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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.

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.

In a classical paper [20] in 2000, Lutwak-Yang-Zhang established the \\(L^p\\) analog of the Petty projection inequality and the \\(L^p\\) analog of the Busemann-Petty centroid inequality. In Section 7 of [20], Lutwak-Yang-Zhang proposed the important \\(L^p\\) projection centroid conjecture. We give a positive solution to the \\(L^p\\) projection centroid conjecture in this work.

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.

The cevian triangle corresponding to an interior point \\(M\\) of a triangle is the triangle determined by the feet of the three cevians concurrent at \\(M\\). It is known that the area of the cevian triangle for an interior point \\(M\\) of a triangle is at most \\(14\\) of the area of the triangle, with maximum attained when \\(M\\) is the triangle's centroid. This can be generalized from triangles to \\(n\\)-dimensional simplices, with \\(14\\) replaced by \\(1n^n\\), using barycentric coordinates. We also use this method to solve two optimization problems about the parts of this simplex.

We present an estimate of the occurrence rate of hot Jupiters (7 R ⊕ ≤ R p ≤ 2 R J, 0.8 ≤ P b ≤ 10 days) around early-type M dwarfs based on stars observed by the Transiting Exoplanet Survey Satellite (TESS) during its primary mission. We adopt stellar parameters from the TESS Input Catalog and construct a sample of 60,819 M dwarfs with 10.5 ≤ T mag ≤ 13.5, effective temperatures 2900 ≤ T eff ≤ 4000 K, and stellar masses 0.45 ≤ M * ≤ 0.65 M ⊙. We conduct a uninformed transit search using a detection pipeline based on the box least square search and characterize the searching completeness through an injection and recovery experiment. We combine a series of vetting steps including light centroid measurement, odd/even and secondary eclipse analysis, rotation and transit period synchronization tests as well as inspecting the ground-based photometric, spectroscopic, and imaging observations. Finally, we find a total of nine planet candidates, all of which are known TESS objects of interest. We obtain an occurrence rate of 0.27% ± 0.09% for hot Jupiters around early-type M dwarfs that satisfy our selection criteria. Compared with previous studies, the occurrence rate of hot Jupiters around early-type M dwarfs is smaller than all measurements for FGK stars, although they are consistent within 1σ–2σ. There is a trend that the occurrence rate of hot Jupiters has a peak at G dwarfs and falls toward both hotter and cooler stars. Combining results from transit, radial velocity, and microlensing surveys, we find that hot Jupiters around early-type M dwarfs possibly show a steeper decrease in the occurrence rate per logarithmic semimajor axis bin ( dN/dlog10a ) when compared with FGK stars.

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.