A syntactic realization theorem for justification logics

Justification logics are refinements of modal logics where modalities are replaced by justification terms. They are connected to modal logics via so-called realization theorems. We present a syntactic proof of a single realization theorem that uniformly connects all the normal modal logics formed from the axioms \$mathsfd\$, \$mathsft\$, \$mathsfb\$, \$mathsf4\$, and \$mathsf5\$ with their justification counterparts. The proof employs cut-free nested sequent systems together with Fitting's realization merging technique. We further strengthen the realization theorem for \$mathsfKB5\$ and \$mathsfS5\$ by showing that the positive introspection operator is superfluous.

Air bells of water spiders are an extended phenotype modified in response to gas composition

The water spider Argyroneta aquatica (Clerck) is the only spider that spends its whole life under water. Water spiders keep an air bubble around their body for breathing and build under-water air bells, which they use for shelter and raising offspring, digesting and consuming prey, moulting, depositing eggs and sperm, and copulating. It is unclear whether these bells are an important oxygen reservoir for breathing under water, or whether they serve mainly to create water-free space for feeding and reproduction. In this study, we manipulated the composition of the gas inside the bell of female water spiders to test whether they monitor the quality of this gas, and replenish oxygen if required. We exchanged the entire gas in the bell either with pure O(2), pure CO(2), or with ambient air as control, and monitored behavioural responses. The test spiders surfaced and replenished air more often in the CO(2) treatment than in the O(2) treatment, and they increased bell building behaviour. In addition to active oxygen regulation, they monitored and adjusted the bells by adding silk. These results show that water spiders use the air bell as an oxygen reservoir, and that it functions as an external lung, which renders it essential for living under water permanently. A. aquatica is the only animal that collects, transports, and stores air, and monitors its property for breathing, which is an adaptive response of a terrestrial animal to the colonization of an aquatic habitat. J. Exp. Zool. 307A:549-555, 2007. (c) 2007 Wiley-Liss, Inc.

Functional equations and the Cauchy mean value theorem

The aim of this note is to characterize all pairs of sufficiently smooth functions for which the mean value in the Cauchy mean value theorem is taken at a point which has a well-determined position in the interval. As an application of this result, a partial answer is given to a question posed by Sahoo and Riedel.

A four moments theorem for Gamma limits on a Poisson chaos

This paper deals with sequences of random variables belonging to a fixed chaos of order q generated by a Poisson random measure on a Polish space. The problem is investigated whether convergence of the third and fourth moment of such a suitably normalized sequence to the third and fourth moment of a centred Gamma law implies convergence in distribution of the involved random variables. A positive answer is obtained for q = 2 and q = 4. The proof of this four moments theorem is based on a number of new estimates for contraction norms. Applications concern homogeneous sums and U-statistics on the Poisson space.