Elementary Proofs of Some Stirling Bounds

We give elementary proofs of several Stirling's precise bounds. We first improve all the precise bounds from the literature and give new precise bounds. In particular, we show that for all $n\ge 8$ $$\sqrt{2\pi n}\left(\frac{n}{e}\right)^n e^{\frac{1}{12n}-\frac{1}{360n^3+103n}} \ge n!\ge \sqrt{2\pi n}\left(\frac{n}{e}\right)^n e^{\frac{1}{12n}-\frac{1}{360n^3+102n}}$$ and for all $n\ge 3$ $$\sqrt{2\pi n}\left(\frac{n}{e}\right)^n e^{\frac{1}{12n+\frac{2}{5n}-\frac{1.1}{10n^3}}} \ge n!\ge \sqrt{2\pi n}\left(\frac{n}{e}\right)^n e^{\frac{1}{12n+\frac{2}{5n}-\frac{0.9}{10n^3}}}.$

Bounds for the Number of Tests in Non-Adaptive Randomized Algorithms for Group Testing

We study the group testing problem with non-adaptive randomized algorithms. Several models have been discussed in the literature to determine how to randomly choose the tests. For a model ${\cal M}$, let $m_{\cal M}(n,d)$ be the minimum number of tests required to detect at most $d$ defectives within $n$ items, with success probability at least $1-\delta$, for some constant $\delta$. In this paper, we study the measures c_{\cal M}(d)=\lim_{n\to \infty} \frac{m_{\cal M}(n,d)}{\ln n} \mbox{ and } c_{\cal M}=\lim_{d\to \infty} \frac{c_{\cal M}(d)}{d}. In the literature, the analyses of such models only give upper bounds for $c_{\cal M}(d)$ and $c_{\cal M}$, and for some of them, the bounds are not tight. We give new analyses that yield tight bounds for $c_{\cal M}(d)$ and $c_{\cal M}$ for all the known models~${\cal M}$