Department of Mathematics. Revised Syllabus of M Phil/P hd Course work In operation w.e.f Academic Session 2017

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1 Department of Mathematics Revised Syllabus of M Phil/P hd Course work In operation w.e.f Academic Session 2017

2 Paper-I Research Methodology-I Unit-I: Introduction and Motivation Meaning of research, research objectives and motivation. Types and significance of research. Qualitative and quantitative research. Research and scientific method. Selecting the problem. Techniques involved in defining a problem. Meaning and need of research design. Important concepts relating to research design. How to design a research synopsis? Unit-II: Literature review Literature review: uses of literature review, sources of information. Plagiarism and its prevention. Importance of communication skills in research. How to write a research paper? Writing research grant proposals. Progress report writing on research topic(s). Searching Math Sci. Net, SCOPUS, SCI and Google Scholar, Research Gate, Web of Science. Unit-III: Mathematical Softwares Basics of Mathematica and MATLAB, solving linear and non-linear equations using these softwares, Latex and various flavours of latex, basics of using latex, latex input files, input file structures, layout of the document, titles, chapter and sections, cross references, foot note, environments, typesetting, building blocks of a mathematical formula, matrices, tables, including encapsulated postscript graphics, bibliography, downloading and installing latex packages. Unit-IV: Statistical Skills Binomial, Poisson, Normal and Chi-square, Gamma, Beta distributions and their mean variance and m.g.f. Correlation and regression Theory, Statistical estimation Theory, Chisquare test and t-tests. Unit-V: Fourier Transform and Laplace Transform Fourier Transformation, Fourier Integrals, Inverse Fourier Transformation, Properties of Fourier Transformation, Fourier Convolution Theorem, Parseval s Relation, Fourier Transformation of Elementary functions, Dirac Delta function, Laplace Transformation, Properties of Laplace Transformation, Derivation of Laplace transformation, Laplace Convolution Theorem, Inverse Laplace Transformation. UNIT-VI: Numerical Methods Solution of algebraic and transcendental equations, Bisection method, Method of false position, Iteration Method, Newton-Raphson Method, Numerical Differentiation and Integration, Trapezoidal rule, Simpson-1/3 rule, Simpson-3/8 rule, Interpolation, Newton s formula for interpolation, Lagrange s formula for interpolation., Books Recommended: 1. Research Methodology: Methods and Techniques, C S Kothari, New age Internationals. 2. Research Methods, Ram Ahuja, Rawat Publications.

3 3. The Mathematica, Stephen Wolfram, Wolfram Media. 4. Latex, Beginner s guide, Stefan, Kottwitz, Packt Publishing Limited. 5. Mathematical Statistics, Hogg and Craig, Pearson. 6. Numerical Methods, S, Sastry, PHI. 7. Advanced Engineering mathematics, R.K Jain, S.R.K. Iyengar

4 Paper-II (Research Specific) Topics in Algebraic Theory of Semigroups Unit-I Basic definitions and examples of semi-groups and sub semi groups, Direct products, Monogenic semi groups, homomorphism and Transformations. Partial orders, semi lattices and lattices, Equivalences and Congruences. Unit-II Homomorphism theorems, Ideals and Riez s Congruences. Lattices of equivalences and congruences. Free semi groups and Monoids: Presentations. Unit-III Green's Equivalences, The structure of D-classes, Green's lemma and its corollaries. Regular D-classes, Regular semi groups, The Sandwich set. Unit-IV Simple, 0-simple and completely 0-simple semi groups. The Rees Theorem Completely simple semi groups Unit-V Completely regular semi groups, Clifford semi groups and semigroup varieties. Inverse semi groups: Representation by injective partial mappings. Unit-VI Systems and Bi-systems. Tensor product of systems. Free products of semigroups and related results. Dominions and Zigzags. Isbell Zigzag Theorem on dominions and related results. Recommended Texts: 1. Fundamentals of semigroup theory, John M Howie Clarendon press. Oxford. 2. The Algebraic theory semigroups, Vol. I, Vol. II, A H Clifford and G B Preston, Mathematical surveys of the American Mathematical Society. 3. Techniques of Semi Group Theory, P M Higgins, Oxford University Press.

5 Paper-II (Research Specific) Geometry of Complex Function Theory Unit 1: Bound estimates of zeros of polynomials Cauchy classical theorem for the zeros of polynomials and some of its generalizations, refinements and extensions, A theorem of Montel and Marty and related results. p-zeros of smallest modulus. Pellet s Theorem. Eneström Kakeya theorem and its various generalizations. Companion matrix, Hadmard s theorem, Greshgorian disc theorem and its applications, Unit II: Critical points of a polynomial Relation between zeros and critical points, Gauss Lucas theorem, Polar Derivative of a polynomial, Laguerre s theorem and its analytical proof by A Aziz. Grace Heawood theorem, Landau s Theorem, Apolar polynomials, Grace s theorem, Walsh s Coincidence theorem, Generalizations of Grace s theorem due to Aziz, composition of functions Szego s Composition Theorem, Egervary s theorem and some related results. Unit: III. Maximum modulus of a polynomial and its derivative Derivative estimates on the unit interval, Trigonometric polynomials, Inequalities of Bernstein and Markov, Bernstein s inequality and some recent developments in extremal properties of polynomials, maximum modulus of a polynomial on a larger/smaller circle in terms of the maximum modulus on a unit circle, A theorem of Ankeny and Rivlin and its various refinements and generalizations, Rivlin s theorem and its various generalizations. Unit: IV. Extremal properties of a class of Meromorphic functions. Some basic results of Aziz, Shah and others on the inequalities concerning the polar derivative of a polynomial, Bound estimates for rational functions, Results of Govil, Mohapatra, Aziz, Xin Li and others. Refinements of these results by using Schwarz s/ Dubinin s Lemma, obtained by Dubinin, Olesov, Shah and others Unit: V. Integral Estimates for Polynomials and Rational Functions Zygmund inequality and de Burgin s theorem for polynomials, Some recent generalizations and refinements of these L p, p 1 inequalities, Extension of these results to the case 0 < p < 1, Theorem of Boas and Rahman and their refinements and various generalizations proved by Aziz and others. Theorem of Malik, Principle of Subordination and its use in Integral inequalities. Integral inequalities for a class of rational functions

6 Unit: VI. Inequalities for functions of exponential type. Entire function of finite degree, Definition of M(r), Order ρ, Type τ and Indicator h f (θ), Bernstein s theorem for entire functions of type τ, Asymmetric functions. Result of Boas concerning asymmetric entire functions of type τ.generalizations of Turán s theorem by Rahman and related results. Polar derivative of a function of exponential type and some related results. Books Recommended: 1. Analytic Theory of Polynomials by Q. I. Rahman and G Schmeisser, Oxford Science Publications. 2. Geometry of Polynomials by M. Marden, 2 nd edition. American Mathematical Society, Providence R.I. 3. Entire Functions, by R P Boas, Jr. Academic Pres New York

Analytic Theory of Polynomials

Analytic Theory of Polynomials Q. I. Rahman Universite de Montreal and G. Schmeisser Universitat Erlangen -Niirnberg CLARENDON PRESS-OXFORD 2002 1 Introduction 1 1.1 The fundamental theorem of algebra