scholarly journals SPECTRA OF SELF-SIMILAR LAPLACIANS ON THE SIERPINSKI GASKET WITH TWISTS

Fractals ◽  
2008 ◽  
Vol 16 (01) ◽  
pp. 43-68 ◽  
Author(s):  
ANNA BLASIAK ◽  
ROBERT S. STRICHARTZ ◽  
BARIS EVREN UGURCAN
Keyword(s):  
Sierpinski Gasket ◽  
Outer Approximation ◽  
Invariant Set ◽  
Self Similarity ◽  
Sierpiński Gasket ◽  
Constant Multiple ◽  
Self Similar ◽  
Definition Of ◽  
Two Parameter ◽  
Present Experimental Evidence

We study the spectra of a two-parameter family of self-similar Laplacians on the Sierpinski gasket (SG) with twists. By this we mean that instead of the usual IFS that yields SG as its invariant set, we compose each mapping with a reflection to obtain a new IFS that still has SG as its invariant set, but changes the definition of self-similarity. Using recent results of Cucuringu and Strichartz, we are able to approximate the spectra of these Laplacians by two different methods. To each Laplacian we associate a self-similar embedding of SG into the plane, and we present experimental evidence that the method of outer approximation, recently introduced by Berry, Goff and Strichartz, when applied to this embedding, yields the spectrum of the Laplacian (up to a constant multiple).

ECCENTRIC DISTANCE SUM OF SIERPIŃSKI GASKET AND SIERPIŃSKI NETWORK

Fractals ◽  
2019 ◽  
Vol 27 (02) ◽  
pp. 1950016 ◽  
Author(s):  
JIN CHEN ◽  
LONG HE ◽  
QIN WANG
Keyword(s):  
Asymptotic Formula ◽  
Complex Networks ◽  
Sierpinski Gasket ◽  
The Self ◽  
Self Similarity ◽  
Sierpiński Gasket ◽  
Similar Measure ◽  
Self Similar ◽  
Eccentric Distance

The eccentric distance sum is concerned with complex networks. To obtain the asymptotic formula of eccentric distance sum on growing Sierpiński networks, we study some nonlinear integral in terms of self-similar measure on the Sierpiński gasket and use the self-similarity of distance and measure to obtain the exact value of this integral.

scholarly journals Self-similar resistive circuits as fractal-like structures

2017 ◽  
Vol 40 (1) ◽  
Author(s):  
Claudio Xavier Mendes dos Santos ◽  
Carlos Molina Mendes ◽  
Marcelo Ventura Freire
Keyword(s):  
Scale Invariance ◽  
Self Similarity ◽  
General Properties ◽  
Modern Physics ◽  
Resistive Networks ◽  
Self Similar ◽  
And Mathematics ◽  
Definition Of ◽  
The Individual

Fractals play a central role in several areas of modern physics and mathematics. In the present work we explore resistive circuits where the individual resistors are arranged in fractal-like patterns. These circuits have some of the characteristics typically found in geometric fractals, namely self-similarity and scale invariance. Considering resistive circuits as graphs, we propose a definition of self-similar circuits which mimics a self-similar fractal. General properties of the resistive circuits generated by this approach are investigated, and interesting examples are commented in detail. Specifically, we consider self-similar resistive series, tree-like resistive networks and Sierpinski’s configurations with resistors.

AVERAGE GEODESIC DISTANCE OF NODE-WEIGHTED SIERPINSKI NETWORKS

Fractals ◽  
2019 ◽  
Vol 27 (07) ◽  
pp. 1950110
Author(s):  
LIFENG XI ◽  
QIANQIAN YE ◽  
JIANGWEN GU
Keyword(s):  
Asymptotic Formula ◽  
Geodesic Distance ◽  
Sierpinski Gasket ◽  
Weight Vector ◽  
Sierpiński Gasket ◽  
Similar Measure ◽  
Self Similar

This paper discusses the asymptotic formula of average distances on node-weighted Sierpinski skeleton networks by using the integral of geodesic distance in terms of self-similar measure on the Sierpinski gasket with respect to the weight vector.

scholarly journals Spectral decimation for families of self-similar symmetric Laplacians on the Sierpiński gasket

2019 ◽  
Vol 7 (1) ◽  
pp. 1-62
Author(s):  
Sizhen Fang ◽  
Dylan King ◽  
Eun Bi Lee ◽  
Robert Strichartz
Keyword(s):  
Sierpinski Gasket ◽  
Sierpiński Gasket ◽  
Self Similar

scholarly journals COMB GRAPHS AND SPECTRAL DECIMATION

2009 ◽  
Vol 51 (1) ◽  
pp. 71-81 ◽  
Author(s):  
JONATHAN JORDAN
Keyword(s):  
Adjacency Matrix ◽  
Spectral Properties ◽  
Sierpinski Gasket ◽  
Self Similarity ◽  
Sierpiński Gasket ◽  
Spectral Similarity ◽  
Similarity Relationship ◽  
Adjacency Matrices ◽  
Similarity Operators ◽  
Math Phys

AbstractWe investigate the spectral properties of matrices associated with comb graphs. We show that the adjacency matrices and adjacency matrix Laplacians of the sequences of graphs show a spectral similarity relationship in the sense of work by L. Malozemov and A. Teplyaev (Self-similarity, operators and dynamics,Math. Phys. Anal. Geometry6(2003), 201–218), and hence these sequences of graphs show a spectral decimation property similar to that of the Laplacians of the Sierpiński gasket graph and other fractal graphs.

Self–similar Energy Forms on the Sierpinski Gasket with Twists

2007 ◽  
Vol 27 (1) ◽  
pp. 45-60 ◽  
Author(s):  
Mihai Cucuringu ◽  
Robert S. Strichartz
Keyword(s):  
Sierpinski Gasket ◽  
Sierpiński Gasket ◽  
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].

scholarly journals Polynomials on the Sierpiński gasket with respect to different Laplacians which are symmetric and self-similar

2020 ◽  
Vol 7 (4) ◽  
pp. 387-444
Author(s):  
Christian Loring ◽  
W. Jacob Ogden ◽  
Ely Sandine ◽  
Robert Strichartz
Keyword(s):  
Sierpinski Gasket ◽  
Sierpiński Gasket ◽  
Self Similar

scholarly journals Intrinsic Metric Formulas on Some Self-Similar Sets via the Code Representation

2019 ◽  
Vol 3 (1) ◽  
pp. 13
Author(s):  
Melis Güneri ◽  
Mustafa Saltan
Keyword(s):  
Sierpinski Gasket ◽  
Intrinsic Metric ◽  
Explicit Formulas ◽  
Sierpiński Gasket ◽  
Intrinsic Metrics ◽  
Geometrical Properties ◽  
Self Similar ◽  
Sierpinski Gaskets

In recent years, intrinsic metrics have been described on various fractals with different formulas. The Sierpinski gasket is given as one of the fundamental models which defined the intrinsic metrics on them via the code representations of the points. In this paper, we obtain the explicit formulas of the intrinsic metrics on some self-similar sets (but not strictly self-similar), which are composed of different combinations of equilateral and right Sierpinski gaskets, respectively, by using the code representations of their points. We then express geometrical properties of these structures on their code sets and also give some illustrative examples.

From 1 → 3 dendritic designs to fractal supramacromolecular constructs: understanding the pathway to the Sierpiński gasket

2015 ◽  
Vol 44 (12) ◽  
pp. 3954-3967 ◽  
Author(s):  
George R. Newkome ◽  
Charles N. Moorefield
Keyword(s):  
Self Assembly ◽  
Sierpinski Gasket ◽  
Building Blocks ◽  
The Self ◽  
Self Similarity ◽  
Sierpiński Gasket

The potential to incorporate dendritic characteristics, such as self-similarity into new fractal-based materials is exemplified in the self-assembly of novel, polyterpyridine-based, building blocks.

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