SYLLABUS

M.A./M./Sc. I SEMESTER

ORDINARY DIFFERENTIAL EQUATIONS : MMM-1006

UNIT-1 Introduction, initial value problem, boundary value problem, linear dependence

equations with constant as well as variable coefficient, Wronskian, variation of

parameter, method of undetermined coefficients, reduction of the order of

equation, method of Laplace’s transform.

UNIT-2 Lipchilz’s condition and Gron Wall’s inequality, Picards theorems, dependence of

solution on initial conditions and on function, Continuation of solutions, Non-

local existence of solutions Systems as vector equations, existence and uniqueness

of solution to systems and existence and uniqueness of solution for linear systems.

UNIT-3 Introduction, Strum-Liouvilles system, Green’s function and its applications to

boundary value problems, some oscillation theorems such as Strum theorem,

Strum comparison theorem and related results.

UNIT-4 Introduction, System of first order equation, fundamental matrix, Non-

homogeneous linear system, Linear system’s with constant as well as periodic

coefficients.

Books Recommended:

1. E.A. Coddington: An introduction to Ordinary Differential Equations, Prentice Hall of

India, New Delhi, 1991.

2. S.C. Deo, Y. Lakshminathan and V. Raghavendra: Text Book of Ordinary Differential

Equation (Second Edition) Tata McGraw Hill, New Delhi (Chapters IV, VII and VIII).

Reference Books:

1. P. Haitman: Ordinary Differential Equations, Wiley, New York, 1964.

2. E.A. Coddington and H. Davinson: Theory of Ordinary Differential Equations,

McGraw Hill, NY, 1955.

SYLLABUS

M.A./M./Sc. I SEMESTER

Advance Theory of Groups and Homological Algebra: MMM-1007

Unit-I:

Relation of conjugacy, conjugate classes of a group, number of elements in a conjugate class of

an element of a finite group, class equation in a finite group and related results, partition of a

positive integer, conjugate classes in Sn, Sylow’s theorems, external and internal direct products

and related results.

Unit-II:

Structure theory of finite abelian groups, subgroup generated by a non-empty subset of a group,

commutator subgroup of a group, subnormal series of a group, refinement of a subnormal series,

length of a subnormal series, solvable groups and related results, n-th derived subgroup, upper

central and lower central series of a group, nilpotent groups, relation between solvable and

nilpotent groups, composition series of a group, Zassenhaus theorem, Schreier refinement

theorem, Jordan-Holder theorem for finite groups.

Unit-III:

Direct products and direct sums of modules, natural injections into the direct sum and natural

projections from direct product with their related results, diagonal and summing

homomorphisms, injective representation of a module as a direct sum, decomposition of a

module into direct sum of submodules, free modules with related results, exact sequences and

short exact sequences with their related results.

Unit-IV:

Splitting sequences with related results, The Four lemma, The Five lemma, semi-exact

sequences, derived module, lower (upper) sequence or chain (cochain)complex of modules, n-

dimensional chain of modules and boundary operators, modules of n-dimensional cycles and n-

dimensional boundaries of a chain complex of modules, n-dimensional homology (cohomology)

module of a lower (upper) sequence, tensor product of modules with related results.

Recommended and Reference Books:

I.N. Herstein: Topics in Algebra.

Surjeet Singh and Qazi Zameeruddin: Modern Algebra.

P.B. Bhattacharya, S.K. Jain, S.R. Nagpaul: Basic Abstract Algebra

S.T. Hu: Introduction to Homological Algebra.

SYLLABUS

M.A./M.Sc. I SEMESTER

Advanced Real Analysis: MMM-1008

Unit-I:

Sequence and series of functions, Pointwise and uniform convergence, Cauchy criterion for

uniform convergence, Weierstrass M-test, Abel’s test, Dirichlet’s test for uniform convergence,

Properties of uniformly convergent sequences and series, Uniform convergence and continuity,

Integrability and differentiability of sequences and series.

Unit-II:

Weierstress approximation theorem, Riemann-Stieltjes integral (revisited) Theorem on

composition of functions and related results, Riemann integration over a rectangle, Theorems on

integrable functions and examples, Fubini’s theorem, Integration of vector-vector functions,

Rectifiable curves.

Unit-III:

Rearrangement of terms of series of numbers, Dirichlet’s theorem, Riemann theorem, The

Cauchy product of series and related results, Power series, Definitions, examples, radius of

convergence, Algebraic operations on power series, Differentiation of power series, Abel’s

theorem, Taylor’s theorem.

Unit-IV:

Functions of several variables, Derivatives in , Matrix representation, Existence of the

derivative, Chain rule, Mean value theorem, Partial derivatives, Interchange of the order of

differentiation, Young’s Theorem, Schwarz’s Theorem, Taylor Theorem, Inverse function

Theorem, Implicit Function Theorem.

Books Recommended:

W. Rudin, Principle of Mathematical Analysis, McGraw-Hill, Book Company, Japan.

John K. Hunter: An introduction to Real Analysis, Department of Mathematics,

University of California at Davis, 2014.

D. Somasundram and B. Choudhary: A First Course in Mathematical Analysis, Narosa,

1999.

S.C. Malik and Savita Arora, Mathematical Analysis, New Age International, 2017.

SYLLABUS

M.A./M./Sc. I SEMESTER

TOPOLOGY-I : MMM-1009

UNIT-1 Definitions and examples of topological spaces, Topology induced by a metric,

closed sets, Closure, Dense subsets, Neighbourhoods, Interior, Exterior and

boundary accumulation points and derived sets, Bases and subbases, Topology

generated by the subbasis, subspaces and relative topology, Alternative methods

of defining a topology in terms of Kuratowski closure operator and

neighbourhood systems, Continuous functions and homomorphism.

UNIT-2 First and second countable spaces, Lindelof spaces, Separable spaces, Second

countability and separability, Separation axioms, T0,T1,T2,T3.5,T4 spaces and their

characterizations and basic properties, Brysohn’s lemma, Tietze extension

theorem.

UNIT-3 Compact spaces and their basic properties, Separation of a space, Connected

spaces, Connected sets in the real line, Totally disconnected spaces, Intermediate

value theorem, path connected, Components, Path components, Locally connected

spaces, Locally path connected spaces, Totally disconnected spaces, Continuous

functions and connected sets.

UNIT-4 Product topology (finite and infinite number of spaces), Tychonoff product

topology in terms of standard sub-base and its characterizations, Projection maps,

Separation axioms and product spaces, Connectedness and product spaces,

Compactness and product spaces (Tychonoll’s theorem), Countability and product

spaces.

Books Recommended:

1. James R. Munkres: Topology, A first course, Prentice Hall of India Pvt. Ltd., New

Delhi, 2000.

SYLLABUS

M.A. /M. Sc. I Semester

Advanced Complex Analysis: MMM-1010

UNIT-I Curves in the complex plane; Properties of complex line integrals; Fundamental theorem of line

integrals (or contour integration); Simplest version of Cauchy’s theorem; Cauchy-Goursat

theorem; Symmetric, starlike, convex and simply connected domains; Cauchy’s theorem for a

disk; Cauchy’s integral theorem; Index of a closed curve; Advanced versions of Cauchy integral

formula and applications; Cauchy’s estimate; Morera’s theorem (Revisited); Riemann’s

removability theorem; Examples.

UNIT-II Convergence of sequences and series of functions; Weierstrass’ M-test; Power series as an analytic

function; Root test; Ratio test; Uniqueness theorem for power series; Zeros of analytic functions;

Identity theorem and related results; Maximum/Minimum modulus principles and theorems;

Schwarz’ lemma and its consequences; Advanced versions of Liouville’s theorem; Fundamental

theorem of algebra; Isolated and non-isolated singularities; Removable singularities; Poles;

Characterization of singularities through Laurent’s series; Examples.

UNIT-III Calculus of residues; Residue at a finite point; Results for computing residues; Residue at the point

at infinity; Cauchy’s residue theorem; Residue formula; Meromorphic functions; Number of zeros

and poles; Argument principle; Evaluation of integrals; Rouche’s theorem; Mittag-Leffer

expansion theorem; Examples.

UNIT-IV Introduction and preliminaries; Conformal mappings; Special types of transformations; Basic

properties of Möbius maps; Images of circles and lines under Mobius maps; Fixed points;

Characterizations of Möbius maps in terms of their fixed points; Triples to triples under Möbius

maps; Cross-ratio and its invariance property; Mappings of half-planes onto disks; Inverse function

theorem and related results; Examples.

Books Recommended

1. John B. Conway; Functions of One Complex Variable, Second Edition, Springer International

Student-Edition, Narosa Publishing House, 1980.

2. Lars V. Ahlfors; Complex Analysis, McGraw-Hill Book Company, Inc., New York, 1986.

3. S. Ponnusamy;Foundations of Complex Analysis, Second Edition, Narosa Publishing House, 2005.

SYLLABUS

M.A/M.Sc. I SEMESTER

FUNCTIONAL ANALYSIS: MMM-1011

UNIT-I Normed spaces; Banach spaces, their examples and properties; Incomplete normed

spaces; Open and Closed spheres in normed spaces; Denseness, separability and closedness;

Completion of normed linear spaces ; Finite dimensional normed spaces and subspaces;

Equivalent norms; Compactness and finite dimension; Riesz`s lemma.

UNIT-II Quotient spaces; Bounded linear operators and bounded linear functional with their

norms and properties; Algebraic and topological (continuous) duals; Examples and properties of

dual spaces, Weak convergence and strong convergence (convergence in norm); Reflexive

normed spaces and different kinds of topologies; Properties of reflexive normed spaces.

UNIT-III Hahn-Banach theorems and their consequences (Analytic and Geometric forms);

Pointwise and uniform boundedness; Uniform boundedness principle and its applications; Open

and closed maps; Open mapping and closed graph theorems, their consequences and

applications; Banach contraction theorem with its applications.

UNIT-IV Inner product space and examples; Parallelogram law; Polarization identity and

related results; Schwartz and triangle inequalities; Orthogonality of vectors, Orthogonal

complements and related results; Projection theorem and related results; Orthogonal projection

and properties.

Text Book:

1. E. Kreyszig: Introductory Functional Analysis with Applications, John Willey, 1978.

Reference Books:

1. M. Thumban Nair: Functional Analysis: A First Course, Prentice Hall of India, New

Delhi, 2002.

2. P.K. Jain, O.P. Ahuja and Khalil Ahmad: Functional Analysis, New Age International (P)

Limited, Publishers, 1995.

SYLLABUS

M.A./M./Sc. II SEMESTER

MEASURE THEORY : MMM-2002

UNIT-1 Lebesgue outer measure, Measurable and non-measurable sets measurable

functions, Borel Lebesgue measureablity.

UNIT-2 Measure and outer measure, Extensions of a measure, Uniqueness of extension,

Completion of measure, integration of non-negative functions, the general

integral.

UNIT-3 Riemann and Lebesgue integrals, The four derivatives, Lebesgue differentiation,

The differentiation and integration, Measure spaces, Convergence in measure.

UNIT-4 The Lp-spaces, Convex functions, Jensen’s inequality, Holder and Minkowski

inequalities, Completeness of Lp.

Books Recommended:

1. H.L. Royden: Real Analysis, Macmillan, 1993.

2. P.R. Halmos: Measure Theory, Van Nostrand, Princeton, 1950.

Reference Books:

1. Inder .K. Rana: An Introduction to Measure and Integration, Narosa, 1997.

2. P.K. Jain and V.P. Gupta: Lebesgue Measure and Integration, New Age International,

1986.

SYLLABUS

M.A./M./Sc. II SEMESTER

PARTIAL DIFFERENTIAL EQUATIONS : MMM-2005

UNIT-1 Classification of seconds order partial differential equation, Laplace’s equation

solution by the method of separation of variables, Fourier series solution,

Applications to two dimensional heat flow, Mean value formulas, Properties of

Harmonic functions, Green’s functions.

UNIT-2 Heat equation-solution by the method of separation of variables, Fourier series

solution, Applications to one dimensional heat flow, Mean value formula,

Properties of solutions.

UNIT-3 Wave equation-solution by the method of separation of variables, Fourier series

solution, Solution by spherical means and Riemann method of solution,

Applications to vibration of strings..

UNIT-4 Numerical solution of partial differential equations, The wave equation, One

dimensional heat flow and Laplace’s equation.

Books Recommended:

1. Elements of Partial Differential Equations by I.N. Sneddon, McGraw Hill Book

Company, 1957.

2. Partial Differential Equations by Phoolan Prasad and Renuka Ravindran, Wiley Eastern

Limited, 1987.

3. Numerical Methods in Science and Engineering by M.K. Venkatraman, The national

Publ. Company, 1990.

4. Calculus of Variations by I.M. Gelfand and S.V. Formin, Prentice Hall, Inc., 1963.

5. Partial Differential Equations by L.C. Ivans, M Graduate Studies in Mathematics,

Volume 19, AMS, 1968

SYLLABUS

M.A./M.Sc. II Semester

Advanced Linear Algebra: MMM-2007

UNIT I: Recall of vector space, basis, dimension and related properties, Algebra of Linear

transformations, Vector space of Linear transformations L(U,V), Dimension of space of linear

transformations, Change of basis and transition matrices, Linear functional, Dual basis,

Computing of a dual basis, Dual vector spaces, Annihilator, Second dual space, Dual

transformations.

UNIT II: Inner-product spaces, Normed space, Cauchy-Schwartz inequality, Pythagorean

Theorem, Projections, Orthogonal Projections, Orthogonal complements, Orthonormality, Matrix

Representation of Inner-products, Gram-Schmidt Orthonormalization Process, Bessel’s

Inequality, Riesz Representation theorem and orthogonal Transformation, Inner product space

isomorphism.

UNIT III: Operators on Inner-product spaces, Isometry on Inner-product spaces and related

theorems, Adjoint operator, selfadjoint operator, normal operator and their properties, Matrix

of adjoint operator , Algebra of Hom(V,V), Minimal Polynomial, Invertible Linear

transformation, Characteristic Roots, Characteristic Polynomial and related results,

UNIT IV: Diagonalization of Matrices, Invariant Subspaces, Cayley-Hamilton Theorem,

Canonical form, Jordan Form. Forms on vector spaces, Bilinear Functionals, Symmetric Bilinear

Forms, Skew Symmetric Bilinear Forms, Rank of Bilinear Forms, Quadratic Forms,

Classification of Real Quadratic forms.

BOOKS RECOMMENDED

1. Kenneth Hoffman and Ray Kunze : Linear Algebra (second Edition)

2. Sheldon Alexer: Linear Algebra Done Right, Springer (Third Edition)

3. I.N. Herstein: Topics in Algebra

SYLLABUS

M.A./M.Sc. II SEMESTER

ALGEBRAIC TOPOLOGY MMM-2008

UNIT-1 Urysohn Metrization Theorem, Partitions of unity, local finiteness, The Nagota

Metrization Theorem, para-compactnes, The Smirnov Metrization Theorem.

UNIT-2 Nets and filters, topology and convergence of nets, Hausdorffness and nets,

compactness and nets, filters and their convergence, canonical way connecting

nets to filters and vice-versa, Ultra filters and compactness.

UNIT-3 Homotopy, relative homotopy, path homotopy, homotopy classes, construction of

fundamental groups for topological spaces and its properties.

UNIT-4 Covering maps, local homomorphism, covering spaces, lifting lemma, The

fundamental group of circle, Torus and punctured plane, The fundamental

Theorem of Algebra.

Books Recommended:

(A) For Units I, III and IV:

1. Topology, A first course by J.M. Munkres 1987, (relevant portion)

(B) For Units II: (relevant portions of the following books)

2. Elementary Topology by M.C. Gemignani

3. Elementary General Topology by Jheral O. Moore

4. Topology by J. Dugundji

5. Topology by Sheldon W. Daves

6. Topology by H. Schubert

SYLLABUS

M.A./M.Sc. II SEMESTER

ADVANCED FUNCTIONAL ANALYSIS: (MMM-2009)

UNIT-1 Hilbert spaces (revisited), Orthonormal sets and sequences, Bessel inequality,

Bessel generalized inequality, Parseval relation and related results, Gram-Schmidt

process, total (complete) orthonormal sets and separability, bounded linear

functionals, Riesz representation theorem, reflexivity of Hilbert space,

sesquilinear form and related results, representation of sesquilinear form.

UNIT-2 Bilinear form, Lax-Miligram lemma, adjoint of bounded linear operators in

normed spaces, Hilbert adjoint operator: Existence: ranges, null spaces and related

results, closed range theorem, self adjoint operators, normal operators, unitary

operators, positive operators, orthogonal projection operators and related results.

UNIT-3 Spectral theory of linear operators in normed spaces: Eigenvalues and

eigenvectors, resolvent operators, spectrum, spectral properties of bounded linear

operators, properties of resolvent and spectrum, spectral mapping theorem

(statement only), spectral radius, compact linear operators on normed spaces,

compactness criterion, finite dimension domain or range, sequence of compact

linear operators.

UNIT-4 Spectral properties of bounded self-adjoint linear operators, unbounded linear

operators, Hellinger-Toeplitz theorem, Hilbert-adjoint operators and related

results, symmetric and self adjoint linear operators, closed linear operators and

closures.

Books Recommended:

1. E. Kreyazig: Introductory Functional Analysis with Applications, John Wiley and Sons,

New York, 1989.

2. G. Bachman and L. Narici: Functional Analysis, Academic Press, New York, 1966.

3. M. Thamban Nair: Functional Analysis, A First Course, PHI Learning Pvt Ltd, New

Delhi, 2010

4. A.H. Siddiqi, K. Ahmad and P. Manchanda: Introduction to Functional Analysis with

Applications, Anamaya Publishers, New Delhi, 2006.

5. P.K. Jain, O.P. Ahuja and K. Ahmad: Functional Analysis, New Age International (P)

Ltd., New Delhi, 1995.

SYLLABUS M.A./M./Sc. II SEMESTER

DIFFERENTIABLE MAINIFOLDS: MMM-2010

UNIT-1 Charts, Atlases, Manifolds, Differentiable structure on a manifold, Smooth maps,

Tangent vectors and Tangent space.

UNIT-2 Vector fields, Lie product of Jacobian of a smooth map, Integral curves on a

manifold, One parameter group of a transformation.

UNIT-3 Cotangent spaces, pullback of l-form, Tensor fields, Differential forms, Exterior

product and derivative, Exterior algebra.

UNIT-4 Connexion, parallelism, Geodesic, Covariant differentiation Torsion, Curvature,

Structure equation of Cartan, Bianchi identities.

Books Recommended:

1. Differentiable Manifolds: K. Matsushima.

2. Lecture Notes on Differentiable Manifolds: S.I. Husain

SYLLABUS

M.A./M./Sc. III SEMESTER

MECHANICS : MMM-3003

UNIT-1 General force system, euipollent force system, equilibrium conditions, Reduction

of force systems, couples, moments and wrenches, Necessary and sufficient

conditions of rigid bodies, General motion of rigid body, Moments and products

of inertia and their properties, Momental ellipse, Kinetic energy and angular

motion of rigid bodies.

UNIT-2 Moving frames of references and frames in general motion, Euler’s dynamical

equations, Motion of a rigid body with a fixed point under no force, Method of

pointset Constraints, Generalized coordinates, D’Alembert’s principle and

Lagrange’s equations, Applications of Lagrangian formulation.

UNIT-3 Hamilton’s principle, Techniques of calculus of variations, Lagrange’s equations

through Hamilton’s principle, Cyclic coordinates and conservation theorems,

Canonical equations of Hamilton, Hamilton’s equations from variational

principle, Principle of least action.

UNIT-4 Galilean transformation, Postulates of special relativity, Lorentz transformation

and its consequences, Length contraction, Time dilation, Addition of velocities,

variation of mass with velocity, Equivalence of mass and energy, Four

dimensional formalism, Relativistic classification of particles, Maxwell’s

equations and their Lorentz invariance.

Books Recommended:

1. J.L. Synge and B.A. Griffith: Principle of Mechanics, McGraw-Hill Book Company

(1970) (relevant portion only).

2. H. Goldstein: Classical Mechanics: Second Edition, Narosa Publishing House (1980),

(relevant portion only).

3. Zafar Ahsan: Lecture Notes on Mechanics, Department of Mathematics, AMU, (1999),

(Chapters III-VI).

SYLLABUS

M.A./M./Sc. III SEMESTER

NONLINEAR FUNCTIONAL ANALYSIS: MMM-3005

Total Lectures: 48

UNIT-I:

Fixed Point Theorems: Banach contraction theorem and its extensions, namely, Boyd and

Wong theorem, Caristi’s fixed point theorem.

Set-Valued Maps: Introduction to set-valued maps, definitions and examples; Lower and upper

semi continuity and their characterizations and examples.

UNIT-II:

Set-Valued Maps (Continue): Hausdorff metric, H-continuity or Hausdorff continuity, Set-

valued Lipschitz maps, Set-valued contraction maps; Nadler’s fixed point theorem.

Ekeland’s Variational Principle: Strong and weak forms of Ekeland’s variational principle,

converse of Ekeland’s variational principle, Applications to Banach contraction theorem,

Caristi’s fixed point theorem; Takahashi’s minimization theorem.

UNIT-III:

Geometry of Banach Spaces: Strict convexity and modulus convexity; Uniform convexity;

Duality mapping; Smoothness, Best approximation in Banach spaces, Retraction mappings.

UNIT-IV:

Iterative Methods for Fixed Points: Demiclosed Principle, Picard iterative method; Mann

iterative methods, Ishikawa iterative method; Helpern iterative method; Browder iterative

method.

Book Recommended:

1. Q.H. Ansari, Metric Spaces: Including Fixed Point Theory and Set-valued Maps, Narosa

Publishing House, New Delhi, 2010.

Sections 7.1, 7.2, 7.3, 8.1, 8.2, 8.3, 9.1, 9.2, 9.3 for unit I and II.

2. Q.H. Ansari, Topics in Nonlinear Analysis and Optimization, World Education, Delhi,

2012.

Chapter 2: Sections 2.4, 2.5, 2.6, 2.7 for unit III.

3. S. Almezel, Q.H. Ansari and M.A. Khamsi, Topics in Fixed Point Theory, Springer, New

York, 2014.

Chapter 8 for unit IV.

Reference Book:

1. V.I. Istratescu, Fixed Point Theory: An Introduction, D. Reidel Publishing Company,

Dordrecht / Boston / London, 1981, ISBN 90-277-1224-7.

2. M.A. Khamsi and W.A. Kirk, Metric Fixed Point Theory, Academic Press, New York.

3. S.P. Singh, B. Watson and P. Srivastava, Fixed Point Theory and Best Approximation:

The KKM-map Principle, Kluwer Academic Publishers, Dordrecht / Boston / London,

1997, ISBN 0-7923-4758-7.

4. W.Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and its Applications,

Yokohama Publishers, Yokohama, Japan, 2000, ISBN 4-946552-04-9.

SYLLABUS

M.A./M./Sc. III SEMESTER

ADVANCED RING THEORY: MMM-3006

UNIT-1

Examples and fundamental properties of rings(Review), Direct and discrete direct sum of rings,

Ideals generated by subsets and their characterizations in terms of elements of the ring under

different conditions, Sums and direct sums of ideals, Ideal products and nilpotent ideals, Minimal

and maximal ideals.

UNIT-2

Complete matrix ring, Ideals in complete matrix ring, Residue class rings, Homomorphisms,

Subdirect sum of rings and its characterizations, Zorn’s Lemma, Subdirectly irreducible rings,

Boolean rings.

UNIT-3

Prime ideals and m-systems, Different equivalent formulation of prime ideals, Semi prime ideals

and n-systems, Equivalent formulation of semi prime ideals, Necessary and sufficient conditions

for an ideal to be a prime ideal, Prime radical of a ring.

UNIT-4

Prime rings and its characterization in terms of prime ideals, Primeness of complete matrix rings,

D.C.C. for ideals and the prime radical, Jacobson radical: Definition and simple properties,

Relationship between Jacobson radical and prime radical of a ring, Primitive rings, Jacobson

radical of primitive rings.

BOOK RECOMMENDED:

N.H. McCoy: The Theory of Rings

BOOKS FOR REFERENCE:

Anderson and Fuller: Rings and Categories of Modules

I.S. Luthar, I.B.S.Passi: Algebra Volume 2: Rings

SYLLABUS

M.A./M./Sc. III SEMESTER

RIEMANNIAN GEOMETRY AND SUBMANIFOLDS : MMM-3007

UNIT-1 Partition of unity, paracompactness, Riemannian matrix of a paracompact

manifold, First fundamental form on a Riemannian manifold, Riemannian

connexion, Riemannian curvature, Ricci and scalar curvature.

UNIT-2 Immersion, Imbedding, Distribution, Submanifold, Submanifold of Riemannian

manifold, Sypersurfaces, Gauss and Weingarten formulae, Equation of Gauss,

Coddazi and Ricci.

UNIT-3 Complex and almost manifolds, Nejenhuis tensor and integrability of a structure,

Almost Hermitian, Kaehler and nearly Kaehler manifolds, Almost contact and

Sasakian manifolds.

UNIT-4 Submanifolds of almost Hermitian manifolds, Invariant and Anti- Invariant

distributions of a Hermitian manifold, C.R.-submanifolds of Kaehler and nearly

Kaehler, Generic and slant submanifolds of Kaehler manifold.

Books Recommended:

1. Riemannian Geometry: R.S. Mishra

2. Geometry of Submanifolds: B.Y. Chen

3. Foundation of Geometry (Volume I): S. Kobayashi and K. Nomizu

4. Lecture Notes on Differentiable Manifolds: S.I. Husain

Optional Paper

Syllabus

M.A./M.Sc. III Semester

Variational Analysis and Optimization: MMM-3015

Unit 1: Convex Set, Hyperplanes, Convex function and its characterizations; Generalized

convex functions and their characterizations, Optimality criteria, Kuhn-Tucker optimality

criteria.

Unit 2: Subgradients and subdifferentials; Monotone and generalized monotone maps, their

generalizations and their relations with convexity.

Unit 3: Variational inequalities and related problems, Existence and uniqueness results, Solution

methods.

Unit 4: Generalized variational inequalities and related topics; Basic existence and uniqueness

results.

Books:

1. Q.H. Ansari, C.S. Lalitha and M. Mehta, Generalized Convexity, Nonsmooth Variational

and Nonsmooth Optimization, CRC Press, Taylor and Francis Group, Boca Raton,

London, New York, 2014.

Optional Paper

Syllabus

M.A./ M.Sc. III-Semester

Wavelet Analysis

Paper Code: MMM-3016 Total Lectures: 48

Unit-I Gabor and Wavelet Transform

Fourier and inverse Fourier transform, Parseval identity, Convolution, Dirac delta function,

Gabor transform, Gaussian function, Centre and width of Gaussian function, Time-frequency

window of Gabor transform, Advantage of Gabor transform over Fourier transform, Continuous

wavelet transform, Time-frequency window of wavelets, Discrete wavelet transform, Haar

wavelet and its Fourier transform, Wavelets by convolution, Mexican hat wavelet, Morlet

wavelet.

Unit-II Multiresolution Analysis and Construction of Wavelets Parseval theorem for wavelet transform, Inversion formula of wavelets, Multiresolution

Analysis, Decomposition and reconstruction algorithm, Filter coefficients and their properties,

Wavelets and Fourier transform, Orthonormality in frequency domain, Numerical evaluation of

scaling function and wavelets.

Unit-III Construction of Wavelets and its Applications Cardinal B-splines and spline wavelets, Franklin wavelets, Battle-Lemari_e wavelets,

Daubechies wavelets, Application of wavelets in Image processing, Wavelet packets, Best basis,

Image compression and denoising.

Unite-IV Concept of Frames and its Applications Concept of Frames in Hilbert space, Properties and related theorems, Characterization of frames,

Frame multiresolution analysis, Gabor frames, Wavelet frames, Wavelet frames by extension

principles, Applications of tight frames in image deblurring.

Recommended Books:

1. C.K. Chui, An Introduction to Wavelets, Academic Press, New York, 1992.

2. I. Daubechies, Ten Lectures on Wavelets, CBS-NSF Regional Conferences in Applied

Mathematics, SIAM, Philadelphia, 1992.

3. O. Christensen, An Introduction to Frames and Riesz bases, Birkh• auser, Boston, 2003.

Reference Books:

1. Y. Meyer, Wavelets: Algorithms and Applications, SIAM, Philadelphia, 1993.

2. L. Debnath, Wavelet Transforms and their Applications, Birkh• auser, Boston, 2002.

3. M.W. Frazier, An Introduction to Wavelets through Linear Algebra, Springer, New

York, 1999.

4. M.K. Ahmad, Lecture Notes on Wavelet Analysis, Seminar Library, Department of

Maths, AMU, 2015.

OPTIONAL

SYLLABUS

M.A./M./Sc. III SEMESTER

LATTICE THEORY AND ALGEBRAIC STRUCTURES : MMM-3017

Total Lectures: 48

UNIT-1 Lattices (12 Lectures) Partially order sets, Lattices, Modular Lattice, Schreier’s theorem, The chain conditions,

Decomposition theorem for lattices with ascending chain condition, Independence,

Complemented modular lattices, Boolean Algebras.

UNIT-2 Modules and Ideals (12 Lectures)

Generators, Unitary Modules, Chain conditions, Hilbert Basis Theorem, Noetherian Rings,

Prime and Primary ideals, Representation of an ideal as intersection of primary ideals,

Uniqueness Theorems, Integral dependence.

UNIT-3 Lie and Jordan Structures in Rings (12 Lectures)

Lie and Jordan ideals in ring R, Jordan simplicity of ring R, Lie structure of [R, R], Subring fixed

by automorphism, Simple rings with involutions, Involution of second kind, Skew elements and

related results.

UNIT-4 Homomorphisms and Derivations (12 Lectures)

Jordan Homomorphisms onto Prime rings, n–Jordan mappings, Derivations, Lie Derivations and

Jordan derivations, Some results of Martindale, Herstein theorem on Jordan derivation.

Books Recommended:

1. Lectures in Abstract Algebra by Nathan Jacobson

2. General Lattice Theory by George Gratzer

3. Topics in Rings Theory by I.N. Herstein

4. Rings with Involutions by I.N. Herstein

SYLLABUS

(Optional)

M.A./M./Sc. III- SEMESTER (CBCS System)

THEORY OF SEMIGROUPS : MMM-3018

Total Lectures: 48

UNIT-1

Basic definitions, Group with zero, Monogenic semigroups, Ordered sets, Semilattices and

lattices, Binary relations, Equivalences and related results.

UNIT-2

Congruences and related results, Free semigroups and monoids, Presentation of semigroups,

Ideals and Rees congruences, Lattices of equivalences and congruences.

UNIT-3

Green's Equivalences and related results, The structure of D-classes, Green's lemma and its

corollaries. Regular D-classes, Regular semi groups, The Sandwich set.

UNIT-4

Simple and 0-simple semigroups, Completely 0-simple semigroups.The Rees Theorem,

Completely simple semigroups, Isomorphism and normalization.

Book Recommended:

1. Fundamentals of semi group theory by John M Howie (Clarendon press. Oxford 1995).

Reference Books:

1. The Algebraic theory of semi groups, Vol. 1 and 2 by A H Clifford and G B Preston

(Mathematical surveys of the AMS- 1961 and 1967).

2. Techniques of Semi Group Theory by P M Higgins (Oxford University Press 1992).

SYLLABUS

M.A./M./Sc. IV SEMESTER

SEQUENCE SPACES: MMM-4014

UNIT-1

Classical sequence spaces, their topological properties, Maddox type sequence spaces, Linear

Metric spaces, Paranormed spaces, Frechet spaces, FK and BK-spaces, Schauder basis, AK-,

AD-, AB-, FH-, and BH-spaces.

UNIT-2

Dual of sequence spaces, Continuous duals, Kothe-Toeplitz, generalized Kothe-Toeplitz and

bounded Kothe-Toeplitz duals, Determination of duals of classical sequence spaces, their

relationships.

UNIT-3

Matrix transformations, matrix transformations between some classical sequence spaces,

Conservative and regular matrices, Schur matrix, Coregular and conull matrices, Matrix Classes

of some FK and BK spaces.

UNIT-4

Banach limit, Almost convergence, Relation between convergence and almost convergence,

Almost regular and almost conservative matrices, Duals of Maddox type sequence spaces and

their matrix transformations.

Books Recommended:

1. Element of Functional Analysis by I.J. Maddox, Cambridge University Press (1970) and

(1988).

2. Elements of Metric Spaces by Mursaleen, Anamaya Publ. Company, 2005.

SYLLABUS

M.A./M.Sc. IV SEMESTER

FIELD AND MODULE THEORY: MMM-4015 UNIT-1 Field extensions, Finite Field extensions, Finitely generated extensions of a field, Simple

extension of a field, Algebraic extension of a Field, Splitting (Decomposition) fields, Multiple

roots, Normal and separable extension of a Field.

UNIT-2 Automorphism and group of automorphisms of a Field, Galois group, Galois group of a

separable polynomial, Galois group of a polynomial of permutation of its roots, Finite Fields and

Galois Fields.

UNIT-3 Modules, Submodules and factor modules, Sum and intersection of sub-modules, Subsets and the

submodules they generate, Homomorphisms of modules, Isomorphisms theorems, Bimodules,

Inverse image of submodules, Annihilators, Torsion and torsion-free modules.

UNIT-4 Direct sums, Internal direct sums, Direct summands, Natural maps, Splitting maps, Projections

and injections, Idempotent endomorphisms, Essential and superfluous submodules, Semi-simple

modules, Socle and radical of modules, Linearly independent sets, Bases and free modules, Rank

of free modules, Divisible modules and their basic properties.

BOOKS RECOMMENDED:

1. I.T. Adamson: Introduction to Field Theory.

2. M.E. Keating: A _rst course in Module Theory.

BOOKS FOR REFERENCE

1. J.S. Milne: Fields and Galois Theory.

2. F.W. Anderson and K.R. Fuller: Rings and Categories of Modules.

SYLLABUS

M.A./M./Sc. IV SEMESTER

STRUCTURES ON MANIFOLDS : MMM-4016

UNIT–I

Almost Complex and Hermitian manifolds. The fundamental two form and the Kaehler structure

on a manifold. Infinitesimal automorphism. Holomorphic vector fields. Characterizations for a

vector field to be an infinitesimal automorphism. Cayley algebra on R8 and almost Hermitian

structure on S6. The space of constant holomorphic sectional curvature.

UNIT –II

Almost contact structure on a smooth manifold. Contact manifolds. Torsion tensor of an almost

contact metric manifold. Sasakian manifold, Kenmotsu manifold. Trans-Sasakian manifold.

Sasakian and Kenmotsu space forms.

UNIT–III:

Invariant submanifolds in contact metric manifold. Semi-invariant submanifolds of a Sasakian

manifold. Umbilical submanifolds of almost contact metric manifolds. Semi-invariant products

in Sasakian manifolds. Some characterizations. Totally contact umbilical semi-invariant

submanifolds of Sasakian manifold. Pseudo-umbilical submanifold.

UNIT–IV

Topological groups. Subgroups and quotient spaces. Homomorphisms of topological groups.

Connected components of a topological group. Lie groups and Lie-algebras. Invariant

differential forms on Lie-groups. One parameter subgroup and exponential map. Example of Lie-

groups.

Books Recommended:

1. Structures on manifolds, Kentaro Yano and Masahiro Kon, World Scientific Press.

2. Differentiable manifolds, Y.Matsushima, Marcel Dekker, inc.

Reference Books:

1. Foundations of Differential Geometry, vol II- S.Kobayashi and K.Nomizu, John Wiley

and sons.

2. Geometry of CR-submanifolds -Aurel Bejancu, D.Reidel publication co.

3. Contact manifolds in Riemannian geometry – D.E. Blair, Lecture notes in Math. 509,

Springer-Verlag.

SYLLABUS

M.A./M./Sc. IV SEMESTER

Non Commutative Rings (MMM-4018)

Unit I

Basic terminology and examples, Free k-rings, Rings with generators and relations, Twisted

polynomial rings, Differential polynomial rings, Group rings, skew group rings.

Unit II

Radical of a ring, Prime radical of a ring, Jacobson radical of a ring, Neotherian rings, Artinian

rings, Simple rings, Semisimple rings, Simple Artinian rings, Semisimple artinian rings.

Unit III

Prime rings, Semiprime rings, Subdirectly irreducible rings, Primitive rings, Density Theorem,

Wedderburn Artin's Theorem.

Unit IV

Regular rings, Some commutativity Theorems, Wedderburn Theorem, Generalizations of

Wedderburn Theorem.

Text Book

Non commutative Rings, I.N. Herstein John Wiley and Sons, INC.

Reference Book

A First Course in Non commutative Rings Springer-Verlag.

SYLLABUS

OPEN ELECTIVE: ELEMENTS OF ELEMENTARY CALCULUS

M.Sc. IV SEMESTER (MATHEMATICS)

PAPER CODE: MMM-4091

4 Credits

M.M.: 100

Sessional Marks: 10

Mid Term Marks: 30

Final Marks: 60

Unit 1: Sets, Function and Limit

Sets, and their properties, Functions and their properties, Some known functions, Domain,

Range, Graph of Functions, Limit and its basic properties.

Unit 2: Continuity and its Basic Properties

Derivative (as rate of change and slope of a tangent), Properties of derivatives, Derivatives of

some known functions, namely, polynomial, logarithmic functions, exponential functions,

trigonometric functions.

Unit 3: Application of Derivative

Rate of change, increasing and decreasing functions, maxima and minima of polynomials and

trigonometric functions (first and second derivative test motivated geometrically) simple

problems (that illustrates basic problems and understanding of the subject as well as real life

situations. Mean Theorem Functions.

Unit 4: Integration and its applications

Indefinite integral, standard formulae of indefinite integral, Definite integral as a limit of sum,

Basic properties and formulae of definite integral of simple functions (without proof)

Applications finding the area of under simple curves, especially lines, area of circle, parabolas,

ellipse (in standard form only).

Books Recommended:

1. Calculus, Finney and Thomas, Addison-Wesley Pub. Company.

2. Mathematics Vol. 1 and 2 Class 12, R.D. Sharma, Dhanpat Rai and Sons.

3. Problems in Calculus of one variable, I.A. Marun, Arihant Publication.