The explicit molecular expansion of the combinatorial logarithm

Just as the power series of $\log (1+X)$ is the analytical substitutional inverse of the series of $\exp (X)-1$, the (virtual) combinatorial species, $\mathrm{Lg} (1+X)$, is the combinatorial substitutional inverse of the combinatorial species, $E(X)-1$, of non-empty finite sets. This $\textit{combinatorial logarithm}$, $\mathrm{Lg} (1+X)$, has been introduced by A. Joyal in 1986 by making use of an iterative scheme. Given a species $F(X)$ (with $F(0)=1$), one of its main applications is to express the species, $F^{\mathrm{c}}(X)$, of $\textit{connected}$ $F$-structures through the formula $F{\mathrm{c}} = \mathrm{Lg} (F) = \mathrm{Lg} (1+F_+)$ where $F_+$ denotes the species of non-empty $F$-structures. Since its creation, equivalent descriptions of the combinatorial logarithm have been given by other combinatorialists (G. L., I. Gessel, J. Li), but its exact decomposition into irreducible components (molecular expansion) remained unclear. The main goal of the present work is to fill this gap by computing explicitly the molecular expansion of the combinatorial logarithm and of $-\mathrm{Lg}(1-X)$, a "cousin'' of the tensorial species, $\mathrm{Lie}(X)$, of free Lie algebras

On extensions of the Newton-Raphson iterative scheme to arbitrary orders

Abstract. The classical quadratically convergent Newton-Raphson iterative scheme for successive approximations of a root of an equation f (t) = 0 has been extended in various ways by different authors, going from cubical convergence to convergence of arbitrary orders. We introduce two such extensions, using appropriate differential operators as well as combinatorial arguments. We conclude with some applications including special series expansions for functions of the root and enumeration of classes of tree-like structures according to their number of leaves. Résumé. Le schéma itératif classiqueà convergence quadratique de Newton-Raphson pour engendrer des approximations successives d'une racine d'uneéquation f (t) = 0 aétéétendu de plusieurs façons par divers auteurs, allant de la convergence cubiqueà des convergences d'ordres arbitraires. Nous introduisons deux telles extensions en utilisant des opérateurs différentiels appropriés ainsi que des arguments combinatoires. Nous terminons avec quelques applications incluant des développements en séries exprimant des fonctions de la racine et l'énumération de classes de structures arborescentes selon leur nombre de feuilles

New combinatorial computational methods arising from pseudo-singletons

Since singletons are the connected sets, the species $X$ of singletons can be considered as the combinatorial logarithm of the species $E(X)$ of finite sets. In a previous work, we introduced the (rational) species $\widehat{X}$ of pseudo-singletons as the analytical logarithm of the species of finite sets. It follows that $E(X) = \exp (\widehat{X})$ in the context of rational species, where $\exp (T)$ denotes the classical analytical power series for the exponential function in the variable $T$. In the present work, we use the species $\widehat{X}$ to create new efficient recursive schemes for the computation of molecular expansions of species of rooted trees, of species of assemblies of structures, of the combinatorial logarithm species, of species of connected structures, and of species of structures with weighted connected components

Closed paths whose steps are roots of unity

We give explicit formulas for the number $U_n(N)$ of closed polygonal paths of length $N$ (starting from the origin) whose steps are $n^{\textrm{th}}$ roots of unity, as well as asymptotic expressions for these numbers when $N \rightarrow \infty$. We also prove that the sequences $(U_n(N))_{N \geq 0}$ are $P$-recursive for each fixed $n \geq 1$ and leave open the problem of determining the values of $N$ for which the $\textit{dual}$ sequences $(U_n(N))_{n \geq 1}$ are $P$-recursive

Influence of Acidic pH on Hydrogen and Acetate Production by an Electrosynthetic Microbiome

Production of hydrogen and organic compounds by an electrosynthetic microbiome using electrodes and carbon dioxide as sole electron donor and carbon source, respectively, was examined after exposure to acidic pH (∼5). Hydrogen production by biocathodes poised at −600 mV vs. SHE increased>100-fold and acetate production ceased at acidic pH, but ∼5–15 mM (catholyte volume)/day acetate and>1,000 mM/day hydrogen were attained at pH ∼6.5 following repeated exposure to acidic pH. Cyclic voltammetry revealed a 250 mV decrease in hydrogen overpotential and a maximum current density of 12.2 mA/cm2 at −765 mV (0.065 mA/cm2 sterile control at −800 mV) by the Acetobacterium-dominated community. Supplying −800 mV to the microbiome after repeated exposure to acidic pH resulted in up to 2.6 kg/m3/day hydrogen (≈2.6 gallons gasoline equivalent), 0.7 kg/m3/day formate, and 3.1 kg/m3/day acetate ( = 4.7 kg CO2 captured).</p

Abundance and Distribution of Enteric Bacteria and Viruses in Coastal and Estuarine Sediments—a Review

The long term survival of fecal indicator organisms (FIOs) and human pathogenic microorganisms in sediments is important from a water quality, human health and ecological perspective. Typically, both bacteria and viruses strongly associate with particulate matter present in freshwater, estuarine and marine environments. This association tends to be stronger in finer textured sediments and is strongly influenced by the type and quantity of clay minerals and organic matter present. Binding to particle surfaces promotes the persistence of bacteria in the environment by offering physical and chemical protection from biotic and abiotic stresses. How bacterial and viral viability and pathogenicity is influenced by surface attachment requires further study. Typically, long-term association with surfaces including sediments induces bacteria to enter a viable-but-non-culturable (VBNC) state. Inherent methodological challenges of quantifying VBNC bacteria may lead to the frequent under-reporting of their abundance in sediments. The implications of this in a quantitative risk assessment context remain unclear. Similarly, sediments can harbor significant amounts of enteric viruses, however, the factors regulating their persistence remains poorly understood. Quantification of viruses in sediment remains problematic due to our poor ability to recover intact viral particles from sediment surfaces (typically <10%), our inability to distinguish between infective and damaged (non-infective) viral particles, aggregation of viral particles, and inhibition during qPCR. This suggests that the true viral titre in sediments may be being vastly underestimated. In turn, this is limiting our ability to understand the fate and transport of viruses in sediments. Model systems (e.g., human cell culture) are also lacking for some key viruses, preventing our ability to evaluate the infectivity of viruses recovered from sediments (e.g., norovirus). The release of particle-bound bacteria and viruses into the water column during sediment resuspension also represents a risk to water quality. In conclusion, our poor process level understanding of viral/bacterial-sediment interactions combined with methodological challenges is limiting the accurate source apportionment and quantitative microbial risk assessment for pathogenic organisms associated with sediments in aquatic environments

Exhaustive generation of atomic combinatorial differential operators

Labelle and Lamathe introduced in 2009 a generalization of the standard combinatorial differential species operator D, by giving a combinatorial interpretation to Ω(X, D)F(X), where Ω(X, T) and F(X) are two-sort and one-sort species respectively. One can show that such operators can be decomposed as sums of products of simpler operators called atomic combinatorial differential operators. In their paper, Labelle and Lamathe presented a list of the first atomic differential operators. In this paper, we describe an algorithm that allows to generate (and enumerate) all of them, subject to available computer resources. We also give a detailed analysis of how to compute the molecular components of Ω(X, D)F(X)