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MSC Math

The document outlines the syllabus and examination scheme for the M.Sc. Mathematics program at Gurugram University, effective from the 2020-21 academic session. It details the program's vision, mission, program outcomes, specific outcomes, qualification requirements, and the structure of courses across four semesters. The evaluation scheme includes internal and external assessments, with a minimum attendance requirement and a credit system for course completion.

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0% found this document useful (0 votes)

33 views81 pages

MSC Math

Uploaded by

Deepali Ahire

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Download as PDF, TXT or read online on Scribd

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Syllabi, Department of Mathematics, GUG w.e.f.

session 2020-21

Gurugram University, Gurugram

(A State University Established under Haryana Act 17 of 2017)

Examination Scheme
&
Syllabus
For
M.Sc. - MATHEMATICS
(SEMESTER- I to IV)
(w.e.f. 2020-21)
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

Background
 The Mathematics Department of the Gurugram University, Gurugram, Haryana - India was
established in 2020.
 Department has started P.G. Program M.Sc. - Mathematics w.e.f. 2020-2021.
 Specialization in pure and applied Mathematics.
 Intake in First session 2020-22 was 20 Students.
 In Session 2021-23 Intake increased up-to 40 Students.

Vision
 The vision of the Department of Mathematics is to promote excellence and innovation in
teaching and research in pure/applied Mathematics and interdisciplinary area for “Mathematics
and Statistics”.

Mission

 To build research and creative activities among faculty and students

 To impact high quality mathematical education for global mathematics competence.

Program Outcomes (POs)

Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

PO1 Knowledge Capable of demonstrating comprehensive disciplinary

knowledge gained during course of study
PO2 Research Aptitude Capability to ask relevant/appropriate questions for identifying,
formulating and analyzing the research problems and to draw
conclusion from the analysis
PO3 Communication Ability to communicate effectively on general and scientific
topics with the scientific community and with society at large
PO4 Problem Solving Capability of applying knowledge to solve scientific and other
problems
PO5 Individual and Capable to learn and work effectively as an individual, and as a
Team Work member or leader in diverse teams, in multidisciplinary settings.
PO6 Investigation of Ability of critical thinking, analytical reasoning and research
Problems based knowledge including design of experiments, analysis and
interpretation of data to provide conclusions
PO7 Modern Tool usage Ability to use and learn techniques, skills and modern tools for
scientific practices
PO8 Science and Ability to apply reasoning to assess the different issues related to
Society society and the consequent responsibilities relevant to the
professional scientific practices
PO9 Life-Long Learning Aptitude to apply knowledge and skills that are necessary for
participating in learning activities throughout life
PO10 Ethics Capability to identify and apply ethical issues related to one’s
work, avoid unethical behaviour such as fabrication of data,
committing plagiarism and unbiased truthful actions in all
aspects of work
PO11 Project Ability to demonstrate knowledge and understanding of the
Management scientific principles and apply these to manage projects
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

Program Specific Outcomes (PSOs)

After successful completion of the programme, a student will be able to:

PSO1 Have deep understanding and knowledge in the core areas of Mathematics and
demonstrate understanding and application of the concepts/theories/principles/
methods/ techniques in different areas of pure and applied Mathematics.
PSO21. Have capability to read and understand mathematical texts, demonstrate and
communicate mathematical knowledge effectively and unambiguously through oral
and/or written expressions and attain skills of computing/programming/using
software tools/formulating models.
PSO3 Attain abilities of critical thinking, logical reasoning, investigating problems,
analysis, problem solving, application of mathematical methods/techniques,
disciplinary knowledge so as to develop skills to solve mathematical problems
having applications in other disciplines and/or in the real world.
PSO42. Have strong foundation in basic and applied aspects of Mathematics so as to venture
into research in different areas of mathematical sciences, jobs in scientific and
various industrial sectors and/or teaching career in Mathematics.

Qualification descriptors
Bachelor degree with Mathematics as one of the subjects with atleast 50% marks (47.5% marks for
SC/ST/ Blind/ Visually and Differently Abled candidates of Haryana only) in aggregate or any other
examination recognized by State Universities of Haryana as equivalent thereto.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

Gurugram University, Gurugram

(A State University Established under Haryana Act 17 of 2017)

Study & Evaluation Scheme

of
M.Sc. Mathematics
Summary

Programme : M.Sc. Mathematics

Duration : Two year full time (Four Semesters)
Medium : English
Minimum Required Attendance : 75%
Total Credits : 114
Assessment/Evaluation
Internal External Total
20 80 100

Internal Evaluation (Theory

Papers) Minor Attendance Assignment Total
Test
10 5 5 20

Duration of Examination External Internal (Minor Test)

3 hrs. 1 ½ hrs.

To qualify the course, a student is required to secure a minimum of 40% marks in

aggregate including the end semester examination and internal evaluation. (i.e., both
internal and external). A candidate who secures less than 40% of marks in a course shall
be deemed to have failed in that course. The student should have at least 40% marks in
aggregate to clear the semester.

Question Paper Structure

1. The question paper shall consist of 9 questions. Out of which, first question shall
be of short answer type and will be compulsory. Question no. 1 shall contain 8
parts representing all units of the syllabus and students shall have to answer all
parts.
2. The remaining 8 questions shall have internal choice. The weightage for each
question shall be 16 marks.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

Gurugram University, Gurugram

Scheme of Examination for M.Sc.-MATHEMATICS
Semester-I (w.e.f. 2020-21) Credits= 29 Marks=800
Sr. Paper Subjects Type Credit Contact Hours Per Week Examination Scheme Total
No. Code of Theory Tutorial Practical Total Theory Tutorial Practical Total Exam Internal Practical Marks
Course /Seminar /Seminar Assessment /Seminar
1 MATH- Abstract C.C. 4 1 -- 5 4 1 -- 5 80 20 -- 100
111 Algebra -I

2 MATH- Ordinary C.C. 4 1 -- 5 4 1 -- 5 80 20 -- 100

112 Differential
Equations
3 MATH- Mechanics C.C. 4 1 -- 5 4 1 -- 5 80 20 -- 100
113
4 MATH- Measure and C.C. 4 1 -- 5 4 1 -- 5 80 20 -- 100
114 Integration

5 MATH- Mathematical C.C. 4 1 -- 5 4 1 -- 5 80 20 -- 100

115 Statistics

6 MATH- Computing S.E.C. 2 2 4 2 4 6 40 10 50 100

116 Lab-I
(Introduction
To MS Excel
and
Programming
in Fortran)
7 MATH- Seminar SEC - 1 -- _ 2 2 _ 20 80 100
117
8 Open O.E.C. 2 -- 2 2 -- 2 80 20 -- 100
Elective-I
Total 24 5 3 31 24 5 6 35 800

C.C. = Core Course S.E.C.=Skill Enhancement Course O.E.C. = Open Elective Course
Note: 1. 20MATH-117 (Seminar) is based on paper 20MATH-111 to 20MATH-115 and two contact hours per week for each paper.
2. For Open Elective-I, Students will have to be chosen a course from the list of open electives offered by other UTDs.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

Gurugram University, Gurugram

Scheme of Examination for M.Sc.-MATHEMATICS
Semester-II (w.e.f. 2020-21) Credits= 28 Marks=750

Sr. Paper Code Subjects Type Credit Contact Hours Per Week Examination Scheme Total
No. of Theory Practical Total Theory Practical Total Exam Internal Practical
Course /Seminar /Seminar Assessment /Seminar
1 MATH-121 Abstract Algebra -II C.C. 4 -- 4 4 -- 4 80 20 -- 100

2 MATH-122 Complex Analysis C.C. 4 -- 4 4 -- 4 80 20 -- 100

3 MATH-123 Topology C.C. 4 -- 4 4 -- 4 80 20 -- 100

4 MATH-124 Operations Research C.C. 4 -- 4 4 -- 4 80 20 -- 100
Techniques
5 MATH-125 Computational C.C. 4 -- 4 4 -- 4 80 20 -- 100
Techniques
6 MATH-126 Computing Lab-II S.E.C. -- 2 2 -- 4 4 -- 20 80 100
(Programming in C)
7 MATH-127 Mathematical Lab –II S.E.C - 5 5 _ 2x5=10 10 _ 20 80 100
& Seminar
8 CS-100 Communication Skills A.E.C.C. -- 1 1 -- 2 2 -- 50 -- 50
Total 28 36 750

C.C. = Core Course S.E.C.= Skill Enhancement Course A.E.C.C.= Ability Enhancement Compulsory Course

Note: 1. 20MATH-127 (Mathematical Lab –II & Seminar) is based on paper 20MATH-121 to 20MATH-125 and two contact hours per week
for each paper.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21
Gurugram University, Gurugram
Scheme of Examination for M.Sc.- MATHEMATICS
Semester-III (w.e.f.2020-21) Credits= 29 Marks=750
Sr. Paper Subjects Type Credit Contact Hours Per Week Examination Scheme Total
No. Code of Theory Tutorial Practical Total Theory Tutorial Practical Total Exam Internal Practical
Course /Seminar /Seminar Assessment /Seminar
1 MATH- Analytical C.C. 4 1 -- 5 4 1 -- 5 80 20 -- 100
231 Number
Theory
2 MATH- Partial C.C. 4 1 -- 5 4 1 -- 5 80 20 -- 100
232 Differential
Equations
3 MATH- Mechanics of C.C. 4 1 -- 5 4 1 -- 5 80 20 -- 100
233 Solids-I
4 MATH- Discipline D.S.E 4 1 -- 5 4 1 -- 5 80 20 -- 100
234 Elective-I
5 MATH- Discipline D.S.E. 4 1 -- 5 4 1 -- 5 80 20 -- 100
235 Elective-II
6 MATH- Computing S.E.C. 2 -- 2 4 2 -- 4 6 40 10 50 100
236 Lab-III
(Programming
in MATLAB)
7 MATH- Seminar S.E.C. -- -- 1 1 _ -- 2 2 _ 10 40 50
237
8 Open O.E.C. 2 -- -- 2 2 -- -- 2 80 20 -- 100
Elective-II
Total 24 5 29 24 5 35 750

C.C. = Core Course, O.E.C. = Open Elective Course, D.S.E. = Discipline Specific Elective, S.E.C.= Skill Enhancement Course
List of papers for Discipline Elective I&II:(a) MATH-I Fluid Dynamics
(b) MATH-II Discrete Mathematics
(c) MATH-III Differential Geometry
(d) MATH-IV Coding Theory .(e)MATH-V Applied Statistics
(f) MATH-VI Financial Mathematics
(g) MATH-VII Bio- Mathematics
Note:1. MATH-236& MATH-237 (Mathematical Lab –III & Seminar) is based on paper MATH-231 to MATH-233&
Discipline Elective-I &Discipline Elective-II and two contact hours per week for each paper.
2.The Discipline elective paper can be offered depending upon the availability of the resources and faculties.
3.For Open Elective-II, Students will have to be chosen a course from the list of open electives offered by other UTDs.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21
Gurugram University, Gurugram
Scheme of Examination for M.Sc. - MATHEMATICS.
Semester-IV (w.e.f. 2020-21) Credits= 28 Marks=750
Sr. Paper Code Subjects Type Credit Contact Hours Per Week Examination Scheme Total
No. of Theory Practical Total Theory Practical Total Exam Internal Practical
Course /Seminar /Seminar Assessment /Seminar
1 MATH-241 Functional Analysis C.C. 4 -- 4 4 -- 4 80 20 -- 100
2 MATH-242 Integral Equations and C.C. 4 -- 4 4 -- 4 80 20 -- 100
Calculus of Variations
3 MATH-243 Mechanics of Solids-II C.C. 4 -- 4 4 -- 4 80 20 -- 100
4 MATH-244 Discipline Elective-III D.S.E. 4 -- 4 4 -- 4 80 20 -- 100
5 MATH-245 Discipline Elective-IV D.S.E. 4 -- 4 4 -- 4 80 20 -- 100
6 MATH-246 Computing Lab-IV S.E.C. -- 2 2 -- 4 4 -- 20 80 100
(Programming in
LaTeX and Gnuplot)
7 MATH-247 Mathematical Lab –IV S.E.C. - 5 5 _ 2x5=10 10 _ 20 80 100
& seminar
8 MATH-248 Self-Study Paper A.E.C.C. -- 1 1 -- 2 2 -- -- 50 50
Total 28 36 750

C.C. = Core Course, D.S.E. = Discipline Specific Elective, S.E.C.= Skill Enhancement Course, A.E.C.C.= Ability Enhancement
Compulsory Course
List of papers for Discipline Elective III&IV:(a) MATH-I Advanced Fluid Dynamics
(b)MATH-II Mathematical Aspects of Seismology
(c) MATH-III Mathematical Modeling
(d)MATH-IV Fourier and Wavelet Analysis.(e)MATH-V Statistical Inference
(f)MATH-VI Advanced Coding Theory
(g)MATH-VII Bio-Mechanics
Note:1. 20MATH-247 (Mathematical Lab –IV& Seminar) is based on paper MATH-241 to MATH-243, Discipline Elective-III,
Discipline Elective-IV and two contact hours per week for each paper.
2. The Discipline elective paper can be offered depending upon the availability of the resources and faculties.
Duration - 2 Years (4 Semesters)
Total Marks – 3100
Total credits-114
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

GENERAL INSTRUCTIONS

I. SELF-STUDY PAPER:

Maximum Marks-50

Objective: This course intends to create habits of reading books and to develop writing skills in a
manner of creativity and originality. The students are to emphasis his/her own ideas/words which he/she
has learnt from different books, journals and newspapers and deliberate the same by adopting different
ways of communication techniques and adopting time scheduling techniques in their respective fields.
This course aims:
- To motivate the students for innovative, research and analytical work
- To inculcate the habit of Self-Study and comprehension
- To infuse the sense of historical back ground of the problems
- To assess intensity of originality and creativity of the students
Students are guided to select topic of their own interest in the given area in consultation with their
teachers/Incharge/Resource Person.

Instructions for Students

1. Choose the topic of your interest in the given areas and if necessary, seek the help of your
teacher.
2. Select a suitable title for your paper.
3. You are expected to be creative and original in your approach.
4. Submit your paper in two typed copies of A4 size 5-6 pages (both sides in 1.5 line spaces in
Times New Roman Font size 12).
5. Organize your paper in three broad steps:
(a) Introductions
(b) Main Body
(c) Conclusions
6. Use headings and sub-headings
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

7. Use graphics wherever necessary

8. Give a list of books/references cited/used
9. The external examiner will evaluate the self-study paper in two ways i.e. Evaluation 30 Marks
and Viva-Voce 20 marks.

Distribution of Marks

1. The evaluation is divided into different segments as under : 30 Marks

The 30 marks evaluation is further subdivided as under:
(i) Selection of Topic - 6 Marks
(ii) Logical Organization of subject matter - 10 Marks
(iii) Conclusions - 10 Marks
(iv) References - 4 Marks
2. Viva-Voce: 20 Marks
The external examiner will hold Viva-Voce based on contents of the student’s Self Study Paper focusing
upon the description by the Candidate.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESTER-I
MATH-111
Abstract Algebra-I
Maximum Marks-100
External Examination-80
Internal Assessment-20
Max. Time- 3 hrs.

Course Outcomes
Students would be able to:
CO1 Describe various canonical types of groups (including cyclic and permutation groups)
CO2 Apply group theoretic reasoning to group actions.
CO3 Learn properties and analysis of solvable & nilpotent groups, Noetherian & Artinian modules and rings.
CO4 Apply Sylow's theorems to describe the structure of some finite groups and use the concepts of isomorphism
and homomorphism for groups and rings.
CO5 Use various canonical types of groups and rings- cyclic groups and groups of permutations, polynomial rings
and modular rings.
CO6 Analyze and illustrate examples of composition series, normal series, subnormal series.
CO7 Study about the Nilpotent transformation, index of nilpotency, invariants of Nilpotent transformations.
CO8 Have knowledge of the Jordan blocks and Jordan forms.

Unit - I
Definition of a group, examples including matrices, permutation groups, groups of symmetry,
roots of unity, Zassenhaus lemma, Normal and subnormal series, Composition series, Jordan-Holder
theorem, Solvable series, Derived series, Solvable groups, Solvability of Sn – the symmetric group of
degree n ≥ 2.
Unit - II
Nilpotent group: Central series, Nilpotent groups and their properties, Equivalent conditions for
a finite group to be nilpotent, Upper and lower central series, Sylow-p sub groups, Sylow theorems with
simple applications. Description of group of order p2 and pq, where p and q are distinct primes (in
general survey of groups up to order 15).

Unit - III
Definition of a ring, examples including congruence classes modulo n, ideals and homomorphism,
quotient rings, polynomial ring in one variable over a ring, units, non-zero divisors, integral domains.
UFD, PID, ED.
Primary decomposition of ideals: Radical of an ideal, Primary ideals, Primary decomposition

Unit -IV
Canonical form : Similarity of matrices, Similarity of linear transformation, Invariant subspaces,
Triangular form, Invariant direct-sum decomposition, nilpotent transformation, Jordan canonical form,
Rational canonical form.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

Suggested Readings:
1. D.S. Malik, J. N. Mordenson, and M. K. Sen, Fundamentals of Abstract Algebra, McGraw Hill,
International Edition, 1997.
2. I. N. Herstein, Topics in Algebra, Wiley Eastern Ltd., New Delhi, 1975.
3. I. S. Luther and I. B. S. Passi, Algebra, Vol. I-Groups, Vol. II-Rings, Narosa Publishing House (Vol. I –
1996, Vol. II –2000).
4. N. Jacobson, Basic Algebra, Vol. I & II, W. H Freeman, 1985 (also published by Hindustan Publishing
Company).
5. P. B. Bhattacharya, S.K. Jain and S.R. Nagpaul, Basic Abstract Algebra (2nd Edition), Cambridge
University Press, Indian Edition, 1997.
6. P. M. Cohn, Algebra, Vols. I, II & III, John Wiley & Sons, 1982, 1989, 1991.
7. S. Lang, Algebra, 3rd editioin, Addison-Wesley, 2015.
8. Vivek Sahai and Vikas Bist, Algebra, Narosa Publishing House, 1999.
9. Joseph A Galian, Contemporary Abstract Algebra Eighth edition Narosa Publishing House,2013.
10. M. Artin, Prentice-Hall of India, 2015.
11. K. B. Datta, Matrix and Linear Algebra, Prentice Hall of India Pvt., New Delhi, 2016.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESTER-I
MATH-112
Ordinary Differential Equations
Maximum Marks-100
External Examination-80
Internal Assessment-20
Max. Time- 3 hrs.
Course Outcomes
Students would be able to:
CO1 Apply differential equations to variety of problems in diversified fields of life.
CO2 Learn use of differential equations for modeling and solving real life problems.
CO3 Interpret the obtained solutions in terms of the physical quantities involved in the original problem under
reference.
CO4 Use various methods of approximation to get qualitative information about the general behaviour of the
solutions of various problems.

Unit - I
Preliminaries: Initial value problem and equivalent integral equation. Approximate solution, Cauchy-
Euler construction of an approximate solution, Equicontinuous family of functions, Ascoli-Arzela
lemma, Cauchy-Peano existence theorem. Uniqueness of solutions, Lipschitz condition, Picard-
𝑑𝑦
Lindelof existence and uniqueness theorem for 𝑑𝑡 = f(t,y), Dependence of solutions on initial conditions
and parameters, Solution of initial-value problems by Picard method.
Unit - II
Sturm-Liouville BVPs, Sturms Separation and Comparison theorems, Lagrange’s identity and Green’s
formula for second order differential equations, Properties of eigenvalues and eigen functions, Pruffer
transformation, Adjoint systems, Self-adjoint equations of second order. Linear systems, Matrix
method for homogeneous first order system of linear differential equations, Fundamental set and
fundamental matrix, Wronskian of a system, Method of variation of constants for a
nonhomogeneous system with constant coefficients, nth order differential equation equivalent to a first
order system.
Unit - III
Nonlinear differential system, Plane autonomous systems and critical points, Classification of critical
points – rotation points, foci, nodes, saddle points. Stability, Asymptotical stability and unstability of
critical points, Dynamical systems and basic notions of dynamical systems such as flows, Rectification
theorem, rest-points and its stability.
Unit - IV
Almost linear systems, Liapunov function and Liapunov’s method to determinestability for nonlinear
systems, Periodic solutions and Floquet theory for periodic systems, Limit cycles, Bendixson non-
existence theorem, Poincare-Bendixson theorem(Statement only), Index of a critical point.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

Publishers Pvt Ltd, 2012.

4. R. G. Bartle, The Elements of Real Analysis, Wiley International Edition. 1976
5. R. R. Goldberg, Methods of Real Analysis, Oxford & IBH Pub. Co. Pvt. Ltd., 2012.
6. Walter Rudin, Principles of Mathematical Analysis (3rd edition) McGraw-Hill,
Kogakusha, 2017, International Student Edition.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESTER-I
MATH-115
Mathematical Statistics
Maximum Marks-100
External Examination-80
Internal Assessment-20
Max. Time- 3 hrs.
Course Outcomes
Students would be able to:
CO1 Understand the mathematical basis of probability and its applications in various fields of life.
CO2 Use and apply the concepts of probability mass/density functions for the problems involving
single/bivariate random variables.
CO3 Have competence in practically applying the discrete and continuous probability distributions along
with their properties.
CO4 Decide as to which test of significance is to be applied for any given large sample problem.

Note: There shall be nine questions in all. Question no. 1 shall be compulsory, consisting of
eight short answer type questions covering the entire syllabus. Two questions will be asked
from each unit. Student will have to attempt one question from each unit. Each question shall
carry equal marks.
Unit - I
Probability: Classical, Statistical and axiomatic approach, Addition theorem, Boole’s
inequality, Conditional probability and multiplication theorem, Independent events, Bayes’
theorem and its applications.
Random variable, discrete and continuous random variables, probability mass and density
functions, distribution function.
Unit - II
Joint, marginal and conditional distributions.Mathematical Expectation and its
properties.Variance, Covariance, Moment generating function. Discrete distributions: Binomial,
Poisson and geometric distributions with their properties.
Unit - III
Continuous distributions: Uniform, Normal, Gamma and Exponential distributions and their
properties. Chebychev’s inequality, Central Limit Theorem. Weak Law of Large Numbers.
Unit - IV
Point and interval estimation, Unbiasedness, Sufficiency,Consistencyand Efficiency. Testing of
Hypothesis, Null and alternative hypotheses, Simple and composite hypotheses,types of
errors,Level of significance, Power of the Test, Critical Region. One tailed and two tailed tests,t-
test, Chi-square test, F-test.
Suggested Readings:
1. A.M. Mood, Graybill F.A. and Boes D.C., Introduction to the theory of
Statistics,McGraw Hill Book Company, 2001.
2. J.E. Freund, Mathematical Statistics, Prentice Hall of India, 2014.
3. M.Speigel, Probability and Statistics, Schaum Outline Series.
4. S.C. Gupta and V.K. Kapoor, Fundamentals of Mathematical Statistics, S. Chand Pub.,
New Delhi,2014.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

5. H.C. Taneja, Statistical Methods for Engineering& Sciences, I. K.International,2009.

6. Gun A.M, Gupta M.K, Dasgupta , Fundamental of Statistics Vol. 1 & 2, World
Press,2016
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESRTER-I
MATH-116
Computing Lab-I
(Introduction to MS Excel and Programming in FORTRAN)
Maximum Marks-100
External Practical Examination-90
Internal Assessment-10
Practical will be conducted Externally by the Department as per the following distribution of
marks:
Writing Programme in FORTRAN and running it on PC
& Proficiency in MS Excel: 50 marks.
Viva voce: 20 marks.
Practical record: 10 marks.
Course Outcomes
Students would be able to:
CO1 Create chart wizard in MS Excel.
CO2 Fit curves and do basic mathematical programs in Excel.
CO3 Demonstrate basic structure of FORTRAN Program.

List of practicals is to be given are as follows:

1. Introduction to MS Excel, features, menus, formatting.
2. Chart Wizard in MS Excel
3. Mathematical and statistical functions in Excel
4. Solving mean, median, standard deviation, variance, correlation in Excel
5. Statistical computing in excel
6. Curve fitting and plotting in MS Excel
7. Demonstrate basic structure of FORTRAN Program.
8. Demonstrates the use of arrays for data analysis-max, min, and average and use of
formatted output. Demonstrates the use of arrays with Fortran 90 specifications.
9. To find largest (or smallest) of three numbers. Computation of LCM and GCD.
10. To find the transpose of a matrix, addition and multiplication of two matrices.
11. Find the roots of quadratic equation, Area, volume of given figure.
12. Uses of control statements, eg, IF, DO loop statement etc.
13. Calculate the SIN and COS of an angle given in degrees.
14. Example of a BUBBLE SORT.
15. Example of a SELECTION SORT using Fortran 90.

Suggested Readings:
1. M. Metcalf, J. Reid and M. Cohen, FORTRAN 90/95 Explained, Oxford University Press;
2nd Revised edition edition,(1999)
2. V. Rajaraman, Computer Programming in FORTRAN 90 and 95, PHI Learning. New Delhi,
(2009).
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

3. Spreadsheet Accounting - Tutorial and Applications, G. E. Anders, C. R. Schaber and R. D.

Fish, Glencoe, . McGraw-Hill, 1995.
4. Microsoft Office 2000 Excel Comprehensive, Shelly Cashman, Course Technology, 2000.
5. Lambert, Joan and Cox, Joyce.: “Microsoft Office Professional 2010 step by step”.
6. Nyhoff, L and Leestma S, Introduction to Fortran 90,Prentice Hall,1999.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESTER-I
MATH-117
Mathematical Lab- I & Seminar
Maximum Marks-100
External Practical Examination-80
Internal Assessment-20

Course Outcomes
Students would be able to:
CO1 Solve mathematical problems using various mathematical techniques.
CO2 Analyze overall performance in all the subjects concerned in the semester.

Mathematical problem Solving Techniques based on paper 20MATH-111 to 20MATH-115

will be taught.
There will be problems based on atleast 5-6 problem solving techniques from each paper.

Note:- Every student will maintain practical record of problems solved during practical
class-work in a file. Examination will be conducted through a question paper set jointly by
the external and internal examiners. The question paper will consists of questions based on
problem solving techniques/algorithm. An examinee will be asked to write the solutions in
the answer book. Evaluation will be made on the basis of the examinee’s performance in
written solutions and presentation with viva-voce and practical record.

Practical will be conducted externally by the department as per the following distribution of
marks: Writing solutions of problems: 40 marks.
Presentation &Viva voce: 20 marks.
Practical record: 20 marks.
Internal Assessment: 20 marks (Attendance=5marks, Assignment=5marks, seminar=10marks)
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESTER-II
MATH-121
Abstract Algebra-II
Maximum Marks-100
External Examination-80
Internal Assessment-20
Max. Time- 3 hrs.
Course Outcomes
Students would be able to:
CO1 Establish the connection between the concept of field extensions and Galois theory.
CO2 Learn about the Nil and Nilpotent ideals in Noetherian and Artinian ring.
CO3 Study about the Nilpotent transformation, index of nilpotency, invariants of Nilpotent
transformations.
CO4 Use diverse properties of field extensions in various areas.
CO5 Compute the Galois group for several classical situations.
CO6 Solve polynomial equations by radicals along with the understanding of ruler and compass
constructions.
Note: There shall be nine questions in all. Question no. 1 shall be compulsory, consisting of
eight short answer type questions covering the entire syllabus. Two questions will be asked
from each unit. Student will have to attempt one question from each unit. Each question shall
carry equal marks.
Unit – I
Module, Sub module & properties of Module, Cyclic modules, Simple and semi-simple modules,
Schur’s lemma, Free modules, Fundamental structure theorem of finitely generated modules
over principal ideal domain and its applications to finitely generated abelian groups.

Unit - II
Neotherian and Artinian modules and rings with simple properties and examples, Nil and
Nilpotent ideals in Neotherian and Artinian rings, Hilbert Basis theorem.
Unit - III
Algebraic extension of a field : Extension of a field, Degree of a field extension, Algebraic
extension of a field, Roots of a polynomial in an extension field, Algebraically closed field
Splitting field : Root fields, Splitting field or decomposition field, Theorems on decomposition
field.
Unit – IV
Normal and Separable Extensions of Fields : Normal extensions, Some theorems on normal
extensions, Multiple roots, Separable and inseparable extension, Some theorems on separability,
Primitive element.
Galois Theory : Introduction, Finite fields or Galois fields, properties of finite fields, Structure of
the multiplicative group of a finite field, Construction of a Galois field, Construction of subfields
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

of GF(pn).

Suggested Readings:
1. D.S. Malik, J.N. Mordenson, and M.K. Sen, Fundamentals of Abstract Algebra, McGraw Hill,
International Edition, 1997.
2. I.N.Herstein, Topics in Algebra, Wiley Eastern Ltd., New Delhi, 1975.
3. I.S. Luther and I.B.S.Passi, Algebra, Vol. I-Groups, Vol. II-Rings, Narosa Publishing House
(Vol. I – 1996, Vol. II –1990).
4. Joseph A Galian, Contemporary Abstract Algebra Eighth edition Narosa Publishing House.
5. M. Artin, Algebra, Prentice-Hall of India, 1991.
6. P.B.Bhattacharya, S.K. Jain and S.R. Nagpaul, Basic Abstract Algebra (2nd Edition), Cambridge
University Press, Indian Edition, 1997.
7. P.M. Cohn, Algebra, Vols. I, II & III, John Wiley & Sons, 1982, 1989, 1991, 2000.
8. T.Y Lam, Lectures on Modules and Rings, GTM Vol. 189, Springer-Verlag, 1999.
9. D. S. Dummit& Foote, Abstract Algebra, Wiley India Pvt., Ltd.
10. T. W. Hungerford, Algebra, Springer-Verlag, 1998.
11. N. Jacobson, Basic Algebra, Volume II, Hindustan Publishing Co., 1989.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESTER-II
MATH-122
Complex Analysis
Maximum Marks-100
External Examination-80
Internal Assessment-20
Max. Time- 3 hrs.
Course Outcomes
Students would be able to:
CO1 Familiarize with complex numbers and their geometrical interpretations.
CO2 Understand the concept of complex numbers as an extension of the real numbers.
CO3 Represent the sum function of a power series as an analytic function.
CO4 Demonstrate the ideas of complex differentiation and integration for solving related problems and
establishing theoretical results.
CO5 Understand concept of residues, evaluate contour integrals and solve polynomial equations.
CO6 Describe various types of singularities and their role in study of complex functions.

Unit –II
Path, Region, Contour, Simply and multiply connected regions, Complex integration.
Cauchy theorem. Cauchy’s integral formula. Poisson’s integral formula. Complex integral as a
function of its upper limit, Morera’s theorem. Cauchy’s inequality. Liouville’s theorem. The
Fundamental theorem of algebra.
Unit –III
Zeroes of an analytic function, Laurent’s series. Singularities. Cassorati- Weierstrass
theorem, Limit point of zeros and poles. Maximum and Minimum modulus principles. Schwarz
lemma. Meromorphic functions. Residues. Cauchy’s residue theorem. Evaluation of improper
integrals.
Unit – IV
The argument principle. Rouche’s theorem, Inverse function theorem. Bilinear transformations,
their properties and classifications. Definitions and examples of Conformal mappings. Space of
analytic functions and their completeness, Riemann mapping theorem.

Suggested Readings:
1. E.C. Titchmarsh, The Theory of Functions, Oxford University Press, London, 1976.
2. E.T. Copson, An Introduction to the Theory of Functions of a Complex Variable, Oxford
University Press, London, 1960.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

3. H.A. Priestly, Introduction to Complex Analysis, Clarendon Press, Oxford, 2008.

4. J.B. Conway, Functions of one Complex variable, Springer-Verlag, International student-
Edition, Narosa Publishing House, 2012.
5. L.V. Ahlfors, Complex Analysis, McGraw Hill, 2013.
6. Liang-shin Hann &Bernand Epstein, Classical Complex Analysis, Jones and Bartlett
Publishers International, London, 1996.
7. S. Lang, Complex Analysis, Addison Wesley, 1985.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESTER-II
MATH-123
Topology
Maximum Marks-100
External Examination-80
Internal Assessment-20
Max. Time- 3 hrs.
Course Outcomes
Students would be able to:
CO1 Get familiar with the concepts of topological space and continuous functions.
CO2 Generate new topologies from a given set with bases.
CO3 Describe the concept of homeomorphism and topological invariants.
CO4 Establish connectedness and compactness of topological spaces and proofs of related theorems.
CO5 Have in-depth knowledge of separation axioms and their properties
Note: There shall be nine questions in all. Question no. 1 shall be compulsory, consisting of
eight short answer type questions covering the entire syllabus. Two questions will be asked
from each unit. Student will have to attempt one question from each unit. Each question shall
carry equal marks.
Unit - I
Definition and examples of topological spaces. Closed sets. Closure. Dense subsets.
Neighbourhoods. Interior, exterior and boundary points of a set. Accumulation points and
derived sets. Bases and sub-bases. Subspaces and relative topology. Alternate methods of
defining a topology in terms of Kuratowski Closure Operator and Neighbourhood Systems.
Unit - II
Continuous functions and homeomorphism. Connected spaces. Connectedness on the real line.
Components. Locally connected spaces.
Unit - III
Compactness,compact sets, Basic properties of compactness. Compactness and finite intersection
property. Sequentially and countably compact sets. Local compactness and one point
compactification. Stone-Cech compactification Compactness in metric spaces. Equivalence of
compactness, countable compactness and sequential compactness in metric spaces.

Unit - IV
First and Second Countable spaces. Lindelof’s theorem. Separable spaces. Second Countability
and Separability. Separation axioms. T0, T1, and T2 spaces. Their characterization and basic
properties. Baire Category Theorem for locally compact Hausdorff spaces. Regular and normal
spaces. Urysohn's Lemma and Tietze Extension theorem.T3 and T4 spaces.Complete regularity
and Complete normality. T3/½ and T5 spaces.

Suggested Readings:
1. G. F. Simmons, Introduction to Topology and Modem Analysis, McGraw-Hill Book
Company, 2004.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

2. J. Dugundji, Topology, Allyn and Bacon, 1966 (Reprinted in India by Prentice Hall of
India Pvt. Ltd.).
3. J. L. Kelley, General Topology, Van Nostrand, Reinhold Co., New York, 2008.
4. J. R. Munkres, Topology, A First Course, Prentice Hall of India Pvt. Ltd., New Delhi,
2000.
5. K. D. Joshi, Introduction to General Topology, Wiley Eastern Ltd., 2017.
6. W. J. Pervin, Foundations of General Topology, Academic Press Inc. New York, 1964.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESTER-II

MATH-124
Operations Research Techniques
Maximum Marks-100
External Examination-80
Internal Assessment-20
Time: 3 hrs.
Course Outcomes
Students would be able to:
CO1 Explain operations research, its scope and utility.
CO2 Know about operations research models, its methodology and classification.
CO3 Formulate and solve LPP’s using Graphical & Simplex methods and understand the concept of
duality in linear programming.
CO4 Classify and handle different types of transportations problems.
CO5 Explain inventory control and inventory models for specific situations in an organization.
CO6 Understand queuing system and various characteristics of single server/multi servers queuing
models with limited and unlimited capacity.
Note: There shall be nine questions in all. Question no. 1 shall be compulsory, consisting of
eight short answer type questions covering the entire syllabus. Two questions will be asked
from each unit. Student will have to attempt one question from each unit. Each question shall
carry equal marks.
Unit - I
Operations Research: Origin, definition and its scope, Linear Programming:
Formulation and Solution of linear programming problems by Graphical and Simplex methods,
Big - M and Two-phase methods, Degeneracy, Duality in linear programming.

Unit - II
Transportation Problems: Basic Feasible Solutions, Optimum solution by stepping
stoneand modified distribution methods, unbalanced and degenerate problems, trans-shipment
problem. Assignment problems: Solution by Hungarian method, unbalanced problem, case of
maximization, travelling salesman and crew assignment problems.

Unit - III
Queuing models: Basic components of a queuing system, General birth-death equations,
steady-state solution of Markovian queuing models with single and multiple servers (M/M/1,
M/M/C, M/M/1/k, M/M/C/k).
Unit - IV
Inventory control models: Economic order quantity (EOQ) model with uniform demand
and with different rates of demands in different cycles, EOQ when shortages are allowed, EOQ
with uniform replenishment, Inventory control with price breaks.

Suggested Readings:-
1. F. Hillier and G.J. Lieberman, Introduction to Operation Research, Holden Day,1990.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

2. H. A. Taha, Operation Research-An Introduction, Printice Hall of India, 2017.

3. J.K. Sharma, Mathematical Model in OperationS Research, Tata McGraw Hill,1989.
4. Kanti Swaroop, P.K. Gupta, Man Mohan, Operations Research, Sultan Chand and Sons,
2010.
5. N.S.Kambo, Mathematical Programming Techniques, McGraw Hill, 2008.
6. P. K. Gupta, and D.S. Hira, Operations Research, S. Chand & Co., 1976.
7. S. D. Sharma, Operation Research, Kedar Nath Ram Nath Publications, 2009.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESTER-II

MATH-125
Computational Techniques
Maximum Marks-100
External Examination-80
Internal Assessment-20
Max. Time- 3 hrs.
Course Outcomes
Students would be able to:
CO1 Learn about interpolation with equal and unequal intervals.
CO2 Apply forward, backward, central and divided difference formulae for interpolation.
CO3 Apply standard probability distributions to the concerned problems.
CO4 Understand the method of numerical differentiation and various methods for finding solution of
Eigen value problems.
CO5 Know how to solve integration and ordinary differential equation using numerical data.
Note: There shall be nine questions in all. Question no. 1 shall be compulsory, consisting of
eight short answer type questions covering the entire syllabus. Two questions will be asked
from each unit. Student will have to attempt one question from each unit. Each question shall
carry equal marks.
Unit -I
Error Analysis: Errors, Absolute errors, Rounding errors, Truncation errors, Inherent Errors,
Major and Minor approximations in numbers.
The Solution of Linear Systems: Gaussian elimination method with pivoting, Algorithm and
convergence of Jacobi iterative Method, Algorithm and convergence of Gauss Seidel Method,
Method of Relaxation.
The Solution of Non-Linear Equation: Bisection Method, Fixed point iterative method,
Newton-Raphson method, Muller’s Method, Secant method, Method of false position,
Algorithms and convergence of these methods,Complex roots by Newton’s method, System of
two Equations by Newton Method and Method of Iteration.

Unit -II
Difference Operators: Forward difference operators, Backward difference operators, Shift
operators, Average and central difference operators and relation between them.
Interpolation: Linear interpolation, Lagrange’s interpolation Polynomial, Divided difference
Table, Interpolation with equidistant Points: Forward Difference Table and Backward Difference
Table, Spline interpolation (Cubic), Chebyshew interpolation Polynomials, Errors and algorithms
of these interpolations.
Unit -III
Numerical Differentiation: Differentiating continuous Functions: Forward Difference quotient,
Central Difference Method, Error analysis, Higher order derivatives. Differentiating tabulated
function:Error analysis and Higher order derivatives, Difference tables, Richardson
Extrapolation.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

Numerical Integration: General Integration formula, Rectangular rule, Trapezoidal rule,

Simpson’s rule, Boole’s rule, Weddle’s rule, Gaussian quadrature formulae, Errors in quadrature
formulae, Newton-Cotes formulae, Romberg Integration.

Unit -IV
Ordinary Differential Equations: Euler’s Method, Modified Euler’s methods with error
analysis, Taylor’s series and Runge-Kutta methods with error analysis, Predictor-corrector
methods for solving initial value problems, Finite Difference & Collocation methods.
Difference Equations:Linear homogeneous and non-homogeneous difference equations with
constant coefficients.
Eigenvalue and eigenvector: The Fadeev-Leverrier method, Evaluating the Eigen Values and
Detremining the Eigen Vectors, Power method.

Suggested Readings:
1. E. Balagurusamy, Numerical Methods, (Tata McGraw-Hill Publishing Company Pvt. Ltd.,
2010)
2. B.P.Demidovich and I.A. Maron, Computational Mathematics, (Medtech-A division of
Scientific Research, 2018)
3. S.S. Sastry, Introductory Methods of Numerical Analysis, (Prentice-Hall of India, 5TH
Edition. 2012)
4. Curtis F. Gerald and Patrick O. Wheatley, Applied Numerical Analysis,
(AddisonEngineering, (Prentice Hall International, 2007)
5. John H. Mathews, Numerical Methods for Mathematics, Science and Publishing
Company,1997.
6. Steven C Chapra and Raymond P Canale, Numerical Methods for Engineers, (Tata McGraw-
Hill Publishing Company Pvt. Ltd., 2015)
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESTER-II
MATH-126
Computing Lab – II
(Programming in C)
Maximum Marks-100
External Practical Examination-80
Internal Assessment-20
Practical will be conducted Externally by the Department as per the followingdistribution of
marks:
Writing Programme inC and running it on PC: 50 marks.
Viva voce: 20 marks.
Practical record: 10 marks.

Course Outcomes
Students would be able to:
CO1 Define basic structure of a C Programs with various Data types and Operators.
CO2 Introduce IDE of C Compilers and designing Flow Charts.
CO3 Implement control structures and conditions.
CO4 Make arrays and pointers with passing of arguments.
CO5 Construct structure and Union with examples.
CO6 Make Input/output with examples.
CO7 Develop functions in C with examples.

Following Programs to be done during the course:

C Programs:
1. Program in C to find magnitude of a vector.
2. Computer GCD and LCM of two positive integer values using recursion.
3. Calculate the Eigen values and Eigen vectors of a symmetric matrix of order three.
4. Program to find largest/smallest of three numbers.
5. Demonstrates the use of Euler and Runge –Kutta methods for solving IVP of ODEs.
6. C program to evaluate the given polynomial equation by the method studied.
7. Program to compute Sin x and Cos x using series and check for error in verifying
sin2x + cos2x=1.
8. To find the inverse of a given non-singular square matrix.
9. To find complex roots of a Quadratic Equation.
10. Program to sort ten numbers in increasing and decreasing order.
11. Other programs may be included in the course as relevant to mathematics.

Suggested Readings:
1. Balagurusamy E: Programming in ANSI C, Third Edition, Tata McGraw-Hill Publishing
Co. Ltd.
2. Brian W. Kernighan & Dennis M. Ritchie, The C Programming Language, Second
Edition (ANSI features), Prentice Hall 1989.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

3. Byron, S. Gottfried: Theory and Problems of Programming with C, Second Edition

(Schaum’s Outline Series), Tata McGraw-Hill Publishing Co. Ltd.
4. Peter A. Darnell and Philip E. Margolis, C: A Software Engineering Approach, Narosa
Publishing House (Springer International Student Edition) 1993.
5. Samuel P. Harkison and Gly L. Steele Jr., C: A Reference Manual, Second Edition,
Prentice Hall, 1984.
6. Venugopal K. R. and Prasad S. R.: Programming with C, Tata McGraw-Hill Publishing
Co. Ltd.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESTER-II
MATH-127
Mathematical Lab- II & Seminar
Maximum Marks-100
External Practical Examination-80
Internal Assessment-20

Course Outcomes
Students would be able to:
CO1 Solve mathematical problems using various mathematical techniques.
CO2 Analyze overall performance in all the subjects concerned in the semester.

Mathematical problem Solving Techniques based on paper 20MATH-121 to 20MATH-

125will be taught.
There will be problems based on atleast 5-6 problem solving techniques from each paper.

Note :-Every student will maintain practical record of problems solved during practical
class-work in a file. Examination will be conducted through a question paper set jointly by
the external and internal examiners. The question paper will consists of questions based on
problem solving techniques/algorithm. An examinee will be asked to write the solutions in
the answer book. Evaluation will be made on the basis of the examinee’s performance in
written solutions and presentation with viva-voce and practical record.

Practical will be conducted Externally by the department as per the following distribution of
marks: Writing solutions of problems: 40 marks.
Presentation &Viva voce: 20 marks.
Practical record: 20 marks.
Internal Assessment: 20 marks (Attendance=5marks, Assignment=5marks, seminar=10marks)
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESTER-II
CS-100
Communication Skills
Maximum Marks-50

Note: One hour of classroom teaching will be devoted to the teaching of theory. In another
hour the students will be engaged in practical activities and the evaluation of their
communication skills will be done by the internal examiner on the basis of classroom
presentations, discussions and assignments.

Course Outcomes
Students would be able to:
CO1 Practice Communication skills to communicate effectively

Objective:
To introduce the theory and practice of communicative skills so as to enable the students to
communicate effectively and conduct themselves graciously in the business of life.

Unit-I
Human Communication, Verbal and Non Verbal Communication, Barriers to communication;
the seven C‟s of effective communication. Preparing for interviews, CV/ Biodata, Group
Discussion, Public Speaking, Mass Communication.

Unit -II
Common Courtesies, Introducing Oneself Formally and Informally; Introducing Oneself on
Social Media; Speaking Strategies: Making Enquiries, Clarifications, Recommendations,
Explanations, Rejections, etc.; Being Diplomatic; Telephonic Communication.
Unit-III
Conversational Practice in Various Situations:
(meeting, parting, asking/talking about daily activities, at railway station, seeking information,
buying at shops, asking about buses, travelling by bus, using expressions of time, talking about
money, identifying people, at the post office, at the bank, at the grocery store, immediate family
and relatives, hiring a taxi, talking about weather/weather conditions, breakfast or lunch at a
restaurant, ordering food, dinner conversations, at the doctors clinic, quitting and finding jobs,
office conversations, conversations about school/ college/ university, the English class, driving a
car).
Students shall develop dialogue-based conversations on the above-mentioned situations.
Unit-IV
Personality Development Skills: Personal Grooming; Assertiveness; Improving Self-Esteem;
Significance of Critical Thinking; Confidence Building; SWOC analysis.
Emotional intelligence: Recognizing and Managing Emotions and Situations; Stress and Anger
Management; Positive Thinking; Developing Sense of Humour.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M. Sc. (MATHEMATICS)
SEMESTER-III
MATH-231
Analytical Number Theory
Maximum Marks-100
External Examination-80
Internal Assessment-20
Time: 3 hrs.
Course Outcomes
Students would be able to:
CO1 Know about the classical results related to prime numbers and get familiar with the irrationality of e
and Π.
CO2 Study the algebraic properties of Un and Qn.
CO3 Learn about the Waring problems and their applicability.
CO4 Learn the definition, examples and simple properties of arithmetic functions and about perfect
numbers.
CO5 Understand the representation of numbers by two or four squares.

Unit -I
Distribution of primes. Fermat’s and Mersenne numbers, Farey series and some results
concerning Farey series. Approximation of irrational numbers by rationals, Hurwitz’s theorem.
Irrationality of e and Π.

Unit -II
Diophantine equations ax + by = c, x2+y2 = z2 and x4+y4 = z4. The representation of number by
two or four squares. Warig’s problem, Four square theorem, the numbers g(k) & G(k). Lower
bounds for g(k) & G(k).Simultaneous linear and non-linear congruencies, Chinese Remainder
Theorem and its extension.

Unit -III
Quadratic residues and non-residues. Legender’s Symbol. Gauss Lemma and its applications.
Quadratic Law of Reciprocity, Jacobi’s Symbol. The arithmetic in Zn. The group Un .
Congruencies with prime power modulus, primitive roots and their existence.

Unit -IV

k(n), u(n), N(n), I(n).Definition and examples and

Suggested Readings:-
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

1. D.M., Burton, Elementary Number Theory.

2. G.H., Hardy and E.M., Wright, An Introduction to the Theory ofNumbers
3. I.Niven, And H.S., Zuckermann, An Introduction to the Theory ofNumbers.
4. J. Mary Jones & Gareth, A. Jones; Elementary Number Theory,Springer Ed. 1998.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESTER-III
MATH-232
Partial Differential Equations
Maximum Marks-100
External Examination-80
Internal Assessment-20
Max. Time- 3 hrs.
Course Outcomes
Students would be able to:
CO1 Establish a fundamental familiarity with partial differential equations and their applications.
CO2 Distinguish between linear and nonlinear partial differential equations.
CO3 Solve boundary value problems related to Laplace and heat equations by various methods.
CO4 Use Green's function method to solve partial differential equations.

Unit -I
First Order P.D.E.:Curves and surfaces, Genesis of first order P.D.E., Classification of
integrals, Compatible systems, Charpit’s method, Integral Surfaces through a Given curve,
Quasi-Linear Equations. Method of separation of variables.

Unit -II
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 Strings, Vibrations of a
Semi-Infinite String, Vibrations of a string of finite length.

Unit -III
Laplace’s Equations. :Boundary Value problems, Maximum and Minimum Principles,
The Cauchy Problem, The Dirichlet Problem for upper half plane, The Neumann Problem for
upper half plane, The Dirichlet problem for a circle, The Dirichlet Exterior problem for a circle,
The Neumann problem for a circle, the Dirichlet problem for a rectangle.

Unit -IV
Heat Conduction problem: Heat conduction- Infinite rod case, Heat conduction- finite
rod case, Duhamel’s principle, Heat Conduction Equation, Classification in the case of n-
variables, Families of Equipotential Surfaces, Kelvin’s Inversion theorem.

Suggested Readings:
1. E.T. Copson: Partial Differential Equations (Cambridge university press 1975)
2. I.N. Sneddon: Elements of partial differential equations (Mc-Graw Hill Book
company1957)
3. Peter V O’Neil: Advanced Engineering Mathematics seventh edition
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

4. T. Amarnath: An elementary course in partial differential equations (Narosa Publishing

House). Second edition 2003.
5. W.E. Williams: Partial Differential equations (Charendon press-oxford 1980)
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M. Sc. (MATHEMATICS)
SEMESTER-III
MATH-233
Mechanics of Solids-I
Maximum Marks-100
External Examination-80
Internal Assessment-20
Time: 3 hrs.
Course Outcomes
Students would be able to:
CO1 Get familiar with Cartesian tensors, as generalization of vectors, and their properties which are used
in the analysis of stress and strain to describe the phenomenon of solid mechanics.
CO2 Analyse the basic properties of stress and strain components, their transformations, extreme values,
invariants and Saint-Venant principle of elasticity.
CO3 Demonstrate generalized Hooke's law for three dimensional elastic solid which provides the linear
relationship between stress components and strain components.
CO4 Use different types of elastic symmetries to derive the stress-strain relationship for isotropic elastic
materials for applications to architecture and engineering
Note: There shall be nine questions in all. Question no. 1 shall be compulsory, consisting of
eight short answer type questions covering the entire syllabus. Two questions will be asked
from each unit. Student will have to attempt one question from each unit. Each question shall
carry equal marks.

Unit-I
Cartesian tensors: Cartesian tensors of different order, Properties of tensors, Symmetric
and skew-symmetric tensors, Isotropic tensors of different orders and relation between them,
Tensor invariants. Eigen-values and eigen vectors of a second order tensor, Scalar, vector, tensor
functions, Comma notation, Gradiant, Divergence and Curl of a tensor field.
Unit-II
Analysis of Stress: Stress vector, stress components. Cauchy equations of equilibrium. Stress
tensor. Symmetry of stress tensor. Stress quadric ofCauchy. Principal stress and invariants.
Maximum normal and shear stresses. Mohr’s diagram. Examples of stress.

Unit-III
Analysis of Strain: Affine transformations. Infinitesimal affine deformation. Geometrical
interpretation of the components of strain. Strain quadric of Cauchy. Principal strains and
invariants. General infinitesimal deformation. Saint-Venant’s equations of Compatibility. Finite
deformations. Examples of uniform dilatation, simple extension and shearing strain.

Unit-IV
Equations of Elasticity: Hooke’s law and its generalization. Hooke’s law in media with one
plane of symmetry, orthotropic and transversely isotropic media, Homogeneous isotropic media.
Elastic moduli for isotropic media. Equilibrium and dynamic equations for an isotropic elastic
solid. Beltrami-Michell compatibility equations. Strain energy function. Clapeyron’s theorem.
Saint-Venant’sPrinciple(statement only).
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

Suggested Readings:-
1. A. E. H. Love, A Treatise on a Mathematical Theory of Elasticity, Dover Pub., New
York.
2. A. S. Saada, Elasticity- Theory and applications, Pergamon Press,New York.
3. D.S. Chandersekhariah and L. Debnath, Continum Mechanics, Academic Press, 1994.
4. H. Jeffreys, Cartesian tensors.
5. I.S. Sokolnikoff, Mathematical Theory of Elasticity, Tata McGraw HillPublishing
Company Ltd., New Delhi, 1977.
6. Shanti Narayan, Text Book of Tensors, S. Chand & Co.
7. Teodar M. Atanackovic and ArdeshivGuran, Theory of Elasticity for Scientists and
Engineers Birkhausev, Boston, 2000.
8. Y.C. Fung, Foundations of Solid Mechanics, Prentice Hall, New Delhi, 1965.
9. R.B. Hetnarski, J. Ignaczak; the Mathematical Theory of Elasticity, CRC Press, 2nd
Edition.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M. Sc. (MATHEMATICS)
SEMESTER-III
MATH-I
Fluid dynamics
Maximum Marks-100
External Examination-80
Internal Assessment-20
Time: 3 hrs.
Course Outcomes
Students would be able to:
CO1 Be familiar with continuum model of fluid flow and classify fluid/flows based on physical properties
of a fluid/flow along with Eulerian and Lagrangian descriptions of fluid motion.
CO2 Derive and solve equation of continuity, equations of motion, vorticity equation, equation of moving
boundary surface, pressure equation and equation of impulsive action for a moving inviscid fluid.
CO3 Calculate velocity fields and forces on bodies for simple steady and unsteady f low including those
derived from potentials.
CO4 Understand the concepts of velocity potential and stream function.
CO5 Represent mathematically the potentials of source, sink and doublets in two dimensions as well as
three-dimensions, and study their images in impermeable surfaces.

Unit-I
Kinematics - Velocity at a point of a fluid. Eulerian and Lagrangian methods. Stream lines, path
lines and streak lines. Velocity potential.Irrotational and rotational motions.Vorticity and
circulation. Equation of continuity. Boundary surfaces. Acceleration at a point of a fluid.
Components of acceleration in cylindrical and spherical polar co-ordinates.

Unit-II
Pressure at a point of a moving fluid. Euler’s and Lagrange’s equations of motion. Equations of
motion in cylindrical and spherical polar co-ordinates. Bernoulli’s equation. Impulsive motion.
Kelvin’s circulation theorem. Vorticity equation. Energy equation for incompressible flow.

Unit-III
Acyclic and cyclic irrotational motions. Kinetic energy of irrotational flow. Kelvin’s minimum
energy theorem. Mean potential over a spherical surface. K.E. of infinite fluid. Uniqueness
theorems. Axially symmetric flows. Liquid streaming part a fixed sphere. Motion of a sphere
through a liquid at rest at infinity. Equation of motion of a sphere. K.E. generated by impulsive
motion.

Unit-IV
Three-dimensional sources, sinks and doublets. Images of sources, sinks and doublets in rigid
impermeable infinite plane and in impermeable spherical surface. Two dimensional motion,
Kinetic energy of acyclic and cyclic irrotational motion. Use of cylindrical polar co-ordinates.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

Stream function. Axi-symmetric flow. Stoke’s stream function. Stoke’s stream function of basic
flows.
Suggested Readings:-
1. F. Chorlton, Text Book of Fluid Dynamics, C.B.S. Publishers, Delhi,1985
2. G.K. Batchelor, An Introduction to Fluid Mechanics, Foundation Books, New Delhi,
1994.
3. O’Neill, M.E. and Chorlton, F., Ideal and Incompressible Fluid Dynamics, Ellis Horwood
Limited, 1986.
4. R.K. Rathy, An Introduction to Fluid Dynamics, Oxford and IBH Publishing Company,
New Delhi, 1976.
5. S.W. Yuan, Foundations of Fluid Mechanics, Prentice Hall of IndiaPrivate Limited, New
Delhi, 1976.
6. W.H. Besaint and A.S. Ramsey, A Treatise on Hydromechanics, Part II,CBS Publishers,
Delhi, 1988.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M. Sc. (MATHEMATICS)
SEMESTER-III
MATH-II
Discrete Mathematics
Maximum Marks-100
External Examination-80
Internal Assessment-20
Time: 3 hrs.
Course Outcomes
Students would be able to:
CO1 Be familiar with fundamental mathematical concepts and terminology of discrete mathematics and
discrete structures.
CO2 Express a logic sentence in terms of predicates, quantifiers and logical connectives.
CO3 Apply the rules of inference and contradiction for proofs of various results.
CO4 Evaluate boolean functions and simplify expressions using the properties of boolean algebra.
Note: There shall be nine questions in all. Question no. 1 shall be compulsory, consisting of
eight short answer type questions covering the entire syllabus. Two questions will be asked
from each unit. Student will have to attempt one question from each unit. Each question shall
carry equal marks.

Unit -I
Graph Theory – Definitions and basic concepts, special graphs, Subgraphs, isomorphism
of graphs, Walks, Paths and Circuits, Eulerian Paths and Circuits, Hamiltonian Circuits, matrix
representation of graphs, Planar graphs, Coloring of Graph.

Unit -II
Directed Graphs, Trees, Isomorphism of Trees, Representation of Algebraic Expressions
by Binary Trees, Spanning Tree of a Graph, Shortest Path Problem, Minimal spanning Trees, Cut
Sets, Tree Searching..

Unit -III
Formal Logic – Statements. Symbolic Representation and Tautologies. Quantifier,
Predicates and Validty. Propositional Logic. Pigeonhole principle, principle of inclusion and
exclusion, derangements.
Lattices- Lattices as partially ordered sets. Their properties. Lattices as Algebraic
systems. Sublattices, Direct products, and Homomorphisms. Some Special Lattices e.g.,
Complete. Complemented and Distributive Lattices. Join-irreducible elements. Atoms and
Minterms.

Unit -IV
Boolean Algebras.Boolean Algebras as Lattices.Various Boolean Identities. The
switching Algebra example. Sub algebras, Direct Products and Homomorphisms. Boolean Forms
and Their Equivalence. Minterm Boolean Forms, Sum of Products Canonical Forms.
Minimization of Boolean Functions. Applications of Boolean Algebra to Switching Theory. The
Karnaugh Map method.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

Suggested Readings:-
1. Babu Ram, Discrete Mathematics, Vinayak Publishers and Distributors, Delhi, 2004.
2. C.L. Liu, Elements of Discrete Mathematics, McGraw-Hilll Book Co.
3. J.L. Gersting, Mathematical Structures for Computer Science, (3rdedition), Computer
Science Press, New York.
4. J.P. Tremblay & R. Manohar, Discrete Mathematical Structures withApplications to
Computer Science, McGraw-Hill Book Co., 1997.
5. Seymour Lipschutz, Finite Mathematics (International edition 1983), McGraw-Hill Book
Company, New York.
6. A. Sehgal, Mathematical Foundation of Computer Science, Jeevansons Publication, New
delhi.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M. Sc. (MATHEMATICS)
SEMESTER-III
MATH-III
Differential Geometry
Maximum Marks-100
External Examination-80
Internal Assessment-20
Time: 3 hrs.
Course Outcomes
Students would be able to:
CO1 Describe the basic features of curves and parametric families of surfaces in Euclidean space.
CO2 Be familiar with the concepts of envelope, edge of regression and developable surfaces.
CO3 Acquire knowledge of the normal curvature of a surface and its connection with the first and second
fundamental forms.
CO4 Have knowledge of geodesics on a surface, their characterization, torsion of geodesic, Bonnet
theorem and its implication for a geodesic triangle.
Note: There shall be nine questions in all. Question no. 1 shall be compulsory, consisting of
eight short answer type questions covering the entire syllabus. Two questions will be asked
from each unit. Student will have to attempt one question from each unit. Each question shall
carry equal marks.

Unit –I
Curves with torsion: Tangent, Principal Normal, Curvature, Binormal, Torsion, Serret-Frenet
formulae,
Locus of centre of Curvature, Spherical Curvature, Locus of centre of Spherical Curvature.

Unit –II
Envelopes: Surfaces, Tangent plane, Envelope, Characteristics, Edge of regression.

Unit –III
Curvilinear Co-ordinates: First order magnitude, Directions on a surface, Second order
magnitudes,
Derivative of unit normal, Principal directions and curvatures.

Unit -IV
Geodesics: Geodesic property, Equations of geodesics, Torsion of a geodesic.

Suggested Readings:
1. C.E., Weatherburn, Differential Geometry of Three Dimensions.
2. J. Thorpe, Elementary Differential Geometry ; Elsevier.
3. M. do Carmo,Differential Geometry of Curves and Surfaces .
4. R. Millman& G. Parker ; Elements of Differential Geometry.
M. Sc. (MATHEMATICS)
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

SEMESTER-III
MATH-IV
Coding Theory
Maximum Marks-100
External Examination-80
Internal Assessment-20
Time: 3 hrs.
Course Outcomes
Students would be able to:
CO1 Design new algorithms for coding.
CO2 Calculate the parameters of given codes and their dual codes using standard matrix and polynomial
operations.
CO3 Compare the error-detecting/correcting facilities of given codes for a given binary symmetric
channel.
CO4 Understand and explain the basic concepts of Hamming codes, perfect and quasi- perfect codes,
Golay codes, Hamming sphere and bounds for various codes.
CO5 Describe the real life applications of codes.

Unit-I
The communication channel, The coding problem, Types of codes, Block codes, Error-detecting
and error-correcting codes, Linear codes, The Ham-Ming metric, Description of linear block
codes by matrices, Dual codes.

Unit-II
Standard array, Syndrome, Step-by-step decoding, Modular representation, Error-correction
capabili-ties of linear codes,
Unit-III
Bounds on minimum distance for block codes, Plotkin bound, Hamming sphere packing bound,
Varshamov-Gilbert-Sacks bound.Bounds for burst-error detecting and correcting codes,

Unit-IV
Important linear block codes, Hamming codes.Golay codes, Perfect codes, Quasi-perfect codes,
Reed-Muller codes, Codes derived from Hadamard matrices, Product codes, Concatenated codes.

Suggested Readings:
1. Raymond Hill, A First Course in Coding Theory, Oxford University Press, 1990.
2. Man Young Rhee, Error Correcting Coding Theory, McGraw Hill Inc., 1989.
3. F.J. Macwilliams and N.J. A. Sloane, The Theory of Error Correcting Codes, North-
Holland, 2006.
4. W.W. Peterson and E.J. Weldon, Jr., Error-Correcting Codes. M.I.T. Press, Cambridge,
Massachusetts, 1972.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M. Sc. (MATHEMATICS)
SEMESTER-III
MATH-V
Applied statistics
Maximum Marks-100
External Examination-80
Internal Assessment-20
Time: 3 hrs.
Course Outcomes
Students would be able to:
CO1 Study linear models and anlyse variance.
CO2 Give complete analysis of completely randomised, randomized block and latin square designs and
solve various related problems.
CO3 Have the skill of solving problems on Factorial designs – 2 2 and 23 designs.
CO4 Study about reliability theory and statistical quality control and discuss its purposes.

Unit-I
Linear Models: Standard Gauss Markov Models, Estimability of Parameters, Best Linear
Unbiased Estimator (BLUE), Method of Least Squares, Gauss-Markov Theorem, Variance-
Covariance Matrix of Blues.
Analysis of Variance for One- Way, Two -Way for Fixed, Mixed and Random Effects Models,
Tukey’s Test for Non- Additively.
Unit-II
Fundamental principles of experimental design, Size and shape of plots and blocks; Layout and
analysis of completely randomized design (CRD), randomized block design (RBD) and Latin
Square Design (LSD); Efficiency of RBD relative to CRD and Efficiency of LSD relative to
RBD and CRD.
Factorial designs – 22 and 23 designs, Illustrations, Main effects, Interaction effects and analysis
of these designs.
Unit-III
Reliability theory: Basic concepts of life testing experiments, reliability, hazard function and
their relationship. System reliability concepts: Parallel system, series system and k out of n
system, Non-series parallel systems, Systems with mixed mode failures. Standby redundancy:
Simple standby system, k-out-of-n standby system.

Unit-IV
Statistical Quality Control and Its Purposes, 3 Sigma Control Limit, Shewart Control Chart.
Control Charts For Variables and Attributes, Natural Tolerance Limits and Specification Limits:
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

Modified Control Limits. Sampling Inspection Plan, Producer’s and Consumer’s Risk OC and
ASN Function, Acceptable Quality Level (AQL), Lot Tolerance Proportion Defective (LTPD)
and Average Total Inspection (ATI).

Suggested Readings:
1. Wu, C. F. J. and Hamada, M. (2009). Experiments, Analysis, and Parameter Design
Optimization 2nd edition, John Wiley.
2. Renchner, A. C. And Schaalje, G. B. (2008). Linear Models in Statistics, 2nd edition,
John Wiley and Sons.
3. Montogomery, D. C. (2009): Introduction to Statistical Quality Control, 7th Edition,
Wiley India Pvt. Ltd.
4. Goon A. M., Gupta M.K. and Dasgupta B. (2002): Fundamentals of Statistics, Vol. I
a.& II,8th Edn. The World Press, Kolkata.
5. Mukhopadhyay, P (2011):Applied Statistics, 2nd edition revised reprint, Books
andAllied(P) Ltd.
6. Cochran, W.G. and Cox, G.M. (1992): Experimental Design. 2nd Edition, Asia
Publishing House.
7. Das, M.N. and Giri, N.C. (2017): Design and Analysis of Experiments. 3rd Edition ,
Wiley Eastern Ltd.
8. Goon, A.M., Gupta, M.K. and Dasgupta, B. (2005): Fundamentals of Statistics.
Vol.II, 8th Edition. World Press, Kolkata.
9. Kempthorne, O. (2005): The Design and Analysis of Experiments. 2nd Edition, John
Wiley.
10. Montgomery, D. C. (2012): Design and Analysis of Experiments, John Wiley.
11. Balagurusami, E. (2002) Reliability Engineering, Tata McGraw Hill, New Delhi.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M. Sc. (MATHEMATICS)
SEMESTER-III
MATH-VI
Financial Mathematics
Maximum Marks-100
External Examination-80
Internal Assessment-20
Time: 3 hrs.

Course Outcomes
Students would be able to:
CO1 Understand the fundamentals of financial mathematics
CO2 Analyze Black Scholes and Asset Price Model.
CO3 Discuss numerical methods and interest rate derivatives.

Unit- I
Fundamentals of Financial Mathematics, Asset Price Model
Unit- II
Black-Scholes Analysis, Variations on Black-Scholes models
Unit- III
Numerical Methods, American Option, Exotic Options
Unit- IV
Path-Dependent Options, Bonds and Interest Rate Derivatives, Stochastic calculus

Suggested Readings:
1. Financial Mathematics: I-Liang Chern Department of Mathematics, National Taiwan
University
2. Sheldon M. Ross, An Introduction to Mathematical Finance, Cambridge Univ. Press.
3. Robert J. Elliott and P. Ekkehard Kopp. Mathematics of Financial Markets, Springer-
Verlag,
4. New York Inc.
5. Robert C. Marton, Continuous-Time Finance, Basil Blackwell Inc.
6. Daykin C.D., Pentikainen T. and Pesonen M., Practical Risk Theory for Actuaries,
Chapman &Hall.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M. Sc. (MATHEMATICS)
SEMESTER-III
MATH-VII
Bio-Mathematics
Maximum Marks-100
External Examination-80
Internal Assessment-20
Time: 3 hrs.
Course Outcomes
Students would be able to:
CO1 Understand population dynamics and discuss various models.
CO2 Discuss discrete and continuous age-structured populations.
CO3 Analyze infectious disease model and associated genetics.
CO4 Evaluate sequence alignment and Brute force alignment.
Note: There shall be nine questions in all. Question no. 1 shall be compulsory, consisting of
eight short answer type questions covering the entire syllabus. Two questions will be asked
from each unit. Student will have to attempt one question from each unit. Each question shall
carry equal marks.

Unit- I
Population Dynamics: The Malthusian growth ; The Logistic equation; A model of species
competition; The Lotka-Volterra predator-prey model, Age-structured Populations : Fibonacci’s
rabbits;The golden ratio Φ; The Fibonacci numbers in a sunflower; Rabbits are an age-structured
population;

Unit-II
Discrete age-structured populations; Continuous age-structured populations; The brood size of a
hermaphroditic worm. Stochastic Population Growth : A stochastic model of population growth;
Asymptotics of large initial populations; Derivation of the deterministic model; Derivation of the
normal probability distribution; Simulation of population growth.

Unit- III
Infectious Disease Modeling: The SI model; The SIS model; The SIR epidemic disease model;
Vaccination ; The SIR endemic disease model ; Evolution of virulence. Population Genetics:
Haploid genetics; Spread of a favored allele; Mutation-selection balance; Diploid genetics;
Sexual reproduction; Spread of a favored allele; Mutation-selection balance.

Unit-IV
Heterosis; Frequency-dependent selection; Linkage equilibrium; Random genetic drift.
Biochemical Reactions: The law of mass action; Enzyme kinetics; Competitive inhibition;
Allosteric, inhibition; Cooperativity. Sequence Alignment: DNA ; Brute force alignment;
Dynamic programming; Gaps; Local alignments; Software.

Suggested Readings:
1. Mathematical Biology, Lecture notes for MATH 4333, (Jeffrey R. Chasnov)
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

2. Mathematical Biology I. An Introduction, Third Edition, (J.D. Murray)

Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESTER-III
MATH-236
Computing Lab – III
(Programming in SCILAB/MATLAB)
Maximum Marks-100
External Practical Examination-80
Internal Assessment-20
Practical will be conducted externally by the Department as per the following distribution of
marks:
Writing Programme in SCILAB/MATLAB and running it on PC: 50 marks.
Viva voce: 20 marks.
Practical record: 10 marks.

Course Outcomes
Students would be able to:
CO1Get introduction to SCILAB/MATLAB: SCILAB/MATLAB overview, Applications to
Mathematics, Mathematical operations.
CO2 Study different types of Variables, Functions, Operators, and Data types.
CO3 Understand Scripts, Control Flow and Operators.
CO4 Perform Input and Output operations.
CO5 Understand graphics and plotting Graphs.
CO6 Understand basic features of programming in SCILAB/MATLAB.

Programs:
1. Basic mathematical computation.
2. Program to generate Matrix with addition, subtraction and multiplication of matrices.
3. Solving System of Linear equations with two variables.
4. Program to use inbuilt functions and creating new functions.
5. Plotting the graph of function of two variables
6. Program to Differentiate and Integrate.
7. Program to study Interpolation and Regression.
8. Program to solve Fourier analysis.
9. Program to solve Ordinary Differential Equations
10. To plot Graph of Various Functions like sin x, cos x, etc.
11. To find Eigenvalues and Eigenvectors of given matrix.
12. Solving polynomial equations with the method studied.
13. Solving Partial Differential equations
14. Solving boundary value problems.
15. Program to solve Newton’s, Bisection methods.
16. Program to solve given optimization problem.
17. Program to solve Euler’s method, Runge-Kutta’s methods of solving ODE.

Suggested Readings:
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

1. JaanKiusalaas, “Numerical Methods in Engineering with MATLAB”, Cambridge

University Press, 2016.
2. Rudra Pratap, “Getting Started with MATLAB: A Quick Introduction for Scientists and
Engineers”, Paperback.
3. S.N. Alam and S.S. Alam, “Understanding Matlab: A Textbook for Beginners”,
Paperback, 2013.
4. Singh Y. Kirani, Chaudhuri B. B., “Matlab Programming”, Paperback.
5. Timmy Siauw and Alexandre Bayen, “An Introduction to MATLAB Programming and
Numerical Methods for Engineers”, Academic Press, 2015
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESTER-III
MATH-237
Mathematical Lab- III& Seminar
Maximum Marks-100
External Practical Examination-80
Internal Assessment-20
Course Outcomes
Students would be able to:
CO1 Solve mathematical problems using various mathematical techniques.
CO2 Analyze overall performance in all the subjects concerned in the semester.

Mathematical problem-Solving Techniques based on paper MATH-231 to MATH-233&

Discipline elective-I, Discipline elective-II will be taught.
There will be problems based on atleast 5-6 problem solving techniques from each paper.

Note :-Every student will maintain practical record of problems solved during practical
class-work in a file. Examination will be conducted through a question paper set jointly by
the external and internal examiners. The question paper will consists of questions based on
problem solving techniques/algorithm. An examinee will be asked to write the solutions in
the answer book. Evaluation will be made on the basis of the examinee’s performance in
written solutions and presentation with viva-voce and practical record.

Practical will be conducted Externally by the department as per the following distribution of
marks:Writing solutions of problems: 40 marks.
Presentation &Viva voce: 20 marks.
Practical record: 20 marks.
Internal Assesment:20 marks (Attendance=5marks, Assignment=5marks, seminar=10marks)
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESTER-IV
MATH-241
Functional Analysis
Maximum Marks-100
External Examination-80
Internal Assessment-20
Time: 3 hrs.
Course Outcomes
Students would be able to:
CO1 Be familiar with the completeness in normed linear spaces.
CO2 Understand the concepts of bounded linear transformation, equivalent formulation of continuity and
spaces of bounded linear transformations.
CO3 Understand Hilbert spaces and related terms.
CO4 Understand uniform boundedness principle and its consequences.
CO5 Understand the spectral representation of bounded self adjoint linear operators, extension of the
spectral theorem to continuous functions.
Note: There shall be nine questions in all. Question no. 1 shall be compulsory, consisting of
eight short answer type questions covering the entire syllabus. Two questions will be asked
from each unit. Student will have to attempt one question from each unit. Each question shall
carry equal marks.

Unit -I
Normed spaces, Banach spaces, Finite dimensional normed space and subspaces, Linear
operators, Bounded and Continuous linear operators, Linear Functionals, Normed spaces of
operators, Dual spaces.
Unit -II
Inner product space and its properties, Hilbert spaces, orthogonal complements and direct
sums, Legendre, Hermite and laguerre polynomials, Representation of Functionals on Hilbert
spaces, Hilbert- Adjoint Opertaor, Self- Adjoint, Unitary and Normal Operators.

Unit -III
Hahn Banach Theorem, Uniform bounded principle, Closed graph Theorem, Open
mapping Theorem, Adjoint Operators, Reflexivity.

Unit -IV
Spectral theory in finite dimensional normed spaces, Spectral properties of Bounded
Linear Operators, Further Properties of Resolvent and Spectrum, Spectral properties of Bounded
Self- Adjoint Linear Operators, Positive Operators.

Suggested Readings:-
1. A. Taylor and D. Lay, Introduction to functional analysis, Wiley, New York, 1980.
2. B.V. Limaye, Functional Analysis, 2nd ed., New Age International, New Delhi, 1996.
3. C. Goffman and G. Pedrick, First course in functional analysis, Prentice-Hall, 1974.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

4. E. Kreyszig, Introductory Functional Analysis with applications, John Wiley & Sons,
NY, 1978.
5. J. B. Conway, A course in Functional Analysis, Springer-Verlag, Berlin, 1985.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESTER-IV
MATH-242
Integral Equations and Calculus of Variations
Maximum Marks-100
External Examination-80
Internal Assessment-20
Time: 3 hrs.
Course Outcomes
Students would be able to:
CO1 Understand the methods to reduce Initial value problems associated with linear differential equations
to various integral equations.
CO2 Categorise and solve different integral equations using various techniques.
CO3 Describe importance of Green's function method for solving boundary value problems associated
with non-homogeneous ordinary and partial differential equations, especially the Sturm-Liouville
boundary value problems.
CO4 Learn methods to solve various mathematical and physical problems using variational techniques.

Unit -I
Linear Integral Equations, Some Basic Identities, Initial value problems reduced to
Volterra integral equations, Method of successive substitution and successive approximation to
solve Volterra Integral equations of second kind, iterated Kernels and Neumann series for
Volterra Equations, Resolvent Kernel as a series, Laplace transform method for a difference
kernel, Solution of aa Volterra integral Equation of the first kind.

Unit -II
Boundary value problems reduced to Fredholm Integral equations, Method of successive
substitution and successive approximation to solve Fredholm Integral equations of second kind,
iterated Kernels and Neumann series for Fredholm Equations, Resolvent Kernel as a sum of
series, Fredholm resolvent Kernel as a ratio of two series, Fredholm equation with sparable
kernels, Approximation of a kernel by a separable kernel, Fredholm Alternative, Non-
Homogenous Fredholm Integrak Equations with degenerate kernels.
.

Unit -III
Green Function, Use of method of variation of parameters to construct the Green function for a
non- homogenous linear second order linear boundary value problems, Basic four properties of
the Green Function, Alternate Procedure for construction of the Green function by using its basic
four properties, Reduction of a boundary value problem to a Fredholm Integral equation with
kernel as Green Function,Hilbert- Schmidt theory of symmetric kernels.

Unit -IV
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

Motivating problems of Calculus of Variations, Shortest Distance, Minimum surface of

Resolution, Brachistochrone problem, Isoperimetric problem, Geodesic. Fundamental Lemma of
Calculus of Variations, Euler equation for one dependent function and its generalization to ‘n’
dependent functions and to higher order derivaties, Conditional Extremum under geometric
Constraints and under integral Constraints.

Suggested Readings:-
1. I. N. Sneddon, Mixed Boundary Value Problems in potential theory, North Holland,
1966.
2. I. Stakgold, Boundary Value Problems of Mathematical Physics Vol.-I, II,
MacMillan, 1969.
3. R. P. Kanwal, Linear Integral Equations, Theory and Techniques, Academic Press,
New York.
4. S. G. Mikhlin, Linear Integral Equations (translated from Russian) Hindustan Book
Agency, 1960.
5. S. K. Pundir and R. Pundir, Integral Equations and Boundary value problems, Pragati
Prakashan, Meerut.
6. A. J. Jerri, Introduction to Integral equations with applications, A WilseyInterscience
Publication, 1991.
7. W.V. Lovvit, Linear Integral equations, McGraw Hill New York.
8. J. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice Hall, New Jersy,
1963.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M. Sc. (MATHEMATICS)
SEMESTER-IV
MATH-243
Mechanics of Solids – II
Maximum Marks-100
External Examination-80
Internal Assessment-20
Time: 3 hrs.
Course Outcomes
Students would be able to:
CO1 Be familiar with the concept of generalized plane stress and solution of two dimensional biharmonic
equations.
CO2 Solve the problems based on thick-walled tube under external and internal pressures.
CO3 Understand the concept of torsional rigidity, lines of shearing stress and solve the problems of
torsion of beams with different cross-sections.
CO4 Describe Ritz method, Galerkin method, Kantrovich method and their applications to the torsional
problems.
CO5 Get familiar with simple harmonic progressive waves, plane waves and wave propagation in two-
dimensions.
Note: There shall be nine questions in all. Question no. 1 shall be compulsory, consisting of
eight short answer type questions covering the entire syllabus. Two questions will be asked
from each unit. Student will have to attempt one question from each unit. Each question shall
carry equal marks.

.
Unit-I
Two-dimensional Problems: Plane strain and Plane stress. Generalized plane stress. Airy stress
function for plane strain problems. General solutions of a Biharmonic equation. Stresses and
displacements in terms of complex potentials.Thickwalled tube under external and internal
pressures. Rotating shaft.

Unit-II
Torsion of Beams: Torsion of cylindrical bars. Torsional rigidity. Torsion and stress functions.
Lines of shearing stress, Simple problems related to circle, ellipse and equilateral triangle cross-
section. Circular groove in a circular shaft.
Extension of Beams: Extension of beams by longitudinal forces. Beam stretched by its own
weight.

Unit-III
Bending of Beams: Bending of Beams by terminal Couples, Bending of a beam bytransverse
load at the centroid of the end section along a principal axis.
Variational Methods:Reciprocal theorem of Betti and Rayleigh. Deflection of elastic string,The
Ritz method-one & two dimensional, The Galerkin method, The method of Kantrovitch.

Unit-IV
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

Waves: Simple harmonic progressive waves, scalar wave equation, progressivetype solutions,
plane waves and spherical waves, stationary type solutions inCartesian and Cylindrical
coordinates.
Elastic Waves: Propagation of waves in an unbounded isotropic elastic solid. P.SV and SH
waves. Surface Waves: frequency equation for Rayleigh and Love waves.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

Liquid-Liquidinterface, Liquid-Solid interface and Solid-Solid interface.Surface waves: Rayleigh

waves, Love waves and Stoneley waves.

Unit -IV
Two dimensional Lamb’s problems in an isotropic elastic solid: Area sources and Line Sources
in anunlimited elastic solid. A normal force acts on the surface of a semi-infinite elastic solid,
tangentialforces acting on the surface of a semi-infinite elastic solid.Three dimensional Lamb’s
problems in an isotropic elastic solid: Area or Volume sources and Pointsources in an unlimited
elastic solid, Area or Volume source and Point source on the surface of semi-infiniteelastic
solid.Haskell matrix method for Love waves in multilayered medium.

Suggested Readings:
1. C.A. Coulson and A. Jefferey, Waves, Longman, New York, 1977.
2. M. Bath, Mathematical Aspects of Seismology, Elsevier Publishing Company, 1968.
3. W.M. Ewing, W.S. Jardetzky and F. Press, Elastic Waves in Layered Media, McGraw
HillBook Company, 1957.
4. C.M.R. Fowler, The Solid Earth, Cambridge University Press, 1990
5. P.M. Shearer, Introduction to Seismology, Cambridge University Press,(UK) 1999.
6. K. E. Bullen, B. A. Bolt, An Introduction to the theory of Seismology, 4th Edi., 1985.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESTER-IV

MATH-III
Mathematical Modeling
Maximum Marks-100
External Examination-80
Internal Assessment-20
Time: 3 hrs.
Course outcomes:
Students would be able to:
CO1 Understand the need/techniques/classification of mathematical modeling through the use of first
order ODEs and their qualitative solutions through sketching.
CO2 Learn to develop mathematical models using systems of ODEs to analyse/predict population growth,
epidemic spreading for their significance in economics, medicine, arm-race or battle/war.
CO3 Attain the skill to develop mathematical models involving linear ODEs of order two or more and
difference equations, for their relevance in probability theory, economics, finance, population
dynamics and genetics.
C04 Develop mathematical models through PDEs for mass-balance, variational principles, probability
generating function, traffic flow problems alongwith relevant initial & boundary conditions.
Note: There shall be nine questions in all. Question no. 1 shall be compulsory, consisting of
eight short answer type questions covering the entire syllabus. Two questions will be asked
from each unit. Student will have to attempt one question from each unit. Each question shall
carry equal marks.
Unit -I
The process of Applied Mathematics; mathematical modeling: need, techniques, classification
and illustrative; mathematical modeling through ordinary differential equation of first order;
qualitative solutions through sketching.
Unit -II
Mathematical modeling in population dynamics, epidemic spreading and compartment models;
mathematical mode1ing through systems of ordinary differential equations; mathematical
mode1ing in economics, medicine, arm-race, battle.

Unit -III
Mathematical modeling through ordinary differential equations of second order. Higher order
(linear) models. Mathematical modeling through difference equations: Need, basic theory;
mathematical modeling in probability theory, economics, finance, population dynamics and
genetics.
Unit -IV
Mathematical modeling through partial differential equations: simple models, mass-balance
equations, variational principles, probability generating function, traffic flow problems, initial &
boundary conditions.
Suggested Readings:
1. J.N. Kapur: Mathematical Modeling, Wiley Eastern Ltd., 1990
2. B. Albright;Mathematical Modeling with Excel, Jones & Bartlett,2010
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESTER-IV

MATH-IV
Advanced Coding Theory
Maximum Marks-100
External Examination-80
Internal Assessment-20
Time: 3 hrs.
Course Outcomes:
Students would be able to:
CO1 Understand tree codes, matrix encoding techniques, polynomial codes and Hamming codes.
CO2 Have deep understanding of finite fields, BCH codes.
CO3 Learn about MDS codes, Hadamard matrices and error locating codes.
Note: There shall be nine questions in all. Question no. 1 shall be compulsory, consisting of
eight short answer type questions covering the entire syllabus. Two questions will be asked
from each unit. Student will have to attempt one question from each unit. Each question shall
carry equal marks.

Unit-I
Tree codes, Convolutional codes, Description of linear tree and convolu-tional codes by
matrices, Standard array, Bounds on minimum distance for convolutional codes, V-G-S bound,
Bounds for burst-error detecting and correcting convolutional codes, The Lee metric, Packing
bound for Hamming code w.r.t. Lee metric, The algebra of polynomials, Residue classes, Galois
fields, Multiplicative group of a Galois field.
Unit-II
Cyclic codes, Cyclic codes as ideals, Matrix description of cyclic codes, Hamming and Golay
codes as cyclic codes, Error detection with cyclic codes, Error-correction procedure for short
cyclic codes, Shortended cyclic codes, Pseudo cyclic codes.

Unit-III
Code symmetry, Invariance of codes under transitive group of permuta-tions, Bose-Chaudhary-
Hocquenghem (BCH) codes, BCH bounds, Reed-Solomon (RS) codes, Majority-logic decodable
codes, Majority-logic de-coding.
Unit-IV
Singleton bound, The Griesmer bound, Maximum-distance separable (MDS) codes, Generator
and parity-check matrices of MDS codes, Weight distri-bution of MDS code, Necessary and
suﬃcient conditions for a linear code to be an MDS code, MDS codes from RS codes, Abramson
codes, Closed-loop burst-error correcting codes (Fire codes), Error locating codes.

Suggested Readings:
1. F.J. Macwilliams and N.J. A. Sloane, Theory of Error Correcting Codes, North- Holland
Publishing Company, 2006.
2. W.W. Peterson and E.J. Weldon, Jr., Error-Correcting Codes, M.I.T. Press, Cambridge,
Massachusetts, 1972.
3. E.R. Berlekamp, Algebraic Coding Theory, McGraw Hill Inc., 1984.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

4. W.C. Huﬀman and V. Pless, The Theory of Error Correcting Codes,

CambridgebUniversity Press, 1998.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M. Sc. (MATHEMATICS)

SEMESTER-IV
MATH-V
Statistical Inference
Maximum Marks-100
External Examination-80
Internal Assessment-20
Time: 3 hrs.
Course Outcomes
Students would be able to:
CO1 Understand the concepts of point estimation and interval estimation.
CO2 Identify good estimators using criterion of good estimators and obtain estimators using method of
maximum likelihood and moments.
CO3 Learn about the chi-square, Students’ t and Snedcor F-statistics and their important applications.
CO4 Carry out different tests of significance for small samples and apply common nonparametric tests to
real life problems.
CO5 Explain and use Neyman-Pearson lemma and likelihood ratio tests.
Note: There shall be nine questions in all. Question no. 1 shall be compulsory, consisting of
eight short answer type questions covering the entire syllabus. Two questions will be asked
from each unit. Student will have to attempt one question from each unit. Each question shall
carry equal marks.
Unit - I
Estimation: Neymann Factorization theorem. Complete statistic, Minimum variance unbiased
estimator (MVUE), Exponential Family of Distributions and its Properties, Rao-Blackwell and
Lehmann-Scheffe theorems and their applications. Cramer-Rao inequality and MVB estimators
(statement and applications).Methods of Estimation: Method of moments, method of maximum
likelihood estimation, method of minimum Chi-square.
Unit – II
Statistical hypotheses, critical region, size and power of a test, most powerful test, randomized
and non randomized test, Neyman Pearson lemma and its applications, uniformly most powerful
unbiased test, power likelihood ratio test and its applications, functions of UMP with simple
illustration.Elements of decision problems: Loss function, risk function, estimation and testing
viewed asdecision problems. Bayes rule.

Unit – III
Non Parametric Theory: Concept of Non Parametric and Distribution Free Methods, Order
Statistics, Their Marginal and Joint Distributions. Distributions of Median, Range and Coverage;
Moments of Order Statistics. Asymptotic Distribution of Order Statistics.

Unit – IV
Non Parametric Tests: One Sample and Paired Sample Problems. Ordinary Sign Test, Wilcoxon
Signed Ranked Test, and Their Comparison. General Problem of Tied Differences. Goodness of
Fit Problem : Chi-Square Test and Kolmogrov – Smirnov One Sample Test, and Their
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

Comparison. Two Sample Problems: K-S Two Sample Test, Wald – Wolfwitz Run Test, Mann –
Whiteney U Test, Median Test.

Suggested Readings:
1. Freund J.E. (1992): Mathematical Statistical, 5th Edition, Prentice Hall of India.
2. Hogg R.V. and Craig A.T. (2012): Introduction of Mathematical Statistics, 7th Edition,
Collier Macmillon Publishers.
3. Mood A.M., Graybill E.A. and Bose D.C. (2011): Introduction to the Theory of Statistics,
3rdEdition, McGraw Hill.
4. Rao, C.R. (2001): Linear Statistical Inference and its Applications, 2nd edition Wiley
Eastern.
5. Rohtagi V.K. and A. K. Md. Ehsanes Saleh (2015): An Introduction to Probability and
Statistics, 3rd Edition, John Wiley and Sons.
6. Goon A.M., Gupta M.K. and Dasgupta B. (2013) : An Outline of Statistical Theory, Vol.
2 The World Press Publishers Pvt. Ltd. Calcutta.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESTER-IV
MATH-VI
Fourier and Wavelet Analysis
Maximum Marks-100
External Examination-80
Internal Assessment-20
Time: 3 hrs.
Course Outcomes:
Students would be able to:
CO1 Have an idea of the finite Fourier transform, convolution on the circle group T, the Fourier
transform and residues and know about continuous analogue of Dini’s theorem and Lipschitz’s test.
CO2 Know about (C,1) summability for integrals, understand the Fejer-Lebesgue inversion theorem,
Parseval’s identities, the L 2 theory, Plancherel theorem and Mellin transform.
CO3 Have understanding of the Discrete and Fast Fourier transforms, and Buneman’s Algorithm.
CO4 Understand Multiresolution Analysis, Mother wavelets; construction of scaling function with
compact support, Shannon wavelets, Franklin wavelets, frames, splines and the continuous wavelet
transform.
Note: There shall be nine questions in all. Question no. 1 shall be compulsory, consisting of
eight short answer type questions covering the entire syllabus. Two questions will be asked
from each unit. Student will have to attempt one question from each unit. Each question shall
carry equal marks.
Unit-I
Fourier Transform: The finite Fourier transform, the circle group T, convolution to T, (L1
(T),+,*) as aBanach algebra, convolutions to products, convolution on T, the exponential form of
Lebesgue’stheorem, Fourier transform : trigonometric approach, exponential form,
Basics/examples.Fourier transform and residues, residue theorem for the upper and lower half
planes, the Abel kernel, theFourier map, convolution on R, inversion, exponential form,
inversion, trigonometric form, criterion forconvergence, continuous analogue of Dini’s theorem,
continuous analogue of Lipschitz’s test, analogueof Jordan’s theorem,

Unit-II
(C,1) summability for integrals, the Fejer-Lebesgue inversion theorem,the continuous Fejer
Kernel, theFourier map is not onto, a dominated inversion theorem, criterion for integrability of
𝑓̂Approximate identity for L1(R), Fourier Sine and Cosine transforms, Parseval’s identities, the
L2theory, Parseval’s identities for L2, inversion theorem for L2 functions, the Plancherel theorem,
Asampling theorem, the Mellin transform, variations.

Unit-III
Discrete Fourier transform, the DFT in matrix form, inversion theorem for the DFT, DFT map as
alinear bijection, Parseval’s identities,cyclic convolution, Fast Fourier transform for
N=2k,Buneman’sAlgorithm, FFT for N=RC, FFT factor form.

Unit-IV
Wavelets : orthonormal basis from one function , Multiresolution Analysis, Mother wavelets
yieldWavelet bases, Haar wavelets, from MRA to Mother wavelet, Mother wavelet theorem,
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

construction ofscaling function with compact support, Shannon wavelets, Riesz basis and MRAs,
Franklin wavelets,frames, splines, the continuous wavelet transform.

Suggested Books

1. G. Bachman, L. Narici and E. Beckenstein : Fourier and Wavelet Analysis, Springer,

2000 Hernandez and G. Weiss : A first course on wavelets, CRC Press, New York, 1996
2. C. K. Chui: An introduction to Wavelets, Academic Press,1992
3. V. Meyer, Wavelets, algorithms and applications SIAM, 1993
4. M.V. Wickerhauser: Adapted wavelet analysis from theory to software, Wellesley,MA,
A.K.
5. Peters, 1994
6. D. F. Walnut: An Introduction to Wavelet Analysis, Birkhauser, 2002
7. K. Ahmad and F.A. Shah: Introduction to Wavelets with
8. Applications, World Education Publishers, 2013.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESTER-IV

MATH-VII
Bio-Mechanics
Maximum Marks-100
External Examination-80
Internal Assessment-20
Time: 3 hrs.
Course Outcomes
Students would be able to:
CO1 Use the mathematics of mechanics to quantify the kinematics and kinetics of human movement
alongwith describing its qualitative analysis.
CO2 Possess knowledge of steady laminar flow in elastic tubes, pulsatile flow and significance of non-
dimensional number affecting the flow.
CO3 Study the problems of external flow around bodies moving in wind and water, in locomotion, flying
and swimming.
CO4 Be familiar with internal flows such as blood flow in blood vessels, gas in lungs, urine in kidneys,
water and other body fluids in interstitial space between blood vessels and cells.
Note: There shall be nine questions in all. Question no. 1 shall be compulsory, consisting of
eight short answer type questions covering the entire syllabus. Two questions will be asked
from each unit. Student will have to attempt one question from each unit. Each question shall
carry equal marks.

Unit-I
Newton equations of motion, Mathematical modeling, Continuum approach, Segmental
movement and vibrations, Lagrange equations, Normal modes of vibration, Decoupling of
equations of motion. Flow around an airfoil, Flow around bluff bodies, Steady state aeroelastic
problems, Transient fluid dynamics forces due to unsteady motion, Flutter.

Unit-II
Kutta-Joukowski theorem, Circulation and vorticity in the wake, Vortex system associated with a
finite wing in nonsteady motion, Thin wing in steady flow. Blood flow in heart, lungs, arteries,
and veins, Field equations and boundary conditions, Pulsatile flow in arteries, Progressive waves
superposed on a steady flow, Reflection andtransmission of waves at junctions.

Unit-III
Velocity profile of a steady flow in a tube, Steady laminar flow in an elastic tube, Velocity
profile of Pulsatile flow, The Reynolds number, Stokes number, and Womersley number,
Systematic blood pressure, Flow in collapsible tubes. Micro-and macrocirculation Rheological
properties of blood, Pulmonary capillary bloodflow, Respiratory gas flow, Intraction between
convection and diffusion, Dynamics of the ventilation system.

Unit-IV
Laws of thermodynamics, Gibbs and Gibbs – Duhem equations, Chemical potential, Entropy in a
system with heat and mass transfer, Diffusion, Filtration, and fluid movement ininterstitial space
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

in thermodynamic view, Diffusion from molecular point of view. Mass transport in capillaries,
Tissues, Interstitial space, Lymphatics, Indicator dilution method, and peristalsis, Tracer motion
in a model of pulmonary microcirculation.

Suggested Readings:
1. Y.C. Fung, Biomechanics: Motion, Flow, Stress and Growth, Springer-Verlag, New York
Inc., 1990.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESTER-IV
MATH-246
Computing Lab – IV
Programming in LaTeX and Gnuplot
Maximum Marks-100
External Practical Examination-80
Internal Assessment-20
Practical will be conducted externally by the Department as per the followingdistribution of
marks:
Writing ProgrammeinLaTeX and Gnuplot running it on PC: 50 marks.
Viva voce: 20 marks.
Practical record: 10 marks.
Objective: This computing lab is designed to introduce practical applications of Advanced
Mathematical Tools like LaTeX and Gnuplot to the students.

Course Outcomes:

Students would be able to:

Introduction to LaTeX: Purpose of LaTeX, Typesetting System, typing your first article, typing
text, structuring your document (sections and paragraphs), different packages, Typesetting math
in LaTeX, Adding a picture, Generating a table of contents, Adding a bibliography, Adding
footnotes, Creating tables with LaTeX, Plots – Plotting and Visualizing your data.

Introduction to Gnuplot: graph plotting and data fitting, creating simple plots, plotting data from
a file, fitting a function to data.

List of Practical’s:
1. Getting started with simple document.
2. Create a numbered list, and try typesetting the following (using either inline or display
environments as appropriate):
(a) You know that e0 = exp 0 = 1 and log 1 = 0, but did you know that e-x → 0 as x →∞?
(b) Write the second part of the sentence above using limit notation, as a display
equation.
(c) If W ∼ χ21 then P(W > 3.84) ≈ 0.05.

𝑛(𝑛+1)
(d) ∑𝑛𝑖=1 𝑖 =
2
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

3. Write the following set of equations in augmented matrix form.

y + 2z = 1
x + 2y + 3z = 2
3x + y + 2z = 4
4. Create a simple table in LaTeX. Add data in table. Add centering, a caption and a label.
5. Add a title to your document. Add sections, one for each practical, like in these notes. If
you have used one large list environment, you will need to split it into several smaller
lists, but this is not difficult.
6. Insert your chosen picture into your document, using a figure environment with
appropriate scaling.
7. Add a caption after the picture and a label after the caption. Add a reference to your
picture somewhere in your text.
8. Plot a blue curve for f(x) = cos x on the same picture, with a label.
9. Plot an orange curve for f(x) = 1 20 exp x on the same picture, with a label.
10. Try using formatting commands (see above), \onslide, \uncover and \pause to liven up the
TikZ picture, equations and table.
11. Plotting data from a file using Gnuplot.
12. Fitting a function to data.
Syllabi, Department of Mathematics, GUG w.e.f. session 2020-21

M.Sc.-MATHEMATICS
SEMESTER-IV
MATH-247
Mathematical Lab- IV & Seminar
Maximum Marks-100
External Practical Examination-80
Internal Assessment-20
Course Outcomes
Students would be able to:
CO1 Solve mathematical problems using various mathematical techniques.
CO2 Analyze overall performance in all the subjects concerned in the semester.

Mathematical problem-Solving Techniques based on paper MATH-241 to MATH-243&

Discipline elective-III, Discipline elective-IV will be taught.
There will be problems based on atleast 5-6 problem solving techniques from each paper.

Note: -Every student will maintain practical record of problems solved during practical
class-work in a file. Examination will be conducted through a question paper set jointly by
the external and internal examiners. The question paper will consists of questions based on
problem solving techniques/algorithm. An examinee will be asked to write the solutions in
the answer book. Evaluation will be made on the basis of the examinee’s performance in
written solutions and presentation with viva-voce and practical record.

Practical will be conducted Externally by the department as per the following distribution of
marks:Writing solutions of problems: 40 marks.
Presentation &Viva voce: 20 marks.
Practical record: 20 marks.
Internal Assesment:20 marks (Attendance=5marks, Assignment=5marks, seminar=10marks)

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MSC Math

Uploaded by

MSC Math

Uploaded by

Syllabi, Department of Mathematics, GUG w.e.f.

Gurugram University, Gurugram

 To build research and creative activities among faculty and students

Program Outcomes (POs)

PO1 Knowledge Capable of demonstrating comprehensive disciplinary

Program Specific Outcomes (PSOs)

Gurugram University, Gurugram

Study & Evaluation Scheme

Programme : M.Sc. Mathematics

Internal Evaluation (Theory

Duration of Examination External Internal (Minor Test)

To qualify the course, a student is required to secure a minimum of 40% marks in

Question Paper Structure

Gurugram University, Gurugram

2 MATH- Ordinary C.C. 4 1 -- 5 4 1 -- 5 80 20 -- 100

5 MATH- Mathematical C.C. 4 1 -- 5 4 1 -- 5 80 20 -- 100

6 MATH- Computing S.E.C. 2 2 4 2 4 6 40 10 50 100

Gurugram University, Gurugram

2 MATH-122 Complex Analysis C.C. 4 -- 4 4 -- 4 80 20 -- 100

3 MATH-123 Topology C.C. 4 -- 4 4 -- 4 80 20 -- 100

Instructions for Students

7. Use graphics wherever necessary

1. The evaluation is divided into different segments as under : 30 Marks

Publishers Pvt Ltd, 2012.

5. H.C. Taneja, Statistical Methods for Engineering& Sciences, I. K.International,2009.

List of practicals is to be given are as follows:

3. Spreadsheet Accounting - Tutorial and Applications, G. E. Anders, C. R. Schaber and R. D.

Mathematical problem Solving Techniques based on paper 20MATH-111 to 20MATH-115

3. H.A. Priestly, Introduction to Complex Analysis, Clarendon Press, Oxford, 2008.

2. H. A. Taha, Operation Research-An Introduction, Printice Hall of India, 2017.

Numerical Integration: General Integration formula, Rectangular rule, Trapezoidal rule,

Following Programs to be done during the course:

3. Byron, S. Gottfried: Theory and Problems of Programming with C, Second Edition

Mathematical problem Solving Techniques based on paper 20MATH-121 to 20MATH-

k(n), u(n), N(n), I(n).Definition and examples and

1. D.M., Burton, Elementary Number Theory.

4. T. Amarnath: An elementary course in partial differential equations (Narosa Publishing

2. Mathematical Biology I. An Introduction, Third Edition, (J.D. Murray)

1. JaanKiusalaas, “Numerical Methods in Engineering with MATLAB”, Cambridge

Mathematical problem-Solving Techniques based on paper MATH-231 to MATH-233&

Motivating problems of Calculus of Variations, Shortest Distance, Minimum surface of

Liquid-Liquidinterface, Liquid-Solid interface and Solid-Solid interface.Surface waves: Rayleigh

4. W.C. Huﬀman and V. Pless, The Theory of Error Correcting Codes,

1. G. Bachman, L. Narici and E. Beckenstein : Fourier and Wavelet Analysis, Springer,

Students would be able to:

3. Write the following set of equations in augmented matrix form.

Mathematical problem-Solving Techniques based on paper MATH-241 to MATH-243&

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