Laplacians on self-similar sets—analysis on fractals

1994 ◽  
pp. 75-93
Author(s):  
Jun Kigami
Keyword(s):  
Analysis On Fractals ◽  
Self Similar

Effective resistances for harmonic structures on p.c.f. self-similar sets

1994 ◽  
Vol 115 (2) ◽  
pp. 291-303 ◽  
Author(s):  
Jun Kigami
Keyword(s):  
Diffusion Processes ◽  
Difference Operator ◽  
Scaling Limit ◽  
Sierpinski Gasket ◽  
Dirichlet Forms ◽  
Sierpiński Gasket ◽  
Know How ◽  
Analysis On Fractals ◽  
Topological Description ◽  
Self Similar

In mathematics, analysis on fractals was originated by the works of Kusuoka [17] and Goldstein[8]. They constructed the ‘Brownian motion on the Sierpinski gasket’ as a scaling limit of random walks on the pre-gaskets. Since then, analytical structures such as diffusion processes, Laplacians and Dirichlet forms on self-similar sets have been studied from both probabilistic and analytical viewpoints by many authors, see [4], [20], [10], [22] and [7]. As far as finitely ramified fractals, represented by the Sierpinski gasket, are concerned, we now know how to construct analytical structures on them due to the results in [20], [18] and [11]. In particular, for the nested fractals introduced by Lindstrøm [20], one can study detailed features of analytical structures such as the spectral dimensions and various exponents of heat kernels by virtue of the strong symmetry of nested fractals, cf. [6] and [15]. Furthermore in [11], Kigami proposed a notion of post critically finite (p.c.f. for short) self-similar sets, which was a pure topological description of finitely ramified self-similar sets. Also it was shown that we can construct Dirichlet forms and Laplacians on a p.c.f. self-similar set if there exists a difference operator that is invariant under a kind of renormalization. This invariant difference operator was called a harmonic structure. In Section 2, we will give a review of the results in [11].

Self-similar collapse with non-radial motions

2006 ◽  
Vol 20 ◽  
pp. 1-4
Author(s):  
A. Nusser
Keyword(s):  
Self Similar

PREDICTING CHARACTERISTICS OF SELF SIMILAR TRAFFIC BY SPLINE-EXTRAPOLATION

2019 ◽  
Author(s):  
Irina Strelkovskay ◽  
Irina Solovskaya ◽  
Anastasija Makoganjuk ◽  
Nikolaj Severin
Keyword(s):  
Quality Characteristics ◽  
Traffic Prediction ◽  
Normative Values ◽  
Spline Functions ◽  
Network Nodes ◽  
Traffic Forecasting ◽  
Network Equipment ◽  
Self Similar ◽  
Accuracy Of Prediction

The problem of forecasting self-similar traffic, which is characterized by a considerable number of ripples and the property of long-term dependence, is considered. It is proposed to use the method of spline extrapolation using linear and cubic splines. The results of self-similar traffic prediction were obtained, which will allow to predict the necessary size of the buffer devices of the network nodes in order to avoid congestion in the network and exceed the normative values ​​of QoS quality characteristics. The solution of the problem of self-similar traffic forecasting obtained with the Simulink software package in Matlab environment is considered. A method of extrapolation based on spline functions is developed. The proposed method has several advantages over the known methods, first of all, it is sufficient ease of implementation, low resource intensity and accuracy of prediction, which can be enhanced by the use of quadratic or cubic interpolation spline functions. Using the method of spline extrapolation, the results of self-similar traffic prediction were obtained, which will allow to predict the required volume of buffer devices, thereby avoiding network congestion and exceeding the normative values ​​of QoS quality characteristics. Given that self-similar traffic is characterized by the presence of "bursts" and a long-term dependence between the moments of receipt of applications in this study, given predetermined data to improve the prediction accuracy, it is possible to use extrapolation based on wavelet functions, the so-called wavelet-extrapolation method. Based on the results of traffic forecasting, taking into account the maximum values ​​of network node traffic, you can give practical guidance on how traffic is redistributed across the network. This will balance the load of network objects and increase the efficiency of network equipment.

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