Interesting identities involving weighted representations of integers as sums of arbitrarily many squares.

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.

Role of dipeptidyl peptidase-4 as a potentiator of activin/nodal signaling pathway.

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.