On a Class of Self-Adjoint Compact Operators in Hilbert Spaces and Their Relations with Their Finite-Range Truncations

This paper investigates a class of self-adjoint compact operators in Hilbert spaces related to their truncated versions with finite-dimensional ranges. The comparisons are established in terms of worst-case norm errors of the composite operators generated from iterated computations. Some boundedness properties of the worst-case norms of the errors in their respective fixed points in which they exist are also given. The iterated sequences are expanded in separable Hilbert spaces through the use of numerable orthonormal bases.

Fixed Points of Closed and Compact Composite Sequences of Operators and Projectors in a Class of Banach Spaces

Some results on fixed points related to the contractive compositions of bounded operators in a class of complete metric spaces which can be also considered as Banach's spaces are discussed through the paper. The class of composite operators under study can include, in particular, sequences of projection operators under, in general, oblique projective operators. In this paper we are concerned with composite operators which include sequences of pairs of contractive operators involving, in general, oblique projection operators. The results are generalized to sequences of, in general, nonconstant bounded closed operators which can have bounded, closed, and compact limit operators, such that the relevant composite sequences are also compact operators. It is proven that in both cases, Banach contraction principle guarantees the existence of unique fixed points under contractive conditions.

Combined Fluorescent-Chromogenic In Situ Hybridization for Identification and Laser Microdissection of Interphase Chromosomes

Chromosome territories constitute the most conspicuous feature of nuclear architecture, and they exhibit non-random distribution patterns in the interphase nucleus. We observed that in cell nuclei from humans with Down Syndrome two chromosomes 21 frequently localize proximal to one another and distant from the third chromosome. To systematically investigate whether the proximally positioned chromosomes were always the same in all cells, we developed an approach consisting of sequential FISH and CISH combined with laser-microdissection of chromosomes from the interphase nucleus and followed by subsequent chromosome identification by microsatellite allele genotyping. This approach identified proximally positioned chromosomes from cultured cells, and the analysis showed that the identity of the chromosomes proximally positioned varies. However, the data suggest that there may be a tendency of the same chromosomes to be positioned close to each other in the interphase nucleus of trisomic cells. The protocol described here represents a powerful new method for genome analysis