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Making use of complex analytic techniques as well as methods involving weight functions, we study a few equivalent conditions of a Hilbert-type integral inequality with nonhomogeneous kernel and parameters. In the form of applications we deduce a few equivalent conditions of a Hilbert-type integral inequality with homogeneous kernel, and we additionally consider operator expressions. [ABSTRACT FROM AUTHOR]

We propose a tree‐based algorithm (μCART) for classification and regression problems in the context of functional data analysis, which allows to leverage measure learning and multiple splitting rules at the node level, with the objective of reducing error while retaining the interpretability of a tree. For each internal node, our main contribution is the idea of learning a weighted functional L2 space by means of constrained convex optimization, which is then used to extract multiple weighted integral features from the functional predictors, in order to determine the binary split. The approach is designed to manage multiple functional predictors and/or responses, by defining suitable splitting rules and loss functions that can depend on the specific problem and can also be combined with additional scalar and categorical predictors, as the tree is grown with the original greedy CART algorithm. We focus on the case of scalar‐valued functional predictors defined on unidimensional domains and illustrate the effectiveness of our method in both classification and regression tasks, through a simulation study and four real‐world applications. [ABSTRACT FROM AUTHOR]

Let L = - Δ + V be a Schrödinger operator with the potential V belonging to the reverse Hölder class B q , q > n / 2 . Denote by CMO θ (ρ) the vanishing mean oscillation type space associated with L. By the aid of the regularity estimates of the fractional heat kernel related with L, we investigate the weighted boundedness and compactness of the commutators of operators generated by fractional heat semigroups related to L and functions belonging to CMO θ (ρ) . [ABSTRACT FROM AUTHOR]

The present article provides a solution to the open problem on the exponential type operators, connected with 1 + x 2 . We are able to achieve the semi-exponential extension of such operators. [ABSTRACT FROM AUTHOR]

We obtain some criteria for a bilinear inequality of a class of operators of fractional integration that imply the corresponding criteria and bilinear inequalities for the Riemann–Liouville, Hadamard, and Erdelyi–Kober operators. [ABSTRACT FROM AUTHOR]

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In the present paper in space the optimal quadrature formulas with derivatives are constructed for approximate solution of a singular integral equation of the first kind with Cauchy kernel. Approximate solution of the singular integral equation is obtained applying the optimal quadrature formulas. Explicit forms of coefficients for the of optimal quadrature formulas are obtained. Some numerical results are presented. [ABSTRACT FROM AUTHOR]

Abstract By means of the weight functions, Hermite–Hadamard’s inequality, and the techniques of real analysis, a new more accurate reverse half-discrete Mulholland-type inequality involving one higher-order derivative function is given. The equivalent statements of the best possible constant factor related to a few parameters, the equivalent forms, and several particular inequalities are provided. Another kind of the reverses is also considered.

The Cardioid (C) distribution is one of the most important models for modeling circular data. Although some of its structural properties have been derived, this distribution is not appropriate for asymmetry and multimodal phenomena in the circle, and then extensions are required. There are various general methods that can be used to produce circular distributions. This paper proposes four extensions of the C distribution based on the beta, Kumaraswamy, gamma, and Marshall–Olkin generators. We obtain a unique linear representation of their densities and some mathematical properties. Inference procedures for the parameters are also investigated. We perform two applications on real data, where the new models are compared to the C distribution and one of its extensions.

External circumferential surface crack is one of the most common defects in offshore pipes and risers. The stress intensity factors (SIFs) for specific crack shapes are essential parameters for assessing the fatigue life of surface-cracked pipes when the linear elastic fracture mechanics approach is adopted. In this paper, a general point load weight function of SIFs for external circumferential surface cracks in pipes is derived. Three-dimensional finite element (FE) analysis is utilized to calculate the SIF solutions for determining the unknown weight function coefficients. The geometric configurations of surface-cracked pipes cover wide ranges of crack aspect ratios, crack depth-to-thickness ratios and thickness-to-inner radius ratios. The obtained weight function is validated against the normalized SIF solutions available in the literature and those obtained from FE analysis for various one-dimensional and two-dimensional stress distributions, showing good accuracy. Moreover, an analytical approach for predicting the surface crack growth in pipes is established based on the developed weight function and validated against the available experimental data from literature. The results demonstrate that the present weight function can be used for SIF calculation and fatigue crack growth prediction for offshore pipes and risers under complicated loading conditions. • A general point load weight function is developed for external circumferential surface cracks in pipes. • Empirical formulas of weight function coefficients are obtained from parametric finite element analysis. • The application range of weight function covers wide ranges of geometric parameters. [ABSTRACT FROM AUTHOR]

Modeling pollutant transport in heterogeneous media is an important task of hydrology. Pollutant transport in a non-homogeneous environment typically exhibits non-local transport dynamics, whose efficient characterization requires a parsimonious model with the non-local feature. This study encapsulates the non-local transport characteristic of pollutants into the peridynamic differential operator (PDDO) and develops a PDDO-based model for quantifying the observed pollutant non-local transport behavior. The simulation results show that the proposed model can describe pollutant non-local transport behavior in various heterogeneous media. The non-local nature of pollutant transport can be adjusted by pre-defined weight function w (| ξ |) and horizon H x . Applications show that the PDDO-based model can better capture pollutant non-local transport behavior than the classical advection–diffusion equation (ADE) model, especially for quantifying the tail of the experimental data late. Analyses further reveal that the PDDO-based model can characterize both normal (Fickian) and anomalous (Lévy) diffusion regimes. [ABSTRACT FROM AUTHOR]

In this paper, by the introduction of several parameters, we construct a new kernel function which is defined in the whole plane and includes some classical kernel functions. Estimating the weight functions with the techniques of real analysis, we establish a new Hilbert-type inequality in the whole plane, and the constant factor of the newly obtained inequality is proved to be the best possible. Additionally, by means of the partial fraction expansion of the tangent function, some special and interesting inequalities are presented at the end of the paper. [ABSTRACT FROM AUTHOR]

A review of some results and open problems in Thermodynamic Formalism of Infinite Alphabet Markov Shifts (TFIAMS) is presented. The authors of main works, thanks to which this mathematical field has appeared, are mentioned, and specific aims of these authors are described. Then basic notions and some basic results of TFIAMS are recalled. In conclusion, a few generalizations and modifications of these results are stated in form of hypotheses. [ABSTRACT FROM AUTHOR]

In this paper, a new family of eighth-order iterative methods for solving simple roots of nonlinear equations is developed. Each member of the proposed family requires four functional evaluations in each iteration that it is optimal according to the sense of Kung-Traub's conjecture. They have four selfaccelerating parameters that are calculated using the adaptive method. The R-order of convergence has increased from 8 to 16 (maximum improvement). [ABSTRACT FROM AUTHOR]

In this paper, we considered the non-uniformly distributed zeros on the unit circle, which are obtained by projecting vertically the zeros of the derivative of Legendre polynomial together with x = 1 and x = -1 onto the unit circle. An interpolating polynomial is considered by prescribing the function on the above said nodes and its second derivative with suitable weight function at all nodes except at x = 1 and x = -1. We obtained the existence, explicit representations, estimation and convergence theorem of that interpolatory polynomial. Such type of interpolation is known as weighted Lacunary interpolation on the unit circle. [ABSTRACT FROM AUTHOR]

Purpose. The main purpose of the work is the development of identification models and a new method of modeling electromagnetic processes in electric traction systems with simultaneous consideration of all its subsystems, as well as several feeder zones of the electrified section. Methodology. To achieve this purpose, the methods of mathematical modelling, the basics of the theory of random processes and the methodology of their probabilistic-statistical processing, the methods for solving integral equations and analysis of electric traction circuits in electric traction systems are used. Findings. The requirements to be met by an adequate, stochastic identification model of electric traction devices are established. The solution of Fredholm’s integral correlation equation of the first kind is performed. The analytical expression of the identification dynamic model of the electric locomotive DE–1 is obtained and its adequacy is checked. The methodology of combined modeling of electromagnetic processes in devices and subsystems of electric traction systems is developed and presented tabularly. Originality. For the first time it is proposed to use the pulse transition function as identification models of traction substation and traction network with electric rolling stock in predictive modeling of electromagnetic and electric power processes in electric traction systems. A new method has been developed, a method of complex modeling of electromagnetic and electric power processes in the system of electric traction with simultaneous consideration of all its subsystems, as well as several inter-substation zones of the electrified section. For the first time, a method of partitioning the correlation functions for solving an integral correlation equation has been proposed, which allows defining a pulse transition function as an identification model of any subsystem of an electric traction system. Practical value. The developed identification models and the method of combined modeling make it possible to predict electromagnetic processes simultaneously in all feeder zones of the electrified section of the electric traction system. The obtained identification model of the electric locomotive DE–1 can be adapted with its subsequent use in modeling processes in the traction circuits of electric locomotives of other types. The method of factorization of correlation functions used in solving the Volterra integral correlation equation of the first kind (convolution type) can be adapted to the solution of other integral equations, which describe the processes in electric traction systems.

This work develops accurate weight functions for a single crack at a hole in a finite width plate for various hole sizes. In order to develop an accurate weight function, we first obtain accurate stress intensity factors, using the finite element method (FEM), for a reference load case of uniform stress on the crack line. Following the earlier approach for developing a weight function suggested by Wu and Carlsson, we fit the reference stress intensity factor data from FEM to a smooth analytic function; however, for the open hole it is necessary to adopt a piecewise polynomial to fit the stress intensity factor data, in place of the single polynomial suggested by Wu and Carlsson. We validate the new weight function for the case of remote uniform applied stress, which induces a stress field on the crack line exhibiting the well-known stress concentration at the hole, and for which we have accepted stress intensity factor solutions. The new weight functions provide stress intensity factors that agree very well with results from two commercial fracture mechanics software packages. Comparing results from the new and earlier weight functions shows good agreement for some crack line stress fields, but errors of a few percent for other stress fields, with the new weight function providing more reasonable results. The improved quality of the new weight functions is due both to the new reference solution for uniform crack line stress and to the piecewise fit to the reference stress intensity data. Trivial changes to the FEM model allow us to provide additional weight functions for the cases of symmetric double cracks at a hole (by adding a symmetry plane to the FEM mesh) and a single crack at a hole in a square plate (by reducing the length of the FEM mesh).

Eigenstrain is a distributed strain field considered in mechanics that is particularly helpful in evaluating residual stress fields in the finite element method, and estimating the stress intensity factor due to residual stress in cracked components. The objective of this paper is to provide a solution for a simple eigenstrain problem in a two-dimensional rectangular domain that can serve as a benchmark for validation of fracture mechanics analysis methods. The solution provides residual stress fields and the stress intensity factor for a single edge crack as a function of crack size. Documenting the benchmark provides opportunities to demonstrate the correlation of different means to determine the stress intensity factor and to highlight details in implementing stress intensity factor calculations.

A Flying Ad-hoc Networks (FANETs) is an autonomous technology that creates a self-organized wireless network via Unmanned Arial Vehicles (UAVs). In this network, all UAVs can communicate within a restricted range of wireless communication in the absence of fixed infrastructure. As a result of high mobility, the limited energy, and the communication range of UAVs, network forming and deformation between them are very frequent that causes packet delivery failure. Therefore, a stable route is always needed to ensure effective data dissemination between source and destination in FANETs. Since it has drastically changing network topology, therefore, to maintain the stable route during packet transmission, there is a need for a suitable routing protocol. This paper proposes an Optimized Location-Aided Routing (O-LAR) protocol which is the modified version of Location-Aided Routing (LAR) protocol. Our protocol’s novelty comes from the fact that it established an optimal route between UAVs for information dissemination towards their respective destination UAV by considering weight function. A weighted function is used to decide the best next-hop node selection based on the parameters like residual energy, distance, and UAV movement direction. The performance of the O-LAR is evaluated mathematically and simulated through the NS-2 simulator. The empirical results attest that O-LAR improves the link duration, network lifetime, packet delivery ratio, and average throughput compared with the state-of-the-art protocols: LEPR, D-LAR, and LAR. Further, the proposed scheme reduces the number of next-hops, routing overhead and end-to-end delay compared to the state-of-the-art protocols.

Abstract By applying the weight functions, the idea of introducing parameters and the technique of real analysis, a new multiple Hilbert-type integral inequality involving the upper limit functions is given. The constant factor related to the gamma function is proved to be the best possible in a condition. A corollary about the case of the nonhomogeneous kernel and some particular inequalities are obtained.

Cracks often appear on the inner surface of metal thick-walled cylinders with multiple interference fits. Considering that no relatively accurate model exists for the cracks on the interface of multi-layered, rotating, thick-walled cylinders, in this paper, the stress intensity factor is established for a radial penetrating crack on the interface of a multi-layered, rotating, thick-walled cylinder. The parameters included in the equation are the rotation speed, the wall thickness ratio, and the interference. First, finite element software is used to calculate the stress intensity factors of two thick-walled cylinders under an interference fit with a crack on the interface. Then, the equation of the stress intensity factor is fitted with the parameters of contact pressure, crack depth, and wall thickness ratio. Next, the weight function is used to calculate the stress intensity factor for radial penetrating cracks on the inner surface of the cylinder's interface. Finally, 2D finite element models of the four-layer cylinder with a crack are established to verify the equation. [ABSTRACT FROM AUTHOR]

In this paper, for a locally compact commutative hypergroup K and for a pair (\Phi _1, \Phi _2) of Young functions satisfying sequence condition, we give a necessary condition in terms of aperiodic elements of the center of K, for the convolution f\ast g to exist a.e., where f and g are arbitrary elements of Orlicz spaces L^{\Phi _1}(K) and L^{\Phi _2}(K), respectively. As an application, we present some equivalent conditions for compactness of a compactly generated locally compact abelian group. Moreover, we also characterize compact convolution operators from L^1_w(K) into L^\Phi _w(K) for a weight w on a locally compact hypergroup K. [ABSTRACT FROM AUTHOR]

We study sufficient conditions on radial and non-radial weight functions that guarantee mean approximation of polyanalytic functions by polyanalytic polynomials in poly-Bergman, poly-Dirichlet, and poly-Besov spaces. [ABSTRACT FROM AUTHOR]

Diffractions carry meaningful information on subsurface discontinuities and are regarded as effective tools for high-resolution seismic exploration. However, diffractions often behave as weak events and, therefore, are masked by strong specular reflections. In this study, the diffraction and reflection behaviors in common image gathers have been elucidated. Fortunately, diffractions are observed to be kinematically different from reflections in dip-angle gathers. Under a suitable velocity, diffractions appear as flat events, whereas reflections appear as curves with stationary phase apexes. Therefore, the diffraction energy has a wide and nearly uniform distribution along the flat events, while the reflection energy is concentrated in the Fresnel zone with the stationary phase point. Consequently, a single-order weight function was proposed and tested by measuring the difference between an individual sample and the mean of the entire data set to eliminate the reflection energy. It is observed that the weight function often leads to a reflection residual, if it is based only on the zero-order amplitude information. Therefore, first-order derivative information is used to build the weight function which is also called the double-order weight function, to further suppress reflections and highlight weak diffractions. Synthetic and field data applications demonstrate the feasibility and effectiveness of the proposed weight function in imaging subsurface discontinuities. [ABSTRACT FROM AUTHOR]

Deblurring images with outliers has always been a significantly challenging problem. Previous methods mainly involved complex operations, such as outlier and light streak detection, or sophisticated image priors for blur-kernel estimation, which increased the difficulty of deblurring images. Therefore, we developed a simple, yet efficient, blind deblurring algorithm in this study for handling images with outliers. To eliminate the impact of outliers during the kernel estimation process, we employed a robust Welsch loss function to characterize the data-fidelity term of our model. We observed that this function could extract significant edges successfully. Therefore, the image regularization term was also described by the same function. Using this unified robust function to describe our model can considerably reduce the complexity of the algorithm. Moreover, we derived a flexible weight function from the Welsch function to further improve the efficiency of our algorithm. To finally obtain accurate latent images, we developed a robust non-blind deblurring approach based on this flexible weight function. The experimental results indicate that our approach outperforms state-of-the-art methods in deblurring images with or without outliers. Compared with the method specifically for outliers, the recovery performance of our method can be improved by 12.9% (considering a dataset with impulse noise), and the execution efficiency is about 1.5 times faster. [ABSTRACT FROM AUTHOR]

The current research is unclear regarding the sensitivity of the output of a magnetic flowmeter analysis to the fidelity of the simulated flow field. This study evaluates the effects of higher fidelity models on magnetic flowmeter analysis. An eddy viscosity model, second‐moment closure model, and a Large‐Eddy simulation were compared to laboratory velocity profile data 0.67D downstream of a 1.58D elbow at a Reynolds number of 34,000. The Large‐Eddy simulation results matched the laboratory velocity profile data best. The authors conclude that the fidelity of the flow field model does cause differences in the analysis of the flowmeter voltage output. [ABSTRACT FROM AUTHOR]

Abstract In this article, by using weight functions, the idea of introducing parameters, the reverse extended Hardy–Hilbert integral inequality and the techniques of real analysis, a reverse Hardy–Hilbert-type integral inequality involving one derivative function and the beta function is obtained. The equivalent statements of the best possible constant factor related to several parameters are considered. The equivalent form, the cases of non-homogeneous kernel and some particular inequalities are also presented.

This paper is aimed at introducing the concept of Spherical Interpolating Moving Least Squares to the problems in geodesy and geophysics. Based on two previously known methods, namely Spherical Moving Least Squares and Interpolating Moving Least Squares, a simple theory is formulated for using Spherical Moving Least Squares as an interpolant. As an application, a case study is presented in which gravity accelerations at sea surface in the Persian Gulf are derived, using both the approximation and interpolation mode of the Spherical Moving Least Squares. The roles of the various elements in the methods-weight function, scaling parameter, and the degree of spherical harmonics as the basis functions-are investigated. Then, the results of approximation and interpolation are compared with the field data at sea surface, collected by shipborne gravimetry approach. Finally, the results are compared with another independent interpolation method-spline interpolation. It is shown that in this particular problem, SMLS approximation and SIMLS interpolation present a better accuracy than spherical splines.

In the present work, the stress intensity factor (SIF) of a small embedded square-like flaw is calculated by means of a procedure based on the Oore-Burns integral. An explicit equation is given to evaluate the SIF along the two axes of symmetry that correspond to the points where the SIF takes its maximum and minimum value on the contour crack. The SIF is calculated in accordance with FE numerical results.

Profiles of atmospheric temperature and water vapor are important for studying atmospheric state and play an important role in the energy balance of earth-atmosphere system. Limb remote sensing is an important means to obtain the profile of atmospheric parameters. The atmospheric radiation ultra-high spectral detector developed by Shanghai Institute of Technical Physics, Chinese Academy of Sciences, has a detection band range of 650-3050 cm-1 and the spectral resolution on limb view is as high as 0.015 cm-1, which will be the highest spectral resolution that the world's Fourier spectral detector can achieve. A method by using information and weighting function linearization are proposed to evaluate the inversion accuracy of the research instrument in advance. Weighting functions of atmospheric temperature and water vapor at 16 different tangent points are simulated and calculated by RFM model. The degree of signal freedom and the entropy reduction are also calculated by the information content method, and the optimal number of inversion channels is determined to be 200 by the stepwise iterative algorithm. Combined with the threshold (0.3 K) of detectable brightness temperature and the linearized weighting function of the instrument, the available spectral channel numbers of atmospheric temperature and volume mixing ratio of water vapor profiles under different inversion accuracy of six atmosphere models (US standard atmosphere, tropical atmosphere, middle-latitude summer atmosphere, middle-latitude winter atmosphere, subarctic summer atmosphere, subarctic winter atmosphere) are calculated and analyzed, and the inversion accuracy is estimated theoretically. On the demanded optimal 200 channels, the inversion accuracy of the whole temperature profile is 0.6 K, but if the inversion accuracy of the temperature profile is required to be 0.5 K, the number of channels available for inversion at a higher tangent height is smaller. Except the tropical atmosphere model, there are enough channels for the other five atmosphere models meeting 5% accuracy demands of the inversion of water vapor volume mixing ratio profiles. However, the inversion of the water vapor profile of the tropical atmosphere has barely enough channels at 16-20 km for 10% relative inversion accuracy of volume mixing ratio. The number of channels usable for atmospheric parameters retrieving increases by the decreasing of inversion accuracy. Among six atmosphere models, the tropical atmosphere is relatively special and its inversion accuracy is lower, which may be related to the unique temperature profile of the tropical atmosphere. There is no isothermal layer in the tropical atmosphere, which may lead to fewer atmospheric parameter inversion channels near the height of sharp temperature transition.

Abstract On the interval [ 0 , 1 ] $[0,1]$ we consider the nth order linear differential equation, the coefficient of the highest derivative of which is equivalent to the power function t μ $t^{\mu }$ when t → 0 $t\rightarrow 0$ . The main aim of the paper is to pose “generalized” Cauchy conditions for the given equation at the point of singularity t = 0 $t=0$ , which would be correct for any μ > 0 $\mu >0$ .

Abstract Using weight functions, we establish a few equivalent statements of two kinds of Hardy-type integral inequalities with nonhomogeneous kernel in the whole plane. The constant factors related to the extended Hurwitz-zeta function are proved to be the best possible. In the form of applications, we deduce some special cases involving homogeneous kernel. We additionally consider some particular inequalities and operator expressions.

In order to overcome the shortcomings of serious arc drawing and low computational efficiency in the crosswell seismic migration method and the problems of the inaccurate velocity model and sparse distribution of reflection points in the traditional stack imaging method, the article proposes an inverse Fresnel beam XSP-CDP stack imaging method based on first-arrival wave velocity tomography combined with the characteristics of crosswell seismic wave field. Firstly, an accurate crosswell velocity model is established by the first-arrival wave tomography inversion method based on the characteristics of high energy and easy pick-up of the first-arrival wave in crosswell seismic. Secondly, the velocity model is optimized, and the energy contribution weights of effective rays to the receiver point are calculated through the crosswell seismic Fresnel beam wave field forward numerical simulation method. Then, the reflected wave field is dynamically migrated to the reflection points within the first Fresnel zone according to the weight function, and the intensive common reflection point (CRP) gather after normal moveout (NMO) correction is generated. Finally, an appropriate bin is selected for stacking. In this article, the inverse Fresnel beam method is used to decompose the single-channel seismic wave field into the effective reflection points in the Fresnel zone, which makes the fold of the reflection point more uniform and improves the imaging accuracy. The model test and actual data processing results proved the validity and robustness of this method.

In this paper, necessary and sufficient conditions for the boundedness and compactness of one class of integral operators with power and logarithmic singularities in weighted Lebesgue spaces are obtained. [ABSTRACT FROM AUTHOR]

Based on local fractional calculus theory, the Hölder double local fractional integral inequality with Weighted is proved. By using the methods of weight function and some analysis techniques on the fractal real set, a local fractional integral inequality with the best constant is given, which is generalization of Yang–Hardy–Hilbert's inequality on the fractal set ℝ + α of fractal dimension α (0 < α ≤ 1) and its equivalent form is considered. [ABSTRACT FROM AUTHOR]

We produce a series of results extending information-theoretical inequalities (discussed by Dembo–Cover–Thomas in 1988–1991) to a weighted version of entropy. Most of the resulting inequalities involve the Gaussian weighted entropy; they imply a number of new relations for determinants of positive-definite matrices. Unlike the Shannon entropy where the contribution of an outcome depends only upon its probability, the weighted (or context-dependent) entropy takes into account a 'value' of an outcome determined by a given weight function φ . An example of a new result is a weighted version of the strong Hadamard inequality (SHI) between the determinants of a positive-definite d × d matrix and its square blocks (sub-matrices) of different sizes. When φ ≡ 1 , the weighted inequality becomes a 'standard' SHI; in general, the weighted version requires some assumptions upon φ . The SHI and its weighted version generalize a widely known 'usual' Hadamard inequality det C ≤ ∏ j = 1 d C jj . [ABSTRACT FROM AUTHOR]