Percolation of entropy functionals on cayley tree graphs as a method of order-disorder character diagnostics of complex structures

Yudin, V. V.; Titov, P. L.; Mikhalyuk, A. N.

doi:10.3103/s1062873809090202

Cited by 7 publications

(8 citation statements)

References 2 publications

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“…This means that the stochastic operator undergoes a stretching or contraction in accordance with the norm of the distribution. Analysis of the percolation problem in entropic and divergent terms constitutes the infodynamical part of our theory [11], [13], [15], [30], [32], [34]. We believe that just the percolation dependences of these infodynamical functionals and accordingly their critical indices will help to obtain the general method of solving the problem of identifying the order type of grid structures and lattices.…”

Section: Percolation Of Infodynamical Functionals On Ctgsmentioning

confidence: 99%

Entropic measure of the order-disorder character in lattice systems in the representation of coordination Cayley tree graphs

2010

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We systematically expound the infodynamical method for analyzing lattice and grid systems. We establish the logic and algorithm for mapping given objects to coordination Cayley tree graphs and present their main properties. Tree graphs of grid systems are complicated objects, and the principle of cluster-type simplicial decomposition can be used to study them. Based on a simplicial decomposition, we construct the enumerating structures, from which we construct entropy-type functionals. We pose the percolation problem on Cayley tree graphs in a nonconventional sense, which may be considered for both enumerating structures and their entropies. The corresponding entropy percolational dependences and their critical indices can be considered sufficiently universal measures of order in lattice systems. The simpliciality also implies an analogy with the fractality principle. We introduce three types of fractal characteristics and give analytic expressions for fractal dimensions for the tangential and streamer representations and for the Mandelbrot shell.

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Section: Percolation Of Infodynamical Functionals On Ctgsmentioning

confidence: 99%

Entropic measure of the order-disorder character in lattice systems in the representation of coordination Cayley tree graphs

2010

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“…If they are equal, then entropy is ultimate. Therefore, the entropy acts in such case as the perfection measure rather than the disorder one [13].…”

Section: Analysis Of the Ordering With The Use Of The Configurational...mentioning

confidence: 99%

“…It should be noted that during entropy calculation each nanotube is represented as a separate point located at the barycenter of its cross-section. Entropy was calculated by Vajda form [13][14][15].…”

Section: Analysis Of the Ordering With The Use Of The Configurational...mentioning

confidence: 99%

Ordering Сharacteristics of Titanium Dioxide Nanotubes in Anodic Coatings

Titov

Shchegoleva

Kondrikov

2019

KEM

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In this paper, the ordering of the arrays of TiO2 nanotubes obtained by the method of anodic oxidation in the fluoro-containing aqueous-nonaqueous electrolytes containing glycerine and surface-active materials is investigated. For analysis of ordering, the two-dimensional Fourier spectrum, do-it-yourself configurational geometrical entropy and section of the two-dimensional autocorrelation function were used. These characteristics allow us to identify a nature of ordering in sufficient detail and to obtain the preliminary quantitative assessments of this order. It is found that, in the systems of titanium-oxide nanotubes, the stable, almost correct short-range order is established within the first coordination sphere. Such order is similar to the amorphous ordering. At the same time, the ordering of nanotubes arrays differs in detail from the amorphous one in the greater expressiveness of the typical scale the sizes of which can be estimated using the Fourier spectra as well as autocorrelation function.

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“…As above, when studying the evolutionary properties of the GBFPIS, we apply the entropy formalism [9], [27]. One nontrivial application example is the method of order-disorder diagnostics for developed nanostructures of quartz and metal glasses, amorphous films, and quasicrystal mosaics in [30], [31], where we also expounded the method of critical exponents for entropy percolation dependences of these media [31]. It turned out that all these dependences on the coordinate Cayley tree graphs of lattice and grid systems of these media belong to the same hyperbolic class of dependences with the critical exponent not exceeding one.…”

Section: The Gbis Of the Square-octagonal Mosaic And Its Morphogenesismentioning

confidence: 99%

“…We use the entropy formalism [27], [30], [31] described in the preceding section. The evolution is realized through the bivector channel in our representation; on the other hand, it occurs on the operatorial orbits of the GBPIS.…”

Section: The Penrose System As the Proper Fibonacci Subsemigroupmentioning

confidence: 99%

The Fibonacci-Penrose semigroup formalism and morphogenetic synthesis of quasicrystal mosaics

Yudin¹,

Startzev²,

Permyakova³

2011

Theor Math Phys

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We construct the group bundle for the inverse Fibonacci semigroups in the axiomatic approach framework. The proper Fibonacci semigroup is the corresponding group bundle for the Penrose semigroups, which can be interpreted as the generating grammar of the morphogenetic synthesis of the pentasymmetric Penrose parquet in the numbers of tiling by golden rhombuses. This morphogenetic synthesis of the Penrose parquet satisfies the scaling principle. Parquet plates are not absolutely rigid, and the relations between their metric characteristics are governed by the golden section and other magic numbers. The characteristic form factors of three-level dual alphabets are the corresponding invariants. We realize the morphogenetic synthesis in the examples of square-octagonal and bihexagonal lattices. We consider cumulative properties of magic series and the evolutionary aspects of semigroup orbits in the entropy representation.

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Percolation of entropy functionals on cayley tree graphs as a method of order-disorder character diagnostics of complex structures

Cited by 7 publications

References 2 publications

Entropic measure of the order-disorder character in lattice systems in the representation of coordination Cayley tree graphs

Entropic measure of the order-disorder character in lattice systems in the representation of coordination Cayley tree graphs

Ordering Сharacteristics of Titanium Dioxide Nanotubes in Anodic Coatings

The Fibonacci-Penrose semigroup formalism and morphogenetic synthesis of quasicrystal mosaics

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