# Syllabus for Master of Science (Mathematics) Academic Year  (2022)

 1 Semester - 2022 - Batch Course Code Course Type Hours Per Week Credits Marks MTH111 RESEARCH METHODOLOGY Skill Enhancement Course 2 2 0 MTH131 REAL ANALYSIS Core Courses 4 4 100 MTH132 ABSTRACT ALGEBRA 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 PYTHON PROGRAMMING FOR MATHEMATICS Core Courses 3 3 50 2 Semester - 2022 - Batch Course Code Course Type Hours Per Week Credits Marks MTH211 MACHINE LEARNING - 2 2 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 MTH251 COMPUTATIONAL MATHEMATICS USING PYTHON - 3 3 50 3 Semester - 2021 - Batch Course Code Course Type Hours Per Week Credits Marks MTH311 TEACHING TECHNOLOGY AND SERVICE LEARNING Skill Enhancement Course 2 2 0 MTH331 MEASURE THEORY AND LEBESGUE INTEGRATION Core Courses 4 4 100 MTH332 NUMERICAL ANALYSIS Core Courses 4 4 100 MTH333 DIFFERENTIAL GEOMETRY Core Courses 4 4 100 MTH341A BOUNDARY LAYER THEORY Discipline Specific Elective 4 4 100 MTH341B ADVANCED GRAPH THEORY Discipline Specific Elective 4 4 100 MTH341C PRINCIPLES OF DATA SCIENCE Discipline Specific Elective 4 4 100 MTH342A MAGNETOHYDRODYNAMICS Discipline Specific Elective 4 4 100 MTH342B THEORY OF DOMINATION IN GRAPHS Discipline Specific Elective 4 4 100 MTH342C NEURAL NETWORKS AND DEEP LEARNING Discipline Specific Elective 4 4 100 MTH351 NUMERICAL METHODS USING PYTHON Core Courses 3 3 50 MTH381 INTERNSHIP Core Courses 2 2 0 4 Semester - 2021 - Batch Course Code Course Type Hours Per Week Credits Marks MTH431 CLASSICAL MECHANICS - 4 4 100 MTH432 FUNCTIONAL ANALYSIS - 4 4 100 MTH433 ADVANCED LINEAR PROGRAMMING - 4 4 100 MTH441A COMPUTATIONAL FLUID DYNAMICS - 4 4 100 MTH441B ATMOSPHERIC SCIENCE - 4 4 100 MTH441C WAVELET THEORY - 4 4 100 MTH441D MATHEMATICAL MODELLING - 4 4 100 MTH442A ALGEBRAIC GRAPH THEORY - 4 4 100 MTH442B STRUCTURAL GRAPH THEORY - 4 4 100 MTH442C APPLIED GRAPH THEORY - 4 4 100 MTH442D ALGORITHMS FOR NETWORKS AND NUMBER THEORY - 4 4 100 MTH443A REGRESSION ANALYSIS - 4 4 100 MTH443B DESIGN AND ANALYSIS OF ALGORITHMS - 4 4 100 MTH451A NUMERICAL METHODS FOR BOUNDARY VALUE PROBLEM USING PYTHON - 3 3 50 MTH451B NETWORK SCIENCE WITH PYTHON AND NETWORKX - 3 3 50 MTH451C PROGRAMMING FOR DATA SCIENCE IN R - 3 3 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  respectively. Special importance is given to the skill enhancement courses: Research Methodology, Machine Learning 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

 Course Code Title CIA (Max Marks) Attendance (Max Marks) ESE (Max Marks) MTH131 Real Analysis 45 5 50 MTH132 Abstract Algebra 45 5 50 MTH133 Ordinary Differential Equations 45 5 50 MTH134 Linear Algebra 45 5 50 MTH135 Discrete Mathematics 45 5 50 MTH151 Python Programming for Mathematics 50 -- -- MTH111 Research Methodology 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 MTH251 Computational Mathematics using Python 50 -- -- MTH211 Machine Learning G -- -- MTH331 Measure Theory and Lebesgue Integration 45 5 50 MTH332 Numerical Analysis 45 5 50 MTH333 Differential Geometry 45 5 50 MTH341A Boundary Layer Theory 45 5 50 MTH341B Advanced Graph Theory 45 5 50 MTH341C Principles of Data Science 45 5 50 MTH342A Magnetohydrodynamics 45 5 50 MTH342B Theory of Domination in Graphs 45 5 50 MTH342C Neural Networks and Deep Learning 45 5 50 MTH351 Numerical Methods using Python 50 -- -- MTH381 Internship G -- -- MTH311 Teaching Technology and Service learning G -- -- MTH431 Classical Mechanics 60 100 4 MTH432 Functional Analysis 60 100 4 MTH433 Advanced Linear Programming 45 5 50 MTH441A Computational Fluid Dynamics 45 5 50 MTH441B Atmospheric Science 45 5 50 MTH441C Wavelet Theory 45 5 50 MTH441D Mathematical Modelling 45 5 50 MTH442A Algebraic Graph theory 45 5 50 MTH442B Structural Graph Theory 45 5 50 MTH442C Applied Graph Theory 45 5 50 MTH443A Regression Analysis 45 5 50 MTH443B Design and Analysis of Algorithms 45 5 50 MTH451A Numerical Methods for Boundary Value Problem using Python 50 -- -- MTH451B Network Science with Python and NetworkX 50 -- -- MTH451C Programming for Data Science in 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 (2022 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  in Mathematics. Also, the students are exposed to the principles, procedures and techniques of planning and implementing a research project and also to the preparation of a research article. 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: Foster a clear understanding about research design that enables students in analyzing and evaluating the published research.CO2: Obtain necessary skills in understanding the mathematics research articles.CO3: 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, 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.

Unit-3
Teaching Hours:10
Type Setting research articles

Package for Mathematics Typing, MS Word, LaTeX, Overleaf, Tikz Library, Origin, Pictures and Graphs, producing various types of documents using TeX.

Text Books And Reference Books:

.

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
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 >

MTH131 - REAL ANALYSIS (2022 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: Determine the Riemann-Stieltjes integrability of a bounded function.

CO2: Recognize the difference between pointwise and uniform convergence of sequence/series of functions.

CO3: Illustrate the effect of uniform convergence on the limit function with respect to continuity, differentiability, and integrability.

CO4: Analyze and interpret the special functions such as exponential, logarithmic, trigonometric and Gamma functions.

CO5: 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, 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.

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

MTH132 - ABSTRACT ALGEBRA (2022 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: demonstrate knowledge of conjugates, the Class Equation and Sylow theorems.

CO2: demonstrate knowledge of polynomial rings and associated properties.

CO3: derive and apply Gauss Lemma, Eisenstein criterion for irreducibility of rationals.

CO4: demonstrate the characteristic of a field and the prime subfield.

CO5: 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

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.

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

MTH133 - ORDINARY DIFFERENTIAL EQUATIONS (2022 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

COBJ 1: Solve adjoint differential equations and understand the zeros of solutions

COBJ 2:Understand the existence and uniqueness of solutions of differential equations and to solve the Strum-Liouville problems.

COBJ 3:Solve the differential equations by power series method and also hypergeometric equations.

COBJ 4:Understand and solve the non-linear autonomous system of equations.

Course Outcome

CO1: Understand concept of linear differential equation, Fundamental set Wronskian.

CO2: Understand the concept of Liouvilles theorem, Adjoint and Self Adjoint equation, Lagrange's Identity, Green?s formula, Eigenvalue and Eigenfunctions.

CO3: Identify ordinary and singular point by Frobenius Method, Hyper geometric differential equation and its polynomial.

CO4: Understand the basic concepts existence and uniqueness of solutions.

CO5: Understand basic concept of solving the linear and non-linear autonomous systems of equations.

CO6: 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 system 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.
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 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

MTH134 - LINEAR ALGEBRA (2022 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: Gain in-depth knowledge on Linear transformations.

CO2: Demonstrate the elementary canonical forms, rational and Jordan forms.

CO3: Apply the inner product space in orthogonality.

CO4: 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.

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 (2022 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 fundamental concepts and tools in discrete mathematics with emphasis on their applications to mathematical writing, enumeration, recurrence relations and analysis of algorithms.

Course Objectives: The course will help the learner to

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

COBJ 2: demonstrate logical reasoning to solve a variety of practical problems.

COBJ 3: implement enumeration techniques in a variety of real-life problems.

COBJ 4: develop efficient algorithms and determine their efficiency.

COBJ 5: communicate the basic and advanced concepts of the subject precisely and effectively.

Course Outcome

CO1: Perform correct mathematical arguments.

CO2: comprehend the fundamental and advanced concepts of relations, functions and discrete structures.

CO3: demonstrate enumeration skills in various real-life problems.

CO4: formulate and solve problems using generating functions and recurrence relations.

CO5: apply mathematical logic to write mathematical proofs and solve problems.

Unit-1
Teaching Hours:15
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, normal forms, rules of inference, predicates, quantifiers, nested quantifiers, arguments, formal proof methods and strategies.

Unit-2
Teaching Hours:15
Enumeration Relations and Functions

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

Unit-3
Teaching Hours:15
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. S. S. Epp, Discrete Mathematics with Applications, Boston:Cengage Learning, 2019 (for Unit-4).

1. C. L. Liu, Introduction to Combinatorial Mathematics, New York: McGraw-Hill Book Co., 1968.
2. K. Erciyes, Discrete Mathematics and Graph Theory, New York: Springer, 2021.
3. K.H. Rosen, Discrete Mathematics with Applications, New York: McGraw-Hill Higher Education, 2019.
4. B. Kolman, R. C. Busby and S. C. Ross, Discrete Mathematical Structures, 6th ed., New Jersey: Pearson Education, 2013.
5. J. P. Tremblay and R. Manohar, Discrete Mathematical Structures with Application to Computer Science, Noida: Tata McGraw Hill Education, 2008.
6. S. Lipschutz and M. Lipson, Discrete Mathematics, New Delhi: Tata McGraw-Hill, 2013.
7. N. L. Biggs, Discrete Mathematics, 2nd ed., New Delhi: Oxford University Press, 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 End Semester Examination Basic, conceptual, and analytical knowledge of the subject 50 Total 100

MTH151 - PYTHON PROGRAMMING FOR MATHEMATICS (2022 Batch)

Total Teaching Hours for Semester:45
No of Lecture Hours/Week:3
Max Marks:50
Credits:3

Course Objectives/Course Description

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

Course objectives: This course will help the learner to

COBJ1. Acquire skill in usage of suitable functions/packages of Python to solve mathematical problems.

COBJ2. Gain proficiency in using Python to solve problems on Differential equations.

COBJ3. The built in functions required to deal withcreating and visualizing Graphs, Digraphs, MultiGraph.

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 the built-in functions to deal with Graphs and Digraphs.

Unit-1
Teaching Hours:15
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:15
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:15
Data Visualization

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

Text Books And Reference Books:
1. Svein Linge & Hans Petter Langtangen, Programming for computations- Python -A gentle Introduction to Numerical Simulations with Python 3.6, Springer Open, Second Edm. 2020.
2. 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)
3. H P Langtangen, A Primer on Scientific Programming with Python, 2nd ed., Springer, 2016.
1. Walter A Strauss, Partial Differential Equations - An Introduction, John Wiley and Sons, 2007.
2. K. H. Rosen and K. Krithivasan, Discrete mathematics and its applications. McGrawHill, 2013.
3. S. Rao, Partial Differential Equations, Prentice Hall of India, 2009.
4. B E Shapiro, Scientific Computation: Python Hacking for Math Junkies, Sherwood Forest Books, 2015.
5. C Hill, Learning Scientific Programming with Python, Cambridge Univesity Press, 2016.
6. Jaan Kiusalaas, Numerical methods in engineering with Python 3, Cambridge University press, 2013.
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 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

MTH211 - MACHINE LEARNING (2022 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 introducing the elementary notions on Machining learning and focuses on some simple application of machine-learning, algorithms on supervised machine learning and unsupervised learning.

Course Objective: This course will help the learner to:

COBJ1. Be proficient on the idea of machine learning

COBJ2. Implement Supervised Machine Learning Algorithms

COBJ3. Handle computational skills related to unsupervised learning and preprocessing

Course Outcome

CO1: Demonstrate some simple applications of Machine learning.

CO2: Use supervised machine learning algorithms on k-nearest neighbor, linear model, decisions trees.

CO3: Showcase the skill using the unsupervised learning and preprocessing.

Unit-1
Teaching Hours:7
Introduction to Machine Learning

Introduction - simple machine learning applications: classifying iris species: meet the data, training and testing data, pair plot of iris dataset - k-nearest neighbours model, evaluating model.

Unit-2
Teaching Hours:13
Supervised Learning

Classification and regression - generalization, overfitting and underfitting - relation of model complexity to dataset size, supervised machine learning algorithms: k-nearest neighbour algorithm: k-neighbors classification, k-neighbors regression, strengths, weakness and parameters of k-NN algorithm, linear models: linear models for regression, linear models for classification, linear models for multiclass classification, strengths, weakness and parameters of linear models, decision trees: building decision trees, controlling complexity of decision trees, analyzing decision trees, strengths, weakness and parameters of decision trees.

Unit-3
Teaching Hours:10
Unsupervised Learning and Preprocessing

Types of unsupervised learning, challenges in unsupervised learning, preprocessing and scaling: different kinds of preprocessing, applying data transformations, scaling training and test data, principal component analysis, non-negative matrix factorization.

Text Books And Reference Books:

A. C. Müller and S. Guido, Introduction to machine learning with Python,  O’Reilly, 2017.

1. D. Julian, Designing machine learning systems with python. Packt Publishing Limited, 2016.
2. D. Cielen, M. A. D. B., and A. Mohamed, Introducing data science: big data, machine learning, and more, using Python tools. Manning., 2016.
3. M. Bowles, Machine Learning in Python: essential techniques for predictive analysis. John Wiley &amp; Sons, 2019.
4. R. Garreta and G. Moncecchi, Learning scikit-learn: machine learning in python. Packard publishing limited, 2013.
5. S. Raschka and V. Mirjalili, Python machine learning machine learning and deep learning with Python, scikit-learn, and TensorFlow. Packt Publishing, 2018.
6. L. P. Coelho and W. Richert, Building machine learning systems with Python. Packt Publishing, 2015
7. G. James, D. Witten, T. Hastie, and R. Tibshirani, An introduction to statistical learning: with applications in R. Springer, 2017.
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 >

MTH231 - GENERAL TOPOLOGY (2022 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. Provide precise definitions and  appropriate examples and counter-examples of  fundamental concepts in general topology.

COBJ2. Acquire knowledge about a generalisation of the concept of continuity and related properties.

COBJ3. Appreciate the beauty of deep mathematical results such as  Uryzohn’s lemma and understand and apply various proof techniques.

Course Outcome

CO1: Define topological spaces, give examples and counter-examples on concepts like open sets, basis and subspaces.

CO2: Establish equivalent definitions of continuity and apply the same in proving theorems.

CO3: Understand the concepts of metrizability, connectedness, compactness and learn the related theorems.

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.

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 (2022 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 , Rouche’s theorem and Hadamard’s 3-circles theorem.

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: Apply the concept and consequences of analyticity and the Cauchy-Riemann equations and of results on harmonic and entire functions including the fundamental theorem of algebra.

CO2: Compute complex contour integrals in several ways: directly using parameterization, using the Cauchy-Goursat theorem, Evaluate complex contour integrals directly and by the fundamental theorem, apply the Cauchy integral theorem in its various versions, and the Cauchy integral formula.

CO3: Represent functions as Taylor, power and Laurent series, classify singularities and poles, find residues and evaluate complex integrals using the residue theorem.

CO4: Use conformal mappings and know about meromorphic functions.

Unit-1
Teaching Hours:18
Power Series

Power series, radius and circle of convergence, power series and analytic functions, Line and contour integration, Cauchy’s theorem, Cauchy integral formula, Cauchy integral formula for derivatives, Cauchy integral formula for multiply connected domains, Morera’s theorem, Gauss mean value theorem, Cauchy inequality for derivatives, Liouville’s theorem, fundamental theorem of algebra, maximum and minimum modulus principles.

Unit-2
Teaching Hours:15
Singularities

Taylor’s series, Laurent’s series, zeros of analytical functions, singularities, classification of singularities, characterization of removable singularities and poles.

Unit-3
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-4
Teaching Hours:12
Meromorphic functions

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

Text Books And Reference Books:
1. M.J. Ablowitz and A.S. Fokas, Complex Variables: Introduction and Applications, Cambridge University Press, 2003.
2. J.B.Conway, Functions of One Complex Variable, 2nd ed., New York: Springer, 2000.
1. J.H. Mathews and R.W. Howell, Complex Analysis for Mathematics and Engineering, 6th ed., London: Jones and Bartlett Learning, 2011.
2. J.W. Brown and R.V. Churchill, Complex Variables and Applications, 7th ed., New York: McGraw-Hill, 2003.
3. L.S. Hahn and B. Epstein, Classical Complex Analysis, London: Jones and Bartlett Learning, 2011.
4. D. Wunsch, Complex Variables with Applications, 3rd ed., New York: Pearson Education, 2009.
5. D.G. Zill and P.D. Shanahan, A First Course in Complex Analysis with Applications, 2nd ed., Boston: Jones and Bartlett Learning, 2010.
6. E.M. Stein and Rami Sharchi, Complex Analysis, New Jersey: Princeton University Press, 2003.
7. 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 (2022 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: Understand the basic concepts and definition of PDE and also mathematical models representing stretched string, vibrating membrane, heat conduction in rod.

CO2: Demonstrate the canonical form of second order PDE.

CO3: Demonstrate initial value boundary problem for homogeneous and non-homogeneous PDE.

CO4: Demonstrate on boundary value problem by Dirichlet and Neumann problem.

UNIT 1
Teaching Hours:10
First Order Partial differential equations order

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.
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 (2022 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: Write precise and accurate mathematical definitions of basics concepts in Graph Theory.

CO2: Provide appropriate examples and counterexamples to illustrate the basic concepts.

CO3: Demonstrate various proof techniques in proving theorems.

CO4: 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.
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 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

MTH235 - INTRODUCTORY FLUID MECHANICS (2022 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 representatives.

COBJ2: Derive and understand the governing equations in fluid mechanics.

COBJ3: Familiarize with two- or three-dimensional incompressible flows.

COBJ4: Determine properties of inviscid and viscous fluids.

COBJ5: Describe and construct analytically standard two- or three-dimensional viscous flows.

Course Outcome

CO1: Confidently calculate and derive tensor expressions using index notation, and use the divergence theorem and the transport theorem.

CO2: Derive the fundamental equations of fluid mechanics and appreciate their physical interpretations.

CO3: Comprehend two and three dimension flows incompressible flows.

CO4: Describe two-dimensional viscous flows and their classifications.

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, stream lines, path lines.

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

Stress and 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 equation of continuity, conservation of mass, equation of motion (Navier-Stokes equations), conservation of momentum, the energy equation, conservation of energy.

Unit-3
Teaching Hours:15
One, Two and Three Dimensional Invisid Incompressible Flow

Bernoulli equation, applications of Bernoulli equation, circulation theorems, circulation concept, Kelvin’s 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.
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

MTH251 - COMPUTATIONAL MATHEMATICS USING PYTHON (2022 Batch)

Total Teaching Hours for Semester:45
No of Lecture Hours/Week:3
Max Marks:50
Credits:3

Course Objectives/Course Description

Course Description: This course aimsto solve mathematical models using differential equations, linear algebra and fluid mechanics using Python libraries.

Course objectives​: This course will help the learner to

COBJ1: Acquire skill in using suitable libraries of Python to solve real-world problems giving rise to differential equations

COBJ2: Gain proficiency in using Python to solve problems on linear algebra.

COBJ3: Build user-defined functions to deal with the problem on fluid mechanics.

Course Outcome

CO1: Demonstrate the use of Python libraries for handling problems on mathematical modelling.

CO2: Compute the problems on linear algebra using Python libraries.

CO3: Handle the Python libraries for solving problems on fluid dynamics.

Unit-1
Teaching Hours:45
Proposed Topics:

1. Linear and non-linear model growth and decay Newton’s law of cooling/warming, mixtures, computational models with quadratic growth, concentration of a nutrients.
2. Lotka-Volterra predator-prey model, competition models.
3. Spring/mass systems: Free undamped motion, free damped motion, driven motion, nonlinear springs, simple pendulum, projectile, double pendulum.
4. Matrices and determinants, operations on matrices, powers and inverse of a matrix.
5. Rank, solving systems of linear equations (Gauss-Elimination method, Gauss-Jordan method, LU decomposition method), eigenvalues, eigenvectors, and diagonalization.
6. Linear combinations, linearly independence and dependence, basis and dimension, linear transformation, orthogonal set, orthogonal projection of a vector, orthonormal, Gram Schmidt process.
7. Stream lines, path lines, vortex lines and their plots.
8. Calculating Rayleigh number for Rayleigh-Bénard convection with external constraints magnetic field, rotation, non-uniform temperature gradients.
9. Solution of Lorenz equations, Nusselt number.
Text Books And Reference Books:
1. D. G. Zill, First Course in Differential Equations with Modelling Applications, 11th ed. 2019.
2. H. P. Langtangen, A Primer on Scientific Programming with Python, 2nd ed., Springer, 2016.
1. H. Fangour, Introduction to Python for computational science and Engineering, 2015.
2. H. P. Langtangen, A premier on scientific programming with Python, 3rd ed., 2012.
3. S. Linge and H. P. Langtangen, Programming for computations - Python, 2nd ed., 2018.
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 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

MTH311 - TEACHING TECHNOLOGY AND SERVICE LEARNING (2021 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 on the use of modern technology in teaching, they are exposed to the principles, procedures and techniques of planning and implementing teaching techniques. Through service learning they will apply the knowledge in real-world situations and benefit 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 situation.

COBJ 4: Enhances academic comprehension through experiential learning.

Course Outcome

CO1: Gain necessary skills on the use of modern technology in teaching.

CO2: Understand the components and techniques of effective teaching.

CO3: Obtain necessary skills in understanding the mathematics teaching.

CO4: Strengthen personal character and sense of social responsibility through service learning module.

CO5: Contribute to the community by addressing and meeting 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.
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 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 >

MTH331 - MEASURE THEORY AND LEBESGUE INTEGRATION (2021 Batch)

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

Course Objectives/Course Description

Course description: The Course covers the basic material that one needs to know in the theory of functions of a real variable and measure and integration theory as expounded by Henri Léon Lebesgue.

Course objectives: This course will help the learner to

COBJ1. Enhance the understanding of the advanced notions from Mathematical Analysis

COBJ2. Know more about the Measure theory and Lebesgue Integration

Course Outcome

CO1: Understand the fundamental concepts of Mathematical Analysis.

CO2: State some of the classical theorems in of Advanced Real Analysis.

CO3: Be familiar with measurable sets and functions.

CO4: Integrate a measurable function.

CO5: Understand the properties of Lp Spaces.

Unit-1
Teaching Hours:15
Lebesgue Measure

Lebesgue Outer Measure, The s-Algebra of Lebesgue Measurable Sets, Outer and Inner Approximation of Lebesgue Measurable Sets, Countable Additivity, Continuity and the Borel-Cantelli Lemma, Nonmeasurable Sets, The Cantor Set and the Canton-Lebesgue Function, Sums, Products and Compositions of Lebesgue Measurable Functions, Sequential Pointwise Limits and Simple Approximation, Littlewood’s three principles, Egoroff’s Theorem and Lusin’s Theorem.

Unit-2
Teaching Hours:15
The Lebesgue Integration

The Lebesgue Integral of a Bounded Measurable Function over a Set of Finite Measure, The Lebesgue Integral of a Measurable Nonnegative Function; The General Lebesgue Integral; Countable Additivity and Continuity of Integration, Uniform Integrability, Uniform Integrability and Tightness, Convergence in measure, Characterizations of Riemann and Lebesgue Integrability.

Unit-3
Teaching Hours:15
Differentiation and Lebesgue Integration

Continuity of Monotone Functions, Differentiation of Monotone Functions, Functions of Bounded Variation, Absolutely Continuous Functions, Integrating Derivatives.

Unit-4
Teaching Hours:15
The Lp Spaces

Normed Linear Spaces, The Inequalities of Young, Hölder and Minkowski, The Lp spaces, Approximation and Separability, The Riesz Representation for the Dual of Lp, Weak Sequential Convergence in Lp, Weak Sequential Compactness, The Minimization of Convex Functionals.

Text Books And Reference Books:

H.L. Royden and P.M. Fitzpatrick, “Real Analysis,” 4th ed. New Jersey: Pearson Education Inc., 2013.