FACULTY OF SCIENCES SYLLABUS FOR M. Sc. Mathematics (CBCEGS) (SEMESTER: I - IV) Examinations: 2016 - 17 GURU NANAK DEV UNIVERSITY AMRITSAR Note: (i) Copy rights are reserved. Nobody is allowed to print it in any form. Defaulters will be prosecuted. (ii) Subject to change in the syllabi at any time. Please visit the University website time to time.
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FACULTY OF SCIENCES
SYLLABUS
FOR
M. Sc. Mathematics (CBCEGS)(SEMESTER: I - IV)
Examinations: 2016 - 17
GURU NANAK DEV UNIVERSITYAMRITSAR
Note: (i) Copy rights are reserved.Nobody is allowed to print it in any form.Defaulters will be prosecuted.
(ii) Subject to change in the syllabi at any time.Please visit the University website time to time.
1M.Sc. Mathematics (CBCEGS) (Semester System)
Programme Code: MTBMaster of Science (Honours) in MathematicsTotal minimum credits required for M. Sc. (Honours) Mathematics are 96.*Out of 96 credits, student will choose minimum 8 credits interdisciplinary courses from otherdepartments by choosing at least three credits course in each of the third and fourth semester.
Programme Code: MTBTotal minimum credits required for M. Sc. (Honours) Mathematics are 96.
SEMESTER – I:
Course No. C/E/I Course Title L T P TotalCredits
Core CoursesMTL 401 C Real Analysis-I 4 0 0 4MTL 402 C Algebra-I 4 0 0 4MTL 403 C Linear Algebra 4 0 0 4MTL 404 C Number Theory 4 0 0 4MTL 405 C Complex Analysis 4 0 0 4MTL 406 C Differential Equations 4 0 0 4
Total Credits: 24 0 0 24
SEMESTER – II:
Course No. C/E/I Course Title L T P TotalCredits
Core CoursesMTL 451 C Real Analysis-II 4 0 0 4MTL 452 C Algebra-II 4 0 0 4MTL 453 C Probability Theory 4 0 0 4MTL 454 C Classical Mechanics and Calculus of
Variations4 0 0 4
MTL 455 C Differential Geometry 4 0 0 4MTL 456 C Mathematical Methods 4 0 0 4
Total Credits: 24 0 0 24
2M.Sc. Mathematics (CBCEGS) (Semester System)
SEMESTER-III
Course No. C/E/I Course Title L T P TotalCredits
Core CoursesMTL 501 C Measure Theory 4 0 0 4MTL 502 C Functional Analysis-I 4 0 0 4MTL 503 C Statistical Inference 3 0 1 4
Elective/Optional Courses (Choose any two courses)MTL 531 E Operations Research-I 4 0 0 4MTL 532 E Discrete Mathematics-I 4 0 0 4MTL 533 E Fluid Dynamics 4 0 0 4MTL 534 E Advanced Numerical
Analysis4 0 0 4
MTL 535 E Stochastic Process 4 0 0 4MTL 537 E Calculus of Several
Variables4 0 0 4
MTL 538 E Commutative Algebra 4 0 0 4MTL 539 E Theory of Wavelets 4 0 0 4MTL 541 E Fourier Analysis 4 0 0 4
*Interdisciplinary CoursesTotal Credits
3M.Sc. Mathematics (CBCEGS) (Semester System)
SEMESTER–IV
Course No. C/E/I Course Title L T P TotalCredits
Core CoursesMTL 551 C Topology 4 0 0 4MTL 552 C Functional Analysis-II 4 0 0 4MTL 553 C Field Extensions and Galois Theory 4 0 0 4
Elective/Optional Courses (Choose any two courses)MTL 581 E Topological Vector Spaces 4 0 0 4MTL 582 E Computer Programming with C 3 0 1 4MTL 583 E Operations Research-II 4 0 0 4MTL 584 E Discrete Mathematics-II 4 0 0 4MTL 586 E Banach Algebra and Operator
Theory4 0 0 4
MTL 588 E Financial Derivatives 4 0 0 4MTL 589 E Theories of Integration 4 0 0 4MTL 590 E Algebraic Topology 4 0 0 4MTL 591 E Theory of Sample Survey 4 0 0 4MTL 592 E Special Functions 4 0 0 4MTL 593 E Representation Theory of Finite
Groups4 0 0 4
MTL-594 E Analytic Number Theory 4 0 0 4*Interdisciplinary Courses
Total Credits
*Student has to choose minimum 8 credits interdisciplinary courses in two semesters(III and IVsemesters) from other departments by choosing at least three credits course in each of
third and fourth semesters.
4M.Sc. Mathematics (CBCEGS) (Semester–I)
MTL 401
REAL ANALYSIS-IL T P4 0 0
Unit-ICountable and uncountable sets. Metric spaces: Definition and examples, open sets, closed sets,
compact sets, elementary properties of compact sets.
Unit-IICompactness of k- cells, Compact subsets of Euclidean space Rk. Heine Borel theorem, Perfect
sets, The Cantor set, Separated sets, connected sets in a metric space, connected subsets of real
line, Components, Functions of Bounded Variation.
Unit-III
Sequences in Metric Spaces: Convergent sequences (in Metric Spaces), subsequences, Cauchy
Unit – IVector Spaces, Subspaces, Basis and Dimension Theorems, Direct sum decomposition, TheAlgebra of linear transformations, Matrices associated with linear transformations, change ofordered bases, Elementary matrix operations and Elementary matrices, Row rank, Column rankand their equality, Condition of consistency of system of Linear Equations (with proof).
Unit-IIEigen values and Eigen Vectors of Linear Operators, Characteristic and minimal polynomials,companion matrix, subspaces invariant under linear operators, triangulation, Diagonalization,Linear functionals, Dual Spaces and dual basis, the double dual, Inner Product Spaces, TheGram-Scmidt Orthogonalization, orthogonal complements.
Unit-IIIThe Adjoint of a Linear operator on an inner product space, Normal and self-Adjoint operators,Unitary and Normal Operators, Spectral Theorem, Bilinear and Quadratic forms, GeneralizedEigen Vectors, Rational and Jordan Canonical forms.
Recommended Books:1. Hoffman, K. and Kunze, R.: Linear Algebra, Second Edition, Prentice Hall.2. Axler, S.: Linear Algebra Done Right, Second Edition,
Springer-Verlag.3. Friedberg, S.H., Insel,A.J.,. Spence, L.E: Lawrence, Linear Algebra, Second Edition
Prentice Hall,1989.4. Lang, S.: Linear Algebra, Springer-Verlag, 2000.
7M.Sc. Mathematics (CBCEGS) (Semester–I)
MTL 404
NUMBER THEORY
L T P
4 0 0
Unit-I
Number theoretic functions and, Multiplicative functions, The Mobius Inversion formula,
Eular’s Phi-function and its properties, Euler’s theorem, Fermat’s Theorem, Lagrange’s
Theorem, Primitive roots, m indices and their applications.
Unit-II
Euler’s criterion, The Legendre symbol and its properties, Gauss Lemma, Quadratic reciprocity
law, Jacobi’s symbol and it’s Properties, Pythagoreon triples, the famous “Last Theorem"
Unit-III
Representation of an integer as a sum of two squares and sum of four squares, finite and infinite
simple continued fractions, Convergents, The fundamental solution of Pell’s Equations,
Applications to Pell’s equations.
BOOKS RECOMMENDED:
1. David M. Burton: Elementary Number Theory, Mc Graw Hill 2002.
2. Hardy and Wright: Theory of Numbers.
8M.Sc. Mathematics (CBCEGS) (Semester–I)
MTL 405
COMPLEX ANALYSISL T P4 0 0
Unit-I
Functions of complex variables, continuity and differentiability, Analytic functions, Conjugate
function, Harmonic function. Cauchy Riemann equations (Cartesian and Polar form).
Construction of analytic functions.
Unit-II
Complex line integral, Cauchy’s theorem, Cauchy’s integral formula and its generalized form.
Cauchy’s inequality. Poisson’s integral formula, Morera’s theorem. Liouville’s theorem. Power
series, Taylor’s theorem, Laurent’s theorem. Fundamental theorem of Algebra and Rouche’s
theorem.
Unit-III
Zeros, Singularities, Residue at a pole and at infinity. Cauchy’s Residue theorem, Jordan’s
lemma. Integration round Unit circle. Evaluation of integrals.
Continuous Distributions: Uniform, Exponential, Normal distributions, Gamma distribution,
Beta distribution, t distribution, F distribution, Chi-square distribution, sampling distribution of
mean and variance of sample from normal distribution.
Books Recommended:
1. Hogg, R.V., Mckean, J.W. and Craig, A.T. : Introduction to Mathematical Statistics.
2. Rohtagi, V. K. and Ehsanes Saleh, A. K. Md. An Introduction to Probability and Statistics
3. Casella, G. and Berger, R. L. : Statistical Inference
13M.Sc. Mathematics (CBCEGS) (Semester–II)
MTL 454CLASSICAL MECHANICS AND CALCULUS OF VARIATIONS
L T P4 0 0
Unit-I
Generalized coordinates and generalized velocities, virtual work, generalized forces, Lagrange’sequations for a holonomic dynamical system, conservative system, holonomic dynamical systemfor impulsive forces and their applications, kinetic energy as a quadratic function of velocities,theory of small oscillations.
Unit-II
Functional, variation of functional and its properties, fundamental lemma of calculus ofvariation, Euler’s equations, necessary and sufficient conditions for extremum, TheBrachistochrone problem, Functionals dependent on higher order derivatives and severaldependent variables, Variational problems with moving boundaries, Transversality conditions,Orthogonality conditions.
Unit-III
Sturm-Liouville’s theorem on extremals, One sided variations, Hamilton’s principle, Theprinciple of least action, Langrange’s equations from Hamilton’s principle. Variational Methods,Direct Methods, Euler’s finite difference method, The Ritz method, Kantorovich Method forBoundary value problems in ODE’s & PDE’s, Isoperimetric Problems.
Recommended Books:
1. Chorlton, F.: Text Book of Dynamics.2. Elsgolts, L: Differential Equations and the Calculus of Variations.3. Gelfand,I.M. and Fomin, S.V.: Calculus of Variations.
14M.Sc. Mathematics (CBCEGS) (Semester–II)
MTL 455
DIFFERENTIAL GEOMETRYL T P4 0 0
Unit-I
Curves in R3:- A simple arc, curves and their parametric representation, arc length and naturalparameter, Contact of curves, tangent, principal normal, binormal, osculating plane, curvatureand torsion, Serret-Frenet Formule, Helics, Evolute and Involute of a parametric curve, sphericalcurves.
Unit-II
Tensor Analysis:- Einstein’s Summation Convention, Transformation laws for vectors, Contra-variant, covariant and mixed Tensors, addition, multiplication, contraction and quotient law oftensors, Differentiation of Cartesian Tensors, metric Tensor, , Christoffel symbols. Covariantdifferentiation of tensors.
Unit-III
Surfaces in R3:- Implicit and Explicit forms for the equation of the surface, the two fundamentalforms of a surface, Family of surfaces, Edge of regression, Envelops, Ruled surface, Developableand skew surfaces, Gauss and Weingarten formulae, Introduction to Geodesics, Geodesicsdifferential equation.
Recommended Books:-
1. A. Pressley: Elementary Differential Geometry, Springer, 2005.2. T.J.Willmore: Introduction to Differential Geometry4. Martin M. Lipschutz: Differential Geometry5. U.C. De; A.A. Shaikh & J. Sengupta: Tensor Calculus6. M.R. Spiegel: Vector Analysis7. D. Somasundaram: Differential Geometry – A First course, Narosa
Publishing House
15M.Sc. Mathematics (CBCEGS) (Semester–II)
MTL 456
MATHEMATICAL METHODSL T P4 0 0
Unit-I
Laplace Transform: Definition, existence, and basic properties of the Laplace transform, Inverse
Laplace transform, Convolution theorem, Laplace transform solution of linear differential equations
and simultaneous linear differential equations with constant coefficients, Complex Inversion
formula.
Unit-II
Fourier Transform: Definition, existence, and basic properties, Convolution theorem, Fourier
transform of derivatives and Integrals, Fourier sine and cosine transform, Inverse Fourier
transform, solution of linear ordinary differential equations and partial differential equations.
Unit-III
Volterra Equations : Integral equations and algebraic system of linear equations. Volterra
equation L2 Kernels and functions. Volterra equations of first & second kind. Volterra integral
equations and linear differential equations. Fredholm equations, solutions by the method of
successive approximations. Neumann’s series, Fredholm’s equations with Pincherte-Goursat
Kernel’s.
Books Recommended:
1. Tricomi, F.G. : Integral Equation (Ch. I and II)
2. Kanwal R, P : Linear Integral Equations
3. Mikhlin : Integral Equations
4. Pinckus, A. and Zafrany, S.: Fourier Series and Integral Transforms
16M.Sc. Mathematics (CBCEGS) (Semester–III)
MTL 501
MEASURE THEORY
L T P4 0 0
Unit-I
Lebesgue Outer Measure, Measurable Sets and their properties, Non Measurable Sets, Outer and
Inner Approximation of the Lebesgue Measurable Sets, Borel Sigma Algebra and The Lebesgue
Sigma Algebra, Countable Additivity, Continuity and the Borel-Cantelli Lemma.
Unit-II
The motivation behind Measurable Functions, various Characterizations and Properties of
Measurable functions: Sum, Product and Composition, Sequential Pointwise Limits and Simple
Approximations to Measurable Functions. Littlewood’s three Principles.
Lebesgue Integral: Lebesgue Integral of a simple function and bounded measurable function over
a set of finite measure. Comparison of Riemann and Lebesgue Integral. Bounded Convergence
Theorem, Integral of a non-negative measurable function, Fatou’s Lemma, Monotone
convergence Theorem.
Unit-III
General Lebesgue Integral, Lebesgue Dominated Convergence Theorem, Countable Additivity
and Continuity of Integration, Vitali Covers and Differentiability of Monotone Functions,
Functions of Bounded Variation, Jordan's Theorem, Absolutely Continuous Functions, Absolute
Continuity and the Lebesgue Integral.
17M.Sc. Mathematics (CBCEGS) (Semester–III)
Books Recommended:
1. Royden, H.L. and Fitzpatrick: Real Analysis (Fourth Edition), Pearsoon Education Inc.New Jersey, U.S.A.(2010).
2. R. A. Gordon, The integrals of Lebesgue, Denjoy, Perron and Henstock, Amer. Math.Soc. Province, RI, (1994).
3. Barra, G De. : Introduction to Measure Theory, Van Nostrand and Reinhold Company.
4. Jain, P.K. and Gupta, V.P.: Lebesgue Measure and Integration.
18M.Sc. Mathematics (CBCEGS) (Semester–III)
MTL 502FUNCTIONAL ANALYSIS – I
L T P4 0 0
Unit-I
Normed linear spaces, Banach spaces, subspaces, quotient spaces. Continuous linear
transformations, equivalent norms.
Unit-II
Finite dimensional normed linear spaces and compactness, Riesz Lemma, The conjugate space
N*. The Hahn-Banach theorem and its consequences. The natural imbedding of N into N**,
reflexivity of normed spaces.
Unit-III
Open mapping theorem, projections on a Banach space, closed graph theorem, uniform
boundedness principle, conjugate operators. Lp-spaces: Holder’s and Minkowski’s Inequalities,
completeness of Lp-spaces.
BOOKS RECOMMENDED:
1. G.F. Simmons: Introduction to Topology and Modern Analysis,
Ch. 9, Ch.10 (Sections 52-55), Mc.Graw-Hill International Book
Company, 1963.
2. Royden, H. L.: Real Analysis, Ch 6 (Sections 6.1 -6.3), Macmillan Co.
1988.
3. Erwin Kreyszig: Introduction to Functional Analysis with Applications,
John Wiley & Sons, 1978.
4. Balmohan V. Limaye: Functional Analysis, New Age International Limited.
5. P.K.Jain and : Functional Analysis, New Age International (P) Ltd.
O.P Ahuja Publishers, 2010.
6. K. Chanrashekhra Rao: Functional Analysis, Narosa, 2002
7. D. Somasundram: A First Course in Functional Analysis, Narosa, 2006.
19M.Sc. Mathematics (CBCEGS) (Semester–III)
MTL 503
STATISTICAL INFERENCE
L T P3 0 1
Unit-I
Point Estimation: Sufficient statistics, Neyman factorization theorem, minimal sufficient
and boundary conditions, use of shell momentum balance to solve laminar flow problems: flow
of a falling film, flow through a circular tube (Hagen-Poiseuille flow), flow through an annulus,
flow of two adjacent immiscible fluids, creeping flow around a sphere.
Unit-III
The Navier-Stokes Equation: The Navier-Stokes equation, use of the Navier-Stokes equation in
solving the following flow problems: Steady flow in a long circular cylinder, falling film with
variable viscosity, The Taylor-Couette flow, Plane Couette flow; Shape of the surface of a
rotating liquid, Flow near a slowly rotating sphere, Steady viscous flow in tubes of uniform cross
sections, viscous flow past a fixed sphere, Dimensional analysis of fluid motion, Prandtl
boundary layer, Time dependent flows of Newtonian fluids.
Reference Books:
1. F. Charlton, Textbook of Fluid Dynamics 1st Edition. (Scope in Ch.2-5, 8)2. R. B. Bird, W. E. Stewart, E. Lightfoot, Transport Phenomena, 2nd Edition.
(Scope in Ch. 1-4)3. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 3rd Edition. (Scope in Ch. I-II)
23M.Sc. Mathematics (CBCEGS) (Semester–III)
(Elective Paper)
MTL 534
ADVANCED NUMERICAL ANALYSIS
L T P4 0 0
Unit-IFinite difference approximation to partial derivatives, parabolic equations: Transformation to
non-dimensional forms, an explicit method, Crank Nicolson Implicit method, solution of implicit
equations by Gauss Elimination, derivative boundary conditions, local truncation error,
Consistency, Convergence and stability. Iterative methods for elliptic equations, Jacobi’s
1. Atiyah, M.F. and Macdonald, I.G.: Introduction to Commutative Algebra
2. Matsumura, H.: Commutative Ring Theory
3. Reid, M: Undergraduate Commutative Algebra
4. Jacobson, N: Basic Algebra-II, Dover Publications, Inc.
5. Gopalakrishnan, N.S.: Commutative Algebra
27M.Sc. Mathematics (CBCEGS) (Semester–III)
(Elective Paper)
MTL 539
THEORY OF WAVELETS
L T P
4 0 0
Unit-I
Orthonormal systems and basic properties, Trigonometric system, Walsh orthonormal system,
Haar system, Generalization of Orthonormal system (Frames and Riesz basis) and their
examples.
Unit-II
Introduction to wavelets: Definition and examples of continuous and discrete wavelet transforms.
Multi-resolution analysis, Properties of translation, dilation and rotation operators, Wavelet and
scaling series.
Unit-III
Wavelets and signal analysis, Denoising, Representation of signals by frames.
BOOKS RECOMMENDED:
1. D.F. Walnut, An Introduction to Wavelet Analysis, Birkhauser, Boston, Basel 2000.
2. F. Schipp, W.R. Wade and P.Simon, Walsh Series: An Introduction to Dyadic Harmonic
Analysis, Adam Hilger, Bristol, New York 1990.
3. O. Christensen, An Introduction to Frames and Reisz Bases, Birkhauser, Basel 2008.
4. K.P. Soman & K.I. Ramachandran. Insight into Wavelets: From Theory to Prachce,
Prentice Hall of India, 2008.
28M.Sc. Mathematics (CBCEGS) (Semester–III)
(Elective Paper)
MTL 541
FOURIER ANALYSISL T P4 0 0
Prerequisite: Real Analysis
Unit ITrigonometric Series, Basic Properties of Fourier Series, Riemann-Lebesgue Lemma, TheDirichlet and Fourier Kernels, Continuous and Discrete Fourier Kernels, Pointwise and UniformConvergence of Fourier Series.
Unit IICesaro and Abel Summability. Fejer's Kernel, Fejer's theorem, a continuous function withdivergent Fourier series, termwise integration, termwise differentiation, Lebesgue’s pointwiseconvergence theorem.
Unit IIIFinite Fourier Transforms, Convolutions, the exponential form of the Lebesgue’s theorem, theFourier Transforms and Residues, inversions of the trigonometric and exponential forms, FourierTransformations of derivatives and integrals.
BOOKS RECOMMENDED:
1. R. Strichartz, A Guide to Distributions and Fourier Transforms, CRC Press.
2. E.M. Stein and R. Shakarchi, Fourier Analysis: An Introduction, Princeton University
Press, Princeton 2003.
3. G. Bachman, L. Narici, E. Beckenstein; Fourier and Wavelet Analysis, Universitext,
Springer-Verlag, New York, 2000. x+505 pp. ISBN: 0-387-98899-8
Programming Basics: Basic Structure of C-Program, Constants, variables, Data types,
Assignments, console I/O statements, Arithmetical, Relational and logical operators, Control
statements: if, switch, while, do while, for, continue, goto and break.
Unit-II
Functions, Arrays and Pointers: Function definition and declaration, Arguments, return
values and their types, Recursion. One and two-dimensional arrays, Initialization, Accessing
array elements, Functions with arrays. Address and pointer variables, declaration and
initialization, pointers and arrays, pointers and functions.
Unit-III
Structure, Union and File Handling: Definition, structure initialization, structure processing,
nested structure, Array of structures, structure and functions. Union. Defining and opening a file,
closing a file, Input/Output operations on files.
Practical: Based on implementation of Numerical and Statistical Techniques Using
C Language.
Solution to non linear equations, a system of linear equations; Numerical integration, Solution to
ordinary differential equations. Measures of central tendency, Correlation and Regression.
BOOKS RECOMMENDED:
1. Byron S. Gottrfried: Programming with C (Schaum’s outline series).2. Stan Kelly-Bootle: Mastering Turbo C.3. Brain Kernighan & Dennis Ritchi: The C Programming Language.4. Yashavant Kanetkar: Let us C.5. E Balagurusamy: Programming in ANSI C.6. R.S. Salaria: Application Programming in C.
34M.Sc. Mathematics (CBCEGS) (Semester–IV)
(Elective Paper)
MTL 583
OPERATIONS RESEARCH-II
L T P4 0 0
Unit-I
Queueing Theory: Introduction, Queueing System, elements of queueing system, distributions
of arrivals, inter arrivals, departure service times and waiting times. Classification of queueing
1. Trambley, J.P. and Manohar,R: Discrete Mathematical Structures with Applications to
Computer Science.
2. Liu C.L.: Elements of Discrete Mathematics.
3. Alan Doerr and Kenneth Levasseur: Applied Discrete Structures for Computer Science
4. Narsingh Deo: Graph Theory with Applications to Engineering and Computer Sciences
36M.Sc. Mathematics (CBCEGS) (Semester–IV)
(Elective Paper)MTL 586
BANACH ALGEBRA AND OPERATOR THEORY
L T P4 0 0
Unit-I
Banach Algebras: Definitions and simple examples. Regular and singular elements. Topological
divisors of zero, Spectrum of an element of a Banach Algebra, formula for spectral radius.
Unit-II
Compact and Bounded Operators: Spectral properties of compact linear operators, spectral
properties of bounded linear operators, operator equations involving compact linear operators,
Spectral radius of a bounded linear operator.
Unit-III
Spectral properties of bounded self adjoint linear operators on a complex Hilbert space. Positive
operators. Monotone sequence theorem for bounded self adjoint operators on a complex Hilbert
space. Square roots of a positive operator. Projection operators, Properties of projection
operators.
BOOKS RECOMMENDED:
1. Simmons, G.F.: Introduction to Topology and Modern Analysis
(Section 6.4-6.8), Mc Graw- Hill (1963) International Book
Company.
2. Kreyszig, E. Introductory Functional Analysis with Applications,
(Sections 8.1-8.5, 9.1-9.6) John Wiley & Sons,
New York, 1978.
37M.Sc. Mathematics (CBCEGS) (Semester–IV)
(Elective Paper)
MTL 588
FINANCIAL DERIVATIVES
L T P4 0 0
Unit-I
Products and Markets: Time Value of money, Periodic and Continuous compoundingCommodities, equities Currencies, Indices, Fixed income securities, Derivatives: BasicConcepts, Pay-off diagrams, Risk and Return, One step Binomial model. Random behaviour ofassets, Time scales, Wiener Process.
Unit-II
Forwards contracts and future contracts. Options, call and Put options, Put-call parity, Bounds onOption Prices, European and American calls. Elementary Stochastic Calculus: Motivation andexamples, Brownian Motion, Mean Square limit..
Unit-IIIIto’s Lemma, Some Pertinent examples, Black Scholes Model: Arbiterage, The derivation ofBlack Schole, Partial differential equation, Reduction of Black Scholes equation to diffusionequation, Numerical solutions of Black Scholes equation.
Recommended Books:
1. M. Capinski and T. Zastawniak: Mathematics for Finance: An Introduction toFinancial Engineering, Springer
2. P.Wilmott: The Theory and Practice of FinancialEngineering, John Willey and Sons, London,1998.
3. P.Wilmott, Sam Howison and Jeff Dewynne: The Mathematics of FinancialDerivatives, Cambridge UniversityPress, 1995.
38M.Sc. Mathematics (CBCEGS) (Semester–IV)
(Elective Paper)MTL 589
THEORIES OF INTEGRATIONL T P4 0 0
Prerequisites: Real Analysis, Measure Theory
Unit- IThe need to extend the Lebesgue integral, The Darboux integral, necessary and sufficientconditions for Darboux integrability, the equivalence of the Riemann and Darboux integrals,tagged divisions and their use in elementary real analysis, Cousin's lemma, the Henstock-Kurzweil and McShane integrals, Saks-Henstock Lemma, the fundamental theorems of Calculusfor the gauge integrals and its consequences.
Unit- IIThe Squeeze theorem, regulated functions and their integrability, Convergence theorems for thegauge integrals, the Hake's Theorem, The McShane integral Vs Lebesgue integral, extensions ofAbsolute Continuity and Bounded variation, the relationship between the function classesACG*, ACG\delta, BVG* and BVG\delta, the Denjoy and Perron integrals.
Unit- IIILocally and globally small Riemann sums and their equivalence, The class of Henstock-Kurzweil integrable functions, advantages of the Henstock-Kurzweil integral over Riemann,Lebesgue, Denjoy, Perron and McShane integrals.
BOOKS RECOMMENDED:
1. R. A. Gordon, The Integrals of Lebesgue, Denjoy, Perron and Henstock, Amer. Math.Soc. Province, RI, (1994).
2. R.G. Bartle, A Modern Theory of Integration, Graduate Studies in Mathematics, 32.Amer. Math. Soc., Province, RI (2001).
3. D. S. Kurtz; C. W. Schwatz, Theories of Integration, the Integrals of Riemann,Lebesgue, Henstock-Kurzweil and McShane, Series in Real Analysis 9, World ScientificPublishing Co., Inc., NJ, (2004).
4. P. Y. Lee; R. Vyborny, Integral: An Easy Approach after Kurzweil and Henstock, Aus.Math. Soc. Lecture Series 14. Cambridge University Press, Cambridge, (2000).
39M.Sc. Mathematics (CBCEGS) (Semester–IV)
(Elective Paper)
MTL 590
ALGEBRAIC TOPOLOGY
L T P
4 0 0
Unit-I
N-manifolds, orientable vs nonorientable manifolds, Compact-connected 2-manifolds,
Classification theorem for compact surfaces, Triangulations of compact surfaces, The Euler
characteristic of a surface, Fundamental group of a space, Fundamental group of Circle and
product spaces, The Brouwer fixed point theorem in dimension 2, Homotopy type and
Homotopy Equivalence of Spaces.
Unit-II
The Seifert Von Kampen Theorem: Weak products of abelian groups, free abelian groups, free
products of groups, free groups, The Seifert Von Kampen theorem and its applications, Structure
of the fundamental group of a Compact surface.
Unit-III
Covering Spaces: Lifting of paths to covering spaces, The funcdamental group of a covering
space, Homomorphism and automorphisms of covering spaces, Regular covering spaces and
Quotient spaces, The Borsuk-Ulam theorem for 2-sphere, The existence theorem for covering
spaces.
BOOKS RECOMMENDED:
1. W.S. Massey. A Basic Course in Algebraic Topology, Springer
(Indian reprint) 2007. (Ch. 1-5)
2. J.R. Munkres. Topology, Prentice Hall of India (India reprint) 2007.
40M.Sc. Mathematics (CBCEGS) (Semester–IV)
(Elective Paper)
MTL 591THEORY OF SAMPLE SURVEY
L T P4 0 0
Unit-IConcepts of population, population unit, sample, sample size, parameter, statistics estimator,biased and unbiased estimator, mean square error, standard error. Census and Sample surveys,Sampling and Non sampling errors,Concepts of Probability and non-probability sampling, sampling scheme and sampling strategy,Introduction of Simple Random Sampling (Use of Lottery Method, Random numbers andPseudo random numbers)
Unit-IISimple Random sampling (with or without replacement); Estimation of population Mean andTotal, Expectation and Variance of these Estimators, unbiased estimators of the variance of theseestimatorsEstimation of Population proportion and Variance of these estimators, estimation of sample sizebased on desired accuracy, Confidence interval for population Mean and Proportion
Unit-IIIConcepts of Stratified population and stratified sample, estimation of population mean and Totalbased on stratified sample. Expectation and variance of estimator of population mean and totalassuming SRSWOR within strata. Unbiased estimator of the variances of these estimators.Proportional Allocation, Optimum allocation (Neyman allocation) with and without varyingcosts, Comparison of simple random sampling and stratified random sampling with proportionaland optimum allocations.
BOOKS RECOMMENDED:
1. Sukhatme P.V., Sukkhatme P.V., Sukhatme S. & Ashok C. (1997): Sampling Theory ofSurveys and Applications-Piyush Publications.
2. Des Raj and P.Chandok (1998): Sample Survey Theory. Narosa Publishing House.