| CODE | SCI3101 | ||||||||
| TITLE | Linear Programming and Fourier Transforms | ||||||||
| UM LEVEL | 03 - Years 2, 3, 4 in Modular Undergraduate Course | ||||||||
| MQF LEVEL | 6 | ||||||||
| ECTS CREDITS | 2 | ||||||||
| DEPARTMENT | Faculty of Science | ||||||||
| DESCRIPTION | Linear Programming: - An Introduction to Mathematical Optimization - Formulation of Linear Programming Problems - Basic Theory and the Fundamental Theorem of Linear Programming - The Graphical Method - The Simplex Method - Interpretation of the Simplex Table - Sensitivity Analysis - Network Problems (Minimum Cost Network Flow, Transportation, Shortest Path). Fourier Transforms: - Fourier's identity - Transform pairs, duality and symmetry - Fourier Sine Transform and Fourier Cosine Transform - The Delta function and the Unit Step function - Properties including linearity, scaling, shift, modulation, and convolution - Transforms of derivatives and integrals - Solution of boundary value problems - Parseval's Theorem. Study-unit Aims: In the first part of this study-unit, the main aim is that students are able to express various (including engineering) optimization problems in the form of Linear Programming (LP) problems. In addition, the students will learn what mathematical modeling is about and its role in taking optimal decisions. The most renowned method to solve LP problems, that is the Simplex Method, will be presented and explained in detail, while the students will be able to appreciate the steps involved in this method since an introduction to the basic theory and concepts of LP will be provided as well. Instructions and appropriate explanations on how to use a number of available software packages for solving LP problems will be given as well. In the second part of this study-unit, the focus is shifted to functions defined on the real domain whose Fourier transform can be shown to exist. The main aim is to equip the students with a powerful tool, namely the Fourier transform, that can be used to handle such functions when addressing engineering problems. Learning Outcomes: 1. Knowledge & Understanding By the end of the study-unit the student will be able to: - Convert certain real-life problems into a mathematical model; - Solve manually LP problems using the Simplex method; - Explain how the Simplex method works based on the fundamental LP theory; - Perform sensitivity analysis on certain parameters involved in an LP problem; - Define and derive the Fourier transform, the Fourier Sine transform, the Fourier Cosine transform and the associated inverse transforms; - Distinguish between the different properties of Fourier transforms and apply them in the appropriate contexts; - Apply Fourier transforms to solve boundary value problems. 2. Skills By the end of the study-unit the student will be able to: - Solve LP problems graphically; - Apply the Simplex Algorithm; - Use various software packages to solve LP problems; - Interpret the results provided by solvers; - Assess importance of LP model parameters by performing sensitivity analysis; - Formulate solutions to problems by using appropriate mathematical techniques; - Address engineering problems by applying appropriate mathematical tools. Main Text/s and any supplementary readings: Main Texts: - Bazaraa, M.S., Jarvis, J.J. and Sherali, H.D. (2010) Linear Programming and Network Flows, Wiley, 4th ed. - Peter V. O’Neil, Advanced Engineering Mathematics, Cengage, 7th ed. . Supplementary Readings: - Luenberger, David G, and Ye, Yinyu. Linear and Nonlinear Programming. Vol. 228. Cham: Springer International AG, 2015. International Ser. in Operations Research & Management Science. Web. - Bracewell R.N., The Fourier Transform and its Applications, McGraw-Hill, New York, 3rd edition, 2000. |
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| RULES/CONDITIONS | In TAKING THIS UNIT YOU CANNOT TAKE MAT2803 | ||||||||
| ADDITIONAL NOTES | Pre-requisite Study-units: MAT1801, MAT1802 | ||||||||
| STUDY-UNIT TYPE | Lecture, Independent Study & Tutorial | ||||||||
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| LECTURER/S | Onur Baysal (Co-ord.) Maria Kontorinaki |
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The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
The availability of optional units may be subject to timetabling constraints. Units not attracting a sufficient number of registrations may be withdrawn without notice. It should be noted that all the information in the description above applies to study-units available during the academic year 2023/4. It may be subject to change in subsequent years. |
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