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Annexure ‘AAB-CD-01’
Course Title: Advance Real Analysis
|L |T |P/S |SW/FW |TOTAL CREDIT UNITS |
|3 |1 | |0 |4 |
Course Code: to be decided later
Credit Units: 4
Level: PG
|# |Course Title |Weightage (%) |
|1 |Course Objectives: | |
| |to understand the basic concepts of real analysis and its physical properties | |
| |to develop fundamental knowledge and understanding of the many techniques in Real variable . | |
| |to make the students aware of General theory of differentiation and integration under the sign | |
| |of integration, question of convergence of series, Dirichlet’s integral, Laplace and Laplace | |
| |Steiltjes transform are employed in the theory of probability distributions. Similarly, | |
| |Bolzano-Weirstrass,Heine Borel theorems etc. are very much useful in Statistical Inference . | |
| |to apply statistical concepts to various fields of statistics to analyze and interpret data. | |
|2 |Prerequisites: | |
| | NIL | |
|3 |Student Learning Outcomes: | |
| |The students will be able to learn various continuity of functions. | |
| |The students will able to acquire knowledge on convergences. | |
| |The students will able to apply the properties of mgf and cf for distributions. | |
| |The students will able to define sequences of the functions. | |
| |The course enables the students to develop the skill set to solve the problems based on real | |
| |life situation. | |
|Course Contents / Syllabus: |
|4 |Module I: |20% Weightage |
| |Monotone functions and functions of bounded variation. Real valued functions, continuous | |
| |functions, Absolute continuity of functions, standard properties, uniform continuity, sequence | |
| |of functions, uniform convergence, power series and radius of convergence. | |
|5 |Module II: |20% Weightage |
| |Riemann-Stieltjes integration, standard properties, multiple integrals and their evaluation by | |
| |repeated integration, change of variable in multiple integration. Uniform convergence in | |
| |improper integrals, differentiation under the sign of integral - Leibnitz rule, Integration | |
| |under the sign of differentiation. Dirichlet integral. | |
|6 |Module III: |30% Weightage |
| |Introduction to n-dimensional Euclidean space, open and closed intervals (rectangles), compact | |
| |sets, Bolzano-Weierstrass theorem, Heine-Borel theorem. Maxima-minima of functions of several | |
| |variables, constrained maxima-minima of functions. | |
|7 |Module IV: Applications of mgf and cf for continuous distributions |30% Weightage |
| |Laplace and Laplace-Steiltjes transforms. Solutitions of linear differential.Properties of | |
| |Laplace transforms, Transforms of derivatives, Transforms of integrals, Evalualtion of integrals| |
| |using Laplace transform, convolution theorem, Applications to differential equations, | |
| |simultaneous linear equations with constant coefficient, unit step functions and Periodic | |
| |functions. | |
|8 |Pedagogy for Course Delivery: | |
| | | |
| |The class will be taught using theory and practical methods using software in a separate Lab | |
| |sessions. In addition to numerical applications, the real life problems and situations will be | |
| |assigned to the students and they are encouraged to get a feasible solution that could deliver | |
| |meaningful and acceptable solutions by the end users. The focus will be given to incorporate | |
| |probability and related measures to develop a risk model for various applications. | |
|9 | | |
| |Assessment/ Examination Scheme: | |
| | | |
| |Theory L/T (%) | |
| |Lab/Practical/Studio (%) | |
| |End Term Examination | |
| | | |
| | | |
| |30% | |
| | | |
| |NA | |
| | | |
| |70% | |
| | | |
| |Theory Assessment (L&T): | |
| |Continuous Assessment/Internal Assessment | |
| |End Term Examination | |
| | | |
| | | |
| | | |
| |Components (Drop down) | |
| | | |
| |Mid-Term Exam | |
| | | |
| |Project | |
| | | |
| |Viva | |
| | | |
| |Attendance | |
| | | |
| | | |
| |Weightage (%) | |
| | | |
| |10% | |
| | | |
| |10% | |
| | | |
| |5% | |
| | | |
| |5% | |
| | | |
| |70% | |
| | | |
| | | |
Text & References:
• Rudin, Walter (1976). Principles of Mathematical Analysis, McGraw Hill.
• Apostol, T. M. (1985). Mathematical Analysis, Narosa, Indian Ed.
• Narayan, S., (2010). Elements of Real Analysis, S. Chand and Sons.
• Miller, K. S. (1957). Advanced Real Calculus, Harper, New York
• Courant, R. and John, F. (1965). Introduction to Calculus and Analysis, Wiley
• Bartle, R.G. (1976): Elements of Real Analysis, John Wiley & Sons.
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