Applied Mathematics

Base Knowledge

Elementar concepts of programming and probability theory.

Teaching Methodologies

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

Learning Results

  • 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

Program

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

Curricular Unit Teachers

Internship(s)

NAO

Bibliography

– 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