# 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 (Modulo 2):

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 (Modulo 1):

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 (Module 2):

This part of the exam is written (duration: 1 h 3m). 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.

The student who is unsatisfied by the mark of this part of the exam must communicate the decision to repeat it to prof Abenda during office hours and in any case within 5 days from the date of publication of the results of the call on Almaesami.

Additional piece of information for on line exams (EOL+ZOOM):

Students who encounter problems in participating in on-line exam calls either for personal or technological reasons must contact   prof.Simonetta Abenda via e-mail as soon as possible and in any case well in advance before the exam call and before enrolling in the exam call.

During the on-line exam call, students must keep the camera and the microphone  on and must share the desktop.

The smartphone must be kept turned off during the exam..

The smartphone can be turned only if the the student needs to photograph the answer sheet to upload the file.

During the exercise part the student may use their notes and books. Books and notes must be visible on the desk.

During the theory part only white sheets and a pen are allowed on the desk. The phone turned off must be visible.

Within the end of the exam call the student must upload the files with texts and solutions to the exercises/the question in .pdf format using the platfomr EOL.

FAQ for exams on -line:

- Before the exam call, please prepare a sufficient number of white sheets with your name, surname and badge number. Do not forget to leave a space where to put our university badge at the moment of photocopying and uploading the file.

- Do not forget to number the pages! Write as clear as possible. Explain your computations, motivate your answers.

- If the student looses the internt connection before the end of the exm, please send immediately an e-mail to prof. Abenda with the photo of the exercises / theoretical question as it is from your Unibo e-mail addrees name.surname@studio.unibo.it alla docente.

- If the camera or the microphone are turned off or the desktop share ends before the end of the exam, the student is withdrawn and can partecipat in a future exam call.

Graph Theory (Module 1)

Both parts of the exam on line will be on Microsoft Teams

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, they must recover it; the dates for recovering  are published on Almaesami.

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. Verbalization is made by prof. Abenda. Prof Abenda signs the grades on Almaesami within 5 days from the completion of the two parts of the exam. The student who does not accept the final grade must communicate their decision to prof. Abenda by e-mail within 5 days from the completion of the exam and, in such case, shall repeat the complete exam (Graph Theory + Mathematical Analysis).

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 Massimo Ferri