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Using the discrete version of finite pattern, we obtain the Wiener sum of fractal network modeled by the level-3 Sierpinski gasket.

We introduce a mathematical framework to describe self-similarity and structural recursion within the active tile self-assembly model, thereby providing a connection between substitution tiling and algorithmic self-assembly. We show that one such structurally recursive assembly system can simulate the dynamics of the self-similar substitution tiling known as the L-shape tiling.

A simple two-dimensional system is introduced which suggests a qualitative dynamical relationship between (1) stock market prices in the presence of nonlinear trend-followers and nonlinear value investors, (2) the world human population with a competition between a population-dependent growth rate and a nonlinear dependence on a finite carrying capacity and (3) the failure of materials subjected to a time-varying stress with a competition between positive geometrical feedback on the damage variable and nonlinear healing. Our model keeps three key ingredients (inertia, nonlinear positive and negative feedbacks) that compete to give rise to singularities in finite time decorated by accelerating oscillations.

We study quantum mechanical scattering by one-dimensional sequence of Gaussian barriers and calculate the transmission through such systems. We illustrate the physical aspects of resonance states and their coupling with the example of two and three barriers. Finally, we consider Fibonacci sequences of such barriers to demonstrate the properties of scaling, multifractality and self-similarity. These results are compared with those of periodic structures. All the calculations are based on the transfer matrix approach resulting from a fine segmentation of each barrier.

Applying the statistical hypothesis testing, we investigate several nonlinear properties embedded in the return series of the Chinese Fund Market (CFM), which suggests the series is non-normal, auto-correlative and heteroskedastic. We hereby analyze the Hurst exponent of the return series in different timescales on the basis of the detrended fluctuation analysis (DFA) algorithm, and discuss the fractal behavior of the CFM. Furthermore, by studying the correlation of different weights in the volatility, we find the persistent long-range power-law correlation exists in the time series. We also provide hints that the above statistical properties are insensitive to the funds kind, and may be irrelevant to the market phases. Our work may reveal the self-similarity characteristics of the financial market and show a better understanding of the CFM.

The fractality of complex networks is studied by estimating the correlation dimensions of the networks. Comparing with the previous algorithms of estimating the box dimension, our algorithm achieves a significant reduction in time complexity. For four benchmark cases tested, that is, the Escherichia coli (E. Coli) metabolic network, the Homo sapiens protein interaction network (H. Sapiens PIN), the Saccharomyces cerevisiae protein interaction network (S. Cerevisiae PIN) and the World Wide Web (WWW), experiments are provided to demonstrate the validity of our algorithm.

In this paper, we use momentum sum rule to compute the fractions of momentum carried by quarks and gluons in a self-similarity based model of proton. Comparison of the results with the prediction of QCD asymptotics is also made.

Momentum sum rule can be used as an inequality to estimate the lower and upper bounds of the momentum fractions of quarks and gluons in a model of proton valid in a limited x range. We compute such bounds in a self-similarity based model of proton structure function valid in the range 6.2 × 10^-7 ≤ x ≤ 10^-2. The results conform to the asymptotic QCD expectations.

In this paper, we revisit the classical mechanism to produce $d$-dimensional spacetimes via Kaluza–Klein compactification. We made the assumption that the $(d+1)$-dimensional bulk geometry ${ℳ}$ admits a Homothetic Vector Field (HVF) $H$ relaxing the existence of a zero i.e. $H\ne 0$ holds on ${ℳ}$ and the homothetic bivector $F_{ab}(H)=0$. Under these circumstances, we identify the origin of a positive cosmological constant in the $d$-dimensional spacetime as the homothetic factor representing a geometrical artifact of late time state of the $(d+1)$-dimensional bulk spacetime.

In this paper we show how to study under the self-similarity hypothesis a perfect fluid Bianchi I model with variables G and Λ, but under the condition div T≠0. We arrive to the conclusion that: G and Λ are decreasing time functions (the sign of Λ depends on the equation of state), while the exponents of the scale factor must satisfy the conditions and , ∀ω ∈ (-1, 1), relaxing in this way the Kasner conditions. We also show the connection between the behavior of G and the Weyl tensor.

Self-similarity of jet production in pp and collisions is studied in the framework of z-scaling. Inclusive jet transverse momentum distributions measured by the STAR Collaboration at RHIC, the CDF and D∅ Collaborations at Tevatron and the CMS and ATLAS Collaborations at LHC are analyzed. The experimental spectra are compared with next-to-leading order QCD calculations in p_T- and z-presentations. It is shown that self-similar features of jet cross-sections manifested by the z-scaling give strong restriction on the scaling function ψ(z) at high z. New results on energy and angular independence and asymptotic behavior of ψ(z) are discussed. The obtained results are considered as confirmation of self-similarity of jet production, fractality of hadron structure and locality of constituent interactions at small scales.

We construct a model for double parton distribution functions (dPDFs) based on the notion of self-similarity, pursued earlier for small x physics at HERA. The most general form of dPDFs contains total 13 parameters to be fitted from data of proton–proton collision at LHC. It is shown that the constructed dPDF does not factorize into two single PDFs in conformity with QCD expectation, and it satisfies the condition that at the kinematic boundary x₁+x₂ = 1 (where x₁ and x₂ are the longitudinal fractional momenta of two partons), the dPDF vanishes.

Sometime back, a self-similarity based model of the proton structure function at small $x$ was proposed by Lastovicka. We make reanalysis of this model with most recent HERA data. No significance difference with the earlier analysis is found. Both the analyses have singularity within the kinematical range of $x$: $0\le x\le 1$. We therefore study the model with the additional assumption that it should be singularity free, imposing positivity conditions on the model parameters. This results in a new model which is, however, phenomenologically valid only in a limited low $Q^2$ range. We therefore make further generalization of the defining self-similar unintegrated Parton Density Function (uPDF) and show that the with proper generalizations and initial conditions on them not only remove the undesired singularity but also results in a structure function with logarithmic growth in $Q^2$ closer to QCD. The phenomenological range of validity is then found to be much larger than the earlier versions. We also extrapolate the models to large $x$ in a parameter-free way. The possibility of incorporation of Transverse Momentum Dependent (TMD) PDF in this approach is explored as well.

Experimental data on transverse momentum spectra of strange particles $(K_S^0,K^{-},K^{\ast 0},\varphi ,\mathrm{\Lambda },{\mathrm{\Lambda }}^{\ast },{\mathrm{\Sigma }}^{\ast },{\mathrm{\Xi }}^{-},\mathrm{\Omega })$ produced in $pp$ collisions at $\sqrt{s}=200\mathrm{\text{GeV}}$ obtained by the STAR and PHENIX collaborations at RHIC are analyzed in the framework of $z$-scaling approach. The concept of the $z$-scaling is based on fundamental principles of self-similarity, locality, and fractality of hadron interactions at high energies. General properties of the data $z$-presentation are studied. Self-similarity of fractal structure of protons and fragmentation processes with strange particles is discussed. A microscopic scenario of constituent interactions developed within the $z$-scaling scheme is used to study the dependence of momentum fractions and recoil mass on the collision energy, transverse momentum and mass of produced inclusive particle, and to estimate the constituent energy loss. We consider that obtained results can be useful in study of strangeness origin, in searching for new physics with strange probes, and can serve for better understanding of fractality of hadron interactions at small scales.

In this paper, we report an analysis of partial momentum fractions carried by quarks and gluons in six alternative phenomenological models of proton structure function valid in limited small $x$ regions: $x_{a_i}\le x\le x_{b_i}$, $i=1$ to 6; the limits being determined by phenomenological range of validity in each model. Since the physics of small $x$ is not completely understood at this point, we have considered both self-similarity-based as well as QCD-based models. The procedure by which one can determine the applicability ranges in $x$ and $Q^2$ of the models is presented. We find that while the self-similarity-based models with linear rise in $Q^2$ has limited phenomenological ranges of validity, an improved version with linear rise in $log{Q}^2$ has a wider phenomenological range. We then compare the partial momentum fractions in all the small $x$ models. Our analysis implies that the role of small $x$ sea quarks in calculating the second moments of parton distribution is minor one. We have also made a comprehensive comparison of all the phenomenological models considered to the available perturbative QCD, lattice QCD as well as Ads/QCD results.

Multiparticle production processes provide valuable information about the mechanism of the conversion of the initial energy of projectiles into a number of secondaries by measuring their multiplicity distributions and their distributions in phase space. They therefore serve as a reference point for more involved measurements. Distributions in phase space are usually investigated using the statistical approach, very successful in general but failing in cases of small colliding systems, small multiplicities, and at the edges of the allowed phase space, in which cases the underlying dynamical effects competing with the statistical distributions take over. We discuss an alternative approach, which applies to the whole phase space without detailed knowledge of dynamics. It is based on a modification of the usual statistics by generalizing it to a superstatistical form. We stress particularly the scaling and self-similar properties of such an approach manifesting themselves as the phenomena of the log-periodic oscillations and oscillations of temperature caused by sound waves in hadronic matter. Concerning the multiplicity distributions we discuss in detail the phenomenon of the oscillatory behavior of the modified combinants apparently observed in experimental data.

Froissart bound implies that the total proton–proton cross-section (or equivalently proton structure function) cannot rise faster than ${log}^{2}s\sim {log}^2\frac{1}{x}$. Compatibility of such behavior with the notion of self-similarity in proton structure function was suggested by us sometime back. In the present work, we generalize and improve it further by considering more recent self-similarity based models of proton structure functions and compare with recent data as well as with the model of Block, Durand, Ha and McKay.

Fractal self-similarity of hadron interactions demonstrated by the $z$-scaling of inclusive spectra is studied. The scaling regularity reflects fractal structure of the colliding hadrons (or nuclei) and takes into account general features of fragmentation processes expressed by fractal dimensions. The self-similarity variable $z$ is a function of the momentum fractions $x_1$ and $x_2$ of the colliding objects carried by the interacting hadron constituents and depends on the momentum fractions $y_a$ and $y_b$ of the scattered and recoil constituents carried by the inclusive particle and its recoil counterpart, respectively. Based on entropy principle, new properties of the $z$-scaling concept are found. They are conservation of fractal cumulativity in hadron interactions and quantization of fractal dimensions characterizing hadron structure and fragmentation processes at a constituent level.

By resorting to measurements of physically characterizing observables of water samples perturbed by the presence of Nafion and by iterative filtration processes, we discuss their scale free, self-similar fractal properties. By use of algebraic methods, the isomorphism is proved between such self-similarity features and the deformed coherent state formalism.

Fractal generally has self-similarity. Using the self-similarity of fractal, we can obtain some important theories about complex networks. In this paper, we concern the Vicsek fractal in three-dimensional space, which provides a natural generalization of Vicsek fractal. Concretely, the Vicsek fractal in three-dimensional space is obtained by repeatedly removing equilateral cubes from an initial equilateral cube of unit side length, at each stage each remaining cube is divided into $n^3$ smaller cubes of which $(3n-2)$ are kept and the rest discarded, where $n$ is odd. In addition, we obtain the skeleton network of the Vicsek fractal in three-dimensional space. Then we focus on weighted average geodesic distance of the Vicsek fractal in three-dimensional space. Take $n=5$ as an example, we define a similar measure on the Vicsek fractal in three-dimensional space by weight vector and calculate the weighted average geodesic distance. At the same time, asymptotic formula of weighted average geodesic distance on the skeleton network is also obtained. Finally, the general formula of weighted average geodesic distance should be applicable to the models when $n$, the base of a power, is odd.