robertotonelli

Giu 232023
 

Boundary value problems with nonsmooth and multivalued terms

Prof. Vasile Staicu
University of Aveiro, Portugal

Abstract

The aim of this course is to present some methods and tools used in the study of nonlinear boundary value problems involving multivalued maps and nonsmooth functions and illustrate those methods with some recent results concerning existence and multiplicity of solutions. The course will be given during the period April 26 – May 16, 2023, under the support of Visiting Professor Program of INdAM.

Outline

We start by introducing some notions of multivalued analysis: multivalued maps and their continuity. Then we introduce some elements of nonsmooth analysis that we use to develop nonsmooth critical point theory succeeded in extending a big part of the smooth theory to nonsmooth functionals: the main tool in the variational method for solving boundary value problems, which consists of trying to find solutions for a given boundary value problem by looking for stationary points of a real functional defined on a space of functions in which the solution of the boundary value problem is assumed to lie.

Degree theory is a basic tool of nonlinear analysis and produces powerful existence and multiplicity results for nonlinear boundary value problems. We present a generalization of Brouwer degree theory to multivalued perturbations of monotone type maps, developed in a joint paper with Aizicovici and Papageorgiou (Mem. Amer. Math. Soc., 196, 2008). Then we consider a nonlinear Dirichlet problem with nonsmooth potential (hemivariational inequality) and use variational method to prove two constant sign solutions (one positive and another negative) and then by using degree theoretical approach we prove the existence of a third one, nontrivial solution, distinct from the previous two.

We present several existence and multiplicity results, with sign information for the solutions of nonlinear boundary value problems driven by the Laplacian, the p-Laplacian and then try to extend such results to nonlinear boundary value problems driven by fractional order differential operators and to nonlocal pseudo-differential inclusions, possibly including obstacles or constraints. It is reasonable to expect that the pairing of nonlocal diffusion operators and set-valued reaction will provide a more realistic model for applications to such problems as quantum mechanics, image restoration, and financial mathematics, which typically present a high degree of uncertainty, rather than elliptic equations with smooth, single-valued reactions.

Schedule

27/4 11-13 28/4 9-11 2/5 11-13 5/5 9-11 9/5 11-13 11/5 11-13 12/5 9-11 16/5 11-13 (room F)

References

Vasile Staicu, Lecture Notes, 2023

Open questions will be considered and analyzed with the Ph. D. students.

 

 Scritto da in 23 Giugno 2023  Senza categoria  Commenti disabilitati su PhD Course: Boundary value problems with nonsmooth and multivalued terms
Gen 042023
 

Matrix functions: Computation and application to Complex Networks analysis

Dott. Caterina Fenu
Dipartimento di Matematica e Informatica
Università degli Studi di Cagliari

Abstract

The aim of the course is to introduce basic concepts related to the computation of matrix functions and their use in the analysis of complex networks. Half of the course will be dedicated to the implementation of some algorithms by using the MATLAB software.

Outline 

– Linear algebra: eigenvalue decomposition, Jordan decomposition, singular value decomposition;
– Introduction to matrix functions and bilinear/quadratic forms: orthogonal polynomials, Lanczos algorithms, quadrature formulas;
– Introduction to complex networks theory: graphs, centrality/communicability indices, network indices;
– Use of the MATLAB software.

Exam

The oral exam will consist of a discussion of a research paper concerning the topics of the course.

References

1. G. H. Golub and G. Meurant, Matrices, Moments and Quadrature with Applications, Princeton University Press, Princeton, 2010.
2. N. J. Higham, Functions of matrices: theory and computation, SIAM, 2008.
3. W. Gautschi, Orthogonal polynomials: computation and approximation, Oxford University Press, 2004.

 

Nov 292022
 

Factorizations in monoids and rings

Laura Cossu
University of Graz

Outline

Many problems in algebra involve the decomposition of certain elements of a ring (or more generally of a monoid) into a product of certain other elements (hereinafter generically referred to as building blocks) that are in some sense minimal. The classical theory of factorization investigates factorizations in which the building blocks are atoms, i.e., non-unit elements of a monoid that are not products of two non-units. For example, it is well known that every non-zero non-unit of a Dedekind domain (more generally, of a Noetherian domain) can be written as a finite product of atoms and that in general such decompositions are not unique. On the other hand, examples of factorizations that lie beyond the scope of the classical theory include additive decompositions into multiplicative units in rings; cyclic decompositions of permutations in the symmetric group of degree n; idempotent factorizations of the “singular elements” of a monoid; and so on.

After an introductory overview of the history and main results in the classical framework, in this course, we combine the language of monoids and preorders (partial orders that are not necessarily antisymmetric) to make first steps towards the construction of a “unified theory of factorization”. In particular, we prove an abstract existence theorem that recover, among others, a classical theorem of Cohn on atomic factorizations in cancellative monoids, a classical result by Anderson and Valdes-Leon on “irreducible factorizations” in commutative monoids, but also a theorem by Erdos on idempotent factorizations of square singular matrices over fields. We also introduce a notion of “minimal factorization”, suitable for the new abstract setting, and we qualify and quantify the related non-uniqueness properties in specific cases but also in some generality. Examples will help to motivate and illustrate the theory.

Prerequisites: Standard knowledge of (basic) algebraic structures and (binary) relations.

Duration of the course: 10 hours.

Schedule

– November 28, 16:00-18:00 (Aula B)
– November 29, 16:00-18:00 (Aula A)
– November 30, 16:00-18:00 (Aula A)
– December 1, 16:00-18:00 (Aula A)
– December 2, 16:00-18:00 (Aula A)

Exam

Written. Writing of a short essay.

References

1. L. Cossu and S. Tringali, Abstract Factorization Theorems with Applications to Idempotent Factorizations, e-print (Aug. 2021), arXiv:2108.12379.
2. L. Cossu and S. Tringali, Factorization under Local Finiteness Conditions, e-print (Aug. 2022), arXiv:2208.05869.
3. A. Geroldinger and F. Halter-Koch, Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Pure Appl. Math. 278, Chapman & Hall/CRC, Boca Raton (FL), 2006.
4. S. Tringali, An Abstract Factorization Theorem and Some Applications, J. Algebra 602 (2022), 352–380.

 Scritto da in 29 Novembre 2022  Senza categoria  Commenti disabilitati su PhD Course: Factorizations in monoids and rings
Mar 302021
 

Introduction to Finite Mixture Models and Latent Class Analysis

Dott.ssa Silvia Columbu
Postdoctoral researcher in Statistics
Department of Mathematics and Computer Science, University of Cagliari

Outline

This course is intended as both a theoretical and practical introduction to finite mixtures as a model based clustering tool. We will introduce the concept of finite mixture of distributions for categorical as well as continuous variables. We will study the problem of maximum likelihood via expectation-maximization (EM) algorithms. We will start from simple data structures and then move to the introduction of regressors and to more complex model structures. We will also explore the problem of model selection and choice of the number of clusters. We will see practical applications with the aid of devoted packages in the statistical software R. Students will learn how to implement an analysis and how to read and interpret outputs and graphical representations of results.

Schedule

Duration: 20 hrs, in the period going from 18th to 26th of February 2021, for a total of 4 CFR

Course schedule

 Scritto da in 30 Marzo 2021  Senza categoria  Commenti disabilitati su PhD Course: Introduction to Finite Mixture Models and Latent Class Analysis
Nov 162020
 

Corso di dottorato (20 ore con esame – 4 CFR)

Il dott. Mirko Marras  terrà, a partire dal 9 dicembre 2020, un corso per il dottorato dal titolo “Python for machine Learning Research”.

Le informazioni sono disponibili al seguente  link

Seminario dottorale (20 ore con esame – 4 CFR)

Il prof. Alessandro Buccini terrà, a partire dal 3 giugno 2020, un corso per il dottorato dal titolo “Regularization of ill-posed linear inverse problems”. Le lezioni si terranno online, su una piattaforma che verrà comunicata in seguito, e includeranno alcune sessioni di laboratorio Matlab.

L’abstract del corso è disponibile a questo link.
 Scritto da in 16 Novembre 2020  Senza categoria  Commenti disabilitati su PhD Courses: Python for machine Learning Research / Regularization of ill-posed linear inverse problems
Gen 112020
 

Il dott. Marco Ortu del DIEE, Università di Cagliari
terrà un corso breve dal titolo:

Leggi di potenza, con applicazioni nell’ingegneria del software

16 Gennaio 2020, ore 9:00 – 14:00 ( ​ Aula C – Dip. di Matematica e Informatica, via Ospedale 72)

Allegata la Locandina

 Scritto da in 11 Gennaio 2020  Senza categoria  Commenti disabilitati su Seminario Dottorale (5 ore con esame – 1 CFU)
Gen 052020
 

I seminari di Matematica previsti per il mese di Gennaio 2020

sono disponibili in allegato (link)

 Scritto da in 5 Gennaio 2020  Senza categoria  Commenti disabilitati su Seminari Matematica Gennaio 2020
Dic 062019
 

Seminari matematica

Martedì 10 dicembre ore 16.50 – Aula B

Prof. Andrea Ratto

APPLICAZIONI ARMONICHE E POLIARMONICHE.

In questo seminario verranno presentati, con un approccio rivolto a chi non e’  uno specialista del settore, diversi aspetti generali relativi alla teoria delle applicazioni armoniche.

In particolare, si cerchera’ dapprima di dare un’idea dei principali concetti di geometria, topologia e calcolo variazionale che confluiscono nello studio di questi argomenti.

Poi, si passera’ all’illustrazione delle ragioni che rendono significativa questa area di ricerca. Infine, verra’ introdotta la nozione di applicazione poliarmonica e saranno illustrati alcuni problemi e risultati che costituiscono oggetto di ricerche in corso

 

 Scritto da in 6 Dicembre 2019  Senza categoria  Commenti disabilitati su
Dic 062019
 

Seminari matematica

Giovedì 19 dicembre alle 15.30
Raffaella Mulas
Titolo: La bellezza dell'operatore di Laplace normalizzato per grafi
Abstract:I network sono ovunque intorno a noi. I grafi offrono un modello perfetto
per i network e, come vedremo in questa presentazione, l'operatore di Laplace
normalizzato è uno strumento molto potente per descrivere i grafi.
 Scritto da in 6 Dicembre 2019  Senza categoria  Commenti disabilitati su
Dic 042019
 

Avviso di seminario:

lunedì 9 dicembre alle ore 17.30 nell’Aula B del Dipartimento di Matematica e Informtatica,

la dott.ssa Maria Infusino, dell’Università di Costanza (Germania), terrà un seminario dal titolo:

Il problema dei momenti: oltre la dimensione finita

 Scritto da in 4 Dicembre 2019  Senza categoria  Commenti disabilitati su
contatti | accessibilità Università degli Studi di Cagliari
C.F.: 80019600925 - P.I.: 00443370929
note legali | privacy