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The fractional Hilbert transform was introduced by Zayed [30, Zayed, 1998] and has been widely used in signal processing. In view of its connection with the fractional Fourier transform, Chen, the first, second and fourth authors of this paper in [6, Chen et al., 2021] studied the fractional Hilbert transform and other fractional multiplier operators on the real line. The present paper is concerned with a natural extension of the fractional Hilbert transform to higher dimensions: this extension is the fractional Riesz transform and is given by multiplication which a suitable chirp function on the fractional Fourier transform side. In addition to a thorough study of the fractional Riesz transform, in this work we also investigate the boundedness of singular integral operators with chirp functions on rotation invariant spaces, chirp Hardy spaces and their relation to chirp BMO spaces, as well as applications of the theory of fractional multipliers in partial differential equations. Through numerical simulation, we provide physical and geometric interpretations of high-dimensional fractional multipliers. Finally, we present an application of the fractional Riesz transforms in edge detection which verifies a hypothesis insinuated in [26, Xu et al., 2016]. In fact our numerical implementation confirms that amplitude, phase, and direction information can be simultaneously extracted by controlling the order of the fractional Riesz transform. [ABSTRACT FROM AUTHOR]

In the setting of quaternionic Heisenberg group Hn−1$\mathcal H^{n-1}$, we characterize the boundedness and compactness of commutator [b,C]$[b,\mathcal {C}]$ for the Cauchy–Szegö operator C$\mathcal {C}$ on the weighted Morrey space Lwp,κ(Hn−1)$L_w^{p,\,\kappa }(\mathcal H^{n-1})$ with p∈(1,∞)$p\in (1, \infty)$, κ∈(0,1)$\kappa \in (0, 1)$, and w∈Ap(Hn−1)$w\in A_p(\mathcal H^{n-1})$. More precisely, we prove that [b,C]$[b,\mathcal {C}]$ is bounded on Lwp,κ(Hn−1)$L_w^{p,\,\kappa }(\mathcal H^{n-1})$ if and only if b∈BMO(Hn−1)$b\in {\rm BMO}(\mathcal H^{n-1})$. And [b,C]$[b,\mathcal {C}]$ is compact on Lwp,κ(Hn−1)$L_w^{p,\,\kappa }(\mathcal H^{n-1})$ if and only if b∈VMO(Hn−1)$b\in {\rm VMO}(\mathcal H^{n-1})$. [ABSTRACT FROM AUTHOR]

This work is devoted to the nonlinear Schrödinger–Kirchhoff-type equation − (a + b ∫ R 3 | ∇ u | 2 d x) Δ u + V (x) u = f (x , u) , in R 3 , where a > 0 , b ≥ 0 , the nonlinearity f (x , ⋅) is 3-superlinear and the potential V is either periodic or exhibits a finite potential well. By the mountain pass theorem, Lions' concentration-compactness principle, and the energy comparison argument, we obtain the existence of positive ground state for this problem without proving the Palais–Smale compactness condition. [ABSTRACT FROM AUTHOR]

We obtain the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type. In addition, with the aid of interpolation theory, we provide weighted version of the commutator theorems by establishing new characterizations of the weighted BMO space. Finally, a concrete example shows that the local version of commutators also has an independent interest. [ABSTRACT FROM AUTHOR]

This paper is devoted to the L p (R) theory of the fractional Fourier transform (FRFT) for 1 ≤ p < 2. In view of the special structure of the FRFT, we study FRFT properties of L 1 functions, via the introduction of a suitable chirp operator. However, in the L 1 (R) setting, problems of convergence arise even when basic manipulations of functions are performed. We overcome such issues and study the FRFT inversion problem via approximation by suitable means, such as the fractional Gauss and Abel means. We also obtain the regularity of fractional convolution and results on pointwise convergence of FRFT means. Finally we discuss L p multiplier results and a Littlewood-Paley theorem associated with FRFT. [ABSTRACT FROM AUTHOR]

This article addresses two characterizations of BMO(ℝn)-type space via the commutators of Hardy operators with homogeneous kernels on Lebesgue spaces: (i) characterization of the central BMO(ℝn) space by the boundedness of the commutators; (ii) characterization of the central BMO(ℝn)-closure of Cc∞(ℝn) space via the compactness of the commutators. This is done by exploiting the center symmetry of Hardy operator deeply and by a more explicit decomposition of the operator and the kernel function. [ABSTRACT FROM AUTHOR]

Under weaker conditions on the kernel functions, we discuss the boundedness of bilinear square functions associated with non-smooth kernels on the product of weighted Lebesgue spaces. Moreover, we investigate the weighted boundedness of the commutators of bilinear square functions (with symbols which are BMO functions and their weighted version, respectively) on the product of Lebesgue spaces. As an application, we deduce the corresponding boundedness of bilinear Marcinkiewicz integrals and bilinear Littlewood-Paley g-functions. [ABSTRACT FROM AUTHOR]

In this paper, the fractional calculus approach is employed in the constitutive relationship of the fluid model and the Darcy's law. The flow of the fractional Jeffreys' fluid induced by the impulsive motion of a flat plate in a porous half space is studied in form of an initial and boundary value problem with fractional derivatives. Using the Laplace transform method, we obtain an exact solution of the model in term of Fox's H -function. As a byproduct, the solutions of the analogous flow for the generalized second grade fluid, the fractional Maxwell fluid and the Newtonian fluid in the porous half space are also deduced. In addition, the influence of the material parameters and the fractional parameters on the fluid motion is investigated, as well as a comparison among the fractional Jeffreys' fluid, the fractional Maxwell fluid and the Newtonian fluid in porous medium is also analyzed by graphical illustrations. [ABSTRACT FROM AUTHOR]

We establish the boundedness and continuity of parametric Marcinkiewicz integrals associated to homogeneous compound mappings on Triebel-Lizorkin spaces and Besov spaces. Here the integral kernels are provided with some rather weak size conditions on the unit sphere and in the radial direction. Some known results are naturally improved and extended to the rough case. [ABSTRACT FROM AUTHOR]

Karamata regular variation is a basic tool in stochastic process and the boundary blow-up problems for partial differential equations (PDEs). Morrey space is closely related to study of the regularity of solutions to elliptic PDEs. The aim of this paper is trying to bring together these two areas and this paper is intended as an attempt at motivating some further research on these areas. A version of Morrey space associated with Karamata regular variation is introduced. As application, some estimates of operators, especially one-sided operators, on these spaces are considered. [ABSTRACT FROM AUTHOR]

Let n≥3, Ω be a strongly Lipschitz domain of [ABSTRACT FROM AUTHOR]

In the setting of the Heisenberg group, based on the rotation method, we obtain the sharp ( p, p) estimate for the Hardy operator. It will be shown that the norm of the Hardy operator on L(ℍ) is still p/( p-1). This goes some way to imply that the L norms of the Hardy operator are the same despite the domains are intervals on ℝ, balls in ℝ, or 'ellipsoids' on the Heisenberg group ℍ. By constructing a special function, we find the best constant in the weak type (1, 1) inequality for the Hardy operator. Using the translation approach, we establish the boundedness for the Hardy operator from H to L. Moreover, we describe the difference between M weights and A weights and obtain the characterizations of such weights using the weighted Hardy inequalities. [ABSTRACT FROM AUTHOR]

We investigate factorizations of modified Helmholtz equations in Clifford algebra $Cl(V_{3,3})$. Using the method of fundamental solutions for modified Helmholtz equations and Clifford calculus, we obtain some integral representation theorem in Clifford analysis. The boundedness of singular integral operators in Hölder space is given. Moreover, we establish solvability conditions of Riemann type problems for modified Helmholtz equations in Clifford analysis. As applications, we solve a kind of singular integral equations. The explicit representation of the solution is also given. [ABSTRACT FROM AUTHOR]

In this paper, we study the weighted norm inequalities for commutators formed by a class of one-sided oscillatory integral operators and functions in one-sided BMO spaces. [ABSTRACT FROM AUTHOR]

The operator norms of weighted Hardy operators on Morrey spaces are worked out. The other purpose of this paper is to establish a sufficient and necessary condition on weight functions which ensures the boundedness of the commutators of weighted Hardy operators (with symbols in BMO(ℝ n)) on Morrey spaces. [ABSTRACT FROM AUTHOR]

Some boundedness results are established for commutators generated by classical fractional Hardy operators and one-sided CMO functions on the homogeneous Herz spaces, which extends some known results. [ABSTRACT FROM AUTHOR]

In this paper, we study the nonlinear Schrödinger–Kirchhoff-type equation with pure power nonlinearities in R 3 by variational methods. By carrying out the constrained minimization on a special manifold which is a combination of the Nehari manifold and Pohozaev manifold, we proved the existence of positive radial solutions of this equation for the power p ∈ (1 , 5). The results of this paper extend some existing conclusions, especially for p ∈ (1 , 2 ]. [ABSTRACT FROM AUTHOR]

Polynomial neural network (PNN) is a flexible self-organizing network model, and there have been lots of improvements to it. However, various and improved models of PNN do not properly balance their performance and efficiency. In this study, we present novel self-organized hybrid fuzzy neural networks (SHFNN). The purpose of this study is to develop a new design methodology of constructing a hybrid fuzzy model through enhanced input strategy (EIS) and probability-based node selection (PNS) to reinforce the performance of the model without sacrificing efficiency. l 2 -norm regularization (l 2) is utilized to mitigate overfitting as well as enhance generalization capability. The key points of SHFNN can be summarized as follows. First, a hybrid network structure is constructed by combining fuzzy radial basis function neural networks (FRBFNN) and PNN. Second, we propose a probability-based node selection strategy for node (neuron) selection. Third, an enhanced input strategy (EIS) is proposed to design the enhanced input set by merging the general candidate input set (GCIS) and the original candidate input set (OCIS). The regularization coefficient estimation method based on l 2 is used in conjunction with EIS and PNS to improve the performance and enhance the stability of the model. The effectiveness of SHFNN is validated using synthetic and machine learning dataset. The experimental results show that the prediction accuracy of the proposed model is improved by 4–22% when compared with some methods reported in the literature. Statistical analysis is also considered to demonstrate the superiority of the proposed model. [ABSTRACT FROM AUTHOR]

We define the new central Morrey space with variable exponent and investigate its relation to the Morrey-Herz spaces with variable exponent. As applications, we obtain the boundedness of the homogeneous fractional integral operator TΩ,σ and its commutator [b,TΩ,σ] on Morrey-Herz space with variable exponent, where Ω∈Ls(Sn-1) for s≥1 is a homogeneous function of degree zero, 0<σ

Letandbe a Muckenhoupt weight. In this article, the authors study the real interpolation of the weighted tent space. For all,,and, the authors show that, whereanddenotes the weighted Lorentz-tent space, which is introduced in this article. As an application, the authors prove a real interpolation result on the weighted Hardy spacesfor alland, which, when, seals a gap existing in the original proof of a corresponding result of Fefferman et al. [Trans. Amer. Math. Soc. 1974;191:75–81]. [ABSTRACT FROM AUTHOR]

Fuzzy rule-based models are widely employed to tackle regression problems due to their simplicity and comprehensibility. Numeric functions (e.g., linear ones) are generally utilized to represent the conclusion part of a rule, but this simple module cannot well describe the behavior of data located within local area defined by the premise part. In this study, a novel reinforced fuzzy clustering-based rule model (RFCRM) is presented to address this issue. First, information granulation (Fuzzy C-Means) approach is used to reveal the structure in the data and divide the input space into local regions, as well as form the premise parts. Second, the augmented Random Vector Functional Link Networks (RVFLNs) are employed as the conclusion parts to strengthen the description and representation of the behavior of data positioned within the local regions. Weighted Least Squares Error estimation with regularization is exploited to estimate the coefficients of the connection weights of RVFLN. Also, an exponential weight method benefited from the weight function theory encountered in harmonic analysis is introduced to improve the penalty strategy of regularization in coefficient estimation. More specifically, an exponential weighted ℓ 2 regularization (EW ℓ 2) strategy equipped with the exponential functions as the penalty terms is proposed to realize the targeted penalty on coefficients. The merit of EW ℓ 2 over ordinary ℓ 2 is that different types of polynomial terms can be identified and penalized separately in coefficient estimation. This strategy not only prevents the decline of generalization ability, but also effectively enhances the prediction potential of the model. By combining weighted LSE based on EW ℓ 2 with FCM partition and the augmented RVFLN leads to better prediction accuracy as well as lower complexity (viz., number of rules). Finally, detailed experimental studies are conducted to demonstrate the effectiveness of the proposed RFCRM. [ABSTRACT FROM AUTHOR]

Let AZ(R) be the infinitesimal asymptotic Teichmüller space of a Riemann surface R of infinite type. It is known that AZ(R) is the quotient Banach space of the infinitesimal Teichmüller space Z(R), where Z(R) is the dual space of integrable quadratic differentials. The purpose of this paper is to study the nonuniqueness of geodesic segment joining two points in AZ(R). We prove that there exist infinitely many geodesic segments between the basepoint and every nonsubstantial point in the universal infinitesimal asymptotic Teichmüller space AZ(D) by constructing a special degenerating sequence. [ABSTRACT FROM AUTHOR]

In this study, a design methodology based on fuzzy relation-based neural networks for stabilized network structure is introduced to cope with over-fitting as well as multi-collinearity problems which generally appear in conventional fuzzy neural networks. The design method of the proposed self-organizing network structure provides an efficient solution to construct the stabilized Fuzzy Relation-based Neural Networks (FRNN) through a synergy of multi-techniques such as ensemble neurons/layers, L2-norm regularization, compromise technique, and multi-optimization, in order to generate the structure of the multi-layered self-organizing network designed with the aid of the learning as well as novel structural design. The overall network structure is realized with the aid of parallel network structure with newly added layers as well as effective node selection method through the combination technique of both a sum of squared coefficients (SSC) and performance index (PI) as a node selection criterion for each layer in FRNN. Ensemble neurons consist of the current inputs and the original inputs, and they serve to reduce the bias of the proposed FRNN by maintaining the information of the original inputs. Also, ensemble layers stand for the combination of the current layer and the front layer, and they play a role for control the variance of the model by taking into account the output of the previous layers. The least square error estimation (LSE)-based learning method with L2-norm regularization is used for constructing the stabilized network architecture, and their ensuring design methodologies result in alleviating the overfitting phenomenon and also enhancing the generalization ability. For the performance enhancement of FRNN directly affected by some parameters such as the number of input variables, collocation of the specific subset of input variables, the number of membership functions per each variable, and the order of polynomial in the consequent parts of the fuzzy rules, multi-particle swarm optimization (MPSO) is exploited for the effectively structural as well as parametric optimization of the proposed networks. That is, the multi-optimization helps achieve a compromise between the better generation performance and the alleviated over-fitting leading to the stabilization of the proposed multi-layered self-organizing network structure realized with the aid of synergistic multi-techniques and ensemble structures such as a) L2-norm regularization-based LSE learning, b) combination technique for effective node selection through the combination of SSC and PI, and c) a novel parallel network structure including newly added layers and node selection method. The performance of the proposed network structure is quantified by comprehensive experiments and comparative analysis. [ABSTRACT FROM AUTHOR]

We study the properties of the Triebel-Lizorkin spaces associated with the multidimensional Hermite expansions on Rd (d≥1). In addition to the endpoint estimates, we give the continuous characterizations of these spaces via harmonic and thermic extensions from Rd into R+d+1. Based on this result, we obtain the boundedness of Riesz transform associated with the Hermite expansions. [ABSTRACT FROM AUTHOR]

Multiple Support Vector Machine (SVM) classifier based on ensemble learning approaches could be enhanced from the view point of accuracy, but the performance of these classifiers closely depends on the initial condition of the partitioning method used in the design. Furthermore, these classifiers are more easily affected by noise and outliers. In this study, a novel fuzzy quasi-linear SVM classifier realized with the aid of a composite kernel function and Fuzzy C-Means (FCM) clustering is proposed. The objective of this approach is to reduce the effect of noise and outliers and also handle the overfitting problem through the synergistic effect of the two methods: First, Fuzzy C-Means (FCM) is used to partition the training dataset into several subsets as a preprocessing phase of the proposed classifier. Second, the composite kernel based on multiple linear kernel expression is considered to avoid overfitting problem. In more detail, each training data is assigned to the corresponding membership degree. Some data which are potential noise or outliers are assigned with lower membership degrees and thus yield a small contribution to the composite kernel function. Then, the composite kernel function for multiple local SVMs is constructed according to the distribution of training data. The designed fuzzy quasi-linear SVM classifier is tested with both artificial and UCI data sets. It is also applied for sorting the problem of black plastic wastes being handled in the practice in order to verify the effective as well as efficient classification improvement. Experimental results demonstrate that the proposed method shows the elevated classification performance when compared to performance produced by the methods studied previously. [ABSTRACT FROM AUTHOR]

This study focuses on two critical design aspects of a progressive fuzzy polynomial neural network (PFPNN): the influence of the gated recurrent unit (GRU) structure and the implementation of fitness-based candidate neuron selection (FCNS) through two probabilistic strategies. The primary objectives are to enhance modeling accuracy and to reduce the computational load associated with nonlinear regression tasks. Compared with the existing fuzzy rule-based modeling architecture, the proposed dynamic model consists of the GRU structure and the hybrid fuzzy polynomial architecture. In the initial two layers of the PFPNN, we introduce three types of polynomial and fuzzy rules into the GRU neurons (GNs) and fuzzy polynomial neurons (FPNs), which can effectively reveal potential complex relationships in the data space. The synergy of the FCNS strategies and the ℓ 2 regularization learning method is to design a progressive regression model adept at melding the GRU structure with a self-organizing architecture. The proposed GRU structure and polynomial-based neurons significantly improve the modeling accuracy for time-series datasets. The rational utilization of FCNS strategies can reinforce the network structure and discover the potential performance of neurons of the network. Furthermore, the inclusion of ℓ 2 norm regularization provides additional stability to the proposed model and mitigates the overfitting issue commonly encountered in many existing learning methods. We validated the proposed neural networks using six time-series, four machine learning, and two real-world datasets. The PFPNN outperformed other models in the comparison studies in 83.3% of the datasets, emphasizing its superiority in terms of developing a stable deep structure from diverse candidate neurons and reducing computational overhead. [ABSTRACT FROM AUTHOR]

Humans maintain and develop interrelationships through various forms of communication, including verbal and nonverbal communications. Gestures, which constitute one of the most significant forms of nonverbal communication, convey meaning through diverse forms and movements across cultures. In recent decades, research efforts aimed at providing more natural, human-centered means of interacting with computers have garnered increasing interest. Technological advancements in real-time, vision-based hand motion recognition have become progressively suitable for human–computer interaction, aided by computer vision and pattern recognition techniques. Consequently, we propose an effective system for recognizing hand gestures using time-of-flight (ToF) cameras. The hand gesture recognition system outlined in the proposed method incorporates hand shape analysis, as well as robust fingertip and palm center detection. Furthermore, depth sensors, such as ToF cameras, enhance finger detection and hand gesture recognition performance, even in dark or complex backgrounds. Hand shape recognition is performed by comparing newly recognized hand gestures with pre-trained models using a YOLO algorithm-based convolutional neural network. The proposed hand gesture recognition system is implemented in real-world virtual reality applications, and its performance is evaluated based on detection performance and recognition rate outputs. Two distinct gesture recognition datasets, each emphasizing different aspects, were employed. The analysis of results and associated parameters was conducted to evaluate the performance and effectiveness. Experimental results demonstrate that the proposed system achieves competitive classification performance compared to conventional machine learning models evaluated on standard evaluation benchmarks. [ABSTRACT FROM AUTHOR]

In this paper, by applying the well-known method for dealing with -Laplace type elliptic boundary value problems, the authors establish a sharp estimate for the decreasing rearrangement of the gradient of solutions to the Dirichlet and the Neumann boundary value problems of a class of Schrödinger equations, under the weak regularity assumption on the boundary of domains. As applications, the gradient estimates of these solutions in Lebesgue spaces and Lorentz spaces are obtained. [ABSTRACT FROM AUTHOR]

• Overall structural framework designed for the restoration of motion-blurred images is proposed with the aid of PSO-based parameter estimation and image quality assessment. • The proposed Image Restoration Framework has a complete function which can effectively enhance the restored image quality. • Blur parameter estimation algorithm based on PSO (BPPO) is employed to optimize the motion-blurred parameter estimation. • A polynomial-based radial basis function neural network is used as image quality evaluation method to evaluate restored image quality for efficient classification. The restoration of motion-blurred images has always been a complex problem in image restoration. The current single blurred image algorithm cannot very well solve the estimation error of motion blur parameters. A comprehensive motion-blurred image restoration framework is proposed, which includes motion-blurred data generation, blur parameter estimation, and image quality assessment of restored images. First, we designed and used four image data sets with different degrees of blurring. We innovatively propose a blur parameter estimation algorithm based on the particle swarm optimization (B-PSO) algorithm. The Naturalness Image Quality Evaluator (NIQE) is used as the fitness function of the PSO algorithm. The framework also introduces a polynomial-based radial basis function neural network (P-RBFNN) as a new image quality assessment (IQA) method , with good image classification performance. Test results from public datasets show that the proposed framework can accurately estimate blur parameters. The peak signal-to-noise ratio (PSNR) reaches 29.976 dB, the structural similarity (SSIM) reaches 0.9044, and the classification rate is 96%. The proposed restoration framework produces the best image restoration results. [ABSTRACT FROM AUTHOR]

• AFS and RS are used to effectively obtain the transmittance spectra of black plastic wastes not available in the existing NIR. • A feature selection method of the peak points is considered based on the analysis of chemical information for low cost spectrometer by using a small amount of input variables. • A method of 2D reconstruction through mapping from 1D signal data based on peak points is proposed to effectively deal with spatial structure information between data. • 2DCNN design driven by 2D information features or chemical peak points data leads to high performance as well as anti-noise capability through effective data mapping. In this study, two types of convolutional neural network (CNN) classifiers are designed to handle the problem of classifying black plastic wastes. In particular, the black plastic wastes have the property of absorbing laser light coming from spectrometer. Therefore, the classification of black plastic wastes remains still a challenging problem compared to classifying other colored plastic wastes using existing spectroscopy (i.e., NIR). When it comes the classification problem of black plastic wastes, effective classification techniques by the laser spectroscopy of Fourier Transform-Infrared Radiation (FT-IR) with Attenuated Total Reflectance (ATR) and Raman to analyze the classification problem of black plastic wastes are introduced. Due to the strong ability of extracting spatial features and remarkable performance in image classification, 1D and 2D CNN through data features are designed as classifiers. The technique of chemical peak points selection is considered to reduce data redundancy. Furthermore, through the selection of data features based on the extracted 1D data with peak points is introduced. Experimental results demonstrate that 2DCNN classifier designed with the help of 2D data feature selection as well as 1DCNN classifier shows the best performance compared with other reported methods for classifying black plastic wastes. [ABSTRACT FROM AUTHOR]

We establish one-sided weighted endpoint estimates for the ϱ -variation ( ϱ > 2 ) operators of one-sided singular integrals under certain priori assumption by applying one-sided Calderón–Zygmund argument. Using one-sided sharp maximal estimates, we further prove that the ϱ -variation operators of related commutators are bounded on one-sided weighted Lebesgue and Morrey spaces. In addition, we also show that these operators are bounded from one-sided weighted Morrey spaces to one-sided weighted Campanato spaces. As applications, we obtain some results for the λ -jump operators and the numbers of up-crossings. Our main results represent one-sided extensions of many previously known ones. [ABSTRACT FROM AUTHOR]