We give an algorithm to classify singular fibers of finite cyclic covering fibrations of a ruled surface by using singularity diagrams. As the first application, we classify all fibers of 3-cyclic covering fibrations of genus 4 of a ruled surface and show that the signature of a complex surface with this fibration is non-positive by computing the local signature for any fiber. As the second application, we classify all fibers of hyperelliptic fibrations of genus 3 into 12 types according to the Horikawa index. We also prove that finite cyclic covering fibrations of a ruled surface have no multiple fibers if the degree of the covering is greater than 3.

We consider the stochastic ranking process with space-time dependent unbounded jump rates for the particles. We prove that the joint empirical distribution of jump rate and scaled position converges almost surely to a deterministic distribution in the infinite particle limit. We assume topology of weak convergence for the space of distributions, which implies that the fluctuations among particles with different jump rates cancel in the limit. The results are proved by first finding an auxiliary stochastic ranking process, for which a strong law of large numbers is applied, and then applying a multi time recursive Gronwall's inequality. The limit has a representation in terms of non-Markovian processes which we call point processes with last-arrival-time dependent intensities. We also prove the propagation of chaos, i.e., the tagged particle processes also converge almost surely.

Harvey-Lawson and Anciaux introduced the notion of austere submanifolds in pseudo-Riemannian geometry. We give an equivalent condition for an orbit of the isotropy representations for semisimple pseudo-Riemannian symmetric space to be an austere submanifold in a pseudo-sphere in terms of restricted root system theory with respect to Cartan subspaces. By using the condition we give examples of austere orbits.

It is known that a contact Riemannian manifold carries a generalized Fefferman metric on a circle bundle over the manifold. We compute the curvature of the metric explicitly in terms of a modified Tanno connection on the underlying manifold. In particular, we show that the scalar curvature descends to the pseudohermitian scalar curvature multiplied by a certain constant. This is an answer to a problem considered by Blair-Dragomir.

We determine a primitive form for a universal unfolding of an affine cusp polynomial. Moreover, we prove that the resulting Frobenius manifold is isomorphic to the one constructed from the Gromov–Witten theory for an orbifold projective line with at most three orbifold points.

Dans cette courte note, on remarque qu’une petite modification dans le calcul effectué dans [5] de l’algèbre du torseur d’isomorphismes entre la réalisation de Betti tangentielle et la réalisation de De Rham résulte en un énoncé du type Kontsevich–Zagier fonctionnel purement algébrique et nettement plus satisfaisant que l’énoncé obtenu dans [5].

In this short note, we remark that a small modification in the computation made in [5] of the algebra of the torsor of isomorphisms between the tangential Betti realisation and the De Rham realisation results in a statement of functional Kontsevich–Zagier type which is purely algebraic and much more satisfactory than the statement obtained in [5].