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1 KADI SARVA VISHWAVIDYALAYA

KADI SARVA VISHWAVIDYALAYA

GANDHINAGAR

Proposed Syllabus for

M.Sc. Mathematics under CBCS

(2 Years Full Time: 4 Semesters Programme)

LDRP Campus, Sector – 15, Nr. KH – 5 Circle,

Gandhinagar - 382015


About the Trust

“Sarva VishwavidyalayaKelavaniMandal” the trust which has been in existence for more than

eight decades is a well reputed prestigious educational trust in North Gujarat. The alumni of

SVKM has managed and nurtured the trust to its present eminence.

The trust was formed in 1919, and commenced its activities with a school and student

residential “Ashram” at Kadi in 1921 through the generous donation from the society and

through the visionary efforts of “Chaganbha” who is the establisher of the Mandal.

The trust has setup as many as 30 different educational institutions, ranging from Primary

schools to postgraduate courses. Engaged in the right pursuit of contributing to the noble

cause of education the trust, which started with a school and a handful of students, has today

to its credit two mega campuses at Kadi and Gandhinagar. More than 50,000 young students

are being groomed at these campuses.

Having provided primary, secondary and higher secondary for almost seven decades, the

trust has started imparting higher education and being sensitive to the needs of environment,

has added technology, management and computer oriented courses to prepare youth of the

region to take up the challenges of the future.

Be it quality of students, quality of faculty or quality of infrastructure at Sarva

Vishwavidhyalaya Kelavani Mandal, nothing would be less par excellence. With the co-

operation from its Alumni settled across the globe, the trust is committed to attain higher and

higher standards of quality education to serve the coming generation.

About Kadi Sarva Vishwavidyalaya(KSV)

Kadi Sarva Vishwavidyalaya (KSV) is a University established vide Gujarat State Government Act 21

of 2007 in May 2007 and approved by UGC (Ref.: F. 9-18/2008(cpp-1) March 19,2009).

The University has been setup by Sarva Vidyalaya Kelavani Mandal, a trust with more than 95 years

of philanthropic existence to achieve the following objectives:

To provide need based education and develop courses of contemporary

relevance.

To be a University of excellence by providing research based activities

which would foster higher economic growth.

To provide education to all irrespective of caste, creed, religion etc

http://svkmgujarat.org.in/


Kadi Sarva Vishwavidhyalaya

MASTER OF SCIENCE (MATHEMATICS)

(1) Learning outcomes (Objectives and Aim) This program leading to this degree provides the opportunities to develop and

demonstrate knowledge and understanding of the fundamental and advanced content

in Mathematics which will be determined by his /her particular choice of courses,

according to his/her particular needs and interests.

Cognitive skills:

After the completion of the degree, the student will be able to:

Understand how to solve some problems using the methods taught

Assimilate complex mathematical ideas and arguments

Develop abstract mathematical thinking

Develop mathematical and physical intuition.

Practical and/or professional skills and Key Skills:

When one has completed this degree he / she will be able to demonstrate the

following skills:

The ability to advance your own knowledge and understanding through

independent learning

Communicate clearly knowledge, ideas and conclusions about mathematics

Develop problem- solving skills and apply them dependently to problems in pure,

applied and applicable mathematics

Communicate effectively in writing about the subject

Improve his/her own learning and performance.

(2) Duration of the course:

The CBCS pattern M.Sc. programme with multidisciplinary approach in Mathematics

is offered on a fulltime basis. The duration of the course is of two academic years

consisting of four semesters each of 15 weeks duration.


(3) Teaching and learning methods:

All relevant material is provided and taught in the course texts through the study of set

books. Various modern resources will be provided to enhance his/her skill. One will

build up knowledge gradually, with sufficient in text examples to support one’s

understanding. He/ She will be able to assess his/her own progress and

understanding by using the in - text problems and exercises at the end of each unit.

Opportunity to engage with what is taught is provided by means of the assignment

questions and understanding will be reinforced by personal feedback from the teacher

in the form of comments based on the answers to one’s assignments, seminars, unit

tests and project.

(4) Course of study:

The curriculum has seven major components:

1. Core/Principal/Fundamental Mathematical courses

2. Pure Mathematical Courses

3. Applied Mathematical Courses

4. Applicable/Application Oriented Mathematical Courses (disciplinary)

5. Soft Skill Based Courses (Inter-disciplinary)

6. Cognitive Skill- Work Based Courses

The following courses are prescribed in the following classification to be studied to

acquire M.Sc. Degree in Mathematics.

(I) Principal / Core / Compulsory Courses (HARD CORE) (MTCG 1 to 8)

All Basic / Core courses carry 4 credits in 4 hours per week teaching and in semester I and II,

any four core courses while in Semester III and IV, any three core courses are to be selected

from the list of MT Core Group (various groups are listed in detail syllabus) with no

repetitions i.e there are total 14 Mathematical Core Courses to be selected from semester- I to

semester IV.

(II) Elective Disciplinary courses (SOFT CORE): (MTEG 1 to 8)

All elective courses carry 4 credits in 4 Hours per week teaching. During the span of the

program, there are one Mathematical Elective Course offered in Sem III and IV covering the

two major components of pure mathematical group and applied mathematical group.


(III) Soft Skill Based Courses: (SSB-1 to 3)

All soft skill based coursed carry 4 credits in 2 hours per week teaching and 4 credits

for Practical in 4 hours per week. There are total 3 different Courses to be chosen

from the list of SSB.

(IV) Cognitive Skills Work Project/Dissertation Work for Research Problem

This is also described in detail syllabus at the end.

(5) Assessment and examination method: A candidate’s understanding of concepts will be assessed through CIA and UE pattern

as follow:

Mid-term Semester Examination (MSE) :

The Internal Assessment (MSE) will be evaluated as:

1. Mid Sem Exam: 40 Marks / 2 = 20 Marks

2. Report Submission / Attendance = 5 Marks

3. Test / Seminar/ Quiz/ Assignments = 5 Marks

University Examination (UE) :

There shall be four semester examinations, one at the end of each semester in each

academic year. A candidate who does not pass the examination in any course (s) in a

semester will be permitted to appear in such failed course (s) also, with subsequent

semester examinations: University Examination (UE) only.

Classifications of MSE & UE for different courses of different credits are:

1. Courses of 4 Credits = 70 (UE) +30 (MSE) = 100 marks. (Theory)

2. Courses of 4 Credits = 100 (UE) = 100 marks. (Practical)

3. Courses of 4 Credits = 100(UE) = 100 marks. (Project)

(6) Rules and regulations

1. Candidates for admission to the Master of Science (Mathematics) must have a Bachelor's

degree with Mathematics as a principle subject of minimum three year duration.

2. The duration of the course will be full time two academic years. The examination for the

Master of Science (Mathematics) course will be conducted under the semester system. For


this purpose the academic year will be divided into two semesters. No candidate will be

allowed to join any other fulltime course simultaneously.

3. No candidates will be admitted to any semester examination for Master of Science

(Mathematics) unless it is certified by the HOD, M.Sc. (Mathematics) that he/she has attended

the courses of study to the satisfaction of the HOD, M.Sc. (Mathematics). For granting the

terms, minimum attendance of 75% of the theory, lectures and practical’s will be required out

of the total number of lectures and practical’s conducted in the terms.

4. Candidates desirous of appearing at any semester examination of the M.Sc.

(Mathematics) course must forward their application in the prescribed form to the Registrar,

through the HOD, M.Sc. (Mathematics) on or before the date prescribed for the purpose under

the relevant intimation of the University.

5. For any Semester, the maximum marks in any subject(s) for the internal and external

assessments shall be shown in the teaching and examination scheme for each individual

subjects. For the purpose of internal assessment, tests, quizzes, assignment or any other

suitable methods of continuous evaluation may be used by the department. If a student keeps

term and does not appear for examinations as well as if he/she fail to reappear in the re-test

(block test) examination in the same academic session, his/her internal in the relevant

subject(s) would be considered as ABSENT (INCOMLETE grade “I”). The department will

submit the internal marks of all subject(s) as per the notification of the University.

6. No candidate will be permitted to reappear at any semester examination, which he/she has

already passed.

7. To obtain the Degree of Master of Science (Mathematics), student should clear all the four

semester examinations within a period of four years from the date of his/her Registration.

Failing which, he/she shall be required to register himself/herself as a fresh candidate and keep

the attendance and appear and pass in the four semester examinations afresh from first

semester onwards in order to obtain the Degree of Master of Science (Mathematics).

8. There shall be an Examination at the end of each of the four semesters to be known as First

semester Examination, Second semester Examination, Third semester Examination and Fourth

semester Examination respectively, at which a student shall appear in that portion of papers

practical and Viva- Voce if any, for which he/she has kept the semester in accordance with the

regulations in this behalf.


A candidate, whose term is not granted for whatsoever reason, shall be required to keep

attendance for that semester or terms when the relevant papers are actually taught at the

department.

9. No candidates will be allowed to reappear in a subject/course in which he/she has already

passed. He /She can reappear only for the examination i.e. Internal or University examination

in which he/she has failed. His/ Her marks in the examination passed will be carried

forwarded.

(7) Rules for Grading for M.Sc. Mathematics Programme (KSV)

(i) Theory Subjects and Practical Subjects are allotted credits as per the hours allocated to them per

week. (i.e. 1 hr/week = 1 Credit = 25 Marks).

(ii) To pass a subject in any Semester a candidate must obtain a minimum of 40% of marks under

each head of the subject and minimum of 40% in the individual subject head.

Courses University

Theory Exam

Mid-Semester

Exam (MSE) +

Continuous

Internal

Assessment

(CIA)

Practical/Viva

Max.

Marks

Passing

Marks

Max.

Marks

Passing

Marks

Max.

Marks

Passing

Marks

M.Sc. 70 28 30 12 50 20

M.Sc. 50 20 25 10 - -

(iii)If a candidate fails in any heads of a subject, he has to appear for that particular head to pass.

(That is, for example if candidate fails in midterm exam of a subject, he has to reappear for

midterm of that subject.)

(iv) The performance of each candidate in all the subjects will be evaluated on 7-point scale in term

of grades as follow:


Grading Scheme Percentage according to Grade

Grade

Points

Qualitative

Meaning

of Grade

1 A+ 90-100 10 Outstanding

2 A 80-89 9 Excellent

3 A- 70-79 8 Very Good

4 B+ 60-69 7 Good

5 B 50-59 6 Average

6 B- 40-49 5 Fair

7 F Less Than 40 0 Fail

8 I Incomplete

Award of Class

The class awarded to a student with his/her M.Sc. (Maths) course is decided by his/her final CPI as per

the following table:

Distinction CPI not less than 7.50

First Class CPI less than 7.50, but not less than 6.50

Second Class CPI less than 6.50, but not less than 5.50

Pass Class CPI less than 5.50

Semester Performance Index (SPI)

The performance of a student in a semester is expressed in terms of the Semester

Performance Index (SPI).

The Semester Performance Index (SPI) is the weighted average of course grade points

obtained by the student in the courses taken in the semester. The weights assigned to

course grade points are the credits carried by the respective courses.

g1 c1 + g2 c2 + ……

SPI =

c1 + c2 +......

Where g1, g2 …… are the grade points obtained by the student in the semester, for

courses carrying credits c1, c2 …… respectively.


Cumulative Performance Index (CPI)

The cumulative performance of a student is expressed in terms of the Cumulative

Performance Index (CPI). This index is defined as the weight age average of course

grade points obtained by the students for all courses taken since his admission to the

program, where the weights are defined in the same way as above.

If a student repeats a course, only the grade points obtained in the latest attempt are

counted towards the Cumulative Performance Index.

(8) For any Semester the maximum marks for the internal and external assessments shall be shown in

the teaching and examination scheme. For the purpose of internal assessment, tests, quizzes,

assignments or any other suitable methods assessment may be used by a department.

(9) Semester Passing Scheme

For each semester examination, a candidate will be considered as pass/clear if

he/she has secured “B-” or above grade in the Internal as well as in the

University Examination separately in each course of theory, practical and project

work.

For each semester examination, a candidate will be considered as fail if he/she has

secured “F” grade in any or all of the subject(s).

If the candidate does not fulfill the subject requirements, he/she will be given I-

grade and the candidate will have to complete the course requirement before the

commencement of the next semester-end examination. If the candidate does not

clear I grade in any subject, he/she will be considered fail – F grade.

Candidate has to clear his / her ‘F’ grade or ‘I’ grade, if any, in the next

examination.

(10) Semester Promotion Scheme

A candidate would be granted admission to the Second Semester irrespective of the

result of First Semester. He / She will be permitted to pursue his/her study of the Second

Semester, provided his/her term for the first semester is granted and applied for the

university examination.

A candidate would be granted admission to the Third Semester if and only if he / she has

cleared all the subjects of First Semester and irrespective of the result of Second Semester.


He/She will be permitted to pursue his/her study of the Third Semester, provided his/her

term for second semester is granted and applied for the university examination.

A candidate would be granted admission to the Fourth Semester if and only if he / she

has cleared all the subjects of Second Semester. He / She will be permitted to pursue

his/her study of the Fourth Semester, provided his/her term for third semester is granted

and applied for the university examination.

The final degree would be awarded to the student on successful completion of all the

Semester.

Promotion Conditions for Promotion

Semester-II Term of semester-I is granted

Semester-III Term of semester-I and semester-II both are granted

Semester-IV Pass in all subjects of semester-I;

and Term of semester-II and semester-III both are granted

(11) Following criteria would be followed for awarding the mark statement of any Semester:

The Grade (Mark) sheet will contain separate grades internal and University examination for

each of compulsory papers (subjects), Practical work, Project Work and overall grade for all

the subjects combined.

It will also contain percentage and the class obtained. The percentage will be calculated on

the basis of cumulative performance index (CPI) obtained by candidate.

CPI will be shown in each semester’s Grade (mark) sheet for each end-semester

examination.

(12) Withdrawal of Exam form:

Student can withdraw exam form with the prior permission of principal.

(13) Punishment Details for Unfair Means:

As per Appendix A

(14) For Physically Challenged / Disabled Candidate for Examination

As per Appendix B


Subject wise Grade and grade points will be calculated based on the Grading

Scheme defined. For example:-

FOR SEMESTER-I

Subjects Total

Marks

(Int+Ext)

Marks

secured

(Int+Ext)

In %

Grade Points

as per

grade

Subje

ct

wise

credit

point

s

Product of

credit

points and

grade

Points

(Total credits)

A 100 75 75.00 A- 8 4 32

B 100 64 64.00 B+ 7 4 28

C 100 82 82.00 A 9 4 36

D 100 54 54.00 B 6 4 24

E 150 73 49.00 B- 5 6 30

F 100 80 80.00 A 9 4 36

G 100 72 72.00 A- 8 4 32

Total 30 218

SPI: 218 / 30 = 7.27, CPI = 7.27

FOR SEMESTER-II

Subject

s

Total

Marks

(Int+Ext)

Marks

secured

(Int+Ext)

In %

Grade Points

as per

grade

Subje

ct

wise

credit

points

Product of

credit

points and

grade

Points

(Total

credits)

A 100 82 82.00 A 9 4 36

B 100 76 76.00 A- 8 4 32

C 100 71 71.00 A- 8 4 32

D 100 65 65.00 B+ 7 4 28

E 150 45 30.00 F 0 6 0

F 100 52 52.00 B 6 4 24

G 100 44 44.00 B- 5 4 20

Total 30 172

SPI: 172 / 30 = 5.73, CPI: 6.50 (As Follow)


Semester Points of Sem (SPI)

Sem-I 7.27

Sem-II 5.73

Total SPI 13.00

CPI (SPI/2) 6.50

In this case, the candidate is failing in one subject i.e. Project-II, and he/she has secured

5.23 SPI for semester II and 7.27 CPI for semester I and II both. Whenever the candidate

clears the subject i.e. Project-II in the next semester examination, the total credits for that

subject will be add to CPI of the candidate.

To calculate the final grade of the course, CPI will be calculated as follows:–

SEMESTER POINTS OF SEM

(SPI)

SEM-I 6.79

SEM-II 5.30

SEM-III 8.33

SEM-IV 5.56

Total SPI 25.98

CPI 6.50

CPI: 6.50

Class of M.Sc. Mathematics Course will be now – ‘First’ as it falls in that range.

(15) Career scope

There are numbers of opportunities in various fields after successfully completing the

program. Mathematics is the basic need of any natural sciences, so this course has

significant role in the society.

Teaching or Research

Tutor or Academician is the foundation of any educational institute, so one can achieve

such profession. There is a need of good Mathematics researcher in Universities and

research institute.

Actuarial science

Actuarial science takes mathematics and statistics and applies them to finance and

insurance. Actuarial science includes a number of interrelating disciplines, including

probability and statistics, finance, and economics.


Computer science

Computer science is the study of the theoretical foundations of information and

computation and their implementation and application in computer systems.

Mathematicians, with the training in logical and precise thinking, are highly prized in

this field.

Biomathematics

Mathematical biology or biomathematics is an interdisciplinary field of study. It models

natural and biological processes using mathematical techniques and tools. Results

have been applied to areas such as cellular neurobiology, epidemic modeling, and

population genetics.

Finance

Finance is a field that studies and addresses the ways in which individuals,

businesses, and organizations raise, allocate, and use monetary resources over time,

taking into account the risks entailed in their projects. Mathematicians can build

models to help explain and predict the behavior of financial markets.

(9) Recognition of lecturer, examiner & evaluator

Experts with the following qualifications and experience shall be eligible to be

recognized to teach, examine and evaluate:

(A) Ph.D. holder in Mathematics or Person having M. Phil degree in Mathematics or

Person having M.Sc. degree in Mathematics and who has cleared NET or SLET

Examination are eligible as full time Assistant Professor.

(B) Person having B+ (Minimum 55%) at M.Sc. in Mathematics and having 5 years

experience of teaching at graduate level are eligible as examiner or evaluator.


BASIC STRUCTURE OF SUBJECTS:

(1) Core/ Principle / Compulsory Courses:

All core courses carry 4 credits in 4 hours per week teaching except project work. Project work

carries 10 credits in 20 hrs per week.

List of Courses

[MT-101] Algebra-I [MT-102] Ordinary Differential Equations

[MT-103] Real Analysis [MT-104] Advanced Linear Algebra

[MT-201] Algebra-II [MT-202] Partial Differential Equations

[MT-203] Special Functions [MT-204] Statistical Methods

[MT-301] Complex Analysis [MT-302] Mathematical Modeling

[MT-303] Research Methodology [MT-401] Number Theory

[MT-402] Functional Analysis [MT-403] Integral Transforms

[MTPW] Project Work

(2) Elective Disciplinary Courses:

All electives carry 4 credits in 4 hours per week teaching. Choice of electives can be done from

the list given below from Group A or Group B:

(a) Group A: Pure Mathematical Group

[MT - 304A] Topology-I

[MT - 404 A] Topology-II

(b) Group B: Applied Mathematical Group

[MT - 304B] Advanced Operations Research

[MT – 404B] Fluid Dynamics

(3) Mathematical Practical’s:

All mathematical practical’s except Mathematical practical-3 carry 4 credits in 8 hours per week

for practical. Mathematical practical-3 carries 6 credits in 12 hours per week for practical.

There are 3 Mathematical Practical’s in the respective semester.

[MT-105] Mathematical Practical-1




(4) Soft Skill Based Courses:

All soft skill based courses carry 4 credits in 2 hours theory and 4 hours of practical.

[SSB-1] Introduction to SCILAB [SSB-2] Introduction to C

[SSB-3] Introduction to PYTHON


KADI SARVA VISHWAVIDYALAYA, GANDHINAGAR

MASTER OF SCIENCE (MATHEMATICS) PROGRAMME

PROPOSED CBCS STRUCTURE FOR M.Sc. (MATHEMATICS)

M.Sc.(MATHEMATICS) SEMESTER - I SYLLABUS W.E.F YEAR: 2017-18

Sr.

No Subject Code Name of Subject

Total

Credit

Teaching

Scheme

(Per

Week)

Examination Scheme

MID External Total

Marks Th. Pr. Th. Th. Pr.

1 MT-101 Algebra-I 4 4 - 30 70 - 100

2 MT-102 Ordinary Differential Equations 4 4 - 30 70 - 100

3 MT-103 Real Analysis 4 4 - 30 70 - 100

4 MT-104 Advanced Linear Algebra 4 4 - 30 70 - 100

5 MT-105 Mathematical Practical-1 4 - 8 - - 100 100

6 SSB-1 Introduction to SCILAB 4 2 4 - 50 50 100

Total 24 18 12 120 330 150 600


M.Sc.(MATHEMATICS) SEMESTER - II SYLLABUS W.E.F YEAR: 2017-18

Sr.

No

Subject

Code Name of Subject

Total

Credit

Teaching

Scheme

(Per Week)

Examination Scheme

MID External Total

Marks

Th. Pr. Th. Th. Pr.

1 MT-201 Algebra-II 4 4 - 30 70 - 100

2 MT-202 Partial Differential Equations 4 4 - 30 70 - 100

3 MT-203 Special Functions 4 4 - 30 70 - 100

4 MT-204 Statistical Methods 4 4 - 30 70 - 100


6 SSB-2 Introduction to C 4 2 4 - 50 50 100

Total 24 18 12 120 330 150 600


M.Sc. (MATHEMATICS) SEMESTER - III SYLLABUS W.E.F YEAR: 2017-18

Sr.

No

Subject

Code Name of Subject

Total

Credit

Teaching

Scheme

(Per Week)

Examination Scheme

MID External Total

Marks

Th. Pr. Th. Th. Pr.

1 MT-301 Complex Analysis 4 4 - 30 70 - 100

2 MT-302 Mathematical Modeling 4 4 - 30 70 - 100

3 MT-303 Research Methodology 4 4 - 30 70 - 100

4 MT-304A /

MT-304B Elective-1 4 4 - 30 70 - 100


6 SSB-3 Introduction to PYTHON 4 2 4 - 50 50 100

Total 26 18 16 120 330 200 650


M.Sc.(MATHEMATICS) SEMESTER - IV SYLLABUS W.E.F YEAR: 2017-18

Sr.

No Subject Code Name of Subject

Total

Credit

Teaching

Scheme

(Per

Week)

Examination Scheme

MID External Total

Marks Th. Pr.

Th. Pr.

1 MT-401 Number Theory 4 4 - 30 70 - 100

2 MT-402 Functional Analysis 4 4 - 30 70 - 100

3 MT-403 Integral Transforms 4 4 - 30 70 - 100

4 MT-404A /

MT-404B Elective-2 4 4 - 30 70 - 100

5 MTPW Project Work 10 - 20 50 - 200 250

Total 26 16 20 170 280 200 650



GANHINAGAR

Syllabus of Master of Science

Mathematics Semester – I


ALGEBRA - I

M.Sc. 1st SEMESTER

SUBJECT CODE: - MT-101

Teaching Scheme (Credits and Hours)

Teaching Scheme Evaluation Scheme

Th.

Hrs /

week

Pr.

Hrs /

week

Course

credit

UE

Hrs

UE

Marks

MSE + CIA

Marks

Total

Marks

4 0 4 3 70 30 100

Learning Objectives:

The objective of this course is

To be familiar with the definition of various types of groups, and a number of examples and

theorems.

To understand and apply the conceptual structure of group theory.

To gain skills in problem solving and critical thinking.

Outline of the Course:

Sr. No. Topic

1 Homomorphism and Isomorphism of Groups

2 Automorphism of Groups

3 Sylow’s Theorem

4 Solvable Groups

Total hours: 48


Detailed Syllabus

Unit

No.

Topics Lectures

(Hours)

Weight

age

1

Homomorphism and Isomorphism of Groups: Definition and basic

examples of Group, Sub group, Normal subgroups, Quotient group and

Cyclic group. Homomorphism of Group, Fundamental Theorem on

homomorphism, Isomorphism of groups, Laws of Isomorphism.

12 25%

2

Automorphism of Groups: Conjugacy relation on a group and its

applications, Class Equation, Automorphism of a group, Inner

Automorphism, External direct product of groups, Inner direct products.

12 25%

3

Sylow’s Theorems: Cauchy’s theorem for abelian groups, Cauchy’s

theorem for finite groups, Sylow’s p-subgroup, Sylow’s Theorems, Finite

Abelian Group.

12 25%

4 Solvable Groups: Subnormal and normal series, Composition series,

Nilpotent group, Solvable groups. 12 25%

Total 48 100%

Instructional Method and Pedagogy

At the start of course, the course delivery pattern, pre-requisite of the subject will be discussed.

Attendance is compulsory in lectures and will be included in the overall internal evaluation.

One internal exam will be conducted as a part of mid semester evaluation.

Assignments based on course content will be given to the student for each unit/topic and will be

evaluated at regular interval and will be included in the overall internal evaluation.

Surprise tests/ Quiz/ Seminar may be conducted and will be included in the overall internal

evaluation.

Student Learning Outcomes:

On successful completion of the course, students should be able to

Differentiate between homomorphism, isomorphism and Automorphism.

Recognize and apply Sylow’s theorem to characterize certain finite groups.

Determine whether a given set is solvable group or not.


Use the skills of proof by contradiction, proof by contraposition, proof of set equality, and proof

using both forms mathematical induction.

Reference Books:

Topics in Algebra by I. N. Herstein, John Wiley and Sons Inc., 2nd Edition.

“Advanced Abstract Algebra” by S.K. Pundir, Krishna Prakashan (P) Ltd., Meerut.

“A First Course in Abstract Algebra” by John B. Fraleigh, Pearson

“Basic Abstract Algebra” by Bhattacharya, Jain and Nagpal, 2nd Edition.

“Algebra” by S. Mcclane and G. Birkhoff, 2nd Edition.

“Basic Algebra” by N. Jacbson, Hind, Pub. Corp, 1984.

“A first course in Abstract Algebra” by John Fraleigh (3rd Edition), Narossa Publishing House,

New Delhi.

“Contemporary Abstract Algebra” Joseph A. Gallian, Narossa Publishing House, New Delhi.


ORDINARY DIFFERENTIAL EQUATIONS

M.Sc. 1st SEMESTER




Th.

Hrs /

week

Pr.

Hrs /

week

Course

credit

UE

Hrs

UE

Marks

MSE + CIA

Marks

Total

Marks

4 0 4 3 70 30 100


Identify an ordinary differential equation and its order

Verify whether a given function is a solution of given ordinary differential equation

Classify ordinary differential equation in linear and non-linear equations

Solve first order linear differential equations

Find solution of separable differential equation and exact differential equation

Find the general solution of second order linear homogeneous equation with constant coefficient


Unit No. Topic

1 Introduction to ODE

2 Non-linear Differential Equations

3 Series Solution

4 Numerical Solution

Total hours: 48


Detailed Syllabus

Unit

No.

Topics Lectures

(Hours)

Weight

age

1

Introduction to ODE: Singular solution and extraneous loci, Review of

Simultaneous Ordinary Differential Equations of First Order, linear

differential equation of second order, Exact linear differential equation of

nth order,

12 25%

2 Non-linear Differential Equations: Non-linear differential equations of

particular forms, Total differential equation. 12 25%

3 Series solution: Ordinary and singular point, Cauchy Euler equation,

Series solution near a regular singular point. 12 25%

4

Numerical Solution: Numerical solution of ordinary differential

equations using Euler’s method, Runge Kutta method (one stage and two

stage)

12 25%

Total 48 100%








evaluation.


On successful completion of the course, students will be able to

Distinguish between linear, non-linear, partial and ordinary differential equations.

Recognize and solve variable separable, homogeneous, exact, linear differential equation.


Find particular solution to initial value problem.

Solve basic application problem described by first order differential equation.

Reference Books:

1. M.D Raisinghania, Ordinary and Partial Differential Equations, S Chand & Co.

2. H.K. Dass, Advanced Engineering Mathematics, S.Chand

3. Differential Equations, Vol II, Bansal, H.L. and Dhami, H.S.

4. Gupta, Malik and Mittal, Differential Equations, Pragati Prakashan

5. Sharma & Gupta, Differential Equations, Krishna Prakashan Media (P) Ltd.


REAL ANALYSIS

M.Sc. 1st SEMESTER




Th.

Hrs /

week

Pr.

Hrs /

week

Course

credit

UE

Hrs

UE

Marks

MSE + CIA

Marks

Total

Marks

4 0 4 3 70 30 100


To provide knowledge of the theory of measurable sets, integration and differentiation of

measurable functions.


Unit No. Topic

1 Lebesgue outer measure

2 Measurable function

3 Integration of non-negative functions

4 Differentiation of measurable functions

Total hours: 48


Detailed Syllabus

Unit

No.

Topics Lectures

(Hours)

Weightage

1

Lebesgue outer measure: Algebra and σ- algebra of sets, σ-

algebra of Borel sets, Lebesgue outer measure on R, Measurable

sets, Lebesgue measure.

12 25%

2

Measurable function: Measurable function, Littlewoods’s

three principles, Egoroff’s theorem, Integral of a simple

function, Lebesgue integral of bounded functions, Comparison

of Reimann and Lebesgue integration, Bounded convergence

theorem.

12 25%

3

Integration of non-negative functions: Integral of non-

negative measurable functions, General Lebesgue (integral),

Fatou’s lemma, Monotone convergence theorem, Lebesgue’s

convergence theorem, Convergence in measure.

12 25%

4

Differentiation of measurable functions: Differentiation of

monotone functions, Functions of bounded variation,

Differentiation of an integral, Absolutely continuous functions

and indefinite integrals.

12 25%

Total 48 100%








evaluation.




Identify and formulate the basic concepts and theorems of sigma algebras, measure and abstract

measure spaces

Synthesize techniques that have been developed in the course to solve particular problems and

explain the basic concepts and main theorems of Lebesgue and different types of convergence

theorems.

Reference Books:

“Real Analysis” by H. L Ryoden, Macmillan Pub. Co 3rd Ed.

“Theory of Functions of a Real Variable”- by I. N. Natansen, Fredrik Pub Co., 1964.

“Measure Theory”- by P. R. Halmos, East and West Press.

“Introduction to Real Variable Theory”- by S.C. Saxena and S. N Shah Prentice Hall of India

1980.

“Real and Complex Analysis”, Rudin, W., 2nd Edition, Tata McGraw- Hill Publishing Co., Ltd

1974.


ADVANCED LINEAR ALGEBRA

M.Sc. 1st SEMESTER




Th. Hrs

/ week

Pr.

Hrs /

week

Course

credit

UE

Hrs

UE

Marks

MSE + CIA

Marks

Total

Marks

4 0 4 3 70 30 100


To provide students with a good understanding of the concepts and methods of linear algebra,

described in detail in the syllabus.

To connect linear algebra to other fields both within and without mathematics.

To develop abstract and critical reasoning by studying logical proofs and the axiomatic method

as applied to linear algebra


Unit No. Topic

- Revision

1 Characteristic roots & Diagonalization of Matrices

2 Triangular canonical forms

3 Decomposition theorem & Jordan canonical forms

4 Rational canonical forms & Determinants

Total hours: 48


Detailed Syllabus

Unit

No.

Topics Lectures

(Hours)

Weight

age

- Revision: Vector spaces, Subspaces, Bases and dimensions, Dual

spaces, Linear transformations.

04 5

1 Characteristic roots & Matrices: The Algebra of Linear

Transformation, Characteristic roots, Characteristic vectors,

Diagonalization of Matrices.

11

25%

2 Triangular canonical forms: Triangular canonical form and its

theorems, Nilpotent linear transformations and its theorems. 11

25%

3 Decomposition theorem & Jordan canonical forms: Trace and

transpose, Decomposition theorem, Jordan canonical forms. 11

25%

4 Rational canonical forms & Determinants: Rational canonical forms,

Determinants. 11

20%

Total 48 100%


At the start of course, the course delivery pattern and the pre-requisite of the subject will be

discussed.






evaluation.



Analyze and evaluate the accuracy of common numerical methods.


Demonstrate understanding of common numerical methods and how they are used to obtain

approximate solutions to otherwise intractable mathematical problems.

Reference Books:

Topics in Algebra”, 2nd edition, by I N Herstein, John Wiley and Sons, Student Edition, New

York (2004).

Lenneth Hoffman, Ray Kunze, Linear Algebra, 2nd edition Prentice Hall of India, New Delhi

(1971).

S. K. Pundir, Advanced Abstract Algebra, Krishna Prakashan Media (P) Ltd. Meerut

P B Bhattacharya, Phani Bhusan Bhattacharya, S K Jain and S R Nagpaul, Advanced linear

algebra, New Age International Ltd Publishers, New Delhi (2008).

Steven Roman, Advanced Linear Algebra, 3rd Edition, Springer (2008).


MATHEMATICAL PRACTICAL-1

M.Sc. 1st SEMESTER




Th.

Hrs /

week

Pr./Tut.

Hrs/

week

Course

credit

UE

Hrs

UE

Marks

MSE + CIA

Marks

Total

Marks

0 8 4 3 100 - 100


To develop skill of students to solve ODE with the help of Scilab.

To provide knowledge to student to find nil potency index of given matrix by using Scilab.

To enhance knowledge of students for finding numerical solutions of ODE using Scilab.


This course contains only problem solving sessions.

Total hours: 96


Detailed Syllabus

Sr.

No.

Topics Practical

(Hours)

Weight

age

1 Examples on Groups and Sub-groups. 4

30%

2 Examples on Group Homomorphism, Isomorphism and

Automorphism. 4

3 Examples on Sylow’s theorems. 5

4 Examples on Solvable group. 4

5 Examples on ODE with initial conditions. 5

30%

6 Examples on ODE using Taylor series method. 5

7 Examples on initial value problems using Euler’s method 5

8 Examples on initial value problems using Modified Euler’s method 5

9 Examples on Numerical solution of ODE using 2nd order RK method. 5

10 Examples on Numerical solution of ODE using 4th order RK method. 5

11 Examples on Vector Space 4

40%

12 Examples on Linear combination, Linear Dependent, Linear span. 4

13 Examples on Linear Transformations. 5

14 Examples on Eigen values of given matrix. 4

15 Examples on Eigen vectors of given matrix. 5

16 Examples on algebraic and Geometric multiplicity. 4

17 Examples Nilpotent Canonical Form. 5

18 Examples on Jordan Canonical Form 5

19 Examples on Rational Canonical Form 5

20 Examples on Minimum Polynomials. 4

21 Examples on Determinants. 4

Total 96 100%




Attendance is compulsory in Lab.

Assignments/Surprise tests/Quiz/Seminar may be conducted.


After finished course the student should be able to use an advanced mathematical tool.

The student should be able to adopt an applied problem and solve it with Scilab

Reference Books:

“Topics in Algebra”, by I. N. Herstein, John Wiley and Sons Inc., 2nd Edition.




“Algebra” by S. Mcclane and G. Birkhoff, 2nd Edition.



New Delhi.

“The Elements of Complex Analysis”, John Duncan, John Wiley and Son Ltd. London (1968).

“Complex Analysis”, L V Ahlfors, ,3rd edition, McGraw Hill, International Ed. (1966).

“Functions of one complex variables”, J B Conway, 2nd edition, Springer Verlag, New York

(1967) [Indian edition: Narosa Publication House, New Delhi (1982)].

“Complex Analysis”, Serge Lang, Addison- Wesley, Publishing Co. (1997).

“The Elements of Complex, Analysis”, B Choudhary, 2nd edition, New Age International Ltd

Publishers, New Delhi (1992).

“Linear Algebra”, Lenneth Hoffman and Ray Kunze, , 2nd edition Prentice Hall of India

New Delhi (1971).


INTRODUCTION TO SCILAB

M.Sc. 1st SEMESTER

SUBJECT CODE: SSB-1



Th.

Hrs /

week

Pr./Tut.

Hrs/

week

Course

credit

UE

Hrs

UE

Marks

MSE +

CIA

Marks

Total

Marks TH. PR. TH. PR.

2 4 4 2 2 50 50 - 100


To develop the knowledge of Import/export data, Create and manipulate variables, Program and

run simple scripts.

Use graphics tools to display data and Use of built-in help features.

To learn the basics of Scilab as a method of solving problems and to see a few solution

techniques you will implement to solve these problems.


Unit No. Topic

1 Introduction

2 Array and Matrices

3 Programming in Scilab

4 Menus and Plots

Total hours: 72 (24 (Th.) + 48 (Pr.))


Detailed Syllabus

Unit

No.

Topics Lectures

(Hours)

Weight

age

1 Introduction: Scilab Environment, Scilab Data types, Scilab Operators,

Scilab Built In Functions.

6 25%

2 Arrays & Matrices: Arithmetic operations with arrays, Polynomial

operation using arrays, Matrices & Sub matrices, Matrix operations,

Working with Polynomials, Working with Linear equations.

6 25%

3 Programming In Scilab: Working with Variables, Assignment

statements, Working with Operators, Input and Output, Flow control/

Branching/ Conditional Statements, Loops, Break and Continue, User

defined functions, Scripts.

6 25%

4 Menus and Plots: Menus and Dialog boxes, Plotting – 2D and 3D plots,

Other graphical primitives, basic statistical functions, Application-

Image processing using Scilab.

6 25%

Total 24 100%

Detailed Practical List

Practical related to the followings

Sr.No. Topics Practical

(Hours)

1 SciLab Environment: SciLab interface, commands & Variables 3Hrs

2 Built in SciLab Functions 3Hrs

3 Using Vectors ,arrays and Metric 4Hrs

4 Algebraic operations on matrices, Transpose of a matrix, Determinants, inverse of

a matrix.

4Hrs

5 Solving System of linear equations 4Hrs

6 Create polynomials of different degrees and hence find its real roots. 4Hrs

7 Programming with Script in Scilab. 6Hrs

8 Problem solving using different looping structure using Scilab 6Hrs


9 Write a script to solve given problem using Scilab 4Hrs

10 User Defined Functions 4Hrs

11 Using plots 4Hrs

12 Image processing using Scilab 2Hrs

Total 48

Instructional Method and Pedagogy (Continuous Internal Assessment (CIA) Scheme)






Solve the problems efficiently using Scilab programming and thus student’s logical skills and

ability will be developed.

Reference Books:

Programming in Scilab 4.1 – Vinu V. Das, New Age International Publishers.

Scilab A Free Software to MATLAB – Er. Hema Ramachandran Achuthsankar S. Nair, S.

Chand.



GANDHINAGAR


Mathematics Semester – II


ALGEBRA – II

M.Sc. 2nd SEMESTER




Th. Hrs

/ week

Pr. Hrs

/ week

Course

credit

UE

Hrs

UE

Marks

MSE + CIA

Marks

Total

Marks

4 0 4 3 70 30 100


The objective of this course is to provide knowledge of Ring theory, Unique factorization domain and

Field extension.


Sr. No. Topic

1 Rings and Fields

2 Euclidean Ring and Unique Factorization Domains.

3 Algebraic Extension of a Field

4 Normal and Separable Extension of Fields

Total hours: 48


Detailed Syllabus

Unit

No.

Topics Lectures

(Hours)

Weight

age

- Revision: Definition of Ring, Subring, Quotient ring, Ring

homomorphism, Integral domain, Ideal, Prime ideal, Maximal ideal,

Polynomial ring

2

1 Ring and Field: Definition and examples of Field, General theorems on

field, Subfield, Necessary and sufficient condition to be a subfield,

Characteristic of a ring, Characteristic of a field, Ordered integral

domain, Principal ideal, Principal ideal ring, Units and Associates,

Embedding of rings

11 25%

2 Euclidean Ring and Unique Factorization Domains: Prime and

Irreducible element, Definition and examples of Euclidean ring,

Properties of Euclidean ring, Unique factorization theorem, Definition of

Unique factorization domain (UFD), Properties of UFD, Polynomial ring

over UFD, Field of quotients of a UFD, Eisenstein’s criterion of

irreducibility.

11 25%

3 Algebraic Extension of a Field: Field, Subfield, Characteristic of a field,

Extension of a field, Simple extension of a field, Algebraic extension of a

field, Algebraically closed field.

12 25%

4 Normal and Separable Extension of Fields: Root fields, Splitting field

or decomposition field, Normal Extension, Separable and inseparable

extensions.

12

25%

Total 48 100%


At the start of course, the course delivery pattern, pre-requisites of the subject will be discussed.







evaluation.



Identify different types of rings and fields

Understand basic knowledge of UFD

Explain the fundamental concepts of field extensions

Reference Books:




“Algebra” by S. Mc Clane and G. Birkhoff, 2nd Edition.



New Delhi.


PARTIAL DIFFERENTIAL EQUATIONS

M.Sc. 2nd SEMESTER




Th. Hrs

/ week

Pr. Hrs

/ week

Course

credit

UE

Hrs

UE

Marks

MSE + CIA

Marks

Total

Marks

4 0 4 3 70 30 100



To understand the concept of second order partial differential equations.

Introductions to boundary value problems.


Sr. No. Topic

1 Introduction

2 Classification of second order partial differential equations

3 Second order partial differential equations with variable coefficients

4 Boundary value problems

Total hours: 48


Detailed Syllabus

Unit

No.

Topics Lectures

(Hours)

Weight

age

1 Introduction: Origin of second order partial differential equations, linear

second order partial differential equations with constant coefficients,

solutions for f(x; y) to be polynomial, exponential, Sin/Cos functions,

general method for homogeneous equations.

12 25%

2 Classification of second ordered partial differential equations:

Classification of second order partial differential equations and canonical

form. Non-linear second order partial differential equations, Solution by

Monge's method, Special case and General case.

12 25%

3 Second order partial differential equations with variable coefficients:

Second order partial differential equations with variable coefficients,

method of changing variables for special type of equations. Separation of

variable Method: Solution of three special equations – Laplace, Wave and

Diffusion equation, Solution of these equations in different coordinate

systems.

12 25%

4 Boundary value problems: Dirichlet boundary value problems,

Neumann boundary value problems, Maximum and minimum principles,

Harnack's theorem, Green's function.

12 25%

Total 48 100%









evaluation.



Recognize some standard types of partial differential equations.

Know the techniques for solving second order partial differential equations.

Identify and solve Dirichlet boundary value problems and Neumann boundary value problems

Reference Books:

Differential Equations, JPH Pub., J.L. Bansal , H.S. Dhami

Elementary Course in Partial Differential Equations, Amarnath, T., Narosa Publ. House, New

Delhi, 1997.

Elements of Partial Differential Equations, Sneddon, I. N., McGraw- Hill Publ. Co., 1957.

Higher Engineering Mathematics, Grewal, B. S. and Grewal, J. S., (36th Edition), Khanna

Publ., New Delhi, 2000.

Advanced Differential Equations, Raisinghania, M. D., S. Chand & Co., 1995.

Partial Differential Equations, Phoolan Prasad and Ravindran, R., Wiley Eastern.


SPECIAL FUNCTIONS

M.Sc. 2nd SEMESTER




Th. Hrs

/ week

Pr. Hrs

/ week

Course

credit

UE

Hrs

UE

Marks

MSE + CIA

Marks

Total

Marks

4 0 4 3 70 30 100


The objective of the course is to introduce some special functions that appear in different areas

of applied mathematics.


Sr. No. Topic

1 Bessel’s Function

2 Legendre’s Function

3 Hypergeometric Function

4 Hermite and Chebyshev Polynomials

Total hours: 48


Detailed Syllabus

Unit

No.

Topics Lectures

(Hours)

Weight age

1 Bessel’s Equations: Definition and general solution of Bessel’s equation, Integration

of Bessel’s equation for n = 0, Definition of Jn(x), Recurrence formulae of Jn(x),

Generating function for Jn(x).

12

25%

2 Legendre’s Equation: Definition and general solution of Legendre’s equation,

Definition of Pn(x) and Qn(x), Generating function of Pn(x), Laplace’s definite

integral for Pn(x), Orthogonal properties of Pn(x), Recurrence formulae for Pn(x),

Beltrami’s result, Christoffel’s expansion, , Christoffel’s summation formula,

Rodrigue’s formula, Legendre’s function of second kind, Recurrence formula for

Qn(x), Relation between Pn(x) and Qn(x), Christoffel’s second summation formula.

12

25%

3 Hypergeometric Function: Definition of Hypergeometric series, Particular cases of

Hypergeometric series, Solution of Hypergeometric equation, Integral formula for

hyper geometric function, Kummer’s theorem, Gauss theorem, Vandermonde’s

theorem, Differentiation of Hypergeometric function, The confluent Hypergeometric

function, Integral representation of the confluent Hypergeometric function,

Differentiation of confluent Hypergeometric function, Continuous Hypergeometric

function.

12

25%

4 Hermite Polynomial: Hermite differential equation, Solution of Hermite equation,

Hermite’s polynomials, Generating function, Other forms of Hermite polynomials,

To find first few Hermite polynomials, Orthogonal properties of Hermite

polynomials, Recurrence formulae for Hermite polynomials. Chebyshev

Polynomials: Chebyshev’s differential equation, Chebyshev polynomials, To prove

that Tn(x) and Un(x) are independent solutions of Chebyshev’s equation, Relation for

Tn(x) and Un(x), To find first few terms of Chebyshev polynomials, Generating

function, Orthogonal properties of Chebyshev polynomials, Recurrence formulae for

Tn(x) and Un(x).

12

25%

Total 48 100%









evaluation.



Apply and understand the application of Bessel’s and Legendre’s functions.

Apply and understand the application of Hypergeometric function.

Apply and understand the application of Hermite and Chebyshev polynomials

Reference Books:

Differential equations with application and historical notes, George F Simmons Tata McGraw –

Hill, Publishing Co. Ltd., New Delhi, 1974.

Special Functions, J. N. Sharma and R. K. Gupta, Krishna Prakashan Media (P) Ltd. Meerut

An Introduction to Ordinary Differential Equations, E.A Coddington., Prentice-Hall of India

Private Ltd., New Delhi, 2001.

Elementary Differential Equations (3rd Edition), W. T Martain and E. Relssner, Addison Wesley

Publishing Company, inc 1995.

Theory of Ordinary Differential Equations, E. A Codington and N Levinson, Tata McGraw hill

Publishing co Ltd., New Delhi, 1999.


STATISTICAL METHODS

M.Sc. 1Ind SEMESTER




Th. Hrs

/ week

Pr. Hrs

/ week

Course

credit

UE

Hrs

UE

Marks

MSE + CIA

Marks

Total

Marks

4 0 4 3 70 30 100



To provide an understanding of statistical concepts like measurements of location and

dispersion, probability, probability distributions, sampling, estimation, hypothesis testing,

regression, correlation analysis, multiple regression and business/economic forecasting.


Unit No. Topic

1 Descriptive Statistics and Correlation

2 Probability & Probability Distribution

3 Statistical Inference

4 Regression Analysis

Total hours: 48


Detailed Syllabus

Unit

No.

Topics Lectures

(Hours)

Weight

age

1 Descriptive Statistics and Correlation: Introduction to Statistics, Applications in

Business & Economics, Data Summarizing Qualitative & Quantitative Data,

Exploratory Data Analysis, The Stem and leaf Display, Cross tabulation & Scatter

Diagrams, Measures of location, Mean, Median, Mode, Percentiles, Quartiles,

Measures of variability, Range, Inter quartile range, Variance, Standard deviation,

Coefficient variation, Measures of distribution shape, Relative location and

detecting outliers, Measures of association between two variables, Covariance,

Correlation.

12 25%

2 Probability & Probability Distribution: Basic probability concepts, Experiment,

Sample space, Events, Exclusive events, Exhaustive events, Independent events,

Dependent events, Methods for assigning probability: Classical method, Relative

frequency method, Subjective method, Events and their Probability, Addition rule

(not to be proved or derived), Conditional probability, Multiplication rule (not to

be proved or derived), Baye’s theorem (statement only not to be proved or

derived), Random variable, Discrete and continuous random variable, Expected

value and variance of random variable, Probability distribution: Binomial

distribution, Poisson distribution, Normal distribution.

12 25%

3 Statistical Inference: Sampling methods, Sampling distribution, Central limit

theorem (statement only), Point and interval estimation, Sampling distribution of

sample mean, Sampling distribution of sample proportion, Hypothesis tests: Null

and alternative hypothesis, Type I & II errors, One and two tails test, Rejection

rule using p- value and critical value approach, Test of hypothesis about

population mean (known), Test of hypothesis about population and proportion,

Sampling distribution and test of hypothesis about difference between two

population means(known and unknown), Sampling distribution and test of

hypothesis about difference between two population and proportions analysis of

variance.

12 25%

4

Regression: Introduction to Regression, Simple linear Regression Model, Least

Square Method, Coefficient of Determination, Correlation Coefficient, Model

12 25%


Assumptions, Residual Analysis, Validating Model Assumptions, Outliers and

Influential Observations, Using the Estimated Regression Equation for Estimation

and Prediction.

Total 48 100%


At the start of course, the course delivery pattern and the pre-requisites of the subject will be

discussed.

Attendance is compulsory in lectures which and will be included in the overall internal

evaluation.





evaluation.


After completing this course the student will learn to perform the following:

How to calculate and apply measures of location and measures of dispersion to grouped and

ungrouped data cases.

How to apply discrete and continuous probability distributions to various business problems.

Perform Test of Hypothesis as well as calculate confidence interval for a population parameter

for single sample and two sample cases.

Learn non-parametric test such as the Chi-Square test for Independence as well as Goodness of

Fit.

Compute and interpret the results of Bivariate and Multivariate Regression and Correlation

Analysis, for forecasting and also perform ANOVA and F-test

Reference Books:

Anderson, Sweeney, Williams, “Statistics for Business and Economics”, 9th edition, Cengage

Publication.

S.P. Gupta, “Statistical Method”, Sultan Chand and Sons, 37th edition (2008).



M.Sc. 2nd SEMESTER




Th. Hrs

/ week

Pr. Hrs

/ week

Course

credit

UE

Hrs

UE

Marks

MSE + CIA

Marks

Total

Marks

0 8 4 3 100 0 100


To develop skills of students to solve PDE with the help of Scilab.

Understand the importance of Laplace, Wave and diffusion equation in different coordinate

systems.

To understand mathematical statistics and perform practical’s on statistical software.



Total hours: 96


Detailed Syllabus


(Hours)

Weight

age

1 Some theorems and examples on Rings 6

25%

2 Some theorems and examples on Field. 4

3 Some theorems and examples on Euclidean ring and unique

factorization domain 4

4 Some theorems and examples on extension of fields 4

5 Examples on solution of first order partial differential equations 4

25%

6 Examples on solution of non-linear second order partial differential

equations 6

7 Examples on second order partial differential equations with variable

coefficients 6

8 Examples on solution of three special equations – Laplace, Wave and

diffusion equation in different coordinate systems 6

9 Examples on Solution of partial differential equations with boundary

value problems 6

10 Examples on Bessel’s function 4

25% 11 Examples on Legendre’s function 4

12 Examples on Hypergeometric function 4

13 Examples on Chebyshev polynomials and Hermite polynomials 6

14 To find measures of central tendency 4

25%

15 To find measures of dispersion 4

16 Examples on Probability. 8

17 Examples on expected value. 4

18 Fitting a binomial distribution 4

19 Fitting a poison distribution 4

20 Fitting a normal distribution 4

Total 96 100%



At the start of course, the course delivery pattern, Pre-requisites of the subject will be discussed.

Attendance is compulsory in lab.

Assignments based on course content will be given to the student for each unit/topic



Understand about the concept of topic and its application on a statistical package.

Developing programs or codes for solving partial differential equation.

Reference Book:

Advanced Abstract Algebra, S.K. Pundir, Krishna Prakashan (P) Ltd., Meerut.

A First Course in Abstract Algebra, John B. Fraleigh, Pearson

Basic Abstract Algebra, Bhattacharya, Jain and Nagpal, 2nd Edition.

Differential Equations, JPH Pub., J.L. Bansal , H.S. Dhami

Amarnath, T., Elementary Course in Partial Differential Equations, Narosa Publ. House, New

Delhi, 1997.

Sneddon, I. N., Elements of Partial Differential Equations, McGraw- Hill Publ. Co., 1957.

Grewal, B. S. and Grewal, J. S., Higher Engineering Mathematics, (36th Edition), Khanna

Publ., New Delhi, 2000.

Raisinghania, M. D. Advanced Differential Equations, S. Chand & Co., 1995.

Special Functions, J. N. Sharma and R. K. Gupta, Krishna Prakashan Media (P) Ltd. Meerut

An introduction to Ordinary Differential Equations, E.A Coddington., Prentice-Hall of India

Private Ltd., New Delhi, 2001.

Elementary Differential Equations (3rd Edition), W. T Martain and E. Relssner, Addison

Wesley Publishing Company, inc 1995.

Anderson, Sweeney, Williams, “Statistics for business and economics “, 9th edition, Cengage

Publication.

S.P. Gupta, Statistical Method, Sultan Chand and Sons 37th edition (2008).


PROGRAMMING IN C

M.Sc. 2nd SEMESTER

SUBJECT CODE: - SSB-2



Th.

Hrs /

week

Pr./Tut

. Hrs/

week

Course

credit

UE

Hrs

UE

Marks

MSE +

CIA

Marks

Total


2 4 4 2 2 50 50 - 100


To develop programming logic and skills for writing programs using C


Unit No. Topic

1 Introduction to 'C' Language

2 Conditional Statements and Loops

3 Arrays and Functions

4 Structures And Unions

Total hours: 72 [24(Theory) + 48 (Practical)]


Detailed Syllabus

Unit

No.

Topics Lectures

(Hours)

Weight age

1 Introduction To 'C' Language: Character Set, Variables and

Identifiers, Built in Data Types, Variable Definition, Arithmetic

Operators and Expressions, Constants and Literals, Simple Assignment

Statement, Basic Input/ Output Statement, Simple 'C' Programs.

6 25%

2 Conditional Statements And Loops: Decision Making Within a

Program, Conditions, Relational Operators, Logical Connectives, If

Statement, If-Else Statement,

Loops: While Loop, Do While, For Loop, Nested Loops, Infinite

Loops, Switch Statement, Structured Programming.

6 25%

3 Arrays: One Dimensional Arrays, Array Manipulation, Searching,

Insertion, Deletion of an Element from an Array, Two Dimensional

Arrays, Strings as Array of Characters,

Functions: Standard Library of C Functions, Prototype of A Function,

Return Type, Function Call, Block Structure, Passing Arguments To A

Function, Call By Reference, Call By Value, Recursive Function.

6 25%

4 Structures And Unions: Structure Variables, Initialization, Structure

Assignment, Unions.

File Processing: Concept of Files, File Opening in Various Modes and

Closing of a File, Reading from a File, Writing onto a File

6 25%

Total 24 100%


Detailed Practical’s List

Practicals related to the following topics:

Sr. No. Topics Practical

(Hours)

1 Write Simple C’ program to learn basic structure, printing and taking

user inputs.

3

2 Write C’ programs to learn identifiers, literals, variables and constants

3 Write C’ programs for basic arithmetic operation between variables

4 Write C’ programs based on different data types 3

5 C Programming Examples on different types of Operators 4

6 Write C’ programs of conditional statements

(if, if…else, switch… case, Ternary operators

4

7 Write C’ programs of Control Loops

(While, do….while, for loop)

4

8 Write C’ programs on Learning single dimensional Arrays 3

9 Write C’ programs on Learning two dimensional Arrays 3

10 Write C’ program to handle Matrix operations 4

11 C Programming Examples on User-define Functions 4

12 Sample C Programming Examples on Strings handling 4

13 C Programming Examples on Mathematical Functions 4

14 Write C Programs based on structure and union 4

15 Write C programs for file handling 4

Total 48


At the start of course, the course delivery pattern and the pre-requisites of the subject will be

discussed.


Assignments/Surprise tests/ Quiz/ Seminar may be conducted.




Solve the problems efficiently using C programming.

Logical ability will be developed.

Reference Books:

Programming in ANSI C - Balaguruswami, TMH

C The Complete Reference - H. Sohildt, TMH

Let us C - Y. Kanetkar, BPB Publication



GANDHINAGAR


Mathematics Semester – III


COMPLEX ANALYSIS

M.Sc. 3rd SEMESTER




Th.

Hrs /

week

Pr.

Hrs /

week

Course

credit

UE

Hrs

UE

Marks

MSE + CIA

Marks

Total

Marks

4 0 4 3 70 30 100


This is an introductory course in complex analysis which provide a working knowledge of the

basic definitions and theorems of the differential and integral calculus of functions of a complex

variable and know the similarities and differences between real and complex analysis.


Sr. No. Topic

1 Revision

2 The Elementary Functions and Complex Integral

3 Some Important Theorems

4 Residue

Total hours: 48


Detailed Syllabus

Unit

No.

Topics Lectures

(Hours)

Weight

age

1

Revision: Complex numbers and its polar and exponential forms, powers and

roots, Regions in the complex plane, Continuity and differentiability of complex

functions, Analytic functions, Cauchy- Riemann equations, Harmonic Functions

of two variables, Infinite series of complex numbers, Power series functions.

12 25%

2

The Elementary Functions and Complex Integral: Exponential,

Trigonometric, Hyperbolic functions, Logarithmic functions and its branches

rectifiable arcs, Complex line integral, Complex contour integral, Cauchy’s

theorem for triangular contours, Anti derivatives.

12 25%

3

Some Important Theorems: Cauchy’s integral formula, Derivative of

analytic functions, Morera’s theorem, Liouville’s theorem, Fundamental

theorem of algebra, Taylor expansions, Laurent expansions.

12 25%

4 Residue: Singularities, Zeros of analytic functions, Poles, Residues, Residue

Theorem, Residue at poles, Evaluations of improper integrals. 12 25%

Total 48 100%








evaluation.



Develop facility with complex numbers and the geometry of the complex plane culminating in

finding the n nth roots of a complex number.

Set up and directly evaluate contour integrals


Identify and classify zeros and singular points of functions.

Reference Books:

John Duncan, The Elements of Complex Analysis, John Wiley and Sons Ltd.London(1968).

L V Ahlfors, Complex Analysis,3rd edition, McGraw Hill, International Ed. (1966).

J B Conway, Functions of One Complex Variables, 2nd edition, Springer Verlag, New York


Serge Lang, Complex Analysis, Addison- Wesley, Publishing Co. (1997).

B Choudhary, The Elements of Complex, Analysis, 2nd edition, New Age International Ltd

Publishers, New Delhi (1992).

Dr. Shailesh S. Patel, Dr. Narendra B. Desai, Complex Analysis and Numerical Techniques,

Volume IV, Atul Prakashan


MATHEMATICAL MODELING

M.Sc. 3rd SEMESTER




Th.

Hrs /

week

Pr.

Hrs /

week

Course

credit

UE

Hrs

UE

Marks

MSE + CIA

Marks

Total

Marks

4 0 4 3 70 30 100


The overall objective of this course is to provide an introduction to the process of mathematical

modeling while giving students an opportunity to develop and construct appropriate models for

various problem situations.

Analyze given models to uncover underlying assumptions.

Investigate particular problems to find out what has already been done toward developing

solutions.


Sr. No. Topic

1 Introduction

2 Two species population models and Epidemic models

3 Biological Models

4 Traffic models

Total hours: 48


Detailed Syllabus

Unit

No.

Topics Lectures

(Hours)

Weight

age

1

Introduction: Introduction to the subject its scope and limitation,

Classification of models, Techniques of mathematical modeling,

Characteristics of mathematical modeling, Dimensional homogeneity, An

arithmetic model of gravity, Simple population growth model, Logistic

population growth model, Decreasing of temperature model, Diffusion

model, Compartment model.

12 25%

2

Two species population models: Prey predator model for population

dynamics, Geometric interpretation and stability of Prey predator model,

Competition model, Epidemic models: SI model, SIS model, ISI Model,

Epidemic model with removal,

12 25%

3

Biological Models: Diffusion of glucose in the blood stream, Genetics

model, Hardy Weinberg law and ratio for genetics, Genetics model for

blood group, Richardson’s model for arms race. Business model, EOQ

model, Even price adjustment model.

12 25%

4

Traffic models: Macroscopic Highway traffic model, Microscopic

Highway traffic model: Linear car following model, Non-Linear car

following model, To find out stopping distance of a car.

12 25%

Total 48 100%









evaluation.



Students learn to use the modeling process to translate problem situations to mathematical

expressions,

Use a variety of mathematical resources and tools to study problem situations.

Use appropriate technology to assist in the problem-solving process.

Take an analytical approach to problems in their future endeavors

Reference Books:

J. N. Kapur, Mathematical Modeling, Wiley Eastern Ltd., 1988.

J. N. Kapur, Mathematical Models in Biology and Medicine, East West press Pvt Ltd., 1992

Braum, Colemem & Drew, Differential Equation Models, Springer Verlag, 1983.

Martin Braun, Differential Equation and their application, Springer Verlag, 1977.

Dym & Lvey, Principles of Mathematics Modeling, Academic Press - 1980.

Haberman, Mathematical Model, Prentice- Hall Inc., 1977.

Bhupendra Sinh and Neenu Aggarwal, Bio Mathematics, Krishna Prakashan.


RESEARCH METHODOLOGY

M.Sc. 3rd SEMESTER




Th.

Hrs /

week

Pr.

Hrs /

week

Course

credit

UE

Hrs

UE

Marks

MSE + CIA

Marks

Total

Marks

4 0 4 3 70 30 100


The purpose of this study was to describe the sample selection, describe the procedure used in

designing the instrument and collecting the data, and provide an explanation of the statistical

procedures used to analyze the data.


Sr. No. Topic

1 Research and types of research

2 Introduction to Statistics

3 Statistical significance

4 Scientific writing

Total hours: 48


Detailed Syllabus

Unit

No.

Topics Lectures

(Hours)

Weight

age

1

What is research / Science and research, Basic and applied research,

Essential steps in research, Characteristic of scientific research, Research

and experimental design.

12 25%

2

Introduction to Statistics: Definition and scope, Data collection,

Classification, Tabulation of data and its graphical and diagrammatic

presentation, Measures of central tendency, Dispersion and standard error,

Probability distributions, Binomial, Poisson and normal distribution.

12 25%

3

Statistical significance: Hypothesis testing, Types of error, Level of

significance, Various test and Chi- square goodness of fit, Simple linear

regression and Correlation analysis.

12 25%

4

Scientific writing, Research proposal, Research paper, Review paper,

Thesis, Conference Report, Book review and project report (any two),

Reference writing, Scientific abbreviations, Preparation and delivery of

scientific presentations, Research report/ Thesis formatting and typing

(computing), Title page, Certificate, Declaration, Acknowledgement, List

of table, Figures, Abbreviations and symbols, Chapters quotations, Table,

Figures, Summary, Appendices, References etc.

12 25%

Total 48 100%








evaluation.




Understand some basic concepts of research and its methodologies

Identify appropriate research topics

Select and define appropriate research problem and parameters

Prepare a project proposal (to undertake a project)

Organize and conduct research (advanced project) in a more appropriate manner

Write a research report and thesis

Write a research proposal (grant)

Reference Books:

How to write and publish a scientific paper by Day, R. A

Guide to write scientific papers by Garson, G. D.

Developing Bioinformatics computer skill, Gibas.

Instrumental methods of analysis, D. A Skoog.


TOPOLOGY-I

M.Sc. 3rd SEMESTER

SUBJECT CODE: - MT-304A



Th.

Hrs /

week

Pr.

Hrs /

week

Course

credit

UE

Hrs

UE

Marks

MSE + CIA

Marks

Total

Marks

4 0 4 3 70 30 100


This course aims to teach the fundamentals of point set topology and constitute an awareness of

need for the topology in Mathematics.


Sr. No. Topic

1 Topological Spaces

2 Continuous Functions

3 Connectedness

4 Compactness

Total hours: 48


Detailed Syllabus

Unit

No.

Topics Lectures

(Hours)

Weight

age

1

Topological Spaces: Topological spaces, Basis and sub basis for a

topology, The order topology, Subspace topology, Closed set, Limit

points.

12 25%

2

Continuous Functions: Continuous functions, Homeomorphisms, The

pasting lemma, Map into products, The metric topology, The sequence

lemma, Uniform limit theorem, The quotient topology.

12 25%

3

Connectedness: Connected spaces, Path connected spaces, Connected

sets in the real line, Components and path components, Locally

connected spaces and path connected spaces.

12 25%

4

Compactness: Compact spaces, Compact sets in the real line, Limit

point compactness, Locally compact spaces, One point compactification.

Note: All results and examples are to be excluded which use the concept

of the product topology of a collection of infinitely many topological

spaces.

12 25%

Total 48 100%








evaluation.




Student will be able to define topology and its construction

Student will be able to distinguish open and closed subset.

Student will be able to construct closure, interior and boundary of a set

Students will be able to identify given space is connected or not.

Students will be able to identify given space is compact or not.

Reference Books:

J. R. Munkres, Topology, Prentice Hall of India, 2nd edition, 2011.

Simmons G. F., Introduction to Topology and Modern Analysis, McGraw-Hill Co., Tokyo,

1963.

Willards, S., General Topology, Addison-Wesley, Reading, 1970.

J. N. Sharma, J. P. Chauhan, Topology, Krishna Prakashan Media (P) Ltd., 43rd edition, 2013.

J. Dugundji, Topology, Prentice- Hall of India, 1975.

C. O Christonson and W. I Voxman, Aspects of Topology, Marcel-Dekker Inc 1977.

J. L. Kelley, D. Van Nostrand, General Topology, 1950.


ADVANCED OPERATIONS RESEARCH

M.Sc. 3rd SEMESTER

SUBJECT CODE: - MT-304B



Th.

Hrs /

week

Pr.

Hrs /

week

Course

credit

UE

Hrs

UE

Marks

MSE + CIA

Marks

Total

Marks

4 0 4 3 70 30 100


The objectives of this course is to

Give idea of basic inventory control models

Develop the knowledge of queuing theory

Formulate and solve different replacement and maintenance models

Study Sequencing techniques.


Sr. No. Topic

1 Deterministic Inventory Control Models

2 Queuing Theory

3 Replacement and Maintenance Models

4 Sequencing Problems

Total hours: 48


Detailed Syllabus

Unit

No.

Topics Lectures

(Hours)

Weight

age

1

Introduction to Operations Research, history, features, approach &

applications of Operations Research, Operations research models,

Introduction to Linear programming problems.

02 5%

Deterministic Inventory Control Models: Functional role of Inventory,

Factors involved in inventory problem analysis, Inventory Model

Building, Single item inventory control models with and without

shortages, Multi item inventory models with constraints, Single item

inventory control models with quantity discounts, Information system for

inventory control.

10 20%

2

Queuing Theory: The structure of a queuing system, Performance

measures of a queuing system, Probability distributions in queuing

system, Classification of queuing models, Single server queuing models,

Multi server queuing models, Finite calling population queuing models,

Multi phase service queuing models, Special purpose queuing models.

12 25%

3

Replacement and Maintenance Models: Types of failure, Replacement

of items whose efficiency deteriorates with time, Replacement of items

that completely fail, Staffing problem, Equipment renewal problem.

12 25%

4

Sequencing Problems: Notations, Terminology and Assumptions,

Processing n jobs through two machines, Processing n jobs through three

machines, Processing n jobs through m machines, Processing m jobs

through two machines.

12 25%

Total 48 100%









evaluation.



Understand the meaning of inventory control as well as various forms and functional role of

inventory.

Calculate EOQ for minimizing for minimizing total inventory cost.

Identify and examine situations that generate queuing problems

Understand various components of a queuing system and description of each of them

Make distinction between several queuing models and derive performance measure for each of

them.

Apply replacement policy for items whose efficiency deteriorates with time.

Derive replacement policy for items whose running cost increasing with time

Use Johnson’s rule of sequencing or scheduling

Solve some specific problems of scheduling jobs on one, two or three machines.

Reference Books:

J. K. Sharma, “Operation Research- Theory and Application”, 4th Edition, Macmillian

Publishers India Ltd.

Hamdy A. Taha, “Operations Research: An Introduction”, 10th Edition, Pearson

Richard Bronson, Schaum's Outline of Theory and Problems of Operations Research



M.Sc. 3rd SEMESTER




Th.

Hrs /

week

Pr./Tut.

Hrs/

week

Course

credit

UE

Hrs

UE

Marks

MSE + CIA

Marks

Total

Marks

0 12 6 3 150 - 150


To understand basic concepts of complex analysis

To understand the concept of Variational problems with fixed and moving boundaries.

Give introductory idea of integral equation

To understand Fredholm integral equation and its solution.

To develop basic concepts of research methodology

To understand basic concepts of topology

Give idea of basic inventory control models

Develop the knowledge of queuing theory

Formulate and solve different replacement and maintenance models

Study sequencing techniques.



Total hours: 144

Detailed Syllabus


Sr.

No.

Topics Practical

(Hours)

Weightage

Compulsory Subjects

1 Examples on complex numbers 18

40% 2 Examples on elementary functions 18

3 Examples on different theorems. 18

4 Examples on residues. 18

5 Examples on hypothesis testing. 12 10%

6 Examples on Mathematical modeling 12 10%

Elective MT-304A

7 A Examples on topological spaces. 12

20% 8 A Examples on continuous functions. 12

9 A Examples on connectedness. 12

10 A Examples on compactness. 12

Elective MT-304B

7 B Examples on deterministic inventory control models 12

20% 8 B Examples on queuing theory 12

9 B Examples on replacement and maintenance models 12

10 B Examples on sequencing problems 12

Total 144 100%






After finished course the student should be able to use an advanced mathematical tool.


Reference Books:

John Duncan, The Elements of Complex Analysis, John Wiley and Sons Ltd.London(1968).

J B Conway, Functions of one complex variables, 2nd edition, Springer Verlag, New York


Dr. Shailesh S. Patel, Dr. Narendra B. Desai, Complex Analysis and Numerical Techniques,

Volume IV, Atul Prakashan

J. N. Kapur, Mathematical Modeling, Wiley Eastern Ltd., 1988.

J. N. Kapur, Mathematical Models in Biology and Medicine, East West press Pvt Ltd., 1992

Braum, Colemem & Drew. Differential Equation Models, Springer Verlag, 1983.

Martin Braun, Differential Equation and their application, Springer Verlag, 1977.

Dym & Lvey, Principles of Mathematics Modeling, Academic Press- 1980.

Haberman, Mathematical Model, Prentice- Hall Inc., 1977.

Bhupendra Sinh and Neenu Agrwal, Bio Mathematics, Krishna Prakashan.

How to write and publish a scientific paper by Day, R. A

Guide to write scientific papers by Garson, G. D.

Developing Bioinformatics computer skill by Gibas.

Instrumental methods of analysis by D. A Skoog.

J. R. Munkres, Topology, Prentice Hall of India, 2nd edition, 2011.

J. N. Sharma, J. P. Chauhan, Topology, Krishna Prakashan Media (P) Ltd., 43rd edition, 2013.

J. K. Sharma, “Operation Research- Theory and Application”, 4th Edition, Macmillian

Publishers India Ltd.


INTRODUCTION TO PYTHON

M.Sc. 3rd SEMESTER

SUBJECT CODE: - SSB-3



Th.

Hrs /

week

Pr./Tut.

Hrs/

week

Course

credit

UE

Hrs

UE

Marks

MSE +

CIA

Marks

Total


2 4 4 2 2 50 50 - 100


Python is a modern language useful for writing compact codes specifically for

programming in the area of Server side Web development, Data Analytics, AI and

scientific computing as well as production tools and game programming. This course

covers the basics and some advanced Python programming to harness its potential for

modern computing requirements.


Unit No. Topic

1 Functions, Scoping and Abstraction

2 Structured Types, Mutability and Higher-order Functions and Exception

3 Classes and object-Oriented Programming and Simple algorithm and Data Structures

4 Testing, Debugging and Advanced Topics

Total hours: 72 (24 (Th.) + 48 (Pr.))


Detailed Syllabus

Unit

No.

Topics Lectures

(Hours)

Weight

age

1 Introduction to python: The basic elements of python, Branching

Program, Control Structures, String and Input, Iteration

Functions, Scoping and Abstraction: Functions and scoping,

specifications, Recursion, Global Variables, Modules, Files, System

Functions and Parameters

6 25%

2 Structured Types, Mutability and Higher-order Functions: Strings,

Tuples, Lists and Dictionaries, List and Mutability, Functions as objects

Exception: Handling Exceptions

6 25%

3 Classes and object-Oriented Programming: Abstract data type and

classes, Inheritance, Encapsulation and Information Hiding

Simple algorithm and Data Structures: Search Algorithms, Sorting

Algorithms, Hash Tables

6 25%

4 Testing, Debugging: Types of testing – Black box and Glass-box,

Debugging

Advanced Topics: Security – Encryption and Decryption, Classical

ciphers, Graphics and GUI Programming – Drawing using Turtle

6 25%

Total 24 100%

Detailed Practical List


Practical related to the followings


(Hours)

1 Develop programs to understand the control structures of python 3

2 Develop programs to learn different types of structures (list, dictionary, tuples) in

python

3

3 Develop programs to learn concept of functions scoping, recursion and list

mutability.

5

4 Develop programs to understand working of exception handling and assertions 5

5 Develop programs for data structure algorithms using python – searching,

sorting and hash tables.

5

6 Develop programs to learn regular expressions using python. 5

7 Learn to plot different types of graphs using PyPlot. 6

8 Implement classical ciphers using python. 6

9 Draw graphics using Turtle. 5

10 Develop programs to learn GUI programming using Tkinter. 5

Total 48







To develop proficiency in creating based applications using the Python Programming Language.

To be able to understand the various data structures available in Python programming

language and apply them in solving computational problems.

To be able to do testing and debugging of code written in Python.

To be able to draw various kinds of plots using PyLab.

To be able to create GUI applications in Python


Reference Books:

John V Guttag. “Introduction to Computation and Programming Using Python”, Prentice Hall

of India

R. Nageswara Rao, “Core Python Programming”, dreamtech

Wesley J. Chun. “Core Python Programming - Second Edition”, Prentice Hall

Michael T. Goodrich, Roberto Tamassia, Michael H. Goldwasser, “Data Structures and

Algorithms in Pyhon”, Wiley

Kenneth A. Lambert, “Fundamentals of Python – First Programs”, CENGAGE Publication

Luke Sneeringer, “Professional Python”, Wrox

“Hacking Secret Ciphers with Python”,

Alweigart,URL:https://inventwithpython.com/hackin

g/chapter



GANDHINAGAR


Mathematics Semester – IV


NUMBER THEORY

M.Sc. 4th SEMESTER




Th.

Hrs /

week

Pr.

Hrs /

week

Course

credit

UE

Hrs

UE

Marks

MSE + CIA

Marks

Total

Marks

4 0 4 3 70 30 100


The purpose of the course is to give a simple account of classical number theory, prepare

students to course in number theory and to demonstrate applications of number theory.


Sr. No. Topic

1 Introduction

2 Linear Congruence and Number Theoretic Functions

3 Euler’s Phi function and Primitive Roots

4 Quadratic Congruence, Fermat’s last theorem and Sum of Squares

Total hours: 48


Detailed Syllabus

Unit

No.

Topics Lectures

(Hours)

Weight

age

1

Introduction: Division algorithm, Greatest common divisor, Euclidean

algorithm, Diophantine equation ax + by = c, Fundamental theorem of

arithmetic.

12 25%

2

Linear Congruence and Number Theoretic Functions: Basic

properties of congruence, Linear congruence, Chinese remainder

theorem, Fermat’s little theorem, Wilson’s theorem, Fermat-Kraitchik

factorization method, Number-Theoretic functions, Mobius inversion

formula, Greatest integer function.

12 25%

3

Euler’s Phi function and Primitive Roots: Euler’s Phi function, Euler’s

theorem, Properties of the Phi function, Order of an integer modulo n,

Primitive roots for primes, Composite numbers having primitive roots,

The theory of indices, Euler’s criterion, Legendre symbol and its

properties.

12 25%

4

Quadratic Congruence, Fermat’s last theorem and Sum of Squares:

Quadratic reciprocity law, Quadratic congruence with composite moduli,

The equation 2 2 2x y z , Fermat’s last theorem, Sum of two squares,

Sum of more than two squares.

12 25%

Total 48 100%








evaluation.




Demonstrate their knowledge of divisibility, prime numbers and the Euclidean algorithm.

Solve linear Diophantine equations and congruence of various types, and use the theory of

congruence in applications.

Prove and apply properties of multiplicative functions such as the Euler phi-function and of

quadratic residues.

Know how to express an integer in sum of two or more squares.

Reference Books:

David M. Burton, Elementary Number Theory (Seventh Edition), McGraw Hill Education

(India) Private Limited, New Delhi

Ivan Nivan, H. S. Zuckermann, H. L. Montgomery, An introduction to the Theory of Numbers,

(5th edition) John Wiley \ & Sons Inc.

Alan Baker, A Concise introduction to the Theory of Numbers, (Cambridge Uni. Press,

Cambridge).

Hari Kishan, Number Theory, Krishna Prakashan Media (P) Ltd., 8th edition, 2014


FUNCTIONAL ANALYSIS

M.Sc. 4th SEMESTER




Th.

Hrs /

week

Pr.

Hrs /

week

Course

credit

UE

Hrs

UE

Marks

MSE + CIA

Marks

Total

Marks

4 0 4 3 70 30 100


To be familiar with Normed Space, Banach Space, Inner Product Space and Hilbert Space.

Study Riesz representation theorem as an application.

To be familiar with operators.


Sr. No. Topic

1 Introduction to Normed and Banach Space

2 Some Theorems on Normed and Banach Space

3 Introduction to Inner Product Space and Hilbert Space

4 Operators

Total hours: 48


Detailed Syllabus

Unit

No.

Topics Lectures

(Hours)

Weight

age

1

Introduction to Normed and Banach Space: Normed linear spaces.

Banach spaces and examples, Quotient space of normed linear spaces

and its completeness, Bounded linear transformations, normed

linear spaces of bounded linear transformations.

12 25%

2

Some Theorems on Normed and Banach Space: Hahn Banach

theorem, dual spaces with examples, second conjugate space. Open

mapping theorem and closed graph theorems.

12 25%

3

Introduction to Inner Product Space and Hilbert Space: Definitions

examples and properties of Inner-Product space and Hilbert spaces,

Orthogonal complements, Orthonormal sets in a Hilbert space,

Bessel’s inequality.

12 25%

4

Operators: Conjugate space, Riesz representation theorem, Operators

on Hilbert space, Adjoint of an operator, Self-adjoint operator, Normal

and unitary operators.

12 25%

Total 48 100%








evaluation.



On successful completion of the course, students will be

Familiar with Normed Space, Banach Space, Inner Product Space and Hilbert Space.

Able to define / give a Norm or Inner product to certain space.

Able to study some applications of Banach Space and Hilbert Space.

Reference Books:

A.H. Siddiqi, Khalil Ahmad, P. Manchanda, Introduction to Functional Analysis with

Applications, Anamaya Publishers, New Delhi, 2007.

Limaye, B.V, Functional Analysis, NewAgeInternationalPubl.Ltd.,NewDelhi,1996.

H.L. Royden, Real Analysis (3rdEdition) Mc.Millan,1998.

Ronald Larsen, Functional Analysis an Introduction, Marcel Dekker, 1973.

Erwin Kreyszig, Introductory Functional Analysis with its applications, John Wiley and Sons,

2007

G. F. Simmons, Introduction to Topology and Modern Analysis, McGraw Hill Book Co., 2004.


INTEGRAL TRANSFORMS

M.Sc. 4th SEMESTER




Th.

Hrs /

week

Pr.

Hrs /

week

Course

credit

UE

Hrs

UE

Marks

MSE + CIA

Marks

Total

Marks

4 0 4 3 70 30 100


The aim of the course is to describe the ideas of Fourier and Laplace Transforms and indicate

their applications in fields such as digital signal processing and differential equations.


Sr. No. Topic

1 Laplace transform

2 Fourier transform

3 Mellin transform

4 Henkel Transform

Total hours: 48


Detailed Syllabus

Unit

No.

Topics Lectures

(Hours)

Weight

age

1

Laplace transform: Definition and its properties, Rules of shifting,

Laplace transforms of derivatives and integrals, Properties of inverse

Laplace transform, Convolution theorem, Complex inversion formula.

12 25%

2

Fourier transform: Definition and properties of Fourier sine, cosine

and complex transforms, Convolution theorem, Inversion theorems,

Fourier transform of derivatives, Application of Fourier transform.

12 25%

3

Mellin transform: Definition and elementary properties, Mellin

transforms of derivative and integrals, Inversion theorem, Convolution

theorem.

12 25%

4

Henkel Transform: Definition and elementary properties, Henkel

transforms of derivative and integrals, Inversion theorem, Convolution

theorem.

12 25%

Total 48 100%








evaluation.




Understand the idea of a principal value integral and the significance of absolute integrability.

Define the Fourier transform and how to compute it for standard examples.

Understand Inversion Theorem and its uses in computing transforms and inverse transforms.

Know applications of Fourier transforms to partial differential equations.

Define Laplace transform and how to compute it for standard examples.

Know applications of Laplace transforms to differential equations..

Reference Books:

“Integral Transforms and Their Applications”, Brian Davies 3rd edition Springer Publication

“Integral Transforms for Engineers” Larry c Andrews, Bhimsen KShivamoggi, published By

SPIER – The international society for optical engineering.

“Applied Integral Transforms” M. Ya. Antimirov, A. A. Kolyshkin, Remi Vaillancourt.

Published by American mathematical Society.

“Integral Transforms”, A. R. Vasishtha, R. K. Gupta, Krishna Prakashan


TOPOLOGY-II

M.Sc. 4th SEMESTER

SUBJECT CODE: - MT-404A



Th.

Hrs /

week

Pr.

Hrs /

week

Course

credit

UE

Hrs

UE

Marks

MSE + CIA

Marks

Total

Marks

4 0 4 3 70 30 100


Topological properties play an important role in most branches of mathematics. The purpose of

the course is to develop further the concepts, which are introduced in Topology I, in more

advanced settings.

This course will provide a firm foundation in topology to enable the student to continue more

advanced study in this area.


Sr. No. Topic

1 Separable Axioms

2 Regular and Normal Spaces

3 Product and Quotient Topology

4 Sequence, Net and Filters in Topological Spaces

Total hours: 48


Detailed Syllabus

Unit

No.

Topics Lectures

(Hours)

Weight

age

1 Separable Axioms: T0-spaces, T1-spaces, T2-spaces, First countable

Space, Second countable space, Lindel of space, Separable spaces. 12 25%

2

Regular and Normal Spaces: Regular spaces, Normal spaces,

Completely normal spaces, Completely regular spaces, One point

compactificatoin.

12 25%

3 Product and Quotient Topology: Weak topologies, Product spaces,

Tychonoff topology, Tychonoff theorem, Quotient topology. 12 25%

4

Sequence, Net and Filters in Topological Spaces: Sequences in

topological spaces, Direct sets, Residual subset, Net, Convergence of a

net in topological space, Ultranet, Subnet, Cluster points of a net, Filters,

Filters generated by collection of sets, Filter base, Ultrafilter and its

characterization, Convergence of filters, Cluster points of a filter.

12 25%

Total 48 100%








evaluation.




Recognize whether or not a topological space is a particular space or not (like First countable,

Second Countable T0 etc.) and be familiar with the basic properties of these spaces and their

proofs;

Understand the Tchonoff topology and quotient topology.

Understand net and filters.

Reference Books:

R. Munkers, Topology, Prentice Hall of India, 2011.

Simmons G. F., Introduction to Topology and Modern Analysis, McGraw-Hill Co., Tokyo,

1963.

J.N. Sharma, J.P. Chauhan, Topology, Krishna Prakashan Media (P) Ltd. 43rd edition, 2013


FLUID DYNAMICS

M.Sc. 4th SEMESTER

SUBJECT CODE: - MT-404B



Th.

Hrs /

week

Pr.

Hrs /

week

Course

credit

UE

Hrs

UE

Marks

MSE + CIA

Marks

Total

Marks

4 0 4 3 70 30 100


Explain the physical properties of a fluid and the consequence of such properties on fluid flow.

Identify the fundamental kinematics of a fluid element.

State the conservation principles of mass, linear momentum, and energy for fluid flow.

Apply the basic applied-mathematical tools that support fluid dynamics.

Create models of inviscid, steady fluid flow over simple profiles and shapes.


Sr. No. Topic

1 Fundamental of Fluid Dynamics

2 Conservation of Momentum

3 Irrotational Motion

4 Motion in Two Dimensions

Total hours: 48


Detailed Syllabus

Unit

No.

Topics Lectures

(Hours)

Weight

age

1

Fundamental of Fluid Dynamics: Basic Concepts, Types of fluid, Fluid

properties, Density, Specific weight, Specific volume, Specific gravity,

Pressure, Viscosity, Temperature, Thermal conductivity, Specific heat,

Surface tension, Vapor pressure, Bulk modulus of Elasticity, Kinematics

of the Flow Field, Lagrangian method, Eulerian method, Relationship

between the Lagrangian and Eulerian method, Velocity of a fluid particle

at a point, Local, convective and material derivatives, Equation of

continuity, Equation of continuity (stream tube concept, Equation of

continuity (Cartesian coordinates), Equation of continuity (spherical polar

coordinates), Equation of continuity (cylindrical polar coordinates),

Equation of continuity (Lagrangian method), Equivalence of the two

forms of the equation of continuity, Velocity potential, Irrotational flow,

Rotational flow, Vorticity, Vorticity vector, Vortex lines, Vortex tube,

Vortex filament, Boundary Surface.

12 25%

2

Conservation of Momentum: Euler’s equation of motion along a stream

line, Equation of motion of an inviscid fluid, Equation of motion of an

inviscid fluid (Cartesian coordinates), Cauchy’s integral, Bernoulli’s

equation (Stream tube method), Conservative field of force, Integration

of Euler’s equation, Helmholtz equations, Symmetrical forms of the

equation of continuity, Spherical symmetry, Cylindrical symmetry,

Impulsive motion of a fluid, Impulsive motion of a fluid (Cartesian

coordinates), Energy equation, Applications of Bernoulli’s Theorem,

Flow over a protuberance in a closed channel, Pitot tube, Venturi tube,

Orifice plate, Weirs.

12 25%

3

Irrotational Motion: General motion of a fluid element, Motion of a

fluid element (Cartesian coordinates), Vorticity, Body forces and surface

forces, Flow and circulation, Stoke’s theorem, Kelvin’s circulation

12 25%


theorem, Connectivity, Cyclic constants, Irrotational motion in multiply-

connected space, Acyclic and Cyclic motion, Green’s theorem,

Deductions from Green’s theorem, Mean value of the velocity potential

over a spherical surface, Motion regarded as due to Sources and Sinks,

Liquid extending to infinity Kelvin’s minimum energy theorem.

4

Motion in Two Dimensions: Stream function (Plane polar coordinates),

Physical interpretation of Stream function, Complex potential and

complex velocity, Uniform flows, Two dimensional Source and Sink,

Strength, Complex potential of a source, Two-dimensional doublet,

Complex potential of a doublet, Images in two-dimension, Image of a

source with regard to a plane, Image of a doublet with regard to a plane,

The circle theorem, Image of a Source with regard to a circle, Image of a

doublet with regard to a circle, Conformal representation, Application to

Fluid Dynamics.

12 25%

Total 48 100%








evaluation.



Classify and exploit fluids based on the physical properties of a fluid.

Compute correctly the kinematical properties of a fluid element.

Apply correctly the conservation principles of mass, linear momentum, and energy to fluid flow

systems with emphasis on aerodynamics.


Reference Books:

Shanti Swarup, Fluid Dynamics, Krishna Prakashan

F. Chorlton, Text Book of fluid dynamics, CBS Publication, Delhi 1985.

R.W. Fox and A.T. Mc Donald, Introduction to fluid mechanics, Wiley, 1985.

E.Krause, Fluid Mechanics with problems and solutions, Springer, 2005.

B.S. Massey, J.W. Smith and A.J.W. Smith, Mechanics of fluids, Taylor and Francis, New

York, 2005.

P.Orlandi, Fluid Flow Phenomena, Kluwer, New York, 2002.

T.Petrila, Basics of Fluid Mechanics and Introduction to Computational Fluids Dynamics,

Springer, Berlin, 2004.


PROJECT WORK

M.Sc. 4th SEMESTER

SUBJECT CODE: - MTPW



Th.

Hrs /

week

Pr.

Hrs /

week

Course

credit

UE

Marks

MSE + CIA

Marks

Total

Marks

0 20 10 200 50 250

Description in Detail

Cognitive skill project to develop student’s cognitive abilities to solve assignment or problem or

problems in a longer time to frame in usual in other courses. Students will learn how to search for

known results and techniques related to the project work. On completion of project work each student

expected to submit a written document describing the results, mathematical developments, background

material bibliographical search etc. Present orally in a seminar setting of the work done in the project

work. The students will meet regularly with the project guide to work out problems that appear and

adjust the goals and time frame accordingly. The project should be carried out individually/jointly are

acceptable only with prior permission of the guide. Cognitive skill work based project carries 8 credits

in at least 16 hours depending on the number of students and number of batches or groups per week.

The project work is to be chosen from the list of following group.

1. Book review

2. Field work project

3. Problem solving work project

4. Foundation of mathematics

5. History of mathematics

6. Mathematics education


Question Paper Format

Kadi Sarva Vishwavidyalaya, Gandhinagar

M.Sc. (Mathematics), End Term Examination, Month- Year

Semester: Total Marks: 70

Subject: Duration: 3hrs

Subject Code: Date: ----------------------------------------------------------------------------------------------------------------------------------- Instructions: (1) All questions are compulsory (2) Figures to the right denote marks. (3) Indicate clearly the options you attempt along with the respective question number.

Section A (35 m)

Section B (35m)

* Note: Section A: Q1 The examiner has to set the questions from unit 1,2 Q2 The examiner has to set the questions from unit 1

Q3 The examiner has to set the questions from unit 2

Section B: Q1 The examiner has to set the questions from unit 3,4 Q2 The examiner has to set the questions from unit 3 Q3 The examiner has to set the questions from unit 4

Q1 Answer the following questions (Short questions) 10

Q2 Attempt any five questions out of seven 15

Q3 Attempt any two questions out of four 10

Q1 Answer the following questions (Short questions) 10

Q2 Attempt any five questions out of seven 15

Q3 Attempt any two questions out of four 10


Question Paper Format

Kadi Sarva Vishwavidyalaya

M.Sc. (Mathematics), End Term Examination, Month-Year Subject:- Duration: 2 Hours Date: Total Marks: 50 Instructions:

1. All questions are compulsory and carry equal weight. 2. Figures to the right indicate full marks. 3. Indicate clearly, the options you attempt along with its respective Question number. 4. Rough work is done in the last page of main supplementary.

Q.1

Section A

Answer the following questions (Short questions)

[7]

Q.2 Answer the following questions.(Any three) [9]

Q.3 Answer the following questions.(Any three) [9]

Section B

Q.1 Answer the following questions(Short questions) [7]

Q.2 Answer the following questions (Any three) [9]

Q 3 Answer the following questions (Any three) [9]

* Note: Section A: Q1 The examiner has to set the questions from unit 1,2 Q2 The examiner has to set the questions from unit 1

Q3 The examiner has to set the questions from unit 2 Section B: Q1 The examiner has to set the questions from unit 3,4

Q2 The examiner has to set the questions from unit 3 Q3 The examiner has to set the questions from unit 4

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