Syllabus for
Master of Science (Mathematics)
Academic Year  (2017)

EXAMINATION AND ASSESSMENTS (Theory)

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.

• gain necessary skills on the use of modern technology in teaching.  To foster a clear understanding about research design that enables students in analyzing and evaluating the published research.
• understand the components and techniques of effective report writing.
• obtain necessary skills in understanding the mathematics research articles
• acquire skills in preparing scientific documents using MS Word, Math type, Open Office Math editor, yEd Graph Editor and LATEX.
• 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, 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?

This course is concerned with the basics of analytical number theory. Topics such as divisibility, congruence’s, quadratic residues and functions of number theory are covered in this course. An introduction to Cryptography is also included.

• Define and interpret the concepts of divisibility, congruence, greatest common divisor, prime, and prime-factorization,
• Solve linear Diophantine equations and congruences of various types, and use the theory of congruences in applications.
• Prove and apply properties of multiplicative functions such as the Euler phi-function and of quadratic residues.
• Apply the Law of Quadratic Reciprocity and other methods to classify numbers as primitive roots, quadratic residues, and quadratic non-residues,
• Produce rigorous arguments (proofs) centered on the material of number theory, most notably in the use of Mathematical Induction and/or the Well Ordering Principal in the proof of theorems.
• Encrypting and Decrypting messages.

The division algorithm, the Greatest Common Divisor, the Euclidean algorithm, Diophantine Equation, the Fundamental Theorem of Arithmetic, the methods to find prime numbers, the Goldbach Conjecture.

Basic Properties of Congruence, Complete residue system modulo m, reduced residue system modulo m, Euler’s φ function, Fermat’s theorem, Euler’s generalization of Fermat’s theorem, Wilson’s theorem, solutions of linear congruences, the Chinese remainder theorem, solutions of polynomial congruences, prime power moduli, power residues.

Legendre symbol, Gauss’s lemma, quadratic reciprocity, the Jacobi symbol, sums of two squares, Greatest integer function, arithmetic functions, the Mobius inversion formula.

From Caesar Cipher to Public Key Cryptography, The Knapsack Cryptosystem, An Application of Primitive Roots to Cryptography.

2. Neal Koblitz, A course in number theory and cryptography, Reprint: Springer, 2010.

3. Gareth A. Jones and J. Mary Jones, Elementary number theory, Reprint, Springer, 2000.

4. Joseph H. Silverman, A friendly introduction to number theory, Pearson Prentice Hall, 2006.

5. Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An introduction to the theory of numbers, John Wiley, 2004.

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

• Develop their abstract thinking skills.
• Provide precise definitions and  appropriate examples  and counter examples of  fundamental  concepts in general topology.
• Acquire knowledge about various types of topological spaces and their properties.
• Appreciate the beauty of deep mathematical results like Uryzohn’s lemma and understand the dynamics of the 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.

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.

• Understand concept of Linear differential equation, Fundamental set Wronskian.
• Understand the concept of Liouvilles theorem, Adjoint and Self Adjoint equation, Langrage’s Identity, Green’s formula, Eigen value and Eigen functions.
• Identify ordinary and singular point by Frobenius Method, Hyper geometric differential equation and its polynomial.
• Understand the basic concepts and definition of PDE and also mathematical models representing stretched string, vibrating membrane, heat conduction in rod.
• Demonstrate on the canonical form of second order PDE.
• Demonstrate initial value boundary problem for homogeneous and non-homogeneous PDE.
• 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. 

This course aims at studying the fundamentals of fluid mechanics such as kinematics of fluid, incompressible flow and boundary layer flows.

• Confidently manipulate tensor expressions using index notation, and use the divergence theorem and the transport theorem.
• Give a common ground for the mechanics of fluids and solids and the connection of these to applications.
• Remain equiped for further studies in mechanics, applied and industrial mathematics, physics, geology, geophysics, and astrophysics.
• Give an introduction to the basic equations and solution methods for mathematical modeling of viscous fluids and elastic matter.

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.

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.

• Be familiar with the history and development of graph theory
• Write precise and accurate mathematical definitions of basics concepts in graph theory
• Provide appropriate examples  and counter examples to illustrate the basic concepts
• Understand and apply various proof techniques in proving theorems in graph theory.
• Acquire 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.

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

• Proficiency in using the software Scilab and Maxima.
• To understand the use of various techniques of the software for effectively doing Mathematics.
• To obtain necessary skills in programming.
• To make students understand the applications of Mathematics.
• Explore and grasp concepts for the future across a wealth of disciplines.
• Can utilize the software knowledge for academic research.

Installation of the software Scilab, Basic syntax, Mathematical Operators, Predefined constants, Built in functions.  Complex numbers, polynomials, Vectors, Matrix. Handling these data structures using built in functions. Programming, Functions, Loops , Conditional statements. Handling .sci files, Graphics handling -2D, 3D, Function plotting, Data plotting.

Installation of the software Maxima, Intoduction to Maxima, Maxima Interface, Maxima Expressions, Numbers, Operators, Constants and Reserved Words, Introduction to plot2d , Parametric Plots ,line Width and Color , Discrete Data Plots: Point Size, Color, and Type Control , More gnuplot preamble Options, Solving Equations - One Equation or Expression: Symbolic Solution or Roots , The Maxima Function , solve with Expressions or Functions & the multiplicities List , General Quadratic Equation or Function  , Cubic Equation or Expression, Trigonometric Equation or, Equation or Expression Containing Logarithmic Functions, Matrix Methods for Linear Equation Sets.

This course will help students understand the concepts of functions of single and several variables. This course includes such concepts as Riemann-Stieltjes integral, sequences and series of functions, Special Functions and The Implicit Function Theorem.

• Integrate functions a real variable in the sense of Riemann – Stieltjes.
• Classify sequences of functions which are pointwise convergent, uniform convergent etc.

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         

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.

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

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.

• Demonstrate knowledge of conjugates, the Class Equation and Sylow  theorems
• Demonstrate knowledge of polynomial rings and associated properties
• Derive and apply Gauss Lemma, Eisenstein criterion for irreducibility of rationals
• Demonstrate the characteristic of a field and the prime subfield;
• 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. 

This course helps the students to understand the basic concepts of heat transfer, types of convection shear and thermal instability of linear and non-linear problems.  This course also includes the analysis Prandtl boundry layer, porous media and Non-Newtonian fluid.

• Recognize how fluid flow theory can be employed in a modern mechanical engineering design environment.
• Discuss the theory of compressible flows, formulate the relevant theory, and solve the related engineering problems.
• Ability to apply fluid mechanics principles to the analysis of real systems.
• Understand the basic laws of heat transfer and understand the fundamentals of convective heat transfer process. 

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.

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

• Understand the basic concepts and fundamental results in matching, domination, coloring and planarity.
• Construct examples and proofs pertaining to the basic theorems.
• Apply the theoretical knowledge and independent mathematical thinking in creative investigation of questions in graph theory.
• Reason from definitions to construct mathematical proofs.
• Write graph theoretic ideas in a coherent and technically accurate manner.
• 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.

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.

• Understand random variables and probability distributions.
• Distinguish discrete and continuous random variables.
• Obtain ability compute Expected value and Variance of discrete random variable.
• Acquire knowledge in using Binomial distribution, Poisson distribution etc.,
• Define inferential statistics.
• 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.  

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

• Develop their abstract thinking skills.
• Provide precise definitions and  appropriate examples  and counter examples of  fundamental  concepts in general topology.
• Acquire knowledge about various types of topological spaces and their properties.
• Appreciate the beauty of deep mathematical results like Uryzohn’s lemma and understand the dynamics of the 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.

This course helps students to have an in-depth knowledge of various advanced methods in numerical analysis. It also introduces MATLAB programming for scientific computations.  This includes solution of algebraic, transcendental, system of equations, and ordinary differential equations.  

• Derive numerical methods for approximating the solution of problems of algebraic, transcendental, linear systems and ordinary differential equations.
• Implement a variety of numerical algorithms using appropriate technology.
• use MATLAB to solve computational problems.

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.

Initial value problems, Runge-Kutta methods of second and fourth order, multistep method, Adams-Moulton method, stability (convergence and truncation error for the above methods), boundary value problems, second order finite difference method, linear shooting method.

The MATLAB environment, basic operations, use of built-in functions, graphics, programming with MATLAB.

Elementary numerical methods with MATLAB, Solution to single equations and multiple non-linear equations in MATLAB.

Linear system of equations, Numerical differentiation and integration in MATLAB , Data fitting in MATLAB , Solution to Ordinary Differential Equations in MATLAB , Numerical differentiation and finite differences.

2.      S.C.Chapra and R.P. Canale , NumerialMethodsforEngineers,5^th Ed.,McGrawHill, 2006.

3.      Beers, Kenneth J. Numerical Methods for Chemical Engineering: Applications in MATLAB®. New York, NY: Cambridge University Press, November 2006.

4.      Recktenwald, Gerald W. Introduction to Numerical Methods with MATLAB®: Implementations and Applications. Upper Saddle River, NJ: Prentice-Hall, 2000.

5.      K. Mishra, A Handbook on Numerical Technique Lab (MATLAB Based Experiments), I.K. International Publishing House Pvt. Limited, 2007.

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.

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

• Derive some classical differential equations by using principles of calculus of variations.
• Knowledge of Variational Problems, Euler-Lagrange Condition, Second Variation,Generalizations of the Variational Problem.
• Able to find maximum or minimum of a functional using calculus of variations technique.
• Able to solve Voterra integral equations and Fredholm integral equations.
• Able to Reduce the differential equations to integral equations.

Maxima and minima, method of Lagrange multipliers, the simplest case, Euler equation, extremals, stationary function, geodesics, Brachistochrone problem, natural boundary conditions and transition conditions, variational notation, the more general case. 

Galerkian Technique, the Rayleigh-Ritz method.

Definitions, integral equation, Fredholm and Volterra equations, kernel of the integral equation, integral equations of different kinds, relation between differential and integral equations, symmetric kernels, the Green’s function. 

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.     

Domination of Graphs, digraph theory, perfect graphs and chromatic graph theory are dealt with in the detail in this course.

• Thorough understanding of the concepts in domination and perfect graphs.
• Familiarity in implementing the acquired knowledge appropriately.
• Mastery in employing proof techniques.

Domination in Graphs, Bounds in terms of Order, Bounds in terms of Order, Degree and Packing, Bounds in terms of Order and Size, Bounds in terms of Degree, Diameter and Girth, Bounds in terms of Independence and Covering.

T-Colourings,  L(2,1)-colourings,  Radio Colourings,  Hamiltonian Colourings, Domination and Colourings.

The Perfect Graph Theorem, Chordal Graphs Revisited, 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.

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

3.      J. Bang-Jensen and G. Gutin, Digraphs. London: Springer, 2009.

4.      G. Chartrand and P. Zhang, Chromatic Graph Theory. New York: CRC Press, 2009.

2.      F. Harary, Graph Theory, New Delhi: Narosa, 2001.

3.              G. Chartrand and P.Chang, Introduction to Graph Theory, New Delhi: Tata McGraw-Hill,2006.

4.      J. A. Bondy and U.S.R. Murty, Graph Theory, Springer, 2008

5.      J. Clark and D.A. Holton, A First Look At Graph Theory, Singapore: World Scientific, 2005.

6.      R. Balakrishnan and K Ranganathan, A Text Book of Graph Theory, New Delhi: Springer, 2008.

7.      R. Diestel, Graph Theory, New Delhi: Springer, 2006.

8.      M. Bona, A walk through combinatorics, World scientific, 2011. 

The objective of this course is to provide the students an opportunity to gain work experience in the relevant institution, connected to their subject of study. The experienced gained in the workplace will give the students a competetive edge in their career.

·         Expose students to the field of their professional interest

·         Give an opportunity to get practical experience in the field of their interest

·         Strengthen the curriculum based on internship feedback where relevant

·         Help student choose their career through practical experience 

M.Sc. Mathematics students have to undertake a mandatory internship of not less than 30 working days at any of the following: reputed research centres, recognized educational institutions, summer research fellowships, programmes like M.T.T.S or any other approved by the P.G. coordinator and H.O.D.  If the internship is based on teaching in any recognized educational institution, a minimum of 45 hours of teaching and 45 hours of teaching assistance are to be completed.

     The internship is to be undertaken at the end of second semester (during first year vacation). The report submission and the presentation on the report will be held during the third semester and the credits will appear in the mark sheet of third semester.   

     The students will have to give an internship proposal with the following details: Organization where the student proposes to do the internship, reasons for the choice, nature of internship, period on internship, relevant permission letters, if available, name of the mentor in the organization, email, telephone and mobile numbers of the person in the organization with whom Christ University could communicate matters related to internship. Typed proposals will have to be given at least one month before the end of the second semester.  

     The coordinator of the programme in consultation with the HOD will assign faculty members from the department as guides at least two weeks before the end of second semester.

     The students will have to be in touch with the guides during the internship period either through personal meetings, over the phone or through email.

     At the place of internship, students are advised to be in constant touch with their mentors.

     At the end of the required period of internship, the candidates will submit a report in a specified format adhering the guidelines of the department. The report should be submitted within first 20 days of the reopening of the University for the third semester.

     Within a month from the day of reopening, the department must hold a presentation by the students. During the presentation the guide or a nominee of the guide should be present and be one of the evaluators. Students should preferably be encouraged to make a beamer presentation of their report. A minimum of 10 minutes should be given for each of the presenter. The maximum limit is left to the discretion of the evaluation committee. Students will get 2 credits on successful completion of internship. If a student fail to comply the aforementioned guidelines, the student has to repeat the internship.

Differential geometry is the study of geometric properties of curves, surfaces, and their higher dimensional analogues using the methods of calculus. On successful completion of this module students will have acquired an active knowledge and understanding of the basic concepts of the geometry of curves and surfaces in three-dimensional Euclidean space and will be acquainted with the ways of generalising these concepts to higher dimensions

• Obtain sound knowledge in understanding the basic concepts in geometry of curves and surfaces in Euclidean space, especially E3.
• Acquire mastery in solving typical problems associated with the theory.
• Gain sufficient knowledge for generalizing these concepts to higher dimensions.

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.

Isometries of E³ - The derivative map of an Isometry - Surfaces in E³ - patch computations - Differential functions and Tangent vectors - Differential forms on a surface - Mappings of Surfaces.

The Shape operator of M in E³ - Normal Curvature - Gaussian Curvature - Computational Techniques - Special curves in a surface - Surfaces of revolution.

This course helps the students to understand the basic concepts of heat transfer, types of convection shear and thermal instability of linear and non-linear problems, dimensional analysis. The flow problems are analyses using finite element method. 

• Understand both flow physics and mathematical properties of governing Navier-Stokes equations and define proper boundary conditions for solution.
• introduce the 'theory and practice' of the finite element method. Experience with writing a simple finite element solver for an ordinary differential equation in MATLAB.
• Understanding of physics of compressible and incompressible fluid flows.
• Have Ability to solve the fluid flow equations using Finite element method.

Review of classification of partial differential equations, classification of boundary conditions,  numerical analysis, basic governing equations of fluid mechanics. 

Derivation of finite difference methods, finite difference method to parabolic, hyperbolic and elliptic equations, finite difference method to nonlinear equations, coordinate transformation for arbitrary geometry, Central schemes with combined space-time discretization-Lax-Friedrichs, Lax-Wendroff, MacCormack methods, Artificial compressibility method, pressure correction method – Lubrication model, Convection dominated flows – Euler equation – Quasilinearization of Euler equation, Compatibility relations, nonlinear Burger equation.

General introduction, Node-centered-control volume, Cell-centered-control volume and average volume, Cell-Centred scheme, Cell-Vertex scheme, Structured and Unstructured FVMs, Second and Fourth order approximations to the convection and diffusion equations (One and Two-dimensional examples).

Introduction to finite element methods, one-and two-dimensional bases functions – Lagrange and Hermite polynomials elements, triangular and rectangular elements, Finite element method for one-dimensional problem: model boundary value problems, discretization of the domain, derivation of elemental equations and their connectivity, composition of boundary conditions and solutions of the algebraic equations. Finite element method for two-dimensional problems: model equations, discretization, interpolation functions, evaluation of element matrices and vectors and their assemblage.

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.

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

This abstract course imparts an in-depth analysis of Banach spaces, Hilbert spaces, conjugate spaces, etc. This course also includes a few important applications of functional analysis to other branches of both pure and applied mathematics.

• Explain the fundamental concepts of functional analysis.
• Understand the approximation of continuous functions.
• Understand concepts of Hilbert and Banach spaces with l2 and lp spaces serving as examples.
• Understand the definitions of linear functional and prove the Hahn-Banach theorem, open mapping theorem, uniform boundedness theorem, etc.
• Define linear operators, self adjoint, isometric and unitary operators on Hilbert spaces.

Normed linear spaces, Banach spaces, continuous linear transformations, isometric isomorphisms, functionals and the Hahn-Banach theorem, the natural embedding of a normed linear space in its second dual.

The open mapping theorem and the closed graph theorem, the uniform boundedness theorem, the conjugate of an operator.

Inner products, Hilbert spaces, Schwarz inequality, parallelogram law, orthogonal complements, orthonormal sets, Bessel’s inequality, complete orthonormal sets. 

The conjugate space, the adjoint of an operator, self-adjoint, normal and unitary operators, projections, finite dimensional spectral theory.

This course is concerned with the fundamentals of mathematical modeling. The coverage includes mathematical modeling through ordinary and partial differential equations.

• Demonstrate a working knowledge of differential equations in other branches of sciences, commerce, medicine etc.
• Become familiar with some of the classical mathematical models.
• Gain the ability to determine the validity of a given model and will be able to construct further improvement in the models independently.
• Formulate, interpret and draw inferences from mathematical models.
• Solve other problems by means of intuition, creativity, guessing, and the experience gained through the study of particular examples and mathematical models.
• Demonstrate competence with a wide variety of mathematical tools and techniques.
• Create mathematical models of empirical or theoretical phenomena in domains such as the physical, natural, or social science.
• Take an analytical approach to problems in their future endeavors.

Definition, Classification, Characteristics and Limitations, Linear and Nonlinear Models,  Compartment Models – Exponential and decay models, Density-dependent growth, Limited growth with harvesting, Lake pollution models, drug assimilation into blood, equilibrium points and stability, case studies (Solutions of the problems through Mathematical Packages)

Phase-plane analysis of epidemic model, Analysis of a battle model,  Analysis of a predator-prey model, Analysis of competing species models, Closed trajectories for the predator-prey, Extended predator-prey models. Case Studies (Solutions of the problems through Mathematical Packages).                            

Mathematical models of vibrating string, vibrating membrane, conduction of heat in solids, gravitational potential. (Solutions of the problems through Mathematical Packages)

Simple models, conservation law , Traffic flow on highway , Flood waves in rivers, shallow water waves,  Convection diffusion –processes Burger’s equation, Fisher’s equation. Telegraph equation of heat transfer in a layered solid.,diffusion systems, travelling waves, pattern formation, tumour growth (Solutions of the problems through Mathematical Packages).

1. B. Barnes and G. R. Fulford, “Mathematical Modelling with case Studies Using Maple and MATLAB’, CRC Press, 2015.
2. Tyn Myint-U and Lokenath Debnath, “Linear Partial Differential Equations for Scientist and Engineers”, Birkhauser Publication house, 2007.