Sparse coding theories suggest that the visual brain is optimized to encode natural visual stimuli to minimize metabolic cost. It is thought that images that do not have the same statistical properties as natural images are unable to be coded efficiently and result in visual discomfort. Conversely, artworks are thought to be even more efficiently processed compared to natural images and so are esthetically pleasing. This project investigated visual discomfort in uncomfortable images, natural scenes, and artworks using a combination of low-level image statistical analysis, mathematical modeling, and EEG measures. Results showed that the model response predicted discomfort judgments. Moreover, low-level image statistics including edge predictability predict discomfort judgments, whereas contrast information predicts the steady-state visually evoked potential responses. In conclusion, this study demonstrates that discomfort judgments for a wide set of images can be influenced by contrast and edge information, and can be predicted by our models of low-level vision, whilst neural responses are more defined by contrast-based metrics, when contrast is allowed to vary.

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.

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.

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.

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.

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.

In this paper, we study the family of algebraic K3 surfaces generated by the smooth intersection of a (1, 1) form and a (2, 2) form in ${\open P}^2\times{\open P}^2$ defined over ${\open C}$ and with Picard number 3. We describe the group of automorphisms ${\cal A}={\rm Aut}(V/{\open C})$ on V. For an ample divisor D and an arbitrary curve C0 on V, we investigate the asymptotic behavior of the quantity $N_{{\cal A}(C_0)}(t)=\#\{C \in {\cal A}(C_0): C\cdot D<t\}$. We show that the limit $\lim_{t\to \infty } {\log N_{{\cal A}(C\,)}(t)\over \log t}=\alpha\fleqno{}$ exists, does not depend on the choice of curve C or ample divisor D, and that .6515<α<.6538.

The fractal properties of premixed transient flames propagating downwards through a near-isotropic turbulent flow field in a fan-stirred cruciform burner were investigated. The long vertical section of the cruciform burner was used to provide a downward propagating premixed flame at 1 atm. The large horizontal vessel equipped with a pair of counter-rotating fans and perforated plates at each end was used to generate near-isotropic turbulence. Turbulent flame front images were obtained using high-speed laser sheet imaging for both methane-air and propane-air mixtures. The nondimensional turbulent intensity (u′/SL), Reynolds number based on the integral length scale, and turbulent Karlovitz number were varied from 1 to 10, from 698 to 6032, and from 0.05 to 1.43, respectively. Hundreds of runs for each experimental condition were carried out to obtain sufficient images of these turbulent transient flame fronts just in the central uniform region. These images were then processed to extract fractal dimension, inner and outer cutoffs using both the circle and the caliper methods. It was found that the mean fractal dimension is only 2.18, nearly independent of u′/SL, in support of recent Bunsen-flame results found by Gülder and his co-workers. This contradicts the findings of many previous studies in which the fractal dimension may approach asymptotically to a value of 2.33 when u′/SL > 3. The inner (εi) and outer (ε0) cutoffs are found to be nearly constant for all flames studied, where ε0 is an order of magnitude greater than εi and it is smaller than the integral length scale of unreacted turbulence. Finally, the present fractal characteristics cannot predict turbulent burning velocities correctly when the available fractal closure model was used, indicating a limit of the fractal analysis on prediction of turbulent burning velocities.

When fractal dimensions are estimated from an observed point pattern, there is some ambiguity as to the interpretation of the quantity being estimated. (The point pattern itself has dimension zero.) Two possible interpretations are described. In the first of these, the observation region is regarded as being held fixed, while observations accumulate with time. In this case, provided the process is stationary and ergodic in time, and the cumulants satisfy certain regularity constraints, the dimension estimates consistently estimate the Rényi moment dimensions of the marginal distribution in space. If the regularity constraints are not satisfied, then different limits can be obtained according to the manner in which the limits are taken.

In the second case, the process is regarded as being stationary and ergodic in its spatial component, time being held fixed. In this case the estimates provide consistent estimates of the initial power-law rates of growth of the moment measures of the Palm distribution, the estimates for successively higher Rényi dimensions estimating the growth rates for successively higher-order moment measures of the Palm distribution.

Several examples are given, to illustrate the different types of behaviour which may occur, including the case where the points are generated by a dynamical system.

This paper is the first part of a two-part survey reviewing some basic concepts and methods of the modern theory of dynamical systems. The survey is introduced by a preliminary discussion of the relevance of nonlinear dynamics and chaos for economics. We then discuss the dynamic behavior of nonlinear systems of difference and differential equations such as those commonly employed in the analysis of economically motivated models. Part I of the survey focuses on the geometrical properties of orbits. In particular, we discuss the notion of attractor and the different types of attractors generated by discrete- and continuous-time dynamical systems, such as fixed and periodic points, limit cycles, quasiperiodic and chaotic attractors. The notions of (noninteger) fractal dimension and Lyapunov characteristic exponent also are explained, as well as the main routes to chaos.

Development of mycelial cord systems of Phanerochaete velutina (DC.: Pers.) Parmasto from 4-cm3 inocula on a nutrient-depleted non-sterile soil was studied in laboratory microcosms using image analysis techniques. Cord systems were ‘baited’ after 13 d growth with either fresh, non-sterile 4-cm3 wood baits or control Perspex® blocks of the same contact area placed behind the foraging mycelial front. After 26 d growth, mycelial ‘patches’ arose by dedifferentiation of consolidated mycelial cords in both wood- and Perspex-baited cord systems. ‘Patches’ comprised fine, highly branched separate hyphae extending radially from points of aggregated hyphae in cords. ‘Patches’ and cords could be readily distinguished by image analysis and the areas covered by patches and cords could be measured and compared. Whilst the total hyphal cover of Perspex- and wood-baited systems did not differ significantly (P>0·05), patch cover in wood-baited systems was up to 10 times greater than in Perspex-baited systems. Patches were temporary structures, regressing more rapidly with age than mycelial cords. Patch development ceased after application of a nutrient solution which replenished phosphate levels in the soil. Wood-baited mycelial systems displayed significant developmental polarity (P[les ]0·05) of both total hyphal cover (patches plus cords) and hyphae in patches towards the ‘baited’ sector of cord systems after 42 d, which corresponded with peak patch development. However, significant (P[les ]0·05) developmental polarity of the mycelial systems along the bait-inoculum line could be detected 8 d before patch formation when assessed by fractal geometry. Radiotracer studies showed that mycelial patches were not sinks for supplied 32P, but that they were sites of increased nutrient uptake capacity compared with that of mycelial cords. We discuss the need for mycelial cord systems to balance allocation of mycelial biomass between the two essential processes of colonizing wood resource units, and the acquisition of soluble inorganic nutrients from soil.