Eduardo Marques de Sá

Matrix realizations of littlewood—richardson sequences

In this paper we consider the problem of characterizing the invariant factors of three matrices A B, and C, such that AB — C Our matrices have entries over a principal ideal domain or over a local domain. In Section 2 we show that this problem is localizablc The above problem lias a well-known solution in terms of Littlewood-Richardson sequences. We introduce the concept of a matrix realization of a Littlewood-Richardson sequence. The main result is an explicit construction of a sequence of matrices which realizes a previously given Littlewood Richardson sequence. Our methods offer a matrix theoretical proof of a well-known result of T, Klein on extensions of p-modules.

Exposed faces and duality for symmetric and unitarily invariant norms

Abstract Let ψ be a unitarily invariant norm on the space of (real or complex) n×m matrices, and g the associated symmetric gauge function: thus ψ(A)g(s(A)), where s(A) is the decreasing sequence of singular values of A. Denote by Bψ and Bg the closed unit balls of ψ and g. In a previous paper we showed a close relationship between the faces of Bψ and those of Bg. In particular, to each face of Bψ we associated a standard face of Bg, and we used this association to completely describe the matrices that are members of an individual face of ψ. In the present paper, we consider the duality operator that transforms each exposed face of Bψ into an exposed face of the unit ball of ψs dual and the duality operator that does the same with exposed faces of Bg. We show that these two operators are very nicely related. Among other results, we prove the following. Given an exposed face E of Bψ, the standard face associated with the dual of E is precisely the dual of the standard face of Bg associated with E . E is an exposed face of Bψ if and only if its associated standard face is an exposed face of Bg. As a by-product we completely determine the subdifferential of ψ in terms of the subdifferential of g, and we completely characterize the matrices that are dual to a given matrix A with respect to ψ.

SINGULAR VALUES AND INVARIANT FACTORS OF MATRIX SUMS AND PRODUCTS

Abstract Some remarks are made concerning the range of individual singular values of sums and products of two complex square matrices, when these are allowed to vary in their orbits under unitary equivalence. An analogous question is considered for invariant factors of products of matrices over a principal ideal domain.

Faces of the unit ball of a unitarily invariant norm

Abstract Let ψ be a unitarily invariant norm on the space of all n×m complex matrices, and let g: R n→ R be the symmetric gauge function associated with ψ. That is to say, we have ψ(A)g(s(A)) for any A, where s(A) is the nonincreasing sequence of singular values of A. In this article we consider the relationship between the facial structures of the closed unit balls of g and ψ, which we denote by Bg and Bψ. Our main result gives a complete characterization of the faces of Bψ in terms of the faces of Bg. For, with each face E of Bψ we associate a standard face F E of Bg (i.e.,F E is a face of Bg whose barycenter is a nonnegative, nonincreasing n-vector). Conversely, each standard face F of Bg originates a set of n×m matrices, say the set M F, which is proven to be a face of Bψ. Our main result also asserts that any face E of Bψ is of the form Q M FR, where Q and R are unitary matrices of appropriate orders and FF E . We then explore the main result and draw some of its consequences. For example, we show that the singular values of the barycenter of a face E of Bψ have the same multiplicities as the barycenter of F E . Formulas are given which relate the dimensions of E and F E . The relative interior and the relative boundary of E are characterized in terms of the relative interior and the relative boundary of F E . We completely describe the face of Bψ generated by a matrix A. Our results also cover the case of real matrices.

Faces and traces of the unit ball of a symmetric gauge function

Abstract We investigate the faces of the closed unit ball, Bg≔{x:g(x)⩽1}, of a symmetric gauge function g: R n→ R +. Firstly, we describe the group of symmetries of a face F of Bg, in terms of its barycenter. We introduce the concept of standard face of Bg, i.e. a face whose relative interior meets the set D + ≔{x∈ R n :x 1 ⩾⋯⩾x n ⩾0}, called here the standard cone of R n, and the concept of trace of a face F, which is the intersection of F with D +. Characterizations of standard faces are given, and we show that the trace of a face (if nonempty) is the trace of a standard face. It is proved that a face F and its trace have the same dimension iff F is a standard face; a relation is found between the dimension of F and the dimension of its symmetry core (i.e. the set of points of F that are fixed by all symmetries of F). Other results of this type are given. The main result has a long and delicate proof. It completely characterizes, in several different ways, the subsets of Bg that are traces of faces of Bg.

Violence during pregnancy and its effects on mother–baby relationship during pregnancy

Background: Human beings have an innate need to form close emotional bonds with significant others. Objective: The purpose of this research was to study the effect of domestic violence during pregnancy in the mother–infant relationship. Method: 204 pregnant outpatients (of the Obstetrics and Gynecology Department of the Hospital Pedro Hispano, Portugal) with a mean age of 29 and in their last 3 months of pregnancy were the participants of this study. To assess the violence level during pregnancy, we used the Conflict Tactic Scale 2. The mother–baby relationship was determined using the Maternal Foetal Attachment Scale and the Maternal Adjustment and Maternal Attitudes questionnaire. Results: In our work, we encountered 107 pregnant victims (52.4% of all outpatients) of domestic violence. The results showed that victims of domestic violence had a poor relationship with the foetus (M = 97.4, SD = 8.1), but not to the point of statistical significance. In these same patients, we also detected minor adjustments and attitude shifts towards pregnancy (M = 117, SD = 14.2) and this result was statistically significant when compared with women who did not experience domestic violence. Conclusion: Our results suggest that women who are victims of domestic violence are more likely to have a lower attachment with the foetus when compared with women who do not suffer domestic violence. Additionally, the victims showed more negative attitudes towards pregnancy and the foetus when compared to non-abused women.

Interlacing and degree conditios for invariat polynomials

It is well known that the invariant factors of a polynomial matrix interlace with ihose of one of its submatrices. If we restrict the degrees of the entries of our matrices in some prescribed way then the invariant factors of such matrices and submatrices satisfy other conditions that may be predicted and, in some cases, even completely described. That is the kind of problem we deal with Our main result is a sort of interlacing theorem for row proper matrices. Convexity and majorization also enter into play.

Common mental disorders during pregnancy and baby’s development in the first year of life

Background: Evidence shows that pregnancy and early postpartum periods are crucial to the development of the mother–baby relationship. Objective: The aim of this study was to evaluate the impact of common mental disorders (CMD) during pregnancy on child’s mental development during the first year. A prospective study was carried out with 204 pregnant women in the third trimester of pregnancy and continuing with their babies to 3.5 and 12 months of age. Method: To assess the presence of CMD, the Brief Symptom Inventory and the Inventory of the Clinical Evaluation of Depression were used. Evaluation of the babies’ mental development and the socio-emotional state was carried out using the Griffiths Mental Scale (0–2) and the Brief Infant Toddler Social and Emotional Assessment (BITSEA). Results: We observed 20 babies born to women with a positive diagnosis for CMD and who presented a positive screen in the BITSEA. We also observed a statistically significant relationship regarding the diminished development in certain Griffith’s subscales of babies whose mothers showed presence of psychotic, anxiety, hostility and depressive symptoms during pregnancy. Conclusion: We conclude that the presence of CMD influences the mental, social and emotional development levels of infants in their first year.

Generalized singular values, interlacing inequalities, and monotonic norms

Abstract We complete the results of an earlier paper by Sodupe, where interlacing properties for generalized singular values of matrices and submatrices are studied. We show that there is a close relationship between these interlacing properties and the ∗orthant-monotonicity of the norms involved in the definition of the generalized singular values. We briefly discuss interlacing properties for other kinds of a s-numbers.

THE STABILITY OF THE UNIT BALLS OF SYMMETRIC AND UNITARILY INVARIANT NORMS

Abstract A compact convex set K is called stable if the midpoint mapping, K × K → K, (x, y) → (x + y) 2 , is open. The main result asserts that the stability of the closed unit ball of a unitarily invariant norm is equivalent to the stability of the closed unit ball of the associated symmetric gauge function. This result, as well as other pointwise related results, are obtained using a recently found close relationship between the facial structures of those two kinds of unit balls.