APPENDIX-I. DEPARTMENT OF MATHEMATICS. University of Calicut. M. Phil (
Mathematics) Course 2012 Admissions Onwards. Syllabus ...
APPENDIX-I
DEPARTMENT OF MATHEMATICS University of Calicut M. Phil (Mathematics) Course 2012 Admissions Onwards. Syllabus
Paper 1: Research Methodology Unit I Introduction to TEX Programming: Controls, Fonts, Grouping, Running, TEX, Characters, Dimension, Boxes, Modes, Typesetting Math Formulas (Chapter 1 to 13 and 16 of Text 1) Introduction to LATEXMath Packages: amthsmath, amssymb etc (Relevant portions of Text 2) Unit II Defining commands and environments. Theorem like environments, label and referencing \ref,\cite,etc. Bibliography database, special packages ( Refer text 2) Unit III Fourier Series: Warm up, Fourier Sine series and cosine series, Smoothness, The Riemann-Lebesgue lemma, The Dirichlet and Fourier kernels, Pointwise convergence of Fourier series. (Chapter IV: Sections 4.1 to 4.6 of Text 3) Unit IV Uniform convergence, The Gibbs Phenomenon, A Divergent Fourier Series, Termwise Intergration, Trignometic vs. Fourier Series, Termwise Differentiation. Fejer Theory. (Chapter IV: Sections 4.7 to 4.12 & 4.15 of Text 3)
Text Books: 1. Donald Knuth, The TEXBook, Addison Wesely (1986) 2. Leslie Lamport, LATEX A Document Preparation System, Addison Wesely (2000). 3. George Bachman Lawrence Narici Edward Beckenstein, Fourier and Wavelet Analysis, Springer Verlag Newyork(2000). 4. John B. Conway, Functions of One Complex Variable, Narosa Publishing House, New Delhi (1973).
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References [1] Goosens, Mittelbach and Samari, The LATEXCompanian, Addison Wesley (2004) [2] Helmut Kopka and Patricla Daly, Guide to LATEX, Addison wesely (2004)
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Paper 2: Advanced Trends in Mathematics Unit I Categories, Functors, and Natural Transformations: Axioms for Categories, Categories, Functors, Natural Transformations, Monics,Epis,and Zeros, Foundations, Large Categories, Hom-sets. Constructions on Categories: Duality, Contravariance and Opposites, Products of Categories, Functor Categories (Chapters 1 and Chapter 2 (sections 1 to 4) from the Text 1) Unit II Module Theory: Modules, Homomorphism and Exact sequences, Free modules and vector spaces, Projective and Injective Modules.(Chapter IV Sections 1, 2, 3, of text 2) Unit III Topological Vector Spaces: Introduction, Separation properties, Linear mappings, Finite dimensional spaces, Metrization, Boundedness and continuity, Seminorms and local convexity , Quotient spaces. (Chapter I: Sections 1.1 to 1.47 of Text 3)
Unit IV Elements of the theory of Topological Groups: Basic definitions and facts, Subgroups and quotient groups, Product groups (Chapter II: sections 4, 5 & 6 from text 4 (Excluding Projective limits and the subsection Miscellaneous theorems and examples of each section)
Text Books 1. Saunders Mac Lane, Catagories for the working Mathematician,Springer verlag, New York ,Berlin (1971). 2. Thomas W.Hungerford, Algebra ,Springer (1974). 3. Walter Rudin, Functional Analysis, TATA McGRA-HILL (1982). 4. Anton Deitmar, Siegfried Echterhoff, Priciples of Harmonic Anlysis , Springer(2009). 5. Edwin Hewitt and Kenneth A. Ross, Abstract Harmonic Analysis, Second Edition, Springer Verlag, Berlin, Gottingen, Heidelberg(1979). 4
References [1] Bodo Pareigis, York(1970).
Categories and Functors,
Academic Press New
[2] Nathan Jacobson,Basic Algebra Vol. I, Hindustan Publishing Corporation (India) New Delhi(1984). [3] Nathan Jacobson, Basic Algebra Vol. II, Hindustan Publishing Corporation (India) New Delhi(1984). [4] C.Musli, Introduction to Rings and Modules , Narasa Publishing House, New Delhi, Madras, Bombay, Calcutta(1994) [5] George Bachman, Elements of Abstract Harmonic Analysis, Academic Press INC, (London) LTD.( 1965). [6] Paul R.Halmos, Measure Theory,Narasa Publishing House, New Delhi (1978).
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Elective I : Algebraic Graph Theory Unit I The spectrum of a graph, regular graphs and line graphs, cycles and cuts (Sections 2 to 4 of Text 1) Unit II Spanning trees and associated structures, the tree number determinant expansions. (Sections 5 to 7 of Text 1) Note: Additional Results in Text 1 are not included in the Syllabus Unit III Vertex Transitive graphs, Edge Transitive graphs, Cayley graphs, Directed cayley graphs with no Hamilton cycles, Retracts, Transpositions.(Sections 3.1, 3.2, 3.7, 3.8,3.9 of Text 2) Unit IV Arc transitive graphs, Arc graphs, Cubic Arc-Transitive graphs, the Peterson Graph, Distance transitive Graphs, The Coxeter Graph. (Sections 4.1 to 4.6 of Text 2)
Text Books: 1. Norman Biggs, Algebraic Graph theory, Cambridge Mathematical Library (Second Edition, Cambridge University Press) 2. Chris Godsil, Gofdon Royle, Algebraic Graph Theory, Springer, New York
References [1] John Meier, Groups, Graphs and Trees An Introduction to the Geometry of Infinite Graphs, London Mathematical Society, Student Text 73 (2008) [2] D. Cvetkovic, P. Rowlinson, S.Simic, Eigen Spaces of Graphs, Cambridge University.
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Elective 2 :Operator Theory Unit I Locally Convex Spaces: Elementary Properties and Examples, Metrizable and Normable Locally Convex Spaces, Some Geometric Consequences of the Hahn-Banach Theorem, Some Examples of the Dual Space of a locally convex space.(Chapter IV sections 1 to 4) Unit II Weak Topologies: Duality, The Dual of a Subspace and a Quotient Space, Alaoglu’s Theorem, Reflexivity Revisited.(Chapter V sections 1 to 4) Unit - III Weak Topologies:(Continued) Separability and Metrizability, The Krein-Milman Theorem, An Application: The Stone-Weierstrass Theorem, The Schauder Fixed-point Theorem. (Chapter V - sections 5, 7,8,9) Unit IV Linear Operators on a Banach Space: The Adjoint of Linear Operator, The Banach-stone Theorem, Compact Operators. (Chapter VI: section 1,2,3.)
Text Books 1. John B Conway , A Course in Functional Analysis, Springer-verlag, New York(1985).
References [1] Walter Rudin, Functional Analysis, TATA McGraw Hill Publishing Company Ltd., New Delhi (1974). [2] S. David Promislow, A First Course in Functional Analysis , A John Wiley & Sons, INC., Publication(2008). [3] Gert K, Pedersen, Analysis Now, Springer( 1989).
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Elective 3 :Topology and Dynamical Systems Unit-I Filters, the Homotopy relation, the Fundamental Group, π1 (S 1 ). Chapter 4, Section: 12 From Text 1. Chapter 8, Sections: 32, 33, 34 from Text 1. Unit-II Diagonal Uniformities, Uniform Covers, Uniform Products and Subspaces; Weak Uniformities, Uniformizability and Uniform Metrisability, Complete Uniform Spaces;Completion. Chapter 9, Sections: 35 to 39 from Text 1. Unit-III The Notion of A Dynamical System, Circle Rotations, Expanding Endomorphisms of The Circle, Shifts and Subshifts, Quadratic Maps, The Gauss Transformation, Hyperbolic Toral Automorphisms, The Horse Shoe, The Solenoid, Suspension and Cross Section, Chaos and Lyapnov Exponents, Attractors. Chapter 1, Sections: 1.1 to 1.9, 1.11 to 1.13 from Text 2. Unit-IV Limit Sets and Recurrence, Topological Transitivity, Topological Mixing, Expansiveness, Topological Entropy, Topological Entropy for Some examples, Equicontinuity, Distality and Proximality. Chapter 2, Sections: 2.1 to 2.7 from Text 2 Text Books: 1 Stephen Willard, General Topology, Addison Wesley Publishing Company (1970). 2 Michael Brin & Garret Stuck, Introduction To Dynamical Systems, Cambridge University Press (2002). References: 1 Richard A Holmgren, A First Course in Discrete Dynamical Systems, springer-Verlag (2000). 2 N.Bourbaki, General Topology-I, Springer-Verlag. 8
3 A. Wilansky, Topology for Analysts, Wiley Interscience. 4 Colin Adams & Robert Franzosa, Introduction To Topology: Pure and Applied, Pearson Education.
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Examination Pattern 1. Marks Distribution for Paper I (Research Methodology) Internal External Practical Record Examination Examination Examination Marks 20 60 15 5 Duration 3 hours 2 hours 2. Marks Distribution for Paper 2 and Electives.
Marks Duration
Internal Examination External Examination 20 80 3 hours
3. Question Paper Pattern for the External Examinations of Paper I, Research Methodology For this paper there will be written examination based on the entire syllabus and practical examination based on I and II units. (a) Question Paper Pattern for the Written Examination In this paper there will be four units. The first two units carry a maximum of 10 marks and the last two units carry a maximum of 20 marks. There will be two parts in each unit, out of which one has to be answered. Each part will have at least two subdivisions. (b) Question Paper Pattern for the Practical Examination The practical question paper shall consist of two parts viz Part A and Part B. The candidate should do either Part A or Part B on a computer and submit the printout of the LATEX source (tex file) and the associated postscript result (PS file) to the examiner. These printouts are to be treated as the answer sheets of the practical examination. The candidate appearing for the practical examination should submit the record to the examiner. The question paper for the practical examination is to be prepared by the examiner. The Department should inform the Controller of Examinations, University of Calicut well in advance in number of candidates likely the to appear for the practical examination and the number of batches required for completing the practical examination on 10
the basis the availability of computers in the Department. In case, due to some technical problems like power failure or system break down, practical could not be conducted on the specified day, the examiners can choose an alternate day to conduct the examination in fresh. But the matter along with the new dates for the conduct of examination at the center should be brought to the notice of the Controller of Examinations. 4. Question paper pattern for the external examination for Paper II, Advanced Trends in Mathematics and the Elective Papers In this case there will be only written examination. There are four units in each paper. Each unit carries 20 marks. There will be two parts in each unit, out of which one has to be answered. Each part will have at least two subdivisions.
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