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Double Degree in Mathematical Engineering + Physics

The advance of science, technology and innovation requires profiles capable of understanding and modelling the most complex phenomena of reality. The combination of Physics and Mathematics is key in sectors such as energy, engineering, research, artificial intelligence and the aerospace industry. Prepare yourself at UAX to become a highly qualified professional and contribute knowledge and value from day one.

Call for applications for 2026 now open

Start September

Length Credits 5 years / 384 ECTS

Modality On-campus

Campus Madrid Chamberí

Faculty Business & Tech

Programme information

Why UAX?
Programme
Facilities
Career opportunities
Senate
Scholarships

Why study Mathematical Engineering and Physics at UAX?

Because it combines two disciplines that are fundamental to understanding and modelling reality, preparing you to tackle complex scientific and technological challenges in key sectors.

What will you learn on the Mathematical Engineering + Physics course?

The convergence of science, data and technology is driving high demand for STEM professionals with strong analytical skills and practical insight. The Double Degree in Engineering Physics and Mathematics provides you with a solid grounding in understanding, modelling and solving complex problems, combining scientific rigour with practical application in fields such as artificial intelligence, data science, quantum computing, energy and the technology sector.

Furthermore, you will develop strategic skills in using data for decision-making and a vision focused on innovation and the business environment.

During the dual degree programme:

You will take part in real-world innovation and research projects with companies, such as the development of a digital twin in collaboration with Avanade by Microsoft.
You will gain certification in key technology and business skills, such as programming, big data and digital business management.
You will build your own portfolio of real-world projects, working with Agile methodologies such as Scrum, Lean or XP.
You will study within a progressive bilingual programme, with the opportunity to undertake work placements and study abroad in Europe, the US or Asia.
You will train in an environment connected to leading companies, in the heart of Madrid, collaborating with firms such as IBM, CaixaBank, Oracle and AWS.

The degree includes the Digital Business Certificate

All fields are required

Predictive models

Use of artificial intelligence techniques to predict working hours in large international engineering projects.

Mercy ships and Phoenix Woman

Collaboration in solidarity projects in the field of diversity, inclusion, gender equality with organisations with wide recognition.

Industry 5.0

Use of Industry 5.0 techniques to build an analytical environment for real-time image processing and offer a unique user experience in the sector.

Cybersecurity

Collaboration in the design and development of an analytical architecture to derive patterns in global cybersecurity-related data.

Automation

Application of mathematical models and data analysis in the design of a health school for patients and families, improving management and communication in the health sector.

Information on the Double Degree in Mathematical Engineering and Physics

Curriculum

Build a solid foundation in Mathematical Engineering and Physics, enabling you to transition into a career where you can make a real difference.

Bachelor’s Degree in Mathematical Engineering + Bachelor’s Degree in Physics

First Year

FIRST FOUR-MONTH PERIOD

Code

Subjects

Character*

ECTS

C0142300

Algebra I

Código: C0142300 Imprimir

Course 1: First-term module. Foundation course. 6 credits.

Profesores

José Antonio Prieto Persiguero - Coordinador

Ver biodata

Objectives

This module, together with Algebra II, forms the subject of Algebra. This module, which forms part of the degree programme’s Basic Training module, aims not only to ensure that students are familiar with the main fundamental theorems of linear algebra, but also to enable them to understand matrix calculus from a conceptual perspective and to apply it to solving problems typical of mathematical engineering; it is, therefore, the foundation for other subjects and modules within the degree programme, such as Numerical Calculus, Operational Research, Stochastic Calculus and Artificial Intelligence.

Prerequisites

No prerequisites have been set.

Learning Outcomes

Not applicable.

Learning outcomes

RK4 – To be familiar with rigorous proofs of some classical theorems in different areas of mathematics. TYPE: Knowledge or content.

RS1 – To use mathematical language to formulate propositions in various fields and construct proofs for the problems posed. TYPE: Skills or competences.

Description of the content

1. SYSTEMS OF LINEAR EQUATIONS.

1.1 Systems of linear equations. Types. Gauss-Jordan method. Discussion of solutions.

1.2 Matrices. Classification. Elementary transformations, Hermitian normal form and rank. Operations: addition, scalar multiplication and matrix multiplication; trace; transposition; inversion. Properties. Regular matrices. Equivalence.

1.3 Matrix notation for systems of linear equations. Rouché–Frobenius theorem.

1.4 Determinants. Properties. The determinant-rank-inversion relationship. Determinants and systems of linear equations: Cramer’s rule.

2. VECTOR SPACES.

2.1 Vector spaces. Further properties of addition and scalar multiplication.

2.2 Linear dependence and independence. Properties. Systems of generators. Bases. Dimension. Coordinates of a vector in a given basis. Change of coordinates.

2.3 Vector subspaces. Vector subspaces of interest: intersection, linear envelope, row and column spaces of a matrix, solutions to a homogeneous system of linear equations. Equations and dimension of a vector subspace. Sum of vector subspaces. Dimension formula. Direct sum. Quotient vector space.

3. CLASSIFICATION OF ENDOMORPHISMS.

3.1 Linear mappings. Properties. Types. Kernel and image.

3.2 Matrix associated with a linear map. Relationship with the kernel and image; the dimension formula. Changes of basis.

3.3 Operations on linear mappings. Properties.

3.4 Linear forms and dual space. Dual basis. Nullator of a vector subspace. Transposed linear map.

4. DIAGONALISATION OF ENDOMORPHISMS.

4.1 Eigenvectors and eigenvalues of an endomorphism. Characteristic and minimal polynomials. Algebraic and geometric multiplicities.

4.2 Diagonalizable matrices. Eigen subspaces. Diagonal form and basis of eigenvectors. Symmetric matrices.

4.3 Non-diagonalisable matrices. Generalised eigenspaces. Maximum subspaces. Canonical form and Jordan basis.

4.4 Complex eigenvalues and eigenvectors. Real canonical form and Jordan basis.

Learning activities

AP1. Lectures.

AP2. Interactive classes.

AP3.1 Project work and problem-solving.

AP4. Independent study, case studies, problem-solving and literature review.

AP5. Tutoring.

AP6. Knowledge assessments.

Assessment system and criteria

Without prejudice to any other requirements that may be specified in the relevant course syllabus, as a general rule, failure to attend more than 70 per cent of the course’s teaching activities—which require the student’s physical or virtual presence—will result in the loss of the right to continuous assessment during the standard examination period. In this case, the examination to be held during the official period set by the University shall be the sole assessment criterion, with the weighting specified in the course syllabus.

----

The assessment process will consist of evaluating the extent to which the student has achieved the learning outcomes associated with the subject.

ASSESSMENT SYSTEMS

The assessment systems for this module are:

 AS1 Practical activities.

 AS1 Final knowledge assessments.

 AS3 Portfolio.

These systems contribute to a greater or lesser extent to the assessment of the learning outcomes assigned to this subject.

ASSESSMENT CRITERIA

The assessment systems described above are set out in the following assessment criteria:

- There are two official examination sessions: the ordinary and the supplementary.

+++REGULAR EXAMINATION PERIOD+++

The final mark for this sitting is the weighted average of a set of assessment tasks detailed below:

-- a case study (SE1), accounting for 30 per cent of the final mark for the ordinary assessment period, to be carried out during the teaching term in small groups (designated by the course coordinator) and which will require both the submission of exercises whilst the case study is being carried out and the submission of a report and its public defence at the end of the teaching term.

-- a non-exemption exam (SE3), to be taken individually during the teaching term, which will account for 20 per cent of the final mark for the standard assessment period.

-- a final examination (SE2) to be taken individually during the ordinary examination period in January (for further information, please consult the virtual campus), which assesses the entirety of the course content and will account for 50 per cent of the final mark for the standard examination session.

***** The weighted average of all these assessment components will be calculated solely and exclusively if the mark obtained in the final exam is 4.0 out of 10 or higher.

Furthermore, only the examinations will be subject to re-marking.

***** A student will have passed the module in the ordinary examination period if, and only if, they obtain a final mark (weighted average of all assessment tests) of 5.0 out of 10 or higher.

Otherwise, the student may pass the module in the supplementary examination session.

********** If a student loses their right to continuous assessment, they must achieve a mark of 10.0 out of 10.0 in the ordinary examination session in order to pass the module.

+++SITTING OUT OF TERM+++

If a student fails to pass the module in the ordinary examination session, they may do so in the supplementary examination session.

This consists of a single examination which will take place during the supplementary examination period, June–July (for further information, please consult the virtual campus), and which assesses the entirety of the course content.

*** The module is considered passed in the supplementary sitting if the final mark is 5.0 or above.

GRADES

Article 5 of Royal Decree 1125/2003 of 5 September establishes the grading system applicable to modules within degree programmes falling within the scope of the European Higher Education Area. This system is as follows:

The award of the corresponding credits is conditional upon passing the associated examinations or assessment tests.

The level of learning achieved by students shall be expressed as numerical marks on a scale of 0 to 10, to one decimal place, to which the corresponding qualitative mark may be added:

- 0–4.9: Fail (SS).

- 5.0–6.9: Pass (AP).

- 7.0–8.9: Good (NT).

- 9.0–10: Distinction (SB).

The distinction ‘Honours’ shall be awarded to students who have obtained a mark of 9.0 or higher. The number of students awarded this distinction may not exceed five per cent of those enrolled on the course in the relevant academic year, unless the number of enrolled students is fewer than 20, in which case only one ‘Honours’ distinction may be awarded.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	SESSION	Introduction to the course.
2	SESSION	Systems of linear equations. Types. Gauss–Jordan elimination.
3	ASSIGNMENT	Discussion of solutions.
4	ASSIGNMENT	Exercises (systems of linear equations).
5	SESSION	Matrices. Classification. Elementary transformations, Hermitian normal form and rank.
6	SESSION	Operations: addition, scalar multiplication and multiplication. Properties.
7	ASSIGNMENT	Operations: trace; transposition; inversion. Properties.
8	ASSIGNMENT	Exercises (elementary transformations and operations with matrices).
9	SESSION	Regular matrices. Equivalence. Determinants. Properties. The determinant-rank-inverse relationship.
10	SESSION	Matrix notation for systems of linear equations. Rouché–Frobenius theorem.
11	ASSIGNMENT	Determinants and systems of linear equations: Cramer’s rule.
12	ASSIGNMENT	Exercises (determinants/Rouché–Frobenius theorem).
13	SESSION	Vector spaces.
14	SESSION	Vector spaces.
15	ASSIGNMENT	Other properties of the sum and the scalar product.
16	HOMEWORK	Exercises (further properties of addition and scalar multiplication).
17	SESSION	Linear dependence and independence. Properties.
18	SESSION	Generator systems. Fundamentals. Dimension.
19	ASSIGNMENT	Coordinates of a vector in a given basis.
20	ASSIGNMENT	Exercises (coordinates).
21	SESSION	Coordinate transformation.
22	SESSION	Change of coordinates.
23	ASSIGNMENT	Vector subspaces. Examples of interest: intersection, linear envelope, row and column spaces of a matrix, solutions to a homogeneous system of linear equations.
24	ASSIGNMENT	Exercises (coordinate transformation/vector subspaces).
25	SESSION	Equations and dimension of a vector subspace.
26	SESSION	Sum of vector subspaces. Dimension formula. Direct sum.
27	ASSIGNMENT	Quotient vector space.
28	ASSIGNMENT	Exercises (equations and dimension of a vector subspace/sum).
29	EV	EXAM Sections 1 and 2 of the syllabus.
30	EV	EXAM Sections 1 and 2 of the syllabus.	20
31	ASSIGNMENT	Linear applications. Properties. Types.
32	ASSIGNMENT	Exercises (linear applications).
33	SESSION	Nucleus and image.
34	SESSION	Matrix associated with a linear transformation.
35	ASSIGNMENT	Relationship with the kernel and the ‘image’ formula for dimensions.
36	ASSIGNMENT	Exercises (matrix associated with a linear transformation).
37	SESSION	Changes of basis.
38	SESSION	Base changes.
39	ASSIGNMENT	Operations on linear applications. Properties.
40	TRAB	Exercises (changes of basis/operations with linear applications).
41	SESSION	Linear forms and dual space. Dual basis.
42	SESSION	Adjoint of a vector subspace.
43	ASSIGNMENT	Transposed linear application.
44	ASSIGNMENT	Exercises (dual space).
45	SESSION	Eigenvectors and eigenvalues of an endomorphism. Characteristic and minimal polynomials. Algebraic and geometric multiplicities.
46	SESSION	Eigenvectors and eigenvalues of an endomorphism. Characteristic and minimal polynomials. Algebraic and geometric multiplicities.
47	ASSIGNMENT	Diagonalised matrices. Eigen subspaces. Diagonal form and basis of eigenvectors.
48	ASSIGNMENT	Exercises (diagonalisation).
49	SESSION	Symmetric matrices.
50	SESSION	Non-diagonalised matrices. Generalised eigenspaces. Maximum subspaces.
51	ASSIGNMENT	Non-diagonalizable matrices. Generalised eigen subspaces. Maximum subspaces.
52	ASSIGNMENT	Exercises (generalised eigenspaces and maximal subspaces).
53	SESSION	Canonical form and Jordan basis
54	SESSION	Canonical form and Jordan basis
55	ASSIGNMENT	Canonical form and Jordan basis
56	ASSIGNMENT	Exercises (canonical form and Jordan basis).
57	SESSION	Complex eigenvalues and eigenvectors. Real canonical form and Jordan basis.
58	SESSION	Exercises (canonical form and real Jordan basis).
59	EV	Presentation of work/projects
60	EV	Exhibition of work/projects	30

Bibliography

Basic:

1. Juan De Burgos Román

Algebra and Geometry. Definitions, Theorems and Results

García Maroto Editores. 2010.

ISBN: 9788492976942

2. Luis Merino and Evangelina Santos

Linear Algebra using Elementary Methods

Paraninfo. 2010.

ISBN: 978-84-9732-4

Supplementary:

3. Gilbert Strang

Introduction to Linear Algebra

Wellesley Cambridge Press. 2008.

ISBN: 8175968117

4. Juan De Burgos Román

Linear Algebra. 80 Useful Problems

García Maroto Publishers. 2007.

ISBN: 9788493601805

Others:

5. Eugenio Hernández

Linear Algebra and Geometry

3rd ed. ADDISON WESLEY. 2012.

ISBN: 9788478291298

6. Jesús Rojo

Linear Algebra

McGraw-Hill. 2001.

ISBN: 8448130162

7. Jesús Rojo

Exercises and Problems in Linear Algebra

2nd ed. McGraw-Hill. 2005.

ISBN: 8448198581

8. Stanley I. Grossman and José Job Flores

Linear Algebra

McGraw-Hill. 2012.

ISBN: 978-607-15-07

C0142301

Statistical analysis

Código: C0142301 Imprimir

Course 1: First-semester module. Foundation course. 6 credits.

Profesores

Johnny Eduardo Meyers Angulo - Coordinador

Ver biodata

Objectives

To provide students with the basic knowledge and tools of statistical analysis, covering both data representation and, above all, statistical inference, which will be essential for tackling related topics in subsequent courses.

Prerequisites

No prior requirements have been set.

Competencies

Not applicable.

Learning outcomes

RC2 – Identify the different phases of the mathematical modelling process, distinguishing between formulation, analysis, solution and interpretation of results. TYPE: Competencies.

RC3 – Apply software to solve mathematical and engineering problems, using the appropriate computing environment for each case. TYPE: Competencies.

RC7 – Develop work and projects related to mathematical engineering that have an impact on interdisciplinary teams or in multicultural contexts. TYPE: Competencies.

RS3 – Propose, analyse, validate and interpret the most appropriate mathematical models and tools in real-world situations, in accordance with the objectives being pursued. TYPE: Skills or competences.

RS5 – Use computer applications for statistical analysis, numerical and symbolic computation, graphical visualisation, optimisation or other purposes to solve problems. TYPE: Skills or competences.

Description of the content

1. Elements of data analysis

2. Descriptive statistics: samples and distribution of sample characteristics

3. Probability distributions

4. Random variables

5. Statistical inference models. Statistics and their basic properties

6. Frequentist approach: point estimation, interval estimation and hypothesis testing

7. Bayesian approach: posterior distribution, credible intervals and Bayesian tests

Training activities

AP1. Lectures.

AP2. Interactive classes.

AP3.1. Project work and problem-solving.

AP4. Independent study, case studies, problem-solving and literature review.

AP5. Tutoring.

AP6. Knowledge assessments.

Assessment system and criteria

Without prejudice to any other requirements that may be specified in the relevant course syllabus, as a general rule, failure to attend more than 70 per cent of the course’s teaching activities requiring physical attendance will result in the loss of the right to continuous assessment during the standard examination period. In this case, the examination to be held during the official period set by the University will be the sole assessment criterion, with the weighting specified in the course syllabus.

----

ASSESSMENT SYSTEMS

The assessment methods for this module are:

- AS1: Practical activities

- AS2: Final knowledge assessments

- AS3: Portfolio

ASSESSMENT CRITERIA

The assessment methods described above are set out in the following assessment criteria:

- There are two official examination sessions: the ordinary and the supplementary.

+++REGULAR SESSION+++

The final mark for this sitting is the weighted average of a set of assessment tasks detailed below:

-- Assignment submissions (SE3), accounting for 20 per cent of the final mark for the ordinary assessment period, to be completed individually or in small groups during class time.

-- Mid-term exam (SE1), to be taken individually during term time, accounting for 25 per cent of the final mark for the ordinary assessment period.

-- Skill School (SE1), accounting for 5 per cent of the final mark for the standard assessment period.

-- Final exam (SE2), to be taken individually during the official ordinary examination period, covering the entire syllabus, and accounting for 50 per cent of the final mark for the ordinary examination period, provided that the minimum mark exceeds 4 out of 10. If this mark is not achieved, the final mark for the module will be a fail in the ordinary examination period.

*** The module is considered passed in the ordinary examination period if the final mark is 5.0 or higher.

+++EXTRAORDINARY EXAMINATION PERIOD+++

If a student fails the module during the ordinary examination period, they may retake it during the extraordinary examination period, which will account for 100 per cent of the mark and cover all course content.

The supplementary examination period will take place during the July examination period (for further information, please consult the virtual campus). It consists of a single examination covering the entire syllabus of the module.

*** The module is considered passed in the supplementary examination if the final mark is 5.0 or above.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	MG	1. Elements of data analysis
2	SM	1. Elements of data analysis
3	MG	1. Elements of data analysis
4	SM	1. Elements of data analysis
5	MG	1. Elements of data analysis
6	SM	1. Elements of data analysis
7	MG	1. Elements of data analysis
8	SM	1. Elements of data analysis
9	MG	2. Descriptive statistics: samples and distribution of sample characteristics
10	SM	2. Descriptive statistics: samples and distribution of sample characteristics
11	MG	2. Descriptive statistics: samples and distribution of sample characteristics
12	SM	2. Descriptive statistics: samples and distribution of sample characteristics
13	MG	2. Descriptive statistics: samples and distribution of sample characteristics
14	SM	2. Descriptive statistics: samples and distribution of sample characteristics
15	MG	2. Descriptive statistics: samples and distribution of sample characteristics
16	SM	2. Descriptive statistics: samples and distribution of sample characteristics	SE3, 20
17	MG	3. Probability distributions
18	SM	3. Probability Distributions
19	MG	3. Probability distributions
20	SM	3. Probability Distributions
21	MG	3. Probability distributions
22	SM	3. Probability distributions
23	MG	3. Probability distributions
24	SM	3. Probability distributions
25	MG	4. Random variables
26	SM	4. Random variables
27	MG	4. Random variables
28	SM	4. Random variables
29	MG	4. Random variables
30	SM	4. Random variables
31	MG	4. Random variables
32	SM	4. Random variables	SE3, 20
33	MG	5. Statistical inference models. Statistics and their basic properties
34	SM	5. Statistical inference models. Statistics and their basic properties
35	MG	5. Statistical inference models. Statistics and their basic properties
36	SM	5. Statistical inference models. Statistics and their basic properties
37	MG	5. Statistical inference models. Statistics and their basic properties
38	SM	5. Statistical inference models. Statistics and their basic properties
39	MG	5. Statistical inference models. Statistics and their basic properties
40	SM	5. Statistical inference models. Statistics and their basic properties	SE1, 25
41	MG	6. Frequentist approach: point estimation, interval estimation and hypothesis testing
42	SM	6. Frequentist approach: point estimation, interval estimation and hypothesis testing
43	MG	6. Frequentist approach: point estimation, interval estimation and hypothesis testing
44	SM	6. Frequentist approach: point estimation, interval estimation and hypothesis testing
45	MG	6. Frequentist approach: point estimation, interval estimation and hypothesis testing
46	SM	6. Frequentist approach: point estimation, interval estimation and hypothesis testing
47	MG	6. Frequentist approach: point estimation, interval estimation and hypothesis testing
48	SM	6. Frequentist approach: point estimation, interval estimation and hypothesis testing	SE3, 20
49	MG	7. Bayesian approach: posterior distribution, credible intervals and Bayesian tests
50	SM	7. Bayesian approach: posterior distribution, credible intervals and Bayesian tests
51	MG	7. Bayesian approach: posterior distribution, credible intervals and Bayesian tests
52	SM	7. Bayesian approach: posterior distribution, credible intervals and Bayesian tests
53	MG	7. Bayesian approach: posterior distribution, credible intervals and Bayesian tests
54	SM	7. Bayesian approach: posterior distribution, credible intervals and Bayesian tests
55	MG	7. Bayesian approach: posterior distribution, credible intervals and Bayesian tests
56	SM	7. Bayesian approach: posterior distribution, credible intervals and Bayesian tests
57	MG	7. Bayesian approach: posterior distribution, credible intervals and Bayesian tests
58	SM	7. Bayesian approach: posterior distribution, credible intervals and Bayesian tests
59	MG	7. Bayesian approach: posterior distribution, credible intervals and Bayesian tests	SE1, 5
60	SM	7. Bayesian approach: posterior distribution, credible intervals and Bayesian tests	SE3, 20

Bibliography

Core:

1. A. García Pérez

Solved Problems in Basic Statistics.

National University of Distance Education. 1998.

ISBN: 978-84-362-37

2. J. Gorgas García, N. Cardiel López, and J. Zamorano Calvo.

Basic Statistics for Science Students.

UCM Publishing. 2011.

ISBN: 978-84-691-89

3. R. Mullor Ibañez

Basic Statistics I: An Introduction to Statistics.

Published by the University of Alicante. 2017.

ISBN: 978-84-9717-4

4. R. Mullor Ibañez

Basic Statistics II. Probability: Random Variables.

Published by the University of Alicante. 2023.

ISBN: 978-84-9717-8

Supplementary:

5. González Rosales, Alfredo

Applied Statistics:

Madrid: García-Maroto, D.L. 2009. 2009.

ISBN: 9788492976416

6. Murray Spiegel

PROBABILITY AND STATISTICS

4th ed. McGraw-Hill Interamericana de España S.L. 2014.

ISBN: 9786071511881

7. Neuhauser, Claudia

Mathematics for Science

2nd ed. Madrid: Pearson-Prentice Hall, 2004. 2004.

ISBN: 9788420542539

C0142302

Algebraic structures

Código: C0142302 Imprimir

Course 1: First-term module. Compulsory. 6 credits.

Profesores

José Luis Guijarro Regalado - Coordinador

Ver biodata

Objectives

This module has a twofold objective: on the one hand, students must learn to recognise that different mathematical tools share a common algebraic structure and therefore function in essentially the same way.

On the other hand, students must learn to prove theorems and properties of mathematical objects using abstract reasoning. In this module, although some numerical calculations will be carried out, the focus is entirely on the use of symbols and their properties.

Prerequisites

No prerequisites have been set.

Competencies

Not applicable.

Learning outcomes

RC9 – Formulate problems from a professional context in mathematical language in such a way as to facilitate their analysis and resolution. TYPE: Competencies.

RK4 – Be familiar with rigorous proofs of some classical theorems in different areas of mathematics. TYPE: Knowledge or content.

RS1 – Use mathematical language to formulate propositions in various fields and construct proofs for the problems posed. TYPE: Skills or abilities.

Description of the content

Groups:

1) Definition of a group and its properties

2) Examples: congruences, permutations, matrices, dihedral groups, the direct product

3) Subgroups

4) Lagrange’s theorem

5) Normal subgroups. Quotient group

6) Group homomorphisms

7) Isomorphism theorems

Rings

1) Definition of a ring and its properties

2) Subrings and ideals

3) Ring homomorphisms

4) The ring of polynomials in one variable, with coefficients in a field

Training activities

AP1. Lectures.

AP2. Interactive classes.

AP3.1. Project work and problem-solving.

AP4. Independent study, case studies, problem-solving and literature review.

AP5. Tutoring.

AP6. Knowledge assessments.

Assessment system and criteria

----

Continuous assessment consists of the following marks:

1) A set of exercises based on the syllabus will be set, or a written assignment will be set in which a topic from the course is explored in greater depth.

These will account for 10 per cent

2) A written test will be held halfway through the course

This mark will account for 40 per cent

3) The official course examination will be held (ordinary session); it will be written and cover the course content.

This mark will account for 50 per cent. Under no circumstances will passing the tests exempt students from any part of this final exam.

Continuous assessment may be forfeited in two circumstances:

1) if class attendance is less than or equal to 70 per cent

2) if the mark in the official examination is less than or equal to 4

Students who, for any reason, fail to achieve the continuous assessment mark will sit the official examination, which will count for 100 per cent of the final mark

For clarity, please note that even if you do not lose your continuous assessment due to attendance, scoring less than a 4 in the official exam will result in a fail for the module

In the resit period, no previous marks will be taken into account. A single exam covering the entire course content will be held.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	MG	Course introduction.
2	MG	Definition of a group
3	MG	Basic properties of a group
4	SM	Examples of groups: congruences
5	MG	Examples of groups: permutations
6	MG	Examples of groups: the dihedral group
7	MG	Subgroups
8	SM	Examples of subgroups
9	MG	Lagrange’s theorem
10	MG	Normal subgroups I
11	MG	Normal subgroups II
12	SM	Examples
13	MG	Quotient group I
14	MG	Group II quotient
15	MG	Group III quotient
16	SM	Examples
17	MG	Group homomorphisms
18	MG	Isomorphism theorems
19	MG	Classification theorem for cyclic groups
20	SM	Examples
21	MG	Classification Theorem for Finite Abelian Groups I
22	MG	Classification Theorem for Finite Abelian Groups II
23	MG	Classification Theorem for Finite Abelian Groups III
24	SM	Examples of classification
25	EV	Collection of exercises	10
26	MG	Classification Theorem for Finitely Generated Abelian Groups I
27	MG	Classification theorem for finitely generated Abelian groups II
28	MG	Classification Theorem for Finitely Generated Abelian Groups III
29	SM	Examples of classification
30	MG	Jordan’s Theorem I
31	MG	Jordan’s Theorem II
32	MG	Jordan’s Theorem III
33	SM	Examples involving matrices over fields other than the real field
34	MG	The Rubik’s group: modelling
35	MG	The Rubik’s group
36	MG	The Rubik’s group: operations
37	SM	Solving the cube
38	EV	Submission of entries	20
39	MG	Generators and relationships I
40	MG	Generators and relationships II
41	MG	Generators and relationships III
42	SM	Examples
43	MG	Generators and IV relationships
44	MG	Rings
45	MG	Properties of rings
46	MG	Subrings
47	SM	Examples
48	MG	Ideals
49	MG	Ring homomorphisms
50	MG	Isomorphism theorems
51	SM	Examples
52	MG	The ring of polynomials
53	MG	The ring of polynomials II
54	SM	Exercises
55	MG	Factoring
56	MG	DIP
57	MG	Euclidean domains
58	SM	Applications to the factorisation into priomos
59	EV	Collection of exercises	10
60	EV	Final exam	60

Reading list

Essential:

1. E. Bujalance, J. Etayo, J.M. Gamboa

Commutative rings and fields

UNED. 2002.

ISBN: 8436244486

2. E. Bujalance, J. Etayo, J.M. Gamboa

Elementary Group Theory

UNED. 2002.

ISBN: 8436244362

3. J. Dorronsoro, E. Hernández

Numbers, Groups and Rings

Addison Wesley, UAM. 1996.

ISBN: 0201653958

C0142303

Fundamentals of Programming and Computing

Código: C0142303 Imprimir

Course 1: First-term module. Foundation course. 6 credits.

Profesores

José Antonio Marcos García - Coordinador

Ver biodata

Jacinto Velasco Rebolledo

Ver biodata

Objectives

- Understanding the basics of programming: Students will be able to understand the fundamentals of programming, including algorithmic logic and the use of control structures.

- Develop skills in Python: Students are expected to learn to programme effectively in Python, applying the principles of object-oriented programming and other key techniques.

- Solve problems using programming: Students should be able to design algorithms and write code to solve a variety of computational problems.

- Apply good coding practices: Students will gain knowledge of writing clean, efficient and modular code, following industry standards and best practices.

- Foster logical and analytical thinking: Throughout the course, students will develop the skills to tackle complex problems in a structured and efficient manner.

- Develop projects applying programming concepts: Students will be able to integrate the knowledge they have acquired into programming projects that solve real or simulated problems, using software design and development techniques.

- Understand and evaluate the different types of storage systems and how they affect the performance of a computer system.

- Introduction to computer architecture and microprocessors.

Prerequisites

No prerequisites have been set.

Competencies

Not applicable.

Learning outcomes

RC2 – Identify the different phases of the mathematical modelling process, distinguishing between formulation, analysis, solution and interpretation of results. TYPE: Competencies.

RC3 – Apply software to solve mathematical and engineering problems, using the appropriate computing environment for each case. TYPE: Competencies.

RS5 – Use computer applications for statistical analysis, numerical and symbolic computation, graphical visualisation, optimisation or other purposes to solve problems. TYPE: Skills or competences.

Course description

This course is designed to introduce students to the fundamental concepts of programming and computational logic, with a specific focus on the Python programming language and SQL. Python and SQL are widely used in industry due to their simple syntax and readability, making them an excellent choice for beginners. Throughout the course, students will learn essential concepts such as control structures, data types, functions and file management. Principles of algorithmic design and good coding practices will also be covered.

Learning activities

AP1. Lectures.

AP2. Interactive classes.

AP3.2 Workshops and practical sessions.

AP4. Independent study, case studies, problem-solving and literature reviews.

AP5. Tutoring.

AP6. Knowledge assessments.

Assessment system and criteria

----

Ordinary Examination Period:

1) Participation and attendance + completion of case studies (50 per cent):

a) Regular attendance at classes and scheduled activities.

b) Active participation in discussions and debates.

c) Correct and complete completion of case studies.

d) If a ULAB is held, it will be assessed as a mid-term exam.

2) Final examination (50%): an examination held during the standard examination period, comprising 50% theoretical questions and 50% case studies.

The calculation of the final mark in the standard examination period will be conditional upon achieving a minimum mark of 4.0 out of 10 in the final examination. Once this requirement has been met, a weighted average will be calculated from all assessment activities and tests, even if any of them have been failed.

If the mark obtained in the final exam is less than 4.0 out of 10, the final mark will be the mark obtained in that exam.

Extraordinary Examination Period (100% Exam):

- In the resit, assessment will be based solely on an exam covering the entire course content.

- The exam will account for 100 per cent of the final mark.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	MG	Presentation
2	LB	Introduction to programming. Structure of a programme
3	MG	Introduction to programming. Structure of a programme
4	MG	Introduction to programming. Structure of a programme
5	MG	Identifiers. Variables. Data types, literals. Operations and expressions
6	LB	Identifiers. Variables. Types, Literals. Operations and expressions
7	MG	Identifiers. Variables. Types, Literals. Operations and expressions
8	MG	Identifiers. Variables. Types, Literals. Operations and expressions
9	MG	Identifiers. Variables. Types, Literals. Operations and expressions
10	LB	Identifiers. Variables. Types, Literals. Operations and expressions
11	MG	Reading data from the keyboard. Utility classes.
12	EV	Reading data from the keyboard. Utility classes.	5%
13	MG	Classes and objects. Attributes, methods, method calls. Aliases.
14	LB	Classes and objects. Attributes, methods, method calls. Aliases.
15	MG	Classes and objects. Attributes, methods, method calls. Aliases.
16	MG	Classes and objects. Attributes, methods, method calls. Aliases.
17	MG	Classes and objects. Constructors. Returning values.
18	LB	Classes and objects. Constructors. Returning values.
19	MG	Classes and objects. Constructors. Returning values.
20	EV	Classes and objects. Constructors. Returning values.	5%
21	MG	Control statements
22	LB	Control rulings
23	MG	Control sentences
24	EV	Control rulings	10%
25	MG	Laboratory practicals – exam preparation
26	LB	Laboratory practicals – exam preparation
27	EV	Exam
28	EV	Exam	25%
29	MG	Exceptions
30	LB	Exceptions
31	MG	Exceptions
32	MG	Exceptions
33	MG	Arrays
34	LB	Arrays
35	MG	Arrays
36	MG	Arrays
37	MG	Arrays. Laboratory practicals
38	EV	Arrays. Laboratory practicals	5%
39	MG	File Input and Output
40	MG	File input and output
41	MG	File input and output
42	LB	File input and output
43	MG	Laboratory practicals.
44	EV	Laboratory practicals.	15%
45	MG	Laboratory practicals – exam preparation
46	LB	Laboratory practicals – exam preparation
47	EV	Exam
48	EV	Exam	20%
49	MG	Fundamental Computer Architecture and Functional Structure
50	LB	Fundamental Computing Architecture and Functional Structure
51	MG	Instruction Sequencing and Microprogramming
52	MG	Instruction Sequencing and Microprogramming
53	MG	Memory Hierarchy, Devices and Interrupts
54	LB	Memory Hierarchy, Devices and Interrupts
55	MG	Storage Systems and Performance Evaluation
56	MG	Storage systems and performance evaluation
57	MG	Storage systems and performance evaluation
58	LB	Storage systems and performance evaluation
59	EV	Examination
60	EV	Exam	15%

Reading list

Core:

1. Al Sweigart

Automate the Boring Stuff with Python, 3rd Edition: Practical Programming for Total Beginners

No Starch Press. 2025.

ISBN: 1718503407

2. Charles Russell Severance

Python for Everyone: Exploring Data with Python 3

Self-published. 2020.

ISBN: 9798633985566

3. Eric Matthes

Python Crash Course, 3rd Edition: A Hands-On, Project-Based Introduction to Programming

No Starch Press. 2023.

ISBN: 1718502702

4. F. Cuesta

Introduction to Programming with Python

Marcombo. 2019.

ISBN: 978-842673616

5. John M. Zelle

Python Programming: An Introduction to Computer Science

Franklin, Beedle & Associates. 2024.

ISBN: 1590282973

6. John V. Guttag

Introduction to Computation and Programming Using Python, third edition: With Application to Computational Modelling and Understanding Data

The MIT Press. 2021.

ISBN: 0262542366

7. Mark Lutz

Learning Python

O’Reilly Media. 2013.

ISBN: 1449355730

C0142304

Mathematical Foundations of Engineering I

Código: C0142304 Imprimir

Course 1: First-semester module. Foundation course. 6 credits.

Profesores

Jacinto Velasco Rebolledo - Coordinador

Ver biodata

Objectives

By the end of the module, students will be able to understand and use mathematical language and notation correctly, expressing concepts, reasoning and results with rigour. They will also be able to analyse real functions, determine their domain and identify their main properties.

Students will be able to calculate limits and study the continuity and differentiability of functions, as well as interpret the meaning of the derivative and calculate it using the appropriate techniques. They will also be able to apply differentiation to the study of functions, to approximation problems and to the solution of optimisation problems.

They will be able to represent and analyse functions graphically, mathematically justifying their main characteristics based on properties such as domain, continuity, monotonicity, extrema and the behaviour of the function.

Furthermore, they will be able to analyse the convergence of sequences and series, applying the relevant concepts and procedures, and to work with complex numbers, understanding both their algebraic treatment and their geometric interpretation and representation.

Across the curriculum, students will develop the ability to solve mathematical problems independently and logically, selecting the appropriate methods, justifying the steps taken and checking the consistency of the results obtained.

Prerequisites

No prerequisites have been established.

Competencies

Students will acquire the ability to use mathematical reasoning and formal language to formulate, analyse and solve problems, employing notation correctly and rigorously justifying the procedures and results obtained.

They will develop skills to analyse real functions and understand their behaviour, identifying domains, properties, limits, continuity and differentiability, as well as to use the tools of differential calculus in the study and representation of functions.

They will be able to apply differentiation techniques to practical problems, particularly in situations involving approximation, analysis of variation and optimisation, selecting the most appropriate mathematical procedure depending on the problem at hand.

They will acquire the skills to analyse sequences and series and study their convergence, as well as to work with complex numbers, handling their various forms of representation and understanding their geometric interpretation.

Finally, students will develop cross-curricular skills relating to independent problem-solving, analytical and critical thinking, and the capacity for abstraction, learning to devise problem-solving strategies, justify the decisions taken and verify the consistency of the results.

Learning outcomes

What you need to do is draft these for your teaching guide whilst adhering to the UAX criteria, but making them somewhat clearer and more assessable; I would phrase them as follows:

LO1. Distinguish between, interpret and operate correctly with different number sets, including real and complex numbers.

LA2. Analyse numerical sequences and series, applying their main properties and criteria to determine their behaviour and convergence.

LA3. Analyse real functions using the fundamental concepts and theorems of limits, continuity and differentiability.

RA4. Calculate and interpret derivatives using appropriate techniques and apply them to the study and representation of functions.

RA5. Apply the knowledge and tools of mathematical analysis to the formulation and reasoned solution of engineering-related problems.

Course content

Topic 1. Real functions of a real variable. Real numbers, intervals, inequalities and absolute value. The concept of a function, domain and range. Properties of functions, composition and inverse functions. Elementary and conic functions.

Topic 2. Limits and continuity. Concept and calculation of limits of functions. Lateral limits, limits at infinity and limits involving infinity. Indeterminacies and equivalences. Continuity and main types of discontinuities.

Topic 3. Differentiation. Concept and interpretation of the derivative. Basic rules of differentiation, the chain rule and successive derivatives. Lateral derivatives, differentials and linear approximation.

Topic 4. Techniques for calculating limits. Methods for resolving limits and indeterminacies. L’Hôpital’s rule, equivalent infinities and infinitesimals, and approximations using elementary expansions.

Topic 5. Applications of differentiation. Study of monotonicity, extrema, concavity and inflection points. Fundamental theorems of differential calculus and solving optimisation problems.

Topic 6. Graphical representation of functions. Comprehensive study of functions through domain, range, intersections with the axes, sign, symmetries and asymptotes. Analysis of variation and graphical representation.

Topic 7. Sequences of real numbers. The concept of a sequence, monotonicity, boundedness and convergence. Calculation of limits of sequences, recurring sequences and the principle of mathematical induction.

Topic 8. Numerical and power series. The concept and convergence of series. Main convergence criteria, alternating series and absolute convergence. Power series and Taylor and Maclaurin expansions.

Topic 9. Complex numbers. Representation and operations with complex numbers in binomial, polar and exponential form. Powers and roots, De Moivre’s formula and Euler’s formula. Introduction to complex logarithms and loci.

Training activities

AP1. Lectures.

AP2. Interactive classes.

AP3.1 Project work and problem-solving.

AP4. Independent study, case studies, problem-solving and literature review.

AP5. Tutoring.

AP6. Knowledge assessments.

Assessment system and criteria

The mark for the module in the standard assessment period will be determined using the following assessment methods:

The final mark will be calculated using the following weighting: 50 per cent of the mark obtained in continuous assessment and 50 per cent of the mark obtained in the final examination.

To pass the module, students must achieve a final mark of 5.0 out of 10 or higher and a minimum mark of 4.0 out of 10 in the final examination.

The weighted average will be calculated regardless of the mark obtained in continuous assessment. However, to pass the module, the minimum mark requirement set for the final examination must be met.

To retain the mark for continuous assessment, students must demonstrate a minimum attendance rate of 70 per cent. If this percentage is not met, the mark for continuous assessment will be 0 marks, whilst the weighting established for the calculation of the final mark will remain unchanged.

+++SUPPLEMENTARY EXAMINATION SESSION+++

If a student has not passed the module in the ordinary examination session, they may sit the extraordinary examination session.

The supplementary examination session will take place during the July examination period (for further information, please consult the Academic Calendar). It consists of a single examination covering the entire syllabus of the module.

*** The module is considered passed in the extraordinary examination period if the final mark is 5.0 or higher.

GRADES

The award of the corresponding credits is conditional upon passing the associated examinations or assessment tests.

The level of learning achieved by students will be expressed as numerical marks on a scale of 0 to 10, to one decimal place, to which the corresponding qualitative grade may be added:

- 0–4.9: Fail (SS).

- 5.0–6.9: Pass (AP).

- 7.0–8.9: Good (NT).

- 9.0–10: Distinction (SB).

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	MG	Course introduction
2	MG	Real numbers
3	MG	Real numbers
4	SM	Problem-solving with real numbers
5	MG	The field of complex numbers.
6	MG	The field of complex numbers.
7	MG	The field of complex numbers.
8	SM	Solving problems involving complex numbers.
9	MG	The field of complex numbers.
10	MG	The field of complex numbers.
11	MG	The field of complex numbers.
12	SM	Solving problems involving complex numbers.
13	MG	Sequences of real numbers.
14	MG	Sequences of real numbers.
15	MG	Sequences of real numbers.
16	SM	Solving problems involving sequences of real numbers.
17	MG	Sequences of real numbers.
18	MG	Sequences of real numbers.
19	MG	Sequences of real numbers.
20	SM	Solving problems involving real number series.
21	MG	Preliminaries on real-valued functions of a real variable
22	MG	Revision of exam topics
23	EV	First set of exercises	10%
24	EV	First mid-term exam	30%
25	MG	Preliminaries on real functions of a real variable
26	MG	Preliminaries on real-valued functions of a real variable
27	MG	Preliminaries on real-valued functions of a real variable
28	SM	Solving problems involving real functions.
29	MG	Preliminaries on real-valued functions of a real variable
30	MG	Preliminaries on real-valued functions of a real variable
31	MG	Introduction to real-valued functions of a real variable
32	SM	Solving problems involving real-valued functions.
33	MG	Limits and continuity of real functions of a real variable
34	MG	Limits and continuity of real-valued functions of a real variable
35	MG	Limits and continuity of real-valued functions of a real variable
36	SM	Problem-solving on limits and continuity of real-valued functions of a real variable
37	SM	Problem-solving: Limits and continuity of real-valued functions of a real variable
38	MG	Limits and continuity of real-valued functions of a real variable
39	MG	Limits and continuity of real-valued functions of a real variable
40	MG	Limits and continuity of real-valued functions of a real variable
41	SM	Problem-solving on limits and continuity of real-valued functions of a real variable
42	MG	Applications of the derivative. Optimisation
43	MG	Applications of the derivative. Optimisation
44	MG	Applications of the derivative. Optimisation
45	SM	Solving problems involving applications of the derivative.
46	MG	Applications of the derivative. Optimisation
47	MG	Applications of the derivative. Optimisation
48	MG	Applications of the derivative. Optimisation
50	MG	Derivatives of real-valued functions of a real variable
51	MG	Derivatives of real-valued functions of a real variable
52	MG	Derivatives of real-valued functions of a real variable
53	SM	Solving problems involving derivatives of real-valued functions of a real variable
54	MG	Derivatives of real-valued functions of a real variable
55	MG	Derivatives of real-valued functions of a real variable
56	MG	Derivatives of real-valued functions of a real variable
57	SM	Solving problems involving derivatives of real-valued functions of a real variable
58	MG	Revision of exam topics
59	EV	Second set of exercises and submission of coursework	10% (exercises) 20% (assignment)
60	EV	Second mid-term exam	30%

Reading list

Core:

1. Michael Spivak

Infinitesimal Calculus

2nd ed. Reverté. 1988.

ISBN: 8429151362

Supplementary:

2.- Roland E. Larson, Robert P. Hostetler, Bruce H. Edwards

Calculus and Analytic Geometry (Volume 1)

McGraw-Hill. 2010.

ISBN: 8448122291

C0142602

Fundamentals of Physics I

Código: C0142602 Imprimir

Course 1: First-term module. Basic training. 6 credits.

Profesores

Yanela Mendez González - Coordinador	Ver biodata Yanela Méndez González holds a PhD in Physical Sciences, specialising in the synthesis and characterisation of ferroelectric, antiferroelectric and multifunctional materials, with applications in emerging technologies such as photovoltaic devices, electrothermal systems and energy storage. Her teaching duties include delivering core modules such as Electromagnetism, Mechanics and Optics at the University of Havana, where she served as an assistant lecturer between 2013 and 2016. She has a solid track record in research, gained at prestigious institutions such as the Polytechnic University of Madrid, the Autonomous University of Madrid, the Institute of Materials Science and Technology (IMRE, Cuba) and research centres in Brazil, France and Mexico. She is a member of several international research groups focused on the study of advanced functional materials. She is the author of more than 45 scientific publications in high-impact indexed journals, such as Applied Physics Letters, Journal of the American Ceramic Society and Current Applied Physics, with significant contributions to the analysis of ceramic thin films and lanthanide-doped systems. Her research work has been recognised with national and international awards. She has participated in projects funded by European programmes such as the Marie Skłodowska-Curie Actions and by Latin American agencies such as FAPEMIG and CAPES (Brazil). He is currently collaborating on the development of radiation-resistant materials for nuclear fusion reactors at the ‘Guillermo Velarde’ Institute of Nuclear Fusion.
Joaquín Bermejo Ortiz	Ver biodata Yanela Méndez González holds a PhD in Physical Sciences, specialising in the synthesis and characterisation of ferroelectric, antiferroelectric and multifunctional materials, with applications in emerging technologies such as photovoltaic devices, electrothermal systems and energy storage. Her teaching duties include delivering core modules such as Electromagnetism, Mechanics and Optics at the University of Havana, where she served as an assistant lecturer between 2013 and 2016. She has a solid track record in research, gained at prestigious institutions such as the Polytechnic University of Madrid, the Autonomous University of Madrid, the Institute of Materials Science and Technology (IMRE, Cuba) and research centres in Brazil, France and Mexico. She is a member of several international research groups focused on the study of advanced functional materials. She is the author of more than 45 scientific publications in high-impact indexed journals, such as Applied Physics Letters, Journal of the American Ceramic Society and Current Applied Physics, with significant contributions to the analysis of ceramic thin films and lanthanide-doped systems. Her research work has been recognised with national and international awards. She has participated in projects funded by European programmes such as the Marie Skłodowska-Curie Actions and by Latin American agencies such as FAPEMIG and CAPES (Brazil). He is currently collaborating on the development of radiation-resistant materials for nuclear fusion reactors at the ‘Guillermo Velarde’ Institute of Nuclear Fusion.
David Herranz Fernández	Ver biodata Yanela Méndez González holds a PhD in Physical Sciences, specialising in the synthesis and characterisation of ferroelectric, antiferroelectric and multifunctional materials, with applications in emerging technologies such as photovoltaic devices, electrothermal systems and energy storage. Her teaching duties include delivering core modules such as Electromagnetism, Mechanics and Optics at the University of Havana, where she served as an assistant lecturer between 2013 and 2016. She has a solid track record in research, gained at prestigious institutions such as the Polytechnic University of Madrid, the Autonomous University of Madrid, the Institute of Materials Science and Technology (IMRE, Cuba) and research centres in Brazil, France and Mexico. She is a member of several international research groups focused on the study of advanced functional materials. She is the author of more than 45 scientific publications in high-impact indexed journals, such as Applied Physics Letters, Journal of the American Ceramic Society and Current Applied Physics, with significant contributions to the analysis of ceramic thin films and lanthanide-doped systems. Her research work has been recognised with national and international awards. She has participated in projects funded by European programmes such as the Marie Skłodowska-Curie Actions and by Latin American agencies such as FAPEMIG and CAPES (Brazil). He is currently collaborating on the development of radiation-resistant materials for nuclear fusion reactors at the ‘Guillermo Velarde’ Institute of Nuclear Fusion.

Objectives

The aim of the module is to equip students with the necessary tools to tackle fundamental problems in the field of physics (historical context, kinematics, dynamics, energy and associated theorems, systems of multiple particles), thereby providing them with the necessary foundation for subsequent modules.

Prerequisites

No prior requirements have been met.

Learning Outcomes

RK1 Understand the most important phenomena and theories in the various branches of physics, as well as their historical context

RK3 To analyse the fundamental concepts and principles of physical systems in order to develop approximations that enable the construction of a simplified model

RS3 Estimate orders of magnitude to interpret laboratory phenomena in the field of Physics and its related disciplines, as well as in Chemistry.

RS4 Apply mathematical and numerical methods to the modelling and explicit solution of problems in physics and related disciplines, selecting the appropriate tools and interpreting results.

RS5 Use appropriate electronic instruments and/or computer tools in modelling to find solutions to physics problems.

Learning outcomes

RA1 Identifies the relevant physical principles and, where necessary, makes simplifications and uses estimates of orders of magnitude in order to model and solve practical problems.

RA2 Handles fundamental concepts such as particles and fields, force, work, etc., with ease, in order to describe physical systems correctly.

LR3 Applies Newton’s laws appropriately to solving problems involving particles and systems of particles, and in relation to oscillatory motion.

RA4 Understands the units of the International System of Units and correctly assigns them to each of the physical quantities studied, as well as other units commonly used in the field of physics.

RC1 Work independently on the management of projects related to the different areas of physics

Description of the content

- Topic 1: Historical introduction.

- Topic 2: Particle kinematics and types of motion.

- Topic 3: Particle dynamics and the causes of motion.

- Topic 4: Work, energy and associated theorems.

- Topic 5: Systems of particles. Geometry of masses.

- Topic 6: Kinematics of a rigid body

- Topic 7: Dynamics of a rigid body

Learning activities

Learning activity No. of hours* Contact hours (8–12)** % of contact time

AP1.- Participatory lectures 50 2.78 100

AP2.- Seminars or practical application classes 30 1.67 100

AP3.- Practical activities (case studies, project work, simulation, etc.) 72 2 50

AP4.- Independent study 180 0 0

AP5. – Tutorials 36 0.6 30

AP6.- Knowledge assessments 8 0.44 100

AP10.- Workshop and/or laboratory activities 74 4.11 100

TOTAL 450 11.60

Assessment system and criteria

The assessment process will consist of verifying and evaluating the student’s acquisition of the required competences.

ASSESSMENT SYSTEMS

The assessment systems for this module are:

- AS1: Practical activities (case studies, problem-solving and challenges, project work, oral presentations, debates, etc.).

- AS2: Final knowledge assessments.

- AS3: Laboratory practical book

ASSESSMENT CRITERIA

The assessment methods described above are set out in the following assessment criteria. There are two official examination sessions: the ordinary and the supplementary.

+++REGULAR EXAMINATION PERIOD+++

The final mark for this sitting is the weighted average of a set of assessment tasks detailed below:

- Practical activities (SE1) 30% of the final mark: Mid-term exam set in advance (20%) + Submission of exercises (10%).

- Final knowledge assessments (SE2) 50% of the final mark: Final exam for the ordinary assessment period, covering the entire course content.

- Laboratory practical logbook (SE3): 20 per cent of the final mark. Final report on the metrology practicals carried out in November.

For continuous assessment (comprising the Practical Activities (SE1) and the Laboratory Practical Logbook (SE3)) to be taken into account, students must achieve a mark of 4.0/10 or higher in the final exam during the ordinary examination period. Otherwise, their mark will correspond directly to that obtained in that exam.

Furthermore, in order for students to benefit from continuous assessment, they must attend at least 70 per cent of the scheduled class hours (SESSION, LAB). If attendance is less than 70 per cent, the module may be passed by sitting a single examination during an official examination period (ordinary or supplementary). In the ordinary examination session, the weighting of this examination will be 60 per cent, in accordance with the ‘SE2 – Final Knowledge Tests’ assessment system set out in the report for the application for degree accreditation, which means that a student who has forfeited their right to continuous assessment may obtain, at most, a mark of 6.0 out of 10 for the module in that sitting; to pass the module, they must achieve a mark of 8.34 out of 10 in the examination.

The module is considered to have been passed in the ordinary examination period if the mark obtained in accordance with the above guidelines is 5.0 or higher.

+++EXTRAORDINARY EXAMINATION SESSION+++

If a student has not passed the module in the ordinary examination session, they may sit the extraordinary examination session.

The supplementary examination period will take place during the June and July examination periods (for further information, please consult the Academic Calendar).

- 100 per cent of the final mark for the module in the supplementary examination period corresponds to the final examination for this period, which assesses the entire syllabus of the module.

The module is considered to have been passed in the extraordinary examination period if the final mark is 5.0 or higher.

GRADES

The award of the corresponding credits is conditional upon passing the associated examinations or assessment tests.

The level of learning achieved by students will be expressed as numerical marks on a scale of 0 to 10, to one decimal place, to which the corresponding qualitative grade may be added:

- 0–4.9: Fail (SS).

- 5.0–6.9: Pass (AP).

- 7.0–8.9: Good (NT).

- 9.0–10: Distinction (SB).

The distinction ‘Honours’ shall be awarded to students who have obtained a mark of 9.0 or higher. The number of such students may not exceed five per cent of those enrolled on the course in the relevant academic year, unless the number of enrolled students is fewer than 20, in which case only one ‘First Class Honours’ may be awarded.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	SESSION	Introduction to the course
2	SESSION	Introduction: physics and its context; physical quantities and units of measurement
3	ASSIGNMENT	Introduction: physical quantities and units of measurement
4	SESSION	Particle kinematics
5	SESSION	Particle Kinematics
6	ASSIGNMENT	Particle kinematics
7	SESSION	Types of motion
8	SESSION	Types of movement
9	WORK	Types of movement
10	SESSION	Particle dynamics
11	SESSION	Particle dynamics
12	ASSIGNMENT	Particle dynamics
13	SESSION	Particle dynamics
14	SESSION	Particle dynamics
15	ASSIGNMENT	Particle dynamics
16	SESSION	Particle dynamics
17	SESSION	Particle dynamics
18	ASSIGNMENT	Particle dynamics
19	SESSION	Work and energy and associated theorems.
20	SESSION	Work and energy and associated theorems.
21	ASSIGNMENT	Work and energy and related theorems.
22	SESSION	Oscillatory motion.
23	SESSION	Oscillatory motion.
24	ASSIGNMENT	Oscillatory motion.
25	EV	Exemption exam – introduction and teaching units 1–4
26	EV	Exemption exam – introduction and teaching units 1–4	29.75
27	LB	Laboratory – metrology
28	LB	Laboratory – metrology
29	LB	Laboratory – metrology
30	SESSION	Particle systems. Mass geometry.
31	SESSION	Particle systems. Mass geometry.
32	ASSIGNMENT	Particle systems. Mass geometry.
33	LB	Laboratory – metrology
34	LB	Laboratory – metrology
35	SESSION	Statics.
36	SESSION	Static.
37	ASSIGNMENT	Static.
38	LB	Laboratory – metrology
39	LB	Laboratory – metrology
40	SESSION	Statics.
41	SESSION	Static.
42	ASSIGNMENT	Static.
43	LB	Laboratory – metrology
44	LB	Laboratory – metrology
45	SESSION	Elasticity.
46	SESSION	Elasticity.
47	ASSIGNMENT	Elasticity.
48	LB	Laboratory – metrology
49	LB	Laboratory – metrology
50	SESSION	Gravity
51	SESSION	Gravitation
52	ASSIGNMENT	Printing
53	LB	Laboratory – metrology
54	LB	Laboratory – metrology
55	SESSION	Gravity
56	SESSION	Gravitation
57	LB	Laboratory – metrology
58	LB	Laboratory – metrology	15
59	EV	Exhibition of work/projects
60	EV	Exhibition of work/projects	25.5

TOTAL:

SECOND FOUR-MONTH PERIOD

Code

Subjects

Character*

ECTS

C0142305

Algebra II

Código: C0142305 Imprimir

Course 1. Second-term module. Foundation course. 6 credits.

Profesores

José Antonio Prieto Persiguero - Coordinador

Ver biodata

Objectives

This module, together with Algebra I, forms the subject Algebra. This module, which forms part of the degree programme’s Basic Training module, aims not only to ensure that students are familiar with the main fundamental theorems of linear algebra, but also to enable them to understand matrix calculus from a conceptual perspective and to apply it to solving problems typical of mathematical engineering; it therefore therefore, the foundation for other subjects and modules within the degree programme, such as Numerical Calculus, Operational Research, Stochastic Calculus and Artificial Intelligence.

Prerequisites

Although no prerequisites have been established, it is advisable to have previously taken the course ‘Algebra I’ or another course with similar learning outcomes.

Competencies

Not applicable.

Learning outcomes

RK4 – To be familiar with rigorous proofs of some classical theorems in different areas of mathematics. TYPE: Knowledge or content.

RS1 - Use mathematical language to formulate propositions in various fields and construct proofs for the problems posed. TYPE: Skills or abilities.

Description of the content

1. QUADRATIC FORMS: CONCEPT AND CLASSIFICATION.

1.1 Bilinear forms. Properties. Associated matrix. Change of basis. Symmetric and antisymmetric bilinear forms. Degeneration and positive definition.

1.2 Quadratic forms. Polar form of a quadratic form. Associated matrix. Conjugation. Signature. Classification. Diagonalisation by congruence. Sylvester’s criterion.

2. EUCLIDEAN VECTOR SPACES.

2.1 Scalar product. Properties. Associated matrix (Gram matrix). Change of basis.

2.2 Angle between two vectors. Orthogonality. Distance between two vectors. Orthogonal projection. Orthogonal complement of a vector subspace.

2.3 Orthogonal and orthonormal bases. Gram–Schmidt algorithm. Change of coordinates between orthonormal bases: orthogonal matrices.

2.4 Vector product.

3. AFFINE SPACES AND EUCLIDEAN AFFINE SPACES.

3.1 Affine spaces. Reference systems and coordinates. Change of reference system.

3.2 Affine varieties. Affine varieties of interest: affine variety generated by a set of points, intersection. Equations and dimension of an affine variety. Sum of affine varieties.

3.3 Relative positions between affine varieties. Orthogonal projection of a point onto an affine variety. Distance from a point to an affine variety. Distance between affine varieties. Metric problems in two- and three-dimensional Euclidean affine spaces.

4. CONIC SECTIONS, QUADRATIC CURVES AND MOTIONS.

4.1 Conic sections. General and reduced equations of a conic. Associated matrix: classification. Calculation of geometric elements. Metric invariants and the reduced equation of conics.

4.2 Quadric surfaces. General and reduced equations of a quadric. Associated matrix: classification. Metric invariants and classification of quadric surfaces by invariants.

4.3 Affine transformations and movements. Examples. Matrix representation. Rigid movements. Fixed points and invariant varieties. Classification of rigid movements in two- and three-dimensional Euclidean affine spaces.

Training activities

AP1. Lectures.

AP2. Interactive classes.

AP3.1 Project work and problem-solving.

AP4. Independent study, case studies, problem-solving and literature review.

AP5. Tutoring.

AP6. Knowledge assessments.

Assessment system and criteria

----

The assessment process will consist of evaluating the extent to which the student has achieved the learning outcomes associated with the subject.

ASSESSMENT SYSTEMS

The assessment systems for this module are:

 AS1 Practical activities.

 AS1 Final knowledge assessments.

 AS3 Portfolio.

These systems contribute to a greater or lesser extent to the assessment of the learning outcomes assigned to this subject.

ASSESSMENT CRITERIA

The assessment systems described above are set out in the following assessment criteria:

- There are two official examination sessions: the ordinary and the supplementary.

+++REGULAR EXAMINATION PERIOD+++

The final mark for this sitting is the weighted average of a set of assessment tests detailed below:

-- a non-exemption exam (SE3), to be taken individually during the teaching term, accounting for 20 per cent of the final mark for the standard assessment period.

***** The weighted average of all these assessment components will be calculated solely and exclusively if the mark obtained in the final exam is 4.0 out of 10 or higher.

Furthermore, only the examinations will be subject to re-marking.

***** A student will have passed the module in the ordinary examination period if, and only if, they obtain a final mark (weighted average of all assessment tests) of 5.0 out of 10 or higher.

Otherwise, the student may pass the module in the supplementary examination session.

********** Should a student lose their entitlement to continuous assessment, they must achieve a mark of 10.0 out of 10.0 in the ordinary examination session in order to pass the module.

+++SITTING OUT OF TERM+++

If a student fails to pass the module in the ordinary examination session, they may do so in the supplementary examination session.

During this session, there will be a single assessment, consisting of an examination to be held during the supplementary examination period in June–July (for further information, please consult the virtual campus), which will cover the entire syllabus of the module.

***** The module will be deemed to have been passed in the supplementary examination period if the mark obtained in that exam is 5.0 out of 10.0 or higher.

GRADES

The award of the corresponding credits will be contingent upon passing the associated examinations or assessment tests.

The level of learning achieved by students shall be expressed as numerical marks on a scale of 0 to 10, to one decimal place, to which the corresponding qualitative mark may be added:

- 0–4.9: Fail (SS).

- 5.0–6.9: Pass (AP).

- 7.0–8.9: Good (NT).

- 9.0–10: Distinction (SB).

The distinction of ‘Honours’ shall be awarded to students who have obtained a mark of 9.0 or higher. The number of such students may not exceed five per cent of those enrolled on the course in the relevant academic year, unless the number of enrolled students is fewer than 20, in which case only one ‘First Class Honours’ may be awarded.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	SESSION	Introduction to the course.
2	SESSION	Bilinear forms. Properties.
3	ASSIGNMENT	Matrix associated with a bilinear form.
4	ASSIGNMENT	Exercises (bilinear forms and associated matrices).
5	SESSION	Change of basis in a bilinear form.
6	SESSION	Symmetric and antisymmetric bilinear forms.
7	ASSIGNMENT	Degeneracy and positive definition.
8	ASSIGNMENT	Exercises (change of basis in a bilinear form / symmetric and antisymmetric linear forms / degeneracy and positive definition).
9	SESSION	Quadratic forms. Polar form of a quadratic form. Associated matrix.
10	SESSION	Conjugation of a quadratic form. Signature. Classification.
11	ASSIGNMENT	Diagonalisation of a quadratic form by congruence. Sylvester’s criterion.
12	ASSIGNMENT	Exercises (matrix associated with a quadratic form/signature and classification/diagonalisation by congruence/Sylvester’s criterion).
13	SESSION	Scalar product. Properties.
14	SESSION	Matrix associated with the scalar product (Gram matrix).
15	ASSIGNMENT	Change of basis in a scalar product.
16	EV	SUBMISSION Section 1 of the programme.	7.5
17	SESSION	Angle between two vectors. Orthogonality. Distance between two vectors.
18	SESSION	Orthogonal projection.
19	ASSIGNMENT	Orthogonal complement of a vector subspace.
20	ASSIGNMENT	Exercises (orthogonal complement of a vector subspace).
21	SESSION	Orthogonal and orthonormal bases. The Gram–Schmidt algorithm.
22	SESSION	Coordinate transformation between orthonormal bases: orthogonal matrices.
23	ASSIGNMENT	Cross product.
24	ASSIGNMENT	Exercises (Gram–Schmidt algorithm/coordinate transformation between orthonormal bases).
25	SESSION	Affine spaces.
26	SESSION	Reference systems and coordinates.
27	ASSIGNMENT	Change of reference system.
28	ASSIGNMENT	Exercises (reference systems and coordinates/change of reference system).
29	SESSION	Affine transformations.
30	SESSION	Exercises (affine varieties).
31	EV	EXAM Sections 1 and 2 of the syllabus.
32	EV	EXAM Sections 1 and 2 of the syllabus.	20
33	SESSION	Affine variety generated by a set of points, intersection.
34	SESSION	Equations and dimension of an affine manifold.
35	ASSIGNMENT	Sum of affine varieties.
36	ASSIGNMENT	Exercises (related varieties, equations and dimension).
37	SESSION	Relative positions between affine varieties.
38	SESSION	Orthogonal projection of a point onto an affine variety. Distance from a point to an affine variety.
39	ASSIGNMENT	Distance between affine varieties.
40	ASSIGNMENT	Exercises (relative positions and distances between related varieties).
41	SESSION	Metric problems in two- and three-dimensional Euclidean affine spaces.
42	SESSION	Metric problems in two- and three-dimensional Euclidean affine spaces.
43	ASSIGNMENT	Metric problems in two- and three-dimensional Euclidean affine spaces.
44	ASSIGNMENT	Exercises (metric problems in two- and three-dimensional Euclidean affine spaces).
45	EV	ASSIGNMENT Section 3 of the syllabus.	7.5
46	SESSION	Conic sections. General and reduced equations of a conic section. Associated matrix: classification.
47	ASSIGNMENT	Calculation of the geometric elements of a conic section.
48	ASSIGNMENT	Exercises (matrix associated with a conic section/calculation of geometric elements).
49	SESSION	Metric invariants of a conic.
50	SESSION	Quadrics. General and reduced equations of a quadric. Associated matrix: classification.
51	ASSIGNMENT	Metric invariants of a quadric.
52	ASSIGNMENT	Exercises (matrix associated with a quadric/calculation of geometric elements).
53	SESSION	Affine transformations and movements. Examples.
54	SESSION	Matrix representation of an affine transformation.
55	ASSIGNMENT	Rigid motions.
56	ASSIGNMENT	Exercises (movements and matrix notation).
57	SESSION	Fixed points and invariant manifolds.
58	ASSIGNMENT	Classification of rigid motions in two- and three-dimensional Euclidean affine spaces.
59	ASSIGNMENT	Exercises (fixed points and invariant manifolds).
60	EV	REPORT Course content.	15

Bibliography

Essential:

1. Juan De Burgos Román

Algebra and Geometry. Definitions, Theorems and Results

García Maroto Editores. 2010.

ISBN: 9788492976942

2. Luis Merino and Evangelina Santos

Linear Algebra using Elementary Methods

Paraninfo. 2010.

ISBN: 978-84-9732-4

Supplementary:

3. Gilbert Strang

Introduction to Linear Algebra

Wellesley Cambridge Press. 2008.

ISBN: 8175968117

4. Juan De Burgos Román

Linear Algebra. 80 Useful Problems

García Maroto Publishers. 2007.

ISBN: 9788493601805

Others:

5. Eugenio Hernández

Linear Algebra and Geometry

3rd ed. ADDISON WESLEY. 2012.

ISBN: 9788478291298

6. Jesús Rojo

Linear Algebra

McGraw-Hill. 2001.

ISBN: 8448130162

7. Jesús Rojo

Exercises and Problems in Linear Algebra

2nd ed. McGraw-Hill. 2005.

ISBN: 8448198581

8. Stanley I. Grossman and José Job Flores

Linear Algebra

McGraw-Hill. 2012.

ISBN: 978-607-15-07

C0142306

Data Structures and Algorithms I

Código: C0142306 Imprimir

Course 1: Second-term module. Compulsory. 6 credits.

Profesores

Fausto Ruiz Madrid - Coordinador

Ver biodata

Objectives

The objectives of this module are:

1) To understand the function and importance of data structures:

- To recognise the importance of selecting and designing appropriate data structures to optimise programme performance.

- To distinguish between different categories of data structures (linear, non-linear, dynamic, etc.) and their fields of application.

2) To design, implement and manipulate fundamental data structures:

- To understand and work with basic structures such as arrays, lists, stacks, queues, trees and graphs.

- Implement essential operations (insertion, deletion, traversal, search) whilst ensuring robustness and clarity in the code.

3) Analyse the computational complexity of algorithms:

- Calculate and compare the time and space complexity of different operations and algorithms.

- Use Big O notation to estimate the performance of solutions and propose improvements.

4) Apply algorithmic problem-solving methodologies:

- Employ techniques such as recursion, divide and conquer or backtracking when solving problems.

- Select the most appropriate algorithmic strategy based on the type of problem and the available resources.

5) Develop programming skills and best practices:

- Use a clear, modular and well-documented programming style.

- Carry out testing and validation to ensure the correctness and reliability of implementations.

6) Promote critical thinking and decision-making skills:

- Evaluate different approaches to the design of data structures and algorithms to determine the most efficient option.

- Justify the choice of a structure or algorithm based on functional requirements, time and space constraints, and possible use cases.

7) Foster the ability to learn independently and work as part of a team:

- Participate in collaborative project work, exchanging ideas and reviewing code constructively.

- Continue to expand knowledge of data structures and algorithms by consulting literature and external resources.

Prerequisites

It is recommended that students have (previously) taken the module ‘Fundamentals of Programming and Computing’ or another module covering similar skills and learning outcomes.

Competencies

Not applicable.

Learning Outcomes

RC3 – Apply programmes to solve mathematical and engineering problems, using the appropriate computational environment for each case. TYPE: Competencies.

RC6 – Work independently and in an organised manner to develop solutions to the various problems that may arise in the field of mathematical engineering, subject to strict time or budgetary constraints. TYPE: Competencies.

RK2 - Master the basic concepts of discrete mathematics, logic, algorithms, coding, operations research and artificial intelligence, and their application to solving engineering problems. TYPE: Knowledge or content.

Description of the content

1. Introduction to algorithm efficiency

- Basic concepts of computational complexity: Notions of Big O, Big Theta and Big Omega. Evaluation of efficiency in terms of time (operations) and space (memory).

- Case analysis: worst-case, average-case and best-case scenarios. Practical examples of simple algorithms (linear search, binary search) in Python to illustrate the concepts.

- Introduction to optimisation: Identifying bottlenecks in Python code and initial optimisation techniques.

2. Abstract Data Type (ADT)

- Definition of ADTs: Understanding data abstraction through defined operations (creation, insertion, deletion, search, etc.), regardless of the underlying implementation.

- Examples of ADTs in Python: Using classes, methods and encapsulation to create ADTs that represent common entities (e.g. complex numbers, fractions, polynomials).

3. Linear and associative ADTs

- Linear TADs: Lists, stacks and queues. Discussion of their fundamental operations, complexity and application in various engineering contexts.

- Implementation in Python: Lists, queues and stacks using `collections.deque`.

- Associative data structures: Hash tables (dictionaries in Python) and sets. Analysis of collisions, hash functions, and average and worst-case complexity.

- Practical examples: Implementation of custom structures (specific stacks and queues), performance evaluation compared to predefined Python structures.

4. Tree Data Structures

- Basic concepts: General trees, binary trees, search trees, balanced trees (AVL, Red-Black), etc.

- Fundamental operations: Insertion, deletion, traversal (in-order, pre-order, post-order), search and rebalancing.

- Implementation in Python: Representation of nodes and pointers; use of classes to encapsulate logic. Analysis of common use cases (file systems, hierarchical organisation of data).

5. Graph Data Structures

- Graph representation: Adjacency matrix and adjacency lists. Advantages and disadvantages of each approach.

- Traversals and basic algorithms: Breadth-first search (BFS) and depth-first search (DFS). Applications in networks, maps and route-finding problems.

- Introduction to more advanced algorithms: Shortest paths (Dijkstra, Floyd-Warshall), minimum spanning trees (Kruskal, Prim), depending on the scope of the course.

- Implementation in Python: Use of dictionaries and lists to represent graph structures, together with functions to perform traversals and calculations.

6. Data structures on disk

- Persistent storage: The concept of secondary storage structures (files, databases, etc.) and their impact on efficiency.

- Introduction to on-disk indexes and trees: B-tree, B+tree. Differences from main memory structures and justification for their design.

- Practical approach in Python: Use of libraries and storage formats (e.g. sqlite3, pickle) to illustrate how to handle data beyond main memory.

7. Applying data structures to problem-solving

- Integration of content: Development of small projects or case studies requiring the selection and implementation of various data structures, analysing their performance with inputs of different sizes.

- Optimisation and refactoring: Code improvement practices, profiling algorithms in Python (e.g. using the cProfile library), and justifying the changes made.

- Collaborative work: Use of version control (Git) and simple agile methodologies (Scrum, Kanban) for carrying out projects.

Training activities

AP1. Lectures.

AP2. Interactive classes.

AP3.1. Project development and problem-solving.

AP4. Independent study, case studies, problem-solving and literature review.

AP5. Tutoring.

AP6. Knowledge assessments.

Assessment system and criteria

----

REGULAR EXAMINATION PERIOD

In the ordinary assessment period, the objective assessment of the student’s learning will be carried out through continuous assessment.

To be eligible for continuous assessment, students must, as indicated above, attend at least 70 per cent of face-to-face sessions (both theoretical and practical).

The weighting of the continuous assessment activities is distributed as follows:

a) Practical 1 (7.5%).

Assessment criteria:

- Correct implementation of the basic data structures (lists, stacks, queues, trees) covered in class.

- Organisation and clarity of the code, including appropriate use of functions and good programming practices.

- Code documentation (docstrings and comments) explaining the logic behind basic operations (insertion, deletion, search) and the chosen data structures.

- Efficiency of operations (basic analysis of time and/or space complexity).

b) Practical 2 (7.5%).

Assessment criteria:

- Use of advanced techniques or more complex structures (balanced trees, hash tables, graphs, etc.), justifying the choice according to the requirements of the proposed problem.

- Correct organisation and modularity of the code (separation of concerns and use of appropriate design patterns).

- Validation of the solution (functionality tests, unit tests) and verification of the correct implementation of the structures.

- Cleanliness, maintainability and readability of the code.

c) Final Assignment (15%).

Assessment criteria:

- Integration of different data structures and algorithms studied throughout the course (e.g. graph search and traversal, sorting algorithms, hierarchical structures).

- Design of an efficient and scalable solution that addresses a more complex problem, applying optimisation strategies and appropriate selection of data structures and algorithms.

- Quality of the project documentation (detailed README, user guides, references to theoretical concepts).

- Presentation of results (performance tests, comparison of complexities between different approaches and evaluation of scalability with increasing input sizes).

d) Non-exemption mid-term exam (30%).

Assessment criteria:

- Understanding of the fundamentals of linear and non-linear data structures (lists, stacks, queues, trees, graphs).

- Ability to design and propose basic algorithmic solutions, justifying the choice of the appropriate data structure.

- Theoretical knowledge of the complexity of elementary operations (insertion, deletion, search) and their implications for performance.

- Problem-solving and short exercises focusing on the correct application of structures and algorithms to simple use cases.

e) Final examination covering the entire module (40%).

This is the standard-session examination, which assesses all the content covered during the semester.

Assessment criteria:

- Comprehensive mastery of the data structures and algorithms covered in the module: theory, application, optimisation and appropriate selection according to context.

- Ability to analyse computational complexity (time and space) and identify potential bottlenecks in the implementation of structures.

- Application of design patterns or best practice in solving more complex problems.

- Both conceptual and practical questions, with an emphasis on identifying and comparing different algorithmic approaches and their optimisation.

IMPORTANT: The marks for the practicals, the mid-term exam and the final exam will only be averaged if the mark for each and every one of these assessment activities is 4.0 out of 10.0 or higher.

In the event of failure to achieve continuous assessment due to unexcused attendance of less than 70 per cent, the final mark for the module in the ordinary examination period will be 40 per cent of the mark obtained in the final exam.

SUPPLEMENTARY EXAMINATION PERIOD

In the supplementary examination period, the objective assessment of the student’s learning will be based on a single examination covering the entire course, which will therefore account for 100 per cent of the final mark.

Assessment criteria:

It will consist of theoretical and practical questions covering the entire syllabus, including:

- Fundamental concepts relating to basic and advanced data structures (lists, queues, stacks, trees, graphs, hash tables).

- Algorithm design and analysis techniques (recursion, divide and conquer, greedy algorithms, etc.).

- Examples of how these structures are used in common problems (sorting, searching, pathfinding in graphs, etc.).

- Complexity analysis and justification of design decisions.

- Assessment will focus on conceptual rigour, the ability to solve complex problems and clarity in justifying proposed solutions.

- A minimum mark of 5 out of 10 is required to pass the module.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	SESSION	Topic 1: Introduction to Parallel and Distributed Programming
2	SESSION	Topic 1: Introduction to Parallel and Distributed Programming
3	ASSIGNMENT	Topic 1: Introduction to Parallel and Distributed Programming
4	ASSIGNMENT	Topic 1: Introduction to Parallel and Distributed Programming
5	SESSION	Topic 2: Scientific Computing: Requirements
6	SESSION	Topic 2: Scientific Computing: Requirements
7	ASSIGNMENT	Topic 2: Scientific Computing: Requirements
8	ASSIGNMENT	Topic 2: Scientific Computing: Requirements
9	SESSION	Topic 2: Scientific Computing: Requirements
10	SESSION	Topic 2: Scientific Computing: Requirements
11	ASSIGNMENT	Topic 3: Distributed Programming
12	ASSIGNMENT	Topic 3: Distributed Programming
13	SESSION	Topic 3: Distributed Programming
14	SESSION	Topic 3: Distributed Programming
15	ASSIGNMENT	Topic 3: Distributed Programming
16	ASSIGNMENT	Topic 3: Distributed Programming
17	SESSION	Topic 4: High-Performance I/O
18	SESSION	Topic 4: High-Performance I/O
19	ASSIGNMENT	Topic 4: High-Performance I/O
20	ASSIGNMENT	Topic 4: High-Performance I/O
21	SESSION	Topic 4: High-Performance I/O
22	SESSION	Topic 4: High-Performance I/O
23	ASSIGNMENT	Topic 4: High-Performance I/O
24	ASSIGNMENT	Topic 4: High-Performance I/O
25	SESSION	Topic 5: Parallel Programming
26	SESSION	Topic 5: Parallel Programming
27	ASSIGNMENT	Topic 5: Parallel Programming
28	ASSIGNMENT	Topic 5: Parallel Programming
29	SESSION	Topic 5: Parallel Programming
30	SESSION	Topic 5: Parallel Programming
31	EV	Practical 1
32	EV	Practical 1	7.5%
33	SESSION	Topic 5: Parallel Programming
34	SESSION	Topic 5: Parallel Programming
35	ASSIGNMENT	Topic 5: Parallel Programming
36	ASSIGNMENT	Topic 5: Parallel Programming
37	EV	Mid-term exam.
38	EV	Mid-term exam.	30%
39	ASSIGNMENT	Topic 6: Model-based/Paradigm-based Parallel Programming Environments
40	ASSIGNMENT	Topic 6: Model- and Paradigm-Based Parallel Programming Environments
41	SESSION	Topic 6: Model- and Paradigm-Based Parallel Programming Environments
42	SESSION	Topic 6: Model- and Paradigm-Based Parallel Programming Environments
43	ASSIGNMENT	Topic 6: Model- and Paradigm-Based Parallel Programming Environments
44	ASSIGNMENT	Topic 6: Model- and Paradigm-Based Parallel Programming Environments
45	SESSION	Topic 6: Model- and Paradigm-Based Parallel Programming Environments
46	SESSION	Topic 6: Model- and Paradigm-Based Parallel Programming Environments
47	ASSIGNMENT	Topic 7: Accelerators
48	ASSIGNMENT	Topic 7: Accelerators
49	SESSION	Topic 7: Accelerators
50	SESSION	Topic 7: Accelerators
51	EV	Practical 2
52	EV	Practical 2	7.5%
53	SESSION	Topic 7: Accelerators
54	SESSION	Topic 7: Accelerators
55	ASSIGNMENT	Topic 7: Accelerators
56	SESSION	Topic 7: Accelerators
57	EV	Final practical
58	EV	Final practical	15%
59	EV	Final exam.
60	EV	Final exam.	40%

Reading list

Core:

1. Michael T. Goodrich, Roberto Tamassia, Michael H. Goldwasser

Data Structures and Algorithms in Python

Wiley. 2013.

ISBN: 978-1-118-293

2. Walter Bel

Algorithms and Data Structures in Python

UADER Publishing. 2020.

ISBN: 978-950-9581-

Supplementary:

3.- Mariona Nadal

Data Structures and Algorithms

Anaya Multimedia. 2022.

ISBN: 978-84-415-45

Others:

4. Kent D. Lee and Steve Hubbard

Data Structures and Algorithms with Python

Springer. 2015.

ISBN: 978-331913071

C0142307

Physical Fundamentals of Engineering

Código: C0142307 Imprimir

Course 1. Second-term module. Foundation course. 6 credits.

Profesores

Yanela Mendez González - Coordinador

Ver biodata

Objectives

The design of electrical circuits, and in particular logic circuits, which enable the processing, storage and transmission of information, is key to both classical and quantum computing. Such design is based not only on the practical applications of electromagnetism, but also, and above all, on the use of semiconductor devices – the technological implementation of solid-state physics, a branch of condensed matter physics which, in turn, on other branches of physics such as quantum mechanics.

Through this module, which forms part of the Physics subject within the degree programme’s Basic Training module, a detailed analysis is carried out of the physical fundamentals of computational electronics, as well as basic logic circuits, with the aim of providing students with a better understanding of computing.

Prerequisites

No prerequisites have been set for this module.

Learning Outcomes

Not applicable.

Learning outcomes

RK1 – To understand the basic concepts of electromagnetism and circuit theory for solving engineering problems. TYPE: Knowledge or content.

Description of the content

o Topic 1: Introduction.

o Topic 2: Electrostatic field.

o Topic 3: Magnetostatic field.

o Topic 4: Electromagnetic induction; basic direct current and alternating current circuits.

o Topic 5: Semiconductor devices.

o Topic 6: Logic circuits.

o Topic 7: Fundamentals of integrated circuits.

o Topic 8: Sequential and combinational circuits.

Teaching activities

AP1. Lectures.

AP2. Interactive classes.

AP3.2 Workshops and laboratory sessions.

AP4. Independent study, case studies, problem-solving and literature review.

AP5. Tutoring.

AP6. Knowledge assessments.

Assessment system and criteria

Without prejudice to any other requirements that may be specified in the relevant course syllabus, as a general rule, failure to attend more than 70 per cent of the course’s teaching activities—which require the student’s physical or virtual presence—will result in the loss of the right to continuous assessment during the standard examination period. In this case, the examination to be held during the official period established by the University shall be the sole assessment criterion, with the corresponding weighting as set out in the course syllabus.

----

The assessment process will consist of evaluating the extent to which the student has acquired the competences associated with the module.

REGULAR EXAMINATION PERIOD

In the regular examination period, the objective assessment of the student will consist of a continuous assessment process and a final examination.

Continuous assessment will consist of the following components:

• Portfolio: an individual assignment, accounting for 10 per cent of the final mark. It will involve solving various problems throughout the term.

• Mid-term exam: this will account for 30% of the final mark. Taking this exam will not reduce or eliminate any content from the final exam. The date will be announced in good time.

As for the final exam, this is the standard exam, and it will assess all the course content. It will account for 60 per cent of the final mark.

A weighted average will only be calculated if the marks for continuous assessment and the final exam are both 4.0 or higher. The mark for continuous assessment will be calculated by weighting the portfolio and the mid-term exam. Furthermore, only the exams may be subject to re-marking."

In order for a student to benefit from continuous assessment, a minimum attendance rate of 70 per cent at scheduled class sessions (SESSION, TRAB) is required. If attendance falls below 70 per cent without a valid reason, the module must be passed by sitting a final examination in an official examination session (ordinary or supplementary). In the ordinary examination session, the final mark will be 60 per cent of the mark obtained in that examination.

The module is considered to have been passed in the ordinary sitting when the final mark is 5.0 or above.

REGULAR EXAMINATION PERIOD

In the supplementary sitting, the student’s assessment will consist of a single examination covering the entire course content, which will account for 100 % of the final mark.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	SESSION	Introduction
2	SESSION	Introduction
3	ASSIGNMENT	Electrostatic field
4	ASSIGNMENT	Electrostatic field
5	SESSION	Electrostatic field
6	SESSION	Electrostatic field
7	ASSIGNMENT	Electrostatic field
8	TRAB	Electrostatic field
9	SESSION	Electrostatic field
10	SESSION	Electrostatic field
11	ASSIGNMENT	Magnetostatic field
12	TRAB	Magnetostatic field
13	SESSION	Magnetostatic field
14	SESSION	Magnetostatic field
15	ASSIGNMENT	Magnetostatic field
16	TRAB	Magnetostatic field
17	SESSION	Magnetostatic field
18	SESSION	Magnetostatic field
19	ASSIGNMENT	Electromagnetic induction; basic direct current and alternating current circuits
20	ASSIGNMENT	Electromagnetic induction; basic direct current and alternating current circuits
21	SESSION	Electromagnetic induction; basic direct current and alternating current circuits
22	SESSION	Electromagnetic induction; basic direct current and alternating current circuits
23	ASSIGNMENT	Electromagnetic induction; basic direct current and alternating current circuits
24	ASSIGNMENT	Electromagnetic induction; basic direct current and alternating current circuits
25	SESSION	Electromagnetic induction; basic direct current and alternating current circuits
26	SESSION	Electromagnetic induction; basic direct current and alternating current circuits
27	EV	Mid-term exam (topics 1–4)
28	EV	Mid-term exam (topics 1–4)	30%
29	SESSION	Semiconductor devices
30	SESSION	Semiconductor devices
31	ASSIGNMENT	Semiconductor devices
32	TRAB	Semiconductor devices
33	SESSION	Semiconductor devices
34	SESSION	Semiconductor devices
35	ASSIGNMENT	Semiconductor devices
36	TRAB	Semiconductor devices
37	SESSION	Logic circuits
38	SESSION	Logic circuits
39	ASSIGNMENT	Logic circuits
40	ASSIGNMENT	Logic circuits
41	SESSION	Logic circuits
42	SESSION	Logic circuits
43	ASSIGNMENT	Logic circuits
44	TRAB	Logic circuits
45	SESSION	Fundamentals of integrated circuits
46	SESSION	Fundamentals of Integrated Circuits
47	ASSIGNMENT	Fundamentals of Integrated Circuits
48	ASSIGNMENT	Fundamentals of Integrated Circuits
49	SESSION	Fundamentals of Integrated Circuits
50	SESSION	Fundamentals of Integrated Circuits
51	ASSIGNMENT	Sequential and Combinational Circuits
52	ASSIGNMENT	Sequential and combinational circuits
53	SESSION	Sequential and combinational circuits
54	SESSION	Sequential and combinational circuits
55	ASSIGNMENT	Sequential and combinational circuits
56	ASSIGNMENT	Sequential and combinational circuits
57	EV	Sequential and combinational circuits
58	EV	Sequential and combinational circuits
59	EV	Portfolio presentation
60	EV	Portfolio presentation	10%

Bibliography

Core:

1. Antonio M. Criado and Fabián Frutos

Introduction to the Physical Foundations of Computer Science

Paraninfo. 1999.

ISBN: 8428326061

2. Wolfgang Bauer and Gary D. Westfall

Physics for Engineering and Science (Volume 2)

McGraw-Hill. 2011.

ISBN: 978-607-15-05

Supplementary:

3. Hugh D. Young and Roger A. Freedman (Francis Sears and Mark Zemansky)

University Physics (Volume 2)

12th ed. Addison-Wesley. 2009.

ISBN: 9780321501219

C0142308

Mathematical Foundations of Engineering II

Código: C0142308 Imprimir

Course 1. Second-term module. Foundation course. 6 credits.

Profesores

Jacinto Velasco Rebolledo - Coordinador

Ver biodata

Objectives

This module, which together with Mathematical Foundations of Engineering I forms part of the Mathematical Analysis I course, which is part of the degree programme’s Basic Training module, aims to provide the fundamentals of one-dimensional Riemann integration, as well as to explore in greater depth the study of power series and the concept of polynomial approximation of functions.

Prerequisites

Although no prerequisites have been set, it is advisable to have previously taken the course ‘Mathematical Foundations of Engineering I’ or another course covering similar skills and learning outcomes.

Competencies

Not applicable.

Learning Outcomes

RK4 – To be familiar with rigorous proofs of some classical theorems in different areas of mathematics. TYPE: Knowledge or content.

RS1 – To use mathematical language to formulate propositions in different fields and construct proofs for the problems posed. TYPE: Skills or competences.

RS3 - To propose, analyse, validate and interpret the most appropriate mathematical models and tools in real-world situations, in accordance with the objectives being pursued. TYPE: Skills or competences.

Description of the content

Topic 0. Elementary functions

Topic 1. Riemann integral.

Topic 2. Integration techniques.

Topic 4. Polynomial approximation of functions: Taylor’s theorem.

Topic 5. Series and sequences.

Learning activities

AP1. Lectures.

AP2. Interactive classes.

AP3.1 Project work and problem-solving.

AP4. Independent study, case studies, problem-solving and literature review.

AP5. Tutoring.

AP6. Knowledge assessments.

Assessment system and criteria

----

The assessment system for the REGULAR EXAMINATION PERIOD consists of the following:

- Continuous assessment (50 per cent):

+ Submission of assignments (10%).

+ Group assessments (20%).

+ Non-exemption mid-term exam (20%).

- Final exam (50%): this is the regular assessment exam, which covers the entire course.

A minimum mark of 4.0 out of 10.0 is required in this exam to be included in the average with the continuous assessment. In this case, the average is calculated even if the continuous assessment mark is a fail.

In the event of failure to achieve the continuous assessment mark due to unexcused attendance of less than 70 per cent, the final mark for the module in the ordinary examination period will be 50 per cent of the mark obtained in the final exam.

The assessment system for the EXTRAORDINARY EXAMINATION PERIOD consists of the following:

- Exam covering the entire course (100 per cent).

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	SESSION	Introduction to the course
2	SESSION	Elementary functions.
3	ASSIGNMENT	Elementary functions.
4	TRAB	Basic functions.
5	SESSION	Basic functions.
6	SESSION	Elementary functions.
7	ASSIGNMENT	The indefinite integral. Calculation of antiderivatives.
8	ASSIGNMENT	The indefinite integral. Calculation of antiderivatives.
9	SESSION	The indefinite integral. Calculation of antiderivatives.
10	SESSION	The indefinite integral. Calculation of antiderivatives.
11	ASSIGNMENT	The indefinite integral. Calculation of antiderivatives.
12	ASSIGNMENT	The indefinite integral. Calculation of antiderivatives.
13	SESSION	The indefinite integral. Calculation of antiderivatives.
14	SESSION	The indefinite integral. Calculation of antiderivatives.
15	ASSIGNMENT	The indefinite integral. Calculation of antiderivatives.
16	ASSIGNMENT	The indefinite integral. Calculation of antiderivatives.
17	SESSION	The indefinite integral. Calculation of antiderivatives.
18	SESSION	The indefinite integral. Calculation of antiderivatives.
19	ASSIGNMENT	The indefinite integral. Calculation of antiderivatives.
20	ASSIGNMENT	The definite integral: the Fundamental Theorem of Calculus.
21	SESSION	The Definite Integral: The Fundamental Theorem of Calculus.
22	SESSION	The Definite Integral: The Fundamental Theorem of Calculus.
23	ASSIGNMENT	The Definite Integral: The Fundamental Theorem of Calculus.
24	ASSIGNMENT	The Definite Integral: The Fundamental Theorem of Calculus.
25	SESSION	The Definite Integral: The Fundamental Theorem of Calculus.
26	SESSION	The Definite Integral: The Fundamental Theorem of Calculus.
27	ASSIGNMENT	Revision of exam topics
28	ASSIGNMENT	Revision of exam topics
29	EV	First Mid-Term Exam	20%
30	SESSION	Applications of integral calculus.
31	ASSIGNMENT	Assignment
32	ASSIGNMENT	Applications of integral calculus.
33	SESSION	Applications of integral calculus.
34	SESSION	Applications of integral calculus.
35	ASSIGNMENT	Applications of integral calculus.
36	ASSIGNMENT	Applications of integral calculus.
37	SESSION	Applications of integral calculus.
38	SESSION	Applications of integral calculus.
39	ASSIGNMENT	Applications of integral calculus.
40	ASSIGNMENT	Applications of integral calculus.
41	SESSION	Applications of integral calculus.
42	SESSION	Applications of integral calculus.
43	ASSIGNMENT	Applications of integral calculus.
44	ASSIGNMENT	Applications of integral calculus.
45	SESSION	Applications of integral calculus.
46	SESSION	Approximation using polynomial functions.
47	ASSIGNMENT	Approximation using polynomial functions.
48	ASSIGNMENT	Approximation using polynomial functions.
49	SESSION	Approximation using polynomial functions.
50	SESSION	Approximation using polynomial functions.
51	ASSIGNMENT	Approximation using polynomial functions.
52	ASSIGNMENT	Sequences and series of functions. Uniform convergence
53	SESSION	Sequences and series of functions. Uniform convergence
54	SESSION	Sequences and series of functions. Uniform convergence
55	ASSIGNMENT	Sequences and series of functions. Uniform convergence
56	ASSIGNMENT	Sequences and series of functions. Uniform convergence
57	SESSION	Sequences and series of functions. Uniform convergence
58	SESSION	Revision of exam topics
59	ASSIGNMENT	Revision of exam topics
60	EV	Submission of exercises and coursework	10% (exercises) and 20% (assignment)

Reading list

Core:

1. Michael Spivak

Infinitesimal Calculus

2nd ed. Reverté. 1988.

ISBN: 8429151362

Supplementary:

2.- Roland E. Larson, Robert P. Hostetler, Bruce H. Edwards

Calculus and Analytic Geometry (Volume 1)

McGraw-Hill. 2010.

ISBN: 8448122291

C0142309

Logic and Discrete Mathematics

Código: C0142309 Imprimir

Course 1: Second-term module. Compulsory. 6 credits.

Profesores

José Luis Guijarro Regalado - Coordinador

Ver biodata

Objectives

Although this course covers several distinct topics, the learning objectives are essentially the same for each topic. These are:

- To be able to prove statements rigorously using logic.

- To be able to use discrete sets such as integers, congruences modulo n or graphs.

Prerequisites

No prerequisites have been set.

Competencies

Not applicable.

Learning outcomes

RC9 – Formulate problems from a professional context in mathematical language in such a way as to facilitate their analysis and resolution. TYPE: Competencies.

RK4 – Be familiar with rigorous proofs of some classical theorems in different areas of mathematics. TYPE: Knowledge or content.

RS1 – Use mathematical language to formulate propositions in various fields and construct proofs for the problems posed. TYPE: Skills or abilities.

Description of the content

To better adapt the content to the duration of the course, it has been divided into four main topics or blocks:

1) Set theory: an introduction to formal logic, Boolean algebra and binary relations.

2) Integers: solving equations involving integers using Euclid’s algorithm and performing operations modulo n.

3) Combinatorics: variations, permutations and combinations with and without repetition.

4) Graph theory: three problems are studied: when are two graphs equal? When is a graph planar? This includes Euler’s formula and, finally, Eulerian and Hamiltonian paths are studied.

Training activities

AP1. Lectures.

AP2. Interactive classes.

AP3.1. Project work and problem-solving.

AP4. Independent study, case studies, problem-solving and literature review.

AP5. Tutoring.

AP6. Knowledge assessments.

Assessment system and criteria

----

REGULAR EXAMINATION PERIOD

Continuous assessment will consist of a test following each of the four content blocks, with each test accounting for 10 per cent of the final mark for the module in the ordinary assessment period. Under no circumstances will these tests exempt students from further study of the material.

Once the teaching period has ended, the ordinary assessment exam (final exam) will be held, accounting for the remaining 60 per cent. This exam covers the entire course. Furthermore, in order for the mark to be included in the average with the continuous assessment, a minimum mark of 4.0 out of 10.0 must be obtained in this exam. In this case, the average is calculated even if the continuous assessment is failed.

In the event of loss of continuous assessment due to unexcused attendance of less than 70 per cent, the final mark for the module in the ordinary examination period will be 60 per cent of the mark obtained in the final exam.

SUPPLEMENTARY EXAMINATION SESSION

In the supplementary examination session, the final mark for the module will be the mark obtained in the examination held during that session, which will assess all the content covered.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	SESSION	Introduction to the course.
2	SESSION	Definition of a set
3	ASSIGNMENT	Basic properties of a set
4	ASSIGNMENT	Set operations: union and intersection
5	SESSION	Morgan’s laws and related properties
6	SESSION	Examples
7	ASSIGNMENT	Proposition Logic
8	ASSIGNMENT	Proposition Algebra
9	SESSION	Binary relations
10	SESSION	Quotient set
11	ASSIGNMENT	Correspondences, functions and applications
12	ASSIGNMENT	Examples
13	SESSION	Boolean Algebras I
14	SESSION	Boolean Algebras II
15	EV	First chapter exam	10%
16	ASSIGNMENT	Introduction to whole numbers
17	SESSION	Axioms of the integers
18	SESSION	Greatest Common Divisor and Least Common Multiple
19	ASSIGNMENT	Euclid’s algorithm
20	ASSIGNMENT	Examples
21	SESSION	The Fundamental Theorem of Arithmetic
22	SESSION	Modular Calculus I
23	ASSIGNMENT	Modular Calculus II
24	ASSIGNMENT	Examples
25	SESSION	Fermat’s Little Theorem
26	SESSION	The Chinese Remainder Theorem
27	ASSIGNMENT	Modular Calculus III
28	ASSIGNMENT	Modular Calculus IV
29	EV	First Chapter Exam	10%
30	SESSION	Introduction to Combinatorics
31	ASSIGNMENT	Variations
32	ASSIGNMENT	Permutations
33	SESSION	Exercises
34	SESSION	Combinations
35	TRAB	With repetition
36	TRAB	Combinatorial numbers
37	SESSION	Exercises and counting formulas
38	SESSION	Combinatorics with symmetries I
39	ASSIGNMENT	Combinatorics with Symmetries II
40	ASSIGNMENT	Combinatorics with Symmetries III
41	SESSION	Exercises
42	SESSION	The beginning of the dovecote
43	ASSIGNMENT	Generating functions
44	TRAB	Applications
45	EV	Chapter 3 Test	10%
46	SESSION	Introduction to graphs and trees
47	ASSIGNMENT	Examples
48	ASSIGNMENT	Roads, connectivity and distance
49	SESSION	Operations between graphs
50	SESSION	Adjacency matrix
51	ASSIGNMENT	Examples
52	TRAB	Euler’s formula
53	SESSION	Eulerian paths
54	SESSION	Exercises
55	ASSIGNMENT	Hamiltonian paths
56	ASSIGNMENT	Circuits in general, graph topology
57	SESSION	Integration of Chapter 4
58	EV	Review of Chapter 4	10%
59	EV	Final exam
60	EV	Final exam	60%

Reading list

Core:

1. José Dorronsoro, Eugenio Hernández

Numbers, Groups and Rings

Addison-Wesley / UAM. 1996.

ISBN: 0201653958

C0142608

Fundamentals of Chemistry

Código: C0142608 Imprimir

Course 1. Second-term module. Foundation course. 6 credits.

Profesores

Tomás Pedregal Garrido - Coordinador

Ver biodata

Objectives

The objectives set out for the ‘Fundamentals of Chemistry’ module within the Physics degree programme uphold effective equality between women and men, as established by Organic Law 3/2207 of 22 March, the principles of equal opportunities, non-discrimination and universal accessibility for people with disabilities, as set out in Act 51/2203 of 2 December, and promote education for peace, non-violence and human rights, as set out in Act 27/2005 of 30 November.

The specific objectives proposed for the ‘Fundamentals of Chemistry’ module are as follows:

To provide training in the sciences, with a particular focus on chemistry, enabling students to undertake the study of technological subjects.

To ensure the acquisition of cross-curricular competences and skills that enable and enhance the application of the knowledge acquired.

To foster innovation and the dissemination of scientific findings.

To foster a commitment to ethical standards in both professional and social contexts.

Prerequisites

No prerequisites.

Competencies

RK2 Understand physically distinct phenomena and their underlying analogies in order to apply known solutions to new problems

RS2 Carry out calculations, assessments, studies, reports and tasks to produce high-quality work in the field of Physics.

RS3 Estimate orders of magnitude to interpret laboratory phenomena in the field of Physics and its sub-disciplines, as well as in Chemistry.

Learning outcomes

RA1 List, at a basic level, the principles that explain the physico-chemical properties of matter.

RA2 Formulate and name simple inorganic compounds.

RA3 Describe the main mechanisms involved in a chemical reaction, identify different types of reaction, determine the quantities of reactants, and use thermodynamic potentials to characterise the reaction energetically.

LA4 Determine the mechanisms responsible for chemical equilibrium and explain how the parameters on which it depends act.

RA5 Determines the acidity of a solution.

RA6 Identifies some of the main organic functional groups and determines their most important chemical reactions.

RA7 Analyses, evaluates and interprets the results obtained when solving problems.

Description of the content

- Atoms and chemical bonding. Inorganic formulae and nomenclature. Intermolecular forces.

- States of matter and phase diagrams. Fundamentals of thermochemistry. Mixtures. Solutions.

- Chemical reactions: mechanisms and rates; stoichiometry.

- Chemical equilibrium: constants; Le Chatelier’s principle. Solubility equilibrium. Acids and bases.

- Oxidation-reduction reactions and electrochemistry. Introduction to organic chemistry.

Teaching activities

AP1.- Participatory lectures

AP2.- Seminars or practical application classes

AP3. Practical activities (case studies, project work, simulations, etc.)

AP4. – Independent study

AP5. Tutoring

AP6. Knowledge assessments

AP10.- Workshop and/or laboratory activities

Assessment system and criteria

REGULAR EXAM SESSION

Both parts, Chemistry and Physics: Same assessment system. The weighted average of both parts must be 5 (with a minimum mark of 4 in each part to be included in the average) to pass.

IT WILL NOT BE POSSIBLE TO PASS THE COURSE BASED ON MID-TERM EXAMS ALONE.

A) CONTINUOUS ASSESSMENT with group assignments

- 15% practicals. Minimum mark of 5.0 to pass.

- 10% Mid-term 1, treated as a group assignment

- 5% class activities (group assignments on the virtual campus)

- 5% inorganic chemistry problem-solving exam

- 5% organic chemistry problem-solving exam

B) FINAL EXAM - Two parts (Chemistry and Physics)

- Students must sit the standard examination for both the Chemistry and Physics sections.

- If a student has failed the laboratory examination for either the inorganic or organic chemistry modules, they may also retake it as part of this final examination.

- The mark obtained in the course activities will be retained for the calculation of the final mark.

SUPPLEMENTARY EXAMINATION SESSION

In the final exam held as part of the supplementary sitting, the entire course (Chemistry, Physics and Practical Work) will be assessed, and the mark for this exam will account for 100 per cent of the final mark for the course.

C0142609

Basic experimental techniques

Código: C0142609 Imprimir

Course 1. Second-term module. Foundation course. 6 credits.

Profesores

Yanela Mendez González - Coordinador

Ver biodata

José Luis Guijarro Regalado

Ver biodata

Objectives

To carry out laboratory measurements in accordance with established protocols, involving the calibration of instruments, the estimation of systematic and random uncertainties and the identification of strategies for their elimination, the collection of data and its mathematical analysis.

To establish measurement protocols, particularly those relating to the safety of the experimenter.

To prepare reports on measurement procedures, the analysis of results and the conclusions drawn.

Prerequisites

No prerequisites.

Competencies

RK1 Understand the most important phenomena and theories in the various branches of physics, as well as their historical context

RK3 Analyse the fundamental concepts and principles of physical systems in order to develop approximations that enable the construction of a simplified model

RS3 Estimate orders of magnitude to interpret laboratory phenomena in the field of physics and its related disciplines, as well as in chemistry.

RS4 Apply mathematical and numerical methods to the modelling and explicit solution of problems in physics and related disciplines, selecting the appropriate tools and interpreting results.

RS5 Use appropriate electronic instruments and/or computer tools in modelling to find solutions to physics problems.

RC1 Work independently on the management of projects related to the different areas of physics

Learning outcomes

RA9 Carries out laboratory measurements in accordance with established protocols, including the calibration of instruments, the estimation of systematic and random uncertainties (identifying strategies for their elimination), the collection of data and the mathematical analysis thereof.

LA10 Follow measurement protocols, particularly those relating to the safety of the experimenter.

RA11 Prepares reports on measurement procedures, the analysis of results and the conclusions drawn.

Description of the content

- The nature of physical phenomena and their measurement.

- Processing of experimental data and calculation of errors.

- General physics laboratory practicals related to Fundamentals of Physics I and Fundamentals of Physics II

Teaching activities

AP1.- Participatory lectures

AP2. Seminars or practical application classes

AP3. Practical activities (case studies, project work, simulations, etc.)

AP4. – Independent study

AP5. Tutoring

AP6. Assessment tests

AP10.- Workshop and/or laboratory activities

Assessment system and criteria

Regular assessment period

VERY IMPORTANT: in order for students to benefit from continuous assessment during the ordinary assessment period, they must attend at least 70 per cent of the scheduled class hours (SESSION, LAB).

Assessment system Weighting

SE1.- Practical activities (case studies, problem-solving and challenges, project work, oral presentations, debates, etc.)  10

SE2. Final knowledge assessments (with a minimum mark of 4/10 to be included in the average with the other assessment components) 50

SE3. – Laboratory practical logbook (students must submit the provided reports on the practicals carried out for assessment) 40

Extraordinary examination session

The final mark for the module will correspond to 100 per cent of the mark obtained in this examination

TOTAL:

Second Year

FIRST FOUR-MONTH PERIOD

Code

Subjects

Character*

ECTS

C0242300

Differential calculus

Código: C0242300 Imprimir

Year 2, Course 2. First term. Foundation module. 6 credits.

Profesores

Hugo Andrea Galindo Beleña - Coordinador

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Objectives

This module, which together with Integral Calculus forms part of Mathematical Analysis II, a module within the degree programme’s Foundation Stage, aims not only to ensure that students are familiar with the main theorems relating to the differential calculus of functions of several variables, but also to enable them to understand differential calculus from a conceptual perspective and to apply it to solving engineering problems.

Prerequisites

None have been specified, although it is strongly recommended that students have previously taken the modules ‘Mathematical Foundations of Engineering I and II’ or other modules covering similar skills and learning outcomes.

Competencies

Not applicable.

Learning Outcomes

RC9 – Formulate problems from a professional context in mathematical language in such a way as to facilitate their analysis and solution. TYPE: Competencies.

RK4 – Be familiar with rigorous proofs of some classical theorems in different areas of mathematics. TYPE: Knowledge or content.

RS1 – Use mathematical language to formulate propositions in different fields and construct proofs for the problems posed. TYPE: Skills or abilities.

Description of the content

The module covers the following topics:

Topic 1. Topological concepts of ℝ^n.

Topic 2. Limits and continuity of functions of several variables.

Topic 3. Directional and partial derivatives. Differentiability.

Topic 4. Higher-order derivatives.

Topic 5. Extremes of functions of several variables.

Topic 6. The inverse and implicit function theorems.

Topic 7. Conditional extrema.

Teaching activities

AP1. Lectures.

AP2. Interactive classes.

AP3.1 Project work and problem-solving.

AP4. Independent study, case studies, problem-solving and literature review.

AP5. Tutoring.

AP6. Knowledge assessments.

Assessment system and criteria

The assessment system for the REGULAR EXAMINATION PERIOD consists of continuous assessment, which accounts for 50 per cent, and a final examination, which accounts for 50 per cent.

- Continuous assessment (50%)

+ Mid-term exam (30%).

+ Group assessments (20%).

- Final exam (50%).

A minimum mark of 4 is required to be included in the average with the continuous assessment.

If the minimum mark is not achieved, the mark obtained in the final exam will be awarded.

In the EXTRAORDINARY EXAMINATION SESSION, the final exam will account for 100% of the mark.

GRADING

The award of the corresponding credits will be contingent upon passing the associated examinations or assessment tests.

The level of learning achieved by students will be expressed as numerical marks on a scale of 0 to 10, to one decimal place, to which the corresponding qualitative grade may be added:

- 0–4.9: Fail (SS).

- 5.0–6.9: Pass (AP).

- 7.0–8.9: Good (NT).

- 9.0–10: Distinction (SB).

The distinction of ‘Honours’ shall be awarded to students who have obtained a mark of 9.0 or higher. The number of students awarded this distinction may not exceed five per cent of those enrolled on the course in the relevant academic year, unless the number of enrolled students is fewer than 20, in which case only one ‘First Class Honours’ may be awarded.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	MG	Course introduction.
2	MG	Introduction to the course.
3	SM	Metric spaces. Topological concepts.
4	SM	Classification of points.
5	MG	Standard spaces
6	SM	Exercise solutions I
7	SM	Exercise solutions II
8	MG	Equivalent rules, Rules in R-n
9	MG	Metric induced by a norm. Euclidean space, associated norm and associated metric.
10	SM	Exercise solutions III
11	SM	Exercise solutions IV
12	MG	Limits and continuity of functions of several variables: Sequences and limits.
13	MG	Limits of functions. Properties
14	SM	Solutions to Exercises V
15	SM	Exercise solutions VI
16	MG	Iterated and curve-based limits.
17	MG	Limits using polar coordinates
18	SM	Exercise solutions VII
19	SM	Exercise solutions VIII
20	MG	Continuity. Properties of continuous functions
21	MG	Differentiability.
22	SM	Solutions to Exercises IX
23	EV	First set of exercises	7.5%
24	EV	First module exam
25	EV	First mid-term exam	15%
26	MG	Directional and partial derivatives
27	MG	Linear application of the differential.
28	SM	Exercise solutions XI
29	SM	Answers to Exercises XII
30	MG	Necessary and sufficient conditions for differentiability.
31	MG	Properties of differentiable functions.
32	MG	Lagrange operator; Chain rule
33	SM	Solutions to Exercises XIII
34	SM	Answers to Exercises XIV
35	MG	Higher-order derivatives.
36	SM	Solutions to Exercises XV
37	SM	Answers to Exercises XVI
38	SM	Answers to Exercises XVII
39	MG	Swarzt’s lemma
40	MG	Hessian matrix
41	SM	Solutions to Exercises XVIII
42	SM	Answers to Exercises XIX
43	SM	Answers to exercises from previous examination sessions
44	MG	Taylor series of vector functions I
45	MG	Taylor series of vector functions II
46	MG	Extremes of functions of several variables. Relative extremes
47	SM	Exercise solutions XX
48	SM	Answers to Exercises XXI
49	SM	Answers to exercises from previous examination sessions
50	MG	Constrained problems: Substitution method and Lagrange multipliers
51	MG	Necessary condition for the existence of a conditional extremum and sufficient condition for the existence of a conditional extremum
52	MG	Inverse Function Theorem I
53	MG	Inverse Function Theorem II
54	MG	Implicit Function Theorem I
55	MG	Implicit Function Theorem II
56	SM	Solutions to Exercises XXII
57	EV	Second set of exercises	7.5%
58	EV	Submission of assignment	15%
59	EV	Second mid-term exam
60	EV	Second mid-term exam	15%

Reading list

Core:

1. Jerrold E. Marsden

Elementary Classical Analysis

W. H. Freeman and Company. 1974.

ISBN: 0716721058

Supplementary:

2. Jerrold E. Marsden, Anthony J. Tromba

Vector Calculus

3rd ed. Addison-Wesley Iberoamericana. 1991.

ISBN: 0201629356

Other:

3. James R. Munkres

Analysis on Manifolds

Addison-Wesley. 1991.

ISBN: 0201315963

4. Michael Spivak

Calculus on Manifolds

Addison-Wesley. 1971.

ISBN: 9780805390216

C0242301

Differential Equations and Difference Equations

Código: C0242301 Imprimir

Year 2, Course 2. First term. Compulsory. 6 credits.

Profesores

Yanela Mendez González - Coordinador

Ver biodata

Objectives

This module aims to contribute to the development of students’ skills and, in particular, to familiarise them with the various techniques for the analytical solution of the most important ordinary differential equations and difference equations.

Prerequisites

No prerequisites have been set for this module. However, it is strongly recommended that students have completed or are currently taking modules on calculus involving one and several real variables.

Competencies

Not applicable.

Learning Outcomes

RC2 – Identify the different stages of the mathematical modelling process, distinguishing between formulation, analysis, solution and interpretation of results. TYPE: Competencies.

RC3 – Apply software to solve mathematical and engineering problems, using the appropriate computational environment for each case. TYPE: Competencies.

RC7 – Carry out work and projects related to mathematical engineering that have an impact on interdisciplinary teams or in multicultural contexts. TYPE: Competencies.

RC9 – Formulate problems from a professional context in mathematical language in such a way as to facilitate their analysis and resolution. TYPE: Competences.

RK4 – Be familiar with rigorous proofs of certain classical theorems in different areas of mathematics. TYPE: Knowledge or content.

RS1 – Use mathematical language to state propositions in different fields and construct proofs for the problems posed. TYPE: Skills or abilities.

RS5 – Use computer applications for statistical analysis, numerical and symbolic calculation, graphical visualisation, optimisation or other purposes to solve problems. TYPE: Skills or competences.

Description of the content

- Methods for solving Ordinary Differential Equations (ODEs): Basic concepts and classification of ODEs. Equations with separable variables, exact equations and the integrating factor.

- First-order differential equations and higher-order linear equations.

- Systems of first-order differential equations: homogeneous and non-homogeneous linear systems.

- Non-linear differential equations and systems. Stability theory.

- Laplace transform and its applications to differential equations.

Teaching activities

AP1. Lectures.

AP2. Interactive classes.

AP3.1. Project work and problem-solving.

AP4. Independent study, case studies, problem-solving and literature review.

AP5. Tutoring.

AP6. Knowledge assessments.

Assessment system and criteria

SE1. Practical activities – 20%

• This comprises the daily assignment of questions and problems not solved in class by the lecturer.

AS2. Final knowledge assessments – 50%

• Final written examination: the entire course content is assessed

SE3. – Portfolio – 30%

• Mid-term exam set in advance (20 per cent)

• Assignments set by the lecturer (10%)

Assessment criteria:

Assessment of the module will be carried out on a continuous and final basis, through a combination of practical activities, knowledge tests and the compilation of a portfolio. The aim is thus to assess both the acquisition of theoretical skills and the practical application of the course content, as well as the student’s individual progress throughout the course.

SE1 Practical activities

Assessment of the level of achievement of the learning outcomes (in terms of knowledge, skills and attitude) attained through the completion of individual and group practical activities (case studies, problem-solving and challenges, project work, oral presentations, debates, etc.) based on assessment rubrics and observation tools previously designed and published by the lecturer.

SE3 Portfolio

Evidence of the student’s learning (in terms of knowledge, performance or output) collected in a dossier, folder or portfolio throughout the course is assessed with the aim of explaining and reflecting on the learning that has taken place.

The weighting we assign to the portfolio will include continuous assessment tests (those taken during term time).

SE2 Final knowledge assessments

Written examinations designed to assess students’ understanding of fundamental theoretical concepts and their ability to apply methods for solving differential equations in different contexts. Mid-term exams will not count towards the final mark; therefore, the final exam will cover the entirety of the course content. The exam must, without exception, be taken on the date set by the University for the regular examination period (virtual campus).

The final mark for the module will be calculated by weighting the three assessment methods described above.

+++REGULAR EXAMINATION PERIOD+++

The calculation of the final mark for the ordinary examination period is conditional upon achieving a minimum mark of 4.0 out of 10 in the final exam. Provided this requirement is met, a weighted average will be calculated from all assessment activities and tests, even if any of them have been failed.

If the mark obtained in the final exam is less than 4.0 out of 10, the final mark will be the mark obtained in that exam.

+++EXTRAORDINARY EXAMINATION PERIOD+++

If a student has not passed the module in the ordinary examination period, they may sit the extraordinary examination period. The extraordinary examination period will take place during the June and July examination periods (for further information, please consult the Academic Calendar).

100 per cent of the final mark for the module in the Extraordinary Examination Session corresponds to the Final Examination for this session, which assesses the entire syllabus of the module.

The module is considered to have been passed in the extraordinary examination period if the final mark is 5.0 or higher.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	MG	Course introduction.
2	SM	Separation of Variables method, linear and homogeneous equations
3	MG	Basic concepts of the theory of linear differential equations
4	MG	Introduction to Hilbert spaces and elementary topology
5	MG	Basic functional analysis
6	SM	Exact equations, the integrating factor and other classifications of non-linear differential equations
7	MG	The Cauchy problem and homogeneous equations
8	MG	Cauchy’s problem and homogeneous equations
9	MG	Solving linear equations: the discriminant
10	SM	Cauchy’s problem
11	MG	The exponential of a matrix and eigenvalue analysis: The logistic equation and the fundamental theorem of separable variables
12	MG	Introduction to the Cauchy–Lipschitz theory
13	MG	Picar-Lindelöf Theorem
14	EV	Submission of Proposed Problems	10%
15	MG	Fundamental theorems of Cauchy–Lipschitz theory
16	MG	Equations of order N greater than 1: fundamental theorem
17	MG	Fundamental theorem and examples
18	SM	Problems
19	MG	Gronwall’s lemma and the Lipschitz dependence theorem
20	MG	Consequences of the theory and examples
21	MG	Peano’s Differentiability Theorem
22	SM	Problems
23	MG	Introduction to Cauchy–Peano theory
24	MG	Ascoli’s Theorem
25	MG	Global Cauchy–Peano theorem
26	SM	Problems
27	MG	Example of uniqueness in the Cauchy–Peano framework
28	MG	Fundamental theorems of Cauchy–Peano theory
29	MG	Fundamental Theorems of Cauchy–Peano Theory
30	SM	Problems: Müller’s counterexample
31	MG	Picard’s iterated functions
32	MG	Kamke’s theory
33	MG	Applications
34	EV	Mid-term exam	20%
35	MG	Kneser’s theorem
36	MG	Kneser’s theorem
37	MG	Approximation of continuous functions by differentiable functions
38	SM	Problems
39	MG	Introduction to Qualitative Theory
40	MG	Types of stable points
41	MG	Phase diagram
42	EV	Submission of Proposed Problems	10%
43	MG	Jordan’s Theorem
44	MG	Consequences of Jordan’s theorem
45	MG	Examples and applications
46	SM	Problems
47	MG	Autonomous systems: trajectories
48	MG	Autonomous systems
49	MG	Autonomous systems
50	SM	Problems
51	MG	Eigenvalue analysis
52	MG	Eigenvalue analysis
53	MG	Address field
54	MG	Newtonian mechanics
55	MG	Fourier–Fick’s Law
56	MG	Phase diagram
57	MG	Advanced methods in qualitative theory
58	MG	Poincaré–Bendixon Theorem
59	EV	Final exam
60	EV	Final exam	60%

Reading list

Core:

1. Dennis G. Zill

Differential Equations with Modelling Applications

Cengage Learning. 2019.

ISBN: 6075266313

2. Frank Ayres, Jr.

Differential Equations

McGraw-Hill. 1996.

ISBN: 970 10 0004 8

3. M. Braun

Differential Equations and their Applications

Grupo Editorial Iberoamerica. 1983.

ISBN: 968 7270 58 6

C0242302

Applied Statistics

Código: C0242302 Imprimir

Year 2, Course 2. First term. Compulsory. 6 credits.

Profesores

Wolfram Rozas Rodriguez - Coordinador

Ver biodata

Wolfram is a quantitative economist with a PhD in industrial technologies from the UNED’s School of Industrial Engineering. His thesis, Artificial Intelligence: Predictive Models for Local Renewable Energy Markets, was awarded the grade ‘outstanding cum laude’ in September 2024. He is an expert in the use of all exponential technologies, including Big Data and Business Analytics, Artificial Intelligence, Cloud Computing, the Internet of Things, Blockchain and Quantum Computing, to achieve strategic business objectives. He has over 25 years’ experience at companies such as PwC and IBM, managing projects in Business Intelligence, Business Analytics and Machine Learning, Deep Learning, and implementing Cognitive Systems in the industrial sectors of Finance, Telecommunications, Retail, Tourism and Transport, Consumer Goods, Energy and Utilities, Chemicals and Oil. He has created business solutions across all these sectors to improve business processes such as marketing and commerce, finance and risk, supply chain, operations and human resources. He is currently a data and AI consultant, carrying out business development work on big data and AI projects.

He has been teaching since 2003. He has contributed as an expert in Business Intelligence, Big Data & Business Analytics and Artificial Intelligence (encompassing machine learning and deep learning) at business schools and universities such as CUNEF, the School of Industrial Organisation, Instituto de Empresa, Pablo de Olavide University, Carlos III University, UNIR, the University of Salamanca and the University of Navarra, amongst others. He has published numerous articles in the field of data analysis. Pablo de Olavide, U. Carlos III, UNIR, U. Salamanca and U. Navarra, amongst others. He has published numerous articles in the field of data analysis. He has also published scientific articles on the application of AI to the field of renewable energy in journals such as Elsevier, Energies and Sustainability.

Objectives

The course aims to provide students with the statistical foundations and tools necessary for the design and analysis of experiments, the modelling of relationships between variables using regression techniques, and the performance of multivariate statistical inference. It also aims to develop students’ ability to apply statistical methods to the analysis and control of processes, as well as to the study of time series, identifying their main components, trends and seasonal variations.

The aim is for students to be able to select and apply appropriate statistical techniques to various data analysis problems, interpret the results obtained and use statistical models for estimation and prediction, thereby contributing to the resolution of problems typical of Mathematical Engineering.

-------------------------

The course aims to provide students with the statistical foundations and tools required for the design and analysis of experiments, the modelling of relationships between variables using regression techniques, and the performance of multivariate statistical inference. It also aims to develop students’ ability to apply statistical methods to process analysis and control, as well as to the study of time series, identifying their main components, trends and seasonal variations.

Students are expected to be able to select and apply appropriate statistical techniques to different data analysis problems, interpret the results obtained, and use statistical models for estimation and prediction, thereby contributing to the solution of problems arising in the field of Mathematical Engineering.

Prerequisites

No prerequisites have been established; however, it is essential that students have previously completed the course Statistical Analysis, or another course with similar competences and learning outcomes.

Competencies

Not applicable.

Learning Outcomes

RC2 – To identify the different phases of the mathematical modelling process, distinguishing between formulation, analysis, solution and interpretation of results. TYPE: Competences.

RC3 – To apply software programmes that solve mathematical and engineering problems, using the appropriate computational environment for each case. TYPE: Competences.

RC4 – To be able to plan and develop projects in the field of Mathematical Engineering. TYPE: Competences.

RC6 – To work independently and in an organised manner to develop solutions to the various problems that may arise in the field of mathematical engineering, subject to strict time or cost constraints./ to work independently and in an organised manner to develop solutions to the various problems that may arise in the field of mathematical engineering, subject to strict time or financial constraints. TYPE: Competences./TYPE: Competences.

RC9 – To formulate problems arising in a professional context using mathematical language, in a way that facilitates their analysis and resolution./ To formulate problems in a professional environment using mathematical language, in a way that facilitates their analysis and resolution. TYPE: Competences./TYPE: Competences.

RK3 – To understand the concepts of statistics and statistical inference in large datasets. TYPE: Knowledge or content.

RK4 – To understand rigorous proofs of some classical theorems in various areas of mathematics. TYPE: Knowledge or content.

RS1 – To use mathematical language to formulate propositions in various fields and to construct proofs for the given problems. TYPE: Skills.

RS3 – To propose, analyse, validate and interpret the most appropriate mathematical models and tools in real-world situations, in accordance with the intended objectives. TYPE: Skills.

RS4 – To plan the resolution of a problem in accordance with the available tools and the constraints of time and resources. TYPE: Skills.

RS5 – To use computer applications for statistical analysis, numerical and symbolic calculation, graphical visualisation, optimisation or other purposes to solve problems. TYPE: Skills.

Description of the content

Experimental designs.

• Experimental designs.

• Experimental strategy.

• Randomised single-factor experiment.

• Analysis of variance.

• Sample size determination for a randomised single-factor experiment.

Regression techniques.

• Linear regression

• Non-linear regression

• Multiple linear regression

Multivariate inferential analysis and multivariate techniques.

• Multidimensional distributions.

• Properties of estimators.

• Maximum likelihood estimation.

• Obtaining an estimator for a distribution.

• Obtaining MV estimators for µ.

• Deriving the ML estimators for θ.

• Deriving ML estimators for σ.

• Multivariate distributions.

• Multivariate inferential analysis.

• Multivariate techniques.

Process control: quality analysis.

• Statistical quality control.

• Statistical process control.

Time series. Basic models.

• Representation.

• Classification of characteristic trends in time series.

• Trend estimation.

• Estimation of seasonal variations.

• Forecasting.

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Experimental Design.

• Experimental design.

• Experimental strategies.

• Single-factor randomised experiments.

• Analysis of variance.

• Determining sample size for single-factor randomised experiments.

Regression techniques.

• Linear regression.

• Non-linear regression.

• Multiple linear regression.

Multivariate Statistical Inference and Multivariate Techniques.

• Multivariate distributions.

• Properties of estimators.

• Maximum likelihood estimation.

• Obtaining an estimator for a distribution

• Obtaining the maximum likelihood estimators for μ

• Obtaining the maximum likelihood estimators for θ

• Obtaining the maximum likelihood estimators for σ

• Multivariate distributions

• Multivariate statistical inference

• Multivariate techniques

Process Control: Quality Analysis.

• Statistical quality control.

• Statistical process control.

Time Series: Basic Models.

• Representation.

• Classification of characteristic trends in time series.

• Trend estimation.

• Estimation of seasonal variations.

• Forecasting.

Teaching activities

AP1. Lectures.

AP2. Interactive classes.

AP3.1. Project development and challenge-based problem solving.

AP4. Independent study, case and problem solving, and literature search.

AP5. Tutorials.

AP6. Knowledge assessment tests.

Assessment system and criteria

REGULAR EXAMINATION PERIOD

Continuous assessment:

- (15% of the mark) First mid-term exam.

- (15% of the mark) Second mid-term exam.

- (30%) Active participation. Completion of exercises.

- (40%) Final exam.

The calculation of the final mark in the standard assessment period will be subject to achieving a minimum mark of 4.0 out of 10 in the final exam. Once this requirement has been met, a weighted average will be calculated for all assessment activities and tests, even if any of them have been failed.

If the mark obtained in the final exam is less than 4.0 out of 10, the final mark will be the mark obtained in that exam.

EXTRAORDINARY EXAMINATION SESSION

In this case, the mark for the module will be that obtained in the supplementary examination.

-------------------------

ORDINARY EXAMINATION PERIOD

Continuous assessment:

- 15% of the final mark: First mid-term examination.

- 15% of the final mark: Second mid-term examination.

- 30% of the final mark: Active participation and problem-solving activities.

- 40 per cent of the final mark: Final examination.

The calculation of the final mark during the standard examination period is conditional upon achieving a minimum mark of 4.0 out of 10 in the final examination. Once this requirement has been met, the weighted average of all assessment activities and tests shall be calculated, even if any of them have been failed.

If the mark obtained in the final examination is below 4.0 out of 10, the final mark shall be the mark obtained in that examination.

EXTRAORDINARY EXAMINATION PERIOD

In this case, the final mark for the course will be the mark obtained in the extraordinary examination.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	SESSION	Introduction to the course.
2	ASSIGNMENT	Review of fundamental concepts in statistical analysis
3	SESSION	Statistical inference
4	ASSIGNMENT	Statistical inference
5	SESSION	Inference Activity
6	ASSIGNMENT	Inference Activity
7	SESSION	Experimental Design
8	ASSIGNMENT	Experimental Design
9	SESSION	Experimental Strategy
10	ASSIGNMENT	Randomised single-factor experiment
11	SESSION	Analysis of variance
12	ASSIGNMENT	Analysis of variance
13	SESSION	Determining sample size for a single-factor randomised experiment
14	ASSIGNMENT	Determining the sample size for a single-factor randomised experiment
15	SESSION	Experimental design activity
16	ASSIGNMENT	Experimental design activity
17	SESSION	Regression techniques
18	ASSIGNMENT	Properties of estimators
19	SESSION	Properties of estimators
20	ASSIGNMENT	Maximum Likelihood Estimation
21	SESSION	Obtaining an estimator for a distribution
22	ASSIGNMENT	Obtaining an estimator for a distribution
23	SESSION	Obtaining the MV estimators for µ
24	ASSIGNMENT	Obtaining the MV estimators for µ
25	SESSION	Obtaining the MV estimators for µ
26	ASSIGNMENT	Linear Regression
27	SESSION	Linear Regression
28	ASSIGNMENT	Non-linear regression
29	SESSION	Non-linear regression
30	EV	Mid-term exam 1
31	EV	Mid-term exam 1	15
32	ASSIGNMENT	Multivariate inferential analysis
33	SESSION	Multidimensional distribution
34	ASSIGNMENT	Multivariate techniques
35	SESSION	Multiple linear regression
36	ASSIGNMENT	Multiple linear regression exercise
37	SESSION	Multiple linear regression exercise
38	ASSIGNMENT	Process control: Quality analysis
39	SESSION	Statistical Quality Control
40	ASSIGNMENT	Statistical process control
41	SESSION	Quality analysis activities
42	ASSIGNMENT	Quality analysis activity
43	SESSION	Time series: Basic models
44	ASSIGNMENT	Representation
45	SESSION	Representation
46	TRAB	Classification of characteristic movements of a ST
47	SESSION	Time Series Analysis
48	ASSIGNMENT	Trend Estimation
49	SESSION	Estimation of seasonal variations
50	TRAB	Estimation of seasonal variations
51	SESSION	Forecasting
52	WORK	Forecast
53	SESSION	Applications to the field of mathematical engineering
54	ASSIGNMENT	Applications to the field of mathematical engineering
55	SESSION	Applications to the field of mathematical engineering
56	ASSIGNMENT	Applications to the field of mathematical engineering
57	SESSION	Applications to the field of mathematical engineering
58	ASSIGNMENT	Applications to the field of mathematical engineering
59	EV	Mid-term exam 2
60	EV	Mid-term exam 2	15

Reading list

Core:

1. Hair, J. F., Black, W. C., Babin, B. J., & Anderson, R. E.

Multivariate Data Analysis (8th ed.)

Cengage. 2019.

ISBN: 978-1-292-314

2. Montgomery, D. C.

Design and Analysis of Experiments (9th ed.)

Wiley. 2017.

ISBN: 978-1-119-469

3. Montgomery, D. C., Peck, E. A., & Vining, G. G.

Introduction to Linear Regression Analysis (6th ed.)

Wiley. 2021.

ISBN: 978-1-119-648

4. Montgomery, Douglas C.

Probability and Statistics Applied to Engineering

: McGraw-Hill. 1996.

ISBN: 9701010175

5. Rencher, A. C., & Christensen, W. F.

Methods of Multivariate Analysis (3rd ed.).

Wiley. 2012.

ISBN: 978-0-470-380

6. Rob J. Hyndman and George Athanasopoulos

Forecasting: Principles and Practice (3rd edition, 2021)

Otexts. 2021.

ISBN: 0987507133

7. Ruiz-Maya Pérez, Luis

Statistics II: Inference

2nd ed. Madrid: AC, 2003. 2003.

ISBN: 8472881962

8. Visauta Vinacua, Bienvenido

Statistical Analysis with SPSS for Windows

2nd ed.: McGraw-Hill. 2003.

ISBN: 8448139933

C0242303

Data Structures and Algorithms II

Código: C0242303 Imprimir

Year 2, Course 2. First term. Compulsory. 6 credits.

Profesores

Fausto Ruiz Madrid - Coordinador

Ver biodata

Objectives

The objectives of this module are:

1) To apply knowledge of algorithms and computational complexity

- To develop the ability to use principles of complexity and algorithm analysis to solve engineering problems, selecting appropriate techniques that optimise performance and computational resources.

2) To identify and propose solutions to efficiency problems

- To strengthen the ability to detect bottlenecks or inefficiencies in the design and execution of algorithms, proposing improvements based on complexity analysis and the selection of suitable data structures.

3) Calculate the efficiency of iterative algorithms by applying calculation rules

- Refine the ability to estimate the computational cost of iterative algorithms (in terms of time and space), using appropriate notations and methods for calculating complexity (for example, Big-O and Big-Theta notations).

4) Design and scale algorithms for environments of varying size and complexity

- Develop the ability to devise algorithmic strategies, adapting their structure and execution methods to contexts with different data volumes and performance requirements.

5) Solve engineering problems by applying knowledge of system architecture and programming

- Equip students to tackle and solve complex problems, effectively combining techniques of analysis, algorithm design and the application of programming principles in Python or other languages used in the professional field.

-------------------------

The objectives of this course are:

1) To apply knowledge of algorithms and computational complexity

- To develop the ability to apply principles of computational complexity and algorithm analysis to solve engineering problems, selecting appropriate techniques to optimise performance and computational resources.

2) Identify and propose solutions to efficiency problems

- Strengthen the ability to identify bottlenecks and inefficiencies in the design and execution of algorithms, proposing improvements based on complexity analysis and the selection of appropriate data structures.

3) Calculate the efficiency of iterative algorithms using complexity analysis techniques

- Develop proficiency in estimating the computational cost of iterative algorithms, in terms of time and space, using appropriate complexity notations and analysis methods (e.g., Big-O and Big-Theta notation).

4) Design and scale algorithms for environments of different sizes and levels of complexity

- Develop the ability to devise algorithmic strategies, adapting their structure and execution methods to contexts involving different data volumes and performance requirements.

5) Solve engineering problems by applying knowledge of systems architecture and programming

- Enable students to address and solve complex problems by effectively combining analysis and algorithm design techniques with programming principles in Python and other programming languages used in professional settings.

Prerequisites

It is essential to have previously completed the modules ‘Fundamentals of Programming and Computers’ and ‘Data Structures and Algorithms I’, or other modules with similar competences and learning outcomes./It is essential that students have previously completed the courses ‘Fundamentals of Programming and Computers’ and ‘Data Structures and Algorithms I’, or other courses with similar competences and learning outcomes.

Competencies

Not applicable.

Learning Outcomes

RC3 – Apply software programmes that solve mathematical and engineering problems, using the appropriate computational environment for each case. TYPE: Competences.

RK2 – Master the basic concepts of discrete mathematics, logic, algorithms, coding, operational research and artificial intelligence, and their application to solving engineering problems./ Master the basic concepts of discrete mathematics, logic, algorithms, coding, operational research and artificial intelligence, and their application to solving engineering problems. TYPE: Knowledge or Content.

Description of the content

1. Analysis of Algorithm Efficiency

- Fundamental concepts: time complexity (Big-O, Big-Theta and Big-Omega notations) and space complexity.

- Analysis tools: recursion, iterative traversals and their impact on performance.

- Practical examples in Python: empirical measurement of execution time (use of modules such as `time` and `timeit`) and preliminary code optimisation.

2. Algorithm Design

- Design methodology: problem identification, definition of steps and structuring of the solution (pseudocode and flowcharts).

- Effective use of data structures: lists, dictionaries, queues, stacks, trees (depending on the required complexity).

- Verification and testing in Python: use of unittest and/or pytest to ensure the correctness and robustness of the solution.

3. Study of Algorithmic Techniques

3.1 Greedy Algorithms

- Basic principles: selection of the best local option in the hope of obtaining the best global solution.

- Classic implementations in Python: currency exchange, the fractional knapsack problem, shortest paths (Prim/Kruskal algorithm for graphs).

- Analysis of cases where the greedy approach works and where it does not.

3.2 Divide and Conquer

- Basic strategy: breaking the problem down into smaller sub-problems, solving them, and combining the results.

- Iconic examples: merge sort, quick sort, binary search, matrix multiplication.

- Complexity analysis: recurrence relations and methods for solving them (recurrence trees).

4. Dynamic Programming

- Concept of overlapping subproblems: when a large problem is solved using solutions to previously computed subproblems.

- Implementation techniques:

- Top-down (memoisation): use of dictionaries or lists in Python to store intermediate results.

- Bottom-up (tabulating): building solutions from the bottom up to the complete problem.

- Case studies: the Fibonacci sequence, the integer knapsack problem, sequence alignment, minimum paths in graphs, etc.

5. Backtracking

- Principle of exploring solution spaces: searching for all feasible solutions and discarding those that do not meet the criteria.

- Implementation in Python:

- Use of recursive functions and auxiliary data structures.

- Pruning to reduce the search space.

- Examples: solving Sudoku puzzles, maze problems, the n-queens problem, etc.

6. Branch and Bound

- Differences from Backtracking: an approach more focused on global optimisation (bound) and ordered exploration of branches.

- Classic implementations:

- The Travelling Salesman Problem.

- Optimised 0/1 knapsack problem.

- Lower and upper bound techniques: use of estimates to guide the search for the optimal solution.

7. Application of Algorithmic Techniques to Problem Solving

- Integrated projects in Python: designing applications or scripts that combine various algorithmic techniques depending on the type of problem.

- Optimisation and fine-tuning: use of appropriate data structures and continuous analysis of complexity for environments of varying sizes.

-------------------------

1. Algorithm Efficiency Analysis

- Fundamental concepts: time complexity (Big-O, Big-Theta and Big-Omega notation) and space complexity.

- Analysis tools: recursion, iterative approaches, and their impact on performance.

- Practical examples in Python: empirical measurement of execution time (using modules such as `time` and `timeit`) and preliminary code optimisation.

2. Algorithm Design

- Design methodology: problem identification, definition of the steps, and structuring of the solution (pseudocode and flowcharts).

- Appropriate use of data structures: lists, dictionaries, queues, stacks and trees, according to the required level of complexity.

- Verification and testing in Python: use of unittest and/or pytest to ensure the correctness and robustness of the solution.

3. Study of Algorithmic Techniques

3.1 Greedy Algorithms

- Basic principles: selecting the best local option with the expectation of obtaining the best global solution.

- Classical implementations in Python: the coin-change problem, the fractional knapsack problem, and shortest-path problems (Prim’s and Kruskal’s algorithms for graphs).

- Analysis of cases in which the greedy approach works and cases in which it does not.

3.2 Divide and Conquer

- Basic strategy: breaking a problem down into smaller subproblems, solving them, and combining the results.

- Classic examples: merge sort, quicksort, binary search, and matrix multiplication.

- Complexity analysis: recurrence relations and methods for solving them (recursion trees).

4. Dynamic Programming

- Concept of overlapping subproblems: solving a larger problem based on solutions to previously computed subproblems.

- Implementation techniques:

- Top-down approach (memoisation): use of dictionaries or lists in Python to store intermediate results.

- Bottom-up approach (tabulation): constructing solutions from the base cases up to the complete problem.

- Case studies: the Fibonacci sequence, the 0/1 knapsack problem, sequence alignment, shortest paths in graphs, etc.

5. Backtracking

- Principle of exploring the solution space: searching for all feasible solutions and discarding those that do not meet the required criteria.

- Implementation in Python:

- Use of recursive functions and auxiliary data structures.

- Pruning to reduce the search space.

- Examples: Solving Sudoku, maze problems, the n-queens problem, etc.

6. Branch and Bound

- Differences from backtracking: a stronger focus on global optimisation through bounding and on the ordered exploration of branches.

- Classical implementations:

- The Travelling Salesman Problem (TSP).

- The 0/1 knapsack problem using optimisation techniques.

- Lower- and upper-bound techniques: the use of estimates to guide the search for the optimal solution.

7. Application of Algorithmic Techniques to Problem Solving

- Integrative Python projects: design of applications or scripts combining several algorithmic techniques according to the type of problem.

- Optimisation and fine-tuning: use of appropriate data structures and continuous complexity analysis for environments of different sizes.

Teaching Activities

AP1. Lectures.

AP2. Interactive classes.

AP3.1. Project development and challenge-based problem solving.

AP4. Independent study, case and problem solving, and literature search.

AP5. Tutorials.

AP6. Knowledge assessment tests.

Assessment system and criteria

1. REGULAR EXAMINATION PERIOD. CONTINUOUS ASSESSMENT

The final mark is the weighted average of the following activities. The course content is cumulative: no assessment is limited to a single topic, except for the oral examination, which is expressly restricted to one module.

SE1 · Practical activities (20 %)

— Individual project with oral presentation … 10 per cent

— Group project with oral presentation … 10 %

SE2 · Final knowledge assessments (50 %)

— Final exam (ordinary sitting) … 50 per cent

SE3 · Portfolio (30 per cent)

— Written mid-term exam … 10 %

— Oral exam on a section of the syllabus … 10 %

— Exercises completed and handed in during class … 10 per cent

a) Individual assignment with oral presentation (10 per cent)

Assessment criteria:

— Algorithmic correctness and efficiency analysis: correct implementation of the solution and a reasoned justification of its time and space complexity in asymptotic notation, including the derivation and solution of the recurrence relation where the algorithm is recursive.

— Appropriateness of the chosen technique: justification of why the technique used (greedy, divide and conquer, dynamic programming, backtracking or branch and bound) is suitable for the problem, and why the others are ruled out.

— Choice of data structures: suitability of the structures used in relation to the cost of the algorithm’s dominant operations.

— Code structure and clarity: modular organisation, appropriate use of functions and adherence to the language’s style conventions (PEP 8 in Python).

— Documentation: docstrings and comments explaining the algorithm’s logic, and an analysis of the cost of the main operations.

— Verification: unit tests using `unittest` or `pytest` to demonstrate the correctness and robustness of the solution, including edge cases.

— Oral presentation: ability to explain design decisions, to reason about computational cost, and to respond to variations on the problem statement raised during the presentation.

b) Group project with oral presentation (10 %)

In addition to the above criteria:

— Integration of various techniques: solving a problem that requires combining more than one algorithmic technique and justifying the allocation between them.

— Empirical comparison: measuring execution times using `time` or `timeit` and comparing the results with the expected theoretical complexity, whilst explaining any deviations.

— Division of labour: each group member is responsible for defending the entire piece of work, not just their own part.

Failure to attend the defence will result in the loss of 50 per cent of the mark for the individual component of the assessment.

c) Final examination (ordinary session) (50 per cent)

This takes place on the date set by the University and assesses the entire course content. The mid-term exam does not exempt students from the final exam.

Assessment criteria:

— Efficiency analysis: calculation of the time and space complexity of given algorithms, in O, Theta and Omega notation, and solving recurrence relations.

— Algorithm design: writing the solution to a new problem in pseudocode, justifying the chosen data structure.

— Mastery of techniques: application of greedy algorithms, divide and conquer, dynamic programming, backtracking, and branch-and-bound to problems not covered in class, including the formulation of the pruning function or bound values where appropriate.

— Selection criteria: the ability to decide which technique is appropriate for a given problem and to justify the rejection of the others.

— Formal correctness: precision in notation, in demonstrating the optimality of a greedy solution, and in formulating the recurrence relation for a dynamic programming problem.

d) Written mid-term exam (10 %)

A handwritten examination, held halfway through the term, focusing on the design of algorithms in pseudocode and on complexity analysis. This examination does not exempt students from the course material.

Assessment criteria:

— Correctness of the pseudocode and flowcharts.

— Accuracy of the complexity analysis and the solution to recurrence relations.

— Clarity of written reasoning, assessed independently of the final result.

e) Oral examination on a section of the syllabus (10 %)

An individual examination limited to one section of the syllabus, announced in advance. It does not cover the entire syllabus.

Assessment criteria:

— Understanding versus rote learning: ability to explain how an algorithm from the section works without referring to the written code.

— Reasoning regarding cost: deducing complexity aloud, step by step.

— Response to follow-up questions: reaction to a change in the question or to a borderline case raised during the examination.

— Terminological accuracy: correct use of subject-specific vocabulary.

f) Exercises completed and handed in during class (10 per cent)

A set of exercises completed during the sessions and uploaded to the campus by the deadline specified for each one.

Assessment criteria:

— Consistency: regular submission throughout the term, not concentrated at the end.

— Correctness of the solution and adherence to the question.

— Progress: incorporation into subsequent exercises of the corrections highlighted in previous ones.

— Traceability of one’s own work: the exercise must be able to be explained in class at the lecturer’s request. An exercise that the student cannot defend will not be marked.

2. REQUIREMENTS FOR CONTINUOUS ASSESSMENT TO BE AVERAGED

— Achieve a mark of 5.0 out of 10 or higher in the final exam during the standard examination period. Below this mark, the course mark will be the mark obtained in that exam.

— Achieve a weighted average of 5.0 out of 10 or higher across all activities.

3. NON-CONTINUOUS ASSESSMENT

Students who do not participate in continuous assessment, or who do not meet the minimum attendance requirement, will be assessed by means of a single examination which accounts for 100 per cent of the mark and covers the entire course content, both theoretical and practical.

4. SESSION EXAM

The mark is derived exclusively from the supplementary examination, which covers the entire syllabus. Continuous assessment activities do not count towards this sitting.

-------------------------------------------------------------------------------------------------------

1. ORDINARY EXAMINATION PERIOD. CONTINUOUS ASSESSMENT

The final mark is the weighted average of the following activities. Course content is cumulative: no assessment is restricted to an isolated topic, with the sole exception of the oral examination, which is expressly limited to one block of the syllabus.

SE1 · Practical activities (20 %)

— Individual assignment with oral defence … 10 %

— Group assignment with oral defence … 10 %

SE2 · Final knowledge tests (50 %)

— Final examination, ordinary sitting … 50 %

SE3 · Portfolio (30 %)

— Written mid-term examination … 10 %

— Oral examination on one section of the syllabus … 10 %

— Exercises completed and submitted in class … 10 %

a) Individual assignment with oral defence (10 %)

Assessment criteria:

— Algorithmic correctness and efficiency analysis: correct implementation of the solution and a reasoned justification of its time and space complexity in asymptotic notation, including the formulation and solution of the recurrence relation where the algorithm is recursive.

— Suitability of the chosen technique: justification of why the technique used (greedy, divide and conquer, dynamic programming, backtracking or branch and bound) is the appropriate one for the problem, and what rules out the others.

— Choice of data structures: suitability of the structures used in relation to the cost of the algorithm’s dominant operations.

— Code structure and clarity: modular organisation, appropriate use of functions and adherence to the language’s style conventions (PEP 8 in Python).

— Documentation: docstrings and comments explaining the logic of the algorithm, together with an analysis of the cost of its main operations.

— Verification: unit tests written using unittest or pytest, demonstrating the correctness and robustness of the solution, including edge cases.

— Oral defence: ability to explain design decisions, to reason about cost and to respond to variations on the problem statement raised during the defence.

b) Group assignment with oral defence (10 %)

In addition to the criteria above:

— Integration of several techniques: solution to a problem requiring more than one algorithmic technique, with a reasoned explanation of how the work is divided between them.

— Empirical comparison: measurement of execution times using `time` or `timeit` and comparison of the results against the predicted theoretical complexity, explaining any deviation.

— Division of labour: each member of the group is accountable for the entire assignment during the defence, not just for their own part.

Failure to attend the oral defence results in the loss of 50 per cent of the mark for that activity on an individual basis.

c) Final examination, ordinary session (50 %)

Held on the date set by the University and covering the entire course content. The mid-term examination does not exempt students from any part of the course content.

Assessment criteria:

— Efficiency analysis: calculation of the time and space complexity of given algorithms in O, Theta and Omega notation, and solution of recurrence relations.

— Algorithm design: pseudocode solution to an unseen problem, with justification of the chosen data structure.

— Mastery of techniques: application of greedy algorithms, divide and conquer, dynamic programming, backtracking and branch and bound to problems not covered in class, including the formulation of the pruning function or of the bounds where applicable.

— Selection criteria: the ability to decide which technique is best suited to a given problem and to explain why the others are ruled out.

— Formal correctness: precision in notation, in proving the optimality of a greedy solution and in formulating the recurrence relation for a dynamic programming problem.

d) Written mid-term examination (10 %)

A handwritten examination held halfway through the semester, focusing on algorithm design in pseudocode and on complexity analysis. It does not exempt students from any course content.

Assessment criteria:

— Correctness of the pseudocode and flowcharts.

— Accuracy of the complexity analysis and of the solution to recurrence relations.

— Clarity of the written reasoning, assessed independently of the final result.

e) Oral examination on one section of the syllabus (10 %)

Individual examination limited to one section of the syllabus, announced in advance. It does not cover the entire syllabus.

Assessment criteria:

— Understanding rather than memorisation: the ability to explain how an algorithm from the section works without relying on written code.

— Reasoning about complexity: step-by-step derivation of the complexity, stated aloud.

— Response to follow-up questions: reaction to a change in the problem statement or to an edge case raised during the examination.

— Terminological precision: correct use of the subject’s vocabulary.

f) Exercises completed and submitted in class (10 %)

A set of exercises completed during the sessions and uploaded to the campus platform by the deadline specified for each one.

Assessment criteria:

— Consistency: regular submission throughout the term, not concentrated at the end.

— Correctness of the solution and adherence to the problem statement.

— Progression: incorporation into later exercises of the corrections noted in earlier ones.

— Traceability of the student’s own work: the student must be able to explain the exercise in class if asked to do so. An exercise that the student cannot defend will be marked as zero.

2. REQUIREMENTS FOR CONTINUOUS ASSESSMENT TO COUNT TOWARDS THE FINAL MARK

— Achieving a mark of 5.0 out of 10 or higher in the final examination held during the regular examination period. If the mark is below this, the mark for the course will be that obtained in that examination.

— Demonstrating a minimum attendance of 70 per cent of the face-to-face sessions.

— Achieving a weighted average of 5.0 out of 10 or higher across all activities.

3. NON-CONTINUOUS ASSESSMENT

Students who do not undertake continuous assessment, or who do not meet the minimum attendance requirement, will be assessed by means of a single examination worth 100 per cent of the mark, covering the entire content of the course, both theoretical and practical.

4. RESIT (EXTRAORDINARY EXAMINATION PERIOD)

The mark is based solely on the resit examination, which covers the entire course content. Continuous assessment activities do not count towards the mark during this period.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	SESSION	Topic 1: Analysis of Algorithm Efficiency
2	ASSIGNMENT	Topic 1: Analysis of Algorithm Efficiency
3	SESSION	Topic 1: Analysis of Algorithm Efficiency
4	ASSIGNMENT	Topic 1: Analysis of Algorithm Efficiency
5	SESSION	Topic 1: Analysis of Algorithm Efficiency
6	ASSIGNMENT	Topic 1: Analysis of Algorithm Efficiency
7	SESSION	Topic 2: Algorithm Design
8	ASSIGNMENT	Topic 2: Algorithm Design
9	SESSION	Topic 2: Algorithm Design
10	ASSIGNMENT	Topic 2: Algorithm Design
11	SESSION	Topic 2: Algorithm Design
12	ASSIGNMENT	Topic 2: Algorithm Design
13	SESSION	Topic 3: Study of Algorithmic Techniques
14	ASSIGNMENT	Topic 3: Study of Algorithmic Techniques
15	SESSION	Topic 3: Study of Algorithmic Techniques
16	ASSIGNMENT	Topic 3: Study of Algorithmic Techniques
17	SESSION	Topic 3: Study of Algorithmic Techniques
18	ASSIGNMENT	Topic 3: Study of Algorithmic Techniques
19	SESSION	Topic 3: Study of Algorithmic Techniques
20	ASSIGNMENT	Topic 3: Study of Algorithmic Techniques
21	SESSION	Topic 3: Study of Algorithmic Techniques
22	ASSIGNMENT	Topic 3: Study of Algorithmic Techniques
23	SESSION	Topic 3: Study of Algorithmic Techniques
24	ASSIGNMENT	Topic 3: Study of Algorithmic Techniques
25	EV	Practical 1
26	EV	Practical 1	7.5%
27	SESSION	Topic 4: Dynamic Programming
28	ASSIGNMENT	Topic 4: Dynamic Programming
29	SESSION	Topic 4: Dynamic Programming
30	ASSIGNMENT	Topic 4: Dynamic Programming
31	SESSION	Topic 4: Dynamic Programming
32	ASSIGNMENT	Topic 4: Dynamic Programming
33	SESSION	Topic 4: Dynamic Programming
34	ASSIGNMENT	Topic 4: Dynamic Programming
35	EV	Mid-term exam.
36	EV	Mid-term exam.	30%
37	SESSION	Topic 5: Backtracking
38	ASSIGNMENT	Topic 5: Backtracking
39	SESSION	Topic 5: Backtracking
40	ASSIGNMENT	Topic 5: Backtracking
41	SESSION	Topic 5: Backtracking
42	ASSIGNMENT	Topic 5: Backtracking
43	SESSION	Topic 5: Backtracking
44	ASSIGNMENT	Topic 5: Backtracking
45	SESSION	Topic 6: Branch and Bound
46	ASSIGNMENT	Topic 6: Branch and Bound
47	SESSION	Topic 6: Branch and Bound
48	ASSIGNMENT	Topic 6: Branch and Bound
49	SESSION	Topic 6: Branch and Bound
50	ASSIGNMENT	Topic 6: Branch and Bound
51	EV	Practical 2
52	EV	Practical 2	7.5%
53	SESSION	Topic 7: Application of Algorithmic Techniques to Problem Solving
54	ASSIGNMENT	Topic 7: Application of Algorithmic Techniques to Problem Solving
55	SESSION	Topic 7: Application of Algorithmic Techniques to Problem Solving
56	ASSIGNMENT	Topic 7: Application of Algorithmic Techniques to Problem Solving
57	EV	Final practical
58	EV	Final practical	15%
59	EV	Final exam.
60	EV	Final exam.	40%

Reading list

Core:

1. George T. Heineman, Gary Pollice, Stanley Selkow

Algorithms in a Nutshell (2nd edition)

O’Reilly Media. 2016.

ISBN: 978-1-4919-01

2. Mariona Nadal

Data Structures and Algorithms

Anaya Multimedia. 2022.

ISBN: 978-844154519

Others:

3. Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest and Clifford Stein

Introduction to Algorithms

McGraw-Hill. 2022.

ISBN: 978-607150285

C0242304

Differential Geometry and Applications

Código: C0242304 Imprimir

Year 2, Course 2. First term. Compulsory. 6 credits.

Profesores

José Luis Guijarro Regalado - Coordinador

Ver biodata

Objectives

The fundamental objective is for students to learn how to perform calculations on curves and surfaces in Euclidean space. It is not so much a matter of developing spatial awareness as of understanding the fundamental quadratic forms that define a surface and being able to deduce from them the curvatures and other differential invariants that characterise them.

This objective focuses on the study of intrinsic and local geometry, as opposed to extrinsic geometry, which studies surfaces embedded within others, and global geometry, which studies the characterisation of the surface as a whole through characteristic classes.

Prerequisites

Although no prerequisites have been established, it is advisable to have previously taken the modules ‘Mathematical Foundations of Engineering I and II’ and ‘Algebra I and II’, and to have taken or be currently taking ‘Differential Calculus’ or other modules with similar competencies and learning outcomes.

Competencies

Not applicable.

Learning Outcomes

RC2 – Identify the different phases of the mathematical modelling process, distinguishing between formulation, analysis, solution and interpretation of results. TYPE: Competencies.

RC3 – Apply software to solve mathematical and engineering problems, using the appropriate computational environment for each case. TYPE: Competencies.

RC9 – Formulate problems from a professional context in mathematical language in such a way as to facilitate their analysis and resolution. TYPE: Competencies.

RK4 – Be familiar with rigorous proofs of some classical theorems in different areas of mathematics. TYPE: Knowledge or content.

RS1 – Use mathematical language to state propositions in different fields and construct proofs for the problems posed. TYPE: Skills or abilities.

RS5 – Use computer applications for statistical analysis, numerical and symbolic calculation, graphical visualisation, optimisation or other purposes to solve problems. TYPE: Skills or competences.

Description of the content

1) Curves in the plane and in space

a) curves parameterised by arc length

b) Frenet’s formulas

2) Surfaces in Euclidean space

a) Parametrised and implicit regular surfaces: the tangent plane

b) the first fundamental form

c) applications: areas, lengths and angles

3) Local intrinsic geometry

a) The Gauss map

b) the second fundamental form

c) applications:

i) classification of points on a surface

ii) principal directions and asymptotes

iii) lines of curvature and asymptotes

Training activities

AP1. Lectures.

AP2. Interactive classes.

AP3.1. Project work and problem-solving.

AP4. Independent study, case studies, problem-solving and literature review.

AP5. Tutoring.

AP6. Knowledge assessments.

Assessment system and criteria

The assessment process will consist of verifying and evaluating the student’s acquisition of the required competences.

- There are two official examination sessions: the ordinary and the supplementary.

REGULAR EXAMINATION PERIOD

In the ordinary assessment period, the final mark for the module is determined by a process of continuous assessment.

Continuous assessment comprises the following assessments:

1) a curve-based exam in October, which will account for 20 per cent

2) a surfaces exam on the last teaching day of the year (approximately 20 December), which will account for 30 per cent

3) the official examination (ordinary assessment period), which will account for 50 per cent. Under no circumstances will passing the assessments exempt students from this final examination.

For continuous assessment to apply, the mark in the official examination must be 4 or above. In other words, if a mark of less than 4 is obtained in the final examination, the module will not be passed, and this mark will be recorded directly on the academic transcript.

REGULAR EXAM SESSION

In the supplementary sitting, no marks will be carried over, and the corresponding examination will be held, accounting for 100 per cent of the mark.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	MG	Course introduction.
2	MG	Definition of a curve
3	MG	Basic properties of a curve
4	SM	Examples of curves
5	MG	Regular parametric curves
6	MG	Arc length
7	MG	Curvature and torsion
8	SM	Frenet equations
9	MG	Examples
10	EV	Curve control	10%
11	MG	Parametrised surfaces
12	SM	Examples
13	MG	The tangent plane
14	MG	The Tangent Plane II
15	MG	The derivative of a function
16	SM	Examples
17	MG	The first fundamental form
18	MG	The first fundamental form II
19	MG	The first fundamental form III
20	SM	Examples
21	MG	Areas
22	MG	Lengths
23	MG	Angles on surfaces
24	SM	Examples
25	EV	Surface examination	20%
26	MG	Parameter changes and diffeomorphisms
27	MG	Examples
28	MG	Examples
29	SM	Examples
30	MG	The Gauss app
31	MG	The Gauss app II
32	MG	The Gauss application III
33	SM	Examples
34	MG	The second fundamental form
35	MG	main directions
36	MG	asymptotic directions
37	SM	Classification of points
38	EV	Submission of work	10%
39	MG	Gauss’s application in coordinates
40	MG	Consequences
41	MG	Gaussian curvature and mean
42	SM	Examples
43	MG	Calculations
44	MG	Calculus II
45	MG	Calculus III
46	MG	Vector fields
47	SM	Examples
48	MG	Computer simulations
49	MG	Computer simulation
50	MG	Problems
51	SM	Examples
52	MG	Introduction to the in-depth topics
53	MG	Geodesics
54	SM	Exercises
55	MG	Gauss’s theorem
56	MG	Intrinsic versus extrinsic geometry
57	MG	Exam exercises
58	SM	Exam exercises
59	SM	Queries and questions
60	EV	Final exam	60%

Reading list

Core:

1. A.M. Amores Lázaro

Basic Course on Curves and Surfaces

Sanz y Torres. 2001.

ISBN: 8488667779

2. Carmo, Manfredo P. do

Differential Geometry of Curves and Surfaces

Madrid: Alianza, 1994. 1994.

ISBN: 8420681350

C0242603

Mechanics and Waves I

Código: C0242603 Imprimir

Year 2, Course 2. First term. Compulsory. 6 credits.

Profesores

Víctor David Boscá Navarro - Coordinador

Objectives

This module, which together with Mechanics and Waves II forms the subject of Mechanics and Waves, has as its main objective to familiarise students with the Newtonian formulation of classical mechanics and to enable them to apply it correctly to the solution of mechanical problems. More specifically, students must understand the basic concepts of oscillatory motion, as well as its associated phenomenology; they must be able to describe both the kinematics and dynamics of a rigid body in a plane and in space; and they must understand the relationship between symmetries and conservation laws in physics.

Prerequisites

None

Competencies

RK1 Understand the most important phenomena and theories in the various branches of physics, as well as their historical context

RK2 Understand physically distinct phenomena and their underlying analogies in order to apply known solutions to new problems

RK3 To analyse the fundamental concepts and principles of physical systems in order to develop approximations that enable the construction of a simplified model

RK5 To understand the scope and limitations of classical physics that led to the formulation of special and general relativity, as well as quantum mechanics, in order to address the new problems arising in modern physics.

RK7 Understand the laws and principles of physics, identifying their logical and mathematical structure, their experimental basis and the phenomena described by them

RK12 Understand the theories, laws and models governing physical phenomena related to mechanics

RS1 Apply the most important knowledge, concepts and methods from the various branches of physics.

RS4 Apply mathematical and numerical methods to the modelling and explicit solution of problems in physics and related disciplines, selecting the appropriate tools and interpreting results.

Learning outcomes

RA1 Applies the Newtonian formulation appropriately to the solution of mechanical problems.

RA2 Understands the basic concepts of wave motion, as well as the basic phenomenology of oscillatory motion, including coupled oscillations and resonance, and is able to solve problems involving free, forced and damped oscillations.

LA3 Describes both the kinematics and dynamics of a rigid body in a plane and in space, and applies this knowledge to solving problems involving rigid bodies.

RA4 Relates symmetries and conservation laws in physics, applying them to solve practical exercises.

Course content

The module comprises the following topics:

1. Particle kinematics.

2. Newtonian dynamics.

3. Non-inertial reference frames.

4. Kinematics of a rigid body.

5. Dynamics of a rigid body.

6. Oscillations and waves.

Learning activities

Learning activity No. of hours* Contact hours

(8–12)** % Face-to-face

AP1. Participatory lectures 48 4 100

AP2.- Seminars or practical application classes 30 2.5 100

AP3.- Practical activities (case studies, project work, simulations, etc.) 72 3 50

AP4. – Independent study 120 0 0

AP5.- Tutorials 24 0.6 30

AP6.- Knowledge assessments 6 0.5 100

TOTAL 300 10.6

Assessment system and criteria

The assessment process will consist of verifying and evaluating the student’s acquisition of the required competences.

ASSESSMENT SYSTEMS

The assessment systems for this module are:

- AS1: Practical activities (case studies, problem-solving and challenges, project work, oral presentations, debates, etc.).

- AS2: Final knowledge assessments.

- AS4: Portfolio.

ASSESSMENT CRITERIA

The assessment methods described above are set out in the following assessment criteria. There are two official examination sessions: the ordinary and the supplementary.

+++REGULAR SESSION+++

The final mark for this sitting is the weighted average of a set of assessment tasks detailed below:

- Practical activities (SE1) 30% of the final mark: Mid-term exam I (15%) + Mid-term exam II (15%).

- Final knowledge assessments (SE2) 60% of the final mark: Final exam for the ordinary assessment period, which covers the entire course content.

- Portfolio (SE4) 10% of the final mark. This assessment system involves completing and submitting exercises carried out in class, with non-trivial modifications designed to encourage creativity.

For continuous assessment (comprising the Practical Activities (SE1) and the portfolio (SE4)) to be taken into account, students must achieve a mark of 4.0/10 or higher in the final exam during the standard examination period. Otherwise, their mark will correspond directly to that obtained in that exam.

The module is considered passed in the ordinary examination period if the mark obtained in accordance with the above guidelines is 5.0 or higher.

+++SUPPLEMENTARY EXAMINATION PERIOD+++

If a student has not passed the module during the ordinary examination period, they may sit the module during the supplementary examination period.

The supplementary examination period will take place during the June and July examination periods (for further information, please consult the Academic Calendar).

100% of the final mark for the supplementary sitting: Final examination for the supplementary sitting, which assesses the entire course content.

The module is considered passed in the extraordinary examination period if the final mark is 5.0 or above.

GRADES

The award of the corresponding credits is conditional upon passing the associated examinations or assessment tests.

The level of learning achieved by students will be expressed as numerical marks on a scale of 0 to 10, to one decimal place, to which the corresponding qualitative mark may be added:

- 0–4.9: Fail (SS).

- 5.0–6.9: Pass (AP).

- 7.0–8.9: Good (NT).

- 9.0–10: Distinction (SB).

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	SESSION	Introduction to the course
2	ASSIGNMENT	Particle kinematics
3	SESSION	Particle Kinematics
4	ASSIGNMENT	Particle Kinematics
5	SESSION	Particle Kinematics
6	ASSIGNMENT	Particle Kinematics
7	SESSION	Newtonian dynamics
8	ASSIGNMENT	Newtonian dynamics
9	SESSION	Newtonian dynamics
10	ASSIGNMENT	Newtonian dynamics
11	SESSION	Newtonian dynamics
12	ASSIGNMENT	Newtonian dynamics
13	SESSION	Newtonian dynamics
14	ASSIGNMENT	Newtonian dynamics
15	SESSION	Newtonian dynamics
16	ASSIGNMENT	Newtonian Dynamics (SUBMISSION)
17	SESSION	Oscillations
18	ASSIGNMENT	Oscillations
19	SESSION	Oscillations
20	WORK	Oscillations
21	SESSION	Oscillations
22	ASSIGNMENT	Oscillations
23	SESSION	Oscillations
24	ASSIGNMENT	Oscillations
25	SESSION	Oscillations
26	ASSIGNMENT	Oscillations (SUBMISSION)
27	EV	First mid-term exam
28	EV	First mid-term exam
29	SESSION	Non-inertial reference systems
30	ASSIGNMENT	Non-inertial reference systems
31	SESSION	Non-inertial reference systems
32	ASSIGNMENT	Non-inertial reference systems
33	SESSION	Non-inertial reference systems
34	ASSIGNMENT	Non-inertial reference systems
35	SESSION	Non-inertial reference systems
36	ASSIGNMENT	Non-inertial reference systems
37	SESSION	Non-inertial reference systems
38	ASSIGNMENT	Non-inertial reference systems
39	SESSION	Non-inertial reference systems
40	ASSIGNMENT	Non-inertial reference systems (ASSIGNMENT)
41	SESSION	Kinematics of a rigid body
42	ASSIGNMENT	Kinematics of a Rigid Body
43	SESSION	Kinematics of a Rigid Body
44	ASSIGNMENT	Kinematics of a rigid body
45	SESSION	Kinematics of a rigid body
46	ASSIGNMENT	Kinematics of a Rigid Body
47	SESSION	Kinematics of a rigid body
48	ASSIGNMENT	Kinematics of a rigid body
49	SESSION	Dynamics of Rigid Bodies
50	ASSIGNMENT	Dynamics of Rigid Bodies
51	SESSION	Dynamics of Rigid Bodies
52	ASSIGNMENT	Dynamics of Rigid Bodies
53	SESSION	Dynamics of Rigid Bodies
54	ASSIGNMENT	Dynamics of Rigid Bodies
55	SESSION	Dynamics of Rigid Bodies
56	ASSIGNMENT	Dynamics of Rigid Bodies
57	SESSION	Dynamics of Rigid Bodies
58	ASSIGNMENT	Dynamics of Rigid Bodies
59	SESSION	Dynamics of Rigid Bodies
60	ASSIGNMENT	Dynamics of Rigid Bodies (SUBMISSION)
61	EV	Second mid-term exam
62	EV	Second mid-term exam
63	SESSION	Q&A session
64	ASSIGNMENT	Clarification of queries (ASSIGNMENT)

Bibliography

Basic:

1. A. Fernández Rañada

Classical Dynamics

Alianza. 1994.

ISBN: 84-206-8133-4

Supplementary:

2. J.R. Taylor

Classical Mechanics

Reverté. 2013.

ISBN: 8429143122

Others:

3. C. Kittel, W.D. Knight, M.A. Ruderman

Mechanics

Reverté. 1968.

ISBN: 978-84-291-42

4. S.T. Thornton, J.B. Marion

Classical Dynamics of Particles and Systems

Reverté. 1975.

ISBN: 9788429140941

TOTAL:

SECOND FOUR-MONTH TERM

Code

Subjects

Character*

ECTS

C0242305

Integral calculus

Código: C0242305 Imprimir

Year 2, Course 2. Second term. Foundation module. 6 credits.

Profesores

Hugo Andrea Galindo Beleña - Coordinador

Ver biodata

Objectives

This module, which together with Differential Calculus forms part of the Mathematical Analysis II course within the degree programme’s Core Module, has as its main objective to provide students with a solid grounding in the Riemann integral of functions of several variables and in the fundamental tools of integral calculus in higher dimensions. In particular, the course will cover the integration of functions of several variables over different types of domains, exploring in depth the conceptual and technical aspects that enable the extension of one-dimensional concepts to the multivariable case. Furthermore, Fubini’s Theorem and its applications to the calculation of iterated integrals will be studied, as well as the change of variable theorem, which allows integrals to be simplified through appropriate transformations. Furthermore, improper integrals in several variables will be analysed, with a focus on convergence criteria and their correct interpretation.

Furthermore, the study of line and surface integrals will be introduced, from both a geometric and an analytical perspective, establishing their connection with scalar and vector fields. Finally, the main integration theorems of vector calculus will be presented; these relate integrals over domains to integrals over their boundaries and constitute fundamental tools in mathematics, physics and engineering. In this way, the course aims not only to ensure that students understand the theoretical foundations of integral calculus in several variables, but also that they are able to apply them rigorously and competently when solving specific problems.

Prerequisites

None have been specified, although it is strongly recommended that students have previously taken the courses ‘Mathematical Foundations of Engineering I and II’ and ‘Differential Calculus’ or other courses covering similar skills and learning outcomes.

Competencies

Not applicable.

Learning Outcomes

RC9 – Formulate problems from a professional context in mathematical language in such a way as to facilitate their analysis and solution. TYPE: Competences.

RK4 – Be familiar with rigorous proofs of some classical theorems in different areas of mathematics. TYPE: Knowledge or content.

RS1 – Use mathematical language to state propositions in different fields and construct proofs for the problems posed. TYPE: Skills or abilities.

Description of the content

The course covers the following topics:

1. Integration of functions of several variables.

2. Fubini’s theorem.

3. The change of variables theorem.

4. Improper integrals.

5. Line and surface integrals.

6. Theorems on vector integration.

Learning activities

AP1. Lectures.

AP2. Interactive classes.

AP3.1 Project work and problem-solving.

AP4. Independent study, case studies, problem-solving and literature review.

AP5. Tutoring.

AP6. Knowledge assessments.

Assessment system and criteria

There are two official examination sessions: the ordinary and the supplementary.

*** Ordinary examination session ***

The final mark for this sitting is the weighted average of the assessment tests detailed below:

- 2 non-exemption mid-term exams: 30% of the final mark (15% for each mid-term exam).

- Completion of exercises: 15% of the final mark.

- Final exam for the ordinary assessment period: 55% of the final mark. This exam will assess all the content covered in the module.

For continuous assessment (comprising the submission of exercises and the two mid-term exams) to be taken into account, students must achieve a minimum mark of 4.0 out of 10.0 in the final exam. In this case, the average will be calculated between the continuous assessment and this exam, even if the former is a fail.

If the mark for the final exam is below 4.0 out of 10.0, the course mark will correspond to 55 per cent of the mark obtained in that exam.

The module is considered to have been passed in the ordinary examination period if the mark obtained in accordance with the above guidelines is 5.0 or higher.

*** Extraordinary examination session ***

If a student has not passed the module in the ordinary examination session, they may sit the extraordinary examination session.

The module is considered to have been passed in the supplementary examination if the final mark is 5.0 or higher.

GRADES

The award of the corresponding credits is conditional upon passing the associated examinations or assessment tests.

The level of learning achieved by students will be expressed as numerical marks on a scale of 0 to 10, to one decimal place, to which the corresponding qualitative mark may be added:

- 0–4.9: Fail (SS).

- 5.0–6.9: Pass (AP).

- 7.0–8.9: Good (NT).

- 9.0–10: Distinction (SB).

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	MG	Course introduction.
2	MG	Double integrals
3	MG	Calculation of double integrals
4	SM	Exercises (Double integrals).
5	MG	Fubini’s theorem.
6	MG	Applications of Fubini’s Theorem.
7	MG	Exercises (double integral).
8	SM	Exercises (double integral).
9	MG	Change of variables
10	MG	Polar coordinates
11	MG	Variable substitution exercises
12	SM	Exercises on changing variables
13	MG	Multiple integration
14	MG	Triple integration
15	MG	Application of Fubini’s theorem and change of variables
16	SM	Multiple integral exercises
17	MG	Multiple integral exercises
18	EV	First set of exercises	7.5%
19	MG	Derivative under the integral sign.
20	SM	Exercises: Derivation under the integral sign.
21	MG	Exercises: Differentiation under the integral sign.
22	MG	Exercises: Differentiation under the integral sign.
23	MG	Improper integrals.
24	SM	Exercises on improper integrals.
25	MG	Improper integral exercises.
26	MG	Improper integral exercises.
27	MG	Multiple integral applications
28	SM	Derivatives under the integral sign.
29	MG	Applications of improper integrals
30	EV	First mid-term exam
31	EV	First mid-term exam	15%
32	SM	Line integrals and scalar fields
33	MG	Applications of the line integral to scalar fields
34	MG	Applications of line integrals to scalar fields
35	MG	Line integral exercises in R²
36	SM	Integral line exercises in R3
37	MG	Line integrals and vector fields
38	MG	Applications of the line integral to vector fields
39	MG	Grenn’s Theorem
40	SM	Conservative fields
41	MG	Exercises on Grenn’s theorem
42	MG	Exercises on integrals in conservative fields
43	MG	Comprehensive exercises on line fields in R2
44	SM	Integral line exercises: conservative fields in R3
45	EV	Second set of exercises	7.5%
46	MG	Surface integral of scalar fields
47	MG	Applications of the surface integral in scalar fields
48	SM	Applications of surface integrals to scalar fields
49	MG	Surface integral exercises
50	MG	Surface integral exercises
51	MG	Surface integrals and vector fields
52	SM	Applications of the surface integral to vector fields
53	MG	Related general theorems: Stokes’ theorem
54	MG	Exercises on Stokes’ theorem
55	MG	Exercises on Stokes’ theorem
56	SM	Exercises on Stokes’ theorem
57	MG	Divergence theorem
58	MG	Exercises on the Divergence Theorem
59	MG	Exercises on the divergence theorem
60	EV	Second mid-term exam	15%

Bibliography

Core:

1. Jerrold E. Marsden

Elementary Classical Analysis

W. H. Freeman and Company. 1974.

ISBN: 0716721058

2. Jerrold E. Marsden, Anthony J. Tromba

Vector Calculus

3rd ed. Addison-Wesley Iberoamericana. 1991.

ISBN: 0201629356

Others:

3.- James R. Munkres

Analysis on Manifolds

Addison-Wesley. 1991.

ISBN: 0201315963

4. Michael Spivak

Calculus on Manifolds

Addison-Wesley. 1971.

ISBN: 9780805390216

C0242306

Technical Communication in English

Código: C0242306 Imprimir

Year 2, Course 2. Second term. Foundation module. 6 credits.

Profesores

Francisco Javier Alonso Matas - Coordinador

Ver biodata

Objectives

To acquire the necessary skills in existing methods to reach a B2/C1 level, with particular emphasis on individual expression (spoken and written), the communicative process (speaking and listening), the correct use of spoken and written language (accuracy, coherence and appropriateness, lexical accuracy, spelling, vocabulary, pronunciation and creativity) and reading texts (reading, comprehension and critical thinking). An introduction to technical English in the fields of engineering, aeronautics and mechanics will also be provided within this level. Students will be familiarised with basic technical vocabulary and introduced to B2-level texts within the scope of their degree programme./Acquisition of the competencies required to use the existing methods to attain a B2/C1 level, with particular emphasis on individual expression (oral and written), the communicative process (speaking and listening), the accurate use of spoken and written language (correctness, coherence and appropriateness, lexical accuracy, spelling, vocabulary, pronunciation and creativity), and the reading of texts (reading, comprehension and critical thinking). The course will also provide an introduction to technical English in the fields of engineering, aeronautics and mechanics at the corresponding level. Students will become familiar with basic technical vocabulary and will be introduced to B2-level texts related to their field of study.

Prerequisites

No prerequisites have been established for this course.

Learning outcomes

Not applicable.

Learning outcomes

RS6 – To analyse texts in the field of mathematics with the aim of becoming familiar with the structure and scientific publications, including those in English. TYPE: Skills.

RODS – To develop effective communication, teamwork, analytical thinking, creativity and ethical leadership from a cross-disciplinary perspective, clearly inspired by democratic principles and values, as well as the Sustainable Development Goals, in order to conduct oneself with integrity in the professional sphere./ To develop effective communication, teamwork, analytical thinking, creativity and ethical leadership from a cross-disciplinary perspective, clearly inspired by democratic principles and values, as well as the Sustainable Development Goals, in order to perform with integrity in the professional sphere. TYPE: Competences.

Course description

The content of this module is designed to enable students to acquire the skills required for reading comprehension, listening comprehension, spoken communication and written communication, allowing them to function effectively in a professional context in English./The content of this course is designed to enable students to develop the skills required for reading comprehension, listening comprehension, spoken communication and written communication, allowing them to function effectively in a professional context in English.

The course will cover a combination of basic English language content, focusing on the study and refinement of language use in various everyday contexts, and technical English, focusing on the study of vocabulary and concepts specific to different areas of specialisation./The course combines general English language content, focusing on the study and development of language use in a range of everyday contexts, with technical English, focusing on the acquisition of vocabulary and concepts specific to different areas of professional specialisation.

Unit 1 – Innovations

1.1 Eureka! (p. 4) – Talking about innovations. Grammar: Past and present perfect continuous

1.2 Smart wells (p. 6) – Clause linking;;;; Reporting on completed tasks. Grammar: Past participle and cohesion

1.3 Lasers (p. 8) – Technical descriptions;;;; Section markers in a talk.

Unit 2 – Design

2.1 Spin-offs (p. 10) – Function of a device;;;; Grammar: Present and past simple passive

2.2 Specifications (p. 12) – Necessity, ability, recommendation;; Grammar: Modals and semi-modals

2.3 Properties (p. 14) – Describing properties. Grammar: Phrases to encourage participation.

Unit 3 – Systems

3.1 Problems (p. 20) – Low probability, reassurance. Grammar: Present continuous passive and phrases suggesting low risk

3.2 Solutions (p. 22) – Summarising, linking. Grammar: Non-defining relative clauses, present participle, ‘although’

3.3 Controls (p. 24) – Contrasting, note-taking. Grammar: Contrastive linkers

Unit 4 – Procedures

4.1 Shutdowns (p. 26) – Past events. Grammar: Two-part phrasal verbs

4.2 Overhaul (p. 28) – Past procedure; instructions. Grammar: Nouns derived from phrasal verbs.

4.3 Instructions (p. 30) – Instructions and simultaneous actions. Grammar: Spoken versus written instructions.

Unit 5 – Processes

5.1 Causes (p. 36) – Cause and effect. Grammar: Verb, noun and prepositional phrases of cause and effect.

5.2 Steps (p. 38) – Explaining a process. Grammar: Choosing between the active and passive voice.

5.3 Stages (p. 40) – Note-taking and writing up. Grammar: Gerunds and nouns as captions; lexical cohesion.

Unit 6 – Planning

6.1 Risk (p. 42) – Degrees of certainty. Grammar: Phrases expressing degrees of certainty.

6.2 Crisis (p. 44) – Immediate and long-term plans. Grammar: Future/future perfect passive

6.3 Projects (p. 46) – Taking part in meetings. Grammar: Phrases used when chairing a meeting.

Learning activities

AP1. Lectures.

AP2. Interactive classes.

AP3.1. Project development and challenge-based problem solving.

AP4. Independent study, case and problem solving, and literature search.

AP5. Tutorials.

AP6. Knowledge assessment tests.

Assessment system and criteria

The assessment process will be carried out with the aim of achieving the learning outcomes set out in the course description.

The assessments carried out will evaluate the four language skills (reading comprehension, listening comprehension, written expression and oral expression).

These tests will be:

• Writing tests.

• Written tests comprising multiple-choice, true or false, fill-in-the-blank and short-answer questions.

• Reading and reading comprehension exercises.

• Vocabulary and grammar exercises.

• Completing and presenting assignments.

• Listening comprehension tests.

• Oral expression tests.

CONTINUOUS ASSESSMENT

Students will be assessed through continuous assessment, as follows:

Mid-term test 1 (35% of total mark): all four skills will be assessed

Final exam (35 per cent): all four skills will be assessed

1 Oral presentations (20%) (If a student passes the oral test with a minimum mark of 5 in class, this mark will be carried forward to the main examination session)

Classwork (behaviour/attitude in class, attendance and active participation, completion of assignments): 10%

IMPORTANT:

Should a student have not sat any of the mid-term tests or fail them, the ordinary examination will account for 100% of the mark. In this case, students will be assessed as follows:

Writing 25%

Reading 25%

Listening 25%

Oral: 25%

Once all continuous assessment tests have been completed, if the overall average mark in any of the skills (reading, writing, listening and/or speaking) is below 2.5, no average mark will be calculated. In this case, the final mark will be a maximum of 3 and, therefore, the student must sit the corresponding ordinary examination session.

FINAL EXAM: REGULAR EXAM SESSION WITHOUT CONTINUOUS ASSESSMENT AND/OR EXTRAORDINARY EXAM SESSION.

Students will be assessed as follows:

Writing 25%

Reading 25%

Listening 25%

Oral 25%

-------------------------

The assessment process will be carried out with the aim of achieving the intended learning outcomes set out in the programme documentation.

The assessment tests will evaluate the four language skills: reading comprehension, listening comprehension, written production and spoken production.

The assessment activities will include:

• Writing tasks.

• Written tests comprising multiple-choice questions, true-or-false questions, gap-filling exercises and short-answer questions.

• Reading and reading comprehension exercises.

• Vocabulary and grammar exercises.

• Preparation and delivery of presentations.

• Listening comprehension tests.

• Speaking tests.

CONTINUOUS ASSESSMENT

Students will be assessed through continuous assessment as follows:

Mid-term Examination 1 (35 per cent): all four language skills will be assessed.

Final Examination (35%): all four language skills will be assessed.

Oral Presentations (20 per cent): if a student passes the oral assessment in class with a minimum mark of 5 out of 10, the mark obtained will be carried forward to the ordinary examination period.

Classwork (10 per cent): including behaviour and attitude in class, attendance and active participation, and completion of assigned tasks.

IMPORTANT

If a student does not sit or fails any of the mid-term examinations, the Ordinary Examination will account for 100 per cent of the final mark. In this case, students will be assessed as follows:

Writing: 25%

Reading: 25%

Listening: 25%

Speaking: 25%

Once all continuous assessment activities have been completed, if the overall average mark in any of the four language skills (Reading, Writing, Listening and/or Speaking) is below 2.5 out of 10, the marks cannot be averaged. In this case, the final mark will be capped at 3 out of 10, and the student will therefore be required to sit the corresponding Ordinary Examination.

FINAL EXAMINATION: ORDINARY EXAMINATION WITHOUT CONTINUOUS ASSESSMENT AND/OR EXTRAORDINARY EXAMINATION

Students will be assessed as follows:

Writing: 25%

Reading: 25%

Listening: 25%

Speaking: 25%

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	MG	Course introduction.
2	MG	Unit 1 (1)
3	MG	Unit 1 (2)
4	MG	Unit 1 (3)
5	MG	Unit 1 (revision + extra activity)
6	MG	Unit 2 (1)
7	MG	Unit 2 (2)
8	MG	Unit 2 Speaking
9	MG	Unit 2 (3)
10	EV	FIRST MID-TERM	35%
11	MG	FIRST MID-TERM
12	MG	Unit 4 (1)
14	EV	First oral presentation	10%
15	MG	First oral presentation
16	MG	Workbook revision
17	MG	Workbook revision
18	MG	Unit 4 (2)
19	MG	Unit 4 (2)
20	MG	Unit 4 (3)
21	MG	Unit 4 (3)
22	EV	Second presentation	10%
23	MG	Second presentation
24	MG	Second presentation
25	MG	Second presentation
26	MG	Correcting Presentations
27	MG	Correcting presentations
28	MG	Unit 5 (1)
29	MG	Unit 5 (2, 3)
30	MG	Unit 5 (revision)
31	MG	Workbook revision
32	MG	Workbook revision
33	MG	Practise writing
34	MG	Practise writing
35	MG	Unit 6 (1)
36	MG	Unit 6 (2)
37	MG	Unit 6 (3 and revision)
38	MG	Practise writing
39	MG	Practise writing
40	MG	Unit 7 (1)
41	MG	Unit 7 (2)
42	MG	Unit 7 (3)
43	MG	Revision and workbook
44	MG	Unit 8 (1)
45	MG	Unit 8 (2)
46	MG	Unit 8 (3)
47	MG	Practise writing skills
48	MG	Practise writing skills
49	MG	Unit 8 (revision)
50	MG	Workbook revision
51	MG	Workbook revision
52	MG	Unit 10(1)
53	MG	Unit 10 (2)
54	MG	Unit 10 (3 and revision)
55	MG	Workbook revision
56	MG	Workbook revision
57	MG	Revision and workbook
58	MG	Revision and workbook
59	EV	Final Exam
60	EV	Final Exam	35%

Reading List

Core:

1. Chris Jacques

Technical English 4

Pearson. 2022.

ISBN: 1292424478

//www.pearson.com

C0242307

Partial Differential Equations

Código: C0242307 Imprimir

Year 2, Second Term. Compulsory. 6 credits.

Profesores

José Luis Guijarro Regalado - Coordinador

Ver biodata

Objectives

This module offers many possible approaches. One could study PDEs in n dimensions, or the properties of solutions to certain types of PDEs, etc., but this course has been designed to focus on the solution of PDEs in two and three variables.

Prerequisites

No prerequisites have been set for this module. However, it is strongly recommended that students have completed or are currently taking the modules on calculus of one and several real variables, as well as Differential Equations and Difference Equations.

Learning Outcomes

Not applicable.

Learning Outcomes

RC2 – Identify the different stages of the mathematical modelling process, distinguishing between formulation, analysis, solution and interpretation of results. TYPE: Competencies.

RC3 – Apply software to solve mathematical and engineering problems, using the appropriate computational environment for each case. TYPE: Competencies.

RC7 – Carry out work and projects related to mathematical engineering that have an impact on interdisciplinary teams or in multicultural contexts. TYPE: Competencies.

RC9 – Formulate problems from a professional context in mathematical language in such a way as to facilitate their analysis and resolution. TYPE: Competences.

RK4 – Be familiar with rigorous proofs of certain classical theorems in different areas of mathematics. TYPE: Knowledge or content.

RS1 – Use mathematical language to state propositions in different fields and construct proofs for the problems posed. TYPE: Skills or abilities.

RS5 – Use computer applications for statistical analysis, numerical and symbolic calculation, graphical visualisation, optimisation or other purposes to solve problems. TYPE: Skills or competences.

Description of the content

Topic 1: Linear PDEs in 2 and 3 variables. Semilinear PDEs. The Pfaffian. General PDE1.

Topic 2: Continuous functions and their completion to a Hilbert space via the Lebesgue integral. The basis theorem. Fourier basis and series expansions.

Topic 3: The Sturm–Liouville problem. Eigenvalues and eigenfunctions.

Topic 4: Solving linear second-order differential equations in two and three variables.

Training activities

AP1. Lectures.

AP2. Interactive classes.

AP3.1. Project work and problem-solving.

AP4. Independent study, case studies, problem-solving and literature review.

AP5. Tutoring.

AP6. Knowledge assessments.

Assessment system and criteria

REGULAR EXAM SESSION

- Two non-exemption tests will be held, each accounting for 15 per cent of the mark.

- A final exam covering the entire course will be held during the ordinary assessment period, accounting for 70% of the mark.

Important: to be included in the average mark, students must achieve a minimum of 4.0 out of 10.0 in the final exam. In this case, the average mark will be calculated even if the continuous assessment mark is a fail.

SUPPLEMENTARY EXAMINATION PERIOD

- A comprehensive exam covering the entire syllabus will be held, accounting for 100% of the mark.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	SESSION	Introduction to the course.
2	SESSION	Method of Separation of Variables
3	ASSIGNMENT	Basic concepts of the theory of partial differential equations
4	ASSIGNMENT	Basic concepts of the theory of partial differential equations
5	SESSION	Basic concepts of the theory of partial differential equations
6	SESSION	Problems
7	ASSIGNMENT	Heat equation
8	ASSIGNMENT	Preliminary concepts of second-order linear PDEs
9	SESSION	The general solution
10	SESSION	Problems
11	ASSIGNMENT	The solution space of the homogeneous equation
12	ASSIGNMENT	The solution space of the homogeneous equation
13	SESSION	The solution space of the homogeneous equation
14	SESSION	Classification of second-order linear PDEs
15	ASSIGNMENT	Classification of second-order linear PDEs
16	TRAB	Classification of second-order linear PDEs
17	SESSION	Introduction to the heat equation
18	SESSION	Problems
19	ASSIGNMENT	Isolated bar with zero temperature at the ends
20	PROBLEM	Method of separation of variables
21	SESSION	Modelling problems
22	SESSION	Problem modelling
23	ASSIGNMENT	The wave equation
24	ASSIGNMENT	The wave equation
25	SESSION	The wave equation
26	SESSION	Problems
27	ASSIGNMENT	Small transverse oscillations of a string
28	EXERCISE	Small transverse oscillations of a string
29	SESSION	Small transverse oscillations of a string
30	EV	Control 1	15%
31	TRAB	Oscillation of a constrained string
32	ASSIGNMENT	Homogeneous equation with homogeneous boundary conditions
33	SESSION	Applications
34	SESSION	Problems
35	ASSIGNMENT	Laplace’s equation
36	ASSIGNMENT	Laplace’s equation in a circle
37	SESSION	Applications
38	SESSION	Problems
39	ASSIGNMENT	Evolutionary EDPs
40	ASSIGNMENT	Poisson’s equation
41	SESSION	Poisson’s equation in different domains
42	SESSION	Problems
43	ASSIGNMENT	Cauchy problem for the heat equation
44	ASSIGNMENT	Cauchy problem for the heat equation
45	SESSION	Cauchy problem for the wave equation
46	SESSION	Problems
47	ASSIGNMENT	Cauchy problem for the wave equation
48	ASSIGNMENT	Laplace transform
49	SESSION	Fourier transform
50	SESSION	Fourier transform
51	ASSIGNMENT	Partial differential equations in bounded domains
52	ASSIGNMENT	Sturm–Liouville theory
53	SESSION	Sturm-Liouville Theory
54	SESSION	Sturm–Liouville Theory
55	ASSIGNMENT	Sturm-Liouville Theory
56	ASSIGNMENT	Applications
57	SESSION	Applications
58	SESSION	Applications
59	WORK	Applications
60	EV	Control 2	15%

Bibliography

Core:

1. Gregorio Orozco and José Luis Guijarro

Partial Differential Equations: A Mathematical Course Focused on Solving Problems in Physics and Engineering

Bellisco. 2011.

ISBN: 8495277166

C0242308

Introduction to Parallel and Distributed Programming / Introduction to Parallel and Distributed Programming

Código: C0242308 Imprimir

Year 2, Second Term. Compulsory. 6 credits.

Profesores

Fausto Ruiz Madrid - Coordinador

Ver biodata

Objectives

The course aims to provide students with the fundamentals and terminology of parallel, concurrent and distributed programming, distinguishing between the main concepts, models and computing paradigms. It also aims to develop students’ ability to analyse and design computational solutions, emphasising the importance of planning and prior analysis of the problem, identifying potential bottlenecks and proposing optimisation strategies aimed at improving efficiency and reducing computation times.

The aim is to equip students to design and implement parallel, concurrent and distributed solutions using Python and its modules and libraries, selecting programming models—such as multithreading and multiprocessing—in accordance with the characteristics of each problem. It also aims to provide an applied perspective on parallel and distributed programming in fields such as scientific computing, simulation, graphics, computer vision, collaborative environments, web applications and the processing of large volumes of data.

Finally, the aim is to integrate the fundamentals of parallel and distributed programming, scientific computing and distributed programming into the development of solutions and projects, covering high-performance input and output techniques, strategies for optimising communication and resource access, as well as different programming environments based on parallel and distributed models and paradigms, and the use of computational accelerators.

-------------------------

The course aims to provide students with the foundations and terminology of parallel, concurrent and distributed programming, distinguishing between the main concepts, models and computing paradigms. It also aims to develop students’ ability to analyse and design computational solutions, recognising the importance of planning and prior problem analysis, identifying potential bottlenecks, and developing optimisation strategies aimed at improving efficiency and reducing computation times.

The course is intended to equip students with the skills required to design and implement parallel, concurrent and distributed solutions using Python and its modules and libraries, selecting programming models such as multithreading and multiprocessing according to the characteristics of each problem. It also provides an applied perspective on parallel and distributed programming in areas such as scientific computing, simulation, graphics, computer vision, collaborative environments, web applications and large-scale data processing.

Finally, the course aims to integrate the foundations of parallel and distributed programming, scientific computing and distributed computing into the development of solutions and projects, addressing high-performance input/output techniques, strategies for optimising communication and resource access, as well as different programming environments based on parallel and distributed models and paradigms and the use of computational accelerators.

Prerequisites

It is essential to have previously completed the modules ‘Fundamentals of Programming and Computers’ and ‘Data Structures and Algorithms I and II’, or other modules with similar competences and learning outcomes./It is essential that students have previously completed the courses ‘Fundamentals of Programming and Computers’ and ‘Data Structures and Algorithms I and II’, or other courses with similar competences and learning outcomes.

Competencies

Not applicable.

Learning Outcomes

RC2 – To identify the different phases of the mathematical modelling process, distinguishing between formulation, analysis, resolution and interpretation of results. TYPE: Competences.

RC3 – To apply software programmes that solve mathematical and engineering problems, using the appropriate computational environment for each case. TYPE: Competences.

RC4 – To be able to plan and develop projects in the field of Mathematical Engineering. TYPE: Competences.

RC7 – To carry out work and projects related to mathematical engineering that have an impact on interdisciplinary teams or in multicultural contexts./ To carry out work and projects related to mathematical engineering that have an impact on interdisciplinary teams or in multicultural contexts. TYPE: Competences./TYPE: Competences.

RS4 – To plan the resolution of a problem in accordance with the available tools and the constraints of time and resources. TYPE: Skills.

RS5 – To use computer applications for statistical analysis, numerical and symbolic calculation, graphical visualisation, optimisation or other purposes to solve problems. TYPE: Skills.

Course description

1) Introduction to Parallel and Distributed Programming

- Basic concepts of concurrency and parallelism: differences and motivations.

- Historical development and rationale for the need for parallel (multiprocessor, multithreaded) and distributed (clusters, clouds) systems.

- Main challenges: synchronisation, communication, scalability and load balancing.

2) Scientific Computing: Requirements

- Issues involving high computational cost in scientific environments (simulations, numerical analysis, big data).

- Methods for utilising available computing capacity: algorithm optimisation, exploitation of hardware architecture (multicore, GPUs).

- Introduction to Python libraries and frameworks for large-scale scientific tasks (NumPy, SciPy, etc.), which are then used with parallel paradigms.

3) Distributed Programming

- Communication models in distributed environments: message passing (MPI), queuing systems, MapReduce and similar.

- Building applications that coordinate multiple nodes (clusters, containers, cloud environments).

- Synchronisation mechanisms and fault management (fault tolerance, data replication, resilience).

4) High-Performance I/O

- Strategies for optimising data flow and reducing bottlenecks in disk or network access.

- Distributed and parallel file systems (HDFS, Lustre, etc.).

- Techniques for buffering, caching and data partitioning for HPC (High-Performance Computing) environments.

5) Parallel Programming

- Parallelism paradigms in Python: threads (threading module), processes (multiprocessing module) and alternatives (futures, asyncio).

- Concurrent programming patterns (producer-consumer, map-reduce, pipelines).

- Security and consistency issues: mutual exclusion, locks, semaphores and risks (race conditions, deadlocks).

6) Model/Paradigm-Based Parallel Programming Environments

- Overview of frameworks and libraries: mpi4py for MPI, Dask and PySpark for distributed data management, amongst others.

- Choosing models based on the type of application (data-intensive vs. computation-intensive).

- Performance validation and scalability metrics (speed-up, efficiency, throughput).

7) Accelerators

- Use of GPUs and coprocessors to accelerate computationally intensive tasks.

- Integration with Python (CUDA-Python, Numba, libraries that utilise the GPU).

- Hybrid models combining multicore CPUs and GPUs to optimise algorithm execution in real-world scenarios.

-------------------------

1) Introduction to Parallel and Distributed Programming

- Basic concepts of concurrency and parallelism: differences and motivations.

- Historical development and rationale for the need for parallel systems (multiprocessors, multithreading) and distributed systems (clusters, cloud computing).

- Main challenges: synchronisation, communication, scalability and load balancing.

2) Scientific Computing: Requirements

- Computationally intensive problems in scientific contexts (simulations, numerical analysis, big data).

- Methods for utilising available computing capacity: algorithm optimisation and exploitation of hardware architectures (multicore, GPUs).

- Introduction to Python libraries and frameworks for large-scale scientific computing (NumPy, SciPy, etc.), subsequently used with parallel programming paradigms.

3) Distributed Programming

- Communication models in distributed environments: message passing (MPI), queue-based systems, MapReduce, and similar approaches.

- Development of applications that coordinate multiple nodes (clusters, containers, cloud environments).

- Synchronisation and fault management mechanisms (fault tolerance, data replication, resilience).

4) High-Performance I/O

- Strategies for optimising data flow and reducing bottlenecks in disk or network access.

- Distributed and parallel file systems (HDFS, Lustre, etc.).

- Buffering, caching and data partitioning techniques for High-Performance Computing (HPC) environments.

5) Parallel Programming

- Parallel programming paradigms in Python: threads (threading module), processes (multiprocessing module), and alternatives (futures, asyncio).

- Concurrent programming patterns (producer–consumer, MapReduce, pipelines).

- Safety and consistency issues: mutual exclusion, locks, semaphores, and risks (race conditions, deadlocks).

6) Parallel Programming Environments Based on Models/Paradigms

- Overview of frameworks and libraries: mpi4py for MPI, Dask and PySpark for distributed data management, amongst others.

- Selection of programming models according to the type of application (data-intensive vs. compute-intensive).

- Performance validation and scalability metrics (speed-up, efficiency, throughput).

7) Accelerators

- Use of GPUs and coprocessors to accelerate computationally intensive tasks.

- Integration with Python (CUDA-Python, Numba, GPU-accelerated libraries).

- Hybrid models combining multicore CPUs and GPUs to optimise algorithm execution in real-world scenarios.

Training activities

AP1. Lectures.

AP2. Interactive classes.

AP3.2 Workshops and computer laboratories.

AP4. Independent study, case and problem solving, and literature search.

AP5. Tutorials.

AP6. Knowledge assessment tests.

Assessment system and criteria

REGULAR EXAMINATION PERIOD

In the regular examination period, the objective assessment of students’ learning will be carried out through continuous assessment.

The weighting of the continuous assessment activities is distributed as follows:

a) Practical 1 (7.5%)

Assessment criteria:

- Correct implementation of the parallelism techniques (multithreading or multiprocessing) covered in class.

- Organisation and clarity of the code, appropriate use of data structures and Python libraries.

- Code documentation (docstrings and comments) explaining the logic and the concurrency patterns chosen.

- Efficiency of the solution in terms of speed improvement compared to the sequential version (where applicable).

b) Assignment 2 (7.5%)

Assessment criteria:

- Use of specific libraries for distributed programming (e.g. MPI, PySpark) and justification for their use.

- Correct architecture of the distributed solution (communication between nodes, load balancing, scalability).

- Validation of results and verification of correct operation in a distributed environment or simulated cluster.

- Cleanliness, maintainability and readability of the code.

c) Final Practical Assignment (15%)

Assessment criteria:

- Integration of knowledge of concurrency (multithreading, multiprocessing) and distribution (clusters, containers, use of HPC libraries).

- Design of an efficient and scalable solution that addresses a complex problem in big data, machine learning or scientific computing, applying parallelism and distribution strategies.

- Quality of project documentation (detailed README, user guides, references to theoretical concepts).

- Presentation of results (performance graphs, tests on different input sizes and assessment of scalability).

d) Non-exemption mid-term exam (30%)

Assessment criteria:

- Understanding of the fundamentals of concurrency in Python (use of threads, processes and thread-safety).

- Ability to design and propose basic parallel solutions.

- Theoretical knowledge of the main distributed programming models (MPI, MapReduce, etc.) and their advantages.

- Problem-solving and short exercises focused on verifying performance and managing shared states.

e) Final examination covering the entire module (40%)

Assessment criteria:

- Comprehensive mastery of parallel and distributed programming: theory of concurrency, synchronisation, deadlocks, scalability and distributed architectures.

- Application of design patterns for concurrent and distributed systems.

- Ability to identify bottlenecks and propose performance improvements.

- Both conceptual and practical questions, with an emphasis on solving real-world problems and optimising parallel and/or distributed systems.

IMPORTANT: an average will only be calculated across the practical assessments, the mid-term exam and the final exam if the mark for each and every one of these assessment activities is 4.0 out of 10.0 or higher.

EXTRAORDINARY EXAMINATION SESSION

In the extraordinary examination session, the objective assessment of the student’s learning will be based on a single examination covering the entire module, which will therefore account for 100 per cent of the final mark.

Assessment criteria:

It will consist of theoretical and practical questions covering the entire syllabus, including:

- Concepts of concurrency in Python (processes, threads, GIL).

- Synchronisation and communication techniques (locks, semaphores, queues).

- Distributed programming libraries (MPI, PySpark, Dask, etc.).

- Design and optimisation of parallel and distributed systems (cluster deployment, scalability, fault tolerance).

Conceptual rigour, the ability to solve complex problems and clarity in justifying proposed solutions will be assessed.

A minimum mark of 5 out of 10 is required to pass the module.

-------------------------

ORDINARY EXAMINATION PERIOD

During the Ordinary Examination Period, the objective assessment of student learning will be carried out through continuous assessment.

The weighting of the continuous assessment activities is distributed as follows:

a) Practical Assignment 1 (7.5%)

Assessment criteria:

- Correct implementation of the parallelisation techniques covered in class (multithreading or multiprocessing).

- Organisation and clarity of the code, and appropriate use of Python data structures and libraries.

- Code documentation (docstrings and comments) explaining the logic and the concurrency patterns selected.

- Efficiency of the solution in terms of speed-up compared with the sequential version, where applicable.

b) Practical Assignment 2 (7.5%)

Assessment criteria:

- Use of specific distributed programming libraries (e.g. MPI, PySpark) and justification for their use.

- Correct architecture of the distributed solution (communication between nodes, workload distribution and scalability).

- Validation of results and verification of correct operation in a distributed environment or simulated cluster.

- Code cleanliness, maintainability and readability.

c) Final Practical Assignment (15%)

Assessment criteria:

- Integration of knowledge of concurrency (multithreading, multiprocessing) and distributed computing (clusters, containers, use of HPC libraries).

- Design of an efficient and scalable solution addressing a complex big data, machine learning or scientific computing problem, applying parallelisation and distributed computing strategies.

- Quality of project documentation (detailed README, user guides, and references to theoretical concepts).

- Presentation of results (performance graphs, tests with different input sizes, and scalability assessment).

d) Non-eliminatory mid-term examination (30%)

Assessment criteria:

- Understanding of the fundamentals of concurrency in Python (use of threads, processes and thread safety).

- Ability to design and propose basic parallel solutions.

- Theoretical knowledge of the main distributed programming models (MPI, MapReduce, etc.) and their advantages.

- Problem-solving and short exercises focusing on performance verification and shared-state management.

e) Final examination covering the entire course (40 per cent)

Assessment criteria:

- Comprehensive understanding of parallel and distributed programming: concurrency theory, synchronisation, deadlocks, scalability and distributed architectures.

- Application of design patterns to concurrent and distributed systems.

- Ability to identify bottlenecks and propose performance improvements.

- Both conceptual and practical questions, with an emphasis on solving real-world problems and optimising parallel and/or distributed systems.

IMPORTANT: The marks for the practical assignments, mid-term examination and final examination will only be averaged if the mark obtained in each and every one of these assessment activities is 4.0/10.0 or higher.

EXTRAORDINARY EXAMINATION PERIOD

During the Extraordinary Examination Period, the objective assessment of student learning will be carried out through a single examination covering the entire course. Consequently, this examination will account for 100 per cent of the final mark.

Assessment criteria:

The examination will consist of theoretical and practical questions covering the entire syllabus, including:

- Concepts of concurrency in Python (processes, threads, GIL).

- Synchronisation and communication techniques (locks, semaphores, queues).

- Distributed programming libraries (MPI, PySpark, Dask, etc.).

- Design and optimisation of parallel and distributed systems (cluster deployment, scalability, fault tolerance).

Conceptual soundness, the ability to solve complex problems, and clarity in justifying the proposed solutions will be assessed.

A minimum mark of 5.0/10.0 is required to pass the course.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	SESSION	Topic 1: Introduction to Parallel and Distributed Programming
2	SESSION	Topic 1: Introduction to Parallel and Distributed Programming
3	ASSIGNMENT	Topic 1: Introduction to Parallel and Distributed Programming
4	ASSIGNMENT	Topic 1: Introduction to Parallel and Distributed Programming
5	SESSION	Topic 2: Scientific Computing: Requirements
6	SESSION	Topic 2: Scientific Computing: Requirements
7	ASSIGNMENT	Topic 2: Scientific Computing: Requirements
8	ASSIGNMENT	Topic 2: Scientific Computing: Requirements
9	SESSION	Topic 2: Scientific Computing: Requirements
10	SESSION	Topic 2: Scientific Computing: Requirements
11	ASSIGNMENT	Topic 3: Distributed Programming
12	ASSIGNMENT	Topic 3: Distributed Programming
13	SESSION	Topic 3: Distributed Programming
14	SESSION	Topic 3: Distributed Programming
15	ASSIGNMENT	Topic 3: Distributed Programming
16	ASSIGNMENT	Topic 3: Distributed Programming
17	SESSION	Topic 4: High-Performance I/O
18	SESSION	Topic 4: High-Performance I/O
19	ASSIGNMENT	Topic 4: High-Performance I/O
20	ASSIGNMENT	Topic 4: High-Performance I/O
21	SESSION	Topic 4: High-Performance I/O
22	SESSION	Topic 4: High-Performance I/O
23	ASSIGNMENT	Topic 4: High-Performance I/O
24	ASSIGNMENT	Topic 4: High-Performance I/O
25	SESSION	Topic 5: Parallel Programming
26	SESSION	Topic 5: Parallel Programming
27	ASSIGNMENT	Topic 5: Parallel Programming
28	ASSIGNMENT	Topic 5: Parallel Programming
29	SESSION	Topic 5: Parallel Programming
30	SESSION	Topic 5: Parallel Programming
31	EV	Practical 1
32	EV	Practical 1	7.5%
33	SESSION	Topic 5: Parallel Programming
34	SESSION	Topic 5: Parallel Programming
35	ASSIGNMENT	Topic 5: Parallel Programming
36	ASSIGNMENT	Topic 5: Parallel Programming
37	EV	Mid-term exam.
38	EV	Mid-term exam.	30%
39	ASSIGNMENT	Topic 6: Model-based/Paradigm-based Parallel Programming Environments
40	ASSIGNMENT	Topic 6: Model- and Paradigm-Based Parallel Programming Environments
41	SESSION	Topic 6: Model- and Paradigm-Based Parallel Programming Environments
42	SESSION	Topic 6: Model- and Paradigm-Based Parallel Programming Environments
43	ASSIGNMENT	Topic 6: Model- and Paradigm-Based Parallel Programming Environments
44	ASSIGNMENT	Topic 6: Model- and Paradigm-Based Parallel Programming Environments
45	SESSION	Topic 6: Model- and Paradigm-Based Parallel Programming Environments
46	SESSION	Topic 6: Model- and Paradigm-Based Parallel Programming Environments
47	ASSIGNMENT	Topic 7: Accelerators
48	ASSIGNMENT	Topic 7: Accelerators
49	SESSION	Topic 7: Accelerators
50	SESSION	Topic 7: Accelerators
51	EV	Practical 2
52	EV	Practical 2	7.5%
53	SESSION	Topic 7: Accelerators
54	SESSION	Topic 7: Accelerators
55	ASSIGNMENT	Topic 7: Accelerators
56	SESSION	Topic 7: Accelerators
57	EV	Final practical
58	EV	Final practical	15%
59	EV	Final exam.
60	EV	Final exam.	40%

Reading list

Core:

1. Giancarlo Zaccone

Python Parallel Programming Cookbook

Packt Pub Ltd. 2015.

ISBN: 1785289586

C0242309

Numerical methods

Código: C0242309 Imprimir

Year 2 Course. Second term module. Compulsory. 6 credits.

Profesores

José Antonio Prieto Persiguero - Coordinador

Ver biodata

Aims

The course ‘Numerical Methods’ introduces students to fundamental methods, together with the associated concepts, developed to solve mathematical problems approximately when it is not possible to obtain an analytical solution. It focuses not only on obtaining an approximate solution but also, and above all, on analysing the error introduced by that approximation, its stability, and the efficiency of the methods applicable to solving a problem. Students’ theoretical and practical learning is complemented by the computational implementation of these methods, thereby strengthening their programming skills and their ability to critically analyse the results obtained.

In this context, the learning objectives are:

- To understand the fundamental principles of numerical methods and their importance in solving mathematical and, consequently, engineering problems.

- To analyse and evaluate the numerical errors associated with the different methods, including their origin, propagation and minimisation.

- To develop the ability to select the most appropriate numerical method for solving a given mathematical problem.

- To implement numerical algorithms using modern computational tools, such as Python or MATLAB.

- To interpret and validate the results obtained using numerical methods, taking into account their accuracy and applicability.

- Apply the knowledge acquired to practical problems related to mathematics and engineering, fostering a critical and reflective approach.

Prerequisites

A solid foundation in differential and integral calculus in one variable (Mathematical Foundations of Engineering I and II), as well as in linear algebra (Algebra I and II), is recommended. Knowledge of Python programming is also advisable.

Competencies

Not applicable.

Learning Outcomes

RC2 – Identify the different phases of the mathematical modelling process, distinguishing between formulation, analysis, solution and interpretation of results. TYPE: Competencies.

RC3 – Apply software to solve mathematical and engineering problems, using the appropriate computational environment for each case. TYPE: Competencies.

RK5 - Be familiar with various simulation models, stochastic simulation, and the management and planning of logistics systems; and use software to solve cases involving production management and planning models. TYPE: Knowledge or content.

RS1 – Use mathematical language to formulate propositions in different fields and construct proofs for the problems posed. TYPE: Skills or competences.

RS4 – Plan the resolution of a problem in accordance with the available tools and the constraints of time and resources. TYPE: Skills or competences.

RS5 – Use computer applications for statistical analysis, numerical and symbolic calculation, graphical visualisation, optimisation or other purposes to solve problems. TYPE: Skills or competences.

Course content

This module is structured around 5 topics:

- Topic 1. Introduction to numerical methods.

- Topic 2. Non-linear equations.

- Topic 3. Numerical interpolation.

- Topic 4. Numerical differentiation and integration.

- Topic 5. Numerical linear algebra.

Teaching activities

AP1. Lectures.

AP2. Interactive classes.

AP3.1. Project work and problem-solving.

AP4. Independent study, case studies, problem-solving and literature review.

AP5. Tutoring.

AP6. Knowledge assessments.

Assessment system and criteria

The assessment process will consist of evaluating the extent to which students have achieved the learning outcomes associated with the module.

ASSESSMENT SYSTEMS

The assessment systems for this module are:

 AS1 Practical activities.

 SE2 Final knowledge assessments.

 AS3 Portfolio.

These systems contribute to a greater or lesser extent to the assessment of the learning outcomes assigned to this subject.

ASSESSMENT CRITERIA

There are two official examination sessions: the ordinary and the supplementary.

+++REGULAR EXAMINATION PERIOD+++

The final mark for this assessment period will be the weighted average of a set of assessment tests

detailed below:

-- a case study (SE1), accounting for 30 per cent of the final mark for the ordinary assessment period, to be carried out

during the teaching period in small groups (assigned by the course coordinator) and for which

will require both the submission of exercises whilst the case study is in progress and the submission of a report at the end of the term.

-- a mid-term exam (SE3) which is not an optional component; this will be held in a classroom on an individual basis during the teaching period and

which will account for 20 per cent of the final mark for the standard assessment period.

-- a final exam (SE2), to be taken individually in a classroom during the

the standard examination period, in May–June (for further information, please consult the virtual campus), which

assesses the entirety of the course content, and which will account for 50 per cent of the

final mark for the standard assessment period, provided that the student achieves a mark equal to or

4.0 out of 10.0. Otherwise (a mark lower than 4.0 out of 10.0), the mark for the module in the ordinary examination period will be that obtained in the final examination.

*** Only the examinations will be subject to re-marking.

***** The module will be deemed to have been passed in the ordinary examination period if the final mark is 5.0 points

out of 10.0 or higher.

+++EXTRA SESSION+++

If a student fails the module in the ordinary examination period, they may retake it in the

extraordinary sitting.

In this sitting, there will be a single assessment, consisting of an exam to be held

during the extraordinary examination period, June–July (for further information, please consult

the virtual campus), which will assess the entire syllabus covered in the module.

***** The module will be deemed to have been passed in the supplementary sitting if the mark obtained in that

exam is 5.0 out of 10.0 or higher.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	SESSION	Introduction to numerical methods.
2	SESSION	Standardised floating-point format. Finite-digit arithmetic: approximations and errors. Arithmetic operations and loss of precision.
3	ASSIGNMENT	Reformulation of calculations and algorithms.
4	ASSIGNMENT	Exercises (sessions 1–3).
5	SESSION	Binary machine numbers (IEEE 754 standard). Types. Numerical range. Precision.
6	SESSION	Exercises (session 5).
7	ASSIGNMENT	Non-linear equations. Multiplicity and parity of roots.
8	ASSIGNMENT	Exercises (session 7).
9	SESSION	Bisection method. Upper bound on error and sufficient number of iterations.
10	SESSION	Exercises (Session 9).
11	ASSIGNMENT	Fixed-point method. Consistency. Fixed-point theorem and convergence. Sufficient number of iterations.
12	ASSIGNMENT	Exercises (session 11).
13	SESSION	Convergence speed and loss of accuracy.
14	SESSION	Exercises (session 13).
15	ASSIGNMENT	Newton-Raphson method.
16	ASSIGNMENT	Exercises (session 15).
17	SESSION	Multiple roots and convergence rate.
18	SESSION	Exercises (session 17).
19	ASSIGNMENT	The secant method. Other methods.
20	ASSIGNMENT	Exercises (session 20).
21	SESSION	Taylor’s polynomial. Weierstrass’s approximation theorem and polynomial interpolation.
22	SESSION	Lagrange interpolation. Truncation error. Neville’s method.
23	ASSIGNMENT	Osculating polynomial. Hermite interpolation. Truncation error.
24	ASSIGNMENT	Exercises (sessions 21–23).
25	SESSION	Differential equations.
26	SESSION	Piece-wise polynomial approximation.
27	ASSIGNMENT	Natural and conditioned splines.
28	ASSIGNMENT	Exercises (sessions 25–27).
29	SESSION	Mid-term exam preparation exercises
30	SESSION	Mid-term exam preparation exercises
31	EV	Mid-term exam
32	EV	Mid-term exam	20%
33	SESSION	Formulas for the first derivative in differences. Truncation error.
34	SESSION	Higher-order derivatives.
35	ASSIGNMENT	Rounding error and stability of the total error.
36	ASSIGNMENT	Exercises (sessions 33–35).
37	SESSION	Simple numerical integration: explicit formulas (trapezium rule, Simpson’s rule, etc.). Truncation error.
38	SESSION	Simple numerical integration: open-form formulas (midpoint rule, etc.). Truncation error.
39	ASSIGNMENT	Closed and open Newton-Cottes formulas.
40	ASSIGNMENT	Exercises (sessions 37–39).
41	SESSION	Compound numerical integration. Rounding error. Stability of the total error.
42	SESSION	Exercises (session 41).
43	ASSIGNMENT	Solving systems of linear equations and rounding error stability. Pivoting strategies (partial, scaled partial and full).
44	ASSIGNMENT	Exercises (session 43).
45	SESSION	Factorisation of a square matrix.
46	SESSION	Exercises (session 45).
47	ASSIGNMENT	Iterative methods: Jacobi, Gauss-Seidel and others. Truncation error.
48	ASSIGNMENT	Exercises (session 47).
49	SESSION	Iterative refinement; conditioning number.
50	SESSION	Exercises (session 49).
51	ASSIGNMENT	Maximum descent method.
52	ASSIGNMENT	Exercises (session 51).
53	SESSION	Numerical methods for diagonalisation by similarity: power, inverse power and inverse power with seed.
54	SESSION	Exercises (session 53).
55	ASSIGNMENT	Householder’s method / QR method.
56	ASSIGNMENT	Examples (session 55).
57	SESSION	Singular value decomposition of a matrix. Linear least squares.
58	SESSION	Pseudoinverse of a matrix.
59	ASSIGNMENT	Exercises (sessions 57–58).
60	EV	Presentation of case study results	30%

Bibliography

Core:

1. Burden, Richard L.

Numerical Analysis

7th ed. Mexico City: Thomson, 2002. 2002.

ISBN: 9706861343

2. Sánchez, Juan Miguel

Problems in Numerical Calculus for Engineers with Applications

Madrid: McGraw-Hill, 2005. 2005.

ISBN: 8448129512

Supplementary:

3.- Vázquez Espí, Carlos

Numerical Analysis /

Madrid: García-Maroto, 2013. 2013.

ISBN: 9788415793069

Others:

4.- Demidóvich, B. P.

Fundamental Numerical Calculus

3rd ed. Madrid: Paraninfo, 1988. 1988.

ISBN: 842830887X

5.- Gerald, Curtis F.

Numerical Analysis with Applications

6th ed. Mexico [etc.]: Pearson Educación, 2000. 2000.

ISBN: 9684443935

C0242604

Thermodynamics

Código: C0242604 Imprimir

Year 2 Course. Second term module. Compulsory. 6 credits.

Profesores

Arantxa Tymowska N/A - Coordinador

Ver biodata

Objectives

The aim of the module is to equip students with the necessary tools to understand thermodynamic processes and the principles and laws that govern them, so that they are able to successfully analyse and solve problems within the field of thermodynamics.

Prerequisites

None

Learning Outcomes

RK1 To be familiar with the most important phenomena and theories in the various branches of physics, as well as their historical context

RK2 Understand physically distinct phenomena and their underlying analogies, enabling the application of known solutions to new problems

RK3 To analyse the fundamental concepts and principles of physical systems in order to develop approximations that enable the construction of a simplified model

RK7 Understand the laws and principles of physics, identifying their logical and mathematical structure, their experimental basis and the phenomena described by them

RS1 Apply the most important knowledge, concepts and methods from the various branches of physics.

RS4 Apply mathematical and numerical methods to the modelling and explicit solution of problems in physics and related disciplines, selecting the appropriate tools and interpreting results.

RS10 Apply the principles and laws of thermodynamics involved in the analysis of physical phenomena.

Learning outcomes

RA1 Identifies the essential aspects of physical phenomena, describing them quantitatively and qualitatively using thermodynamic formalism

RA2 Presents and interprets thermodynamic information (graphs, tables, etc.)

RA3 Understands the First Law as a general principle of energy conservation, with a state function, namely internal energy.

LA4 States the Laws of Thermodynamics, analyses their implications and applies them to problem-solving

RA5 Understands how entropy and its properties account for the thermodynamic behaviour of systems.

RA6 Identifies thermodynamic potentials and analyses the thermodynamic behaviour of systems.

Description of the content

- The zeroth law. The concept of temperature.

- Thermodynamic relations.

- Fundamental thermodynamic equation.

- Thermodynamic processes

- First law: work, internal energy and heat. Enthalpy.

- Second law: entropy.

- Thermodynamic potentials, equilibrium and stability.

- Open systems, phase transitions, critical points.

- Third law.

Training activities

Training activity No. of hours* Contact hours

(8–12)** % Face-to-face

AP1.- Participatory lectures 24 4 100

AP2.- Seminars or practical application classes 15 2.5 100

AP3.- Practical activities (case studies, project work, simulations, etc.) 36 3 50

AP4. – Independent study 60 0 0

AP5.- Tutorials 12 0.6 30

AP6.- Knowledge assessments 3 0.5 100

Assessment system and criteria

Regular assessment period.

Assessment system | Weighting

SE1.- Practical activities | 25. Mid-term exam

SE2.- Final knowledge assessments | 60. Final exam (ordinary examination session)

SE4.- Portfolio | 15. This assessment system involves the daily submission of questions and problems not solved in class by the lecturer. This assessment system will be weighted against the student’s overall attendance for the module: if, at the end of the academic year, the student’s attendance at the scheduled class hours (SESSION, LAB) is 60 per cent or higher, the mark for this system will be that obtained following the lecturer’s marking. If the student’s attendance at all scheduled course sessions (SESSION and LAB) is less than 60 per cent, the mark for this assessment system, as determined following the teacher’s marking, will be reduced in accordance with the level of attendance achieved, in accordance with an attendance criterion defined by the lecturer.

In order for the continuous assessment (comprising Practical Activities (SE1) and the Portfolio (SE4)) to be taken into account, students must

achieve a mark of 5.0/10 or higher in the final exam of the standard examination session. Otherwise, their mark will correspond directly to the

mark obtained in that exam.

The module is considered passed in the ordinary examination period if the final mark is 5.0/10 or higher.

Supplementary examination period

If a student has not passed the module in the ordinary examination session, they may sit the extraordinary examination session. The extraordinary examination session will take place during the June and July examination periods (for further information, please consult the Academic Calendar).

100 per cent of the final mark for the module in the supplementary sitting is based on the final exam for this sitting.

The module is considered passed in the supplementary examination period if the final mark is 5.0/10 or higher.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	MG	Zero principle. Concept of temperature.	No
1	MG	Zero principle. Concept of temperature.	No
2	MG	Zero principle. Concept of temperature.	No
2	MG	Thermodynamic relationships.	No
3	MG	Thermodynamic relationships.	No
3	MG	Thermodynamic relationships.	No
4	MG	Fundamental thermodynamic equation.	No
4	MG	Thermodynamic relationships.	No
5	MG	Fundamental thermodynamic equation.	No
5	MG	Thermodynamic processes	No
6	MG	Thermodynamic processes	No
6	MG	Thermodynamic processes	No
7	MG	Thermodynamic processes	No
7	MG	First law: work, internal energy and heat. Enthalpy.	No
8	MG	First law: work, internal energy and heat. Enthalpy.	No
8	MG	First law: work, internal energy and heat. Enthalpy.	No
9	MG	First law: work, internal energy and heat. Enthalpy.	No
9	MG	First law: work, internal energy and heat. Enthalpy.	No
10	MG	First law: work, internal energy and heat. Enthalpy.	No
10	MG	First law: work, internal energy and heat. Enthalpy.	No
11	MG	First law: work, internal energy and heat. Enthalpy.	No
11	MG	First law: work, internal energy and heat. Enthalpy.	No
12	MG	First law: work, internal energy and heat. Enthalpy.	No
12	MG	First law: work, internal energy and heat. Enthalpy.	No
13	MG	First law: work, internal energy and heat. Enthalpy.	No
13	MG	First law: work, internal energy and heat. Enthalpy.	No
14	MG	First law: work, internal energy and heat. Enthalpy.	Yes
14	MG	First law: work, internal energy and heat. Enthalpy.	If
15	MG	Second law: entropy.	No
15	MG	Second law: entropy.	No
16	MG	Second law: entropy.	No
16	MG	Second law: entropy.	No
17	MG	Second law: entropy.	No
17	MG	Second law: entropy.	No
18	MG	Second law: entropy.	No
18	MG	Second law: entropy.	No
19	MG	Second law: entropy.	No
19	MG	Second law: entropy.	No
20	MG	Second law: entropy.	No
20	MG	Second law: entropy.	No
21	MG	Thermodynamic potentials, equilibrium and stability.	No
21	MG	Thermodynamic potentials, equilibrium and stability.	No
22	MG	Thermodynamic potentials, equilibrium and stability.	No
22	MG	Thermodynamic potentials, equilibrium and stability.	No
23	MG	Thermodynamic potentials, equilibrium and stability.	No
23	MG	Thermodynamic potentials, equilibrium and stability.	No
24	MG	Thermodynamic potentials, equilibrium and stability.	No
24	MG	Thermodynamic potentials, equilibrium and stability.	No
25	MG	Thermodynamic potentials, equilibrium and stability.	No
25	MG	Open systems, phase transitions, critical points.	No
26	MG	Open systems, phase transitions, critical points.	No
26	MG	Open systems, phase transitions, critical points.	No
27	MG	Open systems, phase transitions, critical points.	No
27	MG	Open systems, phase transitions, critical points.	No
28	MG	Open systems, phase transitions, critical points.	No
28	MG	Open systems, phase transitions, critical points.	No
29	MG	Open systems, phase transitions, critical points.	No
29	MG	Third law.	No
30	MG	Third principle.	No
30	MG	Third principle.	No

C0242608

Mechanics and Waves II / Mechanics and Waves II

Código: C0242608 Imprimir

Year 2, Second Term. Compulsory. 6 credits.

Profesores

José Luis Guijarro Regalado - Coordinador

Ver biodata

Objectives

This course, together with ‘Mechanics and Waves I’, forms part of the Mechanics and Waves module. Its main objective is for students to become familiar with Lagrangian mechanics and Hamiltonian mechanics, two reformulations of Newtonian mechanics upon which much of modern fundamental physics is built (including Quantum Field Theory and General Relativity). In addition to understanding the conceptual foundations of these formalisms, students should be able to apply them to solving mechanical problems.

Furthermore, an introduction to the two-body problem will be provided, and the dynamics of particles interacting via central forces and the resulting orbits will be discussed. Finally, the implications of moving away from the classical notion of absolute space and time and adopting the postulates of Special Relativity in the development of a mechanical theory will be explored.

Prerequisites

No prerequisites

Learning Outcomes

RK1 To be familiar with the most important phenomena and theories in the various branches of physics, as well as their historical context

RK2 To understand physically distinct phenomena and their underlying analogies, enabling the application of known solutions to new problems

RK3 To analyse the fundamental concepts and principles of physical systems in order to develop approximations that enable the construction of a simplified model

RK5 Understand the scope and limitations of classical physics that led to the formulation of special and general relativity, as well as quantum mechanics, to address the new problems arising in modern physics.

RK7 Understand the laws and principles of physics, identifying their logical and mathematical structure, their experimental basis and the phenomena described by them

RK12 Understand the theories, laws and models governing physical phenomena related to mechanics

RS1 Apply the most important knowledge, concepts and methods from the various branches of physics.

RS4 Apply mathematical and numerical methods to the modelling and explicit solution of problems in physics and related disciplines, selecting the appropriate tools and interpreting results.

Learning outcomes

RA5 Applies the Lagrangian and Hamiltonian formulations appropriately to the solution of

mechanical problems.

RA6 Formulates the equations of motion for central forces, solving them completely and obtaining the solutions for the motion and the trajectories.

LA7 Formulates and solves the equations of a system that deviates from its equilibrium position, classifying said equilibrium.

RA8 Gains a deeper understanding of the fundamentals of special relativity and its most significant physical consequences, developing proficiency in the study of particle kinematics and dynamics within the context of Minkowski spacetime.

Description of the content

- Lagrangian mechanics.

- The two-body problem. Central forces. Orbits.

- Hamiltonian mechanics.

- Relativistic mechanics.

Teaching activities

AP1.- Interactive lectures

AP2. Seminars or practical application classes

AP3.- Practical activities (case studies, project work, simulation, etc.)

AP4. Independent study

Assessment system and criteria

ING

The assessment process will consist of verifying and evaluating the student’s acquisition of the required competences.

ASSESSMENT SYSTEMS

The assessment systems for this course are:

- SE1: Practical activities. These consist of two mid-term exams taken during the teaching period.

- SE2: Final knowledge assessment. This corresponds to a final exam covering the entire course syllabus.

- SE4: Portfolio. This assessment method involves the completion and submission of approximately four sets of exercises, which may be carried out individually or in small groups depending on the progress of the course.

ASSESSMENT CRITERIA

The assessment methods described above are set out in the following assessment criteria. There are two official examination sessions: the ordinary and the extraordinary.

ORDINARY SESSION

The final mark for this session is the weighted average of the assessment components detailed below:

- Practical activities (SE1): 25 per cent of the final mark (12.5 per cent for each mid-term exam).

- Final knowledge assessment (SE2): 60 per cent of the final mark.

- Portfolio (SE4): 15 per cent of the final mark.

For the continuous assessment (comprising the portfolio and the two mid-term exams) to be taken into account, students must achieve a minimum mark of 4.0 in the final exam of the ordinary session. Otherwise, the final mark will correspond directly to the mark achieved in that exam.

The course is considered passed in the ordinary session if the final mark is 5.0 or higher.

EXTRAORDINARY SESSION

If a student does not pass the course in the ordinary session, they may sit the extraordinary session.

The supplementary session takes place during the July examination period (for further information, please consult the Academic Calendar). It consists of a single examination covering the entire course content. The mark in the supplementary session corresponds directly to the mark obtained in this examination.

The course is considered passed in the extraordinary session if the final mark is 5.0 or higher.

GRADING SYSTEM

Article 5 of Royal Decree 1125/2003, of 5 September, sets out the grading system applicable to modules within degree programmes in the European Higher Education Area.

To obtain the corresponding credits, students must pass the exams or assessment tests associated with the course.

The level of learning achieved by students will be expressed using numerical marks on a scale from 0 to 10, to one decimal place, to which a qualitative grade may be added:

- 0–4.9: Fail (SS).

- 5.0–6.9: Pass (AP).

- 7.0–8.9: Good (NT).

- 9.0–10: Excellent (SB).

The distinction ‘Honours’ (Matrícula de Honor) may be awarded to students who achieve a mark of 9.0 or higher. The number of such distinctions may not exceed five per cent of the students enrolled on the course during the relevant academic year, unless the number of enrolled students is fewer than 20, in which case only one ‘Honours’ distinction may be awarded.

ESP

The assessment process will consist of verifying and evaluating the student’s acquisition of the required competences.

ASSESSMENT SYSTEMS

The assessment methods for this module are:

- AS1: Practical activities. These consist of two mid-term examinations held during the term.

- AS2: Final knowledge assessments. These consist of a final exam covering the entire syllabus of the module.

- AS4: Portfolio. This assessment method involves completing and submitting approximately four sets of exercises, which may be undertaken individually or in small groups depending on the progress of the course.

ASSESSMENT CRITERIA

The assessment systems described above are set out in the following assessment criteria. There are two official examination sessions: the ordinary and the supplementary.

REGULAR EXAMINATION PERIOD

The final mark for this sitting is the weighted average of the assessment tests detailed below:

- Practical activities (SE1): 25 per cent of the final mark (12.5 per cent for each mid-term exam).

- Final knowledge assessments (SE2): 60% of the final mark.

- Portfolio (SE4): 15% of the final mark.

For continuous assessment (comprising the portfolio and the two mid-term exams) to be taken into account, students must achieve a minimum mark of 4.0 in the final exam of the ordinary assessment period. Otherwise, their mark will correspond directly to that obtained in that exam.

The module is considered passed in the ordinary examination period if the final mark is 5.0 or higher.

SUPPLEMENTARY EXAMINATION SESSION

If a student has not passed the module during the ordinary examination period, they may sit the extraordinary examination.

The supplementary examination period will take place during the July examination period (for further information, please consult the Academic Calendar). It consists of a single examination covering the entire syllabus of the module. The mark for the supplementary examination corresponds directly to the mark obtained in this examination.

The module is considered passed in the extraordinary examination session if the final mark is 5.0 or above.

GRADES

The award of the corresponding credits is conditional upon passing the associated examinations or assessment tests.

The level of learning achieved by students will be expressed as numerical marks on a scale of 0 to 10, to one decimal place, to which the corresponding qualitative mark may be added:

- 0–4.9: Fail (SS).

- 5.0–6.9: Pass (AP).

- 7.0–8.9: Good (NT).

- 9.0–10: Distinction (SB).

Bibliography

Essential:

1. A. Fernández Rañada

Classical Dynamics

1st ed. Alianza. 1994.

ISBN: 8420681334

2. H. Goldstein

Classical Mechanics

Addison-Wesley. 1950.

ISBN: 9780201025101

3. J.R. Taylor

Classical Mechanics

Reverté. 2013.

ISBN: 8429143122

4. S.T. Thornton, J.B. Marion

Classical Dynamics of Particles and Systems

Reverté. 1975.

ISBN: 9788429140941

TOTAL:

Third Year

FIRST FOUR-MONTH PERIOD

Code

Subjects

Character*

ECTS

C0242601

Electromagnetism I

Código: C0242601 Imprimir

Year 3, First-term module. Compulsory. 6 credits.

Profesores

Joaquín Bermejo Ortiz - Coordinador

Objectives

Students will learn the fundamental concepts of electromagnetism, with particular emphasis on the effective application of vector calculus to relevant problems in electrostatics, magnetostatics and problems involving boundary conditions.

Prerequisites

None

Learning Outcomes

RA1 Has a firm grasp of the basic description of how electromagnetic fields are generated by charges and currents, and of the effect of these fields on charges.

RA2 Understands how material media behave in the presence of electric and magnetic fields and knows how to calculate these fields.

RA3 Understands and applies analytical and numerical techniques relating to boundary value problems for potential.

RA4 Understands and is able to use Maxwell’s equations in their differential and integral forms.

Learning outcomes

o RK1 Understand the most important phenomena and theories in the various branches of physics, as well as their historical context

o RK2 Understand physically distinct phenomena and their underlying analogies, enabling the application of known solutions to new problems

o RK3 Analyse the fundamental concepts and principles of physical systems in order to develop approximations that enable the construction of a simplified model

o RK7 Understand the laws and principles of physics, identifying their logical and mathematical structure, their experimental basis and the phenomena described by them

o RK11 Understand the fundamentals and basic concepts of electric and magnetic fields, as well as the interrelationship between these fields and their unification in electromagnetism

o RS1 Apply the most important knowledge, concepts and methods from the various branches of physics.

o RS4 Apply mathematical and numerical methods to the modelling and explicit solution of problems in physics and related disciplines, selecting the appropriate tools and interpreting results.

Course content

-Topic 1. Electrostatics

The electric field (discrete and continuous charge distributions), divergence and curl of the electric field (Gauss’s law and irrotationality of electrostatic fields), Electric potential (Poisson’s equation, Laplace’s equation and boundary conditions), Work and energy in electrostatics

-Topic 2. Conducting media

Basic properties, Induced charges, Surface charge and force on a conductor, Capacitors

-Topic 3. Magnetostatics and Electric Current

Magnetic forces (Lorentz forces and steady-state currents), Biot–Savart law, Divergence and curl of the magnetic field (Ampère’s law, non-divergence of the magnetic field)

Training activities

Learning activity | No. of hours (8–12) | Contact hours | % Contact time

AP1.- Participatory lectures | 48 | 4| 100

AP2.- Seminars or practical application classes |30 |2.5 |100

AP3.- Practical activities (case studies, project work, simulations, etc.) |72 |3 |50

AP4. – Independent study |120 |0 |0

AP5. – Tutoring | 24 |0.6 |30

AP6. – Knowledge assessments |6 |0.5 |100

TOTAL 300 10.6

Assessment system and criteria

The assessment process will consist of verifying and evaluating the student’s acquisition of the required competences.

ASSESSMENT SYSTEMS

The assessment systems for this module are:

- AS1: Practical activities (case studies, problem-solving and challenges, project work, oral presentations, debates, etc.).

- AS2: Final knowledge assessments.

- AS4: Portfolio.

ASSESSMENT CRITERIA

The assessment methods described above are set out in the following assessment criteria. There are two official examination sessions: the ordinary and the supplementary.

+++REGULAR EXAMINATION PERIOD+++

The final mark for this sitting is the weighted average of a set of assessment tasks detailed below:

- Practical activities (SE1) 30% of the final mark: A mid-term exam set in advance (20%) + Assignments set by the lecturer (10%).

- Final knowledge assessments (SE2) 60% of the final mark: Final exam for the ordinary assessment period, covering the entire course content.

- Portfolio (SE4): 10% of the final mark. This assessment system involves the daily submission of questions and problems not solved in class by the lecturer. This assessment system will be weighted against the student’s overall attendance for the course: if, at the end of the academic year, the student’s attendance at the scheduled class sessions (SESSION, LAB) is 60 per cent or higher, the mark for this system will be that obtained following the lecturer’s marking. If the student’s attendance at all scheduled course sessions (SESSION and LAB) is less than 60 per cent, the mark for this assessment system, as determined following the lecturer’s marking, will be reduced in accordance with the level of attendance achieved, in accordance with an attendance criterion defined by the lecturer.

In order for the continuous assessment (comprising the Practical Activities (SE1) and the portfolio (SE4)) to be taken into account, students must achieve a mark of 4.0/10 or higher in the final exam of the ordinary examination session. Otherwise, their mark will correspond directly to that obtained in that examination.

The module is considered passed in the ordinary examination period if the mark obtained in accordance with the above guidelines is 5.0 or higher.

+++SUPPLEMENTARY EXAMINATION SESSION+++

If a student has not passed the module in the ordinary examination period, they may sit the extraordinary examination.

The supplementary examination period will take place during the June and July examination periods (for further information, please consult the Academic Calendar).

- 100 per cent of the final mark for the module in the extraordinary sitting corresponds to the final exam for this sitting, which assesses the entire syllabus of the module.

The module is considered to have been passed in the extraordinary examination period if the final mark is 5.0 or higher.

GRADES

The award of the corresponding credits is conditional upon passing the associated examinations or assessment tests.

The level of learning achieved by students will be expressed as numerical marks on a scale of 0 to 10, to one decimal place, to which the corresponding qualitative mark may be added:

- 0–4.9: Fail (SS).

- 5.0–6.9: Pass (AP).

- 7.0–8.9: Good (NT).

- 9.0–10: Distinction (SB).

Bibliography

Essential:

1. Griffiths, David J

Introduction to Electrodynamics / David J. Griffiths

Harlow, UK: Pearson. 2014.

ISBN: 1108420419

2. Richard P. Feynman, Robert B. Leighton, Matthew Sands

The Feynman Lectures on Physics, Vol. II The New Millennium Edition: Mainly Electromagnetism and Matter

Basic Books. 2011.

ISBN: 9780465024940

C0342300

Extension of Numerical Methods / Extension of Numerical Methods

Código: C0342300 Imprimir

Year 3, First-term module. Compulsory. 6 credits.

Profesores

Arantxa Tymowska N/A - Coordinador

Ver biodata

Objectives

The objectives of the course are for students to learn the various numerical methods for solving differential equations and to become proficient in the use of numerical simulation libraries. Furthermore, students will aim to apply these methods to problems of relevance in mathematical engineering./The objectives of the course are for students to learn different numerical methods for solving differential equations and develop proficiency in the use of numerical simulation libraries. In addition, students are expected to be able to apply these methods to problems of relevance in mathematical engineering.

Prerequisites

It is essential that students have previously completed the courses Differential and Difference Equations, Partial Differential Equations, and Numerical Methods, or other courses with similar competences and learning outcomes./It is essential that students have previously completed the courses Differential and Difference Equations, Partial Differential Equations, and Numerical Methods, or other courses with similar competences and learning outcomes.

Competences

Not applicable.

Learning Outcomes

RC2 – To identify the different phases of the mathematical modelling process, distinguishing between formulation, analysis, solution and interpretation of results. TYPE: Competences.

RC3 – To apply software programmes that solve mathematical and engineering problems, using the appropriate computational environment for each case. TYPE: Competences.

RK5 – To understand different simulation models, stochastic simulation, and the management and planning of logistics systems; and to solve cases involving production management and planning models using software. TYPE: Knowledge or content.

RS1 – To use mathematical language to formulate propositions in various fields and to construct proofs for the given problems. TYPE: Skills.

RS3 – To propose, analyse, validate and interpret the most appropriate mathematical models and tools in real-world situations, in accordance with the intended objectives. TYPE: Skills.

RS4 – To plan the resolution of a problem in accordance with the available tools and the constraints of time and resources. TYPE: Skills.

RS5 – To use computer applications for statistical analysis, numerical and symbolic calculation, graphical visualisation, optimisation or other purposes to solve problems. TYPE: Skills.

Course description

The behaviour of linear models and non-linear differential equations is studied, identifying both the stability of the models and their long-term behaviour.

Various numerical methods for solving ordinary linear and non-linear differential equations are defined and understood, with a focus on the computational implementation of several of these methods, such as the Euler method, prediction-correction methods and the fourth-order Runge-Kutta method.

Numerical methods for solving higher-order ODEs and systems of ODEs are also developed. In this context, non-linear ODE systems are also studied, and their main applications in the field of mathematical engineering are explored, such as population dynamics, applications in industrial chemistry or electronics, as well as transport phenomena.

-------------------------

The course examines the behaviour of linear models and non-linear differential equations, focusing on the analysis of model stability and long-term behaviour.

Various numerical methods for solving linear and nonlinear ordinary differential equations (ODEs) are introduced and studied, with particular emphasis on the computational implementation of several of these methods, including Euler’s method, predictor–corrector methods, and the fourth-order Runge–Kutta method.

Numerical methods for solving higher-order ODEs and systems of ODEs are also developed. In this context, nonlinear systems of ODEs are studied, together with their main applications in mathematical engineering, including population modelling, applications in industrial chemistry and electronics, and transport phenomena.

Teaching activities

AP1. Lectures.

AP2. Interactive classes.

AP3.1. Project development and challenge-based problem solving.

AP4. Independent study, case and problem solving, and literature search.

AP5. Tutorials.

AP6. Knowledge assessment tests.

Assessment system and criteria

ASSESSMENT SYSTEMS

The systems designed to assess the extent to which students have achieved the learning outcomes are:

SE1. Practical activities.

Individual and group practical activities (case studies, problem-solving and challenges, project work, oral presentations, debates, etc.).

AS2: Final knowledge assessments.

Objective knowledge assessments. These may be written or oral, open-ended, short-answer or multiple-choice, etc.

SE3. Portfolio.

Evidence of the student’s learning (in terms of knowledge, performance or output) collected in a dossier, folder or portfolio throughout the course is assessed, with the aim of explaining and reflecting on the learning that has taken place. 

These systems are set out in the following assessment criteria:

REGULAR EXAM SESSION

- SE3: 20% Mid-term exam.

- SE1: 30% These are practical case studies assessed through a final report. Generally, between 2 and 4 are carried out during the academic year.

- SE2: 50% These are written knowledge tests, comprising the final exam for the ordinary examination period.

The calculation of the final mark for the ordinary examination period is conditional upon achieving a minimum mark of 5.0 out of 10 in the final exam. Once this requirement has been met, a weighted average of all assessment activities and tests will be calculated.

If the mark obtained in the final exam is less than 5.0 out of 10, the final mark will be the mark obtained in that exam.

Furthermore, the assessment tasks corresponding to SE1 – case studies accounting for 30 per cent of the assessment – must not contain more than 30 per cent of IA; should this limit be exceeded, the task in question will be marked as 0/10.

EXTRAORDINARY EXAMINATION SESSION

The supplementary examination involves sitting a final exam covering the entire module.

The mark for the extraordinary examination will be calculated as follows:

- 100% Mark for the theoretical/practical examination.

-------------------------

ASSESSMENT METHODS

The assessment methods designed to evaluate the extent to which students have achieved the intended learning outcomes are as follows:

SE1 – Practical Activities

Individual and group practical activities (case studies, problem-solving and challenge-based activities, project work, oral presentations, debates, etc.).

SE2 – Final Knowledge Assessments

Objective knowledge assessments. These may be written or oral and may take the form of extended-response, short-answer, multiple-choice or other types of questions.

SE3 – Portfolio

Assessment of the student’s evidence of learning (knowledge, performance or output) collected in a dossier, folder or portfolio throughout the course, with the aim of documenting and reflecting on the learning achieved.

These assessment methods are specified through the following assessment criteria:

REGULAR EXAMINATION PERIOD

SE3: 20% – Mid-term examination.

SE1: 30% – Practical case studies, assessed on the basis of a final report. Generally, between two and four case studies are completed during the course.

SE2: 50% – Written knowledge assessment: final examination during the regular assessment period.

The calculation of the final mark in the ordinary examination period is conditional upon achieving a minimum mark of 5.0 out of 10 in the final examination. Once this requirement has been met, the weighted average of all assessment activities and tests shall be calculated, even if any of them have been failed.

If the mark obtained in the final examination is below 5.0 out of 10, the final mark shall be the mark obtained in that examination.

Furthermore, the assessment activities corresponding to SE1 (practical case studies), which account for 30 per cent of the overall assessment, must not contain more than 30 per cent AI-generated content. If this threshold is exceeded, the assessment activity will receive a mark of 0/10.

EXTRAORDINARY EXAMINATION PERIOD

The Extraordinary Examination Period consists of a final examination covering the entire course.

The mark for the Extraordinary Examination Period will be calculated as follows:

- 100% — Theoretical/practical examination mark.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	MG	Course introduction.
2	SM	Course introduction.
3	MG	Discrete functions. Derivatives and partial derivatives.
4	MG	Discrete functions. Derivatives and partial derivatives.
5	MG	Discrete functions. Derivatives and partial derivatives.
6	SM	Finding roots of functions. Newton’s method.
7	MG	Finding roots of functions. Newton’s method.
8	MG	Basic concepts of the theory of linear differential equations
9	MG	Basic concepts of the theory of linear differential equations
10	EV	Numerical integration. Trapezoids and Simpson’s rule.	SE1: 5%
11	MG	Numerical integration. Trapezoids and Simpson’s rule.
12	MG	Euler’s method for solving differential equations.
13	MG	Euler’s method for solving differential equations.
14	EV	Integral equations. Runge-Kutta methods.	SE1: 5%
15	MG	Integral equations. Runge-Kutta methods.
16	MG	Integral equations. Runge-Kutta methods.
17	MG	Integral equations. Runge-Kutta methods.
18	SM	Integral equations. Runge-Kutta methods.
19	MG	Multi-step methods, prediction-correction.
20	MG	Multi-pass methods, prediction-correction.
21	MG	Multi-pass methods, prediction-correction.
22	EV	Presentation of the work	SE2: 10%
23	MG	Multi-step methods, prediction-correction.
24	MG	Numerical simulation packages.
25	MG	Numerical simulation packages.
26	SM	Numerical simulation packages.
27	MG	Numerical simulation packages.
28	MG	Non-linear differential equations.
29	EV	Mid-term exam 1	SE3: 20%
30	SM	Study of equilibrium and stability.
31	MG	Equilibrium and stability analysis.
32	MG	Study of equilibrium and stability.
33	MG	Study of phase space.
34	SM	Study of phase space.
35	MG	Study of phase space.
36	MG	Introduction to chaotic systems.
37	MG	Introduction to chaotic systems.
38	EV	Introduction to chaotic systems.	SE1: 5%
39	MG	Introduction to chaotic systems.
40	MG	Poincaré cross-section surface.
41	MG	Poincaré cross-section.
42	SM	Poincaré cross-section.
43	MG	Poincaré cross-section.
44	MG	Fourier and Laplace transforms.
45	MG	Fourier and Laplace transforms.
46	EV	Fourier and Laplace transforms.	SE1: 5%
47	MG	Fourier and Laplace transforms.
48	MG	Study of the regularity of a system using the Fourier transform.
49	MG	Study of the regularity of a system using the Fourier transform.
50	EV	Study of the regularity of a system using the Fourier transform.	SE2: 10%
51	MG	Study of the regularity of a system using the Fourier transform.
52	MG	Convolution of functions.
53	MG	Convolution of functions.
54	SM	Presentation of work
55	MG	Applications.
56	MG	Applications.
57	MG	Applications.
58	SM	Seminar
59	MG	Chaos Indicators.
60	EV	Exam.	SE3: 40.00

Bibliography

Core:

1. D. A. Ovalle, M. Á. Bernal-Yermanos and J. A. Posada-Restrepo

Mathematics for Engineering. Numerical Methods with Python

Grancolombiano Polytechnic University. 2017.

ISBN: 978-958-8721-

2. Steven Chapra and Raymond Canale

Numerical Methods for Engineers

McGraw-Hill Interamericana de España S.L.. 2011.

ISBN: 6071504996

C0342301

Object-Oriented Development

Código: C0342301 Imprimir

Year 3, First-term module. Compulsory. 6 credits.

Profesores

Carmen Mansilla Mansilla - Coordinador

Ver biodata

Objectives

This module is divided into a series of modules designed to improve and refine the techniques students use in programming and testing applications. The modules comprising the course are:

1. Advanced Data Structures. This module introduces students to specialised techniques for defining data structures that model reality.

2. Algorithmic Techniques. This module examines different approaches to algorithmic problem-solving that can be applied at various stages of computer-based problem-solving.

3. Programme Testing. The aim of this module is to make students aware of the importance of testing in all development work. Students will also be taught the most common techniques for testing programmes and computer systems.

-------------------------

This course is divided into a series of modules aimed at developing and refining the techniques students use in application programming and testing. The course comprises the following modules:

1. Advanced Data Structures. This module introduces students to specialised techniques for defining data structures that model real-world systems and phenomena.

2. Algorithmic Techniques. This module examines different approaches to solving problems algorithmically, which can be applied at various stages of computer-based problem-solving.

3. Software Testing. The aim of this module is to raise students’ awareness of the importance of testing throughout the software development process. Students will also be introduced to the most common techniques for testing software programmes and computer systems.

Prerequisites

It is essential that students have previously completed the courses Fundamentals of Programming and Computers, Data Structures and Algorithms I and II, and Introduction to Parallel and Distributed Programming, or other courses with similar competences and learning outcomes./It is essential that students have previously completed the courses ‘Fundamentals of Programming and Computers’, ‘Data Structures and Algorithms I and II’, and ‘Introduction to Parallel and Distributed Programming’, or other courses with similar competences and learning outcomes.

Competencies

Not applicable./Not applicable.

Learning Outcomes

RC2 – To identify the different phases of the mathematical modelling process, distinguishing between formulation, analysis, resolution and interpretation of results. TYPE: Competences.

RC3 – To apply software programmes that solve mathematical and engineering problems, using the appropriate computational environment for each case. TYPE: Competences.

RC4 – To be able to plan and develop projects in the field of Mathematical Engineering. TYPE: Competences.

RS4 – To plan the resolution of a problem in accordance with the available tools and the constraints of time and resources. TYPE: Skills.

RS5 – To use computer applications for statistical analysis, numerical and symbolic calculation, graphical visualisation, optimisation or other purposes to solve problems. TYPE: Skills.

Course description

1. Introduction to Object-Oriented Programming (OOP).

1.1. Phases of software development. Methodologies.

1.2. Design diagrams. UML.

2. Classes in Python.

2.1. Imperative programming.

2.2. Objects and classes.

2.3. Encapsulation.

2.4. Modularity.

3. Class inheritance.

3.1. Class hierarchy. Encapsulation.

3.2. Abstract classes and interfaces.

3.3. Error handling. Exceptions.

3.4. Collections and generics. Inner classes.

3.5. Polymorphism. Concurrency. Functional interfaces.

4. User interface design.

4.1. Elements of graphical user interfaces.

4.2. Geometric arrangement of components. Layouts.

4.3. Event handling. Listeners.

4.4. Interface frameworks.

4.5. Use of graphics.

5. Design patterns.

5.1. General concepts of patterns.

5.2. Representative examples in Python.

-------------------------

1. Introduction to Object-Oriented Programming (OOP).

1.1. Software development phases and methodologies.

1.2. Design diagrams and the Unified Modelling Language (UML)

2. Classes in Python.

2.1. Imperative programming.

2.2. Objects and classes.

2.3. Encapsulation.

2.4. Modularity.

3. Class inheritance.

3.1. Class hierarchies and encapsulation.

3.2. Abstract classes and interfaces.

3.3. Error handling and exceptions.

3.4. Collections and generics. Inner classes.

3.5. Polymorphism, concurrency and functional interfaces.

4. User Interface Design.

4.1. Graphical user interface elements.

4.2. Geometric arrangement of components and layouts.

4.3. Event handling and event listeners.

4.4. User interface frameworks.

4.5. Use of graphics.

5. Design Patterns.

5.1. General concepts of design patterns.

5.2. Representative examples in Python.

Learning activities

AP1. Lectures.

AP2. Interactive classes.

AP3.2 Workshops and computer laboratories.

AP4. Independent study, case and problem solving, and literature search.

AP5. Tutorials.

AP6. Knowledge assessment tests.

Assessment system and criteria

The assessment process will take into account the various learning outcomes.

REGULAR EXAMINATION SESSION

The assessment will consist of the following parts:

- 50% theoretical/practical examination.

- 50% laboratory practicals.

The calculation of the final mark for the ordinary assessment period is conditional upon achieving a minimum mark of 4.0 out of 10 in the final examination. Once this requirement has been met, a weighted average will be calculated from all assessment activities and tests, even if any of them have been failed.

If the mark obtained in the final examination is less than 4.0 out of 10, the final mark will be the mark obtained in that examination.

SUPPLEMENTARY EXAMINATION SESSION

The assessment will consist of a single component (100%), comprising a theoretical/practical examination.

-------------------------

The assessment process will take into account the different competencies to be developed.

ORDINARY EXAMINATION PERIOD

Assessment will consist of the following components:

- 50% - Theoretical/practical examination.

- 50% - Laboratory practical assignments.

The calculation of the final mark in the ordinary examination period is conditional upon achieving a minimum mark of 4.0 out of 10 in the final examination. Once this requirement has been met, the weighted average of all assessment activities and tests shall be calculated, even if any of them have been failed.

If the mark obtained in the final examination is below 4.0 out of 10, the final mark shall be the mark obtained in that examination.

SUPPLEMENTARY EXAMINATION PERIOD

Assessment will consist of a single component (100%), a theoretical/practical examination.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	MG	Course introduction.
2	MG	Introduction to the course.
3	SM	Placing Object-Oriented Programming within the framework of software development methodologies
4	SM	Introduce the main elements of OO software design using diagrams UML
5	MG	Introduce the main elements of OO software design using diagrams UML
6	EV	Introduce the fundamental elements common to other non-OO imperative languages 5%	5%
7	SM	Introduce the fundamental elements common to other non-OO imperative languages OO
8	SM	Introduce the concepts of objects and classes
9	MG	Introduce the concepts of objects and classes
10	MG	Incorporate data encapsulation techniques for classes
11	SM	Incorporate data encapsulation techniques for classes
12	SM	Analyse the modularity of data types and their distribution across components
13	MG	OOP exercises
14	MG	OOP exercises
15	SM	OOP practice
16	EV	OOP Practice	5%
17	MG	To introduce the concepts of class inheritance and data encapsulation in the context of inheritance
18	MG	Introduce the concepts of class inheritance and data encapsulation in the context of inheritance
19	SM	Introduce abstract classes and interfaces
20	SM	Introducing abstract classes and interfaces
21	MG	Study error handling using ‘exception’ classes
22	MG	Study error handling using ‘exception’ classes
23	SM	Analysing special classes dedicated to collections of data and types, and internal classes of the language
24	EV	Analyse the special classes dedicated to data and type collections, and internal language classes	10%
25	MG	Present the handling of operation polymorphism and process concurrency
26	MG	Present the treatment of operation polymorphism and process concurrency
27	SM	Practical programme testing
28	SM	Programme testing practice
29	MG	Programme testing and debugging. Exercises
30	MG	Programme testing and debugging. Exercises
31	EV	Mid-term exam.
32	EV	Mid-term exam.	30%
33	MG	To introduce the fundamental concepts used in the design of graphical user interfaces
34	MG	Introduce the fundamental concepts used in the design of graphical user interfaces
35	SM	Introduce the fundamental concepts used in the design of graphical user interfaces
36	SM	To introduce the fundamental concepts used in the design of graphical user interfaces
37	MG	Dynamic Programming
38	MG	Dynamic Programming
39	SM	Dynamic Programming
40	SM	Dynamic Programming
41	MG	Practical Course in Dynamic Programming
42	MG	Dynamic Programming Practicum
43	SM	Dynamic Programming Practise
44	EV	Dynamic Programming Practicum	10%
45	MG	Study the mechanism for distributing components using ‘layouts’
46	MG	Study the mechanism for distributing components using ‘layouts’
47	SM	Study the mechanism for handling events using ‘listeners’
48	SM	To study the event handling mechanism using ‘listeners’
49	MG	Present the main Qt components for user interfaces
50	MG	Introduce the main Qt components for user interfaces
51	SM	Introduce the concept of design patterns in OO
52	SM	Introducing the concept of patterns in OO
53	MG	Representative examples of the concept of a pattern
54	MG	Representative examples of the concept of a pattern
55	SM	Analysing the presentation of graphs and data
56	EV	Analysing the processing of graphs and data	10%
57	MG	Exercises
58	MG	Exercises
59	EV	Mid-term exam.
60	EV	Mid-term exam.	30%

Reading list

Core:

1. David A. Ham

Object-Oriented Programming in Python for Mathematicians (3rd Edition)

Self-published. 2023.

ISBN: 979-886257750

2. Krzysztof Postek, Alessandro Zocca, Joaquim A. S. Gromicho and Jeffrey C. Kantor

Hands-On Mathematical Optimisation with Python

Cambridge University Press. 2025.

ISBN: 1009493507

3. Luciano Ramalho

Fluent Python: Clear, Concise, and Effective Programming (2nd Edition)

O’Reilly Media. 2022.

ISBN: 1492056359

4. Steven F. Lott and Dusty Phillips

Python Object-Oriented Programming (4th Edition)

Packt Publishing. 2021.

ISBN: 1801077266

Links

Python – Official Python documentation

Basic Python tutorials – Python tutorials on W3Schools

C0342302

Operational research

Código: C0342302 Imprimir

Year 3, First-term module. Compulsory. 6 credits.

Profesores

Hugo Andrea Galindo Beleña - Coordinador

Ver biodata

Objectives

The general objectives of the module are for students to:

o Model basic problems in Operational Research.

o Understand the fundamentals of the simplex algorithm and duality.

o Solve linear optimisation problems and interpret the results correctly.

o Understand the classical models of integer optimisation.

o Apply non-linear optimisation conditions in simple cases.

o Use software to solve typical Operational Research problems, particularly those involving linear optimisation.

Prerequisites

No prerequisites have been set for this module. However, it is highly recommended that students have completed the modules Algebra I and II, as well as having prior experience with the programming languages Python and VBA (Visual Basic for Applications).

Learning Outcomes

Not applicable.

Learning Outcomes

RC2 – Identify the different phases of the mathematical modelling process, distinguishing between formulation, analysis, solution and interpretation of results. TYPE: Competencies.

RC3 – Apply programmes to solve mathematical and engineering problems, using the appropriate computational environment for each case. TYPE: Competencies.

RC5 – Demonstrate a critical and self-critical attitude when tackling the challenges of their work as a mathematical engineer, and have the ability to adopt attitudes consistent with the ethical and professional principles that govern scientific innovation and professional practice as a mathematical engineer. TYPE: Competencies.

RC9 – Formulate problems from a professional context in mathematical language in a way that facilitates their analysis and resolution. TYPE: Competences.

RK4 – Be familiar with rigorous proofs of some classical theorems in different areas of mathematics. TYPE: Knowledge or content.

RS1 – Use mathematical language to formulate propositions in different fields and construct proofs for the problems posed. TYPE: Skills or abilities.

RS5 – Use computer applications for statistical analysis, numerical and symbolic calculation, graphical visualisation, optimisation or other purposes to solve problems. TYPE: Skills or competences.

RS6 – Analyse texts in the field of mathematics with the aim of familiarising oneself with their structure and with scientific publications, including those in English. TYPE: Skills or competences.

Description of the content

CLASSIC MODELLING PROBLEMS IN OPERATIONS RESEARCH

LINEAR OPTIMISATION

- Theoretical foundations of the simplex algorithm

- The simplex algorithm

- Initialisation of the algorithm: the penalty method and the two phases

- Theoretical foundations of duality

- Dual simplex algorithm

- Initialisation of the algorithm: the artificial constraint method

- Sensitivity analysis and post-optimisation

INTEGER OPTIMISATION

- Branch-and-bound method

NON-LINEAR OPTIMISATION

- Karush-Kuhn-Tucker conditions

Training activities

AP1. Lectures.

AP2. Interactive classes.

AP3.2 Workshops and computer labs.

AP4. Independent study, case studies, problem-solving and literature review.

AP5. Tutoring.

AP6. Knowledge assessments.

Assessment system and criteria

There are two official examination sessions: the ordinary and the supplementary.

The final mark for the module in the REGULAR EXAMINATION PERIOD will be calculated as a weighted average of projects and examinations as follows:

- Continuous assessment (50%)

+ Mid-term exam (20%)

+ Python project* (30%)

- Final exam (50%)

* To be assessed on the Python project, students must have attended and completed all scheduled laboratory sessions.

The calculation of the final mark in the standard assessment period is conditional upon achieving a minimum mark of 4.0 out of 10 in the final exam. Provided this requirement is met, a weighted average will be calculated from all assessment activities and tests, even if any of them have been failed.

If the mark obtained in the final exam is less than 4.0 out of 10, the final mark will be the mark obtained in that exam.

The final mark for the module in the EXTRAORDINARY EXAMINATION PERIOD will correspond to 100 per cent of the mark obtained in this examination.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	MG	Course introduction.
2	MG	A historical introduction to operations research.
3	MG	Topic 1: Formulation of operational research problems, examples of optimal diet and allocation models.
4	SM	Completing problem-solving exercises.
5	MG	Topic 1: Graphical solution.
6	MG	Topic 1: Graphic resolution II.
7	MG	Topic 1: Models with more than 2 decision variables.
8	SM	Completing exercises involving graphical solution methods.
9	SM	Topic 1: Simplex algorithm I.
10	MG	Topic 1: Simplex algorithm II.
11	MG	Practising Simplex algorithm exercises.
12	SM	Lab 1: Introduction to Linear Programming with Python
13	MG	Lab 1: Introduction to OL with Python
14	MG	Topic 2: Sensitivity analysis using the graphical method I.
15	SM	Sensitivity analysis exercises using the graphical method.
16	MG	Topic 2: Sensitivity analysis using the graphical method II.
17	MG	Topic 2: Sensitivity analysis using the graphical method II.
18	MG	Topic 2: Sensitivity analysis using the simplex algorithm.
19	MG	Topic 2: Sensitivity analysis using the simplex algorithm.
20	SM	Carrying out sensitivity analysis exercises.
21	EV	Mid-term exam on topics 1 and 2	20
22	MG	Topic 3: Formulating optimisation problems, linear relaxation of optimisation problems.
23	MG	Topic 3: Formulation of programming problems, linear relaxation of programming problems.
24	SM	Lab 2: Solving LP problems using Python
25	MG	Lab 2: Solving PL problems with Python
26	MG	Topic 3: Optimum value bounds, optimality tests, optimality gap.
27	SM	Completion of PE exercises.
28	SM	Physical education exercises.
29	MG	Topic 3: Branch-and-Bound Method I.
30	MG	Topic 3: Branch and Bound Method I.
31	SM	Completing ‘Branch and Bound’ exercises.
32	SM	Completing ‘Branch and Limit’ exercises.
33	MG	Topic 3: The ‘Branch and Bounds’ method II.
34	MG	Topic 3: Branch-and-Bound Method II.
35	MG	Topic 3: Branch and Bounds Method II.
36	SM	Completing ‘Branch and Bound’ exercises.
37	SM	Lab 3: Solving PE problems with Python
38	MG	Lab 3: Solving PE problems with Python
39	MG	Topic 4: Formulating NLP problems.
39	MG	Topic 4: Formulating NLP problems.
41	MG	Topic 4: KKT conditions.
42	MG	Topic 4: KKT conditions.
43	MG	Topic 4: KKT conditions.
44	SM	Carrying out NLP exercises.
45	SM	Carrying out NLP exercises.
46	MG	Topic 4: Fritz Jones’ Conditions.
47	MG	Topic 4: Fritz Jones’s Terms.
49	MG	Topic 4: Special cases
50	MG	Topic 4: Special cases
51	MG	Topic 4: Special cases
52	SM	Carrying out NLP exercises.
52	SM	Carrying out NLP exercises.
53	SM	Carrying out NLP exercises.
54	SM	Carrying out NLP exercises.
55	MG	Carrying out NLP exercises.
56	MG	Carrying out NLP exercises.
57	SM	Lab 4: Solving ONL problems with Python
58	SM	Lab 4: Solving ONL problems with Python
59	SM	Completing exams from previous years’ courses.
60	SM	Taking exams for past courses.

Reading list

Essential:

1. José Niño Mora

Introduction to Decision Optimisation: Methods and Models in Operations Research.

Ediciones Pirámide. 2021.

ISBN: 9788436845280

Supplementary:

2.- Hillier, Frederick S.

Operations Research

7th ed.: McGraw-Hill Interamericana. 2002.

ISBN: 9701034864

3. Taha, Hamdy A.

Operations Research

9th ed.: Pearson Education. 2012.

ISBN: 9786073207966

C0342303

Optimisation and Control Techniques

Código: C0342303 Imprimir

Year 3, First-term module. Compulsory. 6 credits.

Profesores

Jacinto Velasco Rebolledo - Coordinador

Ver biodata

Objectives

The course ‘Optimisation and Control’ for mathematical engineers covers key techniques such as the Calculus of Variations, which optimises functionals using the Euler–Lagrange equation, applied to problems such as the brachistochrone problem. In Control, dynamic systems are studied, including PID controllers and optimal control based on Pontryagin’s principle. Dynamic Programming is introduced to solve complex sequential problems, using Bellman’s optimality principle, which is applicable to both discrete and continuous problems.

Prerequisites

We recommend that students have a knowledge of differential and integral calculus, differential equations and statistics. A knowledge of Python programming is advised.

Competencies

Not applicable.

Learning Outcomes

RC2 – Identify the different stages of the mathematical modelling process, distinguishing between formulation, analysis, solution and interpretation of results. TYPE: Competencies.

RC3 – Apply software to solve mathematical and engineering problems, using the appropriate computational environment for each case. TYPE: Competencies.

RC4 – Be able to plan and carry out projects in the field of Mathematical Engineering. TYPE: Competencies.

RC7 – To carry out work and projects related to mathematical engineering that have an impact on interdisciplinary teams or in multicultural contexts. TYPE: Competencies.

RC9 – Formulate problems from a professional context in mathematical language in a way that facilitates their analysis and resolution. TYPE: Competencies.

RK2 – Master the basic concepts of discrete mathematics, logic, algorithms, coding, operations research and artificial intelligence, and their application to solving engineering problems. TYPE: Knowledge or content.

RS1 – Use mathematical language to formulate propositions in different fields and construct proofs for the problems posed. TYPE: Skills or abilities.

RS5 – Use computer applications for statistical analysis, numerical and symbolic calculation, graphical visualisation, optimisation or other purposes to solve problems. TYPE: Skills or competences.

RS6 – Analyse texts in the field of mathematics with the aim of familiarising oneself with their structure and with scientific publications, including those in English. TYPE: Skills or competences.

Description of the content

Topic 1. Calculus of Variations

Topic 2. Optimal control techniques

Topic 3. Linear-quadratic control and the Kalman filter

Topic 4. Dynamic Programming

Upon completion of the course, students will be able to:

- Understand and apply the calculus of variations to optimise functionals, using the Euler–Lagrange principle in path problems and other contexts.

- Master classical and optimal control techniques, including the design and analysis of controllers and the application of Pontryagin’s principle

- Develop skills in dynamic programming to break down and solve complex sequential problems, applying Bellman’s optimality principle in both discrete and continuous scenarios.

- Integrate theoretical and practical knowledge to model, analyse and solve real-world optimisation and control problems in various areas of engineering and the applied sciences.

- Use advanced computational tools to implement and simulate optimal solutions in dynamic systems and decision-making processes.

Training activities

AP1. Lectures.

AP2. Interactive classes.

AP3.2 Workshops and computer labs.

AP4. Independent study, case studies, problem-solving and literature reviews.

AP5. Tutoring.

AP6. Knowledge assessments.

Assessment system and criteria

Regular Assessment Period (Continuous assessment: 50 per cent, exam weighting: 50 per cent)

Continuous assessment will account for 50% of the final mark.

The exam will account for 50 per cent of the final mark.

If the mark obtained in the final exam is less than 4.0 out of 10, the final mark will be the mark obtained in that exam.

Supplementary Examination Period (100% Exam)

In the resit, assessment will be based solely on an examination covering the entire course content.

The exam will account for 100 per cent of the final mark.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	MG	Course introduction.
2	SM	Course introduction.
3	MG	Introduction to Optimisation and Optimal Control.
4	MG	Introduction to Optimisation and Optimal Control.
5	MG	Theoretical Concepts of Optimisation and Optimal Control.
6	SM	Theoretical concepts of optimisation and optimal control.
7	MG	Theoretical concepts of optimisation and optimal control.
8	LB	Theoretical Concepts of Optimisation and Optimal Control.
9	MG	Numerical Methods in Optimisation and Optimal Control.
10	SM	Numerical Methods for Optimisation and Optimal Control.
11	MG	Numerical Methods for Optimisation and Optimal Control.
12	MG	Calculus of Variations: optimisation of dynamic systems.
13	MG	Calculus of Variations: optimisation of dynamic systems.
14	SM	Calculus of Variations: optimisation of dynamic systems.
15	MG	Variational Calculus.
16	MG	Variational calculus.
17	MG	Variational calculus.
18	SM	Problems
19	MG	Continuous-time optimal control. Pontryagin’s minimum principle.
20	MG	Optimal control in continuous time. Pontryagin’s minimum principle.
21	MG	Optimal control in continuous time. Pontryagin’s minimum principle.
22	SM	Pontryagin’s minimum principle.
23	MG	Pontryagin’s minimum principle.
24	MG	Pontryagin’s minimum principle.
25	MG	Numerical methods for maximisation and minimisation.
26	SM	Numerical methods for maximisation and minimisation.
27	MG	Numerical methods for maximisation and minimisation.
28	MG	Numerical methods for maximisation and minimisation.
29	MG	Application: Calculation of the path of minimum energy
30	SM	Application: Calculation of the path of minimum energy
31	MG	The LQR problem: solution and control.
32	MG	The LQR problem: resolution and control.
33	MG	The LQR problem: resolution and control.
34	LB	The LQR problem: resolution and control.
35	MG	Continuous-time state vector estimation
36	MG	Continuous-time state vector estimation
37	MG	Continuous-time state vector estimation of continuous functions via differentiable functions
38	EV	Mid-term exam on topics 1 and 2.	50
39	MG	Topic 3. Introduction to automatic control.
40	MG	Topic 3. System modelling and transfer functions.
41	MG	Topic 3. Transient response analysis and stability.
42	LB	Topic 3. Simulation of automatic control systems.
43	MG	Topic 3. Proportional, integral and derivative control actions.
44	MG	Topic 3. PID controllers: structure and operation.
45	MG	Topic 3. Methods for tuning PID controllers.
46	LB	Topic 3. Design, tuning and simulation of a PID controller.
47	MG	Topic 3. Digital PID control and applications.
48	SM	Topic 3. Troubleshooting automatic control and PID systems.
49	MG	Topic 4. Introduction to optimal control in discrete time.
50	MG	Topic 4. Fundamentals of dynamic programming.
51	MG	Topic 4. Bellman’s optimality principle.
52	LB	Topic 4. Computational problem-solving using dynamic programming.
53	MG	Topic 4. Bellman’s equation and recurrence relations.
54	MG	Topic 4. Deterministic dynamic programming.
55	MG	Topic 4. Stochastic dynamic programming.
56	LB	Topic 4. Implementation of dynamic programming algorithms.
57	MG	Topic 4. Applications of dynamic programming to engineering problems.
58	SM	Topic 4. Case studies and preparation of the final presentation.
59	EV	Presentation and defence of Assignment 1: automatic control and PID.	25
60	EV	Presentation and defence of Assignment 2: dynamic programming.	25

Bibliography

Core:

1. D. O. Anderson and John B. Moore

Optimal Control: Linear Quadratic Methods

Dover Publications. 2007.

ISBN: 9780486457666

2. Donald E. Kirk

Optimal Control Theory: An Introduction

Dover Publications. 2024.

ISBN: 048632432X

3. Frederick S. Hiller and Gerald J. Lieberman

Introduction to Operations Research

McGraw-Hill. 1991.

ISBN: 8486862450

4. Robert Grover Brown and Patrick Y. C. Hwang

Introduction to Random Signals and Applied Kalman Filtering

Wiley. 2012.

ISBN: 0470609699

C0342304

Complex Variables and Fourier Analysis

Código: C0342304 Imprimir

Year 3, First-term module. Compulsory. 6 credits.

Profesores

José Luis Guijarro Regalado - Coordinador

Ver biodata

Objectives

To understand, study and apply the theoretical and practical results of analytic functions, in particular elementary functions and their compositions. Understand and apply the Cauchy–Gousart theorem and the Cauchy integral formula, in its various forms for functions of a complex variable, to the integration of holomorphic functions. Understand and correctly apply Cauchy’s residue theorem and its applications. Understand the discrete Fourier transform and its properties, and apply it to signal theory: the fast Fourier transform and signal filtering; and apply it to image processing and audio compression.

Prerequisites

No prerequisites have been established.

Competencies

Not applicable.

Learning outcomes

RC2 – Identify the different phases of the mathematical modelling process, distinguishing between formulation, analysis, solution and interpretation of results. TYPE: Competencies.

RC3 – Apply software to solve mathematical and engineering problems, using the appropriate computing environment for each case. TYPE: Competencies.

RC7 – Carry out work and projects related to mathematical engineering that have an impact on interdisciplinary teams or in multicultural contexts. TYPE: Competences.

RC9 – Formulate problems from a professional context in mathematical language in such a way as to facilitate their analysis and resolution. TYPE: Competences.

RK4 – Be familiar with rigorous proofs of certain classical theorems in different areas of mathematics. TYPE: Knowledge or content.

RS1 – Use mathematical language to state propositions in different fields and construct proofs for the problems posed. TYPE: Skills or abilities.

RS5 – Use computer applications for statistical analysis, numerical and symbolic calculation, graphical visualisation, optimisation or other purposes to solve problems. TYPE: Skills or competences.

Description of the content

- Introduction: complex numbers.

- Analytic functions.

- Elementary functions.

- Cauchy’s integral theorem and formula for functions of a complex variable.

- Cauchy’s residue theorem and its applications.

Teaching activities

AP1. Lectures.

AP2. Interactive classes.

AP3.1. Project work and problem-solving.

AP4. Independent study, case studies, problem-solving and literature review.

AP5. Tutoring.

AP6. Knowledge assessments.

Assessment system and criteria

The continuous assessment process consists of:

1) A test on calculations with complex numbers, topics 1 and 2 of the syllabus, halfway through the academic year, accounting for 25% of the mark

2) A test on complex integration, topics 3 and 5 of the syllabus, at the end of the academic year, accounting for 25%

3) The official examination, in the ordinary sitting, covering the entire course, accounting for 50% of the mark. Under no circumstances do passing the tests exempt students from this final examination.

For the supplementary sitting, only this exam will be taken into account, accounting for 100% of the mark.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	MG	Course introduction.
2	MG	Introduction to the course.
3	SM	Complex numbers: basic concepts and axioms.
4	SM	Riemann sphere.
5	MG	Applications
6	SM	Exercise solutions I
7	SM	Exercise solutions II
8	MG	Functions of a complex variable.
9	MG	Limits and theorems on limits.
10	SM	Solving exercises III
11	SM	Exercise solutions IV
12	MG	Continuity.
13	MG	Derivatives and differentiation formulas.
14	SM	Solutions to Exercises V
15	SM	Exercise solutions VI
16	MG	Cauchy–Riemann equations
17	MG	Sufficient conditions and polar form
18	SM	Exercise solutions VII
19	SM	Exercise solutions VIII
20	MG	Elementary functions
21	MG	Exponential function and properties
22	SM	Solving exercises IX
23	EV	Submission of exercises, part 1	7.5%
24	EV	First module exam
25	EV	First block exam	35%
26	MG	Logarithmic functions, values and their branches.
27	MG	Trigonometric and hyperbolic functions
28	SM	Solving exercises XI
29	SM	Answers to Exercises XII
30	MG	Inverse trigonometric and hyperbolic functions.
31	MG	Complex exponents
32	MG	Complex integration: contours and line integrals
33	SM	Exercise solutions XIII
34	SM	Answers to Exercises XIV
35	MG	Primitive functions and the Cauchy–Gousart theorem
36	SM	Solutions to Exercises XV
37	SM	Answers to Exercises XVI
38	SM	Answers to Exercises XVII
39	MG	Simply connected and connected domains. Cauchy’s integral formula
40	MG	Morera’s and Liouville’s theorems. The fundamental theorem of algebra
41	SM	Solutions to Exercises XVIII
42	SM	Answers to Exercises XIX
43	SM	Convergence of sequences and series
44	MG	Taylor series.
45	MG	Laurent series
46	MG	Absolute and uniform convergence of power series, integration and differentiation
47	SM	Exercise solutions XX
48	SM	Answers to Exercises XXI
49	SM	Answers to exercises from previous examination sessions
50	MG	Waste and hubs
51	MG	Residue Theorem
52	MG	Applications
53	MG	Discrete Fourier transform and its application to signals.
54	MG	Fast Fourier Transform
55	MG	Applications of signal theory to image processing and audio compression.
56	SM	Solutions to Exercises XXII
57	EV	Submission of Exercises, Part 2	7.5%
58	EV	Submission of assignments	15%
59	EV	Mid-term exam.
60	EV	Mid-term exam.	35%

Reading list

Core:

1. Murray R. Spiegel

Complex Variables

McGraw-Hill. 2011.

ISBN: 6071505518

2. Ruel V. Churchill and James Ward Brown

Complex Variables and Applications

7th ed. McGraw-Hill. 2010.

ISBN: 8448142128

TOTAL:

SECOND FOUR-MONTH TERM

Code

Subjects

Character*

ECTS

C0242605

Electromagnetism II

Código: C0242605 Imprimir

Year 3, Second Term. Compulsory. 6 credits.

Profesores

Joaquín Bermejo Ortiz - Coordinador

Objectives

The aim of this module is to provide students with a basic understanding of electrodynamics and its relationship with special relativity.

Prerequisites

No prerequisites

Learning Outcomes

RK1 To be familiar with the most important phenomena and theories in the various branches of physics, as well as their historical context

RK2 To understand physically distinct phenomena and their underlying analogies, in order to apply known solutions to new problems

RK3 To analyse the fundamental concepts and principles of physical systems in order to develop approximations that enable the construction of a simplified model

RK7 Understand the laws and principles of physics, identifying their logical and mathematical structure, their experimental basis and the phenomena described by them

RK11 Understand the fundamentals and basic concepts of electric and magnetic fields, as well as the interrelationship between these fields and their unification in electromagnetism

RS1 Apply the most important knowledge, concepts and methods from the various branches of physics.

RS4 Apply mathematical and numerical methods to the modelling and explicit solution of problems in physics and related disciplines, selecting the appropriate tools and interpreting results.

Learning outcomes

RA5 Understands the relevant aspects of the propagation of electromagnetic waves in free space and in the presence of boundaries

RA6 Identifies the mechanisms of electromagnetic wave emission.

RA6 Analyses the phenomena of propagation and emission of electromagnetic waves.

LR7 Demonstrates an understanding of the close relationship between electromagnetism and the theory of relativity.

Course content

Topic 1: Conductors in electrostatic equilibrium

Topic 2: Electrodynamics

-Electromotive force

-Electromagnetic induction

-Maxwell’s equations

-Electromagnetic waves

Topic 3: Electromagnetism and relativity.

Topic 4: Electromagnetic radiation and radiating systems.

Teaching activities

AP1.- Interactive lectures

AP2.- Seminars or practical application classes

AP3. Practical activities (case study sessions, project work, simulations, etc.)

AP4. Independent study

AP5. Tutoring

AP6. Knowledge assessments

Assessment system and criteria

Continuous assessment

Activity Weighting (%)

SE1.- Practical activities 40

Mid-term exam (20%).

Submission of questions and problems not solved in class (20%).

SE2. Final knowledge assessments. 60

For component SE1 to count towards the final mark, a minimum mark of 4/10 must be obtained in the final exam during the regular examination period. In any other case, this component will be weighted at 0/10.

Extraordinary final assessment

In the supplementary sitting, the exam mark counts for 100 per cent of the final mark.

Bibliography

Core:

1. Griffiths, David J.

Introduction to Electrodynamics

Pearson. 2014.

ISBN: 9781292021423

2. Jackson,

J.D. Classical Electrodynamics,

Cambridge University Press. 2017.

ISBN: 9781119770763

Supplementary:

3.- R.P. Feynman, R.B. Leighton, M. Sands

The Feynman Lectures on Physics, Vol. II: The New Millennium Edition: Mainly Electromagnetism and Matter: 02

Inter-American Educational Fund. 1972.

ISBN: 9780465040841

C0242606

Experimental Laboratory I

Código: C0242606 Imprimir

Year 3, Second Term. Compulsory. 6 credits.

Profesores

Arantxa Tymowska N/A

Ver biodata

Joaquín Bermejo Ortiz - Coordinador

Ver biodata

Objectives

The aim of this course is to introduce students to experimentation in physics, data analysis and error propagation in the fields of mechanics, electromagnetism and thermodynamics.

Prerequisites

No prerequisites

Learning Outcomes

RK2 Understand physically distinct phenomena and their underlying analogies in order to apply known solutions to new problems

RS2 Carry out calculations, assessments, studies, reports and tasks to produce high-quality work in the field of physics.

RS3 Estimate orders of magnitude to interpret laboratory phenomena in the field of Physics and its sub-disciplines, as well as in Chemistry.

RS5 Use appropriate electronic instruments and/or computer tools in modelling to find solutions to physical problems.

Learning outcomes

RA1 Understands the principles, techniques and measuring instruments, as well as the phenomena of interest in Mechanics and Waves, Thermodynamics and Electromagnetism.

RA2 Uses measuring apparatus (Mechanics and Waves, Thermodynamics and Electromagnetism) appropriately and efficiently, following measurement protocols, particularly those relating to the safety of the experimenter.

LA3 Is able to assess the limitations of measurement methods due to interference, the simplicity of models and the effects that are neglected in the measurement method (Mechanics and Waves, Thermodynamics and Electromagnetism).

RA4 Plots data graphically, extracts information from the plot, analyses the data, models the results and compares them with the physical laws relating to Mechanics and Waves, Thermodynamics and Electromagnetism.

RA5 Documents the measurement process in terms of its basis, the instrumentation required and the conditions under which it is valid, carrying out a complete analysis in accordance with the IMRD format (Mechanics and Waves, Thermodynamics and Electromagnetism).

RC1 Work independently on the management of projects related to the different areas of physics

Course description

Six experiments will be carried out:

Mechanics Labs:

Free fall

Thermodynamics Labs:

2. Ideal gas law

3. Thermal conductivity

4. Adiabatic gas law

Electromagnetism Labs:

5. Magnetic field in coils

6. Faraday’s law of induction

Training activities

AP1. Participatory lectures

AP2. Seminars or practical application classes

AP3. Practical activities (case studies, project work, simulations, etc.)

AP4. Independent study

AP5. Tutoring

AP6. Assessment tests

AP10.- Workshop and/or laboratory activities

Assessment system and criteria

1. Writing of laboratory reports (SE1 + SE3, 40%), one of which must be in article format (SE2, 20%):

a PDF produced using LaTeX with the provided template and written in English.

The remaining reports may be in any format preferred by the student, although LaTeX remains compulsory............. 60%

2. Laboratory notebook (SE3)............................................................................... 20%

3. Oral exam (SE2)............................................................ 20%

C0342305

Stochastic calculus

Código: C0342305 Imprimir

Year 3, Second Term. Compulsory. 6 credits.

Profesores

Hugo Andrea Galindo Beleña - Coordinador

Ver biodata

Objectives

The aim of this module is to provide students with a sound grounding in the analysis and modelling of stochastic processes, equipping them to tackle dynamic problems subject to uncertainty in a variety of contexts. By the end of the module, students will be able to:

- Understand the fundamentals of stochastic processes and their relevance to the modelling of random phenomena.

- Analyse and characterise Markov chains, applying their properties to the solution of mathematical problems and real-world applications.

- Develop a thorough understanding of martingales and their fundamental properties, applying them to the analysis of games of chance, finance and other stochastic systems.

- Model random phenomena using Brownian motion, understanding its properties and its role as the basis for stochastic calculus.

- To be introduced to stochastic calculus through stochastic integration and the application of Itô’s formula, applying it to the solution of stochastic differential equations.

- Formulate and solve stochastic problems using a theoretical and analytical approach, with applications in physics, financial engineering and other fields of mathematical engineering.

Prerequisites

No prerequisites have been set for this module. However, it is strongly recommended that students have taken or are currently taking the statistics modules within the degree programme.

Learning Outcomes

Not applicable.

Learning Outcomes

RC2 – Identify the different phases of the mathematical modelling process, distinguishing between formulation, analysis, solution and interpretation of results. TYPE: Competencies.

RC7 – Carry out work and projects related to mathematical engineering that have an impact on interdisciplinary teams or in multicultural contexts. TYPE: Competencies.

RC8 – Assess the social repercussions and impact of mathematical engineering solutions and proposals, ensuring compliance with quality standards and regulations applicable to the project. TYPE: Competencies.

RC9 – Formulate problems from a professional context in mathematical language in such a way as to facilitate their analysis and resolution. TYPE: Competences.

RK3 – Understand the concepts of statistics and statistical inference in large data sets. TYPE: Knowledge or content.

RS1 – Use mathematical language to formulate propositions in different fields and construct proofs for the problems posed. TYPE: Skills or abilities.

RS4 – Plan the resolution of a problem in accordance with the available tools and the constraints of time and resources. TYPE: Skills or competences.

RS5 – Use computer applications for statistical analysis, numerical and symbolic calculation, graphical visualisation, optimisation or other purposes to solve problems. TYPE: Skills or competences.

Description of the content

Topic 0. Revision of probability theory.

- Probability spaces, random variables and probability distributions.

- Expectation, variance and covariance.

- Generating functions and Laplace transforms.

- Random vectors and joint probability distributions.

- Multivariate normal distribution.

Topic 1. Introduction to stochastic processes.

Definition and classification of stochastic processes.

Discrete-time and continuous-time processes.

Autocorrelation function.

Weak and strong stationarity.

Topic 2. Discrete-time Markov chains.

- Definition and fundamental properties.

- Transition matrices and regular chains.

- Stationary distribution and convergence.

- Applications in mathematical modelling.

Topic 3. Discrete-time martingales.

- Definition and properties of martingales.

- Martingales, submartingales and supermartingales.

- Convergence theorems for martingales.

- Applications in finance.

Topic 4. Continuous-time Markov chains.

- Continuous-time Markov processes.

- Infinitesimal generators and transition semigroups.

- Poisson processes.

- Birth-death processes.

Topic 5. Brownian motion.

- Definition and construction of Brownian motion.

- Fundamental properties: continuity, independence of increments and normal distribution.

- Brownian motion in several dimensions.

- Applications in physics and finance (asset pricing models).

Topic 6. Introduction to stochastic calculus: stochastic integration.

- The Itô integral and its fundamental properties.

- Itô’s formula and applications.

- Applications in financial engineering and the modelling of dynamical systems.

Teaching activities

AP1. Lectures.

AP2. Interactive classes.

AP3.1. Project work and problem-solving.

AP4. Independent study, case studies, problem-solving and literature review.

AP5. Tutoring.

AP6. Knowledge assessments.

Assessment system and criteria

The final mark for the module in the REGULAR EXAMINATION PERIOD will be calculated as a weighted average of projects and examinations as follows:

- Continuous assessment (60 per cent):

+ Submission of exercises (20%).

+ Practical group activities (20 per cent).

+ Non-exemption mid-term exam (20%).

- Final exam covering the entire course (40 per cent): a minimum mark of 4.0 is required to be included in the average.

If the mark for the final exam is 4.0 or above, it is averaged with the continuous assessment, even if the student has failed the continuous assessment.

The final mark for the module in the EXTRAORDINARY EXAMINATION SESSION is based entirely on the mark achieved in the final exam for that session.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	MG	Course introduction
2	MG	Topic 0. Revision of probability theory: Probability spaces, random variables and probability distributions.
3	MG	Topic 0. Revision of probability theory: Expectation, variance and covariance.
4	MG	Topic 0. Review of probability theory: Generating functions and Laplace transforms.
5	MG	Topic 0. Revision of probability theory: Random vectors and joint probability distributions.
6	SM	Solving exercises.
7	SM	Exercise answers.
8	MG	Topic 1. Introduction to stochastic processes: Definition and classification.
9	MG	Topic 1. Introduction to stochastic processes: Discrete-time and continuous-time processes.
10	MG	Topic 1. Introduction to stochastic processes: Properties of a process.
11	MG	Topic 1. Introduction to stochastic processes: Simple random walk.
12	MG	Topic 1. Introduction to stochastic processes: Weak and strong stationarity.
13	MG	Topic 1. Introduction to stochastic processes: Gaussian processes.
14	SM	Working through exercises.
15	SM	Exercise solutions.
16	MG	Topic 2. Discrete-time Markov chains: Definition and fundamental properties.
17	MG	Topic 2. Discrete-time Markov chains: Transition matrices and regular chains.
18	MG	Topic 2. Discrete-time Markov chains: Definition and fundamental properties.
19	MG	Topic 2. Discrete-time Markov chains: Definition and fundamental properties.
20	MG	Topic 2. Discrete-time Markov chains: Stationary distribution and convergence.
21	MG	Topic 2. Discrete-time Markov chains: Stationary distribution and convergence.
22	SM	Exercise solutions.
23	SM	Exercise answers.
24	SM	Group exercise
25	MG	Topic 2. Discrete-time Markov chains: Stop times.
26	MG	Topic 2. Discrete-time Markov chains: Stop times.
27	SM	Work through the exercises.
28	SM	Answers to exercises.
29	MG	Topic 3. Discrete-time martingales: Definition and properties.
30	MG	Topic 3. Discrete-time martingales: Martingales, submartingales and supermartingales.
31	MG	Topic 3. Discrete-time martingales: Convergence theorems for martingales.
32	SM	Solutions to exercises.
33	SM	Answers to exercises.
34	MG	Topic 3. Discrete-time martingales: Applications.
35	MG	Topic 3. Discrete-time martingales: Applications.
36	SM	Group exercise
37	SM	Working through the exercises.
38	EV	Mid-term exam	20%
39	MG	Topic 4. Continuous-time Markov chains: Definition and properties.
40	MG	Topic 4. Continuous-time Markov chains: Infinitesimal generators and transition semigroups.
41	MG	Topic 4. Continuous-time Markov chains: Poisson processes
42	MG	Topic 4. Continuous-time Markov chains: Birth-death processes.
43	SM	Solutions to exercises.
44	SM	Group exercise
45	MG	Topic 5. Brownian motion: Definition and construction of Brownian motion.
46	MG	Topic 5. Brownian motion: Definition and construction of Brownian motion.
47	MG	Topic 5. Brownian motion: Brownian motion in several dimensions.
48	MG	Topic 5. Brownian motion: Applications in engineering.
49	SM	Solutions to exercises.
50	SM	Answers to exercises.
51	MG	Topic 6. Introduction to stochastic calculus: stochastic integration.
52	MG	Topic 6. Introduction to stochastic calculus: the Itô integral and fundamental properties.
53	MG	Topic 6. Introduction to stochastic calculus: Itô’s formula and applications.
54	MG	Topic 6. Introduction to stochastic calculus: Applications in engineering and finance.
55	SM	Exercise solutions.
56	SM	Answers to exercises.
57	SM	Answers to exercises.
58	SM	Group exercise
59	SM	Exercise answers.
60	SM	Answers to exercises

Bibliography

Core:

1. Richard Durrett

Essentials of Stochastic Processes

Springer. 2018.

ISBN: 3319833316

C0342306

Cryptography and Security / Cryptography and Security

Código: C0342306 Imprimir

Year 3, Second Term. Compulsory. 6 credits.

Profesores

Jorge Calvo Martín - Coordinador

Ver biodata

Objectives

In this module, we will lay the foundations of modern cryptography, a discipline that is fundamental to ensuring the security of digital systems and communications. Cryptography is essential for protecting sensitive information, guaranteeing privacy and building trust in the digital age.

Why is cryptography important?

Cryptography is not only essential for security, but also plays a key role in:

• Secure machine learning and privacy-preserving AI.

• Blockchain and distributed systems.

• Secure communications in the Internet of Things (IoT) and cloud computing.

• Digital identity and authentication systems.

• Data privacy and GDPR compliance.

-------------------------

This course introduces the foundations of modern cryptography, a discipline that is essential for ensuring security in digital systems and communications. Cryptography plays a fundamental role in protecting sensitive information, ensuring privacy, and establishing trust in the digital age.

Why is Cryptography Important?

Cryptography is not only essential for security but also plays a key role in:

• Secure machine learning and privacy-preserving AI.

• Blockchain and distributed systems.

• Secure communications in the Internet of Things (IoT) and cloud computing.

• Digital identity and authentication systems.

• Data privacy and GDPR compliance.

Prerequisites

Learning Outcomes

Not applicable.

Learning outcomes

RC2 – To identify the different phases of the mathematical modelling process, distinguishing between formulation, analysis, solution and interpretation of results. TYPE: Competences.

RC3 – To apply software programmes that solve mathematical and engineering problems, using the appropriate computational environment for each case. TYPE: Competences.

RC4 – To be able to plan and develop projects in the field of Mathematical Engineering. TYPE: Competences.

RC5 – To have a critical and self-critical attitude to face the challenges of working as a mathematical engineer, and to demonstrate attitudes consistent with the ethical and deontological principles that govern scientific innovation and professional practice as a mathematical engineer./ to have a critical and self-critical attitude to face the challenges of working as a mathematical engineer, and to demonstrate attitudes consistent with the ethical and deontological principles that govern scientific innovation and professional practice as a mathematical engineer. TYPE: Competences.

RC6 – To work independently and in an organised manner to develop solutions to the various problems that may arise in the field of mathematical engineering, subject to strict time or economic constraints./ to work independently and in an organised manner to develop solutions to the various problems that may arise in the field of mathematical engineering, subject to strict time or financial constraints. TYPE: Competences.

RC8 - To assess the social impact and implications of solutions and proposals in mathematical engineering, ensuring compliance with quality standards and applicable project regulations. TYPE: Competences.

RS4 – To plan the resolution of a problem in accordance with the available tools and the constraints of time and resources. TYPE: Skills.

RS5 - To use computer applications for statistical analysis, numerical and symbolic calculation, graphical visualisation, optimisation or other purposes to solve problems. TYPE: Skills.

Course description

The module is worth 6 ECTS credits and covers:

• Theoretical foundations: mathematical principles underpinning cryptographic systems, including information theory, number theory and computational complexity.

• Symmetric cryptography: stream and block ciphers, including DES and AES, and their modes of operation.

• Hash functions: cryptographic hash functions, their properties and applications in integrity verification and digital signatures.

• Asymmetric cryptography: public-key cryptosystems such as RSA, the Diffie-Hellman key exchange and ElGamal.

• Elliptic curve cryptography (ECC): modern cryptographic systems based on elliptic curves, which offer greater security with smaller key sizes.

• Post-quantum cryptography: an introduction to cryptographic systems resistant to attacks using quantum computing.

-------------------------

The course is worth 6 ECTS credits and covers the following topics:

• Theoretical Foundations: Mathematical principles underlying cryptographic systems, including information theory, number theory and computational complexity.

• Symmetric Cryptography: Stream and block ciphers, including DES and AES, and their modes of operation.

• Hash Functions: Cryptographic hash functions, their properties, and their applications in integrity verification and digital signatures.

• Asymmetric Cryptography: Public-key cryptosystems such as RSA, the Diffie–Hellman key exchange, and ElGamal.

• Elliptic Curve Cryptography (ECC): Modern cryptographic systems based on elliptic curves, which provide enhanced security with smaller key sizes.

• Post-Quantum Cryptography: An introduction to cryptographic systems designed to withstand attacks using quantum computing.

Teaching activities

AP1. Lectures.

AP2. Interactive classes.

AP3.2 Workshops and computer laboratories.

AP4. Independent study, case and problem solving, and literature search.

AP5. Tutorials.

Assessment system and criteria

In the course’s virtual classroom, you will be able to view in detail the activities you are required to complete, as well as the submission deadlines, assessment criteria and rubrics for each one.

REGULAR EXAM SESSION

To pass the module in the regular assessment period, you must achieve a mark of 5.0 out of 10.0 or higher in the final module mark (weighted average) and, in addition:

IMPORTANT: The average mark for all course activities must be 5.0 out of 10.0 or higher in order to be included in the calculation of the final mark. Furthermore, the exam mark must be 4.0 out of 10.0 or higher to be included in the average with the marks for the course activities.

The final mark for this examination session will be based on the following assessment system:

Continuous assessment (40% of the final mark):

• Participation in the forum and daily behaviour in the classroom: 5 per cent of the final mark.

• Practical sessions TP1 and TP2: 15 per cent each (30 per cent in total).

– TP1: Implementation of classical cryptographic systems or mathematical fundamentals.

– TP2: Implementation of attacks on cryptographic protocols.

• Forum participation: 5% of the final mark.

• Attitude and work in the classroom: 5 per cent of the final mark.

– Comprehensive project involving symmetric and/or asymmetric cryptography.

Final exam (60 per cent of the final mark).

The final exam will assess your overall understanding of cryptography, including:

• Theoretical concepts and mathematical foundations.

• Analysis of cryptographic protocols.

• Security evaluation and vulnerability analysis.

• Problem-solving using appropriate cryptographic techniques.

SPECIAL EXAMINATION SESSION

To pass the module in the supplementary examination period, students must achieve a mark of 5.0 or above out of 10.0 in the final module mark (weighted average).

Students must submit any assignments that were not passed during the ordinary assessment period, after receiving the relevant feedback from the lecturer, as well as any assignments that were not previously submitted.

The following assessment criteria apply:

• Both the average mark for the assignments and the exam mark must be ≥ 5.0.

• The final weighted average must be ≥ 5.0.

-------------------------

Detailed information about the assignments to be completed, submission deadlines, assessment criteria and the corresponding rubrics for each assignment can be found on the course’s virtual campus.

ORDINARY EXAMINATION PERIOD

To pass the course during the Ordinary Examination Period, students must achieve a final course mark (weighted average) of 5.0/10.0 or higher and, in addition:

IMPORTANT: The average mark across all course activities must be 5.0/10.0 or higher in order to be included in the calculation of the overall mark. Similarly, the examination mark must be 4.0/10.0 or higher in order to be included in the calculation of the overall mark.

The final mark for this examination period will be based on the following assessment system:

Continuous Assessment (40% of the final mark):

• Participation in the forum and daily engagement in class: 5 per cent of the final mark.

• Practical Assignments TP1 and TP2: 15 per cent each (30 per cent in total).

– TP1: Implementation of classical cryptographic systems or mathematical foundations.

– TP2: Implementation of attacks on cryptographic protocols.

• Forum assignment: 5% of the final mark.

• Classroom participation and coursework: 5% of the final mark.

– Comprehensive project involving symmetric and/or asymmetric cryptography.

Final Examination (60% of the final mark).

The final examination will assess students’ overall understanding of cryptography, including:

• Theoretical concepts and mathematical foundations.

• Analysis of cryptographic protocols.

• Security assessment and vulnerability analysis.

• Problem-solving using appropriate cryptographic techniques.

EXTRAORDINARY EXAMINATION PERIOD

To pass the course during the Extraordinary Examination Period, students must achieve a final course mark (weighted average) of 5.0/10.0 or higher.

Students must submit any assignments that were not passed during the Ordinary Examination Period, after receiving the relevant feedback from the lecturer, as well as any assignments that were not submitted.

The following assessment criteria apply:

• Both the average mark for the assignments and the examination mark must be 5.0/10.0 or higher.

• The final weighted average must be 5.0/10.0 or higher.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	MG	Course introduction.
2	MG	Introduction to the course.
3	SM	Unit 1. Introduction to Cybersecurity
4	LB	1. Basic concepts of security
5	MG	2. Security strategies
6	MG	3. Cryptographic Systems
7	SM	4. Physical and Logical Security
8	LB	Exercises on abstract data types
9	MG	UD2. Protection Systems and Techniques
10	MG	1. Introduction.
11	SM	2. Defence in depth.
12	LB	3. Vulnerabilities.
13	MG	4. Detection of cyber attacks and malicious activity.
14	MG	5. Recovery from cyber-attacks.
15	SM	Exam exercises.
16	LB	Exam exercises.
17	MG	6. Introduction to Forensic Analysis.
18	EV	Practical session.	15
19	EV	Mid-term exam.
20	EV	Mid-term exam.	20
21	MG	Unit 3. Penetration Testing.
22	MG	1. Introduction.
23	SM	2. Penetration Testing Methodology.
24	LB	3. Scanning.
25	MG	4. Enumeration.
26	MG	5. Exploitation of vulnerabilities.
27	SM	6. Practical work. Gathering information about a target from open-source intelligence.
28	LB	UD4. Introduction to Malware
29	MG	1. Introduction to Malware.
30	MG	2. Classification of Malware
31	SM	3. Obtaining Malware. Honeynet.
32	LB	4. Classification
33	MG	5. Static analysis.
34	MG	6. Dynamic analysis.
35	SM	7. Practical work. Analysis of a piece of malware
36	EV	Practical session.	15
37	EV	Mid-term exam.
38	EV	Mid-term exam.	20
39	SM	Unit 5. Risk Analysis.
40	LB	1. Introduction to risk analysis.
41	MG	2. Risk analysis method: identification of all system assets, threats to assets, vulnerabilities, and impacts and risks.
42	MG	3. Identification and implementation of safeguards.
43	SM	4. Risk analysis projects.
44	LB	5. Safety plan.
45	MG	6. Practical assignment. Risk analysis of an organisation.
46	MG	Unit 6: Blockchain: From Theory to Practice
47	SM	Data organisation exercise.
48	LB	1. Introduction
49	MG	2. Initial Approach
50	MG	3. Applying Blockchain: First Steps
51	SM	Blockchain practical
52	LB	Blockchain practice
53	MG	4. Applying Blockchain: Practical Exercise
54	MG	5. Cryptography and Blockchain
55	SM	6. Future and Outlook
56	LB	Blockchain in practice
57	MG	Exercises
58	EV	Exercises	15
59	EV	Mid-term exam.
60	EV	Mid-term exam.	15

Bibliography

Core:

1. Luis Hernández Encinas

Cryptography

Los libros de la Catarata. 2016.

ISBN: 9788490971079

2. María Isabel González Vasco and Ángel Luis Pérez del Pozo

Essential Cryptography: Basic Principles for the Design of Secure Schemes and Protocols

University Press. 2021.

ISBN: 978-958-792-2

3. Pino Caballero Gil

An Introduction to Cryptography

RA-MA S.A. Publishing House. 2002.

ISBN: 9788478975204

C0342307

Data Management

Código: C0342307 Imprimir

Year 3, Second Term. Compulsory. 6 credits.

Profesores

Carmen Mansilla Mansilla - Coordinador

Ver biodata

Objectives

This course aims to provide students with the knowledge and skills required to manage, transform and analyse data effectively, using a range of techniques for data exploration, dimensionality reduction, clustering and knowledge extraction./The course aims to provide students with the knowledge and skills required to manage, transform and analyse data effectively, using a range of techniques for data exploration, dimensionality reduction, clustering and knowledge extraction.

Prerequisites

No prerequisites have been set for this course. However, it is highly recommended that students have previously completed or are currently taking the programming courses included in the degree programme./No prerequisites have been established for this course. However, it is highly recommended that students have previously completed or are currently taking the programming courses included in the degree programme.

Learning Outcomes

Not applicable.

Learning Outcomes

RC1 – To develop data visualisation tools for large volumes of data in order to communicate the results of the analyses carried out on them, adapting them to different audiences, both technical and non-technical/ To develop visualisation tools for large volumes of data in order to communicate the results of the analyses carried out on them, adapting them to different audiences, both technical and non-technical. TYPE: Competences.

RC2 – To identify the different phases of the mathematical modelling process, distinguishing between formulation, analysis, solution and interpretation of results. TYPE: Competences.

RC3 – To apply software programmes that solve mathematical and engineering problems, using the appropriate computational environment for each case. TYPE: Competences.

RC4 – To be able to plan and develop projects in the field of Mathematical Engineering. TYPE: Competences.

RC6 – To work independently and in an organised manner to develop solutions to the various problems that may arise in the field of mathematical engineering, subject to strict time or economic constraints./ to work independently and in an organised manner to develop solutions to the various problems that may arise in the field of mathematical engineering, subject to strict time or financial constraints. TYPE: Competences.

RC9 – To formulate problems arising in a professional context using mathematical language, in a way that facilitates their analysis and resolution. TYPE: Competences.

RK3 – To understand the concepts of statistics and statistical inference in large datasets. TYPE: Knowledge or Content.

RS2 – To apply Data Science methods (management, machine learning) as part of the process of analysing large datasets in computing environments./ To apply Data Science methods (management, machine learning) as part of the process of analysing large datasets in computing environments. TYPE: Skills.

RS3 – To propose, analyse, validate and interpret the most appropriate mathematical models and tools in real-world situations, in accordance with the intended objectives./ To propose, analyse, validate and interpret the most appropriate mathematical models and tools in real-world situations, in accordance with the intended objectives. TYPE: Skills.

RS4 – To plan the resolution of a problem in accordance with the available tools and the constraints of time and resources. TYPE: Skills.

RS5 – To use computer applications for statistical analysis, numerical and symbolic calculation, graphical visualisation, optimisation or other purposes to solve problems. TYPE: Skills.

Course description

The Data Management module comprises the following topics:

o Topic 1. Fundamentals of data management.

o Topic 2. Data Storage I: SQL

o Topic 3. Knowledge extraction techniques

o Topic 4. Data Storage II: NoSQL

Upon completion of the course, students will be able to:

- Manage and store data efficiently, understanding different models and architectures.

- Transform data, correcting errors and handling missing values.

- Explore and select relevant data, applying exploratory analysis and visualisation.

- Reduce dimensionality, using techniques such as PCA and t-SNE to optimise analysis.

- Apply clustering and knowledge extraction methods, such as clustering, classification and association rules.

- Implement solutions in Python, developing models applicable to real-world problems.

-------------------------

The Data Management course covers the following topics:

o Topic 1. Fundamentals of Data Management

o Topic 2. Data Storage I: SQL

o Topic 3. Knowledge Extraction Techniques

o Topic 4. Data Storage II: NoSQL

Upon completion of the course, students will be able to:

- Manage and store data efficiently, with an understanding of different data models and architectures.

- Transform data by correcting errors and handling missing values.

- Explore and select relevant data by applying exploratory data analysis and visualisation techniques.

- Reduce dimensionality using techniques such as PCA and t-SNE to optimise data analysis.

- Apply clustering and knowledge extraction methods, such as clustering, classification and association rules.

- Implement solutions in Python, developing models applicable to real-world problems.

Training activities

AP1. Lectures.

AP2. Interactive classes.

AP3.2 Workshops and computer laboratories.

AP4. Independent study, case and problem solving, and literature search.

AP5. Tutorials.

AP6. Knowledge assessment tests.

Assessment system and criteria

REGULAR EXAM SESSION:

Continuous assessment:

- Participation and attendance + completion of case studies (50 per cent):

- Regular attendance at classes and scheduled activities.

- Active participation in discussions and debates

- Correct and complete completion of case studies.

- Final exam covering the entire course (50%).

The mark for continuous assessment activities must be 5.0 out of 10.0 or higher to be included in the overall mark. Similarly, the mark for the final exam must also be 5.0 out of 10.0 or higher to be included in the overall mark.

SPECIAL EXAMINATION SESSION: 100% exam.

-------------------------

ORDINARY EXAMINATION PERIOD

Continuous Assessment:

- Participation and Attendance + Completion of Use Cases (50 per cent):

- Regular attendance at classes and scheduled activities.

- Active participation in discussions and debates.

- Correct and complete completion of the use cases.

- Final examination covering the entire course (50%).

The continuous assessment mark must be at least 5.0/10.0 in order to be included in the average with the examination mark. Similarly, the final examination mark must also be at least 5.0/10.0 in order to be included in the average with the continuous assessment mark.

RE-SIT EXAMINATION PERIOD: Examination (100%).

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	MG	Fundamentals of data management.
2	MG	Fundamentals of data management.
3	MG	Data Storage I: SQL
4	MG	Data Storage I: SQL
5	MG	Database Concepts
6	MG	Database Concepts
7	MG	ER diagrams
8	MG	ER diagrams
9	MG	ER diagrams
10	MG	ER diagrams
11	MG	Dimensional Model
12	MG	Dimensional Model
13	MG	Dimensional Model
14	MG	Dimensional Model
15	MG	Data Warehouse Concept
16	MG	The Concept of a Data Warehouse
17	MG	Analytical environment
18	MG	Analytical approach
19	MG	SQL: Query Filters
20	MG	SQL: Query hints
21	MG	SQL: Query hints
22	MG	SQL: Query hints
23	MG	Knowledge Extraction Techniques
24	MG	Knowledge Extraction Techniques
25	MG	PCA
26	MG	PCA
27	MG	T-SNE
28	MG	T-SNE
29	MG	LDA
30	MG	LDA
31	MG	Unsupervised models for data analysis
32	MG	Unsupervised models for data analysis
33	MG	Data Storage II: NoSQL
34	MG	Data Storage II: NoSQL
35	MG	Introduction to MongoDB
36	MG	Introduction to MongoDB
37	MG	Cloud Atlas Environments
38	MG	Cloud Atlas Soundtracks
39	MG	SQL vs NoSQL
40	MG	SQL vs NoSQL
41	MG	Queries in NoSQL
42	MG	Queries in NoSQL
43	MG	Queries in NoSQL
44	MG	Non-SQL queries
45	LB	Using SQL and Python for data management
46	LB	Using SQL and Python for data management
47	LB	Using SQL and Python for data management
48	LB	Using SQL and Python for data management
49	LB	Using SQL and Python for data management
50	LB	Using SQL and Python for data management
51	EV	Using SQL and Python for data management	25
52	ASSIGNMENT	USE CASE 1: SQL and knowledge extraction.
53	ASSIGNMENT	USE CASE 1: SQL and knowledge extraction.
54	ASSIGNMENT	USE CASE 1: SQL and knowledge extraction.
55	ASSIGNMENT	ASSIGNMENTUSE CASE 1: SQL and knowledge extraction.
56	ASSIGNMENT	Non-SQL use case
57	ASSIGNMENT	Non-SQL use case
58	TRAB	Non-SQL use case
59	EV	Non-SQL use case	25
60	EV	Knowledge quiz	50

Bibliography

Essential:

1. Anne-Christine BISSON

SQL: Fundamentals of the Language

ENI Editions. 2021.

ISBN: 9782409030376

2. E. Redmond and J. R. Wilson

Seven Databases in Seven Weeks: A Guide to Modern Databases and the NoSQL Movement

Pragmatic Bookshelf. 2012.

ISBN: 9781934356920

3. K. Chodorow

MongoDB: The Definitive Guide (3rd ed.)

O’Reilly Media. 2019.

ISBN: 9781491954461

4. R. Elmasri and S. B. Navathe

Fundamentals of Database Systems (7th ed.)

Pearson. 2015.

ISBN: 9780133970777

C0342308

Numerical simulation

Código: C0342308 Imprimir

Year 3, Second Term. Compulsory. 6 credits.

Profesores

Arantxa Tymowska N/A - Coordinador

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Objectives

The course objectives are divided into two main groups. On the one hand, students will learn the main numerical methods for solving partial differential equations (PDEs). On the other hand, students will learn the main techniques for analysing and processing the solutions obtained.

The first group is divided into explicit and implicit methods for solving PDEs, together with methods for finding special solutions (such as stationary solutions) to PDEs.

The second group is divided into algorithms for handling large matrices and the use of functionals and functional analysis to understand and analyse the solutions to PDEs obtained through the first group of objectives.

-------------------------

The course objectives are divided into two main areas. The first focuses on the study of the main numerical methods for solving partial differential equations (PDEs). The second focuses on the main techniques for analysing and processing the solutions obtained.

The first area covers explicit and implicit methods for solving PDEs, together with methods for finding special solutions to PDEs, such as steady-state solutions.

The second area covers algorithms for handling large matrices, as well as the use of functionals and functional analysis to understand and analyse the solutions to the PDEs obtained using the methods covered in the first area.

Prerequisites

No prerequisites have been established for this course. However, it is highly recommended that students have taken or are currently taking the courses in the Numerical Analysis subject area, as well as the programming courses included in the degree programme./No prerequisites have been established for this course. However, it is highly recommended that students have previously completed or are currently taking the courses in the Numerical Analysis subject area, as well as the programming courses included in the degree programme.

Learning Outcomes

Not applicable.

Learning Outcomes

RC2 – To identify the different phases of the mathematical modelling process, distinguishing between formulation, analysis, solution and interpretation of results. TYPE: Competences.

RC3 – To apply software programmes that solve mathematical and engineering problems, using the appropriate computational environment for each case. TYPE: Competences.

RS1 – To use mathematical language to formulate propositions in various fields and to construct proofs for the given problems. TYPE: Skills.

RS3 – To propose, analyse, validate and interpret the most appropriate mathematical models and tools in real-world situations, in accordance with the intended objectives. TYPE: Skills.

RS4 – To plan the resolution of a problem in accordance with the available tools and the constraints of time and resources. TYPE: Skills.

RS5 – To use computer applications for statistical analysis, numerical and symbolic calculation, graphical visualisation, optimisation or other purposes to solve problems. TYPE: Skills.

Course content

- Introduction to simulation using differential equations. The main solutions to first- and second-order differential equations will be reviewed.

- Finite difference method and the Crank-Nicholson method. Both explicit and implicit finite difference methods and the Crank-Nicholson method will be studied and implemented to obtain solutions for:

- Numerical solution of the wave equation.

- Numerical solution of the heat equation.

- Numerical solution of the Laplace equation.

- Solving partial differential equations with different types of boundary conditions. Both homogeneous and non-homogeneous boundary conditions will be examined.

- Handling of sparse matrices. A study will be carried out on how to numerically implement the solution of sparse matrices and their advantages over conventional methods for solving systems of equations.

- Finite element method. An introduction to the finite element method and variational formulation will be provided. The method will first be implemented for solving ODEs and subsequently for solving second-order PDEs.

-------------------------

- Introduction to simulation using differential equations. The main methods for solving first- and second-order differential equations will be reviewed.

- Finite Difference and Crank–Nicolson Methods. Explicit and implicit finite difference methods, as well as the Crank–Nicolson method, will be studied and implemented to obtain numerical solutions to:

- Numerical solution of the wave equation.

- Numerical solution of the heat equation.

- Numerical solution of Laplace’s equation.

- Solution of partial differential equations with different types of boundary conditions. Homogeneous and non-homogeneous boundary conditions will be considered.

- Sparse matrix techniques. Numerical methods for solving systems involving sparse matrices will be studied, together with their advantages over conventional methods for solving systems of equations.

- Finite Element Method. An introduction to the Finite Element Method and its variational formulation will be provided. The method will first be applied to solving ordinary differential equations (ODEs) and subsequently to solving second-order partial differential equations (PDEs).

Teaching activities

AP1. Lectures.

AP2. Interactive classes.

AP3.1. Project development and challenge-based problem solving.

AP4. Independent study, case and problem solving, and literature search.

AP5. Tutorials.

AP6. Knowledge assessment tests.

Assessment system and criteria

REGULAR EXAM SESSION

Continuous assessment:

- Workshops and assignments (35%).

*** Assignments accounting for more than 25–30% of the overall mark will be weighted with a mark out of 10.

- Ordinary assessment exam (65%): final exam (covering the entire module).

*** A minimum mark of 4.5 out of 10.0 is required in this exam to be included in the overall assessment alongside the workshops and assignments.

The module is considered passed in the ordinary examination session if the final mark is 5.0 out of 10.0 or higher.

SUPPLEMENTARY SESSION

In the supplementary examination, students will be assessed on all the course content in a single examination. The mark for this examination will be the mark obtained in that examination (neither the workshops nor the assignments will be taken into account).

The module is considered passed in the extraordinary sitting if the final mark is 5.0 out of 10.0 or higher.

-------------------------

ORDINARY EXAMINATION PERIOD

Continuous Assessment:

- Workshops and Assignments (35%).

*** Any assignment in which the use of AI exceeds 25–30 per cent will be awarded a mark of 0/10.

- Ordinary Examination (65%): final examination covering the entire course.

*** A minimum mark of 4.5/10.0 must be achieved in this examination for the mark to be averaged with the continuous assessment mark for workshops and assignments.

The course is deemed to have been passed during the Ordinary Examination Period if the final mark is 5.0/10.0 or higher.

EXTRAORDINARY EXAMINATION PERIOD

During the Extraordinary Examination Period, students will be examined on all course content in a single examination. The mark for this examination period will be the mark obtained in that examination; neither workshops nor assignments will be taken into account.

The course is considered passed in the Extraordinary Examination Period if the final mark is 5.0/10.0 or higher.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	MG	Course introduction.
2	SM	Simulation of an EDO
3	MG	Finite difference method
4	MG	Finite difference method
5	MG	Review of the main PDEs
6	SM	Problems
7	MG	Introduction to explicit methods for the numerical solution of PDEs
8	MG	Introduction to explicit methods for the numerical solution of PDEs
9	MG	Introduction to explicit methods for the numerical solution of PDEs
10	EV	Class workshops 1	SE1: 5%
11	MG	Numerical solution of the wave equation
12	MG	Numerical solution of the wave equation
13	MG	Numerical solution of the wave equation (second-order Taylor approximation)
14	SM	Submission of proposed problems
15	MG	Numerical solution of the heat equation
16	MG	Numerical solution of the heat equation
17	MG	Implicit methods for solving PDEs
18	EV	Class workshops 2	SE1: 5%
19	MG	Cranck-Nicolson method
20	MG	Crank-Nicolson method
21	MG	Crank-Nicolson method
22	SM	Submission of Proposed Problems
23	MG	Cauchy boundary conditions
24	MG	Cauchy boundary conditions
25	MG	Laplace’s equation (numerical)
26	SM	Problems
27	MG	Function space
28	MG	Function space
29	MG	Function space
30	EV	Part 1	SE3: 30%
31	MG	Problem on Eigenvalues and Eigenvectors (numerical)
32	MG	Eigenvalues and Eigenvectors Problem (numerical)
33	MG	Eigenvalues and Eigenvectors Problem (numerical)
34	SM	Problems
35	MG	Solving EDPs by diagonalising an operator
36	MG	Solving EDPs by diagonalising an operator
37	MG	Solving EDPs by diagonalising an operator
38	EV	Class 3 workshops	SE1: 5%
39	MG	Steady states of an EDP
40	MG	Steady states of an EDP
41	MG	Steady states of an EDP
42	SM	Problems
43	MG	Finite element method
44	MG	Finite element method
45	MG	Finite element method
46	SM	Problems
47	MG	Clustering method for finding periodic orbits in a system
48	MG	Clustering method for finding periodic orbits in a system
49	MG	Clustering method for finding periodic orbits in a system
50	EV	Class 4 workshops	SE1: 5%
51	MG	Sparse Matrices
52	MG	Sparse Matrices
53	MG	Sparse Matrices
54	SM	Sparse Matrices
55	MG	Wave packet propagation
56	MG	Propagation of wave packets
57	MG	Propagation of wave packets
58	EV	Presentation of work	SE2: 20%
59	MG	General review of concepts
60	EV	Exam	SE3: 30%

Reading list

Core:

1. Juan Carlos Jiménez Bedolla

Numerical Methods using Python

National Autonomous University of Mexico. 2022.

ISBN: 978-607-30-58

2. O. C. Zienkiewicz

The Finite Element Method

Reverté. 2010.

ISBN: 978-84-291-91

3. Steven Chapra and Raymond Canale

Numerical Methods for Engineers

McGraw-Hill Interamericana de España S.L.. 2011.

ISBN: 6071504996

C0342309

Artificial intelligence

Código: C0342309 Imprimir

Year 3, Second Term. Compulsory. 6 credits.

Profesores

Jacinto Velasco Rebolledo - Coordinador

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María González García

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Objectives

The course ‘Introduction to Artificial Intelligence’ is designed to provide Mathematical Engineering students with an overview of the fundamental concepts, techniques and applications of AI. The course ranges from basic principles, such as the definition and impact of AI on different industries, to the development of practical models. Students will learn to implement classification models (both basic and advanced) and regression models, explore data analysis through time series, and be introduced to neural networks and unsupervised learning. Priority is given to a practical approach, using tools and languages such as Python, to solve real-world problems whilst developing a solid theoretical foundation.

Upon completion of the course, students will be able to (learning objectives):

- Acquire a broad understanding of the subject and its most common applications.

- Implement different models using the syntax of the Python programming language.

- Learn and analyse how to train artificial intelligence models

- Apply AI models to a variety of real-world problems.

Prerequisites

We recommend that you have some knowledge of the Python programming language, as this will be used as the main programming language on this course. It is recommended that you have taken modules in numerical analysis, optimisation and control techniques, and linear algebra.

Competencies

Not applicable.

Learning Outcomes

RC1 – Develop tools for visualising large volumes of data in order to communicate the results of the analyses carried out on them, adapting them to different audiences, both technical and non-technical. TYPE: Competencies.

RC2 – Identify the different phases of the mathematical modelling process, distinguishing between formulation, analysis, solution and interpretation of results. TYPE: Competencies.

RC3 - Apply software to solve mathematical and engineering problems, using the appropriate computational environment for each case. TYPE: Competencies.

RC4 – Be able to plan and carry out projects in the field of Mathematical Engineering. TYPE: Competencies.

RC5 – Demonstrate a critical and self-critical attitude when facing the challenges of their work as a mathematical engineer, and demonstrate behaviour consistent with the ethical and professional standards governing scientific innovation and professional practice as a mathematical engineer. TYPE: Competencies.

RC7 – Carry out work and projects related to mathematical engineering that have an impact within interdisciplinary teams or in multicultural contexts. TYPE: Competencies.

RC9 – Formulate problems arising from a professional context in mathematical language, in a way that facilitates their analysis and resolution. TYPE: Competences.

RK3 – Understand the concepts of statistics and statistical inference in large data sets. TYPE: Knowledge or content.

RS2 – Apply data science methods (management, machine learning) as part of the process of analysing large data sets in computing environments. TYPE: Skills or competences.

RS4 – Plan the resolution of a problem in accordance with the available tools and time and resource constraints. TYPE: Skills or competences.

RS5 – Use computer applications for statistical analysis, numerical and symbolic calculation, graphical visualisation, optimisation or other purposes to solve problems. TYPE: Skills or competences.

Course description

The Artificial Intelligence module on the Mathematical Engineering degree programme at Alfonso X el Sabio University comprises the following topics:

- Topic 0. Fundamentals of AI.

- Topic 1. Classification models.

- Topic 2. Regression models.

- Topic 3. Time series and AI.

- Topic 4. Introduction to Neural Networks.

- Topic 5. Unsupervised and Reinforcement Learning Models.

Training activities

AP1. Lectures.

AP2. Interactive classes.

AP3.2 Workshops and computer labs.

AP4. Independent study, case studies, problem-solving and literature review.

AP5. Tutoring.

AP6. Knowledge assessments.

Assessment system and criteria

REGULAR EXAM SESSION (continuous assessment + final exam):

+++ Continuous assessment: participation and attendance + completion of case studies (50%).

- Regular attendance at classes and scheduled activities.

- Active participation in discussions and debates.

- Correct and complete completion of case studies.

+++ Final exam: exam covering all course content in the standard assessment period (50%).

- A minimum mark of 4.0 will be required in this exam for it to be included in the calculation of the final mark

alongside the continuous assessment. In such cases, the average will be calculated even if the mark obtained

in the continuous assessment is below 5.0.

SUPPLEMENTARY SESSION:

In the supplementary sitting, assessment will be based solely on an exam covering all course content.

The exam will account for 100 per cent of the final mark.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	MG	Course introduction.
2	MG	Introduction to the course.
3	MG	Introduction to AI.
4	MG	Definition. History.
5	MG	Definition. History.
6	MG	Types of AI in Industry
7	MG	Machine Learning & Deep Learning
8	MG	AI Ethics
9	MG	Supervised models
10	MG	Unsupervised models
11	MG	Model Evaluation
12	MG	Model Evaluation
13	MG	Classification Models
14	MG	Simple classification models
15	MG	Simple classification models
16	MG	Complex classification models: random forest & boost
17	MG	Complex classification models: random forest & boost
18	MG	Complex classification models: random forest & boost
19	MG	Complex classification models: random forest & boost
20	MG	Training issues, overfitting, underfitting and unbalanced datasets
21	MG	Training problems, overfitting, underfitting and unbalanced datasets
22	MG	Classification models II: decision trees with regression
23	MG	Classification Models II: Decision Trees with Regression
24	MG	Classification models II: regression
25	MG	Classification models II: regression
26	MG	Classification Models II: Ridge and Lasso regression
27	MG	Classification Models II: Ridge and Lasso Regression
28	MG	AI and time series
29	MG	AI and time series
30	MG	Time series analysis. Classical methods
31	MG	Application of AI to time series
32	MG	Attribution models
33	MG	Attribution models
34	MG	Introduction to deep learning
35	MG	Introduction to deep learning
36	MG	Primitive Neural Networks
37	MG	Primitive Neural Networks
38	MG	Multilayer neural networks
39	MG	Multilayer neural networks
40	MG	Multilayer neural networks
41	MG	Multilayer neural networks
42	MG	Application of neural networks in business
43	MG	Application of neural networks to business
44	MG	Big Data Analysis with AI
45	MG	Big Data Analysis with AI
46	LB	Customer segmentation
47	LB	Purchase propensity model using ML
48	LB	Purchase propensity model using ML
49	LB	MMM attribution model
50	LB	MMM attribution model
51	LB	Classification using neural networks applied to the digital world
52	LB	Classification using neural networks applied to the digital world
53	TRAB	Purchase propensity model using ML
54	TRAB	Purchase propensity model using ML	12.5
55	TRAB	MMM Attribution Model
56	TRAB	MMM attribution model	12.5
57	TRAB	Classification using neural networks applied to the digital world
58	ASSIGNMENT	Classification using neural networks applied to the digital world	12.5
59	ASSIGNMENT	Customer segmentation
60	TRAB	Customer segmentation	12.5

Bibliography

Basic:

1. Charu C. Aggarwal

Linear Algebra and Optimisation for Machine Learning

Springer. 2020.

ISBN: 3030403432

2. Eloy Vicente Cestero and Alfonso Mateos Caballero

Artificial Intelligence: Mathematical, Algorithmic and Methodological Foundations

978-84-09-46911-6. 2023.

ISBN: 8409469111

3. John D. Kelleher, Brian Mac Namee and Aoife D’Arcy

Fundamentals of Machine Learning for Predictive Data Analytics, second edition: Algorithms, Worked Examples, and Case Studies

The MIT Press. 2020.

ISBN: 0262044692

4. Peter J. Brockwell (Author), Richard A. Davis

Introduction to Time Series and Forecasting (Springer Texts in Statistics)

3rd ed. Springer International Publishing AG. 2016.

ISBN: 9783319298528

TOTAL:

Year 4

FIRST FOUR-MONTH PERIOD

Code

Subjects

Character*

ECTS

C0342602

Quantum Physics I

Código: C0342602 Imprimir

Course 4. First-semester module. Compulsory. 6 credits.

Profesores

José Antonio Prieto Persiguero - Coordinador

Ver biodata

Objectives

The aim of this module is for students to acquire a rigorous introduction to quantum physics, gaining an understanding of the experimental and conceptual foundations that gave rise to quantum theory and becoming familiar with the wave formulation of quantum mechanics through the Schrödinger equation.

The course provides the necessary physical and mathematical foundations for describing simple quantum systems, developing an understanding of energy quantisation, wave-particle duality, the wave function and the probabilistic interpretation of quantum states.

Upon completion of the module, students are expected to be able to:

• understand the experimental foundations that led to the development of quantum physics;

• interpret the dual nature of radiation and matter;

• apply the basic concepts associated with the wave function and the probabilistic description of quantum systems;

• apply the time-independent Schrödinger equation to simple one-dimensional and three-dimensional problems;

• analyse fundamental quantum systems, such as the harmonic oscillator and the hydrogen-like atom;

• solve basic problems in quantum mechanics using the mathematical tools specific to Schrödinger’s formulation.

The course establishes the conceptual, physical and mathematical foundations necessary for the subsequent study of the general formalism of quantum mechanics, approximation methods, angular momentum theory, multi-electron atoms and other advanced applications covered in the modules ‘Quantum Physics II’ and ‘Structure of Matter’.

Prerequisites

None

Learning outcomes

RK1 Understand the most important phenomena and theories in the various branches of physics, as well as their historical context

RK2 Understand physically distinct phenomena and their underlying analogies, enabling the application of known solutions to new problems

RK3 To analyse the fundamental concepts and principles of physical systems in order to develop approximations that enable the construction of a simplified model

RK6 Understand the principles of mathematics and statistics that underpin the study of physics in classical and quantum systems

RK7 Understand the laws and principles of physics, identifying their logical and mathematical structure, their experimental basis and the phenomena described by them

RK8 Understand the fundamental concepts of quantum physics in modelling phenomena at the atomic and subatomic scales

RS1 Apply the most important knowledge, concepts and methods from the various branches of physics.

Learning outcomes

RA1 Apply the experimental foundations of quantum physics and its postulates to discuss laboratory exercises and/or experiments effectively.

RA2 Applies the mathematical formulation of quantum mechanics appropriately to simple one-dimensional and three-dimensional systems to successfully carry out practical activities.

LA3 Understands the dual nature of microscopic entities and its implications for their characteristics and description.

RA4 Understands the experimental foundations of quantum physics and is proficient in handling the orders of magnitude of various physical quantities at the atomic scale.

Course content

Topic 1: Thermal radiation and Planck’s postulate.

Topic 2: Particle properties of electromagnetic radiation.

Topic 3: de Broglie’s hypothesis and the wave properties of matter.

Topic 4: Classical and semi-classical atomic models.

Topic 5: The wave formulation of quantum mechanics.

Topic 6: The time-independent Schrödinger equation.

Topic 7: Schrödinger’s atomic model.

Learning activities

Learning activity No. of hours* Contact hours

(8–12)** % Face-to-face

AP1.- Participatory lectures 48 4 100

AP2. – Seminars or practical application classes 30 2.5 100

AP3.- Practical activities (case studies, project work, simulations, etc.) 72 3 50

AP4. – Independent study 120 0 0

AP5.- Tutorials 24 0.6 30

AP6.- Knowledge assessments 6 0.5 100

TOTAL 300 10.6

Assessment system and criteria

The assessment process will consist of evaluating the extent to which the student has acquired the competences associated with the module.

REGULAR EXAMINATION PERIOD – CONTINUOUS ASSESSMENT

Continuous assessment will consist of the following components:

-- a case study, accounting for 20 per cent of the final mark for the module, which students will undertake in small groups throughout the term; this practical case study will involve the submission of one or more deliverables, each of which will be assessed and will carry its corresponding weighting in the case study mark. Submission dates will be announced well in advance.

-- a mid-term exam, accounting for 20% of the final mark for the module. The date of the first exam, which will cover topics 1 to 3, will be announced well in advance (it will take place during November).

-- a final examination, accounting for 60 per cent of the final mark for the module, which will cover all the content (topics) taught and will take place on the date set by the university for the ordinary examination session (officially announced at the start of the term).

***** The weighted average of all these assessment tests will be calculated only and exclusively if the mark obtained in the final exam is 4.0 out of 10 or higher.

Furthermore, only the examinations will be subject to re-marking.

***** The student will have passed the module in the ordinary examination session if, and only if, they obtain a final mark (weighted average of all assessment tests) of 5.0 out of 10 or higher.

Otherwise, the student may pass the module in the supplementary examination period.

EXTRAORDINARY EXAMINATION SESSION

In this sitting, the student will be examined on all the content (topics) covered in a final examination to be held on the date set by the university for the supplementary sitting. The mark for this sitting will be solely and exclusively that obtained in this examination, and the module will be deemed to have been passed if the mark is 5.0 out of 10 or higher.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	SESSION	Thermal radiation. Spectral radiance and radiance. Stefan’s law and Wien’s displacement law.
2	ASSIGNMENT	Exercises (thermal radiation).
3	SESSION	Black body. Spectral density. Mean energy. Rayleigh–Jeans law and the ultraviolet catastrophe.
4	ASSIGNMENT	Exercises (black body and the Rayleigh–Jeans law).
5	SESSION	Planck’s postulate and average energy. Planck’s law.
6	ASSIGNMENT	Exercises (black body and Planck’s law).
7	SESSION	Exercises (thermal radiation; Rayleigh–Jeans and Planck’s laws).
8	ASSIGNMENT	Exercises (thermal radiation; Rayleigh–Jeans and Planck’s laws).
9	SESSION	The photoelectric effect and Einstein’s hypothesis.
10	ASSIGNMENT	Exercises (photoelectric effect)
11	SESSION	Compton effect. Relativistic and quantum explanations.
12	ASSIGNMENT	Exercises (Compton effect).
13	SESSION	X-ray and bremsstrahlung generation. Quantum explanation.
14	ASSIGNMENT	Exercises (bremsstrahlung).
15	SESSION	Generation and annihilation of matter-antimatter particles. Quantum explanation.
16	ASSIGNMENT	Exercises (creation and annihilation of matter-antimatter particles).
17	SESSION	Electromagnetic spectrum. Photon energy.
18	ASSIGNMENT	Exercises (corpuscular nature of electromagnetic radiation).
19	SESSION	de Broglie’s hypothesis. The wave nature of matter.
20	ASSIGNMENT	Exercises (de Broglie’s hypothesis).
21	SESSION	Wave properties of matter.
22	ASSIGNMENT	Exercises (wave properties of matter).
23	SESSION	Heisenberg’s uncertainty principle.
24	ASSIGNMENT	Exercises (Heisenberg’s uncertainty principle).
25	EV	EXAM: topics 1, 2 and 3.
26	EV	EXAM: topics 1, 2 and 3.	20%
27	SESSION	Discovery of the electron. Thomson’s atomic model: background, successes and failures. Perrin’s correction.
28	ASSIGNMENT	Exercises (Thomson’s atomic model).
29	SESSION	Rutherford’s atomic model: background, successes and failures.
30	ASSIGNMENT	Exercises (Rutherford’s atomic model).
31	SESSION	Bohr’s atomic model: background, successes and failures.
32	ASSIGNMENT	Exercises (Bohr’s atomic model).
33	SESSION	Sommerfeld’s atomic model: background, successes and failures.
34	ASSIGNMENT	Exercises (Sommerfeld’s atomic model).
35	SESSION	Schrödinger’s theory of quantum mechanics: arguments for plausibility leading to the Schrödinger equation.
36	ASSIGNMENT	Exercises (Einstein’s and de Broglie’s relations; free particle).
37	SESSION	Born’s interpretation of the wave function.
38	ASSIGNMENT	Exercises (probability density).
39	SESSION	Physical observables and expected values.
40	ASSIGNMENT	Exercises (physical observables and expected values).
41	SESSION	Properties of wave functions.
42	ASSIGNMENT	Exercises (properties of wave functions).
43	SESSION	Time-independent Schrödinger equation. Expected energy values.
44	ASSIGNMENT	Exercises (time-independent Schrödinger equation and expected energy values).
45	SESSION	Time-independent Schrödinger equation: zero potential; potential step; potential barrier and tunnelling effect.
46	ASSIGNMENT	Exercises (zero potential; potential step; potential barrier and tunnelling effect).
47	SESSION	Time-independent Schrödinger equation: square-well potential; infinite potential; quantum harmonic oscillator.
48	ASSIGNMENT	Exercises (square-well potential; infinite potential, quantum harmonic oscillator).
49	SESSION	Exercises (time-independent Schrödinger equation in one-dimensional problems).
50	ASSIGNMENT	Exercises (time-independent Schrödinger equation in one-dimensional problems).
51	SESSION	Hydrogen-like atom in the Coulomb approximation. Schrödinger’s equation in spherical coordinates.
52	ASSIGNMENT	Exercises (Schrödinger’s equation in spherical coordinates).
53	SESSION	Quantum numbers in central potentials, expected energy values and degeneracy.
54	ASSIGNMENT	Exercises (hydrogen-like atom in the Coulomb approximation).
55	SESSION	Eigenfunctions and probability densities: orbitals.
56	ASSIGNMENT	Exercises (orbitals).
57	SESSION	Angular momentum in the hydrogen-like atom in the Coulomb approximation.
58	ASSIGNMENT	Exercises (angular momentum).
59	EV	Presentation and defence of a CASE STUDY.
60	EV	Presentation and defence of a CASE STUDY.	20%

Bibliography

Core:

1. Juan José Gómez Cadenas

Quantum Mechanics: Introduction and Applications

Ediciones Paraninfo. 2012.

ISBN: 8498825126

2. Robert Martin Eisberg and Robert Resnick

Quantum Physics: Atoms, Molecules, Solids, Nuclei and Particles

Limusa S.A. de C.V. (Mexico). 1978.

ISBN: 978-968180419

Supplementary:

3. Alberto Galindo and Pedro Pascual

Quantum Mechanics

Reverte Publishers. 1991.

ISBN: 8429158801

4. Ramón Fernández and José Luis Sánchez

100 Problems in Quantum Physics

Alianza Editorial. 2001.

ISBN: 8420686336

Others:

5. David J. Griffiths

Introduction to Quantum Mechanics

Cambridge University Press. 2018.

ISBN: 1107189632

6. J. J. Sakurai

J. J. Sakurai – Modern Quantum Mechanics

Addison-Wesley. 2017.

ISBN: 1108422411

C0442300

Machine Learning

Código: C0442300 Imprimir

Course 4. First-semester module. Compulsory. 6 credits.

Profesores

Jacinto Velasco Rebolledo - Coordinador

Ver biodata

María González García

Ver biodata

Objectives

This course aims to introduce the basic and advanced concepts of Machine Learning in a step-by-step manner. The course begins by covering the fundamentals of supervised machine learning, before moving on to deep learning models. The course will conclude with unsupervised analysis models.

The knowledge required to understand the neural network models presented in this course includes fundamental theoretical concepts, as well as the ability to build your own models using the Python programming language. To achieve this, we will combine theoretical concepts with use cases, which will enable you to verify the results obtained on real-world problems.

Prerequisites

We recommend that you have some knowledge of the Python programming language, as it will be used as the main programming language in this Machine Learning course.

Skills

Not applicable.

Learning Outcomes

RC1 – Develop tools for visualising large volumes of data in order to communicate the results of analyses carried out on them, adapting them to different audiences, both technical and non-technical. TYPE: Competencies.

RC2 – Identify the different phases of the mathematical modelling process, distinguishing between formulation, analysis, solution and interpretation of results. TYPE: Competencies.

RC3 - Apply software to solve mathematical and engineering problems, using the appropriate computational environment for each case. TYPE: Competencies.

RC4 – Be able to plan and carry out projects in the field of Mathematical Engineering. TYPE: Competences.

RC5 – Demonstrate a critical and self-critical attitude when facing the challenges of their work as a mathematical engineer, and possess the ability to adopt attitudes consistent with the ethical and professional standards governing scientific innovation and professional practice as a mathematical engineer. TYPE: Competencies.

RC7 – Carry out work and projects related to mathematical engineering that have an impact within interdisciplinary teams or in multicultural contexts. TYPE: Competencies.

RC9 – Formulate problems from a professional context in mathematical language in a way that facilitates their analysis and resolution. TYPE: Competences.

RK3 – Understand the concepts of statistics and statistical inference in large data sets. TYPE: Knowledge or content.

RS2 – Apply data science methods (management, machine learning) as part of the process of analysing large data sets in computing environments. TYPE: Skills or competences.

RS4 – Plan the resolution of a problem in accordance with the available tools and time and resource constraints. TYPE: Skills or competences.

RS5 – Use computer applications for statistical analysis, numerical and symbolic calculation, graphical visualisation, optimisation or other purposes to solve problems. TYPE: Skills or competences.

Description of the content

Topic 0. Fundamentals of machine learning

Topic 1. Classification and regression models

Topic 2. Neural networks

Topic 3. Convolutional Neural Networks (CNNs)

Topic 4. Recurrent Neural Networks (RNNs)

Topic 5. Natural Language Processing

Topic 6. Unsupervised models

By the end of the course, students will be able to:

- Gain a broad understanding of machine learning and its most common applications.

- Implement different models using the syntax of the Python programming language.

- Learn and analyse how to train machine learning models.

- Apply Machine Learning models to a variety of real-world problems.

- Understand and correctly use the tools and techniques for deploying pre-trained models.

Training activities

AP1. Lectures.

AP2. Interactive classes.

AP3.2 Workshops and computer labs.

AP4. Independent study, case studies, problem-solving and literature review.

AP5. Tutoring.

AP6. Knowledge assessments.

Assessment system and criteria

REGULAR EXAMINATION PERIOD (weighting of case studies: 50 per cent, weighting of the exam: 50 per cent):

1) Participation and attendance + case studies (50 per cent):

a) Regular attendance at classes and scheduled activities.

b) Active participation in discussions and debates

c) Correct and complete completion of the case studies

2) Final examination (50%): an examination in the ordinary assessment period comprising 50% theoretical questions and 50% case studies.

The calculation of the final mark in the standard examination period will be conditional upon achieving a minimum mark of 4.0 out of 10 in the final examination. Once this requirement has been met, a weighted average will be calculated from all activities and assessment tasks, even if any of them have been failed.

If the mark obtained in the final exam is less than 4.0 out of 10, the final mark will be the mark obtained in that exam.

SUPPLEMENTARY EXAMINATION PERIOD (100% Exam):

In the supplementary examination, assessment will be based solely on an examination covering the entire course content.

The exam will account for 100 per cent of the final mark.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	MG	Course introduction.
2	MG	Introduction to the course.
3	MG	Supervised models
4	MG	Unsupervised models
5	MG	Introduction to Machine Learning
6	MG	Review of Classification Models
7	MG	Model Evaluation
8	MG	Simple Neural Networks
9	MG	Complex Neural Networks
10	MG	Backpropagation, loss functions and activation functions
11	MG	Backpropagation, loss functions and activation functions
12	MG	Evaluation of neural networks
13	MG	Convolutional Neural Networks
14	MG	Co-involutional Neural Networks
15	MG	Image processing
16	MG	Image processing
17	MG	Model architecture: pooling and convolution layers
18	MG	Model architecture: pooling and convolution layers
19	MG	Use cases
20	MG	Use cases
21	MG	Recurrent Neural Networks
22	MG	Recurrent Neural Networks
23	MG	LSTM Recurrent Neural Networks
24	MG	LSTM recurrent neural networks
25	MG	Backpropagation principle (BPTT)
26	MG	The principle of backpropagation (BPTT)
27	MG	Time Series with Recurrent Neural Networks
28	MG	Time Series with Recurrent Networks
29	MG	Time Series with Recurrent Networks
30	MG	Introduction to NLP
31	MG	Introduction to NLP
32	MG	Introduction to NLP
33	MG	Introduction to Generative AI
34	MG	Introduction to Generative AI
35	MG	Cost estimation in ML projects
36	MG	Cost estimation in ML projects
37	MG	Unsupervised methods
38	MG	Unsupervised methods
39	MG	K-means and DBSCAN
40	MG	K-means and DBSCAN
41	MG	Reinforcement learning
42	MG	Reinforcement learning
43	MG	Use cases in marketing and finance
44	MG	Use cases in Marketing and Finance
45	MG	Use cases in Marketing and Finance
46	LB	Customer segmentation – Unsupervised
47	LB	Customer segmentation – Unsupervised
48	LB	Time Series using RNN
49	LB	Time Series using RNNs
50	LB	Medical imaging diagnosis of melanomas for classification
51	LB	Imaging diagnosis of melanomas for classification
52	TRAB	Customer segmentation – Unsupervised
53	EV	Customer segmentation – Unsupervised	12.5
54	TRAB	Time Series using RNN
55	EV	Time Series using RNNs	12.5
56	TRAB	Imaging diagnosis of melanomas for classification
57	EV	Imaging diagnosis of melanomas for classification	12.5
58	LB	Propensity model
59	TRAB	Propensity model
60	EV	Propensity model	12.5

Bibliography

Basic:

1. Charu C. Aggarwal

Linear Algebra and Optimisation for Machine Learning

Springer. 2020.

ISBN: 3030403432

2. Eloy Vicente Cestero and Alfonso Mateos Caballero

Artificial Intelligence: Mathematical, Algorithmic and Methodological Foundations

978-84-09-46911-6. 2023.

ISBN: 8409469111

3. John D. Kelleher, Brian Mac Namee and Aoife D’Arcy

Fundamentals of Machine Learning for Predictive Data Analytics, second edition: Algorithms, Worked Examples, and Case Studies

The MIT Press. 2020.

ISBN: 0262044692

4. Peter J. Brockwell (Author), Richard A. Davis

Introduction to Time Series and Forecasting (Springer Texts in Statistics)

3rd ed. Springer International Publishing AG. 2016.

ISBN: 9783319298528

C0442301

Management and Production Models

Código: C0442301 Imprimir

Course 4. First-term module. Compulsory. 6 credits.

Profesores

Johnny Eduardo Meyers Angulo - Coordinador

Ver biodata

Víctor David Boscá Navarro

Ver biodata

Objectives

Management and Production Models offers Mathematical Engineering students a comprehensive overview of project management and production, combining mathematical techniques with agile methodologies such as Scrum and Kanban. The course will explore processes, tools and models applied to the planning, execution and optimisation of projects in various production environments.

Throughout the module, students will develop the skills to:

- Apply mathematical models to project management, inventory optimisation, task sequencing and queue management.

- Implement agile methodologies using tools such as Jira and Figma to organise workflows and improve productivity.

- Model and solve reliability, replacement and maintenance problems in various production contexts.

- Understand and apply simulation models, including the generation of random numbers and variables for analysis and decision-making.

- Integrate mathematical techniques with modern management approaches to improve efficiency and adaptability in production.

This module combines theory and practice through applied case studies and software tools, equipping students to tackle the challenges of project management in industrial and technological environments.

-------------------------

Management and Production Models provides Mathematical Engineering students with a comprehensive perspective on project management and production, combining mathematical techniques with agile methodologies such as Scrum and Kanban. The course explores processes, tools and models applied to project planning, execution and optimisation across different production environments.

Throughout the course, students will develop the ability to:

- Apply mathematical models to project management, inventory optimisation, task scheduling and queueing systems.

- Implement agile methodologies using tools such as Jira and Figma to organise workflows and improve productivity.

- Model and solve reliability, replacement and maintenance problems in different production contexts.

- Understand and apply simulation models, including the generation of random numbers and random variables for analysis and decision-making.

- Integrate mathematical techniques with modern management approaches to improve efficiency and adaptability in production.

The course combines theoretical and practical learning through applied case studies and software tools, preparing students to tackle the challenges of project management in industrial and technological environments.

Prerequisites

To get the most out of the course, it is recommended that students have a basic knowledge of applied mathematics, access to the Internet and proficiency in the use of digital tools, as well as familiarity with collaborative working environments./To make the most of the course, students are expected to have a basic knowledge of applied mathematics, access to the Internet and proficiency in the use of digital tools, as well as familiarity with collaborative working environments.

Learning Outcomes

Not applicable.

Learning outcomes

RC1 – Develop data visualisation tools for large volumes of data in order to communicate the results of the analyses carried out on them, adapting them to different audiences, both technical and non-technical/ To develop visualisation tools for large volumes of data in order to communicate the results of the analyses carried out on them, adapting them to different audiences, both technical and non-technical. TYPE: Competences.

RC2 – To identify the different phases of the mathematical modelling process, distinguishing between formulation, analysis, solution and interpretation of results. TYPE: Competences.

RC3 – To apply software programmes that solve mathematical and engineering problems, using the appropriate computational environment for each case. TYPE: Competences.

RC4 – To be able to plan and develop projects in the field of Mathematical Engineering. TYPE: Competences.

RC6 – To work independently and in an organised manner to develop solutions to the various problems that may arise in the field of mathematical engineering, subject to strict time or economic constraints./ to work independently and in an organised manner to develop solutions to the various problems that may arise in the field of mathematical engineering, subject to strict time or financial constraints. TYPE: Competences.

RC9 – To formulate problems arising in a professional context using mathematical language, in a way that facilitates their analysis and resolution. TYPE: Competences.

RK3 – To understand the concepts of statistics and statistical inference in large datasets. TYPE: Knowledge or content.

RS2 – To apply Data Science methods (management, machine learning) as part of the process of analysing large datasets in computing environments. TYPE: Skills.

RS4 – To plan the resolution of a problem in accordance with the available tools and the constraints of time and resources. TYPE: Skills.

RS5 – To use computer applications for statistical analysis, numerical and symbolic calculation, graphical visualisation, optimisation or other purposes to solve problems. TYPE: Skills.

Course description

This module provides Mathematical Engineering students with a practical insight into project management and delivery using agile methodologies such as Scrum and Kanban. The use of digital tools such as Jira and Figma to plan, execute and monitor projects in production environments will be explored. Key concepts in project management will be covered, including agile planning, team management, process simulation and workflow optimisation, integrating mathematical approaches to decision-making and continuous improvement.

GENERAL CONTENT

Unit 1: Fundamentals of Project Management

Introduction to project management: basic principles and historical development.

Comparison between traditional and agile methodologies.

Key roles in project management: Product Owner, Scrum Master and development team.

Application of mathematical models in project management.

Unit 2: Agile Methodologies and Management Tools

Introduction to Scrum and Kanban: principles and differences.

Using Jira for agile project management.

Organising the backlog, prioritising tasks and planning sprints.

Implementing Kanban and Scrum boards in Jira.

Visual work management and workflow optimisation.

Unit 3: Project Planning with Jira

Creating and managing the backlog: epics, user stories and tasks.

Task estimation and resource allocation.

Developing the project schedule and monitoring progress.

Workload analysis and optimisation of team performance.

Unit 4: Project Execution and Monitoring

Monitoring the team’s work using Jira.

Using burn-down charts and cumulative flow diagrams.

Quality control and risk management in agile projects.

Incident resolution and reporting in Jira.

Unit 5: Design and Prototyping with Figma

Introduction to Figma as a prototyping tool.

Creating wireframes and interactive prototypes.

Team collaboration and real-time prototype review.

Design documentation and preparation for development.

Unit 6: Project Closure and Assessment

Validation of deliverables and project closure.

Evaluation of results and performance metrics.

Reflection on processes and continuous improvement.

Ethics and best practice in agile project management.

METHODOLOGY

The module takes a practical approach, with students directly applying the concepts to real-world projects using Jira and Figma. Active learning will be encouraged through simulations, case studies and teamwork. Theoretical classes will be combined with practical sessions using digital tools, ensuring an applied and dynamic learning experience.

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This course provides Mathematical Engineering students with a practical perspective on project management and production through agile methodologies such as Scrum and Kanban. It explores the use of digital tools such as Jira and Figma to plan, execute and monitor projects in production environments. Key project management concepts are covered, including agile planning, team management, process simulation and workflow optimisation, integrating mathematical approaches to support decision-making and continuous improvement.

GENERAL CONTENTS

Unit 1: Fundamentals of Project Management

Introduction to project management: basic principles and historical development.

Comparison between traditional and agile methodologies.

Key roles in project management: Product Owner, Scrum Master and development team.

Application of mathematical models to project management.

Unit 2: Agile Methodologies and Management Tools

Introduction to Scrum and Kanban: principles and differences.

Use of Jira for agile project management.

Backlog organisation, task prioritisation and sprint planning.

Implementation of Kanban and Scrum boards in Jira.

Visual work management and workflow optimisation.

Unit 3: Project Planning with Jira

Backlog creation and management: epics, user stories and tasks.

Task estimation and resource allocation.

Project scheduling and progress monitoring.

Workload analysis and optimisation of team performance.

Unit 4: Project Execution and Monitoring

Monitoring team work using Jira.

Use of burn-down charts and cumulative flow diagrams.

Quality control and risk management in agile projects.

Issue resolution and reporting in Jira.

Unit 5: Design and Prototyping with Figma

Introduction to Figma as a prototyping tool.

Creating wireframes and interactive prototypes.

Team collaboration and real-time prototype review.

Design documentation and preparation for development.

Unit 6: Project Closure and Evaluation

Validation of deliverables and project closure.

Evaluation of results and performance metrics.

Reflection on processes and continuous improvement.

Ethics and best practice in agile project management.

METHODOLOGY

The course takes a practical approach in which students will directly apply the concepts covered to real-world projects using Jira and Figma. Active learning will be encouraged through simulations, case-based problem-solving and teamwork. The course combines theoretical lessons with practical implementation activities using digital tools, ensuring an applied and dynamic learning experience.

Training activities

AP1. Lectures.

AP2. Interactive classes.

AP3.1. Project development and challenge-based problem solving.

AP4. Independent study, case and problem solving, and literature search.

AP5. Tutorials.

AP6. Knowledge assessment tests.

Assessment system and criteria

ASSESSMENT SYSTEMS

The assessment systems for this module are:

- AS1: Practical activities

- AS2: Final knowledge assessment tests

- AS3: Portfolio

ASSESSMENT CRITERIA

The assessment methods described above are set out in the following assessment criteria:

- There are two official examination sessions: the ordinary and the supplementary.

+++REGULAR SESSION+++

The final mark for this sitting is the weighted average of a set of assessment tasks detailed below:

-- Applied exercises/tests (SE3), accounting for 20 per cent of the final mark for the ordinary assessment period, to be completed individually or in small groups during term time in class.

-- Final project (SE1), to be carried out individually or in small groups during term time, accounting for 30 per cent of the final mark for the ordinary assessment period.

-- Comprehensive examination (SE2), to be taken individually during the official ordinary examination period, covering the entire syllabus, and accounting for 50 per cent of the final mark for the ordinary examination period, provided that the minimum mark exceeds 4 out of 10. If this mark is not achieved, the final mark for the module will be a fail in the ordinary assessment period.

*** The module is considered passed in the ordinary examination period if the final mark is 5.0 or higher.

+++EXTRAORDINARY EXAMINATION PERIOD+++

If a student fails the module in the ordinary examination period, they may retake it in the extraordinary examination period, which accounts for 100 per cent of the mark and covers the entire syllabus.

*** The module is considered passed in the supplementary examination if the final mark is 5.0 or above, with a minimum of 5 in both parts.

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ASSESSMENT SYSTEMS

The assessment systems for this course are:

SE1: Practical activities

SE2: Final knowledge assessments

SE3: Portfolio

ASSESSMENT CRITERIA

The assessment methods described above are implemented through the following assessment criteria:

- There are two official examination periods: the Ordinary Examination Period and the Extraordinary Examination Period.

+++ORDINARY EXAMINATION PERIOD+++

The final mark for this examination period is the weighted average of the following assessment activities:

-- Practical exercises/practical tests (SE3): 20 per cent of the final mark for the Ordinary Examination Period. These activities will be completed individually or in small groups during the teaching period in class.

-- Final project (SE1): 30 per cent of the final mark for the Ordinary Examination Period. The project will be completed individually or in small groups during the teaching period.

-- Comprehensive examination (SE2): 50% of the final mark for the Ordinary Examination Period. The examination will be sat individually during the official Ordinary Examination Period and will cover the entire course content. A minimum mark of 4 out of 10 must be achieved in this examination for the weighted average to be calculated. If this minimum mark is not achieved, the final mark for the course will be a Fail in the Ordinary Examination Period.

*** The course is considered passed in the Ordinary Examination Period if the final mark is 5.0 or higher.

+++EXTRAORDINARY EXAMINATION PERIOD+++

If a student does not pass the course during the Ordinary Examination Period, they may sit the course examination during the Extraordinary Examination Period, in which the examination accounts for 100 per cent of the final mark and covers all course content.

The Extraordinary Examination Period will take place during the July examination period (for further information, please consult the virtual campus). It consists of a single examination covering all the course content.

*** The course is considered passed during the Extraordinary Examination Period if the final mark is 5.0 or higher.

Timetable

Click on this link to view the detailed timetable in Excel

Session	Activity	Description	Assessment
1	MG	UNIT 1. Fundamentals of Project Management
2	SM	UNIT 1. Fundamentals of Project Management
3	MG	UNIT 1. Fundamentals of Project Management
4	SM	UNIT 1. Fundamentals of Project Management
5	MG	UNIT 1. Fundamentals of Project Management
6	SM	UNIT 1. Fundamentals of Project Management
7	MG	UNIT 1. Fundamentals of Project Management
8	SM	UNIT 1. Fundamentals of Project Management
9	MG	UNIT 2. Agile Methodologies and Management Tools
10	SM	UNIT 2. Agile Methodologies and Management Tools
11	MG	UNIT 2. Agile Methodologies and Management Tools
12	SM	UNIT 2. Agile Methodologies and Management Tools
13	MG	UNIT 2. Agile Methodologies and Management Tools
14	SM	UNIT 2. Agile Methodologies and Management Tools
15	MG	UNIT 2. Agile Methodologies and Management Tools
16	SM	UNIT 2. Agile Methodologies and Management Tools
17	MG	UNIT 3. Project Planning with Jira	SE3, 20
18	SM	UNIT 3. Project Planning with Jira
19	MG	UNIT 3. Project Planning with Jira
20	SM	UNIT 3. Project Planning with Jira
21	MG	UNIT 3. Project Planning with Jira
22	SM	UNIT 3. Project Planning with Jira
23	MG	UNIT 3. Project Planning with Jira
24	SM	UNIT 3. Project Planning with Jira
25	MG	UNIT 4. Project Execution and Monitoring
26	SM	UNIT 4. Project Implementation and Monitoring
27	MG	UNIT 4. Project Implementation and Monitoring
28	SM	UNIT 4. Project Implementation and Monitoring
29	MG	UNIT 4. Project Implementation and Monitoring
30	SM	UNIT 4. Project Implementation and Monitoring
31	MG	UNIT 4. Project Implementation and Monitoring
32	SM	UNIT 4. Project Implementation and Monitoring
33	MG	UNIT 5. Design and Prototyping with Figma	SE3, 20
34	SM	UNIT 5. Design and Prototyping with Figma
35	MG	UNIT 5. Design and Prototyping with Figma
36	SM	UNIT 5. Design and Prototyping with Figma
37	MG	UNIT 5. Design and Prototyping with Figma
38	SM	UNIT 5. Design and Prototyping with Figma
39	MG	UNIT 5. Design and Prototyping with Figma
40	SM	UNIT 5. Design and Prototyping with Figma
41	MG	UNIT 5. Design and Prototyping with Figma
42	SM	UNIT 5. Design and Prototyping with Figma
43	MG	UNIT 5. Design and Prototyping with Figma
44	SM	UNIT 5. Design and Prototyping with Figma
45	MG	UNIT 6. Project Wrap-up and Assessment
46	SM	UNIT 6. Project Closure and Evaluation
47	MG	UNIT 6. Project Closure and Evaluation
48	SM	UNIT 6. Project Closure and Evaluation
49	MG	UNIT 6. Project Closure and Evaluation
50	SM	UNIT 6. Project Closure and Evaluation
51	MG	UNIT 6. Project Closure and Evaluation
52	SM	UNIT 6. Project Closure and Evaluation
53	MG	UNIT 6. Project Closure and Evaluation
54	SM	UNIT 6. Project Closure and Evaluation
55	MG	UNIT 6. Project Closure and Evaluation
56	SM	UNIT 6. Project Closure and Evaluation
57	MG	UNIT 6. Project Closure and Evaluation
58	SM	UNIT 6. Project Closure and Evaluation	SE3, 20
59	MG	UNIT 6. Project Closure and Evaluation	SE1, 30
60	SM	UNIT 6. Project Closure and Assessment	SE3, 20

Bibliography

Core:

1. Chopra, Sunil; Meindl, Peter

Supply Chain Management: Strategy, Planning, and Operations

Pearson Education. 2022.

ISBN: 9780132743952

2. Heizer, Jay; Render, Barry; Munson, Chuck

Operations Management: Sustainability and Supply Chain Management (Also known in Spanish as ‘Principles of Operations Management’)

Pearson Education. 2009.

ISBN: 978-607-442-0

3. Nahmias, Steven; Olsen, Tava Lennon

Production and Operations Analysis

Waveland Press. 2021.

ISBN: 978-147864766

C0442302

Network optimisation

Código: C0442302 Imprimir

Course 4. First-semester module. Compulsory. 6 credits.

Profesores

Hugo Andrea Galindo Beleña - Coordinador

Ver biodata

Objectives

- Correctly identifies various situations, such as network problems, and applies the appropriate model.

- Understands and implements the appropriate algorithms to solve network problems.

- Knows how to apply heuristic methods to combinatorial optimisation problems.

Prerequisites

No prerequisites have been set for this module. However, it is strongly recommended that students have completed all the mathematics modules in the degree programme, as well as the programming modules from the first two years.