Indian Institute of Technology, Patna

Courses

B.Tech Courses

First Semester

MA101 Mathematics - I

MA101	Mathematics - I	3-1-0-8	Pre-requisites: nil

Properties of real numbers. Sequences of real numbers, montone sequences, Cauchy sequences, divergent sequences. Series of real numbers, Cauchy’s criterion, tests for convergence. Limits of functions, continuous functions, uniform continuity, montone and inverse functions. Differentiable functions, Rolle's theorem, mean value theorems and Taylor's theorem, power series. Riemann integration, fundamental theorem of integral calculus, improper integrals. Application to length, area, volume, surface area of revolution. Vector functions of one variable and their derivatives. Functions of several variables, partial derivatives, chain rule, gradient and directional derivative. Tangent planes and normals. Maxima, minima, saddle points, Lagrange multipliers, exact differentials. Repeated and multiple integrals with application to volume, surface area, moments of inertia. Change of variables. Vector fields, line and surface integrals. Green’s, Gauss’ and Stokes’ theorems and their applications.

Texts:

G. B. Thomas and R. L. Finney, Calculus and Analytic Geometry, 6th Ed/ 9th Ed, Narosa/ Addison Wesley/ Pearson, 1985/ 1996.
T. M. Apostol, Calculus, Volume I, 2nd Ed, Wiley, 1967.
T. M. Apostol, Calculus, Volume II, 2nd Ed, Wiley, 1969.

References:

R. G. Bartle and D. R. Sherbert, Introduction to Real Analysis, 5th Ed, Wiley, 1999.
J. Stewart, Calculus: Early Transcendentals, 5th Ed, Thomas Learning (Brooks/ Cole), Indian Reprint, 2003.

Second Semester

MA102 Mathematics - II

MA102	Mathematics - II	3-1-0-8	Pre-requisites: nil

Linear Algebra: Vector spaces (over the field of real and complex numbers). Systems of linear equations and their solutions. Matrices, determinants, rank and inverse. Linear transformations. Range space and rank, null space and nullity. Eigenvalues and eigenvectors. Similarity transformations. Diagonalization of Hermitian matrices. Bilinear and quadratic forms.

Ordinary Differential Equations: First order ordinary differential equations, exactness and integrating factors. Variation of parameters. Picard's iteration. Ordinary linear differential equations of n-th order, solutions of homogeneous and non-homogeneous equations. Operator method. Method of undetermined coefficients and variation of parameters.

Power series methods for solutions of ordinary differential equations. Legendre equation and Legendre polynomials, Bessel equation and Bessel functions of first and second kind.

Systems of ordinary differential equations, phase plane, critical point, stability.

Texts:

K. Hoffman and R. Kunze, Linear Algebra, Prentice Hall, 1996.
T. M. Apostol, Calculus, Volume II, 2nd Edition, Wiley, 1969.
S. L. Ross, Differential Equations, 3rd Edition, Wiley, 1984.
E. A. Coddington, An Introduction to Ordinary Differential Equations, Prentice Hall, 1995.
W.E. Boyce and R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 7th Edition, Wiley, 2001.

References:

E. Kreyszig, Advanced Engineering Mathematics, 9th Edition, Wiley, 2005.

Third Semester

MA201 Mathematics - III

MA201	Mathematics - III	3-1-0-8	Pre-requisites: nil

Complex Analysis: Complex numbers, geometric representation, powers and roots of complex numbers. Functions of a complex variable: Limit, Continuity, Differentiability, Analytic functions, Cauchy-Riemann equations, Laplace equation, Harmonic functions, Harmonic conjugates. Elementary Analytic functions (polynomials, exponential function, trigonometric functions), Complex logarithm function, Branches and Branch cuts of multiple valued functions. Complex integration, Cauchy's integral theorem, Cauchy's integral formula. Liouville’s Theorem and Maximum-Modulus theorem, Power series and convergence, Taylor series and Laurent series. Zeros, Singularities and its classifications, Residues, Rouches theorem (without proof), Argument principle (without proof), Residue theorem and its applications to evaluating real integrals and improper integrals. Conformal mappings, Mobius transformation, Schwarz-Christoffel transformation.

Fourier series: Fourier Integral, Fourier series of 2p periodic functions, Fourier series of odd and even functions, Half-range series, Convergence of Fourier series, Gibb’s phenomenon, Differentiation and Integration of Fourier series, Complex form of Fourier series.

Fourier Transformation: Fourier Integral Theorem, Fourier Transforms, Properties of Fourier Transform, Convolution and its physical interpretation, Statement of Fubini’s theorem, Convolution theorems, Inversion theorem

Partial Differential Equations: Introduction to PDEs, basic concepts, Linear and quasi-linear first order PDE, Second order PDE and classification of second order semi-linear PDE, Canonical form.. Cauchy problems. D’ Alemberts formula and Duhamel’s principle for one dimensional wave equation, Laplace and Poisson equations, Maximum principle with application, Fourier method for IBV problem for wave and heat equation, rectangular region. Fourier method for Laplace equation in three dimensions.

Texts:

R. V. Churchill and J. W. Brown, Complex Variables and Applications, 5th Edition, McGraw-Hill, 1990.
K. Sankara Rao, Introduction to Partial Differential Equations, 2nd Edition, 2005.

References:

J. H. Mathews and R. W. Howell, Complex Analysis for Mathematics and Engineering, 3rd Edition, Narosa, 1998.
I. N. Sneddon, Elements of Partial Differential Equations, McGraw-Hill, 1957.
E. Kreyszig, Advanced Engineering Mathematics, 5th / 8th Edition, Wiley Eastern / John Wiley, 1983/1999

Fourth Semester

MA212 Algebra and Number Theory

MA212	Algebra and Number Theory	3-0-0-6	Pre-requisites:Nil

Algebra: Semigroups, groups, subgroups, normal subgroups, homomorphisms, quotient groups, isomorphisms. Examples: group of integers modulo m, permutation groups, cyclic groups, dihedral groups, matrix groups. Sylow's theorems and applications. Basic properties of rings, units, ideals, homomorphisms, quotient rings, prime and maximal ideals, fields of fractions, Euclidean domains, principal ideal domains and unique factorization domains, polynomial rings. Finite field extensions and roots of polynomials, finite fields.

Number Theory: Divisibility, primes, fundamental theorem of arithmetic. Congruences, solution of congruences, Euler's Theorem, Fermat's Little Theorem, Wilson's Theorem, Chinese remainder theorem, primitive roots and power residues. Quadratic residues, quadratic reciprocity. Diophantine equations, equations ax+by=c, x²+y²=z², x⁴+y⁴=z² Simple continued fractions: finite, infinite and periodic, approximation to irrational numbers, Hurwitz's theorem, Pell's equation. Partition functions: Formal power series, generating functions and Euler's identity, Euler's theorem, Jacobi's theorem, congruence properties of p(n). Arithmetical functions: Φ(n), μ(n), d(n), σ(n). A particular Dirichlet series for Riemann Zeta Function.

Texts:

I. N. Herstein. Topics in Algebra, Wiley, 2006
I. Niven, H.S. Zuckerman, H.L. Montgomery. An introduction to the theory of numbers, Wiley, 2000

References:

D.S. Dummit & R.M. Foote. Abstract Algebra, Wiley, 1999
G.H. Hardy, E.M. Wright. An introduction to the theory of numbers, OUP, 2008
T.M. Apostol. Introduction to Analytic Number Theory, Springer, UTM, 1998

MA214 Introduction to Computational Topology

MA214	Introduction to Computational Topology	3-0-0-6	Pre-requisites: nil

Introduction and general notions of point set topology : Open and Closed Sets, Neighbourhoods, Connectedness and Compactness, Separation, Continuity.

An overview of topology and classification of surfaces : Surfaces – orientable and non-orientable, their topology, classification of closed suraces

Combinatorial Techniques : Simplicial complexes, and simplicial maps, triangulations, Euler characteristics, Maps on surfaces.

Homotopy and Homology Groups: Introducing Groups and concept of Homotopy, fundamental group and its calculations, Homology.

Calculating Homology : Computation of homology of closed surfaces.

Topics in Geometry : Delauny triangulations, Voronoi diagrams, Morse functions

Texts:

Afra Zomordian: Topology for Computing, CUP, 2005
H. Edelsbrunner and J. Harer. Computational Topology. An Introduction. Amer. Math. Soc., Providence, Rhode Island, 2009
J. J. Rotman: An introduction to Algebraic Topology, GTM- 119, Springer, 1998

References:

Tomasz K., K. Mischaikow and M. Mrozek, Computational Homology, Springer, 2003
H.Edelsbrunner, Geometry and Topology for Mesh Generation, CUP, 2001
D. Kozlov, Combinatorial Algebraic Topology, Springer, 2008
V. A. Vassiliev, Introduction to Topology, AMS, 2001
R. Messer and P. Straffin, Topology Now, MAA, 2006

MA231 INTRODUCTION TO NUMERICAL METHODS

MA231	INTRODUCTION TO NUMERICAL METHODS	3-0-0-6	Pre-requisites:Nil

Number Representation and Errors: Numerical Errors; Floating Point Representation; Finite Single and Double Precision Differences; Machine Epsilon; Significant Digits.

Numerical Methods for Solving Nonlinear Equations: Method of Bisection, Secant Method, False Position, Newton‐Raphson's Method, Multidimensional Newton's Method, Fixed Point Method and their convergence.

Numerical Methods for Solving System of Linear Equations: Norms; Condition Numbers, Forward Gaussian Elimination and Backward Substitution; Gauss‐Jordan Elimination; FGE with Partial Pivoting and Row Scaling; LU Decomposition; Iterative Methods: Jacobi, Gauss Siedal; Power method and QR method for Eigen Value and Eigen vector.

Interpolation and Curve Fitting: Introduction to Interpolation; Calculus of Finite Differences; Finite Difference and Divided Difference Tables; Newton‐Gregory Polynomial Form; Lagrange Polynomial Interpolation; Theoretical Errors in Interpolation; Spline Interpolation; Approximation by Least Square Method.

Numerical Differentiation and Integration: Discrete Approximation of Derivatives: Forward, Backward and Central Finite Difference Forms, Numerical Integration, Simple Newton‐Cotes Rules: Trapezoidal and Simpson's (1/3) Rules; Gaussian Quadrature Rules: Gauss‐Legendre, Gauss‐Laguerre, Gauss‐Hermite, Gauss‐Chebychev.

Numerical Solution of ODE & PDE: Euler's Method for Numerical Solution of ODE; Modified Euler's Method; Runge‐Kutta Method (RK2, RK4), Error estimate; Multistep Methods: Predictor‐Corrector method, Adams‐Moulton Method; Boundary Value Problems and Shooting Method; finite difference methods, numerical solutions of elliptic, parabolic, and hyperbolic partial differential equations.

Exposure to software package MATLAB.

Texts:

K. E. Atkinson, Numerical Analysis, John Wiley, Low Price Edition (2004).
S. D. Conte and C. de Boor, Elementary Numerical Analysis ‐ An Algorithmic Approach, McGraw‐Hill, 2005.

References:

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, 2nd Edition, Texts in Applied Mathematics, Vol. 12, Springer Verlag, 2002.
J. D. Hoffman, Numerical Methods for Engineers and Scientists, McGraw‐Hill, 2001.
M.K Jain, S.R.K Iyengar and R.K Jain, Numerical methods for scientific and engineering computation (Fourth Edition), New Age International (P) Limited, New Delhi, 2004.
S. C. Chapra, Applied Numerical Methods with MATLAB for Engineers and Scientists, McGraw‐Hill 2008.

MA251 Optimization Techniques

MA251	Optimization Techniques	3-0-0-6	Pre-requisites: nil

Introduction to linear and non-linear programming. Problem formulation. Geo- metrical aspects of LPP, graphical solution. Linear programming in standard form, simplex, Big M and Two Phase Methods. Revised simplex method, special cases of

LP. Duality theory, dual simplex method. Sensitivity analysis of LP problem. Transportation, assignment and traveling salesman problem.

Integer programming problems-Branch and bound method, Gomory cutting plane method for all integer and for mixed integer LP.

Theory of games: Computational complexity of the Simplex algorithm, Karmarkar's algorithm for LP.

Unconstrained Optimization, basic descent methods, conjugate direction and Newton's methods. Acquaintance to Optimization softwares like TORA.

Texts:

Hamdy A. Taha, Operations Research: An Introduction, Eighth edition, PHI, New Delhi (2007).
S. Chandra, Jayadeva, Aparna Mehra, Numerical Optimization with Applications, Narosa Publishing House (2009).
A. Ravindran, Phillips, Solberg, Operation Research, John Wiley and Sons, New York (2005).
M. S. Bazaraa, J. J. Jarvis and H. D. Sherali, Linear Programming and Network Flows, 3rd Edition, Wiley (2004).

References:

D. G. Luenberger, Linear and Nonlinear Programming, 2nd Edition, Kluwer, 2003. S. A. Zenios (editor), Financial Optimization, Cambridge University Press (2002).
F. S. Hiller, G. J. Lieberman, Introduction to Operations Research, Eighth edition, McGraw Hill (2006).

MA225 Probability Theory and Random Process

MA225	Probability Theory and Random Process	3 1 0 8	Pre-requisites: nil

Axiomatic construction of the theory of probability, independence, conditional probability, and basic formulae, random variables, probability distributions, functions of random variables; Standard univariate discrete and continuous distributions and their properties, mathematical expectations, moments, moment generating function, characteristic functions; Random vectors, multivariate distributions, marginal and conditional distributions, conditional expectations; Modes of convergence of sequences of random variables, laws of large numbers, central limit theorems. Definition and classification of random processes, discrete-time Markov chains, Poisson process, continuous-time Markov chains, renewal and semi-Markov processes, stationary processes, Gaussian process, Brownian motion, filtrations and martingales, stopping times and optimal stopping.

Texts:

G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, Oxford University Press, 2001.
P. G. Hoel, S. C. Port and C. J. Stone, Introduction to Probability Theory, Universal Book Stall, 2000.
W. Feller, An Introduction to Probability Theory and its Applications, Vol. 1, 3rd Edition, Wiley, 1968.
K. S. Trivedi, Probability and Statistics with Reliability, Queuing, and Computer Science Applications, Prentice Hall of India, 1998.
A. Papoulis and S. Unnikrishna Pillai, Probabilities, Random Variables and Stochastic Processes, 4th Edition, Tata McGraw-Hill, 2002.
S.M. Ross, Stochastic Processes, 2nd Edition, Wiley, 1996.
J. Medhi, Stochastic Processes, New Age International, 1994.

BS in Mathematics & Computing

BS in Mathematics & Computing

Discipline	Curriculum
BS in Mathematics & Computing Curriculum	2021 Admission (PDF)

M.Sc. in Mathematics

M.Sc Course

Detailed Course

M.Tech. in Mathematics and Computing

M.Tech Course

Detailed course

Ph.D. Courses

Algebra

MA701	Algebra	3-0-0-6

Abstract Algebra:

Elementary set theory. Groups, subgroups, normal subgroups, homomorphisms, quotient groups, automorphisms, groups acting on sets, Sylow theorems and applications, finitely generated abelian groups. Examples: permutation groups, cyclic groups, dihedral groups, matrix groups. Basic properties of rings, units, ideals, homomorphisms, quotient rings, prime and maximal ideals, fields of fractions, Euclidean domains, principal ideal domains and unique factorization domains, polynomial rings. Elementary properties of finite field extensions and roots of polynomials, finite fields.

Linear Algebra:

Vector spaces, Bases and dimensions, Change of bases and change of coordinates, Sums and direct sums, Quotient spaces. Linear transformations, Representation of linear transformations by matrices, The rank and nullity theorem, Dual spaces, Transposes of linear transformations. Trace and determinant, Eigenvalues and eigenvectors, Invariant subspaces, Direct-Sum decomposition, Cyclic subspaces and Annihilators, The minimal polynomial, The Jordan canonical form. Spectral theorem for normal operators, Quadratic forms. Singular value decomposition, polar decomposition.

Text Books:

D. Dummit, R. Foote, Abstract Algebra, 3rd edition, Wiley, 2004.
K. Hoffman and R. Kunze, Linear Algebra, Prentice-Hall, 1996.

Reference Books:

S. Axler, Linear Algebra Done Right, 2nd Edition, UTM Series, Springer, 1997.
M. Artin, Algebra, Prentice Hall, 1994.

Analysis-1

MA711	Analysis-1	3-0-0-6

Topology:

Topological spaces, Basis for a topology, Limit points and closure of a set, Continuous and open maps, Homeomorphisms, Subspace topology, Product and quotient topology. Connected and locally connected spaces, Path connectedness, Components and path components, Compact and locally compact spaces, One point compactification. Countability axioms, Separation axioms, Urysohn’s Lemma, Urysohn’s metrization theorem, Tietze extension theorem, Tychonoff’’s theorem, Completely Regular Spaces, Stone-Cech Compactification.

Functional Analysis:

Banach spaces, Continuity of linear maps, Hahn-Banach theorem, Open mapping and closed graph theorems, Uniform boundedness principle. Duals and Transposes. Compact operators and their spectra. Weak and Weak* convergence, Reflexivity. Hilbert spaces, Bounded operators on Hilbert spaces. Adjoint operators, Normal, Unitary, Self-adjoint operators and their spectra. Spectral theorem for compact self-adjoint operators, statement of spectral theorem for bounded self-adjoint operators.

Text Books:

James R. Munkres, Topology, 2nd Edition, Prentice Hall, 1999.
E. Kreyszig, Introductory Functional Analysis with Applications, Wiley, 1989.

Reference Books:

J. L. Kelley, General Topology, Springer International Edition, Indian Reprint, 2005.
M. A. Armstrong, Basic Toplogy, Springer- Verlag, 1997
M. Thamban Nair, Functional Analysis: A First Course, Prentice Hall of India, 2002.
G. F. Simmons, Introduction to Topology and Modern Analysis, Wiley, 2003.
J. B. Conway, A Course in Functional Analysis, GTM Series, Springer, 1990.

Differental Equation

MA731	Differental Equation	3-0-0-6

Ordinary Differential Equations:

First Order ODE y'=f(x,y)-geometrical Interpretation of solution, Equations reducible to separable form, Exact Equations, Integrating factor, Linear Equations, Orthogonal trajectories, Picard’s Theorem for IVP and Picard’s iteration method, Euler’ Method, Improved Euler’s Method, Elementary types of equations. F(x,y,y') =0; not solved for derivative, Second Order Linear

Differential equations:

fundamental system of solutions and general solution of homogeneous equation. Use of Known solution to find another, Existence and uniqueness of solution of IVP, Wronskian and general solution of non-homogeneous equations. Euler-Cauchy Equation, extensions of the results to higher order linear equations, Power Series Method application to Legendre Eqn., Legendre Polynomials, Frobenious Method, Bessel equation, Properties of Bessel functions, Sturm-Liouville BVPs, Orthogonal functions, Sturm comparison Theorem. Systems of Linear ODEs, Reduction of higher order linear ODEs to first order linear systems, Stability of linear systems.

Transforms:

Fourier Series, Fourier transform and Laplace Transform. Solving Differential Equations using Transform methods.

Partial Differential Equations:

Introduction to PDE, basic concepts, Linear and quasilinear first order PDE, Cauchy-Kowalewski theorem, second order PDE and classification of second order semilinear PDE (Canonical form), D’ Alemberts formula and Duhamel’s principle for one dimensional wave equation, Laplace’s and Poisson’s equations, Maximum principle with application, Fourier Method for IBV problem for wave and heat equation, rectangular region, Fourier method for Laplace’s equation in three dimensions.

Text Books:

G. Birkhoff and G. C. Rota, Ordinary Differential Equations, 4th Edition, Wiley Singapore Edition, 2003.
I. N. Sneddon, Elements of Partial Differential Equations, Dover, 2006.

Reference Books:

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Tata McGraw Hill, 1984.
R. Haberman, Applied Partial Differential Equations, 4th Edition, Prentice Hall, 2003.
L. Perko, Differential Equations and Dynamical Systems, Springer – Verlag, 2006

Analysis-II

MA712	Analysis-II	3-0-0-6

Complex Analysis:

Basic properties of the complex number system. Analytic functions, Cauchy-Riemann equations, elementary functions and their basic properties (rational functions, exponential function, logarithm function, trigonometric functions, roots functions). Cauchy’s theorem and Cauchy’s integral formula, Liouville’s theorem, Morera’s theorem and Maximum Modulus principle. Power series and Laurent series, isolation of zeros. Singularities and classification of isolated singularities (including singularity at infinity). Argument principle, Rouche’s theorem, Casorati-Weierstrass Theorem, Schwarz’s lemma, Residue theorem, evaluation of definite and improper integrals. Linear fractional transformations and mapping properties, Conformal maps. Statement of Riemann Mapping Theorem.

Measure Theory:

Algebras and sigma algebras, measures, outer measures, measurable sets, Lebesgue measure and its properties, non-measurable sets, measurable functions and their properties, Egoroff’s theorem, Lusin’s theorem; Lebesgue Integration: simple functions, integral of bounded functions over a set of finite measure, bounded convergence theorem, integral of nonnegative functions, Fatou’s lemma, monotone convergence theorem, the general Lebesgue integral, Lebesgue convergence theorem, change of variable formula; Differentiation and integration: functions of bounded variation, differentiation of an integral, absolute continuity; Signed and complex measures, Radon-Nikodym theorem. Product measures, constructions, Fubini’s theorem and its applications.

Text Books:

J. B. Conway, Functions of One Complex Variable, 2nd Edition, Springer, 2005
G.de Barra, Measure Theory and Integration, New Age International, 2000.

Reference Books:

T. W. Gamelin, Complex Analysis, Springer, 2001.
D. L. Cohn, Measure Theory, 1st Edition, Birkhauser, 1994.
H. L. Royden, Real Analysis, 3rd Edition, Prentice Hall/Pearson Education, 1988.

Advanced Optimization Techniques

MA751	Advanced Optimization Techniques	3-0-0-6

Nonlinear programming: Convex sets and convex functions, their properties, convex programming problem, generalized convexity, Pseudo and Quasi convex functions, Invex functions and their properties, KKT conditions.

Goal Programming: Concept of Goal Programming, Model Formulation, Graphical solution method.

Separable programming. Geometric programming: Problems with positive coefficients up to one degree of difficulty, Generalized method for the positive and negative coefficients.

Search Techniques:Direct search and gradient methods, Unimodal functions, Fibonacci method, Golden Section method, Method of steepest descent, Newton-Raphson method, Conjugate gradient methods.

Dynamic Programming: Deterministic and Probabilistic Dynamic Programming, Discrete and continuous dynamic programming, simple illustrations.

Multiobjective Programming:Efficient solutions, Domination cones.

Text Books:

Mokhtar S. Bazaaraa, Hanif D. Shirali and M.C.Shetty, Nonlinear Programming, Theory and Algorithms, John Wiley & Sons, New York (2004).

Reference Books:

D. G. Luenberger, Linear and Nonlinear Programming, Second Edition, Addison Wesley (2003).
DR. E. Steuer, Multi Criteria Optimization, Theory, Computation and Application, John Wiley and Sons, New York (1986).

Topology

MA713	Topology	3-0-0-6

Quotient topology, Topological groups, Group Actions, Orbit spaces.

Homotopic maps, Construction of the fundamental group, Fundamental group of circle, Homotopy type, Covering spaces, Borsuk-Ulam and Ham-sandwich theorems, A lifting criterion, Seifert-van kampen theorem, Brouwer fixed point theorem and other applications

Polyhedra, PL maps, PL manifolds, Cell complexes, Subdivisions, Simplicial complexes, Simplicial maps, Triangulations, Derived subdivisions, Pseudomanifolds, Abstract simplicial complexes, isomorphism.

Orientation of complexes, Chains, Cycles and boundaries, Homology groups, Euler-Poincare formula, Barycentric subdivision, Simplicial approximation, Induced homomorphism, Degree and Lefschetz number fixed-point theorem.

Text Books:

C. A. Kosniowski, First course in Algebraic Topology, Cambridge Univ. Press, 2008.
C. P. Rourke, and B. J. Sanderson, Introduction to Piecewise-Linear Topology, Springer-Verlag, 1982.

Reference Books:

J. R. Munkres, Elements of Algebraic Topology, The Benjamin Cummings Pub., Co., 1984.
M. A. Armstrong, Basic Topology, Springer (India), 2004.
K. Janich, Topology, Springer-Verlag (UTM), 1984.

Probability Theory and Statistical Inference

MA 771	Probability Theory and Statistical Inference	3-1-0-8	Pre-requisites: nil

Probability Theory:

Algebra of sets, Measure and probability measure, Random variables, Standard discrete and continuous distributions, Conditional distributions and their independence, Distribution of functions of random variables, Expectation, Variance, Correlation, Moment generating functions and their properties, Convergence of random variables, Characteristic functions and properties, Laws of large numbers, Limit theorems.

Statistical Inference:

Parametric and nonparametric models, Exponential families, Sufficiency, Completeness, Basu's Theorem, Invariance and maximal invariant statistic.

Point Estimation:

Unbiased estimation, maximum likelihood estimation, method of moments, Loss functions, Risk Functions, Bayesian methods, Minimax and admissible estimators, Interval estimation, Equivariance Principle

Hypothesis Testing:

Neyman-Pearson theory, Most Powerful (MP)Test, UMP Test, Unbiased Test, Monotone likelihood ratio property, Likelihood ratio tests, Wald’s Sequential Probability ratio Test (SPRT), Invariant tests.

Text Books:

K.L. Chung, A course in probability theory, Second Edition, Academic Press, 2000
V.K., Rohatgi and Md. Ehsanes Saleh, An introduction to probability and statistics, Second Edition, Wiley India, 2009.

Reference Books:

G. Casella, and R.L. Berger, Statistical inference, Second Edition, Wadsworth, Belmont CA, 2001.
J.Shao, Mathematical statistics, Second Edition, Springer, 2008.
E.L.Lehmann and G.Casella, Theory of point estimation, Second Edition, Springer(India), 2003.

Mathematical Control Theory

MA752	Mathematical Control Theory	3-0-0-6

Control systems and Mathematical modeling, classification of control systems, finite dimensional Deterministic linear control systems, transfer function, state-space representation, computation of transition matrix and solution of linear system, controllability and observability for linear dynamical systems, duality theorem, stability, Liapunov's method, Routh criterion, Nyquist criterion, stabilizability, multivariable system, discrete system, optimal control problem, linear systems with quadratic cost, introduction to calculus of variations and maximum principle .

Text Books:

Zabczyk Jerzy , Mathematical Control Theory: An Introduction, Series: Modern Birkhäuser Classics, 1st ed. 1992. 2nd, corr. printing 1995. Reprint, 2008.
R.G. Cameron and S. Barnett, Introduction to Mathematical Control Theory, Oxford Univ Press, 1990.

Reference Books:

Eduardo D. Sontag, Mathematical Control Theory Deterministic Finite Dimensional Systems, Series: Texts in Applied Mathematics , Vol. 6, Springer, 1998.
D. Subbaram Naidu, Optimal Control Systems, Series: Electrical Engineering Series Volume: 2, CRC Press, 2002.