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The mass and centroid of rolling rings are the key factors affecting the mass and centroid detection of large launch vehicles. However, at present, the detection of rolling ring mass and centroid stays in the use of theoretical parameters or specially designed measuring tables, and the method used is time-consuming and laborious, and the compatibility is poor. To solve this problem, this paper proposes a flexible measurement method for rolling ring mass and centroid. This method does not need a special measuring platform and has the characteristics of strong universality and high precision. Finally, the feasibility of this method is verified by experiments.

To address challenges associated with the traditional drill positioning method, which demands manual walking tracking and imposes stringent environmental conditions, this paper introduces an improved weighted centroid localization (WCL) algorithm based on multiple magnetic beacons. This algorithm alleviates the environmental requirements. Initially, a magnetic beacon measurement model immune to sensor attitude is formulated, followed by the development of a positioning model based on multiple magnetic beacons. The WCL algorithm is then introduced and refined for positioning with multiple magnetic beacons. Finally, the effectiveness of the proposed approach is validated through simulation experiments, revealing an average error of 0.632 m in large-scale positioning. This demonstrates clear advantages over traditional methods, making it highly applicable.

To achieve high-speed and high-precision measurements, this study introduces a height measurement system based on LCCD and optical triangulation that utilizes a gray-scale centroid algorithm for sub-pixel positioning. Ultimately, the system achieves a resolution of 10 nm, a repeatability of 15 nm, and a stability of 7.8 nm, by setting up an experimental system in the laboratory.

Let A be a prime ring, Z(A) its center, Q its right Martindale quotient ring, C its extended centroid, ψ a non-zero b-generalized derivation of A with associated map ξ. In this article, we prove that: (i) If [ψ(x), ψ(y)] = 0 for all x, y ∈ A, then A is either commutative or there exists q ∈ Q such that ξ = ad(q), ψ(x) = -bxq, and qb = 0. (ii) If ψ(x) ◦ ψ(y) = 0 for all x, y ∈ A, then A is either commutative with char(A) = 2 or there exists q ∈ Q such that ψ(x) = -bxq and qb = 0. Additional results are established for cases involving [ξ(x), ψ(x)] = 0 or ξ(x)◦ψ(x) = 0, where char(A) = 2. Furthermore, we give some examples that show the importance of the hypotheses of our theorems. Sea A un anillo primo, Z(A) su centro, Q su anillo de cocientes de Martindale por derecha, C su centroide extendido, ψ una derivada b-generalizada de A con mapa asociado ξ. En este artículo probamos los siguientes resultados: (i) Si [ψ(x), ψ(y)] = 0 para todo x, y ∈ A, entonces o A es conmutativo o existe q ∈ Q tal que ξ = ad(q), ψ(x) = -bxq, y qb = 0. (ii) Si ψ(x) ◦ ψ(y) = 0 para todo x, y ∈ A, entonces o A es conmutativo con char(A) = 2 o existe q ∈ Q tal que ψ(x) = -bxq y qb = 0. También se analizan los casos donde [ξ(x), ψ(x)] = 0 o ξ(x) ◦ ψ(x) = 0, donde char(A) = 2. Se incluyen ejemplos que ilustran la importancia de las hipótesis de los teoremas.

The objective of this article is to establish a condition by which we are able to state that an ellipsoidal fragment formed by a plane cutting the ellipsoid can always contain a sphere in any position inside in it. A method to construct a chain of mutually tangent spheres inscribed in the ellipsoidal segment has been proposed. The locus of the centroid as well as the radii of the mutually tangent spheres have been computed. The prime concern of our work is to explore some geometrical properties of such a chain of spheres which includes the condition of inscribability of a sphere in any position inside the ellipsoid along with the computation of points of tangency between consecutive spheres.

The conventional K‐Means clustering algorithm is widely used for grouping similar data points by initially selecting random centroids. However, the accuracy of clustering results is significantly influenced by the initial centroid selection. Despite different approaches, including various K‐Means versions, suboptimal outcomes persist due to inadequate initial centroid choices and reliance on common normalization techniques like min‐max normalization. In this study, we propose an improved algorithm that selects initial centroids more effectively by utilizing a novel formula to differentiate between instance attributes, creating a single weight for differentiation. We introduce a preprocessing phase for dataset normalization without forcing values into a specific range, yielding significantly improved results compared to unnormalized datasets and those normalized using min‐max techniques. For our experiments, we used five real datasets and five simulated datasets. The proposed algorithm is evaluated using various metrics and an external benchmark measure, such as the Adjusted Rand Index (ARI), and compared with the traditional K‐Means algorithm and 11 other modified K‐Means algorithms. Experimental evaluations on these datasets demonstrate the superiority of our proposed methodologies, achieving an impressive average accuracy rate of up to 95.47% and an average ARI score of 0.95. Additionally, the number of iterations required is reduced compared to the conventional K‐Means algorithm. By introducing innovative techniques, this research provides significant contributions to the field of data clustering, particularly in addressing modern data‐driven clustering challenges.

Depending upon the load intensity and time variation information, we established the indicator system to analyze the of commercial halls, so that the further optimization actions can be taken in practical application scenarios. The representative clustering algorithms including the k-means and AC algorithm are chosen for the comparative analysis, and on this basis, an enhanced k-means algorithm is put forward in this study because the traditional method is sensitive to the initial clustering centroids and lacks the stability. A series of experiments are performed in this work, and the experimental findings reveal the validity of the proposed method. It provides a new idea for the efficiency analysis of commercial halls.

K-Means is a clustering algorithm based on a partition where the data only entered into one K cluster, the algorithm determines the number group in the beginning and defines the K centroid. The initial determination of the cluster center is very influential on the results of the clustering process in determining the quality of grouping. Better clustering results are often obtained after several attempts. The manhattan distance matrix method has better performance than the euclidean distance method. The author making the result of conducted testing with variations in the number of centroids (K) with a value of 2,3,4,5,6,7,8,9 and the authors having conclusions where the number of centroids 3 and 4 have a better iteration of values than the number of centroids that increasingly high and low based on the iris dataset.

K-Means is one of the most popular clustering algorithms because it is easy and simple when implemented. However, clustering results from K-Means are very sensitive to the selection of initial centroid. Better clustering results are often obtained after several experiments. In this study, Sum of Squared Error (SSE) was used as an approach to determine initial centroid of K-Means algorithm. If the SSE value is smaller then the data in one cluster will be more homogeneous and certainly give a good cluster result. In this study, Sum of Squared Error (SSE) was used as an approach to determine initial centroid of K-Means algorithm. Testing was performed on 3 datasets and the number of clusters 2, 3 and 4. From the test, the average value of Davies-Bouldin Index (DBI) for 3 datasets was 0.2427, while the Simple determine initial centroid of K-Means algorithm obtained an average DBI value of 0.2805. These results prove that clustering with method of determining initial centroid of K-Means algorithm based on Sum of Squared Error minimum able to improve clustering result and enhance DBI value obtained by simple determine initial centroid of K-Means algorithm.

An industrial scale experiment on the impacts of process variations on ingot quality of IN718 was carried out by monitoring the associated magnetic fields by VARmetric. The test performed monitored operational variations during the processing of the ingot when different arc gaps were maintained. A short, medium, long, and extra-long arc gap were chosen during the melt, the system was allowed to operate for approximately 2 hours at each arc gap before readjusting to the next arc gap. Magnetic field measurements were used to continuously determine the arc centroid position during the melt. A detailed analysis of the arc position measurements was conducted employing quantitative and statistical tools to assess the overall impact of process variations on the distribution of arcs. The arc rotation and spread were characterized for different arc gaps which clearly shows the transition from a diffuse to a constricted arc mode. A strong correlation was observed between the arc radius, rotations, and arc gap which may be able to be used to monitor arc gap size in lieu of drop shorts.