Self-avoiding walks and connective constants

The connective constant $\mu(G)$ of a quasi-transitive graph $G$ is the asymptotic growth rate of the number of self-avoiding walks (SAWs) on $G$ from a given starting vertex. We survey several aspects of the relationship between the connective constant and the underlying graph $G$. $\bullet$ We present upper and lower bounds for $\mu$ in terms of the vertex-degree and girth of a transitive graph. $\bullet$ We discuss the question of whether $\mu\ge\phi$ for transitive cubic graphs (where $\phi$ denotes the golden mean), and we introduce the Fisher transformation for SAWs (that is, the replacement of vertices by triangles). $\bullet$ We present strict inequalities for the connective constants $\mu(G)$ of transitive graphs $G$, as $G$ varies. $\bullet$ As a consequence of the last, the connective constant of a Cayley graph of a finitely generated group decreases strictly when a new relator is added, and increases strictly when a non-trivial group element is declared to be a further generator. $\bullet$ We describe so-called graph height functions within an account of "bridges" for quasi-transitive graphs, and indicate that the bridge constant equals the connective constant when the graph has a unimodular graph height function. $\bullet$ A partial answer is given to the question of the locality of connective constants, based around the existence of unimodular graph height functions. $\bullet$ Examples are presented of Cayley graphs of finitely presented groups that possess graph height functions (that are, in addition, harmonic and unimodular), and that do not. $\bullet$ The review closes with a brief account of the "speed" of SAW.Comment: Accepted version. arXiv admin note: substantial text overlap with arXiv:1304.721

Osteological and Soft-Tissue Evidence for Pneumatization in the Cervical Column of the Ostrich (Struthio camelus) and Observations on the Vertebral Columns of Non-Volant, Semi-Volant and Semi-Aquatic Birds

© 2015 Apostolaki et al. This is an open access article distributed under the terms of the Creative Commons Attribution License [4.0], which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. The attached file is the published version of the article

Evidence for Avian Intrathoracic Air Sacs in a New Predatory Dinosaur from Argentina

Background: Living birds possess a unique heterogeneous pulmonary system composed of a rigid, dorsally-anchored lung and several compliant air sacs that operate as bellows, driving inspired air through the lung. Evidence from the fossil record for the origin and evolution of this system is extremely limited, because lungs do not fossilize and because the bellow-like air sacs in living birds only rarely penetrate (pneumatize) skeletal bone and thus leave a record of their presence. Methodology/Principal Findings: We describe a new predatory dinosaur from Upper Cretaceous rocks in Argentina, Aerosteon riocoloradensis gen. et sp. nov., that exhibits extreme pneumatization of skeletal bone, including pneumatic hollowing of the furcula and ilium. In living birds, these two bones are pneumatized by diverticulae of air sacs (clavicular, abdominal) that are involved in pulmonary ventilation. We also describe several pneumatized gastralia (‘‘stomach ribs’’), which suggest that diverticulae of the air sac system were present in surface tissues of the thorax. Conclusions/Significance: We present a four-phase model for the evolution of avian air sacs and costosternal-driven lung ventilation based on the known fossil record of theropod dinosaurs and osteological correlates in extant birds: (1) Phase I—Elaboration of paraxial cervical air sacs in basal theropods no later than the earliest Late Triassic. (2) Phase II—Differentiation of avian ventilatory air sacs, including both cranial (clavicular air sac) and caudal (abdominal air sac) divisions, in basal tetanurans during the Jurassic. A heterogeneous respiratory tract wit

Homeotic Evolution in the Mammalia: Diversification of Therian Axial Seriation and the Morphogenetic Basis of Human Origins

Despite the rising interest in homeotic genes, little has been known about the course and pattern of evolution of homeotic traits across the mammalian radiation. An array of emerging and diversifying homeotic gradients revealed by this study appear to generate new body plans and drive evolution at a large scale.This study identifies and evaluates a set of homeotic gradients across 250 extant and fossil mammalian species and their antecedents over a period of 220 million years. These traits are generally expressed as co-linear gradients along the body axis rather than as distinct segmental identities. Relative position or occurrence sequence vary independently and are subject to polarity reversal and mirroring. Five major gradient modification sets are identified: (1)--quantitative changes of primary segmental identity pattern that appeared at the origin of the tetrapods ; (2)--frame shift relation of costal and vertebral identity which diversifies from the time of amniote origins; (3)--duplication, mirroring, splitting and diversification of the neomorphic laminar process first commencing at the dawn of mammals; (4)--emergence of homologically variable lumbar lateral processes upon commencement of the radiation of therian mammals and ; (5)--inflexions and transpositions of the relative position of the horizontal septum of the body and the neuraxis at the emergence of various orders of therian mammals. Convergent functional changes under homeotic control include laminar articular engagement with septo-neural transposition and ventrally arrayed lumbar transverse process support systems.Clusters of homeotic transformations mark the emergence point of mammals in the Triassic and the radiation of therians in the Cretaceous. A cluster of homeotic changes in the Miocene hominoid Morotopithecus that are still seen in humans supports establishment of a new "hominiform" clade and suggests a homeotic origin for the human upright body plan

Visualisation and characterisation of mononuclear phagocytes in the chicken respiratory tract using CSF1R-transgenic chickens

Additional file 2. Location of B cells, T cells and follicular dendritic cells (FDC) in the lung of MacReporter chickens. The BALT region of 5 to 7 week old non-vaccination animals were analysed for B, T and FCD cells. Isotype controls were used to standardise the microscope and examine aspecific binding before acquiring images (A-B). The GC of MacReporter animals are tightly packed with Bu1-CSF1R-eGFP+ FDC cells and Bu1+CSF1R-eGFP- B cells (C) with few Bu1+ B cells found in the parabronchi (F). CD3+ T cells are disperse within and outside the GC (D) and parabronchi (G). CSF1R-eGFP+ FDC cells express Fc receptors and trap immunoglobulin by expressing IgY (E) and CSF1R-eGFP+ IgY+ FDC are rarely detected out with the GC, BALT region of the lung. GC are indicated by white dashed lines