Multiple planar coincidences with N-fold symmetry

Planar coincidence site lattices and modules with N-fold symmetry are well understood in a formulation based on cyclotomic fields, in particular for the class number one case, where they appear as certain principal ideals in the corresponding ring of integers. We extend this approach to multiple coincidences, which apply to triple or multiple junctions. In particular, we give explicit results for spectral, combinatorial and asymptotic properties in terms of Dirichlet series generating functions.Comment: 13 pages, two figures. For previous related work see math.MG/0511147 and math.CO/0301021. Minor changes and references update

Sums of two squares and the tau-function: Ramanujan's trail

Ramanujan, in his famous first letter to Hardy, claimed a very precise estimate for the number of integers that can be written as a sum of two squares. Far less well-known is that he also made further claims of a similar nature for the non-divisibility of the Ramanujan tau-function for certain primes. In this survey, we provide more historical details and also discuss related later developments. These show that, as so often, Ramanujan was an explorer in a fascinating wilderness, leaving behind him a beckoning trail

Cyclotomic polynomials with prescribed height and prime number theory

Given any positive integer $n,$ let $A(n)$ denote the height of the $n^{\text{th}}$ cyclotomic polynomial, that is its maximum coefficient in absolute value. It is well known that $A(n)$ is unbounded. We conjecture that every natural number can arise as value of $A(n)$ and prove this assuming that for every pair of consecutive primes $p$ and $p'$ with $p\ge 127$ we have $p'-p<\sqrt{p}+1.$ We also conjecture that every natural number occurs as maximum coefficient of some cyclotomic polynomial and show that this is true if Andrica's conjecture that always $\sqrt{p'}-\sqrt{p}<1$ holds. This is the first time, as far as the authors know, a connection between prime gaps and cyclotomic polynomials is uncovered. Using a result of Heath-Brown on prime gaps we show unconditionally that every natural number $m\le x$ occurs as $A(n)$ value with at most $O_{\epsilon}(x^{3/5+\epsilon})$ exceptions. On the Lindel\"of Hypothesis we show there are at most $O_{\epsilon}(x^{1/2+\epsilon})$ exceptions and study them further by using deep work of Bombieri--Friedlander--Iwaniec on the distribution of primes in arithmetic progressions beyond the square-root barrier

CAL1 is the Drosophila CENP-A assembly factor

Centromeres are specified epigenetically by the incorporation of the histone H3 variant CENP-A. In humans, amphibians, and fungi, CENP-A is deposited at centromeres by the HJURP/Scm3 family of assembly factors, but homologues of these chaperones are absent from a number of major eukaryotic lineages such as insects, fish, nematodes, and plants. In Drosophila, centromeric deposition of CENP-A requires the fly-specific protein CAL1. Here, we show that targeting CAL1 to noncentromeric DNA in Drosophila cells is sufficient to heritably recruit CENP-A, kinetochore proteins, and microtubule attachments. CAL1 selectively interacts with CENP-A and is sufficient to assemble CENP-A nucleosomes that display properties consistent with left-handed octamers. The CENP-A assembly activity of CAL1 resides within an N-terminal domain, whereas the C terminus mediates centromere recognition through an interaction with CENP-C. Collectively, this work identifies the “missing” CENP-A chaperone in flies, revealing fundamental conservation between insect and vertebrate centromere-specification mechanisms

Evolution of metabolic divergence in <i>Pseudomonas aeruginosa</i> during long-term infection facilitates a proto-cooperative interspecies interaction

The effect of polymicrobial interactions on pathogen physiology and how it can act either to limit pathogen colonization or to potentiate pathogen expansion and virulence are not well understood. Pseudomonas aeruginosa and Staphylococcus aureus are opportunistic pathogens commonly found together in polymicrobial human infections. However, we have previously shown that the interactions between these two bacterial species are strain dependent. Whereas P. aeruginosa PAO1, a commonly used laboratory strain, effectively suppressed S. aureus growth, we observed a commensal-like interaction between the human host-adapted strain, DK2-P2M24-2003, and S. aureus. In this study, characterization by matrix-assisted laser desorption ionization-time of flight (MALDI-TOF) imaging mass spectrometry (IMS) and mass spectral (MS) molecular networking revealed a significant metabolic divergence between P. aeruginosa PAO1 and DK2-P2M24-2003, which comprised several virulence factors and signaling 4-hydroxy-2-alkylquinoline (HAQ) molecules. Strikingly, a further modulation of the HAQ profile was observed in DK2-P2M24-2003 during interaction with S. aureus, resulting in an area with thickened colony morphology at the P. aeruginosa–S. aureus interface. In addition, we found an HAQ-mediated protection of S. aureus by DK2-P2M24-2003 from the killing effect of tobramycin. Our findings suggest a model where the metabolic divergence manifested in human host-adapted P. aeruginosa is further modulated during interaction with S. aureus and facilitate a proto-cooperative P. aeruginosa–S. aureus relationship