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Let Q (x) be a positive definite integral quadratic form with the determinant D being squarefree, and r (n , Q) denote the number of representations of n by the quadratic form Q. In this paper, we apply the Hardy-Littlewood-Kloosterman circle method to derive the asymptotic formula for the shifted convolution sum of the divisor function d (n) and Fourier coefficients r (n , Q). With more efforts, our method should have a number of applications for other multiplicative functions. [ABSTRACT FROM AUTHOR]

Let λ f (n) be the normalized Fourier coefficients of a Hecke-Maass or Hecke holomorphic cusp form f for congruence group Γ 0 (N) with level N and nebentypus χ N. Let Q (x) be a positive definite integral quadratic form, and r (n , Q) denote the number of representations of n by the quadratic form Q. In this paper, we apply Jutila's circle method to derive a sharp bound for the shifted convolution sum of Fourier coefficients λ f (n) and r (n , Q). We generalize and improve previous results without the Ramanujan conjecture. [ABSTRACT FROM AUTHOR]

Let d k (n) denote the k -th divisor function. In this paper, we study the asymptotic formula of the sum ∑ 1 ≤ n 1 , n 2 , ... , n l ≤ x d k (n 1 2 + n 2 2 + ⋯ + n l 2) , where n 1 , n 2 , ... , n l ∈ Z + , k ≥ 4 and l ≥ 3 are integers. Previously only the cases of k = 2 , 3 are studied. [ABSTRACT FROM AUTHOR]

Crustins represent one type of antimicrobial peptides (AMPs) that are key components of the innate immune process of crustaceans. This study successfully identified a novel crustin-like peptide, EcCrustin2, in ridgetail white prawn, Palaemon carinicauda (formerly Exopalaemon carinicauda). EcCrustin2 was found to be 1082 bp in length, with a 378 bp open reading frame (ORF) encoding 125 amino acids. The deduced amino acid sequence of EcCrustin2 exhibited characteristics of crustins in crustacean, including a Cys-rich region at the N-terminus as well as a whey acidic protein domain at the C-terminus. Phylogenetic analysis revealed that the EcCrustin2 was first clustered with Type I crustins, then with other crustins. Expression of EcCrustin2 was mainly detected in immune tissues, including hemocytes, gill and stomach. The expression level of EcCrustin2 was also significantly up-regulated after being exposed to lipopolysaccharide (LPS), lipoteichoic acid (LTA), Vibrio parahaemolyticus and Staphylococcus aureus. EHP infection could also induce EcCrustin2 expression in P. carinicauda. Knockdown of EcCrustin2 with siRNA increased the mortality of V. parahaemolyticus challenged shrimp. Finally, the recombinant EcCrustin2 protein was obtained and demonstrated a wide spectrum of antibacterial activity in vitro. These results indicated that EcCrustin2 takes part in the immune response against bacteria and EHP infection. • The crustin-like peptide exhibited typical characteristics of crustin. • The crustin-like peptide was mainly expressed in immune tissues and could be induced by bacteria and EHP. • Knocking down the gene of crustin-like peptide with siRNA increased the mortality of V. parahaemolyticus challenged shrimp. • The crustin-like peptide demonstrated a wide spectrum of antibacterial activity in vitro. [ABSTRACT FROM AUTHOR]