AU Math MPhil/MSc Syllabi

================================================================= Restructured Syllabi (w.e.f. 2014-15 admitted batch) M.A./M.Sc. Mathematics ============================================================================== Internal Assessment 20% and External Examination 80% Q1 answer ALL 8 bits each for 2 marks; answer one question in each unit for 16 marks. ============================================================================== First Year: First Semester: (All papers are compulsory) M101: Algebra – I M102: Real Analysis – I M103: Topology – I M104: Differential Equations - I M105: Lin. Alg. and Disc. Math. Second Semester: (All papers are compulsory) M201: Algebra – II M202: Real Analysis – II M203: Topology – II M204: Complex Analysis M205: Graph Th and Coding Th (Non Core/CBCS For AU Students) M205: Graph Th and Advanced Coding Th (Also for affiliated Colleges) ============================================================================ Second Year: Third Semester: M301: Functional Analysis (Compulsory) M302: (Stream A) ( For Optional 1) M302(1): Number Theory – I M302(2): Universal Algebra – I M302(3): Fuzzy Set Theory and Applications M303: (Stream B) (For Optional 2) M303(1): Lattice Theory – I M303(2): Operations Research M303(3): Mathematical Biology M304: (Stream C) (For Optional 3) M304(1): Commutative Algebra – I M304(2): Semigroups – I M304(3): Advanced Graph Th. M305: Lin Alg and Numb Th (Non Core/CBCS For AU Students ONLY) Complex Analysis – II (For affiliated Colleges ONLY) Fourth Semester: M401: Measure and Integration (Compulsory) M402: (Stream A) (For Optional 1) M402(1): Number Theory – II (Prerequisite Number Theory – I) M402(2): Universal Algebra – II (Prerequisite Universal Algebra – I) M402(3): Operator Theory M403: (Stream B) (For Optional 2) M403(1): Lattice Theory – II (Prerequisite Lattice Theory – I) M403(2): Formal Languages and Automata Theory M403(3): Banach Algebras M404: (Stream C) (For Optional 3) M404(1): Commutative Algebra–II (Prerequisite Commutative Algebra–I) M404(2): Semigroups – II (Prerequisite Semigroups – I) M404(3): Nonlinear Functional Analysis M405: Partial Differential Equations (Compulsory) ================================================================ ANNEXURE-I ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS I-SEMESTER M101 ALGEBRA – I (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I Group Theory Definition of a Group- Some Examples of Groups- Some preliminary Lemmas- Subgroups- A counting principle- Normal subgroups and Quotient Groups-Homomorphisms- Automorphisms. Chapters 2 sections 2.1 – 2.8. UNIT-II Group Theory Cayley”s Theorem- Permutation Groups- Another counting principle- Sylow’s Theorem- Direct products- Finite Abelian Groups. Chapter 2 sections 2.9 – 2.14 UNIT-III Ring Theory Definition and Examples of Rings- Some special classes of Rings- Homomorphisms- Ideals and Quotient Rings- More Ideals and Quotient Rings- The Field of Quotients of an Integral Domain. Chapter 3 sections 3.1 – 3.6 UNIT-IV Ring Theory Euclidean Rings- A particular Euclidean Ring- Polynomial Rings- Polynomials over the Rational Field- Polynomial Rings over Commutative Rings. Chapter 3 sections 3.7 – 3.11 Prescribed Book: Topics in Algebra: I. N. Herstein , Second edition, John Wiley & Sons ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS I-SEMESTER M102 REAL ANALYSIS-I (Restructured/w.e.f. 2014-15 admitted batch) UNIT–I Basic Topology: Finite, Countable, and Uncountable Sets, Metric spaces, Compact sets, Connected sets. (Chapter 2 of the text book) UNIT–II Numerical Sequences and Series: Convergent sequences, Subsequences, Cauchy sequences, Upper and Lower limits, Some special sequences, Series, Series of non-negative terms , number e , The Root and Ratio tests, Power series , Summation by parts , Absolute Convergence , Addition and Multiplication of series, Rearrangements. (Chapter 3 of the text book) UNIT–III Continuity: Limits of Functions, Continuous Functions, Continuity and Compactness, Continuity and Connectedness, Discontinuities, Monotone Functions, Infinite Limits and Limits at Infinity. (Chapter 4 of the text book) UNIT–IV Differentiation: The Derivative of a Real Function, Mean Value Theorems, The Continuity of Derivatives, L’ Hospital’s Rule, Derivatives of Higher order, Taylor’s theorem, Differentiation of Vector- valued Functions. (Chapter 5 of the text book) Text Book: Principles of Mathematical Analysis by Walter Rudin, International Student Edition, 3rd Edition, 1985. Reference: Mathematical Analysis by Tom M. Apostal, Narosa Publishing House, 2nd Edition, 1985. ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS I-SEMESTER M103 TOPOLOGY-I (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I Sets and Functions: Sets and Set inclusion – The algebra of sets – Functions – Products of sets – Partitions and equivalence relations – Countable sets – Uncountable sets – Partially ordered sets and lattices. Chapter I: Sections 1 to 8. UNIT-II Metric spaces: The definition and some examples – Open sets – Closed sets – Convergence, Completeness and Baire’s theorem – Continuous mappings. Chapter 2: Sections 9 to 13. UNIT-III Metric spaces (Continued): Spaces of continuous functions – Euclidean and unitary spaces. Topological spaces: The definition and some examples – Elementary concepts – Open bases and open sub bases – Weak topologies – The function algebras C(X, R) and C(X, C). Chapter 2: Sections 14,15 and Chapter 3: 16 to 20. UNIT-IV Compactness: Compact spaces – Product of Spaces – Tychonoff’s theorem and locally Compact spaces – Compactness for metric spaces – Ascoli theorem. Chapter 4: Sections 21 to 25. Prescribed book: Introduction to Topology by G.F.Simmons, Mc.Graw-Hill book company. ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS I-SEMESTER M104 DEFFERENTIAL EQUATIONS (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I Second order linear differential equations: Introduction-general solution of the homogeneous equation - Use of a known solution to find another - Homogeneous equation with constant coefficients - method of undetermined coefficients - method of variation of parameters Chapter 3 (Sec 14-19) of prescribed text book. UNIT-II Oscillation theory and boundary value problems: Qualitative properties of solutions - The Sturm comparison theorem - Eigen values, Eigen functions and the vibrating string Chapter 4 (Sec 22-24, Appendix A) of prescribed text book. UNIT-III Power series solutions: A review of power series-series solutions of first order equations-second order linear equations - ordinary points-regular singular points Chapter 5 (Sec 25-29) of prescribed text book. UNIT-IV Systems of first order equations: Linear systems - Homogeneous linear systems with constant coefficients - Existence and Uniqueness of solutions - successive approximations - Picard’s theorem - Some examples. Chapter 7 (Sec 36-38) and Chapter 11(Sec 55-56) of prescribed text book. Text book: George F. Simmons, Differential Equations, Tata McGraw-Hill Publishing Company Limited, New Delhi. ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS I-SEMESTER M105 LINEAR ALGEBRA AND DISCRETE MATHEMATICS (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I Elementary Canonical Forms: Introduction-Characteristic Values-Annihilating Polynomials-Invariant Subspaces. Sections 6.1, 6.2, 6.3, 6.4, of .chapter 6 in Prescribed Text book I UNIT-II Simultaneous Triangulation-Simultaneous Diagonalization-Direct-sum Decompositions-Invariant Direct Sums-The Primary Decomposition Theorem. Sections 6.5, 6.6, 6.7, 6.8 of Chapter 6 in Prescribed Text book I UNIT-III Definitions of lattices, Modular lattices and distributive lattices. Chapter I of text book of II UNIT-IV Basic properties, Boolean polynomials, ideals, minimal forms of Boolean polynomials, Chapter 2 of text book II Text Book I: Linear Algebra second edition By Kenneth Hoffman and Ray Kunze, Prentice-Hall of India Private Limited, New Delhi-110001, 2002 Text Book II: Applied Abstract Algebra by Rudolf Lidl and Gunter Pilz, Published by Springer verlag. ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS II-SEMESTER M201 ALGEBRA – II (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I Fields: Extension Fields- The Transcendence of e – Roots of Polynomials- Construction with Straightedge and Compass. Chapter 5 sections 5.1 -5.4 UNIT-II Fields: More about roots- The elements of Galois Theory- Solvability by Radicals- Galois Groups over the Rationals. Chapter 5 sections 5.5- 5.8 UNIT-III Finite Fields- Wedderburn’s Theorem on Finite Division Rings. Chapter 7 sections 7.1 , 7.2 UNIT-IV A Theorem of Frobenius- Integral Quaternions and the Four-Square Theorem. Chapter 7 sections 7.3, 7.4 Prescribed Book: Topics in Algebra: I. N. Herstein , Second edition, John Wiley & Sons. ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS II-SEMESTER M202 REAL ANALYSIS-II (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I Riemann-Stieltjes Integral: Definition and existence of the Riemann Stieltjes Integral, Properties of the Integral, Integration and Differentiation, the fundamental theorem of calculus – Integral of Vector- valued Functions, Rectifiable curves. (Chapter 6 of the text book) UNIT-II Sequences and Series of the Functions: Discussion on the Main Problem, Uniform Convergence, Uniform Convergence and Continuity, Uniform Convergence and Integration, Uniform Convergence and Differentiation, Equicontinuous families of Functions, the Stone-Weierstrass Theorem. (Chapter 7 of the text book) UNIT-III Power Series: (A section in Chapter 8 of the text book) Functions of Several Variables: Linear Transformations, Differentiation, The Contraction Principle, The Inverse Function theorem. (First Four sections of chapter 9 of the text book) UNIT-IV Functions of several variables Continued: The Implicit Function theorem, The Rank theorem, Determinates, Derivatives of Higher Order, Differentiation of Integrals. (5th to 9th sections of Chapter 9 of the text book) TEXT BOOK: Principles of Mathematical Analysis by Walter Rudin, International Student Edition, 3rd Edition, 1985. REFERENCE: Mathematical Analysis by Tom M. Apostal, Narosa Publishing House, 2nd Edition, 1985. ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS II-SEMESTER M203 TOPOLOGY II (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I Separation: – space and Hausdorff spaces – Completely regular spaces and normal spaces – Urysohn’s lemma and the Tietze extension theorem – The urysohn imbedding theorem – The stone – chech compactification. Chapter 5: Sections 26 to 30 Prescribed text book – 1. UNIT-II Connectedness: Connected spaces – The components of a space – Totally disconnected spaces – Locally connected spaces. Chapter 6: Sections 31 to 34 Prescribed text book – 1. UNIT-III Approximation: The weierstrass approximation theorem – The stone-weierstrass theorems – Locally compact Hausdorff spaces – The extended stone-weierstrass theorems. Chapter 7: Sections 35 to 38 Prescribed text book – 1. UNIT-IV Topological Groups: Neighborhoods of a point in topological group-Isomorphism and local isomorphisms-Subgroups-Quotient groups–Homomorphisms. Chapter 3, sections 1 and 2.1 to 2.8, pages 219-237, of Prescribed text book 2. Prescribed book: 1. Introduction to Topology by G.F.Simmons, Mc.Graw-Hill book company. 2. General Topology by Bourbaki, Wesely publishing company. ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS II-SEMESTER M204 COMPLEX ANALYSIS (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I Elementary properties and examples of analytic functions: Power series- Analytic functions- Analytic functions as mappings, Mobius transformations. ($1, $2, $3 of chapter-III of prescribed text book) UNIT-II Complex Integration: Riemann - Stieltjes integrals - Power series representation of analytic functions- zeros of an analytic function -The index of a closed curve. ($1, $2, $3 $4 of chapter-IV of prescribed text book) UNIT-III Cauchy’s theorem and integral formula - the homotopic version of cauchy’s theorem and simple connectivity- Counting zeros; the open mapping theorem. ($5, $6, $7 of chapter-IV of prescribed text book) UNIT-IV Singularities: Classifications of singularities- Residues- The argument principle. ($1, $2, $3 of chapter-V of prescribed text book) Prescribed text book: Functions of one complex variable by J.B.Conway : Second edition, Springer International student Edition, Narosa Publishing House, New Delhi. ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS II-SEMESTER M205 GRAPH THEORY AND CODING THEORY (Restructured/w.e.f. 2014-15 admitted batch) (Non Core/CBCS, For AU Students) UNIT-I: Graphs, digraphs, network, multi graph, elementary results , structure based on connectivity, characterization, theorems on trees, tree distances, binary trees. Chapters 1, 2 and 3 of Text Book I UNIT-II: Eulerian graphs, Hamiltonian graphs, Spanning trees, Fundamental cycles, Minimal spanning trees, (Chapter 4 of text book I) Kruskal algorithm, Prims algorithm (8.5 of Text Book II) UNIT-III: Introduction to Coding Theory: Introduction, Basic assumptions, correcting and detecting codes, Information rate, The effects of error detection and correction, Finding the most likely code word transmitted, Some basic algebra, Weight and distance, Maximum likelihood decoding, Reliability of M L D. From Chapter 1 of Text Book III UNIT-IV: Error detecting codes, Error corer correcting codes; Linear codes: Linear codes, Two Important subspaces, Independence, Basis, Dimension, Matrices, Bases for C=<S> and C, Generating matrices and Encoding, Patity check matrices. Chapter 2 of Text Book III TEXT BOOK I: Graph Theory applications By L.R.Foulds, Narosa publishing House, New Delhi TEXT BOOK II: Discrete mathematical structures by Kolman and Busby and Sharon Ross Prentice Hall of India-2000, (Third Edition) TEXT BOOK III: Coding Theory by D. G. Hoffman, D. A. Lanonard, C. C. Lindroes M205 GRAPH THEORY AND ADVANCED CODING THEORY (For affiliated P.G. Colleges) (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I: Graphs, digraphs, network, multi graph, elementary results , structure based on connectivity, characterization, theorems on trees, tree distances, binary trees. Chapters 1, 2 and 3 of Text Book I UNIT-II: Eulerian graphs, Hamiltonian graphs, Spanning trees, Fundamental cycles, Minimal spanning trees. (Chapter 4 of text book I) Kruskal algorithm, Prims algorithm (8.5 of Text Book II) UNIT-III: Introduction to Coding Theory: Introduction, Basic assumptions, correcting and detecting codes, Information rate, The effects of error detection and correction, Finding the most likely code word transmitted, Some basic algebra, Weight and distance, Maximum likelihood decoding, Reliability of M L D, Error detecting codes, Error correcting codes. From Chapter 1 and first few sections of Chapter 2 of Text Book III. UNIT-IV: Linear codes: Linear codes, Two Important subspaces, Independence, Basis, Dimension, Matrices, Bases for C=<S> and C, Generating matrices and Encoding, Patity check matrices, Equivalent codes, Distance of a Linear code, Cosets, M L D for Linear codes, Reliability of Linear codes. Rest of Chapter 2 of Text Book III. TEXT BOOK I: Graph Theory applications By L.R.Foulds, Narosa publishing House, New Delhi TEXT BOOK II: Discrete mathematical structures by Kolman and Busby and Sharon Ross Prentice Hall of India-2000, (Third Edition) TEXT BOOK III: Coding Theory by D. G. Hoffman, D. A. Lanonard, C. C. Lindroes ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS III-SEMESTER M301 FUNCTIONAL ANALYSIS (COMPULSORY) (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I Banach spaces: The definition and some examples, continuous linear transformation, The Hahn-Banach theorem, the natural imbedding of N in N**, The open mapping theorem. Sections 46-50, Chapter 9. UNIT-II The conjugate of an operator, Hilbert spaces: The definition and some simple properties, orthogonal complements, orthonormal sets. Section 51, Chapter 9 and Sections 52-54, Chapter 10. UNIT-III (Self) Adjoint, Normal, Unitary Operators: The conjugate space H*, the adjoint of an operator, Self-adjoint operators, Normal and Unitary operators, Projections. Sections 55-59, Chapter 10. UNIT-IV Finite-dimensional spectral theory: Matrices, determinants and the spectrum of an operator, the spectral theorem. A survey of the situation. Sections 60-63, Chapter 11. Text Book: Introduction to Topology and Modern Analysis by G. F. Simmons, McGraw Hill Book Company. Inc-International student edition. STREAM – A ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS III-SEMESTER M302(1)-NUMBER THEORY- I (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I:- ARITHMETICAL FUNCTIONS AND DIRICHLET MULTIPLICATION: Introduction- The Mobius function function (n) – The Euler totient function (n)- A relation connecting and - A product formula for (n)- The Dirichlet product of arithmetical functions- Dirichlet inverses and the Mobius inversion formula- The Mangoldt function (n)- multiplicative functions- multiplicative functions and Dirichlet multiplication- The inverse of a completely multiplicative function-Liouville’s function - The divisor functions . Chapter-2:- Articles 2.1 to 2.14 AVERAGES OF ARITHMETICAL FUNCTIONS: Introduction- The big oh notation. Asymptotic equality of functions- Euler’s summation formula- Some elementary asymptotic formulas-The average order of d(n)- The average order of the divisor functions - The average order of (n). The partial sums of a Dirichlet product- Applications to (n) and (n)- Another identity for the partial sums of a Dirichlet product. Chapter -3:- Articles 3.1 to 3.7 UNIT-II :- The partial sums of a Dirichlet product- Applications to (n) and (n)- Another identity for the partial sums of a Dirichlet product. SOME ELEMENTARY THEOREMS ON THE DISTRIBUTION OF PRIME NUMBERS: Introduction- Chebyshev’s functions and - Relations connecting and - Some equivalent forms of the prime number theorem-Inequalities for - Shapiro’s Tauberian theorem- Applications of Shapiro’s theorem- An asymptotic formula for the partial sums - The partial sums of the Mobius function – The partial sums of the Mobius function. Chapter -3:- Articles 3.10 &3.11 and Chapter-4:- Articles 4.1 to 4.9 UNIT-III :- CONGRUENCES: Definition and basic properties of congruences- Resudue classes and complete residue systems- Linear congruences- Reduced residue systems and the Euler- Fermat theorem- Polynomial congruences modulo p. Lagrange’s theorem- Applications of Lagrage’s theorem- Simultaneous linear congruences. The Chinese remainder theorem- Applications of the Chinese remainder theorem- Polynomial congruences with prime power moduli. Chapter -5:- Articles 5.1 to 5.9 UNIT-IV :- FINITE ABELIAN GROUPS AND THEIR CHARACTERS: Characters of finite abelian groups- The character group- The orthogonality relations- for characters- Dirichlet characters- Sums involving Dirichlet characters-The nonvanishing of L(1,) for real nonprincipal . DIRICHLET’S THEOREM ON PRIMES IN ARITHMETIC PROGRESSIONS: Introduction- Dirichlet’s theorem for primes of the form 4n-1 and 4n+1- The plan of the proof of Dirichlet’s theorem- Proof of Lemma 7.4- Proof of Lemma 7.5- Proof of Lemma 7.6- Proof of Lemma 7.7- Proof of Lemma 7.8- Distribution of primes in arithmetic progressions. Chapter 6:- Articles 6.5 to 6.10 and Chapter 7 :- 7.1 to 7.9 TEXT BOOK: Introduction to Analytic Number Theory- By T.M.APOSTOL-Springer Verlag-New York, Heidalberg-Berlin-1976. STREAM – A ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS III-SEMESTER M302(2) UNIVERSAL ALGEBRA-I (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I:- Lattices: Definitions of Lattices – Isomorphism’s of Lattices and Sub lattices- Distributive and Modular Lattices- Complete lattices- Equivalence relations- Algebraic lattices- Closure operators. (Sections 1, 2, 3,4,5, of Chapter-I of the prescribed text book) UNIT-II:- The Elements of Universal Algebra: Definition and examples of algebras- Isomorphic algebras and sub algebras – Algebraic lattices and sub universes – The irredundant Basis theorem. (Sections 1, 2, 3,4,, of Chapter-II of the prescribed text book) UNIT-III:- Congruences and Quotient algebras- Homomorphisms – The homomorphism and isomorphism theorems. (Sections 5, 6,, of Chapter-II of the prescribed text book) UNIT-IV:- Direct products- Factor congruences and Directly indecomposable algebras- Sub direct products- Subdirectly irreducible algebras and Simple algebras- Class operators- Varieties. (Sections 7, 8, 9,, of Chapter-II of the prescribed text book) Prescribed Book: A course in Universal algebra- Stanley Burris, H.P. Sankappanavar, Springer-Verlag, New York- Heidelberg- Berlin. STREAM – A ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS III-SEMESTER M302(3) FUZZY SET THEORY AND APPLICATIONS (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I: From Clasical(Crisp) sets to fuzzy sets:- Introduction-Crispsets: An overview-fuzzyset:Basic types-Fuzzy sets. Basic Concepts-Characteristics and significance of the paradign shift(CH-1 of (I)). Fuzzysets versus Crisp sets-Additional Properties of a cuts-Representations of Fuzzysets-Extension principle for Fuzzysets(CH-2 of (I)). UNIT-II: Operations on Fuzzysets - Types of Operations - Fuzzy Compliments - Fuzzy Inter sections: t-norms - Fuzzy unions; t-Conorms - Combinations of operations - Agreegation Operations(CH-3 of(I)). UNIT-III: Fuzzy Arithmetic -Fuzzy Numbers - Linguistic variables - Arithmetic operations on intervals - Arithmetic operations on Fuzzy numbers - Lattice of fuzzy numbers - Fuzzy equations(CH-4 of (I)). UNIT-IV: Fuzzy Relations - Crisp versus fuzzy relations - Projections and Cylindric Extensions - Binary Fuzzy Relations - Binary Relations and Singleset - Fuzzy Equivalence Relations - Binary Relations on a single set - Fuzzy Compatibility Relations - Fuzzy Ordering Relations - Fuzzy Morphisms - Sup - Compositions of Fuzzy Relations - Inf - Compositions of fuzzy Relations(CH-5 of (I)). TEXT BOOK: G.J.KLIR and BOYUAN, "Fuzzy sets and Fuzzy Logic, Theory and Applications" Prentice - Hall of India Pvt. Ltd., New Delhi., 2001. STREAM - B ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.Sc. MATHEMATICS-III SEMESTER M303(1) LATTICE THEORY-I (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I:- Partially Ordered sets- Diagrams- Special subsets of a poset –length- lower and upper bounds- the minimum and maximum condition- the Jordan Dedekind chain conditions – Dimention functions. ( Sections 1 to 9 of chapter I of the prescribed text book) UNIT-II:- Algebras-lattices- the lattice theoretic duality principle- semilattices- lattices as posets-diagrams of lattices- semi lattices, ideals-bound elements of Lattices-atoms and dual atoms-complements, relative complements, semi complements-irreducible and prime elements of a lattice- the homomorphism of a lattice-axioms systems of lattices. ( Sections 10 to 21 of chapter II of the prescribed text book) UNIT-III:- Complete lattices- complete sublattices of a complete lattice- conditionally complete lattices- lattices – compact elements, compactly generated lattices- subalgebra lattice of an algebra-closure operations- Galois connections, Dedekind cuts- partially ordered sets as topological spaces. (Sections 22 to 29 of chapter III of the prescribed text book) UNIT-IV:- Distributive lattices-infinitely distributive and completely distributive lattices-modular lattices-characterization of modular and distributive lattices by their sublattices- distributive sublattices of modular lattices- the isomorphism theorem of modular lattices, covering conditions-meet representations in modular and distributive lattices- some special subclasses of the class of modular lattices-preliminary theorems – modular lattices of locally finite length- the valuation of a lattice, metric and quasi metric lattices- complemented modular lattices. (Sections 30 to 40 of Chapters IV and V of the prescribed text book) Prescribed Text Book: Introduction to Lattice Theorey, by Gabor Szasz, Academic Press, New York. Book for reference: General Lattice Theory by G. Gratzer, Academic Press, New York. STREAM – B ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS III-SEMESTER M303(2) OPERATIONS RESEARCH (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I: Linear Programming: The Simplex Method – Overall Idea of the Simplex Method – Development of the Simples Method – Primal Simplex method – Dual Simplex Method – Special cases in Simplex Method Applications – Sensitivity Analysis. Sections 3.1 to 3.7 of the Chapter 3 in the Text Book UNIT-II: Revised Simplex Method and Duality: Mathematical Foundations – Revised (Primal) Simplex Method – Definition of the Dual Problem – Solution to the Dual Problem – Economic Interpretation of the Dual Problem. Sections 4.1 to 4.3 of the Chapter 4 and sections 5.1 to 5.4 of Chapter 5 in the Text Book. UNIT-III: Transportation Model and Net Works: Definitions and Applications of the Transportation – Solution of the Transportation Problem – The Assignment Model – The Transhipment Model – Network Definitions – Minimal Spanning Tree problem – Shortest – Route Problem. Sections 6.1 to 6.5 of the Chapter 6 and sections 8.1 to 8.3 of the Chapter 8 in the Text Book. UNIT-IV: Network Models: - Maximal Flow Problem – Minimum Cost Capacitated Flow Problem Decision Theory and Games: Decisions Under Uncertainty – Game Theory – Optimal solution of Two-Person Zero-sum Games, Mixed strategies. Sections 8.4 to 8.6 of the Chapter 8 and sections 12.3, 12.4.1, 12.4.2 of the Chapter 12 in the Text Book. Text Book: Operations Research, An Introduction: Hamdy A Taha, Maxwell Macmillan International Edition, New York, 1992. STREAM – B ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS III-SEMESTER M303(3) MATHEMATICAL BIOLOGY (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I: Autonomous differential equations - Equilibrium solutions - Stability nature of equilibrium solutions, single species growth models involving exponential, logistic and Gompertz growths. Harvest models – bifurcations and break points. (Sections 1 and 2 of the Text Book) UNIT-II: Lotka Volterra predator – prey model – phase plane analysis, General predator prey systems – equilibrium solutions – classification of equilibria – existence of cycles – Bendixson-Dulac’s negative criterion – functional responses. (Sections 7 and 8 of the text book) UNIT-III: Global bifurcations in predator prey models – Freedman and Wolkowicz model - type IV functional response – Hopf bifurcation – Homoclinic orbits – Global bifurcations using Allee effect in prey – Competition models – Lotka – Voltrrra Competition model – exploitation competition models. (Sections 9 and 12 of the text book) UNIT-IV: Mutualism models – various types of mutualisms – cooperative systems – Harvest models and optimal control theory – open access fishery – sole owner fishery – Pontryagin’s maximum principle – Economic interpretation of Hamiltonian and adjoint variable. (Sections 13 and 14 of the text book) Text book: Mark Kot, 2001, Elements of Mathematical Ecology, Cambridge University Press. Reference: Nisbet and Gurney, 1982, Modelling Fluctuating Populations, John Wiley & Sons. STREAM – C ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS III-SEMESTER M304(1) COMMUTATIVE ALGEBRA- I (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I Rings and ring homomorphism, ideals, quotient rings, zero divisors, Nilpotent elements, units, prime ideals and Maximal ideals, nil radical and Jacobson radical, operations on ideals, Extensions and contractions. UNIT-II Modules and module homomorphisms, Sub modules and quotient modules, operations on submodules, Direct sum and product, finitely generated modules, exact sequences, Tensor product of modules, Restriction and extension of scalars, Exactness properties of the tensor product, algebras, tensor product of algebras. UNIT-III Local Properties, Extended and contracted ideals in rings of fractions. UNIT-IV Primary decompositions Content and extent of chapters 1 to 4 of the prescribed text book. Prescribed text book: Introduction to commutative algebra, By M.F. ATIYAH and I.G. MACDONALD, Addison-Wesley publishing Company, London. STREAM – C ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS III-SEMESTER M304(2) SEMI GROUPS- I (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I Basic definition, monogenic semigroups, ordered sets, semilatttices and lattices, binary relations, equivalences and congruences. UNIT-II Free semigroups, Ideals and Rees’ congruences, Lattices of equivalences and congruences. Green’s equivalences, the structure of D-classes, regular semigroups. UNIT-III Simple and 0-simple semigroups, Principal factors, Rees’ theorem, Primitive idempotents. UNIT-IV Congruences on completely 0-simple semi groups, The lattice of congruences on a completely 0-simple semigroup, Finite congruence free semigroups. Contents of the syllabus-Chapters 1,2 and 3 of the text book. Text Book: An introduction to semi group theory by J.M. Howie, 1976, Academic press, New York. STREAM – C ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS III-SEMESTER M304(3) ADVANCED GRAPH THEORY (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I Matching and Augmenting paths; The marriage problem; The personnel assignment problem; The optimal Assignment problem. (Chapter 4 of the Text Book) UNIT-II Plane and planar graphs; Euler’s formula, The platonic bodies, kurtowski’s Theorem, Non-Hamiltonian plane graphs, The dual of a plane graph. (Chapter 5 of the Text Book) UNIT-III Vertex colouring; vertex colouring Algorithms; Critical graphs; cliques; Edge colouring; Map colouring. (Chapter 6 of the Text Book) UNIT-IV Definitions; Indegree and outdegree; Tournaments; Traffic flow.(Chapter 7 of the Text Book) Text Book: A first look at GRAPH THEORY; John Clark & Derek Allan Holton, Allied Publishers Limited 1995. ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS III-SEMESTER M305 LINEAR ALGEBRA AND NUMBER THEORY (Non Core/CBCS/For A.U. Students ONLY) M305 COMPLEX ANALYSIS II (For Affiliated Colleges ONLY) (Restructured/w.e.f. 2014-15 admitted batch) M305 LINEAR ALGEBRA AND NUMBER THEORY (Non Core/CBCS/For A.U. Students ONLY) UNIT-I: Vector Spaces, Subspaces, Bases and dimension, Coordinates Articles 2.1 to 2.4 of CHAPTER 1 OF TEXT BOOK I UNIT-II: Linear Transformations, The Algebra of Linear Transformations, Isomorphim, Representation of Linear Transformation by Matrices. Articles 3.1 to 3.4 UNIT-III: Divisibility, Greatest Common Divisor, Prime numbers, The Fundamental Theorem of Arithmetic, The Euclidean Algorithm, Chapter 1 of Text book II UNIT-IV: Definition and Basic Properties of Congruences, Residue classes and Complete Residue Systems, Linear Congruences, Reduced Residue Systems, and the Euler Fermat Theorem Chapter 5 0f Text book II TEXT BOOK I: LINEAR ALGEBRA by KENNETH HOFFMAN , RAY KUNZ (Second Edition) Prentice Hall of India. TEXT BOOK II: Introduction to Anaylitic Number Theory by TOM M APOSTOL , Springer Verlag, New York M305 COMPLEX ANALYSIS II (For Affiliated Colleges ONLY) (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I The maximum modulus theorem: The maximum principle-Schwarz’s lemma- Convex functions and Hadamard’s three circles theorem- Phragmen- Lindelof theorem. (Sections 1,2,3,4 of Chapter-VI of the prescribed text book) UNIT-II Compactness and convergence in the Spaces of Analytic Functions: The space of continuous functions C (G, Ω) - Spaces of Analytic functions- Spaces of meromorhic functions- The Riemann Mapping Theorem- Weierstrass Factorization theorem- Factorization of sine functions. (Sections 1, 2, 3,4,5,6 of Chapter-VII of the prescribed text book) UNIT-III Runge’s Theorem: Runge’s Theorem-Simple connectedness- Mittag-Leffler’s Theorem, Analytic Continuation and Riemann Surfaces, Schwarz Reflection Principle- Analytic Continuation Along A Path- Mondromy Theorem. (Sections 1, 2, 3 of Chapter- VIII, Sections 1, 2, 3 of Chapter-IX of the prescribed text book) UNIT-IV Harmonic Functions: Basic properties of Harmonic functions- Harmonic functions on a disk. Jenson’s formula, The genus and the order of an entire function Hadamard’s factorization theorem. (Sections 1, 2, of Chapter- X and Sections 1, 2, 3 of Chapter- XI of the prescribed text book) Prescribed text book: Functions of one complex variables by J. B. Conway: Second edition, Springer International Student Edition. Narosa Publishing House, NEW DELHI. ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS IV-SEMESTER M401 MEASURE and INTEGRATION (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I Lebesgue measure: Introduction, Outer measure, measurable sets and Lebesgue measure, A nonmeasurable set, measurable functions, Littlewood’s three principles. Chapter 3 of the text book UNIT-II The Lebesgue Integral: The Riemann integral, The Lebesgue integral of a bounded function over a set of finite measure, the integral of nonnegative function, the general Lebesgue integral, convergences in measure. Chapter 4 of the text book UNIT-III Differentiation and integration: Differentiation of monotone functions, Functions of bounded variation and differentiation of an integral, Absolute continuity, and convex functions. Chapter 5 of the text book UNIT-IV The classical Banach spaces: The Lp-spaces, The Minkoswki and Holder inequalities, convergence and completeness, approximation in Lp , Bounded linear functionals on the Lp spaces. Chapter 6 of the text book TEXT BOOK: Real Analysis by H. L. Royden, Macmillan Publishing Co. Inc. 3rd Edition, New York, 1988. STREAM – A ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS IV-SEMESTER M402(1) NUMBER THEORY- II (PRE-REQUISITE NUMBER THEORY – I) (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I:- PERIODIC ARITHMETIAL FUNCTIONS AND GAUSS SUMS: Functions periodic modulo k- Existence of finite Fourier series for periodic arithmetical functions- Ramanujan’s sum and generalizations- Multiplicative properties of the sums - Gauss sums associated with Dirichlet characters-Dirichlet characters with nonvanishing Gauss sums. QUADRATIC RESIDUES AND THE QUADRATIC RECIPROCITY LAW: Quadratic residues- Legendre’s symbol and its properties- Evaluation of (-1/p) and (2/p)- Gauss Lemma-The quadratic reciprocity law-Applications of the reciprocity law- The Jacobi symbol-Applications to Diophantine equations- Gauss sums and the quadratic reciprocity law. Chapter 8:- Articles 8.1 to 8.6 and Chapter 9:- Articles 9.1 to 9.9 UNIT-II:- PRIMITIVE ROOTS: The exponent of a number mod m. Primitive roots- Primitive roots and reduced residue systems-The nonexistence of primitive roots mod for - The existence of primitive roots and p for odd primes p. Primitive roots and quadratic residues- The existence of primitive roots mod - The existence of primitive roots mod 2- The non existence of primitive roots in the remaining cases- The number of primitive roots mod m. The index calculus- Primitive roots and Dirichlet characters-Real-valued Dirichlet characters mod -Primitive Dirichlet characters mod . UNIT-III :- DIRICHLET SERIES AND EULER PRODUCTS: Chapter- 11:- Articles 11.1 to 11.7. The half- plane of absolute convergence of a Dirichlet series, The function defined by Dirichlet series, Multiplication of Dirichlet series, Euler Products, The half-plane of convergence of a Dirichlet series, Analytic properties of Dirichlet series, Dirichlet series with non negative coefficients. UNIT-IV :- Chapter- 12:- Articles 12.1 to 12.8. Properties of the gamma function, Integral representation for the Hurwitz zeta function, A contour integral representation for the Hurwitz zeta function, The analytic continuation of the Hurwitz zeta function, Analytic continuation of (s), L(s, ) , Hurwitz’s formula for (s, a), The functional equation for Riemann zeta function. TEXT BOOK: Introduction to Analytic Number Theory- By T.M.APOSTOL- Springer Verlag-New York, Heidalberg-Berlin-1976. STREAM – A ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS IV-SEMESTER M402(2) UNIVERSAL ALGEBRA-II (PRE-REQUISITE: UNIVERSAL ALGERBRA-I) (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I: Terms, Term Algebras and Free algebras- Identities , Free algebras and Birkhoff’s theorem- Mal’cev conditions- The Centre of an algebra. (Sections 10, 11, 12, 13, of Chapter-II of the prescribed text book) UNIT-II: Boolean Algebras- Boolean rings – Filters and ideals- Stone identity. (Sections 1, 2, 3, 4, of Chapter-IV of the prescribed text book) UNIT-III: Boolean Powers- Ultra products and congruence- Distributive varieties- Primal algebras- Boolean Products. (Sections 5, 6, 7, 8, of Chapter-IV of the prescribed text book) UNIT-IV: Discriminator varieties – Quasi primal algebras – Functionally complete algebras – skew-free algebras- Semi simple varieties – Directly represent able varieties. (Sections 9, 10, 11, 12, 13, of Chapter-IV of the prescribed text book Prescribed Book: A course in Universal algebra- Stanley Burris, H.P. Sankappanavar, Springer-Verlag, New York- Heidelberg- Berlin. STREAM – A ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS IV-SEMESTER M402(3) OPERATOR THEORY (Restructured/W.E.F. from 2014-15 admitted batch) UNIT-I Banach fixed point theorem- application of Banach’s theorem to linear equations-application of Banach’s theorem to differential equations-application of Banach’s theorem to integral equations. Chapter 5 of the text book UNIT-II Approximation in normed spaces-Uniqueness, strict convexity-uniform approximation-Chebyshev polynomials – Splines. Sections 6.1 to 6.4 and 6.6 of Chapter 6 of the text book. UNIT-III Spectral theory in finite dimensional Normed spaces-basic concepts-spectral properties of bounded linear operators-further properties of Resovent and spectrum-use of complex analysis in spectral theory. Sections 7.1 to 7.5 of Chapter 7 of the text book UNIT-IV Compact linear operator of normed spaces-Further properties of compact linear operators-Spectral properties of compact linear operators on normed spaces-further spectral properties of compact linear operators. Sections 8.1 to 8.4 of Chapter 8 of the text book. Text book: Introductory Functional Analysis and Applications by Kreyszig, John Wiley and Sons, Delhi, 2001. STREAM B ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS IV-SEMESTER M403(1) LATTICE THEORY-II (PRE-REQUISITE: LATTICE THEORY-I) (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I Boolean algebras, De Morgan formulae- Complete Boolean algebras- Boolean algebras and Boolean rings- The algebra of relations- The lattice of propositions- Valuations of Boolean algebras. (Sections 42 to 47 of chapters VI of the prescribed text book) UNIT-II Birkhoff lattices- Semimodular lattices- Equivalence lattices- Linear dependence- Complemented semimodular lattices. (Sections 48 to 52 of chapters VII of the prescribed text book) UNIT-III Ideals and dual ideals, Ideal chains- Ideal lattices- Distributive lattices and rings of sets. (Sections 53 to 55 of chapters VIII of the prescribed text book) UNIT-IV Congruence relation of an algebra- Permutable equivalence relations- The Schreier refinement theorem in arbitrary algebras- Congruence relations of lattices- Minimal congruence relations of some subsets of a distributive lattice- The connection between ideals and congruence relations of a lattice. (Sections 56 to 61 of chapters IX of the prescribed text book) Prescribed text book: Introduction to Lattice Theory by Gabor Szasz, Academic Press, New York. Books for reference: General Lattice Theory by G. Gratzer, Academic Press, New York. STREAM – B ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS IV-SEMESTER M403(2) Formal Languages and Automata Theory (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I: Sets, relations, strings, alphabets, languages, inductive proofs, finite state systems, nondeterministic finite automata, finite automata e-moves, regular expressions, 2 way finite automata, finite automata with output, applications of finite automata (Chaps. 1 and 2, pgs. 1-54.). UNIT-II: The pumping lemma for regular sets. Closure properties, decision algorithms, The Myhill-Nerode theorem, minimization of finite automata. (Chap. 3, pgs. 55-76.) UNIT-III: Context free grammars, derivation trees, simplification of CF grammars, Chomsky normal form, Greibach normal form, existence of inherently ambiguous CF languages. (Chaps 4, pgs. 77-106) UNIT-IIII: Push down automata, the pumping lemma, closure properties, decision algorithms for CFL’s. (Chaps. 5 and 6, pgs. 107-145.) Content and scope as in Hopcroft and Ullman, Chaps. 1-6, Pages 1-145. Prescribed text book: Hopcroft J. and Ullman J.D., Introduction to Automata Theory, Languages and Computation. STREAM – B ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS IV-SEMESTER M403(3) BANACH ALGEBRAS (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I: General preliminaries on Banach Algebras – The definition and examples – Regular and singular elements – Topological divisors of Zero – The spectrum – The formula for the spectral radius – The radical and the semi – simplicity. UNIT-II: The structure of commutative Banach Algebras - The Gelfand mapping - Applications of the formula r(x) = lim // xn // 1/n – Involutions in Banach Algebras – The Gelfand – Neumark theorem. UNIT-III: Some special commutative Banach Algebras - Ideals in C(x) and the Banach – Stone theorem - The stone – Chech compactification – commutaticve C* - algebras. UNIT-IV: Fixed point theorems and some applications to analysis – Brouwer’s and Schauder’s fixed point theorems (without proofs) Picard’s theorem – Continuous curves – The Hahn – Mazurkiewicz theorem (without proof). Boolean rings – The stone representation theorem. Text Book: Introduction to Topology and Modern Analysis – By G.F. Simmons – International Student edition – McGraw – Hill Kogakusha Ltd. STREAM – C ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS IV-SEMESTER M404(1) COMMUTATIVE ALGEBRA-II (PRE-REQUISITE: COMMUTATIVE ALGERBRA-I) (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I: Integral dependence, the going-up theorem-Integrally closed integral domains, the going-down theorem, valuation rings. UNIT-II: Chain Conditions UNIT-III: Noetherian rings- Primary decomposition of Noetherian rings, Artin rings UNIT-IV: Discrete valuation rings, Dedekind domains, Fractional ideals. Content and extent of Chapters 5 to 9 of the prescribed text book. Prescribed Text Book : Introduction to commutative algebra by M.F.Atiya and I.G. Macdonald, Addison-Wesley Publishing Company, London. STREAM – C ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS IV-SEMESTER M404(2) SEMI GROUPS-II (PRE-REQUISITE SEMIGROUPS – I) (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I: Union of Groups, Semi lattices of groups, bands, free bands, varieties of bands. UNIT-II: Introduction to inverse semi groups, preliminaries, the natural partial order on an inverse semi group, fundamental inverse semi groups, anti-uniform semilattices. UNIT-III: Bi-simple inverse semi groups, simple inverse semi-groups, representation of inverse semigroups. UNIT-IV: Orthodox semigroups, basic properties, the analogue of the Munn semi-group, uniform and anti-uniform bands, the structure of orthodox semi groups. Text Book: An introduction to semigroup theory by J.M.Howie, 1976, Academic press, New York. STREAM – C ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS IV-SEMESTER M404(3) NONLINEAR FUNCTIONAL ANALYSIS (Restructured/w.e.f. 2014-15 admitted batch) UNIT-I:- Various forms of continuity- Geometry in normed spaces and duality mapping, Nemytskii, Hammerstein and Urysohn operators. Chapter 1 of the textbook UNIT-II:- Gateaux and Frechet derivative, Properties of derivative, Taylor’s theorem, Inverse function theorem and Implicit function theorem, Sub differential of convex functions. Chapater 2 of the text book UNIT-III:- Banach’s contraction principle and its genrerliation, Nonexpansive mappings, Fixed point theorems of Brouwer and Schauder. Secions 4.1 to 4.3 of Chapter 4 of the text book. UNIT-IV:- Fixed point theorems for multifunctions, common fixed point theorems, Sequences of contractions, generalized contractions and fixed points. Sections 4.4 to 4.6 of Chapter 4 of the textbook. Prescribed Book: Some topics in Nonlinear functional analysis by Mohan C. Joshi and Ramendra k. Bose, Wiley Eastern limited- Hyderabad, 1985. ANDHRA UNIVERSITY DEPARTMENT OF MATHEMATICS M.A/M.SC MATHEMATICS IV-SEMESTER M405 PARTIAL DIFFERENTIAL EQUATIONS (COMPULSORY) UNIT-I: First Order P.D.E. : Curves and Surfaces – Genesis of First Order P.D.E. – Classification of Integrals – Linear equations of the First Order - Pfaffian Differential Equations – Compatible Systems – Charpit’s Method Jacobi’s Method - Integral Surfaces Through a Given Curve. Chapter 1 – sections 1.1 – 1.9 UNIT-II: Second Order P.D.E. : Genesis of Second Order P.D.E. – Classification of Second Order P.D.E. – One Dimensional Wave equation: Vibrations of an Infinite String – Vibrations of Semi infinite String Vibrations of a String of Finite Length – Riemann’s Method – Vibrations of a String of Finite Length( Method of Seperation of Variables). Chapter 2 – section 2.1 – 2.3. UNIT-III: Laplace’s Equation : Boundary value Problems- Maximum and Minimum Principles- The Cauchy Problem – The Dirichlet Problem for the Upper Half Plane – The Neumann Problem for the Upper Half Plane – Dirichlet Problem for a Circle – The Dirichlet Exterior Problem for a Circle- The Neumann Problem for a Circle – The Dirichlet Problem for a Rectangle- Harnack’s Theorem Laplace’s Equation – Green ‘s Function – The Dirichlet Problem for a Half Plane – The Dirichlet problem for a Circle. Chapter 2 – section 2.4 UNIT-IV: Heat Conduction Problem : Heat Conduction – Infinite Rod Case - Heat Conduction –Finite Rod Case- Duhamel’s Principle –Wave Equation –Heat Conduction Equation – Quasi Linear Equations – Non Linear First Order P.D.E. Chapter 2 – sections 2.5, 2.6 and Chapter 1 –sections 1.10 , 1.11. Text book: T. Amarnath, An Elementary Course in Partial differential equations, Second Edition, Narosa Publishing House . ANNEXURE-II Department of Mathematics Andhra University Pre – Ph. D. / M. Phil. Examinations Paper I – TOPOLOGY Zorn’s Lemma – Zermelo’s Theorem – Ordinals – Comparability of Ordinals – Transfinite Induction and Construction – Ordinal numbers – Cardinal Arithemetic – The Ordinal Number. (Chapter - 2 of the Prescribed Book) Uniformities and Uniform Topology – Uniform Continuity – Product Uniformities – Metrization – Completeness – Completion – Compact spaces – Baire theorem – Localization – of category Uniformly open map. (Chapter – 6 of the Prescribed Book) Function Spaces – Point wise Convergence – Compact open topology and joint continuity – Uniform convergence – Uniform convergence on compacta – Compactness and equi-continuity – Ascoli’s Theorem. (Chapter 7 of the Prescribed Book ) Topological Groups – Neighborhoods of a point in a topological group – Isomorphism and local isomorphisms – Sub groups – Quotient groups – Homomorphisms – Homogenerous spaces – Product groups – Uniform structures on groups. ( Chapter 31 &33 of the Prescribed Book). Prescribed Books: Topology : James Dugundji ,Universal Book Stall, New Delhi. General Topology : John L Kelly, D. Van Nostrand Company, Inc. Princeton, New Jereey . Afflinted East –West Press Pvt. Ltd. General Topology: Bourbaki, Addison – Wesley Publishing Company, London. Department of Mathematics Andhra University Pre – Ph. D. / M. Phil. Examinations Paper II – Boolean Algebras Boolean rings – Boolean algebras – Fields of sets – Regular open Sets – Elementary relations – Order – Infinite operations – Subalgebras – Homomorphisms – Free Algebras – Ideals and filters – The Homomorphism theorem – Boolean -algebras – The countable chain condition – Measure algebras – Atoms – Boolean spaces – The representation theorem – Duality for ideals – Duality for homomorphisms. Text Book: Lectures on Boolean Algebras, by Paul R. Halmos, D. Van Nostrand Company, Inc. Princeton, New Jersey. Department of Mathematics Andhra University Pre – Ph. D. / M. Phil. Examinations Paper II – Fuzzy Algebra Union of two fuzzy sub groups – fuzzy sub group generated by a fuzzy subset – fuzzy normal sub groups fuzzy conjugate subgroups and fuzzy characteristic subgroups – fuzzy syllow subgroups. (Chapter 2 of prescribed book) Some elementary properties of fuzzy ideals – union of fuzzy subrings (fuzzy ideals) fuzzy subring (fuzzy ideal) generated by a fuzzy subset – fuzzy ideals and homomorhisms – fuzzy cosets (Chapter 2 of prescribed book) Fuzzy prime ideals fuzzy maximal ideals – fuzzy semiprime ideals – characterization of regularity (Chapter 4 of prescribed book) Fuzzy primary ideals – fuzzy semiprimary ideals definition and some properties – fuzzy ideals and fuzzy irreducible ideals in Noetherian ring. (Chapter 5 & 6 of prescribed book) Prescribed Text Book: Fuzzy algebra by Rajesh Kumar, University Press University of Delhi, Delhi – 110007. Department of Mathematics Andhra University Pre – Ph. D. / M. Phil. Examinations Paper II - MATHEMATICAL MODELLING Single species models: Exponential, logistic and Gompertz growth Harvest models: bifurcations and break points Interacting populations: A classical predator – prey model To cycle or not to cycle Global bifurcations in predator-prey models Chemostat models Competition models Mutualism models (A.1-2, B.7-10, B.12-13 (excluding discrete dynamic models) of the Text book) Text book: Mark Kot, 2001, Elements of Mathematical Ecology, Cambridge University press. References: Nisbet and Gurney, 1982, Modelling Fluctuating Populations, John Wiley & Sons. Department of Mathematics Andhra University Pre – Ph. D. / M. Phil. Examinations Paper II : Nonlinear Functional Analysis Results based on completeness: Banach Contraction Principle – Elementary domain Invariance - Continuation Methods for Contractive Maps – Nonlinear alternative for contractive Maps – Extensions of the Banach Theorem. Fixed point theorems in complete metric spaces. Extensions of the Banach theorem. (Sections 1 to 5, 6A and 6B of I, § 1 of Text Book). Order- theoretic results: The Knaster-Tarski Theorem – Order and completeness, Theorem of Bishop – Phelps – Fixed Points for set-valued contractive maps – Applications to Geometry of Banach spaces – Applications to the theory of Critical Points. Fixed Points of Partially ordered sets. (Sections 1 to 5 and 6A of I, § 2 of Text Book). Results based on convexity: KKM- Maps and the Geometric KKM- Principle – Theorem of Non Neumann and systems of inequalities – Fixed points of Affine Maps, Markoff – Kakutani theorem – Fixed points for Families of maps, Theorem of Kakutani. (Sections 1 to 4 of I, § 3 of Text Book). Further results and applications of Fixed Point theorems : Nonexpansive Maps in Hilbert space – Applications of the Banach Principle to Integral and Differential equations – Applications of the Elementary Domain Invariance – Elementary KKM – Principle and its applications – Theorems of Mazur – Orlicz and Hahn – Banach. (Sections 1 to 5 of I, § 4 of Text Book). TEXT BOOK: Andrzej Granas and James Dugundji : Fixed point theory, Springer, 2003. Department of Mathematics Andhra University Pre – Ph. D. / M. Phil. Examinations Paper – II Numerical Analysis Polynomial Interpolation – Function approximation, Polynomial interpolation, Finite differences, Newton’s interpolation formulae, Least square curve fitting, Spline interpolation. (Chapter 3, Section 3.1 – 3.4 and Chapter 7 Section 7.2 – 7.4 of the textbook) Numerical solution of ordinary differential equations: Initial value problem: an example – existence and uniqueness of solutions, single and multi step methods, stability study, first order systems, numerical solution of non-linear systems – the problem, Picard iteration, Newton’s method. (Chapter 5 Section 5.1 – 5.5 and Chapter 8.1 – 8.3 of the textbook) Finite difference methods of Partial differential equations: Parabolic type – Introduction, Characteristic and types of second order equations, well posed problems, some basic properties of linear and quasi-linear equations, system of first order equations, finite difference discretization, parabolic type equations, Crank – Nicholson implicit scheme. (Chapter 9 Section 9.1 – 9.9 of textbook) Equation of hyperbolic and elliptic type – hyperbolic type equations, FTCS and other explicit scheme, Lax Wendorf scheme. (Chapter 10 Section 10.1 – 10.3 of the textbook) TEXT BOOK: Numerical Analysis and Algorithms, Pradip Niyogi, Tata McGraw Hill Publishing Company Limited, 2003. Department of Mathematics Andhra University Pre-Ph.D. / M.Phil. Examinations Paper-II – Number Theory and Cryptography Some topics in Elementary Number Theory……………. Time estimates for doing arithmetic – Divisibility and Euclidean algorithm – Congruences – Some applications for factoring. (1 to 4 sections of Chapter-I of prescribed book) Cryptography…………. Some simple Cryptosystems – Enciphering Matrices. (1 to 2 sections of Chapter-III of prescribed book) Pubic Key …………….. The idea of public key Cryptography – R.S.A. – Discrete log. (1, 2, 3 sections of Chapter IV of prescribed book) Elliptic Curves …………….. Basic facts – Elliptic Curve cryptosystems (1, 2 sections of Chapter VI of prescribed book) Prescribed Text Book: A Course in Number Theory and Cryptography Author – Neal Koblitz. Ref: Introduction to Cryptography Author: Johannes A. Buchmann Department of Mathematics Andhra University Pre – Ph. D. / M. Phil. Examinations Paper III – UNIVERSAL ALGEBRA (Reading Course) Definitions of Lattices – Isomorphic lattices and Sublattices – Distributive and Modular Lattices – Complete Lattices, Equivalences and Algebraic Lattices – Closure Operators. Definitions and Examples of Algebras – Isomorphic Algebras and Subalgebras – Algebraic Lattices and Subuniverses – The Irredundant basis theorem – Congruences and Quotient Algebras – Homomorphisms and the Homomorphism and Isomorphism theorems – Direct Products, Factor congruneces and Directly indecomposable algebras – Subdirect products, subdirectly irreducible algebras and simple algebras – Class operators and Varieties – Terms, Term algebras and Free algebras – Identities, Free algebras and Birkhoff Theorem – Malcev Conditions – The center of an algebra. Boolean algebras – Boolean rings – Filters and Ideals – Stone Duality - Boolean Powers – Ultra products and congruences – Distributive varieties – Primal algebras – Boolean products – Discriminator varieties – Quasiprimal algebras Functionally complete algebras and Skew free varieties. Prescribed book: A course in Universal Algebra by Stanley Burris and H.P. Sankappanavar, Springer Verlag Publications. Department of Mathematics Andhra University Pre – Ph. D. / M. Phil. Examinations Paper III – LATTICE THEORY (Reading Course) UNIT– I: Two Definitions of Lattices, How to Describe Lattices, Some Algebraic Concepts, Polynomials, Identities and Inequalities (Sections 1, 2, 3 & 4 of Chapter – I of Prescribed Text Book ) UNIT– II: Free Lattices, Special Elements, Characterization Theorems and Representation Theorems, Congruence Relations (Sections 5.6 of Chapter – I & Sections 1, 2, 3 of Chapter – II of Prescribed Text Book) UNIT– III: Boolean Algebras R- generated by Distributive Lattices, Topological Representation, Distributive Lattices with Pseudo complementation (Sections 4.5 & 6 of Chapter – II of Prescribed Text Book) UNIT–IV: Weak protectivity and Congruences, Distributive, Standard and Neutral elements, Distributive, Standard and Neutral Ideals, Structure theorems (Sections 1,2,3 & 4 of Chapter – III of Prescribed Text Book) Text Book: General Lattice Thoery by George Gratzer, Academic Press, New York, 1978. Department of Mathematics Andhra University Pre – Ph. D. / M. Phil. Examinations Paper III – Semi Groups (Reading Course) Preliminaries: Introduction, Definition of a Semigroup, Special Subsets of Semigroups, Special elements of a semigroup, Relations and Functions on a Semigroup, Examples (Chapter 1 of Textbook) Semilattice Decompositions: Subdirect Products, Completely prime ideals and filters, Completely Semiprime ideals and -subsets, Semilattices of simple Semigroups, Weakly commutative Semigroups, Separative semigroups, -Semigroups (Chapter 2 of Textbook) Ideal Extensions: Extensions and Translations, Extensions of a weakly reductive Semigroup, Strict and pure extensions, Retract extensions, Dense extensions, Extensions of an arbitrary Semigroup, Semillatice composistions. (Chapter 3 of Textbook) Completely Regular Semigroups: Generalities, Completely simple Semigroups, Semillatices of rectangular groups, Strong Semilattices of a completely simple Semigroups, Subdirect Products of a Semilattice and a completely simple Semigroup. (Chapter 4 of Textbook) TEXT BOOK: Introduction to Semigroups, Mario Petrich, Charles E. Merrill Publishing Company, A Bell & Howell Company, Columbus, Ohio. Department of Mathematics Andhra University Pre – Ph. D. / M. Phil. Examinations Paper III – Mathematical Aspects of renewable resource management (Reading Course) Dynamics of exploited populations: Harvest models and optimal control theory. (Section C-14 of Textbook 1) Harvesting in two species models: Harvesting of species in competition. Harvesting of predator prey systems. Intermittent harvesting of Predator-prey systems Economic aspects of harvesting Optimization of harvesting returns Justification of the optimization result A nonlinear optimization problem Economic interpretation of the maximum principle. (Sections 6.1-6.8 of Textbook 2) Text Books: Mark Kot, 2001,Elements of Mathematical Ecology, Cambridge University press. Fred Brauer and Carlos Castilo-Chavez, Mathematical Models in Population Biology and Epidemiology, Texts in Applied Mathematics 40, Springer Verlag, 2001. References: 1. C. W. Clark, 1976, Mathematical Bioeconomics: The optimal management of Renewable resources, John Wiley & Sons 2. Bean-San Goh, 1980, Management and Analysis of Biological Populations, Elsevier Scientific Publishing Company, New York. ________________________________________________________________________________ Department of Mathematics Andhra University Pre – Ph. D. / M. Phil. Examinations Paper III - Iterative Approximation of Fixed Points (Reading Course) The Picard iteration: Banach’s fixed point theorem, Theorem of Nemytzki-Edelstein, Quasi-nonexpansive operators, Maia’s theorem, φ-contractions, Generalized φ-contractions. (Chapter 2 of Text Book.) The Krasnoselskij iteration: Nonexpansive operators in Hilbert spaces, strictly pseudocontractive operators, Lipschitzian and generalized pseudocontractive operators, Pseudo φ-contractive operators. (Chapter 3 of Text Book.) The Mann iteration: The general Mann iteration, nonexpansive and Quasi-nonexpansive Operators, strongly pseudocontractive operators, Quasi-contractive type operators. (Chapter 4 of Text Book.) The Ishikawa iteration: Lipschitzian and pseudo-contractive operators in Hilbert spaces, strongly pseudo-contractive operators in Banach spaces, nonexpansive operators in Banach spaces satisfying Opial condition, Quasi-contractive type operators. (Chapter 5 of Text Book) Other fixed point iteration procedures-Mann and Ishikawa iterations with errors, modified Mann and Ishikawa iterations, Ergodic fixed point iteration procedures. (Chapter 6 of Text Book) Text Book: V. Berinde – Iterative approximation of fixed points, Efemeride, 2002. Department of Mathematics Andhra University Pre – Ph. D. / M. Phil. Examinations Paper III - ADVANCED GRAPH THEORY (Reading Course) Matchings: Matching, Matchings and Colorings in Bipartitegraphis, Perfect Matchings, The Personnel Assignment Problem, The optimal Assignment Problem. Edge Colouring: Edge chromatic Number, Vizing’s Theorem, The time tabling Problem. Independent Sets and Cliques: Independent sets, Ramsay’s Theorem, Turan’s Theorem, Schur’s Theorem, A Geometry Problem Vertex Colourings: Chromatic Number, Brook’s Theorem, Hajo’s Conjecture, Chromatic Polynomials, Girth and Chromatic Number, A strange Problem Planar Graphs: Plane and Planar graphs, Dual graphs, Euler’s formula, Bridges, Kuratowski’s Theorem, The five colour Theorem, Nonhamiltonian Planar graphs, A Planarity Algorithm. Text Book and Content: Graph Theory with applications, J. A. Bondy and U. S. R. Murthy, 1976 edition, Macmillan press Ltd. Chapters: 5, 6, 7, 8, 9 (except the four colour conjecture In 9.6 as the conjecture is settled recently) Department of Mathematics Andhra University Pre – Ph. D. / M. Phil. Examinations Paper III - Decision Theory (Reading Course) Review of basic probability, laws of probability, Random variables and probability distribution, Expectation of a random variable, common probability distributions, Empirical distributions, Decision analysis: Decision making under certainty – Analytic Hierarchy Process (AHP), Decision making under risk, Decision under uncertainty (Chapter 12 Sections 12.1 – 12.5, Chapter 3 Sections 13.1 – 13.3 of the Textbook) Queueing models, elements of a queueing model, Role of exponential distributions, pure birth and death models, generalized Poisson queueing model, specialized Poisson queues. (Chapter 15 Sections 15.1 – 15.6 of the Textbook) (M/G/1): (GD/∞/∞) model – Pollaczek – Khintchine (P–K) formula, other queueing models, Queueing Decision models, Simulation modeling, Monte-Carlo Simulation, types of simulation, Generalization of random numbers, Mechanics of discrete simulation, Simulation languages. (Chapter 15 Sections 15.7 – 15.9, Chapter 16 Sections 16,1 – 16.7 of the Textbook) Markov chains, Definition of Markov chains, absolute and n-step transition probabilities, Classification of states in a M.C., Steady state probability and mean return times of Ergodic chains, First passage time, Analysis of absorbing states. (Chapter 17 Sections 17.1 – 17.6 of the Textbook) TEXT BOOK: Operations Research, An Introduction by Hamdy A. Taha, Eight Edition, Prentice Hall of India, New Delhi, 2006. Reference Book: Fundamentals of Queueing Theory by D. Gross and C.M. Harris, Third Edition, John-Wiely & Sons, Inc., 1998. Department of Mathematics Andhra University Pre – Ph. D. / M. Phil. Examinations Paper III - SELECTED TOPICS IN NUMBER THEORY AND CRYPTOGRAPHY (Reading Course) Finite Fields and Quadratic Residues……………. Finite fields – Quadratic residues and reciprocity. (1, 2 sections of Chapter-II of prescribed book I) Primality and Factoring…………. Pseudoprimes – The rho method – Fermat factorization and factor bases. (1 to 3 sections of Chapter-V of prescribed book I) Discrete Logarithms …………….. The DL Problem – Enumeration – Shanks Baby-Step Giant-Step Algorithm – The Pollard ρ-Algorithm – The Pohlig-Hellman Algorithm – Index calculus – Other Algorithms – Generalization of the Index Calculus Algorithm. (10.1 to 10.8 sections of Chapter 10 of prescribed book II) Digital Signatures …………….. Idea – Security – RSA Signatures – Signatures from Public-Key Systems – ElGamal Signature. (12.1 to 12.5 sections of Chapter 12 of prescribed book II) Prescribed Text Book: A Course in Number Theory and Cryptography Author – Neal Koblitz. Introduction to Cryptography Author: Johannes A. Buchmann ***