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Faculty of Mathematical and Statistical Sciences INSH Faculty of Mathematical and Statistical Sciences INSH PROGRAMME M.Sc. (Mathematics) Programme Learning Objective Mathematics Postgraduate curriculum should such that it offers our student To train qualified, adaptable, motivated, and responsible mathematicians who will contribute to the scientific and technological development. To develop the mathematical skills needed in modeling and solving practical problems. To prepare students to face the present challenges in different field of Mathematics. To develop an outlook and high level study skills that will be hugely valuable whatever career path follow after post graduation. To impart an intensive and in-depth learning to the students in the field of Mathematics. To develop students’ skills in dissertation writing. The programme also provides students with the knowledge and opportunity to progress towards a Doctorate programme. Programme Learning Outcomes After doing Mathematics Postgraduate Course, our student will be able to PLO1: Identify fundamental concepts of higher mathematics as applied to applied mathematics. PLO2: Apply analytic, numerical and computational skill to analyze and solve higher Mathematical problems. PLO3: Apply knowledge of higher Mathematics with integrative approach in diverse fields. PLO4: Acquire appropriate skill in higher Mathematics to handle research oriented scientific problems. PLO5: Gain specific knowledge and understanding will be determined by your particular choice of modules, according to your particular needs and interests. PLO6: Communicate scientific information effectively in written and oral formats. PLO7: The basic and advanced knowledge attained by student will also make him/her enable to inspire them to pursue higher studies. PLO8: Apply their responsibilities in social and environmental context. STUDY & EVALUATION SCHEME (Effective from the session 2017 -2018) STUDY & EVALUATION SCHEME (Session 2019-2020) M. Sc.: Mathematics I Year: I Semester S. Subject Subject L T P CIE ESE Total C No. Code THEORY 1. MMA1001 Algebra-I 3 2 - 40 60 100 5 2. MMA1002 Real Analysis 3 2 - 40 60 100 5 Differential Geometry of 3. MMA1003 3 2 - 40 60 100 5 Manifolds 4. MMA1004 Differential Equations 3 2 - 40 60 100 5 5. MMA1005 Mechanics 3 2 - 40 60 100 5 PRACTICAL/TRAINING/PROJECT 6. MMA1501 MATHEMATICA Lab - - 2 80 20 100 1 7. MMA1502 Seminar - - 2 80 20 100 1 Total 15 10 4 360 340 700 27 L - Lecture T -Tutorial P -Practical CIE -Continuous Internal Evaluation ESE -End Semester Exam C -Credit STUDY & EVALUATION SCHEME (Effective from the session 2017-2018) STUDY & EVALUATION SCHEME (Session 2019-2020) M. Sc.: Mathematics I Year: II Semester S. Subject [[[[[[ Subject L T P CIE ESE Total C No. Code THEORY 1. MMA2001 Algebra-II 3 2 - 40 60 100 5 2. MMA2002 Complex Analysis 3 2 - 40 60 100 5 3. MMA2003 Topology 3 2 - 40 60 100 5 Calculus of Variations and 4. MMA2004 3 2 - 40 60 100 5 Integral Equations 5. MMA2005 Numerical Methods 3 2 - 40 60 100 5 PRACTICAL/TRAINING/PROJECT 6. MMA2501 MATLAB - - 2 80 20 100 1 7. MMA2502 Seminar - - 2 80 20 100 1 Total 15 10 4 360 340 700 27 STUDY & EVALUATION SCHEME (Effective from the session 2017-2018) STUDY & EVALUATION SCHEME (Session 2019-2020) M. Sc.: Mathematics II Year: III Semester S. Subject Subject L T P CIE ESE Total C No. Code THEORY 1. MMA3001 Measure and Integration Theory 3 2 - 40 60 100 5 2. MMA3002 Probability and Statistics 3 2 - 40 60 100 5 3. MMA3003 Fluid Dynamics 3 2 - 40 60 100 5 4. --- Elective-I 3 1 - 40 60 100 4 5. --- Elective-II 3 1 - 40 60 100 4 PRACTICAL/TRAINING/PROJECT 6. MMA3501 LATEX - - 2 80 20 100 1 7. MMA3502 Seminar - - 2 80 20 100 1 Total 15 8 4 360 340 700 25 University Mandatory Non-Credit Course 1. XHUX601 Human Values and Ethics 2 - - 100 - 100 0 STUDY & EVALUATION SCHEME (Session 2019-2020) M. Sc.: Mathematics II Year: IV Semester S. Subject Subject L T P CIE ESE Total C No. Code THEORY 1. MMA4001 Functional Analysis 3 2 - 40 60 100 5 2. MMA4002 Operations Research 3 1 - 40 60 100 4 3. --- Elective-III 3 1 - 40 60 100 4 PRACTICAL/TRAINING/PROJECT 4. MMA4501 Project - - 2 80 20 100 4 5. MMA4502 Comprehensive Viva-Voce - - - 100 - 100 2 Total 9 4 2 120 180 500 19 STUDY & EVALUATION SCHEME (Session 2019-2020) M. Sc.: Mathematics List of Electives S. No. Subject Code Subject Elective-I (Semester-III) 1. MMA3101 Discrete Mathematical Structure 2. MMA3102 Mathematical Biology 3. MMA3103 Stochastic Processes Elective-II (Semester-III) 1. MMA3201 Number Theory and Cryptography 2. MMA3202 Dynamical Systems 3. MMA3203 Fuzzy Sets and Applications Elective-III (Semester-IV) 1. MMA4101 Mathematical Modeling 2. MMA4102 Mathematics of Finance and Insurance 3. MMA4103 Optimization Techniques 4. MMA4104 Fractional Calculus and Nonlinear Dynamics STUDY & EVALUATION SCHEME (Effective from the session 2017-2018) Algebra-I (MMA1001) L T P C 3 2 0 5 (40 Hours) Course Objective (CO): CO1: To introduce some fundamental concepts of Algebra. CO2: To provide some understanding of the concepts like group and ring. CO3: To introduce some understanding of the field and module. CO4: To explore the application based problems of the related topics and use this knowledge in more advanced and complex situations of pure and applied mathematics. UNIT – I: GROUP THEORY (10 Hours) Normal subgroups, Quotient groups, Homomorphism and Isomorphism theorems of groups, Cayley’s theorem, class equations, Direct product of groups (External and Internal), Cauchy’s Theorem for finite abelian groups, p-groups , Sylow subgroups, Sylow’s Theorem, applications of Sylow subgroups. UNIT-II: SYLOW’S THEOREM AND SOLVABILITY (07 Hours) Normal and subnormal series, composition series, Jordan holder theorem, Solvable groups, simplicity of An (n 5) , Nilpotent groups. UNIT-III: ADVANCED RINGS THEORY (08 Hours) Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domain, principal ideal domain, Euclidean domain, Polynomial rings and irreducibility criteria, Eisenstein’s criterion of irreducibility. UNIT-IV: FINITE FIELDS (08 Hours) Extension fields, Finite, algebraic and transcendental extensions, Simple and algebraic field extensions, Splitting fields and normal extensions, algebraically closed fields. UNIT-V: MODULE THEORY (07 Hours) Modules, Submodules, Quotient modules and cyclic modules, Homomorphism and Isomorphism theorems, Simple modules, free modules, Rank of module. TEXT BOOKS: T1. Ramji Lal, Algebra, Vols. I & II, Shail Publications, Allahabad, 2002. T2. P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Basic Abstract Algebra. T3. Vivek Sahai and Vikas Bist, Algebra, Narosa Publishing House 1999. T4. Khanna & Bhamri, “A course in Abstract Algebra”, Vikas Publishing House. REFERENCES BOOKS: R1. I.N.Herstein.. Topics in Algebra, John Wiley & Sons. R2. M. Artin.. Algebra, Prentice Hall of India. R3. D. S. Malik, J. N. Mordeson, and M. K. Sen, Fundamentals of Abstract Algebra, McGraw-Hill International Edition, 1997. R4. N. Jacobson, Basic Algebra, Vol. I, Hindustan Publishing Co., New Delhi, 1984. R5. Joseph & Gallian, “Contemparay Abstract Algebra”, Narosa Publishing House. Course Learning Outcome (CLO): After completing this course, our Student will be able to CLO1: recall, apply and analyze the basic properties of group theory. CLO2: Define, illustrate and interpret about Sylow’s theorem and Solvability. CLO3: understand, analyze and apply advanced Ring theory. CLO4: remember, comprehend, apply and analyze finite fields and module theory. Matching of PLOs and CLOs: PLO1 PLO2 PLO3 PLO4 PLO5 PLO6 PLO7 PLO8 CLO1 H H H M H M L M CLO2 H H L M H M L M CLO3 H M M M H M L L CLO4 H L H M H M L M Real Analysis (MMA1002) L T P C 3 2 0 5 (40 Hours) Course Objective (CO): CO1: To introduce some fundamental ideas about sequence, series and sequence, series of the functions. CO2: To provide some understanding of the basic concept and theories of metric spaces. CO3: To aim that understanding the concept of Riemann-Stieltjes integral and their properties. CO4: To furnish students with mathematical tools that have been found essential in dealing with variety of problems as they arise in the physical world. UNIT-I: SEQUENCES AND SERIES (08 Hours) Sequences and series, Series of arbitrary terms, Convergence, divergence and oscillation, Abel’s and Dirichilet’s tests, Multiplication of series, Rearrangements of terms of a series, Bolzano Weierstrass theorem, Heine-Borel theorem, Continuity, uniform continuity, differentiability, mean value theorem. UNIT-II: SEQUENCES AND SERIES OF FUNCTIONS (08 Hours) Pointwise convergence, uniform convergence on interval, Cauchy’s criterian for uniform convergence, test for uniform convergence, test for uniform convergence of series, Weierstrass M-test, Abel’s and Dirichlet’s test, properties of uniform convergence. UNIT- III: METRIC SPACE-I (08 Hours) Definition and examples of metric spaces, Neighborhoods, closure and interior, boundary points, Limit points, Open and closed sets, Subspaces, Convergent and Cauchy sequences. UNIT- IV: METRIC SPACE-II (08 Hours) Continuous functions, Uniform continuity, Complete metric space and its properties, Cantor’s intersection Theorem, Compact metric space and its properties, Finite intersection property, Balzano- Weierstrass property, Connectedness. UNIT-V: RIEMANN-STIELTJES INTEGRAL (08 Hours) Riemann sums and Riemann integral, Improper Integrals. Monotonic functions, types of discontinuity, functions of bounded variation, Definition and existence of Riemann-Stieltjes integral, Conditions for R- S integrability, Properties of the R-S integral, Integration of vector valued functions. TEXT BOOKS: T1. T.M. Apostol, Mathematical Analysis, Narosa Publishing House , New Delhi, 1985. T2. S.C. Malik., Mathematical Analysis, Wiley Eastern Ltd., New Delhi.. T3. Walter Rudin, Principles of Mathematical Analysis- McGraw Hill International Editions, Mathematics series, Third Edition, 1964. REFERENCE BOOKS: R1. Gabriel Klambauer, Mathematical Analysis, Marcel Dekkar, Inc. New York, 1975. R2. R.R. Goldberg, Real Analysis, Oxford & I.B.H. Publishing Co., New Delhi, 1970. R3. D. Soma Sundaram and B.
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