# 29161 - Mathematical Methods M

• from Sep 17, 2024 to Dec 17, 2024

• from Sep 20, 2024 to Dec 20, 2024

## Learning outcomes

To know and to be able to use some mathematical techniques for the information engineering. Competencies: to know the theory of linear differential equations and systems; to be able to solve constant coefficient linear differential equations and systems; to know the Laplace transform and its use in solving linear differential equations; to have a basic knowledge of dynamical systems. Detailed contents: linear ordinary differential equations, Cauchy problem, existence and uniqueness of solutions. First-order linear equations. Discussion of existence and uniqueness of solutions of first-order differential equations and applications. Higher-order linear differential equations. Numerical solutions of differential equations. Introduction to nonlinear systems. Laplace transform: definition, convergence abscissa; formal properties of the Laplace transform; Laplace transforms of standard functions. Step functions and their transforms. Laplace transforms of some further special functions: the saw-tooth function, the Dirac delta. Applications of Laplace transform to ordinary differential equations: theory and application in solving simple ordinary differential equations with constant coefficients and given boundary conditions. Basic facts about linear transformations; eigenvalues, eigenvectors. Systems of linear differential equations; matrix exponential; dynamical systems, stability; numerical solutions of differential equations. General form of solutions. Transfer function. Stabilization problem.

## 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 (Graph theory)

Graphs and subgraphs. Trees. Connectivity. Euler tours and Hamilton cycles. Matchings. Edge colourings. Independent sets and cliques. Vertex colourings. Planar graphs. Directed graphs. Hints at networks. A detailed program could be found on the e-learning platform Insegnamenti On-line.

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

Graph theory (Modulo 2):

Official textbook

J.A. Bondy and U.S.R. Murty, "Graph theory with applications",

Support textbooks

J.A. Bondy and U.S.R. Murty, "Graph theory",
Springer Series: Graduate Texts in Mathematics, Vol. 244 (2008)

R. Diestel, "Graph theory", Springer Series: Graduate Texts in Mathematics, Vol. 173 (2005)
[http://diestel-graph-theory.com/basic.html] (3 MB).

## Teaching methods

Lessons and exercises

## Assessment methods

Fourier analysis (Modulo 1)

The exam is oral.

The student answers to questions on the topics of the course. The first question is a topic chosen by the student, then the exam continues with questions on different topics of the program.

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. It is possible to refuse the grade of this part only once.

Graph Theory (Module 2)

The exam is made up of two parts: a mid-term test with exercises and a final oral exam. Students are invited to show the University badge before starting either part.

Examples of mid-term test are published on the e-learning platform Insegnamenti On-line and on the program page: http://www.dm.unibo.it/~ferri/hm/progmame.htm The date of the mid-term test will be published on this page: https://www.dm.unibo.it/~ferri/hm/ricapp.htm#app The mid-term test MUST be passed with a score of at least 14 (over 24). If a student doesn't pass, he/she must recover it; the dates for recovering will also be published at this page: https://www.dm.unibo.it/~ferri/hm/ricapp.htm#app

Apply for the final oral exam at AlmaEsami. The final exam is on the whole program published, by the end of the course, on the e-learning platform Insegnamenti On-line and is as follows: two subjects are proposed to the student (each of which is either the title of a long chapter, or the sum of the titles of two short ones); they choose one and write down all what they remember about it, without the help of notes, texts, electronic devices; a discussion on their essay and in general about the chosen subject follows. It is an oral examination, so writing is only a help for the student to gather ideas.
---

Final mark and verbalization

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

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.

Graph theory (Module 2)

Textbook available at http://book.huihoo.com/pdf/graph-theory-With-applications/

Additional material is published on the e-learning platform Insegnamenti On-line and on the program page [http://www.dm.unibo.it/~ferri/hm/progmame.htm]

## Office hours

See the website of Nicola Arcozzi

See the website of Massimo Ferri

### SDGs

This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.