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Rao, H. Toutenburg, Shalabh, and C. Principles of sample surveys; Simple, Stratified and unequal probability Sampling with and without replacement; ratio, product and regression method of estimation; systematic sampling; cluster and subsampling with equal unequal sizes; double sampling; sources of errors in surveys. Sukhatme, B. V Sukhatme, S. Sukhatme and C.


Introduction to ODE; Existence and uniqueness of solution; Continuity and differentiability of solution w. Introduction to PDEs, First order quasilinear and nonlinear equations; Higher order equations and classifications; Solution of wave equations, Duhamel's principle and applications; Existence and uniqueness of solutions; BVPs for Laplace's and Poisson's equations, Green's function, Maximum principle for the Laplace equation; Heat equation, Maximum principle for the heat equation, Uniqueness of solutions of IVPs for heat conduction equation.

Brief review of distribution theory of uni-dimensional random variables. Multi-dimensional random variables random vectors : Joint, marginal, and conditional distribution functions; Independence; Moments and moment generating function; Conditional mean and conditional variance; Some examples of conditional expectations useful in Rao-Blackwellization; Discrete and absolutely continuous random variables distributions ; Multinomial and multivariate normal distributions.

Distribution of functions of random variables including order statistics; Properties of random vectors which are equal in distribution; Exchangeable random variables and their properties. Reference material s :. Dudewicz and S. Rohatgi, A. What is a model?

What is Mathematical modelling? Role of Mathematics in problem solving; Transformation of Physical model to Mathematical model with some illustrations of real world problems; Mathematical formulation, Dimensional analysis, Scaling, Sensitivity analysis, Validation, Simulation, Some case studies with analysis such as exponential growth and decay models, population models, Traffic flow models, Optimization models. Murthy, N. Page and E.

Sean Bohun, S. McCollum and T. Integral Transforms: Fourier and Hankel Transforms with their inverse transforms properties, convolution theorem and application to solve differential equation. Perturbation Methods: Perturbation theory, Regular perturbation theory, Singular perturbation theory, Asymptotic matching. Calculus of Variation: Introduction, Variational problems with functional containing first order derivatives and Euler equations.

Functional containing higher order derivatives and several independent variables. Variational problem with moving boundaries. Boundaries with constraints. Higher order necessary conditions, Weierstrass function, Legendre and Jacobi's condition. Existence of solutions of variational problems. Rayleigh-Ritz method, statement of Ekelands variational principle; Self adjoint, normal and unitary operators; Banach algebras.

Linear least squares problems, existence and uniqueness, sensitivity and conditioning, orthogonalization methods, SVD, Optimization, existence and uniqueness, sensitivity and conditioning, Newton's method, Unconstrained Optimization, Steepest descent, Conjugate gradient method, Constrained optimization optional , Numerical solution to ODE, IVP: Euler's method, One step and linear multistep methods, Stiff differential equations, boundary value problems, Numerical solution to PDEs, review of second order PDEs: hyerbolic, parabolic and elliptic PDEs, Time dependent problems, Time independent problems Reference material s :.

Simulation of random variables from discrete, continuous, multivariate distributions and stochastic processes, Monte Carlo methods. Regression analysis, scatterplot, residual analysis. Graphical representation of multivariate data, Cluster analysis, Principal component analysis for dimension reduction. Hastie, R.

Tibshirani and M. Gilks, S. Richardson, D. Analysis of completely randomized design, randomized block design, Latin squares design; Split plot, 2 and 3-factorial designs with total and partial confounding, two way non-orthogonal experiment, BIBD, PBIBD; Analysis of covariance, missing plot techniques; First and second order response surface designs.

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Sahai and M. Group families; Principle of invariance and equivariant estimators- location family, scale family, location-scale family; General Principle of equivariance; Minimum risk equivariant estimators under location scale and location-scale families; Bayesian estimation; prior distributions; posterior distribution; Bayes estimators; limit of Bayes estimators; hierarchical Bayes estimators; Generalized Bayes estimators; highest posterior density credible regions; Minimax estimators and their relationships with Bayes estimators; admissibility; Invariance in hypothesis testing; Review of convergence in probability and convergence in distributions; consistent estimators; Consistent and Asymptotic Normal CAN estimators; BAN estimator; asymptotic relative efficiency ARE ; Limiting risk efficiency LRE ; Limiting risk deficiency LRD; CRLB and asymptotically efficient estimator; large sample properties of MLE.

Casella : Theory of Point Estimation, Springer. Order statistics, Run tests, Goodness of fit tests, rank order statistics, sign test and signed rank test. General two sample problems, Mann Whitney test, Linear rank tests for location and scale problem, k-sample problem, Measures of association, Power and asymptotic relative efficiency, Concepts of jack knifing, Bootstrap methods. Randles and D. Computer arithmetic. Vector and matrix norms. Condition number of a matrix and its applications. Singular value decomposition of a matrix and its applications.

Linear least squares problem. Householder matrices and their applications. Numerical methods for matrix eigenvalue problem. Numerical methods for systems and control. Mixed methods, Iterative Techniques. Preliminaries: Introduction to algorithms; Analysing algorithms: space and time complexity; growth of functions; summations; recurrences; sets, etc. Greedy Algorithms: General characteristics; Graphs: minimum spanning tree; The knapsack problem; scheduling.

Divide and Conquer: Binary search; Sorting: sorting by merging, quicksort. Dynamic Programming: Elements of dynamic programming; The principle of optimality; The knapsack problem; Shortest paths; Chained matrix multiplication. Number Theoretic Algorithms: Greatest common divisor; Modular arithmetic; Solving modular linear equations. Introduction to cryptography. Computational Geometry: Line segment properties; Intersection of any pair of segments; Finding the convex hull; Finding the closest pair of points.

Introduction to Data Mining; supervised and un-supervised data mining, virtuous cycle. Dimension Reduction and Visualization Techniques; Chernoff faces, principal component analysis. Feature extraction; multidimensional scaling. Cluster Analysis: hierarchical and non-hierarchical techniques. Density estimation techniques; parametric and Kernel density estimation approaches. Statistical Modelling; design, estimation and inferential aspects of multiple regression, Kernel regression techniques.

Tree based methods; Classification and Regression Trees. Neural Networks; multi-layer perceptron, feed-forward and recurrent networks, supervised ANN model building using back-propagation algorithm, ANN model for classification. Genetic algorithms, neuro-genetic models. Self-organizing Maps. Project - I Click to collapse. Project - II Click to collapse. Elementary mathematical models; Role of mathematics in problem solving; Concepts of mathematical modelling; System approach; formulation, Analyses of models; Sensitivity analysis, Simulation approach; Pitfalls in modelling, Illustrations.

Fields: Definition and examples, Irreducibility Criterions, Prime Subfield, Algebraic and transcendental elements and extensions, Splitting field of a polynomial. Personal Details Links Amazon. Product Details. Personal Details. Ostberg, Elementary differential equations with linear algebra. Marsden, Basic complex analysis. Parzynski, Philip W. Zipse, Introduction to mathematical analysis.

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Pecaric, Frank Proschan, Y. Tong, Convex functions, partial orderings, and statistical applications. Duren, Theory of H p spaces. Zachmanoglou, Dale W. Thoe, Introduction to partial differential equations with applications. Bluman, Problem book for first year calculus. Nohel, Ordinary differential equations. Krantz, Function theory of several comple variables.

Weinberger, Partial differential equations. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order. Cheng, analysis of linear systems. Jerome Keisler, Foundations of infinitesimal calculus. Grauert, K. Fritzsche, Several complex variables. Fleming, Functions of several variables. Churchill, Complex variables and applications. Churchill, James W.

Brown, Roger F. Verhey, complex variables and applications. Paliouras, Complex variables for scientists and engineers. Maddox, Elements of functional analysis. Berberian, Introduction to Hilbert space. Kanwal, Generalized functions: Theory and technique. Shubin, Pseudodifferential operators and spectral theory. Reidel, Fixed point theory. Tyrrell Rockafellar, Convex analysis. Corwin, Robert H. Szczarba, Calculus in vector spaces.

Berger, nonlinearity and functional analysis. Folland, Introduction to partial differential equations. Strichartz, The way of analysis. Whyburn, Topological Analysis, Revised Edit. Aliprantis, Principles of Real Analysis, Arnold. Sherbert, Introduction to Real Analysis, Wiley. Advanced Real Calculus. Gordon and Breach. L Lions, Controle Optimal de systems gouvernes par des Equa. Aux derivees partielles. Munroe, Introduction to measure and integration.

McGregor, Elementary partial differential quations. Bezdek, Analysis of fuzzy information Vol. Mathematics and Logic. Kaufmann, Introduction to the theory of fuzzy subsets. Academic press. Kaplansky, Commutative Ring, The Univ.

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Kaplansky, Fields and Rings, The Univ. Cartan and S. Eilenberg, Homological Algebra, Princeton Univ. Northcott, Ideal Theory, Cambridge Univ. Atiyah and I. Hoffman and R. Hall, Jr. Herrlich and G. Gallian, Contemporary Abstract Algebra, D. Akhizer and I. Brown and D.

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Anton, Elementary Linear Algebra 3rd ed. Anton, Elementary Linear Algebra 2nd ed. Borel, Linear Algebraic Groups, W. Passman, Permutation Groups, W. Fogarty, Invariant Theory, W. Fulton, Algebraic Curves, W. Lang, Undergraduate Algebra. Waerden, Algebra Vol. Beckenstein, L. Narici and C. Jerison, Rings of Continuous Functions, D. Van Nostrand, Warner, Topological Fields. Dxion, Problem in Group Theory, 1st. Cohn, F. Meara, B. Wang, Introduction to Vectors and Tensors.

Herstein and D. Dickson, Theory of Numbers Vol. Dudley, Elementary Number Theory, W. Dudley, Elementary Number Theory 2nd ed. Taylor, Measure algebras. Milnor, James D. Stasheff, Characteristic classes. Eldon, Boollean Algebra and Its Applications. Barr, C. Wells, Toposoes, Triples and Theories. Springer-Verlag G. Hadley, Linear Algebra, Addison -Weasley. S Blyth, E. Finkbeiner, Introduction to Matrices and Linear Transformaions 2nd. Mcgraw-Hill Jacobson, basic Algebra1 W.

Rothenberg, Linear algebra with computer applications. Van Nortrand. Van Nostrand. The Macmillan.

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No Topology Max K. Herrlich, Categorical Topology, Bremen Univ. Fairchild, Topology, W. Saunders, Gemignani, Elementary Topology 2nd ed. Reisel, Elementary Theory of Metric Spaces. Press of Virginia, Barr, What is Topology? Crowell Company, Greenberg and J. Nash and S. Guillemin and A. Lin and S. Steen and J.

Seebach, Jr. Singer and J. Montgomery and L. Whyburn and E. Duda, Dynamic Topology, Arkhangel'skill and V. Ponomarev, Fundamentals of General Topology, D. Kunen, Handbook of Set-Theoretic Topology. Nadler, Jr. Saunders Co. Herrlich, and G. Vaught, Set Theory. Enderton, Elements of Set Theory. Gamelin and R. Young, Topology, Fred. Lee and Thomas H. Parker, The Yamable Problem. Arkhangel'skii and V. Topology, 2nd. Topology a First Course. J Pervin, Foundations of General topology.

A Naimpally, B. Vick, Homology Theory Gray. Homotopy Theory. Academic Press Hu.

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Academic Press Choquet. Academic Press Gaal. Point Set Topology. Academic Press N. Cauchy-type curvilinear integrals are then shown to generalize to any number of real variables differential forms, Stokes-type formulas. The fundamentals of the theory of manifolds are then presented, mainly to provide the reader with a "canonical'' language and with some important theorems change of variables in integration, differential equations. A final chapter shows how these theorems can be used to construct the compact Riemann surface of an algebraic function, a subject that is rarely addressed in the general literature though it only requires elementary techniques.

Besides the Lebesgue integral, Volume IV will set out a piece of specialized mathematics towards which the entire content of the previous volumes will converge: Jacobi, Riemann, Dedekind series and infinite products, elliptic functions, classical theory of modular functions and its modern version using the structure of the Lie algebra of SL 2,R. Springer Professional. Back to the search result list. Cauchy Theory Abstract. Let X be a 2-dimensional C 0 manifold in the sense of Chap.