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Let f be a polynomial, a , b be two algebraic numbers, and (a n) n ≥ 1 be a sequence satisfying f (a 1) = a and f (a n) = a n − 1 for all n ≥ 2. We generalize the concept of primitive prime divisors to the shifted sequence (a n − b) n ≥ 1 and study the generalized Zsigmondy set, that is, the set of n such that all prime ideal divisors of a n − b divide at least one prior term. The finiteness of Zsigmondy sets are determined in various cases. [ABSTRACT FROM AUTHOR]

Let b ≥ 2 be an integer and A = (a n) n = 1 ∞ be a strictly increasing subsequence of positive integers with η : = lim sup n → ∞ a n + 1 a n < + ∞. For each irrational real number ξ , we denote by v ˆ b , A (ξ) the supremum of the real numbers v ˆ for which, for every sufficiently large integer N , the equation ‖ b a n ξ ‖ < (b a N ) − v ˆ has a solution n with 1 ≤ n ≤ N. For every v ˆ ∈ [ 0 , η ] , let V ˆ b , A (v ˆ) ( V ˆ b , A ⁎ (v ˆ)) be the set of all real numbers ξ such that v ˆ b , A (ξ) ≥ v ˆ ( v ˆ b , A (ξ) = v ˆ) respectively. In this paper, we give some results of the Hausdorfff dimensions of V ˆ b , A (v ˆ) and V ˆ b , A ⁎ (v ˆ). When η = 1 , we prove that the Hausdorfff dimensions of V ˆ b , A (v ˆ) and V ˆ b , A ⁎ (v ˆ) are equal to (1 − v ˆ 1 + v ˆ ) 2 for any v ˆ ∈ [ 0 , 1 ]. When η > 1 and lim n → ∞ ⁡ a n + 1 a n exists, we show that the Hausdorfff dimension of V ˆ b , A (v ˆ) is strictly less than (η − v ˆ η + v ˆ ) 2 for some v ˆ , which is different with the case η = 1 , and we give a lower bound of the Hausdorfff dimensions of V ˆ b , A (v ˆ) and V ˆ b , A ⁎ (v ˆ) for any v ˆ ∈ [ 0 , η ]. Furthermore, we show that this lower bound can be reached for some v ˆ. [ABSTRACT FROM AUTHOR]

Sulfamethoxazole (SMX), is a ubiquitous antibiotic in the aquatic environment and received concerns on its health hazards, especially its sub-lethal effects on non-target organisms which were remained largely unknown. In the present study, in order to investigate SMX induced tissue damages and reveal underlying mechanisms, marine mussels, Mytilus galloprovincialis were challenged to SMX series (0.5, 50 and 500 μg/L) for six-days followed by six-day-recovery. Comprehensive histopathological alteration (including qualitative, semi-quantitative and quantitative indices), together with transcriptional and (post-) translational responses of key factors (p38, NFκB and p53) in the p38-MAPK signaling pathway were analyzed in gills and digestive glands. Tissue-specific responses were clearly investigated with gills showing more prompt responses and digestive glands showing higher tolerance to SMX. The histopathology showed that SMX triggered inflammatory damages in both tissues and quantitative analysis revealed more significant responses, suggesting its potential as a valuable health indicator. SMX activated expressions of p38, NFκB and p53 at transcriptional and (post-) translational levels, especially after exposed to low level SMX, evidenced by p38 coupled with NFκB/p53 regulation on immunity defense in mussels. Less induction of targeted molecules under severe SMX exposure indicated such signaling transduction may not be efficient enough and can result in inflammatory damages. Taken together, this study expanded the understanding of aquatic SMX induced health risk in marine mussels and the underlying regulation mechanism through p38 signaling transduction. • Sulfamethoxazole triggered inflammation in gill and digestive gland of mussels. • Sulfamethoxazole activated p38, NFκB and p53 at transcriptional and translational level. • Sulfamethoxazole altered inflammation and apoptosis of mussels via p38-MAPK signaling pathway. [ABSTRACT FROM AUTHOR]