Notas para a unidade curricular Tópicos de Matemática Discreta - mestrado integrado em Engenharia Informática

Texto de apoio às aulas teóricas da UC Tópicos de MAtemática Discreta do curso de Mestrado Integrado em Engenharia Informática.info:eu-repo/semantics/draf

Injective linear transformations with equal gap and defect

Suppose V is an infinite-dimensional vector space over a field F and let I(V) denote the inverse semigroup of all injective partial linear transformations on V. Given ß in I(V), we denote the domain and the range of ß by dom ß and im ß, respectively, and we call the cardinals g(ß)=codim(domß) and d(ß)=codim(imß) the `gap' and the `defect' of ß, respectively. In this paper, we study the semigroup A(V) of all injective partial linear transformations with equal gap and defect, and characterise Green's relations and ideals in A(V). This is analogous to work by Sanwong and Sullivan in 2009 on a similarly-defined semigroup for the set case, but we show that these semigroups are never isomorphic.The research of the authors was partially financed by Portuguese Funds through FCT (Fundação para a Ciência e a Tecnologia) within the Projects UIDB/00013/2020 and UIDP/00013/2020

Signals on graphs : transforms and tomograms

Development of efficient tools for the representation of large datasets is a precondition for the study of dynamics on networks. Generalizations of the Fourier transform on graphs have been constructed through projections on the eigenvectors of graph matrices. By exploring mappings of the spectrum of these matrices we show how to construct more general transforms, in particular wavelet-like transforms on graphs. For time-series, tomograms, a generalization of the Radon transforms to arbitrary pairs of non-commuting operators, are positive bilinear transforms with a rigorous probabilistic interpretation which provide a full characterization of the signals and are robust in the preseninfo:eu-repo/semantics/publishedVersio

The completion problem for N-matrices

An $n\times m$ matrix is called an $N$-matrix if all principal minors are negative. In this paper, we are interested in $N$-matrix completion problems, that is, when a partial $N$-matrix has an $N$-matrix completion. In general, a combinatorially or non-combinatorially symmetric partial $N$-matrix does not have an $N$-matrix completion. Here, we prove that a combinatorially symmetric partial $N$-matrix has an $N$-matrix completion if the graph of its specified entries is a 1-chordal graph. We also prove that there exists an $N$-matrix completion for a partial $N$-matrix whose associated graph is an undirected cycle.Fundação para a Ciência e a Tecnologia (FCT) – Programa Operacional “Ciência, Tecnologia, Inovação” (POCTI) DGI - BFM2001-0081-C03-0

Sign pattern matrices that admit P_0-matrices

For sign patterns corresponding to directed or undirected cycles, we identify conditions under which the patterns admit or require P0–matrices.Fundação para a Ciência e a Tecnologi

Sign pattern matrices that admit M-, N-, P- or inverse M-matrices

In this paper we identify the sign pattern matrices that occur among the N–matrices, the P–matrices and the M–matrices. We also address to the class of inverse M–matrices and the related admissibility of sign pattern matrices problem.Fundação para a Ciência e a Tecnologia (FCT)Spanish DGI grant number MTM2007-6447

Some results on B-matrices and doubly B-matrices

A real matrix with positive row sums and all its off-diagonal elements bounded above by their corresponding row means was called in [4] a B-matrix. In [5], the class of doubly B-matrices was introduced as a generalization of the previous class. We present several characterizations and properties of these matrices and for the class of B-matrices we consider corresponding questions for subdirect sums of two matrices (a general ‘sum’ of matrices introduced in [1] by S.M. Fallat and C.R. Johnson, of which the direct sum and ordinary sum are special cases), for the Hadamard product of two matrices and for the Kronecker product and sum of two matrices.Fundação para a Ciência e Tecnologia (FCT)Ministerio de Ciencia y Tecnología (Espanha

Moore-Penrose invertibility in involutory rings: the case aa+=bb+

In this article, we consider Moore-Penrose invertibility in rings with a general involution. Given two von Neumann regular elements a, b in a general ring with an arbitrary involution, we aim to give necessary and sufficient conditions to aa† = bb†. As a special case, EP elements are considered.Fundação para a Ciência e a Tecnologia (FCT)POCT

N_0 completions on partial matrices

An $n\times n$ matrix is called an $N_0$-matrix if all its principal minors are nonpositive. In this paper, we are interested in $N_0$-matrix completion problems, that is, when a partial $N_0$-matrix has an $N_0$-matrix completion. In general, a combinatorially or non-combinatorially symmetric partial $N_0$-matrix does not have an $N_0$-matrix completion. Here, we prove that a combinatorially symmetric partial $N_0$-matrix, with no null main diagonal entries, has an $N_0$-matrix completion if the graph of its specified entries is a 1-chordal graph or a cycle. We also analyze the mentioned problem when the partial matrix has some null main diagonal entries.Fundação para a Ciência e a Tecnologia(FCT) através do programa POCTISpanish DGI grant number MTM2007-6447