Mathematics

Linear algebra is an essential tool to handle multi-component quantities. It is used not only in mathematics but also in natural and social sciences. This course provides basic notions and understanding of linear algebra such as vectors and matrices. 

Stochastic processes find applications in a wide variety of fields and offer a refined and powerful framework to examine and analyze time series. This course presents the basics for the treatment of stochastic signals and time series. Topics covered include models for stochastic dependence; concepts of description of stationary stochastic processes in the time domain including expectation, covariance, and cross-covariance functions; concepts of description of stationary stochastic processes in the frequency domain including effect spectrum and cross-spectrum; Gaussian process, Wiener process, white noise, and Gaussian fields in time and space; Stochastic processes in linear filters including relationships between in- and out-signals, autoregression and moving average (AR, MA, ARMA), and derivation and integration of stochastic processes; the basics in statistical signal processing, estimation of expectations, covariance function, and spectrum; and application of linear filters: frequency analysis and optimal filters.

This course is part of the Laurea Magistrale degree program and is intended for advanced level students. Enrolment is by permission of the instructor. At the end of the course, students know basic numerical methods for evolutive ordinary and partial differential problems, together with their main theoretical and computational properties. In particular, students are able to analyze the properties of numerical methods; constructively examine corresponding computational results; advance their scientific computing education in higher level courses; and employ the acquired numerical skills in a variety of application areas. The two main topics covered are: 1) numerical solution of Ordinary Differential Equations (ODEs): Initial Value Problems; and 2) numerical solution of ODEs: Boundary Value Problems.

Time series analysis concerns the mathematical modeling of time-varying phenomena, e.g., ocean waves, water levels in lakes and rivers, demand for electrical power, radar signals, muscular reactions, ECG signals, or option prices at the stock market. The structure of the model is chosen both concerning the physical knowledge of the process, as well as using observed data. Central problems are the properties of different models and their prediction ability, estimation of the model parameters, and the model's ability to accurately describe the data. Consideration must be given to both the need for fast calculations and the presence of measurement errors. The course gives a comprehensive presentation of stochastic models and methods in time series analysis. Time series problems appear in many subjects and knowledge from the course is used in, i.e., automatic control, signal processing, and econometrics.

This course introduces basic and advanced mathematics used in engineering to develop an awareness and an appreciation of the role of mathematics in engineering. This course deals with mathematical principles, methods, and modeling.  

This course covers a number of fundamental topics concerning groups of graph automorphisms, with an emphasis on group-theoretic notions and results. Topics include fundamentals of graph theory and of group theory; graph automorphisms, transitive graphs; group actions on graphs; Cayley graphs, Schreier graphs; fundamental group of a graph, coverings; free group: definition, elementary properties; subgroups of free groups; and Hanna Neumann conjecture.

The course gives an introduction to group and ring theory with emphasis on finite groups, polynomial rings, and field extensions.

This course offers a study of ordinary differential equations (ODEs). Topics include: origins of ODEs in applications; first order equations; linear order equations, higher order, and linear differential systems; existence, uniqueness, and continuation of solutions; resolution of ODE with power series; nonlinear equations-- autonomous systems, phase plane, classification of critical points, and stability theorems.

This course covers multiple linear regression and least squares methods; generalized linear models; survival regression models; nonlinear effects and basis expansions; parametric, semiparametric, and nonparametric likelihood methods; and aspects of practical regression analysis in R.

The course covers: Countability, measure spaces, σ-algebras, π-systems and uniqueness of extension. Construction of Lebesgue measure on R (proof non-examinable), Independence. The Borel-Cantelli lemmas, measurable functions and random variables, independence of random variables. Notions of probabilistic convergence. Construction of integral and expectation. Integration and limits. Density functions. Product measure and Fubini’s theorem. Laws of large numbers. Characteristic functions and weak convergence, Gaussian random variables. The central limit theorem. Conditional probability and expectation.