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Buchweitz-Greuel-Schreyer conjectured in 1987 a lower bound on the ranks of matrix factorizations over certain local hypersurface rings [Invent. Math. 88 (1987), pp. 165–182]. We study a graded version of this conjecture, and we show that it implies a novel conjecture concerning the cohomology of sheaves over non-Fano projective hypersurfaces. 

Over a Cohen-Macaulay local ring, the minimal number of generators of a maximal Cohen-Macaulay module is bounded above by its multiplicity. In 1984 Ulrich [Math. Z. 188 (1984), pp. 23–32] asked whether there always exist modules for which equality holds; such modules are known nowadays as Ulrich modules. We answer this question in the negative by constructing families of two dimensional Cohen-Macaulay local rings that have no Ulrich modules. Some of these examples are Gorenstein normal domains; others are even complete intersection domains, though not normal. 

Inbreeding depression is a key factor regulating the evolution of self-fertilization in plants. Despite predictions that inbreeding depression should evolve with selfing rates as deleterious alleles are increasingly exposed and removed by selection, evidence of purging the genetic load in wild populations is equivocal at best. This discordance could be explained, in part, if the load underlying inbreeding depression is subject to soft selection, i.e., the fitness of selfed individuals depends on the frequency and density of selfed vs. outcrossed individuals in the population. Somewhat counterintuitively, this means that populations with contrasting mutation load can have similar fitness. Soft selection against selfed individuals may be expected when there is inbreeding depression for competitive ability in density-regulated populations. We tested population-level predictions of inbreeding depression in competitive ability by creating a density series of potted plants consisting of either purely outcrossed, purely selfed, or mixed (50% outcrossed, 50% selfed) seed of the mixed-mating biennialSabatia angularis(Gentianaceae) representing ecological neighborhoods. Focusing on the growth and survival of juveniles, we show that mean plant size is independent of neighborhood composition when resources are limiting, but greatest in outcrossed neighborhoods at low densities. Across a range of densities, this manifests as stronger density-dependence in outcrossed populations compared to selfed or mixed ones. We also found significantly greater size inequalities among individuals in mixed neighborhoods, even at high densities where mean juvenile size converged, a key signature of asymmetric competition between outcrossed and selfed individuals. Our work illustrates how soft selection could shelter the genetic load underlying inbreeding depression and its demographic consequences. 

Ribosomally synthesized and post-translationally modified peptides (RiPPs) are a growing class of natural products biosynthesized from a genetically encoded precursor peptide. The enzymes that install the post-translational modifications on these peptides have the potential to be useful catalysts in the production of natural-product-like compounds and can install non-proteogenic amino acids in peptides and proteins. However, engineering these enzymes has been somewhat limited, due in part to limited structural information on enzymes in the same families that nonetheless exhibit different substrate selectivities. Despite AlphaFold2’s superior performance in single-chain protein structure prediction, its multimer version lacks accuracy and requires high-end GPUs, which are not typically available to most research groups. Additionally, the default parameters of AlphaFold2 may not be optimal for predicting complex structures like RiPP biosynthetic enzymes, due to their dynamic binding and substrate-modifying mechanisms. This study assessed the efficacy of the structure prediction program ColabFold (a variant of AlphaFold2) in modeling RiPP biosynthetic enzymes in both monomeric and dimeric forms. After extensive benchmarking, it was found that there were no statistically significant differences in the accuracy of the predicted structures, regardless of the various possible prediction parameters that were examined, and that with the default parameters, ColabFold was able to produce accurate models. We then generated additional structural predictions for select RiPP biosynthetic enzymes from multiple protein families and biosynthetic pathways. Our findings can serve as a reference for future enzyme engineering complemented by AlphaFold-related tools. 

Abstract This paper extends the results of Boij, Eisenbud, Erman, Schreyer and Söderberg on the structure of Betti cones of finitely generated graded modules and finite free complexes over polynomial rings, to all finitely generated graded rings admitting linear Noether normalizations. The key new input is the existence of lim Ulrich sequences of graded modules over such rings. 

ABSTRACT We present results from a radio survey for variable and transient sources on 15-min time-scales, using the Australian SKA Pathfinder (ASKAP) pilot surveys. The pilot surveys consist of 505 h of observations conducted at around 1 GHz observing frequency, with a total sky coverage of 1476 deg2. Each observation was tracked for approximately 8 – 10 h, with a typical rms sensitivity of ∼30 μJy beam−1 and an angular resolution of ∼12 arcsec. The variability search was conducted within each 8 – 10 h observation on a 15-min time-scale. We detected 38 variable and transient sources. Seven of them are known pulsars, including an eclipsing millisecond pulsar, PSR J2039−5617. Another eight sources are stars, only one of which has been previously identified as a radio star. For the remaining 23 objects, 22 are associated with active galactic nuclei or galaxies (including the five intra-hour variables that have been reported previously), and their variations are caused by discrete, local plasma screens. The remaining source has no multiwavelength counterparts and is therefore yet to be identified. This is the first large-scale radio survey for variables and transient sources on minute time-scales at a sub-mJy sensitivity level. We expect to discover ∼1 highly variable source per day using the same technique on the full ASKAP surveys.