B003242 - MATHEMATICAL METHODS AND PROBABILITY

Italian

I and II ODE. nth ODE. Existence and uniqueness Theorems to the Initial Value Problem (IVP). Qualitative analysis and stability. Functions spaces and Fourier series. Probability spaces. Discrete and continuous random variables of one and many dimensions.

1) G.Borgioli: Lecture Notes, on the webpage http://www.modmat.unifi.it
2) G. Borgioli - Modelli Matematici di Evoluzione ed Equazioni Differenziali - CELID, Torino (1996).
3) M.Bramanti, C.D.Pagani, S.Salsa - Matematica – Zanichelli
4) G. Modica, L. Poggiolini - Note di Calcolo delle Probabilità - Pitagora
5) R. Giuliano, L. Ladelli, P. Baldi, Laboratorio di statistica e probabilità, McGraw-Hill.
6) G.C.Barozzi - Matematica per l'Ingegneria dell'Informazione - Zanichelli
7) M.Marini - Metodi Matematici per lo studio delle Reti Elettriche - CEDAM
8) W. E. Boyce, R. C. DiPrima - Elementary Differential Equations and Boundary Value Problems,John Wiley & Sons, Inc.
9) W. Navidi, Probabilità e statistica per l'ingegneria e le scienze, McGraw-Hill.
10) F. Mugelli, M. Spadini - Metodi Matematici - Esculapio.
11) Sheldon M. Ross, "Calcolo delle probabiltà", Apogeo, Milano
12) Paolo Baldi, "Calcolo delle probabiltà", McGraw-Hill, Milano
13) D. Bertacchi, M. Bramanti, G. Guerra, " Esercizi di Calcolo delle Probabilità e Statistica", Progetto Leonardo, Esculapio, Bologna.

On the webpage http://www.modmat.unifi.itstudents can find:
- Notes and exercises.
- The lectures diary .
- Written tests of past exams.

The aim of the course is providing basic knowledge in dealing Ordinary Differential Equations, in Fourier Analysis and in Probability Theory. Students learn methods to solve EDO of I and II (linear) order. They are able to face simple qualitative analysis of ODE and Fourier series. Moreover students are able to solve problems in Probability, know main Probability distributions of discrete and continuous random variables of one and many dimensions.

The subjects from the course of Analisi Matematica.

Lectures and Exercises

A written test consisting in solving some exercises and an oral exam.

1- ORDINARY DIFFERENTIAL EQUATIONS (ODE)
1.1-Definitions and terminology; the normal form; vector equation y'(x)=f(x,y(x)) as a general representation of n-th order ODE and systems of ODE; the Cauchy problem or Initial Value Problem (IVP); The existence and uniqueness Theorem.
1.2-First order ODE: solving methods for separable equations, homogeneous equations, linear equations, Bernoulli equations, exact equations and integrating factors.
1.3 Second order ODE: equations convertible to first order; linear equations with constant coefficients, homogeneous and nonhomogeneous: the undetermined coefficients method and the variation of constants method.
1.4-Linear equations in R^n: general solutions. Linear spaces of functions: the space generated by the solutions of linear homogeneous ODE.
1.5-Geometric interpretation and phase space analysis for second order ODE and systems of ODE.
1.6-Stability of solutions with respect to the initial conditions: Liapunov stability: Stability of equilibrium and of evolution equations;Asymptotic stability; stability properties of linear ODE and linear systems of ODE; the phase space; phase portrait for two-dimensions equations and systems: center, saddle point, spiral point, node; stability of almost linear systems; nonlinear ODE and systems: Liapunov’s second method and the Liapunov function; the phase portrait and the “energy-balance” plane.
1.7- Mathematical models in Mechanics and in Theory of Electric Circuits: the harmonic oscillator, the linearly damped harmonic oscillator (LDHO), the forced LDHO and the linear resonance;
Mathematical models in Theory of Populations: the Malthus and the logistic model, the prey-predator (Lotka-Volterra) model.
1.8- Laplace transform (introductory elements).

2-FOURIER SERIES (FS)
2.1- Functions Spaces equipped with inner product (Unitary Spaces); norm of a function. Schwartz, Minkowski and Bessel inequalities. Hilbert Space L^2. Space of the functions piecewise continuous on an interval. Trigonometrical polynomials and Fourier polynomials.
Real and complex approximating basis.
2.2 - Sequences of functions: pointwise convergence and uniform convergence. Series of functions, power series, radius of convergence, Abel's criterium.
2.3-Real and complex Fourier series; computation of the coefficients; FS of periodic functions and of functions defined on an interval; quadratic convergence; the Parseval equality; Dirichlet conditions for the pointwise convergence; pointwise convergence of the derivative and the integral series; even and odd functions and thei FS; the Gibbs phenomenon and the uniform convergence.
2.4-Introduction to Partial Differential Equations (PDE) and initial and boundary value problem. One-dimensional Diffusion (Heat) equation, wave equation, Laplace equation. Introduction to Maxwell equations problems.

3 PROBABILITY
3.1 Random experiments, the space of events, elements of combinatorics, definitions of probability, conditional probability, independent events, Bayes' Theorem
3.2. Random variables and probability distributions: discrete and continuous random variables, probability distributions, mathematical expectation, probability and point probability density, joint density, distribution parameters: mean value and variance. Probability density function of the random variable, functions of random variables.
3.3 Discrete Probability distributions: Bernoulli and binomial distributions, Poisson distribution approximation to the binomial distribution with the Poisson distribution.
3.4 Continuous Probability distributions: normal or Gaussian distribution, standard normal distribution. Some applications of the normal distribution, use the tables of the normal distribution. Uniform distribution, the exponential distribution, the relation between the binomial distribution and the normal distribution, the relationship between the normal distribution and the Poisson distribution.