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Lithium–ion batteries are key elements of electric vehicles and energy storage systems, and their accurate State of Charge (SOC) estimation is momentous for battery energy management, safe operation, and extended service life. In this paper, the Multi-Kernel Relevance Vector Machine (MKRVM) and Whale Optimization Algorithm (WOA) are used to estimate the SOC of lithium–ion batteries under different operating conditions. In order to better learn and estimate the battery SOC, MKRVM is used to establish a model to estimate lithium–ion battery SOC. WOA is used to automatically adjust and optimize weights and kernel parameters of MKRVM to improve estimation accuracy. The proposed model is validated with three lithium–ion batteries under different operating conditions. In contrast to other optimization algorithms, WOA has a better optimization effect and can estimate the SOC more accurately. In contrast to the single kernel function, the proposed multi-kernel function greatly improves the precision of the SOC estimation model. In contrast to the traditional method, the WOA-MKRVM has a higher precision of SOC estimation. [ABSTRACT FROM AUTHOR]

In this paper, the static and dynamic analysis of the higher-order shear deformation nanobeam is investigated within the framework of the two-phase local/nonlocal integral model, in which, the stress is described as the integral convolution form between the strain field and a decay kernel function to address the long-range force interactions in the domain. Based on the principle of minimum potential energy, the finite element formulation of the nonlocal higher-order shear deformation theory nanobeams is derived in a general sense through finite element method (FEM). The explicit expressions of the stiffness, geometric stiffness and mass stiffness matrix of the higher-order shear deformation theory nanobeams are derived directly. The efficiency and accuracy of the developed finite element model of higher-order shear deformation nanobeam are validated by conducting a comparation with the existing analysis results in the researches. Furthermore, under different loading and supported conditions, the effect of nonlocal parameter, nonlocal phase parameter and slenderness ratio on the bending, buckling and free vibration responses of higher-order shear deformation theory nanobeams is investigated in detail. [ABSTRACT FROM AUTHOR]

We prove that the Bergman kernel function associated to a smooth measure supported on a piecewise-smooth maximally totally real submanifold 퐾 in C n is of polynomial growth. For example, this holds in dimension one if 퐾 is a finite union of transverse Jordan arcs in ℂ. Our bounds are sharp when 퐾 is smooth. We give an application to the equidistribution of the zeros of random polynomials, which extends a result of Shiffman–Zelditch to the higher-dimensional setting. [ABSTRACT FROM AUTHOR]

We study Jánossy densities of a randomly thinned Airy kernel determinantal point process. We prove that they can be expressed in terms of solutions to the Stark and cylindrical Korteweg–de Vries equations; these solutions are Darboux tranformations of the simpler ones related to the gap probability of the same thinned Airy point process. Moreover, we prove that the associated wave functions satisfy a variation of Amir–Corwin–Quastel's integro-differential Painlevé II equation. Finally, we derive tail asymptotics for the relevant solutions to the cylindrical Korteweg–de Vries equation and show that they decompose asymptotically into a superposition of simpler solutions. [ABSTRACT FROM AUTHOR]

When overdispersion and zero-inflated feature are both concerned, Zero-inflated negative binomial (ZINB) model tends to be more appropriate than the popular Zero-inflated Poisson (ZIP) model most used. Moreover, when some variables that model the mean response and the probability of zero-excess have missing data, ignoring the missing or using inappropriate methods is likely to produce questionable estimates. To remedy this problem, we propose a non-parametric and parametric inverse probability weighting (IPW) estimators to estimate parameters of the ZINB models. In particular, the generalized additive models (GAMs), the indicator or kernel smoothing functions are considered for the non-parametric approaches. Meanwhile, logistic model is used to estimate selection probability in parametric method. The performance of our proposed methods are investigated by conducting several simulation studies and exploring a real example analysis. We aim to develop and compare several IPW estimators in order to recommend the most accurate approaches. Numerical illustrations support that the semiparametric IPW methods which were based on the kernel smoothing or indicator function weights tend to produce the best results, because they tend to outperform the GAMs and parametric IPW estimates in terms of efficiency. Hence we recommend that the first approach to be used in the analysis for the ZINB model should be the semiparametric IPW based on the kernel smoothing or indicator function weights. The GAM and parametric weighting estimators are also quite good, as long as their related estimated weights are correct. • This study proposes the parameter estimations on Zero-inflated Negative Binomial (ZINB) model with missing data. • The inverse probability weighting (IPW) method is considered in this work. • We propose the three different approaches include: parametric, non-parametric and generalized additive models. • We compared the proposed approaches with the easiest method of this topic which is the complete case (CC) method. • Many numerical simulations and a real data sets includes the infected blood cells (IBC) was conducted in this study. [ABSTRACT FROM AUTHOR]

We study the behavior of the shifted convolution sum involving even power of the Fourier coefficients of holomorphic cusp forms with a weight function to be the k -full kernel function for any fixed integer k ≥ 2. [ABSTRACT FROM AUTHOR]

The goal of blind image super-resolution (BISR) is to recover the corresponding high-resolution image from a given low-resolution image with unknown degradation. Prior related research has primarily focused effectively on utilizing the kernel as prior knowledge to recover the high-frequency components of image. However, they overlooked the function of structural prior information within the same image, which resulted in unsatisfactory recovery performance for textures with strong self-similarity. To address this issue, we propose a two stage blind super-resolution network that is based on kernel estimation strategy and is capable of integrating structural texture as prior knowledge. In the first stage, we utilize a dynamic kernel estimator to achieve degradation presentation embedding. Then, we propose a triple path attention groups consists of triple path attention blocks and a global feature fusion block to extract structural prior information to assist the recovery of details within images. The quantitative and qualitative results on standard benchmarks with various degradation settings, including Gaussian8 and DIV2KRK, validate that our proposed method outperforms the state-of-the-art methods in terms of fidelity and recovery of clear details. The relevant code is made available on this link as open source. [ABSTRACT FROM AUTHOR]

Aiming at the problem of inaccurate transformer fault diagnosis in dissolved gas analysis, this paper proposes a novel diagnostic method that integrates an enhanced honey badger algorithm (EHBA) with an ensemble learning-based deep hybrid kernel extreme learning machine (DHKELM). First, kernel principal component analysis (KPCA) was deployed for feature fusion of the gas data, thus extracting more effective features. The DHKELM, combining polynomial and RBF kernel functions, was used as a base learning to build a powerful classifier with Adaboost framework. The EHBA introduces information sharing and firefly perturbation strategies based on HBA. This EHBA was harnessed to optimize the DHKELM's critical parameters, establishing the EHBA-DHKELM-Adaboost transformer fault diagnosis model. Finally, the features garnered by KPCA were fed into the model, simulating and validating various fault diagnosis models. The findings reveal that EHBA-DHKELM-Adaboost achieves 98.75% diagnostic accuracy in transformer faults, surpassing other models. [ABSTRACT FROM AUTHOR]

Social networking is a place where humans can interact using the internet network to be able to disseminate information, discuss, exchange ideas, pour out their hearts, and share activities. Many social networks are popularly used, one of which is Twitter. Information can be received quickly using Twitter. In addition, various government agencies also use Twitter to be able to interact directly with the community so that every government policy is disseminated through this social network. Every government policy neglects to reap the pros and cons of society, both collectively and individually. As a result of the pros and cons, a trial called cyberbullying was recorded. Cyberbullying in various studies has been carried out to change a person's raw material so that with the application of information technology, identifying cyberbullying needs to be carried out further. The problem of cyberbullying is generally detected using the support vector machine (SVM) method. Cyberbullying detection is conducted in dealing with government policy data such as "cipta kerja" by using the SVM method which is optimized using the cubic kernel function. The accuracy value achieved in SVM uses a linear kernel function of 92.3% while using a cubic linear function of 90%. [ABSTRACT FROM AUTHOR]

Due to the restrictive growth and/or monotonicity requirements inherent in their employment, classical iterative fixed-point theorems are rarely used to approximate solutions to an integral operator with Green's function kernel whose fixed points are solutions of a boundary value problem. In this paper, we show how one can decompose a fixed-point problem into multiple fixed-point problems that one can easily iterate to approximate a solution of a differential equation satisfying one boundary condition, then apply a bisection method in an intermediate value theorem argument to meet a second boundary condition. Error estimates on the iterates are also established. The technique will be illustrated on a second-order right focal boundary value problem, with an example provided showing how to apply the results. [ABSTRACT FROM AUTHOR]

Background: One of the limitations in leveraging the potential of artificial intelligence in X‐ray imaging is the limited availability of annotated training data. As X‐ray and CT shares similar imaging physics, one could achieve cross‐domain data sharing, so to generate labeled synthetic X‐ray images from annotated CT volumes as digitally reconstructed radiographs (DRRs). To account for the lower resolution of CT and the CT‐generated DRRs as compared to the real X‐ray images, we propose the use of super‐resolution (SR) techniques to enhance the CT resolution before DRR generation. Purpose: As spatial resolution can be defined by the modulation transfer function kernel in CT physics, we propose to train a SR network using paired low‐resolution (LR) and high‐resolution (HR) images by varying the kernel's shape and cutoff frequency. This is different to previous deep learning‐based SR techniques on RGB and medical images which focused on refining the sampling grid. Instead of generating LR images by bicubic interpolation, we aim to create realistic multi‐detector CT (MDCT) like LR images from HR cone‐beam CT (CBCT) scans. Methods: We propose and evaluate the use of a SR U‐Net for the mapping between LR and HR CBCT image slices. We reconstructed paired LR and HR training volumes from the same CT scans with small in‐plane sampling grid size of 0.20×0.20mm2$0.20 \times 0.20 \, {\rm mm}^2$. We used the residual U‐Net architecture to train two models. SRUNResK$^K_{Res}$: trained with kernel‐based LR images, and SRUNResI$^I_{Res}$: trained with bicubic downsampled data as baseline. Both models are trained on one CBCT dataset (n = 13 391). The performance of both models was then evaluated on unseen kernel‐based and interpolation‐based LR CBCT images (n = 10 950), and also on MDCT images (n = 1392). Results: Five‐fold cross validation and ablation study were performed to find the optimal hyperparameters. Both SRUNResK$^K_{Res}$ and SRUNResI$^I_{Res}$ models show significant improvements (p‐value <$<$ 0.05) in mean absolute error (MAE), peak signal‐to‐noise ratio (PSNR) and structural similarity index measures (SSIMs) on unseen CBCT images. Also, the improvement percentages in MAE, PSNR, and SSIM by SRUNResK$^K_{Res}$ is larger than SRUNResI$^I_{Res}$. For SRUNResK$^K_{Res}$, MAE is reduced by 14%, and PSNR and SSIMs increased by 6 and 8%, respectively. To conclude, SRUNResK$^K_{Res}$ outperforms SRUNResI$^I_{Res}$, which the former generates sharper images when tested with kernel‐based LR CBCT images as well as cross‐modality LR MDCT data. Conclusions: Our proposed method showed better performance than the baseline interpolation approach on unseen LR CBCT. We showed that the frequency behavior of the used data is important for learning the SR features. Additionally, we showed cross‐modality resolution improvements to LR MDCT images. Our approach is, therefore, a first and essential step in enabling realistic high spatial resolution CT‐generated DRRs for deep learning training. [ABSTRACT FROM AUTHOR]

The paper deals with minimum energy problems in the presence of external fields on a locally compact space X with respect to a function kernel κ satisfying the energy and consistency principles. For quite a general (not necessarily lower semicontinuous) external field f, we establish sufficient and/or necessary conditions for the existence of λ A , f minimizing the Gauss functional ∫ κ (x , y) d (μ ⊗ μ) (x , y) + 2 ∫ f d μ over all positive Radon measures μ with μ (X) = 1 , concentrated on quite a general (not necessarily closed or bounded) set A ⊂ X , thereby giving an answer to a question raised by M. Ohtsuka (J. Sci. Hiroshima Univ., 1961). Such results are specified for the Riesz kernels | x - y | α - n , 0 < α < n , on R n , n ⩾ 2 , and are illustrated by some examples. Furthermore, we provide various alternative characterizations of the minimizer λ A , f , and as a by-product we prove the continuity of both λ A , f and the modified Robin constant under the exhaustion of A by compact K ⊂ A . Along with the advantages of using perfect kernels, the analysis performed is substantially based on the author's recent theory of inner balayage on locally compact spaces (Potential Anal., 2022). The results obtained do hold and are new for many interesting kernels in classical and modern potential theory. [ABSTRACT FROM AUTHOR]

This paper studies two nonlocal diffusion problems with a free boundary and a fixed boundary in heterogeneous environment. The main goal is to understand how the evolution of the two species is affected by the heterogeneous environment. We first prove the existence and uniqueness of a global solution for such systems. Then, for models with Lotka–Volterra type competition or predator–prey growth terms, we establish the spreading‐vanishing dichotomy. Sharp criteria of spreading and vanishing are also obtained. Furthermore, we show that accelerated spreading occurs if and only if the kernel function violates the threshold condition. [ABSTRACT FROM AUTHOR]

This study proposes multivariate kernel density estimation by stagewise minimization algorithm based on U-divergence and a simple dictionary. The dictionary consists of an appropriate scalar bandwidth matrix and a part of the original data. The resulting estimator brings us data-adaptive weighting parameters and bandwidth matrices, and provides a sparse representation of kernel density estimation. We develop the non-asymptotic error bound of the estimator that we obtained via the proposed stagewise minimization algorithm. It is confirmed from simulation studies that the proposed estimator performs as well as, or sometimes better than, other well-known density estimators. [ABSTRACT FROM AUTHOR]

The quality of visual media is critically impacted by low illumination and the presence of airborne particulates, leading to challenges in brightness balance, color saturation, and texture clarity which are detrimental to various applications in image processing and computer vision. Addressing these challenges, this study introduces a novel image enhancement algorithm that significantly improves the quality of degraded images. Our proposed method, the multi-channel phase activation and multi-constraint dark channel prior (MMDCP), leverages an innovative approach by integrating the phase-adjusted Gaussian kernel function for brightness channel optimization in the Fourier transform frequency domain. This optimization is enhanced through the application of a saturated dark channel prior, achieving simultaneous brightness enhancement and color fidelity. Furthermore, we refine the dark channel prior deblurring algorithm by incorporating intensity, brightness, and color constraints to correct overexposure issues and color offsets in the reconstructed images. The efficacy of the MMDCP algorithm is demonstrated through extensive experimentation, comparing it against six contemporary image enhancement algorithms using two types of objective indicators and subjective assessments across four public datasets. The MMDCP algorithm consistently outperforms the existing methods, with a notable average improvement of 20% in PSNR and 19.6% in SSIM metrics, substantiating its superiority in enhancing brightness, detail, and color accuracy. This study's results underline the MMDCP algorithm's robustness and versatility in improving image quality across various conditions, including daytime, nighttime, indoor, and outdoor settings. [ABSTRACT FROM AUTHOR]

Gaussian Radial Basis Function Kernels are the most-often-employed kernel function in artificial intelligence for providing the optimal results in contrast to their respective counterparts. However, our understanding surrounding the utilization of the Generalized Gaussian Radial Basis Function across different machine learning algorithms, such as kernel regression, support vector machines, and pattern recognition via neural networks is incomplete. The results delivered by the Generalized Gaussian Radial Basis Function Kernel in the previously mentioned applications remarkably outperforms those of the Gaussian Radial Basis Function Kernel, the Sigmoid function, and the ReLU function in terms of accuracy and misclassification. This article provides a concrete illustration of the utilization of the Generalized Gaussian Radial Basis Function Kernel as mentioned earlier. We also provide an explicit description of the reproducing kernel Hilbert space by embedding the Generalized Gaussian Radial Basis Function as an L 2 − measure, which is utilized in implementing the analysis support vector machine. Finally, we provide the conclusion that we draw from the empirical experiments considered in the manuscript along with the possible future directions in terms of spectral decomposition of the Generalized Gaussian Radial Basis Function. [ABSTRACT FROM AUTHOR]

In this paper, a new approach is presented to fit a fuzzy regression model with the fuzzy coefficients when the explanatory variables and the response variable are as fuzzy numbers. In this approach, a nonlinear function is introduced to predict the response variables based on the kernel function. To estimate the parameters of regression model, the objective function is calculated using the quantile loss function on fuzzy numbers. To evaluate the goodness of fit of the optimal quantile fuzzy regression models, two indices are introduced using the similarity measures. Based on the presented results, the proposed fuzzy regression models have the perfect performances on the original data and also in the presences of different types of outliers. [ABSTRACT FROM AUTHOR]

In this paper, we present an algorithm for clustering multidimensional data, which we named TreeKDE. It is based on a tree structure decision associated with the optimization of the one-dimensional kernel density estimator function constructed from the orthogonal projections of the data on the coordinate axes. Among the main features of the proposed algorithm, we highlight the automatic determination of the number of clusters and their insertion in a rectangular region. Comparative numerical experiments are presented to illustrate the performance of the proposed algorithm and the results indicate that the TreeKDE is efficient and competitive when compared to other algorithms from the literature. Features such as simplicity and efficiency make the proposed algorithm an attractive and promising research field, which can be used as a basis for its improvement, and also for the development of new clustering algorithms based on the association between decision tree and kernel density estimator. [ABSTRACT FROM AUTHOR]

The Nyström method is a convenient strategy to obtain low-rank approximations to kernel matrices in nearly linear complexity. Existing studies typically use the method to approximate positive semidefinite matrices with low or modest accuracies. In this work, we propose a series of heuristic strategies to make the Nyström method reach high accuracies for nonsymmetric and/or rectangular matrices. The resulting methods (called high-accuracy Nyström methods) treat the Nyström method and a skinny rank-revealing factorization as a fast pivoting strategy in a progressive alternating direction refinement process. Two refinement mechanisms are used: alternating the row and column pivoting starting from a small set of randomly chosen columns, and adaptively increasing the number of samples until a desired rank or accuracy is reached. A fast subset update strategy based on the progressive sampling of Schur complements is further proposed to accelerate the refinement process. Efficient randomized accuracy control is also provided. Relevant accuracy and singular value analysis is given to support some of the heuristics. Extensive tests with various kernel functions and data sets show how the methods can quickly reach prespecified high accuracies in practice, sometimes with quality close to SVDs, using only small numbers of progressive sampling steps. [ABSTRACT FROM AUTHOR]

Hierarchical matrices provide a powerful representation for significantly reducing the computational complexity associated with dense kernel matrices. For example, the fast multipole method (FMM) and its variants are highly efficient when the kernel function is related to fundamental solutions of classical elliptic PDEs. For general kernel functions, interpolation-based methods are widely used for the efficient construction of hierarchical matrices. In this paper, we present a fast hierarchical data reduction (HiDR) procedure with O(n) complexity for the memoryefficient construction of hierarchical matrices with nested bases where n is the number of data points. HiDR aims to reduce the given data in a hierarchical way so as to obtain O(1) representations for all nearfield and farfield interactions. Based on HiDR, a linear complexity H² matrix construction algorithm is proposed. The use of data-driven methods enables better efficiency than other general-purpose methods and flexible computation without accessing the kernel function. Experiments demonstrate significantly improved memory efficiency of the proposed data-driven method compared to interpolation-based methods over a wide range of kernels. For the Coulomb kernel, the proposed general-purpose algorithm offers competitive performance compared to FMM and its variants, such as PVFMM. The data-driven approach not only works for general kernels but also leads to much smaller precomputation costs compared to PVFMM. [ABSTRACT FROM AUTHOR]

Thunderstorms are natural disasters that impact people, animals, and the economy. Thunderstorms' detrimental repercussions can be avoided by identifying their occurrence in advance. The current work, in this respect, uses soft computing techniques such as K-Nearest Neighbour (KNN), Decision Tree (DT), Logistic Regression (LR), and Support Vector Machine (SVM) with various kernel functions to categorize the occurrence of thunderstorms over Ranchi, India. These techniques were trained and tested using two data sets: daily average and hourly meteorological datasets. The primary purpose of this study is to find which dataset-classifier combination is optimal for categorizing thunderstorm occurrence in Ranchi. No classifier was found to adequately classify either the Day Average Dataset or the Modified Day Average Dataset. On the other hand, the Hourly Dataset was found to be more balanced in terms of the number of thunderstorms that occurred than the Day Average and Modified Average datasets. The F-Score value of the incidence of thunderstorm incidents after using different classifiers was used to compare the outcomes of these datasets. The results reveal that using SVM with radial basis function. The Hourly Dataset is the best for thunderstorm day classification. For the overall and only incidence of thunderstorms classes, SVM-RBF gets 0.81 and 0.74 F-Scores, respectively. Other approaches, like grid search and Bagging, have been used to increase SVM-RBF performance. Grid search and Bagging are used on SVM-RBF to produce a hybrid Grid-Bag-SVM-RBF classifier with 82.04% accuracy and F-scores of 0.83 and 0.78 for overall and just thunderstorm occurrence, respectively. [ABSTRACT FROM AUTHOR]

We present results for the short-distance window observable of the hadronic vacuum polarization contribution to the muon g – 2, computed via the time-momentum representation (TMR) in lattice QCD. A key novelty of our calculation is the reduction of discretization effects by a suitable subtraction applied to the TMR kernel function, which cancels the leading -behaviour at short distances. To compensate for the subtraction, one must substitute a term that can be reliably computed in perturbative QCD. We apply this strategy to our data for the vector current collected on ensembles generated with 2 + 1 flavours of O(a)-improved Wilson quarks at six values of the lattice spacing and pion masses in the range 130 – 420 MeV. Our estimate at the physical point contains a full error budget and reads = 68.85(14)stat (42)syst·10−10, which corresponds to a relative precision of 0.7%. We discuss the implications of our result for the observed tensions between lattice and data-driven evaluations of the hadronic vacuum polarization. [ABSTRACT FROM AUTHOR]

A method based on kernel extreme learning machine is proposed for accurate localization and recognition of human bones. In this method, the human skeleton is divided into 16 key nodes based on physiology, and the human skeleton model is formed by connecting lines between the nodes. The sequence images of human actions are taken as input and fed into the input and hidden layers of the extreme learning machine. During the learning and training process, a kernel function is used to construct a kernel matrix to complete weight allocation. The experimental study was conducted on the action sequences of figure skaters, and the results showed that our method achieved an accuracy rate of over 90% in recognizing human bones, with individual sequences achieving an accuracy rate of over 95%, which is superior to convolutional neural network (CNN) and long short-term memory (LSTM) methods. [ABSTRACT FROM AUTHOR]

In this paper, the authors study the integral operator S ϕ f (z) = ∫ ℂ ϕ (z , w ¯) f (w) d λ α (w) induced by a kernel function ϕ(z,·) ∈ Fα∞ between Fock spaces. For 1 ≤ p ≤ ∞, they prove that Sϕ: Fα1 → Fαp is bounded if and only if sup a ∈ ℂ ∣ ∣ S ϕ k a ∣ ∣ p , α < ∞ , where ka is the normalized reproducing kernel of Fα2; and, Sϕ: Fα1 → Fαp is compact if and only if l i m ∣ a ∣ → ∞ ∣ ∣ S ϕ k a ∣ ∣ p , α = 0. When 1 < q ≤ ∞, it is also proved that the condition (†) is not sufficient for boundedness of Sϕ: Fαq → Fαp. In the particular case ϕ (z , w ¯) = e α z w ¯ φ (z − w ¯) with φ ∈ Fα2, for 1 ≤ q < p < ∞, they show that Sϕ: Fαp → Fαq is bounded if and only if φ = 0; for 1 < p ≤ q < ∞, they give sufficient conditions for the boundedness or compactness of the operator Sϕ: Fαp → Fαq. [ABSTRACT FROM AUTHOR]

This paper studies the influence of scaling on the behavior of radial basis function interpolation. It focuses on certain central aspects, but does not try to be exhaustive. The most important questions are: How does the error of a kernel-based interpolant vary with the scale of the kernel chosen? How does the standard error bound vary? And since fixed functions may be in spaces that allow scalings, like global Sobolev spaces, is there a scale of the space that matches the function best? The last question is answered in the affirmative for Sobolev spaces, but the required scale may be hard to estimate. Scalability of functions turns out to be restricted for spaces generated by analytic kernels, unless the functions are band-limited. In contrast to other papers, polynomials and polyharmonics are included as flat limits when checking scales experimentally, with an independent computation. The numerical results show that the hunt for near-flat scales is questionable, if users include the flat limit cases right from the start. When there are not enough data to evaluate errors directly, the scale of the standard error bound can be varied, up to replacing the norm of the unknown function by the norm of the interpolant. This follows the behavior of the actual error qualitatively well, but is only of limited value for estimating error-optimal scales. For kernels and functions with unlimited smoothness, the given interpolation data are proven to be insufficient for determining useful scales. [ABSTRACT FROM AUTHOR]

Asymmetry plays a significant role in the transmission dynamics in novel discrete fractional calculus. Few studies have mathematically modeled such asymmetry properties, and none have developed discrete models that incorporate different symmetry developmental stages. This paper introduces a Taylor monomial falling function and presents some properties of this function in a delta fractional model with Green's function kernel. In the deterministic case, Green's function will be non-negative, and this shows that the function has an upper bound for its maximum point. More precisely, in this paper, based on the properties of the Taylor monomial falling function, we investigate Lyapunov-type inequalities for a delta fractional boundary value problem of Riemann–Liouville type. [ABSTRACT FROM AUTHOR]

This article studies the error function and its invariance properties in the convolutional kernel function of bone fractal operators. Specifically, the following contents are included: (1) demonstrating the correlation between the convolution kernel function and error function of bone fractal operators; (2) focusing on the main part of bone fractal operators: p + α 2 -type differential operator, discussing the convolutional kernel function image; (3) exploring the fractional-order correlation between the error function and other special functions from the perspective of fractal operators. [ABSTRACT FROM AUTHOR]

• Mathematical and quantitative methods were used to describe the damaged shape repair and provide the restoration results in the form of a probability distribution. • A specialized triangle mesh topology for Buddha statue head registration was designed, and a semi-automated registration strategy was developed to achieve high-quality shape registration. • Using the deformation field and interpolator, we defined the Gaussian process model, in which every shape sampling from the model is a function defined on the reference shape. • A set of kernel functions was designed to extend the flexibility of the learned model. • A series of experiments was conducted to investigate the effectiveness of our method, and we applied the methods to an actual damaged Buddha statue head repair project. Buddha statues are distributed worldwide, and each piece reflects the aesthetic fashion of a particular region and historical era. Therefore, these sculptures have important cultural, aesthetic, and historical value. Owing to natural or artificial destruction, several Buddha statues have been damaged, with head or facial damage being the most prominent damage type. Because Buddha statue heads exhibit intricate variations in facial expressions and proportions, restoring head or facial damage is the most challenging aspect of damaged sculpture restoration projects. This study provides a new method for the restoration of damaged statues based on a Gaussian process model. The Gaussian process model entails a type of statistical shape modeling that allows it to learn shape distribution properties from a training dataset. During the restoration process, we used the residual parts of the damaged shape for the observations and added them to a Bayesian inference system to obtain the possible restoration result in the form of a posterior distribution. Therefore, the proposed method can be regarded as a quantitative mathematical representation of the subjective deduction process. Compared to those of the traditional manual restoration method, our results were more repeatable and stable, and the restoration process was automatic and rigorous, which reduced the requirements for professional knowledge and experience. Based on the learned model information, the proposed method exhibits strong robustness and anti-noise properties. In contrast to the optimization method, our final restoration result is a probability distribution, applicable to the repair of cultural relics. As our method make the missing shape inference using residual shape information and the regularity learned from training dataset, it is suitable for the restoration of damaged parts with no direct shape evidence. Finally, our model is generative; in other words, we can generate shapes on the posterior distribution and provide the corresponding probability. To demonstrate the effectiveness of our method, a series of experiments were conducted to restore damaged parts under different parameters and damage type settings. Subsequently, a quantitative analysis was performed. Finally, we applied this method to the virtual restoration of a damaged Buddha statue head in Xi 'an Museum. [Display omitted] [ABSTRACT FROM AUTHOR]

Studies have shown that the existence of the anomalous diffusion of porous media leads to the abnormal heat conduction, which can be well described by fractional derivatives. In the present work, memory dependent derivative is introduced into the existing linear thermoelastic theory to analyze the transient thermoelastic responses of a porous half-space subjected to a thermal loading at the boundary. By using the method of Laplace transform and its numerical inversion, the closed form solutions of temperature, volume fraction, displacement, and stress are obtained. The effects of time delay, kernel function and a memory-dependent parameter on the physical fields are discussed. The results show that the memory dependent derivative plays an essential role in controlling the heat transfer process of porous materials. [ABSTRACT FROM AUTHOR]

Based on fractional strain theory and considering microscale memory dependent effect, the transient response of a one-dimensional micro-nano-structured viscoelastic rotating rod under the action of a moving heat source is investigated in the paper. The rod is placed in a magnetic field of constant strength and its outer surface is unconstrained. The effects of the kernel function, time lag factor, fractional order strain parameter, rotational parameter, visco-thermoelastic relaxation time, and nonlocal parameter on the distribution of the physical field in the rod are obtained by solving the governing equations through the Laplace integral transform and its numerical inverse transform. The results show that the fractional strain parameters and rotational angular velocity have little effect on the dimensionless temperature, but have significant effects on the dimensionless displacement, stress, and strain. The kernel function, time delay factor, viscoelastic relaxation time, and nonlocal parameter have different degrees effects on temperature, displacement, stress, and strain. [ABSTRACT FROM AUTHOR]

Snow cover has a significant impact on numerous ecological, climatic, and hydrological processes, particularly in high-latitude areas. Remote sensing data estimates of snow cover ranges than traditional surveying methods. Thus, the present study used Sentinel-2B satellite images to compare the performance of support vector machine (SVM) kernel functions and object-oriented fuzzy operators to estimate the snow cover area (SCA) on Alvand Mountain. The research data included four Sentinel-2B 10 m spatial resolution bands (B2, B3, B4, and B8) that were launched on March 6, 2020. In this study, linear, polynomial, radial, and sigmoid SVM kernel functions, as well as the object-oriented fuzzy operators (AND, OR, MGE, MAR, MGWE, and ALP), were used. The results indicated that AND algorithm, which represents a logical commonality, had the lowest return fuzzy value of 98% compared to other algorithms, thus, providing the overall highest accuracy. Based on these findings, the object-oriented processing method in digital image classification could provide the highest accuracy compared to the SVM kernel functions; that is because a wide range of information, such as texture, shape, position, content, and bandwidth, is associated with the objects in this classification method. Finally, the SCA in Alvand Mountain was estimated to be 604 km2 based on the object-based AND fuzzy operator. [ABSTRACT FROM AUTHOR]

Support vector machine is one of the most used machine learning algorithms with a comprehensive mathematical infrastructure. The power behind the algorithm is the kernel trick that enables the model to overcome non-linear data distributions by using functions that satisfy the Mercer condition. Undoubtedly, the radial basis function (RBF) is among the most widely used of these functions. The RBF kernel, which has a Gaussian curve, performs local boundary surfaces and supports the generalization capability of the model. In this study, a novel kernel function named Logarithmic Kernel Function (LKF), which has a Gaussian-like curve is presented. The crucial contribution of the LKF is that it can model the dataset better than other similarly shaped kernels in the case of a few training samples (10% and 30%). In the study, 6 classifications and 5 regression sets are handled to compare kernels. Instead of giving the hyperparameters needed by the models manually, they are estimated through the Tree Parzen Estimator, which is based on Sequential Modeling Optimization. This estimation process is repeated 10 times and the average results are obtained. The proposed LKF surpasses the most competitive kernels in various classification and regression tasks. • The novel Gaussian-like Logarithmic Kernel Function (LKF) was proposed. • LKF has been shown to work better than other kernels of similar shape at a small amount of training samples. • For a fair comparison, the parameters of kernels were not determined manually. • Tree Parzen Estimator optimization methods was preferred for parameter selection. • In comparisons made with 6 classification and 6 regression tasks. [ABSTRACT FROM AUTHOR]

In this paper, Gaussian Process Regression (GPR)‐based models which use the Bayesian approach to regression analysis problem such as load forecasting (LF) are proposed. The GPR is a non‐parametric kernel‐based learning method having the ability to provide correct predictions with uncertainty in measurements. The proposed model provides an hourly and monthly load forecast for an Australian city and four Indian cities in the Maharashtra state. Twelve GPR models are trained with historical datasets including hourly load and environmental data. To evaluate the trained model, the actual and predicted load demand curve is plotted and mean average percentage error (MAPE) is calculated corresponding to different kernel functions of the GPR model. To the best of the author's knowledge, the prediction of load demand using GPR for Indian cities of Maharashtra state has been made for the first time. The calculated MAPE in LF is 0.15% for Australia and 0.002%, 0.209%, 0.077%, and 0.140% for Indian cities viz. Nasik, Bhusawal, Kolhapur, and Aurangabad, respectively. The test results illustrate that minimum MAPE in load prediction is obtained using the proposed model that is GPR with 'Exponential' kernel functions. Furthermore, the comparative analysis with the existing approaches confirms the dominance of the proposed model. [ABSTRACT FROM AUTHOR]

A mutualist model with nonlocal diffusions and a free boundary is first considered. We prove that this problem has a unique solution defined for t ≥ 0 , and its dynamics are governed by a spreading-vanishing dichotomy. Some criteria for spreading and vanishing are also given. Of particular importance is that we find that the solution of this problem has quite rich longtime behaviors, which vary with the conditions satisfied by kernel functions and are much different from those of the counterpart with local diffusion and free boundary. At last, we extend these results to the model with nonlocal diffusions and double free boundaries. [ABSTRACT FROM AUTHOR]

In this paper, we proposed a new primal-dual second-order corrector interior-point algorithm for semidefinite optimization. The algorithm is based on Darvay–Takács neighbourhood of the central path. In the new algorithm, the search directions are determined by the Darvay–Takács's direction and a second-order corrector direction in each iteration. The iteration complexity bound is $ O(\sqrt {n}L) $ O (n L) for the Nesterov–Todd scaling direction, which coincides with the best-known complexity results for semidefinite optimization. Finally, numerical experiments show that the proposed algorithm is promising. [ABSTRACT FROM AUTHOR]

Objectives: Taylor series expansion is often used in the downward continuation of potential field, and its performance depends on the accuracy and reliability of vertical partial derivatives or radial partial derivatives (RPDs) of potential field parameters. Methods: In order to avoid the singularity on spherical boundary and the uncertainty to the computational results by using the closed analytic kernel function to solve the partial derivative, first, this paper considers the fact that all kinds of gravity observations behave as a type of limited spectrum bandwidth signal after being filtered, and proposes to express the kernel function of the Poisson integral for the external gravity anomaly by a spherical harmonic series expansion, which is truncated into a band-limited summation with the same spectrum range as the gravity observation. Then, we derive a set of band-limited formulas to calculate the high-order RPDs, which are modified and applied to the downward continuation of the gravity anomaly by Taylor series expansion. Results and Con⁃ clusions: The formulas are validated using the ultra-high-degree geopotential model EGM2008 by a twostage procedure. The numerical tests of the band-limited formulas and the Taylor series expansion downward continuation model show that the proposed band-limited formulas have good reliability and validity, and are superior to other models in terms of stability and accuracy. [ABSTRACT FROM AUTHOR]

We consider nonparametric kernel estimation of density functions in the bounded-support setting having known support $ [a,b] $ [ a , b ] using a boundary-adaptive kernel function and data-driven bandwidth selection, where a and b are finite and known prior to estimation. We observe, theoretically and in finite sample settings, that when bounds are known a priori this kernel approach is capable of outperforming even correctly specified parametric models, in the case of the uniform distribution. We demonstrate that this result has implications for modelling a range of densities other than the uniform case. Furthermore, when bounds $ [a,b] $ [ a , b ] are unknown and the empirical support (i.e. $ [\min (x_i),\max (x_i)] $ [ min (x i) , max (x i) ]) is used in their place, similar behaviour surfaces. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the viscoelastic wave equation with variable coefficients and a delay, which is subject to nonlinear and nonlocal boundary dissipation. The existence of strong and weak solutions is obtained by means of Faedo-Galerkin approximation and denseness argument. By introducing an equivalent energy functional and using the Riemannian geometry method, we show the general decay rate of energy when kernel function satisfies some sufficient small conditions that we provide the range accurately. [ABSTRACT FROM AUTHOR]

The formalism of the generalized quantum master equation (GQME) is an effective tool to simultaneously increase the accuracy and the efficiency of quasiclassical trajectory methods in the simulation of nonadiabatic quantum dynamics. The GQME expresses correlation functions in terms of a non-Markovian equation of motion, involving memory kernels that are typically fast-decaying and can therefore be computed by short-time quasiclassical trajectories. In this paper, we study the approximate solution of the GQME, obtained by calculating the kernels with two methods: Ehrenfest mean-field theory and spin-mapping. We test the approaches on a range of spin–boson models with increasing energy bias between the two electronic levels and place a particular focus on the long-time limits of the populations. We find that the accuracy of the predictions of the GQME depends strongly on the specific technique used to calculate the kernels. In particular, spin-mapping outperforms Ehrenfest for all the systems studied. The problem of unphysical negative electronic populations affecting spin-mapping is resolved by coupling the method with the master equation. Conversely, Ehrenfest in conjunction with the GQME can predict negative populations, despite the fact that the populations calculated from direct dynamics are positive definite. [ABSTRACT FROM AUTHOR]

Support Vector Machines (SVMs) can learn from high-dimensional and small amounts of training data, thanks to effective optimization methods and a diverse set of kernel functions (KFs). The adaptability of SVM to numerous real-world problems has increased interest in the SVM method, and studies conducted with this method carry significant weight in various fields. The fixed parameter "AtanK" for KFs must be specified before the SVM model training process. Therefore, determining the appropriate kernel parameter values can be timE−consuming and may lead to slow convergence of the SVM model. On the other hand, the method provides faster and more robust convergence due to the adaptive parameter in the SVM model. In this study, a new Adaptive Arctan (AA) KF, tailored to the characteristics of different datasets, is proposed as an enhancement to the AtanK KF for the SVM algorithm. The proposed AA KF is compared with experimental results on 30 public KEEL and UCI datasets, alongside AtanK, adaptive Gaussian, Radial Basis Function, linear, and polynomial KFs. The results demonstrate that the proposed AA KF outperforms the other KFs, and it exhibits superior learning ability. [ABSTRACT FROM AUTHOR]

Suppose V ν (·) is the pseudo-variance function of the Cauchy–Stieltjes Kernel (CSK) family K - (ν) generated by a non degenerate probability measure ν on the positive real line. Denote by Φ (ν) the law of large numbers for free multiplicative convolution given in [17]. An explicit expression of Φ (ν) is given in [14] in terms of the pseudo-variance function V ν (.) . In this paper, we give explicitly in terms of the pseudo-variance function V ν (.) the limiting distributions Φ (ν ⊠ t) , Φ ((D 1 / s (ν ⊞ s)) ⊠ t) , Φ ((D 1 / r (ν ⊎ r)) ⊠ t) and Φ ((B r (ν)) ⊠ t) for s > 1 , t > 1 and r > 0 , where B r (ν) = (ν ⊞ 1 + r) ⊎ 1 1 + r and D c (ν) denotes the dilation of measure ν by a number c ≠ 0 . Some examples of calculations of Φ (ν ⊠ t) , Φ ((D 1 / s (ν ⊞ s)) ⊠ t) , Φ ((D 1 / r (ν ⊎ r)) ⊠ t) and Φ ((B r (ν)) ⊠ t) are given for probability measures ν of importance in noncommutative probability. [ABSTRACT FROM AUTHOR]

The smoothed particle hydrodynamic technique is strongly based on the proper choice of interpolation functions. This statement is particularly relevant for the study of subsonic fluxes and turbulence, where inherent small errors in the averaging procedures introduce excessive damping on the smallest scales. To mitigate these errors, we can increase both the number of interpolating points and the order of the interpolating kernel function. However, this approach leads to a higher computational burden across all fluid regions. Ideally, the development of a single kernel function capable of effectively accommodating varying numbers of interpolating points in different fluid regions, providing good resolution and minimal errors would be highly desirable. In this work, we revisit and extend the main properties of a family of interpolators called Sinc kernels and compare them with the widely used family of Wendland kernels. We show that a linear combination of low- and high-order Sinc kernels generates good-quality interpolators, which are resistant to pairing instability while maintaining good sampling properties in a wide range of neighbour interpolating points, 60 ≤ nb ≤ 400. We show that a particular case of this linear mix of Sincs produces a well-balanced and robust kernel that improves previous results in the Gresho–Chan vortex experiment even when the number of neighbours is not large, while yielding a good convergence rate. Although such a mixing technique is ideally suited for Sinc kernels owing to their excellent flexibility, it can be easily applied to other interpolating families such as the B-splines and Wendland kernels. [ABSTRACT FROM AUTHOR]

This study focuses on adapting spectral clustering, a numeric data-clustering technique, for categorical and mixed-type data. The method enhances spectral clustering for categorical and mixed-type data with novel kernel functions, showing improved accuracy in real-world applications. Despite achieving better clustering for datasets with mixed variables, challenges remain in identifying suitable kernel functions for categorical relationships. [ABSTRACT FROM AUTHOR]

Spatiotemporally mapping winter wheat is imperative for informing and shaping global food security policies. Traditional mapping methods heavily rely on sufficient and reliable samples obtained through labor-intensive fieldwork and manual sample collection. However, these methods are time-consuming, costly, and lack timely and continuous data collection. To address these challenges and fully leverage remote sensing big data and cloud computing platforms like Google Earth Engine (GEE), this paper developed an algorithm for Auto-Generating Winter Wheat Samples for mapping (AGWWS). The AGWWS utilizes historical samples to determine the optimal migration threshold by measuring Spectral Angle Distance (SAD), Euclidean Distance (ED), and Near-Infrared band Difference Index (NIRDI). This facilitates the auto-generation of winter wheat sample sets for the years 2000, 2005, 2010, 2015, and 2021. Approximately two-thirds of the samples were allocated for training, with the remaining one-third used for validating the mapping method, employing the One-Class Support Vector Machine (OCSVM). The Huang–Huai–Hai (HHH) Plain, a major winter wheat production region, was selected to perform the algorithm and subsequent analysis on. Different combinations of the hyper-parameters, gamma and nu, of the OCSVM based on the Gaussian Radial Basis Function Kernel were tested for each year. Following correlation analysis between the winter wheat area derived from the generated maps and the national statistical dataset at the city level, the map with the highest corresponding R2 was chosen as the AGWWS map for each year (0.77, 0.77, 0.80, 0.86, and 0.87 for 2000, 2005, 2010, 2015, and 2021, respectively). The AGWWS maps ultimately achieved an average Overall Accuracy of 81.65%. The study then explores the Non-Grain Production of Winter Wheat (NGPOWW) by analyzing winter wheat change maps from 2000–2005, 2005–2010, 2005–2010, and 2015–2021 in the HHH Plain. Despite an overall increase in the total planted area of winter wheat, the NGPOWW phenomena has led to concerning winter wheat planting marginalization. Compensatory winter wheat areas are notably situated in mountainous and suburban cultivated lands with low qualities. Consequently, despite the apparent expansion in planted areas, winter wheat production is anticipated to be adversely affected. The findings highlight the necessity for improved cultivated land protection policies monitoring the land quality of the compensation and setting strict quota limits on occupations. [ABSTRACT FROM AUTHOR]

The article proposes a procedure for selecting the blur factor of kernel functions for nonparametric density estimation of a one-dimensional random variable given large amounts of statistical data, for example, obtained via the remote sensing of natural objects. The proposed procedure uses regression density estimation. A procedure is presented for synthesizing a regression density estimate. The estimate synthesis is based on the original sample compression by decomposing the range of a random variable. To this end, the Heinhold-Gaede rule and a formula for choosing the optimal number of sampling intervals are applied. The article considers two approaches to selecting the blur factor of regression density estimation using the conventional method and that proposed by the authors to optimize nonparametric density estimation. The conventional method for optimizing nonparametric density estimation is based on minimizing its standard deviation. In the proposed method, the choice of the blur factors of kernel functions relies on the conditions for the minimum approximation error of regression density estimation. The article analyzes the approximation properties of regression density estimation using two optimization methods. The conditions of their competence in estimating the probability densities of random variables following lognormal distribution are established. The results obtained for a one-dimensional random variable can be used to optimize the regression density estimation of a multi-dimensional random variable. [ABSTRACT FROM AUTHOR]