Jan Hauke

A Further Algebraic Version of Cochran's Theorem and Matrix Partial Orderings*

Abstract A new version of Cochrans theorem for rectangular matrices is established. Being oriented toward partial isometries, the new version parallels corresponding results concerned with arbitrary tripotent matrices and covers results concerned with Hermitian tripotent matrices. A discussion of a related new matrix partial ordering is also given.

Relationships between partial orders of matrices and their powers

Abstract Baksalary and Pukelsheim [Linear Algebra Appl. 151 (1991) 135] considered the problem of how an order between two Hermitian nonnegative definite matrices A and B is related to the corresponding order between the squares A 2 and B 2 , in the sense of the star partial ordering, the minus partial ordering, and the Lowner partial ordering. In the present paper, possibilities of generalizing and strengthening their results are studied from two points of view: by widening the class of matrices considered and by replacing the squares by arbitrary powers.

Partial orderings of matrices referring to singular values or eigenvalues

Abstract A new partial ordering in the set of complex matrices is defined, which is weaker than the star ordering introduced by Drazin in 1978 and stronger than the minus ordering introduced by Hartwig in 1980. This ordering refers to singular values of matrices, and the interest in it was generated by canonical interpretations of the minus and star orderings, given by Hartwig and Styan in 1986. For Hermitian matrices, a similar ordering referring to eigenvalues is also considered, and its connection with a problem concerning distributions of quadratic forms in normal variables is pointed out.

Comparing the BLUEs Under Two Linear Models

In this article, we consider two linear models, ℳ1 = {y, X β, V 1} and ℳ2 = {y, X β, V 2}, which differ only in their covariance matrices. Our main focus lies on the difference of the best linear unbiased estimators, BLUEs, of X β under these models. The corresponding problems between the models {y, X β, I n } and {y, X β, V}, i.e., between the OLSE (ordinary least squares estimator) and BLUE, are pretty well studied. Our purpose is to review the corresponding considerations between the BLUEs of X β under ℳ1 and ℳ2. This article is an expository one presenting also new results.

Evolution of Regional Disparities in Poland

Evolution of Regional Disparities in Poland The article presents the state of and changes in the pattern of regional disparities in Poland over the years 1995-2007. The differences in the level of economic development of regions are examined in a dynamic approach on the basis of per capita income, indices of local government finances, and investment outlays. The effect of development-activating factors on regional income is considered. An answer is sought to the question of whether the growth dynamics had a favourable effect on the evolution of regional disparities in Poland.

Inter- and extrapolatory properties of matrix partial orderings

In this paper we study some matrix partial orderings ⪯? with respect to the preserving the relation A⪯?B for original A or/and B and the linear combinations of A and B. The property that “between” (“above”, respectively) A and (“below”, respectively) B one can locate convex (affine) combinations of original matrices will be called interpolatory (extrapolatory, respectively). Besides “classical” orderings we introduce and discuss in the mentioned aspect the notion of f-orderings.

On partial orderings on the set of rectangular matrices

A new partial ordering defined on the set of rectangular matrices is investigated. Its characterization and comparison with Loewner and star orderings are presented.

Partial orderings, preorderings, and the polar decomposition of matrices

Abstract When a unique decomposition of a complex rectangular matrix into two components is given, then a partial ordering on the domain of one of the components induces a preordering on the whole set of matrices, and partial orderings on the domains of both components induce a partial ordering on the whole set. In this not we consider the three main partial orderings (Lo¨wner, star, and rank subtractivity) on the respective domains of the two components of matrices subjected to polar decomposition ( A. Ben-Israel. T.N.E. Greville, Generalized Inverses: Theory and Applications, Wiley, New York, 1974 , p. 255) and investigate the resulting pre- and partial orderings.

Manual vs. computer-assisted sperm analysis: can CASA replace manual assessment of human semen in clinical practice?

OBJECTIVES The aim of the study was to check the quality of computer-assisted sperm analysis (CASA) system in comparison to the reference manual method as well as standardization of the computer-assisted semen assessment. MATERIAL AND METHODS The study was conducted between January and June 2015 at the Andrology Laboratory of the Division of Infertility and Reproductive Endocrinology, Poznań University of Medical Sciences, Poland. The study group consisted of 230 men who gave sperm samples for the first time in our center as part of an infertility investigation. The samples underwent manual and computer-assisted assessment of concentration, motility and morphology. A total of 184 samples were examined twice: manually, according to the 2010 WHO recommendations, and with CASA, using the program set-tings provided by the manufacturer. Additionally, 46 samples underwent two manual analyses and two computer-assisted analyses. The p-value of p < 0.05 was considered as statistically significant. RESULTS Statistically significant differences were found between all of the investigated sperm parameters, except for non-progressive motility, measured with CASA and manually. In the group of patients where all analyses with each method were performed twice on the same sample we found no significant differences between both assessments of the same probe, neither in the samples analyzed manually nor with CASA, although standard deviation was higher in the CASA group. CONCLUSIONS Our results suggest that computer-assisted sperm analysis requires further improvement for a wider application in clinical practice.

Comparison of experiments via a group majorization ordering

This paper generalizes results of Tong (1989) and Shaked and Tong (1992) on comparison of experiments via dependence of normal variables with a common marginal distribution.