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1 Semester - 2018 - Batch | Course Code |
Course |
Type |
Hours Per Week |
Credits |
Marks |
MTH111 | TEACHING TECHNOLOGY AND RESEARCH METHODOLOGY IN MATHEMATICS AND SERVICE LEARNING | Add On Courses | 3 | 2 | 100 |
MTH131 | NUMBER THEORY AND CRYPTOGRAPHY | - | 4 | 4 | 100 |
MTH132 | GENERAL TOPOLOGY | - | 4 | 4 | 100 |
MTH133 | ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS | - | 4 | 4 | 100 |
MTH134 | FLUID MECHANICS | - | 4 | 4 | 100 |
MTH135 | ELEMENTARY GRAPH THEORY | - | 4 | 4 | 100 |
MTH151 | MATHEMATICS LAB USING FOSS TOOLS | - | 3 | 3 | 50 |
2 Semester - 2018 - Batch | Course Code |
Course |
Type |
Hours Per Week |
Credits |
Marks |
MTH211 | STATISTICS | - | 3 | 2 | 100 |
MTH231 | REAL ANALYSIS | - | 4 | 4 | 100 |
MTH232 | COMPLEX ANALYSIS | - | 4 | 4 | 100 |
MTH233 | ADVANCED ALGEBRA | - | 4 | 4 | 100 |
MTH234 | ADVANCED FLUID MECHANICS | - | 4 | 4 | 100 |
MTH235 | ALGORITHMIC GRAPH THEORY | - | 4 | 4 | 100 |
MTH251 | MATHEMATICS LAB USING PYTHON | - | 3 | 3 | 50 |
3 Semester - 2017 - Batch | Course Code |
Course |
Type |
Hours Per Week |
Credits |
Marks |
MTH311 | STATISTICS | Add On Courses | 3 | 2 | 100 |
MTH331 | MEASURE THEORY AND LEBESGUE INTEGRATION | - | 4 | 4 | 100 |
MTH332 | COMPUTER ORIENTED NUMERICAL METHODS USING MATLAB | - | 4 | 4 | 100 |
MTH333 | CLASSICAL MECHANICS | - | 4 | 4 | 100 |
MTH334 | LINEAR ALGEBRA | - | 4 | 4 | 100 |
MTH335 | ADVANCED GRAPH THEORY | - | 4 | 4 | 100 |
MTH371 | INTERNSHIP IN PG MATHEMATICS COURSE | - | 0 | 2 | 0 |
4 Semester - 2017 - Batch | Course Code |
Course |
Type |
Hours Per Week |
Credits |
Marks |
MTH431 | DIFFERENTIAL GEOMETRY | - | 4 | 4 | 100 |
MTH432 | COMPUTATIONAL FLUID DYNAMICS | - | 4 | 4 | 100 |
MTH433 | FUNCTIONAL ANALYSIS | - | 4 | 4 | 100 |
MTH441 | CALCULUS OF VARIATIONS AND INTEGRAL EQUATIONS | - | 4 | 4 | 100 |
MTH448 | COMBINATORIAL MATHEMATICS | - | 4 | 4 | 100 |
MTH451 | PROJECT | - | 2 | 2 | 100 |
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Introduction to Program: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
The M.Sc. course in Mathematics aims at developing mathematical ability in students with acute and abstract reasoning. The course will enable students to cultivate a mathematician's habit of thought and reasoning and will enlighten students with mathematical ideas relevant for oneself and for the course itself.
COURSE DESIGN: Masters in Mathematics is a two year programme spreading over four semesters. In the first two semesters, focus is on the basic courses in mathematics such as Topology, Algebra, Analysis, Number Theory and ordinary and partial differential equations. Mathematics using FOSS Tools and Mathematics using Python are the two lab oriented courses introduced in first and second semesters respectively. In the third and fourth semesters, focus is on the special courses, elective course and skill-based courses including Functional Analysis, Advanced Fluid Mechanics, Advanced Graph Theory and Computer oriented Numerical Methods using MATLAB. Important feature of the curriculum is that one course on the topic Fluid Mechanics and Graph Theory is offered in each semester with a project on these topics in the fourth semester, which will help the students to pursue the higher studies and research in these topics. Mandatory internship at the end of the first year is another salient feature of the programme to provide exposure to the students in industry or academia. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Assesment Pattern | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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Examination And Assesments | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
EXAMINATION AND ASSESSMENTS (Theory)
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MTH111 - TEACHING TECHNOLOGY AND RESEARCH METHODOLOGY IN MATHEMATICS AND SERVICE LEARNING (2018 Batch) | |
Total Teaching Hours for Semester:45 |
No of Lecture Hours/Week:3 |
Max Marks:100 |
Credits:2 |
Course Objectives/Course Description |
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This course is intended to assist the students in acquiring necessary skills on the use of modern technology in teaching. Also, the students are exposed to the principles, procedures and techniques of planning and implementing the research project. Through service learning they will apply the knowledge in real-world situations and benefit the community.
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Course Outcome |
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On successful completion of the course, the students should be able to |
Unit-1 |
Teaching Hours:10 |
Teaching Technology
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Development of concept of teaching, Teaching skills, Chalk board skills, Teaching practices, Effective teaching, Models of teaching, Teaching aids(Audio-Visual), Teaching aids(projected and non-projected), Communication skills, Feed back in teaching, Teacher’s role and responsibilities, Information technology for teaching.
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Unit-2 |
Teaching Hours:10 |
Research Methodology
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Introduction to research and research methodology, Scientific methods, Choice of research problem, Literature survey and statement of research problem, Reporting of results, Roles and responsibilities of research student and guide.
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Unit-3 |
Teaching Hours:10 |
Mathematical research methodology
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Introducing mathematics Journals, Reading a Journal article, Mathematics writing skills. -Standard Notations and Symbols, Using Symbols and Words, Organizing a paper, Defining variables, Symbols and notations, Different Citation Styles, IEEE Referencing Style in detail.
Package for Mathematics Typing, MS Word, Math Type, Open Office Math Editor, Tex, yEd Graph Editor, Tex in detail, Installation and Set up, Text, Formula, Pictures and Graphs, Producing various types of documents using TeX. | |
Unit-4 |
Teaching Hours:15 |
Service Learning
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Guidelines for service learning: One among the following can be considered as a service learning module:
After deciding, get approval from your respective mentors.
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Text Books And Reference Books: . | |
Essential Reading / Recommended Reading
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Evaluation Pattern . | |
MTH131 - NUMBER THEORY AND CRYPTOGRAPHY (2018 Batch) | |
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
Max Marks:100 |
Credits:4 |
Course Objectives/Course Description |
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This course is concerned with the basics of analytical number theory. Topics such as divisibility, congruence’s, quadratic residues and functions of number theory are covered in this course. An introduction to Cryptography is also included. |
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Course Outcome |
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On successful completion of the course, the students should be able to |
UNIT 1 |
Teaching Hours:15 |
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Divisibility
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The division algorithm, the Greatest Common Divisor, the Euclidean algorithm, Diophantine Equation, the Fundamental Theorem of Arithmetic, the methods to find prime numbers, the Goldbach Conjecture | |||||||||||||||||||||||||||||
UNIT 2 |
Teaching Hours:15 |
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Congruences
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Basic Properties of Congruence, Complete residue system modulo m, reduced residue system modulo m, Euler’s φ function, Fermat’s theorem, Euler’s generalization of Fermat’s theorem, Wilson’s theorem, solutions of linear congruences, the Chinese remainder theorem, solutions of polynomial congruences, prime power moduli, power residues. | |||||||||||||||||||||||||||||
UNIT 3 |
Teaching Hours:15 |
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Quadratic residues and Some functions of number theoretic-functions
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Legendre symbol, Gauss’s lemma, quadratic reciprocity, the Jacobi symbol, binary quadratic forms, equivalence and reduction of binary quadratic forms, sums of two squares, positive definite binary quadratic forms. | |||||||||||||||||||||||||||||
UNIT 4 |
Teaching Hours:15 |
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Introduction to Cryptography (self learning module)
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From Caesar Cipher to Public Key Cryptography, The Knapsack Cryptosystem, An Application of Primitive Roots to Cryptography. | |||||||||||||||||||||||||||||
Text Books And Reference Books: David M. Burton, Elementary Number Theory, 15th Ed. Tata McGraw-Hill, 2016. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading 1. Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, Springer, 2010. 2. Neal Koblitz, A course in number theory and cryptography, Reprint: Springer, 2010. 3. Gareth A. Jones and J. Mary Jones, Elementary number theory, Reprint, Springer, 2000. 4. Joseph H. Silverman, A friendly introduction to number theory, Pearson Prentice Hall, 2006. 5. Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An introduction to the theory of numbers, John Wiley, 2004. | |||||||||||||||||||||||||||||
Evaluation Pattern
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MTH132 - GENERAL TOPOLOGY (2018 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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This course deals with the essentials of topological spaces and their properties in terms of continuity, connectedness, compactness etc. |
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Course Outcome |
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Upon successful completion of this course, students will be able to |
Unit-1 |
Teaching Hours:15 |
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Topological Spaces
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Elements of topological spaces, basis for a topology, the order topology, the product topology on X x Y, the subspace topology, Closed sets and limit points. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Continuous Functions
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Continuous functions, the product topology, metric topology | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Connectedness and Compactness
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Connected spaces, connected subspaces of the Real Line, components and local connectedness, compact spaces, Compact Subspaces of the Real Line, limit point compactness, local compactness. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Countability and Separation Axioms
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The countability axioms, the separation axioms, normal spaces, the Urysohn lemma, the Urysohn metrization theorem, Tietze extension theorem. | |||||||||||||||||||||||||||||
Text Books And Reference Books: J.R. Munkres, Topology, Second Edition, Prentice Hall of India, 2007. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH133 - ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS (2018 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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This helps students understand the beauty of the important branch of mathematics, namely, differential equations. This course includes a study of second order linear differential equations, adjoint and self-adjoint equations, Eigen values and Eigen vectors of the equations, power series method for solving differential equations, second order partial differential equations like wave equation, heat equation, Laplace equations and their solutions by Eigen function method. |
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Course Outcome |
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On successful completion of the course, the students should be able to |
UNIT 1 |
Teaching Hours:20 |
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Linear Differential Equations
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Linear differential equations, fundamental sets of solutions, Wronskian, Liouville’s theorem, adjoint and self-adjoint equations, Lagrange identity, Green’s formula, zeros of solutions, comparison and separation theorems. Legendre, Bessel's, Chebeshev's, Eigen values and Eigen functions, related examples. | |||||||||||||||||||||||||||||
UNIT 2 |
Teaching Hours:10 |
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Power series solutions
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Solution near an ordinary point and a regular singular point by Frobenius method, solution near irregular singular point, hypergeometric differential equation and its polynomial solutions, standard properties.
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UNIT 3 |
Teaching Hours:15 |
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Partial Differential Equations
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Formation of PDE, solutions of first and second order PDE, mathematical models representing stretched string, vibrating membrane, heat conduction in solids and the gravitational potentials, second-order equations in two independent variables, canonical forms and general solution. | |||||||||||||||||||||||||||||
UNIT 4 |
Teaching Hours:15 |
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Solutions of PDE
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The Cauchy problem for homogeneous wave equation, D’Alembert’s solution, domain of influence and domain of dependence, the Cauchy problem for non-homogeneous wave equation, the method of separation of variables for the one-dimensional wave equation and heat equation. Boundary value problems, Dirichlet and Neumann problems in Cartesian coordinates, solution by the method of separation of variables. Solution by the method of eigenfunctions. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH134 - FLUID MECHANICS (2018 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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This course aims at studying the fundamentals of fluid mechanics such as kinematics of fluid, incompressible flow and boundary layer flows. |
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Course Outcome |
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On successful completion of the course, the students should be able to |
Unit-1 |
Teaching Hours:15 |
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Cartesian tensors and continuum hypothesis
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Cartesian tensors: Cartesian tensors, basic properties, transpose, symmetric and skew symmetric tensors, gradient, divergence and curl in tensor calculus, integral theorems. Continuum hypothesis: deformation gradient, strain tensors, infinitesimal strain, compatibility relations, principal strains, material and local time derivatives, transport formulas, stream lines, path lines. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:20 |
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Stress, Strain and basic physical laws
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Stress and Strain: stress components and stress tensor, normal and shear stresses, principal stresses, transformation of the rate of strain and stress, relation between stress and rate of strain. Fundamental basic physical laws: The equation of continuity, conservation of mass, equation of motion (Navier-Stokes equations), conservation of momentum, the energy equation, conservation of energy. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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One, Two and Three Dimensional Invisid Incompressible Flow
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Bernoulli equation, applications of Bernoulli equation, circulation theorems, circulation concept, Kelvin’s theorem, constancy of circulation, Laplace equations, stream functions in two and three dimensional motion. Two dimensional flow: Rectilinear flow, source and sink, the theorem of Blasius.
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Unit-4 |
Teaching Hours:10 |
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Two Dimensional flows of Inviscid fluid
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Flow between parallel flat plates, Couette flow, plane Poiseuille flow, the Hagen-Poiseuille flow, flow between two concentric rotating cylinders.
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Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern Examination and Assessments
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MTH135 - ELEMENTARY GRAPH THEORY (2018 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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This course is an introductory course to the basic concepts of Graph Theory. This includes definition of graphs, vertex degrees, directed graphs, trees, distances, connectivity and paths.
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Course Outcome |
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On successful completion of the course, the students should be able to |
UNIT 1 |
Teaching Hours:15 |
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Introduction to Graphs
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Definition and introductory concepts, Graphs as Models, Matrices and Isomorphism, Decomposition and Special Graphs, Connection in Graphs, Bipartite Graphs, Eulerian Circuits. | |||||||||||||||||||||||||||||
UNIT 2 |
Teaching Hours:15 |
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Vertex Degrees and Directed Graphs
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Counting and Bijections, Extremal Problems, Graphic Sequences, Directed Graphs, Vertex Degrees, Eulerian Digraphs, Orientations and Tournaments. | |||||||||||||||||||||||||||||
UNIT 3 |
Teaching Hours:15 |
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Trees and Distance
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Properties of Trees, Distance in Trees and Graphs, Enumeration of Trees, Spanning Trees in Graphs, Decomposition and Graceful Labellings, Minimum Spanning Tree, Shortest Paths.
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UNIT 4 |
Teaching Hours:15 |
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Connectivity and Paths
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Connectivity, Edge - Connectivity, Blocks, 2 - connected Graphs, Connectivity in Digraphs, k - connected and k-edge-connected Graphs, Maximum Network Flow, Integral Flows.
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Text Books And Reference Books: D.B. West, Introduction to Graph Theory, New Delhi: Prentice-Hall of India, 2011. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern Examination and Assessments
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MTH151 - MATHEMATICS LAB USING FOSS TOOLS (2018 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:45 |
No of Lecture Hours/Week:3 |
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Max Marks:50 |
Credits:3 |
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Course Objectives/Course Description |
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This course aims as introducing the mathematical software packages “WxMaxima” and “Scilab”, for learning basic operation on matrix manipulation, plotting graphs etc.,. Also, these software packages will help students to solve problems / applied problems on Mathematics. |
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Course Outcome |
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On successful completion, a student will be able to Use the basic commands in WxMaxima including the 2D and 3D plots Have a strong command on the inbuilt commands required for the learning and analyzing mathematics Solve problems / applied problems on mathematics by using Scilab. |
Unit-1 |
Teaching Hours:15 |
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Introduction to WxMaxima
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Introduction to WxMaxima Interface - Maxima expressions, numbers, operators, constants and reserved words - input and output in WxMaxima - 2D and 3D plots in WxMaxima - symbolic computations in WxMaxima - Solving Ordinary differential equations in WxMaxima. | |||||||||||||||||||||||||
Unit-2 |
Teaching Hours:20 |
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Introduction to Scilab
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Introduction to Scilab and commands connected with Matrices - Computations with Matrices - 2D and 3D Plots - Script Files and Function Files. | |||||||||||||||||||||||||
Unit-3 |
Teaching Hours:10 |
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Solving problems using Scilab / WxMaxima
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Solving systems of equation and explain consistence - Find the values of some standard trigonometric functions in radians as well as in degree - Create polynomials of different degrees and find its real roots - Display Fibonacci series using Scilab program - Display non-Fibonacci series using Scilab program. | |||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern The course is evaluated based on continuous internal assessment (CIA). The parameters for evaluation under each component and the mode of assessment are given below:
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MTH211 - STATISTICS (2018 Batch) | |||||||||||||||||||||||||
Total Teaching Hours for Semester:45 |
No of Lecture Hours/Week:3 |
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Max Marks:100 |
Credits:2 |
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Course Objectives/Course Description |
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This course aims at teach the students the idea of discrete and continuous random variables, Probability theory, in-depth treatment of discrete random variables and distributions, with some introduction to continuous random variables and introduction to estimation and hypothesis testing. |
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Course Outcome |
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On successful completion of the course, the students should be able to |
Unit-1 |
Teaching Hours:15 |
Random Variables and Expectation
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Discrete and continuous random variables, distribution functions, probability mass and density functions, bivariate distributions, marginal and conditional distributions, expected value of a random variable, independence of random variables, conditional expectations, covariance matrix, correlation coefficients and regression, Chebyshev’s inequality, moments, moment generating functions, characteristic functions. | |
Unit-2 |
Teaching Hours:15 |
Probability Distributions
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Probability: Sample spaces, events, probability of an event, theorems on probability, conditional probability, independent events, Bayes theorem. Boole’s inequality. Discrete Probability Distribution: Introduction, uniform, Bernoulli, Binomial, negative Binomial, geometric, Hypergeometric and Poisson distribution. Continuous Probability Distributions: Introduction, uniform, gamma, exponential, beta and normal distributions. | |
Unit-3 |
Teaching Hours:15 |
Sampling distributions
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t, F and chi-square distributions, standard errors and large sample distributions. | |
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern . | |
MTH231 - REAL ANALYSIS (2018 Batch) | |
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
Max Marks:100 |
Credits:4 |
Course Objectives/Course Description |
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This course will help students understand the concepts of functions of single and several variables. This course includes such concepts as Riemann-Stieltjes integral, sequences and series of functions, Special Functions and The Implicit Function Theorem. |
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Course Outcome |
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On successful completion of the course, the students should be able to |
UNIT 1 |
Teaching Hours:15 |
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The Riemann-Stieltjes Integration
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Definition and Existence of Riemann-Stieltjes Integral, Linearity Properties of Riemann-Stieltjes Integral, The Riemann-Stieltjes Integral as the Limit of Sums, Integration and Differentiation, Integration of Vector-valued Functions, Rectifiable Curves. | |||||||||||||||||||||||||||||
UNIT 2 |
Teaching Hours:15 |
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Sequences and Series of Functions
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Pointwise and uniform convergence, Uniform Convergence: Continuity, Integration and Differentiation, Equicontinuous Families of Functions, The Stone-Weierstrass Theorem | |||||||||||||||||||||||||||||
UNIT 3 |
Teaching Hours:15 |
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Some Special Functions
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Power Series, The Exponential and Logarithmic Functions, The Trigonometric Functions, The Algebraic Completeness of the Complex Field, Fourier Series, The Gamma Function. | |||||||||||||||||||||||||||||
UNIT 4 |
Teaching Hours:15 |
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Functions of Several Variables
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Linear Transformations, Differentiation, The Contraction Principle, The Inverse Function Theorem, The Implicit Function Theorem, The Rank Theorem, Determinants, Derivatives of Higher Order, Differentiation of Integrals
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Text Books And Reference Books: W. Rudin, Principles of Mathematical Analysis, 3rd ed., New Delhi: McGraw-Hill (India), 2016. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH232 - COMPLEX ANALYSIS (2018 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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This course will help students learn about the essentials of complex analysis. This course includes important concepts such as power series, analytic functions, linear transformations, Laurent’s series, Cauchy’s theorem, Cauchy’s integral formula, Cauchy’s residue theorem, argument principle, Schwarz lemma , Rouche’s theorem and Hadamard’s 3-circles theorem. |
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Course Outcome |
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On successful completion of the course, the students should be able to |
Unit-1 |
Teaching Hours:18 |
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Power Series
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Power series, radius and circle of convergence, power series and analytic functions, Line and contour integration, Cauchy’s theorem, Cauchy integral formula, Cauchy integral formula for derivatives, Cauchy integral formula for multiply connected domains, Morera’s theorem, Gauss mean value theorem, Cauchy inequality for derivatives, Liouville’s theorem, fundamental theorem of algebra, maximum and minimum modulus principles. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Singularities
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Taylor’s series, Laurent’s series, zeros of analytical functions, singularities, classification of singularities, characterization of removable singularities and poles. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Mappings
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Rational functions, behavior of functions in the neighborhood of an essential singularity, Cauchy’s residue theorem, contour integration problems, Mobius transformation, conformal mappings. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:12 |
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Meromorphic functions
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Meromorphic functions and argument principle, Schwarz lemma, Rouche’s theorem, convex functions and their properties, Hadamard 3-circles theorem.
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Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH233 - ADVANCED ALGEBRA (2018 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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This course enables students to understand the intricacies of advanced areas in algebra. This includes a study of advanced group theory, Euclidean rings, polynomial rings and Galois theory.
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Course Outcome |
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On successful completion of the course, the students should be able to |
Unit-1 |
Teaching Hours:15 |
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Advanced Group Theory
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Automorphisms, Cayley’s theorem, Cauchy’s theorem, permutation groups, symmetric groups, alternating groups, simple groups, conjugate elements and class equations of finite groups, Sylow theorems, direct products, finite abelian groups, solvable groups. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Rings
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Euclidean Ring, Polynomial rings, polynomials rings over the rational field, polynomial rings over commutative rings, | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Fields
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Extension fields, roots of polynomials, construction with straightedge and compass, more about roots. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Galois theory
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The elements of Galois theory, solvability by radicals, Galois group over the rationals, finite fields. | |||||||||||||||||||||||||||||
Text Books And Reference Books: I. N. Herstein, Topics in algebra, Second Edition, John Wiley and Sons, 2007. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH234 - ADVANCED FLUID MECHANICS (2018 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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This course helps the students to understand the basic concepts of heat transfer, types of convection shear and thermal instability of linear and non-linear problems. This course also includes the analysis Prandtl boundry layer, porous media and Non-Newtonian fluid.
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Course Outcome |
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On successful completion of the course, the students should be able to |
UNIT 1 |
Teaching Hours:15 |
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Dimensional Analysis and Similarity
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Introduction to heat transfer, different modes of heat transfer- conduction, convection and radiation, steady and unsteady heat transfer, free and forced convection. Non-dimensional parameters determined from differential equations – Buckingham’s Pi Theorem – Non-dimensionalization of the Basic Equations - Non-dimensional parameters and dynamic similarity. | |||||||||||||||||||||||||||||
UNIT 2 |
Teaching Hours:20 |
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Heat Transfer and Thermal Instability
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Shear Instability: Stability of flow between parallel shear flows - Squire’s theorem for viscous and inviscid theory – Rayleigh stability equation – Derivation of Orr-Sommerfeld equation assuming that the basic flow is strictly parallel. Basic concepts of stability theory – Linear and Non-linear theories – Rayleigh Benard Problem – Analysis into normal modes – Principle of Exchange of stabilities – first variation principle – Different boundary conditions on velocity and temperature. | |||||||||||||||||||||||||||||
UNIT 3 |
Teaching Hours:10 |
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Prandtl Boundry Layer
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Boundary layer concept, the boundary layer equations in two-dimensional flow, the boundary layer along a flat plate, the Blasius solution. Stagnation point flow. Falkner-Skan family of equations. | |||||||||||||||||||||||||||||
UNIT 4 |
Teaching Hours:15 |
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Porous Media and Non - Newtonian Fluids
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Introduction to porous medium, porosity, Darcy’s Law, Extension of Darcy Law – accelerations and inertial effects, Brinkman’s equation, effects of porosity variations, Bidisperse porous media. Constitutive equations of Maxwell, Oldroyd, Ostwald , Ostwald de waele, Reiner – Rivlin and Micropolar fluid. Weissenberg effect and Tom’s effect.Equation of continuity, Conservation of momentum for non-Newtonian fluids. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH235 - ALGORITHMIC GRAPH THEORY (2018 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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This course helps the students to understand the colouring of graphs, Planar graphs, edges and cycles. |
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Course Outcome |
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On successful completion of the course, the students should be able to |
Unit-1 |
Teaching Hours:15 |
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Colouring of Graphs
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Definition and Examples of Graph Colouring, Upper Bounds, Brooks’ Theorem, Graph with Large Chromatic Number, Extremal Problems and Turan’s Theorem, Colour-Critical Graphs, Counting Proper Colourings. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Matchings and Factors
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Maximum Matchings, Hall’s Matching Condition, Min-Max Theorem, Independent Sets and Covers, Maximum Bipartite Matching, Weighted Bipartite Matching, Tutte’s 1-factor Theorem, Domination. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Planar Graphs
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Drawings in the Plane, Dual Graphs, Euler’s Formula, Kuratowski’s Theorem, Convex Embeddings, Coloring of Planar Graphs, Thickness and Crossing Number. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Edges and Cycles Edge
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Colourings, Characterisation of Line Graphs, Necessary Conditions of Hamiltonian Cycles, Sufficient Conditions of Hamiltonian Cycles, Cycles in Directed Graphs, Tait’s Theorem, Grinberg’s Theorem, Flows and Cycle Covers. | |||||||||||||||||||||||||||||
Text Books And Reference Books: D.B. West, Introduction to Graph Theory, New Delhi: Prentice-Hall of India, 2011. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH251 - MATHEMATICS LAB USING PYTHON (2018 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:45 |
No of Lecture Hours/Week:3 |
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Max Marks:50 |
Credits:3 |
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Course Objectives/Course Description |
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This course aims as introducing the mathematical programming language and its uses in solving problems on discrete mathematics and partial differential equations. |
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Course Outcome |
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On successful completion, a student will be able to: Use the basic commands in Python, including the 2D and 3D plots Use Python in solving problems on PDE Use Python in solving problems on Discrete Mathematics |
Unit-1 |
Teaching Hours:15 |
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Introduction to Python Programming
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Python commands: Comments, Number and other data types, Expressions, Operators, Variables and assignments, Decisions, Loops, Lists, Strings - plotting using “matplotlib” - Basic operations , Simplification, Calculus, Solvers and Matrices using Sympy. | |||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Differential Equations using Python
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Solving ODE’s using Python - Libraries for Differential equations in Python, PDE’s using sympy user functions pde_seperate(), pde_seperate_add(). pde_seperate_mul(), pdsolve(), classify_pde(), checkpdesol() , pde_1st_linear_constant_coeff_homogeneous, pde_1st_linear_constant_coeff, pde_1st_linear_variable_coeff. | |||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Discrete Mathematics using Python
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Creating and visualizing Graphs, Digraphs, MultiGraphs and MultiDiGraph - Python methods for reporting nodes, edges and neighbours of the given graph / digraph - Python methods for counting nodes, edges and neighbours of the given graph / digraph. | |||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern The course is evaluated based on continuous internal assessment (CIA). The parameters for evaluation under each component and the mode of assessment are given below:
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MTH311 - STATISTICS (2017 Batch) | |||||||||||||||||||||||||
Total Teaching Hours for Semester:45 |
No of Lecture Hours/Week:3 |
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Max Marks:100 |
Credits:2 |
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Course Objectives/Course Description |
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This course aims at teach the students the idea of discrete and continuous random variables, Probability theory, in-depth treatment of discrete random variables and distributions, with some introduction to continuous random variables and introduction to estimation and hypothesis testing. |
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Course Outcome |
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On successful completion of the course, the students should be able to |
Unit-1 |
Teaching Hours:15 |
Random Variables and Expectation
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Discrete and continuous random variables, distribution functions, probability mass and density functions, bivariate distributions, marginal and conditional distributions, expected value of a random variable, independence of random variables, conditional expectations, covariance matrix, correlation coefficients and regression, Chebyshev’s inequality, moments, moment generating functions, characteristic functions. | |
Unit-2 |
Teaching Hours:15 |
Probability Distributions
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Probability: Sample spaces, events, probability of an event, theorems on probability, conditional probability, independent events, Bayes theorem. Boole’s inequality. Discrete Probability Distribution: Introduction, uniform, Bernoulli, Binomial, negative Binomial, geometric, Hypergeometric and Poisson distribution. Continuous Probability Distributions: Introduction, uniform, gamma, exponential, beta and normal distributions. | |
Unit-3 |
Teaching Hours:15 |
Sampling distributions
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t, F and chi-square distributions, standard errors and large sample distributions. | |
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern . | |
MTH331 - MEASURE THEORY AND LEBESGUE INTEGRATION (2017 Batch) | |
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
Max Marks:100 |
Credits:4 |
Course Objectives/Course Description |
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The course covers the basic material that one needs to know in the theory of functions of a real variable and measure and integration theory as expounded by Henri Léon Lebesgue. |
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Course Outcome |
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On successful completion of the course, the students should be able to |
Unit-1 |
Teaching Hours:20 |
Lebesgue Measure
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Lebesgue Outer Measure, The s-Algebra of Lebesgue Measurable Sets, Outer and Inner Approximation of Lebesgue Measurable Sets, Countable Additivity, Continuity and the Borel-Cantelli Lemma, Nonmeasurable Sets, The Cantor Set and the Canton-Lebesgue Function, Sums, Products and Compositions of Lebesgue Measurable Functions, Sequential Pointwise Limits and Simple Approximation, Littlewood’s three principles, Egoroff’s Theorem and Lusin’s Theorem. | |
Unit-2 |
Teaching Hours:15 |
The Lebesgue Integration
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The Lebesgue Integral of a Bounded Measurable Function over a Set of Finite Measure, The Lebesgue Integral of a Measurable Nonnegative Function; The General Lebesgue Integral; Countable Additivity and Continuity of Integration, Uniform Integrability, Uniform Integrability and Tightness, Convergence in measure, Characterizations of Riemann and Lebesgue Integrability. | |
Unit-3 |
Teaching Hours:15 |
Differentiation and Lebesgue Integration
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Continuity of Monotone Functions, Differentiation of Monotone Functions, Functions of Bounded Variation, Absolutely Continuous Functions, Integrating Derivatives. | |
Unit-4 |
Teaching Hours:10 |
The Lp Spaces
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Normed Linear Spaces, The Inequalities of Young, Hölder and Minkowski, The Lp spaces, Approximation and Separability, The Riesz Representation for the Dual of Lp, Weak Sequential Convergence in Lp, Weak Sequential Compactness, The Minimization of Convex Functionals. | |
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern CIA - 50% ESE - 50% | |
MTH332 - COMPUTER ORIENTED NUMERICAL METHODS USING MATLAB (2017 Batch) | |
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
Max Marks:100 |
Credits:4 |
Course Objectives/Course Description |
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This course helps students to have an in-depth knowledge of various advanced methods in numerical analysis. It also introduces MATLAB programming for scientific computations. This includes solution of algebraic, transcendental, system of equations, and ordinary differential equations. |
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Course Outcome |
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On successful completion of the course, the students should be able to |
UNIT 1 |
Teaching Hours:20 |
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Solution of algebraic and transcendental equations
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Fixed point iterative method, convergence criterion, Aitken’s -process, Sturm sequence method to identify the number of real roots, Newton-Raphson methods (includes the convergence criterion for simple roots), Bairstow’s method, Graeffe’s root squaring method, Birge-Vieta method, Muller’s method. Solution of Linear System of Algebraic Equations: LU-decomposition methods (Crout’s, Choleky and Delittle methods), consistency and ill-conditioned system of equations, Tri-diagonal system of equations, Thomas algorithm. | |||||||||||||||||||||||||||||
UNIT 2 |
Teaching Hours:15 |
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Numerical solution of ordinary differential equations
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Initial value problems, Runge-Kutta methods of second and fourth order, multistep method, Adams-Moulton method, stability (convergence and truncation error for the above methods), boundary value problems, second order finite difference method, linear shooting method. | |||||||||||||||||||||||||||||
UNIT 3 |
Teaching Hours:10 |
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Fundamentals of MATLAB
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The MATLAB environment, basic operations, use of built-in functions, graphics, programming with MATLAB. | |||||||||||||||||||||||||||||
UNIT 4 |
Teaching Hours:15 |
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Numerical methods with MATLAB
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Elementary numerical methods with MATLAB, Solution to single equations and multiple non-linear equations in MATLAB.
Linear system of equations, Numerical differentiation and integration in MATLAB , Data fitting in MATLAB , Solution to Ordinary Differential Equations in MATLAB , Numerical differentiation and finite differences. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading 1. S. Attaway MATLAB: A Practical Introduction to Programming and Problem Solving, 3rd edition, Elsevier, 2013. 2. S.C.Chapra and R.P. Canale , NumerialMethodsforEngineers,5th Ed.,McGrawHill, 2006. 3. Beers, Kenneth J. Numerical Methods for Chemical Engineering: Applications in MATLAB®. New York, NY: Cambridge University Press, November 2006. 4. Recktenwald, Gerald W. Introduction to Numerical Methods with MATLAB®: Implementations and Applications. Upper Saddle River, NJ: Prentice-Hall, 2000.
5. K. Mishra, A Handbook on Numerical Technique Lab (MATLAB Based Experiments), I.K. International Publishing House Pvt. Limited, 2007. | |||||||||||||||||||||||||||||
Evaluation Pattern
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MTH333 - CLASSICAL MECHANICS (2017 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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This course deals with some of the key ideas of classical mechanics. The concepts covered in the course include generalized coordinates, Lagrange’s equations, Hamilton’s equations and Hamilton - Jacobi theory. |
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Course Outcome |
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On successful completion of the course, the students should be able to |
Unit-1 |
Teaching Hours:12 |
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Introductory concepts
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The mechanical system - Generalised Coordinates - constraints - virtual work - Energy and momentum. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:20 |
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Lagrange's equation
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Derivation and examples - Integrals of the Motion - Small oscillations. Special Applications of Lagrange’s Equations: Rayleigh’s dissipation function - impulsive motion - velocity dependent potentials. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:13 |
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Hamilton's equations
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Hamilton's principle - Hamilton’s equations - Other variational principles - phase space. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Hamilton - Jacobi Theory
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Hamilton's Principal Function – The Hamilton - Jacobi equation - Separability. | |||||||||||||||||||||||||||||
Text Books And Reference Books: Donald T. Greenwood, Classical Dynamics, Reprint, USA: Dover Publications, 2012. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH334 - LINEAR ALGEBRA (2017 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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This course aims at introducing elementary notions on linear transformations, canonical forms, rational forms, Jordan forms, inner product space and bilinear forms.
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Course Outcome |
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On successful completion of the course, the students should be able to Have thorough understanding of the Linear transformations Demonstrate the elementary canonical forms, rational and Jordan forms. Apply the inner product space Express familiarity in using bilinear forms |
Unit-1 |
Teaching Hours:15 |
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Linear Transformations and Determinants
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Vector Spaces: Recapitulation, Linear Transformations: Algebra of Linear Transformations - Isomorphism – Representation of Transformation by Matrices – Linear Functionals – The transpose of a Linear Transformation, Determinants: Commutative Rings – Determinant Functions – Permutation and the Uniqueness of Determinants – Additional Properties of Determinants | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:20 |
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Elementary Canonical Forms, Rational and Jordan Forms
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Elementary Canonical Forms : Characteristic Values – Annihilating Polynomials – Invariant Subspaces – Simultaneous Triangulation and Diagonalization – Direct sum Decomposition – Invariant Dual Sums - The Primary Decomposition Theorem. The Rational and Jordan Forms: Cyclic subspaces and Annihilators – Cyclic Decompositions and the Rational Form – The Jordan Form – Computation of Invariant Factors – Semi-Simple Operators. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Inner Product Spaces
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Inner Products – Inner Product Spaces – Linear Functionals and Adjoints – Unitary Operators – Normal Operators – Forms on Inner Product Spaces – Positive Forms – Spectral Theory – Properties of Normal Operators. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:10 |
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Bilinear Forms
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Bilinear Forms – Symmetric Bilinear Forms – Skew-Symmetric Bilinear Forms – Groups Preserving Bilinear Forms | |||||||||||||||||||||||||||||
Text Books And Reference Books: K. Hoffman and R. Kunze, Linear Algebra, 2nd ed. New Delhi, India: PHI Learning Private Limited, 2011. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH335 - ADVANCED GRAPH THEORY (2017 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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Domination of Graphs, digraph theory, perfect graphs and chromatic graph theory are dealt with in the detail in this course. |
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Course Outcome |
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On successful completion of the course, the students should be able to |
Unit-1 |
Teaching Hours:15 |
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Domination in Graphs
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Domination in Graphs, Bounds in terms of Order, Bounds in terms of Order, Degree and Packing, Bounds in terms of Order and Size, Bounds in terms of Degree, Diameter and Girth, Bounds in terms of Independence and Covering. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Chromatic Graph Theory
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T-Colourings, L(2,1)-colourings, Radio Colourings, Hamiltonian Colourings, Domination and Colourings.
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Unit-3 |
Teaching Hours:15 |
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Perfect Graphs
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The Perfect Graph Theorem, Chordal Graphs Revisited, Other Classes of Perfect Graphs, Imperfect Graphs, The Strong Perfect Graph Conjecture. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Eigenvalues of Graphs
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The Characteristic Polynomial, Eigenvalues and Graph Parameters, Eigenvalues of Regular Graphs, Eigenvalues and Expanders, Strongly Regular Graphs. | |||||||||||||||||||||||||||||
Text Books And Reference Books: 1. D.B. West, Introduction to Graph Theory, New Delhi: Prentice-Hall of India, 2011. 2. T.W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs. Reprint, CRC Press, 2000. 3. J. Bang-Jensen and G. Gutin, Digraphs. London: Springer, 2009.
4. G. Chartrand and P. Zhang, Chromatic Graph Theory. New York: CRC Press, 2009. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading 1. B. Bollabas, Modern Graph Theory, Springer, New Delhi, 2005. 2. F. Harary, Graph Theory, New Delhi: Narosa, 2001. 3. G. Chartrand and P.Chang, Introduction to Graph Theory, New Delhi: Tata McGraw-Hill,2006. 4. J. A. Bondy and U.S.R. Murty, Graph Theory, Springer, 2008 5. J. Clark and D.A. Holton, A First Look At Graph Theory, Singapore: World Scientific, 2005. 6. R. Balakrishnan and K Ranganathan, A Text Book of Graph Theory, New Delhi: Springer, 2008. 7. R. Diestel, Graph Theory, New Delhi: Springer, 2006. 8. M. Bona, A walk through combinatorics, World scientific, 2011. | |||||||||||||||||||||||||||||
Evaluation Pattern
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MTH371 - INTERNSHIP IN PG MATHEMATICS COURSE (2017 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:45 |
No of Lecture Hours/Week:0 |
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Max Marks:0 |
Credits:2 |
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Course Objectives/Course Description |
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The objective of this course is to provide the students an opportunity to gain work experience in the relevant institution, connected to their subject of study. The experienced gained in the workplace will give the students a competetive edge in their career. |
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Course Outcome |
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On successful completion of the course, the students should be able to · Expose students to the field of their professional interest · Give an opportunity to get practical experience in the field of their interest · Strengthen the curriculum based on internship feedback where relevant · Help student choose their career through practical experience |
Unit-1 |
Teaching Hours:45 |
Internship in PG Mathematics course
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M.Sc. Mathematics students have to undertake a mandatory internship of not less than 45 working days at any of the following: reputed research centres, recognized educational institutions, summer research fellowships, programmes like M.T.T.S or any other approved by the P.G. coordinator and H.O.D. The internship is to be undertaken at the end of second semester (during first year vacation). The report submission and the presentation on the report will be held during the third semester and the credits will appear in the mark sheet of third semester. The students will have to give an internship proposal with the following details: Organization where the student proposes to do the internship, reasons for the choice, nature of internship, period on internship, relevant permission letters, if available, name of the mentor in the organization, email, telephone and mobile numbers of the person in the organization with whom Christ University could communicate matters related to internship. Typed proposals will have to be given at least one month before the end of the second semester. The coordinator of the programme in consultation with the HOD will assign faculty members from the department as guides at least two weeks before the end of second semester. The students will have to be in touch with the guides during the internship period either through personal meetings, over the phone or through email. At the place of internship, students are advised to be in constant touch with their mentors. At the end of the required period of internship, the candidates will submit a report as per the guidelines given by the department. The report should be submitted within first 10 days of the reopening of the University for the third semester. Within 20 days from the day of reopening, the department must hold a presentation by the students. During the presentation the guide or a nominee of the guide should be present and be one of the evaluators. Students should preferably be encouraged to make a power point presentation of their report. A minimum of 10 minutes should be given for each of the presenter. The maximum limit is left to the discretion of the evaluation committee. Students will get 2 credits on successful completion of internship. | |
Text Books And Reference Books: . | |
Essential Reading / Recommended Reading . | |
Evaluation Pattern . | |
MTH431 - DIFFERENTIAL GEOMETRY (2017 Batch) | |
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
Max Marks:100 |
Credits:4 |
Course Objectives/Course Description |
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Differential geometry is the study of geometric properties of curves, surfaces, and their higher dimensional analogues using the methods of calculus. On successful completion of this module students will have acquired an active knowledge and understanding of the basic concepts of the geometry of curves and surfaces in three-dimensional Euclidean space and will be acquainted with the ways of generalising these concepts to higher dimensions |
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Course Outcome |
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On successful completion of the course, the students should be able to |
UNIT 1 |
Teaching Hours:15 |
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Calculus on Euclidean Geometry
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Euclidean Space - Tangent Vectors - Directional derivatives - Curves in E3 - 1-Forms - Differential Forms - Mappings. | |||||||||||||||||||||||||||||
UNIT 2 |
Teaching Hours:15 |
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Frame Fields and Euclidean Geometry
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Dot product - Curves - vector field - The Frenet Formulas - Arbitrary speed curves - cylindrical helix - Covariant Derivatives - Frame fields - Connection Forms - The Structural equations. | |||||||||||||||||||||||||||||
UNIT 3 |
Teaching Hours:15 |
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Euclidean Geometry and Calculus on Surfaces
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Isometries of E3 - The derivative map of an Isometry - Surfaces in E3 - patch computations - Differential functions and Tangent vectors - Differential forms on a surface - Mappings of Surfaces. | |||||||||||||||||||||||||||||
UNIT 4 |
Teaching Hours:15 |
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Shape Operators
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The Shape operator of M in E3 - Normal Curvature - Gaussian Curvature - Computational Techniques - Special curves in a surface - Surfaces of revolution. | |||||||||||||||||||||||||||||
Text Books And Reference Books: B.O’Neill, Elementary Differential geometry, 2nd revised ed., New York: Academic Press, 2006. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH432 - COMPUTATIONAL FLUID DYNAMICS (2017 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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This course helps the students to understand the basic concepts of heat transfer, types of convection shear and thermal instability of linear and non-linear problems, dimensional analysis. The flow problems are analyses using finite element method. |
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Course Outcome |
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On successful completion of the course, the students should be able to Understand both flow physics and mathematical properties of governing Navier-Stokes equations and define proper boundary conditions for solution. An introduction to the theory and practice of the finite element method. Experience with writing a simple finite element solver for an ordinary differential equation in MATLAB. Understanding of physics of compressible and incompressible fluid flows. Ability to solve the fluid flow equations using Finite element method. |
Unit-1 |
Teaching Hours:12 |
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Recapitulation
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Review of classification of partial differential equations, classification of boundary conditions, numerical analysis, basic governing equations of fluid mechanics. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:18 |
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Finite Difference Methods
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Derivation of finite difference methods, finite difference method to parabolic, hyperbolic and elliptic equations, finite difference method to nonlinear equations, coordinate transformation for arbitrary geometry, Central schemes with combined space-time discretization-Lax-Friedrichs, Lax-Wendroff, MacCormack methods, Artificial compressibility method, pressure correction method – Lubrication model, Convection dominated flows – Euler equation – Quasilinearization of Euler equation, Compatibility relations, nonlinear Burger equation | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:12 |
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Finite Volume Methods
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General introduction, Node-centered-control volume, Cell-centered-control volume and average volume, Cell-Centred scheme, Cell-Vertex scheme, Structured and Unstructured FVMs, Second and Fourth order approximations to the convection and diffusion equations (One and Two-dimensional examples) | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:18 |
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Finite Element Methods
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Introduction to finite element methods, one-and two-dimensional bases functions – Lagrange and Hermite polynomials elements, triangular and rectangular elements, Finite element method for one-dimensional problem: model boundary value problems, discretization of the domain, derivation of elemental equations and their connectivity, composition of boundary conditions and solutions of the algebraic equations. Finite element method for two-dimensional problems: model equations, discretization, interpolation functions, evaluation of element matrices and vectors and their assemblage. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH433 - FUNCTIONAL ANALYSIS (2017 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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This abstract course imparts an in-depth analysis of Banach spaces, Hilbert spaces, conjugate spaces, etc. This course also includes a few important applications of functional analysis to other branches of both pure and applied mathematics. |
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Course Outcome |
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On successful completion of the course, the students should be able to |
Unit-1 |
Teaching Hours:15 |
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Banach spaces
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Normed linear spaces, Banach spaces, continuous linear transformations, isometric isomorphisms, functionals and the Hahn-Banach theorem, the natural embedding of a normed linear space in its second dual. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:12 |
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Mapping theorems
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The open mapping theorem and the closed graph theorem, the uniform boundedness theorem, the conjugate of an operator. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Inner products
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Inner products, Hilbert spaces, Schwarz inequality, parallelogram law, orthogonal complements, orthonormal sets, Bessel’s inequality, complete orthonormal sets.
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Unit-4 |
Teaching Hours:18 |
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Conjugate space
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The conjugate space, the adjoint of an operator, self-adjoint, normal and unitary operators, projections, finite dimensional spectral theory.
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Text Books And Reference Books: G.F. Simmons, Introduction to topology and modern Analysis, Reprint, Tata McGraw-Hill, 2004. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH441 - CALCULUS OF VARIATIONS AND INTEGRAL EQUATIONS (2017 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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This course concerns the analysis and applications of calculus of variations and integral equations. Applications include areas such as classical mechanics and differential equations. |
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Course Outcome |
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On successful completion of the course, the students should be able to |
Unit-1 |
Teaching Hours:18 |
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Euler equations and variational notations
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Maxima and minima, method of Lagrange multipliers, the simplest case, Euler equation, extremals, stationary function, geodesics, Brachistochrone problem, natural boundary conditions and transition conditions, variational notation, the more general case. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:16 |
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Advanced variational problems
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Galerkian Technique, the Rayleigh-Ritz method. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:12 |
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Linear integral equations
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Definitions, integral equation, Fredholm and Volterra equations, kernel of the integral equation, integral equations of different kinds, relation between differential and integral equations, symmetric kernels, the Green’s function. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:14 |
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Methods for solutions of linear integral equations
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Fredholm equations with separable kernels, homogeneous integral equations, characteristic values and characteristic functions of integral equations, Hilbert-Schmidt theory, iterative methods for solving integral equations of the second kind, the Neumann series. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
R.P. Kanwal, Linear Integral Equations: Theory and Techniques, New York: Birkhäuser, 2013. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH448 - COMBINATORIAL MATHEMATICS (2017 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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Combinatorics deals with the existence of certain configurations in a structure and when it exists it counts the number of such configurations. In this course we deal with the basic concepts such as Permutations and Combinations, Generating Functions, Recurrence Relations, The Principle of Inclusion and Exclusion including Polya’s theory. |
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Course Outcome |
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After completing this course, the student will be able to: Understand the rules of Sum and Product of Permutations and Combinations. Discuss distributions of Distinct Objects into Non-distinct Cells and Partitions of Integers. Identify Solutions by the technique of Generating Functions and Recurrence Relations with Two Indices. Understand the concepts of Permutations with Restrictions on Relative Positions and the Rook Polynomials. Enumerate configuration using Polya’s Theory. |
Unit-1 |
Teaching Hours:20 |
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Permutations, Combinations and Generating Functions
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Introduction - The rules of Sum and Product - Permutations - Combinations - Distributions of Distinct Objects - Distributions of Non distinct Objects. Generating Functions for Combinations - Enumerators for Permutations – Distributions of Distinct Objects into Non distinct Cells - Partitions of Integers - Elementary Relations. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:12 |
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Recurrence Relations
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Introduction - Linear Recurrence relations with Constant Coefficients - Solution by the technique of Generating Functions - Recurrence Relations with Two Indices. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:13 |
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The Principle of Inclusion and Exclusion
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Introduction - The Principle of Inclusion and Exclusion - The General Formula - Derangements - Permutations with Restrictions on Relative Positions - The Rook Polynomials. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Polya?s Theory of Counting
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Introduction - Equivalence Classes under a Permutation Group - Equivalence Classes of Functions -Weights and Inventories of Functions - Polya’s Fundamental Theorem - Generalization of Polya’s Theorem. | |||||||||||||||||||||||||||||
Text Books And Reference Books: C. L. Liu, Introduction to Combinatorial Mathematics, McGraw-Hill Inc., New york,1968. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH451 - PROJECT (2017 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:30 |
No of Lecture Hours/Week:2 |
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Max Marks:100 |
Credits:2 |
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Course Objectives/Course Description |
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The objective of this course is to develop positive attitude, knowledge and competence for the research in Mathematics. |
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Course Outcome |
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Through this project students will develop analytical and computational skills |
Unit-1 |
Teaching Hours:30 |
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PROJECT
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Students are exposed to the mathematical software packages like Scilab, Maxima, Octave, OpenFOAM, Mathematica and Matlab. Students are given a choice of topic either on Fluid Mechanics or Graph theory or any other topic from other fields with the approval of HOD / Coordinator. Each candidate will work under the supervision of the faculty. Coordinator will allot the supervisor for each candidate in consultation with the HOD at the end of third semester. Project need not be based on original research work. Project could be based on the review of advanced text book or advanced research papers. Each candidate has to submit a dissertation on the project topic followed by viva voce examination. The viva voce will be conducted by the committee constituted by the head of the department which will have an external and an internal examiner. The student must secure 50% of the marks to pass the project examination. The candidates who fail must redo the project as per the university regulation. Time line for Project:
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Text Books And Reference Books: . | |||||||||||||||||||
Essential Reading / Recommended Reading . | |||||||||||||||||||
Evaluation Pattern Assessment: Project is evaluated based on the parameters given below:
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