00674 - Mathematics

Learning outcomes

At the end of the course, the student will have the basic knowledge of mathematics, necessary to deal with the other disciplines of the degree programme. In particular, the student will be able to: - understand and use the graph of a function in mathematical models; - understand the use of the tools of differential and integral calculus and linear algebra in applications; - use simple mathematical software to solve equations, draw graphs and study them, perform calculations with derivatives, integrals and matrices. He or she will also have some knowledge of basic statistical methods. In particular, the student acquires familiarity with the scientific method and is able to adopt the most suitable basic statistical analysis methods both in field and laboratory experiments. He or she will possess basic skills of some aspects of computer science. In particular, students will understand Internet basics and will be familiar with software for the management of data bases.

Course contents

Module 1: Mathematics (Marco Trozzo)
Analytic geometry (only Geological Sciences): In the plane: cartesian equation and parametric equations of a line, parallelism and orthogonality, point-line distance, cartesian equation of a circle of given centre and radius, circle-line intersection. In the space: cartesian equation and parametric equations of a line, cartesian equation of a plane, parallelism and orthogonality, point-line distance, point-plane distance.
Linear algebra: Vectors, scalar product, wedge product, matrix algebra, determinant and inverse of a square matrix, linear transformations, linear systems, Rouché-Capelli theorem.
Real functions: Functions, function composition, inverse functions, real functions, in particular linear, polynomial, exponential, logarithmic, trigonometric functions and their graphs, limits and asymptotes, continuity.
Complex numbers: algebra of the complex numbers, polar coordinates.
Differential calculus: Derivatives and differentials, their meaning, differentiation rules, Taylor's formula, de l'Hopital theorem, study of a function.
Integral calculus: Indefinite integral, definite (Riemann) integral and its properties, mean-value theorem for definite integrals, fundamental theorem of calculus, integration techniques (integration by parts and integration by substitution), applications of the definite integral.
Differential equations: solve first order linear differential equations and with separable variables.
Real functions of two variables (only Geological Sciences): partial derivatives, maximum, minimum and saddle points, double integrals.

Module 2: Applied Statistics (Federico Plazzi) (only Natural Sciences)
3rd October: presentation of the course; introduction to descriptive statistics; distribution of the initial questionnaire; measures of central tendency (arithmetic mean, geometric mean, harmonic mean, weighted average, median, mode).
10th October: measures of dispersion (deviation, variance,  standard deviation); concept of statistical distribution of data; normal distribution, standard normal distribution, test Z.
17th October: Test of normality (quantiles and plot quantile-quantile, Shapiro-Wilk test).
24th October: Introduction to inferential statistics; Student's t-test: one-sample t-test.
31st October: LAB: introduction to R; how to write an input file; how to retrieve, view, manage data; test of normality and Student's t-test with R.
7th November: Two-sample t-test of paired samples and of unpaired samples; non normally distributed quantitative variables: Wilcoxon–Mann–Whitney tests.
14th November: Qualitative variables: chi-square test.
21st November: Linear regression and correlation: Pearson's correlation coefficient, significance of a correlation.
28th November: Variance analysis, one-way and two-way ANOVA, Tukey's test.
5th December: LAB: chi-square test, regression and variance analysis with R.

Module 3: Computer Science (self-learning: https://elearning-cds.unibo.it/) (only Natural Sciences)
The structure of the Internet and Internet services; construction of a data-base.

Readings/Bibliography

M. Bramanti, C.D. Pagani, S. Salsa, Matematica. Calcolo infinitesimale e algebra lineare. 2a ed, Zanichelli, Bologna, 2004.

G. Pellacani, G. Pettini, C. Vettori, Istituzioni di Matematica, Clueb, Bologna.

D. Benedetto, M. Degli Esposti, C. Maffei, Dalle funzioni ai modelli. Il calcolo per le Bioscienze. Casa Editrice Ambrosiana, Milano, 2014.

Teaching methods

Lessons and exercises in the classroom. Self-learning via Internet.

Assessment methods

For Natural Sciences: there are independent assessments for each of the three course modules: exams in Mathematics and Statistics and a Computer Science test. The final mark, on a 30-point scale, will be the weighted score with respect to the credit points of Mathematics (6 ECTS) and Statistics (2 ECTS), increased by a possible bonus of one, two or three points if the score of the Computer Science test is between 24 and 26, 27 and 29 or 30 respectively.
The written test in Mathematics is designed to ascertain skills to solve exercises with respect to the topics covered in the course. It serves as a qualifying examination for admission to the oral examination (score at least 15/30), which is compulsory for all.  In Statistics, there is only a written exam.

Teaching tools

Alma Mathematica: an online math-bridge course which with its diagnostic tests offers students the possibility to complete the missing pieces and refresh the material necessary for a successful study of mathematics. This self-study course is complemented by a virtual tutorial where students can get instant help by skype, e-mail and telephone.
Exercises for homework and course material are available at http://www.dm.unibo.it/~marco.trozzo2/.
Software R - a free software environment for statistical computing and graphics, available at http://www.r-project.org/.

Links to further information

http://www.dm.unibo.it/~marco.trozzo2/

Office hours

See the website of Marco Trozzo