scholarly journals HORTON LAW IN SELF-SIMILAR TREES

Fractals ◽  
2016 ◽  
Vol 24 (02) ◽  
pp. 1650017 ◽  
Author(s):  
YEVGENIY KOVCHEGOV ◽  
ILYA ZALIAPIN

Self-similarity of random trees is related to the operation of pruning. Pruning [Formula: see text] cuts the leaves and their parental edges and removes the resulting chains of degree-two nodes from a finite tree. A Horton–Strahler order of a vertex [Formula: see text] and its parental edge is defined as the minimal number of prunings necessary to eliminate the subtree rooted at [Formula: see text]. A branch is a group of neighboring vertices and edges of the same order. The Horton numbers [Formula: see text] and [Formula: see text] are defined as the expected number of branches of order [Formula: see text], and the expected number of order-[Formula: see text] branches that merged order-[Formula: see text] branches, [Formula: see text], respectively, in a finite tree of order [Formula: see text]. The Tokunaga coefficients are defined as [Formula: see text]. The pruning decreases the orders of tree vertices by unity. A rooted full binary tree is said to be mean-self-similar if its Tokunaga coefficients are invariant with respect to pruning: [Formula: see text]. We show that for self-similar trees, the condition [Formula: see text] is necessary and sufficient for the existence of the strong Horton law: [Formula: see text], as [Formula: see text] for some [Formula: see text] and every [Formula: see text]. This work is a step toward providing rigorous foundations for the Horton law that, being omnipresent in natural branching systems, has escaped so far a formal explanation.

Fractals ◽  
2017 ◽  
Vol 25 (02) ◽  
pp. 1750021
Author(s):  
R. K. ASWATHY ◽  
SUNIL MATHEW

Self-similarity is a common tendency in nature and physics. It is wide spread in geo-physical phenomena like diffusion and iteration. Physically, an object is self-similar if it is invariant under a set of scaling transformation. This paper gives a brief outline of the analytical and set theoretical properties of different types of weak self-similar sets. It is proved that weak sub self-similar sets are closed under finite union. Weak sub self-similar property of the topological boundary of a weak self-similar set is also discussed. The denseness of non-weak super self-similar sets in the set of all non-empty compact subsets of a separable complete metric space is established. It is proved that the power of weak self-similar sets are weak super self-similar in the product metric and weak self-similarity is preserved under isometry. A characterization of weak super self-similar sets using weak sub contractions is also presented. Exact weak sub and super self-similar sets are introduced in this paper and some necessary and sufficient conditions in terms of weak condensation IFS are presented. A condition for a set to be both exact weak super and sub self-similar is obtained and the denseness of exact weak super self similar sets in the set of all weak self-similar sets is discussed.


Fractals ◽  
2018 ◽  
Vol 26 (05) ◽  
pp. 1850061
Author(s):  
CHUNTAI LIU

Self-similarity and Lipschitz equivalence are two basic and important properties of fractal sets. In this paper, we consider those properties of the union of Cantor set and its translate. We give a necessary and sufficient condition that the union is a self-similar set. Moreover, we show that the union satisfies the strong separation condition if it is of the self-similarity. By using the augment tree, we characterize the Lipschitz equivalence between Cantor set and the union of Cantor set and its translate.


Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 314
Author(s):  
Tianyu Jing ◽  
Huilan Ren ◽  
Jian Li

The present study investigates the similarity problem associated with the onset of the Mach reflection of Zel’dovich–von Neumann–Döring (ZND) detonations in the near field. The results reveal that the self-similarity in the frozen-limit regime is strictly valid only within a small scale, i.e., of the order of the induction length. The Mach reflection becomes non-self-similar during the transition of the Mach stem from “frozen” to “reactive” by coupling with the reaction zone. The triple-point trajectory first rises from the self-similar result due to compressive waves generated by the “hot spot”, and then decays after establishment of the reactive Mach stem. It is also found, by removing the restriction, that the frozen limit can be extended to a much larger distance than expected. The obtained results elucidate the physical origin of the onset of Mach reflection with chemical reactions, which has previously been observed in both experiments and numerical simulations.


Polymers ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1115
Author(s):  
Dmitry Zimnyakov ◽  
Marina Alonova ◽  
Ekaterina Ushakova

Self-similar expansion of bubble embryos in a plasticized polymer under quasi-isothermal depressurization is examined using the experimental data on expansion rates of embryos in the CO2-plasticized d,l-polylactide and modeling the results. The CO2 initial pressure varied from 5 to 14 MPa, and the depressurization rate was 5 × 10−3 MPa/s. The constant temperature in experiments was in a range from 310 to 338 K. The initial rate of embryos expansion varied from ≈0.1 to ≈10 µm/s, with a decrease in the current external pressure. While modeling, a non-linear behavior of CO2 isotherms near the critical point was taken into account. The modeled data agree satisfactorily with the experimental results. The effect of a remarkable increase in the expansion rate at a decreasing external pressure is interpreted in terms of competing effects, including a decrease in the internal pressure, an increase in the polymer viscosity, and an increase in the embryo radius at the time of embryo formation. The vanishing probability of finding the steadily expanding embryos for external pressures around the CO2 critical pressure is interpreted in terms of a joint influence of the quasi-adiabatic cooling and high compressibility of CO2 in the embryos.


1990 ◽  
Vol 4 (4) ◽  
pp. 447-460 ◽  
Author(s):  
Coastas Courcobetis ◽  
Richard Weber

Items of various types arrive at a bin-packing facility according to random processes and are to be combined with other readily available items of different types and packed into bins using one of a number of possible packings. One might think of a manufacturing context in which randomly arriving subassemblies are to be combined with subassemblies from an existing inventory to assemble a variety of finished products. Packing must be done on-line; that is, as each item arrives, it must be allocated to a bin whose configuration of packing is fixed. Moreover, it is required that the packing be managed in such a way that the readily available items are consumed at predescribed rates, corresponding perhaps to optimal rates for manufacturing these items. At any moment, some number of bins will be partially full. In practice, it is important that the packing be managed so that the expected number of partially full bins remains uniformly bounded in time. We present a necessary and sufficient condition for this goal to be realized and describe an algorithm to achieve it.


Author(s):  
Claudio Xavier Mendes dos Santos ◽  
Carlos Molina Mendes ◽  
Marcelo Ventura Freire

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.


Fractals ◽  
2010 ◽  
Vol 18 (03) ◽  
pp. 349-361 ◽  
Author(s):  
BÜNYAMIN DEMÍR ◽  
ALI DENÍZ ◽  
ŞAHIN KOÇAK ◽  
A. ERSIN ÜREYEN

Lapidus and Pearse proved recently an interesting formula about the volume of tubular neighborhoods of fractal sprays, including the self-similar fractals. We consider the graph-directed fractals in the sense of graph self-similarity of Mauldin-Williams within this framework of Lapidus-Pearse. Extending the notion of complex dimensions to the graph-directed fractals we compute the volumes of tubular neighborhoods of their associated tilings and give a simplified and pointwise proof of a version of Lapidus-Pearse formula, which can be applied to both self-similar and graph-directed fractals.


2013 ◽  
Vol 732 ◽  
pp. 150-165 ◽  
Author(s):  
Harm J. J. Jonker ◽  
Maarten van Reeuwijk ◽  
Peter P. Sullivan ◽  
Edward G. Patton

AbstractThe deepening of a shear-driven turbulent layer penetrating into a stably stratified quiescent layer is studied using direct numerical simulation (DNS). The simulation design mimics the classical laboratory experiments by Kato & Phillips (J. Fluid Mech., vol. 37, 1969, pp. 643–655) in that it starts with linear stratification and applies a constant shear stress at the lower boundary, but avoids sidewall and rotation effects inherent in the original experiment. It is found that the layers universally deepen as a function of the square root of time, independent of the initial stratification and the Reynolds number of the simulations, provided that the Reynolds number is large enough. Consistent with this finding, the dimensionless entrainment velocity varies with the bulk Richardson number as$R{i}^{- 1/ 2} $. In addition, it is observed that all cases evolve in a self-similar fashion. A self-similarity analysis of the conservation equations shows that only a square root growth law is consistent with self-similar behaviour.


2014 ◽  
Vol 11 (S308) ◽  
pp. 542-545 ◽  
Author(s):  
S. Nadathur ◽  
S. Hotchkiss ◽  
J. M. Diego ◽  
I. T. Iliev ◽  
S. Gottlöber ◽  
...  

AbstractWe discuss the universality and self-similarity of void density profiles, for voids in realistic mock luminous red galaxy (LRG) catalogues from the Jubilee simulation, as well as in void catalogues constructed from the SDSS LRG and Main Galaxy samples. Voids are identified using a modified version of the ZOBOV watershed transform algorithm, with additional selection cuts. We find that voids in simulation areself-similar, meaning that their average rescaled profile does not depend on the void size, or – within the range of the simulated catalogue – on the redshift. Comparison of the profiles obtained from simulated and real voids shows an excellent match. The profiles of real voids also show auniversalbehaviour over a wide range of galaxy luminosities, number densities and redshifts. This points to a fundamental property of the voids found by the watershed algorithm, which can be exploited in future studies of voids.


2017 ◽  
Vol 284 (1846) ◽  
pp. 20162395 ◽  
Author(s):  
Kohei Koyama ◽  
Ken Yamamoto ◽  
Masayuki Ushio

Lognormal distributions and self-similarity are characteristics associated with a wide range of biological systems. The sequential breakage model has established a link between lognormal distributions and self-similarity and has been used to explain species abundance distributions. To date, however, there has been no similar evidence in studies of multicellular organismal forms. We tested the hypotheses that the distribution of the lengths of terminal stems of Japanese elm trees ( Ulmus davidiana ), the end products of a self-similar branching process, approaches a lognormal distribution. We measured the length of the stem segments of three elm branches and obtained the following results: (i) each occurrence of branching caused variations or errors in the lengths of the child stems relative to their parent stems; (ii) the branches showed statistical self-similarity; the observed error distributions were similar at all scales within each branch and (iii) the multiplicative effect of these errors generated variations of the lengths of terminal twigs that were well approximated by a lognormal distribution, although some statistically significant deviations from strict lognormality were observed for one branch. Our results provide the first empirical evidence that statistical self-similarity of an organismal form generates a lognormal distribution of organ sizes.


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