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Given a configuration of pebbles on the vertices of a graph G, a pebbling move removes two pebbles from a vertex and puts one pebble on an adjacent vertex. The pebbling number of a graph G is the smallest number of pebbles required such that, given an arbitrary initial configuration of pebbles, one pebble can be moved to any vertex of G through some sequence of pebbling moves. Through constructing a non-tree weight function for Q 4 , we improve the weight function technique, introduced by Hurlbert and extended by Cranston et al., that gives an upper bound for the pebbling number of graphs. Then, we propose a conjecture on weight functions for the n-dimensional cube. We also construct a set of valid weight functions for variations of lollipop graphs, extending previously known constructions. [ABSTRACT FROM AUTHOR]

In the present study, an extension of the idea of dynamic neurons is proposed by replacing the weights with a weight function that is applied simultaneously to all neuron inputs. A new type of artificial neuron called an integral neuron is modeled, in which the total signal is obtained as the integral of the weight function. The integral neuron enhances traditional neurons by allowing the signal shape to be linear and nonlinear. The training of the integral neuron involves finding the parameters of the weight function, where its functional values directly influence the total signal in the neuron's body. This article presents theoretical and experimental evidence for the applicability and convergence of standard training methods such as gradient descent, Gauss–Newton, and Levenberg–Marquardt in searching for the optimal weight function of an integral neuron. The experimental part of the study demonstrates that a single integral neuron can be trained on the logical XOR function—something that is impossible for single classical neurons due to the linear nature of the summation in their bodies. [ABSTRACT FROM AUTHOR]

Quadrature formulas for singular integrals on the integration interval [−1, 1] with certain weight functions are constructed. The formulas use the values of the function and its derivatives at the zeros of the Chebyshev polynomial. The resulting formulas are quadrature formulas of the interpolation type and have an algebraic degree of accuracy . The error estimate is given. [ABSTRACT FROM AUTHOR]

Within the framework of inverse problems of photometry, we study questions on reconstruction of the spatial location and luminosity of a Lambertian curve from a stereo pair of its images. We study sources of ambiguity in reconstruction of the location of such a curve and suggest criteria for existence of a unique solution of the problem for general weight functions. We apply the results to specific classes of weight functions that model the degree of transparency of the medium (including its absorption or scattering). [ABSTRACT FROM AUTHOR]

This work develops a method to predict crack face contact under linear elastic conditions with application to fatigue crack growth in a residual stress bearing body. Residual stress changes the shape of the crack faces in a cracked body and can cause crack face contact that affects fatigue crack growth. Earlier work shows that the weight function can predict crack face shapes for applied or residual stress fields. The present work applies the weight function to compute a system of linear equations that can determine crack face contact pressure via quadratic programming. We validate the new method against the finite element method (FEM) in the context of fatigue crack growth in a single edge cracked beam having residual stress from prior elastic–plastic bending. The new method agrees with FEM over a range of crack sizes. Crack opening profiles agree when crack face contact is ignored (where the crack exhibits unrealistic over-closure) and when crack face contact is included. Stress intensity factors are also in agreement, comprising those due to applied loads, residual stress, and contact pressure. Being more efficient than FEM, the new crack face contract assessment method is useful for predicting residual stress effects in fatigue crack growth.

In this study, we discuss new Hardy-type inequalities for operators involving iteration and provide explicit characterizations of these inequalities. As an application of our results, we consider the problem of the boundedness of the multidimensional Hardy operator from a Lebesgue space to a Morrey-type space.

A new family of fourth‐order iterative methods for solving nonlinear equations is proposed using the weight function procedure. This family is optimal in the sense of the Kung–Traub conjecture, as it requires three function evaluations per iteration. Due to its flexible structure, the new family offers a variety of options, demonstrating that it includes several well‐known and recent methods as special cases. In particular, three new specific methods are designed to achieve better results compared to existing methods within the same family. Various nonlinear functions and engineering problems are used to illustrate the performance of these new specific methods, comparing them with existing ones. Furthermore, the analysis of complex dynamics and basins of attraction shows that the newly proposed methods yield the best results, with wider sets of initial points that lead to convergence. [ABSTRACT FROM AUTHOR]

In the present paper we establish a few equivalent conditions of a reverse Hilbert-type integral inequality with a general non-homogeneous kernel in the whole plane. In the form of applications, we deduce a few equivalent conditions of a reverse Hilbert-type integral inequality with a general homogeneous kernel in the whole plane. We also consider some particular cases and examples. [ABSTRACT FROM AUTHOR]

Intelligent mobile robots have been gradually used in various fields, including logistics, healthcare, service, and maintenance. Path planning is a crucial aspect of intelligent mobile robot research, which aims to empower robots to create optimal trajectories within complex and dynamic environments autonomously. This study introduces an improved A* algorithm to address the challenges faced by the preliminary A* pathfinding algorithm, which include limited efficiency, inadequate robustness, and excessive node traversal. Firstly, the node storage structure is optimized using a minimum heap to decrease node traversal time. In addition, the heuristic function is improved by adding an adaptive weight function and a turn penalty function. The original 8-neighbor is expanded to a 16-neighbor within the search strategy, followed by the elimination of invalid search neighbor to refine it into a new 8-neighbor according to the principle of symmetry, thereby enhancing the directionality of the A* algorithm and improving search efficiency. Furthermore, a bidirectional search mechanism is implemented to further reduce search time. Finally, trajectory optimization is performed on the planned paths using path node elimination and cubic Bezier curves, which aligns the optimized paths more closely with the kinematic constraints of the robot derivable trajectories. In simulation experiments on grid maps of different sizes, it was demonstrated that the proposed improved A* algorithm outperforms the preliminary A* Algorithm in various metrics, such as search efficiency, node traversal count, path length, and inflection points. The improved algorithm provides substantial value for practical applications by efficiently planning optimal paths in complex environments and ensuring robot drivability. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we propose an extended variant of Weighted Generalized Cumulative Residual Entropy (WGCRE), its dynamic version, and a Weighted Generalized Cumulative Kerridge Inaccuracy measure by considering a general continuous non negative weight function. Several properties, including bounds and stochastic ordering, of these measures are obtained under appropriate assumptions. Furthermore, we introduce the empirical extended WGCRE as a non-parametric estimator, demonstrating its almost sure convergence to the proposed measure. A central limit theorem is established, examples are provided for illustrative purposes, and the stability of the empirical measure is examined. Finally, we demonstrate the practicality of this measure using real financial data for risk measurement. [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]

Generalized Rayleigh, a weighted version of power Rayleigh distribution is introduced. The hazard function is increasing upon the values of parameters. The generalized model has been compared with different models for flexibility and efficiency using real life data sets. Further, the parameters of the new model are estimated via classical method of estimation. It is observed that weighted power Rayleigh distribution fits better than power Rayleigh, Rayleigh and weighted Rayleigh distributions. [ABSTRACT FROM AUTHOR]

Within the framework of inverse problems of photometry, we study questions on reconstruction of the spatial location and luminosity of a Lambertian optical surface from its images obtained with the use of a small number of optical systems. We study causes of ambiguity in reconstruction of the location of such a surface. We suggest criteria for existence of a unique solution of the inverse problem on reconstruction of a luminous surface from three images for general weight functions and apply the results to specific classes of weight functions that model the degree of transparency of the medium (including its absorption or scattering). [ABSTRACT FROM AUTHOR]

Nonlocal regularization is frequently used to resolve the mesh dependency issue that is caused by strain softening in finite-element (FE) simulations. Some or all variables that affect strain softening are assumed to depend on the local, neighboring, or both in this method. The weight function is the main component of a regularization method. There are three widely used weight functions, which include the Gaussian distribution (GD), Galavi and Schweiger (GS), and over-nonlocal (ON) functions. All of them could alleviate or eliminate the mesh dependency in simple boundary value problems (BVPs), such as plane strain compression; the evaluation of their performance in real-world BVPs is rare. A detailed comparison of these functions has been carried out based on an anisotropic sand model that accounts for the evolution of anisotropy. The increment of void ratio is assumed nonlocal. All functions give mesh-independent force–displacement relationships in drained and undrained plane strain compression tests. The shear band thickness shows a small variation when the mesh size is smaller than the internal length. None could eliminate the mesh dependency of shear band orientation. The GS method is the most efficient in eliminating the mesh dependency in the strip footing problem. The ON method could give excessive overpredictions of the volume expansion around strip footings, which leads to unrealistic low reaction forces on strip footings at large deformations. All three weight functions give mesh-independent results for the earth pressure that acts on a retaining wall. [ABSTRACT FROM AUTHOR]

In this work, we are interested in to study removability of a singular set in the boundary for some classes of quasilinear elliptic equations. We will approach this question in two different ways: through an asymptotic behavior at the infinity of the solutions, or through conditions in the Sobolev norm of solutions along the direction of the singular set. [ABSTRACT FROM AUTHOR]

Copyright of Journal of South China Normal University (Natural Science Edition) / Huanan Shifan Daxue Xuebao (Ziran Kexue Ban) is the property of Journal of South China Normal University (Natural Science Edition) Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We consider a Hardy-type inequality with Oinarov's kernel in weighted Lebesgue spaces. We give new equivalent conditions for satisfying the inequality, and provide lower and upper estimates for its best constant. The findings are crucial in the study of oscillation and non-oscillation properties of differential equation solutions, as well as spectral properties.

In this paper, we give a necessary and sufficient condition for an almost Jordan algebra, which is also called Lie triple algebra to be an evolution algebra. We study the nilpotency, power associativity of such algebras. We specify the necessary and sufficient conditions for such an algebra to be baric. Along the way we show that train evolution algebras which are almost Jordan algebras are of rank at most 5. We prove that any finite-dimensional non-nil-evolution algebra verifying the identity of almost Jordan algebras has at least one non-zero idempotent. Finally, we study the derivations of this class of algebras. [ABSTRACT FROM AUTHOR]

The paper describes the closure of infinitely differentiable compactly supported functions in certain second-order weighted Sobolev space depending on the degree of singularity of weight functions at zero and at infinity. [ABSTRACT FROM AUTHOR]

In this paper the extension of the weighted Montgomery identity is established by using the integral formula of Pecaric, Matic and Ujevc. Further, by using this extended weighted Montgomery identity for functions whose derivatives of order n-1 are absolutely continunous functions, new inequalities of the weighted Hermite-Hadamard type are obtained. Also, applications of these results are given for various types of weight function. [ABSTRACT FROM AUTHOR]

The Earth's Free Core Nutation (FCN) causes Earth tides and forced nutation with frequencies close to the FCN that exhibit resonance effects. High-precision superconducting gravimeter (SG) and very long baseline interferometry (VLBI) provide good observation techniques for detecting the FCN parameters. However, some choices in data processing and solution procedures increase the uncertainty of the FCN parameters. In this study, we analyzed the differences and the effectiveness of weight function and ocean tide corrections in the FCN parameter detection using synthetic data, SG data from thirty-one stations, and the 10 celestial pole offset (CPO) series. The results show that significant discrepancies are caused by different computing options for a single SG station. The stacking method, which results in a variation of 0.24–5 sidereal days (SDs) in the FCN period (T) and 103-104 in the quality factor (Q) due to the selection of the weighting function and the ocean tide model (OTM), can effectively suppress this influence. The statistical analysis results of synthetic data shows that although different weight choices, while adjusting the proportion of diurnal tidal waves involved, do not significantly improve the accuracy of fitted FCN parameters from gravity observations. The study evaluated a series of OTMs using the loading correction efficiency. The fitting of FCN parameters can be improved by selecting the mean of appropriate OTMs based on the evaluation results. Through the estimation of the FCN parameters based on the forced nutation, it was found that the weight function P1 is more suitable than others, and different CPO series (after 2009) resulted in a difference of 0.4 SDs in the T and of 103 in the Q. We estimated the FCN parameters for SG (T = 430.4 ± 1.5 SDs and Q = 1.52 × 104 ± 2.5 × 103) and for VLBI (T = 429.8 ± 0.7 SDs, Q = 1.88 × 104 ± 2.1 × 103).

In this article, a new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order is obtained, by using the weight functions, the mid-value theorem, and the techniques of real analysis. A few equivalent statements of the best possible constant factor related to several parameters are considered. As applications, the equivalent forms and some particular inequalities are provided.

Abstract In this paper, a new reverse half-discrete Mulholland-type inequality with the nonhomogeneous kernel of the form h ( v ( x ) ln n ) $h(v(x)\ln n)$ and the best possible constant factor is obtained by using the weight functions and the technique of real analysis. The equivalent reverses are considered. As corollaries, we deduce some new equivalent reverse inequalities with the homogeneous kernel of the form k λ ( v ( x ) , ln n ) $k_{\lambda }(v(x),\ln n)$ . A few particular cases are provided. Our new reverse half-discrete Mulholland-type inequality which has a nonhomogeneous kernel is more general than in the previous homogeneous kernel work. The harmonized integration will have more applications.

In this paper, we consider the problem of integral geometry in three-dimensional space over a family of cones with a weight function of a special form. The theorem of uniqueness of the solution is proved, and several examples are considered in order to obtain more precise solutions to the problem of integral geometry. [ABSTRACT FROM AUTHOR]

Large amounts of geographical and property rights information are contained in 3D cadastral property units, although the amount of cadastral information expressed from various perspectives varies. Therefore, it is crucial to choose a perspective that is compatible with human visual perception and conveys the most data as the perspective changes. This paper proposes an optimization method for generating the optimal view of 3D cadastral property units to address the above issues. The construction of candidate perspectives is the first step, followed by the calculation of feature values, including visibility of 3D boundary points under each candidate perspective, visible area ratio, and visual comfort. Then, an improved particle swarm optimization technique is used to determine the optimal view of 3D cadastral property units and the ideal viewpoint for the expected input model. The research results show that the selected optimal viewpoint fully considers human visual comfort, can accommodate a large amount of cadastral information, and can be used as a mapping drawing in social applications of 3D cadastral systems. This viewpoint can effectively supplement front views, side views, and axis side views, enabling more visual information to be transmitted in 3D religion maps and 3D property right maps and improving readability. [ABSTRACT FROM AUTHOR]

We consider a Hardy-type inequality with Oinarov's kernel in weighted Lebesgue spaces. We give new equivalent conditions for satisfying the inequality, and provide lower and upper estimates for its best constant. The findings are crucial in the study of oscillation and non-oscillation properties of differential equation solutions, as well as spectral properties. [ABSTRACT FROM AUTHOR]

In recent years, fractional calculus (FC) has filled in a hole in traditional calculus in terms of the effect of memory, which lets us know things about the past and present and guess what will happen in the future. It is very important to have this function, especially when studying biological models and integral equations. This paper introduces developed mathematical strategies for understanding a direct arrangement of fractional integro-differential equations (FIDEs). We have presented the least squares procedure and the Legendre strategy for discussing FIDEs. We have given the form of the Caputo concept fractional order operator and the properties. We have presented the properties of the shifted Legendre polynomials. We have shown the steps of the technique to display the solution. Some test examples are given to exhibit the precision and relevance of the introduced strategies. Mathematical outcomes show that this methodology is a comparison between the exact solution and the methods suggested. To show the theoretical results gained, the simulation of suggested strategies is given in eye-catching figures and tables. Program Mathematica 12 was used to get all of the results from the techniques that were shown. [ABSTRACT FROM AUTHOR]

In this paper, by the use of weight functions and the technique of real analysis, a new Hilbert-type integral inequality with a general nonhomogeneous kernel as H(xv(y)) is given. A few equivalent statements related to the best possible constant factor and parameters are considered. As applications, the equivalent forms, the operator expressions and some corollaries are obtained. [ABSTRACT FROM AUTHOR]

Opial's inequality and its ramifications play an important role in the theory of differential and difference equations. A sharp unifying generalization of Opial's inequality is presented that contains both its continuous and discrete versions. This generalization, based on distribution functions, is extended to the case of derivatives of arbitrary order. This extension optimizes and improves the constant given in the literature. The special case of derivatives of second order is studied in more detail. Two closely related Opial inequalities with a weight function are presented as well. The associated Wirtinger inequality is studied briefly. [ABSTRACT FROM AUTHOR]

External bonding with fiber-reinforced polymer (FRP) offers a potential solution to mitigate the detrimental effects caused by load impact and corrosion, which can weaken the bond strength of reinforced concrete structures. However, existing models need to be improved in addressing the FRP confinement mechanism and failure modes. As a solution, the proposed model employs stress intensity factor (SIF)-based criteria to determine the internal pressure exerted on the steel-concrete interface during various stages of comprehensive concrete cracking. Critical parameters are evaluated using weight function theory and a finite element model. A bond-slip model is introduced for the FRP-concrete interface and reasonable assumptions on failure plane characteristics. The internal pressure model employed demonstrates that FRP confinement has the ability to generate dual peaks in stress distribution and modify their magnitude as the confinement level increases. The proposed predictive model demonstrates superior performance in failure modes, test methods, and wrap methods for assessing bond strength with FRP confinement. The accuracy of this model is indicated by an integral absolute error (IAE) of 9.6% based on 125 experimental data, surpassing the performance of the other three existing models. Moreover, a new confinement parameter is introduced and validated, showing an upper bound of 0.44 for enhancing FRP bond strength. Additionally, a general expression validating the bond strength model with FRP confinement is established, allowing for the prediction of bond length. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this manuscript, for approximation of solutions to equations that are nonlinear, a new class of two-point iterative structure that is based on a weight function involving two converging power series, is developed. For any method constructed from the developed class of methods, it requires three separate functions evaluation in a complete iteration cycle that is of order four convergence. Also, some well-known existing methods are typical members of the new class of methods. The numerical test on some concrete methods derived from the class of methods indicates that they are effective and competitive when employed in solving a nonlinear equation.

Abstract By means of the weight functions, the idea of introduced parameters, using the transfer formula and Hermite–Hadamard’s inequality, a more accurate half-discrete multidimensional Hilbert-type inequality with the homogeneous kernel as 1 ( x + ∥ k − ξ ∥ α ) λ + m ( x , λ > 0 ) $\frac{1}{(x + \Vert k - \xi \Vert _{\alpha} )^{\lambda + m}}\ (x,\lambda > 0)$ involving one derivative function of m-order is given. The equivalent conditions of the best possible constant factor related to several parameters are considered. The equivalent forms. the operator expressions and some particular inequalities are obtained.

In this paper, we generalize the study of finite sequences of orthogonal polynomials from one to two variables. In doing so, twenty three new classes of bivariate finite orthogonal polynomials are presented, obtained from the product of a finite and an infinite family of univariate orthogonal polynomials. For these new classes of bivariate finite orthogonal polynomials, we present a bivariate weight function, the domain of orthogonality, the orthogonality relation, the recurrence relations, the second-order partial differential equations, the generating functions, as well as the parameter derivatives. The limit relations among these families are also presented in Labelle's flavor. [ABSTRACT FROM AUTHOR]

A family of fifth-order iterative methods is proposed for solving nonlinear equations using the weight function technique. This family offers flexibility through its structure and the choice of weight functions, resulting in a wide range of new specific schemes. It is demonstrated that this proposed family includes several well-known and recent methods as special cases. In addition, several new particular methods are designed to achieve better results than existing methods of the same type. Convergence analysis is conducted, and numerical examples in both real and complex domains are provided for several specific schemes within the proposed family. Comparisons between the existing methods within this family and the newly introduced methods generally indicate improved performance among the new members. Notably, the study of complex dynamics and basins of attraction reveals that our new specific schemes have broader sets of initial points that lead to convergence. [ABSTRACT FROM AUTHOR]

In the conventional finite element method, only a few types of elements can directly use the weight functions to generate a proper, diagonally lumped mass matrix (LMM), which satisfies mass conservation and positive definiteness. This study proposes a new set of weight functions that can be efficiently constructed and used for generating a proper LMM based on classical 6‐, 9‐, and 10‐node triangular elements. Furthermore, we prove that these new elements are all complete and consistent, which ensures the convergence of the numerical solution. Finally, some modal and forced/free vibration analyses are conducted to demonstrate the computational accuracy and validation of the proposed LMM. [ABSTRACT FROM AUTHOR]

In this paper, a new reverse half-discrete Mulholland-type inequality with the nonhomogeneous kernel of the form h (v (x) ln n) and the best possible constant factor is obtained by using the weight functions and the technique of real analysis. The equivalent reverses are considered. As corollaries, we deduce some new equivalent reverse inequalities with the homogeneous kernel of the form k λ (v (x) , ln n) . A few particular cases are provided. Our new reverse half-discrete Mulholland-type inequality which has a nonhomogeneous kernel is more general than in the previous homogeneous kernel work. The harmonized integration will have more applications. [ABSTRACT FROM AUTHOR]

The weight function approach of solving nonlinear equations is presented in this work as a two-step, fourth-order, optimal iterative method. It comes from the idea of the theoretical order of convergence. The performance of the novel method has been assessed on a small number of real-life applications from different domains and contrasted with the fourth-order methods currently in use. Also, we find that this has fewer errors and iterations for the same order than the methods now in use. To examine the dynamical analysis of the proposed method using computer tools, we test some challenging polynomials that explain convergence and other graphical features of the iterative scheme. [ABSTRACT FROM AUTHOR]

The coefficients in the three-term recurrence relation for monic orthogonal polynomials with respect to the Szegő-Bernstein weight functions $ w_\nu(x) = W(x)/(c-x)^\nu $, $ \nu\ge 1 $, on $ (-1, 1) $ are obtained in the explicit form for all Chebyshev cases, i.e., when $ W(x) $ is $ (1-x^2)^{\mp 1/2} $, $ \sqrt{(1+x)/(1-x)} $ and $ \sqrt{(1-x)/(1+x)} $. Chebyshev's method of modified moments is used, as well as the MATHEMATICA package $ {\tt OrthogonalPolinomials}$ developed in [Facta Univ. Ser. Math. Inform. 19 (2004), 17-36] and [Math. Balkanica 26 (2012), 169-184]. [ABSTRACT FROM AUTHOR]

The 3D similarity coordinate transformation is widely used to estimate the transformation parameters for measurement datum transformation. Accurate and reliable transformation parameters are crucial for accurate and reliable data integration. However, the accuracy of the transformation parameters can be significantly affected or even severely distorted when the observed coordinates are contaminated by gross errors. To address this problem, an advanced iteratively weighted least squares solution based on the weighted least squares is proposed. This solution utilizes the singular value decomposition method to obtain the rotation matrix and introduces a novel weight estimation approach based on Gaussian function. This approach enables the weight to be normalized and optimized iteratively. To verify the accuracy and reliability of the proposed algorithm, the root mean square errors from both true and pseudo-observed values are analyzed by simulation experiments. Furthermore, the results of simulated and empirical experiments show that the proposed algorithm can effectively reduce the influence of gross errors to obtain reliable measurement datum transformation parameters. It should be noted that the new algorithm can easily be extended to the 2D/3D affine and rigid transformation cases, such as image matching, point cloud registration, and absolute orientation of photogrammetry. [ABSTRACT FROM AUTHOR]