NOIDA INTERNATIONAL UNIVERSITY

SCHOOL OF SCIENCES

Study & Evaluation Scheme for M.Sc. (Mathematics) 1st Year

Effective from the Session: 2019-2022

SEMESTER-I

SEMESTER-II

S.No Course

Code Subject

Period Evaluation Scheme Subject

Total Credit Sessional Exam External

Exam L T P CA TA Total

1 STPM- 101 Algebra 4 1 0 20 20 40 60 100 5

2 STPM-102 Real Analysis 4 1 0 20 20 40 60 100 5

3 STPM-103

Ordinary

Differential

Equations

4 1 0 20 20 40 60 100 5

4 STPM-104 Mechanics 4 1 0 20 20 40 60 100 5

TOTAL 400 20

S.No Course

Code Subject

Period Evaluation Scheme Subject

Total Credit

Sessional Exam External

Exam L T P CA TA Total

1 STPM-201 Complex Analysis 4 1 0 20 20 40 60 100 5

2 STPM-202 Difference

Equations 4 1 0 20 20 40 60 100 5

3 STPM-203 Partial Differential

Equations 4 1 0 20 20 40 60 100 5

4 STPM-204

Optimization

Techniques and

Control Theory

4 1 0 20 20 40 60 100 5

5 STPM-205 Seminar 2 0 0 15 10 25 75 100 2

TOTAL 500 22

NOIDA INTERNATIONAL UNIVERSITY

SCHOOL OF SCIENCES

Study & Evaluation Scheme for M.Sc. (Mathematics) 2nd

Year

Effective from the Session: 2019-2022

SEMESTER-III

S.No Course

Code Subject

Period Evaluation Scheme Subject

Total Credit Sessional Exam External

Exam L T P CA TA Total

2 STPM-301 Topology 4 1 0 20 20 40 60 100 5

3 STPM-302

Probability

Theory and

Statistics

4 1 0 20 20 40 60 100 5

4 STPM-303 Operation

Research 4 1 0 20 20 40 60 100 5

5 STPM-304

Computer

Fundamentals &

C Programming

4 0 0 20 20 40 60 100 4

PRACTICAL

1 SPPM-304

Computer

Fundamentals &

C Programming

Practical

0 0 2 50 50 100 2

TOTAL 500 21

SEMESTER-IV

SPECIALIZATION IN SEMESTER IV

SPECIAL PAPER I: (Choose Any One Elective Paper)

STPME-1-403 Fluid Dynamics

STPME-2-403 Regression Analysis

STPME-3-403 Algebraic Number Theory

STPME-4-403 Discrete Mathematics

SPECIAL PAPER II: (Choose Any One Elective Paper)

STPME-1-404 Mathematical Modeling

STPME-2-404 Calculus on n

STPME-3-404 Differential Geometry

S.No Course

Code Subject

Period Evaluation Scheme Subject

Total Credit

Sessional Exam External

Exam L T P CA TA Total

1 STPM-401 Integral Equations 4 1 0 20 20 40 60 100 5

2 STPM-402 Functional

Analysis 4 1 0 20 20 40 60 100 5

4 STPME-

1/2/3/4-

403 Special Paper I 4 1 0 20 20 40 60 100 5

5 STPME-

1/2/3/-404 Special Paper II 4 1 0 20 20 40 60 100 5

PRACTICAL

1 SPPM-401 Dissertation 2 0 0 25 75 100 2

TOTAL 500 22

OVER ALL SCHEME

S. No. Semester Theory Total Practical Subject Total Total Credits

1. I 400 - 400 20

2. II 400 100 500 22

3. III 400 100 500 21

4. IV 400 100 500 22

Grand Total 1900 85

SEMESTER – I

ALGEBRA (STPM-101)

L T P

4 1 0

Unit 1:

Class equation for finite groups and its applications, Cauchy`s theorem, Sylow p-subgroups, Sylow's

theorems.

Unit 2:

Direct products, Finite abelian groups. Solvable groups, Insolvability of Sn for n5.

Unit 3:

Modules, Submodules, Simple and Semi-simple modules, Quotient Modules, Cyclic modules, Free

modules.

Unit 4:

Finite fields, Extension fields, Field automorphisms, Splitting fields, Roots of Polynomials, Wedderburn's

theorem on finite division rings.

Unit 5:

Elements of Galois theory, Fundamental Theorem of Galois Theory.

References:

1. P.M. Cohn, Classic Algebra, John Wiley & Sons Ltd., 2000.

2. P.M. Cohn, Basic Algebra: Groups, Rings and Fields, Springer, 2005.

3. Joseph A. Gallian, Contemporary Abstract Algebra, 4th Ed., Narosa Publishing House, New

Delhi, 1999.

4. Joseph J. Rotman, An Introduction to the Theory of Groups, 4th Ed., Springer Verlag, 1995.

5. D.S. Dummit and R.M. Foote, Abstract Algebra, Third Edition, Wiley India Pvt. Ltd., 2011.

6. N. Jacobson, Basic Algebra, Volumes I & II, Second Edition, Dover Publications, 2009.

7. T.W. Hungerford, Algebra, Springer-Verlag, 1981.

8. I.N. Herstein. Topics in Algebra (II Edition) Wiley Eastern Limited, New Delhi, 1975.

REAL ANALYSIS (STPM-102)

L T P

4 1 0

Unit 1:

Improper Integral, Convergence of Improper Integral, Comparison Test, Dirichlet's test and Abel's test -

Rearrangement of series - Riemann's theorem on conditionally convergent series, Differentiation under

Integral Sign.

Unit 2:

Riemann - Stieltjes integral - Linear Properties - Integration by parts- Change of variable in a Riemann -

Stieltjes integral - Reduction to a Riemann Integral - Euler’s summation formula - Monotonically

increasing integrators, Upper and lower integrals - Additive and linearity properties of upper and lower

integrals - Riemann's condition - Comparison theorems.

Unit 3:

Integrators of bounded variation-Sufficient conditions for the existence of Riemann-Stieltjes integrals-

Necessary conditions for the existence of Riemann-Stieltjes integrals- Mean value theorems for Riemann

- Stieltjes integrals - The integrals as a function of the interval - Second fundamental theorem of integral

calculus-Change of variable in a Riemann integral-Second Mean Value Theorem for Riemann integral-

Riemann-Stieltjes integrals depending on a parameter-Differentiation under the integral sign - Lebesgue

criterion for the existence of Riemann integrals.

Unit 4:

Pointwise convergence of sequences of functions - Examples of sequences of real - valued functions -

Definition of uniform convergence - Uniform convergence and continuity - The Cauchy condition for

uniform convergence - Uniform convergence of infinite series of functions - Uniform convergence and

Riemann - Stieltjes integration - Non-uniform Convergence and Term-by-term Integration - Uniform

convergence and differentiation - Sufficient condition for uniform convergence of a series - Mean

convergence.

References:

1. Tom M.Apostol: Mathematical Analysis, 2nd Edition, Addison-Wesley Publishing Company Inc.

New York, 1974.

2. Bartle, R.G. Real Analysis, John Wiley and Sons Inc., 1976.

3. Rudin,W. Principles of Mathematical Analysis, 3rd Edition. McGraw Hill Company, New York,

1976.

4. Malik,S.C. and Savita Arora. Mathematical Anslysis, Wiley Eastern Limited.New Delhi, 1991.

5. E. Hewitt and K. Stromberg, Real and Abstract Analysis: A Modern Treatment of the Theory of

Functions of a Real Variable, Springer, Berlin, 1975.

6. H.L. Royden and P.M. Fitzpatrick, Real Analysis, Fourth Edition, Pearson, 2015.

ORDINARY DIFFERENTIAL EQUATIONS (STPM-103)

L T P

4 1 0

Unit 1:

The existence and uniqueness solution: The method of successive approximation, Picard’s theorem,

system of the second order linear equations, Linear systems, homogeneous linear system with constant

coefficient, Nonlinear system.

Unit 2:

Qualitative properties of solution: Sturm separation theorem, Sturm comparison theorem, The Cauchy

problem, Homogeneous wave equation, Non homogeneous wave equations, Fourier transform, Properties

of Fourier transform.

Unit 3:

Sturm-Liouvill systems, eigen values and eigen functions, Greens function for ordinary differential

equation, construction of green function, Autonomous system, types of critical points, stability critical

points and stability for linear system, stability by Liapunovs direct method.

Unit 4:

Boundary value problem, maximum and minimum principle, uniqueness and continuity theorem,

Dirichlet problem for a circle, Neumann problem for a circle, Dirichlet problem invelvoing the Poisson

equation.

Unit 5:

Greens function and boundary value problem, the Dirac-Delta function Properties of Greens function,

method of Greens function, Dirichlets problem for the Laplace operator, Method of eigen functions.

References:

1. E.A. Coddington, An Introduction to Ordinary Differential Equations, Dover, 1989.

2. T. Myint-U, Ordinary Differential Equations, Elsevier North-Holland, 1978.

3. I. N. Sneddon, Elements of Partial Differential Equations, McGrawHill, 1986.

4. G.F. Simmons, Ordinary Differential Equations with applications and Historical notes,

McGraw-Hill, 1991.

5. T. Myint-U, Linear Partial Differential Equations for Scientists and Engineers, Birkhauser,

(2007).

6. S.L.Ross, Differential Equation, Wiley India, 2004.

MECHANICS - (STPM-104)

L T P

4 1 0

Unit 1:

Newtonian mechanics and its limitations. Constrained motion. Constraints and their classification.

Principle of virtual work. D' Alembert's principle. Generalised coordinates. Deduction of Lagrange's

equations from D' Alembert's Principle. Generalised momenta and energy. Cyclic or ignorable

coordinates. Rayleigh' s dissipation function. Integrals of motion. Symmetries of space and time with

conservation laws.

Unit 2:

Central force, Definition and properties of central force. Two-body central force problem. Stability of

orbits. Conditions for closure. General analysis of orbits. Kepler's laws. Kepler's equation. Artficial

satellites. Rutherford scattering.

Unit 3:

Principle of least action. Hamilton's principle. The calculas of variations. Derivation of Hamilton's

equations of motion for holonomic systems from Hamilton's principle. Hamilton's principle and

characteristic functions.

Unit 4:

Canonical Transformations. Generating functions. Poisson bracket. Poisson's Theorem. Invariance of PB

under canonical transformations. Angular momentum PBs. Hamilton-Jacobi equation. Connection with

canonical transformation. Problems.

Unit 5:

Small Oscillations. Normal modes and coordinates. Problems.

References:

1. N.C. Rana and P.S. Joag, (Tata McGrae-Hill, 1991) : Classical Mechanics.

2. H. Goldstien (Addition Wesley, 1980) : Classical Mechanics

3. A. Sommerfeld : Mechanics.

4. I. Perceival & D.Richards (Cambridge University Press,1982): Introduction to Dynamics.

SEMESTER – II

COMPLEX ANALYSIS (STPM-201)

L T P

4 1 0

Unit 1: Cauchy’s Integral Formula

The Index of a point with respect to a closed curve - The Integral formula - Higher derivatives. Local

Properties of Analytic Functions: Removable Singularities - Taylors’s Theorem - Zeros and poles – The

local Mapping - The Maximum Principle.

Unit 2: The General Form Of Cauchy’s Theorem

Chains and cycles- Simple Continuity - Homology - The General statement of Cauchy’s Theorem - Proof

of Cauchy’s theorem - Locally exact differentials- Multilply connected regions - Residue theorem – The

argument principle.

Unit 3: Evaluation Of Definite Integrals And Harmonic Functions

Evaluation of definite integrals - Definition of Harmonic function and basic properties - Mean value

property - Poisson formula.

Unit 4: Harmonic Functions And Power Series Expansions

Schwarz theorem - The reflection principle - Weierstrass theorem - Taylor’s Series - Laurent series.

Unit 5: Partial Fractions And Entire Functions

Partial fractions - Infinite products - Canonical products - Gamma Function - Jensen’s formula -

Hadamard’s Theorem

References:

1. L.V. Ahlfors, Complex Analysis, Mc Graw Hill Co., Indian Edition, 2017.

2. J.B. Conway, Functions of One Complex Variable, Second Edition, Narosa, New Delhi, 1996.

3. T.W. Gamelin, Complex Analysis, Springer, 2001.

4. L. Hahn, B. Epstein, Classical Complex Analysis, Jones and Bartlett, 1996.

5. D.C. Ullrich, Complex Made Simple, American Mathematical Society, 2008.

DIFFERENCE EQUATIONS (STPM-202)

L T P

4 1 0

Unit 1: Linear Difference Equations of Higher Order

Difference Calculus - General Theory of Linear Difference Equations - Linear Homogeneous

Equations with Constant coefficients - Linear non-homogeneous equations - Method of Undetermined

coefficients, the method of variation of constants - Limiting behavior of solutions.

Unit 2: System of Difference Equations

Autonomous System - The Basic Theory - The Jordan form - Linear periodic system.

Unit 3: The Z-Transform Method

Definition, Example and properties of Z-transform - The Inverse Z-transform and solution of

Difference Equations: Power series method, partial fraction method, the inverse integral method -

Volterra Difference Equation of convolution types - Volterra systems.

Unit 4: Asymptotic Behaviour Of Difference Equation

Tools and Approximations - Poincare’s Theorem - Second order difference equations - Asymptotic

diagonal systems - Higher order Difference Equations.

Unit 5: Oscillation Theory

Three-term difference Equation - Non-linear Difference Equations - Self-Adjoint second order

equations.

References:

1. Saber N. Elaydi, An Introduction to Difference Equations, Springer Verlag, New York, 1996.

2. R.P.Agarwal., Difference Equations and Inequalities, Marcel Dekker, 1999.

3. S. Goldberg, Introduction to Difference Equations, Dover Publications, 1986

4. V. Lakshmi kantham and Trigiante, Theory of Difference Equations, Academic Press, New York,

1988.

5. Peterson, A Difference Equations, An Introduction with Applications, Academic Press, New York,

1991.

PARTIAL DIFFERENTIAL EQUATIONS (STPM-203)

L T P

4 1 0

Unit1:

Partial Differential Equations of First Order

Formation and solution of PDE- Integral surfaces - Cauchy Problem order eqn - Orthogonal surfaces -

First order non-linear - Characteristics - Csmpatible system - Charpit method. Fundamentals:

Classification and canonical forms of PDE.

Unit 2:

Elliptic Differential Equations

Derivation of Laplace and Poisson equation - BVP - Separation of Variables - Dirichlet’s Problem and

Newmann Problem for a rectangle - Interior and Exterior Dirichlets’s problems for a circle - Interior

Newmann problem for a circle - Solution of Laplace equation in Cylindrical and spherical coordinates -

Examples.

Unit 3:

Parabolic Differential Equations

Formation and solution of Diffusion equation - Dirac-Delta function - Separation of variables method -

Solution of Diffusion Equation in Cylindrical and spherical coordinates - Examples.

Unit 4:

Hyperbolic Differential Equations

Formation and solution of one-dimensional wave equation - canocical reduction - IVP- d’Alembert’s

solution - Vibrating string - Forced Vibration - IVP and BVP for two-dimensional wave equation -

Periodic solution of one-dimensional wave equation in cylindrical and spherical coordinate systems -

vibration of circular membrane - Uniqueness of the solution for the wave equation - Duhamel’s Principle

- Examples.

Unit 5:

Green’s Function

Green’s function for laplace Equation - methods of Images - Eigen function Method - Green’s function

for the wave and Diffusion equations. Laplace Transform method: Solution of Diffusion and Wave

equation by Laplace Transform.

References:

1. S, Sankar Rao, Introduction to Partial Differential Equations, 2nd Edition, Prentice Hall of India,

New Delhi. 2005.

2. R.C.McOwen, Partial Differential Equations, 2nd Edn. Pearson Eduction, New Delhi, 2005.

3. I.N.Sneddon, Elements of Partial Differential Equations, McGraw Hill, New Delhi, 1983.

4. R. Dennemeyer, Introduction to Partial Differential Equations and Boundary Value Problems,

McGraw Hill, New York, 1968.

5. M.D.Raisinghania, Advanced Differential Equations, S.Chand & Company Ltd., New Delhi,

2001.

OPTIMIZATION TECHNIQUES AND CONTROL THEORY (STPM-204)

L T P

4 1 0

Unit 1:

Extended real valued functions, Proper convex functions, Subgradients, Directional derivatives

Unit 2:

Conjugate functions, Dual convex programs, Optimality conditions and Lagrange multipliers, Duality

and optimality for standard convex programs, Gradient descent method, Gradient projection method.

Unit 3:

Newton’s method, Conjugate gradient method, Dynamic programming, Bellman’s principle of

optimality, Allocation problem, Stage coach problem.

Unit 4:

Optimal control problem and formulations, Variational approach to the fixed-time free endpoint problem,

Pontryagin’s maximum principle, Dynamic programming and Hamilton-Jacobi-Bellman equation.

References:

1. M. Avriel, Nonlinear Programming: Analysis & Methods, Dover Publications, New York,

2003.

2. O. Güler, Foundations of Optimization, Springer 2010.

3. F.S. Hillier, G.J. Lieberman, P. Nag and P. Basu, Introduction to Operations Research,

Tata McGraw-Hill, 2012.

4. D. Liberzon, Calculus of Variations and Optimal Control Theory: A Concise Introduction,

Princeton University Press, 2012

SEMESTER-III

TOPOLOGY (STPM-301)

L T P

4 1 0

Unit 1: Topological Spaces

Definition and examples of topological spaces, Closed sets, Closure, Dense sets. neighborhoods,

interior, exterior, and boundary, Accumulation points and derived sets, Bases and sub-bases,

Subspaces and relative topology, Alternative methods of defining a topology in terms of Kuratowski

closure operator and neighborhood systems.

Unit 2: Continuous Functions

Continuous functions and homeomorphism, First and second countable space, Lindelöf spaces,

Separablespaces, The separation axioms T0, T1, T2, T3½, T4, their characterizations and basic

properties, Urysohn’s lemma, Tietze extension theorem.

Unit 3: Connectedness

Connected spaces and their basic properties, Connectedness of the real line, Components, Locally

connected spaces.

Unit 4: Compactness

Compactness, Basic properties of compactness, Compactness and finite intersection property,

Sequential, countable, and B-W compactness, Local compactness, One-point compactification.

Unit 5: Product Topology

Tychonoff product topology in terms of standard sub-base and its characterizations, Product topology

and separation axioms, connected-ness and compactness (incl. the Tychonoff’s theorem), product

spaces.

Unit 6: Nets and filters

Nets and filters, their convergence, and interrelation, Hausdorffness and compactness in terms of

net/filter convergence.

References:

1. G.E. Bredon, Topology and Geometry, Springer, 2014.

2. J. Dugundji, Topology, Allyn and Bacon Inc., Boston, 1978.

3. J.L. Kelley, General Topology, Dover Publications, 2017.

4. J.R. Munkres, Topology, Second Edition, Pearson, 2015.

5. T.B. Singh, Elements of Topology, CRC Press, Taylor & Francis, 2013

6. S. Willard, General Topology, Dover Publications, 2004.

PROBABILITY THEORY AND STATISTICS (STPM-302)

L T P

4 1 0

Unit 1: Probability Theory and Characteristic Function

Introduction to Probability Theory, Bayes’ Theorem, Random Variables and Distribution Functions,

Probability mass function, Probability density function Two Dimensional Random Variables - Joint,

Marginal and Conditional Distributions, Independence of Random Variables. Moments of Random

Variables - Expectation, Variance, moment generating function, cumulants generating function,

characteristic generating function.

Unit 2: Distributions

Binomial, Poisson distributions, Negative Binomial Distribution, Polynomial distribution, Hyper

geometric distribution, Uniform(continuous), Normal distribution, Beta-I and Beta-II, Gamma

Distribution, Laplace distributions, distribution of mean and variance, distribution of differences of means

and variances, t distribution, F distribution, Chi-Square distribution.

Unit 3: Survey Sampling

Probability sampling designs, sampling schemes, inclusion probabilities and estimation; Fixed (Design-

based) and Superpopulation (model-based) approaches; Review of important results in simple and

stratified random sampling; Sampling with varying probabilities (unequal probability sampling) with or

without replacement – ps sampling procedures and estimation based on them; Non-negative variance ps

and non-ps, estimation; Two-way stratification, post-stratification, controlled sampling; Estimation

based on auxiliary data (involving one or more auxiliary variables) under design-based and model-based

approaches; Double (two-phase) sampling with special reference to the selection with unequal

probabilities in at least one of the phases; systematic sampling and its application to structured

populations; Cluster sampling (with varying sizes of clusters); Two-stage sampling (with varying sizes of

first-stage units).

Unit 4: Design of Experiments

Review of linear estimation and basic designs. ANOVA: Fixed effect models (2-way classification with

unequal and proportional number of observations per cell), Random and Mixed effect models (2-way

classification with m (>1) observations per cell), Incomplete Block Designs, Concepts of Connectedness,

Orthogonality and Balance, Intrablock analysis of General Incomplete Block design, B.I.B designs with

and without recovery of interblock information.

Books Recommended:

1. Meyer P.L., Introductory Probability and Statistical Applications (Addison Wesley)

2. Goon, A.M., Gupta, M.K. and Dasgupta. B. (1985): An Outline of Statistical Theory, Vol. I

(World Press).

3. Freund J.E. - Mathematical Satistics (Prentice Hall)

4. Mukhopadhyaya P. (1996) Mathematical Statistics (New Central Book Agency)

5. Rohatgi, V.K. and Saleh, A.K. Md. E. (2003): An Introduction to Probability and Statistics,

Second Edn.

6. Feller, W. (1968) : An Introduction to Probability Theory and its Appplications , 3rd Edition, Vol.

I, John Wiley & Sons.

OPERATION RESEARCH (STPME-1-403)

L T P

4 1 0

Unit 1:

Introduction to Operations Research: Basics definition, scope, objectives, phases, models and limitations

of Operations Research. Linear Programming Problem – Formulation of LPP, Graphical solution of LPP.

Simplex Method, Artificial variables, big-M method, two-phase method, degeneracy and unbound

solutions.

Unit 2:

Transportation Problem. Formulation, solution, unbalanced Transportation problem. Finding basic

feasible solutions – Northwest corner rule, least cost method, Assignment model, Hungarian method for

optimal solution. Solving unbalanced problem. Traveling salesman problem and assignment problem.

Unit 3:

Games Theory. Competitive games, rectangular game, saddle point, minimax (maximin) method of

optimal strategies, value of the game. Solution of games with saddle points, dominance principle.

Rectangular games without saddle point – mixed strategy for 2 X 2 games. Replacement Models.

Replacement of Items that Deteriorate whose maintenance costs increase with time without change in the

money value. Inventory models. Inventory costs. Models with deterministic demand and probabilistic

demand

Unit 4:

Integer linear programming: Modeling using pure and mixed integer programming, Branch and bound

technique, Gomory’s cutting plane algorithm, 0-1 programming problem, E-Bala’s additive algorithm,

Linear goal programming: Modeling using goal programming, Archimedean goal programming,

Preemptive goal programming, Graphical method, Lexicographic simplex method.

Linear fractional programming: Generalized convexity verification, Simplex method, Charne’s and

Cooper method, Mathematical programming algorithms: Penalty functions method, Barrier functions

method, Frank and Wolfe’s method, Method of reduced gradient, Convex simplex method.

Unit 5:

Geometric Programming: Constrained and Unconstrained Minimization Problems.

Stochastic Programming: Stochastic Linear and Stochastic Non linear Programming, Netwok Schedulig

by PERT/CPM.

References:

1. Gass, S. I. (1985). Linear programming- methods and applications (5th ed.). New York: McGraw

Hill (Dover edition 2003 is also available).

2. Hadley, G. (2002). Linear programming. New Delhi: Narosa Publishing House.

3. Hillier, F. S., & Lieberman, G. J. (2010). Introduction to operations research- concepts and cases

(9th ed.). New Delhi: Tata McGraw Hill (Indian print).

4. Ravindran, A., Phillips, D. T., & Solberg, J. J. (2005). Operations research- principles and

practice (2nd ed.). New Delhi: Wiley India (P.) Ltd. (Indian print).

5. Taha, H. A. (2007). Operations research-an introduction (8th ed.). New Delhi: Pearson Prentice

Hall (Indian print).

6. Hadley, G., & Whitin, T. M. (1963). Analysis of inventory systems. Prentice-Hall.

7. Bazara, M. S., Sherali, H. D., & Shetty, C. M. (2006). Nonlinear programming-theory and

algorithms (3rd ed.). New Delhi: John Wiley & Sons (Indian print).

COMPUTER FUNDAMENTALS & C - PROGRAMMING

L T P

4 0 2

Unit-I

Fundamental: H/W and S/W part of computer system, Computer Block Diagram, online processing,

time sharing system, real time system, batch system, multiprogramming, multiprocessing, SPOOLING,

distributed data processing. Element of computer file, types of files, file processing activities, file design

factors, access methods, pros and cons of file organization

Unit-II

MS office: Introduction to MS Office; Introduction to MSWord; Features & area of use. Working with

MS Word.; Menus & Commands; Toolbars & Buttons; Shortcut Menus, Wizards & Templates; Creating

a New Document; Different Page Views and layouts; Applying various Text Enhancements; Working

with – Styles, Text Attributes; Paragraph and Page Formatting; Text Editing using various features ;

Bullets, Numbering, Auto formatting, Printing & various print options

Unit-III

MS Excel: Introduction and area of use; Working with MS Excel.; concepts of Workbook &

Worksheets; Using Wizards; Various Data Types; Using different features with Data, Cell and Texts;

Inserting, Removing & Resizing of Columns & Rows; Working with Data & Ranges; Column Freezing,

Labels, Hiding, Splitting etc.; Using different features with Data and Text; Use of Formulas,

Calculations & Functions; Cell Formatting including Borders & Shading; Working with Different Chart

Types; Printing of Workbook.

Unit-IV

Elements of C: C character set, identifiers and keywords, Data types: declaration and definition, storage

classes in C, Type conversion, Types of error, ‘C’ macro, macrovs function.

Operators: Arithmetic, relational, logical, bitwise, unary, assignment and conditional operators and

their hierarchy & associativity. Data input/output.

Control statements: Sequencing, Selection: if and switch statement; alternation, Repetition: for, while,

and do‐while loop; break, continue, goto.

Unit-V

Functions: Definition, prototypes, passing parameters, recursion.

Data Structures: arrays, structure, union, string.

Pointers: Declaration, operations on pointers, array of pointers, pointers to arrays.

String & file handling, Streams, String I/ O, File Operations, Formatted I/O, Character I/ O, Line I/O,

Block I/O, File positioning, File handling.

References

1. Joe Habraken, Microsoft Office 2000, 8 in 1 by, Prentice Hall of India

2. Deitel&Deitel: C How to Program (Prentice Hall), 1996.

3. YashwantKanetker, Let us C, BPB Publications.

4. R. B. Patel, Fundamental of Computers and Programming in C, Khanna Book Publishing

Company PVT.LTD. Delhi, India, 1st edition, 2008, ISBN: 13: 978‐81‐906988‐7‐0, pp. 1‐962.

5. Gottfried, Programming with C, Tata McGraw Hill.

6. Brian W. Kernighan, Dennis M. Ritchie, The C Programming Language, 2nd Ed., Prentice Hall

of India.

SEMESTER-IV

INTEGRAL EQUATIONS (STPM-401)

L T P

4 1 0

Unit 1:

Preliminary concepts of integral equations, Some problems which give rise to integral equations,

Conversion of ordinary differential equations into integral equations, Classification of linear integral

equations.

Unit 2:

Fredholm integral equations of second kind with separable kernels, Eigen Values and Eigen functions,

Reduction to a system of algebraic equations, An approximate Method, Method of successive

approximations, Iterative scheme, Condition of convergence and uniqueness of series solution, Resolvent

kernel and its results, Fredholm theorems.

Unit 3:

Solution of Volterra's integral equations by iterative scheme, Successive approximation, Resolvent kernel,

Integral transform methods: Fourier transform, Laplace transform, Convolution integral, Application to

Volterra integral equations with Convolution type kernels.

Unit 4:

Symmetric kernel, Complex Hilbert space, Orthonormal system of functions, Fundamental properties of

eigen values and eigen functions for symmetric kernels. Expansion in eigen function and bilinear form,

Hilbert Schmidt theorem, Solution of integral equations with symmetric kernels Singular Integral

Equations - Inversion formula for singular integral equation with kernel of type (h(s) - h(t) - a, 0< a <1).

Dirac Delta Function.

References:

1. R.P. Kanwal, Linear Integral Equation. Theory and Techniques, Academic Press, New York, 1971.

2. S.G. Mikhlin, Linear Integral Equations (translated from Russian), Hindustan Book Agency, 1960.

3. Abdul J. Jerri, Introduction to Integral Equations with Applications.

4. Hildebrand. F.B - Method of Applied Mathematics.

5. L.G.Chambers, Integral Equations: A short Course, Int. Text Book Company Ltd. 1976.

6. Harry Hochsdedt, Integral Equations.

FUNCTIONAL ANALYSIS (STPM-402)

L T P

4 1 0

Unit 1: Banach Spaces

Definition - Some examples - Continuous Linear Transformations - The Hahn -Banach Theorem -

The natural embedding of N in N**

Unit 2: Banach Spaces And Hilbert Spaces

Open mapping theorem - conjugate of an operator - Definition and some simple properties -

Orthogonal complements - Orthonormal sets

Unit 3: Hilbert Space

Conjugate space H* - Adjoint of an operator - Self-adjoint operator - Normal and Unitary Operators -

Projections

Unit 4: Preliminaries On Banach Algebras

Definition and some examples - Regular and single elements - Topological divisors of zero -

spectrum - the formula for the spectral radius - the radical and semi-simplicity.

Unit 5: Structure Of Commutative Banach Algebras

Gelfand mapping - Applications of the formula r(x) = lim 7 xn 7

1/n - Involutions in Banach Algebras -

Gelfand-Neumark Theorem.

References:

1. G. F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill, 1963.

2. G. Bachman and L. Narici, Functional Analysis, Academic Press, 1966.

3. A.E.Taylor, Introduction to Functional Analysis, John Wiley, 1958.

4. B.V. Limaye, Functional Analysis, Wiley Eastern.

5. N. Dunford and J. T. Schwartz, Linear Operators, Part-I, Interscience, 1958.

6. R. E. Edwards, Functional Analysis, Holt Rinehart and Winston, 1965.

7. C. Goffman and G. Pedrick, First Course in Functional Analysis, Prentice- Hall of India,

1987.

8. K. K. Jha, Functional Analysis and Its Applications, Students’ Friend, 1986

FLUID DYNAMICS (STPM-303)

L T P

4 1 0

Unit 1:

Kinematics of fluid-Lagrangian and Eulerian methods, Stream lines, Path lines, Streak lines, Velocity

potential, Irrotational and rotational motions. Vortex lines, Equation of Continuity.

Unit 2:

Lagrangian and Eulerian approach, Euler's equation of motion, Bernoulli's theorem, Kelvin circulation

theorem, Vorticity equation, Energy equation for an incompressible flow.

Unit 3:

Boundary conditions, Kinetic energy of liquid, Axially symmetric flows, Motion of a sphere through a

liquid at rest at infinity, Liquid streaming past a fixed sphere, Equation of motion of a sphere, Sources,

Sinks and doublets, Images in a rigid impermeable infinite plane and in impermeable spherical surfaces.

Unit 4:

Two-dimensional irrotational motion produced by motion of circular, co-axial and elliptic cylinders in an

infinite mass of liquid, Stream functions, Stokes stream functions, Complex velocity potential.

Unit 5:

Conformal mapping, Milne-Thomson Circle theorem, Blasius theorem, Vortex Motion and its

elementary properties, Kelvin's proof of permanence, Motion due to rectilinear vortices.

References:

1. F. Chorlton, Text Book of Fluid Dynamics, CBS Publisher, 2005.

2. R.W. Fox, P.J. Pritchard and A.T. McDonald, Introduction to Fluid Mechanics, Seventh Edition,

John Wiley & Sons, 2009.

3. P.K. Kundu, I.M. Cohen, D.R. Dowling, Fluid Mechanics, Sixth Edition, Academic Press, 2016.

DIFFERENTIAL GEOMETRY (STPME-2-403)

L T P

4 1 0

Unit 1: Curves with torsion: Tangent, Principal Normal, Curvature, Binormal, Torsion, Serret Frenet

formulae, Locus of centre of spherical Curvature.

Unit 2: Envelopes: Surfaces, Tangent plane, Envelope, Characteristics, Edge of regression

Unit 3: Curvilinear Co-ordinates: First order magnitude, Directions on a surface, Second order

magnitudes, Derivative of unit normal, Principal directions and curvatures.

Unit 4: Geodesics: Geodesic property, Equations of geodesics, Torsion of a geodesic

References:

1. C.E., Weatherburn, Differential Geometry of Three Dimensions.

ALGEBRAIC NUMBER THEORY (STPME-3-403)

L T P

4 1 0

Unit 1: Primes in certain arithmetical progressions. Fermat numbers and Mersenne numbers.

Approximation of irrational numbers by rationals. Hurwitz's theorem, irrationality of e and π. System of

linear congruences Chinese Remainder Theorem. Quadratic residues and nonresidues. Legendre's

Symbol. Gauss Lemma and its applications. Quadratic Law of Reciprocity jacobi's Symbol.

Unit 2: Riemann Zeta Function ξ(s) and its convergence. Application in prime numbers. ξ(s) as Euler's

product. Evaluation of ξ(2) and ξ(2k). Dirichlet series with simple properties. Dirichlet series as analytic

function and its derivative. Eulers products. Introduction to modular forms.

Unit 3: Euler's summation formula and some elementary asymptotic formula. Average order of the

arithmetical functions d(n), σαa (n), φ(n), μ(n) and Λ(n). Partial sums of a Dirichlet product and their

application to μ(n) and Λ(n).

Unit 4: Chebyshev's functions Ψ(x) and ν(x) and relation between ν(x) and π(x). Shapiro's Tauberian

theorem and its applications. Partial sums of the Mobius function. Selberg's asymptotic formula.

References:

1. T.M. Apostol. Introduction to Analytic number theory (Narosa Publishing House 1980).

2. T.M. Apostol. Modular functions and Dirichlet series in Number Theory (Springer-Verlag 1976).

3. J.P. Serre. A Course in Arithmetic G.T.M. Vol.7 (Springer Verlag 1973).

DISCRETE MATHEMATICS (STPME-4-403)

L T P

4 1 0

Unit 1: Lattices

1. Properties and examples of Lattices

2. Distributive lattices

3. Boolean algebras

4. Boolean polynomials

5. Minimal Forms of Boolean Polynomials.

Unit 2: Applications of lattices

6. Switching Circuits

7. Applications of Switching Circuits

Unit 3: Finite fields and polynomials

8. Finite fields

Unit 4: Finite fields and polynomials

9. Irreducible Polynomials over Finite fields

10. Factorization of Polynomials over Finite fields

Unit 5: Coding theory

11. Linear Codes

12. Cyclic Codes

References:

1. Rudolf Lidl & Gunter Pilz. Applied Abstract Algebra, Second Indian Reprint

2006, Springer Verlag, NewYork.

2. A.Gill, Applied Algebra for Computer Science, Prentice Hall Inc., New Jersey.

3. J.L.Gersting, Mathematical Structures for Computer Science(3rd Edn.), Computer Science Press,

New York.

4. S.Wiitala, Discrete Mathematics- A Unified Approach, McGraw Hill Book Co.

MATHEMATICAL MODELING (STPME-1-404)

L T P

4 1 0

Unit 1:

Introduction, basic steps of Mathematical Modeling, its needs, types of models, limitations. Elementary

ideas of dynamical systems, autonomous dynamical systems in the plane-linear theory. Equilibrium point,

node, saddle point, focus, centre and limit-cycle ideas with simple illustrations and figures.

Unit 2:

Linearization of non-linear plane autonomous systems. Mathematical Modeling in the biological

environment. Blood flow and oxygen transfer. Modeling blood flow, viscousity, Poiseuille law,

mathematical formulation of the problem, solution and interpretation. Oxygen transfer in red cells,

diffusion, mathematical formulation, solution, interpretation, and limitations.

Unit 3:

Single species population models. Basic concepts. Exponential growth model, formulation, solution,

interpretation, and limitations. Compensation and depensation. Logistic growth model, formulation,

solution, interpretation, and limitations.

Unit 4:

Gompertz growth model, formulation, solution, interpretation, and limitations. Two species population

models. Types of interaction between two species. Lotka-VolteiTa prey-predator model, formulation,

solution, interpretation, arid limitations. Lotka-Volterra model of two competing species, formulation,

solution, interpretation, and limitations.

Unit 5:

Mathematical modeling of epidemics. Basic concepts. Simple epidemic model, formulation, solution,

interpretation, and limitations. General epidemic model, formulation, solution, interpretation, and

limitations.

References:

1.Rutherford Aris, Mathematical Modelling Techniques, Dover Publications Inc.; 1994.

CALCULUS ON n (STPME-2-404)

L T P

4 1 0

Unit 1:

Paths and homotopy, homotopy equivalence, contractibility, deformation retracts, Basic constructions:

cones, mapping cones, mapping cylinders, suspension.

Unit 2:

Cell complexes, subcomplexes, CW pairs.

Fundamental groups. Examples and applications, Brouwer Fixed Point Theorem and Borsuk-Ulam

Theorem.

Unit 3:

Van Kampen's Theorem, Covering spaces, lifting properties, deck transformations. universal coverings.

Simplicial complexes, barycentric subdivision, stars and links, simplicial approximation. Simplicial

Homology.

Unit 4:

Singular Homology. Mayer-Vietoris Sequences. Long exact sequence of pairs and triples. Homotopy

invariance and excision, Degree. Cellular Homology.

Unit 5:

Applications of homology: Jordan-Brouwer separation theorem, Invariance of dimension, Hopf's

Theorem for commutative division algebras with identity, Borsuk-Ulam Theorem, Lefschetz Fixed Point

Theorem.

References:

1. Edwin H. Spanier, Algebraic Topology, Springer Verlag, 1966.

REGRESSION ANALYSIS (STPME-3-404)

Unit 1

Simple and multiple linear regression models %G––%@ estimation, tests and confidence regions. Check

for normality assumption. Likelihood ratio test, confidence intervals and hypotheses tests; tests for

distributional assumptions.

Unit 2

Collinearity, outliers; analysis of residuals, Selecting the $(B!H(BBest$(B!I(B regression equation,

transformation of response variables. Ridge's regression.

Unit 3

Nature of econometrics. The general linear model (GLM) and its extension. Ordinary least squares

(OLS) estimation and prediction. Generalized least squares (GLS) estimation and prediction.

Heteroscedastic disturbances.

References:

1. B.L. Bowerman and R.T. O'Connell, Linear Statistical Models: An Applied Approach, PWS-

KENT Pub., Boston, 1990

2. N.R Draper. And H. Smith., Applied Regression Analysis, John Wiley and Sons (Asia) Pvt. Ltd.,

Series in Probability and Statistics, 2003.

3. D.C. Montgomery, E.A. Peck, G.G. Vining, Introduction to Linear Regression Analysis, John

Wiley, NY, 2003

4. A.A. Sen and M. Srivastava, Regression Analysis %G–%@ Theory, Methods & Applications,

5. Springer-Verlag, Berlin, 1990.