course contents full copy ma msc

7/27/2019 Course Contents Full Copy MA MSC

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HAZARA UNIVERSITY, MANSEHRA, NWFP

DEPARTMENT OF MATHEMATICSHAZARA UNI VERSITY MANSEHRA

Brief Introduction

Department of Mathematics started with the start of the uni versity in 2002.The Department

has started M.Sc. in Mathematics since Spring 2005. I t has also intr oduced BSc (Hons)

four years programme approved by HEC. Besides this the staf f of the department teaches

mathematics and statistics in almost al l the discipl ines of the university. The Department

has collaboration with the mathematical institute University of St.Andrews U.K and Gomal

University D.I .Khan. Mathematics has been declared as a core subject for all levels in

physical as well as social and management sciences by H.E.C.

Mathematics has played and is playing a vital role in a rapidly changing world of

science and technology. I t has developed tremendously in the last centur y and these

developments have fur ther been accelerated by the use of i nf ormation technology in every

walk of li fe. Thi s motivated to the discovery of varied new techniques in mathematics.

Modern era of science and technology has proved that not only the natural sciences but

the social and administrative sciences have also been developed to the extent that they too,

need an input of mathematics.

The MSc Mathematics programme started in Hazara University Dhodial M ansehra

has been designed to produce better quali f ied and more competent manpower to meet the

needs of the society in general . To provide expertise in Mathematics, to solve problems in

other areas such as Physics, Chemistry, Engineering, Economics, Defense, Industry etc. I thas further been designed to motivate, Create in terest in Mathematics and thus help in

creating a scienti fi c cultur e in the country.

Department has also planned to start M .Phil and PhD Programme in the near futu re.


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* REQUIREMENTS FOR SELECTION OF COURSES IN TH IRD AND

FOURTH SEMESTERS

A student is required to select five courses in each Semester from a

variety of courses offered, (subject to the availability of expertise). Each course

shall carry 100 marks and will be of three credit hours.In the fourth semester certain students (selected strictly on merit of the

previous semester examinations) will be given an option to write a research

report on a topic given by staff members supervising the reports. These students

shall be required to submit the reports, well in time, before the final semester

examination, so that these can be sent to the external examiners for evaluation.

NOTE: I n Case of students doing MSc in aff il iated Colleges the courses of

F irst and Second Semester wi l l be considered as the cour ses of MSc Previous.

The courses of Thir d and four th semester wil l be the courses of M Sc F inal.

After the final semester examination each student shall have to pass a

comprehensive viva-voce examination carrying 100 marks.


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DETAIL OF INDIVIDUAL COURSES

F IRST SEMESTER

MSCMath-111. REAL ANALYSIS

IReview of the real number system. Properties of real numbers, concepts of supremum andinfimum of Sets of real numbers. Sequences and series; subsequences, cauchy sequences,

completeness, convergence of series. Absolute and conditional convergence of series. Continuity:Properties of continuous functions, continuity and compactness, continuity and connectedness. Typesof discontinuities. Differentiable functions. Mean-value theorems, continuity of derivatives. Implicitfunctions, Jacobians, Implicit function theorem, functional dependence, Taylors theorem for afunction of two variables. Maxima and minima of functions of two and three variables. Method of

lagranges multipliers.

RECOMMENDED BOOKS:1. Addison-Wesley, 1978, Mathematical Analysis, Apostol. T. M.2. Addison-Wesley, 1965, W.Advanced Calculus, Kaplan.3. Rudin. 1976, W. Principles of Mathematical Analysis, 3rd Ed, McGraw Hill.4. S.C.Malik, 1984. Mathematical Analysis, Wiley Eastern Ltd.MSCMath-112. DIFFERENTIAL GEOMETRY

Space curves. Osculating plane. the moving trihedron. Serret-Frenet formulae. Osculating

circle. The concept of surface curves. Spherical and Cylindrical helices. Spherical Indicatrix,Involutes and Evolutes. First fundamental form of a surface. The second fundamental form. Normalcurvature. Principal directions and principal curvatures. Gaussian and mean curvature. Euler's

Theorem. Gauss-Weingarten and Gauss-Godazzi equations.

RECOMMENDED BOOKS:1. E.Weatherburn; 1961 Differential Geometry of Three Dimensions Cambridge Uni., .2. Millman & Parker; 1977 Elements of Differential Geometry, Prentice Hall.3. D.J.Struik; 1962. Lectures on Classical Differential Geometry Addison Wesley.

MSCMath-113. GENERAL TOPOLOGYDefinition of a topological space, open and closed sets. Neighbourhoods. Limit points. Closure

of a set. Comparison of different topologies. Bases and sub-bases. Metric space. First and secondaxiom of countability. Continuous functions and Homeomorphisms. Product spaces. Separationaxioms. Completely regular spaces. Normal spaces. Compactness. local compactness. connectedspaces. Convergence and completeness. Baire's theorem.

RECOMMENDED BOOKS:1. J.R.Munkres; 1975 Topology (A first course), Prentice Hall Inc,.

2. T.Hussain; 1977 Topology and Maps, Plenum Press NY.3. A.Majeed; 1990 Elements of Topology & Functional Analysis, Ilmi Kitab Khana.4. S.Willard; 1970 General Topology, Addison Wesley NY. .

5. F.Simmon; 1963 Introduction to Topology & Modern Analysis, McGraw Hill, NY.


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MSC Math-114. ALGEBRA-IReview of elementary concepts of groups. Cyclic groups, cosets, decomposition of a group,

Lagrange's theorem and its consequences. Normal subgroups, homomorphism of groups. Quotientgroups. Fundamental theorem of homomorphism, Isomorphism theorems, conjugacy classes.Centralisers and normalisers, permutation groups, Cayley's theorem. Endomorphism and

Automorphism of groups, simple groups. (definition and examples), Rings and Fields ideals, quotientrings and homomorphism of rings.

RECOMMENDED BOOKS:1. J.B.Fraleigh, 1976 A First course in Abstract Algebra, Addision Wesley Company.2. I.N.Herstein, 1980 Topics in Algebra, Addison-Wesley.3. N.P.Chaudery, 1983 Abstract Algebra, McGraw Hill Book Co, New Dehli.4. L.J Golstein, 1973 Abstract Algebra, Prentice Hall NJ,5. A. Majeed, 1983 Theory Of Groups, UGC,6. T.S. Blyth & E.F. Robertson, 1986 Essential student Algebra, Vol I-V,

Chapman & Hall,

7. T.S. Blyth & E.F. Robertson, 1984 Algebra Through Practice, Book I-VI,Cambridge University Press.

MSCMath-115. ORDINARY DIFFERENTIAL EQUATIONS

Second order linear differential equations: The fundamental existence theorem (withoutproof) Linear Operators, Fundamental solutions of the homogeneous equations, Reduction of order ofa differential equation, Linear homogeneous equations with constant coefficients, the method of

undetermined coefficients, the method of variation of parameters, the Cauchy Euler equation.Series solution of linear differential equations: Power series solution about an ordinary point

regular singular points, Eulers equation, Series solutions near a regular singular point. Systems offirst order linear differential equations. Boundary value problems, Linear homogeneous boundaryvalue problems, Eigen values and eigen functions.

RECOMMENDED BOOKS:

1. L. L. Pennisi., Elements of Ordinary Differential Equations.2. Boyce & Diprima: 1986 Elemenary Differential Equations and Boundary Value Problems. John

Wiley.3. Robert M. Martin Jr. 1983 Elementary Differential Equations with Boundary Value Problems,

McGraw Hill Book Company Inc.

4. L.R.Shepley: Introduction to Ordinary Differential Equations, John Wiley and Sons.5. Colmb & Shanks, Elements of Ordinary Differential Equations McGraw Hill.


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SECOND SEMESTER

MSCMath-121. REAL ANALYSIS-IIMultiple integrals. Line and surface integrals. Theorems of Gauss. Stoke and Green.

Functions of bounded variation. The Riemann-stieltjes integral: Properties of Riemann-stieltjes

integration Uniform convergence. Difference between pointwise and uniform convergence ofsequences and series of functions, Weierstrass M-Test, Uniform convergence and continuity, Uniformconvergence and integration, Uniform convergence and differentiation.

Improper Riemann-stieltjes integrals: Infinite Riemann-stieltjes integrals, Tests forconvergence of infinite integrals, Infinite series and infinite integrals. Improper Integrals of thesecond kind, uniform convergence of improper integrals.

RECOMMENDED BOOKS:4. Apostol. 1978 T. M. Mathematical Analysis, Addison-Wesley.5. Kaplan . 1965 W.Advanced Calculus ,Addison-Wesley.6. Rudin. N. Y, 1976.W. Principles of Mathematical Analysis, 3rd Ed, McGraw Hill,4. S.C.Malik, 1984. Mathematical Analysis, Wiley Eastern Ltd.

MSCMath-122. DYNAMICSParticle Dynamics: Projectile motion under gravity, constrained particle motion, angular

momentum of a particle.

Orbital Motion: Motion of a particle under a central force, use of reciprocal polarco-ordinates, use of pedal co-ordinates and equations, Kepler's laws of planetary motion.

Motion of a system of Particles: Linear momentum of a system of particles, angular momentumand rate of change of angular momentum of a system, use of centroid, moving origins, impulsive

forces, elastic impact.Introduction to Rigid Body Dynamics: Moments and products of inertia, the theorems of

parallel and perpendicular axes, angular momentum of a rigid body about a fixed point and about

fixed axes, principal axes. Kinetic energy of a rigid body rotating about a fixed point, general motionof a rigid body, momental ellipsoid, equimomental system, coplanar distribution.

RECOMMENDED BOOKS:1. F.Chorlton; 1983.Text book of Dynamics, Ellis Horwood Ltd.2. L.A.Pars; 1953.Introduction to Dynamics, Cambridge Uni. Press.3. A.S.Remsey; 1962 Dynamics Part-I, Cambridge Uni. Press .4. J.L.Synge and B.A.Griffith; 1970 Principle of Mechanics, McGraw Hill Book Co.

MSCMath-123. COMPLEX ANALYSISAnalytic functions. Cauchy-Riemnn equations. Harmonic functions. Contour integration. The

Cauchy integral theorem. The Cauchy integral formula and related theorems. Power series. Uniform

convergence of power series. Taylor's and Laurent's series. The calculus of residues. Singularities.Zeros. Poles and residues. The evaluation of the integrals of certain periodic functions. Indented

contours. Integral involving multiple-valued functions.RECOMMENDED BOOKS:1. L.L.Pennisi; 1976. Elements of Complex Variables, Holt Rinehart & Winston NY.

2. R.V.Churchil; 1960. Complex Variables and Applications, McGraw Hill Co.3. L.V.Ahlfors; 1966. Complex Analysis, McGraw Hill Co.7. S. Levinson and R. Redheffer; 1970. Complex Variables, Holden Day Inc.8. S.B.H.Anthony; Complex Function Theory, Holland N Y.9. M. Iqbal, 1998. Fundamental of Complex Analysis, Ilmi Kitab Khana, Lahore,


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MSCMath-124. ALGEBRAIIReview of elementary concepts of vector spaces. Linear dependence/ independence

of vectors. Vector spaces and subspaces, Direct sum of spaces, Linear transformations, Rank andNullity of linear transformations, Algebra of linear transformations and representation of lineartransformations as matrices. Change of bases, linear functionals. Dual spaces and annihilators, Eigen

vectors and values, Cayley-Hamilton theorem. Diagonalization of matrices, Inner product spaces,Bilinear, Quadratic and Hermitian forms.

RECOMMENDED BOOKS:1. D.T. Finkbeiner, 1978. Introduction to Matrices and Linear Transformations , 3rd. Ed., N.H.

Freeman and company San Francisco,2. A.M.,Tropper,Linear algebra, Thomas Nelson & Sons,.3. 1973. S.Lang.,Linear Algebra, Addison-Wesley.4. K.R.Hoffman and Kunze, 1971 R.Linear Algebra , Prentice Hall.5. I.N.Herstein, 1980.Topics in Algebra, Addison-Wesley,6. K.H.Dar. 1998. First Step to Abstract Algebra, 2nd ed., Feroze Sons Pvt.7. T.S. Blyth & E.F. Robertson, 1986. Essential student Algebra, Vol I-V, Chapman & Hall.8. T.S. Blyth & E.F. Robertson, 1984.Algebra Through Practice, Book I-VI, CUP.

MSCMath-125. SET THEORY AND MATHEMATICAL LOGIC

Set Theory: Functions, Relations, Partially and totally ordered sets, Axiom of choice,Hausdorffs Maximal Principle, Zorns Lemma, Well-Ordering theorem, Zormelos theorem and their

equivalences, Finite, Infinite and Denumerable sets, Cardinal Arithmetic, Ordering of the cardinalnumbers and Schroder-Bernstein theorem, Arithmetic and ordering of the ordinal numbers.

Mathematical Logic: Statement forms and connectives, Tautology and contradiction, Logicalequivalence, Algebra of propositions, Logical Implication, Arguments, Adequate systems ofconnectives.

Applications: Definition and properties of a Boolean algebra ., partial orders in a Boolean

algebra, switching circuit design, Lattices (modular, distributive, complemented).RECOMMENDED BOOKS:1. R.R.Stoll: 1971. Set theory and Logic, Dover Publication Inc. N.Y.2. C.C.Pinter: 1963 Set theory, Addison- Wesley Publishing Company, Inc. N.Y.3. P.R.Halmos: N.J.1968.Nave Set theory. D.van Nostrand Co. Princeton,4. E.Mendelson: 1970 Boolean algebra and Switching Circuits Mc Graw Hill Inc.5. J.D.Monk, 1976 Mathematical Logic, Springer-Verlag N.Y..


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TH IRD SEMESTER

MSCMath-231. NUMERICAL ANALYSIS-I

Numerical Solution of Non-linear Equations: The simple iterative method, the bisection method,

the method of false position, the secant method, the Newton-Raphson method, Rate of convergence ofiterative methods. Numerical solution of polynomial equations by Bairstow method.

Matrix Inversion: Elimination methods (Gauss, Gauss-Jordan and Jordans), CroutsTriangularization method, iterative method, Cholesky method, partition method.

Eigen value problems: Rutishauser method, the power and inverse power method, Rayleigh

quotient method, Jacobi's method, Given's method and House-holder's method.Numerical solutions of simultaneous linear algebraic equations: Solution by matrix inversion

methods and Iterative methods (Jacobi, Gauss-Seidel, successive over relaxation), convergence ofiterative methods.

Interpolation: Lagrange interpolation, Newton divided difference interpolation, Aitkens andinverse interpolations. The error of interpolating polynomials. Finite difference operators(forward, backward, central, average and shift) and tables. Newton's forward and backward

difference formulas.

RECOMMENDED BOOKS:1. W.A.Smith; 1979. Elementary Numerical Analysis, Harper & Row Pub. Int.2. C.E. Froberg, 1974. Inroduction to Numerical Analysis, Addsion-Wesley Co.3. M.K.Jain, 1985.Numerical Methods for Scientific and Engineering Comp., Wiley E. Ltd.

MSCMath-232. MATHEMATICAL STATISTICS-IProbability and its various definitions and laws, Marginal and conditional probability, Bays

formula, law of independence, Discrete and continuous random variables, Mathematical expectation,

density function generating functions, distribution functions, characteristic function. Distribution:uniform, binomial, negative binomial, poisson , geometric and hyper geometric and their moments.

Regression and Correlation, Partial correlation and multiple correlation.RECOMMENDED BOOKS: 4. A.Samad Hirai; Mathematical Statistics, Ilmi KitabKhana, Lahore.

5. S.M.Chaudhry; Introduction to Statistical Theory, Ilmi Kitab Khana, Lahore.

1. Hogg & Craig; 1978. Introduction to Mathematical Statistics, The McMillan Co.2. Mood & Garaybill; 1974. Introduction to Theory of Statistics, McGraw Hill Co.3. John & Walpole; 1980. Mathematical Statistics, Prentice Hall In.,

MSCMath-233. FUNCTIONAL ANALYSIS-INormed linear spaces, the Holder and Minkowski inequalities. Linear Operators. Linear

functionals. Annihilators. Banach spaces. Quotient spaces. Conjugate spaces. Principle of UniformBoundedness, Hahn-Banach theorem in normed spaces, Open Mapping and Closed graph theoremsand their applications. Topological Linear Spaces.

RECOMMENDED BOOKS:1. A.E.Taylor & D.C.Lay; 1980. Introduction to Functional Analysis, John Wiley & Sons.2. E.Kreyszig; 1978. Introductory Functional Analysis With Applications, John Wiley & Sons,3. W.Rudin; 1978. Functional Analysis, McGraw Hill.4. G.F.Simmons; 1963. Introduction to Topology & Modern Analysis, McGraw Hill.5. K.Yosida; 1967 Functional Analysis, Springer Verleg NY


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6. Abdul Majeed; 1997. Elements of Topology and Functional Analysis, Ilmi Kitab Khana, LahoreMSCMath-234. ALGEBRA-III

Direct product of groups, subnormal series of a group, refinement theorem, composition seriesand Jordan Holder theorem, solvable and nilpotene groups, Sylow groups and Sylow theorems, freegroups and finitely generated abelian groups.

RECOMMENDED BOOKS:1. J.B.Fraleigh; 1976. A First Course in Algebra, Addison Wesley Co.2. I.N.Herstein; Topics in Algebra, Ginn & Co.

3. P.M.Cohn; Algebra Vols, I & II, John Wiley & Sons.4. Abdul Majeed; Theory of Groups, University Grant Commission.5. T.S. Blyth & E.F. Robertson, 1986. Essential student Algebra, Vol I-V, Chapman & Hall6. T.S. Blyth & E.F. Robertson, 1984. Algebra Through Practice, Book I-VI, CUPMSCMath-235. ANALYTICAL MECHANICS-I

Generalized Co-ordinates. Constraints. Holonomic and Non-holonomic system. Conservationtheorem. Principle of virtual work. D'Alembert's principle. Konning theorem. Euler dynamical

equation of motion of rigid body about a fixed point. Properties of rigid body under no force. The

rotating earth. Lagrange's equation for holonomic and non-holonomic system and their application.Energy Integral. Application of Noether's theorem. Equation of motion with or without Lagrangemultipliers. Lagrange,s equation for impulsive force. Motion of a symmetrical top. Eulerian angles.

RECOMMENDED BOOKS:1. H.Goldstein; 1987. Classical Mechanics, Addison Wesley Co.2. F.Chorlton; 1983. Text Book of Dynamics, John Wiley & Sons.3. L.A.Pars; A Treatise on Analytical Dynamics, Cambridge Uni. Pr.

4. E.T.Whittaker; 1970. A Treatise on Analytical Dynamics of Particles and Rigid Bodies, CUP.MSCMath-236. QUANTUM MECHANICS-I

Inadequacy of Classical Mechanics. Wave particle duality. Schrodinger's equation. Harmonic

oscillator. One dimensional motion in a potential well. Reflection by and transmission across apotential barrier. Uncertainty principle. Direct delta function. Operator formulism in Quantum

Mechanics. Angular momentum. Pauli exclusion principle. Hydrogen atom.

RECOMMENDED BOOKS:1. R.L.White; 1966.Basic Quantum Mechanics, McGraw Hill Book Co. NY,

2. L.I.Schiff; 1955. Quantum Mechanics, McGraw Hill Kogakusha Ltd.3. P.T.Mathews; 1974. Introduction to Quantum Mechanics, McGraw Hill Book Co.4. Dicke & Wittke; 1966. Introduction to Quantum Mechanics, Addison Wesley Pub. Co. Inc.5. F.Mandl; 1966. Quantum Mechanics, Butterworth, London.7th Impression6. P.M.Mathews, K.Venkatesan; 8th Reprint, 1984. A Text Book Of Quantum Mechanics, Tata

Mc GrawHill Publishing Company Limited, New Delhi,7. P.A.M.Dirac; Introduction to Quantum Mechanics.

8. Riazuddin and Fayyazuddin; 1990. Introduction to Quantum Mechanics, World Scientific.


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MSCMath-237. MEASURE & INTEGRATION-ILebesgue measure, Outer measure. Measurable set and Lebesgue measure, A non-measurable

set, measurable function. The Lebesgue Integral: The Lebesgue integral of a bounded function. Thegeneral Lebesgue integral. Lebesgue integral and its relation to Riemann integral. Convergence in

measure.

RECOMMENDED BOOKS:1. H.L.Royden; 1968.Real Analysis, The McMillian Co.2. D de Barra; 1981. Measure Theory & Integration, Eillis Horwood Ltd.,3. P.R.Halmos. 1950. ,Measure Theory, Von Nostrand NY,4. A.Mukherjea; 1978. Real and Functional Analysis, Plenum and K.Pothoven Press,5. A.Rahim Khan; Introduction to Lebesgue Integration, Ilmi Kitab Khana, Lahore.MSCMath-238. ALGEBRAIC TOPOLOGY-I

Categories. Functions, Paths and path connected spaces, Homotopy, Simplicial complex,Homotopy and homeomorphism of polyhedra. Subdivision and the simplicial approximation theorem .

The Fundamental Group and Covering spaces, the fundamental group of circle, the fundamental

group of covering spaces.RECOMMENDED BOOKS:1. E.H.Spanier; 1966. Algebraic Topology, Tata McGraw Hill2. C.Kosniowski; 1988. A First Course in Alegbraic Topology, Cambridge Uni. Pr.

3. C.R.E.Maunder; 1980. Algebraic Topology, Cambridge Uni. Pr.,4. J.Mayer; Algebraic Topology, Prentice Hall NJ.

MSCMath-239. RELATIVITY-I (SPECIAL RELATIVITY)Historical background. Gallillean transformations. The postulates of special relativity. Lorentz

transformations. Simultaneity in special relativity. Length contraction and time dilation. Velocity

addition formulae. The four-vector formulism and Minkowski space. Time four-vector and energymomentum four-vector. Equivalence of mass and energy. Relativistic Kinematics. Doppler's effect.

Particle production and particle decay. Threshold and binding energies. Accelerational effects inspecial relativity. The principles of equivalence and gravitation effects in special relativity. The clock

paradox. Tachyaons. Electromagnetism in relativity.

RECOMMENDED BOOKS:1. Adler,Bazin & shiffer; 1965 Introduction to General Relativity, McGraw Hill Co.2. Sears & Brehme; 1968 Introduction to the Theoryof Relativity, Addison Wesley.

3. Derek & Lawden; An Introduction to Tensor & Relativity Science Paper Books.4. Synge; Relativity Vol,: I5. W.Rindler; Eessential Relativity.6. A.P.French; Special Relativity, The ELBS Nelson.7. C.Moller; Introduction to Relativity.

8. Asghar Qadir; 1989. Relativity: An Introduction to the Special Relativity, World Scientific,9. M.Saleem & M.Rafique; An Introduction to Special Relativity.


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MSCMath-2310. FLUID MECHANICS-IIntroduction, Fluid and Continuum hypothesis, surface and body forces, stress at a point,

Viscosity, Newtons viscosity law; Viscous and Inviscous flows; Laminar and Turbulent flows;Compressible and Incompressible flows; Lagrangian and Eulerian descriptions; Local, Convective

and Total rates of change; Conservation of mass. Inviscous Fluids, Irrotational motion, Boundaryconditions, Streamlines, Vortex lines and Vortex Sheets; Kelvins minimum enery theorem,Conservation of Linear momentum; Bernoullis theorem and its applications; Circulation: Rate ofchange of circulation (Kelvins theorem): Axially-symmetric motion: Stokes stream function.

Inviscous Fluids: Two-dimensional motion, stream function, complex potential and some

potential flows; sources, sinks and doublets; Circle theorem; Method of images; Blasius theorem;Aerofoil and the theorem of Kutta and Joukowski; Vortex motion; Karmans vortex street.

RECOMMENDED BOOKS:1. I.G.Currie, 1974 Fundamental Mechanics of Fluids, McGraw-Hill Co.2. Nazeer Ahmed, 1987. Fluid Mechanics, Engineering Press Inc.,

3. H. Schlichting, 1979. Boundary Layer Theory, McGraw-Hill Co.4. F.Chorltan, 1985. Fluids Dynamics, CBS Pub. & Dist.

5. G.K.Batchelor; 1969. An Introduction to Fluid Dynamics, Cambridge Uni. Press,

MSCMath-2311. ELECTROMAGNETIC THEORY-IEquations of electrostatic and magnetostatic boundary conditions, Boundary value problems

and methods of solution, Electrostatics and magnetostatics of macroscopic medium. Dipoles andMultipole. Dielectrics. Steady currents and their interaction. Varying Currents. Electromagnetic

induction. Maxwell's equations.

RECOMMENDED BOOKS:1. V.C.A Ferrars; 1950 Electromagnetic Theory, ELBS London,

2. Lorrain & Corson; 1970. Electromagnetic Fields and Waves,Toppan Company Ltd.3. C.A.Coulson; 1951. Electricity, liver & Boyd Edinburgh,4. A.S.Ramsey; 1952. Electricity & Magnetism, Cambridge Uni. Pr.

5. J.R.Reitz, F.J.Milford & Christy; Foundation of Electromagnetic Theory.

MSCMath-2312. ASTRONOMY-IElements of spherical Astronomy. Astronomical Coordinate system. Celestical Navigation.

Time. Planetary Motion. Celestial Mechanics. Artificial Statistics.Solar system, planetary study

(albede, temperature and brightness of planets, planetary Atmospheres. Surface Feature andsatellites).Earth Moon system (Dimensions and figure of Earth ,Earth's Atmosphere, refraction.Moon's Motion, Mass and size Moon's Surface). Suns interior and

Layers. Quiet and Active sun, Interaction with Earth Atmosphere. Meteors, Comets andAsteroids. Inter planetary Medium. Evolution of Solar system.Use of Telescope and AstronomicalHand Books.

RECOMMENDED BOOKS:1. W.M.Smart; 1962. Text Book on Spherical Astronomy Cambridge Uni. pr.2. H.S.Schild; Structure & Evolution of Stars Dover Pub. Pr.3. W.M.Smart; 1957.Foundation of Astronomy, C.U.P4. Jastraw & Thompson; Astronomy.

5. M.Harwit; Astrophysical Concepts.


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MSCMath-2313. HOMOLOGICAL ALGEBRA-ITensor product of modules and groups of homomorphism. Direct and Inverse Limits.

Categories and functions Projective and injective modules. Applications.

RECOMMENDED BOOKS:

1. D.G.Northcott; 1960. An Introduction to Homological Algebra, Cambridge Uni Pr.2. L.O.Barbora; Homological Dimension of Modules, AMS3. P.Hilton; Lectures in Homological Algebra, AMS

4. Carton & Osopaky; 1960. Homological Algebra, Princeton Uni. Pr.5. S.Maclane; 1970. Homology, Springer Verlog.

MSCMath-2314. OPERATIONS RESEARCH-IIntroduction, Formulation and graphical solution of two variables linear programms, Simplex

method, Method of Penality, the two-phase technique, Sensitivity analysis, Dual Simplex method,Duality, Sensitivity and parametric Analysis, Transportation and Assignment Models. LinearProgramming: Advanced Topics.

RECOMMENDED BOOKS:1. H.A.Taha; 1982. Introduction to Operations Research , 3rd Ed. McMillan Pub. Company N.Y.2. G.Hadley; Introduction to Linear Programming.

MSCMath-2315. GRAPH THEORYUndirected graphs, Geometric graphs, Abstract graphs, Isomorphism, Edge progressions chains

and circuits rank and nullity, Degrees, Trees. Bipartite graphs, Unicursal graphs, Hamiltonian Graphs.Directed graphs, Arc Progressions, paths progression and cycle progression.

Partition and distances in graphs, edge partitions, Arc partitions, Hamiltonian chains andcircuits, vertex partitions, radius and diameter, minimal length problem. Foundation of electrical network theory. Matrix representation, the incidence matrix, the circuit matrix, the cut set matrix, the

vertex or adjacency matrix, the path matrix.

RECOMMENDED BOOKS:1. Robert G. Busacker and Thomas L. Seaty; Finite graphs and Networks'An introduction with applications' , McGraw Hill Book Company.

2. Robin J. Wilson; 1985. Introduction to Graph Theory, Longman Scientific and technical,3. Wai-Kaichen; 1976. Applied graph Theory "graphs and Electrical networks, North-Holland Pub.

MSCMath-2316. APPLIED ALGEBRA-IFinite State Machines: Binary devices and states, finite state machines, covering and

equivalence, equivalent states, a minimization procedure. Programming Languages: identifiers arrays ,for statements, block structures in ALGOL.

Boolean polynomials, connections with logic, logical capabilities of ALGOL, Booleanapplications and subalgebras.

Optimization and Computer Design: Computerizing optimization, logical design, theminimization problem.

RECOMMENDED BOOKS:1. G.Birkhoff & T.C.Bartee; 1970 Modern Applied Algebra, Mc Graw Hill Book Co. N.Y.2. M.Minsky; Computation: 1967. Finite and infinite Machines, Prentice-Hall,3. T.C.Bartee, I.L.Lebaw & I.S.Reed: Theory and Design of Digital Machine, Mc Graw Hill.


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MSCMath-2317. HISTORY OF MATHEMATICS-IChinese, Hindu, and Arabic Mathematics; European Mathematics 500 to 1600; The

Seventeenth century Mathematics (the Dawn of modern Mathematics, the Calculus and relatedconcepts).

RECOMMENDED BOOKS:1. Howard Eves; 1990. An Introduction to the History of Mathematics, Saunders College Pub.Philadelphia , Sixth Ed.,

2. Carl B. Boyer And Uta C. Merzbach; 1989. A History of Mathematics, John Wiley & Sons,N.Y., 2nd Ed.,

MSCMath-2318. RIEMANNIAN GEOMETRYThe concept of manifolds. Tensors defined on manifolds. Definite and indefinite metrics.

Covariant derivative. Intrinsic derivative and Lie derivative. Parallel and Lie transport. Killing vectors.Curvature and torsion tensors. Bianchi identities. Geodesic deviation. Conformal transformation, andthe conformal curvature tensor.

RECOMMENDED BOOKS:1. Livelock & Rund; Tensors, Differential Form and Variational Principles, John Wiley2. D.Langwitz; 1977. Differential & Riemannian Geometry, Academic Press,3. L.P.Eisenhart; Riemannian Geometry, Princeton Uni. Pr.

MSCMath-2319. INTEGRAL EQUATIONSClassification of integral equations. Voltera integral equations. Relation between linear

differential equations and Voltera's integral equations. Solution of the integral equation of second

kind in series. The method of successive approximation and substitution. Method of LaplaceTransform. Iterated Kernels. Reciprocal kernel. Voltera's solution of the Fredholm's equation.

Fredholm's two fundamental relations. Fredholm's solution of the integral equation when D()=0,

Solution of the homogeneous equation when D()=0, D/() 0, Solution of the homogeneous integral

equation when D()=0. Characteristic constants and fundamental functions. Associated homogeneous

integral equation. Kernels of the form ai(x) bi(y) Existence of at least one characteristic constant for

a symmetric kernel. Orthogonality of fundamental functions for symmetric kernel. Schmidt's solutionof the non-homogeneous integral equation.

RECOMMENDED BOOKS:1. W.V.Lovitt; 1950. Linear Integral Equations, Dover Pub. Inc.2. A.J.Jerri; 1999. Introduction to Integral Equations with Applications, Marcel Dekker Inc.3. R.P.Kanwal; 1996. Linear Integral Equation, Academic Press.N.Y.4. H.Hochstadt; 1973. Integral Equations, John Wiley N.Y.


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MSCMath-2320. OPTIMIZATION THEORY-IStatement of the problem, condition for optimality, concept of direction of search, alternating

direction and steepest descent methods, conjugate direction method, conjugate gradient method,Newtons method, Quasi-Newton equation, derivation of updating formulae for Quasi-Newton

equation, The Gauss-Newton method, The Levenberg-Marquart method, The corrected Gauss-Newton method, Methods for large scale problems.

RECOMMENDED BOOKS:1. Gill, P.E., Murray, E & Wright, H.H. 1981. Practical Optimization, Academic Press,2. Fletcher,R. 1980. Practical Methods of Optimization Vol.I & II, John Wiley and Sons,3. S.S. Rao., 1984. Optimization Theory and Application., Wiley Eastern Ltd,3. David G. Luenberger, 1968. Optimization by Vector Space Methods, John Wiley & Sons4. David G. Luenberger, 1965. Introduction to Linear & Nonlinear Programming. Addison Wesley

Publishing Co. Sydney,5. Bazaraa, M.S. and Shetty, C.M., 1979. Nonlinear Programming: Theory and Algorithms,

John Wiley,

MSCMath-2321. BIO-MATHEMATICSThe history of a system in the course of irreversibility transformation. The statistical meaningof irreversibility. Evolution conceived as a redistribution. The programme of physical biology. Thefundamental equations of the kinetics of the evolving systems. (General case; equations with onedependent variable; equations with two or three dependent variables). Analysis of the growth function.

RECOMMENDED BOOKS:1. A.J.Lotka; 1956. Elements of Mathematical Biology, Dover Publications, N.Y.


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FOURTH SEMESTER

MSCMath-241. NUMERICAL ANALYSISIINumerical differentiation: Forward difference formulas, Central difference Formulas,

error in numerical differentiation, extrapolation to the limit.Numerical integration: the rectangular, Trapezoidal and Simpson rules. Romberg

integration. Method of undetermined coefficients.Numerical solution of Difference Equations: Formation of difference equations,

numerical solution of linear homogeneous and nonhomogeneous difference equations with

constant coefficients.Numerical solution of Ordinary Differential Equations: Initail Value Problems;

Taylor's methods and truncation error. Eulers method and error estimates. Modified Euler

method. Runge-Kutta methods. Predictor and corrector methods; Milne-Simpson method,

Adams-Bashforth-Moulton method and Hamming's method. Method of undeterminedcoefficients for multisteps methods. Stability analysis of numerical methods. Boundary

value problems; Finite difference method and the Shooting method.Numerical Solution of Partial Differential Equations: Finite difference method for solving Parabolic,

Hyperbolic and Elliptic Equations

RECOMMENDED BOOKS:1. W.A.Smith; 1979. Elementary Numerical Analysis, Harper & Row Pub.,2. C.E. Froberg, 1974. Inroduction to Numerical Analysis, Addsion-Wesley Co.3. M.K.Jain, et al: Numerical Methods for Scientific and Engineering Computation, Wiley Ltd.MSCMath-242. MATHEMATICAL STATISTIC-II

Distributions: Rectangular, Triangular, Laplace, Gamma, Beta, Exponential, Normal and theirmoments. Sampling theory and distribution, transformation of variables, order statistics, theory of

point and interval estimation, limiting distribution, sufficient statistics, statistical hypothesis and tests.

RECOMMENDED BOOKS:1. Hogg & Craig; 1978. Introduction to Mathematical Statistics, The McMillan Co.,2. Mood & Garaybill; 1974. Introduction to Theory of Statistics, McGraw Hill Co.,3. John & Walpole; Mathematical Statistics, 1980. Prentice Hall Int.,

4. A.Samad Hirai; Mathematical Statistics, llmi Kitab Khana, Lahore.5. S.M.Chaudhry; Introduction to Statistical Theory, Ilmi Kitab Khana, Lahore.MSCMath-243. FUNCTIONAL ANALYSIS-II

Inner product spaces, Hilbert spaces, orthonormal bases, convexity in Hilbert spaces,operators in Hilbert spaces, Invariant sub-spaces. Decomposition of Hilbert spaces. Finite

dimensional spectral theory, and spectral mapping theorem.

RECOMMENDED BOOKS:

1.

A.E.Taylor & D.C.Lay; 1980. Introduction to Functional Analysis,John Wiley & Sons,2. E.Kreyszig; 1978. Introductory Functional Analysis With Applications,

John Wiley & Sons,3. W.Rudin; 1978. Functional Analysis, McGraw Hill,4. G.F.Simmons; 1963. Introduction to Topology & Modern Analysis McGraw Hill,

5. K.Yosida; ,1967 Functional Analysis, Springer Verleg NY6. Abdul Majeed; 1997.Elements of Topology and Functional Analysis,

Ilmi Kitab Khana, Lahore,


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MSCMath-244. ALGEBRA-IVOperations on ideals, Isomorphism, Integral domains and fields, Embedability of an integral

domain in a field, Field of quotients, Maximal, Prime and primary ideals and their properties,Divisibility theory in integral Domains, Polynomial rings, Division Algorithm, Remainder theorem.R(X) as a unique factorization Domain, Irreducible Polynomials, Primitive polynomials, Gauss's

Lemma, simple algebraic Extensions, Transcendental extensions, Finite extensions of fields andrelated theorems, splitting field extensions. Elements of Galois theory.

RECOMMENDED BOOKS:1. J.B.Fraleigh; 1976. A First Course in Algebra, Addison Wesley Co.2. I.N.Herstein; Topic in Algebra, Ginn & Co.3. P.M.Cohn; Algebra Vol: I & II, John Wiley & Sons.4. Burton; A First Course in Rings & Ideals, Addison Wesley Co.5. J.Lambek; Lectures on Rings & Modules, Blaisdel.

6. T.S. Blyth & E.F. Robertson, 1986. Essential student Algebra, Vol I-V, Chapman & Hall.7. T.S. Blyth & E.F. Robertson, 1984. Algebra Through Practice, Book I-VI, CUP.MSCMath-245. ANALYTICAL MECHANICS II

Hamiltonian theory, Cyclic co-ordinates and Routh's procedure. Variational principle.Canonical transformation. Lagrange's and Poisson's brackets. Hamilton Jacobi theory. Theory ofsmall oscillations. Normal co-ordinates and normal modes of vibration.

RECOMMENDED BOOKS:1. H.Goldstein; 1987. Classical Mechanics, Addison Wesley Co.2. F.Chorlton; 1983. Text Book of Dynamics, John Wiley & Sons,3. Gupta & Satya; Classical Mechanics, KRN, Meerath.4. L.A.Pars; A Treatise on Analytical Dynamics, Cambridge Uni. Pr.5. E.T.Whittaker; 1970.A Treatise on Analytical Dynamics of Particles and Rigid Bodies. CUP.

MSCMath-246. QUANTUM MECHANICS-IIHeisenberg equations of motion and equivalence of Schrodinger and Heisenberg physical

pictures. Scattering theory. Born approximation. Partial wave analysis. Optical theorem. Timedependent & time independent perturbation theory. Selection rules. Klein-Gordon equation. Dirac'sequation. Spin angular momentum.

RECOMMENDED BOOKS:1. R.L.White; 1966.Basic Quantum Mechanics, McGraw Hill Book Co. NY,2. L.I.Schiff; 1955. Quantum Mechanics, McGraw Hill Kogakusha Ltd.

3. P.T.Mathews; 1974.Introduction to Quantum Mechanics, McGraw Hill Book Co.4. Dicke & Wittke; 1966.Introduction to Quantum Mechanics, Addison Wesley Pub. Co. Inc.5. F.Mandl; 1966. Quantum Mechanics, Butterworth, London. 7th Impression.6. P.M.Mathews, K.Venkatesan; 1984. A Text Book Of Quantum Mechanics, Tata Mc Graw Hill

Publishing Company Limited, New Delhi, 8th Reprint,

7. P.A.M.Dirac; 1990 Introduction to Quantum Mechanics. Riazuddin and Fayyazuddin;Introduction to Quantum Mechanics, World Scientific,


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MSCMath-247. MEASURE & INTEGRATION-IIMeasure spaces, Measurable functions. Integration, General convergence theorems. Signed

measures. The Radon-Nikodym theorem. The L -spaces, Outer measure and measurability, Theextension theorem, The Lebesgue Stieltjes integral, product measure. Inner Measure.

RECOMMENDED BOOKS:

1. H.L.Royden; 1968. Real Analysis, The McMillian Co.,2. G de Barra; 1981.Measure Theory & Integration, Eillis Horwood Ltd.,3. P.R.Halmos; 1950.Measure Theory, Von Nostrand NY,

4. A.Mukherjea; 1978. Real and Functional Analysis, Plenum and K.pothoven Press,5. A.Rahim Khan; Introduction to Lebesgue Integration, Ilmi Kitab Khana, Lahore.

MSCMath-248. ALGEBRAIC TOPOLOGY-IIHomology theory: Homology groups, simplical homology, exact sequences. singular

homology. Cohomology, Duality and Topological manifolds. The Alexander Poincar's dualitytheorem.

General homotopy theory: some geometric construction. Homotopy classes of maps, Exact

sequences, Fibre and cofibre maps.

RECOMMENDED BOOKS:1. E.H.Spanier; Algebraic Topology, Tata McGraw Hill2. C.Kosniowski; 1988. A First Course in Algebraic Topology, Cambridge Uni. Pr.3. C.R.E.Maunder; 1980. Algebraic Topology, Cambridge Uni. Pr.,

4. J.Mayer; Algebraic Topology, Prentice Hall NJ.

MSCMath-249. RELATIVITY-II (GENERAL RELATIVITY)Vector fields in affine and Riemann space. Applications of Tensor analysis. Closed and exact

tensors. Tensor densities. Dual tensors. Vector fields on curves. Geodesics. Maxwell's equations intensor form. Proper time and the equations of motion. Gravity as a metric phenomenon. The red shift.

Criteria for the field equations. The Rieman curvature tensor and its symmetry properties. TheBianchi identities. The Einstein field equation. for free space.The Schwarzchild solution. Geodesics

in the Schwarzchid space time. Black holes.

RECOMMENDED BOOKS:1. Adler, Bazin & shiffer; 1965. Introduction to General Relativity, McGraw Hill Co.2. Sears & Brehme; 1968. Introduction to the Theory of Relativity, Addison Wesley,3. Derek & Lawden; An Introduction to Tensor & Relativity4. W.Rindler; Eessential Relativity.

5. C.Moler; Introduction to Relativity.

MSCMath-2410. FLUID MECHANICS-IIViscous Fluids: Two-dimensional motion, stream function, complex potential and some

potential flows; sources, sinks and doublets; Circle theorem; Method of images; Blasius theorem;

Aerofoil and the theorem of Kutta and Joukowski; Vortex motion; Karmans vortex street. ViscousFluids, constitutive equations; Navier-Stokes equations; Exact solutions of Navier-stokes equations,Steady unidirectional flow; Poiseuille flow, Couette flow, Unsteady Unidirectional flow; sudden

motion of a plane boundary in a fluid at rest, flow due to an oscillatory boundary, Equations ofmotion relative to a rotating system, Ekman flow, Dynamical similarity and the Reynolds number,

Boundary layer concept and its governing equations; Flow over a flat plate (Blasius solution);Reynolds equations of turbulent motion.


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RECOMMENDED BOOKS:1. I.G.Currie, 1974 Fundamental Mechanics of Fluids, McGraw-Hill Co.2. Nazeer Ahmed, 1987.Fluid Mechanics, Engineering Press Inc.3. Schlichting, 1979. Boundary Layer Theory, McGraw-Hill Co.4. F.Chorltan, 1985. Fluids Dynamics, CBS Pub. & Dist.,5. G.K.Batchelor; 1969. An Introduction to Fluid Dynamics, Cambridge Uni. Press,MSCMath-2411. ELECTROMAGNETIC THEORY-II

Energy, momentum (poynting) vectors and stress tensor of electromagnetic fields. Wavepropagation, Waves in a conducting medium, reflection and dispersion. Lorentz formula. Waveguide and cavity resonators. Spherical waves. Field of a uniformly moving charged particle. Field ofan oscillating dipole. Diffraction of a electromagnetic waves.

RECOMMENDED BOOKS:1. V.C.A Ferrars; 1950 Electromagnetic Theory, ELBS London.2. Lorrain & Corson; 1970.Electromagnetic Fields and Waves, Toppan Company Ltd.3. C.A.Coulson; 1951.Electricity, Cliver & Boyd Edinburgh.

4. A.S.Ramsey; 1952. Electricity & Magnetism, Cambridge Uni. Pr.

5. J.R.Reitz, F.J.Milford & Christy; Foundation of Electromagnetic Theory.

MSCMath-2412. ASTRONOMY-IIPlanetary phenomena Astronomical Corrections. Proper motion occultations and eclipse. Stars

in the milky way. Classification of Stars. H-R diagram. Stellar Interiors, Binary variable and peculiarstars. Exterior Galaxies. The physical universe, Cosmology, Use of telescope and astronomical hand

book.

RECOMMENDED BOOKS:1. W.M.Smart; Text Book on Spherical Astronomy, Cambridge Uni. pr., 1962.2. H.S.Schild; Structure and Evolution of Stars Dover Pub. Pr.

3. W.M.Smart; Foundation of Astronomy, C.U.P4. Jastraw & Thompson; Astronomy M.Harwit; Astrophysical Concepts

MSCMath-2413. HOMOLOGICAL ALGEBRA-IIDerived Functions. Homology and Cohomology theory of groups. Theory of homological

dimension. Application.

RECOMMENDED BOOKS:1. D.G.Northcott; 1960. An Introduction to Homological Algebra, Cambridge Uni Pr.2. L.O.Barbora; Homological Dimension of Modules, AMS3. P.Hilton; Lectures in Homological Algebra, AMS4. Carton & Osopaky; 1960. Homological Algebra, Princeton Uni. Pr.,

5. S.Maclane; 1970. Homology, Springer Verlog,

MSCMath-2414.OPERATIONS RESEARCH-IIMatric definition of the standard LP problem, Foundation of Linear Programming, Revised

simplex Method, Bounded variables, Decomposition Algorithm, Parametric Linear Programming.

Application of Integer Programming, Cutting Methods, the Fractional (pure Integer) algorithm,mixed algorithm, Game theory, graphical solutions of two-person zero-sum games, mixed strategies,Graphical solution of (2xn) and (mx2) games, Solution of (mxn) games by linear programming.

RECOMMENDED BOOKS:1. H.A.Taha; 1982. Introduction to Operations Research , 3rd Ed. McMillan Pub. Company N.Y.


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MSCMath- 2415. ADVANCED TOPOLOGYDirected sets and nets, subnets and cluster points, sequences and subsquences, quotient spaces,

the Tychonoff theorm, completely regular spaces, the Stone-Eeih compactification, meterizationtheorms and paracompactness, function spaces.

RECOMMENDED BOOKS:

1. J.L.Kelley, , 1975. General Topology, Springer Verlag2. S.Willard; 1970.General Topology, Addison Wesley Pub. Co.3. J.R.Munkers, 1975.Topology (a first course), Prentice Hill Inc.

MSCMath-2416. APPLIED ALGEBRA-IIBinary Group Codes: Encoding and decoding, block codes, matrix encoding techniques, group

code, decoding tables, hamming codes.Lattices: Lattices and Posets,Lattices as Posets, Lattices and Semilattices, sublattices, distributive

lattices, modular and geometric lattices, Boolean lattices.Polynomial Codes: Polynomial codes, advantageous properties, shift registers, formal

communications

Recurrent Sequences: Redar communications systems, difference codes, difference equations,

formal power series, application to difference codes, recurrent sequences.RECOMMENDED BOOKS:1. G.Birkhoff & T.C.Bartee; 1970. Modern Applied Algebra, McGraw Hill Book Co.2. S.W.Golomb; 1964. Digital Communications, Prentice Hall

3. A.Gill; 1966. Linear Sequential Circuits, McGraw Hill.

MSCMath-2417. HISTORY OF MATHEMATICS-IIThe Eighteenth Century and the Exploitation of the Calculus; The Early Nineteenth Century

and the Liberation of Geometry and Algebra; The Later Nineteenth Century and the Arithmatizationof Analysis into the twentieth Century.

RECOMMENDED BOOKS:

1.

Howard Eves; 1990. An Introduction to the History of Mathematics,Saunders College Pub. Philadelphia etc, Sixth Ed.

2. Carl B. Boyer And Uta C. Merzbach; 1989. A History of Mathematics,John Wiley & Sons, N.Y. 2nd Ed.,

MSCMath-2418. CONTINUOUS AND SYMMETRIC GROUPSDiscrete and continuous groups. Parameterisation of a group. symmetric group and its

representations. symmetric and action groups. Many parameter continuous groups and lie groups.Lie algebra. General linear, orthogonal and unitary groups. Young tableau.

RECOMMENDED BOOKS:1. M.Hammermesh; Group Theory, Addison Wesley

2. E.P.Wigner; Group Theory, Addison Press


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MSCMath-2419. PARTIAL DIFFERENTIAL EQUATIONS:Formulation of Equations. Linear First-order Equations. Quasilinear First-order

Equations, Method of Lagrange. Cauchy Problem for First-order Equations. LinearSecond-order Equations in Two Independent Variables. Normal Forms. Hyperbolic,

Parabolic and Elliptic Equations. Cauchy Problem for Linear Second-order Equations in

Two Independent Variables. Adjoint Operator. Self-adjoint Differential Operator forEquations in Two Independent Variables. Laplaces Equation. Separation of Variables

One-dimensional Wave Equation.

RECOMMENDED BOOKS:1. R.Dennemeyer,. 1968. Introduction to Partial Differential Equations and Boundary Value

Problems McGraw Hill Co. N.Y.2. I.N.Sneddon., 1957. Elements of Partial Differential Equations. McGraw Hill Co.3. C.R.Chester., 1957 Techniques in Partial Differential Equations. McGraw Hill Co.4. R.Haberman, 1993.Elementary Applied Partial Differential Equations with Fourier series and

Boundary Value Problems. Prentice Hall Inc.

5. W.E.Williams: 1980. Partial Differential Equations. Clarendon Press Oxford.MSCMath-2420. OPTIMIZATION THEORY-II

Theory of constrained optimization, condition of optimality, methods for minimizing ageneral function subject to linear equality constraints, active set strategies for linear inequalityconstraints, special forms of the objectives functions, Lagrange multiplier estimates, Changes inworking set, Barries function methods, Penalty functions methods, Methods based on Langrangianfunctions reduced gradient and gradient projection methods.

RECOMMENDED BOOKS:1. Gill, P.E., 1981. Murray, E & Wright, H.H. Practical Optimization, Academic Press,2. Fletcher,R. 1980. Practical Methods of Optimization Vol.I & II, John Wiley and Sons.3. S.S. Rao., 1984. Optimization Theory and Application. Wiley Eastern Limited.4. David G. Luenberger, 1968. Optimization by Vector Space Methods,

John Wiley & Sons, New York.5. David G. Luenberger, 1965. Introduction to Linear & Nonlinear Programming . Addison-WesleyPublishing Co. Sydney.

6. Bazaraa, M.S. and Shetty, C.M., 1979. Nonlinear Programming: Theory and Algorithms, JohnWiley & Sons,

MSCMath-2421. CATEGORY THEORYBasic Concept of category, examples, epimorphisms monomorphisms, retraction,

coretractions, initial, terminal and null objects, category of graphs, equailzers, products and dual

notations, equivalence relations and kernel pairs, pull backs, inverse image and inter sections,constructions with kernel pairs, existence theorem coequilizers, functors, bifuntors, natural

transformation, diagrams limit, colimits, universal problem and adjoint functors.

RECOMMENDED BOOKS:1. Herrifch H and Strecker G.E. Category Theory, Allys and Bacon2. Maclane S. Categories for the working Mathematicians Heidlberge

Barry Mitchell. Teory of Categories, Academic Press.


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MSCMath-2422. RELIABILITY ANALYSIS, QUALITY AND SAFETY1. Need for Reliability, Quality and safety History: Reliability, Quality, Safety2. Introduction to Reliability: Need for Reliability, Bathtub Hazard Rate Curve General and

specific Hazard Rate functions, General and specific Reliability Functions, Mean time offailure, Failure Rate Estimation, Failure Data Collection, Sources and failure rates for

selected items.3. Static Reliability Evaluation Models: Introduction, Series Network, Parallel Network,

Series-Parallel Network, Parallel-Series Network, Bridge Network.4. Dynamic Reliability Evaluation Models: Introduction, Series Network, Parallel Network,

Series-Parallel Network, Parallel-Series Network, Bridge Network.

5. Reliability Evaluation Methods; Reliability Testing; Reliability Management andcosting;

RECOMMENDED BOOKS:

1. Reliability, Quality and safety for Engineers by B.S.Dhillon, CRC Press, N.Y. 2005.

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