RESUMO
We consider the number of ways to write an integer as a sum of squares, a problem with a long history going back at least to Fermat. The previous studies in this area generally fix the number of squares which may occur and then either use algebraic techniques or connect these to coefficients of certain complex analytic functions with many symmetries known as modular forms, from which one may use techniques in complex and real analysis to study these numbers. In this paper, we consider sums with arbitrarily many squares, but give a certain natural weighting to each representation. Although there are a very large number of such representations of each integer, we see that the weighting induces massive cancellation, and we furthermore prove that these weighted sums are again coefficients of modular forms, giving precise formulas for them in terms of sums of divisors of the integer being represented.
RESUMO
DPP4 (dipeptidyl peptidase-4), a highly conserved transmembrane glycoprotein with an exo-peptidase activity, has been shown to contribute to glucose metabolism, immune regulation, signal transduction, and cell differentiation. Here, we show that DPP4 is involved in control of activin/nodal signaling in Xenopus early development. In support of this, gain of function of DPP4 augmented Smad2 phosphorylation as well as expression of target genes induced by activin or nodal signal. In addition, Dpp4 and Xnr1 showed synergistic effect on induction of ectopic dorsal body axis, when co-injected at suboptimal doses in early embryos. Conversely, saxagliptin, a DPP4 inhibitor repressed activin induction of Smad2 phosphorylation. Notably, overexpression of Dpp4 disrupted specification of dorsal body axis of embryo, leading to malformed phenotypes such as spina bifida and a shortened and dorsally bent axis. Together, these results suggest that DPP4 functions as a potentiator of activin/nodal signaling pathway. [BMB Reports 2018; 51(12): 636-641].
