This paper is concerned with the numerical solution of the mixed Volterra-Fredholm integral equations by using a version of the block by block method. This method efficient for linear and nonlinear equations and it avoids the need for spacial starting values. The convergence is proved and finally performance of the method is illustrated by means of some significative examples.

There are several methods in the literature to find the fuzzy optimal solution of fully fuzzy linear programming (FFLP) problems. However, in all these methods, it is assumed that the product of two trapezoidal (triangular) fuzzy numbers will also be a trapezoidal (triangular) fuzzy number. Fan et al. (“Generalized fuzzy linear programming for decision making under uncertainty: Feasibility of fuzzy solutions and solving approach”, Information Sciences, Vol. 241, pp. 12–27, 2013) proposed a method for finding the fuzzy optimal solution of FFLP problems without considering this assumption. In this paper, it is shown that the method proposed by Fan et al. (2013) suffer from errors and to overcome these errors, a new method (named as Mehar method) is proposed for solving FFLP problems by modifying the method proposed by Fan et al. (2013) . To illustrate the proposed method, some numerical problems are solved.

An image adaptive noise reduction enhancement algorithm based on NSCT is proposed to perform image restoration preprocessing on the defocused image obtained under the microscope. Defocused images acquired under micro-nano scale optical microscopy, usually with inconspicuous details, edges and contours, affect the accuracy of subsequent observation tasks. Due to its multi-scale and multi-directionality, the NSCT transform has superior transform functions and can obtain more textures and edges of images. Combined with the characteristics of micro-nanoscale optical defocus images, the NSCT inverse transform is performed on all sub-bands to reconstruct the image. Finally, the experimental results of the standard 500 nm scale grid, conductive probe and triangular probe show that the proposed algorithm has a better image enhancement effect and significantly improves the quality of out-of-focus images.

In this study, the entropy weight method, the α convergence model, the absolute β convergence model and the conditional β convergence model are used to evaluate the 31 provinces’ innovative potential in China from 2011 to 2022. It is found that the innovative potential in nationwide China and in various regions are all increasing year by year, and the innovative potential in the eastern region is obviously better than that in the central region and western region. No matter considering the influence of external factors or not, the gap of innovative potential among provinces in different regions will gradually expand over time, with the largest gap among provinces in the eastern region, followed by the central region and the smallest in the western region. The conclusion of this study is instructive to enhance the innovative potential of China and promote the balanced development of regional innovative potential in China.

Carbon-based hollow structured nanomaterials have become one of the hot areas for research and development of hollow structured nanomaterials due to their unique structure, excellent physicochemical properties and promising applications. The design and synthesis of novel carbon-based hollow structured nanomaterials are of great scientific significance and wide application value. The recent research on the synthesis, structure and functionalization of carbon-based hollow structured nanomaterials and their related applications are reviewed. The basic synthetic strategies of carbon-based hollow structure nanomaterials are briefly introduced, and the structural design, material functionalization and main applications of carbon-based hollow structure nanomaterials are described in detail. Finally, the current challenges and opportunities in the synthesis and application of carbon-based hollow structured nanomaterials are discussed.