The representation dimension of domestic weakly symmetric algebras

Auslander’s representation dimension measures how far a finite dimensional algebra is away from being of finite representation type. In [1], M. Auslander proved that a finite dimensional algebra A is of finite representation type if and only if the representation dimension of A is at most 2. Recently, R. Rouquier proved that there are finite dimensional algebras of an arbitrarily large finite representation dimension. One of the exciting open problems is to show that all finite dimensional algebras of tame representation type have representation dimension at most 3. We prove that this is true for all domestic weakly symmetric algebras over algebraically closed fields having simply connected Galois coverings.     

Approximation,solution operators and quantale-valued metrics

A generalized solution operator is a mapping abstractly describing a computational problem and its approximate solutions. It assigns a set of \(\varepsilon \)-approximations of a solution to the problem instance f and accuracy of approximation \(\varepsilon \). In this paper we study generalized solution operators for which the accuracy of approximation is described by elements of a complete lattice equipped with a compatible monoid structure, namely, a quantale. We provide examples of computational problems for which the accuracy of approximation of a solution is measured by such objects. We show that the sets of \(\varepsilon \)-approximations are, roughly, closed balls with radii \(\varepsilon \) with respect to a certain family of quantale-valued generalized metrics induced by a generalized solution operator.     

On the transposition anti-involution in real Clifford algebras I: the transposition map

A particular orthogonal map on a finite-dimensional real quadratic vector space (V,?Q) with a non-degenerate quadratic form Q of any signature (p,?q) is considered. It can be viewed as a correlation of the vector space that leads to a dual Clifford algebra C?(V*,?Q) of linear functionals (multiforms) acting on the universal Clifford algebra C?(V,?Q). The map results in a unique involutive automorphism and a unique involutive anti-automorphism of C?(V,?Q). The anti-involution reduces to reversion (resp. conjugation) for any Euclidean (resp. anti-Euclidean) signature. When applied to a general element of the algebra, it results in transposition of the element matrix in the left regular representation of C?(V,?Q). We also give an example for real spinor spaces. The general setting for spinor representations will be treated in part II of this work [R. Ab?amowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras II: Stabilizer groups of primitive idempotents, Linear Multilinear Algebra, to appear].     

Noncommuting Domination in Krein Spaces Via Commutators of Block Operator Matrices

The commutators of 2 × 2 block operator matrices with (unbounded) operator entries are investigated. The matrix representation of a symmetric operator in a Krein space is exploited. As a consequence, the domination result due to Cichoń, Stochel and Szafraniec is extended to the case of Krein spaces.     

Supravital fluorometric apoptosis detection in a single mouse embryo using lab-on-a-chip

Detection of apoptosis is one of the main criteria of preimplantation embryo growth potential assessment. Recent developments in lab-on-a-chip techniques has led to apoptosis detection and monitoring on a single cell or embryo level. However, single embryo apoptosis detection without a change in embryo developmental competence and post-examination "recovery" still remains a challenge. In this paper we present a lab-on-a-chip, co-working with miniaturized optical instrumentation, which allows supravital examination of single embryos for the presence of apoptotic blastomers with full after lab-on-a-chip study "recovery" and maintenance of their further developmental capacity.