Prime and semiprime inner functions

Abstract This paper studies the structure of inner functions under the operation of composition, and in particular the notions or primeness and semiprimeness. Results proved include the density of prime finite Blaschke products in the set of finite Blaschke products, the semiprimeness of finite products of thin Blaschke products and their approximability by prime Blaschke products. An example of a nonsemiprime Blaschke product that is a Frostman Blaschke product is also provided.

Equality of P-partition Generating Functions

To every partially ordered set (poset), one can associate a generating function, known as the P-partition generating function. We find necessary conditions and sufficient conditions for two posets to have the same P-partition generating function. We define the notion of a jump sequence for a labeled poset and show that having equal jumpsequences is a necessary condition for generating function equality. We also develop multiple ways of modifying posets that preserve generating function equality. Finally, we are able to give a complete classification of equalities among partially ordered setswith exactly two linear extensions.

Bone morphogenetic protein-7 is a MYC target with prosurvival functions in childhood medulloblastoma

Medulloblastoma (MB) is the most common malignant brain tumor in children. It is known that overexpression and/or amplification of the MYC oncogene is associated with poor clinical outcome, but the molecular mechanisms and the MYC downstream effectors in MB remain still elusive. Besides contributing to elucidate how progression of MB takes place, most importantly, the identification of novel MYC-target genes will suggest novel candidates for targeted therapy in MB. A group of 209 MYC-responsive genes was obtained from a complementary DNA microarray analysis of a MB-derived cell line, following MYC overexpression and silencing. Among the MYC-responsive genes, we identified the members of the bone morphogenetic protein (BMP) signaling pathway, which have a crucial role during the development of the cerebellum. In particular, the gene BMP7 was identified as a direct target of MYC. A positive correlation between MYC and BMP7 expression was documented by analyzing two distinct sets of primary MB samples. Functional studies in vitro using a small-molecule inhibitor of the BMP/SMAD signaling pathway reproduced the effect of the small interfering RNA-mediated silencing of BMP7. Both approaches led to a block of proliferation in a panel of MB cells and to inhibition of SMAD phosphorylation. Altogether, our findings indicate that high MYC levels drive BMP7 overexpression, promoting cell survival in MB cells. This observation suggests the potential relevance of targeting the BMP/SMAD pathway as a novel therapeutic approach for the treatment of childhood MB.

Assessment of post-stroke fatigue: the fatigue scale for motor and cognitive functions

Post-stroke fatigue (PSF) is an important but still controversial issue since knowledge on its nature is still humble. The aim of the present study was to characterize PSF beyond the subacute phase.

Putative functions of extracellular matrix glycoproteins in secondary palate morphogenesis

Cleft palate is a common birth defect in humans. Elevation and fusion of paired palatal shelves are coordinated by growth and transcription factors, and mutations in these can cause malformations. Among the effector genes for growth factor signaling are extracellular matrix (ECM) glycoproteins. These provide substrates for cell adhesion (e.g., fibronectin, tenascins), but also regulate growth factor availability (e.g., fibrillins). Cleft palate in Bmp7 null mouse embryos is caused by a delay in palatal shelf elevation. In contrast, palatal shelves of Tgf-β3 knockout mice elevate normally, but a cleft develops due to their failure to fuse. However, nothing is known about a possible functional interaction between specific ECM proteins and Tgf-β/Bmp family members in palatogenesis. To start addressing this question, we studied the mRNA and protein distribution of relevant ECM components during secondary palate development, and compared it to growth factor expression in wildtypewild type and mutant mice. We found that fibrillin-2 (but not fibrillin-1) mRNA appeared in the mesenchyme of elevated palatal shelves adjacent to the midline epithelial cells, which were positive for Tgf-β3 mRNA. Moreover, midline epithelial cells started expressing fibronectin upon contact of the two palatal shelves. These findings support the hypothesis that fibrillin-2 and fibronectin are involved in regulating the activity of Tgf-β3 at the fusing midline. In addition, we observed that tenascin-W (but not tenascin-C) was misexpressed in palatal shelves of Bmp7-deficient mouse embryos. In contrast to tenascin-C, tenascin-W secretion was strongly induced by Bmp7 in embryonic cranial fibroblasts in vitro. These results are consistent with a putative function for tenascin-W as a target of Bmp7 signaling during palate elevation. Our results indicate that distinct ECM proteins are important for morphogenesis of the secondary palate, both as downstream effectors and as regulators of Tgf-β/Bmp activity.

Intestinal microbes affect phenotypes and functions of invariant natural killer T cells in mice

Invariant natural killer T (iNKT) cells undergo canonical, Vα14-Jα18 rearrangement of the T-cell receptor (TCR) in mice; this form of the TCR recognizes glycolipids presented by CD1d. iNKT cells mediate many different immune reactions. Their constitutive activated and memory phenotype and rapid initiation of effector functions after stimulation indicate previous antigen-specific stimulation. However, little is known about this process. We investigated whether symbiotic microbes can determine the activated phenotype and function of iNKT cells.

Neural progenitor cells regulate microglia functions and activity

We found mouse neural progenitor cells (NPCs) to have a secretory protein profile distinct from other brain cells and to modulate microglial activation, proliferation and phagocytosis. NPC-derived vascular endothelial growth factor was necessary and sufficient to exert at least some of these effects in mice. Thus, neural precursor cells may not only be shaped by microglia, but also regulate microglia functions and activity.

Comparing skew Schur functions: a quasisymmetric perspective

Reiner, Shaw and van Willigenburg showed that if two skew Schur functions sA and sB are equal, then the skew shapes $A$ and $B$ must have the same "row overlap partitions." Here we show that these row overlap equalities are also implied by a much weaker condition than Schur equality: that sA and sB have the same support when expanded in the fundamental quasisymmetric basis F. Surprisingly, there is significant evidence supporting a conjecture that the converse is also true. In fact, we work in terms of inequalities, showing that if the F-support of sA contains that of sB, then the row overlap partitions of A are dominated by those of B, and again conjecture that the converse also holds. Our evidence in favor of these conjectures includes their consistency with a complete determination of all F-support containment relations for F-multiplicity-free skew Schur functions. We conclude with a consideration of how some other quasisymmetric bases fit into our framework.