73545 - Mathematical Methods M

Learning outcomes

In the first part the student is supposed to learn the different types of graphs, their matrix representations, the related invariants and the problems which can find a model and solution in Graph Theory. In the second part, differential equations of the first and second order are studied.

Course contents

Module 1 (Fourier analysis)

Brief introduction to Banach and Hilbert spaces; Fourier series and applications; Fourier transform; FFT and DFT; Wavelets; Applications to ODE and PDE of interest in engineering application. The detailed program is published on the e-learning platform Virtuale.

Module 2

Module 2

The topics of Module 2, taught by Andrea Petracci, include:
- Some topics in Linear algebra: spectral theorem and matrix exponential;
- Basic notions of complex analysis: holomorphic functions, radius of convergence of a power series, Taylor and Laurent series expansions for holomorphic functions, the z-transform and its region of convergence, poles of the z-transform;
- Basic notions of graph theory: basic definitions, properties of the eigenvalues of the Laplacian matrix.
The detailed syllabus and the lecture log will be available on the website: https://www.dm.unibo.it/~andrea.petracci3/2025MathMethodsM/

Prerequisites: linear algebra (solving linear systems, determining bases of vector subspaces, matrix operations, determinants, eigenvalues and eigenvectors, matrix diagonalizability) and calculus (limits, series, partial derivatives, one-variable integrals).

Readings/Bibliography

Fourier analysis (Modulo 1):

Lecture notes of the teacher. The notes (pdf) will be available through the institutional site Virtuale before the lessons. Students may also use the following textbooks:

- Davide Guidetti: Notes of the course Mathematical Methods (Pdf file available on AMS-Campus: Chapters 2 (normed spaces, Fourier series) and Chapter 4 (Fourier transform)

- Erwin Kreyszig: Advanced Engineering Mathematics, 10th Edition J. Wiley (2014) Chapter 11 (Fouries series and Fourier transform ) and Chapter 12 (PDEs)

- Tim Olson: Applied Fourier Analysis: from signal processing to medical imaging, Birkhauser Chapters 1-5, 10

Modulo 2:

Module 2: some references are below. Additional or more precise references will be given during the course and will appear at https://www.dm.unibo.it/~andrea.petracci3/2025MathMethodsM/

Treil, Linear algebra done wrong, https://www.math.brown.edu/streil/papers/LADW/LADW.html

Poole, Linear algebra, a modern introduction, 3rd ed.

Anton, Kaul, Elementary linear algebra, 12th ed., Wiley

Lay, Lay, McDonald, Linear algebra and its applications, Pearson

Damelin, Miller, The mathematics of signal processing, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2012

Esakkirajan, Veerakumar, Subudhi, Digital signal processing, Springer

Kovacevic, Goyal, Vetterli, Foundations of signal processing, Cambridge University Press https://www.fourierandwavelets.org [https://www.fourierandwavelets.org/]

Mathews, Howell, Complex analysis for mathematics and engineering

Moudgalya, Digital control, Wiley

Proakis, Manolakis, Digital signal processing, 4th ed., Prentice Hall

Kreyszig, Advanced engineering mathematics, 10th ed.

Diestel, Graph Theory, 6th edition, Graduate Texts in Mathematics, Springer https://diestel-graph-theory.com/basic.html

Bondy, Murty, Graph theory, Graduate Texts in Mathematics, Springer

Chung, Spectral graph theory, American Mathematical Society

Teaching methods

Lessons and exercises

Assessment methods

Fourier analysis (Modulo 1)

Written exam, with theory and exercises. A mock exam will be given during the course.

Calls are regularly opened on ALMAESAMI.

Exams are in presence. It is obligeatory to enrol in the chosen call on Almaesami.

Students may give the exam at any call and must show their University badge before starting the exam.

The grade of this part of the exam is expressed in X/30s and published on Almaesami.

Module 2

Written exam, with theory and exercises. Calls are regularly scheduled on Almaesami.

Exams are held in person. Registration for the chosen call on Almaesami is mandatory and must be completed at least 4 days in advance.

Students may take the exam during any call and must present their University badge (showing their name and a clear photo) before the exam begins.

The grade for this part of the exam is expressed in X/30 and will appear on Almaesami.

Students with learning disorders and/or temporary or permanent disabilities: please, contact the office responsible (https://site.unibo.it/studenti-con-disabilita-e-dsa/en/for-students ) as soon as possible so that they can propose acceptable adjustments. The request for adaptation must be submitted in advance (15 days before the exam date) to the lecturer, who will assess the appropriateness of the adjustments, taking into account the teaching objectives.

Final mark and verbalization

The final grading is given by the arithmetic average of the grades in the mathematical analysis and graph theory part.

The registration of the grade is made by prof. Arcozzi.

Prof Arcozzi signs the grades on Almaesami within 5 days from the completion of the two parts of the exam.

Teaching tools

Fourier analysis:

Detailed programme, lecture notes, texts and solutions of exercises classes, recordings of the lectures. and instructions for the exam will be made available on the e-learning platform Virtuale.

Module 2

Module 2:
blackboard lectures, summary of lectures, suggested exercises. The material will appear in https://www.dm.unibo.it/~andrea.petracci3/2025MathMethodsM/

Office hours

See the website of Nicola Arcozzi

See the website of Andrea Petracci