Fractal Number Research Articles

Formal Grammars Generating Fractal Descriptions of Molecular Structures

Simple rewriting rules are used to produce alphanumeric strings that embed fractal number sequences and are directly translatable into descriptions of hydrocarbon structures of considerable complexity, featuring hierarchical schemes. Rotations of the alphanumeric strings lead to radical rearrangements of the corresponding structures, which lose their initial schemes and become much less predictable, featuring different topologies of polygonal cycles. This shows that a complex and not necessarily ordered molecular structure may nevertheless have a relatively low algorithmic complexity. The variety and versatility of reorganization in chemical topology, due to the nonlocal representation of bonds in the coding string, may have played a role in prebiotic chemistry.

Non-invasive ultrasound localization microscopy (ULM) in azoospermia: connecting testicular microcirculation to spermatogenic functions.

Rationale: Azoospermia is a significant reproductive challenge. Differentiating between non-obstructive azoospermia (NOA) and obstructive azoospermia (OA) is crucial as each type requires distinct management strategies. Testicular microcirculation plays a profound role in spermatogenic functions. However, current diagnostic methods are limited in their ability to effectively elucidate this crucial connection. Methods: We employed ultrasound localization microscopy (ULM) to visualize testicular microcirculation in NOA and OA patients and quantified the testicular hemodynamic parameters. Pearson correlation analysis was conducted to investigate the inner connection between parameters of testicular microcirculation and clinical spermatogenic functions. We conducted multiple logistic regression analysis to establish a new diagnostic model that integrates follicle-stimulating hormone (FSH) and mean vascular diameter to distinguish NOA from OA. Results: Our findings demonstrated significant differences in vascular parameters between NOA and OA, with NOA characterized by lower mean vascular diameter (p < 0.001), vessel density (p < 0.001), and fractal number (p < 0.001). Testicular volume showed a moderate positive correlation with mean vascular diameter (r = 0.419, p < 0.01) and vessel density (r = 0.415, p < 0.01); Mean vascular diameter exhibited negative correlations with both FSH (r = -0.214, p < 0.05) and age (r = -0.240, p < 0.05); FSH (r = -0.202, p < 0.05) and luteinizing hormone (LH) (r = -0.235, p < 0.05) were negatively correlated with mean blood flow velocity. The diagnostic model demonstrated an area under the curve (AUC) of 0.968. We also reported a method to map the vascular pressure distribution derived from the blood flow velocity generated by ULM. Conclusions: ULM provides a non-invasive and detailed assessment of testicular microvascular dynamics. The ULM-derived vascular parameters are able to connect testicular microcirculation to spermatogenic functions. The combination of FSH and mean vascular diameter enhances diagnostic precision and holds potential for distinguishing NOA from OA.

Complexity of Negation Domains, Swarm Optimization, Fractal Numbers and Experimentation

Complexity of Negation Domains, Swarm Optimization, Fractal Numbers and Experimentation

Complexity of Negation Domains, Swarm Optimization, Fractal Numbers and Experimentation

Complexity of Negation Domains, Swarm Optimization, Fractal Numbers and Experimentation

Fractals and Finite Distributions of Power Sets

In this paper, we are committed to investigating the fractal decomposition of power sets. Our main result is that every power set can be decomposed into a sum of a power set and an isomorphic set that does not intersect with it. For the finite power set, this property can be drawn on the ordinal line by constructing the fractal number axis of the ordinal line, and the fractal distribution and fractal graph of the finite power set can be obtained by using the parallel translation drawing method. Moreover, the distributions do not overlap or cross. The results in this paper provide a new perspective for further investigation of the fractal distribution of power sets.

An efficient surface profile measurement with the scanning path based on Hilbert and Gosper fractal curves

This paper proposes a non-contact efficient measurement method of surface morphology with the scanning path based on the Hilbert and Gosper fractal curves. By comparing the proposed two types of fractals scanning curves with the traditional reciprocating S-curves, it can be found that the plane distribution of measuring points of the fractal curves is much more uniform, and then the measurement performance is much improved. The two fractal curves with different iterative times are introduced as a measurement path of the surface morphology, and the local three-dimensional roughness and overall shape tolerances of the parts are determined. In the view of the measurement point uniformity, different total number of fractals of iterations are compared as the measurement path of the surface morphology. In order to verify the accuracy of the proposed method, a specialized measuring device is developed, and measurement verification is made with the optical profilometer.

Flow and heat transfer characteristics of a fractal microchannel heat sink with three‐dimensional fractal tail

AbstractIn order to solve the problem of weak heat transfer capacity on the peripheral part of the disk‐shaped fractal microchannel heat sink (FMHS), different from the traditional plane fractal, a higher‐order fractal was carried out at the end of the fractal microchannel, and the end of the microchannel was transformed into a three‐dimensional fractal to strengthen the heat transfer in the peripheral part of the FMHS. Different forms of bionic FMHSs were established based on the fractal number and fractal topology at the tail, and the flow and heat transfer characteristics of different fractals were studied. The results showed that with the increase of fractal number, the pressure drop is increased by 5.96–8.29%, the heat transfer effect is enhanced, the surface temperature of the FMHS gradually decreases, and the temperature uniformity is improved. For each topological relationship and fractal number, the heat transfer capacity of the four fractals microchannel with B‐type topological relationship is the strongest, and the comprehensive performance index can reach 1.29 (Re = 200).

Fractal Statistical Study on the Strength of Jointed Rock Mass

The failure probability and strength of jointed and fractured rock mass under water pressure action place important constraints on the safety and stability of rock mass engineering. In this study, two strength calculation models for rock mass joint rupture failure and shear slip failure are established according to different relative sizes and working conditions, and the effects of fractal dimension, water pressure, and number of cracks on the strength and failure probability of rock mass are analyzed. The most unfavorable dip angle under shear-slip conditions is calculated. The proposed models accurately reflect the failure probability and strength of jointed rock mass, and different strength calculation methods should be selected according to the specific working conditions. The effect of water pressure on the strength of jointed rock mass is sensitive to the failure mode. Reasonable drainage should be carried out according to the different failure modes when constructing tunnels with abundant groundwater, and certain drainage or water plugging construction schemes are not necessarily applicable to all cases. The dip angle at which joint fissures are most prone to shear slip is given as π/4 + φ/2, where φ is the jointed rock mass strength parameter. An increase in the fractal dimensions, external stress, and number of fractures will enhance the rock mass failure probability regardless of failure mode. Controlled blasting and advanced grouting reinforcement measures are recommended when a tunnel traverses through a joint fissure zone to reduce the number of cracks generated in the surrounding rock and thus improve its strength and stability.

Investigation of the relief geometry of the membrane of human buccal epithelium cells using atomic force microscopy methods

The analysis of the shape of the Brownian relief of the surface of human buccal epithelium cells was carried out using the method of fractal geometry. The scaling coefficients eta, zeta and the fractal dimension Df were used to estimate their actual areas Sc(fact) and linear characteristics lf. Keywords: atomic force microscopy, buccal epithelium, fractal number.

Analog programing of conducting-polymer dendritic interconnections and control of their morphology

Although materials and processes are different from biological cells’, brain mimicries led to tremendous achievements in parallel information processing via neuromorphic engineering. Inexistent in electronics, we emulate dendritic morphogenesis by electropolymerization in water, aiming in operando material modification for hardware learning. Systematic study of applied voltage-pulse parameters details on tuning independently morphological aspects of micrometric dendrites’: fractal number, branching degree, asymmetry, density or length. Growths time-lapse image processing shows spatial features to be dynamically dependent, and expand distinctively before and after conductive bridging with two electro-generated dendrites. Circuit-element analysis and impedance spectroscopy confirms their morphological control in temporal windows where growth kinetics is finely perturbed by the input frequency and duty cycle. By the emulation of one’s most preponderant mechanisms for brain’s long-term memory, its implementation in vicinity of sensing arrays, neural probes or biochips shall greatly optimize computational costs and recognition required to classify high-dimensional patterns from complex environments.

Detailed numerical simulation and experimental investigation on micromixers with serpentine Koch fractal microchannels

Micromixer is a kind of microfluidic chip for fast mixing and analysis. Mixing in a micromixer is usually a micron scale. At low Reynolds number, the fluid in the channel is laminar flow, which mainly depends on molecular diffusion as the main mixing mode. Fluid mixing in microchannels is very difficult, especially when the viscosity of the fluid is high. In this paper, we design a novel passive micromixer. The effects of fractal number, Koch fractal channel spacing, microchannel depth and cross-section shape on mixing efficiency were studied. Through a large number of numerical simulations, we continue to optimize the structure of the micromixer and improve the mixing efficiency. Finally, through the continuous optimization of the structure of the micromixer, the mixing efficiency of the micromixer can reach more than 95%.

Multi-objective optimization and multi-factors analysis of the thermal/hydraulic performance of the bionic Y-shaped fractal heat sink

Facing the challenge of high heat flux in electronic devices, effective heat dissipation with low consumption has drawn great attention. Based on the constructal theory, the optimal structures of fractal heat sinks are obtained by genetic algorithm (GA) optimization. The optimal branching level is 4. Furthermore, the Taguchi experimental design is employed to conduct the CFD simulation, and the grey analysis and variance analysis are used to investigate the influence rank and contribution of impact factors (branching levels, fractal number, heat flux, and flow rate) on the performance parameters (maximum temperature, pressure drop and comprehensive performance). It is found that the heat flux exhibits the highest contributions of 74.75% and 79.26% on the performance value of maximum temperature and comprehensive performance, respectively. For the pressure drop, the fractal number show the highest contribution of 67.99%. This work provides a novel method and guidance for the design and analysis of heat sinks.

Multi-objective optimizations on thermal and hydraulic performance of symmetric and asymmetric bionic Y-shaped fractal networks by genetic algorithm coupled with CFD simulation

With the high energy output of electronic devices, effective heat dissipation with low energy consumption has drawn great attention. Based on the construction theory, the flow resistance of the generalized bionic Y-shaped fractal networks is theoretically analyzed. It is found that the symmetrical width has a positive effect on reducing the hydraulic resistance, while the symmetrical length plays a negative influence. Then, the optimal designs of symmetrical and asymmetrical bionic Y-shaped fractal heat sinks are obtained by multi-objective optimization with genetic algorithm (GA). The pressure loss in the asymmetrical bionic Y-shaped fractal heat sink is always lower than that of the symmetrical bionic Y-shaped fractal heat sink at the same fractal number. As the constraint of thermal resistance reduces from 0.045 K/W to 0.042 K/W, the pumping power difference between symmetrical and asymmetrical bionic Y-shaped fractal heat sinks increases from 70.1 mW to 99.1 mW. Furthermore, according to the CFD investigation, it is noted that the symmetrical heat sink exhibits a lower temperature difference and more uniform temperature distribution.

Micromixer structure design based on Koch fractal principle and baffle distribution on the lower surface

In this paper, the effect of the baffle on the mixing efficiency is studied when the baffle is distributed on the lower surface of the microchannel. The mixing efficiency of the micromixer was analyzed by numerical simulation. To study the change of mixing efficiency of micromixer, we changed the fractal number of baffles, height of baffles, spacing of baffles, angle of baffles, the number of baffles. Through analysis and comparison, we finally chose a micromixer with fractal number of 2, baffle height of 0.2 mm, baffle spacing of 0.25 mm, baffle angle of 45° and baffle number of 15.

Investigation of convective heat transfer performance in nanochannels with fractal Cantor structures

The micro/nanochannel cooling technology is a promising and flourishing technique of thermal management to dissipate the heat of electronic devices. Here, we apply fractal Cantor structures to construct the self-affine rough surface of the nanochannel and investigate the effect of the fractal Cantor surface on the convective heat transfer performance in nanochannels by using molecular dynamics simulation. We first find out that fractal structures can accelerate the temperature development of the fluid and improve the convective heat transfer of nanochannels in comparison with those of smooth surfaces. With the increase of the fractal number (n) and the surface wettability, both the heat transfer and flow resistance further increase. The largest comprehensive convective heat transfer performance indicator is obtained at n = 3. The results show that the expansion of low potential energy regions along with the increasing fractal number makes more near-wall fluid atoms gather at the wall-fluid interface to act as “phonon bridge” to promote the convective heat transfer in nanochannels. Meanwhile, the generation of narrowest gaps in the fractal structure of n = 3 induces the local hydrophobic effect, which can ameliorate comprehensive thermal and flow characteristics of nanochannels.

Influence of dissolution on long-term frictional properties of carbonate fault gouge

Velocity stepping experiments have been performed on a simulated calcite gouge using an annular shear apparatus to investigate the effect of dissolution on the frictional properties of a carbonate fault. The tested material was put in contact with hydrochloric acid at different concentration in order to provoke grains dissolution prior to the experiments. Particle size analysis shows that the small grains tend to disappear due to the chemical reaction, whereas the distribution of large grains is not much affected. The dissolution process induces a decrease of the fractal number of the grain size distribution and an increase of the roughness of the particles. The study of the rate-and-state parameter of the materials, which is commonly used to characterize the ability of the fault to generate earthquakes, shows a decrease towards more negative values with dissolution for dry specimens and an increase for wet samples. Moreover, a decrease of the steady state friction coefficient is observed with dissolution for wet samples. This decrease of the friction would promote the triggering of slip along an existing fault after the injection of an acid fluid like in the cases of CO2 storage or acid gas disposal projects in a carbonate reservoir, but the increase of the rate and state parameter (a–b) of saturated specimens implies that the seismogenic potential of faults could potentially be reduced.

Chaotic model for COVID-19 growth factor

PurposeThe new COVID-19 pandemic caused by SARS-CoV-2 initiated in the world a large quarantine, due to the exponential capacity of the virus in spreading from human contact. In the present work, the dynamics of such spreading was evaluated by the analysis of the growth factor of the disease.MethodIt was applied the space phase of the time series, the detrended fluctuation analysis (DFA) of the series, and the fractal dimension of the space phase.ResultsIt was possible to notice a strange attractor in the space phase of the growth factor indicating that the process is a deterministic chaos. The value of the alpha coefficient by DFA showed to be less than 0.5, characteristic of anti-persistent long-range memory, in which large events alternate with small events, and vice-versa. The fractal dimension of the phase space set was a fractal number, between 1 and 2, another indicator that the growth factor of the disease is not random.ConclusionThis global analysis is pointing that the spread of COVID-19 is a deterministic chaos, with long-range memory. Understanding such dynamics may help the control of this disease.

Atomistic simulations of phonon behaviors in isotopically doped graphene with Sierpinski carpet fractal structure

Two-dimensional (2D) graphene monolayer has been attached importance because of the fantastic physical properties. In this work, we conduct the atomistic simulations to evaluate the phonon behaviors in isotopically doped graphene with Sierpinski Carpet (SC) fractal structure. The thermal conductivities (k) with different fractal numbers are calculated by molecular dynamics simulation. The relationship between the k and the fractal number from 0 to 8 shows a first decreasing and then stable trend. The maximum reduction ratio of the k in SC fractal structures is 52.37%. Afterwards, we utilize the molecular dynamics simulation, phonon wave packet simulation and lattice dynamics simulation to investigate the phonon density of states (PDOS), energy transmission coefficient (ETC), phonon group velocity and participation ratio (PR) in SC fractal structures. In SC fractal structures, the PDOS increases in the low frequency region and the G-band will soften with the enhanced fractal number. We also observe that the isotopic doping atoms can lead to continuous reflected waves in SC fractal structure regions. Moreover, phonon modes in SC fractal structures possess the lower ETCs, phonon group velocities and PRs in comparison with the pristine graphene monolayer. Therefore, we attribute the lower k in SC fractal structures to the stronger phonon-impurity scattering and the increasing localized phonon modes.

Enhanced Shielding Performance of Layered Carbon Fiber Composites Filled with Carbonyl Iron and Carbon Nanotubes in the Koch Curve Fractal Method.

Layered carbon fiber composites (CFC) with enhanced shielding effectiveness (SE) were prepared with mixed fillers of carbon nanotubes (CNTs) and carbonyl iron powders (CIPs) in the form of a Koch curve fractal. In the layered composite structure, glass fiber (GF) cloth was used in the wave–transmissive layer (WTL), and the carbon fiber (CF) cloth was used in the supporting layer (SL). Between WTL and SL, CNTs and CIPs were distributed in epoxy resin in the form of a Koch curve fractal to act as an absorbing layer (AL), and copper foil was used as a reflective layer (RL) and bonded at the bottom of the whole composites. The layered structure design and excellent interlayer interface integration obviously improved the SE performance of the CFC. The SE of different samples was investigated, and the results show that, with the increase in the number (n) of Koch curve fractals, the SE of the samples enhanced in the low frequency scope (1–5 GHz). The sample with n = 2 has the highest SE value of 73.8 dB at 2.3 GHz. The shielding performance of the fractal sample filled by CNTs and CIPs simultaneously has a comprehensive improvement in the whole scope of 1–18 GHz, especially for the sample with n = 2. The cumulative bandwidth value of the SE exceeding 55 dB is about 14.3 GHz, accounting for 85% of the whole frequency scope, indicating the composite fabricated in this paper is an electromagnetic shielding material with great prospect.

Fractal C60 fullerene aggregation: Equilibrium thermodynamics approach

Using dynamic light scattering (DLS) data we showed that the main physico-chemical parameters characterizing C60 fullerene aggregation in solution, viz. equilibrium aggregation constant and fractal number, may be obtained from equilibrium thermodynamics approach. We implemented the standard isodesmic model of aggregation for approximation of experimental particle size distribution and obtained the magnitude of fractal number, df, previously known from other physico-chemical methods. We propose that the suggested DLS-based approach may be applied as a method of analysis of fractal aggregation of other types of nanoparticles, such as silver, gold, or nanodiamonds, alternative to standard spectroscopic (SAXS/SANS/SLS) methods.