Narrow Results

In this paper, in order to calculate the average path length for unweighted and weighted hierarchical networks, we define the sum of the distances from each node to the vertex and the bottom nodes. For the unweighted network, we show that the average path length grows with the size of $N_t$ as $ln(N_t)$. For the weighted network, we prove its weighted path length approaches a constant related to the weighting factor $r$ and parameter $m$. In particular, for $r=\frac{1}{m}$, the average weighted path length of the network tends to a specific value $\frac{4}{m}$.

In this paper, we discuss the average path length for a class of scale-free modular networks with deterministic growth. To facilitate the analysis, we define the sum of distances from all nodes to the nearest hub nodes and the nearest peripheral nodes. For the unweighted network, we find that whether the scale-free modular network is single-hub or multiple-hub, the average path length grows logarithmically with the increase of nodes number. For the weighted network, we deduce that when the network iteration $t$ tends to infinity, the average weighted shortest path length is bounded, and the result is independent of the connection method of network.

Let ${(t_n)}_{n\ge 0}$ be the well-known $\pm 1$ Thue–Morse sequence

The $p$-adic number field ${\mathbb{Q}}_p$ and the $p$-adic analogue of the complex number field ${ℂ}_p$ have a rich algebraic and geometric structure that in some ways rivals that of the corresponding objects for the real or complex fields. In this paper, we attempt to find and understand geometric structures of general sets in a $p$-adic setting. Several kinds of fractal measures and dimensions of sets in ${ℂ}_p$ are studied. Some typical fractal sets are constructed. It is worthwhile to note that there exist some essential differences between $p$-adic case and classical case.

The Fermat problem is a crucial topological issue corresponding to fractal networks. In this paper, we discuss the average Fermat distance (AFD) of the Vicsek polygon network and analyze structural properties. We construct the Vicsek polygon network based on Vicsek fractal in an iterative way. Given the structure of network, we present an elaborate analysis of the Fermat point under various situations. The special network structure allows a way to calculate the AFD based on average geodesic distance (AGD). Moreover, we introduce the Vicsek polygon fractal and calculate its AGD and AFD. Its relationship with the network enables us to deduce the above two indices of the network directly. The results show that both in network and fractal, the ratio of AFD and AGD tends to 3/2, which demonstrates that both of them can serve as indicators of small-world property of complex networks. In fact, in Vicsek polygon network, the AFD grows linearly with network order, implying that our evolving network does not possess the small-world property.

Let $K$ be a compact metric space and let $\phi :K\to K$ be continuous. We study a $C^{*}$-algebra ${{ℳ}{𝒞}}_{\phi }$ generated by all multiplication operators by continuous functions on $K$ and a composition operator $C_{\phi }$ induced by $\phi$ on a certain $L^2$ space. Let $\gamma =({\gamma }_1,\dots ,{\gamma }_n)$ be a system of proper contractions on $K$. Suppose that ${\gamma }_1,\dots ,{\gamma }_n$ are inverse branches of $\phi$ and $K$ is self-similar. We consider the Hutchinson measure ${\mu }^H$ of $\gamma$ and the $L^2$ space $L^2(K,{\mu }^H)$. Then we show that the $C^{*}$-algebra ${{ℳ}{𝒞}}_{\phi }$ is isomorphic to the $C^{*}$-algebra ${{𝒪}}_{\gamma }(K)$ associated with $\gamma$ under some conditions.

A new self-similar multibarrier system is proposed and used to study transmission of Dirac electrons in graphene. Such system is based on the scaling of the length and energy of the barriers. The use of self-similar structures allows us to compare the transmission in graphene and gallium arsenide (GaAs). The transmission coefficient for charge carriers in graphene shows a surprising scaling behavior structure, which is not seen in GaAs. The scaling properties are established as a function of three parameters: barrier’s energy, the length and the generation of the system.

The simplest infinite sequences that are not ultimately periodic are pure morphic sequences: fixed points of particular morphisms mapping single symbols to strings of symbols. A basic way to visualize a sequence is by a turtle curve: for every alphabet symbol fix an angle, and then consecutively for all sequence elements draw a unit segment and turn the drawing direction by the corresponding angle. This paper investigates turtle curves of pure morphic sequences. In particular, criteria are given for turtle curves being finite (consisting of finitely many segments), and for being fractal or self-similar: it contains an up-scaled copy of itself. Also space-filling turtle curves are considered, and a turtle curve that is dense in the plane. As a particular result we give an exact relationship between the Koch curve and a turtle curve for the Thue–Morse sequence, where until now for such a result only approximations were known.

We have studied sidebranching induced by fluctuations in dendritic growth by means of a phase-field model. We have considered a region where the linear theories are not valid and we have computed the contour length and the area of the dendrite at different distances from the tip. The dependence of the ratio of both magnitudes with the undercooling shows a behaviour in agreement with previous experiments. The derived scaling relation implies that dendrites are self-similar in the considered region.