Radius Of Hypersphere Research Articles

Two-Electron Atomic Systems—A Simple Method for Calculating the Ground State near the Nucleus: Some Applications

A simple method of non-relativistic variational calculations of the electronic structure of a two-electron atom/ion, primarily near the nucleus, is proposed. The method as a whole consists of a standard solution of a generalized matrix eigenvalue equation, all matrix elements of which are reduced to a numerical calculation of one-dimensional integrals. The distinctive features of the method are as follows: The use of the hyperspherical coordinate system. The inclusion of logarithms of the hyperspherical radius R in the basis functions, similar to the Fock expansion. Using a special basis function including the leading angular Fock coefficients to provide the correct behavior of the wave function near the nucleus. The main numerical parameters characterizing the properties of the helium atom and a number of helium-like ions near the nucleus are calculated and presented in tables. Among others, the specific coefficients, a21, of the Fock expansion, which can only be calculated using a wave function with the correct behavior near the nucleus, are presented in table and graphs.

Continuous percolation in a Hilbert space for a large system of qubits

AbstractThe development of percolation theory was historically shaped by its numerous applications in various branches of science, in particular in statistical physics, and was mainly constrained to the case of Euclidean spaces. One of its central concepts, the percolation transition, is defined through the appearance of the infinite cluster, and therefore cannot be used in compact spaces, such as the Hilbert space of an N-qubit system. Here, we propose its generalization for the case of a random space covering by hyperspheres, introducing the concept of a “maximal cluster”. Our numerical calculations reproduce the standard power-law relation between the hypersphere radius and the cover density, but show that as the number of qubits increases, the exponent quickly vanishes (i.e., the exponentially increasing dimensionality of the Hilbert space makes its covering by finite-size hyperspheres inefficient). Therefore the percolation transition is not an efficient model for the behavior of multiqubit systems, compared to the random walk model in the Hilbert space. However, our approach to the percolation transition in compact metric spaces may prove useful for its rigorous treatment in other contexts.

Dynamical integrity estimation in time delayed systems: A rapid iterative algorithm

The robustness of dynamical systems against external perturbations is crucial in engineering; however, it is often overlooked for the lack of methods for rapidly computing it. This paper proposes a novel algorithm for estimating the robustness of systems subject to time delay. More precisely, the algorithm iteratively estimates the so-called local integrity measure (LIM), that is, the radius of the largest hypersphere centered at the fixed point and located within its basin of attraction. Since time-delayed systems are infinite dimensional, initial conditions are restricted to a constrained type of initial functions. The semi-discretization method is used for rapidly simulating the dynamics of the systems, while trajectories are classified as converging or diverging using a subdivision of the reduced phase space into cells. The algorithm was tested on four different mechanical systems, and in all cases it very quickly provided an accurate estimation of the LIM. Moreover, it enabled the study of LIM trends in a multi-dimensional parameter space, which would have been unfeasible with alternative methods. This breakthrough in computational efficiency has important implications for engineering design, allowing for careful consideration of dynamical integrity and enhancing the safety and reliability of engineered systems, especially in the presence of time delays.

A transformed-feature-space data augmentation method for defect segmentation

Data augmentation is widely used in convolutional neural network (CNN) models to improve the performance of downstream tasks. The images generated by traditional data augmentation methods are usually random and can struggle to overcome the range limitation of the sampled data distribution, which can result in meaningless augmented data and poor diversity. As a result, a novel data augmentation algorithm is proposed to generate images that are outside the scope of the sampled data distribution along the feature direction with diversity weights. First, an image is mapped to the latent feature space based on the encoder. Then, the range loss function can restrict the latent variables within a hypersphere of radius k. The feature directions and diversity weights can be found based on the entropy variation in the latent feature space. During the CNN model’s training process, probabilities are constructed based on diversity weights to select the feature direction. Then, the latent variable of the input image is transformed in the feature direction to the region beyond the hypersphere. Finally, the transformed latent variable is mapped to the pixel space based on the decoder to generate images outside the sampled data space to improve diversity. Experimentally, the proposed method considerably improves the performance of defect segmentation in different industrial scenes.

A novel dynamic radius support vector data description based fault diagnosis method for proton exchange membrane fuel cell systems

Timely fault detection is critical to improving the reliability and durability of the proton exchange membrane fuel cell (PEMFC) system. This paper proposes a novel fault diagnosis method, dynamic radius support vector data description (DR-SVDD), to efficiently identify the PEMFC system's faults. Compared to the classic support vector data description (SVDD) and improved SVDDs, this method considers both the SVDD hypersphere radius information and the distribution characteristics of the training set samples to obtain a more accurate and adequate description of the sample data. The cell voltages and the pressure drops at the cathode and anode obtained experimentally under various fault conditions are chosen as the feature variables for the PEMFC fault diagnosis. The comparative results show that the proposed DR-SVDD strategy performs well in fault class identification for a PEMFC system.

Geometric imbalanced deep learning with feature scaling and boundary sample mining

Data imbalance is a significant factor affecting classification performance in computer vision. In particular, data imbalance is harmful to classification learning and representation learning. To address this issue, this paper proposes a geometric deep learning framework combined with Feature Scaling Module (FSM) and Boundary Samples Mining Module (BSMM). Considering the geometric information in sample distributions of training samples, FSM is proposed to scale the features by hypersphere radius of each class, which improves the representation ability of minority classes. Meanwhile, it is noteworthy that the relationships and information between samples are essential for classification. Therefore, BSMM is proposed to mine the boundary samples by Gabriel Graph that takes the relationships into account. Finally, a loss scheduler is designed to adjust the training process of these two modules. With the scheduler, the model first learns representation and then focuses more on minority classes gradually. Extensive experiments on three benchmark datasets demonstrate the advantages of the proposed learning framework over the state-of-the-art models for solving the imbalance problem.

Dynamical integrity assessment of stable equilibria: a new rapid iterative procedure

A new algorithm for the estimation of the robustness of a dynamical system’s equilibrium is presented. Unlike standard approaches, the algorithm does not aim to identify the entire basin of attraction of the solution. Instead, it iteratively estimates the so-called local integrity measure, that is, the radius of the largest hypersphere entirely included in the basin of attraction of a solution and centred in the solution. The procedure completely overlooks intermingled and fractal regions of the basin of attraction, enabling it to provide a significant engineering quantity in a very short time. The algorithm is tested on four different mechanical systems of increasing dimension, from 2 to 8. For each system, the variation of the integrity measure with respect to a system parameter is evaluated, proving the engineering relevance of the results provided. Despite some limitations, the algorithm proved to be a viable alternative to more complex and computationally demanding methods, making it a potentially appealing tool for industrial applications.

A Survey of Recent Variants and Applications of Antlion Optimizer

This work proposes a review of a recently developed swarm intelligence-based metaheuristic algorithm called Antlion Optimizer (ALO), its variants, and applications. The suitable blending of a random walk with an adaptive shrinking of hypersphere radius makes this algorithm more effective and impressive over other recent optimization algorithms. This paper elaborates on the recent variants of ALO by reviewing the concerned publications. It also summarized the applications of ALO for solving real-world complex optimization problems of a wide variety of areas. So, this paper comprises of summarized review of various recently published ALO papers. Firstly, the natural phenomena of ALO and the working principle of its various operators are described. Then the recently developed variants of ALO are described in detail depicting in various categories. The real-world applications using ALO and its variants are also described under global optimization, power and system engineering, electronics and communication engineering, machine learning, environmental engineering, and networking.

A new dynamic radius SVDD for fault detection of aircraft engine

When using traditional support vector data description (SVDD) to deal with classification problems, low accuracy is often achieved, especially in the case of noise interference. The traditional SVDD adopts a fixed hypersphere radius to set classification boundary, which means that it tends to force all normal samples and outliers to be separated by a fixed radius, which is unreasonable to a certain extent. In order to reduce the impact of this defect, a dynamic radius support vector data description (DR-SVDD) is firstly proposed in this paper, which introduces the idea of angle in the kernel space and flexibly selects the relevant decision radius for each testing sample. Then, the feasibility and effectiveness of the proposed algorithm are verified on several benchmark data sets. Finally, DR-SVDD based fault detection is carried out for a certain turboshaft engine, and the expected results are obtained, which fully demonstrates its effectiveness and robustness.

Condensing the solution of support vector machines via radius-margin bound

The paper deals with the problem of how to condense efficiently the used basis vectors in the decision function of support vector machines (SVM). Most existing methods learn the basis vectors and the corresponding coefficients by maximizing the margin between different classes. They ignore the key fact that condensing the solution of SVM is equivalent to constructing the SVM model in the transformation space determined by the basis vectors. Thus, the radius-margin bound, which is related with the generalization ability of SVM, changes with them. In the paper, we propose a novel method called sparse support vector machine guided by radius-margin bound (RMB-SSVM) to learn the condensed solution for SVM. The key characteristic of RMB-SSVM is that it employs a criterion that exploits integratedly the margin and the radius of the minimum hypersphere enclosing the data to guide the selecting of the basis vectors and the learning of the corresponding coefficients. Thus, the learning criterion of RMB-SSVM is directly related with the generalization ability of SVM and so it can yield better performance. We develop an effective method to handle the proposed optimization model, and propose a heuristic scheme to select the basis vectors used in the decision function. Further, we explore how to use the maximum pairwise distance over the pairs of data points to approximate the radius in our methodology. This can simplify the proposed model and reduce the training time while not drastically impairing the performance. Finally, we conduct the comprehensive experiments to demonstrate the effectiveness and superiority of the proposed methods by comparing them with the counterparts.

A hardness of approximation result in metric geometry

We show that it is $\mathsf{NP}$-hard to approximate the hyperspherical radius of a triangulated manifold up to an almost-polynomial factor.

Averaged electron densities of the helium-like atomic systems

Different kinds of averaging of the wavefunctions/densities of the two-electron atomic systems are investigated. Using several fully three-body methods of variational and direct types, the ground state wave functions Ψ of the helium-like atomic systems with nucleus charge 1 ≤ Z ≤ 5 are calculated in a few coordinate systems including the hyperspherical coordinates R,α,θ. The wave functions Ψav(R) of the hyperspherical radius R are calculated numerically by averaging Ψ over the hyperspherical angles α and θ. The exact analytic representations for the relative derivatives Ψav′(0)/Ψav(0) and Ψav′′(0)/Ψav(0) are derived. Analytic approximations very close to Ψav(R) are obtained. Using the Pekeris-like wave functions Ψ, the one-electron densities ρ(r) are calculated as functions of the electron–nucleus distance r. The relevant derivatives ρ′(0)/ρ(0) and ρ″(0)/ρ(0) characterizing the behavior of ρ(r) near the nucleus are calculated numerically. Very accurate analytical approximations, representing the one-electron density both near the nucleus and far away from it, are derived. All the analytical and numerical results are supplemented with tables and graphs.

Clustering applications of IFDBSCAN algorithm with comparative analysis

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.

Geometric probabilistic evolutionary algorithm

In this paper we introduce a crossover operator and a mutation operator, called Bernoulli Reflection Search Operator (BRSO) and Cauchy Distributed Inversion Search Operator (CDISO) respectively, in order to define the search mechanism of a new evolutionary algorithm for global continuous optimisation, namely the Geometric Probabilistic Evolutionary Algorithm (GPEA). Both operators have been motivated by geometric transformations, namely inversions with respect to hyperspheres and reflections with respect to a hyperplanes, but are implemented stochastically. The design of the new operators follows statistical analyses of the search mechanisms (Inversion Search Operator (ISO) and Reflection Search Operator (RSO)) of the Spherical Evolutionary Algorithm (SEA). From the statistical analyses, we concluded that the non-linearity of the ISO can be imitated stochastically, avoiding the calculation of several parameters such as the radius of hypersphere and acceptable regions of application. In addition, a new mutation based on a normal distribution is included in CDISO in order to guide the exploration. On the other hand, the BRSO imitates the mutation of individuals using reflections with respect to hyperplanes and complements the CDISO. In order to evaluate the proposed method, we use the benchmark functions of the special session on real-parameter optimisation of the CEC 2013 competition. We compare GPEA against 12 state-of-the-art methods, and present a statistical analysis using the Wilcoxon signed rank and the Friedman tests. According to the numerical experiments, GPEA exhibits a competitive performance against a variety of sophisticated contemporary algorithms, particularly in higher dimensions.

Analytical tension-distribution computation for cable-driven parallel robots using hypersphere mapping algorithm

Fully restrained cable-driven parallel robots (CDPRs) are known for their low moving inertia, high payload-to-weight ratio, and large workspace, and are widely used in modern industry. For the existence of redundancy, the cable tension-distribution computation is essentially an optimisation problem. To avoid the long computation time of the iterative calculation, and satisfy the real-time demands of control design and trajectory planning, this paper presents a novel measurement index of the magnitude of tensions, the tension hypersphere radius (THSR) for fully restrained CDPRs. This can be used to prevent slackness and excessive tension in the cables. Furthermore, according to non-probabilistic convex uncertainty propagation, the tensions with the minimum THSR can be calculated via an analytical formulation, in which the variables are the geometry, external wrench, and prescribed trajectory. The validity and applicability of presented tension hypersphere mapping algorithm are verified by the comparing with the results of 2-norm quadratic programming.

Mapping the Wall-Region Dynamics of High-Flux Gas-Solid Riser Using Scaling Regions from the Solid Concentration Time Series

An experimental study of the gas-solid flow dynamics in the high-flux CFB riser was accomplished by analysing the scaling regions from solid concentration signals collected from a 76 mm internal diameters and 10 m high riser of a circulating fluidized bed (CFB) system. The riser was operated at 4.0 to 10.0 m/s gas velocity and 50 to 550 kg/m2s solids flux. Spent fluid catalytic cracking (FCC) catalyst particles of 67 μm mean diameter and 1500 kg/m3 density together with 70% to 80% humid air was used. Solid concentration data were analysed using codes prepared in FORTRAN 2008 to get correlation integrals at different embedding dimensions and operating conditions and plot their profiles. Scaling regions were identified by visual inspection method and their location on planes determined. Scaling regions were analysed based on operating conditions and riser spatial locations. It was found that scaling regions occupy different locations on the plane depending on the number of embedding dimensions and operating conditions. As the number of embedding dimensions increases the spacing between scaling regions decreases until it saturates towards higher embedding dimensions. Slopes of scaling regions increases with embedding dimensions until saturation where they become constant. Slopes of scaling regions towards the wall decrease while the number of scaling regions for a particular profile increases. The span of the scaling region is wider at the initial values of hyperspherical radius than its final values. The scaling regions in some flow development sections show multifractal behaviour for each embedding dimension which manifests into visible basin which is defined in this study as multifractal basin. Further, the end points of the scaling region for each correlation integral profile differ from each other as the embedding dimension changes. This study suggests that identification of scaling region by visual inspection method is useful in understanding the gas-solid flow dynamics in the High-Flux CFB riser system. Further studies are recommended on risers of different diameters and heights operated at low and high solid fluxes and different gas velocities for comparison or usage of time series of different signal types like pressure fluctuations.

Mapping Correlation Dimension along the Wall Region of a High-Flux Gas-Solid Riser Using Embedded Solid Concentration Time Series

Analysis of the entrance and wall dynamics of a high-flux gas-solid riser was conducted using embedded solid concentration time series collected from a 76 mm internal diameter and 10 m high riser of a circulating fluidized bed (CFB) system. The riser was operated at 4.0 to 10.0 m/s air velocity and 50 to 550 kg/m2s solids flux of spent fluid catalytic cracking (FCC) catalyst particles with 67 μm mean diameter and density of 1500 kg/m3. Data were analyzed using prepared FORTRAN 2008 code to get correlation integral followed by determination of correlation dimensions with respect to the hyperspherical radius and their profiles, plots of which were studied. It was found that correlation dimension profiles at the centre have single peak with higher values than the wall region profiles. Towards the wall, these profiles have double or multiple peaks showing bifractal or multifractal flow behaviors. As the velocity increases the wall region profiles become random and irregular. Further it was found that, as the height increases the correlation dimension profiles shift towards higher hyperspherical radius at the centre and towards lower hyperspherical radius in the wall region at r/R = 0.81. The established method of mapping correlation dimension profiles in this study forms a suitable tool for analysis of high-flux riser dynamics compared to other analyses approaches. However, further analysis is recommended to other gas-solid CFB riser of different dimensions operated at high-flux conditions using the established method.

The Application of One-Class Classifier Based on CNN in Image Defect Detection

In the field of defect detection, image processing algorithms and feature extraction algorithms have some limitations, owing to their necessity for extracting a large number of different features of diverse products images. Meanwhile, the images of defective products are less and various. Aiming at these problems, we presented a One-Class classifier based on deep convolution neural network to detect the defect images in this paper. We design a loss function with the penalty term based on Euclidean distance to train the deep convolution neural network model. A hypersphere is used as classification decision surface after setting an appropriate hypersphere radius according to the inspection accuracy. It maps the non-defective products into a hypersphere in a high dimensional feature space, while the defect images are mapped somewhere far from the center of hypersphere. Thus, a One-Class classifier based on convolutional neural network(CNN) model is proposed to detect the defects. Experiments show that the proposed method, with less number of iteration, help build the classifier for image defect detection with high generalization ability and high detection precision.

Optimum Channel Equalization with Template Compression

Latent structure in the signals transmitted through dispersive channels (real or complex) was exploited to formulate the maximum a posteriori probability (MAP) equalization in closed form, which allowed robust integer processing and some benchmark results. Emphasis is mostly on the adaptive implementation of the equalizer on unknown channels, with particular attention to complexity and bit error rate (BER) performance. The latent structure, laid as rules underlying the channel templates, allowed further the design of an original robust hyper-sphere-based complexity reduction algorithm wherein the hyper-sphere radius is self-adjusting. Simulation results are provided on the algorithm behavior and on performance of the proposed adaptive equalizer.

Single-epoch GPS L1 phase measurements for determining rigid-body attitudes

As the length of the baseline installed on a vehicle on land, sea, or air is usually very short, it is ideal to work with the global positioning system L1 carrier double-difference phase and pseudorange measurements. Meanwhile, the relative body-fixed coordinates of the baselines should be strictly integrated into an adjustment model. The main objective of this study is to instantaneously determine the roll, pitch, and yaw angles of a vehicular platform spanning at least two baselines that are not aligned collinearly. A novel algorithm is developed to explain how to achieve a diagonally dominant phase ambiguity covariance matrix. Specifically, a hyperspherical radius serves as a threshold to boost search efficiency by restricting the number of trial-and-error integer ambiguities. Short baselines of 1–2 m or 5–6 m in length were used in the experiments. The L1 carrier data-sets were processed on epoch-by-epoch basis for analysis of the angular precision and accuracy. A blueprint for future work is also given in the conclusion section.