Re: Mathematical Logic Discrete Mathematics By Tremblay Manohar Pdf Free 125
Casandro Diveley
Unit I of the syllabus covers propositional logic and counting theory. It introduces concepts such as propositions, logical connectives like conjunction, disjunction, negation, implication and biconditional. It discusses how to represent compound statements using these connectives and their truth tables. The unit also covers topics like predicate logic, methods of proof, mathematical induction and fundamental counting principles like permutations and combinations. It aims to provide the logical foundations for discrete mathematics concepts that will be useful in computer science and information technology.Read less
mathematical logic discrete mathematics by tremblay manohar pdf free 125
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Set Theory - sets and classes, relations and functions, recursive definitions, posets, Zorn - s lemma, cardinal and ordinal numbers; Logic - propositional and predicate calculus, well-formed formulas, tautologies, equivalence, normal forms, theory of inference. Combinatorics - permutation and combinations, partitions, pigeonhole principle, inclusion-exclusion principle, generating functions, recurrence relations. Graph Theory - graphs and digraphs, Eulerian cycle and Hamiltonian cycle, adjacency and incidence matrices, vertex colouring, planarity, trees.
Introduction - the von Neumann architecture, machine language, assembly language, high level programming languages, compiler, interpreter, loader, linker, text editors, operating systems, flowchart; Basic features of programming (Using C) - data types, variables, operators, expressions, statements, control structures, functions; Advance programming features - arrays and pointers, recursion, records (structures), memory management, files, input/output, standard library functions, programming tools, testing and debugging; Fundamental operations on data - insert, delete, search, traverse and modify; Fundamental data structures - arrays, stacks, queues, linked lists; Searching and sorting - linear search, binary search, insertion-sort, bubble-sort, selection-sort; Introduction to object oriented programming.
Groups, subgroups, homomorphism; Group actions, Sylow theorems; Solvable and nilpotent groups; Rings, ideals and quotient rings, maximal, prime and principal ideals; Euclidean and polynomial rings; Modules; Field extensions, Finite fields.
Systems of linear equations, vector spaces, bases and dimensions, change of bases and change of coordinates, sums and direct sums; Linear transformations, matrix representations of linear transformations, the rank and nullity theorem; Dual spaces, transposes of linear transformations; trace and determinant, eigenvalues and eigenvectors, invariant subspaces, generalized eigenvectors; Cyclic subspaces and annihilators, the minimal polynomial, the Jordan canonical form; Inner product spaces, orthonormal bases, Gram-Schmidt process; Adjoint operators, normal, unitary, and self-adjoint operators, Schur's theorem, spectral theorem for normal operators.
Convergence of sequence of real numbers, real valued functions of real variables, differentiability, Taylor's theorem; Functions of several variables - limit, continuity, partial and directional derivatives, differentiability, chain rule, Taylor's theorem, inverse function theorem, implicit function theorem, maxima and minima, multiple integral, change of variables, Fubini's theorem; Metrics and norms - metric spaces, convergence in metric spaces, completeness, compactness, contraction mapping, Banach fixed point theorem; Sequences and series of functions, uniform convergence, equicontinuity, Ascoli's theorem, Weierstrass approximation theorem.
Asymptotic 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, greedyalgorithm, amortized analysis; Elementary graph algorithms, minimum spanning tree, shortest path algorithms.
First order linear and quasi-linear partial differential equations (PDEs), Cauchy problem, Classification of second order PDEs, characteristics, Well-posed problems, Solutions of hyperbolic, parabolic and elliptic equations, Dirichlet and Neumann problems, Maximum principles, Green's functions.
Review of complex numbers; Analytic functions, harmonic functions, elementary functions, branches of multiple-valued functions, conformal mappings; Complexintegration, Cauchy's integral theorem, Cauchy's integral formula, theorems of Morera and Liouville, maximum-modulus theorem; Power series, Taylor's theorem and analytic continuation, zeros of analytic functions, open mapping theorem; Singularities, Laurent's theorem, Casorati-Weierstrass theorem, argument principle, Rouche's theorem, residue theorem and its applications in evaluating real integrals.
Axiomatic definition of probability, probability spaces, probability measures on countable and uncountable spaces, conditional probability, independence; Random variables, distribution functions, probability mass and density functions, functions of random variables, standard univariate discrete and continuous distributions and their properties; Mathematical expectations, moments, moment generating functions, characteristic functions, inequalities; Random vectors, joint, marginal and conditional distributions, conditional expectations, independence, covariance, correlation, standard multivariate distributions, functions of random vectors; Modes of convergence of sequences of random variables, weak and strong laws of large numbers, central limit theorems; Introduction to stochastic processes, definitions and examples.
Mathematical foundations and basic definitions: concepts from linear algebra, geometry, and multivariable calculus. Linear optimization: formulation and geometrical ideas of linear programming problems, simplex method, revised simplex method, duality, sensitivity snalysis, transportation andassignment problems. Nonlinear optimization: basic theory, method of Lagrange multipliers, Karush-Kuhn-Tucker theory, convex optimization. Numerical optimization techniques: line search methods, gradient methods, Newton's method, conjugate direction methods, quasi-Newton methods, projected gradient methods, penalty methods.
Alphabets, languages, grammars; Finite automata, regular languages, regular expressions; Context-free 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; NP-Completeness.
Normed linear spaces, Banach spaces; Continuity of linear maps, Hahn-Banach theorem, open mapping and closed graph theorems, uniform boundedness principle; Duals and transposes, weak and weak* convergence, reflexivity; Spectra of bounded linear operators, compact operators and their spectra; Hilbert spaces, bounded linear operators on Hilbert spaces; Adjoint operators, normal, unitary, self-adjoint operators and their spectra, spectral theorem for compact self-adjoint operators.
Definition and sources of errors, solutions of nonlinear equations; Bisection method, Newton's method and its variants, fixed point iterations, convergence analysis; Newton's method for non-linear systems; Finite differences, polynomial interpolation, Hermite interpolation, spline interpolation; Numerical integration - Trapezoidal and Simpson's rules, Gaussian quadrature, Richardson extrapolation; Initial value problems - Taylor series method, Euler and modified Euler methods, Runge-Kutta methods, multistep methods and stability; Boundary value problems - finite difference method, collocation method.
Fundamentals - overview of matrix computations, norms of vectors and matrices, singular value decomposition (SVD), IEEE floating point arithmetic, analysis ofroundoff errors, stability and ill-conditioning; Linear systems - LU factorization, Gaussian eliminations, Cholesky factorization, stability and sensitivity analysis; Jacobi, Gauss-Seidel and successive overrelaxation methods; Linear least-squares - Gram- Schmidt orthonormal process, rotators and reflectors, QR factorization, stability of QR factorization; QR method linear least-squares problems, normal equations, Moore- Penrose inverse, rank deficient least-squares problems, sensitivity analysis. Eigenvalues and singular values - Schur's decomposition, reduction of matrices to Hessenberg and tridiagonal forms; Power, inverse power and Rayleigh quotient iterations; QR algorithm, implementation of implicit QR algorithm; Sensitivity analysis of eiegnvalues; Reduction of matrices to bidiagonal forms, QR algorithm for SVD.
Finite difference schemes for partial differential equations - explicit and implicit schemes; Consistency, stability and convergence - stability analysis by matrix method and von Neumann method, Lax's 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; CFL conditions; Finite element method for ordinary differential equations - variational methods, method of weighted residuals, finite element analysis of one-dimensional problems.
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.
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