Classification of full exceptional collections of line bundles on three blow-ups of P3\mathbb{P}^{3}

A fullness conjecture of Kuznetsov says that if a smooth projective variety $X$ admits a full exceptional collection of line bundles of length $l$, then any exceptional collection of line bundles of length $l$ is full. In this paper, we show that this conjecture holds for $X$ as the blow-up of $\mathbb{P}^{3}$ at a point, a line, or a twisted cubic curve, i.e. any exceptional collection of line bundles of length 6 on $X$ is full. Moreover, we obtain an explicit classification of full exceptional collections of line bundles on such $X$.Comment: 28 pages. To appear in Journal of the Korean Mathematical Society, A previous version with a different title appeared as [CGP17025] at https://cgp.ibs.re.kr/archive/preprints/201

Electricity-powered artificial root nodule.

Root nodules are agricultural-important symbiotic plant-microbe composites in which microorganisms receive energy from plants and reduce dinitrogen (N2) into fertilizers. Mimicking root nodules using artificial devices can enable renewable energy-driven fertilizer production. This task is challenging due to the necessity of a microscopic dioxygen (O2) concentration gradient, which reconciles anaerobic N2 fixation with O2-rich atmosphere. Here we report our designed electricity-powered biological|inorganic hybrid system that possesses the function of root nodules. We construct silicon-based microwire array electrodes and replicate the O2 gradient of root nodules in the array. The wire array compatibly accommodates N2-fixing symbiotic bacteria, which receive energy and reducing equivalents from inorganic catalysts on microwires, and fix N2 in the air into biomass and free ammonia. A N2 reduction rate up to 6.5 mg N2 per gram dry biomass per hour is observed in the device, about two orders of magnitude higher than the natural counterparts

Genus-gg, nn-point, bb-boundary, cc-crosscap correlation functions of two-dimensional conformal field theory: Definition and general properties

We propose a systematic definition of genus-$g$, $n$-point, $b$-boundary, $c$-crosscap correlation functions $\mathcal{F}_{g,n,b,c}^{x}$ with $x$ boundary operators, for general two-dimensional conformal field theory. The $\mathcal{F}_{g,n,b,c}^{x}$ are defined as the inner products of the surface states with the tensor product of $n$ asymptotic states, $b$ boundary states and $c$ crosscap states, where the $b$ defining boundary states carry a total of $x$ boundary operators. The definition implies that $\mathcal{F}_{g,n,b,c}^{x}$ are infinite linear combinations of genus-$g$, $(n+b+c)$-point functions $\mathcal{F}_{g,(n+b+c)}$. A single pole structure is identified in the expansion coefficients $\mathcal{F}_{g,n,b,c}^{0}$, which is analogous to the single pole structure in the conformal block decomposition. This linear property unifies boundary and bulk CFTs. The consistency conditions of $\mathcal{F}_{g,n,b,c}^{x}$ are discussed: conventional upper-half plane (UHP) and crosscap constraints can be reformulated as extra constraints on a collection of bulk correlation functions.Comment: 41 pages + references (Minor corrections + references added comparing to v1