General Kernel Research Articles

Semiconvexity estimates for nonlinear integro‐differential equations

AbstractIn this paper we establish for the first time local semiconvexity estimates for fully nonlinear equations and for obstacle problems driven by integro‐differential operators with general kernels. Our proof is based on the Bernstein technique, which we develop for a natural class of nonlocal operators and consider to be of independent interest. In particular, we solve an open problem from Cabré‐Dipierro‐Valdinoci. As an application of our result, we establish optimal regularity estimates and smoothness of the free boundary near regular points for the nonlocal obstacle problem on domains. Finally, we also extend the Bernstein technique to parabolic equations and nonsymmetric operators.

Global existence and decay of a viscoelastic wave equation with logarithmic source under acoustic, fractional, and nonlinear delay conditions

This study is concerned with a nonlinear viscoelastic wave equation with logarithmic source term. By considering the nonlinear distributed delay feedback influenced by acoustic and fractional boundary conditions at the boundary coupling, we establish the global existence and general decay results under appropriate assumptions, even in the case of a general kernel. This result extends and complements some previous results.

Exploring the Influence of Generalized Kernels on Green’s Function in Fractional Differential Equations

The basic purpose of this article is to define the Green’s function in order to provide the solution of fractional differential equations in the presence of general analytic kernel. Using the technique of Laplace and Fourier transforms, we construct the Green’s function for ordinary and partial fractional differential equations. The presented results will provide the generalization of some models existing in the literature. Some examples are also provided to prove the results for some particular cases.

Dynamics of a pituitary–adrenal model with distributed time delays

This paper investigates the effects of distributed time delays on the dynamics of the pituitary–adrenal axis. Our two-dimensional model incorporates distributed time delays with general delay kernels, specifically illustrating interactions between adrenocorticotropic hormone (ACTH) and cortisol (CORT). We derive general theoretical results for general delay kernels, exemplified by Dirac and weak Gamma kernels, revealing stability transitions characterized by Hopf bifurcations. We establish conditions for the local asymptotic stability of the unique equilibrium and discuss the existence of periodic solutions. Numerical simulations demonstrate that periodic oscillations appear for appropriate values of the average time delays. Including an external input results in both ultradian and circadian rhythms, highlighting the system’s dynamic responsiveness to external stimuli.

Special functions with general kernel: Properties and applications to fractional partial differential equations

Abstract In this paper, we reconstruct the gamma and beta functions using a general kernel function in their integral representations. We also reconstruct the Gauss and confluent hypergeometric functions using the beta function with general kernel in their series representations. The general kernel function we use here can be chosen as any special function such as exponential function, Mittag-Leffler function, Wright function, Fox-Wright function, Kummer function or M-series. Using different choices of this general kernel function, various of the generalized gamma, beta, Gauss hypergeometric and confluent hypergeometric functions in the literature can be obtained. In this paper, we first obtain the integral representations, functional relations, summation, derivative and transformation formulas and double Laplace transforms of the special functions we construct. Furthermore, we compute the solutions of some fractional partial differential equations involving special functions with general kernel via the double Laplace transform and graph some of the solutions for specific values. Finally, we obtain the incomplete beta function with general kernel by defining the beta distribution with general kernel.

Maximum Local Density-Driven Non-overlapping Radial Basis Function Support Kernel Neural Network

The learning and optimization of kernels in the radial basis function neural network (RBFNN) are crucial. However, in existing methods, there are issues of overfitting when learning kernel parameters. The learned kernels are also sensitive to outliers. This paper proposes a general kernel learning strategy for RBFNN called non-overlapping maximum local density support kernel learning (MLD-SKL), which contains two modules, the non-overlapping maximum local density (MLD) kernel learning module and support kernel learning (SKL) module. In the MLD kernel learning stage, the candidate set of kernels is incrementally determined based on the local density of samples. Meanwhile, it is required that the coverage ranges of kernels from different classes do not overlap with each other. This module is effective in reducing the impact of outliers. In the SKL stage, kernel importance indicator is defined to measure the importance of kernels. The learned support kernels are utilized to construct a maximum local density-driven non-overlapping radial basis function support kernel neural network (MLD-RBFSKNN). The RBFNN constructed through MLD-SKL exhibits a more compact structure. The experiments demonstrate that the proposed MLD-RBFSKNN improves accuracy in recognition task. Furthermore, while achieving superior recognition performance, the final constructed network also has the minimum number of kernels.

Experimental Investigation of Sulfur and Nitrogen Oxide Emissions and Corrosion Propensity During the Combustion of Coals and Biomass Fuels

ABSTRACT The increasing use of biomass in Indonesian coal-fired power plants, mainly through co-firing, presents challenges due to different fuel characteristics, which can affect gas emissions, ash deposition, and combustion efficiency. This study assesses flue gas emissions (SO2 and NOx) and corrosion tendencies of various biomass types and coal. The study begins with characteristic testing and ash analysis of coal and biomass samples, including proximate, ultimate, and calorific value analyses, to determine their suitability as fuels. Preliminary calculations predict flue gas emissions and ash deposit formation, followed by laboratory-scale combustion experiments using a DTF to assess flue gas and ash deposit characteristics. The results show that sulfur in biomass is converted to the gas phase up to 99%, while in coal, it is converted up to 95% at high temperatures. NOx levels are primarily affected by the nitrogen contents of the fuels; for instance, Coal-B at 0.88 wt% produces less NOx than solid recovered fuel (SRF) and refused derived fuel (RDF) with 1.73 wt% and 1.51 wt%, respectively, yet still generates higher emissions than the other three biomass types. In general, wood chips (WC), rise husk (RH), and palm kernel shell (PKS) emit 30–80% less flue gas compared to coal. However, ash deposition results indicate that PKS, SRF, and RDF exhibit corrosion tendencies due to the dominance of albite (NaAlSi3O8) in the mineral phase. Additionally, the high chlorine content in SRF (0.544 wt%) and RDF (0.224 wt%) leads to the formation of sticky ash deposits, which potentially impact heat transfer surfaces. These findings contribute to identifying suitable biomass types for both co-firing or dedicated combustion in coal-fired power plants.

Distributionally Robust Newsvendor Under Stochastic Dominance with a Feature-Based Application

Problem definition: In this paper, we study the newsvendor problem under some distributional ambiguity sets and explore their relations. Additionally, we explore the benefits of implementing this robust solution in the feature-based newsvendor problem. Methodology and results: We propose a new type of discrepancy-based ambiguity set, the JW ambiguity set, and analyze it within the framework of first-order stochastic dominance. We show that the distributionally robust optimization (DRO) problem with this ambiguity set admits a closed-form solution for the newsvendor loss. This result also implies that the newsvendor problem under the well-known infinity-Wasserstein ambiguity set and Lévy ball ambiguity set admit closed-form inventory levels as a by-product. In the application of feature-based newsvendor, we adopt general kernel methods to estimate the conditional demand distribution and apply our proposed DRO solutions to account for the estimation error. Managerial implications: The closed-form solutions enable an efficient computation of optimal inventory levels. In addition, we explore the property of optimal robust inventory levels with respect to the nonrobust version via concepts of perceived critical ratio and mean repulsion. The results of numerical experiments and the case study indicate that the proposed model outperforms other state-of-the-art approaches, particularly in environments where demand is influenced by covariates and difficult to estimate. Funding: X. Li is supported by the Singapore Ministry of Education [Tier 1 Grant 23-0619-P0001, 24-0500-A0001] and National Natural Science Foundation of China [Grant 72331004]. L. Zhang is partially supported by the National Natural Science Foundation of China [Grants 72171156 and 72231002] and the Hong Kong Research Grants Council [Grant 16212419]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.0159 .

A general kernel machine regression framework using principal component analysis for jointly testing main and interaction effects: Applications to human microbiome studies.

The effect of a treatment on a health or disease response can be modified by genetic or microbial variants. It is the matter of interaction effects between genetic or microbial variants and a treatment. To powerfully discover genetic or microbial biomarkers, it is crucial to incorporate such interaction effects in addition to the main effects. However, in the context of kernel machine regression analysis of its kind, existing methods cannot be utilized in a situation, where a kernel is available but its underlying real variants are unknown. To address such limitations, I introduce a general kernel machine regression framework using principal component analysis for jointly testing main and interaction effects. It begins with extracting principal components from an input kernel through the singular value decomposition. Then, it employs the principal components as surrogate variants to construct three endogenous kernels for the main effects, interaction effects, and both of them, respectively. Hence, it works with a kernel as an input without knowing its underlying real variants, and also detects either the main effects, interaction effects, or both of them robustly. I also introduce its omnibus testing extension to multiple input kernels, named OmniK. I demonstrate its use for human microbiome studies.

Physically-informed change-point kernels for variable levels of physical knowledge inclusion in Gaussian processes

The relative balance between physics and data within any physics-informed machine learner is an important modelling consideration to ensure that the benefits of both physics and data-based approaches are maximised. An over reliance on physical knowledge can be detrimental, particularly when the physics-based component of a model may not accurately represent the true modelled system. An under utilisation of physical knowledge potentially wastes a valuable resource, along with benefits in model interpretability and reduced demand for expensive data collection. Although adjusting the relative levels of physics and data reliance within a model is possible through the adaptation of the model structure, in practice, this can be challenging, with the relative balance produced by new model structures not always clear before they are implemented. This paper presents a means of being able to tune the balance of physics and data reliance within a model through the development of physically-informed change-point kernels for Gaussian processes. These combine more structured physical kernels, capable of enforcing physically derived behaviours, with flexible, general purpose kernels, and provide means to dynamically change the relative levels of reliance on physics and data within a model.

Frictional contact analysis between two-dimensional deformable anisotropic magneto-electro-elastic bodies via a semi-analytical method

In this paper, we utilize the surface Green's function of a magneto-electro-elastic (MEE) half-plane as a general analytical kernel to develop a semi-analytical method (SAM). This SAM is designed to solve the two-dimensional contact problems of two dissimilar deformable anisotropic MEE bodies. Using the surface Green's function, the associated influence matrices in SAM can be obtained in a closed mathematical form and then computed analytically, offering significant benefits in terms of computational efficiency and accuracy. This method can be applied to general contact problems where the contact surface is frictional sliding and subjected to four different electric and magnetic conditions. By assuming the elastic and piezoelectric materials as uncoupled MEE materials, the method can also be applied to contact problems in which anisotropic elastic and piezoelectric materials are involved. To demonstrate the accuracy and applicability of the proposed SAM, four different numerical examples are provided, and their results are compared with those obtained by using other approaches, such as analytical solutions and the previous SAM in the literature. Through the numerical results, parametric studies are also performed, allowing us to explore the influences of material properties, loadings, frictional coefficient, and contact surface profile on mechanical/electrical/magnetic contact behaviors.

First step toward a parameter-free, nonlocal kinetic energy density functional for semiconductors and simple metals.

The accuracy of orbital-free density functional theory depends on the approximations made for a Kinetic Energy (KE) functional. Until now, the most accurate KEDFs are based on non-local kernels constructed from the linear response theory of homogeneous electron gas. In this work, we explore beyond the HEG by employing a more general kernel based on the jellium-with-gap model (JGM). The proposed functional incorporates several new features, such as (i) having the correct low momentum(q) limit of the response function for metals and semiconductors without any modeling term, (ii) the underlying kernel is density-independent, and most importantly, (iii) parameter-free. The accuracy and efficiency of the proposed JGM NL-KEDF have been demonstrated for several semiconductors and metals. The encouraging results indicate the utility and predictive power of the JGM kernel for NL KEDF developments. This approach is also physically appealing and practically useful as we have presented a general formalism to incorporate the gap kernel in all existing Lindhard-based functionals.

The approximation capabilities of Durrmeyer-type neural network operators

AbstractIn this paper, a new family of neural network (NN) operators is introduced. The idea is to consider a Durrmeyer-type version of the widely studied discrete NN operators by Costarelli and Spigler (Neural Netw 44:101–106, 2013). Such operators are constructed using special density functions generated from suitable sigmoidal functions, while the reconstruction coefficients are based on a convolution between a general kernel function $$\chi $$ χ and the function being reconstructed, f. Here, we investigate their approximation capabilities, establishing both pointwise and uniform convergence theorems for continuous functions. We also provide quantitative estimates for the approximation order thanks to the use of the modulus of continuity of f; this turns out to be strongly influenced by the asymptotic behaviour of the sigmoidal function $$\sigma $$ σ . Our study also shows that the estimates we provide are, under suitable assumptions, the best possible. Finally, $$L^p$$ L p -approximation is also established. At the end of the paper, examples of activation functions are discussed.

Rates of the Strong Uniform Consistency with Rates for Conditional U-Statistics Estimators with General Kernels on Manifolds

Rates of the Strong Uniform Consistency with Rates for Conditional U-Statistics Estimators with General Kernels on Manifolds

Solving Abel integral equations by regularisation in Hilbert scales

Integral operators of Abel type of order a>0 arise naturally in a large spectrum of physical processes. Their inversion requires care since the resulting inverse problem is ill-posed. The purpose of this work is to devise and analyse a family of appropriate Hilbert scales so that the operator is ill-posed of order a in the scale. We provide weak regularity assumptions on the kernel underlying the operator for the above to hold true. Our construction leads to a well-defined regularisation strategy by Tikhonov regularisation in Hilbert scales. We thereby generalise the results of Gorenflo and Yamamoto for a<1 to arbitrary a>0 and more general kernels. Thanks to tools from interpolation theory, we also show that the a priori information associated to the Hilbert scale formulates in terms of smoothness in usual Sobolev spaces up to boundary conditions, and that the regularisation term actually amounts to penalising derivatives. Finally, following the theoretical construction, we develop a comprehensive numerical approach, where the a priori smoothness is encoded in a single parameter rather than in a full operator. Several numerical examples are shown, both confirming the theoretical convergence rates and showing the general applicability of the method.

UFBSM: unmixing fusion and background sparse dictionary Model for hyperspectral anomaly detection

ABSTRACT In recent years, low-rank sparse representation (LRR) schemes have been applied to the field of hyperspectral image (HSI) anomaly detection (AD) and received attention. However, it is often accompanied by a small amount of noise in the AD, which leads to a high false alarm rate of detection results. In this paper, the HSI-AD algorithm is proposed to address this problem. First, the HSI is unmixed using the deep network to obtain the endmembers and abundance, and the abundance matrix is fused with the HSI to obtain the mixed data, which to a certain extent complements the spatial information of the HSI because the abundance matrix can reflect the subpixel-level information of the target image elements. Then, the mixed data are decomposed based on the model proposed in this paper. To overcome the situation that the sparse part of the traditional model is prone to mix with a small amount of noise, the sparse decomposition of the mixed data into anomaly and noise is used to remove the small amount of noise from the sparse part. A joint dictionary consisting of background and anomaly is used in the above model. In addition, to address the problem of information loss caused by the uniform accumulation of kernel norm over all singular values in the process of model decomposition, this paper designs the γ norm decomposition method to replace the general kernel norm decomposition for a more effective extraction of useful information.

Bridging the gap between models based on ordinary, delayed, and fractional differentials equations through integral kernels

Evolution equations with convolution-type integral operators have a history of study, yet a gap exists in the literature regarding the link between certain convolution kernels and new models, including delayed and fractional differential equations. We demonstrate, starting from the logistic model structure, that classical, delayed, and fractional models are special cases of a framework using a gamma Mittag-Leffler memory kernel. We discuss and classify different types of this general kernel, analyze the asymptotic behavior of the general model, and provide numerical simulations. A detailed classification of the memory kernels is presented through parameter analysis. The fractional models we constructed possess distinctive features as they maintain dimensional balance and explicitly relate fractional orders to past data points. Additionally, we illustrate how our models can reproduce the dynamics of COVID-19 infections in Australia, Brazil, and Peru. Our research expands mathematical modeling by presenting a unified framework that facilitates the incorporation of historical data through the utilization of integro-differential equations, fractional or delayed differential equations, as well as classical systems of ordinary differential equations.

Hypergraph Isomorphism Computation.

The isomorphism problem, crucial in network analysis, involves analyzing both low-order and high-order structural information. Graph isomorphism algorithms focus on structural equivalence to simplify solver space, aiding applications like protein design, chemical pathways, and community detection. However, they fall short in capturing complex high-order relationships, unlike hypergraph isomorphism methods. Traditional hypergraph methods face challenges like high memory use and inaccurate identification, leading to poor performance. To overcome these, we introduce a hypergraph Weisfeiler-Lehman (WL) test algorithm, extending the WL test from graphs to hypergraphs, and develop a hypergraph WL kernel framework with two variants: the Hypergraph WL Subtree Kernel and Hypergraph WL Hyperedge Kernel. The Hypergraph WL Subtree Kernel counts different types of rooted subtrees and generates the final feature vector for a given hypergraph by comparing the number of different types of rooted subtrees. The Subtree Kernel identifies different rooted subtrees, while the Hyperedge Kernel focuses on hyperedges' vertex labels, enhancing feature vector generation. In order to fulfill our research objectives, a comprehensive set of experiments was meticulously designed, including seven graph classification datasets and 12 hypergraph classification datasets. Results on graph classification datasets indicate that the Hypergraph WL Subtree Kernel can achieve the same performance compared with the classical Graph Weisfeiler-Lehman Subtree Kernel. Results on hypergraph classification datasets show significant improvements compared to other typical kernel-based methods, which demonstrates the effectiveness of the proposed methods. In our evaluation, our proposed methods outperform the second-best method in terms of runtime, running over 80 times faster when handling complex hypergraph structures. This significant speed advantage highlights the great potential of our methods in real-world applications.

On generalized fractional differential equation with Sonine kernel on a function space

Sonine kernel is a special type of the general kernels, satisfying some integrability and monotonicity conditions and properties. Sonine kernels generate some convolution polynomials (power functions) used in the construction of the general fractional calculus. The general fractional calculus with Sonine kernel is applied in the setting up and development of operational calculus.Consequently, we research on a kind of a generalized fractional differential equation with a Sonine type kernel: (∗D(κ)Cμ)(t)=ϱϑ(t,μ(t)),t∈(0,T],T<∞,with μ(0)=ω a non-negative and bounded initial data. The function ϑ:(0,T]×C−11(0,T]→R is assumed to be Lipschitz continuous on its second variable, ϱ>0 is a constant, ∗D(κ)C is a generalized Caputo fractional derivative operator and κ is a Sonine kernel belonging to a function space with an integrable singularity at the point of origin. Given some defined properties of the kernel, we establish evidence for the well-posedness of the solution via Banach fixed point theorem and reveal that the solution indicates an exponential growth bound. Moreso, we show that the solution is uniformly continuous in time and has continuous dependence on the initial condition.

A general integration kernel formulation for immersed boundary method

This study proposes a new robust and accurate immersed boundary method for the immersion of solid bodies within a fluid with a Cartesian grid. The present method introduces the signed distance fields to recognize the immersed geometry contours, eliminating the need of Lagrangian points. To fully maximize the advantages offered by signed distance fields, a general integration kernel formulation is introduced into the direct forcing method to replace the conventional regularized delta function. With the combination of signed distance fields and kernel function, an interpolation along the radial direction instead of three-dimensional directions is feasible, which further reduces the extra calculation cost involved by immersed boundary method. The numerical results at low Reynolds numbers are compared to experimental and previous numerical results, which shows the efficiency and accuracy of this new method. Upon thorough validation, the proposed method in this paper demonstrates excellent performance across various scenarios, including static and moving cases as well as two- and three-dimensional configurations. And our method greatly reduces the cost of pretreatment of immersed geometry contours and apparently improves the convenience of the method.