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

Time To Live: Temporal Management of Large-Scale RFID Applications

In coming years, there will be billions of RFID tags living in the world tagging almost everything for tracking and identification purposes. This phenomenon will impose a new challenge not only to the network capacity but also to the scalability of event processing of RFID applications. Since most RFID applications are time sensitive, we propose a notion of Time To Live (TTL), representing the period of time that an RFID event can legally live in an RFID data management system, to manage various temporal event patterns. TTL is critical in the "Internet of Things" for handling a tremendous amount of partial event-tracking results. Also, TTL can be used to provide prompt responses to time-critical events so that the RFID data streams can be handled timely. We divide TTL into four categories according to the general event-handling patterns. Moreover, to extract event sequence from an unordered event stream correctly and handle TTL constrained event sequence effectively, we design a new data structure, namely Double Level Sequence Instance List (DLSIList), to record intermediate stages of event sequences. On the basis of this, an RFID data management system, namely Temporal Management System over RFID data streams (TMS-RFID), has been developed. This system can be constructed as a stand-alone middleware component to manage temporal event patterns. We demonstrate the effectiveness of TMS-RFID on extracting complex temporal event patterns through a detailed performance study using a range of high-speed data streams and various queries. The results show that TMS-RFID has a very high throughout, namely 170,000 - 870,000 events per second for different highly complex continuous queries. Moreover, the experiments also show that the main structure to record the intermediate stages in TMS-RFID does not increase exponentially with the number of events. These illustrate that TMS-RFID not only has a high processing speed, but also has a good scalability

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

SMCB-E-03172004-0141.R2 1 A Multiagent Evolutionary Algorithm for Constraint Satisfaction Problems

Abstract—With the intrinsic properties of constraint satisfaction problems (CSPs) in mind, we divide CSPs into two types, namely, permutation CSPs and non-permutation CSPs. According to their characteristics, several behaviors are designed for agents by making use of the ability of agents to sense and act on the environment. These behaviors are controlled by means of evolution, so that the multiagent evolutionary algorithm for constraint satisfaction problems (MAEA-CSPs) results. To overcome the disadvantages of the general encoding methods, the minimum conflict encoding is also proposed. Theoretical analyses show that MAEA-CSPs has a linear space complexity and converges to the global optimum. The first part of the experiments uses 250 benchmark binary CSPs and 79 graph coloring problems from the DIMACS challenge to test the performance of MAEA-CSPs for non-permutation CSPs. MAEA-CSPs is compared with six well-defined algorithms and the effect of the parameters is analyzed systematically. The second part of the experiments uses a classical CSP, n-queen problems, and a more practical case, job-shop scheduling problems (JSPs), to test the performance of MAEA-CSPs for permutation CSPs. The scalability of MAEA-CSPs along n for n-queen problems is studied with great care. The results show that MAEA-CSPs achieves good performance when n increases from 10 4 to 10 7, and has a linear time complexity. Even for 10 7-queen problems, MAEA-CSPs finds the solutions by only 150 seconds. For JSPs, 59 benchmark problems are used, and good performance is also obtained. Index Terms—Constraint satisfaction problems, evolutionary algorithms, graph coloring problems, job-shop schedulin