65851 - Mathematics and Statistics

Learning outcomes

The main goal is to give to the student the mathematical background to study and analyze phenomena from natural sciences. The student acquires the main basic methods and tools of the quantitative study of collective phenomena and the critical capacity of interpretation and evaluation of statistical information in order to be able to produce and process statistical data.

Course contents

The program is divided into 3 main parts : Basic Concepts (diactic unit 0 ) Linear algebra and study of functions ( diactic units 1-3 ) units and elements of statistics (diactic unit 4) . Some topics, included in round brackets, have secondary priority and therefore they will be developed, even partially, according to the time available. A) Prerequisites basic mathematic B ) Teaching units 0.

Elements of mathematic ( Total 5 hours teaching units )

0.1 Equations and inequalities of the second degree

0.2 Elements of trigonometry

0.3 Logarithms Knowledge acquired in the unit teaching 0: The purpose of this unit is to review with the student some basic mathematical ingredients necessary to learn the concepts and techniques that will be developed in detail during the course.

B) Didactic units

1. Elements of linear algebra ( Total 13 hours teaching units )

1.1 Vector spaces

Vector spaces , vectors and operations with vectors ,

1.2 Matrices

Transformation matrices , operations with matrices , (determinants, eigenvalues and eigenvectors)

Knowledge acquired in the didactic unit 1:

It introduces the student to vector calculus and matrices with special attention to examples and exercises derived from natural science .

 

2 Functions (Total 24 hours teaching units)

2.1 Functions of one variable

The graph of a function, increasing functions and descrescent functions, maximum and minimum

2.2 Polynomial functions

Linear, quadratic and polynomial functions.

2.3 Exponential functions, the logarithmic laws, periodic phenomenas

Knowledge acquired in the didactic unit 2:

At the end of this unit the student is able to understand the concept of function and understand how to model a natural phenomenon using this mathematical tool. Furthermore it becomes able to recognize the main differences between various types of elementary functions . This teaching unit and corroborated by several examples and exercises.

3 Limits and differential and integral calculus ( Total 18 hours teaching units)

3.1 Limits and asymptotes

Calculating the limits, (sequences and series ,recurrence laws)

3.2 Differential calculus

Derived rate of change, rules for computing the derivatvie, (higher order derivatives and Taylor formula

3.3 Integral calculus)

From derivatives to integrals, entropy and order.

Knowledge acquired in the didactic unit 3:

The purpose of this unit is to teach the student many mathematical techniques to study the problem of the asymptotic behavior and the rate of average and instantaneous change by proposing models of evolution. In addition, the student becomes familiar with the concept of integral and its significance within natural sciences.

 

4. Elements of statistics ( Total 20 hours teaching units)

4. Elements of Statistics

 

4.1. Motivations and historical remarks

4.2. Probability thoery:

- Probability spaces and random variables

- Conditional probability and independence

- Discrete and continuous distributions

- Law of large numbers and Central limit theorem

4.3. Inferential statistics:

- Sampling

- Correlation and linear regression

- Statistical tests

 

Readings/Bibliography

Matematica per le scienze della vita, D. Benedetto, M. Degli Esposti, C. Maffei.

Teaching methods

Frontal lessons: learning the math concepts takes place through examples inspired by the reality of natural phenomena. Weekly exercises complete the preparation.

Assessment methods

A written test , and possibly an individual oral test : both primarily dedicated to the resolution of exercises

Teaching tools

Blackboard lectures mainly. Overhead projector, personal computer and projector will be used for some lectures

Office hours

See the website of Emanuele Latini

See the website of Stefano Pagliarani