Lattice-type self-similar sets with pluriphase generators fail to be Minkowski measurable

A long-standing conjecture of Lapidus claims that under certain conditions, self-similar fractal sets fail to be Minkowski measurable if and only if they are of lattice type. The theorem was established for fractal subsets of $\mathbb{R}$ by Falconer, Lapidus and v.~Frankenhuijsen, and the forward direction was shown for fractal subsets of $\mathbb{R}^d$, $d \geq 2$, by Gatzouras. Since then, much effort has been made to prove the converse. In this paper, we prove a partial converse by means of renewal theory. Our proof allows us to recover several previous results in this regard, but is much shorter and extends to a more general setting; several technical conditions appearing in previous versions of this result have now been removed.Comment: 20 pages, 6 figure

A Model for the Development of the Rhizobial and Arbuscular Mycorrhizal Symbioses in Legumes and Its Use to Understand the Roles of Ethylene in the Establishment of these two Symbioses

We propose a model depicting the development of nodulation and arbuscular mycorrhizae. Both processes are dissected into many steps, using Pisum sativum L. nodulation mutants as a guideline. For nodulation, we distinguish two main developmental programs, one epidermal and one cortical. Whereas Nod factors alone affect the cortical program, bacteria are required to trigger the epidermal events. We propose that the two programs of the rhizobial symbiosis evolved separately and that, over time, they came to function together. The distinction between these two programs does not exist for arbuscular mycorrhizae development despite events occurring in both root tissues. Mutations that affect both symbioses are restricted to the epidermal program. We propose here sites of action and potential roles for ethylene during the formation of the two symbioses with a specific hypothesis for nodule organogenesis. Assuming the epidermis does not make ethylene, the microsymbionts probably first encounter a regulatory level of ethylene at the epidermis–outermost cortical cell layer interface. Depending on the hormone concentrations there, infection will either progress or be blocked. In the former case, ethylene affects the cortex cytoskeleton, allowing reorganization that facilitates infection; in the latter case, ethylene acts on several enzymes that interfere with infection thread growth, causing it to abort. Throughout this review, the difficulty of generalizing the roles of ethylene is emphasized and numerous examples are given to demonstrate the diversity that exists in plants

New insights on subsurface energy resources in the Southern North Sea Basin area

The Southern North Sea Basin area, stretching from the UK to the Netherlands, has a rich hydrocarbon exploration and production history. The past, present and expected future hydrocarbon and geothermal exploration trends in this area are discussed for eight key lithostratigraphic intervals, ranging from the Lower Carboniferous to Cenozoic. In the period between 2007 and 2017, a total of 95 new hydrocarbon fields were discovered, particularly in Upper Carboniferous, Rotliegend and Triassic reservoirs. Nineteen geothermal systems were discovered in the Netherlands onshore, mainly targeting aquifers in the Rotliegend and Upper Jurassic/Lower Cretaceous formations. Although the Southern North Sea Basin area is mature in terms of hydrocarbon exploration, it is shown that with existing and new geological insights, additional energy resources are still being proven in new plays such as the basal Upper Rotliegend (Ruby discovery) for natural gas and a new Chalk play for oil. It is predicted that hydrocarbon exploration in the Southern North Sea Basin area will probably experience a slight growth in the coming decade before slowing down, as the energy transition further matures. Geothermal exploration is expected to continue growing in the Netherlands onshore as well as gain more momentum in the UK

Deep sequencing of the Mexican avocado transcriptome, an ancient angiosperm with a high content of fatty acids

Background: Avocado (Persea americana) is an economically important tropical fruit considered to be a good source of fatty acids. Despite its importance, the molecular and cellular characterization of biochemical and developmental processes in avocado is limited due to the lack of transcriptome and genomic information. Results: The transcriptomes of seeds, roots, stems, leaves, aerial buds and flowers were determined using different sequencing platforms. Additionally, the transcriptomes of three different stages of fruit ripening (pre-climacteric, climacteric and post-climacteric) were also analyzed. The analysis of the RNAseqatlas presented here reveals strong differences in gene expression patterns between different organs, especially between root and flower, but also reveals similarities among the gene expression patterns in other organs, such as stem, leaves and aerial buds (vegetative organs) or seed and fruit (storage organs). Important regulators, functional categories, and differentially expressed genes involved in avocado fruit ripening were identified. Additionally, to demonstrate the utility of the avocado gene expression atlas, we investigated the expression patterns of genes implicated in fatty acid metabolism and fruit ripening. Conclusions: A description of transcriptomic changes occurring during fruit ripening was obtained in Mexican avocado, contributing to a dynamic view of the expression patterns of genes involved in fatty acid biosynthesis and the fruit ripening process