B2374 - Mathematical Models for Industrial Engineering

Learning outcomes

The course focuses on the advanced mathematics areas which are most commonly used in mechanical and industrial engineering applications. At the end of the course the student is able to build, use and validate physical-mathematical models.

Course contents

Prerequisites

Basic knowledge of Mathematical Analysis and Linear Algebra acquired during an undergraduate degree program.
In particular, students are expected to have a solid understanding of:
- functions of several real variables,
- limits and continuity,
- differential and integral calculus in multiple variables.

Course Program

1. Elements of Complex Analysis

Review of the topology of the Euclidean plane and the theory of integration of differential 1-forms.
Gauss–Green and divergence theorems in the plane.
Introduction to the theory of holomorphic functions: Cauchy–Riemann equations, mean value property, maximum principle.
Complex integration: Cauchy’s integral theorem, Cauchy’s integral formula and representation theorem.
Complex power series and analytic functions.
Zeros of analytic functions and the analytic continuation theorem.
Residue theorem and main applications, particularly to the computation of improper integrals.

2. Elements of Functional Analysis

Overview of abstract integration theory.
Banach and Hilbert spaces; L^p spaces and spaces of continuous functions.
Projections and orthonormal bases in Hilbert spaces.
Fourier series and their properties: convergence theorems for periodic functions.

3. Fourier and Laplace Transforms

Fourier transform in L^1 and L^2: algebraic and differential properties, relation with convolution.
Introduction to rapidly decreasing functions and distributions.
Laplace transform: definition, region of convergence, and main computation rules.

4. Introduction to Partial Differential Equations (PDEs)

Applications of Fourier and Laplace transforms to solving differential equations.
Examples of transport, diffusion, and wave propagation models.

5. Elements of Probability Theory

Probability calculus: combinatorial methods, Bayes’ theorem, conditional probability, and independent events.
Discrete and continuous random variables; independence of random variables.
Expectation, variance, and covariance.

Readings/Bibliography

Recommended textbooks:

1. R. P. Agarwal, K. Perera and S. Pinelas, An Introduction to Complex Analysis, Springer, 2011.
2. C. Barozzi, Matematica per l'ingegneria dell'informazione, ristampa aggiornata, Zanichelli, 2005.
3. M. Cadegone, L. Lussardi, Metodi matematici per l’ingegneria, seconda edizione, 2021.
4. K. F. Riley, M. P. Hobson, S. J. Bence, Mathematical Methods for Physics and Engineering, Cambridge University Press, 2006.
   
5. F. Gazzola, F. Tomarelli, M. Zanotti, Analytic functions Integral transforms Differential equations, Esculapio, 2023. [italian language edition] F. Gazzola, F. Tomarelli, M. Zanotti, Analisi complessa trasformate Equazioni differenziali, Esculapio, 2023.

Recommended for further study:

1. E. M. Stein, R. Shakarchi, Princeton Lectures in Analysis II: Complex Analysis, Princeton University Press, 2003.
2. E. M. Stein, R. Shakarchi, Princeton Lectures in Analysis I: Fourier Analysis, An Introduction, Princeton University Press, 2003.
3. E. M. Stein, R. Shakarchi, Princeton Lectures in Analysis III: Real Analysis, Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, 2005.
4. E. M. Stein, R. Shakarchi, Princeton Lectures in Analysis IV: Functional Analysis, Introduction to Further Topics in Analysis, Princeton University Press, 2011
5. S. Salsa, G. Verzini, Partial Differential Equations in Action, UNITEX Springer, 2022.
   
6. A. Pascucci, Probability Theory I, UNITEX Springer, 2024.
   [italian language edition] A. Pascucci, Teoria della Probabilità variabili aleatorie e distribuzioni, UNITEX Springer, 2020.

To cover possible gaps in Mathematical Analysis prerequisites and for introductory review:

1. N. Fusco, P. Marcellini, C. Sbordone, Mathematical Analysis, UNITEX Springer, 2022. [italian language edition] N. Fusco, P. Marcellini, C. Sbordone, Lezioni di Analisi Matematica Due, Zanichelli, 2020.

Teaching methods

The course consists of theoretical lectures supplemented by practical exercise sessions, which aim to help students gain familiarity and proficiency with the mathematical tools and methods introduced during the classes.

Assessment methods

Assessment Methods and Criteria

The exam consists of a single written test aimed at evaluating the understanding of the theoretical foundations and the ability to apply the analytical methods covered during the course.

The written exam lasts 3 hours and consists of 7 exercises/questions (worth approximately 5 points each, with possible additional points for more demanding questions or those requiring mathematical proofs). The structure of the exam is as follows:

• 1 exercise on the calculation of real integrals using the method of residues;

• 1 theoretical question on complex analysis;

• 1 exercise on Fourier series or projections;

• 1 theoretical question on functional analysis (metric spaces, Banach and Hilbert spaces, projections, Fourier series);

• 1 exercise and 1 theoretical question on the Fourier transform;

• 1 theoretical question with a direct application of probability.

In the theoretical questions, knowledge and execution of at least one formal proof will be required.

During the exam, students are required to demonstrate:

• knowledge and understanding of the theoretical principles underlying complex analysis, functional analysis, and Fourier analysis;

• the ability to apply knowledge and understanding in the practical resolution of the proposed mathematical problems (e.g., calculation of integrals, series, and transforms);

• making judgments in correctly structuring mathematical proofs and selecting the most appropriate resolution methods;

• communication skills, reflected in the rigor of the mathematical formalism and the logical clarity in the presentation of theoretical concepts;

• learning skills, by critically connecting cross-disciplinary topics of the syllabus.

The total score of the test is calculated out of a maximum of 35 points; honors (30 e lode) are achieved with a final score exceeding 30. The final grade is determined by considering: the accuracy and completeness of the answers, the depth of understanding of the theoretical concepts, the logical-mathematical rigor in the proofs, and the ability to connect analytical methods to applications. To pass the exam, an adequate achievement of the expected learning outcomes is required.

Exam Rules and Academic Integrity

During the exam, it is not permitted to consult any support material (notes, books, handouts) nor to use calculators or IT tools of any kind, under penalty of exclusion from the test.

In full compliance with the University Code of Ethics (Codice Etico di Ateneo), all students are reminded of the obligation to maintain conduct based on the highest standards of honesty and integrity. Consequently, any action capable of altering the correct and regular execution of the assessment tests is strictly prohibited. By way of example but not limited to, it is forbidden to:

• resort to practices of plagiarism or cheating;

• access educational resources or online platforms during the exam;

• use Artificial Intelligence tools not expressly authorized (for the purposes of this learning assessment, the use of AI is completely prohibited - Scenario 1);

• use or possess unauthorized materials, notes, devices, or equipment.

Please note that the mere possession of unauthorized tools or supports during the exam will lead to the immediate invalidation of the submitted work and the simultaneous notification to the competent offices. Furthermore, any behavior in violation of these provisions may trigger disciplinary proceedings against the student and, where criminal implications apply, a report to the competent authorities, with the consequent risk of legal action being initiated.

Additional information and logistics

The exam dates can be found on the AlmaEsami web platform of the University of Bologna.To take part in the exam, each student must register well in advance on the lists available on the platform.On the day of the exam, students will be allowed to take the test only if they present a valid identification document.

Students with special educational needs related to learning disorders and/or disabilities are advised to contact the competent University Office (https://site.unibo.it/studenti-con-disabilita-e-dsa/it) in advance so that possible exam accommodations can be arranged. Any proposed accommodations must be submitted for the instructor’s approval at least 15 days before the exam date, who will assess their appropriateness in relation to the learning objectives of the course.

 

Clarifications on the recording of the exam grade for those who have the  course MATHEMATICAL METHODS FOR ENGINEERING (6 CFU) as a module of the integrated course of NUMERICAL AND MATHEMATICAL METHODS FOR ENGINEERING (12 CFU)

If the MATHEMATICAL METHODS FOR ENGINEERING course (6 CFU) is one of the two modules that, together with NUMERICAL ANALYSIS (6 CFU), constitutes the integrated course of NUMERICAL AND MATHEMATICAL METHODS FOR ENGINEERING (12 CFU), the grade that will be recorded will be calculated with the arithmetic average of the single grades that the student has obtained in the two modules. It should be noted that the result of the average will be rounded to the nearest integer. Only if the resulting average is exactly equidistant between two integers, the grade will be obtained by rounding up to the next highest integer. Lastly, in order to obtain the "30 cum laude" final evaluation, the student must be in one of the two following cases:

- receiving "30 cum laude" in both modules;

- obtaining "30 cum laude" in one module and 30 in the other one.

Lastly, it should be noted that the recording of the final evaluation requires the passing of both the exam of NUMERICAL ANALYSIS (6 CFU) and the exam of MATHEMATICAL METHODS FOR ENGINEERING (6 CFU) in a time interval not exceeding 12 months.

Teaching tools

Pdf files uploaded in the institutional site.

Links to further information

https://virtuale.unibo.it/course/view.php?id=76178

Office hours

See the website of Enzo Maria Merlino