Teaching code

4S02690

Teacher

Roberto Chignola

Coordinator

Roberto Chignola

Credits

12

Language

Italian

Scientific Disciplinary Sector (SSD)

MAT/05 - MATHEMATICAL ANALYSIS

Period

Semester 1 dal Oct 2, 2023 al Jan 26, 2024.

Courses Single

Authorized with reserve

Lessons timetable Moodle Seminars 0

Learning objectives

Mathematics: This course aims at providing the students with the mathematical tools (set-theoretic and algebraic structures, differential and integral calculus in one or several real variables, ordinary differential equations) whose knowledge is indispensable for the achievement of the degree. A particular attention is paid to the concrete application of the learned notions. At the end of the course students should be able to use appropriately the mathematical language and the notions of the syllabus and furnish valid arguments in support of the solution of the proposed problems. Statistics: The aim of the course is to make the students acquainted with basic statistical ideas and mathematical methods and their applications in the correct planning of experiments, data sampling, analysis, and presentation. The course conjugates concepts of basic statistics and probability theory as well as applied mathematics with real situations as they emerge in a standard biotechnology laboratory. The students acquire appropriate skills to understand how biological systems work and more generally to cope with real-life problems in different applied scientific fields. At the end of the course the students are able to: - analyse experimental observations and prepare professional reports - appropriately plan experiments - autonomously acquire new skills in specific fields of applied statistics and mathematics.

Prerequisites and basic notions

Basic mathematical skills (high school level)

Program

MATHEMATICS
- review of some fundamental notions of arithmetics and their application to a biological problem
- introduction to sets, basic set operations
- introduction to logic
- modeling the Lac operons with Boolean logic
- definition of function, graph of a function, examples
- classes of relevant functions for the life sciences: exponential, logarithmic, polynomial, trigonometric functions
- limits: definition, properties, left and right continuity, asymptotes
- derivative of a function, rules of derivation.
- definite integrals
- indefinite integrals
- integrations by parts
- integration by numerical methods: an elementary introduction and application in the life sciences
- differential equations in chemistry, biology and medicine
- first and second order differential equations
STATISTICS
- brief introduction on the scientific method: the philosophical approach of Popper, Khun, and Lakatos and the concept of validation/falsification of hypotheses
- variables and measurements, frequency distribution of data sampled from discrete and continuous variables, displaying data
- elements of probability theory: definition, a brief history of probability, the different approaches to probability, the rules for adding and multiplying probabilities, Bayes' theorem
- discrete probability distributions: the Binomial and the Poisson distributions and the limiting dilution assay with animal cells
- continuous probability distributions: the concept of probability density, the Normal distribution and the Z statistics
- statistical inference: the problem of deducing the properties of an underlying distribution by data analysis; populations vs. samples. The central limit theorem
- the Student distribution and the t statistics. Confidence intervals for the mean. Comparing sample means of two related or independent samples
- mathematical properties of the variance and error propagation theory
- planning experiments and the power of a statistical test
- the χ2 distribution and confidence intervals of the variance
- goodness-of-fit test and χ2 test for contingency tables
- problems of data dredging and the ANOVA test
- correlation and linear regression

Bibliography

Didactic methods

Classroom-taught lessons. One written test, approximately at the end of the course, will allow the students to self-evaluate their knowledge. The results of these tests will have no impact on the final exam.

Learning assessment procedures

Written test. Students will have to solve 6 problems (within a maximum time of 3 hours, and they will be allowed to consult the reference textbook and other didactic material (e.g. slides). Personal computers or other electronic devices used to access the internet are not allowed.
The problems will concern mathematics applied to the life sciences and the analysis of data from real biological and biomedical experiments.
The assessment methods could change according to the academic rules.

Evaluation criteria

Five points are assigned to the solution of each exercise and all points are then summed up. To pass their test students must reach a minimum score of 18 points. To those test that will reach a score of 30 a "laude" might be further assigned. To this end the clarity of presentation of all solutions will be evaluated.

Criteria for the composition of the final grade

Not applicable

Exam language

Italiano