m. a./ m. sc. in mathematics faculty of science second ...€¦ · faculty of science second...

M. A./M. Sc. in MATHEMATICS/Syllabus/IInd Semester/SU

M. A./ M. Sc. in MATHEMATICS FACULTY OF SCIENCE

SECOND SEMESTER (EVEN SEMESTER)

Eligibility

Criteria

(Qualifying

Exams)

Course

Code

Course

Type Course (Paper/Subjects) Credits

Contact

Hours Per

WeeK

EoSE

Duration

(Hrs.)

L T P Thy P

Aft

er a

pp

eari

ng i

n t

he

Fir

st

sem

este

r ex

am

inati

on

irre

spec

tive

of

an

y n

um

ber

of

back

/ arr

ear

pap

ers

MSM 201 CCC Abstract Algebra II 6 4 3 00 3 00

MSM 202 CCC Real Analysis II 6 4 3 00 3 0

MSM 203 CCC Topology II 6 4 3 00 3 0

MSM 221 PRJ/FST/EST SOCIAL OUTREACH AND SKILL DEVELOPMENT 6 00 00 9 00 4

MSM B01 ECC/CB ENVIRONMENTAL AND FOREST LAWS

6 4 3 00 3 00 MSM B02 ECC/CB Advanced Discrete Mathematics (II)

MSM B03 ECC/CB Algebraic Number Theory

MSM B04 ECC/CB Wavelets

TOTAL=

30

M. A./ M. Sc. in MATHEMATICS /Syllabus/IInd Semester/SU

M.A. / M. Sc. (MATHEMATICS) IIND SEMESTER

COURSE CODE: MSM 201 COURSE TYPE: CCC

COURSE TITLE: ABSTRACT ALGEBRA II

CREDIT:6

HOURS:90

THEORY: 6 PRACTICAL:0 THEORY: 90 PRACTICAL: 0

MARKS

THEORY: 100 (30+70) PRACTICAL:0

UN

IT-1

25 h

rs.

Modules - Cyclic modules. Simple modules. Semi-simple modules.

Schuler’s Lemma. Free modules. Noetherian and artinian modules and

rings-Hilbert basis theorem. Wedderburn Artin theorem. Uniform

modules, primary modules, and Noether-Lasker theorem.

UN

IT-2

15 h

rs. Linear Transformations - Algebra of linear transformation, characteristic

roots, matrices and linear transformations.

UN

IT-3

20 h

rs.

Canonical Forms - Similarity of linear transformations. Invariant

subspaces. Reduction to triangular forms. Nilpotent transformations.

Index of nilpotency. Invariants of a nilpotent transformation. The

primary decomposition theorem. Jordan blocks and Jordan forms.

UN

IT-4

18 h

rs.

Smith normal form over a principal ideal domain and rank. Fundamental

structure theorem for finitely generated modules over a Principal ideal

domain and its applications to finitely generated abelian groups.

UN

IT-5

12 h

rs. Rational canonical from. Generalised Jordan form over any field.


SU

GG

ES

TE

D

RE

AD

ING

S

1. P. B. Bhattacharya, S.K.Jain, S.R.Nagpaul : Basic Abstract Algebra, Cambridge University press 2. I. N. Herstein : Topics in Albegra, Wiley Eastern Ltd. 3. Quazi Zameeruddin and Surjeet Singh : Modern Algebra 4. M. Artin, Algeabra, Prentice -Hall of India, 1991. 5. P. M. Cohn, Algebra,Vols. I,II &III, John Wiley & Sons, 1982,1989,1991. 6. N. Jacobson, Basic Algebra, Vols. I & II,W.H. Freeman, 1980 (also published by Hindustan Publishing Company). 7. S. Lang, Algebra, 3rd edition, Addison-Wesley, 1993. 8. I. S. Luther and I.B.S. Passi, Algebra, Vol. I-Groups, Vol.II-Rings, Narosa Publishing House (Vol.l-1996,Vol. II-1999) 9. D. S. Malik, J.N.Mordeson, and M.K.Sen, Fundamentals of Abstract Algebra, Mc Graw-Hill, International Edition,1997. 10. K.B. Datta, Matrix and Linear Algebra, Prentice Hall of India Pvt. Ltd., New Delhi, 2000. 11. S. K. Jain, A. Gunawardena and P. B Bhattacharya, Basic Linear Algebra with MATLAB, Key College Publishing (Springer-Verlag),2001. 12. S. Kumaresan, Linear Algebra, A Geometric Approach, Prentice-Hall of India, 2000. 13. Vivek Sahai and Vikas Bist, Algebra, Narosa Publishing House, 1999.


M.A./ M. Sc. (MATHEMATICS) IIND SEMESTER


COURSE TITLE: REAL ANALYSIS-II

CREDIT:6

HOURS:90

THEORY: 6 PRACTICAL: 0 THEORY: 90 PRACTICAL: 0

MARKS

THEORY: 100 (30+70) PRACTICAL: 0

UN

IT-1

18 h

rs

Definition and existence of Riemann-Stieltjes integral, Properties of the

Integral, integration and differentiation, the fundamental theorem of

Calculus, integration of vector-valued functions, Rectifiable curves.

UN

IT-2

18 h

rs

Lebesgue outer measure. Measurable sets. Regularity. Measurable

functions. Borel and Lebesgue measurability. Non-measurable sets.

Integration of Non-negative functions. The General integral. Integration

of Series.

UN

IT-3

18 h

rs

Measures and outer measures, Extension of a measure. Uniqueness of

Extension. Completion of a measure. Measure spaces. Integration with

respect to a measure. Reimann and Lebesgue Integrals.

UN

IT-4

18 h

rs

The Four derivatives. Lebesgue Differentiation Theorem.

Differentiation

and Integration.

UN

IT-5

18 h

rs

Functions of Bounded variation. The Lp -spaces. Convex functions.

Jensen’s inequality. Holder and Minkowski inequalities. Completeness

of Lp, Convergence in Measure, Almost uniform convergence.


SU

GG

ES

TE

D

RE

AD

ING

S

1. Principle of Mathematical Analysis by W. Rudin 2. Real Analysis by H. L. Roydon 3. T.M. Apostol, Mathematical Analysis, Narosa Publishing House, New Delhi,1985. 4. Gabriel Klambauer, Mathematical Analysis, Marcel Dekkar,Inc. New York,1975. 5. A.J. White, Real Analysis; an introduction, Addison-Wesley Publishing Co.,Inc.,1968. 6. G.de Barra, Measure Theory and Integration, Wiley Eastern Limited, 1981. 6. E. Hewitt and K. Stromberg. Real and Abstract Analysis, Berlin, Springer, 1969. 7. P.K. Jain and V.P. Gupta, Lebesgue Measure and Integration, New Age International (P) Limited Published, New Delhi, 1986 Reprint 2000). 8. I.P. Natanson, Theory of Functions of a Real Variable. Vol. l, Frederick Ungar Publishing Co., 1961. 9. Richard L. Wheeden and Antoni Zygmund, Measure and Integral: An Introduction to Real Analysis, Marcel Dekker Inc.1977. 10. J.H. Williamson, Lebesgue Integration, Holt Rinehart and Winston, Inc. New York. 1962. 11. K.R. Parthasarathy, Introduction to Probability and Measure, Macmillan Company of India Ltd., Delhi, 1977. 12. Inder K. Rana, An Introduction to Measure and Integration, Norosa Publishing House, Delhi, 1997.




COURSE TITLE: Topology II

CREDIT:6

HOURS:90

THEORY: 6 PRACTICAL: 0 THEORY: 90 PRACTICAL:0

MARKS

THEORY: 100 (30+70) PRACTICAL:0

UN

IT-1

18 h

rs

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

characterizations. Projection maps. Separation axioms.

UN

IT-2

18 h

rs

Product spaces. Connectedness and product spaces. Compactness and

product spaces (Tychonoff’s theorem). Countability and product

spaces.

UN

IT-3

18 h

rs

Embedding and metrization. Embedding lemma and Tychonoff

embedding. The Urysohn metrization theorem. Metrization theorems

and Paracompactness-Local finiteness. The Nagata-Smirnov

metrization theorem. Paracompactness. The Smirnov metrization

theorem.

UN

IT-4

18 h

rs

Nets and filter. Topology and convergence of nets. Hausdorffness and

nets. Compactness and nets. Filters and their convergence. Canonical

way of converting nets to filters and vice-versa. Ultra-filters and

Compactness.

UN

IT-5

18 h

rs

The fundamental group and covering spaces-Homotopy of paths. The

fundamental group. Covering spaces. The fundamental group of the

circle and the fundamental theorem of algebra


SU

GG

ES

TE

D

RE

AD

ING

S

1. James R.Munkres, Topology, A First Course, Prentice Hall of India Pvt. Ltd., New Delhi,2000. 2. K.D.Joshi, Introduction to General Topology, Wiley Eastern Ltd., 1983. 3. J. Dugundji, Topology, Allyn and Bacon, 1966 (reprinted in India by Prentice Hall of India Pvt. Ltd.). 4. George F.Simmons, Introduction to Topology and modern Analysis, McGraw-Hill Book Company, 1963. 5. J. Hocking and G Young, Topology, Addison-Wiley Reading, 1961. 6. J. L. Kelley, General Topology, Van Nostrand, Reinhold Co., New York, 1995. 7. L. Steen and J. Seebach, Counter examples in Topology, Holt, Rinehart and Winston, New York, 1970. 8. W.Thron, Topologically Structures, Holt, Rinehart and Winston, New York,1966. 9. N. Bourbaki, General Topology Part I (Transl.),Addison Wesley, Reading, 1966. 10. R. Engelking, General Topology, Polish Scientific Publishers, Warszawa, 1977. 11. W. J. Pervin, Foundations of General Topology, Academic Press Inc. New York, 1964. 12. E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966. 13. S. Willard, General Topology, Addison-Wesley, Reading, 1970. 14. Crump W.Baker, Introduction to Topology, Wm C. Brown Publisher 15. M.J. Mansfield, Introduction to Topology, D.Van Nostrand Co. Inc. Princeton, N. J., 1963.



COURSE CODE : MSMB01 COURSE TYPE : ECC/CB

COURSE TITLE: ENVIRONMENTAL AND FOREST LAWS

CREDIT: 06

THEORY: 06

HOURS : 90

THEORY: 90

MARKS : 100

THEORY: 70 CCA : 30

OBJECTIVE:

- Understands the concept and place of research in concerned subject

- Gets acquainted with various resources for research

- Becomes familiar with various tools of research

- Gets conversant with sampling techniques, methods of research and techniques of analysis of data

- Achieves skills in various research writings

- Gets acquainted with computer Fundamentals and Office Software Package .

UN

IT -

1

1

8 H

rs

EVOLUTION OF FOREST AND WILD LIFE LAWS

a) Importance of Forest and Wildlife

b) Evolution of Forest and Wild Life Laws

c) Forest Policy during British Regime

d) Forest Policies after Independence.

e) Methods of Forest and Wildlife Conservation.

UN

IT -

2

18 H

rs

FOREST PROTECTION AND LAW

a) Indian Forest Act, 1927

b) Forest Conservation Act, 1980 & Rules therein

c) Rights of Forest Dwellers and Tribal

c) The Forest Rights Act, 2006

d) National Forest Policy 1988

UN

IT -

3

18 H

rs

WILDLIFE PROTECTION AND LAW

a) Wild Life Protection Act, 1972

b) Wild Life Conservation strategy and Projects

c) The National Zoo Policy


UN

IT -

4

18 H

rs

CHAPTER – BASIC CONCEPTS

a. Meaning and definition of environment.

b. Multidisciplinary nature of environment

c. Concept of ecology and ecosystem

d. Importance of environment

e. Meaning and types of environmental pollution.

f Factors responsible for environmental degradation.

CHAPTER– INTRODUCTION TO LEGAL SYSTEM

a. Acts, Rules, Policies, Notification, circulars etc

b. Constitutional provisions on Environment Protection

c. Judicial review, precedents

d. Writ petitions, PIL and Judicial Activism

CHAPTER – LEGISLATIVE FRAMEWORK FOR POLLUTION CONTROL LAWS

a) Air Pollution and Law.

b) Water Pollution and Law.

c) Noise Pollution and Law.

UN

IT -

5

1

8 H

rs

CHAPTER- LEGISLATIVE FRAMEWORK FOR ENVIRONMENT PROTECTION

a) Environment Protection Act & rules there under

b) Hazardous Waste and Law

c) Principles of Strict and absolute Liability.

d) Public Liability Insurance Act

e) Environment Impact Assessment Regulations in India

CHAPTER – ENVIRONMENTAL CONSTITUTIONALISM

a. Fundamental Rights and Environment

i) Right to Equality ……….Article 14

ii) Right to Information ……Article 19

iii) Right to Life …………..Article 21

iv) Freedom of Trade vis-à-vis Environment Protection

b. The Forty-Second Amendment Act

c. Directive Principles of State Policy & Fundamental Duties

d. Judicial Activism and PIL


SU

GG

ES

TE

D

RE

AD

ING

S

Bharucha, Erach. Text Book of Environmental Studies. Hyderabad : University Press (India)

Private limited, 2005.

Doabia, T. S. Environmental and Pollution Laws in India. New Delhi: Wadhwa and Company,

2005.

Joseph, Benny. Environmental Studies, New Delhi: Tata McGraw-Hill Publishing Company

Limited, 2006.

Khan. I. A, Text Book of Environmental Laws. Allahabad: Central Law Agency, 2002.

Leelakrishnan, P. Environmental Law Case Book. 2nd

Edition. New Delhi: LexisNexis

Butterworths, 2006.

Leelakrishnan, P. Environmental Law in India. 2nd

Edition. New Delhi: LexisNexis Butterworths,

2005.

Shastri, S. C (ed). Human Rights, Development and Environmental Law, An Anthology. Jaipur:

Bharat law Publications, 2006. Environmental Pollution by Asthana and Asthana, S,Chand Publication

Environmental Science by Dr. S.R.Myneni, Asia law House

Gurdip Singh, Environmental Law in India (2005) Macmillan.

Shyam Diwan and Armin Rosencranz, Environmental Law and Policy in India –

Cases, Materials and Statutes (2nd

ed., 2001) Oxford University Press.

JOURNALS :-

Journal of Indian Law Institute, ILI New Delhi.

Journal of Environmental Law, NLSIU, Bangalore.

MAGAZINES :-

Economical and Political Weekly

Down to Earth.


M.A./ M. Sc. (MATHEMATICS) II ND SEMESTER

COURSE CODE: MSM B02 COURSE TYPE: ECC/CB

COURSE TITLE: Advanced Discrete Mathematics (II)

CREDIT: 6

HOURS: 90


MARKS

THEORY:100 (30+70) PRACTICAL: 0

UN

IT-1

18 h

rs.

Graph Theory-Definition of (Undirected) Graphs, Paths, Circuits, Cycles, & Subgraphs. Induced Subgraphs. Degree of a vertex. Connectivity. Planar Graphs and their properties. Trees. Euler’s Formula for connected planar Graphs. Complete & Complete Bipartite Graphs. Kuratowski’s Theorem (statement only) and its use.

UN

IT-2

18 h

rs.

Spanning Trees, Cut-sets, Fundamental Cut -sets, and Cycle. Minimal Spanning Trees and Kruskal’s Algorithm. Matrix Representations of Graphs. Euler’s Theorem on the Existence of Eulerian Paths and Circuits.

UN

IT-3

18 h

rs.

Graphs. In degree and Out degree of a Vertex. Weighted undirected Graphs. Dijkstra’s Algorithm.. strong Connectivity & Warshall’s Algorithm. Directed Trees. Search Trees. Tree Traversals.

UN

IT-4

18 h

rs. Introductory Computability Theory-Finite State Machines and their

Transition Table Diagrams. Equivalence of finite State Machines. Reduced Machines. Homomorphism.

UN

IT-5

18 h

rs.

Finite Automata. Acceptors. Non-deterministic Finite Automata and equivalence of its power to that of Deterministic Finite Automata. Moore and mealy Machines. Turing Machine and Partial Recursive Functions.


SU

GG

ES

TE

D

RE

AD

ING

S

1. Elements of Discrete Mathematics By C.L.Liu 2. Graph Theory and its application By N. Deo 3. Theory of Computer Science By K.L.P.Mishra and N. Chandrashekaran 4. J.P. Tremblay & R. Manohar, Discrete Mathematical Structures with Applications to Computer Science, McGraw-Hill Book Co., 1997. 5. J.L. Gersting, Mathematical Structures for Computer Science, (3rd edition), Computer Science Press, New York. 6. Seymour Lepschutz, Finite Mathematics (International) edition 1983), McGraw-Hill Book Company, New York. 7. S.Wiitala, Discrete Mathematics-A Unified Approach, McGraw-Hill Book Co. 8. J.E. Hopcroft and J.D Ullman, Introduction to Automata Theory, Languages & Computation, Narosa Publishing House. 9. C.L Liu, Elements of Discrete Mathematics, McGraw-Hill Book Co. 10. N. Deo. Graph Theory with Application to Engineering and Computer Sciences. Prentice Hall of India.


M. A./ M.Sc. (MATHEMATICS ) II ND SEMESTER


COURSE TITLE: ALGEBRAIC NUMBER THEORY

CREDIT: 6

HOURS:90


MARKS

THEORY: 100(30+70) PRACTICAL: 0

UN

IT-1

18 h

rs.

ALGEBRAIC BACKGROUND Rings and Fields- Factorization of Polynomials - Field Extensions - Symmetric Polynomials - Modules - Free Abelian Groups. Chapter 1: Sec. 1.1 to 1.6

UN

IT-2

18 h

rs. ALGEBRAIC NUMBERS

Algebraic numbers - Conjugates and Discriminants - Algebraic Integers - Integral Bases - Norms and Traces - Rings of Integers. Chapters 2: Sec. 2.1 to 2.6

UN

IT-3

18 h

rs.

QUADRATIC AND CYCLOTOMIC FIELDS Quadratic fields and cyclotomatic fields : Factorization into Irreducibles : Trivial factorization - Factroization into irreducibles - Examples of non-unique factorization into irreducibles. Chapter 3: Sec. 3.1 and 3.2 ; Chapter 4: Sec. 4.2 to 4.4

UN

IT-4

18 h

rs.

Prime Factroization - Euclidean Domains - Euclidean Quadratic fields -

Consequences of unique factorization - The Ramanujan -Nagell Theorem.

UN

IT-5

18 h

rs.

Prime Factorization of Ideals - The norms of an Ideal - Non-unique Factorization in Cyclotomic Fields.. Chapter 5 : Sec. 5.2 to 5.4

SU

GG

ES

TE

D

RE

AD

ING

S

I. Steward and D.Tall. Algebraic Number Theory and Fermat’s Last Theorem (3rd Edition) A.K.Peters Ltd., Natrick, Mass. 2002.


M. A. /M.Sc(MATHEMATICS) IIND SEMESTER


COURSE TITLE: WAVELETES

CREDIT: 6

HOURS: 90


MARKS

THEORY: 100 (30+70) PRACTICAL: 0

UN

IT-1

25 h

rs.

AN OVERVIEW Fourier to Wavelets - Integral Wavelet Transform and Time-frequency analysis - Inversion formulas and duals - Classification of Wavelets - Multiresolution analysis - Splines and Wavelets. Fourier Analysis: Fourier and Inverse Fourier Transforms - Continuous time convolution - The delta fnction - Fourier Transform of square integrable functions. Chapter 1: Sections 1.1 to 1.6 Chapter 2: Sections 2.1 to 2.3

UN

IT-2

20 h

rs.

FOURIER ANALYSIS (CONTD) Fourier Series - Basic Convergence Theory - Poisson Summaton Formula. Wavelet Tranforms and Time Frequency Analysis: The Gabor Transform - Short time Fourier Transforms and the uncertainty principle - The integral Wavelet Transform - Dyadic Wavelets - Inversions - Frames - Wavelet Series. Chapter 2: 2.4 and 2.5 Chapter 3: Section 3.1 to 3.6

UN

IT-3

15 h

rs.

SCALING FUNCTIONS AND WAVELETS Multiresolution analysis - Scaling functions with finite two scale relation - Direction sum Decompositions of L2( R ) - Wavelets and their duals. Chapter 5: Sections 5.1 to 5.4 (omit 5.5 and 5.6)

UN

IT-4

15 h

rs. Prime Factroization - Euclidean Domains - Euclidean Quadratic fields -

Consequences of unique factorization - The Ramanujan -Nagell Theorem. Chapter 4: Sec. 4.5 to 4.9 M.Sc. Mathematics : Syllabus (CBCS) 33

UN

IT-5

15 h

rs.

CARDINAL SPLINE WAVELETS Interpolating splines wavelets - Compactly supported spline - Wavelets - Computation of Cardinal spline Wavelets - Euler - Frebenious Polynomials. Orthogonal Wavelets: Examples of orthogonal Wavelets - Identification of orthogonal two scale symbols - Construction of compactly supported orthogonal wavelets. Chapter 6 : Sections 6.1 to 6.4 (omit 6.5 and 6.6) Chapter 7: Sections 7.1 to 7.3 (omit 7.4 and 7.5)


SU

GG

ES

TE

D

RE

AD

ING

S

Recommended Text Charles K.Chui , An Introduction to Wavelets, Academic Press, New York, 1992. Reference Books 1. Chui. C.K. (ed) Approximation theory and Fourier Analysis, Academic Press Boston, 1991.

2. Daribechies,I. Wavelets, CBMS-NSF Series in Appl.. math. SIAM. Philadelphia, 1992.

3. Schumaker,L.L. Spline Functions: Basic Theory , Wiley, New York 1981.

4. Nurnberger, G. Applicatins to Spline Functions, Springer Verlag, New York. 1989.

5. Walnut,D.F. Introduction to Wavelet Analysis,Birhauser, 2004.

m. a./ m. sc. in mathematics faculty of science second ...€¦ · faculty of science second...

Documents