The Abelian Sandpile Model on an Infinite Tree

We consider the standard Abelian sandpile process on the Bethe lattice. We show the existence of the thermodynamic limit for the finite volume stationary measures and the existence of a unique infinite volume Markov process exhibiting features of self-organized criticality

Whole genome sequencing and microsatellite analysis of the Plasmodium falciparum E5 NF54 strain show that the var, rifin and stevor gene families follow Mendelian inheritance

Background: Plasmodium falciparum exhibits a high degree of inter-isolate genetic diversity in its variant surface antigen (VSA) families: P. falciparum erythrocyte membrane protein 1, repetitive interspersed family (RIFIN) and subtelomeric variable open reading frame (STEVOR). The role of recombination for the generation of this diversity is a subject of ongoing research. Here the genome of E5, a sibling of the 3D7 genome strain is presented. Short and long read whole genome sequencing (WGS) techniques (Ilumina, Pacific Bioscience) and a set of 84 microsatellites (MS) were employed to characterize the 3D7 and non-3D7 parts of the E5 genome. This is the first time that VSA genes in sibling parasites were analysed with long read sequencing technology. Results: Of the 5733 E5 genes only 278 genes, mostly var and rifin/stevor genes, had no orthologues in the 3D7 genome. WGS and MS analysis revealed that chromosomal crossovers occurred at a rate of 0–3 per chromosome. var, stevor and rifin genes were inherited within the respective non-3D7 or 3D7 chromosomal context. 54 of the 84 MS PCR fragments correctly identified the respective MS as 3D7- or non-3D7 and this correlated with var and rifin/stevor gene inheritance in the adjacent chromosomal regions. E5 had 61 var and 189 rifin/stevor genes. One large non-chromosomal recombination event resulted in a new var gene on chromosome 14. The remainder of the E5 3D7-type subtelomeric and central regions were identical to 3D7. Conclusions: The data show that the rifin/stevor and var gene families represent the most diverse compartments of the P. falciparum genome but that the majority of var genes are inherited without alterations within their respective parental chromosomal context. Furthermore, MS genotyping with 54 MS can successfully distinguish between two sibling progeny of a natural P. falciparum cross and thus can be used to investigate identity by descent in field isolates

How T-cells use large deviations to recognize foreign antigens

A stochastic model for the activation of T-cells is analysed. T-cells are part of the immune system and recognize foreign antigens against a background of the body's own molecules. The model under consideration is a slight generalization of a model introduced by Van den Berg, Rand and Burroughs in 2001, and is capable of explaining how this recognition works on the basis of rare stochastic events. With the help of a refined large deviation theorem and numerical evaluation it is shown that, for a wide range of parameters, T-cells can distinguish reliably between foreign antigens and self-antigens.Comment: 16 pages, 6 figures; minor revision, new simulations; J Math Biol., in pres

Noncommutative Kn\"{o}rrer periodicity and noncommutative Kleinian singularities

We establish a version of Kn\"{o}rrer's Periodicity Theorem in the context of noncommutative invariant theory. Namely, let $A$ be a left noetherian AS-regular algebra, let $f$ be a normal and regular element of $A$ of positive degree, and take $B=A/(f)$. Then there exists a bijection between the set of isomorphism classes of indecomposable non-free maximal Cohen-Macaulay modules over $B$ and those over (a noncommutative analog of) its second double branched cover $(B^\#)^\#$. Our results use and extend the study of twisted matrix factorizations, which was introduced by the first three authors with Cassidy. These results are applied to the noncommutative Kleinian singularities studied by the second and fourth authors with Chan and Zhang.Comment: Numerous typos fixed, removed unnecessary finite order hypothesi