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I RAN THE WORLD
Sport Aid Videos
Sport Aid '88 Videos
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Chris Long
Do they know it's Christmas?
Chapter 2 Page 27 A few days later Band Aid was announced and shortly [...]
USA for Africa
Chapter 3 page 31 Around this time an American benefit record was also [...]
Cars Video
Chapter 4 Page 39 "What followed was an image of a small African [...]
Everybody wants to Run the World
Chapter 8 Page 84 "We talked about a music promo for Sport Aid and I [...]
Chapter 16 Page 226 The boys appeared on TV to launch the song. We had [...]
Sport Aid 1000 Erasure video
Chapter 15 Page 212 Erasure had agreed to re-record It Doesn't Have to [...]
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I Ran the World is the remarkable autobiography of Chris Long, founder and chairman of Sport Aid.
Chris was searching for something different when, aged 23, he headed to Greece to pursue his dream but events conspired against him and he returned to London broke and broken. Licking his wounds and searching for something to reboot his soul, he watched Michael Buerk's iconic 1984 BBC report on the famine in Ethiopia and thought 'I can help.'
A subsequent meeting with Bob Geldof changed his life. Chris went to Live Aid and thought it was the best thing he'd ever seen – but he believed people could do more than just listen to music and make a donation – they could actively engage in the cause. And what better vehicle than sport?
In May 1986 he staged the biggest mass participation sports event the world has ever seen. The TV show covering it was the most complex ever produced, involving 19.8 million people in 89 countries, all running at the same time, for the same reason. It raised $35 million for UNICEF and the Band Aid Trust and was a huge and historic success.
I Ran the World describes the challenges of getting sports stars, celebrities, heads of state and governments on board, including Bob Geldof and Band Aid, the United Nations and UNICEF, British Airways and the British royals, as well as numerous other organisations.
It also reveals – in unsparing detail – the devastating reasons behind the ultimate failure of Sport 88 and how it nearly drove Chris to suicide.
But it is also a story of hope – and how you can't keep a good man down.
Chris bounced back with Earthdive, a unique and innovative project to protect our precious underwater environment. Still determined to make a difference he developed a citizen science project to help protect our oceans and reefs.
The book is a passionate, sometimes funny and often insightful look into how one man and a small group of like-minded people really made a difference.
Finally, it provides a blueprint for a future global event, this time harnessing technology and by recognising the lifestyle and short attention span of generation millennial.
It begs a simple question – if you could change the world – would you?
© Chris Long 2020
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text:
\section*{Acknowledgment}

We are very grateful to an anonymous reviewer for helpful suggestions and encouraging us to establish the precise constant factors in the analysis, which allowed us to significantly strengthen an earlier version of our main result containing an unspecified constant.

\bibliographystyle{imaiai}

\section{Proofs of Algorithmic Upper Bounds} \label{app:dd}

Under the combinatorial prior, the algorithmic upper bound for Theorem \ref{thm:main1} in the regime $k \le n^{1-\Omega(1)}$ is proved in \cite{Joh16} using the following strategy:
\begin{itemize}
    \item {\bf Test design.} Generate the $T \times n$ test matrix $\mathbf{X}$ according to the {\em near-constant tests-per item} design: For each item $i = 1,\dotsc,n$, select $L = \lfloor \frac{T \ln 2}{k} \rfloor$ tests uniformly at random {\em with replacement}, and set the corresponding entries in the $i$-th column of $\mathbf{X}$ to one.
    \item {\bf Decoding algorithm.} Given the test outcomes, estimate the defective set using the {\em Definite Defectives (DD)} algorithm:
    \begin{itemize}[leftmargin=7ex]
        \item[(i)] Mark all items in negative tests as {\em definitely non-defective}, and all remaining items as {\em possibly defective} (PD);
        \item[(ii)] For any PD item appearing in some (necessarily positive) test without any other PD items, mark it as {\em definitely defective} (DD).
        \item[(iii)] Return the set of DD items as the final estimate.
    \end{itemize}
\end{itemize}
The main result of \cite{Joh16} states that when $k = \Theta(n^{\theta})$ with $\theta \in (0,1)$ and the number of tests satisfies
\begin{equation}
    T \ge \frac{ \max\{\theta,1-\theta\} }{\ln 2} \big(k \log_2 n\big) (1+\epsilon) \label{eq:T_old}
\end{equation}
for some $\epsilon > 0$, the resulting error probability approaches zero as $n \to \infty$.  Our goal is to generalize this result to denser sparsity regimes.

The condition \eqref{eq:T_old} ensures that $L = \lfloor \frac{T \ln 2}{k} \rfloor$ scales as $\omega(1)$.  Hence, the effect of rounding is negligible, in the sense that $L =  \frac{T \ln 2}{k} (1+o(1))$.  As in \cite{Joh16}, we subsequently work with the exact expression $L =  \frac{T \ln 2}{k}$ for notational convenience, since the $o(1)$ term does not affect the final result.

With the regime $k \le n^{1-\Omega(1)}$ having been handled in \cite{Joh16}, it suffices to consider $k = n^{1-o(1)}$.  In this regime, it holds that $T^*(n,k) = \frac{k \log_2 k}{\ln 2}$.  For convenience, we apply the fact that $\log_2 k = (\log_2 n)(1+o(1))$ (whenever $k = n^{1-o(1)}$), meaning that it suffices to show that the success probability approaches one when
\begin{equation}
    T \ge \frac{k \log_2 n}{\ln 2} (1+\epsilon). \label{eq:T_choice}
\end{equation}
Observe that this matches \eqref{eq:T_old}, but with the quantity
\begin{equation}
    m =\max\{\theta,1-\theta\} \label{eq:m_def}
\end{equation}
replaced by $m=1$.

{\bf Analysis.} We start with the following bound which is central to the analysis of \cite{Joh16}, and is conveniently non-asymptotic so can can also be used here: For any defective $i \in S$, denoting the final estimate by $\widehat{S}$, we have
\begin{align}
    \mathbb{P}[i \notin \widehat{S}]
    &\le \underbrace{\sum_{w \in [w_-,w_+]} \mathbb{P}[W^{(S\setminus i)} = w] \sum_{j=0}^L \mathbb{P}[M_i = j | W^{(S\setminus i)} = w] \phi_j(1/w_-, g^* L)}_{:=\Psi_1} \nonumber \\
    &\qquad\qquad\qquad + \underbrace{\mathbb{P}[W^{(S\setminus i)} \notin [w_-,w_+]]}_{:=\Psi_2} + \underbrace{\mathbb{P}[G > g^* | W^{(S\setminus i)} \notin [w_-,w_+]]}_{:=\Psi_3}, \label{eq:three_terms}
\end{align}
where:
\begin{itemize}
    \item $W^{(S\setminus i)}$ denotes the number of (necessarily positive) tests containing at least one item in $S \setminus \{i\}$, i.e., a defective item differing from $i$;
    \item $M_i$ denotes the number of tests containing $i \in S$ and no other defectives;
    \item $G$ denotes the number of non-defectives that do not appear in any negative tests;
    \item $w_-$ and $w_+$ are arbitrary thresholds, but should be chosen to ensure that $W^{(S\setminus i)} \in [w_-,w_+]$ with high probability;
    \item $g^*$ is an arbitrary threshold, but should be chosen to ensure that $G \le g^*$ with high probability;
    \item $\phi_j(s,V) = \sum_{\ell = 0}^j (-1)^{\ell} {j \choose \ell} (1 - \ell s)^V$ is a quantity arising from applying the inclusion-exclusion principle to a union of events in a coupon collector problem \cite[Appendix B]{Joh16}.
\end{itemize}
We set $w_-$, $w_+$, and $g^*$ in the same way as \cite{Joh16}:
\begin{align}
    w_- &= \frac{T}{2} (1-\delta) \label{eq:w-} \\
    w_+ &= \frac{T}{2} (1+\delta) \label{eq:w+} \\
    g^* &= n\Big(\frac{1}{2} + \delta\Big)^L, \label{eq:g*}
\end{align}
for some $\delta > 0$ to be specified later.  The interaction between $\delta$ and $\epsilon$ (see \eqref{eq:T_choice}) turns out to be slightly delicate, and choosing them appropriately is the main difference here compared to \cite{Joh16}.

The analysis of \cite{Joh16} focuses on the case that \eqref{eq:T_old} holds with equality.  This is without loss of generality, since additional tests can only ever help the DD algorithm.  We similarly assume that \eqref{eq:T_choice} holds with equality.
In view of the union bound over the $k$ defectives, the goal is to show that $k \Psi_{\nu} \to 0$ for $\nu \in \{1,2,3\}$ in \eqref{eq:three_terms}.  We proceed as follows:
\begin{enumerate}
    \item For $\Psi_1$, it is shown in \cite[Eq.~(39)]{Joh16} that if $L = m(1+\epsilon) \frac{\ln n}{\ln 2}$ (which holds via $L = \frac{T \ln 2}{k}$ and equality holding in \eqref{eq:T_old}) and $k \le c n^m$ for some constant $c$ (with $m$ given in \eqref{eq:m_def}), then
    \begin{align}
        k\Psi_1 \le c \exp\Big( \frac{L^2}{4w_-} \Big) \exp\bigg( -\Big( \epsilon - \frac{1+\epsilon}{\ln 2} \Big(\delta + \frac{g^* L}{w_-} (1-\delta)\Big)   \Big) m \ln n \bigg). \label{eq:kPsi1}
    \end{align}
    Recall that we are adopting the choice $m=1$; this means that the condition $k \le c n^m$ is trivially satisfied with $c = 1$.  Hence, if we can further establish that $\frac{L^2}{4w_-} = o(1)$ and $\frac{g^* L}{w_-} = o(1)$, it will follow from \eqref{eq:kPsi1} that
    \begin{align}
        k\Psi_1 \le (1+o(1)) \exp\bigg( -\Big( \epsilon - \frac{1+\epsilon}{\ln 2} (\delta + o(1)) \Big) \ln n \bigg).
    \end{align}
    This approaches zero as $n \to \infty$ when $\delta$ is strictly smaller than $\frac{\epsilon \ln 2}{1+\epsilon}$.  For concreteness, we set $\delta = \frac{2}{3}\epsilon$ (note that $\frac{2}{3} < \ln 2$), so that the preceding requirement holds when $\epsilon$ is sufficiently small.

    The above-mentioned requirement $\frac{L^2}{4w_-} = o(1)$ follows immediately from the fact that $L = \Theta(\log n)$ and $w_- = \Theta(T) = \Theta(k \log n)$ (with $k = n^{1-o(1)}$).  As for $\frac{g^* L}{w_-}$, the steps in \cite[Eq.~(31)]{Joh16} turn out to be too loose for our purposes, but are easily modified: Combining $L = \frac{T \ln 2}{k}$ with \eqref{eq:w-} gives $\frac{L}{w_-} = \frac{2 \ln 2}{k(1-\delta)}$, and further combining with \eqref{eq:g*} gives
    \begin{align}
        \ln \frac{g^* L}{w_-} = \ln \frac{n}{k} + L\ln\Big( \frac{1}{2} + \delta\Big) + \ln\frac{2 \ln 2}{1-\delta}. \label{eq:log_ratio}
    \end{align}
    The assumption $k = n^{1-o(1)}$ gives $\ln \frac{n}{k} = o(\log n)$, and combining this with $L = \Theta(\log n)$ and $\ln\big( \frac{1}{2} + \delta\big) < 0$ (for small enough $\delta$), it follows that the right-hand side of \eqref{eq:log_ratio} approaches $-\infty$, and hence $\frac{g^* L}{w_-} = o(1)$ as desired.
    \item For $\Psi_2$, we can directly use the following finding from \cite{Joh16} based on McDiarmid's inequality:
    \begin{equation}
        k \Psi_2 \le k\exp\Big( \frac{\delta^2 T}{4 \ln2} (1+o(1)) \Big).
    \end{equation}
    This approaches zero as $n \to \infty$, since $T = \Theta(k \log n)$.
    \item For $\Psi_3$, we use the following bound \cite[Eq.~(42)]{Joh16} based on Bernstein's inequality, which holds provided that $L \to \infty$ (which we already established) and $\delta \le \frac{1}{4}$:
    \begin{equation}
        k \Psi_3 \le k \exp\bigg( - n \frac{(1/2+\delta)^L}{2/3 + o(1)} \bigg). \label{eq:Psi3_bound}
    \end{equation}
    Recall that $L = \frac{T \ln 2}{k}$; substituting $T$ equaling the right-hand side of \eqref{eq:T_choice} gives $L = (1+\epsilon) \log_2 n$. We proceed by considering the logarithm (base 2) of $n(1/2+\delta)^L$:
    \begin{align}
        \log_2 \Big( n(1/2+\delta)^L \Big)
        &= \log_2 n + L \log_2 \Big(\frac{1}{2} + \delta\Big) \\
        &=  \big(\log_2 n\big) \bigg[ 1 + (1+\epsilon)\log_2\Big(\frac{1}{2} + \delta\Big) \bigg]. \label{eq:log_cond}
    \end{align}
    Using the above choice $\delta = \frac{2}{3} \epsilon$, a simple Taylor expansion yields the following as $\epsilon \to 0$:\footnote{In fact, a visual plot reveals that $1+(1+\epsilon)\log_2\big(\frac{1}{2} + \frac{2}{3}\epsilon\big)$ is positive for all $\epsilon > 0$.}
    \begin{equation}
        (1+\epsilon)\log_2\Big(\frac{1}{2} + \frac{2}{3}\epsilon\Big) = -1 + \epsilon\Big( \frac{2}{3} \cdot \frac{2}{\ln 2} - 1 \Big) + o(\epsilon).
    \end{equation}
    Hence, since $\frac{2}{3} \cdot \frac{2}{\ln 2} \approx 1.92 > 1$, we have for sufficiently small $\epsilon$ that \eqref{eq:log_cond} is positive and scales as $\Theta(\log n)$, and substituting into \eqref{eq:Psi3_bound} gives $k \Psi_3 \le k \exp\big( - n^{\Theta(1)} \big)\to 0$, as desired.
\end{enumerate}
Since the above analysis holds for arbitrarily small $\epsilon > 0$ (and hence arbitrarily small $\delta > 0$ via $\delta = \frac{2}{3}\epsilon$) when the number of tests satisfies \eqref{eq:T_choice} with equality, the upper bound in Theorem \ref{thm:main1} follows.

{\bf Handling the i.i.d.~prior.} While \cite{Joh16} only considers the combinatorial prior with a fixed value of $k$, the analogous result follows essentially immediately for the i.i.d.~prior, in which $k$ is a random variable.  This is because by a simple concentration argument (e.g., Hoeffding's inequality), as long as $np \to \infty$, it holds that $k = np(1+o(1))$ with probability approaching one.  We can therefore replace the choice $L =  \frac{T \ln 2}{k} (1+o(1))$ by $L =  \frac{T \ln 2}{np} (1+o(1))$, and under the high-probability event $k = np(1+o(1))$, the two are equivalent up to a change in the $o(1)$ term.  Since conditioning on any particular value of $k$ under the i.i.d.~prior brings us back to the combinatorial prior, the desired result follows.
\section{Introduction}

The group testing problem was originally studied in the context of testing blood samples for rare diseases \cite{Dor43}, with the key idea being to reduce the required number of tests via pooling.
Group testing has since found applications in communications \cite{Ant11}, information retrieval \cite{Cor05}, compressed sensing \cite{Gil08}, and most recently, COVID-19 testing \cite{Yel20}.

The problem is formally defined as follows: There are $n$ items $[n]=\sett{1,2,\ldots,n}$, a subset $S\subseteq [n]$ of which is defective, with $|S|=k$.  A number of tests are performed, each taking as input a subset of items, and returning positive if and only if the subset contains at least one defective item.  A group testing algorithm specifies the number of tests $T$, the items included in each test, and a decoder that returns an estimate $\hat{S}$ of the defective set given the test outcomes.   We are interested in the required number of tests to attain asymptotically vanishing error probability, i.e., $\lim_{n \to \infty} \mathbb{P}[\hat{S}\ne S] = 0$.

We focus on the non-adaptive setting, in which all tests must be specified prior to observing any outcomes; this is often highly desirable in applications, since it permits the tests to be implemented in parallel.  In this setting, the tests can be represented as a test matrix $\bb{X} \in \{0,1\}^{T \times n}$, where the $(i,j)$-th entry is 1 if and only if the $i$-th test contains the $j$-th item.  The test outcomes are then given by the element-wise ``OR'' of the $k$ columns corresponding to the defective items.  Mathematically, the $i$-th outcome is given by
\begin{align}
    Y^{(i)}=\bigvee_{j\in S}X_j^{(i)}, \label{eq:test_outcome_formula}
\end{align}
where $X_j^{(i)}$ is the $(i,j)$-th entry of $\bb{X}$.

We place a random model on the defective set $S$.  Throughout the majority of the paper, we assume that each item is included in $S$ (i.e., is defective) independently with some probability $p$ (possibly depending on $n$), referred to as the {\em prevalence}.  We also consider a closely-related model in which $k$ is fixed, and $S$ is a uniformly random subset of $[n]$ of cardinality $k$.  We refer to these models as the {\em i.i.d.~prior} and {\em combinatorial prior} respectively.  The two are closely related, since under the i.i.d.~model we have $k = np(1+o(1))$ with probability approaching one as long as $np = \omega(1)$.  See \cite[Sec.~1.A]{Ald19} for a more detailed summary of the connections between these models.

Throughout the paper, we make the mild assumption that $p \le \frac{1}{2}$ (i.i.d.~prior) or $k \le \frac{n}{2}$ (combinatorial prior).  Otherwise, the problem fails to be sparse, and it is already well-established that individual (one-by-one) testing is optimal even when adaptivity is allowed \cite{ungar1960,riccio2000sharper,aldridge2019rates}.  In addition, our analysis applies essentially unchanged when this factor of $\frac{1}{2}$ is replaced by any fixed constant less than one.

\section{Existing Results and Contributions} \label{sec:related}

Here we state the most relevant existing results on probabilistic non-adaptive group testing, and state our own main results in the context of these existing ones.  For consistency with the vast majority of existing works, we express the previously-known results in terms of $k$ and $n$, corresponding to the combinatorial prior.  However, the same results apply under the i.i.d.~prior when $k$ is replaced by $\overbar{k} := np$ throughout, under the mild assumption that $\overbar{k} \to \infty$ as $n \to \infty$.

\subsection{Main Existing and New Results}

A simple counting-based (or entropy-based) argument reveals that the number of tests for the high-probability recovery of $S$ must satisfy $T \ge (1-o(1)) \log_2 {n \choose k}$, or more simply $T = \Omega\big(k \log \frac{n}{k}\big)$ \cite{Mal78,Cha11,Ati12,Bal13}.  Moreover, this scaling is order-optimal in the following widely-considered regimes:
\begin{itemize}
    \item If $k \le n^{1-\Omega(1)}$, then we have $k \log \frac{n}{k} = \Theta(k \log n)$, and thus, the lower bound matches the ubiquitous $O(k \log n)$ upper bound obtained via random testing \cite{Fre75,Mal78,Cha11,Ald14a} or certain explicit designs \cite{Ina19}.
    \item If $k = \Theta(n)$, then we have $k \log \frac{n}{k} = \Theta(n)$, and thus, the lower bound matches the trivial $O(n)$ upper bound corresponding to testing each item individually.
\end{itemize}
While these observations cover the majority of scaling regimes, there remain ``mildly sublinear'' regimes in which the existing upper and lower bounds do not match, namely, $k = \Theta\big( \frac{n}{f(n)} \big)$ for any $f(n)$ satisfying $f(n) = \omega(1)$ and $f(n) = o(n^c)$ for all $c > 0$.  A notable example of such a regime is $k = \frac{n}{ {\rm poly}(\log n) }$.  Our first main result, stated below, closes this gap by showing that the correct scaling is always $\Theta( \min\{ k \log n, n\} )$.

Before stating the result, we introduce the following threshold:
\begin{equation}
    T^*(n,k) = \max\Big\{ k \log_2\frac{n}{k}, \frac{k \log_2 k}{\ln 2}\Big\}. \label{eq:T*}
\end{equation}
While the above discussion focuses on scaling laws, recent refined analyses \cite{Joh16,Ald18,Coj19,Coj19a} have nailed down the precise constants in the above-mentioned scaling regimes:
\begin{itemize}
    \item When $k \le n^{1-\Omega(1)}$, the optimal threshold for non-adaptive group testing is $T^*$ \cite{Coj19a}.  Specifically, there exists a strategy using $T \le (1+\epsilon)T^*$ tests that succeeds with probability approaching one and has decoding time polynomial in $n$, whereas any algorithm requires $T \ge (1-\epsilon)T^*$ to have a success probability bounded away from zero.
    \item When $k = \Theta(n)$, the optimal threshold for non-adaptive group testing is $n$ \cite{Ald18}.  Specifically, with $T = n$ one can trivially use one-by-one testing, whereas any strategy attaining success probability arbitrarily close to one must have $T \ge n-1$.\footnote{The subtraction of one is merely due to the fact that under the combinatorial prior, knowing the status of $n-1$ items also implies knowing the status of the remaining item.}
\end{itemize}
Based on these results, a reasonable guess is that the optimal threshold for group testing in general scaling regimes is $\min\{T^*(n,k),n\}$ (which scales as $\Theta( \min\{ k \log n, n\} )$, consistent with the above discussion).  Our main result, stated as follows, reveals that this is indeed the case.

\begin{thm} \label{thm:main1}
    In the non-adaptive group testing problem with $n$ items and prevalence $p$ (possibly depending on $n$) under the i.i.d.~prior with $p = \omega\big(\frac{1}{n}\big)$ and $p \le \frac{1}{2}$,\footnote{The assumption $p = \omega\big(\frac{1}{n}\big)$ ensures that $\overbar{k} = np = \omega(1)$, and hence the number of defectives concentrates around $\overbar{k}$.  The theorem cannot be true as stated when $p \le O\big(\frac{1}{n}\big)$; for example, even the trivial strategy of declaring every item non-defective has $\Omega(1)$ probability of succeeding, in contrast to the second part of the theorem.  In addition, our main novel contribution is handling the significantly denser regime $\overbar{k} = n^{1-o(1)}$.  } we have the following for any $\epsilon > 0$:
    \begin{itemize}
        \item There exists a test design and polynomial-time decoding algorithm using $T \le \min\{ (1+\epsilon)T^*(n,np), n\}$ tests and having a success probability approaching one as $n \to \infty$.
        \item Any group testing strategy having success probability bounded away from zero as $n \to \infty$ must use at least $T \ge (1-\epsilon)\min\{ T^*(n,np), n \}$ tests.
    \end{itemize}
\end{thm}

While we find it most convenient to establish this result for the i.i.d.~prior, we can use known connections between the two priors to establish the following analog for the combinatorial prior.

\begin{cor} \label{cor:main1}
    In the non-adaptive group testing problem with $n$ items and $k \le \frac{n}{2}$ defectives under the combinatorial prior, we have the following for any $\epsilon > 0$:
    \begin{itemize}
        \item There exists a test design and polynomial-time decoding algorithm using $T \le \min\{ (1+\epsilon)T^*(n,k), n\}$ tests and having a success probability approaching one as $n \to \infty$.
        \item Any group testing strategy having success probability bounded away from zero as $n \to \infty$ must use at least $T \ge (1-\epsilon)\min\{ T^*(n,k), n \}$ tests.
    \end{itemize}
\end{cor}

These results not only show that optimal non-adaptive group testing requires $T = \Theta(\min\{k \log n,n\})$ tests in general sparsity regimes, but also provide the precise underlying constants.

The algorithmic upper bounds in Theorem \ref{thm:main1} and Corollary \ref{cor:main1} are already known in the regime $k \le n^{1-\Omega(1)}$ \cite{Joh16,Coj19a}, and follow trivially from one-by-one testing when $\min\{ (1+\epsilon)T^*, n\} = n$.  Hence, it suffices to establish success using $(1+\epsilon)T^*$ tests in the regime $k = n^{1-o(1)}$.  Fortunately, although the analysis of the Definite Defectives (DD) algorithm in \cite{Joh16} was formally only stated for $k \le n^{1-\Omega(1)}$, the analysis can be adapted to the regime $k = n^{1-o(1)}$ with only minor modifications.  We detail the required changes in Appendix \ref{app:dd}.

As for the algorithm-independent lower bounds, with the regime $k \le n^{1-\Omega(1)}$ having been solved in \cite{Coj19a}, we can again focus on the regime $k = n^{1-o(1)}$ (including $k = \Theta(n)$).  In this case, we were unable to directly infer the desired result from \cite{Coj19a}, and we thus provide a detailed proof in Section \ref{sec:proofs}, though we still naturally re-use the main tools and ideas proposed in \cite{Coj19a}.

{\bf Specialization to dense regimes.} As hinted above, when the $\min\{T^*,n\}$ term is attained by $n$, our results indicate that one-by-one testing is asymptotically optimal.  This is consistent with the above-mentioned result of \cite{Ald18}, but also strengthens it in two ways:
\begin{itemize}
    \item Individual testing is not only asymptotically optimal when the goal is to succeed with probability approaching one, but also when the goal is attaining {\em any} strictly positive target success probability.
    \item Individual testing is not only asymptotically optimal when $k = \Theta(n)$, but also when $k = \omega\big( \frac{n}{\log n} \big)$, or even more generally, when $k > \frac{n \ln 2}{\log_2 n}$.
\end{itemize}
On the other hand, it is worth noting that our result only indicates failure when $T < (1-\epsilon)n$, whereas that of \cite{Ald18} handles the more general scenario $T < n-1$.  This distinction is necessary when establishing high-probability failure and/or handling the regime $k = o(n)$, since otherwise one could consider a strategy that (e.g.) tests the first $n-2$ items one-by-one and then guesses the remaining two to be non-defective.

{\bf Note on partially concurrent work.} In the initial version of our work, we focused only on the lower bound, and provided a weaker result with an unspecified coefficient to the $k \log n$ term in the $\min\{ k \log n, n \}$ scaling.  After releasing the initial version, the important case of $k = \Theta\big( \frac{n}{\log n}\big)$ was studied in more detail in \cite{Flo21}, giving upper and lower bounds with explicit constants.  The updated version of our work was developed in parallel with \cite{Flo21}, and establishes the precise constants.  Our upper bound in fact matches that of \cite{Flo21} (and is proved similarly), whereas a refinement of the main proof technique is needed to obtain our tight lower bound (see the stopping condition of Step 4(a), Procedure \ref{alg:fullalg}).

\subsection{Further Existing Results}

Before proceeding, we provide a brief summary of some further existing works.  Since these are less directly related to our work, we omit the details, and refer the reader to \cite{Du93,Ald19} for more detailed surveys.

For certain variants of group testing, the optimal number of tests is $\Theta\big( k \log \frac{n}{k}\big)$, as opposed to $\Theta(\min\{k \log n,n\})$ under the setup we consider.  Specifically, two notable cases with scaling $\Theta\big( k \log \frac{n}{k}\big)$ are (i) the adaptive setting, in which each test can be designed based on previous outcomes \cite{Hwa72,aldridge2019rates,aldridge2020conservative}, and (ii) the approximate recovery criterion, in which $\Theta(k)$ false positives and $\Theta(k)$ false negatives are allowed in the reconstruction \cite{Sca15b,Sca17}.

In contrast to the noiseless setting that we consider in this paper, in the {\em noisy} setting, the number of tests is at least $\Omega(k \log n)$ even if $k \log n \gg n$, and even if adaptivity is allowed \cite{Sca18}.

Finally, while the focus of our work is on high-probability recovery, extensive results have been established for the stronger guarantee of {\em uniform recovery}, i.e., a single test matrix that uniquely recovers any defective set of cardinality at most $k$, without allowing any error probability (e.g., see \cite{Kau64,Dya82,Du93,Che13a} and the references therein).  This stronger guarantee comes at the price of requiring significantly more tests, with a quadratic dependence on $k$ instead of a linear dependence.  In addition, the associated proof techniques are very different.

\begin{comment}
For adaptive testing in the linear regime, \citep{ungar1960} proved that in the probabilistic setting, individual testing is optimal for $p>\frac{3-\sqrt{5}}{2}\approx 0.382$, and \citep{riccio2000sharper} proved that in the combinatorial setting, individual testing is optimal for $p>1-\log_3 2\approx 0.369$. Outside these ranges, an algorithm by \citep{aldridge2019rates} achieves $T<1.05H_2(p)n$ in the former case and $T<1.11H_2(p)n$ in the latter, which is rather close to the trivial information-theoretic lower bound of $H_2(p)n$, where $H_2(p)=-p\log_2 p-(1-p)\log_2(1-p)$ refers to the binary entropy. The below figure from \citep{aldridge2019rates} illustrates the near-optimality of these algorithms:
\begin{Figure}
\centering
\includegraphics[width=220pt]{ald.png}
\end{Figure}
\end{comment}

\section{Proofs of Algorithm-Independent Lower Bounds} \label{sec:proofs}

We first consider Theorem \ref{thm:main1} regarding the i.i.d.~prior, and then turn to Corollary \ref{cor:main1} regarding the combinatorial prior.

\subsection{Proof of the Lower Bound for Theorem \ref{thm:main1}} \label{sec:pf_thm}

Our analysis builds on the ideas of \cite{Ald18,Coj19a}, both of which identify {\em totally disguised} items (see Definition \ref{def:disguised} below) whose defectivity status can be flipped without changing the test outcomes.  In \cite{Ald18}, one such item suffices for attaining the weak converse (i.e., $\mathbb{P}[\hat{S} \ne S] \not\to 0$) in the linear regime.  To obtain a stronger statement of the form $\mathbb{P}[\hat{S} \ne S] \to 1$ and also handle sublinear sparsity regimes, we follow the idea from \cite{Coj19a} of identifying {\em many} such items.

Specifically, we follow the high-level steps of \cite{Coj19a} and utilize certain auxiliary results therein, but modify the details in order to handle the regime $k = n^{1-o(1)}$ instead of $k \le n^{1-\Omega(1)}$.  The key idea is to identify many items that are disguised {\em independently of one another}.  We then apply an auxiliary result of \cite{Ald18} (see Lemma \ref{lem:Aldridge} below) along with some ``clean-up'' steps to ensure that its assumptions remain valid each time it is invoked.

In the following, we let $q=1-p$ for convenience.  The following useful definition was introduced in \cite{Ald18}.

\begin{defn} \label{def:disguised}
    \cite{Ald18}

    We say that an item $i$ is \textit{disguised} in test $t$ if at least one of the other items in the test is defective. We say that an item is \textit{totally disguised} if it is disguised in every test it is included in. Let $D_i$ denote the event that item $i$ is totally disguised.
\end{defn}

It is noted in \cite{Ald18} that if an item is totally disguised, then it remains totally disguised even if it is changed from defective to non-defective or vice versa.  Thus, under the i.i.d.~prior, the tests do not reveal any information about that item's defectivity status, and we have the following.

\begin{lem} \label{lem:cond_disguised}
    {\em (Implicit in \cite{Ald18} and \cite[Sec.~3]{Coj19a})}
    For any given test matrix $\mathbf{X}$, and a defective set $S$ generated according to the i.i.d.~prior, we have the following: Conditioned on a given item $i$ being totally disguised, that item is defective with conditional probability $p$ (i.e., the same as the prior defectivity probability).
\end{lem}

It follows that for any totally disguised item, the best the algorithm can do is choose the more likely outcome, and succeed with probability $\max\sett{p,1-p} = 1-p$ (recalling that we focus on the case that $p \le \frac{1}{2}$).

The following result from \cite{Ald18} is crucial for characterizing the probability of items being totally disguised.

\begin{lem} \label{lem:Aldridge}
    {\em \cite[Eq.~(1)]{Ald18}}
    Define $\mathcal{L}(p)=\min_{x=2,3,\ldots,n}x\ln(1-q^{x-1})$, where $q = 1-p$. If the test design $\bb{X}$ has no tests with 0 or 1 items, then
    \begin{align}
        \frac{1}{n}\sum_{i=1}^n\ln \mathbb{P}[D_i]\geq\frac{T}{n}\cdot \mathcal{L}(p) \label{eq:AldBound}
    \end{align}
    Hence, there exists an item $i$ with $\ln \mathbb{P}[D_i]\geq\frac{T}{n}\cdot \mathcal{L}(p)$.
\end{lem}

At a high level, this lemma is proved by directly calculating $\mathbb{P}[D_i]$ in terms of the $i$-th test size (which $x$ plays the role of in the definition of $\mathcal{L}(p)$), then averaging over the resulting log-values and applying some simple lower bounding techniques.

In the linear regime (i.e., $k = \Theta(n)$), one has $\mathcal{L}(p) = \Theta(1)$, and consequently, Lemma \ref{lem:Aldridge} directly implies that the success probability is bounded away from one, after removing all tests with 0 or 1 items \cite{Ald18}.
More generally, it is natural to ask whether there are, in fact, {\em many} items $i$ with $\ln \mathbb{P}[D_i]$ close to the the right-hand side of \eqref{eq:AldBound} \cite{Coj19a}. If we can find a ``large'' set $W$ of such items such that these items are totally disguised independently from each other, then we may apply standard binomial distribution concentration bounds to conclude that many totally disguised items exist, with high probability.

Following \cite{Coj19a}, we interpret the testing strategy as a bipartite graph $G_{\bb{X}}$ in which there is a vertex $v_i$ for each item $i$ and a vertex $v_t$ for each test $t$, with an edge between $v_i$ and $v_t$ if item $i$ is placed in test $t$.
Before constructing the desired set (denoted by $W$), we present two simple lemmas (which are analogous to \cite[Lemmas 3.7 and 3.8]{Coj19a}) and two subroutines that will be useful.

\begin{lem}
    \label{nobigtests}
    Let $z=\frac{2}{\ln\frac{1}{q}}$, and suppose that $T \le n$. Then, the probability that there exists a negative test containing more than $z\ln n$ items is at most $\frac{1}{n}$.
\end{lem}

\begin{proof}
    Recalling that $q = 1-p$, a given test containing at least $z\ln n$ items is negative with probability at most
    \begin{align}
        q^{z\ln n}=e^{z\ln q\ln n}=\frac{1}{n^2}
    \end{align}
    by the definition of $z$.  Since $T \le n$, a union bound yields the desired result.
\end{proof}

We henceforth assume that no test contains more than $z\ln n$ items, since Lemma \ref{nobigtests} implies that the decoder may declare all such tests to be positive without increasing the error probability by more than $\frac{1}{n}\to0$.

\begin{lem}
    \label{fewbigitems}
    Fix $\xi > 0$, and define an item to be {\em very-present} if it appears in more than $n^{\xi}$ tests. If $T \le n$ and no test contains more than $z\ln n$ items, then there are no more than $zn^{1-\xi}\ln n$ very-present items.
\end{lem}

\begin{proof}
    We count the number $P$ of pairs $(i,t)$ such that item $i$ is in test $t$. By assumption, $P \leq Tz\ln n \le nz\ln n$. Letting $n_{\mathrm{vp}}$ be the number of very-present items, it follows that $n_{\mathrm{vp}}n^{\xi}\leq P< nz\ln n$, and rearranging yields the desired result.
\end{proof}

\begin{algorithm}[t]
    \caption*{ \manuallabel{sub}{3.1} \textbf{Subroutine 3.1:} $\texttt{Clean}(\bb{X})$. \label{alg:clean} }

    \begin{enumerate}
        \item[1.] Identify the set of tests $T_{\leq1}$ containing 0 or 1 items, and the set of items $I$ contained in at least one test in $T_{\leq1}$.
        \item[2.] Return $\bb{X}_{\geq2}$, defined to be $\bb{X}$ with the rows and columns indexed by $T_{\leq1}$ and $I$ removed.
    \end{enumerate}
\end{algorithm}

\begin{algorithm}[t]
    \caption*{ \manuallabel{subr}{3.2} \textbf{Subroutine 3.2:} $\texttt{Extract}(\bb{X},W)$. \label{alg:extract} }

    \begin{enumerate}
        \item[1.] Let $\widetilde{D}_i$ be the event that $i$ is totally disguised with respect to $\bb{X}$. Let the item with the highest $\mathbb{P}[\widetilde{D}_i]$ be denoted by $i_0$, and set $W_{\rm next} = W \cup \{i_0\}$.
        \item[2.] Let $T_{\textrm{close}}$ and $I_{\textrm{close}}$ denote the sets of tests and items within distance at most 4 from $i_0$ in $G_{\bb{X}}$.
        \item[3.] Set $\bb{X}_{\rm pruned}$ to be $\bb{X}$ with the rows and columns indexed by $T_{\textrm{close}}$ and $I_{\rm close}$ removed.
        \item[4.] Return $(\bb{X}_{\textrm{pruned}},W_{\rm next})$
    \end{enumerate}
\end{algorithm}

We are now in a position to describe the construction of the desired set $W$, namely, a set of items that are disguised independently of one another.  To establish a hardness result, we would like to ensure that the size of $W$ and the probability of each $i \in W$ being disguised are both large enough, so that the resulting error probability is high.

Towards achieving this goal, we introduce Subroutines \ref{alg:clean} and \ref{alg:extract}.  \texttt{Clean} removes all tests with 0 or 1 items, allowing us to apply Lemma \ref{lem:Aldridge}, and \texttt{Extract} adds an item to $W$.  Both will be called multiple times in the construction of $W$, and their calls will reduce the effective $T$ and/or $n$.

\begin{algorithm}[t]
    \caption*{ \manuallabel{algo}{3.1} \textbf{Procedure 3.1:} $\texttt{ConstructSet}(\bb{X})$. \label{alg:fullalg} }

    \begin{enumerate}
        \item[1.] Let $G_0=G_{\bb{X}}$, and let $(n,T)$ be the number of items and tests in $\bb{X}$. Remove all very-present items from $G_0$ to obtain $G_1$. Let $G=G_1$.
        \item[2.] Initialize $W_0=\emptyset$, $i=1$.
        \item[3.] Set $\bb{X}_i\leftarrow$ test design represented by $G_i$. Set $\bb{X}_{\mathrm{tmp},i}\leftarrow\texttt{Clean}(\bb{X}_i)$, and let $(n_i,T_i)$ be the corresponding number of items and tests in $\bb{X}_{\mathrm{tmp},i}$.
        \item[4.] Perform the following:
        \begin{itemize}[leftmargin=3ex]
            \item[(a)] If $n_i > 0$ and $\frac{T_i}{n_i} \le (1+\xi) \frac{T}{n}$, then set $(\bb{X}_{i+1},W_{i}) \leftarrow\texttt{Extract}(\bb{X}_{\mathrm{tmp},i},W_{i-1})$, $G_{i+1}\leftarrow G_{\bb{X}_{i+1}}$, and $i\leftarrow i+1$, and return to Step 3.
            \item[(b)] Otherwise, terminate the procedure and return $W = W_{i-1}$.
        \end{itemize}
    \end{enumerate}
\end{algorithm}

The full procedure for constructing $W$ is described in Procedure \ref{alg:fullalg}, which depends on a generic constant $\xi > 0$; although its use in step 4(a) is not directly related to its used in Lemma \ref{fewbigitems}, we find it sufficient to use the same constant in both cases.
To justify step 1, we momentarily imagine that there exists a ``genie'' that tells the decoder the identity of the very-present items. Let the test results for $G_0$ and $G_1$ be $\bb{y}_0$ and $\bb{y}_1$ respectively; then, knowing $\bb{X}$, we see that $\bb{y}_0$ can be derived from $\bb{y}_1$ and the genie information. If we can prove that the error probability tends to one even with the help of the genie (and knowing $\bb{y}_1$), then it certainly tends to one without it, so step 1 is justified. After step 1, each item is contained in at most $n^{\xi}$ tests.

Let $w_i$ denote the $i$-th item placed in $W$. Let $D_{w_i}$ be the event that $w_i$ is totally disguised with respect to $\bb{X}_1$, and let $\widetilde{D}_{w_i}$ be the event that $w_i$ is totally disguised with respect to $\bb{X}_{{\rm tmp},i}$ (see Procedure \ref{alg:fullalg} for the definitions of  $\bb{X}_1$ and $\bb{X}_{{\rm tmp},i}$).
Since the totally disguised event $D_{w_i}$ only depends on the 2-neighborhood of $w_i$ in $G_1$, and the 2-neighborhoods of items in $W$ are pairwise disjoint by construction (due to the $\texttt{Extract}$ subroutine), the events $\sett{D_{w}:w\in W}$ are independent (this independence property for nodes having distance greater than $4$ was also used in \cite{Coj19a}).

Next, we state the following simple lemma relating the events $D_{w_i}$ and $\widetilde{D}_{w_i}$, both of which represent events of being totally disguised, but with  respect to different test matrices.

\begin{lem} \label{lem:EtoD}
    Under the preceding setup, we have $\mathbb{P}[D_{w_i}] \geq \mathbb{P}[\widetilde{D}_{w_i}]$.
\end{lem}
\begin{proof}
    In each \texttt{Clean}/\texttt{Extract} step, whenever we remove a test, we remove all of its items. It follows that $w_i$ is contained in $\textit{the same tests}$ in $\bb{X}_1$ and $\bb{X}_{\mathrm{tmp},i}$, except that each such test in $\bb{X}_{\mathrm{tmp},i}$ has \textit{fewer items}.  Since a disguised item always remains disguised when further items are added to its tests, it follows that $\widetilde{D}_{w_i}$ implies $D_{w_i}$.
\end{proof}

In addition, we have the following lower bound on $|W|$, the total number of extracted items.  Here and subsequently, we recall that to prove Theorem \ref{thm:main1}, it suffices to consider the regime $p = n^{-o(1)}$, since for any smaller $p$ (i.e. $p = n^{-\Omega(1)}$), Theorem \ref{thm:main1} was already established in \cite{Coj19a}.

\begin{lem} \label{lem:num_extracted}
    Under the preceding setup, if $p = n^{-o(1)}$ and $T \le (1-\epsilon)n$, then the size of the set $W$ returned by Procedure \ref{alg:fullalg} satisfies the following:
    \begin{equation}
        |W|\, \ge n^{1-3\xi}. \label{eq:W_asymp}
    \end{equation}
\end{lem}
\begin{proof}
    We first count the number of removed items as follows:
    \begin{itemize}
        \item No more than $T$ items alone in some test are removed by \texttt{Clean}.
        \item Lemma \ref{fewbigitems} implies that we removed at most $zn^{1-\xi}\ln n$ very-present items, and this scales as $o(n)$ due to the fact that $z = \frac{2}{\ln\frac{1}{1-p}} = \Theta\big(\frac{1}{p}\big) = n^{o(1)}$ (by the assumption $p = n^{-o(1)}$).
        \item By the assumption stated following Lemma \ref{nobigtests} and the removal of very-present items, each call to \texttt{Extract} removes at most $z^2n^{2\xi}\ln^2 n$ items.
    \end{itemize}
    We now argue by contradiction that \eqref{eq:W_asymp} must hold.  Suppose to the contrary that Procedure \ref{alg:fullalg} terminates at some iteration $i^* \le n^{1-3\xi}$.  Then, the above calculations imply that
    \begin{align}
        n_{i^*} &\ge n - T - zn^{1-\xi}\ln n- n^{1-3\xi} \cdot  z^2n^{2\xi}\ln^2n \label{eq:n_i_0} \\
        &\ge \epsilon n - o(n), \label{eq:n_i}
    \end{align}
    where we used the fact that $T \le (1-\epsilon)n$ and $z = n^{o(1)}$.  This means that the stopping condition met in step 4(a) cannot have been $n_i$ reaching zero, so it must have been $\frac{T_{i^*}}{n_{i^*}}$ exceeding $(1+\xi) \frac{T}{n}$.

    While \eqref{eq:n_i} indicates that the majority of items could eventually be removed in principle, this is only due to the subtraction of $T$ in \eqref{eq:n_i_0}; the other two terms behave as $o(n)$, and we conclude that the removal of very-present items and the calls to \texttt{Extract} collectively only remove $o(n)$ items.  Any further removal of items can only be due to the tests containing one item in \texttt{Clean}; removing these causes $T_i$ and $n_i$ to be reduced by the {\em same amount}.  However, as long as $\frac{T_i}{n_i} < 1$ (which holds by assumption for $i=0$, and subsequently for all $i \le i^*$ due to the stopping condition), reducing $T_i$ and $n_i$ by the same amount can {\em only make the ratio smaller} (i.e., $\frac{T_i - c}{n_i - c} \le \frac{T_i}{n_i}$ for any $c \in [0,T_i]$).

    More formally, suppose that up to index $i^*$, a total of $c$ items and tests are removed due to tests containing a single item, and a total of $c'$ items are removed for the other reasons mentioned above.  Then, we have
    \begin{equation}
        \frac{T_{i^*}}{n_{i^*}} \le \frac{ T - c }{ n - c - c' }. \label{eq:ratio}
    \end{equation}
    As established above, we have $0 \le c \le T \le n(1-\epsilon)$ and $c' = o(n)$.  However, since $\frac{T}{n} < 1$ by assumption, substituting these findings into \eqref{eq:ratio} reveals that $\frac{T_{i^*}}{n_{i^*}} \le \frac{T}{n}(1+o(1))$, which gives the desired contradiction to the stopping condition $\frac{T_{i^*}}{n_{i^*}} > (1+\xi) \frac{T}{n}$.

\end{proof}

Recall that all of the extracted items have independent totally disguised events, each with probability lower bounded according to Lemma \ref{lem:Aldridge}.
We need to consider applying this lemma with possibly smaller choices of $T$ and $n$ than the original values (namely, $T_i$ and $n_i$), but the stopping condition in step 4(a) of Procedure \ref{alg:fullalg} ensures that $\frac{T_i}{n_i} \le (1+\xi) \frac{T}{n}$.
As a result, Lemmas \ref{lem:Aldridge} and \ref{lem:EtoD} guarantee for any extracted item $i$ that
\begin{align}
    \mathbb{P}[D_i]\geq\exp\bigg( \frac{T(1+\xi)}{n}\cdot \mathcal{L}(p) \bigg), \label{eq:Di_bound1}
\end{align}
where we recall that $\mathcal{L}(p)=\min_{x=2,3,\ldots,n}x\ln(1-q^{x-1})$ with $q = 1-p$.  Note that $x\ln(1-q^{x-1}) < 0$, so this minimum is to be interpreted as ``most negative''. This minimum is characterized in the following lemma, which is similar to \cite[Claim 3.12]{Coj19a}.

\begin{lem} \label{lem:x_opt}
    For any $p \le \frac{1}{2}$ satisfying $p = n^{-o(1)}$, we have the following: (i) If $p = o(1)$, then $-\mathcal{L}(p) = \frac{(\ln 2)^2}{p} (1+o(1))$; (ii) If $p = \Theta(1)$, then  $-\mathcal{L}(p) = \Theta(1)$.
\end{lem}
\begin{proof}
    We provide a simple generalization of the argument from \cite[Claim 3.12]{Coj19a}, which focuses on the regime $k \le n^{1-\Omega(1)}$.
    We first write
    \begin{equation}
        -\mathcal{L}(p)=\max_{x=2,3,\ldots,n}x\ln\frac{1}{1-q^{x-1}}.
    \end{equation}
    This quantity is lower bounded by the argument corresponding to $x = \lceil \frac{1}{p}\rceil + 1$, which readily yields $\ln\frac{1}{1-q^{x-1}} = \Theta(1)$ and hence an $\Omega\big(\frac{1}{p}\big)$ lower bound on $-\mathcal{L}(p)$.

    For the upper bound, we note that for $x = o\big( \frac{1}{p} \big)$ we have $q^{x-1} = (1-p)^{x-1} = 1-\Theta(px)$, so the objective function behaves as $O(x \ln \frac{1}{px})$, which is  $o\big(\frac{1}{p}\big)$ (since $px \ln \frac{1}{px} \to 0$ as $px \to 0$).  On the other hand, if $x = \omega\big(\frac{1}{p}\big)$, then $q^{x-1} = (1-p)^{x-1} \to 0$, so the objective behaves as $O(x (1-p)^{x-1}) = O(x e^{-px})$, which is $o\big(\frac{1}{p}\big)$  (since $px e^{-\Theta(px)} \to 0$ as $px \to \infty$).  Hence, the optimal choice of $x$ must scale as $\Theta\big(\frac{1}{p}\big)$, and in this case, we have $\ln\frac{1}{1-q^{x-1}} = \Theta(1)$, yielding an $O\big(\frac{1}{p}\big)$ upper bound on $-\mathcal{L}(p)$.

    The second part of the lemma follows immediately, whereas for the first part, a refined analysis is needed.  For $p = o(1)$ and $x = \Theta\big( \frac{1}{p} \big)$, we have
    \begin{align}
        x\ln\frac{1}{1-(1-p)^{x-1}}
        &= x\ln\frac{1}{1-e^{-px}(1+O(p))} \\
        &= -x\ln(1-e^{-px}) + O(px)
    \end{align}
    by standard Taylor expansions.  Since $O(px) = O(1)$ is asymptotically negligible compared to the $\Theta\big( \frac{1}{p} \big)$ scaling derived above, it suffices to consider maximizing the first term.  As noted in \cite{Coj19a}, we can define $d = px$ and write this term as $\frac{1}{p}\big( - d \ln(1-e^{-d}) \big)$, and it is a simple differentiation exercise to verify that $-d \ln(1-e^{-d})$ is maximized at $d = \ln 2$, with maximum value $(\ln 2)^2$.  While the corresponding choice $x = \frac{d}{p}$ may not be integer-valued, the effect of rounding is asymptotically negligible for $p = o(1)$ by the continuity of the function $-d \ln(1-e^{-d})$.
\end{proof}

Combining Lemma \ref{lem:x_opt} with \eqref{eq:Di_bound1}, we obtain for some $c_p = \Theta(1)$ that
\begin{align}
    \mathbb{P}[D_i]\geq\exp\bigg( - \frac{(1+\xi) c_p T}{n p} \bigg), \label{eq:Di_bound2}
\end{align}
and moreover, when $p = o(1)$ we specifically have $c_p = (\ln 2)^2 (1+o(1))$.

Since the events $\{D_i\}_{i \in W}$ are mutually independent by construction, and $|W| \ge n^{1-3\xi}$ according to Lemma \ref{lem:num_extracted}, we deduce that the number of totally disguised items is stochastically dominated by ${\rm Binomial}( n^{1-3\xi}, e^{-\frac{(1+\xi) c_p T}{ np }} )$.  In particular, the average number of totally disguised items is
\begin{equation}
    n^{1-3\xi} e^{-\frac{(1+\xi) c_p T}{ np }},
\end{equation}
and by simple re-arrangements, this is lower bounded by $n^{\xi}$ whenever
\begin{equation}
    T \le \frac{np(1-4\xi)}{(1+\xi) c_p } \ln n. \label{eq:T_bound2}
\end{equation}
Thus, by the multiplicative form of the Chernoff bound, the actual number is at least $N_{\min} := \frac{1}{2} n^{\xi}$ with probability approaching one when \eqref{eq:T_bound2} holds.

By Lemma \ref{lem:cond_disguised} and the assumption $p \le \frac{1}{2}$, for any item that is disguised, the optimal algorithm can do no better than declare it to be non-defective, and the resulting probability of being correct is at most
\begin{equation}
    (1-p)^{N_{\min}} \le e^{-p N_{\min}} = e^{-\frac{p}{2} n^{\xi} } = o(1),
\end{equation}
where the last step follows from the assumption $p =  n^{-o(1)}$.  Thus, we have proved that $\mathbb{P}[\hat{S} = S] = o(1)$ whenever $T \le (1-\epsilon) n$ and \eqref{eq:T_bound2} holds.

When $p = \Theta(1)$ (or more generally $p = \omega\big( \frac{1}{\log n}\big)$), the stricter of these two conditions is $T \le (1-\epsilon) n$, and the constant $c_p$ in \eqref{eq:T_bound2} is inconsequential.  On the other hand, when $p = o(1)$, we have established that $c_p = (\ln 2)^2 (1+o(1))$.  Since $\xi$ can be arbitrarily small, it follows that \eqref{eq:T_bound2} reduces to $T \le (1-\epsilon') \frac{np}{ (\ln 2)^2 } \ln n$ for arbitrarily small $\epsilon' > 0$.  Finally, since $\frac{\ln n}{\ln 2} = \log_2 n$, and the assumption $p = n^{-o(1)}$ implies that $\log_2 n = (\log_2 k)(1+o(1))$, we obtain the desired threshold corresponding to the second term of $T^*$ in \eqref{eq:T*}.

\subsection{Proof of the Lower Bound for Corollary \ref{cor:main1}} \label{sec:pf_cor}

We utilize an approach from \cite[Lemma 3.6]{Coj19a} for transferring the key auxiliary results on the number of disguised items from the i.i.d.~prior to the combinatorial prior.  Despite the high level of similarity, we provide the main details for completeness.

The idea is to show that with too few tests, the number of totally disguised defectives and totally disguised non-defectives both grow unbounded with high probability.  When this occurs, interchanging the statuses among these items would not impact the test results, and hence, there exist an unbounded number of candidate defective sets of cardinality $k$ consistent with the test outcomes.  The decoder cannot do any better than guess one of these at random, failing with high probability. This intuition is easily made precise \cite{Coj19a}, giving the following.

\begin{lem} \label{lem:cond_error}
    {\em \cite[Facts 3.1 and 3.3]{Coj19a}}
    Under the combinatorial prior, the conditional error probability of any group testing strategy given that there are $\tilde{n}_0$ totally disguised non-defectives and $\tilde{n}_1$ totally disguised defectives is at least $1 - \frac{1}{\tilde{n}_0 \tilde{n}_1}$.   In particular, if $\tilde{n}_0 = \omega(1)$ and $\tilde{n}_1 = \omega(1)$, then the conditional error probability is $1-o(1)$.
\end{lem}

Consider the combinatorial prior with $n^{1-o(1)} \le k \le \frac{n}{2}$, where the condition $k \ge n^{1-o(1)}$ is safe to assume since Corollary \ref{cor:main1} is already well-known when $k \le n^{1-\Omega(1)}$ (see Section \ref{sec:related}).  We consider generating $S$ according to the following procedure:
\begin{enumerate}
    \item Let $S_{0} \subseteq [n]$ include each item independently with probability $p_0 = \frac{k - \sqrt{k} \ln n}{n}$.  That is, $S_0$ follows the i.i.d.~prior with parameter $p_0$.
    \item Form $S$ by adding $\max\{k - |S_0|, 0\}$ elements of $[n] \setminus S_0$ to $S_0$, chosen uniformly at random.
\end{enumerate}
By the symmetry of this construction, conditioned on the event $|S_0| \le k$, the resulting set $S$ is indeed distributed according to the combinatorial prior.  While $|S_0| >  k$ has a non-zero probability, for the purposes of proving a converse, we can simply assume that this event always leads to successful recovery.  Since we assume that $k \ge n^{1-o(1)}$, a simple concentration argument (e.g., the Chernoff bound or central limit theorem) gives with probability $1-o(1)$ that
\begin{equation}
    k - 2\sqrt{k} \ln n \le |S_0| \le k, \label{eq:conc}
\end{equation}
so the resulting contribution to the success probability is asymptotically negligible.

We now introduce the terminology that an item $i$ is {\em totally disguised in the first step} if the defectives from $S_0$ alone are enough to disguise $i$ in every test it is included in.  Clearly, being totally disguised in the first step is sufficient for being totally disguised after the second step, since the second step only involves marking more items as defective.

Hence, trivially, the number of totally disguised defective items only increases (or stays the same) after the second step.  The number of totally disguised non-defectives may in principle decrease due to non-defectives being changed to defective, but conditioned on \eqref{eq:conc}, any given non-defective is only changed with probability $O\big( \frac{\sqrt{k} \ln n}{n} \big) = o(1)$.  As a result, if there are $\omega(1)$ totally disguised non-defectives, the same still remains true with probability $1-o(1)$ after the second step.

Hence, in accordance with Lemma \ref{lem:cond_error}, it suffices to show that under the i.i.d.~prior with parameter $p_0 = \frac{k - \sqrt{k} \ln n}{n}$, the number of totally disguised defectives and totally disguised non-defectives both behave as $\omega(1)$ with probability $1-o(1)$.   Note that the assumption $n^{1-o(1)} \le k \le \frac{n}{2}$ ensures that $n^{-o(1)} \le p_0 \le \frac{1}{2}$, as was assumed in the later parts of Section \ref{sec:pf_thm}.

We already argued that when \eqref{eq:T_bound2} holds, the average number of totally disguised items is at least $n^{\xi}$.  Since $n^{-o(1)} \le p_0 \le \frac{1}{2}$, it follows that the average number of totally disguised defectives and totally disguised non-defectives are both at least $n^{\xi-o(1)}$ on average.  Again, the multiplicative form of the Chernoff bound implies the same with high probability, and we have the desired $\omega(1)$ scaling.  This establishes that the condition on $T$ from Theorem \ref{thm:main1} with $p_0$ in place of $p$ is necessary for attaining a success probability bounded away from zero, and since $np_0 = k(1+o(1))$ by definition, Corollary \ref{cor:main1} follows.

\section{Conclusion} \label{sec:conclusion}

We have proved that the optimal number of tests for probabilistic noiseless non-adaptive group testing is $\Theta(\min\{k \ln n,n\})$, as well as establishing the precise underlying constant factors.  This closes gaps exhibited by existing bounds in the in the case that $k$ is ``mildly'' sublinear in $n$, so that the optimal thresholds are now known for arbitrary scaling regimes.  Perhaps the main challenge remaining in this setting is to understand how the number of tests increases when the target error probability decreases to zero at a given rate depending on $n$.

redpajama_set_name: RedPajamaArXiv


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By engaging in simple everyday activities and play, young children develop physical, social, intellectual and emotional skills as well as build creativity and imagination. Participants will explore budget friendly strategies to promote curiosity in young learners and integrate S.T.E.A.M throughout the early childhood day. Participants will also learn how to create engaging activities that encourage innovative thinking, problem-solving, and social-emotional development.
FREE for members of the Early Childhood Services & PreK Cooperative; Teachers of Early Head Start & Head Start (for which Region 10 is the grantee); & PPCD Teachers. $100 for Non-Cooperative & Out-of-Region.


5. Hmmm ... If dogs serve humans, and humans serve cats, why can't we cats ever get these stupid dogs to do anything for us?
7. Hey -- no kidding, I'm sure that's the can opener.
8. Would humans have built a vast and complex civilization of their own if we cats hadn't given them a reason to invent sofas and can openers in the first place?
10. If that really was the can opener, I'll play finicky just to let them know who's boss!
У котов есть хвост. Если хвост морковкой – к зайцам.


text:
Q: How to change text within a dataset (div) using javascript (CLASS DOESN'T WORK) <div data-e2e="topBalanceTotal" class="templatesuccess__Wrapper-sc-4pme9a-0 dWpLPl">$0.00</div>
I want to change the inner text of this class, I use the following to try and do so but the console outputs undefined.
document.getElementByClassName('templatesuccess__Wrapper-sc-4pme9a-0 dWpLPl').innerHTML = '$100';
How do I get the following text? Maybe by getting the element by using data-e2e? I'm confused.

redpajama_set_name: RedPajamaStackExchange


why must bacteria have ribosomes when they lack other organelles?
Describe mechanism of sympathetic control of kidney function?
Why does water flow out of collecting duct when aquaporin channels?
What is job of aquaporin channel and where are the channel found?


text:
We have now entered our 9th decade of helping special needs and disadvantaged children.
We had a ball in January 2019 with our annual 'Mad Hatter's Tea Party', where our guest stars Donchez Dacres, Calum Courtney and Lydia Lucy sang and entertained the children. It was another huge success.
On Valentine's Day 150 of our supporters, benefactors, fundraisers and volunteer London licensed taxi drivers all made there way to Buckingham Palace for a reception, hosted by HRH The Duchess of Cornwall, to celebrate our rebrand and 90 years of history.
On Thursday 6 June, we held a fundraiser at the Russell Square cabmen's shelter to raise money for our 2019 Southend-on-Sea outing, when we took over 250 very excited special needs and disadvantaged children, along with their carers, to the seaside for the day.
If you are a licensed London taxi driver and would like to volunteer for our outings, then please get in touch via our Get Involved page.
Whilst we are well known for our trips and outings, our appeals work is the core of the charity.
Our appeals officers work with the rest of the committee to determine our involvement and how we can help.
To date our funds have provided essential medical equipment, technology and recreational aids to truly amazing children.
We have delivered play areas and playgrounds, provided funds for hospice renovations and sourced specialist mobility aids and to do this, we rely heavily on your help.
Get involved in one of the many ways to raise funds for the charity and be a part of the fantastic work we do.
GET INVOLVED NOW
The committee consists of working taxi drivers and associate members, and any time spent raising money for events is done voluntarily. The members of the committee do not draw any money from the fund and this is how the running costs are kept down to a bare minimum.
The committee has overall responsibility for strategic direction of the charity and its financial overview. They meet every month to plan for future outings and events and during the meeting the appeals received for financial help are reviewed.
The charity is indebted to all its supporters and benefactors and specifically the London taxi drivers who give their time freely taking vulnerable and disadvantaged children and their carers on outings to the seaside, zoos and other venues.
Michael Bernard Son BEM
Born: London 1943
Michael joined the ranks of the 'world's finest' in 1965 and still enjoys the freedom of taxi driving and interacting with so many different people. He joined the charity in 1986 and has taken on various roles and responsibilities including Chairman, Secretary, Public Relations and Appeals Officer and continues to be a very active member of the team. To raise funds and increase the profile of the charity he has produced classical music concerts in some iconic venues including St Paul's Cathedral. Michael received the British Empire Medal in 2016 for his services to charity.
Malcolm Shaffron
Malcolm became a taxi driver in 1969 and within two years signed up for his first outing and has been on all of the outings since 1971. He joined the committee in 1975 becoming Secretary four years later in 1979. Malcolm served as Chairman for 3 years before resuming his role as Secretary a role he performs to this day.
Gerry Dunn MBE
Born: Buckinghamshire 1944
Gerry joined the charity as a driver soon after he got his green badge in 1966 before being asked to join the committee. He has been the Treasurer on and off for over 17 years and has been Chairman, Sponsor organiser, Asst. PRO, Appeals Officer, and raffle and tombola organiser. Gerry was also the Master of the Fellowship of Hackney Carriage Drivers and a founder member of the LTDA. He has organised many charity trips to the Romanian orphanages and since 1966 has been a volunteer driver for the Taxi Charity for Military Veterans. He was awarded the MBE for his services to charity. One of the reasons Gerry wanted to get involved with the charity taking orphans from the Norwood Orphanage, was because his father was an orphan there himself.
Alan Cohen MBE
Born: Derbyshire 1942
Alan got his taxi licence in 1966 and joined the committee in 1968. He served as Chairman, as PRO for over 20 years, and instigated the Diamond Jubilee outing to Buckingham Palace. Alan has organised the Woburn Taxi Safari and trips to Chessington, Lego Land, Brighton and a party at Buckingham Palace, and has led the Southend outing more times than he can remember. He organised two film premieres, one of which was attended by HM The Queen along with Princes Andrew and Edward. Alan was awarded the MBE for services to charity in 2004 and is currently the longest serving member on the committee.
Claire Zazzara
Claire became a taxi driver in February 2012 and attended her first Southend outing as a driver in 2014. She loved decorating her taxi for the trips and won the Best Dressed Taxi competition in 2016, and again in 2018. Claire joined the committee in 2019 and is keeping our social media platforms up to date with news stories and events. She is also getting stuck in to fundraising for the charity, as she is part of the Cabbies Do Kili team who are aiming to climb Mount Kilimanjaro in October.
Colin Greaves
Minutes Secretary
Having supported the charity by attending numerous outings as a 'driver' following qualifying as a Licensed London Taxi Driver in 2003, Colin joined the Committee of the LTCFC in April 2015. He has served as Minutes Secretary since joining the Committee and has also served as the Appeals Officer from 2016-19. He recently organised, co-ordinated and participated in the charity's cycle team that completed the 2019 London to Brighton Cycle Ride, which raised over £8000 for the charity.
Born: Essex 1970
Driver Liaison
John has worked in the taxi industry for over 30 years. In 2000 he got his taxi licence and started driving for the charity. John was invited to join the committee in 2018 and has been the Driver Liaison since then.
Child Liaison
Paul became a taxi driver in 1993 and within six months did his first Southend outing, joining the committee two years later. Previously organising the Appeals for the charity, Paul went on to become the Child Liaison officer and has continued in the role to this day. His role involves arranging for children from varied backgrounds and with different needs to attend our events. Every child is special regardless of creed, colour or physical ability, and as a parent once said to him, "you not only give the children memories, you give the parents memories for the children with life limiting illnesses."
Steve became a Taxi Driver in 2000 and signed up to be a driver for the charity on his way home from collecting his badge. He joined the committee in 2011 and was Driver Liaison for 6 years, and has also dealt with Sponsorship and Entertainment too. With a creative streak and a very vivid imagination Steve won the Southend Outing Best Dressed Taxi competition 5 years running and is still happy to dress up taxis for fundraising opportunities.
Larry Abrahams
Born: Cambridge 1942
Associate Committee Member
Larry became a London Taxi Driver in 1966 just before the World Cup and got involved with the charity in the 90's. Always ready and available to do anything as required, he really enjoyed being involved with organising the Sponsorship side of the charity.
Simeos Yiannikaris
Appeals Officer
Sim having completed the knowledge in 1993 volunteered as a driver in the spring and summer outings the following year. In 1995 he joined the committee and has had the honour of serving as Chairman and Treasurer over the years. Sim is currently undertaking the role of Appeals officer.
Andy is an Associate Committee Member and not from a Taxi Driver background. Andy's involvement with the Charity started through a Company Charity sponsorship scheme a number of years ago and has continued to support the Charity ever since, joining the Committee just over a year ago.
Wendy Bell
Wendy joined as an associate member in 2019 but has been involved with the charity since husband Steven Bell joined the committee in 2011. With a background in clerical, Wendy aims to assist the charity with this function including keeping the website and procedure manuals up to date.


Below find works completed by President of Abe Art, Nicole Titus. This is merely Ms. Titus' personal portfolio and not a reflection of the hundreds of pieces available for commercial art purposes. Some artwork below is available for purchase.
…fabulous, with exquisite taste and fresh originality–and heart!


text:
Home Tags Keith Simpson
Tag: Keith Simpson
Corrections gradually embraces innovation in access control
June 20, 2012David McNabb
Biometrics, hand-held alarms, more reliable perimeter detection and other advanced technologies for protecting, identifying and tracking humans are finding their way into North America's prisons, jails and detention centers despite the corrections industry's reputation for foot-dragging when it comes to embracing cutting-edge security."Corrections is not exactly a leader" in adopting the latest innovation, said Martha Lee, vice president of sales at Ottawa-based Senstar, a global player...
ASTM, Brian Mikiten, corrections, Ian Francisco, Jerry Forstater, Keith Simpson, Martha Lee, Professional Systems Engineering, Secure Control Systems, Security Systems News
Simpson Security Systems finishing $7m prison job
ALEXANDRIA, La.—Shrinking profit margins, a common lament among integrators, are something Keith Simpson, CEO of Simpson Security Systems, says he's experienced on smaller commercial jobs, but larger projects remain nicely profitable for his company.Why? There's less competition for "true integration" jobs, Simpson told Security Systems News."You don't get today [what you used to get] for an eight-camera job," Simpson said. "There's a broad range of competition...
Federal Bureau of Prisons, Keith Simpson, Kevin Simpson, Simpson Security Systems


text:
A critique of a method to determine long-term decline of coral reef ecosystems
Ridd, Peter V. (2007) A critique of a method to determine long-term decline of coral reef ecosystems. Energy and Environment, 18 (6). pp. 783-796.
PDF (Published Version)
View at Publisher Website: http://www.multi-science.co.uk/ee.htm
The supposedly already-degraded state of coral reef ecosystems is sometimes claimed to be a reason why anthropogenic global warming will have a major impact on the reefs, i.e. they are already close to extinction and can easily be tipped over the edge. Recently published work by Pandolfi et al. (2003) in Science has outlined a method for measuring the decline of coral reef ecosystems throughout the world according to which the outer and inner Great Barrier Reef (GBR) are claimed to be 28% and 36% respectively, down the path towards ecological extinction. This is a highly significant claim given the important status of the GBR, so the result deserves attention and objective scrutiny. This paper sets out to scrutinise the methodology used by Pandolfi et al. (2003) under four headings: (i) the guilds are poorly weighted and focus largely on human target species, rather than species that are ecologically important to reefs; (ii) the numerical scale used to "measure" the state of the reefs is not well-founded and hence distorts the result; (iii) the analysis fails to recognize that the GBR is of relatively recent origin and therefore never existed in the pre-human/pristine cultural period as defined by PAN; and (iv) in many cases it is doubtful that the literature cited demonstrates the claimed decline in ecological state. It is concluded that the work of Pandolfi et al. (2003) cannot be used as justification that the Great Barrier Reef has lost significant resilience, or that it is particularly susceptible to global warming because of its present supposedly degraded state.
coral reef; ecosystem decline; anthropogenic effects; Great Barrier Reef
Reproduced with permission from Energy and Environment. This publication was originally published in Energy and Environment, Year 2007, Vol. 18, Issue 6.
96 ENVIRONMENT > 9606 Environmental and Natural Resource Evaluation > 960699 Environmental and Natural Resource Evaluation not elsewhere classified @ 100%


text:
Caratteristiche tecniche
È un difensore centrale.

Carriera

Club
Comincia a giocare nella Guadalupa, al Red Star. Nel 2006 passa all'Evolucas. Nel gennaio 2010 si trasferisce in Francia, al Vannes. Nella stagione 2010-2011 passa alla seconda squadra del Vannes. Nell'estate 2012 viene acquistato dalla seconda squadra del Caen. Nel gennaio 2013 passa al Cherbourg. Nell'estate 2013 si accasa all'Uzès. Nel 2014 viene acquistato dall'.

Nazionale
Ha debuttato in Nazionale il 18 gennaio 2008, nell'amichevole Dominica-Guadalupa (0-3). Ha partecipato, con la Nazionale, alla Gold Cup 2009. Ha collezionato in totale, con la maglia della Nazionale, 13 presenze.

Collegamenti esterni

Calciatori della Nazionale guadalupense

redpajama_set_name: RedPajamaWikipedia


text:
Best Binoculars
Best Scopes
Squirrel Proof
Bird Feeder Plans
Hummingbirds - Nectar Recipe, Gardens, Migration, Feeders, Fun Facts and More!
Who doesn't like hummingbirds? Mean people, we guess?! Really, have you ever heard anyone say, "Bah, hummingbirds!"
Zippin' and hummin' around our flowers and feeders, these tiny little birds are an endless source of joy and amusement for all. Let's explore the world of our beloved hummers and learn some interesting facts about these little flying jewels.
Depending if you're a taxonomic lumper or splitter, there are somewhere between 325-340 species of hummingbirds in the world, making them the second largest family of birds after flycatchers.
The family is divided into two sub-families: Phaethornithinae (Hermits) currently composed of 34 species, and the larger group, Trochilinae (Trochilines or "typical" hummingbirds) composed of the remaining 291-306 species.
Order: Trochiliformes
Family: Trochilidae
Hummingbird Range
Hummingbirds are only found in North, Central, and South America. Their range extends from southern Canada and Alaska in the north, to Tierra del Fuego in the south. They are also found in the islands of the West Indies.
The majority of hummingbird species occur in Central and South America with the greatest diversity within the latitudes of 10° North and 10° South of the equator. The largest number of species are found in Ecuador, Columbia & Peru.
Of the 21 species of hummingbirds that reach the U.S., 16 regularly breed here, one is a common visitor, and four others are classified as vagrants - only occasionally present.
The most common hummingbird east of the Mississippi in the U.S. is the Ruby-throated Hummingbird (Archilochus colubris).
Common Western U.S. Hummingbirds
Black-chinned hummingbird (Archilochus alexandri)
Costa's hummingbird (Calypte costae)
Anna's hummingbird (Calypte anna)
Broad-tailed Hummingbird (Selasphorus platycercus)
Rufous Hummingbird (Selasphorus rufus)
Allen's Hummingbird (Selasphorus sasin)
Calliope Hummingbird (Stellula calliope)
These species are typically found in the extreme southwestern U.S. (parts of AZ, NM, & TX)
Violet-crowned Hummingbird (Amazilia violiceps)
Lucifer Hummingbird (Calothorax lucifer)
Broad-billed Hummingbird (Cynanthus latirostris)
Blue-throated Hummingbird (Lampornis clemenciae)
Magnificent Hummingbird (Eugenes fulgens)
So what makes a hummingbird a hummingbird?
Hummingbirds are unique for their:
Agility in flight,
Extremely brilliant, iridescent plumage,
Long, specialized bills,
And general small size
Let's explore some of these & other unique characteristics:
Hummingbird Flight
In flight, a hummingbird's wing is only visible as a blur, because of the speed of its wing beat, between 10 to 80 times per second for normal flight. The speed depends on the size of the bird, with the highest wing-beat rates occurring in the smaller species.
The normal flight speed of hummingbirds is between 25-30 mph. During display dives, wing beats may be as high as 200 beats per second and birds reaching speeds of up to 65mph!
Hummingbirds have many skeletal and flight muscle adaptations that allow for hovering and high maneuverability in flight.
They have compact, strongly muscled bodies. Muscles make up 25-30% of their body weight - like tiny little Schwarzeneggers!
They have rather long, bladelike wings that, unlike the wings of other birds, articulate (connect) to the body only from the shoulder joint. It is at this joint that the wing rotates almost 180 degrees to enable the bird to fly in its distinctive way.
Other birds can generate power only on the down stroke, but hummingbirds can move freely in any direction.
Their wing architecture permits hummingbirds to fly not only forward but also straight up and down, sideways, and backwards and to hover in front of flowers as they obtain nectar and insects.
Hummingbirds are one of the few birds that can lift themselves straight up into the air. When they leave their perch, they rise into the air entirely on their own power and do not use the twig or branch as a brace to push against.
Although you hardly ever see them - yes they do have feet! They use their tiny appendages mostly for perching, unlike other birds that may hop or run about on the ground.
What do I do if I found a hummingbird?
Glenda Asked: I found a hummingbird alive on the ground, still can make a flying noise, but the tail feathers are short and can't get far up in the air. I am giving nectar to the bird but I don't know how often and how much to give. Is it possible the bird will fly again?
Bliss Said: The hummingbird may have hit a window or something similar and was stunned and may take awhile to fly again.
It's best to leave the hummingbird alone in a shady location or, if it can, leave it perched on a branch in a tree.
If it is still there after a hour or two, then there is probably something more seriously wrong with it, beyond what you can do. At that point, the only thing you can do is leave it or take it to your local wildlife rehabilitator.
Hummingbird tail feathers are typically very short when full length.
Even if all the tail feathers were missing, the hummingbird should be able to fly as the majority of the propulsion comes from the wing flight feathers; the tail is used more for steering.
Hummingbird Bill
Hummingbird bills vary greatly in length & shape according to the flowers at which they feed. In fact, co-evolution of flower shape and bill shape is a well-studied phenomenon.
Some bills are slightly curved, some long & straight, some short. The main purpose of the bill is to enable them to reach nectar inside slim, tubular flowers.
Like woodpeckers, hummingbirds have a bony structure called a hyoid apparatus attached to the tongue. This pair of bony coils serves to extend the tongue for considerable lengths and pull it back in again like a spring.
Near its base, the tongue is flat and bears a slight groove along the middle.
Near its halfway point, it divides completely lengthwise, and each tip is pointed.
These 2 tongue sections roll up inwardly to create two tubes when the bird feeds. Nectar is not sucked up but held in the tubular portion and swallowed when the tongue is returned to the mouth.
The tongue collects nectar by means of a licking motion at a rate of about 13 licks per second, regardless of bird size.
Approximately 3.1 to 7.9 grams of nectar are consumed by a hummingbird at at feeder at one time.
Larger hummingbirds have larger tongues which will gather more nectar per lick. How many licks does it take to get to the center of a corolla?
We use the bill to help determine hummingbird age when mist-netting at our MAPS (Monitoring Avian Productivity & Survivorship) station.
Juvenile hummingbirds have soft, & deeply corrugated bills. In the first 5-9 months after fledging, the bill hardens & the corrugations are lost due to the hardening process and wear.
Hummingbird Plumage
The brilliant, iridescent colors of their feathers have earned hummingbirds such names as "flying jewels." Audubon called the hummingbird a "glittering fragment of the rainbow."
Hummingbird plumage is extremely varied. In general, upperparts are usually a glittering green, while the underparts are often paler, particularly in females.
In the Hermit subfamily, plumage color is generally brownish, grayish, or reddish and without iridescence.
Trochilines often have brilliant patches of iridescent feathers of metallic green, blue, purple, red or yellow usually on the crown, throat or upper breast.
The most typical patch is the gorget (think "gorgeous"), a bib of iridescent feathers that changes in color with viewing angle.
Females & young hummingbirds lack the distinctive gorgets of adult males. Before the first molt, which takes place at the end of the breeding season, young males have a mixed plumage, and may show some flecks of color.
All gorgets will appear black or dark if light does not reflect off them at a favorable angle. The sun, viewer and bird must be aligned properly to behold the sparkling plumage at its best.
Males may also have other adornments such as crests, "pantaloons" (tufts of puffy thigh feathers) or greatly elongated tails.
Feather counts on hummingbird range from about 940-1,500, depending on molt condition & season period.
As an extreme comparison, a whistling swan may have more than 25,000 feathers.
The hummingbird actually possesses more feathers per gram of body weight than the swan, which is to be expected since small bodies have relatively more heat-losing surface per unit of weight than large bodies.
Hummingbird Diet
Hummingbirds are nectarivores and insectivores; approximately 90% of their diet comes from nectar and 10% from arthropods (files, wasps, spiders, beetles, and ants).
A hummingbird requires an adequate diet of protein, carbohydrates, fats, vitamins and minerals.
Nectar is a dilute solution of sugars with trace amounts of amino acids and electrolytes.
These sugars are easily digested and quickly converted into energy, supplying a hummingbird's carbohydrate requirements.
Learn how to make your own hummingbird food recipe so that its sugar concentration nectar is as close to what is naturally found in hummingbird pollinated flowers.
Insects are also part of a hummingbird's diet.
Insects provide protein, minerals, vitamins and fats and are digested more slowly than nectar.
Nevertheless, it takes only 10 minutes for the remains of some insects to pass completely through the hummingbird's body.
Some hummingbirds will snatch insects with their long bills right out of the air (hawking or hover-gleaning).
Other pick them from both inside and outside flowers and from underneath leaves.
They are also often hunted along walls, trees or flower bushes, and filmy webs are sometimes picked for tiny spiders or their prey.
Most straight-billed hummingbirds use both methods of hawking or gleaning, but the curved-billed hermits catch most of their insects & spiders by picking them off the tips, edges and undersides of leaves.
Because they are so small and operate at extremely high speeds, hummingbirds require a lot of fuel to maintain their body temperature and their high rate of metabolism.
Food intake varies with temperature and activity but typically a hummingbird can consume 50% of its weight in sugar each day and not get fat. If only we could do the same with chocolate!
Compared to other birds, the metabolic rates of hummingbirds are extremely high.
For example, a 4-gram hummingbird has a basal metabolic rate of 1,400 calories per kilogram.
Compare that to a 121 gram mourning dove which has a rate of 127 calories per kilogram.
If humans had as high a weight-specific metabolic rate as a hummingbird, our daily intake of food would have to be approximately twice our body weight.
Also, our temperature would be over 750?F and we would use up 155,000 calories per day!
If a hummingbird needs to conserve energy, it becomes torpid at night.
This enables it to slow their metabolic & heart rates and reduce body temperature.
During periods of torpor the heartbeat slows from as many as 1,260 beats per minute to only 50 beats per minute.
They may appear dead during torpor states, but will quickly revive once the morning light & warmth arrives & are able to find food again.
Hummingbird Nest
Only the female hummingbird builds the nest, incubates the eggs, and feeds the young while the male is out goofing around. The male's only role is to contribute his genes during mating.
Below is footage of a Anna's hummingbird building a nest over a 10-day period.
Each species of hummingbird has its own particular type of nest, usually measuring about 2 inches in diameter. Some are cone-shaped and other look like tiny cups.
Locations vary. Some nests hang from ledges, while others are built on branches and still others can be found clinging to rocks. Some are even suspended by tendrils from nearby foliage.
Females begin by building a platform of various soft and strong materials, gluing them together with spider webs and other fibers. They will use these same fibers to attach the nest to its base as well.
The outside of the nest is camouflaged with bits of bark, moss, grass, lichen, and/or small roots so that is virtually indistinguishable, resembling a small bump on a branch.
The interior of the egg cups are extremely soft and thick, often made of cottony down, feathers, and/or leaves.
The female sits in the nest to shape it, turning frequently & pushing the sides with her breast. To shape the rim, she holds it between her chin & breast & moves from side to side.
It can take a hummingbird from one day to two weeks to build a nest depending on weather conditions and difficulty of gathering material.
Hummingbird Eggs
It should come as no surprise that hummingbirds lay the smallest eggs.
Elliptical in shape, they usually measure less than 1/2 inch and weigh less than 1/2 gram (not much bigger than a jelly bean!).
For a fun comparison, the ostrich egg is 6 1/2 inches long and weighs 1,400 grams.
Relative to a female hummingbird's size, the eggs she lays are quiet large.
Eggs may weigh 10-20% or more of her own body weight, compared to only 2-4% in other larger birds. I bet she's feeling relieved when they finally come out.
Anna's Hummingbird Eggs Hatching in Nest
Hummingbirds nearly always lay 2 eggs.
They tend to lay their eggs in the morning, and there is generally a 2-day spread between laying of eggs. Eggs are incubated for approximately 15-22 days.
A female hummingbird will generally sit on her eggs from 60-80% of every day.
She will not leave her eggs when it's cold or rainy, and on these days she will fluff out her feathers and press herself especially close to give the eggs as much warmth as possible.
When the weather is hot, she may hold her body off the eggs slightly or stand inside the nest to shade them from the burning sun.
Hummingbird Nestlings
Newborn hummingbirds are altricial. They are blind, nearly featherless, and virtually helpless and in need of constant care.
These hatchlings rely entirely on the mother for warmth until about 8-12 days when they are then able to maintain their own body temperatures.
The female sits on her nestlings (brooding) from 12-18 days, during which babies are fed from 1 to 3 times every hour.
Brooding periods vary with weather. Where there is abundant food and the weather is milder, the young may grow faster and leave the nest earlier.
The nestlings are fed by regurgitation, usually while the female is hovering.
The mother inserts her bill into the nestling's throat and pumps in nectar & insects.
Eventually, the nest becomes too small to hold the growing youngsters and they will perch on the rim.
Fledgling occurs in approximately 20-28 days. After leaving the nest, fledgling hummingbirds may be fed and preened by their mothers for as long as 20-40 days.
Like most mothers, hummingbird females are fearless when it comes to defending their nests and young.
They have been known to attack such large predators as hawks, crows, chipmunks & snakes.
Hummingbird Size Extremes
Smallest Hummingbird
The Bee Hummingbird (Mellisuga helenae) is the smallest of all hummingbirds, as well as the smallest of all birds, and one of the smallest warm-blooded vertebrates in the world.
It is found in Cuba, measuring a mere 2.25 inches (~6cm) in length and weighing only ~2 grams or 0.07; lighter than a penny or 2 paper clips! Awe, now that's cute!
Largest Hummingbird
The Giant Hummingbird (Patagona gigas) is the largest member of the hummingbird family, weighing 18-20 grams (6/10 - 7/10 of an ounce).
It's not fat, just big-boned! It measures 8.5 inches in length (~22cm), which is about the length of a an Eastern Kingbird, Gray Catbird, or European Starling.
It is found between 2,000 and 3,500 meters (6,500-11,000 feet) above sea level in the Andes of South America, from northern Ecuador to central Chile and Argentina.
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Sure, everyone has heard of Prince Fielder and Tony Gwynn Jr., but the Nationals' recent acquisition of Cutter Dykstra prompted me to think about other players with ties to former big-leaguers. Cutter is the son of Lenny Dykstra. "Nails" was part of the bad-boy Mets who made the unlikely comeback in the 1986 World Series. His son Cutter hasn't exactly bowled over scouts with his minor league start, barely cracking the Mendoza line in '09. Although last year he showed signs of improvement, hitting .312 in 420 plate appearances with the Single-A Wisconsin Timber Rattlers. Seems the Nationals thought enough about his potential to acquire him for proven outfielder Nyjer Morgan. There is a descendant of another '86 Series participant making some noise within the Houston organization. In 2004, the Astros drafted a Texas first baseman, Roger's son Koby Clemens. He had mild success in high-level Single-A ball and seems to project more as a catcher or 3B for Corpus Christi. His average dipped in Double-A, and it seems he may have ridden his father's coattails as far as he can. From what I can tell, the highlight of his career has been cracking a home run off his father in an exhibition game. Beau Mills is currently in the Cleveland system and was recently assigned back to the minors. His father Brad is known more for his time on the pine as a bench coach and manager than his stint with the Expos. Beau has been somewhat of journeyman himself, spending time at Fresno State, the Alaskan Baseball League and Lewis-Clark State College before he was selected in the 1st round of the 2007 draft. Unfortunately, the youngster has found himself in a variety of off-field situations that have been a distraction, from academic problems at Fresno to an alleged assault just last year. On a positive note, his RBI totals are an indication that his high draft slot was not completely unwarranted. Former 2B Delino DeShields, best known for flying around the base paths, has a son that is no slouch in the speed department. Delino Jr., another top-10 overall pick, is currently in the Houston Astros system. He would be the least-proven from this list, having just been taken in the last draft. Until he is tested further, I'd say it's hard to evaluate anything more than his quickness. The material I came across indicates he'll spend the season in Single-A ball. One other note, the teenage phenom also had a brush with the law, being charged with a DUI early this year. Probably the most heralded of the list is Kyle Drabek, son of the former Cy-Young-award-winner Doug Drabek (of Pittsburgh Pirates fame). He was, arguably, the centerpiece of the prospect package that landed "Doc" Halladay in Philly. He continued his tear down on the farm upon joining Toronto, including a no-hitter for Double-A New Hampshire. The Jays' brass were compelled to give Kyle a shot at the bigs last September, where he came crashing back to earth going 0-3 in three starts. With a spot already secured in this year's rotation, it will be exciting to watch how he performs in his first full year in the Show. This list is probably just scratching the surface. If you have any generational players to add to the list, please share in the comments.


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One of the last free-flowing rivers on the East Coast, the Delaware River has remained an amazing natural resource through government oversight and protection. Campers can find many great recreational opportunities just a beautiful drive away. From rafting to kayaking, fishing to historical sites, the Upper Delaware River is full of intriguing spots and sights.
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The activities of radioCona in the last few years opened up the possibility to collaborate with Botanical garden. It seems right that radioCona in the new decade (it intervenes at frequency 88.8 MHz since 2008) enters the Steklenik gallery at Tivoli Greenhouse. To sharpen our hearing and find our place in the context of nature and art, we consciously unite both environments. In the warm conditions of Tivoli Greenhouse we will be able to submerge ourselves into remote listening of lonesome winter landscape and crystalline water structures. In this way we transfer experiences of transposed realities of hybrid conditions from nearby and afar.
ICEmeltings FM is taking place in the context of radioCona, that was founded as project occasional radio station for contemporary art. Cona Institute uses the project to expose that radio frequency is also a form of public space and therefore could be used as an exhibition space. For several years we were scheduling the annual FM exhibition in winter time to bring sound and radio art to homes of listeners instead of just to classical venues. This year we will not organise a FM exhibition, visitors will be able to listen to group sound work ICEbreakings at Steklenik.
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