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Extended kalman filter (EKF) is widely used to integrated navigation system of global navigation satellite system (GNSS) and strapdown inertial navigation system (SINS). However, non-Gaussian noise and the uncertainty of measurement noise can seriously reduce the performance of EKF, so it is difficult to obtain an optimal GNSS/INS integration solution. At present, non-Gaussian noise processing is still a difficult issue in filter research. In this paper, an adaptive and robust maximum correntropy extended kalman filter (MCEKF) method based on Cauchy kernel function is proposed to solve the above problem. Thanks to the excellent properties of Cauchy kernel function, the proposed method can effectively avoid filter faults and has better stability. However, the performance of MCEKF depends on the select of kernel bandwidth parameter, which is a difficult problem in the engineering application of MCEKF. In this paper, the switching kernel bandwidth algorithm is employed to adaptively estimate the kernel bandwidth parameter to solve the tradeoff between convergence rate and steady-state misalignment in the MCEKF algorithm under constant kernel bandwidth. Finally, the performance of the proposed algorithm is verified by two sets of vehicle-mounted precise point positioning (PPP) and SINS tightly coupled integration experiments in urban environment. The results show that compared with EKF, the proposed method can significantly improve the positioning accuracy, that is, the PPP/SINS positioning accuracy based on GE, GR and GRE system is improved by 25.1 %, 2.5 % and 17.0 % in D196 experiment, and 23.5 %, 16.5 % and 31.8 % in D107 experiment, respectively. The velocity and attitude accuracy of the two methods are similar. The velocity errors of all schemes can reach cm/s level. The roll, pitch and heading errors of navigation-grade inertial sensor are all less than 0.12 degree. The roll and pitch errors of MEMS are less than 1.0 degree, the heading error of the G, GE, GR, and GRE systems are 1.719, 1.464, 1.676, and 1.475 degree, respectively. [ABSTRACT FROM AUTHOR]

This special issue of the Journal of Optimization Theory & Applications focuses on the work of Professors Cornelis Roos and Florian A. Potra in the field of optimization. The issue includes a biographical sketch of both professors, highlighting their achievements. The papers in the special issue cover various topics such as conic optimization, interior point methods, proximal methods, convex optimization, and stochastic optimal control. The authors of the papers have close connections to Professors Roos and Potra, either as students, visitors, or co-authors. The document is a compilation of research papers on interior-point methods (IPMs) and their applications in optimization and conic programming. The papers cover theoretical developments of IPMs, algorithms for linear complementarity problems (LCPs), weighted LCPs, and sufficient linear complementarity problems. There are also papers on new computational approaches for IPMs, applications of IPMs in different fields such as mechanics and support vector machines, and the theory and application of conic optimization. The document also includes papers on proximal methods for convex and non-convex problems, as well as papers on convexity and convex optimization. The volume concludes with acknowledgments to the authors, reviewers, and editor-in-chief. [Extracted from the article]

This paper is devoted to the study of the multi-dimensional integral transform with the Fox H-function in the kernel in weighted spaces with integrable functions in the domain R + n with positive coordinates. Due to the generality of the Fox H-function, many special integral transforms have the form studied in this paper, including operators with such kernels as generalized hypergeometric functions, classical hypergeometric functions, Bessel and modified Bessel functions and so on. Moreover, most important fractional integral operators, such as the Riemann–Liouville type, are covered by the class under consideration. The mapping properties in Lebesgue-weighted spaces, such as the boundedness, the range and the representations of the considered transformation, are established. In special cases, it is applied to the specific integral transforms mentioned above. We use a modern technique based on the extensive use of the Mellin transform and its properties. Moreover, we generalize our own previous results from the one-dimensional case to the multi-dimensional one. The multi-dimensional case is more complex and needs more delicate techniques. [ABSTRACT FROM AUTHOR]

We determine the spreading speed of an epidemic model with nonlocal diffusion and free boundary. The model is evolved from a degenerate reaction-diffusion model of Capasso and Maddalena (J Math Biol 13:173–184, 1981), and was studied in Zhao et al. (Commun Pure Appl Anal 19:4599–4620, 2020) recently, where it was shown that as time goes to infinity, the population of the infective agents either vanishes or spreads successfully. In this paper, we show that when spreading is successful, the asymptotic spreading speed is finite or infinite depending on whether a threshold condition is satisfied by the kernel function governing the spatial dispersal of the agents. The proof relies on a rather complete understanding of the associated semi-wave problem and traveling wave problem. For free boundary models, the case of infinite spreading speed, also known as accelerated spreading, is only recently shown to happen in Du et al. (J Math Pure Appl 154:30–66, 2021) for a single species Fisher-KPP model; this paper is the first to show that it happens to a very different two species model with free boundary. This suggests that accelerated spreading is a rather common phenomenon for free boundary problems with nonlocal diffusion. In contrast, for the corresponding models with local diffusion, the spreading can only proceed with finite speed. [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]

In this paper, the well‐posedness of Cauchy's problem of fractional Schrödinger equations with a power‐type nonlinearity on n‐dimensional manifolds with nonnegative Ricci curvature is studied. Under suitable volume conditions, the local solution with initial data in H[n2]+1 will blow up in finite time no matter how small the initial data is, which follows from a new weight function and ODE inequalities. Moreover, the upper‐bound of the lifespan can be estimated. [ABSTRACT FROM AUTHOR]

Twin support vector machine (TSVM) is an effective machine learning tool for classification problems. However, TSVM classifier works on empirical risk principle only and also while training, each sample contributes equally, even if it is a noise or an outlier. It does not incorporate the uncertainties associated with data into modeling and hence its generalization ability declines. To address these issues, intuitionistic fuzzy regularized least square twin support vector machine having intuitionistic fuzzy network has been proposed in this paper. The non-parallel classifiers are obtained by solving two systems of linear equations only rather than the solution of two quadratic programming problems as in TSVM, which leads to speed up the training process. Moreover, the method follows both structural risk and empirical risk minimization principles. In order to de-escalate the effect of pollutant patterns, their contribution of the patterns into learning the decision function has been made according to their importance in the classification. The significance of the training patterns is measured in terms of intuitionistic fuzzy numbers based on their geometrical locations and surroundings. The method is further extended to find non-parallel decision planes in the feature space using nonlinear kernel function, which also gives rise to the solution of two systems of linear equations. To show the efficacy of the proposed method, computer simulations on fourteen standard and six big UCI datasets using linear and Gaussian kernels are performed and their results have been compared with well-established methods in the literature. The experimental results are represented in terms of accuracy, computational time, F-measure, sensitivity and specificity rates. The outcomes demonstrate that the proposed method outperforms the existing methods and is also feasible for big datasets. The comparison and statistical inferences using two non-parametric: Friedman and Nemenyi tests, conclude that the proposed approach is fast and yields better generalization. [ABSTRACT FROM AUTHOR]

In this paper, we study the weak strong uniqueness of the Dirichlet type problems of fractional Laplace (Poisson) equations. We construct the Green’s function and the Poisson kernel. We then provide a somewhat sharp condition for the solution to be unique. We also show that the solution under such condition exists and must be given by our Green’s function and Poisson kernel. In doing these, we establish several basic and useful properties of the Green’s function and Poisson kernel. Based on these, we obtain some further a priori estimates of the solutions. Surprisingly those estimates are quite different from the ones for the local type elliptic equations such as Laplace equations. These are basic properties to the fractional Laplace equations and can be useful in the study of related problems. [ABSTRACT FROM AUTHOR]

A multistage machining process is employed to machine the blades with low stiffness. Nonetheless, machining errors can be transferred and accumulate throughout the multistage machining process, complicating the precise prediction of the final accuracy of thin-walled blades. Consequently, this paper introduces a machining accuracy prediction model for thin-walled blades that takes into account initial error. The machining error prediction model of thin-walled blades is developed using Gaussian process regression optimized by the sparrow search algorithm (SSA-GPR) with the initial contour error, depth of cut, feed per tooth, and spindle speed as inputs, and the machining error as the output. And the results show that the prediction accuracy of the SSA-GPR is 6.73 % higher than that of the Gaussian process regression (GPR), 13.73 % higher than that of the back propagation neural network (BPNN), and 32.32 % higher than that of the support vector machine regression (SVR). The influence of the initial error and milling parameters on the machining error is analyzed through the length-scales of the Gaussian kernel function. The findings indicate that the depth of cut, feed per tooth and initial error significantly affect the machining error, whereas the spindle speed has a minor impact on the machining error. Furthermore, the 3D graph based on the SSA-GPR shows that the increase of the initial error will increase the machining error of thin-walled blades. This research provides a theoretical foundation for the process optimization of thin-walled blades. • A machining accuracy prediction method considering initial error is proposed. • Combine SSA with GPR to improve the prediction accuracy of GPR. • The correlation is conveniently examined via the hyperparameters of the GPR. • The initial error significantly affects the machining error of thin-walled blades. [ABSTRACT FROM AUTHOR]

This paper presents the reconstruction of the quintessence dark energy potential in a model- independent way. Reconstruction relies on a Gaussian process and on available expansion rate data. Specifically, 40-point values of H(z) are used, consisting of a 30-point sample deduced from a differential age method and an additional 10-point sample obtained from the radial BAO method. Results are obtained for two kernel functions and for three different values of H0. This sheds light on the H0 tension problem indicating that it is not just a numerical problem. The model-independent reconstruction of the potential can serve as a reference to constraint available models and construct new ones. Various possibilities, including V(ϕ) ~ e-λϕ , are compared with the reconstructions here obtained, which is notably the first truly model-independent reconstruction of the quintessence dark energy potential. This allows the selection of new models that can be interesting for cosmology. The method can be extended to reconstruct the potential of related dark energy models, to be considered in future work. [ABSTRACT FROM AUTHOR]

In this paper, we develop an integrodifference equation model that incorporates spatial memory and learning so that each year, a fraction of the population use the same dispersal kernel as the previous year, and the remaining individuals return to where they bred or were born. In temporally static environments, the equilibrium of the system corresponds to an ideal free dispersal strategy, which is evolutionarily stable. We prove local stability of this equilibrium in a special case, and we observe convergence towards this equilibrium in numerical computations. When there are periodic or stochastic temporal changes in the environment, the population is less able to match the environment, but is able to do so to some extent depending on the parameters. Overall, the mechanism proposed in this model shows a possible way for the dispersal kernel of a population to evolve towards an ideal free dispersal kernel. [ABSTRACT FROM AUTHOR]

Kernel methods are popular in nonlinear and nonparametric regression due to their solid mathematical foundations and optimal statistical properties. However, scalability remains the primary bottleneck in applying kernel methods to large-scale data regression analysis. This paper aims to improve the scalability of kernel methods. We combine Nyström subsampling and the preconditioned conjugate gradient method to solve regularized kernel regression. Our theoretical analysis indicates that achieving optimal convergence rates requires only (n) memory and (n n) time (up to logarithmic factors). Numerical experiments show that our algorithm outperforms existing methods in time efficiency and prediction accuracy on large-scale datasets. Notably, compared to the FALKON algorithm [A. Rudi, L. Carratino and L. Rosasco, Falkon: An optimal large scale kernel method, in Advances in Neural Information Processing Systems (Curran Associates, 2017), pp. 3891–3901], which is known as the optimal large-scale kernel method, our method is more flexible (applicable to non-positive definite kernel functions) and has a lower algorithmic complexity. Additionally, our established theoretical analysis further relaxes the restrictive conditions on hyperparameters previously imposed in convergence analyses. [ABSTRACT FROM AUTHOR]

• Double bias fields are introduced into fidelity term to approximate image intensity inhomogeneity. • The proposed energy function is strictly convex, and allows flexible initialization. • A TV (total variation) regularization term is introduced to keep convex level-set function smooth. • The proposed method is robust against to noise and intensity inhomogeneity. The existing active contour models (ACMs) based on bias field (BF) correction mostly rely on a single BF assumption and lack in-depth discussion on the convexity of the energy functional, often leading to the problem of local minima. To address this issue, this paper introduces a dual BF and proposes a convex level-set (LS) method based on multiplicative-additive (MA) model to achieve global minima. Firstly, a MA model is adopted as the fidelity term, and a kernel function is introduced to adjust the size of the intensity inhomogeneous neighborhood, enhancing the adaptability to intensity inhomogeneity. Then, the convex LS function is embedded in the variational framework to ensure convexity of each variable in the energy functional. This transformation turns the segmentation problem into a convex optimization problem. By introducing the total variation regularization term to smooth the LS function, the model's resistance to noise is effectively enhanced. Finally, by minimizing the proposed energy functional, image segmentation and BF correction are successfully achieved. Experimental results validate the global minima property of our model, while also demonstrating good flexibility in the initial contour. The proposed model achieves superior segmentation results compared to other classical ACMs on various types of images. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the nonlocal‐in‐time Allen‐Cahn equation (NiTACE), which incorporates a nonlocal operator in time with a finite nonlocal memory. Our objective is to examine the well‐posedness of the NiTACE by establishing the maximal Lp$$ {L}^p $$ regularity for the nonlocal‐in‐time parabolic equations with fractional power kernels. Furthermore, we derive a uniform energy bound by leveraging the positive definite property of kernel functions. We also develop an energy‐stable time stepping scheme specifically designed for the NiTACE. Additionally, we analyze the discrete maximum principle and energy dissipation law, which hold significant importance for phase field models. To ensure convergence, we verify the asymptotic compatibility of the proposed stable scheme. Lastly, we provide several numerical examples to illustrate the accuracy and effectiveness of our method. [ABSTRACT FROM AUTHOR]

Bourgain in his seminal paper of 1986 about the analysis of maximal functions associated to convex bodies has estimated in a sharp way the |$L^{2}$| -operator norm of the maximal function associated to a kernel |$K\in L^{1},$| with differentiable Fourier transform |$\widehat{K}.$| We formulate the extension to Bourgain's |$L^{2}$| -estimate in the setting of maximal functions on graded Lie groups. Our criterion is formulated in terms of the group Fourier transform of the kernel. We discuss the application of our main result to the |$L^{p}$| -boundedness of maximal functions on graded Lie groups. [ABSTRACT FROM AUTHOR]

A type of general fractional calculus with respect to another function is introduced in this paper. The kernel function within the fractional derivative and integral is specified as the Exponential function in a typical way. Based on the excellent property of the newly defined general fractional calculus, we proposed two viscoelastic models in the fractional manner compared with the classical calculus. [ABSTRACT FROM AUTHOR]

In this paper, by replacing the exponential memory kernel function of a tabu learning single-neuron model with the power-law memory kernel function, a novel Caputo's fractional-order tabu learning single-neuron model and a network of two interacting fractional-order tabu learning neurons are constructed firstly. Different from the integer-order tabu learning model, the order of the fractional-order derivative is used to measure the neuron's memory decay rate and then the stabilities of the models are evaluated by the eigenvalues of the Jacobian matrix at the equilibrium point of the fractional-order models. By choosing the memory decay rate (or the order of the fractional-order derivative) as the bifurcation parameter, it is proved that Hopf bifurcation occurs in the fractional-order tabu learning single-neuron model where the value of bifurcation point in the fractional-order model is smaller than the integer-order model's. By numerical simulations, it is shown that the fractional-order network with a lower memory decay rate is capable of producing tangent bifurcation as the learning rate increases from 0 to 0.4. When the learning rate is fixed and the memory decay increases, the fractional-order network enters into frequency synchronization firstly and then enters into amplitude synchronization. During the synchronization process, the oscillation frequency of the fractional-order tabu learning two-neuron network increases with an increase in the memory decay rate. This implies that the higher the memory decay rate of neurons, the higher the learning frequency will be. [ABSTRACT FROM AUTHOR]

In this paper, we present a re-established functional fractal circuit model of arterial blood flow that incorporates the shunt effect of the branch vessels. Under the background of hemodynamics, we abstracted a family of fractal operators and investigate the kernel function and properties thereof. Based on fractal operators, the intrinsic relation between Bessel function and Struve function was revealed, and some new special functions were found. The results provide mathematical tools for biomechanics and automatic control. [ABSTRACT FROM AUTHOR]

In this paper, we consider a kernel-based full-Newton step feasible interior-point method (IPM) for P ∗ (κ) -Weighted Linear Complementarity Problem (WLCP). The specific eligible kernel function is used to define an equivalent form of the central path, the proximity measure, and to obtain search directions. Full-Newton steps are adopted to avoid the line search at each iteration. It is shown that with appropriate choices of the parameters, and a certain condition on the starting point, the iterations always lie in the defined neighborhood of the central path. Assuming strict feasibility of P ∗ (κ) -WLCP, it is shown that the IPM converges to the ε -approximate solution of P ∗ (κ) -WLCP in a polynomial number of iterations. Few numerical results are provided to indicate the computational performance of the algorithm. [ABSTRACT FROM AUTHOR]

We propose new short-step interior-point algorithms (IPAs) for solving P ∗ (κ) -linear complementarity problems (LCPs). In order to define the search directions, we use the algebraic equivalent transformation (AET) technique of the system describing the central path. A novelty of the paper is that we introduce a whole, new class of AET functions for which a unified complexity analysis of the IPAs is presented. This class of functions differs from the ones used in the literature for determining search directions, like the class of concave functions determined by Haddou, Migot and Omer, self-regular functions, eligible kernel and self-concordant functions. We prove that the IPAs using any member φ of the new class of AET functions have polynomial iteration complexity in the size of the problem, in starting point's duality gap, in the accuracy parameter and in the parameter κ . [ABSTRACT FROM AUTHOR]

This paper mainly concerns the biholomorphic invariance of p-essential normality of Hilbert modules on bounded symmetric domains. By establishing new integral formulas concerning rational function kernels for the Taylor functional calculus, we prove a biholomorphic invariance result related to the p-essential normality. Furthermore, for quotient analytic Hilbert submodules determined by analytic varieties, we develop an algebraic approach to proving that the p-essential normality is preserved invariant if the coordinate multipliers are replaced by arbitrary automorphism multipliers. Moreover, the Taylor spectrum of the compression tuple is calculated under a mild condition, which gives a solvability result of the corona problem for quotient submodules. As applications, we extend the recent results on the equivalence between \infty-essential normality and hyperrigidity. [ABSTRACT FROM AUTHOR]

Sea surface target detection is a key stage in a typical target detection system and directly influences the performance of the whole system. As an effective discriminator, the one-class support vector machine (OCSVM) has been widely used in target detection. In OCSCM, training samples are first mapped to the hypersphere in the kernel space with the Gaussian kernel function, and then, a linear classification hyperplane is constructed in each cluster to separate target samples from other classes of samples. However, when the distribution of the original data is complex, the transformed data in the kernel space may be nonlinearly separable. In this situation, OCSVM cannot classify the data correctly, because only a linear hyperplane is constructed in the kernel space. To solve this problem, a novel one-class classification algorithm, referred to as ensemble one-class support vector machine (En-OCSVM), is proposed in this paper. En-OCSVM is a hybrid model based on k -means clustering and OCSVM. In En-OCSVM, training samples are clustered in the kernel space with the k -means clustering algorithm, while a linear decision hyperplane is constructed in each cluster. With the combination of multiple linear classification hyperplanes, a complex nonlinear classification boundary can be achieved in the kernel space. Moreover, the joint optimization of the k -means clustering model and OCSVM model is realized in the proposed method, which ensures the linear separability of each cluster. The experimental results based on the synthetic dataset, benchmark datasets, IPIX datasets, and SAR real data demonstrate the better performance of our method over other related methods. [ABSTRACT FROM AUTHOR]

In this paper we discuss some basic properties of octonionic Bergman and Hardy spaces. In the first part we review some fundamental concepts of the general theory of octonionic Hardy and Bergman spaces together with related reproducing kernel functions in the monogenic setting. We explain how some of the fundamental problems in well-defining a reproducing kernel can be overcome in the non-associative setting by looking at the real part of an appropriately defined para-linear octonion-valued inner product. The presence of a weight factor of norm 1 in the definition of the inner product is an intrinsic new ingredient in the octonionic setting. Then we look at the slice monogenic octonionic setting using the classical complex book structure. We present explicit formulas for the slice monogenic reproducing kernels for the unit ball, the right octonionic half-space and strip domains bounded in the real direction. In the setting of the unit ball we present an explicit sequential characterization which can be obtained by applying the special Taylor series representation of the slice monogenic setting together with particular octonionic calculation rules that reflect the property of octonionic para-linearity. [ABSTRACT FROM AUTHOR]

Very recently one has started to study Bergman and Szegö kernels in the setting of octonionic monogenic functions. In particular, explicit formulas for the Bergman kernel for the octonionic unit ball and for the octonionic right half‐space as well as a formula for the Szegö kernel for the octonionic unit ball have been established. In this paper, we extend this line of investigation by developing explicit formulas for the Szegö kernel of strip domains of the form S:={z∈핆|0<ℜ(z)

This paper introduces a novel approach to obtaining analytical solutions for fuzzy fractional differential equations in the context of weighted Caputo-type generalized Hukuhara derivatives. The paper proposes the use of non-singular kernels to improve the accuracy of fractional calculus in fuzzy space and establishes the uniqueness of solutions for fuzzy fractional differential equations. The paper also introduces the concept of fuzzy Laplace transforms to facilitate the solution of these equations. Practical examples, such as the fuzzy fractional Newton’s law of heating and cooling, are provided to demonstrate the effectiveness of the proposed method. Overall, this paper contributes to the development of practical solutions for real-world problems in fuzzy space and enhances the accuracy of fractional calculus in this context. [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]

To avoid the expensive quadratic programming in the L2-loss support vector regression (SVR) model, smooth approximation and iteratively reweighted least square (IRLS) techniques were introduced in literature, resulting in smoothed SVR (SSVR) and IRLS-SVR. However, for nonlinear models, SSVR and IRLS-SVR both involve operations with matrices and vectors of the same size as the training set. Thus, as the training set becomes large, nonlinear SSVR and IRLS-SVR both need long training time and large memory. To further alleviate the training cost, this paper projects the original data into a low dimensional space via random Fourier feature. The inner product of the random Fourier features of two data points is approximately the same as the kernel function evaluated at these two data points. Hence, it is possible to use a linear model in the new low dimensional space to approximate the original nonlinear model, and consequently the time/memory efficient linear training algorithms could be applied. This paper applies the idea of random Fourier features to nonlinear SSVR and IRLS-SVR, and our testing results on real-world datasets show that, the introduction of random Fourier features makes SSVR and IRLS-SVR achieve similar prediction accuracy as the original nonlinear version with substantially higher time efficiency. [ABSTRACT FROM AUTHOR]

The paper is essentially motivated by the demonstrated potential for applications of the presented results in numerous widespread research areas, such as the mathematical, physical, engineering, and statistical sciences. The main object is to introduce and investigate a class of fractional integral operators involving a certain general family of multiindex Mittag-Leffler functions in their kernel. Among other results obtained in the paper, we establish several interesting expressions for the composition of well-known fractional integral and fractional derivative operators, such as (e.g.) the Riemann–Liouville fractional integral and fractional derivative operators, the Hilfer fractional derivative operator, and the above-mentioned fractional integral operator involving the general family of multiindex Mittag-Leffler functions in its kernel. Our main result is a generalization of the results obtained in earlier investigations in this field. We also present some potentially useful integral representations for the product of two members of the general family of multiindex Mittag-Leffler functions in terms of the well-known Fox–Wright hypergeometric function pΨ q with p numerator and q denominator parameters. [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]

In this paper we consider the homogenization of the evolution problem associated with a jump process that involves three different smooth kernels that govern the jumps to/from different parts of the domain. We assume that the spacial domain is divided into a sequence of two subdomains A n ∪ B n and we have three different smooth kernels, one that controls the jumps from A n to A n , a second one that controls the jumps from B n to B n and the third one that governs the interactions between A n and B n . Assuming that χ A n (x) → X (x) weakly in L ∞ (and then χ B n (x) → 1 - X (x) weakly in L ∞ ) as n → ∞ and that the initial condition is given by a density u 0 in L 2 we show that there is an homogenized limit system in which the three kernels and the limit function X appear. When the initial condition is a delta at one point, δ x ¯ (this corresponds to the process that starts at x ¯ ) we show that there is convergence along subsequences such that x ¯ ∈ A n j or x ¯ ∈ B n j for every n j large enough. We also provide a probabilistic interpretation of this evolution equation in terms of a stochastic process that describes the movement of a particle that jumps in Ω according to the three different kernels and show that the underlying process converges in distribution to a limit process associated with the limit equation. We focus our analysis in Neumann type boundary conditions and briefly describe at the end how to deal with Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

The paper presents research on measuring and the possibility of prediction of noise levels on the left and right sides of the operator within the cabin of an agricultural tractor when moving across various agrotechnical surfaces, considering movement velocity and tire pressures while employing machine learning techniques. Noise level measurements were conducted on a LANDINI POWERFARM 100 type tractor, and aligned with standards (HRN ISO 5008, HRN ISO 6396 and HRN ISO 5131). The obtained noise values were divided into two data sets (left and right set) and processed using multiple linear regression (mlr) and three machine learning methods (gradient boosting machine (gbm); support vector machine using radial basis function kernel (svmRadial); monotone multi-layer perceptron neural network (monmlp)). The most accurate method, considering surfaces, from the left side data set—(R2 0.515–0.955); (RMSE 0.302–0.704); (MAE 0.225–0.488)—and the right side—(R2 0.555–0.955); (RMSE 0.180–0.969); (MAE 0.139–0.644)—was monmlp predominantly, and to a lesser extent svmRadial. On analyzing the total data sets from the left and right sides regarding surfaces, gbm emerged as the most accurate method. The application of machine learning methods demonstrated data accuracy, yet in future research, measurements on certain surfaces may need to be repeated multiple times potentially to improve accuracy further. [ABSTRACT FROM AUTHOR]

This paper gives the controllability and Ulam–Hyers–Rassias (U–H–R) stability results for non-instantaneous impulsive stochastic multiple delays system with nonpermutable variable coefficients. The solution for nonlinear non-instantaneous impulsive stochastic systems is presented without the assumption of commutative property on delayed matrix coefficients. The kernel function of the solution operator is defined by sum of noncommutative products of delayed matrix constant coefficients. Sufficient conditions for controllability of linear and nonlinear non-instantaneous impulsive stochastic multiple delays system are established by using the Mönch fixed-point theorem under the proof that the corresponding linear system is controllable. Thereafter, U–H–R stability result is proved. Finally, the theoretical results are verified by a numerical example. [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]

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]

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, 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]

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]

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]

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]

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]

Hierarchical matrices are dense but data-sparse matrices that use low-rank factorizations of suitable submatrices to reduce the storage and computational cost to linear-polylogarithmic complexity. In this paper, we propose a new approach to efficiently compute QR factorizations in the hierarchical matrix format based on block Householder transformations. To prevent unnecessarily high ranks in the resulting factors and to increase speed and accuracy, the algorithm meticulously tracks for which intermediate results low-rank factorizations are available. We also use a special storage scheme for the block Householder reflector to further reduce computational and storage costs. Numerical tests for two- and three-dimensional Laplacian boundary element matrices, different radial basis function kernel matrices, and matrices of typical hierarchical matrix structures but filled with random entries illustrate the performance of the new algorithm in comparison to some other QR algorithms for hierarchical matrices from the literature. [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]

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]

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]

Rotor imbalances in wind turbines present safety risks and lead to economic losses, and a method to diagnose rotor imbalances is urgently needed. A diagnostic method for rotor imbalances is proposed in this paper. First, a signal reconstruction method is proposed, and a novel index is used to determine the number of components used in signal decomposition in order to effectively address the interference by noise on the sensor. Second, an entropy calculation method is proposed, and the Gaussian kernel function is used to replace the fuzzy functions. The results indicate significant differences for different types of rotor imbalances. Moreover, it exhibits good noise robustness and a low dependence on the data length. Third, a support vector machine with multiscale kernels is proposed, and kernel functions with various characteristics and scales are combined. It has a well-distributed hyperplane and better classification performance, and it is robust to wind conditions. Finally, the method is tested and verified with varying levels of noise and turbulence. The results demonstrate satisfactory performance because the proposed method can effectively identify rotor imbalances under different noise and wind conditions. [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 proposes a novel meshless local integral equation (LIE) method for numerical solutions of two and three-dimensional Poisson equations. The proposed method can be regarded as a new variant of the meshless local Petrov-Galerkin (MLPG) method which already has six variants, and so the proposed method can be called MLPG7. A variant of MLPG called local boundary integral equation (LBIE) method, uses a localized fundamental solution of Laplace equation as test function. This test function vanishes on local boundaries and therefore, the derived local equations do not involve the gradient of field variables. The motivation for new formulation is using an average of LBIE equation over radius of local sub-domains instead of a single sub-domain. A new kernel is introduced which is a new modification of fundamental solutions of Laplace equation. It is proved that using the new kernel as test function leads to the same local equations as averaging. Infact, the new formulation is a modification of the LBIE method for which the boundary integral is replaced with a domain integral. It is proved that new local weak formulation is equivalent with strong form. The convergence of the method for the case of regularly spaced nodal points is proved. For trial approximation, this paper uses Gaussian radial basis functions (Gaussian RBF), however, derivative free property of the method allows the use of any non-differentiable basis functions. Stability and convergence of the method are numerically studied by considering both 2D and 3D problems with regular and irregular nodal points. The method is compared with LBIE, finite differences (FD) and RBF-FD methods and the numerical results reveal the significance of the proposed method. [ABSTRACT FROM AUTHOR]

The present paper deals with the study of the approximation properties of the well-known sampling Kantorovich (SK) operators in "Sobolev-like settings". More precisely, a convergence theorem in case of functions belonging to the usual Sobolev spaces for the SK operators has been established. In order to get such a result, suitable Strang-Fix type conditions have been required on the kernel functions defining the above sampling type series. As a consequence, certain open problems related to the convergence in variation for the SK operators have been solved. Then, we considered the above operators in a fractional-type setting. It is well-known that, in the literature, several notions of fractional Sobolev spaces are available, such as, the Gagliardo Sobolev spaces (GSs) defined by means of the Gagliardo semi-norm, or the weak Riemann-Liouville Sobolev spaces (wRLSs) defined by the weak (left and right) Riemann-Liouville fractional derivatives and so on. Here, in order to face the above convergence problem, we introduced a new definition of fractional Sobolev spaces, that we called the tight fractional Sobolev spaces (tfSs) and generated as the intersection of the GSs and the symmetric Sobolev spaces (i.e., that given by the intersection of the left and the right wRLSs). In the latter setting, we obtain one of the main results of the paper, that is a convergence theorem for the SK operators with respect to a suitable norm on tfSs. [ABSTRACT FROM AUTHOR]