Applied Mathematics

Elementary concepts of programming and probability theory.

The course content is delivered through guided theoretical exposition, with emphasis on the mathematical formalisation of the fundamental concepts (networks, queueing systems and image processing).

The practical component focuses on problem-solving in class, computer-based simulation, and critical discussion of results, promoting active and collaborative learning.

In-class tasks integrate the critical use of Artificial Intelligence tools, encouraging validation of results, identification of limitations, and reflection on ethical implications in engineering contexts.

The course fosters the development of the mathematical competences proposed by Niss: thinking, reasoning, modelling, problem-solving, and communicating mathematically, bridging theoretical rigour with practical application to real engineering problems.

By the end of this course unit, students should be able to:

Think and reason mathematically about network optimisation problems, analysing different algorithms (shortest/longest path, maximum/minimum capacity, maximum capacity among shortest paths) and their conditions of applicability.

Model mathematically real engineering situations in the context of queueing systems (M/M/1, M/M/S, M/M/1/k, M/M/S/k, finite capacity models, combinations of models), translating stochastic processes into formal structures. 

Represent and manipulate mathematical entities in binary images through mathematical morphology, using symbols and formal operations (dilation, erosion, opening, closing, gradient, thickening/thinning).

Pose and solve mathematical problems applied to concrete engineering contexts, combining formal rigour with the interpretation of results.

Communicate in, with and about mathematics, clearly presenting reasoning, results and implications in technical reports and collaborative settings.

Make use of digital tools and technologies, including optimisation algorithms, simulation software and image processing, critically and responsibly, recognising their limitations and ethical implications.

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

Periodic

Continuous assessment consists of three in-class tasks (with critical use of AI) and two individual written tests (without consultation), as described below:

In-class tasks (6 points in total):

• Task 1 – Network optimisation (2 points, 2h, with AI)
• Task 2 – Queueing systems (2 points, 2h, with AI)
• Task 3 – Binary image operations (2 points, 2h, with AI)

May be carried out individually or in pairs.

Written tests (14 points in total):

• Test 1 – Network optimisation (7 points, 2h, without consultation)
• Test 2 – Queueing systems and image operations (4 + 3 points, 2h, without consultation)

Individual and conducted during the semester.

Passing requirements:

Each written test requires a minimum score of 3.5 points (Test 1 – 3.5 points; Test 2 – 3.5 points, i.e. 2 + 1.5) to pass through continuous assessment.

In-class tasks must include a record of the AI tool used, the prompt, and the consultation date, and will be assessed based on their critical and ethical analysis.

A student who does not pass through continuous assessment, or who does not take part in it, may sit the final examination in the periods defined in the ISEC academic calendar. The exam also consists of two parts, each worth 10 points, with a minimum of 4.75 points in each part. In the second part, the minimum distribution is 2.5 points for Queueing Systems and 1.25 points for Image Operations. This choice excludes the marks obtained in the in-class tasks.

If, in continuous assessment, the student does not achieve the minimum score in one of the parts, they may, if they wish, repeat in the final exam only the missing part, with the same distribution of marks as in this component of assessment in continuous assessment.

Proposed dates for the written tests during the semester:

• 1st Test: 7th teaching week (20.10.2025)
• 2nd Test: last teaching week (15.12.2025)

Both conducted during class hours.

Assessment Rubrics

In-class tasks (2 points each): Criteria (total 2 points):

Technical accuracy/validation (0.8 points): correct calculations, appropriate application of algorithms (e.g. network problems, appropriate queueing system formulas, morphological operations in images).

Critical analysis of AI (0.8 points): identification of strengths and weaknesses in the AI's response, limits of explainability, discussion of errors or simplifications.

Clarity and presentation (0.4 points): organised structure, correct mathematical language, synthesis in 1–2 pages.

Written tests (7 points each): Criteria (total 7 points):

Correct formulation/modelling (2.0 points): clear definition of variables, objective functions, constraints or parameters.

Step-by-step resolution (3.5 points): The appropriate application of methods (e.g., algorithms for network problems, formulas for queueing system models, and matrix operations in image processing).

Interpretation and analysis (1.5 points): interpretation of results in the context of the problem, sensitivity to parameter changes, link to engineering practice.

NAO