Narrow Results

We study the number of acyclic orientations on the generalized two-dimensional Sierpinski gasket SG_2,b(n) at stage n with b equal to two and three, and determine the asymptotic behaviors. We also derive upper bounds for the asymptotic growth constants of SG_2,b and d-dimensional Sierpinski gasket SG_d.

We consider an unbiased random walk on a finite, nth generation Sierpinski gasket (or "tower") in d = 3 Euclidean dimensions, in the presence of a trap at one vertex. The mean walk length (or mean number of time steps to absorption) is given by the exact formula

The generalization of this formula to the case of a tower embedded in an arbitrary number d of Euclidean dimensions is also found, and is given by

This also establishes the leading large-n behavior that may be expected on general grounds, where N_n is the number of sites on the nth generation tower and is the spectral dimension of the fractal.

We discuss the behavior of the dynamic dimension exponents for families of fractals based on the Sierpinski gasket and carpet. As the length scale factor for the family tends to infinity, the lattice approximations to the fractals look more like the tetrahedral or cubic lattice in Euclidean space and the fractal dimension converges to that of the embedding space. However, in the Sierpinski gasket case, the spectral dimension converges to two for all dimensions. In two dimensions, we prove a conjecture made in the physics literature concerning the rate of convergence. On the other hand, for natural families of Sierpinski carpets, the spectral dimension converges to the dimension of the embedding Euclidean space. In general, we demonstrate that for both cases of finitely and infinitely ramified fractals, a variety of asymptotic values for the spectral dimension can be achieved.

We show that the restriction of an eigenfunction of the Laplacian on the Sierpinski Gasket (SG) to any segment inside the SG is monotone on finite pieces, i.e. there is a subdivision of the segment, such that the function is monotone on all subintervals.

The solution to a deceptively simple combinatorial problem on bit strings results in the emergence of a fractal related to the Sierpinski Gasket. The result is generalized to higher dimensions and applied to the study of global dynamics in Boolean network models of complex biological systems.

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).

In contrast to the classical situation, it is known that many Laplacian operators on fractals have gaps in their spectrum. This surprising fact means there can be no limit in the Weyl counting formula and it is a key ingredient in proving that the convergence of Fourier series on fractals can be better than in the classical setting. Recently, it was observed that the Laplacian on the Sierpinski gasket has the stronger property that there are intervals which contain no ratios of eigenvalues. In this paper we give general criteria for this phenomena and show that Laplacians on many interesting classes of fractals satisfy our criteria.

Piezoelectric ultrasonic transducers typically employ composite structures to improve their transmission and reception sensitivities. The geometry of the composite is regular with one dominant length scale and, since these are resonant devices, this dictates the central operating frequency of the device. In order to construct a wide bandwidth device it would seem natural therefore to utilize resonators that span a range of length scales. In this article we consider such a device and build a theoretical model to predict its performance. A fractal medium is used as this contains a wide range of length scales and yields to a renormalization approach. The propagation of an ultrasonic wave in this heterogeneous medium is then analyzed and used to construct expressions for the electrical impedance, and the transmission and reception sensitivities of this device as a function of the driving frequency. The results presented show a marked increase in the reception sensitivity of the device.

Many natural systems are irregular and/or fragmented, and have been interpreted to be fractal. An important parameter needed for modeling such systems is the fractal dimension, D. This parameter is often estimated from binary images using the box-counting method. However, it is not always apparent which fractal model is the most appropriate. This has led some researchers to report different D values for different phases of an analyzed image, which is mathematically untenable. This paper introduces a new method for discriminating between mass fractal, pore fractal, and Euclidean scaling in images that display apparent two-phase fractal behavior when analyzed using the traditional method. The new method, coined "bi-phase box counting", involves box-counting the selected phase and its complement, fitting both datasets conjointly to fractal and/or Euclidean scaling relations, and examining the errors from the resulting regression analyses. Use of the proposed technique was demonstrated on binary images of deterministic and stochastic fractals with known D values. Traditional box counting was unable to differentiate between the fractal and Euclidean phases in these images. In contrast, bi-phase box counting unmistakably identified the fractal phase and correctly estimated its D value. The new method was also applied to three binary images of soil thin sections. The results indicated that two of the soils were pore-fractals, while the other was a mass fractal. This outcome contrasted with the traditional box counting method which suggested that all three soils were mass fractals. Reclassification has important implications for modeling soil structure since different fractal models have different scaling relations. Overall, bi-phase box counting represents an improvement over the traditional method. It can identify the fractal phase and it provides statistical justification for this choice.

The average geodesic distance is concerned with complex networks. To obtain the limit of average geodesic distances on growing Sierpinski networks, we obtain the accurate value of integral in terms of average geodesic distance and self-similar measure on the Sierpinski gasket. To provide the value of integral, we find the phenomenon of finite pattern on integral inspired by the concept of finite type on self-similar sets with overlaps.

Piezoelectric ultrasonic transducers have the ability to act both as a receiver and a transmitter of ultrasound. Standard designs have a regular structure and therefore operate effectively over narrow bandwidths due to their single length scale. Naturally occurring transducers benefit from a wide range of length scales giving rise to increased bandwidths. It is therefore of interest to investigate structures which incorporate a range of length scales, such as fractals. This paper applies an adaptation of the Green function renormalization method to analyze the propagation of an ultrasonic wave in a series of pre-fractal structures. The structure being investigated here is the Sierpinski carpet. Novel expressions for the non-dimensionalized electrical impedance and the transmission and reception sensitivities as a function of the operating frequency are presented. Comparisons of metrics between three new designs alongside the standard design (Euclidean structure) and the previously investigated Sierpinski gasket device are performed. The results indicate a significant improvement in the reception sensitivity of the device, and improved bandwidth in both the receiving and transmitting responses.

In this paper, we examine the number of geodesics between two points of the Sierpinski Gasket ($S$) via code representations of the points and as a main result we show that the maximum number of geodesics between different two points with respect to the intrinsic metric on $S$ is five.

The divisible sandpile model is a growth model on graphs that was introduced by Levine and Peres [Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile, Potential Anal. 30(1) (2009) 1–27] as a tool to study internal diffusion limited aggregation. In this work, we investigate the shape of the divisible sandpile model on the graphical Sierpinski gasket $\mathsf{S}\mathsf{G}$. We show that the shape is a ball in the graph metric of $\mathsf{S}\mathsf{G}$. Moreover, we give an exact representation of the odometer function of the divisible sandpile.

It is of great interest to analyze geodesics in fractals. We investigate the structure of geodesics in $n$-dimensional Sierpinski gasket $F_n$ for $n\ge 3$, and prove that there are at most eight geodesics between any pair of points in $F_n$. Moreover, we obtain that there exists a unique geodesic for almost every pair of points in $F_n$.

For the self-affine measures ${\mu }_{M,D}$ generated by a diagonal matrix $M$ with entries $p_1,p_2,p_3\in \mathbb{Z}\setminus \{0,\pm 1\}$ and the digit set $D=\{{(0,0,0)}^t,{(1,0,0)}^t,{(0,1,0)}^t,{(0,0,1)}^t\}$, Li showed that there exists an infinite orthogonal exponential functions set in $L^2({\mu }_{M,D})$ if and only if at least two of the three numbers $p_1,p_2,p_3$ are even, and conjectured that there exist at most four mutually orthogonal exponential functions in $L^2({\mu }_{M,D})$ for other cases [J.-L. Li, Non-spectrality of self-affine measures on the spatial Sierpinski gasket, J. Math. Anal. Appl. 432 (2015) 1005–1017]. This conjecture was disproved by Wang and Li through constructing a class of the eight-element orthogonal exponential functions [Q. Wang and J.-L. Li, There are eight-element orthogonal exponentials on the spatial Sierpinski gasket, Math. Nachr. 292 (2019) 211–226]. In this paper, we will show that there are any number of orthogonal exponential functions in $L^2({\mu }_{M,D})$ if two of the three numbers $\left|p_1\right|,\left|p_2\right|,\left|p_3\right|$ are different odd and the other is even.

We consider criteria for the differentiability of functions with continuous Laplacian on the Sierpiński Gasket and its higher-dimensional variants ${\mathrm{\text{SG}}}_N$, $N>3$, proving results that generalize those of Teplyaev [Gradients on fractals, J. Funct. Anal. 174(1) (2000) 128–154]. When ${\mathrm{\text{SG}}}_N$ is equipped with the standard Dirichlet form and measure $\mu ,$ we show there is a full $\mu$-measure set on which continuity of the Laplacian implies existence of the gradient $\nabla u$, and that this set is not all of ${\mathrm{\text{SG}}}_N$. We also show there is a class of non-uniform measures on the usual Sierpiński Gasket with the property that continuity of the Laplacian implies the gradient exists and is continuous everywhere in sharp contrast to the case with the standard measure.

Let $I$ be the unit matrix and $D=\left\{\right.\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill \hfill \end{array}\right),\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \\ \hfill \hfill \end{array}\right),\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \\ \hfill \hfill \end{array}\right),\left(\begin{array}{c}\hfill -1\hfill \\ \hfill -1\hfill \\ \hfill \hfill \end{array}\right)\left\}\right.$. In this paper, we consider the self-similar measure ${\mu }_{\rho I,D}$ on ${ℝ}^2$ generated by the iterated function system ${\{{\varphi }_d(x)=\rho I(x+d)\}}_{d\in D}$ where $0<\left|\rho \right|<1$. We prove that there exists $\mathrm{\Lambda }$ such that $E_{\mathrm{\Lambda }}=\left\{e^{2\pi i<\lambda ,x>}:\lambda \in \mathrm{\Lambda }\right\}$ is an orthonormal basis for $L^2({\mu }_{\rho I,D})$ if and only if $\left|\rho \right|=1/(2q)$ for some integer $q>0$.

As a kind of classical fractal network models, Sierpinski Gasket and its extended network models have attracted the attention of many scholars because of their many applications in real-world networks. Based on the classical Sierpinski Gasket, this paper proposes a joint Sierpinski Gasket model, which can be used not only to study the topology and dynamic properties of Sierpinski Gasket itself, but also to study the Sierpinski Gasket under special conditions, such as the residual Sierpinski Gasket after damage. Since the mean time to absorption (MTA) on the network is a basic dynamic property related to the random walk, in this paper, we study the expression of MTA on the proposed general joint Sierpinski Gasket model, and find that the MTA is determined by the number of iterations $g$ and two variables determined by the mode variable $S$, which can be solved by matrix algorithm. Therefore, using the expression of MTA on the general joint Sierpinski Gasket model, we calculate the analytical expressions of the MTA on a general example of the joint Sierpinski Gasket model and three Sierpinski Gaskets with varying degrees of damage and do the corresponding numerical simulation to verify the correctness of the conclusion.

Under certain continuity conditions, we estimate upper and lower box dimensions of the graph of a function defined on the Sierpiński gasket. We also give an upper bound for Hausdorff dimension and box dimension of the graph of a function having finite energy. Further, we introduce two sets of definitions of bounded variation for a function defined on the Sierpiński gasket. We show that fractal dimension of the graph of a continuous function of bounded variation is $\frac{log3}{log2}.$ We also prove that the class of all bounded variation functions is closed under arithmetic operations. Furthermore, we show that every function of bounded variation is continuous almost everywhere in the sense of $\frac{log3}{log2}$-dimensional Hausdorff measure.

In this paper, we investigate the Zagreb eccentricity index of the level-$n$ Sierpinski gasket $K_n$. Based on the self-similarity and finite pattern, we compute the Zagreb eccentricity index of $K_n,$ which is the integral of square of eccentricity in terms of the self-similar singular measure.