MMA402DMS Differential Manifolds

Transcription

1 TEACHING AND EXAMINATION SCHEME Semester IV Sr. No. Subject Code Subject Name Credit Hours (per week) Theory Practical Lecture(DT) Practical(Lab.) Lecture(DT) Practical(Lab.) CE SEE Total CE SEE Total L TU Total P TW Total L TU Total P TW Total 1 MMA401ATH Algebraic Topology MMA402DMS Differential Manifolds MMA403AOR Advance Operation Research 4 MMA404PIC Programming in C MMB405PSM Problem Solving in Mathematics Total Page 1 of 6

2 Subject code MMA401ATH Subject Name Algebraic Topology Concept of Topology, Group theory After successful completion of the course, students shall be able to construct fundamental groups of the topological spaces as well as a path or a loop in the topological space. 1 Homotopy theory: Homotopy of paths and loops, Product of two loops, fundamental group, Homomorphism induced by homotopy, retraction. 2 Covering spaces, the fundamental groups of the circle, lifting of a path, Path lifting theorem, Lifting correspondence, generator and order, Retractions and fixed points, no retraction theorem, Brouwer fixed point theorem 3 & 4 The fundamental theorem of algebra, The Borsuk-Ulam theorem, Deformation retracts and homotopy type, first fundamental group of doubly punctured plane and theta-space. Homotoy equivalence, the fundamental group of the punctured plane, the n-sphere. 5 Fundamental group of some surface: figure eight, tourus and double tourus, projective Plane, The Jordan Separation Theorem and Nulhomotopy lemma for 1 Elements of Algebraic Topology, James R. Munkres, Addison wesley Pub.Co Basic concepts of Algebraic Topology by Fred H. Croom Springer, Verlag Algebraic Topology: An Introduction by W. S. Massey, Springer Verlag Homotopy Theory by S. T. Hu, Holden-Day, Inc. San Francisco, Algebraic Topology by C. R. F. Maunder Van Nostrand Reinhold Co., Algebraic Topology by E. H. Spanier, McGraw Hill Book Co., Aspects of Topology by Charles O. Christenson and William L. Voxman, Marcel Dekker Inc. 8 Algebraic Topology: An Introduction by W. S. Massey Harcourt Brace Jovanovich, Algebraic Topology by E. H. Spanier, McGraw-Hill Book Co Page 2 of 6

3 Subject code MMA402DMS Subject Name Differential Manifolds Concept of Differential and Integral Calculus, Group theory, Tensor algebra. After successful completion of the course, students shall be able to construct Manifold as well as Lie Group. 1 Quick review of differential calculus, directional derivatives, derivatives of linear transformations, derivatives in matrix spaces, higher derivatives. 2 Smooth functions with compact supports in, differential manifolds, examples. Smoothmaps, diffeomorphisms, Lie groups 3 Tangent spaces to a manifold, basis theorem, derivatives of smooth maps, vector field, sub manifolds 4 Ows and exponential maps, Lie groups and Lie algebras, Lie algebra of a Lie group. 5 Multilinear algebra, tensors on a vector space, tensor product of vector spaces, the Exterior algebra. 1 S. Kumaresan, A course on Differential Geometry and Lie Groups. TRIM 22. Hindustan Book Agency, F. Warner, Foundation of Differentiable Manifolds and Lie Groups, Springer-Verlag, M. Spiwak, A Comprehensive Introduction to Differential Geometry, Published on Perish, Page 3 of 6

4 Subject code MMA403AOR Subject Name Advance Operation Research Concept Linear programming, Game theory. After successful completion of the course, students shall be able to classify and formulate integer programming problems as well as dynamic programming problems. 1 Introduction to optimization - Statement of an Optimization Problem - Classification of Optimization Problems - Optimization Techniques advance topic in LPP: Introduction - Revised Simplex Method - Sensitivity or Post optimality Analysis - Karmarkar s Interior Method - Quadratic Programming. 2 Classical Optimization Techniques - Single variable Optimization - Multivariable Optimization with No Constraints - Multivariable Optimization with Equality Constraints - Multivariable Optimization with Inequality Constraints. 3 INTEGER PROGRAMMING: Integer Linear Programming - Gomory s Cutting Plane Method - Integer Nonlinear programming - Branch and Bound Method - Sequential Linear Discrete Programming. 4 & 5 DYNAMIC PROGRAMMING: Introduction - Multistage Decision Process - Concept of Suboptimization and Principle of Optimality - Computational Procedure in Dynamic Programming - Example illustrating the Calculus Method of Solution, the Tabular Method of Solution - Conversion of a Final Value Problem into an Initial Value Problem J. K. Sharma, Operation Research - Theory and Application, 4 th Edition, Macmillan Publishers India Ltd. 2 N. H. Shah, Ravi Gor, Hardik Soni, Operation Research, PHI 3 Singiresu S. Rao, Engineering Optimization: Theory and Practice, 4 th Edition, John Wiley & Sons. Inc. 4 Stephen Boyd and Lieven Vandenberghe, Convex Optimization, CAMBRIDGE UNIVERSITY PRESS Page 4 of 6

5 Subject code MMA404PIC Subject Name Programming in C Credit Theory Hours Practical Computer Programming terminologies. After successful completion of the course, students shall be able to design an algorithmic solution for a given problem as well as C program for simple applications of real life using structures and files. 1 Structure of a C program: the concept of function; preprocessors in C, include statement, function prototype error, comments in C, arithmetic and relational operators in C, data types in C, integer family, oat family, character family, type casting of variables, automatic casting in C, Input output function, I/O format string, precision of numbers, field width, assignment operators, Mathematics expressions, logical expressions, precedence of operators, standard library functions, define statement, common programming errors, arrays in C. 2 branches in C, if, if-else, else-if statement, go to statement, switch statement, Loops, while, do while, for, break and continue statements, nesting of loops. 3 Functions: global and local variable, functions definition, calling a function, recursive functions, passing by values and passing by reference, function prototype-forward reference. 4 & 5 elementary problems of number theory such as sum of digits of a number, reverse order of digits of a number, primes, perfect, Fibonacci numbers, factorization of a number, Roots of quadratic equation, maximum/minimum and average of n numbers, values of sin(x), cos(x) etc. Solution of f(x) = 0 by using numerical methods. 1 P. B. Mahapatra, Thinking in C including objects orientated programming with C++, Wheeler Publishing, New Delhi 2 B. W. Kernighan and D. M. Ritchie, The C programming Language. Prentice Hall of India Pvt. Ltd., V. Rajaraman, Computer Programming in C, Prentice Hall of India Pvt. Ltd., Page 5 of 6

6 Subject code MMB405PSM Subject Name Problem Solving in Mathematics Students should have theoretical knowledge of subject mentioned in below content. After successful completion of the course, students shall be able to improve problem solving skill as well as implement the Mathematical concept to solve real world problems. Content Students will be required to prepare him/ her for any two of the following courses at the level up to M.Sc. second semester of Ganpat University for problems and exercises. The Regular teaching involves intensive problem session followed by problem assignments; and the examination would consist of problems only. 1.Algebra 2.Topology 3.Real Analysis 4.Complex Analysis 5.Linear Algebra 6.Ordinary Differential Equations 7.Partial Differential Equations Hrs 60 Page 6 of 6