MS/M.Phil and Ph.D Program - University of Sargodha · compact sets in locally convex spaces,...

2011-Onward

MS/M.Phil and Ph.D Program

Department of Mathematics

1

M.Phil and Ph.D in Mathematics Curriculum

(w.e.f Sessions 2011)

Scheme of Studies of Ph.D Mathematics

1st Semester 2nd Semester

Course

Code Course Title

Cr.

Hrs.

Course

Code Course Title

Cr.

Hrs.

MATH-

918 Symmetries of Spacetimes 03

MATH-

906 Applications of Inequalities 03

MATH-

919

Convex Analysis and

Applications 03

MATH-

907

Propagation of Waves in

Different Media 03

MATH-

760 Integral Equations 03

MATH-

729

Introduction to Subdivision

Scheme 03

Total Cr. Hrs. 09 Total Cr. Hrs. 09

Scheme of Studies of MS/M.Phil Mathematics

1st Semester 2nd Semester

Course

Code Course Title

Cr.

Hrs.

Course

Code Course Title

Cr.

Hrs.

MATH-

701 Representation Theory-I 03

MATH-

719

Computer Aided Geometric

Design 03

MATH-

724 Elastodynamics-I 03

MATH-

732 Representation Theory-II 03

MATH-

726 General Relativity 03

MATH-

755 Elastodynamics-II 03

MATH-

729

Introduction to Subdivision

Scheme 03

MATH-

760 Integral Equations 03

Total Cr. Hrs. 12 Total Cr. Hrs. 12

3rd and 4th Semesters

M.Phil Thesis 06

• MS/M.Phil students will have to pass the 24 credit hours courses, which can be opted from the

approved list of M.Phil courses.

• A PhD student will have to complete 18 credit hours course work, which can be opted from the

approved list of PhD courses.

• Department can offer any course from the list of approved M.Phil/Ph.D courses on the

availability of resources.

2

• The supervisor may recommend a Ph.D student opt courses of his/her relevant field from

approved courses of M.Phil/Ph.D offered by Department of Physics or Department of Computer

Science & IT to fulfill his/her Ph.D coursework condition.

• The supervisor may recommend a PhD student to opt courses of M.Phil to fulfill his/her Ph.D

coursework condition if he/she didn’t study these courses during his/her M.Phil and vice versa.

LIST OF COURSES

M.PHIL COURSES (3 CREDIT HOURS FOR EACH)

MATH-701 Representation Theory-I

MATH-702 Semigroup Theory

MATH-703 Near Rings-I

MATH-704 Advanced Ring Theory-I

MATH-705 Theory of Group Actions

MATH-706 Graph Theory

MATH-707 Topological Vector Spaces

MATH-708 Banach Algebras

MATH-709 The Classical Theory of Fields

MATH-710 Advanced Complex Analysis-I

MATH-711 One Parameter Semigroups

MATH-712 Fixed Point Theory

MATH-713 Approximation Theory

MATH-714 Topological Algebras

MATH-715 Commutative Algebra-I

MATH-716 Homological Algebra-I

MATH-717 Theory of Semirings

MATH-718 Partial Differential Equations

MATH-719 Computer Aided Geometric Design

MATH-720 Magnetohydrodynamics-I

MATH-721 Electrodynamics-I

MATH-722 Advance Fluid Mechanics

MATH-723 Advanced Analytical Dynamics-I

MATH-724 Elastodynamics-I

MATH-725 Plasma Theory-I

MATH-726 General Relativity

MATH-727 Numerical Solutions of Ordinary Differential Equation

MATH-728 Heat Transfer

MATH-729 Introduction to Subdivision Scheme

MATH-730 Nilpotent and Solvable Groups

MATH-731 Convex Analysis

MATH-732 Representation Theory-II

MATH-733 La-Semigroups

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MATH-734 Near Rings-II

MATH-735 Advanced Ring Theory-II

MATH-736 Theory of Group Graphs

MATH-737 Advanced Complex Analysis-II

MATH-738 Non-Standard Analysis

MATH-739 Ordered Vector Spaces

MATH-740 C*-Algebras

MATH-741 Extension of Symmetric Operators

MATH-742 Banach Lattices

MATH-743 Loop Group

MATH-744 Variational Inequalities

MATH-745 Field Extensions and Galois Theory

MATH-746 Theory of Complex Manifolds

MATH-747 Commutative Algebra-II

MATH-748 Commutative Semigroup Rings

MATH-749 Homological Algebra-II

MATH-750 Cosmology

MATH-751 Magnetohydrodynamics-II

MATH-752 Electrodynamics-II

MATH-753 Mathematical Techniques for Boundary Value Problems

MATH-754 Advanced Analytical Dynamics-II

MATH-755 Elastodynamics-II

MATH-756 The Classical Theory of Fields

MATH-757 Plasma Theory-II

MATH-758 Design Theory

MATH-759 Acoustics

MATH-760 Integral Equations

MATH-761 Combinatorics

MATH-762 Theory of Majorization

MATH-763 Inequalities Involving Convex Functions

MATH-764 Harmonic Analysis

MATH-765 Research Methodology

MATH-766 Integral Transform

MATH-767 Advance Numerical Analysis

MATH-768 Generalized Special Functions

MATH-769 Scientific Computation

MATH-770 Mathematical Modeling-I

MATH-771 Mathematical Modeling-II

MATH-772 Computer Graphics

MATH-773 Dynamic Inequalities on Time Scales

MATH-774 Set-Valued Analysis

MATH-775 Fractional Calculus

MATH-776 Perturbation Methods-I

MATH-777 Perturbation Methods-II

MATH-778 Viscous Fluids-I

MATH-779 Viscous Fluids-II

MATH-780 Fuzzy Algebra

PhD COURSES (3 CREDIT HOURS FOR EACH)

MATH-901 Lie Algebras

MATH-902 Numerical Solutions of Integral Equations

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MATH-903 Multiresolution Analysis in Geometric Modeling

MATH-904 Advanced Graph Theory

MATH-905 Strict Convexity

MATH-906 Applications of Inequalities

MATH-907 Propagation of Waves in Different Media

MATH-908 Scattering and Diffraction of Elastic Waves

MATH-909 Lattice Theory

MATH-910 Gravitational Collapse and Black Holes

MATH-911 Spectral Theory in Hilbert Spaces- I

MATH-912 Spectral Theory in Hilbert spaces – II

MATH-913 Multivariate Analysis-I

MATH-914 Multivariate Analysis-II

MATH-915 Spacetimes Foliations

MATH-916 Teleparallel Theory of Gravity

MATH-917 Homotopy Theory

MATH-918 Symmetries of Spacetimes

MATH-919 Convex Analysis and Applications

MATH-920 Numerical Solutions of Partial Differential Equations

MATH-921 Representation Theory and the Symmetric Groups

MATH-922 Non-Newtonian Fluid Mechanics

MATH-923 Orthogonal Polynomials

MATH-924 Numerical Spline Techinques-I

MATH-925 Numerical B-Spline Techinques-II

MATH-926 Computational Geometry

MATH-927 Special Functions and Statistical Distribution

MATH-928 Fuzzy Analysis

MATH-929 Advanced Partial Differential Equation

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MATH -701: REPRESENTATION THEORY-I

Preliminaries from group theory, Group representations, FG-modules, FG-submodules and reducibility,

Group algebras, FG-homeomorphisms, Maschke’s Theorem, Schur’s lemma, Irreducible modules and

the group algebra, Conjugacy classes, Characters, Inner products of characters, The number of

irreducible characters.

RECOMMENDED BOOKS:

1. Gordow James and Martin Liebech, Representations and Characters of Groups, 2nd Edition,

Cambridge University Press, 2001.

2. Tullio Cecherini-Silberstein, Fabio Scarabotti and Filipoo Tolli, Harmonic Analysis on Finite

Groups: Representation Theory, Gelfand Pairs and Markov Chains, Cambridge University

Press, 2008.

3. Charles W. Curtis and Irving Reiner,Representation Theory of Finite Groups and Associative

Algebras, American Mathematical Society, 2006.

4. Steven H. Weintraub,Representation Theory of Finite Groups: Algebra and

Arithmetic,American Mathematical Society, 2003.

5. William Fulton,Young Tableaux: With Applications to Representation Theory and Geometry,

by, Cambridge University Press, 1997.

6. I. G. Macdonald,Symmetric Functions and Hall Polynomials,Oxford University Press,1999.

MATH -702: SEMIGROUP THEORY

Introductory ideas: Basic definitions, cyclic semigroups; Ordered sets, semi lathces and lattices, Binary

relations; Equivalences Congruences; Free semigroups; Green’s Equivalanes; L,R,H,J and D Regular

semi groups, O-Simple semigroups; Simple and O-Simple semi groups; Rees’s theorem; Primitive

idempotents; Completely, O-Simple semi groups; Finite congruence free semigroups, Union of groups;

Bands; Free bands varieties of bands Inverse semigroups. Congruences on Inverse semigroups;

Fundamental inverse semi groups; Bisimple and simple inverse semigroups.Orthodox semigroups;

Basic properties; the structure of orthodox semi groups.

RECOMMENDED BOOKS:

1. Books Llc, Semigroup Theory: Monoid, Semigroup, Special Classes of Semigroups, Inverse

Semigroup, Light's Associativity Test, C0-Semigroup, Books LLC, 2010.

2. A.H. Clifford and G.B. Preston. The Algebraic Theory of Semigroups; Vol. I& II, AMS Math.

Surveys, 1967.

3. J.M. Howie, An Introduction to Semigroup Theory, Academic Press, 1967.

4. Related Research Papers.

MATH -703: NEAR RINGS-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, Redicals of near-rings.

RECOMMENDED BOOKS:

1. Mikhail Chebotar, Rings and Nearrings, De Gruyter Proceedings in Mathematics, Walter de

Gruyter, 2007.

2. G. Pilz,Near Rings, 2nd Edition, North Holland, 1983.

MATH -704: ADVANCED RING THEORY-I

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.

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RECOMMENDED BOOKS:

1. Dinh Van Huynh,Advances in Ring Theory, Birkhäuser/Springer Basel AG, 2010.

2. R. Wiegandt,Radical an Semisimple Classes of Rings, Queen’s Papers in Pure and Applied

Mathematics No. 37, Queen’s University, Kingston, Ontario, 1974.

MATH -705: THEORY OF GROUP ACTIONS

Preliminaries, the theory of group actions, Coset spaces, Multiplicative group of a finite fields,

Extensions of finite fields, Projective line over finite fields, projective and linear groups through actions.

RECOMMENDED BOOKS:

1. Wilhelm Magnus, Abraham Karrass, Donald Solitar,Combinatorial Group Theory:

Presentations of Groups in Terms of Generators and Relations, Dover Publications, 2004.

2. John S. Rose, A Course on Group Theory (Dover Books on Advanced Mathematics), Dover

Publications, 1994.

3. Harold S.M. Coxeter, William O. J. Moser, Generators and Relations for Discrete Groups, 4th

Edition, Springer, 1980.

MATH -706: GRAPH THEORY

Fundamentals.Definition.Paths cycles and trees.Hamilton cycles and Euler circuits. Planer graphs.

Flows, Connectivity and Matching Network flows. Connectivity and Menger’s theorem.External

problems paths and Complete Subgraphs.Hamilton path and cycles.Colouring.Vertex colouring Edge

colouring. Graph on surfaces.

RECOMMENDED BOOKS:

1. Adrian Bondy and U.S.R. Murty, Graph Theory, Springer, 2010.

2. J.L. Gross and J. Yellen, Graph Theory and its Applications, Chapman and Hall, 2005.

3. B. Bollobas, Modern Graph Theory, Springer Verlag, NY, 2002.

4. B. Bollobas, Graph Theory, Springer Verlag, NewYork, 1979.

5. R. J. Wilson, Introduction to Graph Theory, Longman London, 1979.

MATH -707: TOPOLOGICAL VECTOR SPACES

Vector spaces: Balanced sets, absorbent sets, convex sets, linear functionals, linear manifolds, sublinear

functionals and extension of linear functionals.

Topological vector spaces: Definitions and general properties, product spaces, quotient spaces,

bounded and totally bounded sets, convex sets and compact sets in topological vector spaces, closed

hyperplanes and separation of convex sets, complete topological vector spaces, mertizable topological

vector spaces, normed vector spaces, normable topological vector spaces and finite dimensional spaces.

Locally convex spaces: General properties, subspaces, product spaces, quotient spaces, Convex and

compact sets in locally convex spaces, bornological spaces, barreled spaces, spaces of continuous

functions, spaces of indefinitely differentiable function, the notion of distributions, nuclear spaces,

montal spaces, Schwartz spaces, (DF)-spaces and Silva spaces.

RECOMMENDED BOOKS:

1. Lawrence Narici, Edward Beckenstein, Topological Vector Spaces, 2nd Edition, Chapman and

Hall/CRC, 2010.

2. H.H. Schaefer, M.P. Wolff, Helmut H. Schaefer, Topological Vector Spaces, 2nd Edition,

Springer, 1999.

3. R. Cristescu, Topological Vector Spaces, Noordhoff International Publishing Netherlands,

1977.

4. F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press New York,

1967.

5. H. Schaefer, Topological Vector Spaces, Springer-Verlag, 1966.

7

MATH -708: BANACH ALGEBRAS

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. Theodore W. Palmer, Banach Algebras and the General Theory of *-Algebras, Cambridge

University Press, 2009.

2. W. Rudin, Functional Analysis, McGraw Hill Publishing Company Inc. New York, 1991.

3. M.A. Naimark, Normed Algebras, Wolters Noordhoff Publishing Groningen, The Netherlands,

1972.

4. W. Zelazko, Banach Algebras, American Elsevier Publishing Company Inc. New York, 1973.

MATH -709: THE CLASSICAL THEORY OF FIELDS

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 the 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 Electrodynmamics (John Wiley & Sons, 1999).

4. Misner, C.W., Thorne, K.S. and Wheeler, J.A.: Gravitation (W.H. Freeman and Co., 1973).

5. Carroll, S.M.: An Introduction to General Relativity: Spacetime and Geometry (Addison Wesley,

2004).

MATH -710: ADVANCED COMPLEX ANALYSIS-I

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. E. Hille, Analytic Function Theory, American Mathematical Society, 2006.

2. Roland Schinzinger, Patricio A. A. Laura, Conformal Mapping: Methods and Applications,

Dover Publications, 2003.

3. Z. Nehari, Conformal Mappings, Dover Publications, 1982.

4. G. Sansone, J. Gerretsen, Lectures on the Theory of Functions of a Complex Variable, Vol. 2,

Wolters-Noordhoff Publishing, Gröningen, 1969.

MATH -711: ONE PARAMETER SEMIGROUPS

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.

8

RECOMMENDED BOOKS:

1. Einar Hille, Functional Analysis And Semi-Groups, Dutt Press, 2008.

2. A. Gheondea, D. Timotin, F.H. Vasilescu, Operator Extensions, Interpolation of Functions and

Related Topics (Operator Theory: Advances and Applications), Birkhäuser Basel, 1993.

3. E. B. Davies, One-parameter Semigroups, Academic Press, 1980.

MATH -712: FIXED POINT THEORY

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. Andrzej Granas and James Dugundji,Fixed Point Theory, 1st Edition, Springer, 2010.

2. Ravi P. Agarwal, Maria Meehan and Donal O'Regan, Fixed Point Theory and Applications,


3. K. Goebel and W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press,

1990.

4. J. Dugundji and A. Granas,Fixed Point Theory, Polish Scientific Publishers, Warszawa, 1982.

5. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry and Nonexpansive

Mapping, Marcel Dekker Inc. 1984.

MATH -713: APPROXIMATION THEORY

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. N.I. Achieser, Theory of Approximation, Dover Publications, 2004.

2. E. W. Cheney, Introduction to Approximation Theory, 2nd Edition, Amer Mathematical

Society, 2000.

3. M. D. Powell, Approximation Theory and Mehtods, Cambridge University Press, 1981.

4. R.B. Holmes, A Course on Optimization and Best Approximation, Lecture Notes in

Mathematics No. 257, Springer-Verlag, 1971.

MATH -714: TOPOLOGICAL ALGEBRAS

Definition of Topological algebra and its examples.Adjumetion 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 algebra Gelfand-

Naimark theorem l.m.c algebras.

RECOMMENDED BOOKS:

1. Alexander Arhangel'skii and Mikhail Tkachenko, Topological Groups and Related Structures:

An Introduction to Topological Algebra, Atlantis Press, 2008.

2. A. Mallios, Topological Algebras, Selected Topics, North-Holland Company, 1986.

3. T. Hussain, Multiplicative Functions on Topological Algebras, Pitman Advanced Publishing

Program, 1983.

4. E. Beckenstein, L. Narici and C. Suffel, Topological Algebras, North-Holland Company, 1977.

9

MATH -715: COMMUTATIVE ALGEBRA-I

Commutative Rings: Definition and examples, Integral domains, unit, irreducible and prime elements

in ring, Tyupes 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 n

indeterminate, i.e and 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.

Dimension 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 a domain, integrally closed domain, and completely integrally

closed domain.

Valuation Rings: Valuation map and value group, Rank of 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 has

dimension one.

RECOMMENDED BOOKS:

1. Gregor Kemper, A Course in Commutative Algebra, Springer, 2010.

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

Co.,1969.

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

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

MATH -716: HOMOLOGICAL ALGEBRA-I

Revision of basic concepts of Ring theory and Module Theory, Modules, Homomorphism and exact

sequences.Product and co-product of Modules.Comparision 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 Semi-simple Modules.

RECOMMENDED BOOKS:


2. Joseph J. Rotman, An Introduction to Homological Algebra (Universitext), 2nd Edition,

Springer, 2008.

3. Friedrich Kasch and Adolf Mader, Rings, Modules, and the Total (Frontiers in Mathematics),

Birkhäuser Basel, 2005.

4. Frank W. Anderson and Kent R. Fuller, Rings and Categories of Modules, Springer, 2nd

Edition, 1998.

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

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

MATH -717: THEORY OF SEMIRINGS

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.

10

RECOMMENDED BOOKS:

1. J.S. Golan, Semirings and Affine Equations over Them: Theory and Applications (Mathematics

and Its Applications), 1st Edition, Springer, 2011.

2. K. Glazek, A Guide to the Literature on Semirings and their Applications in Mathematics and

Information Sciences: With Complete Bibliography, Springer, 2010.

3. U Hebisch and H.J. Weinert, Semirings Algebraic Theory and Applications in Computer

Science, World Scientific, 1998.

4. J.S. Golan, The Theory of Semirings and Application in Mathematics and Theoretical Computer

Science, Longman Scientific &Technical John Wiley & Sons New York, 1992.

MATH -718: PARTIAL DIFFERENTIAL EQUATIONS

Cauchy’s problems for linear second order equations in n-independent variables.Cauchy Kowalewski

Theorem.Characteristics surfaces.Adjoint operations, Bicharacteristics Spherical and Cylindrical

Waves.Heat equation.Wave equation.Laplace equation. Maximum-Minimum Principle, Integral

Transforms.

RECOMMENDED BOOKS:

1. Lawrence C. Evans, Partial Differential Equations, 2nd Edition, Amer Mathematical Society,

2010.

2. J. David Logan, Applied Partial Differential Equations, 2nd Edition, Springer-Verlag New

York, 2004.

3. Tyn Myint-U and Lokenath Debnath, Linear Partial Differential Equations for Scientists and

Engineers, 4th Edition, Birkhäuser Boston, 2006.

4. Jürgen Jost, Partial Differential Equations, Springer, 2002.

MATH -719: COMPUTER AIDED GEOMETRIC DESIGN

Linear interpolation, piecewise linear interpolation blossoms, barycentric coordinates in the plane, the

de Casteljau algorithm, properties of Bezier curves, Bernstein polynomials, composite Bezier curves,

degree elevation, the variation diminishing property, degree reduction, Polynomial curve constructions:

Aitken’s Algorithm, Lagrange Polynomials, Lagrange interpolation, cubic Hermite interpolation, Point-

normal interpolation, B-Spline curves: B-spline segments, curves, Knot insertion, degree elevation,

Greville Abscissae, smoothness. Constructing Splines Curves: Greville interpolation, modifying B-

Spline curves, cubic spline interpolation, the minimum property, piecewise cubic interpolation.

Rational Bezier and B-Spline Curves: Rational Bezier curves, Rational Cubic B-spline curves

RECOMMENDED BOOKS:

1. Gerald Farin,Curves and Surfaces for CAGD, A Practical Guide, 5th Edition, Morgan

Kaufmann Publishers, 2002.

2. Josef Hoschek and Dieter Lasser, Fundamentals of Computer Aided Geometric Design,A K

Peter, Ltd, 1993.

3. Gerald Farin, Josef Hoschek and Myung Sookim, Hand Book of Computer Aided Geometric

Design, Elsevier Science, 2002.

MATH -720: MAGNETOHYDRODYNAMICS-I

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 ad shock waves in relativistic magnetohydrodynamics,

11

stability and strucure of shock waves, discontinuities in various quantities, piston problem, oblique

shock waves.

RECOMMENDED BOOKS:

1. J.P. Hans Goedbloed and Stefaan Poedts, Principles of Magnetohydrodynamics: With

Applications to Laboratory and Astrophysical Plasmas, Cambridge University Press, 2004.

2. P.A. Davidson, An Introduction to Magnetohydrodynamics (Cambridge Texts in Applied

Mathematics), Cambridge University Press, 2001.

3. George W. Sutton and Arthur Sherman, Engineering Magnetohydrodynamics, Dover

Publications, 2006.

4. A.I. Akhiezer,Plasma Electrodynamics, Pergamon Press, 1975.

5. J.E. Anderson, Magnetohydrodynamics, Shock Waves, M.I.T Press,1975.

MATH -721: ELECTRODYNAMICS-I

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 cylindrical waves, inhomogeneous wave equation,

Retarded potentials, Lenard-Wiechart potentials, field of uniformly moving point charge, radiation from

a group of moving charges, field of oscillating dipole, field of an accelerated point charge.

RECOMMENDED BOOKS:

1. F.W. Hehl and Yuri N. Obukhov, Foundation of Classical Electrodynamics, 2003.

2. Fulvio Melia, Electrodynamics, University Of Chicago Press, 2001.

3. John R. Reitz, Frederick J. Milford and Robert W. Christy, Foundations of Electromagnetic

Theory, 4thd Edition, Addison Wesley, 2008.

MATH -722: ADVANCE FLUID MECHANICS

Navier-Stoke’s equation and exact solutions, dynamical similarity and Reynold’s number, Turbulent

flow, Boundary layer concept and governing equations,laminar flat plate boundary layer: exact solution,

momentum, integral equation, use of momentum integral equation for flow with zero pressure gradient,

pressure gradient in boundary-layer flow, Reynold’s equations of turbulent motion.

Magnetohydrodynamics, MHD equations, fluid drifts, stability and equilibrium problems.

RECOMMENDED BOOKS:

1. M. Rahman and C. A. Brebbia, Advances in Fluid Mechanics VII, WIT Press, 2008.

2. G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 2000.

3. J.A. Shercliff, The Theory of Electromagnetic Flow-Measurement, Cambridge University

Press, 1987.

4. Francis F. Chen, Introduction to Plasma Physics and Controlled Fusion, Springer, 2010.

5. N.A. Krall and A.W. Trivelpiece, Principles of Plasma Physics, San Francisco Press,

Incorporated, 1986.

MATH -723: ADVANCED ANALYTICAL DYNAMICS-I

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. Edmund Taylor Whittaker, A Treatise On the Analytical Dynamics of Particles and Rigid

Bodies: With an Introduction to the Problem of Three Bodies, FQ Books, 2010.

2. R.B. Bhat and Antonio Lopez-Gomez,Advanced Dynamics, 1st Edition, Narosa, 2001.

3. L.A. Pars, A Treatise on Analytical Dynamics, Ox Bow Pr., 1981.

12

MATH -724: ELASTODYNAMICS-I

Tensor Analysis, Cartesian tensors, Orthogonal rotation of axes, Transformation equations. Translation

and rotation, Different orders of tensors.Algebra of tensors, contraction of tensors, Inner and outer

multiplication of tensors, Symmetric and anti-symmetric tensors. Different types of tensors, Tensor

Calculus. Differentiation and integration of tensors, application to vector analysis, Integral theorems in

tensor form.

Deviators, types of solid Material, Stress vector and stress tensor, Analysis of strain, displacement

vector, Lagrangian strain tensor, Physical interpretation of strain components. Basic equation of theory

of Elasticity, .Generalized Hooke’s law.Types of bodies. Physical interpretation of Lame’s constants.

Navier’s equation.

RECOMMENDED BOOKS:

1. N.A. Shah, Vector and Tensor Analysis, A-One publisher, Urdu Bazar Lahore, 2005.

2. F.D. Zaman, An introduction to Elastodynamic, National Academy of Higher Education,

Islamabad, 1987.

3. K. F. Graff, Wave Motion in Elastic Solids, Dover Publication Inc. New York, 1991.

MATH - 725: PLASMA THEORY-I

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: Atmospheric source of magnetospheric plasma and its temperature; Plasma from Jupiter.

RECOMMENDED BOOKS:

1. J.P. Friedberg, Plasma physics and Fusion energy,Cambridge University Press, 2008.

2. I.H. Hutchinson, Principles of Plasma Diagnostics, Cambridge University Press, 2005.

3. A. Nishida,Magnetospheric Plasma Physics, Springer, 1982.

4. R.O. Dendy, Plasma Physics (An Introductory Course), Cambridge University Press,1995.

5. F.F. Chen,Introduction to Plasma Physics, Plenum Press, New York, 1974.

6. D.B. Melrose,Plasma Astrophysics, Gordon and Breach Science Publishers, 1980.

MATH-726: GENERAL RELATIVITY

Review of special relativity, tensors and field theory. The principles on which general relativity is based.

Einstein’s field equations obtained from geodesic deviation. Vacunm equation.The Schwarzschild

exterior solution.Solution of the Einstein-Maxwall 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.Isometrics.Conformal transformations. Problems of “quantum gravity”

RECOMMENDED BOOKS:

1. James J. Callahan, The Geometry of Spacetime: An Introduction to Special and General

Relativity, Springer, 2011.

2. Abhay Ashtekar, 100 Years of Relativity: Space-time Structure Einstein And Beyond, World

Scientific Pub Co Inc, 2006.

3. A. Qadir, Relativity: An Introduction to the Special Theory, World Scientific, 1990.

4. C.W. Misner, K.S. Thorne and J.A. Wheeler,Gravitation, W.H. Freeman, 1974.

5. S.W. Hawking and G.F.R. Ellis,The Large Scale Structure of Spacetime, Academic Press, 1972.

13

MATH-727: NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATION.

Theory and implementation of numerical methods for initial and boundary value problems in ordinary

differential equations. One-step, linear multi-step, Runge-Kutta, and Extrapolation methods;

convergence, stability, error estimates, and practical implementation, Study and analysis of shooting,

finite difference and projection methods for boundary value problems for ordinary differential equation.

RECOMMENDED BOOKS:

1. Donald Greanspan, Numerical Solution of Ordinary Differential Equations. Wiley-VCH, 2008.

2. Lawrence F. Shampine Numerical Solution of Ordinary Differential Equation, Chapman &

Hall Mathematics CRC Press Publisher, 1994.

3. L.F. Shampine,Numerical Solutions of Ordinary Differential Equations, Springer, 1994.

MATH-728: HEAT TRANSFER

Thermodynamics systems, work and heat, first law of thermodynamics applied to closed and open

systems, propertie of vapours in ideal gasses; second law of thermodynamics and the concept of entropy.

The external problem with reference to the vertical flat plate and horizontal circular cylinder for

isothermal surface condition and constant hear flux, limiting velocity and thermal fields for small and

large Prandilt numbers. Exact solutions for free convecation from a point or line source of heat, The

turbulent plume. The internal problem with reference to flow in Cavities, Lighthill’s thermosyphon and

the cooling of a turbine blade.Batchelor’s work on double glazing.Analytical solutions of ostrich on

fully developed combined free and forced convection in vertical tubes, the effects of viscous dissipation;

effects of non-uniform convection on forced flow in a uniformly heated horizontal circular tube

following work by Monton.Some simple unsteady free convection problems with analytical solution.

RECOMMENDED BOOKS:

1. J.P. Holman, Heat Transfer, 10th Edition, McGraw-Hill Science/Engineering/Math, 2010.

2. W.M. Kays and M.E. Crowfard,Convective Heat and Mass Transfer, 4th Edition, McGraw-

Hill, New York, 2005.

3. F.P. Incropera and D.P. Dewitt,Fundamentals of Heat and Mass transfer, John Wiley & Sons,

New York, 2006.

MATH-729: INTRODUCTION TO SUBDIVISION SCHEME

Tensor product patches: Bilinear interpolation, The direct de Casteljau Algorithm, the tensor product

approach and its properties, degree elevation, constructing polynomial patches: Ruled surfaces, Coons

patches, translational surfaces, tensor product interpolation, bicubic Hermite patches, Composite

Surfaces: smoothness and subdivision, tensor product B-spline surfaces, bicubic spline interpolation,

Rational Bezier and B-spline surfaces, surface of revolution, volume deformations, CONS and trimmed

surface. Bezier Triangles: The de Casteljau algorithm, triangular blossoms, Bernstein polynomial,

derivatives, subdivision, differentiability, degree elevation, nonparametric patches, the multivariate

case, S-Patches. Surfaces with Arbitrary Topology: Recursive subdivision curves

RECOMMENDED BOOKS:

1. Gerald Farin, Curves and Surfaces for CAGD. A Practical Guide, 5th Edition, Morgan

Kaufmann Publishers, 2002.

2. Gerald Farin, Josef Hoschek and Myung sookim, Hand Book of Computer Aided Geometric


3. Josef Hoschek and Dieter Lasser, Fundamentals of Computer Aided Geometric Design,A K

Peter Ltd, 1993.

MATH-730: NILPOTENT AND SOLVABLE GROUPS

Normal and subnormal series, Abelian and central series, direct products, finitely generated Abelian

groups, splitting theorems, solvable and Nilpotent groups, commutators subgroup, derived series, the

14

lower and upper central series, characterzation of finite Nilpotent groups, Fitting subgroup, Frattini

subgroup, Dedekind groups

supersolvable groups, solvable 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. Hans Kurzweil and Bernd Stellmacher, The Theory of Finite Groups: An Introduction,

Springer, 2010.

2. Simon R. Blackburn, Peter M. Neumann OBE, Geetha Venkataraman, Enumeration of Finite

Groups, Cambridge University Press, 2007.

3. Derek J.S. Robinson, A Course in the Theory of Groups, 2nd Edition, Springer, 1995.

4. Klaus Doerk and Trevor Hawkes, Finite Soluble Groups (De Gruyter Expositions in

Mathematics), Walter De Gruyter Inc, 1992.

MATH-731: CONVEX ANALYSIS

Convex functions on the real line, Continuity and differentiability of convex functions,

Characterizations, Differences of convex functions, Conjugate convex functions, Convex sets and affine

sets, Convex functions on a normed linear space, Continuity of convex functions on normed linear

space, Differentiable convex function on normed linear space, The support of convex functions,

Differentiability of convex function on normed linear space.

RECOMMENDED BOOKS:

1. Jonathan M. Borwein and Adrian S. Lewis, Convex Analysis and Nonlinear Optimization:

Theory and Examples (CMS Books in Mathematics), 2nd Edition, Springer, 2010.

2. A. W. Roberts and D. E. Varberg, Convex Functions, Academic Press, New York, 1973.

3. C. P. Niculescu, L-E. Persson, Convex Functions and Their Applications, Springer verlag New

York, 2006.

4. R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N.J., 1970.

MATH -732: REPRESENTATION THEORY-II

Character tables and orthogonality relations, Normal subgroups and lifted characters, Some elementary

character tables, Tensor products, Restriction to a subgroup, induced modules and characters, Algebraic

integers, Real representations, Summary of properties of character tables Characters of groups of order

pq, Characters of some p-groups, character tables of the sample group of order 168, Character table of

GL (2,q), Permutations and characters.

RECOMMENDED BOOKS:

1. Gordow James and Martin Liebech, Representations and Characters of Groups,2nd Edition,


2. Tullio Cecherini-Silberstein, Fabio Scarabotti and Filipoo Tolli, Harmonic Analysis on Finite

Groups: Representation Theory, Gelfand Pairs and Markov Chains, Cambridge University

Press, 2008.

3. Charles W. Curtis and Irving Reiner,Representation Theory of Finite Groups and Associative

Algebras, American Mathematical Society,2006.

4. Steven H. Weintraub,Representation Theory of Finite Groups: Algebra and

Arithmetic,American Mathematical Society, 2003.

5. William Fulton,Young Tableaux: With Applications to Representation Theory and Geometry,

by, Cambridge University Press, 1997.

6. I. G. Macdonald,Symmetric Functions and Hall Polynomials,Oxford University Press,1999.

15

MATH -733: LA-SEMIGROUPS

LA-semigroups and basic results, connection with other algebraic structures, Medial and exponential

properties, LA-semigroups defined by cummulative inverse semigroups, Homomorphism theorems for

LA-semigroups, Abelian groups defined by LA-semeigroups, Embedding theorem for LA-semigroups,

Structural properties of LA-semigroups, LA semigroups as samilattice 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-

semigroup.

RECOMMENDED BOOKS:

1. Books Llc, Semigroup Theory: Monoid, Semigroup, Special Classes of Semigroups, Inverse

Semigroup, Light's Associativity Test, C0-Semigroup, Books LLC, 2010.

2. A.H. Clifford and G.B. Preston. The Algebraic Theory of Semigroups; Vol. I& II, AMS Math.

Surveys, 1967.


MATH -734: NEAR RINGS-II

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.

RECOMMENDED BOOKS:

1. Mikhail Chebotar, Rings and Nearrings, De Gruyter Proceedings in Mathematics, Walter de

Gruyter, 2007.

2. G. Pilz,Near Rings, 2nd Edition, North Holland, 1983.

MATH -735: ADVANCED RING THEORY-II

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. Dinh Van Huynh,Advances in Ring Theory, Birkhäuser/Springer Basel AG, 2010.

2. R. Wiegandt,Radicalan Semisimple Classes of Rings, Queen’s Papers in Pure and Applied

Mathematics No. 37, Queen’s University, Kingston, Ontario, 1974.

MATH -736: THEORY OF GROUP GRAPHS

Graphs, graphs of group actions, projective special linear group and its action on real, rational and

irrational fields, Graphical representations of mobius, Orthogonal Affine and Euclidean groups.

RECOMMENDED BOOKS:



2. John S. Rose, A Course on Group Theory (Dover Books on Advanced Mathematics), Dover

Publications, 1994.

3. Harold S.M. Coxeter, William O. J. Moser, Generators and Relations for Discrete Groups, 4th

Edition, Springer, 1980.

MATH -737: ADVANCED COMPLEX ANALYSIS-II

Holomorphic functions: Review of 1-variable theory, Real and complex differentiability, Power series,

Complex differentiable Functions. Cauchy integral formula for a polydise, Chauchy inequalities The

maximum principle.

16

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. Kunihiko Kodaira, Complex Analysis (Cambridge Studies in Advanced Mathematics),


2. Klaus Fritzsche, Hans Grauert, From Holomorphic Functions to Complex Manifolds, Springer,

2010.

3. M. Field, Several Complex Variables and Complex Manifolds, Cambridge University Press,

1982.

4. H. Grauert and K. Fritsche, Several Complex Variables, Springer Verlag, 1976.

MATH -738: NON-STANDARD ANALYSIS

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 numbrers, 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 Equieontinuity, Compact mapping.

RECOMMENDED BOOKS:

1. Martin Davis, Applied Non standard Analysis, Dover Publications, 2005.

2. A. Robinson, Non standard Analysis (Studies in Logic and the Foundations of Mathematics),

North Holland, 1974.

3. M. Machover and J. Hirschfled, Lectures on Non-standard Analysis, Springer-Verlag, 1969.

MATH-739: ORDERED VECTOR SPACES

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 Archimedean spaces, Vector Lattice, Decomposition of a

vector lattice, Concrete spaces, Topological ordered vector spaces.

RECOMMENDED BOOKS:

1. Alfred Göpfert, Hassan Riahi, Christiane Tammer, Constantin Zalinescu, Variational Methods

in Partially Ordered Spaces, Springer, 2011.

2. T. Cristescu, Ordered Vector Spaces and Linear Operators, Abacus Press, England, 1976.

3. A.L. Peressini, Ordered Topological Vector Spaces, Harper and Row, New York, 1967.

MATH -740: C*-ALGEBRAS

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:



2. W. Rudin, Functional Analysis, McGraw Hill Publishing Company Inc. New York, 1991.

17

3. J. Dixmier, C*-Algebras, North Holland Publishing Company, 1977.

4. M.A. Naimark, Normed Algebras, Wolters Noodhoff Publishing Groningen, The Netherlands,

1972.

MATH -741: EXTENSION OF SYMMETRIC OPERATORS

Deficiency Indices.Neumann formula, spactra of self-adjoint extensions of symmetric operators.Self-

adjoint extensions to larger spaces.Applications to differential operators.

RECOMMENDED BOOKS:

1. Jussi Behrndt, Karl-Heinz Förster and Carsten Trunk, Recent Advances in Operator Theory in

Hilbert and Krein Spaces, Birkhäuser Basel, 2009.

2. A. Gheondea, D. Timotin, F.H. Vasilescu, Operator Extensions, Interpolation of Functions and

Related Topics (Operator Theory: Advances and Applications), Birkhäuser Basel, 1993.

3. N.I. Akhiezer and I.M. Clazman, Theory of Linear Operators; Vol. II, Frederick Ungar

Publishing Co. 1963.

MATH -742: BANACH LATTICES

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:



2. H. H. Schaeff, Banach Lattices and Positive Operators, Springer-Verlag, 1974.

MATH -743: LOOP GROUP

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 ( ) 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. Edward Frenkel, Langlands Correspondence for Loop Groups, Cambridge University Press,

2007.



3. A. Pressley and G. Segal: Loop Groups, Oxford University Press, 1986.

4. V.G. Kac, Infinite Dimensional Lie Algebras, Birkhäuser, 1983.

MATH -744: VARIATIONAL INEQUALITIES

Variational problems, existence results for the general implicit variational problems, implicit Ky Fan’s

inequality for monotone functions, Jartman stampacchia theorem for monotone for compact operators,

Selection of fixed points by monotone functions, Variational and quasivariational inequalities for

monotone operators.

18

RECOMMENDED BOOKS:

1. C.j. Goh, Duality in Optimization and Variational Inequalities (Optimization Theory &

Applications), 1st Edition, Taylor & Francis, 2000.

2. C. Baiocchi, A. Capelo, Variational and Quasi-Variational Inequalities, Wiley, 1984.

3. V. Mosco, Implicit Variational Problems and Quasi Variational Inequalities, Lecture Notes in

Mathematics-543, Springer-Verlage, Berlin, 1976.

MATH -745: FIELD EXTENSIONS AND GALOIS THEORY

Extension of Fields Elementary properties, Simple extensions, Algebraic extensions, Factorization of

polynomials, Splitting fields, algebraically closed fields, Separable extensions. Galois Theory

Automorphism of fields, Normal extensions, the fundamental theorem of Galois theory, Norms and

traces, the primitive element theorem Lagrange’s theorem, Nominal bases. Applications Finite fields,

cyclotomic extension of rational number field, cyclic extensions, Wedderburn’s Theorem, Ruler-and-

Compasses Construction, Solution by Radicals.

RECOMMENDED BOOKS:

1. Jean-Pierre Serre, Galois Groups and Fundamental Groups, Cambridge University Press,

2003.

2. Steven H. Weintraub, Galois Theory: Universitext, 2nd Edition, Springer, 2009.

3. Joseph J. Rotman ,AnIntroduction to the Theory of Groups, 2nd Edition, Springer,1995.

MATH -746: THEORY OF COMPLEX MANIFOLDS

Algebraic preliminaries; Almost complex manifolds and complex manifolds, connections in almost

complex manifolds; Hermitian metrics and Kaehler metrics; Kaehler metric in local coordinate systems;

examples of Kaehler manifolds; Holomorphic sectional curvature: De Rham de composion of Kaehler

manifolds; curvature of Kaehler submanifolds; topology of Kaehler manifolds with positive curvature.

Hermitian connections in Hermitian vector boundless. 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. Kunihiko Kodaira, Complex Manifolds and Deformation of Complex Structures (Classics in

Mathematics), Springer, 2004.

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, 1994.

MATH -747: COMMUTATIVE ALGEBRA-II

Unique Factorization Domains: Basics and examples, Guass Theorem, Quotient of a UFD, Nagata

Theorem.

Class Groups: Divisor classes, 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 the 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, characterization of BFD

through length function.Polynomial extension of BFDs, Noetherian and Krull domains and BFDs.

Half Factorial Domains: Class number of a Field, Carlitz Theorem, examples and basic results, Dedkind

and Krull examples, inetegrabiltiy and HFD, on polynomial and polynomial like extensions.

19

Finite Factorization Domains: Group of divisibility G(D) of a domain D, G(D) and FFD, Atomic idf-

domain is FFD.

RECOMMENDED BOOKS:



Springer, 2008.

3. S.T. Chapman & Sara Glaz, Non Noetherian Commutative Ring Theory, Mathematics & its

Application series Vol.520, Kluwar Acaademic Publishers, 2000.


MATH -748: COMMUTATIVE SEMIGROUP RINGS

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, principle ideal domains and unique factorization domains. Definition

and examples of DVRs, Dedkind and Krull domains.

Commutative Semigroups: Basic notions, Cyclic semigoups, 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:

1. Scott T. Chapman, Arithmetical Properties of Commutative Rings and Monoids (Lecture Notes

in Pure and Applied Mathematics), CRC Press, 2005.

2. R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, New York, 1972.


4. R. Gilmer, Commutative Semigroup Rings, TheUniversity of Chicago Press, Chicago, 1984.

MATH -749: HOMOLOGICAL ALGEBRA-II

Tensor products of modules, Singular Homology flate Modules.Categories and factors cogenerator.

Finitely related (finitely presented) Modules. Pure 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:



Springer, 2008.

3. Friedrich Kasch and Adolf Mader, Rings, Modules, and the Total (Frontiers in Mathematics),

Birkhäuser Basel, 2005.

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

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

MATH -750: COSMOLOGY

Review of Relativity, historical background: Astronomy; Astrophysics Cosmology, the cosmological

principle and its strong form. The Einstein and DeSitter universe models.Measurement of cosmic

distance.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

20

standard model of cosmology. Bianchi space-time’s. Mixmaster models.Inflationary

cosmology.Further developments of inflationary models.Kaluza-Klein cosmologies. Review of

material.

RECOMMENDED BOOKS:

1. Steven L. Weinberg, Cosmology, Oxford University Press, USA, 2008.

2. Peter Schneider, Extragalactic Astronomy and Cosmology: An Introduction, Springer, 2010.

3. P.J.E. Peebles, Principles of Physical Cosmology, Princeton University Press, 1993.

4. M.P. Jr. Ryan and L.C. Shepley,Homogeneous Relativistic Cosmologies, Princeton University

Press, 1975.

5. E.W. Kolb and M.S. Tarner,The Early Universe, Addison Wesley, 1990.

6. L.F. Abbott and S.Y. Pi, Inflationary Cosmology, World Scientific 1986.

MATH -751: MAGETOHYDRODYNAMICS-II

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.

RECOMMENDED BOOKS:

1. J.P. Hans Goedbloed and Stefaan Poedts, Principles of Magnetohydrodynamics: With

Applications to Laboratory and Astrophysical Plasmas, Cambridge University Press, 2004.

2. P.A. Davidson, An Introduction to Magnetohydrodynamics (Cambridge Texts in Applied

Mathematics), Cambridge University Press, 2001.

3. George W. Sutton and Arthur Sherman, Engineering Magnetohydrodynamics, Dover

Publications, 2006.

4. A.I. Akhiezer,Plasma Electrodynamics, Pergamon Press, 1975.

5. J.E. Anderson, Magnetohydrodynamics, Shock Waves, M.I.T Press,1975.

MATH -752: ELECTRODYNAMICS-II

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 o f 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.

RECOMMENDED BOOKS:

1. F.W. Hehl and Yuri N. Obukhov, Foundation of Classical Electrodynamics, 2003.

2. Fulvio Melia, Electrodynamics, University Of Chicago Press, 2001.

3. John R. Reitz, Frederick J. Milford and Robert W. Christy, Foundations of Electromagnetic

Theory, 4thd Edition, Addison Wesley, 2008.

MATH -753: MATHEMATICAL TECHNIQUES FOR BOUNDARY VALUE PROBLEMS

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. A.H. Nayfeh, Perturbation Methods, 1st Edition, Wiley-VCH, 2000.

2. I. Stakgold, Boundary Value Problems of Mathematical Physics, 2 Vol Set, SIAM, 1987.

21

3. B. Noble, Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential

Equations, 2nd ed. American Math. Society, 1998.

4. A.H. Nayfeh, Introduction to Perturbation Techniques, John Wiley Inc., 1993.

MATH -754: ADVANCED ANALYTICAL DYNAMICS-II

Groups of continuous transformation and Poincare’s equations.Systems with one degree of

freedom.Singular points, cyclic characteristics of systems with a degree of freedom.Ergodie theorem,

metric indecompossability stability of motion.

RECOMMENDED BOOKS:

1. Edmund Taylor Whittaker, A Treatise On the Analytical Dynamics of Particles and Rigid

Bodies: With an Introduction to the Problem of Three Bodies, FQ Books, 2010.

2. R.B. Bhat and Antonio Lopez-Gomez,Advanced Dynamics, 1st Edition, Narosa, 2001.

3. L.A. Pars, A Treatise on Analytical Dynamics, Ox Bow Pr., 1981.

MATH -755: ELASTODYNAMICS-II

Derivation of equation of motion, Helmotz theorem, components of displacement in terms of potentials.

Strain components, stress components, Waves and vibrations in strings. Waves in long string,

Reflection and transmission at boundaries.Free vibration of a finite string.Forced vibration of a

string.The string on an elastic base dispersion. Pulses ina dispersive media. The string on a viscous sub

grade.

RECOMMENDED BOOKS:

1. N.A. Shah, Vector and Tensor Analysis, A-One publisher, Urdu Bazar Lahore, 2005.

2. F.D. Zaman, An introduction to Elastodynamic, National Academy of Higher Education,

Islamabad, 1987.

3. K. F. Graff, Wave Motion in Elastic Solids, Dover Publication Inc. New York, 1991.

MATH -756: THE CLASSICAL THEORY OF FIELDS

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.Diffranction 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 equation.

RECOMMENDED BOOKS:

1. Davison E. Soper, Classical Field Theory, Dover Publications, 2008.

2. Michele Maggiore, A Modern Introduction to Quantum Field Theory, Oxford University Press,

USA, 2005.

3. L.A. Scipio, Principles of Continua with Applications, John Wiley, New York, 1969.

4. L.D. Landau and M. Lifshitz, The Classical Theory of Fields, Pergamon Press, 1980.

5. J.D. Jackson, Classical Electrodynamics, 3rd Edition, Wiley, 1998.

MATH-757: PLASMA THEORY-II

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 Valsov 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.

22

Fluctuations, Correlations and Radiations: Shielding of a moving test charge, electric field fluctuations

in Maxwellian and Nonmaxvellian plasmas, emission of electrostatic waves; electromagnetic

fluctuations, emission of radiation from plasma; black body radiation; cyclotron radiation.

RECOMMENDED BOOKS:

1. Jaffrey P. Friedberg ,Plasma physics and Fusion energy,Cambridge University Press, 2008.

2. I.H. Hutchinson, Principles of Plasma Diagnostics, Cambridge University Press, 2005.

3. A. Nishida,Magnetospheric Plasma Physics, Springer, 1982.

4. R.O. Dendy, Plasma Physics (An Introductory Course), Cambridge University Press,1995.

5. F.F. Chen,Introduction to Plasma Physics, Plenum Press, New York, 1974.

6. D.B. Melrose,Plasma Astrophysics, Gordon and Breach Science Publishers, 1980.

MATH-758: DESIGN THEORY

Basic definitions and properties, related structure.The incidence matrix, graphs, residual structures.The

Bruck-Ryser-Chowla theorem. Singer groups and difference sets. Arithmetical relations and Hadamard

2- designs.Projective and affine planes. Latin squares, nets. Hadamard matrices and Hadamard 20

design. Biplanes, strongly regular graphs. Cameron’s theorem and Hadamard 3-desings. Steiner triple

systems. The Mathieu groups.

RECOMMENDED BOOKS:

1. T. Beth,D. Jungnickel and H. Lenz,Design Theory, 2nd edition, CambridgeUniversity Press,

2000.

2. K. Quinn,B. Webb, C. Rowley and F.C. Holroyal,Combinatorial Designs and

Their Applications, Chapman and Hall, 1999.

3. P.J. Cameron and J.H. Lint,Designs, Graphs, Codes and Their Links, Cambridge University

Press, 1991.

4. P.J. Cameron,Permutation Groups, Cambridge University Press, 1999.

MATH-759: ACOUSTICS

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, equatin of cout, Euler;s equations, 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 waves guides; underwater acoustics.

RECOMMENDED BOOKS:

1. F. Alton Everest, Ken Pohlmann, Master Handbook of Acoustics, 5th Edition, McGraw-

Hill/TAB Electronics, 2009.

2. L.E. Kinsler, A.R. Frey, A.B. Coppens and J.V. Sanders, Fundamentals of Acoustics, John

Wiley & Sons,1981.

3. Philip M. Morse, K. Uno Ingard, Theoretical Acoustics, Princeton University Press, 1987.

1. : An Introduction for science & engineering students, Juta & Co. Ltd., 1996.

MATH -760: INTEGRAL EQUATIONS

Existence theorems, intergral equations with Kernels; Initial and boundary value problems, conversion

of differential equations into integral equations, methods to solve of the integral equations. Applications

to partial differential equations. Integral transforms.

RECOMMENDED BOOKS:

1. Raisinghania, M. D., Integral Equations and Boundary Value Problems, 3rdEdition, S. Chand

and Company, 2010.

23

2. George C. Hsiao and Wolfgang L. Wendland, Boundary Integral Equations, 2nd Edition,

Springer, 2010.

3. Ricardo Estrada and Ram P. Kanwal, Singular Integral Equations, Birkhäuser Boston, 1999.

4. Carlo Bardaro, Julian Musielak, Gianluca Vinti, Nonlinear Integral Operators and Applications,

Walter de Gruyter-Berlin, New York, 2003.

MATH-761: COMBINATORICS

Elementary concepts of several combinatorial structures.Recurrence relations and generating

functions.Principle of inclusion and exclusion. Latin squares and SDRs. Steiner systems. A direct

construction.A recursive construction.Paking and covering.

Linear algebra over finite fields.Gaussian coefficients.The pigeonhole Principle.Some special

cases.Ramsey’s theorem.Bounds for Ramsey numbers and applications.Automorphism groups and

permutation groups.Enumeration under group action.

RECOMMENDED BOOKS:

1. M. Lothaire, Algebraic Combinatorics on Words, Cambridge University Press, 2002.

2. Richard P. Stanley, Combinatorics and Commutative Algebra, 2nd Edition, Birkhäuser Boston,

2004.

3. Richard A. Brualdi, Introductory Combinatorics, 5th Edition, Prentice Hall, 2009.

MATH-762: THEORY OF MAJORIZATION

Motivation and Basic Definitions, Majorization as a Partial Ordering, Order-Preserving Functions,

Partial Orderings Induced by Convex Cones, Partial Orderings Generated by Groups of

Transformations, Majorization for Vectors of Unequal Length, Majorization for Infinite Sequences,

Majorization for Matrices, Lorenz Ordering, Majorization and Dilations, Complex Majorization.

RECOMMENDED BOOKS:

1. Albert W. Marshall, Ingram Olkin, Barry Arnold, Inequalities: Theory of Majorization and Its

Applications, 2nd Edition, Springer, 2011.

2. R. Bhatia,Matrix Analysis, Springer-Verlag, New York, 1997.

3. J. Pecaric, F. Proschan and Y. C. Tong, Convex functions, Partial Orderings and Statistical

Applications, Vol. 187 of Mathematics in Science and Engineering, Academic Press, Boston,

Mass, USA, 1992.

4. Latest Research Papers.

MATH-763: INEQUALITIES INVOLVING CONVEX FUNCTIONS

Jensen’s and related inequalities, Some general inequalities involving convex functions, Hadamard’s

inequalities, Inequalities of Hadamard type I, Inequalities of Hadamard type II, Some inequalities

iInvolving concave functions, Miscellaneous inequalities

RECOMMENDED BOOKS:

1. B.G. Pachpatte, Mathematical Inequalities, North-Holland Mathematical Library, Vol. 67,

Elsevier, 2005.

2. J. Pečarić, F. Proschan and Y. C. Tong, Convex functions, Partial Orderings and Statistical

Applications, Vol. 187 of Mathematics in Science and Engineering, Academic Press, Boston,

Mass, USA, 1992.

3. D. S. Mitrinovic, J. Pečarić and A.M. Fink, Classical and New Inequalities in Analysis, Kluwer

Academic Publishers, The Netherlands, 1993.

MATH-764: HARMONIC ANALYSIS

Topology.Sets and Topologies.Separation axioms and related theorems.The Stone- Weierstrass

theorem.Cartesian products and weak topology.Banach spaces.Normed linear spaces.Bounded linear

24

transformations.Linear functionals.The weak topology for X*.Hilbert space.Involution on ß

(H).Integration.The Daniell integral.Equivalence and measurability. The real LP -spaces. The conjugate

space of LP.Integration on locally compact Hausdorff spaces. The complex LP –spaces. Banach

Algebras.Definition and examples. Function algebras. Maximal ideals.Spectrum; adverse Banach

algebras; elementary theory.The maximal ideal space of a commutative Banach algebra. Some basic

general theorems

RECOMMENDED BOOKS:

1. Anton Deitmar and Siegfried Echterhoff, Principles of Harmonic Analysis (Universitext),

Springer, 2008.

2. Yitzhak Katznelson, An Introduction to Harmonic Analysis, Cambridge University Press, 2004.

3. J. Lindenstrauss and L. Tasfriri, Classical Banach Spaces-I, Springer Verlag, 1977.

4. J. Lindenstrauss and L. Tasfriri, Classical Banach Spaces-II, Springer Verlag, 1979.

MATH-765: RESEARCH METHODOLOGY

Scientific statements, hypothesis, model, Theory & Law, Types of research, Problem definition,

objectives of the research, research design, data collection, data analysis, Interpretation of results,

validation of results, Limitation of Science, calibration, Sensitivity, Least count and reproducibility,

Stability and objectivity, Difference between accuracy and precision, Literature search, defining

problem, Feasibility study, pilot projects / field trials, Formal research proposal, budgeting and funding,

Progress report, final technical and fiscal report, Purpose of experiment, good and bad experiments,

Inefficient experiments, null and alternative hypothesis, Alpha and beta errors, Relationship of alpha

and beta errors to sensitivity and specificity, Designing efficient experiments, Simple random sampling,

systematic sampling, Stratified sampling, cluster sampling, Convenience sampling, judgment sampling,

quota sampling, snow ball sampling , Identifying variables of interest and their interactions, Operating

characteristic curves, power curves, Surveys and field trials, Submission of a paper, role of editor, Peer-

review process, importance of citations, impact factor, Plagiarism, protection of your work from misuse,

Simulation, need for simulation, types of simulation, Introduction to algorithmic research, algorithmic

research problems, types of algorithmic research, problems, types of solution procedure.

BOOKS RECOMMENDED:

1. Dr Ranjit Kumar, Research Methodology: A Step-by-Step Guide for Beginners, 3rd Edition,

Sage Publications Ltd, 2010.

2. R. Harre, Great Scientific Experiments: Twenty Experiments that Changed the World, Dover,

New York, 2002.

3. R.A. Day, How to Write and Publish a Scientific Paper, ISI Press, Philadelphia, USA,1979.

4. W.J. Diamond, Practical Experiment Designs for Scientists and Engineers, 2nd Edition, John

Wiley, New York. 1989.

MATH-766: INTEGRAL TRANSFORM

Laplace transform, Application to integral equations, Fourier transforms, Fourier sine and cosine

transform, Inverse transform, Application to differentiation, Convolutions theorem, Application to

partial differential equations, Hankel transform and its applications, Application to integration, Mellin

transform and its applications.

RECOMMENDED BOOKS:

1. Brian Davies, Integral Transforms and Their Applications, Third Edition (Texts in Applied

Mathematics),Springer, 2002.

2. Roald M. Trigub and Eduard S. Belinsky, Fourier Analysis and Approximation of Functions,

Springer, 2010.

3. Allan Pinkus and Samy Zafrany, Fourier Seriesand Integral Transforms, Cambridge University

Press, 1997.

4. R. Vasistha, R. K. Gupta, Integral Transform, Krisna Prakashan Media Pvt. Ltd., India, 2007.

25

MATH-767: ADVANCE NUMERICAL ANALYSIS

Introduction.Euler’s method.The improved and modified Euler’s method.Runge-Kutta method.Milnes

method. Hammign’s methods.Initial value problem. The special cases when the first derivative is

missing. Boundary value problems.The simultaneous algebraic equations method.Iterative methods for

linear equations.Gauss-Siedel method.Relaxation methods.Vector and matrix norms.Sequences and

series of matrices. Graph Theory. Directed graph of a matrix.Strongly connected and irreducible

matrices.Grerschgoin theorem.Symmetric and positive definite matrices. Cyclic-Consistently ordered

matrices. Choice of optimum value for relaxation parameter.

RECOMMENDED BOOKS

1. Richard L. Burden, J. Douglas Faires, Numerical Analysis, 9th Edition, Brooks Cole, 2010.

2. Stanislaw Rosloniec, Fundamental Numerical Methods for Electrical Engineering (Lecture

Notes in Electrical Engineering), Springer, 2010.

3. C. F. Gerald and P.O. Wheatley,Applied Numerical Analysis, Addison-Wesley Publishing

Company, 1994.

MATH-768: GENERALIZED SPECIAL FUNCTIONS

Infinite Products, the gamma and beta functions, the hypergeometric functions, generalized

hypergeometric functions, Bessel functions, the confluent hypergeometric functions, introduction to q-

series, k- hypergeometric functions, generalized k-hypergeometric functions, confluent k-

hypergeometric functions

RECOMMENDED BOOKS

1. Rainville, E. D., Special Functions, The Macmillan Company, New York, 3rd Edition, 1965.

2. Andrews, G. E., Richard, A. and Roy, R., Special Functions, Cambridge University Press, 1st

Edition, 2000.

3. Mathai, A. M., A Hand Book of Generalized Special Functions for Statistical and Physical

Sciences, University Press Inc., New York, 1993.

MATH-769: SCIENTIFIC COMPUTATION

Vector and matrix operation, Building exploratory environments, Floating point arithmetic, Error

analysis, The interpolating polynomial, Piecewise linear interpolation, Piecewise cubic Hermite

interpolation, Different degree spline interpolations, Shape-preserving interpolants, Bisection method,

Newton’s method, Second method, Inverse quadratic interpolation, Quasi-Newton’s method, Basic

quadrature rules, Adaptive quadrature, Least squares data fitting, Models and data curve fitting, Norms,

The QR factorization, Pseudo inverse, Eigenvalues and singular values, Symmetric and Hermitian

matrices, Eigenvalue and singular value decompositions, Eigenvalue sensitivity and accuracy, Singular

sensitivity and accuracy, Principle components, Parallel computing, Matrix-matrix product.

Recommended Books

1. D. Moler, Numerical Computing with Matlab, SIAM, (2004)

2. B. I. Kvasov, Method of Shape-preserving spline approximation, World Scientific

Publishing Co. Pte. Ltd. (2000)

3. P.M. Prenter, Splines and Variational Methods, John Wiley & Sons, (1989).

4. R.L. Burdern, J.D. Faires, Numerical Analysis, eighth ed., Brooks Cole, (2004).

26

5. S.S. Sastry, Introductory Methods of Numerical Analysis, fourth ed., PHI

Learning, (2009).

6. J. D. Hoffman, Numerical methods for Engineers and Scientists, second ed.,

Marcel Dekker, Inc, New York,.Basel, (1992).

7. G. Farin, Curves and Surfaces for CAGD: A Practical Guide, 5th ed., Morgan

Kaufman, (2002).

8. Recent research papers

MATH-770: MATHEMATICAL MODELLING-I

Introduction to Modelling. Collection and interpretation of data. Setting up and developing models.

Checking models. Consistency of models. Dimensional analysis. Discrete models. Multivariable

models. Matrix models. Continuous models. Modelling rates of changes. Limiting models. Graphs of

functions as models. Periodic models. Modelling with difference equations. Linear, Quadratic and Non-

Linear Models.

Recommended Books

1. Edwards, D. and Hamson, M.: Mathematical Modelling Skills Macmillan Press

Ltd., (1996).

2. Giordano, F.R., Weir, M.D. and Fox, W.P.: A First Course in Mathematical

Modelling Thomson Brooks/Cole, 2003).

3. Law, A.M. and Kelton, W.D.: Simulation Modelling and analysis McGraw-Hill,

(1982).

4. Spriet, J.A. and Vnsteenkiste, G.C.: Computer Aided Modelling and Simulation,

Academic Press, (1982).

5. Aris, R.: Mathematical Modelling Techniques Dover Publication, (1995).

MATH-771: MATHEMATICAL MODELLING-II

Modeling with Differential Equations: Exponential growth and decay. Linear, non-linear

systems of differential equations. Modeling with integration. Modeling with random numbers:

Simulating qualitative random variables. Simulating discrete random variables. Standard

models. Monte Carlo simulation. Fitting models to data. Bilinear interpolation and Coons patch.

27

Recommended Books

1. Edwards, D. and Hamson, M.: Mathematical Modelling Skills Macmillan Press

Ltd., (1996).

2. Giordano, F.R., Weir, M.D. and Fox, W.P.: A First Course in Mathematical

Modelling Thomson Brooks/Cole, (2003).

3. Law, A.M. and Kelton, W.D.: Simulation Modelling and analysis (McGraw-Hill,

1982).

4. Spriet, J.A. and Vnsteenkiste, G.C.: Computer Aided Modelling and Simulation,

Academic Press, (1982).

5. Aris, R.: Mathematical Modelling Techniques Dover Publication, (1995).

MATH-772: COMPUTER GRAPHICS

Introduction to computer graphics and its applications. Overview of raster graphics and

transformation pipeline, i.e. transformations between different coordinate systems which

involve modelling coordinate system. Device coordinate system. World coordinate system.

Normalized coordinate system. Display window coordinate system and screen coordinate

system. Graphics output primitives in drawing of lines, polygons, triangles, etc. Draw polylines

with different line joining methods. Attributes of graphics primitives like color, line style and

fill style. 2D and 3D transformations and viewing. Describing and using viewing parameters to

change the shape of the object, using viewport to change the ratio of clipping window.

Differences in viewing and modelling transformations. Window clipping by Cohen-Sutherland

algorithm.

Recommended Books

1. Donald, H. and Baker, M. P.: Computer Graphics with OpenGL Prentice Hall, (2003).

2. James, D. Foley et al.: Introduction to Computer Graphics Addison-Wesley, (1993).

3. Richard, S. Wright, Benjamin Lipchak: OpenGL SuperBible Sams, (2004).

MATH-773: DYNAMIC INEQUALITIES ON TIME SCALES

Time scale calculus; Basic definitions, differentiation, examples and applications,

integration, chain rule, polynomials, further basic results. Dynamic inequalities;

Gronwall inequality, Holder and Minkowski’s inequalities, Jensen’s inequality.

28

Recommended Books

1. Martin Bohner and Allan Peterson, Dynamic Equations on Time Scales, Birkhauser Boston,

Mass, USA, (2001).

2. Martin Bohner and Allan Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser

Boston, Mass, USA, (2003).

3. V. Lakshmikatham, T. G. Bhaskar, and J. V. Devi, Theory of Set Differential equations in Metric

Spaces, Cambridge Scientic, Cambridge, UK, (2006).


MATH-774: SET-VALUED ANALYSIS

Preliminaries, Hausdorff - Pompeiu metric, upper and lower semicontinuous

multifunctions, Hausdorff-Pompeiu continuity, closed multifunctions, continuous

Selections, measurable multifunctions, Aumann integral, Hukuhara derivative.

Recommended Books

1. Jean Pierre Aubin, Hélène Frankowska, Set-Valued Analysis (Systems & Control: Foundations

& Applications), Birkhauser Boston, (1990).

2. Enayet U Tarafdar and Mohammad S R Chowdhury, Topological Methods for Set-Valued

Nonlinear Analysis, World Scientific Publishing Co. Pte. Ltd., (2008).

3. Guang-ya Chen, Xuexiang Huang, Xiaogi Yang, Vector optimization: Set-valued and

variational analysis, Springer-Verlag Berlin Heidelberg (2005).



5. Aubin J.-P., Frankowska H., Set-Valued Analysis, Birkhäuser, Berlin, (1990).

6. Related research papers.

MATH-775: FRACTIONAL CALCULUS

Introduction; motivation, basics, application of fractional calculus. Riemann–Liouville,

differential and integral operators; Riemann–Liouville integrals, Riemann–Liouville

http://www.google.com.pk/search?tbo=p&tbm=bks&q=inauthor:%22Jean+Pierre+Aubin%22

http://www.google.com.pk/search?tbo=p&tbm=bks&q=inauthor:%22H%C3%A9l%C3%A8ne+Frankowska%22

http://www.amazon.com/Set-Valued-Analysis-Systems-Control-Applications/dp/0817634789/ref=sr_1_1?s=books&ie=UTF8&qid=1416380648&sr=1-1&keywords=9780817634780

http://www.amazon.com/Set-Valued-Analysis-Systems-Control-Applications/dp/0817634789/ref=sr_1_1?s=books&ie=UTF8&qid=1416380648&sr=1-1&keywords=9780817634780

29

derivatives, relations between Riemann–Liouville integrals and derivatives, Grunwald–

Letnikov operators. Caputo’s approach; definition and basic properties, non-classical

representations of Caputo operators.

Recommended Books

1. Kai Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag Berlin

Heidelberg (2010).

2. Anatoly A. Kilbas, Hari M. Srivastava and Juan J. Trujillo, Theory and Applications of

Fractional Differential Equations, North-Holland Mathematics Studies 204 Elsevier, (2006).

3. Keith B. Oldham and Jerome Spanier, The Fractional Calculus: Theory and Applications of

Differentiation and Integration to Arbitrary Order, Dover Books on Mathematics, (2006).

4. Related research papers.

MATH-776: PERTURBATION METHODS-I

Parameter perturbations, coordinate perturbations, order symbols and gauge functions,

asymptotic series and expansions. Asymptotic expansion of integrals, integration by parts

Laplace’s method and Watson’s lemma, method of stationary phase and method of

steepest descent. Straightforward expansions and sources of nonuniformity, the Duffing

equation, small Reynolds number flow past a sphere, small parameter multiplying the

highest derivative. The method of strained coordinates, the Lindstedt – Poincare’ method,

renormalization method. Variation of parameters and method of averaging, examples

Method of Multiple scales with examples.

Recommended Books

1. Nafeh, A.H, Perturbation Methods, John Wiley & Sons, (2000).

2. Nafeh, A.H, Problems in Perturbation, John Wiley & Sons, (1985).

3. Alan W. Bush, Pertubation Methods for Engineers and Scientists, CRC Press London, (1992)

MATH-777: PERTURBATION METHODS – II

Approximate Solution of Linear Differential Equations Approximate Solution of Nonlinear

30

Differential Equations, Perturbation Series, Regular and Singular Perturbation Theory

Perturbation Methods for Linear Eigenvalue Problems, Asymptotic Matching Boundary

Layer Theory Mathematical Structure of Boundary Layers: Inner, Outer, and Intermediate

Limits Higher-Order Boundary Layer Theory Distinguished Limits and Boundary Layers of

Thickness WKB Theory Exponential Approximation for Dissipative and Dispersive

Phenomena, Conditions for Validity of the WKB approximation, Patched Asymptotic

Approximations: WKB Solution of Inhomogeneous Linear equations. Matched Asymptotic

Approximation: Solution of the One-Turning-Point Problem.

Recommended Books

1. Nafeh, A.H, Perturbation Methods, John Wiley & Sons, (2000).

2. Nafeh, A.H, Problems in Perturbation, John Wiley & Sons, (1985).

3. Alan W. Bush, Pertubation Methods for Engineers and Scientists, CRC Press London, (1992)

MATH-778: VISCOUS FLUIDS-I

Some examples of viscous flow phenomena; properties of fluids; boundary conditions.

Equation of continuity; the navier stokes equations; the energy equation; boundary

conditions; orthogonal coordinate systems; dimensionless parameters; velocity

considerations; two dimensional considerations, and the stream functions. Coutte flows;

paisville flow; unsteady duct flows; similarity solutions; some exact analytic solution from

the papers. Introduction; laminar boundary layers equations; similarity solutions; two

dimensional solutions; thermal boundary layer. Some exposure will also be given from the

recent literature appearing in the journals.

Recommended Books

1. Viscous Fluid Flow, F.M. White, McGraw Hill Inc. (1991)

2. Boundary Layer Theory, H. Schlichting & K. Gertsen, Springer (1991)

3. An Introduction to Magnetohydrodynamics, P.A. Davidson, Cambridge University Press

(2001).

31

MATH-779: VISCOUS FLUIDS- II

Introduction: The concept of small disturbance stability; linearized stability; parametric effects in the

linear stability theory; transition to turbulences. Boundary layer equation in plane flow; general solution

and exact solutions of the boundary layer equations. Thermal boundary layers without coupling of

velocity field to the temperature field: Boundary layer equations for the temperature field; forced

convection; effect of Pr number; similar solution of the thermal boundary layers Thermal boundary

layer with coupling of velocity field to the temperature field: Boundary layer with moderate wall heat

transfer; natural convection effect of dissipation; indirect natural convection; mixed convection.

Different kinds of boundary layer control; continuous suction and blowing; massive suction and

blowing; similar solutions.

Recommended Books

1. Viscous Fluid Flow, F.M. White, McGraw Hill Inc. (1991).

2. Boundary Layer Theory, H. Schlichting & K. Gertsen, Springer (1991).

3. An Introduction to Magnetohydrodynamics, P.A. Davidson, Cambridge University Press

(2001).

MATH-780: FUZZY ALGEBRA

Introduction

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

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

(1999).

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

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

4. John N. Mordeson, D.S. Malik and Nobuki Kurok, Fuzzy Semigroups, Springer-Verlage,

(2003).

32

MATH-901: LIE ALGEBRAS

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. Karin Erdmann and Mark J. Wildon, Introduction to Lie Algebras, Springer, 2006.

2. J.E. Humphreys, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction,

Springer, 2003.

3. B. O’Neill, Semi-Riemannian Geometry, Academic Press, 1983.

4. H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, E. Herlt,Exact Solutions of Einstein’s

Fields Equations, Cambridge University Press, 1980.

5. J. Lepowsky and G.W.Mecollum, Elementary Lie Algebra Theory, Yale University, 1974.


MATH-902: NUMERICAL SOLUTIONS OF INTEGRAL EQUATIONS

Numericaland approximate solutions of Fredholm integral equaaiton of the second kind (both linear and

nonlinear).Approximation of integral operators and quadrature methods.Nystrom method.Method of

degenerate kernels.Collectively compact operator approximations.Numerical methods of Volterra

integral equations.Methods of collocation, Galerkin, moments, and Spline approximations for integral

equations.Iterative methods for linear and nonlinear integral equations.Eigenvalue problems.

RECOMMENDED BOOKS:

1. Kendall E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind.


2. Michael A. Globerg, Numerical Solution of Integral Equations, Springer, 2001.


MATH-903: MULTIRESOLUTION ANALYSIS IN GEOMETRIC MODELING

Interpolatory subdivision schemes: The univariate stationary case, non-stationary univariate

interpolatory schemes exact for exponentials, tensor product interpolatory schemes for surfaces, the

butterfly scheme. Analysis of convergence and smoothness by the formalism of Laurent polynomials:

introduction, analysis of univariate schemes, analysis of bivariate scheme with factorizable symbols. A

non-stationary subdivision scheme for curve interpolation, , A non-stationary subdivision scheme for

generalizing trigonometric spline surfaces to arbitrary meshes, A non-stationary uniform tension

controlled interpolating 4-point scheme reproducing conics, First Generation Wavelets: Multiresolution

analysis, Second Generation Wavelets: multiresolution analysis.

RECOMMENDED BOOKS:

1. Armin Iske, Ewald Quak, Michael S. Floater, Tutorials on Multiresolution in Geometric

Modeling. Summer School Lecture Notes Springer-Berlin, 2002.

2. Gerald Farin, Josef Hoschek and Myung sookim ,Hand Book of Computer Aided Geometric


3. Wim Sweldens and Petere Schroder, Building Your Own Wavelet at Home, Online Lecture

Notes.


MATH-904: ADVANCED GRAPH THEORY

Graph Spectrum Theory: Matrices associated to a graph, the spectrum of various graphs, diameter,

regular

graphs, spanning trees, complete graphs, complete bipertite graphs, Calay graphs.

33

Some Modules on the Applications of Spectral Graph Theory: Introduction to spectral geometry of

graphs; Courant-Fischer theorem and graph colorings; Inequalities and bounds on eigenvalues; graph

approximations; Cheeger's inequalities; Diffusion on graphs; Discretizations of heat kernels.

Energy of Graph: Laplacian Energy, Signless Energy, Distance Energy, Normalized Laplacian Energy.

He-matrix for HoneyComb Graph: Honeycomb graph, He matrix, Spectral radius of He-matrix and

bounds, He-matrix with Integer spectrum, number of triangles of He-matrix.

Energy of He-matrix for Hexagonal System: Energy of He matrix, Upper bounds for the energy, Energy

of Coalescence of graphs and various theorems.

Degree Sequence: Properties of Degree Sequences, Degree Sequences of Edge colored Graphs.

RECOMMENDED BOOKS:

1. C. Godsil and G. Royle, Algebraic Graph Theory, Springer, 2004.

2. Edgar G. Goodaire and Michael M. Parmenter, Discrete Mathematics with Graph Theory, 3rd

Edition, Prentice Hall, 2005.

3. N. L. Biggs, Algebraic Graph Theory, Cambridge University, 1993.

4. Lecture Notes and Research Papers.

MATH-905: STRICT CONVEXITY

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. Lars Hörmander, Notions of Convexity, Birkhäuser Boston, 2006.

2. Constantin Niculescu and Lars-Erik Persson, Convex Functions and their Applications: A

Contemporary Approach (CMS Books in Mathematics), Springer, 2005.

3. V. I. Istratescue, Strict Convexity and Complex Strict Convexity,CRC Press, 1984.

4. J. Diestel, Geometry of Banach Spaces, Springer, 1975.


MATH-906: APPLICATIONS OF INEQUALITIES

Jensen’s and related inequalities, Some general inequalities involving convex functions, Hadamard’s

inequalities, Inequalities of Hadamard type I, Inequalities of Hadamard type II, Some inequalities

involving concave functions, Miscellaneous inequalities

RECOMMENDED BOOKS:

5. B.G. Pachpatte, Mathematical inequalities, (North-Holland Mathematical Library,Vol.67),

Elsevier, 2005.

6. Constantin Niculescu and Lars-Erik Persson, Convex Functions and their Applications: A

Contemporary Approach (CMS Books in Mathematics), Springer, 2005.

7. J. Pečarić, F. Proschan and Y. C. Tong, Convex Functions, Partial Orderings and Statistical

Applications, vol. 187 of Mathematics in Science and Engineering, Academic Press, Boston,

Mass, USA, 1992.

8. D. S. Mitrinovic, J. Pečarić and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer

Academic Publishers, The Netherlands, 1993.


MATH-907: PROPAGATION OF WAVES IN DIFFERENT MEDIA

Waves in infinite Media: Wave types, The governing equations, dilatational and distortional waves,

Plane waves, waves generated by body forces, certain classical solutions, simple SH wave source, cavity

source problems, Harmonic dilatational waves from a spherical cavity, Dilatational waves from a step

pulse, General case of dilatational waves with spherical symmetry.

34

Harmonic waves from a cylindrical cavity, Transient waves from a cylindrical cavity. Propagation in a

semi-infinite media: Propagation and reflection of plane waves in a half-space, Governing equations,

Waves at oblique incidence, waves at grazing incidence, Surface waves, Wave reflection under mixed

boundary conditions.

RECOMMENDED BOOKS:

1. E.B. Magrab, Vibrations of Elastic Structural Members (Mechanics of Structural Systems),

Springer, 2010.

2. H. Kolskey,Stress Waves in Solids, (Dover Phoenix Editions), Dover Publications, 2003.

3. Marc A. Meyers,Dynamic Behavior of Materials, Wiley-Interscience, 1994.

4. Karl F.Graff,Wave Motion in Elastic Solids, Dover Publications, New York, 1991.

5. J.D. Achenbach,Wave Propagation in Elastic Solids, North Holland Series, Elsevier 1984.

MATH-908: SCATTERING AND DIFFRACTION OF ELASTIC WAVES

SH waves source-method of steepest descent. The SH wave- formal solution. SH wave source solution

by steepest decent. Surface source problems, waves from harmonic, normal line force, Transient normal

loading on a half-space, waves in layered media. Two semi-infinite media in contact-plane waves. Love

waves, experimental studies on waves in semi-infinite media, surface waves on a half-space, other

studies on surface waves .

Scattering of waves by cavities, Scattering of SH waves by a cylindrical cavity, Scattering of

Compressional waves by a spherical obstacle. Diffraction of plane waves.Discussion of greens function

approach, the sommerfeld diffraction problem.

RECOMMENDED BOOKS:

1. John G. Harris, Linear Elastic Waves (Cambridge Texts in Applied Mathematics), Cambridge


2. E.B. Magrab, Vibrations of Elastic Structural Members (Mechanics of Structural Systems),

Springer, 2010.

3. H. Kolskey,Stress Waves in Solids, (Dover Phoenix Editions), Dover Publications, 2003.

4. Karl. F. Graff,Wave Motion in Elastic Solids, Dover Publications., New York, 1991.

5. Marc. A. Meyers,Dynamic Behavior of Materials, Wiley-Interscience, 1994.


MATH-909: LATTICE THEORY

Elementary Concepts: Definition of lattice. Some algebraic

concepts.Polynomials.Identities.Inequalities.Free lattices.Special elements.

Distributive lattices: Distributive lattices, Characterization and representation theorems. Polynomials

and freeness.Congruence relations.Boolean algebra.

RECOMMENDED BOOKS:

1. T.S. Blyth and G. Birkhoff,Lattices and Ordered Algebraic Structures, Springer, 2005.

2. G. Grazer,Lattice Theory, W.H. Freeman and Company, New York, 1971.

3. B.A. Davey and H.A. Priestley,Introduction to Lattices and Order, Cambridge University

Press, 2002.

4. T. Donnellan,Lattice Theory, Elsevier Science Ltd., 1968.


MATH-910: GRAVITATIONAL COLLAPSE AND BLACK HOLES

Sigularity, kinds of singularity, black holes, Schwarzschild singularity, the Schwarzschild solutions in

other coordinates systems, Schwarzschild solution as a black hole, gravitational collapse, collapse to a

black hole, the evolutionary phases of a sphericaally symmetric stars, the critical mass of star,

gravittaional collapse of spherically symmetric dust.

35

Rotating black holes, the Kerr solution, gravitational collapse ( the possible life history of a rotating

star), some properties of black holes.

RECOMMENDED BOOKS:

1. H. Stephani,Relativity, An Introduction to Special and General Relativity, 3rd edition,


2. W. Rindler,Relativity, Special and Cosmology, 2nd edition, Oxford University Press, 2007.

3. J. B. Hartle,Gravity, An introduction to Einstein's General Relativity, (1st edition), Pearson

Education, 2006.


MATH-911: SPECTRAL THEORY IN HILBERT SPACES–I

The concept of Hilbert spaces.Finite dimensional Euclidean spaces.Inner product spaces.Normed linear

spaces.The Hilbert spaces.The specific geometry of Hilbert spaces.Subspaces. Othogonal subspaces.

Bases. Polynomial bases in L2 Spaces. Isomorphisms.Bounded linear operators.Bounded linear

mappings.Linear operators. Bilinear forms. Adjoint operators.Projection operators.The Fourier-

Plancherel operators.General theory of linear operators.Adjoint operators (general case).Differentiation

operators in L2 Spaces.Multiplication operators in L2 Spaces.Closed linear operators. Invariant

subspaces of a linear operator. Eigenvalues of a linear operator.The Spectrum of a linear operator. The

spectrum of a self-adjoint operators.

RECOMMENDED BOOKS:

1. L. Debnath and P. Mikusinski,Introduction to Hilbert Spaces with Applications, 3rd edition,

Saurabh Printers Noida, 2005.

2. G. Helmberg,Introduction to Spectral Ttheory in Hilbert Spaces, Dover Publications, 2008.

3. F. Riesz and B. S. Nagay,Functional Analysis, Ungar Publishing Co., New York, 1955.

4. N. I. Akhiezer, I. M. Glazman,Theory of Linear Operators in Hilbert Spaces, Dover

Publications, 1993.

5. W. Rudin, Functional Analysis, 2nd Edition, McGraw-Hill Science/Engineering/Math,1991.


MATH-912: SPECTRAL THEORY IN HILBERT SPACES–II

Spectral analysis of compact linear operators.Compact linear operators.Weakly converging

sequences.The spectrum of a compact linear operator.The spectral decomposition of a compact self-

adjoint operator.Fredholm integral equations.Spectral analysis of bounded linear operators.The order

relation for bounded self-adjoint operators.Polynomials in a bounded linear operator. Continuous

functions of a bounded self-adjoint operator. Step functions of a bounded self-adjoint operator. The

spectral decomposition of a bounded self-adjoint operator.Functions of a unitary operator.The spectral

decomposition of a unitary operator.The spectral decomposition of a bounded normal operator.Spectral

analysis of unbounded self-adjoint operators. The Cayley transform. The spectral decomposition of an

unbounded self-adjoint operator. Limit points of a spectrum. Perturbation of the spectrum by the

addition of a completely continuous spectrum.Continuous perturbation.Analytic perturbations.

RECOMMENDED BOOKS:

1. L. Debnath and P. Mikusinski,Introduction to Hilbert Spaces with Applications, 3rd edition,

Saurabh Printers Noida, 2005.

2. G. Helmberg,Introduction to Spectral Ttheory in Hilbert Spaces, Dover Publications, 2008.

3. F. Riesz and B. S. Nagay,Functional Analysis, Ungar Publishing Co., New York, 1955.

4. N. I. Akhiezer, I. M. Glazman,Theory of Linear Operators in Hilbert Spaces, Dover

Publications, 1993.

5. W. Rudin, Functional Analysis, 2nd Edition, McGraw-Hill Science/Engineering/Math,1991.


36

MATH-913: MULTIVARIATE ANALYSIS-I

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. Alvin C. Rencher, Methods of Multivariate Analysis, 2nd Edition, Wiley-Interscience, 2002.

2. Joseph F. Hair, William C. Black, Barry J. Babin and Rolph E. Anderson, Multivariate Data

Analysis, 7th Edition, Prentice Hall, 2009.

3. K. V. Mardia, J. T. Kent and J. M. Bibby,Multivariate Analysis, Academic Press, London,

1982.

4. A. M. Kshirsagar, Multivariate Analysis, Marcell Dekker, New York, 1972.


MATH-914: MULTIVARIATE ANALYSIS-II

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. Alvin C. Rencher, Methods of Multivariate Analysis, 2nd Edition, Wiley-Interscience, 2002.

2. Joseph F. Hair, William C. Black, Barry J. Babin and Rolph E. Anderson, Multivariate Data

Analysis, 7th Edition, Prentice Hall, 2009.

3. K. V. Mardia, J. T. Kent and J. M. Bibby,Multivariate Analysis, Academic Press, London,

1982.

4. A. M. Kshirsagar, Multivariate Analysis, Marcell Dekker, New York, 1972.


MATH-915: SPACETIMES FOLIATIONS

Foliation, foliation in relativity.foliation and frame of references, Hamiltonian formalism and

Hamiltonian equation, Qadir and Siddiqui flat foliation. flat foliation of Schwarzschild and Rieznor-

Nordstorm spacetimes by spacelike hypersurfaces, geodesics and foliation of extreme Rieznor-

Nordstorm spacetime, existance of KS-type coordinates, free fall geodesics in the extreme Rieznor-

Nordstorm spacetime foliating hypersurfaces, duality between geodesics and hypersurfaces and its use

for continuation of hypersurfaces, K sslicing or York slicing of different spacetimes.

RECOMMENDED BOOKS:

1. A. Qadir,Symmetries of Spacetimes (Pre-Print), 2007.

37

2. H. Stephani,Relativity, An Introduction to Special and General Relativity, (3rd edition),


3. W. Rindler,Relativity, Special, and Cosmology, (2nd edition), Oxford University Press, 2007.

4. J. B. Hartle, Gravity, An introduction to Einstein's General Relativity, (1st edition), Pearson

Education, 2006.


MATH-916: TELEPARALLEL THEORY OF GRAVITY

Tetrads, linear Connections; linear transformations; orthogonal transformations; connections revisited;

Back to equivalence; two gates into gravitation; Preliminaries; general concepts; gauge transformations;

spacetime geometric structure; Lagrangian and field equations; equivalence with General Relativity;

Energy-momentum density of gravitation; Bianchi identities.

Gravitational Lorentz force; torsion force equation; Curvature geodesic equation; weak equivalence

principle; equivalence of the action; teleparallel spin connection; teleparallel coupling prescription;

application to fundamental field.

RECOMMENDED BOOKS:

1. R. Aldrovandi andJ. G. Pereira,An Introduction to Gravitation Theory (Pre-Print), 2001.





Education, 2006.


MATH-917: HOMOTOPY THEORY

Paths and path connected spaces. Homotopy of continuous mappings.Homotopy of paths.Homotopy

classes.The fundamental group of a circle.Higher fundamental groups.The fundamental group of

covering spaces.Torus.Orbit spaces. Punctured plane and

surfaces.

RECOMMENDED BOOKS:

1. Paul Selick, Introduction to Homotopy Theory (Fields Institute Monographs), American

Mathematical Society, 2008.

2. Allen Hatcher, Algebraic Topology, Cambridge University Press, 2001.

3. C. Kosniowski,A First Course in Algebraic Topology, Cambridge University Press, 1980.

4. J. R. Munkres,Topology,2nd Edition, Prentice Hall of India, 2000.

5. G. W. Whitehead, Elements of Homotopy Theory, Springer, 1995.

MATH-918: SYMMETRIES OF SPACETIMES

Spacetime symmetry, geometry and matter, spacetime and matter symmetry, use of symmetries to

simplify equations, physical significance of symmetries, conservation laws corresponding to isometries,

homotheties and their significance, collineation vectors, conformal symmetries, killing vectors(KV's),

curvature collineations (CC's), Ricci collineations (RC's), matter collinations (MC's), Weyl

collineations (WC's), relation between homotheties, KV's, CC's, RC's, MC's, WC's.

RECOMMENDED BOOKS:




38


Education, 2006.


MATH-919: CONVEX ANALYSIS AND APPLICATIONS

Convex functions on the real line, Continuity and differentiability of convex functions,

Characterizations, Differences of convex functions, Conjugate convex functions, Convex sets and affine

sets.

Convex functions on a normed linear space, Continuity of convex functions on normed linear space,

Differentiable convex function on normed linear space, The support of convex functions,

Differentiability of convex function on normed linear space,

RECOMMENDED BOOKS:

1. P. Niculescu, L-E. Persson, Convex Functions and Their Applications, Springer Verlag, New

York, 2006.

2. Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal, Fundamentals of Convex Analysis, Springer,

2004.

3. W. Roberts and D. E. Varberg, Convex Functions, Academic Press, New York, 1973.

4. R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton, N.J., 1970.

5. Latest Research Papers Related to Applications of Convex Functions.

MATH-920: NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS

Boundary and initial conditions, Polynomial approximations in higher dimensions,

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.

Finite Difference Method: Finite difference approximations.

Application to solution of linear and non-linear partial differential equations appearing in physical

problems.

RECOMMENDED BOOKS:

1. G. Strang, and G. Fix, An Analysis of Fintie Element Method, 2nd Edition, Wellesley-

Cambridge, 2008.

2. Daryl L. Logan, A First Course in the Finite Element Method, 5th Edition, CL-Engineering,

2011.

3. George Buchanan, Schaum's Outline of Finite Element Analysis, McGraw-Hill, 1994.

4. Myron B. Allen, Ismael Herrera, G. F. Pinder,Numerical Modeling in Science and Engineering,

John Wiley & Sons, Inc.,New York, 1988.


MATH-921: Representation Theory and the Symmetric Groups

Introduction to group representations: matrix representation, the group algebra, reducibility, Maschke’s

Theorem, Schur’s Lemma, group characters.

Representations of the symmetric group (using Specht modules).

Combinatorial algorithms in representation theory: Robinson-Schensted-Knuth algorithm, Novelli-Pak-

Stoyanovskii hook formula, Frobenius-Young determinantal formula, Schutzenberger’s jeu du taquin,

Introduction to Symmetric functions: Schur functions, Littlewood-Richardson and Murnaghan-

Nakayama Rules.

Applications: Stanley’s theory of differential posets, Fomin’s concept of growths, unimodality results,

Stanley’s symmetric function analogue of the chromatic polynomial of a graph.

RECOMMENDED BOOKS:

39

1. Bruce E. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and

Symmetric Functions, 2nd Edition, Springer, 2001.

2. William Fulton, Young Tableaux: With Applications to Representation Theory and Geometry

, Cambridge University Press, 1997.

MATH-922: NON-NEWTONIAN FLUID MECHANICS

Classification of non-Newtonian fluids, Rheological formulae (time-independent fluids, thixotropic

fluids and viscoelastic fluids), variable viscosity fluids, cross viscosity fluids, the deformation rate,

viscoelastic equation, materials with short memories, time dependent viscosity, the Rivlin-Ericksen

fluid, basic equations of motion in rheological models. The linear viscoelastic liquid, Couette flow,

Poiseuille flows, the current semi-infinite field.Axial oscillatory tube flow, angular oscillatory motion,

periodic transients, basic equations in boundary layer theory, orders of magnitude, truncated solutions

for viscoelastic flow, similarity solutions, turbulent boundary layers, stability analysis.

RECOMMENDED BOOKS:

1. R. Glowinski, Jinchao Xu, Philippe G. Ciarlet, Numerical Methods for Non-Newtonian Fluids,

Volume 16: Special Volume , North Holland, 2011.

2. Clayton T. Crowe, Donald F. Elger, John A. Roberson, Barbara C. Williams, Engineering Fluid

Mechanics, 9th Edition, Wiley, 2008.

3. R.B. Bird, R.C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Vol. I, 2nd ed.,

John Wiley & Sons, New York, NY, 1987.

4. W.R. Schowalter, Mechanics of Non-Newtonian Fluids, Pergamon Press, New York, 1978.

5. G. Astarita and G. Merrucci, Principles of Non-Newtonian Fluid mechanics, McGraw-Hill,

1974.


MATH-923: ORTHOGONAL POLYNOMIALS

Generating functions, orthogonal polynomials, Legendre polynomials, Hermit polynomials, Laguerre

polynomials, Jacobi polynomials, application of special functions to probability distributions, special

functions and distributional integral transform, special functions of several variables.

RECOMMENDED BOOKS:

1. Mathai, A. M. and Houbold, H. J., Special Functions for Applied Scientists, Springer Science

and Business Media, LLC, New York, 2008.

2. Singh, U. P. and Denis, R. Y., Special Functions and Their Applications, Dominant Publishers

and Distributors, 2001.

3. Rainville, E. D., Special Functions, The Macmillan Company, New York, 3rd Edition, 1965.

4. Andrews, G. E., Richard, A. and Roy, R., Special Functions, Cambridge University Press, 1st

Edition, 2000.

MATH-924: NUMERICAL SPLINE TECHINQUES-I

B-spline representation in terms of divided differences. The B-spline representation of spline

functions. Computational considerations, the representation of B-splines (Method based on the

recursive definition of divided differences). Classification of linear/non-linear second orders

PDEs in two and more variables. Laplacian Equations. Heat Equation. Review for the

representation of B-splines (Polynomial B-spline, Extended B-spline, exponential B-spline,

40

Trigonometric B-spline) (Method based on the recursive definition of divided differences). B-

spline numerical methods of solving Laplace, Heat and Wave equations. Finite difference and

finite element methods. Error analysis of methods. Trigonometric/Exponential/Extended B-

spline Numerical Solutions for Elliptic, Parabolic and Hyperbolic PDEs. B-spline approach for

solving system of linear/non-linear PDEs, Irregular Boundaries. Error and stability analysis of

B-spline methods.

Recommended Books

1. de Boor, C.: A Practical Guide to Splines, Springer Verlag, (2001).

2. Smith, D.G.: Numerical Solution of Partial Differential Equations: Finite Difference

Methods Oxford Press, (1990).

3. Ames, W.F.: Numerical Methods for Partial Differential Equations, Academic Press,

(1997).

4. Henwood, D. and Bonet, J.: Finite Elements, Macmillan Press, (1996).

5. Bickford, W.B.: A First Course in Finite Elements, Irwin Inc., (1994).



8. S.S. Sastry, Introductory Methods of Numerical Analysis, fourth ed., PHI Learning,

(2009).

9. J. D. Hoffman, Numerical methods for Engineers and Scientists, second ed., Marcel

Dekker, Inc, New York,.Basel, (1992).

10. S. R. K. Iyengar, R. K. Jain, Numerical methods, New AGE international Publishers,

New Delhi, India, (2009).

11. DU. V. Rosenberg, Methods for solution of partial differential equations, vol. 113. New

York: American Elsevier Publishing Inc.; (1969)


MATH-925: NUMERICAL B-SPLINE TECHINQUES-II

Classification of linear/non-linear second orders ODEs. Discuss the types of boundary

conditions, Theory and implementation of numerical methods for initial and boundary value

problems in ordinary differential equations. One-step, linear multi-step, Runge-Kutta, and

Extrapolation methods; convergence, stability, B-spline collocation methods for numerical

solution of initial and boundary values problems, B-spline for solving class of singular and non-

singular problems, Trigonometric B-spline approach for solving second order system of

linear/non-linear ODEs, B-spline solution for nonlinear differential equation arising in general

relativity, Bratu’s problem, Perturbation’s problem.

Recommended Books

1. de Boor, C.: A Practical Guide to Splines, Springer Verlag, (2001).

41



4. S.S. Sastry, Introductory Methods of Numerical Analysis, fourth ed., PHI Learning,

(2009).

5. J. D. Hoffman, Numerical methods for Engineers and Scientists, second ed., Marcel

Dekker, Inc, New York,.Basel, (1992).

6. Randall J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential

Equations, Siam, Society for Industrial and Applied Mathematics, Philadelphia, (2007).

7. Kendall E. Atkinson, Weimin Han, and David, E. Stewart , Numerical Solution of

Ordinary Differential Equations, John Wiley & Sons, Inc. (2009).

8. C. Henry Edwards, David E. Penney, David Calvis, Elementary Differential Equations,

Pearson Education, Inc . Upper Saddle River, New Jersey, (2008).

9. Strikwerda, John C, Finite difference schemes and Partial differential equations, Siam,

Society for Industrial and Applied Mathematics, Philadelphia, (2004).


MATH-926: COMPUTATIONAL GEOMETRY

Introduction: polygon trapezoidation, Geometry of convex hull, Voronoi diagram and Delaunay

triangulation, Algorithms for their construction, Arrangements of curves, surfaces and lines,

Theory of oct-trees, kd-trees and BSP trees, Applications, Special topics

Recommended Books

1. M. de Berg, M. van Kreveld, M. Overmars and O. Schwarzkopf, Computational

Geometry: Algorithms and Applications, Springer-Verlag, (2000).

2. Recent research papers.

MATH-927: SPECIAL FUNCTIONS AND STATISCAL DISTRIBUTION

Introduction, frequency distribution, mean, mode, median, harmonic mean, mean

deviation, moments, variance, standard deviation, properties of variance and standard

deviation, co-efficient of variation, continuous and discrete distribution, symmetrical

distribution, skewness and kurtosis, moments of continuous and discrete distribution,

probability and probability distribution, Expected values, random variable, binomial

distribution, uniform distribution, moments generating function, cummulants and

cumulative function, exponential distribution, Poisson distribution, Poisson

approximation to the binomial, hypergeometric distribution, geometric

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distribution, normal distribution, gamma function and gamma distribution, beta function

and beta distributions of first and second kind, chi square distribution, students

T-distribution, F distribution, k-distributions.

Recommended Books

1. M.G. Kendall and A. Stuart, The Advanced Theory of Statistics, Vol. 2, Charles Griffin

and Company Limited, London (1961).

2.C. Walac, A Hand Book on Statistical Distributions for Experimentalists, Particle

Physics Group Fysikum. University of Stockholm last modification 10 September (2007).

3.N.A.J. Hasting and J.B. Peacock: Statistical distributions, Butterworth and Company

Ltd, (1975).

4. E.D. Rainville, Special Functions, Macmillan Company, New York, 1960.press, (1990).

MATH-928: FUZZY ANALYSIS

Prerequisite: Set-Valued Analysis

Fuzzy Sets; Fuzzy Sets, level sets, special types of fuzzy sets, Zadeh’s extension principle, fuzzy

functions, sup-min extension principle, interval arithmetic, fuzzy numbers and fuzzy arithmetic. fuzzy

metric spaces; fuzzy metric spaces, inner product, support function, embedding results, continuous

fuzzy functions, measurable fuzzy functions, integrable fuzzy functions, differentiable fuzzy functions.

Recommended Books

1. J.-P. Aubin, H. Frankowska, Set-Valued Analysis, Birkhäuser, Berlin, (1990).



3. V. Lakshmikantham, R. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions,

Taylor - Francis, London, (2003).

4. A. D. R. Choudary, V. Lupulescu, Fuzzy Sets and Fuzzy Differential Equations, Abdus Salam

SMS Lahore, (2010).

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5. C. Negoita, D. Ralescu, Applications of Fuzzy Sets to Systems Analysis, Wiley, New York,

(1975).

MATH-929:ADVANCED PARTIAL DIFFERENTIAL EQUATION

Classification of Partial Differential Equations; Canonical form; Laplace, Wave and Diffusion

Equations. Partial Differential Equations with at least 3 independent variables; Nonhomogeneous

problems; Green function for time independent problems; Infinite domain problems; Green function for

time dependent problems, Wave equation and the method of characteristics. Mathematical modeling.

Recommended Books

1. Elementary Applied Differential Equation; Richard Haberman, Prentice hall, inc. England

Clifts, (1983).

2. Partial Differential Equations, J. Kevorkian, Wadsworth & Books (1989).

3. Partial Differential Equations of Mathematical Physics, Tyn Myint- U, Elsevier Publishing

Company, New York, (1973).

MS/M.Phil and Ph.D Program - University of Sargodha · compact sets in locally convex spaces,...

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