives, Differentiation of an Integral.[Scope as in Chapter-3,4,5( ) of RR-1] TEXT BOOKS: 1. H.L. Royden: Real analysis, Macmillan Pub. co. Inc. 4 th Ed., New York, Chapters 3, 4, 5 and Sections 1 to 4 of Chapter T. M. Apostal, Mathematical Analysis, 2 nd Ed., Narosa Publishing House. 3. Walter Rudin: Real and Complex Analysis, 3 rd edition, Mc GrawHill, Education, 2017, Indian Edition. Chapter 9 (Excluding Sections 9.30 to 9.43)

4 CORE COURSE MM 102: ALGEBRA - I Internal Assessment: 30 INSTRUCTIONS FOR THE PAPER-SETTER Candidates are required to attempt three questions in all selecting two questions from each SECTION-A Review of groups, Normal and subnormal series, Solvable groups, Nilpotent groups, Composition Series, Jordan-Holder theorem for groups. Group action, Stabilizer, orbit, Class equation and its applications permutation groups, cyclic decomposition, conjugacy classes in permutation groups. Alternating group An, Simplicity of An. SECTION-B Structure theory of groups, Fundamental theorem of finitely generated abelian groups, Invariants of a finite abelian group, Groups of Automorphisms of cyclic groups, homomorphism between two cyclic groups, Sylow s theorems, Groups of order p 2, pq. Review of rings and homomorphism of rings, Ideals, Algebra of Ideals, Maximal and prime ideals, Ideal in Quotient rings, Field of Quotients of integral Domain, Matrix Rings and their ideals; Rings of Endomorphisms of Abelian Groups. TEXT BOOK: 1. Bhattacharya, Jain & Nagpaul: Basic Abstract Algebra, Cambridge University Press, 2 nd Ed., (Ch. 6, 7, 8, 10) 2. Surjeet Singh & Qzai Zimeeruddin : Modern Algebra, 8 th Ed., Vikas Publishing House, I. N. Herstein: Topics in Algebra, Second Ed., Wiley, J. B. Fraleigh: A First Course In Abstract Algebra, 7 th Ed., Pearson, 2002.

5 CORE COURSE MM 103: DIFFERENTIAL GEOMETRY Internal Assessment: 30 INSTRUCTIONS FOR THE PAPER-SETTER Candidates are required to attempt three questions in all selecting two questions from each SECTION-A A simple arc, Curves and their parametric representation, are length and natural parameter, contact of curves, Tangent to a curve, osculating plane, Frenet trihedron, Curvature and Torsion, Serret Frenet formulae, fundamental theorem for spaces curves, helices, contact between curves and surfaces. Evolute and involute, Bertrand Curves, spherical indicatrix, implicit equation of the surface, Tangent plane, the first fundamental form of a surface, length of tangent vector and angle between two tangent vectors, area of a surface. SECTION-B The second fundamental form, Gaussian map and Gaussian curvature, Gauss and Weingarten formulae, Codazzi equation and Gauss theorem, curvature of a curve on a surface, geodesic curvature. Geodesics, Canonical equations of geodesic, Normal properties of geodesics. Normal Curvature, principal curvature, Mean Curvature, principal directions, lines of curvature, Rodrigue formula, asymptotic Lines, conjugate directions, envelopes, developable surfaces associated with spaces curves, minimal surfaces, ruled surfaces. TEXT BOOKS: 1. A. Goetz: Introduction to differential geometry, Addison-Wesley Educational Publishers Inc, T. J. Willmore: An introduction to differential geometry, Oxford University Press India, Erwin Kreyszig: Differential Geometry, Dover Publications Inc., A. Pressley: Elementary Differential Geometry, 4 th Ed., Springer, 2009.

6 CORE COURSE MM 104-LINEAR ALGEBA Internal Assessments: 30 INSTRUCTIONS FOR THE PAPER SETTER The question paper will consist of five sections: A, B and C. Sections A and B will have four Candidates are required to attempt five questions in all selecting two questions from each SECTION-A Linear Transformations: Introduction to Linear Transformation, the Algebra of Linear Transformation, Isomorphism, Representation of Transformations by Matrices, Linear Functional, the Transpose of Linear Transformation. Elementary Canonical forms: Characteristic values, Annihilating polynomials, Invariant subspaces, simultaneous triangulation, simultaneous diagonalization, direct sum decompositions, Invariant direct sums, the primary decomposition theorem. (Chapter 3 and 6 of text book 1). SECTION-B The rational and Jordan forms: Cyclic subspaces and Annihilators, Cyclic decomposition and the rational forms, The Jordan form, computation of Invariant factors. Inner product space: Introduction to Inner product space, Cauchy-Schwarz inequality, Holder inequality, Gram-Schmidt orthogonalization process, Bessel inequality. (Chapter 7 and section 8.1 and 8.2 of chapter 8 of text book 1). TEXT BOOK: 1. K. Hoffmann & R. Kunze: Linear algebra, 2 nd Ed., PHI. 2. Vivek Sahai & VikasBist: Linear Algebra, Narosa Publishing House, 3. P. R. Halmos: Finite Dimensional Vector Space. 4. Serge Lang: Linear Algebra, Springer-Verlag Undergraduate Text in Mathematics.

7 ELECTIVE COURSE MM 105-OPTIMIZATION TECHNIQUES Internal Assessments: 30 INSTRUCTIONS FOR THE PAPER SETTER The question paper will consist of five sections: A, B and C. Sections A and B will have four Candidates are required to attempt five questions in all selecting two questions from each SECTION-A Introduction, definition of operation research, models in operation research, general methods for solving O.R. models Elementary theory of convex sets, Linear programming problems, examples of LPPs, mathematical formulation of the mathematical programming problems, Graphical solution of the problem. Simplex method, Big M method, Two Phase method, problem of degeneracy. Duality in linear programming: Concept of duality, fundamental properties of duality, duality theorems, complementary slackness theorem, duality and simplex method, dual simplex method. Sensitivity Analysis: Discrete changes in the cost vector, in the requirement vector and in the coefficient matrix. SECTION-B Transportation Problem: Introduction, mathematical formulation of the problem, initial basic feasible solution, optimum solution, degeneracy in transportation problems, transportation algorithm, unbalanced transportation problems. Assignment Problems: Introduction, mathematical formulation of an assignment problem, assignment algorithm, unbalanced assignment problems. Integer Programming: Introduction, Gomory's all-ipp method, Gomory's mixed-integer method, Branch and Bound method. Games and Strategies : Introduction, Two person zero sum games, Maximum, Minimum, Principle; Games without saddle points, Mixed Strategies, Graphical solution, Dominance property, Reducing the game problem to a LPP. TEXT BOOKS: 1. Kanti Swarup, P. K. Gupta and Man Mohan: Operations Research, Sultan Chand and Sons, New Delhi, Chander Mohan and Kusum Deep: Optimization Techniques, New Age International, H. Taha: Operation Research An Introduction, Pearson, G. Sriniwasan: Operation Research Principle & Application, Printice Hall.

8 ELECTIVE COURSE MM 106: NUMBER THEORY Internal Assessment : 30 Time Allowed : 3 hours Total : 100 INSTRUCTIONS FOR THE PAPER - SETTER Candidates are required to attempt five questions in all selecting two questions from each SECTION - A Arithmetical functions: Mobius function, Euler's totient function, Mangoldt function, Liouville's function, The divisor functions, Relation connecting ϕ and, product formula for ϕ(n), Dirichlet product of arithmetical functions, Dirichlet inverses and Mobius inversion formula, Multiplicative functions, Dirichlet multiplication, The inverse of a completely multiplicative function, Generalized convolutions. Averages of arithmetical functions: The big oh notation, Asymptotic equality of functions, Euler's summation formula, Elementary asymptotic formulas, Average order of d(n), ϕ(n), (n), (n), The Partial sums of a Dirichlet product, Applications to (n) and (n), Legendre's Identity. [TEXT 1: chapter 2(Sections 2.1 to 2.14), Chapter 3(Sections 3.1 to 3.7, 3.10 to 3.12)] SECTION - B Some elementary theorems on the distribution of prime numbers: Chebyshev's functions (x) & (x), Relation connecting (x) and (x), Abel's identity, equivalent forms of Prime number theorem, inequalities for (n) and P n, Shapiro's Tauberian theorem, applications of Shapiro's theorem, Asymptotic formula for the partial sums p x (1/p). Elementary properties of groups, Characters of finite abelian groups, The character group, Orthogonality relations for characters, Dirichlet characters, Dirichlet's theorem for primes of the form 4n-1 and 4n+1, Dirichlet's theorem in primes on Arithmetical progression. [TEXT 1: chapter 4(Sections 4.1 to 4.8), Chapter 6, Chapter 7(Sections 7.1 to 7.8)] TEXTBOOKS:

9 1. T. M. Apostol: Introduction to Analytic Number Theory, 8 th Edition, Narosa publishing House, G. H. Hardy and E. M. Wright: An Introduction to Theory of Numbers, Oxford University Press, 6th Ed., I. Niven, H. S. Zuckerman and H. L. Montgomery: An Introduction to the Theory of Numbers, John Wiley and Sons, (Asia), 5th Ed., G. E. Andrews: Number Theory, Dover Books, Z. I. Borevich & I. R. Schafarevich: Number Theory Academic Press, 1966.

10 SYLLABUS M.Sc. Mathematics (Part-I) Session Semester II Paper Code Paper Name Credits L T P Core course MM 201 Algebra-II (rings and modules) Maximum Marks Internal Marks External Marks 51 0 (6) Core Course MM 202 Core Course MM 203 Core Course MM 204 Complex 51 0 (6) Analysis I Topology-I 51 0 (6) Differential Equations 51 0 (6) CHOOSE ANY ONE OF THE FOLLOWING ELECTIVE COURSES Elective Course MM 205 Numerical Methods 51 0 (6) Elective Course MM 206 Discrete Mathematics 51 0 (6) Seminar (1) Total

11 CORE COURSE MM 201: ALGEBRA-II (RINGS AND MODULES) Internal Assessment: 30 INSTRUCTIONS FOR THE PAPER-SETTER Candidates are required to attempt three questions in all selecting two questions from each SECTION-A Unique Factorization Domains, Principal Ideal Domains, Euclidean Domains, Polynomial Rings over UFD, Rings of Fractions. (RR1: Ch. 11 and Section 1 of Chapter 12). Modules: Definition and Examples, Submodules, Direct sum of submodules, Free modules, Difference between modules and vector spaces, Quotient modules, Homomorphism, Simple modules, Modules over PID. (RR2: Chapter 5) SECTION - B Modules with chain conditions: Artinian Modules, Noetherian Modules, Artinian Implies Noetherian in Rings, Composition series of a module, Length of a module, Hilbert Basis Theorem (RR2: Chapter 6). Cohen Theorem, Radical Ideal, Nil Radical, Jacobson Radical, Radical of an Artinian ring. Nil Radical and Jacobson Radical of Polynomial Rings R[x], R commutative. (RR2: Chapter 6) TEXT BOOKS: 1. Bhattacharya, Jain and Nagpaul: Basic Abstract Algebra, Second Ed., Cambridge University Press, C. Musili: Introduction to Rings and Modules, Second Revised Ed., Narosa Publishing House, 1997

12 CORE COURSE MM 202: COMPLEX ANALYSIS-I Internal Assessment: 30 INSTRUCTIONS FOR THE PAPER-SETTER Candidates are required to attempt three questions in all selecting two questions from each SECTION-A Function of complex variable, Analytic function, Cauchy-Riemann equations, Harmonic function and Harmonic conjugates, Branches of multivalued functions with reference to arg z, logz and z c, Conformal Mapping, Bilinear transformation. Complex Integration, Cauchy s theorem, Cauchy Goursat theorem Cauchy integral formula, Morera s theorem, Liouville's theorem, Fundamental theorem of Algebra, Maximum Modulus Principle. Schwarz lemma. SECTION-B Taylor s theorem. Laurent series in an annulus. Singularities, Meromorphic function. Cauchy s theorem on residues. Application to evaluation of definite integrals. Principle of analytic continuation, General definition of an analytic function. Analytic continuation by power series method, Natural boundary, Harmonic functions on a disc, Schwarz Reflection principle, Mittag-Leffler s theorem (only in case when the set of isolated singularities admits the point at infinity alone as an accumulation point). TEXT BOOKS: 1. L. V. Ahlfors: Complex Analysis, 3 rd Ed., McGraw Hill, E. T. Copson: An introduction to Theory of Functions of a Complex Variable, Oxford University Press, H. S. Kasana: Complex Variables, Prentice Hall of India, Herb Silverman: Complex Variables, Houghton Mifflin Company Boston, 1975.

13 CORE COURSE MM 203: TOPOLOGY I Internal Assessment: 30 INSTRUCTIONS FOR THE PAPER-SETTER Candidates are required to attempt three questions in all selecting two questions from each SECTION A Cardinals: Equipotent sets, Countable and Uncountable sets, Cardinal Numbers and their Arithmetic, Bernstein s Theorem and the Continumm Hypothesis. Topological Spaces: Definition and examples, Euclidean spaces as topological spaces, Basis for a given topology, Topologizing of Sets; Sub-basis, Equivalent Basis. Elementary Concepts: Closure, Interior, Frontier and Dense Sets, Topologizing with preassigned elementary operations. Relativization, Subspaces. Maps and Product Spaces: Continuous Maps, Restriction of Domain and Range, Characterization of Continuity, Continuity at a point. Open Maps and Closed Maps, Homeomorphisms and Embeddings. SECTION B Cartesian Product Topology, Elementary Concepts in Product Spaces, Continuity of Maps in Product Spaces and Slices in Cartesian Products. Connectedness: Connectedness and its characterizations, Continuous image of connected sets, Connectedness of Product Spaces, Applications to Euclidean spaces. Components, Local Connectedness and Components, Product of Locally Connected Spaces. Path Connectedness. Compactness and Countability: Compactness and Countable Compactness, Local Compactness, One-point Compactification, T0, T1, and T2 spaces, T2 spaces and Sequences and Hausdorfness of One-Point Compactification. TEXT BOOKS: 1. W. J. Pervin: Foundations of General Topology, Academic Press Inc., Ch. 2 (Sections 2.1, 2.2), Section 4.2, and Ch 5 (Sec 5.1 to 5.3). 2. James Dugundji: TOPOLOGY, William C Brown Pub, (Relevant Portions from Ch.III (excluding Sec 6 and Sec 10), Ch IV; (Sections 1-3) and Ch V) 3. James Munkers: Topology, 2 nd Ed., Printice Hall, Indian Learning Pvt. Ltd. 4. Sheldon W. Davis: Topology; Tata McGraw-Hill Ed., 1899.

14 CORE COURSE MM 204: DIFFERENTIAL EQUATIONS Internal Assessment: 30 INSTRUCTIONS FOR THE PAPER-SETTER Candidates are required to attempt three questions in all selecting two question from each SECTION- A Existence of solution of ODE of first order, initial value problem, Ascoli s Lemma, Gronwall s inequality, Cauchy Peano Existence Theorem, Uniqueness of Solutions. Method of successive approximations, Existence and Uniqueness Theorem. System of differential equations, nth order differential equation, Existence and Uniqueness theorems for system and higher order equations, [Text 1: Chapter 1 (Sections 1, 2, 3, 4, 5 and 6), Text 2: Chapter 10 (Sections 10.2, 10.3 and 10.4)] SECTION- B Linear system of equations (homogeneous & non homogeneous). Superposition principle, Fundamental set of solutions, Fundamental Matrix, Wronskian, Abel Liouville formula, Reduction of order, Adjoint systems and self adjoint systems of second order, Floquet Theory. Linear 2 nd order equations, preliminaries, Sturm s separation theorem, Sturm s fundamental comparison theorem, Sturm Liouville boundary value problem, Characteristic values & Characteristic functions, Orthogonality of Characteristic functions, Expansion of a function in a series of orthonormal functions. [Text 2: Chapter 11, Chapter 12 (Sections 12.1, 12.2 and 12.3)] TEXT BOOKS: 1. Earl A. Coddington & Norman Levinson: Theory of Ordinary Differential Equations, Tata Mc-Graw Hill, India., 9th Ed., S. L. Ross: Differential Equations, 3 rd Ed., John Wiley & sons, 1984 Asia. 3. D. A. Sanchez: Ordinary Differential Equations & Stability Theory, Freeman & company A. C. King, J. Billingham & S. R. Otto: Differential Equations, Linear, Nonlinear, Ordinary, Partial, Cambridge University Press, W.E. Boyce & R.C. DiPrima, Elementary Differential Equations, 9 th Ed., Wiley,2008.

15 ELECTIVE CORSE MM 205: NUMERICAL METHODS Internal Assessment: 30 INSTRUCTIONS FOR THE PAPER-SETTER Candidates are required to attempt three questions in all selecting two question from each SECTION-A Error Analysis, Bisection, Regula-Falsi, Secant, Newton-Raphson, Muller, Chebyshev and General Iteration Methods and their rate of convergence, Aitken Method for acceleration of the Convergence, Methods for multiple roots. [Ref.1 Chap ] Direct Methods: Gauss elimination method, Gauss-Jordan Elimination methods, Decomposition methods (LU and Cholskey), Partition method and their error analysis. Iterative Methods: Jacobi iterative method, Gauss-Seidel iterative method, Successive over relaxation iterative method, Convergence Analysis of iterative methods. Eigen Value Problems: Gerschgirun Theorem, Jacobi, Givens methods Householder s method for Symmetric matrices, Ruthishauser, Power and Inverse Power methods. [Ref.1 Chap 3] SECTION-B Lagrange s interpolation, Newton Interpolation, Finite Difference Operators, Piecewise and Spline Interpolation, Interpolating Polynomials using Finite Differences and Hermite Interpolation. Least square approximation. [Ref.1 Chap ] Numerical Differentiation, Error in Numerical Differentiation, Cubic Spline method, Maximum and Minimum values of a tabulated function, Numerical Integration: Trapezoidal Rule, Simpson s 1/3-Rule, Simpson s 3/8-Rule, Boole s and Weddle s Rule, Integration using Cubic Splines, Romberg Integration, Newton Cotes formulae. Taylor s Series method, Picard s Method, Euler s and modified Euler s methods, Runge Kutta methods [Ref. 3 Chap , Chap ] TEXT BOOKS: 1. M. K Jain, S. R. K lyenger and R. K Jain: Numerical Methods for Scientific and Engineering Computations, 6th Edition, New Age Intenational (P) Limited, Publishers, New Delhi. 2. Kendall E Atkinson: An introduction to Numerical Analysis, 2nd Edition John Wiley &

16 Sons, Printed in India by Replika Pvt. Ltd., S. S. Sastry: Introductory Methods of Numerical Analysis, 4th Edition (2010), Prentice Hall of India Pvt. Ltd., New Delhi. 4. F. B Hilderbrand: Introduction to Numerical Analysis, 2nd Edition, Dover Publication Inc, New York, 1987.

17 ELECTIVE COURSE MM 206: DISCRETE MATHEMATICS Internal Assessment: 30 INSTRUCTIONS FOR THE PAPER-SETTER Candidates are required to attempt three questions in all selecting two questions from each SECTION-A Graphs, Konisberg seven bridges problem. Finite and infinite graphs. Incidence vertex. Degree of a vertex. Isolated and pendant vertices. Null graphs. Isomosphism of graphs. Subgraphs, walks, paths and circuits. Connected and disconnected graphs. Components of a graph. Euler graphs. Hamiltonian paths and circuits. The traveling salesman problem. Trees and their properties. Pendant vertices in a tree. Rooted and binary tree. Spanning tree and fundamental circuits. Spanning tree in a weighted graph. (Chapter 1,2,3 of the book given at Sr. No. 1). Cutsets and their properties. Fundamental circuits and cutsets. Connectivity and separability. Network flows. Planner graphs. Kuratowski s two graphs. Representation of planner graphs. Euler formula for planner graphs. Incidence matrix A(G) of a graph G, Submatrices of A(G), Circuit matrix, Fundamental circuit matrix, and its rank, Cutset matrix, path matrix and adjacency matrix of a graph. (Chapter 4, Theorems 5.1 to 5.6 of chapter 5, 6 & 7 of the book given at Sr. No. 1). SECTION-B Partially ordered sets and lattices. Lattice as an algebraic system. Sublattices. Isomorphism of lattices. Distributive and modular lattices. Lattices as intervals. Similar and projective intervals. Chains in lattices. Zassenhaus s Lemma and Schreier Theorem, Composition chain and Jordan Holder Theorem. Chain conditions. Fundamental dimensionality relation for modular lattices. Decomposition theory for lattices with ascending chain conditions, i.e. reducible and irreducible elements. Independent elements in lattices. (Relevant portion of the chapter 7 and chapter 12 of the books given at Sr. No. 2 & 3). Points (atoms) of a lattice. Complemented lattices. Chain conditions and complemented lattices. Boolean algebras. Conversion of a Boolean algebra into a Boolean ring with unity and vice versa. Direct product of Boolean algebras. Uniqueness of finite Boolean algebras. Boolean

18 functions and Boolean expressions. (Relevant portion of the chapter 7 and chapter 12 of the books given at Sr. No. 2 & 3). TEXT BOOKS: 1. Narsingh Deo: Graph Theory with application to Engineering and Computer Science, Prentice Hall of India, Nathan Jacobson: Lectures in Abstract Algebra Vol-I, D. Van Nostrand Company, Inc. 3. L. R. Vermani: A course in discrete Mathematical structures(imperial College Shalini Press London, 2011

RANI DURGAVATI UNIVERSITY, JABALPUR

RANI DURGAVATI UNIVERSITY, JABALPUR SYLLABUS OF M.A./M.Sc. MATHEMATICS SEMESTER SYSTEM Semester-II (Session 2012-13 and onwards) Syllabus opted by the board of studies in Mathematics, R. D. University