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1
DEPARTMENT OF MATHEMATICS
Dr. B. R. AMBEDKAR UNIVERSITY – SRIKAKULAM
Outcome Based Curriculum
Mathematics Programme
(With Effect From 2019-20 Admitted Batch)
Dr. B.R. Ambedkar University, Srikakulam
Etcherla– 532410
2
INDEX
S.No Content Page
1 About Department 5
2 University Vision, Mission
and Department Vision, Mission
6
3 SWOT/C Analysis 7
4 Strategic Plans 8
5 Short and Long Term Goals 9
6 Programme Educational Objectives and Programme
Outcomes 10
7 General Regulations 12-15
8 First Year – First semester credit system 16
9 First Year – Second semester credit system 17
10 Second Year – Third semester credit system 18
11 Second Year – Fourth semester credit system 19
12 M101 Algebra - I 20-21
13 M102 Real Analysis – I 22-23
14 M103 Topology 24-25
15 M104 Differential Equations 26-27
16 M105 Linear Algebra 28-29
17 M201 Algebra – II 30-31
18 M202 Real Analysis – II 32-33
19 M203 Probability 34-35
20 M204 Complex Analysis 36-37
3
21 M205 Graph Theory 38-39
22 M301 Functional Analysis 40-41
23 M302 Number Theory 42-43
24 M303 Classical Mechanics 44-45
25 M304(1) Operations Research 46-47
26 M304(2) Mathematical Biology 48-49
27 M304(3) Commutative Algebra - I 50-51
28 M304(4) Banach Algebras 52-53
29 M305(1) Numerical Analysis 54-55
30 M305(2) Mathematical Software 56-57
31 M305(3) Fuzzy Set Theory 58-59
32 M305(4) Universal Algebra 60-61
33 M401 Measure and Integration 62-63
34 M402 Partial Differential Equations 64-65
35 M403 Mathematical Methods 66-67
36 M404(1) Lattice Theory 68-69
37 M404(2) Theory of Computations 70-71
38 M404(3) Commutative Algebra-II (Prerequisite
Commutative Algebra – I)
72-73
39 M404(4) Theory of Linear Operators 74-75
40 M405(1) Wavelet Analysis 76-77
41 M405(2) Programming in C 78-79
42 M405(3) Semi Groups 80-81
4
43 M405(4) Financial Mathematics 82-83
44 Model Papers
5
ABOUT DEPARTMENT
Department of Mathematics was established in the academic year 1988-1989 and it offers
opportunities for the education and research in a wide spectrum of areas in Mathematics and
Applied Mathematics. The Department started offering CBCS elective courses for
postgraduate programs and also advanced courses for Ph. D. program. The Department of
Mathematics strives to be recognized for excellence among academic institutions in India.
The Department wishes to focus on providing a comprehensive curriculum at postgraduate
levels and career opportunities in India. The department is committed to train the students to
make them motivated and dedicated teaching and scientific research.
The department is offering two programmes one is M.ScMathematics and other one is
M.ScApplied Mathematicsandhas been introduced choice based credit system curriculum.
The new syllabus that is proposed for the M.Sc Mathematics and M.Sc Applied Mathematics
curriculum has been prepared keeping in view of the guidelines presented in UGC model
curriculum for PG programme. The proposed syllabus covers 95% of the content suggested
by UGC. This also covers the syllabus that is needed to clear the examinations conducted at
national level such as CSIR-NET/SET examination for research fellowships, eligibility for
lecturer-ship, GATE examination for higher studies in technical education, also employed in
various government and non government organizations.
This syllabus is oriented to create suitable workforce to support teaching community in
Mathematics as well as in Applied Mathematics. The courses involved in the curriculum have
tremendous potential for applications in the industry. Some of these courses are useful in
designing algorithms and performing computations needed in several real world applications.
6
Vision of the Department
To make the department as a global centre
for excellence in Mathematics to
coordinate the growth of Science and
Technology.
University Mission
Mitigating the economic and social
sufferings of the region by invoking the
strengths of faculty through community
oriented actions by optimal usage of
human resources.
University Vision
Creation of an enabling environment
where in universities would act as agents
of social change and transformation
through innovativeness and outreaching
and make it a “People’s University”.
Mission of the Department
To impart quality education and scientific
researchin Mathematics through updated
curriculum, effective teaching learning
process. To inculcate innovative skills,
theme of team-work and ethical practices
among students so as to meet societal
expectations.
7
SWOT/C Analysis
Strengths
1. The major strength of the Department of Mathematics is the quality of the faculty. A
majority of the faculty have strong and established research programs with strengths in the
areas of nonlinear analysis, applied analysis, mathematical modelling and simulation,
Mathematical Biology, Lattice Theory.
2. Quality teaching has been a priority for mathematics faculty, and several faculties have
received outstanding research awards.
3. Strong faculty qualification, committed, talented and dedicated, knowledgeable and
cooperative faculty, great collaboration and good communication among faculty.
4. Faculty-initiated activitiesoutside class like weeklyseminar and studentresearch.
Weaknesses
1. Difficult to communicate to higher authorities in proper way.
2. Failed to eliminate chalkboards for teaching.
3. Lack of regular faculty.
Opportunities
1. Economy encouraging more students to choose Ship, raising enrolment.
2. Current trends in improvingSTEM education provide grantopportunities.
3. Promote more than math minimum
4. Competitive enrolment (better prerequisite enforcement)
5. Better collaboration with other disciplines.
Threats
1. Student attitude towards mathematics
2. Lack of student dedication
3. Lack of technology (hardware /software)
4. Lack of computer lab & library in the department
5.Administration fails to involve proper individuals before making decisions
8
Strategic Plans
The Faculty of Mathematics Strategic Plan 2019-2024 was developed through extensive
consultation with Mathematics faculty, students, staff, university administrators, national
researchers, employers and alumni. The purpose is to formulate objectives for the
department's future research profile and to bring new study programs and new educational
approaches into the department's educational profile.
1. Fill open and anticipated positions with faculty who excel in both research and teaching.
2. Continue to make connections with industries, labs, and government agencies that can
offer our students internships and regular employment. Bring speakers and recruiters to
campus when possible.
3. Introduce more innovative teaching methods to some of our coursework (blended
[online/classroom] learning, the Moore method, etc.) to encourage more intellectual
engagement.
4. Continue to emphasize the importance of external funding.
5. Be one of the top research departments in the world, and be widely recognized as such.
6. Develops world class facilities for collaborative research and learning in mathematics.
7. Maintain an existing, dynamic, collaborative environment supporting activities at he
frontiers of the mathematical sciences.
9
Short Term Goals
1. To get regular faculty
2. To start a new post graduates programme in Mathematics.
3. To build on inter disciplinary programmes with other departments within the University.
4. Prepare graduate students for leadership in both academic and non-academic career paths
in an increasingly interdisciplinary world.
Long term Goals
1. To attract and retain academics of a high calibre on the Department’s faculty.
2. To attract motivated and talented students to the master’s and doctoral programmes of the
Department.
3. To provide the best possible facilities for our faculty and students, particularly in the areas
of computer facilities, library facilities and administrative support.
4. To create an environment that supports outstanding research.
5. To pursue collaborative programmes with highly reputed national institutions.
6. To provide a simulating teaching environment for the post graduate students of the
department.
10
M.Sc Mathematics Programme Educational Objectives
The objectives of the M.Sc. Mathematics program are to develop students with the following
capabilities
1. To produce PG students with high integrity and good ethics and to train students to deal
with the problems faced by software industry through knowledge of mathematics and
scientific computational techniques.
2. The Department wishes to focus on providing a comprehensive curriculum at postgraduate
levels and career opportunities in India.
3. The graduates will work and communicate effectively in intra-disciplinary,
interdisciplinary environment, either independently or in team, and demonstrate leadership
quality in area of Mathematics.
M.Sc Mathematics Programme Outcomes
The successful completion of this program will enable the students to
1. Apply a wide range of mathematical techniques and application of mathematical
methods/tools in other scientific and engineering domains.
2. Gain the knowledge of contemporary issues in the field of Mathematics and applied
sciences.
3. Understand the scientific theories and methods, gain experience in working independently
with scientific questions and engineering problems, and clearly express their opinion on
academic issues.
11
Dr. B. R. AMBEDKAR UNIVERSITY, SRIKAKULAM
General Regulations relating to
POST GRAUDATE AND PROFESSIONAL COURSES
Syllabus under Credit Based Semester System
(With effect from 2019-2020 admitted batch)
1. Candidates seeking admission for the Masters/Professional Degree Courses shall be required to have passed the qualifying examination prescribed for the course of any University recognized by Dr. B.R. Ambedkar University, Srikakulam as equivalent there to.
2. The course and scope shall be as defined in the Scheme of Instruction and syllabus
prescribed.
3. The course consists of 2/4/6 semesters, @ two semesters/year, unless otherwise
specified.
4. The candidates shall be required to take an examination at the end of each semester of the study as detailed in the Scheme of Examination.
i. (a). Each semester theorypaper in M.Sc Mathematics/M.Sc Applied
Mathematics programme except M305(2) Mathematical Software (3rd semester) and M405(2) Programming in C (4th semester) carries a maximum of 100 marks, of which 75 marks shall be for semester-end theory examination of the paper of three hours duration and 25 marks shall be for internal assessment.
(b). M305(2) Mathematical Software (3rd semester) and M405(2) Programming in C (4th semester) carries a maximum of 100 marks, of which 50 marks shall be for semester-end theory examination paper of the of 3hrs duration, 25 marks shall be for internal assessment and 25 marks of which 5 marks shall be for Lab, 5 marks shall be for observation, 5 marks shall be for record work and 10 marks shall be for viva examination (External).
ii. Internal Assessment for 25 Marks: Three mid-term exams, two conventional
(descriptive) for 15 marks and the third – ‘on-line’ with multiple choice questions for 5 marks for each theory paper shall be conducted and 5 marks for student Assignment submission for each course. The average of these first two mid-term and the marks in the online mid exams shall be taken as marks obtained for the paper under internal assessment. If any candidate appears for only one mid-term exam, the average mark, dividing by two shall be awarded. If any candidate fails to appear for all the midterm exams of a paper, only marks obtained in the theory paper shall be taken into consideration for declaring the result. Each mid-term exam shall be conducted only once.
iii. Candidates shall be declared to have passed each theory paper if he/she
obtains not less than E Grade ie., an aggregate of 40 % of the total marks inclusive of semester-end and internal assessment marks in each paper.
12
5. A candidate appearing for the whole examination shall be declared to have passed the examination if he/she obtains a Semester Grade Point (SGP) of 5.0 and a CGPA of 5.0 to be declared to have passed the Course.
6. Notwithstanding anything contained in the regulations, in the case of Project
Report/Dissertation/ Practical/Field Work/Viva-voce etc., candidates shall obtain not less than D grade, i.e., 50% of marks to be declared to have passed the examination.
7. ATTENDANCE: Candidates shall put in attendance of not less than 75% of attendance, out of the total number of working periods in each semester. Only such candidates shall be allowed to appear for the semester-end examination.
(a) A candidate with attendance between 74.99% and 66.66% shall be allowed to appear for the semester-end examination and continue the next semester only on medical and other valid grounds, after paying the required condonation fee.
(b) In case of candidates who are continuously absent for 10 days without prior permission on valid grounds, his/her name shall automatically be removed from the rolls.
(c) If a candidate represents the University at games, sports or other officially organized extra-curricular activities, it will be deemed that he/she has attended the college on the days/periods
8 Candidates who put in a minimum of 50% attendance shall also be permitted to continue for the next semester. However, such candidates have to re-study the semester course only after completion of the course period for which they are admitted. The candidate shall have to meet the course fees and other expenditure.
9 Candidates who have completed a semester course and have fulfilled the necessary
attendance requirement shall be permitted to continue the next semester course irrespective of whether they have appeared or not at the semester-end examination, at their own cost.
Such candidates may be permitted to appear for the particular semester-end
examination only in the following academic year; they should reregister/ reapply for the Semester examination.
The above procedure shall be followed for all the semesters
10. Candidates who appear and pass the examination in all the papers of each and every
semester at first appearance only are eligible for the award of Medals/Prizes/Rank Certificates
11. BETTERMENT: Candidates declared to have passed the whole examination may reappear for the same examination to improve their SGPA, with the existing regulations without further attendance, paying examination and other fees. Such reappearance shall be permitted only with in 3 consecutive years from the date of first passing the final examination. Candidates who wish to appear thereafter should take the whole examination under the regulations then in vogue.
12. The semester-end examination shall be based on the question paper set by an external paper-setter and there shall be double valuation for post-Graduate courses. The concerned Department has to submit a panel of paper-setters and examiners approved by the BOS and the Vice-chancellor nominates the paper-setters and examiners from the panel.
13
13. In order to be eligible to be appointed as an internal examiner for the semester-end
examination, a teacher shall have to put in at least three years of service. Relaxation of service can be exempted by the Vice-Chancellor in specific cases.
14. If the disparity between the marks awarded in the semester-end examination by
internal and external examiners is 25% or less, the average marks shall be taken as the mark obtained in thepaper. If the disparity happens to be more, the paper shall be referred to another examiner for third valuation. In cases of third valuation, of the marks obtained either in the first or second valuation marks, whichever is nearest to the third valuation marks are added for arriving at the average marks.
15. Candidates can seek revaluation of the scripts of the theory papers by paying the
prescribed fee as per the rules and regulations in vogue.
16. The Project Report/Dissertation/ Practical/Field Work/Viva-voce etc shall have
double valuation by internal and external examiners.
17. A Committee comprising of the HOD, one internal teacher by nomination on rotation
and one external member, shall conduct viva-voce examination. The department has to submit the panel, and the Vice-chancellor nominates viva-voce Committee.
18. Grades and Grade Point Details (with effect from 2019-20 admitted batches)
S.No. Range of Marks% Grade Grade Points
1. > 90 ≤100 O 10.0 Out Standing
2. > 80 ≤ 90 A+ 9.0 Excellent
3. > 70 ≤80 A 8.0 Very Good
4. > 60 ≤70 B+ 7.0 Good
5. > 55 ≤60 B 6.0 Above Average
6. > 50 ≤55 C 5.0 Average
7. ≥ 40 ˂50 D 4.0 Pass
8. ˂40 F 0.0 Fail
9. 0.0 Absent
19. Calculation of SGPA (Semester Grade Point Average) & CGPA (Cumulative Grade
Point Average):
For example, if a student gets the grades in one semester A,A,B,B,B,D in six subjects having credits 2(S1), 4(S2), 4(S3), 4(S4), 4(S5), 2(S6), respectively. The SGPA is calculated as follows:
{ 9(A)x2(S1)+9(A)x4(S2)+8(B)x4(S3)+8(B)x4(S4)+8(B)x4(S5)+6(D)x2(S6)} 162
SGPA = --------------------------------------------------------------------------- = ------ = 8.10
{2(S1) +4(S2) +4(S3) +4(S4) +4(S5) +2(S6)} 20
i. A student securing ‘F’ grade thereby securing 0.0 grade points has to appear and
secure at least ‘E’ grade at the subsequent examination(s) in that subject.
ii. If a student gets the grades in another semester D, A, B, C, A, E, A, in seven subjects
having credits 4(S1),
14
2(S2), 4(S3), 2(S4), 4(S5), 4(S6), 2(S7) respectively,
{6(D)x4(S1)+9(A)x2(S2)+8(B)x4(S3)+7(C)x2(S4)+9(A)x4(S5)+5(E)x4(S6)+9(A)x2(S7)} 162
SGPA = --------------------------------------------------------------------------------------------------------- = ------ = 7.36
{4(S1) +2(S2) +4(S3) +2(S4) +4(S5) +4(S6) +2(S7)} 22
(9x2+9x4+8x4+8x4+6x2+6x4+9x2+8x4+7x2+9x4+5x4+9x2) 324
CGPA = ------------------------------------------------------------------------------ = -------- = 7.71
(20+22) 42
a) A candidate has to secure a minimum of 5.0 SGPA for a pass in each semester in case
of all PG and Professional Courses. Further, a candidate will be permitted to choose any paper(s) to appear for improvement in case the candidate fails to secure the minimum prescribed SGPA/CGPA to enable the candidate to pass at the end of any semester examination.
b) There will be no indication of pass/fail in the marks statement against each individual
paper.
c) A candidate will be declared to have passed if a candidate secures 5.0 CGPA for all
PG and Professional Courses.
d) The Classification of successful candidates is based on CGPA as follows:
i) Distinction –CGPA 8.0 or more;
ii) First Class –CGPA 6.5 or more but less than 8.0
iii) Second Class –CGPA 5.5 or more but less than 6.5
iv) Pass –CGPA 5.0 or more but less than 5.5
e) Improving CGPA for betterment of class will be continued as per the rules in vogue.
f) CGPA will be calculated from II Semester onwards up to the final semester. CGPA
multiplied by gives“10” aggregate percentage of marks obtained by a candidate.
15
DEPARTMENT OF MATHEMATICS
Dr. B. R. AMBEDKAR UNIVERSITY – SRIKAKULAM CURRICULUM STRUCTURE FOR CHOICE BASED CREDIT SYSTEM
(CBCS)
(W.E.F. 2019-20 ADMITTED BATCH)
.
M.Sc Mathematics FIRST YEAR – FIRST SEMESTER
Paper
Code
Title of the paper
Core papers
Credit
value
Lectures
per
week
Max
Marks
External
Assessment
Internal
Assessment
Lab
Assess
ment
M 101 Algebra – I 4 6 100 75 25 0
M 102 Real Analysis – I 4 6 100 75 25 0
M 103 Topology 4 6 100 75 25 0
M 104 Differential
Equations 4 6 100 75 25 0
M 105 Linear Algebra 4 6 100 75 25 0
SD 106 Communication
Skills - I 2 3 100 75 25 0
FW-1 Extension Activity I 2 4 25 0 25 0
Internship 1 0 - - - -
Total 25 - 625 - - -
16
M.Sc Mathematics FIRST YEAR – SECOND SEMESTER
Paper
Code
Title of the
paper
Core papers
Credit
value
Lectures
per
week
Max
Marks
External
Assessment
Internal
Assessment
Lab
Assessment
M 201 Algebra – II 4 6 100 75 25 0
M 202 Real Analysis –
II 4 6 100 75 25 0
M 203 Probability 4 6 100 75 25 0
M 204 Complex
Analysis 4 6 100 75 25 0
M 205 Discrete
Mathematics 4 6 100 75 25 0
SD 206 Communication
Skills - II 2 3 100 75 25 0
FW-2 Extension
Activity II 2 4 25 0 25 0
MOOCS MOOCS 2 0 - - - -
Internship 1 0 - - - -
Total 27 - 625 - - -
17
M.Sc Mathematics SECOND YEAR – THIRD SEMESTER
Paper
Code
Title of the paper
Core papers
Credit
value
Lectures
per week
Max
Marks
External
Assessment
Internal
Assessment
Lab
assessment
M 301 Functional Analysis 4 6 100 75 25 0 M 302 Number Theory 4 6 100 75 25 0 M 303 Classical Mechanics
4 6 100
75 25 0
Elective Papers
(Stream
A)
M 304
M304(1) Operations
Research
4
6
100
75
25
0
M304(2) Mathematical
Biology
M304(3) Commutative
Algebra-I
M304(4) Banach
Algebras
(Stream
B)
M 305
M305(1)Numerical
Analysis
4
6
100
75
25
0 M305(2)Mathematical
Software
50
25
25 M305(3) Fuzzy Set
Theory
75
25
0 M305(4) Universal
Algebra
75
25
0 SD 306 Communication
Skills - III 2 3 100 75 25 0
FW-3 Extension Activity III 2 4 25 0 25 0 MOOCS MOOCS 2 0 - - - -
Internship 1 0 - - - - Total 27 - 625 - - -
18
M.Sc Mathematics SECOND YEAR – FORTH SEMESTER
Paper
Code
Title of the paper
Core papers
Credit
value
Lectures
per week
Max
Marks
External
Assessment
Internal
Assessment
Lab
Assessment
M 401 Measure and Integration 4 6 100 75 25 0 M 402 Partial Differential
Equations 4 6 100 75 25 0
M403 Mathematical Methods 4 6 100 75 25 0 Elective Papers
(Stream
A)
M 404
M404(1) Lattice Theory
4
6
100
75
25
0
M404(2) Commutative
Algebra-II (Prerequisite
Commutative Algebra –
I)
M404(3) Theory Of
Computations
M404(4) Theory of
Linear Operators
(StreamB)
M 405
M405(1) Wavelet
Analysis
4
6
100
75
25
0 M405(2)Programming in
C 50 25 25
M405(3)Semi Groups
75
25
0 M405(4)Financial
Mathematics 75 25 0
SD 406 Communication
Skills - IV
2 3 100 75 25 0
FW-4 Extension Activity IV 2 4 25 0 25 0 MOOCS MOOCS 2 0 - - - -
Internship 1 0 - - - - Viva-voce 4 0 100 100 - - Total 31 - 725 - - -
Total Marks: 2600
Total Credits: 110
19
M.Sc Mathematics FIRST YEAR – FIRST SEMESTER
Course Code
&Course Name M 101
Algebra - I
Objectives
1. The primary goal in Algebra - I is to help students transfer their concrete mathematical
knowledge to more abstract algebraic generalizations.
2. Algebra - I is designed to give students a foundation for all future mathematics courses.
3. Students will explore: Basic set theory. Groups- Some Examples of Groups- Some preliminary
Lemmas- Subgroups-A counting principle- Normal subgroups and Quotient Groups,
homomorphism, Isomorphism, automorphism and its applications.
4. Throughout the course, Common Core standards are taught and reinforced as the student learns
how to apply the concepts in real-life situations.
5. The concept of groups, rings, fields and vector spaces are essential building blocks of Modern
algebra and are an integral part of any post graduate course.
SYLLABUS
Unit- I
Learning
Out Comes
Group Theory
Definition of a Group- Some Examples of Groups- Some preliminary Lemmas-
Subgroups-A counting principle- Normal subgroups and Quotient Groups
On successful completion of this unit, students should be able to
1. Learn to concepts of group, Sub group, Normal sub group and Quotient group.
2. Determine whether a given set and binary operation form a group by checking
group axioms.
3. Carry out calculations in quotient groups.
Unit II
Learning
Out Comes
Group Theory Continued..
Homomorphisms – Automorphisms - Cayley”sTheorem- Permutation Groups-
Another counting principle
On successful completion of this unit, students should be able to
1. Understand the importance of Homomorphism, automorphisms in group theory
and their applications.
2. Prove properties of homomorphism and understand the connection to normal
subgroups.
3. Define permutation groups and State Cayley’s theorem and its generalization.
Unit III
Learning
Out Comes
Group Theory Continued...
Sylow’s Theorem- Direct products- Finite Abelian Groups.
On successful completion of this unit, students should be able to
1. Acquire knowledge of Sylow’s Theorems and finite abelian groups.
2. Apply Sylow theorems to rule out existence of simple groups of certain orders.
3. Express a given finite cyclic group as the direct product of cyclic groups of
prime power order and given two direct products of cyclic groups, determine
whether or not they are isomorphic.
Unit IV
Learning
Ring Theory
Definition and Examples of Rings- Some special classes of Rings-
Homomorphisms- Ideals and Quotient Rings- More Ideals and Quotient Rings-
The Field of Quotients of an Integral Domain.
On successful completion of this unit, students should be able to
20
Out Comes 1. Learn fundamental concepts of Ring theory that help understand other algebra
courses.
2. Recognize and apply action of rings on sets in geometric and abstract contexts.
3. Understand the applications of integral domain and its properties.
Unit V
Learning
Out Comes
Ring Theory Continued…
Euclidean Rings- A particular Euclidean Ring- Polynomial Rings- Polynomials
over the Rational Field- Polynomial Rings over Commutative Rings.
On successful completion of this unit, students should be able to
1. Learn certain Euclidean ring,Polynomial ring and Polynomial ring over
Commutative Ring and their properties.
2. To recognize the reducible & irreducible Polynomial.
3. learn applications of ring theory.
Prescribed
Text Book
Reference
Books
Online
Sources
Topics in Algebra by I. N. Herstein, Second edition, John Wiley & Sons.
1. Modern Algebra by P.B.bhattacharya, S.K.Jain and SR Nagpaul.
2. Group theory byMaroofSamhan and Fadwa Abu MoryafaPublisher Dar Al
Khraiji1st edition 1427 H.
3. A First Course in Abstract Algebra by John B. FraleighPublisher: Pearson, 7th
edition, 2013
4. Contemporary Abstract algebra by J. Gallian, Brooks/Cole Pub Co; 8 edition (13
July 2012).
5. J. A. Gallian, Contemporary Abstract Algebra, Brooks/Cole Cengage Learning, 2010.
1. http://www.supermath.info/AlgebraInotes_2016.pdf
2. http://www-users.math.umn.edu/~garrett/m/algebra/notes/Whole.pdf
3. https://www.cse.iitk.ac.in/users/rmittal/prev_course/s15/notes/complete_notes.pdf.
Course
Out Comes
After completion of this course, students will be able to
1. Understand the concepts of group, subgroups, Normal subgroup and quotient
group and gain knowledge to write some properties of these concepts.
2. Recognize, formulate, classify and solve problems in a mathematical context.
3. Formulate mathematical hypotheses and have an understand of how such
hypotheses can be verified using mathematical methods.
4. Demonstrate knowledge and understanding of fundamental concepts including
groups, subgroups, normal subgroups, homeomorphisms and isomorphism.
5. Students grasp the fundamental principles and theory concerning basic algebraic
structures such as groups, rings, integral domains.
21
Course Code
&Course Name M 102
Real Analysis - I
Objectives1. This course will focus on the proofs of basic theorems of analysis, as appeared in
one variable calculus. Along the way to establish the proofs, many new concepts will be
introduced. Understanding them and their properties are important for the development of the
present and further courses.
2. The course includes axioms of real number systems, convergence of sequences and series,
Continuous functions, uniform continuity and Differentiation and Mean Value theorems.
3. The Fundamental Theorem of Calculus. Series. Power series and Taylor series.
4. Apply mathematical methodologies to open-ended real-world problems
5. Construct rigorous mathematical proofs of basic results in real analysis.
SYLLABUS
Unit I
Learning
Out Comes
Basic Topology: Finite, Countable, and Uncountable Sets, Metric spaces,
Compact sets, Connected sets.
On successful completion of this unit, students will be able to
1. Understand the concept of finite sets, countable and uncountable sets.
2. Describe the real line as a complete, ordered field
3. Use the definitions of convergence as they apply to sequences, series, and
functions.
Unit II
Learning
Out Comes
Numerical Sequences and Series: Convergent sequences, Subsequences, Cauchy
sequences, Upper and Lower limits, Some special sequences, Series, Series of non-
negative terms, number e.
On successful completion of this unit, students will be able to
1. Identify the convergence sequences, sub sequences and Cauchy sequences.
2. Understand the concept of upper limit and lower limits, some special sequences
and their applications.
3. Demonstrate an understanding of limits and how they are used in sequences,
series, differentiation and integration
Unit III
Learning
Out Comes
The Root and Ratio tests, Power series, Summation by parts, Absolute
Convergence, Addition and Multiplication of series, Rearrangements.
On successful completion of this unit, students will be able to
1. Write the proofs of the root and ratio tests.
2. Apply the power series, summations by parts, absolute convergence.
3. Appreciate how abstract ideas and rigorous methods in mathematical analysis
can be applied to important practical problems.
Unit IV
Learning
Out Comes
Continuity: Limits of Functions, Continuous Functions, Continuity and
Compactness, Continuity and Connectedness, Discontinuities, Monotone
Functions, Infinite Limits and Limits at Infinity.
On successful completion of this unit, students will be able to
1. Understand the concepts of limit of functions, continuous functions.
2. Write the simple proofs of the concepts continuity and connectedness, monotone
functions.
3. Identify the infinite limits and limits at infinity.
Unit V
Differentiation:The Derivative of a Real Function, Mean Value Theorems, The
Continuity of Derivatives, L’ Hospital’s Rule, Derivatives of Higher order,
22
Learning
Out Comes
Taylor’s theorem, Differentiation of Vector- valued Functions.
On successful completion of this unit, students will be able to
1. Determine the derivative real valued functions, mean value theorems.
2. Apply the Mean Value Theorem and the Fundamental Theorem of Calculus to
problems in the context of real analysis
3. Learn applications of the Taylors theorem, differentiation of rector valued
function.
Prescribed
Text
Reference
Books
Online
Source
Principles of Mathematical Analysis by Walter Rudin, International Student
Edition, 3rd Edition, 1985.
1. Mathematical Analysis by Tom M. Apostal, Narosa Publishing House, 2nd
Edition, 1985.
2. Real Analysis by H.L.Royden, Pearson, 4th Ed., 2010.
1. https://math.berkeley.edu/~gbergman/ug.hndts/m104_Rudin_exs.pdf
(Solutions to Principles of Mathematical Analysis by Walter Rudin)
2. https://www.math.lsu.edu/~sengupta/4031f06/IntroRealAnalysNotes.pdf
3. http://www.math.iitb.ac.in/~ars/week1.pdf
4. http://www.rupp.edu.kh/news/documents/A_Course_in_Real_Analysis.pdf
Course Out
Comes
After completing this course students will
1. gain knowledge of concepts of modern analysis, such as convergence,
continuity, completeness, compactness and convexity in the setting of Euclidean
spaces and more general metric spaces
2. develop a higher level of mathematical maturity combined with the ability to
think analytically
3. be able to write simple proofs on their own and study rigorous proofs
4. be able to follow more advanced treatments of real analysis and study its
applications in disciplines such as economics.
5. Demonstrate an understanding of limits and how that are used in sequences,
series and differentiation.
23
CourseCode
&Course Name M 103
Topology
Objectives1. Topology is the study of properties of spaces invariant under continuous
deformation. For this reason it is often called ``rubber-sheet geometry''
2. This is an introductory course in topology, or the study of shape.
3. Student will to have knowledge on point set topology, topological spaces, Quotient spaces,
Product spaces and metric spaces, sequences, continuity of functions, connectedness and
compactness, homotopy and covering spaces.
4. To explain how to distinguish spaces by means of simple topological invariants (compactness,
connectedness and the fundamental group)
5. Apply your knowledge to solve problems and prove theorems.
SYLLABUS
Unit I
Learning
Out Comes
Topological Spaces: The definition and some examples, interior, closure, and
boundary, Basis and sub basis, Continuity and Topological Equivalence,
Subspaces.
On successful completion of this course, students will be able to
1. Learnthebasic concepts of Topology, basis, subspaces Continuity.
2. Write the proofs of some results on these concepts
3. Apply these concepts in real life problems.
Unit II
Learning
Out Comes
Connectedness: Connected and disconnected spaces, theorems on connectedness,
connected subsets of the real line, applications of connectedness, path connected
spaces, locally connected and locally path connected spaces.
On successful completion of this course, students will be able to
1. Understand the concepts of connectedness and disconnectedness.
2. Write the solutions of some problems on these concepts.
3. Identify the connected spaces and path connected spaces.
Unit III
Learning
Out Comes
Compactness: Compact spaces and subspaces, compactness and continuity,
properties related to compactness, one- pointcompactification, the cantor set.
On successful completion of this course, students will be able to
1. Study the conceptsof compactness and one- point compactification.
2. Write the simple proofs of some results on these concepts.
3. Distinguish the compact and one point compactification.
Unit IV
Learning
Out Comes
Product and Quotient spaces: Finite products, arbitrary products, comparison of
topologies, quotient spaces.
On successful completion of this course, students will be able to
1. Learn the concept of quotient spaces, product spaces.
2. Write the simple proofs of some results on these concepts.
3. Understand the proofs of these concepts.
Unit V
Learning
Out Comes
Separation properties and Metrization: T0 , T1, and T2 – spaces, regular spaces,
normal spaces, separation by continuous functions, metrization, the stone-cech
compactification.
On successful completion of this course, students will be able to
1. Study the concept of T0 , T1, and T2 – spaces, regular spaces, normal spaces.
2. Write the simple proofs of some results on these concepts.
3. Learn applications of Urysohn lemma, TheUrysohnmetrization theorem.
24
Prescribed
Text Book
Reference
Text Book
Online
Source
1.Principles of Topology by Fred H. Croom, Cengage learning india private
Limited, Alps building, 1st Floor, %6- Janpath, New Delhi 110001.
1. Topology by James R. Munkers, Second edition, Pearson education Asia – Low
price edition
2. Topology by Dugundji, McGraw-Hill Inc.,US (1 April 1988)
3. Elements of General Topology by Hu, Holden-Day, 1964.
1. https://www.math.colostate.edu/~renzo/teaching/Topology10/Notes.pdf.
2. http://www.pdmi.ras.ru/~olegviro/topoman/part1.pdf.
3. http://www.math.hcmus.edu.vn/~hqvu/n.pdf.
Course Out
Comes
After completion of this course, students will be able to
1. know the definitions of standard terms in topology.
2. Students will know how to read and write proofs in topology.
3. Students will know a variety of examples and counterexamples in topology.
4. Students will know about the fundamental group and covering spaces.
5. Students will understand the machinery needed to define homology and co
homology.
6. Students will understand computations in and applications of algebraic
topology.
25
Course Code
&Course Name M 104
Differential Equations
Objectives
1. Describe a collection of methods and techniques used to find solutions to several types of
differential equations, including first order scalar equations, second order linear equations, and
systems of linear equations.
2. Study qualitative techniques for understanding the behavior of solutions.
3. Learn to construct differential equations, corresponding to different ecological or physical
systems.
4. Student will explore: boundary value problems, Sturm-Liouville problems, and Fourier Series.
SYLLABUS
Unit I
Learning
Out Comes
Second order linear differential equations: Introduction-general solution of the
homogeneous equation - Use of a known solution to find another - Homogeneous
equation with constant coefficients - method of undetermined coefficients -
method of variation of parameters
On successful completion of this unit, students will be able to
1. Explain the meaning of solution of the homogenous equation.
2. Solve the homogenous second order equation with constant coefficient.
3. Applies the method of undetermined coefficient and method of variation of
parameters to find the solution of second order linear differential equation with
variable coefficients.
Unit II
Learning
Out Comes
Oscillation theory and boundary value problems: Qualitative properties of
solutions - The Sturm comparison theorem - Eigen values, Eigen functions and the
vibrating string.
On successful completion of this unit, students will be able to
1. Solve the qualitative properties of solutions.
2. Expresses the Sturm comparison theorem.
3. Solve the Eigen values, eigen functions and the vibrating string.
Unit III
Learning
Out Comes
Power series solutions: A review of power series-series solutions of first order
equations-second order linear equations - ordinary points-regular singular points
On successful completion of this unit, students will be able to
1. Solve the power series solutions of first order equations.
2. Solve the power series solutions of second order equations with ordinary points.
3. Solve the regular singular point.
Unit IV
Learning
Out Comes
Systems of first order equations: Linear systems - Homogeneous linear systems
with constant coefficients
On successful completion of this unit, students will be able to
1. Learn the system of first order equations.
2. Determines the types of linear differential equation systems.
3. Solve the homogenous linear systems with constant coefficient.
Unit V
Learning
Out Comes
Existence and Uniqueness of solutions - successive approximations - Picard’s
theorem - Some examples
On successful completion of this unit, students will be able to
1. Learn the existence and uniqueness of solutions.
2. Apply the method of successive approximations.
26
3. Expresses the Picard’s theorem of differential equations.
Prescribed
Text Book
Reference
Books
Online
Source
1. George F. Simmons, Differential Equations, Tata McGraw-Hill Publishing
Company Limited, New Delhi.
1. Ordinary Differential Equations and stability theory by S. G. Deo and V.
Raghavendra, TATA Mc Graw Hill Ltd, 1980.
2. Theory of Ordinary Differential Equations by Coddington and Levinson,
Krieger Pub Co (June 1984).
1. https://www.math.ust.hk/~machas/differential-equations.pdf.
2. http://www.math.toronto.edu/selick/B44.pdf.
3. https://www.mat.univie.ac.at/~gerald/ftp/book-ode/ode.pdf.
Course Out
Comes
After completion of this course, students will be able to
1.The study of Differential focuses on the existence and uniqueness of solutions
and also emphansizes the rigorous justification of methods for approximating
solutions in pure and applied mathematics.
2.It plays an important role in modelling virtually every physically technical or
biological process from celestial motion to bridge design to interactions between
neurons.
3.Theory of differential equations is widely used in formulating many fundamental
laws of physics and chemistry.
4. Theory of differential equation is used in economics and biology to model the
behavior of complex systems.
5. Differential equations have a remarkable ability to predicts the world around us.
They can describe exponential growth and decay population growth of species or
change in investment return over time.
27
Course Code
&Course Name M105
Linear Algebra
Objectives
1. To make the students familiar with principles and techniques of Linear algebra and their
applications.
2. To effective communicative ideas and explain procedures.
3. To effectively interpret results and solutions in written.
4. To identify and develop system equation models.
5. To make effective techniques in Linear algebra.
SYLLABUS
Unit I
Learning
Out Comes
The Elementary Canonical Forms: Introduction-Characteristic Values-Annihilating
Polynomials-Invariant Subspaces
On successful completion of this unit, students will be able to
1. Determine relationship between coefficient matrix invertibility and solutions to
a system of linear equations and the inverse matrices.
2. Determine the concept of Annihilating Poly, Invariant Subspaces.
3. Learn Applications of Annihilating Poly, Invariant Subspaces.
Unit II
Learning
Out Comes
Simultaneous Triangulation-Simultaneous Diagonalization-Direct-sum
Decompositions,Invariant Direct Sums- Derive The Primary Decomposition
Theorem.
On successful completion of this unit, students will be able to
1. Understand linear independence and dependence.
2. Find basis and dimension of a vector space, and understand change of basis.
3. Determine the Applications of Primary Decompositon.
Unit III
Learning
Out Comes
Cyclic subspaces and annihilators, Derive Cyclicdecompositions and the rational
form, Evaluate the Jordon form, Computation of Invariant factors.
On successful completion of this unit, students will be able to
1. Understand the concept of Cyclic subspaces, annihilators.
2. Derive Cyclicdecompositions and the rational form.
3. Learn the systems of linear equations using various methods.
Unit IV
Learning
Out Comes
Semi-Simple operators and Bilinear forms.
On successful completion of this unit, students will be able to
1.Understand the concept of Semi-Simple operators,
2. Understand the concept of Bilinear forms.
3. Determine the applications of different operators.
Unit V
Learning
Out Comes
Evaluate Symmetric Bilinear form, Skew Symmetric Bilinear form, and Group
Preserving Bilinear form.
On successful completion of this unit, students will be able to
1. Understand the concept of Symmetric and Skew Bilinear form.
2. Understand the concept of Group Preserving Bilinear form.
3. Determine the applications of Bilinear forms.
Prescribed
Text Book
1. Linear Algebra second edition By Kenneth Hoffman and Ray Kunze, Prentice-
Hall of India Private Limited, New Delhi-110001, 2002
28
Reference
Books
Online
Source
1. A. RamachandraRao and P. Bhimsankaram. Linear Algebra, Hindustan Book
Agency; 2nd Revised edition edition (15 May 2000.
2. S. Kumaresan-Linear Algebra, Prentice Hall India Learning Private Limited;
New title edi- tion (2000).
1. http://joshua.smcvt.edu/linearalgebra/book.pdf
2.http://linear.ups.edu/download/fcla-3.40-tablet.pdf
3.http://omega.albany.edu:8008/mat220/LAbook.pdf
4.http://www.math.nagoya-u.ac.jp/~richard/teaching/f2014/Lin_alg_Lang.pdf
Course
Out Comes
After completion of this course, students will be able to
1. Use computational techniques and algebraic skills essential for the study of
systems of linear equations, matrix algebra, vector spaces, eigenvalues and
eigenvectors, orthogonality and diagonalization.
2. Evaluate determinants and use them to discriminate between invertible and non-
invertible matrices.
3. Identify linear transformations of finite dimensional vector spaces and compose
their matrices in specific bases.
4. Use visualization, spatial reasoning, as well as geometric properties and
strategies to model, solve problems, and view solutions, especially in R2 and R3,
as well as conceptually extend these results to higher dimensions.
5. Critically analyze and construct mathematical arguments that relate to the study
of introductory linear algebra.
29
M.Sc Mathematics FIRST YEAR – SECOND SEMESTER
CourseCode&Cours
e Name M 201
Algebra - II
Objectives1. As a second course in algebra the objective of this course is to have a complete
understanding of fields and linear transformations.
2. The concept of Galois theory in fields is central to theory of equations and is a must for all
mathematics students.
3. Student will explore: Fields, linear transformations, finite fields.
4. Appraise how to use the computer skills and library.
5. The knowledge on this course will provide the basis for further studies in advanced algebra like
commutative algebra, linear groups, modules etc., which forms the basics of higher mathematics.
SYLLABUS
Unit I
Learning
Out Comes
Fields: Extension Fields- The Transcendence of e – Roots of Polynomials
(Chapter 5 sections 5.1 -5.4)
On successful completion of this Unit, students should be able to :
1. Learn to concepts of Fields, extension fieldsandTranscendence numbers.
2. Determine whether a given set and binary operation form a filed by checking
field axioms.
3. Carry out calculations in polynomials roots.
Unit II
Learning
Out Comes
Fields: Construction with Straightedge and Compass, more about roots.
(Chapter 5 sections 5.1 -5.4)
After studying this unit, students should be able to :
1. Understand importance of More about roots of polynomials,
2. Explain what is meant by a symmetry of a plane figure.
3. Describe the symmetries of some bounded three-dimensional figures.
4. Use the straightedge and compass to identify and construct examples of the
reducible and irr-reducible polynomials.
Unit III
Learning
Out Comes
Fields:The elements of Galois Theory- Solvability by Radicals- Galois Groups
over the Rationals.
(Chapter 5 sections 5.5- 5.8)
After studying this unit, the student is expected to be able to:
1. Acquire knowledge of Galois Theory
2. Apply Galois Theoryto rule out existence of simple extension of fixed fields.
3. Determine the elementary symmetric functions of symmetric polynomials.
Unit IV
Learning
Out Comes
Finite Fields: Wedderburn’s Theorem on Finite Division Rings.(Chapter 7
sections 7.1 , 7.2)
Upon successful completion of this unit, students will be able to:
1.Learn concept of Wedderburn’s Theorem
2. Give examples of skew field and prove their simplicity.
3. Prove properties of finite division rings and understand the connection to fields
Unit V
Finite Fields: A Theorem of Frobenius- Integral Quaternions and the Four-Square
Theorem.(Chapter 7 sections 7.3, 7.4)
30
Learning
Out Comes
Upon successful completion of this unit, students will be able to:
1. Prove the Frobenius theorem
2. Determine the Lagrange identity and left division algorithm.
3. Prove the Four-Square Theorem and their applications.
Prescribed
Text Book
Reference
Books
Online
Source
Topics in Algebra: I. N. Herstein , Second edition, John Wiley & Sons.
1. Modern Algebra by P.B.bhattacharya, S.K.Jain and SR Nagpaul.
2. Group theory byMaroofSamhan and Fadwa Abu MoryafaPublisher Dar Al
Khraiji1st edition 1427 H.
3. A First Course in Abstract Algebra by John B. FraleighPublisher: Pearson, 7th
edition, 2013
4. Contemporary Abstract algebra by J. Gallian, Brooks/Cole Pub Co; 8 edition (13
July 2012).
1. http://www.supermath.info/AlgebraInotes_2016.pdf
2. http://www-users.math.umn.edu/~garrett/m/algebra/notes/Whole.pdf
3. https://www.cse.iitk.ac.in/users/rmittal/prev_course/s15/notes/complete_notes.pdf
Course
Out Comes
After completion of this course, students will be able to
1. Recognize technical terms and appreciate some of the uses of algebra
2. Demonstrate insight into abstract algebra with focus on axiomatic theories
3. Apply algebraic ways of thinking
4. Demonstrate knowledge and understanding of fundamental concepts including
fields, extension fields and finite fields.
5. Students grasp the fundamental principles and theory concerning basic algebraic
structures such as fields and extension fields
31
Course Code
&Course Name M 202
Real Analysis - II
Objectives1. This course will focus on the proofs of basic theorems of analysis, as appeared in
one variable calculus. Along the way to establish the proofs, many new concepts will be
introduced. Understanding them and their properties are important for the development of the
present and further courses.
2. The course includes axioms of real number systems, uniform convergence of sequences and
series, Continuous functions, uniform continuity and Differentiation and Mean Value theorems.
3. The Fundamental Theorem of Calculus. Series. Power series and Taylor series.
4. Investigate the consequences of different types of convergence for sequences and series of
functions, the interchange of limits with other operations such as integrals and derivatives
5. Also develop some of the calculus of functions of several variables including the general
inverse and implicit function theorems.
SYLLABUS
Unit I
Learning
Out Comes
Riemann-Stieltjes Integral: Definition and existence of the Riemann Stieltjes
Integral, Properties of the Integral, Integration and Differentiation, the fundamental
theorem of calculus – Integral of Vector- valued Functions.
(Chapter 6 of the prescribed text book)
On successful completion of this unit, students will be able to
1.Determine the Riemann integrability of a bounded function and prove a selection
of theorems concerning integration.
2. Understand the applications of Riemann-Stieltjes Integral
3. Apply these concepts in real world problems.
Unit II
Learning
Out Comes
Sequences and Series of the Functions: Discussion on the Main Problem, Uniform
Convergence, Uniform Convergence and Continuity, Uniform Convergence and
Integration, Uniform Convergence and Differentiation
(First five sections of Chapter 7 of the prescribed text book)
On successful completion of this unit, students will be able to
1. Understand the concepts of Uniform Convergence, Uniform Convergence and
Continuity, Uniform Convergence and Integration
2. Recognize the difference between point wise and uniform convergence of a
sequence of functions
3. Illustrate the effect of uniform convergence on the limit function with respect to
continuity, differentiability, and integrability.
Unit III
Learning
Out Comes
Equicontinuous families of Functions, the Stone-Weierstrass Theorem.
(6th& 7thsections of Chapter 7 of the text book)
Power Series: (A section in Chapter 8 of the text book)
On successful completion of this unit, students will be able to
1. Understand the concepts of Equicontinuous families of Functions, the Stone-
Weierstrass Theorem.
2. Identify the series and power series.
3. Studying applications of Equicontinuous families of Functions, the Stone-
Weierstrass Theorem.
Unit IV
Functions of Several Variables: Linear Transformations, Differentiation, The
Contraction Principle, The Inverse Function theorem. (First Four sections of
chapter 9 of the text book)
32
Learning
Out Comes
On successful completion of this unit, students will be able to
1. Illustrate the convergence properties of power series.
2. Studying the concept of contraction principle, properties and their applications.
3. To apply the acquired knowledge in signals and Systems, Digital Signal
Processing. Etc
Unit V
Learning
Out Comes
Functions of several variables Continued: The Implicit Function theorem, The
Rank theorem, Determinates, Derivatives of Higher Order, Differentiation of
Integrals.
(5th to 9th sections of Chapter 9 of the text book)
On successful completion of this unit, students will be able to
1.Studying the concept of implicit function theorem and its applications.
2. Understand the concepts of Rank theorem, Determinates, Derivatives of Higher
Order, Differentiation of Integrals.
3. Learn to apply these concepts to real problems
Prescribed
Text Book
Reference
Books
Online
Source
Principles of Mathematical Analysis by Walter Rudin, International Student
Edition, 3rd Edition, 1985.
1. Mathematical Analysis by Tom M. Apostal, Narosa Publishing House, 2nd
Edition, 1985.
2. Real Analysis by H.L.Royden, Pearson, 4th Ed., 2010.
1. https://math.berkeley.edu/~gbergman/ug.hndts/m104_Rudin_exs.pdf
(Solutions to Principles of Mathematical Analysis by Walter Rudin)
2. https://www.math.lsu.edu/~sengupta/4031f06/IntroRealAnalysNotes.pdf
3. http://www.math.iitb.ac.in/~ars/week1.pdf 4. http://www.rupp.edu.kh/news/documents/A_Course_in_Real_Analysis.pdf
Course
Out Comes
After completion of this course, students will be able to
1. Have sound mathematical knowledge. They have a profound overview of the
contents of fundamental mathematical disciplines and are able to identify their
correlations.
2. Write simple proofs of the results and study the applications.
3. Demonstrate an understanding of limits ad how that is used in sequences of
functions, series of functions and differentiation.
4. Construct rigorous mathematical proofs of basic results in real analysis.
5. Appreciate how abstract ideas and regions methods in mathematical analysis can
be applied to important practical problems.
33
Course Code
&Course Name M 203
Probability
Objectives
1.This course provides an elementary introduction to probability and statistics with applications.
2. Student will explore: basic probability models; combinatorics; random variables; discrete and
continuous probability distributions; statistical estimation and testing; confidence intervals; and
an introduction to linear regression.
3. Be able to use the addition law and be able to compute the probabilities of events using
conditional probability and the multiplication law.
4. Be able to use new information to revise initial (prior) probability estimates using Bayes'
theorem.
5. Apply problem-solving techniques to solving real-world events.
SYLLABUS
Unit I
Learning
Out Comes
Combinatorial analysis: Introduction,The Basic Principle of counting,
Permutations, Combinations, Multinomial Coefficients,, The number of integer
solutions of equations. Axioms of Probability: Introduction,Sample space and
events, Axioms of Probability, Some Simple Propositions, Sample Spaces Having
Equally Likely Outcomes, Probability as a Continuous Set Function.
Section 1.1 to 1.6 of Chapter 1 and 2.1 to 2.6 of Chapter 2
On successful completion of this unit, students will be able to
1.Understanding the basic concepts of basic principle of counting, Permutations,
Combinations, Multinomial Coefficients, Sample space and events, Axioms of
Probability, Some Simple Propositions
2. Understand the meaning of probability and probabilistic experiment
3. Obtain an understanding of the role probability information plays in the decision
making process.
Unit II
Learning
Out Comes
Conditional Probability and Independence:Introduction, Conditional Probabilities,
Bayes’s Formula, Independent Events, P(·|F) Is a Probability.
Section 3.1 to 3.5 of chapter 3
On successful completion of this unit, students will be able to
1. Understand the meaning of conditional probability, conditioning and reduced
simple space.
2. Study applications of Conditional Probabilities, Bayes’s Formula.
3. Distinguish independent events and dependent events.
Unit III
Learning
Out Comes
Random Variables: Random Variables, Discrete Random Variables,Expected
Value, Expectation of a Function of a Random, The Bernoulli and Binomial
Random,Properties of Binomial Random Variables,Computing the Binomial
Distribution Function.
Section 4.1 to 4.6 of Chapter 4
On successful completion of this unit, students will be able to
1. Demonstrate understanding the random variable, expectation, variance and
distributions.
2. Partially characterize a distribution using a expected value, variance and
moments.
3. Derive distributive functions of a random variable.
Unit IV
Poisson Random, Computing the Poisson Distribution Function, Expected Value
of Sums of Random Variables, Properties of the Cumulative Distribution Function.
34
Learning
Out Comes
Section 4.7 & 4.9 to 4.10 of Chapter 4
On successful completion of this unit, students will be able to
1. Understand the concepts of Poisson Random, Computing the Poisson Distribution
Function, Expected Value of Sums of Random Variables, Properties of the
Cumulative Distribution Function.
2. Calculate various moments of common random variables including at least
means, variances and standard deviations.
3. Identify important types of distribution functions such as Poisson distribution
function, Binomial distribution function and Cumulative Distribution Function.
Unit V
Learning
Out Comes
Limit Theorems:Introduction, Chebyshev’s Inequality and the Weak Law of Large
Numbers, TheCentralLimitTheorem,The Strong Law of Large Numbers.Additional
Topics in Probability, The Poisson Process, MarkovChains.
Section 8.1 to 8.4 Chapter 8 and 9.1 to 9.2 Chapter 9
On successful completion of this unit, students will be able to
1. Understand limit theorems such as Weak Law of Large Numbers,
TheCentralLimitTheorem,The Strong Law of Large Numbers
2. Learn applications of law of large numbers and the central limit theorem and
how these concepts are used to model various random phenomena.
3. Extend the concept of a random variable to a random process and understand the
basic concept of random process.
Prescribed
Text Book
Reference
Books
Online
Source
A First course in Probability by Sheldon Ross, Eight Edition 2010.
Pearson Publications
1. Modern probability theory by B.R. Bhat, Wiley, 1985.
2. Probability and measure by P. Billingsley, Wiley, 1986.
3. A graduate course in probability theory by H.G.Tucker, 1967, (AP)
1. http://julio.staff.ipb.ac.id/files/2015/02/Ross_8th_ed_English.pdf
2. https://www.stat.berkeley.edu/~aldous/134/gravner.pdf
3. http://www.iiserpune.ac.in/~ayan/MTH201/Sahoo_textbook.pdf.
Course
Out Comes
Upon completion of this course, the student will be able to
1. Compute the probabilities of composite events using the basic rules of probability.
2. Demonstrate understanding the random variable, expectation, variance and
distributions.
3. Compute the sample mean and sample standard deviation of a series of independent
observations of a random variable.
4. Apply the concepts of multiple random variables to engineering applications.
5. Analyze the correlated data and fit the linear regression models.
35
Course Code
&Course Name M 204
COMPLEX ANALYSIS
Objectives1. This course is aimed to provide an introduction to the theories for functions of a
complex variable.
2. Student will explore: Elementary properties and examples of analytic functions: Power series-
Analytic functions- Analytic functions as mappings.
3. It begins with quick review on the exploration of the algebraic, geometric and topological
structures of the complex number field.
4. The concepts of analyticity and mapping properties of function of a complex variable will be
illustrated.
5. Complex integration and complex power series are presented.
SYLLABUS
Unit I
Learning
Out Comes
Elementary properties and examples of analytic functions: Power series- Analytic
functions- Analytic functions as mappings, Mobius transformations
($1, $2, $3 of chapter-III of prescribed text book)
On successful completion of this unit, students will be able to
1. Determine the elementary properties and examples of analytic functions.
2. Evaluate the power series analytic functions as mappings.
3. Understanding and evaluate analytic functions and Mobius transformation
functions.
Unit II
Learning
Out Comes
Complex Integration: Riemann- Stieltjes integrals- Power series representation of
analytic functions
($1, $2 of chapter-IV of prescribed text book)
On successful completion of this unit, students will be able to
1.Understanding the concept of Riemann-Stieltjescomplex integrals and power
series with applications.
2. Expresses the cauchy’s integral formula.
3. Determine the Riemann stieltjes integrals with applications.
Unit III
Learning
Out Comes
Zeros of analytic functions- The index of a closed curve, Cauchy’s theorem and
integral formula.
($3 $4, $5 of chapter-IV of prescribed text book).
On successful completion of this unit, students will be able to
1. Explaining the concepts, proof of Cauchy’s theorem.
2. Determine the zeros of analytic functions.
3. Expresses the Cauchy’s theorem, Cauchy’s integral formula and Moreara’s
theorem.
Unit IV
Learning
Out Comes
The homotophic version of Cauchy’s theorem and simple connectivity- Counting
zeros; the open mapping theorem.
($6, $7 of chapter-IV of prescribed text book)
On successful completion of this unit, students will be able to
1. Understand the concept of homotophic version of Cauchy’s theorem and simple
connectivity- Counting zeros; and proof of the open mapping theorem.
2. Determine the concept of homotopic version of Cauchy’s theorem and simple
connectivity.
3. Evaluate the counting zeros of analytic functions.
Unit V Singularities: Classifications of singularities- Residues- The argument principle.
36
Learning
Out Comes
($1, $2, $3 of chapter-V of prescribed text book)
On successful completion of this unit, students will be able to
1.Classification of singularities.
2. Evaluate the Residue and expresses the residue theorem.
3. Learn the concept singularities and their applications.
Prescribed
Text Book
Reference
Books
Online
Resource
Functions of one complex variables by J.B.Conway : Second edition, Springer
International student Edition, Narosa Publishing House, New Delhi.
1. Complex Variable and Applications by R.V. Churchill and J. W. Brown, Tata
McGraw Hill, 2008.
2. Complex Analysis by L.V.Ahlfors, Tata McGraw Hill, 1979.
3. Foundation of Complex Analysis by S. Ponnuswamy,Narosa Publishing House,
2007.
4. Complex Variables: Theory and Applications by H.S. Kasana, PHI, 2006. 1. http://www.maths.lth.se/matematiklu/personal/olofsson/CompHT06.pdf.
2.https://www.if.ufrj.br/~tgrappoport/aulas/metfis1/complex.pdf
3. http://math.sfsu.edu/beck/papers/complex.pdf.
Course
Out Comes
After completion of this course, students will be able to
1. Analyze limits and continuity for functions of complex variables, understand
about the Cauchy-Riemann equations, analytic functions, entire functions
including the fundamental theorem of algebra,
2. Evaluate complex contour integrals and apply the Cauchy integral theorem in its
various versions, and the Cauchy integral formula,
3. Analyze sequences and series of analytic functions and types of convergence,
4. Represent functions as Taylor and Laurent series, classify singularities and
poles, find residues and evaluate complex integrals using the residue theorem’
5. Understand the conformal mapping and apply real world problems
37
Course Code &
Course Name M 205
Graph Theory
Objectives
1. Graphs are used to model networking problems in physical and biological sciences etc.
2. As an essential tools in computer and information sciences, the concepts in Graph Theory
address problems of social media, linguistics, chemical bonds, computational neuro science,
market and financial analysis, communication system, data organisation, flows and links.
3. Graphs are mathematical constructions used to describe collections of objects some pairs of
which are related to each other. For example a family tree is a collection of people in which some
are related to others by parentage.
4. To understand and apply the fundamental concepts in graph theory.
5. To apply graph theory based tools in solving practical problems.
SYLLABUS
Unit I
Learning
Out Comes
Basic concepts, Isomorphism, Euclidian and Hamilton Graphs
On successful completion of this unit, students will be able to
1. Learn the concepts of connectedness, walk, path circuits.
2.Write examples of connected graphs, walk, path circuits Euler graph, etc.
3. Apply these concepts in real life problems
Unit II
Learning
Out Comes
Trees, Properties of Trees, Spanning Trees, Connectivity and Separability,
Network flows.
On successful completion of this unit, students will be able to
1. Determine the concept of Graph theory and different types of graph.
2. Understand the concept of Fundamental circuits and cut-sets
3. Describe the network flows and isomorphism
Unit III
Learning
Out Comes
Planar graphs, Kuratowski’s two graphs, Different representations of planar
graphs, Detection of Planarity, Geometric and Combinational Duals of a graph,
Vector spaces of a Graph.
On successful completion of this unit, students will be able to
1. Will be able to learn the concept of fundamental cycles spanning concept
2. Describe the concept of geometric dual, spanning concept.
3. Solving the application of fundamental cycles spanning concept.
Unit IV
Learning
Out Comes
Matrix representation of graphs, Incidence and circuit matrices of a graph,
Fundamental Circuit matrix, Application to a Switching network, Cut set and Path
Matrices, Adjacency matrices, Directed Graphs, Trees with directed Edges,
Incidence and adjacency matrix of a digraph.
On successful completion of this unit, students will be able to
1. Determine the concept of Coloring problems.
2. Describe the concept of chromatic number.
3. Solve the application of four color problem.
Unit V
Learning
Coloring, Covering and Partitioning, Chromatic number, Chromatic Partitioning,
Chromatic polynomial, Matchings, Coverings, The form color problem,
Applications of graph theory inOperations Research.
On successful completion of this unit, students will be able to
38
Out Comes 1. Describe the concept of Directed graphs.
2. Determine the concept of binary relations.
3. Solve the concept of Euler digraphs.
Prescribed
Text Book
Reference
Books
Online
Source
Graph Theory and its Application to Engineering and Computer Science by
NarsingDeo, PHI, 1979.
1. Graph Theory by F. Harary, Addison Wesley Publishing company, 1969.
2. Introduction to Graph Theory by R. J. Wilson, Longman Group Ltd., 1985 .
3.Graph Theory with applications by Bond JA and Murthy USR, North Holland,
New York.
4.Discrete Mathematics and Graph Theory (by BhavanariSatyanarayana and
KunchamSyam Prasad), Second Edition (2014), PHI Learning Private Limited,
New Delhi, ISBN : 978-81-203-4948-3.
5.Mathematical Foundation of Computer Science (by Dr.BhavanariSatyanarayana,
Dr.TumurukotaVenkata Pradeep Kumar, Dr. Shaik Mohiddin Shaw), BS
Publications, Hyderabad, 2016 (ISBN: 978-93-83635-81-8) and CRC Press,
England, 2019 (ISBN-13: 978-0-367-3681-0).
1. http://www.zib.de/groetschel/teaching/WS1314/BondyMurtyGTWA.pdf.
2. http://math.tut.fi/~ruohonen/GT_English.pdf.
3. https://www.maths.ed.ac.uk/~v1ranick/papers/wilsongraph.pdf.
Learning
Out Comes
On successful completion of this course, students will be able to
1. Key concepts from the text and questions arising
2. Investigation of questions posed for seminar discussion;
3. Occasional presentation by students of key items from the SYLLABUS;
4. Strategies for thinking about graph theory and about mathematics generally;
5. Proofs in graph theory.
6. Mathematical writing.
Course
Out Comes
After completion of this course, students will be able
1. Demonstrate knowledge of the SYLLABUS material;
2. Write precise and accurate mathematical definitions of objects in graph theory;
3.Use mathematical definitions to identify and construct examples and to
distinguish examples from non-examples;
4. Validate and critically assess a mathematical proof;
5.Use a combination of theoretical knowledge and independent mathematical
thinking in creative investigation of questions in graph theory;
6. Reason from definitions to construct mathematical proofs;
7. Write about graph theory in a coherent and technically accurate manner.
39
M.Sc Mathematics SECOND YEAR – THIRD SEMESTER
Course Code
&Course Name M 301
FUNCTIONAL ANALYSIS
Objectives1. The objective of the course is the study of the main properties of bounded operators
between Banach and Hilbert spaces.
2. The basic results associated to different types of convergences in normed spaces and the
spectral theorem and some of its applications.
3. The course gives an introduction to functional analysis, which is a branch of analysis in which
one develops analysis in infinite dimensional vector spaces.
4. The central concepts which are studied are normed spaces with emphasis on Banach and
Hilbert spaces, and continuous linear maps (often called operators) between such spaces.
5. Spectral theory for compact operators is studied in detail, and applications are given to integral
and differential equations.
SYLLABUS
UNIT I
Learning
Out Comes
Banach spaces: The definition and some examples, continuous linear
transformation, The Hahn-Banach theorem, the natural imbedding of N in N**.
(Sections 46 to 49 of Chapter 9 of the text book 1)
On successful completion of this unit,students will be able to
1. understand the concept of Banach space and applications of some results on
Banach spaces. 2. Analyze the normed linear spaces, Banach space and Dual spaces
3. Role of completeness through the Baire category theorem and its consequences
for operators on Banach spaces.
UNIT II
Learning
Out Comes
The open mapping theorem, The conjugate of an operator, Hilbert spaces: The
definition and some simple properties, orthogonal complements.
(Sections 50, 51 of Chapter 9 & Sections 52, 53 of Chapter 10 of the text book 1.)
On successful completion of this unit,students will be able to
1. Understand the open mapping theorem and its applications. 2. Understand inner product spaces, orthogonality and Hillbert spaces.
3. State and apply the Banach Isomorphism Theorem and Closed Graph Theorem
to determine whether operators are bounded.
UNIT III
Learning
Out Comes
Orthonormal sets, The conjugate space H*, the adjoint of an operator, Self-
adjointoperators.
(Sections 54 to 57 of Chapter 10 of the text book 1.)
On successful completion of this unit,students will be able to
1.Understand the concept of Orthonormal sets, adjoint operators and its
applications
2. Have a demonstrable knowledge of the properties of a Hilbert space, including
orthogonal complements, orthonormal sets, complete orthonormal sets together
with related identities and inequalities
3.Familiar with the theory of linear operators on a Hilbert space, including adjoint
operators, self-adjoint and unitary operators with their spectra.
UNIT IV
Normal and Unitary operators, Projections, Finite-dimensional spectral theory:
Matrices.
(Sections 58, 59 of Chapter 10 & Section 60 of Chapter 11 of the text book 1.)
40
Learning
Out Comes
On successful completion of this unit,students will be able to
1.Understand the concept of normal and unitary operators and its
applications 2. Determine the finite and infinite dimensional spaces.
3. Determine whether linear operators are continuous, invertible, self-adjoint,
compact etc, and determine adjoints.
UNIT V
Learning
Out Comes
Determinants and the spectrum of an operator, the spectral theorem.A survey
of the situation.
(Sections 61 to 63 of Chapter 11 of the text book 1.)
On successful completion of this unit, students will be able to
1. Analyze the concept of spectrum of an operator and the real world
applications on this. 2. Apply linear operators in the formulation of differential and integral equations.
3. Define the spectrum of an operator, and derive basic properties
Prescribed
Text Book
Reference
Books
Online
Source
Introduction to Topology and Modern Analysis by G. F. Simmons, McGraw Hill
Book Company. Inc-International student edition
1. Functional Analysis A First Course by M.Thamban Nair
2. A Course in Functional Analysis by Conway, John B. Springer.
2. Functional Analysis by B. V. Limaye, Willey Eastern Limited, Bombay 1981.
3. First Course in Functional Analysis, C. Goffman and George Pedrick,
Prentice Hall of India Private Limited, New Delhi-110001.
4. E. Kreyszig, Introductory Functional Analysis with applications, Wiley Eastern,
1989.
5. Functional Analysis by Bachmen and Narici, Dover Publications Inc.; 2nd
edition edition (28 March 2003).
1. http://www.math.kit.edu/iana1/lehre/funcana2012w/media/fa-lecturenotes.pdf
2. https://users.math.msu.edu/users/jeffrey/920/920notes.pdf
3. https://www.mimuw.edu.pl/~aswiercz/AnalizaF/lecture.pdf
Course
Outcomes
1. The fundamental properties of normed spaces and of the transformations
between them.
2. To be acquainted with the statement of the Hahn-Banach theorem and its
corollaries.
3. To understand the notions of dot product and Hilbert space.
4. To apply the spectral theorem to the resolution of integral equations
Sturm-Liouville problems.
5. Demonstrate capacity for mathematical reasoning through analysing proving
and explaining concepts from functional analysis.
41
Course Code
&Course Name M 302
NUMBER THEORY
Objectives
1. The objective of the course is the study of the main properties of Primes, Divisibility,
Fundamental Theorem of Arithmetic, Greatest Common Divisor and Euclidean Algorithm
2. The basic results associated to Euler’s summation formula, asymptotic formulas and the
average order of d(n), ( )n , (n) and its applications.
3. The course gives an introduction to strong background of the Mobius function and the partial
sums of the Mobius function with their applications.
4. The central concepts which are studied properties of congruences, resudueclasses, complete
residue systems, linear congruences and reduced residue systems
5. To develop their skills in the programming of Characters in finite abelian groups, the
orthogonality relations,Dirichlet characters.
SYLLABUS
Unit I
Learning
Out Comes
Arithmetical Functions And Dirichlet Multiplication: Introduction- The Mobius
function function (n) – The Euler totient function (n)- A relation connecting
and - A product formula for (n)- The Dirichlet product of arithmetical
functions- Dirichlet inverses and the Mobius inversion formula- The Mangoldt
function (n)- multiplicative functions- multiplicative functions and Dirichlet
multiplication- The inverse of a completely multiplicative function-Liouville’s
function ( )n - The divisor functions ( )n . Chapter-2:- Articles 2.1 to 2.14
On successful completion of this Unit, students should be able to :
1. Prove properties of Mobius functions, Euler totient function and understand the
connection to
and
.
2. Determine the dirichlet product of arithmetical functions
3. Explian what is meant by a multiplicative and completely multiplicative of a
arithmetical functions.
Unit II
Learning
Out Comes
Averages of arithmetical functions: Introduction- The big oh notation. Asymptotic
equality of functions- Euler’s summation formula- Some elementary asymptotic
formulas-The average order of d(n)- The average order of the divisor functions
( )n - The average order of (n). The partial sums of a Dirichlet product-
Applications to (n) and (n)- Another identity for the partial sums of a
Dirichletproduct.Chapter -3:- Articles 3.1 to 3.7
After studying this unit, students should be able to:
1.Apply the Euler summation formula in asymptotic equalities and find the
average order of arithmetical functions
2. Determine the partial sums of dirichlet product.
3. Express applications of (n) and (n).
Unit III
some elementary theorems on the distribution of prime numbers:Introduction-
Chebyshev’s functions ( )x and ( )x - Relations connecting ( )x and ( )x -
Some equivalent forms of the prime number theorem-Inequalities for ( ) and pnn -
Shapiro’s Tauberian theorem- Applications of Shapiro’s theorem- An asymptotic
formula for the partial sums (1/ )p x
p
- The partial sums of the Mobius function –
The partial sums of the Mobius function. Chapter -3:- Articles 3.10 &3.11 and
Chapter-4:- Articles 4.1 to 4.9
42
Learning
Out Comes
On successful completion of this unit, students should be able to
1. Acquire knowledge of prime number theorem and Shapiro’s Tauberian theorem
with their Applications.
2. Understand the connection between chebyshev’s functions
3. Apply the Shapiro’s Tauberian theorem in asymptotic formulas.
Unit IV
Learning
Out Comes
CONGRUENCES: Definition and basic properties of congruences- Resudue
classes and complete residue systems- Linear congruences- Reduced residue
systems and the Euler- Fermat theorem- Polynomial congruences modulo p.
Lagrange’s theorem- Applications of Lagrage’s theorem- Simultaneous linear
congruences. The Chinese remainder theorem- Applications of the Chinese
remainder theorem- Polynomial congruences with prime power moduli. Chapter -
5:- Articles 5.1 to 5.9
On successful completion of this unit, students should be able to:
1.Determine thecongruences,
2. Prove theEuler- Fermat theorem, Lagrage’stheoremand Chinese remainder
theorem.
3. Give examples of linear congruences and their RRS properties.
Unit V
Learning
Out Comes
FINITE ABELIAN GROUPS AND THEIR CHARACTERS:
Characters of finite abelian groups- The character group- The orthogonality
relations- for characters- Dirichlet characters- Sums involving Dirichlet characters-
The nonvanishing of
L(1, ) for real nonprincipal .
DIRICHLET’S THEOREM ON PRIMES IN ARITHMETIC
PROGRESSIONS:
Introduction- Dirichlet’s theorem for primes of the form 4n-1 and 4n+1- The plan
of the proof of Dirichlet’s theorem- Proof of Lemma 7.4- Proof of Lemma 7.5-
Proof of Lemma 7.6- Proof of Lemma 7.7- Proof of Lemma 7.8- Distribution of
primes in arithmetic progressions.
Chapter 6:- Articles 6.5 to 6.10 and Chapter 7 :- 7.1 to 7.9
After studying this unit, students will be able to
1. Find the Characters of finite abelian groups
2. Prove dirichlet’s theorem and their properties.
3.Express orthogonality relations of characters and dirichlet characters.
Prescribed
Text Book
Reference
Books
Online
Source
Introduction to Analytic Number Theory- By T.M.APOSTOL-Springer Verlag-
New York, Heidalberg-Berlin-1976.
1. A Course in Number Theory and Cryptography Neal Koblitz, Graduate Texts in
Mathematics, New-York: Springer-Verlag, 1987.
1. http://homepages.warwick.ac.uk/staff/J.E.Cremona/courses/MA257/ma257.pdf
2. http://www2.math.uu.se/~astrombe/talteori2016/lindahl2002.pdf
Course
OutComes
After studying this course, you should be able to:
1. find quotients and remainders from integer division
2. apply Euclid’s algorithm and backwards substitution
3. understand the definitions of congruences, residue classes and least residues
4. add and subtract integers, modulo n, multiply integers and calculate powers,
modulo n
5. determine multiplicative inverses, modulo n and use to solve linear congruences.
43
Course Code
&Course Name M 303
Classical Mechanics
Objectives1. Beginning with a review of Newton's Laws applied to systems of particles.
2. The course moves on to rotational motion, dynamical gravity (Kepler's Laws) and motion in
non-inertial reference frames.
3. Students will know the concepts of classical mechanics and demonstrate a proficiency in the
fundamental concepts in this area of science.
4. Employ conceptual understanding to make predictions, and then approach the problem
mathematically. 5. Students will be able to solve problems using their knowledge and skills in modern physics.
SYLLABUS
Unit I
Learning
Out Comes
Introductory Ideas: Introduction, Space and Time, Newton’s Laws of Motion,
Inertial Frames, Gravitational Mass, Mechanics of a Particle: Conservation Laws,
Mechanics of a System of Particles, Lagrangian Dynamics: Introduction, Basic
Concepts, Constraints, Generalized Coordinates.
(Sections 1.1 to 1.7, 2.1 to 2.4 of the prescribed book.)
Having successfully completed this module, student will be able to
1. Understand the linear motion of systems of particles
2. Identify angular momentum for a particle and a system
3. Analyze, synthesize and process information.
Unit II
Learning
Out Comes
D’Alembert’s Principle, Lagrange’s Equations from D’Alembert’s principle,
Procedure for formation of Lagranges’s Equations, Lagrange’s Equations in
presence of Non-conservative forces, Generalized Potential – Lagrangian for a
Charged Practicle, Hamilton’s Priniciple and Lagrange’s Equations, Superiority of
Lagrangian Mechanics over Newtonian Approach, Guage Invariance of the
Lagrangian, Symmetry Properties of Space and Time and Conservation Laws,
Invariance under Galilean Transformation.
(Sections 2.6 to 2.14 of the prescribed text book.)
Having successfully completed this module, you will be able to
1. Describe and understand the motion of a mechanical system using Lagrange-
Hamilton formalism.
2. Analyze, synthesize and process information.
3. Utilize appropriate mathematical tools to analyze and solve a system’s equations
Unit III
Learning
Out Comes
Hamiltonian Dynamics: Introduction, Generalized momentum and cyclic
coordinates, Conservation Theorems, Hamiltonian Function H and Conservation of
Energy: Jacobi’s integral, Hamilton’s Equations, Hamilton’s Equations in different
coordinate systems, Examples in Hamiltonian Dynamics. (Sections 3.1 to 3.7 of
the prescribed text book.)
Having successfully completed this module, you will be able to
1. Discuss the linear motion of systems of particles (e.g. rocket motion)
2. Define angular momentum for a particle and a system
3. Define moment of inertia and use it in simple problems
Unit IV
Variational principles: Introduction, The Caclculus of Variations and Euler-
Lagranges’s Equations, Deduction of Hamilton’s Principle from D’Alembert’s
Principle, Modified Hamilton’s Principle, Deduction of Hamilton’s Equations
from modified Hamilton’s principle (Variational Principle), Deduction of
Lagrange’s Equations form Variational Principle for non-conservative systems,
44
Learning
Out Comes
Physical significance of Lagrange’s Multipliers λ, ∆-Variation, Principle of Least
Action, Other Forms of Principle of Least Action.
(Sections 5.1 to 5.8, 5.10 to 5.12 of the prescribed text book.)
Having successfully completed this module, you will be able to
1. Demonstrate knowledge and understanding of Lagrangian and Hamiltonian
formulation of mechanics.
2. Describe and understand the vibrations of discrete and continuous mechanical
systems.
3. Apply the Lagrangian formalism to analyze problems in Mechanics
Unit V
Learning
Out Comes
Dynamics of Rigid Body: Generalized Coordinates of a Rigid Body, Body and
Space Reference Systems, Euler’s Angles, Infinitesimal Rotations as Vectors-
Angular Velocity, Components of Angular Velocity, Angular Momentum and
Inertia Tensor, Principle Axes Principle Moments of Inertia, Rotational kinetic
Energy of a Rigid Body, Toque-Free Motion for a Rigid Body.
(Sections 10.1 to 10.12 of the prescribed text book.)
Having successfully completed this module, you will be able to
1. To demonstrate knowledge and understanding the dynamics of system of
particles,motion of rigid body.
2. Describe and understand planar and spatial motion of a rigid body.
3. Translate physical problems into appropriate mathematical language and apply
appropriate mathematical tools
Prescribed
Text Book
Reference
Books
Online
Source
Classical mechanics by J.C. Upadhyaya, Himalaya Publishing House Pvt. Ltd.
1. Classical mechanics by H. Goldstein, 2nd edition, Narosa Publishing House.
2. Classical Mechanics by Gupta, Kumar and Sharma
1. http://math.ucr.edu/home/baez/classical/texfiles/2005/book/classical.pdf
2. http://courses.physics.ucsd.edu/2010/Fall/physics200a/LECTURES/200_COURSE.pdf
Course
Outcomes
Upon successful completion, students will have the knowledge and skills to
1. Solve complicated physical problems using the principle of least action.
2. Describe the role of the wave equation and appreciate the universal nature of
wave motion in a range of physical systems.
3. Use Fourier theory and diffraction to describe properties of waves.
Understand the fundamentals of the mechanics of continuous systems.
4. Model and analyze the dynamics of physical systems using computational
methods.
5. Through the lab course, understand the principles of measurement and error
analysis and develop skills in experimental design.
45
Course Code
&Course Name M304(1)
OPERATIONS RESEARCH
Objectives
1. To make the students familiar with principles and techniques of operations research and their
applications in decision making.
2.This course aims at familiarizing the students with quantitative tools and techniques, which are
frequently applied to business decision-making
3. To provide a formal quantitative approach to problem solving and an intuition about situations
where such an approach is appropriate.
4. To effective communicative ideas and explain procedures.
5. To effectively interpret results and solutions in written.
SYLLABUS
Unit I
Learning
Out Come
Linear Programming:DetermineThe Simplex Method – Overall Idea of the
Simplex Method – Development of the Simples Method – Primal Simplex method
– Dual Simplex Method – Special cases in Simplex Method Applications –
Sensitivity Analysis.
On successful completion of this unit, students will be able to
1.Explain the importance and scope of operations research
2. Formulate linear programming problems for resource allocation.
3. Solve linear programming problems using appropriate techniques.
Unit II
Learning
Out Come
Revised Simplex Method and Duality: Mathematical Foundations – Revised
(Primal) Simplex Method – Definition of the Dual Problem – Solution to the Dual
Problem – Economic Interpretation of the Dual Problem.
On successful completion of this unit, students will be able to
1. Solve linear programming problems using Revised Simplex method.
2. Solve linear programming problems using Dual Problem.
3. Applications of Economic Interpretation of the Dual Problem.
Unit III
Learning
Out Come
Determine the Transportation Model, Net Works and Applications of the
Transportation – Solution of the Transportation Problem – The Assignment Model
– The Transhipment Model.
On successful completion of this unit, students will be able to
1. Choose the method for initial solution of transportation problem.
2. Solve the transportation problem using optimum solution.
3. Solve the assignment problems using different methods.
Unit IV
Learning
Out Come
Network Definitions – Minimal Spanning Tree problem – Shortest – Route
Problem Network Models: - Maximal Flow Problem
On successful completion of this unit, students will be able to
1. Understand the concept of Network model
2. Applications of Shortest route network models
3. Allications of maximal flow models.
Unit V
Learning
Out Come
The Minimum Cost Capacitated Flow Problem Decision Theory and
Games:Decisions Under Uncertainty – Game Theory – Optimal solution of Two-
Person Zero-sum Games, Mixed strategies.
On successful completion of this unit, students will be able to
1. Understand the concept of Flow problems.
46
2. Applications of decision theory.
3. Applications of game theory.
Prescribed
Text Book
Reference
Books
Online
Source
1.Operations Research, An Introduction: Hamdy A Taha, Maxwell Macmillan
International Edition, New York, 1992.
1. Operations Research by Hira and Gupta, S.Chand Company and PVT LTd.
2. Operations Research by D N Mishra , S K Agarwal, PoojaSinha, World Press,
Luknow,U.P,India.
1. https://www.pdfdrive.com/operations-research-books.html
2. www.pondiuni.edu.in/storage/dde/downloads/mbaii_qt.pdf
3. https://easyengineering.net/operations-research-p-ramamurthy/
4. www.cs.toronto.edu/~stacho/public/IEOR4004-notes1.pdf
Course
Out Comes
At the end of the course students should be able to
1. Understand the mathematical tools that are needed to solve optimization.
2. Choose the method for effective allocation.
3. Interpret the relation between initial to optimum solution in transportation.
4. Evaluate the right person to the right job.
5. Choose the best strategy for business decision making.
47
Course Code
&Course Name M304(2)
MATHEMATICAL BIOLOGY
Objectives
1.This course is aimed to be accessible both to Master's students of biology who have a good
understanding of the introductory course to mathematical biology and to Master's students in
Mathematics looking to broaden their application areas.
2. The course extends the range of usage of mathematical models in biology, ecology and
evolution.
3. Biologically, the course looks at models in evolution, population genetics and biological
invasions.
4. Mathematically the course involves the application of multivariable calculus, ordinary
differential equations and partial differential equations.
5. Formulation and analysis of ordinary differential equation (ODE) models for the population of
a single species, finding equilibrium populations and determining how their stability depends on
parameters.
SYLLABUS
Unit I
Learning
Out Comes
Autonomous differential equations - Equilibrium solutions - Stability nature of
equilibrium solutions, single species growth models involving exponential, logistic
and Gompertz growths. Harvest models – bifurcations and break points.
(Sections 1 and 2 of the Text Book1)
Upon completing this unit, students will be able to
1. Understand the concept of single species growth models involving exponential,
logistic and Gompertz growths. Harvest models – bifurcations and break points.
2. Convert verbal descriptions of biological systems into appropriate mathematical
models amenable to quantitative and qualitative analysis.
3. Develop the ability to explain mathematical results in language understandable
by biologists.
Unit II
Learning
Out Comes
LotkaVolterra predator – prey model – phase plane analysis, General predator prey
systems – equilibrium solutions – classification of equilibria – existence of cycles
– Bendixson-Dulac’s negative criterion – functional responses.
(Sections 7 and 8 of the text book1)
Upon completing this unit, students will be able to
1. Understand the concept of LotkaVolterra predator – prey model – phase plane
analysis and applications of Bendixson-Dulac’s negative criterion.
2. Identify the equilibrium points and study the phase portrait analysis of predator
prey model.
3. Perform elementary mathematical analysis of models introduced and interpret
conditions obtained from the analysis - usually taking the form of relationships
between model parameters - that correspond to specific model behavior, and
express the ramifications for the biological process being considered.
Unit III
Learning
Out Comes
Global bifurcations in predator prey models – Freedman and Wolkowicz model -
type IV functional response – Hopf bifurcation – Homoclinic orbits – Global
bifurcations using Allee effect in prey – Competition models.
(Sections 9 and 10 of the prescribed text book1)
Upon completing this unit, students will be able to
1. Understand the concept of Global bifurcations in predator prey models, Global
bifurcations using Allee effect in prey – Competition models and applications.
48
2. Understand and apply the concept of stability of a fixed point solution of a
system of ordinary differential equations.
3. Analyze the model with graphical representation and give biological
interpretation.
Unit IV
Learning
Out Comes
Lotka – Voltrrra Competition model – exploitation competition models. Mutualism
models – various types of mutualisms – cooperative systems – Harvest models and
optimal control theory (Sections 11 and 12 of the text book1)
Upon completing this unit, students will be able to
1. Analyze ODE models for the populations of two interacting species.
2. Identify equilibrium points and using information about their linear stability to
characterize the long-term behavior of the system.
3. Analyze the model with graphical representation and give biological
interpretation.
Unit V
Learning
Out Comes
Open access fishery – sole owner fishery – Pontryagin’s maximum principle –
Economic interpretation of Hamiltonian and adjoint variable.
(Sections 13 and 14 of the prescribed text book)
Upon completing this unit, students will be able to
1. Understand the concepts of Open access fishery, sole owner fishery
2. Apply Pontryagin’s maximum principle to Open access fishery, sole owner
fishery
3. Analyze economical interpretation for sole owner fishery.
Prescribed
Text Book
Reference
Books
Online
Source
Elements of Mathematical Ecology by Mark Kot, Cambridge University Press,
2001.
1. Nisbet and Gurney, 1982, Modelling Fluctuating Populations, John Wiley &
Sons.
2. Modeling through Differential Equation by D. N. Burghes Ellis Horwood and
John Wiley.
3. Principle of Mathematical Modeling by C. Dyson and E. Levery, Academic
Press New York.
4. A First Course in Mathematical Modeling by Giordano, Weir, Fox 2nd Edition,
Brooks/ Cole Publishing Company, 1997.
5. Mathematical Modeling by J. N. Kapur, Wiley Eastern Ltd. 1994.
6. Mathematical Modeling with Case Studies B. Barnes, G. R. Fulford, A
Differential Equation Approach using Maple and Matlab, 2nd Ed., Taylor and
Francis group, London and New York, 2009.
1. https://www.math.mun.ca/~zhao/ARRMSschool/MathBioNotes2011.pdf
2. http://www.di.univr.it/documenti/OccorrenzaIns/matdid/matdid262230.pdf
Course Out
Comes
On successful completion of this course unit students will be able to
1. Read, situate, and understand research papers in the area of mathematical
biology.
2. Prepare to discuss specific biological systems with life scientists, and in
particular communicate efficiently how values of model parameters can impact the
qualitative behavior of the system.
3. Solve mathematically and interpret biologically simple problems involving one-
and two-species ecosystems, epidemics and biochemical reactions.
4. Analyze the model with graphical representation and give biological
interpretation for competition models, mutualism models.
5. Analyze economical interpretation for open access fishery, sole owner fishery
models using Pontryagin’s maximum principle.
49
Course Code
&Course Name M304(3)
Commutative Algebra I
Objectives
1.The course develops the theory of commutative rings.
2.These rings are of fundamental significance since geometric.
3.Number theoretic ideas is described algebraically by commutative rings
4.Knows basic definitions concerning elements in rings, classes of rings, and ideals in
commutative rings.
5.Can use algebraic tools which are important for many problems and much theory development
in algebra.
SYLLABUS
Unit I
Learning
Out Comes
Rings and ring homomorphism, ideals, quotient rings, zero divisors, Nilpotent
elements,units, prime ideals and Maximal ideals.
On successful completion of this unit, students will be able to
1. Understand the concept of Ring theory.
2. Understand the concept of different idels.
3. Applications of Ring theory.
Unit II
Learning
Out Comes
Nil radical and Jacobson radical, operations on ideals, Extensions and
contractions-Modules and module homomorphisms, Sub modules and quotient
modules, operations onsubmodules, Direct sum and product, finitely generated
modules.
On successful completion of this unit, students will be able to
1. Express the concept of radical and Jacobson radical.
2. Understand the concept of Modules and its applications.
3. Understand the concept of Direct sum and product.
Unit III
Learning
Out Comes
Exact sequences, Tensor product of modules, Restriction and extension of
scalars,Exactness properties of the tensor product, algebras, tensor product of
algebras.
On successful completion of this unit, students will be able to
1. Understand the concept of Exact sequence, Tensor product.
2. Express the properties of Exactness and algebra.
3. Understand the concept of tensor product of algebras.
Unit IV
Learning
Out Comes
Local Properties- Extended and contracted ideals in rings of fractions.
On successful completion of this unit, students will be able to
1. Understand the concept of Local properties.
2. Application of the concept of Local properties.
3. Understand the Extended &contracted ideals in rings.
Unit V
Learning
Out Comes
The Primary decompositions.
On successful completion of this unit, students will be able to
1. Understand the concept of Decomposition.
2. Understand the concept of Primary Decomposition.
3. Application of Primary Decomposition.
Prescribed
Text Book
Introduction to commutative algebra, By M.F. ATIYAH and I.G.
MACDONALD, Addison-Wesley publishing Company, London.
50
Reference
Books
Online
Source
1. Basic Commutative Algebra by Balwant Singh, World scientific Publishing Co.
Pte. Ltd.
2. Commutative Algebra by N.S.Gopal Krishna, Second Edition.
1. web.mit.edu/18.705/www/13Ed.pdf
2. math.uga.edu/~pete/integral.pdf
3.https://www.jmilne.org/math/xnotes/CA.pdf
4. www.math.toronto.edu/jcarlson/A--M.pdf
Course
Out Comes
After studying this course, you should be able to
1. Knows basic definitions concerning elements in rings, classes of rings, and
ideals in commutative rings.
2. Know constructions like tensor product and localization, and the basic theory for
this.
3. Know basic theory for noetherian rings and Hilbert basis theorem.
4. Know basic theory for integral dependence, and the Noether normalization
lemma.
5. Have insight in the correspondence between ideals in polynomial rings, and the
corresponding geometric objects: affine varieties.
51
Course Code
&Course Name M304(4)
Banach Algebras
Objectives: 1. Banach algebras have a lot of structure, combining the topological
features of a Banach space with the algebraic features of a ring.
2. Although we shall see many other examples, our main focus will be onexaminingBanach
algebras consisting of continuous linear operatorsonHilbert and Banach spaces.
3. Students will able to learn the structure of commutative Banach Algebras.
4. Knows basic definitions concerning elements in rings, classes of rings, and ideals in
commutative rings.
5. Can use algebraic tools which are important for many problems and much theory development
in banachalgebra.
SYLLABUS
Unit I
Learning
Out Come
General preliminaries on Banach Algebras – The definition and examples –
Regular and singular elements – Topological divisors of Zero – The spectrum
On successful completion of this unit, students will be able to
1. Understand the concept of General preliminaries on Banach Algebras.
2. Application of the concept of Local properties and Regular and singular
elements.
3. Understand the Topological divisors of Zero.
Unit II
Learning
Out Come
The formula for the spectral radius – The radical and the semi – simplicity.
On successful completion of this unit, students will be able to
1. Understand the concept of Banach Algebras.
2. Application of the concept of formula for the spectral radius
3. Understand the radical and the sem-simplicity.
Unit III
Learning
Out Come
The structure of commutative Banach Algebras - TheGelfand mapping -
Applications of the formula r(x) = lim // xn // 1/n – Involutions in Banach Algebras
– The Gelfand – Neumark theorem.
On successful completion of this unit, students will be able to
1. Understand the concept of commutative Banach Algebras.
2. Application of the concept of TheGelfand mapping.
3. Understand the Involutions in Banach Algebras.
Unit IV
Learning
Out Come
Some special commutative Banach Algebras - Ideals in C(x) and the Banach –
Stone theorem - The stone – Cechcompactification – commutaticve C* - algebras.
On successful completion of this unit, students will be able to
1. Understand the concept of Ideals in C(x) and the Banach – Stone theorem.
2. Application of the concept of The stone – Cechcompactification.
3. Understand the commutaticve C* - algebras.
Unit V
Learning
Out Come
Fixed point theorems and some applications to analysis – Brouwer’s and
Schauder’s fixed point theorems (without proofs) Picard’s theorem – Continuous
curves – The Hahn – Mazurkiewicz theorem (without proof). Boolean rings – The
stone representation theorem.
On successful completion of this unit, students will be able to
1. Understand the concept of Fixed point theorems and some applications to
analysis.
52
2. Application of the concept of Picard’s theorem – Continuous curves.
3. Understand The Hahn – Mazurkiewicz theorem.
Prescribed
Text Book
Reference
Books
Online
Source
1.Introduction to Topology and Modern Analysis – By G.F. Simmons –
International Student edition – McGraw – Hill Kogakusha Ltd.
1. W.B. Arveson, A Short Course in Spectral Theory. (Chapters 1 and 2).
2. J.B. Conway, A Course in Functional Analysis. (Chapters 7, 8 and 9).
3.P.R. Halmos, A Hilbert Space Problem Book, (2nd ed.), Springer-Verlag, 1982.
1. http://www.math.nagoya-u.ac.jp/~richard/teaching/s2014/Course_Wilde.pdf 2. http://www.math.lmu.de/~petrakis/INTRODUCTION%20TO%20BANACH%20ALGEBRAS.pdf
3. https://www.math.uni-hamburg.de/home/khomskii/papers/Bachelor_Thesis_Yurii_Khomskii.pdf
Course
Out Comes
After completion of this course, student will be able to
1. Students taking this course will develop an appreciation of the basic concepts of
Functional Analysis, including the study of operator theory and the study of
topological function spaces.
2.These methods will be useful for further study in arange of other fields, e.g.
Quantum Theory, Stochastic calculus and Harmonicanalysis.
3. Use advanced theoretical and practical knowledge gained in the field.
4. Play a role in decision-making process based on the arising problems while
working with different disciplines.
53
Course Code
&Course Name M 305(1)
NUMERICAL ANALYSIS
Objectives
1.This course is aimed to be accessible both to Master's students of Mathematics who have a good
understanding of the introductory course to Numerical Analysis and to Master's students in
Mathematics looking to broaden their application areas.
2. The course extends the range of several available solutions of equations in one and more
variable.
3. Formulation and analysis of the several available methods to Solve the simultaneous equations.
4. Mathematically the course involves of Single step methods, multi step Method and obtaining
numerical solutions to problems of mathematics.
5. Analyze and evaluate numerical methods for various mathematical operations and tasks, such
as interpolation, differentiation, integration, the solution of linear and nonlinear equations, and the
solution of differential equations.
SYLLABUS
Unit I
Learning
Out Comes
Interpolation and Approximation: Introduction-Lagrange and Newton
interpolation-Finite difference operators- Interpolating polynomials using finite
differences – Hermite Interpolations.
Section 4.1 to 4.5 of Chapter-IV of the prescribed text book 1.
On successful completion of this unit, students should be able to:
1. Construct a function which closely fits given n-points in the plane by using
interpolation method.,
2. Find the lagrange polynomial passing through the given points.
3. Find the hermite polynomial passing through the given points.
4. Find the cubic spline passing through the given points.
Unit II
Learning
Out Comes
Differentiations and Integration: Introduction – Numerical differentiation –
optimum choice of step length – extrapolation methods.
Section 5.1 to 5.4 of Chapter V of the prescribed text book 1.
After studying this unit, students should be able to:
1. Describing and understanding of the several errors and approximation in
numerical methods.
2. Find the optimum choice of step length problems
3. Demonstrate understanding of common numerical extrapolation methods.
Unit III
Learning
Out Comes
Numerical Integration – Methods based on interpolation – method based on
undetermined coefficients – composite Integration methods - Romberg Integration.
Section 5.6 to 5.10 of Chapter V of the prescribed text book 1.
On successful completion of this unit, students should be able to:
1. Express the method based interpolation
2. Understanding of several available solutions of equations in one and more
variable.
3. Find the solution of an equation by the composite Integration methods -
Romberg Integration.
Unit IV
Learning
Ordinary differential Equations: Introduction – Numerical Method – single step
methods( Taylor series Method)
Section 6.1 to 6.3 of Chapter VI of the prescribed text book 1.
After studying this unit, students should be able to:
54
Out Comes 1. Analyse and evaluate the accuracy of common numerical methods.
2. Find the determined function using Taylor series Method.
3. Find the solution of an equation by the single step methods.
Unit V
Learning
Out Comes
Ordinary differential Equations:Single step methods (Runge - Kutta2nd order
and 4th order Methods, Estimation of Local truncation error, system of Equations)
– Multi step Method.
Section 6.3 to 6.4 of Chapter VI of the prescribed text book 1.
After studying this unit, students should be able to:
1. Understand thefundamental concepts of Single step methods and multi
stepMethod
2. Find the solution of an equation by the Runge – Kutta2nd order and 4th order
Methods.
3. Derive numerical methods for various mathematical operations and tasks such
as interpolation, difference of linear and nonlinear equations and the solutions of
differential equations.
Prescribed
Text Book
Reference
Books
Online
Source
1.Numerical Methods, by M.K. Jain, S.R.K.Iyengar, R.K. Jain, 3rd Edition
New Age International (P) Limited, Publishers.
1.Numerical Methods, by M.K. Jain, S.R.K.Iyengar, R.K. Jain, 6th Edition
New Age International (P) Limited, Publishers.
2. Introductory Methods of Numerical Analysis by S.S. Sastry, 4th edition PHI
Publication.
3. K. E. Atkinson, An Introduction to Numerical Analysis (2nd edition), Wiley-
India, 1989.
4. S. D. Conte and Carl de Boor, Elementary Numerical Analysis - An Algorithmic
Approach (3rd edition), McGraw-Hill, 1981.
1. https://www.math.ust.hk/~machas/numerical-methods.pdf
2. http://www.math.iitb.ac.in/~baskar/book.pdf
Course
Out Comes
After completion of this course, student will be able to
1. Have advanced theoretical and practical knowledge to comprehend textbooks,
scientific papers etc. containing current information
2. Use advanced theoretical and practical knowledge gained in the field.
3. Update the theoretical and practical knowledge depending on the current
conditions.
4. Play a role in decision-making process based on the arising problems while
working with different disciplines.
55
Course Code
&Course Name M 305 (2)
Mathematical Software
Objectives
1. This course provides mathematical software to write mathematical notes, projects, and articles,
solving equations and plotting functions of one, two and three variables.
2. Student will explore: Latex, Matlab for numerical computation and Maple
3. To enable the student on how to approach for solving Engineering problems using simulation
tools.
4. Numeric and symbolic tools for discrete and continuous calculus including definite
and indefinite integration, definite and indefinite summation, automatic differentiation and
continuous and discrete integral transforms
5. Plotting of function of one, two and three variables using maple.
SYLLABUS
Unit I
Learning
Out Comes
LATeX introduction- Installation – Math symbols and tables – TeX symbol and
tables – Matrix and lists – Typing Math and text – Text environments.
On successful completion of this unit, students will be able to
1. Know about latex software.
2.Learn how to write math symbols in latex software
3. Write list of mathematical symbols and some paragraph.
Unit II
Learning
Out Comes
Document structure – Latex Documents – The AMS articles document class –
Bemer Presentation and PDF documents – Long Documents – BibteX – Make
index – Books in LateX- Colours and Graphics – TeXCAD – LATeX CAD.
On successful completion of this unit, students will be able to
1. Know about document structure, bemer presentation.
2. Learn how to write Document structure – Latex Documents – The AMS articles
document class.
3. Write books in latex – colours and graphics.
Unit III
Learning
Out Comes
Starting with MATLAB- Variables Vectors, Matrices – Creating Array in
MATLAB –Menu, Workspace, working Directory, Command window, Diary,
Printing- Built-in function, User defined functions, Script M-files- Complex
Arithmetic, Eigen values and Eigen vectors – Two and three dimensional Plots.
On successful completion of this unit, students will be able to
1. Know about MATLAB software.
2. Learn Variables Vectors, Matrices – Creating Array in MATLAB.
3. Create M-files and Plot a data diagram.
Unit IV
Learning
Out Comes
Getting around with maple – Maple input and output - Programming in Maple.
On successful completion of this unit, students will be able to
1. Know about maple software.
2. Learn input and output data, programming in maple.
3. Plot a data diagram.
Unit V
Learning
Maple: Abstract algebra – Linear algebra – Calculus on numbers – Variables-
Complex Arithmetic, Eigen values and Eigen vectors – Two and three dimensional
plots.
On successful completion of this unit, students will be able to
56
Out Comes 1. Know about maple software and write maple code.
2. Calculate arithmetic expressions, eigenvalues and eigenvectors.
3. Plot a data diagram.
Prescribed
Text Book
Reference
Books
Online
Source
1. G. Gratzer, More Math Into LATEX, 4th edition, Springer, (2007).
2. AMOS Gilat, MATLAB an introduction with application, WILEY India
Edition, (2009).
3. Brain R Hunt, Ronald L Lipsman,A Guide to MATLAB for beginners and
Experienced users, Cambridge University Press. (2003)
4. Ander Heck, Introduction in Maple, Springer, (2007)
1. LaTeX Beginner's Guide Kindle Editionby Stefan Kottwitz
2. MATLAB: A Practical Introduction to Programming and Problem Solving by
Stormy Attaway.
3. Understanding maple by Ian Thompson.
1. http://www.docs.is.ed.ac.uk/skills/documents/3722/3722-2014.pdf
1.http://page.math.tu-berlin.de/~chern/notes/MatlabLectureNote.pdf
2.http://mayankagr.in/images/matlab_tutorial.pdf
4. https://www.eecs.umich.edu/dco/docs/maple/intropg.pdf
Course Out
Comes
After completing this course students will be able to
1. Gain knowledge of Latex software and learn to write mathematical symbols in
Latex.
2. Write mathematical projects in Latex software and prepare mathematics notes.
3. Know uses of Matlab software and solve some mathematical expressions using
codes in Matlab.
4. Plot diagrams by writing codes in Matlab.
5. Solve for systems of equations, ODEs, PDEs and recurrence relations writing
codes in maple.
57
Course Code
&Course Name M 305(3)
Fuzzy Set Theory
Objectives
1. Provide an understanding of the basic mathematical elements of the
theory of fuzzy sets.
2. Provide an emphasis on the differences andsimilarities between fuzzy
sets and classical sets theories.
3. The mainobjective of this course is to establish thorough background
knowledgeon evolutionary algorithms in post graduate students.
4. The students topursue individual research in solving real world
optimization problemslike Constrained, Multimodal, Multi objective and
CombinatorialOptimizations.
5. Students will explore about Combinations of operations - Aggregation
Operations
SYLLABUS
Unit I
Learning
Out Comes
From Classical(Crisp) sets to fuzzy sets:- Introduction-Crispsets: An overview-
fuzzyset:Basic types-Fuzzy sets. Basic Concepts-Characteristics and significance
of the paradigm shift (Chapter-1).
Fuzzysets versus Crisp sets-Additional Properties of a𝛼-cuts-Representations of
Fuzzysets-Extension principle for Fuzzysets (Chapter-2).
After studying this unit, student will able to
1. Learn Basic types-Fuzzy sets
2. Know about Fuzzysets versus Crisp sets-Additional Properties of a cuts-
Representations of Fuzzysets
Unit II
Learning
Out Comes
Operations on Fuzzysets - Types of Operations - Fuzzy Compliments - Fuzzy
Inter sections: t-norms - Fuzzy unions; t-Conorms - Combinations of operations -
Agreegation Operations (Chapter-3).
After studying this unit, student will able to
1. Know aboutFuzzy Compliments - Fuzzy Inter sections
2. LearnFuzzy unions; t-Conorms
3. Know about Combinations of operations - Agreegation Operations
Unit III
Learning
Out Comes
Fuzzy Arithmetic -Fuzzy Numbers - Linguistic variables - Arithmetic operations
on intervals - Arithmetic operations on Fuzzy numbers - Lattice of fuzzy numbers -
Fuzzy equations (Chapter-4).
After studying this unit, student will able to
1. Know about Fuzzy Numbers - Linguistic variables
2. Know about Arithmetic operations on intervals
3. Learn Lattice of fuzzy numbers - Fuzzy equations
Unit IV
Learning
Out Comes
Fuzzy Relations - Crisp versus fuzzy relations - Projections and Cylindric
Extensions - Binary Fuzzy Relations - Binary Relations and Singleset - Fuzzy
Equivalence Relations. (5.1-5.5 in Chapter-5)
After studying this unit, student will able to
1. Know about Crisp versus fuzzy relations
58
2. Know about Projections and Cylindric Extensions
3. LearnBinary Relations and Singleset - Fuzzy Equivalence Relations.
Unit V
Learning
Out Comes
Binary Relations on a single set - Fuzzy Compatibility Relations - Fuzzy Ordering
Relations - Fuzzy Morphisms - Sup - Compositions of Fuzzy Relations - Inf -
Compositions of fuzzy Relations. (5.6-5.10 in Chapter-5)
After studying this unit, student will able to
1. Learn about Binary Relations on a single set.
2. Know about Fuzzy Morphisms - Sup - Compositions of Fuzzy Relations.
3. Express Inf - Compositions of fuzzy Relations
Prescribed
Text Book
Reference
Books
Online
Source
G.J.KLIR and BOYUAN "Fuzzy sets and Fuzzy Logic, Theory and Applications"
Prentice - Hall of India Pvt. Ltd., New Delhi., 2001.
1. H.J. Zimmermann, “Fuzzy set theory and its Applications “Allied Publishers
Ltd., New Delhi,1991 (For Units I & II).
2. Yu, Xinjie, Gen, Mitsuo, “Introduction to Evolutionary Algorithms”, Spinger,
ISBN 978- 1-84996-129-5.
1. T.J. Ross, John Wiley & Sons, Fuzzy Logic with Engineering Applications”,
IInd Ed., 2005.
2. M.C. Bhuvaneswari, “Application of Evolutionary Algorithms for Multi-
objectiveOptimization in VLSI and Embedded Systems”, Spinger,2014.
3. Ashlock, D. (2006), “Evolutionary Computation for Modeling and
Optimization”, Springer, ISBN0-387-22196-4.
1. https://www.mv.helsinki.fi/home/niskanen/zimmermann_review.pdf
2. https://cours.etsmtl.ca/sys843/REFS/Books/ZimmermannFuzzySetTheory2001.pdf
3. http://logica.dipmat.unisa.it/lucaspada/wp-content/uploads/foligno_handout.pdf
Course Out
Comes
1. Students taking this course will develop an appreciation of the basic concepts of
Functional Analysis, including the study of operator theory and the study of
topological function spaces.
2. These methods will be useful for further study in a range of other fields, e.g.
Quantum Theory, Stochastic calculus and Harmonic analysis.
3. Awareness of advanced theoretical and applied information supported by the
statistical sources.
4. Awareness of the significance and impact of statistical methods on the social
dimensions of interdisciplinary studies.
5. The ability to share solutions related with the problems encountered in the field
with experts and non-experts by supporting quantitative and qualitative data.
59
Course Code
&Course Name M 305(4)
Universal Algebra
Objectives:1.One of the aims of universal algebra is to extract the common elements of
seemingly different types of algebraic structures such as groups, rings or lattices.
2.Doing so one discovers general concepts, constructions, and results which unify and generalize
the known special situations.
3.Applications of universal algebra can be found in logic through the interface of algebraic logic.
4.The course will introduce the students to the basic concepts and theory of universal algebra.
SYLLABUS
Unit I
Learning
Out Comes
Definitions of Lattices – Isomorphisms of Lattices and Sub lattices- Distributive
andModular Lattices- Complete lattices- Equivalence relations- Algebraic lattices.
Sections 1 to 4 of Chapter 1 of the prescribed text book.
After the completion of the unit, Students will be able to
1. Distinguish isomorphism of lattices and sub lattices.
2. Classify the modular and distributive lattices.
3. Learn applications equivalence relations.
Unit II
Learning
Out Comes
Closure operators, Definition and examples of algebras- Isomorphic algebras and
subalgebras – Algebraic lattices and sub universes.
Section 5 of Chapter 1 & Sections 1 to 3 of Chapter 2 of the prescribed text book.
After the completion of the unit, Students will be able to
1.Classify Algebras-lattices- closure operators.
2. Determine the isomorphic algebras and sub algebras.
3. Understand the concept of algebraic lattices and sub universes.
Unit III
Learning
Out Comes
The irredundant Basis theorem- Congruences and Quotient algebras.
Homomorphisms –The homomorphism and isomorphism theorems, Direct
products- Factor congruences –Directly indecomposable algebras.
Sections 4 to 7 of Chapter 2 of the prescribed text book.
After the completion of the unit, Students will be able to
1. Determine the concept of irredundant basis theorems.
2. Understand the concept of homomorphism and isomorphism theorems.
3. Understand the concept of directly indecomposable algebras.
Unit IV
Learning
Out Comes
Sub direct products- Subdirectly irreducible algebras- Simple algebras- Class
operators-Varieties.Terms- Term algebras- Free algebras.
Sections 8 to 10 of Chapter 2 of the prescribed text book.
1.Understand the concept of sub direct products.
2.Determine the concept of subdirectly irreducible algebras , simple algebras.
3. Understand the concept of terms ,term algebras ,free algebras.
60
Unit V
Learning
Out Comes
Identities and Free algebras- Birkhoff’s theorem- Malcev conditions- The Centre
of analgebra.
Sections 11 to 13 of Chapter 2 of the prescribed text book.
1. Understand the concept of Ideals and dual ideals.
2. Determine the concept of Ideal chains- Ideal lattices
3. Understand the concept of Distributive lattices and rings of sets.
Prescribed
Text Book
Reference
Books
Online
Source
A course in Universal algebra- Stanley Burris, H.P. Sankappanavar,Springer-
Verlag, New York- Heidelberg- Berlin.
1. A Course in. Universal Algebra. H. P. Sankappanavar. Stanley Burris.
2. Universal Algebra: Fundamentals and Selected Topics Book by Clifford
Bergman.
3. Universal Algebra Book by Paul Cohn.
1. https://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra2012.pdf
2. https://link.springer.com/book/10.1007/978-0-387-77487-9 3.https://kluedo.ub.unikl.de/frontdoor/deliver/index/docId/1493/file/universal_algebra.pdf
Course Out
Comes
1.Recognise technical terms and appreciate some of the uses of algebra
2.Collect like terms and simplify expressions term by term
3.Multiply out brackets
4.Simplify some formulas
5.Solve simple linear equations.
61
M.Sc Mathematics SECOND YEAR – FORTH SEMESTER
Course Code &
Course Name M 401
MEASURE & INTEGRATION
Objectives
1. In this course we will learn the basic concepts of Measure theory and integration theory, with
related discussions on applications in classical Banach spaces.
2. The solution methods studied in this course will include the basic curriculum since it is crucial
for understanding the theoretical basis of probability and statistics.
3. Introduce students to how to prove bounded convergence theorem, Fatcous lemma and
monotone convergence theorem.
4. The course helps students develop skills to think quantitatively and analyse problems in absolute
continuity, convex functions and Bounded variation functions
5. This course will be useful for majors in such as harmonic analysis, ergodic theory, theory of
partial differential equations and probability theory.
SYLLABUS
Unit I
Learning
outcomes
Lebesgue measure: Introduction, Outer measure, measurable sets and Lebesgue
measure, A non-measurable set, measurable functions, Littlewood’s three
principles.
Chapter 3 of the text book
After the unit the students are expected to be able to:
1. Elaborate how to demonstrate a property for measurable sets and Lebesgue measure
2. Demonstrate understanding of non-measurable sets
3.Express the Littlewood’s three principles
Unit II
Learning
outcomes
The Lebesgue Integral: The Riemann integral, The Lebesgue integral of a
bounded function over a set of finite measure, the integral of nonnegative function.
Chapter 4 of the text book
By the end of the unit, the students must be able to:
1. Define and understand basic notations in riemann integral.
2.Determine conditions for the convergence of integrals
3.Apply integral convergence theorem to obtain approximate solutions to
mathematical problems
Unit III
Learning
outcomes
The Lebesgue Integral:The general Lebesgue integral, convergences in measure,
Differentiation and integration: Differentiation of monotone functions,
Chapter 5 of the text book
By the end of the unit, the students will be able to:
1.Describe the notion of convergences in measure
62
2.Establish basic properties for monotone functions,
3. Prove the bounded convergence theorem, Fatcous lemma and monotone
convergence theorem.
Unit IV
Learning
outcomes
Differentiation and integration:Functions of bounded variation and
differentiation of an integral, Absolute continuity, and convex functions.
Chapter 5 and Chapter 6 of the text book
1. Acquire knowledge of bounded variation functions
2. Define and understand the concept of Absolute continuity, and convex
functions.
3.Learn these apply for real world problem.
Unit V
Learning
outcomes
The classical Banach spaces: The Lp-spaces, The Minkoswki and Holder
inequalities, convergence and completeness, approximation in Lp, Bounded linear
functionals on the Lp spaces.
Chapter 6 of the text book
1. Understand the definition of Lp spaces.
2. Apply holders and minkowskis inequalities.
3. describeRiesz representation theorem.
Prescribed
Text Book
Reference
Books
Online
Source
1. Real Analysis by H. L. Royden, Macmillan Publishing Co. Inc. 3rd Edition, New
York, 1988.
1.Jones, Frank. Lebesgue Integration on Euclidean Space. Boston: Jones & Bartlett
Publishers, February 1, 1993.
2.Evans, Lawrence C., and Ronald F. Gariepy. Measure Theory and Fine
Properties of Function. Boca Raton, Florida: CRC Press, December 18, 1991.
ISBN: 0849371570.
1. https://people.math.ethz.ch/~salamon/PREPRINTS/measure.pdf.
2. http://www.math.utoronto.ca/almut/MAT1000/LL-1.pdf.
3. http://www.math.nagoya-u.ac.jp/~richard/teaching/s2017/Nelson_2015.pdf.
Course
Out Comes
After completion of this course, student will be able to
1. Computation of Lebesgue measures.
2. Establishing measurability or non-measurability of sets and functions. 3.
Approximating measurable functions by simple and step functions.
4. Computation of Lebesgue integrals, applications to volume calculations and
Fourier analysis.
5. Deciding under which conditions the fundamental theorem of calculus is
applicable in the context of Lebesgue integration.
63
Course Code
&Course Name M 402
PARTIAL DIFFERENTIAL EQUATIONS
Objectives 1. In this course we will study first and second order partial differential equations.
2. The solution methods studied in this course will include the method of characteristics,
separation of variables.
3. Introduce students to how to solve linear Partial Differential with different methods.
4. Technique of separation of variables to solve PDEs and analyze the behavior of solutions in
terms of eigenfunction expansions.
5. This course will be useful for majors in economics, mathematical finance, engineering and
physics.
SYLLABUS
Unit I
Learning
Out Comes
First Order P.D.E.: Curves and Surfaces – Genesis of First Order P.D.E. –
Classification of Integrals – Linear equations of the First Order - Pfaffian
Differential Equations – Compatible Systems.
Chapter 1 – sections 1.1 – 1.6
After the completion of the unit, Students will be able to
1. Distinguish ordinary differential equations and partial differential equation and
understand the concepts of first order -Pfaffian differential equations – compatible
Systems.
2. Classify the first order partial differential equations.
3. Learn applications of First order partial differential equations.
Unit II
Learning
Out Comes
Charpit’sMethod Jacobi’s Method - Integral Surfaces Through a Given Curve–
Quasi Linear Equations.
Chapter 1 – sections 1.7 – 1.11
After the completion of the unit, Students will be able to
1. Use the method of undetermined coefficients to solve second order, linear
homogeneous equations with constant coefficients
2. Use the method of reduction of order to find a second linearly independent
solution of a second order, linear homogeneous equation when one solution is
given.
3. Understand the notion of linear independence and the notion of a fundamental
set of solutions
Unit III
Second Order P.D.E.: Genesis of Second Order P.D.E. – Classification of Second
Order P.D.E. – One Dimensional Wave equation: Vibrations of an Infinite String –
Vibrations of Semi infinite String Vibrations of a String of Finite Length –
Riemann’s Method – Vibrations of a String of Finite Length (Method of
Seperation of Variables).
64
Learning
Out Comes
Chapter 2 – section 2.1 – 2.3.
After the completion of the unit, Students will be able to
1. Classify partial differential equations and transform into canonical form.
2. Use the method of variation of parameters to find particular solutions of second
order, linear homogeneous equations
3. Analyzestring of finite length, string of infinite length.
Unit IV
Learning
Out Comes
Laplace’s Equation: Boundary value Problems- Maximum and Minimum
Principles- The Cauchy Problem – The Dirichlet Problem for the Upper Half Plane
– The Neumann Problem for the Upper Half Plane – Dirichlet Problem for a Circle
– The Dirichlet Exterior Problem for a Circle- The Neumann Problem for a Circle
– The Dirichlet Problem for a Rectangle- Harnack’s Theorem Laplace’s Equation
– Green ‘s Function – The Dirichlet Problem for a Half Plane – The Dirichlet
problem for a Circle.
Chapter 2 – section 2.4
After the completion of the unit, Students will be able to
1. Understand the concepts of boundary value problems,maximum and minimum
principles, the Cauchy Problem.
2. Construct the Green’s function for Partial differential equations.
3. Apply Neumann problem for the upper half plane, for a circle.
Unit V
Learning
Out Comes
Heat Conduction Problem: Heat Conduction – Infinite Rod Case - Heat
Conduction –Finite Rod Case- Duhamel’s Principle –Wave Equation –Heat
Conduction Equation
Chapter 2 – sections 2.5, 2.6.
After the completion of the unit, Students will be able to
1. Understanding the heat conduction equation.
2. Apply Duhamel’s Principle for wave equation, heat conduction equation.
3. Apply partial derivative equation techniques to predict the behaviour of certain
phenomena.
Prescribed
Text Book
Reference
Books
Online
Source
An Elementary Course in Partial differential equations by T. Amarnath, Second
Edition, Narosa Publishing House, 1997.
1. Elements of Partial Differential Equations by Ian Sneddon, International
Students Edition.
2. Partial Differential Equations by Phoolan Prasad and RenukaRavindran, New
Age International, 1985.
3. Partial Differential Equations by F. John, Springer-Verlag, New York, 1978. 20
4. Partial Differential Equations by Tyn-Myint-U, North Holland Publication, New
York, 1987.
5.Partial Differential Equations for Engineers and Scienistsby J. N. Sharma, K.
Singh, Narosa, 2nd Edition.
1. http://www.math.tifr.res.in/~publ/ln/tifr70.pdf
2. https://www.iist.ac.in/sites/default/files/people/PDE-Notes2.pdf
3. https://www.math.uni-leipzig.de/~miersemann/pdebook.pdf
Course
Out Comes
Upon successful completion of the course, students will have the knowledge and
skills to
1. Apply a range of techniques to find solutions of standard Partial Differential
Equations (PDE).
2. Understand basic properties of standard PDE's.
3. Demonstrate accurate and efficient use of Fourier analysis techniques and their
65
applications in the theory of PDE's.
4. Demonstrate capacity to model physical phenomena using PDE's (in particular
using the heat and wave equations).
5. Apply specific methodologies, techniques and resources to conduct research and
produce innovative results in the area of specialisation.
Course Code
&Course Name M 403
MATHEMATICAL METHODS
Objectives
1. This course aims to develop a basic understanding of a range of mathematics tools with
emphasis on engineering applications.
2. It is intended for students to solve problems with techniques from advanced linear algebra,
ordinary differential equations and multi-variable differentiation.
3. Laplace transforms are also introduced.
4. The course helps students develop skills to think quantitatively and analyse problems critically.
5. This course addresses a number of important mathematical methods often used in physics.
SYLLABUS
Unit I
Learning
Out Comes
Laplace Transforms: Introduction – a few Remarks on the theory – Application to
differential equation – derivatives and intigrals of Laplace Transforms –
convolutions and Abel’s mechanical problem.
Sections 50 to 54 of Chapter 10 of the prescribed text book I.
On completion of this module, the learner will be able to
1. Describe several areas of Laplace transformation and applications.
2. Solving and model applied problems.
3.Determinederivatives and intigrals of Laplace Transforms and Abel’s mechanical
problem.
Unit II
Learning
Out Comes
Volterra Integral Equations: Basic concepts – Relationship between linear
differential equation and volterra Integral Equations – Resolvent kernel of Volterra
Integral Equations. Solution of Integral Equations by Resolvent kernel – The
Method of successive approximations – convolution type equation – solution of
integro- differential equation with the aid of the Laplace transformation
Sections 1 to 6 of Chapter I of the prescribed text book II.
On completion of this module, the learner will be able to
1. Describe several areas of Integral Equations
2.Determine the Method of successive approximations and convolution type
equation
3.Understanding the concept of Integro- differential equation with the aid of the
Laplace transformation
Unit III
Fredholm integral equations: Fredholm integral equations of 2nd kind.
Fundamentals – the method of Fredholm determinants – iterated kernals.
Constructing the resolvent kernel with the aid of iterated kernals- integral
equations with degenatekernals. Hammerstein type equation- characterstic
numbers and eigen functions.
Sections 12 to 16 of Chapter II of the prescribed text book II.
66
Learning
Out Comes
On completion of this module, the learner will be able to
1. Understanding the concept Fredholm integral equations
2. Describe the concept of resolvent kernel with the aid of iterated kernals
3. Solving the applications of characterstic numbers and eigen functions.
Unit IV
Learning
Out Comes
Solution of homogeneous of integral equations with degenate kernel -Non
homogeneous symmetric equations - Fredholm alternative – construction of greens
function for ordinary differential equations – using Green’s function in the solution
of boundary value.
Sections 17 to 20 of Chapter II of the prescribed text book II.
On completion of this module, the learner will be able to
1. Describe homogeneous of integral equations with degenate kernel
2. Understanding the concept of Fredholmalternative,construction of greens
function for ODE.
3.Solving applications of Green’s function in the solution of boundary value.
Unit V
Learning
Out Comes
Calculus of variance: Introduction some typical problems of the subject – Euler
differential equation for an extremal – Isopermetric Problems.
Sections 47 to 49 of Chapter 9 of the prescribed text book I.
On completion of this module, the learner will be able to
1. Solving problems of the Calculus of variance.
2. Determine concept of Euler differential equation for an extremal
3. Solving Isopermetric Problems.
Prescribed
Text Book
Reference
Books
Online
Source
1. Differential equation with Application historical notes by G.F.Simmons.
2.Integral Equation by Krasanov.
1. Sneddon I.N.,The Use of Integral Transforms, Tata McGraw Hill (1985).
2. GelfandI.M. andFominS.V., Calculus of Variations, Prentice Hall (1963).
3. Kenwal Ram P., Linear Integral Equations: Theory and Techniques, Academic
Press (1971).
1. https://www.researchgate.net/publication/267866066_Mathematical_Methods/download
2. https://physics.bgu.ac.il/~gedalin/Teaching/Mater/am.pdf.
3. https://www.elsevier.com/books/mathematical-methods/korevaar/978-1-4832-2813-6
Course Out
Comes
At the end of the course, students will
1. Laplace Transformation to solve initial and boundary value problems.;
2. To learn Fourier transformation and Z transformation and their applications to
relevant problems.;
3. To understand Hankel's Transformation to solve boundary value problem.;
4. Find solutions of linear integral equations of first and second type (Volterra and
Fredhlom)
5. Understand theory of calculus of variations to solve initial and boundary value
problems.
67
Course Code &
Course Name M 404(1)
Lattice Theory
Objectives
1. Lattice theory is the study of sets of objects known as lattices.
2. It is an outgrowth of the study of Boolean algebras.
3. Provides a framework for unifying the study of classes or ordered sets in mathematics.
4. Mathematical Logic, Boolean Algebra and its Applications, Switching circuit & Logic Gates,
Graphs and Trees.
5. Important theorems with constructive proofs, real life problems & graph theoretic algorithms.
SYLLABUS
Unit I
Learning
Out Comes
Partially Ordered sets- Diagrams- Special subsets of a poset -length- lower and
upper bounds- the minimum and maximum condition- the Jordan Dedekind chain
conditions -
Dimention functions.
Chapter I(sections 1 to 9) of the prescribed text book.
After the completion of the unit, Students will be able to
1. Distinguish Partially Ordered sets with diagrams
2. Classify the minimum and maximum conditions.
3. Learn applications Jordan Dedekind chain conditions.
Unit II
Learning
Out Comes
Algebras-lattices- the lattice theoretic duality principle- semilattices- lattices as
posets-diagrams of lattices- sub lattices, ideals
Chapter II(sections 10 to 16) of the prescribed text book
After the completion of the unit, Students will be able to
1.Classify Algebras-lattices- the lattice theoretic duality principle- semilattices
2. Determine the diagrams of lattices- sub lattices.
3. Understand the concept of ideals.
Unit III
Learning
Out Comes
Bound elements of Lattices-atoms and dualatomscomplements, relative
complements, semi complements-irreducible and prime elements of a lattice- the
homomorphism of a lattice-axioms systems of lattices.
Chapter II (sections 17 to 21) of the prescribed text book.
After the completion of the unit, Students will be able to
1. Determine the concept of atoms and dual atoms.
2. Understand the concept of compliments,semi compliments, relative
compliments.
3. Understand the concept of the homomorphism of a lattice-axioms systems of
lattices.
Unit IV
Boolean algebras, De Morgan formulae- Complete Boolean algebras- Boolean
algebras and Boolean rings- The algebra of relations- The lattice of propositions-
68
Learning
Out Comes
Valuations of Boolean algebras.
Chapters VI(sections 42 to 47) of the prescribed text book .
1.Understand the concept of Boolean algebras, Demorgan formulae.
2.Determine the concept of Complete Boolean algebras- Boolean algebras and
Boolean rings-
3. Understand the concept of Valuations of Boolean algebras
Unit V
Learning
Out Comes
Ideals and dual ideals- Ideal chains- Ideal lattices- Distributive lattices and rings of
sets.
Chapter VIII(sections 53 to 55) of the prescribed text book .
1.Understand the concept of Ideals and dual ideals.
2.Determine the concept of Ideal chains- Ideal lattices
3. Understand the concept of Distributive lattices and rings of sets.
Prescribed
Text Book
Reference
Books
Online
Source
Introduction to Lattice Theory by Gabor Szasz, Academic Press, New York.
General Lattice Theory by G. Gratzer, Academic Press, New York.
1. http://www.math.hawaii.edu/~jb/lat1-6.pdf
2. http://www.math.ucla.edu/~yy26/works/Lattice%20Talk.pdf
3. http://boole.stanford.edu/cs353/handouts/book1.pdf
Course
Out Comes
After completion of this course, students will be able to
1. Understand the concepts of Partially Ordered sets- Diagrams- Special subsets of
a poset -length- lower and upper bounds.
2 Understand the concepts of graph theory, Lattices, and Boolean Algebra in
analysis of various computer science applications.
3. Apply the knowledge of Boolean algebra in computer science for its wide
applicability in switching theory, building basic electronic circuits and design of
digital computers
4. Demonstrate knowledge and understandingthe concept of Complete Boolean
algebras- Boolean algebras and Boolean rings.
5. Understand the concept of Ideals and dual ideals, Distributive lattices and rings
of sets.
69
Course Code
&Course Name M404(2)
Theory Of Computations
Objectives: The goal of this course is to provide students with an understanding of basic
concepts in the theory of computation. At the end of this course students will:
1. Be able to construct finite state machines and the equivalent regular expressions.
2. Be able to prove the equivalence of languages described by finite state machines and regular
expressions.
3. Be able to construct pushdown automata and the equivalent context free grammars.
4. Be able to prove the equivalence of languages described by pushdown automata and context
free grammars.
5. Be able to prove the equivalence of languages described by Turing machines and Post
machines
SYLLABUS
Unit I
Learning
Out Comes
Sets, Relations, Special types of binary relations, logic preliminaries, finite-infinite
sets, fundamental proof techniques, alphabets, languages and their representations.
On successful completion of this unit, students will be able to
1. Write the Sets, Relations, Special types of binary relations .
2. Understand the concept of logic preliminaries, finite-infinite sets, fundamental
proof techniques.
3. Application of alphabets, languages and their representations.
Unit II
Learning
Out Comes
Deterministic finite automata, their equivalence, properties of languages accepted
by finite automata
On successful completion of this unit, students will be able to
1. Write the Deterministic finite automata.
2. Understand the concept of finite automata, their equivalence, properties.
3. Application of properties of languages accepted by finite automata.
Unit III
Learning
Out Comes
Regular expressions, (non) regular languages, Context free grammars.
On successful completion of this unit, students will be able to
1. Write the Regular expressions.
2. Understand the concept of regular languages.
3. Application of Context free grammars.
Unit IV
Learning
Out Comes
Context free languages, properties, push down automata, determinism and parsing.
On successful completion of this unit, students will be able to
1. Write the Context free languages .
2. Understand the concept of push down automata.
3. Application of determinism and parsing.
Unit V Turing machine, computing with Turing machines, combining Turing machines,
70
Learning
Out Comes
Extensions of Turing machines, nondeterministic Turing machines.
On successful completion of this unit, students will be able to
1. Write the Turing machine, computing with Turing machines.
2. Understand the concept ofcombining Turing machines.
3. Application ofExtensions of Turing machines, nondeterministic Turing
machines.
Prescribed
Text Book
Reference
Books
Online
Source
Hopcroft J. and Ullman J.D., Introduction to Automata Theory, Languages and
Computation.
1. Peter Linz, An Introduction to Formal Languages and Automata, Third
Edition, Jones and Bartlett, 2001.
2. Papadimitriou, Elements of the Theory of Computation, Prentice-Hall, 1998
3. John E. Hopcroft, Rajeev Motwani, Jeffrey D. Ullman, "Introduction to
Automata Theory,Languages, and Computation", 2nd Edition, Prentice-Hall, 2001
4. Peter Dehning, Jack B. Dennis, “Machines, Languages and Computation”,
Second Edition, Prentice-Hall, 1978.
Course
Out Comes
1.Students will learn about a variety of issues in the mathematical development of
computer science theory, particularly finite representations for languages and
machines, as well as gain a more formal understanding of algorithms and
procedures.
2. In order to improve the pedagogy of this course, interactive animations of the
various automata using available simulators are recommended.
3. Acquire a full understanding and mentality of Automata Theory as the basis of
all computer science languages design
4. Have a clear understanding of the Automata theory concepts such as RE's,
DFA's, NFA's, Stack's, Turing machines, and Grammars.
5. Be able to minimize FA's and Grammars of Context Free Languages.
71
Course Code
&Course Name M404(3)
Commutative Algebra II
Objectives
1. The course develops the theory of commutative rings.
2.These rings are of fundamental significance since geometric
3. Numbertheoretic ideas is described algebraically by commutative rings.
4. Knows basic definitions concerning elements in rings, classes of rings, and ideals in
commutative rings.
5. Can use algebraic tools which are important for many problems and much theory development
in algebra
SYLLABUS
Unit I
Learning
Out Comes
Integral dependence, the going-up theorem-Integrally closed integral domains.
On successful completion of this unit, students will be able to
1. Understand the concept of Integral dependence,
2. Write the going-up theorem
3. Understand the concept Integrally closed integral domains.
Unit II
Learning
Out Comes
The going-down theorem, valuation rings.
On successful completion of this unit, students will be able to
1. Write the going –down theorem.
2. Understand the concept of valuation rings.
3. Application of valuation rings.
Unit III
Learning
Out Comes
Chain Conditions.
On successful completion of this unit, students will be able to
1. Understand the concept of Chain conditions.
2. Write Ascending chain condition.
3. Write Descending chain condition.
Unit IV
Learning
Out Comes
Noetherian rings- Primary decomposition of Noetherian rings, Artin rings.
On successful completion of this unit, students will be able to
1. Understand the concept ofNoetherian rings.
2. Understand the concept of Primary decomposition of N.R.
3. Understand the concept ofArtin rings.
Unit V
Learning
Out Comes
Discrete valuation rings, Dedekind domains, Fractional ideals.
On successful completion of this unit, students will be able to
1. Express the concept of Discrete valuation rings.
2. Understand the concept of Dedekind domains.
3. Express the concept of Fractional ideals.
Prescribed
Text Book
1.Introduction to commutative algebra by M.F.Atiya and I.G.
Macdonald, Addison-Welsey Publishing Company, London.
72
Reference
Books
Online
Source
1. Basic Commutative Algebra by Balwant Singh, World scientific Publishing Co.
Pte. Ltd.
2. Commutative Algebra by N.S.Gopal Krishna, Second Edition.
1. web.mit.edu/18.705/www/13Ed.pdf
2. math.uga.edu/~pete/integral.pdf
3.https://www.jmilne.org/math/xnotes/CA.pdf
4. www.math.toronto.edu/jcarlson/A--M.pdf
Course
Out Comes
On Completion of this module, the learner will be able to
1. Knows basic definitions concerning elements in rings, classes of rings, and
ideals in commutative rings.
2. Know constructions like tensor product and localization, and the basic theory for
this.
3. Know basic theory for noetherian rings and Hilbert basis theorem.
4. Know basic theory for integral dependence, and the Noether normalization
lemma.
5. Have insight in the correspondence between ideals in polynomial rings, and the
corresponding geometric objects: affine varieties.
73
Course Code
&Course Name M404(4)
Theory Of Linear Operators
Objectives: 1. Brief review of basic fact and terminology related to complete normed spaces and
linear functionals.
2. Finite, infinite and block matrices as linear operators. Schur test for boundedness. The adjoint
and the hermitianadjoint of an operator.
3. Finite rank and compact linear operators. Examples of integral operators. Operators with
compact resolvent. Application to some differential equations.
4. Hilbert-Schmidt operators. Examples of integral operators.
5. Spectral representation of compact selfadjoint operators in Hilbert spaces.
SYLLABUS
Unit I
Learning
Out Comes
Spectral Theory in Finite Dimensional Normed Spaces, Basic Concepts, Spectral
Properties of Bounded Linear Operators, Further Properties of Resolvent and
Spectrum, Use of Complex Analysis in Spectral Theory.
On successful completion of this unit, students will be able to
1. Express the concept of Spectral Theory in Finite Dimensional Normed Spaces
2. Understand the concept of Basic Concepts, Spectral Properties of Bounded
Linear Operators .
3. Express the concept of Resolvent and Spectrum, Use of Complex Analysis in
Spectral Theory.
Unit II
Learning
Out Comes
Banach Algebras, Further Properties of Banach Algebras, Compact Linear
Operators on Normed Spaces and Their Spectrum, Compact Linear Operators on
Normed Spaces, Further Properties of Compact Linear Operators, Spectral
Properties of Compact Linear Operators on Normed Spaces.
On successful completion of this unit, students will be able to
1. Express the concept of banach Algebras.
2. Understand the concept of Compact Linear Operators on Normed Spaces and
Their Spectrum.
3. Express the concept of Compact Linear Operators on Normed Spaces.
Unit III
Learning
Out Comes
Further Spectral Properties of Compact Linear Operators, Operator Equations
Involving Compact Linear Operators, Further Theorems of Fredholm
Type,Fredholm Alternative.
On successful completion of this unit, students will be able to
1. Express the concept of spectral Properties of Compact Linear Operators.
2. Understand the concept ofCompact Linear Operators.
3. Express the concept ofTheorems of Fredholm Type, Fredholm Alternative.
Unit IV
Spectral Theory of Bounded Self-Adjoint Linear Operators, Spectral Properties of
Bounded Self-Adjoint Linear Operators, Further Spectral Properties of Bounded
Self-Adjoint Linear Operators, Positive Operators, Square Roots of a Positive
Operator, Projection Operators, Further Properties of Projections
74
Learning
Out Comes
On successful completion of this unit, students will be able to
1. Express the concept of Spectral Theory of Bounded Self-Adjoint Linear
Operators.
2. Understand the concept ofSpectral Properties of Bounded Self-Adjoint Linear
Operators.
3. Express the concept ofPositive Operators, Square Roots of a Positive Operator.
Unit V
Learning
Out Comes
Spectral Family, Spectral Family of a Bounded Self-Adjoint Linear Operator,
Spectral Representation of Bounded Self-Adjoint Linear Operators, Extension of
the Spectral Theorem to Continuous Functions,Properties of the Spectral Family of
a Bounded Self- Adjoint Linear Operator.
On successful completion of this unit, students will be able to
1. Express the concept of Spectral Family of a Bounded Self-Adjoint Linear
Operator.
2. Understand the concept ofSpectral Representation of Bounded Self-Adjoint
Linear Operators.
3. Express the concept ofProperties of the Spectral Family of a Bounded Self-
Adjoint Linear Operator.
Prescribed
Text Book
Reference
Books
Online
Source
E. Kreyszig, Introductory Functional Analysis with Applications, JohnWiley&
Sons, New York, 1978.
1. P.R. Halmos, Introduction to Hilbert Space and the Theory of Spectral
Multiplicity, Second-Edition, Chelsea Publishing Co., New York, 1957.
2. N. Dunford and J.T. Schwartz, Linear Operators -3 Parts, Interscience/Wiley,
New York, 1958-71.
3. G. Bachman and L. Narici, Functional Analysis, Academic Press, York, 1966.
1.http://www-personal.acfr.usyd.edu.au/spns/cdm/resources/Kreyszig%20-
%20Introductory%20Functional%20Analysis%20with%20Applications.pdf
2.
Course
Out Comes
1. On successful completion of the course, students can opt for courses like
Operator Theory, Spectral Theory, Representation Theory etc.
2. Applications of spectral Thorem for compact operators. Polar decomposition.
3. Examples of spectral measures and applications of the Spectral Theorem.
4. Elementary properties of closed unbounded operators and solved applications of
differential operators. Cayley transform of symmetric oper
5. Spectral theorem for unbounded operators and Spectra of functions of operators.
75
Course Code
&Course Name M 405(1)
Wavelet Analysis
Objectives
1. The objective of this course is to establish the theory necessary to understand and use wavelets
and related constructions.
2. A particular emphasis will be put on constructions that are amenable to efficient algorithms,
since ultimately these are the ones that are likely to have an impact.
3. Study applications in signal processing, communications, and sensing where time-frequency
transforms like wavelets play an important role.
4. A particular emphasis will be put on constructions that are amenable to efficient algorithms,
since ultimately these are the ones that are likely to have an impact.
5. Study applications in signal processing, communications, and sensing where time-frequency
transforms like wavelets play an important role.
SYLLABUS
Unit I
Learning
Out Comes
An Overview: From Fourier analysis to wavelet analysis, The integral wavelet
transform and time-frequency analysis, Inversion formulas and duals,
Classification of wavelets, Multiresolution analysis, splines, and wavelets, Wavelet
decompositions and reconstructions.
On Completion of this module, the learner will be able to
1. Understand the concepts Fourier, Time-frequency window of wavelets, Discrete
wavelet transform, Haar wavelet and its Fourier transform.
2. Ability to distinguish Fourier and wavelet transforms.
3. Ability to write simple for some results on wavelet transforms.
Unit II
Learning
Out Comes
Fourier Analysis:Fourier and inverse Fourier transforms, Continuous-time
convolution and the delta function,Fourier transform of square-
integrablefunctions,Fourierseries,Basic convergence theory and Poisson's
summation formula
On Completion of this module, the learner will be able to
1. Understand Fourier and inverse Fourier transforms, continuous-time
convolution and the delta function.
2. Learn Fourier transform of square-integrable functions, Fourier series.
3. Ability to write proofs for some properties of wavelets.
Unit III
Learning
Out Comes
Wavelet Transforms and Time-Frequency Analysis:The Gabor transform,Short-time
Fourier transforms and the Uncertainty Principle,The integral wavelet
transform,Dyadic wavelets and inversions,Frames, Wavelet series
On Completion of this module, the learner will be able to
1. Understand the concepts of The Gabor transform, Short-time Fourier transforms
and the Uncertainty Principle.
2. Prepare Decomposition and reconstruction algorithm.
3. Ability to write proofs for some properties of wavelets.
Unit IV
Cardinal Spline Analysis: Cardinal spline spaces, B-splines and their basic properties,
The two-scale relation and an interpolatory
76
Learning
Out Comes
graphical display algorithm,B-net representations and computation of cardinal
splines,Construction of spline approximation formulas, Construction of spline
interpolation formulas.
On Completion of this module, the learner will be able to
1. Learn Orthonormality in frequency domain, Numerical evaluation of scaling
function and wavelets.
2. Prepare Construction of spline approximation formulas, Construction of spline
interpolation formulas.
3. Ability to write proofs for some properties of wavelets.
Unit V
Learning
Out Comes
Scaling Functions and Wavelets:Multiresolutionanalysis,Scaling functions with
finite two-scale relations, Direct-sum decompositions of L2(IR),Wavelets and their
duals,Linear –phase filtering, Compactly supported wavelets.
On Completion of this module, the learner will be able to
1. Understand the concepts of Multiresolution analysis,Scaling functions with
finite two-scale relations.
2. Prepare Decomposition and reconstruction algorithm.
3. Ability to write proofs for some properties of wavelets.
Prescribed
Text Book
Reference
Text Books
Online
Source
1. C.K. Chui, An Introduction to Wavelets, Academic Press, New York, 1992.
2. I. Daubechies, Ten Lectures on Wavelets, CBS-NSF Regional Conferences in
Applied Mathematics, SIAM, Philadelphia, 1992.
3. O. Christensen, An Introduction to Frames and Riesz bases, Birkh• auser,
Boston, 2003.
1. Y. Meyer, Wavelets: Algorithms and Applications, SIAM, Philadelphia, 1993.
2. L. Debnath, Wavelet Transforms and their Applications, Birkh• auser, Boston,
2002.
3. M.W. Frazier, An Introduction to Wavelets through Linear Algebra, Springer,
New York, 1999.
4. M.K. Ahmad, Lecture Notes on Wavelet Analysis, Seminar Library, Department
of Maths, AMU, 2015.
5. A First course on Waveletsby Eugenio Hernandez and weiss, Ist edition CRC
press
1. http://math.bu.edu/people/mkon/Wavelets.pdf
2. http://web.iitd.ac.in/~sumeet/WaveletTutorial.pdf
3. https://inside.mines.edu/~whereman/talks/UIA-00-Wavelet-Lectures.pdf
Course
Out Comes
Upon completion of this course, you should be able to
1. Understand the terminology that is used in the wavelets literature.
2. Explain the concepts, theory, and algorithms behind wavelets from an
interdisciplinary perspective that unifies harmonic analysis (mathematics), filter
banks (signal processing), and multiresolution analysis (computer vision).
3. Master the modern signal processing tools using signal spaces, bases, operators
and series expansions.
4. Apply wavelets, filter banks, and multiresolution techniques to a problem at
hand, and justify why wavelets provide the right tool.
5. Research, present, and report a selected project within a specified time.
77
Course Code
&Course Name M 405(2)
PROGRAMMING IN C
Objectives
1.The course is designed to provide complete knowledge of C language.
2. Students will be able to develop logics which will help them to create programs, applications in
C.
3. Also by learning the basic programming constructs they can easily switch over to any other
language in future.
4. The course is oriented to those who want to advance structured and procedural programming
understating and to improve C programming skills.
5. The major objective is to provide students with understanding of code organization and
functional hierarchical decomposition with using complex data types.
SYLLABUS
Unit I
Learning
Out Comes
Definition of Algorithms x- Flow Charts writing Algorithms – some simple
examples to illustrate these concepts like finding the sum, GCD of two numbers –
swapping two variables, simple interest, area of a circle given its radius, area of a
triangle given all its sides, Largest of given three numbers, sowing a given
quadratic equation, sum of first ‘n’ natural numbers, Generation of Fibonacci
sequence - Given integer is prime or not.
(First four units of the prescribed text book I)
On completion of this module, the learner will be able to
1.Illustrate the flowchart and design and algorithm for a given problem and to
develop IC programs using operators.
2. Ability to understand the concepts like finding the sum, GCD of two numbers –
swapping two variables, simple interest and etc.
3. Ability to write flow charts and algorithms for finding the sum, GCD of two
numbers – swapping two variables, simple interest and etc.
Unit II
Learning
Out Comes
Constants, Variables, and Data Types: Introduction - character set –C tokens –
keywords and identifiers constant – variables – Data types – Declaration of
variables – Assigning values to variables
Operators and Expression: Introduction - Arithmetic operators – Relational
operators – logical operators – Assignment operators - increment and decrement
operators – conditional operators – bitwise operators – special operators
Lab: writing C programmes which are related to problems on mathematics
(Unit 2 to Unit 3.9 of the prescribed text book II)
On Completion of this module, the learner will be able to
1. Understand the preliminaries for C language.
2. Ability to write the simple C programs for these concepts.
3. Ability to write C programs which are related to problems on mathematics
Unit III
Arithmetic expressions, evaluation of expressions, precedence of arithmetic
operators, some computational problems, type conversions in expressions, operator
precedence and associativity, Mathematical functions.Managing input and output
operations: Reading a character – writing a character formatted input – output
Lab: writing C programmes which are related to problems on mathematics
78
Learning
Out Comes
(Unit 3.10 to Unit 4 of the prescribed text book II)
On Completion of this module, the learner will be able to
1. Understand the mathematical preliminaries for C language.
2. Ability to write the simple C programs for these concepts.
3. Ability to write C programs which are related to problems on mathematics
Unit IV
Learning
Out Comes
Decision making and branching: Decision making with if statement – simple if
statement – the if else statement - nesting of if –else statement – The else of ladder
– the switch statement – the? Operators – the GO TO Decision Making and
Looping: Introduction - the while statement – the Do statement – the FOR
statement – jumps in loops
Lab: writing C programmes which are related to problems on mathematics
(Unit 5 to Unit 6 of the prescribed text book II)
On Completion of this module, the learner will be able to
1. Understand the concepts decision making statements, loops and nested loops.
2. Ability to write the simple C programs for these concepts.
3. Ability to write C programs which are related to problems on mathematics
Unit V
Learning
Out Comes
Arrays: Introduction - One dimensional arrays – two dimensional arrays –
initializing two- dimensional arrays – multidimensional arrays. Introduction to
pointers.
Lab: writing C programmes which are related to problems on mathematics.
(Unit 7 of the prescribed text book II)
On Completion of this module, the learner will be able to
1. Understand the declaration and implementation of arrays, pointers, functions
and structures.
2. Inscribe C programs using pointers and to allocate memory using dynamic
memory management functions.
3. Ability to write C programs which are related to problems on mathematics using
array and pointers
Prescribed
Text Book
Reference
Books
Online
Source
1. Programming techniques through ‘c’ by M.G. VenkateshMutry
2. C programming in Ans1 ‘C’ by E Balaguruswamy (Unit 2,3,4,) (second edition)
1. C Programming Absolute Beginner’s Guide (3rd Edition)’ by Greg Perry and
Dean Miller
2. The C Programming Language’ by Brian W. Kernighan and Dennis M. Ritchie.
3. C Programming: A Modern Approach (2nd Edition)’ by K. N. King
1. http://www.vssut.ac.in/lecture_notes/lecture1424354156.pdf
2. http://www.vssut.ac.in/lecture_notes/lecture1422486950.pdf
3. http://www.kciti.edu/wp-content/uploads/2017/07/cprogramming_tutorial.pdf
Course
Out Comes
After course completion the students will be able to
1. Develop conditional and iterative statements to write C programs.
2. Exercise user defined functions, data types, including structures and unions to
solve real time problems.
3. Exercise files concept to show input and output of files in C
4. To understand the file operations, character I/O, String I/O, file pointers and
importance of pre-processor directives.
5. Understand the declaration and implementation of arrays, pointers, functions
and structures and ability to write C programs which are related to problems on
mathematics using array and pointers.
79
Course Code
&Course Name M405(3)
Semi Groups
• Objectives :To introduce the Concepts of Semigroups, monogenic Semigroups, Free
Semigroups, Ideals, Regular Semigroups, Simple and Q-Simple Semigroups, and their related
theories to develop working knowledge on these concepts and moreover
• 1. This course aims to expose the students to more liberal and powerful tools of Algebra that are
applicable in the present-day life.
• 2.To familiarise students with the elementary notions of semigroup theory.
• 3.To illustrate abstract ideas by applying them to a range of concrete examples of semigroups.
• 4.To study Green's relations and how these may be used to develop structure theorems for
semigroups.
• 5.To develop problem solving skills and to acquire knowledge on basic concepts of Semi groups,
Ideals and Rees’ congruencesand the structure of D-classes.
SYLLABUS
Unit I
Learning
Out Comes
Basic definition, monogenic semigroups, ordered sets, semilatttices and lattices,
binary relations, equivalences and congruences. Free semigroups, Ideals and Rees’
congruences.( Sections 1 to 4 of Chapter- I).
On successful completion of this unit, students will be able to
1.Understand basic definitions of Semigroups, Semilattices and Lattices, and their
basic Results.
2.student able to appreciate the importance of semigroup theory in abstract
algebra;
3.Familiar with the most important classes of semigroups and have an
understanding of the structure of important examples, such as the most famous
transformation semigroups.
Unit II
Learning
Out Comes
Lattices of equivalences and congruences, Green’s equivalences, the structure of
D-classes, regular semigroups- Simple and Q- simple semigroups. . ( Sections 5 to
8 of Ch. I ).
Upon completion of this unit, the student will be able to:
1.Understandcongrancesand Green’s equivalences, and also find Structure of D-
Classes.
2.The basic ideas of the subject, including Green’s relations, and be able to handle
the algebra of semigroups in a comfortable way.
3.The role of structure theorems, and be able to use Rees' theorem for completely
Q-simple semigroups.
80
Unit III
Learning
Out Comes
Principal factors, Rees’ theorem, Primitive idempotents.Congruences on
completely Q-simple semi groups.(Sections 1 to 3 of Chapter III ).
Upon completion of this unit, the student will be able to:
1. Analyze Simple and Q-Simple Semigroups, and Rees’s Theorem.
2.Demonstrate knowledge of the primitive idempotents;
3.Write precise and accurate mathematical definitions of objects in Semi groups;
Unit IV
Learning
Out Comes
The lattice of congruences on a completely 0-simple semigroup, Finite congruence
free semigroups.(Sections 4 to 6 of Chapter III ).
Upon completion of this unit, the student will be able to:
1. Describe Congruences on Completely O-Simple Semigroups and Finite
Congruences.
2.Identify and develop free semigroup models from the verbal description of the
real system.
3. Understand the definitions of (completely) (0)-simple semigroups and the proofs
of some of the main theorems in this section.
Unit V
Learning
Out Comes
Union of Groups, Semi lattices of groups, bands, free bands, varieties of bands.
(Sections 1 to 5 of Chapter IV ).
Upon completion of this unit, the student will be able to:
1. Students will demonstrate knowledge and comprehension of basic principles of
semi-lattice of groups.
2. Students will be able to apply basic principles of bands and varieties of bands in
simple mathematical problem solving involving mathematical structures.
3.Use mathematical definitions to identify and construct examples and to
distinguish examples from non-examples;
Prescribed
Text Book
Reference
Books
Online
Source
An introduction to semi group theory by J.M. Howie, 1976, Academic press,
New York.
1.John M. Howie: Fundamentals of semigroup theory, Clarendon press, Oxford,
1995.
2. A. H. Clifford and G. B. Preston: The Algebraic theory of semi groups, Vol. 1,
and 2, Mathematical surveys of the AMS, 1961 and 1967.
3. P. M. Higgins: Techniques of Semi Group Theory, Oxford University Press,
1992.
1. https://pdfs.semanticscholar.org/3193/9dfde70be855c8919462216c0801b5d4a8de.pdf
2. https://www.ams.org/books/surv/007.1/surv007.1-endmatter.pdf
Course
Out Comes Having successfully completed this course student will be able to:
1. the basic properties of Green's relations and use these in an appropriate way
2. Construct new semigroups using congruences.
3.To appreciate the importance of semigroup theory in abstract algebra.
4.To learn and feel that learnig further advance tools of this discipline will equip
them to apply these tools to the huge world of Automata, Languages and
Machines.
5. The student realizes the richness of properties enjoyed by Semigroups, an
81
algebraic structure with fewer facilities than Groups.
Course Code
&Course Name M 405(4)
Financial Mathematics
Objectives
1. This is an introductory course in Financial Mathematics.
2. Student will learn about the different types of interest (simple interest, discount interest,
compound interest), annuities, debt retirement methods, investing in stocks and bonds. Time
permitting, more advanced topics will also be covered.
3. Apply logical thinking to problem solving in context.
4. Use appropriate technology to aid problem solving.
5. Define interest rate risk in terms of duration and convexity of fixed interest products.
SYLLABUS
Unit I
Learning
Out Comes
The Measurement of Interest: Introduction, The accumulation and amount
functions, The effective rate of interest, Simple interest, Compound interest,
Present value, The effective rate of discount, Nominal rates of interest and
discount, Forces of interest and discount, Varying interest, Summary of results.
Solution of Problems in Interest: Introduction, The basic problem, Equation of
value, Unknown time, Unknown rate of interest, Determining time periods,
Practical examples.
On completion of this module, the learner will be able to
1. Define and recognize the definitions of the following terms: interest rate (rate of
interest), simple interest, compound interest, accumulation function, future value,
current value, present value, etc.
2. Give any one of the effective interest rate, the nominal interest rate.
3. Understand the concepts.
Unit II
Learning
Out Comes
Basic Annuities: Introduction, Annuity-immediate, Annuity-due, Annuity values
on any date, Perpetuities, Unknown time, Unknown rate of interest, Varying
interest, Annuities not involving compound interest.
On completion of this module, the learner will be able to
1. Understand the concepts Unknown rate of interest, Varying interest.
2.Write the equation of value given a set of cash flows and an interest rate.
3.Apply these concepts in real world problems.
Unit III
Learning
Out Comes
More General Annuities: Introduction, Differing payment and interest conversion
periods, Annuities payable less frequently than interest convertible, Annuities
payable more frequently than interest convertible, Continuous annuities, Payments
varying in arithmetic progression, Payments varying in geometric progression,
More general varying annuities, Continuous varying annuities, Summary of
results.
On completion of this module, the learner will be able to
1. Understand the concepts carefully.
2.Describe in detail the various types of annuities and perpetuities and use them to
solve financial transaction problems.
3. Apply these concepts in real world problems.
Unit IV Amortization Schedules and Sinking Funds: Introduction, Finding the outstanding
82
Learning
Out Comes
loan balance, Amortization schedules, Sinking funds, Differing payment periods
and interest conversion periods, Varying series of payments, Amortization with
continuous payments, Step-rate amounts of principal.
On completion of this module, the learner will be able to
1.Understand the concepts
2.Explain the details of arbitrage and its use in the valuation of forward contracts.
Employ term structure of interest rates to calculate forward and spot rates.
3. Apply these concepts in real world problems.
Unit V
Learning
Out Comes
Bonds and Other Securities: Introduction, Types of securities, Price of a bond,
Premium and discount, Valuation between coupon payment dates, Determination
of yields rates, Callable and putable bonds, Serial bonds, some generalizations,
other securities, Valuation of securities.
Yield Rates: Introduction, Discounted cash flow analysis, Uniqueness of the yield
rate, Reinvestment rates, Interest measurement of a fund, Time-weighted rates of
interest, Portfolio methods and investment year methods, Short sales, Capital
budgetingbasic technique and other technique.
On completion of this module, the learner will be able to
1.Define interest rate risk in terms of duration and convexity of fixed interest
products.
2. Find Employ term structure of interest rates to calculate forward and spot rates.
3. Apply these methods in real life.
Prescribed
Text Book
Reference
Books
Online
Source
1. Stephen G. Kellison, The Theory of Interest, 3rd Edition. McGraw Hill
International Edition (2009).
2. R. J. Elliott and P. E. Kopp, Mathematics of Financial Markets, Springer (1999)
Harshbarger, R.J. & Reynolds, J.J., Mathematical Applications for the
Management, Life and Social Sciences 12th ed.
1. http://www1.maths.leeds.ac.uk/~jitse/math1510/notes-all.pdf
2. https://people.kth.se/~lang/finansmatte/fin_note.pdf
Course
Out Comes
At the end of the course students will be expected to
1. Determine and select the most appropriate standard mathematical, statistical and
computing methods appropriate for specifying mathematical problems in banks
and other financial institutions through a critical understanding of the relative
advantages of these methods, and to develop extensions to these methods
appropriate for the solution of non-standard problems;
2. Know the main features of models commonly applied in financial firms, be able
to express these mathematically and be able to appraise their utility and
effectiveness;
3. Explain and critically appraise the rationale for the selection of mathematical
tools used in the analysis of common financial problems;
4. Be able to demonstrate the appropriateness of modelling or numerical solutions
in analysing common problems in banks and other financial institutions;
5. Be able to select and apply numerical solutions in some areas of finance.