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

The topics of Module 1, taught by Diego Ribeiro Moreira, include:

• Orthogonality and Fourier series;
• Fourier sine and cosine series, even and odd functions, and half-range expansions;
• Basic convergence and approximation properties of Fourier series;
• Fourier integral, Fourier sine and cosine transforms, and Fourier transform on the real line;
• Basic properties of the Fourier transform;
• Discrete Fourier transform and fast Fourier transform;
• Elementary applications of Fourier methods to partial differential equations, including selected examples involving the heat equation, the wave equation, and Laplace's equation.

Prerequisites: elementary linear algebra, calculus of one and several variables, basic ordinary differential equations, and elementary complex numbers.

Module 2

The topics of Module 2, taught by Andrea Petracci, include:
- Some topics in Linear algebra: diagonalization and spectral theorem;
- 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.petracci3/2026MathMethodsM/

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

 

Modulo 1

Main reference:

Kreyszig, Advanced engineering mathematics, 10th ed., selected topics from Chapters 11 and 12.

Additional lecture notes and exercises may be provided during the course.

Modulo 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.petracci3/2026MathMethodsM/

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

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

Modules 1 and 2

Both modules consist of lessons and exercises.

Assessment methods

Modules 1 and 2

Each module has a 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 each module exam during any call and must present their University badge (showing their name and a clear photo) before the exam begins.

The grade for each module 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 obtained in Module 1 and Module 2, rounded up to the nearest integer.

The registration of the grade is made by prof. Diego Ribeiro Moreira. He signs the grades on Almaesami a few days after the completion of the two parts of the exam.

Teaching tools

Module 1

blackboard lectures, lecture recordings, lecture notes, suggested exercises. The material will appear on Virtuale.


Module 2

blackboard lectures, summary of lectures, suggested exercises. The material will appear in https://www.dm.unibo.it/~andrea.petracci3/2026MathMethodsM/

Office hours

See the website of Diego Ribeiro Moreira

See the website of Andrea Petracci