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Journal articles on the topic 'Multidimensional Hypersphere'

Author: Grafiati

Published: 6 September 2023

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XU, JINQUAN, ERIC S. CARLSON, and VISHAL V. VORA. "Multidimensional Finite Differencing (MDFD) with Hypersphere-Close-Pack Grids." Chemical Engineering Communications 192, no. 8 (August 2005): 984–1016. http://dx.doi.org/10.1080/009864490517296.

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Sokolov, E. N. "Four-dimensional color space." Behavioral and Brain Sciences 20, no. 2 (June 1997): 207–8. http://dx.doi.org/10.1017/s0140525x9747142x.

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Multidimensional scaling of subjective color differences has shown that color stimuli are located on a hypersphere in four-dimensional space. The semantic space of color names is isomorphic with perceptual color space. A spherical four-dimensional space revealed in monkeys and fish suggests the primacy of common neuronal basis.

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Terada, Yoshikazu, and Hiroshi Yadohisa. "Multidimensional scaling with the nested hypersphere model for percentile dissimilarities." Procedia Computer Science 6 (2011): 364–69. http://dx.doi.org/10.1016/j.procs.2011.08.067.

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Kovalov, Sergiy, and Oleksandr Mostovenko. "SOME PROPERTIES OF THE HYPERSPHERE IN N-DIMENSIONAL SPACE." APPLIED GEOMETRY AND ENGINEERING GRAPHICS, no. 100 (May 24, 2021): 153–61. http://dx.doi.org/10.32347/0131-579x.2021.100.153-161.

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The study of the properties of surfaces contributes to the expansion of their use in solving various practical problems, especially if such properties can be generalized to manifolds of n-dimensional space. The most thoroughly studied are the properties of the simplest surfaces, including the properties of a sphere. That is why the simplest surfaces are most often used in practice. Each property not covered in the existing literature expands the indicated possibilities. Therefore, the purpose of this article is to identify the properties of the hypersphere unknown from the literature. Most of the properties of a circle and a sphere have been known since ancient times [1, 4, 5]. The generalized concept of a sphere into multidimensional spaces is based on the general principles of multidimensional geometry [3]. In [4], eleven basic properties of the sphere are listed and analyzed. In works [8, 10] it is shown that a circle can be considered as an isoline, and a sphere as an isosurface when modeling energy fields. In geometric modeling of energy fields with point energy sources, an essential role is played by the distances from the points of the field to the given energy sources [6, 7]. In [9], two schemes are given for determining the parameter t, taking into account the effect of the distance from the points of the field to the point sources of energy on the potentials of the points of the field. In a particular case, if this parameter is determined according to a simplified scheme with f(l)=al2, then the formula for calculating the potential of an arbitrary point of the energy field is a mathematical model of the energy field generated by the number n of point energy sources. The geometric model of the field will be a manifold that can be foliated into a one-parameter set of isospheres [8, 10]. Abstracting from the physical nature of the field, simplifying the equation for calculating the potential of an arbitrary point of the energy field and generalizing it to n-dimensional space, we can formulate the following properties: Property 1. A hypersphere can be considered as a locus of points, the sum of the squared distances from which to n given points is a constant value. Property 2. Arbitrary coefficients ki at distances li affect the parameters of the hypersphere without changing the type of surface.

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Maeda, Satoshi, Yu Watanabe, and Koichi Ohno. "A scaled hypersphere interpolation technique for efficient construction of multidimensional potential energy surfaces." Chemical Physics Letters 414, no. 4-6 (October 2005): 265–70. http://dx.doi.org/10.1016/j.cplett.2005.08.063.

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Unver, Mustafa, and Nihal Erginel. "Clustering applications of IFDBSCAN algorithm with comparative analysis." Journal of Intelligent & Fuzzy Systems 39, no. 5 (November 19, 2020): 6099–108. http://dx.doi.org/10.3233/jifs-189082.

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Density Based Spatial Clustering of Application with Noise (DBSCAN) is one of the mostly preferred algorithm among density based clustering approaches in unsupervised machine learning, which uses epsilon neighborhood construction strategy in order to discover arbitrary shaped clusters. DBSCAN separates dense regions from low density regions and simultaneously assigns points that lie alone as outliers to unearth the hidden cluster patterns in the datasets. DBSCAN identifies dense regions by means of core point definition, detection of which are strictly dependent on input parameter definitions: ε is distance of the neighborhood or radius of hypersphere and MinPts is minimum density constraint inside ε radius hypersphere. Contrarily to classical DBSCAN’s crisp core point definition, intuitionistic fuzzy core point definition is proposed in our preliminary work to make DBSCAN algorithm capable of detecting different patterns of density by two different combinations of input parameters, particularly is a necessity for the density varying large datasets in multidimensional feature space. In this study, preliminarily proposed DBSCAN extension is studied: IFDBSCAN. The proposed extension is tested by computational experiments on several machine learning repository real-time datasets. Results show that, IFDBSCAN is superior to classical DBSCAN with respect to external & internal performance indices such as purity index, adjusted rand index, Fowlkes-Mallows score, silhouette coefficient, Calinski-Harabasz index and with respect to clustering structure results without increasing computational time so much, along with the possibility of trying two different density patterns on the same run and trying intermediary density values for the users by manipulating α margin.

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Wu, Guangjun, Bingqing Zhu, Jun Li, Yong Wang, and Yungang Jia. "H2SA-ALSH: A Privacy-Preserved Indexing and Searching Schema for IoT Data Collection and Mining." Wireless Communications and Mobile Computing 2022 (April 18, 2022): 1–12. http://dx.doi.org/10.1155/2022/9990193.

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Currently, smart devices of Internet of Things generate massive amount of data for different applications. However, it will expose sensitive information to external users in the process of IoT data collection, transmission, and mining. In this paper, we propose a novel indexing and searching schema based on homocentric hypersphere and similarity-aware asymmetric LSH (H2SA-ALSH) for privacy-preserved data collection and mining over IoT environments. The H2SA-ALSH collects multidimensional data objects and indexes their features according to the Euclidean norm and cosine similarity. Additionally, we design a c - k -AMIP searching algorithm based on H2SA-ALSH. Our approach can boost the performance of the maximum inner production (MIP) queries and top- k queries for a given query vector using the proposed indexing schema. Experiments show that our algorithm is excellent in accuracy and efficiency compared with other ALSH-based algorithms using real-world datasets. At the same time, our indexing scheme can protect the user’s privacy via generating similarity-based indexing vectors without exposing raw data to external users.

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Karbauskaitė, Rasa, and Gintautas Dzemyda. "Optimization of the Maximum Likelihood Estimator for Determining the Intrinsic Dimensionality of High–Dimensional Data." International Journal of Applied Mathematics and Computer Science 25, no. 4 (December 1, 2015): 895–913. http://dx.doi.org/10.1515/amcs-2015-0064.

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AbstractOne of the problems in the analysis of the set of images of a moving object is to evaluate the degree of freedom of motion and the angle of rotation. Here the intrinsic dimensionality of multidimensional data, characterizing the set of images, can be used. Usually, the image may be represented by a high-dimensional point whose dimensionality depends on the number of pixels in the image. The knowledge of the intrinsic dimensionality of a data set is very useful information in exploratory data analysis, because it is possible to reduce the dimensionality of the data without losing much information. In this paper, the maximum likelihood estimator (MLE) of the intrinsic dimensionality is explored experimentally. In contrast to the previous works, the radius of a hypersphere, which covers neighbours of the analysed points, is fixed instead of the number of the nearest neighbours in the MLE. A way of choosing the radius in this method is proposed. We explore which metric—Euclidean or geodesic—must be evaluated in the MLE algorithm in order to get the true estimate of the intrinsic dimensionality. The MLE method is examined using a number of artificial and real (images) data sets.

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Sharmin, Dmitrii V., Tamara N. Sharmina, and Valentin G. Sharmin. "Curvature Tensor of the n-Surface and Its Spherical Image in En+k." UNIVERSITY NEWS. NORTH-CAUCASIAN REGION. NATURAL SCIENCES SERIES, no. 1 (217) (March 31, 2023): 29–34. http://dx.doi.org/10.18522/1026-2237-2023-1-29-34.

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The main task of classical multidimensional differential geometry is to study the properties of various n-surfaces. Often these studies use torsion coefficients that are defined for any n-surface with codimension k &gt; 1 in (n + k)-dimensional Euclidean space. For hypersurfaces, the torsion coefficients are not defined.Another important concept used to study the properties of n-surfaces is the spherical Gaussian mapping. The Gaussian mapping defined on submanifolds of Euclidean and pseudo-Euclidean spaces allows one to study the external properties of a submanifold immersed in a Euclidean or pseudo-Euclidean space. In a number of papers, the properties of the Gaussian mapping are studied, as well as the geometric characteristics of the images of submanifolds under a spherical mapping, which are submanifolds of a hypersphere or a Grassmannian.In this article, we study the local properties of the spherical image of a regular n-surface of arbitrary codimension. The spherical mapping is defined for n-surfaces with codimension greater than one in Euclidean space by means of a regular vector field. Each vector of this field at a point of the submanifold is orthogonal to the tangent space of the submanifold at the chosen point.The article uses the methods of differential and Riemannian geometry, as well as tensor analysis to study n-surfaces with a codimension greater than one. Under some additional conditions, a connection is established between the curvature tensor of a given surface and the curvature tensor of its spherical image. Under the same additional conditions, some geometric characteristics of the points of the spherical image of the original n-surface are studied.

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Stoyan, Yuriy, Georgiy Yaskov, Tatiana Romanova, Igor Litvinchev, Sergey Yakovlev, and José Manuel Velarde Cantú. "Optimized packing multidimensional hyperspheres: a unified approach." Mathematical Biosciences and Engineering 17, no. 6 (2020): 6601–30. http://dx.doi.org/10.3934/mbe.2020344.

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Vazquez-Leal, H., V. M. Jimenez-Fernandez, B. Benhammouda, U. Filobello-Nino, A. Sarmiento-Reyes, A. Ramirez-Pinero, A. Marin-Hernandez, and J. Huerta-Chua. "Modified Hyperspheres Algorithm to Trace Homotopy Curves of Nonlinear Circuits Composed by Piecewise Linear Modelled Devices." Scientific World Journal 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/938598.

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We present a homotopy continuation method (HCM) for finding multiple operating points of nonlinear circuits composed of devices modelled by using piecewise linear (PWL) representations. We propose an adaptation of the modified spheres path tracking algorithm to trace the homotopy trajectories of PWL circuits. In order to assess the benefits of this proposal, four nonlinear circuits composed of piecewise linear modelled devices are analysed to determine their multiple operating points. The results show that HCM can find multiple solutions within a single homotopy trajectory. Furthermore, we take advantage of the fact that homotopy trajectories are PWL curves meant to replace the multidimensional interpolation and fine tuning stages of the path tracking algorithm with a simple and highly accurate procedure based on the parametric straight line equation.

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AKHMETYANOV, R. F., and E. S. SHIKHOVTSEVA. "REPRESENTATION OF THE PAIRED INTERACTION POTENTIAL IN THE FORM OF MULTIDIMENSIONAL RATIONAL SERIES IN JACOBI VARIABLES FOR MANY-BODY PROBLEMS." Izvestia Ufimskogo Nauchnogo Tsentra RAN, no. 4 (December 13, 2021): 9–15. http://dx.doi.org/10.31040/2222-8349-2021-0-4-9-15.

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Scalar power functions of the form x1 + + xN -v Î are in some cases found in physical problems and applications, especially in many-body problems with paired interactions. There are known decompositions for two vectors in three-dimensional space. In this paper, we consider analogous decompositions with any number of N arbitrary M-dimensional vectors in Euclidean space as a product of a multidimensional rational series with respect to spatial variables and hyperspheric functions on the unit sphere SM-1. Such an advantage of expansion arises in three-body problems when solving the Faddeev equation, where it is known that the main problem is the approximate choice of approximation of interaction potentials, in which the t-matrix scattering elements acquired a separable form.

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Huang, Mingfeng, Jianping Sun, Kang Cai, and Qiang Li. "MEMD-Based Hybrid Modal Identification for High-Rise Structures with Multi-Sensor Vibration Measurements." Applied Sciences 12, no. 16 (August 20, 2022): 8345. http://dx.doi.org/10.3390/app12168345.

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Although widely used in various fields due to its powerful capability of signal processing, empirical mode decomposition has to decompose signals separately, which limits its application for multivariate data such as the structural monitoring data recorded by multiple sensors. In order to avoid this shortcoming, a multivariate extension of empirical mode decomposition is proposed to deal with the multidimensional signals synchronously by employing a real-valued projection on hyperspheres. This study presents a hybrid modal identification method combining the multivariate empirical mode decomposition with stochastic subspace identification and fast Bayesian FFT methods to more conveniently and accurately identify structural dynamic parameters from multi-sensor vibration measurements. Deployed as a preprocessing tool, the multivariate signals are decomposed into several aligned intrinsic mode functions, which contain only a dominant component in the frequency domain. Then, the modal parameters can be identified by advanced fast Bayesian FFT and stochastic subspace identification directly. The combined method is first validated by a numerical illustration of a frame structure and then is applied in a shaking table test and a full-scale measurement under nonstationary earthquake excitation. Compared with the finite element method, the peak–pick, the half-power bandwidth methods, and Hilbert–Huang transform method, the results show that this hybrid method is more robust and reliable in the modal parameters identification. The main contribution of this paper is to develop a more effective integrated approach for accurate modal identification with the output-only multi-dimensional nonstationary signal.

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Grishentsev, A. Yu. "Numerical solution of multidimensional Thomson Problem for vectors packaging on hypersphere in broadband radiocommunication problems." Scientific and Technical Journal of Information Technologies, Mechanics and Optics, July 1, 2019, 730–39. http://dx.doi.org/10.17586/2226-1494-2019-19-4-730-739.

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Watanabe, Junya. "Detecting (non)parallel evolution in multidimensional spaces: angles, correlations and eigenanalysis." Biology Letters 18, no. 2 (February 2022). http://dx.doi.org/10.1098/rsbl.2021.0638.

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Parallelism between evolutionary trajectories in a trait space is often seen as evidence for repeatability of phenotypic evolution, and angles between trajectories play a pivotal role in the analysis of parallelism. However, properties of angles in multidimensional spaces have not been widely appreciated by biologists. To remedy this situation, this study provides a brief overview on geometric and statistical aspects of angles in multidimensional spaces. Under the null hypothesis that trajectory vectors have no preferred directions (i.e. uniform distribution on hypersphere), the angle between two independent vectors is concentrated around the right angle, with a more pronounced peak in a higher-dimensional space. This probability distribution is closely related to t - and beta distributions, which can be used for testing the null hypothesis concerning a pair of trajectories. A recently proposed method with eigenanalysis of a vector correlation matrix can be connected to the test of no correlation or concentration of multiple vectors, for which simple test procedures are available in the statistical literature. Concentration of vectors can also be examined by tools of directional statistics such as the Rayleigh test. These frameworks provide biologists with baselines to make statistically justified inferences for (non)parallel evolution.

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