Linear Mathematics: A Practical Approach (Dover Books on Mathematics)

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  3. Matrix and Linear Algebra
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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. The structure of cyclic codes, Reed Mueller codes, Simplex codes. Weight distribution of codes, Generalized BCH codes including the BCH bound and decoding methods , Self-dual codes and invariant theory, MacWilliams identities and Gleason's theorems on self-dual codes, Covering radius problem, Convolutional codes.

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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. Introduction to elliptic curves, Group structure, Rational points on elliptic curves, Elliptic Curve Cryptography.

Applications in cryptography and factorization, Known attacks.

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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. 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, Minkowski's theory, computation of class group of number fields. Dirichlet Unit Theorem, fundamental units, units in quadratic fields, Pell's equation.

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Cyclotomic fields. 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, Ramanujan's 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 mtrix-tree theorem, enumeration and patterns, Polya-Redfield theorem, cycle index, symmetries on N and R, polyominoes.

Scope of Parallel Computing - limits to parallelizability, 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.

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. The course aims to introduce database management systems from application perspective.

This is expected to helpful for M. Mathematics and Computing students who wish to pursue career in software industry.


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Algebraic numbers, Primes and factoring, Trapdoors and public key, Pseudo-random numbers. The finite Fourier transforms. The fast Fourier transform. 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.

Matrix and Linear Algebra

Self-similarity, Scaling, Similarity dimension, Box-counting dimension, Information dimension, Capacity dimension. 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. Solution of nonlinear systems by Newton's method, Local error estimation.

Fourier series: Definition, examples and uniqueness of Fourier series. Convolution, Cesaro summability and Abel summability, the Poisson kernel and Dirichlet's problem in the unit disc. Mean-square and pointwise convergence of Fourier series, applications of Fourier series.

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Fourier transform: Elementary theory of Fourier transforms, the Schwartz space, the Fourier inversion, the Plancherel formula. Theory of distributions, Fourier transform of a tempered distribution. Poisson summation formula, Heisenberg's uncertainty principle, Paley-Wiener theorem, Wiener's theorem, Wiener-Tauberian theorem. 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. 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, Euler's equation of motion, Bernoulli's equation.


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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, Weiss's theorem and its applications.

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.

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.

Kenschaft is the author of: [4]. Kenschaft was the winner of the Louise Hay Award of the Association for Women in Mathematics "in recognition of her long career of dedicated service to mathematics and mathematics education" and for her work "writing about, speaking about, and working for mathematics and mathematics education in the areas of K—12 education, the environment, affirmative action and equity, and public awareness of the importance of mathematics in society".

From Wikipedia, the free encyclopedia. However, see MR for the full title. July , The Mathematical Gazette , 88 : —, doi : Explore Plus. Did you find what you were looking for?

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Concepts of Physics 2 1st PROS:- 1 Cheap. Vivek Anand Certified Buyer Jan, Explanations, examples and exercises are very useful. Recommended for everyone.