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COURSES OF STUDIES

Courses of Studies ( Choice Based Credit System)B.Sc. (Hons.) Mathematics

CORE COURSES

B.Sc.(Honours)-Mathematics

CREDIT : 06 each

KHALLIKOTE CLUSTER UNIVERSITY

BERHAMPUR,GANJAM, ODISHA-760001

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Semester – I

C-1.1: Calculus –I(Total Marks; 80+20)

5 Lectures, 1 Tutorial per week Unit-I

Curvature, Asymptotes, Singular points and Curve tracing. Unit-II

Rectification, Quadrature, Volume and surface area of the solid surface of revolution. Unit-III

Vector calculus, Limits, continuity and differentiability of vector function, Point functions (scalar and vector), Directional derivative, gradient and curl of point functions and their properties. Unit-IV

Sphere, Cone and Cylinder. Books Prescribed

1. Elementary Calculus by Panda and Satapathy. Chapters: 1,2,5,6.

2. Analytical Geometry of Quadratic Surfaces by B.P.Acharya and D.P.Sahu, Kalyani Publishers Chapters: 2, 3 Books for reference

1. Text book of Calculus, Part-II by Shantinarayan, S Chand & Co.2. Text book of Calculus, Part-III by Shantinarayan, S Chand & Co.3. Calculus by G.B.Thomas, Pearson Education, Delhi

4. Analytical Solid Geometry by S.Narayan and Mittal, S.Chand Co.

C-1.2: (Abstract Algebra) Algebra –I(Total Marks; 80+20)


Group theory: Definition of a group, Some examples of groups, Some preliminaries lemmas, Subgroups, A counting principle, Normal subgroups and quotient groups, Homomorphism. Unit-II

Ring theory: Definition and examples of rings, some special class of rings, Homomorphism, Ideals and quotient rings. Unit-III

Theory of equation: Preliminary, Properties of equations, Descartes’ rule of signs, relation between roots and coefficients, Transformation of equations, Multiple roots, Sums of powers of roots, Reciprocal equations Unit-IV

Cubic and Biquadratic equations: Solution of the cubic, Nature of the roots of a cubic, Expressing the cubic as a difference of two cubes, Solution by Symmetric functions of roots, Solution of the bi-quadratic, solution by redicals. Numerical solution of equations: Numerical equations, limits of the roots of equations, Integer roots, Newton’s method of approximation, Horner’s method, Books Prescribed

1.Topics in algebra by I.N.Herstein, Wiley Eastern Limited, India, 1975 Chapters:2(up to 2.7), 3(up to 3.4) 2.Text book of algebra and theory of equation( Tenth edition) by Chandrika Prasad, Pothishala Private limited.Chapters: XI, XII, XII . Books for reference

1. A first course in Abstract Algebra by J.B.Fraleigh, Pearson, 2002.2. An Introduction to theory of Groups by J.J.Rotman, Springer Verlag.3. Contemporary Abstract Algebra by J.A.Gallian, Narosa Pub. House.

Semester – II

C-2.1: Analysis –I(Total Marks; 80+20)


Field structure and order structure, Bounded and unbounded sets, (excluding Dedikinds form of completeness property), completeness in the set of real numbers, Absolute value of a real number. Neighborhood of a point, Interior point, Limit point, Open set, Closed set, Dense set, Perfect set, Bolzano-Weierstrass’s theorem, Countable and Uncountable sets. Unit-II

Sequences, Limit points f a sequence, Limit inferior and superior, Convergent sequence, Non-convergent sequence, Cauchy’s general principle of convergence. Algebra of sequences, Some important theorems, Monotonic sequence. Unit-III

Infinite series, Positive term series, Comparison test for positive term series, Cauchy’s root test, D’Alemberts root test, Raabe’s test, Logarithemic test, Integral test, Series with arbitrary terms, Rearrangement of the terms. Unit-IV

Power series and Fourier series.

Books Prescribed

Mathematical Analysis by S.C.Malik and Savita Arora, New-Age Pvt. Ltd.Chapters: 1,2,3,4,13,14. Books for reference

1. Fundamentals of Real Analysis by S.L.Gupta and Nisha Rani.2. Fundamentals of Mathematical Analysis by G.Das and S.Pattanayak.3. Elementary Analysis: The theory of calculus by K.A.Ross, Springer.4. Introduction to Analysis by A.Mattuck, Prentice Hall.

C-2.2: Ordinary Differential Equation(Total Marks; 80+20)


Basic concepts of differential equation, formation of differential equation, solution of first order first degree and first order, Equations first order but higher degree. Unit-II

Linear equation with constant coefficient, Differential equation with Variable Coefficients Unit-III

Series solutions and Special function Unit-IV

Laplace Transformation.

Books Prescribed

A Course of ordinary and partial differential equation by J.Sinha Roy and S.Padhy, Kalyani PublisherChapters: 1,2(2.1-2.7),3,4(4.1-4.7),5,7(7.1-7.4),9(9.1-9.5,9.10,9.11,9.13) Books for reference

1. Text book of Differential equation by N.M.Kapur, 2. Advanced Engineering Mathematics by Erwin Kreyzig, John-Wiley 3. Differential Equation and their Apllications by Martin Braun, Springer

International.

Semester – III

C-3.1: Theory of Real functions(Analysis –II)(Total Marks; 80+20)

5 Lectures, 1 Tutorial per week

Unit-I

Functions of single variable, Limits, Continuous functions, Continuous functions on closed intervals, Uniform continuity. Unit-II

Derivatives, Increasing and decreasing functions, Darboux’s theorem, Rolle’s theorem, Langrage’s Mean value theorem, Cauchy’s Mean value theorem, Higher order derivatives. Unit-III

Power series, Exponential functions, Logarithmic functions, Trigonometric functions, Functions of Bounded variation. Unit-IV

Uniform convergence, Point-wise convergence, Uniform convergence on an interval, Test of uniform convergence, Weierstrass Approximation theorem. Books Prescribed

Mathematical Analysis by S.C.Malik and Savita Arora, New-Age Pvt. Ltd.Chapters: 5,6,8(except 8.7),12(except 12.4). Books for reference

1. Fundamentals of Real Analysis by S.L.Gupta and Nisha Rani.2. Fundamentals of Mathematical Analysis by G.Das and S.Pattanayak.3. Elementary Analysis: The theory of calculus by K.A.Ross, Springer.4. Introduction to Analysis by A.Mattuck, Prentice Hall.

C-3.2: (Linear Algebra) Algebra –II(Total Marks; 80+20)

5 Lectures, 1 Tutorial per week Unit-I .Vector space: Definitions and examples, Subspaces, Span of a set, More about subspaces, Linear dependence and independence, Dimension and Basis. Unit-II

Linear transformation: Definition and examples, Range and kernel of a linear map, Rank and nullity, The space L(U,V), Composition of linear maps, Matrix Associated with a linear map, Linear map associated with a matrix, Linear operation, Matrix multiplication, Rank and nullity of a matrix, Transpose of a matrix. Unit-III

Elementary row operations, System of linear equations, Matrix inversion, Determinants, Minor and cofactors, Rank of a matrix, Product of determinants, Application to linear equations, Eigen value and Eigen vectors. Unit-IV

Similarity of Matrices, Inner Product space, Orthogonal and Unitary matrices, Application to reduction to Quadrics. Books Prescribed

An Introduction to Linear Algebra by V. Krishnamurthy, V.P.Mainra ad J.L.Arora, Affiliated East-West Press Pvt. Ltd.Chapters: 3,4,5,6,7. Books for reference

1. Linear Algebra- A Geometric Approach by S.Kumarsen, Prentice Hall.2. Linear Algebra by Hoffman and Kunze, Prentice Hall

C-3.3: Partial Differential Equation (Total Marks; 80+20)


Simultaneous Differential equations: Simultaneous equations with constant coefficients, Simultaneous equations with variable coefficients, Method of solution of equation in symmetrical form, Method of introduction of new variable.Total differential equations: Condition of integrability, Method of obtaining primitive, Solution by inspection, Homogeneous equations, Unit-II

Partial differential equation of first order: Classification of integrals, Complete, Singular and General integrals and their geometrical

interpretations, Formation of partial differential equations, Linear partial differential equations and its solutions by Lagrange’s method, Non-linear partial differential equations and its solutions by Charpit’s method, Standard Forms: I,II,III,IV of Non-linear partial differential equations and their solutions. Unit-III

Linear partial differential equations with constant coefficients: Solutions of Homogeneous linear equations with constant coefficients, Methods for finding complementary functions and Particular integrals of Linear partial differential equations with constant coefficients.Linear partial differential equations order two with constant coefficients: Methods of solutions of Linear partial differential equations with constant coefficients of Type I,II,III,IV. Laplaces transformation for reducing Linear partial differential equations with constant coefficients to standard types. Unit-IV

Monge’s method of solving non-linear partial differential equations of order two of the form Rr + Ss + Tt = V.Application of partial differential equation: Mathematical model for Wave equation, Solution of wave equation by separation of variables method using Fourier series, D’Alembert’s solutions of wave equation, Solution of Heat equation by Fourier series. Books Prescribed

1.Text book of Differential equation by N.M.Kapur, Chapters: 8,9(up to 9.7),10,11(up to 11.8),12(up to 12.9).2.Advanced Engineering Mathematics by Erwin Kreyzig, John-Wiley . Chapter: 11(11.2,11.3,11.4,11.5.) Books for reference

1. A Course of ordinary and partial differential equation by J.Sinha Roy and S.Padhy, Kalyani Publisher.

2. Differential Equations by S.L.Ross, John Wiley & Sons.3. Linear Partial Differential Equations For Scientist and Engineers by Tyn

Myint-U and L.Debnath, Springer.

Semester – IV

C-4.1: Numerical Methods

(Total Marks; 80+20) 5 Lectures, 1 Tutorial per week

Unit-I

Source of Error, Significant figures: Absolute, Relative and Percentage errors, Generation and propagation of Round of errors, Significance errors, Estimation of errors, Power series for evaluating Transcendental functions, Binary, octal and hexadecimal number system, Floating point Arithmetic. Solution of system of Linear equations: Gaussian elimination method, Gauss-Jordan elimination method, Gauss elimination method to compute the inverse of a matrix, Method of Matrix Factorization, Solution of Tri-Diagonal systems Method of Iteration(Jacobi and Gauss-Seidel) Unit-II

Interpolation: Introduction, Lagrangian Interpolating formula, Error in the Interpolating Polynomial, Advantage and disadvantage of Lagrangiian interpolation, Divided differences and their properties, Newton’s fundamental Interpolation formula, Equivalence of Lagrangian and Newtonian Interpolation, Finite difference, Forward and Backward difference and their relationship, Symbolic operators and their relation, Differences of a polynomial, Factorial Power functions, Difference of a Factorial Power function, Newton’s Forward and Backward Interpolation Formulas and errors, Gaussian, Stirling’s, and Bessel’s Interpolation formulas. Unit-III

Numerical Differentiation and Integration: Errors in Numerical differentiation, Differentiation based on Newton’s forward and backward interpolation formula, Differentiation based on Strling’s formula, Integration of Lagrangian Interpolation polynomial, Newton’s-Cote’s Quadrature formula, Trapezoidal and Simpson’s one-third formulas and their errors, Derivation of Trapezoidal and Simpson’s one-third formula using Newton’ forward difference formula, Trapezoidal, Simpsons’s one-third and Weddle’s rule of Integration,Numerical Solutions of ordinary differential equations: Picard’s method of successive approximation, Euler’s method, Taylor’s series method and Runga-Kutta methods of solving ordinary differential equations. Unit-IV

Solutions of Algebraic and Transcendental equations: Methods of finding out crude approximation of a real root, Method of bisection, Method of iteration, Regula-Falsi method, Newton-Raphson method, Convergence of Newton-Raphson method, Curve fitting: Method of least square, Curve fitting with an exponential curve, Fitting a trigonometric function.

Books Prescribed

Introductory Numerical Analysis by N.Datta and R.N.Jana, Shreedhar Prakashani, Calcutta.Chapters: 1,2(up to 2.19),3(3.1-3.4, 3.7-3.15), 4(up to 4.7), 5, 6(up to 6.5), 7(7.1-7.4). Books for reference

1. Numerical methods for Scientific and Engineering Computations by Jain Iyengar and Jain, New Age International

2. A course on Numerical Analysis by B.P.Acharya and R.N.Das, Kalyani Publisher.

3. A friendly Introduction to Numerical Analysis by Brian Braie, Pearson Educaton.

C-4.2: Riemann Integration and Complex Analysis(Analysis-III)(Total Marks; 80+20)


Riemann Integrals: Definitions of the integral, Refinement of partitions, Darboux’s theorem, Conditions of integrability, Integrability of sum and difference of integrable functions, Integral as limit of sum, Some integrable functions, Integration and differentiation, Fundamental theorem of calculus. Unit-II

Improper integrals: Integration of unbounded functions with finite limits of integration, Comparison tests for convergence, Infinite range of integration, Beta and Gamma functions and their convergences. Unit-III

Complex numbers, Complex plane, Polar form of complex numbers, powers and roots, Derivative, analytic functions, Cauchy-Riemann equation, Laplace’s equation. Unit-IV

Geometry of Analytic functions: Conformal mapping, Exponential, Trigonometric, Hyperbolic, Logarithmic functions, Linear fractional transformations. Books Prescribed

1.Mathematical Analysis by S.C.Malik and Savita Arora, New-Age Pvt. Ltd.Chapters: 9,11.

3. Advanced Engineering Mathematics by Erwin Kreyzig, John-WileyChapter: 12.

Books for reference

1. Fundamentals of Real Analysis by S.L.Gupta and Nisha Rani.2. Fundamentals of Mathematical Analysis by G.Das and S.Pattanayak.3. Complex Variables and Applications by J.W.Brown and R.V.Churchill,

McGraw Hill International Edition.4. Functions of one complex variable by J.B.Conway, Springer.

C-4.3: (ANSI- C) Computer fundamentals (Total Marks; 80+20)


Overview of C; Costants, Variables and Data types; Operators and expressions. Unit-II

Decision making and branching, Decision making and looping. Unit-III

Arrays, Character Arrays and Strings. Unit-IV

User-defined Functions. Books Prescribed

Programming in ANSI C, Third Edition, by Balagurusamy,Tata McGraw-Hill Publishing company Limited.Chapters:1,2,3,5,6,7,8,9. Books for reference

Applications Programming in ANSI C by R. Johnsonbaugh and L. Ljoie, Pearson Education.

Semester – V

C-5.1: Multivariate Calculus(Calculus –II)(Total Marks; 80+20)


Functions of several variables, Explicit and implicit functions, Continuity, Partial derivatives, Differentiability, Partial derivative and differentials of higher order, Functions of functions, Change of variables. Unit-II

Taylors theorem of function of two variables, Extreme values(Maxima and minima), Jacobians, Stationary values under subsidiary conditions. Unit-III

Integration on R2 . Unit-IV

Integration on R3 . Books Prescribed

Mathematical Analysis by S.C.Malik and Savita Arora, New-Age Pvt. Ltd.Chapters: 15,16,17,18. Books for reference

1. Fundamentals of Real Analysis by S.L.Gupta and Nisha Rani.2. Fundamentals of Mathematical Analysis by G.Das and S.Pattanayak3. Calculus by G.B.Thomas and R.L.Finney, Pearson Education.

C-5.2: Practical

(Total Marks: 100) 2 Practical classes per week

A student is required to perform Four experiments two from each Group. The duration of Practical examination is six hours. A student has to perform at least 75% of the number of prescribed experiments. Two experiments from Group-A 30 Marks Two experiments from Group-A 30 Marks Practical record( 10+10) 20 Marks Viva-voce ( 10+10) 20 Marks A student shall be required to maintain two records, one for each group, certified by the teacher and produce them at the time of examination.

List of ExperimentsGroup – A

i. Graphical solution of Linear Programming problems.ii. Solution of Linear Programming problems by simplex

method.iii. Solution of Transportation problems.iv. Solution of Assignment problems by Hungarian method.v. Tracing of Curves: Catenaries, Beroulli’s Lemniscates,

Cissoids, Asteroid, Cardioids and Descarte’s Folium.vi. Estimation of a function using Lagrange’s, Newton’s

forward, Newton’s backward Interpolation formulas.vii. Numerical solutions of transcendental equations by

Bisection method, Regula-Falsi method, Newton-Raphson method and Iteration methods.

viii. Numerical Integration by Composite Trapezoidal and Composite Simpson’s one-third rule.

ix. Fitting of exponential and Logarithmic function.x. To find the correlation coefficient between tw variable

and also the line of regression.

Group – B( using any soft-ware)Lab-work to be performed on a computer

i. Writing a program for searching primes less than or equal to a specific number N.

ii. Writing a program for arranging a given set of numbers in a specific order.

iii. Writing a program for the Numerical solutions of transcendental equations by Bisection method, Regula-Falsi method, Newton-Raphson method

iv. Writing a program to solve quadratic equationv. Writing a program to find the numerical solution of first

order and first degree differential equations by Euler's method.

vi. Writing a program to find out the product of two matrices.

vii. Writing a program to find the Armstrong number between 1 to 100.

viii. Writing a program to find out sum of n odd/even numbers.

ix. Writing a program to evaluate the series 1 + x +x2 + x3 +. .. .. . .. . for −1 < x <1 to 0.01%.

x. Write a program for Numerical Integration by Composite Trapezoidal and Composite Simpson’s one-third rule.

Semester – VI

C-6.1: Metric Space(Analysis –IV)(Total Marks; 80+20)


Metric Spaces : Definitions and examples, Open and closed sets, Convergence and completeness. Unit-II

Continuity and uniform continuity, Cnnectedness. Unit-III

Compactness. Unit-IV

Measurable sets, Sets of measure zero, Measurable functions, Lebesgue Integral and its properties for bounded measurable functions Books Prescribed

Mathematical Analysis by S.C.Malik and Savita Arora, New-Age Pvt. Ltd.Chapters: 19, 20. Books for reference

1. Metric spaces by Jain and Ahmad.2. Method of Real Analysis by R.G.Goldberg.3. Metric spaces by Satish Shirali and H.L.Vasudeva, Springer Verlag.

C-6.2: Probability and Statistics (Total Marks; 80+20)


Axiomatic approach of Probability theory: Finite sample space, Conditional probability, Independent events. Unit-II

Random variables: The concept of random variables, Distribution function, Moments of the random variables. Unit-III

Some Probability distributions: Binomial distribution, Poisson distribution, Uniform distribution, Normal distribution, Generating functions, Moment generating functions, Characteristic function. Unit-IV

Jointly distributed random variables: Joint distribution functions and density functions, Conditional densities of continuous random variables, conditional expectation and variance. Books Prescribed

Probability and Random Processes by S.K.Srinivasan and K.M.Meheta, McGraw_Hill Publishing Company Ltd.Chapters: 2,3,4,5,6. Books for reference

1. Elements of Probability theory by D.Prathihari and S.P.Mohanty2. Modern probability theory by B.R.Bhat. 3. Probability by Kai Lai Chung,

Discpline Specific Electives(DES)Credit : 06 each

DSE-ITotal Marks : 100

Theory: 80 Marks + Mid-Sem: 20 Marks5 Lectures, 1 Tutorial per week

(Any one of the following)

1. Linear Programming Unit-I Linear programming problem: Formulation and Graphical solution, Solution of Linear programming problem by Simplex method, Fundamental properties of solution, Computational procedure, Use of artificial variable.

Unit-II

Duality in linear programming: General primal-Dual pair, Formulating a dual problem, Primal-Dual pair in matrix form, Complementary slackness theorem, Duality and simplex method, Dual simplex method.

Unit-IIITransportation problem: LP Formulation of transportation problem, Existence of solution in Transportation problem, Duality in Transportation problem, Transportation table, Loops in Transportation table, Triangular basis in Transportation problem, Finding initial basic feasible solution, Test for optimality, Economic interpretation of u’s and v’s, Transportation algorithm(Modi method),Stepping Stone solution method.Assignment Problem: Mathematical formulation and solution, Special cases in Assignment Problem. Unit-IVGames and Strategies: Two person Zero-sum game, Some basic terms, Maximin and Minimax principle, Games without saddle point, Graphic solution of games, Dominance property, Arithmetic method for n x n games. Books PrescribedOperations research by Kant Swarup, P.K.Gupta and Man Mohan, Sultan Chand & sons.

Chapters: 2,3,4(up to 4.4),5(except 5.8),1010,11,17(up to 17.8) Books for reference

1. Operations research, An Introduction by H.A.Taha, Prentice-Hall.2. Linear Programming by G.Hadley, Narosa publishing House.

2. Finite Element Methods Unit-I

Introduction to finite element methods, comparison with finite difference method, Methods of weighted residuals, collections, least squares and Galerkin’s method, Variational formulation of boundary value problems equivalence of Galerkin and Ritz method.

Unit-II

Applications to solving simple problems of ordinary differential equations, Linear, quadratic and higher order elements in one dimensional and assembly, Solution of assembled system.

Unit-IIISimplex elements in two and three dimensions, quadratic triangular elements, ectangular elements, serendipity elements and isoperimetric elements and their assembly, discretization with curved boundaries.

Unit-IVInterpolation functions, Numerical integrations, Modeling considerations, Solution of two dimensional partial differential equations under different Geometric conditions. Books Prescribed1. Introduction to Finite element Methods by J.N.Reddy, Tata McGraw – Hill, 2003

Books for reference

1. Finite Element Analysis by George R. Buchanan, McGraw-Hill, 19942. The Finite Element method: Linear Static and Dynamic Finite

Element Analysis by T.J.R. Huhes, Dover Publication

DSE-IITotal Marks : 100



1. Discrete Mathematics Unit-I

Proportional equivalence, Introduction to proofs and methods of proof, Mathematical induction, Strong Induction and well ordering, Recursive Definitions and structural induction , The basic counting, The Pigeon-hole principle, Generalized Permutations and Combinations Unit-II

Recurrence relations, counting using recurrence relations, Solving linear recurrence relations with constant coefficients, Generating functions, Solving recurrence relations using generating functions,. Relations and their properties, n-ary relations and their applications, Representing relations

Unit-III

Graphs: Basic concepts and graph terminology, Representing graphs and graph isomorphism, Distance in a graph, Cut-vertices and cut-edges, Euler and Hamiltonian path, Shortest path problem,Planar graphs, Graph coloring.

Unit-IVIntroduction to Trees, Applictions of trees, Tree travel, Spnning trees, Minimum spanning trees, Boolean function. Books PrescribedDiscrete Mathematics and Applictions by K.H.Rosen, Tata McGraw-Hill Publications.Chapters: 1(1.2,1.5,1.6), 4(4.1, 4,2, 4.3), 5(5.1, 5.2, 5.5), 6(6.1, 6.2, 6.4) 7(7.1 - 7.3), 8, 9, 10(10.1).

Books for reference

1. Fundamentals of Discrete Mathematical Structures by K.R. Chowdhary, Eastern Economy Edition(PHI Learning Pvt. Ltd.).2. Discrete Mathematics with Graph Theory by Edgar G. Goodaire and Michael M. Parmenter, Pearson Education(Singapore) Pte. Ltd.3. Discrete Mathematics: Theory & Applications by D.S. Malik,Cengage Learning India Pvt. Ltd.4. Discrete mathematical Structures by Kevin Ferland, Cengage Learning India Pvt. Ltd.

2. Econometrics

Unit-I Statistical concepts: Normal distribution, Chi-square distribution, t and F distribution, Estimation of parameters, properties of estimators, Testing of hypothesis, defining statistical hypothesis, distribution of test statistics, testing of hypothesis related to population parameters, Type-I and Type-II errors, Power of a test, Test for comparing parameters from two samples.

Unit-II

Linear regression Models: Two variable Case estimation of model by method of ordinary least squares, Properties of estimators, Goodness of fit, Test of hypothesis, scaling and units of measurement, Confidence intervals, Gauss-Markov theorem, Forecasting.

Unit-III

Multiple Linear regression Models: Estimation of parameters, Properties of OLS estimators, Goodness of fit-R and adjusted R , Partial regression correlation, Testing hypothesis- individual and joint, Functional forms of regression models, Qualitative(dummy) independent variables. Unit-IV

Violation of classical assumptions: Consequences, Direction and remedies multicollinearity, heteroscedasticity, serial correlation, Specification Analysis omission of a relevant variable, Inclusion of irrelevant variable, tests of specification errors.

Books Prescribed1. Probabilty and statistics for engineers by Jay L. Devore, Cengage

Learnig,2010,2. Mathematical Statistics by John E. Freund, Prentice-Hall.3. Essentials of Econometrics by D.N.Gujarati and D.C.Porter,

McGraw Hill.4. Introduction to Econometrics by Christopher Dougherty, Oxford

University Press.

DSE-IIITotal Marks : 100



1. Mathematical Modeling Unit-I

Mathematical Modeling through ordinary differential equations of first order: Linear growth and decay model, Non-linear growth and decay model, Compartment model, Mathematical modeling in Dynamics through ordinary differential equations of first order, Mathematical modeling of geometrical problems through ordinary differential equations of first order. Unit-II

Mathematical Modeling through systems of ordinary differential equations of the first order: Mathematical Modeling in Population Dynamics, Mathematical Modeling of epidemics through system of ordinary differential equations of first order, Compartment models through system of ordinary differential equations of first order, Mathematical modeling in economics through system of ordinary differential equations of first order, Mathematical Models in Medicine, Arms Race, Battles, international trades in terms of system of ordinary differential equations, Mathematical modeling in Dynamics through system of ordinary differential equations of first order, Unit-III

Mathematical Modeling through systems of ordinary differential equations of the second order: Mathematical Modeling of planetary motion, Mathematical Modeling of circular motion and motion of satellites, Mathematical Modeling through linear differential equations of second order, Miscellaneous Mathematical Models through ordinary differential equations of the second order.

Unit-IV

Mathematical Modeling through difference equations: Basic theory of linear difference equations with constant coefficients, Mathematical Modeling through difference equations in economics and finance, Mathematical Modeling through difference equations in population Dynamics and genetics.

Books PrescribedMathematical Modelling by J.N.Kapur, New-Age international Publishers.Chapters: 2,3,4,5.

2. Mathematical Finance

Unit-I

Basic Principles: Comparison, arbitrage and risk aversion, Interest(simple and compound, discrete and continuous), time value of money, inflation, net present value, internal rate of return(calculation by bisection and Newton-Raphson methods), Comparison of NPV and IRR Bonds, bond prices and yields, Macaulay and modified duration, term structure of interest rates, Spot and forward rates, explanations of term structure, running present value, floating-rate bonds, Immunization, Convexity, putable and callable bonds. Unit-II

Asset return, short selling, portfolio return, brief introduction to expectation, variance, covariance, and correlation, Random returns, Portfolio mean return and variance, Diversification, Portfolio diagram, Feasible set, Markowitz model( review of Lagrange multipliers for 1 and 2 constraints), Two fund theorem, risk free assets, One fund theorem, Capital market line, Sharpe index, Capital Asset Pricing Model(CAPM), Betas of stocks and portfolios, Security market line, use of CAPM in investment analysis and as a pricing formula, Jensens Index,

Unit-IIIForwards and futures , Marking to market, value of a forward/futures contract, replicating portfolios, futures on assets with known income or divided yield, currency futures, hedging (short, long, cross, rolling), optional hedge ratio, hedging with stock index futures, interest rate futures,swaps. Unit-IVLognormal distribution, Lognormal model/Geometric Brownian Motion for stock prices, Binomial tree model for stock prices, parameter estimation, comparison of the model, Options, Types of options: put/ call, European/ American, pay off of an option, factors affecting option prices, put call parity.

Books Prescribed1. Investment Science by David G. Luenberger, Oxford University

Press Chapters: 1,2,3,4,5,6,7,8(8.5-8.8),10(excet10.11,10.12), 11(except 11.2, 11.11.8)2. Options, Futures and Other Derivatives by John C. Hull, Pretice-

Hall.Chapters: 3, 5, 6, 7(except 7.10, 7.11), 8, 9.

DSE-IV

Total Marks : 100 (Any one of the following)

1. Project Work Project : 75 Marks, Viva-Voce : 25 Marks 2.Differential Geometry


Unit-I

Theory of space curves: Space curves, Planer curves, Curvature, Torsion, and Serret - Frenet formula. Unit-II

Osculating circles and spheres, Existence of space curves, Evolutes and involutes of curves.

Unit-IIIDevelopables: Developables associated with space curves and curveson surfaces, Minimal surfaces. Unit-IVTheory of surfaces: Parametric curves on surfaces, Direction coefficients, First and second Fundamental forms, Principal Gaussian curvature, Lines of curvature, Euler’s theorem, Rodrigues formula, Conjugate and asymptotic lines. Books PrescribedDifferential Geometry of three Dimensions by C.E.Weatherburn, Cambridge University Press.Chapters: 1(1-4,7,8,10), 2(13, 14, 16, 17), 3, 4(29-31,35, 37, 38)

Books for reference

1. An Introduction to Differential Geometry by T.J.Wilmore, Dover Publications, 2012.

2. Elementary Differential Geometry by A.N.Pressley, Springer.

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Generic Electives/Interdisciplinary Credit: 06 each, Marks: 100

GE-1:Calculus

(Total Marks; 80+20) 5 Lectures, 1 Tutorial per week

Curvature, Asympotes, Singular points and Curve tracing. Unit-II

Rectification, Quadrature, Volume and surface area of he solid surface of revolution. Unit-III

Vector calculus, Limits, continuity and differentiability of vector function, Point functions(scalar and vector), Directional derivative , gradient and curl of point functions and their properties. Unit-IV

Sphere, Cone and Cylinder. Books Prescribed

1. Elementary Calculus by Panda and Satapathy. Chapters: 1,2,5,6.

2. Analytical Geometry of Quadratic Surfaces by B.P.Acharya and D.P.Sahu, Kalyani Publishers

Chapters: Books for reference1. Text book of Calculus, Part-II by Shantinarayan, S Chand & Co.2. Text book of Calculus, Part-III by Shantinarayan, S Chand & Co.3. Calculus by G.B.Thomas, Pearson Education, Delhi4. Analytical Solid Geometry by S.Narayan and Mittal, S.Chand Co.

GE-2: Ordinary Differential Equation(Total Marks; 80+20)


Introduction and some basic concept of ordinary differential equation, formation of differential equation, solution of first order first degree and first order but not of first degree differential equations.

Unit-II

Higher order differential equations, Solution of higher order linear differential equations with constant and variable coefficients. Unit-III

Power series solution of second order linear differential equations about ordinary points and singular points. Solutions of Legendre and Bessel’s equations and some of their simple properties. Unit-IV

Laplace Transformation.

Books Prescribed

Text book of Differential equation by N.M.Kapur, Chapters: 1,2,3,4,13,14,19. Books for reference

1. A Course of ordinary and partial differential equation by J.Sinha Roy and S.Padhy, Kalyani Publisher.

2. Advanced Engineering Mathematics by Erwin Kreyzig, John-Wiley and co.

3. Differential Equation and their Apllications by Martin Braun, Springer International.

------------------------------------------------------------------------------------------Skill Enhancement Courses (SEC)(Credit: 2 each, Total Marks: 50)

SEC-ICommunicative English and Writing Skill (compulsory)

SEC-II(Any one of the following)

1. Graph Theory

Definition, examples and basic properties of graph, Pseudo graphs, Completes graphs, Bi-partite graphs, Isomorhism of graphs, Paths and circuits, Eulerian circuits, Hamiltonian cycles, Adjacency matrix, Weighted graph, Travelling salesman’s problem, Shortest path, Dijkstra’s algorithm, Floyd-Warshall algorithm.

Books Recommended1.Introduction to Lattices and Order, by B.A.Davey and H.A.Priestley, Cambridge University Press2. Discrete Mathematics withGraph theory by E.G.Goodaier and M.M.Parmenter, Pearson Education

2. Bio-MathmaticsMathematical Biology and the modeling process: An overview, Continuous models: Malthus model, Logistic growth, Allee effect, Gompertz growth,Michaelis-Menten Kinetics, Holling type growth, Bacterial growth in a Chemostat, Harvesting a single natural population, Epidemic model(SI, SIR, SIRS, SIC), Activator-inhibitor system, Inset outbreak model: Spruce Budworm, Numerical solution of the models and its graphical representation, Qualitative analysis of continuous models: Steady state solutions, Stability and linearization, Multiple series communities and Routh-Hurwitz criteria, Phase plane methods and qualitative solutions, Bifurcations and limit cycles with examples in the contest of biological scenario.

Books Recommended1.Mathematical Model in Biology by L.E.Keshet, SIAM, 19882.Mathematical Biology by J.D.Murray3.Biomechanics by Y.C.Fung, Springer, 2008.

3. Computer Graphics

Development of computer graphics: Raster Scan and Random Scan graphics storages, Display processors and character generators, Colour display techniques, Interactive input/outout devices, Points, lines, and curves, Scan conversion,Line-drawing algorithms, Circle and ellipse generation, Cnic-section generation, Polygon filling anti-aliasing , Two-dimensional viewing, Coordiate systems, Liear transformations, line and polygon clipping algorithm.

Books Recommended1. Computer Graphics by D.Hearn and M.PBaker , Prentice_Hall,2004.2. Computer Graphics: Principals and Practices by J.D.Foley, A van Dam,

S.K.Feiner and J.F.Hughes, Addison-Wesley, MA 1990.

4. Industrial Mathematics

Medical Imaging and Inverse Problems: The content is based on Mathematics of X-ray on the knowledge of calculus, elementary differential equations, complex numbers and matrices.Introduction to Inverse Problems: Why should we teach inverse problem? Illustration of Inverse problems through problems taught

in Pre-Calculus, calculus, Matrices and Differential equations. Geological anomalies in Earth’s interior from measurements at its surface(Inverse problems for Natural disaster) and Tomography.X-ray: Introduction, X-ray behavior and Bears law(the fundament question of image construction), Lines in the place.Random Transform: Definition and examples, Linearity, Phantom(Shepp-Phantom-Mathematical Phantoms)Back Projection: Definition properties and examples.

Books Recommended1. The Mathematics of Medical Imaging, A Beginners Guide by

T.G.Feeman, Springer, 20102. Inverse Problems by C.W.Grotesch, The Mathematical Association

of America, 1999.

5. Logic and Sets

Introduction, propositions, truth table, negation, conjunction and disjunctions. Implications, biconditional propositions, converse, contra positive and inverse propositions and precedence of logical operators. Propositional equivalence: Logical equivalence. Predicates and quantifiers: Introduction, Quantifiers, binding variables and negations. Set, Subsets, Set operations and the laws of Set theory and Venn diagrams. Examples of finite and infinite sets, Finite sets and counting principle, Empty set, properties of empty set, Standard set operations, Classes of sets, Power set of a set. Difference and symmetric difference of two sets, Set identities, Generalized union and intersections. Relation: Product set, Composition of relations, Types of relations, Partitions, Equivalence relations with example of congruence modulo relation, Partial ordering relations, n-ary relations.

Books Recommended

1. Naive Set Theory by P.R.Halmos, Springer, 1974.2. Theory of sets by E.Kamke, Dover Publication,1950.

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