Orthogonalities and functional equations

In this survey we show how various notions of orthogonality appear in the theory of functional equations. After introducing some orthogonality relations, we give examples of functional equations postulated for orthogonal vectors only. We show their solutions as well as some applications. Then we discuss the problem of stability of some of them considering various aspects of the problem. In the sequel, we mention the orthogonality equation and the problem of preserving orthogonality. Last, but not least, in addition to presenting results, we state some open problems concerning these topics. Taking into account the big amount of results concerning functional equations postulated for orthogonal vectors which have appeared in the literature during the last decades, we restrict ourselves to the most classical equations

Regulation of monocyte induced cell migration by the RNA binding protein, FXR1

<p>FXR1 belongs to a family of RNA-binding proteins that play critical roles in post-transcriptional regulation of gene expression in immunity, development and cancer. FXR1 is associated with regulation of specific mRNAs in myocytes and macrophages. In quiescent cells (> 24 h of extended serum-starvation, ∼30-48 h or more), a spliced isoform of FXR1, FXR1a, promotes translation of the cytokine TNFα, independent of the effects of RNA levels. Here we examined the role of FXR1 in THP1 human monocytic leukemic cells that were grown in serum, as well as in early (24 h) serum-starvation conditions that demonstrates differences in gene expression mechanisms and is distinct from quiescent (> 24 h extended serum-starvation) cells. Global RNA profiling, conducted to investigate the role of FXR1 on mRNA levels, revealed that FXR1 affects levels of specific mRNAs in serum-grown and in early 24 h serum-starvation conditions. FXR1 decreases levels of several mRNAs, including as previously identified, CDKN1A (p21CIP1 or p21) mRNA in serum-grown cells. Interestingly, we find that FXR1 positively regulates mRNA levels of specific cytokines and chemokines in serum-grown and in early 24 h serum-starvation conditions. These include IL1β and CCL2 that control cell migration. Accordingly, depletion and overexpression of FXR1 decreased and increased levels of CCL2 mRNA. Consistent with the reduced levels of IL1β, CCL2 and other chemokines upon FXR1 depletion, our data reveal that depletion of FXR1 decreases the ability of these cells to induce cell migration of neighboring monocytic cells. These data reveal a new role of FXR1 in controlling induction of monocyte migration.</p

Non-polyconvexity of the stored energy function of a Saint Venant-Kirchhoff material

summary:A direct proof of the non-polyconvexity of the stored energy function of a Saint Venant-Kirchhoff material is given by means of a simple counter-example