Department of Mathematics, University of Delhi

Semester-VI
BMATH613: Complex Analysis
Total Marks: 150 (Theory: 75, Internal Assessment: 25 and Practical: 50)
Workload: 4 Lectures, 4 Practicals (per week), Credits: 6 (4+2)
Duration: 14 Weeks (56 Hrs. Theory + 56 Hrs. Practical) Examination: 3 Hrs.

Course Objectives: This course aims to introduce the basic ideas of analysis for complex
functions in complex variables with visualization through relevant practicals. Emphasis has
been laid on Cauchy’s theorems, series expansions and calculation of residues.
Course Learning Outcomes: The completion of the course will enable the students to:
i) Learn the significance of differentiability of complex functions leading to the
understanding of Cauchy−Riemann equations.
ii) Learn some elementary functions and valuate the contour integrals.
iii) Understand the role of Cauchy−Goursat theorem and the Cauchy integral formula.
iv) Expand some simple functions as their Taylor and Laurent series, classify the nature
of singularities, find residues and apply Cauchy Residue theorem to evaluate
integrals.

Unit 1: Analytic Functions and Cauchy− −Riemann Equations

Functions of complex variable, Mappings; Mappings by the exponential function, Limits,
Theorems on limits, Limits involving the point at infinity, Continuity, Derivatives,
Differentiation formulae, Cauchy−Riemann equations, Sufficient conditions for
differentiability; Analytic functions and their examples.

Unit 2: Elementary Functions and Integrals

Exponential function, Logarithmic function, Branches and derivatives of logarithms,
Trigonometric function, Derivatives of functions, Definite integrals of functions, Contours,
Contour integrals and its examples, Upper bounds for moduli of contour integrals,

Unit 3: Cauchy’s Theorems and Fundamental Theorem of Algebra

Antiderivatives, Proof of antiderivative theorem, Cauchy−Goursat theorem, Cauchy
integral formula; An extension of Cauchy integral formula, Consequences of Cauchy
integral formula, Liouville’s theorem and the fundamental theorem of algebra.

Unit 4: Series and Residues

Convergence of sequences and series, Taylor series and its examples; Laurent series and its
examples, Absolute and uniform convergence of power series, Uniqueness of series
representations of power series, Isolated singular points, Residues, Cauchy’s residue
theorem, residue at infinity; Types of isolated singular points, Residues at poles and its
examples.

Reference:
1. Brown, James Ward, & Churchill, Ruel V. (2014). Complex Variables and
Applications (9th ed.). McGraw-Hill Education. New York.

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Department of Mathematics, University of Delhi

Additional Readings:
i. Bak, Joseph & Newman, Donald J. (2010). Complex Analysis (3rd ed.).
Undergraduate Texts in Mathematics, Springer. New York.
ii. Zills, Dennis G., & Shanahan, Patrick D. (2003). A First Course in Complex Analysis
with Applications. Jones & Bartlett Publishers, Inc.
iii. Mathews, John H., & Howell, Rusell W. (2012). Complex Analysis for Mathematics
and Engineering (6th ed.). Jones & Bartlett Learning. Narosa, Delhi. Indian Edition.

Practical / Lab work to be performed in Computer Lab:

Modeling of the following similar problems using Mathematica/Maple/MATLAB/Maxima/
Scilab etc.
1. Make a geometric plot to show that the nth roots of unity are equally spaced points
that lie on the unit circle Š 0 = wm ∶ m = 1y and form the vertices of a regular
polygon with n sides, for n = 4, 5, 6, 7, 8.
2. Find all the solutions of the equation m  = 89 and represent these geometrically.
3. Write parametric equations and make a parametric plot for an ellipse centered at the
origin with horizontal major axis of 4 units and vertical minor axis of 2 units.
t
Show the effect of rotation of this ellipse by an angle of Πradians and shifting of the
centre from (0,0) to (2,1), by making a parametric plot.
4. Show that the image of the open disk v −1 − 9 = wm ∶ m + 1 + 9 < 1y under the
linear transformation w = f(z) = (3 – 4i)z + 6 + 2i is the open disk:
D5(–1 + 3i) = {w: |w + 1 – 3i| < 5}.
5. Show that the image of the right half plane Re z = x > 1 under the linear
transformation w = (–1 + i)z – 2 + 3i is the half plane B >  + 7, where u = Re(w),
etc. Plot the map.

6. Show that the image of the right half plane A = {z : Re z ≥ } under the mapping
C
 
 = Um = is the closed disk v 1 = {w : |w – 1| ≤ 1} in the w- plane.
Ž
 
7. Make a plot of the vertical lines x = a, for  = −1, − C , C , 1 and the horizontal lines
 
y = b, for  = −1, − C , C , 1. Find the plot of this grid under the mapping  = Um =

.
Ž
8. Find a parametrization of the polygonal path C = C1 + C2 + C3 from –1 + i to 3 – i,
where C1 is the line from: –1 + i to –1, C2 is the line from: –1 to 1 + i and C3 is the
line from 1 + i to 3 – i. Make a plot of this path.
t
9. Plot the line segment ‘L’ joining the point A = 0 to B = 2 + 9 and give an exact
j

calculation of 1 & Ž 2m.
10. Plot the semicircle ‘C’ with radius 1 centered at z = 2 and evaluate the contour
 
integral 1‘ 2m.
Ž@C
 
11. Show that 1‘ m2m = 1‘ m2m = 4 + 29 where C1 is the line segment from –1 – i to
T \
3 + i and C2 is the portion of the parabola x = y2 + 2y joining –1 – i to 3 + i. Make
plots of two contours C1 and C2 joining –1 – i to 3 + i .

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Department of Mathematics, University of Delhi

 
12. Use ML inequality to show that W1‘ \ 2mW ≤ , where C is the straight line
Ž + C;u
segment from 2 to 2 + i. While solving, represent the distance from the point z to the
points i and – i, respectively, i.e. m − 9 and m + 9 on the complex plane ℂ.
hŽ
13. Show that 1‘ T[\ , where z1/2 is the principal branch of the square root function and
CŽ
C is the line segment joining 4 to 8 + 6i. Also plot the path of integration.
14. Find and plot three different Laurent series representations for the function Um =

\
, involving powers of z.
C+Ž@Ž

15. Locate the poles of Um = and specify their order.
uŽ “ +CŒŽ \ +u
t ”`•tŽ
16. Locate the zeros and poles of ^m = Ž\
and determine their order. Also justify
that Res(g, 0) = −π 2 3.
 C
17. Evaluate 1‘ X˜ exp YŽZ 2m, where Š+ 0 denotes the circle {z : |z| = 1} with positive
T

orientation. Similarly evaluate 1‘ X˜ Ž “ +Ž ] @CŽ \ 2m.
T

ote: For practicals: Sample materials of files in the form Mathematica/Maple 2011.zip,
www.jblearning.com/catalog/9781449604455/.

Teaching Plan (Theory of BMATH613: Complex Analysis):

Week 1: Functions of complex variable, Mappings, Mappings by the exponential function.
[1] Chapter 2 (Sections 12 to 14).
Week 2: Limits, Theorems on limits, Limits involving the point at infinity, Continuity.
[1] Chapter 2 (Sections 15 to 18).
Week 3: Derivatives, Differentiation formulae, Cauchy-Riemann equations, Sufficient conditions for
differentiability.
[1] Chapter 2 (Sections 19 to 22).
Week 4: Analytic functions, Examples of analytic functions, Exponential function.
[1] Chapter 2 (Sections 24 and 25) and Chapter 3 (Section 29).
Week 5: Logarithmic function, Branches and Derivatives of Logarithms, Trigonometric functions.
[1] Chapter 3 (Sections 30, 31 and 34).
Week 6: Derivatives of functions, Definite integrals of functions, Contours.
[1] Chapter 4 (Sections 37 to 39).
Week 7: Contour integrals and its examples, upper bounds for moduli of contour integrals.
[1] Chapter 4 (Sections 40, 41 and 43).
Week 8: Antiderivatives, proof of antiderivative theorem.
[1] Chapter 4 (Sections 44 and 45).
Week 9: State Cauchy−Goursat theorem, Cauchy integral formula.
[1] Chapter 4 (Sections 46 and 50).
Week 10: An extension of Cauchy integral formula, Consequences of Cauchy integral formula,
Liouville’s theorem and the fundamental theorem of algebra.
[1] Chapter 4 (Sections 51 to 53).
Week 11: Convergence of sequences, Convergence of series, Taylor series, proof of Taylor’s
theorem, Examples.
[1] Chapter 5 (Sections 55 to 59).
Week 12: Laurent series and its examples, Absolute and uniform convergence of power series,
uniqueness of series representations of power series.
[1] Chapter 5 (Sections 60, 62, 63 and 66).
Week 13: Isolated singular points, Residues, Cauchy’s residue theorem, Residue at infinity.
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Department of Mathematics, University of Delhi

[1]: Chapter 6 (Sections 68 to 71).

Week 14: Types of isolated singular points, Residues at poles and its examples.
[1] Chapter 6 (Sections 72 to 74).

Facilitating the Achievement of Course Learning Outcomes

Unit Course Learning Outcomes Teaching and Learning Assessment Tasks
o. Activity
1. Learn the significance of (i) Each topic to be • Presentations and
differentiability of complex explained with class discussions.
functions leading to the illustrations. • Assignments and
understanding of Cauchy−Riemann (ii) Students to be class tests.
equations. encouraged to discover • Student
2. Learn some elementary functions the relevant concepts. presentations.
and valuate the contour integrals. (iii) Students to be given • Mid-term
3. Understand the role of homework/assignments. examinations.
Cauchy−Goursat theorem and the (iv) Discuss and solve the • Practical and viva-
Cauchy integral formula. theoretical and practical voce examinations.
4. Expand some simple functions as problems in the class. • End-term
their Taylor and Laurent series, (v) Students be encouraged examinations.
classify the nature of singularities, to apply concepts to real
find residues and apply Cauchy world problems.
Residue theorem to evaluate
integrals.

Keywords: Analytic functions, Antiderivatives, Cauchy−Riemann equations,

Cauchy−Goursat theorem, Cauchy integral formula, Cauchy's inequality, Cauchy's residue
theorem, Closed contour, Contour integrals, Fundamental theorem of algebra, Liouville's
theorem, Morera's theorem, Poles, Regions in complex plane, Residue, Singular points,
Taylor’s and Laurent’s series.

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Department of Mathematics, University of Delhi

BMATH614: Ring Theory and Linear Algebra-II

Total Marks: 100 (Theory: 75, Internal Assessment: 25)
Workload: 5 Lectures, 1 Tutorial (per week) Credits: 6 (5+1)
Duration: 14 Weeks (70 Hrs.) Examination: 3 Hrs.

Course Objectives: This course introduces the basic concepts of ring of polynomials and
irreducibility tests for polynomials over ring of integers, used in finite fields with applications
in cryptography. This course emphasizes the application of techniques using the adjoint of a
linear operator and their properties to least squares approximation and minimal solutions to
systems of linear equations.
Courses Learning Outcomes: On completion of this course, the student will be able to:
i) Appreciate the significance of unique factorization in rings and integral domains.
ii) Compute the characteristic polynomial, eigenvalues, eigenvectors, and eigenspaces, as
well as the geometric and the algebraic multiplicities of an eigenvalue and apply the
basic diagonalization result.
iii) Compute inner products and determine orthogonality on vector spaces, including
Gram−Schmidt orthogonalization to obtain orthonormal basis.
iv) Find the adjoint, normal, unitary and orthogonal operators.

Unit 1: Polynomial Rings and Unique Factorization Domain (UFD)

Polynomial rings over commutative rings, Division algorithm and consequences,
Principal ideal domains, Factorization of polynomials, Reducibility tests, Irreducibility tests,
Eisenstein’s criterion, Unique factorization in ℤ; Divisibility in integral domains,
Irreducibles, Primes, Unique factorization domains, Euclidean domains.
Unit 2: Dual Spaces and Diagonalizable Operators
Dual spaces, Double dual, Dual basis, Transpose of a linear transformation and its matrix in
the dual basis, Annihilators; Eigenvalues, Eigenvectors, Eigenspaces and characteristic
polynomial of a linear operator; Diagonalizability, Invariant subspaces and
Cayley−Hamilton theorem; Minimal polynomial for a linear operator.
Unit 3: Inner Product Spaces
Inner product spaces and norms, Orthonormal basis, Gram−Schmidt orthogonalization
process, Orthogonal complements, Bessel’s inequality.
Unit 4: Adjoint Operators and Their Properties
Adjoint of a linear operator, Least squares approximation, Minimal solutions to systems of
linear equations, Normal, self-adjoint, unitary and orthogonal operators and their properties.
References:
1. Friedberg, Stephen H., Insel, Arnold J., & Spence, Lawrence E. (2003). Linear
Algebra (4th ed.). Prentice-Hall of India Pvt. Ltd. New Delhi.
2. Gallian, Joseph. A. (2013). Contemporary Abstract Algebra (8th ed.). Cengage
Learning India Private Limited. Delhi. Fourth impression, 2015.
Additional Readings:
i. Herstein, I. N. (2006). Topics in Algebra (2nd ed.). Wiley Student Edition. India.
ii. Hoffman, Kenneth, & Kunze, Ray Alden (1978). Linear Algebra (2nd ed.). Prentice-
Hall of India Pvt. Limited. Delhi. Pearson Education India Reprint, 2015.
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Department of Mathematics, University of Delhi

iii. Lang, Serge (1987). Linear Algebra (3rd ed.). Springer.

Teaching Plan (BMATH614: Ring Theory and Linear Algebra-II):

Week 1: Polynomial rings over commutative rings, Division algorithm and consequences, Principal
ideal domains.
[2] Chapter 16.
Weeks 2 and 3: Factorization of polynomials, Reducibility tests, Irreducibility tests, Eisenstein’s
criterion, Unique factorization in ℤ.
[2] Chapter 17.
Weeks 4 and 5: Divisibility in integral domains, Irreducibles, Primes, Unique factorization domains,
Euclidean domains.
[2] Chapter 18.
Week 6: Dual spaces, Double dual, Dual basis, Transpose of a linear transformation and its matrix in
dual basis, Annihilators.
[1] Chapter 2 (Section 2.6).
Weeks 7 and 8: Eigenvalues, Eigenvectors, Eigenspaces and characteristic polynomial of a linear
operator; Diagonalizability, Invariant subspaces and Cayley−Hamilton theorem; Minimal
polynomial for a linear operator.
[1] Chapter 5 (Sections 5.1, 5.2 and 5.4), Chapter 7 (Section 7.3, Statement of Theorem 7.16)
Week 9: Inner product spaces and norms.
[1] Chapter 6 (Section 6.1).
Weeks 10 and 11: Orthonormal basis, Gram−Schmidt orthogonalization process, Orthogonal
complements, Bessel’s inequality.
[1] Chapter 6 (Section 6.2).
Week 12: Adjoint of a linear operator and its properties, Least squares approximation, Minimal
solutions to systems of linear equations.
[1] Chapter 6 (Section 6.3, Statement of Theorem 6.13 with applications).
Weeks 13 and 14: Normal, self-adjoint, unitary and orthogonal operators and their properties.
[1] Chapter 6 (Sections 6.4, and 6.5, up to Theorem 6.21, Page 385).

Facilitating the Achievement of Course Learning Outcomes

Unit Course Learning Outcomes Teaching and Learning Assessment
o. Activity Tasks
1. Appreciate the significance of unique (i) Each topic to be • Student
factorization in rings and integral domains. explained with presentations.
2. Compute the characteristic polynomial, examples. • Participation in
eigenvalues, eigenvectors, eigenspaces, as (ii) Students to be discussions.
well as the geometric and the algebraic involved in discussions • Assignments
multiplicities of an eigenvalue and apply and encouraged to ask and class tests.
the basic diagonalization result. questions. • Mid-term
3. Compute inner products and determine (iii) Students to be given examinations.
orthogonality on vector spaces, including homework/assignment. • End-term
Gram−Schmidt orthogonalization to obtain (iv) Students to be examinations.
orthonormal basis. encouraged to give
4. Find the adjoint, normal, unitary and short presentations.
orthogonal operators.

Keywords: Bessel's inequality, Cayley−Hamilton theorem, Eigenvalues and eigenvectors,

Eisenstein’s criterion, Euclidean domains, Inner product spaces, Orthonormal basis, Principal
ideal domains, Unique factorization domains, Normal, self-adjoint and unitary operators.
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Department of Mathematics, University of Delhi

Discipline Specific Elective (DSE) Course - 3

Any one of the following (at least two shall be offered by the college):
DSE-3 (i): Mathematical Finance
DSE-3 (ii): Introduction to Information Theory and Coding
DSE-3 (iii): Biomathematics

DSE-3 (i): Mathematical Finance

Total Marks: 100 (Theory: 75 + Internal Assessment: 25)
Workload: 5 Lectures, 1 Tutorial (per week) Credits: 6 (5+1)
Duration: 14 Weeks (70 Hrs.) Examination: 3 Hrs.

Course Objectives: This course is an introduction to the application of mathematics in

financial world, that enables the student to understand some computational and quantitative
techniques required for working in the financial markets and actuarial mathematics.
Course Learning outcomes: On completion of this course, the student will be able to:
i) Know the basics of financial markets and derivatives including options and futures.
ii) Learn about pricing and hedging of options, as well as interest rate swaps.
iii) Learn about no-arbitrage pricing concept and types of options.
iv) Learn stochastic analysis (Ito formula, Ito integration) and the Black−Scholes model.
v) Understand the concepts of trading strategies and valuation of currency swaps.
Unit 1: Interest Rates
Interest rates, Types of rates, Measuring interest rates, Zero rates, Bond pricing, Forward rate,
Duration, Convexity, Exchange traded markets and OTC markets, Derivatives--forward
contracts, Futures contract, Options, Types of traders, Hedging, Speculation, Arbitrage.
Unit 2: Mechanics and Properties of Options
No Arbitrage principle, Short selling, Forward price for an investment asset, Types of
options, Option positions, Underlying assets, Factors affecting option prices, Bounds on
option prices, Put-call parity, Early exercise, Effect of dividends.
Unit 3: Stochastic Analysis of Stock Prices and Black-Scholes Model
Binomial option pricing model, Risk neutral valuation (for European and American options
on assets following binomial tree model), Lognormal property of stock prices, Distribution of
rate of return, expected return, Volatility, estimating volatility from historical data, Extension
of risk neutral valuation to assets following GBM, Black−Scholes formula for European
options.
Unit 4: Hedging Parameters, Trading Strategies and Swaps
Hedging parameters (the Greeks: Delta, Gamma, Theta, Rho and Vega), Trading strategies
involving options, Swaps, Mechanics of interest rate swaps, Comparative advantage
argument, Valuation of interest rate swaps, Currency swaps, Valuation of currency swaps.
Reference:
1. Hull, J. C., & Basu, S. (2010). Options, Futures and Other Derivatives (7th ed.).
Pearson Education. New Delhi.
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Additional Readings:
i. Luenberger, David G. (1998). Investment Science, Oxford University Press. Delhi.
ii. Ross, Sheldon M. (2011). An elementary Introduction to Mathematical Finance (3rd
ed.). Cambridge University Press. USA.

Teaching Plan (DSE-3 (i): Mathematical Finance):

Weeks 1 and 2: Interest rates, Types of rates, Measuring interest rates, Zero rates, Bond pricing,
Forward rate, Duration, Convexity.
[1] Chapter 4 (Section 4.1 to 4.4, 4.6, 4.8 and 4.9).
Weeks 3 and 4: Exchange traded markets and OTC markets, Derivatives- forward contracts, Futures
contract, Options, Types of traders, Hedging, Speculation, Arbitrage.
[1] Chapter 1 (Sections 1.1 to 1.9).
Week 5: No Arbitrage principle, Short selling, Forward price for an investment asset.
[1] Chapter 5 (Sections 5.2 to 5.4).
Week 6: Types of options, Option positions, Underlying assets, Factors affecting option prices.
[1] Chapter 8 (Sections 8.1 to 8.3), and Chapter 9 (Section 9.1).
Week 7: Bounds on option prices, Put-call parity, Early exercise, Effect of dividends.
[1] Chapter 9 (Sections 9.2 to 9.7).
Week 8: Binomial option pricing model, Risk neutral valuation (for European and American options
on assets following binomial tree model).
[1] Chapter 11 (Sections 11.1 to 11.5).
Weeks 9 to 11: Lognormal property of stock prices, Distribution of rate of return, expected return,
Volatility, estimating volatility from historical data. Extension of risk neutral valuation to assets
following GBM (without proof), Black−Scholes formula for European options.
[1] Chapter 13 (Sections 13.1 to 13.4, 13.7 and 13.8).
Week 12: Hedging parameters (the Greeks: Delta, Gamma, Theta, Rho and Vega).
[1] Chapter 17 (Sections 17.1 to 17.9).
Week 13: Trading strategies Involving options.
[1] Chapter 10 (except box spreads, calendar spreads and diagonal spreads).
Week 14: Swaps, Mechanics of interest rate swaps, Comparative advantage argument, Valuation of
interest rate swaps, Currency swaps, Valuation of currency swaps
[1] Chapter 7 (Sections 7.1 to 7.4 and 7.7 to 7.9).

Facilitating the Achievement of Course Learning Outcomes

Unit Course Learning Outcomes Teaching and Learning Assessment
o. Activity Tasks
1. Know the basics of financial markets and (i) Each topic to be • Student
derivatives including options and futures. explained with examples. presentations.
Learn about pricing and hedging of (ii) Students to be involved • Participation in
options, as well as interest rate swaps. in discussions and discussions.
2. Learn about no-arbitrage pricing concept encouraged to ask • Assignments
and types of options. questions. and class tests.
3. Learn stochastic analysis (Ito formula (iii) Students to be given • Mid-term
and Ito integration) and the homework/assignments. examinations.
Black−Scholes model. (iv) Students to be • End-term
4. Find the adjoint, normal, unitary and encouraged to give short examinations.
orthogonal operators. presentations.

Keywords: Black−Scholes model, Forward contracts, Futures contract, Options, Hedging,

Speculation, Arbitrage, Put-call parity, Short sellings, Swaps.
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Department of Mathematics, University of Delhi

DSE-4 (ii): Linear Programming and Applications

Total Marks: 100 (Theory: 75 and Internal Assessment: 25)
Workload: 5 Lectures, 1 Tutorial (per week) Credits: 6 (5+1)
Duration: 14 Weeks (70 Hrs.) Examination: 3 Hrs.

Course Objectives: This course develops the ideas underlying the Simplex Method for
Linear Programming Problem, as an important branch of Operations Research. The course
covers Linear rogramming with applications to transportation, assignment and game problem.
Such problems arise in manufacturing resource planning and financial sectors.
Course Learning Outcomes: This course will enable the students to:
i) Learn about the graphical solution of linear programming problem with two variables.
ii) Learn about the relation between basic feasible solutions and extreme points.
iii) Understand the theory of the simplex method used to solve linear programming
problems.
iv) Learn about two-phase and big-M methods to deal with problems involving artificial
variables.
v) Learn about the relationships between the primal and dual problems.
vi) Solve transportation and assignment problems.
vii) Apply linear programming method to solve two-person zero-sum game problems.

Unit 1: Introduction to Linear Programming

Linear programming problem: Standard, Canonical and matrix forms, Graphical solution;
Convex and polyhedral sets, Hyperplanes, Extreme points; Basic solutions, Basic feasible
solutions, Reduction of feasible solution to a basic feasible solution, Correspondence
between basic feasible solutions and extreme points.

Unit 2: Methods of Solving Linear Programming Problem

Simplex method: Optimal solution, Termination criteria for optimal solution of the linear
programming problem, Unique and alternate optimal solutions, Unboundedness; Simplex
algorithm and its tableau format; Artificial variables, Two-phase method, Big-M method.

Unit 3: Duality Theory of Linear Programming

Motivation and formulation of dual problem; Primal-Dual relationships; Fundamental
theorem of duality; Complimentary slackness.

Unit 4: Applications
Transportation Problem: Definition and formulation; Methods of finding initial basic feasible
solutions; Northwest-corner rule. Least- cost method; Vogel’s approximation method;
Algorithm for solving transportation problem.
Assignment Problem: Mathematical formulation and Hungarian method of solving.
Game Theory: Basic concept, Formulation and solution of two-person zero-sum games,
Games with mixed strategies, Linear programming method of solving a game.

References:
1. Bazaraa, Mokhtar S., Jarvis, John J., & Sherali, Hanif D. (2010). Linear
Programming and etwork Flows (4th ed.). John Wiley and Sons.
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2. Hadley, G. (1997). Linear Programming. Narosa Publishing House. New Delhi.

3. Taha, Hamdy A. (2010). Operations Research: An Introduction (9th ed.). Pearson.

Additional Readings:
i. Hillier, Frederick S. & Lieberman, Gerald J. (2015). Introduction to Operations
Research (10th ed.). McGraw-Hill Education (India) Pvt. Ltd.
ii. Thie, Paul R., & Keough, G. E. (2014). An Introduction to Linear Programming and
Game Theory. (3rd ed.). Wiley India Pvt. Ltd.

Teaching Plan (DSE-4 (ii): Linear Programming and Applications):

Week 1 : Linear programming problem: Standard, Canonical and matrix forms, Graphical solution.
[1] Chapter 1 (Section 1.1).
[2] Chapter 1 (Sections 1.1 to 1.4 and 1.6).
Weeks 2 and 3: Convex and polyhedral sets, Hyperplanes, Extreme points; Basic solutions, Basic
feasible solutions; Reduction of any feasible solution to a basic feasible solution; Correspondence
between basic feasible solutions and extreme points.
[2] Chapter 2 (Sections 2.16, 2.19 and 2.20), and Chapter 3 (Sections 3.4 and 3.10).
[1] Chapter 3 (Section 3.2).
Week 4: Simplex Method: Optimal solution, Termination criteria for optimal solution of the linear
programming problem, Unique and alternate optimal solutions, Unboundedness.
[1] Chapter 3 (Sections 3.3 and 3.6).
Weeks 5 and 6: Simplex algorithm and its tableau format.
[1] Chapter 3 (Sections 3.7 and 3.8).
Weeks 7 and 8: Artificial variables, Two-phase method, Big-M method.
[1] Chapter 4 (Sections 4.1 to 4.3).
Weeks 9 and 10: Motivation and formulation of dual problem; Primal-dual relationships.
[1] Chapter 6 (Section 6.1 and 6.2, up to Example 6.4).
Week 11: Statements of the fundamental theorem of duality and complimentary slackness theorem
with examples.
[1] Chapter 6 (Section 6.2).
Weeks 12 and 13: Transportation problem, Assignment problem.
[3] Chapter 5 (Sections 5.1, 5.3 and 5.4).
Week 14: Game Theory: Basic concept, Formulation and solution of two-person zero-sum games,
Games with mixed strategies, Linear programming method of solving a game.
[2] Chapter 11 (Sections 11.12 and 11.13).

Facilitating the Achievement of Course Learning Outcomes

Unit Course Learning Outcomes Teaching and Learning Assessment
o. Activity Tasks
1. Learn about the graphical solution of (i) Each topic to be • Student
linear programming problem with two explained with presentations.
variables. examples. • Participation in
Learn about the relation between basic (ii) Students to be involved discussions.
feasible solutions and extreme points. in discussions and • Assignments
2. Understand the theory of the simplex encouraged to ask and class tests.
method used to solve linear programming questions. • Mid-term
problems. (iii) Students to be given examinations.
Learn about two-phase and big-M homework/assignments. • End-term
methods to deal with problems involving (iv) Students to be examinations.
artificial variables. encouraged to give short

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Department of Mathematics, University of Delhi

3. Learn about the relationships between the presentations.

primal and dual problems.
4. Solve transportation and assignment
problems.
Apply linear programming method to
solve two-person zero-sum game
problems.

Keywords: Artificial variables, Big-M method, Duality, Extreme points and basic feasible
solutions, Simplex method, Two-phase method, Vogel’s approximation method.

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