Double Degree in Mathematical Engineering + Physics

Objectives

This module, together with Algebra II, forms the subject of Algebra. This module, which forms part of the degree programme’s Core Curriculum, 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

No prerequisites have been set.

Competencies

Basic and general competences:

CB1 – Students have demonstrated that they possess and understand knowledge in an area of study building on the foundations of general secondary education; this is typically at a level which, whilst drawing on advanced textbooks, also includes some aspects requiring knowledge from the cutting edge of their field of study.

CB2 – Students should be able to apply their knowledge to their work or profession in a professional manner and possess the skills typically demonstrated through the formulation and defence of arguments and the resolution of problems within their field of study.

CB3 – Students should be able to gather and interpret relevant data (usually within their field of study) in order to form judgements that include reflection on relevant social, scientific or ethical issues.

CB4 – Students should be able to communicate information, ideas, problems and solutions to both specialist and non-specialist audiences.

Cross-disciplinary competences:

CT2 – The ability to draft and produce reports, papers and other documents in the field of Mathematical Engineering, communicating them clearly and effectively both in writing and orally.

Specific competences:

CE1 – To understand and use mathematical language. To acquire the ability to formulate propositions in different fields of mathematics, to construct proofs and to convey the mathematical knowledge acquired.

CE2 – To be familiar with rigorous proofs of some classical theorems in different areas of mathematics.

Learning outcomes

- Is familiar with the main basic theorems of linear algebra.

- Understands matrix calculus from the conceptual perspective provided by vector and affine spaces.

- Applies knowledge of linear algebra to solve problems that may arise in engineering.

- Apply basic concepts of linear systems to solve engineering problems.

Course content

1. SYSTEMS OF LINEAR EQUATIONS.

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

1.2 Matrices. Classification. Elementary row operations, Hermitian normal form and rank. Operations: addition, scalar multiplication and ‘trace’, ‘transposition’ and ‘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 with linear maps. 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

AF1: Presentation of the concepts related to the modules comprising each subject and the resolution of case studies enabling students to understand how to approach them, as well as other face-to-face group sessions such as discussion classes, group work, etc.

AF2: Practical activities of increasing difficulty that enable students to gradually acquire the ability to solve problems independently.

AF3: Independent study, report writing, practical work, etc., carried out by individual students or groups of students.

AF4: Assessment tests.

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 will 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 acquired the competences associated with the module.

ASSESSMENT SYSTEMS

The assessment methods for this module are:

- AS1: Various types of exercises in which the student must answer different questions.

- AS2: Reports on case studies presented throughout the course.

- AS3: Exams covering the full range of learning activities.

These systems contribute to a greater or lesser extent to the assessment of the basic and general competences (CB1 to CB4), cross-curricular competences (CT2) and specific competences (CE1 and CE2) 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 SESSION+++

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

-- a case study, accounting for 30% of the final mark for the ordinary assessment period, to be carried out during the teaching period 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 (SE1) and the submission of a report and its public defence (SE2) at the end of the teaching term.

-- a non-exemption exam (SE3), to be taken individually during the teaching period, which will account for 20% of the final mark for the ordinary examination session.

-- a final examination (SE3) 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% of the final mark for the standard assessment period.

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

+++EXTRAORDINARY EXAMINATION PERIOD+++

If a student does not pass the module during the ordinary examination period, they may do so during the extraordinary examination period.

This consists of a single exam which will take place during the supplementary examination period, June–July (for further information, please consult the virtual campus), and which assesses the entire syllabus covered in the module.

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

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 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).

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 students enrolled 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

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

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 prerequisites have been set.

Competencies

Basic and general competences:

CB1 – Students have demonstrated that they possess and understand knowledge in an area of study building on the foundations of general secondary education; this is typically at a level which, whilst drawing on advanced textbooks, also includes some aspects requiring knowledge from the cutting edge of their field of study.

CB2 – Students should be able to apply their knowledge to their work or profession in a professional manner and possess the skills typically demonstrated through the formulation and defence of arguments and the resolution of problems within their field of study.

CB3 – Students should be able to gather and interpret relevant data (usually within their field of study) in order to form judgements that include reflection on relevant social, scientific or ethical issues.

CB4 – Students should be able to communicate information, ideas, problems and solutions to both specialist and non-specialist audiences.

CG3 – Ability to carry out work and projects related to mathematical engineering, either individually, in interdisciplinary teams or in multicultural contexts.

Cross-cutting competences:

CT1 – The ability to apply acquired knowledge flexibly and creatively, as well as to adapt it to new contexts and situations.

CT2 – The ability to draft and produce reports, written work and other documents in the field of Mathematical Engineering, communicating them clearly and effectively both in writing and orally.

Specific competences:

CE3 – To propose, analyse, validate and interpret the most appropriate mathematical models and tools in real-world situations, in accordance with the objectives being pursued.

CE5 – Identify the different phases of the mathematical modelling process, distinguishing between formulation, analysis, solution and interpretation of results.

CE7 – Use computer applications for statistical analysis, numerical and symbolic calculation, graphical visualisation, optimisation and other purposes to solve problems.

CE8 – Understand and use software programmes that solve mathematical problems with engineering applications, utilising the appropriate computing environment for each case.

Learning outcomes

- Applies the techniques, methods of representation and summarisation, and measures characteristic of Descriptive Statistics and Inferential Statistics.

- Determines whether a dataset allows a specific hypothesis to be accepted or rejected, and the error involved in doing so.

- Determine and quantify the degree of association between statistical variables.

Course 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

Teaching activities

AF1: Presentation of concepts related to the topics comprising each subject and the resolution of case studies enabling students to understand how to approach them, as well as other face-to-face group sessions such as discussion classes, group work, etc.

AF2: Practical activities of increasing difficulty that enable students to gradually acquire the ability to solve problems independently.

AF3: Independent study, report writing, practical work, etc., carried out by individual students or groups of students.

AF4: Assessment tests.

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 will be the sole assessment criterion, with the weighting specified in the course syllabus.

----

ASSESSMENT SYSTEMS

The assessment methods for this module are:

- AS1: Assignment sheet to be handed in.

- SE2: Problem sheet with an oral presentation of the problems.

- AS3: Exams covering the full range of course activities.

ASSESSMENT CRITERIA

The assessment methods described above are specified 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 set of exercises (SE1), accounting for 10% of the final mark for the ordinary assessment period, to be completed individually or in small groups during the term (for further information, please refer to the timetable).

-- a set of exercises and a presentation (SE2), accounting for 10% of the final mark for the ordinary assessment period, to be completed individually or in small groups at the end of the term (for further information, please refer to the timetable).

-- a mid-term exam (SE3), to be taken individually during the teaching term (for further information, please refer to the timetable), which will account for 20% of the final mark for the ordinary assessment period.

-- a comprehensive exam (SE3) to be taken individually during the official February (ordinary) examination period, covering the entire syllabus, and accounting for 60% 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.

+++SUPPLEMENTARY EXAMINATION SESSION+++

If a student fails the course during the ordinary examination period, they may retake it during the supplementary examination period.

The supplementary examination session 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 sitting if the final mark is 5.0 or higher.

Timetable

Click on this link to view the detailed timetable in Excel

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

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

Basic and general competences:

CB2 – Students should be able to apply their knowledge to their work or profession in a professional manner and possess the competences typically demonstrated through the development and defence of arguments and the resolution of problems within their field of study.

CB3 – Students should be able to gather and interpret relevant data (usually within their field of study) in order to make judgements that include reflection on relevant social, scientific or ethical issues.

CB4 – Students should be able to communicate information, ideas, problems and solutions to both specialist and non-specialist audiences.

Cross-disciplinary competences:

CT2 – The ability to draft and produce reports, papers and other documents in the field of Mathematical Engineering, communicating them clearly and effectively both in writing and orally.

Specific competences:

CE1 – To understand and use mathematical language. To acquire the ability to formulate propositions in different fields of mathematics, to construct proofs and to convey the mathematical knowledge acquired.

CE2 – To be familiar with rigorous proofs of some classical theorems in different areas of mathematics.

CE4 – Formulate problems from a professional context, using mathematical language, in a way that facilitates their analysis and resolution.

CE11 – Master the basic concepts of discrete mathematics, logic, algorithms, coding, operational research and artificial intelligence, and their application to solving engineering problems.

Learning outcomes

- Understands the basic concepts of group and ring theory.

- Recognises basic structures in practical situations, such as: finitely generated Abelian groups, alternating and dihedral symmetric groups, the ring of integers, and the rings of polynomials in one or several variables with coefficients in an arbitrary ring.

- Applies the knowledge acquired to real-world situations.

Course 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

Learning activities

AF1: Presentation of the concepts related to the topics comprising each subject and the resolution of case studies enabling students to understand how to approach them, as well as other face-to-face group sessions such as discussion classes, group work, etc.

AF2: Practical activities of increasing difficulty that enable students to gradually acquire the ability to solve problems independently.

AF3: Independent study, report writing, practical work, etc., carried out by individual students or groups of students.

AF4: Assessment tests.

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 will be the sole assessment criterion, with the weighting specified in the course syllabus.

----

Continuous assessment consists of the following marks:

1) A set of exercises based on the syllabus covered at that time will be set.

These will account for 20 per cent

3) A written assignment will be submitted, in which a topic from the course is explored in greater depth.

This assignment will account for 20 per cent

The deadline for submission is the day of the official exam for this module

4) The official written exam for the subject, covering the course content, will be held.

This mark will account for 60%

If you fail the continuous assessment, the ordinary examination will count for 100 per cent

In the supplementary assessment, 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

Bibliography

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

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.

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

- To 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

BASIC AND GENERAL COMPETENCIES:

CB2 – Students should be able to apply their knowledge to their work or profession in a professional manner and possess the skills typically demonstrated through the development and defence of arguments and the resolution of problems within their field of study.

CB4 – Students should be able to communicate information, ideas, problems and solutions to both specialist and non-specialist audiences.

CROSS-CURRICULAR COMPETENCIES:

CT2 – The ability to draft and produce reports, written pieces and other documents in the field of Mathematical Engineering, communicating them clearly and effectively both in writing and orally.

SPECIFIC COMPETENCIES:

CE5 - To identify the different phases of the mathematical modelling process, distinguishing between formulation, analysis, solution and interpretation of results.

CE7 – Use computer applications for statistical analysis, numerical and symbolic calculation, graphical visualisation, optimisation and other purposes to solve problems.

SC8 – Be familiar with and use software programmes that solve mathematical problems with engineering applications, utilising the appropriate computing environment for each case.

Learning outcomes

- Understand and apply the fundamentals of programming:

Students will be able to explain and correctly apply the basic concepts of programming, such as variables, data types, operators, flow control (conditional statements and loops), and functions.

- Develop programmes in Python to solve specific problems:

They will be able to design and write programmes in Python that solve specific problems, using structured and object-oriented programming techniques.

-Implement fundamental data structures:

Students will know how to use lists, tuples, dictionaries and sets to manage and manipulate data efficiently.

-Develop the ability to think algorithmically:

They will be able to break down complex problems into simple steps and develop algorithms to solve them, using a logical and systematic approach.

-Apply good programming practices:

Students will write clean, readable code, following Python style conventions (PEP 8), with particular attention to modularity, code reusability and clear documentation.

-Apply code debugging and testing techniques:

Students will know how to identify, diagnose and correct errors in their programmes using debugging tools, and carry out tests to ensure the reliability of the software.

-Develop small applications and projects:

They will be able to create functional applications that incorporate the concepts learnt, such as small games, automation tools or data analysis programmes.

-Understand the basic use of files and databases:

They will be able to read from and write to files from their programmes, as well as perform basic operations on databases using standard Python libraries.

-Collaborate on programming projects:

Students will learn to work as part of a team, using version control (such as Git) to collaborate on programming projects, managing code versions and working collaboratively.

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 the 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

AF1: An introduction to the concepts related to the topics covered in each module and the resolution of case studies that enable students to understand how to approach them, as well as other face-to-face group sessions such as discussion classes, group work, etc.

AF2: Practical activities of increasing difficulty, enabling students to gradually develop the ability to solve problems independently.

AF3: Independent study, report writing, practical work, etc., carried out by individual students or groups of students.

AF4: Assessment tests.

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 will be the sole assessment criterion, with the weighting specified in the course syllabus.

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Ordinary Examination Period:

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

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

An average will be calculated from the continuous assessment and the final exam, even if the former is below 5.

Extraordinary sitting (100% exam):

- In the supplementary sitting, 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

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

Objectives

This module, which together with Mathematical Foundations of Engineering II forms part of Mathematical Analysis I, a module within the degree programme’s Basic Training module, aims to provide the mathematical foundations necessary to understand, interpret and apply various concepts and theories, which are fundamental for a graduate in mathematics.

Prerequisites

No prerequisites have been set.

Competencies

BASIC AND GENERAL LEARNING OUTCOMES:

CB1 – Students have demonstrated that they possess and understand knowledge in a field of study building on the foundations of general secondary education; this is typically at a level which, whilst drawing on advanced textbooks, also includes some aspects requiring knowledge from the cutting edge of their field of study.

CB2 – Students should be able to apply their knowledge to their work or profession in a professional manner and possess the skills typically demonstrated through the formulation and defence of arguments and the resolution of problems within their field of study.

CB3 – Students should be able to gather and interpret relevant data (usually within their field of study) in order to form judgements that include reflection on relevant social, scientific or ethical issues.

CB4 – Students should be able to communicate information, ideas, problems and solutions to both specialist and non-specialist audiences.

CROSS-CURRICULAR COMPETENCIES:

CT2 – The ability to draft and produce reports, written work and other documents in the field of Mathematical Engineering, communicating them clearly and effectively both in writing and orally.

SPECIFIC COMPETENCIES:

CE1 - To understand and use mathematical language. To acquire the ability to formulate propositions in different fields of mathematics, to construct proofs and to convey the mathematical knowledge acquired.

CE2 – Be familiar with rigorous proofs of some classical theorems in different areas of mathematics.

CE3 – Propose, analyse, validate and interpret the most appropriate mathematical models and tools in real-world situations, in accordance with the objectives being pursued.

Learning outcomes

- Distinguishes between and handles different sets of numbers.

- Is familiar with the main basic theorems on numerical sequences and series.

- Understands the main basic theorems relating to limits, continuity and differentiability.

- Calculates derivatives.

- Applies knowledge of mathematical analysis to solve problems that may arise in engineering.

Course description

The content to be covered in this module is as follows:

Topic 1: Real numbers

Topic 2: Complex numbers

Topic 3: Numerical sequences

Topic 4: Numerical series

Topic 5: Limits and continuity

Topic 6: Derivatives

Topic 7: Applications of the derivative

Topic 8: Graphing functions

Learning activities

LA1: Presentation of concepts related to the topics comprising each subject and the resolution of case studies enabling students to understand how to approach them, as well as other face-to-face group sessions such as discussion classes, group discussions, etc.

LA1: Practical activities of increasing difficulty that enable students to gradually develop the ability to solve problems independently.

LA3: Independent study, report writing, practical work, etc., carried out by individual students or groups of students.

AF4: Assessment tests.

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 will be the sole assessment criterion, with the weighting specified in the course syllabus.

----

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: Various types of exercises in which the student must answer different questions.

- AS2: Reports on case studies presented throughout the course.

- AS3: Exams covering the full range of learning activities.

These systems contribute to a greater or lesser extent to the assessment of the basic and general competences (CB1 to CB4), cross-curricular competences (CT2) and specific competences (CE1 to CE3) assigned to this module.

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

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:

- Assignment submissions (SE1), accounting for 20% of the final mark, to be completed individually or in small groups during the term.

- Submission of a project (SE2), accounting for 20% of the final mark, to be completed individually or in small groups at the end of the term.

- Two mid-term exams (SE3), to be taken individually during the term. Each will account for 30% of the final mark.

*** The module is considered passed in the ordinary assessment period through continuous assessment if the final mark is 5.0 or above.

*** Otherwise, the student must sit the final exam in the ordinary examination period. Their mark for the ordinary examination period will correspond to the mark obtained in the final exam.

+++EXTRAORDINARY EXAMINATION PERIOD+++

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 module is considered passed in the supplementary sitting if the final mark is 5.0 or higher.

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 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).

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 students enrolled 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

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

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

- Historical introduction.

- Particle kinematics. Types of motion.

- Particle dynamics.

- Work and energy, and associated theorems.

- Oscillatory motion.

- Systems of particles. Geometry of masses.

- Statics.

- Elasticity.

Training activities

Teaching 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, simulations, 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

Assessment system Weighting (%)

SE1.- Practical activities (case studies, problem-solving and challenges, project work, oral presentations, debates, etc.) 30

AS2.- Final knowledge assessments 50 50

SE3.- Laboratory practical booklet 20

Timetable

Click on this link to view the detailed timetable in Excel

Objectives

This module, together with Algebra I, forms the subject Algebra. This module, which forms part of the degree programme’s Core 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 within the degree programme, such as Numerical Calculus, Operational Research, Stochastic Calculus and Artificial Intelligence.

Prerequisites

Although no prerequisites have been set, it is advisable to have previously taken the course Algebra I or another course covering similar skills and learning outcomes.

Competencies

BASIC AND GENERAL COMPETENCIES:

CB1 – Students have demonstrated that they possess and understand knowledge in an area of study building on the foundations of general secondary education; this is typically at a level which, whilst drawing on advanced textbooks, also includes some aspects requiring knowledge from the cutting edge of their field of study.

CB2 – Students should be able to apply their knowledge to their work or vocation in a professional manner and possess the skills typically demonstrated through the development and defence of arguments and the resolution of problems within their field of study.

CB3 – Students should be able to gather and interpret relevant data (usually within their field of study) in order to form judgements that include reflection on relevant social, scientific or ethical issues.

CB4 – Students should be able to communicate information, ideas, problems and solutions to both specialist and non-specialist audiences.

CROSS-CURRICULAR COMPETENCIES:

CT2 – The ability to draft and produce reports, written work and other documents in the field of Mathematical Engineering, communicating them clearly and effectively both in writing and orally.

SPECIFIC COMPETENCIES:

CE1 - To understand and use mathematical language. To acquire the ability to formulate propositions in different fields of mathematics, to construct proofs and to convey the mathematical knowledge acquired.

CE2 – Be familiar with rigorous proofs of some classical theorems in different areas of mathematics.

Learning outcomes

- Is familiar with the main basic theorems of linear algebra.

- Understands matrix calculus from the conceptual perspective provided by vector and affine spaces.

- Applies knowledge of linear algebra to solve problems that may arise in engineering.

- Apply basic concepts of linear systems to solve engineering problems.

Course content

1. QUADRATIC FORMS: CONCEPT AND CLASSIFICATION.

1.1 Bilinear forms. Properties. Associated matrix. Change of basis. Symmetric and antisymmetric bilinear forms. Degeneracy and positive definiteness.

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.

Learning activities

AF1: Presentation of the concepts related to the topics comprising each subject and the resolution of case studies enabling students to understand how to approach them, as well as other face-to-face group sessions such as discussion classes, group work, etc.

AF2: Practical activities of increasing difficulty that enable students to gradually acquire the ability to solve problems independently.

AF3: Independent study, report writing, practical work, etc., carried out by individual students or groups of students.

AF4: Assessment tests.

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 will 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 acquired the competences associated with the module.

ASSESSMENT SYSTEMS

The assessment methods for this module are:

- AS1: Various types of exercises in which the student must answer different questions.

- AS2: Reports on case studies presented throughout the course.

- AS3: Exams covering the full range of learning activities.

These systems contribute to a greater or lesser extent to the assessment of the basic and general competences (CB1 to CB4), cross-curricular competences (CT2) and specific competences (CE1 and CE2) 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 SESSION+++

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

-- a case study, accounting for 30% of the final mark for the ordinary assessment period, to be carried out during the teaching period 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 (SE1) and the submission of a report (SE2) at the end of the teaching period.

-- a mid-term exam (SE3), which is not a pass/fail assessment; this will be held in a classroom on an individual basis during the teaching period and will account for 20% of the final mark for the ordinary examination session.

-- a final exam (SE3) to be taken individually in the classroom during the ordinary examination period in May–June (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 assessment period, provided that the student achieves a mark of 4.0 out of 10.0 or higher. Otherwise (a mark below 4.0 out of 10.0), the mark for the module in the ordinary examination period will be that obtained in the final exam.

*** Only the examinations will be subject to review.

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

******* If a student loses 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.

+++EXTRAORDINARY EXAMINATION PERIOD+++

If a student fails to pass the module during the ordinary examination period, they may do so during the extraordinary examination period.

During this session, there will be a single assessment, consisting of an exam to be held during the extraordinary exam 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

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 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 ‘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 students enrolled 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

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

Objectives

The objectives of this module are:

1) To understand the role 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 to solve 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

BASIC AND GENERAL COMPETENCES:

CB2 - Students should be able to apply their knowledge to their work or profession in a professional manner and possess the skills typically demonstrated through the development and defence of arguments and the resolution of problems within their field of study.

CB3 – Students should be able to gather and interpret relevant data (usually within their field of study) in order to make judgements that include reflection on relevant social, scientific or ethical issues.

CB4 – Students should be able to communicate information, ideas, problems and solutions to both specialist and non-specialist audiences.

CB5 – Students should have developed the learning skills necessary to undertake further study with a high degree of autonomy.

CG2 – Ability to work independently and in an organised manner to develop solutions subject to strict time or budgetary constraints.

CROSS-CURRICULAR COMPETENCIES:

CT1 – Ability to apply acquired knowledge flexibly and creatively, and to adapt it to new contexts and situations.

CT2 – Ability to draft and produce reports, written work and other documents in the field of Mathematical Engineering, communicating them clearly and effectively both in writing and orally.

CT3 – Ability to generate new ideas and incorporate them into day-to-day work.

SPECIFIC COMPETENCIES:

CE8 – Knowledge of and ability to use software programmes that solve mathematical problems with applications in engineering, utilising the appropriate computing environment for each case.

CE11 – Mastery of the basic concepts of discrete mathematics, logic, algorithms, coding, operations research and artificial intelligence, and their application to solving engineering problems.

CE14 – Develop and use 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.

Learning outcomes

This module shares learning outcomes with Data Structures and Algorithms II, albeit at a basic level. These learning outcomes are as follows:

- Apply knowledge of algorithms and basic computational complexity to solve problems that may arise in engineering.

- Identify and propose basic solutions to problems relating to algorithmic efficiency.

- Calculate the efficiency of basic iterative algorithms by applying the appropriate calculation rules.

- Design and scale basic algorithms for environments of varying size and complexity.

- Solves problems that may arise in engineering by applying basic knowledge relating to the structure and programming of computer systems.

Course 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 against Python’s predefined 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 (for example, 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.

Learning activities

AF1: Presentation of concepts related to the modules comprising each subject and the resolution of case studies enabling students to understand how to approach them, as well as other face-to-face group sessions such as discussion classes, group discussions, etc.

AF2: Practical activities of increasing difficulty that enable students to gradually acquire the ability to solve problems independently.

AF3: Independent study, report writing, practical work, etc., carried out by individual students or groups of students.

AF4: Assessment tests.

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 will be the sole assessment criterion, with the weighting specified in the course syllabus.

----

REGULAR EXAMINATION PERIOD

During 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 problem set.

- Correct organisation and modularity of the code (separation of concerns and use of appropriate 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-pass/fail 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 basic operations (insertion, deletion, search) and their implications for performance.

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

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

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

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 practices 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 SESSION

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

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

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.

Competencies

BASIC AND GENERAL COMPETENCIES:

CB1 – Students have demonstrated that they possess and understand knowledge in an area of study building on the foundations of general secondary education; this is typically at a level which, whilst drawing on advanced textbooks, also includes some aspects requiring knowledge from the cutting edge of their field of study.

CB2 – Students should be able to apply their knowledge to their work or profession in a professional manner and possess the skills typically demonstrated through the formulation and defence of arguments and the resolution of problems within their field of study.

CB3 – Students should be able to gather and interpret relevant data (usually within their field of study) in order to form judgements that include reflection on relevant social, scientific or ethical issues.

CB4 – Students should be able to communicate information, ideas, problems and solutions to both specialist and non-specialist audiences.

CROSS-CURRICULAR COMPETENCIES:

CT2 – The ability to draft and produce reports, written work and other documents in the field of Mathematical Engineering, communicating them clearly and effectively both in writing and orally.

SPECIFIC COMPETENCIES:

CE10 - To have a thorough grasp of the basic concepts of electromagnetism and circuit theory for solving engineering problems.

Learning outcomes

• Solves problems relating to the physical fundamentals of Computer Science of varying complexity

• Applies the knowledge acquired to real-world situations

• Carries out and verifies experiments on real-world cases

• Carries out research projects on specific topics.

Course 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

LA1: Presentation of concepts related to the subjects comprising each module and the resolution of case studies enabling students to understand how to approach them, as well as other face-to-face group sessions such as discussion classes, group work, etc.

LA2: Practical activities of increasing difficulty that enable students to gradually acquire the ability to solve problems independently.

LA3: Independent study, report writing, practical work, etc., carried out by individual students or groups of students.

AF4: Assessment tests.

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 will 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 acquired the competences associated with the module.

REGULAR EXAMINATION PERIOD

In the ordinary assessment 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% 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 cover 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 exams may be subject to re-marking."

In order for students to benefit from continuous assessment, a minimum attendance rate of 70 per cent at scheduled class sessions (SESION, TRAB) is required. If attendance falls below 70 per cent without a valid reason, the module must be passed by sitting a final examination during an official examination period (ordinary or supplementary). During the ordinary examination period, 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 examination period when the final mark is 5.0 or higher.

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

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

Objectives

This module, which together with Mathematical Foundations of Engineering I forms part of the Mathematical Analysis I course within the degree programme’s Basic Training module, aims to provide the fundamentals of Riemann integration in one dimension, 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

BASIC AND GENERAL COMPETENCIES:

CB1 – Students have demonstrated that they possess and understand knowledge in an area of study building on the foundations of general secondary education; this is typically at a level which, whilst drawing on advanced textbooks, also includes some aspects requiring knowledge from the cutting edge of their field of study.

CB2 – Students should be able to apply their knowledge to their work or profession in a professional manner and possess the competences typically demonstrated through the formulation and defence of arguments and the resolution of problems within their field of study.

CB3 – Students should be able to gather and interpret relevant data (usually within their field of study) in order to form judgements that include reflection on relevant social, scientific or ethical issues.

CB4 – Students should be able to communicate information, ideas, problems and solutions to both specialist and non-specialist audiences.

CROSS-CURRICULAR COMPETENCIES:

CT2 – The ability to draft and produce reports, written work and other documents in the field of Mathematical Engineering, communicating them clearly and effectively both in writing and orally.

SPECIFIC COMPETENCIES:

CE1 - To understand and use mathematical language. To acquire the ability to formulate propositions in different fields of mathematics, to construct proofs and to communicate the mathematical knowledge acquired.

CE2 – Be familiar with rigorous proofs of some classical theorems in different areas of mathematics.

CE3 – Propose, analyse, validate and interpret the most appropriate mathematical models and tools in real-world situations, in accordance with the objectives being pursued.

Learning outcomes

- Is familiar with the main basic theorems concerning sequences and series of functions.

- Is familiar with the main basic theorems of the integral calculus of real functions.

- Calculates antiderivatives and improper integrals.

- Applies knowledge of mathematical analysis to solve problems that may arise in engineering.

Course content

Topic 0. Elementary Functions

Topic 1. Riemann integral.

Topic 2. Integration techniques.

Topic 2. Integration techniques.

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

Topic 5. Series and sequences.

Learning activities

LA1: Presentation of concepts related to the subjects comprising each module and the resolution of case studies enabling students to understand how to approach them, as well as other face-to-face group sessions such as discussion classes, group work, etc.

LA2: Practical activities of increasing difficulty that enable students to gradually acquire the ability to solve problems independently.

AF3: Independent study, report writing, practical work, etc., carried out by individual students or groups of students.

AF4: Assessment tests.

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 will be the sole assessment criterion, with the weighting specified in the course syllabus.

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

- Continuous assessment (50%):

+ 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 SESSION consists of the following:

- Exam covering the entire course (100%).

Timetable

Click on this link to view the detailed timetable in Excel

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

Objectives

Although this course covers several distinct topics, the 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

BASIC AND GENERAL COMPETENCES:

CB2 – Students should be able to apply their knowledge to their work or profession in a professional manner and possess the skills typically demonstrated through the development and defence of arguments and the resolution of problems within their field of study.

CB3 – Students should be able to gather and interpret relevant data (usually within their field of study) in order to make judgements that include reflection on relevant social, scientific or ethical issues.

CB4 – Students should be able to communicate information, ideas, problems and solutions to both specialist and non-specialist audiences.

CROSS-CURRICULAR COMPETENCIES:

CT2 – The ability to draft and produce reports, written work and other documents in the field of Mathematical Engineering, communicating them clearly and effectively both in writing and orally.

SPECIFIC COMPETENCIES:

CE1 - To understand and use mathematical language. To acquire the ability to formulate propositions in different fields of mathematics, to construct proofs and to convey the mathematical knowledge acquired.

CE2 – Be familiar with rigorous proofs of some classical theorems in different areas of mathematics.

CE4 – Formulate problems from a professional context in mathematical language in a way that facilitates their analysis and resolution.

CE11 – Master the basic concepts of discrete mathematics, logic, algorithms, coding, operational research and artificial intelligence, and their application to solving engineering problems.

Learning outcomes

- Solves logical problems of varying complexity.

- Solves problems in discrete mathematics of varying complexity.

- Applies the knowledge acquired to real-world situations

Course 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 modulo n operations.

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.

Learning activities

AF1: Presentation of the concepts related to the topics comprising each subject and the resolution of case studies that enable students to understand how to approach them, as well as other face-to-face group sessions such as discussion classes, group discussions, etc.

AF2: Practical activities of increasing difficulty that enable students to gradually develop the ability to solve problems independently.

AF3: Independent study, report writing, practical work, etc., carried out by individual students or groups of students.

AF4: Assessment tests.

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 will be the sole assessment criterion, with the weighting specified in the course syllabus.

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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 is a comprehensive exam covering the entire course. Furthermore, in order for the mark to be averaged with the continuous assessment, a minimum mark of 4.0 out of 10.0 must be achieved 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 unjustified 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.

EXTRAORDINARY 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 cover all the content taught.

Timetable

Click on this link to view the detailed timetable in Excel

Reading list

Core:

1. José Dorronsoro, Eugenio Hernández

Numbers, Groups and Rings

Addison-Wesley / UAM. 1996.

ISBN: 0201653958

Objectives

The objectives set out for the ‘Fundamentals of Chemistry’ module within the Bachelor’s Degree in Physics respect 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 science, 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 the capacity for 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 physicochemical 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 in terms of energy.

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.

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

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. Assessment tests

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 exam 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 this in the 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 for 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.