On post-critically finite self-similar sets, whose walk dimensions of diffusions are in general larger than 2, we find a sharp region where two classes of Besov spaces, the heat Besov spaces $B^{p,q}_\sigma (K)$ and the Lipschitz–Besov spaces $\Lambda ^{p,q}_\sigma (K)$, are identical. In particular, we provide concrete examples that $B^{p,q}_\sigma (K)=\Lambda ^{p,q}_\sigma (K)$ with $\sigma>1$. Our method is purely analytical, and does not involve heat kernel estimate.

Animal welfare assessment commonly involves behavioural and physiological measurements. Physiological measures have become increasingly sophisticated over the years, while behavioural measurements, for example duration or frequency, have changed little. Although these measures can undoubtedly contribute to our assessment of an animal's welfare status, a more complex analysis of behavioural sequences could potentially reveal additional and valuable information. One emerging methodology that could provide such information is fractal analysis, which calculates measures of complexity in continuous time series. Its previous application in medical physiology suggested that it could reveal ‘hidden’ information in a dataset beyond that identified by traditional analyses. Consequently, we asked if fractal analysis of behaviour might be a useful non-invasive measure of acute and chronic stress in laying hens and in pigs. Herein, we outline our work and briefly review some previous applications of fractal analysis to animal behaviour patterns. We successfully measured novel aspects of complexity in the behavioural organisation of hens and pigs and found that these were stress-sensitive in some circumstances. Although data collection is time consuming, the benefit of fractal analysis is that it can be applied to simple behavioural transitions, thereby reducing subjective interpretation to a minimum. Collectively, the work to date suggests that fractal analysis — by providing a novel measure of behavioural organisation — could have a role in animal welfare assessment. As a method for extracting extra information from behavioural data, fractal analysis should be more widely examined in animal welfare science.

The aim of this study is to determine the ability and consequent significance of fractal and lacunarity analysis together with computational morphometric and gray-level co-occurrence matrix (GLCM) analysis in detecting subtle initial UVB-induced chromatin and cytosolic changes in neutrophil granulocytes. In addition, the direction and potential significance of the observed changes is speculated. Feulgen-stained neutrophils are pictured and their digitalized images are analyzed in specialized software for digital image processing and ImageJ analysis. Significant statistical difference is observed (p<0.01) between UV-treated neutrophils and controls in the vast majority of cytoplasmic morphometric parameters. Nuclear GLCM analysis was very sensitive with just GLCM homogeneity as non-significant between UV-treated cells and controls (p>0.05). For other parameters there was mostly high statistical significance (p>0.05). Significant unmatched correlations were found as sensitive markers of early morphological changes in cells exposed to UV light. In addition, the correlation between nuclear area and entropy was determined and was highly significant (p<0.001). UVB light, due to its high absorbance by DNA molecules, leads to double behavior of the cells. On one hand, cells start to rearrange but on the other UV light starts very early to immediately damage the cell. All these processes are very subtle in their intensity and GLCM analysis and computational imaging methods based on fractal geometry, i.e. fractal and morphometric analysis, in particular their combination, are very sensitive for detecting and describing these early chromatin changes.

Apart from their effector functions in allergic disorders, tissue-resident mast cells (MC) are gaining recognition as initiators of inflammatory events through their distinctive ability to secrete many bioactive molecules harbored in cytoplasmic granules. Activation triggers mediator release through a regulated exocytosis named degranulation. MC activation is still substantiated by measuring systemic levels of MC-restricted mediators. However, identifying the anatomical location of MC activation is valuable for disease diagnosis. We designed a computer-assisted morphometric method based on image analysis of methylene blue (MB)-stained normal mouse skin tissue sections that quantitates actual in situ MC activation status. We reasoned MC cytoplasm could be viewed as an object featuring unique relative mass values based on activation status. Integrated optical density and area (A) ratios were significantly different between intact and degranulated MC (p<0.001). The examination of fractal characteristics is of translational diagnostic/prognostic value in cancer and readily applied to quantify cytoskeleton morphology and vasculature. Fractal dimension (D), a measure of their comparative space filling capacity and structural density, also differed significantly between intact and degranulated MC (p<0.001). Morphometric analysis provides a reliable and reproducible method for in situ quantification of MC activation status.

Hearts of mice with reduction of function mutation in STAT3 (SA/SA) develop fibrotic collagen foci and reduced systolic function with hypertension. This model was used to determine if fractal dimension and image analysis can provide a quantitative description of myocardial fibrosis using routinely prepared trichome-stained material. Collagen was characterized by relative density [integrated optical density/area (IOD/A)] and fractal dimension (D), an index of complexity. IOD/A of collagen in wild type mice increased with hypertension while D decreased, suggesting tighter collagen packing that could eventually stiffen the myocardium as in diastolic heart failure. Reduced STAT3 function caused modest collagen fibrosis with increased IOD/A and D, indicating more tightly packed, but more disorganized collagen than normotensive and hypertensive controls. Hypertension in SA/SA mice resulted in large regions where myocytes were lost and replaced by fibrotic collagen characterized by decreased density and increased disorder. This indicates that collagen associated with reparative fibrosis in SA/SA hearts experiencing hypertension was highly disorganized and more space filling. Loss of myocytes and their replacement by disordered collagen fibers may further weaken the myocardium leading to systolic heart failure. Our findings highlight the utility of image analysis in revealing importance of a cellular protein for normal and reparative extracellular matrix deposition.

Theories have been put forward on the etiology of sialoliths; however, a comprehensive understanding of their growth mechanisms is lacking. In an attempt to fill this gap, the current study has evaluated the internal architecture and growth patterns of a set of 30 independent specimens of sialoliths characterized at different scales by computed microtomography and electron microscopy. Tomography reconstructions showed cores in most of the sialoliths. The cores were surrounded by concentric or irregular patterns with variable degrees of mineralization. Regardless of the patterns, at finer scales the sialoliths consisted of banded and globular structures. The distribution of precipitates in the banded structures is compatible with a Liesegang–Ostwald phenomenon. On the other hand, the globular structures appear to arise from surface tension effects and to develop self-similar features as a result of a viscous fingering process. Electron diffraction patterns demonstrated that Ca- and P-based electrolytes crystallize in a structure close to that of hydroxyapatite. The organic matter contained sulfur with apparent origin from sulfated components of secretory material. These results cast new light on the mechanisms involved in the formation of sialoliths.

The current prognostic parameters, including tumor volume, biochemistry, or immunohistochemistry, are not sufficient to reflect the properties of cancer cells that distinguish them from normal cells. Our focus is to evaluate the effects of a combination of microtubule-polymerizing Taxol® and -depolymerizing colchicine on IAR20 PC1 liver cells by measuring the surface fractal dimension as a descriptor of two-dimensional vascular geometrical complexity. The fractal dimension offers a rapid means of assessing cell shape. Furthermore, we show correlations of fractal dimensions of cell contours with the latent factors from our previously employed cell shape analysis.

Tumors are supported by the development of a unique vascular bed. We used fractal dimension (Db) and image analysis to quantify differences in the complexity of the vasculature in normal intestinal submucosa and intestinal polyps. ApcMin/+ mice and wild-type mice were perfused with a curable latex compound, intestines sectioned, and images collected via confocal microscopy. The images were analyzed and area (A), perimeter (P), and integrated optical density (IOD) of the normal and tumor vascular beds were measured. The Db, a quantitative descriptor of morphological complexity, was significantly greater for the polyp vasculature from ApcMin/+ mice than controls. This indicates that the polyp microvasculature is more chaotic than that of the controls, while the IOD and average vascular density values displayed no differences. This suggests the mass of blood volume is equivalent in normal and polyp microvasculature. The lower vascular area-perimeter ratios expressed by the polyp microvasculature suggest it is composed of smaller, more tortuous vessels. These data demonstrate that fractal analysis is applicable for providing a quantitative description of vascular complexity associated with angiogenesis occurring in normal or diseased tissue. Application of Db, IOD, and average density provides a clearer quantification of the complex morphology associated with tissue microvasculature.

The respiratory emission of CO2 from roots is frequently proposed as an attractant that allows soil-dwelling insects to locate host plant roots, but this role has recently become less certain. CO2 is emitted from many sources other than roots, so does not necessarily indicate the presence of host plants, and because of the high density of roots in the upper soil layers, spatial gradients may not always be perceptible by soil-dwelling insects. The role of CO2 in host location was investigated using the clover root weevil Sitona lepidus Gyllenhall and its host plant white clover (Trifolium repens L.) as a model system. Rhizochamber experiments showed that CO2 concentrations were approximately 1000 ppm around the roots of white clover, but significantly decreased with increasing distance from roots. In behavioural experiments, no evidence was found for any attraction by S. lepidus larvae to point emissions of CO2, regardless of emission rates. Fewer than 15% of larvae were attracted to point emissions of CO2, compared with a control response of 17%. However, fractal analysis of movement paths in constant CO2 concentrations demonstrated that searching by S. lepidus larvae significantly intensified when they experienced CO2 concentrations similar to those found around the roots of white clover (i.e. 1000 ppm). It is suggested that respiratory emissions of CO2 may act as a ‘search trigger’ for S. lepidus, whereby it induces larvae to search a smaller area more intensively, in order to detect location cues that are more specific to their host plant.