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 200Dept. Elective/Institute ElectiveElective I2106Elective II210Elective III21MA 593 Project I (Optional)---- ----6Elective IV21066Elective V210606Elective VI2106-4Elective VII2106---6MA 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 528 Hyperplane Arrangements3006MA 556 Differential Geometry2106MA 530 Nonlinear Analysis2106MA 581 Elements of Differential Topology2106MA 532 Analytic Number Theory2106SI 5073108MA 534 Modern Theory of PDE2106MA5101 Algebra II2106MA 540 Numerical Methods for PDE2106MA5103 Algebraic Combinatorics2106MA 562 Mathematical Theory of Finite Elements2106MA5105 Coding Theory2106SI 416Optimization2106MA5107 Continuum Mechanics2106SI 527Introduction to Derivative Pricing2106MA5109 Graph Theory2106MA5102 Basic Algebraic Topology2106MA5111 Theory of Finite Semigroups2106MA5104 Hyperbolic Conservation Laws2106MA5115 Hopf Algebras2106MA5106 Introduction to Fourier Analysis2106MA5108 Lie Groups and Lie Algebra2106MA5110 Noncommutative AlgebraMA5112 Introduction to Math. Methods2 12 10066MA5116 Species and Operads206Numerical Analysis1
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.Texts / References:Supplementary Reading Materials (SelectedBook Chapters and Papers)N.N. Biswas, FORTRAN IV ComputerProgramming, Radiant Books, 1979.HS 200 Environmental Studies 3 0 0 3K.D. Sharma, Programming in Fortran IV,Affiliated East West, 1976.ES 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.Multidisciplinary 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.Texts / References:Text / References:W. P. Cunningham and M. A. Cunningham,Principles of Environmental Science, TataMcGraw-Hill Publishing Company, 2002.N. Agar, Life's Intrinsic Value, ColumbiaUniversity Press, 2001.J. A. Nathanson, Basic EnvironmentalTechnology:WaterSupplyWasteManagement and Pollution Control, 4thEdition, Prentice Hall of India, 2002.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.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.K. Hoffman and R. Kunze, Linear Algebra,Pearson Education (India), 2003.R. Guha and M. Gadgil, Ecology and Equity:The Use and Abuse of Nature inContemporary India, Penguin, 1995.S. Lang, Linear Algebra, UndergraduateTexts in Mathematics, Springer-Verlag, NewYork, 1989.N. Hanley, J. F. Shogren and B. White,Environmental Economics in Theory andPractice, MacMillan, 2004.P. Lax, Linear Algebra, John Wiley & Sons,1997.H.E. Rose, Linear Algebra, Birkhauser, 2002.A. Naess and G. Sessions, Basic Principles ofDeep Ecology, Ecophilosophy, Vol. 6 (1984).M. Redclift and G. Woodgate (Eds.),International Handbook of EnvironmentalSociology, Edward Edgar, 1997.S. Lang, Algebra, 3rd Edition,Springer (India), 2004.O. Zariski and P. Samuel, CommutativeAlgebra, Vol. I, Springer, 1975.MA 401 Linear Algebra 3 1 0 8MA 403 Real AnalysisVector spaces over fields, subspaces, basesand dimension.Systems of linear equations, matrices, rank,Gaussian elimination.Linear transformations, representation oflinear transformations by matrices, ranknullity theorem, duality and transpose.3108Review of basic concepts of real numbers:Archimedean property, Completeness.Metric spaces, compactness, connectedness,(with emphasis on Rn).Continuity and uniform continuity.Determinants, Laplace expansions, cofactors,adjoint, Cramer's Rule.Monotonic functions, Functions of boundedvariation; Absolutely continuous functions.Derivatives of functions and Taylor'stheorem.Eigenvalues and eigenvectors, characteristicpolynomials, minimal polynomials, CayleyHamilton Theorem, triangulation, diagonallization, rational canonical form, Jordancanonical form.Riemann integral and its properties,characterization of Riemann integrablefunctions. Improper integrals, Gammafunctions.Inner product spaces, Gram-Schmidt orthonormalization, orthogonal projections, linearfunctionals and adjoints, Hermitian, selfadjoint, unitary and normal operators,Spectral Theorem for normal operators.Sequences and series of functions, uniformconvergence and its relation to continuity,differentiation and integration. Fourier series,pointwise convergence, Fejer's theorem,Weierstrass approximation theorem.Bilinear forms, symmetric and skewsymmetric bilinear forms, real quadraticforms, Sylvester's law of inertia, positivedefiniteness.Texts / References:T. Apostol, Mathematical Analysis, 2ndEdition,Narosa,2002.Texts / References:M. Artin, Algebra, Prentice Hall of India,1994.
K. Ross, Elementary Analysis: The Theoryof Calculus, Springer Int. Edition, 2004.W. Rudin, Principles of MathematicalAnalysis, 3rd Edition, McGraw-Hill, 1983.MA 406 General Topology 3 1 0 8Prerequisites: MA 403 (Real Analysis)K. D. Joshi, Introduction to GeneralTopology, New Age International, 2000.K. D. Joshi, Introduction to GeneralTopology, New Age International, 2000.J. L. Kelley,Nostrand, 1955.GeneralTopology, VanJ. R. Munkres, Topology, 2nd Edition,Pearson Education (India), 2001.Topological Spaces: open sets, closed sets,neighbourhoods, bases, sub bases, limitpoints, closures, interiors, continuousfunctions, homeomorphisms.G. F. Simmons, Introduction to Topologyand Modern Analysis, McGraw-Hill, 1963.Examples of topological spaces: subspacetopology, product topology, metric topology,order topology.Prerequisites: MA 403 (Real Analysis)Quotient Topology: Construction of cylinder,cone, Moebius band, torus, etc.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,One-point Compactification.Complete metric spaces and function spaces,Characterization of compact metric spaces,equicontinuity, Ascoli-Arzela Theorem,Baire Category Theorem. Applications: spacefillingcurve,nowheredifferentiablecontinuous function.Optional Topics: Topological Groups andorbit spaces, Paracompactness and partitionof unity, Stone-Cech Compactification, Netsand filters.Texts / ReferencesM. A. Armstrong, Basic Topology, Springer(India),2004.MA 408 Measure Theory 3 1 0 8Semi-algebra, Algebra, Monotone class,Sigma-algebra, Monotone class theorem.Measure spaces.Outline of extension of measures fromalgebras to the generated sigma-algebras:Measurable sets; Lebesgue Measure and itsproperties.Measurable functions and their properties;Integration and Convergence theorems.Introduction to Lp-spaces, Riesz-Fischertheorem; Riesz Representation theorem for L2spaces. Absolute continuity of measures,Radon-Nikodym theorem. Dual of Lp-spaces.Product measure spaces, Fubini's theorem.Fundamental Theorem of Calculus forLebesgue Integrals (an outline).Texts / 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.
MA 410 Multivariable Calculus2106Prerequisites: 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.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.Texts / References:Guillemin and A. Pollack, DifferentialTopology, Prentice-Hall Inc., EnglewoodCliffe, New Jersey, 1974.V.Fleming, Functions of Several Variables,2nd Edition, Springer-Verlag, 1977.W.J.R. Munkres, Analysis on Manifolds,Addison-Wesley, 1991.W. Rudin, Principles of MathematicalAnalysis, 3rd Edition, McGraw-Hill, 1984.M. Spivak, Calculus on Manifolds, AModern Approach to Classical Theorems ofAdvanced Calculus, W. A. Benjamin, Inc.,1965.MA 412 Complex Analysis 3 1 0 8Complex numbers and the point at infinity.Analytic functions.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.Zeroes and poles, Maximum ModulusPrinciple, Argument Principle, Rouche'stheorem.Texts / References:J.B. Conway, Functions of One ComplexVariable, 2nd Edition, Narosa, NewDelhi, 1978.T.W. Gamelin, Complex Analysis, SpringerInternational Edition, 2001.R. Remmert, Theory of Complex Functions,Springer Verlag, 1991.A.R. Shastri, An Introduction toComplex Analysis, Macmilan India, NewDelhi, 1999.MA 414 Algebra I3108Prerequisite: MA 401 Linear Algebra,MA 419 Basic AlgebraFields, 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.Galois groups, Fundamental Theorem ofGalois Theory, Composite extensions,Examples (including cyclotomic extensionsand extensions of finite fields).Norm, trace and discriminant. Solvability byradicals, Galois' Theorem on solvability.Cyclic extensions, Abelian extensions,Polynomials with Galois groups Sn.Transcendental extensions.
Texts / References:M. Artin, Algebra, Prentice Hall ofIndia, 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.M. Hirsch, S. Smale and R. Deveney,Differential Equations, Dynamical Systemsand Introduction to Chaos, Academic Press,2004M. Hirsch, S. Smale and R. Deveney,Differential Equations, Dynamical Systemsand Introduction to Chaos, Academic Press,2004Perko, Differential Equations andDynamical Systems, Texts in AppliedMathematics, Vol. 7, 2nd Edition, SpringerVerlag, New York, 1998.L.Rama Mohana Rao, Ordinary DifferentialEquations: Theory and Applications.Affiliated East-West Press Pvt. Ltd., NewDelhi, 1980.M.MA 417 Ordinary DifferentialEquations 3 1 0 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: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's Theorem, homomorphisms, normalsubgroups. Quotients of groups, Basicexamples of groups: symmetric groups, matrixgroups, group of rigid motions of the plane and
finite groups of motions. Cyclic groups,generators and relations, Cayley's Theorem,group actions, Sylow Theorems. Directproducts, Structure Theorem for finiteabelian 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.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 Functional Analysis,2nd Edition, Springer, Berlin, 1990.C. Goffman and G. Pedrick, A First Coursein Functional 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.A. Gallian, Contemporary AbstractAlgebra, 4th Edition, Narosa, 1999.J.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.K.A. Taylor and D. Lay, Introduction toFunctional Analysis, Wiley, New York, 1980.T. T. Moh, Algebra, World Scientific, 1992.MA 504 Operators on HilbertSpaces 2 1 0 6S. Lang, Undergraduate Algebra, 2ndEdition, Springer, 2001.Prerequisites:Analysis)S. Lang, Algebra, 3rd Edition, Springer(India), 2004.Adjoints of bounded operators on a Hilbertspace, Normal, self-adjoint and unitaryoperators, their spectra and numerical ranges.D. Joshi, Foundations of DiscreteMathematics, Wiley Eastern, 1989.J. Stillwell, Elements of Algebra, Springer,1994.MA 503 Functional Analysis3108Prerequisites: MA 401 (Linear Algebra),MA 408 (Measure Theory)Normed spaces. Continuity of linear maps.Hahn-Banach Extension and SeparationMA503(FunctionalCompact operators on Hilbert spaces.Spectral theorem for compact self-adjointoperators.Application to Sturm-Liouville Problems.Texts / References:B.V. Limaye, Functional Analysis, 2ndEdition, New Age International, 1996. J.B.
Conway, A Course in Functional Analysis,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.W. Fulton, Algebraic Curves, Benjamin,1969.J. Harris, Algebraic Geometry: A FirstCourse, Springer-Verlag, 1992.M.Reid,UndergraduateAlgebraicGeometry, Cambridge University Press,Cambridge, 1990.I.R. Shafarevich, Basic Algebraic Geometry,Springer-Verlag, Berlin, 1974.R.J. Walker, Algebraic Curves, SpringerVerlag, Berlin, 1950.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.Texts / References:S.S. Abhyankar, Algebraic Geometry forScientists and Engineers, American MathematicalSociety,1990.MA 515 Partial DifferentialEquations 3 1 0 8Prerequisites: MA 417 (OrdinaryDifferential Equations), MA 410(Multivariable Calculus)Cauchy Problems for First Order HyperbolicEquations: method of characteristics, Mongeco
Course Curricula : M.Sc. Mathematics) First Semester Second Semester Course Name L T P C Course Name L T P C CS101 Computer Programmin
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 Programming
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.
First Contact Practitioners and Advanced Practitioners in Primary Care: (Musculoskeletal) A Roadmap to Practice 12.9 Tutorial record 75 12.10 Tutorial evaluation 76 12.11 Multi-professional Supervision in Primary Care for First Contact & Advanced Practitioners - course overview 77
