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Abstract In 2021, Chen proved that the size of any connected component of the Markoff mod$$p$$ $p$graph is divisible by$$p$$ $p$. In combination with the work of Bourgain, Gamburd, and Sarnak, Chen’s result resolves a conjecture of Baragar for all but finitely many primes: the Markoff mod$$p$$ $p$graph is connected. In particular, strong approximation for Markoff triples holds for all but finitely many primes. We provide an alternative proof of Chen’s theorem. 

Axion-like particles (ALPs) arise from well-motivated extensions to the Standard Model and could account for dark matter. ALP dark matter would manifest as a field oscillating at an (as of yet) unknown frequency. The frequency depends linearly on the ALP mass and plausibly ranges from 10⁻²²to 10 eV/c². This motivates broadband search approaches. We report on a direct search for ALP dark matter with an interferometer composed of two atomic K-Rb-³He comagnetometers, one situated in Mainz, Germany, and the other in Kraków, Poland. We leverage the anticipated spatio-temporal coherence properties of the ALP field and probe all ALP-gradient-spin interactions covering a mass range of nine orders of magnitude. No significant evidence of an ALP signal is found. We thus place new upper limits on the ALP-neutron, ALP-proton and ALP-electron couplings reaching belowg_aNN < 10⁻⁹ GeV⁻¹,g_aPP < 10⁻⁷ GeV⁻¹andg_aee < 10⁻⁶ GeV⁻¹, respectively. These limits improve upon previous laboratory constraints for neutron and proton couplings by up to three orders of magnitude. 

Abstract For 𝓞 an imaginary quadratic ring, we compute a fundamental polyhedron in hyperbolic 3-space for the action of PE₂(𝓞), the projective elementary subgroup of PSL₂(𝓞). This allows for new, simplified proofs of theorems of Cohn, Nica, Fine, and Frohman. Namely, we obtain a presentation for PE₂(𝓞), show that it has infinite index and is its own normalizer in PSL₂(𝓞), and split PSL₂(𝓞) into a free product with amalgamation that has PE₂(𝓞) as one of its factors. 

We propose and demonstrate a general method to calibrate the frequency-dependent response of selfcompensating noble-gas–alkali-metal comagnetometers to arbitrary spin perturbations. This includes magnetic and nonmagnetic perturbations such as rotations and exotic spin interactions. The method is based on a fit of the magnetic field response to an analytical model. The frequency-dependent response of the comagnetometer to arbitrary spin perturbations can be inferred using the fit parameters. We demonstrate the effectiveness of this method by comparing the inferred rotation response to an experimental measurement of the rotation response. Our results show that experiments relying on zero-frequency calibration of the comagnetometer response can over- or underestimate the comagnetometer sensitivity by orders of magnitude over a wide frequency range. Moreover, this discrepancy accumulates over time as operational parameters tend to drift during comagnetometer operation. The demonstrated calibration protocol enables accurate prediction and control of comagnetometer sensitivity to, for example, ultralight bosonic dark-matter fields coupling to electron or nuclear spins, as well as accurate monitoring and control of the relevant system parameters. 

ABSTRACT Histone-like nucleoid structuring (H-NS) and H-NS-like proteins serve as global gene silencers and work with antagonistic transcriptional activators (counter-silencers) to properly coordinate the expression of virulence genes in pathogenic bacteria. InBrucella, MucR has been proposed as a novel H-NS-like gene silencer, but direct experimental evidence is lacking. Here, we show that MucR serves as an H-NS-like silencer of theBrucella abortusgenes encoding the polar autotransporter adhesins BtaE and BmaC, the c-di-GMP-specific phosphodiesterase BpdB, and the quorum-sensing regulator BabR. We also demonstrate that the MarR-type transcriptional activator MdrA can displace MucR from thebtaEpromoter, supporting the existence of MucR counter-silencers inBrucella. Moreover, our chromatin immunoprecipitation (ChIP)-seq analysis identified 546 MucR enrichment peaks along the genome, including in the promoters of the genes encoding the Type IV secretion machinery and effectors and the quorum-sensing regulator VjbR. Importantly, MucR ChIP-seq peaks overlap with the previously described binding sites for the transcriptional activators VjbR, BvrR, and CtrA suggesting that these regulators serve as MucR counter-silencers and work in concert with MucR to coordinate virulence gene expression inBrucella. In addition, using chromosome conformation capture (Hi-C), we show that like H-NS inEscherichia coli, MucR alters the global structure of theBrucellanucleoid. Finally, a copy of theE. coli hnsrescues the distinctive growth defect and elevatedbtaEexpression of aB. abortus mucRmutant. Together, these findings solidify the role of MucR as a novel type of H-NS-like protein and suggest that MucR’s gene-silencing properties play a key role in virulence inBrucella. IMPORTANCEHistone-like nucleoid structuring (H-NS) and H-NS-like proteins coordinate host-associated behaviors in many pathogenic bacteria, often through forming silencer/counter-silencer pairs with signal-responsive transcriptional activators to tightly control gene expression.Brucellaand related bacteria do not encode H-NS or homologs of known H-NS-like proteins, and it is unclear if they have other proteins that perform analogous functions during pathogenesis. In this work, we provide compelling evidence for the role of MucR as a novel H-NS-like protein inBrucella. We show that MucR possesses many of the known functions attributed to H-NS and H-NS-like proteins, including the formation of silencer/counter-silencer pairs to control virulence gene expression and global structuring of the nucleoid. These results uncover a new role for MucR as a nucleoid structuring protein and support the importance of temporal control of gene expression inBrucellaand related bacteria.