1.2.2

M. Sc Mathematics

Name of the School: School of Basic and Applied Sciences

Division: Mathematics

Year: 2017-18 

 

 

     

     

List of electives

Master of Science (Mathematics)

2017-19 Employability

Skill Development

Elective-I

Sl. No Course Code

Name of the Course L T P C

1 MSCM6004 Module Theory 4 0 0 4

2 MSCM6005 Measure and Probability Theory 4 0 0 4

3 MSCM6006

Analytical Number Theory

4 0 0 4

4 MSCM6007 Harmonic Analysis 4 0 0 4

Elective-II

Sl. No Course Code

Name of the Course L T P C

1 MSCM6008 Algebraic Topology 4 0 0 4

2 MSCM6009 Dynamical systems 4 0 0 4

3 MSCM6010 Fluid Mechanics 4 0 0 4

4 MSCM6011 Discrete Structures 4 0 0 4

Elective-III

Sl. No Course Code

Name of the Course L T P C

1 MSCM6015 Manifolds and Applications 4 0 0 4

2 MSCM6016 Mathematical Modelling 4 0 0 4

3 MSCM6017 Financial Mathematics 4 0 0 4

4 MSCM6018 Coding Theory 4 0 0 4

Elective-IV

Sl. No Course Code

Name of the Electives L T P C

1 MSCM6019 Finite Element method 4 0 0 4

2a. MSCM6020 Computational Fluid Dynamics 3 0 0 3

     

     

2b.

MSCM6021

Computational Fluid Dynamics

Lab

0 0 2 1

3 MSCM6022 Stochastic Processes 4 0 0 4

4 MSCM6023 Automata & Formal Languages 4 0 0 4

5 MSCM6024 Cryptography 4 0 0 4

Detailed Syllabus

Elective-I

Name of The Course Module Theory

Course Code MSCM6004Prerequisite Corequisite Antirequisite L T P C 4 0 0 4

Course Objectives: Students will be able to understand Module theory as linear algebra over general rings. They will have knowledge of special classes of modules such as free modules, projective modules, flat modules to name a few. Students will have knowledge of theory of modules over PID and its application to Jordan and Rational canonical forms.

Course Outcomes

CO1 To Summarize various types of modules and module homomorphism. CO2 To describe different properties of modules and tensor product.CO3 To define rings and ideals and understand various properties.CO4 To explain Noetherian Rings, Primary Decomposition and different theorems

related to them.

Text Book(s)

1. Lang, S., Algebra, Addison-Wesley, 1993. Lam, T.Y., A First Course in Non-Commutative Rings, Springer Verlag. Hungerford, T.W., Algebra, Springer.

2. Jacobson, N., Basic Algebra, II, Hindusthan Publishing Corporation, India.

     

     

Reference Book(s)

3. Dummit, D.S., Foote, R.M., Abstract Algebra, Second Edition, John Wiley & Sons, Inc., 1999.

4. Atiyah, M., MacDonald, I.G., Introduction to Commutative Algebra, Addison-Wesley, 1969.

5. Malik, D.S., Mordesen, J.M., Sen, M.K., Fundamentals of Abstract Algebra, The McGraw-Hill Companies, Inc.

6. Curtis, C.W., Reiner, I., Representation Theory of Finite Groups and Associated Algebras, Wiley-Interscience, NY.

Unit-1 10 Hours Units and Module Homomorphisms, Submodules and Quotient Modules, Operations on submodules, Direct Sum and Product, Finitely Generated Modules, Free Modules. Unit-2 10 Hours Tensor Products of modules, Universal Property of the tensor product, Restriction and Extension of Scalars, Algebras. Exact Sequences, Projective, Injective and Flat Modules, Five Lemma, Projective Modules and HomR(M,-), injective modules and HomR(-,M), Flat modules and M x R - . Unit-3 10 Hours

Rings and Modules of Fractions, Local Properties, Extended and contracted ideals in rings of fractions. Nilradical and Jacobson radical, Nakayama’s Lemma, Operations on Ideals, Prime Avoidance, Chinese Remainder Theorem, Extension and Contraction of ideals. Unit-4 10 Hours Noetherian Rings, Primary Decomposition in Noetherian Rings. Integral Dependence, Lying-Over Theorem, Going-Up Theorem, Integrally Closed Domains, Going-Down Theorem, Noether Normalization, Hilbert Nullstellensatz. Transcendence Base, Separably Generated Extensions, Schmidt and Lüroth Theorems.

Continuous Assessment Pattern

Internal Assessment (IA)

Mid Term Test (MTE)

End Term Test (ETE)

Total Marks

20 30 50 100

     

     

Name of The Course Measure and Probability TheoryCourse Code MSCM6005Prerequisite Corequisite Antirequisite L T P C 4 0 0 4

Course Objectives: The aim of this course is to learn the basic elements of Measures and Probability Theory. It provides a foundation for many branches of mathematics such as harmonic analysis, theory of partial differential equations and probability theory.

Course Outcomes

CO1 Summarizes the basic ideas of measure, Lebesgue measure, probability measure and random variables with variety of examples.

CO2 Solve challenging problems, develop proofs of theorems on their own and present those proofs clearly and coherently with appropriate illustrative examples.

CO3 Define and illustrate the concept of measurable functions, Borel and Lebesgue measurability, the 𝐿 - space.

CO4 Define and illustrate the concept of probability space, limit of events, random vectors, distribution and expectation.

CO5 Define the concepts of sequence of random variables, moment generating function and modes of convergence.

CO6 Prove a selection of theorems concerning Weak and strong laws of large number, continuity theorem and central limit theorem.

Text Book(s)

1. P. Billingsley, Probability and Measure, 3rd ed., John Wiley & Sons, New York, 1995

2. G. De Barra, Measure theory and Integration, New age international publishers, 2012 Reference Book(s)

3. J. Rosenthal, A First Look at Rigorous Probability, World Scientific, Singapore, 2000.

4. A.N. Shiryayev, Probability, 2nd ed., Springer, New York, 1995.

5. K.L. Chung, A Course in Probability Theory, Academic Press, New York, 1974. Unit-1 10 Hours Measure Theory: Measures and outer measures. Measure induced by an outer measure, Extension of a measure.Uniqueness of Extension, Completion of a measure. Lebesgue outer measure. Measurable sets. NonLegesgue measurable sets. Regularity. Measurable

     

     

functions. Borel and Lebesgue measurability. Integration of non-negative functions. The general integral. Convergence theorems. Riemann and Lebesgue integrals. The LP -space. Convex functions. Jensen’s inequality. Holder and Minkowski inequalities. Unit-2 10 Hours Probability measure, probability space, construction of Lebesgue measure, extension theorems, limit of events, Borel-Cantelli lemma.Unit-3 10 Hours

Random variables, Random vectors, distributions, multidimensional distributions, independence. Expectation, change of variable theorem, convergence theorems. Unit-4 10 Hours Sequence of random variables, modes of convergence. Moment generating function and characteristics functions, inversion and uniqueness theorems, continuity theorems, Weak and strong laws of large number, central limit theorem.

Continuous Assessment Pattern

Internal Assessment (IA)

Mid Term Test (MTE)

End Term Test (ETE)

Total Marks

20 30 50 100

     

     

Course Objectives: Students will develop an understanding of Analytic aspects and methods used to study the distribution of prime numbers, Arithmetic functions and their utility in the analytic theory of numbers including the distribution of primes.

Course Outcomes

CO1 Introduction to various special kind of functions like Moebius function, Euler phi (totient) function, Von Mangoldt function, divisor and sum-of-divisors functions to name a few.

CO2 Exposure to Riemann zeta function, Partial sums of the Euler phi function and averages of the Moebius functions.

CO3 Develop understanding and knowledge of Dirichlet series and its analytic properties.

CO4 Ability to understand the proofs of basic theorems of Analytical Number Theory.

Text Book(s)

1. A.J. Hildebrand : Introduction to Analytic Number Theory Math 531 Lecture Notes, Fall 2005 , http://www.math.uiuc.edu/~hildebr/ant

Reference Books(s)

2. Harold Davenport: Multiplicative number theory, third ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000,

3.T. Apostol : Introduction to analytic number theory, New York: Springer-Verlag, 1976

Unit-1 10 Hours Review of: Primes and the Fundamental Theorem of Divisibility and primes , The Fundamental Theorem of Arithmetic , The infinitude of primes , Elementary theory of Arithmetic functions, Introduction and basic examples .Additive and multiplicative functions. The Moebiusfunction . The Euler phi (totient) function. The von Mangoldt function. The divisor and sum-of-divisors functions.

Name of The Course Analytical Number TheoryCourse Code MSCM6006Prerequisite Basic concepts of number theoryCorequisite Antirequisite L T P C 4 0 0 4

     

     

Arithmetic functions II: Asymptotic estimates : Big oh and small oh notations, asymptotic equivalence , Basic definitions, Extensions and Examples, The logarithmic integral, Sums of smooth functions: Euler’s summation formula, Statement of the formula , Partial sums of the harmonic series, Partial sums of the logarithmic function and Stirling’s formula. Unit-2 10 Hours Integral representation of the Riemann zeta function. Removing a smooth weight function from a sum: Summation by parts, The summation by parts formula, Kronecker’s Lemma , Relation between different notions of mean values of arithmetic functions , Dirichlet series and summatory functions , Approximating an arithmetic function by a simpler arithmetic function: The convolution method , Description of the method, Partial sums of the Euler phi function , The number of squarefree integers below x, Wintner’s mean value theorem , A special technique: The Dirichlet hyperbola method, Sums of the divisor function, Distribution of primes I: Elementary results , Chebyshev type estimates, Mertens type estimates, Elementary consequences of the PNT, The PNT and averages of the Moebius function. Unit-3 10 Hours Arithmetic functions III: Dirichlet series and Euler products , Algebraic properties of Dirichlet series , Analytic properties of Dirichlet series, Dirichlet series and summatory functions, Mellin transform representation of Dirichlet series , Analytic continuation of the Riemann zeta function, Lower bounds for error terms in summatory functions , Evaluation of Mertens’ constant , Inversion formulas.Unit-4 10 Hours Distribution of primes II: Proof of the Prime Number Theorem :Introduction , The Riemann zeta function, I: basic properties , The Riemann zeta function, II: upper bounds , The Riemann zeta function, III: lower bounds and zero free region, Proof of the Prime Number Theorem.

Continuous Assessment Pattern

Internal Assessment (IA)

Mid Term Test (MTE)

End Term Test (ETE)

Total Marks

20 30 50 100

     

     

Course Objectives: Harmonic analysis is equated with the study of Fourier series and integrals. In this course we will study Fourier Transforms on the Line, Fourier Analysis on Locally Compact Abelian Groups , Commutative Banach Algebras and Spectral synthesis in regular algebras.

Course Outcomes

CO1 To define Fourier transform in various function spacesCO2 To explain Fourier Analysis on Locally Compact Abelian Groups and Haar

measure. CO3 To define Commutative Banach Algebras.CO4 To perform spectral synthesis in different Algebras.

Text Book(s)

1. Yitzhak Katznelson ., : An Introduction to Harmonic Analysis, Third Edition , Cambridge University Press

2. Henry Helson. - Harmonic analysis, Springer Verlag- 1995. Reference book(s)

3. T. W. Kroner,: Fourier Analysis, Cambridge University Press.

Unit-1 10 Hours Fourier Transforms on the Line : Fourier transforms for L 1, Fourier–Stieltjes transforms , Fourier transforms in L p (R) , 1 < p ≤ 2 , Tempered distributions and pseudo-measures , Almost- Periodic functions on the line , The weak-star spectrum of bounded functions , The Paley– Wiener theorems , The Fourier–Carleman transform, Kronecker’s theorem. Unit-2 10 Hours Fourier Analysis on Locally Compact Abelian Groups : Locally compact abelian groups, The Haar measure, Characters and the dual group, Fourier transforms, Almost-periodic functions and the Bohr compactificationUnit-3 10 Hours

Commutative Banach Algebras : Definition, examples, and elementary properties

Name of The Course Harmonic AnalysisCourse Code MSCM6007Prerequisite Corequisite Antirequisite L T P C 4 0 0 4

     

     

Maximal ideals and multiplicative linear functionals , The maximal-ideal space and the Gelfand representation , Homomorphisms of Banach algebras , Regular algebras, Wiener’s general Tauberian theorem. Unit-4 10 Hours Spectral synthesis in regular algebras, Functions that operate in regular Banach algebras , The algebra M (T) and functions that operate on Fourier–Stieltjes coefficients, The use of tensor products.

Continuous Assessment Pattern

Internal Assessment (IA)

Mid Term Test (MTE)

End Term Test (ETE)

Total Marks

20 30 50 100

     

     

Elective-II

Name of The Course Algebraic TopologyCourse Code MSCM6008Prerequisite Corequisite Antirequisite L T P C 4 0 0 4

Course Objectives: This course focuses on the computation of homotopy invariants of topological spaces, in particular the fundamental group, the homology groups and the co homology ring.

Course Outcomes

CO1 To define Homotopy of paths, contractibility and the fundamental group of circle.CO2 To explain different homology groups, their properties and homomorphism induced

by continuous map. CO3 To define covering projections and homomorphism.CO4 To describe Singular homology, the Excision Theorem, and Mayer-Vietoris

sequence.

Texts Book(s)

1. S. Deo, Algebraic Topology, Hindustan book agency, India, 2003 2. A. Hatcher, Algebraic Topology, Cambridge Univ. Press, Cambridge, 2002.

References Book(s)

3. W. Massey, A Basic Course in Algebraic Topology, Springer-Verlag, Berlin, 1991

4. J.J. Rotman, An Introduction to Algebraic Topology, Springer (India), 2004. Unit-1 10 Hours Homotopy of paths, homotopy equivalence, contractibility, deformation retracts. Fundamental groups and its properties, The fundamental group of circle. Unit-2 10 Hours Simplicial complexes and simplicial maps; Homology groups; Barycentric subdivision; The simplicial approximation theorem. Simpilicial homology,simplicial chain complex and homology, Properties of integral homology groups, invariance of homology groups, subdivision chain map, homomorphism induced by continuous map, homotopy invariance, Lefschetz fixed point theorem, The Borsuk-Ulam theorem. Unit-3 10 Hours

Covering projections and its properties, Application of homotopy lifting theorem, lifting of an arbitrary map, covering homomorphisms, Universal covering spaces.

     

     

Unit-4 10 Hours Singular homology, singular chain complex, one dimensional Homology, Homotopy axiom for singular homology, The Excision Theorem, Homology and cohomology theories, Mayer-Vietoris sequence.

Continuous Assessment Pattern

Internal Assessment (IA)

Mid Term Test (MTE)

End Term Test (ETE)

Total Marks

20 30 50 100

     

     

Name of The Course Dynamical SystemsCourse Code MSCM6009Prerequisite Differential EquationsCorequisite NA Antirequisite NA L T P C 4 0 0 4

Course Objectives: The course objectives to introduce the main features of dynamical systems, particularly as they arise from systems of ordinary differential equations as models in applied mathematics. The topics presented will include phase space, fixed points and stability analysis, bifurcations, Hamiltonian systems and dissipative systems. Discrete dynamical systems will also be discussed briefly.

Course Outcomes

CO1 Describe the main features of dynamical systems and their realisation as systems of ordinary differential equations

CO2 Identify fixed points of simple dynamical systems, and study the local dynamics around these fixed points, in particular to discuss their stability and bifurcations

CO3 Use a range of specialised analytical techniques which are required in the study of dynamical systems

CO4 Prove simple theoretical results about abstract dynamical systems CO5 Analyze the chaotic behaviour of any dynamical system.

Text Book(s)

1. M. W. Hirsch & S. Smale – Differential Equations, Dynamical Systems and Linear Algebra (Academic Press 1974)

2. L. Perko – Differential Equations and Dynamical Systems (Springer – 1991) Reference Book(s)

3. Ferdinand Verhulst : Nonlinear differential equations and dynamical systems: 2nd Edition, Springer, 1996.

4. D.W. Jordan and P. Smith : Nonlinear Ordinary Differential Equations: An Introduction to Dynamical Systems, 4th Edition, (Oxford University Press, 2007).

5. J.K. Hale and H. Kocak : Dynamics and Bifurcations: (Springer, 1991). 6. I.P. Glendinning : (Cambridge Stability, Instability and Chaos : University Press

1994).

Unit-1 10 Hours Introduction: Phase variables and Phase space, continuous and discrete time systems, flows(vector fields), maps (discrete dynamical systems), orbits, asymptotic states, fixed

     

     

(equilibrium) points periodic points, concepts of stability and SDIC (sensitive dependence of initial conditions) chaotic behaviour, dynamical system as a group.Unit-2 10 Hours Linear systems: Uncoupled Linear Systems, Diagonalization, Fundamental theorem and its application. Properties of exponential of a matrix, Exponential of operators ,linear systems in R, Complex eigenvalues, multiple eigenvalues generalized eigenvectors of a matrix, nilpotent matrix, Jordan Canonical Forms , stability theory ,stable, unstable and center subspaces, hyperbolicity, contracting and expanding behaviour. Non-homogeneous Linear systems. Unit-3 10 Hours Nonlinear Systems: Local Theory, Fundamental existence theorem dependence on initial conditions and parameters, the maximal interval of existence, Flow defined by a differential equation. Linearization, stable manifold theorem,Unit-4 10 Hours Nonlinear Vector Fields : Stability characteristics of an equilibrium point. Liapunov and asymptotic stability. Source, sink, basin of attraction. Phase plane analysis of simple systems, homoclinic and heteroclinic orbits, hyperbolicity, statement of Hartmann-Grobman theorem and stable manifold theorem and their implications. Liapunov function and Liapunov theorem.Statement of Lienard’s theorem and its application to vander Pol equation, Poineare-Bendixsom theorem (statement and applications only), structural stability and bifurcation through examples of saddle-node, pitchfork and Hopf bifurcations.

Continuous Assessment Pattern

Internal Assessment (IA)

Mid Term Test (MTE)

End Term Test (ETE)

Total Marks

20 30 50 100

     

     

Name of The Course Fluid MechanicsCourse Code MSCM6010Prerequisite Corequisite Antirequisite L T P C 4 0 0 4

Course Objectives: To give fundamental knowledge of fluid, its properties and behavior under various conditions of internal and external flows.To develop understanding about hydrostatic law, principle of buoyancy and stability of a floating body and application of mass, momentum and energy equation in fluid flow.

Course Outcomes

CO1 Know basic definition, about fluid motion, equation of continuity.

CO2 Study Bernoulli’s equation, the irrotational motion, cyclic motions, Vortex motion, sources and sinks and some related theorems.

CO3 Study motion of circular and elliptic cylinders, theorem of Kutta and Juokowski, some special transformation, Source, sinks, doublets and their images with regards to a plane and sphere.

CO4 Learn about the Vortex motion in detail.

Text Book(s)

1. A. S. Ramsay, “Hydrodynamics: A Treatise on Hydromechanics – Part II ”, Bell, 1913.

2. L. D. Landau and E. M. Lifshitz, “Fluid Mechanics”, Pergamon Press,1959. Reference Book(s)

3.H. Lamb, “Hydrodynamics”, Cambridge University Press, 1932. 4. L. M. MilneThomson, “Theoretical Hydrodynamics”, MacMillan, 1955. 5.S. Swaroop, “Fluid Dynamics”, Krishna Prakashan, 2000.

Unit-1 10 Hours Lagrange’s and Euler’s methods in fluid motion. Equation of motion and equation of continuity, Boundary conditions and boundary surface stream lines and paths of particles. Irrotational and rotational flows, velocity potential. Bernoulli’s equation. Impulsive action. Equations of motion and equation of continuity in orthogonal curvilinear co-ordinates. Euler’s momemtum theorem and D’Alemberts paradox.Unit-2 10 Hours

     

     

Theory of irrotational motion flow and circulation. Permanence irrotational motion. Connectivity of regions of space. Cyclic constant and acyclic and cyclic motion. Kinetic energy. Kelvin’s minimum. Energy theorem. Uniqueness theorem. Complex potential, sources. sinks, doublets and their images circle theorem. Theorem of Blasius. Unit-3 10 Hours

Motion of circular and elliptic cylinders. Steady streaming with circulation. Rotation of elliptic cylinder. Theorem of Kutta and Juokowski. Conformal transformation. Juokowski transformation. Schwartz-chirstoffel theorem. Motion of a sphere. Stoke’s stream function. Source, sinks, doublets and their images with regards to a plane and sphere. Unit-4 10 Hours Vortex motion. Vortex line and filament equation of surface formed by stream lines and vortex lines in case of steady motion. Strength of a filament. Velocity field and kinetic energy of a vortex system. Uniqueness theorem rectilinear vortices. Vortex pair. Vortex doublet. Images of a vortex with regards to plane and a circular cylinder. Angle infinite row of vortices. Karman’s vortex sheet.

Continuous Assessment Pattern

Internal Assessment (IA)

Mid Term Test (MTE)

End Term Test (ETE)

Total Marks

20 30 50 100

     

     

Name of The Course Discrete StructuresCourse Code MSCM6011Prerequisite Corequisite Antirequisite L T P C 4 0 0 4

Course Objectives: Discrete structure is the study of mathematical structures that are

fundamentally discrete rather than continuous. The objective of this course is to teach students

how to think logically and mathematically.

Course Outcomes

CO1 Apply advance counting techniques to solve a variety of problems . Apply the Rules of Inference in solving variety of problems including the validity of an argument.

CO2 Apply various methods to solve recurrence relations.

CO3 Understand Posets and Lattices and their various types. CO4 Understand the concept of graph theory and is various applications.

Text Book(s)

1. Kenneth Rosen,” Discrete Mathematics and it’s Applications”, 7th Ed,Mc Graw Hill publications, 2012.

2. Y.N Singh, Discrete Mathematical Structures,Willey India, New Delhi, 1st edition,2010.

Reference Book(s)

3. Narsingh Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice-Hall of India,1984.

4. Liu and Mohapatra,”Elements of Discrete Mathematics”, Mc Graw Hill,

Unit-1 10 Hours Basic counting principles: Permutations and Combinations (with and without repetitions), Binomial theorem, Multinomial theorem, Counting subsets, Set-partitions, Stirling numbers,Principle of Inclusion and Exclusion, Derangements, Inversion Formulae. Unit-2 12 Hours Recurrence Relation and Generating functions: Algebra of formal power series, Generating function models, Calculating generating functions, Exponential generating

     

     

functions, Recurrence relations: Recurrence relation models, Divide and conquer relations, Solution of recurrence relations, Solutions by generating functions

Unit-3 10 Hours Lattices: Partial order sets, Hasse diagram, Lattices: definition, properties of lattices, bounded lattices, complemented lattices, modular lattices, modular and complete lattices, morphism of lattices. Unit-4 8 Hours Graph Theory: Definitions of different types of graphs, degree, sub-graph, intersection of graphs, homeomorphism and isomorphism of graphs. Computer representation of graphs and diagraphs. Adjacency and incidence matrices of a graph and a diagraph. Walks, trails and paths, cycles, connectedness. Trees, forests and spanning trees. Euler graph, postman problem, Moor’s, Bellman’s and Dijkstra’s algorithms for shortest path.

Continuous Assessment Pattern

Internal Assessment (IA)

Mid Term Test (MTE)

End Term Test (ETE)

Total Marks

20 30 50 100

     

     

Elective-III

Name of The Course Manifolds and ApplicationsCourse Code MSCM6015Prerequisite Corequisite Antirequisite L T P C 4 0 0 4

Course Objective: The course will be an introduction to differentiable manifolds with an eye towards Lie groups and Lie bracket. We will start with the basics of differentiable manifolds (tangent spaces, vector fields, Lie brackets, etc.) and come to grips with differential forms and tensors. Course also contains Riemannian manifolds, Submanifolds and Hypersurfaces and Almost Complex manifolds.

Course Outcomes

CO1 Understand basic concepts of manifolds.CO2 Explain Riemannian manifolds and tensors.CO3 Understand Submanifolds and HypersurfacesCO4 Understand Almost Complex manifolds, Nijenhuis tensor, Contravariant and

covariant almost analytic vector fields

Text Book(s)

1. Serge Lang, Introduction to Differentiable Manifolds, Springer-verlag, 2002.

2. R. S. Mishra, Structures on a differentiable manifold and their applications, Chandrama Prakashan, Allahabad, 1984.

Reference Book(s)

3. R. S. Mishra, A course in tensors with applications to Riemannian Geometry, Pothishala (Pvt.) Ltd., 1965.

4. K. Yano and M. Kon, Structure of Manifolds, World Scientific Publishing Co. Pvt. Ltd., 1984.

5. B. B. Sinha, An Introduction to Modern Differential Geometry, Kalyani Publishers, New Delhi, 1982.

6. U. C. De and A. A. Shaikh, Differential Geometry of Manifolds, Narosa Publishing House Pvt. Ltd., 2007.

7. Gerardo F. Torres Del Castillo, Differentiable Manifolds: A Theoretical Physics Approach, Birkhauser Boston, 2011.

     

     

Unit-1 10 Hours Definition and examples of differentiable manifolds, Vector fields and Tangent spaces, Jacobian map, One parameter group of transformations, Lie bracket, Covariant, Lie and Exterior derivative. Unit-2 10 Hours Riemannian manifolds, Riemannian connection, Torsion tensor, Curvature tensors, Ricci tensor, scalar curvature, Sectional Curvature, Schur’s theorem, Geodesics in a Riemannian manifold, Projective curvature tensor and conformal curvature tensor.Unit-3 10 Hours Submanifolds and Hypersurfaces, Normals, Gauss’ formulae, Weingarten equations, Lines of Curvature, Generalized Gauss and Mainardi-Codazzi equations.Unit-4 10 Hours Almost Complex manifolds, Nijenhuis tensor, Contravariant and covariant almost analytic vector fields, F-connection.

Continuous Assessment Pattern

Internal Assessment (IA)

Mid Term Test (MTE)

End Term Test (ETE)

Total Marks

20 30 50 100

     

     

Name of The Course Mathematical ModellingCourse Code MSCM6016Prerequisite Linear algebra & CalculusCorequisite NA Antirequisite NA L T P C 4 0 0 4

Course Objectives: The overall objectives of this course is to enable students to build mathematical models of real-world systems, analyze them and make predictions about behaviour of these systems. Variety of modelling techniques will be discussed with examples taken from physics, biology, chemistry, economics and other fields. The focus of the course will be on seeking the connections between mathematics and physical systems, studying and applying various modelling techniques to creating mathematical description of these systems, and using this analysis to make predictions about the system’s behaviour. Course Outcomes

CO1 Assess and articulate what type of modelling techniques are appropriate for a given real world system

CO2 Construct a mathematical model of a given real world system and analyze it, CO3 Make predictions of the behaviour of a given real world system based on the

analysis of its mathematical model.CO4 Recognise the power of mathematical modelling and analysis and be able to apply

their understanding to their further studies.CO5 Apply Sensitivity analysis and find Pitfalls in mathematical models

Text Book(s) 1. Kapur , J.N.,”Mathematical Modelling”,New Age international publisher, 1988.

2. Burghes D.N , “Modelling with differential equations”, Ellis Horwood and

John Wiley,1991 Reference Book(s)

3. Burghes, D.N.,” Mathematical Modelling in the Social Management and Life Science”,

Ellie Herwood and John Wiley. 4. Charlton, F.,” Ordinary Differential and Difference Equations”, Van Nostrand.

5. Brauer, Castillo-Chavez ,”Mathematical Models in Population Biology and

Epidemiology”.

Unit-1 10 Hours

Need, Techniques and classification: Linear growth and decay model, Non Linear growth and decay model, compartment model, some simple models.

     

     

Unit-2 10 Hours

Modelling through Ordinary Differential Equations: Basic theory, Models in Economics and finance, Population dynamics and Genetics.

Unit-3 10 Hours Modelling through Partial Differential Equations: Mass balance approach, Momentum Balance approach, Models for traffic flow on highway, BOD-DO models Unit-4 10 Hours Modelling through Graphs: Directed graphs, signed graphs, weighted digraphs, Linear programming models in forest management, Transportation and assignment models. Analyses of models: Sensitivity analysis, Pitfalls in modelling, Illustrations

Continuous Assessment Pattern

Internal Assessment (IA)

Mid Term Test (MTE)

End Term Test (ETE)

Total Marks

20 30 50 100

     

     

Name of The Course Financial MathematicsCourse Code MSCM6017Prerequisite Corequisite Antirequisite L T P C 4 0 0 4

Course Objectives:. To make students to understand Interest rates, annuities and mortgages, bonds and bond market structure.

Course Outcomes

CO1 Summarize the concepts of time value of money using simple interest and discounting

CO2 Able to apply compound interest model with effect of investment CO3 Able to apply discounted cash flow techniques in different project appraisal CO4 Calculate the price of a forward contract CO5 Applying hedging in the contract

Text Book (s)

1. Suresh Chandra, S. Dharmaraja, Aparna Mehra, R. Khemchandani, Financial Mathematics: An Introduction, Narosa Publication House, 2012

Reference Book (s)

2. D.G. Luenberger, Investment Science, Oxford University Press, Oxford, 1998.

3. J.C. Hull, Options, Futures and Other Derivatives, 4th ed., Prentice-Hall, New York, 2000.

4. J.C. Cox and M. Rubinstein, Options Market, Englewood Cliffs, N.J.: Prentice Hall, 1985.

Unit-1 10 Hours Interest rates, Simple interest rates, Present value of a single future payment. Discount factors, effective and nominal interest rates. Real and money interest rates. Unit-2 10 Hours Compound interest rates. Relation between the time periods for compound interest rates and the discount factor. Compound interest functions. Annuities and perpetuities. Unit-3 10 Hours Loans. Introduction to fixed-income instruments. Generalized cash flow model. Net present value of a sequence of cash flows. Equation of value. Internal rate of return. Investment project appraisal. Cash flow, present value of a cash flow, securities, fixed income securities, types of markets. Unit-4 10 Hours

     

     

Forward and futures contracts, options, properties of stock option prices, trading strategies involving options, option pricing using Binomial trees, Black – Scholes model, Black-Scholes formula, Risk-Neutral measure, Delta – hedging, options on stock indices, currency options.

Continuous Assessment Pattern

Internal Assessment (IA)

Mid Term Test (MTE)

End Term Test (ETE)

Total Marks

20 30 50 100

     

     

Name of The Course Coding TheoryCourse Code MSCM6018Prerequisite Corequisite Antirequisite L T P C 4 0 0 4

Course Objective: The course aims an introduction to traditional and modern coding theory. It provides an overview of various encoding and decoding methods and their application.

Course Outcomes

CO1 Explain basic concepts of coding theory.CO2 Understand various types of coding and decoding techniques .CO3 Understand the Golay Codes, Codes and Lattices, Weight Enumerators. CO4 Apply a Double-Error Correcting Decimal Code and Introduce to BCH Codes.

Text Book(s) 1. Raymond Hill: A First Course in Coding Theory, Oxford Applied

Mathematics and Computing Science Series.1990 2. W. Wesley Peterson and E. J. Weldon: Error Correcting Code, 2nd ed., MIT

Press. 1972 Reference Book(s)

3. Mac Williams and Sloane: The Theory and Practice of Error-Correcting Codes, North Holland Pub Company

4. Van Lint, J. H. Introduction to coding theory, Third edition. Graduate Texts in Math-ematics, 86. Springer-Verlag, Berlin, 1999.

5. Huffman, W. C. and Pless, V. Fundamentals of error-correcting codes. Cambridge University Press, Cambridge, 2003.

Unit-1 10 Hours Introduction to error correcting codes, Minimum distance, types and properties of codes, linear and non linear codes, Repetition Codes, Main coding theorem problem, Shannon's Noisy Channel Coding Theorem. Review of number theory, arithmetics in Finite Fields and Vector Spaces over Finite fields Unit-2 10 Hours Introduction to Linear Codes, Encoding and Decoding with a Linear Code, The Dual Code, the Parity-Check Matrix, and Syndrome Decoding, Bounds on Codes, The Hamming Codes, Perfect Codes

Unit-3 10 Hours

     

     

The Golay Codes, Codes and Lattices, Weight Enumerators and the MacWilliams Theorem, MDS Codes Unit-4 10 Hours A Double-Error Correcting Decimal Code and an Introduction to BCH Codes, Cyclic Codes, Hadamard Codes, Reed-Solomon Codes

Continuous Assessment Pattern

Internal Assessment (IA)

Mid Term Test (MTE)

End Term Test (ETE)

Total Marks

20 30 50 100

     

     

Elective-IV

Name of The Course Finite Element MethodCourse Code MSCM6019Prerequisite Corequisite Antirequisite L T P C 4 0 0 4

Course Objectives: To make students to learn basic principles of finite element analysis procedure, to learn the theory and characteristics of finite elements that represent engineering structures. Course Outcomes

CO1 Understand the fundamental theory of the FEA method. CO2 Understand the use of the basic finite elements for different structural problems. CO3 Develop the ability to generate the governing FE equations for systems govern

by ordinary and partial differential equations.CO4 Demonstrate the ability to evaluate and interpret FE analysis results for design and

evaluation purposes. CO5 d Develop a basic understanding of the limitations of the FE method and understand

possible error sources in its use.

Text Book (s)

1. Reddy J.N., “Introduction to the Finite Element Methods”, Tata McGraw-Hill. 2003 2. Bathe K.J., Finite Element Procedures”, Prentice-Hall. 2001

Reference Book (s)

3. Cook R.D., Malkus D.S. and Plesha M.E., “Concepts and Applications of Finite Element Analysis”, John Wiley.2002

4. Thomas J.R. Hughes “The Finite Element Method: Linear Static and Dynamic Finite Element Analysis”. 2000

5. George R. Buchanan “Finite Element Analysis”, 1994

Unit-1 10 Hours Introduction to finite element methods, comparison with finite difference methods. Methods of weighted residuals, collocations, least squares and Galerkin’s method, Variational formulation of boundary value problems equivalence of Galerkin and Ritz methods. Unit-2 10 Hours Applications to solving simple problems of ordinary differential equations, Linear, quadratic and higher order elements in one dimensional and assembly, solution of assembled system.

     

     

Unit-3 10 Hours Simplex elements in two and three dimensions, quadratic triangular elements, rectangular elements, serendipity elements and isoperimetric elements and their assembly, discretization with curved boundaries.Unit-4 10 Hours Interpolation functions, numerical integration, and modelling considerations, Solution of

two dimensional partial differential equations under different geometric conditions.

Continuous Assessment Pattern

Internal Assessment (IA)

Mid Term Test (MTE)

End Term Test (ETE)

Total Marks

20 30 50 100

     

     

Name of The Course Computational Fluid DynamicsCourse Code MSCM6020Prerequisite Corequisite Antirequisite L T P C 3 0 0 3

Course Objectives: The course aims to shape the attitudes of learners regarding the field of Computational Fluid Dynamics and its application.

Course Outcomes

CO1 Identify mathematical characteristics of partial differential equations CO2 Explain the basic properties of computational methods – accuracy, stability,

consistency CO3 Apply computational solution techniques for various types of partial differential

equations CO4 Apply computational method to solve Euler and Navier-Stokes equations

Text Book (s) 1. C. A. J. Fletcher, “Computational Techniques for Fluid Dynamics”, Vol-I

and Vol-II, Springer, 1988. 2. J. C. Tanehill, D. A. Anderson, R. H. Pletcher, “Computational Fluid

Mechanics and Heat Transfer”, Taylor & Francis, 1997. Reference Book (s)

3. P. Niyogi, S. K. Chakraborty and M. K. Laha, “Introduction to Computational Fluid Dynamics”, Pearson Education, Delhi, 2005.

4. R. Peyret and T. D. Taylor, “Computational Methods for Fluid Flow”, Springer, 1983.

5. J. F. Thompson, Z.U.A Warsi and C. W. Martin, “Numerical Grid Generation, Foundations and Applications”, Prentice Hall, 1985.

6. J.D. Anderson, “Computational Fluid Dynamics”, Mc Graw Hill, 1995.

Unit-1 10 Hours Classification of 2 order partial differential equations - parabolic, hyperbolic and elliptic types. Governing equations of fluid dynamics, Introduction to finite difference discretization. Explicit and Implicit schemes. Truncation error, consistency, convergence and stability analysis. Unit-2 10 Hours Thomas algorithm. ADI method for 2-D heat conduction problem. Splitting and approximate factorization for 2-D Laplace equation. Multigrid method. Upwind scheme, CFL stability condition. Lax-Wendroff and MacCormack schemes.Unit-3 10 Hours

     

     

Finite Volume method: Preliminary concepts. Flux computation across quadrilateral cells. Reduction of a BVP to algebraic equations. Illustrative example like, solution of Dirichlet problem for 2-D Laplace equation. Conservation principles of fluid dynamics. Basic equations of viscous and inviscid flow. Basic equations in conservative form. Associated typical boundary conditions for Euler and Navier-Stokes equations. Grid generation using elliptic partial differential equations.Unit-4 10 Hours Incompressible viscous flow field computation: Stream function vorticity formulation, Staggered grid, MAC method, SIMPLE algorithm.

Continuous Assessment Pattern

Internal Assessment (IA)

Mid Term Test (MTE)

End Term Test (ETE)

Total Marks

20 30 50 100

     

     

Name of The Course Computational Fluid Dynamics LabCourse Code MSCM6021Prerequisite

Corequisite Antirequisite L T P C 0 0 2 1

Course Objectives: The course aims to shape the attitudes of learners regarding the field of Computational Fluid Dynamics and its application.

Course Outcomes

CO1 Understand the classification of PDECO2 Understand the concept of solving Heat Conduction equation CO3 Apply the concept of Finite volume method. CO4 Understand the concept of viscous and inviscous flow. CO5 Apply the concept of vorticity.

Text Book (s) 1. C. A. J. Fletcher, “Computational Techniques for Fluid Dynamics”, Vol-I and Vol-II, Springer, 1988. 2. J. C. Tanehill, D. A. Anderson, R. H. Pletcher, “Computational Fluid Mechanics and Heat Transfer”, Taylor & Francis, 1997. 3. P. Niyogi, S. K. Chakraborty and M. K. Laha,“Introduction to Computational Fluid Dynamics”, Pearson Education, Delhi, 2005.

Reference Book (s) 4. R. Peyret and T. D. Taylor,“Computational Methods for Fluid Flow”, Springer, 1983.

5. J. F. Thompson, Z.U.A Warsi and C. W. Martin,“Numerical Grid Generation, Foundations and Applications”, Prentice Hall, 1985. 6. J.D. Anderson,“Computational Fluid Dynamics”, Mc Graw Hill, 1995. S. No. Experiment

1. Installation of the Scilab, Overview, Basic syntax, Mathematical Operators, Predefined constants, Built in functions.

2. Determination of vector differential operators for the given tensors 3. Plotting of stream lines and plot lines.

     

     

4. Plots of solution curves/surfaces to both ODE and PDE.

5. Demonstration of plane-Couette flow. 6. Demonstration of the flow of a viscous and inviscous incompressible fluid

between two vertical plates placed at a finite distance7. Demonstration of the radially symmetric incompressible steady flow between

two cylinders. 8. Finite Volume method: Preliminary concepts. Flux computation across

quadrilateral cells.

9. Demonstration of the pressure distribution on an idealized underwater vehicle as it moves along near the ocean bottom

10. Demonstration of Laminar flow of an incompressible viscous fluid between two parallel plates.

Continuous Assessment Pattern

Internal Assessment Lab (IA)

End Term Lab Test (ETE)

Total Marks

50 50 100

     

     

Name of The Course Stochastic ProcessesCourse Code MSCM6022Prerequisite Corequisite Antirequisite L T P C 4 0 0 4

Course Objectives: The aim of this course is to make students understand the concept

of Random process and its applicability.

Course Outcomes

CO1 explain basic concepts of probability and stochastic processCO2 apply various stochastic processes .CO3 explain Discrete parameter Markov Chains and apply it to their application. CO4 explain Continuous parameter MarkovChains and apply it to their application.

Text Book (s)

1. J. Medhi, Stochastic Processes, 3rd Edition, New Age International, 2009. 2. Liliana Blanco Castaneda, Viswanathan Arunachalam and S. Dharmaraja, Introduction to Probability and Stochastic Processes with Applications, Wiley, 2012.

Reference Book (s)

3. S.M. Ross, Stochastic Processes, 2nd Edition, Wiley, 1996. 4. S Karlin and H M Taylor, A First Course in Stochastic Processes, 2nd edition,

Academic Press, 1975. 5. Kishor S. Trivedi, Probability, Statistics with Reliability, Queueing and

Computer Science Applications, 2nd edition, Wiley, 2001. 6. S. E. Shreve, Stochastic Calculus for Finance, Vol. I & Vol. II, Springer, 2004. 7. V. G. Kulkarni, Modelling and Analysis of Stochastic Systems, Chapman & Hall,

1995. 8. G. Sankaranarayanan, Branching Processes and Its Estimation Theory, Wiley,

1989.

Unit-1 10 Hours Introduction to Stochastic Processes (SPs): Definition and examples of SPs, classification of random processes according to state space and parameter space, types of SPs, elementary problems. Stationary Processes: Weakly stationary and strongly stationary processes, moving average and auto regressive processes.Unit-2 10 Hours

     

     

Discrete-time Markov Chains (DTMCs): Definition and examples of MCs, transition probability matrix, Chapman-Kolmogorov equations; calculation of n-step transition probabilities, limiting probabilities, classification of states, ergodicity, stationary distribution, transient MC; random walk and gambler’s ruin problem, applications. Unit-3 10 Hours Continuous-time Markov Chains (CTMCs): Kolmogorov- Feller differential equations, infinitesimal generator, Poisson process, birth-death process, stochastic Petri net, applications to queueing theory and communication networks. Martingales: Conditional expectations, definition and examples of martingales. Unit-4 10 Hours Brownian Motion: Wiener process as a limit of random walk; process derived from Brownian motion, stochastic differential equation, stochastic integral equation, Ito formula, Some important SDEs and their solutions, applications to finance.

Continuous Assessment Pattern

Internal Assessment (IA)

Mid Term Test (MTE)

End Term Test (ETE)

Total Marks

20 30 50 100

     

     

Name of The Course Automata & Formal LanguagesCourse Code MSCM6023Prerequisite Corequisite Antirequisite L T P C 4 0 0 4

Course Objectives: Introduce students to the mathematical foundations of computation including automata theory; the theory of formal languages and grammars; the notions of algorithm, decidability, complexity, and computability. Course Outcomes

CO1 Understand basic concepts of mathematical prilimnaries and finite automata and their applications.

CO2 Understand the concepts as well as applications of regular expressions and regular languages and their applications.

CO3 Understand the concepts context-free languages and pushdown automata. CO4 Understant the basic concepts of Turing machines and their applications.

Text Book(s)

1. D. Kelly, Automata and Formal Languages: An Introduction, Prentice-Hall, 1995.

2. P. Linz, An Introduction to Formal Languages and Automata, 3rd Edition, Narosa, 2002.

References Book(s)

3. J. E. Hopcroft, R. Motwani, and J.D. Ullman, Introduction to Automata, Languages, and

Computation (2nd edition), Pearson Edition, 2001.

Unit-1 10 Hours Alphabets and Languages: Alphabets, words, and languages. Operations on strings and languages. Regular Languages and Automata: Regular languages and regular expressions. Deterministic finite automata. DFAs and languages. Nondeterministic finite automata. Unit-2 10 Hours Equivalence of NFA and DFA. ε-Transitions. Minimization and equivalence of finite automata. Finite automata with outputs. Moore and Mealy machines. Finite automata and regular expressions. Properties of regular languages. Pumping lemma.Unit-3 10 Hours Context-free Languages: Grammars. Regular grammars. Regular grammars and regular languages. Context-free grammars. Derivation or parse tree and ambiguity. Simplifying context-free grammars. The Chomsky normal form. Properties of context-free languages. Pumping lemma for context-free languages. The CYK algorithmUnit-4 10 Hours

     

     

Turing Machines: Basic definitions. Turing machines as language acceptors. Modifications to Turing machines. Universal Turing machines. Turing Machines and Languages: Languages accepted by Turing machines. Regular, context-free, recursive, and recursively enumerable languages. Unrestricted grammars and recursively enumerable languages. Context-sensitive languages and the Chomsky hierarchy.

Continuous Assessment Pattern

Internal Assessment (IA)

Mid Term Test (MTE)

End Term Test (ETE)

Total Marks

20 30 50 100

Name of The Course CryptographyCourse Code MSCM6024Prerequisite

     

     

Corequisite Antirequisite L T P C 4 0 0 4

Course Objectives: 1. To develop a mathematical foundation for the study of cryptography. 1. To Understand Number Theory and Algebra for design of cryptographic algorithms 2. To understand the role of cryptography in communication over an insecure channel. 3. Analyse and compare symmetric-key encryption public-key encryption schemes

based on different security models

Course Outcomes

CO1 Describe modern concepts related to cryptography and cryptanalysis.

CO2 Describe and implement the specifics of some of the prominent techniques for public-key cryptosystems and digital signature schemes (e.g., Rabin, RSA, ElGamal, DSA, Schnorr)

CO3 Explain the notions of public-key encryption and digital signatures, and sketch their formal security definitions

CO4 Explain the notions of public-key encryption and digital signatures, and sketch their formal security definitions

Text Book(s)

1. Douglas R. Stinson: Cryptography: Theory and Practice, Third Edition, CRC Press.2006 2. Alfred Menezes, Paul C. van Oorschot: et. al., : Handbook of Applied

Cryptography, 5th ed. CRC Press, 2001 Reference Book(s)

3.Bruce Schnier: Applied Cryptography (2nd Edition):, John Wiley and Sons.

Unit-1 10 Hours Introduction to Cryptography and Cryptanalysis. Features of Cryptography. Classical methods and modern methods. Cryptographic Protocols and standards. Fiestel Ciphers, Block Ciphers and Stream Cihpers. Symmetric key algorithms, Asymmetric Key Algorithms. Key Exchange algorithms and protocols. Digital Signatures. CA. Review of number theory and finite field arithmetics. Random-Sequence and Random number generators. Unit-2 10 Hours Stream Ciphers: RC4, RC5. Symmetric Key Algorithms: DES, AES, Asymmetric Key Algorithms: RSA, El-Gamal,Unit-3 10 Hours

     

     

Key Exchange Algorithms, Public-key, Private Key, Signature Schemes, Introduction, The ElGamal Signature Scheme, The Digital Signature Standard , One-time Signatures , Undeniable Signatures , Fail-stop SignaturesUnit-4 10 Hours Hashing Functions: Signatures and Hash Functions ,Collision-free Hash Functions, The Birthday Attack , The MD5, SHA1. Hash Function, Introduction to Stenography, Timestamping , Zero-knowledge Proofs , Interactive Proof Systems , Perfect Zero-knowledge Proofs , Bit Commitments , Computational Zero-knowledge Proofs , Zero-knowledge Arguments , Elliptic Curve Cryptosystems. A Discrete Log Hash Function, Extending Hash Functions, Hash Functions From Cryptosystems ,

Continuous Assessment Pattern

Internal Assessment (IA)

Mid Term Test (MTE)

End Term Test (ETE)

Total Marks

20 30 50 100