1 st Semester Paper Code Paper Name Marks MTMPG- 101 Real Analysis 50 MTMPG- 102 Linear Algebra 50 MTMPG- 103 Classical Mechanics 50 MTMPG- 104 Complex Analysis 50 MTMPG- 105 Group – A: Graph Theory 25 Group – B: Discrete Mathematics 25 MTMPG- 106 Computer Programming 30 Computer Lab 20
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1st Semester
Paper Code Paper Name Marks
MTMPG- 101 Real Analysis
50
MTMPG- 102 Linear Algebra
50
MTMPG- 103
Classical Mechanics
50
MTMPG- 104 Complex Analysis
50
MTMPG- 105
Group – A: Graph Theory
25
Group – B: Discrete Mathematics
25
MTMPG- 106
Computer Programming
30
Computer Lab
20
Paper – MTMPG-101
REAL ANALYSIS 50
Functions of Bounded Variation, Riemann Integration, Riemann Stieltjes Integration in
R.
Lebesgue Integral.
Functions of Several Variables, Continuity, Differentiations-Directional Derivatives
and Frechet Derivative, Mean value Theorem, Maxima, Minima, Inverse and Implicit
Function Theorem.
Riemann Integration in , Parametric Hypersurfaces, tangent Plane, Normal and
Surface Measure, Surface Integrals and Divergence Theorem
References
1. S. Lang-Analysis
2. Charles Chapman Pagh- Real Mathematical Analysis
3. I. P. Natanson – Theory of Functions of a Real Variable, Vol. I, Fedrick Unger
Publi. Co., 1961.
4. Lusternik and Sovolev-Functional Analysis
5. A.H. Siddiqui- Functional Analysis with applications, TMG Publishing Co. Ltd, New Delhi
6. K.K. Jha- Functional Analysis, Student’s Friends,1986 7. Vulikh- Functional Analysis 8. G. Bachman & L. Narici- Functional Analysis, Academic Press,1966 9. A.E. Taylor- Functional Analysis, John Wiley and Sons, New York,1958
10. E. Kreyszig-Introductory Functional Analysis with Applications, Wiley Eastern,1989
11. L.V. Kantorvich and G.P. Akilov-Functional Analysis, Pergamon Press,1982 12. G.F. Simmons- Introduction to Topology and Modern Analysis ,Mc Graw Hill, New
York, 1963 13. B.V. Limaye- Functional Analysis, Wiley Easten Ltd 14. Burkil & Burkil – A second Course of Mathematical Analysis, CUP, 1980. 15. Goldberg – Real Analysis, Springer-Verlag, 1964 16. Royden – Real Analysis, PHI, 1989 17. T.M. Apostol – Mathematical Analysis, Narosa Publi. House, 1985. 18. Titchmarsh – Theory of Functions, CUP, 1980
Paper – MTMPG-102
LINEAR ALGEBRA 50
Vector Spaces, Complex and Real Euclidean Spaces, Linear Transformations and
Matrix representation of linear transformations. Change of basis, canonical forms,
diagonal forms, triangular forms, Jordan forms. Inner product spaces, Spectral
Theorem, Orthonormal basis. Quadratic forms, reduction and classification of
quadratic forms
References:
1. Hoffman and Kunze, Linear Algebra (Prentice Hall).
4. Graph Theory Applications, L.R. Foulds, Narosa Publishing House, New
Delhi,1993.
5. Graph Theory with Applications, J.A. Bondy and U.S.R. Murty, Elsevier science,
1976.
6. Graphs and Digraphs, G. Chartrand and L. Lesniak, Chapman & Hall, 1996.
7. Theory of Graphs, O. Ore, AMS Colloq. 38, Amer.Math.Soc., 1962.
Paper – MTMPG-105 (Group - B)
DISCRETE MATHEMATICS 25
Mathematical induction. Principle of inclusion and exclusion. Pigeon hole principle. Finite combinatorics. Generating functions. Par-titions. Recurrence relations. Linear difference equations with constant coefficients. Partial and linear orderings. Chains and antichains. Lattices. Distributive lattices.
Complementation.
Alphabets and strings. Formal languages and phrase structure grammars. BNF notation. Derivations. Language generated by a gram-mar. The Chomsky hierarchy: Regular, Context-free, Context-sensitive, and arbi-trary grammars.
Finite state machines. Nondeterministic finite automata. Regular languages. Closure properties.
Kleene's theorem. Regular expressions. Pumping lemma. Al-gorithms for regular
grammars.
Introduction to the theory of Context-free languages, push-down automata, and
parsing.
References:
1. Introduction to Graph Theory, Douglas B. West, Prentice-Hall of India Pvt. Ltd.,
1. Working with matrix: Generating matrix, Concatenation, Deleting rows and columns. Symmetric matrix, matrix multiplication, Test the matrix for singularity, magic matrix. Matrix analysis using function: norm,
Element-by-element division, Element-by-element left division, Element-by-element power. Multidimensional Arrays, Cell Arrays, Characters and
Text in array, 3. Graph Plotting: Plotting Process, Creating a Graph, Graph Components,
Figure Tools, Arranging Graphs Within a Figure, Choosing a Type of
Graph to Plot, Editing Plots, Plotting Two Variables with Plotting Tools, Changing the Appearance of Lines and Markers, Adding More Data to the
Graph, Changing the Type of Graph, Modifying the Graph Data Source, Annotating Graphs for Presentation, Exporting the Graph.
4. Using Basic Plotting Functions: Creating a Plot, Plotting Multiple Data
Sets in One Graph, Specifying Line Styles and Colors, Plotting Lines and Markers, Graphing Imaginary and Complex Data, Adding Plots to an Existing Graph, Figure Windows, Displaying Multiple Plots in One
Figure, Controlling the Axes , Adding Axis Labels and Titles, Saving Figures.
5. Programming: Conditional Control – if, else, switch, Loop Control – for, while, continue, break, Error Control – try, catch, Program Termination – return.
6. Scripts and Functions: Scripts, Functions, Types of Functions, Global Variables, Passing String Arguments to Functions, The eval Function,
Function Handles, Function Functions, Vectorization, Preallocation. 7. Data Analysis: (i) Preprocessing Data : Loading the Data, Missing Data,
Outliers, Smoothing and Filtering, (ii)Summarizing Data: Measures of
Location, Measures of Scale, Shape of a Distribution, (iii) Visualizing Data: 2-D Scatter Plots, 3-D Scatter Plots, Scatter Plot Arrays, Exploring Data in Graphs, (iv) Modeling Data: Polynomial Regression, General
Linear Regression, 8. Linear Algebra: Systems of Linear Equations, Inverses and
Determinants, Factorizations, Powers and Exponentials, Eigenvalues, Singular Values.
9. Polynomials: Polynomial functions in the MATLAB® environment,
inequality. Parallelogram law. Projection theorem. Inner product is a continuous operator.
Relation between IPS and NLS. Bessel’s inequality. Parseval’s identity Strong and Weak
convergence of sequence of operators. Reflexivity of Hilbert space. Riesz Representation
Theorem for bounded linear functional on a Hilbert space.
Definition of self adjoint operator, Normal, Unitary and Positive operators, Related simple
theorem.
Applications in differential and integral equations
Ref. Books:
1. E. Kreyszig-Introductory Functional Analysis with Applications,Wiley Eastern,1989. 2. Joseph Mascut, Functional Analysis, Springer
3. B.Chowdhary, Sudarsan Nanda, Functional Analysis with Applications, Wiley Eastern ltd, 1991. 4. G.F. Simmons- Introduction to Topology and Modern Analysis ,Mc Graw Hill, New York, 1963.
5. A.E. Taylor- Functional Analysis, John wiley and Sons, New York,1958.
Paper – MTMPG-302
PROBABILITY & MEASURE THEORY 50
Probability: Borel’s Normal Number Theorem, Probability Measures, Existence and
Extension, Denumerable Probabilities, Simple Random Variables, The law of Large
Numbers, Gambling Systems, Markov Chains, Large Deviations and the Law of the
Iterated Logarithm.
Measure: General Measures, Outer Measures, Measures in Euclidean Space,
Measurable Functions and Mappings, Distribution Functions.
Random Variables and Expected Values: Random Variables and Distribution,
Expected Values, Sums of Independent Random Variables, The Poisson Process, The
Ergodic Theorem.
Convergence of Distribution: Weak Convergence, Characteristic Function, The
Central Limit Theorem, Infinitely Divisible Distributions, Limit Theorems in Rk, The
Method of Moments.
Derivatives and Conditional Probability: Derivatives on the Line, The Random –
M/M/C for finite and infinite queue length, Non-Poisson queue – M/G/1, Machine-
Maintenance (Steady State).
Network: PERT and CPM: Introduction, Basic difference between PERT and CPM,
Steps of PERT/CPM Techniques, PERT/CPM network components and precedence
relationships, Critical path analysis, Probability in PERT analysis, Project Time – Cost,
Trade-off, Updating of the project, Resource allocation – resource smoothing and
resource leveling.
Replacement and Maintenance Models: Introduction, Failure Mechanism of items,
Replacement of items deteriorates with time, Replacement policy for equipments when
value of money changes with constant rate during the period, Replacement of Item
that fail completely – individual replacement policy and group replacement policy,
Other replacement problems – staffing problem, equipment.
Simulation: Introduction, Steps of simulation process, Advantages and disadvantages
of simulation, Stochastic simulation and random number – Monte Carlo simulation,
Random number, Generation, Simulation of Inventory Problems, Simulation of
Queuing problems, Role of computers in Simulation, Applications of Simulations.
4th SEMESTER
Paper Code Paper Name Marks
MTMPG-401
A. Calculus of Variations
B. Electrodynamics
20
30
MTMPG-402 Fluid Mechanics
50
MTMPG-403
A. Cryptography
B. Soft Computing
25
25
MTMPG-404
Project Work
50
MTMPG-405
IIIA
SPECIAL PAPER III
Advance Optimization And Operation
Research
50
IIIB
SPECIAL PAPER III
Algebraic Topology
50
MTMPG-
406
IVA:
SPECIAL PAPER IV Operational Research Modeling – I
50
IVB:
SPECIAL PAPER IV
Techniques of Differential Topology in Relativity
50
Paper – MTMPG-401
A. Calculus of Variations 20
Calculus of Variation : Variation of a functional, Euler-Lagrange equation, Necessary and
sufficient conditions for extrema. Variational methods for boundary value problems in ordinary
and partial differential equations.
References:
1. Gupta, A.S. Calculus of Variations, PHI
B. Electrodynamics 30
1. Field of moving charges and radiations: Retarded potentials, Lienard Wichert potentials, Field produced by arbitrarily moving charged particle & uniformly moving charged particle, radiation from an accelerated charged particle at low velocity and at high velocity, angular distribution of radiated power. Radiation from an oscillating dipole, radiation from a linear antenna
2. Radiation in material media: Cherenkov effect, Thomson and Rayleigh Scattering, dispersion and absorption, Kramer Kronig dispersion relation.
3. Relativistic electrodynamics: Transformation equations for field vectors and. Covariance of Maxwell equations in 4 vector form, Covariance of Maxwell equations in 4-tensor forms; Covariance and transformation law of Lorentz force. Self energy of electron
4. Radiation loss of energy by the free charges of plasma: Radiation by excited atoms and ions. Cyclotron or Betatron radiation, Bremsstrahlung, Recombination radiation, Transport of radiation.
Oscillations and waves in Plasma: Mechanisms of Plasma oscillations, Electron plasma
oscillations, ion oscillations and waves
References:
1. Griffiths, D.J., Classical Electrodynamics,
Paper – MTMPG-402
Fluid Mechanics 50
Irrotational Motion in Two Dimensions: General motion of a cylinder in two dimensions.
Motion of a cylinder in a uniform stream, Liquid streaming past a fixed circular cylinder and two
coaxial cylinders. Equations of motion of a circular cylinder. Circulation about a moving cylinder.
Conjugate function. Elliptic cylinder. Liquid streaming past a fixed elliptic cylinder. Elliptic
cylinder rotating in an infinite mass of liquid at rest at infinity. Circulation about an elliptic
cylinder. Kinetic energy. Blasius theorem and its application. Kutta and Joukowski theorem,
D'Alemberts paradox. Application of conformal mapping.
Vortex Motion: Vortex line, Vortex tube, Properties of the vortex, Strength of the vortex,
Rectilinear vortices, Velocity component, centre of vortices. A case of two vortex filaments,
vortex pair. Vortex doublet. Image of vortex filament with respect to a plane. An infinite single
row of parallel rectilinear vortices of same strength. Two infinite row of parallel rectilinear