31586 - Integrational Elements of Mathematical Analysis and Elements of Probability Calculation T

Learning outcomes

At the end of the course, the student will possess basic knowledge in the differential and integral calculus for functions in more than one variable and their applications. The student will also possess basic knowledge in probability calculus and in some continuous distributions (uniform and normal distributions).

Course contents

Mathematical Analysis

Differential Calculus in more than one variable Introduction to topology, metric spaces, Banach spaces. Functions from R^n to R^m (n,m=1,2,3). Limits and continuity. Bolzano and Weierstrass theorems. Functions from R^n to R:partial derivatives, directional derivatives,properties of the gradient function, higher order derivatives, Hessian, Schwarz lemma, Taylor formula at second order, tangent plane. Functions with vector values: the Jacobian, the Jacobian of composite functions.

Applications of differential calculus Local minima and maxima. Fermat theorem. Definition and classification of quadratic forms associated to symmetric matrices, Sylvester theorem, classification of critical points: necessary/sufficient conditions for C^2 functions. Regular varieties in implicit form. Normal and tangent spaces. Dini theorem and the local parametrization of a variety. Conditioned extrema. Fermat theorem in such case. Lagrange multiplier theorem.

Measure and integration Peano-Jordan measure. Riemann integration for function from R^n to R. Properties of integration: additivity, linearity, monotonicity. The integral average theorem. Reduction theorems for double and triple integrals in normal domains. Cavalieri principle and Cavalieri theorem. The change of variable in the integral. Polar, spherical and cylindrical coordinates.

Curves in parametric form and curvilinear integrals Regular curves. Piecewise regular curves. Orientation. Curvilinear integrall on non oriented curves: length, curvilinear integral of a function (mass, barycenter, inertia moments). Vector fields and differential forms. Curvilinear integral of a differential form and work. Exact differential forms and conservative vector fields. Closed differential forms and irrotational vector fields. Potential of a conservative vector field. Poincarè lemma.

Parametric surfaces and integrals on surfaces. Regular surfaces. Tangent plane and normal vector. Orientation. Area of a surface and integration of scalas functions on non-oriented surfaces (mass, barycenter, inertia momenta). Regular surfaces with boundary. Canonical orientation of the boundary. Piecewise regular surfaces. Stokes theorem. Gauss theorem.

Probability

Numerical series Definition of convergent series in C and R. The geometric series in C. A necessary condition for the convergence of the series. Criteria of convergence of numerical series with positive terms: report, comparison, comparison and asymptotic integral criterion. Convergence criteria for the series in terms of alternating signs: criteria of Leibnitz and Dirichlet. Definition of absolutely convergent series. Riemann-Dini. Stability with respect to rearrangements and exchanges of summations of the series is absolutely convergent.

Probability spaces Definition of sigma-algebra, function definition of probability and probability space. Kolmogorov axioms. Bayes' formula. Formula of total probability. Definition of conditional probability. Definition of independent events. Combinatorics. Cardinality of finite and countable. Injective functions between finite sets and their cardinality. Cardinality of the set of subsets of finite sets. Newton's binomial formula.

Discrete models Discrete random variables. Definition of density function for discrete random variables. Density of Bernoulli, geometric, modified geometric, hypergeometric and Poisson. Independent random variables. Mathematical expectation of discrete random variables, moments of order k of discrete random variables, variance and covariance of random variables. Standard deviation. Chebyshev's inequality. The theorem on the law of large numbers. Multi-dimensional random variables. Definition of joint probability density function of random variables. Definition of marginal density of random variables. Relation between joint density and marginal. Mathematical expectation of discrete random variables as compositions of functions and vector random variables. Correlation and independence of random variables. The regression line and its meaning. Calculation of the regression line

Continuous models Continuous random variables. Definition of absolutely continuous random variable and density. Definition of distribution function. Quantile of order \ alpha. The main continuous density: uniform, exponential, normal, N (\ mu, \ sigma ^ 2). Random variables are independent. Mathematical expectation, moments of order k, centered moments and variance of continuous random variables. Calculation of density of random variables. Relations between random variables and density. Random variables and multidimensional definition of joint density of multidimensional continuous variables. Definitions of marginal density of continuous random variables. Density of the sum of two random variables. Definition of convolution of functions.

Approximation The Central Limit Theorem. Outline of the concept of normal approximation..

Readings/Bibliography

Mathematical Analysis:

Simonetta Abenda, Analisi Matematica, Ed. Esculapio (Bologna)

Simonetta Abenda: Esercizi di Analisi Matematica, Ed. Esculapio (Bologna)

Nicola Fusco, Paolo Marcellini e Carlo Sbordone, Elementi di Analisi Matematica Due, Liguori Editore.

Marco Bramanti, Carlo Domenico Pagani e Sandro Salsa: Analisi matematica 1. Ed. Zanichelli, 2008.

Probability Calculus:

Paolo Baldi, Calcolo delle probabilità e statistica, McGraw-Hill
Giovanni Prodi, Metodi matematici e statistici, McGraw-Hill
G. Klimov, Probability theory and Mathematical statistics

Please refer to the official web pages of proff. Abenda and Grammatico for any further piece of information.

Teaching methods

Traditional lessons and exercise classes

Assessment methods

The exam is written and it consists of two parts. It is obligeatory to enrol via Almaesami to both parts.

The dates of the exam are published in Almaesami. Facsimiles of the Mathematical Analysis exercises are published in Alma Campus.

The final rank is given by the algebraic sum of the ranks obtained in part A and Part B. It is published on Almaesami. If the student obtains a final rank greater than 30, on Almaesami it will appear as 30/30 cum laude.

The student may check his work before the signature of marks on Almaesami during an office hour dedicated to such purpose.

Further piece of information concerning this exam are available at the official web pages of proff. Abenda and Grammatico.

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Part A (3h). The student must solve exercises of Mathematical Analysis (both multiple choice and with open answer) and solve exercises and answer to theoretical questions of Calculus of Probability. The student may only use their own book of Mathematical Analysis. It is forbidden to use personal notes, books of Calculus of Probability and to use electronic devices. The maximal rank for each sub-part (Mathematical Analysis and Calculus of Probability) is 12.

The student passes part A if he/she obtains at least 5/12 in each sub-part (Mathematical Analysis, Probability).

Part B (1h). It concerns only the Mathematical Analysis part of the program. The students may use just his/her own pen and the sheets given by the professors. The student must write the solution of one of the multiple choice exercises of part A and answer to two theoretical questions following the proposed draft. The maximal rank for this part is 12.

Office hours

See the website of Simonetta Abenda

See the website of Cataldo Grammatico