Syllabus for
Master of Science (Mathematics)
Academic Year  (2021)

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 Algebra, Topology, Analysis, Discrete Mathematics and Number Theory/Cryptography 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. Classical Mechanics, Linear Algebra and Numerical Analysis. 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 in these topics. To gain proficiency in software skills, Mathematics Lab papers are introduced in each semester. Special importance is given to the skill enhancement courses Teaching Technology and Research Methodology in Mathematics and service learning, Introduction to Free and Open-Source Software (FOSS) Tools: (GNU Octave) and Statistics.

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:

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, 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

Package for Mathematics Typing, MS Word, LaTeX, Overleaf, Tikz Library, Origin, Pictures and Graphs, Producing various types of documents using TeX.

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. Demonstrate the ability to manipulate and use of special functions

COBJ5. Use and operate 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, The Rank Theorem, Determinants, Derivatives of Higher Order, Differentiation of Integrals

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, 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.

Course objectives : This course will help the learner to

COBJ1. Solve adjoint differential equations, hypergeometric differential equation and power series.

COBJ2. Solve partial differential equation of the type heat equation, wave equation and Laplace equations.

COBJ3. Also solving initial boundary value problems.

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, Eigenvalues and Eigenfunctions, related examples. 

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. 

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 

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

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 on 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 aims at introducing the fundamentals of fluid mechanics.  This course aims at imparting the knowledge 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 representative

COBJ2. Physics and mathematics behind the basics of fluid mechanics

COBJ3. Familiar with two or three dimensional incompressible flows

COBJ4. Classifications of non-Newtonian fluids

COBJ5. Familiar with standard two or three dimensional viscous flows

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.

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.     

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.

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

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

COBJ1. Know the history and development of graph theory

COBJ2. Understand all the elementary concepts and proof techniques  in Graph Theory

Definition and introductory concepts, Graphs as Models, Matrices and Isomorphism, Decomposition and Special Graphs, Connection in Graphs, Bipartite Graphs, Eulerian Circuits.

Counting and Bijections, Extremal Problems, Graphic Sequences, Directed Graphs, Vertex Degrees, Eulerian Digraphs, Orientations and Tournaments.

Properties of Trees, Distance in Trees and Graphs, Enumeration of Trees, Spanning Trees in Graphs, Decomposition and Graceful Labellings, Minimum Spanning Tree, Shortest Paths.

Connectivity, Edge - Connectivity, Blocks, 2 - connected Graphs, Connectivity in Digraphs, k - connected and k-edge-connected Graphs, Maximum Network Flow, Integral Flows.

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

Course objectives: This course will help the learner to

COBJ1.Acquire skill in usage of suitable functions/packages of Python to solve mathematical problems.

COBJ2.Gain proficiency in using Python to solve problems on Differential equations. 

COBJ3. The built in functions required to deal withcreating and visualizing Graphs, Digraphs, MultiGraph.

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 Nonlinear equations, List, Tuples and Arrays.

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

Course Description: This course aims at introducing the elementary notions on Machining learning and focuses on some simple application of machine-learning, algorithms on supervised machine learning and unsupervised learning.

Course Objective: This course will help the learner to:

COBJ1. Be proficient on the idea of machine learning

COBJ2. Implement Supervised Machine Learning Algorithms

COBJ3. Handle computational skills related to unsupervised learning and Preprocessing

Introduction - Simple Machine Learning Applications: Classifying Iris Species: Meet the data, Training and Testing Data, Pair Plot of Iris dataset - k-nearest neighbours model, Evaluating model.

Classification and Regression - Generalization, Overfitting and Underfitting - Relation of Model Complexity to Dataset Size, Supervised Machine Learning Algorithms: k-Nearest Neighbour algorithm: k-Neighbors classification, k-neighbors regression, Strengths, Weakness and parameters of k-NN algorithm, Linear Models: Linear models for regression, Linear models for classification, Linear models for multiclass classification, Strengths, Weakness and parameters of linear models, Decision Trees: Building decision trees, controlling complexity of decision trees, Analyzing decision trees, Strengths, Weakness and parameters of decision trees.

Types of unsupervised learning, Challenges in unsupervised learning, Preprocessing and scaling: Different kinds of preprocessing, Applying Data transformations, Scaling training and test data, Principal component analysis, Non-negative matrix factorization.

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

COBJ2. Acquire knowledge about generalization of the concept of continuity and related properties

COBJ3. Appreciate the beauty of deep mathematical results such as  Uryzohn’s lemma 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 , Rouche’s theorem and Hadamard’s 3-circles theorem.

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.

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..

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.

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 Objective: This course will help learner to

COBJ1. Gain proficiency on the theories of Linear Algebra

COBJ2. Enhance problem solving skills in Linear Algebra

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.

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 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 mathematical modelling of nano-liquids

Course objectives: This course will help the learner to

COBJ 1: Understand the different modes of heat transfer and their applications.

COBJ 2: Understand the importance of doing the non-dimensionalization of basic equations.

COBJ 3: Understand the boundary layer flows.

COBJ 4: 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 – Non-dimensionalization 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.

Buongiorno Model (Two phase model): Nanoparticle/Fluid Slip : Inertia, Brownian Diffusion, Thermophoresis, Diffusiophoresis, Magnus Effect, Fluid Drainage, Gravity, Relative importance of the Nanoparticle Transport Mechanisms. Conservation Equation for two phase Nanoliquids: The Continuity equation, The Momentum equation and The energy equation.

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.

Course description: This course helps the students to understand the colouring of graphs, Planar graphs, edges and cycles.

Course objectives: This course shall help the learner to

COBJ1. construct examples and proofs pertaining to the basic theorems

COBJ2. apply the theoretical knowledge and independent mathematical thinking in creative investigation of questions in graph theory

COBJ3. write graph theoretic ideas in a coherent and technically accurate manner.

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.

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.

Drawings in the Plane, Dual Graphs, Euler’s Formula, Kuratowski’s Theorem, Convex Embeddings, Coloring of Planar Graphs, Thickness and Crossing Number

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

Course Description: This course aims at introducing Python programming using the libraries of Python programming language for Mathematical modelling, Linear Algebra and Fluid Mechanics.

Course objectives​: This course will help the learner to

COBJ1: Acquire skill in using suitable libraries of Python to solve problems on Mathematical modelling.

COBJ2: Gain proficiency in using Python to solve problems on Linear Algebra.

COBJ3: Build user-defined functions to deal with the problem on Fluid Mechanics.

Linear and non-Linear Model – Growth and decay, Half-life, Newton’s law of cooling / warming, Mixtures, Computational Models with Quadratic Growth, A Predator-Prey Model, Lotka-Volterra predator-prey model, competition models, Concentration of a Nutrient, Spring/Mass Systems: Free Undamped Motion, Free Damped Motion, Driven Motion, Nonlinear Springs, Simple Pendulum, Projectile, Double Pendulum

Matrix construct, eye, zeros matrices, Addition, Subtraction, Multiplication of matrices, powers and invers of a matrix. Accessing Rows and Columns, Deleting and Inserting Rows and Columns, Determinant, reduced row echelon form, nullspace, column space, Rank, Solving systems of linear equations (Gauss Elimination Method, Gauss Jordan Method, LU- decomposition Method), Eigenvalues, Eigenvectors, and Diagonalization, Linear combinations, Linearly independence and dependence, basis and dimension, Linear Transformation, Orthogonal set, orthogonal projection of a vector, Orthonormal, Gram-Schmidt Process.

Stream Lines, Path lines, Vortex lines and their plots, Calculating Rayleigh number for Rayleigh-Benard convection with external constraints magnetic field, rotation, non -uniform temperature gradients, solution of Lorenz equations – Nusselt number.

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

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 >

Course description: The Coursecovers 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

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.

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.

Normed Linear Spaces, The Inequalities of Young, Hölder and Minkowski, The L^p spaces, Approximation and Separability, The Riesz Representation for the Dual of L^p, Weak Sequential Convergence in L^p, Weak Sequential Compactness, The Minimization of Convex Functionals.

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

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

COBJ2. To 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), 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, Cholesky and Doolittle's methods), consistency and ill-conditioned system of equations, Tridiagonal system of equations, Thomas algorithm.

Lagrange, Hermite, Cubic-spline’s (Natural, Not a Knot and Clamped) - with uniqueness and error term, for polynomial interpolation. 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.

Course description:: Differential geometry is the study of geometric properties of curves, surfaces, and their higher dimensional analogues using the methods of calculus. This course helps learners to acquire active knowledge and understanding of the basic concepts of the geometry of curves and surfaces in Euclidean space.

Course objectives: This course will help the learner to

CO BJ1. Write proofs for the theorems on Curves and Surfaces in R3.

COBJ2. Implement the properties of curves and surfaces in solving problems described in terms of tangent vectors / vector fields / forms etc.

Euclidean Space – Tangent Vectors – Directional derivatives – Curves in E3 – 1-Forms – Differential Forms – Mappings.

Dot product – Curves – Vector field - The Frenet Formulas – Arbitrary speed curves – Cylindrical helix – Covariant Derivatives – Frame fields – Connection Forms - The Structural equations.

First fundamental quadratic form of a surface - Angle of two intersecting curves in a surface - element of area - Family of curves in a surface - principle directions - isometric surfaces - The Riemannian curvature tensor, the Gaussian curvature of a surface.

Second fundamental form of a surface - equation of Gauss and equations of Codazzi - Normal curvature of surface - lines of curvature of a surface - Isometric conjugate nets-Dupin indicatrix.

Course Description: This course helps the students to understand the basic concepts of boundary layer theory and gains the knowledge of the flow of air and other fluids with small viscosity and its engineering applications.

Course objectives: This course will help the learner to
COBJ 1:Understands the boundary layer flows and its applications.
COBJ 2:Understands the findings of exact and approximation solution of two-dimensional boundary layer equations.
COBJ 3:Understands the axial symmetry boundary layer flow.
COBJ 4:Understands the three dimensional boundary layer flow.
COBJ 5:Understands the unsteady boundary layer flow and its applications.

Boundary layer concept, Boundary layer equation for two dimensional incompressible flow, Separation of boundary layer, skin – friction, Boundary layer along a plate – The Blasius solution, Boundary layer of higher order, Similar solutions.

Exact solution: Flow past wedge, Flow in a convergent channel, Flow past cylinder, Flow on a flat plate at zero incidence, Gortler series method, Plane free jet, Prandtl-Mises transformation and its applications of plane free jet. Approximate solution: Von Karman’s integral equation – Momentum integral equation – Energy integral equation, Applications of Von Karman’s integral equation – absence of pressure gradient – with pressure gradient – Van Karman- Pohlhausen method.

Axially symmetrical boundary layer on a body of revolution, Mangler’s transformation, Three dimensional boundary layer – boundary layer flow on yawed cylinder and on other bodies.

Unsteady boundary layer equation, Method of successive approximations, Boundary layer after impulsive start of motion, Periodic boundary layer flow. Free convection from heated vertical plate, Thermal –Energy integral equation – approximate solution.

Course description:Theory of intersection graphs, perfect graphs, chromatic graph theory and eigenvalues of graphs are dealt with in the detail in this course.

Course objectives: This course will help the learner to

COBJ1. Understand the advanced topics in Graph Theory.

COBJ2. Enhance the understanding of techniques of writing proofs for advanced topics in Graph Theory.

T-Colourings, L-colourings, Radio Colourings, Hamiltonian Colourings, Domination and Colourings.

Intersection Graphs, clique graphs, line graphs, hypergraphs, interval graphs, chordal graphs, weakly and strongly chordal graphs

Vertex multiplication, Perfect graphs, The Perfect Graph Theorem, Other Classes of Perfect Graphs, Imperfect Graphs, the Strong Perfect Graph Conjecture.

The Characteristic Polynomial, Eigenvalues and Graph Parameters, Eigenvalues of Regular Graphs, Eigenvalues and Expanders, Strongly Regular Graphs.

Data Science is an interdisciplinary, problem-solving oriented subject that learns to apply scientific techniques to practical problems. This course provides 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.