CHRIST (Deemed to University), Bangalore

DEPARTMENT OF MATHEMATICS

School of Business and Management

Syllabus for
Master of Science (Mathematics)
Academic Year  (2024)

 
1 Semester - 2024 - Batch
Course Code
Course
Type
Hours Per
Week
Credits
Marks
MTH111 RESEARCH METHODOLOGY Skill Enhancement Courses 2 2 0
MTH112 STATISTICS Skill Enhancement Courses 2 2 0
MTH131 ABSTRACT ALGEBRA Core Courses 4 4 100
MTH132 REAL ANALYSIS Core Courses 4 4 100
MTH133 ORDINARY DIFFERENTIAL EQUATIONS Core Courses 4 4 100
MTH134 LINEAR ALGEBRA Core Courses 4 4 100
MTH135 DISCRETE MATHEMATICS Core Courses 4 4 100
MTH151 INTRODUCTORY COURSE ON PYTHON PROGRAMMING Core Courses 2 2 50
2 Semester - 2024 - Batch
Course Code
Course
Type
Hours Per
Week
Credits
Marks
MTH211 TEACHING TECHNOLOGY AND SERVICE LEARNING - 2 2 0
MTH212 RESEARCH AND DEVELOPMENT IN MATHEMATICS - 2 1 0
MTH231 GENERAL TOPOLOGY - 4 4 100
MTH232 COMPLEX ANALYSIS - 4 4 100
MTH233 PARTIAL DIFFERENTIAL EQUATIONS - 4 4 100
MTH234 GRAPH THEORY - 4 4 100
MTH235 INTRODUCTORY FLUID MECHANICS - 4 4 100
MTH236 PRINCIPLES OF DATA SCIENCE - 4 4 100
3 Semester - 2023 - Batch
Course Code
Course
Type
Hours Per
Week
Credits
Marks
MTH311 PRACTICE TEACHING Skill Enhancement Courses 2 2 0
MTH312 CALCULUS OF VARIATIONS AND INTEGRAL EQUATIONS Discipline Specific Elective Courses 2 2 0
MTH331 MEASURE THEORY AND LEBESGUE INTEGRATION Core Courses 4 4 100
MTH332 NUMERICAL ANALYSIS Core Courses 4 4 100
MTH333 NUMBER THEORY AND CRYPTOGRAPHY Core Courses 4 4 100
MTH334 MACHINE LEARNING AND ARTIFICIAL INTELLIGENCE Core Courses 4 4 100
MTH341A ADVANCED FLUID MECHANICS Discipline Specific Elective Courses 4 4 100
MTH341B ADVANCED GRAPH THEORY Discipline Specific Elective Courses 4 4 100
MTH341C NUMERICAL LINEAR ALGEBRA Discipline Specific Elective Courses 4 4 100
MTH381 INTERNSHIP Skill Enhancement Courses 4 3 0
4 Semester - 2023 - Batch
Course Code
Course
Type
Hours Per
Week
Credits
Marks
MTH431 CLASSICAL MECHANICS - 4 4 100
MTH432 FUNCTIONAL ANALYSIS - 4 4 100
MTH433 DIFFERENTIAL GEOMETRY - 4 4 100
MTH434 NEURAL NETWORKS AND DEEP LEARNING - 4 4 100
MTH441A COMPUTATIONAL FLUID DYNAMICS - 4 4 100
MTH441B ALGEBRAIC GRAPH THEORY - 4 4 100
MTH441C ADVANCED ANALYSIS - 4 4 100
MTH451 PROGRAMMING FOR DATA SCIENCE USING R - 2 2 50
MTH481 PROJECT - 4 4 100
    

    

Introduction to Program:

The MSc course in Mathematics aims at developing mathematical ability in students with acute and abstract reasoning. The course will enable students to cultivate a mathematician’s habit of thought and reasoning and will enlighten students with mathematical ideas relevant for oneself and for the course itself.

Course Design: Masters in Mathematics is a two year programme spreading over four semesters. In the first two semesters focus is on the basic courses in mathematics such as Algebra, Topology, Analysis and Graph Theory along with the basic applied course ordinary and partial differential equations. In the third and fourth semester focus is on the special courses, elective courses and skill-based courses including Measure Theory and Lebesgue Integration, Functional Analysis, Computational Fluid Dynamics, Advanced Graph Theory, Numerical Analysis  and courses on Data Science . Important feature of the curriculum is that students can specialize in any one of areas (i) Fluid Mechanics, (ii) Graph Theory and (iii) Data Science, with a project on these topics in the fourth semester, which will help the students to pursue research in these topics or grab the opportunities in the industry. To gain proficiency in software skills, four Mathematics Lab papers are introduced, one in each semester. viz. Python Programming for Mathematics, Computational Mathematics using Python, Numerical Methods using Python and Numerical Methods for Boundary Value Problem using Python / Network Science with Python and NetworkX / Programming for Data Science in R / Numerical Linear Algebra using MATLAB respectively. Special importance is given to the skill enhancement courses: Research Methodology, Machine Learning (during 2024-2025 for 2023-2024 batch) and Teaching Technology and Service learning.

Programme Outcome/Programme Learning Goals/Programme Learning Outcome:

PO1: Engage in continuous reflective learning in the context of technology and scientific advancement

PO2: Identify the need and scope of the Interdisciplinary research

PO3: Enhance research culture and uphold the scientific integrity and objectivity

PO4: Understand the professional, ethical and social responsibilities

PO5: Understand the importance and the judicious use of technology for the sustainability of the environment

PO6: Enhance disciplinary competency, employability and leadership skills

Programme Specific Outcome:

PSO1: Attain mastery over pure and applied branches of Mathematics and its applications in multidisciplinary fields

PSO2: Demonstrate problem solving, analytical and logical skills to provide solutions for the scientific requirements

PSO3: Develop critical thinking with scientific temper

PSO4: Communicate the subject effectively and express proficiency in oral and written communications to appreciate innovations in research

PSO5: Understand the importance and judicious use of mathematical software's for the sustainable growth of mankind

PSO6: Enhance the research culture in three areas viz. Graph theory, Fluid Mechanics and Data Science and uphold the research integrity and objectivity

Assesment Pattern

Assessment Pattern

 

Course Code

Title

CIA (Max Marks)

Attendance (Max Marks)

ESE (Max Marks)

MTH131

Abstract Algebra

45

5

50

MTH132

Real Analysis

45

5

50

MTH133

Ordinary Differential Equations

45

5

50

MTH134

Linear Algebra

45

5

50

MTH135

Discrete Mathematics

45

5

50

MTH151

Introductory course on Python Programming 

50

--

--

MTH111

Research Methodology

G

--

--

MTH112

Statistics

G

--

--

 

Holistic Education

G

--

--

MTH231

General Topology

45

5

50

MTH232

Complex Analysis

45

5

50

MTH233

Partial Differential Equations

45

5

50

MTH234

Graph Theory

45

5

50

MTH235

Introductory Fluid Mechanics

45

5

50

MTH236

Principles of Data Science 

45

5

50

MTH211

Teaching Technology and Service Learning

G

--

--

MTH212

Fuzzy Mathematics

G

--

--

MTH213

Research and Development in Mathematics

G

--

--

 

Holistic Education

G

--

--

MTH331

Measure Theory and Lebegue Integration

45

5

50

MTH332

Numerical Analysis

45

5

50

MTH333

Number Theory and Cryptography

45

5

50

MTH334

Machine Learning and Artificial Intelligence

45

5

50

MTH341A

Advanced Fluid Mechanics 

45

5

50

MTH341B

Advanced Graph Theory

45

5

50

MTH351C

Numerical Linear Algebra

45

5

50

MTH381

Internship

G

--

--

MTH311

Practice Teaching

G

--

--

MTH312

Calculus of Variations and Integral Equations

G

--

--

MTH431

Classical Mechanics

45

5

50

MTH432

Functional Analysis

45

5

50

MTH433

Differential Geometry

45

5

50

MTH434

Neural Networks and Deep Learning

45

5

50

MTH441A

Computational Fluid Dynamics

45

5

50

MTH441B

Algebraic Graph theory

45

5

50

MTH441C

Advanced Analysis

45

5

50

MTH451

Programming for Data Science using R

50

--

--

MTH481

Project

100

--

--

Examination And Assesments

EXAMINATION AND ASSESSMENTS (Theory)

Component

Mode of Assessment

Parameters

Points

CIA I

Written Assignment

Reference work

Mastery of the core concepts

 

10

CIA II

Mid-semester Examination

Basic, conceptual and analytical knowledge of the subject

 

25

CIA III

Written Assignment

Class Test

Problem solving skills

Familiarity with the proof techniques

10

Attendance

Attendance

Regularity and Punctuality

05

ESE

 

Basic, conceptual and analytical knowledge of the subject

50

Total

100

 

EXAMINATION AND ASSESSMENTS (Practicals)

The course is evaluated based on continuous internal assessment (CIA). The parameters for evaluation under each component and the mode of assessment are given below:

 

Component

Parameter

Mode of assessment

Maximum points

CIA I

Mastery of  the fundamentals

Lab Assignments

10

CIA II

Familiarity with the commands and execution of them in solving problems. Analytical and Problem Solving skills

Lab Work

Problem Solving

10

CIA III

Conceptual clarity and analytical skills in solving Problems using Mathematical Package / Programming

Lab Exam based on the Lab exercises

25

Attendance

Regularity and Punctuality

Lab attendance

05                  

              =100%:5

    97 – <100% :4

    94 – < 97%  :3

    90 – <94%  :2

    85 – <90%  :1

              <85% :0

Total

50

MTH111 - RESEARCH METHODOLOGY (2024 Batch)

Total Teaching Hours for Semester:30
No of Lecture Hours/Week:2
Max Marks:0
Credits:2

Course Objectives/Course Description

 

Course Description: This course is intended to assist students in acquiring necessary skills on the use of research methodology. Also, the students are exposed to the principles, procedures and techniques of planning and implementing a research project.

Course Objectives: This course will help the learner to

COBJ 1. Know the general research methods.

COBJ 2. Get hands on experience in methods of research that can be employed for research in mathematics.

Course Outcome

CO1: On successful completion of the course, the students should be able to foster a clear understanding about research design that enables students in analyzing and evaluating the published research.

CO2: On successful completion of the course, the students should be able to obtain necessary skills in understanding the mathematics research articles.

CO3: On successful completion of the course, the students should be able to acquire skills in preparing scientific documents using MS Word, Origin, LaTeX and Tikz Library.

Unit-1
Teaching Hours:10
Research Methodology
 

Introduction to research and research methodology, Scientific methods, Choice of research problem, Literature survey and statement of research problem, Reporting of results, Roles and responsibilities of research student and guide.

Unit-2
Teaching Hours:10
Mathematical research methodology
 

Introducing mathematics Journals, reading a Journal article, Ethics in Research and publications, Mathematics writing skills - Standard Notations and Symbols, Using Symbols and Words, organizing a paper, Defining variables, Symbols and notations, Different Citation Styles, IEEE Referencing Style in detail, Tools for checking Grammar and Plagiarism, IPR, Patents/Trademarks and Copyrights, Procedure to apply Patents/Trademarks and Copyrights, The Patent/Trademarks Agent Examinations.

Unit-3
Teaching Hours:10
Type Setting research articles
 

Packages for Documentation - MS Word, LaTeX, Overleaf, Tikz Library, Origin, Pictures and Graphs, producing various types of documents using TeX.

Text Books And Reference Books:

C. R. Kothari and G Garg, Research methodology methods and techniques, 4 th ed., New Age International Publishers, New Delhi, 2019.

Essential Reading / Recommended Reading
  1. E. B. Wilson, An introduction to scientific research, Reprint, Courier Corporation, 2012.R. Ahuja, Research Methods, Rawat Publications, 2001.
  2. G. L. Jain, Research Methodology, Mangal Deep Publications, 2003.
  3. B. C. Nakra and K. K. Chaudhry, Instrumentation, measurement and analysis, TMH Education, 2003.
  4. L. Radhakrishnan, Write Mathematics Right: Principles of Professional Presentation, Exemplified with Humor and Thrills, Alpha Science International, Limited, 2013.
  5. G. Polya, How to solve it: a new aspect of mathematical method. Princeton, N.J.: Princeton University Press, 1957.
  6. R. Hamming, You and your research, available at https://www.cs.virginia.edu/~robins/YouAndYourResearch.html
  7. T. Tao, Advice on writing papers, https://terrytao.wordpress.com/advice-on-writing-papers/
  8. Intellectual property rights- laws and practice, The Institute of Company Secretaries of India, New Delhi, 2018.
  9. https:// ipindia.gov.in (Official website of Intellectual Property India), 2024.
  10. S. Shukla, J.P. George, K. Tiwari and J.V. Kureethara (2022). Intellectual Property Right - Copyright. In: Data Ethics and Challenges. SpringerBriefs in Applied Sciences and Technology(). Springer, Singapore. https://doi.org/10.1007/978-981-19-0752-4_5
Evaluation Pattern

The course is evaluated based on continuous internal assessment (CIA). The parameters for evaluation under each component and the mode of assessment are given below:

Component

Parameter

Mode of assessment

Maximum points

CIA I

Mastery of  the fundamentals

Assignments

10

CIA II

Analytical and Problem Solving skills

Problem Solving (or) Assessment on software skills (if any)

10

CIA III

Conceptual clarity and analytical skills in solving Problems (using Mathematical Package / Programming, if any)

Problem Solving (or) Assessment on software skills (if any)

25

Attendance

Regularity and Punctuality

Attendance

05                  

               =100%:5

     97 – <100% :4

     94 – < 97%  :3

     90 – <94%  :2

     85 – <90%  :1

               <85% :0

Total

50

< marks to be converted to credits >

MTH112 - STATISTICS (2024 Batch)

Total Teaching Hours for Semester:30
No of Lecture Hours/Week:2
Max Marks:0
Credits:2

Course Objectives/Course Description

 

Course Description:

This course aims at teaching the students the idea of discrete and continuous random variables, Probability theory, in-depth treatment of discrete random variables and distributions, with some introduction to continuous random variables and introduction to estimation and hypothesis testing.

Course Objectives:

This course will help the learner to

COBJ1: be proficient in understanding and solving problems on random variables

COBJ2: efficiently solve problems involving probability distributions.

COBJ3: Acquire proficiency in hypothesis testing.

Course Outcome

CO1: On successful completion of the course, the students should be able to understand random variables and probability distributions.

CO2: On successful completion of the course, the students should be able to distinguish between discrete and continuous random variables.

CO3: On successful completion of the course, the students should be able to acquire knowledge in using Binomial distribution, Poisson distribution.

Unit-1
Teaching Hours:10
Random variables and Distributions
 

Introduction to Random variables: Distribution functions, probability mass function and probability density function, Chebyshev's inequality, law of large numbers, central limit theorem, moments and moment generating functions, Binomial, Poisson, Negative binomial, Geometric, Hypergeometric, Discrete uniform. Uniform, Exponential, Gamma, Beta, Weibull, Normal, Lognormal and replacement, t , chi-square, F distribution.

Unit-2
Teaching Hours:10
Theory of estimation
 

Basic concepts of estimation, point estimation, methods of estimation, method of moments, method of interval estimation, Maximum likelihood estimates.

Unit-3
Teaching Hours:10
Testing of hypothesis
 

Null and alternative hypothesis, type I and II errors, power function, t, chi-square, F test method of finding tests, likelihood ratio test, Neyman Pearson lemma, uniformly most powerful tests, some results based on normal population.

Text Books And Reference Books:

Gupta S.C. and Kapoor V.K., Fundamentals of Mathematical Statistics, Sultan Chand and Sons, New Delhi, 2001.

Essential Reading / Recommended Reading
  1. E. Freund John, Mathematical Statistics, 5th Ed., Prentice Hall of India, 2000.
  2. Paul G. Hoel, Introduction to Mathematical Statistics, Wiley, 2000.
  3. Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, Probability and Statistics for Engineers and Scientists, Pearson Prentice Hall, 2006.
  4. D. Wackerly, W. Mendenhall and R. L. Scheaffer, Mathematical Statistics with Applications, Duxburry Press, 2007. 
  5. Neil Weiss, Introductory Statistics, Addison-Wesley, 2002.
  6. S. M. Ross, A first course in probability, Pearson Prentice Hall, 2005.
Evaluation Pattern

SKILL ENHANCEMENT COURSE

The course is evaluated based on continuous internal assessment (CIA). The parameters for evaluation under each component and the mode of assessment are given below:

Component

Mode of Assessment

Parameters

Points

CIA I

Mastery of the fundamentals

Mastery of the core concepts / Problem Solving

10

CIA II

Conceptual Clarity/ Analytical and Problem-Solving skills

Mastery of the core concepts / Problem Solving

10

CIA III

Skill Assessment

Problem solving skills

25

Attendance

Regularity and Punctuality

Attendance

05

      =100%     :5

97= <100% :4

94= <97%   :3

90= <94%   :2

85= <90%   :1

<85% :0

Total

50

MTH131 - ABSTRACT ALGEBRA (2024 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course Description: This course enables students to understand the intricacies of advanced areas in algebra. This includes a study of advanced group theory, Euclidean rings, polynomial rings and Galois theory.

Course objectives​: This course will help the learner to

COBJ1. Enhance the knowledge of advanced-level algebra.

COBJ2. Understand the proof techniques for the theorems on advanced group theory, rings and Galois theory.

Course Outcome

CO1: On successful completion of the course, the students should be able to demonstrate knowledge of conjugates, the Class Equation and Sylow theorems.

CO2: On successful completion of the course, the students should be able to demonstrate knowledge of polynomial rings and associated properties.

CO3: On successful completion of the course, the students should be able to derive and apply Gauss Lemma, Eisenstein criterion for the irreducibility of rationals.

CO4: On successful completion of the course, the students should be able to demonstrate the characteristic of a field and the prime subfield.

CO5: On successful completion of the course, the students should be able to demonstrate factorisation and ideal theory in the polynomial ring; the structure of primitive polynomials; field extensions and characterization of finite normal extensions as splitting fields; the structure and construction of finite fields; radical field extensions; Galois group and Galois theory.

Unit-1
Teaching Hours:15
Advanced Group Theory
 

Automorphisms, Cayley’s theorem, Cauchy’s theorem, permutation groups, symmetric groups, alternating groups, simple groups, conjugate elements and class equations of finite groups, Sylow theorems, direct products, finite Abelian groups, solvable groups.

Unit-2
Teaching Hours:15
Rings
 

Euclidean Ring, polynomial rings, polynomials rings over the rational field, polynomial rings over commutative rings.

Unit-3
Teaching Hours:15
Fields
 

Extension fields, roots of polynomials, construction with straightedge and compass, more about roots.

Unit-4
Teaching Hours:15
Galois theory
 

The elements of Galois theory, solvability by radicals, Galois group over the rationals, finite fields.

Text Books And Reference Books:

I. N. Herstein, Topics in algebra, Second Edition, John Wiley and Sons, 2007.

Essential Reading / Recommended Reading
  1. S. Lang, Algebra, 3rd revised ed., Springer, 2002.  
  2. S. Warner, Modern Algebra, Reprint, Courier Corporation, 2012.
  3. G. Birkhoff and S.M. Lane, A Survey of ModernAlgebra, 3rd ed., A K Peters/CRC Press, 2008.
  4. J. R. Durbin, Modern algebra: An introduction, 6th ed., Wiley, 2008.
  5. N. Jacobson, Basic algebra – I, 2nd ed., Dover Publications, 2009.
  6. J. B. Fraleigh, A first course in abstract algebra, 7th ed., Addison-Wesley Longman, 2002.  
  7. D.M. Dummit and R.M.Foote, Abstract Algebra, 3rd  ed., John Wiley and Sons, 2003.
Evaluation Pattern

Component

Mode of Assessment

Parameters

Points

CIA I

Written Assignment

Reference work 

Mastery of the core concepts 

 

10

CIA II

Mid-semester Examination

Basic, conceptual, and analytical knowledge of the subject

 

25

CIA III

Written Assignment

Class Test

Problem-solving skills

Familiarity with the proof techniques

10

Attendance

Attendance

Regularity and Punctuality

05

End Semester Examination

 

Basic, conceptual, and analytical knowledge of the subject

50

Total

100

MTH132 - REAL ANALYSIS (2024 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course Description: This course will help students to understand the concepts of functions of single and several variables. This course includes such concepts as Riemann-Stieltjes integral, sequences and series of functions, Special Functions, and the Implicit Function Theorem.

 

Course objectives​: This course will help the learner to

COBJ1. develop in a rigorous and self-contained manner the elements of real variable functions

COBJ2. integrate functions of a real variable in the sense of Riemann – Stieltjes

COBJ3. classify sequences and series of functions which are pointwise convergent and uniform Convergent

COBJ4. explore the properties of special functions

COBJ5. understand and apply the functions of several variables.

Course Outcome

CO1: On successful completion of the course, the students should be able to determine the Riemann-Stieltjes integrability of a bounded function.

CO2: On successful completion of the course, the students should be able to recognize the difference between pointwise and uniform convergence of sequence/series of functions.

CO3: On successful completion of the course, the students should be able to illustrate the effect of uniform convergence on the limit function with respect to continuity, differentiability, and integrability.

CO4: On successful completion of the course, the students should be able to analyze and interpret the special functions such as exponential, logarithmic, trigonometric and Gamma functions.

CO5: On successful completion of the course, the students should be able to gain in depth knowledge on functions of several variables and the use of Implicit Function Theorem.

UNIT 1
Teaching Hours:15
The Riemann-Stieltjes Integration
 

Definition and Existence of Riemann-Stieltjes Integral, Linearity Properties of Riemann-Stieltjes Integral, The Riemann-Stieltjes Integral as the Limit of Sums, Integration and Differentiation, Integration of Vector-valued Functions, Rectifiable Curves.

UNIT 2
Teaching Hours:15
Sequences and Series of Functions
 

Pointwise and uniform convergence, Uniform Convergence: Continuity, Integration and Differentiation, Equicontinuous Families of Functions, The Stone-Weierstrass Theorem

UNIT 3
Teaching Hours:15
Some Special Functions
 

Power Series, The Exponential and Logarithmic Functions, The Trigonometric Functions, The Algebraic Completeness of the Complex Field, Fourier Series, The Gamma Function.

UNIT 4
Teaching Hours:15
Functions of Several Variables
 

Linear Transformations, Differentiation, The Contraction Principle, The Inverse Function Theorem, The Implicit Function Theorem.

Text Books And Reference Books:

W. Rudin, Principles of Mathematical Analysis, 3rd ed., New Delhi: McGraw-Hill (India), 2016.

Essential Reading / Recommended Reading
  1. T.M. Apostol, Mathematical Analysis, New Delhi: Narosa, 2004.
  2. E.D. Bloch, The Real Numbers and Real Analysis, New York: Springer, 2011.
  3. J.M. Howie, Real Analysis, London: Springer, 2005.
  4. J. Lewin, Mathematical Analysis, Cambridge: Cambridge University Press, 2003.
  5. F. Morgan, Real Analysis, New York: American Mathematical Society, 2005.
  6. S. Ponnusamy, Foundations of Mathematical Analysis, illustrated ed., Birkhauser, 2012.
  7. S.C. Malik and S. Arora, Mathematics Analysis, 4th ed., New Age International, 2012.
Evaluation Pattern

Component

Mode of Assessment

Parameters

Points

CIA I

Written Assignment

Reference work 

Mastery of the core concepts 

 

10

CIA II

Mid-semester Examination

Basic, conceptual and analytical knowledge of the subject

 

25

CIA III

Written Assignment

Class Test

Problem solving skills

Familiarity with the proof techniques

10

Attendance

Attendance

Regularity and Punctuality

05

ESE

 

Basic, conceptual and analytical knowledge of the subject

50

Total

100

MTH133 - ORDINARY DIFFERENTIAL EQUATIONS (2024 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course Description:

This helps students understand the beauty of the important branch of mathematics, namely, differential equations. This course includes a study of second order linear differential equations, adjoint and self-adjoint equations, existence and uniqueness of solutions, Eigenvalues and Eigenvectors of the equations, power series method for solving differential equations. Non-linear autonomous system of equations.

Course Objectives​:

This course will help the learner to

COBJ1. solve adjoint differential equations and understand the zeros of solutions.
COBJ2. understand the existence and uniqueness of solutions of differential equations and to solve the Strum-Liouville problems.
COBJ3. solve the differential equations by power series method and hypergeometric equations.
COBJ4. understand and solve the non-linear autonomous system of equations.

 

Course Outcome

CO1: On successful completion of the course, the students should be able to understand concept of linear differential equation, Fundamental set Wronskian.

CO2: On successful completion of the course, the students should be able to understand the existence and uniqueness of solutions of differential equations and to solve the Strum-Liouville problems.

CO3: On successful completion of the course, the students should be able to identify ordinary and singular points by Frobenius Method, Hyper geometric differential equation and its polynomial.

CO4: On successful completion of the course, the students should be able to understand the basic concepts of the existence and uniqueness of solutions.

CO5: On successful completion of the course, the students should be able to understand basic concept of solving the linear and non-linear autonomous systems of equations.

CO6: On successful completion of the course, the students should be able to understand the concept of critical point and stability of the system.

Unit-1
Teaching Hours:15
Linear Differential Equations
 

Linear differential equations, fundamental sets of solutions, Wronskian, Liouville’s theorem, adjoint and self-adjoint equations, Lagrange identity, Green’s formula, zeros of solutions, comparison and separation theorems.

Unit-2
Teaching Hours:15
Existence and Uniqueness of solutions
 

Fundamental existence and uniqueness theorem, Dependence of solutions on initial conditions, existence and uniqueness theorem for higher order and system of differential equations, Eigenvalue Problems, Strum-Liouville problems, Orthogonality of eigenfunctions.

Unit-3
Teaching Hours:15
Power series solutions
 

Ordinary and singular points of the differential equations, Classification of singular points, Solution near an ordinary point and a regular singular point by Frobenius method, solution near irregular singular point, Hermite, Laguerre, Chebyshev and Hypergeometric differential equation and its polynomial solutions, standard properties.

Unit-4
Teaching Hours:15
Linear and non-linear Autonomous differential equations
 

Linear system of homogeneous and non-homogeneous equations, Non-linear autonomous sysem of equations, Phase plane, Critical points, Stability, Liapunov direct method, limit cycle and periodic solutions, Bifurcation of plane autonomous systems.

Text Books And Reference Books:
  1. G. F. Simmons, Differential equations with applications and historical notes, Tata McGraw Hill, 2003.
  2. S. J. Farlow, An Introduction to Differential Equations and their Applications, reprint, Dover Publications Inc., 2012.
Essential Reading / Recommended Reading
  1. K. F. Riley, M. P. Hobson and S. J. Bence, Mathematical Methods for Physics and Engineering, Cambridge, 2005.
  2. E. Penney, Differential Equations and Boundary Value Problems, Pearson Education, 2005.
  3. E. A. Coddington, Introduction to ordinary differential equations, Reprint: McGraw Hill, 2006.
  4. M. D. Raisinghania, Advanced Differential Equations, S Chand & Company, 2010.
  5. M. D. Raisinghania, Ordinary and Partial Differential Equations, S Chand Publishing, 2013.
Evaluation Pattern

Component

ModeofAssessment

Parameters

Points

CIAI

Written AssignmentReferencework

Masteryofthe coreconcepts

10

CIAII

Mid-semesterExamination

Basic,conceptualandanalyticalknowledgeof thesubject

25

CIAIII

Written AssignmentClassTest

Problemsolvingskills

Familiaritywiththeprooftechniques

10

Attendance

Attendance

RegularityandPunctuality

05

ESE

 

Basic,conceptualandanalyticalknowledgeof thesubject

50

Total

100

MTH134 - LINEAR ALGEBRA (2024 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course Description: This course aims at introducing elementary notions on linear transformations, canonical forms, rational forms, Jordan forms, inner product space and bilinear forms.

Course Objectives: This course will help the learner to

COBJ 1. have thorough understanding of Linear transformations and its properties.

COBJ 2. understand and apply the elementary canonical forms, rational and Jordan forms in real life problems.

COBJ 3. gain knowledge on Inner product space and the orthogonalisation process.

COBJ 4. explore different types of bilinear forms and their properties.

Course Outcome

CO1: On successful completion of the course, the students should be able to gain in-depth knowledge on Linear transformations.

CO2: On successful completion of the course, the students should be able to demonstrate the elementary canonical forms, rational and Jordan forms.

CO3: On successful completion of the course, the students should be able to apply the inner product space in orthogonality.

CO4: On successful completion of the course, the students should be able to gain familiarity in using bilinear forms.

Unit-1
Teaching Hours:15
Linear Transformations and Determinants
 

Linear transformations, algebra of linear transformations, isomorphism, representation of transformation by matrices, linear functionals, the transpose of a linear transformation, determinants: commutative rings, determinant functions, permutation and the uniqueness of determinants, additional properties of determinants.

Unit-2
Teaching Hours:20
Elementary Canonical Forms, Rational and Jordan Forms
 

Elementary canonical forms: characteristic values, annihilating polynomials, invariant subspaces, simultaneous triangulation and diagonalization, direct sum decomposition, invariant dual sums, the primary decomposition theorem. the rational and Jordan forms: cyclic subspaces and annihilators, cyclic decompositions and the rational form, the Jordan form, computation of invariant factors, semi-simple operators.

Unit-3
Teaching Hours:15
Inner Product Spaces
 

Inner products, Inner product spaces, Linear functionals and adjoints, Unitary operators – Normal operators, Forms on Inner product spaces, Positive forms, Spectral theory, Properties of Normal operators.

Unit-4
Teaching Hours:10
Bilinear Forms
 

Bilinear forms, Symmetric Bilinear forms, Skew-Symmetric Bilinear forms, Groups preserving Bilinear forms.

Text Books And Reference Books:

K. Hoffman and R. Kunze, Linear Algebra, 2nd ed. New Delhi, India: PHI Learning Private Limited, 2011.

Essential Reading / Recommended Reading
  1. S. Lang, Introduction to Linear Algebra, Undergraduate Texts in Mathematics, 2nd ed. New York: Springer, 1997.
  2. P. D. Lax, Linear Algebra and its Applications, 2nd ed., John Wiley and Sons, 2013.
  3. S. Roman, Advanced Linear Algebra, 3rd ed., Springer Science and Business Media, 2013.
  4. G. Strang, Linear Algebra and its Applications, 15th Re-print edition, Cengage Learning, 2014.
  5. S. H. Friedberg, A. J. Insel and L. E. Spence, Linear Algebra, 4th ed., Prentice Hall, 2003.
  6. J. Gilbert, L. Gilbert, Linear Algebra and Matrix Theory, Thomson Brooks/Cole, 2004.
Evaluation Pattern

 

Component

Mode of Assessment

Parameters

Points

CIA I

Written Assignment

Reference work  

Mastery of the core concepts  

10

CIA II

Mid-semester Examination

Basic, conceptual and analytical knowledge of the subject

25

CIA III

Written Assignment

Class Test

Problem solving skills

Familiarity with the proof techniques

10

Attendance

Attendance

Regularity and Punctuality

05

ESE

 

Basic, conceptual and analytical knowledge of the subject

50

Total

100

MTH135 - DISCRETE MATHEMATICS (2024 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course Description: This course will discuss the fundamental concepts and tools in discrete mathematics with emphasis on their applications to mathematical writing, enumeration and recurrence relations.

Course Objectives: The course will help the learner to

COBJ 1. develop logical foundations to understand and create mathematical arguments..

COBJ 2. implement enumeration techniques in a variety of real-life problems.

COBJ 3. analyze the order and efficiency of algorithms.

COBJ 4. communicate the basic and advanced concepts of the topic precisely and effectively.

 

Course Outcome

CO1: On successful completion of the course, the students should be able to demonstrate mathematical logic to write mathematical proofs and solve problems.

CO2: On successful completion of the course, the students should be able to apply the concepts of sets, relations, functions and related discrete structures in practical situations.

CO3: On successful completion of the course, the students should be able to understand and apply basic and advanced counting techniques in real-life problems

CO4: On successful completion of the course, the students should be able to analyse algorithms, determine their efficiency and gain proficiency in preparing efficient algorithms

Unit-1
Teaching Hours:15
Set Theory and Mathematical Logic
 

Sets: Cardinality and countability, recursively defined sets, relations, equivalence relations and equivalence classes, partial and total ordering, representation of relations, closure of relations, functions, bijection, inverse functions.

Logic: Propositions, logical equivalences, rules of inference, predicates, quantifiers, nested quantifiers, arguments, formal proof methods and strategies.

Unit-2
Teaching Hours:15
Enumeration Techniques
 

Fundamental principles, pigeon-hole principle, permutations – with and without repetitions, combinations- with and without repetitions, binomial theorem, binomial coefficients, principle of inclusion and exclusion, derangements, arrangements with forbidden positions, rook polynomial.

Unit-3
Teaching Hours:15
Generating Functions and Recurrence Relations
 

Ordinary and exponential generating functions, recurrence relations, first order linear recurrence relations, higher order linear homogeneous recurrence relations, non-homogeneous recurrence relations, solving recurrence relations using generating functions.

Unit-4
Teaching Hours:15
Analysis of Algorithms
 

Real-valued functions, big-O, big-Omega and big-Theta notations, orders of power functions, orders of polynomial functions, analysis of algorithm efficiency, the sequential search algorithm, exponential and logarithmic orders, binary search algorithm.

Text Books And Reference Books:

  1. R. P. Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction,5th ed., New Delhi: Pearson, 2014.

  2. C L Liu, Introduction to Combinatorial Mathematics, Mcgraw Hill,1968.

  3. S. S. Epp, Discrete Mathematics with Applications, Boston: Cengage Learning, 2019 (for Unit-4).

 

Essential Reading / Recommended Reading

  1. K.H. Rosen, Discrete Mathematics with Applications, New York: McGraw-Hill Higher Education, 2019.

  2. K. Erciyes, Discrete Mathematics and Graph Theory, New York: Springer, 2021.

  3. B. Kolman, R. C. Busby and S. C. Ross, Discrete Mathematical Structures, 6th ed., New Jersey: Pearson Education, 2013.

  4. S. Sridharan and R. Balakrishnan R,  Discrete Mathematics, Bocca Raton: CRC Press, 2020.

  5. J. P. Tremblay and R. Manohar, Discrete Mathematical Structures with Application to Computer Science, Noida: Tata McGraw Hill Education, 2008.

Evaluation Pattern

Component

Mode of Assessment

Parameters

Points

CIA I

Written Assignment

Reference work 

Mastery of the core concepts 

 

10

CIA II

Mid-semester Examination

Basic, conceptual, and analytical knowledge of the subject

 

25

CIA III

Written Assignment

Class Test

Problem-solving skills

Familiarity with the proof techniques

10

Attendance

Attendance

Regularity and Punctuality

05

End Semester Examination

 

Basic, conceptual, and analytical knowledge of the subject

50

Total

100

MTH151 - INTRODUCTORY COURSE ON PYTHON PROGRAMMING (2024 Batch)

Total Teaching Hours for Semester:30
No of Lecture Hours/Week:2
Max Marks:50
Credits:2

Course Objectives/Course Description

 

This course aims at introducing the programming language Python andits uses in solving problems on discrete mathematics and differential equations.

Course Objectives​: This course will help the learner to

COBJ1. gain proficiency in using Python for programming.

COBJ2. acquire skills in usage of suitable functions/packages of Python to solve mathematical problems.

COBJ3. acquaint with Sympy and Numpy packages for solving concepts of calculus, linear algebra and Differential equations.

COBJ4. illustrates use of built-in functions of Pandas and Matplotlib packages for visualizing of data and plotting of graphs.

Course Outcome

CO1: acquire proficiency in using different functions of Python to compute solutions of basic mathematical problems.

CO2: demonstrate the use of Python to solve differential equations along with visualize the solutions.

CO3: be familiar with manipulating and visualizing data using pandas.

Unit-1
Teaching Hours:10
Basic of Python
 

Installation, IDE, variables, built-in functions, input and output, modules and packages, data types and data structures, use of mathematical operators and mathematical functions, programming structures (conditional structure, the for loop, the while loop, nested statements)

Unit-2
Teaching Hours:10
Symbolic and Numeric Computations
 

Use of Sympy package, Symbols, Calculus, Differential Equations, Series expressions, Linear and non-linear equations, List, Tuples and Arrays.

Unit-3
Teaching Hours:10
Data Visualization
 

Standard plots (2D, 3D), Scatter plots, Slope fields, Vector fields, Contour plots, streamlines, Manipulating and data visualizing data with Pandas, Mini Project.

Text Books And Reference Books:

Svein Linge & Hans Petter Langtangen, Programming for computations- Python -A gentle Introduction to Numerical Simulations with Python 3.6, Springer Open, Second Edn. 2020.

Essential Reading / Recommended Reading
  1. B E Shapiro, Scientific Computation: Python Hacking for Math Junkies, Sherwood Forest Books, 2015.
  2. C Hill, Learning Scientific Programming with Python, Cambridge Univesity Press, 2016.
  3. Hans Fangohr, Introduction to Python for Computational Science and Engineering (A beginner’s guide), University of Southampton, 2015 (https://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for Computational-Science-and-Engineering.pdf)
  4. H P Langtangen, A Primer on Scientific Programming with Python, 2nd ed., Springer, 2016.
  5. Jaan Kiusalaas, Numerical methods in engineering with Python 3, Cambridge University press, 2013.
Evaluation Pattern

The course is evaluated based on continuous internal assessments (CIA) and the lab e-record. The parameters for evaluation under each component and the mode of assessment are given below.

Component

Parameter

Mode of Assessment

Maximum

Points

CIA I

Mastery of the concepts

Lab Assignments

20

CIA II

Conceptual clarity and analytical skills 

Lab Exam - I

10

Lab Record

Systematic documentation of the lab sessions.

e-Record work 

07

Attendance

Regularity and Punctuality

Lab attendance

03

95-100% : 3

90-94%   : 2

85-89%   : 1

CIA III

Proficiency in executing the commands appropriately.

Lab Exam - II

10

Total

50

MTH211 - TEACHING TECHNOLOGY AND SERVICE LEARNING (2024 Batch)

Total Teaching Hours for Semester:30
No of Lecture Hours/Week:2
Max Marks:0
Credits:2

Course Objectives/Course Description

 

Course Description: This course is intended to assist the students in acquiring necessary skills for the use of modern technology in teaching. They are exposed to the principles, procedures, techniques of planning and implementing teaching techniques. Through service learning, they will apply the knowledge in real-world situations and serve the community.

Course objectives: This course will help the learner to

COBJ 1: understand the pedagogy of teaching.

COBJ 2: able to use various ICT tools for effective teaching.

COBJ 3: apply the knowledge in real-world situations.

COBJ 4: enhances academic comprehension through experiential learning.

Course Outcome

CO1: On successful completion of the course, the students should be able to gain necessary skills on the use of modern technology in teaching.

CO2: On successful completion of the course, the students should be able to understand the components and techniques of effective teaching.

CO3: On successful completion of the course, the students should be able to obtain necessary skills for pursuing mathematics teaching.

CO4: On successful completion of the course, the students should be able to strengthen personal character and attain the sense of social responsibility through service-learning module.

CO5: On successful completion of the course, the students should be able to contribute to the community by addressing and meeting the community needs.

Unit-1
Teaching Hours:10
Teaching Technology
 

Development of concept of teaching, Teaching skills, Chalk board skills, Teaching practices, Effective teaching, Models of teaching, Teaching aids (Audio-Visual), Teaching aids (projected and non-projected), Communication skills, Feedback in teaching, Teacher’s role and responsibilities, Information technology for teaching.

Unit-2
Teaching Hours:5
Service Learning
 

Concept of difference between social service and service learning, Case study of best practices, understanding contemporary societal issues, Intervention in the community, Assessing need and demand of the chosen community.

Unit-3
Teaching Hours:15
Community Service
 

A minimum of fifteen (15) hours documented service is required during the semester. A student must keep a log of the volunteered time and write the activities of the day and the services performed. A student must write a reflective journal containing an analysis of the learning objectives.

Text Books And Reference Books:
  1. R. Varma, Modern trends in teaching technology, Anmol publications Pvt. Ltd., New Delhi 2003.
  2. U. Rao, Educational teaching, Himalaya Publishing house, New Delhi 2001.
  3. C. B. Kaye, The Complete Guide to Service Learning: Proven, Practical Ways to Engage Students in Civic Responsibility, Academic Curriculum, & Social Action, 2009.
Essential Reading / Recommended Reading
  1. J. Mohanthy, Educational teaching, Deep & Deep Publications Pvt. Ltd., New Delhi 2001.
  2. K. J. Sree and D. B. Rao, Methods of teaching sciences, Discovery publishing house, 2010.
  3. D. Butin, Service-Learning in Theory and Practice-The Future of Community Engagement in Higher Education, Palgrave Macmillan US., 2010.
Evaluation Pattern

The course is evaluated based on continuous internal assessment (CIA). The parameters for evaluation under each component and the mode of assessment are given below:

Component

Parameter

Mode of assessment

Maximum points

CIA I

Mastery of  the fundamentals

Assignments

10

CIA II

Analytical and Problem Solving skills

Assessment on writing lesson plans and teaching technology

 

10

CIA III

Conceptual clarity and analytical skills in solving Problems (using Mathematical Package / Programming, if any)

Project on service learning

25

Attendance

Regularity and Punctuality

Attendance

05                  

               =100%:5

     97 – <100% :4

     94 – < 97%  :3

     90 – <94%  :2

     85 – <90%  :1

               <85% :0

Total

50

 < marks to be converted to credits >

MTH212 - RESEARCH AND DEVELOPMENT IN MATHEMATICS (2024 Batch)

Total Teaching Hours for Semester:30
No of Lecture Hours/Week:2
Max Marks:0
Credits:1

Course Objectives/Course Description

 

 

This course aims at strengthening students by providing exposure to analyze and understand the literature from mathematics journals of their choice and present it among peers.

Course Outcome

CO1: On successful completion of the course, the students should be able to demonstrate their ability to understand the research articles published in journals.

CO2: On successful completion of the course, the students should be able to explain the research articles to the mathematics fraternity.

CO3: On successful completion of the course, the students should be able to handle the queries on the research articles that are presented.

Unit-1
Teaching Hours:30
.
 

.

Text Books And Reference Books:

.

Essential Reading / Recommended Reading

.

Evaluation Pattern

Assessment Criteria:

 

No.

Criteria

Marks

1

Preparation of the material- its content- coverage & quality

50

2

Presentation

        -Contents

        -Confidence

        -Convincing

40

3

Summation Question Answers

10

 

Total

100

MTH231 - GENERAL TOPOLOGY (2024 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course Description: This course deals with the essentials of topological spaces and their properties in terms of continuity, connectedness, compactness etc.

Course objectives​: This course will help the learner to:

COBJ1. understand precise definitions and appropriate examples and counter examples of fundamental concepts in general topology.

COBJ2. acquire knowledge about generalization of the concept of continuity, product topology, metric topology and related results.

COBJ3. appreciate the beauty of deep mathematical results such as Uryzohn’s metrization theorem and understand and apply various proof techniques.

Course Outcome

CO1: On successful completion of the course define topological spaces, give examples and counterexamples on core concepts like open sets, basis and subspaces and other related concepts in topology.

CO2: On successful completion of the course, establish equivalent definitions of continuity and apply the same in proving theorems, analyse product/metric spaces.

CO3: On successful completion of the course, understand the concepts of connectedness and compactness and prove the related theorems. Analyze the proof techniques involved in proving Urysohn Metrization Theorem and Titetze Extension Theorem

Unit-1
Teaching Hours:15
Topological Spaces
 

Elements of topological spaces, basis for a topology, the order topology, the product topology on X x Y, the subspace topology, Closed sets and limit points.

Unit-2
Teaching Hours:15
Continuous Functions
 

Continuous functions, the product topology, metric topology.

Unit-3
Teaching Hours:15
Connectedness and Compactness
 

Connected spaces, connected subspaces of the Real line, components and local connectedness, compact spaces, Compact Subspaces of the Real line, limit point compactness, local compactness.

Unit-4
Teaching Hours:15
Countability and Separation Axioms
 

The countability axioms, the separation axioms, normal spaces, the Urysohn lemma, the Urysohn metrization theorem, Tietze extension theorem.

Text Books And Reference Books:

J.R. Munkres,Topology, Second Edition, Prentice Hall of India, 2007.

Essential Reading / Recommended Reading
  1. G.F.Simmons, Introduction to topology and modern analysis, Tata McGraw Hill Education, 2004.
  2. J. Dugundji, Topology, Prentice Hall of India, 2000.
  3. S. Willard, General topology, Courier-Corporation, 2012.
  4. C. W. Baker, Introduction to topology, Krieger Publishing Company, 2000.
Evaluation Pattern

Component

Mode of Assessment

Parameters

Points

CIA I

Written Assignment

Reference work 

Mastery of the core concepts 

 

10

CIA II

Mid-semester Examination

Basic, conceptual and analytical knowledge of the subject

 

25

CIA III

Written Assignment

Class Test

Problem solving skills

Familiarity with the proof techniques

10

Attendance

Attendance

Regularity and Punctuality

05

ESE

 

Basic, conceptual and analytical knowledge of the subject

50

Total

100

MTH232 - COMPLEX ANALYSIS (2024 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course Description:This course will help students learn about the essentials of complex analysis. This course includes important concepts such as power series, analytic functions, linear transformations, Laurent’s series, Cauchy’s theorem, Cauchy’s integral formula, Cauchy’s residue theorem, argument principle, Schwarz lemma and theorems on meromorphic functions.

Course objectives​: This course will help the learner to

COBJ1. enhance the understanding the advanced concepts in complex Analysis

COBJ2. acquire problem solving skills in complex Analysis.

Course Outcome

CO1: On successful completion of the course, the students should be able to apply the concept and consequences of analyticity and related theorems.

CO2: On successful completion of the course, the students should be able to represent functions as Taylor and Laurent series, classify singularities and poles, find residues, and evaluate complex integrals using the residue theorem and understand conformal mappings.

CO3: On successful completion of the course, the students should be able to understand meromorphic functions and simple theorems concerning them.

CO4: On successful completion of the course, the students should be able to understand advanced theorems on meromorphic functions.

Unit-1
Teaching Hours:15
Analytic functions and singularities
 

Morera’s theorem, Gauss mean value theorem, Cauchy inequality for derivatives, Liouville’s theorem, fundamental theorem of algebra, maximum and minimum modulus theorems.  Taylor’s series, Laurent’s series, zeros of analytical functions, singularities, classification of singularities, characterization of removable singularities and poles.

Unit-2
Teaching Hours:15
Mappings
 

Rational functions, behavior of functions in the neighborhood of an essential singularity, Cauchy’s residue theorem, contour integration problems, Mobius transformation, conformal mappings.

Unit-3
Teaching Hours:15
Meromorphic functions - 1
 

Meromorphic functions and argument principle, Schwarz lemma, Rouche’s theorem, convex functions and their properties, Hadamard 3-circles theorem.

Unit-4
Teaching Hours:15
Meromorphic functions - 2
 

Phragmen-Lindelöf theorem, Riemann mapping theorem, Weierstrass factorization theorem, Harmonic functions, Poisson formula, Poisson integral formula, Jensen’s formula, Poisson-Jensen formula.

Text Books And Reference Books:

J. B. Conway, Functions of One Complex Variable, 2nd ed., New York: Springer, 2000.

Essential Reading / Recommended Reading
  1. M. J. Ablowitz and A. S. Fokas, Complex Variables: Introduction and Applications, Cambridge University Press, 2003.
  2. J. H. Mathews and R. W. Howell, Complex Analysis for Mathematics and Engineering, 6th ed., London: Jones and Bartlett Learning, 2011.
  3. J. W. Brown and R. V. Churchill, Complex Variables and Applications, 7th ed., New York: McGraw-Hill, 2003.
  4. L. S. Hahn and B. Epstein, Classical Complex Analysis, London: Jones and Bartlett Learning, 2011.
  5. D. Wunsch, Complex Variables with Applications, 3rd ed., New York: Pearson Education, 2009.
  6. D. G. Zill and P. D. Shanahan, A First Course in Complex Analysis with Applications, 2nd ed., Boston: Jones and Bartlett Learning, 2010.
  7. E. M. Stein and Rami Sharchi, Complex Analysis, New Jersey: Princeton University Press, 2003.
  8. T. W. Gamblin, Complex Analysis, 1st ed., Springer, 2001.
Evaluation Pattern

Component

Mode of Assessment

Parameters

Points

CIA I

Written Assignment

Reference work 

Mastery of the core concepts 

 

10

CIA II

Mid-semester Examination

Basic, conceptual and analytical knowledge of the subject

 

25

CIA III

Written Assignment

Class Test

Problem solving skills

Familiarity with the proof techniques

10

Attendance

Attendance

Regularity and Punctuality

05

ESE

 

Basic, conceptual and analytical knowledge of the subject

50

Total

100

MTH233 - PARTIAL DIFFERENTIAL EQUATIONS (2024 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course Description: This helps students understand the beauty of the important branch of mathematics, namely, partial differential equations. This course includes a study of first and second order linear and non-linear partial differential equations, existence and uniqueness of solutions to various boundary conditions, classification of second order partial differential equations, wave equation, heat equation, Laplace equations and their solutions by Eigenfunction method and Integral Transform Method.

Course Objectives: This course will help the learner to

COBJ 1. understand the occurrence of partial differential equations in physics and its applications.

COBJ 2. solve partial differential equation of the type heat equation, wave equation and Laplace equations.

COBJ 3. also solving initial boundary value problems.

Course Outcome

CO1: On successful completion of the course, the students should be able to understand the basic concepts and definition of PDE and mathematical models representing stretched string, vibrating membrane, heat conduction in rod.

CO2: On successful completion of the course, the students should be able to demonstrate the canonical form of second order PDE.

CO3: On successful completion of the course, the students should be able to demonstrate initial value boundary problem for homogeneous and non-homogeneous PDE.

CO4: On successful completion of the course, the students should be able to demonstrate on boundary value problem by Dirichlet and Neumann problem.

UNIT 1
Teaching Hours:10
First Order Partial differential equations
 

Formation of PDE, initial value problems (IVP), boundary value problems (BVP) and IBVP, solutions of first, methods of characteristics for first order PDE, linear and quasi, linear, method of characteristics for one-dimensional wave equations and other hyperbolic equations.

UNIT 2
Teaching Hours:15
Second order Partial Differential Equations
 

Origin of second order PDE, Classification of second order PDE, Initial value problems (IVP), Boundary value problems (BVP) and IBVP, Mathematical models representing stretched string, vibrating membrane, heat conduction in solids, second-order equations in two independent variables. Cauchy’s problem for second order PDE, Canonical forms, General solutions.

UNIT 3
Teaching Hours:15
Solutions of Parabolic PDE
 

Occurrence of heat equation in Physics, resolution of boundary value problem, elementary solutions, method of separation of variables, method of eigen function expansion, Integral transforms method, Green’s function.

UNIT 4
Teaching Hours:20
Solutions of Hyperbolic and Elliptic PDE
 

Occurrence of wave and Laplace equations in Physics, Jury problems, elementary solutions of wave and Laplace equations, methods of separation of variables,, the theory of Green’s function for wave and Laplace equations.

Text Books And Reference Books:
  1. C. Constanda, Solution Techniques for Elementary Partial Differential Equations, New York: Chapman & Hall, 2010.
  2. I. N. Sneddon, Elements of Partial Differential Equations, Dover Publications, 2010.
Essential Reading / Recommended Reading
  1. K. F. Riley, M. P. Hobson and S. J. Bence, Mathematical Methods for Physics and Engineering, Cambridge, 2005.
  2. J. D. Logan, Partial Differential Equations, 2nd ed., New York: Springer, 2002.
  3. A. Jeffrey, Applied Partial Differential Equations: An Introduction, California: Academic Press, 2003.
  4. M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, 2nd ed., New York: Springer, 2004.
  5. L. C. Evans, Partial Differential Equations, 2nd ed., American Mathematical Society, 2010.
  6. K. Sankara Rao, Introduction to Partial Differential Equations, 2nd ed., New Delhi: Prentice Hall of India, 2006.
  7. R. C. McOwen, Partial Differential Equations: Methods and Applications, 2nd ed., New York: Pearson Education, 2003.
  8. T. Myint-U and L. Debnath, Linear Partial Differential Equations, Boston: Birkhauser, 2007.
  9. M. D. Raisinghania, Ordinary and Partial Differential Equations, S Chand Publishing, 2013.
Evaluation Pattern

Component

Mode of Assessment

Parameters

Points

CIA I

Written Assignment

Reference work 

Mastery of the core concepts 

 

10

CIA II

Mid-semester Examination

Basic, conceptual and analytical knowledge of the subject

 

25

CIA III

Written Assignment

Class Test

Problem solving skills

Familiarity with the proof techniques

10

Attendance

Attendance

Regularity and Punctuality

05

ESE

 

Basic, conceptual and analytical knowledge of the subject

50

Total

100

MTH234 - GRAPH THEORY (2024 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course Description: This course is an introductory course to the basic concepts of Graph Theory. This includes definition of graphs, vertex degrees, directed graphs, trees, distances, connectivity and paths.

Course objectives: This course will help the learner to

COBJ 1: know the history and development of Graph Theory

COBJ 2: understand all the elementary concepts and results

COBJ 3: learn proof techniques and algorithms in Graph Theory

Course Outcome

CO1: On successful completion of the course, the students should be able to write precise and accurate mathematical definitions of basics concepts in Graph Theory.

CO2: On successful completion of the course, the students should be able to provide appropriate examples and counterexamples to illustrate the basic concepts.

CO3: On successful completion of the course, the students should be able to demonstrate various proof techniques in proving theorems.

CO4: On successful completion of the course, the students should be able to use algorithms to investigate Graph theoretic parameters.

Unit-1
Teaching Hours:15
Introduction to Graphs
 

Graphs as models, degree sequences, classes of graphs, matrices, isomorphism, distances in graphs, connectivity, Eulerian and Hamiltonian graphs, Chinese postman problems, travelling salesman problem and Dijkstra’s algorithm.

Unit-2
Teaching Hours:15
Trees
 

Properties of trees, rooted trees, spanning trees, algorithms on trees- Prufer’s code, Huffmans coding, searching, and sorting algorithms, spanning tree algorithms.

Unit-3
Teaching Hours:15
Planarity
 

Graphical embedding, Euler’s formula, platonic bodies, homeomorphic graphs, Kuratowski’s theorem, geometric duality.

Unit-4
Teaching Hours:15
Graph Invariants
 

Vertex and edge coloring, chromatic polynomial and index, matching, decomposition, independent sets and cliques, vertex and edge covers, clique covers, digraphs and networks.

Text Books And Reference Books:
  1. D. B. West, Introduction to Graph Theory, New Delhi: Prentice-Hall of India, 2011.
  2. R. Diestel, Graph Theory 5th ed., New York: Springer, 2018.
Essential Reading / Recommended Reading
  1. F. Harary, Graph Theory, New Delhi: Narosa, 2001.
  2. N. Deo, Graph Theory with applications to engineering and computer science, Delhi: Prentice Hall of India, 1979.
  3. G. Chartrand, L Lesniak, P Zhang, Graphs and Digraphs, Bocca Raton: CRC Press, 2011.
  4. G. Chartrand and P. Zhang, Introduction to Graph Theory, New Delhi: Tata McGraw Hill, 2006.
  5. J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, North-Holland, Amsterdam, 1976.
  6. J L Gross, J Yellen, M Andersen, Graph Theory and Its Applications, CRC Press, Bocca Raton, 2019.
Evaluation Pattern

Component

Mode of Assessment

Parameters

Points

CIA I

Written Assignment

Test

Mastery of the core concepts 

 

10

CIA II

Mid-semester Examination

Basic, conceptual and analytical knowledge of the subject

 

25

CIA III

Written Assignment

Test

Problem solving skills

Familiarity with the proof techniques

10

Attendance

Attendance

Regularity and Punctuality

05

ESE

 

Basic, conceptual and analytical knowledge of the subject

50

Total

100

MTH235 - INTRODUCTORY FLUID MECHANICS (2024 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Course Description: This course aims at introducing the fundamental aspects of fluid mechanics. They will have a deep insight and general comprehension on tensors, kinematics of fluid, incompressible flow, boundary layer flows and classification of non-Newtonian fluids.

Course Objectives: This course will help the learner to

COBJ1: understand the basic concept of tensors and their representations.
COBJ2: applies the Physics and Mathematics concepts for derivations and interpretations of concepts of fluid mechanics.
COBJ3: familiarize with two- or three-dimensional incompressible flows and viscous flows.
COBJ4: classify Newtonian and non-Newtonian Fluids.

Course Outcome

CO1: On successful completion of the course, the students should be able to confidently manipulate tensor expressions using index notation and use the divergence theorem and the transport theorem.

CO2: On successful completion of the course, the students should be able to understand the basics laws of Fluid mechanics and their physical interpretations.

CO3: On successful completion of the course, the students should be able to comprehend two and three dimension flows incompressible flows.

CO4: On successful completion of the course, the students should be able to appreciate the concepts of the viscous flows, their mathematical modelling and physical interpretations.

Unit-1
Teaching Hours:15
Cartesian tensors and continuum hypothesis
 

Cartesian tensors: Cartesian tensors, basic properties, transpose, symmetric and skew symmetric tensors, gradient, divergence and curl in tensor calculus, integral theorems.  Continuum hypothesis: deformation gradient, strain tensors, infinitesimal strain, compatibility relations, principal strains, material and local time derivatives, transport formulas, streamlines, path lines.

Unit-2
Teaching Hours:20
Stress, Strain and basic physical laws
 

Stress and Rate of Strain: stress components and stress tensor, normal and shear stresses, principal stresses, transformation of the rate of strain and stress, relation between stress and rate of strain.  Fundamental basic physical laws: The equations of conservation of mass, linear momentum (Navier-Stokes equations), and energy.

Unit-3
Teaching Hours:15
One-, two- and three-Dimensional inviscid incompressible Flow
 

Bernoulli equation, applications of Bernoulli equation, Concept of circulation, Kelvin circulation theorem, constancy of circulation, Laplace equations, stream functions in two- and three-dimensional motion. Two dimensional flow: Rectilinear flow, source and sink, the theorem of Blasius.

Unit-4
Teaching Hours:10
Two-dimensional flows of viscous fluid
 

Flow between parallel flat plates, Couette flow, plane Poiseuille flow, the Hagen Poiseuille flow, flow between two concentric rotating cylinders.

Text Books And Reference Books:
  1. S. W. Yuan, Foundations of fluid mechanics, Prentice Hall of India, 2001.
  2. M. D. Raisinghania, Fluid Dynamics, S. Chand and Company Ltd., 2010.
Essential Reading / Recommended Reading
  1. D. S. Chandrasekharaiah and L. Debnath, Continuum mechanics, Academic Press, 2014 (Reprint).
  2. P. K. Kundu, Ira M. Cohen and David R. Dowling, Fluid Mechanics, Fifth Edition, 2010.
  3. G. K. Batchelor, An introduction to fluid mechanics, Cambridge University Press, 2000.
  4. F. Chorlton, Text book of fluid dynamics, New Delhi: CBS Publishers & Distributors, 2004.
  5. J. F. Wendt, J. D. Anderson, G. Degrez and E. Dick, Computational fluid dynamics: An introduction, Springer-Verlag, 1996.
  6. F. M. White, Fluid Mechanics, Tata Mcgraw Hill. 2010.
Evaluation Pattern

Examination and Assessments

Component

Mode of Assessment

Parameters

Points

CIA I

Written Assignment

Reference work 

Mastery of the core concepts 

 

10

CIA II

Mid-semester Examination

Basic, conceptual and analytical knowledge of the subject

 

25

CIA III

Written Assignment

Class Test

Problem solving skills

Familiarity with the proof techniques

10

Attendance

Attendance

Regularity and Punctuality

05

ESE

 

Basic, conceptual and analytical knowledge of the subject

50

Total

100

 

 

MTH236 - PRINCIPLES OF DATA SCIENCE (2024 Batch)

Total Teaching Hours for Semester:60
No of Lecture Hours/Week:4
Max Marks:100
Credits:4

Course Objectives/Course Description

 

Data Science is an interdisciplinary, problem-solving oriented subject that learns to apply scientific techniques to practical problems. This course provides a strong foundation for data science and application area related to information technology and understand the underlying core concepts and emerging technologies in data science.

Course Outcome

CO1: On successful completion of the course, the students should be able to have the managerial understanding of the tools and techniques used in Data Science process.

CO2: On successful completion of the course, the students should be able to analyze data analysis techniques for applications handling large data.

CO3: On successful completion of the course, the students should be able to apply techniques used in Data Science and Machine Learning algorithms to make data driven, real time, day to day organizational decisions.

CO4: On successful completion of the course, the students should be able to present the inference using various Visualization tools.

CO5: On successful completion of the course, the students should be able to learn to think through the ethics surrounding privacy, data sharing and algorithmic decision-making

UNIT 1
Teaching Hours:12
Introduction to Data Science
 

Definition, big data and data science hype, why data science, getting past the hype, the current landscape, who is data scientist? - data science process overview, defining goals, retrieving data, data preparation, data exploration, data modeling, presentation.

UNIT 2
Teaching Hours:12
Big Data
 

Problems when handling large data, General techniques for handling large data, Case study, Steps in big data, Distributing data storage and processing with Frameworks, Case study.

UNIT 3
Teaching Hours:14
Machine Learning
 

Machine learning, modeling process, training model, validating model, predicting new observations, supervised learning algorithms, unsupervised learning algorithms. introduction to deep learning.

UNIT 4
Teaching Hours:12
Data Visualization
 

The characteristic polynomial, eigenvalues and graph parameters, eigenvalues of regular graphs, eigenvalues and expanders, strongly regular graphs.

Unit-5
Teaching Hours:10
Ethics and Recent Trends
 

Data Science Ethics – Doing good data science – Owners of the data - Valuing different aspects of privacy - Getting informed consent - The Five Cs – Diversity – Inclusion – Future Trends.

Text Books And Reference Books:
  1. D. Cielen, A. D. B. Meysman and M. Ali, Introducing Data Science, Manning Publications Co., 1st ed., 2016
  2. G. James, D. Witten, T. Hastie and R. Tibshirani, An Introduction to Statistical Learning: with Applications in R, Springer, 1st ed., 2013
  3. Y. Bengio, A. Courville, Deep Learning, Ian Goodfellow, MIT Press, 1st ed., 2016
  4. D. J. Patil, H. Mason, M. Loukides, O’ Reilly, Ethics and Data Science, 1st ed., 2018
Essential Reading / Recommended Reading
  1. J. Grus, Data Science from Scratch: First Principles with Python, O’Reilly, 1st ed., 2015.
  2. C. O'Neil and R. Schutt, Doing Data Science, Straight Talk from the Frontline, O’Reilly, 1st ed., 2013.
  3. J. Leskovec, A. Rajaraman, J. D. Ullman, Mining of Massive Datasets, Cambridge University Press, 2nd ed., 2014.
Evaluation Pattern

Component

Mode of Assessment

Parameters

Points

CIA I

Written Assignment

Reference work 

Mastery of the core concepts 

 

10

CIA II

Mid-semester Examination

Basic, conceptual and analytical knowledge of the subject

 

25

CIA III

Written Assignment

Class Test

Problem solving skills

Familiarity with the proof techniques

10

Attendance

Attendance

Regularity and Punctuality

05

ESE

 

Basic, conceptual and analytical knowledge of the subject

50

Total

100

MTH311 - PRACTICE TEACHING (2023 Batch)

Total Teaching Hours for Semester:30
No of Lecture Hours/Week:2
Max Marks:0
Credits:2

Course Objectives/Course Description

 

This course is designed to prepare students for real class room situation under the supervision of faculty mentors. It provides experiences in the actual teaching and learning environment.

·       Fifteen hours of teaching assignments for UG classes shall be undertaken by each student during the 3rd and 4th semester.

·       Each student shall be under the supervision of a faculty mentor/guide.

·       The 15 hours may be distributed among 1 or 2 subjects- which shall be a combination of theory and problem based papers.

·       A Structured Plan stating the Topic- Objectives- Methodology and Evaluation shall be prepared in advance by the student for each class session and submitted to the faculty mentor/guide.

·       Faculty guides shall maintain an assessment register for their respective students and record assessment for each session on the given parameters.

Course Outcome

CO1: On successful completion of the course, the students should be able to demonstrate and use various teaching pedagogies.

CO2: On successful completion of the course, the students should be able to develop content and material for classroom teaching.

CO3: On successful completion of the course, the students should be able to manage classroom sessions effectively.

CO4: On successful completion of the course, the students should be able to assist the teachers in internal assessments.

CO5: On successful completion of the course, the students should be able to articulate and communicate in an effective way.

Unit-1
Teaching Hours:15
Practice Teaching
 

This course is designed to prepare students for real class room situation under the supervision of faculty mentors. It provides experiences in the actual teaching and learning environment.

  • Fifteen hours of teaching assignments for UG classes shall be undertaken by each student during the 3rd and 4th semester.
  • Each student shall be under the supervision of a faculty mentor/guide.
  • The 30 hours may be distributed among 1 or 2 subjects- which shall be a combination of theory and problem based papers.
  • A Structured Plan stating the Topic- Objectives- Methodology and Evaluation shall be prepared in advance by the student for each class session and submitted to the faculty mentor/guide.
  • Faculty guides shall maintain an assessment register for their respective students and record assessment for each session on the given parameters.
Text Books And Reference Books:

NA

Essential Reading / Recommended Reading

NA

Evaluation Pattern

 

No.

Criteria

Marks

1

Preparation of the material- its content- coverage & quality

50

2

Presentation

        -Contents

        -Confidence

        -Convincing

40

3

Summation Question Answers

10

 

Total

100

MTH312 - CALCULUS OF VARIATIONS AND INTEGRAL EQUATIONS (2023 Batch)

Total Teaching Hours for Semester:30
No of Lecture Hours/Week:2
Max Marks:0
Credits:2

Course Objectives/Course Description