Abstract:
In this paper we obtain $\Omega$ and $\Omega_\pm$ estimates for a wide class of error terms $\Delta(x)$ appearing in Perron summation formula. We revisit some classical $\Omega$ and $\Omega_{\pm}$ bounds on $\Delta(x)$, and obtain $\Omega$ bounds for Lebesgue measure of the following types of sets: \begin{align*} \A_+&:=\{T\leq x \leq 2T: \Delta(x)> \lambda x^{\alpha}\},\\ \A_-&:=\{T\leq x \leq 2T: \Delta(x)< -\lambda x^{\alpha}\},\\ \A~&:=\{T\leq x \leq 2T: |\Delta(x)|>\lambda x^{\alpha}\}, \end{align*} where $\alpha, \lambda>0$. We also prove that if Lebesgue measure of $\A$ is $\Omega(T^{1-\delta})$ then \[\Delta(x)=\Omega_\pm(x^{\alpha-\delta})\] for any $0<\delta<\alpha$.

Abstract:
By the Chinese remainder theorem, the canonical map \[\Psi_n: R[X]/(X^n-1)\to \oplus_{d|n} R[X]/\Phi_d(X)\] is an isomorphism when $R$ is a field whose characteristic does not divide $n$ and $\Phi_d$ is the $d$th cyclotomic polynomial. When $R$ is the ring $\mathbf{Z}$ of rational integers, this map is injective but not surjective. In this paper, we give an explicit formula for the elementary divisors of the cokernel of $\Psi_n$(when $R=\mathbb{Z}$) using the prime factorisation of $n$. We also give a pictorial algorithm using Young Tableaux that takes $O(n^{3+\epsilon})$ bit operations for any $\epsilon > 0$ to determine a basis of Smith vectors (see Definition 3.1) for the codomain of $\Psi_n$. In general when $R$ is an integral domain, we prove that the determinant of $\Psi : R[X]/(\prod_j f_j) \to \bigoplus_j R[X]/(f_j)$ written with respect to the standard basis is $\prod_{1 \leqslant i < j \leqslant n} \mathcal{R}(f_j, f_i)$, where $f_i$'s are pairwise relatively prime monic polynomials and $\mathcal{R}(f_j, f_i)$ is the resultant of $f_j$ and $f_i$.

Abstract:
Highly pathogenic avian influenza virus type H5N1 represents one of the major causes of morbidity and mortality of poultry in both developed and developing countries. However, little is known about the transmission of this virus in developing countries that usually raise poultry as family-based farming. The study was conducted at 10 of total 64 administrative districts of Bangladesh that experienced H5N1 virus outbreaks since 2007. Trained field workers visited 30 rural families at each district to check family poultry management system. The collected data were transcribed and coded according to the standardized mutual performance of the field workers. Approximately two-third of farmers (67%) were rearing only chickens and remaining (33%) both chickens and ducks. Most of the farmers provided night shelter to their birds inside their living room (24%) or close proximate (69%). Usually ducks were scavenged in water land (58.6%) or paddy field (18.2%). The majority of owners (93%) also shared the same water land with migratory/wild birds for their daily necessity. The marketing system of poultry was characterized by comprehensive interactions among family poultry and commercial birds for prolonged duration. Unsold or newly bought birds were brought back to farmer’s house in almost all instances (97.8%). Findings from this study indicated that interactions of domestic chickens and ducks with their owners (through contaminated agricultural and fisheries tools or clothing) are partially, if not solely, responsible for wide spread transmission of Avian influenza virus type H5N1.