• Communications for Statistical Applications and Methods
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• 제31권3호
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• pp.337-347
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• 2024

Kernel density estimation is a prevalent technique employed for nonparametric density estimation, enabling direct estimation from the data itself. This estimation involves two crucial elements: selection of the kernel function and the determination of the appropriate bandwidth. The selection of the bandwidth plays an important role in kernel density estimation, which has been developed over the past decade. A range of methods is available for selecting the bandwidth, including the plug-in bandwidth. In this article, the proposed plug-in bandwidth is introduced, which leverages shifted Chebyshev series-based approximation to determine the optimal bandwidth. Through a simulation study, the performance of the suggested bandwidth is analyzed to reveal its favorable performance across a wide range of distributions and sample sizes compared to alternative bandwidths. The proposed bandwidth is also applied for kernel density estimation on real dataset. The outcomes obtained from the proposed bandwidth indicate a favorable selection. Hence, this article serves as motivation to explore additional plug-in bandwidths that rely on function approximations utilizing alternative series expansions.

• Kyungpook Mathematical Journal
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• 제52권3호
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• pp.347-357
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• 2012

When Newton's method, or Halley's method is used to approximate the pth root of 1-z, a sequence of rational functions is obtained. In this paper, a beautiful formula for these rational functions is proved in the square root case, using an interesting link to Chebyshev's polynomials. It allows the determination of the sign of the coefficients of the power series expansion of these rational functions. This answers positively the square root case of a proposed conjecture by Guo(2010).

• Nuclear Engineering and Technology
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• 제45권7호
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• pp.895-910
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• 2013

The reflood test data from the rod bundle heat transfer (RBHT) test facility showed that the grids in the upper portion of the rod bundle could become wet well before the arrival of the quench front and that the sizes of liquid droplets downstream of a wet grid could not be predicted by the droplet breakup models for a dry grid. To investigate the water droplet generation from a wet grid spacer, a viscous linear temporal instability model of the water sheet issuing from the trailing edge of the grid with the surrounding steam up-flow is developed in this study. The Orr-Sommerfeld equations along with appropriate boundary conditions for the flow are solved using Chebyshev series expansions and the Tau-Galerkin projection method. The effects of several physical parameters on the water sheet oscillation are studied by determining the variation of the temporal growth rate with the wavenumber. It is found that a larger relative steam velocity to water velocity has a tendency to destabilize the water sheet with increased dynamic pressure. On the other hand, a larger ratio of steam boundary layer to the half water sheet thickness has a stabilizing effect on the water sheet oscillation. Droplet diameters downstream of the spacer grid predicted by the present model are found to compare reasonably well with the data obtained at the RBHT test facility as well as with other data recently reported in the literature.

• Nonlinear Functional Analysis and Applications
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• 제28권3호
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• pp.647-657
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• 2023

The aim of this paper is to study certain subclass ${\tilde{S^q_{\Sigma}}}({\lambda},\,{\alpha},\,t,\,s,\,p,\,b)$ of analytic and bi-univalent functions which are defined by using symmetric q-derivative operator. We estimate the second and third coefficients of the Taylor-Maclaurin series expansions belonging to the subclass and upper bounds for Feketo-Szegö inequality. Furthermore, some relevant connections of certain special cases of the main results with those in several earlier works are also pointed out.