Class Test 5 X 10 = 50 Seminar and Vivavoce: 25 + 25 =50 TOTAL MARKS 400 x 4 = 1600 Course Details SEMESTER 1 Paper: Full marks Paper I: Abstract Algebra I 60 Paper II: Real Analysis 60 Paper III: Topology I 60 Paper IV: Complex Analysis 60 Paper V: Ordinary Differential Equations 60 Class Test 5 x 10 = 50 Seminar and vivavoce: 25 + 25 =50 Paper I: Abstract Algebra I
References
Paper II : Real Analysis Definition and existence of RiemannStieltjes integral, Properties of the Integral, Integration and differentiation, the fundamental theorem of Integral Calculus, integration of vectorvalued functions, Rectifiable curves. Rearrangements of terms of a series, Riemann's theorem. Sequences and series of functions, pointwise and uniform convergence, Cauchy criterion for uniform convergence, Weierstrass Mtest, Abel's and Dirichlet's tests for uniform convergence, uniform convergence and continuity, uniform convergence and RiemannStieltjes integration, uniform convergence and differentiation, Weierstrass approximation theorem, Power series, uniqueness theorem for power series, Abel's and Tauber's theorem. Lebesgue outer measure, Measurable sets, Regularity, Measurable functions, Borel and Lebesgue measurability, Nonmeasurable sets. Integration of nonnegative functions, the General Integral, Integration of Series, Riemann and Lebesgue Integrals. The Four derivatives, Young’s Theorem, Functions of Bounded variation, Absolute continuity, Differentiation and Integration. Measures and outer measures, Extension of a measure, Uniqueness of Extension, Completion of a measure, Measure spaces, Integration with respect to a measure. The Lpspaces. Convex functions, Jensen's inequality. Holder and Minkowski inequalities Completeness of Lp, Convergence in Measure, Almost uniform convergence. References: 1. Walter Rudin, Principles of Mathematical analysis (3rd edition) McGrawHill, Kogakusha, 1976, International student edition. 2. T. M. Apostol, Mathematical Analysis, Narosa Publishing House , New Delhi, 1985. 3. Gabriel Klambauer, Mathematical Analysis, Marcel Dekkar, Inc. New York, 1975 4. A. J. White, Real Analysis; an introduction, AddisonWesley Publishing Co., Inc., 1968. 5. G. de Barra, Measure Theory and Integration, Wiley Eastern Limited, 1981. 6. E. Hewitt and K. Stromberg. Real and Abstract Analysis, Berlin , Springer, 1969. 7. P. K. Jain and V. P. Gupta, Lebesgue Measure and Integration, New Age International (P) Limited Published, New Delhi, 1986 (Reprint 2000). 8. I. P. Natanson,Theory of Functions of a Real Variable.Vol. I,Frederick Ungar Publishing Co. 1961. 9. H. L. Royden, Real analysis, Macmillan Pub. Co. Inc. 4th edn. N.Y. 1993. 10. Richard L. Wheeden and Antoni Zygmund, Measure and Integral: An introduction to Real Analysis , Marcel Dekker Inc. 1977. 11. J. H. Williamson, Lebesgue Integration, Holt Rinehart and Winston, Inc. N.Y. 1962. 12. A. Friedman, Foundations of Modern Analysis, Holt, Rinehart and Winston, Inc., NY, 1970 13. P. R. Halmos, Measure Theory, Van Nostrand, Princeton, 1950 14. T. G. Hawkins, Lebesgue's Theory of Integration: Its Origins and Development, Chelsea, NY., 1979. 15. R. G. Bartle, The Elements of Integration, John Wiley & Sons, Inc. NY, 1966. 16. Serge Lang, Analysis I & II , AddisonWesley Publishing Co., Inc. 1969. 17. Inder K. Rana, An Introduction to Measure and Integration, Narosa Publishing House, Delhi, 1997. 18. Walter Rudin, Real & Complex Analysis, Tata McGrawHill Publishing Co. Ltd. Paper III: Topology 1
References: 1. James R, Munkres, Topology, A first course, Prentice Hall of India Pvt. Ltd., New Delhi, 2000. 2. J. Dugundji, Topology, Allyn and Bacon, 1966 (Reprinted in India by Prentice Hall of India Pvt.Ltd. 3. George F. Simmons, Introduction to Topology and Modern Analysis, McGrawHill Book Compoany, 1963f. 4. K.D. Joshi, Introduction to General Topology, Wiley Eastern Ltd., 1983. 5. J. Hocking and G. Young, Topology, AddisonWeslley, Reading, 1961 6. J. L. Kelley, General Topology, Van Nostrand, Reinhold Co., NY, 1995. 7. L. Steen and J. Seebach, Counter examples in Topology, Holt, Rinehart and Winston, New York, 1970. 8. W. Thron, Topological Structures, Holt, Rinehart and Winston, New York, 1966. 9. N. Bourbaki, General Topology, Part I (Transl). Addison Wesley, Reading, 1966. 10. R. Engelking, General Topology, Polish Scientific Publishers, Warszawa, 1977. 11. W. J. Pervin, Foundations of General Topology, Academic Press Inc. NY. 1964. 12. E. H. Spanier, Algebraic Topology. Academic Press Inc. NY, 1966. 13. S. Willard, General Topology, AddisonWesley, Reading , 1970. 14. Crump W. Baker, Introduction to Topology, Wm C. Brown Publisher, 1991 15. SzeTsen Hu, Elements of General Topology, HoldenDay. Inc. 1965. 16. D. Bushaw, Elements of General Topology, John Wiley & Sons, NY, 1963. Paper IV: Complex Analysis Review of basic concepts of functions of a single Complex variable. Complex integration., CauchyGoursat Theorem (for convex region), Cauchy's integral formula, Higher order derivatives, Morera's Theorem, Cauchy's inequality and Liouville's theorem, The fundamental theorem of algebra, Maximum modulus principle, Taylor’s theorem, Schwarz lemma, Laurent's series, Isolated singularities, Casorattiweierstrass theorem, Meromorphic functions, The argument principle, Rouche's theorem, inverse function theorem. Residues. Cauchy's residue theorem, Evaluation of integrals, Branches of many valued functions with special reference to arg z , log z and z^{a}., Riemann surfaces. Bilinear transformations, their properties and classifications, Definitions and examples of Conformal mappings. Analytic continuation, Uniqueness of direct analytic continuation, Uniqueness of analytic continuation along a curve, Power series method of analytic continuation. References 1. H. A. Priestly, Introduction to Complex Analysis, Clarendon Press Oxford, 1990 2. J. B. Conway, Functions of one Complex variable. SpringerVerlag. International Student Edition, Narosa Pub. House. 1980. 3. Liangshin Hahn & Bernard Epstein, Classical Complex Analysis. Jones and Bartlett Pub. International London, 1996. 4. L. V. Ahlfors. Complex Analysis, McGrawHill. 5. S. Lang. Complex Analysis, Addison Wesley. 1970. 6. D. Sarason, Complex Function Theory , Hindustan Book Agency, Delhi, 1994 7. Mark J. Ablowitz and A. S. Fokas, Complex Variables: Introduction and Applications, Cambridge Univ. Press, South Asian edn. 1998. 8. E. Hille, Analytic Function Theory (2 vols) , Gonn & Co, 1959. 9. W.H.J. Fuchs, Topics in the Theory of Functions of one complex variable, D. Van Nostrand Co. , 1967. 10. C. Caratheodory. Theolry of ;Functions (2 vols) Chelsea Publishing Company, 1964. 11. M. Heins, Complex Function Theory. Academic Press, 1968. 12. Walter Rudin, Real and Complex Analysis, McGraw  Hill Book Co, 1966. 13. S. Saks and A. Zygmund, Analytic Functions, Monografie Matematyczne, 1952 14. E. C. Titchmarsh, The Theory of Functions, Oxford Univ. Press, London. 15. W. A. Veech, A Second Course in Complex Aanlysis. W. A. Benjamin, 1967. 16. S. Ponnusamy, Foundations of Complex Analysis, Narosa Pub. House, 1997. Paper V: Ordinary Differential Equations Picard’s existence and uniqueness theorem, CauchyLipschitz method, Peano’s theorem, Linear second order ODE in Complex plane, singular points , Frobenius methods, Special functions – Hypergeometric, Legendre, Bessel, etc functions, recurrence relations, integral representations and other properties, System of 1st order linear diff. equations. Comparison and Separation theorems, Adjoint and Selfadjoint equations, Green’s functions, SturmLiouville boundary value problem, Eigen function expansions. Nonlinear diff. Equations, phase plane analysis, plane autonomous systems, sink, source, foci, nodes, saddle points, limit cycles through simple examples. References 1. P. Hartman, Ordinary Differential equations, John Wiley & sons, 1964 2. E.A.Coddington and N. Levinson, Theory of ordinary differential equations, McGrow Hill (1955) 3. G. F. Simmons, Differential equations , TataMcGrowHill. 4. M.Braun, Differential equations and their applications: an introduction to applied maths., Applied Mathematical Sciences ( 3^{rd} edn) Vol. 15 (springer ) 1983. 5.W.T. Reid, Ordinary differential equations, J. Wiley (1971) 8. Treaties on special functions, B. Sen (Allied Pub) 9. Ordinary Difference Equations, D. Somasundaram, Narosa. SEMESTER 2 Paper: Full marks Paper VI: Abstract Algebra II 60 Paper VII: Calculus on R^{n} 60 Paper VIII: Topology II 60 Paper IX: Partial Diff. Eqns. & Integral Eqns. 60 Paper X: Differential Geometry 60 Class Test: 5 x 10 = 50 Seminar and vivavoce: 25 + 25 =50 ^ Field extension – Algebraic and transcendental Extensions. Separable and Inseparable extensions. Perfect fields, Normal extensions. Finite fields. Primitive elements. Algebraically closed fields. Galois extensions. Fundamental theorem of Galois theory. Solutions of polynomial equations by radicals. Insolvability of the general equation of degree 5 by radicals. Module theory – Modules, sub modules, quotient modules; homomorphism and isomorphism theorems; cyclic modules, simple modules, semisimple modules. ^ Vector spaces, subspaces, linear dependence, basis, dimension, algebra of linear transformations. Algebra of matrices, rank and determinant of matrices, linear equations. Eigenvalues and eigenvectors, CayleyHamilton theorem. Matrix representation of linear transformations. Change of basis, canonical forms, diagonal forms, triangular forms, Jordan forms. Inner product spaces, orthonormal basis. Quadratic forms, reduction and classification of quadratic forms. References: 1. I.N. Herstein, Topics in Algebra, Wiley Eastern Ltd., New Delhi, 1975 2. P. B. Bhattacharyya, S. K. Jain and S. R. Nagpaul, Basic Abstract Algebra (2e), Camb.Univ. Press, Indian Edition , 1997. 3. M. Artin, Algebra, Perentice Hall of India, 1991. 4. P.M. Cohn, Algebra, vols, I,II, & III, John Wiley & Sons, 1982, 1989, 1991. 5. N. Jacobson, Basic Algebra, vols. I & II, W. H. Freeman, 1980 (also published by Hindustan Publishing Company) 6. S. Lang. Algebra, 3rd edn. AddisonWeslley, 1993. 7. I.S. Luther and I.B.S. Passi, Algebra, Vol.IIIModules, Narosa Publishing House. 8. D. S. Malik, J. N. Modrdeson, and M. K. Sen, Fundamentals of Abstract Algebra, Mc GrawHill, International Edition, 1997. 9. Vivek Sahai and Vikas Bist, Algebra, Narosa Publishing House, 1999 10. I. Stweart, Galois Theory, 2nd edition, Chapman and Hall, 1989. 11. J.P. Escofier, Galois theory, GTM Vol.204, Springer, 2001. 12. T.Y. Lam, Lectures on Modules and Rings, GTM Vol. 189, springerverlag 13. K. Hauffman and R. Kunz, Linear Algebra, Pearson Education (INDIA), 2003. 14. S. Lang, Linear Algebra, Springer, 1989. Paper VII: Calculus on R^{n} Functions of several variables, linear transformations, Derivatives in an open subset of R^{n}, Chain rule, Partial derivatives, interchange of the order of differentiation, Derivatives of high orders, Taylor's theorem, Inverse function theorem, Implicit function theorem, Jacobians, extremum problems with constraints, Lagrange's multiplier method, Differentiation of integrals, Partition of unity, Differential forms, Stoke's theorem. References: 1. Walter Rudin, Principles of Mathematical analysis (3rd edition) McGrawHill, Kogakusha, 1976, International student edition. 2. T. M. Apostol, Calculus, Volume II , Wiley Eastern (2e). 3. E. Hewitt and K. Stromberg. Real and Abstract Analysis, Berlin , Springer, 1969. 4.. H. L. Royden, Real analysis, Macmillan Pub. Co. Inc. 4th edn. N.Y. 1993. 5. Richard L. Wheeden and Antoni Zygmund, Measure and Integral: An introduction to Real Analysis , Marcel Dekker Inc. 1977. 6. A. Friedman, Foundations of Modern Analysis, Holt, Rinehart and Winston, Inc., NY, 1970 7. Serge Lang, Analysis I & II , AddisonWesley Publishing Co., Inc. 1969. Paper VIII: Topology II
(iii) Metrizable & locally metrizable topological spaces. (v) Introduction to Uniform spaces, Fundamental Group and Covering Spaces. References: 1. James R, Munkres, Topology, A first course, Prentice Hall of India Pvt. Ltd., New Delhi, 2000. 2. J. Dugundji, Topology, Allyn and Bacon, 1966 ( Reprinted in India by Prentice Hall of India Pvt.Ltd. 3. George F. Simmons, Introduction to Topology and Modern Analysis, McGrawHill Book Compoany, 1963f. 4. K.D. Joshi, Introduction to General Topology, Wiley Eastern Ltd., 1983. 5. J. Hocking and G. Young, Topology, AddisonWeslley, Reading, 1961 6. J. L. Kelley, General Topology, Van Nostrand, Reinhold Co., NY, 1995. 7. L. Steen and J. Seebach, counter examples in topology, Holt, Rinehart and Winston, New York, 1970. 8. W. Thron, Topological Structures, Holt, Rinehart and Winston, New York, 1966. 9. N. Bourbaki, General Topology, Part I (Transl). Addison Wesley, Reading, 1966. 10. R. Engelking, General Topology, Polish Scientific Publishers, Warszawa, 1977. 11. W. J. Pervin, Foundations of General Topology, Academic Press Inc. NY. 1964. 12. E. H. Spanier, Algebraic Topology. Academic Press Inc. NY, 1966. 13. S. Willard, General Topology, AddisonWesley, Reading , 1970. 14. Crump W. Baker, Introduction to Topology, Wm C. Brown Publisher, 1991 15. SzeTsen Hu, Elements of General Topology, HoldenDay. Inc. 1965. 16. D. Bushaw, Elements of General Topology, John Wiley & Sons, NY, 1963. Paper IX : Partial Differential Equations & Integral Equations First order nonlinear PDE, Cauchy problem, Method of characteristics, Classifications of second order semilinear PDE, D’Alembert’s method for wave equation, Separation of variables for Laplace equations in two and three dimensions, Wave equation and Heat equation in one and two dimensions, applications to initial and boundary value problems, well posed boundary value problems. Integral transforms, Fourier and Laplace transforms, Mellin & Hanckels transform, Inversion formulae, Bromwich Integral, Convolutions and applications, Distributions and their transforms. Applications to Wave, Heat and Laplace equations. Integral equations, classifications, successive approximations, separable kernels, Fredholm alternative, HilbertSchemidt theory of symmetric kernels, Construction of Green’s function, Convoluted Kernels, Abels equations and solutions. References: 1. T. Amarnath, Partial differential equations, Narosa 2. I. N. Sneddon, Partial differential equations, McGrow Hill. 3. E. Evans, Partial Differential equations, Academic 4. Phoolan Prasad and Ravindran Partial Differential equations, Narosa.
Paper X: Differential Geometry Curvilinear Coordinates, Elementary arc length, Length of a vector, Angle between two nonnull vectors, Reciprocal Base system, Intrinsic Differentiation, Parallel Vector Fields. Geometry of Space Curves: SerretFrenet formulae, Equation of a Straight Line, Helix, Bertrand Curves. Geometry of a Surface: Nature of surface coordinates, First Fundamental Form, Direction Parameter, Angle between two vectors on a surface, esystems & εtensors, SerretFrenet formulae for a surface curve. Geodesics and geodesic coordinates. Isometric surfaces. Gaussian curvature and total curvature. Developable surface. Normal line to a surface. Second Fundamental form. References:
SEMESTER 3 Paper Full marks Paper XI: Integration Theory 60 Paper XII: Functional Analysis 60 Paper XIII: Compute Aided Numerical Analysis Theory 60 SPECIAL PAPERS: Papers XIV and XV (Two sets of papers to be taken out of the following. The students will have to inform their choices before the beginning of 3^{rd} semester. However combinations will be decided by the Departmental Council before the beginning of the Academic Session every year on the basis of the availability of Faculty Members. The corresponding papers are to be continued in the next semester also)
Class Test 5 X 10 = 50 Seminar and Vivavoce: 25 + 25 =50 Paper XI: Integration Theory
References: 1. H. L. Royden, Real analysis, Macmillan Publishing Co., Inc. 4th Edition, 1993. 2. S. K. Berberian. Measure and integration. Chelsea Publishing Company, NY, 1965 3. G. de Barra, Measure Theory and integration, Wiley Eastern Ltd. 1981. 4. P. K. Jain and V. P. Gupta, Lebesgue Measure and Integration, New Age International (P) Ltd. New Delhi. 5. Richard L. Wheeden and Antoni Zygmund, Measure and Integral: An Introduction to Real Analysis, Marcel Dekker Inc. 1977. 6. J.H. Williamson, Lebesgue Integration, Holt Rinehart and Winston, Inc, New ;York. 1962. 7. P.R. Halmos, Measure Theory. Van Nostrand. Princeton 1950 8. T. G. Hawkins. Lebesgue's Theory of Integration: Its Origins and Development , Chelsea New York 1979 9. K. R. Parthasarathy , Introduction to Probability and Measure, Macmillan Co. India Ltd.Delhi  1977. 10. R. G. Bartle, The Elements of Integration, John Wiley & Sons, Inc. New York, 1966 11. Serge Lang. Aanlysis I & II, AddisonWesley Publishing Co. Inc. 1967. 12. Inder K. Rana, An Introduction to Measure and Integration, Narosa Publishing House, Delhi 1997 Paper XII :Functional Analysis Normed linear spaces. Banach spaces and examples. Quotient space of normed linear spaces and its completeness, equivalent norms. Riesz Lemma, basic properties of finite dimensional normed linear spaces and compactness. Weak convergence and bounded linear transformations, normed linear spaces of bounded linear transformations, dual spaces with examples. Uniform boundedness theorem and some of its consequences. Open mapping and closed graph theorems. HahnBanach theorem for real linear spaces, complex linear spaces and normed linear spaces. Reflexive spaces. Weak Sequential compactness. Compact Operators. Solvability of linear equations in Banach spaces. The closed Range Theorem. Inner product spaces. Hilbert spaces. Orthonormal sets. Bessel's inequality. Comlplete orthonormal sets and Parseval's identity. Structure of Hilbert spaces. Projection theorem. Riesz representation theorem. Adjoint of an operator on a Hilbert space. Reflexivity of Hilbert spaces. Selfadjoint operators, Positive, projection, normal and unitary operators. References: 1. G. Bachman and L. Narici, Functional Analysis, Academic Press, 1966 2. N. Dunford and J. T. Schwartz, Linear Operators, Part I, Interscience, New York, 1958 3. R. E. Edwards, Functional Analysis. Holt Rinehart and Winston, New York, 1965. 4. C. Goffman and G. Pedrick, First Course in Functional Analysis, Prentice Hall of India, New Delhi, 1987 5. P. K. Jain, O. P. Ahuja and Khalil Ahmad, Functional Analysis, New Age International (P) Ltd. Wiley Eastern Ltd. N. Delhi 1997 6. R. B. Holmes, Geometric Functional Analysis and its Applications, SpringerVerlag 1975 7. K. K. Jha, Functional Analysis, Students Friends, 1986 8. L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, 1982. 9. K. Kreyszig , Introductory Functional Analysis with Applications, John Wiley & Sons New York, 1978 10. B. K. Lahi;ri, Elements of Functional Analysis, The World Press Pvt. Ltd. Calcutta, 1994 11. B. Choudhury and Sudarsan Nanda, Functional Analysis, with Applications, Wiley Eastern Ltd., 1989. 12. B. V. Limaye, Functional Analysis, Wiley Eastern Ltd. 13. L. A. Lustenik and V. J. Sobolev, Elements of Functional Aanlysis, Hindustan Pub. Corpn. N.Delhi 1971. 14. G. F. Simmons, Introduction to Topology and Modern Aanlysis, McGraw Hill Co. New York , 1963. 15. A. E. Taylor, Introduction to Functional Analysis, John Wiley and Sons, New York, 1958 16. K. Yosida, Functional Analysis, 3rd edition Springer  Verlag, New York 1971. 17. J. B. Conway, A course in functional analysis, SpringerVerlag, New York 1990 Paper XIII: Computer Aided Numerical Analysis A. Programming in C –30 Marks An overview of computer programming languages., C programming Essentials, Program control flow, Basic inputoutput, Arrays, Pointers, Functions, Storage classes Derived data types, File handling , Preprocessor Standard C library functionsdetails of each of these major topics shall be explained through suitable examples. B. ^ System of linear equations and eigenvalue problem: Operational counts for direct methods of solving linear algebraic equations, Gaussian operational count for inversion of a matrix, eqigenvalue problem, General Iterative method, Jacobi & Gauss, Seidel method, Relaxation method, Necessary & sufficient conditions for convergence, speed of convergence, SOR & SUR methods, Gerschgorin’s circle theorem, Determination of eigenvalue by iterative methods, Ill conditioned system. ^ Newtons’s method, existence of roots, stability & convergence under variation of initial approx., general iterative method for system x=g(x) and its sufficient condition for convergence, the method of steepest descent.
