Lagrange-type Algebraic Minimal Bivariate Fractal Interpolation Formula
Ildikó Somogyi,
Anna Soós
Issue:
Volume 12, Issue 5, October 2023
Pages:
109-113
Received:
28 July 2023
Accepted:
14 September 2023
Published:
25 September 2023
Abstract: Fractal interpolation methods became an important method in data processing, even for functions with abrupt changes. In the last few decades it has attracted several authors because it can be applied in various fields. The advantage of these methods are that we can generalize the classical approximation methods and also we can combine these methods for example with Lagrange interpolation, Hermite interpolation or spline interpolation. The classical Lagrange interpolation problem give the construction of a suitable approximate function based on the values of the function on given points. These method was generalized for more than one variable functions. In this article we generalize the so-called algebraic maximal Lagrange interpolation formula in order to approximate functions on a rectangular domain with fractal functions. The construction of the fractal function is made with a so-called iterated function system. This method it has the advantage that all classical methods can be obtained as a particular case of a fractal function. We also use the construction for a polynomial type fractal function and we proof that the Lagrange-type algebraic minimal bivariate fractal function satisfies the required interpolation conditions. Also we give a delimitation of the error, using the result regarding the error of a polynomial fractal interpolation function.
Abstract: Fractal interpolation methods became an important method in data processing, even for functions with abrupt changes. In the last few decades it has attracted several authors because it can be applied in various fields. The advantage of these methods are that we can generalize the classical approximation methods and also we can combine these methods...
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Research Article
Trapping Issues for Weight-dependent Walks in the Weighted Extended Cayley Networks
Dandan Ye,
Fei Zhang,
Yiteng Qin,
Xiaojuan Zhang,
Ning Zhang,
Jin Qin,
Wei Chen,
Yingze Zhang
Issue:
Volume 12, Issue 5, October 2023
Pages:
114-139
Received:
31 August 2023
Accepted:
25 September 2023
Published:
1 November 2023
Abstract: The weighted extended Cayley networks are an extension of extended Cayley networks, which are the structures constructed by introducing power spaces into traditional Cayley trees. The weighted extended Cayley networks are constructed depending on two structural parameters of the network m, n and a weight factor r. Firstly, we used a new calculation method to calculate the exact analytic formula of the average weighted shortest path (AWSP). The obtained results show that: (1) For very large systems, the AWSPs for different value of weight factor r are less affected by the parameter m. (2) The AWSPs are less affected by the weight factor r when r is greater than 0 and less than or equal to n, while the AWSPs depend on the scaling factor r when r is greater than n. We have presented a trapping issue of weight-dependent walks in the weighted extended Cayley networks, focusing on a specific case with a perfect trap located at the central node. Then, the scaling expression of the average trapping time (ATT) is derived based on the layering of weighted extended Cayley networks. It was surprisingly found that (1) Regardless of the relationship between m and n, the dominant terms of ATTs are consistent. (2) ATTs are less affected by the structural parameter m and the weight factor r when r is less than or equal to the ratio of n to m−1, indicating that the efficiency of the trapping process is independent of m and r. (3) When r is greater than the ratio of n to m−1, the efficiency of the trapping process depends on three main parameters: two structural parameters of the networks m, n and a weight factor r, which means that the smaller the multiplier of three numbers r, n and m − 1 is, the more efficient the trapping process is. Therefore, the trapping efficiency of the weighted extended Cayley networks is not only affected by the underlying structures of the networks m and n, but also by the weight factor r.
Abstract: The weighted extended Cayley networks are an extension of extended Cayley networks, which are the structures constructed by introducing power spaces into traditional Cayley trees. The weighted extended Cayley networks are constructed depending on two structural parameters of the network m, n and a weight factor r. Firstly, we used a new calculation...
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