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Mathematical methods
04LSIOD, 02LSILP
A.A. 2023/24
Course Language
Inglese
Degree programme(s)
1st degree and Bachelor-level of the Bologna process in Ingegneria Fisica - Torino
1st degree and Bachelor-level of the Bologna process in Electronic And Communications Engineering (Ingegneria Elettronica E Delle Comunicazioni) - Torino
Course structure
| Teaching | Hours |
|---|---|
| Lezioni | 80 |
| Esercitazioni in aula | 20 |
Lecturers
| Teacher | Status | SSD | h.Les | h.Ex | h.Lab | h.Tut | Years teaching |
|---|---|---|---|---|---|---|---|
| Como Giacomo | Professore Ordinario | IINF-04/A | 40 | 5 | 0 | 0 | 6 |
Co-lectures
Context
| SSD | CFU | Activities | Area context | MAT/05 MAT/06 |
6 4 |
A - Di base A - Di base |
Matematica, informatica e statistica Matematica, informatica e statistica |
|---|
2023/24
The course aims at completing the students' education in basic mathematics. It introduces the theory of complex and analytic functions, distributions, Fourier and Laplace transforms, and discrete and continuous probability. Such topics play an essential role in engineering applications. Examples and motivation will be drawn from problems in engineering, offering further insights.
a) Knowledge and understanding Students are taught some basic mathematical notions and tools for solving various problems ranging from signals analysis to the study of random phenomena. The theory of distributions provides a general language which enables to deal with signals arising in impulsive or discontinuous phenomena: this theory is the natural setting for the study of the Fourier and Laplace transforms. Students learn the techniques for the computation of the transforms of the main distributions: delta Dirac, Dirac comb, and piecewise regular functions included. Complex analysis is the proper setting for the theory of the Laplace transform and is the advanced tool for the analysis of singular phenomena and for the computation of integrals. Moreover, students are provided with the main probabilistic tools necessary for solving problems under uncertainty. They learn how to deal with random phenomena and with the variables involved in them.
b) Applying knowledge and understanding At the end of the course students will be able to apply the analytical techniques required for the analysis of the signals of any nature (impulsive, erratic, etc.). Also, they will be expected to have acquired the skills to evaluate the probability of outcomes and extrapolate information useful in solving problems in electronic and telecommunication engineering. The ability to apply the gained knowledge will be verified through class exercises.
Students are required to be familiar with the notions and tools of the mathematics courses of the first two years. These include differential and integral calculus of one or several variables.
1. (27h) Function theory of complex variable: differentiability, Cauchy-Riemann equations, line integrals. Cauchy theorem, Cauchy integral formula, Taylor series of analytic functions, Laurent series. Residue theorem, computation of residues and application to the calculation of integrals.
2. (15h) Theory of distributions: definitions and basic operations (algebraic operations, translation, rescaling, derivatives), Dirac delta, p.v.(1/t), Dirac comb. Convolution of functions and distributions.
3. (18h) Fourier and Laplace transforms of functions and tempered distributions: definitions and properties, inverse transforms, inversion formula. Notable transforms.
4. (15h) Combinatorics, probability measures and related elementary properties. Conditional probability and independence. 5. (15h) Discrete and continuous random variables. Notable examples. Expected values.
6. (10h) Joint distribution, independence and correlation.
Exercises will cover the topics of the lectures. Some of them will be carried out by the teacher, others will actively involve the students.
Exercises will cover the topics of the lectures. Some of them will be carried out by the teacher, others will actively involve the students.
Lecture notes and slides will be made available on the course web page. Recommended textbook in probability: Ross, S. 'A first course in Probability', Pearson Ed., 2014 (or any other edition).
Slides; Esercizi; Esercizi risolti; Video lezioni tratte da anni precedenti;
Lecture slides; Exercises; Exercise with solutions ; Video lectures (previous years);
Modalità di esame: Prova scritta (in aula); Prova orale facoltativa;
Exam: Written test; Optional oral exam;
...
The final exam is in written form. An oral exam is optional on the student' or the course responsible's request. The exam duration is 1,5 hours.
During the exam, students can only use a printed copy of the formula sheet made available by the teachers on the Portale della Didattica, along with pen and white paper. The use of calculator is limited to the one made available in the Exam platform. The use of any other device is strictly forbidden.
The written exam comprises two parts:
1. ten multiple-choice questions, six of which on the Mathematical Analysis part and the remaining four on the Probability part of the course;
2. two exercises, one of which on the Mathematical Analysis part and the other one on the Probability part of the course, both involving several questions.
Each of the ten multiple choice questions has four alternative answers, only one of which is correct. A correct answer gives one point, a wrong answer gives 0 points. Hence, the maximum achievable score in this part of the exam is 10 points.
Each exercise will involve several questions, either with open answer or with possible answers TRUE or FALSE. Each of this questions has an associated number of points and a wrong answer is penalised with -50% of the points associated to the question. If no answer is given, then there is no penalisation (0 points). The maximum total score achievable in the mathematical analysis exercise is 13 points, the maximum score achievable in the probability exercise is 9 points: both maximum scores can be achieved by answering correctly to ALL questions.
The exam is passed if the total score is greater than or equal to 18, with at least 6 points scored in the mathematical analysis part and at least 4 points scored in the probability part.
The final score corresponds to the final grade if its is less than or equal to 30. Score 31 corresponds to grade 30, score 32 to grade 30 e lode.
The score will not be immediately available to the students at the end of the exam as the teachers will have to check the regularity of the exam. Any misconduct will be notified to the Commissione Disciplinare.
Teachers have the possibility to ask for a supplementary oral exam at their discretion. Students who have passed the written exam with a score of 18 or higher can ask to be admitted to the oral exam. If an oral exam is asked and performed, it becomes part of the evaluation together with the written part. During an oral exam both theoretical questions (including, e.g., any definition, theorem, and proof covered during the course) and exercises may be asked. Upon an oral exam the student's grade might be increased, kept as is, or decreased, depending on her/his performance.
Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.
Exam: Written test; Optional oral exam;
The final exam is in written form. An oral exam is optional on the student's or the course responsible's request. The exam duration is 1.5 hours. During the exam, students can only use a printed copy of the formula sheet made available by the teachers on the Portale della Didattica, along with pen and white paper. The use of calculator is limited to the one made available in the Exam platform. The use of any other device is strictly forbidden.
The written exam comprises two parts:
1. ten multiple-choice questions, six of which on the Mathematical Analysis part and the remaining four on the Probability part of the course;
2. two exercises, one of which on the Mathematical Analysis part and the other one on the Probability part of the course, both involving several questions.
Each of the ten multiple choice questions has four alternative answers, only one of which is correct. A correct answer gives one point, a wrong answer gives 0 points. Hence, the maximum achievable score in this part of the exam is 10 points.
Each exercise will involve several questions, either with open answer or with possible answers TRUE or FALSE. Each of this questions has an associated number of points and a wrong answer is penalised with -50% of the points associated to the question. If no answer is given, then there is no penalisation (0 points). The maximum total score achievable in the mathematical analysis exercise is 13 points, the maximum score achievable in the probability exercise is 9 points: both maximum scores can be achieved by answering correctly to ALL questions.
The exam is passed if the total score is greater than or equal to 18, with at least 6 points scored in the mathematical analysis part and at least 4 points scored in the probability part.
The final score corresponds to the final grade if its is less than or equal to 30. Score 31 corresponds to grade 30, score 32 to grade 30 e lode.
The score will not be immediately available to the students at the end of the exam as the teachers will have to check the regularity of the exam. Any misconduct will be notified to the Commissione Disciplinare.
Teachers have the possibility to ask for a supplementary oral exam at their discretion. Students who have passed the written exam with a score of 18 or higher can ask to be admitted to the oral exam. If an oral exam is asked and performed, it becomes part of the evaluation together with the written part. During an oral exam both theoretical questions (including, e.g., any definition, theorem, and proof covered during the course) and exercises may be asked. Upon an oral exam the student's grade might be increased, kept as is, or decreased, depending on her/his performance.
In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.