PORTALE DELLA DIDATTICA

### Mathematical methods for Computer Science

01UROLM

A.A. 2022/23

Course Language

Inglese

Degree programme(s)

1st degree and Bachelor-level of the Bologna process in Ingegneria Informatica (Computer Engineering) - Torino

Course structure
Teaching Hours
Lezioni 100
Lecturers
Teacher Status SSD h.Les h.Ex h.Lab h.Tut Years teaching
D'Onofrio Giuseppe   Ricercatore a tempo det. L.240/10 art.24-B MAT/06 45 0 0 0 1
Co-lectuers
Context
SSD CFU Activities Area context
MAT/05
MAT/06
SECS-S/01
2
4
4
A - Di base
A - Di base
C - Affini o integrative
Matematica, informatica e statistica
Matematica, informatica e statistica
Attivit� formative affini o integrative
2022/23
The course aims at completing the students' education in basic mathematics, by introducing the theory of 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.
The course aims at completing the education of a Computer Engineering student in basic mathematics, by introducing tools of mathematical statistics and probability, as well as some basic notions from the theory of analytic functions, distributions, Fourier and Laplace transforms. Such topics are of fundamental importance in a Bachelor Degree in Computer Engineering. In fact, they play a central role in data analysis, when dealing with uncertainty and related applications. Besides, they provide the necessary mathematical tools for the study of signals and systems theory. Lectures will be enriched with examples and motivations 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.
The goal of the course is to introduce the tools of probability, mathematical statistics and mathematical analysis for producing, selecting and processing information and for time-frequency signals analysis. The students learn the methodology and the logic for modeling and evaluating random phenomena, data analysis and statistical inference, and also the basic techniques for the computation of the transforms for impulsive phenomena analysis. Students are expected to acquire the skills to solve the most frequent practical problems. Students' mastery of concepts as well as the ability to apply them will be ascertained through discussions and in-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.
Students are required to be familiar with the notions and tools of the mathematics courses of the first year and of the first semester of the second year: these include differential and integral calculus of one or several variables and complex variables functions theory.
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.
1. Descriptive statistics: frequency tables and graphs. Central tendency and variability indexes, percentiles and correlation coefficient (5h). 2. Elementary probability: elements of combinatorics; probability definitions; conditional probability, independence (15h). 3. Discrete and absolutely continuous random variables: distributions, expected values and variances. Notable examples (15h). 4. Jointly distributed random variables: marginals and conditional distributions. Independence, correlation e conditional independence, notable multivariate distributions (15h). 5. Convergence in probability and in distribution: weak law of large numbers and central limit theorem (5h). 6. Statistical inference: sampling from a population, sampling statistics and their distribution; parameter estimation, estimators and related properties, maximum likelihood estimators, pseudo-random numbers and examples of related generating alghoritms (10h). 7. Confidence intervals: confidence intervals for the mean and for proportions, asymptotic confidence intervals. Basics of hypothesis testing ​(10h). 8. Multiple linear regression and least square estimation (5h). 9. 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 (8h). 10. Fourier and Laplace transforms of complex valued functions and of distributions: definitions and properties, inverse transforms, inversion formula. Notable transforms (12h). Exercises will cover the topics of the lectures. Some of them will be carried out by the teacher at the blackboard, others will actively involve the students.
Exercises will cover the topics of the lectures. Some of them will be carried out by the instructor at the blackboard, others will actively involve the students.
Lecture notes will be available in the course web page. Recommended textbook in probability: Ross, S. 'A first course in Probability', Pearson Ed., 2014 (or any other edition).
Lecture notes will be available in the course web page. Recommended textbook in probability and mathematical statistics: Ross, S. 'Probability and Statistics for Engineers and Scientists', Academic Press 5th Ed., 2014 (or any other edition). For analysis, lecture notes and texts will be suggested by the instructor.
Modalit� di esame: Prova scritta (in aula); Prova orale facoltativa;
Exam: Written test; Optional oral exam;
... The final exam is written. An oral exam is optional on students' request or at the discretion of the instructor. The written exam is two hour long. Students are allowed to use only a non-programmable calculator and the formulae sheets provided by the instructors. The written exam is composed of two parts: 1. ten multiple-choice quizzes, six of which in analysis and four in probability; 2. two exercises, one in analysis and one in probability, composed of different questions. For each quiz, four possible answers are shown, only one of which is correct. The goal of the multiple choice test is to verify the understanding of the fundamental basic concepts of the analysis and probability parts. Each answer to the test is evaluated 1 point if correct and 0 otherwise. Thus the maximum score to the test is 10. The scope of the exercises of the second part is to verify the knowledge and capability to treat problems involving complex analysis, distributions theory, Fourier and Laplace transforms, probability, random variables and expected values. The exercise in analysis is evaluated maximum 13 points, the one in probability 9 points. To pass the written part of the exam students have to totalize at least 18/30, with at least 4/30 in probability and at least 6/30 in analysis. If the sum of the two parts of the exam is less or equal to 30, it represents the final mark. If it is 31 or 32, the final mark is 30 or 30 with honor (30L) respectively. Only students who passed the written exam can ask to be admitted to the oral exam. In particular, if an oral exam is asked and performed, it becomes part of the evaluation together with the written part. Depending on the performance of the student, the final mark could be less, equal or greater than the total score of the written exam.
Gli studenti e le studentesse con disabilit� 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'Unit� Special Needs, al fine di permettere al/la docente la declinazione pi� idonea in riferimento alla specifica tipologia di esame.
Exam: Written test; Optional oral exam;
Assessment methods: Written individual open and/or closed answers questions closed books exam. The oral exam is optional on students' request or at the discretion of the instructor. The exam is 2 hour long. Students are allowed to use only a pen, white paper, the calculator and the formulae sheets provided by the instructors and available on the course website. The exam is composed of two parts: 1. ten multiple-choice quizzes, two of which in analysis, four in probability and four in statistics; 2. three exercises, one in analysis, one in probability and one in statistics (each composed of different questions). For each quiz, four possible answers are shown, only one of which is correct. Each answer to the test is evaluated 1 point if correct and 0 otherwise. Thus the maximum score to the test is 10. The goal of the multiple choice test is to verify the understanding of the fundamental basic concepts of each of the course modules. The goal of these exercises is to verify the knowledge and understanding of the students main tools for solving the problems taught during the lectures and practice sessions. Each of the three exercises is composed . The maximum score of the analysis exercise is 5 points, whereas the maximum score of the probability and statistics exercises is 9 points each. If the sum of the two parts scores of the exam is less or equal to 30, it represents the final mark. If it is 31, the final mark is 30 and, if it is 31 or 32, the final mark is 30 with honor (30L). An oral exam can be required at the discretion of the instructor in the case a further investigation is needed to ascertain students' mastery of the concepts delivered in the course. Only students who passed the exam can ask to be admitted to the oral exam. In particular, if an oral exam is asked and performed, it becomes part of the evaluation together with the written part. Depending on the performance of the student, the final mark could be less, equal or greater than the total score of the written exam.
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.