M.A./M.Sc. MATHEMATICS SEMESTER-1 Syllabus Pdf Download

In this article, we are sharing M.sc Mathematics semester 1 Syllabus pdf / MA Mathematics semester 1 Syllabus. Hope you like it. Let's see.

M.A./M.Sc. MATHEMATICS SEMESTER-1 Paper -1 Advanced Absetract Algebra Syllabus

Unit 1- Normal and Subnormal series of groups, Composition series, Jordan-Holder series. 

Unit 2- Solvable and Nilpotent groups. 

Unit 3- Extension fields, Roots of polynomials. Algebraic and transcendental extensions, Splitting fields, Separable and inseparable extensions. 

Unit 4- Perfect fields, Finite fields, Primitive elements, Algebraically closed fields. 

Unit 5- Automorphism of extensions, Galois extensions, Fundamental theorem of Galois theory, Solution of polynomial equations by radicals, Insolvability of general equation of degree 5 by radicals. 

Recommended Books: 

(1] I.N. Herstein, Topics in Algebra, Wiley Eastern Ltd. New Delhi, 1975. 

[2] Vivek Sahai and Vikas Bist, Algebra, Narosa Publishing House, 1999. 

[3] P.B. Bhattacharya, S.K.Jain and S.R.Nagpaul, Basic Abstract Algebra (2nd Edition). Cambridge University Press, Indian Edition, 1997. 

Reference Books: 

[1] N. Jacobson, Basic Algebra, Vols. I & II, W.H. Freeman, 1980(also published by Hindustan Publishing Company). 

[2] S. Lang, Algebra, Addison-Wesley. 

[3] I.S. Luther and I.B.S. Passi, Algebra, Vol.I-Groups, Vol. II-Rings, Narosa Publishing House (VO1. I-1996, Vol. II- 1999).

M.A./M.Sc. MATHEMATICS SEMESTER-1 Paper - II Real Analysis Syllabus

Unit 1- Definition and existence of Ricmann-Stieltjes integral and its properties, Integration and differentiation, The fundamental theorem of calculus. 

Unit 2- Integration of vector-valued functions., Rectifiable curves, Rearrangement of terms of series, Riemann's theorem.

Unit 3- Sequence and series of functions, Pointwise and uniform convergence, Cauchy criterion for uniform convergence, Weierstrass M-test, Abel's and Dirichlet's test for 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. 

Unit 4- Functions of several variables, Linear transformations, Derivatives in an open subset of Rr, Chain rule, Partial derivatives, Interchange of the order of differentiation, Derivatives of higher orders, Taylor's theorem, Inverse function theorem. 

Unit 5- Implicit function theorem, Jacobians, Extremum problems with constraints, Lagrange's multiplier method, Differentiation of integrals, Partitions of unity, Differential forms, Stoke's theorem. 

Recommended Books: 

[1] Walter Rudin, Principles of Mathematical Analysis (3rd Edition), McGraw-Hill, Kogakusha, 1976, International Student Edition. 

Reference Books: 

[1] T.M.Apostol, Mathematical Analysis, Narosa Publishing House, New Delhi, 1985. 

[2] H.L.Roden, Real Analysis, Macmillan Publishing Co. Inc., 4th Edition, New York, 1993.

M.A./M.Sc. MATIIRMATICS SEMESTER-I Paper - III Topology - 1 Syllabus

Unit 1- nders and its arithmetic. Schroeder-Bernsteln theorem, Cantor's theorem and the continuum hypothesis, Zorn's lemmarWell-ordering theorem. Countable and uncountable sets. Infinite sets and the axlom of choice, Cardinal 

Unit 2- Definition and examples of topological spaces, Closed sets, Neighbourhoods, Glosure, Dense sets, Interior, Exterlor and boundary, Accumulation points and derived sets, Bases and sub-bases, Subspaces and relative topology. 

Unit 3- Alternate methods of defining a topology in terms of Kuratowski closure operator and Neighbourhood systems, Continuous functions and homeomorphism. 

Unit 4- First and second countable spaces, Separable spaces, Second countability and Separability. 

Unit 5- Connected spaces, Connectedhess on real line, Components, Locally connected spaces, Path-connectedness. 

Recommended Books: 

[1] James R. Munkres, Topology: A First Course, Prentice-Hall of India Pvt. Ltd. New Delhi, 2000. 

Reference Books: 

[1] K.D.Joshi, Introduction to General Topology, Willey Eastern Limited, 1983. 

[2] G.F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill Book Company, 1963. 

[3] J.Dugundji, Topology, Allyn and Bacon, 1966(Reprinted in India by Prentice-Hall of India Pvt. Ltd.). 

[4] N. Bourbaki, General Topology Part-1 (Transl.) Addition Wesley Reading 1966.

M.A./M.Sc. MATHEMATICS SEMESTER-1 Theory Paper - IV Complex Analysis - 1 Syllabus

Unit 1- complex Integration, Cuachy-Gaursat theorem, Cauchy integral formula, Higher order derivatives. 

Unit 2- Morera's theorem, Cauchy's inequality, Liouville's theorem, The fundamental theorem of algebra, Taylor's theorem. 

Unit 3- Laurent's series, The maximum modulus principle, Schwartz lemma, Singularities, Meromorphic functions, The argument principle, Rouche's theorem, Inverse function theorem. 

Unit 4- Residues, Cauchy's residue theorem, Evaluation of integrals, Branches of many valued functions with special reference to arg z, log z, z^a. 

Unit 5- Uniform convergence of sequence, series and power series, General principle of uniform convergence, Weierstrass's M-test, Hardy's test for uniform convergence, Infinite products, General principle of uniform convergence of infinite product, Absolute and uniform convergence of an infinite product, Theorems on infinite product. 

Recommended Books: 

[1] J.B.Convey, Functions of One Complex Variable, Springer-Verlag. International Student Edition, Narosa Publishing House,1980. 

Reference Books: 

[1] S.Ponnusamy, Foundations of Complex Analysis, Narosa Publishing House,1997. 

[2] L.V. Ahlfors, Complex Analysis, McGraw-Hill, 1979. 

[3] J.W.Brown and R.V.Churchill, Complex Variables and Applications, 2004.

M.A./M.Sc. MATHEMATICS SEMESTER-I Paper V Optlonal (1) Advanced Discrete Mathematics-1 Syllabus

Unit 1- Subgroups and Monolds-Definitions and examples of Semigroups and Monoids (including those pertaining to concatenation operation). Homomorphism of sernigroups and Monoids. Congruence relation and quotient semigroups. Subsemigroup and submonoids. Direct products, Basic homomorphism theorem. 

Unit 2- 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. 

Unit 3- Boolean Algebras- Boolean Algebras as lattices, Various Boolean identities, The switching algebra example, Subalgebras, Direct products and homomorphisms, Join-irreducible elements, Atoms and Minterms, Bogolcan forms and their cauivalence, Minterm Boolean forms, Sum of products canonical forms. Minimization of Boolean functions, Application of Boolean algebra to switching theory (using AND, OR & NOT gates), The Karnaugh map method. 

Unit 4- Graph theory- Definition of (undirected) graphs, Paths, Circuits, Cycles and subgraphs, Induced subgraphs, Degree of vertex, Connectivity, Planer graphs and their properties, Trees. 

Unit 5- Euler's formula for connected planer graphs, Complete and Complete Bipartite graphs, Kurtowski's theorem (statement only) and its use. Spanning trees, Cut-Sets, Fundamental Cut-Sets and Cycles, Minimal spanning trees and Kruskal's algorithm, Matrix representation of graphs. 

Recommended Books: 

[1] J.P.Trambly and R. Manohar, Discrete Mathematical Structures with Application to Computer Science, McGraw-Hill book Co., 1997. 

[2] N. Deo, Graph Theory with Application to Engineering and Computer Sciences, Prentice Hall of India. 

Reference Books: 

[1] J.L. Gersting, Mathematical Structure for Computer Science (3rd Edition), Computer Science Press, New York. 

[2] Seymour Lepschutz, Finite Mathematics (International Edition, 1983), McGraw Hill Co., New York. 

[3] S.Wiitala, Discrete Mathematics-A Unified Approach, McGraw Hill Book Co. 

[4] J.E. Hopcroft and J.D. Ullman, Introduction to Automata Theory Languages and Compulation, Narosa Publishing House. 

[5] C.L. Liu, Elements of Discrete Mathematics, McGraw Hill Book Co.

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