Glória Cravo

Eigenvalues of matrices with several prescribed blocks, II

Abstract A previous paper by D. Hershkowitz [Linear and Multilinear Algebra 14 (1983) 315–342] described the possible eigenvalues of an n×n matrix when 2n−3 entries are fixed and the others vary. This paper describes the possible eigenvalues of a pk×pk matrix, partitioned into k×k blocks of size p×p when 2k−3 blocks are fixed and the others vary.

Characteristic polynomials and controlability of partially prescribed matrices

Abstract In a previous paper it was proved that n−1 arbitrary entries and the characteristic polynomial of a n×n matrix over a field F can be arbitrarily prescribed, except if all the nonprincipal entries of a row or column are prescribed equal to zero and the characteristic polynomial does not have a root in F. This paper describes the possible characteristic polynomials of a pk×pk matrix, partitioned into k×k blocks of size p×p when k−1 blocks are fixed and the others vary. It also studies the possibility of a pair of matrices (A 1 ,A 2 ), where A 1 is square and [ A 1 A 2 ] is partitioned into k×(k+1) blocks of size p×p, being completely controllable when some of the blocks are prescribed and the others vary.

Applications of propositional logic to workflow analysis

Abstract In this paper our main goal is to describe the structure of workflows. A workflow is an abstraction of a business process that consists of one or more tasks to be executed to reach a final objective. In our approach we describe a workflow as a graph whose vertices represent workflow tasks and the arcs represent workflow transitions. Moreover, every arc ( t k , t l ) (i.e., a transition) has attributed a Boolean value to specify the execution/non-execution of tasks t k , t l . With this attribution we are able to identify the natural flow in the workflow. Finally, we establish a necessary and sufficient condition for the termination of workflows. In other words, we identify conditions under which a business process will be complete.

Matrices with Prescribed Eigenvalues and Blocks, II

Let F be a field and let n, p1, p2, p3 be positive integers such that n=p1+p2+p3. Let where the blocks Ci,j are of type pi× pj (i, j∈ {1,2,3}) and C1,1, C2,2, C3,3 are square submatrices. In this paper we describe conditions for which it is possible to prescribe arbitrarily the eigenvalues of C, when three arbitrary positions are prescribed and the remaining are free.

The number of nonconstant invariant polynomials of matrices with several prescribed blocks

Abstract Consider a n × n matrix partitioned into k × k blocks: C =[ C i , j ], where C 1,1 ,…, C k , k are square. This paper studies the possible numbers of nonconstant invariant polynomials of C when a diagonal of blocks C i , j is fixed and the others vary.

Workflow Modelling and Analysis Based on the Construction of Task Models

We describe the structure of a workflow as a graph whose vertices represent tasks and the arcs are associated to workflow transitions in this paper. To each task an input/output logic operator is associated. Furthermore, we associate a Boolean term to each transition present in the workflow. We still identify the structure of workflows and describe their dynamism through the construction of new task models. This construction is very simple and intuitive since it is based on the analysis of all tasks present on the workflow that allows us to describe the dynamism of the workflow very easily. So, our approach has the advantage of being very intuitive, which is an important highlight of our work. We also introduce the concept of logical termination of workflows and provide conditions under which this property is valid. Finally, we provide a counter-example which shows that a conjecture presented in a previous article is false.

A Note on Matrix Completion Problems

Matrix completion problems are an important subclass of problems in matrix theory. An important question in matrix completion problems was posed by Oliveira in 1975, where the author proposed the description of the characteristic polynomial of a partitioned matrix of the form A = [Ai,j], i, j ∈ {1,2} (whose entries are in a field and A1,1, A2,2 are square submatrices), when some of the blocks Ai,j are prescribed and the others are unknown. The analysis of this problem gave rise to several subproblems, according to the location of the prescribed submatrices. Many authors have considered this list of subproblems. In this note we provide a new proof of a result obtained by Oliveira inserted in this class of subproblems.

Controllability of partially prescribed matrices

AbstractLetF be an infinite field and letn,p1,p2,p3 be positive integers such thatn =p1 +p2 +p3. Let

Verifying the Logical Termination of Workflows

C_{1,2} \in F^{p_1 \times p_2 }