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Prolonged periods of extreme heat and drought are changing forest health and forest structure. Loss of vitality, signs of dieback, and mortality of trees are reported in many regions of the world. There is a need for information on the quantification of drought effects on forest structure in order to detect drought stress at an early stage. Furthermore, forest structure is linked to growth processes that result from single tree and stand dynamics. In this study, we used mobile laser scanning to objectively assess forest structure in European beech (Fagus sylvatica L.) stands differently affected by drought. We used the box-dimension as a holistic measure of structural complexity that jointly describes both distribution and amount of plant material in a forest, and canopy cover as a measure for overall defoliation. We observed large canopy gaps in heavily damaged stands, but the overall structural complexity of the forest stands remained unchanged. However, when dividing the 3D forest stand model from mobile laser scanning into 5 m height layers, we noticed a “structural flip” with reduced structural complexity in the upper parts of the forests and increased structural complexity in the lower strata. This indicates that the forests are responding to drought with increased mortality of mature trees, resulting in increased light availability in the understory and consequently increased growth of the understory trees. The rather indifferent overall structural complexity of the investigated forests was due to the reorganization of structures by shifting the major foliage layer from top-to-bottom and is interpreted as a successful ecosystem-scale response to the drought events. However, it is unclear how these forests will respond to repeated droughts. The “structural flip” should therefore be regarded as an early warning signal pointing to increased ecosystem stress, though the stress level has not yet exceeded the adaptive capacity in the investigated forest.

Today, creating or maintaining forest structural complexity is a management paradigm in many countries due to the positive relationships between structural complexity and several forest functions and services. In this study, we tested whether the box-dimension (Db), a holistic and objective measure to describe the structural complexity of trees or forests, can be used to quantify the structural complexity of 14 European beech (Fagus sylvatica L.) dominated forest plots by means of mobile laser scanning (MLS). The goal of this study was to explore the potential of this approach for quantifying the effect of leaves (summer vs winter) and management (lately unmanaged vs managed) on forest structural complexity. The findings suggest that repeated measurements on the same site and at the same time yielded consistent results if the measuring scheme is standardized. The results also showed that standardized measurement protocols allowed quantifying differences in forest structural complexity due to season. The highest stand structural complexity was found in leaf-on condition during summer, with the complexity being significantly higher than in winter condition. Also, in case of our beech-dominated plots, managed forests were more complex in structure than formerly managed but now unmanaged forests. This study illustrates the potential of MLS for monitoring the changes in forest structural complexity and allows correcting stand structural information for seasonality.

Abstract Aboveground tree architecture is neither fully deterministic nor random. It is likely the result of mechanisms that balance static requirements and light‐capturing efficiency. Here, we used terrestrial laser scanning data to investigate the relationship between tree architecture, here addressed using the box‐dimension (Db), and the architectural benefit‐to‐cost ratio, the light availability, and the growth of trees. We detected a clear relationship between Db and the benefit‐to‐cost ratio for the tested three temperate forest tree species (Fagus sylvatica L., Fraxinus excelsior L., and Acer pseudoplatanus L.). In addition, we could also show that Db is positively related to the growth performance of several tropical tree species. Finally, we observed a negative relationship between the strength of competition enforced on red oak (Quercus rubra L.) trees and their Db. We therefore argue that Db is a meaningful and integrative measure that describes the structural complexity of the aboveground compartments of a plant as well as its relation to structural efficiency (benefit‐to‐cost ratio), productivity, and growing conditions (competition or availability of light).

We introduce the notion of uniform distribution on a metric compactum. The desired distribution is defined as the limit of a sequence of the classical uniform distributions on finite sets which are uniformly distributed on the compactum in the geometric sense. We show that a uniform distribution exists on the metrically homogeneous compacta and the canonically closed subsets of a Euclidean space whose boundary has Lebesgue measure zero. If a compactum (satisfying some metric constraints) admits a uniform distribution then so does its every canonically closed subset that has zero uniform measure of the boundary. We prove that compacta, admitting a uniform distribution, are dimensionally homogeneous in the sense of box-dimension. [ABSTRACT FROM AUTHOR]

Trees have a fractal-like branching architecture that determines their structural complexity. We used terrestrial laser scanning technology to study the role of foliage in the structural complexity of urban trees. Forty-five trees of three deciduous species, Gleditsia triacanthos, Quercus macrocarpa, Metasequoia glyptostroboides, were sampled on the Michigan State University campus. We studied their structural complexity by calculating the box-dimension (Db) metric from point clouds generated for the trees using terrestrial laser scanning, during the leaf-on and -off conditions. Furthermore, we artificially defoliated the leaf-on point clouds by applying an algorithm that separates the foliage from the woody material of the trees, and then recalculated the Db metric. The Db of the leaf-on tree point clouds was significantly greater than the Db of the leaf-off point clouds across all species. Additionally, the leaf removal algorithm introduced bias to the estimation of the leaf-removed Db of the G. triacanthos and M. glyptostroboides trees. The index capturing the contribution of leaves to the structural complexity of the study trees (the ratio of the Db of the leaf-on point clouds divided by the Db of the leaf-off point clouds minus one), was negatively correlated with branch surface area and different metrics of the length of paths through the branch network of the trees, indicating that the contribution of leaves decreases as branch network complexity increases. Underestimation of the Db of the G. triacanthos trees, after the artificial leaf removal, was related to maximum branch order. These results enhance our understanding of tree structural complexity by disentangling the contribution of leaves from that of the woody structures. The study also highlighted important methodological considerations for studying tree structure, with and without leaves, from laser-derived point clouds.

Aboveground tree architecture is neither fully deterministic nor random. It is likely the result of mechanisms that balance static requirements and light‐capturing efficiency. Here, we used terrestrial laser scanning data to investigate the relationship between tree architecture, here addressed using the box‐dimension (Db), and the architectural benefit‐to‐cost ratio, the light availability, and the growth of trees. We detected a clear relationship between Db and the benefit‐to‐cost ratio for the tested three temperate forest tree species (Fagus sylvatica L., Fraxinus excelsior L., and Acer pseudoplatanus L.). In addition, we could also show that Db is positively related to the growth performance of several tropical tree species. Finally, we observed a negative relationship between the strength of competition enforced on red oak (Quercus rubra L.) trees and their Db. We therefore argue that Db is a meaningful and integrative measure that describes the structural complexity of the aboveground compartments of a plant as well as its relation to structural efficiency (benefit‐to‐cost ratio), productivity, and growing conditions (competition or availability of light). [ABSTRACT FROM AUTHOR]

The unprecedented drought between 2018 and 2020 had a significant impact on European beech (Fagus sylvatica L.) forests in Central Europe. The role of different forest structures in mitigating drought stress remains controversial. This contentious debate prompted our study, in which we aimed to quantify the effect of forest structure on drought stress in beech forests in two ecoregions of northern Bavaria, Germany. Using a mobile laser scanner, we surveyed 240 plots in drought-stressed forests. We analyzed the responses of beech trees to the drought period through radial growth, wood-derived δ13C signal, and crown defoliation. Results revealed significant responses of beech forests in both regions to the drought event, including increased crown defoliation, reduced tree growth, and altered δ13C signatures compared to pre-drought conditions. Our results show a relationship between crown closure and crown defoliation in beech, suggesting an increased vulnerability of beech to drought in more open canopies. However, the potential for silvicultural intervention to mitigate drought stress, as measured by BAI and δ13C signal, appears limited. Neighboring trees and forest structure had little influence on average drought resistance. The δ13C signal showed minimal responsiveness to variations in canopy openness, as well as to distinctions between single and multi-layered forests. However, increased structural complexity within stands tended to increase resistance due to the compensatory effects of understory trees. Future forest management strategies could focus on promoting structural diversity, selecting resilient individuals but also actively enrich the forests with more drought-adapted species to increase the adaptive capacity of beech-dominated forests in the face of changing climate conditions. • We analyzed the response of 240 plots in European beech forests to the 2018-2020 drought period. • We found significant responses of beech forests to the drought event, but the responses were highly variable. • Sub-canopy trees were less affected by the drought event. • Silvicultural decisions can only mitigate drought events to a certain extent. • We emphasize the importance of promoting structural diversity but also tree species diversity through active enrichment with drought-tolerant species. [ABSTRACT FROM AUTHOR]

One of the main goals of modern silviculture is to emulate the structural complexity of old-growth forests. In this context, it is of advantage to identify a target state of structural complexity at the stand level and to analyze the spatial characteristics that led to the desired complexity of forest structures in primary forest references. In this study, we used 3D forest scenes captured by terrestrial laser scanning (TLS) to identify spatial patterns of structural complexity of differently managed and unmanaged European forests dominated by beech (Fagus sylvatica L.). We scanned in managed even-aged and uneven-aged stands, as well as in formerly managed forests (National Parks) and primary forests. For three different forest strata, representing the understory, the midstory, and the overstory of a forest stand, we determined the structural complexity mathematically using fractal analysis. Beyond that, we analyzed the density, as well as the horizontal and vertical distribution of plant material. For all three forest strata, we observed differences in structural complexity between the different forest types. Within the lower and middle strata, the investigated primary forests showed a random to regular distribution of plant material, as well as a complex understory structure as a result of pronounced natural decay. Compared to the primary forests, the managed uneven-aged stands showed quite similar spatial patterns of distribution of plant material, but on average a higher space occupation in the lower and middle forest stratum. Our results suggest that single tree or group selection cutting is a useful management tool to imitate old-growth structures of undisturbed beech-dominated forests.

We propose here three expansions of real numbers in [0,1). By using these expansions, we define three functions of Cantor type. We then determine the non-differentiability set of these functions and then we show that the dimension of these sets is all 0. [ABSTRACT FROM AUTHOR]

Relation between microstructures and electric properties of 3-mol%-yttria-doped zirconia (3Y-TZP)/Ba 0.5 Sr 0.5 Fe 12 O 19 composites prepared by two different methods were investigated. Ba 0.5 Sr 0.5 Fe 12 O 19 ferrite powders were firstly prepared by solid reaction and sol–gel process, respectively. They were then combined with 3Y-TZP. Finally, the mixture powders were molded and sintered by a conventional ceramic method. The electric properties of the composites were studied through the impedance and dielectric spectroscopy. A corresponding equivalent-circuit model including the constant phase element of the composites was obtained using Zview software. The grain boundary morphology of the both composites was analyzed using fractal theory. A mathematical evaluation of the grain boundary morphology of the both composites can be given by the box-dimension. The information of the microstructure features of the composites can be indirectly reflected by the obtained parameters of the equivalent-circuit model and dielectric loss peak. [ABSTRACT FROM AUTHOR]

Abstract: The frost crystals present needle-like shapes and dendrite growth patterns at the initial period of the frost formation on cold surfaces. They gradually turn into plane-like shapes with time disappear. After frost crystals cover the whole cold surface, the frost formation steps into the frost layer growth period. During this period, as the frost layer becomes thicker, its surface temperature can be high enough to result the frost crystals melt and fall down periodically. It is the growth pattern of the frost crystals that results in a multiple-step ascending tendency of the fractal dimension with time. The physical meaning of the fractal dimension and the volume fraction in describing the frost layer characteristics are also discussed in this paper and it is pointed out that it is necessary to introduce a fractal parameter in modeling frost formation process. The fractal dimension may well define the frost crystal shape and structure features; the volume fraction is directly proportional to the frost layer density, and has a very weak relation with the frost crystal shape and structure features. [Copyright &y& Elsevier]

The classical digital speckle, or digital image, correlation method of deformation measurement is based on gray level correlation between unformed and deformed digital images. The pattern of artificial random speckles and the natural textures on some object''s surfaces have fractal characteristics, and their fractal dimensions represent both gray and morph information. Furthermore, the fractal dimensions are stable feature parameters of the patterns. The digitized images of the patterns are confirmed to be also fractals. By this fact, a new method of displacement measurement is developed in the paper, based on the fractal dimensions correlation. The in-plane displacement fields of a body can be acquired. In order to verify the validity of the new method, an experiment has been designed and the results have been compared with those obtained from the classical digital image correlation method. The validity of the new method is not less than that of classical method. Further discussions about the traits and the developing vista of the method are given at the end. [Copyright &y& Elsevier]

The scale-invariant behavior of air pollutant concentration (APC) time structure was investigated by applying the box counting method to APC time series. One-year series of hourly average APC observations, including O3, CO, SO2, NO, NO2, and PM10 which were obtained from urban, traffic, and national park air monitoring station at Taipei (Taiwan), were transferred into a useful compact form through this method, namely, the box-dimension (DB)-threshold (Th) and critical scale (CS)-threshold (Th) plots. The validity of this approach was supported with the result that the practical implications of DB-Th (or CS-Th) plots could be interpreted in terms of traditional statistical parameters. Since the dependences of both DB and CS on the Th values were closely related to the variation of APC in time, they were used to characterize the temporal distribution of APC. The analysis confirmed the existence of scale invariance in those investigated APC time series. Moreover, the DB (CS) was shown to be a decreasing (increasing) function of the threshold level, implying multifractal characteristics, i.e. the weak and intense regions scale differently. Some practical applications based on the box counting method were also discussed. [Copyright &y& Elsevier]

Trees have a fractal-like branching architecture that determines their structural complexity. We used terrestrial laser scanning technology to study the role of foliage in the structural complexity of urban trees. Forty-five trees of three deciduous species, Gleditsia triacanthos, Quercus macrocarpa, Metasequoia glyptostroboides, were sampled on the Michigan State University campus. We studied their structural complexity by calculating the box-dimension (Db) metric from point clouds generated for the trees using terrestrial laser scanning, during the leaf-on and -off conditions. Furthermore, we artificially defoliated the leaf-on point clouds by applying an algorithm that separates the foliage from the woody material of the trees, and then recalculated the Db metric. The Db of the leaf-on tree point clouds was significantly greater than the Db of the leaf-off point clouds across all species. Additionally, the leaf removal algorithm introduced bias to the estimation of the leaf-removed Db of the G. triacanthos and M. glyptostroboides trees. The index capturing the contribution of leaves to the structural complexity of the study trees (the ratio of the Db of the leaf-on point clouds divided by the Db of the leaf-off point clouds minus one), was negatively correlated with branch surface area and different metrics of the length of paths through the branch network of the trees, indicating that the contribution of leaves decreases as branch network complexity increases. Underestimation of the Db of the G. triacanthos trees, after the artificial leaf removal, was related to maximum branch order. These results enhance our understanding of tree structural complexity by disentangling the contribution of leaves from that of the woody structures. The study also highlighted important methodological considerations for studying tree structure, with and without leaves, from laser-derived point clouds. [ABSTRACT FROM AUTHOR]

One of the main goals of modern silviculture is to emulate the structural complexity of old-growth forests. In this context, it is of advantage to identify a target state of structural complexity at the stand level and to analyze the spatial characteristics that led to the desired complexity of forest structures in primary forest references. In this study, we used 3D forest scenes captured by terrestrial laser scanning (TLS) to identify spatial patterns of structural complexity of differently managed and unmanaged European forests dominated by beech (Fagus sylvatica L.). We scanned in managed even-aged and uneven-aged stands, as well as in formerly managed forests (National Parks) and primary forests. For three different forest strata, representing the understory, the midstory, and the overstory of a forest stand, we determined the structural complexity mathematically using fractal analysis. Beyond that, we analyzed the density, as well as the horizontal and vertical distribution of plant material. For all three forest strata, we observed differences in structural complexity between the different forest types. Within the lower and middle strata, the investigated primary forests showed a random to regular distribution of plant material, as well as a complex understory structure as a result of pronounced natural decay. Compared to the primary forests, the managed uneven-aged stands showed quite similar spatial patterns of distribution of plant material, but on average a higher space occupation in the lower and middle forest stratum. Our results suggest that single tree or group selection cutting is a useful management tool to imitate old-growth structures of undisturbed beech-dominated forests. [ABSTRACT FROM AUTHOR]

The present paper proposes a reconstruction of the epidemic curves from the fractal interpolation point of view. Looking at the epidemic curves as fractal structures might be an efficient way to retrieve missing pieces of information due to insufficient testing and predict the evolution of the disease. A fractal approach of the epidemic curve can contribute to the assessment and modeling of other epidemics. On the other hand, we have considered the spread of the epidemic in countries like Romania, Italy, Spain, and Germany and analyzed the spread of the disease in those countries based on their fractal dimension., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2020 Elsevier Ltd. All rights reserved.)