Syllabus for
Master of Science (Mathematics)
Academic Year  (2019)

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 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.

Course Objective: This course will help the learner to:

COBJ1. acquire skills in using technology effectively for teaching

COBJ2. know the general research method and methods of research that can be employed for research in Mathematics

COBJ3. experience service learning

CO1. gain necessary skills on the use of modern technology in teaching.

CO2. foster a clear understanding about research design that enables students in analyzing and evaluating the published research.

CO3. understand the components and techniques of effective report writing.

CO4. obtain necessary skills in understanding the mathematics research articles

CO5. acquire skills in preparing scientific documents using MS Word, Mathtype, Open Office Math editor, yEd Graph Editor and LaTeX.

CO6. strengthen personal character and sense of social responsibility through service learning module.

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.

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.

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.

Guidelines for service learning:

One among the following can be considered as a service learning module:

• Tie up with schools for teaching elementary mathematics in an easier way.
• Developing e-content for particular topics which will be a Vehicle for Teaching Curriculum Theory, Assessment, and Design (as per the requirements).
• Math Exhibition: To strengthen students' math skills, a mathematics camp can be organised in the school premises. Students will participate in challenging academic coursework of math, make projects related to mathematical concepts, explore many inventions and historical aspects in mathematics. Students can strengthen and expand their scientific and mathematical knowledge while having fun.
• Students can create a website for the Department of Mathematics/the project area, putting all the information about the activities and events coming up.
• Students can assist in statistical research(based on its needs), in developing a survey tool, organizing and/or conducting the survey, compiling and analyzing data, or some combination of these or some other statistical undertakings.
• Develop a mathematical model and should also be able to provide a solution for an existing real-world problem.

After deciding, get approval from your respective mentors.

• Each student will develop a learning/lesson plan composed of three (3-4) measurable learning objectives. Examples of learning objectives are:
• Improve algebraic/problem solving skills.
• Improve methods of communicating mathematics to others effectively.
• Identify common mistakes and misconceptions that mathematics students make.
• A minimum of fifteen (15) hours documented service is required during the semester.
• A student must keep a log of the volunteered time.
• A student must write a diary containing an analysis of the activities of the day and the services performed.
• A student must write a reflective journal containing an analysis of the learning objectives.

You will be graded on the quality of your work rather than on the quantity of hours.

You need to provide with the following items:

• The log of all 15 hours with dates and services performed.
• A brief summary (typed, Times new Roman (heading 16, sub-heading 14, content 12, doubly spaced, and each not more than one page) of the activities of the day and how these activities are related to the objectives. This summary must be updated every time you do the service (Like a Diary).
• A summary journal (typed, doubly spaced, and not less Three pages), of what your experience meant to you and how especially to your objectives.

Perform the service. Make sure you answer each of the following questions every time you perform the service. Write your diary based on the following:

• How did you feel at the beginning of the visit?
• What was the activity of the day?
• What was the student’s attitude toward the activity?
• Did the student benefit from the activity? If so, how? If not, why not?
• What did you gain mathematically from the experience?
• How did you feel at the end of the visit?
• What was the most difficult and most satisfying parts of this experience?
• Describe your progress toward each of your learning objectives. Do you need to update or revise your learning objectives?

Service-Learning Student Diary Format

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.

CO1. determine the Riemann-Stieltjes integrability of a bounded function.

CO2. recognize the difference between pointwise and uniform convergence of sequence/series of functions.

CO3. illustrate the effect of uniform convergence on the limit function with respect to continuity, differentiability, and integrability.

CO4. analyze and interpret the special functions such as exponential, logarithmic, trigonometric and Gamma functions.

CO5. gain in depth knowledge on functions of several variables and the use of Implicit Function Theorem.

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.

CO1. Understand concept of Linear differential equation, Fundamental set Wronskian.

CO2. Understand the concept of Liouvilles theorem, Adjoint and Self Adjoint equation, Lagrange's Identity, Green’s formula, Eigen value and Eigen functions.

CO3. Identify ordinary and singular point by Frobenius Method, Hyper geometric differential equation and its polynomial.

CO4. Understand the basic concepts and definition of PDE and also mathematical models representing stretched string, vibrating membrane, heat conduction in rod.

CO5. Demonstrate on the canonical form of second order PDE.

CO6. Demonstrate initial value boundary problem for homogeneous and non-homogeneous PDE.

CO7. Demonstrate on boundary value problem by Dirichlet and Neumann problem.

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.

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 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

CO1. confidently manipulate tensor expressions using index notation, and use the divergence theorem and the transport theorem.

CO2. able to understand the basics laws of Fluid mechanics and their physical interpretations.

CO3. able to understand two or three dimension flows incompressible flows.

CO4. able to understand the viscous flows, their mathematical modelling and physical interpretations.

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

CO1. write precise and accurate mathematical definitions of basics concepts in graph theory

CO3. provide appropriate examples and counterexamples to illustrate the basic concepts

CO3. demonstrate and apply various proof techniques in proving theorems in graph theory

CO4. showcase mastery in using graph drawing tools

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 providing the necessary background for the students to gain proficiency in abstraction, notation, and critical thinking in the Discrete Mathematics.

Course objectives:This course will help the learner to be familiar with:

COBJ1. mathematical logic and the rules of inference

COBJ2. concepts of sets, relation and functions

COBJ3. the combinatorial techniques of enumeration

COBJ4. generating functions, recurrence relations and their applications

CO1. construct mathematical arguments using logical connectives and quantifiers.

CO2. verify the correctness of an argument using propositional and predicate logic and truth tables.

CO3. construct proofs using direct proof, proof by contraposition, proof by contradiction, proof by cases, and mathematical induction.

CO4. understand the counting techniques and apply

CO5. gain proficiency in handling the properties connected to the relations.

Propositional Logic, Propositional Equivalences, Predicates and Quantifiers, Nested Quantifiers, Rules of Inference, Introduction to Proofs, Proof Methods and Strategy.

The Basics of Counting, The Pigeonhole Principle, Permutations and Combinations, Binomial Coefficients, Generalized Permutations and Combinations, Generating Permutations and Combinations.

Recurrence Relations, Solving Linear Recurrence Relations, Divide-and-Conquer Algorithms, Generating Functions, Inclusion-Exclusion, Applications of Inclusion-Exclusion

Relations and Their Properties, n-ary Relations and Their Applications, Representing Relations, Closures of Relations, Equivalence Relations, Partial Orderings  

Course Description: 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. Some of the applications of the said concepts are also included.

Course objectives:This course will help the learner to be familiar with:

COBJ1. all the elementary concepts and proof techniques in Number Theory

CO1. Define and interpret the concepts of divisibility, congruence, greatest common divisor, prime, and prime-factorization,

CO2. Solve linear Diophantine equations and congruences of various types, and use the theory of congruences in applications.

CO3. Prove and apply properties of multiplicative functions such as the Euler phi-function and of quadratic residues.

CO4. Apply the Law of Quadratic Reciprocity and other methods to classify numbers as primitive roots, quadratic residues, and quadratic non-residues,

CO5. Produce rigorous arguments (proofs) centered on the material of number theory, most notably in the use of Mathematical Induction and/or the Well Ordering Principle in the proof of theorems.

The division algorithm, the Euclidean algorithm, the unique factorization theorem, Euclid’s theorem, linear Diophantine equations.

Definitions and properties, 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, number theory from algebraic point of view, groups, rings and fields.

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

Greatest integer function, arithmetic functions, the Mobius inversion formula.

Course Description: Cryptography is the science of encrypting and decrypting any information. This is one of the finest applications of Number Theory. In this course, the fundamentals of cryptography are dealt with. As a piece of information is expressed through symbols, representing it in a way that only the intended party would know it is the best part of encryption and decryption. As the world is flooded with information, generation, transfer and acquisition of it is very important. Students with basic background in Number Theory can take up this course.

Course objectives:This course will help the learner to be familiar with:

COBJ1. the fundamental notions in cryptography

COBJ2. the foundational Number Theory required for handling encryption and decryption techniques in Cryptography

CO1. use Number Theory for encryption and decryption

CO2. encrypt and Decrypt message

CO3. know the difference between private key and public key cryptographies

CO4. understand a number of privacy mechanisms

Elementary concepts of number theory, time estimates for doing arithmetic, divisibility and the Euclidian algorithm, congruences, some applications to factoring. Finite fields and quadratic residues: Finite fields, quadratic residues and reciprocity.

Some simple cryptosystems, enciphering matrices.

The idea of public key cryptography, RSA, discrete log., knapsack, zero-knowledge protocols and oblivious transfer..

Basic facts, elliptic curve cryptosystems, elliptic curve primality test, elliptic curve factorization.

Course Description: This course aims as introducing LaTeX and the mathematical software packages “WxMaxima” and “Scilab”, for learning basic operation on matrix manipulation, plotting graphs etc.,. These software packages will also help students to solve problems / applied problems on  Mathematics.

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

COBJ1. understand the basics of LaTeX and its implementation

COBJ2. use the FOSS tools WxMaxima and Scilab effectively

CO1. use LaTeX effectively

CO2. Use the basic commands in WxMaxima including the 2D and 3D plots

CO3. Have a strong command on the inbuilt commands required for the learning and analyzing mathematics

CO4. Solve problems / applied problems on mathematics by using Scilab.

Document Markup - The absolute basics of LaTeX - The TeX conceptual model of typesetting - Text elements - Tables and Figures - Math - References.

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 - Introduction to Scilab and commands connected with Matrices - Computations with Matrices - 2D and 3D Plots - Script Files and Function Files

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.

Course Description: 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.

Course Objective: 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

CO1. Understand random variables and probability distributions.

CO2. Distinguish discrete and continuous random variables.

CO3. Obtain ability compute Expected value and Variance of discrete random variable.

CO4. Acquire knowledge in using Binomial distribution, Poisson distribution etc.,

CO5. Define inferential statistics.

CO6. Effectively use sampling distributions in inferential statistics.

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.  

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.

t, F and chi-square distributions, standard errors and large sample distributions.  

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

CO1. define topological spaces, give examples and counterexamples on concepts like open sets, basis and subspaces

CO2. establish equivalent definitions of continuity and apply the same in proving theorems

CO3. understand the concepts of metrizability, connectedness, compactness and learn the related theorems

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.

he 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.

CO1. apply the concept and consequences of analyticity and the Cauchy-Riemann equations and of results on harmonic and entire functions including the fundamental theorem of algebra

CO2. compute complex contour integrals in several ways: directly using parameterization, using the Cauchy-Goursat theorem Evaluate complex contour integrals directly and by the fundamental theorem, apply the Cauchy integral theorem in its various versions, and the Cauchy integral formula

CO3. represent functions as Taylor, power and Laurent series, classify singularities and poles, find residues and evaluate complex integrals using the residue theorem

CO4. use conformal mappings and know about meromorphic functions

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 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

CO1. Demonstrate knowledge of conjugates, the Class Equation and Sylow  theorems

CO2. Demonstrate knowledge of polynomial rings and associated properties

CO3. Derive and apply Gauss Lemma, Eisenstein criterion for irreducibility of rationals

CO4. Demonstrate the characteristic of a field and the prime subfield;

CO5. Demonstrate Factorization and ideal theory in the polynomial ring; the structure of a primitive polynomials; Field extensions and characterization of finite normal extensions as splitting fields; The structure and construction of finite fields; Radical field extensions;Galois group 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 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. familiarity with porous medium and non-Newtonian fluids

CO1. understand the basic laws of heat transfer and understand the fundamentals of convective heat transfer process.

CO2. solve Rayleigh - Benard problem and their physical phenomenon.

CO3. solve and understand different boundary layer problems

CO4. give an introduction to the basic equations with porous medium and solution methods for mathematical modeling of viscous fluids and elastic matter

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.

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.

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

Course objectives: This course will 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.

CO1. understand the basic concepts and fundamental results in matching, domination, coloring and planarity.

CO2. reason from definitions to construct mathematical proofs

CO3. obtain a solid overview of the questions addressed by graph theory and will be exposed  to emerging areas of research

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 as 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

CO1. Acquire proficiency in using different functions of Python to compute solutions of basic mathematical problems

CO2. Demonstrate the use of Python to solve differential equations along with visualize the solutions

CO3. Be familiar with the built-in functions to deal with  Graphs and Digraphs

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.

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.

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.

Course Description: This course is a foundation for introducing to Free and Open-Source Software (FOSS) Tools (Octave). It enables the students to explore mathematical concepts and verify mathematical facts through the use of software and also enhance the skills in programming.

Course Objective: This course will the learner to

COBJ1. use FOSS tool GNU Octave to effectively calculate the solutions of problems on Mathematics

CO1. express proficiency in using Octave,

CO2. understand the use of various techniques of the software for effectively doing Mathematics.

CO3. obtain necessary skills in Octave programming.

CO4. use octave for applications of Mathematics

Getting started with GNU Octave - vector operations - Projections - Matrix operations - Plotting: plotting options - saving plots - Matrices and Linear systems: solution of linear system using Gaussian elimination, left division, LU decomposition - Polynomial curve fitting - Matrix transformations.

Limits, sequences and series - Numerical integration using Quadrature - Numerical integration using approximate sums - parametric and polar plots - special functions.

Three dimensional graphs: space curves - surfaces - solids of revolution - Multiple integrals - Vector fields - Differential Equations: slope fields - Euler’s method - The Livermore solver

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

CO1. understand the fundamental concepts of Mathematical Analysis.

CO2. tate some of the classical theorems in of Advanced Real Analysis.

CO3. be familiar with measurable sets and functions.

CO4. integrate a measurable function

CO5. understand the properties of Lp Spaces

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 to develop the basic understanding of the construction of numerical algorithms, and perhaps more importantly, the applicability and limits of their appropriate use. The learners will be 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

CO1. Derive numerical methods for approximating the solution of problems of algebraic and transcendental equations, ordinary differential equations and boundary value problems.

CO2. Implement a variety of numerical algorithms appropriately in various situations

CO3. interpret, analyse and evaluate results from numerical computations

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.

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, Gelarkin Method.

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.

• Understand and use the 3N-coordinate system made up of N-spacial coordinates, N-velocity coordinates and N-acceleration coordinates.
• Understand the motion of mechanical systems with constraints using Lagrangian description.
• Use Hamilton’s principle and gain proficiency in solving equations of motions.
• Understand the use of Hamilton–Jacobi theory in solving equations of motions.

The mechanical system - Generalised Coordinates - constraints - virtual work - Energy and momentum.

Derivation and examples - Integrals of the Motion - Small oscillations. Special Applications of Lagrange’s Equations: Rayleigh’s dissipation function - impulsive motion - velocity dependent potentials.

Hamilton's principle - Hamilton’s equations - Other variational principles - phase space.

Hamilton's Principal Function – The Hamilton - Jacobi equation - Separability.

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

CO1. Have thorough understanding of the Linear transformations

CO2.Demonstrate the elementary canonical forms, rational and Jordan forms.

CO3. Apply the inner product space

CO4. Express familiarity in using bilinear forms

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