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1 Semester - 2023 - Batch | Course Code |
Course |
Type |
Hours Per Week |
Credits |
Marks |
MTH111 | RESEARCH METHODOLOGY | Skill Enhancement Courses | 2 | 2 | 0 |
MTH131 | ABSTRACT ALGEBRA | Core Courses | 4 | 4 | 100 |
MTH132 | REAL ANALYSIS | Core Courses | 4 | 4 | 100 |
MTH133 | ORDINARY DIFFERENTIAL EQUATIONS | Core Courses | 4 | 4 | 100 |
MTH134 | LINEAR ALGEBRA | Core Courses | 4 | 4 | 100 |
MTH135 | DISCRETE MATHEMATICS | Core Courses | 4 | 4 | 100 |
MTH151 | PYTHON PROGRAMMING FOR MATHEMATICS | Core Courses | 3 | 3 | 50 |
2 Semester - 2023 - Batch | Course Code |
Course |
Type |
Hours Per Week |
Credits |
Marks |
MTH211 | TEACHING TECHNOLOGY AND SERVICE LEARNING | - | 2 | 2 | 0 |
MTH231 | GENERAL TOPOLOGY | - | 4 | 4 | 100 |
MTH232 | COMPLEX ANALYSIS | - | 4 | 4 | 100 |
MTH233 | PARTIAL DIFFERENTIAL EQUATIONS | - | 4 | 4 | 100 |
MTH234 | GRAPH THEORY | - | 4 | 4 | 100 |
MTH235 | INTRODUCTORY FLUID MECHANICS | - | 4 | 4 | 100 |
MTH251 | COMPUTATIONAL MATHEMATICS USING PYTHON | - | 3 | 3 | 50 |
3 Semester - 2022 - Batch | Course Code |
Course |
Type |
Hours Per Week |
Credits |
Marks |
MTH311 | TEACHING TECHNOLOGY AND SERVICE LEARNING | Skill Enhancement Courses | 2 | 2 | 0 |
MTH331 | MEASURE THEORY AND LEBESGUE INTEGRATION | Core Courses | 4 | 4 | 100 |
MTH332 | NUMERICAL ANALYSIS | Core Courses | 4 | 4 | 100 |
MTH333 | DIFFERENTIAL GEOMETRY | Core Courses | 4 | 4 | 100 |
MTH341A | ADVANCED FLUID MECHANICS | Discipline Specific Elective Courses | 4 | 4 | 100 |
MTH341B | ADVANCED GRAPH THEORY | Discipline Specific Elective Courses | 4 | 4 | 100 |
MTH341C | PRINCIPLES OF DATA SCIENCE | Discipline Specific Elective Courses | 4 | 4 | 100 |
MTH341D | NUMERICAL LINEAR ALGEBRA | Discipline Specific Elective Courses | 4 | 4 | 100 |
MTH342A | MAGNETOHYDRODYNAMICS | Discipline Specific Elective Courses | 4 | 4 | 100 |
MTH342B | THEORY OF DOMINATION IN GRAPHS | Discipline Specific Elective Courses | 4 | 4 | 100 |
MTH342C | NEURAL NETWORKS AND DEEP LEARNING | Discipline Specific Elective Courses | 4 | 4 | 100 |
MTH342D | FRACTIONAL CALCULUS | Discipline Specific Elective Courses | 4 | 4 | 100 |
MTH351 | NUMERICAL METHODS USING PYTHON | Core Courses | 3 | 3 | 50 |
MTH381 | INTERNSHIP | Core Courses | 2 | 2 | 0 |
4 Semester - 2022 - Batch | Course Code |
Course |
Type |
Hours Per Week |
Credits |
Marks |
MTH411 | TEACHING PRACTICE | - | 1 | 1 | 0 |
MTH431 | CLASSICAL MECHANICS | - | 4 | 4 | 100 |
MTH432 | FUNCTIONAL ANALYSIS | - | 4 | 4 | 100 |
MTH433 | ADVANCED LINEAR PROGRAMMING | - | 4 | 4 | 100 |
MTH441A | COMPUTATIONAL FLUID DYNAMICS | - | 4 | 4 | 100 |
MTH441B | ATMOSPHERIC SCIENCE | - | 4 | 4 | 100 |
MTH441C | MATHEMATICAL MODELLING | - | 4 | 4 | 100 |
MTH442A | ALGEBRAIC GRAPH THEORY | - | 4 | 4 | 100 |
MTH442B | STRUCTURAL GRAPH THEORY | - | 4 | 4 | 100 |
MTH442C | APPLIED GRAPH THEORY | - | 4 | 4 | 100 |
MTH443A | REGRESSION ANALYSIS | - | 4 | 4 | 100 |
MTH443B | DESIGN AND ANALYSIS OF ALGORITHMS | - | 4 | 4 | 100 |
MTH444A | RIEMANNIAN GEOMETRY | - | 4 | 4 | 100 |
MTH444B | FUZZY MATHEMATICS | - | 4 | 4 | 100 |
MTH444C | ADVANCED ANALYSIS | - | 4 | 4 | 100 |
MTH451A | NUMERICAL METHODS FOR BOUNDARY VALUE PROBLEM USING PYTHON | - | 3 | 3 | 50 |
MTH451B | NETWORK SCIENCE WITH PYTHON AND NETWORKX | - | 3 | 3 | 50 |
MTH451C | PROGRAMMING FOR DATA SCIENCE IN R | - | 3 | 3 | 50 |
MTH451D | NUMERICAL LINEAR ALGEBRA USING MATLAB | - | 3 | 3 | 50 |
MTH481 | PROJECT | - | 4 | 4 | 100 |
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Introduction to Program: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
The MSc course in Mathematics aims at developing mathematical ability in students with acute and abstract reasoning. The course will enable students to cultivate a mathematician’s habit of thought and reasoning and will enlighten students with mathematical ideas relevant for oneself and for the course itself. Course Design: Masters in Mathematics is a two year programme spreading over four semesters. In the first two semesters focus is on the basic courses in mathematics such as Algebra, Topology, Analysis and Graph Theory along with the basic applied course ordinary and partial differential equations. In the third and fourth semester focus is on the special courses, elective courses and skill-based courses including Measure Theory and Lebesgue Integration, Functional Analysis, Computational Fluid Dynamics, Advanced Graph Theory, Numerical Analysis and courses on Data Science . Important feature of the curriculum is that students can specialize in any one of areas (i) Fluid Mechanics, (ii) Graph Theory and (iii) Data Science, with a project on these topics in the fourth semester, which will help the students to pursue research in these topics or grab the opportunities in the industry. To gain proficiency in software skills, four Mathematics Lab papers are introduced, one in each semester. viz. Python Programming for Mathematics, Computational Mathematics using Python, Numerical Methods using Python and Numerical Methods for Boundary Value Problem using Python / Network Science with Python and NetworkX / Programming for Data Science in R / Numerical Linear Algebra using MATLAB respectively. Special importance is given to the skill enhancement courses: Research Methodology, Machine Learning (during 2024-2025 for 2023-2024 batch) and Teaching Technology and Service learning. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Programme Outcome/Programme Learning Goals/Programme Learning Outcome: PO1: Engage in continuous reflective learning in the context of technology and scientific advancementPO2: 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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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 (2023 Batch) | |
Total Teaching Hours for Semester:30 |
No of Lecture Hours/Week:2 |
Max Marks:0 |
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 in Mathematics. Also, the students are exposed to the principles, procedures and techniques of planning and implementing a research project and also to the preparation of a research article. Course Objectives: This course will help the learner to COBJ 1: Know the general research methods COBJ 2: Get hands on experience in methods of research that can be employed for research in mathematics |
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Course Outcome |
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CO1: Foster a clear understanding about research design that enables students in analyzing and evaluating the published research. CO2: Obtain necessary skills in understanding the mathematics research articles. CO3: Acquire skills in preparing scientific documents using MS Word, Origin, LaTeX and Tikz Library. |
Unit-1 |
Teaching Hours:10 |
Research Methodology
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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, Ethics in Research and publications, Mathematics writing skills. - Standard Notations and Symbols, Using Symbols and Words, Organizing a paper, Defining variables, Symbols and notations, Different Citation Styles, IEEE Referencing Style in detail, Tools for checking Grammar and Plagiarism. | |
Unit-3 |
Teaching Hours:10 |
Type Setting research articles
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Package for Mathematics Typing, MS Word, LaTeX, Overleaf, Tikz Library, Origin, Pictures and Graphs, producing various types of documents using TeX. | |
Text Books And Reference Books: . | |
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: < marks to be converted to credits > | |
MTH131 - ABSTRACT ALGEBRA (2023 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 enables students to understand the intricacies of advanced areas in algebra. This includes a study of advanced group theory, Euclidean rings, polynomial rings and Galois theory. Course objectives: This course will help the learner to COBJ1. Enhance the knowledge of advanced-level algebra. COBJ2. Understand the proof techniques for the theorems on advanced group theory, rings and Galois theory. |
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Course Outcome |
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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 the irreducibility of rationals. CO4: demonstrate the characteristic of a field and the prime subfield. CO5: demonstrate factorisation and ideal theory in the polynomial ring; the structure of primitive polynomials; field extensions and characterization of finite normal extensions as splitting fields; the structure and construction of finite fields; radical field extensions; Galois group and Galois theory. |
Unit-1 |
Teaching Hours:15 |
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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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MTH132 - REAL ANALYSIS (2023 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 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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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. | |||||||||||||||||||||||||||||
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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MTH133 - ORDINARY DIFFERENTIAL EQUATIONS (2023 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, existence and uniqueness of solutions, Eigenvalues and Eigenvectors of the equations, power series method for solving differential equations. Non-linear autonomous system of equations. Course Objectives: This course will help the learner to COBJ 1: Solve adjoint differential equations and understand the zeros of solutions COBJ 2:Understand the existence and uniqueness of solutions of differential equations and to solve the Strum-Liouville problems. COBJ 3:Solve the differential equations by power series method and also hypergeometric equations. COBJ 4:Understand and solve the non-linear autonomous system of equations. |
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Course Outcome |
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CO1: Understand concept of linear differential equation, Fundamental set Wronskian. CO2: Understand the concept of Liouvilles theorem, Adjoint and Self Adjoint equation, Lagrange's Identity, Green?s formula, Eigenvalue and Eigenfunctions. CO3: Identify ordinary and singular point by Frobenius Method, Hyper geometric differential equation and its polynomial. CO4: Understand the basic concepts existence and uniqueness of solutions. CO5: Understand basic concept of solving the linear and non-linear autonomous systems of equations. CO6: Understand the concept of critical point and stability of the system. |
UNIT 1 |
Teaching Hours:15 |
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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. | |||||||||||||||||||||||||||||
UNIT 2 |
Teaching Hours:15 |
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Existence and Uniqueness of solutions
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Fundamental existence and uniqueness theorem, Dependence of solutions on initial conditions, existence and uniqueness theorem for higher order and system of differential equations, Eigenvalue Problems, Strum-Liouville problems, Orthogonality of eigenfunctions. | |||||||||||||||||||||||||||||
UNIT 3 |
Teaching Hours:15 |
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Power series solutions
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Ordinary and singular points of the differential equations, Classification of singular points, Solution near an ordinary point and a regular singular point by Frobenius method, solution near irregular singular point, Hermite, Laguerre, Chebyshev and Hypergeometric differential equation and its polynomial solutions, standard properties. | |||||||||||||||||||||||||||||
UNIT 4 |
Teaching Hours:15 |
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Linear and non-linear Autonomous differential equations
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Linear system of homogeneous and non-homogeneous equations, Non-linear autonomous system of equations, Phase plane, Critical points, Stability, Liapunov direct method, limit cycle and periodic solutions, Bifurcation of plane autonomous systems. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH134 - LINEAR ALGEBRA (2023 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 Objectives: This course will help the learner to COBJ 1: Have thorough understanding of Linear transformations and its properties. COBJ 2: Understand and apply the elementary canonical forms, rational and Jordan forms in real life problems. COBJ 3: Gain knowledge on Inner product space and the orthogonalisation process. COBJ 4: Explore different types of bilinear forms and their properties. |
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Course Outcome |
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CO1: Gain in-depth knowledge on Linear transformations. CO2: Demonstrate the elementary canonical forms, rational and Jordan forms. CO3: Apply the inner product space in orthogonality. CO4: Gain familiarity in using bilinear forms. |
Unit-1 |
Teaching Hours:15 |
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Linear Transformations and Determinants
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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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MTH135 - DISCRETE MATHEMATICS (2023 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 discuss the fundamental concepts and tools in discrete mathematics with emphasis on their applications to mathematical writing, enumeration and recurrence relations. Course Objectives: The course will help the learner to COBJ 1: develop logical foundations to understand and create mathematical arguments.. COBJ 2: implement enumeration techniques in a variety of real-life problems. COBJ 3: analyze the order and efficiency of algorithms. COBJ 4: communicate the basic and advanced concepts of the topic precisely and effectively.
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Course Outcome |
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CO1: demonstrate mathematical logic to write mathematical proofs and solve problems. CO2: apply the concepts of sets, relations, functions and related discrete structures in practical situations. CO3: understand and apply basic and advanced counting techniques in real-life problems CO4: analyse algorithms, determine their efficiency and gain proficiency in preparing efficient algorithms |
Unit-1 |
Teaching Hours:15 |
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Set Theory and Mathematical Logic
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Sets: Cardinality and countability, recursively defined sets, relations, equivalence relations and equivalence classes, partial and total ordering, representation of relations, closure of relations, functions, bijection, inverse functions. Logic: Propositions, logical equivalences, normal forms, rules of inference, predicates, quantifiers, nested quantifiers, arguments, formal proof methods and strategies. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Enumeration Relations and Functions
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Fundamental principles, pigeon-hole principle, permutations – with and without repetitions, combinations- with and without repetitions, binomial theorem, binomial coefficients, the principle of inclusion and exclusion, derangements, arrangements with forbidden positions, rook polynomial. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Generating Functions and Recurrence Relations
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Ordinary and exponential generating functions, recurrence relations, first-order linear recurrence relations, higher-order linear homogeneous recurrence relations, non-homogeneous recurrence relations, solving recurrence relations using generating functions. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Analysis of Algorithms
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Real-valued functions, big-O, big-Omega and big-Theta notations, orders of power functions, orders of polynomial functions, analysis of algorithm efficiency, the sequential search algorithm, exponential and logarithmic orders, binary search algorithm. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH151 - PYTHON PROGRAMMING FOR MATHEMATICS (2023 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: gain proficiency in using Python for programming. |
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Course Outcome |
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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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Basic of Python
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Installation, IDE, variables, built-in functions, input and output, modules and packages, data types and data structures, use of mathematical operators and mathematical functions, programming structures (conditional structure, the for loop, the while loop, nested statements) | |||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Symbolic and Numeric Computations
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Use of Sympy package, Symbols, Calculus, Differential Equations, Series expressions, Linear and non-linear equations, List, Tuples and Arrays. | |||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Data Visualization
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Standard plots (2D, 3D), Scatter plots, Slope fields, Vector fields, Contour plots, stream lines, Manipulating and data visualizing data with Pandas, Mini Project. | |||||||||||||||||||||||||
Text Books And Reference Books:
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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 - TEACHING TECHNOLOGY AND SERVICE LEARNING (2023 Batch) | |||||||||||||||||||||||||
Total Teaching Hours for Semester:30 |
No of Lecture Hours/Week:2 |
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Max Marks:0 |
Credits:2 |
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Course Objectives/Course Description |
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Course Description: This course is intended to assist the students in acquiring necessary skills on the use of modern technology in teaching, they are exposed to the principles, procedures and techniques of planning and implementing teaching techniques. Through service learning they will apply the knowledge in real-world situations and benefit the community. Course objectives: This course will help the learner to COBJ 1: Understand the pedagogy of teaching. COBJ 2: Able to use various ICT tools for effective teaching. COBJ 3: Apply the knowledge in real-world situations. COBJ 4: Enhances academic comprehension through experiential learning. |
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Course Outcome |
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CO1: Gain necessary skills on the use of modern technology in teaching. CO2: Understand the components and techniques of effective teaching. CO3: Obtain necessary skills in understanding the mathematics teaching. CO4: Strengthen personal character and sense of social responsibility through service learning module. CO5: Contribute to the community by addressing and meeting the community needs.
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Unit-1 |
Teaching Hours:10 |
Teaching Technology
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Development of concept of teaching, Teaching skills, Chalk board skills, Teaching practices, Effective teaching, Models of teaching, Teaching aids (Audio-Visual), Teaching aids (projected and non-projected), Communication skills, Feedback in teaching, Teacher’s role and responsibilities, Information technology for teaching. | |
Unit-2 |
Teaching Hours:5 |
Service Learning
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Concept of difference between social service and service learning, Case study of best practices, understanding contemporary societal issues, Intervention in the community, Assessing need and demand of the chosen community. | |
Unit-3 |
Teaching Hours:15 |
Community Service
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A minimum of fifteen (15) hours documented service is required during the semester. A student must keep a log of the volunteered time and write the activities of the day and the services performed. A student must write a reflective journal containing an analysis of the learning objectives. | |
Text Books And Reference Books:
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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: < marks to be converted to credits > | |
MTH231 - GENERAL TOPOLOGY (2023 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 a generalisation of the concept of continuity and related properties. COBJ3. Appreciate the beauty of deep mathematical results such as Uryzohn’s lemma and understand and apply various proof techniques. |
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Course Outcome |
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CO1: Define topological spaces, give examples and counter-examples on concepts like open sets, basis and subspaces. CO2: Establish equivalent definitions of continuity and apply the same in proving theorems. CO3: Understand the concepts of metrizability, connectedness, compactness and learn the related theorems. |
Unit-1 |
Teaching Hours:15 |
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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 (2023 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 and theorems on meromorphic functions. Course objectives: This course will help the learner to COBJ1. Enhance the understanding the advanced concepts in Complex Analysis COBJ2. Acquire problem solving skills in Complex Analysis.
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Course Outcome |
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CO1: Apply the concept and consequences of analyticity and related theorems. CO2: Represent functions as Taylor and Laurent series, classify singularities and poles, find residues, and evaluate complex integrals using the residue theorem and understand conformal mappings.
CO3: Understand meromorphic functions and simple theorems concerning them.
CO4: Understand advanced theorems on meromorphic functions. |
Unit-1 |
Teaching Hours:15 |
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Analytic functions and singularities
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Morera’s theorem, Gauss mean value theorem, Cauchy inequality for derivatives, Liouville’s theorem, fundamental theorem of algebra, maximum and minimum modulus theorems. Taylor’s series, Laurent’s series, zeros of analytical functions, singularities, classification of singularities, characterization of removable singularities and poles. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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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-3 |
Teaching Hours:15 |
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Meromorphic functions - 1
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Meromorphic functions and argument principle, Schwarz lemma, Rouche’s theorem, convex functions and their properties, Hadamard 3-circles theorem. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Meromorphic functions - 2
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Phragmen-Lindelöf theorem, Riemann mapping theorem, Weierstrass factorization theorem, Harmonic functions, Poisson formula, Poisson integral formula, Jensen’s formula, Poisson-Jensen formula. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH233 - PARTIAL DIFFERENTIAL EQUATIONS (2023 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, partial differential equations. This course includes a study of first and second order linear and non-linear partial differential equations, existence and uniqueness of solutions to various boundary conditions, classification of second order partial differential equations, wave equation, heat equation, Laplace equations and their solutions by Eigenfunction method and Integral Transform Method. Course Objectives: This course will help the learner to COBJ 1: Understand the occurrence of partial differential equations in physics and its applications. COBJ 2: Solve partial differential equation of the type heat equation, wave equation and Laplace equations. COBJ 3: Also solving initial boundary value problems. |
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Course Outcome |
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CO1: Understand the basic concepts and definition of PDE and also mathematical models representing stretched string, vibrating membrane, heat conduction in rod. CO2: Demonstrate the canonical form of second order PDE. CO3: Demonstrate initial value boundary problem for homogeneous and non-homogeneous PDE. CO4: Demonstrate on boundary value problem by Dirichlet and Neumann problem. |
UNIT 1 |
Teaching Hours:10 |
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First Order Partial differential equations order
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Formation of PDE, initial value problems (IVP), boundary value problems (BVP) and IBVP, solutions of first, methods of characteristics for first order PDE, linear and quasi, linear, method of characteristics for one-dimensional wave equations and other hyperbolic equations. | |||||||||||||||||||||||||||||
UNIT 2 |
Teaching Hours:15 |
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Second order Partial Differential Equations
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Origin of second order PDE, Classification of second order PDE, Initial value problems (IVP), Boundary value problems (BVP) and IBVP, Mathematical models representing stretched string, vibrating membrane, heat conduction in solids, second-order equations in two independent variables. Cauchy’s problem for second order PDE, Canonical forms, General solutions. | |||||||||||||||||||||||||||||
UNIT 3 |
Teaching Hours:15 |
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Solutions of Parabolic PDE
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Occurrence of heat equation in Physics, resolution of boundary value problem, elementary solutions, method of separation of variables, method of eigen function expansion, Integral transforms method, Green’s function. | |||||||||||||||||||||||||||||
UNIT 4 |
Teaching Hours:20 |
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Solutions of Hyperbolic and Elliptic PDE
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Occurrence of wave and Laplace equations in Physics, Jury problems, elementary solutions of wave and Laplace equations, methods of separation of variables,, the theory of Green’s function for wave and Laplace equations. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH234 - GRAPH THEORY (2023 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 COBJ 1: Know the history and development of Graph Theory COBJ 2: Understand all the elementary concepts and results COBJ 3: Learn proof techniques and algorithms in Graph Theory |
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Course Outcome |
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CO1: Write precise and accurate mathematical definitions of basics concepts in Graph Theory. CO2: Provide appropriate examples and counterexamples to illustrate the basic concepts. CO3: Demonstrate various proof techniques in proving theorems. CO4: Use algorithms to investigate Graph theoretic parameters. |
Unit-1 |
Teaching Hours:15 |
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Introduction to Graphs
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Graphs as models, degree sequences, classes of graphs, matrices, isomorphism, distances in graphs, connectivity, Eulerian and Hamiltonian graphs, Chinese postman problems, travelling salesman problem and Dijkstra’s algorithm. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Trees
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Properties of trees, rooted trees, spanning trees, algorithms on trees- Prufer’s code, Huffmans coding, searching and sorting algorithms, spanning tree algorithms. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Planarity
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Graphical embedding, Euler’s formula, platonic bodies, homeomorphic graphs, Kuratowski’s theorem, geometric duality. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Graph Invariants
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Vertex and edge coloring, chromatic polynomial and index, matching, decomposition, independent sets and cliques, vertex and edge covers, clique covers, digraphs and networks. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH235 - INTRODUCTORY FLUID MECHANICS (2023 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 fundamental aspects of fluid mechanics. They will have a deep insight and general comprehension on tensors, kinematics of fluid, incompressible flow, boundary layer flows and classification of non-Newtonian fluids. Course Objectives: This course will help the learner to
COBJ1: understand the basic concept of tensors and their representations. |
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Course Outcome |
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CO1: Confidently manipulate tensor expressions using index notation and use the divergence theorem and the transport theorem. CO2: understand the basics laws of Fluid mechanics and their physical interpretations. CO3: comprehend two and three dimension flows incompressible flows. CO4: appreciate the concepts of 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 Rate of Strain: stress components and stress tensor, normal and shear stresses, principal stresses, transformation of the rate of strain and stress, relation between stress and rate of strain. Fundamental basic physical laws: The equations of conservation of mass, linear momentum (Navier-Stokes equations), and energy. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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One, Two and Three Dimensional Invisid Incompressible Flow
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Bernoulli equation, applications of Bernoulli equation, Concept of circulation, Kelvin circulation theorem, constancy of circulation, Laplace equations, stream functions in two- and three-dimensional motion. Two dimensional flow: Rectilinear flow, source and sink, the theorem of Blasius. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:10 |
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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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MTH251 - COMPUTATIONAL MATHEMATICS USING PYTHON (2023 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 aimsto solve mathematical models using differential equations, linear algebra and fluid mechanics using Python libraries. Course objectives: This course will help the learner to COBJ1: Acquire skill in using suitable libraries of Python to solve real-world problems giving rise to differential equations COBJ2: Gain proficiency in using Python to solve problems on linear algebra. COBJ3: Build user-defined functions to deal with the problem on fluid mechanics. |
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Course Outcome |
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CO1.: Demonstrate the use of Python libraries for handling problems on mathematical modelling CO2.: Compute the problems on linear algebra using Python libraries. CO3.: Handle the Python libraries for solving problems on fluid dynamics. |
Unit-1 |
Teaching Hours:45 |
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Proposed Topics:
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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 - TEACHING TECHNOLOGY AND SERVICE LEARNING (2022 Batch) | |||||||||||||||||||||||||
Total Teaching Hours for Semester:30 |
No of Lecture Hours/Week:2 |
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Max Marks:0 |
Credits:2 |
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Course Objectives/Course Description |
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Course Description: This course is intended to assist the students in acquiring necessary skills on the use of modern technology in teaching, they are exposed to the principles, procedures and techniques of planning and implementing teaching techniques. Through service learning they will apply the knowledge in real-world situations and benefit the community. Course objectives: This course will help the learner to COBJ 1: Understand the pedagogy of teaching. COBJ 2: Able to use various ICT tools for effective teaching. COBJ 3: Apply the knowledge in real-world situations. COBJ 4: Enhances academic comprehension through experiential learning. |
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Course Outcome |
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CO1: Gain necessary skills on the use of modern technology in teaching. CO 2: Understand the components and techniques of effective teaching. CO 3: Obtain necessary skills in understanding the mathematics teaching. CO4: Strengthen personal character and sense of social responsibility through service learning module. CO5: Contribute to the community by addressing and meeting community needs. |
Unit-1 |
Teaching Hours:10 |
Teaching Technology
|
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Development of concept of teaching, Teaching skills, Chalk board skills, Teaching practices, Effective teaching, Models of teaching, Teaching aids (Audio-Visual), Teaching aids (projected and non-projected), Communication skills, Feedback in teaching, Teacher’s role and responsibilities, Information technology for teaching. | |
Unit-2 |
Teaching Hours:5 |
Service Learning
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Concept of difference between social service and service learning, Case study of best practices, understanding contemporary societal issues, Intervention in the community, Assessing need and demand of the chosen community. | |
Unit-3 |
Teaching Hours:15 |
Community Service
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A minimum of fifteen (15) hours documented service is required during the semester. A student must keep a log of the volunteered time and write the activities of the day and the services performed. A student must write a reflective journal containing an analysis of the learning objectives. | |
Text Books And Reference Books:
| |
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: < marks to be converted to credits > | |
MTH331 - MEASURE THEORY AND LEBESGUE INTEGRATION (2022 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: The Course covers the basic material that one needs to know in the theory of functions of a real variable and measure and integration theory as expounded by Henri Léon Lebesgue. Course objectives: This course will help the learner to COBJ1. Enhance the understanding of the advanced notions from Mathematical Analysis COBJ2. Know more about the Measure theory and Lebesgue Integration |
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Course Outcome |
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CO1: understand the fundamental concepts of mathematical analysis. CO2: State some of the classical theorems in of Advanced Real Analysis. CO3: be familiar with measurable sets and functions. CO4: Integrate a measurable function. CO5: understand the properties of Lebesgue Normed Linear Spaces. |
Unit-1 |
Teaching Hours:15 |
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Lebesgue Measure
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Introduction to Analysis, Lebesgue Outer Measure, The sigma-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 | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Lebesgue Measurable Functions
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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-3 |
Teaching Hours:15 |
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Lebesgue Integration
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The Riemann Integral; 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-4 |
Teaching Hours:15 |
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Differentiation, Lebesgue Integration and Lp-Spaces
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Continuity of Monotone Functions, Differentiation of Monotone Functions, Functions of Bounded Variation, Absolutely Continuous Functions, Integrating Derivatives, Convex Functions, Normed Linear Spaces, The inequalities of Young, Holder and Minkowski | |||||||||||||||||||||||||||||
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 (2022 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. COBJ2. To become familiar with the methods which will help to obtain solution of algebraic and transcendental equations, linear system of equations, finite differences, interpolation numerical integration and differentiation, numerical solution of differential equations and boundary value problems. COBJ3. Understand accuracy, consistency, stability and convergence of numerical methods. |
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Course Outcome |
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CO1.: 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), Graeffe’s root squaring method. Solution of Linear System of Algebraic Equations: LU-decomposition methods (Cholesky method), consistency and ill-conditioned system of equations, Tridiagonal system of equations, Thomas algorithm. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Interpolation and Numerical Integration
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Lagrange, Hermite, Natural Cubic-spline interpolation - with uniqueness and error term. 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, Galerkin 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 - DIFFERENTIAL GEOMETRY (2022 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 learners to acquire active knowledge and understanding of the basic concepts and properties of the geometry of curves and surfaces in Euclidean space. Also, this course aims at connecting geometric properties of curves, surfaces, and their higher dimensional analogues using the methods of calculus. Course Objectives: This course will help the learner to COBJ 1: understand the calculus on the Euclidean geometry of E3 COBJ 2: implement the properties of curves and surfaces in solving problems described in terms of tangent vectors / vector fields / forms. COBJ 3: derive and understand about the intrinsic geometry of the surfaces and curves on surfaces. COBJ 4: interpret the structure of surfaces using the first and second fundamental forms. |
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Course Outcome |
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CO1: Express sound knowledge on the basic concepts in geometry of curves and surfaces in Euclidean space. CO2: Demonstrate mastery in solving typical problems associated with the theory. |
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, and 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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Intrinsic geometry of Surface
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First fundamental quadratic form of a surface, angle of two intersecting curves in a surface, element of area, family of curves in a surface, principal directions, isometric surfaces, the Riemannian curvature tensor, the Gaussian curvature of a surface | |||||||||||||||||||||||||||||
UNIT 4 |
Teaching Hours:15 |
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Surfaces in Space
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Second fundamental form of a surface, equation of Gauss and equations of Codazzi, normal curvature of surface, lines of curvature of a surface, isometric conjugate nets-Dupin indicatrix. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH341A - ADVANCED FLUID MECHANICS (2022 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 mathematical modelling of nano-liquids Course objectives: This course will help the learner to COBJ 1: Understand the different modes of heat transfer and their applications. COBJ 2: Understand the importance of doing the non-dimensionalization of basic equations. COBJ 3: Understand the boundary layer flows. COBJ 4: Familiarity with porous medium and non-Newtonian fluids |
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Course Outcome |
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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 Classification and the basic equations of non Newtonian Fluids 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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Mathematical Modelling of Nano-liquids for Thermal Applications
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Buongiorno Model (Two phase model): Nanoparticle/Fluid Slip : Inertia, Brownian Diffusion, Thermophoresis, Diffusiophoresis, Magnus Effect, Fluid Drainage, Gravity, Relative importance of the Nanoparticle Transport Mechanisms. Conservation Equation for two phase Nanoliquids: The Continuity equation, The Momentum equation and The energy equation. | |||||||||||||||||||||||||||||
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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MTH341B - ADVANCED GRAPH THEORY (2022 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: Theory of intersection graphs, perfect graphs, chromatic graph theory and eigenvalues of graphs are dealt with in this course. Course objectives: This course will help the learner to COBJ 1: understand and apply the fundamental and advanced concepts in distance-related graph coloring problems. COBJ 2: understand and apply the fundamental and advanced concepts in color connections and disconnections. COBJ 3: understand and apply the fundamental and advanced concepts in spectral properties of graphs. COBJ 4: enhance the skill of proof-writing techniques. |
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Course Outcome |
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CO1: implement concepts and principles of distance-related graph coloring in real-life problems. CO2: implement the concepts and principles of color connections and disconnections in practical problems. CO3: understand and apply the concepts and principles of spectral graph theory in practical situations. CO4: demonstrate the ability to communicate the subject in a meaningful and efficient way. |
Unit-1 |
Teaching Hours:15 |
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Distance and Coloring
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Channel assignment problem, T-colourings, L(2,1)-colourings, radio colourings, Hamiltonian colourings. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Color Connections
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Rainbow connections, strong rainbow connections, rainbow connectivity, proper connection, rainbow disconnection. | |||||||||||||||||||||||||||||
Unit-3 |
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, incidence matrix of oriented graphs, definite and semidefinite matrices, strongly regular graphs. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Laplacian of Graphs
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Laplacian matrix of a graph, representations, energy and eigenvalues, connectivity, interlacing, the generalised Laplacian. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH341C - PRINCIPLES OF DATA SCIENCE (2022 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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Data Science is an interdisciplinary, problem-solving oriented subject that learns to apply scientific techniques to practical problems. This course provides strong foundation for data science and application area related to information technology and understand the underlying core concepts and emerging technologies in data science. |
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Course Outcome |
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CO1: The managerial understanding of the tools and techniques used in Data Science process. CO2: Analyze data analysis techniques for applications handling large data.
CO3: Apply techniques used in Data Science and Machine Learning algorithms to make data driven, real time, day to day organizational decisions CO4: Present the inference using various Visualization tools CO5: Learn to think through the ethics surrounding privacy, data sharing and algorithmic decision-making |
UNIT 1 |
Teaching Hours:12 |
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Introduction to Data Science
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Definition, big data and data science hype, why data science, getting past the hype, the current landscape, who is data scientist? - data science process overview, defining goals, retrieving data, data preparation, data exploration, data modeling, presentation. | |||||||||||||||||||||||||||||
UNIT 2 |
Teaching Hours:12 |
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Big Data
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Problems when handling large data, General techniques for handling large data, Case study, Steps in big data, Distributing data storage and processing with Frameworks, Case study. | |||||||||||||||||||||||||||||
UNIT 3 |
Teaching Hours:14 |
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Machine Learning
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Machine learning, modeling process, training model, validating model, predicting new observations, supervised learning algorithms, unsupervised learning algorithms. Introduction to deep learning. | |||||||||||||||||||||||||||||
UNIT 4 |
Teaching Hours:12 |
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Data Visualization
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The characteristic polynomial, eigenvalues and graph parameters, eigenvalues of regular graphs, eigenvalues and expanders, strongly regular graphs. | |||||||||||||||||||||||||||||
Unit-5 |
Teaching Hours:10 |
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Ethics and Recent Trends
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Data Science Ethics – Doing good data science – Owners of the data - Valuing different aspects of privacy - Getting informed consent - The Five Cs – Diversity – Inclusion – Future Trends. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH341D - NUMERICAL LINEAR ALGEBRA (2022 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 computational aspects of Linear Algebra. Course Objectives: This course will help the learner to COBJ1. Demonstrate the computational ability in handling matrices, norms and method of least squares. COBJ2. Solve systems of equations using various methods of numerical linear algebra.. |
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Course Outcome |
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CO1: Understand computational aspects of multiplying matrices, finding inverse and determinant of square matrices, finding various norms. CO2: Learn QR factorization, orthogonalization and handle least squares problems. CO3: Gain the skill set to solve large system of equations using elimination and LU factorization. CO4: compute eigenvalues of large linear systems and compute the SVD. |
Unit-1 |
Teaching Hours:15 |
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Fundamentals
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Matrix-vector multiplication, inverse, determinant, Orthogonal vectors and matrices, norms, Singular value decomposition. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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QR factorization and least squares
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Projectors, QR factorization, Gram-Schmidt orthogonalization, Householder triangularization, Least squares problems. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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System of Equations
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Gauss elimination, pivoting, stability, Stability of Gaussian elimination, Cholesky factorization | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Eigenvalues
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Eigenvalues and algorithms, Reduction to Heissenberg or tridiagonal form, Rayleigh-Quotient method, Computing the SVD. | |||||||||||||||||||||||||||||
Text Books And Reference Books: L. N. Trefethen and D. Bau, Numerical Linear Algebra, SIAM, 1997. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH342A - MAGNETOHYDRODYNAMICS (2022 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 shall help the students to COBJ1: understand mathematical form of Gauss’s law, Faraday’s law and Ampere’s law and their 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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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 Solution
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Hartmann flow, generalized Hartmann flow, velocity distribution, expression for induced current and magnetic field, temperature distribution, Hartmann Couette flow, magnetostatic-force free magnetic field, abnormality parameter, Chandrasekhar 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 corona, 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. | |||||||||||||||||||||||||||||
Evaluation Pattern
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MTH342B - THEORY OF DOMINATION IN GRAPHS (2022 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 covers a large area of domination in graphs. This course discusses different types of dominations with their applications in real-life situations, the relation of domination related parameters with other graph parameters such as vertex degrees, chromatic number, independence number, packing number, matching number etc.
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Course Outcome |
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CO1: Have a thorough understanding on the concepts domination in graphs. CO2: Apply the domination theory in various practical problems. CO3: Gain mastery over the reasoning and proof writing techniques. |
Unit-1 |
Teaching Hours:15 |
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Domination in Graphs
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Dominating sets in graphs, mathematical history of domination in graphs, chessboard problems, applications to real-life situations, total domination, independent domination, connected domination, other variants of domination. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Bounds of Domination Number
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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-3 |
Teaching Hours:15 |
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Domination, Independence & Irredundance
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Hereditary and super hereditary properties, independent ad dominating sets, irredundant sets, Domination Chain. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Changing and Unchanging Domination
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Classes of changing graphs: Vertex removal (CVR), Edge Removal (CER) and Edge Addition (CEA), classes of unchanging graphs, relation between classes of changing and unchanging graphs. | |||||||||||||||||||||||||||||
Text Books And Reference Books: T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs, Reprint, CRC Press, 2000. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH342C - NEURAL NETWORKS AND DEEP LEARNING (2022 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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The main aim of this course is to provide fundamental knowledge of neural networks and deep learning. On successful completion of the course, students will acquire fundamental knowledge of neural networks and deep learning, such as Basics of neural networks, shallow neural networks, deep neural networks, forward & backward propagation process and build various research projects. |
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Course Outcome |
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CO1: Understand the major technology trends in neural networks and deep learning. CO2: Build, train and apply neural networks and fully connected deep neural networks. CO3: Implement efficient (vectorized) neural networks for real time application. |
Unit-1 |
Teaching Hours:12 |
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Introduction to Artificial Neural Networks
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Neural Networks-Application Scope of Neural Networks- Fundamental Concept of ANN: The Artificial Neural Network-Biological Neural Network-Comparison between Biological Neuron and Artificial Neuron-Evolution of Neural Network. Basic models of ANN-Learning Methods-Activation Functions-Importance Terminologies of ANN. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:12 |
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Supervised Learning Network
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Shallow neural networks- Perceptron Networks-Theory-Perceptron Learning Rule-Architecture-Flowchart for training Process-Perceptron Training Algorithm for Single and Multiple Output Classes. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:12 |
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Convolutional Neural Network
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Introduction - Components of CNN Architecture - Rectified Linear Unit (ReLU) Layer -Exponential Linear Unit (ELU, or SELU) - Unique Properties of CNN -Architectures of CNN -Applications of CNN | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:12 |
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Recurrent Neural Network
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Introduction- The Architecture of Recurrent Neural Network- The Challenges of Training Recurrent Networks- Echo-State Networks- Long Short-Term Memory (LSTM) - Applications of RNN | |||||||||||||||||||||||||||||
Unit-5 |
Teaching Hours:12 |
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Auto Encoder And Restricted Boltzmann Machine
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Introduction - Features of Auto encoder Types of Autoencoder Restricted Boltzmann Machine- Boltzmann Machine - RBM Architecture -Example - Types of RBM | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH342D - FRACTIONAL CALCULUS (2022 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: The main aim of this course is to provide fundamental knowledge of fractional calculus. This course includes a study of special functions and different fractional differential and integral operators and their properties. This course also helps to know how to solve fractional differential equations using different methods.
COBJ1.fundamentals and properties of special functions and their properties. COBJ2.familiarize with the difference between different fractional derivatives. COBJ3. analyze and develop problem-solving skills for fractional differential equations by various methods. |
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Course Outcome |
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CO1: familiarize with basic concepts and properties of special functions. CO2.: Understand the basics of the fractional calculus of different operators. CO3.: Apply methods to solve fractional differential equations. |
Unit-1 |
Teaching Hours:10 |
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Special Functions of the Fractional Calculus
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Gamma function: Definition, properties and contour integral representation. Mittag-Leffler function-Definition, relation to some other functions, derivative of the Mittag-Leffler function. Wright function and their properties. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:20 |
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Fractional Derivatives and Integrals
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History of fractional calculus, Grunwald-Letnikov and Reimann-Liouville differential and integral operators, Caputo fractional derivative. Left and right fractional derivatives, Properties of fractional derivatives – linearity, Leibniz rule, and fractional derivative of a composite function. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:10 |
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Existence Uniqueness Theorems
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Fractional Green's function-Definiation and properties. The linear and general form of fractional differential equations, Existence uniqueness as a method of solutions. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:20 |
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Methods for fractional differential equations
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Laplace transform - Reimann-Liouville, Grunwald-Letnikov, and Caputo fractional derivatives. Mellin transform - Reimann-Liouville, Miller-Ross, Caputo fractional derivatives. Power series and numerical methods solve fractional differential equations. | |||||||||||||||||||||||||||||
Text Books And Reference Books: I. Podlubny, Fractional differential equations, Academic Press, (1998). | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH351 - NUMERICAL METHODS USING PYTHON (2022 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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CO1: Formulate and solve Linear Programming Problems using graphical and simplex method. CO2: Solve Transportation problems by using Modified distribution method. CO3: Solve assignment problems by using Hungarian technique. CO4: Solve simple two person zero sum games. |
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
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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 H. 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 (2022 Batch) | |||||||||||||||||||||||||
Total Teaching Hours for Semester:30 |
No of Lecture Hours/Week:2 |
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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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CO1: Expose to the field of their professional interest CO2: Explore an opportunity to get practical experience in the field of their interest CO3: Strengthen the research culture. |
Unit-1 |
Teaching Hours:30 |
Internship in PG Mathematics course
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M.Sc. Mathematics students have to undertake a mandatory internship in Mathematics for a period of not less than 45 working days. Students can chose to their internship in reputed research centers, recognized educational institutions, or participate in training or fellowship program offered by research institutes or organization subject to the approval of program coordinator and the Head of the department. 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 Head of the Department will assign faculty members from the department as supervisors 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 end of the required period of internship, the student 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. Within 20 days from the day of reopening, the department will conduct a presentation by the student followed by a Viva-Voce. During the presentation, the supervisor or a nominee of the supervisor should be present and be one of the evaluators. 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 within one month of commencement of 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. The final evaluation includes a presentation by the students followed by the Viva-Voce examination. | |
Text Books And Reference Books: . | |
Essential Reading / Recommended Reading . | |
Evaluation Pattern . | |
MTH411 - TEACHING PRACTICE (2022 Batch) | |
Total Teaching Hours for Semester:15 |
No of Lecture Hours/Week:1 |
Max Marks:0 |
Credits:1 |
Course Objectives/Course Description |
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This course is designed to prepare students for real class room situation under the supervision of faculty mentors. It provides experiences in the actual teaching and learning environment. · Fifteen hours of teaching assignments for UG classes shall be undertaken by each student during the 3rd and 4th semester. · Each student shall be under the supervision of a faculty mentor/guide. · The 15 hours may be distributed among 1 or 2 subjects- which shall be a combination of theory and problem based papers. · A Structured Plan stating the Topic- Objectives- Methodology and Evaluation shall be prepared in advance by the student for each class session and submitted to the faculty mentor/guide. · Faculty guides shall maintain an assessment register for their respective students and record assessment for each session on the given parameters. |
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Course Outcome |
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CO1: Demonstrate and use various teaching pedagogies. CO2: Develop content and material for class room teaching. CO3: Manage classroom sessions effectively. CO4: Assist the teachers in internal assessments. CO5: Articulate and communicate in an effective way. |
Unit-1 |
Teaching Hours:15 |
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Teaching Practice
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This course is designed to prepare students for real class room situation under the supervision of faculty mentors. It provides experiences in the actual teaching and learning environment. · Fifteen hours of teaching assignments for UG classes shall be undertaken by each student during the 3rd and 4th semester. · Each student shall be under the supervision of a faculty mentor/guide. · The 15 hours may be distributed among 1 or 2 subjects- which shall be a combination of theory and problem based papers. · A Structured Plan stating the Topic- Objectives- Methodology and Evaluation shall be prepared in advance by the student for each class session and submitted to the faculty mentor/guide. · Faculty guides shall maintain an assessment register for their respective students and record assessment for each session on the given parameters. | ||||||||||||||||
Text Books And Reference Books: NA | ||||||||||||||||
Essential Reading / Recommended Reading NA | ||||||||||||||||
Evaluation Pattern This course is CIA based. Only grade is given.
Assessment Criteria:
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MTH431 - CLASSICAL MECHANICS (2022 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 some of the key ideas of classical mechanics like generalized coordinates, Lagrange's equations and Hamilton's equations. Also, this course aims at introducing the Lagrangian Mechanics and Hamiltonian mechanics on Manifolds. Course Objectives: This course will help the learner to COBJ 1: derive necessary equations of motions based on the chosen configuration space. COBJ 2: gain sufficient skills in using the derived equations in solving the applied problems in Classical Mechanics. COBJ 3: effectively use Lagrangian mechanics on the manifolds. COBJ 4: deal with the Hamiltonian mechanics on the manifolds. |
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Course Outcome |
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CO1: interpret mechanics through the configuration space. CO2: solve problems on mechanics by using Lagrange's and Hamilton's principle. CO3: demonstrate the Lagrangian and Hamiltonian Mechanics on Manifolds. |
Unit-1 |
Teaching Hours:10 |
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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 and Hamilton's equations
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Derivation of Lagrange's equations: Kinetic energy, Lagranges equations, form of equations of motion, non-holonomic systems, Examples: Spherical pendulum, Double pendulum, Lagrangian multiplier and constraint forces, Particle in a Whirling tube, Particle with moving support, Rheonomic constraint system, Integrals of Motion: Ignorable coordinates, Examples: the Kepler problem, Routhian function, Conservative systems, Natural systems, Liouvillie' system and examples. Hamilton's principle, Hamilton's equations. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Lagrangian Mechanics on Manifolds
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Introduction to differentiable manifolds, Lagrangian system on a manifold, Lagrangian system with holonomic constraints, Lagrangian non-autonomous system, Noether's theorem, equivalence of D'Alembert-Lagrange principle and the variational principle, Linearization of the Lagrangian system, small oscillations. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Hamiltonian Mechanics on Manifolds
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Hamiltonian vector fields, Hamiltonian Phase flows, Integral invariants, Law of conservation of energy, Lie algebra of Hamiltonian functions, Locally Hamiltonian vector fields. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH432 - FUNCTIONAL ANALYSIS (2022 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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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:15 |
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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. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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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. | |||||||||||||||||||||||||||||
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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MTH433 - ADVANCED LINEAR PROGRAMMING (2022 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, goal programming, decision analysis and games, CPM - PERT methods and dynamic programming. Course Objectives: This course will help the students to COBJ 1: 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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CO1: apply the notions of linear programming in solving transportation problems.
CO2: acquire knowledge in formulating Tax planning problem and use goal programming algorithms.
CO3: make decisions using decision analysis under certainty and uncertainty.
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.
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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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Goal Programming and Decision Analysis
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Goal programming: formulation, tax planning problem, goal programming algorithms, the weights method, preemptive method. Decision Analysis and Games: Decision making under certainty (AHP), Decision making under risk, Decision under uncertainty, Game theory. | |||||||||||||||||||||||||||||
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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MTH441A - COMPUTATIONAL FLUID DYNAMICS (2022 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 to solve the Burger’s equations using finite difference equations, 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 using different shape functions. |
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Course Outcome |
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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
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Evaluation Pattern
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MTH441B - ATMOSPHERIC SCIENCE (2022 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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CO1: Model the atmospheric flows mathematically CO2: Understand the atmospheric waves and instabilities in atmosphere
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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
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MTH441C - MATHEMATICAL MODELLING (2022 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 students to COBJ 1: Interpret the real-world problems in the form of ordinary and partial differential equations. COBJ 2: 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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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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MTH442A - ALGEBRAIC GRAPH THEORY (2022 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: The theory of the automorphism of graphs, permutation groups, and different transitive graphs is discussed in this course. Course Objectives: This course will help the learner to COBJ 1: understand the fundamental and advanced concepts in automorphism groups of graphs. COBJ 2: apply the concepts in permutation groups and related concepts. COBJ 3: understand the concepts in different types of transitivity of graphs. COBJ 4: develop critical thinking, communication, and empirical and quantitative skills.
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Course Outcome |
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CO1: implement the concepts and principles of algebraic properties of graphs in a meaningful way. CO2: implement the concepts and principles of the theory of transitive graphs in practical situations. CO3: demonstrate the ability to communicate the subject in a meaningful way. CO4: have acquaintance with emerging areas of research in the topics concerned. |
Unit-1 |
Teaching Hours:15 |
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Morphisms of Graphs
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Graphs, Automorphisms and homomorphisms of graphs, circulant graphs, Johnson graphs and Kneser graphs, line graphs. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Permutation Groups and Graphs
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Permutation groups, orbits and stabilizers, Burnside’s theorem, asymmetric graphs, orbits in pairs, primitive groups, primitivity and connectivity. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Transitive Graphs
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Vertex transitive graphs, edge transitive graphs, connectivity, Hamilton paths and cycles, Cayley graphs, retracts and cores. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Arc-Transitive Graphs
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Arc transitive graphs, arc graphs, cubic arc transitive graphs, distance transitive graphs, the Petersen graph, the Coxeter graph, Tutte’s 8-cage. | |||||||||||||||||||||||||||||
Text Books And Reference Books: C. Godsill and G. Royle, Algebraic Graph Theory, Springer, 2001. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH442B - STRUCTURAL GRAPH THEORY (2022 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 covers the topics in intersection graphs, interval graphs, chordal graphs and perfect graphs. Course objectives: This course will help the learner to COBJ 1: apply the concepts of topological indices in problems related to chemical or biological structures. COBJ 2:understand the concepts of degree-based topological indices in network-related problems. COBJ 3: apply the fundamental and advanced concepts in structural properties of graphs. COBJ 4: enhance the skills of writing proofs. |
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Course Outcome |
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CO1: use the concepts of the topological aspects of graphs.
CO2: implement the topics concerned in practical situations related to chemical, social, and biological networks. CO3: demonstrate the ability to communicate the subject in a meaningful way.
CO4: get acquainted with emerging research areas in the topics concerned. |
Unit-1 |
Teaching Hours:15 |
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Intersection Graphs
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Intersection graphs, intersection classes, clique graphs, line graphs, hypergraphs, interval graphs, interval hypergraphs, proper interval graphs, unit interval graphs. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Chordal Graphs
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Chordal graphs, perfect elimination ordering, chordal graphs as intersection graphs, weakly and strongly chordal graphs, comparability graphs, split graphs. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Perfect Graphs
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Vertex multiplication, perfect graphs, the perfect graph theorem, classes of perfect graphs, imperfect graphs, strong perfect graph theorem. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Topological Representations
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Subgraphs, subdivisions, homeomorphism, Cartesian products, edge-complements, suspensions, amalgamations, regular quotients and coverings, orientable and non-orientable surfaces, imbeddings. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH442C - APPLIED GRAPH THEORY (2022 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: Theory of automorphism of graphs, permutation groups, transitive graphs and eigenvalues and Laplacian eigenvalues of graphs are dealt with in this course. Course Objectives: This course will help the students to COBJ 1: apply the concepts of topological indices in problems related to chemical or biological structures. |
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Course Outcome |
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CO1: use the concepts of the topological aspects of graphs. CO2: implement the topics concerned in practical situations related to chemical, social, and biological networks. CO3: demonstrate the ability to communicate the subject in a meaningful way. CO4: get acquainted with emerging research areas in the topics concerned. |
Unit-1 |
Teaching Hours:15 |
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Distance Related Topological Indices
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Distance based indices, Wiener index, properties related to distance, extremal problems in general graphs, Wiener index of trees, Wiener index of graph classes, inverse problems. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Degree Related Topological Indices
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Degree based indices, Randic index, degree-based indices of trees, Zagreb indices, more on ABC indices. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Connectivity and Independence Based Indices
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Zagreb group indices, connectivity index, connectivity ID number, Z-index, Hosoya index, elementary properties of M-S index and Hosoya index, extremal problems, graph transformations. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Graph Spectra and Graph Energy
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Graph spectra, graph energy, elementary properties, bounds for graph energy, extremal problems in trees, energy like invariants, other invariants of graph spectra. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH443A - REGRESSION ANALYSIS (2022 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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This course aims to provide the grounding knowledge about the regression model building of simple and multiple regression. |
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Course Outcome |
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CO1: Develop a deeper understanding of the linear regression model. CO2: Learn about R-square criteria for model selection CO3: Understand the forward, backward and stepwise methods for selecting the variables CO4: Understand the importance of multicollinearity in regression modelling CO5: Ability to use and understand generalizations of the linear model to binary and count data
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Unit-1 |
Teaching Hours:15 |
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Simple Linear Regression
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Introduction to regression analysis: Modelling a response, overview and applications of regression analysis, major steps in regression analysis. Simple linear regression (Two variables): assumptions, estimation and properties of regression coefficients, significance and confidence intervals of regression coefficients, measuring the quality of the fit. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Multiple Linear Regression
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Multiple linear regression model: assumptions, ordinary least square estimation of regression coefficients, interpretation and properties of regression coefficient, significance and confidence intervals of regression coefficients. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:10 |
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Criteria for Model Selection
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Mean Square error criteria, R2 and R2 (R bar square) criteria for model selection; Need of the transformation of variables; Box-Cox transformation; Forward, Backward and Stepwise procedures. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:10 |
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Residual Analysis
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Residual analysis, Departures from underlying assumptions, Effect of outliers, Collinearity, Non-constant variance and serial correlation, Departures from normality, Diagnostics and remedies. | |||||||||||||||||||||||||||||
Unit-5 |
Teaching Hours:10 |
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Non-linear Regression
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Introduction to nonlinear regression, Least squares in the nonlinear case and estimation of parameters, Models for binary response variables, estimation and diagnosis methods for logistic and Poisson regressions. Prediction and residual analysis. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH443B - DESIGN AND ANALYSIS OF ALGORITHMS (2022 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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This course aims to introduce the methods to analyze and evaluate the performance of an algorithm. It introduces the different design techniques for designing efficient algorithms. |
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Course Outcome |
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CO1: Demonstrate their ability to apply appropriate Data Structures to solve problems. CO2: Design and develop algorithms using various design techniques. CO3: Evaluate the efficiency of Algorithms by analyzing the running time of algorithms for problems in various domain |
UNIT 1 |
Teaching Hours:12 |
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Introduction
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Algorithm Specification, Analysis of Insertion sort, Performance Analysis, Space complexity, Time Complexity, Asymptotic notations, Amortized Analysis | |||||||||||||||||||||||||||||
UNIT 2 |
Teaching Hours:12 |
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Divide & Conquer and Greedy Approach
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Divide and Conquer, Binary search, Quick sort, Strassen’s Matrix Multiplication Greedy Approach, Knapsack problem, Minimum cost spanning tree, PRIM’s and Kruskal’s Algorithm, single source shortest path. | |||||||||||||||||||||||||||||
UNIT 3 |
Teaching Hours:12 |
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Dynamic Programming and Backtracking
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Dynamic Programming, All pairs shortest path, longest common sequence, The general method, 8 Queens Problem, Sum of subsets. | |||||||||||||||||||||||||||||
UNIT 4 |
Teaching Hours:12 |
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Branch and Bound Techniques
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Branch and Bound, 0/1 knapsack problem, Travelling salesperson problem. | |||||||||||||||||||||||||||||
Unit-5 |
Teaching Hours:12 |
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NP Hard and NP Complete Problems
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Basic Concepts, NP hard Graph problems, NP Hard Scheduling problems, NP hard code generation problems, Approximation Algorithms, Polynomial time approximation schemes, PRAM Algorithms, Computational model, merge sort, Mesh Algorithms, Computational model. | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH444A - RIEMANNIAN GEOMETRY (2022 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 basic Riemannian geometry that covers the topics tangent spaces, tensor fields, Riemannian metrics, curvature using Levi-Civita connection, Geodesics and Riemannian immersions. Course objectives: This course will help the learner to COBJ1. understand all the basic concepts and theory in Riemannian Geometry. |
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Course Outcome |
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CO1: On successful completion of the course, the students should be able to understand the basics of Riemannian Geometry.
CO2: On successful completion of the course, the students should be able to demonstrate the notion of Levi-Civita Connection and Curvature. CO3: On successful completion of the course, the students should be able to use the theory of Riemannian Geometry to understand the notion of Geodesics. CO4: On successful completion of the course, the students should be able to apply the theory of Riemannian Geometry to understand the notion of Immersions.
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Unit-1 |
Teaching Hours:20 |
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Basic Riemannian Geometry
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Differential Manifolds, Smooth Maps and Diffeomorphisms, Tangent spaces to a Manifold, Derivatives of Smooth Maps, Immersions and Submersions, Submanifolds, Tensor fields, Covariant Differentiation, Riemannian metrics. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Curvature
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The Levi-Civita Connection, Gauss Theory of Surfaces in R^3, Curvature and Parallel Transport, The Curvature tensor. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:10 |
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Geodesics
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Vector fields along Maps, Two Parameter Maps, Geodesics: Definition and Existence, Examples of Geodesics, Geodesics by Calculus of Variations. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:15 |
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Riemannian Immersions
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Exponential Map, Riemannian Submanifolds, Minimal submanifolds and convex hypersurfaces, Totally Geodesic submanifolds. | |||||||||||||||||||||||||||||
Text Books And Reference Books: S Kumaresan, Riemannian Geometry: Concepts, Examples and Techniques, Techno world, Kolkata, 2020. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH444B - FUZZY MATHEMATICS (2022 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 Fuzzy sets, operations on fuzzy sets, Fuzzy numbers, Arithmetic Fuzzy numbers, Fuzzy equations, Fuzzy relations, Projection of Fuzzy relations.
Course objectives: This course will help the learner to COBJ1. understand the notion of Fuzzy sets and their operations. COBJ2. demonstrate the different types of fuzzy numbers. COBJ3. perform arithmetic on Fuzzy numbers. COBJ4. handle fuzzy equations, Crisp Relations, and Fuzzy Relations. |
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Course Outcome |
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CO1: Understand different types of Fuzzy sets and operations on/among the Fuzzy sets. CO2: Demonstrate the use of Fuzzy numbers and arithmetic of Fuzzy numbers. CO3: Solve fuzzy equations and use Crisp Relations, Fuzzy Relations. |
Unit-1 |
Teaching Hours:20 |
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Fuzzy Sets and their Operations
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Vagueness and Fuzzy Mathematics, Algebra of truth values, Posets, Lattices, Frames, Zadeh’s Fuzzy sets, α-Cuts of Fuzzy sets, Interval-valued and Type 2 Fuzzy sets, Triangular Norms and Conorms, L-fuzzy sets, “Intuitionistic” Fuzzy sets and their extensions, the extension principle, Boolean-valued sets, Axiomatic Fuzzy set theory. | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:10 |
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Fuzzy Numbers
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Triangular Fuzzy Numbers, Trapezoidal Fuzzy Numbers, Gaussian Fuzzy Numbers, Quadratic Fuzzy Numbers, Exponential Fuzzy Numbers, L-R Fuzzy Numbers, Generalized Fuzzy Numbers. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:10 |
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Arithmetic of Fuzzy Numbers
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Interval Arithmetic and α-Cuts, Fuzzy Arithmetic and the Extension Principle, Fuzzy Arithmetic of Triangular Fuzzy Numbers, Fuzzy Arithmetic of Generalized Fuzzy Numbers, Comparing Fuzzy Numbers. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:20 |
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Fuzzy Equations, Relations and Projection of Fuzzy Relations
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Fuzzy Equations: Solving the Fuzzy Equation A . X + B = C, Solving Fuzzy Equation A . X^2+B . X+C= D; Crisp Relations: Properties of Relations, New Relations and old ones, Representing Relations using Matrices, Representing Relation using Matrices, Transitive closure of relations, Equivalence Relations; Fuzzy Relations; Projection of Fuzzy Relations, cylindrical extension; Fuzzy binary relation on a set; Fuzzy orders. | |||||||||||||||||||||||||||||
Text Books And Reference Books: A. Syropoulos, T. Grammenos, A Modern introduction to Fuzzy Mathematics, 1st ed., John Wiley and Sons, 2020. | |||||||||||||||||||||||||||||
Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH444C - ADVANCED ANALYSIS (2022 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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This course aims at introducing advanced mathematical analysis on space of summable functions and its relation with partial differential equations, convex sets ad convex functions, Random measures in infinite dimensional space, matrix monotone function, matrix means, matrix power mean and Karcher mean.
This course will help the learner to COBJ1: understand the use of advanced mathematical analysis in PDE, Convex sets, convex functions, random measures in infinite-dimensional space. COBJ2: demonstrate the applications of random measures in infinite-dimensional space, matrix inequalities via matrix means. |
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Course Outcome |
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CO1: understand the relationship between space of summable functions and PDE. CO2: demonstrate the notion of convex sets, convex functions and its applications.
CO3: apply RKHSs
CO4: use matrix monotone function, matrix means, matrix power and Karcher mean.
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Unit-1 |
Teaching Hours:15 |
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The Space of Summable Functions and PDE
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The Laplace, Heat and Wave Equations, The method of separation of variables, Lebesgue’s spaces, The existence theorem for PDE’s | |||||||||||||||||||||||||||||
Unit-2 |
Teaching Hours:20 |
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Convex Sets and Convex Functions
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Convex Sets, Proper Convex Functions, Convex Duality, inequalities, Action and Energy, the thermodynamic equilibrium, Polyhedral sets, Convex optimization, Stationery states for discrete-time Markov Process, Linear Programming, Minimax theorems and the theory of games, A general approach to convexity. | |||||||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Random Measures in infinite-dimensional dynamics
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Introduction, RKHSs in measurable category, L2loc(λ) vs L2(λ), continuous networks, Applications of RKHSs. | |||||||||||||||||||||||||||||
Unit-4 |
Teaching Hours:10 |
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Extensions of Some Matrix Inequalities via Matrix Means
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Matrix monotone function, Matrix means, Matrix power mean, Karcher mean | |||||||||||||||||||||||||||||
Text Books And Reference Books:
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Essential Reading / Recommended Reading
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Evaluation Pattern
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MTH451A - NUMERICAL METHODS FOR BOUNDARY VALUE PROBLEM USING PYTHON (2022 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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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. | |||||||||||||||||||||||||
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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MTH451B - NETWORK SCIENCE WITH PYTHON AND NETWORKX (2022 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 NetworkX module of Python to study about the Networks, Affiliation networks and Communities.
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Course Outcome |
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CO1: create diagramatic representations of networks using Python CO2: effectively use the in-built functions of Python on Affiliation networks and measure various parameters of a network CO3: identify the various global properties of networks and communities. |
Unit-1 |
Teaching Hours:15 |
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Generating Networks using Python/NetworkX
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Introduction to Networks - Working with Networks in NetworkX: The 'Graph' class for undirected networks - Adding attributes to nodes and edgss - Adding edge weights - The 'DiGraph' class - The 'MultiGraph' and 'MultiDiGraph' classes - Reading and writing Network files - Creating a network with code. | |||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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NetworkX for Affiliation networks and Centrality
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Nodes and Affiliations - Affiliation networks in NetworkX - Projections on affiliation networks using NetworkX - Centrality, Betweenness centrality, Eigenvector centrality, Closeness centrality and Local clustering using NetworkX. | |||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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NetworkX for Global properties of networks and Communities
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Global properties of whole network - Diameter and shortest paths in a network - Global clustering/transitivity to quantify interconnections of nodes in a network - Measring resilience - Minimum cuts - Connectivity - Centralization and inequality - Networks within Networks (communities) - Community detection and visualizing - Online social networks - Girvan-Newman betweenness-based communities - Cliques, k-Cores. | |||||||||||||||||||||||||
Text Books And Reference Books: 1. E. L. Platt, Network Science with Python and NetworkX Quick Start Guide, Birmingham-Mulbai, Packt Publishing, 2019. | |||||||||||||||||||||||||
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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MTH451C - PROGRAMMING FOR DATA SCIENCE IN R (2022 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 packages in R that are necessary for visualizing, transforming, analyzing, reading, writing, processing the data and hence construct predictive models. Course Objectives: This course will help the learner to COBJ 1: Acquire skills in using R packages/functions in visualizing, transforming and analyzing data. COBJ 2: Apply the packages/functions of R in reading, writing and processing data. COBJ 3: Understand the use of inbuilt functions/packages of R in handling data and build models based on data analysis. |
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Course Outcome |
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CO1: Use R packages in handling data for visualizing, transforming and analizing it. CO2: Effectively use the in-built functions of R in reading, writing and processing data. CO3: Effectively use the in-built functions of R in reading, writing and processing data. |
Unit-1 |
Teaching Hours:12 |
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Data Visualization, Transformation and Analysis
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Data Visualization with ggplot2 - Workflow: Basics - Data Transformation with dplyr - Workflow: Scripts - Exploratory Data Analysis - Workflow: Projects | |||||||||||||||||||||||||
Unit-2 |
Teaching Hours:18 |
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Reading, Writing and Processing Data
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Tibbles with tibble - Data import with readr - Tidy data with tidyr - Relational data with dplyr - Factors with forcats | |||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Vectors, Functions and Models
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Pipes with magrittr - Functions - vectors - Iteration with purr - Model basics with modelr | |||||||||||||||||||||||||
Text Books And Reference Books: H. Wickham and G. Grolemund, R for Data Science - Import, Tidy, Transform, Visualize and Model data, 1st ed., O'Reilly Media Inc., 2016. | |||||||||||||||||||||||||
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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MTH451D - NUMERICAL LINEAR ALGEBRA USING MATLAB (2022 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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This course aims at introducing the functions in MATLAB that are necessary for numerical linear algebra computations.
This course will help the learner to
COBJ1: use MATLAB functions / codes to represent matrices and handle computations of matrices. COBJ2: make fundamental analysis of matrices using MATLAB functions / codes. COBJ3: perform elementary row operations and matrix decompositions using MATLAB functions / code. |
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Course Outcome |
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CO1.: understand various MATLAB functions to handle matrices with numerical and symbolic entries.
CO2.: analyze the matrices involved in numerical linear algebra.
CO3.: compute using fundamental transformation and decomposition of matrices.
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Unit-1 |
Teaching Hours:15 |
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Representation and fundamental computations of matrices
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Input of an ordinary matrix; Input of special matrices: Matrices of zeros, ones, and identity matrices, Matrices with random elements, Hankel matrix, Diagonal matrices, Hilbert matrix and its inverse, Companion matrices, Vandermonde matrices; Symbolic matrix input: Input of special symbolic matrices, Generating arbitrary constant matrices, Arbitrary matrix function input; Input of sparse matrices; Fundamental matrix computations: Handling complex matrices, Transposition and rotation of matrices, Algebraic computation of matrices, Kronecker products and sums; Calculus computations with matrices: - Matrix derivatives, Integrals of matrix functions, Jacobian matrices of vector functions , Hessian matrices. | |||||||||||||||||||||||||
Unit-2 |
Teaching Hours:15 |
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Fundamental analysis of matrices
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Determinants: Determinants of small-scale matrices, MATLAB solution of determinant problems, Determinants of special matrices of any sizes, Cramer’s rule in linear equations, Positive and totally positive matrices; Simple analysis of matrices: The traces, Linear independence and matrix rank, Norms, Vector space; Inverse and generalized inverse matrices: Inverse matrices, Derivatives of inverse matrix, MATLAB based inverse matrix evaluation, Reduced row echelon form; Characteristic polynomials and eigenvalues: Characteristic polynomials, Finding the roots of polynomial equations, Eigenvalues and eigenvectors, Matrix polynomials: Solutions of matrix polynomial problems, Conversions between symbolic and numerical polynomials. | |||||||||||||||||||||||||
Unit-3 |
Teaching Hours:15 |
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Fundamental transformation and decomposition of matrices
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Similarity transform and orthogonal matrices: Similarity transform, Orthogonal matrices and orthonormal bases; Elementary row transforms: Three types of elementary row transforms, Inverse through elementary row transforms, Computing inverses with pivot methods; Triangular decomposition of matrices: Gaussian elimination of linear equations, Triangular factorization and implementation, MATLAB triangular factorization solver, Cholesky factorization: Cholesky factorization of symmetric matrices, Quadratic forms of symmetric matrices, Positive definite and regular matrices, Cholesky factorization of nonpositive definite matrices, Companion and Jordan transforms: Companion form transformation, Matrix diagonalization, Jordan transform; Singular value decomposition: Singular values and condition numbers, Singular value decomposition of rectangular matrices, SVD based simultaneous diagonalization | |||||||||||||||||||||||||
Text Books And Reference Books: D. Xue, Linear Algebra and Matrix Computations with MATLAB, Tsinghua University Press Ltd. And Walter de Gruyter GmbH, Berlin/Boston 2020. | |||||||||||||||||||||||||
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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MTH481 - PROJECT (2022 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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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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CO1: Through this project students will develop analytical and computational skills along with research 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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