Attractors of directed graph IFSs that are not standard IFS attractors and their Hausdorff measure

For directed graph iterated function systems (IFSs) defined on R, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known

Parametrization of global attractors experimental observations and turbulence

This paper is concerned with rigorous results in the theory of turbulence and fluid flow. While derived from the abstract theory of attractors in infinite-dimensional dynamical systems, they shed some light on the conventional heuristic theories of turbulence, and can be used to justify a well-known experimental method. Two results are discussed here in detail, both based on parametrization of the attractor. The first shows that any two fluid flows can be distinguished by a sufficient number of point observations of the velocity. This allows one to connect rigorously the dimension of the attractor with the Landau–Lifschitz ‘number of degrees of freedom’, and hence to obtain estimates on the ‘minimum length scale of the flow’ using bounds on this dimension. While for two-dimensional flows the rigorous estimate agrees with the heuristic approach, there is still a gap between rigorous results in the three-dimensional case and the Kolmogorov theory. Secondly, the problem of using experiments to reconstruct the dynamics of a flow is considered. The standard way of doing this is to take a number of repeated observations, and appeal to the Takens time-delay embedding theorem to guarantee that one can indeed follow the dynamics ‘faithfully’. However, this result relies on restrictive conditions that do not hold for spatially extended systems: an extension is given here that validates this important experimental technique for use in the study of turbulence. Although the abstract results underlying this paper have been presented elsewhere, making them specific to the Navier–Stokes equations provides answers to problems particular to fluid dynamics, and motivates further questions that would not arise from within the abstract theory itself

Generalised dimensions of measures on almost self-affine sets

We establish a generic formula for the generalised q-dimensions of measures supported by almost self-affine sets, for all q>1. These q-dimensions may exhibit phase transitions as q varies. We first consider general measures and then specialise to Bernoulli and Gibbs measures. Our method involves estimating expectations of moment expressions in terms of `multienergy' integrals which we then bound using induction on families of trees

Emergent behavior of soil fungal dynamics:influence of soil architecture and water distribution

Macroscopic measurements and observations in two-dimensional soil-thin sections indicate that fungal hyphae invade preferentially the larger, air-filled pores in soils. This suggests that the architecture of soils and the microscale distribution of water are likely to influence significantly the dynamics of fungal growth. Unfortunately, techniques are lacking at present to verify this hypothesis experimentally, and as a result, factors that control fungal growth in soils remain poorly understood. Nevertheless, to design appropriate experiments later on, it is useful to indirectly obtain estimates of the effects involved. Such estimates can be obtained via simulation, based on detailed micron-scale X-ray computed tomography information about the soil pore geometry. In this context, this article reports on a series of simulations resulting from the combination of an individual-based fungal growth model, describing in detail the physiological processes involved in fungal growth, and of a Lattice Boltzmann model used to predict the distribution of air-liquid interfaces in soils. Three soil samples with contrasting properties were used as test cases. Several quantitative parameters, including Minkowski functionals, were used to characterize the geometry of pores, air-water interfaces, and fungal hyphae. Simulation results show that the water distribution in the soils is affected more by the pore size distribution than by the porosity of the soils. The presence of water decreased the colonization efficiency of the fungi, as evinced by a decline in the magnitude of all fungal biomass functional measures, in all three samples. The architecture of the soils and water distribution had an effect on the general morphology of the hyphal network, with a "looped" configuration in one soil, due to growing around water droplets. These morphologic differences are satisfactorily discriminated by the Minkowski functionals, applied to the fungal biomass

Control of pore geometry in soil microcosms and its effect on the growth and spread of <i>Pseudomonas </i>and <i>Bacillus</i> sp.

Simplified experimental systems, often referred to as microcosms, have played a central role in the development of modern ecological thinking on issues ranging from competitive exclusion to examination of spatial resources and competition mechanisms, with important model-driven insights to the field. It is widely recognized that soil architecture is the key driver of biological and physical processes underpinning ecosystem services, and the role of soil architecture and soil physical conditions is receiving growing interest. The difficulty to capture the architectural heterogeneity in microcosms means that we typically disrupt physical architecture when collecting soils. We then use surrogate measures of soil architecture such as aggregate size distribution and bulk-density, in an attempt to recreate conditions encountered in the field. These bulk-measures are too crude and do not describe the heterogeneity at microscopic scales where microorganisms operate. In the current paper we therefore ask the following questions: (i) To what extent can we control the pore geometry at microscopic scales in microcosm studies through manipulation of common variables such as density and aggregate size?; (ii) What is the effect of pore geometry on the growth and spread dynamics of bacteria following introduction into soil? To answer these questions, we focus on Pseudomonas sp. and Bacillus sp. We study the growth of populations introduced in replicated microcosms packed at densities ranging from 1.2 – 1.6 g cm-3, as well as packed with different aggregate sizes at identical bulk-density. We use X-ray CT and show how pore geometrical properties at microbial scales such as connectivity and solid-pore interface area, are affected by the way we prepare microcosms. At a bulk-density of 1.6 g cm-3 the average number of Pseudomonas was 63% lower than at a bulk-density of 1.3 g cm-3. For Bacillus this reduction was 66 %. Depending on the physical conditions, bacteria in half the samples took between 1.62 and 9.22 days to spread 1.5 cm. Bacillus did spread faster than Pseudomonas and both did spread faster at a lower bulk-density. Our results highlight the importance that soil physical properties be considered in greater detail in soil microbiological studies than is currently the case

An ERTS-1 investigation for Lake Ontario and its basin

The author has identified the following significant results. Methods of manual, semi-automatic, and automatic (computer) data processing were evaluated, as were the requirements for spatial physiographic and limnological information. The coupling of specially processed ERTS data with simulation models of the watershed precipitation/runoff process provides potential for water resources management. Optimal and full use of the data requires a mix of data processing and analysis techniques, including single band editing, two band ratios, and multiband combinations. A combination of maximum likelihood ratio and near-IR/red band ratio processing was found to be particularly useful

Combining X-ray CT and 3D printing technology to produce microcosms with replicable, complex pore geometries

Measurements in soils have been traditionally used to demonstrate that soil architecture is one of the key drivers of soil processes. Major advances in the use of X-ray Computed Tomography (CT) afford significant insight into the pore geometry of soils, but until recently no experimental techniques were available to reproduce this complexity in microcosms. This article describes a 3D additive manufacturing technology that can print physical structures with pore geometries reflecting those of soils. The process enables printing of replicated structures, and the printing materials are suitable to study fungal growth. This technology is argued to open up a wealth of opportunities for soil biological studies

A Two-sided-Loop X-Ray Solar Coronal Jet and a Sudden Photospheric Magnetic-field Change, Both Driven by a Minifilament Eruption

Most of the commonly discussed solar coronal jets are of the type consisting of a single spire extending approximately vertically from near the solar surface into the corona. Recent research of a substantial number of events shows that eruption of a miniature filament (minifilament) drives at least many such single-spire jets, and concurrently generates a miniflare at the eruption site. A different type of coronal jet, identified in X-ray images during the Yohkoh era, are two-sided-loop jets, which extend from a central excitation location in opposite directions, along two opposite low-lying coronal loops that are more-or-less horizontal to the surface. We observe such a two-sided-loop jet from the edge of active region (AR) 12473, using data from Hinode XRT and EIS, and SDO AIA and HMI. Similar to single-spire jets, this two-sided-loop jet results from eruption of a minifilament, which accelerates to over 140 km/s before abruptly stopping upon striking overlying nearlyhorizontal magnetic field at 30,000 km altitude and producing the two-sided-loop jet via interchange reconnection. Analysis of EIS raster scans show that a hot brightening, consistent with a small flare, develops in the aftermath of the eruption, and that Doppler motions (40 km/s) occur near the jet-formation region. As with many single-spire jets, the trigger of the eruption here is apparently magnetic flux cancelation, which occurs at a rate of 410^18 Mx/hr, comparable to the rate observed in some single-spire AR jets. This example of a two-sided jet, along with numerous examples of single-spire jets, supports that essentially all coronal jets result from eruptions of minifiaments, and frequently the eruption of the minifilment is triggered by magnetic flux cancelation. (Details are in Sterling et al. 2019, ApJ, 871, 220.