M.PHIL. MATHEMATICS PROGRAMME New Syllabus (with effect from Academic Year) Scheme of the Programme. of Credits

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1 I Semester II Semester M.PHIL. MATHEMATICS PROGRAMME New Syllabus (with effect from Academic Year) Scheme of the Programme Subject Subject Number Exam Internal External Total Code of Duration Credits Hrs Research methodology Functional Analysis with Applications General Skills in Fourier * Analysis TOTAL 18 * For written exam 75 marks and for oral exam 25 marks Subject Code Subject Number of Credits Exam Duration Hrs Internal External Total Elective Paper TOTAL Dissertation & Viva Code No Elective Paper Commutative Algebra Domination in Graphs Measure Theory For each paper (i) Internal Marks: 25 (ii) External Marks: 75 Passing minimum 50 marks out of total 100 marks, external 30 marks out of 75 marks and internal 10 marks out of 25 marks For Dissertation Evaluation Internal 75 marks, External 75 marks & Viva Voce 50 marks Question Paper Pattern No. of questions Choices if any Marks per questions Total (75) 5 (either (a) or (b) Answer (a) or (b) type in each question

2 Program: M. Phil. (Mathematics) Semester-I Course Title: RESEARCH METHODOLOGY Course Objectives The objective of this course is to Provide the overview of research methodology. Introduce the required mathematical research foundations of the probability theory and random processes. Develop their skills in Latex. Course Description UNIT I: Meaning of Research - Objectives of Research - Motivation in Research - Types of Research- Research Approaches - Significance of Research - Research Methods versus Methodology - Research and Scientific Method - Importance of Knowing How Research is Done- Research Process - Criteria of Good Research.. UNIT II: Definition of Stochastic Processes Markov chains: definition, order of a Markov chain Higher transition probabilities Classification of states and chains denumerable number of states and reducible chains. UNIT III: Markov process with discrete state space: Poisson process and related distributions Properties of Poisson process, Generalizations of Poisson processes Birth and death processes Continuous time Markov chains. Stochastic processes in Queueing Systems: Concepts Queueing model M/M/1 transient behavior of M/M/1 model. UNIT IV: Contents of a LATEX source file, Document class, Page style, Parts of the document, Changing font, Centering and indenting, Lists, Theorem-like declarations, Tables, UNIT V: Mathematical environments, Main elements of math mode, Mathematical symbols, Additional elements, Fine-tuning mathematics, processing parts of a document, In-text references, Bibliographies. Course Learning Outcomes After successful completion of this course, students will be able to Understand research methods Possess the basic knowledge about stochastic processes in the time domain. Acquire more detailed knowledge about Markov processes with a discrete state space, including Markov chains, Poisson processes & birth and death processes. Typeset mathematical document in Latex.

3 Text Book: 1. Research Methodology: Methods and Techniques - C. R. Kothari, 2nd Edition, New Age International Publishers, J. Medhi Stochastic Processes Fourth Edition - New age international Private limited, H. Kopka and P.W. Daly - A Guide to LATEX - Fourth Edition - Addision Wesley, London, Unit I : Chapter 1 Unit II : Chapter 2 Chapter 3 Unit III : Chapter r 4 : : : Sections 2.1 to 2.3 & Sections 3.1 to 3.9 Sections 4.1 to 4.5 Chapter 10 : Sections 10.1 to 10.2 Unit IV: Sections 1.52, 3.1, 3.2, 3.3, 4.1, 4.2, 4.3, 4.5, 4.8. Unit V : Sections 5.1, 5.2, 5.3, 5.4, 5.5, 9.1, 9.2, 9.3. Reference Books 1. Gurumani, N. (2010), Thesis Writing and Paper Presentation, MJP Scientific Publishers. 2. E. Cinlar, Introduction to Stochastic Processes, Prentice Hall, Inc, New Jersey, Robert G. Gallager, Stochastic Processes: Theory for Applications, Cambridge University Press, S. Kottwitz, Latex Beginners Guide, Packt publishing, 2011.

4 Program: M. Phil. (Mathematics) Semester-I Course Title: Functional Analysis with Applications Course Objectives: The objective of the course is : To introduce Topological vector space and study the basic properties of topological vector spaces. To introduce the standard topology on finite dimensional spaces and study boundedness and continuity of linear transformations. To introduce the completeness and discuss the Baire category theorem, and other theorems derivable from Baire category theorem. To study the structure of weak topologies, compact convex sets and holomorphic functions on locally convex spaces. To study duality in Banach spaces, Adjoints and compact operators. Also To provide applications including of two fixed point theorems. Course Description Unit I Topological Vector Spaces: normed spaces - vector Spaces topological spaces - topological vector spaces - invariance - types of topological vector spaces - separation properties - Linear mapping - finite dimensional spaces metrization - cauchy sequences - boundedness and continuity bounded linear transformation - semi norms and local convexity - quotient spaces - semi norms and quotient spaces. Unit II Completeness: Baire category - Baire s theorem-the Banach - Steinhaus Theorem - the open mapping theorem - the closed graph theorem - bilinear mapping. Unit III Convexity: The Hahn - Banach theorems - weak topologies - the weak topology of a topological vector space-the weak * topology of a dual space-compact convex sets- extreme points-the Krein Milman theorem- Milman s theorem-vector valued integration- Holomorphic functions. Unit IV Duality in Banach Spaces: the normed dual of a normed space-duality- the second dual of a Banach space-annihilators-duals of subspaces and of quotient spaces-adjoints-compact operators

5 Unit V Some Applications: A continuity theorem-closed subspaces of L p -spaces-the range of a vector valued measure-a generalized stone- weierstrass theorem-bishop s theorem-two interpolation theorems- Kakutani s fixed point theorem-haar measure on compact groups projections - groups of linear operators two more fixe d poi n t t he orems-an i n va ria nt H a h n - Ba n a ch t h eo re m. Course Learning Outcomes After the successful completion of this course, students will be able to Students understand the concept of topological vector spaces. Student will recall and understand fundamental concepts in functional analysis. Students will understand the concepts of boundedness and continuity, seminorms and local convexity. Student will understand the concepts of weak topologies, compact convex sets and holomorphic functions Students will understand nature of duality in Banach spaces, Adjoints and compact operators. Also some applications. Text Book 1. Walter Rudin, Functional Analysis, Second edition, Tata McGraw Hill. Unit-I : Chapter 1 Unit-II : Chapter 2 Unit-III : Chapter 3 Unit-IV : Chapter 4 Unit-V : Chapter 5

6 Program: M. Phil (Mathematics) Semester : I Course Title: GENERAL SKILLS IN FOURIER ANALYSIS Course Objectives The objective of the course is To introduce the Concepts of Fourier series, Hilbert spaces and Fourier transforms for classical functions. To introduce the Fourier transform for distributions, and Fourier transforms on abstract dual groups along with general convolutions. To introduce the Fundamentals in Abstract Fourier Analysis. Course Description Unit I Periodic functions-exponentials-the Bessel Inequality- Convergence in the 2 - Norm-Uniform Convergence of Fourier series-periodic functions Revisited. Unit II Unit III Unit IV Unit V Pre-Hilbert and Hilbert spaces- 2 spaces- Orthonormal Bases and Completion- Fourier series Revisited. Convergence Theorem- Convolution-The Fourier Transform- The inversion Formula-Plancherel s theorem-the Poisson Summation Formula-Theta series. Definition of Distributions The derivative of a distribution- Tempered Distributions - Fourier Transform for distributions. Dual groups-the Fourier transform on dual groups- Convolution. Course Learning Outcomes Students will get skills in understanding Fundamentals in Fourier analysis. Students will be able to go through abstract harmonic analysis for classical functions and for distributions. Text Book Anton Deitmar, A first course in Harmonic Analysis, Second Edition, Springer, Unit-I : Chapter 1 Unit-II : Chapter 2 Unit-III : Chapter 3 Unit-IV : Chapter 4 Unit-V : Chapter 5

7 Program: M. Phil (Mathematics) Elective Course Title: COMMUTATIVE ALGEBRA Course Objectives: Introduce the concept of free module, projective modules, tensor products and Flat modules as the generalization of a vector space. Study different types of ideals on local rings, localization and its algebraic applications. Establish Noetherian modules on commutative ring, Primary decomposition and Artinian modules. Develop the theory of integral elements, integral extension and finiteness of integral closure. Discuss Valuation rings, discrete valuation rings and Dedekind domains. Course Description Unit I: Free modules- projective modules-tensor products- Flat modules. Unit II: Ideals-Local rings-localization-applications. Unit III: Noetherian modules- Primary decomposition-artin on modules-length of a modules. Unit IV: Integral elements-integral extensions-integral closed domains-finiteness of integral closure. Unit V: Valuation rings-discrete valuation rings-dedikind domains. Course Learning Outcomes: Allocate features to free modules and demonstrate variety of examples. Access properties implied by different ideals on local rings and localizations. Analyze the Noetherian modules and Artinian modules by giving some illustration Determine integral elements, integral extensions and finiteness of integral closure. Formulating and demonstrating valuation rings and discrete valuation rings.

8 Prescribed Book 1. N.S. Gopalakrishnan, Commutative Algebra, Second Edition University press, Unit I : Chapter 1 Unit II : Chapter 2 Unit III : Chapter 3 Unit IV : Chapter 4 Unit V : Chapter 5

9 Program: M.Phil (Mathematics) Elective Course Title: DOMINATION IN GRAPHS Course Objectives: To discuss the NP-completeness of the dominating set.. To calculate the bounds on the domination number interms of order and size. To develop the interesting problem of domination numbers of planer graphs with small diameter To discuss the concept independent sets and irredundant set of the graph. To apply the neighborhood knockout with replacement process. Course Description Unit I : Dominating Queens Dominating Sets in Graphs Sets of Representatives School Bus routing Computer Communication Networks Social Network Theory NP Completeness NP Completeness of the Domination Problem. Unit II : Packing- Bounds in terms of order Bounds in terms of order, Degree and Bounds in terms of order and size. Unit III: Bounds in terms of Degree, Diameter and Girth Bounds in terms of independence and covering Product graphs and Vizing s conjecture. Unit IV: Hereditary and Super hereditary properties Independent sets Dominating sets Irredundant sets. Unit V: The Dominating Chain Extensions using Maximality and Minimality. Course Learning Outcomes: (Non-trivial proofs of results without proofs are excluded from the syllabus) Demonstrate a through knowledge of the NP completeness of the Domination Problem. To identify the total domination number of graphs To solve the irredundant number and matching number. To Prove results for hereditary and super hereditary properties Find the neighborhood knockout number and replacement.

10 Text Book: J.W. Haynes, S. T. Hedetriemi and P. J. Slater - Fundamentals of Domination in Graphs - Marcel Dekker Inc., UNIT I: Chapter 1 - Sections 1.1, 1.2, 1.3, 1.4, 1.5, 1.8, 1.11, UNIT II: Chapter 2 - Sections 2.1, 2.2, 2.3. UNIT III: Chapter 2 - Sections 2.4, 2.5, 2.6. UNIT IV: Chapter 3 - Sections 3.1, 3.2, 3.3, 3.4. UNIT V: Chapter 3 - Sections 3.5, 3.6.

11 Program: M. Phil (Mathematics) Elective Course Title: MEASURE THEORY Course Objectives To introduce the concept of abstract measures and measure integration and to derive classical Lebesgue measure and Lebesgue integration as particular cases. To introduce Lebesgue spaces L p as normed spaces. To provide representation theorems and duals of L p spaces. Te derive Fubini s theorem and to introduce convolution. Course Description Unit I: Abstract Integration: Set- theoretic notations and terminology-the concept of measurability-simple functions-elementary properties of measures-arithmetic in [0, ]- integration of positive measure-integration of complex functions- the role played by sets of measurable functions. Unit II: Positive Borel Measures: Vector spaces - topological preliminaries - the Riesz representation theorem - regularity properties of Borel measures - Lebesgue measure - continuity properties of measurable functions. Unit III: L p spaces: Convex functions and inequalities- the L p spaces-approximation by continuous functions. Unit IV: Complex Measure: Total variations - absolute continuity - consequences of the Radon - Nikodym theorem - bounded linear functionals on L p - the Riesz representation theorem Unit V: Integration on Product Measures: Measurability on cartesian products-product measures - the Fubini theorem - completion of product measure convolution - distribution functions.

12 Course Learning Outcomes : Students will be able to unify classical Lebesgue integration and classical summation. Students will be able to provide examples for Banach spaces and their duals through Lebesgue spaces Students will be able to study Fourier analysis through convolution products. Text Book: Walter Rudin - Real and Complex Analysis Third Edition - Mc Graw Hill. Unit I : Chapter 1 Unit II : Chapter 2 Unit III : Chapter 3 Unit IV : Chapter 6 Unit V : Chapter 8