Mathematics Major Syllabus Gauhati University(G.U): CBCS for Honors: Recently Gauhati University has changed its syllabus and since there has been a lot of confusion about he syllabus. But our team has got the syllabus bought the syllabus fro Mathematics for B.SC students fro Honors. You can download the PDF.

Contents

- 1 Mathematics Major Syllabus Gauhati University:SEMESTER-I
- 2 Mathematics Major Syllabus Gauhati University:SEMESTER-II
- 3 Mathematics Major Syllabus Gauhati University:SEMESTER-III
- 4 Mathematics Major Syllabus Gauhati University:SEMESTER-IV
- 5 Mathematics Major Syllabus Gauhati University:SEMESTER-V
- 6 6th Semester:Mathematics Major Syllabus Gauhati University

## Mathematics Major Syllabus Gauhati University:SEMESTER-I

Gauhati High Court Question paper.

MAT-HC-1016: Calculus

Course Objectives: The primary objective of this course is to introduce the basic tools of calculus and Geometric properties of different conic sections which are helpful in understanding their applications in planetary motion, design of telescope and to the real world problems. Also, computer lab will help to have a deep conceptual understanding of the above tools in true sense.

Course Learning Outcomes: This course will enable the students to:

- i) Learn first and second derivative tests for relative extremum and apply the knowledge in problems in

Business, economics and life sciences.

- ii) Sketch curves in a plane using its mathematical properties in different coordinate systems.

iii) Compute area of surfaces of revolution and the volume of solids by integrating over cross-sectional areas.

- iv) Understand the calculus of vector functions and its use to develop the basic principles of planetary motion.

UNIT 1: Higher order derivatives, Leibnitz rule and its applications to problems of type e

ax+b sinx, e ax+b

cosx, (ax+b)n sinx, (ax+b) n cosx, concavity and inflection points, asymptotes, curve tracing in Cartesian coordinates, tracing in polar coordinates of standard curves, L’Hopital’s rule, applications in business, economics and life sciences.

UNIT 2: Reduction formulae, derivations and illustrations of reduction formulae of the type ∫sin

nx dx, ∫cos n x dx, ∫tan n x dx, ∫sec n x dx, ∫(log x) n dx, ∫sin n x cos m x dx, volumes by slicing, disks and washers methods, volumes by cylindrical shells, parametric equations, parameterizing a curve, arc length, arc length of parametric curves, area of surface of revolution. numerical methods))

UNIT 3: Triple product, introduction to vector functions, operations with vector-valued functions, limits and continuity of vector functions, differentiation and integration of vector functions, tangent and normal components of acceleration, modelling ballistics and planetary motion, Kepler’s second law.

Practical / Lab work to be performed on a computer:

List of the practical to be done using Matlab / Mathematica / Maple / Scilab / Maxima etc.

(i). Plotting the graphs of the following functions: ax x , [ ] (greatest integer function),

ax b ax b c ax b x x n Z + + ± + n n ∈ ± ,| |, | |, , , 1 ( ) ( ) 1 , sin 1 , sin 1 , and , for 0. , log( ), 1/ ( ), sin( ), cos( ), | sin( ) |, | cos( ) |. X ax b xx x x x e x e ax b ax b ax b ax b ax b ax b ± + ≠ ++ + + + +

Observe and discuss the effect of changes in the real constants a, b and c on the graphs.

(ii). Plotting the graphs of polynomial of degree 4 and 5, the graphs of their first and second derivatives, and analysis of these graphs in context of the concepts covered in Unit 1.

(iii). Sketching parametric curves, e.g., Trochoid, Cycloid, Epicycloid and Hypocycloid.

(iv). Tracing of conic in cartesian coordinates.

(v). Obtaining surface of revolution of curves.

(vi). Graph of hyperbolic functions.

(vii). Computation of limit, Differentiation, Integration and sketching of vector-valued functions.

(viii). Complex numbers and their representations, Operations like addition, Multiplication, Division, Modulus,Graphical representation of polar form.

(ix). Find numbers between two real numbers and plotting of finite and infinite subset of R

Course Objectives: The primary objective of this course is to introduce the basic tools of set theory, functions,

induction principle, theory of equations, complex numbers, number theory, matrices and determinant to understand their connection with the real-world problems. Course Learning Outcomes: This course will enable the students to:

- i) Employ De Moivre’s theorem in a number of applications to solve numerical problems.
- ii) Learn about equivalent classes and cardinality of a set.

iii) Use modular arithmetic and basic properties of congruences.

- iv) Recognize consistent and inconsistent systems of linear equations by the row echelon form of the augmented matrix.
- v) Learn about the solution sets of linear systems using matrix method and Cramer’s rule

UNIT-1: Polar representation of complex numbers, nth roots of unity, De Moivre’s theorem for rational indices and its applications.

[1]: Chapter 2UNIT-2: Statements and logic, statements with quantifier, compound statements, implications, proofs in

Mathematic; Sets, operations on sets, family of sets, power sets, Cartesian product; Functions, one-one, onto functions and bijections, Composition of functions, Inverse of a function, Image and Inverse image of subsets; Relation, Equivalence relations, Equivalence classes and partitions of a set, congruence modulo n in integers;Induction Principles, the well-ordering principle, greatest common divisor of integers.

[2] Chapters 1 – 5.UNIT 3: Systems of Linear Equations, row reduction and echelon forms, vector equations, the matrix equation

Ax =b, solution sets of linear systems, linear independence, introduction to linear transformations, the matrix of a linear transformation; Matrix operations, inverse of a matrix, characterizations of invertible matrices; Determinants, Cramer’s rule

[3]: Chapter 1 (Sections 1.1 – 1.9); Chapter 2 (Sections, 2.1 – 1.3); Chapter 3 (Sections 3.1 – 3.3)

GENERIC ELECTIVE PAPERS

MAT-HG-1016/ MAT-RC-1016:Calculus

Course Objectives: Calculus is referred as ‘Mathematics of change’ and is concerned with describing the precise way in which changes in one variable relate to the changes in another. Through this course, students can understand the quantitative change in the behavior of the variables and apply them on the problems related to the environment. Course Learning Outcomes: The students who take this course will be able to:

- i) Understand continuity and differentiability in terms of limits.
- ii) Describe asymptotic behavior in terms of limits involving infinity.

iii) Use derivatives to explore the behavior of a given function, locating and classifying its extrema, and graphing the function.

- iv) Understand the importance of mean value theorems.

Unit 1: Graphs of simple concrete functions such as polynomial, Trigonometric, Inverse trigonometric,

Exponential and logarithmic functions

Unit 2: Limits and continuity of a function including approach, Properties of continuous functions including

Intermediate value theorem.

[2] Chapter 1Unit 3: Differentiability, Successive differentiation, Leibnitz theorem, Recursion formulae for higher derivatives.

Unit 4: Rolle’s theorem, Lagrange’s mean value theorem with geometrical interpretations and simple

applications, Taylor’s theorem, Taylor’s series and Maclaurin’s series, Maclaurin’s series expansion of

functions such as heir use in polynomial approximation and error estimation.

Unit 5: Functions of two or more variables, Graphs and level curves of functions of two variables, Partial differentiation up to second order.

MAT-HG-1026: Analytic Geometry

Course Objectives: The primary objective of this course is to introduce the basic tools of two dimensional coordinate systems, general conics, and three dimensional coordinates systems. Also, introduces the vectors in coordinate systems with geometrical properties Course Learning Outcomes: This course will enable the students to:

- i) Transform coordinate systems, conic sections
- ii) Learn polar equation of a conic, tangent, normal and related properties

iii) Have a rigorous understanding of the concept of three dimensional coordinate systems

- iv) Understand geometrical properties of dot product, cross product of vectors

UNIT 1: Transformation of coordinates, pair of straight lines. Parabola, parametric coordinates, tangent and normal, ellipse and its conjugate diameters with properties, hyperbola and its asymptotes, general conics: tangent, condition of tangency, pole and polar, centre of a conic, equation of pair of tangents, reduction to standard forms, central conics, equation of the axes, and length of the axes, polar equation of a conic, tangent and normal and properties.

UNIT 2: Three-Dimensional Space: Vectors

Rectangular coordinates in 3-space, Spheres and Cylindrical surfaces, Vector viewed geometrically,Vectors in coordinates system, Vectors determine by length and angle, Dot product, Cross product and their geometrical properties, Parametric equations of lines in 2-space and 3-space.

## Mathematics Major Syllabus Gauhati University:SEMESTER-II

MAT-HC-2016: Real Analysis

Course Objectives: The course will develop a deep and rigorous understanding of real line and of defining terms to prove the results about convergence and divergence of sequences and series of real numbers. These concepts have wide range of applications in real life scenario. Course Learning Outcomes: This course will enable the students to:

- i) Understand many properties of the real line R, including completeness and Archimedean properties.
- ii) Learn to define sequences in terms of functions from N to a subset of R.

iii) Recognize bounded, convergent, divergent, Cauchy and monotonic sequences and to calculate their limit superior, limit inferior, and the limit of a bounded sequence.

- iv) Apply the ratio, root, alternating series and limit comparison tests for convergence and absolute convergence of an infinite series of real numbers.

UNIT 1: Algebraic and order properties of R, absolute value and real line, bounded sets, supremum and infimum, completeness property of R, the Archimedean property, the density theorem, intervals, nested interval theorem.

UNIT-2: Real sequences, limit of a sequence, convergent sequence, bounded sequence, limit theorems, monotone sequences, monotone convergence theorem, subsequences, monotone subsequence theorem, Bolzano Weierstrass theorem for sequences, Cauchy sequences, Cauchy’s convergence criterion, properly divergence sequences.

UNIT 3: Infinite series, convergence and divergence of infinite series, Cauchy criterion, Tests for convergence: comparison test, limit comparison test, ratio test, root test, integral test, Absolute convergence, rearrangement theorem, alternating series, Leibniz test, conditional (nonabsolute) convergence.

MAT-HC-2026: Differential Equations(including practical)

Course Objectives: The main objective of this course is to introduce the students to the exciting world of

differential equations, mathematical modeling and their applications.

Course Learning Outcomes: The course will enable the students to:

- i) Learn basics of differential equations and mathematical modeling.
- ii) Formulate differential equations for various mathematical models.

iii) Solve first order non-linear differential equations and linear differential equations of higher order using various techniques.

- iv) Apply these techniques to solve and analyze various mathematical models.

UNIT 1: Differential equations and mathematical models. General, particular, explicit, implicit and singular solutions of a differential equation. Exact differential equations and integrating factors, separable equations and equations reducible to this form, linear equation and Bernoulli equations, special integrating factors and transformations.

(Section 2.3). [3] Chapter 3 (Section 3.3, A and B with Examples 3.8, 3.9)

UNIT 2: Introduction to compartmental model, exponential decay model, exponential growth of population, limited growth of population, limited growth with harvesting.

UNIT 3: General solution of homogeneous equation of second order, principle of super position for homogeneous equation, Wronskian: its properties and applications, Linear homogeneous and non- homogeneous equations of higher order with constant coefficients, Euler’s equation, method of undetermined coefficients, method of variation of parameters.

List of Practical (using any software)

- Plotting of second order solution family of differential equation.
- Plotting of third order solution family of differential equation.
- Growth model (exponential case only).
- Decay model (exponential case only).
- Lake pollution model (with constant/seasonal flow and pollution concentration).
- Case of single cold pill and a course of cold pills.
- Limited growth of population (with and without harvesting).

GENERIC ELECTIVE PAPERS

MAT-HG-2016/MAT-RC-2016: Algebra

Course Objectives: The primary objective of this course is to introduce the basic tools of theory of equations, complex numbers, number theory, matrices, determinant, along with algebraic structures like group, ring and vector space to understand their connection with the real-world problems. Course Learning Outcomes: This course will enable the students to:

- i) Learn how to solve the cubic and biquadratic equations, also learn about symmetric functions of the roots for cubic and biquadratic.
- ii) Employ De Moivre’s theorem in a number of applications to solve numerical problems.

iii) Recognize consistent and inconsistent systems of linear equations by the row echelon form of the augmented matrix. Finding inverse of a matrix with the help of Cayley-Hamilton theorem

- iv) Recognize the mathematical objects that are groups, and classify them as abelian, cyclic and permutation groups, ring etc.
- v) Learn about the concept of linear independence of vectors over a field, and the dimension of a vector space.

Unit 1: Theory of Equations and Expansions of Trigonometric Functions:

Fundamental Theorem of Algebra, Relation between roots and coefficients of nth degree equation, Remainder and Factor Theorem, Solutions of cubic and biquadratic equations, when some conditions on roots of the equation are given, Symmetric functions of the roots for cubic and biquadratic; De Moivre’s theorem (both integral and rational index), Solutions of equations using trigonometry and De Moivre’s theorem, Expansion for in terms of powers of in terms of cosine and sine of multiples of x.

Unit 2: Matrices:

Types of matrices, Rank of a matrix, Invariance of rank under elementary transformations, Reduction to normal form, Solutions of linear homogeneous and nonhomogeneous equations with number of equations and unknowns up to four; Cayley-Hamilton theorem, Characteristic roots and vectors.

Unit 3: Groups, Rings and Vector Spaces:

Integers modulo n, Permutations, Groups, Subgroups, Lagrange’s theorem, Euler’s theorem,Symmetry Groups of a segment of a line, and regular n-gons for n = 3, 4, 5, and 6; Rings and subrings in the context of C[0,1] and Definition and examples of a vector space, Subspace and its properties, Linear independence, Basis and dimension of a vector space.

MAT-HG-2026: Discrete Mathematics

Course Objectives: The course aims at introducing the concepts of ordered sets, lattices, sublattices and homomorphisms between lattices. It also includes introduction to modular and distributive lattices along with complemented lattices and Boolean algebra. Then some important applications of Boolean algebra are discussed in switching circuits. Course Learning outcomes: After the course, the student will be able to:

- i) Understand the notion of ordered sets and maps between ordered sets.
- ii) Learn about lattices, modular and distributive lattices, sublattices and homomorphisms between lattices.

iii) Become familiar with Boolean algebra, Boolean homomorphism, Karnaugh diagrams, switching circuits and their applications.

Unit 1: Ordered Sets

Definitions, Examples and basic properties of ordered sets, Order isomorphism, Hasse diagrams, Dual of an ordered set, Duality principle, Maximal and minimal elements, Building new ordered sets, Maps between ordered sets.

Unit 2: Lattices

Lattices as ordered sets, Lattices as algebraic structures, Sublattices, Products and homomorphisms; Definitions, Examples and properties of modular and distributive lattices, The M3 – N5 Theorem with applications, Complemented lattice, Relatively complemented lattice, Sectionally complemented lattice. homomorphisms.

Unit 3: Boolean Algebras and Switching Circuits

Boolean Algebras, De Morgan’s laws, Boolean homomorphism, Representation theorem; Boolean polynomials, Boolean polynomial functions, Disjunctive normal form and conjunctive normal form, Minimal forms of Boolean polynomial, Quinn-McCluskey method, Karnaugh diagrams, Switching circuits and applications of switching circuits.

## Mathematics Major Syllabus Gauhati University:SEMESTER-III

MAT-HC-3016: Theory of Real Functions

Course Objectives: It is a basic course on the study of real valued functions that would develop an analytical ability to have a more matured perspective of the key concepts of calculus, namely; limits, continuity, differentiability and their applications Course Learning Outcomes: This course will enable the students to:

- i) Have a rigorous understanding of the concept of limit of a function.
- ii) Learn about continuity and uniform continuity of functions defined on intervals.

iii) Understand geometrical properties of continuous functions on closed and bounded intervals.

- iv) Learn extensively about the concept of differentiability using limits, leading to a better understanding for applications.
- v) Know about applications of mean value theorems and Taylor’s theorem.

UNIT 1: Cluster point or limit point of a set, limits of a function (ε-δ approach), sequential criterion for limits, divergence criteria,limit theorems, one sided limits, infinite limits and limits at infinity.

UNIT 2: Continuous functions, sequential criterion for continuity and discontinuity,algebra of continuous

functions,continuous functions on intervals, maximum-minimum theorem, intermediate value theorem, location of roots theorem, preservation of intervals theorem,uniform continuity, uniform continuity theorem.

UNIT 3: Differentiability of a function at a point and in an interval, Caratheodory’s theorem, chain rule,

derivative of inverse function, Rolle’s theorem, mean value theorem, Darboux’s theorem, Cauchy mean value theorem, L’Hospital’s rules, Taylor’s theorem and applications to inequalities, Taylor’s series expansions of exponential and trigonometric functions, ln(1 + x), 1/(ax+b) and n (1+ x) .

MAT-HC-3026: Group Theory – I

Course Objectives: The objective of the course is to introduce the fundamental theory of groups and their homomorphisms. Symmetric groups and group of symmetries are also studied in detail. Fermat’s Little theorem is studied as a consequence of the Lagrange’s theorem on finite groups.

Course Learning Outcomes: The course will enable the students to:

- i) Recognize the mathematical objects that are groups, and classify them as abelian, cyclic and permutation groups, etc.
- ii) Link the fundamental concepts of groups and symmetrical figures.

iii) Analyze the subgroups of cyclic groups and classify subgroups of cyclic groups.

- iv) Explain the significance of the notion of cosets, normal subgroups and factor groups.
- v) Learn about Lagrange’s theorem and Fermat’s Little theorem.
- vi) Know about group homomorphisms and group is omorphisms.

UNIT 1:Symmetries of a square, Dihedral groups, definition and examples of groups including permutation.

groups and quaternion groups (illustration through matrices), elementary properties of groups. Subgroups and examples of subgroups, centralizer, normalizer, center of a group, product of two subgroups. Properties of cyclic groups, classification of subgroups of cyclic groups.

UNIT 2: Cycle notation for permutations, properties of permutations, even and odd permutations, alternating

group, properties of cosets, Lagrange’s theorem and consequences including Fermat’s Little theorem. External direct product of a finite number of groups, normal subgroups, factor groups, Cauchy’s theorem for finite abelian groups.

UNIT 3:Group homomorphisms, properties of homomorphisms, Cayley’s theorem, properties of

Is omorphisms, First, Second and Third isomorphism theorems.

MAT-HC-3036: Analytical Geometry

Course Objectives: The primary objective of this course is to introduce the basic tools of two dimensional coordinates systems, general conics, and three dimensional coordinate systems. Course Learning Outcomes: This course will enable the students to:

- i) Learn conic sections and transform co-ordinate systems.
- ii) Learn polar equation of a conic, tangent, normal and properties.

iii) Have a rigorous understanding of the concept of three dimensional coordinates systems.

UNIT 1: Transformation of coordinates, pair of straight lines. Parabola, parametric coordinates, tangent and normal, ellipse and its conjugate diameters with properties, hyperbola and its asymptotes, general conics: tangent, condition of tangency, pole and polar, centre of a conic, equation of pair of tangents, reduction to standard forms, central conics, equation of the axes, and length of the axes, polar equation of a conic, tangent and normal and properties.

UNIT 2 : Plane, straight lines and shortest distance. Sphere, cone and cylinder, central conicoids, ellipsoid, hyperboloid of one and two sheets, diametral planes, tangent lines, director sphere, polar plane, section with a given centre

MAT-SE-3014: Computer Algebra Systems and Related Software

Course Objectives: This course aims at familiarizing students with the usage of mathematical softwares

(/Mathematica/MATLAB/Maxima/Maple) and the statistical software R. The basic emphasis is on plotting and working with matrices using CAS. Data entry and summary commands will be studied in R. Graphical representation of data shall also be explored. Course Learning Outcomes: This course will enable the students to:

- i) Use of softwares; Mathematica/MATLAB/Maxima/Maple etc. as a calculator, for plotting functions and animations
- ii) Use of CAS for various applications of matrices such as solving system of equations and finding eigenvalues and eigenvectors.

iii) Understand the use of the statistical software R as calculator and learn to read and get data into R.

- iv) Learn the use of R in summary calculation, pictorial representation of data and exploring relationship between data.
- v) Analyze, test, and interpret technical arguments on the basis of geometry

Unit 1: Introduction to CAS and Applications:

Computer Algebra System (CAS), Use of a CAS as a calculator, Computing and plotting functions in 2D, Plotting functions of two variables using Plot3D and Contour Plot, Plotting parametric curves surfaces, Customizing plots, Animating plots, Producing tables of values, working with piecewise defined functions, Combining graphics.

Unit 2: Working with Matrices:

Simple programming in a CAS, Working with matrices, Performing Gauss elimination, operations (transpose, determinant, inverse), Minors and cofactors, Working with large matrices, Solving system of linear equations, Rank and nullity of a matrix, Eigenvalue, eigenvector and diagonalization.

Practical:

Six practicals should be done by each student. The teacher can assign practical from the exercises from [1,2].

MAT-SE-3024: Combinatorics and Graph Theory

Course Objectives: This course aims to provide the basic tools of conuting principles, pigeonhole principle. Also introduce the basic concepts of graphs, Eulerian and Hamiltonian graphs, and applications to dominoes, Diagram tracing puzzles, Knight’s tour problem and Gray codes. Course Learning Outcomes: This course will enable the students to:

- i) Learn about the counting principles, permutations and combinations, Pigeonhole principle
- ii) Understand the basics of graph theory and learn about social networks, Eulerian and Hamiltonian graphs, diagram tracing puzzles and Knight’s tour problem.

Unit 1: Basic counting principles, Permutations and combinations, the inclusion-exclusion principle, Pigeonhole principle.

Unit 2: Graphs, Diagraphs, Networks and subgraphs, Vertex degree, Paths and cycles, Regular and bipartite graphs, Four cube problem, Social networks, Exploring and travelling, Eulerian and Hamiltonian graphs, Applications to dominoes, Diagram tracing puzzles, Knight’s tour problem, Gray codes.

GENERIC ELECTIVE PAPERS

MAT-HG-3016/MAT-RC-3016: Differential Equations

Course Objectives: The main objective of this course is to introduce the students to the exciting world of

ordinary differential equations, mathematical modeling and their applications.

Course Learning Outcomes: The course will enable the students to:

- i) Learn basics of differential equations and mathematical modeling.
- ii) Solve first order non-linear differential equations and linear differential equations of higher order using various techniques.

Unit 1: First Order Ordinary Differential Equations

First order exact differential equations, Integrating factors, Rules to find an integrating factor

Linear equations and Bernoulli equations, Orthogonal trajectories and oblique trajectories; Basictheory of higher order linear differential equations, Wronskian, and its properties; Solving differential equation by reducing its order.

Unit 2: Second Order Linear Differential Equations

Linear homogenous equations with constant coefficients, Linear non-homogenous equations, The method of variation of parameters, The Cauchy-Euler equation; Simultaneous differential equations.

MAT-HG-3026: Linear Programming

Course Objectives: This course develops the ideas underlying the Simplex method. The course covers Linear Programming problems with applications to transportation, assignment and game problem. Such problems arise in manufacturing resource planning and financial sectors. Course Learning Outcomes: This course will enable the students to:

- i) Learn about the graphical solution of linear programming problem with two variables.
- ii) Learn about the relation between basic feasible solutions and extreme points.

iii) Understand the theory of the simplex method used to solve linear programming problems.

- iv) Learn about two-phase and big-M methods to deal with problems involving artificial variables.
- v) Learn about the relationships between the primal and dual problems.
- vi) Solve transportation and assignment problems.

vii) Apply linear programming method to solve two-person zero-sum game problems.

Unit 1: The Linear Programming Problem: Standard, Canonical and matrix forms, Graphical solution.

Hyperplanes, Extreme points, Convex and polyhedral sets. Basic solutions; Basic Feasible Solutions; Reduction of any feasible solution to a basic feasible solution; Correspondence between basic feasible solutions and extreme points.

Unit 2: Simplex Method: Optimal solution, Termination criteria for optimal solution of the Linear

Programming Problem, Unique and alternate optimal solutions, Unboundedness; Simplex Algorithm and its Tableau Format; Artificial variables, Two-phase method, Big-M method.

Unit 3: Motivation and Formulation of Dual problem; Primal-Dual relationships; Fundamental Theorem of Duality; Complimentary Slackness.

Unit 4: Applications

Transportation Problem: Definition and formulation; Methods of finding initial basic feasible solutions; North West corner rule. Least cost method; Vogel’s Approximation method; Algorithm for solving Transportation

Problem.

Assignment Problem: Mathematical formulation and Hungarian method of solving. Game Theory: Basic concept, Formulation and solution of two-person zero-sum games, Games with mixed strategies, Linear Programming method of solving a game.

## Mathematics Major Syllabus Gauhati University:SEMESTER-IV

MAT-HC-4016: Multivariate Calculus

Course Objectives: To understand the extension of the studies of single variable differential and integral calculus to functions of two or more independent variables. Also, the emphasis will be on the use of Computer Algebra Systems by which these concepts may be analyzed and visualized to have a better understanding. This course will facilitate to become aware of applications of multivariable calculus tools in physics, economics, optimization, and understanding the architecture of curves and surfaces in plane and space etc. Course Learning Outcomes: This course will enable the students to:

- i) Learn the conceptual variations when advancing in calculus from one variable to multivariable discussion.
- ii) Understand the maximization and minimization of multivariable functions subject to the given constraints

iii) Learn about inter-relationship amongst the line integral, double and triple integral formulations.

- iv) Familiarize with Green’s, Stokes’ and Gauss divergence theorems

UNIT 1:Functions of several variables, Level curves and surfaces, Limits and continuity, Partial differentiation, Higher order partial derivative, Tangent planes, Total differential and differentiability, Chain rule, Directional derivatives, The gradient, Maximal and normal property of the gradient, Tangent planes and normal lines.

UNIT 2:Extrema of functions of two variables, Method of Lagrange multipliers, Constrained

optimization problems; Definition of vector field, Divergence and curl.

UNIT 3:Double integration over rectangular and nonrectangular regions, Double integrals in polar coordinates, Triple integral over a parallelepiped and solid regions, Volume by triple integrals,triple integration in cylindrical and spherical coordinates, Change of variables in doubleandtriple integrals.

UNIT 4: Line integrals, Applications of line integrals: Mass and Work, Fundamental theorem for line integrals, Conservative vector fields, Green’s theorem, Area as a line integral; Surface integrals, Stokes’ theorem, The Gauss divergence theorem.

MAT-HC-4026: Numerical Methods (including practical)

Course Objectives: To comprehend various computational techniques to find approximate value for possible root(s) of non-algebraic equations and to find the approximate solutions of system of linear equations and ordinary differential equations. Also, use of Computer Algebra System (CAS) by which the numerical problems can be solved both numerically and analytically, and to enhance the problem solving skills. Course Learning Outcomes: The course will enable the students to:

- i) Learn some numerical methods to find the zeroes of nonlinear functions of a single variable and solution of a system of linear equations, up to a certain given level of precision.
- ii) Know about methods to solve system of linear equations, such as False position method, Fixed point iteration method, Newton’s method, Secant method and LU decomposition.

iii) Interpolation techniques to compute the values for a tabulated function at points not in the table.

- iv) Applications of numerical differentiation and integration to convert differential equations into difference equations for numerical solutions.

UNIT 1:Algorithms, Convergence, Bisection method, False position method, Fixed point iteration method, Newton’s method, Secant method, LU decomposition.

UNIT 2:Lagrange and Newton interpolation: linear and higher order, finite difference operators.

UNIT 3:Numerical differentiation: forward difference, backward difference and central difference. Integration:trapezoidal rule, Simpson’s rule, Euler’s method. Note: Emphasis is to be laid on the algorithms of the above numerical methods.

Practical / Lab work to be performed on a computer:

Use of computer aided software (CAS), for example Matlab / Mathematica / Maple / Maxima etc., for

developing the following Numerical programs:

(i) Calculate the sum 1/1 + 1/2 + 1/3 + 1/4 + ———-+ 1/ N.

(ii) To find the absolute value of an integer.

(iii) Enter 100 integers into an array and sort them in an ascending order.

(iv) Any two of the following

(a) Bisection Method

(b) Newton Raphson Method

(c) Secant Method

(d) RegulaiFalsi Method

(v) LU decomposition Method

(vi) Gauss-Jacobi Method

(vii) SOR Method or Gauss-Siedel Method

(viii) Lagrange Interpolation or Newton Interpolation

(ix) Simpson’s rule.

Note: For any of the CAS Matlab / Mathematica / Maple / Maxima etc., Data types-simple data types, floating data types, character data types, arithmetic operators and operator precedence, variables and constant declarations, expressions, input/output, relational operators, logical operators and logical expressions, control statements and loop statements, Arrays should be introduced to the students.

MAT-HC-4036: Ring Theory

Course Objectives: The objective of this course is to introduce the fundamental theory of rings and their

corresponding homomorphisms. Also introduces the basic concepts of ring of polynomials and irreducibility tests for polynomials over ring of integers. Courses Learning Outcomes: On completion of this course, the student will be able to:

- i) Appreciate the significance of unique factorization in rings and integral domains.
- ii) Learn about the fundamental concept of rings, integral domains and fields.

iii) Know about ring homomorphism and isomorphism theorems of rings.

- iv) Learn about the polynomial rings over commutative rings, integral domains, Euclidean domains, and UFD.

UNIT 1: Definition and examples of rings, properties of rings, subrings, integral domains and fields,

characteristic of a ring. Ideals, ideal generated by a subset of a ring, factor rings, operations on ideals, prime and maximal ideals. Ring homomorphisms, properties of ring homomorphisms, Isomorphism theorems I, II and III, field of quotients.

UNIT 2: Polynomial rings over commutative rings, division algorithm and consequences, principal ideal

domains, factorization of polynomials, reducibility tests, irreducibility tests, Eisenstein criterion, unique

factorization in Z[x]. Divisibility in integral domains, irreducibles, primes, unique factorization domains,

Euclidean domains.

MAT-SE-4014: R Programming

Total marks: 100 (Theory 60, Internal assessment 20, Practical 20)

Per week: 2 Lectures 2 Practical, Credits 4(2+2)

Course Objectives: The purpose of this course is to help using R, a powerful free software program for doing statistical computing and graphics. It can be used for exploring and plotting data, as well as performing statistical tests.

Course Learning Outcomes: This course will enable the students to:

- i) Become familiar with R syntax and to use R as a calculator.
- ii) Understand the concepts of objects, vectors and data types.

iii) Know about summary commands and summary table in R.

- iv) Visualize distribution of data in R and learn about normality test.
- v) Plot various graphs and charts using R.

MAT-HG-4026: Numerical Analysis

Course Objectives: To comprehend various computational techniques to find approximate value for possible root(s) of non-algebraic equations, to find the approximate solutions of system of linear equations and Quadratic equations.

Course Learning Outcomes: The course will enable the students to:

- i) Learn some numerical methods to find the zeroes of nonlinear functions of a single variable and solution of a system of linear equations, up to a certain given level of precision.
- ii) Know about iterative and non-iterative methods to solve system of linear equations.

iii) Know interpolation techniques to compute the values for a tabulated function at points not in the table.

- iv) Integrate a definite integral that cannot be done analytically.
- v) Find numerical differentiation of functional values.
- vi) Solve differential equations that cannot be solved by analytical methods.

Unit 1: Gaussian elimination method (with row pivoting), Gauss-Jordan method; Iterative methods:Jacobian.method, Gauss-Seidel method; Interpolation: Lagrange form, Newton form, Finite difference operators,Gregory-Newton forward and backward difference interpolations, Piecewise polynomial interpolation (Linearand Quadratic).

Unit 2:Numerical differentiation: First and second order derivatives; Numerical integration: Trapezoid rule, Simpson’s rule; Extrapolation methods: Richardson extrapolation, Romberg integration;Ordinary differentialequation: Euler’s method, Modified Euler’s methods (Heun and Mid-point).

## Mathematics Major Syllabus Gauhati University:SEMESTER-V

MAT-HC-5016: Riemann Integration and Metric spaces

Course Objectives: To understand the integration of bounded functions on a closed and bounded interval and its extension to the cases where either the interval of integration is infinite, or the integrand has infinite limits at a finite number of points on the interval of integration. Up to this stage, students do study the concepts of analysis which evidently rely on the notion of distance. In this course, the objective is to develop the usual idea of distance into an abstract form on any set of objects, maintaining its inherent characteristics, and the resulting consequences.

Course Learning Outcomes: The course will enable the students to:

- i) Learn about some of the classes and properties of Riemann integrable functions, and the applications of the Fundamental theorems of integration.
- ii) Know about improper integrals including, beta and gamma functions.

iii) Learn various natural and abstract formulations of distance on the sets of usual or unusual entities. Become aware one such formulations leading to metric spaces.

- iv) Analyse how a theory advances from a particular frame to a general frame.
- v) Appreciate the mathematical understanding of various geometrical concepts, viz. Balls or connected sets etc. in an abstract setting.
- vi) Know about Banach fixed point theorem, whose far-reaching consequences have resulted into an

Independent branch of study in analysis, known as fixed point theory.

vii) Learn about the two important topological properties, namely connectedness and compactness of metric spaces.

UNIT 1: Riemann integration: upper and lower sums; Darbouxintegrability, properties of integral, Fundamental theorem of calculus, mean value theorems for integrals, Riemann sum and Riemann integrability, Riemann integrability of monotone and continuous functions on intervals, sum of infinite series as Riemann integrals, logarithm and exponential functions through Riemann integrals, improper integrals, Gamma functions.

UNIT 2: Metric spaces: definition and examples, sequences in metric spaces, Cauchy sequences, complete metric spaces. Open and closed balls, neighbourhood, open set, interior of a set. Limit point of a set, closed set, diameter of a set, Cantor’s theorem. Subspaces, dense sets, separable spaces.

UNIT 3:Continuous mappings, sequential criterion and other characterizations of continuity. Uniform continuity. Homeomorphism, Contraction mappings, Banach contraction mapping principle. Connectedness, connected subsets of R, connectedness and continuous mappings.

MAT-HC-5026: Linear Algebra

Course Objectives: The objective of this course is to introduce the fundamental theory of vector spaces, also emphasizes the application of techniques using the adjoint of a linear operator and their properties to least squares approximation and minimal solutions to systems of linear equations. Course Learning Outcomes: The course will enable the students to:

- i) Learn about the concept of linear independence of vectors over a field, and the dimension of a vector space.
- ii) Basic concepts of linear transformations, dimension theorem, matrix representation of a linear

transformation, and the change of coordinate matrix.

iii) Compute the characteristic polynomial, eigenvalues, eigenvectors, and eigenspaces, as well as the geometric and the algebraic multiplicities of an eigenvalue and apply the basic diagonalization result.

- iv) Compute inner products and determine orthogonality on vector spaces, including Gram−Schmidt orthogonalization to obtain orthonormal basis.
- v) Find the adjoint, normal, unitary and orthogonal operators.

Unit 1: Vector spaces and subspaces, null space and column space of a matrix, linear transformations, kernel and range, linearly independent sets, bases, coordinate systems, dimension of a vector space, rank, change of basis.

Unit 2: Eigenvectors and eigenvalues of a matrix, the characteristic equation, diagonalization, eigenvectors of a linear transformation, complex eigenvalues,

Invariant subspaces and Cayley-Hamilton theorem.

Unit 3: Inner product, length, and orthogonality, orthogonal sets, orthogonal projections, the Gram–Schmidt process, inner product spaces; Diagonalization of symmetric matrices,the Spectral Theorem..

DISCIPLINE SPECIFIC ELECTIVE PAPERS

distributions, Expectation of function of two random variables, Joint momentgenerating function, Conditional distributions and expectations.

UNIT-4: The Correlation coefficient, Covariance, Calculation of covariance from joint moment generating function, Independent random variables, Linear regression for two variables, The method of least squares, Bivariate normal distribution, Chebyshev’s theorem, Strong law of large numbers, Central limit theorem and weak law of large numbers.

MAT-HE-5046: Linear Programming

Course Objectives: This course develops the ideas underlying the Simplex Method for Linear Programming Problem, as an important branch of Operations Research. The course covers Linear rogramming with applications to transportation, assignment and game problem. Such problems arise in manufacturing resource planning and financial sectors. Course Learning Outcomes: This course will enable the students to:

- i) Learn about the graphical solution of linear programming problem with two variables.
- ii) Learn about the relation between basic feasible solutions and extreme points.

iii) Understand the theory of the simplex method used to solve linear programming problems.

- iv) Learn about two-phase and big-M methods to deal with problems involving artificial variables.
- v) Learn about the relationships between the primal and dual problems.
- vi) Solve transportation and assignment problems.

vii) Apply linear programming method to solve two-person zero-sum game problems.

Unit 1: The Linear Programming Problem: Standard, Canonical and matrix forms, Graphical solution.

Hyperplanes, Extreme points, Convex and polyhedral sets. Basic solutions; Basic Feasible Solutions; Reductionof any feasible solution to a basic feasible solution; Correspondence between basic feasible solutions and extreme points.

Unit 2: Simplex Method: Optimal solution, Termination criteria for optimal solution of the Linear

Programming Problem, Unique and alternate optimal solutions, Unboundedness; Simplex Algorithm and its Tableau Format; Artificial variables, Two-phase method, Big-M method.

Unit 3: Motivation and Formulation of Dual problem; Primal-Dual relationships; Fundamental Theorem of Duality; Complimentary Slackness.

Unit 4: Applications

Transportation Problem: Definition and formulation; Methods of finding initial basic feasible solutions; North West corner rule. Least cost method; Vogel’s Approximation method; Algorithm for solving Transportation

Problem.

Assignment Problem: Mathematical formulation and Hungarian method of solving.

Game Theory: Basic concept, Formulation and solution of two-person zero-sum games, Games with mixed strategies, Linear Programming method of solving a game.

MAT-HE-5056: Spherical Trigonometry and Astronomy

Course Objectives: This main objective of this course is to provide the spherical triangles, Napier’s rule of

circular parts and Planetary motion Course Learning Outcomes: This course will enable the students to:

- i) Learn about the properties of spherical and polar triangles
- ii) know about fundamental formulae of spherical triangles

iii) learn about the celestial sphere, circumpolar star, rate of change of zenith distance and azimuth

- iv) learn about Keplar’s law of planetary motion, Cassini’s hypothesis, differential equation for fraction

Unit 1: Section of a sphere by a plane, spherical triangles, properties of spherical and polar triangles, fundamental formulae of spherical triangles, sine formula, cosine formula, sine-cosine formula, cot formula, Napier’s rule of circular parts.

Unit2: The standard (or geometric) celestial sphere, system of coordinates, conversion of one coordinate

system to the another system, diurnal motion of heavenly bodies, sidereal time, solar time(mean), rising and setting of stars, circumpolar star, dip of the horizon, rate of change of zenith distance and azimuth, examples.

Unit3: Planetary motion: annual motion of the sun, planetary motion, synodic period, orbital period, Keplar’s law of planetary motion, deduction of Keplar’s law from Newton’s law of gravitation, the equation of the orbit, velocity of a planet in its orbit,components of linear velocity perpendicular to the radius vector and to the major axis, direct and retrograde motion in a plane, laws of refraction: refraction for small zenith distance, generalformula for refraction, Cassini’s hypothesis, differential equation for fraction, effect of refraction on sunrise,

sunset, right ascension and declination, shape of the disc of the sun.

MAT-HE-5066: Programming in C (including practical)

Course Objectives: This course introduces C programming in the idiom and context of mathematics and imparts a starting orientation using available mathematical libraries, and their applications. Course Learning Outcomes: After completion of this paper, student will be able to:

- i) Understand and apply the programming concepts of C which is important to mathematical investigation and problem solving.
- ii) Learn about structured data-types in C and learn about applications in factorization of an integer and understanding Cartesian geometry and Pythagorean triples.

iii) Use of containers and templates in various applications in algebra.

- iv) Use mathematical libraries for computational objectives.
- v) Represent the outputs of programs visually in terms of well formatted text and plots.

Unit 1: Variables, constants, reserved words, variable declaration, initialization, basic data types, operators and

expression (arithmetic, relational, logical, assignment, conditional, increment and decrement), hierarchy of operations for arithmetic operators, size of and comma operator, mixed mode operation and automatic (implicit) conversion, cast (explicit) conversion, library functions, structure of a C program, input/output functions and statements.

Unit 2:Control Statements: if-else statement (including nested if-else statement), switch statement. Loop control Structures (for and nested for, while and do-while). Break, continue, go to statements, exit function.

Unit 3: Arrays and subscripted variables: One and Two dimensional array declaration, accessing values in an array, initializing values in an array, sorting of numbers in an array, addition and multiplication of matrices with the help of array. Functions: function declaration, actual and formal arguments, function prototype, calling a function by value, recursive function.

Programmes for practical:

To find roots of a quadratic equation, value of a piecewise defined function (single variable), factorial of a given positive integer, Fibonacci numbers, square root of a number, cube root of a number, sum of different algebraic and trigonometric series, a given number to be prime or not, sum of the digits of any given positive integer, solution of an equation using N-R algorithm, reversing digits of an integer. Sorting of numbers in an array, to find addition, subtraction and multiplication of matrices. To find sin(x), cos(x) with the help of

functions.

## 6th Semester:Mathematics Major Syllabus Gauhati University

Unit 1: Ordered Sets

Definitions, Examples and basic properties of ordered sets, Order isomorphism, Hasse diagrams, Dual of an ordered set, Duality principle, Maximal and minimal elements, Building new ordered sets, Maps between ordered sets.

Unit 2: Lattices

Lattices as ordered sets, Lattices as algebraic structures, Sublattices, Products and homomorphisms; Definitions, Examples and properties of modular and distributive lattices, The M3 – N5 Theorem with applications, Complemented lattice, Relatively complemented lattice, Sectionally complemented lattice.

Unit 3: Boolean Algebras and Switching Circuits

Boolean Algebras, De Morgan’s laws, Boolean homomorphism, Representation theorem;

Boolean polynomials, Boolean polynomial functions, Disjunctive normal form and conjunctive normal form, Minimal forms of Boolean polynomial, Quinn-McCluskey method, Karnaugh diagrams, Switching circuits and applications of switching circuits.

Unit 4: Introduction: Alphabets, strings, and languages. Finite Automata and Regular Languages: deterministic and non-deterministic finite automata, regular expressions, regular languages and their relationship with finite automata, pumping lemma and closure properties of regular languages. Context Free Grammars and Pushdown Automata: Context free grammars (CFG), parse trees, ambiguities in grammars and languages, pushdown automaton (PDA) and the language accepted by PDA, deterministic PDA, Non- deterministic PDA, properties of context free languages; normal forms, pumping lemma, closure properties, decision properties.

MAT-HE-6026: Bio-Mathematics

Course Objectives: The focus of the course is on scientific study of normal functions related to living systems. The emphasis is on exposure to nonlinear differential equations with examples such as heartbeat, chemical reactions and nerve impulse transmission. The basic concepts of the probability to understand molecular evolution and genetics have also been applied. Course Learning outcomes: Apropos conclusion of the course will empower the student to:

- i) Learn the development, analysis and interpretation of bio mathematical models such as population growth, cell division, and predator-prey models.
- ii) Learn about the mathematics behind heartbeat model and nerve impulse transmission model.

iii) Appreciate the theory of bifurcation and chaos.

- iv) Learn to apply the basic concepts of probability to molecular evolution and genetics.

Unit 1: Basic concepts and definitions, Mathematical model, properly posed mathematical problems, System of differential equation, Existence theorems, Homogeneous linear systems, Non-homogeneous linear systems, Linear systems with constant coefficients, Eigenvalues and eigenvectors, Linear equation with periodic coefficients. Population growth model, administration of drug and epidemics, Cell division Predator Prey Model, Chemical reactions and enzymatic catalysis.

Unit 2: Stability and Modeling of Biological phenomenon

The Phase Plane, Local Stability, Autonomous Systems, Stability of Linear Autonomous Systems with

Constant Coefficients, Linear Plane Autonomous Systems, Method of Lyapunov for Non–Linear Systems, Limit Cycles, Forced Oscillations. Mathematics of Heart Physiology: The local model, The Threshold effect, The phase plane analysis and the Heart beat model, Physiological considerations of the Heart beat model, A model of the Cardiac pace-maker. Mathematics of Nerve impulse transmission: Excitability & repetitive firing, Travelling waves.

Unit 3: Bifurcation and Chaos

Bifurcation and chaos: Bifurcation, Bifurcation of a limit cycle, Discrete bifurcation, Chaos, Stability, The Poincare plane.

Unit 4: Modelling Molecular Evolution and Genetics

ModellingMolecularEvolution: Matrixmodels of base substitutionsfor DNAsequences, The Jukes-Cantor Model, the Kimura Models, Phylogeneticdistances. Constructing Phylogenetictrees:Unweightedpair-groupmethodwitharithmeticmeans (UPGMA), Neighbour- Joining Method, Maximum Likelihood approaches. Genetics:Mendelian Genetics, Probability distributions in Genetics, Linked genes and

MAT-HE-6036: Mathematical Modelling (including practical)

Course Objectives: The main objective of this course is to teach students how to model physical problems using differential equations and solve them. Also, the use of Computer Algebra Systems (CAS) by which the listed problems can be solved both numerically and analytically. Course Learning Outcomes: The course will enable the students to:

- i) Know about power series solution of a differential equation and learn about Legendre’s and Bessel’s

equations.

- ii) Use of Laplace transform and inverse transform for solving initial value problems.

iii) Learn about various models such as Monte Carlo simulation models, queuing models, and linear

programming models.

Unit 1: Power series solution of a differential equation about an ordinary point, Solution about a regular

singular point, The method of Frobenius; Legendre’s and Bessel’s equation.

Unit 2: Laplace transform and inverse transform, application to initial value problem up to second order.

Unit 3: Monte Carlo Simulation Modelling: Simulating deterministic behaviour (area under a curve, volume under a surface); Generating Random Numbers: Middle square method, Linear congruence; Queuing Models: Harbor system, Morning rush hour.

Practical / Lab work to be performed in Computer Lab:

Modelling of the following problems using Mathematica/MATLAB/Maple /Maxima/Scilab etc.

(i) Plotting of Legendre polynomial for n = 1 to 5 in the interval [0, 1]. Verifying graphically that all the roots of Pn(x) lie in the interval [0, 1].

(ii) Automatic computation of coefficients in the series solution near ordinary points.

(iii) Plotting of the Bessel’s function of first kind of order 0 to 3.

(iv) Automating the Frobenius Series Method.

(v) Random number generation and then use it for one of the following:

(a) Simulate area under a curve. (b) Simulate volume under a surface.

(vi) Programming of either one of the queuing model:

(a) Single server queue (e.g. Harbor system). (b) Multiple server queue (e.g. Rush hour).

(vii) Programming of the Simplex method for 2 / 3 variables

MAT-HE-6046: Hydromechanics

Course Objectives: The main objectives of this course are to teach students about fluid pressure on plane surfaces, curved surfaces and Gas law. Also, introduces velocity of a fluid at a point, Eulerian and Lagrangian method, velocity potential and acceleration of a fluid at a point. Course Learning Outcomes: The course will enable the students to:

- i) Know about Pressure equation, rotating fluids.
- ii) Learn about Fluid pressure on plane surfaces, resultant pressure on curved surfaces, Gas law, mixture of gases.

iii) Learn about the Eulerian and Lagrangian method.

- iv) Learn about equation of continuity, examples, acceleration of a fluid at a point

Unit 1: Hydrostatics

Pressure equation, condition of equilibrium, lines of force, homogeneous and heterogeneous fluids, elastic fluids, surface of equal pressure, fluid at rest under action of gravity, rotating fluids. Fluid pressure on plane surfaces, centre of pressure, resultant pressure on curved surfaces. Gas law, mixture of gases, internal energy, adiabatic expansion.

Unit 2 Hydrodynamics

Real and ideal fluid, velocity of a fluid at a point, Eulerian and Lagrangian method, stream lines and path lines, steady and unsteady flows, velocity potential, rotational and irrotational motions, material local, convective derivatives, local and particle rate of change, equation of continuity, examples, acceleration of a fluid at a point.

MAT-HE-6056: Rigid Dynamics

Course Objectives: The main objectives of this course is to introduce moments and products of inertia,

theorem of six constants, D’Alembert’s principle, Motion of a body in two dimension and Lagrange’s

equations. Course Learning Outcomes: The course will enable the students to:

- i) Know how to find the moments and products of inertia.
- ii) Learn about the motion of the centre of inertia

iii) Learn about the D’Alembert’s principle and Lagrange’s equations

- iv) Learn about motion of a body in two dimension

Unit1: Moments and products of inertia, parallel axes theorem, theorem of six constants, the momental

ellipsoid, equimomental systems, principle axes.

Unit2: D’Alembert’s principle, the general equation of motion of a rigid body, motion of the centre of inertia and motion relative to the centre of inertia.

Unit3: Motion about a fixed axis, the compound pendulum, centre of percussion. Motion of a body in two dimension under finite and impulsive forces.

Unit4: Conservation of momentum and energy, generalized coordinates, Lagrange’s equations, initial motions.

MAT-HE-6066: Group Theory II

Course Objectives: The course will develop an indepth understanding of one of the most important branch of the abstract algebra with applications to practical real-world problems. Classification of all finite abelian groups (up to isomorphism) can be done.

Course Learning Outcomes: The course shall enable students to:

- i) Learn about automorphisms for constructing new groups from the given group.
- ii) Learn about the fact that external direct product applies to data security and electric circuits.

iii) Understand fundamental theorem of finite abelian groups.

- iv) Be familiar with group actions and conjugacy in Sn.
- v) Understand Sylow theorems and their applications in checking non-simplicity.

Unit 1: Isomorphismsm, automorphisms, inner automorphisms, Automorphisms groups; External direct

products of groups and their properties; the group of units modulo n as an external direct product

Unit 2:Normal subgroups, factor groups and their applications, Internal direct products, of subgroups,

Fundamental theorem of finite Abelian groups, isomorphism classes of finite abelian groups.

Unit 3: Conjugacy classes, The class equation, Conjugacy classes in the symmetric group Sn, p-groups, The Sylow theorems and their applications.

Unit 4: Finite simple groups, nonsimplicity tests; Generalized Cayley’s theorem, Index theorem, Embedding theorem and applications. Simplicity of A5.

MAT-HE-6076: Mathematical Finance

Course Objectives: This course is an introduction to the application of mathematics in financial world, that

enables the student to understand some computational and quantitative techniques required for working in the financial markets and actuarial mathematics.

Course Learning outcomes: On completion of this course, the student will be able to:

- i) Know the basics of financial markets and derivatives including options and futures.
- ii) Learn about pricing and hedging of options, as well as interest rate swaps.

iii) Learn about no-arbitrage pricing concept and types of options.

- iv) Learn stochastic analysis (Ito formula, Ito integration) and the Black−Scholes model.
- v) Understand the concepts of trading strategies and valuation of currency swaps.

Unit 1: Interest Rates: Types of rates, Measuring interest rates, Zero rates, Bond pricing, Forward

rate,Duration, Convexity, Exchange traded markets and OTC markets, Derivatives–Forward

contracts, Futures contract, Options, Types of traders, Hedging, Speculation, Arbitrage.

Unit 2: Mechanics and Properties of Options: No Arbitrage principle, Short selling, Forward price for an

investment asset, Types of Options,Option positions, Underlying assets, Factors affecting option prices, Bounds on option prices,Put-call parity, Early exercise, Effect of dividends.

Unit 3: Stochastic Analysis of Stock Prices and Black-Scholes Model

Binomial option pricing model, Risk neutral valuation (for European and American options on assets following binomial tree model), Lognormal property of stock prices, Distribution of rate of return, expected return, Volatility, estimating volatility from historical data, Extension of risk neutral valuation to assets following GBM, Black-Scholes formula for European options.

Unit 4: Hedging Parameters, Trading Strategies and Swaps

Hedging parameters (the Greeks: Delta, Gamma, Theta, Rho and Vega), Trading strategies involving options, Swaps, Mechanics of interest rate swaps, Comparative advantage argument, Valuation of interest rate swaps, Currency swaps, Valuation of currency swaps.