Program Announcement 2013 - 14 - Institute of Business ...

Description of Courses174MTS412 Functional Analysis-2This is an add-on course to Functional Analysis-1 andbrings the subject of Functional Analysis to a sufficientlysophisticated standard, suitable for senior levelstudents. In brief, topics incorporate affine spaces,Banach spaces, and Hilbert spaces.Calculus onmanifolds, homomorphic spaces, Urysohn's lemma,and Baire category theorem. Metrization of spaces ,and spaces with measure, measurable functions, ideaof ??fields. Students who successfully finish this courseare expected to apply the knowledge for furtherresearch in this and related fields.Pre-requisite: MTS411MTS413 Abstract Algebra-IIThis course aims to teach Abstract Algebra at a seniorlevel, and requires Abstract Algebra 1 as a prerequisite.Topics in brief include, review of rings and?elds, introduction to integral domain, ideals, primeand maximal ideals, ?eld of quotients of an integraldomain, and ?eld extensions. Advanced topicsincorporate free Abelian groups, Z-modules, scissorscongruence group, simplicial complexes, homologygroups, and their computations. Splitting ?elds, andGalois theory is treated last.Pre-requisite: MTS305MTS414 Scientific Computing for Linear PDEs.Linear PDEs model a wide variety of interestingproblems in science and finance. This course aims toimpart the theoretical foundation as well as the practicalimplementation of some of the methods used forsolving PDEs. The numerical solution of importantPDEs, such as the Poisson and the Helmholtz PDE isdealt in this course. Discretization techniques, andmultilevel iterative solution methods are the mainhighlights. Implementation of the algorithms is anothersalient feature.Pre-requisite: MTS303 & MTS306MTS431 Numerical Solutions of PDEsThis course aims at developing computational skillsrequired for numerically solving partial differentialequations (second order). Different discretizationmethods are introduced from the scratch and differentsolution strategies, some for linear PDEs and somefor non-linear PDEs are built. Discretization methodsinclude, the Finite Difference Methods and the FiniteVolume Method. The resulting linear / linearizedsystems are solved directly using suitable LUdecompositions. Successful completetion of this courseimplies the skill set required for direct numericalsolutions of PDEs.Pre-requisite: MTS306MTS432 Integral EquationsThe major topics discussed in this course includeLinear integral equations, Fredholm integral equations,Eigenvalue problems, Volterra integral equations,Singular integral equations, Systems of linear integralequations, Nonlinear integral equations, Miscellaneousspecial kernels, Integro-ordinary differential equations,Integro-partial differential equations, Qualitativebehavior, Abstract integral equations, integral equationsin abstract spaces, Integral operators, Inverseproblems, Random integral equationsPre-requisite: MTS303MTS433 Advanced Numerical Analysis IThe main aim of this course is to teach moderntechniques of solving a system of linear equations,obtained from PDE discretizations of various kinds.The solution method used in this course would beiterative, in particular, the Krylov subspace family ofiterative methods prototyped by Conjugate Gradients.Students taking this course should be able to contributeto the development of these iterative solvers as wellas solving any consistent system of linear equations.Pre-requisite: MTS301 & MTS306MTS434 Advanced Numerical Analysis IIThis is an advanced course for the iterative solution ofdiscretized Partial Differential equations of the secondorder.The iterative solution technique taught here ismultigrid. Concepts related to solving PDEs on multipleresolutions that correspond to each other are builthere. Different multigrid components are constructedand the students are encouraged to explore thecomparative merits of these components. Multigridpreconditioned Krylov methods are also taught, whichform some of the fastest known solvers. The skill setdeveloped in this course would include solving PDEsthrough multigrid based solvers.Pre-requisite: MTS453MTS435 Differential GeometryThe course introduces the basic language ofdifferentiable manifolds, important in understandinggeometric objects independent of being embedded inany ambient space. Differential forms and its integrationare introduced to emphasize the intrinsic notion ofcalculus. By introducing tangent and cotangent bundles,the aim is to lay down a strong foundation for a generaltheory of vector bundles, very important for the gaugetheory of mathematical physics. This is followed topicssuch as Lie bracket, Lie derivative of vector fields andintroduction to Riemann geometry. On completion ofthis course, student is expected to have the basic toolsof modern geometric methods applicable in modernphysics.Pre-requisite: MTS241 & MTS301MTS437 Fluid Dynamics IThe study of fluid mechanics helps to understand manyother aspects of engineering and applied sciences forexample food industry, energy conversion systems andfluid control systems. Fluid mechanics deals with fluidseither in motion or at rest. We concentrate on fluid inmotion and the forces on them (fluids include liquids,gases, and plasmas).The aim of this course is to studyfamous principle of classical mechanics such asNewton's laws of motion, conservations and basicequations used in fluid mechanics. We will study manyuseful situations by using simple idealized fluid modelsand classical principle of mechanics. On completionof this course, students will be able to formulatesolutions to flow problems, including those based ondifferential analysis, using appropriate fluid properties,flow conditions, and coordinate representations.Pre-requisite: MTS303MTS438 Fluid Dynamics IIThis course is a follow up course of fluid dynamics I.In this course we study about rheology and some ofits properties like shear thinking and shear