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1 Semester - 2020 - Batch | Course Code |
Course |
Type |
Hours Per Week |
Credits |
Marks |
MTH111 | RESEARCH METHODOLOGY | Skill Enhancement Courses | 2 | 2 | 50 |
MTH131 | REAL ANALYSIS | Core Courses | 4 | 4 | 100 |
MTH132 | ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS | Core Courses | 4 | 4 | 100 |
MTH133 | ADVANCED ALGEBRA | Core Courses | 4 | 4 | 100 |
MTH134 | FLUID MECHANICS | Core Courses | 4 | 4 | 100 |
MTH135 | ELEMENTARY GRAPH THEORY | Core Courses | 4 | 4 | 100 |
MTH151 | INTRODUCTION TO FOSS TOOLS | Core Courses | 3 | 3 | 50 |
2 Semester - 2020 - Batch | Course Code |
Course |
Type |
Hours Per Week |
Credits |
Marks |
MTH211 | MACHINE LEARNING | Skill Enhancement Courses | 2 | 2 | 50 |
MTH231 | GENERAL TOPOLOGY | Core Courses | 4 | 4 | 100 |
MTH232 | COMPLEX ANALYSIS | Core Courses | 4 | 4 | 100 |
MTH233 | LINEAR ALGEBRA | Core Courses | 4 | 4 | 100 |
MTH234 | ADVANCED FLUID MECHANICS | Core Courses | 4 | 4 | 100 |
MTH235 | ALGORITHMIC GRAPH THEORY | Core Courses | 4 | 4 | 100 |
MTH251 | PYTHON PROGRAMMING FOR MATHEMATICS | Core Courses | 3 | 3 | 50 |
3 Semester - 2019 - Batch | Course Code |
Course |
Type |
Hours Per Week |
Credits |
Marks |
MTH311 | INTRODUCTION TO FREE AND OPEN-SOURCE SOFTWARE (FOSS) TOOLS : (GNU OCTAVE) | Add On Courses | 3 | 2 | 50 |
MTH331 | MEASURE THEORY AND LEBESGUE INTEGRATION | - | 4 | 4 | 100 |
MTH332 | NUMERICAL ANALYSIS | - | 4 | 4 | 100 |
MTH333 | CLASSICAL MECHANICS | - | 4 | 4 | 100 |
MTH334 | LINEAR ALGEBRA | - | 4 | 4 | 100 |
MTH335 | ADVANCED GRAPH THEORY | - | 4 | 4 | 100 |
MTH351 | NUMERICAL METHODS USING PYTHON | - | 3 | 3 | 50 |
MTH381 | INTERNSHIP | - | 0 | 2 | 0 |
4 Semester - 2019 - 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 |
MTH4401 | CALCULUS OF VARIATIONS AND INTEGRAL EQUATIONS | - | 4 | 4 | 100 |
MTH4402 | MAGNETOHYDRODYNAMICS | - | 4 | 4 | 100 |
MTH4403 | WAVELET THEORY | - | 4 | 4 | 100 |
MTH4404 | MATHEMATICAL MODELLING | - | 4 | 4 | 100 |
MTH4405 | ATMOSPHERIC SCIENCE | - | 4 | 4 | 100 |
MTH4406 | ADVANCED LINEAR PROGRAMMING | - | 4 | 4 | 100 |
MTH4407 | DESIGN AND ANALYSIS OF ALGORITHMS | - | 4 | 4 | 100 |
MTH4408 | INTRODUCTION OF THEORY OF MATROIDS | - | 4 | 4 | 100 |
MTH4409 | TOPOLOGICAL GRAPH THEORY | - | 4 | 4 | 100 |
MTH4410 | ALGEBRAIC GRAPH THEORY | - | 4 | 4 | 100 |
MTH451 | NUMERICAL METHODS FOR BOUNDARY VALUE PROBLEM USING PYTHON | - | 3 | 3 | 50 |
MTH481 | PROJECT | - | 2 | 2 | 100 |
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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 Algebra, Topology, Analysis, Discrete Mathematics and Number Theory/Cryptography along with the basic applied course ordinary and partial differential equations. In the third and fourth semester focus is on the special courses, elective courses and skill-based courses including Measure Theory and Lebesgue Integration, Functional Analysis, Computational Fluid Dynamics, Advanced Graph Theory. Classical Mechanics, Linear Algebra and Numerical Analysis. 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 in these topics. To gain proficiency in software skills, Mathematics Lab papers are introduced in each semester. Special importance is given to the skill enhancement courses Teaching Technology and Research Methodology in Mathematics and service learning, Introduction to Free and Open-Source Software (FOSS) Tools: (GNU Octave) and Statistics. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Programme Outcome/Programme Learning Goals/Programme Learning Outcome: PO1: Engage in continuous reflective learning in the context of technology and scientific advancement.PO2: Identify the need and scope of the Interdisciplinary research. PO3: Enhance research culture and uphold the scientific integrity and objectivity PO4: Understand the professional, ethical and social responsibilities PO5: Understand the importance and the judicious use of technology for the sustainability of the environment PO6: Enhance disciplinary competency, employability and leadership skills Programme Specific Outcome: PSO1: Attain mastery over pure and applied branches of Mathematics and its applications in multidisciplinary fieldsPSO2: Demonstrate problem solving, analytical and logical skills to provide solutions for the scientific requirements PSO3: Develop critical thinking with scientific temper. PSO4: Communicate the subject effectively and express proficiency in oral and written communications to appreciate innovations in research PSO5: Understand the importance and judicious use of mathematical software's for the sustainable growth of mankind PSO6: Enhance the research culture in three areas viz. Graph theory, Fluid Mechanics and Data Science and uphold the research integrity and objectivity | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Assesment Pattern | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Assessment Pattern
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Examination And Assesments | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
EXAMINATION AND ASSESSMENTS (Theory)
EXAMINATION AND ASSESSMENTS (Practicals) The course is evaluated based on continuous internal assessment (CIA). The parameters for evaluation under each component and the mode of assessment are given below:
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MTH111 - RESEARCH METHODOLOGY (2020 Batch) | |
Total Teaching Hours for Semester:30 |
No of Lecture Hours/Week:2 |
Max Marks:50 |
Credits:2 |
Course Objectives/Course Description |
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Course Description: This course is intended to assist students in acquiring necessary skills on the use of research methodology. Also, the students are exposed to the principles, procedures and techniques of planning and implementing the research project. Course Objective: This course will help the learner to: COBJ1. know the general research methods COBJ2. get hands on experience in methods of research that can be employed for research in Mathematics |
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Course Outcome |
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After completing this course, the student will be able to: CO1. foster a clear understanding about research design that enables students in analyzing and evaluating the published research CO2. obtain necessary skills in understanding the mathematics research articles CO3. acquire skills in preparing scientific documents using MS Word, Mathtype, Open Office Math editor, yEd Graph Editor and LaTeX. |
Unit-1 |
Teaching Hours:10 |
Research Methodology
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Introduction to research and research methodology, Scientific methods, Choice of research problem, Literature survey and statement of research problem, Reporting of results, Roles and responsibilities of research student and guide. | |
Unit-2 |
Teaching Hours:10 |
Mathematical research methodology
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Introducing mathematics Journals, Reading a Journal article, Mathematics writing skills. -Standard Notations and Symbols, Using Symbols and Words, Organizing a paper, Defining variables, Symbols and notations, Different Citation Styles, IEEE Referencing Style in detail, Tools for checking Grammar and Plagiarism. | |
Unit-3 |
Teaching Hours:10 |
TypeSetting research articles
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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. | |
Text Books And Reference Books: . | |
Essential Reading / Recommended Reading
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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: | |
MTH131 - REAL ANALYSIS (2020 Batch) | |
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
Max Marks:100 |
Credits:4 |
Course Objectives/Course Description |
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Course Description: This course will help students to understand the concepts of functions of single and several variables. This course includes such concepts as Riemann-Stieltjes integral, sequences and series of functions, Special Functions, and the Implicit Function Theorem.
Course objectives: This course will help the learner to COBJ1. Develop in a rigorous and self-contained manner the elements of real variable functions COBJ2. Integrate functions of a real variable in the sense of Riemann – Stieltjes COBJ3. Classify sequences and series of functions which are pointwise convergent and uniform Convergent COBJ4. Demonstrate the ability to manipulate and use of special functions COBJ5. Use and operate functions of several variables. |
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Course Outcome |
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On successful completion of the course, the students should be able to CO1. determine the Riemann-Stieltjes integrability of a bounded function. CO2. recognize the difference between pointwise and uniform convergence of sequence/series of functions. CO3. illustrate the effect of uniform convergence on the limit function with respect to continuity, differentiability, and integrability. CO4. analyze and interpret the special functions such as exponential, logarithmic, trigonometric and Gamma functions. CO5. gain in depth knowledge on functions of several variables and the use of Implicit Function Theorem. |
UNIT 1 |
Teaching Hours:15 |
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The Riemann-Stieltjes Integration
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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 |
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Sequences and Series of Functions
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Pointwise and uniform convergence, Uniform Convergence: Continuity, Integration and Differentiation, Equicontinuous Families of Functions, The Stone-Weierstrass Theorem | |||||||||||||||||||||||||||||
UNIT 3 |
Teaching Hours:15 |
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Some Special Functions
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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 |
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Functions of Several Variables
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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
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Evaluation Pattern
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MTH132 - ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS (2020 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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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 objectives : This course will help the learner to COBJ1. Solve adjoint differential equations, hypergeometric differential equation and power series. COBJ2. Solve partial differential equation of the type heat equation, wave equation and Laplace equations. COBJ3. Also solving initial boundary value problems. |
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Course Outcome |
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On successful completion of the course, the students should be able to CO1. Understand concept of Linear differential equation, Fundamental set Wronskian. CO2. Understand the concept of Liouvilles theorem, Adjoint and Self Adjoint equation, Lagrange's Identity, Green’s formula, Eigen value and Eigen functions. CO3. Identify ordinary and singular point by Frobenius Method, Hyper geometric differential equation and its polynomial. CO4. Understand the basic concepts and definition of PDE and also mathematical models representing stretched string, vibrating membrane, heat conduction in rod. CO5. Demonstrate on the canonical form of second order PDE. CO6. Demonstrate initial value boundary problem for homogeneous and non-homogeneous PDE. CO7. Demonstrate on boundary value problem by Dirichlet and Neumann problem. |
UNIT 1 |
Teaching Hours:20 |
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Linear Differential Equations
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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, Eigenvalues and Eigenfunctions, related examples. | |||||||||||||||||||||||||||||
UNIT 2 |
Teaching Hours:10 |
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Power series solutions
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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 |
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Partial Differential Equations
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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 |
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Solutions of PDE
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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:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH133 - ADVANCED ALGEBRA (2020 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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Course Description: This course enables students to understand the intricacies of advanced areas in algebra. This includes a study of advanced group theory, Euclidean rings, polynomial rings and Galois theory. Course objectives: This course will help the learner to COBJ1. enhance the knowledge on advanced level algebra COBJ2. understand the proof techniques for the theorems on advanced group theory, Rings and Galois Theory |
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Course Outcome |
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On successful completion of the course, the students should be able to CO1. Demonstrate knowledge of conjugates, the Class Equation and Sylow theorems CO2. Demonstrate knowledge of polynomial rings and associated properties CO3. Derive and apply Gauss Lemma, Eisenstein criterion for irreducibility of rationals CO4. Demonstrate the characteristic of a field and the prime subfield; CO5. Demonstrate 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 |
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Advanced Group Theory
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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 |
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Rings
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Euclidean Ring, Polynomial rings, polynomials rings over the rational field, polynomial rings over commutative rings. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Fields
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Extension fields, roots of polynomials, construction with straightedge and compass, more about roots. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Galois theory
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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
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Evaluation Pattern
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MTH134 - FLUID MECHANICS (2020 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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Course Description: This course aims at introducing the fundamentals of fluid mechanics. This course aims at imparting the knowledge on tensors, kinematics of fluid, incompressible flow, boundary layer flows and classification of non-Newtonian fluids.
Course objectives: This course will help the learner to COBJ1. understand the basic concept of tensors and their representative COBJ2. physics and mathematics behind the basics of fluid mechanics COBJ3. familiar with two or three dimensional incompressible flows COBJ4. classifications of non-Newtonian fluids COBJ5. familiar with standard two or three dimensional viscous flows |
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Course Outcome |
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On successful completion of the course, the students should be able to CO1. confidently manipulate tensor expressions using index notation, and use the divergence theorem and the transport theorem. CO2. able to understand the basics laws of Fluid mechanics and their physical interpretations. CO3. able to understand two or three dimension flows incompressible flows. CO4. able to understand the viscous flows, their mathematical modelling and physical interpretations. |
Unit-1 |
Teaching Hours:15 |
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Cartesian tensors and continuum hypothesis
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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 |
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Stress, Strain and basic physical laws
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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 |
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One, Two and Three Dimensional Invisid Incompressible Flow
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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 |
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Two Dimensional Flows of Viscous Fluid
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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:
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Essential Reading / Recommended Reading
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Evaluation Pattern Examination and Assessments
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MTH135 - ELEMENTARY GRAPH THEORY (2020 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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Course Description: This course is an introductory course to the basic concepts of Graph Theory. This includes definition of graphs, vertex degrees, directed graphs, trees, distances, connectivity and paths. Course objectives:This course will help the learner to COBJ1. know the history and development of graph theory COBJ2. understand all the elementary concepts and proof techniques in Graph Theory |
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Course Outcome |
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Course outcomes: The successful completion of this course the students will be able to: CO1. write precise and accurate mathematical definitions of basics concepts in graph theory CO3. provide appropriate examples and counterexamples to illustrate the basic concepts CO3. demonstrate and apply various proof techniques in proving theorems in graph theory CO4. showcase mastery in using graph drawing tools |
Unit-1 |
Teaching Hours:15 |
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Introduction to Graphs
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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 |
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Vertex Degrees and Directed Graphs
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Counting and Bijections, Extremal Problems, Graphic Sequences, Directed Graphs, Vertex Degrees, Eulerian Digraphs, Orientations and Tournaments. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Trees and Distance
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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 |
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Connectivity and Paths
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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
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Evaluation Pattern
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MTH151 - INTRODUCTION TO FOSS TOOLS (2020 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:45 |
No of Lecture Hours/Week:3 |
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Max Marks:50 |
Credits:3 |
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Course Objectives/Course Description |
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Course Description: This course aims at introducing the mathematical software packages “WxMaxima” and “Scilab”, for learning basic operations on matrix manipulation, plotting graphs etc.,. These software packages will also help students to solve problems / applied problems on Mathematics. Course objectives: This course will help the learner to: COBJ1. use the FOSS tool WxMaxima effectively. COBJ2. use the FOSS tool Scilab effectively |
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Course Outcome |
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On successful completion of the course, the students should be able to CO1. Use the basic commands in WxMaxima including the 2D and 3D plots CO2. Have a strong command on the inbuilt commands required for the learning and analyzing mathematics CO3. Solve problems / applied problems on mathematics by using Scilab. |
Unit-1 |
Teaching Hours:15 |
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Introduction to WxMaxima
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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:15 |
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Introduction to Scilab
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Introduction to Scilab and commands connected with Matrices - Computations with Matrices - 2D Plots: plot, plot2d, plot2d2, plot2d3, Histplot, Matplot, Grayplot, 3D Plots: plot3d,, plot3d1, contour, hist3d- Script Files and Function Files | |||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Solving problems using Scilab / WxMaxima
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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 - Display non-Fibonacci series | |||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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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:
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MTH211 - MACHINE LEARNING (2020 Batch) | |||||||||||||||||||||||||
Total Teaching Hours for Semester:30 |
No of Lecture Hours/Week:2 |
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Max Marks:50 |
Credits:2 |
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Course Objectives/Course Description |
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Course Description: This course aims at introducing the elementary notions on Machining learning and focuses on some simple application of machine-learning, algorithms on supervised machine learning and unsupervised learning. Course Objective: This course will help the learner to: COBJ1. Be proficient on the idea of machine learning COBJ2. Implement Supervised Machine Learning Algorithms COBJ3. Handle computational skills related to unsupervised learning and Preprocessing |
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Course Outcome |
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After completing this course, the student will be able to: CO1. Demonstrate some simple applications of Machine learning. CO2. Use supervised machine learning algorithms on k-nearest neighbor, linear model, decisions trees. CO3. Showcase the skill using the unsupervised learning and preprocessing. |
Unit-1 |
Teaching Hours:7 |
Introduction to Machine Learning
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Introduction - Simple Machine Learning Applications: Classifying Iris Species: Meet the data, Training and Testing Data, Pair Plot of Iris dataset - k-nearest neighbours model, Evaluating model. | |
Unit-2 |
Teaching Hours:13 |
Supervised Learning
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Classification and Regression - Generalization, Overfitting and Underfitting - Relation of Model Complexity to Dataset Size, Supervised Machine Learning Algorithms: k-Nearest Neighbour algorithm: k-Neighbors classification, k-neighbors regression, Strengths, Weakness and parameters of k-NN algorithm, Linear Models: Linear models for regression, Linear models for classification, Linear models for multiclass classification, Strengths, Weakness and parameters of linear models, Decision Trees: Building decision trees, controlling complexity of decision trees, Analyzing decision trees, Strengths, Weakness and parameters of decision trees. | |
Unit-3 |
Teaching Hours:10 |
Unsupervised Learning and Preprocessing
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Types of unsupervised learning, Challenges in unsupervised learning, Preprocessing and scaling: Different kinds of preprocessing, Applying Data transformations, Scaling training and test data, Principal component analysis, Non-negative matrix factorization. | |
Text Books And Reference Books: A. C. Müller and S. Guido, Introduction to machine learning with Python, O’Reilly, 2017. | |
Essential Reading / Recommended Reading
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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: | |
MTH231 - GENERAL TOPOLOGY (2020 Batch) | |
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
Max Marks:100 |
Credits:4 |
Course Objectives/Course Description |
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Course Description: This course deals with the essentials of topological spaces and their properties in terms of continuity, connectedness, compactness etc.,. Course objectives: This course will help the learner to: COBJ1. provide precise definitions and appropriate examples and counter examples of fundamental concepts in general topology COBJ2. acquire knowledge about generalization of the concept of continuity and related properties COBJ3. appreciate the beauty of deep mathematical results such as Uryzohn’s lemma and understand and apply various proof techniques |
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Course Outcome |
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On successful completion of the course, the students should be able to CO1. define topological spaces, give examples and counterexamples on concepts like open sets, basis and subspaces CO2. establish equivalent definitions of continuity and apply the same in proving theorems CO3. understand the concepts of metrizability, connectedness, compactness and learn the related theorems |
Unit-1 |
Teaching Hours:15 |
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Topological Spaces
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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 |
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Continuous Functions
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Continuous functions, the product topology, metric topology. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Connectedness and Compactness
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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 |
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Countability and Separation Axioms
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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
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Evaluation Pattern
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MTH232 - COMPLEX ANALYSIS (2020 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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Course Description: This course will help students learn about the essentials of complex analysis. This course includes important concepts such as power series, analytic functions, linear transformations, Laurent’s series, Cauchy’s theorem, Cauchy’s integral formula, Cauchy’s residue theorem, argument principle, Schwarz lemma , Rouche’s theorem and Hadamard’s 3-circles theorem. Course objectives: This course will help the learner to COBJ1. enhance the understanding the advanced concepts in Complex Analysis COBJ2. Acquire problem solving skills in Complex Analysis. |
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Course Outcome |
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On successful completion of the course, the students should be able to CO1. apply the concept and consequences of analyticity and the Cauchy-Riemann equations and of results on harmonic and entire functions including the fundamental theorem of algebra CO2. compute complex contour integrals in several ways: directly using parameterization, using the Cauchy-Goursat theorem Evaluate complex contour integrals directly and by the fundamental theorem, apply the Cauchy integral theorem in its various versions, and the Cauchy integral formula CO3. represent functions as Taylor, power and Laurent series, classify singularities and poles, find residues and evaluate complex integrals using the residue theorem CO4. use conformal mappings and know about meromorphic functions |
Unit-1 |
Teaching Hours:18 |
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Power Series
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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 |
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Singularities
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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 |
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Mappings
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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 |
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Meromorphic functions
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Meromorphic functions and argument principle, Schwarz lemma, Rouche’s theorem, convex functions and their properties, Hadamard 3-circles theorem. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH233 - LINEAR ALGEBRA (2020 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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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 Objective: This course will help learner to COBJ1. gain proficiency on the theories of Linear Algebra COBJ2. enhance problem solving skills in Linear Algebra |
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Course Outcome |
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On successful completion of the course, the students should be able to CO1. Have thorough understanding of the Linear transformations CO2.Demonstrate the elementary canonical forms, rational and Jordan forms. CO3. Apply the inner product space CO4. Express familiarity in using bilinear forms |
Unit-1 |
Teaching Hours:15 |
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Linear Transformations and Determinants
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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 |
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Elementary Canonical Forms, Rational and Jordan Forms
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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 |
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Inner Product Spaces
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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 |
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Bilinear Forms
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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
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Evaluation Pattern
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MTH234 - ADVANCED FLUID MECHANICS (2020 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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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 Prandtlboundry layer, porous media and Non-Newtonian fluid. Course objectives: This course will help the learner to COBJ1. understand the different modes of heat transfer and their applications. COBJ2. understand the importance of doing the non-dimensionalization of basic equations. COBJ3. understand the boundary layer flows. COBJ4. familiarity with porous medium and non-Newtonian fluids |
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Course Outcome |
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On successful completion of the course, the students should be able to CO1. understand the basic laws of heat transfer and understand the fundamentals of convective heat transfer process. CO2. solve Rayleigh - Benard problem and their physical phenomenon. CO3. solve and understand different boundary layer problems CO4. give an introduction to the basic equations with porous medium and solution methods for mathematical modeling of viscous fluids and elastic matter |
UNIT 1 |
Teaching Hours:15 |
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Dimensional Analysis and Similarity
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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 |
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Heat Transfer and Thermal Instability
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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 |
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Prandtl Boundry Layer
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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 |
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Porous Media and Non - Newtonian Fluids
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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:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH235 - ALGORITHMIC GRAPH THEORY (2020 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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Course description: This course helps the students to understand the colouring of graphs, Planar graphs, edges and cycles. Course objectives: This course will help the learner to COBJ1. Construct examples and proofs pertaining to the basic theorems COBJ2. Apply the theoretical knowledge and independent mathematical thinking in creative investigation of questions in graph theory COBJ3. Write graph theoretic ideas in a coherent and technically accurate manner. |
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Course Outcome |
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On successful completion of the course, the students should be able to CO1. understand the basic concepts and fundamental results in matching, domination, coloring and planarity. CO2. reason from definitions to construct mathematical proofs CO3. 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 |
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Colouring of Graphs
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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 |
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Matchings and Factors
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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 |
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Planar Graphs
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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 |
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Edges and Cycles Edge
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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
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Evaluation Pattern
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MTH251 - PYTHON PROGRAMMING FOR MATHEMATICS (2020 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:45 |
No of Lecture Hours/Week:3 |
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Max Marks:50 |
Credits:3 |
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Course Objectives/Course Description |
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Course description: This course aims at introducing the programming language Python and its uses in solving problems on discrete mathematics and differential equations. Course objectives: This course will help the learner to COBJ1.Acquire skill in usage of suitable functions/packages of Python to solve mathematical problems. COBJ2.Gain proficiency in using Python to solve problems on Differential equations. COBJ3. The built in functions required to deal withcreating and visualizing Graphs, Digraphs, MultiGraph. |
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Course Outcome |
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By the end of the course the learner will be able to: CO1. Acquire proficiency in using different functions of Python to compute solutions of basic mathematical problems. CO2. Demonstrate the use of Python to solve differential equations along with visualize the solutions. CO3. Be familiar with the built-in functions to deal with Graphs and Digraphs. |
Unit-1 |
Teaching Hours:15 |
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Introduction to Python Programming
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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 |
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Differential Equations using Python
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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 |
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Discrete Mathematics using Python
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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:
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Essential Reading / Recommended Reading
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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:
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MTH311 - INTRODUCTION TO FREE AND OPEN-SOURCE SOFTWARE (FOSS) TOOLS : (GNU OCTAVE) (2019 Batch) | |||||||||||||||||||||||||
Total Teaching Hours for Semester:45 |
No of Lecture Hours/Week:3 |
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Max Marks:50 |
Credits:2 |
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Course Objectives/Course Description |
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Course Description: This course is a foundation for introducing to Free and Open-Source Software (FOSS) Tools (Octave). It enables the students to explore mathematical concepts and verify mathematical facts through the use of software and also enhance the skills in programming. Course Objective: This course will help the learner to: COBJ1. use FOSS tool GNU Octave to effectively calculate the solutions of problems on Mathematics |
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Course Outcome |
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Course outcomes: After completing this course, the student will be able to: CO1. express proficiency in using Octave, CO2. Understand the use of various techniques of the software for effectively doing Mathematics. CO3. Obtain necessary skills in Octave programming. CO4. Use octave for applications of Mathematics |
Unit-1 |
Teaching Hours:15 |
Introduction to GNU Octave
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Getting started with GNU Octave - vector operations - Projections - Matrix operations - Plotting: plotting options - saving plots - Matrices and Linear systems: solution of linear system using Gaussian elimination, left division, LU decomposition - Polynomial curve fitting - Matrix transformations. | |
Unit-2 |
Teaching Hours:15 |
Calculus using Octave
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Limits, sequences and series - Numerical integration using Quadrature - Numerical integration using approximate sums - parametric and polar plots - special functions. | |
Unit-3 |
Teaching Hours:15 |
3D Graphs, Multiple Integrals, Vector fields and Differential Equations
|
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Three dimensional graphs: space curves - surfaces - solids of revolution - Multiple integrals - Vector fields - Differential Equations: slope fields - Euler’s method - The Livermore solver | |
Text Books And Reference Books: J. Lachniet, Introduction to GNU Octave: a brief tutorial for linear algebra and calculus students. Jason Lachniet, 2017. | |
Essential Reading / Recommended Reading
| |
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:
| |
MTH331 - MEASURE THEORY AND LEBESGUE INTEGRATION (2019 Batch) | |
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
Max Marks:100 |
Credits:4 |
Course Objectives/Course Description |
|
Course description: The Coursecovers the basic material that one needs to know in the theory of functions of a real variable and measure and integration theory as expounded by Henri Léon Lebesgue. Course objectives: This course will help the learner to COBJ1. enhance the understanding of the advanced notions from Mathematical Analysis
COBJ2. know more about the Measure theory and Lebesgue Integration |
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Course Outcome |
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On successful completion of the course, the students should be able to CO1. understand the fundamental concepts of Mathematical Analysis. CO2. tate some of the classical theorems in of Advanced Real Analysis. CO3. be familiar with measurable sets and functions. CO4. integrate a measurable function CO5. understand the properties of Lp Spaces |
Unit-1 |
Teaching Hours:20 |
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Lebesgue Measure
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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 |
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The Lebesgue Integration
|
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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 |
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Differentiation and Lebesgue Integration
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Continuity of Monotone Functions, Differentiation of Monotone Functions, Functions of Bounded Variation, Absolutely Continuous Functions, Integrating Derivatives. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:10 |
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The Lp Spaces
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Normed Linear Spaces, The Inequalities of Young, Hölder and Minkowski, The Lp spaces, Approximation and Separability, The Riesz Representation for the Dual of Lp, Weak Sequential Convergence in Lp, Weak Sequential Compactness, The Minimization of Convex Functionals. | |||||||||||||||||||||||||||||
Text Books And Reference Books: H.L. Royden and P.M. Fitzpatrick, “Real Analysis,” 4th ed. New Jersey: Pearson Education Inc., 2013. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH332 - NUMERICAL ANALYSIS (2019 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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Course description: This course deals with the theory and application of various advanced methods of numerical approximation. These methods or techniques help us to approximate the solutions of problems that arise in science and engineering. The emphasis of the course will be the thorough study of numerical algorithms to understand the guaranteed accuracy that various methods provide, the efficiency and scalability for large scale systems and issues of stability. Course objectives: This course will help the learner
COBJ1. to develop the basic understanding of the construction of numerical algorithms, and perhaps more importantly, the applicability and limits of their appropriate use. |
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Course Outcome |
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On successful completion of the course, the students should be able to CO1. Derive numerical methods for approximating the solution of problems of algebraic and transcendental equations, ordinary differential equations and boundary value problems. CO2. Implement a variety of numerical algorithms appropriately in various situations CO3. interpret, analyse and evaluate results from numerical computations |
Unit-1 |
Teaching Hours:20 |
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Solution of algebraic and transcendental equations
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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 |
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Interpolation and Numerical Integration
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Lagrange, Hermite, Cubic-spline’s (Natural, Not a Knot and Clamped) - with uniqueness and error term, for polynomial interpolation. Chebychev and Rational function approximation. Gaussian quadrature, Gauss-Legendre, Gauss-Chebychev formulas. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Numerical solution of ordinary differential equations
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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. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:10 |
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Boundary Value Problems
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Numerical solutions of second order boundary value problems (BVP) of first, second and third types by shooting method, Rayleigh-Ritz Method, Gelarkin Method. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH333 - CLASSICAL MECHANICS (2019 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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Course description: Classical Mechanics is the study of mechanics using Mathematical methods. 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 objectives: This course will help the learner to
COBJ1. derive necessary equations of motions based on the chosen configuration space. |
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Course Outcome |
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On successful completion of the course, the students should be able to: CO1. Interpret mechanics through the configuration space. |
Unit-1 |
Teaching Hours:12 |
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Introductory concepts
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The mechanical system - Generalised Coordinates - constraints - virtual work - Energy and momentum. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:20 |
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Lagrange's equation
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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 |
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Hamilton's equations
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Hamilton's principle - Hamilton’s equations - Other variational principles - phase space. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Hamilton - Jacobi Theory
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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
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Evaluation Pattern
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MTH334 - LINEAR ALGEBRA (2019 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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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 Objective: This course will help learner to COBJ1. gain proficiency on the theories of Linear Algebra COBJ2. enhance problem solving skills in Linear Algebra |
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Course Outcome |
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On successful completion of the course, the students should be able to CO1. Have thorough understanding of the Linear transformations CO2. Demonstrate the elementary canonical forms, rational and Jordan forms. CO3. Apply the inner product space CO4. Express familiarity in using bilinear forms |
Unit-1 |
Teaching Hours:15 |
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Linear Transformations and Determinants
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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 |
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Elementary Canonical Forms, Rational and Jordan Forms
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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 |
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Inner Product Spaces
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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 |
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Bilinear Forms
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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
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Evaluation Pattern
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MTH335 - ADVANCED GRAPH THEORY (2019 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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Course description:Domination of Graphs, perfect graphs, chromatic graph theory and Eigenvalues of Graphs are dealt with in the detail in this course. Course objectives: This course will help the learner to COBJ1. understand the advanced topics in Graph Theory COBJ2. enhance the understanding of techniques of writing proofs for advanced topics in Graph Theory |
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Course Outcome |
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By the end of the course the learner will be able to: CO1. have thorough understanding of the concepts in domination and perfect graphs CO2. familiarity in implementing the acquired knowledge appropriately CO3. mastery in employing proof techniques |
Unit-1 |
Teaching Hours:15 |
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Domination in Graphs
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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 |
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Chromatic Graph Theory
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T-Colourings, L(2,1)-colourings, Radio Colourings, Hamiltonian Colourings, Domination and Colourings. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Perfect Graphs
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The Perfect Graph Theorem, Chordal Graphs Revisited, Other Classes of Perfect Graphs, Imperfect Graphs, The Strong Perfect Graph Conjecture | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Eigenvalues of Graphs
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The Characteristic Polynomial, Eigenvalues and Graph Parameters, Eigenvalues of Regular Graphs, Eigenvalues and Expanders, Strongly Regular Graphs | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH351 - NUMERICAL METHODS USING PYTHON (2019 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:45 |
No of Lecture Hours/Week:3 |
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Max Marks:50 |
Credits:3 |
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Course Objectives/Course Description |
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Course description: In this course programming Numerical Methods in Python will be focused. How to program the numerical methods step by step to create the most basic lines of code that run on the computer efficiently and output the solution at the required degree of accuracy. Course objectives: This course will help the learner to COBJ1. Program the numerical methods to create simple and efficient Python codes that output the numerical solutions at the required degree of accuracy. COBJ2. Use the plotting functions of matplotlib to visualize the results graphically. COBJ3. Acquire skill in usage of suitable functions/packages of Python to solve initial value problems numerically. |
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Course Outcome |
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By the end of the course the learner will be able to: CO1. Acquire proficiency in using different functions of Python to compute solutions of system of equations. CO2. Demonstrate the use of Python to solve initial value problem numerically along with graphical visualization of the solutions . CO3. Be familiar with the built-in functions to deal with numerical methods. |
Unit-1 |
Teaching Hours:15 |
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Introduction to Python and Roots of High-Degree Equations
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Introduction and Simple Iterations Method, Finite Differences Method | |||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Systems of Linear Equations
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Introduction & Gauss Elimination Method: Algorithm, Gauss Elimination Method, Jacobi's Method, Gauss-Seidel's Method, Linear System Solution in NumPy and SciPy & Summary | |||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Numerical differentiation, Integration and Ordinary Differential Equations
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Introduction & Euler's Method, Second Order Runge-Kutta's Method, Fourth Order Runge-Kutta's Method, Fourth Order Runge-Kutta's Method: Plot Numerical and Exact Solutions. | |||||||||||||||||||||||||
Text Books And Reference Books: J. Kiusalaas, Numerical methods in engineering with Python 3. Cambridge University Press, 2013. | |||||||||||||||||||||||||
Essential Reading / Recommended Reading Hans Fangohr, Introduction to Python for Computational Science and Engineering (A beginner’s guide), University of Southampton, 2015. | |||||||||||||||||||||||||
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:
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MTH381 - INTERNSHIP (2019 Batch) | |||||||||||||||||||||||||
Total Teaching Hours for Semester:45 |
No of Lecture Hours/Week:0 |
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Max Marks:0 |
Credits:2 |
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Course Objectives/Course Description |
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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. |
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Course Outcome |
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Course Outcomes On successful completion of the course, the students should be able to COBJ1. Expose to the field of their professional interest |
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 centers, 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. In the present scenario of COVID 19 pandemic, the students unable to do internship in an organization, have to complete one MOOC in Mathematics that suits the academic interest of the student in consultation with the assigned internship supervisors and a dissertation based on a detailed review of two research articles. The duration of the course has to be at least 30 hours and should be completed on or before 20 June 2020. However, if a student chooses to go ahead with the internship, then they should complete at least 25 working days in the organization on or before 31 May 2020, in which case submission of the dissertation is not necessary. 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 the 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 in a specified format adhering to department guidelines. The report should be submitted within the first 10 days of the reopening of the University for the third semester. The students doing the MOOCs are expected to prepare course notes on their own using all the resources accessible and this is to be given as the first part of the internship report. Within 20 days from the day of reopening, the department must hold a presentation by the students. During the presentation, the supervisor or a nominee of the supervisor should be present and be one of the evaluators. Students should preferably be encouraged to make a presentation of their report. A minimum of 10 minutes should be given for each of the presenters. 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 (2019 Batch) | |
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
Max Marks:100 |
Credits:4 |
Course Objectives/Course Description |
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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 objectives: This course will help the learner to CO BJ1. write proofs for the theorems on Curves and Surfaces in R3. COBJ2. implement the properties of curves and surfaces in solving problems described in terms of tangent vectors / vector fields / forms etc., |
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Course Outcome |
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On successful completion of the course, the students should be able to CO1. express sound knowledge on the basic concepts in geometry of curves and surfaces in Euclidean space, especially E3. CO2. demonstrate mastery in solving typical problems associated with the theory. CO3. extend the knowledge in generalizing the concepts learned to higher dimensions. |
UNIT 1 |
Teaching Hours:15 |
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Calculus on Euclidean Geometry
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Euclidean Space - Tangent Vectors - Directional derivatives - Curves in E3 - 1-Forms - Differential Forms - Mappings. | |||||||||||||||||||||||||||||
UNIT 2 |
Teaching Hours:15 |
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Frame Fields and Euclidean Geometry
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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 |
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Euclidean Geometry and Calculus on Surfaces
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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 |
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Shape Operators
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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
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Evaluation Pattern
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MTH432 - COMPUTATIONAL FLUID DYNAMICS (2019 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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Course Description: This course helps the students to learn the solutions of partial differential equations using finite difference and finite element methods. This course also helps them to know how solve the Burger’s equations using finite difference equation, quasi-linearization of non-linear equations. Course objectives: This course will help the students to COBJ1. be familiar with solving PDE using finite difference method and finite element method COBJ2. Understand the non-linear equation Burger’s equation using finite difference method COBJ3.Understand the compressible fluid flow using ACM, PCM and SIMPLE methods COBJ4.Solve differential equations using finite element method usingdifferent shape functions |
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Course Outcome |
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On successful completion of the course, the students should be able to CO1. Solve both linear and non-linear PDE using finite difference methods CO2. Understand both physics and mathematical properties of governing Navier-Stokes equations and define proper boundary conditions for solution CO3. Understanding of physics of compressible and incompressible fluid flows CO4. Write the programming in MATLAB to solve PDE using finite difference method |
Unit-1 |
Teaching Hours:15 |
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Numerical solution of elliptic partial differential equations
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Review of classification of partial differential equations, classification of boundary conditions, numerical analysis, basic governing equations of fluid mechanics. Difference methods for elliptic partial differential equations, difference schemes for Laplace and Poisson’s equations, iterative methods of solution by Jacobi and Gauss-Siedel, solution techniques for rectangular and quadrilateral regions. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Numerical solution of parabolic and hyperbolic partial differential equations
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Difference methods for parabolic equations in one-dimension, methods of Schmidt, Laasonen, Crank-Nicolson and Dufort-Frankel, stability and convergence analysis for Schmidt and Crank-Nicolson methods, ADI method for two-dimensional parabolic equation, explicit finite difference schemes for hyperbolic equations, wave equation in one dimension. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Finite Difference Methods for non-linear equations
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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-4 |
Teaching Hours:15 |
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Finite Element Methods
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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 and two-dimensional problems: model equations, discretization, interpolation functions, evaluation of element matrices and vectors and their assemblage. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
| |||||||||||||||||||||||||||||
Evaluation Pattern
| |||||||||||||||||||||||||||||
MTH433 - FUNCTIONAL ANALYSIS (2019 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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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 Objective. This course will help learner to COBJ1: know the notions behind Functional Analysis COBJ2. enhance the problem solving ability in Functional Analysis |
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Course Outcome |
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On successful completion of the course, the students should be able to CO1. Explain the fundamental concepts of functional analysis. CO2. Understand the approximation of continuous functions. CO3. Understand concepts of Hilbert and Banach spaces with l2 and lp spaces serving as examples. CO4. Understand the definitions of linear functional and prove the Hahn-Banach theorem, open mapping theorem, uniform boundedness theorem, etc. CO5. Define linear operators, self adjoint, isometric and unitary operators on Hilbert spaces. |
Unit-1 |
Teaching Hours:15 |
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Banach spaces
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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 |
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Mapping theorems
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The open mapping theorem and the closed graph theorem, the uniform boundedness theorem, the conjugate of an operator. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Inner products
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Inner products, Hilbert spaces, Schwarz inequality, parallelogram law, orthogonal complements, orthonormal sets, Bessel’s inequality, complete orthonormal sets.
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Unit-4 |
Teaching Hours:18 |
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Conjugate space
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The conjugate space, the adjoint of an operator, self-adjoint, normal and unitary operators, projections, finite dimensional spectral theory.
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Text Books And Reference Books: G.F. Simmons, Introduction to topology and modern Analysis, Reprint, Tata McGraw-Hill, 2004. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH4401 - CALCULUS OF VARIATIONS AND INTEGRAL EQUATIONS (2019 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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Course description: This course introduces fundamental concepts and some standard results of calculus of variations and the integral equations. It plays an important role for solving various engineering sciences problems. Therefore, it has tremendous applications in diverse fields in engineering sciences. Course objectives: This course will help the learners to study extrema of functional, the Brachistochrone problem, Euler’s equation, variational derivative and invariance of Euler’s equations. It also contains Fredholm and Volterra integral equations and their solutions using various methods such as Neumann series, resolvent kernels, Green’s function approach and transform methods |
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Course Outcome |
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On successful completion of the course, the students should be able to CO1. Derive some classical differential equations by using principles of calculus of variations. CO2. Knowledge of Variational Problems, Euler-Lagrange Condition, Second Variation,Generalizations of the Variational Problem. CO3. find maximum or minimum of a functional using calculus of variations Technique, solve Volterra integral equations and Fredholm integral equations CO4. Reduce the differential equations to integral equations. |
Unit-1 |
Teaching Hours:18 |
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Euler equations and variational notations
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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 |
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Advanced variational problems
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Galerkian Technique, the Rayleigh-Ritz method. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:12 |
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Linear integral equations
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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 |
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Methods for solutions of linear integral equations
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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:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH4402 - MAGNETOHYDRODYNAMICS (2019 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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Course Description: This course provides the fundamentals of Magnetohydrodynamics, which include theory of Maxwell’s equations, basic equations, exact solutions and applications of classical MHD. Course objectives: This course will help the students to COBJ1.Understand mathematical form of Gauss’s Law, Faraday’s Law and Ampere’s Law and corresponding boundary conditions COBJ2. Derive the basic governing equations and boundary conditions of MHD flows. COBJ3. Finding the exact solutions of MHD governing equations. COBJ4. Understand the Alfven waves and derive their corresponding equations. |
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Course Outcome |
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On successful completion of the course, the students should be able to CO1. Derive the MHD governing equations using Faraday’s law and Ampere’s law. CO2. Solve the Fluid Mechanics problems with magnetic field. CO3.Understand the properties of force free magnetic field. CO4. Understand the application of Alfven waves, heating of solar corona, earth’s magnetic field. |
Unit-1 |
Teaching Hours:12 |
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Electrodynamics
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Outline of electromagnetic units and electrostatics, derivation of Gauss law, Faraday’s law, Ampere’s law and solenoidal property, dielectric material, conservation of charges, electromagnetic boundary conditions. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:13 |
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Basic Equations
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Outline of basic equations of MHD, magnetic induction equation, Lorentz force, MHD approximations, non-dimensional numbers, velocity, temperature and magnetic field boundary conditions. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:20 |
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Exact Solutions
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Hartmann flow, generalized Hartmann flow, velocity distribution, expression for induced current and magnetic field, temperature discribution, Hartmann couette flow, magnetostatic-force free magnetic field, abnormality parameter, Chandrashekar theorem, application of magnetostatic-Bennett pinch. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Applications
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Classical MHD and Alfven waves, Alfven theorem, Frozen-in-phenomena, Application of Alfven waves, heating of solar corana, earth’s magnetic field, Alfven wave equation in an incompressible conducting fluid in the presence of an vertical magnetic field, solution of Alfven wave equation, Alfven wave equation in a compressible conducting non-viscous fluid, Helmholtz vorticity equation, Kelvin’s circulation theorem, Bernoulli’s equation. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading D. J. Griffiths, Introduction to electrodynamics, 4th ed., Prentice Hall of India, 2012.
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Evaluation Pattern
| |||||||||||||||||||||||||||||
MTH4403 - WAVELET THEORY (2019 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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Course Description: This course aims at studying the fundamentals of wavelet theory. This includes the concept on the continuous and discrete wavelet transform and wavelet packets like construction and measure of wavelet sets and construction of wavelet spaces. Course objectives: This course will help the students to COBJ1. understand Fourier series, Fourier transform and Wavelet transformation and their interdependence. COBJ2. construct the wavelet transforms. COBJ3. learn the applications of Wavelet transforms |
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Course Outcome |
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On successful completion of the course, the students should be able to CO1. construct the Euler’s formula of complex exponential function and convolutions CO2. understand the discrete wavelet theory in Haar transforms CO3. understand applications of wavelet transform |
Unit-1 |
Teaching Hours:15 |
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Introduction
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Limitations of Fourier Series and Transforms, need of wavelet theory, Complex numbers and basic operation, the space L2(R), inner products, bases and projections, Euler’s formula and complex exponential function, Fourier series, Fourier transforms, Convolutions and B-Splines, the wavelet, requirements for wavelet. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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The Continuous wavelet transform
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The wavelet transform, the inverse wavelet transform, wavelet transform in terms of Fourier transform, Complex wavelets: the Morlet wavelet. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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The discrete wavelet transform
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Frames and orthogonal wavelet bases, Haar space, general Haar space, Haar wavelet space, general Haar wavelet space, discrete Haar wavelet transforms and applications. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Wavelet packets
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The construction of wavelet sets, the measure of the closure of a wavelet set, constructing wavelet packet spaces, wavelet packet spaces | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH4404 - MATHEMATICAL MODELLING (2019 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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Course description: This course is concerned with the fundamentals of mathematical modeling. It deals with finding solution to real world problems by transforming into mathematical models using ordinary and partial differential equations. Course objectives: This course will help the learner to interpret the real-world problems in the form of ordinary and partial differential equations. They shall become familiar with some of the classical mathematical models in the fields such as physics, biology, chemistry, finance and economics. |
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Course Outcome |
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On successful completion of the course, the students should be able to CO1. Create mathematical models of empirical or theoretical phenomena in domains such as the physical, natural, or social science; CO2. Gain the ability to determine the validity of a given model and will be able to construct further improvement in the models independently CO3. formulate, interpret and draw inferences from mathematical models CO4. solve other problems by means of intuition, creativity, guessing, and the experience gained through the study of particular examples and mathematical models CO5. demonstrate competence with a wide variety of mathematical tools and techniques CO6. take an analytical approach to problems in their future endeavours |
UNIT 1 |
Teaching Hours:15 |
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Concept of mathematical modeling
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Definition, Classification, Characteristics And Limitations, Mathematical Modelling
Through Ordinary Differential Equations Of First Order: Linear And Nonlinear Growth and Decay Models Compartment Models, Dynamics Problems, Geometrical Problems, Simulation and Random number generation. | |||||||||||||||||||||||||||||
UNIT 2 |
Teaching Hours:12 |
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Mathematical modelling through systems of ordinary differential equations of first order
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Population Dynamics, Epidemics, Compartment Models, Economics, Medicine, Arms Race, Battles and International Trade and Dynamics | |||||||||||||||||||||||||||||
UNIT 3 |
Teaching Hours:13 |
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Mathematical modelling through ordinary differential equations of second order
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Modeling Of Planetary Motions – Circular Motion Of Satellites, Mathematical Modelling Through Linear Differential Equations Of Second Order, Miscellaneous Mathematical Models | |||||||||||||||||||||||||||||
UNIT 4 |
Teaching Hours:20 |
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Mathematical Modelling leading to linear and nonlinear partial differential equations
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Simple models, conservation law – Traffic flow on highway – Flood waves in rivers – glacier flow, roll waves and stability, shallow water waves – Convection diffusion –processes Burger’s equation, Convection – reaction processes – Fisher’s equation. Telegraph equation heat transfer in a layered solid. Chromatographic models sediment Transport in rivers reaction-diffusion systems, travelling waves, pattern formation, tumour growth. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH4405 - ATMOSPHERIC SCIENCE (2019 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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Course Description: This course provides an introduction to the dynamic meteorology, which includes the essentials of fluid dynamics, atmospheric dynamics and atmosphere waves and instabilities.
Course objectives: This course will help the students to COBJ1. explain the physical laws governing the structure and evolution of atmospheric phenomena spanning a broad range of spatial and temporal scales COBJ2. apply mathematical tools to study atmospheric processes |
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Course Outcome |
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On successful completion of the course, the students should be able to CO1. Model the atmospheric flows mathematically CO2. Understand the atmospheric waves and instabilities in atmosphere |
UNIT 1 |
Teaching Hours:15 |
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Essential Fluid Dynamics
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Thermal wind, geostrophic motion, hydrostatic approximation, consequences, Taylor-Proudman theorem, Geostrophic degeneracy, dimensional analysis and non-dimensional numbers. Physical Meteorology: Atmospheric composition, laws of thermodynamics of the atmosphere, adiabatic process, potential temperature, the Classius-Clapyeron equation, laws of black body radiation, solar and terrestrial radiation, solar constant, Albedo, greenhouse effect, heat balance of earth-atmosphere system. | |||||||||||||||||||||||||||||
UNIT 2 |
Teaching Hours:15 |
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Atmosphere Dynamics
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Geostrophic approximation, pressure as a vertical coordinate, modified continuity equation, balance of forces, non-dimensional numbers (Rossby, Richardson, Froude, Ekman etc.), scale analysis for tropics and extra-tropics, vorticity and divergence equations, conservation of potential vorticity, atmospheric turbulence and equations for planetary boundary layer. | |||||||||||||||||||||||||||||
UNIT 3 |
Teaching Hours:15 |
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General Circulation of the Atmosphere
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Definition of general circulation, various components of general circulation, zonal and eddy angular momentum balance of the atmosphere, meridional circulation, Hadley-Ferrel and polar cells in summer and winter, North-South and East-West (Walker) monsoon circulation, forces meridional circulation due to heating and momentum transport, available potential energy, zonal and eddy energy equations. | |||||||||||||||||||||||||||||
UNIT 4 |
Teaching Hours:15 |
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Atmospheric Waves and Instability
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Wave motion in general, concept of wave packet, phase velocity and group velocity, momentum and energy transports by waves in the horizontal and vertical, equatorial, Kelvin and mixed Rossby gravity waves, stationary planetary waves, filtering of sound and gravity waves, linear barotropic and baroclinic instability. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
| |||||||||||||||||||||||||||||
MTH4406 - ADVANCED LINEAR PROGRAMMING (2019 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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Course Description: This course is about the analysis and applications of transportation and assignment models, game theory, CPM - PERT methods and dynamic programming. Course objectives: This course will help the students to
COBJ1. acquire and demonstrate the implementation of the necessary algorithms for solving advanced level linear programming problems |
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Course Outcome |
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On successful completion of the course, the students should be able to CO1. Apply the notions of linear programming in solving transportation problems. CO2. Understand the theory of games for solving simple games. CO3. Acquire knowledge in formulating Tax planning problem and use goal programming algorithms. CO4. Use linear programming in the formulation of shortest route problem and use algorithmic approach in solving various types of network problems. CO5. Know the use of dynamic programming in various applications |
Unit-1 |
Teaching Hours:16 |
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Transportation Model
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Determination of the Starting Solution – Iterative computations of the transportation algorithm. Assignment Model: - The Hungarian Method – Simplex explanation of the Hungarian Method – The trans-shipment Model. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:16 |
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Game Theory
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Game Theory: Optimal solution of two person zero – sum games – Solution of Mixed strategy Games (both graphical and Linear programming solution) – Goal Programming: - Formulation – Tax Planning Problem – Goal programming algorithms – The weights method – preemptive method. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:16 |
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Network Models
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Linear programming formulation of the shortest-route Problem. Maximal Flow model:- Enumeration of cuts – Maximal Flow Algorithm – Linear Programming Formulation of Maximal Flow Model. CPM and PERT:- Network Representation – Critical path computations – Construction of the Time Schedule – Linear Programming formulation of CPM – PERT calculations. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:12 |
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Dynamic Programming
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Dynamic Programming: Recursive nature of computations in DP – Forward and Backward Recursion – Knapsack / Fly Away / Cargo-Loading Model - Equipment Replacement Model. | |||||||||||||||||||||||||||||
Text Books And Reference Books: A.H. Taha, “Operations research”, 7th Ed, Pearson Education, 2003. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH4407 - DESIGN AND ANALYSIS OF ALGORITHMS (2019 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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Course Description: This course is an extensive study of algorithms in a few domains of mathematics. This includes Elementary Graph Algorithms, Algorithms in Maximum Flow and Linear Programming, Number-Theoretic Algorithms and Approximation Algorithms Course objectives:This course will help the student to learn COBJ1. about the algorithms and their efficiency in implementation COBJ2. to use the appropriate algorithms to problems in Graph Theory and Number Theory |
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Course Outcome |
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After completing this course, the student will be able to: CO1. Analyze the running time of algorithms for problems in various domains. CO2. Apply the algorithms to solve problems in Graph theory and Number theory. CO3. Design efficient algorithms using the various approaches for real world problems. |
Unit-1 |
Teaching Hours:15 |
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Growth of Functions and Elementary Graph Algorithms
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Growth of Functions: Asymptotic notation, Standard notations and common functions. Elementary Graph Algorithms: Representations of graphs, Breadth-first search, Depth-first search, Topological sort, strongly connected components. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Network and Linear Programming Algorithms
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Maximum Flow: Flow networks, The Ford-Fulkerson method, Maximum bipartite matching, Push-relabel algorithms, The relabel-to-front algorithm. Linear Programming: Standard and slack forms, Formulating problems as linear programs, The simplex algorithm, Duality ,The initial basic feasible solution. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Number-Theoretic Algorithms
|
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Elementary number-theoretic notions, Greatest common divisor, modular arithmetic, Solving modular linear equations, The Chinese remainder theorem, Powers of an element , The RSA public-key cryptosystem, Primality testing , Integer factorization. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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NP-Completeness and Recurrence Theory
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NP-Completeness: Polynomial time, Polynomial-time verification, NP-completeness and reducibility, NP-completeness proofs, NP-complete problems. Recurrence Theory: Recurrences, methods for solving recurrences, technicalities in recurrences, running times of divide and conquer algorithm. | |||||||||||||||||||||||||||||
Text Books And Reference Books: Cormen T H, Leiserson C E, Rivest R L and Stein, Clifford, Introduction to algorithms, PHI, 2nd Edition, 2009. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern
| |||||||||||||||||||||||||||||
MTH4408 - INTRODUCTION OF THEORY OF MATROIDS (2019 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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Course Description: This course is an introduction to matroids, with most of its focus on graphic matroids and the graph-theoretic extensions and implications associated to graphs. Course objectives: This course will help the student to learn COBJ1. to capture the concept of independence. COBJ2. the terminology and concepts of duality, representability and connectivity of matroids COBJ3. The geometrical representations and construction of matroids. |
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Course Outcome |
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Course outcomes: After completing this course, the student will be able to: CO1. understand about the matroids and its applications to different fields. CO2. understand and apply basic matroid concepts and theorems. CO3. identify the role of matroid theory in many areas of research such as coding theory, graph theory, tropical geometry, optimization, topological and algebraic combinatorics. |
Unit-1 |
Teaching Hours:15 |
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Introduction to Matroids
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Independent Sets and Circuits - bases - rank - closure - geometric representation - transversal matroids - the lattice of flats - the greedy algorithm. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Duality and connectivity of Matroids
|
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Duality - basic principles - duals of representable matroids - duals of graphic matroids - duals of transversal matroids - contraction - minors - projections - flats - scum theorem - connectivity for graphs and matroids - properties of matroid connectivity. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Graphic and Representable Matroids
|
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Representability - serial-parallel networks - projective geometries - affine geometries - matroid representations and construction - representability over finite fields. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Constructions and higher connectivity
|
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Regular matroids - Algebraic matroids - characteristic sets - modularity. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
J G Oxley, “Matroid Theory”, 2/e, Oxford Univ. Press, 2011. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern
| |||||||||||||||||||||||||||||
MTH4409 - TOPOLOGICAL GRAPH THEORY (2019 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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Course Description: In this course, an introduction to basic and advanced concepts of topological graph theory. The course exposes students to techniques in combinatorial topology, embeddings and their combinatorial descriptions. Course objectives: This course will help the student to learn COBJ.1 the concepts of graph embeddings on surfaces. COBJ2. the concepts and results on voltage graphs. COBJ3. The terminology and results on map colorings and genus of groups. |
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Course Outcome |
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After completing this course, the student will be able to: CO1. deal with the ways to represent the geometric realisation of graphs. CO2. understand and apply the concepts of voltage graphs covering graphs and imbeddings. CO3. understand and apply concepts of imbedded voltage graphs and map colorings. CO4. understand and apply concepts of genus of groups |
Unit-1 |
Teaching Hours:15 |
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Introduction
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Graph representations: drawings, matrices, isomorphism – automorphisms – graph classes- new graphs from old – subgraphs, subdivisions, homeomorphisms – complements, regular quotients and coverings – embedding and planarity - traversability. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Voltage Graphs and Imbeddings
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Ordinary voltages - irregular covering graphs - permutation voltage graphs - subgroups of the voltage group - surfaces and imbeddings - simplicial complexes - band decompositions and graph imbeddings - the classification of surfaces. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Imbedded Voltage Graphs
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The derived imbedding - branched coverings of surfaces - regular branched coverings and group actions - current graphs - voltage-current duality - map colorings | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Genus of a group
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The genus of abelian groups - symmetric genus - groups of small symmetric genus - groups of small genus. | |||||||||||||||||||||||||||||
Text Books And Reference Books: Gross J., Tucker T, “Topological Graph Theory”, | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH4410 - ALGEBRAIC GRAPH THEORY (2019 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:60 |
No of Lecture Hours/Week:4 |
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Max Marks:100 |
Credits:4 |
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Course Objectives/Course Description |
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Course Description: This course introduces the learner to the theory of algebraic graphs. The course handles graph theory using abstract/linear algebraic techniques. Course objectives: This course will help the student to learn COBJ1. to relate the concepts of graph theory and algebra each other . COBJ2. the terminology and results of transitive graphs. COBJ3. to extend the theory of matrices to the matrices associated with graphs. COBJ4. The terminology and results on the Laplacian of graphs |
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Course Outcome |
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After completing this course, the student will be able to: CO1. solve routine problems in algebraic graph theory. CO2. basic algebraic techniques used in the analysis of networks and their complexity. CO3. apply the topics studied to the real-world/practical problems of graph theory. |
Unit-1 |
Teaching Hours:15 |
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Introduction
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Graphs- Automorphisms and homomorphisms - circulant graphs - johnson graphs-line graphs -planar graphs - groups - permutation groups - asymmetric graphs - primitivity and connectivity. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Transitive graphs
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Vertex transitive graphs - edge transitive graphs - connectivity - Hamilton paths and cycles - arc transitive graphs - cubic arc transitive graphs - distance transitive graphs - the coxeter graph -Tutte's 8-cage. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Matrix Theory and Eigenvalues
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Graph matrices - incidence matrix of oriented graphs - symmetric matrices - eigenvectors - definite and semidefinite matrices- rank of a symmetric matrix - generalised line graphs - reflections - generating set - strongly regular graphs. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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The Laplacian of a Graph
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Laplacian matrix - representations - energy and eigenvalues - connectivity - interlacing - the generalised Laplacian - multiplicities - embeddings. | |||||||||||||||||||||||||||||
Text Books And Reference Books: C Godsill, G Royle., “Algebraic Graph Theory”, Springer., 2001. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH451 - NUMERICAL METHODS FOR BOUNDARY VALUE PROBLEM USING PYTHON (2019 Batch) | |||||||||||||||||||||||||||||
Total Teaching Hours for Semester:45 |
No of Lecture Hours/Week:3 |
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Max Marks:50 |
Credits:3 |
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Course Objectives/Course Description |
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Course description: This course helps students to have an in-depth knowledge of Python in solving Boundary Value problems This includes solution of Two-point boundary value problems using core Python. This course also introduces students to FEniCS, an extension of Python for solving various PDE’s and boundary problems. Course objectives: This course will help the learner to COBJ1.Program Python codes to solve two-point boundary value problems at the required degree of accuracy. COBJ2.Use the plotting functions of matplotlib to visualize the solution of BVP’s. COBJ3.Acquire skill in usage of suitable functions/packages of Python to solve partial differential equations. |
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Course Outcome |
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By the end of the course the learner will be able to: CO1. acquire proficiency in using different functions of Python and writing user defined functions to compute solutions of two-point boundary value problems CO2. demonstrate the use of Python to solve ODEs numerically using shooting method with graphical visualization. CO3. be familiar with the built-in functions to deal with solution of PDE’s. |
Unit-1 |
Teaching Hours:15 |
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Two-Point Boundary Value Problems
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Introduction to two-point boundary value Problems - Shooting Method: second order differential equations - Higher order differential equations - solution of second order differential equation using Finite Difference Method, solution of fourth order differential equation using Finite Difference Method. | |||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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FEniCS and Finite element Solvers
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Implementation of FEniCS - The heat equation - A nonlinear Poisson equation - Equation of linear elasticity - The Navier - Stokes equations - A system of advection-diffusion-reaction equations | |||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Subdomains and Boundary conditions
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Combining Dirichlet and Neuman conditions - Setting multiple Dirichlet conditions - defining subdomains - setting up multiple Dirichlet, Neumann, and Robin conditions - Generating meshes with subdomains.
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Text Books And Reference Books:
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Essential Reading / Recommended Reading J. Crank, H. G. Martin, and D. M. Melluish, Non-Linear Ordinary Differential Equations. Oxford University Press. | |||||||||||||||||||||||||
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:
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MTH481 - PROJECT (2019 Batch) | |||||||||||||||||||||||||
Total Teaching Hours for Semester:30 |
No of Lecture Hours/Week:2 |
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Max Marks:100 |
Credits:2 |
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Course Objectives/Course Description |
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The objective of this course is to develop positive attitude, knowledge and competence for research in Mathematics |
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Course Outcome |
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Through this project students will develop analytical and computational skills |
Unit-1 |
Teaching Hours:30 |
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PROJECT
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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 textbook of 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 examination. The candidates who fail must redo the project as per the university regulation. Time line for Project:
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Text Books And Reference Books: . | |||||||||||||||||||
Essential Reading / Recommended Reading . | |||||||||||||||||||
Evaluation Pattern Assessment: Project is evaluated based on the parameters given below:
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