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In this paper, we investigate the average geodesic distance on Sierpiński torus networks. We construct Sierpiński torus networks based on the classic Sierpiński carpet in an iterative way. By applying finite patterns on integrals, we deduce the exact value of the average geodesic distance of the Sierpiński torus. Furthermore, the asymptotic formula for the average geodesic distance of the corresponding networks can be obtained.

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.

Different fractality in hadron–hadron and e⁺e^- collisions and its experimental observation are reviewed. The discussions on related underlying dynamics are presented.

Self-similarity is a significant property for complex networks. Box coverage method and dimension calculation are vital tools to study the characteristic of complex networks. In this paper, we propose an outside-in (OSI) box covering method for the Sierpinski networks, and it is attested that this coverage algorithm is superior to CBB algorithm. In addition, we deduce the optimal box recurrence formula of weighted and unweighted networks theoretically, and the result is the same as that of the algorithm value. We also obtain the information dimension of weighted network, which verifies the validity and feasibility of our method.

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.

Nowadays, it is desirable to generate traffic which can reflect realistic network traffic environment for performance evaluation of network equipments. Existing traffic generation solutions mainly include special test equipments, software traffic generators and field programmable gate array (FPGA)-based traffic generators. However, special test equipments are generally too expensive, software traffic generators could not achieve high data rates and FPGA-based traffic generators mostly lack flexibility. This paper presents a novel traffic generation solution according to an aggregated process-based model to overcome the weakness of above methods. The traffic generator can generate real-time Poisson, two-state Markov-modulated Poisson process (MMPP-2) and self-similar traffic by hardware. The main structure of the traffic generator is presented and statistical properties of the generated traffic have been evaluated. Experiment results indicate that the proposed traffic generation solution can achieve better performance of the generated traffic compared with existing FPGA-based traffic generators while the required date rates can be up to Gbps line rate.

This work develops a preliminary method for coding random self-similar patterns as a series of numbers and investigates the corresponding algorithm to calculate the topological distance between starting point and the link in the generated fractal pattern from the code series. With reference to the wide range of stochastic property in natural patterns, a process for generating fractal patterns with various generating probabilities of the pattern links denoted as separately random self-similar generation or separately random fractal is proposed. To assess the adaptability of the process, the coding method is applied to the generation of a random self-similar river network and the corresponding algorithm for calculating topological distance of the links is used to determine the width function of the pattern. The width function-based geomorphologic instantaneous unit hydrograph (WF-GIUH) model is then applied to estimate the runoff of the Po-bridge watershed in northern Taiwan. The results show that the separately random self-similar generating algorithm can be implemented successfully to calculate hydrologic responses.

This work discusses the random self-similar tree generation by employing multiple basic patterns, and investigates the character code and standardization algorithm for multiple basic patterns. With reference to the wide range of various basic patterns in natural shapes, the general coding method and the corresponding algorithm to calculate the topological distance is developed for random self-similar tree with multiple basic patterns. To assess the adaptability of the process, the general coding method is applied to transfer the generated river network to a code series and the corresponding algorithm for calculating topological distance of the links is used to determine the width function of the pattern. Finally, the width-function based geomorphologic instantaneous unit hydrograph (WF-GIUH) model is then applied to estimate the runoff of the Po-bridge watershed in northern Taiwan. The results reveal that the random self-similar tree with multiple basic patterns proposed in this study can be implemented successfully to calculate hydrologic responses.

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.

The multifractal analysis of stochastic processes deals with the fine scale properties of the sample paths and seeks for some global scaling property that would enable extracting the so-called spectrum of singularities. In this paper, we establish bounds on the support of the spectrum of singularities. To do this, we prove a theorem that complements the famous Kolmogorov’s continuity criterion. The nature of these bounds helps us to identify the quantities truly responsible for the support of the spectrum. We then make several conclusions from this. First, specifying global scaling in terms of moments is incomplete due to possible infinite moments, both of positive and negative orders. The divergence of negative order moments does not affect the spectrum in general. On the other hand, infinite positive order moments make the spectrum of self-similar processes nontrivial. In particular, we show that the self-similar stationary increments process with the nontrivial spectrum must be heavy-tailed. This shows that for determining the spectrum it is crucial to capture the divergence of moments. We show that the partition function is capable of doing this and also propose a robust variant of this method for negative order moments.

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.

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 $\frac{4}{m}$.

This paper is devoted to the study of self-similar fractal rep tiles of the Euclidean planes ${ℝ}^2$ and ${ℝ}^3$ using integer matrices. We construct many examples appearing in the literature, as well as new examples by integer matrices. Several exotic and radially symmetric self-similar rep tiles and their variations are obtained with change of bases extending the work by Bandt and others. We also present a few examples of polyhedral self-similar rep tiles including variations.

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

In real world, many networks have self-similar structures at different length scales. In this paper, we propose a family of growing symmetrical networks based on toy-model from a deterministic initial graph. We aim to analyze its topological properties, such as the average path length, clustering coefficient and degree distribution, which uses iterative calculations and recursive methods.

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.

In this paper, we consider the networks modeled by several self-affine sets based on the Bedford–Mcmullen carpet. We calculate three properties of the networks, including the cumulative degree distribution, the average clustering coefficient and the average path length. We show that such networks have scale-free and small-world effects.

A two-dimensional (2D) steady boundary layer flow along with heat transfer of a self-similar biomagnetic fluid over a permeable moving flat plate has been taken into consideration in this work. The flow is contemplated to be embedded by a magnetic dipole of sufficient magnetic strength. Transpiration as well as movement along the wall is also regarded. By imposing the appropriate similarity technique, the governing equations are converted into a system of coupled nondimensional equations. An efficient numerical technique has been incorporated to solve these dimensionless coupled nonlinear ordinary differential equations. The existence of dual solutions along with their stability has been established with the consideration of stability analysis. We discovered from our analysis that two solutions exist (one stable and another unstable) for the arbitrary values of transpiration, movement velocity and biomagnetic interaction parameters on flow and physical parameters. The attained results are demonstrated graphically and in tabular form. For the validity of our numerical scheme, we compared our findings with others previously published and found significant agreement.

The Skyrme model is a geometric field theory and a quasilinear modification of the Nonlinear Sigma Model (Wave Maps). In this paper, we study the development of singularities for the equivariant Skyrme Model, in the strong-field limit, where the restoration of scale invariance allows us to look for self-similar blow-up behavior. After introducing the Skyrme Model and reviewing what’s known about formation of singularities in equivariant Wave Maps, we prove the existence of smooth self-similar solutions to the $5+1$-dimensional Skyrme Model in the strong-field limit, and use that to conclude that the solution to the corresponding Cauchy problem blows-up in finite time, starting from a particular class of everywhere smooth initial data.