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St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .

ST. PHILOMENA’S COLLEGE(AUTONOMOUS)

MYSORE-570015

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Sl.

No. CONTENTS Page No.

1 Preamble 3-3

2 Guidelines/Regulations 4-9

3 Scheme for CBCS curriculum / Credit distribution across Courses 10-10

4

FIRST SEMESTER MATHEMATICS Algebra-I Real Analysis-I Real Analysis-II Complex Analysis-I Linear Algebra-I

Combinatorics and Graph Theory

Numerical Analysis

11-17

5

SECOND SEMESTER MATHEMATICS Algebra-II Real Analysis-III Complex Analysis-II Linear Algebra-II

Representation Theory of Finite Groups

Theory of Partitions

Fundamentals of Mathematics(OE)

18-24

6

THIRD SEMESTER MATHEMATICS Elements of Functional Analysis Topology-I Graph Theory Commutative Algebra

Algebraic Number Theory

Galois Theory

Mathematical Computation (Interdisciplinary-Mathematics And computer science-student and teacher sharing)

Fundamentals of Mathematics (OE)

25-32

7

FOURTH SEMESTER MATHEMATICS Measure and Integration Topology-II Project Work Ordinary and Partial Differential Equation Theory of Numbers Advanced Graph Theory Deferential Geometry

33-39

8 Blue print of the question paper 40


PREAMBLE

The course was commenced during the academic year 2016-17, This is the first revision of syllabus, the

department has introduced one interdisciplinary soft core paper in the third semester. In this revised

syllabus the paper Linear Algebra has been split in to two papers which pave the way for depth

analyzing. The department has introduced new soft-core paper Numerical Analysis for the first

semester. Teaching method includes the Collaboration of lecture method, problem solving and

discussion.

This syllabus is framed to give sound knowledge with understanding of Mathematics to Post graduate

students of two years of M.Sc. degree course. This program endeavors to provide students a broad

based training in Mathematics with a basic knowledge as well as exposing them to the exciting

advancements in the field.

Mathematics is a discipline of multiple perspectives. The beauty of Mathematics lies in its simplicity

and freedom. The field of Mathematics Education looks into the pertinent gap between Essence of

Mathematics and Teaching of Mathematics.

The syllabi give the foundation of Mathematics and evolution of Mathematics Education. The goal of

the syllabus is to make the study of Mathematics, interesting and encouraging to the students to study

in-depth which helps them for research. The syllabus is based on a basic and applied approach with

vigor and depth. At the same time precaution is taken to make the syllabus comparable to the syllabi of

other universities and the needs of research and its applications. Also, this syllabus is bounded with the

research projects.

The syllabus is prepared after discussion at length with number of faculty members of the subject from

different universities and research fields. The units of the syllabus are well defined, taking into

consideration the level and capacity of students.


ST. Philomena’s College (Autonomous) Mysore

COURSE – MSc in Mathematics

Two-year four semesters Choice Based Credit System (CBCS) and

Continuous Assessment Grading Pattern (CAGP) Under Post Graduate

Programme under Autonomous Structure FROM THE ACADEMIC YEAR- 2018 Onwards

Guidelines/Regulations

1. Eligibility for Admission Candidates shall have studied Mathematics as Major/Core subject in any Bachelor degree course of

any recognized University with not less than 45 % ( 40% for SC/ST and Category - I Candidates) of the

aggregate marks of all the years of course of study. The qualification and the percentage marks for

admission shall be as per the guidelines issued by University of Mysore from time to time

Note: - In case of candidates who have taken more than three years to complete their Bachelors

Degree, the percentage of mark is arrived as per the guidelines issued by University of Mysore from

time to time

2. Duration of the Programs The duration of Programme shall extend over 4 semesters (two academic years) of 20 weeks each

including instructions and examinations.

3. Maximum Period for Completion of the Programs The candidates shall complete the Programme within 4 years from the date of admission. The term

completing the Programme means passing all the prescribed examinations of the programme to become

eligible for the degree. No candidate shall be permitted to appear for the examinations after the

prescribed period for completing the Programme. Whenever a candidate opts for blank semesters/

dropped papers, he/she have to study the prevailing papers offered by the department when he /she

continues his /her studies.

4. Medium of Instruction The medium of instruction shall be English.

5. Hours of Instruction per Week There shall be 24-30 hours of instructions per week in subjects without practicals/field-work and 28-34

hours of instructions per week in subjects with practicals/field-work. These hours may be distributed

for lectures, seminars, tutorials, practicals, project-work and other modes of instruction which

individual courses may demand.

6. Attendance Each paper (theory/practical) shall be treated as an independent unit for the purpose of attendance. A

student shall attend a minimum of 75% of the total instruction hours in a paper (theory/practical)

including tutorials and seminars in each semester. There shall no provision for condonation of shortage

of attendance and a student who fails to secure 75% attendance in a paper shall be required to repeat

that semester with the payment of semester fees.

7. Guidelines to Implement CBCS & CAGP Masters Degree Programme Course: Every paper offered will have three components associated with the teaching-learning process,

namely

(a) L - Lecture (b) T - Tutorial (c) P - Practicals


Where

L - Stands for Lecture session.

T - Stands for Tutorial session consisting participatory discussion/self study/desk work/ brief seminar

presentations by students and such other novel methods that make a student to absorb and assimilate

more effectively the contents delivered in the Lecture classes.

P -Stands for Practical session and it consists of Hands on experience / Laboratory Experiments/Field

Studies/Case studies that equip students to acquire the much required skill component.

In terms of credits, every one hour session of L per week amounts to one credit per semester

and a minimum of two hour session of T or P per week amounts to one credit per semester,

over a period of one semester of 16 weeks for teaching – learning process. The total duration of a

semester is 20 weeks inclusive of semester end examination.

A paper shall have either one or two or all the three components. That means a may have only lecture

component, or only practical component or combination of any two or all the three components.

The total credit earned by a student at the end of the semester upon successfully completing the course

is equal to L + T + P of each paper.

Different papers of study are labeled and defined as follows:

Hard Core Paper

A paper which should compulsorily be studied by a candidate as a core requirement is termed as a

Hard Core Paper.

Soft Core Paper

If there is a choice or an option for the candidate to choose a paper from a pool of papers from the main

discipline subject of study or from a sister/related discipline / subject which supports the main

discipline/ subject is termed as a Soft Core Paper.

Elective Paper

Generally a paper which can be chosen from a pool of papers and which may be very specific or

specialized or advanced or supportive to the discipline / subject of study or which provides an extended

scope or which enables an exposure to some other discipline / subject / domain or nurtures the

candidate’s proficiency / skill is called an Elective Paper. Elective papers may be offered by the main

discipline / subject of study or by sister / related discipline / subject of study. A Soft Core paper may

also be considered as an elective.

Open Elective

An elective paper chosen generally from an unrelated discipline / subject, with an intention to seek

exposure is called an open elective. A core paper offered in a discipline / subject may be treated as

an elective by other discipline / subject and vice versa.

Project work / Dissertation work

It is a special paper involving application of knowledge in solving / analyzing / exploring a real life

situation / difficult problem.

Minor Project Work

A project work up to 4 credits is called Minor Project work.


Major Project Work

A project work of 6 to 8 credits is called Major Project Work.

Dissertation Work A project work can be of 10 – 12 credits. A Project /Dissertation work may be a hard core or a soft

core as decided by the BOS concerned.

8. Scheme of Instruction

8.1 A candidate has to earn a minimum of 76 credits, for successful completion of a Master’s Degree

with a distribution of credits for different papers as given in the following table.

Paper Type Credits

Hard Core A minimum of 42, but not exceeding 52

Soft Core A minimum of 16

Open Elective A minimum of 08

8.2 A candidate can enroll for a maximum of 24 credits per semester.

8.3 Only such candidates who register for a minimum of 18 credits per semester and complete

successfully 76 credits in 4 successive semesters shall be considered for declaration of ranks,

medals and are eligible to apply for student fellowship, scholarship, free ships and hostel

facilities.

9. Continuous Assessment, Earning of Credits and Award of Grades

The evaluation of the candidate shall be based on continuous assessment. The structure for evaluation

is as follows:

9.1 Assessment and evaluation processes happen in a continuous mode. However, for reporting

purposes, a semester is divided into 3 distinct components identified as

C1, C2, and C3

9.2 The performance of a candidate in a paper will be assessed for a maximum of 100 marks as

explained below.

a) The first component (C1) of assessment is for 15 marks. This will be based on test,

assignment, seminar and attendance (Class Participation).During the first half of the semester,

the first 50% of the syllabus will be completed. This shall be consolidated during the 8th week

of the semester. Beyond 8th week, making changes in C1 is not permitted. The marks for the

class participation - 91-100 % -05 marks, 81-90% - 04 marks and 75-80% -03 marks.

b) The second component (C2) of assessment is for 15 marks. This will be based on test,

assignment, seminar and attendance (Class Participation). The continuous assessment and

scores of second half of the semester will be consolidated during the 16th week of the

semester. During the second half of the semester, the remaining units in the paper will be

completed. The marks for the class participation- 91-100 % -05 marks, 81-90% 04 marks and

75-80% -03 marks

c) The outline for continuous assessment activities for Component – I (C1) and Component – II

(C2) will be proposed by the teacher(s) concerned before the commencement of the semester

and will be discussed and decided in the respective Departmental Council. The students

should be informed about the modalities well in advance. The evaluated papers / assignments

during component - I (C1) and component - II (C2) of assessment are immediately returned to


the candidates after obtaining acknowledgement in the register maintained by the concerned

teacher for this purpose.

d) During the 18th – 20th week of the semester, a semester end examination of 2 hours duration

shall be conducted for each paper. This forms the third/final component of assessment (C3)

and the maximum marks for the final component will be 70.

10. Setting Question Papers and Evaluation of Answer Scripts. a) Question papers in three sets shall be set one by the internal and two by the external

examiners. While selecting the examiners the University Guidelines to be followed.

Whenever there are no sufficient internal examiners, the Chairman of Board of Examination

[BOE] shall get the question papers set by external examiners.

b) The Board of Examiners shall scrutinize and approve the question papers and scheme of

valuation.

c) There shall be single valuation for all theory papers by external examiners. A detailed scheme

of valuation to be prepared by the department and to be provided to the external examiner

along with the answer scripts

d) The examination for Practical Work / Field Work / Project Work will be conducted jointly by

internal and external examiners. However, the BOE on its discretion can also permit two

internal examiners from the College.

e) If a paper is full of (L = 0): T: (P=0) type, then the examination for C3 component will be as

decided by the BOS concerned.

i) The details of continuous assessment are summarized in the following Table

Component Syllabus in a paper Weightage Period of continuous

assessment

C1 First 50%

of the Syllabus 15%

First half of the semester

To be consolidated by 8th week

C2 Remaining 50%

of the Syllabus 15%

Second half of the semester

To be consolidated by

16th week

C3 Semester-end examination ( all

units of the paper) 70%

To be completed during

18th – 20th Week

Final grades to be announced latest by 24th week

k) A candidate’s performance from all 3 components will be in terms of scores, and the sum of all

three scores will be for a maximum of 100 marks (15 + 15 + 70).

m) Finally, awarding the grades should be completed latest by 24th week of the Semester.

11. Minor / Major Project Evaluation Right from the initial stage of defining the problem, the candidate has to submit the progress

reports periodically and also present his / her progress in the form of seminars in addition to the

regular discussion with the guide. Components of evaluation are as follows:

Component – I (C1): Periodic Progress and Progress Reports ( 15)

Component – II (C2): Results of Work and Draft Report (15)

Component – III(C3): Final Viva Voce and evaluation (70). The report evaluation is for 40 and

the Viva –voce examination is for 30.

12. In case a candidate secures less than 30% in C1 and C2 put together in a paper, the candidate is

said to have DROPPED that paper, and such a candidate is not allowed to appear for C3 in that

paper.

In case a candidate’s class attendance in a paper is less that 75% or as stipulated by the College,

the candidate is said to have DROPPED that paper, and such a candidate is not allowed to

appear for C3 in that paper.

Teachers offering the papers will place the above details in the P G Department Council

meeting during the last week of the semester, before the commencement of C3 and subsequently

a notification pertaining to the above will be brought out by the Principal before the


commencement of C3 examination. A copy of this notification shall also be sent to the office of

the Controller of Examinations.

12.1 In case a candidate secures less than 30% in C3 he/she may choose DROP/MAKEUP option.

In case a candidate secures more than or equal to 30% in C3 but his/her grade (G) = 4, as per

section 12.5 below, then he/she may be declared to have been conditionally successful in that

paper, provided that such a benefit of conditional clearance based on G = 4 shall not be availed

for a maximum of 8 credits for the entire programme of Master’s Degree of two years.

A candidate exercising his/her option to MAKEUP examination shall be declared passes if

he/she secures more than or equal to 30% in C3 provided he/she fulfils the conditions mentioned

in the Para 12.1 & 12.5. To a candidate who does not pass in MAKE UP examination , no

separate MAKEUP examination shall be conducted. Such a candidate has to appear for the

examination as and when the C3 component examination is conducted for Odd & Even semester

of that academic year along with the regular candidates..

12.2 A candidate has to re-register for the DROPPED paper when the paper is offered again by the

department if it is a hard core paper. The candidate may choose the same or an alternate

core/elective in case the dropped paper is soft core/ elective paper. A candidate who is said to

have DROPPED project work has to re-register for the same subsequently within the stipulated

period. The details of any dropped paper will not appear in the grade card.

12.3 The tentative / provisional grade card will be issued by the Registrar (Evaluation) at the end of

every semester indicating the papers completed successfully. This statement will not contain the

list of DROPPED papers.

12.4 Upon successful completion Master’s degree a final grade card consisting of grades of all

papers successfully completed by the candidate will be issued by the Registrar (Evaluation).

12.5 The Grade (G) and the Grade Point (GP) earned by the candidate in the subject will be as given

below.

P G GP = V x G

90 – 100 10 V x 10

80 – 89 9 V x 9

70 – 79 8 V x 8

60 – 69 7 V x 7

50 – 59 6 V x 6

40 – 49 5 V x 5

30 – 39 4 V x 4

0 – 30 0 V x 0

Here, P is the percentage of marks P= [(C1+C2) + C3] secured by a candidate in a paper which is

rounded to nearest integer. V is the credit value of paper. G is the Grade and GP is the Grade

Point.

12.6 A candidate can withdraw any paper within ten days from the date of notification of final results

of that semester. Whenever a candidate withdraws a paper, he/she has to register for the same

paper in case it is hard core paper, the same paper or an alternate paper if it is soft core/open

elective.

A DROPPED paper is automatically considered as a paper withdrawn.

12.7 The Semester Grade Point Average (SGPA) of a candidate after successful completion the

required number of credits (76) is given by

SGPA = creditsofnumberTotal

GP

12.8 The Final Semester Grade Point Average (SGPA) of a candidate after successful completion the

required number of credits (76) is given by


CGPA = SemesterstheallofCredits

SemestersfourtheallofGP

13. Classification of results The Final Cumulative Grade Point (FGP) to be awarded to the student is based on CGPA

secured by the candidate and is given as follows:

CGPA FGP

Numerical Index Qualitative Index

4< = CGPA < 5 5 SECOND CLASS

5< = CGPA < 6 6

6< = CGPA < 7 7 FIRST CLASS

7< = CGPA < 8 8

8< = CGPA < 9 9 DISTINCTION

9< = CGPA < 10 10

Overall percentage = 10 x CGPA or is said to be 50% in case CGPA < 5


ST. PHILOMENA’S COLLEGE (Autonomous), MYSURU-570 015

M Sc - Mathematics - Course Structure[credits distribution] I Semester Total

Credits Sl. No Code Title Type

(HC/SC)

L:T P Credit

1 Algebra-I HC 4:0:0 4

20

2 Real Analysis-I HC 4:0:0 4

3 Real Analysis-II HC 4:0:0 4

4 Complex Analysis-I HC 4:0:0 4

Any One of the following

5 Linear Algebra-I SC 4:0:0 4

6 Combinatorics and Graph Theory SC 4:0:0 4

7 Numerical Analysis SC 4:0:0 4

II Semester

1 Algebra-II HC 4:0:0 4

20

2 Real Analysis-III HC 4:0:0 4

3 Complex Analysis-II HC 4:0:0 4

Any One of the following

4 Linear Algebra-II SC 4:0:0 4 5 Representation Theory of Finite Groups SC 4:0:0 4 6 Theory of Partitions SC 4:0:0 4 Open Elective

– offered from other departments

7 OE 4:0:0 4

III Semester

1 Elements of Functional Analysis HC 4:0:0 4

20

2 Topology-I HC 4:0:0 4

Any two of the following

3 Graph Theory SC 4:0:0 4 4 Commutative Algebra SC 4:0:0 4

5 Algebraic Number Theory SC 4:0:0 4

6 Galois Theory SC 4:0:0 4

7 Mathematical Computation (Interdisciplinary -Mathematics And computer science-student

and teacher sharing))

SC 3:0:2 4

Open Elective

– offered from other departments

8 OE 4:0:0 4

IV Semester

1 Measure and Integration HC 4:0:0 4

20

2 Topology-II HC 4:0:0 4

Any Three of the following 4 3 Project Work C 4:0:0 4 4 Ordinary and Partial Differential Equation SC 4:0:0 4 5 Theory of Numbers SC 4:0:0 4

6 Advanced Graph Theory SC 4:0:0 4

7 Deferential Geometry SC 4:0:0 4

Open Elective

– offered from our department in II and III sem.

Fundamentals of Mathematics OE 4:0:0 4

HC-44 + SC-28 + OE-8 = 80 80


FIRST SEMESTER

TITLE: ALGEBRA-I

Type: Hard Core

CLASS DURATION – 04 HOURS PER WEEK 64 Hours

Marks-Theory - 70 + Internal Assessment -30= 100

OBJECTIVE: To motivate the students to develop important algebraic concepts through group theory

LEARNING OUTCOME: After completing this paper, the student will be able to get the glimpse of

Number theory and to identify the concept of Normal, Quotients and Permutation groups along with

Sylow's theorem and its applications.

Unit-1 Number Theory 16hrs 1.1 Congruences, residue classes, 1.2 Fermat theorem, Euler and Wilson theorem 1.3 Linear congruences, 1.4 Elementary arithmetical functions, 1.5 Primitive roots

1.6 Quadratic residues and the law of quadratic reciprocity

Unit-2 Groups 16hrs

2.1 Binary operation, definition of algebraic structure and groups

2.2 Subgroups and cosets

2.3 Lagrange’s Theorem, cyclic subgroups

2.4 Normal subgroups and factor groups

Unit-3 Isomorphism 16hrs

3.1 Homomorphism- kernel and image 3.2 The fundamental theorem of homomorphism,

3.3 two laws of isomorphism

Unit-4 Permutation Groups 16hrs

4.1 Group of permutations

4.2 Cayley’s theorem,

4.3 Sylow's theorems.

Reference Books:

1 D. M. Burton – Elementary Number Theory, Tata McGraw-Hill, New Delhi, 6th Ed., 2 I. Niven, H. S. Zuckerman and H. L. Montgomery – An Introduction to the Theory of

Numbers, New York, John Wiley Sons, Inc., 2004, 5th Ed. 3 G. A. Jones and J. M. Jones – Elementary Number Theory, Springer, 1998. 4 Thomas W. Hungerford – Algebra, Springer International Edition, New York. 5 Michael Artin – Algebra, Prentice-Hall of India, New Delhi 6 J. A. Gallian – Contemporary Abstract Algebra, Narosa Publishing House, 4th Ed., 7 D. S. Dummit and R. M. Foote – Abstract Algebra, John Wiley and Sons, 1999. 8 I. N. Herstein – Topics in Algebra, Vikas Publishing House, New Delhi. 9 J. B. Fraleigh – A First course in Abstract Algebra, Addison-Wesley,

10 N. S. Gopalakrishnan – University Algebra, New Age International, 2nd Ed.,


FIRST SEMESTER

TITLE: REAL ANALYSIS-I

Type: Hard Core



OBJECTIVE: This course is concerned with basic properties of the field of real numbers, and the

concept of sequences , series and their infinite products.

LEARNING OUTCOME: Students will be able to understand and explain the basic concepts of

sequences, series and their infinite products

Unit-1 Properties of Real Numbers 16hrs 1.1 The extended real number system 1.2 n-dimensional Euclidean space 1.3 Binomial inequality 1.4 The inequality of the power means 1.5 Cauchy’s Holder’s and Minkowski’s inequality

Unit-2 Sequences of Real Numbers 16hrs

2.1 Numerical sequences

2.2 Convergent sequences

2.3 Cauchy sequences

Unit-3 Series of Real Numbers-I 16hrs 3.1 Series of real numbers,

3.2 Series of non-negative terms,

3.3 The number `e’, and test of convergence

Unit-4 Series of Real Numbers-II 16hrs

4.1 Multiplications of series,

4.2 Re-arrangements.

4.3 Double Series, infinite products

Reference Books:

1 W. Rudin – Principles of Mathematical Analysis, Int. Student edition, McGraw Hill, 3rd Ed

2 T. M. Apostal – Mathematical Analysis, Addison Wesley, Narosa, New Delhi, 2nd Ed. 3 R. R. Goldberg – Methods of real Analysis, Oxford and IBH, New Delhi 4 Torence Tao – Analysis I and Analysis II, Hindustan Book Agency, India, 2006. 5 Kenneth A. Ross – Elementary Analysis: The Theory of Calculus, Springer Inte, Edition,

2004.


FIRST SEMESTER

TITLE: REAL ANALYSIS-II

Type: Hard Core



OBJECTIVE: This course is concerned with the fundamentals of mathematical analysis which makes

the students to understand the concept of countable sets, Riemann - Stieltjes Integration and to work

comfortably with continuity and differentiability of functions.

Learning Outcome: Students will attain the knowledge of basic topology and the elements of

calculus and Riemann- Stieltje’s Integral

Unit-1 Basic Topology 16hrs 1.1 Finite sets, 1.2 Countable and uncountable sets, 1.3 The topology of the real line

Unit-2 Limits and Continuity 16hrs

2.1 Limit of a function

2.2 Continuous functions,

2.3 Properties of continuous functions, discontinuities,

2.4 Monotonic functions.

Unit-3 Differentiation 16hrs

3.1 Differentiability,

3.2 mean value theorems,

3.3 L’ Hospital rule, 3.4 Taylor’s theorem,

3.5 Maxima and minima,

3.6 Functions of bounded variation.

Unit-4 Riemann-Stieltje’s Integral 16hrs

4.1 Definition and existence of integral.

4.2 Properties of the integral,

4.3 Integration and differentiation.

4.4 First and second mean value theorems.

Reference Books:

1 W. Rudin – Principles of Mathematical Analysis, Internatl Student edition, McGraw Hill, 3rd

Ed

2 T. M. Apostal – Mathematical Analysis, Addison Wesley, Narosa, New Delhi, 2nd Ed. 3 R. R. Goldberg – Methods of real Analysis, Oxford and IBH, New Delhi 4 Torence Tao – Analysis I and Analysis II, Hindustan Book Agency, India, 2006. 5 Kenneth A. Ross – Elementary Analysis: The Theory of Calculus, Springer Inte, Edition,

2004.


FIRST SEMESTER

TITLE: COMPLEX ANALYSIS-I

Type: Hard Core



OBJECTIVE: This course is aimed to provide an introduction with the exploration of the algebraic,

geometric and topological structures of the complex number field. The concepts of analyticity, Cauchy-

Riemann relations and harmonic functions are then introduced.

LEARNING OUTCOME: Students will be able to understand and explain the basic concepts of

complex analysis such as- sequence, series, continuity, analytic functions and their differentiation and

integration.

And will be able to explain the proofs of Cauchy’s theorems

Unit-1 Fundamentals of Complex Numbers 16hrs 1.1 Algebra of complex numbers, 1.2 Geometric representation of complex numbers. 1.3 Riemann sphere and Stereographic projection, 1.4 Lines, Circles. Limits and Continuity.

Unit-2 Sequence and Series 16hrs

2.1 Analytic functions

2.2 Cauchy-Riemann equations

2.3 Harmonic functions

2.4 Elementary theory of power series – sequences

2.5 Uniform convergence of power series

2.6 Abel’s limit theorem

Unit-3 Topology and Complex Integration 16hrs

3.1 Topology of the complex plane. 3.2 Linear fractional transformations, Cross-ratio, Symmetry

3.3 Elementary conformal mappings.

3.4 Complex integration – Line integrals

Unit-4 Cauchy’s Theorems 16hrs

4.1 Cauchy’s theorem for a rectangle.

4.2 Cauchy’s theorem in a Circular disk,

4.3 Cauchy’s integral formula.

Reference Books: 1 L. V. Ahlfors – Complex Analysis, McGraw-Hill, Kogakusha, 1979. 2 J. B. Conway – Functions of one complex variable, Narosa, New Delhi. 3 R. P. Boas – Invitation to Complex Analysis, The Random House, 1987 4 B. C. Palka – An Introduction to Complex Function Theory, Springer, 1991. 5 S. Ponnusamy – Foundations of Complex Analysis, Narosa, 1995.


FIRST SEMESTER

TITLE: LINEAR ALGEBRA-I

Type: Soft Core



OBJECTIVE: The main objective is to introduce basic notions in linear algebra that are often used in

mathematics and other sciences. It develops the basic ideas of vector spaces and provides strong

background of linear transformations, linear functional, linear operator and matrix theory.

LEARNING OUTCOME: At the end of the course Student will be able to explain the elements of

vector space, linear transformations and matrix theory.

Unit-1 Vector Spaces 16hrs 1.1 Definitions and examples of Vector Spaces, Subspaces

1.2 Linear Combinations and Systems of Linear Equations

1.3 Linear Dependence and Linear Independence, 1.4 Bases and Dimension, 1.5 Maximal Linearly Independent and Minimal Generating sets;

Unit-2 Linear Transformations 16hrs

2.1 Definitions, examples of Linear Transformations

2.2 Null Spaces, and Ranges,

2.3 The Matrix Representation of a Linear Transformation

2.4 Composition of Linear Transformations and Matrix Multiplication

2.5 Invertibility and Isomorphisms

2.6 The Change of Coordinate Matrix and The Dual Space

Unit-3 Matrix Operations 16hrs

3.1 Elementary Matrix Operations and Elementary Matrices

3.2 The Rank of a Matrix and Matrix Inverses

3.3 Systems of Linear Equations.

3.4 Properties of Determinants and Cofactor Expansions

Unit-4 Eigenvalues and Eigenvectors of Matrices 16hrs

4.1 Elementary Operations and Cramer’s Rule

4.2 Eigenvalues and Eigenvectors, Diagonalizability

4.3 Invariant Subspaces and the Cayley-Hamilton Theorem

Referenc e Books:

1 S. Friedberg, A. Insel, and L. Spence - Linear Algebra, Fourth Edition, PHI, 2009. 2 Jimmie Gilbert and Linda Gilbert – Linear Algebra and Matrix Theory, Academic Press, 3 I. N. Herstein – Topics in Algebra, Vikas Publishing House, New Delhi. 4 Hoffman and Kunze – Linear Algebra, Prentice-Hall of India, 1978, 2nd Ed., 5 P. R. Halmos – Finite Dimensional Vector Space, D. Van Nostrand, 1958. 6 S. Kumeresan – Linear Algebra, A Geometric approach, Prentice Hall India, 2000. 7 Seymour Lipschutz and Marc Lipson, Schaum's Outline of Linear Algebra, McGraw-Hill

Companies, Inc. 4th Ed

https://www.amazon.in/Seymour-Lipschutz/e/B000APW4UE/ref=dp_byline_cont_book_1

https://www.amazon.in/Marc-Lipson/e/B001IOFIYA/ref=dp_byline_cont_book_2


FIRST SEMESTER TITLE: COMBINATORICS AND GRAPH THEORY

Type: Soft Core



OBJECTIVE: This course gives the idea of partially ordered sets, Lattices, Boolean Algebra,

Permutations, Combinations and basic concepts of Graph Theory.

LEARNING OUTCOME: At the end of the course students will be capable to understand partially

ordered sets, Lattices, Boolean Algebra, Permutations, Combinations and the fundamentals of Graph

Theory.

Unit-1 Basics of Lattices. 16hrs 1.1 Partially ordered sets, Lattices, Complete lattices, Distributive Lattices, 1.2 Complements, Boolean Algebra, Boolean expressions, 1.3 Application to switching circuits.

Unit-2 Permutations and Combinations, 16hrs

2.1 Permutations and Combinations,

2.2 Pigeon-hole principle,

2.3 Principle of inclusion and exclusion

Unit-3 Basics of Graphs 16hrs

3.1 Definition, types of graphs, 3.2 Walks and connectedness, Degrees,

3.3 Operations on graphs,

Unit-4 Blocks and acyclic graphs 16hrs 4.1 Blocks - Cutpoints, bridges

4.2 Block graphs and Cutpoint graphs.

4.3 Tree-Elementary properties of trees

Reference Books:

1 C. L. Liu – Elements of Discrete Mathematics, McGraw-Hill, 1986. 2 Kenneth H. Rosen – Discrete Mathematics and its Applications, McGraw-Hill, 2002. 3 F. Harary – Graph Theory, Addition Wesley Reading Mass, 1969 4 K. R. Parthasarathy – Basic Graph Theory, Tata McGraw-Hill, New Delhi, 1994

5 G. Chartand and L. Lesniak – Graphs and Diagraphs, wadsworth and Brooks, 2nd Ed. 6 Clark and D. A. Holton – A First Look at Graph Theory, Allied publishers 7 D. B. West – Introduction to Graph Theory, Pearson Education Inc.,2001, 2nd Ed., 8 J. A. Bondy and U. S. R. Murthy – Graph Theory with applications, Elsevier, 1976


FIRST SEMESTER

TITLE: NUMERICAL ANALYSIS.

Type: Soft Core



OBJECTIVE: To describe different approaches to solve a polynomial equation and to solve systems of linear equations

LEARNING OUTCOME: Student will be able to solve Polynomial Equations ,System of Linear

Algebraic Equations and Eigen value problems

Unit-1 Transcendental and Polynomial Equations 16hrs

1.1 Introduction, The bisection method, Iteration methods based on first degree equation

1.2 Iteration methods based on second degree equation, 1.3 Rate of convergence, Rate of convergence of Secant and Newton-Raphson

method, Iteration methods, first order method, second order method, higher order methods.

1.4 Polynomial equations, Descartes’ Rule of Signs, The Birge-Vieta method.

Unit-2 Solutions of System of Linear Equations by Direct Methods 16hrs

2.1 Gauss elimination method,

2.2 Gauss-Jordan method

2.3 Triangularization method

2.4 Cholesky method

Unit-3 Solutions of System of Linear Equations by Iteration methods 16hrs

3.1 Jacobi iteration method,

3.2 Gauss-Seidel iteration method,

3.3 Convergence analysis, Eigenvalues and eigenvectors

Unit-4 Interpolation and Approximation 16hrs 4.1 Introduction, Lagrange and Newton interpolations

4.2 Linear and Higher order interpolation

4.3 Finite difference operators

4.4 Interpolating polynomials using finite differences

4.5 Hermite interpolation, Approximations

Reference Books:

1 M. K. Jain, S. R. K. Iyengar – R. K. Jain, Numerical Methods for Scientific and Engineering

Computation, New Age International, 6th Edition 2012. 2 C. F. Gerald and P. O. Wheatly – Applied Numerical Analysis, Pearson Education, Inc., 1999 3 A. Ralston and P. Rabinowitz – A First Course in Numerical Analysis, 2nd Edition,

McGraw - Hill, New York, 1978 4 K. Atkinson – Elementary Numerical Analysis, 2nd Edition, John Wiley and Sons, Inc., 1994 5 P. Henrici – Elements of Numerical Analysis, John Wiley and Sons, Inc., New York, 1964


SECOND SEMESTER TITLE: ALGEBRA-II

Type: Hard Core



OBJECTIVE: This course is concerned with the ring theory and field extensions to develop

computational skill in abstract algebra which makes student to have a good mathematical maturity and

enables to build mathematical thinking and skill.

LEARNING OUTCOME: Student will gain the knowledge in ring theory and field extensions.

Unit-1 Rings 16hrs 1.1 Rings, Integral domains and Fields, 1.2 Homomorphisms, 1.3 Ideals and Quotient Rings, 1.4 Prime and Maximal ideals.

Unit-2 Ideal 16hrs

2.1 Euclidean and principal ideal rings,

2.2 Polynomials, Zeros of a polynomial,

2.3 Factorization,

2.4 Irreducibility criterion.

Unit-3 Fileds 16hrs

3.1 Adjunction of roots,

3.2 Algebraic and transcendental extensions

3.3 Finite fields.

Unit-4 Extensions of fields 16hrs

4.1 Separable and inseparable extensions

4.2 Perfect and imperfect fields

4.3 Theorem on the primitive element.

Reference Books:

1 Thomas W. Hungerford – Algebra, Springer International Edition, New York. 2 Michael Artin – Algebra, Prentice-Hall of India, New Delhi.

3 Joseph A. Gallian – Contemporary Abstract Algebra, Narosa, 4th Ed., 4 D. S. Dummit and R. M. Foote – Abstract Algebra, John Wiley and Sons,1999, 2nd Ed., 5 I. N. Herstein – Topics in Algebra, Vikas Publishing House, New Delhi. 6 J. B. Fraleigh – A First course in Abstract Algebra, Addison-Wesley 7 N. S. Gopalakrishnan – University Algebra, New Age International, 2nd ed.


SECOND SEMESTER

TITLE: REAL ANALYSIS-III

Type: Hard Core



OBJECTIVE: This course will focus on the sequences and series functions and their convergence,

uniform convergence, differentiation and helps to understand the concept of functions of several along

with proofs of Taylor`s theorem

LEARNING OUTCOME: At the end of this course Student will get the strong base in sequence and

series of function. And will be capable to analyze some special functions and functions of several

variable

Unit-1 Sequences and series of functions 16hrs

1.1 Sequences and series of functions 1.2 Discussions of main problem 1.3 Uniform convergence, 1.4 Uniform convergence and continuity 1.5 Uniform convergence and integration 1.6 Uniform convergence and differentiation

Unit-2 Special functions. 16hrs

2.1 Power series,

2.2 The exponential and logarithmic functions,

2.3 The trigonometric functions.

2.4 Improper integrals and their convergence.

Unit-3 Functions of two Variables. 16hrs

Functions of two variables.

3.1 Partial derivatives

3.2 Continuity and differentiability, 3.3 The chain rule,

3.4 Jacobians.

Unit-4 Implicit Function and Taylor's Theorem 16hrs

4.1 The Implicit function theorem,

4.2 Taylor's theorem,

4.3 Maxima and Minima,

4.4 Lagrange's multipliers.

Reference Books:

1 W. Rudin – Principles of Mathematical Analysis, International Student edition, McGraw-

Hill, 3rd Ed., 2 T.M. Apostal – Mathematical Analysis, Addison Wesley, Narosa, 2nd Ed., 3 R.R. Goldberg – Methods of Real Analysis, Oxford and IBH, New Delhi. 4 D.V. Widder – Advanced Calculus, Prentice Hall of India, New Delhi, 2nd Ed.,

5 Torence Tao – Analysis I, Hindustan Book Agency, India, 2006. 6 Torence Tao – Analysis II, Hindustan Book Agency, India, 2006. 7 Kenneth A. Ross – Elementary Analysis: The Theory of Calculus, Springer International

Edition, 2004.


SECOND SEMESTER

TITLE: COMPLEX ANALYSIS-II

Type: Hard Core



OBJECTIVE: The objective of this course is to introduce the fundamental ideas of the calculus of

residues, harmonic functions and power series expansion also helps to understand the concept of the

gamma and Beta functions, Taylor series, Laurents series in complex form.

LEARNING OUTCOME: After learning this course students will be able to understand the calculus

of residues, harmonic functions and power series expansion, the gamma and Beta functions, Taylor

series, Laurents series in complex form.

Unit-1 The Calculus of Residues 16hrs

1.1 The residue theorem, 1.2 Argument principle, 1.3 Evaluation of definite integrals

Unit-2 Harmonic Functions 16hrs

2.1 Definition and basic properties,

2.2 Mean value property,

2.3 Poisson’s formula,

2.4 Schwarz’s theorem

2.5 Reflection principle.

Unit-3 Power Series Expansions 16hrs 3.1 The Weierstrass theorem,

3.2 The Taylor series,

3.3 The Laurent series.

3.4

Unit-4 Entire functions 16hrs

4.1 Mittag - Leffer’s theorem,

4.2 Infinite products, Canonical products,

4.3 The Gamma and Beta functions,

4.4 Sterling’s formula.

4.5 Entire functions – Jensen’s formula, 4.6 Hadamard’s theorem.

Reference Books:

1 L. V. Ahlfors – Complex Analysis, McGraw-Hill, Kogakusha, 1979. 2 J. B. Conway – Functions of one complex variable, Narosa, New Delhi. 3 R. P. Boas – Invitation to Complex Analysis, The Random House, 1987. 4 B. C. Palka – An Introduction to the Complex Function Theory, Springer, 1991. 5 S. Ponnusamy – Foundations of Complex Analysis, Narosa, 1995.


SECOND SEMESTER

TITLE: LINEAR ALGEBRA-II

Type: Soft Core



OBJECTIVE: To provide sound knowledge in theory and applied part of linear algebra

LEARNING OUTCOME: After studying this course student will be able to discuss the Inner

Products Space and illustrate Bilinear, Canonical and Quadratic Forms.

Unit-1 Inner Products Space 16hrs 1.1 Inner Products and Norms, 1.2 The Gram-Schmidt Orthogonalization Process 1.3 Orthogonal Complements

Unit-2 Linear Operator 16hrs

2.1 The Adjoint of a Linear Operator

2.2 Normal and Self-Adjoint Operators

2.3 Unitary and Orthogonal Operators and Their Matrices

Unit-3 Orthogonal Projections 16hrs

3.1 Orthogonal Projections

3.2 The Spectral Theorem

3.3 Bilinear and Quadratic Forms

Unit-3 Canonical Forms 16hrs

4.1 The Diagonal form, The Triangular form

4.2 The Jordan Canonical Form

4.3 The Minimal Polynomial; The Rational Canonical Form

Reference Books:

1 S. Friedberg, A. Insel, and L. Spence - Linear Algebra, Fourth Edition, PHI, 2009. 2 Jimmie Gilbert and Linda Gilbert – Linear Algebra and Matrix Theory, Academic Press, 3 I. N. Herstein – Topics in Algebra, Vikas Publishing House, New Delhi. 4 Hoffman and Kunze – Linear Algebra, Prentice-Hall of India, 1978, 2nd Ed., 5 P. R. Halmos – Finite Dimensional Vector Space, D. Van Nostrand, 1958.

6 S. Kumeresan – Linear Algebra, A Geometric approach, Prentice Hall India, 2000. 7 Seymour Lipschutz and Marc Lipson, Schaum's Outline of Linear Algebra, McGraw-Hill

Companies, Inc.

https://www.amazon.in/Seymour-Lipschutz/e/B000APW4UE/ref=dp_byline_cont_book_1

https://www.amazon.in/Marc-Lipson/e/B001IOFIYA/ref=dp_byline_cont_book_2


SECOND SEMESTER TITLE: REPRESENTATION THEORY OF FINITE GROUPS

Type: Soft Core



OBJECTIVE: To give the idea of classical groups, general linear group, orthogonal group, simplistic

group, unitary group conjugate representation, number of irreducible representations.

LEARNING OUTCOME: A sound knowledge in classical groups, general linear group, orthogonal

group, simplistic group, unitary group conjugate representation, number of irreducible representations

is attained.

Unit-1 Classical Groups 16hrs 1.1 General linear group 1.2 Orthogonal group, 1.3 Symplectic group, 1.4 Unitary group.

Unit-2 Group Representation 16hrs

2.1 Group representation,

2.2 Conjugate representation,

2.3 G-invariant spaces – irreducible representations – Schur’s lemma.

Unit-3 Applications 16hrs

3.1 Maschke’s theorem – characters. 3.2 Orthogonality relations for characters

3.3 Number of irreducible representations

Unit-IV Permutation representations 16hrs

4.1 Regular representation.

4.2 Representations of Symmetric groups.

4.3 Representation of Finite abelian groups – Dihedral groups.

Reference Books:

1 Alperin, J. L.; Bell R. B., “Groups and Representations”, Graduate Texts in Mathematics,

162, Springer-Verlag, New York, 1995 2 Curtis C. W.; Reiner I., “Representation theory of finite groups and associative algebras”,

Pure and Applied Mathematics, Vol. XI Interscience Publishers, a division of John Wiley &

Sons, New York-London 1962 3 Dummit D. S.; Foote R. M. “Abstract algebra”, Third edition, John Wiley & Sons, Inc.,

Hoboken, NJ, 2004. 4 Fulton; Harris, “Representation theory: A first course” Graduate Texts in

Mathematics, 129, Readings in Mathematics, Springer-Verlag, New York, 1991. 5 James, Gordon; Liebeck, Martin, “Representations and characters of groups”, Second edition,

Cambridge University Press, New York, 2001 6 Musili C. S., “Representations of finite groups” Texts and Readings in Mathematics,

Hindustan Book Agency, Delhi, 1993


SECOND SEMESTER OPEN ELECTIVE PAPER

TITLE: FUNDAMENTALS OF MATHEMATICS

Type: Elective


Marks-Theory- 70 + Internal Assessment- 30 = 100

OBJECTIVE: This course is concerned with the Foundations of mathematics which makes the

students to understand the concept of Permutations and Combinations, theory of matrices and gives an

idea about probability.

LEARNING OUTCOME: Student will attain the knowledge of Permutations and Combinations,

theory of matrices and gives an idea about probability

Unit-1 Logarithms and Progression 14hrs 1.1 Definitions and properties of Logarithms and problems thereon 1.2 Arithmetic, Geometric and Harmonic progression: Properties and problems

thereon

Unit-2 Matrix Algebra 18hrs

2.1 Types of matrices : Definitions, Examples.

2.2 Transpose of a matrix and problems

2.3 Definition and Properties of determinants (without proof).

2.4 Singular & non-singular matrix and problems.

2.5 Co-factors matrix, Cramer’s Rule,

2.6 Adjoint matrix, inverse of a matrix,

Unit-3 Permutations and Combinations 16hrs 3.1 Permutations and Combinations,



Unit-4 Probability 16hrs

4.1 Probability of events

4.2 Condition probability

4.3 Baye`s theorem

4.4 Distribution function: Binomial

Reference Books:

1 Shirali, Shailesh, A Primer on Logarithms, Hyderabad: Universities Press, (2002) 2 Kate, S.K.; Bhapkar, H.R. (2009), Basics Of Mathematics, Pune: Technical Publications 3 Gerard Meurant, Ronald Lewis Graham, Martin Grotschel , Laszlo Lovasz , Handbook of

Combinatorics, North Holland 4 S. Lipschutz and M. Lipson: Theory and Problems of Discrete Mathematics. Schaum Series.

2nd Ed. Tata McGraw Hill 5 A.K. Md. Ehsanes Saleh Vijay K. Rohatgi, An Introduction to Probability and Statistics

6 S.C. Gupta, Fundamentals of Mathematical Statistics, Sultan Chand & Sons (2014)

https://books.google.com/books?id=0b0igbb3WaQC&printsec=frontcover#v=onepage&q&f=false

https://books.google.com/books?id=v4R0GSJtEQ4C&pg=PR1#v=onepage&q&f=false

http://www.amazon.in/s/ref=dp_byline_sr_book_1?ie=UTF8&field-author=Gerard+Meurant&search-alias=stripbooks

http://www.amazon.in/s/ref=dp_byline_sr_book_2?ie=UTF8&field-author=Ronald+Lewis+Graham&search-alias=stripbooks

http://www.amazon.in/s/ref=dp_byline_sr_book_3?ie=UTF8&field-author=Martin+Grotschel&search-alias=stripbooks

http://www.amazon.in/s/ref=dp_byline_sr_book_4?ie=UTF8&field-author=Laszlo+Lovasz&search-alias=stripbooks

http://www.amazon.in/s/ref=dp_byline_sr_book_1?ie=UTF8&field-author=A.K.+Md.+Ehsanes+Saleh+Vijay+K.+Rohatgi&search-alias=stripbooks

http://www.amazon.in/s/ref=dp_byline_sr_book_1?ie=UTF8&field-author=S.C.+Gupta&search-alias=stripbooks


SECOND SEMESTER

TITLE: THEORY OF PARTITIONS

Type: Soft Core



OBJECTIVE: This course gives the idea of partition of numbers, generating function, Jacobi’s triple

product, summations formula and their applications. Also it paves the way to study identities like

Rogers-Ramanujan, Euler’s, Gauss, Heine’s, Jacobi’s identities.

LEARNING OUTCOME: Student will gain the knowledge in concepts of partition of numbers,

generating function, Jacobi’s triple product, summations formula and their applications.

Unit-1 Partitions 16hrs 1.1 Partitions of numbers,

1.2 The generating function of p(n) 1.3 Other generating functions

Unit-2 Euler theorems and its applications 16hrs

2.1 Two theorems of Euler

2.2 Jacobi's triple product identity and its applications

Unit-3 Summation Formula and its Applications 16hrs

3.1 1ψ1 - summation formula and its applications,

3.2 Combinatorial proofs of Euler's identity

3.3 Euler's pentagonal number theorem

3.4 Franklin's combinatorial proof

Unit-4 Congruence Properties 16hrs

4.1 Congruence properties of partition function,

4.2 The Rogers - Ramanujan Identities

Reference Books:

1 G. H. Hardy and E. M. Wright – An Introduction to Theory of Numbers, Oxford University

Press, 1979, 5th Ed., 2 I. Niven, H. S. Zuckerman and H. L. Montgomery – An Introduction to the Theory of

Numbers, New York, John Wiley and Sons, Inc., 2004, 5th Ed 3 Bruce C. Berndt – Ramanujan's Note Books Volumes-1 to 5 4 G. E. Andrews – The Theory of Partitions, Addison Wesley, 1976 5 A. K. Agarwal, Padmavathamma, M. V. Subbarao – Partition Theory, Atma Ram & Sons,

Chandigarh, 2005


THIRD SEMISTER

TITLE: ELEMENTS OF FUNCTIONAL ANALYSIS

Type: Hard Core



OBJECTIVE: This course imparts an in-depth analysis of Banach spaces, Hilbert spaces, etc. This

course also includes a few important applications of functional analysis to other branches of both pure

and applied mathematics.

LEARNING OUTCOME: Student will get thorough knowledge on basics concepts of Banach

spaces and Hilbert spaces.

Unit-1 Metric Completion. 16hrs 1.1 Metric completion. 1.2 Banach's contraction mapping theorem and applications, 1.3 Baire' category theorem, 1.4 Ascoli - Arzela theorem

Unit-2 Normed Linear Space 16hrs

2.1 Linear spaces and linear operators,

2.2 Norm of a bounded operator

2.3 The Hahn – Banach extension theorem

2.4 Stone - Weirstrass theorem

Unit-3 Banach Space 16hrs

3.1 Open mapping theorem

3.2 Closed Graph theorems 3.3 The Banach - Steinhaus Principle of Uniform Boundedness.

Unit-4 Hilbert Spaces 16hrs

4.1 The orthogonal projection

4.2 Nearly orthogonal elements,

4.3 Riesz's lemma,

4.4 Riesz's representation theorem.

Reference Books:

1 G. F. Simmons – Introduction to Topology and Modern Analysis, Tata McGraw-Hill,

NewDelhi. 2 A. E. Taylor – Introduction to Functional Analysis, Wiley, New York, 1958. 3 A. Page and A. L. Brown – Elements of Functional Analysis 4 George Bachman and Lawrence Narici – Functional Analysis, Dover Publications, Inc.,

Mineola, New York 5 J. B. Conway – A Course in Functional Analysis, GTM, Vol. 96., Springer, 1985.


THIRD SEMISTER

TITLE: TOPOLOGY-I

Type: Hard Core



OBJECTIVE: This course deals with the essentials of topological spaces and their properties in terms

of continuity, connectedness and compactness.

LEARNING OUTCOME: Student will get a strong base in the basics of topological spaces and their

properties in terms of continuity, connectedness, and compactness.

Unit-1 Topological Space. 16hrs 1.1 Definitions and examples, 1.2 Basis for a topology, 1.3 The order topology, 1.4 The product topology on X x X, 1.5 The subspace topology. 1.6 Closed sets and limit points,

Unit-2 Continuous Functions 16hrs

2.1 Continuous functions,

2.2 The product topology,

2.3 The metric topology,

2.4 The quotient topology.

Unit-3 Connectedness 16hrs

3.1 Connected spaces,

3.2 Connected sets on the real line,

3.3 Path connectedness.

Unit-4 Compactness 16hrs

4.1 Compact spaces,

4.2 Compact sets on the real line,

4.3 Limit point compactness,

4.4 Local compactness.

Reference Books: 1 J. R. Munkres – A First Course in Topology, Prentice Hall India, 2000, 2nd Ed., 2 G. F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill, Kogakusha,

1968. 3 S. Willard – General Topology, Addison Wesley, New York, 1968. 4 J. Dugundji – Topology, Allyn and Bacon, Boston, 1966. 5 J. L. Kelley – General Topology, Van Nostrand and Reinhold Co., New York, 1955.


THIRD SEMISTER

TITLE: GRAPH THEORY

Type: Soft Core



OBJECTIVE: This course is an introductory course to the basic concepts of Graph Theory. This

includes study of graphs, directed graphs, trees, distances, connectivity, path, planar graphs, coloring of

graphs.

LEARNING OUTCOME: Student will get the strong ground on the basic concepts of graph theory

such as directed graphs, trees, distances, connectivity, path, planar graphs, and coloring of graphs.

Unit-1 Graphs 16hrs

The Königsberg bridge problem 1.1 Varieties of graphs 1.2 Walks and connectedness 1.3 Graph Operations and Graph isomorphism 1.4 Degree sequence of a graph

Unit-2 Trees 16hrs

2.1 Characterization of trees, Spanning Tree, Centers and centroids.

2.2 Cutpoinls, bridges, and blocks

2.3 Block graphs and cutpoint graphs

Unit-3 Connectivity and Traversability 16hrs 3.1 Connectivity and line-connectivit

3.2 Graphical variations of Menger's theorem

3.3 Euler graphs and Hamiltonian graphs

Unit-4 Planar graphs and colorability 16hrs

4.1 Planar graphs and Euler’s formula

4.2 Vertex colouring, 4.3 Chromatic number and Five color theorem

Reference Books:

1 F. Harary – Graph Theory, Addition Wesley Reading Mass, 1969. 2 N. Deo – Graph Theory With Applications to Engineering and Computer Science, Prentice

Hall of India, 1987 3 K. R. Parthasarathy – Basic Graph Theory, Tata McGraw-Hill, New Delhi, 1994. 4 G. Chartand and L. Lesniak – Graphs and Diagraphs, Qwadsworth and Brooks, 2nd Ed 5 Clark and D. A. Holton – A First Look at Graph Theory, Allied publishers 6 D. B. West – Introduction to Graph Theory, Pearson Education Inc., 2001, 2nd Ed., 7 J. A. Bondy and U. S. R. Murthy – Graph Theory with applications, Elsevier, 1976. 8 V. R. Kulli, Collage Graph Theory, Vishwa International Publications, 2012.


THIRD SEMISTER

TITLE: COMMUTATIVE ALGEBRA Type: Soft Core



OBJECTIVE: To give the idea of rings, ideals, nilpotent, units, nil radicals, modules, Artinian and Noetherian rings

and Artinian and Noetherian modules.

LEARNING OUTCOME: Student will be able to explain the basic properties of rings, ideals and

modules. Also able to visualize the algebraic structure like Artinian and Noetherian rings, Artinian and

Noetherian modules.

Unit-1 Rings and ideal 16hrs 1.1 Rings and ring homomorphisms, 1.2 Ideals 1.3 Quotient rings 1.4 Zero-divisors, nilpotent elements and units, 1.5 Prime ideals and maximal ideals

Unit-2 Radicals 16hrs

2.1 The prime spectrum of a ring

2.2 The nil radical and Jacobson radical

2.3 Operation on ideals

2.4 Extension and contraction

Unit-3 Modules 16hrs

3.1 Modules and modules

3.2 Homomorphisms

3.3 Submodules and quotient modules

Unit-4 Direct sums and Free module 16hrs 4.1 Direct sums

4.2 Free modules Finitely generated modules

4.3 Nakayama Lemma 4.4 Simple modules, Exact sequences of modules

Reference Books:

1 M. F. Atiyah and I. G. Macdonald – Introduction to Commutative Algebra, Addison-Wesle

2 C. Musili – Introduction to Rings and Modules, Narosa Publishing House 3 Miles Reid – Under-graduate Commutative Algebra, Cambridge University Press 4 N. S. Gopalakrishnan, Commutative Algebra, Oxonian Press


THIRD SEMISTER

TITLE: ALGEBRAIC NUMBER THEORY.

Type: Soft Core



OBJECTIVE : To give the idea of quadratic fields, Diophantine equation, factorization of ideals - Dedekind domains, finiteness of the class group, class number computations.

LEARNING OUTCOME: Student will be able to understand the concepts of quadratic fields, Diophantine equation, factorization of ideals - Dedekind domains, finiteness of the class group, class number computations.

Unit-1 Number Theoretical Applications 16hrs 1.1 Number theoretical applications of unique factorization

1.2 Algebraic integers

Unit-2 Quadratic Fields, 16hrs

2.1 Quadratic fields

2.2 Certain Euclidean rings of algebraic integers,

2.3 Diophantine equations,

2.4 Ramanujan - Nagell theorem

Unit-3 Factorization 16hrs

3.1 Factorization of Ideals - Dedekind domains

3.2 Fractional ideals 3.3 Invertible ideals

3.4 Prime factorization of ideals

Unit-4 Class group and Class number 16hrs

4.1 Class group and Class number

4.2 Finiteness of the Class group

4.3 Class number computations.

Reference Books: 1 Karlheinz Spindler – Abstract Algebra with Applications, Vol. II, Rings and Fields, Marcel

Dekker, Inc. 2 I. N. Stewart and David Tall – Algebraic Number Theory, Chapman and Hall 3 Jody Esmonde and M. Ram Murthy – Problems in Algebraic Number Theory, Springer

Verlag 4 I. S. Luthar and I. B. S. Passi – Algebra Vol. II: Rings, Narosa Publishing House.


THIRD SEMISTER

TITLE: GALOIS THEORY

Type: Soft Core



OBJECTIVE: This course gives the idea of Galois Theory in abstract algebra it also extends the concept of Galois Theory in a field.

LEARNING OUTCOME: Students will get the idea of Galois Theory and field extension

Unit-1 Algebraic study of fields 16hrs 1.1 Algebraically closed fields and algebraic closures 1.2 The existence of an algebraic closure 1.3 The basic isomorphisms of algebraic field theory

Unit-2 Algebraic study of fields 16hrs

2.1 Automorphisms and fixed fields

2.2 The Frobenius automorphism

2.3 The isomorphism extension theorem

Unit-3 Field extension 16hrs

3.1 The index of a field extension

3.2 Splitting fields 3.3 Separable extensions

3.4 Perfect fields

3.5 Normal extensions

Unit-4 Galois theory 16hrs

4.1 The main theorem of Galois theory,

4.2 Galois groups over finite fields,

4.3 Symmetric functions

4.4 Cyclotomic extensions 4.5 Constructible numbers

Reference Books:

1 J. B. Fraleigh – A First Course in Abstract Algebra, Narosa Publishing House.

2 Ian Steward – Galois Theory, Chapman and Hall.

3 Joseph Rotman – Galois Theory, Universitext Springer, 1998. 4 Michael Artin – Algebra, Prentice-Hall of India, New Delhi. 5 Joseph A. Gallian – Contemporary Abstract Algebra, Narosa Publishing House, 4th Ed., 6 D. S. Dummit and R. M. Foote – Abstract Algebra, John Wiley and Sons, 1999 7 I. N. Herstein – Topics in Algebra, Vikas Publishing House, New Delhi.

8 N. S. Gopalakrishnan – University Algebra, New Age International, 2nd Ed.,


THIRD SEMISTER

TITLE: MATHEMATICAL COMPUTATION

Type: Interdisciplinary paper



OBJECTIVES : To understand fundamental concepts in graph theory, lattices, matrices and

Boolean algebra and to introduce MATLAB programming with few examples

LEARNING OUTCOME : After successful completion of this course, students are expected to have

the ability to Solve problems using algebraic properties, Identify bounded and complete lattice and to

Use MATLAB for solving problems on vectors, matrices, plotting data etc

UNIT 1 Graph Theory And Lattices Theory 16hrs

1.1 Partially ordered sets, lattices, complete lattices, distributed lattices

1.2 Complements, Boolean algebra, Boolean expressions, application to switching

circuits

1.3 Graphs, vertices of graphs, walks and connectedness, degrees, operations on

graphs

1.4 Trees: elementary properties of trees

UNIT 2 Matrix Algebra 16hrs

2.1 Matrix definition, types of matrix, transpose of matrix

2.2 Determinants, properties of determinants, co factors matrix

2.3 Cramer’s rule, adjoint matrix, inverse of a matrix

2.4 Problems on singular and non-singular matrix

UNIT 3 Introduction to Matlab 32hrs

3.1 Basics of MATLAB programming: Reading data from files, Plotting data

Calculating statistics, Exporting graphics,

3.2 Array operations: Performing calculations with vectors, Creating multiple plots

3.3 Loops and execution control : Programming constructs, User interaction, Flow

control, Loops

3.4 Functions: Creating functions, Calling functions, Setting the MATLAB path,

Debugging

Reference Books:

1 Gratzer, George A: General Lattice theory, 2nd Ed .Birkhauser publisher, 1998. 2 K.R.Parthasarathy: Basic graph theory, Tata McGraw Hill, New Delhi, 1994 3 L.Liu: Elements of discrete mathematics, McGraw Hill, 1986 4 Lancaster and Tismenetsky: The theory of matrices with applications

5 Marc E Herniter: Programming in MATLAB 6 Rudra Pratap: Getting started with Matlab, oxford publisher, 7 Kate S.K Bhapkar H.R: Basics of mathematics, Pune: Technical publications 8 S.Lipschutz and M.Lipson: Theory and problems of discrete mathematics – Schaum series

2nd ed, tat McGraw hill


THIRD SEMISTER

OPEN ELECTIVE PAPER

TITLE: FUNDAMENTALS OF MATHEMATICS

Type: Elective



OBJECTIVE: This course is concerned with the Foundations of mathematics which makes the

students to understand the concept of Permutations and Combinations, theory of matrices and gives an

idea about probability.

LEARNING OUTCOME: Student will capable to understand the concept of Permutations and

Combinations, theory of matrices and gives an idea about probability.

Unit-1 Logarithms and Progression 14hrs 1.1 Definitions and properties of Logarithms and problems thereon 1.2 Arithmetic, Geometric and Harmonic progression: Properties and problems

thereon

Unit-2 Matrix Algebra 18hrs

2.1 Definition, types of matrix,

2.2 Transpose of a matrix and problems

2.3 Definition and Properties of determinants (without proof).

2.4 Singular & non-singular matrix and problems.

2.5 Co-factors matrix, Cramer’s Rule,

2.6 Adjoint matrix, inverse of a matrix,

Unit-3 Permutations and Combinations 16hrs

3.1 Permutations and Combinations,



Unit-4 Probability 16hrs

4.1 Probability of events

4.2 Condition probability

4.3 Baye`s theorem

4.4 Distribution function: Binomial

Reference Books:

1 Shirali, Shailesh, A Primer on Logarithms, Hyderabad: Universities Press, (2002) 2 Kate, S.K.; Bhapkar, H.R. (2009), Basics Of Mathematics, Pune: Technical Publications 3 Gerard Meurant, Ronald Lewis Graham, Martin Grotschel , Laszlo Lovasz , Handbook of

Combinatorics, North Holland 4 S. Lipschutz and M. Lipson: Theory and Problems of Discrete Mathematics. Schaum Series.

2nd Ed. Tata McGraw Hill 5 A.K. Md. Ehsanes Saleh Vijay K. Rohatgi, An Introduction to Probability and Statistics

6 S.C. Gupta, Fundamentals of Mathematical Statistics, Sultan Chand & Sons (2014)

https://books.google.com/books?id=0b0igbb3WaQC&printsec=frontcover#v=onepage&q&f=false

https://books.google.com/books?id=v4R0GSJtEQ4C&pg=PR1#v=onepage&q&f=false

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http://www.amazon.in/s/ref=dp_byline_sr_book_1?ie=UTF8&field-author=S.C.+Gupta&search-alias=stripbooks


FOURTH SEMESTER

TITLE: MEASURE AND INTEGRATION

Type: Hard Core



OBJECTIVE: The objective of this course is to generalize the concept of integration using measures

and it helps to develop the concept of analysis in abstract situations.

LEARNING OUTCOME: At the end of the course student will explain the limitations of Riemann

integration and the extension of it by Lebesgue measure and integration .

Unit-1 Lebesgue Measure 16hrs 1.1 Lebesgue outer measure, 1.2 Measurable sets 1.3 Lebesgue measure 1.4 A non-measurable set, 1.5 Measurable functions

Unit-2 The Lebesgue Integral 16hrs

2.1 Lebesgue Integral of a bounded function over as set of finite measure,

2.2 The integral of a non-negative function

2.3 The general Lebesgue integral

Unit-3 Differentiation and Integration 16hrs

3.1 Differentiation of monotonic functions

3.2 Functions of bounded variation,

3.3 Differentiation of an integral 3.4 Absolute continuity.

Unit-4 Measure and Integration 16hrs

4.1 Measure spaces,

4.2 Measurable functions, integration

4.3 Signed measures, the Radon - Nikodym theorem,

4.4 Measure and outer measure,

4.5 Outer measure and measurability.

Reference Books:

1 H. L. Royden – Real Analysis, Prentice Hall, 3rd Ed. 2 G. de Barra – Measure Theory and Integration, Wiley Eastern Limited. 3 Inder K. Rana – An Introduction to Measure and Integration, Narosa, 1997


FOURTH SEMESTER

TITLE: TOPOLOGY-II

Type: Hard Core



OBJECTIVE: The objective of this course is to generalize the concept of integration using measures

and it helps to develop the concept of analysis in abstract situations.

LEARNING OUTCOME: Student will explain Countability and Separation Axioms and applications

of it.

Unit-1 Countability and Separation Axioms 16hrs 1.1 The countability axioms 1.2 The separation axioms, 1.3 Normality of a compact Hausdorff space

Unit-2 Applications of Countability and Separation Axioms 16hrs

2.1 Urysohn's lemma

2.2 Tietze's extension theorem,

2.3 Urysohn's metrization theorem

2.4 Partitions of unity

Unit-3 Tychonoff's Theorem 16hrs

3.1 Tychonoff's theorem on the product of compact spaces

3.2 Local finiteness,

3.3 Paracompactness, 3.4 Normality of a paracompact space.

Unit-4 The Fundamental Group 16hrs

4.1 Definition of fundamental group

4.2 The Fundamental group of a circle,

4.3 The Fundamental group of the punctured plane

4.4 Essential and Inessential Maps,

4.5 The Fundamental Theorem of Algebra

Reference Books:

1 James R. Munkres - A First Course in Topology , Prentice Hall India, 2000, 2nd 2 G. F. Simmons – Introduction to Topology and Modern Analysis, McGraw-Hill, Kogakusha,

1968.

3 S. Willard – General Topology, Addison Wesley, New York, 1968. 4 J. Dugundji – Topology, Allyn and Bacon, Boston, 1966 5 J. L. Kelley – General Topology, Van Nostrand and Reinhold Co., New York, 1955


FOURTH SEMESTER

Project Work

Type: Minor Project

A project work, which is compulsory for every student, involves self study to be carried out by the

student (on a research problem of current interest or on an advanced topic not covered in the syllabus)

under the

guidance of a faculty member. Project work shall be initiated in the third semester itself and the

project report (dissertation) shall be submitted at the end of the fourth semester.

Project Evaluation

Right from the initial stage of defining the problem, the candidate has to submit the progress

reports periodically and also present his / her progress in the form of seminars in addition to the

regular discussion with the guide. Components of evaluation are as follows:

Component – I (C1): Periodic Progress and Progress Reports (15)

Component – II (C2): Results of Work and Draft Report (15)

Component – III (C3): Final Viva Voce and evaluation (70). The report evaluation is for 40 and

the Viva –voce examination is for 30.


FOURTH SEMESTER

TITLE: ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS

Type: Soft Core



OBJECTIVE: This course is to develop strong background on finding solutions to linear differential

equations with constant and variable coefficients and to introduce to the students the various types of

partial differential equations special function, boundary value problems.

Learning outcome: Students will be able to solve the various types of partial differential equations

special function, boundary value problems.

Unit-1 Linear Second Order Equations 16hrs 1.1 Initial value problem, 1.2 Existence and Uniqueness by Picard’s Theorem,

1.3 Wronskian, separation and comparison theorems 1.4 Poincare phase plane 1.5 variation of parameters

Unit-2 Power series solutions and Applications 16hrs

2.1 Solution near ordinary and regular singular point

2.2 Convergence of the formal power series

2.3 Legendre differential equations with their properties

Unit-3 Types differential equations their properties

3.1 Bessel differential equations with their properties

3.2 Hermite differential equations with their properties

3.3 Laguerre and hypergeometric differential equations with their properties

Unit-3 Partial differential equation 16hrs

4.1 Classification of Second order PDE’s

4.2 Reduction to canonical forms 4.3 Derivation of the equations of mathematical physics and their solutions by

separation of variables.

Reference Books:

1 E. A. Coddington and N. Levinson – Theory of Ordinary Differential equations, Tata

McGraw-Hill, New Delhi.

2 R. Courant and D. Hilbert – Methods of Mathermatical Physics, Vol. I. & II, Tata McGraw-

Hill, New Delhi, 1975

3 G. F. Simmons – Differential Equations with applications and Historical Notes, Tata

McGraw-Hill, New Delhi, 1991 4 I. N. Sneddon – Theory of Partial differential equations, McGraw-Hill, International Student

Edition 5 S. G. Deo and V. Raghavendra – Ordinary Differential Equations and Stability Theory, Tata

McGraw-Hill, New Delhi


FOURTH SEMESTER

TITLE: THEORY OF NUMBERS

Type: Soft Core



OBJECTIVE: This course provides the basic knowledge of numbers, functions like arithmetical,

Mobius function, Euler function. It will give the glimpse of Fibonacci and Lucas series and describes

the concept of continued fraction.

LEARNING OUTCOME: Student will attain the basic knowledge of numbers, functions like

arithmetical, Mobius function, Euler function.

Unit-1 Prime Numbers and Farey series 16hrs 1.1 Prime numbers 1.2 The Fundamental theorem of Arithmetic 1.3 The series of Reciprocals of primes 1.4 The Euclidean Algorithm. Fermat and Mersenne numbers

1.5 Farey series, Farey dissection of the continuum 1.6 Irrational numbers-Irrationality of mth root of N, e and π

Unit-2 Arithmetical Functions 16hrs

2.1 Arithmetical Functions – The Mobius function, The Euler' function and Sigma function,

2.2 The Dirichlet product of Arithmetical functions

2.3 Multiplicative functions

2.4 Averages of Arithmetical functions – Euler summation formula, Some

elementary asymptotic formulas,

2.5 The average orders of d(n), σ(n), φ(n), µ(n)

2.6 An application to the distribution of lattice points visible from the origin

Unit-3 Continued fractions-I 16hrs

3.1 Finite continued fractions,

3.2 Convergent of a continued fraction,

3.3 Continued fractions with positive quotients.

3.4 Simple continued fractions(SCF),

3.5 The representation of an irreducible rational fraction by a SCF

Unit-4 Continued fractions-II 16hrs

4.1 The continued fraction algorithm and Euclid's algorithm.

4.2 The difference between the fraction and its convergents,

4.3 Infinite simple continued fractions

4.4 the representation of an irrational number by an infinite continued fraction

4.5 Equivalent numbers and periodic continued fractions, some special quadratic

surds.

Reference Books:

1 G. H. Hardy and E. M. Wright – An Introduction to Theory of Numbers, Oxford

University Press, 1979, 5th 2 I. Niven, H. S. Zuckerman and H. L. Montgomery – An Introduction to the Theory of

Numbers, New York, John Wiley and Sons, Inc., 2004, 5th Ed., 3 Bruce C. Berndt – Ramanujan's Note Books Volume-1 to 5, Springer. 4 G. E. Andrews – Number Theory, Dover Books, 1995. 5 T. M. Apostol – Introduction to Analytic Number Theory, Narosa Publishing House, New

Delhi.


FOURTH SEMESTER

TITLE: ADVANCED GRAPH THEORY

Type: Soft Core



OBJECTIVE: Through the basic knowledge of graph theory it describes idea of traversability, line

graphs, factorization, colorablity, relations between graphs and matrices.

LEARNING OUTCOME: Students will be able to understand the fundamentals of concepts of

graph theory. Also able to describe the idea of line graphs, distance concept, colorablity, relations

between graphs and matrices.

Unit-1 Line Graphs 16hrs 1.1 Definition and Some properties of line graphs 1.2 Characterization of line graphs 1.3 Line graphs and traversability.

Unit-2 Algebraic Graph Theory 16hrs

2.1 Matrices – The adjacency matrix

2.2 The incidence matrix 2.3 The cycle matrix

2.4 Adjacency Eigen Values

Unit-3 Distances in graphs 16hrs

3.1 Distances in graphs and its applications,. 3.2 Distance matrix

3.3 Characteristic polynomial and Distance Eigen Values

Unit-4 Domination Theory 16hrs

4.1 Definition, types of Domination .

4.2 Domination numbers -Some elementary properties

Reference Books: 1 F Harary – Graph Theory, Addition Wesley Reading Mass, 1969 2 N. Deo – Graph Theory With Applications to Engineering and Computer Science, Prentice

Hall of India, 1987 3 K. R. Parthasarathy – Basic Graph Theory, Tata McGraw-Hill, New Delhi, 1994 4 G. Chartand and L. Lesniak – Graphs and Diagraphs, Qwadsworth and Brooks, 2nd Ed 5 Clark and D. A. Holton – A First Look at Graph Theory, Allied publishers 6 D. B. West – Introduction to Graph Theory, Pearson Education Inc., 2001, 2nd Ed. 7 J. A. Bondy and U. S. R. Murthy – Graph Theory with applications, Elsevier, 1976 8 V. R. Kulli, Domination Theory, Vishwa International Publications, 2012.


FOURTH SEMESTER

TITLE: DIFFERENTIAL GEOMETRY

Type: Soft Core



OBJECTIVE: This course introduces space curves and their intrinsic properties of a surface and

geodesics. Further it deals with conformal mapping of surface and elaborates the concepts of Gaussian

curvature and Gauss map.

LEARNING OUTCOME: Students willable to find the properties of a surface and geodesics.

Unit-1 Introduction to Differential Geometry. 16hrs 1.1 Plane curves and Space curves – Frenet-Serret Formulae. 1.2 Global properties of curves – Simple closed curves 1.3 The isoperimetric inequality

Unit-2 Four Vertex theorem and its applications 16hrs

2.1 The Four Vertex theorem

2.2 Surfaces in three dimensions – Smooth surfaces

2.3 Tangents, Normals and Orientability, Quadric surfaces.

Unit-3 First fundamental Form of Curves. 16hrs

3.1 The First Fundamental form – The lengths of curves on surfaces,

3.2 Isometries of surfaces

3.3 Conformal mappings of surfaces

3.4 Surface area

3.5 Equiareal Maps and a theorem of Archimedes

Unit-4 Second Fundamental Form of Curves 16hrs

4.1 Curvature of surfaces

4.2 The Second Fundamental form

4.3 The Curvature of curves on a surface 4.4 Normal and Principal Curvatures

Reference Books:

1 A. Pressley – Elementary Differential Geometry, Under-graduate Mathematics Series,

Springer. 2 T. J. Willmore – An Introduction to Differential Geometry, Oxford University Press 3 D. Somasundaram – Differential Geometry: A First Course, Narosa, 2005


St. Philomena’s College (Autonomous), Mysore

M. Sc-Mathematics (CBCS)

I/II/III/IV- Semester Examination: 2018-19

Subject:

Time: 3 Hours Max Marks: 70

Blue Print

Sl. No Answer all the questions,

All question carries equal marks

Marks

1 a 2

b 2

c 2

d 2

e 2

f 2

g 2

2 14

OR

3 14

4 14

OR

5 14

6 14

OR

7 14

8 14

OR

9 14

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