On the Convergence of Ritz Pairs and Refined Ritz Vectors for Quadratic Eigenvalue Problems

For a given subspace, the Rayleigh-Ritz method projects the large quadratic eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar to the Rayleigh-Ritz method for the linear eigenvalue problem, the Rayleigh-Ritz method defines the Ritz values and the Ritz vectors of the QEP with respect to the projection subspace. We analyze the convergence of the method when the angle between the subspace and the desired eigenvector converges to zero. We prove that there is a Ritz value that converges to the desired eigenvalue unconditionally but the Ritz vector converges conditionally and may fail to converge. To remedy the drawback of possible non-convergence of the Ritz vector, we propose a refined Ritz vector that is mathematically different from the Ritz vector and is proved to converge unconditionally. We construct examples to illustrate our theory.Comment: 20 page

Genetic signs of multiple colonization events in Baltic ciscoes with radiation into sympatric spring- and autumn-spawners confined to early postglacial arrival

Presence of sympatric populations may reflect local diversification or secondary contact of already distinct forms. The Baltic cisco (Coregonus albula) normally spawns in late autumn, but in a few lakes in Northern Europe sympatric autumn and spring- or winter-spawners have been described. So far, the evolutionary relationships and taxonomic status of these main life history forms have remained largely unclear. With microsatellites and mtDNA sequences, we analyzed extant and extinct spring- and autumn-spawners from a total of 23 Swedish localities, including sympatric populations. Published sequences from Baltic ciscoes in Germany and Finland, and Coregonus sardinella from North America were also included together with novel mtDNA sequences from Siberian C.sardinella. A clear genetic structure within Sweden was found that included two population assemblages markedly differentiated at microsatellites and apparently fixed for mtDNA haplotypes from two distinct clades. All sympatric Swedish populations belonged to the same assemblage, suggesting parallel evolution of spring-spawning rather than secondary contact. The pattern observed further suggests that postglacial immigration to Northern Europe occurred from at least two different refugia. Previous results showing that mtDNA in Baltic cisco is paraphyletic with respect to North American C.sardinella were confirmed. However, the inclusion of Siberian C.sardinella revealed a more complicated pattern, as these novel haplotypes were found within one of the two main C.albula clades and were clearly distinct from those in North American C.sardinella. The evolutionary history of Northern Hemisphere ciscoes thus seems to be more complex than previously recognized

Functional interplay between p63 and p53 controls RUNX1 function in the transition from proliferation to differentiation in human keratinocytes

The interfollicular epidermis is continuously renewed, thanks to a regulated balance between proliferation and differentiation. The ΔNp63 transcription factor has a key role in the control of this process. It has been shown that ΔNp63 directly regulates Runt-related transcription factor 1 (RUNX1) transcription factor expression in mouse keratinocytes. The present study showed for the first time that RUNX1 is expressed in normal human interfollicular epidermis and that its expression is tightly regulated during the transition from proliferation to differentiation. It demonstrated that ΔNp63 directly binds two different RUNX1 regulatory DNA sequences and modulates RUNX1 expression differentially in proliferative or differentiated human keratinocytes. It also showed that the regulation of RUNX1 expression by ΔNp63 is dependent on p53 and that this coregulation relies on differential binding and activation of RUNX1 regulatory sequences by ΔNp63 and p53. We also found that RUNX1 inhibits keratinocyte proliferation and activates directly the expression of KRT1, a critical actor in early keratinocyte differentiation. Finally, we described that RUNX1 expression, similar to ΔNp63 and p53, was strongly expressed and downregulated in basal cell carcinomas and squamous cell carcinomas respectively. Taken together, these data shed light on the importance of tight control of the functional interplay between ΔNp63 and p53 in regulating RUNX1 transcription factor expression for proper regulation of interfollicular epidermal homeostasis

The human keratins: biology and pathology

The keratins are the typical intermediate filament proteins of epithelia, showing an outstanding degree of molecular diversity. Heteropolymeric filaments are formed by pairing of type I and type II molecules. In humans 54 functional keratin genes exist. They are expressed in highly specific patterns related to the epithelial type and stage of cellular differentiation. About half of all keratins—including numerous keratins characterized only recently—are restricted to the various compartments of hair follicles. As part of the epithelial cytoskeleton, keratins are important for the mechanical stability and integrity of epithelial cells and tissues. Moreover, some keratins also have regulatory functions and are involved in intracellular signaling pathways, e.g. protection from stress, wound healing, and apoptosis. Applying the new consensus nomenclature, this article summarizes, for all human keratins, their cell type and tissue distribution and their functional significance in relation to transgenic mouse models and human hereditary keratin diseases. Furthermore, since keratins also exhibit characteristic expression patterns in human tumors, several of them (notably K5, K7, K8/K18, K19, and K20) have great importance in immunohistochemical tumor diagnosis of carcinomas, in particular of unclear metastases and in precise classification and subtyping. Future research might open further fields of clinical application for this remarkable protein family