Teaching code

4S02690

Credits

12

Coordinator

Language

Italian

Scientific Disciplinary Sector (SSD)

MAT/05 - MATHEMATICAL ANALYSIS

Moodle Seminars 0

The teaching is organized as follows:

Learning outcomes

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MM: Matematica
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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.
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MM: Statistica
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The aim of the course is to make the students acquainted with basic statistical ideas and 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 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

Program

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MM: Matematica
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1) Some notions of set theory. 2) The complete ordered field of the real numbers. 3) Euclidean distance and induced topology on the real line. Absolute value of a real number. 4) Cartesian plane. 5) Real functions of one real variable. 6) Polynomials and polynomial functions. Power, exponential and logarithmic functions. Trigonometric functions. 7) Limit of a function of one real variable. 8) Continuity of a function of one real variable at one point. Fundamental theorems on continuos functions. 9) Derivative of a function. Derivation rules. Fundamental theorems on differentiable functions. 10) Monotonicity of a function. Local and global minima and maxima of a function. 11) Convex functions. 12) Riemann integral. Integration rules. Improper integrals. 13) Ordinary differential equations. 14) Linear algebra. Matrices and operations on them. Determinant of a square matrix. 15) Distance between two points in the plane and geometrical loci. Conics. 16) Functions of more variables. Level curves and level sets. 17) Topology in R^2. Continuity of a function of 2 variables. 18) Differentiable functions of 2 variables. Partial derivatives. 19) Local and global minima and maxima of a function of more variables.
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MM: Statistica
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Each class introduces basic concepts of probability theory and applied statistics through combination of lectures and exercises. The exercises focus on the analysis of real experimental data collected in the teacher's lab or in other biotechnology labs. Lectures - 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 The program follows the topics listed in the textbook up to chapter 17 (included) with the following extras: key aspects in probability theory, probability distributions in the biotechnology lab (practical examples), error propagation theory Reference textbook: Michael C. Whitlock, Dolph Schluter. Analisi Statistica dei dati biologici. Zanichelli, 2010. ISBN: 978-88-08-06297-0 Lecture slides are available at: http://profs.scienze.univr.it/~chignola/teaching.html

Bibliography

Examination Methods

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MM: Matematica
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Written exam. The exam consists in 6 exercises to be solved in 3 hours. The minimum pass mark for Mathematics unit is 18 out of 30.
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MM: Statistica
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At the end of the course students are expected to master the basic concepts of probability theory and of validation/falsification of hypotheses, and to apply these concepts to the analysis of experimental data collected in a generic biotechnology laboratory. To pass the final written test, students are asked to solve 4 exercises within a maximum of 2 hours. The exercises concern the analysis of problems as they are found in a biotechnology laboratory. During the test, students are allowed to use learning resources such as books, lecture slides, handouts, but the use of personal computers or any other electronic device with an internet connection is not allowed. Four 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. The final score of the whole course in Mathematics and Statistics is calculated as the weighted mean of the marks obtained by students in both tests by taking into account the number of credits assigned to each course as weights: final grade = (2/3) x1 + (1/3) x2 where x1 and x2 are the marks obtained by students in their tests of Mathematics and Statistics, respectively.