Applied Mathematics

Elementar concepts of programming and probability theory.

Teaching method is essencialy expositive on the theoretical parts and collaborative on the resolution of exercises and application problems.

• Learn essential and basic concepts about Network Optimization, Queues and Image processing (black and white)
• Ability to use mathematical techniques
• Develop the ability to perceive concepts, abstract reasoning, interpret results and solve problems
• Understanding the specificities of the concepts studied for problem solving
• Think mathematically
• To reason mathematically
• Identify and solve problems
• Model mathematically
• Represent mathematical entities
• Manipulate mathematical symbols and use formal language
• Communicate mathematically
• Use mathematical tools and resources

I– Network Optimization: Representation of a network in the form of emerging and immersive arcs; Algorithms for the shortest path, longest path, maximum capacity, minimum capacity, shortest maximum capacity, maximum capacity in the set of shortest routes. Application to concrete problems.

II- Queueing theory: Structure and elementary concepts; Queues modeling; Life and death processes; Fundamental relations of waiting lines; Classification of waiting lines; Models with one or more servers of unlimited and limited lengths; Combination of models; Application to concrete problems.

III– Image Processing – Mathematical Morphology: Basic operations on binary images: dilation and erosion; Properties; Conditional Dilatation; Morphological Gradient; Opening and closing an image: Properties; Hit and miss; Fattening and Thinning (thickening/thinning).

IV – Neural Networks: basic concepts

NAO

– Martins, E.Q.V., Pascoal, M.M.B., Rasteiro, D.M.L.D., Santos, J.L.E.. (Junho 1999). The Optimal Path Problem, Investigação Operacional, Vol 19, no 1, pp. 43-60. (available at curricular unit Moodle webpage)

– Dias Rasteiro, D.M.L. , 9 – Shortest path problem and computer algorithms. (2020). Editor(s): Jesús Martín-Vaquero, Michael Carr, Araceli Queiruga-Dios, Daniela Richtáriková, In Mathematics in Science and Engineering, Calculus for Engineering Students, Academic Press. Pages 179-195, ISSN 00765392, ISBN 9780128172100, https://doi.org/10.1016/B978-0-12-817210-0.00016-3. (available at curricular unit Moodle webpage)

– Syllabus elaborated by the lecturer. (available at curricular unit Moodle webpage)

– Hillier, F. , Lieberman, G. . (2004). “Introduction to Operations Research”, McGraw Hill. Location at ISEC’s Library: 3-9-56 (ISEC) – 09160

Additional bibliography:

– Romão, M. C. , Pinto, L. S. , Simões, O. , Valente, J. e Vaz Pato, M. . (2011). Investigação Operacional – Exercícios e Aplicações, Verlag Dashofer.

– Nielsen, M. A. (2018). Neural Networks and Deep Learning [misc]. Determination Press