Syllabus for
MSc (Mathematics)
Academic Year  (2024)

EXAMINATION AND ASSESSMENTS (Theory)

EXAMINATION AND ASSESSMENTS (Practicals)

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:

Department of Mathematics, CHRIST (Deemed to be University) is one of the oldest departments of the University. It offers programmes in Mathematics at the under graduate level, post graduate level as well as Ph.D. The department aims to:

* enhance the logical, reasoning, analytical and problem solving skills of students.

* cultivate a research culture in young minds.

* foster aesthetic appreciation for mathematical thinking.

* encourage students for pursuing higher studies in mathematics.

* motivate students to uphold scientific integrity and objectivity in professional endeavors.

Vision: Excellence and Service

Mission: To organize, connect, create and communicate mathematical ideas effectively, through 4D’s:Dedication, Discipline, Direction and Determination.

The MSc 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 Algebra, Topology, Analysis and Graph Theory along with the basic applied course ordinary and partial differential equations. In the third and fourth semester focus is on the special courses, elective courses and skill-based courses including Measure Theory and Lebesgue Integration, Functional Analysis, Computational Fluid Dynamics, Advanced Graph Theory, Numerical Analysis  and courses on Data Science . Important feature of the curriculum is that students can specialize in any one of areas (i) Fluid Mechanics, (ii) Graph Theory and (iii) Data Science, with a project on these topics in the fourth semester, which will help the students to pursue research in these topics or grab the opportunities in the industry. To gain proficiency in software skills, four Mathematics Lab papers are introduced, one in each semester. viz. Python Programming for Mathematics, Computational Mathematics using Python, Numerical Methods using Python and Numerical Methods for Boundary Value Problem using Python / Network Science with Python and NetworkX / Programming for Data Science in R / Numerical Linear Algebra using MATLAB respectively. Special importance is given to the skill enhancement courses: Research Methodology, Machine Learning (during 2024-2025 for 2023-2024 batch) and Teaching Technology and Service learning.

Programme Outcome/Programme Learning Goals/Programme Learning Outcome:

PO2: Identify the need and scope of the Interdisciplinary research

PO3: Enhance research culture and uphold the scientific integrity and objectivity

PO4: Understand the professional, ethical and social responsibilities

PO5: Understand the importance and the judicious use of technology for the sustainability of the environment

PO6: Enhance disciplinary competency, employability and leadership skills

Programme Specific Outcome:

PSO2: Demonstrate problem solving, analytical and logical skills to provide solutions for the scientific requirements

PSO3: Develop critical thinking with scientific temper

PSO4: Communicate the subject effectively and express proficiency in oral and written communications to appreciate innovations in research

PSO5: Understand the importance and judicious use of mathematical software's for the sustainable growth of mankind

Course Description: This course is intended to assist students in acquiring necessary skills on the use of research methodology. Also, the students are exposed to the principles, procedures and techniques of planning and implementing a research project.

Course Objectives: This course will help the learner to

COBJ 1. Know the general research methods.

COBJ 2. Get hands on experience in methods of research that can be employed for research in mathematics.

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.

Introducing mathematics Journals, reading a Journal article, Ethics in Research and publications, 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, Tools for checking Grammar and Plagiarism, IPR, Patents/Trademarks and Copyrights, Procedure to apply Patents/Trademarks and Copyrights, The Patent/Trademarks Agent Examinations.

Packages for Documentation - MS Word, LaTeX, Overleaf, Tikz Library, Origin, Pictures and Graphs, producing various types of documents using TeX.

Course Description:

This course aims at teaching 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.

Course Objectives:

This course will help the learner to

COBJ1: be proficient in understanding and solving problems on random variables

COBJ2: efficiently solve problems involving probability distributions.

COBJ3: Acquire proficiency in hypothesis testing.

Introduction to Random variables: Distribution functions, probability mass function and probability density function, Chebyshev's inequality, law of large numbers, central limit theorem, moments and moment generating functions, Binomial, Poisson, Negative binomial, Geometric, Hypergeometric, Discrete uniform. Uniform, Exponential, Gamma, Beta, Weibull, Normal, Lognormal and replacement, t , chi-square, F distribution.

Basic concepts of estimation, point estimation, methods of estimation, method of moments, method of interval estimation, Maximum likelihood estimates.

Null and alternative hypothesis, type I and II errors, power function, t, chi-square, F test method of finding tests, likelihood ratio test, Neyman Pearson lemma, uniformly most powerful tests, some results based on normal population.

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:

Course Description: 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.

Course objectives​: This course will help the learner to

COBJ1. Enhance the knowledge of advanced-level algebra.

COBJ2. Understand the proof techniques for the theorems on advanced group theory, rings and Galois theory.

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.

Euclidean Ring, polynomial rings, polynomials rings over the rational field, polynomial rings over commutative rings.

Extension fields, roots of polynomials, construction with straightedge and compass, more about roots.

The elements of Galois theory, solvability by radicals, Galois group over the rationals, finite fields.

Course Description: This course will help students to 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.

Course objectives​: This course will help the learner to

COBJ1. develop in a rigorous and self-contained manner the elements of real variable functions

COBJ2. integrate functions of a real variable in the sense of Riemann – Stieltjes

COBJ3. classify sequences and series of functions which are pointwise convergent and uniform Convergent

COBJ4. explore the properties of special functions

COBJ5. understand and apply the functions of several variables.

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.

Pointwise and uniform convergence, Uniform Convergence: Continuity, Integration and Differentiation, Equicontinuous Families of Functions, The Stone-Weierstrass Theorem

Power Series, The Exponential and Logarithmic Functions, The Trigonometric Functions, The Algebraic Completeness of the Complex Field, Fourier Series, The Gamma Function.

Linear Transformations, Differentiation, The Contraction Principle, The Inverse Function Theorem, The Implicit Function Theorem.

Course Description:

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, existence and uniqueness of solutions, Eigenvalues and Eigenvectors of the equations, power series method for solving differential equations. Non-linear autonomous system of equations.

Course Objectives​:

This course will help the learner to

COBJ1. solve adjoint differential equations and understand the zeros of solutions.
COBJ2. understand the existence and uniqueness of solutions of differential equations and to solve the Strum-Liouville problems.
COBJ3. solve the differential equations by power series method and hypergeometric equations.
COBJ4. understand and solve the non-linear autonomous system of equations.

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.

Fundamental existence and uniqueness theorem, Dependence of solutions on initial conditions, existence and uniqueness theorem for higher order and system of differential equations, Eigenvalue Problems, Strum-Liouville problems, Orthogonality of eigenfunctions.

Ordinary and singular points of the differential equations, Classification of singular points, Solution near an ordinary point and a regular singular point by Frobenius method, solution near irregular singular point, Hermite, Laguerre, Chebyshev and Hypergeometric differential equation and its polynomial solutions, standard properties.

Linear system of homogeneous and non-homogeneous equations, Non-linear autonomous sysem of equations, Phase plane, Critical points, Stability, Liapunov direct method, limit cycle and periodic solutions, Bifurcation of plane autonomous systems.

Course Description: This course aims at introducing elementary notions on linear transformations, canonical forms, rational forms, Jordan forms, inner product space and bilinear forms.

Course Objectives: This course will help the learner to

COBJ 1. have thorough understanding of Linear transformations and its properties.

COBJ 2. understand and apply the elementary canonical forms, rational and Jordan forms in real life problems.

COBJ 3. gain knowledge on Inner product space and the orthogonalisation process.

COBJ 4. explore different types of bilinear forms and their properties.

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.

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.

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.

Bilinear forms, Symmetric Bilinear forms, Skew-Symmetric Bilinear forms, Groups preserving Bilinear forms.

Course Description: This course will discuss the fundamental concepts and tools in discrete mathematics with emphasis on their applications to mathematical writing, enumeration and recurrence relations.

Course Objectives: The course will help the learner to

COBJ 1. develop logical foundations to understand and create mathematical arguments..

COBJ 2. implement enumeration techniques in a variety of real-life problems.

COBJ 3. analyze the order and efficiency of algorithms.

COBJ 4. communicate the basic and advanced concepts of the topic precisely and effectively.

Sets: Cardinality and countability, recursively defined sets, relations, equivalence relations and equivalence classes, partial and total ordering, representation of relations, closure of relations, functions, bijection, inverse functions.

Logic: Propositions, logical equivalences, rules of inference, predicates, quantifiers, nested quantifiers, arguments, formal proof methods and strategies.

Fundamental principles, pigeon-hole principle, permutations – with and without repetitions, combinations- with and without repetitions, binomial theorem, binomial coefficients, principle of inclusion and exclusion, derangements, arrangements with forbidden positions, rook polynomial.

Ordinary and exponential generating functions, recurrence relations, first order linear recurrence relations, higher order linear homogeneous recurrence relations, non-homogeneous recurrence relations, solving recurrence relations using generating functions.

Real-valued functions, big-O, big-Omega and big-Theta notations, orders of power functions, orders of polynomial functions, analysis of algorithm efficiency, the sequential search algorithm, exponential and logarithmic orders, binary search algorithm.

1. R. P. Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction,5th ed., New Delhi: Pearson, 2014.

2. C L Liu, Introduction to Combinatorial Mathematics, Mcgraw Hill,1968.

3. S. S. Epp, Discrete Mathematics with Applications, Boston: Cengage Learning, 2019 (for Unit-4).

1. K.H. Rosen, Discrete Mathematics with Applications, New York: McGraw-Hill Higher Education, 2019.

2. K. Erciyes, Discrete Mathematics and Graph Theory, New York: Springer, 2021.

3. B. Kolman, R. C. Busby and S. C. Ross, Discrete Mathematical Structures, 6th ed., New Jersey: Pearson Education, 2013.

4. S. Sridharan and R. Balakrishnan R,  Discrete Mathematics, Bocca Raton: CRC Press, 2020.

5. J. P. Tremblay and R. Manohar, Discrete Mathematical Structures with Application to Computer Science, Noida: Tata McGraw Hill Education, 2008.

This course aims at introducing the programming language Python andits uses in solving problems on discrete mathematics and differential equations.

Course Objectives​: This course will help the learner to

COBJ1. gain proficiency in using Python for programming.

COBJ2. acquire skills in usage of suitable functions/packages of Python to solve mathematical problems.

COBJ3. acquaint with Sympy and Numpy packages for solving concepts of calculus, linear algebra and Differential equations.

COBJ4. illustrates use of built-in functions of Pandas and Matplotlib packages for visualizing of data and plotting of graphs.

Installation, IDE, variables, built-in functions, input and output, modules and packages, data types and data structures, use of mathematical operators and mathematical functions, programming structures (conditional structure, the for loop, the while loop, nested statements)

Use of Sympy package, Symbols, Calculus, Differential Equations, Series expressions, Linear and non-linear equations, List, Tuples and Arrays.

Standard plots (2D, 3D), Scatter plots, Slope fields, Vector fields, Contour plots, streamlines, Manipulating and data visualizing data with Pandas, Mini Project.

Course Description: This course is intended to assist the students in acquiring necessary skills for the use of modern technology in teaching. They are exposed to the principles, procedures, techniques of planning and implementing teaching techniques. Through service learning, they will apply the knowledge in real-world situations and serve the community.

Course objectives: This course will help the learner to

COBJ 1: understand the pedagogy of teaching.

COBJ 2: able to use various ICT tools for effective teaching.

COBJ 3: apply the knowledge in real-world situations.

COBJ 4: enhances academic comprehension through experiential learning.

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, Feedback in teaching, Teacher’s role and responsibilities, Information technology for teaching.

Concept of difference between social service and service learning, Case study of best practices, understanding contemporary societal issues, Intervention in the community, Assessing need and demand of the chosen community.

A minimum of fifteen (15) hours documented service is required during the semester. A student must keep a log of the volunteered time and write the activities of the day and the services performed. A student must write a reflective journal containing an analysis of the learning objectives.

 < marks to be converted to credits >

This course aims at strengthening students by providing exposure to analyze and understand the literature from mathematics journals of their choice and present it among peers.

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Course Description: This course deals with the essentials of topological spaces and their properties in terms of continuity, connectedness, compactness etc.

Course objectives​: This course will help the learner to:

COBJ1. understand precise definitions and appropriate examples and counter examples of fundamental concepts in general topology.

COBJ2. acquire knowledge about generalization of the concept of continuity, product topology, metric topology and related results.

COBJ3. appreciate the beauty of deep mathematical results such as Uryzohn’s metrization theorem and understand and apply various proof techniques.

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.

Continuous functions, the product topology, metric topology.

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.

The countability axioms, the separation axioms, normal spaces, the Urysohn lemma, the Urysohn metrization theorem, Tietze extension theorem.

Course Description: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 and theorems on meromorphic functions.

Course objectives​: This course will help the learner to

COBJ1. enhance the understanding the advanced concepts in complex Analysis

COBJ2. acquire problem solving skills in complex Analysis.

Morera’s theorem, Gauss mean value theorem, Cauchy inequality for derivatives, Liouville’s theorem, fundamental theorem of algebra, maximum and minimum modulus theorems.  Taylor’s series, Laurent’s series, zeros of analytical functions, singularities, classification of singularities, characterization of removable singularities and poles.

Rational functions, behavior of functions in the neighborhood of an essential singularity, Cauchy’s residue theorem, contour integration problems, Mobius transformation, conformal mappings.

Meromorphic functions and argument principle, Schwarz lemma, Rouche’s theorem, convex functions and their properties, Hadamard 3-circles theorem.

Phragmen-Lindelöf theorem, Riemann mapping theorem, Weierstrass factorization theorem, Harmonic functions, Poisson formula, Poisson integral formula, Jensen’s formula, Poisson-Jensen formula.

Course Description: This helps students understand the beauty of the important branch of mathematics, namely, partial differential equations. This course includes a study of first and second order linear and non-linear partial differential equations, existence and uniqueness of solutions to various boundary conditions, classification of second order partial differential equations, wave equation, heat equation, Laplace equations and their solutions by Eigenfunction method and Integral Transform Method.

Course Objectives: This course will help the learner to

COBJ 1. understand the occurrence of partial differential equations in physics and its applications.

COBJ 2. solve partial differential equation of the type heat equation, wave equation and Laplace equations.

COBJ 3. also solving initial boundary value problems.

Formation of PDE, initial value problems (IVP), boundary value problems (BVP) and IBVP, solutions of first, methods of characteristics for first order PDE, linear and quasi, linear, method of characteristics for one-dimensional wave equations and other hyperbolic equations.

Origin of second order PDE, Classification of second order PDE, Initial value problems (IVP), Boundary value problems (BVP) and IBVP, Mathematical models representing stretched string, vibrating membrane, heat conduction in solids, second-order equations in two independent variables. Cauchy’s problem for second order PDE, Canonical forms, General solutions.

Occurrence of heat equation in Physics, resolution of boundary value problem, elementary solutions, method of separation of variables, method of eigen function expansion, Integral transforms method, Green’s function.

Occurrence of wave and Laplace equations in Physics, Jury problems, elementary solutions of wave and Laplace equations, methods of separation of variables,, the theory of Green’s function for wave and Laplace equations.

Course Description: 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.

Course objectives: This course will help the learner to

COBJ 1: know the history and development of Graph Theory

COBJ 2: understand all the elementary concepts and results

COBJ 3: learn proof techniques and algorithms in Graph Theory

Graphs as models, degree sequences, classes of graphs, matrices, isomorphism, distances in graphs, connectivity, Eulerian and Hamiltonian graphs, Chinese postman problems, travelling salesman problem and Dijkstra’s algorithm.

Properties of trees, rooted trees, spanning trees, algorithms on trees- Prufer’s code, Huffmans coding, searching, and sorting algorithms, spanning tree algorithms.

Graphical embedding, Euler’s formula, platonic bodies, homeomorphic graphs, Kuratowski’s theorem, geometric duality.

Vertex and edge coloring, chromatic polynomial and index, matching, decomposition, independent sets and cliques, vertex and edge covers, clique covers, digraphs and networks.

Course Description: This course aims at introducing the fundamental aspects of fluid mechanics. They will have a deep insight and general comprehension on tensors, kinematics of fluid, incompressible flow, boundary layer flows and classification of non-Newtonian fluids.

Course Objectives: This course will help the learner to

COBJ1: understand the basic concept of tensors and their representations.
COBJ2: applies the Physics and Mathematics concepts for derivations and interpretations of concepts of fluid mechanics.
COBJ3: familiarize with two- or three-dimensional incompressible flows and viscous flows.
COBJ4: classify Newtonian and non-Newtonian Fluids.

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, streamlines, path lines.

Stress and Rate of 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 equations of conservation of mass, linear momentum (Navier-Stokes equations), and energy.

Bernoulli equation, applications of Bernoulli equation, Concept of circulation, Kelvin circulation 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.

Flow between parallel flat plates, Couette flow, plane Poiseuille flow, the Hagen Poiseuille flow, flow between two concentric rotating cylinders.

Data Science is an interdisciplinary, problem-solving oriented subject that learns to apply scientific techniques to practical problems. This course provides a strong foundation for data science and application area related to information technology and understand the underlying core concepts and emerging technologies in data science.

Definition, big data and data science hype, why data science, getting past the hype, the current landscape, who is data scientist? - data science process overview, defining goals, retrieving data, data preparation, data exploration, data modeling, presentation.

Problems when handling large data, General techniques for handling large data, Case study, Steps in big data, Distributing data storage and processing with Frameworks, Case study.

Machine learning, modeling process, training model, validating model, predicting new observations, supervised learning algorithms, unsupervised learning algorithms. introduction to deep learning.

The characteristic polynomial, eigenvalues and graph parameters, eigenvalues of regular graphs, eigenvalues and expanders, strongly regular graphs.

Data Science Ethics – Doing good data science – Owners of the data - Valuing different aspects of privacy - Getting informed consent - The Five Cs – Diversity – Inclusion – Future Trends.

This course is designed to prepare students for real class room situation under the supervision of faculty mentors. It provides experiences in the actual teaching and learning environment.

·       Fifteen hours of teaching assignments for UG classes shall be undertaken by each student during the 3rd and 4th semester.

·       Each student shall be under the supervision of a faculty mentor/guide.

·       The 15 hours may be distributed among 1 or 2 subjects- which shall be a combination of theory and problem based papers.

·       A Structured Plan stating the Topic- Objectives- Methodology and Evaluation shall be prepared in advance by the student for each class session and submitted to the faculty mentor/guide.

·       Faculty guides shall maintain an assessment register for their respective students and record assessment for each session on the given parameters.

This course is designed to prepare students for real class room situation under the supervision of faculty mentors. It provides experiences in the actual teaching and learning environment.

• Fifteen hours of teaching assignments for UG classes shall be undertaken by each student during the 3rd and 4th semester.
• Each student shall be under the supervision of a faculty mentor/guide.
• The 30 hours may be distributed among 1 or 2 subjects- which shall be a combination of theory and problem based papers.
• A Structured Plan stating the Topic- Objectives- Methodology and Evaluation shall be prepared in advance by the student for each class session and submitted to the faculty mentor/guide.
• Faculty guides shall maintain an assessment register for their respective students and record assessment for each session on the given parameters.

Course description: This course concerns the analysis and applications of calculus of variations and integral equations.  Applications include areas such as classical mechanics and differential equations.  

Course objectives: This course will help the learners to

COBJ1. introduce them to the concepts of calculus of variations and Integral equations.

COBJ2. develop various techniques to derive and solve variational problems.

COBJ3. familiarize the applications in real world problems.

Finding minima of a function: Classical method – Finding minima of a function: Direct methods – Comparison of the two methods – Functions to functionals: A brief historical tour of the origins of the calculus of variations. Euler-Lagrange’s equations, examples of classical problems: The Brachistochrone, the Tautochrone, Fermat’s principle of least time, Euler equation, extremals, stationary function, geodesics, least action principle, isoperimetric problems, minimal surfaces of revolutions, minimal surfaces.

Integral equation and its classification: 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. 

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. 

Course description: 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.

Course objectives: This course will help the learner to 

COBJ1. Enhance the understanding of the advanced notions from Mathematical Analysis 

COBJ2. Know more about the Measure theory and Lebesgue Integration

Introduction to Analysis, Lebesgue Outer Measure, The sigma-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.

The Riemann Integral; 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.

Continuity of Monotone Functions, Differentiation of Monotone Functions, Functions of Bounded Variation, Absolutely Continuous Functions, Integrating Derivatives, Convex Functions, Normed Linear Spaces, The inequalities of Young, Holder and Minkowski

Course Description: This course deals with the theory and application of various advanced methods of numerical approximation. These methods or techniques help us to approximate the solutions of problems that arise in science and engineering. The emphasis of the course will be the thorough study of numerical algorithms to understand the guaranteed accuracy that various methods provide, the efficiency and scalability for large scale systems and issues of stability.

Course Objectives: This course will help the learner to

COBJ1. develop the basic understanding of the construction of numerical algorithms, and perhaps more importantly, the applicability and limits of their appropriate use.

COBJ2. become familiar with the methods which will help to obtain solution of algebraic and transcendental equations, linear system of equations, finite differences, interpolation numerical integration and differentiation, numerical solution of differential equations and boundary value problems.

COBJ3. understand accuracy, consistency, stability and convergence of numerical methods.

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), Graeffe’s root squaring method. Solution of Linear System of Algebraic Equations: LU-decomposition methods (Cholesky method), consistency and ill-conditioned system of equations, Tridiagonal system of equations, Thomas algorithm.

Lagrange, Hermite, Natural Cubic-spline interpolation - with uniqueness and error term. Chebychev and Rational function approximation. Gaussian quadrature, Gauss-Legendre, Gauss-Chebychev formulas.

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.

Numerical solutions of second order boundary value problems (BVP) of first, second and third types by shooting method, Rayleigh-Ritz Method, Galerkin Method.

This course introduces basic and advanced topics of Analytical Number Theory. Topics such as divisibility, congruences and number-theoretic functions are discussed in this course. 

*The course is transacted through a learner-centred approach to facilitate self-directed learning.

Course Objectives:

This course will help the learner to

COBJ1. gain awareness of the fundamental and advanced topics in modular arithmetic.
COBJ2. demonstrate understanding of number theoretic functions and their asymptotic properties.
COBJ3. apply number theoretic concepts in encrypting and decrypting.

Divisibility, Greatest common divisor, distribution of primes, fundamental theorem of arithmetic, series of reciprocals of primes, Euclidean Algorithm.

Mobius function, Euler’s totient function, Dirichlet product of arithmetic functions, Mobius inverse formula, Mangoldt function, multiplicative functions, Liouville’s functions, divisor functions, generalised convolutions, formal power series, derivatives, The Selberg Identity, Averages of arithmetic functions.

ongruences, residue systems, Euler-Fermat Theorem, Polynomial congruences, Lagrange’s Theorem, Simultaneous Linear Congruences, Chinese Remainder Theorem and Applications, polynomial congruences with power moduli, reduced residue systems and their decomposition property.

Quadratic reciprocity, encryption and decryption, public key cryptosystems, RSA algorithm, The Knapsack cryptosystems, primitive roots, application of primitive roots to cryptography.

Course Description: This course aims at introducing the basic aspects of machine learning, algorithms for machine learning, decision tree learning, artificial neural network and their extensions.

Course Objectives​: This course will help the learner to

COBJ1. master the basic concepts of machine learning.
COBJ2. analyze and use the machine learning algorithms.
COBJ3. use the decision tree algorithms.
COBJ4. handle artificial neural networks and its algorithms.
COBJ5. understand the functioning of the artificial neural networks, recurrent networks and dynamically changing networks.

Introduction to Machine Learning, Designing a Learning System, Perspective and issues in Machine Learning, A concept learning Task, Concept learning as search.

The LIST-THEN-ELIMINATE Algorithm, CANDIDATE-ELIMINATION Learning Algorithm, Remarks on Version Spaces and CANDIDATE -ELIMIINATION,

Decision Tree Representation, appropriate problems in decision tree learning, basic decision tree learning algorithm, hypothesis space search in decision tree learning, Inductive bias in Decision tree learning, issues in decision tree learning.

Neural Network Representation, Appropriate problems for Neural Networks Learning, Multilayer networks and the BACKPROPOGATION Algorithm, Remarks on BACKPROPOGATION Algorithm, Application to Face Recognition.

Alternative Error Functions, Alternative error minimization procedure, Recurrent Networks, dynamically modifying Network Structure.

Course Description: 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 Prandtlboundry layer, porous media and non-Newtonian fluid.

Course Objectives: This course will help the learner to
COBJ1. understand the different modes of heat transfer and their applications.
COBJ2. understand the importance of doing the non-dimensionalization of basic equations.
COBJ3. understand the boundary layer flows.
COBJ4. have familiarity with porous medium and non-Newtonian fluids

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 – Nondimensionalization of the Basic Equations - Non-dimensional parameters and dynamic similarity.

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,

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.

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.

2. S. Chardrasekhar,Hydrodynamic and hydrodmagnetic stability, Oxford University Press, 2007 (RePrint).

2. F. M White, Fluid Mechanics, Tata Mcgraw Hill. 2011.

3. D. A. Nield and Adrian Bejan, Convection in Porous Media”, Third edition, Springer, 2006

Course Description: The theory of edge-colouring and colour connections in graphs, fundamentals of domination, theory of intersection graphs, and perfect graphs are discussed in this course.

Course Objectives​: This course will help the students to

COBJ1. Gain knowledge in the theory of graph colouring and colour connections.

COBJ2. Know the advanced topics of domination in graphs.

COBJ3. Know the concept and structural characteristics of intersection graphs and interval graphs.

COBJ4. Know the fundamental and advanced concepts such as chordality and perfection of graphs.

Edge colouring of graphs, rainbow connections, proper connections, Hamiltonian connections, colour connectivity, disconnections.

Dominating sets, types of domination, domination chain, covering and packing, domination-related vertex partition, coalition.

Intersection graphs, line graphs, hypergraphs, Helly property, Johnson graphs, clique graphs, weighted intersection graphs, interval graphs, unit interval graphs, proper interval graphs.

Chordal graphs, MCS Algorithm, weakly and strongly chordal graphs, vertex duplication and vertex multiplication, perfect graphs, perfectly orderable graphs, imperfect graphs, partitionable graphs, strong perfect graphs.

2.  T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs, Reprint, CRC Press, 2000.

3.  T. A. McKee and F.R. McMorris, Topics in Intersection Graph Theory, SIAM, 1999.

4. D. B. West, Introduction to Graph Theory, New Delhi: Prentice-Hall of India, 2011.

2.T. W. Haynes and S. T. Hedetniemi, M.A. Henning, Topics in Domination in Graphs, Springer, 2020

3.M.C. Golumbic, Algorithmic graph theory and perfect graphs, 2nd ed., Elsevier, Amsterdam. 

4.B. Bollobás, Modern Graph Theory, Springer, New Delhi, 2005. 

5. V. Chvátal, and C. Berge, Topics in perfect graphs, Elsevier, 1984.