General Weight Function Research Articles

Existence of positive solutions to negative power nonlinear integral equations with weights

This paper is devoted to the existence and non-existence of positive solutions to the following negative power nonlinear integral equation related to the sharp reversed Hardy–Littlewood–Sobolev inequality: fq−1(x)=∫ΩK(x)f(y)K(y)|x−y|n−αdy+λ∫ΩG(x)f(y)G(y)|x−y|n−α−βdy,f≥0,x∈Ω‾,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ f^{q-1}(x)= \\int _{\\varOmega }\\frac{K(x)f(y)K(y)}{ \\vert x-y \\vert ^{n-\\alpha }}\\,dy+ \\lambda \\int _{\\varOmega }\\frac{G(x)f(y)G(y)}{ \\vert x-y \\vert ^{n-\\alpha -\\beta }}\\,dy, \\quad f\\geq 0, x\\in \\overline{ \\varOmega }, $$\\end{document} where 0< q<1, alpha >n, 0<beta <alpha -n, lambda in mathbb{R}, Ω is a smooth bounded domain, K(x), G(x) are positive continuous functions in Ω̅. For Kequiv Gequiv 1, the existence and non-existence of positive solutions to the equation have been studied by Dou–Guo–Zhu (2019). In this paper we consider the existence and non-existence of positive solutions to the above integral equation with the general weight functions K(x), G(x).

Treatment of rats with spinal cord injury using human bone marrow-derived stromal cells prepared by negative selection

BackgroundSpinal cord injury (SCI) is a highly debilitating pathology without curative treatment. One of the most promising disease modifying strategies consists in the implantation of stem cells to reduce inflammation and promote neural regeneration. In the present study we tested a new human bone marrow-derived stromal cell preparation (bmSC) as a therapy of SCI.MethodsSpinal cord contusion injury was induced in adult male rats at thoracic level T9/T10 using the Infinite Horizon impactor. One hour after lesion the animals were treated with a sub-occipital injection of human bmSC into the cisterna magna. No immune suppression was used. One dose of bmSC consisted, on average, of 2.3 million non-manipulated cells in 100 μL suspension, which was processed out of fresh human bone marrow from the iliac crest of healthy volunteers. Treatment efficacy was compared with intraperitoneal injections of methylprednisolone (MP) and saline. The recovery of motor functions was assessed during a surveillance period of nine weeks. Adverse events as well as general health, weight and urodynamic functions were monitored daily. After this time, the animals were perfused, and the spinal cord tissue was investigated histologically.ResultsRats treated with bmSC did not reject the human implants and showed no sign of sickness behavior or neuropathic pain. Compared to MP treatment, animals displayed better recovery of their SCI-induced motor deficits. There were no significant differences in the recovery of bladder control between groups. Histological analysis at ten weeks after SCI revealed no differences in tissue sparing and astrogliosis, however, bmSC treatment was accompanied with reduced axonal degeneration in the dorsal ascending fiber tracts, lower Iba1-immunoreactivity (IR) close to the lesion site and reduced apoptosis in the ventral grey matter. Neuroinflammation, as evidenced by CD68-IR, was significantly reduced in the MP-treated group.ConclusionsHuman bmSC that were prepared by negative selection without expansion in culture have neuroprotective properties after SCI. Given the effect size on motor function, implantation in the acute phase was not sufficient to induce spinal cord repair. Due to their immune modulatory properties, allogeneic implants of bmSC can be used in combinatorial therapies of SCI.

Multidimensional Hilbert-Type Inequalities Obtained via Local Fractional Calculus

In this paper we give a unified treatment of multidimensional fractal Hilbert-type inequalities. More precisely, we establish a Hilbert-type inequality with a general local fractional continuous kernel and weight functions. A particular emphasis is dedicated to a class of inequalities with a homogeneous kernel. Namely, we impose some weak conditions for which the constants appearing on the right-hand sides of such Hilbert-type inequalities are the best possible. As an application, we discuss some particular choices of homogeneous kernels and power weight functions.

Stable High Order Quadrature Rules for Scattered Data and General Weight Functions

Numerical integration is encountered in all fields of numerical analysis and the engineering sciences. By now, various efficient and accurate quadrature rules are known; for instance, Gauss-type quadrature rules. In many applications, however, it might be impractical---if not even impossible---to obtain data to fit known quadrature rules. Often, experimental measurements are performed at equidistant or even scattered points in space or time. In this work, we propose stable high order quadrature rules for experimental data, which can accurately handle general weight functions.

Dimensional Reduction for 6D Vibrotactile Display.

The human hand detects high-frequency vibrations in all directions but cannot distinguish the direction, which suggests a multi-dimensional vibrotactile stimulus is haptically equivalent to some one-dimensional (1D) stimulus. In this article, we explore how a 6D vibrotactile stimulus rendered at the haptic interaction point (HIP) of a kinesthetic haptic interface, with the stylus held in a precision pen-hold grasp, is mapped to an equivalent 1D stimulus normalized by the detection threshold. We gather a large human-subjects data set in which we determine detection thresholds for 45 distinct combinations of three orthogonal forces and three orthogonal torques rendered at the HIP, at a single frequency of 108Hz corresponding to the peak sensitivity in our prior study. Using this data set, we find a general quadratic weighting function to predict the 1D normalized stimulus for a given 6D vibrotactile stimulus. We find that including just seven (out of a possible 21) independent parameters in the symmetric weighting matrix is sufficient to capture the non-obvious coupling between forces and torques rendered at the HIP for dimensional reduction from 6D to 1D.

General Euler-Poisson-Darboux Equation and Hyperbolic B-Potentials

In this work, we develop the theory of hyperbolic equations with Bessel operators. We construct and invert hyperbolic potentials generated by multidimensional generalized translation. Chapter 1 contains necessary notation, definitions, auxiliary facts and results. In Chapter 2, we study some generalized weight functions related to a quadratic form. These functions are used below to construct fractional powers of hyperbolic operators and solutions of hyperbolic equations with Bessel operators. Chapter 3 is devoted to hyperbolic potentials generated by multidimensional generalized translation. These potentials express negative real powers of the singular wave operator, i. e. the wave operator where the Bessel operator acts instead of second derivatives. The boundedness of such an operator and its properties are investigated and the inverse operator is constructed. The hyperbolic Riesz B-potential is studied as well in this chapter. In Chapter 4, we consider various methods of solution of the Euler-Poisson-Darboux equation. We obtain solutions of the Cauchy problems for homogeneous and nonhomogeneous equations of this type. In Conclusion, we discuss general methods of solution for problems with arbitrary singular operators.

Fredholm Toeplitz operators with VMO symbols and the duality of generalized Fock spaces with small exponents

AbstractWe characterize Fredholmness of Toeplitz operators acting on generalized Fock spaces of the n-dimensional complex space for symbols in the space of vanishing mean oscillation VMO. Our results extend the recent characterizations for Toeplitz operators on standard weighted Fock spaces to the setting of generalized weight functions and also allow for unbounded symbols in VMO for the first time. Another novelty is the treatment of small exponents 0 &lt; p &lt; 1, which to our knowledge has not been seen previously in the study of the Fredholm properties of Toeplitz operators on any function spaces. We accomplish this by describing the dual of the generalized Fock spaces with small exponents.

Distributed accelerating of quantized second‐order consensus with bounded input

AbstractIn this paper, the convergence speed of consensus for a second‐order integrator with the fixed undirected graph is investigated. Additionally, the quantized information and bounded control input are applied to the system. To accelerate the convergence speed, a distributed variable adjacency matrix is proposed where the links' weight are functions defined based on the distance to the neighbors. The stability of the whole system for both of the quantized and non‐quantized consensus protocol considering a general weight function is shown using Lyapunov's direct approach. Furthermore, it is mentioned that the consensus value of the position depends on the structure of the quantizer.

A unified approach to fractal Hilbert-type inequalities

In the present study we provide a unified treatment of fractal Hilbert-type inequalities. Our main result is a pair of equivalent fractal Hilbert-type inequalities including a general kernel and weight functions. A particular emphasis is devoted to a class of homogeneous kernels. In addition, we impose appropriate conditions for which the constants appearing on the right-hand sides of the established inequalities are the best possible. As an application, our results are compared with some previously known ones from the literature.

On Spectral Approximations with Nonstandard Weight Functions and Their Implementations to Generalized Chaos Expansions

In this manuscript, we analyze the expansions of functions in orthogonal polynomials associated with a general weight function in a multidimensional setting. Such orthogonal polynomials can be obtained, e.g, by Gram–Schmidt orthogonalization. However, in most cases, they are not eigenfunctions of some singular Sturm–Liouville problem, as is the case for known polynomials, such as the Jacobi polynomials. Therefore, the standard convergence theorems do not apply. Furthermore, since in general multidimensional cases the weight functions are not a tensor product of one-dimensional functions, the orthogonal polynomials are not a product of one-dimensional orthogonal polynomials, as well. This work provides a way of estimating the convergence rate using a comparison lemma. We also present a spectrally convergent, multidimensional, integration method. Numerical examples demonstrate the efficacy of the proposed method. We also show that the use of non-standard weight functions can allow for efficient integration of singular functions. We demonstrate the use of this method to uncertainty quantification problem using Generalized Polynomial Chaos Expansions in the case of dependent random variables, as well.

Modified Kaplan–Meier Estimator and Nelson–Aalen Estimator with Geographical Weighting for Survival Data

The Kaplan–Meier and Nelson–Aalen estimators are universally used methods in clinical studies. In a public health study, people often collect data from different locations of the medical services provider. When some studies need to consider survival curves from different locations, traditional estimators simply estimate the marginal survival curves using stratification. In this article, we use the idea from geographically weighted regression to add geographical weights to the observations to get modified versions of the Kaplan–Meier and Nelson–Aalen estimators which can represent the local survival curve and cumulative hazard. We use counting process methods to derive these modified estimators and to estimate their variances. In addition, we discuss some general spatial weighting functions which can be used in computing these estimators. Furthermore, we present simulation results to illustrate the performance of the modified estimators. Finally, we apply our method to prostate cancer data from the SEER cancer registry for the state of Louisiana.

Novel weight functions and stress intensity factors for quarterelliptical cracks in lug attachments

In this paper, a general weight function was developed to calculate the stress intensity factors (SIFs) for quarter-elliptical cracks in a wide range of lug attachment family. For this purpose, a series of finite element analyses were conducted to achieve this weight function. Finally, using this unique extracted weight function, the influence of the pin loading model and crack parameters (aspect ratio and relative depth of the quarter-elliptical crack) on the SIFs was evaluated in the cracked lug attachments. The results of the present work were compared with those in the literature, which are compatible with them in those cases.

A generalized weight function for cracks emanated from sharp and blunt V-notches

This paper presents a generalized weight function which applicable to cracks initiated from V-notches with sharp or blunt tips. The proposed function is based on the composition solution of the weight function of an edge crack in a semi-infinite plate. For obtaining the proper relation, two correction factors are used to express the effects of the notch depth, root radius, and opening angle. Each factor can be applied separately to present only the effects of the notch depth or root radius to improve solutions that proposed in the earlier studies. This method can be used for studying the effects of the notch parameters and loading conditions on the stress intensity factors of the notch cracks. The accuracy of the proposed function was verified by comparing the results with finite element data for a case study.

Performance Differences of an Electromagnetic Flow Sensor With Nonideal Electrodes Based on Different-Dimensional Weight Functions

In the strict sense, the electrodes of the electromagnetic flow sensor (EFS) are nonideal, and the selection of models with different dimensions can result in a relatively large difference in the calculation of the weight function, which further affects the estimation of the output signal. The aim of this paper is to explore the difference in the characteristics of the 2-D and 3-D EFSs with nonideal electrodes. In this paper, the analytical expressions of weight function for both 2-D and 3-D cases are deduced mathematically, and a general weight function methodology based on simulation data and fitting is presented to solve the mixed boundary problem. The effectiveness of the proposed method is verified by the comparison with the numerical simulation. Meanwhile, the distributions of weight function for 3-D EFSs with different electrodes in cross sections are obtained by means of the method mentioned above, and the difference between the 2-D model and the 3-D model is revealed intuitively. Finally, the output characteristics of the EFS with different dimensions and electrode shapes are calculated by coupling the data of three fields, including the weight field, the flow field, and the magnetic field. Based on the comparison between the experimental data and the calculated values, the direct influence of the dimension and the electrode shape on the output characteristics of the EFS is discussed, and the theoretical analysis is made for the structure design and optimization of the EFS.

Universal Weight Function Approach to Determine Stress Intensity Factors at Blade Mounting Locations in Steam Turbine Rotor System

The evaluation of a crack requires the calculation of the stress intensity factors. To calculate the stress intensity factors, the general procedure is to represent the stress distribution in the crack with a polynomial equation. In many cases, a cubic polynomial is used to fit the stress distribution. However, the resultant polynomial equation may not always fit optimally for the actual stress distribution. Weight function method makes a more accurate representation of the stress distribution for the calculation of the stress intensity factor. Even though weight function procedure gives good results, it is desirable to incorporate the advantages of both weight function method and polynomial equation representation. In this paper, universal weight function method is proposed which gives the same values of stress intensity factors that are obtained using the general weight function method. First, an analytical approach to calculate the radial stresses is given which provides the radial stress distribution. This method is applied to determine the radial stress distribution at the blade mounting locations of steam turbine rotor system. A semi elliptical crack is considered at the blade mounting locations and using the data of stresses at discrete points on the crack, Mode I stress intensity factors are determined using the universal weight function approach.

Stokes’ second problem of viscoelastic fluids with constitutive equation of distributed-order derivative

The steady-state periodic flow of Stokes’ second problem for viscoelastic fluids with constitutive equation in terms of the distributed-order derivative was considered. The distributed-order derivative involves an integration with respect to the order of fractional derivative, and the order is associated with a weight function p(α). With a general weight function p(α), the flow velocity was obtained. The amplitude, the penetration depth and the wavelength were given analytically. Results of Newtonian fluid and single fractional constitutive equation were derived as special cases of weight function p(α). Also we considered other three cases of weight function p(α): linear combination of Dirac delta functions, constant and parameterized exponential function.

Approximate stress intensity factors for a semi-circular crack in an arbitrary structure under arbitrary mode I loading

Approximate Stress Intensity Factors (SIFs) for a semi-circular surface crack in arbitrary elastic finite and infinite bodies are presented. Utilizing General Point Load Weight Function (GPLWF) concept, an explicit expression is derived to determine SIFs for semi-circular cracks subjected to uniform, linear, and nonlinear loads. The presented formulation not only provides a solution for cracked bodies subjected to arbitrary two-dimensional stress distribution on the crack faces, but also could present SIFs for any point on the crack front. The results of the presented formulation for special cases, i.e. semi-circular cracks in a finite thickness plate subjected to complicated loadings, central and non-central semi-circular cracks in finite-length thick-walled cylinders subjected to a uniform internal pressure, and central semi-circular cracks in an infinite-length thick-walled cylinder subjected to a non-uniform internal pressure are compared with the available results in the literature and those obtained through FEM, indicating a very good accuracy. The proposed procedure will be generalized for semi-elliptical cracks in arbitrary structures and mixed mode loading in future.

Relationship between the weighted distributions and some inequality measures

ABSTRACTIn this paper, we study the relationships between the weighted distributions and the parent distributions in the context of Lorenz curve, Lorenz ordering and inequality measures. These relationships depend on the nature of the weight functions and give rise to interesting connections. The properties of weighted distributions for general weight functions are also investigated. It is shown how to derive and to determine characterizations related to Lorenz curve and other inequality measures for the cases weight functions are increasing or decreasing. Some of the results are applied for special cases of the weighted distributions. We represent the reliability measures of weighted distributions by the inequality measures to obtain some results. Length-biased and equilibrium distributions have been discussed as weighted distributions in the reliability context by concentration curves. We also review and extend the problem of stochastic orderings and aging classes under weighting. Finally, the relationships between the weighted distribution and transformations are discussed.

When the extension property does not hold

A complete extension theorem for linear codes over a module alphabet, equipped with the symmetrized weight composition, is proven. It is also shown that the extension property with respect to a general weight function does not hold for module alphabets, that have a noncyclic socle.

Axiomatic generalization of the membership degree weighting function for fuzzy C means clustering: Theoretical development and convergence analysis

For decades practitioners have been using the center-based partitional clustering algorithms like Fuzzy C Means (FCM), which rely on minimizing an objective function, comprising of an appropriately weighted sum of distances of each data point from the cluster representatives. Numerous generalizations in terms of choice of the distance function have been introduced for FCM since its inception. In a stark contrast to this fact, to the best of our knowledge, there has not been any significant effort to address the issue of convergence of the algorithm under the structured generalization of the weighting function. Here, by generalization we mean replacing the conventional weighting function uijm (where uij indicates the membership of data xi to cluster Cj and m is the real-valued fuzzifier with 1≤ m < ∞) with a more general function g(uij) which can assume a wide variety of forms under some mild assumptions. In this article, for the first time, we present a novel axiomatic development of the general weighting function based on the membership degrees. We formulate the fuzzy clustering problem along with the intuitive justification of the technicalities behind the axiomatic approach. We develop an Alternative Optimization (AO) procedure to solve the main clustering problem by solving the partial optimization problems in an iterative way. The novelty of the article lies in the development of an axiomatic generalization of the weighting function of FCM, formulation of an AO algorithm for the fuzzy clustering problem with the extension to this general class of weighting functions, and a detailed convergence analysis of the proposed algorithm.