Weak randomness in graphons and theons

Call a hereditary family $\mathcal{F}$ of graphs strongly persistent if there exists a graphon $W$ such that in all subgraphons $W'$ of $W$, $\mathcal{F}$ is precisely the class of finite graphs that have positive density in $W'$. The first result of the paper is a complete characterization of the hereditary families of graphs that are strongly persistent as precisely those that are closed under substitutions. In fact, we prove an analogous characterization for the natural extension of these properties to structures in finite relational languages. We call graphons (or more generally theons) with the self-similarity property above weakly random. Weak randomness can be seen as a weakening of graph quasirandomness: while the latter requires densities of any finite graph to be the same in any subgraphon, the former only requires the set of finite graphs with positive density to be invariant across subgraphons. A hereditary family $\mathcal{F}$ is said to have the weakly random Erd\H{o}s--Hajnal property (WR) if every graphon that is a limit of graphs in $\mathcal{F}$ has a weakly random subgraphon; this includes as a special case the approximate Erd\H{o}s--Hajnal property (AEHP), where the subgraphon is required to be an almost clique or almost anti-clique. Among families of graphs that are closed under substitutions, we completely characterize the families that belong to WR as those with "few" prime graphs. The classes AEHP and WR can be seen as measuring the complexity of hereditary families in terms of structural variation in their limit objects (here specifically in terms of unavoidable sub-objects), and this work can be seen as initiating a classification theory for hereditary families of finite structures.Comment: 62 pages, 7 figures, 1 tabl

Classification of Extensions of Classifiable C*-algebras

We classify extensions of certain classifiable C*-algebras using the six term exact sequence in K-theory together with the positive cone of the K_0-groups of the distinguished ideal and quotient. We then apply our results to a class of C*-algebras arising from substitutional shift spaces.Comment: 22 pages, Reordered some sections, an application involving graph algebras is adde

Refinement Types for Logical Frameworks and Their Interpretation as Proof Irrelevance

Refinement types sharpen systems of simple and dependent types by offering expressive means to more precisely classify well-typed terms. We present a system of refinement types for LF in the style of recent formulations where only canonical forms are well-typed. Both the usual LF rules and the rules for type refinements are bidirectional, leading to a straightforward proof of decidability of typechecking even in the presence of intersection types. Because we insist on canonical forms, structural rules for subtyping can now be derived rather than being assumed as primitive. We illustrate the expressive power of our system with examples and validate its design by demonstrating a precise correspondence with traditional presentations of subtyping. Proof irrelevance provides a mechanism for selectively hiding the identities of terms in type theories. We show that LF refinement types can be interpreted as predicates using proof irrelevance, establishing a uniform relationship between two previously studied concepts in type theory. The interpretation and its correctness proof are surprisingly complex, lending support to the claim that refinement types are a fundamental construct rather than just a convenient surface syntax for certain uses of proof irrelevance

The infinite random simplicial complex

We study the Fraisse limit of the class of all finite simplicial complexes. Whilst the natural model-theoretic setting for this class uses an infinite language, a range of results associated with Fraisse limits of structures for finite languages carry across to this important example. We introduce the notion of a local class, with the class of finite simplicial complexes as an archetypal example, and in this general context prove the existence of a 0-1 law and other basic model-theoretic results. Constraining to the case where all relations are symmetric, we show that every direct limit of finite groups, and every metrizable profinite group, appears as a subgroup of the automorphism group of the Fraisse limit. Finally, for the specific case of simplicial complexes, we show that the geometric realisation is topologically surprisingly simple: despite the combinatorial complexity of the Fraisse limit, its geometric realisation is homeomorphic to the infinite simplex.Comment: 33 page

The molecular genetics and cellular mechanisms underlying pulmonary arterial hypertension

Pulmonary arterial hypertension (PAH) is an incurable disorder clinically characterised by a sustained elevation of mean arterial pressure in the absence of systemic involvement. As the adult circulation is a low pressure, low resistance system, PAH represents a reversal to a foetal state. The small pulmonary arteries of patients exhibit luminal occlusion resultant from the uncontrolled growth of endothelial and smooth muscle cells. This vascular remodelling is comprised of hallmark defects, most notably the plexiform lesion. PAH may be familial in nature but the majority of patients present with spontaneous disease or PAH associated with other complications. In this paper, the molecular genetic basis of the disorder is discussed in detail ranging from the original identification of the major genetic contributant to PAH and moving on to current next-generation technologies that have led to the rapid identification of additional genetic risk factors. The impact of identified mutations on the cell is examined, particularly, the determination of pathways disrupted in disease and critical to pulmonary vascular maintenance. Finally, the application of research in this area to the design and development of novel treatment options for patients is addressed along with the future directions PAH research is progressing towards

Mutations in the EXT1 and EXT2 genes in Spanish patients with multiple osteochondromas

Multiple osteochondromas is an autosomal dominant skeletal disorder characterized by the formation of multiple cartilage-capped tumours. Two causal genes have been identified, EXT1 and EXT2, which account for 65% and 30% of cases, respectively. We have undertaken a mutation analysis of the EXT1 and EXT2 genes in 39 unrelated Spanish patients, most of them with moderate phenotype, and looked for genotype-phenotype correlations. We found the mutant allele in 37 patients, 29 in EXT1 and 8 in EXT2. Five of the EXT1 mutations were deletions identified by MLPA. Two cases of mosaicism were documented. We detected a lower number of exostoses in patients with missense mutation versus other kinds of mutations. In conclusion, we found a mutation in EXT1 or in EXT2 in 95% of the Spanish patients. Eighteen of the mutations were novel.Fil: Sarrión, P.. Universidad de Barcelona; EspañaFil: Sangorrin, A.. Hospital Sant Joan de Déu; EspañaFil: Urreizti, R.. Universidad de Barcelona; EspañaFil: Delgado, María Andrea. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Córdoba; ArgentinaFil: Artuch, R.. Hospital Sant Joan de Déu; EspañaFil: Martorell, L.. Hospital Sant Joan de Déu; EspañaFil: Armstrong, J.. Hospital Sant Joan de Déu; EspañaFil: Anton, J.. Hospital Sant Joan de Déu; EspañaFil: Torner, F.. Hospital Sant Joan de Déu; EspañaFil: Vilaseca, M. A.. Hospital Sant Joan de Déu; EspañaFil: Nevado, J.. Hospital Universitario La Paz; EspañaFil: Lapunzina, P.. Hospital Universitario La Paz; EspañaFil: Asteggiano, Carla Gabriela. Universidad Nacional de Córdoba; Argentina. Universidad Católica de Córdoba; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Balcells, S.. Universidad de Barcelona; EspañaFil: Grinberg, D.. Universidad de Barcelona; Españ