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.
a) Sets and operations with sets.
b) Numerical sets. Bounded and unbounded sets. Minimum, maximum, infimum and supremum of numerical sets.
c) Natural numbers N, integers Z, rationals Q, and reals R. Intervals. Distance.
d) Monomials, polynomials and polynomial decomposition.
e) Absolute values. Powers with natural, rational and real exponent. The polynomial functions x^a, irrational, exponential a^x, logarithmic.
g) Entire, rational, irrational, with absolute values and systems of equalities and inequalities.
h) Exponential, logarithmic and systems of inequalities.
i) Analytical geometry in the Cartesian plane: distances between points, lines, circumference, parabola, ellipse and hyperbola. Mutual positions and geometric problems.
a) Functions of real variable, plot, domain and image.
b) Compounded functions.
c) Inverse functions.
d) Monotonic functions.
e) Bounded and unbounded functions.
f) Maxima and minima, suprema and infima of functions.
g) Signs and zeros of a function.
h) Operations with plots, translations and symmetries.
a) Distance and neighborhoods, right and left neighborhoods. Limit with functions. Continuity at one point. Elementary limits. Limits algebra. Limits of composed functions. Squeeze theorem. Indeterminate forms. Comparison between infinites and between infinitesimals. Horizontal, vertical, and obliquos asymptotes.
b) Continuous functions and their fundamental properties. Weierstrass theorem.
a) Definition of derivative at one points. Left and right derivative. Tangent line to a plot. Derivative function.
b) Derivatives of elementary functions. Derivation rules.
c) Derivability and continuity.
d) Critical points. Fermat theorem, Rolle theorem and Lagrange theorem. Implications of Lagrange theorem: derivable functions with null derivative, derivable functions with equal derivative, sign of first derivative and monotonicity intervals of a function. Detection and classification of relative maxima and minima through derivative's sign.
e) Second derivative, its sign and convexity.
f) Higher order derivatives. Local approximation of functions with polynomials. De l’Hopital theorem. Taylor's series and Taylor's theorem. Determine limits by using Taylor's theorem.
a) Primitive functions (indefinite integrals). Elementary integrals. Definition of definite integral. Fundamental theorem of integral calculus.
b) Computing areas using integrals.
c) Overview of improper integrals on unlimited intervals.
6. DIFFERENTIAL EQUATIONS.
a) Definitions of differential equations (in normal and nonnormal form) and of order of a differential equation.
b) Solution and general solution of a differential equation. Examples of differential equations. Cauchy problem.
7. ALGEBRA LINEARE.
a) Vectors and vectors in R^n. Real valued matrices. Product between matrices.
and its properties. Linear systems in matrix form Ax = b. Solving linear systems with Gauss method.
b) Rank of a matrix. Determinant of square matrices. Rouché-Capelli's theorem. Cramer Teorema. Inverse of a square matrix.
c) Scalar product and its properties. Norm of a vector. Orthogonal vectors.
a) Bivariate functions, domain, detection and classification of critical points.
b) Complex numbers, operations with complex numbers, Euler formula.
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
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 evaluation 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.
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 mean of the marks obtained by students in both tests.