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

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 methods and their ap-plications in the correct planning of experiments, data sampling, analysis, and presentation. The course conjugates con-cepts 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. PRELIMINARY. a) Sets and operations on sets. b) Natural numbers N, integers Z and rational Q c) Real numbers. Operations and arrangement in R. Sets of limited or unlimited real numbers. Upper extremity and lower extremity of sets of real numbers. Intervals. Distance. d) Polinomial and irrational equations and inequalities, with modules and systems e) Analytical geometry in the Cartesian plane: distances between points, line, circumference, parabola, ellipse and hyperbola. Mutual positions and geometric problems 2. ELEMENTARY FUNCTIONS. a) Real functions of real variable, graph, domain, image. Composition of functions. Inverse function. Monotone functions. Limited functions and unlimited functions. Maxima and minima. Upper and lower extremes of functions. Sign and zeros of a function. Diagram operations: translations, symmetries. b) Absolute values. Powers with natural, rational and real exponent. The polynomial functions x ^ a, irrational, exponential a ^ x, logarithmic. The trigonometric functions. c) Algebric, exponential and logarithmic inequalities, systems of inequalities. 3. CONTINUOUS LIMITS AND FUNCTIONS. a) Distance and surroundings, right and left surroundings. Function limits. Continuity in one point. Elementary limits. Algebra of limits. Limits of compound functions. Comparison theorem. Some indeterminate forms. Comparison between infinites and infinitesimal. Horizontal, vertical, oblique asymptotes. b) Continuous functions and their fundamental properties. Theorem of the zeros. Weierstrass theorem. . 4. DERIVATIVES AND APPLICATIONS. a) Definition of derivative in a point. Right derivative and left derivative. Straight tangent to the graph. Derived function. Derivatives of elementary functions. Rules of derivation of sum, product, quotient, compound function, inverse function. Derivability and continuity. Relative maximum and minimum points. Fermat, Rolle and Lagrange theorems. Consequences of the Lagrange theorem: derivable functions with null derivative, derivable functions with the same derivative, sign of the first derivative and intervals of monotony of the function. Search for points of maximum or minimum relative through the sign of the derivative. Second derivative, its sign and convexity. b) Qualitative study of the graph of a function. c) Subsequent derivatives. Local approximation of functions with polynomials. Theorems of De l'Hospital. Taylor's polynomial and Taylor's theorem. Use of the theorem for determining limits. 5. INTEGRAL. a) Primitive functions (indefinite integrals). Elemental integrals. Definition of definite integral. Fundamental theorem of integral calculus. b) Calculation of areas through the use of integrals. c) Overview of improper integrals on unlimited intervals. 6. DIFFERENTIAL EQUATIONS. Definitions of differential equation (in normal and non-normal form) and of order of a differential equation. Solution and general solution of a differential equation. Examples of differential equations. Cauchy problem. 7. LINEAR ALGEBRA. a) Geometric vectors. Vectors in R ^ n. Matrixes with real coefficients. Produced between matrices and its property. Linear systems in matrix form Ax = b. Systems resolution with the Gauss method. b) Rank (or characteristic of A). Determinant of square matrices. Rouché-Capelli theorem. Cramer's theorem. Inverse of a square matrix. c) The scalar product and its properties. Standard (or form) of a vector. Orthogonal vectors. Elements of analytical geometry. Vector product in R ^ 3.
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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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The final exam is written and must be completed in 3 hours. Neither midterm tests nor oral exams will take place. The exam paper consists of 6 exercises. The total of the marks of the exam paper is 30. Any topic dealt with during the lectures can be examined. Students are not allowed to use books, notes or electronic devices during the exam. The mark of any exercise will take into consideration not only the correctness of the results, but also the method adopted for the solution and the precise references to theoretical results (e.g. theorems) taught during the lectures. The pass mark for the exam of the Mathematics module is 18.
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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. Eight 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.