On Arnold's Transversality Conjecture for the Laplace-Beltrami Operator

This paper is concerned with the structure of the set of Riemannian metrics on a connected manifold such that the corresponding Laplace-Beltrami operator has an eigenvalue of a given multiplicity. We introduce a simple geometric condition on metrics and their corresponding Laplace eigenfunctions, and show it is equivalent to the strong Arnold hypothesis. This hypothesis essentially posits that multiple Laplace eigenvalues split up under perturbation like those of symmetric matrices. We prove our condition is satisfied except on a set of infinite codimension, and use this to obtain non-crossing rules for Laplace eigenvalues. Furthermore, we show that our condition is satisfied on all metrics admitting eigenvalues of multiplicity at most six, and exhibit examples of metrics violating it.Comment: 34 pages, changes in v2.1: Updated ArXiv abstrac

Shining Light on Periodic Dominating Sets in Bounded-Treewidth Graphs

For the vertex selection problem $(\sigma,\rho)$-DomSet one is given two fixed sets $\sigma$ and $\rho$ of integers and the task is to decide whether we can select vertices of the input graph, such that, for every selected vertex, the number of selected neighbors is in $\sigma$ and, for every unselected vertex, the number of selected neighbors is in $\rho$. This framework covers Independent Set and Dominating Set for example. We investigate the case when $\sigma$ and $\rho$ are periodic sets with the same period $m\ge 2$, that is, the sets are two (potentially different) residue classes modulo $m$. We study the problem parameterized by treewidth and present an algorithm that solves in time $m^{tw} \cdot n^{O(1)}$ the decision, minimization and maximization version of the problem. This significantly improves upon the known algorithms where for the case $m \ge 3$ not even an explicit running time is known. We complement our algorithm by providing matching lower bounds which state that there is no $(m-\epsilon)^{pw} \cdot n^{O(1)}$ unless SETH fails. For $m = 2$, we extend these bound to the minimization version as the decision version is efficiently solvable

Origin of facultative heterochromatin in the endosperm of Gagea lutea (Liliaceae)

Peer reviewedPostprin

Genome size diversity and its impact on the evolution of land plants

Genome size is a biodiversity trait that shows staggering diversity across eukaryotes, varying over 64,000-fold. Of all major taxonomic groups, land plants stand out due to their staggering genome size diversity, ranging ca. 2400-fold. As our understanding of the implications and significance of this remarkable genome size diversity in land plants grows, it is becoming increasingly evident that this trait plays not only an important role in shaping the evolution of plant genomes, but also in influencing plant community assemblages at the ecosystem level. Recent advances and improvements in novel sequencing technologies, as well as analytical tools, make it possible to gain critical insights into the genomic and epigenetic mechanisms underpinning genome size changes. In this review we provide an overview of our current understanding of genome size diversity across the different land plant groups, its implications on the biology of the genome and what future directions need to be addressed to fill key knowledge gaps

Multiple Origins and Nested Cycles of Hybridization Result in High Tetraploid Diversity in the Monocot Prospero

This work was supported by the Austrian Science Fund (FWF P21440-B03 to HW-S)

Next-generation sequencing reveals the impact of repetitive DNA in phylogenetically closely related genomes of Orobanchaceae

We used next-generation sequencing to characterize the genomes of nine species of Orobanchaceae of known phylogenetic relationships, different life forms, and including a polyploid species. The study species are the autotrophic, nonparasitic Lindenbergia philippensis, the hemiparasitic Schwalbea americana, and seven nonphotosynthetic parasitic species of Orobanche (Orobanche crenata, Orobanche cumana, Orobanche gracilis (tetraploid), and Orobanche pancicii) and Phelipanche (Phelipanche lavandulacea, Phelipanche purpurea, and Phelipanche ramosa). Ty3/Gypsy elements comprise 1.93%–28.34% of the nine genomes and Ty1/Copia elements comprise 8.09%–22.83%. When compared with L. philippensis and S. americana, the nonphotosynthetic species contain higher proportions of repetitive DNA sequences, perhaps reflecting relaxed selection on genome size in parasitic organisms. Among the parasitic species, those in the genus Orobanche have smaller genomes but higher proportions of repetitive DNA than those in Phelipanche, mostly due to a diversification of repeats and an accumulation of Ty3/ Gypsy elements. Genome downsizing in the tetraploid O. gracilis probably led to sequence loss across most repeat types

Introgression of apomixis into sexual species is inhibited by mentor effects and ploidy barriers in the Ranunculus auricomus complex

Background and Aims Apomictic plants maintain functional pollen, and via pollination the genetic factors controlling apomixis can be potentially transferred to congeneric sexual populations. In contrast, the sexual individuals do not fertilize apomictic plants which produce seeds without fertilization of the egg cells. This unidirectional introgressive hybridization is expected finally to replace sexuality by apomixis and is thought to be a causal factor for the wide geographical distribution of apomictic complexes. Nevertheless, this process may be inhibited by induced selfing (mentor effects) of otherwise self-incompatible sexual individuals. Here whether mentor effects or actual cross-fertilization takes place between diploid sexual and polyploid apomictic cytotypes in the Ranunculus auricomus complex was tested via experimental crosses. ¿ Methods Diploid sexual mother plants were pollinated with tetra- and hexaploid apomictic pollen donators by hand, and the amount of well-developed seed compared with aborted seed was evaluated. The reproductive pathways were assessed in the well-developed seed via flow cytometric seed screen (FCSS). ¿ Key Results The majority of seed was aborted; the well-developed seeds have resulted from both mentor effects and cross-fertilization at very low frequencies (1.3 and 1.6% of achenes, respectively). Pollination by 4x apomictic pollen plants results more frequently in cross-fertilization, whereas pollen from 6x plants more frequently induced mentor effects. ¿Conclusions It is concluded that introgression of apomixis into sexual populations is limited by ploidy barriers in the R. auricomus complex, and to a minor extent by mentor effects. In mixed populations, sexuality cannot be replaced by apomixis because the higher fertility of sexual populations still compensates the low frequencies of potential introgression of apomixi

Genome size variation in diploid and tetraploid wild wheats

Low but significant intraspecific genome size variations were found in diploid and tetraploid wild wheats. This limited variation is not correlated with geographical and climate variables. It can be concluded that the genome size of Triticum species is generally stable, despite of the presence of many potentially active retroelements

Eukaryotic genome size databases

Three independent databases of eukaryotic genome size information have been launched or re-released in updated form since 2005: the Plant DNA C-values Database (), the Animal Genome Size Database () and the Fungal Genome Size Database (). In total, these databases provide freely accessible genome size data for >10 000 species of eukaryotes assembled from more than 50 years' worth of literature. Such data are of significant importance to the genomics and broader scientific community as fundamental features of genome structure, for genomics-based comparative biodiversity studies, and as direct estimators of the cost of complete sequencing programs

Molecular and morphological diversity in the /Rhombisporum clade of the genus Entoloma with a note on E. cocles

A combined morphological and molecular genetic study of the European species within the /Rhombisporum clade of the genus Entoloma reveals a high species diversity. This group comprises typical grassland species with pronounced and welldifferentiated cheilocystidia, and a wide range of spore shapes varying from rhomboid to five-angled. To fix the concept of the classical species E. rhombisporum, a neotype is designated. Nine species are described as new to science based on the result of nrDNA ITS phylogeny with additional gap coding, and morphological characterization: E. caulocystidiatum, E. lunare, E. pararhombisporum, E. pentagonale, E. perrhombisporum, E. rhombiibericum, E. rhombisporoides, E. sororpratulense, and E. subcuboideum. The ITS sequences of the holotypes of previously described species belonging to the /Rhombisporum clade, viz., E. laurisilvae and E. pratulense have also been generated and are published here for the first time. Since many of the above-mentioned species have been misidentified as E. cocles, it seemed opportune to also study this species and to designate a neotype to fix its current concept. A key including European species is presented. As most of the species are potentially important indicators for threatened grassland communities, the 130 ITS barcodes newly generated for this study may be useful as a reference in conservation and metabarcoding projects. Agaricales . Conservation . Endangered grassland communities . Entolomataceae semi-cryptic diversity . Taxonomy . TricholomatinaepublishedVersio