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This article addresses the identification problem of the input nonlinear closed-loop system with unknown time delay (INCTD). An identification algorithm of the INCTD system which combines the wavelet packet decomposition (WPD), the variance Gaussian kernel function (VGKF) and the grouped method of data handling (GMDH) network is proposed. The data collected by the sensor is decomposed into intrinsic mode function with different nonlinear characteristics using WPD. The GMDH network is applied to mine bidirectional strong nonlinear relationship between input and output data. The VGKF criterion selects the optimal complexity model to enhance the extrapolation performance, and further improves the identification accuracy. Thus, the WPD-GMDH-VGKF identification algorithm is derived. The numerical simulation result verifies that the WPD-GMDH-VGKF identification method can effectively identify the INCTD system, and it is superior to GMDH and WPD-GMDH in terms of identification accuracy. The application case verifies the feasibility of applying the WPD-GMDH-VGKF identification method to the battery management system (BMS). [ABSTRACT FROM AUTHOR]

Summary: This article presents an improved least mean square (LMS) algorithm‐based control approach for a three‐phase grid interfaced solar photovoltaic (GISPV) system for smart building applications. The SPV is primarily utilized to supply household loads, and the absence or intermittency of SPV is curbed by utilizing power from the grid. The majority of household loads are nonlinear loads that affect the grid power quality (PQ). However, the effects of both linear and nonlinear loads on grid PQ have been analyzed in this article. This article proposes a Gaussian Kernel function‐based normalized LMS (GKNLMS) algorithm to calculate the active peak weight component of the load current, thereby utilizing it to obtain a sinusoidal reference current. Considering a higher step size and utilizing the Gaussian Kernel function for reduced mean square error (MSE) improves both the convergence rate and accuracy of estimation in the conventional LMS. The proposed control technique effectively addresses system anomalies caused by dynamic solar irradiance, load variation, and unbalanced loading conditions. Furthermore, the system response with the GKNLMS algorithm is assessed in MATLAB/Simulink and validated by developing a laboratory setup of the system for experimentation. [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]

The goal of this paper is to establish Green function estimates for a class of purely discontinuous symmetric Markov processes with jump kernels degenerate at the boundary and critical killing potentials. The jump kernel and the killing potential depend on several parameters. We establish sharp two-sided estimates on the Green functions of these processes for all admissible values of the parameters involved. Depending on the regions where the parameters belong, the estimates on the Green functions are different. In fact, the estimates have three different forms. As applications, we prove that the boundary Harnack principle holds in certain region of the parameters and fails in some other region of the parameters. Together with the main results of our previous paper [Potential Anal., online, 2021], we completely determine the region of the parameters where the boundary Harnack principle holds. [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]

Kernel methods are often used for nonlinear regression and classification in statistics and machine learning because they are computationally cheap and accurate. The wavelet kernel functions based on wavelet analysis can efficiently approximate any nonlinear functions. In this article, we construct a novel wavelet kernel function in terms of random wavelet bases and define a linear vector space that captures nonlinear structures in reproducing kernel Hilbert spaces (RKHS). Based on the wavelet transform, the data are mapped into a low-dimensional randomized feature space and convert kernel function into operations of a linear machine. We then propose a new Bayesian approximate kernel model with the random wavelet expansion and use the Gibbs sampler to compute the model's parameters. Finally, some simulation studies and two real datasets analyses are carried out to demonstrate that the proposed method displays good stability, prediction performance compared to some other existing methods. [ABSTRACT FROM AUTHOR]

Conventional classification assumes a balanced sample distribution among classes. However, such a premise leads to biased performance over the majority class (with the highest number of instances). The Twin Support Vector Machines (TWSVM) obtained great prominence due to their low computational burden compared to the standard SVM. Besides, traditional machine learning seeks methods whose solution depends on a convex problem or semi-positive definite similarity matrices. Yet, this kind of matrix cannot adequately represent many real-world applications. The above defines the need to use non-negative measures as an indefinite function in a Reproducing Kernel Kreĭn Space (RKKS). This paper proposes a novel approach called Kreĭn Twin Support Vector Machines (KTSVM), which appropriately incorporates indefinite kernels within a TWSVM-based gradient optimization. To code pertinent input patterns within an imbalanced data discrimination, our KTSVM employs an implicit mapping to a RKKS. Also, our approach takes advantage of the TWSVM scheme's benefits by creating two parallel hyperplanes. This promotes the KTSVM optimization in a gradient-descent framework. Results obtained on synthetic and real-world datasets demonstrate that our solution performs better in terms of imbalanced data classification than state-of-the-art techniques. • Kreĭn Twin Support Vector Machines (KTSVM) enhances nonlinear TWSVM with indefinite kernels for imbalanced classification. • KTSVM represents high dimensions with indefinite kernels, improving imbalanced classification. • Our approach transforms the traditional TWSVM's convex dual problem into a gradient-based optimization. • KTSVM builds dual hyperplanes in RKKS, refined with gradient descent. • KTSVM excels over SVM, KSVM, and TWSVM in binary imbalanced classification. [ABSTRACT FROM AUTHOR]

In this paper, we study the Neumann problem for a class of nonlocal dispersal models with inhomogeneous kernel function. We investigate the existence, uniqueness, and limit of solutions when the inhomogeneous diffusion kernel is rescaled. Our result exhibits that the inhomogeneous nonlocal dispersal equation is analogous to local problem without convection in the limit case. [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]

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]

In this paper, we address the challenge of ensuring stability in bipedal walking robots and exoskeletons. We explore the feasibility of real-time implementation for the Predicted Step Viability algorithm (PSV), a complex multi-step optimization criterion for planning future steps in bipedal gait. To overcome the high computational cost of the PSV algorithm, we performed an analysis using 11 classification algorithms and a stacking strategy to predict if a step will be stable or not. We generated three datasets of increasing complexity through PSV simulations to evaluate the classification performance. Among the classifiers, k Nearest Neighbors, Support Vector Machine with Radial Basis Function Kernel, Decision Tree, and Random Forest exhibited superior performance. Multi-Layer Perceptron also consistently performed well, while linear-based algorithms showed lower performance. Importantly, the use of stacking did not significantly improve performance. Our results suggest that the feature vector applied with this approach is applicable across various robotic models and datasets, provided that training data is balanced and sufficient points are used. Notably, by leveraging classifiers, we achieved rapid computation of results in less than 1 ms, with minimal computational cost. [ABSTRACT FROM AUTHOR]

Efficient global optimization is a widely used method for optimizing expensive black-box functions. In this paper, we study the worst-case oracle complexity of the efficient global optimization problem. In contrast to existing kernel-specific results, we derive a unified lower bound for the oracle complexity of efficient global optimization in terms of the metric entropy of a ball in its corresponding reproducing kernel Hilbert space. Moreover, we show that this lower bound nearly matches the upper bound attained by non-adaptive search algorithms, for the commonly used squared exponential kernel and the Matérn kernel with a large smoothness parameter ν . This matching is up to a replacement of d/2 by d and a logarithmic term log R ϵ , where d is the dimension of input space, R is the upper bound for the norm of the unknown black-box function, and ϵ is the desired accuracy. That is to say, our lower bound is nearly optimal for these kernels. [ABSTRACT FROM AUTHOR]

Shrimp fry counting is an important task for biomass estimation in aquaculture. Accurate counting of the number of shrimp fry in tanks can not only assess the production of mature shrimp but also assess the density of shrimp fry in the tanks, which is very helpful for the subsequent growth status, transportation management, and yield assessment. However, traditional manual counting methods are often inefficient and prone to counting errors; a more efficient and accurate method for shrimp fry counting is urgently needed. In this paper, we first collected and labeled the images of shrimp fry in breeding tanks according to the constructed experimental environment and generated corresponding density maps using the Gaussian kernel function. Then, we proposed a multi-scale attention fusion-based shrimp fry counting network called the SFCNet. Experiments showed that our proposed SFCNet model reached the optimal performance in terms of shrimp fry counting compared to CNN-based baseline counting models, with MAEs and RMSEs of 3.96 and 4.682, respectively. This approach was able to effectively calculate the number of shrimp fry and provided a better solution for accurately calculating the number of shrimp fry. [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]

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 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]

Distributed learning has attracted considerable attention in recent years due to its power to deal with big data in various science and engineering problems. Based on a divide-and-conquer strategy, this paper studies the distributed robust regression algorithm associated with correntropy losses and coefficient regularization in the scheme of kernel networks, where the kernel functions are not required to be symmetric or positive semi-definite. We establish explicit convergence results of such distributed algorithm depending on the number of data partitions, robustness and regularization parameters. We show that with suitable parameter choices the distributed robust algorithm can obtain the optimal convergence rate in the minimax sense, and simultaneously reduce the computational complexity and memory requirement in the standard (non-distributed) algorithms. [ABSTRACT FROM AUTHOR]

The alternating active-phase algorithm (AAPA) is widely recognized as being efficient for multi-material topology optimization. It has the advantages of easy implementation and broad applicability, but faces shortcomings, such as a uniform punishment intensity for all elements, excessive gray elements, and slow optimization convergence. To overcome these shortcomings, proposed here is an enhanced AAPA (EAAPA), the core ideas of which are as follows. (1) A new equal-scale Heaviside projection function (EHPF) is proposed, with the scale factor introduced to adjust the physical density adaptively to overcome the challenges of the invariance of the density sum and the uniqueness of projecting faced by the traditional Heaviside projection function. This adaptation reduces the number of gray elements and accelerates the convergence. (2) An innovative solid isotropic material with exponential sigmoid function (SIMESF) is constructed, with a penalty threshold and a steepness factor introduced to divide different penalty intensity regions and control their transition steepness. This improves the applicability and flexibility of the interpolation model, addressing the issue of a single punishment intensity in the solid isotropic material with penalization. The parameter values of the SIMESF model are explored via the design of a bridge, and the performances of AAPA and EAAPA are evaluated via the designs of simply supported and cantilever beams. The results show clearly that EAAPA has significant advantages in terms of optimization effectiveness and convergence speed. [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]

Purpose: Financial mathematics is one of the most rapidly evolving fields in today's banking and cooperative industries. In the current study, a new fractional differentiation operator with a nonsingular kernel based on the Robotnov fractional exponential function (RFEF) is considered for the Black–Scholes model, which is the most important model in finance. For simulations, homotopy perturbation and the Laplace transform are used and the obtained solutions are expressed in terms of the generalized Mittag-Leffler function (MLF). Design/methodology/approach: The homotopy perturbation method (HPM) with the help of the Laplace transform is presented here to check the behaviours of the solutions of the Black–Scholes model. HPM is well known for its accuracy and simplicity. Findings: In this attempt, the exact solutions to a famous financial market problem, namely, the BS option pricing model, are obtained using homotopy perturbation and the LT method, where the fractional derivative is taken in a new YAC sense. We obtained solutions for each financial market problem in terms of the generalized Mittag-Leffler function. Originality/value: The Black–Scholes model is presented using a new kind of operator, the Yang-Abdel-Aty-Cattani (YAC) operator. That is a new concept. The revised model is solved using a well-known semi-analytic technique, the homotopy perturbation method (HPM), with the help of the Laplace transform. Also, the obtained solutions are compared with the exact solutions to prove the effectiveness of the proposed work. The different characteristics of the solutions are investigated for different values of fractional-order derivatives. [ABSTRACT FROM AUTHOR]

Seismic data reconstruction is a crucial process involving the restoration of missing or corrupted traces to create a uniform dataset for subsequent data processing. Various factors such as equipment failures, and surface obstacles, result in irregularly located or corrupted traces. The absence of these traces can compromise the quality and accuracy of the resulting image. To address this issue, the Nonuniform Fast Fourier Transform (NUFFT) method is employed to reconstruct missing traces in datasets with non-uniformly sampled data. It works by interpolating the non-uniformly sampled data onto a regular grid, enabling the traditional Fast Fourier Transform application for data recovery. This interpolation process is adjusted using a kernel function to account for non-uniform sampling and reduce aliasing artifacts. The outcome is a collection of Fourier coefficients that can be utilized to reconstruct missing or incomplete parts of data. This problem is transformed into a linear constraint problem, which is efficiently solved using the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA). In this study, we explore various techniques aimed at improving the convergence of FISTA, collectively referred to as improved FISTA methods. To validate the NUFFT+FISTA method for data reconstruction, we conducted numerical tests using 3D and 2D synthetic datasets, as well as field data. These tests show the advantages of the Greedy-FISTA in terms of convergence rate and affirm the accuracy of this approach in filling missing data traces. [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]

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]

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]

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]

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]

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]

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]

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]

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]

The use of tunable metasurface technology to realize the underwater tracking function of submarines, which is one of the hotspots and difficulties in submarine design. The structure-to-sound-field metasurface design approach is a highly iterative process based on trial and error. The process is cumbersome and inefficient. Therefore, an inverse design method was proposed based on parallel deep neural networks. The method took the global and local target sound field feature information as input and the metasurface physical structure parameters as output. The deep neural network was trained using a kernel loss function based on a radial basis kernel function, which established an inverse mapping relationship between the desired sound field to the metasurface physical structure parameters. Finally, the sound field intensity modulation at a localized target range was achieved. The results indicated that within the regulated target range, this method achieved an average prediction error of less than 5 dB for 92.9% of the sample data. [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, 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]

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]

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]

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]

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]

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]

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]

• 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]

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]

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]

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]