CHRIST (Deemed to University), Bangalore

DEPARTMENT OF mathematics-and-statistics

sciences

Syllabus for
Master of Science (Mathematics)
Academic Year  (2018)

 
1 Semester - 2018 - Batch
Course Code
Course
Type
Hours Per
Week
Credits
Marks
MTH111 TEACHING TECHNOLOGY AND RESEARCH METHODOLOGY IN MATHEMATICS AND SERVICE LEARNING Add On Courses 3 2 100
MTH131 NUMBER THEORY AND CRYPTOGRAPHY - 4 4 100
MTH132 GENERAL TOPOLOGY - 4 4 100
MTH133 ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS - 4 4 100
MTH134 FLUID MECHANICS - 4 4 100
MTH135 ELEMENTARY GRAPH THEORY - 4 4 100
MTH151 MATHEMATICS LAB USING FOSS TOOLS - 3 3 50
2 Semester - 2018 - Batch
Course Code
Course
Type
Hours Per
Week
Credits
Marks
MTH211 STATISTICS - 3 2 100
MTH231 REAL ANALYSIS - 4 4 100
MTH232 COMPLEX ANALYSIS - 4 4 100
MTH233 ADVANCED ALGEBRA - 4 4 100
MTH234 ADVANCED FLUID MECHANICS - 4 4 100
MTH235 ALGORITHMIC GRAPH THEORY - 4 4 100
MTH251 MATHEMATICS LAB USING PYTHON - 3 3 50
3 Semester - 2017 - Batch
Course Code
Course
Type
Hours Per
Week
Credits
Marks
MTH311 STATISTICS Add On Courses 3 2 100
MTH331 MEASURE THEORY AND LEBESGUE INTEGRATION - 4 4 100
MTH332 COMPUTER ORIENTED NUMERICAL METHODS USING MATLAB - 4 4 100
MTH333 CLASSICAL MECHANICS - 4 4 100
MTH334 LINEAR ALGEBRA - 4 4 100
MTH335 ADVANCED GRAPH THEORY - 4 4 100
MTH371 INTERNSHIP IN PG MATHEMATICS COURSE - 0 2 0
4 Semester - 2017 - Batch
Course Code
Course
Type
Hours Per
Week
Credits
Marks
MTH431 DIFFERENTIAL GEOMETRY - 4 4 100
MTH432 COMPUTATIONAL FLUID DYNAMICS - 4 4 100
MTH433 FUNCTIONAL ANALYSIS - 4 4 100
MTH441 CALCULUS OF VARIATIONS AND INTEGRAL EQUATIONS - 4 4 100
MTH448 COMBINATORIAL MATHEMATICS - 4 4 100
MTH451 PROJECT - 2 2 100
    

    

Introduction to Program:
The M.Sc. 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 Topology, Algebra, Analysis, Number Theory and ordinary and partial differential equations. Mathematics using FOSS Tools and Mathematics using Python are the two lab oriented courses introduced in first and second semesters respectively. In the third and fourth semesters, focus is on the special courses, elective course and skill-based courses including Functional Analysis, Advanced Fluid Mechanics, Advanced Graph Theory and Computer oriented Numerical Methods using MATLAB. Important feature of the curriculum is that one course on the topic Fluid Mechanics and Graph Theory is offered in each semester with a project on these topics in the fourth semester, which will help the students to pursue the higher studies and research in these topics. Mandatory internship at the end of the first year is another salient feature of the programme to provide exposure to the students in industry or academia.
Assesment Pattern

 

SEMESTER

COURSE CODE

COURSE TITLE

CIA (Max Marks)

Attendance (Max Marks)

ESE (Max Marks)

1

         

MTH131

Number Theory and Cryptography

45

5

50

MTH132

General Topology

45

5

50

MTH133

Ordinary and Partial Differential Equations

45

5

50

MTH134

Fluid Mechanics

45

5

50

MTH135

Elementary Graph Theory

45

5

50

MTH151

Mathematics Lab using FOSS tools

     

2

MTH231

Real Analysis

45

5

50

MTH232

Complex  Analysis

45

5

50

MTH233

Advanced Algebra

45

5

50

MTH234

Advanced Fluid Mechanics

45

5

50

MTH235

Algorithmic Graph Theory

45

5

50

MTH251

Mathematics Lab using Python

     
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

MTH111 - TEACHING TECHNOLOGY AND RESEARCH METHODOLOGY IN MATHEMATICS AND SERVICE LEARNING (2018 Batch)

Total Teaching Hours for Semester:45
No of Lecture Hours/Week:3
Max Marks:100
Credits:2

Course Objectives/Course Description

 

This course is intended to assist the students in acquiring necessary skills on the use of modern technology in teaching. Also, the students are exposed to the principles, procedures and techniques of planning and implementing the research project. Through service learning they will apply the knowledge in real-world situations and benefit the community.

 

Course Outcome

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

  • gain necessary skills on the use of modern technology in teaching.  To foster a clear understanding about research design that enables students in analyzing and evaluating the published research.
  • understand the components and techniques of effective report writing.
  • obtain necessary skills in understanding the mathematics research articles
  • acquire skills in preparing scientific documents using MS Word, Math type, Open Office Math editor, yEd Graph Editor and LATEX.
  • strengthen personal character and sense of social responsibility through service learning module.

 

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, Feed back in teaching, Teacher’s role and responsibilities, Information technology for teaching.

 

Unit-2
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-3
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.

 

Package for Mathematics Typing, MS Word, Math Type, Open Office Math Editor, Tex, yEd Graph Editor, Tex in detail, Installation and Set up, Text, Formula, Pictures and Graphs, Producing various types of documents using TeX.

Unit-4
Teaching Hours:15
Service Learning
 

Guidelines for service learning:

One among the following can be considered as a service learning module:

  • Tie up with schools for teaching elementary mathematics in an easier way.
  • Developing e-content for particular topics which will be a Vehicle for Teaching Curriculum Theory, Assessment, and Design (as per the requirements).
  • Math Exhibition: To strengthen students' math skills, a mathematics camp can be organised in the school premises. Students will participate in challenging academic coursework of math, make projects related to mathematical concepts, explore many inventions and historical aspects in mathematics. Students can strengthen and expand their scientific and mathematical knowledge while having fun.
  • Students can create a website for the Department of Mathematics/the project area, putting all the information about the activities and events coming up.
  • Students can assist in statistical research(based on its needs), in developing a survey tool, organizing and/or conducting the survey, compiling and analyzing data, or some combination of these or some other statistical undertakings.
  • Develop a mathematical model and should also be able to provide a solution for an existing real-world problem.

After deciding, get approval from your respective mentors.

  • Each student will develop a learning/lesson plan composed of three (3-4) measurable learning objectives. Examples of learning objectives are:
    • Improve algebraic/problem solving skills.
    • Improve methods of communicating mathematics to others effectively.
    • Identify common mistakes and misconceptions that mathematics students make.
  • A minimum of fifteen (15) hours documented service is required during the semester.
  • A student must keep a log of the volunteered time.
  • A student must write a diary containing an analysis of 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:

.

Essential Reading / Recommended Reading
  1. R.Varma, Modern trends in teaching technology, Anmol publications Pvt.Ltd., New Delhi 2003.
  2. Usha Rao, Educational teaching, Himalaya Publishing house, New Delhi 2001.
  3. J. Mohanthy, Educational teaching, Deep & Deep Publications Pvt.Ltd., New Delhi 2001.
  4. K. J. Sree and D. B. Rao, Methods of teaching sciences, Discovery publishing house, 2010.
  5. E. B. Wilson, An introduction to scientific research, Reprint, Courier Corporation, 2012.
  6. R. Ahuja, Research Methods, Rawat Publications, 2001.
  7. G. L. Jain, Research Methdology, Mangal Deep Publictions, 2003.
  8. B. C. Nakra and K. K. Chaudhry, Instrumentation, measurement and analysis, TMH Education, 2003.
  9. 9. L. Radhakrishnan, Write Mathematics Right: Principles of Professional Presentation, Exemplified with Humor and Thrills, Alpha Science International, Limited, 2013.
  10. Cathryn Berger Kaye, The Complete Guide to Service Learning: Proven, Practical Ways to Engage Students in Civic Responsibility, Academic Curriculum, & Social Action, 2009.
  11. Butin, D , Service-Learning in Theory and Practice -The Future of Community Engagement in Higher Education , Palgrave Macmillan US., 2010.

 

 

 

 

 

 

 

 

Evaluation Pattern

.

MTH131 - NUMBER THEORY AND CRYPTOGRAPHY (2018 Batch)

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

Course Objectives/Course Description

 

This course is concerned with the basics of analytical number theory. Topics such as divisibility, congruence’s, quadratic residues and functions of number theory are covered in this course. An introduction to Cryptography is also included.

Course Outcome

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

  • Define and interpret the concepts of divisibility, congruence, greatest common divisor, prime, and prime-factorization,
  • Solve linear Diophantine equations and congruences of various types, and use the theory of congruences in applications.
  • Prove and apply properties of multiplicative functions such as the Euler phi-function and of quadratic residues.
  • Apply the Law of Quadratic Reciprocity and other methods to classify numbers as primitive roots, quadratic residues, and quadratic non-residues,
  • Produce rigorous arguments (proofs) centered on the material of number theory, most notably in the use of Mathematical Induction and/or the Well Ordering Principal in the proof of theorems.
  • Encrypting and Decrypting messages.

 

 

 

 

UNIT 1
Teaching Hours:15
Divisibility
 

The division algorithm, the Greatest Common Divisor, the Euclidean algorithm, Diophantine Equation, the Fundamental Theorem of Arithmetic, the methods to find prime numbers, the Goldbach Conjecture

UNIT 2
Teaching Hours:15
Congruences
 

 

Basic Properties of Congruence, Complete residue system modulo m, reduced residue system modulo m, Euler’s φ function, Fermat’s theorem, Euler’s generalization of Fermat’s theorem, Wilson’s theorem, solutions of linear congruences, the Chinese remainder theorem, solutions of polynomial congruences, prime power moduli, power residues.

UNIT 3
Teaching Hours:15
Quadratic residues and Some functions of number theoretic-functions
 

 

Legendre symbol, Gauss’s lemma, quadratic reciprocity, the Jacobi symbol, binary quadratic forms, equivalence and reduction of binary quadratic forms, sums of two squares, positive definite binary quadratic forms.

UNIT 4
Teaching Hours:15
Introduction to Cryptography (self learning module)
 

 

From Caesar Cipher to Public Key Cryptography, The Knapsack Cryptosystem, An Application of Primitive Roots to Cryptography.

Text Books And Reference Books:

David M. Burton, Elementary Number Theory, 15th Ed. Tata McGraw-Hill, 2016.

Essential Reading / Recommended Reading

1. Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, Springer, 2010.

2. Neal Koblitz, A course in number theory and cryptography, Reprint: Springer, 2010.

3. Gareth A. Jones and J. Mary Jones, Elementary number theory, Reprint, Springer, 2000.

4. Joseph H. Silverman, A friendly introduction to number theory, Pearson Prentice Hall, 2006.

5. Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An introduction to the theory of numbers, John Wiley, 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

MTH132 - GENERAL TOPOLOGY (2018 Batch)

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

Course Objectives/Course Description

 

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

Course Outcome

Upon successful completion of this course, students will be able to

  • Develop their abstract thinking skills.
  • Provide precise definitions and  appropriate examples  and counter examples of  fundamental  concepts in general topology.
  • Acquire knowledge about various types of topological spaces and their properties.
  • Appreciate the beauty of deep mathematical results like Uryzohn’s lemma and understand the dynamics of the proof techniques.

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, Reprint, Prentice Hall of India, 2000.
  3. S. Willard, General topology, Courier-Corporation, 2012.
  4. C. W. Baker, Introduction to topology, Reprint, 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

MTH133 - ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS (2018 Batch)

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

Course Objectives/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, Eigen values and Eigen vectors of the equations, power series method for solving differential equations, second order partial differential equations like wave equation, heat equation, Laplace equations and their solutions by Eigen function method.

Course Outcome

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

  • Understand concept of Linear differential equation, Fundamental set Wronskian.
  • Understand the concept of Liouvilles theorem, Adjoint and Self Adjoint equation, Langrage’s Identity, Green’s formula, Eigen value and Eigen functions.
  • Identify ordinary and singular point by Frobenius Method, Hyper geometric differential equation and its polynomial.
  • Understand the basic concepts and definition of PDE and also mathematical models representing stretched string, vibrating membrane, heat conduction in rod.
  • Demonstrate on the canonical form of second order PDE.
  • Demonstrate initial value boundary problem for homogeneous and non-homogeneous PDE.
  • Demonstrate on boundary value problem by Dirichlet and Neumann problem.

 

UNIT 1
Teaching Hours:20
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. Legendre, Bessel's, Chebeshev's, Eigen values and Eigen functions, related examples. 

UNIT 2
Teaching Hours:10
Power series solutions
 

Solution near an ordinary point and a regular singular point by Frobenius method, solution near irregular singular point, hypergeometric differential equation and its polynomial solutions, standard properties. 

 

UNIT 3
Teaching Hours:15
Partial Differential Equations
 

Formation of PDE, solutions of first and second order PDE, mathematical models representing stretched string, vibrating membrane, heat conduction in solids and the gravitational potentials, second-order equations in two independent variables, canonical forms and general solution.      

UNIT 4
Teaching Hours:15
Solutions of PDE
 

The Cauchy problem for homogeneous wave equation, D’Alembert’s solution, domain of influence and domain of dependence, the Cauchy problem for non-homogeneous wave equation, the method of separation of variables for the one-dimensional wave equation and heat equation. Boundary value problems, Dirichlet and Neumann problems in Cartesian coordinates, solution by the method of separation of variables. Solution by the method of eigenfunctions. 

Text Books And Reference Books:
  1. Christian Constanda, Solution Techniques for Elementary Partial Differential Equations, New York: Chapman & Hall, 2010.

  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 andEngineering, Cambridge, 2005.

  2. Edwards Penney, Differential Equations and Boundary Value Problems, Pearson Education, 2005.

  3. J. David Logan, Partial Differential Equations, 2nd ed., New York: Springer, 2002.

  4. Alan Jeffrey, Applied Partial Differential Equations: An Introduction, California: Academic Press, 2003.

  5. M. Renardy and R.C. Rogers, An Introduction to Partial Differential Equations, 2nd ed., New York: Springer, 2004.

  6. L.C. Evans, Partial Differential Equations, 2nd ed., American Mathematical Society, 2010.

  7. K. Sankara Rao, Introduction to Partial Differential Equations, 2nd ed., New Delhi: Prentice-Hall of India, 2006.

  8. R.C. McOwen, Partial Differential Equations: Methods and Applications, 2nd ed., New York: Pearson Education, 2003.

  9. E. A. Coddington, Introduction to ordinary differential equations, Reprint: McGraw Hill, 2006.

  10. G. F. Simmons, Differential equations with applications and historical notes, Tata McGraw Hill, 2003.

  11. Tyn Myint-U and L. Debnath, Linear Partial Differential Equations, Boston: Birkhauser, 2007.

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 - FLUID MECHANICS (2018 Batch)

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

Course Objectives/Course Description

 

This course aims at studying the fundamentals of fluid mechanics such as kinematics of fluid, incompressible flow and boundary layer flows.

Course Outcome

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.
  • Give a common ground for the mechanics of fluids and solids and the connection of these to applications.
  • Remain equiped for further studies in mechanics, applied and industrial mathematics, physics, geology, geophysics, and astrophysics.
  • Give an introduction to the basic equations and solution methods for mathematical modeling of viscous fluids and elastic matter.

 

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 Inviscid 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. 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

 

 

MTH135 - ELEMENTARY GRAPH THEORY (2018 Batch)

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

Course Objectives/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 Outcome

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

  • Be familiar with the history and development of graph theory
  • Write precise and accurate mathematical definitions of basics concepts in graph theory
  • Provide appropriate examples  and counter examples to illustrate the basic concepts
  • Understand and apply various proof techniques in proving theorems in graph theory.
  • Acquire mastery in using graph drawing tools

 

UNIT 1
Teaching Hours:15
Introduction to Graphs
 

Definition and introductory concepts, Graphs as Models, Matrices and Isomorphism, Decomposition and Special Graphs, Connection in Graphs, Bipartite Graphs, Eulerian Circuits. 

UNIT 2
Teaching Hours:15
Vertex Degrees and Directed Graphs
 

Counting and Bijections, Extremal Problems, Graphic Sequences, Directed Graphs, Vertex Degrees, Eulerian Digraphs, Orientations and Tournaments.

UNIT 3
Teaching Hours:15
Trees and Distance
 

Properties of Trees, Distance in Trees and Graphs, Enumeration of Trees, Spanning Trees in Graphs, Decomposition and Graceful Labellings, Minimum Spanning Tree, Shortest Paths.

 

UNIT 4
Teaching Hours:15
Connectivity and Paths
 

Connectivity, Edge - Connectivity, Blocks, 2 - connected Graphs, Connectivity in Digraphs, k - connected and k-edge-connected Graphs, Maximum Network Flow, Integral Flows.

 

Text Books And Reference Books:

 D.B. West, Introduction to Graph Theory, New Delhi: Prentice-Hall of India, 2011.

Essential Reading / Recommended Reading
  1. B. Bollabas, Modern Graph Theory, Springer, New Delhi, 2005.

  2. F. Harary, Graph Theory, New Delhi: Narosa, 2001.

  3.  G. Chartrand and P.Chang, Introduction to Graph Theory, New Delhi: Tata McGraw-Hill, 2006.

  4. G. Chatrand and L. Lesniak, Graphs and Digraphs, Fourth Edition, Boca Raton: CRC Press, 2004.

  5.  J. A. Bondy and U.S.R. Murty, Graph Theory, Springer, 2008

  6. J. Clark and D.A. Holton, A First Look At Graph Theory, Singapore: World Scientific, 2005.

  7. R. Balakrishnan and K Ranganathan, A Text Book of Graph Theory, New Delhi: Springer, 2008.

  8. R. Diestel, Graph Theory, New Delhi: Springer, 2006.

  9. V. K. Balakrishnan Graph Theory, Schaum’s outlines, New Delhi:Tata Mcgrahill, 2004.

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

 

 

MTH151 - MATHEMATICS LAB USING FOSS TOOLS (2018 Batch)

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

Course Objectives/Course Description

 

This course aims as introducing the mathematical software packages “WxMaxima” and “Scilab”, for learning basic operation on matrix manipulation, plotting graphs etc.,. Also, these software packages will help students to solve problems / applied problems on  Mathematics.

Course Outcome

On successful completion, a student will be able to

 

  • Use the basic commands in WxMaxima including the 2D and 3D plots

  • Have a strong command on the inbuilt commands required for the learning and analyzing mathematics

  • Solve problems / applied problems on mathematics by using Scilab.

Unit-1
Teaching Hours:15
Introduction to WxMaxima
 

Introduction to WxMaxima Interface - Maxima expressions, numbers, operators, constants and reserved words - input and output in WxMaxima - 2D and 3D plots in WxMaxima - symbolic computations in WxMaxima - Solving Ordinary differential equations in WxMaxima.

Unit-2
Teaching Hours:20
Introduction to Scilab
 

Introduction to Scilab and commands connected with Matrices - Computations with Matrices - 2D and 3D Plots - Script Files and Function Files.

Unit-3
Teaching Hours:10
Solving problems using Scilab / WxMaxima
 

Solving systems of equation and explain consistence - Find the values of some standard trigonometric functions in radians as well as in degree - Create polynomials of different degrees and find its real roots  - Display Fibonacci series using Scilab program - Display non-Fibonacci series using Scilab program.

Text Books And Reference Books:
  1. C. Gomez, C. Bunks, J. P. Chancelier, F. Delebecque, M. Goursat, R. Nikouhah and S. Steer, Engineering and scientific computing with scilab. Birkhauser, 2013.

  2. Z. Hannan, wxMaxima for calculus I. Zachary Hannan, 2015.

  3. Z. Hannan, wxMaxima for calculus II. Zachary Hannan, 2015.

Essential Reading / Recommended Reading
  1. E.L.Woollett, Maxima by Example, Online resource(https://web.csulb.edu/~woollett/), 2009.

  2. Sourceforge,  Introduction to Graphs,  Online resource (http://maxima.sourceforge.net/docs/manual/de/maxima_50.html#SEC290).

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 - STATISTICS (2018 Batch)

Total Teaching Hours for Semester:45
No of Lecture Hours/Week:3
Max Marks:100
Credits:2

Course Objectives/Course Description

 

This course aims at teach 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 Outcome

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

  • Understand random variables and probability distributions.
  • Distinguish discrete and continuous random variables.
  • Obtain ability compute Expected value and Variance of discrete random variable.
  • Acquire knowledge in using Binomial distribution, Poisson distribution etc.,
  • Define inferential statistics.
  • Effectively use sampling distributions in inferential statistics.

Unit-1
Teaching Hours:15
Random Variables and Expectation
 

Discrete and continuous random variables, distribution functions, probability mass and density functions, bivariate distributions, marginal and conditional distributions, expected value of a random variable, independence of random variables, conditional expectations, covariance matrix, correlation coefficients and regression, Chebyshev’s inequality, moments, moment generating functions, characteristic functions.  

Unit-2
Teaching Hours:15
Probability Distributions
 
Probability: Sample spaces, events, probability of an event, theorems on probability, conditional probability, independent events, Bayes theorem. Boole’s inequality.
Discrete Probability Distribution:  Introduction, uniform, Bernoulli, Binomial, negative Binomial, geometric, Hypergeometric and Poisson distribution. Continuous Probability Distributions: Introduction, uniform, gamma, exponential, beta and normal distributions.
 
Unit-3
Teaching Hours:15
Sampling distributions
 

t, F and chi-square distributions, standard errors and large sample distributions.  

Text Books And Reference Books:
  1. E. Freund John, Mathematical Statistics, 5th Ed., Prentice Hall of India, 2000.
  2. Gupta S.C. and Kapoor V.K., Fundamentals of mathematical Statistics, Sultan Chand and Sons, New Delhi, 2001.
  3. Ronald E. Walpole, Raymond H. Myers and Sharon L. Myers, Probability and Statistics for Engineers and Scientists, Pearson Prentice Hall, 2006.
Essential Reading / Recommended Reading
  1. Paul G. Hoel, Introduction to mathematical Statistics, Wiley, 2000.
  2. M. Spiegel, Probability and statistics, Schaum’s Outline Series, 2000.
  3. Neil Weiss, Introductory Statistics, Addison-Wesley, 2002.
  4. S. M. Ross, A first course in probability, Pearson Prentice Hall, 2005.
  5. D. Wackerly, W. Mendenhall and R. L. Scheaffer, Mathematical Statistics with Applications, Duxburry Press, 2007.    
Evaluation Pattern

.

MTH231 - REAL ANALYSIS (2018 Batch)

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

Course Objectives/Course Description

 

This course will help students 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 Outcome

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

  • Integrate functions a real variable in the sense of Riemann – Stieltjes.
  • Classify sequences of functions which are pointwise convergent, uniform convergent etc.

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, The Rank Theorem, Determinants, Derivatives of Higher Order, Differentiation of Integrals         

 

 

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

MTH232 - COMPLEX ANALYSIS (2018 Batch)

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

Course Objectives/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 Outcome

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

  • 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.
  • 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, and
  • Represent functions as Taylor, power and Laurent series, classify singularities and poles, find residues and evaluate complex integrals using the residue theorem.
  • 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 ApplicationsCambridge University Press, 2003.
  2. J.B.Conway, Functions of One Complex Variable, 2nd ed., New York: Springer, 2000.
Essential Reading / Recommended Reading
  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. David 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 - ADVANCED ALGEBRA (2018 Batch)

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

Course Objectives/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 Outcome

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

  • Demonstrate knowledge of conjugates, the Class Equation and Sylow  theorems
  • Demonstrate knowledge of polynomial rings and associated properties
  • Derive and apply Gauss Lemma, Eisenstein criterion for irreducibility of rationals
  • Demonstrate the characteristic of a field and the prime subfield;
  • Demonstrate Factorization and ideal theory in the polynomial ring; the structure of a 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 Modern Algebra, 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

ESE

 

Basic, conceptual and analytical knowledge of the subject

50

Total

100

MTH234 - ADVANCED FLUID MECHANICS (2018 Batch)

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

Course Objectives/Course Description

 

This course helps the students to understand the basic concepts of heat transfer, types of convection shear and thermal instability of linear and non-linear problems.  This course also includes the analysis Prandtl boundry layer, porous media and Non-Newtonian fluid.

 

Course Outcome

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

  • Recognize how fluid flow theory can be employed in a modern mechanical engineering design environment.
  • Discuss the theory of compressible flows, formulate the relevant theory, and solve the related engineering problems.
  • Ability to apply fluid mechanics principles to the analysis of real systems.
  • Understand the basic laws of heat transfer and understand the fundamentals of convective heat transfer process. 

 

UNIT 1
Teaching Hours:15
Dimensional Analysis and Similarity
 

Introduction to heat transfer, different modes of heat transfer- conduction, convection and radiation, steady and unsteady heat transfer,  free and forced convection. Non-dimensional parameters determined from differential equations – Buckingham’s Pi Theorem –  Non-dimensionalization of the Basic Equations - Non-dimensional parameters and dynamic similarity.  

UNIT 2
Teaching Hours:20
Heat Transfer and Thermal Instability
 

Shear Instability: Stability of flow between parallel shear flows -  Squire’s theorem for viscous and inviscid theory – Rayleigh stability equation – Derivation of Orr-Sommerfeld equation assuming that the basic flow is strictly parallel. Basic concepts of stability theory – Linear and Non-linear theories – Rayleigh Benard Problem – Analysis into normal modes – Principle of Exchange of stabilities – first variation principle – Different boundary conditions on velocity and temperature. 

UNIT 3
Teaching Hours:10
Prandtl Boundry Layer
 

Boundary layer concept, the boundary layer equations in two-dimensional flow, the boundary layer along a flat plate, the Blasius solution. Stagnation point flow. Falkner-Skan family of equations.

UNIT 4
Teaching Hours:15
Porous Media and Non - Newtonian Fluids
 

Introduction to porous medium, porosity, Darcy’s Law, Extension of Darcy Law – accelerations and inertial effects, Brinkman’s equation, effects of porosity variations, Bidisperse porous media. Constitutive equations of Maxwell, Oldroyd, Ostwald , Ostwald de waele, Reiner – Rivlin and Micropolar fluid.  Weissenberg effect and Tom’s effect.Equation of continuity, Conservation of momentum for non-Newtonian fluids.

Text Books And Reference Books:
  1. Drazin and Reid, Hydrodynamic instability, Cambridge University Press, 2006.

  2. S. Chardrasekhar, Hydrodynamic and hydrodmagnetic stability, Oxford University Press, 2007 (RePrint).

Essential Reading / Recommended Reading
  1. P. K. Kundu, Ira M. Cohen and David R Dowling, Fluid Mechanics, 5th ed., Academic Press, 2011.
  2. F. M White, Fluid Mechanics, Tata Mcgraw Hill. 2011.
  3. D. A. Nield and Adrian Bejan, Convection in Porous Media”, Third edition, Springer, 2006.

 

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 - ALGORITHMIC GRAPH THEORY (2018 Batch)

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

Course Objectives/Course Description

 

This course helps the students to understand the colouring of graphs, Planar graphs, edges and cycles.

Course Outcome

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

  • Understand the basic concepts and fundamental results in matching, domination, coloring and planarity.
  • Construct examples and proofs pertaining to the basic theorems.
  • Apply the theoretical knowledge and independent mathematical thinking in creative investigation of questions in graph theory.
  • Reason from definitions to construct mathematical proofs.
  • Write graph theoretic ideas in a coherent and technically accurate manner.
  • Obtain a solid overview of the questions addressed by graph theory and will be exposed to emerging areas of research.

Unit-1
Teaching Hours:15
Colouring of Graphs
 

Definition and Examples of Graph Colouring, Upper Bounds, Brooks’ Theorem, Graph with Large Chromatic Number, Extremal Problems and Turan’s Theorem, Colour-Critical Graphs, Counting Proper Colourings.

Unit-2
Teaching Hours:15
Matchings and Factors
 

Maximum Matchings, Hall’s Matching Condition, Min-Max Theorem, Independent Sets and Covers, Maximum Bipartite Matching, Weighted Bipartite Matching, Tutte’s 1-factor Theorem, Domination.

Unit-3
Teaching Hours:15
Planar Graphs
 

Drawings in the Plane, Dual Graphs, Euler’s Formula, Kuratowski’s Theorem, Convex Embeddings, Coloring of Planar Graphs, Thickness and Crossing Number.

Unit-4
Teaching Hours:15
Edges and Cycles Edge
 

Colourings, Characterisation of Line Graphs, Necessary Conditions of Hamiltonian Cycles, Sufficient Conditions of Hamiltonian Cycles, Cycles in Directed Graphs, Tait’s Theorem, Grinberg’s Theorem, Flows and Cycle Covers.

Text Books And Reference Books:

D.B. West, Introduction to Graph Theory, New Delhi: Prentice-Hall of India, 2011.

Essential Reading / Recommended Reading
  1. B. Bollabas, Modern Graph Theory, Springer, New Delhi, 2005.
  2. F. Harary, Graph Theory, New Delhi: Narosa, 2001.
  3. G. Chartrand and P.Chang, Introduction to Graph Theory, New Delhi: Tata McGraw-Hill, 2006.
  4. G. Chatrand and L. Lesniak, Graphs and Digraphs, Fourth Edition, Boca Raton: CRC Press, 2004.
  5. J. A. Bondy and U.S.R. Murty, Graph Theory, Springer, 2008.
  6. J. Clark and D.A. Holton, A First Look At Graph Theory, Singapore: World Scientific, 2005.
  7. R. Balakrishnan and K Ranganathan, A Text Book of Graph Theory, New Delhi: Springer, 2008.
  8. R. Diestel, Graph Theory, New Delhi: Springer, 2006.
  9. V. K. Balakrishnan Graph Theory, Schaum’s outlines, New Delhi:Tata Mcgrahill, 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

MTH251 - MATHEMATICS LAB USING PYTHON (2018 Batch)

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

Course Objectives/Course Description

 

 This course aims as introducing the mathematical programming language and its uses in solving problems on discrete mathematics and partial differential equations.

Course Outcome

On successful completion, a student will be able to:

 

  • Use the basic commands in Python, including the 2D and 3D plots

  • Use Python in solving problems on PDE

  • Use Python in solving problems on Discrete Mathematics

Unit-1
Teaching Hours:15
Introduction to Python Programming
 

Python commands: Comments, Number and other data types, Expressions, Operators, Variables and assignments, Decisions, Loops, Lists, Strings - plotting using “matplotlib” -  Basic operations , Simplification, Calculus, Solvers and Matrices using Sympy.

Unit-2
Teaching Hours:15
Differential Equations using Python
 

 

Solving ODE’s using Python - Libraries for Differential equations in Python, PDE’s using sympy user functions pde_seperate(), pde_seperate_add(). pde_seperate_mul(), pdsolve(), classify_pde(), checkpdesol() , pde_1st_linear_constant_coeff_homogeneous, pde_1st_linear_constant_coeff, pde_1st_linear_variable_coeff.

Unit-3
Teaching Hours:15
Discrete Mathematics using Python
 

 

Creating and visualizing Graphs, Digraphs, MultiGraphs and MultiDiGraph - Python methods  for reporting nodes, edges and neighbours of the given graph / digraph - Python methods for counting nodes, edges and neighbours of the given graph / digraph.

Text Books And Reference Books:

 

  1. I. N. SNEDDON, Elements of PDE’s , McGraw Hill Book company Inc. 2009.

  2. L DEBNATH , Nonlinear PDE’s for Scientists and Engineers, Birkhauser, Boston, 2008.

  3. C. L. Liu: Elements of Discrete Mathematics, Tata McGraw-Hill, 2000.

Essential Reading / Recommended Reading

 

  1. J.P. Tremblay and R.P. Manohar : Discrete Mathematical Structures with applications to computer science, McGraw Hill, 1997.

  2. K. H. Rosen and K. Krithivasan, Discrete mathematics and its applications. McGraw-Hill, 2013.

  3. S. Rao, Partial Differential Equations, Prentice Hall of India, 2009.

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 - STATISTICS (2017 Batch)

Total Teaching Hours for Semester:45
No of Lecture Hours/Week:3
Max Marks:100
Credits:2

Course Objectives/Course Description

 

This course aims at teach 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 Outcome

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

  • Understand random variables and probability distributions.
  • Distinguish discrete and continuous random variables.
  • Obtain ability compute Expected value and Variance of discrete random variable.
  • Acquire knowledge in using Binomial distribution, Poisson distribution etc.,
  • Define inferential statistics.
  • Effectively use sampling distributions in inferential statistics.

Unit-1
Teaching Hours:15
Random Variables and Expectation
 

Discrete and continuous random variables, distribution functions, probability mass and density functions, bivariate distributions, marginal and conditional distributions, expected value of a random variable, independence of random variables, conditional expectations, covariance matrix, correlation coefficients and regression, Chebyshev’s inequality, moments, moment generating functions, characteristic functions.  

Unit-2
Teaching Hours:15
Probability Distributions
 
Probability: Sample spaces, events, probability of an event, theorems on probability, conditional probability, independent events, Bayes theorem. Boole’s inequality.
Discrete Probability Distribution:  Introduction, uniform, Bernoulli, Binomial, negative Binomial, geometric, Hypergeometric and Poisson distribution. Continuous Probability Distributions: Introduction, uniform, gamma, exponential, beta and normal distributions.
 
Unit-3
Teaching Hours:15
Sampling distributions
 

t, F and chi-square distributions, standard errors and large sample distributions.  

Text Books And Reference Books:
  1. E. Freund John, Mathematical Statistics, 5th Ed., Prentice Hall of India, 2000.
  2. Gupta S.C. and Kapoor V.K., Fundamentals of mathematical Statistics, Sultan Chand and Sons, New Delhi, 2001.
  3. Ronald E. Walpole, Raymond H. Myers and Sharon L. Myers, Probability and Statistics for Engineers and Scientists, Pearson Prentice Hall, 2006.
Essential Reading / Recommended Reading
  1.  

    1.  Paul G. Hoel, Introduction to mathematical Statistics, Wiley, 2000.

    2.  M. Spiegel, Probability and statistics, Schaum’s Outline Series, 2000.

    3. Neil Weiss, Introductory Statistics, Addison-Wesley, 2002.

    4.  S. M. Ross, A first course in probability, Pearson Prentice Hall, 2005.

    5. D. Wackerly, W. Mendenhall and R. L. Scheaffer, Mathematical Statistics with Applications, Duxburry Press, 2007.     
Evaluation Pattern

.

MTH331 - MEASURE THEORY AND LEBESGUE INTEGRATION (2017 Batch)

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

Course Objectives/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 Outcome

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

  • Understand the fundamental concepts of Mathematical Analysis
  • State some of the classical theorems in of Advanced Real Analysis
  • Be familiar with measurable sets and functions
  • Integrate a measurable function
  • Understand the properties of Classical Banach Spaces

 

Unit-1
Teaching Hours:20
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:10
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:
  1. H.L. Royden and P.M. Fitzpatrick, Real Analysis, 4th ed. New Jersey: Pearson Education Inc., 2013.
Essential Reading / Recommended Reading
  1. P. R. Halmos, Measure Theory, Springer, 2014.

  2. M.E. Munroe, Introduction to measure and integration, Addison Wesley, 1959.

  3. G. de Barra, Measure theory and integration, New Age, 1981.

  4. P.K. Jain and V.P. Gupta, Lebesgue measure and integration, New Age, 1986.

  5. F. Morgan, Geometric measure theory – A beginner’s guide, Academic Press, 1988.

  6. F. Burk, Lebesgue measure and integration: An introduction, Wiley, 1997.

  7. D.H. Fremlin, Measure theory, Torres Fremlin, 2000.

  8. M.M. Rao, Measure theory and integration, Second Edition, Marcel Dekker, 2004. 

Evaluation Pattern

CIA - 50%

ESE - 50%

MTH332 - COMPUTER ORIENTED NUMERICAL METHODS USING MATLAB (2017 Batch)

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

Course Objectives/Course Description

 

This course helps students to have an in-depth knowledge of various advanced methods in numerical analysis. It also introduces MATLAB programming for scientific computations.  This includes solution of algebraic, transcendental, system of equations, and ordinary differential equations.  

Course Outcome

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

  • Derive numerical methods for approximating the solution of problems of algebraic, transcendental, linear systems and ordinary differential equations.
  • Implement a variety of numerical algorithms using appropriate technology.
  • use MATLAB to solve computational problems.

UNIT 1
Teaching Hours:20
Solution of algebraic and transcendental equations
 

Fixed point iterative method, convergence criterion, Aitken’s -process, Sturm sequence method to identify the number of real roots, Newton-Raphson methods (includes the convergence criterion for simple roots), Bairstow’s method, Graeffe’s root squaring method, Birge-Vieta method, Muller’s method. Solution of Linear System of Algebraic Equations: LU-decomposition methods (Crout’s, Choleky and Delittle methods), consistency and ill-conditioned system of equations, Tri-diagonal system of equations, Thomas algorithm.

UNIT 2
Teaching Hours:15
Numerical solution of ordinary differential equations
 

 

Initial value problems, Runge-Kutta methods of second and fourth order, multistep method, Adams-Moulton method, stability (convergence and truncation error for the above methods), boundary value problems, second order finite difference method, linear shooting method.

UNIT 3
Teaching Hours:10
Fundamentals of MATLAB
 

 

The MATLAB environment, basic operations, use of built-in functions, graphics, programming with MATLAB.

UNIT 4
Teaching Hours:15
Numerical methods with MATLAB
 

Elementary numerical methods with MATLAB, Solution to single equations and multiple non-linear equations in MATLAB.

 

Linear system of equations, Numerical differentiation and integration in MATLAB , Data fitting in MATLAB , Solution to Ordinary Differential Equations in MATLAB , Numerical differentiation and finite differences.

Text Books And Reference Books:
  1. S. C. Chapra, Applied Numerical Methods with MATLAB for Engineers and Scientists, 3rd ed., Mc Graw Hill, 2012.

  2. L.V.  Fausett, Applied Numerial Analysis Usng MATLAB, 2nd ed., Pearson Education, 2007.

  3. Rudra Pratap, Getting started with MATLAB – a quick introduction to scientists and engineers, Reprint. USE: Oxford university press, 2005.

Essential Reading / Recommended Reading

1.      S. Attaway MATLAB: A Practical Introduction to Programming and Problem Solving, 3rd edition, Elsevier, 2013.

2.      S.C.Chapra and R.P. Canale , NumerialMethodsforEngineers,5th Ed.,McGrawHill, 2006.

3.      Beers, Kenneth J. Numerical Methods for Chemical Engineering: Applications in MATLAB®. New York, NY: Cambridge University Press, November 2006.

4.      Recktenwald, Gerald W. Introduction to Numerical Methods with MATLAB®: Implementations and Applications. Upper Saddle River, NJ: Prentice-Hall, 2000.

 

5.      K. Mishra, A Handbook on Numerical Technique Lab (MATLAB Based Experiments), I.K. International Publishing House Pvt. Limited, 2007.

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

MTH333 - CLASSICAL MECHANICS (2017 Batch)

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

Course Objectives/Course Description

 

This course deals with some of the key ideas of classical mechanics. The concepts covered in the course include generalized coordinates, Lagrange’s equations, Hamilton’s equations and Hamilton - Jacobi theory.

Course Outcome

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

  • Understand and use the 3N-coordinate system made up of N-spacial coordinates, N-velocity coordinates and N-acceleration coordinates.
  • Understand the motion of mechanical systems with constraints using Lagrangian description.
  • Use Hamilton’s principle and gain proficiency in solving equations of motions.
  • Understand the use of Hamilton–Jacobi theory in solving equations of motions.

Unit-1
Teaching Hours:12
Introductory concepts
 

The mechanical system - Generalised Coordinates - constraints - virtual work - Energy and momentum.

Unit-2
Teaching Hours:20
Lagrange's equation
 

Derivation and examples - Integrals of the Motion - Small oscillations. Special Applications of Lagrange’s Equations: Rayleigh’s dissipation function - impulsive motion - velocity dependent potentials.

Unit-3
Teaching Hours:13
Hamilton's equations
 

Hamilton's principle - Hamilton’s equations - Other variational principles - phase space.

Unit-4
Teaching Hours:15
Hamilton - Jacobi Theory
 

Hamilton's Principal Function – The Hamilton - Jacobi equation - Separability.

Text Books And Reference Books:

Donald T. Greenwood, Classical Dynamics, Reprint, USA: Dover Publications, 2012.

Essential Reading / Recommended Reading
  1. H. Goldstein, Classical Mechanics, Second edition, New Delhi : Narosa Publishing House,  2001.
  2. N.C. Rana and P.S. Joag, Classical Mechanics, 29th Reprint, New Delhi: Tata McGraw- Hill, 2010.
  3. J.E. Marsden, R. Abraham, Foundations of Mechanics, 2nd ed., American Mathematical Society, 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

ESE

 

Basic, conceptual and analytical knowledge of the subject

50

Total

100

MTH334 - LINEAR ALGEBRA (2017 Batch)

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

Course Objectives/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 Outcome

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

 

  • Have thorough understanding of the Linear transformations

  • Demonstrate the elementary canonical forms, rational and Jordan forms.

  • Apply the inner product space

  • Express familiarity in using bilinear forms

Unit-1
Teaching Hours:15
Linear Transformations and Determinants
 

 

Vector Spaces: Recapitulation, 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.

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

MTH335 - ADVANCED GRAPH THEORY (2017 Batch)

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

Course Objectives/Course Description

 

Domination of Graphs, digraph theory, perfect graphs and chromatic graph theory are dealt with in the detail in this course.

Course Outcome

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

  • Thorough understanding of the concepts in domination and perfect graphs.
  • Familiarity in implementing the acquired knowledge appropriately.
  • Mastery in employing proof techniques.

Unit-1
Teaching Hours:15
Domination in Graphs
 

Domination in Graphs, Bounds in terms of Order, Bounds in terms of Order, Degree and Packing, Bounds in terms of Order and Size, Bounds in terms of Degree, Diameter and Girth, Bounds in terms of Independence and Covering.

Unit-2
Teaching Hours:15
Chromatic Graph Theory
 

T-Colourings,  L(2,1)-colourings,  Radio Colourings,  Hamiltonian Colourings, Domination and Colourings.

 

 

 

Unit-3
Teaching Hours:15
Perfect Graphs
 

The Perfect Graph Theorem, Chordal Graphs Revisited, Other Classes of Perfect Graphs, Imperfect Graphs, The Strong Perfect Graph Conjecture.

Unit-4
Teaching Hours:15
Eigenvalues of Graphs
 

The Characteristic Polynomial, Eigenvalues and Graph Parameters, Eigenvalues of Regular Graphs, Eigenvalues and Expanders, Strongly Regular Graphs.

Text Books And Reference Books:

1.      D.B. West, Introduction to Graph Theory, New Delhi: Prentice-Hall of India, 2011.

2.      T.W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs. Reprint, CRC Press, 2000.

3.      J. Bang-Jensen and G. Gutin, Digraphs. London: Springer, 2009.

 

4.      G. Chartrand and P. Zhang, Chromatic Graph Theory. New York: CRC Press, 2009.

Essential Reading / Recommended Reading

1.      B. Bollabas, Modern Graph Theory, Springer, New Delhi, 2005.

2.      F. Harary, Graph Theory, New Delhi: Narosa, 2001.

3.              G. Chartrand and P.Chang, Introduction to Graph Theory, New Delhi: Tata McGraw-Hill,2006.

4.      J. A. Bondy and U.S.R. Murty, Graph Theory, Springer, 2008

5.      J. Clark and D.A. Holton, A First Look At Graph Theory, Singapore: World Scientific, 2005.

6.      R. Balakrishnan and K Ranganathan, A Text Book of Graph Theory, New Delhi: Springer, 2008.

7.      R. Diestel, Graph Theory, New Delhi: Springer, 2006.

8.      M. Bona, A walk through combinatorics, World scientific, 2011. 

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

MTH371 - INTERNSHIP IN PG MATHEMATICS COURSE (2017 Batch)

Total Teaching Hours for Semester:45
No of Lecture Hours/Week:0
Max Marks:0
Credits:2

Course Objectives/Course Description

 

The objective of this course is to provide the students an opportunity to gain work experience in the relevant institution, connected to their subject of study. The experienced gained in the workplace will give the students a competetive edge in their career.

Course Outcome

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

·         Expose students to the field of their professional interest

·         Give an opportunity to get practical experience in the field of their interest

·         Strengthen the curriculum based on internship feedback where relevant

 

·         Help student choose their career through practical experience 

Unit-1
Teaching Hours:45
Internship in PG Mathematics course
 

M.Sc. Mathematics students have to undertake a mandatory internship of not less than 45 working days at any of the following: reputed research centres, recognized educational institutions, summer research fellowships, programmes like M.T.T.S or any other approved by the P.G. coordinator and H.O.D.  

    The internship is to be undertaken at the end of second semester (during first year vacation). The report submission and the presentation on the report will be held during the third semester and the credits will appear in the mark sheet of third semester.    

    The students will have to give an internship proposal with the following details: Organization where the student proposes to do the internship, reasons for the choice, nature of internship, period on internship, relevant permission letters, if available, name of the mentor in the organization, email, telephone and mobile numbers of the person in the organization with whom Christ University could communicate matters related to internship. Typed proposals will have to be given at least one month before the end of the second semester.   

    The coordinator of the programme in consultation with the HOD will assign faculty members from the department as guides at least two weeks before the end of second semester.

    The students will have to be in touch with the guides during the internship period either through personal meetings, over the phone or through email.

    At the place of internship, students are advised to be in constant touch with their mentors.

    At the end of the required period of internship, the candidates will submit a report as per the guidelines given by the department. The report should be submitted within first 10 days of the reopening of the University for the third semester.

      Within 20 days from the day of reopening, the department must hold a presentation by the students. During the presentation the guide or a nominee of the guide should be present and be one of the evaluators. Students should preferably be encouraged to make a power point presentation of their report. A minimum of 10 minutes should be given for each of the presenter. The maximum limit is left to the discretion of the evaluation committee. Students will get 2 credits on successful completion of internship.

Text Books And Reference Books:

.

Essential Reading / Recommended Reading

.

Evaluation Pattern

.

MTH431 - DIFFERENTIAL GEOMETRY (2017 Batch)

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

Course Objectives/Course Description

 

Differential geometry is the study of geometric properties of curves, surfaces, and their higher dimensional analogues using the methods of calculus. On successful completion of this module students will have acquired an active knowledge and understanding of the basic concepts of the geometry of curves and surfaces in three-dimensional Euclidean space and will be acquainted with the ways of generalising these concepts to higher dimensions

Course Outcome

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

  • Obtain sound knowledge in understanding the basic concepts in geometry of curves and surfaces in Euclidean space, especially E3.
  • Acquire mastery in solving typical problems associated with the theory.
  • Gain sufficient knowledge for generalizing these concepts to higher dimensions.

UNIT 1
Teaching Hours:15
Calculus on Euclidean Geometry
 

Euclidean Space - Tangent Vectors  - Directional derivatives - Curves in E3 - 1-Forms - Differential Forms - Mappings.

UNIT 2
Teaching Hours:15
Frame Fields and Euclidean Geometry
 

Dot product - Curves - vector field - The Frenet Formulas - Arbitrary speed curves -  cylindrical helix - Covariant Derivatives - Frame fields - Connection Forms - The Structural equations.

UNIT 3
Teaching Hours:15
Euclidean Geometry and Calculus on Surfaces
 

Isometries of E3 - The derivative map of an Isometry - Surfaces in E3 - patch computations - Differential functions and Tangent vectors - Differential forms on a surface - Mappings of Surfaces.

UNIT 4
Teaching Hours:15
Shape Operators
 

The Shape operator of M in E3 - Normal Curvature - Gaussian Curvature - Computational Techniques - Special curves in a surface - Surfaces of revolution.

Text Books And Reference Books:

B.O’Neill, Elementary Differential geometry, 2nd revised ed., New York: Academic Press, 2006.

Essential Reading / Recommended Reading
  1. J.A. Thorpe, Elementary topics in differential geometry, 2nd ed., Springer, 2004.
  2. A. Pressley, Elementary differential geometry, 2nd ed., Springer, 2010.
  3. Mittal and Agarwal, Differential geometry, 36th ed., Meerut: Krishna Prakashan Media (P) Ltd., 2010.
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

MTH432 - COMPUTATIONAL FLUID DYNAMICS (2017 Batch)

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

Course Objectives/Course Description

 

This course helps the students to understand the basic concepts of heat transfer, types of convection shear and thermal instability of linear and non-linear problems, dimensional analysis. The flow problems are analyses using finite element method.

Course Outcome

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

 

  • Understand both flow physics and mathematical properties of governing Navier-Stokes equations and define proper boundary conditions for solution.

  • An introduction to the theory and practice of the finite element method. Experience with writing a simple finite element solver for an ordinary differential equation in MATLAB.

  • Understanding of physics of compressible and incompressible fluid flows.

  • Ability to solve the fluid flow equations using Finite element method.

Unit-1
Teaching Hours:12
Recapitulation
 

 

Review of classification of partial differential equations, classification of boundary conditions,  numerical analysis, basic governing equations of fluid mechanics.

Unit-2
Teaching Hours:18
Finite Difference Methods
 

Derivation of finite difference methods, finite difference method to parabolic, hyperbolic and elliptic equations, finite difference method to nonlinear equations, coordinate transformation for arbitrary geometry, Central schemes with combined space-time discretization-Lax-Friedrichs, Lax-Wendroff, MacCormack methods, Artificial compressibility method, pressure correction method – Lubrication model, Convection dominated flows – Euler equation – Quasilinearization of Euler equation, Compatibility relations, nonlinear Burger equation

Unit-3
Teaching Hours:12
Finite Volume Methods
 

General introduction, Node-centered-control volume, Cell-centered-control volume and average volume, Cell-Centred scheme, Cell-Vertex scheme, Structured and Unstructured FVMs, Second and Fourth order approximations to the convection and diffusion equations (One and Two-dimensional examples)

Unit-4
Teaching Hours:18
Finite Element Methods
 

Introduction to finite element methods, one-and two-dimensional bases functions – Lagrange and Hermite polynomials elements, triangular and rectangular elements, Finite element method for one-dimensional problem: model boundary value problems, discretization of the domain, derivation of elemental equations and their connectivity, composition of boundary conditions and solutions of the algebraic equations. Finite element method for two-dimensional problems: model equations, discretization, interpolation functions, evaluation of element matrices and vectors and their assemblage.

Text Books And Reference Books:

 

  1. T. Chung, Computational Fluid Dynamics, Cambridge University Press, 2003.

  2. J. Blazek, Computational Fluid Dynamics, Elsevier Science, 2001.

  3. H. Lomax, T. Pulliam and D. Zingg, Fundamentals of Computational Fluid Dynamics, NASA Report, 2006.

Essential Reading / Recommended Reading

 

  1. C. Fletcher, Computational Techniques for Fluid Dynamics 1, Springer Berlin Heidelberg, 1991.

  2. C. Fletcher, Computational Techniques for Fluid Dynamics 2, Springer Berlin Heidelberg, 1991.

  3. D. Anderson, R. Pletcher, J. Tannehill and, Computational Fluid Mechanics and Heat Transfer, McGraw Hill Book Company, 2010.

  4. K. Muralidhar and T. Sundararajan, Computational Fluid Flow and Heat Transfer, Narosa Publishing House, 2010.

  5. W. Ames, Numerical Method for Partial Differential Equation, Academic Press, 2008.

  6. T. Cebeci and P. Bradshaw, Physical and Computational Aspects of Convective Heat Transfer, Springer-Verlag, 2005.

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

MTH433 - FUNCTIONAL ANALYSIS (2017 Batch)

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

Course Objectives/Course Description

 

This abstract course imparts an in-depth analysis of Banach spaces, Hilbert spaces, conjugate spaces, etc. This course also includes a few important applications of functional analysis to other branches of both pure and applied mathematics.

Course Outcome

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

  • Explain the fundamental concepts of functional analysis.
  • Understand the approximation of continuous functions.
  • Understand concepts of Hilbert and Banach spaces with l2 and lp spaces serving as examples.
  • Understand the definitions of linear functional and prove the Hahn-Banach theorem, open mapping theorem, uniform boundedness theorem, etc.
  • Define linear operators, self adjoint, isometric and unitary operators on Hilbert spaces.

Unit-1
Teaching Hours:15
Banach spaces
 

Normed linear spaces, Banach spaces, continuous linear transformations, isometric isomorphisms, functionals and the Hahn-Banach theorem, the natural embedding of a normed linear space in its second dual.

Unit-2
Teaching Hours:12
Mapping theorems
 

The open mapping theorem and the closed graph theorem, the uniform boundedness theorem, the conjugate of an operator.

Unit-3
Teaching Hours:15
Inner products
 

Inner products, Hilbert spaces, Schwarz inequality, parallelogram law, orthogonal complements, orthonormal sets, Bessel’s inequality, complete orthonormal sets. 

 

Unit-4
Teaching Hours:18
Conjugate space
 

The conjugate space, the adjoint of an operator, self-adjoint, normal and unitary operators, projections, finite dimensional spectral theory.

 

Text Books And Reference Books:

G.F. Simmons, Introduction to topology and modern Analysis, Reprint, Tata McGraw-Hill, 2004.

Essential Reading / Recommended Reading
  1. K. Yoshida, Functional analysis, 6th ed., Springer Science and Business Media, 2013.
  2. Kreyszig, Introductory functional analysis with applications, 1st ed., John Wiley, 2007.
  3. B.V. Limaye, Functional analysis, 3rd ed., New Age International, 2014.
  4. W. Rudin, Functional analysis, 2nd ed., McGraw Hill, 2010.
  5. S. Karen, Beginning functional analysis, Reprint, Springer, 2002.
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

MTH441 - CALCULUS OF VARIATIONS AND INTEGRAL EQUATIONS (2017 Batch)

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

Course Objectives/Course Description

 

This course concerns the analysis and applications of calculus of variations and integral equations. Applications include areas such as classical mechanics and differential equations. 

Course Outcome

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

  • Derive some classical differential equations by using principles of calculus of variations.
  • Knowledge of Variational Problems, Euler-Lagrange Condition, Second Variation,Generalizations of the Variational Problem.
  • Able to find maximum or minimum of a functional using calculus of variations technique.
  • Able to solve Voterra integral equations and Fredholm integral equations.
  • Able to Reduce the differential equations to integral equations.

Unit-1
Teaching Hours:18
Euler equations and variational notations
 

Maxima and minima, method of Lagrange multipliers, the simplest case, Euler equation, extremals, stationary function, geodesics, Brachistochrone problem, natural boundary conditions and transition conditions, variational notation, the more general case. 

Unit-2
Teaching Hours:16
Advanced variational problems
 

Galerkian Technique, the Rayleigh-Ritz method.

Unit-3
Teaching Hours:12
Linear integral equations
 

Definitions, integral equation, Fredholm and Volterra equations, kernel of the integral equation, integral equations of different kinds, relation between differential and integral equations, symmetric kernels, the Green’s function. 

Unit-4
Teaching Hours:14
Methods for solutions of linear integral equations
 

 

Fredholm equations with separable kernels, homogeneous integral equations, characteristic values and characteristic functions of integral equations, Hilbert-Schmidt theory, iterative methods for solving integral equations of the second kind, the Neumann series.     

Text Books And Reference Books:
  1. F.B. Hildebrand, Methods of Applied Mathematics, New York: Dover, 1992.

R.P. Kanwal, Linear Integral Equations: Theory and Techniques, New York: Birkhäuser,  2013.

Essential Reading / Recommended Reading
    1. B. Dacorogna, Introduction to the Calculus of Variations, London: Imperial College Press, 2004.

    2. F. Wan, Introduction to the Calculus of Variations and Its Applications, New York: Chapman/Hall, 1995.

    3. J. Jost and X. Li-Jost, Calculus of Variations, Cambridge: Cambridge University Press, 2008.

    4. C. Corduneanu, Integral Equations and Applications, Cambridge: Cambridge University Press, 2008.  

    5. A. J. Jerri, Introduction to integral equations with applications. Sampling Publishing, 2007.

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

MTH448 - COMBINATORIAL MATHEMATICS (2017 Batch)

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

Course Objectives/Course Description

 

Combinatorics deals with the existence of certain configurations in a structure and when it exists it counts the number of such configurations. In this course we deal with the basic concepts such as Permutations and Combinations, Generating Functions, Recurrence Relations, The Principle of Inclusion and Exclusion including Polya’s theory.

Course Outcome

After completing this course, the student will be able to:

  • Understand the rules of Sum and Product of Permutations and Combinations.

  • Discuss distributions of Distinct Objects into Non-distinct Cells and Partitions of Integers.

  • Identify Solutions by the technique of Generating Functions and Recurrence Relations with Two Indices.

  • Understand the concepts of Permutations with Restrictions on Relative Positions and the Rook Polynomials.

  • Enumerate configuration using Polya’s Theory.

Unit-1
Teaching Hours:20
Permutations, Combinations and Generating Functions
 

Introduction - The rules of Sum and Product - Permutations - Combinations - Distributions of Distinct Objects - Distributions of Non distinct Objects.  Generating Functions for Combinations - Enumerators for Permutations – Distributions of Distinct Objects into Non distinct Cells - Partitions of Integers - Elementary Relations.

Unit-2
Teaching Hours:12
Recurrence Relations
 

 

Introduction - Linear Recurrence relations with Constant Coefficients - Solution by the technique of Generating Functions - Recurrence Relations with Two Indices.

Unit-3
Teaching Hours:13
The Principle of Inclusion and Exclusion
 

 

Introduction - The Principle of Inclusion and Exclusion - The General Formula - Derangements - Permutations with Restrictions on Relative Positions - The Rook Polynomials.

Unit-4
Teaching Hours:15
Polya?s Theory of Counting
 

Introduction - Equivalence Classes under a Permutation Group - Equivalence Classes of Functions -Weights and Inventories of Functions - Polya’s Fundamental Theorem - Generalization of Polya’s Theorem.

Text Books And Reference Books:

C. L. Liu, Introduction to Combinatorial Mathematics, McGraw-Hill Inc., New york,1968.

Essential Reading / Recommended Reading
  1. J. H. Van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge University Press, 2001.

  2. Titu Andreescu and ZumingFeng, A Path to Combinatorics, Springer Science & Business Media, 2004.

  3. R. L. Graham, D. E. Kunth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed., Addision Wesley, 1994.

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

MTH451 - PROJECT (2017 Batch)

Total Teaching Hours for Semester:30
No of Lecture Hours/Week:2
Max Marks:100
Credits:2

Course Objectives/Course Description

 

The objective of this course is to develop positive attitude, knowledge and competence for the research in Mathematics.  

Course Outcome

Through this project students will develop analytical and computational skills

Unit-1
Teaching Hours:30
PROJECT
 

Students are exposed to the mathematical software packages like Scilab, Maxima, Octave, OpenFOAM, Mathematica and Matlab. Students are given a choice of topic either on Fluid Mechanics or Graph theory or any other topic from other fields with the approval of HOD / Coordinator. Each candidate will work under the supervision of the faculty.  Coordinator will allot the supervisor for each candidate in consultation with the HOD at the end of third semester.

Project need not be based on original research work. Project could be based on the review of advanced text book or advanced research papers.

Each candidate has to submit a dissertation on the project topic followed by viva voce examination. The viva voce will be conducted by the committee constituted by the head of the department which will have an external and an internal examiner. The student must secure 50% of the marks to pass the project examination.  The candidates who fail must redo the project as per the university regulation.

Time line for Project:

Third Semester

Fourth Semester

June

July

August - October

November

December

January

Allotment of Guides

Selecting 3 journal papers

Literature

Review

Reviewing the 3 journal papers

Presentation on complete review of the 3 papers

Extension and Report Writing

Final Presentation and Viva

 

 

Text Books And Reference Books:

.

Essential Reading / Recommended Reading

.

Evaluation Pattern

Assessment:

Project is evaluated based on the parameters given below:

 

Parameter

Maximum points

Project related literature is communicated / published

05

Extension

10

Journal Club:

Attendance - 4 marks

Presentation and Questions - 6 marks

10

Progress Presentation

20

Project Report

30

Viva

25

Total

100