CO
UNION CHRISTIAN COLLEGE, ALUVA
DEPARTMENT OF MATHEMATICS
UNDERGRADUATE COURSES
INDEX
1.  Complementary Course – Physics/ Chemistry – First  PARTIAL DIFFERENTIATION,MATRICES,TRIGONOMETRY 
2.  COMPLEMENTARY COURSE –
PHYSICS/CHEMISTRY – Second

INTEGRAL CALCULUS AND DIFFERENTIAL EQUATIONS

3.  COMPLEMENTARY COURSE –
PHYSICS/CHEMISTRY – Third

VECTOR CALCULUS, ANALYTIC GEOMETRY AND ABSTRACT ALGEBRA 
4.  Complementary Course – Physics/ Chemistry Fourth  FOURIER SERIES, LAPLACE TRANSFORM AND COMPLEX ANALYSIS. 
5.  Complementary Course – ECONOMICS – First  MM1CMT04 – GRAPHING FUNCTIONS, EQUATIONS, DIFFERENTIAL CALCULUS AND LOGARITHMIC AND EXPONENTIAL FUNCTIONS. 
6.  Complementary Course – ECONOMICS – Second

Matrix, Linear Programming and Integral Calculus 
7.  Complementary Course – Computer Science  Discrete Mathematics 1 
8.  Complementary Course – Computer Science  Discrete Mathematics 2 
9.  Core Course Mathematics – First  Foundation of Mathematics 
10.  Core Course Mathematics – Second  ANALYTIC GEOMETRY, TRIGONOMETRY AND DIFFERENTIAL CALCULUS 
11.  Core Course Mathematics – Third  CALCULUS 
12.  Core Course Mathematics – Fourth  Vector Calculus, Theory of Numbers and Laplace Transform 
13.  Core Course Mathematics – Fifth  MATHEMATICAL ANALYSIS 
14.  Core Course Mathematics – Fifth  Differential Equations 
15.  Core Course Mathematics – Fifth  Environmental Mathematics and Human Rights 
16.  Open Course Fifth  Applicable Mathematics 
17.  Core Course Mathematics – Fifth  Abstract Algebra 
18.  Core Course Mathematics – Sixth  Linear Algebra 
19.  Core Course Mathematics – Sixth  REAL ANALYSIS 
20.  Core Course Mathematics – Sixth  Complex Analysis 
21.  Core Course Mathematics – Sixth  GRAPH THEORY AND METRIC SPACES 
22.  Core Course Mathematics – Sixth  OPERATIONS RESEARCH 
Department  MATHEMATICS 
Name of Faculty  
Programme Name  Complementary Course – Physics/ Chemistry 
Level of study  UG 
Semester  FIRST 
Course Name/Subject
Name 
MM1CMT01: PARTIAL DIFFERENTIATION,MATRICES,TRIGONOMETRY 
Total Hours  72 
Course Outcomes
CO
Number 
Description  CO Evaluation methods 
CO1  Understand and familiarize a real valued function of several variables.  Assignment,Test 
CO2  Learn to differentiate functions of several variables.  Assignment,Test 
CO3  Learn the concept of rank of a matrix and how to compute rank.  Assignment,Test, Seminar 
CO4  Understand how matrices could be used to solve systems of equations that are derived based on practical applications.  Assignment, Test 
CO5  Learn to find characteristic roots, vectors and equations.  Assignment,Test,Seminar 
CO6  Learn to expand trigonometric functions, separate real and imaginary parts and sum infinite series whose terms involve trigonometric functions.  Assignment, Viva, Test 
CO7  Analyze the approximate roots of equations, by either bracketing a root or without bracketing a root.  Assignment, Test 
Module 1 Hours : 14  
Syllabus: Partial Differentiation – Functions Of Several Variables (Definitions and simple graphs
only), Partial derivatives, The Chain Rule.


Sl.no  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  1, 2  Functions Of Several Variables  4  Lecture, Demonstration 
2  1,2  Partial Differentiation – concept and problems  8  Lecture, Demonstration 
3  2  The Chain Rule  2  Lecture 
Module 2 Hours : 21  
Syllabus: Matrices – Rank Of a Matrix, Elementary transformations of a matrix, Reduction to Normal form, Employment of only row (column) transformations, System of Linear Homogeneous Equations, Systems of linear nonhomogeneous equations, Characteristic roots and characteristic vectors of a square matrix, Characteristic matrix and Characteristic equation of a matrix, Cayley Hamilton theorem, Expression of the inverse of a nonsingular matrix A as a polynomial in A with scalar coefficients


Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  3  Rank Of a Matrix, Elementary transformations of a matrix  3  Lecture 
2  3  Reduction to Normal form, Employment of only row (column) transformations  4  Lecture 
3  3, 4  System of Linear Homogeneous Equations  5  Lecture 
4  3, 4  Systems of linear nonhomogeneous equations  5  Lecture 
5  5  Characteristic roots and characteristic vectors  2  Lecture 
6  5  Characteristic equation of a matrix,  2  Lecture 
7  5  Cayley Hamilton theorem, Expression of the inverse of a nonsingular matrix A as a polynomial in A with scalar coefficients

2  Lecture 
Module 3 Hours : 23  
Syllabus: Expansions ofsinnθ, cos nθ,tan nθ, sin^nθ, cos^nθ, sin^nθcos^mθ, circular and hyperbolic
Functions, inverse circular and hyperbolic function,Separation Into Real And Imaginary Parts.


Slno  CO
Number 
Topic/Activity  No of
hours 
Instructional methods to be used 
1  6  Expansions of sinnθ, cos nθ, tan nθ  7  Lecture 
2  6  Expansions of sin^nθ, cos^nθ, sin^nθcos^mθ  7  Lecture 
4  6  Circular and hyperbolic
Functions 
4  Lecture 
5  6  Inverse circular and hyperbolic function  1  Lecture 
6  6  Separation Into Real And Imaginary Parts  4  Lecture 
Module 4 Hours : 14  
Syllabus: Numerical MethodsBisection Method, Method of False Position, Iteration Method, NewtonRaphsonMethod.


Slno  CO
Number 
Topic/Activity  No of
hours 
Instructional methods to be used 
1  7  Bisection Method,  2  Lecture 
2  7  Method of False Position  3  Lecture 
3  7  Iteration Method  4  Lecture 
4  7  NewtonRaphsonMethod.

5  Lecture 
Department  Mathematics 
Name of Faculty  
Programme Name  B. Sc. Degree Programme 
Level of study  UG 
Semester  II 
Course Name/Subject Name  MATHEMATICS COMPLEMENTARY COURSE TO
PHYSICS/CHEMISTRY MM2CMT01: INTEGRAL CALCULUS AND DIFFERENTIAL EQUATIONS

Total Hours  72 
Course Outcomes
CO Number  Description  CO Evaluation methods 
CO1  Use the tools of integration to find volume, arc length, area of surface of revolution  Assignment, Test 
CO2  Find the area and volume by applying the techniques of double and triple integrals  Assignment, Test 
CO3  Find solutions to Ordinary Differential Equations like variable separable, Linear and Bernoulli equations  Assignment, Test 
CO4  Generate Partial Differential Equations  Assignment, Test 
CO5  Solve the differential equation  Assignment, Test 
CO6  Use Lagrange’s method for solving the first order linear partial differential equation  Assignment, Test 
Module 1 : Integral Calculus  Hours : 15  
Syllabus :
Volumes using CrossSections, Volumes using Cylindrical shells, Arc lengths, Areas of surfaces of Revolution. 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used  
1  CO1  Volumes using CrossSections  4  Lecture, problem solving  
2  CO1  Volumes using Cylindrical shells  4  Lecture, problem solving  
3  CO1  Arc lengths  3  Lecture, problem solving  
4  CO1  Areas of surfaces of Revolution.  4  Lecture, problem solving  
Module 2 : Multiple Integrals  Hours : 17  
Syllabus:
Double and iterated integrals over rectangles, Double integrals over general regions, Area by double integration, Triple integrals in rectangular coordinates. 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used  
1  CO2  Double and iterated integrals over rectangles  4  Lecture, problem solving  
2  CO2  Double integrals over general regions  4  Lecture, problem solving  
3  CO2  Area by
double integration 
5  Lecture, problem solving  
4  CO2  Triple integrals in rectangular coordinates  4  Lecture, problem solving  
Module 3 : Ordinary Differential Equations  Hours : 20  
Separable Variables, Exact Differential Equation, Equations reducible to exact form, Linear
Equations, Solutions by Substitutions, Homogeneous equations and Bernoulli’s Equations. 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used  
1  CO3  Separable Variables  3  Lecture, problem solving  
2  CO3  Exact Differential Equation  4  Lecture, problem solving  
3  CO3  Linear Equations  4  Lecture, problem solving  
4  CO3  Homogeneous equations  5  Lecture, problem solving  
5  CO3  Bernoulli’s Equations  4  Lecture, problem solving  
Module 4 : Partial Differential Equations  Hours : 20  
Syllabus:
Surfaces and Curves in three dimensions, Solution of equations of the form Origin of first order and second order partial differential equations, Linear equations of the first order, Lagrange’s method. 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used  
1  CO4  Surfaces and Curves in three dimensions  4  Lecture, problem solving  
2  CO4, CO5  Origin of first order and second order partial differential equations  8  Lecture, problem solving  
3  CO6  Linear equations of the first order, Lagrange’s method.  8  Lecture, problem solving  
Department  Mathematics 
Name of Faculty  
Programme Name  B. Sc. Degree Programme 
Level of study  UG 
Semester  III 
Course Name/Subject Name  MATHEMATICS COMPLEMENTARY COURSE TO
PHYSICS/CHEMISTRY MM3CMT01:VECTOR CALCULUS, ANALYTIC GEOMETRY AND ABSTRACT ALGEBRA 
Total Hours  90 
Course Outcomes
CO Number  Description  CO Evaluation methods 
CO1  Differentiate vector valued functions  Assignment, Test 
CO2  Find arc length and unit tangent vector, curvature and the unit normal vector, tangential and normal components of acceleration  Assignment, Test 
CO3  Find directional derivatives, gradient vectors, tangent planes and normal lines  Assignment, Test 
CO4  Familiarize line integrals and surface integrals  Assignment, Test 
CO5  Find work, circulation and flux, conservative fields and potential functions  Assignment, Test 
CO6  Apply Green’s theorem, Stokes’ theorem and Divergence theorem  Assignment, Test 
CO7  Sketch conics and solve problems in conic sections  Assignment, Test 
CO8  Familiarize basic concepts of Abstract Algebra like Groups , Subgroups and Homomorphism  Assignment, Test 
Module 1 : Vector valued Functions  Hours : 15  
Syllabus :
Curves in space and their tangents, Arc length in space, Curvature and Normal Vectors of a curve, Directional Derivatives and Gradient Vectors. 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used  
1  CO1  Curves in space and their tangents  3  Lecture, problem solving  
2  CO2  Arc length in space  4  Lecture, problem solving  
3  CO2  Curvature and Normal Vectors of a
curve 
4  Lecture, problem solving  
4  CO3  Directional Derivatives and Gradient Vectors  4  Lecture, problem solving  
Module 2 : Integration in Vector Fields  Hours : 25  
Syllabus:
Line Integrals, Vector fields and line integrals: Work, Circulation and Flux. Path independence, Conservation Fields and Potential Functions , Green’s theorem in Plane (Statement and problems only), Surface area and Surface integral, Stoke’s theorem( Statement and Problems only), the Divergence theorem and a Unified theory ( Statement and simple problems only). 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used  
1  CO4  Line Integrals  4  Lecture, problem solving  
2  CO5  Vector fields and line integrals  6  Lecture, problem solving  
3  CO6  Green’s theorem in Plane  4  Lecture, problem solving  
4  CO6  Surface area and Surface integral  4  Lecture, problem solving  
5  CO6  Stoke’s theorem  4  Lecture, problem solving  
6  CO6  Divergence
theorem and a Unified theory 
3  Lecture, problem solving  
Module 3 : Analytic Geometry  Hours : 25  
Polar coordinates, Conic sections, Conics in Polar coordinates.  
Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used  
1  CO7  Polar coordinates  7  Lecture, problem solving  
2  CO7  Conic sections  9  Lecture, problem solving  
3  CO7  Conics in Polar coordinates  9  Lecture, problem solving  
Module 4 : Abstract algebra  Hours : 25  
Syllabus:
Groups, Subgroups, Cyclic groups, Groups of Permutations, Homomorphism. 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used  
1  CO8  Groups  7  Lecture, problem solving  
2  CO8  Subgroups  6  Lecture, problem solving  
3  CO8  Cyclic groups  7  Lecture, problem solving  
4  CO8  Groups of Permutations  5  Lecture, problem solving  
Department  MATHEMATICS 
Programme Name  B.Sc. PHYSICS / CHEMISTRY 
Level of study  UG 
Semester  FOURTH 
Course Name  MM4CMT01: FOURIER SERIES, LAPLACE TRANSFORM AND COMPLEX ANALYSIS. 
Total hours  90 
Course Outcomes
CO Number  Description  Co Evaluation methods 
CO 1  Find Fourier series of functions.  Assignment and Test 
CO 2  Solve problems involving Fourier Series and Legendre polynomials.  Assignment and Test 
CO 3  Apply Power series method to solve differential equations.  Assignment and Test 
CO 4  Familiarize Laplace transform and its properties.  Assignment and Test 
CO 5  Apply Laplace transforms to solve differential equations.  Assignment and Test 
CO 6  Conceive the concept of analytic functions.  Assignment and Test 
CO 7  Familiar with the theory and techniques of complex integration.  Assignment and Test 
Module I Hours: 25  
Syllabus: Fourier Series and Legendre Polynomials
Periodic Functions, Trigonometric Series, Fourier Series, Functions of any period p = 2L, Even and Odd functions, Half range Expansions. A brief introduction to power series and power series method for solving Differential equations, Legendre equation and Legendre polynomials.


Sl. No.  CO Number  Topic/ Activity  No. of hours  Instructional methods to be used 
1  CO 1  Periodic Functions, Trigonometric Series, Fourier Series, Functions of any period p = 2L, Even and Odd functions, Half range Expansions.  10  Lecture, Problem Solving 
2  CO 2  Legendre equation and Legendre polynomials.  6  Lecture, Problem Solving 
3  CO 3  A brief introduction to power series and power series method for solving Differential equations  9  Lecture, Problem Solving 
Module II Hours: 20  
Syllabus: Laplace Transform
Laplace Transform, Inverse Laplace transform, Linearity, Shifting, transforms of Derivatives and Integrals, Differential Equations, Differentiation and Integration of Transforms, Laplace transform general Formula (relevant formulae only), Table of Laplace Transforms (relevant part only) 

Sl. No.  CO Number  Topic/ Activity  No. of hours  Instructional methods to be used 
1  CO 4  Laplace Transform, Inverse Laplace transform, Linearity, Shifting, transforms of Derivatives and Integrals.  8  Lecture, Problem Solving 
2  CO 5  Differential Equations, Differentiation and Integration of Transforms  7  Lecture, Problem Solving 
3  CO 4  Laplace transform general Formula (relevant formulae only), Table of Laplace Transforms (relevant part only  5  Lecture, Problem Solving 
Module III Hours: 25  
Syllabus: Complex Numbers and Functions
Complex Numbers, Complex Plane, Polar form of Complex Numbers, Powers and Roots, Derivative, Analytic Functions, CauchyRiemann Equations, Laplace’s Equation, Exponential Function, Trigonometric Functions, Hyperbolic Functions, Logarithm, General Power. 

Sl. No.  CO Number  Topic/ Activity  No. of hours  Instructional methods to be used 
1  CO 6  Complex Numbers, Complex Plane, Polar form of Complex Numbers, Powers and Roots.  5  Lecture, Problem Solving 
2  CO 6  Derivative, Analytic Functions, CauchyRiemann Equations, Laplace’s Equation.  8  Lecture, Problem Solving 
3  CO 6  Exponential Function.  5  Lecture, Problem Solving 
4  CO 6  Trigonometric Functions, Hyperbolic Functions.  4  Lecture, Problem Solving 
5  CO 6  Logarithm, General Power.  3  Lecture, Problem Solving 
Module IV Hours: 20  
Syllabus: Complex Integration
Line Integral in the Complex Plane, Cauchy’s Integral Theorem, Cauchy’s Integral Formula, Derivatives of Analytic functions. 

Sl. No.  CO Number  Topic/ Activity  No. of hours  Instructional methods to be used 
1  CO 7  Line Integral in the Complex Plane.  10  Lecture, Problem Solving and Demonstration. 
2  CO 7  Cauchy’s Integral Theorem, Cauchy’s Integral Formula.  5  Lecture, Problem Solving 
3  CO 7  Derivatives of Analytic functions  5  Lecture, Problem Solving 
Department  MATHEMATICS 
Programme Name  B.A. ECONOMICS 
Level of study  UG 
Semester  FIRST 
Course Name  MM1CMT04 – GRAPHING FUNCTIONS, EQUATIONS, DIFFERENTIAL CALCULUS AND LOGARITHMIC AND EXPONENTIAL FUNCTIONS. 
Total hours  108 
Course Outcomes
CO Number  Description  Co Evaluation methods 
CO 1  Familiarize linear equations, functions and graphing functions.  Assignment and Test 
CO 2  Find solutions to quadratic equations and system of linear equations  Assignment and Test 
CO 3  Understand the basic concepts of differential calculus and its applications  Assignment and Test 
CO 4  Familiarize exponential and logarithmic functions  Assignment and Test 
CO 5  Compute simple and compound interest  Assignment and Test 
CO 6  Apply the above theories in business and economics  Assignment and Test 
Module I Hours: 20  
Syllabus: Equations and Graphs Equations
Review – (Exponents, polynomials, factoring, fractions, radicals, order of mathematical operations.) Cartesian Coordinate system, linear equations and graphs slopes intercepts. The slope intercept form. Determining the equation of a straight line. Applications of line equations in business and economics. 

Sl. No.  CO Number  Topic/ Activity  No. of hours  Instructional methods to be used 
1  CO 1  Review – Exponents, polynomials, factoring, fractions, radicals, order of mathematical operations.  5  Lecture, Problem Solving 
2  CO 1  Cartesian Coordinate system, linear equations and graphs slopes intercepts.  4  Lecture, Problem Solving 
3  CO 1  The slope intercept form.  2  Lecture, Problem Solving 
4  CO 1  Determining the equation of a straight line.  4  Lecture, Problem Solving 
5  CO 6  Applications of line equations in business and economics.  5  Lecture, Problem Solving 
Module II Hours: 23  
Syllabus: Functions Concepts
Functions Concepts and definitions graphing functions. The algebra of functions. Applications of linear functions for business and economics. Solving quadratic equations Facilitating nonlinear graphing. Application of nonlinear functions in business and economics. System of equations Introduction, graphical solutions. Supplydemand analysis. Breakeven analysis. Elimination and substitution methods. ISLM analysis. Economic and mathematical modeling. Implicit functions and inverse functions. 

Sl. No.  CO Number  Topic/ Activity  No. of hours  Instructional methods to be used 
1  CO 1  Functions Concepts and definitions graphing functions. The algebra of functions.  3  Lecture, Problem Solving 
2  CO 6  Applications of linear functions for business and economics.  4  Lecture, Problem Solving 
3  CO 2  Solving quadratic equations  2  Lecture, Problem Solving 
4  CO 2  Facilitating nonlinear graphing.  2  Lecture, Problem Solving 
5  CO 6  Application of nonlinear functions in business and economics.  3  Lecture, Problem Solving 
6  CO 2  System of equations Introduction, graphical solutions.  2  Lecture, Problem Solving 
7  CO 6  Supplydemand analysis
Breakeven analysis. Elimination and substitution methods ISLM analysis. Economic and mathematical modeling Implicit functions and inverse functions. 
2  Lecture, Problem Solving 
Module III Hours: 40  
Syllabus: Differential Calculus
Limits and continuity. Evaluation of limit of a function. Algebraic limit. The derivative and the rules of differentiation: The slope of curvilinear function. Derivative notation. Rules of differentiation. Higher order derivatives. Derivative of Implicit functions. Applications of derivatives. Increasing and decreasing functions. Concavity and convexity. Relative extrema. 147 Inflection points. Curve sketching. Optimization of functions. The successive derivative test. Marginal concepts in economics. Optimizing economic functions of business. Relation among total, marginal and average functions. 

Sl. No.  CO Number  Topic/ Activity  No. of hours  Instructional methods to be used 
1  CO 3  Limits and continuity. Evaluation of limit of a function. Algebraic limit.  5  Lecture, Problem Solving 
2  CO 3  The derivative and the rules of differentiation: The slope of curvilinear function. Derivative notation. Rules of differentiation. Higher order derivatives. Derivative of Implicit functions.  10  Lecture, Problem Solving 
3  CO 6  Applications of derivatives.  5  Lecture, Problem Solving 
4  CO 3  Increasing and decreasing functions. Relative extrema. Inflection points. Curve sketching. Optimization of functions. The successive derivative test.  10  Lecture, Problem Solving and Demonstration. 
5  CO 6  Marginal concepts in economics. Optimizing economic functions of business. Relation among total, marginal and average functions.  10  Lecture, Problem Solving 
Module IV Hours: 25  
Syllabus: Exponential functions and Logarithmic functions
Exponential functions. Logarithmic functions. Properties of exponents and logarithms. Natural exponential and logarithmic functions. Solving natural exponential and logarithmic functions. Logarithmic transformation of nonlinear functions. Derivatives of natural exponential and logarithmic functions. Interest compounding. Estimating growth rates from data points. 

Sl. No.  CO Number  Topic/ Activity  No. of hours  Instructional methods to be used 
1  CO 4  Exponential functions. Logarithmic functions. Properties of exponents and logarithms. Natural exponential and logarithmic functions. Solving natural exponential and logarithmic functions. Logarithmic transformation of nonlinear functions.  10  Lecture, Problem Solving and Demonstration. 
2  CO 4  Derivatives of natural exponential and logarithmic functions.  5  Lecture, Problem Solving 
3  CO 5  Interest compounding.  5  Lecture, Problem Solving 
4  CO 6  Estimating growth rates from data points.  5  Lecture, Problem Solving 
Department  Mathematics 
Name of Faculty  
Programme Name  BA Economics 
Level of study  UG 
Semester  Second 
Course Name  Matrix, Linear Programming and Integral Calculus 
Total hours  108 
Course Outcomes
CO Number  Description  CO Evaluation methods 
Upon completion of this course, the students will be able to:  
CO1  Understand the concept of matrix and basic operations of matrices.  Assignment, viva, Seminar, Test 
CO2  Apply matrix theory for solving linear equations and in business and economics problems.  Assignment, viva, Seminar, Test 
CO3  Find Mathematical formulation of Linear Programming Problem.  Assignment, viva, Seminar, Test 
CO4  Solve maximization and minimization problems using graphical method.  Assignment, viva, Seminar, Test 
CO5  Evaluate indefinite integral of functions.  Assignment, viva, Seminar, Test 
CO6  Evaluate definite integral of functions and familiarize its properties.  Assignment, viva, Seminar, Test 
CO7  Evaluate area under a curve and area between two curves by applying definite integral and optimize multi variable functions in Business and Economics by using the techniques of partial derivatives.  Assignment, viva, Seminar, Test 
Module: 1 Hours: 30  
Syllabus:
Matrix Algebra Introduction. Definition and terms. Addition and subtraction of matrices. Scalar multiplication. Vector multiplication. Multiplication of matrices. Matrix expression of a system of linear equations. Augmented matrix. Row operation. Gaussian method of solving linear equations. Solving linear equations with. Matrix algebra Determinants and linear independence. Third order determinants. Cramer’s rule for solving linear equations. Inverse matrices. Gaussian method of finding an inverse matrix. Solving linear equations with an inverse matrix. Business and Economic applications. Special determinants. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO1  Introduction. Definition and terms. Addition and subtraction of matrices. Scalar multiplication. Vector multiplication. Multiplication of matrices.  6  Lecture, Problem solving 
2  CO2  Matrix expression of a system of linear equations. Augmented matrix. Row operation. Gaussian method of solving linear equations.  6  Lecture, Problem solving 
3  CO2  Solving linear equations with. Matrix algebra Determinants and linear independence. Third order determinants. Cramer’s rule for solving linear equations.  6  Lecture, Problem solving 
4  CO2  Inverse matrices. Gaussian method of finding an inverse matrix. Solving linear equations with an inverse matrix.  6  Lecture, Problem solving 
5  CO2  Business and Economic applications. Special determinants.  6  Lecture, Problem solving 
Module: 2 Hours: 20  
Syllabus:
Linear programming Linear programming problem (LPP), Mathematical Formulation of LPP. Basic solution, Feasible solution and Region of feasible solution of an LPP. The extreme point theorem. Solving Maximisation and Minimisation problems using graphical method. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO3  Linear programming problem (LPP), Mathematical Formulation of LPP.  5  Lecture, Problem solving 
2  CO3  Basic solution, Feasible solution and Region of feasible solution of an LPP  5  Lecture, Problem solving 
3  CO4  The extreme point theorem. Solving Maximisation and Minimisation problems using graphical method.  10  Lecture, Problem solving 
Module: 3 Hours: 35  
Syllabus:
Integral calculus Integration rules for indefinite integrals. Integration by substitution. Integration by parts. The definite integral. The fundamental theorems of calculus. Properties of definite integrals. Area under a curve. Area between curves. Present value of cash flow consumers and producers surplus. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO5  Integration rules for indefinite integrals. Integration by substitution. Integration by parts.  10  Lecture, Problem solving 
2  CO6  The definite integral. The fundamental theorems of calculus. Properties of definite integrals.  10  Lecture, Problem solving 
3  CO7  Area under a curve. Area between curves.  10  Lecture, Problem solving 
4  CO7  Present value of cash flow consumers and producers surplus.  5  Lecture, Problem solving 
Module: 4 Hours: 23  
Syllabus:
Calculus of Multivariable functions Functions of several independent variables. Partial derivatives. Rules of partial differentiation. Second order partial derivatives. Optimization of multivariable functions. Constrained optimization with Lagrange Multipliers. Income determination Multipliers. Optimization of multivariable functions in business and economics constrained optimization of multivariable economic functions. Constrained optimization of Cobb Douglas production functions. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO7  Functions of several independent variables. Partial derivatives. Rules of partial differentiation. Second order partial derivatives.  9  Lecture, Problem solving 
2  CO7  Optimization of multi variable functions. Constrained optimization with Lagrange Multipliers.  9  Lecture, Problem solving 
3  CO7  Optimization of multivariable functions in business and economics constrained optimization of multivariable economic functions. Constrained optimization of Cobb Douglas production functions.  5  Lecture, Problem solving 
Department  MATHEMATICS 
Name of Faculty  
Programme Name  B.Sc. Computer Science 
Level of study  UG 
Semester  1 
Course Name/Subject
Name 
Discrete Mathematics 1 
Total Hours  72 
Course Outcomes
CO
Number 
Description  CO Evaluation methods 
CO1  To familiarize with the basic concepts of logic and to develop logical ability.  Assignment, Test 
CO2  To use logical reasoning to analyse any mathematical argument/ problem.  Assignment, Test 
CO3  To obtain basic knowledge about sets, functions, sequences and summations.  Assignment, Test 
CO4  To familiarize with the graphs of some important functions.  Assignment, Test 
CO5  To introduce number theory and some applications..  Assignment, Test 
CO6  To get basic concepts of relations , its properties, representation and types of relations  Assignment,Test 
Module 1 Hours : 18  
Syllabus: Logic
Propositional Logic, Propositional Equivalence, Predicates and Quantifiers and Rules of Inference 

Sl no.  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  CO1  Propositional Logic,, Predicates and Quantifiers  10  Lecture 
2  CO2  Propositional Equivalence, Rules of Inference  8  Lecture 
Module 2 Hours : 15 
Syllabus:
Basic Structures Sets, Set Operations, Functions, Sequences and Summations 

Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  CO3  Sets, Set Operations, Functions, Sequences ,Summations  12  Lecture 
1  CO1  Propositional Logic,, Predicates and Quantifiers  10  Lecture 
2  CO2  Propositional Equivalence, Rules of Inference  8  Lecture 
Module 2 Hours : 15  
Syllabus:
Basic Structures Sets, Set Operations, Functions, Sequences and Summations


Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  CO3  Sets, Set Operations, Functions, Sequences ,Summations  12  Lecture 
2  CO4  Graphs of functions  3  
Module 3 Hours : 20  
Syllabus: Number Theory and Cryptosystem
The Integers and Division, Primes and Greatest Common Divisors, Applications of Number Theory 
Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to used 
1.  CO5  The Integers and Division, Primes ,Greatest Common Divisors, Applications of Number Theory

20  Lecture 
Module 4 Hours : 19  
Syllabus:
Relations Relations and Their Properties, Representing Relations, Equivalence Relations, Partial Ordering 

Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  CO5  Relations, its representations, types of relations  19  Lecture, Seminar 
Department  MATHEMATICS 
Name of Faculty  
Programme Name  B.Sc. Computer Science 
Level of study  UG 
Semester  2 
Course Name/Subject
Name 
Discrete Mathematics 2 
Total Hours  72 
Course Outcomes
CO
Number 
Description  CO Evaluation methods 
CO1  To familiarize with graph terminologies and different types of graphs.  Assignment, Test 
CO2  Representation of graph in matrix form.  Assignment, Test 
CO3  To get knowledge about trees.  Assignment, Test 
CO4  To familiarize with Boolean functions and logic gates .  Assignment, Test 
CO5  Conceive the basic concepts of matrices such as rank of a matrix, Characteristic equation, Characteristic roots, and characteristic vectors of a square matrix

Assignment, Test 
CO6  To solve system of linear equations using matrices  Assignment, Test 
Module 1 Hours : 18  
Syllabus:
Graphs and Graph Models, Graph Terminology and Special types of Graphs,Representing Graphs and Graph Isomorphism, Connectivity, Euler and Hamilton Paths. 

Sl no.  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  CO1  Graph terminologies,graph models,graph isomorphism,Euler and Hamiltonian paths.  1  Lecture 
2  CO2  Representation of graphs in matrix form.  8  Lecture 
Module 2 Hours : 17  
Syllabus: Trees
Introduction to Trees, Application of Trees, Tree Traversal, and Spanning Trees. 

Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  CO3  Trees, Application of trees, Spanning trees  17  Lecture 
Module 3 Hours : 17  
Syllabus: Boolean Algebra
Boolean Function, Representing Boolean Functions and Logic Gates 
Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to used 
1  CO4  Boolean functions, Logic gates  17  Lecture 
Module 4 Hours : 20  
Syllabus:
Matrices Definitions and examples of Symmetric, Skewsymmetric, Conjugate, Hermitian, Skew hermitian matrices. Rank of Matrix , Determination of rank by Row Canonical form and Normal form , Linear Equations, Solution of non homogenous equations using Augmented matrix and by Cramers Rule , Homogenous Equations, Characteristic Equation, Characteristic roots and Characteristic vectors of matix , Cayley Hamilton theorem and applications.


Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  CO5  Types of matrices, rank, linear equations, Characteristic roots, Characteristic vectors  8  Lecture 
2  CO6  Solving system of linear equations, Finding Characteristic roots, vectors, Verifying Cayley Hamilton Theorem  12  Lecture 
Department  Mathematics 
Name of Faculty  
Programme Name  B. Sc. Mathematics 
Level of study  UG 
Semester  1 
Course Name/Subject Name  MM1CRT01: Foundation of Mathematics 
Total Hours  72 
Course Outcomes
CO Number  Description  CO Evaluation methods 
CO1  Familiarize the concepts of mathematical logic and methods of proofs  Assignment, Test 
CO2  Conceive the concepts of sets and functions  Assignment, Test 
CO3  Learn about relations and partial orderings  Assignment, Test 
CO4  Understand the basic concepts of theory of equations  Assignment, Test 
Module 1 : Basic Logic  Hours : 20  
Syllabus :
Propositional logic, Propositional equivalences, Predicates and quantifiers, Rules of inference, Introduction to proofs


Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO1  Propositional logic  4  Lecture 
2  CO1  Propositional equivalences  4  Lecture 
3  CO1  Predicates and quantifiers  5  Lecture 
4  CO1  Rules of inference  4  Lecture 
5  CO1

Introduction to proofs  3  Lecture 
Module 2 : Set theory  Hours : 12  
Syllabus:
Sets, Set operations, Functions


Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO2  Sets  3  Lecture 
2  CO2  Set operations  4  Lecture 
3  CO2  Functions  5  Lecture 
Module 3 : Relations  Hours : 20  
Syllabus:
Relations and their properties, Representing relations, Equivalence relations, Partial orderings


Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO3  Relations and their properties  5  Lecture 
2  CO3  Representing relations  3  Lecture 
3  CO3  Equivalence relations  6  Lecture 
4  CO3  Partial orderings  6  Lecture 
Module 4 : Theory of equations  Hours : 20  
Syllabus:
Roots of Equations, Relation Connecting the roots and coefficients of an equation, Transformation of equations, Special Cases, The Cubic equation, The Biquadratic Equation, Character and Position of the Roots of an Equation, Some General Theorems, Descartes’s Rule of Signs, Corollaries, Reciprocal Equations


Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO4  Roots of Equations, Relation Connecting the roots and coefficients of an equation  4  Lecture 
2  CO4  Transformation of equations, Special Cases  3  Lecture 
3  CO4  The Cubic equation  3  Lecture 
4  CO4  The Biquadratic Equation  3  Lecture 
5  CO4  Character and Position of the Roots of an Equation, Some General Theorems, Descartes’s Rule of Signs, Corollaries  4  Lecture 
6  CO4  Reciprocal Equations  3  Lecture 
Department  MATHEMATICS 
Name of Faculty  
Programme Name  B.Sc. MATHEMATICS 
Level of study  UG 
Semester  SECOND 
Course Name/Subject
Name 
MM2CRT01: ANALYTIC GEOMETRY, TRIGONOMETRY AND DIFFERENTIAL CALCULUS 
Total Hours  72 
Course Outcomes
CO
Number 
Description  CO Evaluation methods 
CO1  To identify a correspondence between geometric curves and algebraic equations. To identify vertex, focus, directrix and sketch the graph of the corresponding equation.  Assignment,Test 
CO2  Understand the terminology used in analyzing curves like Chord, Tangent, Normal, Orthoptic locus, pole, Polar…etc  Assignment,Test 
CO3  Learn about polar coordinates and to translate whatever learnt about conic sections in Cartesian coordinates to polar coordinates.  Assignment,Test, Seminar 
CO4  Understand and analyze the Relations connecting Circular and hyperbolic functions and to separate functions of complex variables to real and imaginary parts  Assignment, Test 
CO5  Factorization of ? ^{?} − 1 ,? ^{?} + 1 ,? ^{2?} − 2?^{ ? }?^{ ?} ????? + ?^{ 2?} . To sum an infinite series by ? + ?? method  Assignment,Test,Seminar 
CO6  Find the higher order derivatives  Assignment, Viva, Test 
CO7  Understand indeterminate forms and evaluate limits of indeterminate forms  Assignment, Test 
Module 1 Hours : 22  
Syllabus: Conic Sections – Tangent and Normals of a Conic (Cartesian and Parametric form), Orthoptic Locus,Chords in terms of given points, Pole and Polar and Conjugate diameters of Ellipse  
Sl.no  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  1, 2  Conic Sections – Introduction  4  Lecture, Demonstration 
2  1,2  Tangent and Normals of a Conic Tangents in terms of slope of a line  3  Lecture, Demonstration 
3  2  Orthoptic Locus  2  Lecture 
4  1,2  Parametric Coordinates – Parabola Ellipse, Hyperbola 1  3  Lecture, Demonstration 
5  1,2  Chords in terms of given points – Chord of Contact,Chord with a given Midpoint  3  Lecture 
6  2  Equation of the polar of a given point Pole of a given line  3  Lecture 
7  2  Conjugate lines, Conjugate diameters of Ellipse  4  Lecture 
Module 2 Hours : 15  
Syllabus: Polar Coordinates – Polar Coordinates, Polar Equation of a line , Polar Equation of Circle, Polar Equation of Conic Polar Equations of tangents and Normals , Chords of Conic Sections.  
Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  1,3  Polar Coordinates – Introduction  2  Lecture, Demonstration 
2  1,3  Polar Equation of a line, Polar Equation of Circle  3  Lecture, Demonstration 
3  1,3  Polar Equation of Conic  4  Lecture, Demonstration 
4  1,3  Polar Equations of tangents and Normals  4  Lecture 
5  1,3  Chords of Conic  2  Lecture 
Module 3 Hours : 17  
Syllabus: Trigonometry Circular and Hyperbolic functions of complex variables, Separation of fun variables into real and imaginary parts, Factorization of ? ^{?} − 1 ,? ^{?} + 1 ,? ^{2?} − 2?^{ ? }?^{ ?} ????? + ?^{ 2?} . summation of an infinite series by ? + ?? method


Slno  CO
Number 
Topic/Activity  No of
hours 
Instructional methods to be used 
1  4  Circular functions of complex variables  1  Lecture, Demonstration 
2  4  Hyperbolic functions of complex variables  2  Lecture, Demonstration 
3  4  Relations connecting Circular and hyperbolic functions, Inverse of hyperbolic functions  3  Lecture 
4  4  Separation of functions of complex variables into real and imaginary parts  3  Lecture 
5  5  Factorization of ? ^{?} − 1 ,? ^{?} + 1 ,? ^{2?} − 2?^{ ? }?^{ ?} ????? + ?^{ 2?}  3  Lecture 
6  5  Summation of infinite series by ? + ?? method – based on geometric series, binomial series, exponential series

5  Lecture 
Module 4 Hours : 18  
Syllabus: Differential Calculus Successive Differentiation and Indeterminate forms  
Slno  CO
Number 
Topic/Activity  No of
hours 
Instructional methods to be used 
1  6  Higher order derivative Calculation of ??ℎ derivative. Some standard results  3  Lecture 
2  6  Determination of ??ℎderivative of rational functions  2  Lecture 
3  6  The ??ℎ derivative of the product of the power of sines and cosines, Leibnitz‟s theorem.  4  Lecture 
4  6  The ??ℎ derivative of the product of two functions  3  Lecture 
5  7  The Indeterminate Forms 0/0 ∞/∞ 0. ∞ ∞ − ∞  3  Lecture 
6  7  The Indeterminate Forms 0^{ ?} , 1 ^{∞}, ∞^{ 0}

3  Lecture 
Department  MATHEMATICS 
Name of Faculty  ELDO VARGHESE 
Programme Name  B.Sc. Mathematics 
Level of study  UG 
Semester  Three 
Course Name/Subject Name  MM3CRT03: CALCULUS 
Total Hours  90 
Course Outcomes
CO Number  Description  CO Evaluation methods 
CO1  Understand how to expand functions using Maclaurin’s Theorem and Taylor’s theorem  Assignments, Viva, Written Examinations 
CO2  To find the concavity and point of inflexion of curves  Assignments, Viva, Written Examinations 
CO3  Evaluate the curvature, radius of curvature and center of curvature  Assignments, Viva, Written Examinations 
CO4  Compute the length of arcs  Assignments, Viva, Written Examinations 
CO5  Determine the evolutes and involutes and analyze their properties  Assignments, Viva, Written Examinations 
CO6  Determine the asymptotes and envelopes  Assignments, Viva, Written Examinations 
CO7  Define and compute partial derivatives  Assignments, Viva, Written Examinations 
CO8  Apply the chain rule for partial differentiation  Assignments, Viva, Written Examinations 
CO9  Determine the extreme values and saddle points using the method of Lagrange multipliers  Assignments, Viva, Written Examinations 
CO10  Apply integration to evaluate volumes using crosssections and cylindrical shells  Assignments, Viva, Written Examinations 
CO11  Determine arc lengths using integration  Assignments, Viva, Written Examinations 
CO12  Evaluate areas of surfaces of revolution  Assignments, Viva, Written Examinations 
CO13  Compute double and triple integrals over rectangular regions  Assignments, Viva, Written Examinations 
CO14  Determine areas of regions using double integrals  Assignments, Viva, Written Examinations 
CO15  Compute volumes of solid regions using double integrals  Assignments, Viva, Written Examinations 
CO16  Apply substitution in multiple integrals  Assignments, Viva, Written Examinations 
Module 1  Hours: 27  
Syllabus : Differential Calculus
Expansion of functions using Maclaurin’s theorem and Taylor’s theorem, Concavity and points of inflexion. Curvature and Evolutes. Length of arc as a function derivative of arc, radius of curvature – Cartesian equations only. (Parametric, Polar, Pedal equation and Newtonian Method are excluded) Centre of curvature, Evolutes and Involutes, properties of evolutes. Asymptotes and Envelopes. 

Sl.No  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO1  Expansion of functions using Maclaurin’s theorem and Taylor’s theorem  4  Lecture, Problem Solving 
2  CO2  Concavity and points of
inflexion 
4  Lecture, Problem Solving 
3  CO3 & CO5  Curvature and Evolutes  4  Lecture, Problem Solving 
4  CO4  Length of arc as a function derivative of arc  3  Lecture, Problem Solving 
5  CO3  Radius of curvature, Centre of curvature  4  Lecture, Problem Solving 
6  CO5  Evolutes and Involutes, properties of evolutes  4  Lecture, Problem Solving 
7  C06  Asymptotes and Envelopes.  4  Lecture, Problem Solving 
Module 2  Hours: 18  
Syllabus: Partial Differentiation
Partial derivatives, The Chain rule, Extreme values and saddle points, Lagrange multipliers. 

Sl.No  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO7  Partial derivatives  4  Lecture, Problem Solving 
2  CO8  The Chain rule  4  Lecture, Problem Solving 
3  CO9  Extreme values and saddle points  4  Lecture, Problem Solving 
4  CO9  Lagrange multipliers  6  Lecture, Problem Solving 
Module 3  Hours : 20  
Syllabus: Integral Calculus
Volumes using Crosssections, Volumes using cylindrical shells, Arc lengths, Areas of surfaces of Revolution. 

Sl.No  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO10  Volumes using Crosssections  6  Lecture, Problem Solving 
2  CO10  Volumes using cylindrical shells  6  Lecture, Problem Solving 
3  CO11  Arc lengths  3  Lecture, Problem Solving 
4  CO12  Areas of surfaces of Revolution  5  Lecture, Problem Solving 
Module 4  Hours: 25  
Syllabus: Multiple Integrals
Double and iterated integrals over rectangles, Double integrals over general regions, Area by double integration, Triple integrals in rectangular coordinates, Triple integrals in cylindrical and spherical coordinates, Substitutions in multiple integrals. 

Sl.No  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO13  Double and iterated integrals over rectangles  3  Lecture, Problem Solving 
2  CO13  Double integrals over general regions  4  Lecture, Problem Solving 
3  CO14  Area by double integration  5  Lecture, Problem Solving 
4  CO13 & CO15  Triple integrals in rectangular coordinates  5  Lecture, Problem Solving 
5  CO15 & CO16  Triple integrals in cylindrical and spherical coordinates  5  Lecture, Problem Solving 
6  CO16  Substitutions in multiple integrals  2  Lecture, Problem Solving 
Department  Mathematics 
Name of Faculty  
Programme Name  BSc Mathematics 
Level of study  UG 
Semester  Fourth 
Course Name  Vector Calculus, Theory of Numbers and Laplace Transform 
Total hours  90 
Course Outcomes
CO Number  Description  CO Evaluation methods 
Upon completion of this course, the students will be able to:  
CO1  Find vector and cartesian equation for lines and planes  Assignment, viva, Seminar, Test 
CO2  Analyze vector functions to find limits, derivatives, velocity and acceleration vectors, tangent vector, arc length, curvature, unit normal vector.  Assignment, viva, Seminar, Test 
CO3  Find tangential and normal components of acceleration and calculate directional derivatives and gradients.  Assignment, viva, Seminar, Test 
CO4  Find tangent planes and normal lines to a surface and vealuate line integrals and differentiate vector fields.  Assignment, viva, Seminar, Test 
CO5  Calculate work, circulation, flux, potential function and verify path independence and evaluate line integrals, surface area and surface integrals.  Assignment, viva, Seminar, Test 
CO6  Apply Green’s theorem, Stoke’s theorem and Divergence theorem. Define and interpret the concepts of divisibility, congruence, greatest common divisor, primefactorization and Euler’s phi function.  Assignment, viva, Seminar, Test 
CO7  Apply Fermat’s theorem, Wilson’s theorem and familiarize Laplace transforms, its properties and analyze transforms of derivatives, solve ordinary differential equations & initial value problems by using Laplace transform.  Assignment, viva, Seminar, Test 
Module: 1 Hours: 25  
Syllabus:
Vector Differentiation A vector equation and Parametric equations for lines and equation for a plane in space, Vector functions, Arc length and Unit tangent vector, Curvature and the Unit normal vector, Tangential and Normal Components of Acceleration, Directional derivatives and Gradient vectors, tangent planes and Normal lines. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO1  A vector equation and Parametric equations for lines and equation for a plane in space.  3  Lecture, Problem solving 
2  CO2  Vector functions, Arc length and Unit tangent vector.  4  Lecture, Problem solving 
3  CO2  Curvature and the Unit normal vector.  5  Lecture, Problem solving 
4  CO3  Tangential and Normal Components of Acceleration.  5  Lecture, Problem solving 
5  CO3  Directional derivatives and Gradient vectors.  4  Lecture, Problem solving 
6  CO4  Tangent planes and Normal lines .  4  Lecture, Problem solving 
Module: 2 Hours: 30  
Syllabus:
Vector Integration Line integrals, Vector fields, Work, Circulation and Flux, Path Independence, Conservative Fields and Potential Functions, Green’s theorem in the plane, Surfaces and Area: Parameterisations of surfaces, Implicit surfaces, Surface integrals, Stokes’ theorem, Divergence theorem. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO4  Line integrals, Vector fields.  4  Lecture, Problem solving 
2  CO5  Work, Circulation and Flux, Path Independence, Conservative Fields and Potential Functions.  7  Lecture, Problem solving 
3  CO6  Green’s theorem in the plane.  5  Lecture, Problem solving 
4  CO5  Surfaces and Area, Parameterisations of surfaces, Implicit surfaces, Surface integrals.  7  Lecture, Problem solving 
5  CO6  Stokes’ theorem  4  Lecture, Problem solving 
6  CO6  Divergence theorem  3  Lecture, Problem solving 
Module: 3 Hours: 15  
Syllabus:
Theory of Numbers Basic properties of congruence, Fermat’s theorem, Wilson’s theorem, Euler’s phi function. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO6  Basic properties of congruence  3  Lecture, Problem solving 
2  CO7  Fermat’s theorem  5  Lecture, Problem solving 
3  CO7  Wilson’s theorem  5  Lecture, Problem solving 
4  CO6  Euler’s phi function.  2  Lecture, Problem solving 
Module: 4 Hours: 20  
Syllabus:
Laplace transforms Laplace transform, Linearity of Laplace transform, First shifting theorem, Existence of Laplace transform, Transforms of derivatives, Solution of ordinary differential equation & initial value problem, Laplace transform of the integral of a function, Convolution and Integral equations. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO7  Laplace transform, Linearity of Laplace transform, First shifting theorem.  5  Lecture, Problem solving 
2  CO7  Existence of Laplace transform, Transforms of derivatives.  5  Lecture, Problem solving 
3  CO7  Solution of ordinary differential equation & initial value problem.  5  Lecture, Problem solving 
4  CO7  Laplace transform of the integral of a function, Convolution and Integral equations.  5  Lecture, Problem solving 
Department  MATHEMATICS 
Name of Faculty  
Programme Name  B.Sc. MATHEMATICS 
Level of study  UG 
Semester  FIFTH 
Course Name/Subject
Name 
MM5CRT01 : MATHEMATICAL ANALYSIS 
Total Hours  108 
Course Outcomes
CO
Number 
Description  CO Evaluation methods 
CO1  Familiarize the classification of sets as finite, countably infinite and infinite  Assignment,Test 
CO2  Understand the basic properties of the real numbers.  Assignment,Test 
CO3  Understand and familiarize the concept of sequences, limit of a sequence and its applications  Assignment, Test, Viva 
CO4  Understand an Infinite series and its nature  Assignment, Viva, Test 
C05  Analyze the convergence, absolute convergence of a series  Assignment, Test 
CO6  Familiarize the concept of limits  Assignment, Test 
Module 1 Hours : 30  
Syllabus: REAL NUMBERS – Finite and Infinite Sets, The Algebraic and Order Properties of R, Absolute Value and Real Line, The Completeness Property of R, Applications of the Supremum Property, Intervals.  
Sl.no  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  1  Finite and infinite sets, Countable sets 1 Cantor’s set  4  Lecture 
2  2  Algebraic properties of R  2  Lecture 
3  2  The order properties of R  2  Lecture 
4  2  Inequalities, Absolute value and real line  4  Lecture 
5  2  The completeness property of R Applications of supremum property  5  Lecture 
6  2  The Archimedean property, Density of Rational numbers in R  5  Lecture 
7  2  Intervals, Characterization of Intervals,Nested Intervals  5  Lecture 
8  2  The uncountability of R

3  Lecture 
Module 2 Hours : 30  
Syllabus: SEQUENCES – Sequences and their Limits, Limit Theorems, Monotone Sequences, Subsequences and the Bolzano Weierstrass Theorem, The Cauchy Criterion, Properly Divergent Sequences.  
Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  3  ^{Sequences – Introduction The limit of a sequence }  3  Lecture, Demonstration 
2  3  ^{Tails of sequences,Limit theorems}  3  Lecture 
3  3  ^{Monotone sequences , The calculation of square roots}  5  Lecture, Demonstration 
4  3  ^{Euler’s number ,Subsequences}  5  Lecture 
5  3  ^{The existence of monotone subsequences The BolzanoWeierstrass theorem}  5  Lecture 
6  3  ^{Limit superior and limit inferior, The Cauchy criterion }  5  Lecture 
7  3  ^{Properly divergent sequences}  4  Lecture 
Slno  CO
Number 
Topic/Activity  No of
hours 
Instructional methods to be used 
1  4  ^{Series Introduction to infinite series }  3  Lecture 
2  4,5  ^{Tests for convergence root and ratio test, Raabe’s test, comparison tests, integral test}  8  Lecture 
3  4,5  ^{Absolute convergence}  3  Lecture 
4  4,5  ^{Grouping of series}  3  Lecture 
5  4,5  ^{Rearrangements of series}  3  Lecture 
6  4,5  ^{ Tests for absolute and non absolute convergence}  4  Lecture 
7  4,5  ^{Alternating series, The Dirichlet and Abel tests}  3  Lecture 
Module 4 Hours : 24  
Syllabus: LIMITSLimits of Functions, Limit Theorems, Some Extensions of the Limit Concept.  
Slno  CO
Number 
Topic/Activity  No of
hours 
Instructional methods to be used 
1  6  ^{Limits of functions The definition of the limit }  5  Lecture 
2  6  ^{Sequential criterion for limits & Divergence criteria }  7  Lecture 
3  6  ^{Limit theorems & Some extensions of the limit concept}  7  Lecture 
4  6  ^{ Infinite limits, Limits at infinity}

5  Lecture 
Department  Mathematics 
Name of Faculty  
Programme Name  BSc Mathematics 
Level of study  UG 
Semester  Fifth 
Course Name  Differential Equations 
Total hours  108 
Course Outcomes
CO Number  Description  CO Evaluation methods 
Upon completion of this course, the students will be able to:  
CO1  Solve first order differential equations utilizing the standard techniques for separable, linear, exact, homogeneous, or Bernoulli cases.  Assignment, viva, Seminar, Test 
CO2  Obtain an integrating factor which may reduce a given differential equation into an exact one and eventually provide its solution.  Assignment, viva, Seminar, Test 
CO3  Familiarize the orthogonal trajectory of the system of curves on a given surface.

Assignment, viva, Seminar, Test 
CO4  Find the complete solution of a non homogeneous differential equation as a linear combination of the complementary function and a particular solution.  Assignment, viva, Seminar, Test 
CO5  Find the complete solution of a non homogeneous differential equation with constant coefficients by the method of undetermined coefficients.  Assignment, viva, Seminar, Test 
CO6  Find the complete solution of a differential equation with constant coefficients by variation of parameters and find power series solutions of differential equations.  Assignment, viva, Seminar, Test 
CO7  Familiarize the origin of partial differential equation and solve first order linear partial differential equation by using Lagrange’s method.  Assignment, viva, Seminar, Test 
Module: 1 Hours: 26  
Syllabus:
What is a differential equation The nature of solutions, Separable equations, First order linear equations, Exact equations, Orthogonal trajectories and families of curves, Homogeneous equations, Integrating factors, Reduction of orderdependent variable missingindependent variable missing. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO1  The nature of solutions, Separable equations, First order linear equations, Exact equations.  8  Lecture, Problem solving 
2  CO3  Orthogonal trajectories and families of curves  5  Lecture, Problem solving 
3  CO1, CO2  Homogeneous equations, Integrating factors  8  Lecture, Problem solving 
4  CO2  Reduction of orderdependent variable missingindependent variable missing.  5  Lecture, Problem solving 
Module: 2 Hours: 26  
Syllabus:
Second order linear equations Second order linear equations with constant coefficients , The method of undetermined coefficients, The method of variation of parameters, The use of a known solution to find another, Higher order linear equations. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO4  Second order linear equations with constant coefficients  5  Lecture, Problem solving 
2  CO5  The method of undetermined coefficients.  5  Lecture, Problem solving 
3  CO6  The method of variation of parameters  6  Lecture, Problem solving 
4  CO6  The use of a known solution to find another.  5  Lecture, Problem solving 
5  CO4, CO5  Higher order linear equations.  5  Lecture, Problem solving 
Module: 3 Hours: 26  
Syllabus:
Power Series solutions and special functions Introduction and review of power series, Series solutions of first order differential equations, Second order linear equations: ordinary points, Regular singular points, More on regular singular points. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO6  Introduction and review of power series.  4  Lecture, Problem solving 
2  CO6  Series solutions of first second order differential equations.  11  Lecture, Problem solving 
3  CO6  More on regular singular points.  11  Lecture, Problem solving 
Module: 4 Hours: 30  
Syllabus:
Partial Differential equations Methods of solution of = = , origin of first order partial differential equations, Linear equations of the first order, Lagrange’s method, Integral surfaces passing through a given curve. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO7  Methods of solution of = = .  10  Lecture, Problem solving 
2  CO7  Origin of first order partial differential equations.  5  Lecture, Problem solving 
3  CO7  Linear equations of the first order, Lagrange’s method.  10  Lecture, Problem solving 
4  CO7  Integral surfaces passing through a given curve.  5  Lecture, Problem solving 
Department  MATHEMATICS 
Name of Faculty  
Programme Name  B.Sc. Mathematics 
Level of study  UG 
Semester  5 
Course Name/Subject
Name 
Environmental Mathematics and Human Rights 
Total Hours  72 
Course Outcomes
CO
Number 
Description  CO Evaluation methods 
CO1  Encourage students to research, investigate how and why things happen, and make their own decisions about complex environmental issues. By developing and enhancing critical and creative thinking skills, it helps to foster a new generation of informed consumers, workers, as well as policy or decision makers.

Assignment,Test,Seminar 
CO2  Understand how their decisions and actions affect the environment, builds knowledge and skills necessary to address complex environmental issues, as well as ways we can take action to keep our environment healthy and sustainable for the future, encourage character building, and develop positive attitudes and values.

Assignment,Test,Seminar 
CO3  Develop the sense of awareness among the students about the environment and its various problems and to help the students in realizing the interrelationship between man and environment for protecting the nature and natural resources.

Assignment,Test,Seminar 
CO4  Acquire the basic knowledge about environment and to inform the students about the social norms that provide unity with environmental characteristics and create positive attitude about the environment

Assignment,Test,Seminar 
CO5  To familiarize the students different environmental and daily life situations where Mathematics appears  Assignment,Test,Seminar 
CO6  Acquire basic knowledge about Human rights, its history, Human rights in Indian context.  Assignment,Test,Seminar 
Module 1 Hours : 10  
Syllabus:
Unit 1 :Multidisciplinary nature of environmental studies Definition, scope and importance Need for public awareness. Unit 2 : Natural Resources : Renewable and nonrenewable resources : Natural resources and associated problems. a) Forest resources : Use and overexploitation, deforestation, case studies. Timber extraction, mining, dams and their effects on forest and tribal people. b) Water resources : Use and overutilization of surface and ground water, floods, drought, conflicts over water, damsbenefits and problems. c) Mineral resources : Use and exploitation, environmental effects of extracting and using mineral resources, case studies. d) Food resources : World food problems, changes caused by agriculture and overgrazing, effects of modern agriculture, fertilizerpesticide problems, water logging, salinity, case studies. e) Energy resources: Growing energy needs, renewable and non renewable energy sources, use of alternate energy sources, Case studies. f) Land resources: Land as a resource, land degradation, man induced landslides, soil erosion and desertification Role of individual in conservation of natural resources. Equitable use of resources for sustainable lifestyles. 

Sl no.  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  CO1  Definition, scope and importance of Environmental studies,Need for public awareness  1  Lecture 
2  CO3  Renewable and nonrenewable resources : Natural resources and associated problems.

8  Lecture 
3  CO1  Role of individual in conservation of natural resources.
Equitable use of resources for sustainable lifestyles 
1  Lecture 
Module 2 Hours : 14  
Syllabus:
Environmental Pollution Definition Causes, effects and control measures of: – Air pollution ,Water pollution ,Soil pollution, Marine pollution, Noise pollution, Thermal pollution ,Nuclear hazards Solid waste Management: Causes, effects and control measures of urban and industrial wastes. Role of an individual in prevention of pollution Pollution case studies Disaster management: floods, earthquake, cyclone and landslides. (8hrs) Social Issues and the Environment Urban problems related to energy Water conservation, rain water harvesting, watershed management Resettlement and rehabilitation of people: its problems and concerns, Case studies Environmental ethics: Issues and possible solutions Climate change, global warming, acid rain, ozone layer depletion , nuclear accidents and holocaust, Case studies Consumerism and waste products


Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  CO2  Definition of pollution, causes, effects, control measures  7  Lecture 
2  CO3  Solid waste management, Disaster management  2  Lecture 
3  CO4  Urban problems, Water conservation, Watershed management, Environmental ethics  5  Lecture, Demonstration 
Module 3 Hours : 15  
Syllabus:Fibonacci Numbers in nature
The rabbit problem, Fibonacci numbers, recursive definition, Lucas numbers, Different types of Fibonacci and Lucas numbers. Fibonacci numbers in nature : Fibonacci and the earth, Fibonacci and flowers, Fibonacci and sunflower, Fibonacci, pinecones, artichokes and pineapples, Fibonacci and bees, Fibonacci and subsets, Fibonacci and sewage treatment, Fibonacci and atoms, Fibonacci and reflections, Fibonacci, paraffins and cycloparaffins, Fibonacci and music, Fibonacci and compositions with 1’s and 2’s.


Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1.  CO3  The rabbit problem, Fibonacci numbers  4  Lecture,Seminar 
2.  CO5  Fibonacci and the earth, Fibonacci
and flowers, Fibonacci and sunflower, Fibonacci, pinecones, artichokes and pineapples, Fibonacci and bees, Fibonacci and subsets, Fibonacci and sewage treatment 
11  Lecture, Seminar 
Module 4 Hours : 15  
Syllabus: The golden ratio, mean proportional, a geometric interpretation, ruler and compass construction,
Euler construction, generation by Newton’s method. The golden ratio revisited, the golden ratio and human body, golden ratio by origami, Differential equations, Gattei’s discovery of golden ratio, centroids of circles,


Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  CO3  The golden ratio and human body, golden ratio by origami  4  Lecture, Seminar 
2  CO5  The golden ratio, mean proportional, a geometric interpretation, ruler and compass construction, Euler construction, generation by Newton’s method,Differential equations, Gattei’s discovery of golden ratio, centroids of circles  11  Lecture, Seminar 
Module 5 Hours : 18  
Syllabus: Unit1Human Rights– An Introduction to Human Rights, Meaning, concept and
development, Three Generations of Human Rights (Civil and Political Rights; Economic, Social and Cultural Rights). Unit2 Human Rights and United Nations – contributions, main human rights related organs – UNESCO,UNICEF, WHO, ILO, Declarations for women and children, Universal Declaration of Human Rights. Human Rights in India – Fundamental rights and Indian Constitution, Rights for children and women, Scheduled Castes, Scheduled Tribes, Other Backward Castes and Minorities 

Slno  CO
Number 
Topic/Activity  No of
hours 
Instructional methods to be used 
1  CO6  Human rights,related organisations, human rights in India  18  Lecture 
Department  Mathematics 
Name of Faculty  
Programme Name  B. Sc. Mathematics 
Level of study  UG 
Semester  V 
Course Name/Subject Name  MM5OPT02: Applicable Mathematics 
Total Hours  72 
Course Outcomes
CO Number  Description  CO Evaluation methods 
CO1  Helps to acquire problem solving skills for competitive examinations  Test, Seminar 
CO2  Familiarize to problem solving in quadratic equations, permutations and combinations, and trigonometry  Test, Seminar 
CO3  Learn to solve problems related to simple interest, compound interest, time and work, work and wages, time and distance and exponential and logarithmic series  Test, Seminar 
CO4  Understand the basic concepts and develop problem solving skills in elementary mensuration, elementary algebra and differential calculus  Test, Seminar 
Module 1  Hours : 18  
Syllabus :
Types of numbers, HCF & LCM of integers, Fractions, Simplifications (VBODMAS rule), Squares and square roots, Ratio and proportion, Percentage, Profit & loss 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO1  Types of numbers, HCF & LCM of integers  3  Lecture 
2  CO1  Fractions, Simplifications (VBODMAS rule)  4  Lecture 
3  CO1  Squares and square roots  3  Lecture 
4  CO1  Ratio and proportion  4  Lecture 
5  CO1

Percentage, Profit & loss  4  Lecture 
Module 2  Hours : 18  
Syllabus:
Quadratic equations (Solution of quadratic equations with real roots only), Permutations and combinations – simple applications, Trigonometry introduction, values of trigonometric ratios of 0^{0}, 30^{0}, 45^{0}, 60^{0} & 90^{0}, Heights and distances 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO2  Quadratic equations (Solution of quadratic equations with real roots only)  3  Lecture 
2  CO2  Permutations
and combinations – simple applications 
6  Lecture 
3  CO2  Trigonometry introduction, values of trigonometric
ratios of 0^{0}, 30^{0}, 45^{0}, 60^{0} & 90^{0}, Heights and distances 
9  Lecture 
Module 3  Hours : 18  
Syllabus:
Simple interest, Compound interest, Time and work, Work and wages, Time and distance, Exponential series and logarithmic series 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO3  Simple interest, Compound interest  8  Lecture 
2  CO3  Time and work, Work and wages, Time and distance  8  Lecture 
3  CO3  Exponential series and logarithmic series  2  Lecture 
Module 4  Hours : 18  
Syllabus:
Elementary mensuration – Area and perimeter of polygons, Elementary Algebra, monomial, binomial, polynomial (linear, quadratic & cubic), simple factorization of quadratic and cubic polynomials, Differential Calculus – Differentiation – Standard results (derivatives), Product rule, Quotient rule and function of function rule (without proof) and simple problems


Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO4  Elementary mensuration – Area and perimeter of polygons  6  Lecture 
2  CO4  Elementary Algebra, monomial ,
binomial, polynomial (linear, quadratic & cubic), simple factorization of quadratic and cubic polynomials 
6  Lecture 
3  CO4  Differential Calculus – Differentiation – Standard results (derivatives), Product rule, Quotient rule and function of function rule (without proof) and simple problems  6  Lecture 
Department  Mathematics 
Name of Faculty  
Programme Name  B. Sc. Mathematics 
Level of study  UG 
Semester  V 
Course Name/Subject Name  MM5CRT03 : ABSTRACT ALGEBRA 
Total Hours  90 
Course Outcomes
CO Number  Description  CO Evaluation methods 
CO1  Familiarize with Groups and subgroups, Isomorphic binary structures, elementary properties of groups, finite groups and group tables  Assignment, Test, Seminar 
CO2  Identify different types of groups normal subgroup, simple group, cyclic group, Construct group tables and subgroup diagrams.  Assignment, Test, Seminar 
CO3  Familiarize with permutations and its properties  Assignment, Test, Seminar 
CO4  Study Cayley’s Theorem, Theorem of Lagrange, Fundamental homomorphism Theorem.  Assignment, Test, Seminar 
CO5  Understand the concepts of Homomorphism and Factor groups  Assignment, Test, Seminar 
CO6  Conceive the concepts of Rings, fields, Integral domains  Assignment, Test, Seminar 
Module 1 : Groups and subgroups  Hours : 25  
Syllabus :
Binary operations, Isomorphic binary structures, Groupsdefinition and examples, elementary properties of groups, finite groups and group tables, subgroups, cyclic subgroups, cyclic groups, elementary properties of cyclic groups. 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO1  Binary Operations  4  Lecture, problem solving 
2  CO1  Isomorphic Binary Structures  4  Lecture, problem solving 
3  CO1  Groups  5  Lecture, problem solving 
4  CO1, CO2  Subgroups  5  Lecture, problem solving 
5  CO2

Cyclic Subgroups  7  Lecture, problem solving 
Module 2 : Permutations, cosets, and direct products  Hours : 25  
Syllabus:
Groups of permutations, Cayley’s theorem, orbits, cycles and the alternating groups, cosets and the theorem of Lagrange, direct products. 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO3  Permutations  8  Lecture, problem solving 
2  CO3, CO4  Orbits, Cycles and Alternating groups  8  Lecture, problem solving 
3  CO4  Cosets and the theorem of Lagrange  9  Lecture, problem solving 
Module 3 : Homomorphisms and Factor groups  Hours : 20  
Syllabus:
Homomorphisms, properties of homomorphisms, factor groups, The Fundamental Homomorphism theorem, normal subgroups and inner automorphisms, simple groups. 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO5  Homomorphisms.  5  Lecture, problem solving 
2  CO4,CO5  Factor groups  6  Lecture, problem solving 
3  CO2,CO5  Normal subgroups and inner
automorphisms 
5  Lecture, problem solving 
4  CO2  Simple groups  4  Lecture, problem solving 
Module 4 : Rings and fields  Hours : 20  
Syllabus:
Definitions and basic properties, homomorphisms and isomorphisms, Integral domains divisors of zero and cancellation, integral domains, the characteristic of a ring. Ideals and factor rings. Homomorphisms and factor rings.


Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO6  Rings and fields  7  Lecture, problem solving 
2  CO6  Integral domains  6  Lecture, problem solving 
3  CO6  Ideals and factor rings  7  Lecture, problem solving 
Department  Mathematics 
Name of Faculty  
Programme Name  B. Sc. Mathematics 
Level of study  UG 
Semester  VI 
Course Name/Subject Name  MM6CRT04 : LINEAR ALGEBRA 
Total Hours  90 
Course Outcomes
CO Number  Description  CO Evaluation methods 
CO1  Understand the theory and concepts of matrices in a broader sense  Assignment, Test, Viva 
CO2  Solve systems of linear equations using matrices  Assignment, Test, Viva 
CO3  Familiarise with vector spaces, subspaces, linear combination of vectors, spanning set, linear independence and basis.  Assignment, Test, Viva 
CO4  Conceive the concepts of Linear transformations and Linear isomorphism.  Assignment, Test, Viva 
CO5  Understand the application of matrices in vector spaces  Assignment, Test, Viva 
CO6  Familiarise with Eigen values, Eigenvectors and Eigen space.  Assignment, Test, Viva 
Module 1  Hours : 25  
Syllabus :
A review of algebra of matrices is followed by some applications of matrices, analytic geometry, systems of linear equations and difference equations. Systems of linear equations: elementary matrices, the process of Gaussian elimination, Hermite or reduced rowechelon matrices. Linear combinations of rows (columns), linear independence of columns, row equivalent matrices, rank of a matrix, column rank, normal form, consistent systems of equations. 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO1  Algebra of Matrices  6  Lecture, problem solving 
2  CO1  Some Applications of Matrices  3  Lecture, problem solving 
3  CO2  System of Linear Equations  16  Lecture, problem solving 
Module 2  Hours : 25  
Syllabus:
Invertible matrices, left and right inverse of a matrix, orthogonal matrix, vector spaces, subspaces, linear combination of vectors, spanning set, linear independence and basis. 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO1  Invertible Matrices  10  Lecture, problem solving 
2  CO3  Vector Spaces  15  Lecture, problem solving 
Module 3 :  Hours : 25  
Syllabus:
Linear mappings: Linear transformations, Kernel and range, Rank and Nullity, Linear isomorphism. Matrix connection: Ordered basis, Matrix of f relative to a fixed ordered basis, Transition matrix from a basis to another, Nilpotent and index of nilpotency. 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO4  Linear mappings  15  Lecture, problem solving 
2  CO5  Matrix connection  10  Lecture, problem solving 
Module 4 :  Hours : 15  
Syllabus:
Eigenvalues and eigenvectors: Characteristic equation, Algebraic multiplicities, Eigen space, Geometric multiplicities, Eigenvector, diagonalisation, Tridiagonal matrix. 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO6  Eigenvalues and eigenvectors  15  Lecture, problem solving 
Department  MATHEMATICS 
Name of Faculty  
Programme Name  B.Sc. MATHEMATICS 
Level of study  UG 
Semester  SIXTH 
Course Name/Subject
Name 
MM6CRT01 REAL ANALYSIS 
Total Hours  90 
Course Outcomes
CO
Number 
Description  CO Evaluation methods 
CO1  Understand the concept of continuity its definition, geometry.To analyze continuity of functions, understand the various properties of continuous functions, especially its behavior on closed bounded intervals.  Assignment,Test 
CO2  Understand and analyze uniform continuity of functions  Assignment,Test 
CO3  Understand the definition, meaning and physical significance of derivatives. To apply theorems on differentiation  Assignment,Test, Seminar 
CO4  Define Riemann Integrals and understand its geometric interpretation.Understand and familiarize theorems related to integrability  Assignment, Test, Viva 
CO5  Define sequence and series of functions  Assignment, Viva, Test 
CO6  To apply the properties of uniformly convergent sequences and series  Assignment,Test, Seminar 
Module 1 Hours : 25  
Syllabus:Continuous Functions
Continuous Functions – Sequential Criterion for Continuity, Combinations of Continuous Functions, Composition of Continuous Functions, Continuous Functions on Intervals, Boundedness Theorem, MaximumMinimum Theorem, Location of Roots Theorem, Bolzano’s Intermediate Value Theorem, Preservation of Intervals Theorem. Uniform continuity, Non uniform Continuity Criteria, Uniform Continuity Theorem, Lipschitz Functions. Monotone and Inverse Functions, Continuous Inverse Theorem, The nth Root Function


Sl.no  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  1  Continuous Functions & Sequential Criterion for Continuity  5  Lecture 
2  1  Combinations of Continuous Functions  2  Lecture 
3  1  Composition of Continuous Functions  1  Lecture 
4  1  Continuous Functions on closed bounded Intervals  6  Lecture 
5  2  Uniform/ Non Uniform continuity  4  Lecture 
6  2  Lipschitz Functions, Monotone and Inverse Functions  4  Lecture 
7  2  The nth root function  3  Lecture 
Module 2 Hours : 20  
Syllabus: DIFFERENTIATION The Derivative, Caratheodory’s Theorem, Chain Rule, Derivatives of Inverse Functions. The Mean Value Theorem, Interior Extremum Theorem, Rolle’s Theorem, First Derivative Test for Extrema, Applications of the Mean Value Theorem, The Intermediate Value Property of Derivatives, Darboux’s Theorem, Indeterminate Forms, Cauchy Mean Value Theorem,
L’ Hospital’s Rules.


Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  3  The Derivative  3  Lecture, Demonstration 
2  3  Caratheodory’s Theorem, Chain Rule,  3  Lecture 
3  3  Derivatives of Inverse Functions.  5  Lecture 
4  3  Some theorems on derivatives  5  Lecture 
5  3  Indeterminate Forms,  5  Lecture 
6  3  Cauchy Mean Value Theorem,

5  Lecture 
7  3  L’ Hospital’s Rules.

4  Lecture 
Module 3 30 hours  
Syllabus: THE RIEMANN INTEGRAL – The Riemann Integral, Examples and Properties, Boundedness Theorem, Riemann Integrable Functions, Cauchy Criterion, Squeeze Theorem, Classes of Riemann Integrable Functions, Additivity Theorem, The Fundamental Theorem of Calculus (First Form), The Fundamental Theorem of Calculus (Second Form), Substitution Theorem


Slno  CO
Number 
Topic/Activity  No of
hours 
Instructional methods to be used 
1  4  The Riemann Integral  3  Lecture 
2  4  Riemann Integration – Examples and Properties  8  Lecture 
3  4  Riemann Integrable Functions  3  Lecture 
4  4  Cauchy Criterion, Squeeze Theorem  3  Lecture 
5  4  Classes of Riemann Integrable Functions  3  Lecture 
6  4  Additivity Theorem, The Fundamental Theorem of Calculus (First Form), The Fundamental Theorem of Calculus (Second Form),  4  Lecture 
7  4  Substitution Theorem

3  Lecture 
Module 4 Hours : 6  
Syllabus: Pointwise and Uniform convergence, Interchange of Limits.  
Slno  CO
Number 
Topic/Activity  No of
hours 
Instructional methods to be used 
1  5,6  Pointwise convergence  2  Lecture 
2  5,6  Uniform convergence  2  Lecture 
3  5,6  Interchange of Limits.  2  Lecture 
Department  MATHEMATICS 
Name of Faculty  
Programme Name  B.Sc. Mathematics 
Level of study  UG 
Semester  6 
Course Name/Subject
Name 
Complex Analysis 
Total Hours  90 
Course Outcomes
CO
Number 
Description  CO Evaluation methods 
CO1  Conceive the concept of analytic functions

Assignment,Test,Seminar 
CO2  Familiar with the elementary complex functions and their properties

Assignment,Test,Seminar 
CO3  Familiar with the theory and techniques of complex integration  Assignment,Test,Seminar 
CO4  Familiar with the theory and application of the power series expansion of analytic functions

Assignment,Test,Seminar 
CO5  Identify and classify Singular points to use in Complex integrals  Assignment,Test,Seminar 
Module 1 Hours : 32  
Syllabus: Functions of a complex variable, limits, theorems on limits, continuity, derivatives,
differentiation formulas, CauchyRiemann equation, sufficient condition for differentiability, analytic functions, examples, harmonic functions. Elementary functions, the Exponential function, logarithmic function, complex exponents, trigonometric functions, hyperbolic functions, inverse trigonometric and hyperbolic functions. 

Sl no.  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  CO1  Functions of complex variable,Differentiability, Analytic functions, CR equations  22  Lecture  
2  CO2  Elementary functions  10  Lecture  
Module 2 Hours : 25  
Syllabus: Derivatives of functions, definite integrals of functions, contours, contour integrals, some
examples, upper bounds for moduli of contour integrals, antiderivates , CauchyGoursat theorem (without proof ), simply and multiply connected domains, Cauchy’s integral formula, an extension of Cauchy’s integral formula, Liouville’s theorem and fundamental theorem of algebra, maximum modulus principle.


Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used  
1  CO3  Definite integrals of functions,contour integrals,Cauchy’s integral formula,Liouville’s theorem  25  Lecture  
Module 3 Hours :15  
Syllabus:
Convergence of sequences and series, Taylor’s series, proof of Taylor’s theorem, examples, Laurent’s series (without proof), examples. 

Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used  
1.  CO4  Convergence of sequences and series, Taylor’s series,
Laurent’s series

15  Lecture,Seminar  
Module 4 Hours : 18  
Syllabus:
Isolated singular points, residues, Cauchy’s residue theorem, three types of isolated singular points, residues at poles, examples. Applications of residues, evaluation of improper integrals, examples


Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  CO5  Isolated singular points, residues, Cauchy’s residue theorem, three types of isolated singular
points. Applications of residues, evaluation of improper integrals,

18  Lecture, Seminar 
Department  MATHEMATICS 
Programme Name  B.Sc. MATHEMATICS 
Level of study  UG 
Semester  SIXTH 
Course Name  MM6CRT02: GRAPH THEORY AND METRIC SPACES 
Total hours  108 
Course Outcomes
CO Number  Description  Co Evaluation methods 
CO 1  Familiarize with graphs, sub graphs, paths and cycles  Viva and Test 
CO 2  Represent graphs in matrix form  Viva and Test 
CO 3  Conceive the ideas of trees, Bridges, Spanning trees, Cut vertices and Connectivity.  Viva and Test 
CO 4  Familiarize with Euler graphs and Hamiltonian graphs  Viva and Test 
CO 5  Conceive the concepts of Metric Spaces, Open sets, Closed Sets  Viva and Test 
CO 6  Understand convergence in metric spaces and will be familiar with completeness  Viva and Test 
Module I Hours: 36  
Syllabus: Graph Theory
An introduction to graph. Definition of a Graph, More definitions, Vertex Degrees, Sub graphs, Paths and cycles, the matrix representation of graphs. 

Sl. No.  CO Number  Topic/ Activity  No. of hours  Instructional methods to be used 
1  CO 1  An introduction to graph. Definition of a Graph, More definitions.  10  Lecture, Problem Solving 
2  CO 1  Vertex Degrees, Sub graphs, Paths and cycles.  6  Lecture, Problem Solving 
3  CO 2  The matrix representation of graphs.  9  Lecture, Problem Solving 
Module II Hours: 30  
Syllabus: Graph Theory
Trees. Definitions and Simple properties, Bridges, Spanning trees. Cut vertices and Connectivity. Euler’s Tours, the Chinese postman problem. Hamiltonian graphs & the travelling salesman problem. 

Sl. No.  CO Number  Topic/ Activity  No. of hours  Instructional methods to be used 
1  CO 3  Trees. Definitions and Simple properties, Bridges, Spanning trees. Cut vertices and Connectivity.  12  Lecture, Problem Solving and Demonstration. 
2  CO 4  Euler’s Tours, the Chinese postman problem.  9  Lecture, Problem Solving 
3  CO 4  Hamiltonian graphs & the travelling salesman problem.  9  Lecture, Problem Solving 
Module III Hours: 18  
Syllabus: Metric Spaces
Metric Spaces – Definition and Examples, Open sets, Closed Sets, Cantor set. 

Sl. No.  CO Number  Topic/ Activity  No. of hours  Instructional methods to be used 
1  CO 5  Metric Spaces – Definition and Examples.  3  Lecture, Problem Solving 
2  CO 5  Open sets.  8  Lecture, Problem Solving 
3  CO 5  Closed Sets and Cantor set.  7  Lecture, Problem Solving 
Module IV Hours: 24  
Syllabus: Metric Spaces
Convergence, Completeness, Continuous Mapping (Baire’s Theorem included). 

Sl. No.  CO Number  Topic/ Activity  No. of hours  Instructional methods to be used 
1  CO 6  Convergence  7  Lecture, Problem Solving and Demonstration. 
2  CO 6  Completeness  8  Lecture, Problem Solving 
3  CO 6  Continuous Mapping and Baire’s Theorem.  9  Lecture, Problem Solving 
Department  Mathematics 
Name of Faculty  
Programme Name  B. Sc. Mathematics 
Level of study  UG 
Semester  VI 
Course Name/Subject Name  MM6CBT01: OPERATIONS RESEARCH 
Total Hours  72 
Course Outcomes
CO Number  Description  CO Evaluation methods 
CO1  Familiarize the concepts of linear programming (Model formulation and solution by the graphical
method and the simplex method) 
Test, Viva 
CO2  Explain duality in linear programming  Test, Viva 
CO3  Learn about transportation and assignment problems  Test, Viva 
CO4  Understand the basic concepts of game theory  Test, Viva 
Module 1 : Linear Programming: Model formulation and solution by the graphical
method and the simplex method 
Hours : 20  
Syllabus :
General Mathematical Model of LPP, Guidelines on linear Programming model formulation and examples of LP Model formulation, Introduction to graphical method, Definitions, Graphical solution methods of LP Problems, Special cases in linear Programming, Introduction to simplex method, Standard form of an LPP, Simplex algorithm (Maximization case), Simplex algorithm (Minimization case), The Big M Method, Some complications and their resolution, Types of Linear Programming solutions 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO1  General Mathematical Model of LPP, Guidelines on linear Programming model formulation
and examples of LP Model formulation 
4  Lecture 
2  CO1  Introduction to graphical method, Definitions, Graphical
solution methods of LP Problems, Special cases in linear Programming 
5  Lecture 
3  CO1  Introduction to simplex
method, Standard form of an LPP, Simplex algorithm (Maximization case) 
5  Lecture 
4  CO1  Simplex algorithm
(Minimization case), The Big M Method, Some complications and their resolution, Types of Linear Programming solutions 
6  Lecture 
Module 2 : Duality in Linear Programming  Hours : 12  
Syllabus:
Introduction, Formulation of Dual LPP, Standard results on duality, Advantages of Duality, Theorems of duality with proof 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO2  Introduction, Formulation of Dual LPP  4  Lecture 
2  CO2  Standard results on duality, Advantages of Duality  4  Lecture 
3  CO2  Theorems of duality with proof  4  Lecture 
Module 3 : Transportation and Assignment Problems  Hours : 22  
Syllabus:
Introduction, Mathematical model of Transportation Problem, The Transportation Algorithm, Methods for finding Initial solution, Test for optimality, Variations in Transportation Problem, Maximization Transportation problem, Introduction and mathematical models of Assignment problem, Solution methods of Assignment problem, Variations of the assignment problem 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO3  Introduction, Mathematical model of Transportation Problem, The Transportation Algorithm,
Methods for finding Initial solution, Test for optimality 
9  Lecture 
2  CO3  Variations in Transportation Problem, Maximization Transportation problem  3  Lecture 
3  CO3  Introduction and mathematical models of Assignment
problem, Solution methods of Assignment problem 
8  Lecture 
4  CO3  Variations of the assignment problem  2  Lecture 
Module 4 : Theory of Games  Hours : 18  
Syllabus:
Introduction, Twoperson zero sum games, Pure strategic (Minimax and Maximin principles), Games with saddle point, Mixed strategies, Games without saddle point, The rules of dominance, Solution methods: Games without saddle point (Arithmetic method, Matrix method, Graphical method and Linear programming method)


Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO4  Introduction, Twoperson zero sum games, Pure strategic (Minimax and Maximin principles), Games with saddle point, Mixed strategies, Games without saddle point, The rules of dominance  8  Lecture 
2  CO4  Solution methods: Games without saddle point (Arithmetic method, Matrix method,
Graphical method and Linear programming method) 
10  Lecture 
UNION CHRISTIAN COLLEGE, ALUVA
DEPARTMENT OF MATHEMATICS
POST GRADUATE COURSES
INDEX
1.  Semester 1  Graph Theory 
2.  Semester 1  Linear Algebra 
3.  Semester 1  BASIC TOPOLOGY 
4.  Semester 1  ABSTRACT ALGEBRA 
5.  Semester 1  REAL ANALYSIS 
6.  Semester 2  ADVANCED ABSTRACT ALGEBRA 
7.  Semester 2  ADVANCED TOPOLOGY 
8.  Semester 2  Complex Analysis 
9.  Semester 2  Numerical Analysis with Python 
10.  Semester 2  MEASURE AND INTEGRATION 
11.  Semester 3  MULTIVARIATE CALCULUS AND INTEGRAL TRANSFORMS 
12.  Semester 3  Functional Analysis 
13.  Semester 3  Advanced Complex Analysis 
14.  Semester 3  OPTIMIZATION TECHNIQUE 
15.  Semester 3  PARTIAL DIFFERENTIAL EQUATIONS 
16.  Semester 4  ANALYTIC NUMBER THEORY 
17.  Semester 4  Spectral Theory 
18.  Semester 4 ELECTIVE  OPERATIONS RESEARCH 
19.  Semester 4 ELECTIVE  PROBABILITY THEORY 
20.  Semester 4 ELECTIVE  CODING THEORY 
Department  Mathematics 
Name of Faculty  
Programme Name  MSc Mathematics 
Level of study  PG 
Semester  First 
Course Name  Graph Theory 
Total hours  90 
Course Outcomes
CO Number  Description  CO Evaluation methods 
Upon completion of this course, the students will be able to:  
CO1  Explain the fundamental concepts of graph theory.  Assignment, viva, Seminar, Test 
CO2  Perform operations on graphs.  Assignment, viva, Seminar, Test 
CO3  Familiarize directed graph and tournaments.  Assignment, viva, Seminar, Test 
CO4  Identify connectivity, Vertex cuts, edge cuts, blocks. cyclical edge connectivity and spanning trees of a graph.  Assignment, viva, Seminar, Test 
CO5  Model and solve real world problems using graph theory.  Assignment, viva, Seminar, Test 
CO6  Identify Eulerian and Hamiltonian graphs and its characterization.  Assignment, viva, Seminar, Test 
CO7  Solve problems involving vertex and edge coloring, planarity and familiarize the spectrum of a graph.  Assignment, viva, Seminar, Test 
Module: 1 Hours: 20  
Syllabus:
Introduction, Basic concepts. Sub graphs. Degrees of vertices. Paths and Connectedness, Automorphism of a simple graph, line graphs, Operations on graphs, Graph Products. Directed Graphs: Introduction, basic concepts and tournaments. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO1  Introduction, Basic concepts. Sub graphs. Degrees of vertices. Paths and Connectedness.  5  Lecture, Problem solving 
2  CO1  Automorphism of a simple graph, line graphs  5  Lecture, Problem solving 
3  CO2  Operations on graphs, Graph Products.  5  Lecture, Problem solving 
4  CO3  Directed Graphs: Introduction, basic concepts and tournaments.  5  Lecture, Problem solving 
Module: 2 Hours: 25  
Syllabus:
Connectivity : Introduction, Vertex cuts and edge cuts, connectivity and edge connectivity, blocks, Cyclical edge Connectivity of a graph. Trees; Introduction, Definition, characterization and simple properties, centres and cancroids, counting the number of spanning trees, Cayley’s formula. Applications 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO4  Connectivity : Introduction, Vertex cuts and edge cuts, connectivity and edge connectivity, blocks, Cyclical edge Connectivity of a graph.  9  Lecture, Problem solving 
2  CO4  Trees; Introduction, Definition, characterization and simple properties, centres and cancroids, counting the number of spanning trees  8  Lecture, Problem solving 
3  CO5  Cayley’s formula, Applications.  8  Lecture, Problem solving 
Module: 3 Hours: 20  
Syllabus:
Eulerian and Hamiltonian Graphs: Introduction, Eulereian Hamiltonian Graphs, Hamiltonian around’ the world’ game graphs, Graph Colorings: Introduction, Vertex Colorings, Applications of Graph Coloring, Critical Graphs, Brooks’ Theorem. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO6  Eulerian and Hamiltonian Graphs: Introduction, Eulereian Hamiltonian Graphs, Hamiltonian around’ the world’ game graphs.  8  Lecture, Problem solving 
2  CO7  Graph Colorings: Introduction, Vertex Colorings, Applications of Graph
Coloring. 
8  Lecture, Problem solving 
3  CO7  Critical Graphs, Brooks’ Theorem.  4  Lecture, Problem solving 
Module: 4 Hours: 25  
Syllabus:
Planarity: Introduction, Planar and Non planar Graphs, Euler Formula and Its Consequences, K_{5} and K_{3,3} are Nonplanar Graphs, Dual of a Plane Graph, The FourColor Theorem and the Heawood FiveColor Theorem . Spectral Properties of Graphs: Introduction, The Spectrum of a Graph, Spectrum of the Complete Graph Kn, Spectrum of the Cycle C_{n}. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO7  Planarity: Introduction, Planar and Non planar Graphs, Euler Formula and Its Consequences, K_{5} and K_{3,3} are Nonplanar Graphs, Dual of a Plane Graph.  10  Lecture, Problem solving 
2  CO7  The FourColor Theorem and the Heawood FiveColor Theorem .  6  Lecture, Problem solving 
3  CO7  Spectral Properties of Graphs: Introduction, The Spectrum of a Graph, Spectrum of the Complete Graph Kn, Spectrum of the Cycle C_{n}.  9  Lecture, Problem solving 
Department  MATHEMATICS 
Name of Faculty  
Programme Name  M.Sc. Mathematics 
Level of study  PG 
Semester  1 
Course Name/Subject
Name 
Linear Algebra 
Total Hours  90 
Course Outcomes
CO
Number 
Description  CO Evaluation methods 
CO1  To generalize the concept of vectors to n dimensional spaces.

Assignmnent,Test,Seminar 
CO2  Analyze finite and infinite dimensional vector
spaces and subspaces over a field and their properties, including the basis structure of vector space. 
Assignment,Test,Seminar 
CO3  To understand matrix as a linear transformation  Assignment,Test,Seminar 
CO4  Use the definition and properties of linear
transformations and matrices of linear transformations and change of basis, including kernel, range and isomorphism

Assignment,Test,Seminar 
CO5  To understand determinant functions

Assignment,Test,Seminar 
CO6  Compute with the characteristic polynomial,
eigenvectors, eigenvalues and eigenspaces, as well as the geometric and the algebraic multiplicities of an eigenvalue and apply the basic diagonalization Result.

Assignment,Test,Seminar 
CO7  Understand the basic theory of Simultaneous
triangulations, Direct sum decompositions and Invariant direct sums .

Assignment,Test,Seminar 
Module 1 Hours : 20  
Syllabus:
Vector spaces, subspaces, basis and dimension, Coordinates, summary of rowequivalence, Computations concerning subspaces 

Sl no.  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  CO1  Vector spaces, subspaces  10  Lecture, Seminar 
2  CO2  Coordinates, summary of rowequivalence, Computations concerning subspaces  10  Lecture, Seminar 
Module 2 Hours : 25  
Syllabus:
Linear transformations, the algebra of linear transformations, isomorphism, representation of transformations by matrices, linear functional, double dual, transpose of a linear transformation. 

Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  CO3  Linear transformations, Algebra of linear transformations  10  Lecture, Seminar 
2  CO4  Representation of transformations by matrices, linear functional, double dual,
transpose of a linear transformation 
15  Lecture, Seminar 
Module 3 Hours : 20  
Syllabus:
Determinants: Commutative Rings, Determinant functions, Permutation and uniqueness of determinants, Additional properties of determinants. 

Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1.  CO5  Determinant functions, Permutation and
uniqueness of determinants, Additional properties of determinants.

20  Lecture,Seminar 
Module 4 Hours : 25  
Syllabus:
Introduction to elementary canonical forms, characteristic values, annihilatory Polynomials, invariant sub spaces, Direct sum Decomposition 

Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  CO6  Canonical forms, Characteristic values  10  Lecture, Seminar 
2  CO7  Annihilatory Polynomials, invariant sub spaces, Direct sum Decomposition

15  Lecture, Seminar 
Department  Mathematics 
Name of Faculty  
Programme Name  M. Sc. Mathematics 
Level of study  PG 
Semester  I 
Course Name/Subject Name  ME010103 – BASIC TOPOLOGY 
Total Hours  90 
Course Outcomes
CO Number  Description  CO Evaluation methods 
CO1  Familiarize topological spaces, bases and subbases, subspaces  Assignment, Test, Seminar 
CO2  Understand Closures , Neighbourhoods, Interior and Accumulation points  Assignment, Test, Seminar 
CO3  Concieve the concepts of continuous functions and Quotient spaces  Assignment, Test, Seminar 
CO4  Identify spaces with special properties like compactness and Lindelloff ness, second countability and their properties  Assignment, Test, Seminar 
CO5  Understand Connectedness, Local connectedness and Path connectedness of spaces  Assignment, Test, Seminar 
CO6  Acquire basic concepts of Separation axioms and understand hierarchy of separation axioms  Assignment, Test, Seminar 
Module 1 : Topological Spaces  Hours : 25  
Syllabus:
Definition of a topological space – Examples of topological spacesBases and subbases – subspaces. 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO1  Definition of a topological space  6  Lecture, problem solving 
2  CO1  Examples of topological spaces  7  Lecture, problem solving 
3  CO1  Bases and subbases  7  Lecture, problem solving 
4  CO1  subspaces  5  Lecture, problem solving 
Module 2 : Basic concepts  Hours : 25  
Syllabus:
Closed sets and Closures – Neighbourhoods, Interior and Accumulation points – Continuity and Related Concepts – Making functions continuous , Quotient spaces 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO2  Closed sets and Closures  7  Lecture, problem solving 
2  CO2  Neighbourhoods, Interior and Accumulation points  6  Lecture, problem solving 
3  CO3  Continuity and Related Concepts  7  Lecture, problem solving 
4  CO3  Making functions continuous , Quotient spaces  5  Lecture, problem solving 
Module 3 : Spaces with special properties  Hours : 20  
Syllabus:
Smallness conditions on a space, Connectedness. 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO4  Smallness conditions on a space  10  Lecture, problem solving 
2  CO5  Connectedness  10  Lecture, problem solving 
Module 4  Hours : 20  
Syllabus:
Spaces with special properties: – Local connectedness and Paths Separation axioms: Hierarchy of separation axioms 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO5  Local connectedness and Paths  10  Lecture, problem solving 
2  CO6  Hierarchy of separation axioms  10  Lecture, problem solving 
Department  Mathematics 
Name of Faculty  
Programme Name  M. Sc. Mathematics 
Level of study  PG 
Semester  1 
Course Name/Subject Name  ME010101 – ABSTRACT ALGEBRA 
Total Hours  90 
Course Outcomes
CO Number  Description  CO Evaluation methods 
CO1  Familiarize the concepts of finitely generated abelian groups, factor groups and group action on a set  Assignment, Seminar, Test 
CO2  Understand and apply Isomorphism theorems and Sylow theorems  Assignment, Seminar, Test 
CO3  Understand and apply Fermat’s Theorem and learn about rings of polynomials  Assignment, Seminar, Test 
CO4  Conceive the basic concepts of factor rings and ideals  Assignment, Seminar, Test 
Module 1 :  Hours : 25  
Syllabus :
Direct products and finitely generated abelian groups, Fundamental theorem, Applications, Factor groups, Fundamental homomorphism theorem, Normal subgroups and inner automorphisms, Group action on a set, Isotropy subgroups, Applications of G sets to counting


Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO1  Direct products and finitely generated abelian groups, Fundamental theorem, Applications  8  Lecture 
2  CO1  Factor groups, Fundamental homomorphism theorem, Normal subgroups and inner automorphisms  8  Lecture 
3  CO1

Group action on a set, Isotropy subgroups  7  Lecture 
4  CO1

Applications of G sets to counting  2  Lecture 
Module 2 :  Hours : 25  
Syllabus:
Isomorphism theorems, Sylow theorems, Applications of the Sylow theory 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO2  Isomorphism theorems  8  Lecture 
2  CO2  Sylow theorems  8  Lecture 
3  CO2  Applications of the Sylow theory  9  Lecture 
Module 3 :  Hours : 20  
Syllabus:
Fermat’s and Euler Theorems, The field of quotients of an integral domain, Rings of polynomials, Factorisation of polynomials over a field 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO3  Fermat’s and Euler Theorems  5  Lecture 
2  CO3  The field of quotients of an integral domain  5  Lecture 
3  CO3  Rings of polynomials  5  Lecture 
4  CO3  Factorisation of polynomials over a field  5  Lecture 
Module 4 :  Hours : 20  
Syllabus:
Non commutative examples, Homeomorphisms and factor rings, Prime and Maximal Ideals


Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO4  Non commutative examples  5  Lecture 
2  CO4  Homeomorphisms and factor rings  7  Lecture 
3  CO4  Prime and Maximal Ideals  8  Lecture 
Department  MATHEMATICS 
Name of Faculty  
Programme Name  M.Sc. Mathematics 
Level of study  PG 
Semester  One 
Course Name/Subject Name  ME010104 – Real Analysis 
Total Hours  90 
Course Outcomes
CO Number  Description  CO Evaluation methods 
CO1  To know about the concept of functions of bounded variations and their properties. Also get and idea about how such functions are related to the class of monotonic functions.  Assignments, Seminar, Written Examinations 
CO2  Familiarize the concept of rectifiable curves and establish the relationship between them  Assignments, Seminar, Written Examinations 
CO3  To introduce the concept of Stieltjes integrals as an extension of Riemann integrals and how the results of Riemann integrals are rephrased in the case of Stieltjes integrals  Assignments, Seminar, Written Examinations 
CO4  Extend the concept of sequences and series of numbers to that of functions, establish the relationships between convergence and operations of integration &differentiation  Assignments, Seminar, Written Examinations 
CO5  Understand the concepts of uniform boundedness, equicontinuity Express real and complex valued functions as the uniform limit of a sequence of real or complex polynomials using Weierstrass Extension theorem and StoneWeierstrass theorem.  Assignments, Seminar, Written Examinations 
CO6  Familiarize the concepts and properties of Power series, Exponential functions, Logarithmic functions and trigonometric functions. Also, to know about the concept of algebraic completeness of complex field.  Assignments, Seminar, Written Examinations 
Module 1  Hours : 20  
Syllabus: Functions of bounded variation and rectifiable curves
Introduction, properties of monotonic functions, functions of bounded variation, total variation, additive property of total variation, total variation on as a function of , functions of bounded variation expressed as the difference of increasing functions, continuous functions of bounded variation, curves and paths, rectifiable path and arc length, additive and continuity properties of arc length, equivalence of paths, change of parameter.


Sl.No  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO 1  Introduction, properties of monotonic functions  1  Lecture 
2  CO 1  Functions of bounded variation  2  Lecture, Problem Solving 
3  CO1  Total variation, additive property of total variation  4  Lecture 
4  CO1  Total variation on as a function of , functions of bounded variation expressed as the difference of increasing functions  4  Lectures 
5  CO1  Continuous functions of bounded variation  2  Lectures 
6  CO2  Curves and paths, rectifiable path and arc length  2  Lectures 
7  C02  Additive and continuity properties of arc length  2  Lectures 
8  CO2  Equivalence of paths, change of parameter.  3  Lectures 
Module 2  Hours : 20  
Syllabus: The RiemannStieltjes Integral
Definition and existence of the integral, properties of the integral, integration and differentiation, integration of vector valued functions. 

Sl.No  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO3  Definition and existence of the integral  4  Lectures 
2  CO3  Properties of the integral  10  Lectures 
3  CO3  Integration and differentiation  3  Lectures 
4  CO3  Integration of vector valued functions.  3  Lectures 
Module 3  Hours : 25  
Syllabus: Sequence and Series of Functions
Discussion of main problem, Uniform convergence, Uniform convergence and Continuity, Uniform convergence and Integration, Uniform convergence and Differentiation. 

Sl.No  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO4  Discussion of main problem  10  Lectures, Illustrations using examples 
2  CO4  Uniform convergence  8  Lectures 
3  CO4  Uniform convergence and Continuity  2  Lectures

4  CO4  Uniform convergence and Integration  2  Lectures 
5  CO4  Uniform convergence and Differentiation.  3  Lectures 
Module 4  Hours : 25  
Syllabus: Weierstrass Approximation &Some Special Functions
Equicontinuous families of functions, the Stone – Weierstrass theorem, Power series, the exponential and logarithmic functions, the trigonometric functions, the algebraic completeness of complex field. 

Sl.No  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO5  Equicontinuous families of functions  6  Lectures 
2  CO5  The Stone – Weierstrass theorem  2  Lectures 
3  CO5  Power series  5  Lectures 
4  CO5  The exponential and logarithmic functions  5  Lectures 
5  CO5  The trigonometric functions  5  Lectures 
6  CO5  The algebraic completeness of complex field.  2  Lectures 
Department  Mathematics 
Name of Faculty  
Programme Name  M. Sc. Mathematics 
Level of study  PG 
Semester  2 
Course Name/Subject Name  ME010201 – ADVANCED ABSTRACT ALGEBRA 
Total Hours  90 
Course Outcomes
CO Number  Description  CO Evaluation methods 
CO1  Familiarize the concepts of extension fields and understand the theory of finite fields  Assignment, Seminar, Test 
CO2  Acquire knowledge about unique factorization domains, Euclidean domains and multiplicative norms  Assignment, Seminar, Test 
CO3  Understand and apply isomorphism extension theorem  Assignment, Seminar, Test 
CO4  Understand Galois theory and its applications  Assignment, Seminar, Test 
Module 1 :  Hours : 20  
Syllabus :
Introduction to extension fields, Algebraic extensions, Geometric Constructions, Finite fields


Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO1  Introduction to extension fields  5  Lecture 
2  CO1  Algebraic extensions  5  Lecture 
3  CO1

Geometric Constructions  5  Lecture 
4  CO1

Finite fields  5  Lecture 
Module 2 :  Hours : 20  
Syllabus:
Unique factorization domains, Euclidean domains, Gaussian integers and multiplicative norms 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO2  Unique factorization domains  7  Lecture 
2  CO2  Euclidean domains  6  Lecture 
3  CO2  Gaussian integers and multiplicative norms  7  Lecture 
Module 3 :  Hours : 25  
Syllabus:
Automorphism of fields, The isomorphism extension theorem , Splitting fields 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO3  Automorphism of fields  8  Lecture 
2  CO3  The isomorphism extension theorem  9  Lecture 
3  CO3  Splitting fields  8  Lecture 
Module 4 :  Hours : 25  
Syllabus:
Separable extensions, Galois Theory, Illustrations of Galois Theory, Cyclotomic Extensions 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO4  Separable extensions  10  Lecture 
2  CO4  Galois Theory, Illustrations of Galois Theory  10  Lecture 
3  CO4  Cyclotomic Extensions  5  Lecture 
Department  Mathematics 
Name of Faculty  
Programme Name  M. Sc. Mathematics 
Level of study  PG 
Semester  II 
Course Name/Subject Name  ME010202 – ADVANCED TOPOLOGY 
Total Hours  90 
Course Outcomes
CO Number  Description  CO Evaluation methods 
CO1  Conceive more on compactness and Separation axioms  Assignment, Test, Seminar 
CO2  Understand and apply the Urysohn Characterisation of normality and Tietze Characterisation of normality  Assignment, Test, Seminar 
CO3  Familiarize the product space and product topology  Assignment, Test, Seminar 
CO4  Identify productive properties  Assignment, Test, Seminar 
CO5  Understand and apply embedding lemma, Tychonoff Embedding and The Urysohn Metrisation Theorem  Assignment, Test, Seminar 
CO6  Identify different forms of compactness  Assignment, Test, Seminar 
CO7  Understand the basics of Nets and Filters  Assignment, Test, Seminar 
CO8  Familiarise the idea of Homotopy of paths.  Assignment, Test, Seminar 
Module 1 : Separation axioms  Hours : 20  
Syllabus:
Compactness and Separation axioms , The Urysohn Characterisation of normality –Tietze Characterisation of normality. 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO1  Compactness and Separation axioms  6  Lecture, problem solving 
2  CO2  The Urysohn Characterisation of normality  7  Lecture, problem solving 
3  CO2  Tietze Characterisation of normality  7  Lecture, problem solving 
Module 2 : Products and Coproducts  Hours : 25  
Syllabus:
Cartesian products of families of sets – The product topology Productive properties 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO3  Cartesian products of families of sets  7  Lecture, problem solving 
2  CO3  The product topology  9  Lecture, problem solving 
3  CO4  Productive properties  9  Lecture, problem solving 
Module 3 : Embedding and Metrisation  Hours : 25  
Syllabus:
Evaluation functions into products – Embedding lemma and Tychonoff Embedding – The Urysohn Metrisation Theorem – Variation of compactness . 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO5  Evaluation functions into products  5  Lecture, problem solving 
2  CO5  Embedding lemma and Tychonoff Embedding  7  Lecture, problem solving 
3  CO5  The Urysohn Metrisation Theorem  7  Lecture, problem solving 
4  CO6  Variation of compactness  6  Lecture, problem solving 
Module 4  Hours : 20  
Syllabus:
Definition and convergence of nets, Homotopy of paths 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO7  Definition and convergence of nets  10  Lecture, problem solving 
2  CO8  Homotopy of paths  10  Lecture, problem solving 
Department  MATHEMATICS 
Name of Faculty  
Programme Name  M.Sc. Mathematics 
Level of study  PG 
Semester  2 
Course Name/Subject
Name 
Complex Analysis 
Total Hours  90 
Course Outcomes
CO
Number 
Description  CO Evaluation methods 
CO1  Identify analytic functions as mappings

Assignment,Test,Seminar 
CO2  To introduce complex numbers as points on a sphere.

Assignment,Test,Seminar 
CO3  Evaluate complex Integration  Assignment,Test,Seminar 
CO4  Determining the nature of singularities and
calculating residues

Assignment,Test,Seminar 
CO5  Understand the general form of Cauchy‟s
theorem . 
Assignment,Test,Seminar 
CO6  Evaluate definite integrals.

Assignment,Test,Seminar 
Module 1 Hours : 25  
Syllabus:
The spherical representation of complex numbers , Riemann Sphere, Stereographic projection, Distance between the stereographic projections Elementary Theory of power series,Abel’s Theorem on convergence of the power series, Hadamard’s formula, Abel’s limit Theorem Arcs and closed curves, Analytic functions in regions, Conformal mappings, Length and area ,Linear transformations , The cross ratio, Symmetry, Oriented circles, Families of circles.


Sl no.  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  CO2  The spherical representation of complex numbers , Riemann Sphere, Stereographic projection, Distance between the stereographic projections  8  Lecture, Seminar 
2  CO1  Arcs and closed curves, Analytic functions in regions, Conformal mappings,
Length and area ,Linear transformations , The cross ratio, Symmetry, Oriented circles, Families of circles. 
17  Lecture, Seminar 
Module 2 Hours : 20  
Syllabus:
Fundamental theorems on complex integration: line integrals, rectifiable arcs, line integrals as functions of arcs, Cauchy’s theorem for a rectangle, Cauchy’s theorem in a disk, Cauchy’s integral formula: the index of a point with respect to a cloud curve, the integral formula. 

Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  CO3  Rectifiable arcs, Theorems on complex integration, Cauchy’s theorems, Index of a point, Cauchy’s integral formula.  20  Lecture, Seminar 
Module 3 Hours : 20  
Syllabus:
Higher derivatives. Differentiation under the sign of integration, Morera’s Theorem, Liouville’s Theorem, Fundamental Theorem, Cauchy’s estimate Local properties of analytical functions: removable singularities, Taylor’s theorem, zeroes and poles,Weirstrass Theorem on essential singularity, the local mapping, the maximum principle.Schwarz lemma. 

Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1.  CO3  Higher derivatives. Differentiation under the sign of integration, Morera’s
Theorem, Liouville’s Theorem, Fundamental Theorem, Cauchy’s estimate. 
9  Lecture,Seminar 
2  CO4  Removable singularities, Taylor’s
theorem, zeroes and poles,Weirstrass Theorem on essential singularity, the local mapping, the maximum principle.Schwarz lemma 
11  Lecture,Seminar 
Module 4 Hours : 25  
Syllabus:
The general form of Cauchy’s theorem: chains and cycles, simple connectivity, homology, general statement of Cauchy’s theorem, proof of Cauchy’s theorem, locally exact differentiation, multiply connected regions Calculus of Residues: the residue theorem, the argument principle, evaluation of definite integrals. 

Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  CO5  The general form of Cauchy’s theorem: chains and cycles, simple connectivity, homology, general statement of Cauchy’s theorem, proof of
Cauchy’s theorem, locally exact differentiation, multiply connected regions . 
13  Lecture, Seminar 
2  CO4  Calculus of Residues: the residue theorem, the argument principle.  6  Lecture, Seminar 
3  CO6  Evaluation of definite integrals.  6  Lecture,Seminar 
Department  Mathematics 
Name of Faculty  
Programme Name  MSc Mathematics 
Level of study  PG 
Semester  Second 
Course Name  Numerical Analysis with Python 
Total hours  90 
Course Outcomes
CO Number  Description  CO Evaluation methods 
Upon completion of this course, the students will be able to:  
CO1  Build basic programs using fundamental programming constructs like variables, conditional logic, looping, and functions.  Test, Assignment, Lab exam 
CO2  Use lists, tuples, functions and dictionaries in Python programs.  Test, Assignment, Lab exam 
CO3  Use indexing and slicing to access data in Python
programs. 
Test, Assignment, Lab exam 
CO4  Use exception handling in Python applications for error
handling. 
Test, Assignment, Lab exam 
CO5  Write python code for solving calculus problems.  Test, Assignment, Lab exam 
CO6  Write python code for solving numerical problems like interpolation, curve fitting and numerical integration.  Test, Assignment, Lab exam 
CO7  Write python code for finding roots of equation and solving system of linear equations  Test, Assignment, Lab exam 
Module: Basics of Python Hours: 15  
Syllabus:
Calculations and variables, creating strings, lists are more powerful than strings, tuples, If statements, ifthenelse statements, if and elif statements, combining conditions, the difference between strings and numbers, using for loops, while we are talking about looping, using functions, parts of a function, using modules. The functions abs, float, int, len, max, min, range, sum, complex numbers. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO1, CO2  Calculations and variables, creating strings, lists are more powerful than strings, tuples  4  Lecture, Hands on training in computer lab. 
2  CO1, CO2  If statements, ifthenelse statements, if and elif statements, combining conditions  4  Lecture, Hands on training in computer lab. 
3  CO1  Using for and while loops.  4  Lecture, Hands on training in computer lab. 
4  CO2, CO3  Using functions, parts of a function, using modules. The functions abs, float, int, len, max, min, range, sum, complex numbers.  3  Lecture, Hands on training in computer lab. 
Module: 1 Hours: 20  
Syllabus:
Defining Symbols and Symbolic Operations, Working with Expressions, Solving Equations and Plotting Using SymPy, problems on factor finder, summing a series and solving single variable inequalities. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO1, CO3  Defining Symbols and Symbolic Operations, Working with Expressions.  5  Lecture, Hands on training in computer lab. 
2  CO3, CO4  Solving equations and Plotting Using SymPy.  5  Lecture, Hands on training in computer lab. 
3  CO2,CO4  Problems on factor finder, summing a series.  5  Lecture, Hands on training in computer lab. 
4  CO1  Solving single variable inequalities.  5  Lecture, Hands on training in computer lab. 
Module: 2 Hours: 20  
Syllabus:
Finding the limit of functions, finding the derivative of functions, higherorder derivatives and finding the maxima and minima and finding the integrals of functions are to be done. In the section programming challenges, the following problems – verify the continuity of a function at a point, area between two curves and finding the length of a curve. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO5  Finding the limit of functions, finding the derivative of functions, higherorder derivatives.  8  Lecture, Hands on training in computer lab. 
2  CO5  Finding the maxima and minima and finding the integrals of functions are to be done  7  Lecture, Hands on training in computer lab. 
3  CO5  Verify the continuity of a function at a point, area between two curves and finding the length of a curve.  5  Lecture, Hands on training in computer lab. 
Module: 3 Hours: 25  
Syllabus:
Interpolation and Curve Fitting – Polynomial Interpolation – Lagrange’s Method, Newton’s Method and Limitations of Polynomial Interpolation, Roots of Equations – Method of Bisection and NewtonRaphson Method. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO6  Interpolation and Curve Fitting – Polynomial Interpolation, Lagrange’s Method.  10  Lecture, Hands on training in computer lab. 
2  CO6  Newton’s Method and Limitations of Polynomial Interpolation.  5  Lecture, Hands on training in computer lab. 
3  CO7  Roots of Equations – Method of Bisection and NewtonRaphson Method.  10  Lecture, Hands on training in computer lab. 
Module: 4 Hours: 25  
Syllabus:
Gauss Elimination Method, Doolittle’s Decomposition Method only from LU Decomposition Methods Numerical Integration, NewtonCotes Formulas, Trapezoidal rule, Simpson’s rule and Simpson’s 3/8 rule. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO7  Gauss Elimination Method, Doolittle’s Decomposition Method.  12  Lecture, Hands on training in computer lab. 
2  CO6  Numerical Integration, NewtonCotes Formulas, Trapezoidal rule, Simpson’s rule and Simpson’s 3/8 rule.  13  Lecture, Hands on training in computer lab. 
COURSE PLAN
Department  MATHEMATICS 
Name of Faculty  
Programme Name  M.Sc. Mathematics 
Level of study  PG 
Semester  Two 
Course Name/Subject Name  ME010205 – Measure Theory and Integration 
Total Hours  90 
Course Outcomes
CO Number  Description  CO Evaluation methods 
CO1  Understand the concept of measures, Lebesgue outer measure, Lebesgue measure, Measurable sets and their properties  Assignments, Seminar, Written Examinations 
CO2  To know about measurable functions and their properties  Assignments, Seminar, Written Examinations 
CO3  Extend the concept of Riemann integrals from intervals to arbitrary measurable sets  Assignments, Seminar, Written Examinations 
CO4  Conceive the idea of measures defined for subsets of an arbitrary set, their properties.  Assignments, Seminar, Written Examinations 
CO5  Understand the concept of measurable functions defined on arbitrary sets and extend integrals of such functions to arbitrary measurable functions  Assignments, Seminar, Written Examinations 
CO6  Familiarize the concept of measures defined on product spaces and Fubini’s theorem for integration on product spaces  Assignments, Seminar, Written Examinations 
Module 1  Hours: 25  
Syllabus: Lebesgue Measure
Introduction, Lebesgue outer measure, The algebra of Lebesgue measurable sets, Outer and inner approximation of Lebesgue measurable sets, Countable additivity, continuity and BorelCantelli Lemma – Non measurable sets – The Canter set and Canter Lebesgue function 

Sl.No  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO 1  Introduction, Lebesgue outer measure  4  Lecture, Problem solving 
2  CO 1  The algebra of Lebesgue measurable sets  2  Lecture 
3  CO1  Outer and inner approximation of Lebesgue measurable sets  4  Lecture 
4  CO1  Countable additivity, continuity and BorelCantelli Lemma  8  Lectures, Problem solving 
5  CO1  Non measureable sets  4  Lectures 
6  CO1  The Canter set and Canter Lebesgue function  3  Lectures 
Module 2  Hours: 25  
Syllabus: Lebesgue Measurable Functions and Lebesgue Integration
Sums, products and compositions – Sequential pointwise limits and simple approximation – The Riemann Integral – The Lebesgue integral of a bounded measurable function over a set of finite measure – The Lebesgue integral of a measurable non negative function – The general Lebesgue integral. 

Sl.No  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO2  Sums, products and compositions  4  Lectures, Problem solving 
2  CO2  Sequential pointwise limits and simple approximation  5  Lectures 
3  CO3  The Riemann Integral – The Lebesgue integral of a bounded measurable function over a set of finite measure  5  Lectures 
4  CO3  The Lebesgue integral of a measurable non negative function  6  Lectures, Problem solving 
5  CO3  The general Lebesgue integral  4  Lectures, Problem solving 
Module 3  Hours: 20  
Syllabus: General Measure Space and Measurable Functions
Measures and measurable sets – Signed Measures: The Hahn and Jordan decompositions – The Caratheodory measure induced by an outer measure – Measurable functions 

Sl.No  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO4  Measures and measurable sets  4  Lectures, Problem solving 
2  CO4  Signed Measures  4  Lectures 
3  CO4  The Hahn and Jordan decompositions  5  Lectures

4  CO4  The Caratheodory measure induced by an outer measure  4  Lectures 
5  CO4  Measurable functions  3  Lectures, Problem solving 
Module 4  Hours: 20  
Syllabus: Integration over General Measure Space and Product Measures
Integration of nonnegative measurable functions – Integration of general measurable functions – The Radon Nikodym Theorem – Product measure: The theorems of Fubini and Tonelli 

Slno  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO5  Integration of nonnegative measurable functions  2  Lectures 
2  CO5  Integration of general measurable functions  4  Lectures 
3  CO5  The Radon Nikodym Theorem  3  Lectures 
4  CO6  Product measure  3  Lectures 
5  CO6  The theorems of Fubini and Tonelli  8  Lectures 
Department  MATHEMATICS 
Name of Faculty  
Programme Name  M.Sc. MATHEMATICS 
Level of study  PG 
Semester  THIRD 
Course Name/Subject
Name 
ME010303 – MULTIVARIATE CALCULUS AND INTEGRAL TRANSFORMS 
Total Hours  90 
Course Outcomes
CO
Number 
Description  CO Evaluation methods 
1  Familiarize other forms of Fourier series.  Assignment,Test 
2  Understand theorems like Fourier integral theorem, the exponential form of the Fourier integral theorem  Assignment,Test 
3  Understand Integral transforms and the convolution theorem for Fourier transforms  Assignment,Test 
4  Understand the concept of directional derivatives and total derivative and see how total derivative becomes the generalization for a multivariable function  Assignment,Test, Seminar 
5  Find the Jacobian matrix of a linear function and understand the matrix form of the chain rule

Assignment, Test, Viva 
6  Understand the mean value theorem for differentiable functions  Assignment, Test, Seminar 
7  Derive sufficient condition for differentiability  Assignment, Viva, Test 
8  To find extrema of multivariable functions  Assignment, Viva, Test 
9  Understand the inverse function theorem and the implicit function theorem

Assignment, Test 
10  Familiarize integration of Differential Forms

Assignment, Viva, Test 
Module 1 Hours : 20  
Syllabus: The Weierstrass theorem, other forms of Fourier series, the Fourier integral theorem, the exponential form of the Fourier integral theorem, integral transforms and convolutions, the convolution theorem for Fourier transforms  
Sl.no  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  2  The Weierstrass theorem  3  Lecture 
2  1  Other forms of Fourier series  3  Lecture 
3  2  The Fourier integral theorem  4  Lecture 
4  2  The exponential form of the Fourier integral theorem,  4  Lecture 
5  3  Integral transforms and convolutions  3  Lecture 
6  3  The convolution theorem for Fourier transforms

3  Lecture 
Module 2 Hours : 22  
Syllabus: Multivariable Differential Calculus The directional derivative, directional derivatives and continuity, the total derivative, the total derivative expressed in terms of partial derivatives, An application of complex valued functions, the matrix of a linear function, the Jacobian matrix, the matrix form of the chain rule. Implicit functions and extremum problems, the mean value theorem for differentiable functions  
Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  4  The directional derivative  3  Lecture 
2  4  The total derivative  3  Lecture 
3  4  An application of complex valued functions  3  Lecture 
4  5  The matrix of a linear function, the Jacobian matrix  3  Lecture 
5  5  The chain rule  3  Lecture 
6  8  Implicit functions and extremum problems  4  Lecture 
7  6  The mean value theorem for differentiable functions

3  Lecture 
Module 3 28 hours  
Syllabus: A sufficient condition for differentiability, a sufficient condition for equality of mixed partial derivatives, functions with nonzero Jacobian determinant, the inverse function theorem ,the implicit function theorem, extrema of real valued functions of one variable, extrema of real valued functions of several variables.  
Slno  CO
Number 
Topic/Activity  No of
hours 
Instructional methods to be used 
1  7  Sufficient condition for differentiability  4  Lecture 
2  7  Sufficient condition for equality of mixed partial derivatives  3  Lecture 
3  7  Functions with nonzero Jacobian determinant  3  Lecture 
4  8  The inverse function theorem  5  Lecture 
5  8  The implicit function theorem,  5  Lecture 
6  9  Extrema of real valued functions of one variable  3  Lecture 
7  9  Extrema of real valued functions of several variables.

5 
Module 4 Hours : 20  
Syllabus: Integration of Differential Forms Integration, primitive mappings, partitions of unity, change of variables, differential forms.  
Slno  CO
Number 
Topic/Activity  No of
hours 
Instructional methods to be used 
1  10  Integration  3  Lecture 
2  10  Flips & primitive mappings  5  Lecture 
3  10  Partitions of unity  3  Lecture 
4  10  Change of variables  3  Lecture 
5  10  Differential forms  6  Lecture 
Department  Mathematics 
Name of Faculty  
Programme Name  MSc Mathematics 
Level of study  PG 
Semester  Third 
Course Name  Functional Analysis 
Total hours  90 
Course Outcomes
CO Number  Description  CO Evaluation methods 
Upon completion of this course, the students will be able to:  
CO1  Understand how functional analysis uses and unifies ideas from vector spaces and the theory of metrics spaces.  Assignment, viva, Seminar, Test 
CO2  Understand and apply fundamental theorems from the theory of normed and Banach spaces.  Assignment, viva, Seminar, Test 
CO3  Understand the theory of bounded linear operators and bounded linear functionals.  Assignment, viva, Seminar, Test 
CO4  Realize the role of inner product space and apply ideas from the theory of Hilbert spaces to other areas.  Assignment, viva, Seminar, Test 
CO5  Realize the important role Zorn’s Lemma and its applications including the Hahn Banach Theorems.  Assignment, viva, Seminar, Test 
CO6  Understand different types of operators.  Assignment, viva, Seminar, Test 
Module: 1 Hours: 25  
Syllabus:
Examples, Completeness proofs, Completion of Metric Spaces, Vector Space, Normed Space, Banach space, Further Properties of Normed Spaces, Finite Dimensional Normed spaces and Subspaces, Compactness and Finite Dimension. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO1  Examples, Completeness proofs, Completion of Metric Spaces  5  Lecture, Problem solving 
2  CO2  Vector Space, Normed Space, Banach space, Further Properties of Normed Spaces.  5  Lecture, Problem solving 
3  CO2  Finite Dimensional Normed spaces and Subspaces  8  Lecture, Problem solving 
4  CO2  Compactness and Finite Dimension.  7  Lecture, Problem solving 
Module: 2 Hours: 20  
Syllabus:
Linear Operators, Bounded and Continuous Linear Operators, Linear Functionals, Linear Operators and Functionals on Finite dimensional spaces, Normed spaces of operators, Dual space. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO3  Linear Operators, Bounded and Continuous Linear Operators  5  Lecture, Problem solving 
2  CO3  Linear Functionals, Linear Operators and Functionals on Finite dimensional spaces  10  Lecture, Problem solving 
3  CO3  Normed spaces of operators, Dual space.  5  Lecture, Problem solving 
Module: 3 Hours: 25  
Syllabus:
Inner Product Space, Hilbert space, Further properties of Inner Product Space, Orthogonal Complements and Direct Sums, Orthonormal sets and sequences, Series related to Orthonormal sequences and sets, Total Orthonormal sets and sequences, Representation of Functionals on Hilbert Spaces 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO4  Inner Product Space, Hilbert space, Further properties of Inner Product Space  7  Lecture, Problem solving 
2  CO4  Orthogonal Complements and Direct Sums, Orthonormal sets and sequences, Series related to Orthonormal sequences and sets.  8  Lecture, Problem solving 
3  CO4  Total Orthonormal sets and sequences.  5  Lecture, Problem solving 
4  CO4  Representation of Functionals on Hilbert Spaces.  5  Lecture, Problem solving 
Module: 4 Hours: 20  
Syllabus:
HilbertAdjoint Operator, SelfAdjoint, Unitary and Normal Operators, Zorn’s lemma, Hahn Banach theorem, Hahn Banach theorem for Complex Vector Spaces and Normed Spaces, Adjoint Operators. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO6  HilbertAdjoint Operator, SelfAdjoint, Unitary and Normal Operators.  5  Lecture, Problem solving 
2  CO5  Zorn’s lemma, Hahn Banach theorem, Hahn Banach theorem for Complex Vector Spaces and Normed Spaces  10  Lecture, Problem solving 
3  CO6  Adjoint Operators.  5  Lecture, Problem solving 
Department  Mathematics 
Name of Faculty  
Programme Name  M. Sc. Mathematics 
Level of study  PG 
Semester  3 
Course Name/Subject Name  ME010301 – Advanced Complex Analysis 
Total Hours  90 
Course Outcomes
CO Number  Description  CO Evaluation methods 
CO1  Familiarize the concepts of harmonic and subharmonic functions  Assignment, Seminar, Test 
CO2  Understand the theory and applications of the power series expansions and partial fractions  Assignment, Seminar, Test 
CO3  Familiarize the concepts of Riemann zeta function and normal families of functions  Assignment, Seminar, Test 
CO4  Understand and apply the Riemann mapping theorem and learn about the Weierstrass’s theory  Assignment, Seminar, Test 
Module 1 :  Hours : 25  
Syllabus :
Harmonic Functions – Definitions and Basic Properties, The MeanValue Property, Poisson’s Formula, Schwarz’s Theorem, The Reflection Principle, A closer look at Harmonic Functions – Functions with Mean Value Property, Harnack’s Principle, The Dirichlet’s Problem – Subharmonic Functions, Solution of Dirichlet’s Problem ( Proof of Dirichlet’s Problem and Proofs of Lemma 1 and 2 excluded ) 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO1  Harmonic Functions – Definitions and Basic Properties, The MeanValue Property, Poisson’s Formula  9  Lecture 
2  CO1  Schwarz’s Theorem, The Reflection Principle  5  Lecture 
3  CO1  A closer look at Harmonic Functions – Functions with Mean Value Property, Harnack’s Principle  6  Lecture 
4  CO1

The Dirichlet’s Problem – Subharmonic Functions, Solution of Dirichlet’s Problem  5  Lecture 
Module 2 :  Hours : 25  
Syllabus:
Power Series Expansions – Weierstrass’s theorem, The Taylor Series, The Laurent Series, Partial Fractions and Factorization – Partial Fractions, Infinite Products, Canonical Products, The Gamma Function, Entire Functions – Jensen’s Formula, Hadamard’s Theorem ( Hadamard’s theorem – proof excluded) 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO2  Power Series Expansions – Weierstrass’s theorem, The Taylor Series, The Laurent Series  10  Lecture 
2  CO2  Partial Fractions and Factorization – Partial Fractions, Infinite Products, Canonical Products, The Gamma Function  10  Lecture 
3  CO2  Entire Functions – Jensen’s Formula, Hadamard’s Theorem  5  Lecture 
Module 3 :  Hours : 20  
Syllabus:
The Riemann Zeta Function – The Product Development, The Extension of ζ(S) to the Whole Plane, The Functional Equation, The Zeroes of the Zeta Function, Normal Families – Normality and Compactness, Arzela’s Theorem 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO3  The Riemann Zeta Function – The Product Development, The Extension of ζ(S) to the Whole Plane  7  Lecture 
2  CO3  The Functional Equation, The Zeroes of the Zeta Function  6  Lecture 
3  CO3  Normal Families – Normality and Compactness, Arzela’s Theorem  7  Lecture 
Module 4 :  Hours : 20  
Syllabus:
The Riemann Mapping Theorem – Statement and Proof, Boundary Behaviour, Use of the Reflection Principle, The Weierstrass’s Theory – The Weierstrass’s ρ function, The functions ζ(s) and σ(z), The Differential Equation 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO4  The Riemann Mapping Theorem – Statement and Proof  5  Lecture 
2  CO4  Boundary Behaviour, Use of the Reflection Principle  5  Lecture 
3  CO4  The Weierstrass’s Theory – The Weierstrass’s ρ – function, The functions ζ(s) and σ(z), The Differential Equation  10  Lecture 
Department  MATHEMATICS 
Name of Faculty  
Programme Name  M.Sc. Mathematics 
Level of study  PG 
Semester  Three 
Course Name/Subject Name  ME010302 – PARTIAL DIFFERENTIAL EQUATIONS 
Total Hours  90 
Course Outcomes
CO Number  Description  CO Evaluation methods 
CO1  Familiarize the methods of solutions of systems of ordinary differential equations  Assignments, Seminar, Written Examinations 
CO2  To know about Pfaffian differential equations and the methods of solution of them  Assignments, Seminar, Written Examinations 
CO3  To know how first order P.D.E. are originated, Linear and Nonlinear first order P.D.E. and their solution methods  Assignments, Seminar, Written Examinations 
CO4  Conceive the idea of compatible systems of equations, Charpit’s and Jacobi’s methods to solve such equations  Assignments, Seminar, Written Examinations 
CO5  Introduce the origin of second order P.D.E, linear and nonlinear second order P.D.E. with constant and variable coefficients and their solution methods  Assignments, Seminar, Written Examinations 
CO6  To know about Laplace equations, Families of equipotential surfaces and to establish the relation of Logarithmic potential to the Theory of Functions.  Assignments, Seminar, Written Examinations 
Module 1  Hours: 20  
Syllabus:
Methods of solutions of . Orthogonal trajectories of a system of curves on a surface. Pfaffian differential forms and equations. Solution of Pfaffian differential equations in three variables, Partial differential equations. Origins of first order partial differential equation. 

Sl.No  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO 1  Methods of solutions of
. 
4  Lecture, Problem solving 
2  CO 1  Orthogonal trajectories of a system of curves on a surface.  2  Lecture, Problem solving 
3  CO2  Pfaffian differential forms and equations  4  Lecture 
4  CO2  Solution of Pfaffian differential equations in three variables  8  Lectures, Problem solving 
5  CO3  Origins of first order partial differential equation  2  Lectures, Problem solving 
Module 2  Hours: 25  
Syllabus:
Linear equations of first order. Integral surfaces passing through a given curve. Surfaces orthogonal to a given system of surfaces. Nonlinear partial differential equation of the first order. Compatible systems of first order equations. Charpits Method. Special types of first order equations. Solutions satisfying given conditions. 

Sl.No  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO3  Linear equations of first order  5  Lectures, Problem solving 
2  CO3  Integral surfaces passing through a given curve. Surfaces orthogonal to a given system of surfaces  3  Lectures, Problem solving 
3  CO3  Nonlinear partial differential equation of the first order.  4  Lectures, Problem solving 
4  CO4  Compatible systems of first order equations  2  Lectures, Problem solving 
5  CO4  Charpits Method  6  Lectures, Problem solving 
6  CO4  Special types of first order equations  3  Lectures, Problem solving 
7  CO4  Solutions satisfying given conditions  3  Lectures, Problem solving 
Module 3  Hours: 20  
Syllabus:
Jacobi’ s method. The origin of second order equations. Linear partial differential equations with constant coefficients. Equations with variable coefficients. 

Sl.No  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO4  Jacobi’ s method  4  Lectures, Problem solving 
2  CO5  The origin of second order equations  2  Lectures, Problem solving 
3  CO5  Linear partial differential equations with constant coefficients  8  Lectures, Problem solving 
4  CO5  Equations with variable coefficients  6  Lectures, Problem solving 
Module 4  Hours: 25  
Syllabus:
Separation of variables. Nonlinear equations of the second order. Elementary solutions of Laplace equation. Families of equipotential surfaces. The two dimensional Laplace Equation Relation of the Logarithmic potential to the Theory of Functions. 

Slno  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO5  Separation of variables  2  Lectures, Problem solving 
2  CO5  Nonlinear equations of the second order  4  Lectures, Problem solving 
3  CO6  Elementary solutions of Laplace equation  6  Lectures, Problem solving 
4  CO6  Families of equipotential surfaces  4  Lectures, Problem solving 
5  CO6  The twodimensional Laplace Equation  4  Lectures, Problem solving 
6  CO6  Laplace Equation Relation of the Logarithmic potential to the Theory of Functions.  5  Lectures, Problem solving 
Department  Mathematics 
Name of Faculty  
Programme Name  M. Sc. Mathematics 
Level of study  PG 
Semester  4 
Course Name/Subject Name  ME010402 – ANALYTIC NUMBER THEORY 
Total Hours  90 
Course Outcomes
CO Number  Description  CO Evaluation methods 
CO1  Learn about arithmetical functions and averages of arithmetical functions  Assignment, Seminar, Test 
CO2  Understand some elementary theorems on the distribution of prime numbers  Assignment, Seminar, Test 
CO3  Acquire deep knowledge on the theory of congruences  Assignment, Seminar, Test 
CO4  Familiarize the concepts of quadratic residues and primitive roots  Assignment, Seminar, Test 
Module 1 :  Hours : 30  
Syllabus :
Arithmetical functions – Introduction, The Möbius function μ(n), The Euler totient function ϕ(n), A relation connecting μ and ϕ, A product formula for ϕ(n), The Dirichlet product of arithmetical functions, Dirichlet inverses and the Möbius inversion formula, The Mangoldt function ∧(n), Multiplicative functions, Multiplicative functions and Dirichlet Multiplication, The inverse of a completely multiplicative function, The Liouville’s function λ(n), The divisor function ??(n), Generalized convolutions Averages of Arithmetical functions – Introduction, The big oh notation, Asymptotic equality of functions, Euler’s summation formula, Some elementary asymptotic formulas, The average order of d(n), The average order of the divisor functions ??(n), The average order of ϕ(n), An application to the distribution of lattice points visible from the origin, The average order of μ(n) and of ∧(n), The partial sums of a Dirichlet product, Applications to μ(n) and of ∧(n) 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO1  Arithmetical functions – Introduction, The Möbius function μ(n), The Euler totient function ϕ(n), A relation connecting μ and ϕ, A product formula for ϕ(n)  5  Lecture 
2  CO1  The Dirichlet product of arithmetical functions, Dirichlet inverses and the Möbius inversion formula  5  Lecture 
3  CO1  The Mangoldt function ∧(n), Multiplicative functions, Multiplicative functions and Dirichlet Multiplication, The inverse of a completely multiplicative function  6  Lecture 
4  CO1

Liouville’s function λ(n), The divisor function ??(n), Generalized convolutions  3  Lecture 
5  CO1  Averages of Arithmetical functions – Introduction, The big oh notation, Asymptotic equality of functions, Euler’s summation formula, Some elementary asymptotic formulas  5  Lecture 
6  CO1

The average order of d(n), The average order of the divisor functions ??(n), The average order of ϕ(n), An application to the distribution of lattice points visible from the origin, The average order of μ(n) and of ∧(n), The partial sums of a Dirichlet product, Applications to μ(n) and of ∧(n)  6  Lecture 
Module 2 :  Hours : 15  
Syllabus:
Some Elementary Theorems on the Distribution of Prime Numbers – Introduction, Chebyshev’s functions ψ(x) and ϑ(x), Relation connecting ϑ(x) and π(x), Some equivalent forms of the prime number theorem, Inequalities for π(n) and Pn , Shapiro’s tauberian theorem, Applications of Shapiro’s theorem, An asymptotic formula for the partial sum 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO2  Some Elementary Theorems on the Distribution of Prime Numbers – Introduction, Chebyshev’s functions ψ(x) and ϑ(x), Relation connecting ϑ(x) and π(x)  5  Lecture 
2  CO2  Some equivalent forms of the prime number theorem, Inequalities for π(n) and Pn  5  Lecture 
3  CO2  Shapiro’s tauberian theorem, Applications of Shapiro’s theorem, An asymptotic formula for the partial sum  5  Lecture 
Module 3 :  Hours : 25  
Syllabus:
Congruences – Definitions and basic properties of congruences, Residue classes and complete residue system, Linear congruences, Reduced residue systems and EulerFermat theorem, Polynomial congruences modulo p, Lagrange’s theorem, Applications of Lagrange’s theorem, Simultaneous linear congruences, The Chinese remainder theorem, Applications of the Chinese remainder theorem 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO3  Congruences – Definitions and basic properties of congruences, Residue classes and complete residue system  5  Lecture 
2  CO3  Linear congruences, Reduced residue systems and EulerFermat theorem  6  Lecture 
3  CO3  Polynomial congruences modulo p, Lagrange’s theorem, Applications of Lagrange’s theorem  7  Lecture 
4  CO3  Simultaneous linear congruences, The Chinese remainder theorem, Applications of the Chinese remainder theorem  7  Lecture 
Module 4 :  Hours : 20  
Syllabus:
Quadratic residues – Quadratic residues, Legendre’s symbol and its properties, Evaluation of (1p) and (2p), Gauss’ Lemma, The quadratic reciprocity law, Applications of the reciprocity law, Primitive Roots – The exponent of a number mod m, Primitive roots, Primitive roots and reduced residue systems, The nonexistence of primitive roots mod 2^{α} for ?≥3, The existence of primitive root mod p for odd primes p, Primitive roots and quadratic residues 

Sl no  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO4  Quadratic residues, Legendre’s symbol and its properties, Evaluation of (1p) and (2p), Gauss’ Lemma  5  Lecture 
2  CO4  The quadratic reciprocity law, Applications of the reciprocity law  5  Lecture 
3  CO4  The exponent of a number mod m, Primitive roots, Primitive roots and reduced residue systems  4  Lecture 
4  CO4  The nonexistence of primitive roots mod 2^{α} for ?≥3, The existence of primitive root mod p for odd primes p, Primitive roots and quadratic residues  3  Lecture 
Department  Mathematics 
Name of Faculty  
Programme Name  MSc Mathematics 
Level of study  PG 
Semester  Fourth 
Course Name  Spectral Theory 
Total hours  90 
Course Outcomes
CO Number  Description  CO Evaluation methods 
Upon completion of this course, the students will be able to:  
CO1  Understand category theorem, Uniform Boundedness theorem, Open Mapping Theorem, Closed Graph Theorem and Banach fixed point theorem,  Assignment, viva, Seminar, Test 
CO2  Familiarize different types of convergence of Sequences of Operators and Functionals  Assignment, viva, Seminar, Test 
CO3  Familiarize Spectral Properties of Bounded Linear Operators.  Assignment, viva, Seminar, Test 
CO4  Understand the role of Complex Analysis in Spectral Theory.  Assignment, viva, Seminar, Test 
CO5  Familiarize Banach Algebras and its properties.  Assignment, viva, Seminar, Test 
CO6  Understand and apply fundamental theorems from the theory of Compact linear operators and their spectrum.  Assignment, viva, Seminar, Test 
CO7  Understand Spectral Properties of Bounded Self adjoint linear operators and familiarize Projection and positive Operators and their properties.  Assignment, viva, Seminar, Test 
Module: 1 Hours: 20  
Syllabus:
Reflexive Spaces, Category theorem, Uniform Boundedness theorem, Strong and Weak Convergence, Convergence of Sequences of Operators and Functionals, Open Mapping Theorem, Closed Linear Operators, Closed Graph Theorem. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO1  Reflexive Spaces, Category theorem, Uniform Boundedness theorem.  5  Lecture, Problem solving 
2  CO2  Strong and Weak Convergence, Convergence of Sequences of Operators and Functionals.  6  Lecture, Problem solving 
3  CO1  Open Mapping Theorem, Closed Linear Operators, Closed Graph Theorem.  9  Lecture, Problem solving 
Module: 2 Hours: 25  
Syllabus:
Banach Fixed point theorem, Spectral theory in Finite Dimensional Normed Spaces, Basic Concepts, Spectral Properties of Bounded Linear Operators, Further Properties of Resolvent and Spectrum, Use of Complex Analysis in Spectral Theory 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO1  Banach Fixed point theorem.  3  Lecture, Problem solving 
2  CO3  Spectral theory in Finite Dimensional Normed Spaces, Basic Concepts, Spectral Properties of Bounded Linear Operators, Further Properties of Resolvent and Spectrum.  15  Lecture, Problem solving 
3  CO4  Use of Complex Analysis in Spectral Theory.  7  Lecture, Problem solving 
Module: 3 Hours: 25  
Syllabus:
Banach Algebras, Further Properties of Banach Algebras, Compact Linear Operators on Normed spaces, Further Properties of Compact Linear Operators, Spectral Properties of compact Linear Operators on Normed spaces, Further Spectral Properties of Compact Linear Operators. 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO5  Banach Algebras, Further Properties of Banach Algebras.  5  Lecture, Problem solving 
2  CO6  Compact Linear Operators on Normed spaces, Further Properties of Compact Linear Operators  10  Lecture, Problem solving 
3  CO6  Spectral Properties of compact Linear Operators on Normed spaces, Further Spectral Properties of Compact Linear Operators.  10  Lecture, Problem solving 
Module: 4 Hours: 20  
Syllabus:
Spectral Properties of Bounded Self adjoint linear operators, Further Spectral Properties of Bounded Self Adjoint Linear Operators, Positive Operators, Projection Operators, Further Properties of Projections 

Sl. No  CO Number  Topic / Activity  No. of hours  Instructional methods to be used 
1  CO7  Spectral Properties of Bounded Self adjoint linear operators, Further Spectral Properties of Bounded Self Adjoint Linear Operators  8  Lecture, Problem solving 
2  CO7  Positive Operators  5  Lecture, Problem solving 
3  CO7  Projection Operators, Further Properties of Projections.  7  Lecture, Problem solving 
Department  MATHEMATICS 
Name of Faculty  
Programme Name  M.Sc. MATHEMATICS 
Level of study  PG 
Semester  FOURTH 
Course Name/Subject
Name 
ME810402 OPERATIONS RESEARCH 
Total Hours  90 
Course Outcomes
CO
Number 
Description  CO Evaluation methods 
1  Understand Dynamic Programming(DPP) and to use DPP in solving problems  Assignment,Test 
2  Understand and familiarize the theory and various recursive approaches to solving DPP.  Assignment,Test 
3  Learn to solve systems with more than 1 constraint and to apply DPP to continuous systems  Assignment,Test 
4  Understand a continuous time random process or stochastic process  Assignment,Test, Seminar 
5  Learn to analyze the long run/ steady state behavior of a continuous time stochastic process  Assignment, Test, Viva 
6  Understand in detail two commonly used random process birth death process, Poisson Process  Assignment, Test, Seminar 
7  Familiarize General Characteristics of Queueing Systems

Assignment, Viva, Test 
8  Learn to analyze Markovian Queueing systems using the theory of stochastic processes  Assignment, Test 
9  Familiarize some deterministic and probabilistic inventory models

Assignment, Viva, Test 
10  Learn to analyze some simple inventory models using the theory of stochastic process  Assignment,Test, Seminar 
11  Solve the problems using what they study.  Assignment,Test, Seminar 
Module 1 Hours : 25  
Syllabus: Dynamic Programming Introduction , Problem 1 Minimum path problem, Problem 2 Single additive constraint, additively separable return, Problem 3– Single multiplicative constraint, additively separable return, Problem 4 Single additive constraint, multiplicatively separable return, Computational economy in DP , Serial multistage model, Examples of failure ,Decomposition , Backward and forward recursion , Systems with more than one constraints, Applications of D.P to continuous systems


Sl.no  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  1,11  Dynamic Programming & Minimum path problem  3  Lecture 
2  1,11  DPP with single additive constraint, additively separable return  3  Lecture 
3  1,11  DPP with Single multiplicative constraint, additively separable return,  3  Lecture 
4  1,11  DPP with Single additive constraint, multiplicatively separable return,  3  Lecture 
5  2,11  Computational economy in DP &Serial multistage model  6  Lecture 
6  2,11  Examples of failure & Decomposition  2  Lecture 
7  3,11  Systems with more than one constraints & Applications of DP to continuous systems

3  Lecture 
8  2,11  Backward and forward recursion  2  Lecture 
Module 2 Hours : 20  
Syllabus: Continuous time random processes An example, Formal definitions and theory, the assumptions reconsidered, Steady state probabilities, Birth death processes, The Poisson process.  
Slno  CO
Number 
Topic /Activity  No of
hours 
Instructional methods to be used 
1  4,11  Continuous time random processes  4  Lecture, Demonstration 
2  4,11  Formal definitions and theory  4  Lecture 
3  5,11  Steady state probabilities  4  Lecture 
4  6,11  Birth death processes,  4  Lecture 
5  6,11  The Poisson process

4  Lecture 
Module 3 25 hours  
Syllabus: Queueing Systems Introduction, An example, General Characteristics, Performance Measures, Relations Among the performance Measures, Markovian Queueing Models, The M/M/1 Model, Limited Queue Capacity, Multiple Servers, An example, Finite Sources.  
Slno  CO
Number 
Topic/Activity  No of
hours 
Instructional methods to be used 
1  7,11  Queueing Systems – Introduction, General Characteristics,  5  Lecture 
2  7,11  Performance Measures  3  Lecture 
3  8,11  Markovian Queueing Models  6  Lecture 
4  8,11  Markovian Queueing Models with restricted queueing capacity  4  Lecture 
5  8,11  Multi server queueing systems  4  Lecture 
6  8,11  Queueing systems with finite sources  3  Lecture 
Module 4 Hours : 20  
Syllabus: Inventory Models Introduction The classical Economic Order Quantity, A Numerical example, Sensitivity Analysis, Non Zero lead Time, The EOQ with shortages allowed The Production Lot size (PLS) models ,The Newsboy Problem (a single period model) ,A Lot size reorder point model, Variable lead times, The importance of selecting the right model.  
Slno  CO
Number 
Topic/Activity  No of
hours 
Instructional methods to be used 
1  9,11  Inventory Models Introduction  3  Lecture 
2  9,10,11  Economic Order Quantity Model  3  Lecture 
3  9,10,11  The EOQ with shortages allowed  2  Lecture 
4  9,10,11  The Production Lot size (PLS) models  2  Lecture 
5  9,10,11  The Newsboy Problem (a single period model)  3  Lecture 
6  9,10,11  A Lot size reorder point model  3  Lecture 
7  9,11  Variable lead times, The importance of selecting the right model.

4  Lecture 
Department  MATHEMATICS 
Name of Faculty  
Programme Name  M.Sc. Mathematics 
Level of study  PG 
Semester  Four 
Course Name/Subject Name  ME810403 : CODING THEORY 
Total Hours  90 
Course Outcomes
CO Number  Description  CO Evaluation methods 
CO1  Understand the basics of data transmissions over channels, the problem of data loses  Assignments, Seminar, Written Examinations 
CO2  To know about basic definitions in coding of data for proper data transmission  Assignments, Seminar, Written Examinations 
CO3  Familiarize the different coding methods like Golay Codes  Assignments, Seminar, Written Examinations 
CO4  Introduce the construction of a field of 16 elements and using it in BCH codes  Assignments, Seminar, Written Examinations 
CO5  Conceive the idea of finite fields and their importance in coding of data  Assignments, Seminar, Written Examinations 
CO6  To get indepth idea about Cyclic codes and BCH codes  Assignments, Seminar, Written Examinations 
Module 1  Hours: 25  
Syllabus:
Introduction Basic Definitions Weight, Maximum Likelihood decoding, Syndrome decoding, Perfect Codes, Hamming codes, Sphere packing bound, more general facts. 

Sl.No  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO 1  Introduction  4  Lecture 
2  CO 2  Basic Definitions: Weight, Maximum Likelihood decoding, Syndrome decoding  8  Lecture 
3  CO2  Perfect Codes  3  Lecture 
4  CO2  Hamming codes  6  Lectures 
5  CO2  Sphere packing bound, more general facts.  4  Lectures 
Module 2  Hours: 20  
Syllabus:
Self dual codes, The Golay codes, A double error correction BCH code and a field of 16 elements. 

Sl.No  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO3  Selfdual codes  4  Lectures 
2  CO3  The Golay codes  4  Lectures4 
3  CO4  A double error correction BCH code and a field of 16 elements  12  Lectures, Problem solving 
Module 3  Hours: 20  
Syllabus:
Finite fields 

Sl.No  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO5  Finite fields  20  Lectures, Problem solving 
Module 4  Hours: 25  
Syllabus:
Cyclic Codes, BCH codes 

Slno  CO Number  Topic /Activity  No of hours  Instructional methods to be used 
1  CO6  Cyclic Codes  12  Lectures, Problem solving 
2  CO6  BCH codes  13  Lectures, Problem solving 