text:
\section{Introduction}
The binomial random graph model $G(n,p)$, introduced by Gilbert \cite{G59}, is a random variable on the subgraphs of the complete graph $K_n$ whose distribution is given by including each edge in the subgraph independently with probability $p$. Since its introduction, this model has been extensively studied. A particularly striking feature of this model is the `phase transition' that it undergoes at $p = \frac{1}{n}$, exhibiting vastly different behaviour when $p = \frac{1 - \varepsilon}{n}$ to when $p = \frac{1+\varepsilon}{n}$ (where $\varepsilon$ is a positive constant). For more background on the theory of random graphs, see \cite{B01,JLR00,FK16}.

More recently, the following generalisation of this model has been the object of study: Suppose $G$ is an arbitrary graph with minimum degree $\delta(G)$ at least $d$ and let $G_p$ denote the random subgraph of $G$ obtained by retaining each edge of $G$ independently with probability $p$. When $G= K_{d+1}$, the complete graph on $d+1$ vertices, we recover the model $G(d+1,p)$.

If we take the probability $p$ to be a function of (the lower bound $d$ for) the minimum degree of $G$, rather than the order (i.e., number of vertices) of $G$, it has been shown that the model $G_p$ shares many properties with $G(d+1,p)$. In particular, some of the complex behaviour which occurs whp\footnote{Throughout the paper, all asymptotics will be considered as $d \to \infty$ and so, in particular, whp (with high probability) means with probability tending to one as $d \to \infty$} in $G(d+1,p)$ once we pass the critical point of $p=\frac{1}{d}$ also occurs whp in $G_p$ in the same regime of $p$.
For example, when $p= \frac{1+\varepsilon}{d}$ for $\epsilon > 0$, it has been shown that whp $G_p$ contains a path or cycle of length linear in $d$, see \cite{KS13,KS14,EJ18}. Furthermore, for this range of $p$ it has been shown by Frieze and Krivelevich \cite{FK13} that whp $G_p$ is non-planar and by Erde, Kang and Krivelevich \cite{EKK20} that in fact, whp $G_p$ contains a complete minor of order\footnote{The notation $\tilde{\Omega}\left(\cdot\right)$ here is hiding a polylogarithmic factor in $d$.} $\tilde{\Omega}\left(\sqrt{d}\right)$ and hence has genus $\tilde{\Omega}(d)$. Similarly, when $p = \omega\left(\frac{1}{d}\right)$, whp $G_p$ contains a path or cycle of length $(1-o(1))d$, as shown in \cite{R14,KLS15}. In addition, when $p = (1+\varepsilon)\frac{\log d}{d}$ for $\epsilon >0$, whp $G_p$ contains a path of length $d$ and in fact even a cycle of length $d+1$, see \cite{KLS15,GNS17} respectively. All of these results generalise known results about the binomial model $G(d+1,p)$.

The model $G(d+1,p)$ shows that these results are optimal when $G$ can be an \emph{arbitrary} graph, but for \emph{specific} graphs $G$, they may be far from the truth. One particular graph for which the model $G_p$ has been studied extensively is the \emph{hypercube} $Q^d$, the graph with vertex set $V\left(Q^d\right) = \{0,1\}^d$ and in which two vertices are adjacent if they differ in exactly one coordinate. Throughout this paper we will write $n := 2^d$ for the order of the hypercube and we note that $|E\left(Q^d\right)| = \frac{nd}{2}$.

Random subgraphs of the hypercube were first studied by Saposhenko \cite{S67} and Burtin \cite{B77}, who showed that $Q^d_p$ has a threshold for connectivity at $p=\frac{1}{2}$; for a fixed constant $p < \frac{1}{2}$ whp $Q^d_p$ is disconnected and for a fixed constant $p > \frac{1}{2}$ whp $Q^d_p$ is connected, and this result was strengthened by Erd\H{o}s and Spencer \cite{ES79} and by Bollob\'{a}s \cite{B83}. Bollob\'{a}s \cite{B90} also showed that $p=\frac{1}{2}$ is the threshold for the existence of a perfect matching in $Q^d_p$. Very recently, answering a longstanding open problem, Condon, Espuny D{\'\i}az, Girao, K{\"u}hn and Osthus \cite{CDGKO21} showed that $p=\frac{1}{2}$ is also the threshold for the existence of a Hamilton cycle in $Q^d_p$.

It was conjectured by Erd\H{o}s and Spencer \cite{ES79} that $Q^d_p$ should undergo a similar phase transition at $p=\frac{1}{d}$ as $G(d+1,p)$ does: It is relatively easy to see, by a coupling with a branching process, that when $p = \frac{1-\epsilon}{d}$ for $\epsilon>0$, whp all components of $Q^d_p$ have order $O(d)$, but they conjectured that when $p= \frac{1+\epsilon}{d}$, whp $Q^d_p$ contains a unique `giant' component whose order is linear in $n=2^d$. This conjecture was confirmed by Ajtai, Koml\'{o}s and Szemer\'{e}di \cite{AKS81}.

\begin{theorem}[\cite{AKS81}]\label{t:AKS}
Let $\epsilon > 0$ be a constant and let $p= \frac{1+\epsilon}{d}$. Then there exists a $\gamma >0$ such that whp the largest component of $Q^d_p$ has order at least $\gamma n$.
\end{theorem}

Ajtai, Koml\'{o}s and Szemer\'{e}di also indicated that it is possible to show that there is a unique `giant' component of linear order in $n$, whose order is $\left(\gamma(\epsilon) +o(1)\right) n$ where $\gamma(\epsilon)$ is the survival probability of the Po$(1+\epsilon)$ branching process. These results were later extended to a wider range of $p$, describing more precisely the component structure of $Q^d_p$ when $p = \frac{1+\epsilon}{d}$ with $\epsilon=o(1)$ by Bollob\'{a}s, Kohayakawa and {\L}uczak \cite{BKL92}, by Borgs, Chayes, van der Hofstad, Slade and Spencer \cite{BCVSS06}, and by Hulshof and Nachmias \cite{HN20}, with the correct width of the critical window in this model being only recently identified by van der Hofstad and Nachmias \cite{HN17}. For a more detailed background on the phase transition in this model, see the survey of van der Hofstad and Nachmias \cite{HN14}.

If we consider the structure of $Q^d_p$ above this critical threshold of $p=\frac{1}{d}$, then we can conclude from the results in the $G_p$ model mentioned earlier that whp $Q^d_p$ contains a path and cycle of length $\Omega(d)$, has genus $\tilde{\Omega}(d)$ and contains a complete minor of order $\tilde{\Omega}(\sqrt{d})$. However, these results seem far from optimal. Indeed, already above this threshold whp $Q^d_p$ contains a component whose order is linear in $n=2^d$, and so exponentially large in $d$. Hence, it is possible that $Q^d_p$ could contain a path and cycle of length linear in $n$. Similarly, it is known that the genus of $Q^d$ is exponentially large in $d$. In fact, the genus of $Q^d$ has been shown to be precisely $(d-4)2^{d-3}+1$ for $d \geq 2$, see \cite{R55,BH65}, and the size of the largest complete minor in $Q^d$ was shown to be between $2^{\lfloor\frac{d-1}{2}\rfloor}$ and $\sqrt{d}2^{\frac{d}{2}} + 1$ in \cite{CS07}. Hence, it is possible that whp the genus of $Q^d_p$ in the supercritical regime is linear in $n$ and whp the size of the largest complete minor in $Q^d_p$ is of order $\Omega\left(\sqrt{n}\right)$.

The main aim of this paper is to show that both of these statements are true, at least up to a polylogarithmic factor in $n$. More precisely, we prove the following theorems concerning the length of the longest cycle and the size of the largest complete minor in $Q^d_p$.

	\begin{theorem}\label{t:longcycle}
	Let $\epsilon > 0$ be a constant and let $p = \frac{1+\epsilon}{d}$. Then whp $Q^d_p$ contains a cycle of length $\Omega \left( \frac{2^d}{d^3(\log d)^3 } \right)$.
	\end{theorem}

	\begin{theorem}\label{t:Hadwiger}
	Let $\epsilon > 0$ be a constant and let $p = \frac{1+\epsilon}{d}$. Then whp $Q^d_p$ contains a $K_t$-minor with $t = \Omega \left( \frac{2^{\frac{d}{2}}}{d^3(\log d)^3 } \right)$.
	\end{theorem}
We note that here, and elsewhere in the paper, there may be a dependence on the parameter $\epsilon$ in the implicit constants in the bounds.

In fact, both of these theorems follow from a more general statement, of independent interest, which says that whp the largest component of $Q^d_p$ has \emph{good edge-expansion} properties. That is, every large enough subset of the largest component `expands', in the sense that it has a large edge boundary. The notion of graph expansion has turned out to have fundamental importance in diverse areas of discrete mathematics and computer science: For a comprehensive introduction to expander graphs, see the survey of Hoory, Linial and Widgerson \cite{HLW06}. In particular, notions of expansion have turned out to be a powerful tool in the study of random structures: See for example the survey paper of Krivelevich \cite{K19}.

In what follows, given a subset $S \subseteq V\left(Q^d\right)$, we will write $\partial_v(S)$ and $\partial_e(S)$ for the \emph{vertex} and \emph{edge} boundary of $S$ in $Q^d$ respectively. That is, $\partial_v(S)$ is the set of vertices in $V\left(Q^d\right) \setminus S$ which have a neighbour in $S$ and $\partial_e(S)$ is the set of edges between $S$ and $V\left(Q^d\right)\setminus S$. Furthermore, we will write $\partial_{v,p}(S)$ and $\partial_{e,p}(S)$ for the vertex and edge boundary of $S$ in $Q^d_p$ respectively. Also, given constants $\alpha,\beta >0$ and a statement $A$, we will write `Let $\alpha \ll \beta$. Then $A$ holds' to indicate that there is some fixed, implicit function $f$ such that $A$ holds for all $\alpha \leq f(\beta)$.

\begin{theorem}\label{t:edgeexpansion}
	Let $0 < \alpha \ll \epsilon$ be constants, let $p = \frac{1+\epsilon}{d}$ and let $L_1=L_1\left(Q^d_p\right)$ be the largest component of $Q^d_p$. Then there exists a $\beta >0$ such that whp every subset $S \subseteq L_1$ satisfying ${\alpha n \leq |S| \leq\frac{|V(L_1)|}{2}}$ has an edge boundary satisfying $\left|\partial_{e,p}(S)\right| \geq \beta \frac{n}{d^3 (\log d)^2}$.
	\end{theorem}

As an immediate corollary, since $Q^d$ is $d$-regular, and so every set of $m$ edges in $Q^d$ must span at least $\frac{m}{2d}$ vertices, we can deduce that whp the largest component of $Q^d_p$ also has \emph{good vertex-expansion}. In fact, since in $Q^d_p$ we do not expect significantly fewer than $m$ vertices to span $m$ edges, we can do slightly better.

	\begin{corollary}\label{c:vertexexpansion}
	Let $0 < \alpha \ll \epsilon$ be constants, let $p = \frac{1+\epsilon}{d}$ and let $L_1=L_1\left(Q^d_p\right)$ be the largest component of $Q^d_p$. Then there exists a $\beta >0$ such that whp every subset $S \subseteq L_1$ satisfying ${\alpha n \leq |S| \leq\frac{|V(L_1)|}{2}}$ has a vertex boundary satisfying $\left|\partial_{v,p}(S)\right| \geq \beta \frac{n}{d^3(\log d)^3 }$.
	\end{corollary}

We note that, whilst it is perhaps not immediately clear that we cannot improve Theorem \ref{t:edgeexpansion} to show constant edge-expansion for linear sized subsets of $L_1 = L_1\left(Q^d_p\right)$, it is clear that we cannot hope to show constant vertex-expansion. Indeed, $Q^d$ contains a separator of size $O\left(\frac{n}{\sqrt{d}}\right)$, namely the middle layer, and it is not hard to show that whp this separator splits $L_1$ roughly in half, and so whp there is a linear sized subset of $L_1$ whose vertex boundary has size $O\left(\frac{n}{\sqrt{d}}\right)$. However, it then follows from a similar argument as in Corollary \ref{c:vertexexpansion} that whp $L_1$ contains some linear sized subset whose edge boundary has size $O\left(\frac{n \log d}{\sqrt{d}}\right)$.

In particular, this implies that we cannot deduce optimal versions of Theorems \ref{t:longcycle} or \ref{t:Hadwiger}, that is, ones without the polynomial factors in $d$, straightforwardly from the expansion properties of the largest component of $Q^d_p$.

In fact, it is perhaps surprising that the expansion properties of $L_1\left(Q^d_p\right)$ are useful at all, since random subgraphs of the hypercube have been used by Moshkowitz and Shapira \cite{MS18} to construct graphs which have in some way the worst possible expansion properties.

Whilst Theorem \ref{t:Hadwiger} already implies that the genus of $Q^d_p$ is $\Omega\left(\frac{n}{d^6 (\log d)^6}\right)$, with a more careful argument we are able to give a better bound.

\begin{theorem}\label{t:genus}
Let $\epsilon > 0$ be a constant and let $p = \frac{1+\epsilon}{d}$. Then whp the genus of $Q^d_p$ is $\Omega(n)$.
\end{theorem}

Note that, since whp $\left|E\left(Q^d_p\right)\right| = \Theta(n)$, and clearly the genus of a graph is at most the number of edges, Theorem \ref{t:genus} is optimal up to the value of the leading constant.

The paper is structured as follows. In Section \ref{s:prelim} we collect some lemmas which will be useful in the rest of the paper. Then, in Section \ref{s:expansion} we prove our main results on the expansion properties of the largest component in $Q^d_p$ (Theorem \ref{t:edgeexpansion} and Corollary \ref{c:vertexexpansion}), and use them to deduce Theorems \ref{t:longcycle} and \ref{t:Hadwiger}. In Section \ref{s:genus} we give a proof of Theorem \ref{t:genus}, and then in Section \ref{s:discussion} we mention some open problems.

	\section{Preliminaries}\label{s:prelim}
For real numbers $x,y,z$ we will write $x=y\pm z$ to mean that $y-z \leq x \leq y+z$.

	We will want to use the following simple lemma, which is a slight adaptation of a result in \cite{KN06}, to decompose a tree into roughly equal sized parts.
	\begin{lemma}\label{l:treedecomp}
	Let $T$ be a tree such that $\Delta(T) \leq C_1$, all but $r$ vertices of $T$ have degree at most $C_2\leq C_1$ and $|V(T)| \geq \ell$ for some $C_1,C_2,\ell,r >0$. Then there exist disjoint vertex sets $A_1,\ldots, A_s \subseteq V(T)$ such that
		\begin{itemize}
			\item $V(T) = \bigcup_{i=1}^s A_i$;
			\item $T[A_i]$ is connected for each $1 \le i \le s$;
			\item $\ell \leq |A_i| \leq  C_1 \ell$ for each $1 \le i \le r$; and
			\item $\ell \leq |A_i| \leq  C_2 \ell$ for each $r < i \le s$.
		\end{itemize}
	\end{lemma}
	\begin{proof}
We choose an arbitrary root $w$ for $T$. For a vertex $v$ in a rooted tree $S$, let us write $S_v$ for the subtree of $S$ rooted at $v$.

We construct the vertex sets $A_i$ inductively. Let us start by setting $T(0)=T$. Given a tree $T(i)$ rooted at $w$ such that $|V\left(T(i)\right)| \geq \ell$, let $v_i$ be a vertex of maximal distance from $w$ such that $\left|V\left(T(i)_{v_i}\right)\right| \geq \ell$. We take $A_{i+1} = V\left(T(i)_{v_i}\right)$ and let $T(i+1) = T(i) \setminus T(i)_{v_i}$. We stop when $|V\left(T(i)\right)| < \ell$, and in that case we add $V\left(T(i)\right)$ to the final $A_i$. Finally, let us re-order the sets $A_i$ so that they are non-increasing in size.

We claim that the sets $A_1, A_2, \ldots, A_s$ satisfy the conclusion of the lemma. Indeed, the first two properties are clear by construction. Also, we note that by our choice of $v_i$, $|T(i)_x| < \ell$ for every child $x$ of $v_i$. Hence, if $v_i \neq w$, then $v_i$ has $d(v_i)-1$ children and so, since $A_i$ is the union of the vertices of $T(i)_x$ over all children $x$ of $v_i$, together with a set of size less than $\ell$, it follows that $|A_{i+1}| \leq d(v_i) \ell$. Furthermore, if $v_i=w$, then we note that $A_{i+1}=V(T(i))$ and so $|A_{i+1}| \leq d(w)(\ell-1) + 1 \leq d(w) \ell$. Therefore, since all but $r$ vertices of $T$ have degree at most $C_2$, the third and fourth properties also hold.
	\end{proof}

We will also need the following results, which allow us to deduce the existence of a long cycle and a large complete minor from vertex-expansion properties of a graph. For wider context on properties of expanding graphs, see the survey of Krivelevich \cite{K19}.

\begin{theorem}[{\cite[Theorem 1]{K19a}}]\label{t:cycleexpander}
Let $k \geq 1, t \geq 2$ be integers. Let $G$ be a graph on more than $k$ vertices satisfying
\[
\left|\partial_v (W)\right| \geq t, \qquad \text{for every } W \subset V \text{ with } \frac{k}{2} \leq |W| \leq k.
\]
Then $G$ contains a cycle of length at least $t+1$.
\end{theorem}

The following theorem allows us to deduce the existence of a large complete minor in a graph without any small separators. It is easy to see that graphs with good vertex-expansion do not contain any small separators, and in fact it is known (see \cite[Section 5]{K19}) that the converse is true, in the sense that graphs without small separators must contain large induced subgraphs with good vertex-expansion.

\begin{theorem}[{\cite[Theorem 1.2]{KR10}}]\label{t:minorexpander}
Let $G$ be a graph with $N$ vertices and with no $K_t$-minor. Then $G$ contains a subset $S$ of order $O\left(t \sqrt{N}\right)$ such that each connected component of $G \setminus S$ has at most $\frac{2}{3}N$ vertices.
\end{theorem}

We will use the following Chernoff type bounds on the tail probabilities of the binomial distribution, see e.g. \cite[Appendix A]{AS}.
\begin{lemma}\label{l:Chernoff}
Let $N \in \mathbb{N}$, let $p \in [0,1]$ and let $X \sim \text{Bin}(N,p)$.

\begin{enumerate}[(i)]
\item\label{i:chernoff1} For every positive $a$ with $a \leq \frac{Np}{2}$,
\[
\mathbb{P}\left(\left|X -Np \right| > a\right) < 2 \exp\left(-\frac{a^2}{4Np} \right).
\]
\item\label{i:chernoff2} For every positive $b$,
\[
\mathbb{P}\left(X > bNp \right) \leq \left(\frac{e}{b}\right)^{b Np}.
\]
\end{enumerate}
\end{lemma}

In particular, the following consequence of Lemma \ref{l:Chernoff} on the edge boundary of large sets in $Q^d_p$ will be useful.
\begin{lemma}\label{l:edgeboundary}
Let $k,c,\epsilon>0$ be constants and let $p=\frac{1+\epsilon}{d}$. Then whp every subset $X \subseteq V\left(Q^d\right)$ of size $|X| \geq c n d^{-k}$ has edge boundary in $Q^d_p$ such that
\[
\left|\partial_{e,p}(X) \right| \leq |X| \log d.
\]
\end{lemma}
\begin{proof}
Since any subset $W \subseteq V\left(Q^d\right)$ satisfies $\left|\partial_e(W)\right| \leq d|W|$, it follows from Lemma \ref{l:Chernoff} \eqref{i:chernoff2} that
\begin{align*}
\mathbb{P}\left(\left|\partial_{e,p}(W)\right| \geq |W| \log d\right) &\leq \mathbb{P}\left(\text{Bin}(d |W|,p) \geq |W|\log d\right)\\
&\leq \left(\frac{e(1+\epsilon)}{\log d}\right)^{|W|\log d} \leq d^{-(k+1)|W|}.
\end{align*}
Hence, the probability that there exists a subset $X\subseteq V\left(Q^d_p\right)$ not satisfying the conclusion of the lemma is at most
\[
\sum_{i \geq c n d^{-k}} \binom{n}{i} d^{-(k+1)i} \leq \sum_{i \geq c n d^{-k}}   \left( \frac{en}{d^{k+1}i}\right)^i \leq \sum_{i \geq c n d^{-k}}\left( \frac{e}{cd}\right)^i = o(1).
\]

\end{proof}
We will also need to use the following correlation inequality, which is a consequence of an inequality of Harris \cite{H60} and is itself a special case of the FKG-inequality: See for example \cite[Section, 6]{AS}.

\begin{lemma}\label{l:Harris}
Let $\mathcal{A}$ and $\mathcal{B}$ be families of subgraphs of $Q^d$ which are closed under taking supergraphs. Then
\[
\mathbb{P}\left(Q^d_p \in \mathcal{A} \cap \mathcal{B}\right) \geq \mathbb{P}\left(Q^d_p \in \mathcal{A} \right)\mathbb{P}\left(Q^d_p \in \mathcal{B}\right).
\]
\end{lemma}

Finally, we will use the following result on vertex-isoperimetry in the hypercube, which is a useful strengthening of Harper's \cite{H66} well-known isoperimetric inequality.

\begin{theorem}[{\cite[Corollary 2]{BL97}}]\label{t:vtxiso}
Let $A \subseteq V\left(Q^d\right)$ and let $0 \leq \lambda < 1$ and $k \in \mathbb{N}$ be such that
\[
|A| = \sum_{i=0}^k \binom{d}{i} + \lambda \binom{d}{k+1} \leq \frac{n}{2}.
\]
Then there is a matching from $A$ to its complement $A^c$ of size at least $(1-\lambda)\binom{d}{k} + \lambda \binom{d}{k+1}$.

In particular, for any subset $A \subset V\left(Q^d\right)$ such that both $A$ and $A^c$ have size $\Omega(n)$, there is a matching from $A$ to $A^c$ of size $\Omega\left(\frac{n}{\sqrt{d}}\right)$.
\end{theorem}

\section{Edge-expansion in the largest component}\label{s:expansion}
We will need some properties of the largest component of $Q^d_p$ given in \cite{AKS81}, where the statement below is claimed, although the details of the proof are not spelled out. For completeness we include a proof in Appendix \ref{a:AKS}.
\begin{theorem}[\cite{AKS81}]\label{t:AKSfine}
Let $0 < \delta_2 \ll \delta_1$, let $p=\frac{1+\delta_1}{d}$ and let $\gamma_1$ be the survival probability of the ${\text{Po}(1+\delta_1)}$ branching process. Then there exist constants $\delta_3,\delta_4,\delta_5>0$, which can be chosen such that $\delta_3 \ll \delta_2$, such that, if $Y$ is the set of vertices of $Q^{d}_p$ contained in components of order at least $\delta_3 d^2$, then whp:
\begin{enumerate}[(i)]
\item\label{i:largecomponent} there is a unique component $L_1$ of order at least $\delta_2 n$ in $Q^d_p$ and $|V(L_1)| = (\gamma_1 \pm \delta_2) n$;
\item\label{i:largevtcs} $| V(L_1) \triangle Y| \leq \delta_2 n$;
\item\label{i:badvtcs} the set of vertices
\[
X = \left\{ x \in V\left(Q^d\right) \colon \left|\partial_v(x) \cap Y \right| \leq \delta_4 d \right\}
\]
satisfies $|X| \leq 2^{(1-\delta_5)d}$.
\end{enumerate}
\end{theorem}

With this theorem in hand, let us briefly sketch the strategy to prove Theorem \ref{t:edgeexpansion} on the edge-expansion properties of the largest component of $Q^d_p$. We will use a sprinkling argument, viewing $Q^d_p$ as the union of two independent random subgraphs $Q^d_{q_1}$ and $Q^d_{q_2}$ where $q_1$ is a supercritical probability chosen sufficiently close to $p$ such that, if we denote by $L_1'$ and $L_1$ the largest component in $Q^d_{q_1}$ and $Q^d_{p}$ respectively, then whp $L_1'$ contains most of the vertices in $L_1$, which we can guarantee by Theorem \ref{t:AKSfine}.

After exposing $Q^d_{q_1}$, it will follow from Property \eqref{i:badvtcs} of Theorem \ref{t:AKSfine} that whp most of the vertices contained in components of $Q^d_{q_1}$ of order $\Omega\left(d^2\right)$ are in $L'_1$. Using Lemma \ref{l:treedecomp}, we can split the subgraph of $Q^d_{q_1}$ consisting of the union of these components into a collection $\mathcal{C}$ of connected pieces which all have polynomial size in $d$, specifically almost all the pieces will have size around $d^{\frac{3}{2}}$. We then show that whp for any partition $\cc{C} = \cc{C}_A \cup \cc{C}_B$ of $\cc{C}$ into two parts, each covering a positive proportion of the vertices in $L'_1$, there are exponentially many edge-disjoint paths between vertices in the two partition classes in $Q^d_{q_2}$. We do so via a union bound, showing that the probability that any partition fails to have this property is much smaller than the total number of possible partitions. For partitions in which the two partition classes have many shared neighbours we will use property \eqref{i:badvtcs} of Theorem \ref{t:AKSfine} to find these paths and otherwise we will additionally use Theorem \ref{t:vtxiso}.

Given this, we note that any large subset $S \subseteq V(L_1)$ must either split apart exponentially many of the pieces in $\cc{C}$, in which case for each piece which is split there will be some edge of $Q^d_{q_1}$ present in $\partial_{e,p}(S)$, or there must be a partition $\cc{C} = \cc{C}_A \cup \cc{C}_B$ of $\cc{C}$ into two parts, each covering a positive proportion of $L'_1$, such that each piece in $\cc{C}_A$ is contained in $S$ and almost every piece in $\cc{C}_B$ is disjoint from $S$. Our claim then implies that there are many edge disjoint paths in $Q^d_{q_2}$ between the two partition classes, almost all of which must contribute at least one edge to $\partial_{e,p}(S)$.

\begin{proof}[Proof of Theorem \ref{t:edgeexpansion}]
During the proof we will introduce a series of constants $c_1,c_2,\ldots,$ with the understanding that each $c_i$ will be as small as is necessary for the argument in terms of $\alpha, \epsilon$ and all $c_j$ with $j < i$.

Let us fix $c_2 \ll c_1$, let $\gamma$ be the survival probability of a Po$(1+\epsilon)$ branching process and let us choose $\delta_1 < \epsilon$ such that $\gamma_1$, the survival probability of a Po$(1+\delta_1)$ branching process, is such that $\gamma-\gamma_1 \leq c_2$. We take $\delta_1$ and $\delta_2=c_1$ in Theorem \ref{t:AKSfine} to find $c_2 \gg c_3=\delta_3$, $c_4 = \delta_4$ and $c_5 = \delta_5$. Note that we may assume that $c_5 \ll c_4 \ll c_3$.

Our plan is to expose $Q^d_p$ in two steps, with $q_1= \frac{1+\delta_1}{d}$ and $q_2 = \frac{p-q_1}{1-q_1}$, where we may assume that $q_2 \geq \frac{c_6}{d}$. Let us write $Q_1 = Q^d_{q_1}$ and $Q_2 = Q_1 \cup Q^d_{q_2}$ where the two random subgraphs are chosen independently, so that $Q_2$ has the same distribution as $Q^d_p$.

By Theorem \ref{t:AKSfine}, whp the largest component $L'_1$ in $Q_1$ is such that $|V(L'_1)| = (\gamma_1 \pm c_1) n = (\gamma \pm 2c_1)n$. Furthermore, if $Y$ is the set of vertices of $Q_1$ contained in components of order at least $c_3 d^2$ then whp $|V(L'_1) \triangle Y| \leq c_1 n$ and
\begin{equation}\label{e:X}
X = \left\{ x \in V\left(Q^d\right) \colon |\partial_v(x) \cap Y| \leq c_4 d \right\}
\end{equation}
satisfies $|X| \leq 2^{(1-c_5)d}$.

Recall that $L_1$ is the largest component of $Q^d_p$. We first show that whp the symmetric difference of $V(L_1)$ and $Y$ is small.
\begin{claim}\label{c:giant}
Whp $|V(L_1) \triangle Y| \leq 4c_1n$.
\end{claim}
\begin{proof}[Proof of Claim \ref{c:giant}]
We apply Theorem \ref{t:AKSfine} with $\delta_1 = \epsilon$ and $\delta_2 = c_1$ to $Q_2 = Q^d_p$. It follows that whp there is a unique component $L_1$ in $Q_2$ of order at least $c_1 n$ and that $|V(L_1)| = (\gamma \pm c_1) n$. However, since $Q_1 \subseteq Q_2$ and $|V(L_1')| \geq (\gamma - 2c_1) n \geq c_1 n$, it follows that $L'_1 \subseteq L_1$. Hence, ${|V(L_1) \setminus V(L'_1)| \leq 3 c_1 n}$, and so $|V(L_1) \triangle Y| \leq 4 c_1 n$.
\end{proof}

Next, we want to use Lemma \ref{l:treedecomp} to split each component of $Q_1$ of order at least $c_3 d^2$ into connected pieces of roughly equal size. In order to do so we need the following bound on the degree sequence of $Q_1$.
\begin{claim}\label{c:degrees}
Whp $Q_1$ contains at most $nd^{-5}$ vertices of degree at least $\log d$.
\end{claim}
\begin{proof}[Proof of Claim \ref{c:degrees}]
For any fixed vertex $v \in V\left(Q^d\right)$, the degree of $v$ in $Q_1 = Q^d_{q_1}$ is distributed as Bin$(d,q_1)$, and so by Lemma \ref{l:Chernoff} \eqref{i:chernoff2} we have that
\[
\mathbb{P}\left( d_{Q_1}(v) \geq \log d  \right) \leq \left(\frac{e(1+\delta_1)}{ \log d } \right)^{ \log d } \leq d^{-\frac{\log \log d}{2}}.
\]

It follows that the expected number of vertices in $Q_1$ with degree at least $\log d$ is at most $nd^{-\frac{\log \log d}{2}}$. Hence, by Markov's inequality, whp there at most $nd^{-5}$ vertices with degree at least $\log d$.
\end{proof}
We note that the above argument is rather unoptimised and with a little more care, the bound on the degree of the exceptional vertices could be improved from $\log d$ to $\frac{C \log d}{\log \log d}$ for some suitably large constant $C$. However, for ease of presentation we have not attempted to optimise the logarithmic factors in the proof.

Assuming that Claim \ref{c:degrees} holds, by applying Lemma \ref{l:treedecomp} with $\ell = \frac{1}{c_{10}} d^{\frac{3}{2}}$ to a spanning tree of each component of $Q_1$ of order at least $c_3 d^2$, noting that $\Delta\left(Q_1\right) \leq \Delta\left(Q^d\right) = d$, we can split the collection of these components into connected pieces such that at most $nd^{-5}$ of the pieces have size between $\ell = \frac{1}{c_{10}} d^{\frac{3}{2}}$ and $\ell d = \frac{1}{c_{10}} d^{\frac{5}{2}}$ and the rest have size between $\ell$ and $\ell \log d = \frac{1}{c_{10}} d^\frac{3}{2}\log d$. Let us call this collection of pieces $\cc{C}$, noting that $V\big(\bigcup \cc{C}\big) = Y$.

We next show that whp for any partition $\cc{C}=\cc{C}_A \cup \cc{C}_B$ of $\cc{C}$ into two parts, each covering a positive proportion of the vertices of $Y$, we expose exponentially many edge-disjoint paths between $A$ and $B$ in the second sprinkling step. This argument will require that the pieces in $\cc{C}$ are not significantly smaller than $d^{\frac{3}{2}}$, whereas later bounds will be optimised by taking the pieces in $\cc{C}$ as small as possible, which motivates our choice of $\ell$ in the above.

\begin{claim}\label{c:manypaths}
Whp for any partition $\cc{C} = \cc{C}_A \cup \cc{C}_B$ of $\cc{C}$ into two parts, with $A = V\big(\bigcup \cc{C}_A\big)$ and $B = V\big(\bigcup \cc{C}_B \big)$, such that $\min\{|A|,|B|\} \geq c_7 n$, there are at least $c_{11} nd^{-\frac{3}{2}}$ edge-disjoint paths between $A$ and $B$ in  $Q^d_{q_2}$.
\end{claim}
\begin{proof}[Proof of Claim \ref{c:manypaths}]
The total number of pieces in $\cc{C}$ is at most $c_{10}nd^{-\frac{3}{2}}$ and so the total number of possible partitions is at most $2^{c_{10}nd^{-\frac{3}{2}}}$. Hence, if we can show that the claim does not hold for a fixed partition with probability at most $\exp \left(- c_{10}nd^{-\frac{3}{2}}\right)$, then whp the claim holds for all partitions by the union bound.

Given a fixed partition $\cc{C}_A \cup \cc{C}_B$ with $A$ and $B$ as above, let $N(A) = A \cup \partial_{v,q_1} (A)$ be the inclusive neighbourhood of $A$ in $Q_1$ and similarly let $N(B)$ be the inclusive neighbourhood of $B$ in $Q_1$. Let $D = N(A) \cap N(B)$. We split into two cases:
\begin{enumerate}[(i)]
\item\label{i:Dlarge} $|D| \geq c_8 nd^{-\frac{1}{2}}$;
\item\label{i:Dsmall} $|D| < c_8 nd^{-\frac{1}{2}}$.
\end{enumerate}

{\bf Case \eqref{i:Dlarge}:} Let us suppose first that $|D| \geq c_8 nd^{-\frac{1}{2}}$. In this case, let $D' = D \setminus X$, where $X$ is as in \eqref{e:X}, so that $|D'| \geq \frac{|D|}{2}$, since $|X| \leq n^{1-c_5}$.

Since each $x \in D'$ is in $D$, $x$ is either in $A$ or has a neighbour in $A$ and similarly $x$ is either in $B$ or has a neighbour in $B$. Moreover, since $x \not\in X$ and $A \cup B = Y$, it follows that $x$ has at least $\frac{c_4 d}{2}$ many neighbours in one of $A$ or $B$. For each $x \in D'$, let us fix a set $E_x$ of edges witnessing this, consisting of $\frac{c_4 d}{2}$ edges from $x$ to one of the vertex sets $A$ or $B$ together with a single edge from $x$ to the other vertex set if $x$ is not already an element of that set (see Figure \ref{f:case1}).

\begin{figure}[ht!]
\center
\scalebox{0.8}{
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1cm,y=1cm, scale=0.8]
\draw [line width=2pt] (-3,0) circle (3.973726211001457cm);
\draw [line width=2pt] (3,0) circle (4.014635724446242cm);
\draw [line width=2pt] (-5,0) circle (1.0018482919085105cm);
\draw [line width=2pt] (5,0) circle (0.9800510190801291cm);
\draw (-.2,1.7) node[anchor=north west] {$x$};
\draw (-0.3,4.4) node[anchor=north west] {\large $D$};
\draw [line width=1pt] (0,2.33)-- (0,3.66);
\draw (-5.9,0.3) node[anchor=north west] {\Large $A$};
\draw (5.1,0.3) node[anchor=north west] {\Large $B$};
\draw [rotate around={0:(-0.06,-2.44)},line width=1.2pt,color=ffqqqq] (-0.06,-2.44) ellipse (2.0716606002141353cm and 0.5401644587341691cm);
\draw [color=ffqqqq](-0.4,-3.8) node[anchor=north west] {\large $X$};
\draw [line width=1pt,color=ffqqqq] (0,-2.4)-- (0,-3.8);
\draw [line width=2pt,color=qqqqff] (0,1)-- (4.38,0);
\draw [line width=2pt,color=qqqqff] (0,1)-- (-4.926,0.474);
\draw [line width=2pt,color=qqqqff] (0,1)-- (-5,0);
\draw [line width=2pt,color=qqqqff] (0,1)-- (-4.838,-0.428);
\draw [color=qqqqff](-0.34273971341581727,0.6420739200889095) node[anchor=north west] {\large $E_x$};
\draw (-7,4.35) node[anchor=north west] {\large $N(A)$};
\draw (6,4.35) node[anchor=north west] {\large $N(B)$};
\draw [line width=1pt] (-6.4062538711037,3.5956574004507864)-- (-4.647077751919517,2.598790932913084);
\draw [line width=1pt] (6.421071997947632,3.580997599457585)-- (4.603256674790643,2.628110534899487);
\begin{scriptsize}
\draw [fill=ttqqqq] (-5,0) circle (2.5pt);
\draw [fill=black] (0,1) circle (2.5pt);
\draw [fill=ttqqqq] (4.38,0) circle (2.5pt);
\draw [fill=ttqqqq] (-4.926,0.474) circle (2.5pt);
\draw [fill=ttqqqq] (-4.838,-0.428) circle (2.5pt);
\end{scriptsize}
\end{tikzpicture}
}
\caption{A vertex $x \in D'=D\setminus X$ with corresponding edge set $E_x$ consisting of $\frac{c_4 d}{2}$ edges from $x$ to $A$ and an edge from $x$ to $B$}\label{f:case1}
\end{figure}

Let us choose a subset $D'' \subseteq D'$ of size at least $\frac{|D'|}{2}$ such that all $x \in D''$ have the same parity, and hence the edge sets $E_x$ are disjoint for each $x \in D''$.

Then, for each $x \in D''$, there is a path between $A$ and $B$ using edges from $E_x$ in $Q^d_{q_2}$ with probability at least
\[
q_2 \left(1 - (1-q_2)^{\frac{c_4 d}{2}}\right)\geq q_2 \left( 1- \exp\left( -\frac{q_2 c_4 d}{2}\right)\right) \geq \frac{c_6\left(1- \exp\left( - \frac{c_4 c_6}{2} \right)\right)}{d} \geq \frac{2c_9}{d},
\]
since $1-x \leq e^{-x}$ for all $x>0$ and $q_2 \geq \frac{c_6}{d}$.

Since the sets $E_x$ are disjoint, these events are independent for different $x \in D''$ and so, by Chernoff type bounds (Lemma \ref{l:Chernoff} \eqref{i:chernoff1}), we have that there are less than $\frac{c_9}{d} |D''| \geq  c_{11} nd^{-\frac{3}{2}}$ such paths with probability at most $2\exp \left( - \frac{c_8c_9 nd^{-\frac{3}{2}}}{32}\right) \leq \exp \left(- c_{10}nd^{-\frac{3}{2}}\right)$.

{\bf Case \eqref{i:Dsmall}:} Let us now deal with the second case, where $|D| < c_8 nd^{-\frac{1}{2}}$. In this case we note that $|N(A)| \geq |A| \geq c_7 n$ and $|N(A) \cap B| \leq |D| \leq c_8  n d^{-\frac{1}{2}}$, and so $|N(A)^c| \geq |B| - |D| \geq \frac{c_7}{2} n$. Therefore, it follows from Theorem \ref{t:vtxiso} that there is a matching $F$ in $Q^d$ of size at least $3 c_8 nd^{-\frac{1}{2}}$ from $N(A)$ to $N(A)^c$.

Since $|X| \ll |D| < c_8 nd^{-\frac{1}{2}}$, if we let $F'$ be those edges in $F$ which do not have an endpoint in either $D$ or $X$ then $|F'| \geq c_8 nd^{-\frac{1}{2}}$. For each edge $uv=e$ in $F'$, where without loss of generality $u \in N(A) \setminus (D \cup X)$, either $u \in A$ or $u$ has at least $c_4 d$ many neighbours in $A$ and similarly either $v \in B$ or $v$ has at least $c_4 d$ many neighbours in $B$. For each $e \in F'$, let us fix a set of edges $E_e$ witnessing this, consisting of $e$ together with  $c_4 d$ edges from $u$ to $A$ if $u \not\in A$ and $c_4d$ edges from $v$ to $B$ if $v \not\in B$ (see Figure \ref{f:case2}).

\begin{figure}[ht!]
\center
\scalebox{0.85}{
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1cm,y=1cm, scale=0.8]
\clip(-7.316184854367966,-5.036459304646435) rectangle (9.465115794760276,5.897698261161853);
\draw [line width=2pt] (3,0) circle (3.3842874192853127cm);
\draw [line width=2pt] (-3,0) circle (3.381421569359371cm);
\draw (1.25,.2) node[anchor=north west] {\large $D$};
\draw [line width=1pt] (1.3,-.1)-- (0,.3);
\draw [rotate around={0:(0,-1.5)},line width=1.2pt,color=ffqqqq] (0,-1.5) ellipse (1.2cm and 0.4cm);
\draw [color=ffqqqq](1.45,-1.8) node[anchor=north west] {\large $X$};
\draw [line width=1pt,color=ffqqqq] (1.5,-2.05)-- (0.6,-1.44);
\draw [line width=2pt,color=qqqqff] (-1.3816226920812198,2.4970837563675694)-- (1.4117266172991565,2.4939408427056775);
\draw (-1.4,2.3) node[anchor=north west] {$u$};
\draw (0.9,2.3) node[anchor=north west] {$v$};
\draw [line width=2pt] (-5,0) circle (1.014059040974722cm);
\draw [line width=2pt] (5,0) circle (0.9973217286593371cm);
\draw (-5.9,0.4) node[anchor=north west] {\Large $A$};
\draw (5.2,0.4) node[anchor=north west] {\Large $B$};
\draw [line width=2pt,color=qqqqff] (-1.3816226920812198,2.4970837563675694)-- (-5.003044898313415,0.6256235048101039);
\draw [line width=2pt,color=qqqqff] (-5,0)-- (-1.3816226920812198,2.4970837563675694);
\draw [line width=2pt,color=qqqqff] (-1.3816226920812198,2.4970837563675694)-- (-4.989038340517655,-0.6069535812168142);
\draw [line width=2pt,color=qqqqff] (1.4117266172991565,2.4939408427056775)-- (4.997637367859544,0.5836038314228226);
\draw [line width=2pt,color=qqqqff] (1.4117266172991565,2.4939408427056775)-- (5,0);
\draw [line width=2pt,color=qqqqff] (1.4117266172991565,2.4939408427056775)-- (4.983630810063783,-0.5789404656252933);
\draw [color=qqqqff](-0.3,3.4) node[anchor=north west] {\large $E_e$};
\draw (-6.2,3.75) node[anchor=north west] {\large $N(A)$};
\draw (5,3.75) node[anchor=north west] {\large $N(B)$};
\draw [line width=1pt] (-5.5,3)-- (-4.5,2.5);
\draw [line width=1pt] (5.5,3)-- (4.5,2.5);
\begin{scriptsize}
\draw [fill=ttqqqq] (-1.3816226920812198,2.4970837563675694) circle (2.5pt);
\draw [fill=ttqqqq] (1.4117266172991565,2.4939408427056775) circle (2.5pt);
\draw [fill=ttqqqq] (-5.003044898313415,0.6256235048101039) circle (2.5pt);
\draw [fill=ttqqqq] (-5,0) circle (2.5pt);
\draw [fill=ttqqqq] (-4.989038340517655,-0.6069535812168142) circle (2.5pt);
\draw [fill=ttqqqq] (4.997637367859544,0.5836038314228226) circle (2.5pt);
\draw [fill=ttqqqq] (5,0) circle (2.5pt);
\draw [fill=ttqqqq] (4.983630810063783,-0.5789404656252933) circle (2.5pt);
\end{scriptsize}
\end{tikzpicture}
}
\caption{An edge $uv=e \in F'$ with corresponding edge set $E_e$ consisting of $c_4d$ edges from $u$ to $A$ and $c_4d$ edges from $v$ to $B$.}\label{f:case2}
\end{figure}

Then, for each $uv=e \in F'$, there is a path in $Q^d_{q_2}$  between $A$ and $B$ using edges from $E_e$ with probability at least
\[
q_2 \left(1-(1-q_2)^{c_4 d}\right)^2 \geq q_2 \left(1-\exp\left(-q_2c_4 d \right)\right)^2 \geq \frac{c_6 \left(1-\exp\left(-c_4 c_6\right)\right)^2}{d} \geq \frac{2 c_9}{d}.
\]
Since the sets $E_e$ are disjoint, these events are independent for different $e\in F'$ and so, by Chernoff type bounds (Lemma \ref{l:Chernoff} \eqref{i:chernoff1}), we have that there are less than $\frac{c_9}{d} |F'| \geq c_{11} nd^{-\frac{3}{2}}$ such paths with probability at most $2\exp \left( - \frac{c_8 c_9 nd^{-\frac{3}{2}}}{16} \right) \leq \exp \left(- c_{10}n d^{-\frac{3}{2}}\right)$.
\end{proof}

Assuming Claims \ref{c:giant} and \ref{c:manypaths}, our aim is to show that the edge boundary of every subset $S \subseteq L_1$ satisfying ${\alpha n \leq |S| \leq \frac{|V(L_1)|}{2}}$ is large. Let us fix some such subset $S$.

\begin{claim}\label{c:largeedgeboundary}
Whp $\left| \partial_{e,p}(S) \right| \geq \frac{c_{12} n}{d^3 (\log d)^2}$.
\end{claim}
\begin{proof}[Proof of Claim \ref{c:largeedgeboundary}]
We first note that by Claim \ref{c:giant} whp $|S \cap  Y| \geq (\alpha- 4c_1)n$. Let us write
\begin{align*}
\cc{C}_0 &= \{ C \in \cc{C} \colon S \cap C \neq \emptyset \text{ and } C \setminus S \neq \emptyset \},\\
\cc{C}_A &= \{C \in \cc{C} \colon C \subset S\},\\
\cc{C}_B &= \{C \in \cc{C} \colon S \cap C = \emptyset\},
\end{align*}
and let $A = V\big(\bigcup \cc{C}_A\big)$ and $B =V\big(\bigcup \cc{C}_B \cup \bigcup \cc{C}_0 \big)$.

Since each $C \in \cc{C}$ is connected in $Q_1\subseteq Q_2$, there are at least $|\cc{C}_0|$ edges in the edge boundary of $S$ in $Q_2$. We split into two cases:
\begin{enumerate}[(i)]
\item\label{i:Clarge} $|\cc{C}_0| \geq \frac{c_{12} n}{ d^3 (\log d)^2}$;
\item\label{i:Csmall} $|\cc{C}_0| < \frac{c_{12} n}{ d^3 (\log d)^2}$.
\end{enumerate}

{\bf Case \eqref{i:Clarge}:} Suppose first that $|\cc{C}_0| \geq \frac{c_{12} n}{ d^3 (\log d)^2}$. Since each $C \in \cc{C}_0$ is connected in $Q_1$ and meets both $S$ and $V\left(Q^d\right) \setminus S$, it follows that there are at least $\frac{c_{12} n}{d^3 (\log d)^2}$ many edges in the edge boundary of $S$ in $Q_1$ and hence, at least this many in the edge boundary of $S$ in $Q_2$.

{\bf Case \eqref{i:Csmall}:} Suppose instead that $|\cc{C}_0| < \frac{c_{12} n}{ d^3 (\log d)^2}$. Since there are at most $nd^{-5}$ pieces in $\cc{C}$ of size between $\frac{1}{c_{10}} d^{\frac{3}{2}}$ and $\frac{1}{c_{10}} d^{\frac{5}{2}}$ and the rest have size at most $\frac{1}{c_{10}} d^{\frac{3}{2}} \log d$, it follows that, if we let $Z:=V\big(\bigcup \cc{C}_0\big)$, then
\begin{equation}\label{e:Zbound}
|Z| \leq \frac{2 c_{12} n}{c_{10} d^{\frac{3}{2}} \log d}.
\end{equation}

In particular, we have that $|A| \geq |S| - |Z| \geq c_7 n$ and $|B| \geq |Y| - |S| \geq c_7 n$, and so, by Claim \ref{c:manypaths}, whp there is a family $\cc{P}$ of at least $c_{11} nd^{-\frac{3}{2}}$ many edge disjoint paths between $A$ and $B$ in $Q^d_{q_2}$.

Then, by Lemma \ref{l:edgeboundary} applied with $k = \frac{5}{2}$, whp either $|Z| \leq \frac{c_{11}}{2}nd^{-\frac{5}{2}}$, or $Z$ has edge boundary satisfying
\[
\left|\partial_{e,p}(Z)\right| \leq |Z| \log d \leq \frac{2c_{12}}{c_{10}} n d^{-\frac{3}{2}} \leq \frac{c_{11}}{2}nd^{-\frac{3}{2}},
\]
where we used \eqref{e:Zbound} to bound $|Z|$ from above.
In either case, since every vertex in $Z$ has degree at most $d$, at most half of the paths in $\cc{P}$ meet $Z$, and every other path in $\cc{P}$ must contribute at least one edge to the boundary of $S$ in $Q_2$, since $A \subseteq S$ and $(B \setminus Z) \cap S = \emptyset$. It follows that the edge boundary of $S$ in $Q_2$ has size at least $\frac{c_{11}}{2}nd^{-\frac{3}{2}} \geq \frac{c_{12} n}{d^3 (\log d)^2}$.
\end{proof}

It follows from Claim \ref{c:largeedgeboundary} that the theorem holds with $\beta = c_{12}$.
\end{proof}

\begin{proof}[Proof of Corollary \ref{c:vertexexpansion}]
Let $\beta$ be given by Theorem \ref{t:edgeexpansion}. By Lemma \ref{l:edgeboundary}, we have that whp every subset $X \subseteq V\left(Q^d\right)$ of size $|X| \geq n d^{-5}$ has edge boundary in $Q^d_p$ such that
\[
\left|\partial_{e,p}(X)\right| \leq |X| \log d.
\]

So, we may assume that the above and the conclusion of Theorem \ref{t:edgeexpansion} hold. Let $S \subseteq L$ be such that ${\alpha n \leq |S| \leq \frac{|V(L_1)|}{2}}$ and let $X = \partial_{v,p}(S)$ be the vertex boundary of $S$ in $Q^d_p$. Then, by our choice of $\beta$, we have $\left|\partial_{e,p}(S)\right| \geq \beta  \frac{n}{d^3 (\log d)^2}$ and so, since $Q^d$ is $d$-regular, it follows that $|X| \geq \frac{n}{d^4 (\log d)^2} \geq n d^{-5}$. Hence, $\left|\partial_{e,p}(X)\right| \leq  |X| \log d$.

However, since each edge in the edge boundary of $S$ is in the edge boundary of $X$ we obtain
\[
  |X| \log d \geq \left|\partial_{e,p}(X)\right| \geq \left|\partial_{e,p}(S)\right| \geq \beta \frac{n}{d^3 (\log d)^2}.
\]
It follows that $|X| \geq \beta \frac{n}{d^3 (\log d)^3}$, and so the corollary follows
\end{proof}

As corollaries, we can deduce Theorems \ref{t:longcycle} and \ref{t:Hadwiger}.

\begin{proof}[Proof of Theorem \ref{t:longcycle}]
By Corollary \ref{c:vertexexpansion} given $\alpha \ll \epsilon$ there exists a $\beta >0$ such that whp every subset $S$ of the largest component $L_1$ of $Q^d_p$ with $\alpha n \leq |S| \leq \frac{|V(L_1)|}{2}$ satisfies $\left|\partial_{v,p}(S)\right| \geq  \frac{\beta n}{d^3 (\log d)^3}$. Hence, applying Theorem \ref{t:cycleexpander} with $k = \frac{|V(L_1)|}{2}$ and $t = \frac{\beta n}{d^3 (\log d)^3}$ we can conclude that $L_1$ contains a cycle of length $\Omega\left( \frac{ n}{d^3 (\log d)^3} \right)$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{t:Hadwiger}]
If the largest component $L_1$ of $Q^d_p$ does not contain a $K_t$-minor, then by Theorem \ref{t:minorexpander} there is some constant $C >0$ such that $L_1$ contains a subset $S$ of size at most $C t \sqrt{|V(L_1)|} \leq Ct \sqrt{n}$, such that each component of $G \setminus S$ has order at most $\frac{2|V(L_1)|}{3}$. It follows that we can find some subset of $L_1$ of size between $\frac{|V(L_1)|}{3}$ and $\frac{|V(L_1)|}{2}$ whose vertex boundary is contained in $S$, and hence has size at most $Ct\sqrt{n}$.

However, by Corollary \ref{c:vertexexpansion}, whp every subset of $L_1$ of size between $\frac{1}{3}|V(L_1)|$ and $\frac{1}{2}|V(L_1)|$ has a vertex boundary of size at least $\frac{\beta n}{d^3 (\log d)^3}$. It follows that $t \geq\frac{\beta \sqrt{n}}{ Cd^3 (\log d)^3 } = \Omega \left( \frac{\sqrt{n}}{d^3 (\log d)^3}\right)$.
\end{proof}

\section{Genus of the largest component}\label{s:genus}
Given a graph $G$ let us write $g(G)$ for the \emph{genus} of $G$, the smallest $k \in \mathbb{N}$ such that $G$ can be embedded on an orientable surface of genus $k$.

Since the genus of $K_t$ is known to be $\Omega\left(t^2\right)$, and the genus of a graph is a \emph{minor monotone function}, that is, if $H$ is a minor of $G$ then $g(H) \leq g(G)$, it follows from Theorem \ref{t:Hadwiger} that whp $g\left(Q^d_p\right) = \Omega\left(\frac{n}{d^3 (\log d)^3} \right)$ in the supercritical regime. However, if we are more careful, we can improve this to a bound of optimal asymptotic order.

\begin{proof}[Proof of Theorem \ref{t:genus}]
Euler's formula tells us that the genus of a connected graph $L$ satisfies
\begin{equation}\label{e:euler}
g(L) = \frac{1}{2}\left(e(L) - v(L) - f(L) +2\right) = \frac{1}{2}\left(\text{excess}(L) - f(L) +2\right),
\end{equation}
where $e(L)$ and $v(L)$ are the number of edges and vertices of $L$ respectively, excess$(L):= e(L) - v(L)$ is the \emph{excess} of $L$ and $f(L)$ is the number of faces of $L$ when embedded on a surface of minimal genus. We will show that whp the largest component $L_1$ of $Q^d_p$ satisfies
\begin{enumerate}[(i)]
\item\label{i:excess}excess$(L_1) = \Omega(n)$; and
\item\label{i:faces}$f(L_1) = o(n)$,
\end{enumerate}
from which it follows, from \eqref{e:euler}, that whp $g(L_1)$ is also $\Omega(n)$.

{\bf Property \eqref{i:excess}:} To prove that whp excess$(L_1) = \Omega(n)$, we again argue via a two-round exposure. Let $q_1 = \frac{1+\frac{\epsilon}{2}}{d}$, so that $q_2 = \frac{p-p_1}{1-p_1} \geq \frac{\epsilon}{2d}$ and let us write as before $Q_1 = Q^d_{q_1}$ and $Q_2 = Q_1 \cup Q^d_{q_2}$ where the two random subgraphs are independent, so that $Q_2$ has the same distribution as $Q^d_p$.

By Theorem \ref{t:AKS} there exists $\alpha>0$ such that whp there is a component $L'_1$ of $Q_1$ with ${|V(L_1')| \geq \alpha n}$. Since $L'_1$ is connected it contains at least $|V(L_1')| -1$ edges of $Q_1$.

However, if we let $\beta \ll \alpha$ and let $\cc{A}$ be the event that there exists a set of vertices ${W \subseteq V\left(Q^d\right)}$ of size at least $\alpha n$ such that $W$ spans at most $\beta d |W|$ edges in $Q^d$ and $W$ spans at least $|W|-1 \geq \frac{|W|}{2}$ edges of $Q_1$, then it follows, by Lemma \ref{l:Chernoff} \eqref{i:chernoff2}, that
\begin{align*}
\mathbb{P}(\cc{A}) &\leq \sum_{i \geq \alpha n} \binom{n}{i}\mathbb{P}\left(\text{Bin}\left(\left\lfloor \beta d i \right\rfloor, \frac{2}{d}\right) \geq \frac{i}{2}\right) \leq \sum_{i \geq \alpha n} \left(\frac{en}{i}\right)^i (2\beta e)^{\frac{i}{2}} = \sum_{i \geq \alpha n} \left(\frac{\sqrt{2} e^{\frac{3}{2}} \sqrt{\beta}}{\alpha} \right)^i =o(1).
\end{align*}

Hence, we can conclude that whp $L'_1$ spans at least $\beta d |V(L'_1)| \geq \alpha \beta d n$ many edges of $Q^d$.  Let $T$ be a spanning tree of $L'_1$ in $Q_1$, then $L'_1$ spans at least $\alpha \beta d n - |V(T)| \geq \frac{\alpha \beta d n}{2}$ edges in $Q^d - T$.

Hence, again by Chernoff type bounds (Lemma \ref{l:Chernoff} \eqref{i:chernoff1}), we have that the number of edges spanned by $L'_1$ in $Q^d_{q_2} \setminus T$, which we denote by $Y$, is such that
\[
\mathbb{P}\left(Y \leq \frac{\alpha \beta \epsilon n}{8}\right) \leq \mathbb{P}\left( \text{Bin}\left( \left\lceil\frac{\alpha \beta d n}{2}\right\rceil, \frac{\epsilon}{2d} \right) \leq \frac{\alpha \beta \epsilon n}{8}\right) \leq 2\exp\left(-\frac{\alpha \beta \epsilon n}{32}\right) = o(1).
\]

So, we can conclude that whp there is some component $L_1 \supseteq L'_1$ in $Q_2 = Q_1 \cup Q^d_{q_2}$ which has excess at least $\frac{\alpha \beta \epsilon n}{8}$. Note that, by Theorem \ref{t:AKSfine} whp there is a unique linear sized component in $Q_2$ and so whp $L_1$ is in fact the largest component in $Q_2$. Summing up, whp
\begin{equation}\label{e:excess}
\text{excess}(L_1) \geq \frac{\alpha \beta \epsilon n}{8}.
\end{equation}
We note that the typical existence of a component of large excess in $Q^d_p$ can also be deduced in a relatively straightforward manner from Theorem \ref{t:AKSfine}. Indeed, by Theorem \ref{t:AKSfine} almost every vertex in $Q^d$, and so almost every vertex in the giant component $L$ of $Q_1$, has linearly many (in $d$) neighbours in $L$. It will then follow that whp after sprinkling we add linearly many (in $n$) edges to the vertex set of $L$ in $Q_2$.

{\bf Property \eqref{i:faces}:} To prove that whp $f(L_1) = o(n)$, let us consider the expected number of cycles of length $s$ in $Q^d_p$ for a fixed $s$. We note that we can naively bound the number of cycles of length $s$ in $Q^d$ by $2nd^{s-2}$. Indeed, for any vertex $x$ there are at most $d^{s-2}$ many paths of length $s-2$ starting at $x$. However, since the codegree of any pair of vertices in $Q^d$ is at most two, there are at most two ways to complete any path of length $s-2$ to a cycle of length $s$. Since there are at most $n$ many ways to choose the initial vertex $x$, the bound follows. We mention that this bound is far from tight and indeed, without too much effort, it can be shown that the number of cycles of length $s$ in $Q^d$ is at most $2^d\left(\frac{sd}{2}\right)^{\frac{s}{2}}$ (see \cite[Claim 3.2]{MS18}).

Since each such cycle is in $Q^d_p$ with probability $p^s$, and $p = \frac{1+\epsilon}{d}$, it follows that the number of cycles of length $s$, which we denote by $X_s$, satisfies
\[
\bb{E}(X_s) \leq 2nd^{s-2} p^s = 2n d^{-2}(1+\epsilon)^s.
\]
Hence, the expected number of cycles of length at most $\sqrt{\log d}$ in $Q^d_p$ is at most
\[
\sum_{s=1}^{\sqrt{\log d}} \bb{E}(X_s) \leq 2nd^{-2}\sum_{s=1}^{\sqrt{\log d}} (1+\epsilon)^s \leq 2nd^{-2}\sqrt{\log d} 2^{\sqrt{\log d}}\leq nd^{-1}.
\]
Hence, by Markov's inequality, whp the number of cycles of length at most $\sqrt{\log d}$ in $Q^d_p$ is $o(n)$. It follows that whp the number of faces of length at most $\sqrt{\log d}$ in any embedding of $L_1$ is $o(n)$. Furthermore, since in any embedding each edge is in at most two faces, the number of faces of length at least $\sqrt{\log d}$ in any embedding of $L_1$, which we denote by $Y_{\geq \sqrt{\log d}}$, is at most $\frac{2e(L_1)}{\sqrt{\log d}} \leq \frac{2e\left(Q^d_p\right)}{\sqrt{\log d}}$. However, again by Lemma \ref{l:Chernoff}, we have that whp $e\left(Q^d_p\right) \leq dnp = (1+\epsilon)n$ and so whp
\[
Y_{\geq \sqrt{\log d}} \leq \frac{2e\left(Q^d_p\right)}{\sqrt{\log d}} \leq \frac{2(1+\epsilon)n}{\sqrt{\log d}} = o(n).
\]
Hence, whp $f(L_1) = o(n)$ and so, by Euler's formula \eqref{e:euler} applied to $L_1$, we have
\[
g\left(Q^d_p\right) \geq g(L_1) \geq \frac{1}{2}\left(\text{excess}(L_1) - f(L_1) +2\right) \stackrel{\eqref{e:excess}}{\geq} \frac{1}{2}\left(\frac{\alpha \beta \epsilon n}{8} - o(n) \right) = \Omega(n).
\]
\end{proof}

\section{Discussion}\label{s:discussion}
In Theorems \ref{t:longcycle} and \ref{t:Hadwiger} we have shown that in the supercritical regime when $p=\frac{1+\epsilon}{d}$ for a positive constant $\epsilon$, whp $Q^d_p$ contains a cycle whose length is \emph{almost linear} in $n=2^d$, that is, up to some polylogarithmic term in $n$, and similarly a complete minor of order almost $\sqrt{n}$. However, it seems unlikely that these results are best possible, and analogous to the case of $G(d+1,p)$, it is natural to conjecture that for supercritical $p$ whp $Q^d_p$ contains a cycle whose length is in fact \emph{linear} in $n$, and a complete minor of order $\sqrt{n}$.
\begin{question}Let $\epsilon>0$ and $p=\frac{1+\epsilon}{d}$.
\begin{enumerate}
\item Is it the case that whp $Q^d_p$ contains a cycle of length $\Omega(n)$?
\item Is it the case that whp $Q^d_p$ contains a complete minor of order $\Omega\left(\sqrt{n}\right)$?
\end{enumerate}
\end{question}
Furthermore, in both questions it would also be interesting to know the dependence of the results on $\epsilon$. For example, in $G(d+1,p)$ it is known that for $p = \frac{1+\epsilon}{d}$ the largest component is of order $(2\epsilon + o(\epsilon))d$, the length of the longest cycle is of order $\Theta\left(\epsilon^2\right) d$ (see, for example, \cite[Theorem 5.7]{JLR00}) and the order of the largest complete minor is $\Theta\left(\epsilon^\frac{3}{2}\right) \sqrt{d}$ (see \cite{FKO09}).

As indicated in the introduction, it seems unlikely that such results can be proven simply by considering the expansion properties of the largest component. However, it is still an interesting question precisely how strong expansion properties of the largest component we can guarantee whp. For example, Krivelevich \cite{K18} showed that in the supercritical regime $p= \frac{1+\epsilon}{d}$ in $G(d+1,p)$, whp the largest component contains a subgraph of order $\Theta(d)$ which is a bounded degree $\alpha$-expander for some $\alpha =\alpha(\epsilon) > 0$, where a graph $G$ is an \emph{$\alpha$-expander} if for every subset $W \subseteq V(G)$ with $|W| \leq \frac{|V(G)|}{2}$ we have that $\left|\partial_v(W)\right| \geq \alpha |W|$.

\begin{question}\leavevmode
Let $\epsilon >0$ and $p = \frac{1+\epsilon}{d}$. For what $\alpha =\alpha(d)$ is it true that whp the largest component of $Q^d_p$ contains a subgraph of order $\Theta(n)$ which is an $\alpha$-expander?
\end{question}
We note that by a similar argument as in \cite[Lemma 2.4]{K19}, Corollary \ref{c:vertexexpansion} implies that we can take $\alpha(d) = \Omega\left(\frac{1}{d^3 (\log d)^3}\right)$, however it seems unlikely that this is optimal.

A careful examination of the proof of Theorem \ref{t:genus} shows that whp the genus of the largest component $L_1$ of $Q^d_p$ is asymptotically equal to half of the excess of $L_1$. In the case of ${G(d+1,p)}$, in the supercritical regime, it can be shown that if we delete the largest component, what is left has approximately the distribution of a subcritical random graph, and so whp all other components are unicyclic or trees. From this it follows that the genus of $G(d+1,p)$ is asymptotically equal to the genus of its largest component. Furthermore, by Euler's formula for an arbitrary graph $G$
\[
g(G) = \frac{1}{2}\left(e(G) - v(G) - f(G) + \kappa(G) + 1\right),
\]
where $\kappa(G)$ is the number of components of $G$, and by counting carefully the number of tree-components it is possible to determine asymptotically the genus of $G(d+1,p)$ in this regime, and also the excess of the giant component (see \cite[Theorems 1.2 and 1.3]{DKK20}).

However, it is not the case that whp $Q^d_p$ is planar for an arbitrary subcritical $p$. For example, it is easy to see that whp $Q^4 \subseteq Q^d_p$ when $p= \Theta\left(\frac{1}{d}\right)$, and $Q^4$ is non-planar. So, even if such a `symmetry rule' were to hold in this model, it would not be immediate that the main contribution to the genus of $Q^d_p$ is coming from the excess of the largest component. It would be interesting to determine asymptotically the genus of $Q^d_p$ in the supercritical regime.

There are some interesting open questions about the model $Q^d_p$ in the paper of Condon, Espuny D{\'\i}az, Girao, K{\"u}hn and Osthus \cite{CDGKO21}. In particular, they used as a crucial part of their proof the fact that whp $Q^d_{\frac{1}{2}}$ contains an `almost spanning' path, that is, a path containing $(1-o(1))n$ vertices, and they showed that this property is in fact true for $Q^d_p$ for any constant $p$.

However, analogous to the case of $G(d+1,p)$, we should perhaps expect such a path to exist for much smaller values of $p$. In particular, if we expect the sparse random subgraph $Q^d_{p}$ with $p=\frac{c}{d}$ to contain a path of linear length $f(c)n$ for some function $f(c)$ when $c > 1$, it is natural to conjecture that $f(c) \to 1$ as $c \rightarrow \infty$.

\begin{question}
Let $p = \omega\left(\frac{1}{d}\right)$. Is it true that whp $Q^d_p$ contains a path of length $(1-o(1))n$?
\end{question}

Finally, other notions of random subgraphs of the hypercube  $Q^d$ have also been studied. In particular, if we let $Q^d(p)$ denote a random \emph{induced} subgraph of $Q^d$ obtained by retaining each \emph{vertex} independently with probability $p$, then the typical existence of a giant component in $Q^d(p)$ when $p=\frac{1+\epsilon}{d}$ for a fixed $\epsilon >0$ was shown by Bollob\'{a}s, Kohayakawa and {\L}uczak \cite{BKL94}, and this was extended to a broader range of $p$ with $\epsilon=o(1)$ by Reidys \cite{R09}. It would be interesting to know if the giant component in $Q^d(p)$ whp also has good expansion properties.

	\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


Our capable and motivated team is always way ahead in serving our clients.
We are seeking for team-oriented individuals. We hope you will consider joining our dynamic team – one that values broad cross-cultural perspective, takes great pride in its work and shares an unwavering commitment to excellence and success.
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Currently we don't have any open positions but if you are interested in possible future opportunities please send your CV to hr@msiair.com.


Useful road map of Newtownards district, town centre and surroundings.
Newtownards Maps: With this easy to print map, you can see local districts of Newtownards, its central area and surroundings within the county of County Down, Northern Ireland in the United Kingdom. Using this detailed Newtownards map, you can find your way around the town and locate streets, roads and facilities in Newtownards. The view of the map shown on this page is of the town centre area of Newtownards, use the minus ( - ) button (bottom right of map) to view districts and surrounding areas, villages, towns and points of interest. Newtownards district map. By viewing this map you can judge roughly how far Newtownards is from these nearby places and see the distance between.
Things you can view include: The International Bar, Ards Shopping Centre, The Pub Bar & Restaurant, Ards Hospital, Scrabo Hall, Movilla Abbey Methodist Church, South Eastern Regional College, Newtownards First Presbyterian Church, The A21, Kiltonga Wildlife Reserve, The A48, The Link Family and Community Centre, The Web Theatre, Sleepy Hollow B&B, Ards Airport, Newtownards High Street, Conway Square, Londonderry Park, Greenwell Street Presbyterian Church, Newtownards Police Station, St Mark's Parish Church, Newtownards Congregational Church, The A20, Strangford Lough and more (you may need to utilise "view larger map link").
Northern Ireland Maps: Newtownards, County Down street map. Printable map of Newtownards Northern Ireland.
Newtownards districts, Newtownards streets, Newtownards roads, Newtownards attractions.


text:
package com.googlecode.openbeans;

import com.googlecode.openbeans.DefaultPersistenceDelegate;
import com.googlecode.openbeans.Encoder;
import com.googlecode.openbeans.Expression;
import com.googlecode.openbeans.Statement;

import javax.swing.JTabbedPane;

class SwingJTabbedPanePersistenceDelegate extends
		DefaultPersistenceDelegate {
    @Override
    @SuppressWarnings({ "nls", "boxing" })
    protected void initialize(Class<?> type, Object oldInstance,
            Object newInstance, Encoder enc) {
        // Call the initialization of the super type
        super.initialize(type, oldInstance, newInstance, enc);
        // Continue only if initializing the "current" type
        if (type != oldInstance.getClass()) {
            return;
        }

        JTabbedPane pane = (JTabbedPane) oldInstance;
    	int count = pane.getTabCount();
    	for (int i = 0; i < count; i++) {
			Expression getterExp = new Expression(pane.getComponentAt(i), pane,
					"getComponentAt", new Object[] { i });
			try {
                // Calculate the old value of the property
                Object oldVal = getterExp.getValue();
                // Write the getter expression to the encoder
                enc.writeExpression(getterExp);
                // Get the target value that exists in the new environment
                Object targetVal = enc.get(oldVal);
                // Get the current property value in the new environment
                Object newVal = null;
                try {
                	JTabbedPane newJTabbedPane = (JTabbedPane) newInstance;
					newVal = new Expression(newJTabbedPane.getComponent(i),
							newJTabbedPane, "getComponentAt",
							new Object[] { i }).getValue();
				} catch (ArrayIndexOutOfBoundsException ex) {
					// The newInstance has no elements, so current property
					// value remains null
				}
                /*
				 * Make the target value and current property value equivalent
				 * in the new environment
				 */
                if (null == targetVal) {
                    if (null != newVal) {
                        // Set to null
                    	Statement setterStm = new Statement(oldInstance, "addTab",
                                new Object[] { pane.getTitleAt(i), oldVal });
                        enc.writeStatement(setterStm);
                    }
                } else {
                    Statement setterStm = new Statement(oldInstance, "addTab",
                            new Object[] { pane.getTitleAt(i), oldVal });
                    enc.writeStatement(setterStm);
                }
            } catch (Exception ex) {
                enc.getExceptionListener().exceptionThrown(ex);
            }
		}
	}
}

redpajama_set_name: RedPajamaGithub


An HTML5 Gothic Bullet Hell in-development atm. This game features a protagonist who is a nun, blowing away evil creatures to save little orphan children.
Nuns with Guns is a gothic, dungeon-crawler-bullet-hell that celebrates blowing away evil monsters with extreme prejudice (in order to rescue wee orphan children from demons). Eventually, our hero will blast her way through 8 stages of cartoony gore and violence in order to rescue her retinue of kids--recently kidnapped from St. Germane's school for the innocent.


Where are the gritters now?
Winter weather can be fun for some, but sadly more people are at risk of illness and death during winter than at any other time of the year and may need extra help from us all. As the weather turns colder spare a thought and a little time for your Neighbours. Neighbours may need your help, especially if they are elderly, disabled or have a chronic illness. Cold weather is especially dangerous for these groups - and symptoms don't just get worse during a cold spell, it can last for a few days afterwards.
Whilst loss of power is thankfully a rare occurrence, being ready for such an event is something we all need to do. SP Energy Networks have outlined steps you can take to ensure you are prepared should there be an unexpected interruption to your electricity supply.
SP Energy Networks keep a Priority Services Register so that they can contact their most vulnerable customers if they experience a power interruption. People can be included on the register if they are dependent on medical equipment, chronically sick, disabled, or have some other special need that may require a special level of assistance.


Recently I was privileged to taste some of the great Hospices de Beaune (Burgundy) wines. These were the 2012 vintages, so not only are they "babies," they are still infants.
Which made them all the more remarkable for how present they already are and how their destinies of future greatness are unmistakable.
For sentimental reasons, it was a very special experience for me to taste this wine. My grandmother, all her adult life, sat by the fire in the evenings and knitted bandages that she would donate for the patients at the Hospice. Her name was Charlotte.
Tasting Notes: Deep brilliant ruby red. Light funghi on the bouquet, mineral, ripe red berry fruit finish. Altogether too drinkable. Open late 2013 and keep for up to 10 years and longer.
When the grapes arrived at the wineries, they were in perfect health. We knew it was going to be a good year as soon as we tasted the musts.
[Excerpted from Christie's Auction Catalog]: This gift to the Hospices was made in gothic lettering on parchment, in 1534. "For the cure and salvation of her soul," the childless Charlotte Dumay, wife of the King's Keeper of the Mint in Dijon, gave 100 ouvrées of vineyard and fallow land in the territory of Aloxe, with a house in stone. Today, this is one of the Hospice's longest-lived, most sumptuously structured wines.
This was this year's "La Piece des Presidents" which is the auction item donated entirely to an elected charity. This year it was First Lady Carla Bruni-Sarkozy's foundation.
Tasting Notes: Philippe Le Bon was the Duke of Burgundy (1419-1467). This is one of the most, if not the most, popular Burgundy White in Japan. 2012 vintage has top-notes of white flowers, tint of butter. Delicate citrus and white flowers on the mouth. Drink from late 2013 on or keep 3 – 5 years or more.
[Excerpted from Christie's Auction Catalog]: This Cuvée pays homage to Philippe le Bon, Duke of Burgundy, who gave his backing and authority to Chancellor Nicolas Rolin, during the latter's establishment of the Hôtel-Dieu [that's "hospital" in French], from 1443. It is regularly one of the finest whites, and fiercely contested at auction.
Tasting Notes: Golden robe with brilliant reflections. Elegant mineral nose. Citrus, green apple. Creamy mouth, terroir, mineral, taking on butter hint. Drink from mid 2013 on; Best to keep 3 – 8 years (or longer even).
[Excerpted from Christie's Auction Catalogue]: This vineyard was planted in 1974. It was acquired by the Hospices in 1989, and named after the Ladies of Flanders, in commemoration of the origins of the community of Sisters who have cared for the sick of the Hôtel-Dieu, down the centuries. The vines lie in the commune of Chassagne-Montrachet. They regularly produce a great white wine which is striking for its minerality, its length of flavour, its capacity to age and its exceptional complexity.


This course is compulsory for people who wish to become L2 Driving Assessors for the Registration and Licensing Services, Department of State Growth in Tasmania.
Applicants must be approved by the Department of State Growth to attend this course.
from the Certificate IV in Training and Assessment (TAE40110).
Only Participants who do not have the above two units need to do the two day course.
This course is delivered in Hobart and Launceston when a minimum of six (6) Participants enrol.
The course involves pre-reading of three (3) Learner Guides that cover the ten (10) areas of study above and has a formal assessment Multiple choice and Written questions, 2 Projects and a practical on road driving assessment conducted by an Assessor from the Department of State Growth.


And so on this day, waiting for news about my wife who was undergoing a serious surgical procedure, I penned a lengthy note to her. Then, surrounded by nervous chatter and oblivious children, I listened for the quiet voice of God.
In such times of grief or uncertainty, the honesty of the psalms makes them a natural place to turn. Psalm 31 was the heart-cry of David, who endured so much that he wrote, "My life is consumed by anguish" (v. 10). Compounding that grief was the pain of abandonment by his friends and neighbors (v. 11).
But David had the bedrock of faith in the one true God. "I trust in you, Lord; I say, 'You are my God.' My times are in your hands" (vv. 14-15). His lament concludes with resounding encouragement and hope. "Be strong and take heart, all you who hope in the Lord" (v. 24).
This time in the waiting room, the surgeon gave us good news: My wife could expect a full and complete recovery. Of course we're relieved and grateful! But even if she hadn't been "okay," our times still remain in God's capable hands.
Lord, we give You our deepest grief and pain as well as our joy. Thank You for Your constant love and presence no matter what today holds for us. You alone are faithful!
When we put our problems in God's hands, He puts His peace in our hearts.


text:
Home Missions My experience doing missions in Uganda- Odunaiya, TACN pastor
My experience doing missions in Uganda- Odunaiya, TACN pastor
by Church Times August 30, 2021
written by Church Times August 30, 2021
His gentle disposition does not give him away as a warrior. But he is a warrior sort of; only that he fights more on the mission field.
For Pastor Wilberforce Olukunle Odunaiya, working on strange land seems to be a passion. Though from Ijebu, Ogun State, he has been representing The Apostolic Church of Nigeria in queer places.
He put in about six years as a missionary in Katsina before he moved to Uganda; where God is currently doing great things through him.
He sees going to Uganda after his stay in Katsina as God's doing. "Before I volunteered to go to Uganda, God already prepared my mind for the place. In 2016 precisely, God showed me a vision that I will be on missions outside the country. So when the prophecy came at the ministers' conference of TACN LAWNA that year; that God needed people for missions, I was the first person to volunteer."
By then he was already a District pastor in Katsina. The Uganda opening was, however, according to him, another great opportunity to demonstrate the love of God to the world. Since he moved to that East African Country, God has been glorifying himself.
The challenges notwithstanding, Pastor Odunaiya has recorded incredible results. He got to Uganda in December of 2018 and he was first confronted with extremely cold weather, He also had to face the language barrier.
He found out that apart from English being their lingua franca, they also speak Swahili and several other local languages. So, that was one big hurdle for him. But he soon found a way around the challenge as some indigenous converts now help with interpretation.
The most depressing of the challenges for him however is that those in the big cities of Uganda are not open to the gospel. "Those who live in the cities are used to this epicurean lifestyle. Their conservative way of life has been eroded. Civilisation has taken a toll on them. They also don't believe in raising their children by themselves. Rather, they commit them to teachers. There is high moral decadence in the cities. There is also a high percentage of nominal Christians there," he said.
The people according to him are also proud of their traditional heritage. "They are faithful to African values that are contrary to scriptures. We have so many people who claim to be Christians but still go back to their idols. Some would publicly own up to be witches and still be members of the church. In Uganda, the Catholics don't believe they serve the same God as those who claim to be born again. Catholics are grouped separately. There is the Catholic Church and the born again churches to which I belong to. I belong to the Born Again Christian Association there Just like we have the Christian Association of Nigeria. It is now we are trying to bring all together."
Uganda is made up of so many tribes like Nigeria. When you don't understand their local languages, Odunaya says, "they build a fence". Language is a major barrier.
Odunaya: There is a massive harvest of souls
Wilberforce Odunaya
But Odunaiya is getting results. The church now has indigenous ministers. Some of the crusades have enjoyed massive attendance. The impact has been so much that many church founders in Uganda have submitted their churches to TACN. "We have 16 of such churches that have been submitted to us." Odunaya disclosed.
The initial experience of city dwellers being averse to the gospel did not deter him. He soon discovered that villagers in Uganda are open to the gospel. "In the villages, I found out that people are open to the gospel. If you stand in a place, clapping your hands and praising God, before you know it, people will join you. So, by God's grace, we have been able to make a lot of inroads to the villages in the country."
One thing he finds strange about the Ugandans is that they believe when you come with the gospel, you must bring welfare. Amazingly, this attitude is common among those in the cities who should know better.
"When you talk about Church; the first thing they think of is charity. If you are not giving them anything, they are not likely to listen to you. But many of the people in the rural areas are open to the gospel irrespective of any pecuniary gain," he noted
The gospel is however being preached and souls are coming to Jesus "I can say that God is helping us to reach the heart of many. Souls are being won in their hundreds and lives are being transformed. Some of the churches there have experienced revival through the conferences we hold for pastors and church workers. One of the greatest testimonies for us in Uganda is how God is impacting church leaders and workers of various denominations through our conferences.
"I discovered that many of the pastors from other churches attend our conferences and they get blessed and just decide to stay with us. Apart from pastors who handed over churches to the mission some handed over schools/orphanages too. It's simply mind-blowing for us to experience such." he said
Now, TACN according to Odunaiya has 18 branches in Uganda. The literature the church distributes during their conferences has made a lot of impact in the country. He said after each ministers' conference he receives calls from distant places in Uganda from people who want to identify with the church.
He also noted that "There are places in Uganda that are not motorable. You have to climb mountains for hours to reach the people. But the joy of it all is that they are receptive to the gospel. There is a kind of joy that one experiences when the gospel is being shared with the villagers."
While noting that there is a need for a lot of resources to carry out evangelical work in the Ugandan communities, Odunaiya said, God has been faithful. "I must confess that God has not left us without testimony. For instance, a man came to one of our programmes and he was mightily blessed.
"He insisted that we should come and start a church in his village. The man went ahead to make available one acre of land for the church free of charge. He said he liked the way we prayed and shared the word of God. The irony of it is that land is very expensive to get in Uganda."
For Odunaiya, who spoke with Church Times during the recent 45th annual convention of the LAWNA Territory of TACN, there is a lot of souls waiting to be harvested in Uganda. "The people like I said are eager for the word of God. Many of them especially those in the villages have no inhibition to the gospel. They only need genuine ministers who have passion for souls."
"As homosexual, I was using about four pampers daily, how God delivered me"
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Ibukunoluwa Ademola August 31, 2021 - 7:41 am
It is well in Jesus name

redpajama_set_name: RedPajamaCommonCrawl


I arrived a day early at EWTN to get settled in and begin my adventure steeped in prayer at the EWTN chapel, but just seeing the EWTN sign had me jumping up and down like a kid arriving at Disney World! The satellites are huge! I had to keep reminding myself that I was supposed to be prayerful while here.
I felt Mother Angelica's presence so profoundly as this Global Network is in existence because of a single nun, listening to God's will and acting upon those nudges. Outside the gift shop is a picture of Mother Angelica with roses from her funeral Mass still by it!
For dinner, I met Fr. Mitch Pacwa, who obviously was trying to make this Texas girl feel welcome in his straw hat. Fr. Mitch did not want to discuss vocations before the show so that I would not think we already had discussed a topic during the show.
He was a hoot! He shared with me about his wild animal shoots on his many travels. We spoke about his time as a new seminarian with the Jesuits in Chicago when he was a community organizer! What? I had no idea.
Another EWTN personality joined us, asking Fr. Mitch about the Crusades for an upcoming special. He was having trouble finding the answer until he asked Fr. Mitch, who spent 20 minutes explaining the answer! I was learning more than I anticipated on my trip to EWTN.


text:
ZIMBABWE SAFARI
Are you looking for luxury Zimbabwe safari holidays? Well, you have come to the right place! Here at Ubon Safari, we provide luxury Zimbabwe holidays and exquisite African safari tours. Zimbabwe is a beautiful country which offer lots of wilderness animals, outstanding cuisine, attractions, national parks, games reserves and so much more! Our safari luxury holidays in Zimbabwe has never been more exciting.
Zimbabwe is a landlocked country in the southern part of Africa with South Africa bordering its south, Zambia to its north, Botswana to its west and Mozambique to its east. With Bulawayo and Harare being the two main cities, the Zimbabwe divides into 8 provinces. The country was formerly known as Southern Rhodesia. In 1961, the country got its name Zimbabwe.
It remains one of the most intriguing destinations in the world. Our Zimbabwe safari holidays offer a large variety of scenery, make it one of the African continent's finest and largest concentrations of wildlife and perfect climate with an average of 7 hours sunshine daily throughout the year. Also, Zimbabwe is home to one of the 7 natural wonders of the world, the magnificent Victoria Falls, this avalanche of water beckons invitingly in the heat of the African sun.
CAPITAL CITY: HARARE
AIRPORT: HARARE INTERNATIONAL AIRPORT
CURRENCY: US DOLLAR (US$)
LANGUAGE: ENGLISH, SHONA AND SINDEBELE
VOLTAGE: 220V/50Hz UK Plug
DRY SEASON:
APRIL TO OCT – GREAT FOR WILDLIFE SPOTTING
JUNE TO SEPT – GREAT FOR VICTORIA FALLS
WET SEASON:
NOV TO MARCH – GREEN SCENERY, CALVING SEASON, GREAT BIRDWATCHING, THICK VEGETATION
ROADS CAN BE IMPASSABLE
BEST TIMES:
MAY TO SEPTEMBER
HIGH SEASON – JULY – OCTOBER (VICTORIA FALLS IS BUSY)
LOW SEASON – NOVEMBER – APRIL (SOME LODGES ARE CLOSED)
BEST WEATHER:
APRIL, MAY, SEPTEMBER
Go to our Visa page to find out whether you need a visa to enter Zimbabwe
http://thetravelvisacompany.co.uk/ubonsafari/zimbabwe/
Find out more about Zimbabwe on the largest travel guidebook online
https://www.lonelyplanet.com/zimbabwe
DO I NEED A VISA
AUSTRALIA – YES / EUROPEAN COUNTRIES – YES / UNITED KINGDOM – YES / USA – YES
Please enquire with the Zimbabwean Embassy in your country of Residence as we recommend you get you visa before you before you travel.
There is an option for a KAZA UNIVISA which allows you to cross between Zambia and Zimbabwe only, multiple times and is valid for 30 days.
You can purchase this Visa in arrival (only at the border points between these countries) and has to be paid in cash. For more information www.kazavisa.info
It is not mandatory to tip however we do recommend you to tip in Restaurants, Bars something that would suit you. For Drivers, Guides and Porters we recommend a minimum of USD 5 per person per day.
Most places have WIFI / internet, however in camps and lodges you may be limited to some specified areas, in camping safaris thereis no internet.
MOBILE PHONE ACCESS
Most of the locations in Zimbabwe have great mobile coverage especially in cities and towns. There's limited access in some of the rural areas. A local sim would be easily available for purchase.
WHAT ARE THE TOILETS LIKE?
Western style Toilets are available in most hotels, lodges and camps. Some areas may have squat toilets.
Drinking tap water is not recommended. Mineral water is always available.
ARE CREDIT CARDS ACCEPTED?
Please inform your Bank before you travel to ensure you will not have problems using your cards. Credit card facilities are available in some areas only.
Carry small denominations of US Dollars and South African Rand as Zimbabwean Dollar is no longer acceptable.
ATMS:
ATMs are available in major cities and towns.
DO I NEED A TRAVEL INSURANCE?
It is mandatory you should have Health / Travel Insurance
Please see http://www.worldtravelguide.net/zimbabwe/public-holidays
ELECTRICITY / PLUGS:
Electricity in Zimbabwe has 220V and 3 Pin flat plugs although some new hotels are now having round pin plugs so carry your travel plug.
DOES ZIMBABWE HAVE MALARIA:
Yes, visit your GP before you travel for your vaccinations and Malaria pills.
Dress respectfully, shoulders to knees should be covered especially in when entering places of worship.
Respect local traditions, customs and religions
Please do not buy coral, ivory or any animal products
Dispose of litter appropriately in bins
Do not throw cigarette butts especially in the parks as it may cause fire and destroy wildlife and habitats.
Don't be afraid to learn the local language and smile.
Please support the local shops and people
When taking photos of people please ask permission
At Ubon Safari, we decide our time and efforts to provide our customers and clients with outstanding safari tours and luxury African beach holidays. If you would like to book a trip of a lifetime for our luxury Zimbabwe safari holidays, email info@ubonsafari.com and we will respond to your enquiry as soon as possible.
ZIMBABWE PARKS
Described by the Kololo tribe living in the area in the 1800's as Mosi-oa-Tunya, meaning 'The smoke that thunders', the Victoria Falls is a spectacular awe-inspiring beauty on the Zambezi River forming the border between Zambia and Zimbabwe. River Zambezi is the fourth largest river in Africa, at 2,574 kilometres long, it rises in Zambia and flows through Angola along the border of Namibia, Botswana then along the border between Zambia and Zimbabwe up to Mozambique where it makes its way to empty into the Indian Ocean.
The noise of this powerful Victoria Falls can be heard from as far as 40 kilometres and the spray and mist of the falling water can be seen from as far as 50 kilometres. This is probably why it was called 'The smoke that Thunders'.
David Livingstone, a Scottish missionary and an explorer is believed to be the first European to have discovered the falls and named it in the honour of the Queen of Britain; Victoria in 1855.
The water levels vary throughout the year with the peak in April and the lows in October and November. Walks through parts of the waterfall can be done during the dry season when the water levels are low. Other adventure activities such as Bungee jumping, river surfing, kayaking, Rafting, River cruises, fishing, elephant rides can also be done here.
February – June – Peak flood Season – Best time for viewing the falls
October – November – No water falls at all in some places
Formerly known as Wankie Game Reserve, Hwange is the largest game reserve in Zimbabwe occupying roughly 14,650 square kilometres. It is located in the west of Zimbabwe between Bulawayo and Victoria Falls, just about one hour from the Falls.
Due to its size, the park has different vegetation, scenery and game, from semi-desert scrubs on its south; where the park is close to the edge of the Kalahari Desert, a region with very little water and sparse vegetation and on the north it is mostly mopane woodlands valleys, forests and granite hills.
Hwange was the royal hunting grounds of the Ndebele warrior-king Mzilikazi in the early 19th Century and was later set aside as a National Park in 1929.
Home to over 100 mammal species including Wild dogs, lions, leopards, rhino and hyena; Hwange is popular for its largest elephant population in the world.
Nearly 400 bird species recorded are recorded in the park, which include 19 large herbivores and 8 large carnivores. Some of the commonly spotted birds are Southern ground hornbill, Yellow billed kite, Dickenson's kestrel, Kori bustard and Carmine bee-eater.
August – October and early November – Dry Season – Best time for game viewing
November – April – Rainy (Green) season – Limited game viewing
LAKE KARIBA AND THE MATUSADONA NATIONAL PARK
Lake Kariba, is the largest man-made lake in the world at over 223 kilometres long and upto 40 kilometres wide covering an area of 5,580 square kilometres.
Matusadona National Park is a game reserve park located on the shores of Lake Kariba.
Previously a game reserve, Matusadona became a national park in 1975. The park is a unique combination of rugged wilderness and water frontage of Lake Kariba.
The Lake Kariba offers an excellent opportunity for photography of sunset views with Matusadona mountains in the background.
The Matusadona national park lies about 20 kilometres across the lake from the town of Kariba. The wildlife around Lake Kariba and Matusadona National Park includes Elephants, buffalo, black rhino, lion, wild dog, cheetah, leopard, impala, waterbuck, zebra, giraffe and hyena.
Exceptional birdlife is found in the park of over 240 species include woodland birds, raptors, waterfowl and a healthy fish-eagle population.
July – October – Dry Season – Best time for game viewing
November – April – Rainy (Green) season – Good for bird viewing
MANA POOLS NATIONAL PARK
Mana Pools is a wildlife conservation area and a national park in the north of Zimbabwe in the lower Zambezi valley. Mana is 'four' in the local language, Shona. Over many years the wide and slow Zambezi River has carved through the mineral-rich volcanic soils forming the 'four' large pools, which are the remnants ox-bow lakes.
The National park feels very laid back and tropical on the calm River Zambezi. On the opposite side of the River is Zambia's Lower Zambezi National Park.
Covering an area of 2,196 square kilometres Mana Pool is part of the 10,500 square kilometre Parks and Wildlife Estate that runs from Kariba Dam in the west to Mozambique border in the east.
The park is known as one of the best National parks for Walking safaris. Many of the game
species are fairly relaxed when encountered on foot. Mana Pools National park is UNESCO
World Heritage Site since 1984. The game in the park includes Elephant, eland, buffalo, impala, warthog, baboon, monkey, waterbuck and hippos. Also found are lion, leopard, cheetah and hyena.
More than 380 species of birds are found in Mana Pools including Egyptian and spurwing geese, cormorants, storks, goliath herons, kingfishers and bee-eaters.
Also found are the vultures, livingstone's flycatcher, plovers, fish eagle and white collared pratincoles.
The fertile and rich black cotton soil is a drawback for tourism during the rainy season in
Zimbabwe. All lodges close during the rainy season as accessibility becomes difficult.
GONAREZHOU NATIONAL PARK
Covering an area of 5,053 square kilometres, Gonarezhou is Zimbabwe's second largest national park located in South East corner of the country adjoining with the borders of Mozambique and South Africa. Gonarezhou means 'elephant's tusk' in Shona, the local language of Zimbabwe.
The Gonarezhou National Park links with Kruger National Park in South Africa and Limpopo National Park forming the Great Limpopo Transfrontier Park. Collectively, the area covered by Great Limpopo is 35,000 square kilometres.
Due to its remoteness, the park cannot be reached as easily as the other national parks
hence it attracts very few travellers. Good game viewing opportunities for the peace loving
travellers.
Birdlife has specials including bat hawk, thick-billed cuckoo, racket tailed roller and African finfoot.
MATOBO HILLS NATIONAL PARK
Covering an area of 424 square kilometres, Matobo Hills National Park is part of the Matobo Hills. Matobo or Matopos mean 'bald heads' in local language matebele. The hills were considered sacred by the Ndebele people; the name Matobo 'bald heads' was given by the great Ndebele King, Mzilikazi. He was buried in the hills just a short distance from the Matobo Hills National Park.
The Matobo area is rich in history and has great spiritual and cultural significance to the local people. Over 2000 years ago, the Bushmen lived in this area and have left behind an outstanding rock art collection.
Located just 34 kilometres from south of Bulawayo, makes it an easy half an hour drive and the most accessible national park.
Home to a good variety of animal species, the park is popular for it's black and white rhinos. Also found in the park are duiker, klipspringer, sable, kudu, eland, waterbuck, wildebeest, giraffe, baboon, monkey, crocodile, cheetah and leopard.
The park doesn't have any lions or elephants; this makes it easy for the walking safaris.
Famous for the large concentration of fish eagles, the park has a diverse birdlife including secretary bird, francolin, weavers and Egyptian geese.
Zimbabwe Map
We create tailor-made safari itineraries
Simply send us an email on info@ubonsafari.com and we will get back to you.
CLASSIC VICTORIA FALLS
4 NIGHTS / 5 DAYS DRIVING SAFARI VICTORIA FALLS | ZAMBEZI NATIONAL PARK ARRIVE / DEPART: VICTORIA FALLS INTERNATIONAL AIRPORT
HWANGE NATIONAL PARK SAFARI
5 NIGHTS / 6 DAYS DRIVING SAFARI VICTORIA FALLS | HWANGE NATIONAL PARK ARRIVE / DEPART: VICTORIA FALLS INTERNATIONAL AIRPORT
Imvelo Green Season Victoria Falls and Hwange Classic Safari
9 NIGHTS / 10 DAYS VIC FALLS | HWANGE | VIC FALLS ARRIVE: VICTORIA FALLS DEPART: VICTORIA FALLS INCLUDES Little
Imvelo Authentic Zimbabwe and Victoria Falls Safari
11 NIGHTS / 12 DAYS VIC FALLS | HWANGE | VIC FALLS ARRIVE: VICTORIA FALLS DEPART: VICTORIA FALLS INCLUDES Little
Imvelo Ultimate Zimbabwe Safari Experience
&BEYOND MATETSI PRIVATE GAME RESERVE
MATUSADONA NATIONAL PARK
MALILANGWE WILDLIFE RESERVE
Horse Riding Safaris


August 13 - Air Products has taken delivery of two Shuttlelift DB 70 double-beam mobile gantry cranes for its facilities in Wilkes-Barre, Pennsylvania and Palmetto, Florida.
The Shuttlelift mobile gantry cranes will be responsible for putting products on railcars or trucks, and moving parts on site, both indoors and outdoors.
Air Products' facility in Pennsylvania had short, narrow doorways, which were not suitable for accommodating a 70-tonne capacity mobile gantry crane, so the company approached Shuttlelift to see if it could design a DB 70 that was able to move through the existing doorway.
While a typical Shuttlelift DB 70 has a ground-to-hook height of 30 ft (9.14 m) and an inside clear width of approximately 40 ft (12.19 m), the new crane would have to fit within a 20 ft (6.09 m) x 20 ft (6.09 m) space.
"We knew this was going to be a unique challenge for our engineers, because the crane would have to fit the doorways and be able to lift the product," said Shuttlelift's sales manager, Dan Reinholtz.
Another challenge, explained Shuttlelift, was that Air Products' facilities have equipment tracks on the floor, which the crane cannot touch.
Since Air Products did not want to move all of the tracks, Shuttlelift found another way to solve the problem. "We took our normal wheel frames and put a special cut in them to give us 20 inches (50 cm) of clearance off the ground," said Reinholtz.


Vertex Modelling teamed up with Illusive to create a 3D model for a 4D simulation used at the new attraction in County Hall, London. The animation is featured in a children's attraction consisting of a walk through as well as a ride in which the visitors can fly around London on a double decker bus.
This textured 3D model is based on our LOD4 model. As all our LOD4 models are upgraded LOD3 Models, the model fits seamlessly with our Wide Area 3D London model.
The model has been created using ultra high-res photogrammetry and textured to very high standard. As the animation was focusing on embankment, rear facing textures are 8k, south facing and courtyard textures are of lower quality as they would not feature as much in the animation.
Scenes created for the animation include flying across the London Eye and skimming the River Thames. During this time you get a full view of the textured front façade of County Hall. The model was captured using high-red aerial photogrammetry, using high-res aerial imagery, in order to create a model of London to millimetre accuracy including building façade and terrain detail.
We were very proud to have our model feature in such a popular London attraction and to show what talents our modellers and photogrammetrists have here at Vertex.
We also offer other iconic London landmarks which have been modelled and textured very accurately. All of these models fit into our wider London model and can be ordered.


In the 18th century abortion was a taboo and it was considered pre meditated murder. Women never used contraceptives nor any birth control methods and they conceived all the children inside their wombs and delivered them full term natural birth.
Even though terminating pregnancy seems as a normal practice today, this doesn't rule out the fact that it is an act of committing murder to the unborn infant. It is within the mother's rights of what to do with her body, but the innocent infant in the womb is being robbed of their rights as human beings that are still in the developing process.
The advantage of abortion is that bringing an unwanted new born into an unprivileged environment is indeed unfair. Should the child fail to get what is necessary to raise them, they agree easily bound to turn into criminals. When the population is very high for the government to sustain, it is also more burdens when new born are delivered here and there with no family planning.
On the other hand, abortion can be very dangerous to the mother herself health wise because should she be less careful the process could easily cost her life. Abortion could easily leave side effects on the mother and this includes damage to her womb. When an embryo is being aborted, it is ripped off the womb walls aggressively which leads to the womb failing to hold an embryo tight as normal in future. This damage causes the mother to start having miscarriage whenever she carries later embryos.
Abortion might have been acceptable today but there are several damages caused not to mention defect babies and premature. Contraceptives also have side effects. What contraceptives do is cause an uncomfortable environment around the spot where the egg sticks after being fertilized in order for it to proceed with growth. This contraceptive process could also lead to damaging the womb in a similar manner as the abortion.
To avoid all these complications due to all these abortion and contraceptive methods, i would advice mothers to opt for condoms. Good quality condoms and not these that are being given away freely in public toilets, would be a very clean and healthy method from unwanted pregnancy as well as infections. Throwing sperm away isn't murder but after that sperm cell is fertilized with the female's egg, it is a being destroyed.


Hi there! I graduated from National Laser Institute in February of 2007. I own my own equipment and work half time as a cosmetologist and half time as a laser technican in White Bear Lake MN. I LOVE my job! I love people and have done hair for 20 years trying to make people feel good about themselves from the inside out. This is the same only tenfold! I LOVE IT!!! Thanks National Laser Institute for rocking my world!


September 5, 2009. Tags: A Nightmare on Elm St, Freddie Krueger, Heather Langenkamp, Johnny Depp, NoES, Wes Craven. Entertainment, Fantasy, Film, gaming, Horror, Popular Culture, Sci Fi, Supernatural, Suspense. 1 comment.
WTH is happening in Hollywood? Lack of creativity? How many YARMS (yet another remake) do we have to endure?
And how do you remake WES CRAVEN well? I don't want Lucas-esque remakes of films already done perfectly!! did they learn nada from the sequels? havent they seen Scream? lol..
Word on the street — Elm street, to be exact — is that Terminator: The Sarah Connor Chronicles star Thomas Dekker is in "final negotiations" to join the cast of Platinum Dunes' A Nightmare on Elm Street remake.
The report comes from horror site Shock Till You Drop, which adds that Dekker will play a character named Jesse, "a jock on the swim team," if cast. Watchmen's Jackie Earle Haley and The Haunting in Connecticut's Kyle Gallner have already signed on for the film, which will offer up a fresh take on the tale of the peerless Freddy Krueger..
April 22, 2009. Tags: A Nightmare on Elm St, Freddy Kreuger, Heather Langencamp, Heather Langenkamp, Johnny Depp, New Line Cinema, Robert Englund, Wes Craven. Entertainment, Film, Horror, Popular Culture, Uncategorized. 1 comment.


At this November's midpoint, I saw my 42nd birthday. An actuary, I suppose, would say I am at least half dead. But that dark and clear night, driving in my car to give an after dinner speech, I was watching the news. Not on a television: out my window I noticed gaudy Jupiter rising in the east, with less gaudy though no less beautiful Saturn in its wake, both chasing a fine half moon near its zenith. I watched the news at 11 later that week, when the Leonid meteor shower gave me something to see.
I am embarrassed to be a man who appreciates so little of this wonderful entertainment, and I am resolved to cultivate myself. To lack an ear for opera, a tongue for wine, are things I can endure. But how have I come to know so little of the sky and all its heights? - The early dark of late fall is a gift of a providential order; it makes it easy to remember that man is not the most important thing in the universe.
I recall a sign on a colleague's door: "Literature is news that stays news". Then a star must be news that has always been news, and always will be. The next time you are tempted by a small lie, teach a child how to find a constellation. And if you are tempted by a big lie, buy a telescope, and remember where you are.
Have a happy new year, and when the year is old, then have a happy day, and do not fret if none of the best things in it are new.