We compute the second variation of the λ-invariant, recently introduced by S. Zhang, on the complex moduli space Mg of curves of genus g ≥ 2, using work of N. Kawazumi. As a result we prove that (8g + 4)λ is equal, up to a constant, to the β-invariant introduced some time ago by R. Hain and D. Reed. We deduce some consequences; for example we calculate the λ-invariant for each hyperelliptic curve, expressing it in terms of the Petersson norm of the discriminant modular form.

Mobile apps are very popular. However, this is not true for every app, with some apps receiving millions of downloads, while other apps are mostly ignored. We investigate the popularity of apps in terms of downloads by focusing on two salient cues: (a) online recommendations (e.g., presence and valence of online reviews) and (b) visual characteristics of app icons (e.g., use of visual metaphors and anthropomorphism). Study 1 was a field study in which we content-analyzed 500 apps from the "transportation" subcategory of the Google Play Store. We found that the presence and valence of online reviews, as well as the presence of visual metaphors in app icons were positively related to the number of app downloads. Study 2 was an experiment in which we presented participants with different app icons containing different types of visual metaphors. We again found that app icons with visual metaphors led to more positive attitudes

Around 2008 N. Kawazumi and S. Zhang introduced a new fundamental numerical invariant for compact Riemann surfaces. One way of viewing the Kawazumi-Zhang invariant is as a quotient of two natural hermitian metrics with the same first Chern form on the line bundle of holomorphic differentials. In this paper we determine precise formulas, up to and including constant terms, for the asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces. As a corollary we state precise asymptotic formulas for the beta-invariant introduced around 2000 by R. Hain and D. Reed. These formulas are a refinement of a result Hain and Reed prove in their paper. We illustrate our results with some explicit calculations on degenerating genus two surfaces. ℓ>0 2 λ ℓ h m,n=1 M φ ℓ ω mωn 2 2010 Mathematics Subject Classification. Primary 14H15, secondary 14D06, 32G20.

On donne une borne supérieure explicite en fonction de K , S, g pour la hauteur de Faltings de la jacobienne d'une courbe C de genre g, définie sur un corps de nombres K et ayant bonne réduction en dehors d'un ensemble fini S de places de K , pourvu que C puisse s'écrire comme un revêtement cyclique de degré premier de la droite projective. La preuve repose sur le fait que les birapports des points de branchement du revêtement sont des S-unités, donc de hauteur bornée, et donnent un modèle plan de C.We give an explicit upper bound in terms of K , S, g for the Faltings height of the jacobian of a curve C of genus g, defined over a number field K and with good reduction outside a finite set S of places of K under the condition that C can be written as a cyclic cover of prime order of the projective line. The proof rests on the fact that the cross ratios of the branch points of the cover are S-units, thus of bounded height, and give a plane model of C.

We determine the asymptotic behavior of the Arakelov metric, the Arakelov-Green's function, and the Faltings delta-invariant for arbitrary oneparameter families of complex curves with semistable degeneration. The leading terms in the asymptotics are given a combinatorial interpretation in terms of S. Zhang's theory of admissible Green's functions on polarized metrized graphs.