On some properties of birational derived splinters

A Noetherian reduced ring $A$ is called a birational derived splinter if for all proper birational maps $X\to\operatorname{Spec}(A)$, the canonical map $A\to Rf_*\mathcal{O}_X$ splits. In equal characteristic zero this property characterizes rational singularities, but much less can be said in positive or mixed characteristics. In this paper, we prove some fundamental properties of this notion, including the behavior under localization, taking a pure subring, taking direct limit, and along an \'etale extension. In particular, direct limit of rational singularities in characteristic zero has rational singularities. Then, we study residue extensions (in arbitrary characteristic), and openness and regular extensions in positive characteristic, parallel to Datta-Tucker and the author's previous works on splinters.Comment: 23 page

3-D Velocity Regulation for Nonholonomic Source Seeking Without Position Measurement

We consider a three-dimensional problem of steering a nonholonomic vehicle to seek an unknown source of a spatially distributed signal field without any position measurement. In the literature, there exists an extremum seeking-based strategy under a constant forward velocity and tunable pitch and yaw velocities. Obviously, the vehicle with a constant forward velocity may exhibit certain overshoots in the seeking process and can not slow down even it approaches the source. To resolve this undesired behavior, this paper proposes a regulation strategy for the forward velocity along with the pitch and yaw velocities. Under such a strategy, the vehicle slows down near the source and stays within a small area as if it comes to a full stop, and controllers for angular velocities become succinct. We prove the local exponential convergence via the averaging technique. Finally, the theoretical results are illustrated with simulations.Comment: submitted to IEEE TCST;12 pages, 10 figure

Root microbiota functions in mitigating abiotic and biotic stresses in Arabidopsis

In nature, plants face both biotic and abiotic stresses while at the same time engaging in complex interactions with a vast diversity of commensal microorganisms comprising bacteria, fungi, and oomycetes. This so-called plant microbiota is thought to promote resistance to pathogens and tolerance to specific environmental constraints, likely driving local adaptation in natural plant populations. Reductionist approaches with synthetic microbial communities assembled from microbial culture collections and gnotobiotic plant systems now allow detailed dissection of microbiota-plant-stress interactions under strictly controlled laboratory conditions. Mechanistic understanding into how the root microbiota promotes mineral nutrition and pathogen protection in plants is now emerging. However, whether belowground response to microbial root commensals and aboveground response to abiotic stresses are connected remains largely unexplored. By reconstituting a synthetic, multi-kingdom root microbiota with different microbial input ratios in two gnotobiotic systems (the calcined-clay system and the FlowPot system) (Chapter I), I first showed that distinct input ratios of bacteria, fungi, and oomycetes converge into a similar output community composition, with stable effects on Arabidopsis growth. By testing different abiotic and biotic stresses in three gnotobiotic plant systems (the FlowPot system, the calcined-clay system, and the white sand system) (Chapter I), I provided evidence that salt, drought, and shade stresses negatively affected plant growth across all three systems, whereas nutritional stress affected on plant performance in a system-dependent manner. Moreover, I demonstrated that a synthetic multi-kingdom root microbiota rescued Arabidopsis growth under salt, drought and light limitation stresses in the FlowPot system and the white sand system (Chapter I). Given the importance of light for plant growth, in chapter II, I further dissected the extent to which response to the synthetic root microbiota and light are interconnected. By manipulating light conditions (low photosynthetically active radiation, LP; end of day far red-light treatment, EODFR) in the FlowPot system, I demonstrated that microbial root commensals confer Arabidopsis tolerance to light limitation stresses and that reciprocally, modification in aboveground light condition shifts the composition of root microbial communities. Notably, this shift in the structure of root bacterial community significantly explains the microbiota-induced growth rescue under LP. Arabidopsis transcriptome analysis revealed that immune responses in root and systemic defense responses in shoot were induced in the presence of the root microbiota under normal light conditions. These host responses were largely shut down under light limiting conditions and were correlated with increased susceptibility to unrelated leaf pathogens, implying that root microbiota-induced systemic defense responses were modulated by light. Through an extensive Arabidopsis mutant screen, I demonstrated that root microbiota-mediated plant survival under LP depends on jasmonic acid biosynthesis and signaling, cryptochromes and brassinosteroids. Furthermore, I present genetic evidence that orchestration of this light-dependent growth-defense trade-off requires the transcriptional regulator MYC2. The data suggest that plants can take advantage of root commensals to activate either growth or defense depending on aboveground light conditions

Formal lifting of dualizing complexes and consequences

We show that for a Noetherian ring $A$ that is $I$-adically complete for an ideal $I$, if $A/I$ admits a dualizing complex, so does $A$. We discuss several consequences of this result. We also consider a generalization of the notion of dualizing complexes to infinite-dimensional rings and prove the results in this generality. In addition, we give an alternative proof of the fact that every excellent Henselian local ring admits a dualizing complex, using ultrapower.Comment: 19 pages. Comments welcome

Productivity and farm size in Australian agriculture: reinvestigating the returns to scale

Higher productivity among large farms is often assumed to be a result of increasing returns to scale. However, using farm-level data for the Australian broadacre industry, it was found that constant or mildly decreasing returns to scale is more typical. On examining the monotonic change in marginal input returns as farm operating size increases, it was found that large farms achieve higher productivity through changes in production technology rather than through changes in scale. The results highlight the disparity between ‘returns to scale’ and ‘returns to size’ in Australian agriculture. They also suggest that improving productivity in smaller farms would depend more on their ability to access advanced technologies than their ability to simply expand. The implications for ongoing structural adjustment in Australian agriculture are discussed.returns to scale, returns to size, production function, technology progress, structural adjustment, Australian agriculture, Agricultural and Food Policy,