Basic properties and applications of graded fractal bundles related to Clifford structures: An introduction

Using the central extension of the Cuntz C∗-algebra, we study the periodicity for corresponding fractals.З допомогою центрального розширення C*-алгебри Кунца вивчається періодичність для відповідних фракталів

The genus <i>Elaphomyces </i>(<i>Ascomycota</i>, <i>Eurotiales</i>):a ribosomal DNA-based phylogeny and revised systematics of European 'deer truffles'

Elaphomyces (‘deer truffles’) is one of the most important ectomycorrhizal fungal genera in temperate and subarctic forest ecosystems, but also one of the least documented in public databases. The current systematics are mainly based on macromorphology, and is not significantly different from that proposed by Vittadini (1831). Within the 49 species recognised worldwide, 23 were originally described from Europe and 17 of these were described before the 20th century. Moreover, very recent phylogenetic treatments of the genus are mainly based on a few extra-European species and most common European species are still poorly documented. Based on an extensive taxonomic sampling mainly made in the biogeographically rich Cantabrian area (Spain), complemented with collections from France, Greece, Italy, Norway, Portugal and Sweden, all currently recognized species in Europe have been sequenced at the ITS and 28S of the rDNA. Combined phylogenetic analyses yielded molecular support to sections Elaphomyces and Ceratogaster (here emended), while a third, basal lineage encompasses the sections Malacodermei and Ascoscleroderma as well as the tropical genus Pseudotulostoma. Species limits are discussed and some taxa formerly proposed as genuine species based on morphology and biogeography are re-evaluated as varieties or forms. Spore size and ornamentation, features of the peridial surface, structure of the peridium, and the presence of mycelium patches attached to the peridial surface emerge as the most significant systematic characters. Four new species: E. barrioi, E. quercicola, E. roseolus and E. violaceoniger, one new variety: E. papillatus var. sulphureopallidus, and two new forms: E. granulatus forma pallidosporus and E. anthracinus forma talosporus are introduced, as well as four new combinations in the genus: E. muricatus var. reticulatus, E. muricatus var. variegatus, E. papillatus var. striatosporus and E. morettii var. cantabricus. Lectotypes and epitypes are designated for most recognised species. For systematic purposes, new infrageneric taxa are introduced: E. sect. Ascoscleroderma stat. nov., E. subsect. Sclerodermei stat. nov., E. subsect. Maculati subsect. nov., E. subsect. Muricati subsect. nov., and E. subsect. Papillati subsect. nov. Lastly, E. laevigatus, E. sapidus, E. sulphureopallidus and E. trappei are excluded from the genus and referred to Rhizopogon roseolus, Astraeus sapidus comb. nov., Astraeus hygrometricus and Terfezia trappei comb. nov. (syn.: Terfezia cistophila), respectively

Semantic concept schema of the linear mixed model of experimental observations

In the information age, smart data modelling and data management can be carried out to address the wealth of data produced in scientific experiments. In this paper, we propose a semantic model for the statistical analysis of datasets by linear mixed models. We tie together disparate statistical concepts in an interdisciplinary context through the application of ontologies, in particular the Statistics Ontology (STATO), to produce FAIR data summaries. We hope to improve the general understanding of statistical modelling and thus contribute to a better description of the statistical conclusions from data analysis, allowing their efficient exploration and automated processing.</p

Structure fractals and para-quaternionic geometry

It is well known that starting with real structure, the Cayley-Dickson process gives complex, quaternionic, and octonionic (Cayley) structures related to the Adolf Hurwitz composition formula for dimensions $p = 2, 4$ and $8$, respectively, but the procedure fails for $p = 16$ in the sense that the composition formula involves no more a triple of quadratic forms of the same dimension; the other two dimensions are $n = 2^7$. Instead, Ławrynowicz and Suzuki (2001) have considered graded fractal bundles of the flower type related to complex and Pauli structures and, in relation to the iteration process $p \to p + 2 \to p + 4 \to ...$, they have constructed $2^4$-dimensional “bipetals” for $p = 9$ and $2^7$-dimensional “bisepals” for $p = 13$. The objects constructed appear to have an interesting property of periodicity related to the gradating function on the fractal diagonal interpreted as the “pistil” and a family of pairs of segments parallel to the diagonal and equidistant from it, interpreted as the “stamens”. The first named author, M. Nowak-Kepczyk, and S. Marchiafava (2006, 2009a, b) gave an effective, explicit determination of the periods and expressed them in terms of complex and quaternionic structures, thus showing the quaternionic background of that periodicity. In contrast to earlier results, the fractal bundle flower structure, in particular petals, sepals, pistils, and stamens are not introduced ab initio; they are quoted a posteriori, when they are fully motivated. Physical concepts of dual and conjugate objects as well as of antiparticles led us to extend the periodicity theorem to structure fractals in para-quaternionic formulation, applying some results in this direction by the second named author. The paper is concluded by outlining some applications