Combinatorics: Fundamental techniques of enumeration and construction of combinatorial structures, pigeon-hole principle, permutation, combination, summations. Introduction to recurrence relation and generating function, principle of inclusion-exclusion

Probability: Classical and axiomatic definitions of probability, Addition rule and conditional probability, Multiplication rule, Bayes' Theorem and independence, Discrete and continuous random variables: Uniform, Binomial, Geometric, Poisson, Exponential, Gamma, Normal distributions etc, Jointly distributed random variables. Mathematical expectation, Moments, Probability- and Moment-generating functions, Chebyshev's inequality, the strong and weak law of large numbers, Functions of Random Variables and their probability distributions

Statistics: The Central Limit Theorem, Distributions of the sample mean and the sample variance for a normal population, Sampling Distributions: Chi-Square, t and F distributions, Point and interval Estimation, maximum likelihood estimation, Null and alternative hypotheses, The critical and acceptance regions, Neyman-Pearson Fundamental Lemma, Tests for one sample and two samples problems for normal populations.

Probability: Axiomatic definition, Properties. Conditional probability, Bayes rule and independence of events. Random variables, Distribution function, Probability mass and density functions, Expectation, Moments, Moment generating function, Chebyshev’s inequality. Special distributions: Bernoulli, Binomial, Geometric, Negative Binomial, Hypergeometric, Poisson, Uniform, Exponential, Gamma, Normal, Joint distributions, Marginal and conditional distributions, Moments, Independence of random variables, Covariance, Correlation, Functions of random variables, Weak law of large numbers, P. Levy’s central limit theorem (i.i.d. finite variance case), Normal and Poisson approximations to binomial.

Statistics: Introduction: Population, Sample, Parameters. Point Estimation: Method of moments, MLE, Unbiasedness, Consistency, Comparing two estimators (Relative MSE). Confidence interval estimation for mean, difference of means, variance, proportions, Sample size problem, Test of Hypotheses:-N-P Lemma, Examples of MP and UMP tests, p-value, Likelihood ratio test, Tests for means, variance, Two sample problems, Test for proportions, Relation between confidence intervals and tests of hypotheses, Chi-square goodness of fit tests, Contingency tables, SPRT, Regression Problem:- Scatter diagram, Simple linear regression, Least squares estimation, Tests for slope and correlation, Prediction problem, Graphical residual analysis, Q-Q plot to test for normality of residuals, Multiple regression, Analysis of Variance: Completely randomized design and randomized block design, Quality Control: Shewhart control charts and Cusum charts.

Calculus: Rolle’s theorem, Lagranges theorem, Cauchy’s mean value theorem (Taylor’s and Maclaurin theorems with remainders), Indeterminate forms, Concavity and convexity of a curve, points of inflexion, maximum, minimum of a function, 2nd derivative test for max min, Asymptotes and curvature, Cartesian curve tracing, polar curve tracing.

Calculus of Several Variables: Limit, continuity and differentiability of functions of several variables, partial derivatives and their geometrical interpretation, differentials, derivatives of composite and implicit functions, derivatives of higher order and their commutativity, Euler’s theorem on homogeneous functions, harmonic functions, Taylor’s expansion of functions of several variables, maxima and minima of functions of several variables, Lagrange’s method of multipliers.

Vector Calculus: Double and triple integrals, Scalar and vector fields, level surfaces, directional derivative, Gradient, Curl, Divergence, line and surface integrals, theorems of Green, Gauss and Stokes. Beta and Gamma functions.

Ordinary Differential Equations: First order differential equations, exact, linear and Bernoulli’s form, second order differential equations with constant coefficients, Euler’s equations, particular integrals by: variation of parameters, undetermined coefficients, operator method, system of differential equations.

Biological and Artificial Neuron, Perceptron model, Adaline model, Different types of Activation functions, Learning Techniques: Supervised and Unsupervised, Multilayered feed forward Networks, Back propagation algorithm and its improvements, Applications of Back propagation algorithm to statistical pattern recognition, classification and regression problems, Advantages of Neural Networks over statistical classification techniques, Performance Surfaces and Optimum Points, Steepest Descent, Stable learning Rates, Minimizing Along a line ,Newton’s method and conjugate method. Recurrent networks, Radial Basis Function Networks as an interpolation model, Time delay neural networks for forecasting problems, Probabilistic Neural Networks, Kohonen’s self organizing map, Self organizing maps with quadratic functions and its applications medical imaging, Adaptive Resonance, Theory model, Applications of Art model for knowledge acquisition, Extensive sessions in MATLAB for solving statistical pattern recognition, classification, regression and prediction problems using different kinds of Neural Network models.

Probability: Classical and axiomatic definitions of probability, Addition rule and conditional probability, Multiplication rule, Bayes' Theorem and independence, Discrete and continuous random variables: Uniform, Binomial, Geometric, Poisson, Exponential, Gamma, Normal distributions etc, Jointly distributed random variables. Mathematical expectation, Moments, Probability- and Moment-generating functions, Chebyshev's inequality, the strong and weak law of large numbers, Functions of Random Variables and their probability distributions

Statistics: The Central Limit Theorem, Distributions of the sample mean and the sample variance for a normal population, Sampling Distributions: Chi-Square, t and F distributions, Point and interval Estimation, maximum likelihood estimation, Null and alternative hypotheses, The critical and acceptance regions, Neyman-Pearson Fundamental Lemma, Tests for one sample and two samples problems for normal populations.

Stochastic Processes: Definition and classification of stochastic processes, Discrete time Markov chains, Random walk, Gambler’s ruin, Branching Processes, Time reversible Markov chains, Markov chain Monte Carlo methods, Poisson Process, Non-homogeneous and compound Poisson Process, General continuous time Markov chains, Birth and Death Process, Uniformization, Renewal process, Regenerative process, Semi Markov process, Application to queues, Brownian and stationary process.

Rings and Algebra, Monotone classes. Measures and outer measures. Measurable sets; Lebesgue Measure and its properties. Measurable functions and their properties, Convergence in measure. Integration: Sequence of integrable functions; Signed measures, Hahn and Jordan decomposition, Absolute continuity of measures, Radon-Nikodym theorem; Product measures, Fubini's theorem; Transformations and functions: The isomorphism theorem, Lp-spaces, Riesz-Fischer theorem; Riesz Representation theorem for L2 spaces, Dual of Lp-spaces; Measure and Topology: Baire and Borel sets, Regularity of Baire and Borel measures, Construction of Borel measures, Positive and bounded linear functionals.

Eigenvalues, eigenvectors and similarity, Unitary equivalence and normal matrices, Schur’s theorem, Spectral theorems for normal and Hermitian matrices; Jordan canonical form, Application of Jordan canonical form, Minimal polynomial, Companion matrices, Functions of matrices; Variational characterizations of eigenvalues of Hermitian matrices, Rayleigh-Ritz theorem, Courant-Fischer theorem, Weyl theorem, Cauchy interlacing theorem, Inertia and congruence, Sylvester's law of inertia; Matrix norms, Location and perturbation of eigenvalues Gerschgorin disk theorem; Positive semidefiniteness, Singular value decomposition, Polar decomposition, Schur and Kronecker products; Positive and nonnegative matrices, Irreducible nonnegative matrices.

Linear Algebra: Vector spaces, subspaces, span, Linear dependence, independence of vectors, basis, dimension, linear transformations, range, kernel, rank, nullity of linear transformation, space of all linear transformations, Operator equations, matrix associated with a linear map, linear map associated with a matrix, elementary row operations, solution of algebraic equations, consistency conditions. Matrix inversion by row operations, Eigenvalues and eigenvectors, Hermitian and skew Hermitian matrices, orthogonal and unitary matrices, application to reduction of quadrics.

Complex Analysis: Limit, continuity, differentiability and analyticity of functions Cauchy-Riemann equations (cartesian and polar), Harmonic functions, Elementary complex functions, Line integrals, upper bounds for moduli of contour integrals, Cauchy’s integral theorem, Cauchy’s integral formula, derivatives of analytic functions, Power series, Taylor’s series, Laurent’s series, Zeros and singularities, Residue theorem, evaluation of improper integrals by residue theorem.

Double integral, Triple integral, Moments and Centers of Mass, Arc Length in Space, Curvature and Normal Vectors of a Curve, Tangential and Normal Components of Acceleration, Line Integrals, Vector Fields and Line Integrals: Work, Circulation, and Flux, Path Independence, Conservative Fields. Potential Functions, Green’s Theorem in the Plane, Surfaces and Area, Surface Integrals, Stokes’ Theorem, The Divergence Theorem and a Unified Theory

Discussion: Calculations with complex numbers; encoding linear systems, Fundamental theorem of algebra, Vector spaces and subspaces, Direct sum, linear span, Bases and dimensions of vector spaces, Linear independence, homogenous linear systems, Gaussian elimination, Null space and range of linear maps, Dimension formula for a linear map, Matrix of a linear map, Invertibility, Eigenvalues and eigenvectors, Upper triangular matrix representation, Diagonalization (2x2) and applications, Permutations and the determinant, Properties of the determinant, Inner product spaces, Cauchy-Schwarz, triangle inequality, Orthonormal bases, Gram-Schmidt procedure, Change of bases, Self-adjoint and normal operators, Spectral theorem for normal maps (complex), Diagonalization, Positive operators, polar and singular value decompositions

Cartesian plane, distance formula, midpoint formula, graphs, intercepts, circles, and lines, Functions, composition of functions, and inverse, Limits, Vertical asymptotes and finite limits; horizontal asymptotes and limits of infinity, Continuity, Slope of the tangent line, definition of the derivative, differentiability and continuity, Constant rule, power rule, constant multiple rule, sum and differences rules, Average rate change, instantaneous rate of change, velocity, marginals in economics, Product and quotient rules, Derivatives of trig functions, Chain rule, general power rule, Higher order derivatives, acceleration, Implicit differentiation, Increasing and decreasing functions, critical numbers, Relative extrema, the first-derivative test, absolute extrema, Concavity, points of inflection, the second-derivative test, Optimization problems, Sketching graphs, Differentials

Continuity: Continuous functions, Properties of continuous functions, Uniform continuity, Limits to functions

Sequences and Series: Power series, Uniform convergence, Differentiation of power series

Differentiation: Derivatives, Mean Value Theorem, Taylor’s Theorem

Metric Spaces: Metric Spaces and Continuity, Metric Spaces and connectedness

Discussion: Calculations with complex numbers; encoding linear systems, Fundamental theorem of algebra, Vector spaces and subspaces, Direct sum, linear span, Bases and dimensions of vector spaces, Linear independence, homogenous linear systems, Gaussian elimination, Null space and range of linear maps, Dimension formula for a linear map, Matrix of a linear map, Invertibility, Eigenvalues and eigenvectors, Upper triangular matrix representation, Diagonalization (2x2) and applications, Permutations and the determinant, Properties of the determinant, Inner product spaces, Cauchy-Schwarz, triangle inequality, Orthonormal bases, Gram-Schmidt procedure, Change of bases, Self-adjoint and normal operators, Spectral theorem for normal maps (complex), Diagonalization, Positive operators, polar and singular value decompositions

Double integral, Triple integral, Moments and Centers of Mass, Arc Length in Space, Curvature and Normal Vectors of a Curve, Tangential and Normal Components of Acceleration, Line Integrals, Vector Fields and Line Integrals: Work, Circulation, and Flux, Path Independence, Conservative Fields. Potential Functions, Green’s Theorem in the Plane, Surfaces and Area, Surface Integrals, Stokes’ Theorem, The Divergence Theorem and a Unified Theory

Convergence in probability, Moment generating functions, Computing probabilities and expectations by conditioning, Discrete time Markov chains, Branching processes, Poisson process

Fourier series: Motivation and definition of Fourier series, illustrative examples, convergence in $L^{2}$, pointwise and uniform convergence, convolution, decay of Fourier coefficients, and applications (heat and/or wave equation).

$L^{p}$ Spaces: Monotone Convergence Theorem, Dominated Convergence Theorem, examples illustrating convergence of sequences and integrals, definition and completeness of $L^{p}$ spaces, approximation by simple and smooth functions, and Fubini’s theorem.

Fourier transform (FT): Definition of the Fourier transform, Fourier transforms of Gaussian and other examples, Schwartz space, inverse Fourier transform, translation and differentiation properties, Fourier transform in $L^{2}$, Plancherel theorem, tempered distributions, convolutions, and applications (e.g., Poisson equation and Green’s functions).

Spectral theory: Definitions of spectrum and resolvent sets, basic spectral properties, illustrative examples, spectrum of self adjoint operators, classification of spectra, multiplication operators, unitary operators, spectral theorem for self-adjoint operators (proof omitted), spectral theory of the discrete Laplacian using Fourier series, and spectral theory of compact operators (optional).

Review of analysis on metric spaces: Definitions and ex- amples of metric spaces, $p$-norms on $\mathbb{R}^{n}$, Cauchy-Schwarz and Minkowski inequalities, continuity, open and closed sets, and completeness. Definition of topological spaces.

Compactness: Definitions and examples of sequential compactness and the open-set definition of compactness, equivalence of norms in finite-dimensional normed vector spaces, motivating examples of com-pactness in infinite-dimensional spaces, Riesz’s Lemma, and proof that the closed unit ball in an infinite-dimensional normed vector space is not compact.

Space of continuous functions: definition of $C(K)$ for $K$ compact, uniform convergence and completeness, definition and examples of equicontinuity, Arzel'a-Ascoli theorem, applications of the Arzel'a-Ascoli theorem, and Stone-Weierstrass theorem (proof optional or only the Weierstrass approximation theorem in 1D).

Banach spaces: Definition of Banach spaces, examples of Banach spaces $(C^{n}(K), c, c_{0}, \ell^{p}, L^{p})$, space of bounded and compact linear operators, dual spaces, Hahn-Banach theorem for extension of bounded linear functionals (proof optional).

Hilbert spaces: Definition of Hilbert spaces, examples of Hilbert spaces $(\ell^{2}, L^{2}, W^{1, 2})$ , orthogonality and projections, orthonormal bases, Riesz representation theorem, characterization of compact sets, and the approximation property of Hilbert spaces.

Weak convergence: Definitions of weak and weak* convergence, examples of weak versus strong convergence, definition and examples of weak sequential compactness, Banach-Alaoglu theorem (the closed unit ball is weakly sequentially compact in a reflexive Banach space; other versions optional), and applications.

Combinatorial probability, Axioms of probability and inclusion-exclusion formula, Conditional probability and independence, Discrete Random Variables, Continuous Random Variables, Joint distributions and independence, Sums of random variables, Law of large numbers, and Central Limit Theorem

Vectors and linear combinations, Lengths and dot products, Matrices, Vectors and linear equations, The idea of elimination, Elimination using matrices, Rules for matrix operations, Inverse matrices, Elimination = Factorization: A = LU, Transposes and permutations, Spaces and vectors, Nullspace of A: Solving Ax = 0, The Rank and the Row Reduced Form, The complete solution to Ax = b, Independence, basis, and dimension, Dimensions of the Four Subspaces, Orthogonality of the Four Subspaces, Projections, Least squares approximations, Orthogonal bases and Gram-Schmidt, The properties of determinants, Permutations and cofactors, Introduction to eigenvalues, Diagonalizing a matrix, Symmetric matrices, Positive definite matrices

Cartesian plane, distance formula, midpoint formula, graphs, intercepts, circles, and lines, Functions, composition of functions, and inverse, Limits, Vertical asymptotes and finite limits; horizontal asymptotes and limits of infinity, Continuity, Slope of the tangent line, definition of the derivative, differentiability and continuity, Constant rule, power rule, constant multiple rule, sum and differences rules, Average rate change, instantaneous rate of change, velocity, marginals in economics, Product and quotient rules, Derivatives of trig functions, Chain rule, general power rule, Higher order derivatives, acceleration, Implicit differentiation, Increasing and decreasing functions, critical numbers, Relative extrema, the first-derivative test, absolute extrema, Concavity, points of inflection, the second-derivative test, Optimization problems, Sketching graphs, Differentials

Interval notation, absolute value, factoring, completing the square, quadratic formula, distance formula, midpoint formula, lines, symmetry, circles, solving inequalities, functions, difference quotients, Emphasize finding domains and simplifying difference quotients, graphs of functions, shapes of graphs, average rate of change, translations, reflections, composition of functions, inverse functions, quadratic functions, setting up functions in applications, max-min problems, graphing polynomials, graphing rational functions, Give examples of polynomial division when covering slanted asymptotes, exponential functions, logarithmic functions, properties of logarithms, solving equations with, logarithms and exponentials, Omit inequalities, trig functions of acute angles, right-triangle applications, radian measure and geometry, trig functions of real numbers, graphs of sine and cosine, addition formulas, double-angle formulas, inverse trig functions, Pascal's triangle, Binomial Theorem, Factor Theorem, Rational Roots Theorem, exponential growth and decay