In this paper, an explicit exact  nite-difference Scheme for the Huxley equation is presented based on the nonstandard  nite-difference (NSFD) Scheme. Afterwards, an NSFD Scheme is proposed for the numerical solu-tion of the Huxley equation. The positivity and boundedness of the Scheme is discussed. It is shown through analysis that the proposed Scheme is consis-tent, stable, and convergence. The numerical results obtained by the NSFD Scheme is compared with the exact solution and some available methods, to verify the accuracy and efficiency of the NSFD Scheme.

Background: A Scheme, named SUV_Shape, for the tumor delineation was evaluated in this study.Materials and Methods: Seven vacuous plastic balls filled with [18F] -FDG solution as tumor were acquired by a PET/CT. The volume of each ball was measured by SUV_Shape and a balance, labeled as GTVs and GTVt. Five rabbits implanted VX2 tumors were acquired by [18F] -FDG PET/CT. These rabbits were mercy killed within 24h after acquisition. VX2 tumors’ volumes were measured by SUV_Shape, and caliper, labeled as GTVs and GTVt. The correlation coefficient between GTVs and GTVt were done.Results: The volume of each ball confirmed by the balance was 0.56, 1.26, 5.88, 11.74, 21.30, 64.56, 179.50 mL. Twelve VX2 tumors were measured; their GTVt ranged from 0.11mL to 29.26mL. The Spearman’s rho between GTVt and GTVs were 0.96 and 0.89 for phantom and animal tumor models, respectively.Conclusions: The SUV_Shape could delineate tumors based on their radiopharmaceutical-avid PET images.

Scheduling problems have been considered by researchers from different aspects, and it has been widely addressed in the literature. Most scheduling problems, and almost all that arise in real-world applications, are NP-hard, therefore finding a proper and feasible solution, a solution that is performed in a reasonable time (polynomial time), is difficult. Approximation Scheme, among different methods, is one approach that can overcome difficulties. Polynomial-time Approximation and differential Approximation Schemes are categorized as Approximation approaches based on comparing the proposed algorithm with optimum and worst-case solutions. As the first attempt, the paper investigates differential Approximation Schemes for the minimization of maximum delivery time. In this paper, the main methods and tools for proofing the existence of an Approximation Scheme are presented. To do so, a differential Approximation Scheme is proposed by employing a polynomial-time Approximation Scheme (PTAS). The paper also reviews efficient Approximation Schemes on single machine problems with objective functions minimization of maximum delivery time, and minimizing the total weighted completion time.

Scheduling problems have been considered by researchers from different aspects, and it has been widely addressed in the literature. Most scheduling problems, and almost all that arise in real-world applications, are NP-hard, therefore finding a proper and feasible solution, a solution that is performed in a reasonable time (polynomial time), is difficult. Approximation Scheme, among different methods, is one approach that can overcome difficulties. Polynomial-time Approximation and differential Approximation Schemes are categorized as Approximation approaches based on comparing the proposed algorithm with optimum and worst-case solutions. As the first attempt, the paper investigates differential Approximation Schemes for the minimization of maximum delivery time. In this paper, the main methods and tools for proving the existence of an Approximation Scheme are presented. To do so, a differential Approximation Scheme is proposed by employing a polynomial-time Approximation Scheme (PTAS). The paper also reviews efficient Approximation Schemes on single machine problems with objective functions minimization of maximum delivery time, and minimizing the total weighted completion time. Also, in this paper, using the polynomial time Approximation Scheme (PTAS), a differential Approximation Scheme has been studied for the problem of minimizing the maximum delivery time of works, which has not been investigated so far.

‎‎‎Given a fuzzy normed space, we will introduce the notion of fuzzy near best Approximation as a generalization of the notion of fuzzy best Approximation. Some basic properties are characterized and also many examples for illustration are presented. Also, the hereditary properties of the fuzzy near best Approximation on direct sum and tensor product of linear spaces are discussed.

Over the last decade or so, wavelets have had a growing impact on signal processing theory and practice, both because of their unifying role and their successes in applications. Filter banks, which lie at the heart of wavelet-based algorithms, have become standard signal processing operators, used routinely in applications ranging from compression to modems. The contributions of wavelets have often been in the subtle interplay between discrete-time and continuous-time signal processing. The purpose of this article is to look at recent wavelet advances from a signal processing perspective. In particular, Approximation results are reviewed, and the implication on compression algorithms is discussed. New constructions and open problems are also addressed. Finding a good basis to solve aproblem dates at least back to Fourier and his investigation of the heat equation. The series proposed by Fourier has several distinguishing features: The series is able to represent any finite energy function on an interval. The basis functions are eigenfunctions of linear shift invariant systems or, in other words, Fourier series diagonalize linear, shift invariant operators.

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