Relationship between Nichols braided Lie algebras and Nichols algebras

We establish the relationship among Nichols algebras, Nichols braided Lie algebras and Nichols Lie algebras. We prove two results: (i) Nichols algebra $\mathfrak B(V)$ is finite-dimensional if and only if Nichols braided Lie algebra $\mathfrak L(V)$ is finite-dimensional if there does not exist any $m$-infinity element in $\mathfrak B(V)$; (ii) Nichols Lie algebra $\mathfrak L^-(V)$ is infinite dimensional if $D^-$ is infinite. We give the sufficient conditions for Nichols braided Lie algebra $\mathfrak L(V)$ to be a homomorphic image of a braided Lie algebra generated by $V$ with defining relations.Comment: LeTex 18 pages, need JOLT-macros to compile. To appear in Journal of Lie Theor

On Nichols (braided) Lie algebras

We prove {\rm (i)} Nichols algebra $\mathfrak B(V)$ of vector space $V$ is finite-dimensional if and only if Nichols braided Lie algebra $\mathfrak L(V)$ is finite-dimensional; {\rm (ii)} If the rank of connected $V$ is $2$ and $\mathfrak B(V)$ is an arithmetic root system, then $\mathfrak B(V) = F \oplus \mathfrak L(V);$ and {\rm (iii)} if $\Delta (\mathfrak B(V))$ is an arithmetic root system and there does not exist any $m$-infinity element with $p_{uu} \not= 1$ for any $u \in D(V)$, then $\dim (\mathfrak B(V) ) = \infty$ if and only if there exists $V'$, which is twisting equivalent to $V$, such that $\dim (\mathfrak L^ - (V')) = \infty.$ Furthermore we give an estimation of dimensions of Nichols Lie algebras and two examples of Lie algebras which do not have maximal solvable ideals.Comment: 29 Pages; Substantially revised version; To appear in International Journal of Mathematic

Structures of Nichols (braided) Lie algebras of diagonal type

Let $V$ be a braided vector space of diagonal type. Let $\mathfrak B(V)$, $\mathfrak L^-(V)$ and $\mathfrak L(V)$ be the Nichols algebra, Nichols Lie algebra and Nichols braided Lie algebra over $V$, respectively. We show that a monomial belongs to $\mathfrak L(V)$ if and only if that this monomial is connected. We obtain the basis for $\mathfrak L(V)$ of arithmetic root systems and the dimension for $\mathfrak L(V)$ of finite Cartan type. We give the sufficient and necessary conditions for $\mathfrak B(V) = F\oplus \mathfrak L^-(V)$ and $\mathfrak L^-(V)= \mathfrak L(V)$. We obtain an explicit basis of $\mathfrak L^ - (V)$ over quantum linear space $V$ with $\dim V=2$.Comment: 23 pages. Version to appear in Journal of Lie Theor

Validation of Reference Genes for RT-qPCR Studies of Gene Expression in Preharvest and Postharvest Longan Fruits under Different Experimental Conditions

Reverse transcription quantitative PCR (RT-qPCR), a sensitive technique for quantifying gene expression, relies on stable reference gene(s) for data normalization. Although a few studies have been conducted on reference gene validation in fruit trees, none have been done on preharvest and postharvest longan fruits. In this study, 12 candidate reference genes, namely, CYP, RPL, GAPDH, TUA, TUB, Fe-SOD, Mn-SOD, Cu/Zn-SOD, 18SrRNA, Actin, Histone H3 and EF-1a, were selected. Expression stability of these genes in 150 longan samples was evaluated and analyzed using geNorm and NormFinder algorithms. Preharvest samples consisted of seven experimental sets, including different developmental stages, organs, hormone stimuli (NAA, 2,4-D and ethephon) and abiotic stresses (bagging and girdling with defoliation). Postharvest samples consisted of different temperature treatments (4 and 22 °C) and varieties. Our findings indicate that appropriate reference gene(s) should be picked for each experimental condition. Our data further showed that the commonly used reference gene Actin does not exhibit stable expression across experimental conditions in longan. Expression levels of the DlACO gene, which is a key gene involved in regulating fruit abscission under girdling with defoliation treatment, was evaluated to validate our findings. In conclusion, our data provide a useful framework for choice of suitable reference genes across different experimental conditions for RT-qPCR analysis of preharvest and postharvest longan fruits

3D Instances as 1D Kernels

We introduce a 3D instance representation, termed instance kernels, where instances are represented by one-dimensional vectors that encode the semantic, positional, and shape information of 3D instances. We show that instance kernels enable easy mask inference by simply scanning kernels over the entire scenes, avoiding the heavy reliance on proposals or heuristic clustering algorithms in standard 3D instance segmentation pipelines. The idea of instance kernel is inspired by recent success of dynamic convolutions in 2D/3D instance segmentation. However, we find it non-trivial to represent 3D instances due to the disordered and unstructured nature of point cloud data, e.g., poor instance localization can significantly degrade instance representation. To remedy this, we construct a novel 3D instance encoding paradigm. First, potential instance centroids are localized as candidates. Then, a candidate merging scheme is devised to simultaneously aggregate duplicated candidates and collect context around the merged centroids to form the instance kernels. Once instance kernels are available, instance masks can be reconstructed via dynamic convolutions whose weights are conditioned on instance kernels. The whole pipeline is instantiated with a dynamic kernel network (DKNet). Results show that DKNet outperforms the state of the arts on both ScanNetV2 and S3DIS datasets with better instance localization. Code is available: https://github.com/W1zheng/DKNet.Comment: Appearing in ECCV, 202

DoF-NeRF: Depth-of-Field Meets Neural Radiance Fields

Neural Radiance Field (NeRF) and its variants have exhibited great success on representing 3D scenes and synthesizing photo-realistic novel views. However, they are generally based on the pinhole camera model and assume all-in-focus inputs. This limits their applicability as images captured from the real world often have finite depth-of-field (DoF). To mitigate this issue, we introduce DoF-NeRF, a novel neural rendering approach that can deal with shallow DoF inputs and can simulate DoF effect. In particular, it extends NeRF to simulate the aperture of lens following the principles of geometric optics. Such a physical guarantee allows DoF-NeRF to operate views with different focus configurations. Benefiting from explicit aperture modeling, DoF-NeRF also enables direct manipulation of DoF effect by adjusting virtual aperture and focus parameters. It is plug-and-play and can be inserted into NeRF-based frameworks. Experiments on synthetic and real-world datasets show that, DoF-NeRF not only performs comparably with NeRF in the all-in-focus setting, but also can synthesize all-in-focus novel views conditioned on shallow DoF inputs. An interesting application of DoF-NeRF to DoF rendering is also demonstrated. The source code will be made available at https://github.com/zijinwuzijin/DoF-NeRF.Comment: Accepted by ACMMM 202

Architecture engineering of carbonaceous anodes for high‐rate potassium‐ion batteries

The limited lithium resource in earth's crust has stimulated the pursuit of alternative energy storage technologies to lithium‐ion battery. Potassium‐ion batteries (KIBs) are regarded as a kind of promising candidate for large‐scale energy storage owing to the high abundance and low cost of potassium resources. Nevertheless, further development and wide application of KIBs are still challenged by several obstacles, one of which is their fast capacity deterioration at high rates. A considerable amount of effort has recently been devoted to address this problem by developing advanced carbonaceous anode materials with diverse structures and morphologies. This review presents and highlights how the architecture engineering of carbonaceous anode materials gives rise to high‐rate performances for KIBs, and also the beneficial conceptions are consciously extracted from the recent progress. Particularly, basic insights into the recent engineering strategies, structural innovation, and the related advances of carbonaceous anodes for high‐rate KIBs are under specific concerns. Based on the achievements attained so far, a perspective on the foregoing, and proposed possible directions, and avenues for designing high‐rate anodes, are presented finally