Branch-depth: Generalizing tree-depth of graphs

We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs as follows. For a graph $G = (V,E)$ and a subset $A$ of $E$ we let $\lambda_G (A)$ be the number of vertices incident with an edge in $A$ and an edge in $E \setminus A$. For a subset $X$ of $V$, let $\rho_G(X)$ be the rank of the adjacency matrix between $X$ and $V \setminus X$ over the binary field. We prove that a class of graphs has bounded tree-depth if and only if the corresponding class of functions $\lambda_G$ has bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions $\rho_G$ has bounded branch-depth, which we call the rank-depth of graphs. Furthermore we investigate various potential generalizations of tree-depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by the restriction.Comment: 34 pages, 2 figure

Branch-depth: Generalizing tree-depth of graphs

We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs as follows. For a graph $G = (V,E)$ and a subset $A$ of $E$ we let $\lambda_G (A)$ be the number of vertices incident with an edge in $A$ and an edge in $E \setminus A$. For a subset $X$ of $V$, let $\rho_G(X)$ be the rank of the adjacency matrix between $X$ and $V \setminus X$ over the binary field. We prove that a class of graphs has bounded tree-depth if and only if the corresponding class of functions $\lambda_G$ has bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions $\rho_G$ has bounded branch-depth, which we call the rank-depth of graphs. Furthermore we investigate various potential generalizations of tree-depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by the restriction.Comment: 36 pages, 2 figures. Final versio

Rôle des acides humiques dans le transfert du pyrène entre les minéraux argileux et l'eau

Les substances humiques se fixent en surface des minéraux argileux et modifient les sites d'adsorption des polluants organiques hydrophobes (POH). Parallèlement, les substances humiques dissoutes complexent les POH non ioniques selon des mécanismes de liaison réversible généralement évalués par le coefficient de partage Koc. Les solutés humiques seraient ainsi responsables du transport des POH dans les écosystèmes aquatiques par solubilisation de la fraction adsorbée sur le sédiment. Notre étude a pour objectif la quantification et la modélisation de ces phénomènes pour des composés modèles disponibles commercialement : kaolinite, pyrène et acide humique Aldrich purifié.L'isotherme d'adsorption de l'acide humique (AH) sur la kaolinite à pH 6,5 et à force ionique 10-2 M suit une loi de Freundlich, dont les valeurs du modèle sont KF =0,79 et nF =1,90. Dans les mêmes conditions expérimentales, le pyrène, composé modèle à quatre noyaux du groupe des hydrocarbures aromatiques polycycliques (HAP), semble s'adsorber en formant des microcristaux à la surface de l'argile. La présence d'acide humique fixé sur la kaolinite modifie l'adsorption du pyrène, qui se modélise par une isotherme de Freundlich de paramètres K'F =0,30 et n'F =2,00.Les interactions entre le pyrène et l'acide humique dissous ont été quantifiées par spectroscopie d'extinction de fluorescence et relation de Stern-Volmer. La fraction humique dissoute non adsorbée sur le minéral argileux se caractérise par de plus fortes valeurs de Koc que l'acide humique initial (2,95.105 L.kg-1 contre 1,86.105 L.kg-1). L'hypothèse de fractionnement des substances humiques lors de l'adsorption sur l'argile est confirmée par les variations du poids moléculaire moyen des AH, déterminé par chromatographie d'exclusion stérique, et des variations des intensités de fluorescence des fractions chromatographiées.Les résultats obtenus pour la quantification du pyrène désorbé du solide argileux dans l'eau pure et dans une solution d'AH (50 mg.L-1) tendent à minimiser le rôle de la matière organique naturelle dissoute dans le transfert des polluants hydrophobes du milieu solide au milieu liquide.Humic substances become fixed to the surface of clay minerals and modify the adsorption sites of hydrophobic organic pollutants (HOP). The dissolved humic substances complex the HOP according to reversible binding mechanisms normally assessed by the Koc distribution coefficient. The humic solutes could thus be responsible of transporting HOP in aquatic ecosystems by solubilising the adsorbed fraction on sediment. Our study aims to quantify and model these phenomena using commercially available compounds:- pyrene, a polycyclic aromatic hydrocarbon (PAH) generally used as fluorescence probe and able to bind strongly to humic acid (AH); - a commercially available humic acid (Aldrich), with a high molecular weight and high aromaticity, which can easily adsorb onto colloidal matter and strongly bind PAH; - kaolinite clay, a ubiquitous mineral in aquatic systems, with a particle size distribution in aqueous solution similar to that observed in lake waters.We also have chosen physico-chemical conditions representative of freshwater: pH (6.5) and ionic strength (10-2 M). The adsorption isotherm of the humic acid (HA) on kaolinite at pH=6.5 and at ionic strength 10-2 M follows a Freundlich isotherm, for which model values are fixed at KF =0.79 and nF =1.90. The carboxylic groups of the dissolved HA, which represented more than half the total acidity, were not protonated under these conditions. Despite the negative surface charge of the particles, kaolinite was able to adsorb about 0.2 mgC.g-1 of HA. A decrease in the solution pH occurred as the adsorbed HA quantity increased, suggesting a chemisorption of HA onto clay. The high-pressure size exclusion chromatogram (HPSEC) of the non-adsorbed HA, isolated after centrifugation of the colloidal solution, reflects the selective adsorption of the HA. The higher the molecular weight of the HA (>70 000 Da), the better the adsorption. The hypothesis of the selective adsorption of the humic substances onto clay was also confirmed by the fluorescence variations of the chromatographed fractions.Using the same experimental conditions, pyrene seemed to adsorb and generate micro-crystals on the surface of the clay. The first part of the isotherm fits the pyrene monomer adsorption onto the hydrophobic siloxane surface of the clay. In the second part, the dramatic increase in the adsorption isotherm could be attributed to the formation of pyrene micro-crystals on the solid surface. For a residual pyrene concentration equal to 40 nmol.L-1, the PAH quantity adsorbed onto kaolinite clay (7 nmol.g-1) was about twenty times lower than the quantity adsorbed onto montmorillonite clay (150 nmol.g-1). This could be explained by the inaccessibility of the interlayer cavity of kaolinite to any solute.Natural colloids are often coated by a humic layer. This coating can thus modify HOP adsorption onto the particles. The presence of humic acid fixed on kaolinite effectively modifies pyrene adsorption. This adsorption can be modelled by a Freundlich isotherm with the parameters K'F =0.30 and n'F =2.00. The adsorbed HA molecules seem to limit the formation of pyrene micro-crystals, even if the pyrene adsorption is not competitive with the HA fixation. Indeed, no HA desorption occurs during pyrene fixation onto the organic coated kaolinite.The interactions between pyrene and dissolved humic acid have been quantified by extinction fluorescence spectroscopy and the Stern-Volmer relationship. The unabsorbed, dissolved humic fraction is characterised by Koc values higher than the initial humic acid (2.95 x 105 L.kg-1 compared to 1.86 x 105 L.kg-1). These two values are of the same magnitude as the published value obtained for the non-purified Aldrich HA (2.3±0.3 x 105). The Koc value obtained for the non-adsorbed fraction of HA is unexpected because of the low molecular weight of these molecules. This Koc value can be related either to a better accessibility of the HA hydrophobic cavity for pyrene, or rather to a more efficient fluorescence quenching of pyrene by the humic solutes.The dissolved humic acid may complex the non-ionic PAH according to reversible bonding mechanisms. Therefore, HA can act as a carrier of hydrophobic contaminants if it is able to desorb HOP from the surface of the particles. The results obtained for the quantification of the desorbed pyrene in pure water and in a HA solution (50 mg.L-1) tend to minimise the role of dissolved natural organic matter in the transport of hydrophobic pollutants from the solid to the liquid medium. The values of pyrene concentration in solution obtained after 24 hours contact with kaolinite clay sorbed pyrene with pure water or with the HA solution were very low (respectively 2.4 nM and 12.9 nM) and below the water solubility of pyrene (802 nM). It has been previously demonstrated that the desorption kinetics of pyrene from quartz material are not influenced by the organic matter content of the solution (SCHMITT, 1999). Our results demonstrate that the HA content of the solution did not greatly modify the quantity of pyrene desorbed from clay minerals

Marshall University Music Department Presents the Faculty Woodwind Quintet

https://mds.marshall.edu/music_perf/1079/thumbnail.jp

Marshall University Music Department Presents the Marshall Faculty Woodwind Quintet

https://mds.marshall.edu/music_perf/1091/thumbnail.jp

Femtosecond nonlinear ultrasonics in gold probed with ultrashort surface plasmons

Fundamental interactions induced by lattice vibrations on ultrafast time scales become increasingly important for modern nanoscience and technology. Experimental access to the physical properties of acoustic phonons in the THz frequency range and over the entire Brillouin zone is crucial for understanding electric and thermal transport in solids and their compounds. Here, we report on the generation and nonlinear propagation of giant (1 percent) acoustic strain pulses in hybrid gold/cobalt bilayer structures probed with ultrafast surface plasmon interferometry. This new technique allows for unambiguous characterization of arbitrary ultrafast acoustic transients. The giant acoustic pulses experience substantial nonlinear reshaping already after a propagation distance of 100 nm in a crystalline gold layer. Excellent agreement with the Korteveg-de Vries model points to future quantitative nonlinear femtosecond THz-ultrasonics at the nano-scale in metals at room temperature

The genotype of barley cultivars influences multiple aspects of their associated microbiota via differential root exudate secretion

Plant-associated microbe play vital roles in promoting plant growth and health, with plants secreting root exudates into the rhizosphere to attract beneficial microbes. Exudate composition defines the nature of microbial recruitment, with different plant species attracting distinct microbiota to enable optimal adaptation to the soil environment. To more closely examine the relationship between plant genotype and microbial recruitment, we analysed the rhizosphere microbiomes of landrace (Chevallier) and modern (NFC Tipple) barley (Hordeum vulgare) cultivars. Distinct differences were observed between the plant associated microbiomes of the 2 cultivars, with the plant-growth promoting rhizobacterial genus Pseudomonas substantially more abundant in the Tipple rhizosphere. Striking differences were also observed between the phenotypes of recruited Pseudomonas populations, alongside distinct genotypic clustering by cultivar. Cultivar-driven Pseudomonas selection was driven by root exudate composition, with the greater abundance of hexose sugars secreted from Tipple roots attracting microbes better adapted to growth on these metabolites and vice versa. Cultivar-driven selection also operates at the molecular level, with both gene expression and the abundance of ecologically relevant loci differing between Tipple and Chevallier Pseudomonas isolates. Finally, cultivar-driven selection is important for plant health, with both cultivars showing a distinct preference for microbes selected by their genetic siblings in rhizosphere transplantation assay