# 29161 - Mathematical Methods M

## 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 2 (Introduction to functional 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. A detailed program is published on the e-learning platform Insegnamenti On-line

Module1 (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.

Introduction to functional analysis:

Lecture notes of the teacher. The notes (pdf) will be available through the institutional site AMS-Campus 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:

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)

## Teaching methods

Lectures and exercises

## Assessment methods

Introduction to functional analysis:

This part of the exam is written (about 2 hours). Exam calls are published on Almaesami. Students enrol to an exam call on Almaesami and are invited to show the University badge before starting the exam. It is strictly forbidden the use of any electronic equipment connected to Internet.

During the first part of the exam, students solve three exercises of the same kind as those studied at the exercise classes and may use their own notes or texts for consultation.

In the second part, each student chooses one topic from a list and answers to the questions of the topic. During this part it is forbidden to consult one's notes or text and it is also forbidden the use of any electronic equipment.

The whole list of exam topics is published on AmsCampus.

The final mark for this part is expressed in X/thirties and published on Almaesami.

Graph Theory

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 both the parts.

Examples of mid-term test are published on the e-learning platform Insegnamenti On-line. During the mid-term test the students may use their own notes or texts for consultation, but it is forbitten to use electronic devices. The date of the mid-term test is published at AlmaEsami where students can enroll. The mid-term test MUST be passed with a score of at least 14 (over 24). If a student don't pass, he must recover it; the dates for recovering coincide with the dates of the written exams of Istituzioni di Matematica, Ingegneria Meccatronica (see the list available on the e-learning platform Insegnamenti On-line). For doing so the student has to communicate, by e-mail to alessia.cattabriga@unibo.it, WHEN he wants to take it before the final exam.

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); he chooses one and writes down all what he remembers about it, without the help of notes, texts, electronic devices; a discussion on his 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 aritmetic average of the grades in the mathematical analysis and graph theory part. Verbalization is made by prof. Abenda. After completing the exam the student must send an e-mail to prof. Abenda stating that they passed both exams and confirming acceptation of the grade.

## Teaching tools

Introduction to functional analysis:

Detailed programme, lecture notes, texts and solutions exercises classes, samples of examination texts with solutions and list of the questions is available  on the e-learning platform Insegnamenti On-line.

Graph theory:

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

Additional material is published on the e-learning platform Insegnamenti On-line.

## Office hours

See the website of Simonetta Abenda

See the website of Alessia Cattabriga