## R Syllabus TYBSc BA Mathematics Syllabus Mumbai University by munotes

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AC – 17/05/2022

Item No. 6.9 (R)

UNIVERSITY OF MUMBAI

Revised Syllabus for T.Y.B. Sc./B.A.

(Mathematics)

Sem – V & VI

(Choice Based Credit System)

(With effect from the academic year 2022 -23)

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Dean (Science and Technology)

Prof. Anuradha Majumdar (Dean, Science and Technolo gy)

Prof. Shivram Garje (Associate Dean, Science)

Chairperson Board of Studies of Mathematics

Prof. Vinayak Kulkarni

Members of the Board of Studies of Mathematics

Prof. R. M. Pawale

Prof. P. Veeramani

Prof. S. R. Ghorpade

Prof. Ajit Diwan

Dr. S. Aggarwal

Dr. Amul Desai

Dr. S. A. Shende

Dr. Shridhar Pawar

Dr. Sanjeevani Gharge

Dr. Abhaya Chitre

Dr. Mittu Bhattacharya

Dr. Sushil Kulkarni

Dr. Rajiv Sapre

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CONTENTS

1. Preamble

2. Aims and Objectives

3. Programme Outcomes

4. Course Outcomes

5. Course structure with minimum credits and Lectures/ Week

6. Teaching Pattern for semester V & VI

7. Scheme of Evaluation

8. Consolidated Syllabus for semester V & VI

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1. Preamble

The University of Mumbai has brought into force the revised syllabi as p er the Choice Based

Credit System (CBCS) for the Third year B. Sc / B. A. Programme in Mathe matics from the

academic year 2022-2023. Mathematics has been fundamental to the development of science and

technology. In recent decades, the extent of application of Mathematics to real world problems

has increased by leaps and bounds. Taking into consideration the rapid c hanges in science

and technology and new approaches in diﬀerent areas of mathematics and relate d subjects like

Physics, Statistics and Computer Sciences, the board of studies in Mathematics with concern of

teachers of Mathematics from diﬀerent colleges aﬃliated to University of Mumbai has prepared

the syllabus of T.Y.B. Sc. / T. Y. B. A. Mathematics. The present syllab i of T. Y. B. Sc.

for Semester V and Semester VI has been designed as per U. G. C. Model cu rriculum so that

the students learn Mathematics needed for these branches, learn bas ic concepts of Mathematics

and are exposed to rigorous methods gently and slowly. The syllabi of T. Y. B. Sc. / T. Y. B.

A. would consist of two semesters and each semester would comprise of fou r courses and two

practical courses for T. Y. B. Sc / T.Y.B.A. Mathematics.

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2. Aims and Objectives:

(i) Give the students a suﬃcient knowledge of fundamental princi ples, methods and a clear

perception of innumerous power of mathematical ideas and tools and know h ow to use

them by modeling, solving and interpreting.

(ii) Reﬂecting the broad nature of the subject and developing mathem atical tools for

continuing further study in various ﬁelds of science.

(iii) Enhancing students’ overall development and to equip them with mathematical modeling

abilities, problem solving skills, creative talent and power of comm unication necessary for

various kinds of employment.

(iv) A student should get adequate exposure to global and local concerns t hat explore them

many aspects of Mathematical Sciences.

3. Programme Outcomes:

(i) Enabling students to develop positive attitude towards mathem atics as an interesting and

valuable subject

(ii) Enhancing students overall development and to equip them wi th mathematical modeling,

abilities, problem solving skills, creative talent and power of comm unication.

(iii) Acquire good knowledge and understanding in advanced areas of math ematics and physics.

4. Course outcomes:

(i) Multivariable Calculus II (Sem V): In this course students will learn the basic ideas,

tools and techniques of integral calculus and use them to solve problem s from real-life ap-

plications including science and engineering problems involving areas, volumes, centroid,

Moments of mass and center of mass Moments of inertia. Examine vector ﬁeld s and deﬁne

and evaluate line integrals using the Fundamental Theorem of Line Integr als and Green’s

Theorem; compute arc length.

(ii) Complex Analysis (Sem VI): Students Analyze sequences and series of analytic func-

tions and types of convergence, Students will also be able to evaluate c omplex contour

integrals directly and by the fundamental theorem, apply the Cauchy i ntegral theorem in

its various versions, and the Cauchy integral formula, they will also be able to represent

functions as Taylor, power and Laurent series, classify singularitie s and poles, ﬁnd residues

and evaluate complex integrals using the residue theorem.

(iii) Group Theory, Ring Theory (Sem V, Sem VI) Students will have a working knowl-

edge of important mathematical concepts in abstract algebra such as deﬁnit ion of a group,

order of a ﬁnite group and order of an element, rings, Euclidean domain, Pri ncipal ideal

domain and Unique factorization domain. Students will also understand th e connection

and transition between previously studied mathematics and more advanc ed mathematics.

The students will actively participate in the transition of importan t concepts such homo-

morphisms & isomorphisms from discrete mathematics to advanced abstr act mathematics.

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(iv) Topology of metric spaces (Sem V), Topology of metric spaces and real analysis

(Sem VI):

This course introduces students to the idea of metric spaces. It extends the ideas of

open sets, closed sets and continuity to the more general setting of me tric spaces along

with concepts such as compactness and connectedness. Convergence con cepts of sequences

and series of functions, power series are also dealt with. Formal proof s are given a lot of

emphasis in this course. This course serves as a foundation to advanced courses in analysis.

Apartfromunderstandingtheconceptsintroduced, thetreatmentof thiscoursewillenable

the learner to explain their reasoning about analysis with clarity and r igour.

(v) Partial Diﬀerential equations (Sem V: Paper IV: Elective A):

a. Students will able to understand the various analytical methods for solving ﬁrst order

partial diﬀerential equations.

b. Students will able to understand the classiﬁcation of ﬁrst order p artial diﬀerential

equations.

c. Students will able to grasp the linear and non linear partial diﬀeren tial equations.

(vi) Integral Transforms (Sem VI: Paper IV- Elective A):

a. Students will able to understand the concept of integral transforms and their corre-

sponding inversion techniques.

b. Students will able to understand the various applications of integr al transforms.

(vii) Number Theory and its applications I and II (Sem V, Sem VI):

The student will be able to

a. Identify and apply various properties of and relating to the integers including primes,

unique factorization, the division algorithm, and greatest common divisor s.

b. Understandtheconceptofacongruenceandusevariousresultsrelate dtocongruences

including the Chinese Remainder Theorem. Investigate Pseudo-pr imes , Carmichael

number, primitive roots.

c. Identify how number theory is related to and used in cryptography . Learn to encrypt

and decrypt a message using character ciphers. Learn to encrypt and de crypt a

message using Public-Key cryptology.

d. Express a rational number as a ﬁnite continued fraction and hence sol ve a linear

diophantine equation. Express a given repeated continued fraction in terms of a

surd. Expand a surd as an inﬁnite continued fraction and hence ﬁnd a con vergent

which is an approximation to the given surd to a given degree of accuracy . Solve a

Pell equation from a continued fraction expansion

e. Solve certain types of Diophantine equations. Represent a Primit ive Pythagorean

Triples with a unique pair of relatively prime integers.

f. Identify certain number theoretic functions and their propert ies. Investigate perfect

numbers and Mersenne prime numbers and their connection. Explore the use of

arithmetical functions, the Mobius function, and the Euler functi on.

(viii) Graph Theory (Sem V: Paper IV- Elective C)

Upon successful completion of Graph Theory course, a student will be ab le to:

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a. Demonstrate the knowledge of fundamental concepts in graph theory, in cluding prop-

erties and characterization of graphs and trees.

b. Describe knowledgeably special classes of graphs that arise freque ntly in graph theory

c. Describe the concept of isomorphic graphs and isomorphism invariant p roperties of

graphs

d. Describe and apply the relationship between the properties of a m atrix representation

of a graph and the structure of the underlying graph

e. Demonstrate diﬀerent types of algorithms including Dijkstra’s, BFS, DFS, MST and

Huﬀman coding.

f. Understand the concept of Eulerian graphs and Hamiltonian graphs.

g. Describe real-world applications of graph theory.

(ix) Graph Theory and Combinatorics (Sem VI: Paper IV -Elective C)

a. Understand and apply the basic concepts of graph theory, including colou ring of

graph, to ﬁnd chromatic number and chromatic polynomials for graphs

b. Understand the concept of vertex connectivity, edge connectivit y in graphs and Whit-

ney’s theorem on 2-vertex connected graphs.

c. Derive some properties of planarity and Euler’s formula, develop th e under-standing

of Geometric duals in Planar Graphs

d. Know the applications of graph theory to network ﬂows theory.

e. Understand diﬀerent applications of system of distinct represen tative and matching

theory.

f. Use permutations and combinations to solve counting problems with se ts and multi-

sets.

g. Set up and solve a linear recurrence relation and apply the inclusion /exclusion prin-

ciple.

h. Compute a generating function and apply them to combinatorial problem s.

(x) Basic concepts of probability and random variables (Sem V: Paper IV: Elec tive

D)

Students will be able to understand the role of random variables in the statistical anal-

ysis and use them to apply in the various probability distributions i ncluding Binomial

distribution, Poisson distribution and Normal distribution. Moreove r students will able to

apply the concepts of expectations and moments for the evaluation of various statistical

measures

(xi) Operations research (Sem VI: Paper IV: Elective D)

Students should able to formulate linear programming problem and apply t he graphical

and simplex method for their feasible solution. Moreover students should understand

various alternative operation research techniques for the feasible sol ution of LPP.

(5) Course structure with minimum credits and Lectures/ Week

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SEMESTER V

Multivariable Calculus II

Course Code UNITTOPICS Credits L/Week

USMT 501, UAMT 501 IMultiple Integrals

2.5 3 II Line Integrals

III Surface Integrals

Group Theory

USMT 502 ,UAMT 502 IGroups and Subgroups

2.5 3II Normal subgroups, Direct products and

Cayley’s theorem

III Cyclic Groups and Cyclic Subgroups

Homomorphism

Topology of Metric Spaces

USMT 503, UAMT503 IMetric spaces

2.5 3 II Sequences and Complete metric spaces

III Compact Spaces

Partial Diﬀerential Equations(Elective A)

USMT5A4 ,UAMT 5A4 IFirst Order Partial

2.5 3Diﬀerential Equations.

II Compatible system

of ﬁrst order PDE

III Quasi-Linear PDE

Number Theory and Its applications I (Elective B)

USMT5B4 ,UAMT 5B4 ICongruences and Factorization

2.5 3 II Diophantine equations and their

& solutions

III Primitive Roots and Cryptography

Graph Theory (Elective C)

USMT5C4 ,UAMT 5C4 IBasics of Graphs

2.5 3 II Trees

III Eulerian and Hamiltonian graphs

Basic Concepts of Probability and Random Variables (Elective D)

USMT5D4 ,UAMT 5D4 IBasic Concepts of Probability and

2.5 3 Random Variables

II Properties of Distribution function,

Joint Density function

III Weak Law of Large Numbers

PRACTICALS

USMTP05/UAMTP05 Practicals based on

3 6 USMT501/UAMT 501 and

USMT 502/UAMT 502

USMTP06/UAMTP06 Practicals based on

3 6USMT503/ UAMT 503 and

USMT5A4/ UAMT 5A4 OR

USMT5B4/ UAMT 5B4 OR

USMT5C4/ UAMT 5C4 OR

USMT5D4/ UAMT 5D4

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SEMESTER VI

BASIC COMPLEX ANALYSIS

Course Code UNITTOPICS Credits L/Week

USMT 601, UAMT 601 IIntroduction to Complex Analysis

2.5 3II Cauchy Integral Formula

III Complex power series, Laurent series and

isolated singularities

Ring Theory

USMT 602 ,UAMT 602 IRings

2.5 3 II Ideals and special rings

III Factorization

Topology of Metric Spaces and Real Analysis

USMT 603 / UAMT 603 IContinuous functions on

2.5 3Metric spaces

II Connected sets

Sequences and series of functions

Integral Transforms(Elective A)

USMT6A4 ,UAMT 6A4 IThe Laplace Transform

2.5 3II The Fourier Transform

III Applications of Integral Transforms

Number Theory and Its applications II (Elective B)

USMT6B4 ,UAMT 6B4 IQuadratic Reciprocity

2.5 3 II Continued Fractions

III Pell’s equation, Arithmetic function

& and Special numbers

Graph Theory and Combinatorics (Elective C)

USMT6C4 ,UAMT 6C4 IColourings of Graphs

2.5 3 II Planar graph

III Combinatorics

Operations Research (Elective D)

USMT6D4 ,UAMT 6D4 IBasic Concepts of Probability and

2.5 3Linear Programming I

II Linear Programming II

III Queuing Systems

PRACTICALS

USMTP07/ UAMTP07 Practicals based on

3 6 USMT601/UAMT 601 and

USMT 602/UAMT 602

USMTP08/UAMTP08 Practicals based on

3 6USMT603/ UAMT 603 and

USMT6A4/ UAMT 6A4 OR

USMT6B4/ UAMT 6B4 OR

USMT6C4/ UAMT 6C4 OR

USMT6D4/ UAMT 6D4

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Note: i . USMT501/UAMT501, USMT502/UAMT502, USMT503/UAMT503 are compul-

sory courses for Semester V.

ii . CandidatehastooptoneElectiveCoursefromUSMT5A4/UAMT5A4,USMT5B4/UAMT5B4,

USMT5C4/UAMT5C4 and USMT5D4/UAMT5D4 for Semester V.

iii . USMT601/UAMT601, USMT602/UAMT602, USMT603/UAMT603 are compulsory

courses for Semester VI.

iv . CandidatehastooptoneElectiveCoursefromUSMT6A4/UAMT6A4,USMT6B4/UAMT6B4,

USMT6C4/UAMT6C4 and USMT6D4/UAMT6D4 for Semester VI.

v . Passing in theory and practical and internal exam shall be separate.

(6) Teaching Pattern for T.Y.B.Sc/B.A.

i. Three lectures per week per course (1 lecture/period is of 48 mi nutes duration).

ii. One practical of three periods per week per course (1 lecture/p eriod is of 48 minutes

duration).

(7)Consolidated Syllabus for semester V & VI

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SEMESTER V

MULTIVARIABLE CALCULUS II

Course Code: USMT501/UAMT501

ALL Results have to be done with proof unless otherwise stated.

Unit I: Multiple Integrals (15L)

Deﬁnition of double (resp: triple) integral of a function and bounded on a rectangle (resp:box).

Geometric interpretation as area and volume. Fubini’s Theorem over r ectangles and any closed

bounded sets, Iterated Integrals. Following basic properties of doub le and triple integrals proved

using the Fubini’s theorem:

(1) Integrability of the sums, scalar multiples, products, and (und er suitable conditions)

quotients of integrable functions. Formulae for the integrals of sums and scalar multiples

of integrable functions.

(2) Integrability of continuous functions. More generally, Integrabili ty of functions with a

“small” set of (Here, the notion of “small sets” should include ﬁnite uni ons of graphs of

continuous functions.)

(3) Domain additivity of the integral. Integrability and the integral ove r arbitrary bounded

domains. Change of variables formula (Statement only).Polar, cylindric al and spherical

coordinates, and integration using these coordinates. Diﬀerentiati on under the integral

sign. Applications to ﬁnding the center of gravity and moments of inert ia.

Unit 2: Line Integrals (15L)

Review of Scalar and Vector ﬁelds on Rn, Vector Diﬀerential Operators, Gradient, Curl,

Divergence.

Paths (parametrized curves) in Rn(emphasis on R2and R3), Smooth and piecewise smooth

paths. Closed paths. Equivalence and orientation preserving equival ence of paths. Deﬁnition of

the line integral of a vector ﬁeld over a piecewise smooth path. Basic properties of line integrals

including linearity, path-additivity and behaviour under a change of parameters. Examples.

Line integrals of the gradient vector ﬁeld, Fundamental Theorem of Calcul us for Line Integrals,

Necessary and suﬃcient conditions for a vector ﬁeld to be conservative . Green’s Theorem (proof

in the case of rectangular domains). Applications to evaluation of line inte grals.

Unit 3: Surface Integrals (15 L)

Parameterized surfaces. Smoothly equivalent parameterizations. Are a of such surfaces.

Deﬁnition of surface integrals of scalar-valued functions as well as of vec tor ﬁelds deﬁned on a

surface. Curl and divergence of a vector ﬁeld. Elementary identiti es involving gradient, curl and

divergence. Stoke’s Theorem (proof assuming the general from of Green ’s Theorem). Examples.

Gauss’ Divergence Theorem (proof only in the case of cubical domains). E xamples.

Reference Books:

1. Apostol, Calculus, Vol. 2, Second Ed., John Wiley, New York, 1969 Section 1.1 to 11.8

2. James Stewart, Calculus with early transcendental Functions - Sec tion 16.5 to 16.9

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3. Marsden and Jerrold E. Tromba, Vector Calculus, Fourth Ed., W.H. Free man and Co.,

New York, 1996 Section 6.2 to 6.4.

Other References :

1. T. Apostol, Mathematical Analysis, Second Ed., Narosa, New Delhi. 1947.

2. R. Courant and F.John, Introduction to Calculus and Analysis, Vol.2, Sp ringer Verlag,

New York, 1989.

3. W. Fleming, Functions of Several Variables, Second Ed., Springer-V erlag, New York, 1977.

4. M. H. Protter and C.B.Morrey Jr., Intermediate Calculus, Second Ed ., Springer-Verlag,

New York, 1995.

5. G. B. Thomas and R.L Finney, Calculus and Analytic Geometry, Ninth Ed. (ISE Reprint),

Addison- Wesley, Reading Mass, 1998.

6. D. V. Widder, Advanced Calculus, Second Ed., Dover Pub., New York. 1989.

Course: Group Theory

Course Code: USMT502/UAMT502

Unit 1: Groups and Subgroups (15L)

(1) Deﬁnition and elementary properties of a group. Order of a group. Subgr oups. Criterion

for a subset to be a subgroup. Abelian groups. Center of a group. Homomorphism s and

isomorphisms.

(2) Examples of groups including Z,Q,R,C,Klein 4-group, symmetric and alternating groups,

S1(= the unit circle in C), GL n(R),SL n(R),On(= the group of n×nnonsingular upper

triangular matrices), B n(= the group of n×nnonsingular upper triangular matrices),

and groups of symmetries of plane ﬁgures.

(3) Order of an element. Subgroup generated by a subset of the group.

Unit 2: Normal subgroups, Direct products and Cayley’s Theorem (15L)

(1) Cosets of a subgroup in a group. Lagrange’s Theorem. Normal subgroups. Alternat ing

group An. Listing normal subgroups of A4,S 3. Quotient (or Factor) groups. Fundamental

Theorem of homomorphisms of groups.

(2) External direct products of groups. Examples. Relation with inte rnal products such as

HK of subgroups H,K of a group.

(3) Cayley’s Theorem for ﬁnite groups.

Unit 3: Cyclic groups and cyclic subgroups (15L)

(1) Examplesofcyclicgroupssuchas Zandthegroup µnofthe n−throotsofunity. Properties

of cyclic groups and cyclic subgroups.

(2) Finite cyclic groups, inﬁnite cyclic groups and their generators. Properties of generators.

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(3) The group Z/n Zof residue classes (mod n). Characterization of cyclic groups (as being

isomorphic to Zor Z/n Zfor some n∈N).

Recommended Books.

1. I. N. Herstein, Topics in Algebra, Wiley Eastern Limied, Second editi on.

2. P. B. Bhattacharya, S.K. Jain, S. Nagpaul. Abstract Algebra, Second edition, Foundation

Books, New Delhi, 1995.

3. N. S. Gopalkrishnan, University Algebra, Wiley Eastern Limited.

4. M. Artin, Algebra, Prentice Hall of India, New Delhi.

5. J. B. Fraleigh, A ﬁrst course in Abstract Algebra, Third edition, Narosa, Ne w Delhi.

6. J. Gallian. Contemporary Abstract Algebra. Narosa, New Delhi.

Additional Reference Books

1. T. W. Hungerford. Algebra, Springer.

2. D. Dummit, R. Foote. Abstract Algebra, John Wiley & Sons, Inc.

3. I. S. Luther, I.B.S. Passi. Algebra. Vol. I and II.

Course: Topology of Metric Spaces

Course Code: USMT503/UAMT503

Unit I: Metric spaces (15 L)

Deﬁnition and examples of metric spaces such as R,R2,Rnwith its Euclidean, sup and sum

metrics. C(complex numbers). l1and l2spaces of sequences. C[a,b ] the space of real valued

continuous functions on [ a,b ]. Discrete metric space. Metric induced by the norm. Translation

invariance of the metric induced by the norm. Metric subspaces. Pr oduct of two metric spaces.

Open balls and open sets in a metric space. Examples of open sets in vari ous metric spaces.

Hausdorﬀ property. Interior of a set. Properties of open sets. Structu re of an open set in R.

Equivalent metrics.

Distance of a point from a set, Distance between sets. Diameter of a se t. Bounded sets. Closed

balls. Closed sets. Examples. Limit point of a set. Isolated point. Clos ure of a set. Boundary

of a set.

Unit II: Sequences and Complete metric spaces (15L)

Sequences in a metric space. Convergent sequence in metric space . Cauchy sequence in a metric

space. Subsequences. Examples of convergent and Cauchy sequences i n diﬀerent metric spaces.

Characterization of limit points and closure points in terms of sequenc es. Deﬁnition and exam-

plesofrelativeopenness/closenessinsubspaces. Densesubsetsi nametricspaceandSeparability.

Deﬁnition of complete metric spaces. Examples of complete metric sp aces. Completeness prop-

erty in subspaces. Nested Interval theorem in R. Cantor’s Intersection Theorem. Applications

of Cantors Intersection Theorem:

(i) The set of real Numbers is uncountable.

(ii) Density of rational Numbers.

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(iii) Intermediate Value theorem.

Unit III: Compact spaces (15L)

Deﬁnition of a compact metric space using open cover. Examples of compac t sets in diﬀerent

metric spaces such as R,R2,Rnwith Euclidean metric. Properties of compact sets: A compact

set is closed and bounded, (Converse is not true ). Every inﬁnite bou nded subset of compact

metric space has a limit point. A closed subset of a compact set is compac t. Union and

Intersection of Compact sets.

Equivalent statements for compact sets in Rwith usual metric:

(i) Sequentially compactness property.

(ii) Heine-Borel property.

(iii) Closed and boundedness property.

(iv) Bolzano-Weierstrass property.

Reference books:

1. S. Kumaresan; Topology of Metric spaces.

2. E. T. Copson; Metric Spaces; Universal Book Stall, New Delhi, 1996.

3. P. K. Jain, K. Ahmed; Metric Spaces; Narosa, New Delhi, 1996.

Other references :

1. T. Apostol; Mathematical Analysis, Second edition, Narosa, New Delhi, 1974

2. R. R. Goldberg; Methods of Real Analysis; Oxford and IBH Pub. Co., New De lhi 1970.

3. D. Gopal, A. Deshmukh, A. S. Ranadive and S. Yadav; An Introduction to Me tric Spaces,

Chapman and Hall/CRC, New York, 2020.

4. W. Rudin; Principles of Mathematical Analysis; Third Ed, McGraw- Hill, Auckland, 1976.

5. D. Somasundaram; B. Choudhary; A ﬁrst Course in Mathematical Analysis. Nar osa, New

Delhi

6. G. F. Simmons; Introduction to Topology and Modern Analysis; McGraw- Hi, New York,

1963.

7. Expository articles of MTTS programme.

Course: Partial Diﬀerential Equations (Elective A)

Course Code: USMT5A4/UAMT5A4

Unit I: First Order Partial Diﬀerential Equations. (15L)

Curves and Surfaces, Genesis of ﬁrst order PDE, Classiﬁcation of ﬁrst order PDE, Classiﬁcation

of integrals, The Cauchy problem, Linear Equation of ﬁrst order, Lagrange’s eq uation, Pfaﬃan

diﬀerential equations. (Ref Book: An Elementary Course in Partial Diﬀ erential Equations by

T. Amaranath, 2nd edition, Chapter 1: 1.1, 1.2, 1.3, Lemma 1.3.1, 1.3.2, 1.3.3, 1.4, Theorem

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1.4.1,1.4.2,1.5,Theorem1.5.1,Lemma1.5.1,Theorem1.5.2,Lemma1.5.2andrelatedexam ples)

Unit II: Compatible system of ﬁrst order Partial Diﬀerential Equation s. (15L)

Deﬁnition, Necessary and suﬃcient condition for integrability, Charp it’s method, Some stan-

dard types, Jacobi’s method, The Cauchy problem. (Ref Book: An Eleme ntary Course in

Partial Diﬀerential Equations by T. Amaranath, 2nd edition, Chapter 1: 1.6, T heorem 1.6.1,

1.7, 1.8 Theorem 1.8.1, 1.9 and related examples)

Unit III: Quasi-Linear Partial Diﬀerential Equations. (15L)

Semi linear equations, Quasi-linear equations, ﬁrst order quasi-lin ear PDE, Initial value prob-

lem for quasi-linear equation, Non linear ﬁrst order PDE, Monge cone, Analyt ic expression for

Monge’s cone, Characteristics strip, Initial strip. (Ref Book: An El ementary Course in Partial

Diﬀerential Equations by T. Amaranath, 2nd edition, Chapter 1: 1.10, Theorem 1.10.1, 1.11,

Theorem 1.11.1, Preposition 1.11.1, 1.11.2 and related examples)

Reference Books

1. T. Amaranath; An Elementary Course in Partial Diﬀerential Equations; 2nd edition,

Narosa Publishing house.

2. Ian Sneddon; Elements of Partial Diﬀerential Equations; McGraw Hill book.

3. RaviP.AgarwalandDonalO’Regan; OrdinaryandPartialDiﬀerentialEquation s; Springer,

First Edition (2009).

4. W. E. Williams; Partial Diﬀerential Equations; Clarendon Press, Ox ford, (1980).

5. K. Sankara Rao; Introduction to Partial Diﬀerential Equations; Third Edition, PHI.

Course: Number Theory and its applications I (Elective B)

Course Code: USMT5B4 / UAMT5B4

Unit I: Congruences and Factorization (15L)

Review of Divisibility, Primes and The fundamental theorem of Arith metic.

Congruences : Deﬁnition and elementary properties, Complete resid ue system modulo m, Re-

duced residue system modulo m, Euler’s function and its properties, Fermat’s little Theorem,

Euler’s generalization of Fermat’s little Theorem, Wilson’s theorem , Linear congruence, The

Chinese remainder Theorem, Congruences of Higher degree,

Unit II: Diophantine equations and their solutions (15L)

The linear equations ax +by =c. The equations x2+y2=p, where pis a prime. The

equation x2+y2=z2, Pythagorean triples, primitive solutions, The equations x4+y4=z2and

x4+y4=z4have no solutions ( x;y;z) with xyz /\e}atio\slash= 0. Every positive integer ncan be expressed

as sum of squares of four integers, Universal quadratic forms x2+y2+z2+t2. Assorted examples

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:section 5.4 of Number theory by Niven- Zuckermann-Montgomery.

Unit III: Primitive Roots and Cryptography (15L)

Order of an integer and Primitive Roots. Basic notions such as encrypti on (enciphering) and

decryption (deciphering), Cryptosystems, symmetric key cryp tography, Simple examples such

as shift cipher, Aﬃne cipher, Hill cipher, Vigenere cipher. Concep t of Public Key Cryptosystem;

RSA Algorithm. An application of Primitive Roots to Cryptography.

Reference Books:

1. Niven, H. Zuckerman and H. Montogomery; An Introduction to the Theory of Num bers;

John Wiley & Sons. Inc.

2. David M. Burton; An Introduction to the Theory of Numbers; Tata McGr awHillll Edition.

3. G. H. Hardy and E.M. Wright; An Introduction to the Theory of Numbers; Low priced

edition; The English Language Book Society and Oxford University Press, 1981.

4. Neville Robins. Beginning Number Theory; Narosa Publications.

5. S.D. Adhikari; An introduction to Commutative Algebra and Number Theor y; Narosa

Publishing House.

6. N. Koblitz; A course in Number theory and Cryptography; Springer.

7. M. Artin; Algebra; Prentice Hall.

8. K. Ireland, M. Rosen; A classical introduction to Modern Number Th eory; Second edition,

Springer Verlag.

9. William Stalling; Cryptology and network security.

Course: Graph Theory (Elective C)

Course Code: USMT5C4/UAMT5C4

Unit I: Basics of Graphs (15L)

Deﬁnition of general graph, Directed and undirected graph, Simple and m ultiple graph, Types

of graphs- Complete graph, Null graph, Complementary graphs, Regular graphs Su b graph of a

graph, Vertex and Edge induced sub graphs, Spanning sub graphs. Basic te rminology- degree of

a vertex, Minimum and maximum degree, Walk, Trail, Circuit, Path, C ycle. Handshaking the-

orem and its applications, Isomorphism between the graphs and consequen ces of isomorphism

between the graphs, Self complementary graphs, Connected graphs, Conn ected components.

Matrices associated with the graphs – Adjacency and Incidence matrix of a graph- properties,

Bipartite graphs and characterization in terms of cycle lengths. Degree sequence and Havel-

Hakimi theorem, Distance in a graph- shortest path problems, Dijkstra’ s algorithm.

Unit II: Trees (15L)

Cut edges and cut vertices and relevant results, Characterization of c ut edge, Deﬁnition of a

tree and its characterizations, Spanning tree, Recurrence relation of spanning trees and Cayley

## Page 19

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formula for spanning trees of Kn, Algorithms for spanning tree-BFS and DFS, Binary and

m-ary tree, Preﬁx codes and Huﬀman coding, Weighted graphs and minimal sp anning trees -

Kruskal’s algorithm for minimal spanning trees.

Unit III: Eulerian and Hamiltonian graphs (15L)

Eulerian graph and its characterization- Fleury’s Algorithm-(Chinese postman problem), Hamil-

tonian graph, Necessary condition for Hamiltonian graphs using G\Swhere Sis a proper subset

of V(G), SuﬃcientconditionforHamiltoniangraphs-Ore’stheoremandDirac’st heorem, Hamil-

tonian closure of a graph, Cube graphs and properties like regular, biparti te, Connected and

Hamiltonian nature of cube graph, Line graph of graph and simple results.

Reference Books:

1. Bondy and Murty; Grapgh Theory with Applications.

2. Balkrishnan and Ranganathan; Graph theory and applications.

3. Douglas B. West, Introduction to Graph Theory, 2nd Ed., Pearson, 2000

Additional Reference Book:

1. Behzad and Chartrand; Graph theory.

2. Choudam S. A.; Introductory Graph theory.

Course: Basic Concepts of Probability and Random Variables (Elective D)

Course Code: USMT5D4 / UAMT5D4

Unit I: Basic Concepts of Probability and Random Variables.(15 L)

Basic Concepts: Algebra of events including countable unions and inte rsections, Sigma ﬁeld F,

Probability measure Pon F, Probability Space as a triple (Ω ,F,P ), Properties of Pincluding

Subadditivity. Discrete Probability Space, Independence and Con ditional Probability, Theorem

of Total Probability. Random Variable on (Ω ,F,P ) – Deﬁnition as a measurable function, Clas-

siﬁcation of random variables - Discrete Random variable, Probability fun ction, Distribution

function, Density function and Probability measure on Borel subsets of R, Absolutely contin-

uous random variable. Function of a random variable; Result on a random variabl e R with

distribution function Fto be absolutely continuous, Assume Fis continuous everywhere and

has a continuous derivative at all points except possibly at ﬁnite numb er of points, Result on

density function f2of R2where R2=g(R1),h jis inverse of gover a ‘suitable’ subinterval

f2(y)+ n/summationdisplay

i=1 f1(hj(y)) |h′

j(y)|under suitable conditions.

Reference for Unit 1, Sections 1.1-1.6, 2.1-2.5 of Basic Probability theory by Robert Ash, Dover

Publication, 2008.

Unit II: Properties of Distribution function, Joint Density func tion (15L)

Properties of distribution function F,F is non-decreasing, lim

x−→∞ F(x) = 1 ,lim

x−→−∞ F(x) = 0,

Right continuity of F, lim

x−→ x0F(x) = P({R < x o},P ({R=xo}) = F(xo)−F(x0). Joint distri-

bution, Joint Density, Results on Relationship between Joint and In dividual densities, Related

## Page 20

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result for Independent random variables. Examples of distributions like Binomial, Poisson and

Normal distribution. Expectation and k−th moments of a random variable with properties.

Reference for Unit II:

Sections 2.5-2.7, 2.9, 3.2-3.3,3.6 of Basic Probability theory by Robert Ash, Dov er Publication,

2008.

Unit III: Weak Law of Large Numbers

Joint Moments, Joint Central Moments, Schwarz Inequality, Bounds on C orrelation Coeﬃcient ρ

,Resulton ρasameasureoflineardependence,Var /parenleftBign/summationdisplay

i=1 Ri/parenrightBig

=n/summationdisplay

i=1 Var (Ri)+2 n/summationdisplay

i=1 ≤i

law of Large numbers.

Reference for Unit III

Sections3.4, 3.5, 3.7, 4.1-4.4ofBasicProbabilitytheorybyRobertAsh, Dover Publication, 2008.

Additional Reference Books. Marek Capinski, Probability through Problems, Springer.

Course: Practicals (Based on USMT501 / UAMT501 and USMT502 / UAMT502)

Course Code: USMTP05 / UAMTP05

Suggested Practicals (Based on USMT501 / UAMT501)

1. Evaluation of double and triple integrals.

2. Change of variables in double and triple integrals and applications

3. Line integrals of scalar and vector ﬁelds

4. Green’s theorem, conservative ﬁeld and applications

5. Evaluation of surface integrals

6. Stoke’s and Gauss divergence theorem

7. Miscellaneous theory questions on units 1, 2 and 3.

Suggested Practicals (Based on USMT502 / UAMT502)

1. Examples of groups and groups of symmetries of equilateral triangle, squar e and rectangle.

2. Examples of determining centers of diﬀerent groups. Examples of su bgroups of various

groups and orders of elements in a group.

3. Left and right cosets of a group and Lagrange’s theorem.

4. Normal subgroups and quotient groups. Direct products of groups.

5. Finite cyclic groups and their generators

## Page 21

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6. Inﬁnite cyclic groups and their properties.

7. Miscellaneous Theory Questions

Course: Practicals (Based on USMT503 / UAMT503 and USMT5A4 OR

USMT5B4 OR USMT5C4 OR USMT5D4)

Course Code: USMTP06 / UAMTP06

Suggested Practicals USMT503 / UAMT503:

1. Examples of Metric Spaces, Normed Linear Spaces,

2. Sketching of Open Balls in R2, Open and Closed sets, Equivalent Metrics

3. Subspaces, Interior points, Limit Points, Dense Sets and Separabil ity, Diameter of a set,

Closure.

4. Limit Points ,Sequences , Bounded , Convergent and Cauchy Sequence s in a Metric Space.

5. Complete Metric Spaces and Applications.

6. Examples of Compact Sets.

7. Miscellaneous Theory Questions.

Suggested Practicals on USMT5A4/UAMT5A4

1. Find general solution of Langrange’s equation.

2. Show that Pfaﬃan diﬀerential equation are exact and ﬁnd corresponding i ntegrals.

3. Find complete integral of ﬁrst order PDE using Charpit’s Method.

4. Find complete integral using Jacobi’s Method.

5. Solve initial value problem for quasi-linear PDE.

6. Find the integral surface by the method of characteristics.

7. Miscellaneous Theory Questions.

Suggested Practicals based on USMT5B4/UAMT5B4

1. Congruences.

2. Linear congruences and congruences of Higher degree.

3. Linear diophantine equation.

4. Pythagorean triples and sum of squares.

5. Cryptosystems (Private Key).

6. Cryptosystems (Public Key) and primitive roots.

7. Miscellaneous theoretical questions based on full USMT5B4 .

## Page 22

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Suggested Practicals based on USMT5C4/UAMT5C4

1. Handshaking Lemma and Isomorphism.

2. Degree sequence and Dijkstra’s algorithm

3. Trees, Cayley Formula

4. Applications of Trees

5. Eulerian Graphs.

6. Hamiltonian Graphs.

7. Miscellaneous Problems.

Suggested Practicals based on USMT5D4/UAMT5D4

1. Basic concepts of Probability (Algebra of events, Probability space, P robability measure,

combinatorial problems)

2. Conditional Probability, Random variable (Independence of events. D eﬁnition, Classiﬁca-

tion and function of a random variable)

3. Distribution function, Joint Density function.

4. Expectation of a random variable, Normal distribution.

5. Method of Indicators, Weak law of large numbers.

6. Conditional density, Conditional expectation.

7. Miscellaneous Theoretical questions based on full paper.

SEMESTER VI

BASIC COMPLEX ANALYSIS

Course Code: USMT601/UAMT601

Unit I: Introduction to Complex Analysis (15 L)

Review of complex numbers: Complex plane, polar coordinates, exponen tial map, powers and

roots of complex numbers, De Moivre’s formula, Cas a metric space, bounded and unbounded

sets, point at inﬁnity-extended complex plane, sketching of set i n complex plane (No questions

to be asked).

convergence of sequences of complex numbers and related results. Lim it of a function f:C−→

C, real and imaginary part of functions, continuity at a point and algebra of cont inuous func-

tions. Derivative of f:C−→ C, comparison between diﬀerentiability in real and complex sense,

Cauchy-Riemann equations, suﬃcient conditions for diﬀerentiabili ty, analytic function, if f,g

analytic then f+g,f −g,fg and f/g are analytic, chain rule.

Theorem: If f(z) = 0 everywhere in a domain D, then f(z) must be constant throughout D.

Harmonic functions and harmonic conjugate.

Unit II: Cauchy Integral Formula (15 L)

Evaluation the line integral /integraldisplay

f(z)dz over |z−z0|=rand Cauchy integral formula.

## Page 23

21

Taylor’s theorem for analytic function. Mobius transformations: deﬁni tion and examples.

Exponential function, its properties. trigonometric functions and h yperbolic functions.

Unit III: Complex power series, Laurent series and isolated singul arities. (15 L)

Power series of complex numbers and related results. Radius of conve rgences, disc of conver-

gence, uniqueness of series representation, examples.

Deﬁnition of Laurent series , Deﬁnition of isolated singularity, statem ent (without proof) of ex-

istence of Laurent series expansion in neighbourhood of an isolated singu larity, type of isolated

singularities viz. removable, pole and essential deﬁned using Laure nt series expansion, examples

Statement of Residue theorem and calculation of residue.

Reference Books:

1. J.W. Brown and R.V. Churchill, Complex analysis and Applications : Se ctions 18, 19, 20,

21, 23, 24, 25, 28, 33, 34, 47, 48, 53, 54, 55 , Chapter 5, page 231 section 65, deﬁne residue

of a function at a pole using Theorem in section 66 page 234, Statement of Cauchy ’s

residue theorem on page 225, section 71 and 72 from chapter 7.

Other References:

1. Robert E. Greene and Steven G. Krantz, Function theory of one complex variable

2. T.W. Gamelin, Complex analysis

Course: Ring Theory

Course Code: USMT602 / UAMT602

Unit I. Rings (15L)

(1) Deﬁnition and elementary properties of rings (where the deﬁniti on should include the

existence of unity), commutative rings, integral domains and ﬁelds. E xamples, including

Z,Q,R,Z/n Z,C,M n(R),Z[i],Z[√

2] ,Z[√−5] ,Z[X],R[X],C[X],(Z/n Z)[ X].

(2) Units in a ring. The multiplicative group of units in a ring R[ and, in particular, the

multiplicative group F∗of nonzero elemets of a ﬁeld F]. Description of the units in Z/n Z.

Results such as: A ﬁnite integral domain is a ﬁeld. Z/p Z, where pis a prime, as an

example of a ﬁnite ﬁeld.

(3) Characteristic of a ring. Examples. Elementary facts such as: the c haracteristic of an

itegral domain is either 0 or a prime number.

(Note: From here on all rings are assumed to be commutative with unity).

Unit II. Ideals and special rings(15L)

(1) Ideals in a ring. Sums and products of ideals. Quotient rings. Examp les. Prime ideals and

maximal ideals. Characterization of prime ideals and maximal ideals in a com mutative

ring in terms of their quotient rings. Description of the ideals and t he prime ideals in

Z,R[X] and C[X].

(2) Homomorphisms and isomorphism of rings. Kernel and the image of a homomorphi sm.

Fundamental Theorem of homomorphism of a ring.

## Page 24

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(3) Construction of the quotient ﬁeld of an integral domain (Emphasis on Z,Q). A ﬁeld

contains a subﬁeld isomorphic to Z/p Zor Q.

(4) Notions of euclidean domain (ED), principal ideal domain (PID). Exampl es such as Z,Z[i],

and polynomial rings. Relation between these two notions (ED = ⇒PID ).

Unit III. Factorization (15L)

(1) Divisibility in a ring. Irreducible and prime elements. Exam ples.

(2) Division algorithm in F[X] (where Fis a ﬁeld). Monic polynomials, greatest common

divisorof f(x),g (x)∈F[X](notboth0). Theorem: Given f(x)and g(x)/\e}atio\slash= 0 ,in F[X]then

their greatest common divisor d(x)∈F[X] exists; moreover, d(x) = a(x)f(x)+ b(x)g(x)

for some a(x),b (x)∈F[X]. Relatively prime polynomials in F[X], irreducible polynomial

in F[X]. Examples of irreducible polynomials in ( Z/p Z)[ X] ( pprime), Eisenstein Criterion

(without proof).

(3) Notion of unique factorization domain (UFD). Elementary properties. Ex ample of a non-

UFD is Z[√−5] (without proof). Theorem (without proof). Relation between the th ree

notions (ED = ⇒PID = ⇒UFD). Examples such as Z[X] of UFD that are not PID.

Theorem (without proof): If Ris a UFD, then R[X] is a UFD.

Reference Books

1. N. Herstein; Topics in Algebra; Wiley Eastern Limited, Second edition .

2. P. B. Bhattacharya, S. K. Jain, and S. R. Nagpaul; Abstract Algebra; Second edi tion,

Foundation Books, New Delhi, 1995.

3. N. S. Gopalakrishnan; University Algebra; Wiley Eastern Limited.

4. M. Artin; Algebra; Prentice Hall of India, New Delhi.

5. J. B. Fraleigh; A First course in Abstract Algebra; Third edition, Narosa, New Delhi.

6. J. Gallian; Contemporary Abstract Algebra; Narosa, New Delhi.

Additional Reference Books:

1. S. Adhikari; An Introduction to Commutative Algebra and Number theory; Narosa Pub-

lishing House.

2. T.W. Hungerford; Algebra; Springer.

3. D. Dummit, R. Foote; Abstract Algebra; John Wiley & Sons, Inc.

4. I.S. Luthar, I.B.S. Passi; Algebra; Vol. I and II.

5. U. M. Swamy, A. V. S. N. Murthy; Algebra Abstract and Modern; Pearson.

6. Charles Lanski; Concepts Abstract Algebra; American Mathematical Societ y.

7. Sen, Ghosh and Mukhopadhyay; Topics in Abstract Algebra; Universities press.

## Page 25

23

Course: Topology of Metric Spaces and Real Analysis

Course Code: USMT603/ UAMT603

Unit I: Continuous functions on metric spaces (15 L)

Epsilon-delta deﬁnition of continuity of a function at a point from one m etric space to another.

Characterization of continuity at a point in terms of sequences, open se ts and closed sets and ex-

amples. Algebra of continuous real valued functions on a metric space. Con tinuity of composite

function. Continuous image of compact set is compact, Uniform continuity i n a metric space,

examples (emphasis on R). Results such as: every continuous functions from a compact metric

space is uniformly continuous. Contraction mapping and ﬁxed point theor em. Applications.

Unit II: Connected spaces (15L)

Separated sets- Deﬁnition and examples. Connected and disconnected sets. Connected and

disconnected metric spaces. Results such as: A subset of Ris connected if and only if it is an

interval. A continuous image of a connected set is connected.

Characterization of a connected space, viz. a metric space is connecte d if and only if every con-

tinuous function from Xto {1,−1}is a constant function. Path connectedness in Rn, deﬁnition

and examples. A path connected subset of Rnis connected, convex sets are path connected.

Connected components. An example of a connected subset of Rn which is n ot path connected.

Unit III : Sequence and series of functions(15 lectures)

Sequence of functions - pointwise and uniform convergence of sequenc es of real- valued functions,

examples. Uniform convergence implies pointwise convergence, examp le to show converse not

true, series of functions, convergence of series of functions, Weie rstrass M-test (statement only).

Examples. Properties of uniform convergence: Continuity of the unif orm limit of a sequence of

continuous function, conditions under which integral and the derivat ive of sequence of functions

converge to the integral and derivative of uniform limit on a closed and bou nded interval (state-

ments only). Examples. Consequences of these properties for serie s of functions, term by term

diﬀerentiation and integration(statements only). Power series in Rcentered at origin and at

some point in R, radius of convergence, region (interval) of convergence, uniform conve rgence,

term by-term diﬀerentiation and integration of power series, Exampl es. Uniqueness of series

representation, functions represented by power series, classi cal functions deﬁned by power series

such as exponential, cosine and sine functions, the basic properties of these functions.

Reference books:

1. R. R. Goldberg; Methods of Real Analysis; Oxford and International Book Hou se (IBH)

Publishers, New Delhi.

2. S. Kumaresan; Topology of Metric spaces.

3. E. T. Copson; Metric Spaces; Universal Book Stall, New Delhi, 1996.

4. Robert Bartle and Donald R. Sherbert; Introduction to Real Analysis; Second Edition,

John Wiley and Sons.

## Page 26

24

Other references:

1. W. Rudin; Principles of Mathematical Analysis.

2. T. Apostol; Mathematical Analysis; Second edition, Narosa, New Delhi, 1974

3. E. T. Copson; Metric Spaces; Universal Book Stall, New Delhi, 1996.

4. P. K. Jain. K. Ahmed, Metric Spaces. Narosa, New Delhi, 1996.

5. W. Rudin, Principles of Mathematical Analysis; Third Ed, McGraw- Hill, Auckland, 1976.

6. D. Somasundaram, B. Choudhary; A ﬁrst Course in Mathematical Analysis. Nar osa, New

Delhi

7. G. F. Simmons; Introduction to Topology and Modern Analysis, McGraw- Hi, New York,

1963.

8. Sutherland. Topology.

Course: Intergral Transforms(Elective A)

Course Code: USMT6A4/ UAMT6A4

Unit I: The Laplace Transform (15L)

Deﬁnition of Laplace Transform, theorem, Laplace transforms of some element ary functions,

Properties of Laplace transform, LT of derivatives and integrals, Initial and ﬁnal value theorem,

Inverse Laplace Transform, Properties of Inverse Laplace Transform, Conv olution Theorem, In-

verse LT by partial fraction method, Laplace transform of special functi ons: Heaviside unit step

function, Dirac-delta function and Periodic function.

Unit II: The Fourier Transform

Fourier integral representation, Fourier integral theorem, Fourier S ine & Cosine integral rep-

resentation, Fourier Sine & Cosine transform pairs, Fourier transform of elementary functions,

Properties of Fourier Transform, Convolution Theorem, Parseval’s Iden tity.

Unit III: Applications of Integral Transforms

Relation between the Fourier and Laplace Transform. Application of Laplace tr ansform to eval-

uation of integrals and solutions of higher order linear ODE. Applications of L T to solution

of one dimensional heat equation & wave equation. Application of Fourier tran sforms to the

solution of initial and boundary value problems, Heat conduction in solids ( one dimensional

problems in inﬁnite & semi inﬁnite domain).

Reference Books:

1. LokenathDebnathandDambaruBhatta, IntegralTransformsandtheirApplicat ions, CRC

Press Taylor & Francis.

2. I. N. Sneddon, Use of Integral Transforms, Tata-McGraw Hill.

## Page 27

25

3. L. Andrews and B. Shivamogg, Integral Transforms for Engineers, Prentice Hall of India.

Course: Number Theory and its applications II (Elective B)

Course Code: USMT6B4/ UAMT6B4

Unit I: Quadratic Reciprocity (15 L)

Quadratic residues and Legendre Symbol, Gauss’s Lemma, Theorem on Legendr e Symbol /parenleftBig2

p/parenrightBig

,

the result: If pis an odd prime and ais an odd integer with ( a,p ) = 1 then

/parenleftBiga

p/parenrightBig

= ( −1) twhere t=p−1

2/summationdisplay

k=1 /bracketleftBigka

p/bracketrightBig

, Quadratic Reciprocity law. Theorem on Legendre Symbol

/parenleftBig3

p/parenrightBig

.The Jacobi Symbol and law of reciprocity for Jacobi Symbol. Quadratic Congr uences with

Composite moduli.

Unit II: Continued Fractions (15 L)

Finite continued fractions. Inﬁnite continued fractions and repre sentation of an irrational num-

ber by an inﬁnite simple continued fraction, Rational approximations t o irrational numbers and

order of convergence, Best possible approximations. Periodic contin ued fractions.

Unit III: Pell’s equation, Arithmetic function and Special numbe rs (15 L)

Pell’s equation x2−dy 2=n, where dis not a square of an integer. Solutions of Pell’s equation

(The proofs of convergence theorems to be omitted). Arithmetic func tions of number theory:

d(n) (or τ(n),) σ(n),σ k(n),ω (n)) and their properties, µ(n) and the M¨ obius inversion formula.

Special numbers: Fermat numbers, Mersenne numbers, Perfect n umbers, Amicable numbers,

Pseudo primes, Carmichael numbers.

Reference Books:

1. Niven, H. Zuckerman and H. Montogomery; An Introduction to the Theory of Num bers;

John Wiley & Sons. Inc.

2. David M. Burton; An Introduction to the Theory of Numbers; Tata McGr aw-Hill Edition.

3. G. H. Hardy and E.M. Wright; An Introduction to the Theory of Numbers; Low priced

edition; The English Language Book Society and Oxford University Press, 1981.

4. Neville Robins; Beginning Number Theory; Narosa Publications.

5. S. D. Adhikari; An introduction to Commutative Algebra and Number Theor y; Narosa

Publishing House

6. N. Koblitz; A course in Number theory and Crytopgraphy. Springer.

7. M. Artin; Algebra. Prentice Hall.

8. K. Ireland, M. Rosen; A classical introduction to Modern Number Th eory. Second edition,

Springer Verlag.

## Page 28

26

9. William Stalling; Cryptology and network security.

Course: Graph Theory and Combinatorics (Elective C)

Course Code: USMT6C4 /UAMT6C4

Unit I: Colorings of graph (15L)

Vertex coloring- evaluation of vertex chromatic number of some standard grap hs, critical graph.

Upper and lower bounds of Vertex chromatic Number- Statement of Brooks t heorem. Edge

colouring- Evaluation of edge chromatic number of standard graphs such as compl ete graph,

completebipartitegraph, cycle. StatementofVizingTheorem. Chromat icpolynomialofgraphs-

Recurrence Relation and properties of Chromatic polynomials. Vertex an d edge cuts, vertex and

edge connectivity and the relation between vertex and edge connectiv ity. Equality of vertex and

edge connectivity of cubic graphs. Whitney’s theorem on 2-vertex conn ected graphs.

Unit II: Planar graph (15L)

Deﬁnition of planar graph. Euler formula and its consequences. Non planari ty of K5;K(3;3).

Dual of a graph. Polyhedran in R3and existence of exactly ﬁve regular polyhedron- (Platonic

solids) Colorabilty of planar graphs- 5 color theorem for planar graphs, stateme nt of 4 color

theorem. ﬂows in Networks, and cut in a network- value of a ﬂow and the cap acity of cut in a

network, relation between ﬂow and cut. Maximal ﬂow and minimal cut in a network and Ford-

Fulkerson theorem.

Unit III: Combinatorics (15L)

Applications of Inclusion Exclusion Principle- Rook polynomial, Forbi dden position problems.

Introduction to partial fractions and Newton’s binomial theorem for real p ower series, series

expansion of some standard functions. Forming recurrence relation and ge tting a generating

function. Solving a recurrence relation using ordinary generating f unctions. System of Distinct

Representatives and Hall’s theorem of SDR.

Recommended Books.

1. Bondy and Murty; Grapgh Theory with Applications.

2. Balkrishnan and Ranganathan; Graph theory and applications.

3. Douglas B. West, Introduction to Graph Theory, 2nd Ed., Pearson, 2000

4. Richard Brualdi; Introduction to Combinatorics.

Additional Reference Book.

1. Behzad and Chartrand; Graph theory.

2. Choudam S. A.; Introductory Graph theory; 3 Cohen, Combinatorics.

## Page 29

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Course: Operations Research (Elective D)

Course Code: USMT6D4 / UAMT6D4

Unit I: Linear Programming-I (15L)

Prerequisites: Vector Space, Linear independence and dependenc e, Basis, Convex sets, Dimen-

sion of polyhedron, Faces.

Formation of LPP, Graphical Method. Theory of the Simplex Method- Stand ard form of

LPP, Feasible solution to basic feasible solution, Improving BFS, Op timality Condition, Un-

bounded solution, Alternative optima, Correspondence between BFS and extreme points. Sim-

plex Method – Simplex Algorithm, Simplex Tableau.

Unit II: Linear programming-II (15L)

Simplex Method – Case of Degeneracy, Big-M Method, Infeasible solut ion, Alternate solution,

Solution of LPP for unrestricted variable. Transportation Problem: Format ion of TP, Con-

cepts of solution, feasible solution, Finding Initial Basic Feasible Solution by North West Corner

Method, Matrix Minima Method, Vogel’s Approximation Method. Opti mal Solution by MODI

method, Unbalanced and maximization type of TP.

Unit III: Queuing Systems (15L)

Elements of Queuing Model, Role of Exponential Distribution. Pure Birth and Death Mod-

els; Generalized Poisson Queuing Model. Specialized Poisson Queu es: Steady- state Measures

of Performance, Single Server Models, Multiple Server Models, S elf- service Model, Machine-

servicing Model.

Reference for Unit III:

1. G. Hadley; Linear Programming; Narosa Publishing, (Chapter 3).

2. G. Hadley; Linear Programming; Narosa Publishing, (Chapter 4 and 9).

3. J. K. Sharma; Operations Research; Theory and Applications, (Chapter 4, 9).

4. J. K. Sharma, Operations Research, Theory and Applications.

5. H. A. Taha, Operations Research, Prentice Hall of India.

Additional Reference Books:

1. Hillier and Lieberman, Introduction to Operations Research.

2. RichardBroson, SchaumSeriesBookinOperationsResearch, TataMcGr awHillPublishing

Company Ltd.

Course: Practicals (Based on USMT601 / UAMT601 and USMT602 / UAMT602)

Course Code: USMTP07 / UAMTP07

Suggested Practicals (Based on USMT601 / UAMT601):

1. Limit continuity and derivatives of functions of complex variables.

2. Steriographic Projection , Analytic function, ﬁnding harmonic conju gate.

3. Contour Integral, Cauchy Integral Formula ,Mobius transformations.

## Page 30

28

4. Taylors Theorem , Exponential , Trigonometric, Hyperbolic functions .

5. Power Series , Radius of Convergence, Laurents Series.

6. Finding isolated singularities- removable, pole and essential, Cau chy Residue theorem.

7. Miscellaneous theory questions.

Suggested Practicals (Based on USMT602 / UAMT602)

1. Examples of rings (commutative and non-commutative), integral domains an d ﬁelds

2. Units in various rings. Determining characteristics of rings.

3. Prime Ideals and Maximal Ideals, examples on various rings.

4. Euclidean domains and principal ideal domains (examples and non-exampl es)

5. Examples if irreducible and prime elements.

6. Applications of division algorithm and Eisenstein’s criterion.

7. Miscellaneous Theoretical questions on Unit 1, 2 and 3.

Course: Practicals (Based on USMT603 / UAMT603 and USMT6A4 / UAMT6A4

OR USMT6B4 / UAMT6B4 OR USMT6C4 / UAMT6C4 OR USMT6D4 / UAMT6D4)

Course Code: USMTP08 / UAMTP08

Suggested practicals Based on USMT603 / UAMT603:

1 Continuity in a Metric Spaces

2 Uniform Continuity, Contraction maps, Fixed point theorem

3 Connected Sets , Connected Metric Spaces

4 Path Connectedness, Convex sets, Continuity and Connectedness

5 Pointwise and uniform convergence of sequence functions, properti es

6 Point wise and uniform convergence of series of functions and properti es

7 Miscellaneous Theory Questions.

Suggested Practicals based on USMT6A4 / UAMT6A4

1 Find the Laplace transform of diﬀerential and integral equations.

2 Find the inverse Laplace transform by the partial fraction method.

3 Find the Fourier integral representation of given functions.

4 Find the Fourier Sine / Cosine integral representation of given func tions.

5 Solve higher order ODE using Laplace transform.

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6 Solve one dimensional heat and wave equation using Laplace transform. Solv e initial and

boundary value problems using Fourier transform.

7 Miscellaneous Theory Questions.

Suggested Practicals based on USMT6B4 / UAMT6B4

1 Legendre Symbol.

2 Jacobi Symbol and Quadratic congruences with composite moduli.

3 Finite continued fractions.

4 Inﬁnite continued fractions.

5 Pell’s equations and Arithmetic functions of number theory.

6 Special Numbers.

7 Miscellaneous Theoretical questions.

Suggested Practicals based on USMT6C4 / UAMT6C4

1 Coloring of Graphs

2 Chromatic polynomials and connectivity.

3 Planar graphs

4 Flow theory.

5 Application of Inclusion Exclusion Principle, rook polynomial. Recu rrence relation.

6 Generating function and SDR.

7 Miscellaneous theoretical questions.

Suggested Practicals based on USMT6D4 / UAMT6D4

All practicals to be done manually as well as using software TORA / EXCEL s olver.

1 LPP formation, graphical method and simple problems on theory of simplex method

2 LPP Simplex Method

3 Big-M method, special cases of solutions.

4 Transportation Problem

5 Queuing Theory; single server models

6 Queuing Theory; multiple server models

7 Miscellaneous Theory Questions.

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(8) Scheme of Evaluation

Scheme of Examination (75:25)

The performance of the learners shall be evaluated into two parts.

❼Internal Assessment of 25 percent marks for each paper.

❼Semester End Examination of 75 percent marks for each paper.

I. Internal Evaluation of 25 Marks:

T.Y.B.Sc. :

(i) One class Test on unit I of 20 marks of duration one hour to be conducted d uring

Practical session.

Paper pattern of the Test:

Q1: Deﬁnitions/ Fill in the blanks/ True or False with Justiﬁcation (04 M arks).

Q2: Multiple choice 5 questions. (10 Marks: 5 ×2)

Q3: Attempt any 2 from 3 descriptive questions. (06 marks: 2 ×3)

(ii) Active participation in routine class: 05 Marks.

OR

Students who are willing to explore topics related to syllabus, de aling with applica-

tions historical development or some interesting theorems and their applications can

be encouraged to submit a project for 25 marks under the guidance of teach ers.

T.Y.B.A. :

(i) One class Test on unit I of 20 marks to be conducted during Tutorial s ession.

Paper pattern of the Test:

Q1: Deﬁnitions/ Fill in the blanks/ True or False with Justiﬁcation (04 M arks).

Q2: Multiple choice 5 questions. (10 Marks: 5 ×2)

Q3: Attempt any 2 from 3 descriptive questions. (06 marks: 2 ×3)

(ii) Journal : 05 Marks.

OR

Students who are willing to explore topics related to syllabus, de aling with applica-

tions historical development or some interesting theorems and their applications can

be encouraged to submit a project for 25 marks under the guidance of teach ers.

II. Semester End Theory Examinations : There will be a Semester-end external Theory

examinationof75marksforeachofthecoursesUSMT501/UAMT501,USMT502/UAMT502,

USMT503 and USMT5A4 OR USMT5B4 OR USMT5C4 OR USMT 5D4 of Semester V

andUSMT601/UAMT601,USMT602/UAMT602,USMT603andUSMT6A4ORUSMT6B4

OR USMT 6C4 OR USMT 6D4 of semester VI to be conducted by the University.

1. Duration: The examinations shall be of 2 1

2Hours duration.

2. Theory Question Paper Pattern:

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a) There shall be FOUR questions. The ﬁrst three questions Q1, Q2, Q3 s hall be

of 20 marks, each based on the units I, II, III respectively. The fourt h question

Q4 shall be of 15 marks based on the entire syllabus.

b) All the questions shall be compulsory. The questions Q1, Q2, Q3, Q4 sh all have

internal choices within the questions. Including the choices, t he marks for each

question shall be 30-32.

c) The questions Q1, Q2, Q3, Q4 may be subdivided into sub-question s as a, b, c,

d & e, etc and the allocation of marks depends on the weightage of the topic.

III. Semester End Practical Examinations :

There shall be a Semester-end practical examinations of three hours d uration and 100

marks for each of the courses USMTP05/UAMTP05, USMTP06/UAMTP056 of Semester

V and USMTP07/UAMTP07, USMTP08/UAMTP08 of semester VI.

InsemesterV,thePracticalexaminationsforUSMTP05/UAPTP05andUSMTP06/UAMTP06

are conducted by the college.

InsemesterVI,thePracticalexaminationsforUSMTP07/UAMTP07andUSMTP08/UAMTP08

are conducted by the University.

Question Paper pattern:

Paper pattern: The question paper shall have two parts A, B.

Each part shall have two Sections.

Section I Objective in nature: Attempt any Eight out of Twelve multiple choic e ques-

tions. (8 ×3 = 24 Marks)

Section II Problems: Attempt any Two out of Three. (8 ×2 = 16 Marks)

Practical Part A Part B Marks duration

Course out of

USMTP05/UAMTP05 Questions from Questions from 80 3 hours

USMT501/UAMT501 USMT502/UAMT502

USMTP06/UAMTP06 Questions from Questions from 80 3 hours

USMT503/UAMT503 USMT504/UAMT504

USMTP07/UAMTP07 Questions from Questions from 80 3 hours

USMT601/UAMT601 USMT602/UAMT602

USMTP08/UAMTP08 Questions from Questions from 80 3 hours

USMT603/UAMT603 USMT604/UAMT604

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Marks for Journals and Viva:

For each course USMT501/UAMT501, USMT502/UAMT502, USMT503/UAMT503,

USMT504/UAMT504,USMT601/UAMT601,USMT602/UAMT602USMT603/UAMT603,and

USMT604/UAMT604:

1. Journals: 5 marks.

2. Viva: 5 marks.

Each Practical of every course of Semester V and VI shall contain 10 (ten) pr oblems out of

which minimum 05 (ﬁve) have to be written in the certiﬁed journal .

xxxxx