Multani Mal Modi College, Patiala Unit Planning M.Sc Mathematics

2019

-20

Department of Mathematics

Unit Planning

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Unit Planning

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MULTANI MAL MODI COLLEGE, PATIALA UNIT PLAN

Class – MSc I (Semester I) Mathematics Subject : Alge bra-I Subject Code: MM 401 Subject Teacher : Ms. Rajvinder Kaur Session : 2019-20

S.No. Syllabus/Topics Reference Mode of Transactions

Additional Resources*

July/August 2019

1

Review of Groups, Subgroups, Lagrange’s Theorem, Normal Subgroups, Cyclic groups, Quotient Group, permutation Group, Isomorphism theorems

1.Bhattacharya, Jain and Nagpaul: Basic Abstract Algebra(Chapter-4 and 5) 2.Khanna and Bhambri,A course in Abstract Algebra(Chapter 2 and 3)

Lecture, Discussion

2

Introducing the concept of simple group, maximal Normal Group and realtion between them, Subnormal Series, Normal Series, Composition Series

1.Bhattacharya, Jain and Nagpaul: Basic Abstract Algebra (Chapter-6) 2.Surjeet singh, QaziZameeruddin: Modern Algebra(Chapter-5)

Lecture, Hand Written notes

Assignment-1

3

Examples of composition series related to cyclic groups and permutation Group

Lecture, Discussions

4

ZassenHaus Lemma Jordan Holder Theorem For finite groups, Schreier’s refinement Theorem,Jordan Holder Theorem For General groups and Jordan Holder Theorem implies fundamental Theorem of Arithmetic

Lecture, Discussions and Hand Written notes

5 Assignment-1 Discussions Seminar/Discussions

September 2019

6

Solvable Groups ,Nilpotent Groups

1.Bhattacharya, Jain, Nagpaul: Basic AbstractAlgebra(Ch-6) 2.Surjeet singh, QaziZameeruddin: Modern Algebra(Chapter-5)

Lecture, Hand Written Notes

7

Permutation Groups, Cyclic Decomposition, Alternating group An, Simplicity of An, related

1.Bhattacharya, Jain and Nagpaul: Basic Abstract Algebra(Chapter4

Lecture/Discussions

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examples and 7)

8

Group action, Stabilizer, orbit, Class equation and its applications, Conjugacy classes in permutation groups.

1.Bhattacharya, Jain and Nagpaul: Basic Abstract Algebra(Chapter-5)

Lecture Video Lecture-1-By prof. Krishna

Hanumanthu,Chennai Mathematical

institute

9

Cauchy theorem for finite groups, Sylow –p-group, Sylow’s theorems and examples related to simplicity of groups, Groups of order p2, pq.

1.Bhattacharya, Jain and Nagpaul: Basic Abstract Algebra (Chapter-8) 2.Khanna and Bhambri, A course in Abstract Algebra(Chapter 4 and 5)

Lecture,Discussions Video Lecture-2

video Lecture-3 By prof. Krishna Hanumanthu,Chennai Mathematical institute

October 2019

10 Structure theory of

groups, Fundamental

theorem of finitely

generated abelian groups,

Invariants of a finite

abeliangroup,

Bhattacharya, Jain and Nagpaul: Basic Abstract Algebra(Chapter-8) Bhattacharya, Jain and Nagpaul: Basic Abstract Algebra(Chapter-8)

Lecture/Notes Assignment-2

11 GroupsofAutomorphisms

of cyclic groups

homo.between two cyclic

groups.

Lecture/Discussions

12 Review of Rings, Subrings,

Ideals, Quotient Rings,

Ring Homo and

Isomorphism Theorems

Bhattacharya, Jain and Nagpaul: Basic Abstract Algebra (Chapter-9 and 10)

Lecture

13 Assignment-2,Discussions Seminar/Discussions

November 2019

14 Algebra of Ideals, Maximal and prime ideals, Ideal in Quotient rings,

1.Bhattacharya, Jain and Nagpaul: Basic Abstract Algebra(Chapter-10) 2. Surjeetsingh, QaziZameeruddin: Modern Algebra(Chapter-7 and 8)

Lecture

15 Field of Quotients of integral Domain, Matrix Rings and their ideals; Rings of Endomorphisms of Abelian Groups.

Lecture/Discussions

16 Seminar and class tests on Important Topics

Seminar/Discussions

17 Discussion ( OLD University Papers and Problems)

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MULTANI MAL MODI COLLEGE, PATIALA

UNIT PLAN

Class – MSc I (Semester I) Mathematics

Subject: MATHEMATICAL ANALYSIS Subject Code:- MM 402 Subject Teacher :- Chetna Rani Gupta Session :- 2019-20

S.No. Syllabus/Topics Reference Mode of Transactions

Additional Resorces*

July/August 2019

1 Review of extended Real Number System

H.L.Royden,Real Analysis pearson 4thed(chapter 2&3)

Lecture, discussion

2 Algebras, σ- algebra, their properties

Lecture ,discussion

3

General measurable spaces, measure spaces, properties of measure, Complete measure

Lecture, discussion

video lecture by Prof I.K.Rana ,IIT Bombay

September 2019

4

Lebesgue outer measure and its properties measurable sets and Lebesque measure

H.L.Royden,Real Analysis pearson 4thed(chapter 2&3)

Lecture, discussion

5 A non measurable set

Lecture, discussion

Assignment1MA

6 Measurable function w.r.t. general measure.

Lecture, discussion

7 Borel and Lebesgue measurability.

Lecture, discussion

Assignment 2MA

Octuber 2019

8

Integration of non-negative measurable functions

H.L.Royden,Real Analysis pearson 4thed (Chapter 4,5&6)

Lecture,discussion

9 Fatou’s lemma, Monotone convergence theorem

Lecture, discussion

10

Lebesgue convergence theorem, The general integral, Integration of series, Riemann and lebesgue integrals.

Lecture, discussion

11 Differentiation; Vitalis Lemma, The Dini derivatives

Lecture, discussion

12 Functions of bounded

Lecture,

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variation discussion

13 Differentiation of an Integral

Lecture, discussion

14 Absolute Continuity

Lecture, discussion

15 Convex Fucntions and Jensen’s inequality

Lecture, discussion

Assignment 3 MA

November 2019

16

Linear transformations Walter Rudin, Principles of Mathematical Analysis ,third edition(Chapter 9)

Lecture, discussion

17 Derivatives in an open subset of Rn , Chain Rule

Lecture, discussion

18 Partial derivatives, Interchange of the order of differentiation

Lecture, discussion

19 Derivatives of higher orders,Taylor’s theorem

Lecture, discussion

20

Inverse function theorem

Lecture, discussion

video lecture by Prof Sudipta Dutta IIT kanpur

21 Implicit function theorem.

Lecture, discussion

Assignment 4 MA

22 Seminar on Important Topics

23 Discussion ( Previous University Papers and Problems)

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MULTANI MAL MODI COLLEGE, PATIALA

UNIT PLAN Class – M.Sc.- I (Semester I) Mathematics

Subject : Topology-1 Subject Code: MM-403 Subject Teacher : Dr.Chetna Session : 2019-20 S.No. Syllabus Covered Suggested Reading/

Reference Books

Mode of Transactions

Additional Resources*

July/August 2019

1

Cardinals: Equipotent sets, Countable and Uncountable sets, Cardinal Numbers and their Arithmetic, Bernstein’s Theorem and the Continumm Hypothesis.

W.J. PervinGeneralTopology, Ch. 2,5 James Dugundji; James Munkres TOPOLOGY, Ch. 3,4,5

Lecture

2

Topological Spaces: Definition and examples, Euclidean spaces as topological spaces.

Lecture PPT

3

Basis for a given topology W.J. Pervin Foundations of General Topology, Ch. 5 James Dugundji; James Munkres TOPOLOGY, Ch. 3

Lecture

4

Topologizing of Sets; Sub-basis, Equivalent Basis.

W.J. Pervin Foundations of General Topology, Ch. 2,5 James Dugundji; James Munkres TOPOLOGY, Ch. 3,4,5

Lecture

5

Examples discussed Discussion/Seminar Vedio Lecture by Prof Veeramani; IIT Madras.

September 2019

6

Elementary Concepts, Topologizing with pre-assigned elementary operations. Relativization, Subspaces.

W.J. Pervin Foundations of General Topology, Ch. 2,5 James Dugundji; James Munkres TOPOLOGY, Ch. 3,4,5

Lecture

Assignment

7

Maps and Product Spaces: Continuous Maps, Characterization of Continuity, Continuity at a point.

W.J. Pervin : Foundations of General Topology. Ch-2,5

Lecture

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8

Piecewise definition of Maps and Nhd finite families.

James Dugundji; James Munkres TOPOLOGY, Ch. 3,4,5

Lecture

9

Open Maps and Closed Maps,

Homeomorphisms and

Embeddings.

James Dugundji; James Munkres TOPOLOGY, Ch. 3,4,5

Lecture

10 Assignment Discussion Lecture

October 2019

11

Cartesian Product Topology, Elementary Concepts in Product Spaces, Continuity of Maps in Product Spaces and Cartesian Products.

James Dugundji; James Munkres TOPOLOGY, Ch. 3,4,5

Lecture

12

Connectedness: Connectedness and its characterizations, Continuous image of connected sets, Connectedness of Product Spaces.

W.J. Pervin Foundations of General Topology, Ch. 2,5.Munkres TOPOLOGY, Ch. 3,4,5

Lecture

Vedio Lecture by Prof veeramani; IIT, Madras

13

Applications to Euclidean spaces. Components, Local Connectedness and Components, Product of Locally Connected Spaces. Path Connectedness.

W.J. Pervin Foundations of General Topology, Ch. 2,5.Munkres TOPOLOGY, Ch. 3,4,5

Lecture

November 2019

14

Compactness and Countable Compactness, Local Compactness T0, T1, and T2 spaces, T2 spaces and Sequences One-Point Compactification.

W.J. Pervin Foundations of General Topology, Ch. 2,5. Munkres TOPOLOGY, Ch. 3,4,5

Lecture

15

Axioms of Countablity and Separability, Equivalence of Second axiom Separable and Lindelof in Metric Spaces. Equivalence of Compact and Countably Compact Sets in Metric Spaces.

W.J. Pervin Foundations of General Topology, Ch. 2,5. Munkres TOPOLOGY, Ch. 3,4,5

Lecture

16 Discussion ( OLD University Papers and Problems)

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MULTANI MAL MODI COLLEGE, PATIALA

UNIT PLAN Class – MSc I (Semester I) Mathematics

Subject :- Differential Geometry Subject Code:- MM 710 Subject Teacher :- Dr. Varun Jain Session :- 2019-20

S.No. Syllabus Covered Reference Mode of Transactions

Additional Resources*

JULY/AUGUST2019

1 Basic definition and review of Curves in the planes and in space.

Andrew Pressley, Elementary Differential Geometry

Chapter 1

Lecture, Discussion

Assignment-I

2 Definition and Arc length, reparametrization, curvature

Lecture

SEPTEMBER2019

3 Serret-Frenet formulae.space curves, torsion,

Andrew Pressley, Elementary Differential Chapter 2

Chapter 4

Lecture,

lectures of Prof. Hari Shankar Mahto, Department of Mathematics , IIT Kharagpur (NPTL) Lecture-1,

Lecture-2

4 osculatingcircles,evolutes and involutes of curves

Lecture

5 TheoryofSurfaces, smooth surfaces Lecture

6 Tangents, Normals and Orientability Lecture

7 Quadric surfaces, Chapter 5 Lecture

8 First and the second fundamental theorem

Chapters 6 and 7 Lecture

9 Euler’s theorem, Rodrigue’sformula, Gaussian Curvature

Andrew Pressley, Elementary Differential Geometry Chapter 8

Lecture

10

Gauss map ,Gaussian and mean curvatures

Lecture

11 The pseudosphere, flat surfaces Lecture

12 Surfaces of constant mean curvature Lecture

(Assignment-I Discussion)

OCTOBER2019

13 Basic properties of Geodesic,

Geodesic equations

Andrew Pressley, Elementary Differential Geometry

Chapter 9

Lecture/ Seminar

Assignment-II

14 Geodesics of surfaces of revolution Lecture

15 Geodesics as shortest paths, geodesic

coordinates.

Lecture

16 The Gauss and Codazzi–Mainardi equations, Gauss’ remarkable theorem

Andrew Pressley, Elementary Differential Geometry

Chapter 10

Lecture

17 Surfaces of constant Gaussian curvature, Geodesic mappings

Lecture/ Seminar

18 Minimal Surface and Examples Andrew Pressley Chapter 12

Lecture 19 Plateau’s problem Lecture

20 Assignment -2 Discussion

NOVEMBER2019

22 Gauss map of a minimal surface Andrew Pressley, Elementary

Lecture 23 Conformal parametrization of minimal Lecture

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surfaces Differential Geometry

Chapter 12 24

Minimal surfaces and holomorphic functions

Lecture

25 Seminar on Important Topics Lecture and Discussion

26 OLD University Papers and Problems

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Semester-II

Unit Planning

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MULTANI MAL MODI COLLEGE, PATIALA UNIT PLAN

Class – MSc I (Semester II) Mathematics

Subject : Alge bra-II Subject Code: MM 501

Subject Teacher : Ms. Rajvinder Kaur Session : 2019-20 S.No. Syllabus/Topics Reference Mode of

Transactions

Additional Resources*

January 2020

1

Review of topics related to ring theory, Characteristic of a ring, Integral Domain

1.Bhattacharya, Jain and Nagpaul: Basic Abstract

Algebra(Chapter-11)

2.Surjeet Singh, QaziZameeruddin(Chapter-

10)

3.Khanna ,Bhambri, A course In Abstract algebra(Chapter-8)

Lecture, Discussion

2

Divisibility, Associates, Irreducible element, Prime Element in Integral Domain, examples and counter Examples generalization of integers to other domains

Lecture, Discussions

Assignment-1

3 Principle Ideal Domain Lecture,

4

Factorization Domain, necessary condition to be FD

Lecture, Discussions

5

Unique Factorization Domain, examples and counter Examples

Lecture

6

Lemma Developing Relation Between UFD and PID

1.Bhattacharya, Jain and Nagpaul: Basic Abstract

Algebra(Chapter-11)

2.Surjeet Singh, QaziZameeruddin(Chapter-

10) 3.Khanna ,Bhambri, A

course In Abstract algebra(Chapter-8)

Lecture, Discussions

7

Euclidean Domain and related examples and

Lecture, Discussions Video Lecture-1By Matthew Macauley, Clemson University

8

Polynomial Rings, Polynomial Ring over UFD

Lecture

February 2021

9

Gauss Lemma and Gauss Theorem, Rings of Fractions.

1.Bhattacharya, Jain and Nagpaul: Basic Abstract

Algebra(Chapter-11 and 12)

2.Surjeet Singh, QaziZameeruddin(Chapter-

10) 3.Khanna ,Bhambri, A

course In Abstract

Lecture, Hand Written Notes

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algebra(Chapter-8)

10

Review of topics Vector Spaces Over fields, introducing the concept of Modules over Ring

1.Musili C., Introduction to Rings and Modules(Chapter-

5)

2.Bhattacharya, Jain and Nagpaul: Basic Abstract

Algebra(Chapter-14)

Lecture,

11

Definitions and examples of modules,

1.Musili C., Introduction to Rings and Modules(Chapter-

5)

2.Bhattacharya, Jain and Nagpaul: Basic Abstract

Algebra(Chapter-14)

Lecture, Hand written Notes,Discussions

12

Review the concept of opposite ring, rings of endomorphisms and related Theorem

Lecture

13

Submodules, union and intersection of submodules, Linear Sum and Direct Sum of Submodules

Seminar/Discussions Video Lecture-2By Prof. A.V. Jayanthan,IIT Madras

14

Free Modules ,Vector Spaces, Relation Between Vector Spaces and modules and related examples

Lecture/Discussions

15

Quotient Modules, Homomorphisms

Lecture/Discussions Video Lecture-3 By Prof. A.V. Jayanthan,IIT Madras

16 Simple Modules and Modules over PID

Lecture

March 2020

17 Artinian Modules and Artinian Rings

1.Musili C., Introduction to Rings and Modules(Chapter-

6)

2.Bhattacharya, Jain and Nagpaul: Basic Abstract

Algebra(Chapter-19)

Lecture/Notes Assignment-

2

18 Noetherian Modules and Noetherian Rings

Lecture/Notes

19 Review of the concept Composition Series of

Lecture

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groups and Jordan Holder Theorem, (Assignment-2)

20 Modules of finite Length

1.Musili C., Introduction to Rings and Modules(Chapter-

6) 2.Bhattacharya, Jain and Nagpaul: Basic Abstract

Algebra(Chapter-19)

Lecture/Discussions

21 Hilbert basis Theorem Lecture

22 Cohen Theorem 1.Musili C., Introduction to Rings and Modules(Chapter-6) 2.Bhattacharya, Jain and Nagpaul: Basic Abstract Algebra(Chapter-19)

23 Assignment-2 Discussions

Seminar/Discussions

April 2020

24 Radical Ideal,Nil Radical

1.Musili C., Introduction to Rings and Modules(Chapter-6) 2.Bhattacharya, Jain and Nagpaul: Basic Abstract Algebra(Chapter-19)

Lecture

25 Jacobson Radical,Nakayama Lemma

Lecture/Discussions

26 Radical of an Artinian Ring

Lecture

27 Radical of Polynomial Rings R[x], R commutative

Lecture

28 Seminar and class tests on Important Topics

Seminar/Discussions

29 Discussion ( OLD University Papers and Problems)

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MULTANI MAL MODI COLLEGE, PATIALA UNIT PLAN

Class – M.Sc.- I (Semester II) Mathematics

Subject : Topology-2 Subject Code: MM-502

Subject Teacher : Dr.Chetna Sharma Session : 2019-20 S.No. Syllabus Covered Suggested

Reading/ Reference Books

Mode of Transactions

Additional Resources*

January 2020

1

Introduction, Regular, Completely Regular, Normal and Completely Normal Spaces.

W.J. Pervin : Foundations of General Topology,Sec-2.3 to 2.5 James Dugundji : TOPOLOGY, Chapter 6,7

Lecture

Video Lecture by Prof Veeramani; IIT Madras.

2

Metric Spaces as Completely Normal T2 Spaces in Higher Separation Axioms.

Lecture Video Lecture by Prof Veeramani; IIT Madras.

3 Urysohns Lemma and The Tietze Extension Theorem.

W.J. Pervin : Foundations of General Topology. James Dugundji : TOPOLOGY, Chapter 6,7

Lecture

4 Examples with respect to different spaces.

Lecture Assignment 1

5

Products of first countable, Regular, T2 and Completely Regular Spaces.

Discussion/Seminar Assignment 2

February 2020

6

Non invariance of normality under products.

W.J. Pervin : Foundations of General Topology. James Dugundji : Chapter 6,7 TOPOLOGY

Lecture

7

Embedding of Tichonov spaces into parallelotope and the Stone Cech Compactification

W.J. Pervin : Foundations of General Topology. James Dugundji : Chapter 6,7 TOPOLOGY

Lecture

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*(Demonstration/case study/suggested reading links/images/animations/pdf/ppt)

8

Filter and filterbase, convergence and clustering, filter characterization of closure

Lecture

9

continuity and filter

convergence, ultrafilters,

filter characterization of

compactness and the

Tychonoff Theorem.

W.J. Pervin : Foundations of General Topology. James Dugundji : TOPOLOGY, Chapter 6,7

Lecture

10 Assignment –I Discussion Lecture

March-2020

11

Categories: Definition and Examples, The Arrow Category, Congruence in a Category

Joseph J. Rotman: An Introduction to Algebraic Topology, Chapter 0 and 1

Lecture

Video Lecture by Steven Roman

12

Quotient Category, Functors, Duality, Contravariance and Duality, Homotopy as Congruence in Top, The Category hTop

Joseph J. Rotman: An Introduction to Algebraic Topology, Chapter 0 and 1

Lecture

13

homotopy equivalence, nullhomotopy, convexity, contractibility and cones, the path component functor, invariance of path components under homotopy type.

Joseph J. Rotman: An Introduction to Algebraic Topology, Chapter 0 and 1

Lecture

Assignment 3

April 2021

14

Identification Topology, Identification Map, Subspaces, General Theorem

James Dugundji : TOPOLOGY, Chapter 6,7

Lecture

15

Transgression, Transitivity Spaces with Equivalance Relation, Quotient Spaces.

James Dugundji : TOPOLOGY, Chapter 6,7

Lecture

16 Discussion ( OLD University Papers and Problems)

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MULTANI MAL MODI COLLEGE, PATIALA UNIT PLAN

Class – MSc I (Semester II) Mathematics Subject : Differential Equations- I Subject Code: MM 503 Subject Teacher : Dr.AnuBala Session : 2019-20

S.No. Syllabus Covered Suggested Reading/ Reference Books

Mode of Transactions

Additional Resources*

January 2020

1

Existence of solution of ODE of first order, initial value problem.

1.E. Coddington& N. Levinson, Theory of Ordinary Differential Equations, Tata Mc-Graw Hill, India. (Chapter 1) 2. A.C. King, J. Billingham, S.R. Otto, Differential Equations, Linear, Nonlinear, Ordinary, Partial, Cambridge University Press. (Chapter 8) 3. S.L. Ross,

Differential Equations, 3rd edition, John Wiley & sons (Asia). (Chapter

Lecture, Discussion

2

Ascoli’s Lemma, Gronwall’s inequality, Cauchy Peano Existence Theorem, Uniqueness of Solutions.

Lecture, Discussion

3

Method of successive approximations, Existence and Uniqueness Theorem.

Lecture, Discussion

Assignment –I

4

System of differential equations, nth order differential equation, Existence and Uniqueness of solutions.

Lecture, Discussion

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10)

February 2020

5

Dependence of solutions on initial conditions, Dependence of solutions on parameters.

S..L. Ross, Differential Equations, 3rd edition, John Wiley & sons Chapter -11

Lecture, Discussion

6

Linear system of equations (homogeneous & non homogeneous). Superposition principle, Fundamental set of solutions, Fundamental Matrix, Wronskian.

S.L. Ross, Differential Equations, 3rd edition, John Wiley & sons Chapter -11

Lecture, Discussion

7

Abel Liouville formula, Reduction of order.

Lecture, Discussion

Video Lecture by Prof Srinivasa Rao Manam ( IIT Madras)

March 2020 8 Adjoint systems and

self adjoint systems of second order.

S..L. Ross, Differential Equations, 3rd edition, John Wiley & sons (Asia). (Chapter 11, 12)

Lecture,

Discussion

9 Floquet Theory. Linear 2nd order equations.

Lecture,

Discussion

10 Sturm’s separation theorem, Sturm’s fundamental comparison theorem.

Lecture,

Discussion

April 2020

11 Sturm Liouville boundary value problem.

S.L. Ross,

Differentia

Lecture,

Discussion

Video Lecture by

Prof

Dr.BhaskarDasgupta

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MULTANI MAL MODI COLLEGE UNIT PLAN

CLASS MSc I-(semester II) Mathematics

Subject :- Functional AnalysisSubject Code:- MM-504

Subject Teacher :- Chetna Rani GuptaSession :- 2019-20

S.No. Syllabus Covered Reference Mode of Transactions

Additional Resources

January 2020

1

Review and Basics of Vector Spaces and Metric Spaces

George F,SimmonsTopology and modern Analysis (Chapter 9)

Lecture, Discussion, ICT

PPT slide Share Elementary Linear Algebra By R. larsen (6th Edition)

2

Normed Linear spaces, Banach spaces, Examples of Banach spaces and subspaces.

Lecture, discussion

3 Continuity of Linear maps

Lecture, discussion

February 2020

4 Equivalent Norms. George Lecture, Video

l Equations, 3rd edition, John Wiley & sons (Asia). (Chapter 12)

( IIT Kanpur)

12 Characteristic values & Characteristic functions, Orthogonality of Characteristic functions.

Lecture,

Discussion

Assignment –II

13 Expansion of a function in a series of orthonormal functions.

Lecture,

Discussion

14 Seminar on Important Topics

Lecture,

Discussion

15 Discussion ( Old University Question Papers and Problems)

Lecture,

Discussion

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Normed spaces of bounded linear maps.,Bounded Linear functional

F,SimmonsTopology and modern Analysis (Chapter 9)

discussion Lecture-1 Video

Lecture -2

by Prof P.D. Srivastava ,IIT Kharagpur

5

Hahn-Banach theorem

in Normed linear

spaces and its

applications

Lecture, discussion

6 Uniform boundedness

principle Lecture, discussion

7 Open mapping theorem Lecture,

discussion

Video

lecture by Prof P.D. Srivastava ,IIT Kharagpur

8

Projections on Banach

spaces

Lecture, discussion

9 Closed graph theorem.

Lecture, discussion

Assignment 1 FA

March 2020

10

The conjugate of an operator. Dual spaces of lp and C[a,b], Reflexivity

GeorgrF,SimmonsTopology and modern Analysis ( Chapter 10 )

Lecture, discussion

video

lectureby Prof P.D. Srivastava ,IIT Kharagpur Video

lecture by Prof P.D. Srivastava ,IIT Kharagpur

11 Hilbert spaces, examples

Lecture, discussion

12

Orthogonality, Orthonormal sets,Bessel's inequality, Parseval's theorem.

lecture , discussion

13 The conjugate space of a Hilbert spaces.

Lecture, discussion

14 Adjoint operators, Self-adjoint operators,

lecture , discussion

15 Normal and unitary operators.

lecture , discussion

16 Projection operators. lecture ,

discussion

Assignment 2 FA

April 2020

17

Review and Basics ofmatrices,eigen values ,eigen vectors

GeorgrF,SimmonsTopology and modern Analysis( Chapter 11) and Appendix one) Balmohan V LimayeFunctional Analysis

New Age International Publishers(Appendix A)

lecture , discussion

18

Spectrum of an operator, Spectral Theorem,

lecture , discussion

19 Banach Fixed Point Theorem

lecture , discussion

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20 Brower's Fixed Point Theorem

lecture , discussion

21 Schauder Fixed Point Theorem

lecture , discussion

22 Picards Theorem lecture ,

discussion

23

Applications of Fixed point theorem in differential equations and integral equations.

lecture , discussion

Assignment 3 FA

24 Seminar on Important Topics

25 Discussion ( Previous University Papers and Problems)

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MULTANI MAL MODI COLLEGE, PATIALA UNIT PLAN

Class – MSc I (Semester II) Mathematics

Subject : Complex Analysis Subject Code: MM 505

Subject Teacher : Dr. Varun Jain Session : 2019-20

S.No. Syllabus Covered Reference Mode of

Transactions

AdditionalResources*

January 2020

1

Review of Complex Numbers, Argand plane, Roots of Complex numbers Function of complex variable.

L.V.Ahlfors, Complex Analysis,Chapter 1& 2

H.S.Kasana, Complex Variables chapter-3

Lecture, Discussion ICT

2

Function of Complex variables , Examples and Real and Imaginary parts of various Complex Fnx.

Lecture,

February 2020

3

Definition of Analytic Function, Singularities, Examples, Necessary and Sufficient condition.

H.S. Kasana, Complex Variables Chapter-2

Lecture, PDF Hand written Notes

Video Lectureby

Prof V.Balakrish

anan

( IIT Madras)

4 C-R equations, M.T. Method

Lecture

5 Problems of finding Analytic function when real or imaginary Parts is given or vice versa

Lecture

6 Harmonic Functions and Harmonic Conjugates and related Theorems.

L.V.Ahlfors, Complex Analysis, Chapter -4

H.S. Kasana, Complex Variables Chapter-4

Lecture

Assignment –I

7

Complex Integration, Cauchy Goursat theorem Cauchy integral formula

Lecture/

PDF Hand Written Notes

8 Cauchy integral formula of Derivatives , Problems

Drill / Discussion

9

Morera’s theorem, Liouville's theorem, Fundamental theorem of Algebra.

Lecture/ PDF Hand Written Notes

10

Maximum Modulus Principle. Schwarz lemma

Lecture/ Hand Notes

March 2020

11 Taylor’s Theorem Proof and questions

L.V.Ahlfors, Complex Analysis, Chapter -4

H.S. Kasana, Complex Variables

Chapter-7 & 8

Lecture/ PDF Hand Notes /Drill

12 Laurentz Theorem proof and Questions

13

Types of Singularities, Residue at Various singularities. (Assignment -2)

Lecture/ PDF Hand Written Notes

14

Cauchy’s theorem on residues and its application to four types of integrals.

Lecture/ PDF Hand Written Notes

15

General definition and Principle of analytic continuation.

L.V.Ahlfors, Complex Analysis,Chapter -8

Lecture/PDF Hand written Notes

PPT

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*Demonstration/case study/suggested reading links/images/animations/pdf/ppt etc.

MULTANI MAL MODI COLLEGE, PATIALA UNIT PLAN

Class – MSc II (Semester III) Mathematics Subject :- Differentiable Manifolds Subject Code:- MM 601 Subject Teacher :- Dr. Varun Jain Session :- 2019-20

S.No. Syllabus Covered Reference Mode of

Transactions Additional

Resources

July /August 2019

1

Definition Differentiable

Manifolds, examples of

differentiable manifolds.

U.C. De : Differential

Geometry of Manifolds

Chapter 2

Lecture,

Assignment-1

2 Differentiable maps on manifolds Lecture

3 Tangent vectors and tangent space Lecture

4 Cotangent space. Lie-bracket of

vector fields. Lecture

5 Jacobian map, pull back map, Lecture

6 Tensors, Exterior product, Forms U.C. De : Differential

Geometry of Manifolds

Chapter 3

Lecture Lectures for Reference by Robert Davie Lecture-1, 2

7

Exterior derivative, Contraction Lecture

September 2019

16 Analytic continuation by power series method

L.V.Ahlfors, Complex Analysis,Chapter -8

Lecture PPT

17 Harmonic Function on a Disc Lecture

18 Natural boundary, Schwarz Reflection L.V.Ahlfors, Complex

Analysis,Chapter -8

Lecture

19 Mittag-Leffler’s theorem L.V.Ahlfors, Complex

Analysis (Page-185) Lecture

20 Branches of multivalued functions (

cz ,arg

z, logz)

L.V.Ahlfors, Complex Analysis, Chapter -3

H.S. Kasana, Complex Variables Chapter-9

Lecture

Video Lecture 1

21

Definition of Conformal, isogonal, Translational, Magnification, Inversion and Rotational mapping and related questions

Lecture/ PPT

April 2020

22

Cross ratio, Bilinear Mapping and related question

L.V.Ahlfors, Complex Analysis, Chapter -3

H.S. Kasana, Complex Variables Chapter-9

Lecture

Video Lecture 2

23 Seminar on Important Topics

Lecture & Discussions

24 Discussion ( OLD University Papers and Problems)

Unit Planning

25 | P a g e

8 Lie-derivative. Affine Connection, U.C. De : Differential

Geometry of Manifolds

Chapter-4

Lecture

9 Difference tensor, Lecture

10 Covariant derivative of tensors. Lecture

11 Torsion tensor and curvature tensor

of a connection, U.C. De : Differential

Geometry of Manifolds

Chapter-6

Lecture

12 Properties of torsion and curvature

tensor, Lecture

Assignment-1- Discussions Problem

session

October 2019

13 Bianchi's identities, the

Riemannian metric, U.C. De : Differential

Geometry of Manifolds

Chapter-7

Lecture/

Seminar

( Assignment -2)

14 Riemannian manifolds, Lecture

15 Fundamental theoremof

Riemannian geometry, Lecture

16 Riemannian connection, Christoffel

symbols, Lecture

17 Riemannian curvature tensor and

its properties.

Lecture/

Seminar

18 Sectional curvature, Thm of Schur. Lecture

November 2019

19

Sub-manifolds and hyper-surfaces,

induced connection, U.C. De: Differential

Geometry of Manifolds

Chapter-8

Lecture Basic lectures for more understanding by Robert Davie Lecture -3 20

Assignment -2 Discussion

22

Gauss and Weingarten formulae

and their applications

U.C. De : Differential

Geometry of Manifolds

Chapter-8

Lecture

25 Seminar on Important Topics Lecture and

Discussions

26 OLD University Papers and

Problems

Discussion

MULTANI MAL MODI COLLEGE, PATIALA

UNIT PLAN Class – MSc II (Semester III) Mathematics

Subject : Field Theory Subject Code: MM 602

Subject Teacher : Ms. Rajvinder Kaur Session : 2019-20

S.N

o.

Syllabus/Topi

cs

Reference Mode of

Transactions

Additional

Resources*

July/August2019

Unit Planning

26 | P a g e

1

Review of

topics

:Fields,Irreduci

ble element,

Prime Element,

PID, UFD,

Irreducible

Polynomials,

Polynomial

Ring Gauss

Lemma

1.Bhattacharya, Jain and Nagpaul: Basic Abstract Algebra(Chapter-15)

2.Surjeet Singh,

QaziZameeruddin(Chapter-13)

3. I.N.Herstein(Chapter-5)

Lecture,

Discussion

Video Lecture

By Matthew

Macauley,Cle

mson

University

2

Eisenstein

Criterion

,Examples

related to

check

irreduciblity of

Polynomials

over Rational

Numbers

Lecture,

Discussions

3

Field

Extensions,

Degree of

field, Tower

theorem,

Adjuction of

roots,

Kronecker

Theorem,

Algebraic

Extensions,

Numerical

problems to

find degree of

algebraic

extension

Lecture/Notes Video Lecture-1By

Matthew

Macauley,Cle

mson

University

4

Algebraically

closed Fields,

Algebraic

Closure and

theorems

related to

embedding of

field into an

algebraically

closed field

Lecture,

Discussions Video Lecture-2 By

Matthew

Macauley,Cle

mson

University

September 2019

6

Splitting

Fields,

Uniqueness of

splitting Field

and Numerical

1.Bhattacharya, Jain and Nagpaul: Basic Abstract Algebra(Chapter-16)

2.SurjeetSingh,QaziZameeruddin

(Chapter-13 and 14)

Lecture,

Discussions Assignment-1

Unit Planning

27 | P a g e

problems to

find the

splitting Field

7

Normal

Extensions,

Seperable

Extensions,

Inseperable

Extensions

Lecture

8

Perfect Field,

Finite fields,

Primitive

Elements,

Lagrange’s

theorem on

Primitive

Elements,Num

erical problems

related to

primitive

element

Lecture,

Discussions

9

Automorphism

Groups and

Fixed Fields:

Dedekind

Lemma,

Numerical

problems

related to

Galois Group

and Fixed field

1.Bhattacharya, Jain and Nagpaul: Basic Abstract Algebra(Chapter-17)

3.I.N.Herstein(Chapter-5)

Lecture

10 Assignment-1

Discussions

Seminar/Discus

sions

October 2019

11 Galois

extensions,

Fundamental

theorem of

Galois theory,

Fundamental

theorem of

algebra

1.Bhattacharya, Jain and Nagpaul: Basic Abstract Algebra(Chapter-17 and 18)

2.SurjeetSingh,QaziZameeruddin

(Chapter- 14)

Lecture/Notes

12 Roots of unity

and cyclotomic

polynomials.

Cyclic

extension

Lecture

13 Radical

extension,

Lecture

Unit Planning

28 | P a g e

Solvability by

Radicals

14 Problems

Related to

Galois Theory

and Solvability

by Radicals

Lecture/Discuss

ions

Assignment-2

November 2019

15 Symmetric

functions,

cyclotomic

extension,

Ruler and

Compass

construction

1.Bhattacharya, Jain and Nagpaul: Basic Abstract Algebra(Chapter-18) 2.Surjeet Singh,

QaziZameeruddin(Chapter- 14)

Lecture

16 Assignment-2

Discussions

Seminar/Discus

sions

17 Quintic

equation and

solvability by

radicals

1.M. Artin:Algebra(Chapter16) Lecture

18 Seminar and

class tests on

Important

Topics

Seminar/Discus

sions

19 Discussion (

OLD

University

Papers and

Problems)

Unit Planning

29 | P a g e

MULTANI MAL MODI COLLEGE, PATIALA UNIT PLAN

Class – M.Sc.- II (Semester III) Mathematics

Subject : Differential Equations-II Subject Code:-

MM-603

Subject Teacher :- Dr.Chetna Session :- 2019-20 S.No. Syllabus Covered Suggested

Reading/

Reference Books

Mode of

Transactions Additional

Resources*

July/August 2019

1 Review of Linear order

differential equations E.Coddington&

N. Levinson,

Theory of

Ordinary

Differential

Equations, Tata

Mc-Graw Hill;

ch-1

ch-2

Lecture

video Lecture through Nptel

2

Existence and uniqueness

of solutions of first order

differential equations for

complex systems.

Lecture Assignment –I

3 Maximum and minimum

solution Lecture

4 Caratheodory theorem. Lecture

5 Continuation of solution. Discussion/

Seminar

September 2019

6

Uniqueness of solutions E.Coddington&

N. Levinson,

Theory of

Ordinary

Differential

Equations; ch-1,2

Lecture

7

Method of Successive

approximations. E.Coddington&

N. Levinson,

Theory of

Ordinary

Differential

Equations, Tata

Mc-Graw Hill;

ch-1,2

Lecture

8 Problems discussed Lecture

8 Problems discussed Lecture

9 Variation of Solutions. Lecture

10 Assignment –Discussion Lecture

October 2019

Unit Planning

30 | P a g e

11

Review of Partial

Differential Equations Sneddon I.N.,

Elements of

Partial

Differential

Equations, ch-4

Lecture

Vedio Lecture Through Nptel

12

Occurrence and elementary

solution of Laplace

equation. Family of Equi-

potential surface.

Sneddon I.N.,

Elements of

Partial

Differential

Equations, ch-4

Lecture

13

Interior and exterior Dirichlet

boundary value problem for

Laplace equation.

Sneddon I.N.,

Elements of Partial

Differential

Equations, ch-4

Lecture

November 2019

14

Separation of Variables. Axial

symmetry, Kelvin’s inversion

theorem.

Sneddon I.N.,

Elements of Partial

Differential

Equations, ch-4

Lecture

Assignment

15

Green’s function for Laplace

equation. Dirichlet’s problem

for semi-infinite space and for a

sphere. Copson’s Theorem

(Statement only)

Sneddon I.N.,

Elements of Partial

Differential

Equations, ch-4

Lecture

16 Discussion ( OLD University

Papers and Problems)

Unit Planning

31 | P a g e

MULTANI MAL MODI COLLEGE, PATIALA UNIT PLAN

Class – MSc II (Semester III) Mathematics

Subject : Classical Mechanics-I Subject Code: MM 607

Subject Teacher : Dr.AnuBala Session : 2019-20 S.No. Syllabus Covered Suggested

Reading/

Reference

Books

Mode of

Transactions Additional

Resources*

July-August 2019

1

Basic Principles:

Mechanics of a Particle

and a System of Particles,

Constraints.

Herbert

Goldstein:

Classical

Mechanics.

(Chapter 1)

Lecture,

Discussion

2

Generalized Coordinates,

Holonomic and Non-

Holonomic Constraints.

Lecture,

Discussion

Video Lecture by

Prof.Debmalya

Banerjee ( IIT

Kharagpur)

3

D’Alembert’s Principle

and Lagrange’s Equations,

Velocity Dependent

Potentials and the

Dissipation Function.

Lecture,

Discussion Video Lecture by

Prof.Debmalya

Banerjee ( IIT

Kharagpur)

Video Lecture by

Prof.Debmalya

Banerjee ( IIT

Kharagpur)

4

Simple Applications of the

Lagrangian formulation.

Lecture,

Discussion

5

Variational Principles

and Lagrange’s

Equations: Hamilton’s

Principle, Derivation of

Lagrange’s Equations

from Hamilton’s

Principle.

Herbert

Goldstein:

Classical

Mechanics.

(Chapter 2)

Lecture,

Discussion

6.

Extension of Hamilton’s

Principle to Non-

Holonomic Systems.

September 2019

Unit Planning

32 | P a g e

7

Conservation Theorems

and Symmetry

Properties: Cyclic

Coordinates, Canonical

Momentum and its

Conservation, The

Generalized Force, and

Angular Momentum

Conservation Theorem.

Herbert

Goldstein:

Classical

Mechanics.

(Chapter 2)

Lecture,

Discussion

8

The Two-Body Central

Force Problem:

Reduction to the

Equivalent One-Body

Problem,

Herbert

Goldstein:

Classical

Mechanics.

(Chapter 3)

Lecture,

Discussion

9

The Equation of Motion,

The Equivalent One

Dimensional Problem and

the Classification of

Orbits,

Lecture,

Discussion

10

The Virial Theorem,

Conditions for Closed

Orbits, Bertrand’s

Theorem.

Lecture,

Discussion

October 2019

11 The Kepler Problem:

Inverse Square Law of

Force, The Motion in

Time in the Kepler

Problem.

Herbert

Goldstein:

Classical

Mechanics.

(Chapter 3)

Lecture,

Discussion

12 Kepler’s Laws, Kepler’s

Equation, The Laplace-

Runge-Lenz Vector.

Lecture,

Discussion

Assignment –I

13 Scattering in a Central

Force Field: Cross

Section of Scattering,

Rutherford Scattering

Cross Section.

Lecture,

Discussion

14 Total Scattering Cross

Section, Transformation

of the Scattering Problem

to Laboratory

Coordinates.

Lecture,

Discussion

November 2019

Unit Planning

33 | P a g e

15 The Kinematics of Rigid

Body Motion: The

Independent Coordinates

of Rigid Body, The

Transformation Matrix,

The Euler Angles, The

Cayley-Klein Parameters

and Related Quantities,

Herbert

Goldstein:

Classical

Mechanics.

(Chapter 4)

Lecture,

Discussion

16 Euler’s Theorem on the

Motion of Rigid Bodies,

Finite Rotations,

Infinitesimal Rotations,

The Coriolis Force.

Lecture,

Discussion

17 Seminar on Important

Topics

Lecture,

Discussion

18 Discussion ( Old

University Question

Papers and Problems)

Lecture,

Discussion

MULTANI MAL MODI COLLEGE, PATIALA UNIT PLAN

Unit Planning

34 | P a g e

Class – MSc II (Semester III) Mathematics Subject :-OPTIMIZATION TECHNIQUES-ISubject Code:-MM 609

Subject Teacher :- Chetna Rani Gupta Session :- 2019-20

S.No. Syllabus Covered Reference Mode of

Transactions Additional

Resources*

July/August2019

1

Introduction: Definition of operation research Models in operation research,

General methods for solving

O.R. models Elementary theory of convex

sets.

KantiSwarup,P.K.

Gupta,ManMohan:operationResearch,Sultan

Chand &Sons (chapter 0&1)

Lecture, discussion

Video lecture LPP by Prof Kusumdeep Gupta IIT Roorkee

Viedo lecture Dualility Prof Kusumdep Gupta IIT Rorkee

Assignment :Convex

2

Linear Programming

Problems: Definition of LPP

Examples of LPPs Mathematicalformulation of the

Mathematical programming

problems

Graphical solution of the

problem

KantiSwarup,P.K.

Gupta,ManMohan:operationResearch,Sultan

Chand &Sons (chapter 2&3)

Assignment: LPP, Graphic

September 2019

3

Simplex method

Big M method Two Phase method

problem of degeneracy

KantiSwarup,P.K.

Gupta,ManMohan:operationResearch,Sultan

Chand &Sons (chapter 4)

Assignment :Simplex

4

Duality in linear

programming Concept of duality Fundamental properties of

duality Dualitytheoremscomplementary

slackness theoremduality and

simplex method dual simplex method

KantiSwarup,P.K.

Gupta,ManMohan:operationResearch,Sultan

Chand &Sons (chapter 5)

Lecture, discussion

Assignment :Duality

Assignment: Dual Simplex

5

Sensitivity Analysis: Discrete changes in the cost

vector

Discrete changes inrequirement

KantiSwarup,P.K.

Gupta,ManMohan:operationResearch,Sultan

Chand &Sons (chapter 6)

Lecture,

discussion Assignment : Sestivity Analysis

Unit Planning

35 | P a g e

vector Discrete changes in co-

efficientmatrix

Addition of a new variable Deletion of a variable Addition of new constraint Deletion of aconstraint.

6

Integer Programming: Introduction Gomory's all-IPP method Gomory's mixed-integer

method Branch and Bound method.

KantiSwarup,P.K.

Gupta,ManMohan:operationResearch,Sultan

Chand &Sons (chapter 7)

Lecture, discussion

Assignment :IPP

October 2019

7

Transportation Problem: Introduction, mathematical formulation of

the problem Initial basic feasiblesolution

using North West Corner Method

,Least Cost Method and

Vogel'sApproximation Method

Optimal solution using MODI

method

Degeneracy in transportation

problems

Some exceptional cases in

transportation problems

KantiSwarup,P.K.

Gupta,ManMohan:operationResearch,Sultan

Chand &Sons (chapter 10)

Lecture, discussion

video lecture Transportation by Prof Kusumdeep Gupta IIT Rorkee

8

Assignment Problems: Introduction

Mathematical formulation of an

assignment problem Assignment algorithm

Unbalanced assignment

problems

Travelling Salesman problem

KantiSwarup,P.K.

Gupta,ManMohan:operationResearch,Sultan

Chand &Sons (chapter 11 )

Lecture, discussion

Video lecture Assignment by Prof Kusumdeep Gupta IITRorkee

Assignment :Transportation &Assignment

November 2019

9

Games & Strategies:

Definition & characteristics of

Games Two person zero sum games

Maximin-minimax principle Games without saddle points

Mixed Strategies

Graphical methodfor solving

and games,

KantiSwarup,P.K.

Gupta,ManMohan:operationResearch,Sultan

Chand &Sons (chapter 17)

Lecture,

discussion

Video lecture Game theory by Prof Kusumdeep Gupta IIT Rorkee

Assignment

Unit Planning

36 | P a g e

SEMESTER-IV

Concept of Dominance Reducing the game problem to

LPP Limitations

:Game

10 Seminar on Important Topics

11

Discussion ( Previous

University Papers and

Problems)

Unit Planning

37 | P a g e

MULTANI MAL MODI COLLEGE, PATIALA UNIT PLAN

Class – MSc II (Semester IV) Mathematics

Subject : Algebraic Coding Theory Subject

Code: MM 709 Subject Teacher : Ms. Rajvinder Kaur

Session : 2019-20 S.No. Syllabus/Topics Reference Mode of

Transactions Additional

Resources*

January 2020

1

Introduction to

communication

system, source

coding, channel

coding and related

examples

1.San Ling and

Chaoping Xing-

Coding

Theory,(Chapter-

I and II) 2. Raymond Hill-

Introduction to

Error Correcting

Codes,(Chapter-I

)

Lecture, Discussion

2

Communicatiuon

Channels, Binary

Symmetric

Channel

Lecture,

Discussions PPT By Prof. Josef

Gruska

3

Decoding rule

:Maximum

Likelihood

decoding,

Hamming Distance

and distance of

code

Lecture,

4

Decoding rule

:Nearest

Neighbour

Decoding,

Relation Between

two decoding rules

over BSC

Lecture, Discussions

February 2020

5

Review of Field

theory, Finite

Fields, Assignment

to check previous

knowledge on

Fields

1.San Ling and

Chaoping Xing-

Coding

Theory,(Chapter-

III) 2. Raymond Hill-

Introduction to

Error Correcting

Codes,(Chapter-

III )

Lecture, Hand

Written Notes

Assignment-1

6

Polynomial Rings,

Structure of finite

fields, Examples

related to structure

of finite fields

Lecture,

7

Minimal

polynomials and

Cyclotomiccosets

,examples related

Lecture, Discussions

Unit Planning

38 | P a g e

to topic

8

Numerical

problems related to

primitive element,

minimal

polynomials and

cyclotomiccoset

Lecture

9

Review of topic

Vector Spaces

over finite fields,

Introduction to

Linear Codes,

Dual Codes

1.San Ling and

Chaoping Xing-

Coding

Theory,(Chapter-

IV) 2. Raymond Hill-

Introduction to

Error Correcting

Codes,(Chapter-

IV To VII)

Seminar/Discussions

10

Hamming weight

and realtion

between distance

of a code and

weight ,algorithms

to find bases of

linear code and

related numerical

problems

Lecture/Discussions

11

Algorithm to find

bases of dual code,

Generator Matrix,

Parity Check

Matrix,equivalence

of linear code

Lecture/Discussions Video Lecture-1By Prof.Adrish Banerjee, IIT Kanpur

12

Encoding and

decoding with

linear codes,

Assignment-2 on

Linear codes

Lecture Assignment-2

13 Syndrome

Decoding

Video Lecture-2 By Prof.AdrishBanerjee,IIT Kanpur

14

Numerical

problems related to

encoding and

decoding of linear

codes

March 2020

15 ISBN codes and

related examples Raymond Hill-

Introduction to

Error Correcting

Codes,(Chapter-

III)

Lecture/Notes

Unit Planning

39 | P a g e

16 New codes from

old W. C. Huffman

and Vera Pless –

Fundamentals of

Error Correcting

Codes(Chapter-

I)

Lecture/Notes

17 Assignment-2

Discussions

Seminar/Discussions

18 Main coding

theory problem,

Lower bounds,

Sphere Covering

Bound and sphere

packing bound

San Ling and

Chaoping Xing-

Coding

Theory(chapter-

V)

Lecture/Discussions

19 Gilbert Varshamov

bound Lecture

20 Binary Hamming

codes,q-ary

hamming code

San Ling and

Chaoping Xing-

Coding

Theory(Chapter-

V)

Lecture

21 Decoding with

binary and q-ary

hamming codes

and related

examples

Seminar/Discussions

22 Golay Codes,

Properties of

Extended Binary

Golay code

23 Singleton bound

and MDS codes,

April 2020

24 Simplex Codes,

Plotkin bound San Ling and

Chaoping Xing-

Coding

Theory(Chapter-

V)

Lecture

25 Griesmer bound,

Linear

Programming

bounds

Lecture/Discussions

26 Propagation Rules,

Reed-Muller

codes,.

San Ling and

Chaoping Xing-

Coding

Theory(Chapter-

VI)

Lecture

27 Seminar and class

tests on

Important Topics

Seminar/Discussions

28 Discussion ( OLD

University Papers

and Problems)

Unit Planning

40 | P a g e

MULTANI MAL MODI COLLEGE, PATIALA UNIT PLAN

Class – MSc II (Semester IV) Mathematics

Subject :- Commutative Algebra Subject Code:- MM 710

Subject Teacher :- Dr. Varun Jain Session :- 2019-20

S.No. Syllabus Covered Reference Mode of

Transactions Additional

Resources

JANUARY-2020

1 Review and Basic definition of Ring

Ideals Field,PID,Zero Divisors M.F. Atiyah, I.G

MacDonald :

Introduction to

Commutative

Algebra Chapter 1

Lecture, Discussion

Assignment-I

2

Nil radical and Jacobson radical of

Ring and theorems

Lecture

FEBRUARY-2020

3 Operation Of Ideals, Comaximal

Ideals, Prime Avoidence Lemma M.F. Atiyah, I.G

MacDonald :

Introduction to

Commutative

Algebra Chapter 1

Lecture, .

4 Colon Ideal, Radical of Ideal

Anhilators and related theorems Lecture

5 Extensions and Contraction of Ideals ,

definition and Related Theorems

Lecture

6 Zariski Topology, Assignment Lecture

7 Assessment Discussion

8 Review of Basic Definitions of

Modules and sub Modules

M.F. Atiyah, I.G

MacDonald :

Introduction to

Commutative

Algebra Chapter 2

Lecture,

Discussion

Assignment-II

9 Operation on Modules, Related Thm. Lecture

10 Nakayama lemma, Module Homo.,

First fundamental Theorem. Lecture

11 Direct product and Direct Sum, Free

Module and Related Theorems

Lecture

12 Exact Sequences, Related theorem Lecture

13 Tensor Product of Modules, maps and

its Exactness

Lecture

MARCH-2020

14 Restrictions and Extensions of Scalars

M.F. Atiyah, I.G

MacDonald :

Introduction to

Commutative

Algebra Chapter 2

and 3

Lecture/ Seminar

15 Multiplicative closed subset and

related theorems

Lecture

16 Module of fractions Lecture

17 Extended and contracted ideals in ring

of fractions

Lecture

18 Examples Lecture/ Seminar

19 Primary Ideals Lecture

20 Primary decompositions of Ideals,

Assignment

Lecture

21 Assignment -2 Discussion

APRIL-2020

22 Primary ideals in ring of Fractions, M.F. Atiyah, I.G Lecture

Unit Planning

41 | P a g e

First Uniqueness Theorem MacDonald :

Introduction to

Commutative

Algebra Chapter 4

23 Isolated prime ideals, Second

Uniqueness Theorem

Lecture/ PPT

24 Behavior of primary ideals under

localozation Lecture

25 Seminar on Important Topics

Lecture and

Discussions

26 Discussion ( OLD University Papers

and Problems)

Unit Planning

42 | P a g e

MULTANI MAL MODI COLLEGE, PATIALA UNIT PLAN

Class – MSc II (Semester IV) Mathematics

Subject : Operations Research

Subject Code: MM 711 Subject Teacher : Dr.AnuBala

Session : 2019-20 S.No. Syllabus Covered Suggested

Reading/

Reference Books

Mode of

Transactions Additional

Resources*

January 2020

1

Inventory Models:

Introduction, Costs

involved in inventory

problems, variables in

inventory problems.

1. Sharma, S.D:

Operation

Research,

KedarNath and

Co., Meerut.

(Chapter 20, 21) 2. KantiSwarup,

P.K. Gupta and

Man Mohan:

Operations

Research, Sultan

Chand and Sons.

(Chapter 19)

Lecture,

Discussion

Video Lecture by

Prof G. Srinivasan (

IIT Madras)

2

Classification of inventory

models, deterministic

inventory model, (DIM),

Basic economic order

quantity, (EOQ).

Lecture,

Discussion

Video Lecture by

Dr.Inderdeep Singh

( IIT Roorkee)

3 Models with no shortages:

Model I(a), I(b), I(c).

Lecture,

Discussion

4 DIM with shortages:

Model II(a), II(b), II(c). Lecture,

Discussion

5

Multi item deterministic

inventory models: Models

III(a), III(b) III(c).

Lecture,

Discussion Assignment –I

6. Introduction to Stochastic

inventory models.

February 2020

7

Queueing Problems:

Characteristics of

queueing system.

Distribution in queueing

systems.

1. Sharma, S.D:

Operation

Research,

KedarNath and

Co., Meerut.

(Chapter 23) 2. KantiSwarup,

P.K.Gupta and

Man

Mohan:Operations

Research, Sultan

Chand and Sons.

(Chapter 21)

Lecture,

Discussion

Video Lectureby

Prof G. Srinivasan (

IIT Madras)

8

Poission arrivals and

exponential Service time.

Transient and steady state.

Lecture,

Discussion

9

Probabilistic Queueing

Models (Model I

(M/M/1)(∞/FCFS), Model

II A(General Erlang

Queueing Model), Model

II B(M/M/1):(∞/SIRO),

Measures and their

solutions.

Lecture,

Discussion

Video Lectureby

Prof G. Srinivasan (

IIT Madras)

10

Model III (M/M/1) :

(N/FCFS) , Model-IV

(M/M/S):(∞/FCFS),

Model V (M/M/S);

Lecture,

Discussion

Video Lectureby

Prof G. Srinivasan (

IIT Madras)

Unit Planning

43 | P a g e

March 2020

(N/FCFS), Measures and

their solutions.

11

Model VI A(M/ Ek/1):(

∞/FCFS),Model VIB(M/

Ek/1):( 1/FCFS),

Measures and their

solutions.

Lecture,

Discussion Assignment –II

Unit Planning

44 | P a g e

12 Replacement &

Maintenance Problems: Replacement policy when

money value changes.

1. Sharma, S.D:

Operation

Research,

KedarNath and

Co., Meerut.

(Chapter 22) 2. KantiSwarup,

P.K. Gupta and

Man Mohan:

Operations

Research, Sultan

Chand and Sons.

(Chapter 18)

Lecture,

Discussion

13 Replacement policy when

money value does not

change with time.

Lecture,

Discussion

14 Group replacement of item

that fails suddenly.

Lecture,

Discussion

15 The general renewal

process. Lecture,

Discussion

16 Network Analysis: Introduction to Networks,

Minimal Spanning Tree

problem

1. Kasana H.S.

and Kumar, K.D.:

Introductory

Operation

Research,

Springer.

(Chapter 8) 2. KantiSwarup,

P.K. Gupta and

Man Mohan:

Operations

Research, Sultan

Chand and Sons.

(Chapter 24)

Lecture,

Discussion

17 Shortest path problems,

Dijkastra’s algorithm,

Floyd’s Algorithm.

Lecture,

Discussion

18 Maximum Flow problem.

Lecture,

Discussion

April 2020

19 Project Management:

Critical Path method,

critical path computations.

1. Sharma, S.D:

Operation

Research,

KedarNath and

Co., Meerut.

(Chapter 25) 2. KantiSwarup,

P.K. Gupta and

Man Mohan:

Operations

Research, Sultan

Chand and Sons.

(Chapter 25)

Lecture,

Discussion

20 optimal Scheduling by

CPM, Project Cost

Analysis.

Lecture,

Discussion

21 PERT, Distinction.

Lecture,

Discussion

22 Seminar on Important

Topics

Lecture,

Discussion

23 Discussion ( Old University

Question Papers and

Problems)

Lecture,

Discussion

Unit Planning

45 | P a g e

Unit Planning

46 | P a g e

MULTANI MAL MODI COLLEGE, PATIALA UNIT PLAN

Class – MSc II (Semester IV) Mathematics

Subject : Operations Research

Subject Code: MM 711 Subject Teacher : Dr.AnuBala

Session : 2019-20 S.No. Syllabus Covered Suggested

Reading/

Reference Books

Mode of

Transactions Additional

Resources*

January 2020

1

Inventory Models:

Introduction, Costs

involved in inventory

problems, variables in

inventory problems.

1. Sharma, S.D:

Operation

Research,

KedarNath and

Co., Meerut.

(Chapter 20, 21) 2. KantiSwarup,

P.K. Gupta and

Man Mohan:

Operations

Research, Sultan

Chand and Sons.

(Chapter 19)

Lecture,

Discussion

Video Lecture by

Prof G. Srinivasan (

IIT Madras)

2

Classification of inventory

models, deterministic

inventory model, (DIM),

Basic economic order

quantity, (EOQ).

Lecture,

Discussion

Video Lecture by

Dr.Inderdeep Singh

( IIT Roorkee)

3 Models with no shortages:

Model I(a), I(b), I(c).

Lecture,

Discussion

4 DIM with shortages:

Model II(a), II(b), II(c). Lecture,

Discussion

5

Multi item deterministic

inventory models: Models

III(a), III(b) III(c).

Lecture,

Discussion Assignment –I

6. Introduction to Stochastic

inventory models.

February 2020

7

Queueing Problems:

Characteristics of

queueing system.

Distribution in queueing

systems.

1. Sharma, S.D:

Operation

Research,

KedarNath and

Co., Meerut.

(Chapter 23) 2. KantiSwarup,

P.K.Gupta and

Man

Mohan:Operations

Research, Sultan

Chand and Sons.

(Chapter 21)

Lecture,

Discussion

Video Lectureby

Prof G. Srinivasan (

IIT Madras)

8

Poission arrivals and

exponential Service time.

Transient and steady state.

Lecture,

Discussion

9

Probabilistic Queueing

Models (Model I

(M/M/1)(∞/FCFS),

Model II A(General

Erlang Queueing Model),

Model II

B(M/M/1):(∞/SIRO),

Measures and their

solutions.

Lecture,

Discussion

Video Lectureby

Prof G. Srinivasan (

IIT Madras)

10

Model III (M/M/1) :

(N/FCFS) , Model-IV

(M/M/S):(∞/FCFS),

Lecture,

Discussion

Video Lectureby

Prof G. Srinivasan (

Unit Planning

47 | P a g e

Model V (M/M/S);

(N/FCFS), Measures and

their solutions.

IIT Madras)

11

Model VI A(M/ Ek/1):(

∞/FCFS),Model VIB(M/

Ek/1):( 1/FCFS),

Measures and their

solutions.

Lecture,

Discussion Assignment –II

March 2020

12 Replacement & Maintenance

Problems: Replacement

policy when money value

changes.

1. Sharma, S.D:

Operation Research,

KedarNath and Co.,

Meerut. (Chapter

22) 2. KantiSwarup,

P.K. Gupta and

Man Mohan:

Operations

Research, Sultan

Chand and Sons.

(Chapter 18)

Lecture,

Discussion

13 Replacement policy when

money value does not change

with time.

Lecture,

Discussion

14 Group replacement of item

that fails suddenly. Lecture,

Discussion

15 The general renewal process. Lecture,

Discussion

16 Network Analysis: Introduction to Networks,

Minimal Spanning Tree

problem

1. Kasana H.S. and

Kumar, K.D.:

Introductory

Operation Research,

Springer. (Chapter

8) 2. KantiSwarup,

P.K. Gupta and

Man Mohan:

Operations

Research, Sultan

Chand and Sons.

(Chapter 24)

Lecture,

Discussion

17 Shortest path problems,

Dijkastra’s algorithm, Floyd’s

Algorithm.

Lecture,

Discussion

18 Maximum Flow problem.

Lecture,

Discussion

April 2020

19 Project Management: Critical

Path method, critical path

computations.

1. Sharma, S.D:

Operation Research,

KedarNath and Co.,

Meerut. (Chapter

25) 2. KantiSwarup,

P.K. Gupta and

Man Mohan:

Lecture,

Discussion

20 optimal Scheduling by CPM,

Project Cost Analysis.

Lecture,

Discussion

Unit Planning

48 | P a g e

21 PERT, Distinction.

Operations

Research, Sultan

Chand and Sons.

(Chapter 25)

Lecture,

Discussion

22 Seminar on Important

Topics

Lecture,

Discussion

23 Discussion ( Old University

Question Papers and

Problems)

Lecture,

Discussion

Unit Planning

49 | P a g e

MULTANI MAL MODI COLLEGE, PATIALA UNIT PLAN

Class – M.Sc.- I (Semester II) Mathematics

Subject :- MATHEMATICAL METHODS Subject

Code:- MM-716

Subject Teacher :- Dr.Chetna Session :- 2019-

20 S.No. Syllabus Covered Suggested

Reading/

Reference Books

Mode of

Transactions Additional

Resources*

January 2020

1

Linear integral equations

of first and second kind,

Abel’s problem

M.D. Rai

Singhania,

Integral

equations &

BVP,ch-1

Lecture

2

Relation between linear

differential equation and

Volterra’s equation

Lecture

3

Non linear and Singular

equations, Solution by

successive substitutions

M.D. Rai

Singhania,

Integral

equations &

BVP,ch-1,2,5

Lecture video Lecture

4 Problems discussed. Lecture

5

Volterra’s equation

iterated and reciprocal

functions, Volterra’s

solution of Fredholm’s

equation.

Discussion/Seminar

February 2020

6

Fredholm’s equation as

limit of finite system of

linear equations,

Fredholm’s two

fundamental relations

M.D. Rai

Singhania,

Integral

equations &

BVP,ch-3,4

Lecture

7

Hadamard’s theorem,

convergence proof M.D. Rai

Singhania,

Integral

equations &

BVP,ch-3,4

Lecture

8

Fredholm’s solution of

integral equation when

D()0,Fredholm’s

solution of Dirichlet’s

problem and Neumann’s

problem.

M.D. Rai

Singhania,

Integral

equations &

BVP,ch-5

Lecture Assignment

9

Lemmas on iterations of

symmetric kernel,

Schwarz’s inequality and

its applications

M.D. Rai

Singhania,

Integral

equations &

BVP,ch-6,7

Lecture

10 Assignment -Discussion Lecture

Unit Planning

50 | P a g e

March 2020

11

Simple variational

problems, Necessary

condition for an

extremum, Euler’s

equation, End point

problem.

Pundir;

Pragatiprakashan;

ch-1

Lecture

12

Variational derivative,

Invariance of Euler’s

equation, Fixed end point

problem for nunknown

functions, Variational

problem in parametric form,

Functionals depending on

higher order derivatives

Pundir;

Pragatiprakashan;

ch-1

Lecture

Video Lecture by IIT Kanpur

13

Variational derivative,

Invariance of Euler’s

equation, Fixed end point

problem for nunknown

functions

Pundir;

Pragatiprakashan;

ch-1

Lecture

April 2020

14

Variational problem in

parametric form, Functionals

depending on higher order

derivatives, Geodesics

Pundir;

Pragatiprakashan;

ch-1

Lecture

Assignment

15

Brachistochrone from a

given curve to a fixedpoint,

Snell’s law, Fermat’s

principle and calculus of

variations.

Pundir;

Pragatiprakashan;

ch-1,2

Lecture

16 Discussion ( OLD University

Papers and Problems)