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syllabus of Mathematics And Computing Engineering (MCE) in DCE
Dear Manish,
Course Syllabus (2010 Onwards)
Systems of linear equations and their solutions; vector space Rn and its subspaces; spanning set and linear independence; matrices, inverse and determinant; range space and rank, null space and nullity, eigenvalues and eigenvectors; diagonalization of matrices; similarity; inner product, Gram-Schmidt process; vector spaces (over the field of real and complex numbers), linear transformations.
Convergence of sequences and series of real numbers; continuity of functions; differentiability, Rolles theorem, mean value theorem, Taylors theorem; power series; Riemann integration, fundamental theorem of calculus, improper integrals; application to length, area, volume and surface area of revolution.
Texts:
References:
Vector functions of one variable - continuity and differentiability; functions of several variables - continuity, partial derivatives, directional derivatives, gradient, differentiability, chain rule; tangent planes and normals, maxima and minima, Lagrange multiplier method; repeated and multiple integrals with applications to volume, surface area, moments of inertia, change of variables; vector fields, line and surface integrals; Greens, Gauss and Stokes theorems and their applications.
First order differential equations - exact differential equations, integrating factors, Bernoulli equations, existence and uniqueness theorem, applications; higher-order linear differential equations - solutions of homogeneous and nonhomogeneous equations, method of variation of parameters, operator method; series solutions of linear differential equations, Legendre equation and Legendre polynomials, Bessel equation and Bessel functions of first and second kinds; systems of first-order equations, phase plane, critical points, stability.
Complex numbers and elementary properties. Complex functions - limits, continuity and differentiation. Cauchy-Riemann equations. Analytic and harmonic functions. Elementary functions. Anti-derivatives and path (contour) integrals. Cauchy-Goursat Theorem. Cauchys integral formula, Moreras Theorem. Liouvilles Theorem, Fundamental Theorem of Algebra and Maximum Modulus Principle. Taylor series. Power series. Singularities and Laurent series. Cauchys Residue Theorem and applications. Mobius transformations.
First order partial differential equations; solutions of linear and nonlinear first order PDEs; classification of second-order PDEs; method of characteristics; boundary and initial value problems (Dirichlet and Neumann type) involving wave equation, heat conduction equationi, Laplaces equations and solutions by method of separation of variables (Cartesian coordinates); initial boundary value problems in non-rectangular coordinates.
Laplace and inverse Laplace transforms; properties, convolutions; solution of ODE and PDE by Laplace transform; Fourier series, Fourier integrals; Fourier transforms, sine and cosine transforms; solution of PDE by Fourier transform.
Set theory - sets, relations, functions, countability; Logic - formulae, interpretations, methods of proof, soundness and completeness in propositional and predicate logic; Number theory - division algorithm, Euclids algorithm, fundamental theorem of arithmetic, Chinese remainder theorem, special numbers like Catalan, Fibonacci, harmonic and Stirling; Combinatorics - permutations, combinations, partitions, recurrences, generating functions; Graph Theory - paths, connectivity, subgraphs, isomorphism, trees, complete graphs, bipartite graphs, matchings, colourability, planarity, digraphs; Algebraic Structures - semigroups, groups, subgroups, homomorphisms, rings, integral domains, fields, lattices and Boolean algebras.
Formal properties of integers, equivalence relations, congruences, rings, homomorphisms, ideals, integral domains, fields; Groups, homomorphisms, subgroups, cosets, Lagranges theorem , normal subgroups, quotient groups, permutation groups; Groups actions, orbits,stabilizers, Cayleys theorem, conjugacy, class equation, Sylows theorems and applications; Principal ideal domains, Euclidean domains, unique factorization domains, polynomial rings; Characteristic of a field, field extensions, algebraic extensions, separable extensions, finite fields, algebraically closed field, algebraic closure of a field.
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.
Metrics and norms - metric spaces, normed vector spaces, convergence in metric spaces, completeness; Functions of several variables - differentiability, chain rule, Taylors theorem, inverse function theorem, implicit function theorem; Lebesgue measure and integral - sigma-algebra of sets, measure space, Lebesgue measure, measurable functions, Lebesgue integral, dominated convergence theorem, monotone convergence theorem, L-p spaces.
Principles of Monte Carlo, generation of random numbers from a uniform distribution: linear congruential generators and its variations, inverse transform and acceptance-rejection methods of transformation of uniform deviates, simulation of univariate and multivariate normally distributed random variables: Box-Muller and Marsaglia methods, variance reduction techniques, generation of Brownian sample paths, quasi-Monte Carlo: Low discrepancy sequences.
Asymtotic notation; Sorting - merge sort, heap sort, priortiy queue, quick sort, sorting in linear time, order statistics; Data structures - heap, hash tables, binary search tree, balanced trees (red-black tree, AVL tree); Algorithm design techniques - divide and conquer, dynamic programming, greedy algorithm, amortized analysis; Elementary graph algorithms, minimum spanning tree, shortest path algorithms.
The tutorials will be based on object-oriented programming concepts such as classes, objects, methods, interfaces, packages, inheritance, encapsulation, and polymorphism. Programming laboratory will be set in consonance with the material covered in MA 252. This will include assignments in a programming language like C++ in GNU Linux environment.
Overview of financial engineering, financial markets and financial instruments; Interest rates, present and future values of cash flow streams; Riskfree assets - bonds and bonds pricing, yield, duration and convexity, term structure of interest rates, spot and forward rates; Risky assets - risk-reward analysis, mean variance portfolio optimization, Markowitz model and efficient frontier, CAPM and APT; Discrete time market models ? assumptions, portfolios and trading strategies, replicating portfolios, No-arbitrage principle; Derivative securities ? forward and futures contracts, hedging strategies using futures, pricing of forward and futures contracts, interest rate futures, swaps;General properties of options, trading strategies involving options; Binomial model, risk neutral probabilities, martingales, valuation of European contingent claims, Cox-Ross-Rubinstein (CRR) formula, American options in binomial model, Black-Scholes formula derived as a continuous-time limit; Options on stock indices, currencies and futures, overview of exotic options.
Errors; Iterative methods for nonlinear equations; Polynomial interpolation, spline interpolations; Numerical integration based on interpolation, quadrature methods, Gaussian quadrature; Initial value problems for ordinary differential equations - Euler method, Runge-Kutta methods, multi-step methods, predictor-corrector method, stability and convergence analysis; Finite difference schemes for partial differential equations - Explicit and implicit schemes; Consistency, stability and convergence; Stability analysis (matrix method and von Neumann method), Lax equivalence theorem; Finite difference schemes for initial and boundary value problems (FTCS, Backward Euler and Crank-Nicolson schemes, ADI methods, Lax Wendroff method, upwind scheme).
General probability spaces, filtrations, conditional expectations, martingales and stopping times, Markov processes; Brownian motion and its properties; Itôintegral and its extension t classes of integrands, isometry and martingale properties of Itôintegral, Itôocesses, Itôeblin formula; Derivation of the Black-Scholes-Merton differential equation, Black-Scholes-Merton formula, the Greeks, put-call parity, multi-variable stochastic calculus; Risk-neutral valuation ? risk-neutral measure, Girsanovs theorem for change of measure, martingale representation theorems, representation of Brownian martingales, the fundamental theorems of asset pricing; Stochastic differential equations, existence and uniqueness of solutions, Feynman-Kac formula and its applications.
Classification and general theory of optimization; Linear programming (LP): formulation and geometric ideas, simplex and revised simplex methods, duality and sensitivity, interior-point methods for LP problems, transportation, assignment, and integer programming problems; Nonlinear optimization, method of Lagrange multipliers, Karush-Kuhn-Tucker theory, numerical methods for nonlinear optimization, convex optimization, quadratic optimization; Dynamic programming; Optimization models and tools in finance.
Alphabets, languages, grammars; Finite automata, regular languages, regular expressions; Contextfree languages, pushdown automata, DCFLs; Context sensitive languages, linear bounded automata; Turing machines, recursively enumerable languages; Operations on formal languages and their properties; Chomsky hierarchy; Decision questions on languages.
Continuous time financial market models, Black-Scholes-Merton model, Black-Scholes PDE and formulas, risk-neutral valuation, change of numeraire, pricing and hedging of contingent claims, Greeks, implied volatility, volatility smile; Options on futures, European, American and Exotic options; Incomplete markets, market models with stochastic volatility, pricing and hedging in incomplete markets; Bond markets, term-structures of interest rates, bond pricing; Short rate models, martingale models for short rate (Vasicek, Ho-Lee, Cox-Ingersoll-Ross and Hull-White models), multifactor models; Forward rate models, Heath-Jarrow-Morton framework, pricing and hedging under short rate and forward rate models, swaps and caps; LIBOR and swap market models, caps, swaps, swaptions, calibration and simulation.
This course will focus on implementation of the financial models such as CAPM, binomial models, Black-Scholes model, interest rate models and asset pricing based on above models studied in MA 271 and MA 373. The implementation will be done using S-PLUS/MATLAB/C++.
Floating point computations, IEEE floating point arithmetic, analysis of roundoff errors; Sensitivity analysis and condition numbers; Linear systems, LU decompositions, Gaussian elimination with partial pivoting; Banded systems, positive definite systems, Cholesky decomposition - sensitivity analysis; Gram-Schmidt orthonormal process, Householder transformation, Givens rotations; QR factorization, stability of QR factorization. Solution of linear least squares problems, normal equations, singular value decomposition(SVD), polar decomposition, Moore-Penrose inverse; Rank deficient least-squares problems; Sensitivity analysis of least-squares problems; Review of canonical forms of matrices; Sensitivity of eigenvalues and eigenvectors. Reduction to Hessenberg and tridiagonal forms; Power, inverse power and Rayleigh quotient iterations; Explicit and implicit QR algorithms for symmetric and nonsymmetric matrices; Reduction to bidiagonal form; Golub- Kahan algorithm for computing SVD.
Introduction to statistical packages (R / S-Plus / MATLAB / SAS) and data analysis - financial data, exploratory data analysis tools, kernel density estimation; Basic estimation and testing; Random number generator and Monte Carlo samples; Financial time series analysis - AR, MA, ARMA. ARIMA, ARCH and GARCH models, identification, inference, forecasting, stochastic volatility time series models for term structure of interest rates; Linear regression ?-least squares estimation, inference, model checking; Multivariate data analysis - multivariate normal and inference, Copulae and random simulation, examples of copulae family, fitting Copulas, Monte Carlo simulation with Copulas, dimension reduction techniques, principal component analysis; Risk management - riskmetrics, quantiles, Q-Q plots, quantile estimation with Cornish-Fisher expansion, VaR, expected short fall, time-to-default modeling, extreme value theory (generalized extreme value (GEV), generalized Pareto distribution (GPD); Block Maxima, and Hill methods).
Models of computation - Turing machine, RAM, ìecursive function, grammars; Undecidability - Rices theorem, Post correspondence problem, logical theories; Complexity classes - P, NP, coNP, EXP, PSPACE, L, NL, ATIME, BPP, RP, ZPP, IP.
Review of financial models for option pricing and interest rate modeling, Black -Scholes PDE; Finite difference methods, Crank-Nicolson method, American option as free boundary problems, computation of American options, pricing of exotic options, upwind scheme and other methods, Lax-Wendroff method; Monte-Carlo simulation, generating sample paths, discretization of SDE, Monte-Carlo for option valuation and Greeks, Monte-Carlo for American and exotic options; Term-structure modeling, short rate models, bond prices, multifactor models; Forward rate models, implementation of Heath-Jarrow-Morton model; LIBOR market model, Volatility structure and Calibration.
Electives:
MA424 Advanced Linear Algebra L-T-P-C :3-0-0-6
Review of vector spaces, bases and dimensions, direct sums; Linear transformations, ranknullity theorem, matrix representation of linear transformations, trace and determinant; Eigenvalues and eigenvectors, invariant subspaces, upper-triangular matrices, invariant subspaces on real vector spaces, generalized eigenvectors, characteristic and minimal polynomials, triangulation, diagonalization, Jordan canonical form; Norms and innerproducts, orthonormal bases, orthogonal projections, linear functional and adjoints, selfadjoint and normal operators, Schur decomposition, spectral theorems for selfadjoint, unitary and normal operators, positive definite operators, isometry, polar and singular value decompositions.
Mean-variance portfolio theory, asset return, portfolio mean and variance, Markowitz model, efficient frontier calculation algorithm, single-index and multi-index models; Capital Asset Pricing Model (CAPM), Capital market line, pricing model, security market line, systematic and nonsystematic risk, pricing formulas, investment implications, empirical tests, performance evaluation; Multifactor models, CAPM as a factor model, arbitrage pricing theory (APT), multifactor models in continuous time, data statistics, estimation of parameters; Utility functions, risk aversion, utility functions and the mean-variance criterion, linear pricing, portfolio choice, risk neutral pricing; Optimal portfolio growth, continuous-time growth, log-optimal pricing and the Black-Scholes equation; Multiperiod securities, risk neutral pricing, buying price analysis, continuous time evaluation; Fixed Income Security investment, modeling yield curves, managing a bond portfolio, performance analysis.
Scope of Parallel Computing - limits to parallelizability, NC-reductions, P-completeness; parallel programming platforms; parallel algorithm design - decomposition, task and ineractions; communication models - synchronous and asynchronous; analytical modeling of parallel programs; programming using message passing paradigm and shared address space - threads, MPI, unstructured communications; parallel algorithms and applications - matrix algorithms, sorting, graph algorithms and discrete optimization problems.
Isomorphism, incidence and adjacency matrices, Sperner lemma, Trees, Cayley formula, connector problem, connectivity, constructing reliable communication network, Euler tours, Hamilton cycle, Chinese postman and traveling salesman problems, matchings and coverings, perfect matchings, edge colouring, Vizing Theorem, time table problem, Independent sets, Ramsey theorem, Turan theorem, Schur theorem, vertex colouring, Brook theorem, Hajos conjecture, chromatic polynomials, storage problem, planarity, dual graphs, Euler formula, Kuratowski theorem, five colour theorem, history of four colour theorem, nonhamiltonian planar graphs, planarity algorithm, directed graphs, job sequencing, one way road system, ranking participants in tournaments.
Counting principles, multinomial theorem, set partitions and Striling numbers of the second kind, permutations and Stirling numbers of the first kind, number partitions, Lattice paths, Gaussian coefficients, Aztec diamonds, formal series, infinite sums and products, infinite matrices, inversion of sequences, probability generating functions, generating functions, evaluating sums, the exponential formula, more on number partitions and infinite products, Ramanujans formula, hypergeometric sums, summation by elimination, infinite sums and closed forms, recurrence for hypergeometric sums, hypergeometric series, Sieve methods, inclusion-exclusion, Mobius inversion, involution principle, Gessel-Viennot lemma, Tutte matrix-tree theorem, enumeration and patterns, Polya-Redfield theorem, cycle index, symmetries on N and R, polyominoes
Time and Space complexity, various complexity classes, oracle Turing machine, hierarchy theorems, relations among complexity measures, Savitchs theorem, Borodins Gap theorem, Blums speed-up theorem, the union theorem, axiomatic complexity theory, intractable problems, PSPACE-completeness, EXPSPACE-completeness, QBF problem, provably intractable problems, P = NP?, alternating time and space.
Basics of financial risk management, identification of major financial risks, VaR and risk management, interest rate risk, foreign exchange risk; Credit derivatives: CDS, valuation of CDS, CDO and valuations of synthetic CDO, quantitative models of credit risk; First-passage time models: Black and Cox model, dependent defaults; Hazard process: hazard function of random time, martingale hazard process; Reduced form modeling: intensity based valuation of defaultable claims; Pricing credit derivatives and portfolio optimization under credit risk.
Texts/References:
Combinatorial surface, classical Euler characteristic, Barycentric subdivision; Topological surfaces, Rados theorem on the triangulability of compact surfaces, classification theorem of compact, connected surfaces; Simplicial homology, Betti numbers, Poincares theorem on Euler characteristic; Differential surfaces, tangent spaces, vector fields, Poincare index theorem; Topological and differential manifolds.
Basics of Computational Geometry - convex hull, line segment intersection, triangulation, linear programming, simplex range searching, voronoi diagram (nearest and farthest), arrangement and duality, visibility; Applications of geometric data structures and algorithms - geographic information system (GIS), robot motion planning, physical design in VLSI.
Lie algebras and Lie algebra homomorphisms (definition and examples), solvable and nilpotent Lie algebras, Engels theorem; Semisimple Lie algebras - Lies theorem, Cartans criterion, Jordan-Chevalley decomposition, killing form, complete reducibility of representations, Weyls theorem, irreducible representations of the Lie algebra SL(2), weights and maximal vectors, root space decomposition; Root systems - definition and examples, simple roots and the Weyl group, Cartan matrix of a root system, Dynkin diagrams, classification theorem.
Logic and reasoning; Predicate logic - terms, formulae and clauses, clausal form of formulae, types of clauses, Horn clauses, substitution, unification algorithm, resolution, SLD-refutation; Introduction to Prolog, structure of logic programs, syntax and meaning, controlling backtracking, negation in logic programs and implementation issues, lists, operators, arithmetic, input and output, built-in predicates, operations on data structures, meta-programming; Constraint logic programming.
Introduction to probability distributions. Basics of estimation and testing of hypothesis ( frequentist approach, bayesian approach ). Different censoring schemes: Type-I , Type-II, hybrid, progressive. Different models and EM algorithm: mixture model; bivariate distributions; cure rate model; competing risk model. Generating random sample: discrete and continuous multivariate distributions (multinomial, multivariate normal, multivariate exponential ); acceptance rejection principle; monte carlo markov chain ( metropolis hastings algorithm, gibbs sampler ); Convergence of MCMC : Harris irreducibility, recurrence, minorization, limit theory for harris recurrent markov chains. Resampling techniques: jackknife; bootstrap. Hidden Markov Model ( forward-backward algorithm, viterbi algorithm, baum-welch algorithm). Artificial Neural Network: framework, topology ( feed forward neural network, recurrent neural network), training of ANN ( supervised, unsupervised, reinforced learning ) , robustness. Genetic Algorithm: single objective GA, multi-objective NSGA.
Review of probability theory; Sampling distributions; Point estimation - estimators, sufficiency, completeness, minimum variance unbiased estimation, maximum likelihood estimation, method of moments, Cramer-Rao inequality, consistency; Interval estimation; Testing of hypotheses - tests and critical regions, Neymann-Pearson lemma, uniformly most powerful tests, likelihood ratio tests; Basic non-parametric tests.
Review of probability theory; Discrete-time Markov chains, renewal and regenerative processes, Poisson processes, continuous-time Markov chains, martingales, Brownian motion; Application to queues, communications, finance, biology and manufacturing.
Review of Hilbert spaces, orthonormal bases, weak convergence; Bounded operators on Hilbert spaces, adjoints of bounded operators, algebra of bounded operators; Orthogonal projections, isometric and unitary operators, finite rank and compact operators, Hilbert-Schmidt operators, selfadjoint and normal operators; Spectra of bounded operators, invariant and reducing subspaces; Spectral theorem for compact operators, polar and singular value decompositions, Schatten class operators; Spectral theorem for bounded selfadjoint and normal operators.
Basic concepts of finite element methods; Elements of function spaces, Lax-Milgram theorem, piecewise polynomial approximation in function spaces, Galerkin orthogonality and Cea?s lemma, Bramble-Hilbert lemma, Aubin-Nitsche duality argument; Applications to elliptic, parabolic and hyperbolic equations, a priori error estimates, variational crimes; A posteriori error analysis ? reliability, efficiency and adaptivity.
Graph notations and computer representations, Applications to various disciplines, Worst-case complexity. Shortest paths, Label setting and label correcting algorithms, Maximum flows, Augmenting path and pre flow push algorithms, Minimum cost flows. Pseudopolynomial and polynomial time algorithms, Assignments and matching, Bipartite and nonbipartite matchings, Minimum spanning trees, Convex cost flows and generalized flows, Emphasis on real-life time applications of network flows and state-of-the art algorithms.
Optimization problems, Convex sets and convex functions, Important combinatorial optimization problems, The fundamental algorithms, efficiency and the digital computer. Convex hulls, Polytopes, Facets, Integral polytopes, Total Unimodularity, Total dual integrality, Cutting plane algorithms and bounds, Separation and optimization, Computational complexity. Matroids, Greedy algorithm, Properties, Axioms and constructions of matroids, Matroid Intersection problem, Applications of matroid intersection, Weighted matroid intersection. Heuristics and analysis of heuristics, Heuristics for TSP, Data structure for combinatorial optimization problems.
Binary group codes, Hamming codes, Polynomial codes, Block codes, Linear codes,Generator and check matrices, Sphere packing, Gilbert-Varshamov and Griesmer bounds, Syndrome decoding. The structure of cyclic codes, Reed Mueller codes, Simplex codes. Nonlinear codes, Golay, Hadamard, Justeen, Kerdock, Nordstorm-Robinson codes. Weight distribution of codes, Generalized BCH codes (including the BCH bound and decoding methods), Self-dual codes and invariant theory, MacWilliams identities and Gleasons theorems on self-dual codes, Covering radius problem, Convolutional codes. Reed-Solomon codes, Quadratic-residue codes and perfect codes. The group of a code, permutation and monomial groups, Mathieu groups, General linear and affine groups, Connections with design theory, Steiner systems, Projective and affine planes.
Congruence, Chinese Remainder Theorem, Primitive Roots, Quadratic reciprocity, Finite fields, Arithmetic functions Primality Testing and factorization algorithms, Pseudo-primes, Fermat?s pseudo-primes, Pollard?s rho method for factorization, Continued fractions, Continued fraction method Hash Functions, Public Key cryptography, Diffie-Hellmann key exchange, Discrete logarithm-based crypto-systems, RSA crypto-system, Signature Schemes, Digital signature standard, RSA Signature schemes, Knapsack problem. Introduction to elliptic curves, Group structure, Rational points on elliptic curves, Elliptic Curve Cryptography. Applications in cryptography and factorization, Known attacks.
Interval numbers, Interval arithmetic, Multilevel interval numbers. Fuzzy numbers, Fuzzy numbers in the set of integers, Arithmetic with fuzzy numbers. Definition of fuzzy sets, Fuzzy sets and fuzzy numbers, Basic operations on fuzzy sets, Extension principle of fuzzy sets. Fuzzy relations, Basic properties of fuzzy relations, Fuzzy relations and approximate reasoning. Fuzzy logic, Linguistic variables, Linguistic modifiers, Truth, Propositions of fuzzy logic, Uncertainty based information, Approximate reasoning. Fuzzy decision making, Multicriteria decision making, Multistage decision making, Fuzzy ranking methods. Fuzzy modeling of control parameters, Washing machine, Fuzzy logic control for a predator-prey system.
Cryptanalysis; Shannons Theory; Block Cipher: Data Encryption Standard, Advanced Encryption Standard, Linear and Differential Cryptanalysis; Primality Tests, Factoring Integers, Discrete Logarithm Problem; Public Key Cryptosystem: RSA; Cryptographic Hash Functions; Key Distribution and Key Agreement; Signature Schemes.
Algebraic numbers, transcendental numbers, minimal polynomial, conjugates. Number fields, primitive element, real and complex embeddings, norm, trace and discriminant. Algebraic integers, ring of integers in a number field, integral basis. Dedekind domain, ideal factorization, fractional ideal, ideal class group. Lattices, Minkowskis theory, computation of class group of number fields. Dirichlet Unit Theorem, fundamental units, units in quadratic fields, Pells equation. Cyclotomic fields.
Elements of a Programming language: Defining Syntax; BNF; Conditional Statements; Iterative Statements; Enumerated and Elementary Data Types; Features of Functional and Imperative languages. Elements of Mathematics: Partial and Multi Functions; Isomorphism, Duality, Zero Objects, Products, Co-Products from Category Theory; Term Algebras. Semantics: Operational, Axiomatic and Denotational Semantics of Procedural Languages; Partially Additive Semantics; Recursive Specification; Order Semantics of Recursion; Fixed- Point Semantics; Algebraic Semantics of Abstract Data Types
Functional programming: functions as first class objects, laziness, data-types and pattern matching, classes and overloading, side-effects, description in languages like ML or Haskell; Lambda calculus: syntax, conversions, normal forms, Church-Rosser theorem, combinators; Implementation issues: graph reduction; Logic programming: logic and reasoning, logic programs, Prolog syntax, Horn clauses, resolution-refutation, constraint logic programming.
Algebraic numbers, Primes and factoring, Trapdoors and public key, Pseudo-random numbers. The finite Fourier transforms. The fast Fourier transform., Polynomial rings in several variables, Complexity with respect to multiplication, Shift registers and coding, Finite Boolean algebras, Equivalence classes of switching functions, Monoids and automata.
Reviews of Fourier analysis and LP spaces. Wavelets and atomic decomposition of functions, Multiresolution signal decomposition, Multiresolution analysis and the construction of wavelets, Examples of wavelets,QMF and fast wavelet transform, Localization, Regularity and approximation properties of wavelets. Construction of compactly support wavelets, Orthonormal bases of compactly supported wavelets, Wavelets sampling techniques, Convergence of Wavelet expansion, Time-frequency analysis for signal processing, Applications of wavelets in image and signal processing.
Software Support: MATLAB, MATHEMATICA
Classical Fractals, Cantor set, Sierpinski triangle, Von Koch curve, Hilbert and Peano curves, Weierstrass function. Self-similarity, Scaling, Similarity dimension, Box-counting dimension, Information dimension, Capacity dimension. Foundations of iterated function systems (IFS), Classical fractals generated by IFS, Contractions mapping principle, Collage theorems, Fractal image compression, Image encoding and decoding by IFS. Iteration of quadratic polynomials, Julia sets, Fatou sets, Mandelbrot set, Characterization of Julia sets, Dynamics of functions ez , sin z and cos z, Bifurcation and chaotic burst.
Software Support: MATLAB, MATHEMATICA, GNUPLOT
Texts / References:
Classification of DAEs, Solvability and Index, Linear constant coefficient DAEs, Linear time varying DAEs, Nonlinear systems, Index reduction and constraint stabilization. Runge-Kutta method for DAEs, Classes of implicit Runge-Kutta methods, Convergence analysis for Index 1, Index 2, and Index 3 systems. Solution of nonlinear systems by Newtons method, Local error estimation. Multistep methods, BDF convergence, Semi explicit index 1 systems, Fully implicit index 1 systems, Semi explicit index 2 systems, Index 3 systems of Hessenberg form, BDF methods, Stiff Problems and applications.
Software Support: DASSL, RADAU5
Convolutions ? Fourier Transform ? Two Domains - Fourier Transform properties: scaling, shifting, convolution, correlation theorems ? Parseval?s theorem ? Sampling Theorem ? Discrete Fourier Transform ? Fast Fourier Transform - Tempered Distributions ? Fourier Transform on Tempered Distributions.
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.
Algebras and ólgebras, 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, Lp -spaces and their dual; Product measures, constructions, Fubini?s theorem and its applications.
Review of gradient, divergence and curl. Elementary idea of tensors. Velocity of fluid, Streamlines and path lines, Steady and unsteady flows, Velocity potential, Vorticity vector, Conservation of mass, Equation of continuity. Equations of motion of a fluid, Pressure at a point in fluid at rest, Pressure at a point in a moving fluid, Eulers equation of motion, Bernoullis equation. Singularities of flow, Source, Sink, Doublets, Rectilinear vortices. Complex variable method for two-dimensional problems, Complex potentials for various singularities, Circle theorem, Blasius theorem, Theory of images and its applications to various singularities. Three dimensional flow, Irrotational motion, Weisss theorem and its applications. Viscous flow, Vorticity dynamics, Vorticity equation, Reynolds number, Stress and strain analysis, Navier-Stokes equation, Boundary layer Equations
Model and its different types, Finite models, Statistical models, Stochastic models, Formulation of a model, Laws and conservation principles, Discrete and continuous models, Manipulation into its most respective form, Evaluation of a model. Case studies, Continuum model, Transport phenomena, Diffusion and air pollution models, Microwave heating, Communication and Information technology.
Software Support : MATHEMATICA, LSODE, GNUPLOT, MATLAB.
The concept of variation and its properties, Variational problems with fixed boundaries, The Euler equation, Variational problems in parametric form. Variational problems with moving boundaries, Reflection and refraction extremals. Sufficient conditions for an extremum, Canonical equations and variational principles, Complementary variational principles, The Hamilton-Jacobi equation. Direct methods for variational problems, Rayleigh-Ritz method, Galerkin method. Introduction to optimal control problems, Controllability and optimal control, Isoperimetric problems, Bolza problem, Optimal time of transit, Rocket propulsion problem, Linear autonomous time-optimal control problem, Existence theorems for optimal control problems, Necessary conditions for Optimal controls, The Pontryagin maximum principle.
General solution of Bessel equation, Recurrence relations, Orthogonal sets of Bessel functions, Modified Bessel functions, Applications. General solution of Legendre equation, Legendre polynomials, Associated Legendre polynomials, Rodrigues formula, Orthogonality of Legendre polynomials, Application. Concept and calculation of Greens function, Approximate Greens function, Greens function method for differential equations, Fourier Series, Generalized Fourier series, Fourier Cosine series, Fourier Sine series, Fourier integrals. Fourier transform, Laplace transform, Z-transform, Hankel transform, Mellin transform. Solution of differential equation by Laplace and Fourier transform methods.
Large sparse linear systems, Storage schemes, Review of stationary iterative process, Krylov subspace methods, Conjugate gradients(CG), BiCG, MINRES and GMRES, The Lanczos iteration, From Lanczos to Gauss quadrature, Preconditioning, Error bounds for CG and GMRES, Effects of finite precision arithmetic, Multigrid methods, Multigrid as a preconditioner for Krylov subspace methods. Nonlinear systems, Newtons method and some of its variants, Newton GMRES, Continuation methods, Conjugate direction method, Davidon-Fletcher-Powell Algorithms.
Software Support: HOMPACK, LAPACK.
System Support: PARAM.
Asymptotic expansion and approximations, Asymptotic solution of algebraic and transcendental equations, Introduction to the asymptotic solution of differential equations. Regular Perturbations, Perturbed second order differential equations, Dimensional analysis, Initial and boundary value problems, Partial differential equations, Error estimation. Multiple scales, Overview of multiple scales and averaging, The first order two-scale approximation, Higher order approximations. Methods of WKB type, Introductory examples, The formal WKB expansion without turning points, Ray methods
Incompressible plane flows, Stream function and vorticity equations, Conservative form and normalizing systems, Method for solving vorticity transport equation, Basic finite difference forms, Conservative property, Convergence and stability analysis, Explicit and implicit methods, Stream function equation and boundary conditions, Schemes for advective diffusion equation, Upwind differencing and artificial vorticity, Solution for primitive variables.
Software Support: CFD Software Packages.
Review of sampling distributions. Point and interval-estimation, Hypothesis testing, Likelihood ratio procedure, Bayesian methods. Introduction to decision theory. Regression methods, Linear, Multilinear and polynomial regression. Model checking. Time series analysis, Introduction to autocorrelation function, linear stationary models like autoregressive, integrated moving average processes, Yule-Walker equations and partial auto correlations, Forecasting.
Software Support: Statistical packages like SAS and SPSS.
Decision functions, Risk functions, Utility and subjective probability, Randomization, Optimal decision rules. Admissibility and completeness, Existence of Bayes decision rules, Existence of a minimal complete class, Essential completeness of the class of nonrandomized rules. The minimax theorem. Invariant statistical decision problems. Multiple decision problems. Sequential Decision problems.
Quadratic and nonlinear programming solution methods applied to chance constrained programming problems. Stochastic linear and nonlinear programming problems arising in inventory control and other industrial applications. Queuing models of computer networks, Information processing under uncertainty, Two stage and multi stage solution techniques. Use of Monte carlo, Probabilistic and heuristic algorithms, Genetic algorithms and neural networks for adaptive optimization.
Basic ideas of hedging and pricing by arbitrage. Basic concepts from probability theory and stochastic processes, conditional expectation, martingales, random walk, Markov processes, Brownian motion. Stochastic integration, Itôintegral, Itôformula. Stochastic differential equations. Risk-neutral pricing, Black-Scholes-Merton option pricing model, Girsanov?s Theorem, American derivative securities, term-structure models. Jump processes and their application to option pricing.
Review of probability, random variables, distributions, generating functions; Poisson, Markov, renewal and semi-Markov processes; Characteristics of queueing systems, Little?s law, Markovian and non-Markovian queueing systems, embedded Markov chain applications to M/G/1, G/M/1 and related queueing systems; Networks of queues, open and closed queueing networks; Queues with vacations, priority queues, queues with modulated arrival process, discrete time queues, introduction to matrix-geometric methods, applications in manufacturing, computer and communication networks.
Probability, random variables, probability distributions, expectations, martingales, Brownian motion, Itôtegral, Itôformula; Financial markets and financial instruments, forward and futures contracts and determination of their prices, options, mechanism of options markets, put-call parity, European and American options, risk-neutral valuation, Cox-Ross-Rubinstein model, Black-Scholes-Merton model; Numerical methods for European and American options.
Local theory of plane and space curves, Curvature and torsion formulas, Serret-Frenet formulas, Fundamental Theorem of space curves. Regular surfaces, Change of parameters, Differentiable functions, Tangent plane, Differential of a map. First and second fundamental form. Orientation, Gauss map and its properties, Euler?s Theorem on principal curvatures. Isometries, and Gauss? Theorema Egregium. Parallel transport, Geodesics, Gauss-Bonnet theorem and its applications to surfaces of constant curvature. Hopf-Rinow?s theorem, Bonnet?s theorem, Jacobi fields, Theorems of Hadamard. Riemanns Habilitationsvortrag.
Brief review of rings and ideals, nilradical and Jacobson radicals, extension and contraction; basic theory of modules: submodules and quotient modules, module homomorphisms, annihilators, torsion submodules, irreducible modules, Schurs lemma, direct sum and product of modules, free modules, localization, Nakayamas lemma; Exact sequences, short and split exact sequences, projective modules, injective modules, Baers criterion for injective modules; tensor product of modules, universal property of tensor product, exactness property of tensor products, flat modules; chain conditions on rings, Noetherian rings, Hilbert basis theorem; Artinian rings, discrete valuation rings, Dedekind domains, fractional ideals, ideal class groups.
Field extensions, algebraic extensions, minimal polynomials, separable and normal extensions; Automorphism groups, fixed fields, Galois extensions, Galois groups; Fundamental theorem of Galois theory, Galois closure; Galois groups of finite fields; Cyclotomic extensions, abelian extensions over rationals, Kronecker-Weber theorem; Galois groups of polynomials, symmetric functions, discriminant, Galois groups of quadratic, cubic and quartic polynomials; solvable extensions, radical extensions, solution of polynomial equations in radicals, insolvability of the quintic; cyclic extensions, Kummer theory, Artin-Schreier extensions; Galois groups over rationals, transcendental extensions, infinite Galois groups.
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Filtrations, graded rings, graded modules; Krull dimension, depth, going up and going down, primary ideals, prime avoidance theorem, primary decomposition; Hilbert functions, regular local rings, Cohen-Macaulay rings; Zariski topology, Hilbert Nullstellensatz, spectrum of a ring, affine algebraic sets, affine variety, projective variety; Noether normalization, integral closure; algebraic curves, function field of a curve, divisors, principal divisors, Picard group, divisor class group, genus; monomial orderings, division algorithm, Groebner basis, syzygies.
Overview of distributed computing, Basic algorithms in message-passing systems, Coordination algorithms, Causality and logical time, Distributed snapshot and global states, Distributed algorithms for graphs, fault and fault-tolerance, Distributed mutual exclusion, distributed transactions, distributed consensus and agreement algorithms, Self-stabilization, Applications in wireless sensor networks.
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