Masters Programs
MS Mathematics
 To inculcate habits of creative thinking, critical analysis.
 To make the student appreciate the uniqueness of mathematics as tool of the tools, having the power of generalization and the power of application.
 To develop ability in students to formulate a problem using the language of mathematics.
 To equip students with the mathematical techniques and solutions to indigenous problems applied in industrial, business and financial organizations.
 To strengthen academiaprofessionalworld bonding by tailoring the courses and the trainings offered according to needs of the enduser.
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 Teaching and research at university/post graduate college level in the departments of Pure Mathematics, Applied Mathematics and Computational Mathematics and so on.
 Research and Development in public and private sector organizations.
 Product development in public and private sector organizations.
 Higher studies and research in Mathematics and the relevant fields.
Program Structure and Course Contents
MS Mathematics is spread over a minimum of 4 semesters and maximum of 6 semesters. Each semester has at least 18 weeks including one week for mid semester examination and one week for final examination. MS Mathematics program has 30 credit hours in total including 24 credit hours of course work and 6 credit hours for research thesis. The scheme of studies is given as under: Session (20182020)



Note:
 MS students will have to pass 24 credit hours courses and 6 credit hours thesis.
 Department can offer any course from the list of approved courses on the availability of resources.
 Summer semester will not be offered.
 Internal assessments include a seminar, quizzes and assignments of every student in each subject. At least one seminar per student per subject is compulsory.
 Number of assessment activities is double to the number of credit hours of each subject.
List of Courses
Course Contents
MA 5001: Commutative AlgebraI
Integral domains, unit, irreducible and prime elements in ring, Types of ideals, Quotient rings, Rings of fractions, Ring homomorphism, Euclidean domains. Construction of formal power series ring R[[X]] and polynomial ring R[X] in one indeterminate. Polynomial extension of Noetherian domains, Quotient ring of Noetherian rings, Ring of fractions of Noetherian rings. Valuation map and Valuation rings.
MA 5002: Homological AlgebraI
Revision of basic concepts of Ring theory and Module Theory, Modules, Homomorphism and exact sequences. Product and coproduct of Modules. Comparison of free Modules and Vector Spaces Projective and injective Modules. Hom and Duality Modules over Principal ideal Domain Notherian and Artinian Module and Rings Radical of Rings and Modules Semisimple Modules.
MA 5003: Commutative AlgebraII
Guass Theorem, Quotient of a UFD, Nagata Theorem. Divisor classes, Divisor class monoid, divisor class group, Divisorial ideals, divisors, Krull rings, Atomic Domains, Domains Satisfying ACCP, Bounded Factorization Domains, Half Factorial Domains, Finite Factorization Domains: Group of divisibility G(D) of a domain D, G(D) and FFD.
MA 5004: Homological AlgebraII
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 coefficient theorems. Hilbert Syzygy theorem, Serre’s theorem, mixed identities.
MA 5005: 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. GelfandMazur Theorem, Symbolic calculus: differentiation, Analytic functions. Integration of AValued functions. Normed rings. Gelfand Naimark theorem.
MA 5006: Advanced Complex AnalysisI
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.
MA 5007: Advanced Complex AnalysisII
Holomorphic functions, Extension of analytic functions, Leviconvexity: The Levi form, Geometric interpretation of its signature, E.E. Levi’s theorem, Connections with Kahlerian geometry, Elementary properties of plurisub harmonic functions. Cohomology, complex manifolds. The d.operators, the Poincare Lemma and the Dolbeaut Lemma, The Cousin problems, introduction to Sheaf theory.
MA 5008: Topological Vector Spaces
Vector spaces, Topological vector spaces, 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: 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.
MA5009: Riemannian Geometry
Definition and examples of manifolds. Differential maps. Submanifolds. Tangents. Coordinate vector fields. Tangent spaces. Dual spaces. Multilinear functions. Algebra of tensors. Vector fields. Tensor fields. Integral curves. Flows. Lie derivatives. Brackets. Differential forms. Introduction to integration theory on manifolds. Riemannian and semi Riemannian metrics. Flat spaces. Affine connection. Parallel translations. Covariant differentiation of tensor fields. Curvature and Torsion tensors. Connection of a semiRiemannian tensor. Killing equation and Killing vector fields. Geodesics. Conformal transformations and the Weyl tensor.
MA 5010: Integral Equations
Existence theorems, intergral equations with Kernels.Applications to partial differential equations. Integral transforms, WienerHopf techniques.
MA 5011: 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.
MA 5012: 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 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
MA 5013: Partial Differential Equations
Cauchy’s problems for linear second order equations in nindependent variables. Cauchy Kowalewski Theorem. Characteristics surfaces. Adjoint operations, Bicharacteristics Spherical and Cylindrical Waves. Heat equation. Wave equation. Laplace equation. MaximumMinimum Principle, Integral Transforms.
MA 5014: Numerical Solutions of Ordinary Differential Equations
Theory and implementation of numerical methods for initial and boundary value problems in ordinary differential equations. Onestep, linear multistep, RungeKutta, 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.
MA 5015: General RelativityI
Original formulation of Special Relativity. The null cone. Review of Electromagnetism. The principles of General Relativity. The Einstein field equations. The stressenergy momentum tensor. The vacuum Einstein equations. Birkhoff’s theorem. The ReissnerNordstrom solution. The Kerr and the KerrNewmann solution. The Newtonian limit of Relativity. The Schwarzschild exterior solution and relativistic equations of motion. The classical tests of Relativity. Linearized gravity and gravitational waves. Foliations. Symmetries of Spacetimes.
MA 5016: Graph Theory
Fundamentals. Definition. Paths cycles and trees. Hamilton cycles and Euler circuits. Planer graphs. Flows, Connectivity and Matching Network flows. Connectivity and Menger’stheorem. External problems paths and Complete Subgraphs. Hamilton path and cycles.Colouring. Vertexcolouring Edge colouring. Graph on surfaces.
MA 5017: 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. Packing and covering. Linear algebra over finite fields. Gaussian coefficients. The pigeonhole Principle. Some special cases. Ramsey’s theorem and applications. Automorphism groups and permutation groups. Enumeration under group action.
MA 5018: 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, Literature search, Formal research proposal, budgeting and funding, 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, Peerreview 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.
MA 5019: NonNewtonian Fluid Mechanics
Classification of nonNewtonian fluids, Rheological formulae (timeindependent fluids, thixotropic fluids and viscoelastic fluids), variable viscosity fluids, cross viscosity fluids, the deformation rate, viscoelastic equation, time dependent viscosity, the RivlinEricksen fluid, basic equations of motion in rheological models. The linear viscoelastic liquid, Couette flow, Poiseuille flows, the current semiinfinite 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.
MA5020: Advanced Analytical DynamicsI
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 systems. Equations of Hamilton and Appell. HamiltonJacobi theorem. Separable systems. Holder’s variational principle and its consequences.
MA 5021: 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. Applications to solutions of linear and nonlinear partial differential equations appearing in physical problems.
MA 5022: Functional Analysis
Separation properties. HahnBanach theorem. Banach algebras theorem (Introduction). Linear mappings. Finite dimensional spaces. Metrization. Boundedeness and continuity. Seminorms and local convexity. Baire category theorem. The BanachSteinhaus theorem. The open mapping theorem. The closed graph theorem. Bilinear mappings. The normed dual of normed spaces. Adjoints.
MA 5023: Advanced Numerical Analysis
Introduction. Euler’smethod. The improved and modified Euler’s method.RungeKutta method. Milne’s method. Hamming’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. GaussSiedel 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. CyclicConsistently ordered matrices. Choice of optimum value for relaxation parameter.
MA5024: Mathematical Techniques
Green’s function method with applications to wavepropagation. Solution of algebraic equations by perturbation methods. Evaluation of integrals by expansion of integrands. Laplace methods. The method of stationary phase. The methods of steepest descent. Solution of the linear damped oscillator equation by perturbation methods. The WKB approximation. Variational problems with variable end points. Corner conditions. Sufficient conditions for minimum. The Ritz method and its applications. A survey of transform techniques. WienerHopf technique with applications to diffraction problems.
MA5025: ODEs and Computational Linear Algebra
Introduction. Euler’s method. The improved and modified Euler’s method. RungeKutta 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. GaussSiedel 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. CyclicConsistently ordered matrices. Choice of optimum value for relaxation parameter.
MA5026: Group Theory
Elementary concepts. Symmetric and alternating groups of finite degree. Order of a permutation. Orbits of the symmetric and alternating groups. Stabilizer subgroups and transitive groups. Free products of group. Group amalgams and their embeddability in groups. Generalized free product of groups. Permutational product of groups. Cartesian product of groups. Wreath product of groups. Multiplicative group of a finite field. Projective line over finite fields. Projective and linear groups through action.
MA5027: Advanced Mathematical Physics
Nonlinear ordinary differential equations, Bernoulli’s equation, Riccati equation, Lane Emden equation, Nonlinear Pendulum, Duffing’s equation, Pinney’s equation, Perturbation theory, BogoliubovKrilov method. Linear partial differential equations, classification, initial and boundary values problems, Fourier analysis, Heat equation, Wave equation, Laplace equation etc. Integral equations, classification, integral transform separable kernels, singular integral equations, WiencrHopf equations, Fredholm theory, series solutions. Variational methods, The EulerLagrange equations, Solutions to some famous problems, SturmLiouville Problem and variational principles, RayleighRitz Methods for partial differential equations. Matrix algebra, method of Faddeev, Caley Hamilton’ theorem function of matrices. Functions of matrices, Kronecker and Tensor product, special matrices.
MA5028: Theory of Spline FunctionsI
Parametric Curves: Affine Maps: Translation, Rotation, Reflection, Stretching, Scaling and shear. Barycentric combination. Convex combination. Convex Hull. Forms of parametric curves: Algebraic form, Hermite form, Control point form, Bernstein Bezier form and their matrix forms. Algorithm to compute Bernstein Bezier form. Properties of Bernstein Bezier form: Convex Hull property. Affine invariance property, Variation diminishing property. Rational quadratic form. Rational cubic form. Tensor product surface.
Spline Functions: Natural splines. Cardinal splines. Periodic splines on uniform mesh. Representation of spline and its different forms. Natural spline and periodic spline in terms of polynomials and power truncated functions. Odd degree spline. Existence theorem. Existence and uniqueness of natural and periodic spline. Remainder theorems.
MA5029: Theory of Spline FunctionsII
Interpolatory cubic splines. The representation of s in terms of the values Mi=s(2)(xi), i=0,1,2,…,k. The representation of s in terms of the values mi=s(1)(xi), i=0,1,2,…,k. Quadratic Hermite spline. Theorems regarding error analysis. Theorems regarding to Convergence of the D1, D2, natural and periodic splines. End conditions for cubic Hermite spline interpolation. E(α)cubic splines.
MA5030: Mathematical ModelingI
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 NonLinear Models.
MA5031: Mathematical ModelingII
Modeling with Differential Equations: Exponential growth and decay. Linear, nonlinear 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.
MA5032: Design Theory
Basic definitions and properties, related structure. The incidence matrix, graphs, residual structures. The BruckRyserChowla 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 3desings. Steiner triple systems. The Mathieu groups.
MA5033: Minimal Surfaces
Regular surfaces: Differentiable functions on surfaces. The tangent plane. Geometric definition of area. Gaussian and mean curvature. Curvature in local coordinates. Ruled and minimal surfaces: Historical survey and introduction to the theory of minimal surfaces. Basic minmal surface properties. Topological and physical properties. Stable and unstable minimal surfaces. Two dimensional minimal surfaces in three dimentional space. Helecoid, catenoid and conoid. Harmonic approximation to area. Nambu Goto action.
MA5034: General RelativityII
Black holes. Coordinate and essential singularities. Horizons. Coordinates passing through horizons. The Kruskal and the CarterPenrose (CP) diagrams for the Schwarzschild geometry. The maximal extension. The EinsteinRosen bridge. Wormholes. The CP diagram for the RN metric. The nohair and cosmic censorship hypotheses. Gravitational forces about black holes. Black hole thermodynamics. Observational status and central black holes. KaluzaKlein theory. Problems of quantum gravity. Quantization in curved space backgrounds and Hawking radiation. Isometries. Homotheties and their significance in Relativity.
MA5035: Classical Field Theory
Review of continuum mechanics. Solid and fluid media. Constitutive equations and conservation equations. The concept of a field. The fourdimensional formulation of fields and the stressenergy momentum tensor. The scalar field. Linear scalar fields and the KleinGordon equation. Nonlinear scalar fields and fluids. The vector fields. 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 fields. The massless tensor field and the Einstein field equations. Gravitational waves. The massive tensor fields. Coupled field equations.
MA5036: ElectrodynamicsI
Maxwell’s equations. Electromagnetic wave equation. Boundary conditions. Waves in conducting and nonconducting 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.
MA5037: ElectrodynamicsII
General angular and frequency distributions of radiation from accelerated charges. Thomson scattering. Cherenkov radiation. Fields and radiation localized oscillating sources. Electric dipole fields and radiation. Magnetic dipole and electric quadruple fields. Multipole fields. Multipole expansion of the electromagnetic fields. Angular distribution sources of muiltipole radiation. Spherical wave expansion of a vector plane wave. Scattering of electromagnetic wave by a conducting sphere.
MA5038: MagnetohydrodynamicsI
Equations of electrodynamics. Equations of Fluid Dynamics. Ohm’s law. Equations of magnetohydrodynamics. Motion of a viscous electrically conducting fluid with linear current flow. Steady state motion along a magnetic field. Wave motion of an ideal fluid. Magnetosonic waves. Alfven’s waves. Damping and excitation of MHD waves. Characteristics lines and surfaces. Kinds of simple waves. Distortion of the profile of a simple wave. Discontinuities. Simple and shock waves in relativistic magnetohydrodynamics. Stability and structure of shock waves. Discontinuities in various quantities. Piston problem. Oblique shock waves.
MA5039: MagnetohydrodynamicsII
Flow of an ideal fluid past magnetized bodies. Fluid of finite electrical conductivity flow past a magnetized body. Theory. Bllard’s Theory. Earth’s field. Turbulent motion and dissipation. Vorticity anology. Effects of molecular structure. Currents in a fully ionized gas. Partially ionized gases. Interstellar fields. Dissipation in hot and cool clouds.
MA5040: Quantum Field Theory
Classical field theory, Lagrangian mechanics, variational principle, vibrating stings, classical field theory, Lorentz transformations, Lorentz group, classical scalar fields, KleinGordon equation, complex scalar fields, energymomentum tensor, electromagnetic field, Maxwell’s equations, spinor field, Dirac equation, symmetries and conservation laws, Noether’s theorem, translation invariance. Quantization of fields, canonical quantization of fields, quantization of scalar fields, particle interpretation of quantum field theory. Interacting Quantum Fields, perturbation theory, time ordering, decay rate of an unstable particle, higher order perturbation theory, Wick’s theorem second order perturbation theory, renormalization.
MA5041: Lie Algebra & Lie Groups
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 JordanChevalley decomposition Killing forms Criterion for semisimplicity, product of Lie algebras; Classification of Lie algebras upto dimension 4; Applications of Lie algebras.
MA5042: Computer Aided Geometric Designing
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, Pointnormal interpolation, BSpline curves: Bspline segments, curves, Knot insertion, degree elevation, Greville Abscissae, smoothness. Constructing Splines Curves: Greville interpolation, modifying BSpline curves, cubic spline interpolation, the minimum property, piecewise cubic interpolation. Rational Bezier and BSpline Curves: Rational Bezier curves, Rational Cubic Bspline curves.
MA5043: ElastodynamicsI
Cartesian tensors, Orthogonal rotation of axes, Transformation equations. Translation and rotation, Different orders of tensors. Algebra of tensors, Inner and outer multiplication of tensors, Symmetric and antisymmetric 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.
MA5044: ElastodynamicsII
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.
MA5045: 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.
MA5046: Fluid Dynamics
Euler’s equation of motion. Viscosity. NavierStoke’s equations and exact solutions. Dynamical similarity and Reynold’s number. Turbulent flow. Boundary layer concept and governing equations. Reynold’s equations of turbulent motion. Magnetohydrodynamics. MHD equations. Fluid drifts. Stability and equilibrium problems.
MA5047: Fluid Mechanics
NavierStoke’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 boundarylayer flow, Reynold’s equations of turbulent motion. Magnetohydrodynamics, MHD equations, fluid drifts, stability and equilibrium problems.
MA5048: Mathematical Techniques for BVPs
Green’s function method with applications to wavepropagation. Regular and singular perturbation techniques with applications variational methods. A survey of transform techniques: WienerHopf technique with applications to diffraction problems.
MA5049: Advanced Analytical Dynamics
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. HamiltonJacobi theorem. Separable systems. Holder’s variational principle and its consequences.
MA5050: 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.
MA5051: 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.
MA5052: Inequalities involving Convex Functions
Jensen’s and related inequalities, 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.
MA5053: Structural Dynamics
Formulation of Equation of Motion, Free Vibration of Undamped SDOF Systems, Free Vibration of Damped SDOF Systems, SDOF System Characterization, Undamped Harmonic Response, Damped Harmonic Response, Identification of Structural Damping, Harmonic Base Motion and Accelerometers, Period Loads and the Fourier Series, Impulsive Loads and Shock Spectra, Numerical Integration Methods, Earthquakes and Response Spectra, Seismic Design Spectra, Multiple DegreeofFreedom Systems (MDOF), Modeling Distributed Parameter Systems, Static Condensation and Consistent Mass Matrices, Generalized Eigenvalue Problems, Rayleigh Quotient and Orthonormality of Modes, Modal Superposition, Damping in MDOF Systems, Modal Participation and Contributions, Seismic Response of MDOF Systems.
MA5054: Special Topics in Advance MathematicsI
The course contents should be specified from time to time by the resource person with consultation of the Chairman, Department of Applied Sciences.
MA5055: Special topics in Advanced MathematicsII
The course contents should be specified from time to time by the resource person with consultation of the Chairman, Department of Applied Sciences.
TEX 5078: Functional Textile
Basics of textiles and raw materials, Preparatory processes of Spinning, Types of yarns and spinning, Mathematical Modeling regarding fiber and yarn properties, Woven Fabric Production, Knitted Fabric Production, Mathematical Modeling regarding fiber, yarn and woven fabric properties, Mathematical Modeling regarding fiber, yarn and knitted fabric properties, Nonwoven fabric formation and operations, Introduction to textile processing, Pretreatment and dyeing of textiles, Printing and finishing of textiles, Application of mathematical modelling in textile processing, Clothing Product design and development, Clothing preparatory processes, Clothing manufacturing processes, Applications of mathematical modeling in clothing.
M.Sc./BS in Pure Mathematics/Applied Mathematics/Computational Mathematics (minimum sixteen year education) or its equivalent with a minimum CGPA 2.5/4 in semester system or 60% in annual system/term system from an HEC recognized institute/university.
The candidate will have to pass NTU GAT Test. Further the applicant must not be already registered as a student in any other academic program in Pakistan or abroad.
Admission Process
 The MS program shall be advertised in the beginning of each academic session.
 An applicant shall apply on a prescribed form within the due date given in the advertisement for admission.
 The completed application form, along with required documents, shall be submitted in the admission office.
 The applications shall be evaluated according to the following criteria for making the merit list.
MS Mathematics  
Intermediate  10% weightage 
BS Mathematics/B.Sc + M.Sc  40/20+20% weightage 
NTU GAT (General)  40% weightage 
Interview  10% weightage 
Fee Head  1st  2nd  3rd  4th 

Tuition Fee  27000  27000  18000  18000 
Admission Fee  20000       
Degree Fee        5000 
Certificate Verification Fee  2000       
Processing Fee    5000     
University Security  5000       
Red Crescent Donation  100       
University Card Fee  300       
Library Fee  1000  1000  1000  1000 
Examination Fee  3000  3000  3000  3000 
Medical Fee  2000  2000  2000  2000 
Student Activity Fund  2000  2000  2000  2000 
Endowment Fund  1000  1000  1000  1000 
Transport Fee*  5000  5000  5000  5000 
TOTAL  68400  46000  32000  37000 
* There is no Transport Fee for Hostel Resident but they will pay hostel charges