Course Curricula: M.Sc. (Mathematics)
Course Curricula: M.Sc. (Mathematics)First SemesterSecond SemesterCourse NameL T P CCourse NameL T PCCS1012026MA 406 General Topology3108MA 401 Linear Algebra3108MA 408 Measure Theory3108MA 403 Real Analysis3108MA 410 Multivariable Calculus2106MA 417 Ordinary Differential Equations3108MA 412 Complex Analysis3108MA 419 Basic Algebra3108MA 414 Algebra I3108Computer Programming & UtilizationTotal Credits14 4 2 38Total CreditsThird Semester0 38Fourth semesterMA 503 Functional Analysis3108ES 200/ Environmental Science/MA 515 Partial differential Equations3108HS 200Elective I2106Elective II210Elective III21MA 593 Project I (Optional)--------6Elective IV21066Elective V210606Elective VI2106-4Elective VII2106---6Dept. Elective/Institute ElectiveMA 598 Project II/Dept. Elective/Institute ElectiveTotal Credits14 512 5 0 34Total CreditsGrand TotalElectives I – III11 40 3651 182 140Electives IV – VIIMA 521 Theory of Analytic Functions2106MA 504 Operators on Hilbert Spaces2106MA 523 Basic Number Theory2106MA 510 Introduction to Algebraic Geometry2106MA 525 Dynamical Systems2106MA 518 Spectral Approximation2106MA 533 Advanced Probability Theory2106MA 524 Algebraic Number Theory2106MA 538 Representation Theory of Finite Groups2106MA 526 Commutative Algebra2106MA 539 Spline Theory and Variational Methods2106MA 530 Nonlinear Analysis2106MA 556 Differential Geometry2106MA 532 Analytic Number Theory2106MA 581 Elements of Differential Topology2106MA 534 Modern Theory of PDE2106SI 5073108MA 540 Numerical Methods for PDE2106MA5101 Algebra II2106MA 562 Mathematical Theory of Finite Elements2106MA5103 Algebraic Combinatorics2106SI 416Optimization2106MA5105 Coding Theory2106SI 527Introduction to Derivative Pricing2106MA5107 Continuum Mechanics2106MA5102 Basic Algebraic Topology2106MA5109 Graph Theory2106MA5104 Hyperbolic Conservation Laws2106MA5106 Introduction to Fourier Analysis2106MA5108 Lie Groups and Lie Algebra2106MA5110 Noncommutative Algebra2106MA5112 Introduction to Math. Methods2106Numerical Analysis
COURSE CONTENTSCS 101 Computer ProgrammingUtilization2026&Functional organization of computers, algorithms, basic programming concepts,FORTRAN language programming. Programtestinganddebugging,Modularprogramming subroutines: Selected examplesfrom Numerical Analysis, Game playing,sorting/ searching methods, etc.G.M. Masters, Introduction to EnvironmentalEngineering and Science, Second IndianReprint, Prentice-Hall of India, 2004.M. L. Davis and D. A. Cornwell,Introduction to Environmental Engineering,2nd Edition, McGraw Hill, 1998.R. T. Wright, Environmental Science:Towards a Sustainable Future, 9th Edition,Prentice Hall of India, 2007.Supplementary Reading Materials (SelectedBook Chapters and Papers)Texts / References:N.N. Biswas, FORTRAN IV ComputerProgramming, Radiant Books, 1979.K.D. Sharma, Programming in Fortran IV,Affiliated East West, 1976.HS 200 Environmental Studies 3 0 0 3Social issues and the environment, Publicawareness and Human rights, Indicators ofsustainability, Governance of NaturalResources - Common pool resources: issuesand management.ES 200 Environmental Studies 3 0 0 3Multidisciplinary nature of environmentalproblems; Ecosystems, Biodiversity and itsconservation; Indicators of environmentalpollution; Environment and human health;Utilization of natural resources and environmental degradation. Sustainable development; Environmental policy and law;Environmental impact assessment; Pollutionof lakes, rivers and groundwater. Principlesof water and wastewater treatment; Solid andhazardous waste management. Air Pollution:sources and effects, Atmospheric transport ofpollutants; Noise pollution; Global issues andclimate change: Global warming, Acid rain,Ozone layer depletion.Environmental ethics, Religion and environment, Wilderness and Developing Trends,Environmental movements and tal justice.Environmental economics, Trade and environment, Economics of environmental regulation, Natural resource accounting, GreenGDP.Environment and development, Resettlementand rehabilitation of people, Impacts ofclimate change on economy and society,Vulnerability and adaptation to climatechange.Text / References:Texts / References:W. P. Cunningham and M. A. Cunningham,Principles of Environmental Science, TataMcGraw-Hill Publishing Company, 2002.J. A. Nathanson, Basic EnvironmentalTechnology:WaterSupplyWasteManagement and Pollution Control, 4thEdition, Prentice Hall of India, 2002.N. Agar, Life's Intrinsic Value, ColumbiaUniversity Press, 2001.P. Dasgupta and G. Maler, G. (Eds.), TheEnvironment and Emerging DevelopmentIssues, Vol. I, Oxford University Press, ”inandA.
Raghuramaraju (Ed.), Debating on Gandhi,Oxford University Press, 2006.M. Artin, Algebra, Prentice Hall of India,1994.R. Guha and M. Gadgil, Ecology and Equity:The Use and Abuse of Nature inContemporary India, Penguin, 1995.K. Hoffman and R. Kunze, Linear Algebra,Pearson Education (India), 2003.N. Hanley, J. F. Shogren and B. White,Environmental Economics in Theory andPractice, MacMillan, 2004.A. Naess and G. Sessions, Basic Principles ofDeep Ecology, Ecophilosophy, Vol. 6 (1984).S. Lang, Linear Algebra, UndergraduateTexts in Mathematics, Springer-Verlag, NewYork, 1989.P. Lax, Linear Algebra, John Wiley & Sons,1997.H.E. Rose, Linear Algebra, Birkhauser, 2002.M. Redclift and G. Woodgate (Eds.),International Handbook of EnvironmentalSociology, Edward Edgar, 1997.MA 401 Linear Algebra 3 1 0 8S. Lang, Algebra, 3rd Edition, Springer(India), 2004.O. Zariski and P. Samuel, CommutativeAlgebra, Vol. I, Springer, 1975.Vector spaces over fields, subspaces, basesand dimension.MA 403 Real AnalysisSystems of linear equations, matrices, rank,Gaussian elimination.Linear transformations, representation oflinear transformations by matrices, ranknullity theorem, duality and transpose.Determinants, Laplace expansions, cofactors,adjoint, Cramer's Rule.Eigenvalues and eigenvectors, characteristicpolynomials, minimal polynomials, CayleyHamilton Theorem, triangulation, diagonallization, rational canonical form, Jordancanonical form.Inner product spaces, Gram-Schmidt orthonormalization, orthogonal projections, linearfunctionals and adjoints, Hermitian, selfadjoint, unitary and normal operators,Spectral Theorem for normal operators.Bilinear forms, symmetric and skewsymmetric bilinear forms, real quadraticforms, Sylvester's law of inertia, positivedefiniteness.Texts / References:310 8Review of basic concepts of real numbers:Archimedean property, Completeness.Metric spaces, compactness, connectedness,(with emphasis on Rn).Continuity and uniform continuity.Monotonic functions, Functions of boundedvariation; Absolutely continuous functions.Derivatives of functions and Taylor'stheorem.Riemann integral and its properties,characterization of Riemann integrablefunctions. Improper integrals, Gammafunctions.Sequences and series of functions, uniformconvergence and its relation to continuity,differentiation and integration. Fourier series,pointwise convergence, Fejer's theorem,Weierstrass approximation theorem.Texts / References:T. Apostol, Mathematical Analysis, 2ndEdition, Narosa, 2002.
K. Ross, Elementary Analysis: The Theoryof Calculus, Springer Int. Edition, 2004.K. D. Joshi, Introduction to GeneralTopology, New Age International, 2000.W. Rudin, Principles of MathematicalAnalysis, 3rd Edition, McGraw-Hill, 1983.J. L. Kelley, GeneralNostrand, 1955.MA 406 General Topology 3 1 0 8J. R. Munkres, Topology, 2nd Edition,Pearson Education (India), 2001.Prerequisites: MA 403 (Real Analysis)Topology,VanG. F. Simmons, Introduction to Topology andModern Analysis, McGraw-Hill, 1963.Topological Spaces: open sets, closed sets,neighbourhoods, bases, sub bases, limitpoints, closures, interiors, continuousfunctions, homeomorphisms.MA 408 Measure Theory 3 1 0 8Examples of topological spaces: subspacetopology, product topology, metric topology,order topology.Semi-algebra, Algebra, Monotone class,Sigma-algebra, Monotone class theorem.Measure spaces.Quotient Topology: Construction of cylinder,cone, Moebius band, torus, etc.Outline of extension of measures fromalgebras to the generated sigma-algebras:Measurable sets; Lebesgue Measure and itsproperties.Connectedness and Compactness: Connectedspaces, Connected subspaces of the real line,Components and local connectedness,Compact spaces, Heine-Borel Theorem,Local -compactness.Separation Axioms: Hausdorff spaces,Regularity, Complete Regularity, Normality,Urysohn Lemma, Tychonoff embedding andUrysohn Metrization Theorem, TietzeExtension Theorem. Tychnoff Theorem, Onepoint Compactification.Complete metric spaces and function spaces,Characterization of compact metric spaces,equicontinuity, Ascoli-Arzela Theorem, BaireCategory Theorem. Applications: spacefillingcurve,nowheredifferentiablecontinuous function.Optional Topics: Topological Groups andorbit spaces, Paracompactness and partitionof unity, Stone-Cech Compactification, Netsand filters.Prerequisites: MA 403 (Real Analysis)Measurable functions and their properties;Integration and Convergence theorems.Introduction to Lp-spaces, Riesz-Fischertheorem; Riesz Representation theorem forL2-spaces. Absolute continuity of measures,Radon-Nikodym theorem. Dual of Lp-spaces.Product measure spaces, Fubini's theorem.Fundamental Theorem of CalculusLebesgue Integrals (an outline).forTexts / References:P.R. Halmos, Measure Theory, Graduate Textin Mathematics, Springer-Verlag, 1979.Inder K. Rana, An Introduction to Measureand Integration (2nd Edition), NarosaPublishing House, New Delhi, 2004.H.L. Royden, Real Analysis, 3rd Edition,Macmillan, 1988.Texts / ReferencesMA 410 Multivariable CalculusM. A. Armstrong, Basic Topology, Springer(India), 2004.2106Prerequisites: MA 403 (Real Analysis),MA 401 (Linear Algebra)
Functions on Euclidean spaces, continuity,differentiability; partial and directionalderivatives, Chain Rule, Inverse FunctionTheorem, Implicit Function Theorem.Riemann Integral of real-valued functions onEuclidean spaces, measure zero sets, Fubini'sTheorem.Cauchy-Riemann conditions. Mappings byelementary functions. Riemann surfaces.Conformal mappings.Contour integrals, Cauchy-Goursat Theorem.Uniform convergence of sequences andseries. Taylor and Laurent series. Isolatedsingularities and residues. Evaluation of realintegrals.Partition of unity, change of variables.Integration on chains, tensors, differentialforms, Poincaré Lemma, singular chains,integration on chains, Stokes' Theorem forintegrals of differential forms on chains.(general version). Fundamental theorem ofcalculus.Differentiable manifolds (as subspaces ofEuclidean spaces), differentiable functions onmanifolds, tangent spaces, vector fields,differential forms on manifolds, orientations,integration on manifolds, Stokes' Theorem onmanifolds.Zeroes and poles, Maximum ModulusPrinciple, Argument Principle, Rouche'stheorem.Texts / References:J.B. Conway, Functions of One ComplexVariable, 2nd Edition, Narosa, New Delhi,1978.T.W. Gamelin, Complex Analysis, SpringerInternational Edition, 2001.R. Remmert, Theory of Complex Functions,Springer Verlag, 1991.Texts / References:V. Guillemin and A. Pollack, DifferentialTopology, Prentice-Hall Inc., EnglewoodCliffe, New Jersey, 1974.A.R. Shastri, An Introduction to ComplexAnalysis, Macmilan India, New Delhi,1999.MA 414 Algebra IW. Fleming, Functions of Several Variables,2nd Edition, Springer-Verlag, 1977.J.R. Munkres, AnalysisAddison-Wesley, 1991.on3108Prerequisite: MA 401 Linear Algebra,MA 419 Basic AlgebraManifolds,W. Rudin, Principles of MathematicalAnalysis, 3rd Edition, McGraw-Hill, 1984.M. Spivak, Calculus on Manifolds, A ModernApproach to Classical Theorems ofAdvanced Calculus, W. A. Benjamin, Inc.,1965.Fields, Characteristic and prime subfields,Field extensions, Finite, algebraic andfinitely generated field extensions, Classicalruler and compass constructions, Splittingfields and normal extensions, algebraicclosures. Finite fields, Cyclotomic fields,Separable and inseparable extensions.MA 412 Complex Analysis 3 1 0 8Galois groups, Fundamental Theorem ofGalois Theory, Composite extensions,Examples (including cyclotomic extensionsand extensions of finite fields).Complex numbers and the point at infinity.Analytic functions.Norm, trace and discriminant. Solvability byradicals, Galois' Theorem on solvability.
Cyclic extensions, Abelian extensions,Polynomials with Galois groups Sn.Transcendental extensions.M. Hirsch, S. Smale and R. Deveney,Differential Equations, Dynamical Systemsand Introduction to Chaos, Academic Press,2004Texts / References:M. Artin, Algebra, Prentice Hall of India,1994.D.S. Dummit and R. M. Foote, AbstractAlgebra, 2nd Edition, John Wiley, 2002.J.A. Gallian, Contemporary AbstractAlgebra, 4th Edition, Narosa, 1999.N. Jacobson, Basic Algebra I, 2nd Edition,Hindustan Publishing Co., 1984, W.H.Freeman, 1985.MA 417 Ordinary Differential Equations310 8Review of solution methods for first order aswell as second order equations, Power Seriesmethods with properties of Bessel functionsand Legendré polynomials.Existence and Uniqueness of Initial ValueProblems: Picard's and Peano's Theorems,Gronwall's inequality, continuation ofsolutions and maximal interval of existence,continuous dependence.Higher Order Linear Equations and linearSystems: fundamental solutions, Wronskian,variation of constants, matrix exponentialsolution, behaviour of solutions.Two Dimensional Autonomous Systems andPhase Space Analysis: critical points, properand improper nodes, spiral points and saddlepoints.Asymptotic Behavior: stability (linearizedstability and Lyapunov methods).Boundary Value Problems for Second OrderEquations:Green'sfunction,Sturmcomparison theorems and oscillations,eigenvalue problems.Texts / References:L. Perko, Differential Equations andDynamical Systems, Texts in AppliedMathematics, Vol. 7, 2nd Edition, SpringerVerlag, New York, 1998.M. Rama Mohana Rao, Ordinary DifferentialEquations: Theory and Applications.Affiliated East-West Press Pvt. Ltd., NewDelhi, 1980.D. A. Sanchez, Ordinary DifferentialEquations and Stability Theory: AnIntroduction, Dover Publ. Inc., New York,1968.MA 419 Basic Algebra3108Review of basics: Equivalence relations andpartitions, Division algorithm for integers,primes, unique factorization, congruences,Chinese Remainder Theorem, Euler ϕfunction.Permutations, sign of a permutation,inversions, cycles and transpositions.Rudiments of rings and fields, elementaryproperties, polynomials in one and ls, Division algorithm, RemainderTheorem, Factor Theorem, Rational ZerosTheorem, Relation between the roots andcoefficients,Newton'sTheoremonsymmetric functions, Newton's identities,Fundamental Theorem of ion, unique factorization ofpolynomials in several variables, Resultantsand discriminants.Groups, subgroups and factor groups,Lagrange'sTheorem,homomorphisms,normal subgroups. Quotients of groups,Basic examples of groups: symmetric groups,matrix groups, group of rigid motions of theplane and finite groups of motions.
Cyclic groups, generators and relations,Cayley's Theorem, group actions, SylowTheorems. Direct products, StructureTheorem for finite abelian groups.Simple groups and solvable groups, nilpotentgroups, simplicity of alternating groups,composition series, Jordan-Holder Theorem.Semidirect products. Free groups, freeabelian groups.Rings, Examples (including polynomialrings, formal power series rings, matrix ringsand group rings), ideals, prime and maximalideals, rings of fractions, Chinese RemainderTheorem for pairwise comaximal ideals.Euclidean Domains, Principal Ideal Domainsand Unique Factorizations Domains.Polynomial rings over UFD's.Theorems. Banach spaces. Dual spaces andtransposes.Uniform Boundedness Principle and itsapplications. Closed Graph Theorem, OpenMapping Theorem and their applications.Spectrum of a bounded operator. Examplesof compact operators on normed spaces.Inner product spaces, Hilbert spaces.Orthonormal basis. Projection theorem andRiesz Representation Theorem.Texts / References:J.B. Conway, A Course in FunctionalAnalysis, 2nd Edition, Springer, Berlin, 1990.C. Goffman and G. Pedrick, A First Course inFunctional Analysis, Prentice-Hall, 1974.Texts / References:M. Artin, Algebra, Prentice Hall of India,1994.D. S. Dummit and R. M. Foote, AbstractAlgebra, 2nd Edition, John Wiley, 2002.J. A. Gallian, Contemporary AbstractAlgebra, 4th Edition, Narosa, 1999.K. D. Joshi, Foundations of DiscreteMathematics, Wiley Eastern, 1989.E. Kreyzig, Introduction to FunctionalAnalysis with Applications, John Wiley &Sons, New York, 1978.B.V. Limaye, Functional Analysis, 2ndEdition, New Age International, New Delhi,1996.A. Taylor and D. Lay, Introduction toFunctional Analysis, Wiley, New York, 1980.MA 504 Operators on Hilbert Spaces2106T. T. Moh, Algebra, World Scientific, 1992.S. Lang, UndergraduateEdition, Springer, 2001.rdS. Lang, Algebra, 3(India), 2004.Algebra,2ndEdition, SpringerJ. Stillwell, Elements of Algebra, Springer,1994.MA 503Functional oints of bounded operators on a Hilbertspace, Normal, self-adjoint and unitaryoperators, their spectra and numerical ranges.Compact operators on Hilbert spaces.Spectral theorem for compact self-adjointoperators.3108Application to Sturm-Liouville Problems.Prerequisites: MA 401 (Linear Algebra),MA 408 (Measure Theory)Texts / References:Normed spaces. Continuity of linear maps.Hahn-Banach Extension and SeparationB.V. Limaye, Functional Analysis, 2ndEdition, New Age International, 1996.
J.B. Conway, A Course in FunctionalAnalysis, 2nd Edition, Springer, 1990.C. Goffman and G. Pedrick, First Course inFunctional Analysis, Prentice Hall, 1974.I. Gohberg and S. Goldberg, Basic OperatorTheory, Birkhaüser, 1981.E. Kreyzig, Introduction to FunctionalAnalysis with Applications, John Wiley &Sons, 1978.S. G. Mikhlin, Variation Methods in Mathematical Physics, Pergaman Press, Oxford1964.J. A. Murdock, Perturbations Theory andMethods, John Wiley and Sons, 1991.P. D. Miller, Applied asymptotic analysis,American Mathematical Society, 2006.M. L. Krasnov et.al., Problems and exercisesin the calculus of variations, Mir Publishers,1975.M. Krasnov et. al., Problems and exercises inintegral equations, Mir Publishers, 1971.MA 510 IntroductionGeometry 2 1 0 6toAlgebraicPrerequisites: MA 414 (Algebra 1)Varieties: Affine and projective varieties,coordinate rings, morphisms and rationalmaps, local ring of a point, function fields,dimension of a variety.Curves: Singular points and tangent lines,multiplicities and local rings, intersectionmultiplicities, Bezout's theorem for planecurves, Max Noether's theorem and some ofits applications, group law on a nonsingularcubic, rational parametrization, branches andvaluations.W. Fulton, Algebraic Curves, Benjamin,1969.J. Harris, Algebraic Geometry: A FirstCourse, Springer-Verlag, 1992.M. Reid, Undergraduate Algebraic Geometry,Cambridge University Press, Cambridge,1990.I.R. Shafarevich, Basic Algebraic Geometry,Springer-Verlag, Berlin, 1974.R.J. Walker, Algebraic Curves, SpringerVerlag, Berlin, 1950.MA 515 Partial Differential Equations3108Prerequisites: MA 417 (OrdinaryDifferential Equations), MA 410(Multivariable Calculus)Cauchy Problems for First Order HyperbolicEquations: method of characteristics, Mongecone.Classification of Second Order PartialDifferential Equations: normal forms andcharacteristics.Initial and Boundary Value Problems:Lagrange-Green's identity and uniqueness byenergy methods.Stability theory, energy conservation anddispersion.Laplace equation: mean value property, weakand strong maximum principle, Green'sfunction, Poisson's formula, Dirichlet'sprinciple, existence of solution using Perron'smethod (without proof).Texts / References:Heat equation: initial value problem,fundamental solution, weak and strongmaximum principle and uniqueness results.S.S. Abhyankar, Algebraic Geometry forScientists and Engineers, American Mathematical Society, 1990.Wave equation: uniqueness, D'Alembert'smethod, method of spherical means andDuhamel's principle.
Methods of separation of variables for heat,Laplace and wave equations.MA 521 The Theory of Analytic Functions2106Prerequisites: MA 403 (Real Analysis),MA 412 (Complex Analysis)Texts / References:DifferentialMaximum Modulus Theorem. SchwarzLemma. Phragmen-Lindelof Theorem.L.C. Evans, Partial Differential Equations,Graduate Studies in Mathematics, Vol. 19,American Mathematical Society, 1998.Riemann Mapping Theorem. WeierstrassFactorization Theorem.E.DiBenedetto,PartialEquations, Birkhaüser, 1995.F. John, Partial Differential Equations, 3rdEdition, Narosa, 1979.Runge's Theorem. Simple connectedness.Mittag-Leffler Theorem.Schwarz Reflection Principle.E. Zauderer, Partial Differential Equations ofApplied Mathematics, 2nd Edition, JohnWiley and Sons, 1989.Basic properties of harmonic functions.Picard Theorems.MA 518 Spectral Approximation2106Prerequisite: MA 503 Functional AnalysisSpectral decomposition. Spectral sets offinite type. Adjoint and product spaces.Convergence of operators: norm, collectivelycompact and ν convergence. Error estimates.Finite rank approximations based onprojections and approximations for integraloperators.A posteriori error estimates.Texts / References:L. Ahlfors, Complex Analysis, 3rd Edition,McGraw-Hill, 1979.J.B. Conway, Functions of One ComplexVariable, 2nd Edition, Narosa, 1978.T.W. Gamelin, Complex Analysis, SpringerInternational, 2001.R. Narasimhan, Theory of Functions of OneComplex Variable, Springer (India), 2001.W. Rudin, Real and Complex Analysis, 3rdEdition, Tata McGraw-Hill, 1987.Matrix formulations for finite rank operators.Iterative refinement of a simple eigenvalue.Numerical examples.MA 523 Basic Number TheoryTexts / References:Prerequisites: MA 419 (Basic Algebra)M. Ahues, A. Largillier and B. V. Limaye,SpectralComputationsforBoundedOperators, Chapman and Hall/CRC, 2000.Infinitude of primes, discussion of the PrimeNumber Theorem, infinitude of primes inspecific arithmetic progressions, Dirichlet'stheorem (without proof).F. Chatelin, Spectral Approximation ofLinear Operators, Academic Press, 1983.T. Kato, Perturbation Theory of LinearOperators, 2nd Ed., Springer-Verlag, 1980.2106Arithmetic functions, Mobius inversionformula. Structure of units modulo n, Euler'sphi function
Congruences, theorems of Fermat and Euler,Wilson's theorem, linear congruences,quadratic residues, law of quadraticreciprocity.The ideal class group, finiteness of the idealclass group, Dirichlet units theorem.Binary quadratics forms, equivalence,reduction, Fermat's two square theorem,Lagrange's four square theorem.K. Ireland and M. Rosen, A ClassicalIntroduction to Modern Number Theory, 2ndEdition, Springer-Verlag, Berlin, 1990.Continued fractions, rational approximations,Liouville's theorem, discussion of Roth'stheorem,transcendentalnumbers,transcendence of e and π.S. Lang, Algebraic Number Theory, AddisonWesley, 1970.Diophantine equations: Brahmagupta'sequation (also known as Pell's equation), theThe equation, Fermat's method of descent,discussion of the Mordell equation.D. A. Marcus, Number Fields, SpringerVerlag, 1977.MA 525 Dynamical Systems2106Prerequisites: MA 417 (OrdinaryDifferential Equations)Texts / ReferencesW.W. Adams and L.J. Goldstein, Introductionto the Theory of Numbers, 3rd Edition, WileyEastern, 1972.A. Baker, A Concise Introduction to theTheory of Numbers, Cambridge UniversityPress, 1984.I. Niven and H.S. Zuckerman, AnIntroduction to the Theory of Numbers, 4thEdition, Wiley, 1980.MA 524 Algebraic Number Theory 2 1 0 6Prerequisites: MA 414 Algebra I (Exposure)Algebraic number fields.discrete valuation rings.Texts / ReferencesLocalisation,Integral ring extensions, Dedekind domains,unique factorisation of ideals. Action of theGalois group on prime ideals.Valuations and completions of number fields,discussion of Ostrowski's theorem, Hensel'slemma, unramified, totally ramified andtamely ramified extensions of p-adic fields.Discriminants and Ramification. Cyclotomicfields, Gauss sums, quadratic reciprocityrevisited.Linear Systems: Review of stability forlinear systems of two equations.Local Theory for Nonlinear PlanarSystems: Flow defined by a differentialequation, Linearization and stable manifoldtheorem, Hartman-Grobman theorem,Stability and Lyapunov functions, Saddles,nodes, foci, centers and nonhyperboliccritical points. Gradient and Hamiltoniansystems.Global Theory for Nonlinear PlanarSystems: Limit sets and attractors, Poincarémap, Poincaré Benedixson theory andPoincare index theorem.Bifurcation Theory for Nonlinear Systems:Structural stability and Peixoto's theorem,Bifurcations at nonhyperbolic equilibriumpoints.Texts / References:L. Perko, Differential Equations andDynamical Systems, Springer Verlag, 1991.M. W. Hirsch and S. Smale, DifferentialEquations, Dynamical Systems and LinearAlgebra, Academic Press, 174.P. Hartman, Ordinary Differential Equations,2nd edition, SIAM 2002.
C. Chicone, Ordinary Differential Equationswith Applications, 2nd Edition, Springer,2006.MA 526 Commutative Algebra 2 1 0 6Prerequisites: MA 505 (Algebra II)Dimension theory of affine n lemma, dimension andtranscendence degree, catenary property ofaffine rings, dimension and degree of theHilbert polynomial of a graded ring, Nagata'saltitude formula, Hilbert's Nullstellensatz,finiteness of integral closure.Associated primes of modules, degree of theHilbert polynomial of a graded module,Hilbert series and dimension, ativity formula for multiplicity,Complete local rings: Basics of completions,Artin-Rees lemma, associated graded rings offiltrations, completions of modules, regularlocal ringsBasic Homological algebra: Categories andfunctors, derived functors, Hom and tensorproducts, long exact sequence of homologymodules, free resolutions, Tor and Ext,Koszul complexes.Texts in Mathematics 150, Springer-Verlag,2003.H. Matsumura, Commutative ring theory,CambridgeStudiesinAdvancedMathematics No. 8, Cambridge UniversityPress, 1980.W. Bruns and J. Herzog, Cohen-MacaulayRings, Revised edition, Cambridge Studies inAdvanced Mathematics No. 39, CambridgeUniversity Press, 1998.MA 530 Nonlinear Analysis 2 1 0 6Prerequisites: MA 503 (Functional Analysis)Fixed Point Theorems with Applications:Banach contraction mapping theorem,Brouwer fixed point theorem, LeraySchauder fixed point theorem.Calculus in Banach spaces: Gateaux as wellas Frechet derivatives, chain rule, Taylor'sexpansions, Implicit function theorem withapplications, subdifferential.Monotone Operators: maximal monotoneoperators with properties, surjectivitytheorem with applications.Degree theory and condensing operators withapplications.Texts / References:Cohen-Macaulay rings: Regular sequences,quasi-regular sequences, Ext and depth,grade of a module, Ischebeck's theorem,Basic properties of Cohen-Macaulay rings,Macaulay's unmixed theorem, HilbertSamuel multiplicity and Cohen-Macaulayrings, rings of invariants of finite groups.M.C. Joshi and R.K. Bose, Some Topics inNonlinear Functional Analysis, WileyEastern Ltd., New Delhi, 1985.Optional Topics: Face rings of simplicialcomplexes, shellable simplicial complexesand their face rings. Dedekind Domains andValuation Theory.MA 532 Analytic Number Theory 2 1 0 6Texts / References:The Wiener-Ikehara Tauberian theorem, thePrime Number Theorem.D. Eisenbud, Commutative Algebra (with aview toward algebraic geometry), GraduateE. Zeilder, Nonlinear Functional Analysisand Its Applications, Vol. I (Fixed PointTheory), Springer Verlag, Berlin, 1985.Prerequisites: MA 414 (Algebra I), MA 412(Complex Analysis)
Dirichlet's theorem forArithmetic Progression.primesinanZero free regions for the Riemann-zetafunction and other L-functions.Euler products and the functional equationsfor the Riemann zeta function and DirichletL-functions.Modular forms for the full modular group,Eisenstein series, cusp forms, structure of thering of modular forms.Hecke operators and Euler product formodular forms.The L-function of a modular form, functionalequations. Modular forms and the sums offour squares.Optional topics: Discussion of L-functionsof number fields and the Chebotarev DensityTheorem. Phragmen-Lindelof Principle,Mellin inversion formula, Hamburger'stheorem. Discussion of Modular forms forcongruence subgroups. Discussion of Artin'sholomorphyconjectureandhigherreciprocity laws.Discussion of ellipticcurves and the Shimura-Taniyama conjecture(Wiles' Theorem)Texts / References:S. Lang, Algebraic Number Theory, AddisonWesley, 1970.J.P. Serre, A Course in Arithmetic, SpringerVerlag, 1973.T. Apostol, Introduction to Analytic NumberTheory, Springer-Verlag, 1976.MA 533 Advanced Probability Theory2106Probability measure, probability space,construction of Lebesgue measure, extensiontheorems, limit of events, distributions, multidimensional distributions,independence.Expectation, change of variable theorem,convergence theorems.Sequence of random variables, modes ofconvergence. Moment generating functionand characteristics functions, inversion anduniqueness theorems, continuity theorems,Weak and strong laws of large number,central limit theorem.Radon Nikodym theorem, definition andproperties of conditional expectation,conditional distributions and expectations.Texts / References:P. Billingsley, Probability and Measure, 3rdEdition, John Wiley & Sons, New York,1995.J. Rosenthal, A First Look at RigorousProbability, World Scientific, Singapore,2000.A.N. Shiryayev, Probability, 2nd Edition,Springer, New York, 1995.K.L. Chung, A Course in Probability Theory,Academic Press, New York, 1974.MA 534 Modern Theory of PartialDifferential Equa
Course Curricula: M.Sc. (Mathematics) First Semester Second Semester Course Name L T P C Course Name L T P C CS101 Computer Programming
as HSC Year courses: (in increasing order of difficulty) Mathematics General 1 (CEC), Mathematics General 2, Mathematics (‘2 Unit’), Mathematics Extension 1, and Mathematics Extension 2. Students of the two Mathematics General pathways study the preliminary course, Preliminary Mathematics General, followed by either the HSC Mathematics .
Course Curricula : M.Sc. Mathematics) First Semester Second Semester Course Name L T P C Course Name L T P C CS101 Computer Programmin
IBDP MATHEMATICS: ANALYSIS AND APPROACHES SYLLABUS SL 1.1 11 General SL 1.2 11 Mathematics SL 1.3 11 Mathematics SL 1.4 11 General 11 Mathematics 12 General SL 1.5 11 Mathematics SL 1.6 11 Mathematic12 Specialist SL 1.7 11 Mathematic* Not change of base SL 1.8 11 Mathematics SL 1.9 11 Mathematics AHL 1.10 11 Mathematic* only partially AHL 1.11 Not covered AHL 1.12 11 Mathematics AHL 1.13 12 .
2. 3-4 Philosophy of Mathematics 1. Ontology of mathematics 2. Epistemology of mathematics 3. Axiology of mathematics 3. 5-6 The Foundation of Mathematics 1. Ontological foundation of mathematics 2. Epistemological foundation of mathematics 4. 7-8 Ideology of Mathematics Education 1. Industrial Trainer 2. Technological Pragmatics 3.
1st Grade Mathematics Course Description Course descriptions provide an overview for a course and designate which standards are in that course. The 1st grade mathematics course description includes resources for all 36 standards within the 1st grade mathematics course. Mathematics Formative Assessment System (MFAS)
Enrolment By School By Course 5/29/2015 2014-15 100 010 Menihek High School Labrador City Enrolment Male Female HISTOIRE MONDIALE 3231 16 6 10 Guidance CAREER DEVELOPMENT 2201 114 73 41 CARRIERE ET VIE 2231 32 10 22 Mathematics MATHEMATICS 1201 105 55 50 MATHEMATICS 1202 51 34 17 MATHEMATICS 2200 24 11 13 MATHEMATICS 2201 54 26 28 MATHEMATICS 2202 19 19 0 MATHEMATICS 3200 15 6 9
2021-2022 Graduate Degree Curricula DEGREE CURRICULA NOTES A student is responsible for knowing when his/her required courses are offered. General Remarks Each degree curriculum in this document is provided as a reference showing all course requirements. All degrees leading to USCG licensure also require passing the relevant license exams.
accounting items are presumed in law to give a true and fair view. 8 There is no explicit requirement in the Companies Act 2006 or FRS 102 for companies entitled to prepare accounts in accordance with the small companies regime to report on the going concern basis of accounting and material uncertainties. However, directors of small companies are required to make such disclosures that are .
