Let $\omega (n)$ denote the number of distinct prime factors of a natural number n. In 1940, Erdős and Kac established that $\omega (n)$ obeys the Gaussian distribution over natural numbers, and in 2004, the third author generalized their theorem to all abelian monoids. In this article, we extend her theorem to any subsets of an abelian monoid satisfying some additional conditions, and apply this result to the subsets of h-free and h-full elements. We study generalizations of several arithmetic functions, such as the prime counting omega functions and the divisor function in a unified framework. Finally, we apply our results to number fields, global function fields, and geometrically irreducible projective varieties, demonstrating the broad relevance of our approach.

In this article, we establish a function field analog of Jacobi’s theorem on sums of squares and analyze its moments. Our approach involves employing two distinct techniques to derive the main results concerning asymptotic formulas for the moments. The first technique utilizes Dirichlet series framework to derive asymptotic formulas in the limit of large finite fields, specifically when the characteristic of $\mathbb {F}_q[T]$ becomes large. The second technique involves effectively partitioning the set of polynomials of a fixed degree, providing asymptotic formulas in the limit of large polynomial degree.

Trematodes of the genus Ogmocotyle are intestinal flukes that can infect a variety of definitive hosts, resulting in significant economic losses worldwide. However, there are few studies on molecular data of these trematodes. In this study, the mitochondrial (mt) genome of Ogmocotyle ailuri isolated from red panda (Ailurus fulgens) was determined and compared with those from Pronocephalata to investigate the mt genome content, genetic distance, gene rearrangements and phylogeny. The complete mt genome of O. ailuri is a typical closed circular molecule of 14 642 base pairs, comprising 12 protein-coding genes (PCGs), 22 transfer RNA genes, 2 ribosomal RNA genes and 2 non-coding regions. All genes are transcribed in the same direction. In addition, 23 intergenic spacers and 2 locations with gene overlaps were determined. Sequence identities and sliding window analysis indicated that cox1 is the most conserved gene among 12 PCGs in O. ailuri mt genome. The sequenced mt genomes of the 48 Plagiorchiida trematodes showed 5 types of gene arrangement based on all mt genome genes, with the gene arrangement of O. ailuri being type I. Phylogenetic analysis using concatenated amino acid sequences of 12 PCGs revealed that O. ailuri was closer to Ogmocotyle sikae than to Notocotylus intestinalis. These data enhance the Ogmocotyle mt genome database and provide molecular resources for further studies of Pronocephalata taxonomy, population genetics and systematics.

Chronic inflammation exerts pleiotropic effects in the aetiology and progression of chronic obstructive pulmonary disease (COPD). Glucosamine is widely used in many countries and may have anti-inflammatory properties. We aimed to prospectively evaluate the association of regular glucosamine use with incident COPD risk and explore whether such association could be modified by smoking in the UK Biobank cohort, which recruited more than half a million participants aged 40–69 years from across the UK between 2006 and 2010. Cox proportional hazards models with adjustment for potential confounding factors were used to calculate hazard ratios (HR) as well as 95 % CI for the risk of incident COPD. During a median follow-up of 8·96 years (interquartile range 8·29–9·53 years), 9016 new-onset events of COPD were documented. We found that the regular use of glucosamine was associated with a significantly lower risk of incident COPD with multivariable adjusted HR of 0·80 (95 % CI, 0·75, 0·85; P < 0·001). When subgroup analyses were performed by smoking status, the adjusted HR for the association of regular glucosamine use with incident COPD were 0·84 (0·73, 0·96), 0·84 (0·77, 0·92) and 0·71 (0·62, 0·80) among never smokers, former smokers and current smokers, respectively. No significant interaction was observed between glucosamine use and smoking status (Pfor interaction = 0·078). Incident COPD could be reduced by 14 % to 84 % through a combination of regular glucosamine use and smoking cessation.

Let $k\geqslant 1$ be a natural number and $\omega _k(n)$ denote the number of distinct prime factors of a natural number n with multiplicity k. We estimate the first and second moments of the functions $\omega _k$ with $k\geqslant 1$. Moreover, we prove that the function $\omega _1(n)$ has normal order $\log \log n$ and the function $(\omega _1(n)-\log \log n)/\sqrt {\log \log n}$ has a normal distribution. Finally, we prove that the functions $\omega _k(n)$ with $k\geqslant 2$ do not have normal order $F(n)$ for any nondecreasing nonnegative function F.

We construct a shifted version of the Turán sieve method developed by R. Murty and the second author and apply it to counting problems on tournaments. More precisely, we obtain upper bounds for the number of tournaments which contain a fixed number of restricted $r$-cycles. These are the first concrete results which count the number of cycles over “all tournaments”.

A kind of novel Ni–P gradient coating/stannate conversion film was deposited on AZ91D magnesium alloy (AZ91D alloy) by an integrative method involved stannate conversion and electroless plating. The results indicated that using sodium hypophosphite concentrations varied as 5, 10, 22, 46, and 60 g/L in the bath, the electroless Ni–P gradient coating with typical cell morphologies was successfully prepared, and the structures transited from crystalline → microcrystalline → amorphous were obtained as increasing P content from 3.31 to 12.58 wt%. Furthermore, the corrosion morphologies, polarization curves, and the electrochemical impedance spectroscopy result indicated that the corrosion resistance of AZ91D alloy substrate was significantly improved and the corrosion resistance of Ni–P gradient coating was superior than that of stannate conversion film, which might be attributed to the gradient structure and rising P content with unique function.

Let ${{\mathbb{F}}_{q}}\left[ t \right]$ denote the polynomial ring over the finite field ${{\mathbb{F}}_{q}}$ . We employ Wooley's new efficient congruencing method to prove certain multidimensional Vinogradov-type estimates in ${{\mathbb{F}}_{q}}\left[ t \right]$ . These results allow us to apply a variant of the circle method to obtain asymptotic formulas for a system connected to the problem about linear spaces lying on hypersurfaces defined over ${{\mathbb{F}}_{q}}\left[ t \right]$ .

We have conducted a survey of deuterium fractionation of N2H+, RD(N2H+) ≡ N(N2D+)/N(N2H+), with the Arizona Radio Observatory (ARO) Submillimeter Telescope (SMT) to assess the use of RD(N2H+) as an evolutionary tracer among massive protostellar/cluster cores in early stages. Our sample includes 32 dense cores in various evolutionary stages, from high-mass starless cores (HMSCs), high-mass protostellar objects (HMPOs), to ultra-compact (UC) HII regions, in infrared dark clouds (IRDCs) and high infrared extinction clouds. The results show a decreasing trend in deuterium fractionation with evolutionary stage traced by gas temperature and line width (Fig. 1). A moderate increasing trend of deuterium fractionation with the CO depletion factor is also found among cores in IRDCs and HMSCs. These suggest a general chemical behavior of deuterated species in low- and high-mass protostellar candidates. Upper limits to the ionization degree are also estimated to be in the range of 4 × 10−8 − 5 × 10−6.

Pancreases obtained from 1–9-day-old rabbits were digested with 1 mg/ml ?? collagenase and cultivated with low-glucose Dulbecco's modified Eagle's medium (DMEM). The cells were subcultured almost until confluence. They were identified as pancreatic stem/progenitor cells using two methods: an immunohistochemical staining technique and the response of glucose-stimulated insulin secretion after β-cell induction. Results showed that the cells of rabbits acquired strong proliferation ability, and expressed the markers of pancreatic stem/progenitor cells: PDX-1, CK19 and nestin. After β-cell induction for 2 days, 20% of cells were dithizone (DTZ) positive, rapidly increasing to 80% after 6 days. Insulin secretion was stimulated by a high glucose concentration (26 mmol/l). These powerful tests demonstrated that isolated and cultivated cells in rabbits were pancreatic stem/progenitor cells. This result has not yet been reported elsewhere.

A joule level of XeF(C-A) laser optically pumped by a sectioned surface discharge was developed. The irradiative intensity of pumping source was diagnosed by calculating XeF2 photo-dissociation wave evolvement which was photographed by a framing camera. The photon flux in the wavelength region of 140 to 170nm is about 5 × 1023 photon s−1cm−2, that corresponds to the irradiative brightness temperature of more than 25000 K. The laser experiments were carried out in different conditions. The maximum laser output energy of 2.5 J was obtained with the total conversion efficiency of 0.1%.

let $e/\mathbb{q}$ be an elliptic curve. for a prime $p$ of good reduction, let $e(\mathbb{f}_p)$ be the set of rational points defined over the finite field $\mathbb{f}_p$. denote by $\omega(\#e(\mathbb{f}_p))$ the number of distinct prime divisors of $\#e(\mathbb{f}_p)$. for an elliptic curve with complex multiplication, the normal order of $\omega(\#e(\mathbb{f}_p))$ is shown to be $\log \log p$. the normal order of the number of distinct prime factors of the exponent of $e(\mathbb{f}_p)$ is also studied.

We axiomatize the main properties of the classical Turán Theorem in order to apply it to a general context. We provide applications in the cases of number fields, function fields, and geometrically irreducible varieties over a finite field.

We axiomatize the main properties of the classical Erdös-Kac Theorem in order to apply it to a general context. We provide applications in the cases of number fields, function fields, and geometrically irreducible varieties over a finite field.