In large-scale galaxy surveys, particularly deep ground-based photometric studies, galaxy blending was inevitable. Such blending posed a potential primary systematic uncertainty for upcoming surveys. Current deblenders predominantly depended on analytical modelling of galaxy profiles, facing limitations due to inflexible and imprecise models. We presented a novel approach, using a U-net structured transformer-based network for deblending astronomical images, which we term the CAT-deblender. It was trained using both RGB and the grz-band images, spanning two distinct data formats present in the Dark Energy Camera Legacy Survey (DECaLS) database, including galaxies with diverse morphologies in the training dataset. Our method necessitated only the approximate central coordinates of each target galaxy, sourced from galaxy detection, bypassing assumptions on neighbouring source counts. Post-deblending, our RGB images retained a high signal-to-noise peak, consistently showing superior structural similarity against ground truth. For multi-band images, the ellipticity of central galaxies and median reconstruction error for r-band consistently lie within $\pm$0.025 to $\pm$0.25, revealing minimal pixel residuals. In our comparison of deblending capabilities focused on flux recovery, our model showed a mere 1% error in magnitude recovery for quadruply blended galaxies, significantly outperforming SExtractor’s higher error rate of 4.8%. Furthermore, by cross-matching with the publicly accessible overlapping galaxy catalogs from the DECaLS database, we successfully deblended 433 overlapping galaxies. Moreover, we have demonstrated effective deblending of 63 733 blended galaxy images, randomly chosen from the DECaLS database.

We prove that the multiplicity of a filtration of a local ring satisfies various convexity properties. In particular, we show the multiplicity is convex along geodesics. As a consequence, we prove that the volume of a valuation is log convex on simplices of quasi-monomial valuations and give a new proof of a theorem of Xu and Zhuang on the uniqueness of normalized volume minimizers. In another direction, we generalize a theorem of Rees on multiplicities of ideals to filtrations and characterize when the Minkowski inequality for filtrations is an equality under mild assumptions.

We prove an algebraic version of the Hamilton–Tian conjecture for all log Fano pairs. More precisely, we show that any log Fano pair admits a canonical two-step degeneration to a reduced uniformly Ding stable triple, which admits a Kähler–Ricci soliton when the ground field .

Soft magnetic robots have attracted tremendous interest owning to their controllability and manoeuvrability, demonstrating great prospects in a number of industrial areas. However, further explorations on the locomotion and corresponding deformation of magnetic robots with complex configurations are still challenging. In the present study, we analyse a series of soft magnetic robots with various geometric shapes under the action of the magnetic field. First, we prepared the matrix material for the robot, that is, the mixture of silicone and magnetic particles. Next, we fabricated a triangular robot whose locomotion speed and warping speed are approximately 1.5 and 9 mm/s, respectively. We then surveyed the generalised types of robots with other shapes, where the movement, grabbing, closure and flipping behaviours were fully demonstrated. The experiments show that the arching speed and grabbing speed of the cross-shaped robot are around 4.8 and 3.5 mm/s, the crawling speed of the pentagram-shaped robot is 3.5 mm/s, the pentahedron-shaped robot can finish its closure motion in 1 s and the arch-shaped robot can flip forward and backward in 0.5 s. The numerical simulation based on the finite element method has been compared with the experimental results, and they are in excellent agreement. The results are beneficial to engineer soft robots under the multi-fields, which can broaden the eyes on inventing intellectual devices and equipment.

We show that the K-moduli spaces of log Fano pairs $\left(\mathbb {P}^1\times \mathbb {P}^1, cC\right)$ , where C is a $(4,4)$ curve and their wall crossings coincide with the VGIT quotients of $(2,4)$ , complete intersection curves in $\mathbb {P}^3$ . This, together with recent results by Laza and O’Grady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of $(4,4)$ curves on $\mathbb {P}^1\times \mathbb {P}^1$ and the Baily–Borel compactification of moduli of quartic hyperelliptic K3 surfaces.

Tian’s criterion for K-stability states that a Fano variety of dimension n whose alpha invariant is greater than ${n}{/(n+1)}$ is K-stable. We show that this criterion is sharp by constructing n-dimensional singular Fano varieties with alpha invariants ${n}{/(n+1)}$ that are not K-polystable for sufficiently large n. We also construct K-unstable Fano varieties with alpha invariants ${(n-1)}{/n}$ .

We show that the anti-canonical volume of an $n$-dimensional Kähler–Einstein $\mathbb{Q}$-Fano variety is bounded from above by certain invariants of the local singularities, namely $\operatorname{lct}^{n}\cdot \operatorname{mult}$ for ideals and the normalized volume function for real valuations. This refines a recent result by Fujita. As an application, we get sharp volume upper bounds for Kähler–Einstein Fano varieties with quotient singularities. Based on very recent results by Li and the author, we show that a Fano manifold is K-semistable if and only if a de Fernex–Ein–Mustaţă type inequality holds on its affine cone.

In this study, TiO2 photoanodes doped with samarium ions via a method of hydrothermal treatment were used to fabricate dye-sensitized solar cells (DSSCs). Different doping concentrations were investigated on the effects of the cell’s performance. Some techniques including XRD, scanning electron microscopy, HRTEM, XPS, UV-Vis, photoluminescence were used to characterize the morphology, structure, and optic properties of the prepared photoanodes. The photovoltaic performance of the fabricated cells was further evaluated by measuring the current density–voltage (J–V) curves. It was found that: (1) The down-conversion luminescence effect derived from samarium doping could enhance the light-harvesting ability. (2) Compared with the undoped sample, the samarium-doped cells exhibited enhanced photovoltaic performance. Among the cells with different doping concentrations, the cell TiO2:0.015 Sm showed the best power conversion efficiency of 6.08% with a high open-circuit voltage (V oc) and a short-circuit current density (J sc).

The optical emission spectra of the plasma generated by a 1.06-μm Nd:YAG laser irradiation of Al target in air was recorded and analyzed in a spatially resolved manner. Electron temperatures and densities in the plasma were obtained using the relative emission intensities and the Stark-broadened linewidths of Al(I) emission lines, respectively. The dependence of the electron density and temperature on the distance from the target surface and on the laser irradiance were manifested. We also discussed how the air takes part in the plasma evolution process and confirmed that the ignition of the air plasma was by the collisions between the energetic electrons and the nitrogen atoms through a cascade avalanche process.