SYLLABUSFor M.A./ M.Sc. in Mathematics (Semester System) ( Effective from the Academic Session 2007-2008 and onwards) THE UNISERSITY OF BURDWAN DIRECTORATE OF DISTANCE EDUCATION Golapbag, Burdwan-713 104 West Bengal 2 THE UNIVERSITY OF BURDWAN Syllabus For M.A./M.Sc. in Mathematics Duration of P.G course of studies in Mathematics shall be two years with Semester- I, Semeste-II, Semester-III and Semester-IV each of six months duration leading to Semester-I, Semester-II, Semester-III and Semester-IV examination in Mathematics at the end of Each Semester. Syllabus for P.G courses in Mathemitics at the end of each Semester. Syllabus for P.G courses in Mathematics is hereby framed according to the following schemes and structures. Scheme: Total Marks= 1000 with 250 Marks in each Semester comprising of five papers in each Semester with 50 marks in each paper. In each theoretical paper 10% Marks is alloted for Internal Assessment. There are four theoretical papers and one practical paper in Semester-III. All students admitted to P.G course in mathematics shall take courses of Semester-I and Semester-II; and from the Semester-II they will opt for either Pure Stream or Applied Stream in Mathematics. The option norm is to be framed by the Department in each year. In each stream of Semester-III, the first three papers are general and the last two papers are special papers. In each stream of Semester-IV, the first two papers are general and the next two papers are special papers and the fifth paper is the term paper. In the Semester-III and Semester-IV, the Department will offer a cluster of special papers and the students will have to choose one according to the norms to be decided by the Department. The term paper is related with the respective special papers and the mark distribution is 30 for written submission, 15 for Seminar Presentation and 5 for Viva-Voice. It should be noted that some special papers will be included in future as per discretion of the Department. 1. For semester III & IV of DDE, the students will be divided eqally between pure and applied streams and the same policy will be followed in respect of special papers to the corresponding branches. The norm for the allotment of students to the streams and special papers will be as follows. (i) The marks in Mathematics only in Hons or Pass will be considered. (ii) Hons students will be given preference over the pass students. 3 COURSE STRUCTURE SEMESTER STRUCTURE OF M.SC.II (27+3) 30 Principle of Mechanics-II (18+2) 20 MCG204: Computer Programming (27+3) 30 Continuum Mechanics.I (18+2) 20 MCG105 Principle of Mechanics-I (27+3) 30 Numerical Analysis (18+2) 20 Semester-II Duration : 6 Months Total Marks: 250 Total No. II. SYLLABUS IN MATHEMATICS (SEMESTER-I. of Lecturers : 50 Hours per paper Each Theoretrical Paper Containing 45 Marks and 5 Marks for Internal Assesment Paper Marks MCG201: Complex Analysis-II(27+3) 30 Real Analysis -II (18+2) 20 MCG202: Partial Differential Equations (27+3) 30 Differential Geometry (18+2) 20 MCG203: Operations Research.I (18+2) 20 MCG205: Computer Aided Numerical Practical 50 4 . IV) Semester-I Duration 6 Months Total Marks: 250 Total No. III.A/M. of Lectures : 50 Hours per paper Each Theoretical Paper Containing 45 Marks and 5 Marks for Internal Assesment Paper Marks MCG101: Functional Analysis-I (27+3) 30 Real Analysis-I (18+2) 20 MCG102: Linear Algebra (27+3) 30 Modern Algebra (18+2) 20 MCG103 Elements of General Topology (27+3) 30 Complex Analysis-I (18+2) 20 MCG104 Ordinary Differential Equations & Special Functions (27+3) 30 Operations Research. IV (45+5) 50 MPT405 Term Paper (30+15+5) 50 Term Paper MPT405 is related with the Special papers and the Marks distribution is 30 Marks for written submission and 15 Marks for Seminar Presentaion and 5 Marks for Viva-voice. of Lecturers : 50 Hours per paper Each Theretical Paper Containing 45 Marks and 5 Marks for Internal Assessment Paper Marks MPG301: Modern Algebra -III (27+3) 30 General Topology-I(18+2) 20 MPG302: Graph Theory (36+4) 40 Mathematical Logic-I(9+1) 10 MPG303: Set Theory & Mathematical Logic-II(27+3) 30 Functional Analysis.II (18+2) 20 MPS304: Special Paper-I (45+5) 50 MPS305: Special Paper-II (45+5) 50 Semester-IV : Pure Stream Duration : 6 Months Total Marks : 250 Total No. of Lectures : 50 Hours per paper Each Theoretical Paper Containing 45 Marks and 5 Marks for Internal Assessment Paper Marks MPG401: Modern Algebra-III (45+5) 50 MPG402: General Topology-II (27+3) 30 Functional Analysis-III (18+2) 20 MPS403: Special Paper.III (45+5) 50 MPS404: Special Paper. Semester III & IV Seperated for Pure and Applied Stream Semester-III: Pure Stream Duration : 6 Months Total Marks : 250 Total No. 5 . Cluster of Special Paper-I & III For Semester-III & IV respectively each of 50 marks (Pure Stream) : A Differential Geometry of Manifolds.II (45+5) 50 Semester-IV : Applied Stream Duration : 6 Months Total Marks: 250 Total No.I & II H Applied Functional Analysis.I & II F Non-linear Optimization in Banach Spaces .I & II E Theory of Rings and Algebra.I & II C Advanced Functional Analysis.I & II B Advanced Real Analysiss.III (45+5) 50 MAS404: Special Paper-IV (45+5) 50 MAT405: Term Paper (30+15+5) 50 Term Paper MAT405 is related with the Special papers and the Marks distribution is 30 Marks for written submission and 15 Marks for Seminar Presentation and 5 Marks for Viva-voice. of Lecturers: 50 Hours per paper Each Theoretical Paper Containing 45 Marks and 5 marks for Internal Assesmsnt Paper Marks MAG301: Methods of Applied Mathematics-I(45+5) 50 MAG302: Methods of Applied Mathematics-II(27+3) 30 Theory of Electro Magnetic Fields(18+2) 20 MAG303: Continuum Mechanics-II (27+3) 30 Dynamical System (18+2) 20 MAS304: Special Paper -I (45+5) 50 MAS305: Special Paper.I & II D Rings of Continuous Function.I & II G Harmonic Analysis. Semester-III : Applied Stream Duration : 6 Months Total Marks: 250 Total No. of Lecturers : 50 Hours per paper Each Theotetical Paper Contaioning 45 Marks and 5 Marks for Internal Assesment Paper Marks MPG401: Continuum Mechanics-III (45+5) 50 MAG402: Elements of Quantum Mechanics (27+3) 30 Chaos and Fractals (18+2) 20 MAS403: Special Paper.I & II 6 . I & III For Semestar-III & IV respectively each of 50 marks (Applied Stream): A Viscous Flows.1 & II C Algebric Topology.1 & II G Proximities. The clusters of special papers to be offered in a particular year shall be decided by the Department. Students of Applied Mathematics stream may also opt Applied Functional Analysis.I & III for Semester.I & II D Applied Functional Analysis.I & II As the Special paper . 7 .I & II C Elasticity and Theoretical Seismology.1 & II B Operator Theory and Applications. if they so desire.I & II as the Special paper-II and Special paper-IV respectively.1& II D Computational Fluid Dynamics.III & IV from the respective clusters of special papers for Semester-III and Semester-IV respsctively.1 & II Cluster of Special Paper.I & II and Special paper.I & II B Elasticity.1 & II C Inuiscid Compressible Flows and Turbulence .1& II E Advanced Operations Research-1 & II F Geometric Functional Analysis . Nearness and Extensions of Topographical Spaces-1 & II H Advanced Complex Analysis.II & IV for Semester -III & IV respectively from the same topic. Boundary Layer Theory and Magneto Hydrodynamics.1 & II D Lattice Theory.1 & II B Advanced Operations Research.Cluster of Special Paper-II & IV For Semester-III & IV respectively each of 50 marks (pure Stream): A Measure and Integration . Students of Pure Mathematics Stream may also opt the Advanced Operations Research. and the Special paper.1 & II * All the students will have to take the Special paper. if they so desire.I & II Cluster of Special Paper.I and Special paper-III respectively.III & IV and Special Paper.II & IV For Semestar-II & IV respectively each of 50 marks (Applied Stream): A Quantum Mechanics. C-S inequality.b] (with supmetric). Co. 1966 7. L. Banach contraction Principle.Theory of Functions of a Real Variable. K. Alilov-Functional Analysis. A. 1989 9. TMG Publishing Co. simple function and measurable function. boundedness and continuity.P. continuity and measurability. Jha. DETAILED SYLLABUS SEMESTER-I Paper. examples such as 12 spaces. A. Pergamon Press.H Siddiqui. operation on collection of measurable functions.. equivalent functions. measurable functions. Hilbert spaces. E. Inner product. Vulikh. I. continuity of norm function. Ltd. existence. Functions of bounded variation on an on an interval. Bachman & L.Functional Analysis. 1p (1# p # 4) etc. C [a. Linear operator. Narici. 1986 5. Lusternik and Sovolev .K. measurable sets and their properties.E Taylor.P Natanson. Banach spaces. Lebesgue outer measure. 1982 8 . Kantorvich and G. c0. 12 [a. measurability of Supremum and Infimum. 1958 8. Riemann Stieltjes integral. examples of bounded and unbounded linear operators. G. lebesgue measure. Vol. New Delhi 4. Wiley Eastern. Normed linear spaces. pointwise limit of sequence of measurable functions. Academic Press. continuity and derivatives of functions from Rm to Rn. countable subadditivity.30) Baire category theorem. New York. Fedrick Unger Publi.application to Picards existence tneorem and Implicit function theorem. 1961 2. References 1. convergence problem and other properties.Functional Analysis 6. Minkowski inequality. Group-B Real Analysis.b] etc.Functional Analysis with application. Pythagorean law.V. Kreyszing. I. their properties. Spaces Cn. john wiley and Sons.Functional Analysis 3. Parallelogram law. monotonocity and measurability.Functional Anaysis.I (Marks-20) Monotone functions and their discontinuities.Functional Analysis.MCG101 (Functional Analysis-I & Real Anlysis-I) Group-A Functional Analysis-I (Marks.Introductory Functional Analysis with Applications. reduction of a matrix to normal form. Lahiri. Euclidean domain. Isomorphism of groups. Mc. & Roy . simple groups.V. diagonalization. Linear transformation in finite dimensional spaces.A Second Course of Mathematical Analysis. Normal sub groups and correspondence theorem for groups. Apostol. 1980 19. maximal ideal. Quaternions. Rings: Ring. Jordan Canonical from. House.Graw Hill. othonormal basis.. 14. Kolkata. Boolean ring. rank and nullity.MCG102 (Linear Algebra & Modern Algebra-I) Group. Burkil & Burkil. commutative rings with identity. idempotent element. 1885 18. PHI. 1994 11. The World Press Pvt.Principle of Mathematical Analysis. 1963 12.I (Marks-20) Groups: Homomorphism. Euclidean space. matrix of linear.30) Vector spaces. Springer-Verlag. Wiley Eastern Ltd 13. Prime ideal. CUP. Simmons. Ltd.Real Analysis. Student Edn. W. Unitary space. 1994. CUP. 1980. Unique factorization domain.Mathematical Analysis. Paper. Lahiri. Eigen vectors. 1991. B. 1989 16. Group-B Modern Algebra. Polynomical rings. Sylvesters law of inertia. spaces spanned by eigen vectors. Quadratic form. B.K. groups of order 4 and 6.Elements of Functional Analysis. 9 . Limaye. automorphisms. Mc Graw Hill. reduction to normal form.Theory of Functions. New York. G. Integration and Function Spaces. Isomorphism theorems.Functional Analysis. diagonalization of symmetric and Hermitian matrices. World Scientific.Real Analysis. Charles Swartz: Measure.1964 15. ideals. similar and congruent matrices. Gram-Schmidt orthogonalization process. characterstic polynominal. minimal polynomial.Real Analysis. Cayley-Hamilton theorem. relation between Prime and maximal ideal.Introduction to Topology and Modern Analysis.A Linear Algebra (Marks. Titchmarsh. Rudin. Principal ideal domain. simulaneous reduction of two quadratic forms. World Press. Prime & irreducible elements division ring. 10. Narosa Publi. Goldberg. First and Second isomorphism theorems. aplication to Geometry & Mechanics. annihilator fo subset of a vector space. Royden.F.e 17. 20. General Topology 5.T3½. homeomorphism and topological invariants.Fundamentals of Abstract Algebra (Tata MaGraw-Hill) 5. interior points. 9. P.Linear Algebra. their simple properties and their relationship.J Thron . definition. S. Alternative way of defining a topology using Kuratowski closure operators and neighbourhood systems. S. Group-B Complex Analysis-I (Marks-20) Complex integration. P.Linear Algebra.MCG103 (Elements of General Topology & Complex Analysis-I) Group . J. subspace. 10. Ghosh & Mukhopadhay. W. L.Basic Abstract Algebra (Cambridge) 3.Linear Algebra. S.Basic Algebra.Topology 3.W.Topological Structires 4. limit points. closure. Joshi. Lang. Moreras theorem. subbasis. Jain & S.Linear Algebra (Premtice Hall). neighbourhood. Continuous functions. Malik.Simmons -Introduction to topology & Modern Analysis 2.Linear Algebra 11. Herstein. Simply connected region and primitives of analytic functions. S. lang.A Elements of General Topology (Marks-30) Topological spaces. References 1. 2.Topics in Algebra (Vikas). Cohn.General Topology 10 . M.N. Munkesh. 4. separable spaces and their relationship. Kelly. Fundamental theorem of algebra. Mordeson & Sen. Kumareson. Cauchys fundamental theorem. Hoffman & Kunze.K. . line integral and its fundamental properties. Lendel of spaces. denseness. Introduction to connectedness and compactness. I. Rao & Bhimsankaran. First and second countable spaces. T1. Separation axioms: T0. power series expansion of analytic functions. Hungerford. Sen.References: 1. Liouvilles theorem. basis. Paper. 8. Cauchys integral formula and higher derivatives. open sets. Bhattacharya.T2.Topics in Abstract Algebra (University Press).R.T4 spaces. Noyapal. derived sets. 7. Zeros of analytic functions and their limit points. T.B. entire functions. closed sets. 6.Algebra (Springer). separating and supporting plane). Churchil. Aplication of O. Definition of O. 6. critical points. Its orthogonality. B. Adjoint equations of n-the order: Lagranges identity.P.. stability. Operations Research-I) Group. simple illustrations. Peanos and Picards theorems (statement only) Linear systems.Functions of Complex variable 12.R.30) Ordinary Differential Equations First order sysyem of Equations: Well-posed problems. phase plane analysis. solution about an ordinary point. Scope of O.. solution of Hermite equation as an example. Rodrigues formula. Drawbacks in definition.T. Legendre.P.B Operations Research. from the solution of its adjoint equation.Functions of one Complex Variable Narosa] 7. along with the geometry in n. solution of equation. Special Functions Series Solution : Ordinary point and singularity of a second order linear differential equation in the complex plane. recurrence relations and differential equations satisfied by it. Copson. Ash . Linearization. solutions of hypergeometric.MCG104 (Ordinary Differential Equations & Special Functions.Complex Variable 10.I (marks -20) Introduction.R.R. J.Theory of Functional of Complex variable 11.Functions of Complex variable Paper. Punoswamy. Fuchs theorem. Laguerre and Bessels equations as examples. undamped pendulum.Functions of Complex Variable 9. Regular singularity. Existence and uniqueness of the solution.solution about a regular singularity. self-adjoint equation. O.dimensional Euclidean space (hyperplane.Complex Variable . E.A Ordinary Differential Equations & Special Functions (Marks.P] 8. Gupta & Gupta. Greens function. R. B. Forbeniuss method.R. Philips. non-linear autonomous system. Computer application in O.R in different sectors. Liapunov stability. Conway . W. Applications ro biological system and ecological sysytem. Group. [A. Legendre polynomical: its generating functions. 11 . and decision making. expamsion of a function in a series of Legendre Polynomials. Fundamental theorem of L. Rouths process for the ignoration of co-ordinates.A Principle of Mechanics . Maximization case in Assignment problem. E. Rota. Swarup. D. Principle of least action. Wagner. Hungarian method.Special Functions [ Macmillan] 14. A. Comparison of simplex method and revised simplex method. N. I.Theory and Applications 4. Burkill. Codington and Levinson. Sneddon. Unbalanced Assignment problem. G. Birkhoff & G. Rayleighs dissipations function. Sensitivity analysis. Travelling salesman problem. Constraint. Restriction on Assignment. Bounded variable method. Schaums Outline Series. References: 1.Standard forms of revised simplex method. Optimality condition. Taha.Operations Research 8.Theory of Ordinary Differential Equations [ TMH] 10. Lebedev. N.Special Functions of mathematical Physics & Chemistry [ Oliver & Boyd.Lagrange differential equations.I (Marks-30) Generalised co-ordinates: Degrees of freedom. Rainville. 12 . Computational procedure. Sasivevir.Essentials of Ordinary differential Equations [MGH] 12. Hillie & Liberman. Hamiltons cannocical equations of motion. E. the Primal Dual Method.K Sharma. Configuration space and system point. Friendman. Hamiltons principle. C. J.Ordinary Differential Equation [Ginn] 13. Principle of Virtual Work. Ignoration of coordinates. Calculus of variation and Euler.Operations Research 6.Operations Research. Lagrangian formulation of Dynamics: Lagranges equations of motion for holonomic and non.I & Numerical Analysis) Group. Brachistochrone problem. P.The Theory of Ordinary Differential 9.Principles of Operations Research (PH) 2. Gupta & Manmohan . R.holonomic system. Canonical Transformations. J.C Gupta .Special Functions and Their Applicaitons [PH] Paper . N. Poisson Bracket.Operations Research: Methods and Problems (JW) 3.Operations Research 5. Yaspan.Introducing to OperationsResearch 7. Agarwal & R. Mathematical formulation of Assignment Problem.MCG105 (Principle of Mechanics. Principle of energy. London] 15. Errors and mininizing errors. Goldstein . Gupta and S. Operators and their inter-relationships: Shift. Average operators. Raulstan. Differential operators and differential coefficients. B.A text Book of Dynamic 2. Elliptic. Smith.Computers and Programming (Schaums series) 11.Numerical Analysis 13 . References : 1. Gupta . A. C.Classical Mechanics of Particles and Rigid Bodies.Numerical Analysis 15. Iyengar and Jain.B Numerical Analysis (Marks-20) Numerical Methods: Algorithm and Numerical stability. F. Tchebyshev polynomials. Graffaes root squaring method and Bairstows method for the determination of the roots of a real polynimial equation. Piece-wise polynominal approximation.Numerical Solutionof Partial Dieeerential Equations (Oxford) 12. Best uniformapproximations. T Whittaker. Scheid. Boundary value and Eigen-value problem for second order O. Jain. Cubic splines. Chorlton. Demidovitch and Maron.A Treatise on the Analytical Dynamics of PArticile and Rigid Bodies 5.Numerical Analysis 14. D. convergence & numerical stability. Green Wood.D.Numerical Methods for Scientific and Engineering Computation 13.Corrector method by Adam-Bashforth. Polynomical Approximation : Polynomial interpolation. Central differences. Initial Value Problems for First and Second order O. Scarborough . Gantmacher.Lectures in Analytical Mechanics 7. Evalution of singular integral. E.C Basu.Classical Mechanics 8.D. H.E. Adam-Moulton and Milnes method.E (for two independent variables) by finite difference method. 6. Synge and Griggith. by finite difference method and shooting method. Hildebrad.E. Richardson extrapolation and Rombergs integration method. simple examples.K method (ii) RKF4-method (iii) Predictor. F.Introduction to Numerical Anslysis 9. Forward. F.Classical Dynamics 4. G. D. parabolic and hyperbolic P. Gauss theory of quadrature. Group. Concept of errors. by (i) 4th order R. K.Computational Mathematics 10.Numerical Analysis 16.Principles of Mechanics 3. F. Atkinson.T.D. Backward. Gupta & Gupta .Functions of Complex Variable 4. Lebesgue criterion of Riemannian integrability. Rouches theorem and its application.Reals Analysis 12. Integrable functions and their simple properties. Laurents series expansion and cllasification of isolated singularities.Complex Variable 5. SEMESTER . Charles Scwarz. Cauchys residue theorem and evaluation of improper integrals. Comparison of Lebesges integral and Riemann integral. Integrable functions.20) Lebesgue integral of a simple function. Lebesegue intergal of functions of arbitary sign.Vol. Referene : 1. Apostol . Introduction to Analytic continuation. Conay. Royden. I 6. Integral of point wise limit of sequence of measurable functions. Dirichlets kernel. Singularities.Complex Variable [ A. 14 . Arguement principle.II (Marks.Theory of Functions 13.Theory of Functions of a Real Variable 11. C. Leb esgue dominated convergence theorem. basic properties of the integral. J.30) Open mapping theorem. R.Real Functions 7. P. Group.B Real Analysis. Goldberg .Real Analysis 9. I. B. P] 3. Lebesgue integral of a non-negative (bounded or unbounded) measurable function. Fourier series. B. Titvhmarsh. essential singylarities and Casorati Weierstrasss theorem.Theory of Functions of a Real Variable 8. Schwarzs Lemma and its consequence. Punoswamy.Monotone econvergence theorem and its consequences.Functions of one Complex Variable [Narosa] 2.MCG201 (Complex Analysiss .Real Analysis 10.Measure. Lahiri & Roy . Conformal mappings.II Paper . Burkil & Burkil . Riemann-Lebesgue theorem Pointwise convergence of Fourier series of functions of bounded variation. Maximum modulus theorem. Fatous lemma. Natanson.II (Marks.II & Real Analysis -II) Group -A Complex Analysis.Theory of Functions of a Real Variable. Integration and Function Spaces. Goffman. Ash. Amarnath Partial Differential Equation 2. angle between two curves lying on the surface. Paper. 8. Andrew Pressely.D. M.Elementary Differential Geometry 15 . Geodesics on a surface. M.E.. Gausss formula and second fundamental form of the surface. Solution of Non-linear first order P. Postrikov Lectures in Geometry.Geometry of Curves 11. reduction to normalform. Singular solution.MCG202 (Partial Differential Equations & Differential Geometry) Group -A Partial Differential Equations (Marks-30) General solution and complete integral of a partial differential equation. P.D.E by cauchys method of characteristics.Elementary Differential Geometry 10. Do Carmo.E. N. Intrinsic derivative. Solution of equations with constant coefficients by (i) factorization of operators. T. Tensor derivative. Charpits method (application only). (ii) separation of variables. Its differential equation. So-lution of Laplace equation un Cylindrical and spherical polar co-ordinates. First fundamental form of the surface. Sneddon Partial Differential Equation 3. Meusnier theorem and Eulers theorem.Differential Geometry of Curves and Surfaces in E3. Serret Frenet formula. References: 1. Rutter. Phoolan Prasad & R. 7. Integral surface passing through a curve and circumscribing a surface.D. C. B. Ravichandan Partial Differential Equations 6. Geodesic curvature of a surface curve. Linear Algebra and Differential Geometry.B Differential Geometry (Marks -20) Reciprocal base system. Formulation of Initial and Boundary Value Problem of P. First order P. Solution of Dirichlets and Numanns problem of Laplaces equation for a circle. Cauchys problem. De. D. H. Anamaya Publi.E. Goldstein Classical Mechanics 4. I. ONeill.. Parallel vector field along a curve Space Curve. P. Second order linear P. Solution of one-dimentional wave equation and diffusion equation.D.Differential Geometry of Curves and surfaces 9. U.E : Classification. Group.2007. : Characteristics of a linear first order P. Metric tensor of the surface. E. Project Crashing. Wagner . Features of Inventory System.Operations Research: Methods and Problems (JW) 10. Basic difference between PERT and CPM. Branch and bound method. The Algorithm. Yaspan.Mathematical Theory of Elasticity 7. Gomorys mixed integer method. Processing of two jobs through m machines. Stability of top motion. Gomorys all integer cutting plane method. Classification of Inventories. Torque free motion. D. Friedman . Taha . Processing of n jobs through two machines. Critical Path analysis.Operations Research 12. Energy equation E= me2. Gupta . Einsteins velocity addition theorem.Theory and Applications 11. Motion of a top on a perfectly rough floor.T. J. Swarup.Operations Research 13. Fitz-Gerald contraction and time-dilation.II (Marks. Paper-MCG203 (Operations Research-II & Principle of Mechanics-II) Group .A Operations Research-II (Marks-30) Deterministic Inventory control Models. T. I. S. Special Lorentz Transformation. References : 1.Operations Research.Classical Dynamics 4. Resource Allocation. Eulers dynamical equations of motion of a rigid body about a fixed point. K. Whittaker. Green Wood . Time cost Trade-off-procedure. Normal co-ordinates. Foucaults pendulum. PERT/CPM network Components and Precedence Relationships. Project scheduling by PERT/CPM: Introduction. The concept of cutting plane. C. Steps of PERT/CPM Techniques. Introduction.Introduction to Operations Research 14. Merovitch . Sharma . Advantage of Carrying Inventory. K. Special Theory of Relativity: Postulates.Principles of Mechanics 3.Principles of Operations Research (PH) 9. Deterministic inventory models including price breaks. F. Sokolnikoff . Chorlton . Group. Updating of the Project. Gupta & Manmohan .Operations Research 16 .A Text Book of Dynamic 2.20) Theory of small oscillations. Processing of n jobs through m machines. Motion of a particle relative to rotating earth. Relativistic mechanics of a particle.A Treatise on the Analytical Dynamics of Particles and Rogid Bodies 5.A Treatise on dynamics.Classical Mechanics of Particles and Rigid Bodies 6. Standard form of Integer Programming. Schaums Outline Series . Synge and Griffith.B Principle of Mechanics. Probability in PERT analysis. 8. Sasievir. Hillie & Lieberman . Paper - MCG204 (Computer Programming & Continuum Mechanics-I) Group- A Computer Programming (Marks-30) Structured Programming in FORTRAN 77; Subscripted variables,Type declaration, DIMENSION, DATA, COMMON, EQUIVALENCE, EXTERNAL statements. Function and subroutine sub programs; Programs in FORTRAN77 Programming in C: Introduction, Basic structures, Character set, Keywards, Identifiers, Constants, Variable-type declaration, Operators: Arithmetic, Relational, Logical, assignment, Increment, decrement, Conditional. Operator precedence and associativity, Arithmetic expression, Evaluation and type conversion, Character reading and writing, Formatted input and output, Decision making (branching and looping) Simple and nested IF,IFELSE, WHILEDo, For. Arrays-one and two dimension, String handling with arraysone and two dimension String handling with arrays reading and writing, Concatenation, Comparison, String handling function, User defined functions. Group -B Continuum Mechanics- I ( Marks -20) Continuous media, Deformation, Lagrangian and Eulerian approach; Analysis of strain (infinitesimal theory);. Analysis of stress; Invariants of stress and strain tensors. Principle of conservation of mass; Principle of balance of linear and angular monentum; Stress equation of motion. Necessity of constitutive equations. Hookes law of elasticity, displacement equation of motion. Newtons law ofvidcosity (statement only). References: 1. Ram Kumar Programming With Fortran 77 2. P. S. Grover Fortran 77/90 3. Jain & Suri FORTRAN Programming Language including FORTRAN 90. 4. G. C. Layek, A. Samad and S.Pramanik Computer Fundamentals, Fortran 77 and Numerical Problems including C. S. Chand & Co. 5. Xavier, C. C Language and Numerical Methods, (New Age International (P) Ltd. Pub.) 6. Gottgried, B.S. Programming with C (TMH) 7. Balaguruswamy, E. Programming ANSI C (TMH) 8. F. Scheid Computers and Programming ( Schaums series) 9. T. J. Chang Continuum Mechanics (Prentice Hall) 10. Truedell Continuum Mechanics (Schaum Series) 11. Mollar theory of Relativity 12. F. Gantmacher Lectures in Analytical Mechanics 13. J. L. Bansal Vicscous Fluid Dynamics ( Oxford) 14. R. N. Chatterjee Contunuum Mechanics 15. H. Goldstein Classical Mechanics 17 Paper MCGP205 Computer Aided Numerical Practical (Marks 50) (Numerical Practical Using FORETRAN 77) [Problem 25, Viva 10 & Sessional 15] (Viva to be conducted on Paper MCG105 (B) & MCG204 (A) 1. Inversion of a non-singular square matrix. 2. Solution of a system of linear equations by Gauss Seidel method. 3. Integration by Rombergs method. 4. Initial Value problems for first and second order O.D.E by (i) Milnes method (First order) (ii) 4th order Runge Kutta method (Second order) 5. Dominant Eigen pair of a real matrix by power method (large and least). 6. B.V. P. for second order O. D. E by finite difference method and Shooting method. 7. Parabolic equation (in two variables) by two layer explicit formula and Crank Nickolson inplict formula. 8. Solution of one dimensional wave equation by finite difference methoed. References : 1. Ram Kumar Programming With Fortran77 2. P. S. Grover Fortran 77/90 3. Jain & Suri FORTRAN 77 Programminh Language including FORTRAN90 4. G. C. Layek, A. Samad and S. Pramanik Computer Fundamentals, Fortran 77 and Numerical Problems including C, S. Chand & Co.. 5. Xavier, C. C Language and Numerical Methods, (New Age International (P) Ltd. Pub.) 6. Gottfried, B.S. Programming with C (TMH) 7. Balaguruswamy, E. Programming in ANSI C (TMH) SEMESTER -III PURE MATHEMATICS STREAM Paper MPG301 (Modern Algebra-II & Set Theory-I) Group -A Modern Algebra- II (Marks- 30) Groups : Direct product (internal and external), Group action on a set, Conjugacy classes and conjugacy class equation, p- groups, Cauchys theorem, converse of Languages theorem for finite commutative groups, Syllow theorems and applications, Normal series, solvable series, solvable and Nilpotent groups, Jrrdan-Holder Theorem, Finitely generated Abelian groups, Free Abelian groups. 18 Rings : Unique factorization domain; Factorization of polynomials over a field; Maximal Prime and primary ideal; Noetherian and Artinian Rings; Hilbert basis theorem. References : 1. I. N. Herstein Topics In Algebrs (Wiley Eastern Ltd, New Delhi) 2. M. Artin Algebra (P.H.I). 3. P.M. Cohn Alegebra. Vol. I, II, & III. (john wiley & sons) 4. N. Jacobson Basic Algebra Vol. I, II. 5. D. S. Malik, J. N. Mordeson & M.K. Sen Fundamentals of Abstract Algebra ( Mc.Graw Hill, International Edition). 6. J. B. Eraleigh A first Course in Abstract Algebra (Narosa) 7. M. Gray A Radical Approach to Algebra ( Addision Wesley Publishing Company). 8. Hungerford Algebra. 9. S. Lang Algebra. 10. N. H McCoy The theory of Rings. 11. Burton Ring Theory. 12. Gallian Algebra. Group B General Topology - I (Marks - 20) Normal spaces, Urysohns lemma anf Tietzes extension theorem. Product spaces, embedding lemma, Tychonoff spaces and characterization of Tychonoff spaces as subspaces of cubes. Nets, filters subnets and convergence. Compactness, Compactness and continuity, countable compactness, sequential compactness, BW compactness and their relationship, Local compactness, Tychonoff theorem (on Product of Compact Spaces). References : 1. J. Dugundji Topology ( Allayn and Bacon, 1966) 2. J. L. Kelly General Topology (Van Nostrand, 1955) 3. J. R. Munkers Topology A first Course (Prentice-hall of India, 1978) 4. G. F Simmons Introduction to Topology and Modern Analysis (McGraw- Hill, 1963) 5. W. J. Thron Topological Structures (Holt Reinhurt and Winston, 1966) 19 Wesley.40) Graph. Mengers Theorem. Adjacency matrix. Cycle rank and co-cycle rank. References: 1. Statement of 4 colour theorem. radius and center. Harary Graph Theoru (Addison. domination number. Cut vertices and cut edges. Pathsarthy Basic Graph Theory ( TMH. General Cartesian product of sets. 5 colour theorem. Distance.I (Marks. Murty Graph Theory with Applications (Macmillan. Planner graphs. Sierpienski Cardinal and Ordinal Numbers 6. M. Centres of trees. R. Isomprphism. I. Degree sequence. Fundamental Cycles with respect to Spanning tree and Calyleys theorem on tree.I) Group -A Graph Theory (Marks . Bondy U. Complement. K. Binary relations directed paths.10) Axiom of choice Zorns Lemma. Nar Sing Deo Graph Theory ( Prentice. Schroder. Independence number. Dual of a planar Graph. Havel-Hakimi Theorem (Statement only) Trees. Mendelson Introduction to Methematical Logic 3. Well-ordering theorem and their equivalence. Copi Symbolic Logic 5. Chromatic polynominal. Isomorphism properties fo Graphs. R. Incidence matrix. Konig Theorem. A. K. 1969) 4. Blocks. S . Paper-MPG302 (Graph Theory & Set Theory. G Hamilton Logic for Mathematics (Cambridge University Press) 20 . Kuratowski Introduction to set theory and Topology 2. Vertex and edge conserving number. Subgraph. E. R.. Travelling Salesman Problem. cycles. Hausdorff maximality principle. F. Adjacency matrix of a Diagraph. Spanning trees. edge colouring number. Matching number. Stoll Set Theory and Logic 4. Clique Number. Paths. Vertex and Edge connectivities. fundamental Circuits in Digraphs. W .R.Hall. 1974) 3. Eulers formula. A. 1976) 2.Bernstein theorem. Hamiltonian Graphs. Brokes Theorem. Walks. J. Mycielski Construction. Chromatic number. Ramsayss Theorem. Bipartite graph. Directed Graph.B Set Theory. statement of Kuratowski Theorem. connected components. Cardinal numbers and their ordering. Eulerian and Semi Eulerian Graphs. Diameter. 1994) Group . References: 1. ordering of ordinal numbers. A.II) Group . M Copi Symbolic Logic 5. 1986 21 . R. Properties of normed linear spaces. the cardinal numbers N0 and C and their relation.Functional Analysis 2. References : 1.Steinaus). Convex sets in linear spaces. Siddiqui . statement form.. R. Functional Analysis. order type. Memdelson Introduction to Mathematical Logic 3. Universal and existential. Rieszs Lemma. Formal axiomatic theory. E. Compactness in C [0. Deduction theorem. K. Equivalent Norms and their properties. Quantifiers. Equicontinuous family of Functions. Principle of Uniform Boundedness (Banach. truth tables.II (Marks .1] (Arzela-Ascolis Theorem). addition and multiplication of ordinal numbers. Students Friends. Lusternik and Sovolev . Sierpienski Cardinal and Ordinal Numbers 6. Kuratowski Introduction to Set Theory and Topology 2. L.20) Completion of Metric space. Convergence in Banach Spaces. well-ordered sets. Mathematical Logic Statement calculus : Propositional connectives. A. G Hamiltom Logic for Mathematicians (Cambridge University Press) Group-B Functional Analysis . TMG Publishing Co. adequate sets of connectives.30) Set Theory -II Addition. initial segments. K Jha . Extension of continuous linear mapping. Arguments : Proving validity rule of conditional proof. contradiction.H. Transfinite induction. Totally ordered sets. W. sets of ordinal numbers. multiplication and exponentitation of cardinal numbers. Finite dimentional normed linear spaces.A Set Theory-II & Mathematical Logic (Marks. symbolizing everyday language. Closed graph theorem. Open Mapping theorem. Ltd. Paper MPG303 (SetS Theory-II & Mathematical Logic. Consequences. and its application in Branch Spaces. Fromal statements calculus. New Delhi. 3.Functional Analysis with applications. Tautologies. References : 1. K. I. truth functions.Functional Analysis. Stoll Set Theory and Logic 4. ordinal numbers. Functional Analysis 5. Kreyszing .MPS304 (Special Paper. 5. Associated fibre bundle. F Simmons . New York. Narosa Publishing House Pvt.Functional Analysis. Narici . 1984. Homomorphism and Isomorphism. Ltd.Functional Analysis. Ltd. E Taylor . Topological groups. Pothishala (Pvt. 2007 22 . R. Exterior algebra.. G. A. Ltd. De and A. 3. 1965 2. Structures on differentiable manifold and their applications. Tangent spaces. Chandrama Prakashan. New Delhi. Kalyani Publishers. Bachman & L. S. Product of two Liegroups. R. Differential Geometry of Manifolds. B. Sinha. Bundle hommorphisms.I) (Marks-50) A: Differential Geometry of Manifolds. P. New York. References: 1.I Definition and examples of differential mainfolds.. Mishra. K. Yano and M. Vector bundle. John wiley and sons.) Ltd. Lie groups and Lie Algebra. 1963 11. Limaye .1994 10. Immersions and imbedding. 1984. Example of Liegroups. Tangent bundle. A. U. Vulikh .. World Ccientific Publishing Co. 1966 6. Linear frame bundle. V. Kantorvich abd G. Jacobian map.Functional Analysis. Academic press. S. One parameter subgroups and exponential maps. Mishra. Shaikh. Kon. Pergamon Press.Functional Analysis. 4. V. Exterior derivative. One parameter group of transformations. G. Pvt. K Lahiri. L. induced bundle. Allahabad. 1958 7. Kolkata. 1982 9. General linear groups. An Introduction to Modern Differential Geometry. 1982 4. B. The world Press Pvt. B.Introductory Functional Analysis with Applications. Akilov . Wiley Eastern. 1989 8. E. B. Lie derivatives. Wiley Easten Ltd Paper .. Principal fibre bundle. C.Introduction to Topology and Modern Analysis. Distributions/. Structure of Manifolds. Lie tramsformation groups. Acourse in tensions with applications to Rienamnian Geometry.Elements of Functional Analysis. Mc Graw Hill. Convex Hull and representation Theorem. Separation properties. Barreled spaces and Bornological spaces. Academic Press. E. Fundamental theorem of integral colculus for Lebesfue intergal. Minkowski functionals. H. W. Munroe Measure and Integration 6.b] References: 1. P. E.. Pointwise differentitation of functions of linear intervals. Example of a continuous nowwhere differentiable function. Univorm Convexity of a Hilbert Space. Absolutely continuous functions and singular functions. Springerm 2nd Edn. E. Dini derivates and their properties. Criterion for Locally convex topological vector spaces to be (i) Barreled and (ii) Bornological. I & II C: Advanced Functional Analysis -I Topological vector spaces. Natanson Theory of Functions of a Real Variable. Balanced sets. A. I. Hewitt and Stormberg Real and Abstract Analysis 2. Kreyszing Introductory Functional Analysis with Applications. 1989 23 . Vols. Bounded sets in topological vector space. B: Advanced Real Analysis -I Upper anf lower limits of real function and their properties. Ltd. Vols. Royden Real Analysis 3. Locally compact topological vector space and its dimension. Hyperplanes. Strict convexity and Uniform convexity of a Branch space. Saks Theory of Integrals 4. I & II 7. 1966 4. W. Characterization of absolutely continuous functions (Bansch-Zaricki theorem). Rudin Real and Abstract Analysis 5. Rudin Functional Analysis. Criterion for normability. Local base and its properties. Bachman & L. Extreme points. TMG Publishing Co.. M. G. Lunsins condition (N). Cantor ternery set and cantor function. New Delhi. absorbing sets. Wiley Eastern. 1973 2. Boundedness and continuity of Linear operators. Hobson Theory of Functions of a Real Variable. Derivates and derivtives. Generting family of seminorms in locally convex topological vector spaces. Krein-Milman Theorem on extreme points. Vital covering theorem in one diemension. Monotonicity theorem. Symmetric sets. Seminorms. L. Measurability od derivates. Weierestrass approximation theorem in C [a. Differentiability of monotone functions. W. Locally convex topological vector spaces. Reflexivity of uniformly convex Branch space. References: 1. Indefinite Lebesgue Intergral. A. Linear operators over topological vector space. Differentation of real functions. Separation of convex sets by Hyperplanes. Schaffer Topolical Vector Spaces. 1991 3. Narici Functional Analysis. Holt Rinchart and Wilson. New York and Basel. Yosida Functional Analysis. Ideals. P points and their properties. 1985 7. Spring Verlog. A Wilamsky Functional Analysis. Convergence of Z -filters.. C. Ltd. characterization of P spaces. the residue class rings modulo fixed maximaln ideals in C (X) and C* (X) and the field of reals. New Delhi. Graw Hill.First Course in Functional Analysis. PHI. 1963 8. Narici & Beckerstein Topological Vector spaces. Pedrick . P spaces. F Simmons Introduciton to topology and Modern Analysis. completely separateds sets. Horvorth Topological Vector spaces and Distributions. Mc. their Lattice structure. G. New Delhi 1987 15. Zero. New York. New York. Diestel Application of Geometry of Banach Spaces 6. prime ideals. Urysohns extension theorem and C-Embedding. cluster points. its subrings.. the subring C* (X) . Brown and Page Elements of Functional Analysis. E Edwards Functional Analysis. Taylor Functional Analysis. 1966 14. Srone-Eachs therem concerning adequacy of tychonoff spaces X for investigation of C(X) and C* (X). locally compact spaces and their properties. compact subsets and C embedding. Z. Marcel Dekker Inc. fixed maximal ideals of C (X) and C*(X). NewYork 1958 9. Addison . C* - embedding. their unions and intersection. TMG Publishing Co. Lipscitz General Tropology. Completely regular spaces and the zero-sets. Goffman and G. prime filters and their relation. Compactness and fuxed ideals. their characterizations. Pseudocompactness and internal characterization of Pseudocompact spaces. 1970 12. Holmes Geometric Functional Analysis and its Application 13. Prime Z filters convergence and fixed Z-filters. 5. Von Nostrand Reinhold Publishing Co. Relation between fixed maximal indeal in C(X) and C*(X). 24 . R.maximal ideals. 1965 16. New York. John Wiley and Sons. Schaum Series 10.1990 11.Wesley Publishing Co..filters. Fixed ideals and compactness. properties of P spaces. 1973 D : Rings of Continuous Function I The Ring C(X) of the real valued continuous function on a topological space X. 3rd Edn. A. E. K. weak topologies determined by C(X) and C* (X). J.Sets cozero-sets. ring homomorphisms and lattice homomorphism. V. Precupanu Convexity and Optimization in Banach Spaces ( Editura Acad. Artinian & Noethrian rings. J Divisky Rings and Radicals 9. Ideal. Balakrichnan Applied Functional Analysiss ISpringed. M. N. Non Linear Optimization in Banch Spaces . Direct sum decompostitions. V.Wesley Publishing Company) 2. Minimal and maximal conditions. Ideals in semisimple rings. 1976). Applications to approximations and optimal control problems. Gillmen and M.References : 1. N. Set of minimal points. 3. Hahn- Banch theorem and partially ordered linear spaces. Gateaux and Frechet derivatives. N. Polynomical rings. Herstein Noncommutative Rings 8. H. New York. Bucaresti. projective modules. Richard E. Lam Noncommucative Rings (Springer -verlag0 6. reflexivity of Banach spaces. The Jacoson Radical : Primitive rings. New York. Jerison. Hausdorff Compactifications (Marcel Dekker. L. L. A. N. Bartu and T. Quotient rings and homomorphisms.Verlag) 25 . E. Modules : Preliminaries. Tensor products. 1966). The radical of related rings. H Rowen Ring Theory (Academic Press) 5. The Density Theorem. Modules and applications F. Matrix rings. T. 1986). Berlin) 4. R. Existence Theorems for Minimal Points Problem formulation. 1994) 2.I Rings and Ideals :Definations. Johannes John Introduction to thes Theory of Nonlinear Optimization (Springer- Verlag. Rings. Subdifferential Quasififferential Clarke derivative. The field of quotients. Chandler. The Classical Radical : Nilpotent ideals and the radical. Structure theorems. Primary decomposition. 3. The Wedderburn theorem. Adhikari Groups. Ernst August Behrems ( Translared by Clive Reis) Ring heory (Academic Press. field and matrix representations. P Sankappanvar A course in Universal Algebra (Springer- verlag. Y. Direct sums and free modules. Inc. Berlin) 3. Mary Gray A Radical Approach to Algebra (Addion. References: 1. Rings of Continuous Functions (Von Nostrand. Mc Coy The Theory of Rings 10. Theory of Ringsd And Algebras . Algebras. Topological Structures (Halt Reinhurt and Winston. Existence theorems. Stanley Burris & H. 1960). Jacolson BAsic Alegebra II 7. Generalised Dervatives- Directional derivative. I.I Review of Weak Convergence in normed spaces. References : 1. 2. Farkas theorem. Maximal ideal space of L1 (R). Monotone class of sets. 2. E. T.TZ. Fejers theorem. Analyticity Theorems. References : 1. quotient group and connected groups. subgroups. R.50) A: Measure and Integration .Countably additive ser function. Fó and Gä Sets. Dunford and Schwartz Linear operators.I Basic properties of topology groups. G. Springer-Verlag. I (Springer- Verlag. Vol. G: Harmonic Analysis . Kernels of R. Measure on ó algebra. Spectral theory of operators. Examples of conjugate function series. A. Henry Helson Hermonic Analysis (Hindustan Pub. Outer measire and mearurability.V. and some simple matrix groups. Approximate identities. L1 (T) and L1 (Z). Special Theory of compact operations. Hewitt and K. L2 spaces over Hillbert spaces. K. R. Operations on a separable Hilbert space. L1 (T). 3. Koosis Introduction of HP Spaces ( Cambridge Univ. Borel sets. Extension of measure spaces. L1 (G) and convolution with special emphasis on L1. Non-linear Volterra operators. Discussion of Haar Measure without proof on on R. 1993) 3. Press) 5. Construciton of outer measures. 1994) 2. The Plancherel theorem on R Planchere measure on R. Y.MPS305 (Special Paper. Balakrishnan Applied Functional Analysis. Ross Abstract Harmonic Analysis Vol. L1(Z). Multilinear forms. The Fouries transform. 4. P. Fatous Theorem. Goldberg Fourier transforms 6. (R).. References : 1. S. Paper . Z. 1968 4. Convex programming supports functional of a convex set. Kerin-Linear Differential Equations in a Branch space. Kuhn-Tucker Theorem. Katznelson An Introduction to Harmonic Analysis. 1 & 11. Fejers Poissons and Dirichlets summability in norm and point wise summability. Separation Theorem.II) Marks . Regular 26 . The inequalities of Hausdorff and Young. A. The Classical kernals. Corp.I Algebra and o-algebra of sets. Huissain Introduction topological groups H : Applied Functional Analysis -I Review of Basic properties of Hillbert spaces. Fourier series. T. Yosida-Functional Analysis. Minimax theorem. Minkowski functional. Krein factorization theorem for contunuous kernels and its consequences. (John Wiley. Monotone sequence of positive operators. their algebraic properties. New York. 1990 27 . Vitali convergence theorem. Approximation of measurable functions by simple functions.outer measure. New York. Yosida . Their sum and product. D Barra Measure and Intregration 3. Academic Press. V. Bachman & L. P. G. Narici . Fredholm alternative. Edm. I. G. (Lebesgue). Compact extensions. G. Kolkata 1994 5. Rana Measure and Integration 2. Rudin Real and Abstract Analysis 7.. B. Mc. Measurable functions. Limaye . normal operators in Hilbert spaces. Integration and Function spaces. Hewitt and Stotmberg Real and abstract analysis 4. Monotone convergence theorem. Graw Hill . I. Lahiri . World Scientific Publi. square-root of positive operator. Reference : 1. Operator Theory And Application -I Adjoint operators over normed linear spaces. Springer Verlog. Singapore. operator equation involving compact operators. K. H. Properties of Integrals and Integrable functions. Limit of monotone incresing sequence of Projection operators.Functional Analysis Wiley Easten Ltd 3. Projection operators. 1966 2. Example of non- measurable sets.Measure. Munroe Measure and Intregration 8. E. Weakly compact-operators. Inregrals of simple functions. Adjont operators on Hilbert-spaces. their algebric properties. References : 1.Functional Analysis. prouduct. Compact operators on normal linear spaces.Fundamental Analysis. F. E. Dominated convergence Theorem. Convergence in measure. Ltd. Egoroffs Theorem.Introductory Functional Analysis with applications. positivity norms of Projection operators. positive operators. M. Lusins theorem. B. I & II 10. Taylor Measure and Intregration 9.Introduction to topology and Modern Analysis. K . Idempotent operators.. Kreyszig . Self-adjoint operators. Wiley Eastern. 1963 7. Unitary operators. Fatous Lemma. L Royden Real Analysis 5. 1989 4. W. thier-sum. Charles Schwartz .. 1994 B. Natanson Theory of Functions of a Real Variable Vols. Intergal of measurable functions. Lebesgue Stieltjes measures and distribution function. Saks Theory of Integrals 6. P. 3 rd. Simmons.Elements of Functional Analysis The World Press Pvt. Sequence of Compact operators. Definition of an algebra. Vulikh . Szasz Lattice Theory 2. transposed intervals. R. C. lifting of paths to a covering space.I Homotopy : Definition and some examples of homotopies. graphs. fundamental group of a product space.Elements of Functional Analysis. atoms. John wiley and sons. Birkhoff Lattice Theory 3. deck transformation group. Taylor . modularity and distributive criterion. complements. L. S. E. 1958 10. Pergammon Press. Spanier Algebric Topology 4. Brower fixed point rheorem in dimension 2. 1996 13. Maximal minimal condition. order isomorphism. Closure operation. New york. Lattices as partially ordered sets sublatttices ideals. homomorphism and automorphisms of covering spaces. Dedekind cuts.Functional Analysis 12. 1970 9. Schaum Series C : Algebraic Topology . D. the effect of a continuous mapping on the fundamental group. References: 1. E.Functional Analysis. Gray Homotopy Theory An Introduction to Algebric Topology 5. irreducible and prime elements. 8.Functional Analysis CRC Pres Inc. S. Jordan-Dedekind chain condition. Rutherford Lattive Theory 28 . Bredon Geometry and Topology D : Lattice Theory . meet representation in modular and distributive lattices/ References : 1. Morphisims homomorphisims. factor and natural transformation. H. dimension function. Universal cover. semicomplements. Borsuk Ulam theorem for S2. J. calculation of fundamental groups using covering space. Brown and Page . Von Nostrand Reinhold Co. its existence. B. interval topology. Ideals direct products. Massey Singular Homology Theory 3. fundamental gropus of a circle. G.P. akilov. W. 1982 11.General Topology.I Introduction : Partially ordered sets.space. distributive sublattices of modular lattices. homotopy type and homotopy equivalent spaces. Lipschitz . Lattices as algebras. G. H. retraction and deformation. Massey Algebric Tropology 2. Category : Definitions and some examples of category. Dedekind condition. E. A. Fundamental Groups and covering spaces : Definitaion of the fundamental group of a space. Projection space and tours..Functional Analysis. Completion. Distributive and moduylar lattices. B. Dernkowicz. Arnold Logic and Boolean Algebra 4. V Kantorvich and G. H. density principle. Tinsley Oden & LEszek F. notion of covering spaces. W. Separable convex programming . Wiley Eastern Ltd. minimum cost flows Minflow max-cut theorem. S. McGraw Hill. New Delhi 9.A. Generalized Max Flow min-cut theorem. Moudgalaya. R Bector. Separable Programming Algorithm. Karlins constraunt qualification. Dano : Non-Linear and Dynamic Programming. 5. Optinality criterion with differentiability and convexity separation theorems. Bhatti. Taha Operations Reseach An Introduction. Practical optimization Methods. L. O. X. NEw Delhi 4.I Non-linear Optimization : Local and global minima and maxima. Springer verlag.Linear Programming. Chandra and J. S Rao Optimization theory and Applications. M. Kuhn-Tuckers sufficient optimality theorem. M. Davidon. Hadly Non-Linear and Dynamic Programming. New York 8.. N. Quadratic Programming : Wolfes Modified Simplex method. Beales method.. Optimization theory and Practice. Slater constraint qualification. Narosa Publishing House. Dutta. Ltd. A. Addision -Wesley. New York. Narosa Publishing House. 12. Principles of optimization Methods. New Delhi 11. Gradient Methods : Steeopest descent Quasi-Newtons method. 7. G. Reading Mass. Sultan Chand & Sons. 2. New Delhi 10. Golden Section search. H.. Inc. Conjugate direction method (Fletecher-Reeves method). convex functions and their properties. Unconstrated Optimization : Search method : Fibonacci search. Affiliated East-West Press Pvt. Kuhn-Tuckers Saddle point optimality condidions. C.. S. 29 . Mangasarian Non. M. References : 1. Optimality in absence of diierentiability. Macmillan Publishig Co. 3. Hadly Linear Programming. C Joshi and K. Method of Lagrange multiplier. Luenberger Introduction to Linear and Non-Linear Programming 6. S. G. Swarup Gupta & Manmohan Operations Research. E : Advanced Operations Research . Springer-Verlag. Network Flow : Max-flow min-cut theorem. Kambo MAthematical Programming Rechniques. Linear Programming interpretaion of max-flow min-cut theorem.Fletecher-Powell method. symmetric Eech closure operator. trace system. exponent of vonvergence of zeros of an integral function. Some Application External sets. Cambridge. Convexcity. Hadamards factorization theorem.I Analytic function. and orderings. S. E. clusters and relation between them. Nearness and Extensions of Topology Spaces -I Eech closure operator closure spaces. The set of principal T0 extensions of a T0 topological spaces is a partially ordered set and its consequence for the class of T2 com pactification of a Tychonoff space. Extensions of closure spaces. D. Wanack Proximity Spaces (Cambridge Track No. References : 1. proximity spaces. W. L. genus. Convex function and Handamard three-circle theorem. their characterization and relation between them. Harmonic function Mean value property. J. Weak Topology. Efremovie (EF-) proximities. construction of an intergal functions with given zeros Weierstrass theorem. Borels theorems. C : Geometric Functional Analysis -I Convexity in Linear spaces. New York. Clams. principle (strict) Extensions. proximal neighbourhoods. 1966) H : Advanced Complex Analysiss . References: 1. the function M (r) and A (r). order. canonical product. Classification of basic proximities : Riesez (RI-) proximities . References : 1. continuity. Picards first second theorms. Naimpully and B. Eech Topological Structures ( English Transl. W. Pharagmen-Lindelof theorem. Representation of principal T0 extension of a T0 topological space with a given trace system. Convex functions Basic separation Theorems. induced closure operators. (Basic) proximates. Rudin Real and Complex analysis 30 . 1966) 3. Locally convexity and Topology. Harmonic functions on a disk. Maximum principle. Lattice structure of the basic proximities compatible with a symmetric closure space. Holmes Geometric Functional Analysis and Its Applications G : Proximates. Jensens inequality. homeomorphisms and their properties. p-continuous functions and ttheir properties. Dirichlets problem. Alternate formulation of the the separationPrinciple. Basic proximities are clan generated structures. J Thron Topological Structure (Halt Reinhurt and Winster. Poisson jenson formula. Wiley. Ahlfors Complex Analysis 3.1970) 2. ordering of extensions. Lodato (LO-) proximities. Theorem of Borel and Caratheodary. Hamaeks inequality. V. 59. Integral function. B. A. Linkage compact topological spaces and their relation with compact topological spaces in presence of regularity condition. Conway Functions of One Complex Variable 2. C. Reisz Fischer theorem. Generalised function as the limit of a sequence of good functions. Reduction of differential equation to integral equation. E. Uniqueness and iterative solution of Fredholm and Volterra Integral equations. Hilbert spaces. closedness. T. continuity. Fourier expansion. Convolution Theorem.MAG301 (Methods of Applied Mathematics . Applications. I MArkudevich Theory of Functions of a Complex Variables. M. P. 4. 31 . Antiderivative. Titchmarsh Theory of Functions 5. Simple examples. Isometric isomorphism between a separable Hilbert space and 12. Linear operators on Hilbert space. Examples. T. Differentation of generalized function. Solution of Abels integram equation. Generalised Functions Generalised function. examples.properties. orthonormality.III APPLIED MATHEMATICS STREAM Paper . Laplace transform of generalized function.I (Marks . I & II 9. Parsevals relation. E. Examples. Convolution Theorem. Operator Equations on Hilbert Spaces Inner products spaces.. Elementary properties. Examples. Fourier Transform of generalised function. Solution of Fredholm integral equation for degenerate kernel. Invesion by analytic method and by Bromwitch path. Vol. H. Cartan Analytic Functions 8. Convergence of a sequence of generalised functions .I) Method of Applied Mathematics. SEMESTER . Inversion formula of F. Dutta and Lokenath Debnath Elliptic Functions. Integral Equations Linear Integral Equation. A. (proof not reqd). Existence. Applications. Addition. laplaces Transform and its properties. Multiplication. R. Transformation of variables. Regularisation of divergent intergal : Simple Example. and completeness of sets. Faulting type (closed cycle type) integral equations. Outline of Finite Fourier transform and its inversion formula. Copson Function of a Complex Variable 6. Lerchs Theorem.50) Integral Transforms Fourier Transform and its . singular integral equation. Boas Entire Functions 7. Two-point boundary value problem for a second-order linear O. Greens functions and its bilinear expansion.boundedness. S. solution of Cauchy problem using Dirac Delta function and Fourier transform. Possible discontunuites of solutions . Fredholm alternatives. Eigen value problems. particular integral. F. Linear Integral Equations (Dover Publishers) 9.E. linear transformations. Application to Regular Sturm-Liouvile problem. G. D. necessity of classification and canonical forms Invariance of nature of an equation and its charecterstics under coordinate transformation. transformation of semilinear 2 nd order PDE in two indepedent variables. Arnold Ordinary Differential Equations Paper . Cauchy- Kowalasky theorem (statement only) reason for restriction on cauchy ground curve. wellposedness. Sneddon Fourier Transforms (MacGraw Hill) 3. and linear PDEs with more than two indepedent variables. Analogy between linear simultaneous algebraic equations and Linear differential equation. F. Theory of Electro Magnetic Fields) Group . Solvability of operator equations. Linear huperbolic PDEs in two independent variables Cauchy problem. References : 1.MAG302 (Methods of Applied Mathematics . Compactness. WE. Williams Partial Differential Equations 11. Integral equations with Hilbert-Schmidt kernel. John Partial Differential Equations 10. N. Fundamental solution. Tricomi Integral Equations (Interscience Publishers) 8. Linear parabolic equations : Heat equation in two independent variables. G.A Methods of Applied Mathematics . Epstein Partial Differential Equations 12. Rieman-Green function. Mikhlin Integral Equations (Pergamon Press) 7. self-adjointness. Gelfand & Shilov Generalised Functions (Acdemic Press) 2. adjointness. illustrative examples. boundedness and unboundedness of inverse. Lovit. V. d Alemberts solution and meaning of generalized solution. inverse of a differential operator. Domain of dependence and influence. Mathematical models and initial boundary value problems of 2nd order partial differential equation (PDE). generalized solution.II (Marks -30) Linear ordinary differential equations. Special theorem for compact self-adjoint operators. Churchill Operational Methods 4. invertibility. I. maximum principle forinitial 32 .II. Erwin Lareyizy Introductory Functional Analysis with Applications 6. Chester Partial Differential Equations 13. implications on Laplace operatior. R. Lusternik & Sobolev Functional Analysis 5. V. formulation of eigenvalue problems related to wave. uniqueness and stability of solution. Method of images. John Wiley & Sons. Rayleigh-Ritz method. J. H. D. Greens function for Dirichlet problem in Laplace eqn.20) Empirical basis of Maxwells Equations : Coulombs law. Method of conformal mapping for 2. Electromagnetic energy. Electromagnetic potential. F. 1965) 33 . uniqueness and stability of solutions. J. Field of a point charge in uniform motion. Milford Foundations of Electromagnetic Theory (Addison - Wesley Pub.B Theory of Electro Magnetic Fields (Marks . Group velocity and phase velocity. Co. heat and Laplace equations. C Duff and D Naylor Differential quations of Applied Mathmativs (Wiley Intermational) 2. Material equations. Equation of continuity of charge. its properties and methods of construction.. Greens formulas involving Laplacian. W. Phillips Classical Electricity and Magnetism (Addison .) 4. Methods of Eigen function expansion and Greens function. D.. 2. Faradays law. mean value theorem. Inc 1966) 3. Neumann and Robin. References: 1. Ins. Maxwells equations for electromagnetic field and their empirical basis. 1962). Tikhnov & Samarski Equations of Mathematical Physics Group . References : 1.Wesley Pub. Separation of variables. boundary value (for Dirichlet boundary condition) problem. Griffct Applied Functional Analysis (Ellis Horwood Ltd.Inc.) 3. S Vladiminov Equations of MAthematical Physics ( Marcel Danker. Stedy current. H Panofsky & M. G . Retarded and accelerated potentials. V. Griffth Classical Electrodynamics (Wiley Eastern. Amperes law. Stakgold Greems Functions and Boundary Value Problems (John Wiley & Sons. J. Co.Y. Gauss law.K. Linear elliptic equation : Laplace equation: boundary value problem of Dirichlet. Magnetic induction.dimentional problem with problem of a unit circle as an example Bilinear expansion for Greens funcion for heat equation by the method of Eigen function expansion and Bilinear expansion for Dirac Delta function. Reflection and Refraction of plane waves at the plane boundary between two dielectrics. Poyniting theorem. N. dielectric and conducting media. Reitz & F. Dirchlet principle. Application of Maxwells Equations : Plane electromagnetic Waves in vacuo.) 5. Electrostatic potential. Biot-Savarts law. Conditions at an interface. maximum principle. R. C . Leigh. D.Hill) 10. -Encyclopaedha of Physics Vol III/ 3.Theroretical Hydrodynamics> 7.MAG303 (Continuum Mechanics . Material and spatial coordinates. stream lines and streak lines. Dynamical Systems) Group -A Continuum Mechanics-II (Marks -30) General : Body. Reynolds transport theorem for volume property.A Text Book of Fluid Mechanics 11.C.C. Truesdell. Flugge) 5.Viscous Flow Theory 8. I. principle stretches. Eriengen. Cauchys laws of motion : Stress equation of motion and symmetry of stress tensor for non-polar material.L .Thomson. wave propagation in an infinite elastic medium. L. Reference configuration. S. Chung. axiom of continuity. Material and spatial time derivatives relation between them. configuration. 1965 (Ed. Pai. polar decomposition theorem (Statement). Fundamental boundary value problems of elasticity and uniqueness of their solution (statement only). -Non-Linear Contiuum Mechantcs (MacGraw Hill) 2. relative stretch tensors and relative rotation tensor.J.II. A. . Eulers laws of motion. Classical theory of infinitesimal deformations. Sokolnikoff. Compatibility equations. Saint-Venants principle solution of simple problems. Waves of distotiom and dilatation.M. Elastic modulii. Rate-of-strain tensor-its principal values and invariants rate-of rotation tensor.. Motion of a body. F. Schilichting H . (Bounding surface). References : 1. Kolin Kebel & Roze. T. Body forces and contact forces. Deformation gradient tensor.. velocity gradient tensor. Lagranges criterion for material surface. General principles of momenta balance. Trusdell. Relative deformation gradient tensor. C and Nol. W.vorticity vector.Velocity and acceleration.Non-linear Theory of Continuous Media (Mac Graw. S.Fluid Mechanics 34 . deformarion. Constitutive equation (stress-strain relations) for isotropic elastic solid.Coniumum Mechanics (Prentice Hall) 4. polar decomposition of deformation theorem(statment) polar decemposition gradient tensor stretch and rotation tensors.Continutum Mechunics 3.Boundary Layer Theory 9. Equations of equilibrium and motion in terms of displacement. S. Principle of conservation pf mass Path lines. Beltrami Michel compatibility equations for stresses. . Material surface. chorlton . Strain-energy function.Theroretical Hydrodynamics 12. Physical components of acceleration in general curvilinear coordinates in cylindrical coordinates and spherical polar coordinatess. Besand & Ramsey. Miline. Paper . Energy balance first and second laws of thermodynamics.Mathematical Therory of Elasticity 6. Oseen approximation. pitchfork and Hopf bifurcations. contracting and expanding behaviour. the flow near a rotating disk. L. hydrodynamical theory of lubrication. Stokess flow past a sphere and a cylinder : Stokes paradox. Perko Differential Equations and Dynamical System (Springer 1991) 5. Reynolds number. hyperbolicity. Boundary Layer Theory and Magneto Hydrodynamics-I Viscous Flows Some exact solutions of Navier . References : 1. P. sink. Bansal -Viscous Flow Theary Group -B Dynamical Systems (Marks : 20) Dynamical Systems : Phase variables and phase space.MAS304 (Special Paper-I) Marks: 50 A: Viscous Flows. Phase plane analysis of simple system. stable. M. Strogartz Non-linear Dynamics 3. Dynamical System and Linear Algebra IAcademic Press 1974) 4.Stokes Equations: the flow due to suddenly accelerated plane wall. 13. Liapunov and asymptotic stability. Gledinning Stability. Souce. basin of attraction. statement of Hartmann-Gorbnam theorem and stable manifold theorem and their implications. Paper . concepts of stability and SDIC (sensitive dependence of initial conditions) chaotic behaviour . unstable and center subspaces. Statement of Lienards theorem and its application to vander Pol equation. dynamical system as a group. Bodewadt flow. Creeping motion. Liapunov function and liapunov theorem. maps (discrete dynamical systems). Linear systems : Fundamental theorem and its applications. Poineare Bendixxsom theorem (statement and applications only) structural stability and bifurcation through examples of saddle-node. fixed (equilibrium) points periodic points. Properties of exponential of a matrix. Hirsch & Smale Differential Equations. homoclinic and heteroclinic orbits. Instability and Chaos (Cambridge University Press 1994) 2. J. Oseens solution for a sphere. orbits asymptotic states. hyperbolicity. Arnold Ordinary Differential Equations. plane stagnation point flow (Hiemenz flow). Navier-Stokes equations in non-dimensional form. Laminar Boundary Layer Theory 35 . continuous and discrete time systems flows (vector fields). nilpotent matrix. Hele-shaw flow. the flow near an oscillating plane wall. generalized eigenvectors of a matrix. Periodic solutions limit cycles and their stability concepts. Nonlinear Vector Fields : Stability characteristics of an equilibrium point. W. L.Theory of Elasticity (MIR Pub. A. Formulation of torsion problem as an internal Neumann problem. V. W. L. flow past a circular cylinder. 4. References : 1. 1963. flow along the wall of a convergent channel. J.L. Generalised Hookes law Orthotropic and transversely isotropic media. The spread of a jet : (i) plane free jet (two-dimensional jet). Theorem of min. Solution of flexure problem for simple sections. Blasius-Topfer solution. 1969 2. 3. equations : B. Saint-Venants semi-inverse method of solution (Statement).Pohlhausen method : simple applications. Press. Pergamon Press. Basic equations. Solution in terms of complex analytic function. A. Sherelift : A text Book of Magnetrohydrodynamics. Bansal : Viscous Fluid Dynamics. Landau and E. H. 1961. Lifshitz: Fluid Mechanics. B. A. Prandtls stress function shearing stress in torsion problem. PHI. Two dimensional B. Mc Graw-Hill Book Comp.) 2. E. Karman . 36 . 2004. Potential energy of deformation. L. of torsion problem by conformal mapping. A.D. Dirichlets problem and Poissons problem. Plane problem : plane strain. References : 1. Equations for flow over a plane wall : Boundary layer on a flat plate . flow past a wedge. H. Potential energy. Solution of torsion problem for simple sections Method of sol. 1959. Y. L.. J. B: Elasticity-I 1. Schlichting: Boundary Layer Theory. L. Formulation of torsion problem and the equations satisfied by the torsion function and the boundary condition. Prandtl-Mises transformation : Karman momentum integral equation. 3. plane stress.A treatise on the Mathematical Theory of Elasticity. 6. Similar solutions of the B. CUP. Flexure problem : Reduction of flexure problem to Neumann problem. generalised plane stress. Amenzade . Oxford Univ. 5. Stress- strain relations in isotropic elastic solid. Airys stress function. Separation of boundary layer. Ferraro and C. M. C. Reciprocal theorem of Betti and Rayleigh. Love . 4. L. (ii) circular jet (axisymmetric jet). Plumpton : An Introduction to magneto fluid Mechanics.Concept of boundary layer : Prandtls assumptions. S. Yuan: Foundations of Fluid Mechanics. B. 2. 2003. (10L) 5. 1965. Kuhn-Tucker Theorem. Goodies. S. Amenzade . Operators on a 37 . Separation Theorem.Some Basic Problems of the mathematical theory of Elasticity. 6. I. 3. Timoshenk and N. 1965. Convex programming-support functional of a convex set. C: Elasticity and Theoretical Seismology-I Elasticity Generalished Hookes law.) 2. A.Foundations of Solid Mechanics. Torsion problem. I. Timoshenk and N. Muskhelishvili . Mc Grwa Hill Co. Stress-strain relation in Thermo-elasticity. A. S.. 3. References : 1. P. Mc Grwa Hill Co. CUP. Nowacki . Goodies. W. Transversely isotropic media. 1970. Thermo-elasticity : Thermal stress. 4. Differential Equation and the boundary condition. Theorem of minimum potential energy.Some Basic Problems of the mathematical theory of Elasticity. 4. Tata Mc Graw Hill Co. Max shearing stress. Equation satisfied by torsion function and the boundary condition.Theory of Elasticity (MIR Pub. S. Minkowski functional. Theory of Elasticity. 1963. Stress-strain relations in isotropic media. 1970. W. I.. 1963. P. Y. Basic equations.Mathematical Theory of Elasticity. Spectral theory of operators.. Tata Mc Graw Hill Co. Potential energy of deformation. 1977.Mathematical Theory of Elasticity. Sokolnikoff . 1977. Sokolnikoff . D: Applied Functional Analysis-I Review of basic properties of Hilbert spaces. Y. Noordhoff Ltd.Thermoelasticity (Addison Wesley) 5. S. 6. Prandtls stress function. Muskhelishvili . Farkas theorem. PHI. 1963. Airys stress function. Flexure problem..Thermoelasticity (Addison Wesley) 5.. Solution of torsion problem for simple sections. Nowacki . Love . plane stress. 7.Foundations of Solid Mechanics. 7. C. I. Minimax theorem. Fung . C. Spectral Theory of compact operations. N. Noordhoff Ltd.A treatise on the Mathematical Theory of Elasticity.. Sajnt-Venants semi-inverse method of solution. Y. Solution for simple sections. Plane strain. PHI. H. Fung . E. N. Reciprocal theorem of Betti and Rayleigh. Theory of Elasticity. MAS305 (Special Paper-II) (Marks: 50) A: Quantum Mechanics-I 1. Pauli exclusion principle. Diracs Bra and Ket notation. Non-linear Volterra operators. Transformation of matrix elements. Krein factorization theorem for continuous kernels and its consequences. N. 4. A. Elements of Second Quantization of A System : Creation and Annihilation operator. Change of basis. Messiah . Ground state energy of helium atom.Applied Functional Analysis. Joachain . Unitary transformation. J.. I & II (North ..Atomic Collision Theory (W..separable Hilbert space. Dirac equation for a free particle and its Lorentz convariance. Addition of two angular momenta. Hermitian operator. Stark effect in hydrogen atom.Quantum Collision Theory (North-Holland Pub. 1989). 5.Potential Scattering in Atomic Physics (Plenum Press. Common set of eigen functions of two operators. L2 spaces over Hilbert spaces. 1962). Relativistic Kinematics : Kelin-Gordon equation. 2. Rotation groups. 1977) 4. G.Introduction to Quantum Mechanics (Oxford University Press. A. Vol. Transformation Theory : Adjoint operator. S.Holland Pub. Co. Analyticity Theorems. Degeneracy. 2. Identical particles. Compatibility of observables. 1975) 5. Zeeman effect. Springer-Verlag. P. A. Invariance and conservation theorems. B. Balakrishnan. Y. Joachain . 3. Benjamin Inc. Yosida-Functional Analysis. Projection operator. Relation with Statistics . Bransden & C. C. V. 4. 1 & 11. Completeness and normalization of eigen functions. 3.Quantum Mechanics. Symmetries and Invariance : Angular momentum eigenvalues and eigenfunctions. K. H. References : 1. G. Spin. References : 1. New York. Co. 1970) 38 . Commutation and Anti-commutation rules. Matrix representation of wave functions and operators. Hole theory and positron. Paper . J. Burke .. 2.Bosons and Fermions. Dunford and Schwartz-Linear operators. Krein-Linear Differential Equations in a Branch space. 3. Multikinear forms. vol. Bransden . Electron spin and magnetic moment. Approximation Methods (time-independent) Rayleigh-Schrodinger perturbation method. An harmonic oscillator. B. H. Ltd. Mott & H. Dano . Geltman .Non-Linear and Dynamic Programming.M. H. 1966) B: Advanced Operations Research-I Non-linear Optimization : Local and global minima and maxima. Golden Section search. Conjugate direction method (Fletecher-Reeves method). New Delhi 10. New York. Joshi and K. F. Watson . Newton .S. Kambo. Optimality criterion with differentiabilityand convexity. Kuhn-Tuckers Saddle point optimality conditions. G.6. G. Minimim cost flows. Narosa Publishing House. 1969) 7. Optimization theory and Practice. Y. 3. 6. (Clarendon Press.Operations Research. Generalized Max flow min-cut theorem.. S.Linear Programming. Mc Graw Hill.Theory of Atomic collisions (3rd ed. Separable convex programming. Goldberger & K. Inc. Unconstrained Optimization : Search method : Fibonacci search. T.. Hadly . Olmura . Network Flow : Max-flow min-cut theorem. Slater constraint qualification. N.Quantum Theory of Scattering (Prentice Hall.Non-Linear and Dynamic Programming.Mathematical Programming Techniques. Taha .Hill. S. W. Luenberger . 7. 1964) 10. Affiliated East-West Press Pvt. O. Beales method.Introduction to Linear and Non-Linear Programming. Linear Programming interpretation of Max-flow min-cut theorem. 4. Rao . A. Quadratic Programming : Wolfes Modified Simplex method. Method of Lagrange multiplier. M.Collision Theory (Wiley.). Macmillan Publishing Co. Addision-Wesley. References : 1. Massey ..Scattering Theory of Waves and Particles (McGraw . M. 1962) 8. Narosa Publishing House.Operations Research . 1965) 9. 2. S. New Delhi. Minflow max-cut theorem. separation theorems. S. New Jersey. New Delhi 39 . 8. C. Wiley Eastern Ltd. Y. Hadly .Non-Linear Programming. convex functions and their properties.An Introduction. New York.Optimization theory and Applications. Swarup. Sultan Chand & Sons.. S. N.. Oxford. L. New Delhi. Kuhn-Tuckers sufficient optimality theorem. M. Reading Mass. Optimality conditions : Kuhn-Tucker conditions . Wu and T.non negative constraints. Separable Programming Algorithm. Gupta & Manmohan . R. 9.Topics in Atomic Collision Theory (Academic Press. Davidon-Fletecher-Powell method. Mangasarian . Gradient Methods : Steepest descent Quasi-Newtons method. N. Optimality in absence of differentiability. Moudgalya. 5. Karlins constraint qualification. G. L. E.Frankel scheme. 1969 2.C.Plumpton : An Introduction to magneto fluid Mechanics. Dutta. Peyret and T. Press. Subsonic and supersonic flows. Convergence rates for F. Solution of a one-dimansional linear prabolic equation. Schlichtings : Boundary Layer Theory.11. Mc Graw-Hill Book Comp. for unsteady flow : Stream function for steady two-dimensiona motion.. C. D: Computational Fluid Dynamics-I Finite Difference methods : Solutions of O. unsteady one -dimensional flow. A.. Method of characteristics. S. Bhatti. Steady isentropic motion : Bernoulli. Stability of F. Bector. 1961.W. Explicit splitting method for two dimensinal equation. Circulation theorem. 2003. Steady flows through stream tubes. 5. upwind corrected schems. Landuu and E.E. Finite Element Methods : Variational formulation of of operator eqquations and Galerkins method. Peter Linz.s eqn. 3. the construction of the finite elements. R. propagation of a small distrurbance : Sound waves. A Introduction to Advance Technique (Jhon Wiley and Sons. Pergamon Press. Bernoulli. Yuam : Foundations of Fluid Mechanics. Polytropic gases. Legendre transformation . J. Hodograph method : Molenbroek transformation. The method of factoriazation. M. Eulers equations of Motion.R.Computational Methods for fluid flow (Springer- 40 .M. PHI. Oxfrd Univ. De Leval Nozzle.A. Principles of optimization Theory. iterative methods.Theoretical Numerical Analysis. References : 1.) 2. prandtl- mayer flows. New Delhi 12. Linerrization by the method of small perturbation : Prandtl . Subsonic and supersonic flow past thin bodies. Spectral method some simple applications of Fluid Dynamics Problems. Normal and oblique Shock relations.D. Shercliff : A text Book of Magnetrohydrodynamics 6.L Bansal : Viscous Fluid Dynamics.s eqn. Practical Optimization Methods. Limit lines . Hermitian method . Noncentered schemes. Leapfrog Dufort. J.D.s eqution. S. Chandra and J. Narosa Publishing House. Rayleigh Janzens method for flow past a circular cylinder. Irrotational flow : velocity potential . Shock polar diagram. Explcit and Implicit methods. The Adi method. 4.M Lifshitz : Fluid Mechanics. H. Solution of one dimensional Non-linear parabolic andhyperbolic equations.D Taylor. L.Glauert transformation..M Elementry ideas of Finite Volume method. 1959. Solution of Chaplygine. Conservation of energy. V.A Ferraro and C. References : 1. Exact solution for two-dimensional Steady motions : Redial flow : vortex flow . Springer Verlag C: Inviscid Compressible Flows and Turbulence-I Basic thermodynamics Equations of state. 2004.E. Computational Fluid Dynamics : The Basics With Applications (Mc Graw. Circular link vortex. Circle theorem. lift formula. Kelvins theorem on circulation. Complex potential.possible wavelengths in a rectangular tank. vortex tube and vortex filament. Paths of particles for different waves. Vortex motion vortex surface. Method of Inmages.Verlag). Anderson. Fundamental properties (Helmholtz properties) of vortex motion. Irrotational motion. I and II (Springer. Vol. Velocity field due to a distribution of Vorticity velocity field due to a closed vortex filament. Progressive waves.s minimum energy theorem. Verlag) 3. Gravity waves (Water waves) Surface condition.Jhon C . Cisott. Two dimensional motion Stream function.s theorem ) amd Uniqueness> General equation for impulsive motion and properties of motion under surface impulse Kelvin. Pletcher. Consstitutive eqution for inviscid fluid Eulers equation of motion. Three dimensional source sink and doublet (definitions only). Dale A.D Anderson . Sta- tionary waves. Group velocity and its dy- namical significance. Flow past a circular cylinder with circulartion : Kutta Joukowski. velocity potential acyclic irrotational motion. Tanehill. Navier Stokes equations-vector invariant from Boundary conditions.Principles of Computationonal Fluid Dynamics (Springer- Verlag.2000) 4. Linearly viscous incompressible fluid : Constitutive eqations (stress-rate of strain relations) for linerly viscous fluid. Helmholtz equations for 41 . 1998) 5. P.Hill. Wesseling . Axi-symmetic motio n.Computational Fluid Mechanics and Heat Transfer (Hemisphere Publishing Corporation) SEMESTER-IV APPLIED MATHEMATICS STREAM Paper . D Alemberts paradox Helmhorty equation for vorticity circulation .vetor in varient form Steady motion . C. line doublet and line vortex. Stokes Stream function. 6. Complex potential for line source (sink). Eulers momentum theorem. Energy for progressive Waves and Stationary wave. per- manence of irrotational motion some properties of acyclic irrotatioal motion ( using Green. Blasius theorem for thrust on an obstacle-applications to circular cylindrical boundary.Bernoullis equation and other consequences.MAG401 (Continuum Mechanics-III) (Marks: 50) Inviscid incompressible fluid : Definitions.A.J Fletcher .cases of deep and shallow water.s equa tion for complex polential of small height gravity waves. Richard H. velocity potential acyclic irrotational motion. J.Computational Techniques for Fluid Dynamics. Photoelectric effect and Compton effect.Thomson.Mathematical Theory of Elasticity 19. 1965 (Ed.C. Besand & Ramsey. Flugge) 5. Bansal -Viscous Flow Theary 14.Encyclopaedia of Physics. Trusdell.. Pai. 2. Chung. -Non-Linear Contiuum Mechantcs (MacGraw Hill) 2. Viscomet ric flow Couettee circular motion. Sokolnikoff.Boundary Layer Theory 9. Schilichting H . S. W.MAG402 (Elements of Quantum Mechanics. Sokolnikoff. D..Boundary Layer Theory.C. Miline. C .Coniumum Mechanics (Prentice Hall) 4. Non-dimensional form of N. III/ 3. C and Nol.Continuum Mechanics 16. F. Atomic Spectra.. L. Leigh. 1965 (Ed. Matter waves. S. References : 1. Trusdell.diffusion of vorticity Dissipation of energy. C and Nol. Wave-particle duality.A Text Book of Fluid Mechanics 11. Hagen Poiseule flow through a circular pipe. Schilichting H. Basic Concepts : 42 .Theroretical Hydrodynamics> 7..l. chorlton .Continuum Mechanics (Prentice-Hall) 17. S. L.M.Continutum Mechunics 3. equations. Bohr Sommerfeld theory.M. C.L . Reynoldsnumber simple exact soluttions of N-s equations: Paral- lel flow.Non-linear Theory of Continuous Media (Mac Graw.Limear Continuum Mechanics (Mac Graw Hill) 15.J.Hill) 10. Chung.Fluid Mechanics 13. Pai.S. Eriengen. Vol. The old quantum theory. plane Poiseuille flow and simple Couette flow. Truesdell. . Electron diffraction experiment.. Origin of the Quantum Theory : Black-body radiation. T. truesdell.S. W. Indequacy of clasical theory.Theroretical Hydrodynamics 12.Is.Non. . I. C . J.J.Viscous Flow Theory 8. Principle of similitude.Mathematical Therory of Elasticity 6. Kolin Kebel & Roze.Thomson. A. Miline . T. Paper . -Encyclopaedha of Physics Vol III/ 3.Viscous Flow Theory 21. Flugge) 18. S. Chaos and Fractals) Group-A Elements of Quantum Mechanics (Marks: 30) 1. D. Generalized Couette flow plane. Leigh. . -Theoretical Hydrodynamics 20. One parameter family of maps and bifurcations (through examples only) topological conjugacy. Potential barrier . F. Spherically Symmetric potential. Cantor sets.Introduction to Quantum Mechanics [MacGraw Hill. P. 2.A First Course In Chaotic Dynamical Systems. 1968] Group-B Chaos and Fractals (Marks .s equation of motion for operators References : 1. The virial theorem. A. Horse shoe and the theorem : period 3 implies chaos. 4. binary decimal representation of numbers and saw tooth map. Linear har- monic oscillator. The minimum uncertainty product. Glendinning . R. Discrete and continuous spectra Commutativity of operators. O. 1991) 5. H. Uncertainty and Complementarity principles.Quantum Mechanics [Butterworths Sci.Stability. Dirac . Expectaion values of observables. phase diagrams fixed and periodic points stable and unstable sets smooth maps and conditions for stable and unstable periodic points hyperbolicity. 1930] 2. Gedanken experiments. R. wave packet. T. Hydrogen atom bound state problems. Formal solution of the Scrodinger Equation.. 3. L. Heisenbergs Uncertaity relations.The Principles of Quantum Mechanics [Oxford University Press. A. Holmgren .20) Chaos And Fractals : Examples. 43 . Heisenberg . M. definitions of topological and capacity dimensions. Mandl . Strogartz . Eigenfunction and eigenvalues of operators. L. Wave function of a free particle. Mathews . Devaney . Logistic map.Heisenberg.An Introduction To Chaotic Dynamical System (Addison-Wesley 1987) 6. period doubling route to chaos. Instability and Chaos (Cambridge University Press 1994). Schiff .Fractals and Chaos. SDIC. Three dimen sional box potential. topological transivity (mixing) and Devanceys definition of chaos..Quantum Mechanics [MacGraw Hill. Statistical interpretatotion of the wave function. 1957] 4. P. References : 1.A First Course In Discrete Dynamical Systems (Springer. Devaney . P. examples of fractals. L.1963] 5. Peitgen . graphical analysis. R. Dynamical Variables and Operators : Operators corresponding to physical observables. orbits. Square well Potential.Non-linear Dynamics 3. Pub. 4. Simple Applications (exact solutions) : One dimensional potential step.The Physical Principles of the Quantum Theory [Dover Pub. 1981] 3. I. Schrodinger wave equation. London. A treatise on the Mathematical Theory of Elasticity. Energy equation: Viscous and joule dissipation. Steady viscous incompressible flows: unidirectional flow under a transverse magnetic field: decoupling of MHD equations.Foundations of Solid Mechanics. MHD waves: propagation of small disturbances. A. Fung . Nowacki . A. Mc Graw-Hill Book Comp. Referances: 1. cowlings theorem. Alfvens Theorem. W. 1965. 2003. CUP. existence of finite amplitude MHD waves. Boundary Layer Theory & Magneto-Hydrodynamics-II Electromagnetic equations for moving media. 1977. Thermoelsticity: Stress-strain relations in Thermo elasticity. Oxforde Univ. 1969 2. 4. J. Love . Y. Rayleighs problem. plane waves. 4. Tata Mc. J. Magneto-elasticity: Interaction between mechanical and magnetic field Basic equations Linearisation of the equations.) 2. A. C. Lundquists criterion. Plumpton: An Introduction to magneto fluid Mechanics. Paper . PHI. Alfven waves. 5. Press. General solutions for a force-free field. Ferraros law of isorotation. Boundary conditions. MHD approximations. Y. Bansal: Viscous Fluid Dynamics. L. 2004. Flow through a rectangular duct. L. Graw Hill Co. Pergamon Press. non- existence of force-free field of finite extent. Reflection and transmission of plane harmonic waves. 3. I. Unstady incompressible flows.Mathematical Theory of Elasticity. 44 . Magnetohydrostatics. H. Yuan: Foundations of Fluid Mechanics. Hartmann flow. Coupling of strain and temperature fields. References: 1. Landau and E. their nondimensional forms.Theory of Elasticity (MIR Pub. Equations of motioins and induction.. S. Ohms law including Hall current. Dynamo problem. Alfven waves with ohmic damping. Schlichting: Boundary Layer Theory. Stress-tensor formulation of Lorentz force. M. Poynting theorem. Lorentz force. Amenzade . D. Torsional vibration of a solid circular cylinder and a solid sphere. dimensionless parameteres. 1963. 1961. H. A. S. V. Lifshitz: Fluid Mechanics. Couette flow. PHI. equilibrium-configurations. B: Elasticity-II Vibrations problems: Longitudinal vibration of thin rods. E. Basic equations in dynamic thermo elasticity. 1959. special cases. C. Reductions of stastical thermo- elestic problem to a problem of isothermal elasticity. Shercliff: A Text Book Of Magnetrohydrodynamics 6. pinch effect. Sokolnikoff . force-free fields.Thermoelasticity (Addison Westey) 5. skin effect. Free Rayleigh and Love waves.MAS403 (Special Paper-III) Marks: 50 A: Viscous Flows. 3. W. Ferraro and C. forzen-in- magnetic field. Tata Mc Grand Hill Co. Y. B. 6. S.Seismic wave. 4. 7. 9. PHI. I. Dunford and Schwartz-Linear ooperators. W. A. 3. L. References : 1. Elementary examples of semigroups. Savarensky . Kirchoffs solutions of inhomogeneous wave equation.Some Basic Problems of the mathematical theory of Elasticity. Muskhelishvili .A treatise on the Mathematical Theory of Elasticity. C. 4. 1977. Theory of Elasticity.) 2. E. N. State reductions. I. Ovservability. Kennett.Foundations of Solid Mechanics. 1965.Kinematical conditions and dynamical conditions. 2. C: Elasticity and Theoretical Seismology-II Theoretical Seismology Theory of elastic waves. 1963. K. Cauchy Problem. 1970 7. 1963.Theory of Elasticity (MIR Pub. vol. Controllability. Nowacki . Timoshenk and N. Krein-Linear Differential Equations in a Branch space. An Introduction to the theory of Seismic waves and Sources. 1 & 11. Springer-Verlag. Sokolnioff . Holomorphic semigroups. 1970. A. Compact semigroups. H. E.. Timoshenk and N. S. Motion of a ssurface of discontinuity . Some problems : Application of pressure and twist on the walls of a spherical cavity in an elastic medium. Seismic wave propagation in Stratified Media.. S. S. K. Mc Grwa Hill Co. Surface waves : Rayleigh. Extensions Differential Equations. Y.. 6. 8. Theory of Elasticity. Hard constraints Final value control. Time optimal control problems. Line source and point source on the surface of a semi-infinite elastic medium. Goodies.Mathematical Theory of Elasticity. Springer Verlag. Noordhoff Ltd. 45 . 1963. Balakrishnan-Applied Functional Analysis. P. I. Bullen. Muskhelishvili . Dispersion and Group Velocity of elastic body waves. N. Love and Stonely waves. Stability and stabilizability. Amenzade . Optimal Control Theory-Linear quadratic regulator problem. 3. CUP.Some Basic Problems of the mathematical theory of Elasticity. Love . The same problem with infinite time interval..Thermoelasticity (Addison Wesley) 5. Reflections and refractions of elastic body waves.. Fung . E. G. D : Applied Functional Analysis-II Semigroups of linear operators-general properties of semigroups. CUP 10. Goodies. P. Evolution equations. Noordhooff Ltd. A. V. N. MIR Pub. References : 1. Generation of semigroups. Mc Grwa Hill Co. Yoasida-Functionnal Analysis. Oxford. Bransden & C. Joachain. Vol. Kohn-Hulthen and Sxhwinger variational methods. Model -1: Single additive constraint. Analytic Properties of Sxattering Amplitudw : Jost function. 1962) 8. Semi-Classical Approximations : WKB approximation. Co.Scattering Theory of Waves and Particles (Mcgraw .Theory of Atomic collisions (3rded. Watson . Paper-MAS404 (Special Paper . N.Quantum Collision Theory (North-Holland Pub. N. Olmura . Y. J. Co. Goldberger & K. C. 1969) 7. 1964) 10. seprable return.Hill. Scattering of a particle by a short-range potential. 1989). Rutherford scattering. Bellmans principle of optimality (Mathematical formulation). H.time independent and time-dependent. Geltman . Validity of Born approximation.Atomic Collision Theory (W. 1965) 9. additively seprable return. 2. G. 1962). Variational Principles in the Theory of Collisions: General formulation of the variational principle. Intergral tepresentation of the scattering amplitude.. Scattering by complex potential. Joachain . 3. Scattering by screened Cou-lomb field.1977) 4.A. Model -2: Single additive constraint. S. Messiah-Quantum Mechanice.IV) (Marks .. Bound (minimum) principles. New York. A. Dispersinon relations. N. multiplicative seprable return. 1966) B: Advanced Operations Research-II Dynamic Programming : Characteristics of Dynamic Program-ming problems. 1970) 6.. Massy . Model -3: Single a multiplicative Constraint. Benjamin Inc. Levinson theo-rem. Dynamic 46 . T. Effective range theory. M. Mott & H. Hulthen.50) A : Quantum Mechanics -II Collision Theory : Basic concepts. M.Collision Theory (Wiley. (Clarendon Press. F. additively separable return. Convergence of the Gorn Series. Transition probabilities and cross setions. B. Cross sections. New Jersey. Determination of Phase shifts. Eikonal approximation. Bound states and resonances. 1975) 5. Y. H. Laboratory and center-of-mass coordinates. W. R.Introduction to Quantum Mechamicw (Oxford University Press. P. S. Newton .Y. Burke . G. Wu and T. References : 1. I & II (North Holland Pub. J.).Potential Scattering in Atomic Physics (Plenum Press. L. Scattering matrix. Integral Equation Formulation: Lippmann-Schwinger integral Equation and its formal solutions. Multistage decision process Forward and Backward recursive approach. Scattering length and scattering amplitude for central force problems.Topics in Atomic Collision Theory (Academic Press. Quantum mechanical formulation . B.Quantum Theory of Scatterring (Prentice Hall. Bransden . J. A. 5. Lifshitz: Fluid Mechanecs.Non-Linear and Dynamic Programming. Plumpton: An Introduction to magneto fluid Mechanics. (i) symmetrical cylin-der (ii) a row of parallel rods. 47 . Queuing Theory : Introduction. Schlichting: Boundary Layer Theory. 5. 2. Probability distribution in Queuing systems.. Eulerian correlation with respect to time. New York. Ferraro-and C. Gupta & Manmohan Operations Research. Introductory concept. S. Statistical approach. References : 1. 1961. Classification of Queuing models. H. A. Affili-ated East-West Press Pvt. 2003. New Delhi. karmans similarity hypoth-esis. Reynolds stress- tensor. Swarup. Reynolds equations of mean motion for turbulent flow.Programming approach for solving linear and non-linear programming problems. Reading Mass. Press. Poisson and-non Poisson queuing model. Energy relations in turbulent flows. Macmillan Publishing Co. Inc. PHI. 4.D. Addision Wesley. Mc Graw-Hill Book Comp. Mixing length Prandtls momentum transfer theory. C: Inviscid Compressible Flows and Turbulence-II Turbulence: Introduction. L. References : 1.. Geometric Programming : Elementary properties of Geometric Programming and its applications. G. Seventh power velocity distribution low. correlations in homogeneous turbulence.. Oxford Univ. Application Singleitem N-period deterministic inventory model. Ltd. Velocity distribution in channel flow under constant pres-sure gradient Spread of turbulence: Mixing zone between two parallel flows. 2004 3. Sultan Chand & Sons New Delhi. S. Taylors vorticity theory. Bansal: Viscous Fluid Dynamics. S. Pergamon Press. V. longitudinal and lateral. W.A. Yuam: Foundations of Fluid Mechanics. Turbulent flow through smooth circular pipes. Landau and E. characteristic of Queuing systems. M.L. 1969 2. C. Taha Operations Research An Introduction. Taylors one-dimensional energy spectrum. Phenomenologi-cal theories. N. 1959. Dano Non-Linear and Dynamic Programming 3. J. Hadly . turbulent boundary layer on a flat plate. (two-dimensional) turbulent wake behind. double correlation between velocity components. 4. Shercliff: A text Book of Magnetrohydrodynamics 6. Kambo Mathematical Programming Techniques. eddy viscosity. H. operating characteristics of Queuing system. Anderson. Vol. John C. Development of the MAC Method for NS equa-tions. Anderson Computationasl Fluid Dynamics : The Basics With Applications (McGraw-Hill. Numerical Solution of NS Equations in General Co-ordinates. D. Incompress-ible Navier Stokes (NS) equations Boundary conditions. D. J. Solution of Euler Equations in General Co-ordinates. An Introduction To Advance Techmique (John Wiley & Sons. 2000) 4. Tanehill. 6. Implementation of boundary conditions. 1998) 5. Gried Generation by Algebraic mapping : One-dimensional stretching functions. Dala A. peyret and T. Peter Linz Theoretical Numerical Analysis. Wesseling Principles of Computational Fluid Dynamics (Springer-Verlag. A. p. Taylor Computational Methods for fluid Flow (Springer Verlag) 3. C.) 2. D: Computational Fluid Dynamics-II Multigrid method. I and II (Springer-Verlag). R. Conjugate Gradient method. Richard H. Boundary Filted Coordinate Systems : Elliptic Frid generation. Pletcher Computational Fluid mechanics and Heat Transfer (Hemisphere publishing Corporation) 48 . Fletcher Computational Techniques for Fluid Dynamics. J. References : 1. Spatial and temporal discretization on collocated and on stag-gered grids. Paper MAT 405 Term Paper Marks : 50 FOR REGULAR STUDENTS Term Paper MAT 405 is related with the special papers of the applied stream offfered by the department in each session and the topic of the term paper will also be decided by the depeartment in each session. a comprehension test will be held. FOR DDE STUDENTS Instead of Term Paper for the regular students of a paper of 50 Marks. It will be based on his special papers from which 5 questions may be set and the candidates will be required to answer 3 of them and in answering these the candidates may take help from any book or study materials within the examination hall but not from the outside. However the mark distribution is 30 marks for written submission. it is an open book examination system. that is for this paper. 15 marks for seminer presentation and 5 marks for viva-voice. 49 . uniform continuity. Valuation. Algebric and Transcedental ex- tensions.Insovability of the general equation of degree five or more by radicals. Uniform structure. separable exten- sions. total boundedness.Abstract Algebra. Mordeson & M. Paper MPG 402 (General Topology . Modules : Artinian and Noetherian Modules. Fundamental Theorem of Galois theory. Simple extensions. I. Galois field. S. Homotopy of paths.D and its application to finitely generated Abelian goups. D. S. Splitting fields.Algebra (P. structure of open sets in locally conntected second countable spaces. Sen . 7. local connectedness. J. Completion. primitive elements. Dommit & Foote .I.S. Cauchy filter.Algebra. Soulution of polynomial equations by raducals.Fundamentals of Galois Theory.Fundamentals of Abstract ALgebra ( McGraw. T. Properties of Compact metric spaces Urysohns metrization theorem. Hungeford .MPG 401 (Modern Algebra. M. Connected subsets of the real line. Lang .Definition of the fundamental group of the circle. Stone .II (Marks 30) Connectedness and charecterization of connected subsets. components.I) 2. Ordered field. Galois extension. K.III) (Marks -50) Field Theory : Extension of fields.ALgebra 6.Eech compactification (Wihtout proof) Compactness in metric spaces. uniform topology. union of connected subsets. Fundamental Structure Theorem for fi- nitely generated modules over a P. 3.Introduction to Field Theory -( CAmbridge University Press) 5. uniform spaces. References : 1. Perfect field. completeness and compactness. Lang .Hill) 4. N. SEMESTER IV PURE MATHEMATICS STREAM Paper . Algebraically closed fields. Automorphism of fields. Cyclotomic extensions. connectedness of the product spaces One-point Compactification.II) Group A General Topology . M. normal extension.H. Postnikov . Malik. fundamental group. Adamson . covering spaces. 50 .II & Functionl Analysis . Analysis (Mcgraw - Hill. New York. Narosa puglishing House Pvt. Minimization of norm problem.Functional Analysis. Siddiqui.Functional Analysis. Taylor . R. E. Kolkata. Bessels inequality. Riesz-Fischer theorem. Reflexive spaces Properties of strong and weak convergence. Ltd. The world Press Pvt. 1963) 5. New Delhi 3. A. 1955) 3. 1986 8. 1982 9. 1986 4. New York. Lahiri . 1994 10. F. Kelley General Topology (Van Nostrand. G. L. Lusternik and Sovolev-Functional Analysis 2. K. John wiley and Sons. L. Simmons Introduction to Topology and Modern. B. H. Hahn-Banach theorem and its applications.K. Simmons-Introduction to Topology and Moden Analysis Mc Graw Hill. Persevals identity. Thron Topological Structures (Holt Reinhurt and Winston. New Delhi Group B Functional Analysis . J. J. Riesz representation theorem for bounded linear functionals on Hilbert spaces. G.Algebraic topology. Adjoint (Conjugate) operators and their properties. Vulikh . V. Lp [a.b] (1 £ p £) Continuity of inner product. Wiley Eastern. Linear functionals. Akilov-Functional Analysis. 1966 6. A.P. Narici . 1956 7. Munkres Topolgy A First Course (Prentice-Hall of India. 1966) 2. Dugundji Topology (Allayn and Bacon. Lahiri-Elements of Functional Analysis. Orthogonal set. K. Jha Functional Analysis. Kantorvich and G. 1966) 6. Bachman & L. F.II (Marks 20) Invertible Mappings and their properties. B. Kreyszig-Introductory Functional Analysts with Applications.Functional Analysis 5. Pergamon Press. 1978) 4. W. Conjugate spaces. 1963 51 . TMG Publishing Co. Hilbert Spaces. References : 1. G. Academic press. Convergence. Orthogonality and orthogonal decomposition of a Hilbert Space. E.References : 1. Complete orthonormal set.Functional Analysis with applications. J. Ltd. J. Students Friends. Ltd. K. V. Yano and M. Limaye-Functional Analysis. Narosa Publishing House pvt. Geodesice in a Riemannian manifold. Nijenhuis tensor. Ltd. Lebesgues tehorem Completeness of Trigonometric system. C. Chandrama Prakashan. Gauss formulae. Sets of ht 1st and of the 2nd Categories. Allahabad. Contravariant and covariant almost analysis vector fielda. 2. World Scientific Publishing Co.II Density of arbitrary linear sets. Wiley Easten Ltd. S. Bounded approximately continuous function over [a. Comparison with Lebesgue integral and Newton integral. New Delhi. Hewitt and Stormberg . 5.. Sectional Curvature. Almost Complex manifolds. Pvt. F-connection. Structure of Manifolds. Fejers theorem.) Ltd. R. 1984. I). Lebsegue Density theorem. Properties of approximately continuous functions. Paper MPS 403 (Special Paper -III) (Marks 50) A: Differential Geometry of Manifolds-II Riemannian manifolds.b] and exact derivative. The Perron integral : Definitions and basic properties. 1984. Baires theprem. Shaikh. B: Advanced Real Analysis .. Pothishala (Pvt. Trigonomeric system and trigonometric Fourier series. 1982 4. Conformal curvatre tensor. B. Schurs theorem. Submanifolds and Hypersurfaces. Riemannian connection. Projective curvaturre tensor. Generalized Gauss and Mainardi-Codazzi equations. Mishra. 11. Lines of curvature. Curvature tensors. References: 1.Real and Abstract Analysis 52 .. 1965. Sinha. R. Ltd. An Introduction to Modern Differential Geometry. 3. B. B. Approximate continuity. Semicontinuous functions. 2007. Differential Geometry of Mani-folds. K. Strctures on a differentiable manifold and their applications. Summability of Fourier series by (C. Functions of Baire class one. Mishra. De and A. Baires theorem for Go residual and perfect sets points of condensatia of a set. Kon. S. Weingarten equations. Kalyani Publshers. means. References : 1. U. A. Normals. Baire classification of functrions. A course in tensors with applications to Riemannian Geometry. Lipschitz-General Topology. New Delhi. New York. 1966 14. 1987 53 . W. Representation theorem for bounded linear functionals on C[a. Hobson . F. Approxmation Theory in Normal Linear space. Rudin-Functional Analysis. Separable Hilbert Space. I. 2nd Edn. L Royden . 1966 4. Banach Algebra. Taylor. Compactness of Spectrum. A.Theory of Functions of a Real Variable.W. 1991 3.II Stone. H. weak topology. 1958 9. weak opology. Vols.s Munroe . 1970 12.R) and C (X. Narici. Uniqueness Crite- rion. Horvoth-Topological Vector spaces and Distributions. 3rd Edn. New York. Bachman & L. I & II. Identity element. W.Theory of Integrals 4. G. P. G. References : 1. Natanson . (1£p<µ) consequences of uniform boundedness prin- ciple. analytic property of resolvent Operator. F. Schaum Series 10.b. M. 2. and Page-Elements of Functional Analysis. 1963 8. Simmons Intoduction to topology and Modern Analysis Mc Graw Hill. 7. Goffman and G. Pedrlck-Frist Course in Functional Analysis. Von Nostrand Reinhold Co. C.Functional Analysis. E.Functional Analysis. I & II C: Advanced Functional of a Analysis . Narici & Beckerstein Topological Vector spaces.Theory of Function of a Real Variable.Measure and Integration 6. Addison-Wesley Publishing Co. (1£p<µ) and Lp[a. Wikey Eastern. 1989 5. K. Schaffer-Topological vector spaces. TMG Publishing Co. Saks. 1990 11. C) where X is a compact Hausforff space. Gelfand Neumark Theorem. Spectral radius and Spectral mapping Theorem for polynomials. Brown. Yosida-Functional Analysis Springer Veflog.b]. Best approximation. PHI. Kreyszig-Introductory Functional Analysis with Applications.Weierestrass W-capital theorem in C(X. E. John wiley and Sons.Real and Abstract Analysis 5. 1973 2. New York. Banach-Alaoglu theorem. A. Rudin . A. E. Academic Press..]1p. 1985 7. J. Ltd. New York and Basel.Real analysis 3. Marcel Dekker Inc. Vols. R) and C(X. New Delhi. Holmes-Geometric Functional Analysis and its Application 13. Diestel Application of Geometry of Banach Spaces 6. Springe. Gelfand Theory on representation of Banach Algebra. New York. Jerison. Nilpotenoy and Solvability. H. Jordan Algebras. Gillmen and M. convex ideals. Mary GrayA Radical Approach to Algebra (Addison-Wesley Publishing Company). Rowen Ring Theory (Academic Press) 54 . Emst August Behrems (Translated by Clive Reis) Ring Theory (Academic Press. A structure theorem for nonassociative algebras. Objects and morphisms. Functions. Construction of Stone-Eech compactification. elements of K(X) and the subsets of C* (X). R. 3. Near rings. Berlin) 4. Lie & Jordan Algebras : Definitions. hyper-real ideals in C(X). SankappanvarA Course in Universal Algebra (Springer- Verlag. Characteization of maximal ideas in C*(X) and C(X). Richard E. E : Theory of Rings And Algebras-II Other Radicals and Radical Properties : The Levitzki Radical. Groups. 1966). Holt Reinchart and wilson. References : 1. Limit ordinal. P. Chandler. Relations among the radicals. Inc. 1973 D: Rings of Continuous Function-II Partially ordered rings. Generalizations of the notions of radicals to other systems: Algebras. Category Theory : Definition. properties of convex ideals. Rings of Continuous Functions (Von Nostrand. Hausdorff Compactifications (Marcel Dekker. Topological Structures (Halt Reinhurt and Winston. 3. L. GelfandKolmogorov theorem. New York. Applications of sheaf theopy to the study of rings. New York. Kernels and images. A. Partial ordered set K(X) of the T2 Compactifications of a Partial ordered set X. 1976). 1960). Edwards. Redical subcategories. Characterization of real ideals. Wilansky-Functional Analysis. TMG Publishing Co. lattice otdered rings. Strcture space of a commutative ringanother description of bX. E. Local compactness and the complete lattice K(X). Products & limits. tolat orderedness of the residue class rings modulo prime ideals in C(X) and C(X) real ideals. Characterization of C* . 1965 16. Stanley Burris & H. Exact Categories. Ltd. Elements of Universal Algebra. Lie Algebras.embedded dense subset of a Tychonoff space. London). L. absolutely convex ideals. first uncountable ordinals space and its one point compactification and relation between the rings of continuous function on them. 15. Amitsurs properties. abelian Categories. The Banach-Stone theorem.Functional Andysis. Brown-Mcloy redicals. Cluster point and convergence of Z-filters on a dense subset of a Tychonoff space. More specific properties of bN and bQ and bR. Group Algebras. Lattices. non- limit ordinals compactness of the spaces of the Ordinals. 2. 2. New Delhi. Simple Lie and Jordan algebras. References : 1. and M. R. V. 1993) 3. Introduction to topological grouops 55 . Optimality Conditions. A. Maximal functions of Hardy and Littlewood Translation. Application to optimal control problems. N. Riesz theorem. Structure of inner and outer functions. Y. 5. Conjugate functions. Wieners Jauberian Theorem. Rings. The Titchmarsh Convolution Theorem. F. References : 1. E. H. Modulses and applications. R. The Theorems of Wiener and Beurling. Spectral sets of bounded functions. Duality theorem Saddle point theorems. T. Riesz & Zygmund on conjugate functions. p. Theorems of Beurings and Szego in milltiplication operator form. Katznelson An Introduction to Harmonic Analysis (JohnWiley. A. Some special optimization peoblems-Linear quadratic optimal control problems. M. Mc Coy The Theory of Rings 10. Lam Nocommutative Rings (Springer-Verlag) 6. R. 5. Generalized Lagrange Multiplier Rule Problem formulation. Linear problems. Corp 1994) 2. N. Non Linear Optimization In Banach Spaces-II Tangent Cones-Definition and properties. Y. I (Springer-Verlag. 7. Duality-Problem formulation. The Inequalities of Hardy and Hilbert. N. T. T. References : 1. 3. Goldberg Fourier transforms 6. Precupanu Convexity and Optimization in Banach Spaces (Editura Acad.Harmonic Analysis (Hindustan Pub. Ross Abstract Harmonic Analysis Vol. Statement of the Burkholder- Gundy Silverstein Theorem on. Theorems of Kolmogorov & Zygmund and M. jacolson Basic Algebra II I. Invariant subspaces. Application to approximation problems. Proof of the F. Koosis Introduction of Hp Spaces (Cambridge Univ Press). Necessary and Sufficient optimality conditions. N. 1986). Huissain. Time optimal control problems. Herstein Noncommutative Rings 8. Divisky Rings and Radicals 9. The conjugate function as a singulasr integular integral. Henry Helson . 1994) 2. Hewitt and K. J. Factoring. Lyusternik theorem. Bucaresti. Bartu and T. Adhikari Groups. 1968) 4.II Hardy spaces on the unit circle. V Balakrishnan Applied Functional Analysis (Springer-Verlag) G: Harmonic Analysis . Johames John Intoduction to the Theory of Nonlinear Optimization (Springer- Verlag. Holomorphic semigroups.b] spaces and their representations. Royden Real Analysis 5. H : Applied Functional Analysis -II Semigroups of linear operators-general properties of semigroups. signed measure and complex measure. P. BalakrishnanApplied Functional Analysis. Hewitt and StormbergReal and Abstract Analysis 4. Paper MPS404 (Special Paper -IV) (Marks 50) A: Measure and Integration . Fubinis theorem.II Signed measures. vol 1 & 11. Lebesgue decomposition theorem. Optimal Control Theory-Linear quadratic regulator problem.. Yosida-Functional Analysis. b] spaces. 3. Natanson Theory of Functions of a Real Variable. References : 1. Vols. M. Completeness and other properties of Lp [a. 9. The same problem with infinite time interval. D. Charles Schwartz-Measure. Cauchy Problem. Taylor Measure and Integration 8.t. Radon-Nikodym derivative. Measurable Rectangles. I. Generation of semigroups. Elementary sets. Elementary examples of semigroups. I & II. S. 4. Product measures. Springer-Verlag. 2. Time optimal control problems. 1994. P. V. K. Munroe Measure and Integration 7. Integrability of fuctions w.r. L. Dense subspaces of Lp [a. Dissipative semigroups Compact semigroups. Jordan decomposition theorem. Bounded linear functionals on Lp [a. Lp [a. Hahn-decomposition theorem. G. Controllability. Dunford and Schwartz-Linear operators. H. World Scientific Publi. Barra Measure and Integration 3. Hard constraintsl Final value control. Extension. G. Krein-Linear Differential Equations in a Banach space. Integration and Functions Spaces. Stability and Stabilizability Evolution equations. W. Rana Measure and Integrdition 2. Differential Equations. References : 1. A. Holder and Minkowski inequality. State reduction. Rudin Real and Abstract Analysis 6. Ovservability. Complex measure.b] spaces (1 £ p £ ¥). I.b] spaces. E. Radon Nidodym theorem. 56 . Bachman & L. Vol-I&II. Singular cohomololgy ring. 1990 8. Generalisation of Cauchy-Schwarz inequality. 1958 7. References: 1. real property of spectrum of self-adjoint operators. Eigenvalues. A. F. Resolvant sets.Functional Analysis. Yosida-Functional Analysis. Sesquilinear functionals. property of boundedness and symmetry. B. regular value. G. Unbounded-operators and their adjoint in Hilbert spaces. New York. Dernkowicz-Functional Analysis. Pergamon Press. Brown and Page-Elements of Functional Analysis. N. T. 1982 11.. B. 1970 9. fibre products. 1963 6. Fibre bundles : Definitions and examples of bundles and vector bundles and their morphisms. New York. spectral radius. Dunford and J. Wiley Easten Ltd 3. resolvant of operator. Kantorvich and G. 1989 4. 1996 13.p. calculation of cohomology rint for projective spaces. Existence and Uniqueness of singular homology and cohomology theory.Interscience. akilov-Functional Analysis.Functional Analysis 12. 1958 10.K. eigen-vectors of self-adjoint operators in Hilbert space. J. New York. closure property and boundedness property of spectrum. Kolkata. 57 . Vulikh. L. Lipschitz-General Topology. Springer Verlog. Lahiri-Elements of Functional Analysis. Simmons-Introduction to topology and Modern Analysis Mc Graw Hill. fibre products. Homology exact sequence of a fibre bundle. cross sections. K. 1994 5. E. B: Operator Theory And Applications-II Spectral properties of bounded . Schaum Series C: Algebraic Topology -II Introduction of singular homology and cohomology group by Eilenberg and steentod axioms. Ltd.V. induced bundles and vector bundles and their morphisoms. Academic Press. range of spectrum.. Narici-Functional Analysis.V. Projective spaces torus relation between H1(X) and p1(X). New York. Spectral-theorem for compact normal operators. homotopy properties of vector bundles. John Wiely and Sons. Orthogonal direct sum of Hilbert space. Tinsley Oden & Leszek F. induced bundles and vector bundles. Limaye-Functional Analysis. Schwartz-Linear Opertors.Linear operators in normed linear space. Taylor. The world Press Pvt. 3rd Edn. CRC Press Inc. Spectrum. Von Nostrand Reinhold Co. cross sections. Calculation of homology and chomology groups for circle. G. Dreyszig-Introductory Functional Analysis with Applications. 1966 2.E. Wiley Eastern. Birkhoff Lattice Theory 3.II Covering condition in modular lattices of locally finite length. Ideal Lattices. metric and quasimetric lattice. B. Semimodular lattices. G. operating characteristics of Queuing system. conditionally complete Lattices. Compactly generated lattices.Non-Linear and Dynamic Programming. Model 2 : Single additive constraint. References : 1. Reading Mass. valuation of a lattice. W. B. Massey Algebraic Topology 2. complete Boolean algebras. 2. Model 3 : Single multiplicative constraint. Gray Homotopy Theory An Introduction to Algebraic Topology 5. G. Boolean algebras and Boolean rings. S. W. Congruence relations. Arnold Logic and Boolean Algebra 4. multiplicative separable return. Complete Lattice. D. H. C. Spanier Algebraic Topology 4. Dynamic Programming approach for solving linear and non-linear programming problems. Probability distribution in Queuing systems. Classification of Queuing models. additively separable return. Massey Singular Homology Theory 3. Birkhoff lattices. Queuing theory : Introduction. Dano . Complemented modular lattices. G. subalgebra lattices. Ideals and congruence relations. Bredon Geometry and Topology D: Lattice Theory . H. Boolean algebras. Multistage decision process Forward and Backward recursive approach. Bellmans principle of optimality (Mathematical formulation) Model 1 : Single additive constraint.Non-Linear and Dynamic Programming 58 . Szasz Lattice Theory 2. Rutherford Lattice Theory E: Advanced Operations Research-II Dynamic Programming : Characteristics of Dynamic Programming problems. Application Singleitem N-period deterministic inventory model.References : 1. Fix point theorem.S. Poisson and non-Poisson queuing models.S. References : 1. Ideal chains. Addision . E. Distributive lattices and ring of sets. Complemented semimodular lattices.Wesley. Hadly . E. R. additively separable return. characteristic of Queuing systems. Geometric Programming : Elementary properties of Geometric Programming and its applications. Nearnesses and Extensions of Topology Spaces-II Separated proximities. Affiliated East-West Press Pvt. EF-) proximities. contigual families.Operations Research . Macmillan Publishing Co. Lodato (LO -) nearnesses and Efremovie (EF -) nearnesses.. Some More Applications: The category Theorems. H. Separated nearnesses. their characterization and relationship between them.nearnesses on a Tychonoff space X and the T2 compactifications of X.nearnesses on X. Isomorphism of certain conjugate space. merotopic spaces. Correspondence between the principle (strict) T1 extensions of a T1 topological space X and the cluster generated compatible LO . the correspondence between the principal T1 linkage compactifications of X and the compatible proximal Lo . The Smulian Theorems. New Delhi.(LO-. Classification of basic nearness : Riesz (RI -) nearnesses.An Introduction.Mathematical Programming Techniques. contigual nearness spaces and proximal nearness spaces and relation between them. clusters and cluster generated (concrete) nearnesses. Nearness preserving maps. Holmes . F: Geometric Functional Analysis-II Extreme points.. 3. Universal spaces. The correspondence between EF . A. The class of basic nearnesses compatible with a symmetric closure space (a proximity space. (Basic) nearness. EF-) proximities compatible with a suitable closure operator.S. 4. The Lattice structure of the class of RI . Nearness spaces. a contiguity space) and their Lattice structure. Ltd. Nearnesses are not clan generated structures. the correspondence between principal T1 compactification of X and the compatible contigual LO . merotopic spaces. Some further Application. Kambo. near families. closure operators. Convex functions and optimization. cluster generated nearness spaces.. N.nearnesses on X.nearnesses on X.. Swarup Gupta & Manmohan . New York. References : 1. 5. Taha . 59 .Operations Research. Separated nearnesses. G : Proximities. New Delhi. Nearness preserving maps. proximities and contiguities induced by a (basic) nearnesses. The theorem of James. Inc.Geometric Functional Analysis and Its Applications. Support Points and smooth points. Sultan Chand & Sons. separation axioms satisfied by the closure operators induced by RI . contiguities.(LO -. Clans. 15 marks for seminar presentation and 5 marks for viva-voice. S. Spaces of Analytic functions. Naimpully and B. Paper MPT405 Term Paper Marks : 50 FOR REGULER STUDENTS Term Paper MPT405 related with the speacial papers of the pure stream offered by the department in each seession and the topic of the term paper will also be decided by the department in each session. A. t. E. E. It be based on his special papers from which 5 questions may be set and the candidates will be required to answer 3 of them and in answering these the the candidates may take help from any book or study materials within the examination hall but not from the outside.Topological Structures (English Transl Wiley. However the mark distribution in 30 marks for written submission. Markusevich Theory of Funtions able. I. 1966) H: Advanced Complex Analysis-II Spaces of continuous functions. FOR DDE STUDENTS Instead of Term Paper for the regular students of a paper of 50 marks. W. E. Trichmarsh Theory of Funtions 5. Boas Entire Funtions 7. R. weirstrasss elliptic function p(z).the numbers e1. e2. C. 59. P. addition theorem for p(z). Thron . Wanack . that is for this paper . Eech . 1966) 3. e3- References : 1. L. Ascoli-Arzela theorem. Meromorphic function. J. 1970) 2. Ahlfors Complex Analysis 3. Cambridge. W. differentisl equation satisfied by p(z). Rudin Real and Complex Analysis 4. Conway Functions of One Complex Variable 2. B. M. A.Topological Structures (Halt Reinhurt and Winster. it is an open book examination system. a compre- hension test will be held . New York. 9.References : 1. Mittag-letters theorem. Dutta and Lokenath Debnath Elliptic Functions. Hurwitzs theorem.Proximity Spaces (Cambridge Track No. 60 . Elliptic functiom. Vol. Coposon Function of Complex Variable 6. Riemann mapping theorem. D. Catarn Entire Funtions 8. J. V. H. I & II.