Calculus

Unit: Calculus:
Numbers: natural, rational and real numbers. The field R and its operations. The simbols of infinity. Inequalities in R.
Functions: 1:1, onto and bijective functions; composition and invertibility. Real functions of one real variable, graphics, monotonicity. Elementary functions (powers, logarithms, exponentials, ...).
Linear Algebra: vectors, matrices and their operations. Determinants, inversion, and ranks. Linear systems. Cramer's and Rouché's theorems.
Limits: definitions and basic properties; uniqueness; monotonicity. Limits for elementary functions. Uncertainties. Asymptotics and comparison results. Eulero's number "e" and some fundamental limits. Divergent and vanishing functions. Continuity and related proprties: Darboux and Weierstrass' results.
Differential calculus: first derivative, geometric and mechanical meaning, tangent line. Operations with derivatives: chain rule. Differentiation of elementary functions. Optimization and the classical theorems (Fermat, Rolle, Lagrange), monotonicity. Convexity. Hopital's rule. Qualitative study for the graph of a real valued function. Antiderivatives and some methods to compute indefinite integrals.
Integral calculus: definite integrals and their main properties. The Fundamental Theorem of Integral Calculus. Area of plane regions.

Formal prerequisites
None

Suggested textbook:
P. Marcellini, C. Sbordone, Calcolo - edizione aggiornata per i nuovi corsi di laurea, Liguori editore (2004).
Online course "Matematica Assistita", http://ariel.unimi.it/User/

Previous knowledge and exam
Knowledge of elementary algebra and plane trigonometry, use of logarithms in the real field.
We intend to check the achievements of the stated goals for the course through a written exam (mainly on exercises but not exclusively) and an oral exam (mainly on theory questions but not exclusively). Students are given the opportunity to replace the final exams with two tests on specific topics.

Teaching methods:
Traditional

Teaching language
Italian

Web page http://users.mat.unimi.it/users/paleari

Descriptive Statistics.

1) Sampling from a population. Types of data and variables.
2) Classes of data and frequency tables. Histograms/bar graphs.
3) Statistical indices: centrality indices (mean, modal value, median, midrange), variability indices (range, standard deviation, variance), percentiles, quartiles. Outliers. Boxplot.

Probability and random variables.

4) Sample space, events, probability of events.
5) Probability of the union and the intersection of events. Complement of an event. Independent events. Conditional probability. The Bayes Theorem. Factorials and binomial coefficients.
6) Random Variables. Expectation, variance and standard deviation of discrete random variables.
7) Discrete random variables: binomial and Poisson distributions. Continuous random variables: uniform and normal distributions.
8) Reduction to standard random variables and computations with the normal distribution. Normal approximation of the binomial distribution.

Inferential statistics.

9) General concepts: population, sample, parameters, statistics, estimators. Behaviour of the sample mean: law of large numbers and central limit theorem. Point estimates.
10) Confidence intervals: general concepts. Confidence interval for a proportion.
11) Confidence interval for the mean, with known/unknown variance. Student's t distribution.
12) Hypothesis testing. General concepts: null and alternative hypothesis, errors of the first/second kind, level of significance, power, p-value, test statistics, critical region.
13) Hypothesis tests on a proportion. Hypothesis tests on the mean (with known/unknown variance).
14) Inference for two samples. Inference on two proportions. Inference on two means, either for independent samples or for coupled samples.

Linear regression and non-parametric statistics.

15) Covariance and correlation coefficient. Linear regression (with one predictor and the intercept). Tests on the regressione coefficients and model validation.
16) Test of independence. Test of goodness-of-fit. Chi-square distribution.


Textbook: Textbook: Sheldon Ross, Probabilità e statistica per l'ingegneria e le scienze (terza edizione), Maggioli Editore (2015)


Prerequisites: elements of algebraic manipulation and basic use of a scientific calculator.


Exam: written exam consisting in exercise solving.


Recommended Prerequisites: to attend the course Matematica Generale scheduled in the same semester


Teaching methods: classroom lectures; attendance is highly suggested. Lectures are delivered in italian.


Further information and WEB pages: more detailed information on the course will be available on the Ariel webpage.

PART I - Introduction to Computer Science
G.1. Introduction to Computer Science
G.2. Data representation
G.3. Computer hardware
G.4. Software
G.5. Computer networks

PART II - Data analysis using spreadsheets
F.1. Spreadsheets
F.2. Mathematical functions in Excel
F.3. Statistical functions in Excel
F.4. Graphics in Excel

PART III - Data management and databases
B.1. Data management
B.2. Storing data in databases
B.3. Data models
B.4. Relational databases
B.5. Creation of databases using Access
B.6. Query in Access
B.7. Databases on the web

PART IV - Internet and the Web
I.1. Internet
I.2. Web architecture
I.3. Standard for the Web
I.4. Markup languages
I.5. Client-side applications
I.6. Search engines

PART V - Computer Science and Biology
1. Use of PubMed and UNIMI Library Network for bibliographic search
2. Biological databases
3. Notions of Bioinformatics: sequence alignment, protein folding, molecular docking, bio-inspired algorithms

Teaching resources
Teaching resources consist in lecture notes, slides and spreadsheets exercises, all available through the online software platform.

Previous knowledge and exam
Before taking the examination, students are required to compile the self-assessment questionnaires that are available through the online software platform. The examination consists in a multiple answer test.

Unit: Calculus:
Numbers: natural, rational and real numbers. The field R and its operations. The simbols of infinity. Inequalities in R.
Functions: 1:1, onto and bijective functions; composition and invertibility. Real functions of one real variable, graphics, monotonicity. Elementary functions (powers, logarithms, exponentials, ...).
Linear Algebra: vectors, matrices and their operations. Determinants, inversion, and ranks. Linear systems. Cramer's and Rouché's theorems.
Limits: definitions and basic properties; uniqueness; monotonicity. Limits for elementary functions. Uncertainties. Asymptotics and comparison results. Eulero's number "e" and some fundamental limits. Divergent and vanishing functions. Continuity and related proprties: Darboux and Weierstrass' results.
Differential calculus: first derivative, geometric and mechanical meaning, tangent line. Operations with derivatives: chain rule. Differentiation of elementary functions. Optimization and the classical theorems (Fermat, Rolle, Lagrange), monotonicity. Convexity. Hopital's rule. Qualitative study for the graph of a real valued function. Antiderivatives and some methods to compute indefinite integrals.
Integral calculus: definite integrals and their main properties. The Fundamental Theorem of Integral Calculus. Area of plane regions.

Formal prerequisites
None

Suggested textbook:
P. Marcellini, C. Sbordone, Calcolo - edizione aggiornata per i nuovi corsi di laurea, Liguori editore (2004).
Online course "Matematica Assistita", http://ariel.unimi.it/User/

Previous knowledge and exam
Knowledge of elementary algebra and plane trigonometry, use of logarithms in the real field.
We intend to check the achievements of the stated goals for the course through a written exam (mainly on exercises but not exclusively) and an oral exam (mainly on theory questions but not exclusively). Students are given the opportunity to replace the final exams with two tests on specific topics.

Teaching methods:
Traditional

Teaching language
Italian

Web page http://users.mat.unimi.it/users/paleari

Descriptive Statistics:
1) Sampling from a population. Types of data and variables.
2) Classes of data and frequency tables. Histograms/bar graphs.
3) Statistical indices: centrality indices (mean, modal value, median, midrange), variability indices (range, standard deviation, variance), percentiles, quartiles. Outliers. Boxplot.

Probability and random variables:
4) Sample space, events, probability of events.
5) Probability of the union and the intersection of events. Complement of an event. Independent events. Conditional probability. The Bayes Theorem. Factorials and binomial coefficients.
6) Random Variables. Expectation, variance and standard deviation of discrete random variables.
7) Discrete random variables: binomial and Poisson distributions. Continuous random variables: uniform and normal distributions.
8) Reduction to standard random variables and computations with the normal distribution. Normal approximation of the binomial distribution.

Inferential statistics:
9) General concepts: population, sample, parameters, statistics, estimators. Behaviour of the sample mean: law of large numbers and central limit theorem. Point estimates.
10) Confidence intervals: general concepts. Confidence interval for a proportion.
11) Confidence interval for the mean, with known/unknown variance. Student's t distribution.
12) Hypothesis testing. General concepts: null and alternative hypothesis, errors of the first/second kind, level of significance, power, p-value, test statistics, critical region.
13) Hypothesis tests on a proportion. Hypothesis tests on the mean (with known/unknown variance).
14) Inference for two samples. Inference on two proportions. Inference on two means, either for independent samples or for coupled samples.

Linear regression and non-parametric statistics:
15) Covariance and correlation coefficient. Linear regression (with one predictor and the intercept). Tests on the regressione coefficients and model validation.
16) Test of independence. Test of goodness-of-fit. Chi-square distribution.



Reference Material: Textbook: Sheldon Ross, Probabilità e statistica per l'ingegneria e le scienze (terza edizione), Maggioli Editore (2015).


Prerequisites: elements of algebraic manipulation and basic use of a scientific calculator.


Examination procedures: Written exam consisting in exercise solving.


Teaching Methods: classroom lectures; attendance is highly suggested


Language of instruction: Italian


Recommended Prerequisites: to attend the course Matematica Generale scheduled in the same semester


Program information: more detailed information on the course will be available on the Ariel webpage

PART I - Introduction to Computer Science
G.1. Introduction to Computer Science
G.2. Data representation
G.3. Computer hardware
G.4. Software
G.5. Computer networks

PART II - Data analysis using spreadsheets
F.1. Spreadsheets
F.2. Mathematical functions in Excel
F.3. Statistical functions in Excel
F.4. Graphics in Excel

PART III - Data management and databases
B.1. Data management
B.2. Storing data in databases
B.3. Data models
B.4. Relational databases
B.5. Creation of databases using Access
B.6. Query in Access
B.7. Databases on the web

PART IV - Internet and the Web
I.1. Internet
I.2. Web architecture
I.3. Standard for the Web
I.4. Markup languages
I.5. Client-side applications
I.6. Search engines

PART V - Computer Science and Biology
1. Use of PubMed and UNIMI Library Network for bibliographic search
2. Biological databases
3. Notions of Bioinformatics: sequence alignment, protein folding, molecular docking, bio-inspired algorithms

Teaching resources
Teaching resources consist in lecture notes, slides and spreadsheets exercises, all available through the online software platform.

Previous knowledge and exam
Before taking the examination, students are required to compile the self-assessment questionnaires that are available through the online software platform. The examination consists in a multiple answer test.