A quick review of analysis of several variables, The Alternating Algebra: Multilinear maps, Alternating multilinear maps, Exterior product. Differential forms: Exterior derivative, Pull-back of forms. de Rham cohomology: Poincaré lemma, Chain complexes and their cohomology, Long exact sequences, Homotopy, Application of de Rham cohomology, Smooth manifolds, Differential forms on smooth manifolds, Integration on manifolds.
Linear Algebra: Vectors in Rn; Vector subspaces of Rn; Basis of vector subspace; Systems of Linear equations; Matrices and Gauss elimination; Determinants and rank of a matrix; Abstract vector spaces, Linear transformations, Matrix of a linear transformation, Change of basis and similarity, Rank-nullity theorem; Inner product spaces, Gram-Schmidt process, Orthonormal bases; Projections and least-squares approximation; Eigenvalues and eigenvectors, Characteristic polynomials, Eigenvalues of special matrices; Multiplicity, Diagonalization, Spectral theorem, Quadratic forms. Differential Equations: Exact equations, Integrating factors and Bernoulli's equation; Orthogonal trajectories; Lipschitz condition, Picard’s theorem; Wronskians; Dimensionality of space of solutions, Abel-Liouville formula; Linear ODE’s with constant coefficients; Cauchy-Euler equations; Method of undetermined coefficients; Method of variation of parameters; Laplace transforms, Shifting theorems, Convolution theorem.
Research, Statistics, & Policy Analysis
Thermodynamics vs kinetics; Homogeneous and heterogeneous reactions - chemical reaction control rate equation, reaction rate constant, reaction order, non-elementary reactions; Solid State Diffusion -Fick’s Law, mechanisms of diffusion, uphill diffusion, Kirkendall effect, steady and transient diffusion; External mass transfer -fluid flow and its relevance to mass transfer, general mass transport equation, concept of mass transfer coefficient, models of mass transfer -film theory and Higbie’s penetration theory; Internal mass transfer-ordinary and Knudsen diffusion, mass transfer with reaction; Adsorption –physical adsorption vs. chemisorption, adsorption isotherms - Langmuir, BET; Adsorption as the rate limiting step examples - gasification of C by CO2, dissolution of N2 in molten steel; Porous solids - specific surface area and pore size distribution; Reactor design -batch vs continuous reactors, ideal stirred tank and plug flow reactors; Mass balance in ideal reactors, residence time distribution; Models of industrial reactors; Electrochemical kinetics-concept of polarization, activation over potential, Butler-Volmer and Tafel’s equation, applications in electro-deposition and corrosion.
Basic of Macro Economics 406 - [DOC Document]
Introduction to Optimization; Formulation of Various Process Optimization Problems and their Classification; Basic Concepts of Optimization-Convex and Concave Functions, Necessary and sufficient conditions for Stationary Points; Optimization of one-dimensional Functions; Unconstrained Multivariable Optimization- Direct Search Methods. Indirect First Order and Second Order Methods; Linear Programming and its Applications; Constrained Multivariable Optimization-Necessary and Sufficient Conditions for Constrained Optimum, Quadratic Programming, Sequential Quadratic Programming; Optimization of Staged and Discrete Processes, Dynamic Optimization, Integer and Mixed Integer Programming. Additional topics such as convexification, Global Optimization, and mixed integer programming will be discussed.
Calculating income-expenditure multiplier
Also, a form of punishment atmilitary training facilities wherein the student performs all ofthe proper DRILL and MARCHing routines, usually with a rifle,that will sequentially cover each of the compass points in aquadrangle or on a PARADE GROUND; a CADET "boxes the compass" todiscount his demerits; see ADY, GIG, TURD, SHIT LIST, ATTITUDEADJUSTMENT, FASHION SHOW, DROP, FRONT LEANING REST, AIRMANALIGNMENT TOOL, BLANKET PARTY, SQUEEZE.