On the tower factorization of integers

Under the fundamental theorem of arithmetic, any integer $n>1$ can be uniquely written as a product of prime powers $p^a$; factoring each exponent $a$ as a product of prime powers $q^b$, and so on, one will obtain what is called the tower factorization of $n$. Here, given an integer $n>1$, we study its height $h(n)$, that is, the number of "floors" in its tower factorization. In particular, given a fixed integer $k\geq 1$, we provide a formula for the density of the set of integers $n$ with $h(n)=k$. This allows us to estimate the number of floors that a positive integer will have on average. We also show that there exist arbitrarily long sequences of consecutive integers with arbitrarily large heights.Comment: 8 pages. Accepted for publication in the Amer. Math. Monthl

Counting Involutions on Multicomplex Numbers

We show that there is a bijection between real-linear automorphisms of the multicomplex numbers of order $n$ and signed permutations of length $2^{n-1}$. This allows us to deduce a number of results on the multicomplex numbers, including a formula for the number of involutions on multicomplex spaces which generalizes a recent result on the bicomplex numbers and contrasts drastically with the quaternion case. We also generalize this formula to $r$-involutions and obtain a formula for the number of involutions preserving elementary imaginary units. The proofs rely on new elementary results pertaining to multicomplex numbers that are surprisingly unknown in the literature, including a count and a representation theorem for numbers squaring to $\pm 1$

Bounds for the counting function of the Jordan-Pólya numbers

summary:A positive integer $n$ is said to be a Jordan-Pólya number if it can be written as a product of factorials. We obtain non-trivial lower and upper bounds for the number of Jordan-Pólya numbers not exceeding a given number $x$

Eurasian-Origin Gene Segments Contribute to the Transmissibility, Aerosol Release, and Morphology of the 2009 Pandemic H1N1 Influenza Virus

The epidemiological success of pandemic and epidemic influenza A viruses relies on the ability to transmit efficiently from person-to-person via respiratory droplets. Respiratory droplet (RD) transmission of influenza viruses requires efficient replication and release of infectious influenza particles into the air. The 2009 pandemic H1N1 (pH1N1) virus originated by reassortment of a North American triple reassortant swine (TRS) virus with a Eurasian swine virus that contributed the neuraminidase (NA) and M gene segments. Both the TRS and Eurasian swine viruses caused sporadic infections in humans, but failed to spread from person-to-person, unlike the pH1N1 virus. We evaluated the pH1N1 and its precursor viruses in a ferret model to determine the contribution of different viral gene segments on the release of influenza virus particles into the air and on the transmissibility of the pH1N1 virus. We found that the Eurasian-origin gene segments contributed to efficient RD transmission of the pH1N1 virus likely by modulating the release of influenza viral RNA-containing particles into the air. All viruses replicated well in the upper respiratory tract of infected ferrets, suggesting that factors other than viral replication are important for the release of influenza virus particles and transmission. Our studies demonstrate that the release of influenza viral RNA-containing particles into the air correlates with increased NA activity. Additionally, the pleomorphic phenotype of the pH1N1 virus is dependent upon the Eurasian-origin gene segments, suggesting a link between transmission and virus morphology. We have demonstrated that the viruses are released into exhaled air to varying degrees and a constellation of genes influences the transmissibility of the pH1N1 virus

Sums of random multiplicative functions over function fields with few irreducible factors

We establish a normal approximation for the limiting distribution of partial sums of random Rademacher multiplicative functions over function fields, provided the number of irreducible factors of the polynomials is small enough. This parallels work of Harper for random Rademacher multiplicative functions over the integers.Comment: 10 pages. Simplification of the proof of Lemma 5 and typos corrected, one reference adde

Detection of Airborne Lactococcal Bacteriophages in Cheese Manufacturing Plants▿

The dairy industry adds starter bacterial cultures to heat-treated milk to control the fermentation process during the manufacture of many cheeses. These highly concentrated bacterial populations are susceptible to virulent phages that are ubiquitous in cheese factories. In this study, the dissemination of these phages by the airborne route and their presence on working surfaces were investigated in a cheese factory. Several surfaces were swabbed, and five air samplers (polytetrafluoroethylene filter, polycarbonate filter, BioSampler, Coriolis cyclone sampler, and NIOSH two-stage cyclone bioaerosol personal sampler) were tested. Samples were then analyzed for the presence of two Lactococcus lactis phage groups (936 and c2), and quantification was done by quantitative PCR (qPCR). Both lactococcal phage groups were found on most swabbed surfaces, while airborne phages were detected at concentrations of at least 103 genomes/m3 of air. The NIOSH sampler had the highest rate of air samples with detectable levels of lactococcal phages. This study demonstrates that virulent phages can circulate through the air and that they are ubiquitous in cheese manufacturing facilities

Feather corticosterone during non-breeding correlates with multiple measures of physiology during subsequent breeding in a migratory seabird.

International audienceCarry-over effects in migratory birds are likely mediated by physiological processes that are activated in response to environmental variation. Such processes affect body condition and/or reproductive success, and can include corticosterone (CORT) because this hormone responds to environmental stressors and influences energy balance. Few studies have considered how CORT levels during non-breeding relate to a broader physiological profile during subsequent breeding, and fewer still have considered measures other than body condition. To explore CORT's potential role in carry-over effects, we investigated the relationship between CORT and foraging ecology of northern gannets (Morus bassanus) during the non-breeding period, and tested for associations between these factors and variation in a suite of physiological and biochemical metrics during subsequent breeding. Northern gannets are the largest seabird top predator in the North Atlantic and were among the hardest hit by the Deepwater Horizon oil blowout in the Gulf of Mexico in 2010. We used light-level geolocators to confirm winter origins of individuals in our study. No interrelationships were found among levels of CORT from feathers grown during non-breeding (CORTf) and variation in foraging ecology, measured by stable isotopes of carbon (δ(13)C) and nitrogen (δ(15)N) from the same feathers. CORTf was correlated negatively with hematocrit and positively with triglyceride measured during subsequent incubation, and explained more variation in these variables than did body mass during incubation. These findings provide support for the hypothesis that energy management, measured using CORTf, during non-breeding carries over to influence physiological measures other than body condition. Gannets that previously wintered within the Gulf of Mexico in the years following the Deepwater Horizon oil blowout had higher levels of CORTf compared to birds that wintered along the Atlantic coast, suggesting an increased energetic cost associated with visiting the Gulf of Mexico. Our results indicate that CORT during non-breeding is associated with a broader physiological profile during subsequent breeding than previously reported in birds

Impact of Blueberry Consumption on the Human Fecal Bileacidome: A Pilot Study of Bile Acid Modulation by Freeze-Dried Blueberry

Cholesterol-derived bile acids (BAs) affect numerous physiological functions such as glucose homeostasis, lipid metabolism and absorption, intestinal inflammation and immunity, as well as intestinal microbiota diversity. Diet influences the composition of the BA pool. In the present study, we analyzed the impact of a dietary supplementation with a freeze-dried blueberry powder (BBP) on the fecal BA pool composition. The diet of 11 men and 13 women at risk of metabolic syndrome was supplemented with 50 g/day of BBP for 8 weeks, and feces were harvested before (pre) and after (post) BBP consumption. BAs were profiled using liquid chromatography coupled with tandem mass spectrometry. No significant changes in total BAs were detected when comparing pre- vs. post-BBP consumption samples. However, post-BBP consumption samples exhibited significant accumulations of glycine-conjugated BAs (p = 0.04), glycochenodeoxycholic (p = 0.01), and glycoursodeoxycholic (p = 0.01) acids, as well as a significant reduction (p = 0.03) in the secondary BA levels compared with pre-BBP feces. In conclusion, the fecal bileacidome is significantly altered after the consumption of BBP for 8 weeks. While additional studies are needed to fully understand the underlying mechanisms and physiological implications of these changes, our data suggest that the consumption of blueberries can modulate toxic BA elimination

Oxazolo[5,4-f]quinoxaline-type selective inhibitors of glycogen synthase kinase-3α (GSK-3α): Development and impact on temozolomide treatment of glioblastoma cells

International audienceThe 2-(3-pyridyl)oxazolo[5,4-f]quinoxalines CD-07 and FL-291 are ATP-competitive GSK-3 kinase inhibitors. Here, we investigated the impact of FL-291 on neuroblastoma cell viability and showed that treatment at 10 μM (i.e. ∼500 times the IC50 against the GSK-3 isoforms) has no significant effect on the viability of NSC-34 motoneuron-like cells. A study performed on primary neurons (non-cancer cells) led to similar results. The structures co-crystallized with GSK-3β revealed similar binding modes for FL-291 and CD-07, with their hinge-oriented planar tricyclic system. Both GSK isoforms show the same orientations for the amino acids at the binding pocket except for Phe130 (α) and Phe67 (β), leading to a larger pocket on the opposite side of the hinge region for the α isoform. Calculations of the thermodynamic properties of the binding pockets highlighted the required features of potential ligands; these should have a hydrophobic core (which could be larger in the case of GSK-3β) surrounded by polar areas (a little more polar in the case of GSK-3α). A library of 27 analogs of FL-291 and CD-07 was thus designed and synthesized by taking advantage of this hypothesis. While the introduction of substituents at different positions of the pyridine ring, the replacement of the pyridine by other heterocyclic moieties, or the replacement of the quinoxaline ring by a quinoline moiety did not lead to any improvement, the replacement of the N-(thio)morpholino of FL-291/CD-07 by a slightly more polar N-thiazolidino led to a significant result. Indeed, the new inhibitor MH-124 showed clear selectivity for the α isoform, with IC50 values of 17 nM and 239 nM on GSK-3α and GSK-3β, respectively. Finally, the efficacy of MH-124 was evaluated on two glioblastoma cell lines. Although MH-124 alone did not have a significant impact on cell survival, its addition to temozolomide (TMZ) significantly reduced the TMZ IC50 values on the cells tested. The use of the Bliss model allowed a synergy to be evidenced at certain concentrations