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We investigate Ramanujan congruences for the function which counts the overpartitions of n with restricted odd differences. In particular, we show that only one such congruence exists. Our method involves using the theory of modular forms to prove a more general theorem which bounds the number of primes possible for Ramanujan congruences in certain eta-quotients. This generalizes work done by Jonah Sinick. We also provide two congruences modulo 5 for this function., 14 pages

The evolutionary radiation of birds has produced incredible morphological variation, including a huge range of skull form and function. Investigating how this variation arose with respect to non-avian dinosaurs is key to understanding how birds achieved their remarkable success after the Cretaceous–Paleogene extinction event. Using a high-dimensional geometric morphometric approach, we quantified the shape of the skull in unprecedented detail across 354 extant and 37 extinct avian and non-avian dinosaurs. Comparative analyses reveal fundamental differences in how skull shape evolved in birds and non-avian dinosaurs. We find that the overall skull shape evolved faster in non-avian dinosaurs than in birds across all regions of the cranium. In birds, the anterior rostrum is the most rapidly evolving skull region, whereas more posterior regions—such as the parietal, squamosal, and quadrate—exhibited high rates in non-avian dinosaurs. These fast-evolving elements in dinosaurs are strongly associated with feeding biomechanics, forming the jaw joint and supporting the jaw adductor muscles. Rapid pulses of skull evolution coincide with changes to food acquisition strategies and diets, as well as the proliferation of bony skull ornaments. In contrast to the appendicular skeleton, which has been shown to evolve more rapidly in birds, avian cranial morphology is characterised by a striking deceleration in morphological evolution relative to non-avian dinosaurs. These results may be due to the reorganisation of skull structure in birds—including loss of a separate postorbital bone in adults and the emergence of new trade-offs with development and neurosensory demands. Taken together, the remarkable cranial shape diversity in birds was not a product of accelerated evolution from their non-avian relatives, despite their frequent portrayal as an icon of adaptive radiations., Using high-density morphometrics to quantify skull shape in birds and non-avian dinosaurs, this study reveals that the skull evolves at significantly slower rate in birds compared to other dinosaurs. The macroevolutionary history of the dinosaur skull reflects the mosaic functional, structural, and developmental changes in the skull across the origin of birds.

Major transformations in brain size and proportions, such as the enlargement of the brain during the evolution of birds, are accompanied by profound modifications to the skull roof. However, the hypothesis of concerted evolution of shape between brain and skull roof over major phylogenetic transitions, and in particular of an ontogenetic relationship between specific regions of the brain and the skull roof, has never been formally tested. We performed 3D morphometric analyses to examine the deep history of brain and skull-roof morphology in Reptilia, focusing on changes during the well-documented transition from early reptiles through archosauromorphs, including nonavian dinosaurs, to birds. Non-avialan taxa cluster tightly together in morphospace, whereas Archaeopteryx and crown birds occupy a separate region. There is a one-to-one correspondence between the forebrain and frontal bone and the midbrain and parietal bone. Furthermore, the position of the forebrain–midbrain boundary correlates significantly with the position of the frontoparietal suture across the phylogenetic breadth of Reptilia and during the ontogeny of individual taxa. Conservation of position and identity in the skull roof is apparent, and there is no support for previous hypotheses that the avian parietal is a transformed postparietal. The correlation and apparent developmental link between regions of the brain and bony skull elements are likely to be ancestral to Tetrapoda and may be fundamental to all of Osteichthyes, coeval with the origin of the dermatocranium. Brain and skull development are intimately related across tetrapods. Here, the authors show a close relationship between brain and skull roof across evolutionary transitions and ontogenetic stages of reptiles.

Background: Our motor actions normally generate sensory events, but how do we know which events were self generated and which have external causes? Here we use temporal adaptation to investigate the processing stage and generality of our sensorimotor timing estimates. Methodology/Principal Findings: Adaptation to artificially-induced delays between action and event can produce a startling percept—upon removal of the delay it feels as if the sensory event precedes its causative action. This temporal recalibration of action and event occurs in a quantitatively similar manner across the sensory modalities. Critically, it is robust to the replacement of one sense during the adaptation phase with another sense during the test judgment. Conclusions/Significance: Our findings suggest a high-level, supramodal recalibration mechanism. The effects are well described by a simple model which attempts to preserve the expected synchrony between action and event, but only when causality indicates it is reasonable to do so. We further demonstrate that this model successfully characterises related adaptation data from outside the sensorimotor domain.

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