The course of study
and syllabi for various degrees of the university shall be submitted
by the respective board of studies/Board of faculties to the academic
council and the syndicate for approval.Such courses and syllabi
shall beome effective from the date of approval by the syndicate
or such other date as the syndicate may determine.

Introductory ideas: Basic definitions, Cyclic semigroups; Ordered sets, semi lattices and lattices. Binary relations; Equivalences; Congruences; Free semigroups; Green’s Equivalances; L,R,H,J and D; Regular semigroups, O-Simple semigroups; Simple and O-Simple semigroups; Rees’s theorem; Primative idempotents; Completely O-Simple semigroups; Finite congruence-free semigroups, Union of groups; Bands; Free bands; varieties of bands, Inverse semigroups, Congruences on inverse semigroups; Fundamental inverse semigroups; Bisimple and simple inverse semigroups. Orthodox semigroups; Basic properties; The structure of orthodox semigroups.

RECOMMENDED BOOKS:

1. A.H. Clifford and G.B. Preston, The Algebraic Theory of Semigroups; Vol.I & II. AMS Math. Surveys, 1961 and 1967.
2. J.M. Howie, An Introduction to Semigroup Theory, Academic Press 1967.

LA-semigroups and basic results, Connection with other algebraic structures, Medial and exponential properteis, LA-semigroups defined by commutative inverse semigroups, Homomorphism theorems for LA-semigroups, Abelian groups defined by LA-semigroups, Embedding theorem for LA-semigroups, Structural properties of LA-semigroups, LA-semigroups as a semilattice of LA-subsemigroups, Locally associative LA-semigroups, Relations on locally associative LA-semigroups, Maximal separative homomorphic images of locally associative LA-semigroups, Decomposition of locally associative LA-semigroups.

RECOMMENDED BOOKS:

1. Clifford, A.H. and G.B. Preston., The Algebraic Theory of Semigroups, Vols. I & II, Amer. Math. Soc. Surveys, 7, Providence, R.I.

Near Rings, Ideals of Near-rings, Isomorphism Theorems, Near Rings on finite groups, Near-ring modules. Isomorphism theorem for R-modules, R-series of modules, Jorden-Holder- Schrier Theorem, Type of Representations, Primitive near-rings R-centralizers, Density theorem, Radicals of near-rings.

RECOMMENDED BOOKS:
1. Pilz, G., Near Rings, North Holland.

Distributively generated near-rings, ideals isomorphism theorems, Free d.g. near rings, Representations of d.g. near-rings, Types of representations, upper and lower faithful d.g. near rings, Endomorphism near-rings of groups.

Radical classes, semisimple classes, the upper radical, semisimple images, the lower radical, hereditariness of the lower radical class and the upper radical class. Partitions of simple rings.

RECOMMENDED BOOKS:

1. Wiegandt, R., Radical and Semisimple classes of Rings, Queen’s papers in Pure and Applied Mathematics No.37, Queen’s University, Kingston, Ontario, 1974.

Minimal left ideals, Wedderburn-Artin structure theorem, The Brown-McCoy radical, The Jacobson radical, Connections among radical classes, Homomorphically closed semisimple classes.

RECOMMENDED BOOKS:

1. Wiegandt, R., Radical and Semisimple classes of Rings, Queen’s papers in Pure and Applied Mathematics No.37, Queen’s University, Kingston, Ontario, 1974.

Survey of theory of group actions, Applications of group actions, Transitivity and k-transitivity, Primitivity, Finite fields and their extensions, Projective line over finite fields, Finite geometries, Projective spaces and their groups, Actions of PGL (n,q) and PSL (n,q) on PG (n-I,q), Simplicity of projective special linear groups over finite fields, Modular group, Parameterization of action of the extended modular group on projective lines over finite fields. Projective and linear groups through actions.

RECOMMENDED BOOKS:

1. Coxeter, H.S.M. and Moser, W.O., Generators and Relations for Discrete Groups, Springer-Verlag.
2. J.S. Rose, S., A Course in Group Theory, Cambridge University Press. 1978
3. Johnson, D.L., Presentation of Groups, Cambridge Lecture Notes, 1976.

Generators and relations, Factor groups, Direct Products, Automorphisms, Finite Presentations of Groups, Tiezte transformations, Coset enumerations, Graphs, Cayley diagram, Schrier’s cost diagrams, Coset diagrams for the modular group, Action of the modular group on finite sets, Symmetry in the diagrams, Composition of soset diagrams, Action of the modular group on real projective line, Action of the modular group on finite projective lines over finite fields.

RECOMMENDED BOOKS:

1. Coxeter, H.S.M. and Moser, W.O., Generators and relations for discrete Groups, Springer-Verlag.1965.
2. Rose, S., A course in group theory, Cambridge University Press. 1980.
3. Magnus, W., Karrass, A and Solitar, D., Combinatorial group theory, Dover Publications, 1976.

Definitions and Examples of Lie algebras; ideals and quotients; Simple, solvable and nilpotent Lie algebras; radical of a Lie algebra, Semisimple Lie algebras; Engel’s nilpotency criterion; Lie’s and Cartan theorems; Jordan-Chevalley decomposition; Killing forms; Criterion for semisimplicity; product of Lie algebras; Classification of Lie algebras upto dimension 4; Applications of Lie algebras.

RECOMMENDED BOOKS:

1. Humphreys, J.E., Introduction to Lie Algebras and Representation Theory, SpringerVerlag, 1972.
2. Lepowsky, J. and Mccollum, G.W., Elementary Lie algebra Theory, Yale University, 1974.
3. Jacobson, N., Lie algebras, Interscience, New York, 1962.
4. Miller, W. Jr., Symmetry Groups and their applications, Academic Press, 1972.
5. Kramer, D. Stephani, H., Herlt, E and MacCallum, M., Exact Solutions of Einstein’s Fields Equations, Cambridge University Press, 1980.
6. O’Neill, B., Semi-Riemannian Geometry, Academic Press, 1983.

Holomorphic functions: Review of 1-variable theory, Real and complex differentiability, Power series, Complex differentiable functions, Cauchy integral formula for a polydisc, Cauchy inequalities, The maximum principle. Extension of analytic functions: Hartogs figures, Hartogs theorem, Domains of holomorphy, Holomorphic convexity, Theorem of Cartan Thullen. Levi-convexity: The Levi form, Geometric interpretation of its signature, E.E. Levi’s theorem, Connections with Kahlerian geometry, Elementary properties of plurisubharmonic functions. Introduction to Cohomology: Definition and examples of complex manifolds. The d, ¶, ¶ operators, The Poincare Lemma and the Dolbeaut Lemma, The Cousin problems, Introduction to Sheaf theory.

RECOMMENDED BOOKS:

1. J. Morrow and K. Kodaira, Complex Manifolds, Holt, Rinehart and Winston, New York, 1971.
2. L. Hormander, An Introduction to Complex Analysis in Several Variables, D. Van Nostrand, New York, 1966.
3 H. Grauert and K. Fritsche, Several Complex Variables, Springer Verlag, 1976.
4. M. Field, Several Complex Variables and Complex Manifolds, Cambridge University Press, 1982.

Universe and Languages: Set relations, Filters, Individuals and super structures, Universes, Languages, Semantics, Los Theorem, Concurrence, Infinite Integers, Internal sets.
Ordered Fields, Non-standard Theory of Archimedean Fields, The hyperreal numbers, Real sequences and Functions. Prolongation Theorems. Non-standard Differential calculus, Additivity, The existence of Non-measurable sets.
Topological spaces, Mapping and products, Topological Groups, The existence of Haar Measure, Metric Spaces, Uniform continuity and Equicontinuity, Compact mapping. RECOMMENDED BOOKS:
1. Machover, M and Hirschfled, J., Lectures on Non-standard Analysis, Springer-Verlag.
2. Martin, D., Applied Non-standard Analysis, John Wiley and Sons.
3. Robinson, A., Non-standard Analysis, Studies in Logic and the Foundations of Mathematics, North Holland.

General facts about ordered sets, lattices, convergence, with respect to the order relation.
Topological vector spaces, locally convex spaces, uniform convergence, topologies in spaces of linear continuous operators, Duality between vector spaces.
Ordered vector spaces, Directed spaces and Arehimedean spaces, Vector Lattice, Decomposition of a vector lattice, Concrete spaces, Topological ordered vector spaces.

RECOMMENDED BOOKS:

1.Peressini, A.L., Ordered Topological Vector Spaces, Harper and Row.
2. Cristescu, T., Ordered Vector Spaces and Linear Operators, Abacus Press, England.

Banach Algebra: Ideals, Homomorphisms, Quotient algebra, Wiener’s lemma. Gelfand’s Theory of Commutative Banach Algebras: The notions of Gelfand’s Topology, Radicals, Gelfand’s Transforms.
Basic properties of spectra. Gelfand-Mazur Theorem, Symbolic calculus: differentiation, analytic functions, integration of A-Valued functions. Normed rings. Gelfand-Naimark theorem.

RECOMMENDED BOOKS:

1. Rudin, W., Functional Analysis; McGraw Hill Publishing Company Inc. New York.
2. M.A. Naimark, M., Normed Algebras; Wolters Noordhoff Publishing Groningen. The Netherlands 1972.
3. Zelazko, W., Banach Algebras; American Elsevier Publishing Company Inc.New York, 1973.
4. Rickart, C.E., Banach Algebras; D. Van Nostrand Company Inc. New York 1960.

Involutive Algebras, Normed Involutive algebra, C*-Algebras, Gelfand-Naimark theorem, Positive functions, A characterization of C*-Algebras, Positive forms and representations, Applications of C*-Algebras to differential operators.

RECOMMENDED BOOKS:

1. Dixmier, J., C*-Algebras; North Holland Publishing Company 1977.
2. Rudin, W., Functional Analysis; McGraw Hill Publishing Company Inc. New York.
3.Naimark, M.A., Normed Algebras; Wolters. Noordhoff Publishing Groningen. The Netherlands 1972.

Spectral analysis of unitary and self-adjoint operators: resultion of the indentity, integral representations. The Caley transform. Spectral types, commutative operators. Rings of bounded self-adjoint operators and their examples.

RECOMMENDED BOOKS:

1. Akhiezer and Clazman., Theory of linear operators: Vol. II, Frederick Ungar Publishing Co., 1963.
2. Naimark, M., Theory of Differential Operators, George Harrapand Co., 1967.

Deficiency Indices. Neumann formula, spectra of self-adjoint extensions of symmetric operators. Self-adjoint extensions to larger spaces. Applications to differential operators.

RECOMMENDED BOOKS:

1. Dunford, N., and Schwartz, J.T., Linear Operators, Interscience, 1958.
2. Akhiezer and Gulzman, Theory of Linear Operators, Frederick Ungar Publishing Co.,1963.

Analytic continuation, equicontinuity and uniform boundedness, normal and compact families of analytic functions, external problems, harmonic functions and their properties, Green’s and von Neumann functions and their applications, harmonic measure, conformal mapping and the Riemann mapping theorem, the kernel function, functions of several complex variables.

RECOMMENDED BOOKS:

1. Nehari. Z., Conformal mappings, Constable and Company.
2. Hille, E., Analytic function theory, Chelsea.
3. Sansone, G., and Gewetsen, J., Lectures on the theory of function of a complex variable, Wolters-Noordhoff, Vol.II.

Semigroups, generators and their basic properties, continuity conditions for semigroups, norm continuity, semigroups on dual spaces, Differentiable and analytic vectors, spectral theory, resolvants, classification of generators, Bounded and holomorphic semigroups, convergence of generators. Positive semigroups, criteria for positivity and irreducibility, point spectrum, spectral subspaces, the spectral theorems.

RECOMMENDED BOOKS:

1.E.B. Davies., One-parameter semigroups, Academic Press 1980.
2.Hille, E., and Philips, R.S., Functional analysis and semigroups, American Mathematical Society, Collected Publications, No.31, American Math. Soc.
3.Yosida, K., Functional analysis, Springer-Verlag, Berlin, New York, 1980. (Sixth Printing)

The weak - and strong topologies, Elementary properties of Von Neumann Algebras, Commutant and bicommutant, the density theorems, comparison of projections, introduction to the classification of factors, Normal states and the predual, Gelfand-Naimark-Siegal construction (GNS-constructions).

RECOMMENDED BOOKS:

1.Dixmier, J., Von Neumann Algebras, North Holland, 1977.
2.Dixmier, J., C*-Algebras, North Holland, 1977.
3.Schwartz, J., W*-Algebras, Gordon and Breach, New York, 1967.
4.Sakai, S., C*-Algebras and W*-Algebras, Springer-Verlag, 1971.

Numerical range in normed algebras, Numerical radius, Vidav’s theorem and applications to C*-algebras, The spatial numerical range, spectral properties, second dual of a Banach algebra, spectral states.

RECOMMENDED BOOKS:

1. Bonsall, F.F., and Duncan, J., Numerical ranges of operators on normed spaces and of elements of normed algebras, LMS lecture note series 2, Cambridge University Press,1971.

Locally covex spaces, Banach spaces, basic theorems of linear functional analysis, strict convex spaces, product and quotient spaces and strict convexity, interpolation and strict convexity, modulus of convexity, strict convexity and approximation theory, strict convexity and fixed point theory.

RECOMMENDED BOOKS:

1. Istratescue, V.I., Strict convexity and complex strict convexity, 1984.
2. Day, M.M., Normed linear spaces, 1985.
3. Diestel, J., Geometry of Banach spaces, 1975.
4. Dunford, N., and Schwartz, J.T., Linear operators-I, 1958.

Banach’s contraction principle, Nonexpansive mappings, Sequential approximation techniques for nonexpansive mappings, Properties of fixed point sets and minimal set, Multivalued mappings, Brouwer’s fixed point theorem.

RECOMMENDED BOOKS:

1. K. Goebel and W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, 1990.
2. J. Dugundji and A.Granas, Fixed Point Theory, Polish Scientific Publishers, Warszawa, 1982.
3. V.I. Istratescu, Fixed Point Tjepru. D. Reidel Publication Company, 1981.
4. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mapping, Marcel Dekker Inc. 1984.

Vector lattices over the real field, ideals, bands and projections, maximal and minimal ideals vector lattices of finite dimension, duality of vector lattices, normed vector lattices, abstract M-spaces, abstract L-spaces, duality of AL- and AM-spaces.

RECOMMENDED BOOKS:

1. Schaeff, H.H., Banch lattices and positive operators, 1971.
2. Schaeff, H.H., Banch lattices and positive operators, 1984.

Finite dimensional Lie groups: Complex Groups, Compact Groups, Root Systems, Weyl Groups, Complex Homogeneous Spaces, Borel-Weil theorem. Groups of Smooth maps: Infinite dimensional manifolds, Groups of maps as infinite dimensional Lie groups, The Loop group L(G) = Maps (S1 ,G) and its basic properties. Central extensions: Lie algebra extensions, the Co-adjoint action of the loop group on its Lie algebra, Kirillov method of orbits, group extension of simply connected Lie groups, Circle bundles, Connections and curvature. Kac-Moody Lie algebras: The affine Weyl group and its root system, Generators and relations. RECOMMENDED BOOKS:

1.A. Pressley and G. Segal., Loop Groups, Oxford University Press, 1986.
2. V.G. Kac, Infinite Dimensional Lie Algebras, Birkhauser, 1983.

Best approximation in metric and normed spaces, Least square approximation, Rational approximation, Haar condition and best approximation in function spaces, Interpolation, Stone-Weierstrass theorem for scalar-and vector-valued functions, Spline approximation.

RECOMMENDED BOOKS:

1. E.W. Cheney., Introduction to Approximation Theory, McGraw-Hill, 1966.
2. I. Singer., Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Springer-Verlag, 1970.
3. J.R. Rice., The Approximation of Functions I, II, Addison-Wesley, 1964.
4. R.B. Holmes., A Course on Optimization and Best Approximation, Lecture Notes in Mathematics No.257, Springer-Verlag, 1971.
5. M.D. Powell., Approximation Theory and Methods, Cambridge University Press, 1981.

Definition of a Topological algebra and its Examples. Adjunction of Unity, Locally Convex Algebras, Idempotent and m-convex sets, Locally Multicatively convex (l.m.c) algebras, Q-algebras, Frechet algebras, Spectrum of an element, Spectral radius, Basic theorems on Spectrum, Gelfand-Mazur Theorem. Maximal ideals, Quotient algebras, Multiplicative linear functionals and their continuity, Gelfand transformations, Radical of an algebra, Semi-simple algebras, Involutive algebras, Gelfand-Naimark theorem l.m.c. algebras.

Variational Problems, Existence results for the general implicit variational problems, Implicit Ky Fan’s inequality for monotone functions, Jartman Stampacchia theorem for monotone compact operators, Selection of fixed points by monotone functions, Variational and quasivariational inequalities for monotone operators.

RECOMMENDED BOOKS:

1. J.L. Liions., and G. StamPacchia, Variational Inequalities, Comm. Pure Appl. Math 20, 1967.
2.V. Mosco., Implicit Variational Problems and Quasi Variational Inequalities, Lecture Notes in Mathematics-543, Springer-Verlag, Berlin, 1976.
3. C. Baiocchi and A. Capelo, Variational and Quasi-variational Inequalities, Wiley,1984.

Normal and Subnormal Series, Abelian and Central Series, Direct Products, Finitely Generated Abelian Groups, Splitting Theorems, Solube and Nilpotent Groups, Commutators Subgroup, Derived Series, The Lower and Upper Central Series, Characterization of Finite Nilpotent Groups, Fitting Subgroup, Frattini Subgroup, Dedekind Groups, Supersoluble Groups, Solube Groups with Minimal Condition. Subnormal Subgroups, Minimal Condition on Subnormal Subgroups, The Subnormal Socle, the Wielandt Subgroup and Wielandt Series, T-Groups, Power Automorphisms, Structure and Construction of Finite Soluble T-Groups.

RECOMMENDED BOOKS:

1. Robinson, D.J.S., A Course in the Theory of Groups, Graduate Textes in Mathematics 80, Springer, New York, 1982.
2. Doerk, K. and Hawkes, T., Finite Soluble Groups, De Gruyter Expositions in Mathematics 4, Walter De Gruyter, Berlin, 1992.

Algebraic preliminaries; Almost complex manifolds and complex manifolds; connections in almost complex manifolds; Hermitian metrics and Kaehler matrics; Kaehler metric in local coordinate systems; Examples of Kaehler manifolds; Holomorphic sectional curvature; De Rham decomposition of Kaehler manifolds; Curvature of Kaehler submanifolds; Topology of Kaehler manifolds with positive curvature. Hermitian connections in Hermitian vector bundles. Homogeneous spaces: Structure theorems on homogeneous complex manifolds; Invariant connections on homogeneous spaces. Invariant connections on reductive homogeneous spaces; invariant indefinite Riemannian metrics; holonomy groups of invariant connections; the deRham decomposition and irreducibility; Invariant almost complex structures.

RECOMMENDED BOOKS:

1. Shoshichi Kobayashi and Katsumi Nomizu, Foundations of Differential Geometry, Vol.II, Interscience Publishers, John Wiley & Sons, 1969.
2. Shabat, B.V., Introduction to Complex Analysis, Part II, American Mathematical Society, 1992.
3. Griffiths and Harris, Principles of Algebraic Geometry, Wiley and Sons, Inc., 1994.

Commutative Rings: Definition and examples, Integral domains, unit, irreducible and prime elements in ring, Types of ideals, quotient rings, Rings of fractions, Ring homomorphism, Definitions and examples of Euclidean Domains, Principal ideal domains and Unique Factorization domains. Polynomial and Formal Power series Rings: Construction of Formal Power series ring R[[X]] and Polynomial ring R[X] in one indeterminate. Formal power series and Polynomial rings in a indeterminate, i.e. R[[X1, X2,, ..,Xn]] and R[X1, X2,, .., Xn], Factorization in polynomial rings, Irreducibility Criteria. Noetherian Rings: Definition and examples. Polynomial extension of Noetherian domains, Quotient ring of Noetherian rings, Ring of Fractions of Noetherian rings. Dimeasion of Rings: Chain of prime ideals in a domain, Length of chain of prime ideals, Dimension of ring, Dimension of Polynomial rings. Integral Dependence: Ring extension, Integral element, Almost integral element, Integral closure of a domain, Complete integral closure of domain, integrally closed domain. Completely integrally closed domain. Valuation Rings: Valuation map and value group, Rank of a valuation, Definition and examples of valuation rings, Valuation map and valuation ring, Valuation ring is integrally closed. Discrete Valuation Rings and Dedekind domains: Fractional ideals, finitely generated fractional ideals, invertible fractional ideals, Discrete valuation rings and its examples.
Definitions and examples of Dedekind domains. Dedekind domain is integrally closed, Noetherian and has dimension one.

RECOMMENDED BOOKS:

O. Zariski and P. Samual, Commutative. Algebra, Vol. l, Springer-Verlag, New York, 1958.

M. F. Anayah and L. G. Macdonald, Introduction to Commutative Algebra, Addison Wesley Pub. Co., 1969.

R. Gilmer, Multipllcative Ideal Theory, Marcell Dekker, New York, 1972.

H. Matsumura, Commutative Ring theory, Cambridge University Press, 1986

Unique Factorization Domains: Basics and examples, Guass Theorem, Quotient of a UFD, Nagata Theorem. Class Groups: Divisor Cllasses, Divisor Class monoid, Divisor Class group. Krull Rings and Factorial Ring: Divisorial ideals, Divisors, Krull rings, Stability properties, Two classes of Krull rings, Divisor class groups, Application of a Theorem of Nagata, Examples of Factorial Rings. Atomic Domains: Definition and examples, Polynomial extension of Atomic domains. Domains Satisfying ACCP: Definition and examples, Polynomial extension of domains satisfying ACCP. Connection of domains satisfying ACCP and Atomic domains. Bounded Factorization Domains: Definition and examples Length function, Charecterization of BFD through length function. Polynomial extension of BFDs, Noetherian and Krull domains are BFDs. Half Factorial Domains: Class number of a Field, Carlitz Theorem, Examples and basic results, Dedekind and krull examples, Integrability and HFD, On polynomial and polynomial like extensions. Finite Factorization Domains: Group of Divisibility G(D) of a domain D, G(D) and FFD, Atomic idf-domain is FFD,

RECOMMENDED BOOKS:

P. Samuel, Lecture Notes on Unique Factorization Domains, Tata Institute of Fundamental Research, Bombay, 1964.

R. Gilmer, Multiplicative ideal Theory, Marcel Dekker, New York, 1972.

R. M. Fossum, Divisor Class group of a krull Domain, Spriger Verlag, 1973.

D. D. Anderson, Factorization in Integral Domains, Lecture Notes in Pure & Applied Mathematics, Marcel Dekker, New York, Vol. 189, 1997.

S. T. Chapman & Sara Glaz, Non Noetherian Commutative Ring Theory, Mathematics & its Applications series Vol. 520, Kluwar Academic Publishers, 2000.

Commutative Rings: Definition and examples, Integral domains, unit, irreducible and prime elements in ring, Types of ideals, Quotient rings, Rings of fractions, Ring homomorphism, Definitions and examples of Euclidean Domains, Principal ideal domains and Unique Factorization domains. Definition and Examples of DVRs, Dedekind and Krull Domains. Commutative Semigroups: Basic notions, Cyclic Semigroups, Numerical Monoids,Ordered Semigroups, Congruences, Noetherian Semigroups, Factorization in Commutative Monoids. Semigroup Ring and its Distinguished Elements: Introduction of Polynomial Rings in one indeterminate including its elements of distinct behaviours, Structure of Semigroup ring, Zero Divisors, Nilpotent Elements, Idempotents, Units. Ring Theoretic Properties of Monoid Domains: Integral Dependence for Domains and Monoid Domains, Monoid Domains as Factorial Domains, Monoid Domains as Krull Domains, Divisor Class Group of a Krull Monoid Domain.

RECOMMENDED BOOKS:

M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison Wesley Pub. Co., 1969.

R. Gilmer, Multiplicative Ideal Theory, Marcell Dekker, New York, 1972.

H. Matsumura, Commutative Ring Theory, Cambridge University Press, 1986.

R. Gilmer, Commutative Semigroup Rings, The University of Chicago Press, Chicago, 1984.

Revision of basic concepts of Ring Theory. Modules, Homomorphisms and Exact Sequences, Product and co-product of Modules. Comparison of Free Modules and Vector Spaces Projective and Injective Modules. Hom and Duality Modules over Principal ideal Domain Notherian and Artinian Module and Rings Radical of Rings and Modules Semisimple Modules.

RECOMMENDED BOOKS:

K. R. Fuller and F.W. Anderson: Rings and Categories of Modules, Stringer Verlag 1973.

J. Lambek: lectures on Rings and Modules, New York, 1966.

F. Kasch: Modules and Rings. Academic Press, 1982.

T.W. Hungerford: Algebra, Holt, Rinehart and Winston, Inc. New York, 1974.

J. J. Rotman, An Introduction to Homological Algebra, Academic Press, New York, 1979.

O. Zariski and P. Samual, Commutative Algebra, Vol. I, Springer-Verlag, New York, 1958.

O. Zariski and P. Samual, Commutative Algebra, Vol. II, Springer-Verlag, New York, 1960

M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison Wesley Pub. Co. 1969. .

Tensor Products of Modules, Singular Homology Flat Modules. Categories and Functors Cogenerator. Finitely related (finitely presented) Modules. Ure Ideals of a ring Pure submodules and Pure Exact sequences. Hereditary and Semihereditary Rings. Ext. and extensions, Axioms Tor and Torsion, Universal co-efficient Theorems. Hilbert Syzygy Theorem, Serre’s Theorem, Mixed identities.

RECOMMENDED BOOKS:

J. Fuller and F.W. Anderson: Rings and Categories of Modules, Stringer Verlag, 1973.

J. Lambek: Lectures on Rings and Academic Modules, New York, 1966.

F. Kasch: Modules and Rings. Academic Press, 1982.

T. W. Hungerford: Algebra, Holt, Rinehart and Winston, Inc. New York, 1974.

J. J. Rotman, An Introduction to Homological Algebra, Academic Press, New York, 1979.

O. Zariski and P. Samual, Commutative Algebra, Vol. I, Springer-Verlag, New York, 1958.

O. Zariski and P. Samual, Commutative Algebra, Vol. II, Springer-Verlag, New York, 1960

M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison Wesley Pub. Co. 1969. .

Hemirings and Semirings: definitions and examples. Building new semirings from old. Complemented elements in semrings. Ideals in semirings. Prime and semiprime ideals in semirings. Factor semirings. Morphisms of semirings. Regular semirings. Semimodules over semirings. Morphisms of semimodules. Factor semimodules. Free, projective, and injective semimodules.

RECOMMENDED BOOKS:

J. S. Golan, The Theory of Semirings and Applications in Mathematics and Theoretical Computer Science, Longman Scientific & Technical John Wiley & sons New York, 1992.

U Hebisch and H. J. Weinert, Semirings Algebraic Theory and Applications inComputer Science, Word Scientific Singapore, New Jersey London Hong Kong, 1998.

The Concept of Fuzziness Examples, Mathematical Modeling, Operations of fuzzy sets, Fuzziness as uncertainty.

Algebra of Fuzzy Sets

Boolean Algebra and lattices, Equivalence relations and partions, Composing mappings, Alpha-cuts, Images of alpha-level sets, Operations on fuzzy sets.

Fuzzy Relations

Definition and examples, Binary Fuzzy relations Operations on Fuzzy relations, fuzzy partitions.

Fuzzy Semigroups

uzzy ideals of semigroups, Fuzzy quasi-ideals, Fuzzy bi-ideals of Semigroups, Characterization of different classes of semigroups by the properties of their fuzzy ideals fuzzy quasi-ideals and fuzzy bi-ideals.

Fuzzy Rings

Fuzzy ideals of rings, Prime, semiprime fuzzy ideals, Characterization of rings using the properties of fuzzy ideals

RECOMMENDED BOOKS:

Hung T. Nguyen and A First course in Fuzzy Logic, Chapman and Hall/CRC Elbert A. Walker 1999.

M. Ganesh, Introduction to Fuzzy Sets and Fuzzy Logic, Prentice-Hall of India, 2006.

John N. Mordeson and Fuzzy Commutative algebra, World Scientific, 1998.D.S. Malik,

John N. Mordeson, Fuzzy Semigroups, Springer-Verlage, 2003. D.S. Malik and Nobuki Kuroki

Algebraic Numbers: Algebraic Numbers and Number Fields, Discriminant, Norms and Traces, Algebraic integers and Integral Bases, Factorization and Divisibility, Applications of UFD.

Arthmetic’s Number Fields: Quadratic Dields, Cyclotomic Fields, Units in Number rings.

Ideals Theory: Properties of Ideals, PIDs and UFDs, Dedekind rings, Norms of ideals, Class group and Class Numbers of Quadratic Fields.

Valuations: Definitions and First properties of valuations, Valuation rings, DVRs, P-adic valuation.

RECOMMENDED BOOKS:

Richard A. Molin, “Algebraic Number Theory”, Chapman & Hall, Washington D. C., (2005)

A.N. Parshin and I.R. Shafarevich, “Number Theory I, Fundamental Problems, Ideas and Theories” Springer-Varlag, Berlin Heidelbers, (1995)

G.J. Janusz, “Algebraic Number Fields”. Academic Press, New York and London (1973).

Algebraic Numbers: Algebraic Numbers and Number Fields, Discriminant, Norms and Traces, Algebraic integers and Integral Bases, Factorization and Divisibility, Applications of UFD.

Arthmetic’s Number Fields: Quadratic Dields, Cyclotomic Fields, Units in Number rings.

Ideals Theory: Properties of Ideals, PIDs and UFDs, Dedekind rings, Norms of ideals, Class group and Class Numbers of Quadratic Fields.

Valuations: Definitions and First properties of valuations, Valuation rings, DVRs, P-adic valuation.

RECOMMENDED BOOKS:

Richard A. Molin, “Algebraic Number Theory”, Chapman & Hall, Washington D. C., (2005)

A.N. Parshin and I.R. Shafarevich, “Number Theory I, Fundamental Problems, Ideas and Theories” Springer-Varlag, Berlin Heidelbers, (1995)

G.J. Janusz, “Algebraic Number Fields”. Academic Press, New York and London (1973).

Course overview, History of cryptography, Basic terminology of cryptography, Mathematical background of encryption and decryption, Transposition ciphers, Substitution ciphers.

Preliminary Algebraic Concepts

Groups and semigroups, Ring, Subrings, ideals and factor rings, Polynomial rings, Irreducible polynomials, Primitive element, Ringhomorphisms, Euclidean domains and PIDs, Extended Euclidean algorithm, Ring of classes of risidue modulo.

Introduction to Finite Fields

Finite fields, Existence of finite fields of prime order, Polynomials over finite fields, Construction of Galois fields, Examples of Galois fields, The characteristic of a finite field, Algebra of finite fields.

Block Ciphers

Block Ciphers, DES (Data Encryption Standard), AES (Advanced Encryption Standard), Evaluation criteria for AES, Correlations and Walsh Transforms, Cryptographic Criteria for S-Boxes: Nonlinearity, Strict Avalanche Criterion, Balanced Criterionand Bijective Criterion.

References

Alexander Stanoyevitch, Introduction to Cryptography with Mathematical Foundations and Computer, Chapman and Hall/CRC, 2010.

S.R. Nagpaul and S.K. Jain, Topics in applied Abstract Algebra, Thomson, UK, US, 2005.

Douglas Robert Stinson, Cryptography Theory & Practice, CRC Press, 1995.

Bruce Schnier, Applied Cryptography, Jon Wiley & Sons, 1996.

H.C.A van Tilborg, An Introduction to Cryptology, Kluwer Academic Publisher, Boston, 1988.

Cauchy’s Problems for Linear Second Order Equations in n-Independent Variables. Cauchy Kowalewski Theorem. Characteristic surfaces. Adjoint operations, Bicharacteristics. Spherical and Cylinderical Waves. Heat equation, Wave equation, Laplace equation, Maximum-Minimum Principle, Integral Transforms.

RECOMMENDED BOOKS:

1. Dennemyer, R., Introduction to Partial Differential Equations and Boundary Value Problems, McGraw-Hill Book Company, 1968.
2. Chester, C.R., Techniques in Partial Differential Equations, McGraw-Hill Book Company, 1971.

Existence Theorems, Integral Equations with L2 Kernels. Applications to partial differential equations. Integral transforms, Wiener-Hopf Techniques.

RECOMMENDED BOOKS:

1. Harry Hoch Stadl, Integral Equations, John Wiley, 1973.
2. Stakgold, I., Boundary Value Problems of Mathematical Physics, Macmillan, New York, 1968.

Basic Equations: Equations of electrodynamics, Equations of Fluid Dynamics, Ohm’s law equations of magnetohydrodynamics. Motion of an Incompressible Fluid: Motion of a viscous electrically conducting fluid with linear current flow, steady state motion along a magnetic field, wave motion of an ideal fluid. Small Amplitude MHD Waves: Magneto-sonic waves. Alfve’s waves, damping and excitation of MHD waves, characteristics lines and surfaces. Simples Waves and Shock Waves in Magnetohydrodynamics: Kinds of simple waves, distortion of the profile of a simple wave, discontinuities, simple and shock waves in relativistic magnetohydrodynamics, stability and structure of shock waves, discontinuities in various quantities, piston problem, oblique shock waves.

Flow of Conducting Fluid Past Magnetized Bodies: Flow of an ideal fluid past magnetized bodies, Fluid of finite electrical conductivity flow past a magnetized body. Dynamo Theories: Elsasser’s Theory, Bullard’s Theory, Earth’s field Turbulent motion and dissipation, vorticity anology. Ionized Gases: Effects of molecular structure, Currents in a fully ionized gas, partially ionized gases, interstellar fields, dissipation in hot and cool clouds.

Maxwell’s equations, Electromagnetic wave equation, Boundary conditions, Waves in conducting and non-conducting media, Reflection and polarization, Energy density and energy flux, Lorentz formula, Wave guides and cavity Resonators, Spherical and cylinderical waves, Inhomogeneous wave equation, Retarded potentials, Lenard-Wiechart potentials, Field of uniformly moving point charge, Radiation from a gruop of moving charges, Field of oscillating dipole, Field of an accelerated point charge.

RECOMMENDED BOOKS:

1. Reitz, J.R., and Milford, F.J., Foundations of Electromagnetic Theory, Addison- Wesley, 1969.
2. Panofsky, K.H., and Philips, M., Classical Electricity and Magnetism, Addison- Wesley, 1962.
3. Corson, D., and Lorrain, P., Introduction to Electromagnetic Fields and Waves, Freeman, 1962.
4. Jackson, D.W., Classical Electrodynamics, John-Wiley.

General angular and frequency distributions of radiation from accelerated charges, Thomson scattering, Cherenkov radiation, Fields and radiation of localized oscillating sources, Electric dipole fields and radiation, Magnetic dipole and electric quadruple fields, Multipole fields, Multipole expansion of the electromagnetic fields; Angular distributions sources of multipole radiation; Spherical wave expansion of a vector plane wave; Scattering of electromagnetic wave by a conducting sphere.

Equations of dynamic and its various forms. Equations of Langrange and Euler. Jacobi’s elliptic functions and the qualitative and quantitative solutions of the problem of Euler and Poisson. The Problems of Langrange and Poisson. Dynamical system. Equations of Hamilton and Appell. Hamilton-Jacobi theorem. Separable systems. Holder’s variational principle and its consequences.

RECOMMENDED BOOKS:

1.Pars, L.A., Analytical Dynamics, Heinmann, London.
2.Whittaker, E.T., Atreatise on Dynamics of Rigid Bodies and Particles, Cambridge University Press.

Groups of continuous transformations and Poincare’s equations. Systems with one degree of freedom, Singular points, Cyclic characteristics of systems with a degree of freedom. Ergodic theorem, Metric indecompossability. Stability of motion.

RECOMMENDED BOOKS:

1.Pars, L.A., Analytical Dynamics, Weinemann London.
2.Whittaker, E.T., Atreatise on the Dynamics of Rigid Bodies and Particles, Cambridge University Press.

Green’s function method with applications to wave-propagation. Perturbation method: regular and singular perturbation techniques with applications. Variational methods. A survey of transform techniques; Wiener-Hopf technique with applications to diffraction problems.

RECOMMENDED BOOKS:

1. Nayfeh, A., Perturbation methods.
2. Stakgold, I., Boundary value problems of Mathematical Physics.
3. Noble, B., Methods based on the Wiener-Hopf technique for the solution of Partial Differential Equations.
4. Mitra, R., and Lee, S.W., Analytical Techniques in the Theory of Guided Waves.

Introduction: Definition of plasma; temperature; Debye shielding, the plasma parameter; criteria for plasmas; introduction to controlled fusion. Fluid description of plasma: Wave propagation in plasma; derivation of dispersion relations for simple electrostatic and electromagnetic modes.
Equilibrium and stability (with fluid model); Hydromagnetic equilibrium/diffusion of magnetic field into a plasma; classification of instabilities; two-stream instability; the gravitational instability; resistive drift waves. Space plasma: Atomospheric source of magnetospheric plasma and its temperature; plasma from Jupiter.

RECOMMENDED BOOKS:

1. Chen, F.F., Introduction to Plasma Physics, Plenum Press, New York, 1974.
2. Krall, N.A. and Trivelpiece, A.W., Principles of Plasma Physics, McGraw-Hill Book Company, 1973.
3. Glasstone, S., and Lovberg, R.H., Controlled thermonuclear reactions, Van Nartrand Company, 1960.
4. Nishida, A., Magnetospheric Plasma Physics, D. Reidel Publishing Compnay, 1982.
5. Melrose, D.B., Plasma Astrophysics, Gordon and Breach Science Publishers, 1980.

The Plasma Theory of Waves: Solution of localized Vlasov equation Vlasov theory of small amplitude waves in field free uniform/nonuniform magnetized cold/hot plasmas; the theory of instability. The nonlinear Vlasov theory of plasma waves and instabilities: Conservation of particles, momentum and energy in quasilinear theory; Landau damping; the gentle-bump and two-stream instability in quasilinear theory; plasma wave echoes; nonlinear wave-particle interaction. Fluctuations, correlations and radiations: Shielding of a moving test charge, electric field fluctuations in maxwellian and nonmaxwellian plasmas, emission of electrostatic waves; electromagnetic fluctuations, emission of radiation from plasma; black body radiation; cyclotron radiation. RECOMMENDED BOOKS:
1. Krall, N.A., and Trivelpiece, A.W., Principles of Plasma Physics, McGraw-Hill Book Company, 1973.
2. Hasegawa, A., Plasma instabilities and nonlinear effects, Springer Verlag, 1975.

Review of special relativity, tensors and field theory. The principles on which General Relativity is based. Einstein’s field equations, obtained from geodesic deviation. Vacuum equation. The Schwarzschild exterior solution. Solution of the Einstein-Maxwell field equations and the Schwarzschild interior solution. The Kerr-Newmann solution (without derivation). Foliations. Relativistic corrections to Newtonian gravity. Black holes, the Kruskal and Penrose diagrams. The field theoretic derivation of Einstein’s equations. Weak field approximations and gravitational waves. Kaluza-Klein theory. Isometries. Conformal transformations. Problems of “quantum gravity”.

RECOMMENDED BOOKS:

1.Qadir, A., Relativity: An Introduction to the Special Theory, World Scientific,1989.
2.Misner, C.W., Thorne, K.S. and Wheeler, J.A., Gravitation, W.H. Freeman,1974.
3.Hawking, S.W. and Ellis, G.F.R., The Large Scale Structure of Spacetime, Academic Press,1972.

Review of Relativity. Historical background: Astronomy; Astrophysics; Cosmology. The cosmological principle and its strong form. The Einstein and DeSitter universe models. Measurement of cosmic distances. The Hubble law and the Friedmann models. Steady state models. The hot big bang model. The microwave background. Discussion of significance of a start of time. Fundamentals of high energy physics. The chronology and composition of the Universe. Non-baryonic dark matter. Problems of the standard model of cosmology. Bianchi spacetimes. Mixmaster models. Inflationary cosmology. Further developments of inflationary models. Kaluza-Klein cosmologies. Review of material.

RECOMMENDED BOOKS:

1.Peebles, P.J.E., Principles of Physical Cosmology, Princeton University Press 1993.
2.Ryan, M.P.Jr. and Shepley, L.C., Homogeneous Relativistic Cosmologies, Princeton University Press 1975.
3.Kolb, E.W. and Turner, M.S., The Early Universe, Addison Wesley 1990.
4.Abbott, L.F. and Pi, S.Y., Inflationary Cosmology, World Scientific 1986.

Static stellar structure and the equilibrium conditions. Introduction to stellar modelling. The Hertzprung-Russell diagram and stellar evolution. Gravitational collapse and degenerate stars. White dwarfs, neutron stars and black holes. Systems of stars, irregular and globular clusters, galaxies superclusters and filaments. Astrophysical dark matter and galactic haloes.

RECOMMENDED BOOKS:

1.Chandrasekhar, S., An Introduction to the study of Stellar Structure. Dover Publications, Inc. 1967.
2.Richard, L., and Deeming, T., Astrophysics, Vol.I and II, Jones and Bartlett Publishers, Inc., 1984.
3.Schwarzschild, M., Structure and Evolution of Stars, Dover Publications, New York, 1965.
4. Misner, C.w., Thorne, K.S., and Wheeler, J.A., Gravitation, W.H., Freeman & Co. 1973.

Review of continuum mechanics; solid and fluid media; constitutive equations and conservation equations. The concept of a field. The four dimensional formulation of fields and the stress-energy momentum tensor. The scalar field. Linear scalar fields and the Klein-Gordon equation. Non-linear scalar fields and fluids. The vector field. Linear massless scalar fields and the Maxwell field equations. The electromagnetic energy-momentum tensor. Electromagnetic waves. Diffraction of waves. Advanced and retarded potentials. Multipole expansion of the radiation field. The massive vector (Proca) field. The tensor field. The massless tensor field and Einstein field equations. Gravitational waves. The massive tensor field. Coupled field equations.

RECOMMENDED BOOKS:

1. Scipio, L.A., Principles of Continua with Applications, John Wiley, New York, 1969.
2. Landau, L.D., and Lifshitz, M., The Classical Theory of Fields, Pergamon Press, 1980.
3. Jackson, J.D., Classical Electrodynamics, John Wiley, New York, 1975.
4. Misner, C.W., Thorne, K.S., and Wheeler, J.A., Cravitation, W.H. Freeman, 1973.
5. Romen, P., Introduction to quantum field theory, John Wiley, New York, 1969.

1. Boundary and intial conditions, Polynomial approximations in higher dimensions.

2. Finite Element Method: The Galerkin method in one and more dimensions, Error bound on the Galarki method, The method of collocation, Error bounds on the collocation method, Comparison of efficiency of the finite difference and finite element method.

4. Application to solution of linear and non-linear Partial Differential Equations appearing in Physical Problems.

RECOMMENDED BOOKS:

1. Strang G., and Fix G., An Analysis of Fintie Element Method, Prentice Hall, New Jersey 1973.
2. David S. Burnett., Finite Element Analysis from Concepts to Applications, Addision Wesley, 1987.
3. Myron B. Allen., Ismael Herrera and George F., Pinder Numerical Modeling in Science and Engineering.
4. Desai, G.S., Elementary Finite Element Method, Prentice Hall, Inc. 1988.

Fundamentals of vibrations, Energy of vibration, damped and free oscillations, transient response of an oscillator. Vibrations of strings, Membrances and plates, Forced vibrations, Normal modes, Acoustic waves equation and its solution, Equation of state, Equation of cont, Euler’s equation, Linearized wave equation, Speed of sound in fluid, Energy density, Acoustic intensity, Specific acoustic impedance, Spherical waves, Transmission; Transmission from one fluid to another (Normal incidence) reflection at a surface of solid (normal and oblique incidence). Absorption and attenuation of sound waves in fluids, Pipes Cavities, Wave guides; Underwater acoustics.

RECOMMENDED BOOKS:

1. Kinsler, L.E., Frey, A.R., Coppens, A.B., Sanders, J.V., Fundamentals of Acoustics, John Wiley & Sons, 1981.
2. Junger, M.C., Feit, D., Sound Structures and their Interaction.
3. Morse, P.M., Ingard, K.U., Theoretical Acounts, McGraw-Hill Book Company, 1960.
4. Morse, P.M., Vibration and Sound, McGraw-Hill Book Company, 1948.

Strain potential, Galerkin vector, vertical load on the horizontal surface of a half space, Love’s strain function, Biharmonic functions, Lamb’s problem, Cagniard-de Hoop transformation.
Transient waves in a layer, forced shear motion of a layer.
Thermoelasticity: thermal stresses Chadwick’s solution of thermoelastic solutions.
Piezoelectricity. Tensor formultion of piezoelectricity, elastic waves in a piezoelectric solid, Bleustein-Gnlayev waves.

RECOMMENDED BOOKS:

1.Dieulesant D. and Royer, F., Elastic Waves in Solids, John Wiley and Sons, New York, 1980.
2.Fung, Y.C., Foundations of Solid Mechanics, Prentice-Hall, Englewood Cliffs, 1995.
3.Achenbach, Waves Propagation in Elastics Solids, North-Holland, Amsterdam, 1990.

Equal probability sampling: Simple random sampling and general formula for the derivation of variance and variance estimater. Derivation of variance and estimated variance for the proportion in one class to a group of classes. Derivation of variance and variance estimator for quantitative and qualitative characteristics. Stratified sampling. Optimum allocation. Effect of deviation from optimum allocation. Two way stratification. Controlled selection. Construction of strate, Gain due to stratification. Unequal probability sampling: Probability proportional to size sampling. Sampling with replacement, Cumulative method, Hansen-Hurwitz (multinomial), Pathak, Lahiri selection procedures. Sampling with unequal probability sampling, Yates-Grundy, Midzuno-Sen-Ikeda, Brewer, Sampford, Dubin, Raj, Murthy, Rao-Hartly-Cochran selection procedures.
Practicals based on the above topics.

Ratio and regression estimators with applications in equal and unequal probabilities. Best linear unbiased estimator (BLUE). Model based versus design based estimator. Condition under which ratio estimator is BLUE. Derivation of variance and variance estimator under a model for ratio and regression estimator.

Multistage sampling. Derivation of variance and variance estimator for equal and unequal probability sampling for two-stage sampling. Multistage rules - Durbin’s rule, Raj’s rule, Raj’s modifie rule, Brewer-Hanif rule.

Practicals based on the above topics.

RECOMMENDED BOOKS:

1.Cochran, W.g., Sampling Techniques, John Wiley & Sons, New York, 3rd Ed., 1977.
2.Sukatme, P.V., and Sukatme, B.V., Sampling Theory of Surveys with Applications, Iowa State University Press, Ames., USA, 1970.
3.Raj, D., Sampling Theory, McGraw-Hill Book Company Ltd., New York, 1968.
4.Raj, D., Design of Survey, McGraw-Hill Book Company Ltd., New York, 1970.
5.Kish, I., Survey Sampling, John Wiley & Sons, New York, 1965.

Reveiw of basic designs. Methods of analyzing data arising from these designs when the subclass numbers are unequal. Split plot and split block designs. Analysis of 2n, 3n and mixed factorial experiments. Confounding. Fractional replication. Analysis of series of experiments in time and space. Dealing with missing data. Response surface analysis.
Practicals based on the above topics.

Construction and analysis of balanced and partially balanced incomplete block designs, their types and existance criteria. Competition designs and their analysis.

Practicals based on the above topics.

RECOMMENDED BOOKS:

1.Cox, D.R., Planning of Experiments, John Wiley and Sons, New York, 1958.
2.John, W.M., Statistical Design and Analysis of Experiments, The MacMillan Co., New York, 1971.
3.Davies, O.L., Design and Analysis of Industrial Experiments, Oliver and Boyd, London, 1954.
4.Kempthorn, O., Design and Analysis of Experiments, John Wiley and Sons, New York, 2nd Ed., 1957.
5.Cochran, W.C., and Cox., G.M., Experimental Designs, John Wiley & Sons, New York, 2nd Ed., 1957.
6.Sheffe, H., Analysis of Variance, John Wiley and Sons, New York, 1973.
7.Mead, R., The Design of Experiments - Sttistical Principles for Practical Applications, Cambridge University Press, Cambridge, 1988.
8.Zar, J.H., Biostatistical Analysis, Englewood Cliffs; Prentice Hall, 1984.
9.Street, A.P., and Street, D.J., Combinatories of Experimental Design, Clarendon Press, Oxford, 1987.

Methods of decomposing time-series into its component parts and measuring their effects. Stationary stochastic processes and their properties in the time domain. Frequency domain- Cyclical trend, spectral representation of a stochastic process. Properties of ARMA process and linear filters. Multivariate spectral analysis. Estimation of ARMA models. Model building and forecasting.
Practicals based on the above topics.

RECOMMENDED BOOKS:

1. Harvey, A.C., Time Series Models, Philip Allan Publishers Ltd., 1981.
2. Zurbenko, I.G., The Spectral Analysis of Time Series, Elsevier Science Publishers B.V., Amsterdam, 1986.
3. Hannan, E.J., Time Series Analysis, Methuen, London, 1960.
4. Kendall, M.G., Time-Series, Griffin, London, 1973.

Introduction: Some multivariate problems and techniques. The data matrix. Summary statistics.

Normal distribution theory: Characterization and properties. Linear Forms. The Wishart distribution. The Hotelling T2-dustribution. Distributions related to the multionormal.

Estimation and Hypothesis testing: Maximum likelihood estimation and other techniques. The Behrens-Fisher problem. Simultaneous confidence intervals. Multivariate hypothesis testing.
Design matrices of degenerate rank. Multiple correlation. Least squares estimation. Discarding of variables.

RECOMMENDED BOOKS:

1. Mardia, K.V., Kent, J.T., and Bibby, J.M., Multivariate Analysis, Academic Press, London, 1982.
2. Kshirsagar, A.M., Multivariate Analysis, Marcell Dekker, New York, 1972.

Principal component analysis: Definition and properties of principal components. Testing hypotheses about principal components. Correspondence analysis. Discarding of variables. Principal component analysis in regression.

Factor analysis: The factor model. Relationships between factor analysis and principal component analysis.

Canonical correlation analysis: Dummy variables and qualittive data. Qualitative and quantitative data.

Discriminant analysis: Discrimination when the populations are known. Fisher’s linear discriminant function. Discrimination under estimation.

Multivariate analysis of variance: Formulation of multivariate one-way classification. Testing fixed contrasts. Canonical variables and test of dimensionality. Two-way classification.

RECOMMENDED BOOKS:

1. Mardia, K.V., Kent, J.T., and Bibby, J.M., Multivariate Analysis, Academic Press, London, 1982.
2. Kshirsagar, A.M., Multivariate Analysis, Marcell Dekker, New York, 1972.

Introduction: Basic definitions and concepts. Statistical problems associated with mixtures.
Applications of finite mixture models.
Mathematical aspects of mixtures: Identifiability, multimodality, properties of general mixtures.
Parameters of a mixture: Various methods of determining the parameters of a mixture.

RECOMMENDED BOOKS:

1. Everitt, B.S., and Hand, D.J., Finite Mixture Distributions, Chapman and Hall, London, 1981.
2. Titterington, D.M., Smith, A.F.M., and Makov, U.E., Statistical Analysis of Finite Mixture Distributions, John Wiley and Sons, New York, 1985.

Components of a mixture: Various informal and formal techniques of determining the number of components of a mixture. The structure of modality. Assessment of modality. Discriminant analysis.

Sequential problems and procedures: Introduction to unsupervised learning problems. Approximate solutions to unknown mixing parameters, unknown component distribution parameters, unknown mixing and component parameters and dynamic linear models.

RECOMMENDED BOOKS:

1. Everitt, B.S., and Hand, D.J., Finite Mixture Distributions, Chapman and Hall, London, 1981.
2 .Titterington, D.M., Smith, A.F.M., and Makov, U.E., Statistical Analysis of Finite Mixture Distributions, John Wiley and Sons, New York, 1985.

Basic concepts of groups of transformation; parameter lie group of transformation (LGT); Infinitesimal transformation (I.T); Infinitesimal generators; Lie’s first fundamental theorem; Invariance; Canonical coordinates; Prolongations; Multi-parameter lie group of transformations (MLGT); Lie algebra; Solvable lie algebra; Lie’s second and third fundamental theorems.

Invariance of ODE’s under (LGT) and (MLGT); Mappings of solutions to other solutions from invariance of an ODE and PDE; Determining equations for (I.T) of an n-th order ODE and a system of PDE’s. Determination of n-th order ODE invariant under a given group; Reduction of order by canonical coordinates and differential invariants; Invariant solutions of ODE’s and PDE’s; Separatrices and envelops.

Noether’s theorem and Lie-Backlund symmetries; Potential symmetries; Mappings of differential equations.

RECOMMENDED BOOKS:

1.G.W., Bluman and Sokeyuki Kumei., Symmetries and differential equations, Springer- Verlag, N.Y. 1989.
2. James M. Hill., Differential equations and group methods, CRC Press, Inc. N.Y. 1992.
3.I.P., Eisenhart., Continuous groups of transformations, Dover Publications, Inc. N.Y. 1961.

Introduction: Three-Body Problem; Introduction; Mathematical Description; Numerical Method; Flowchart of the Program; Nature of the Orbits; Practical Examples.

Polytropes: Introduction; Physical Background; The Equation of Hydrostatic Equilibrium; Numerical Discretization Procedure; Program Listings.

Homogeneous Stellar Models: Equation of State. Luminosity function; Boundaries and Polytrope Fitting Procedure; The Polytrope Index in the Core; The Position of the Boundary of the Convective Core. Initial Conditions and Central Values; The Convective Core Iteration; Medel of a Star with arbitrary Mass; Program Listings.

Stellar Atmospheres: The Absorption Coefficient; Optical Depth; Temperature formulae; Ideal and Non-ideal Gas Pressure Formulae; Mean Molecular Weight; Radiation Pressure; Equation of Hydrostatic Equilibrium in Normal Stars.

The Structure of White Dwarfs: Introduction; Electron Degneracy; Equations of Structure; Equation of State; Hydrostatic Equilibrium and Continuity of Mass; Detailed Model of a White Dwarf. The Chandrasekhar Limit; Program Listings.

Star Formation: Introduction. Simple Model for Protostar Formation; Equations of Interaction Between Components; Evolution Towards a Stationary State; Evolution Towards a Limit Cycle; Validity of Different Numerical Methods; Program Listings.

Cosmological Models for the Universe: Introduction; Physical and Mathematical Background; Types and Classes of Zero-Pressure Models; Newtonian Cosmology; Friedmann model; FRW (Fermi-Robertson-Walker) model; Numerical Approximations.

References

P. Bodenheimer, G. Laughlin: An Introduction to Numerical Methods in Astrophysics by. (Institute of Physics, 2005)

S. M. Miyama. K. Tomisaka: Numerical Astrophysics, (Springer, 1999).

M. A. Celia, W. G. Gray: Numerical Methods For Differential Equations. (Prentice Hall, 1991)

J. Franco, S. Lizano, L. Aguilar: Numerical Simulations in Astrophysics. (Cambridge University Press, 1994)

T. Passot: Turbulence and Magnetic Fields in Astrophysics.(Springer Verlag, 2003).

P. Hellings: Astrophysics with a PC, An Introduction to Computational Astrophysics. (Willmann-Bell, 1994)

Microstates; Macrostates; Multiplicity; The second law of thermodynamics; Microcanonical Ensemble; Indistinguishability; Free Energy and Chemical Potential; Gibbs free energy; Chemical Potential Dilute Solutions and Chemical Equilibrium.

Review of Quantum Mechanics (Schoedinger Equation; Angular Momentum; Systems of Many particles); The Gibbs Factor; Grand Canon9ical Ensemble; Bosons and Fermions; The Distribution Functions; Degenerate Fermi Gas.

Systems of Interacting Particles: Weakly Interacting Gases; Partition function; configuration integral; Cluster Expansion; Second Virial Coefficient.

Applications:

Blackbody Radiation; Debye Theory of Solids; Bose-Einstein Condensation; Non-Equilibrium Systems and Chaos; Application of Degeneracy to White Dwarfs and Neutron Stars.

References:

D. Chandler: Introduction to Modern Statistical Mechanics. (Oxford University Press, 1987).

A. I. Khinchin: Mathematical Foundations of Statistical Mechanics. (Dover Publications, 1960).

R. Bowley and M. Sanchez: Introductory Statistical Mechanics. (Oxford University Press, 1999).

L. D. Landau and Statistical Physics. (Butterworth-Heinemann, 1984). E. M. Lifshitz:

Leo Kadanoff, Gordon Baym: Quantum Statistical Mechanics. (Westview Press, 2001).

William G. Hoover: Computational Statistical Mechanics, (Elsevier Science Publishers, 1991),

J. R. Dorfman: An Introduction to Chaos in Nonequilibrium Statistical c Mechanics. (Cambridge University Press, 2003).

Mathematical methods of classical mechanics: Lagrangian and Hamiltonian Formalisms; Hamilton-Jacobi Equations; Noether Theorem; Symmetries and Conservation laws; Lorentz Invariance and Relativistic Mechanics.

Quantum mechanics in Hilbert Space: Operators in Banach Space and Operator Calculus; Applications to Quantum Computing and Information Theory; Representation Theory (including Heisenberg; Schroedinger and Holomorphic Representations); Deformation Quantization.

The Dirac Theory: Classical Field Theory; Examples of Quantized Field Theories; Dirac Equation and Spinor Formulation; Electron Spin; Field Theoretic Methods in Quantum Statistics.

Relativistic Scattering Theory: Free Particle Scattering Problems; General Theory of Free Particle Scattering; Scattering by a Static Potential; Scattering Problems and Born Approximation.

Functional Intergration and Feynman Path Integration: Feynman Path Integral Formalism and Related Wiener theory of Functional Integration; Perturbation theory and Feynman Diagrams; Regularized Determinants of Elliptic Operators Supersymmetry and Path Integral Formalism for Fermions.

References:

J. J. Sakurai: Advanced Quantum Mechanics. (Addison-Wesley, 1967).

A. Messiah: Quantum Mechanics. (John Wiley & Sons Inc. 1961).

P.A. M. Dirac: The Principles of Quantum Mechanics. (Oxford at the Clarendon Press, 1958).

J. von Neumann: Mathematical Foundations of Quantum Mechanics.(Princeton University Press, 1955).

L. D. Landau and E. M. Lifshitz:. Quantum Mechanics: Non-Relativistic Theory (Pergamon Press, 1977).

L. D. Landau, E. M. Lifshitz and L.P. Pitaevskii: Relativistic Quantum Theory. (Pergamon Press, 1977).

Ikenberry: Quantum Mechanics for Mathematicians and Physicists. (Oxford University Press, 1962).

S. Weinberg: The Quantum Theory of Fields, Vol. 1. (Cambridge University Press, 1995).

Fundamental Concepts: Introduction to Robot (Fundamental notions and Definitions), Jacobians: Transformations and Jacobians, Manipulator. Kinematics: Kinematics (Forward and Inverse) of manipulator, Manipulator Dynamics, Trajectory Generation, Manipulator Mechanism, Manipulator Design. Linear Control: Linear Control of Minipulator, Non-linear Control of Manipulator, Forced Control of Manipulator, Multivariable Control: Multivariable control, Feedback linearization, Variable structure and Adaptive Control.

Recommended Books

1) John, J. Craig, Introduction to Robotics, Addison-Wesley Publishing Company Inc. (1999)
2) Mark, W. Sponge and M. Vidyasagar, Robot Dynamics and Control, John Wiley and Sons Inc (2004)
3) Gene Franklin, J. David Powell and Feed-back Control of Dynamic Systems, Abbas Emami-Naeini, Addison-Wesley Publishing Company Inc.(1989)
4) Stainley M. Shinners, Modern Control System Theory and Applications, Addison-Wesley Publishing Company Inc. (1987)
5) John, J. Craig, Adaptive Control of Mechanical Manipulators, Addison-Wesley Publishing Company Inc. (1997)

Classical Control Theory: Background and review; Highlights of Classical Control Theory; State Variables and the State Space Description of Dynamic Systems. Linear Vector Space: Fundamentals of Matrix Algebra; Vectors and Linear Vector Spaces; Simultaneous Linear Equations; Eigenvalues and Eigenvectors; Functions of Square Matrices and the Cayley-Hamilton Theorem. Analysis of System: Analysis of Continuous and Discrete Time State Equations; Stability; Controllability and Observability for Linear Systems. Optimal Control System: The Relationship between state Variable and Transfer Function Descriptions of Systems; Design of Linear Feedback Control Systems; An Introduction to Optimal Control Theory; An Introduction to Nonlinear Control Systems.

Recommended Books

1) W. Brogen Modern Control Theory
Prentice-Hall (1990)
2) P. Gopal Modern Control System
New Age Publishers (1994)
3) J. Kailath Linear Systems
Prentice-Hall (1979)
4) W.J. Rugh Linear Systems Theory
Prentice-Hall (1995)
5) C. T. Chen Linear Systems Theory and Design
Oxford University Press (1999)

Non-Linear System: Introduction; Why Nonlinear Control? Nonlinear Systems Analysis; Phase Plane Analysis; Fundamentals of Lyapunov Theory; Performance Analysis; Control Design Based on Lyapunov’s Direct Method. Advanced Stability Theory: Advanced Stabiltiy Theory; Barbalat’s Lemma; Positive Linear Systems. Absolute Stability: Absolute Stability; Establishing Boundedness of Signals; Existence and Unicity of Solutions; Describing Function Analysis, Nonlinear controdesign; Feedback Linearization and The Cononical Form; Input-State. Linearization: Linearization; Sliding Control; Continuous Approximations of Switching Control Laws; The Modeling/Performance Trade-Offs. Adaptive Control: Adaptive Control; Adaptive Control of First-Order Systems; Adaptive Control of Nonlinear Systems; Robustness of Adaptive Control Systems; On-Line Parameter Estimation. Control of Multi-Input System: Control of Multi-Input Physical Systems; Robotics as a Prototype; Adaptive Robot Trajectory Control; Conservative and Dissipative Dynamics; Spacecraft Control.

Recommended Books

1) W. Brogen Modern Control Theory
Prentice-Hall (1990)
2) J. J. Slotine Weiping Li Applied Nonlinear Control
Prentice-Hall (1991)
3) A. Isidori Nonlinear Control in the year 2000.
Springer (2001)
4) M. Gopal Modern Control System Theory
New Age Publishers (1994)

Symmetry Groups: Introduction, What is symmetry?; The group concept; Transformations of coordinates and points in IR2; Transformations in IR2; Discrete and continuous groups; One-parameter groups in the plane. Invariant of the System: Infinitesimal operator in the plane; Group orbits; Invariants; Differential invariants; Extended transformations; Symmetries of scalar ODEs; Determining equations; First-order scalar ODEs; Canonical variables and invariant approach for first-order scalar ODEs. Higher-Order ODEs: Higher-order ODEs; Reduction of order by one; Notions on Lie algebras; Consecutive reduction of order of second and third-order ODEs; Canonical forms for second-order ODEs; Algebraic and invariant linearization criteria for second-order ODEs; Noether symmetries; Other topics References
1) Mahomed, FM, 2007 Symmetry group classification of ordinary differential
equations: Survey of some results, Math. Methods in Applied
Sciences 30.
2) Ibragimov, NH Editor, CRC Handbook of Lie Group Analysis of Differential
Equations, vols I to III, CRC Press, Boca Raton, 1994-1996.

Lie Group of Transformation: Introduction; Mathematical idea of symmetry; Local solvability for systems; Maximal rank condition, Symmetry transformations, Lie group transformations in IRn+m . Canonical Parameter: Canonical parameter for a group; Infinitesimal transformations in IRn+m . Lie Algebra System: Lie equations; Exponential map; Symmetry groups of differential equation systems; Prolongation formulas; Invariant points; Invariant functions; Canonical variables; Infinitesimal criteria for invariance for systems; Lie algebras. Multi-group and Classification: Multi-parameter groups; Symmetries of partial differential equations; Construction of exact solutions; Group classification; Other topics.

References
1) Ibragimov, NH, Kara AH and Lie-Backlund and Noether symmetries with
Mahomed FM 1998.applications, Nonlinear Dynamics, 15.
2) Ibragimov, NH Editor CRC Handbook of Lie Group Analysis of
Differential Equations, vols I to III, CRC Press, Boca Raton, 1994-1996.