Adsorption studies of aromatic hydrocarbons of various molecular sizes on organo-clays in aqueous solution were carried out for characterizing the surface heterogeneity of organo-clays. Benzene, toluene, p-xylene, ethylbenzene and n-propylbenzene adsorption by a smectite with 5 different exchange degrees of trimethylphenylammonium (TMPA) cations for Ca2+ was measured. The Langmuir isotherm equation did not adequately describe the experimental data, especially for small molecules, whereas the Dubinin-Radushkevich (DR) equation combined with a gamma-type adsorption energy distribution function described all experimental data well, suggesting the surface and structural heterogeneity of TMPA-smectites. The calculated adsorption energy distributions indicated that the apparent heterogeneity depends on the molecular size of adsorbates. Small adsorbate molecules such as benzene explore a highly heterogeneous surface of TMPA-smectites while large molecules such as n-propylbenzene detect a relatively homogeneous surface. The surface fractal dimension was dependent on the extent of TMPA exchange for Ca2+. When TMPA content is less than 75% of the cation exchange capacity (CEC) of the smectite, the heterogeneity decreases as TMPA content increases; it increases with TMPA content thereafter. These results are related to the size distribution of micropores in TMPA-smectites, which are defined by the 2 semi-infinite aluminosilicate sheets and the interlayer cations. The micropore size distributions and, hence, heterogeneity are created in part by the inhomogeneity of the charge density of clay surfaces and the tendency for cation segregation in these systems.

Critical-zone reactions involve inorganic and biogenic colloids in a cation-rich environment. The present research defines the rates and structure of purified Mg-montmorillonite aggregates formed in the presence of monovalent (K+) and divalent (Ca2+, Mg 2+) cations using light-extinction measurements. Time evolution of turbidity was employed to determine early-stage aggregation rates. Turbidity spectra were used to measure the fractal dimension at later stages. The power law dependence of the stability ratios on cation concentration was found to vary with the reciprocal of the valence rather than the predicted reciprocal of valence-squared, indicating that the platelet structure may be a factor influencing aggregation rates. The critical coagulation concentrations (CCC) (3 mM for CaCl2, 4 mM for MgCl2, and 70 mM for KCl) were obtained from the stability ratios. At a later time and above a minimal cation concentration, turbidity reached a quasi-stable state, indicating the formation of large aggregates. Under this condition, an approximate turbidity forward-scattering correction factor was applied and the fractal dimension was determined from the extinction spectra. For the divalent cations, the fractal dimensions were 1.65 ± 0.3 for Ca2+ and 1.75 ± 0.3 for Mg2+ and independent of cation concentrations above the CCC. For the monovalent cation, the fractal dimension increased with K+ concentration from 1.35 to 1.95, indicating a transition to a face-to-face geometry from either an edge-to-edge or edge-to-face orientation.

The study of confined water dynamics in clay minerals is a very important topic in aluminosilicate-surface chemistry. Aluminosilicates are among the most technologically versatile materials in industry today. Dielectric spectroscopy is a very useful method for investigating the structure and dynamics of water adsorbed on solid matrix surfaces and water in the vicinity of ions in solutions. Use of this method for the study of clay minerals has been underutilized to date, however. The main goal of the present research was to understand the relaxation mechanisms of water molecules interacting with different hydration centers in clay minerals, with a view to eventually control this interaction. Two types of natural layered aluminosilicates (clay minerals) — montmorillonite with exchangeable K+, Co2+, and Ni2+ cations and kaolinite with exchangeable K+ and Ba2+ cations — were examined by means of dielectric spectroscopy over wide ranges of temperature (from -121°C to +300°C) and frequency (1 Hz–1 MHz). An analysis of the experimental data is provided in terms of four distributed relaxation processes. The low-temperature relaxation was observed only in montmorillonites and could be subdivided into two processes, each related to a specific hydration center. The cooperative behavior of water at the interface was observed in the intermediate temperature region, together with a proton percolation. The dielectric properties of ice-like and confined water structures in the layered clay minerals were compared with the dielectric response observed in porous glasses. The spatial fractal dimensions of the porous aluminosilicates were calculated by two separate methods — from an analysis of the fractality found in photomicrographs and from the dielectric response.

Bentonite, biotite, illite, kaolin, muscovite, vermiculite and zeolite were acidified or alkalized with HCl orNaOH of concentrations 0.0, 0.1, 1.0 and 5.0 mole dm−3 at room temperature for 2 weeks and converted into Ca homoionic forms. Low-temperature nitrogen and room-temperature water-vapor adsorption-desorption isotherms were used to characterize the mineral pores of radii between 1 and 30 nm. Nanopore volumes, size distributions, average radii and fractal dimensions were calculated. Values calculated from the nitrogen isotherms differed from those derived from water-vapor data. With an increase of the acid-treatment concentration, the pore volumes measured using both adsorption techniques increased markedly for all minerals. The pore radii measured from nitrogen isotherms appeared to decrease for all minerals except zeolite, while the pore radius calculated from water-vapor data increased in most cases. The fractal dimension measured from water vapor isotherms decreased in all cases indicating smoothing of the mineral surfaces and decrease in pore complexity. No well defined trends in any of the pore parameters listed above were noted under alkaline treatment. In the reaction of each mineral with acid and alkali treatments, the individual character of the mineral and the presence of impurities seems important.

We prove that, up to topological conjugacy, every Smale space admits an Ahlfors regular Bowen measure. Bowen’s construction of Markov partitions implies that Smale spaces are factors of topological Markov chains. The latter are equipped with Parry’s measure, which is Ahlfors regular. By extending Bowen’s construction, we create a tool for transferring the Ahlfors regularity of the Parry measure down to the Bowen measure of the Smale space. An essential part of our method uses a refined notion of approximation graphs over compact metric spaces. Moreover, we obtain new estimates for the Hausdorff, box-counting and Assouad dimensions of a large class of Smale spaces.

One of the main concerns about the fast spreading coronavirus disease 2019 (Covid-19) pandemic is how to intervene. We analysed severe acute respiratory syndrome-coronavirus-2 (SARS-CoV-2) isolates data using the multifractal approach and found a rich in viral genome diversity, which could be one of the root causes of the fast Covid-19 pandemic and is strongly affected by pressure and health index of the hosts inhabited regions. The calculated mutation rate (mr) is observed to be maximum at a particular pressure, beyond which mr maintains diversity. Hurst exponent and fractal dimension are found to be optimal at a critical pressure (Pm), whereas, for P > Pm and P < Pm, we found rich genome diversity relating to complicated genome organisation and virulence of the virus. The values of these complexity measurement parameters are found to be increased linearly with health index values.

Order, in the sense of arrangement, structure, coordination, or sequence, arises in many fields and at all scales. It is found with varying degrees of perfection in many if not most subjects for microscopy, can take many forms, and can be challenging to recognize or to measure. A variety of examples are provided that are intended to make the reader more aware of the possibilities.

In a high-level radioactive waste repository, bentonite may react with the alkaline solution produced by cement degradation. In this study, bentonite was mixed with alkaline solution in a closed system and reacted for 3–24 months. Furthermore, swelling tests were conducted on the alkaline-dissolved bentonite immersed in distilled water. The swelling deformation decreased significantly with increases in the concentration of NaOH solution and reaction time, and this was mainly due to montmorillonite dissolution. The fractal e–p relationship (e is the void ratio and p is the vertical pressure) with two calculation coefficients (the swelling coefficient and the fractal dimension) was employed to determine the swelling of alkaline-dissolved bentonite. The fractal dimension increased slightly with increasing reaction time or concentration of NaOH solution, as the dissolution traces caused by the alkaline solution favoured an increase in the irregularity and fractality of the bentonite surface. The swelling coefficient decreased linearly with decreasing montmorillonite content. In addition, the swelling coefficient and the fractal dimension were related exponentially to the reaction time in alkaline solution. A relationship between the swelling of alkaline-dissolved samples and the reaction time was proposed, which might be used to assess the swelling properties of bentonite barriers that would be affected by long-term dissolution of the alkaline solution in a closed repository.

Previous evidence suggested that lymphocytic thyroiditis (LT) was a variant of Hashimoto's thyroiditis (HT), thus the aim of the current study is to quantify structural changes in histological specimens taken from HT and LT patients. A total of 600 images containing a single lymphocyte nucleus (300 nuclei per group) were obtained from 20 patients with HT and LT. In order to quantify changes in the nuclear architecture of investigated lymphocytes, the fractal dimension (FD) and some gray-level co-occurrence matrix texture parameters (angular second moment, inverse difference moment, contrast, entropy, and correlation) were calculated for each nucleus. A statistically significant difference in the FD of the “binary-outlined” nucleus and that of the corresponding “black-and-white” nucleus was detected between HT and LT lymphocyte nuclei. In addition, there was also a statistically significant difference in contrast and correlation between HT and LT lymphocyte nuclei. In conclusion, the results of this study suggested that there was a difference in structural complexity between investigated lymphocyte nuclei; additionally, LT lymphocytes possessed probably more complex texture and larger variations as well as more asymmetrical nuclei compared with HT lymphocytes. Accordingly, these findings indicate that LT is probably not a variant of HT; however, more complex studies are necessary to estimate differences between these types of thyroiditis.

The effects of betaine on hepatocytes chromatin architecture changes were examined by using fractal and gray-level co-occurrence matrix (GLCM) analysis in methionine/choline-deficient (MCD) diet-induced, nonalcoholic fatty liver disease (NAFLD). Male C57BL/6 mice were divided into groups: (1) Control: standard diet; (2) BET: standard diet and betaine supplementation through drinking water (solution 1.5%); (3) MCD group: MCD diet for 6 weeks; (4) MCD+BET: fed with MCD diet + betaine for 6 weeks. Liver tissue was collected for histopathology, immunohistochemistry, and determination of fractal dimension and GLCM parameters. MCD diet induced diffuse micro- and macrovesicular steatosis accompanied with increased Ki67-positive hepatocyte nuclei. Steatosis and Ki67 immunopositivity were less prominent in the MCD+BET group compared with the MCD group. Angular second moment (ASM) and inverse difference moment (IDM) (textural homogeneity markers) were significantly increased in the MCD+BET group versus the MCD group (p<0.001), even though no difference between the MCD and the control group was evident. Heterogeneity parameters, contrast, and correlation were significantly increased in the MCD group versus the control (p<0.001). On the other hand, betaine treatment significantly reduced correlation, contrast, and entropy compared with the MCD group (p<0.001). Betaine attenuated MCD diet-induced NAFLD by reducing fat accumulation and inhibiting hepatocyte proliferation. Betaine supplementation increased nuclear homogeneity and chromatin complexity with reduction of entropy, contrast, and correlation.

Biomimetic hierarchical surface structures that exhibit features having multiple length scales have been used in many technological and engineering applications. Their surface topographies are most commonly analyzed using scanning electron microscopy (SEM), which only allows for qualitative visual assessments. Here we introduce fractal and lacunarity analyses as a method of characterizing the SEM images of hierarchical surface structures in a quantitative manner. Taking femtosecond laser-irradiated metals as an example, our results illustrate that, while the fractal dimension is a poor descriptor of surface complexity, lacunarity analysis can successfully quantify the spatial texture of an SEM image; this, in turn, provides a convenient means of reporting changes in surface topography with respect to changes in processing parameters. Furthermore, lacunarity plots are shown to be sensitive to the different length scales present within a hierarchical structure due to the reversal of lacunarity trends at specific magnifications where new features become resolvable. Finally, we have established a consistent method of detecting pattern sizes in an image from the oscillation of lacunarity plots. Therefore, we promote the adoption of lacunarity analysis as a powerful tool for quantitative characterization of, but not limited to, multi-scale hierarchical surface topographies.

Notoungulates, native South American fossil mammals, have been recently objective of several palaeoecological studies. Ecomorphology and biomechanics of the masticatory apparatus, together with micro and mesowear analyses on tooth enamel, were applied in order to understand their palaeobiology. In particular, the relationship between some dental traits (hypsodonty, occlusal surface area and complexity) and body mass is still poorly understood. These features were measured by means of the hypsodonty index (HI), occlusal surface area (OSA) and tooth area (OTA), enamel crest complexity (ECC) and length (OEL). The relationships between these indices were evaluated in five pan-contemporaneous Santacrucian Notoungulata genera from Patagonia: Adinotherium and Nesodon (Toxodontia), Interatherium, Protypotherium and Hegetotherium (Typotheria). While OSA, OTA and OEL were size dependent and strongly correlated, HI and ECC were size independent. All notoungulates analysed have very hypsodont teeth, indicating high rates of tooth wear in response to an increase of abrasives consumed with the food; their tooth occlusal area and complexity could be related to chewing efforts associated with the toughness of the plants consumed. HI, OSA and ECC were considered useful for palaeoecological reconstructions, but the results presented here show that these three features are integrated as a complex, so should not be evaluated separately.

Gaussian particles provide a flexible framework for modelling and simulating three-dimensional star-shaped random sets. In our framework, the radial function of the particle arises from a kernel smoothing, and is associated with an isotropic random field on the sphere. If the kernel is a von Mises-Fisher density, or uniform on a spherical cap, the correlation function of the associated random field admits a closed form expression. The Hausdorff dimension of the surface of the Gaussian particle reflects the decay of the correlation function at the origin, as quantified by the fractal index. Under power kernels we obtain particles with boundaries of any Hausdorff dimension between 2 and 3.

We investigated the structural complexity and texture of the cytoskeleton and nucleus in human mesenchymal stem cells during early phase differentiation into osteoblasts according to the differentiation–induction method: mechanical and/or chemical stimuli. For this, fractal dimension and a number of parameters utilizing the gray-level co-occurrence matrix (GLCM) were calculated based on single-cell images after confirmation of differentiation by immunofluorescence staining. The F-actin and nuclear fractal dimensions were greater in both stimulus groups compared with the control group. The GLCM values for energy and homogeneity were lower in fibers of the F-actin cytoskeleton, indicating a dispersed F-actin arrangement during differentiation. In the nuclei of both stimulus groups, higher values for energy and homogeneity were calculated, indicating that the chromatin arrangement was chaotic during the early phase of differentiation. It was shown and confirmed that combined stimulation with mechanical and chemical factors accelerated differentiation, even in the early phase. Fractal dimension analysis and GLCM methods have the potential to provide a framework for further investigation of stem cell differentiation.

The pullback asymptotic behavior of the solutions for 2D Nonau-tonomous G-Navier-Stokes equations is studied, and the existence of its L2-pullback attractors on some bounded domains with Dirichlet boundary conditions is investigated by using the measure of noncompactness. Then the estimation of the fractal dimensions for the 2D G-Navier-Stokes equations is given.

Tropical forests of the Amazon Basin are being rapidly converted to agricultural land uses and fallow land, resulting in accelerating rates of forest loss in one of the world's most biodiverse ecoregions. This process has been extensively described and modelled, but as yet there has been no formal test of how the spatial patterns of deforested and fragmented areas change with the spatial scale of forest clearings. It was hypothesised that different land-use practices are driving small and large clearings, with small-scale cultivators often creating small, irregularly shaped clearings and large-scale ranchers and soy farmers creating larger, more regular-shaped clearings. To quantitatively test this hypothesis, Mandelbrot's theory of fractals was applied to deforested areas in the Brazilian Amazon to test for scale-invariance in deforestation patterns. The spatial pattern of deforestation differed between small and large clearings, with the former creating more complex landscapes and with a threshold occurring at c. 1200 ha in area. As a consequence, the sizes and shapes of forest clearings, and hence the relative vulnerability of the remaining forest to edge, area and isolation effects, may differ systematically between landscapes with different deforestation drivers. Further tests of this hypothesis are needed to assess its efficacy in other tropical landscapes and geographical locations.

Scanning electron microscopy (SEM) is widely used in surface studies and continuous efforts are carried out in the search of estimators of different surface characteristics. By using the variogram, we developed two of these estimators that were used to characterize the surface roughness from the SEM image texture. One of the estimators is related to the crossover between fractal region at low scale and the periodic region at high scale, whereas the other estimator characterizes the periodic region. In this work, a full study of these estimators and the fractal dimension in two dimensions (2D) and three dimensions (3D) was carried out for emery papers. We show that the obtained fractal dimension with only one image is good enough to characterize the roughness surface because its behavior is similar to those obtained with 3D height data. We show also that the estimator that indicates the crossover is related to the minimum cell size in 2D and to the average particle size in 3D. The other estimator has different values for the three studied emery papers in 2D but it does not have a clear meaning, and these values are similar for those studied samples in 3D. Nevertheless, it indicates the formation tendency of compound cells. The fractal dimension values from the variogram and from an area versus step log–log graph were studied with 3D data. Both methods yield different values corresponding to different information from the samples.

The distribution and formation of foraging trails have largely been neglected as factors explaining harvesting patterns of leaf-cutting ants. We applied fractal analysis, circular, and conventional statistics to published and newly recorded trail maps of seven Atta colonies focusing on three aspects: permanence, spatio-temporal plasticity and colony life stage. In the long term, trail patterns of young and mature Atta colonies revealed that foraging activities were focused on distinct, static sectors that made up only parts of their potentially available foraging range. Within these foraging sectors, trails were typically ephemeral and highly variable in space and time. These ephemeral trails were concentrated around permanent trunk trails in mature and around nest entrances in young colonies. Besides these similarities, the comparison of trail systems between the two life stages indicated that young colonies exploited fewer leaf sources, used smaller and less-complex systems of foraging trails, preferred different life forms as host plants, and switched hosts more often compared with mature colonies. Based on these analyses, we propose a general hypothesis which describes the foraging pattern in Atta as a result of initial foraging experiences, spatio-temporal distribution of suitable host plants, energetic constraints, and other factors such as seasonality and interspecific predation.

A three-week algal meal supplementation of the cows' basal diet resulted in an increase in the firmness of milk fat crystallized isothermally at 5 °C for 24 h – the apparent elastic constant increased from 100 to 224 N/mm. This was accompanied by a decrease in solid fat content, from 47·7% to 44·4%. The crystallization behaviour of milk fat was also modified significantly. The rate constant of crystallization (Avrami constant) of the enriched milk fat at 19 °C was ∼20 times higher than that of control milk fat. A shorter induction time of nucleation was also observed in the temperature range [20, 27 °C]. These effects were attributed to a higher degree of supersaturation of the enriched milk fat. Enriched milk fat nucleated in a more stable β' polymorphic form at 5 °C, while control milk fat nucleated in the metastable α form, as determined by powder X-ray diffraction and differential scanning calorimetry. Changes in the microstructure of the material were observed by polarized light microscopy at 5 °C. The enriched milk fat displayed a greater amount of crystal clustering than the control. This effect was reflected in a decrease in the box-counting mass fractal dimension (Db) of the fat crystal network from 1·853 to 1·809. The decrease in Db closely predicted the observed 2·2-fold increase in the elastic constant of the fat. These changes in mechanical properties, crystallization behaviour and microstructure were driven by an increase in the 18[ratio ]1trans and a decrease in the 18[ratio ]1cis fatty acid content of the enriched milk fat.