LO1. Apply the concepts and fundamental principles of computational mathematics in industrial and business contexts, assimilating how these are applied to solve practical problems and optimize processes.
LO2. Analyse project proposals for master's theses presented by faculty, evaluating the relevance, feasibility, and potential scientific contribution of each proposed project.
LO3. Identify the key stages and essential components of a scientific research project, acquiring a clear understanding of the research process from the formulation of the initial question to the conclusions.
LO4. Apply methodologies for conducting a systematic and critical literature review, learning to synthesize the state of the art in order to adequately underpin a research project.
LO5. Apply the fundamentals in the logical structuring and writing of research papers and master's dissertations.

PC1: The importance of computational mathematics in industrial and business contexts
PC2: Case study of computational mathematics applications
* Real examples of how computational mathematics is used in process optimization.
* Discussion of case studies of computational mathematical application in business/industrial contexts.
PC3: Research poject proposals
* Presentation of dissertation project proposals by faculty members.
* Criteria for evaluating the relevance, feasibility, and scientific contribution of research projects.
PC4: Scientific research methodology
* Identification of the phases of a research project and methodological approaches for defining objectives an tasks.
PC5: Literature review
* Methodologies for an effective and systematic literature review.
* Synthesis of the state of the art and critical analysis of existing literature.
PC6: Seminars and dissertation proposal presentations
* Presentation of research proposals by students.
* Discussion of the proposals.

Assessment throughout the semester:
1. Participation and Discussion in Seminar (20%):
* Assessment throughout the semester of the student's contribution in seminar sessions, including engagement, constructive feedback, and skills in applying theoretical knowledge in discussions, corresponding to LO1 and LO3.
2. Project Work (30%):
* Students will develop a research project idea, where they will apply the methods and principles discussed in seminar sessions. This work will be evaluated for its originality, structure, and indication of the practical application of mathematical concepts to real-world problems, reflecting LO1, LO2, and LO5.
3. Literature Review (30%):
* Critical and systematic review of the literature relevant to their research project, demonstrating the skills acquired in LO4 and the ability to establish a theoretical base for their work, aligned with LO3.
4. Oral Presentation (20%):
* Presentation of the research work proposal and discussion of its structure and literature review, allowing students to demonstrate communication and argumentation skills, aligning with LO2 and LO5. Each assessment component is strategically designed to evaluate different aspects of the course unit's learning objectives and ensure that students not only absorb theoretical knowledge but are also able to apply it in a context of practical research and scientific communication.
There isn't exam assessment.

LO1: Know the fundamental concepts and principles of computational error theory and their importance in the analysis of numerical methods.
LO2: Apply numerical methods for solving nonlinear equations, systems of equations, function approximation, numerical integration and linear algebra problems.
LO3: Identify and choose the most appropriate method based on the characteristics of the problem at hand, understanding its applicability and limitations.
LO4: Apply numerical methods in difference equations, ordinary differential equations and numerical optimization, integrating theoretical knowledge with practical experiences to solve complex and real problems in various areas.
LO5: Implement numerical algorithms using the Python programming language, promoting the practical understanding of the methods studied and the ability to apply effective numerical solutions in various contexts.

PC1: Error Theory
A-stability, zero-stability, consistency and global convergence.
PC2: Numerical Methods for Nonlinear Equations and Systems of Equations
Simple iteration methods and convergence acceleration.
PC3: Function Approximation and Numerical Integration
Hermite interpolation, spline interpolation, complexa and trigonometric interpolation, least squares method and numerical quadrature for function approximation and integration.
PC4: Numerical Methods for Linear Algebra and Numerical Optimization
Numerical methods for solving large-scale systems of linear equations.
Implementation of algorithms for LU, QR, and SVD decomposition.
Application of numerical linear algebra in real-world problems, such as data analysis and machine learning.
PC5: Implementation in Python
Development of efficient algorithms in Python to solve complex numerical problems.
Practical case studies, promoting a deep understanding of theoretical concepts through practical application.

1. Assessment Methods
- Assessment throughout the semester:
* Completion of two projects throughout the semester, one individual and one group project, and taking an in-person exam on the date of the 1st Examination Period.
* Each project will be analyzed according to a structure defined by the professor and will be accompanied by a discussion.
* Each project contributes 30% to the final grade.
* The in-person exam contributes 40% to the final grade.
* Minimum required grade for each project and the in-person exam: 7 points (in 20).
* Necessary average for continuous assessment: equal to or higher than 9.5 points (in 20).
- Examination Assessment (1st, 2nd Exam Periods and Special Examination Period)
* In-person exam that accounts for 100% of the final grade.

Possibility of holding oral discussions.

LO1. To know the importance of differential equations in mathematical modeling, and to know how to model simple systems.
LO2. To develop the capacity of classification of Ordinary Differential Equations (ODEs) and the appropriate methods for their resolution.
LO3. Apply the techniques discussed in solving ODEs and critically analyze the behavior of the solutions obtained.
LO4. Know and classify second-order Partial Differential Equations (PDEs).
LO5. Apply separation of variables and Fourier series to solve second-order PDEs.
LO6. Apply numerical methods for solving ODEs and PDEs in Python.

I - Mathematical Modeling.
1. The use of differential equations in model building.
2. Classic examples of dynamical and equilibrium models.
3. Classification of differential equations.

II - Ordinary Differential Equations (ODEs).
1. Notable examples of ODEs.
2. Existence and uniqueness of solutions (Picard-Lindelöf).
3. First-order ODEs.
4. Qualitative methods for autonomous ODEs.
5. Equilibrium and stability.
6. Systems of linear first-order ODEs.
7. Linear ODEs of higher order.
8. Homogeneous equation, characteristic equation, and the method of variation of parameters.
9. Numerical methods for solving ODEs (Euler and Runge-Kutta).

III - Partial Differential Equations (PDEs).
1. Notable examples of PDEs.
2. Classification of PDEs (heat equation, wave equation, Laplace equation).
3. Separation of variables.
4. Fourier series and convergence.
5. The Laplace transform.
6. Numerically solving the heat equation using finite differences.

Assessment throughout the semester
- 3 sets of exercises carried out throughout the semester, with a weight of 60% in the final grade (20% each set), done individually.
- 2 individual research projects carried out throughout the semester (with delivery of a written report and an implementation in Python), with a weight of 40% (20% each) in the final grade. The presented projects are subject to discussion.

Assessment by exam (1st Exam Period in case of student's choice,2nd Exam Period and Special Season Period):
Exam (100% of the final grade).

LO1: Identify the main classes of mathematical optimization problems and their properties
LO2: Identify real optimization problems and formalise them as mathematical models
LO3: Integrate computational tools (Python code) with optimization theory in practical problems
LO4: Explain the ideas behind optimization algorithms and the techniques that guarantee their convergence
LO5: Understand the applicability, advantages and limitations of optimization methods and techniques
LO6: Indentify multi-objective problems based on linear programming
LO7: Decide about the suitability of different optimization algorithms for a given problem and apply them
LO8: Apply heuristics and metaheuristics to complex optimization problems
LO9: Identify the most suitable computational tools (Python code) for solving a problem and know how to implement them
LO10: Criticize practical results and compare them with theoretical expectations

PC1. Fundamentals of deterministic optimization
Concepts and optimality conditions. Modelling techniques. Classification of problems
PC2. Convex problems
Convexity. Particular case of linear programming. Duality theory. Post-optimal sensitivity analysis. Goal-orientated programming
PC3. Unrestricted non-linear optimization
Linear search and confidence region statistics. Gradient and 2nd order numerical methods in Python code. Wolfe conditions. Convergence and rate
PC4. Constrained non-linear optimization
Karush-Kuhn-Tucker optimality conditions. Quadratic programming. Penalty and barrier methods. Algorithms based on decomposition
PC5. Discrete modelling
Combinatorial and integer optimization. Branch-and-bound and branch-and-cut techniques. Hamilton and Euler cycles. Network flows. Large-scale or NP-hard problems. Heuristics and meta-heuristics. Iterated local search. Genetic algorithms and evolutionary strategies. Classical examples

Assessment throughout the semester:
Analysis of 2 case studies throughout the semester and a test on the date of the 1st Exam Period. One of the case studies is individual and the other is carried out in a group. Each case study (which will be analysed using a structure pre-defined by the professor) and the respective discussion will have a weight of 25% in the final grade, with a minimum grade of 75,. The average obtained in the case studies must be equal to or greater than 9.5 points. The test has a weight of 50% in the final grade and a minimum grade of 7,5. Possibility of oral proof if appropriate.

Exam assessment (1st Exam Period if chosen by the student, 2nd Exam Period, and Special Exam Period):
Individual face-to-face exam (100% of the final grade).

LO1: To apply the basic concepts of measure theory.
LO2: To use the concepts and results of probability theory.
LO3: To know Brownian motion and stochastic calculus.
LO4: Applying stochastic differential equations to modeling problems of dynamic phenomena.
LO5: To analyze and discuss possible research areas to be developed in a project.
LO6: To acquire autonomy and critical thinking in the use of these and other concepts, particularly in a classroom context.

PC1. Measure theory and probability
1.1 Introduction to measure theory
1.2 Probabilistic Concepts
PC2. Brownian Motion and ""White Noise""
2.1 Motivation and Definitions
2.2 Construction of Brownian Motion
PC3. Stochastic Integrals, Itô's Formula
3.1 Motivation
3.2 Definition and Properties of the Itô Integral
3.3 Itô's Formula
PC4. Stochastic Differential Equations
4.1 Definitions and Examples
4.2 Existence and Uniqueness of Solutions
4.3 Linear Stochastic Differential Equations
PC5. Applications
5.1. Application of stochastic differential equations to dynamic phenomena.

Assessment throughout the semester:
- Four online quizzes completed individually throughout the semester, accounting for 20% of the final grade.
- Individual research project (with the submission of a written report and an implementation component in Python), weighted at 30% of the final grade. The research project should apply the knowledge of stochastic differential equations covered in the course and other relevant tools for modeling specific dynamic phenomena.
- Final in-person test accounting for 50% of the final grade, with a minimum grade of 7.5.
- The evaluation elements are subject to discussion.

Exam assessment (1st Exam Period if chosen by the student, 2nd Exam Period, and Special Period): In-person exam (100% of the final grade).

LO1. Know the main unsupervised learning methods
LO2: Evaluate, validate and interpret the results of unsupervised models
LO3: Develop projects from data using unsupervised learning models
LO4. Apply unsupervised algorithms in practical case studies
LO5: Use software (R or Python) in the context of unsupervised methods

S1: Introduction to unsupervised learning methods: fundamental concepts, types of algorithms and practical applications;
S2: Principal component analysis (PCA): fundamental concepts, steps and practical applications;
S3: Clustering techniques: hierarchical clustering and probabilistic clustering, exploration of algorithms and practical applications;
S4: Association rules: frequency of items and association rules, Apriori algorithm and practical applications.

Passing the course requires a mark of 10 (scale 0-20) in any of the assessment methods.
Assessment throughout the semester:
- group work with a minimum mark of 8 (30%)
- individual assignment(s) with a minimum mark of 8 (30%)
- individual face-to-face test with a minimum mark of 8 (30%)
- autonomous work throughout the semester (10%)
All components (assignments and tests) are compulsory. Passing requires a minimum mark of 10 (scale 0-20).
EXAMS (1st Exam Period, if the student chooses, 2nd Exam Period and Special Period):
The Final Exam corresponds to a face-to-face written exam (100% of the final grade). Students must obtain a minimum mark of 10 to pass.

LO1. Understand the fundamental principles of complex systems, including emergence, self-organisation, and non-linear dynamics.
LO2. Model and analyse continuous and discrete dynamical systems, studying stability, bifurcations, and chaos using mathematical and computational methods.
LO3. Structure and interpret complex networks by applying metrics and models for structural and functional analysis, including neural networks and Graph Neural Networks (GNNs).
LO4. Develop and implement agent-based models, exploring applications in social, economic, and biological systems.
LO5. Utilise computational tools such as Python and NetLogo for the simulation and analysis of complex systems.

PC1. Introduction to Complex Systems
Definition and characteristics
Concepts of emergence and self-organisation
Examples from natural and artificial systems
PC2. Modelling of Dynamical Systems
Applied differential equations
Stability analysis and bifurcations
Introduction to chaos and non-linear systems
PC3. Graphs and Complex Networks
Fundamentals and structural and functional metrics
Network models: random, small-world, and scale-free networks
Introduction to neural networks and GNNs
PC4. Agent-Based Modelling
Concepts and applications of agent-based models
Implementation of models using platforms such as NetLogo
Case studies in social and biological systems
PC5. Computational Tools
Programming in Python for modelling and simulation
Use of specialised libraries for analysis and prediction
Data analysis and visualisation
PC6. Interdisciplinary Applications
Ecological, economic, and financial systems
Population dynamics and epidemic modelling

Approval in the course results from obtaining 10 values (0-20 scale) in one of the assessment modalities.

Assessment throughout the semester:
Resolution of four tasks throughout the semester: two individual and two group tasks. Each task (whose analysis will be developed with a pre-defined structure by the professor) and their respective results and discussion carry a weight of 20% in the final grade. Additionally, there is a written in-person exam with a weight of 20%. All components (tasks and exam) require mandatory completion, and the average grade must be above 9.5 values (0-20 scale) for approval.

Exam evaluation (1st Exam Period, if chosen by the student, 2nd Exam Period, and Special Period):
In-person exam (100% of the final grade)

LO1: Identify the various components of a deep learning model with neural networks.
LO2: Implement stand-alone programs, using python, of simplified versions of the components implicit in LG1.
LO3. Relate different architectures to solve different problems.
LO4. Apply regularization techniques to improve the performance of deep learning models.
LO5. Identify in detail and apply the DQN algorithm in the context of reinforcement learning.
LO6. Use the Keras library to implement deep learning models to solve problems in image recognition, natural language processing and reinforcement learning.
LO7. Identify fundamental theorems about asymptotic neural networks and apply these results in the critical analysis of DL models.
LO8. Evaluate and discuss possible research areas to be developed in the project.

1. Mathematical models of neurons and activation functions.
2. Hardware: AI accelerators for machine learning.
3. Efficient algorithms for matrix manipulation and tensor algebra.
4. Backpropagation and automatic differentiation.
5. Classification and regression problems with feedforward neural nets.
6. Universal aproximation theorems and asymptotic analysis.
7. Convolution networks and image recognition.
8. Regularization techniques.
9. Deep reinforcement learning.
10. Models for sustainable AI.

Assessment throughout the semester:
- Four online quizzes completed individually throughout the semester, accounting for 20% of the final grade.
- Individual research project conducted throughout the semester (including submission of a written report and a Python implementation component), accounting for 30% of the final grade. The research project should correspond to the development of a free extension of the course program, with a mandatory implementation component.
- Final in-person test accounting for 50% of the final grade, with a minimum grade of 7.5.

Exam assessment (1st Exam Period if chosen by the student, 2nd Exam Period, and Special Period): In-person exam (100% of the final grade).

LO1. Understand the fundamentals of Measure and Lebesgue Integration Theory, including the concepts of measurability, measure, and integration.
LO2. Apply the fundamental concepts of Functional Analysis in Banach and Hilbert spaces.
LO3. Study Fourier series and transforms, analyzing their algebraic, analytical and computational properties and applications in different contexts.
LO4. Develop mathematical reasoning and abstraction skills.
LO5. Apply tools from Functional Analysis and Fourier Transforms to model and solve problems in areas such as signal processing and differential equations.
LO6. Implement computational algorithms using the Python programming language.

CP1. Teoria da medida e integração
- Conjuntos mensuráveis e medida de Lebesgue
- Funções mensuráveis
- Integral de Lebesgue
- Teoremas de convergência
- Espaços Lp e desigualdades fundamentais
- Teorema de Radon- Nikodym
CP2. Espaços de Banach e Espaços de Hilbert
- Espaços normados
- Completude e espaços de Banach
- Dualidade
- Teorema de Hahn-Banach
- Espaços de Hilbert
- Teorema da representação de Riesz
- Operadores lineares limitados
- Teorema espectral de operadores compactos e auto-adjuntos
CP3. Séries e Transformadas de Fourier
- Séries de Fourier
- Transformada de Fourier
- Aplicações: processamento de sinal e equações diferenciais

Assessment Throughout the Semester:
- Six individual exercise sheets distributed over the semester, accounting for 20% of the final grade.
- Individual project developed throughout the semester, including a written report, an implementation component in Python, and a final presentation, contributing 30% to the final grade.
- Final in-person test, weighted 50% of the final grade, with a minimum passing score of 8.5 out of 20.
- All assessment components are subject to discussion.

Assessment by Examination:
(1st Examination Period if chosen by the student, 2nd Examination Period, and Special Examination Period)
- In-person exam accounting for 100% of the final grade.

LO1. To know different working methods with a view to obtaining original contributions and recognize their importance in the context of applying mathematics to solving real problems in different areas.
LO2. To know some bibliographic research sources and how to consult them.
LO3. To analyze the international peer review system and the corresponding publication of original contributions.
LO4. To apply the contents of this UC in the selection of a research problem and in the review of the literature.
LO5. To acquire autonomy and critical thinking in the use of these and other concepts, particularly when writing a research proposal.

PC1. The stages of the research process
I. Types of thesis
II. Identification of research problems
III. Planning of work phases
PC2. Preparation of literature review and bibliography sources
PC3. Identify and use computational mathematics tools appropriate to the research problem
PC4. Research topics in applied computational mathematics
PC5. Preparation and presentation of the research project

Assessment throughout the semester:
- analysis of four case studies throughout the semester: two individual and two group case studies. Each case study (whose analysis will be developed with a predefined structure by the professor) and its discussion carry a weight of 25% in the final grade, with a minimum grade of 7.5 values (0-20 scale). The average grade of the case study analysis must be equal to or higher than 9.5 values (0-20 scale).

There isn't exam assessment.

LO1. To know controllability results for linear and non-linear systems and their importance in the context of optimal control theory.
LO2. To apply fundamental concepts of optimal control theory and the influence of existence theorems and Pontryagin's Maximum Principle on problem solving in real contexts.
LO3. To know the practical applications of dynamic programming in the problems resolution.
LO4. To apply suitable numerical methods for different types of optimal control problems.
LO5. To develop critical thinking in the use of the theoretical and numerical results in the resolution of optimal control problems applied to different areas.

PC1.Controllability
I. Introduction and framework
Classic examples
II. Controllability
Controllability of linear systems
Controllability of autonomous linear systems: case without control restrictions - Kalman condition; case with restrictions on control
Controllability of non-autonomous linear systems
Controllability of nonlinear systems
PC2.Optimal control
I. Existence
Existence theorems
II. Necessary conditions for optimality
Pontryagin maximum principle
Particular cases and examples: minimum time problem; linear quadratic problem; examples of linear optimal control (Zermelo problem, the Brachistochrone problem).
PC3.Dynamic programming
General concepts in discrete and continuous time contexts
Dynamic Programming Algorithms
Examples
PC4. Numerical methods for solving optimal control problems
I. Direct methods
Algorithms
Examples
II. Indirect methods
Algorithms
Examples

Assessment throughout the semester:
- four online quizzes carried out throughout the semester with a weight of 20% in the final grade;
- individual research project (with delivery of a written report and an implementation component in Matlab/Octave or Python), with a weight of 30% in the final grade. The research project should apply the knowledge of optimal control theory covered in the course and other relevant tools for solving concrete optimal control problems.
- final test with a weight of 50% in the final grade, with a minimum score of 7.5;
- the evaluation elements are subject to discussion.

Assessment by exam (1st Exam Period in case of student's choice,2nd Exam Period and Special Season Period): Exam (100% of the final grade).

LO1. To be able to understand and apply good bibliographical research and state-of-the-art review practices on a subject;
LO2. Have the ability to formulate research questions and design the appropriate methodology for constructing the answers;
LO3: Define, plan, communicate and carry out original work of appropriate complexity and size;
LO4. Have the capacity for critical reasoning and logical-scientific argumentation on complex issues and to apply knowledge in new situations;
LO5. Have the ability to communicate in writing and orally, to convey their knowledge and present unequivocal conclusions, with conceptual rigor and respecting the requirements of academic writing.

S1: Introduction to research
S2: Formulating the problem and research objectives and planning
S3: Good practices for developing the state of the art
S4: Good practice in scientific writing and presentation of research work
S5: Writing the dissertation

The dissertation is presented in accordance with the rules and within the deadlines established by ISCTE-IUL. The dissertation will be assessed by a jury in public examinations, after the supervisor has confirmed that it has been completed and is in a position to be presented in public examinations. The assessment will be based on the scientific merit of the study, its theoretical and methodological adequacy and its discussion.
- Mid-term assessment includes: project proposal, introductory chapter, literature review, planning of the next stages of the work and a presentation on the work in progress.
- The final assessment will take into account the technical/scientific quality of the work, based on the dissertation document (A), the quality of the presentation and public discussion (B), weighted as follows: Final grade = 0.6*A + 0.4*B.