**Program Educational Objectives (PEOs)**

- Apply fundamental principles of science that underlie mathematics for the solution of relevant scientific problems for the industry.
- Achieve professional success by practicing ethical behavior, sustainability, and diversity with effective communication in individual and team.
- Adopt innovative approaches and pursue career growth undertaking professional training and/or studies in mathematical and physical sciences.

- Graduates will be able to summarize major themes and current research problems in their area of specialization.
- Graduates will be able to communicate the major tenets of their field and their work orally and in writing for students, peers, and the lay public.
- Graduates will be able to identify areas where ethical issues may arise in their work or discipline, and articulate strategies for dealing with ethical issues in the profession.
- Graduates will be able to explain and identify open problems and areas needing development in their fields.
- Graduates will have carried out and presented an original work of research in their discipline.

**Program Structure and Course Contents**

MS Mathematics is spread over a minimum of 4 semesters and maximum of 8 semesters. Each semester has at least 18 weeks including one week for mid semester examination and one week for final semester examination. MS Mathematics program has 32 credit hours in total including 26 credit hours of course work and 6 credit hours for research thesis. The scheme of studies is given as under:

Code | Course Title | Credit Hours |
---|---|---|

MA-5013 | Partial Differential Equations | 3 |

MA-5009 | Riemannian Geometry | 3 |

MA-5023 | Advanced Numerical Analysis | 3 |

MA-50XX | Core Course-I | 3 |

Total |
12 |

Code | Course Title | Credit Hours |
---|---|---|

MA-50XX | Elective Course-I | 3 |

MA-50XX | Elective Course-II | 3 |

MA-50XX | Elective Course-III | 3 |

MA-50XX | Elective Course-IV | 3 |

TEX-5078 | Functional Textile | 2 |

Total |
14 |

Code | Course Title | Credit Hours |
---|---|---|

MA-5090 | Research Thesis | 6(3+3) |

Total Credit Hours of the Programme |
30 |

**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 Algebra-I**

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 Algebra-I **

Revision of basic concepts of Ring theory and Module Theory, Modules, Homomorphism and exact sequences. Product and co-product of Modules. Comparison of free Modules and Vector Spaces Projective and injective Modules. Hom and Duality Modules over Principal ideal Domain Notherian and Artinian Module and Rings Radical of Rings and Modules Semi-simple Modules.

**MA -5003: Commutative Algebra-II**

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 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.

** **

**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. Gelfand-Mazur Theorem, Symbolic calculus: differentiation, Analytic functions. Integration of A-Valued functions. Normed rings. Gelfand Naimark theorem.

**MA**** -5006: 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.

** **

**MA**** -5007: Advanced Complex Analysis-II**

Holomorphic functions, Extension of analytic functions, Levi-convexity: 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.

**MA-5009: ****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 semi-Riemannian 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, Wiener-Hopf 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 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.

**MA -5014: Numerical Solutions of Ordinary Differential Equations**

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.

**MA -5015: General Relativity-I**

Original formulation of Special Relativity. The null cone. Review of Electromagnetism. The principles of General Relativity. The Einstein field equations. The stress-energy momentum tensor. The vacuum Einstein equations. Birkhoff’s theorem. The Reissner-Nordstrom solution. The Kerr and the Kerr-Newmann 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, 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.

**MA -5019:**** 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, 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.

**MA-5020: 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 systems. Equations of Hamilton and Appell. Hamilton-Jacobi 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 non-linear partial differential equations appearing in physical problems.

**MA**** -5022:**** Functional Analysis**

Separation properties. Hahn-Banach theorem. Banach algebras theorem (Introduction). Linear mappings. Finite dimensional spaces. Metrization. Boundedeness and continuity. Seminorms and local convexity. Baire category theorem. The Banach-Steinhaus 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.Runge-Kutta 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. 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.

**MA-5024: Mathematical Techniques**

Green’s function method with applications to wave-propagation. 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. Wiener-Hopf technique with applications to diffraction problems.

** **

**MA-5025: 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.

** **

**MA-5026: 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, Bogoliubov-Krilov 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, Wiencr-Hopf equations, Fredholm theory, series solutions. Variational methods, The Euler-Lagrange equations, Solutions to some famous problems, Sturm-Liouville Problem and variational principles, Rayleigh-Ritz 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.

** **

**MA-5027: Theory of Spline Functions-I **

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.

** **

**MA-5028: Theory of Spline Functions-II**

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.

** **

**MA-5029: Mathematical Modeling-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.

** **

**MA-5030: Mathematical Modeling-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.

** **

**MA-5031: 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.

** **

**MA-5032: 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.

**MA-5033: General Relativity-II **

Black holes. Coordinate and essential singularities. Horizons. Coordinates passing through horizons. The Kruskal and the Carter-Penrose (CP) diagrams for the Schwarzschild geometry. The maximal extension. The Einstein-Rosen bridge. Wormholes. The CP diagram for the RN metric. The no-hair and cosmic censorship hypotheses. Gravitational forces about black holes. Black hole thermodynamics. Observational status and central black holes. Kaluza-Klein theory. Problems of quantum gravity. Quantization in curved space backgrounds and Hawking radiation. Isometries. Homotheties and their significance in Relativity.

** **

**MA-5034: Classical Field Theory**

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 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.

** **

**MA-5035: 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.

** **

**MA-5036: Electrodynamics-II**

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.

** **

**MA-5037: Magnetohydrodynamics-I**

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. Magneto-sonic 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.

** **

**MA-5038: Magnetohydrodynamics-II**

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.

** **

**MA-5039: Quantum Field Theory**

Classical field theory, Lagrangian mechanics, variational principle, vibrating stings, classical field theory, Lorentz transformations, Lorentz group, classical scalar fields, Klein-Gordon equation, complex scalar fields, energy-momentum 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.

** **

**MA-5040: 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 Jordan-Chevalley decomposition Killing forms Criterion for semisimplicity, product of Lie algebras; Classification of Lie algebras upto dimension 4; Applications of Lie algebras.

** **

**MA-5041: 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, 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.

** **

**MA-5042: Elastodynamics-I**

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 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.

**MA-5043: 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.** **

** **

**MA-5044: 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.

**MA-5045: 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.

**MA-5046: 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.

**MA-5047: 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.

** **

**MA-5048: 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 Degree-of-Freedom 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.

** **

**MA-5049: Special Topics in Advance Mathematics-I**

The course contents should be specified from time to time by the resource person with consultation of the Chairman, Department of Applied Sciences.

** **

**MA-5050: Special topics in Advanced Mathematics-II**

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.

MSc/BS in Pure Mathematics/Applied Mathematics/Computational Mathematics (minimum 16-year education) or equivalent degree with minimum CGPA 2.00/4.00 in semester system or 60% in annual system/term system from an HEC recognized institute/university.

- The applicant must pass NTS-GAT (General) test with minimum
**50/100**marks prior to apply. - The applicant must not be already registered as a student in any other academic program in Pakistan or abroad.
- Result waiting applicants may apply for admission, however their merit will be finalized only on submission of final BS/M.Sc or equivalent official transcript or degree.
- The student will submit his/her publication from his/her thesis research work and submit to his/her supervisor. Final defense will be held after the submitted publication of student will be notified as “Under Review” or “Under Consideration” by a journal. It will be compulsory for graduate students to include his/her Supervisor’s name in his/her publication.
- Relevant Admission Committee will determine relevancy of terminal degree and decide deficiency course/s (if any) at the time of admission interview, the detail of which will be provided to the student in his/her admission letter/email.
- Deficiency course/s will be treated as non-credit and qualifying course/s for which student will also pay extra dues as per fee policy. Those course/s will neither be mentioned in student’s final transcript nor will be included for calculation of CGPA. However, the student may obtain his/her a separate transcript for completion of deficiency course/s.

**Merit Criteria**

Admission merit will be prepared according to the following criteria:

BS/MSc or Equivalent | 60% weightage |

NTS-GAT (General) Test | 30% weightage |

Interview | 10% weightage |

Fee Head | 1st | 2nd | 3rd | 4th |
---|---|---|---|---|

Tuition Fee | 30,000 | 30,000 | 21,000 | 21,000 |

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 | 3000 | 3000 | 3000 | 3000 |

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 |

TOTAL |
68400 |
46000 |
32000 |
37000 |