The $c_2$ invariant is an arithmetic graph invariant related to quantum field theory. We give a relation modulo p between the $c_2$ invariant at p and the $c_2$ invariant at $p^s$ by proving a relation modulo p between certain coefficients of powers of products of particularly nice polynomials. The relation at the level of the $c_2$ invariant provides evidence for a conjecture of Schnetz.

The spatial distribution of Neotropical birds at local scales is often associated with plant successional stages and unique microhabitats such as bamboo stands and treefall gaps. Only 15% of the Atlantic Forest remains, yet this habitat is home to more than 200 endemic birds. Understanding the distribution of bird species within the Atlantic Forest can improve the test of ecological hypotheses related to habitat use and habitat selection. This study analyses the distribution of birds among different types of relief (lowlands, ravines, hillsides, and hilltops) in 41 forest fragments of different sizes (1 to 384.5 ha) within the Atlantic Forest, south east Brazil. The presence and absence data were analysed with dynamic occupancy models to determine the influence of area and topography in the distribution of bird species. Of the 169 species detected, we were able to estimate an occupancy probability of 67. Among them, relief type was an important variable for 26 species (39%). Most species showed a preference for lowlands and ravines, and some showed an association of combinations of these two reliefs with hillsides and hilltops. These preferences might be related to distribution of preferred food, nesting microhabitats, and thermoregulation. For the endemic and threatened Atlantic Forest mountain birds, the ecological association with areas of relief is a strong, and previously unsuspected, spatial distribution pattern. Our results suggest that the ecology and distribution of birds in tropical forests across hilly regions could be better understood if topography is considered. This pattern should be better investigated among other taxa in tropical forests.

The Seychelles black parrot Coracopsis (nigra) barklyi, endemic to the Seychelles islands, is the only surviving parrot on the archipelago. Although originally classified as a subspecies of the lesser vasa parrot Coracopsis nigra evidence now indicates that the Seychelles population may be a distinct species, in which case its conservation status also requires reassessment. Here, we address the status of the C. (n.) barklyi population on the islands of its current and likely historical range, Praslin and Curieuse, assess the effect of habitat type on relative abundance, and identify the most appropriate point count duration for monitoring the population. We conducted point count distance sampling at 268 locations using habitat type as a covariate in the modelling of the detection function. Density on Praslin was estimated to be 0.14–0.24 individuals per ha, resulting in an estimate of a total of 520–900 individuals (95% confidence interval). The highest densities occurred in endemic palm and mixed forests. Habitats with a high proportion of palms were more likely to support parrots than habitats without palms. As no parrots were detected on Curieuse the number estimated on Praslin is the global wild population. The small size of this population necessitates its categorization as Vulnerable on the IUCN Red List. Our main recommendation for the conservation of C. (n.) barklyi, in addition to continued monitoring, is the preservation and expansion of the parrot's endemic palm forest habitat.