dr raafftt coff spprooppoosseedd cbbccss syy … · 2019-04-30 · dr raafftt coff spprooppoosseedd...
TRANSCRIPT
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 1
ST. PHILOMENA’S COLLEGE(AUTONOMOUS)
MYSORE-570015
DDRRAAFFTT OOFF PPRROOPPOOSSEEDD CCBBCCSS SSYYLLLLAABBUUSS
FFOORR MM..SSCC..MMAATTHHEEMMAATTIICCSS
FFOORR TTHHEE AACCAADDEEMMIICC YYEEAARR 22001188 OONNWWAARRDDSS
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 2
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
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 3
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) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 4
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
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 5
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.
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 6
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
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 7
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
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 8
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
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 9
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. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 10
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
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 11
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.,
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 12
FIRST SEMESTER
TITLE: REAL ANALYSIS-I
Type: Hard Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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.
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 13
FIRST SEMESTER
TITLE: REAL ANALYSIS-II
Type: Hard Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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.
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 14
FIRST SEMESTER
TITLE: COMPLEX ANALYSIS-I
Type: Hard Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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.
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 15
FIRST SEMESTER
TITLE: LINEAR ALGEBRA-I
Type: Soft Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 16
FIRST SEMESTER TITLE: COMBINATORICS AND GRAPH THEORY
Type: Soft Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 17
FIRST SEMESTER
TITLE: NUMERICAL ANALYSIS.
Type: Soft Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 18
SECOND SEMESTER TITLE: ALGEBRA-II
Type: Hard Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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.
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 19
SECOND SEMESTER
TITLE: REAL ANALYSIS-III
Type: Hard Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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.
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 20
SECOND SEMESTER
TITLE: COMPLEX ANALYSIS-II
Type: Hard Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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.
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 21
SECOND SEMESTER
TITLE: LINEAR ALGEBRA-II
Type: Soft Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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.
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 22
SECOND SEMESTER TITLE: REPRESENTATION THEORY OF FINITE GROUPS
Type: Soft Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 23
SECOND SEMESTER OPEN ELECTIVE PAPER
TITLE: FUNDAMENTALS OF MATHEMATICS
Type: Elective
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
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,
3.2 Pigeon-hole principle,
3.3 Principle of inclusion and exclusion
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)
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 24
SECOND SEMESTER
TITLE: THEORY OF PARTITIONS
Type: Soft Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 25
THIRD SEMISTER
TITLE: ELEMENTS OF FUNCTIONAL ANALYSIS
Type: Hard Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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.
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 26
THIRD SEMISTER
TITLE: TOPOLOGY-I
Type: Hard Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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.
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 27
THIRD SEMISTER
TITLE: GRAPH THEORY
Type: Soft Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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.
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 28
THIRD SEMISTER
TITLE: COMMUTATIVE ALGEBRA Type: Soft Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 29
THIRD SEMISTER
TITLE: ALGEBRAIC NUMBER THEORY.
Type: Soft Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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.
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 30
THIRD SEMISTER
TITLE: GALOIS THEORY
Type: Soft Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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.,
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 31
THIRD SEMISTER
TITLE: MATHEMATICAL COMPUTATION
Type: Interdisciplinary paper
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory- 70 + Internal Assessment- 30 = 100
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
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 32
THIRD SEMISTER
OPEN ELECTIVE PAPER
TITLE: FUNDAMENTALS OF MATHEMATICS
Type: Elective
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
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 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,
3.2 Pigeon-hole principle,
3.3 Principle of inclusion and exclusion
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)
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 33
FOURTH SEMESTER
TITLE: MEASURE AND INTEGRATION
Type: Hard Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 34
FOURTH SEMESTER
TITLE: TOPOLOGY-II
Type: Hard Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 35
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.
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 36
FOURTH SEMESTER
TITLE: ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS
Type: Soft Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 37
FOURTH SEMESTER
TITLE: THEORY OF NUMBERS
Type: Soft Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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.
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 38
FOURTH SEMESTER
TITLE: ADVANCED GRAPH THEORY
Type: Soft Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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.
St. Philomena’s College (Autonomous) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 39
FOURTH SEMESTER
TITLE: DIFFERENTIAL GEOMETRY
Type: Soft Core
CLASS DURATION – 04 HOURS PER WEEK 64 Hours
Marks-Theory - 70 + Internal Assessment -30= 100
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) Mysuru. M.Sc., Mathematics Syllabus 2018-19 onwards .Page 40
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