text:
\section{Introduction}
It is a long-standing question of evolutionary biology to decide about the relative importance of evolutionary factors such as selection versus genetic drift to shape
 patterns of diversity.
Today this topic is studied based on DNA variation data taken from a sample of a population.
Important work on the effects of positive selection on patterns in DNA data was made by \citet{MaynardSmithHaigh1974}. They showed, that neutral variation linked to a beneficial
 allele also increases in frequency. This is called the hitchhiking effect and the resulting reduction of neutral variation is termed a selective sweep.
When a beneficial allele fixes in a population, this allele can have a
 single or several origins, i.e.\ it can be brought to the population
 by a single or by several mutants. If several individuals, called founders, account
 for the fixation of a beneficial allele, we will talk as Pennings and
 Hermisson about a soft sweep and else about a hard sweep.

There are various reasons for a soft sweep. Adaptation can occur from recurrent migration, mutation or act on standing genetic variation.
 We treat here the case of recurrent
mutation, which also applies to migration in a special case. Realistic models for recurrent migration in general can lead to more complex scenarios due to population structure.

It has been shown by Hermisson and Pennings in \citep{HermissonPennings2005,PenningsHermisson2006a}, that soft sweep events become frequent, if the scaled
 mutation rate $\theta_s= 4Nu_s$ (where $N$ is the diploid population size and $u_s$ the mutation probability to the beneficial allele per individual per generation) is sufficiently high.
While hard sweeps dominate for $\theta_s<0.01$, both hard sweeps and soft sweeps occur in the range $0.01<\theta_s < 1$. For $\theta_s>1$ almost
 all adaptive substitutions will result in soft sweeps. Soft sweeps become likely for populations with large population sizes $N$ or for alleles with high recurrent mutation rates $u_s$.
 For example, most pathogens have extremely high population sizes. Therefore their genomes are good candidates for the detection of
soft selective sweeps, see e.g. \citep{NairEtAl2007} for research on soft sweeps in malaria parasites.
 \citet{Karasov2010} concluded lately that in Drosophila melanogaster there should exist a huge amount of soft sweeps due to tremendous short-term effective population
sizes relevant for adaptation. \citet{Schlenke2005} located some of these regions. Recent research by \citet{Scheinfeldt2009} shows, that the DNA pattern around the human
 gene ALMS1, causing the Alstroem Syndrome which presents with early childhood obesity and insulin resistance leading to Type 2 diabetes,
 may also be the result of a soft sweep.
Further \citet{TishkoffEtAl2007} found out, that different SNPs in the human genome all lying in the same short genome region of 110 bp are responsible
 for the human lactase persistence in the African and European human populations. In their studies
 of LD (measured by the D' value and the LOD score) the pattern
of a soft sweep can be recognized. Ongoing research argues for the importance of soft sweeps and polygenic adaptation, see for a review about this discussion e.g. \citep{Pritchard2010}.

In order to detect soft sweeps it is important to understand the footprints they leave in DNA data.
For this purpose it is necessary to make
 statistical predictions available, which allow us to find targets
 of recent positive selection.
\citet{PenningsHermisson2006a} showed that tests based on haplotype structure have high power to detect
soft sweeps. If a soft sweep occurred, the population can be divided into several haplotype groups, one for each founder. Without mutation and recombination during the sweep the genomes of the groups differ at
the same loci as the founders differed in the beginning of the sweep. Especially, in the case of two founders each allelic
variant of a SNP locus is always linked to a single haplotype group. So high linkage disequilibrium of
two neutral loci in a neighborhood of the selected locus should be found.
This gives rise to the conjecture, that linkage disequilibrium is a useful quantity to detect soft sweeps.

\smallskip
LD has been computed under neutrality by
\citet{Ohta1969}.
\citet{Stephan2006}, \citet{McVean2007} and \citet{PfaffelhuberLehnertStephan2008} gave analytical expressions for measures
of LD after a hard selective sweep.
\citet{Kim2004} developed a composite-likelihood method for detecting hard sweeps incorporating information
from measures of linkage disequilibrium based on simulation studies.
The aim of this article is to give analytical expressions for linkage disequilibrium under a
selective sweep with recurrent mutation to the beneficial allele (see Theorem \ref{smallthm} and Theorem \ref{bigthm}).
To determine the linkage disequilibrium we use an extended star-like
 approximation for the genealogy of the selected site, see Section \ref{GeneaDefi}. A similar approach was applied by \citet{PfaffelhuberLehnertStephan2008} to obtain
the linkage disequilibrium in the case of a hard sweep. We will see, that soft sweeps
produce a different signal than hard sweeps. In Section \ref{sim} we compare our computations with simulations.

\begin{figure}
  \text{
    \parbox{1.5in}{\beginpicture
      \setcoordinatesystem units <1.3cm,1.1cm>
      \setplotarea x from 0 to 1.7, y from 0 to 2
      \plot 0 1.5 2 1.5 /
      \plot 0 1.6 0 1.4 /
      \plot 0.6 1.6 0.6 1.4 /
      \plot 2 1.6 2 1.4 /
      \put{$S$} [cC] at 0 1.8
      \put{$L$} [cC] at 0.6 1.8
      \put{$R$} [cC] at 2 1.8
      \plot 0 1.3 0 1.1 /
      \plot 0 0.6 0 1.3 /
      \plot 2 0.6 2 1.3 /
      \multiput {\tiny $\bullet$} at 0 1.5 *200 .01 .0 /
      \arrow <0.2cm> [0.375,1] from 0 1.2 to .6 1.2
      \arrow <0.2cm> [0.375,1] from 0.6 1.2 to 0 1.2
      \arrow <0.2cm> [0.375,1] from 2 1.2 to .6 1.2
      \arrow <0.2cm> [0.375,1] from 0.6 1.2 to 2 1.2
      \arrow <0.2cm> [0.375,1] from 2 0.7 to 0 0.7
      \arrow <0.2cm> [0.375,1] from 0 0.7 to 2 0.7
      \put{$\rho_{SL}$} [cC] at .3 1
      \put{$\rho_{LR}$} [cC] at 1.3 1
      \put{$\rho_{SR}$} [cC] at 1 .5
      \put{$(a)$} [cC] at 1 2.5
      \endpicture}}
  \qquad
  \text{
    \parbox{1.5in}{\beginpicture
      \setcoordinatesystem units <1.3cm,1.1cm>
      \setplotarea x from 0 to 2, y from 0 to 2
      \plot 0 1.5 2 1.5 /
      \plot 0 1.6 0 1.4 /
      \plot 1.2 1.6 1.2 1.4 /
      \plot 2 1.6 2 1.4 /
      \put{$L$} [cC] at 0 1.8
      \put{$S$} [cC] at 1.2 1.8
      \put{$R$} [cC] at 2 1.8
      \plot 0 1.3 0 0.6 /
      \plot 1.2 1.3 1.2 1.1 /
      \plot 2 1.3 2 0.6 /
      \multiput {\tiny $\bullet$} at 0 1.5 *200 .01 .0 /
      \arrow <0.2cm> [0.375,1] from 0 1.2 to 1.2 1.2
      \arrow <0.2cm> [0.375,1] from 1.2 1.2 to 0 1.2
      \arrow <0.2cm> [0.375,1] from 2 1.2 to 1.2 1.2
      \arrow <0.2cm> [0.375,1] from 1.2 1.2 to 2 1.2
      \arrow <0.2cm> [0.375,1] from 2 0.7 to 0 0.7
      \arrow <0.2cm> [0.375,1] from 0 0.7 to 2 0.7
      \put{$\rho_{LS}$} [cC] at .6 1
      \put{$\rho_{SR}$} [cC] at 1.5 1
      \put{$\rho_{LR}$} [cC] at 1 .5
      \put{$(b)$} [cC] at 1 2.5
      \endpicture}}
  \caption{\label{fig:geom}The two possible geometries of the selected locus
    ($S$) and the two neutral loci ($L$ and $R$). The scaled
    recombination rates between the two loci are given by $\rho_{SL}$,
    $\rho_{LR}$, $\rho_{LS}$ and $\rho_{SR}$.}
\end{figure}

\section{Model and measures of linkage disequilibrium}
\subsection{The frequency of the selected locus during a sweep}\label{freq}
We consider a DNA region of a population of $N$ diploid individuals and concentrate on the neighborhood of a bi-allelic selected locus S
 with a wild-type allele $b$ and beneficial allele $B$. The new beneficial allele $B$ with fitness advantage $s$ enters the population recurrently by mutation
 and is assumed to fix eventually.
 The population reproduces at the beneficial locus according to the Moran model in continuous time with selection and recurrent
mutation to the beneficial allele, i.e. denoting by $(X^{N}(t))_{t\geq0}$ the frequency of the individuals
 carrying the beneficial allele in a population of size $N$,
$(X^{N}(t))_{t\geq0}$ is a jump Markov process with transition rates from

\begin{align}
& i/2N \textrm{ to } (i + 1)/2N \textrm{ at rate } u_s (2N-i) + (2N-i) \frac{i}{2N}(1+s)  \\
& i/2N \textrm{ to } (i-1)/2N \textrm{ at rate }  i \frac{(2N-i)}{2N},
\end{align}
with $u_s, s \geq  0$.
The rate $u_s (2N-i)$ is the mutation rate, the rates $(2N-i) \frac{i}{2N}(1+s)$,  $i \frac{(2N-i)}{2N}$, respectively, are resampling rates which change
the frequency of the beneficial allele by plus $\frac{1}{2N}$, minus $\frac{1}{2N}$, respectively. Of course, resampling events inside
the beneficial and wild-type locus, which do not change the frequency of the beneficial allele, are also possible.

The frequency of the beneficial allele can be approximated for large $N$ by a differential equation:
\begin{satz}\label{kurzthm}
Denote by $X^N (t)$ the frequency of the beneficial allele in the Moran model with constant diploid population size $N$ at time $t$.
Let $X^{N}(0)= \frac{\lceil \epsilon2N \rceil}{2N}$. Then the frequency of the beneficial allele
$X^N (t)$ converges for $N\rightarrow \infty$ to the solution of the differential equation:
\begin{equation}\label{diffeq}
\dot{X}(t) = u_s (1-X(t))+ s X(t)(1-X(t))
\end{equation}
with initial condition
$$X(0)= \epsilon,$$
in the sense, that for all $\delta >0$ and all $t$
$$\lim \limits_{N \rightarrow \infty} P(\sup \limits_{s\leq t} |X^N(s) -X(s)| >\delta) =0.$$

\end{satz}
The proof is an easy application of Theorem 3.1 in \citep{Kurtz1971}.

Equation (\ref{diffeq}) has the solution
\begin{equation}
 X(t)= \frac{(\epsilon s + u_s)s e^{s t} - (s - \epsilon s)
    u_s e^{
     -u_s t}}{ s ((\epsilon s + u_s )
      e^{s t} + (s - \epsilon s) e^{-u_s t})}.
\end{equation}
In this approximation we say that the allele fixes in the population,
 if $X_{T}=1-\epsilon$. For the above equation,
 this happens at time $T= \frac{1}{s+u_s} \log\Big(\frac{(1-\epsilon)(u_s+s(1-\epsilon))}{\epsilon(\epsilon s+u_s)}\Big)$.
With $\epsilon= 1/\alpha$, we obtain $T=\frac{1}{s+u_s} \log\Big(\frac{(\alpha-1)(\alpha-1+(\theta_s/2))}{1+(\theta_s/2)}\Big)$, denoting
by $\alpha:=2Ns$ and $\theta_s:=4Nu_s$. For small $u_s \ll s$ and large $N$, such that $\alpha\gg 1$, the fixation time $T$ is approximately
$2(\log{\alpha})/s$.
The fixation time will be relevant below.

\subsection{Measures of linkage disequilibrium}
Our aim is to provide analytical results for the linkage disequilibrium of two neutral loci in a neighborhood of the selected locus.
 Different quantities have been proposed to measure the association of two loci. We will compute two of them here.
 Consider two neutral loci $L$ and $R$, linked to the selective locus $S$.
 The neutral loci can either lie both on the same side of the selected locus or the selected locus
 lies between the neutral loci, see Figure \ref{fig:geom}. (If both neutral loci lie on the left side of the selected locus, we name the leftmost locus $R$-locus
and the locus in the middle $L$-locus, i.e. we have the ordering $R$ $L$ $S$.)
We consider only loci with exactly two allelic
variants. Denote them by $L/\ell$ and $R/r$ and their allelic frequencies by $q_{\ell}, q_L$, etc.

\begin{defi}[$D_{\ell,r}$ and $\widehat{D_{\ell,r}}$ ]

The simplest approach to measure linkage disequilibrium between the allele $\ell$ of the L-locus and the allele $r$ of the R-locus is to compute the quantity
\begin{equation}
 D_{\ell,r} :=  q_{\ell r} - q_{\ell} q_r.
\end{equation}
If $D_{\ell, r}$ is zero the alleles $\ell$ and $r$ are said to be in linkage equilibrium, else in linkage disequilibrium.

In practice, the population frequencies $q_{\ell},  q_r$, etc.~are often not available, but only the allelic frequencies
 in a sample, $\widehat{q}_{\ell}$,$\widehat{q}_r$, etc.
In samples LD can be measured by
$$\widehat{D_{\ell,r}}:= \widehat{q}_{\ell r} - \widehat{q}_{\ell} \widehat{q}_r.$$
\end{defi}
\begin{bem} \label{D}
\emph{An easy calculation shows, that
\begin{equation*}
D_{\ell, r}= D_{L,R}= - D_{\ell,R}=- D_{L,r}
\end{equation*}
and analogous equalities hold for $\widehat{D}$.}
\end{bem}

Averaging $D_{\ell,r}$ over all allelic variants gives zero due to Remark \ref{D}.
Hence it makes sense to consider
 $D_{\ell,r}^2$.
Since $D_{\ell,r}^2= D_{\ell,R}^2= D_{L,R}^2= D_{L,r}^2$ the quantity $D_{\ell,r}^2$ actually does not depend on the allelic
 variant $\ell$,$r$. Therefore we write $D^2$ instead.
However $D^2$ depends strongly on the size of the allelic variants: If $q_{\ell}$ and $q_r$ are small, $D^2$ is also small, whenever the
allelic variants may be not in association at all.
Therefore the so called standard linkage disequilibrium, introduced by \citet{Ohta1969}, is often considered.
\begin{defi}[$\sigma^2_D$ and $\widehat{\sigma^2_D}$]
The standard linkage disequilibrium $\sigma^2_D$, $\widehat{\sigma^2_D}$ in the sample, respectively, is given by
$$ \sigma^2_D = \frac{\mathbbm{E}[D_{l,r}^2]}{\mathbbm{E}[q_{\ell} (1-q_{\ell}) q_r(1-q_r)]},$$ and
$$\widehat{\sigma^2_D} = \frac{\mathbbm{E}[\widehat{D_{l,r}}^2]}{\mathbbm{E}[ \widehat{q}_{\ell} (1-\widehat{q}_{\ell}) \widehat{q}_r (1-\widehat{q}_r)]},$$
respectively.
\end{defi}

\begin{rems}
\emph{
\begin{itemize}
\item Note, that $\frac{\mathbbm{E}[\widehat{D_{l,r}}^2]}{\mathbbm{E}[ \widehat{q}_{\ell} (1-\widehat{q}_{\ell}) \widehat{q}_r(1-\widehat{q}_r)]}$ does not
depend on the particular allelic variant $\ell, r$, too. Therefore it makes sense to write $\widehat{\sigma^2_D}$, instead of $\widehat{\sigma^2_{D_{\ell,r}}}$
\item We compute linkage disequilibrium during the sweep. So, if time is important, we will write $D_{\ell, r}(t):=  q_{\ell r}(t) - q_{\ell}(t) q_r(t)$, etc.
\item \citet{PfaffelhuberLehnertStephan2008} have computed $\mathbbm{E}[D_{\ell, r}(0)| D_{\ell,r}(T)]$ and $\widehat{\sigma^2_D}$ for a hard sweep, i.e. the case $u_s=0$.
See Figure \ref{fig:Theo} for a plot of $\widehat{\sigma^2_D}$ under neutrality and for $\theta_s=0$ and $\theta_s= 0.1$.
\item Naturally one would consider the quantity $$r^2:=  \mathbbm{E}\Big[\frac{\widehat{D}^2}{\widehat{q}_L(1-\widehat{q}_L) \widehat{q}_R(1-\widehat{q}_R)}\Big].$$ But this
quantity is less attractive for analytical studies, because it is mathematically difficult to handle. However, see the recent paper of \citet{SongSong2007} for an analytical computation of $r^2$ under
neutrality.
\end{itemize}}
\end{rems}

\subsection{Genealogies: Motivation}
We want to compute $\mathbbm{E}[D_{\ell,r}(0)|D_{\ell,r}(T)]$ and $\widehat{\sigma^2_D}$ at
 the end of the sweep assuming small sample sizes $n\ll N$ for the computation of $\widehat{\sigma^2_D}$.
We will use the 1-1-correspondence between the probability to draw two pairs of heterozygous neutral loci and $\widehat{\sigma^2_D}$, (see
step 3 of the proof of Theorem \ref{bigthm}). The probability to draw a heterozygous pair at the end of the sweep differs from the probability to draw a heterozygous pair
at the beginning of the sweep due to the change of the genealogy during the sweep. We shall start with heterozygous pairs of a sample
taken from the population at the end of the sweep and follow the lines of the pairs till the beginning of
the sweep. In our notation time is running backwards starting from time $T$ of fixation, i.e. if $t_2> t_1$ the time $t_2$ lies further back in the past then the time $t_1$.  For example
$\mathbbm{E}[D_{\ell,r}(0)|D_{\ell,r}(T)]$ is the expected value of $D_{\ell,r}$ at the end of the sweep given $D_{\ell,r}$ at the beginning of the sweep.

To define the genealogies of two neutral loci in the neighborhood of a selected locus in a Moran model we would have to extend the Moran model as introduced in
Section \ref{freq} to a full three-locus model. However, multi-locus genealogies under such a Moran model are very complex. Under
certain conditions star-like genealogies approximate the genealogies
 of the Moran model quite well and allow a computation of the above probabilities due to independent genealogical lines.
In the following we introduce such star-like genealogies and justify why it is reasonable to use them in our setting.

We suppose, that neutral mutations occur according to a Poisson Process with rates of order $\mathcal{O}(1/N)$.
Since the sweep takes only of order $\log(2Ns)/s$ time units, we can ignore neutral mutations during the sweep.
Moreover, back-mutations are rapidly sorted out as they have no fitness advantage. Hence we will ignore back-mutations, too.

\begin{figure}

   \subfigure
             {
              \includegraphics[scale=0.55]{CoalMut1.pdf}
             }
   \subfigure
             {
            \includegraphics[scale=0.55]{CoalMut2.pdf}
             }
   \caption{The figure on the left shows two lines which coalesce before they mutate. The figure on the right shows two lines which mutate before they coalesce (further back in the past).}
    \label{fig:CoalMut1}
\end{figure}

\subsubsection*{Coalescent and mutations to the beneficial allele}
\begin{itemize}
 \item The rate of coalescence of two lines at time $t$ under the condition, that the two lines are in the beneficial background at time $t_{-}$ and the
 frequency of beneficial allele is $X_{t_{-}}= \frac{i}{2N}$ is equal to $$\frac{\frac{1}{(2N)(2N-1)} \frac{i(i-1)}{2N}}{ \frac{i(i-1)}{(2N)(2N-1)}} +
 \frac{\frac{1}{(2N)(2N-1)} \frac{(i-1)(2N-i+1)(1+s)}{2N}}{\frac{i(i-1)}{(2N)(2N-1)}}
= \frac{(1+s)(2N+1)}{2Ni} -\frac{ s}{2N} $$ for $i>1$. The parents of the beneficial offspring
are either both from the beneficial background or one is from the wild-type and the second from the beneficial background; for similar calculations see \citep{BartonEtheridgeSturm2004}, Lemma 2.4.
 For large $N$ this rate is approximately
\begin{equation}\label{coalrate}
\frac{1+s}{2NX(t)},
\end{equation} since for large $N$ the
 frequency $X^{N}(t)$ is well
approximated by the solution $X(t)$ of the differential Equation (\ref{diffeq}) by Proposition \ref{kurzthm}.

\begin{figure}
 \begin{center}
\includegraphics[scale=0.6]{starApproximation3.pdf}
\end{center}
  \caption{\label{fig:starApproximation3}
Possible lines in the time interval $ [T/2,T)$ (backward in time)
 for geometry (a).
 The corresponding probabilities are given in Table
     \ref{tab:geoalpha}.
}
\end{figure}

\item If an individual mutates to the beneficial type, the genealogical line of this individual jumps (forward in time)
 from the wild-type background to the beneficial background.
Backward in time the line is located at time $t_{-}$ in the beneficial background and at time $t$, after the mutation event,
 in the wild-type background.

The rate of mutation to the beneficial background of a line at time $t$ under the
 condition, that the frequency
of the beneficial allele is $X_{t_{-}}= \frac{i}{2N}$ at time $t_{-}$, is equal to
$$\frac{u_s \frac{2N-i}{2N}}{\frac{i}{2N}},$$

Analogous argumentations as for the coalescence rate yield, that this rate is approximately
 \begin{equation}\label{mutrate}
\frac{u_s(1-X(t))}{ X(t)}\end{equation}
 for large N.
\end{itemize}

Both rates scale with $\frac{1}{X(t)}$, which means that the coalescent and mutation rates
 are high, if the frequency $X(t)$ is small.
 Hence it makes sense to assume that all mutations to the beneficial allele and all coalescent events occur at time $t=T$, i.e. at the beginning of the sweep.
This scaling of the backward mutation rate shows that the star-like approximation, which has before been used for the classical
 hard sweeps case, should also be appropriate for soft sweeps.

With the approximate mutation and coalescent rates (\ref{mutrate}) and (\ref{coalrate}) the probability for a hard sweep in a sample of two can be bounded, see
also \citep{PenningsHermisson2006a} for a similar calculation in a Wright-Fisher-model formulation.
The probability for a hard sweep of two lines equals the probability, that the coalescent event happens before the mutation event.
 The mutation rate in a sample of two lines is approximately $2\frac{u_s(1-X(t))}{ X(t)} = \frac{\theta_s (1-X(t)}{2N X(t)}$,
 if terms of order $(2 N u_s)^2$ are ignored.

\begin{figure}
 \begin{center}
\includegraphics[scale=0.6]{starApproximation4.pdf}
\end{center}
  \caption{\label{fig:starApproximation4}
Same as Figure
     \ref{fig:starApproximation3}
 for geometry (b). The lines (i), (ii) and (v) are not shown, as they are the same as in Figure  \ref{fig:starApproximation3}.
  The corresponding probabilities are given in Table \ref{tab:geobeta}.
}
\end{figure}

Let $C=(C_t)_{0\leq t \leq T}$, $M=(M_t)_{0 \leq t \leq T}$ be two independent Poisson processes with rates $\lambda(t)= \nolinebreak \frac{1+s}{2N X(T-t)},
 \mu(t) = \frac{ \theta_s(1-X(T-t))}{2N X(T-t)}$  respectively. Denote by
$S_1$ the first jump time of $C$ and by $T_1$ the first jump time of $M$.
Then the probability for a hard sweep in a sample of two is
$$P_{hard,2}:= P(S_1< T_1).$$
If the initial frequency of the beneficial allele $X(0)= \epsilon$ is small,
\begin{align*}
P(S_1< T_1) & = \int \limits_0^{\infty} P(t<T_1) f_{S_1}(t) dt =
\int \limits_{0}^{T} \exp\Big(-\int \limits_{0}^{t} \mu(\tau)d\tau\Big) \lambda(t) \exp\Big(-\int \limits_0^t \lambda(\tau) d\tau \Big) dt = \\
&=\int \limits_{0}^{T} \frac{1+s}{2N X(T-t)} \exp\Big(-\int \limits_{0}^{t} \frac{1+s+\theta_s(1-X(T-\tau))}{2N X(T-\tau)} d\tau \Big)dt= \\
& = \frac{1+s}{1+s+\theta_s} \Bigg( \int \limits_{0}^{T}\frac{1+s + \theta_s(1-X(T-t))}{2N X(T-t)}
 \exp\big(- \int \limits_{0}^{t} \frac{ 1+ s+\theta_s(1-X(T-\tau))}{2N X(T-\tau)} d\tau \big)dt\\
&\quad + \frac{\theta_s X(T-s)}{2N X(T-s)}\int \limits_{0}^{T}
 \exp\big(-\int \limits_{0}^{t} \frac{ 1+s+\theta_s(1-X(T-\tau))}{2N X(T-\tau)} d\tau \big)dt \Bigg) \approx \\
& \approx \frac{1+s}{1+s+\theta_s}(1+ u_s \mathcal{T}),
\end{align*}
where $f_{S_1}(t)$ denotes the density of the probability measure induced by $S_1$ and
 $\mathcal{T}$ denotes the expected time for the first coalescent or mutation event.
In the penultimate equation the first summand is the probability for a coalescence or a mutation event, this probability is approximately 1 for $\epsilon$ small.

\begin{table}
  \begin{center}
    \begin{tabular}{|c|c|c|}\hline
      case & event & probability \\ \hline\hline
      \rule[-3mm]{0cm}{.7cm}(i) & \parbox{7cm}{no recombination event} & $p_{SL}p_{LR}$ \\\hline
      \rule[-8mm]{0cm}{1.8cm} (ii) & \parbox{7cm}{a $LR$-recombination event makes the allele at the $R$-locus escape the sweep without the allele at the $L$-locus} & $p_{SL}(1-p_{LR})$ \\\hline
      \rule[-8mm]{0cm}{1.8cm}  (iii) & \parbox{7cm}{by a $SL$-recombination event the line escapes the sweep and the alleles at the $L$- and $R$-locus stay linked} & $(1-p_{SL})p_{LR}$ \\\hline
      \rule[-22mm]{0cm}{4.8cm}  \parbox{.6cm}{(iv)\\[5ex](v)} & \parbox{7cm}{a $SL$-recombination event brings the alleles at the $L$- and $R$-loci linked into the wild-type background; here, the ancestry of both alleles is split by a $LR$-recombination\\[.5ex]a $LR$- and a $SL$-recombination event bring first the allele at the $R$-locus and then the allele at the $L$-locus into the wild-type background} & \parbox{3.8cm}{$\mathbb P[\text{(iv) or (v)}]$ \\ $\;\;=(1-p_{SL})(1-p_{LR})$} \\\hline
    \end{tabular}
  \end{center}
  \caption{\label{tab:geoalpha}Probabilities of several events happening
    between times $T/2$ and $T_{-}$ for geometry (a); see Figure
    \ref{fig:starApproximation3}. All events are described
    backwards in time.}
\end{table}

The time $\mathcal{T}$ lies approximately between 0 and the fixation time $T$ for $\epsilon$ small. So for $s\ll \theta_s$ we can (approximately) bound the probability for
 a hard sweep in a sample of two by
$$\frac{1}{1+\theta_s}\leq  P_{hard,2} \leq \frac{1}{1+\theta_s}(1+ u_s T)$$
Hence, the probability for a soft sweep in a sample of two can be (approximately) bounded by
$$\frac{\theta_s}{1+\theta_s} \geq   P_{soft,2} \geq \frac{\theta_s}{1+\theta_s}(1-u_s T) . $$

We can generalize this approach to obtain the (approximate) distribution of the number
of founders and offspring in a sample of size $n$.
It is given by Ewens sampling formula:

In a sample of size $n$ at time 0, the probability, that there are $a_j$ founders of the sweep (with respect to the selected locus)
 which have $j$ offspring for $j \in \{1, ..., n\}$ is given by Ewens sampling formula
\begin{equation}\label{Ewenssamp}
 \frac{n!}{\theta_s(n)}  \prod \limits_{j=1}^{n} \frac{(\theta_s/j)^{a_j}}{a_j!}
\end{equation}
where $\theta_s(n) = \theta_s \cdot (\theta_s+1) \cdot \dots \cdot (\theta_s+n-1).$
See \citep{PenningsHermisson2006b} for a derivation of this formula in a Wright-Fisher-model formulation.

We will assume in our approximation of the genealogy, that the number of founders and the number of their offspring is Ewens distributed as given in Equation
(\ref{Ewenssamp}).

\begin{table}
  \begin{center}
    \begin{tabular}{|c|c|c|}\hline
      line & event & probability \\ \hline\hline
      \rule[-3mm]{0cm}{.7cm}(i) & \parbox{7cm}{no recombination event} & $p_{LS}p_{SR}$ \\\hline
      \rule[-8mm]{0cm}{1.8cm} (ii) & \parbox{7cm}{a $SR$-recombination event makes the allele at the $R$-locus escape the sweep without the allele at the $L$-locus} & $p_{LS}(1-p_{SR})$ \\\hline
      \rule[-8mm]{0cm}{1.8cm}  (iii) & \parbox{7cm}{a $LS$-recombination event makes the allele at the $L$-locus escape the sweep without the allele at the $R$-locus} & $(1-p_{LS})p_{SR}$ \\\hline
      \rule[-17mm]{0cm}{3.5cm}  \parbox{.6cm}{(iv)\\[7ex](v)} & \parbox{7cm}{a $LS$-recombination event followed by a $SR$-recombination event bring the alleles at the $L$- and $R$-locus into the wild-type background\\[.5ex]same as (iv) but in reverse order of the $LS$- and $SR$-recombination events} & \parbox{3.8cm}{$\mathbb P[\text{(iv) or (v)}]$ \\ $\;\;=(1-p_{LS})(1-p_{SR})$} \\\hline
    \end{tabular}
  \end{center}
  \caption{\label{tab:geobeta}Probabilities for events happening between time $T/2$ and $T_{-}$ for geometry (b); see Figure
    \ref{fig:starApproximation4}. All events backward in time}
\end{table}
\subsubsection*{ Recombination events}
Forward in time at a recombination event two lines merge into one. If a recombination event occurs
 between two neighboring loci $L_1$ and $L_2$ (we will write $L_1 L_2$-recombination event, for short), such that $L_1$ lies on the left side of $L_2$, the
 offspring carries
at all loci left of the locus $L_1$ including the locus $L_1$ the alleles of the first parent and at the remaining loci the alleles of the
 second parent (with $L_1,L_2 \in \{L, R, S\}$).
Backward in time at a recombination event one line splits up into two lines.

Since coalescence events occur at the beginning of the sweep,
 one can assume, that each
 recombination event affects only a single line. The probability for no recombination event in the time interval $[t_1,t_2]$
is given by the probability, that the first jump time of a Poisson process started at time $t_1$ with rate $r(t)$ does not occur until
time $t_2$. The rate $r(t)$ depends on the different kinds of recombination events and is specified in the following.
\begin{itemize}
\item The frequency $X_t$ stays between backward time $0$ and $T/2$ almost the whole time near 1 and is certainly greater 1/2. (The larger $\alpha$
the longer $X_t$ remains in a small neighborhood of 1.)
So, in the first half, recombination between the backgrounds is not frequent.
 Furthermore if $L$, $R$ and $S$ are arranged according to geometry
(a) $SL$-recombination events inside the beneficial background cannot be
seen in the DNA-data. The only events that can be recognized in the DNA-data and occur at a non negligible amount are $LR$-recombination
 events in the beneficial
background.
If the loci are arranged according to geometry (b) all recombination events in the beneficial locus may be seen in the data.

The rate of recombination events between loci $L_1$ and $ L_2$ (with $L_1, L_2 \in \{L,R,S\}$ for geometry (b) and $L_1=L$ and $L_2=R$ for geometry (a))
 in the beneficial background is approximately $\frac{r_{L_1 L_2} X(T-t) X(T-t)}{X(T-t)}$ with $r_{L_1 L_2} \geq 0$.
Therefore the probability for no $L_1 L_2$-recombination event is given by
\begin{equation}\label{prob}
\exp \Big(-\int_{0}^{T/2} r_{L_1 L_2} X(T-t) dt \Big).
\end{equation}
 As long as $u_s$ is small, the differential Equation (\ref{diffeq}) is only a small perturbation of the differential equation
$  \dot{X}(t) = \nolinebreak[10] s X(t)(1-X(t)).$  For this equation the integral in Equation (\ref{prob}) is equal
 to $ r_{L_1 L_2} \log(\alpha)/s + r_{L_1 L_2} \log(2)/s.$
Since the second summand is small, we approximate (\ref{prob}) by $$p_{L_1 L_2}:= \exp(- \rho_{L_1 L_2} \log(\alpha)/\alpha), $$
where $\rho_{L_1 L_2}:= 2N r_{L_1 L_2}$ denotes the recombination rate between the locus $L_1$ and $L_2$.

\begin{figure}
 \begin{center}
\includegraphics[scale=0.5]{ZeitintervallI3.pdf}
\end{center}
  \caption{\label{fig:ZeitintervallI3}
Possible split of two linked neutral loci: Two alleles at the neutral loci linked to the beneficial allele either (i)
have a common ancestor at time $T/2$ or (ii) have two different ancestors that are both linked to a beneficial allele.}
\end{figure}

\item In the time interval $[T/2,T]$ all recombination events with offspring in the beneficial background
 except recombination events inside the beneficial locus are probable to be seen in the data. Similar
arguments as above lead to the following assumption:
The probability for no recombination between locus $L_1$ and $L_2$ in the time interval $[T/2,T]$ is given by
$$p_{L_1 L_2} := \exp(- \rho_{L_1 L_2} \log(\alpha)/\alpha). $$

\end{itemize}

See Figure \ref{fig:starApproximation3} - \ref{fig:ZeitintervallI3}
 for an illustration of the different types of recombination events possible in the time intervals $[0,T/2)$ and $[T/2,T)$.

With this motivation, we can define an extended star-like genealogy:

\subsection{Genealogies: Definition} \label{GeneaDefi}

The joint genealogy of two neutral loci in the neighborhood of the selected locus can be defined as a structured partition-valued process.
Denote by $\Xi_A:= \{ \xi_A | \xi_A \textrm{ partition of } A \}$ the set of partitions of a set
$A$.
A partition $\xi_A = \{\xi_1, ..., \xi_m\} $ is called finer than a partition $\xi^{'}_A =\{\xi^{'}_1, ..., \xi^{'}_k\}$, iff
for each $j\in\{1, ..., m\}$ exists an $i \in\{1, ..., k\}$, such that $\xi_{j} \subseteq \xi^{'}_i$.
We write  $\xi_A \preccurlyeq \xi^{'}_A$, if $\xi_A$ is finer than $\xi^{'}_A$.
Let $\xi= \{\xi_1, ..., \xi_m\} \in \Xi_A$ and $\eta= \{\eta_1, ..., \eta_k\}  \in \Xi_{\{1, ..., m\}}$, $m\geq k$,
then the composition $$\eta \circ \xi := \bigg\{ \bigcup_{j\in \eta_1} \xi_j, ..., \bigcup_{j \in \eta_k} \xi_j \bigg\}.$$
A structured partition of $A$ is a tuple $(\xi_A^{B}, \xi_A^{b})$
 with $\{\xi^{B}_A \cup \xi^{b}_A\} \in  \Xi_A$ and $\xi_A^{B} \cap \xi_A^{b}= \emptyset$.
Partition elements in $\xi^{B}_A$ ($\xi^{b}_A$) are called beneficial (wild-type).
 Denote by $$\Xi^{B,b}_A:= \{ (\xi_A^{B}, \xi_A^{b}) | (\xi_A^{B}, \xi_A^{b}) \textrm{ is a structred partition of } A \}$$
the set of structured partitions of the set A. Elements of a structured partition $(\xi_A^{B}, \xi_A^{b})$ are of the form $(\xi_1, \xi_2)$ with
$\xi_1 \in \xi_A^{B}$ and $\xi_2\in \xi_A^{b}$.

Define $\dickm \ell:= \{1, ...., n \}$ the set of the L-loci and $\dickm r:= \nolinebreak[10] \{n+1, ..., 2n\}$ the set of the R-loci of a sample of size $n$ of the population.
We are interested in the structured partitions of $\dickm \ell \cup \dickm r$.

For geometry (a) the different kinds of recombination events can change the structured partition $(\xi^{B}, \xi^{b})$ to
\begin{itemize}
 \item $\left(\xi^B\setminus\{\xi^B_j\}, \xi^b\cup \{\xi^B_j\}\right),$ if an $SL$-recombination event happens between a wild-type and beneficial line
 and the offspring carries at the selected locus the beneficial allele (thus at the $L$- and $R$-locus the individual carries the alleles of the wild-type line). \\
 \item $\left((\xi^B\setminus\{\xi^B_j\}) \cup \{\xi^B_j\cap \dickm \ell\}, \xi^b\cup \{\xi^B_j\cap\dickm r\} \right)$, if an $LR$-recombination event happens
between an individual of the beneficial background and an individual of the wild-type background and at the $S$ and $L$-locus the beneficial line is carried
 on (forward in time).\\
 \item $\left((\xi^B\setminus \{\xi^B_j\}) \cup \{\xi^B_j\cap \dickm \ell, \xi^B_j\cap \dickm r\}, \xi^b\}\right)$, if an $LR$-recombination event
 happens between two individuals of the beneficial background.\\
  \item $\left(\xi^B, (\xi^b \setminus\{\xi^b_k\}) \cup \{\xi^b_k\cap \dickm \ell, \xi^b_k\cap\dickm r\}\right),$ if an $LR$-recombination event happens between
 two individuals of the wild-type background.\\
 \item  $\left(\xi^B\cup \{\xi^b_k\}, \xi^b\setminus\{\xi^b_k\}\right),$ if an $SL$-recombination event happens between a beneficial and a wild-type line
 and the offspring carries the beneficial allele.\\
 \item $\left(\xi^B\cup \{\xi^b_k\cap\dickm r\}, (\xi^b\setminus\{\xi^b_k\}) \cup \{\xi^b_k\cap \dickm \ell\}\right),$ if an $LR$-recombination event happens
between a beneficial and wild-type line and at the selected and $L$-locus the wild-type is carried on (forward in time).
\end{itemize}

For geometry (b) the partitions change in an analogous manner.

Before we give the definition of an extended star-like genealogy we define genealogies and  samples.
\begin{defi}
The genealogy of a set $A$ is a four-time step Markov chain
$$(\xi_t)_{t\in\{0,T/2,T_{-},T\}} =((\xi^{B}_{0},\xi^{b}_{0}), (\xi^{B}_{T/2}, \xi^{b}_{T/2}),(\xi^{B}_{T_{-}},\xi^{B}_{T_{-}}), (\xi^{B}_{T}, \xi^{b}_{T}))$$ with
 state space $\Xi^{B,b}_{A}$.

A set $\dickm \ell \cup \dickm r$ with $\dickm \ell:= \{1,..., n\}$ and $\dickm r:=\{n+1, ..., 2n\}$, $n\in \mathbbm{N}$, is a
 sample at two loci $L$ and $R$ taken from the population at time $t=0$, if
the genealogy of the sample is at time $t=0$ given by $\xi_0= \big(\big\{\{1, n+1\}, \dots, \{n, 2n\}\big\}, \{\emptyset\} \big)$.
\end{defi}

With this we can define an extended star-like genealogy as a four-time step random experiment:
\begin{defi}\label{stargene}
An extended star-like genealogy of a sample $\dickm \ell \cup \dickm r$ with $\dickm \ell:= \{1,..., n\}$ and $\dickm r:=\{n+1, ..., 2n\}$
 at two loci $L$ and $R$ in the neighborhood of a selected locus $S$
 arranged according to geometry (a) (resp. geometry (b)) is a four-time step Markov chain
 $$(\xi_t)_{t\in\{0,T/2,T_{-},T\}} =((\xi^{B}_{0},\xi^{b}_{0}), (\xi^{B}_{T/2}, \xi^{b}_{T/2}),(\xi^{B}_{T_{-}},\xi^{B}_{T_{-}}), (\xi^{B}_{T}, \xi^{b}_{T}))$$ with
 state space $\Xi^{B,b}_{\dickm \ell \cup \dickm r}$ with the following properties:
\begin{itemize}
\item At time $T/2$:
   \begin{itemize}
    \item Structured partition elements are stochastically independent
    \item No recombination events between the backgrounds and no mutations to the beneficial allele, i.e. $P\big( \xi^{b}_{T/2}=\emptyset\big)=1$
    \item No coalescence events, i.e. $P\big(\xi^{B}_{T/2}  \preccurlyeq \xi^{B}_{0}  \big)=1$
    \item For geometry (a): No $LR$-recombination events in the beneficial background with probability $p_{LR}$, i.e. for $j \in \dickm l$
          a structured partition element at time $t=0$ of the form $(\{\{j, j+n\}\}, \{\emptyset\})$
          \begin{itemize}
          \item is kept at time $T/2$ with probability $p_{LR}$
          \item  and changed to $(\{\{j\}, \{j+n\}\}, \{\emptyset\})$ with probability $1-p_{LR}$
          \end{itemize}
    \item For geometry (b): Neither a $LS$-recombination events nor a $SR$-recombination events in the beneficial background happens
          with probability $p_{LS}p_{SR}$, i.e. for $j \in \dickm l$
          a structured partition element at time $t=0$ of the form $(\{\{j, j+n\}\}, \{\emptyset\})$
         \begin{itemize}
         \item is kept till time $T/2$ with probability $p_{LS}p_{SR}$
         \item and changed to $(\{\{j\}, \{j+n\}\}, \{\emptyset\})$ with probability $1-p_{LS}p_{SR}$
         \end{itemize}
\end{itemize}
\item At time $T_{-}$:
\begin{itemize}
 \item Structured partition elements are stochastically independent
 \item No coalescence events, i.e. $P(\xi^{B}_{T_{-}} \cup \xi^{b}_{T_{-}}  \preccurlyeq \xi^{B}_{T/2}  )=1$
 \item For geometry (a): For $j \in \dickm l$
\begin{itemize}
\item a partition element at time $T/2$ of the form $(\{\{j, j+n\}\}, \{\emptyset\})$ \\
         is kept at time $T_{-}$ with probability $p_{SL}p_{LR}$, \\
         changed to $( \{\emptyset\}, \{ \{j, j+n\}\})$ with probability $(1-p_{SL})p_{LR}$, \\
           changed to $(\{ \{j\}\} , \{\{j+n\}\})$ with probability $p_{SL}(1-p_{LR})$ \\
          and changed to $(\emptyset, \{\{j\}, \{j+n\}\})$ with probability $(1-p_{SL})(1-p_{LR}).$
\item a partition element at time $T/2$ of the form $(\{j\}, \{\emptyset\})$ \\
             is kept at time $T_{-}$ with probability $p_{SL}$ \\
      and changed to $(\{\emptyset\}, \{j\})$ with probability $1-p_{SL}.$
\item a partition element at time $T/2$ of the form $(\{j+n\}, \{\emptyset\})$ \\
             is kept at time $T_{-}$ with probability $p_{SR}$ \\
      and changed to $(\{\emptyset\}, \{j+n\})$ with probability $1-p_{SR}.$
\end{itemize}
\item For geometry (b):
\begin{itemize}
\item A partition element at time $T/2$ of the form $(\{\{j, j+n\}\}, \{\emptyset\})$ \\
         is kept at time $T_{-}$ with probability $p_{LS}p_{SR}$, \\
            changed to $(\emptyset, \{\{j\}, \{j+n\}\})$ with probability $(1-p_{LS})(1-p_{SR}).$ \\
            changed to $(\{\{j\}\} , \{\{j+n\}\})$ with probability $p_{SL}(1-p_{SR})$ \\
          and changed to $(\{\{j+n\}\}, \{\{j\}\})$ with probability $(1-p_{SL})p_{SR}.$
\item A partition element at time $T/2$ of the form $(\{j\}, \{\emptyset\})$ \\
             is kept at time $T_{-}$ with probability $p_{LS}$ \\
      and changed to $(\{\emptyset\}, \{j\})$ with probability $1-p_{LS}.$
\item A partition element at time $T/2$ of the form $(\{j+n\}, \{\emptyset\})$ \\
             is kept at time $T_{-}$ with probability $p_{SR}$ \\
      and changed to $(\{\emptyset\}, \{j+n\})$ with probability $1-p_{SR}.$
\end{itemize}
\end{itemize}
\item At time $t=T$: \\
 At the beginning of the sweep all coalescence and mutation events happen:
Let $ m \in \mathbbm{N}$ and $ a_j\in \{0, ..., m\}$ with $\sum_{j=1}^{m}j a_j=m.$ Denote by
$M^{m} :=\{(\xi^B, \xi^b) \in \Xi^{B,b}_{\dickm \ell \cup \dickm r } ; |\xi^B|=m \}$ the set of structured partition of $\dickm \ell \cup \dickm r$
which beneficial partitions consist of m elements and by $N^{(a_1, ..., a_m)}:= \{ \eta= (\eta_1,..., \eta_k) \in \Xi_{\{1, ..., m\}};
\# \{\eta_l ; |\eta_l|=j\}=a_j \}$ the set of partitions of $\{1, ..., m\}$ containing $a_j$ partition elements of size $j$.
Then for  $\eta\in N^{(a_1, ..., a_m)}$
\begin{equation}\label{probfounder}
P( \xi_T= (\{\emptyset\}, (\eta \circ \xi^{B}) \cup \xi^{b}) | \xi_{T_{-}}=(\xi^{B}, \xi^{b}) \in M^{m}) = \frac{m!}{\theta_s(m)} \prod \limits_{j=1}^{m} \frac{(\theta_s/j)^{a_j}}{a_j!}.
\end{equation}
\end{itemize}
We say, that a population evolved according to an extended star-like genealogy, if the genealogy of each sample of the population is extended star-like.
\end{defi}

\begin{bem}
\emph{
If we are interested in the genealogy of a subset $M$ of a sample $\dickm \ell \cup \dickm r$, the
 genealogy of $M$
 fulfills
all conditions of Definition \ref{stargene}.
In particular, at time $t=T$ the number of founders together with the number of their offspring is Ewens distributed, since Ewens sampling formula is consistent.
At time $t=0$, the genealogy of $M$ is given by $$\xi_0=  \big(\big\{\{1, n+1\}\cap M, \dots, \{n, 2n\} \cap M\big\}, \{\emptyset\} \big).$$
}
\end{bem}
In accordance to the possible recombination events during the time interval $[T/2,T)$ we obtain the ancestral lines
shown in Figure \ref{fig:starApproximation3} for geometry (a) and Figure
\ref{fig:starApproximation4} for geometry (b). The probabilities for these events are listed in Table \ref{tab:geoalpha} for geometry
 (a), in Table \ref{tab:geobeta} for geometry (b). For the time interval $[0,T/2)$ the possible ancestral lines are
 shown in Figure \ref{fig:ZeitintervallI3}. In Figure \ref{fig:CoalMut1} the left picture shows two lines which coalesce first and
mutate then, in the right picture the lines mutate first and coalesce afterwards.

\section{Results}
Our main result is the computation of the linkage disequilibrium at the end of the sweep
 measured by $\mathbbm{E} [D_{\ell, r}(0) |D_{\ell, r}(T)]$ for two fixed allelic variants
$\ell$ and $r$ and $\widehat{\sigma^2_D}$ for two neutral
loci in a neighborhood of the selected locus (backward in time).

We apply the procedure of \citet{PfaffelhuberLehnertStephan2008} to compute $\mathbbm{E}[D_{\ell, r}(0) |D_{\ell, r}(T)]$. The main
 difference between our model and the hard sweep model is, that two
lines do not have to coalesce, since both lines may mutate to the beneficial allele.
\begin{thm}\label{smallthm}
Assume, that the population evolved in a DNA-region containing the two neutral loci $L$ and $R$ and the selected locus $S$ according to an extended star-like
 genealogy and both loci carry exactly two allelic variants $\ell/L$ and $r/R$. Then the linkage
disequilibrium of the allelic variants $\ell$ and $r$ measured by $\mathbbm{E}[D_{\ell,r}(0)|D_{\ell, r}(T) ]$ at the end of the sweep
is given by
\begin{equation}\label{Geoa}
  \mathbbm{E}[D_{\ell, r}(0) |D_{\ell, r}(T)] = p_{LR}^2(1-\frac{1}{1+\theta_s} p_{SL}^2)D_{\ell,r}(T),
\end{equation}
if the two neutral loci are arranged according to geometry (a)
and
\begin{equation}\label{Geob}
 \mathbbm{E}[D_{\ell, r}(0) |D_{\ell, r}(T)] = p_{LR}^2(1-\frac{1}{1+\theta_s})D_{\ell,r}(T),
\end{equation}
for L and R arranged according to geometry (b).
\end{thm}
\begin{proof}
Indeed, consider the genealogy $(\xi)_{t\in\{0,T/2,T_{-},T\}}$ of an $L$-locus $\{1\}$ and an $R$-locus $\{2\}$, i.e. $\xi_t \in \Xi^{B,b}_{\{1,2\}}$.
Denote by $d$ the probability, that the pair $\{1\},\{2\}$
 was linked at the beginning of the sweep, if it is linked at the end of the sweep, i.e.~let $d:= P(\xi_T=(\{\emptyset\}, \{1,2\})| \xi_0=\nolinebreak[30] (\{\{1,2\}\}, \{\emptyset\})).$
   Analogously, denote by $e$ the probability, that
the pair has been linked at the beginning, if it is unlinked at the end of the sweep.
 I.e.~$e:= P(\xi_T=(\{\emptyset\},\{\{1,2\}\})| \xi_0=(\{\{1\},\{2\}\},\{\emptyset\}))$. Then we can write
\begin{equation*}
 \mathbbm{E}[q_{\ell r}(0)| q_{\ell r}(T), q_\ell(T),q_r(T)] = d q_{\ell r}(T) + (1-d)q_\ell(T)q_r(T)
\end{equation*}
\begin{equation*}
 \mathbbm{E}[q_{\ell}(0) q_{r}(0)| q_{\ell r}(T), q_\ell(T), q_r(T)] = e q_{\ell r}(T) + (1-e) q_\ell(T)q_r(T)
\end{equation*}
with $q_{\ell r}\in [0,1]$ and $q_r,q_\ell \in (0,1)$
and so
\begin{equation*}
 \mathbbm{E}[D_{\ell, r}(0) |D_{\ell, r}(T)=x] = (d-e) D_{\ell, r}(T).
\end{equation*}
The probabilities $d$ and $e$ are for geometry (a) and (b) given by
\begin{equation*} a) \ e = \frac{1}{1+ \theta_s} p_{SL}p_{SR} = \frac{1}{1+\theta_s}  p_{SL}^2 p_{LR} \quad  b) \ e= \frac{1}{1+ \theta_s} p_{LS}p_{SR}= \frac{1}{1+ \theta_s} p_{LR}
 \end{equation*}
and
\begin{equation*} a) \ d = e(1- p_{LR}) + p_{LR}p_{LR} \quad b) \ d= e(1- p_{LR}) + p_{LR}p_{LS}p_{SR} =  p_{LR}.
 \end{equation*}
In words, a pair is unlinked at the end of the sweep when it was linked at the beginning, iff
 the pair just coalesces, i.e.
 neither a recombination event between the $S$ and the $L$ locus neither
a recombination event between the $S$ and $R$ locus occurred and the two loci coalesced before they mutated.
And a pair which is linked at the end of the sweep is also linked in the beginning, iff either nothing happens or the pair is divided by a LR-recombination
event first and then linked again by coalescence event.

From this easily follows Equation (\ref{Geoa}) for geometry (a) and Equation (\ref{Geob}) for geometry (b).
\end{proof}

To compute the quantity $\widehat{\sigma^2_D}$ consider the three quantities:

\begin{equation} \label{XYZ}
\begin{aligned}
 \mathcal{X}_t & := \mathbbm{E}[q_L(t)(1-q_L(t))q_R(t)(1-q_R(t))] \\
 \mathcal{Y}_t &:= \mathbbm{E}[D(t)(1-2q_L(t))(1-2q_R(t))] \\
 \mathcal{Z}_t &:= \mathbbm{E}[(D(t))^2]
\end{aligned}
 \end{equation}
for $0\leq t \leq T$.

\begin{thm}\label{bigthm}
Given $\mathcal{X}_{T}, \mathcal{Y}_{T}$ and $\mathcal{Z}_{T}$ at the beginning of the sweep and a sample of size $n$ of a population at the end of the sweep, i.e.~a set of $L$-loci
$\dickm \ell:=\{1, ...,n\}$ and a set of $R$-loci $\dickm r:= \{n+1, ..., 2n\}$, assume
 that the genealogy of the sample is extended star-like. Then the standard linkage disequilibrium $\widehat{\sigma^2_D}$ of this sample of
 two neutral loci at the end of a sweep equals:
\begin{equation}\label{sigma}
\widehat{\sigma^2_D}= \widehat{\mathcal{Z}_0}/\widehat{\mathcal{X}_0}.
\end{equation}
with
\begin{equation} \label{big1}
\begin{aligned}
\widehat{ \mathcal{Z}_0}& =  p_{LR}^4 (p_{SL}-1)^2 \big( p_{SL}^2(\mathcal{X}_T+\mathcal{Y}_T)+(1+2p_{SL})\mathcal{Z}_T \big) \\
& \quad + \theta_s p_{SR}^2 \Big(\frac{\mathcal{X}_T}{3} \big( p_{LR}(11p_{SR}-2-6p_{LR})+p_{SR}(2-4p_{SR}) \big) \\
& \quad + \frac{\mathcal{Y}_T}{12} \big( p_{SR}(2-21p_{SR}) + p_{LR}(38p_{SR}-15p_{LR}-2) \big) + \frac{\mathcal{Z}_T}{3}\big( 9p_{LR}^2-9p_{SR}p_{LR}+p_{SR}^2 \big) \Big)
\end{aligned}
\end{equation}
and
\begin{equation}\label{big2}
\begin{aligned}
\widehat{ \mathcal{X}_0} & = (1-p_{SR})(p_{SL}-1) \big(\mathcal{X}_T(1+p_{SR}+p_{SL})+(\mathcal{X}_T+\mathcal{Y}_T)(p_{SL}p_{SR})\big) \\
& \quad + \frac{\theta_s p_{SL}^2}{3}\Big(\mathcal{X}_T(3p_{LR}^2-5p_{LR}+3 +2p_{SR}p_{LR}+2p_{SR}-4p_{SR}^2)  \\
& \quad + 5\mathcal{Y}_T(p_{SR}p_{LR}+p_{SR}- \frac{17}{20}p_{LR})+  p_{SR}^2(\mathcal{Z}_T-\frac{21}{4}\mathcal{Y}_T)\Big)
\end{aligned}
\end{equation}
 for geometry (a)
and with
\begin{equation}\label{big3}
\begin{aligned}
\widehat{\mathcal{Z}_0} & = \theta_s p_{LR}^3 \Big(\frac{\mathcal{X}_T}{3}\big(1-p_{LS}+2p_{LR}-p_{SR}\big) \\
& \quad + \frac{\mathcal{Y}_T}{12} \big(3p_{LR}-p_{SR}-p_{LS}+1 \big)+ \frac{\mathcal{Z}_T}{3}p_{LR} \Big)
\end{aligned}
\end{equation}
and
\begin{equation}\label{big4}
\begin{aligned}
\widehat{\mathcal{X}_0}&=\mathcal{X}_T \big((1-p_{LS}^2)(1-p_{SR}^2) \big)+ \mathcal{Y}_T \big(p_{LR}(1-p_{LS})(1-p_{SR})\big)\\
& \quad + \theta_s \Big( \frac{\mathcal{X}_T}{3}\big(3p_{LS}^2+2p_{LR}p_{SR}-4p_{LR}^2-5p_{LR}+2p_{LR}p_{LS}+3p_{SR}^2 \big) \\
& \quad + \frac{\mathcal{Y}_T p_{LR}}{12} \big(20 p_{LS}-21p_{LR}-17+20 p_{SR} \big) + \frac{\mathcal{Z}_T}{3}p_{LR}^2 \Big)
\end{aligned}
\end{equation}
for geometry (b),
if we ignore in both geometries terms of order $\theta_s^2$ and $1/n$.
\end{thm}
For a proof of this theorem see Section \ref{sec:proof}.

\begin{bem}\label{bemtheo}
\emph{
\begin{itemize}
 \item In the supporting online material you find a Mathematica-notebook for computing the exact values of the standard linkage disequilibrium
measured by $\widehat{\sigma^2_D}$ without ignoring terms of order $\theta_s^2$ and $1/n$.
 \item If the population evolves neutrally till the beginning of the sweep, \citet{Ohta1969} have shown, that
\begin{align*}\label{XYZ_0}
\mathcal{X}_T & = \frac{1}{4} \frac{\theta^2}{\theta+1} \cdot
  \frac{5+2\theta+\rho_{LR})(3+2\theta+2\rho_{LR})-4}{(1+\theta)(3+2\theta +2 \rho_{LR})(5+ 2\theta +\rho_{LR})- 2(3+ 2\theta)} \\
\mathcal{Y}_T & =  \frac{\theta^2}{\theta+1} \cdot
 \frac{1}{(1+\theta)(3+2\theta +2 \rho_{LR})(5+ 2\theta +\rho_{LR})- 2(3+ 2\theta)} \\
\mathcal{Z}_T & = \frac{1}{4} \frac{\theta^2}{\theta+1} \cdot
 \frac{2\theta +\rho_{LR} +5}{(1+\theta)(3+2\theta +2 \rho_{LR})(5+ 2\theta +\rho_{LR})- 2(3+ 2\theta)},
\end{align*}
where $\theta:= 4Nu$ is the neutral mutation rate. For a comparison of the theoretical results with simulations
 we assume that the population evolved neutrally till the beginning of the sweep.
\item See Figure \ref{fig:Theo} for a plot of the theoretical values of $\widehat{\sigma^2_D}$ for different values of $\theta_s$.
Here we assumed as well neutral evolution till the beginning of the sweep.
 \item Note, that for $\theta_s=0$ we obtain $\widehat{\sigma^2_D}$ for a hard sweep, compare \citep{PfaffelhuberLehnertStephan2008}.
\end{itemize}
}
\end{bem}

\begin{figure}
 \begin{center}
\includegraphics[scale=0.35]{Papertheo1.pdf} \end{center}
  \caption{\label{fig:Theo}
Theoretical values of $\widehat{\sigma^2_D}$ in the neutral setting, for $\theta_s=0$ and $\theta_s=0.1$.
 The distance between the neutral loci is $0.2$ kb, the selection strength $\alpha=1000$, the population size $N=10^6$,
the recombination rate between the neutral loci $\rho_{LR}= 5$ and the neutral mutation rate $\theta=0.005$.
}
\end{figure}

\section{Simulations}\label{sim}
\begin{figure}
\subfigure{
\includegraphics[scale=0.35]{Papertheta01.pdf}
}
\subfigure{
\includegraphics[scale=0.35]{Papertheta05.pdf}
}
 \caption{Left figure:
Plot of $\widehat{\sigma^2_D}$ for a neutral mutation rate $\theta= 0.005$, recombination rate $\rho= 0.025$, selection strength $\alpha=1000$,
recurrent mutation rate to the beneficial allele $\theta_s=0.1$, a distance of 200 bp between the two neutral loci and a DNA-stretch of
length 5 kb based on $10^4$ draws. Right figure:
Plot of $\widehat{\sigma^2_D}$ with the same parameters as in the left figure except for the
recurrent mutation rate to the beneficial allele $\theta_s=0.5$.
}
\label{fig:Plot1}

\end{figure}

\begin{figure}
 \begin{center}
\includegraphics[scale=0.5]{Papersinglesample.pdf} \end{center}
  \caption{\label{fig:Singlesample}
The full linkage disequilibrium spectrum for a single sample of a soft sweep with two founders with respect to the beneficial locus.
}
\end{figure}

We simulated sequence samples with the new program msms (for ms mit Selektion (German: with selection)) of Greg Ewing, see \citep{Ewing} to compare
our theoretical linkage disequilibrium values with linkage disequilibrium values obtained from simulated genealogies assuming neutral evolution till
 the beginning of the sweep.
The program msms generates sequence samples for a single selected locus of a population reproducing according to the Wright-Fisher-model
 with the possibility of recurrent mutation to the beneficial allele. The frequency of the beneficial
allele is simulated stochastically conditioned on fixation.
In Section 2 we argued, that the star-like genealogies approximates the Moran model genealogies well.
For large population sizes the Moran model and Wright-Fisher model deliver similar genealogies, if the parameters are appropriately scaled.
So instead of comparing the theoretical results with results obtained from Moran model simulations, we can check the theoretical results against results
obtained from Wright-Fisher model simulations.

We consider a 5-kb stretch of DNA in a sample of $n=20$ taken at time of fixation
of the beneficial allele.
 We divide the stretch into 50 bins, each of length 0.1kb and measure LD between SNPs of two different bins averaged over $10^4$ draws.
Figure \ref{fig:Plot1} shows the results for a recurrent mutation rate $\theta_s=0.1$ and $\theta_s=0.5$, respectively.
The neutral mutation rate $\theta=0.005$, the recombination rate $\rho= 0.025 $ for two neighboring loci, the distance between neighboring neutral loci $L$ and $R$ is 200bp,
the selection strength $\alpha=1000$ and the population size $N=10^6$ in both plots. These parameter values are realistic for example for Drosophila melanogaster
samples.

As we see in Figure \ref{fig:Plot1} there is a good fit between simulated and theoretical values of $\widehat{\sigma^2_D}$.
For small sample sizes the extended star-like genealogies approximate the simulated Wright-Fisher-genealogies well. The linkage disequilibrium
is for theoretical and simulated values high, if $\theta_s \neq0$ and the distance to the selected locus is small, independently of
the geometry of the considered neutral loci.
Due to recombination linkage disequilibrium decreases with increasing distance to the selected locus.

The differences between the theoretical and simulated values of $\widehat{\sigma^2_D}$ are due to the approximation of the genealogy.
The approach has three effects on LD.

First, a star-like genealogy assumes independent recombinants. But of course in the simulated
genealogies may also occur coalescence events before recombination events, in particular
may arise early recombinants (see \citep{DurrettSchweinsberg2004a} or for slightly different
 models \citep{EtheridgePfaffelhuberWakolbinger2006}).
 For geometry (a) it is important, that recombinants with
 offspring lead to less ``independent'' variation, which can be seen in higher LD values of the simulated data.
For geometry (b) early recombinants become noticeable, because they produce patterns similar to soft sweep patterns:
Recombinants spread through the populations act as additive founders of the sweep.
For this reason our approximation of the genealogy assumes less founders of the sweep than the simulated genealogies have.
 Therefore the LD-patterns of simulated data should look like the LD-pattern of the theoretical values with a slightly
higher $\theta_s$ value. Higher $\theta_s$ values produce in geometry (b) less
$\widehat{\sigma^2_D}$, compare the pictures in Figure \ref{fig:Plot1}.
This effect becomes negligible for increasing $\theta_s$.
On the one hand the fixation of the beneficial allele gets faster, on the other hand for intermediate and high values of $\theta$
also extended star-like genealogies assume in average more than two founders. By measuring linkage disequilibrium one can distinguish well
between the existence of one or two founders of the sweep, but not between the existence of three or four founders.

Second, the star-like approximation of the genealogy is in general longer
than the simulated genealogy, since the beneficial allele spreads faster through the population, if the lines are dependent.
 Therefore, more recombination events are assumed to fall on the theoretical genealogies than on the simulated genealogies.
For loci in a small neighborhood of the selected locus this means, that more SNPs can be found for the
star-like genealogy due to recombination. Third, SNPs of simulated data are noisier, they may exist also due to neutral mutation during the sweep.
Both effects can in geometry (a) be recognized by higher theoretical LD values in a small neighborhood of the selected locus. But at a certain point,
the effect turns over: More recombination brings more ''independent`` variation into the sample: The theoretical values of LD in geometry (a) lie below
 the simulated values.

Often one is interested in the case of a single sample. We simulated a single sample of a soft sweep with two founders and computed linkage
disequilibrium with that data. The result is plotted in Figure \ref{fig:Singlesample}. In that case the pattern is very clear.
In the neighborhood of the selected locus, high linkage disequilibrium can be found independent of the geometry of the loci.
However, such clear patterns cannot be expected in general, even if the sweep has two founders. It is likely, that the number of offspring is not
 distributed equally between the founders. For example it may happen, that in a sample of 20 individuals with respect
to the beneficial allele 2 individuals are offspring of one founder and the remaining 18 individuals are offspring of the second founder.
For such unbalanced cases stochastic effects caused by recombination and mutation destroy the pattern easily.

\section{Discussion}
Soft sweeps have been introduced by Pennings and Hermisson in their series of papers \citep{HermissonPennings2005}, \citep{PenningsHermisson2006a}, \citep{PenningsHermisson2006b}.
They argued, that tests based on haplotype structure have high power to detect soft sweeps.
Linkage disequilibrium is a test sensitive to haplotype structure. If allelic variants are tightly linked to a haplotype, LD
 is high for pairs of such alleles.
We have seen, that linkage disequilibrium under a non vanishing recurrent mutation rate differs sufficiently
 from linkage disequilibrium under neutrality and hard sweeps, see Figure \ref{fig:Theo}.

We computed $\widehat{\sigma^2_D}$ to understand
the interplay of haplotype formation due to a soft sweep and recombination.
The main reason to compute $\widehat{\sigma^2_D}$ instead of $\mathbbm{E}[r^2]$ is its mathematical manageability. However, former studies show
 (and the present study
 does that also), that also $\widehat{\sigma^2_D}$ measures what intuitively is understood under linkage disequilibrium and gives a possibility
to distinguish between different population genetics
 scenarios.

When a soft sweep occurs, recombination breaks up the linkage of loci due to haplotype structure. Under hard sweeps recombination causes linkage of
loci lying on one side of the selected locus. In Figure \ref{fig:Theo} theoretical values of $\widehat{\sigma^2_D}$ are plotted
for different values of $\theta_s$ and under neutrality.
The behavior can be explained roughly in the following manner:

For small values $\theta_s$ we see for both geometries high values of $\widehat{\sigma^2_D}$
 in a small neighborhood of the selected locus decreasing with increasing distance to the selected locus.
If $\theta_s$ is relatively small, \citet{PenningsHermisson2006a} have shown, that soft sweeps are not very likely, most sweeps will be hard.
LD of hard sweeps depends on
recombination. Only recombination brings variation into the sample which is necessary to compute linkage disequilibrium.

After a hard sweep we can see the following pattern of LD due to recombination.
Recombination between the $L$-locus and the $S$-locus includes for geometry (a)
 always a recombination between the $R$-locus and $S$-locus or between the $L$-locus and the $R$-locus, i.e.
the $L$-locus recombines not independently of the
$R$-locus. Therefore LD is high for geometry (a) for a hard sweep.
In geometry (b) an $LS$-recombination event does not cause a $SR$-recombination event and vice versa. So with respect to recombination
 the $L$-locus is
 independent of the $R$-locus. Hence $\widehat{\sigma^2_D}$ is small. If the sample is not finite, $\widehat{\sigma^2_D}$ is
 even zero, see Remark \ref{bemtheo}.

If a soft sweep occurred, different founders of the sweep bring the variation into the sample - recombination is not necessary.
 If there are exactly
two founders and there exist loci with two allelic variants, such that one allelic variant is carried by one haplotype and the other allele by the other haplotype,
 two of such loci are tightly linked, only recombination can break up this linkage.
 Therefore after a soft sweep with only a few number of founders LD is high in a small neighborhood of the selected locus,
independent of the geometry. But the more founders the soft sweep has, the more variation is in the sample not linked to single founder. This reduces
LD.

For $r^2$ we expect for very small values of $\theta_s$ patterns of LD similar to hard sweeps, because for very small
values of $\theta_s$ soft sweeps are rare. But $\widehat{\sigma^2_D}$ shows even for very small values of $\theta_s$ high values in a small neighborhood
of the selected locus.
This comes from the fact, that small values of $D^2$ expected after a hard sweep in geometry (b) have a smaller effect on the numerator of $\widehat{\sigma^2_D}$ than
 higher values of $D^2$ expected after a soft sweep in geometry (b). An analogous statement holds for the denominator of $\widehat{\sigma^2_D}$.

For biological studies often the pattern of a single selective sweep is of interest.
After a soft sweep we expect to find high LD of two neutral loci lying in a neighborhood
 of the selected locus, but almost neutral variation. It can be found haplotype structure, where each founder of the sweep gives rise to
 one haplotype.
 In each haplotype group a hard sweep occurred, i.e. almost no variation
can be found, low LD for neutral loci lying on different sides of the selected locus and high LD for loci lying on the same site of the
 selected locus.
In Figure \ref{fig:Singlesample} simulation results of a single sample of a soft sweep with two ancestors with
 respect to the selected locus are shown. As well as \citet{TishkoffEtAl2007} found a comparable clear linkage disequilibrium pattern of a soft sweep in
 their studies of the human DNA when analyzing the human lactase persistence in African and European human populations.

An adaptation process may not only be initiated by mutation, but also through recurrent migration or from standing genetic
variation during an environmental change. A two-island model with the beneficial allele fixed in one of the islands and migration
 from this island to the
other coincides with our model for recurrent mutation.
A more realistic model assumes, that the beneficial allele is not fixed in both islands and that the allelic
frequencies $q_L$, $q_{R}$, etc. do not coincide on both islands. Unfortunately such (simple) modifications make the
 calculations in the proof of Theorem \ref{bigthm}, especially of matrix $A$ and $B$,
quite complicated.

An improvement of the results could be made by approximating the genealogy not by a star-like approximation but by a marked Yule process
 with immigration. It has been shown by \citet{HermissonPfaffelhuber2008}, that the joint genealogy of the population is better approximated
by these processes. However, explicit calculations
become with this approximation complicated, since recombination is not independent along lines during the sweep.

\section{Proof of Theorem \ref{bigthm}}\label{sec:proof}
We proceed in five steps. The quantities $\mathcal{X}_t,\mathcal{Y}_t, \mathcal{Z}_t$ can be expressed
in pairwise heterozygosities. In step 1 we will give this connection. In step 2 we show, how pairwise heterzygosities are transformed to sample heterozygosities.
In step 3 and 4 we show, how pairwise heterozygosities at time $t=T$ are transformed to pairwise heterozygosities at time $t=0$. In step 5 we collect everything together.

 \medskip

\noindent$\textrm{\textbf{ Step 1}}$:  Link between the pairwise heterozygosities $f_t,g_t,h_t$ and $\mathcal{X}_t$, $\mathcal{Y}_t$, $\mathcal{Z}_t$ \\

\medskip

The quantities $\mathcal{X}_t,\mathcal{Y}_t, \mathcal{Z}_t$ can be expressed in terms of probabilities for pairwise heterozygosities.

 Denote for this purpose by $f_t$ the probability that two pairs heterozygous in both loci are linked, by $g_t$ the probability, that
 exactly one pair of the two pairs
 is linked and
 the other pair is unlinked and by $h_t$ the probability that both pairs are unlinked at time $t$. We can express these probabilities in terms of
structured partitions: Let $\ell_1, \ell_2 $ be two $L$-loci and $r_1, r_2$ be two $R$-loci taken from the population. Let $\xi_t= (\xi^B_t, \xi^b_t)$ be the genealogy of
$\{\ell_1, \ell_2, r_1,r_2\}$ at time $t$, then
$$f_t = P(\xi_t \textrm{ is heterozygous and } \xi^B_t \cup \xi^b_t=\{\{\ell_1,r_1\}, \{\ell_2,r_2\}\}),$$
$$g_t = P(\xi_t\textrm{ is heterozygous and } \xi^B_t \cup \xi^b_t=\{\{\ell_1,r_1\}, \{\ell_2\}, \{r_2\}\}),$$
$$h_t = P(\xi_t\textrm{ is heterozygous and } \xi^B_t \cup \xi^b_t=\{\{\ell_1\},\{r_1\}, \{\ell_2\},\{r_2\}\})$$

From an easy calculation (see for details also \citep{PfaffelhuberLehnertStephan2008}, Equation (A3)) it follows, that

$$\left(\begin{array}{cc}
	\mathcal{X}_t \\
	\mathcal{Y}_t \\
	\mathcal{Z}_t
	\end{array}\right) =  \frac{1}{4} \underbrace{ \left( \begin{array}{clcr}
               0 & 0    & 1      \\
               0  & 4  &  -4   \\
               1  & -2 & 1
          \end{array}   \right)}_{=:E}   \left(\begin{array}{cc}
	f_t \\
	g_t \\
	h_t
	\end{array}\right)  $$

\medskip

\noindent$\textrm{\textbf{ Step 2}}$: Link between pairwise heterozygosities $f,g,h$ and sample
 heterozygosities $\widehat{f}$, $\widehat{g}$, $\widehat{h}$ \\

\medskip
 Denote
 by $\widehat{f}_t$, $\widehat{g}_t$
 and $\widehat{h}_t$ the corresponding sample probabilities, i.e. $\ell_1, \ell_2 \in \dickm \ell$ and $r_1, r_2 \in \dickm r$. It is possible to pick the same individual twice in a sample. Therefore the following relationships hold:
\begin{align*}
\widehat{f}_t & = \left( 1-\frac{1}{n} \right) f_t \\
\widehat{g}_t & = \left( 1-\frac{1}{n} \right) \left( 1-\frac{2}{n} \right) g_t+ \left( 1-\frac{1}{n} \right) \frac{1}{n} f_t \\
\widehat{h}_t & = \left( 1-\frac{1}{n} \right) \left( 1-\frac{2}{n} \right) \left( 1-\frac{3}{n} \right) h_t+
 \left( 1-\frac{1}{n} \right) \frac{4}{n} \left( 1-\frac{2}{n} \right) g_t + \left( 1-\frac{1}{n} \right) \frac{2}{n}\frac{1}{n}f_t.
\end{align*}
Denoting
$$F:=  I + \frac{1}{n}  \left( \begin{array}{clcr}
               -1 & 0    & 0      \\
               1  & -3   & 0   \\
               0  & 4 & -6
          \end{array}   \right)   + \frac{1}{n^2}  \left( \begin{array}{clcr}
               0 & 0    & 0      \\
               -1  & 2   & 0   \\
               2  & -12 & 11
          \end{array}   \right)  + \frac{1}{n^3}  \left( \begin{array}{clcr}
               0 & 0    & 0      \\
               0  & 0   & 0   \\
               -2  & 8 & -6
          \end{array}   \right) ,$$

this is equivalent to
$\left(\begin{array}{cc}
	\widehat{f_t} \\
	\widehat{g_t} \\
	\widehat{h_t}
	\end{array}\right) = F  \left(\begin{array}{cc}
	f_t \\
	g_t \\
	h_t
	\end{array}\right)$

For example, two linked pairs of one allele at the $L$- and one allele at the
$R$-locus each taken at random (with replacement) from a sample are
heterozygous, if we did not pick the same individual twice and the
resulting two different lines are heterozygous at both loci.

\medskip

In the next two steps we compute how to find $f_0,g_0$ and $h_0$ given $f_T,g_T$ and $h_T$, respectively.\\

\medskip

\noindent$\textrm{\textbf{ Step 3}}$ : From $f_{T/2},g_{T/2}$ and $h_{T/2}$ to $f_0,g_0$ and $h_0$, respectively \\

\medskip

For both geometries we have
 $$\left(\begin{array}{cc}
	f_0 \\
	g_0 \\
	h_0
	\end{array}\right) = C  \left(\begin{array}{cc}
	f_{T/2} \\
	g_{T/2} \\
	h_{T/2}
	\end{array}\right)$$ with
$$C =  \left( \begin{array}{clcr}
               {p_{LR}}^2 & 2p_{LR}(1-p_{LR})    & (1-p_{LR})^2      \\
               0  & p_{LR}   & 1-p_{LR}   \\
               0  & 0 & 1
          \end{array}   \right)$$

Our model assumptions coincide with the model assumptions of \citet{PfaffelhuberLehnertStephan2008} in the time interval $[T,T/2)$, so that we obtain the same results here. \\

\medskip

\noindent$\textrm{\textbf{ Step 4}}$ : From $f_{T},g_{T}$ and $h_{T}$ to  $f_{T/2},g_{T/2}$ and $h_{T/2}$, respectively\\

\medskip

For this time step it is important to note, that it has to be paid attention not only on the two neutral loci, but also
on the selected locus. We use Ewens sampling formula (see Equation \ref{probfounder}) to compute the probabilities, if the ancestral lines of the pairs
 share with respect to the selected locus a
common ancestors or different ancestors. With this we get the following relationships:

 $$\left(\begin{array}{cc}
	f_{T/2} \\
	g_{T/2} \\
	h_{T/2}
	\end{array}\right) = A  \left(\begin{array}{cc}
	f_{T} \\
	g_{T} \\
	h_{T}
	\end{array}\right) \qquad \textrm{ for geometry (a) } $$
and

 $$\left(\begin{array}{cc}
	f_{T/2} \\
	g_{T/2} \\
	h_{T/2}
	\end{array}\right) = B  \left(\begin{array}{cc}
	f_{T} \\
	g_{T} \\
	h_{T}
	\end{array}\right) \qquad
\textrm{ for geometry (b) }$$

with matrix $$A= (a_{ij})_{1\leq i,j\leq 3}$$  given by

\begin{align*}a_{11} & = \frac{\theta_s}{\theta_s +1} p_{LR}^2 + \frac{1}{\theta_s +1}(1-p_{SL}^2)p_{LR}^2  \\
a_{12} & = \frac{\theta_s}{\theta_s +1} 2p_{LR}(1-p_{LR})+ \frac{1}{\theta_s +1} 2 (1-p_{SL}^2)p_{LR} (1-p_{LR}) \\
a_{13} & = \frac{\theta_s}{\theta_s +1} (1-p_{LR})^2  + \frac{1}{\theta_s +1} (1-p_{SL}^2)(1-p_{LR})^2\\
a_{21} & = \frac{\theta_s}{(\theta_s+1)(\theta_s+2)} p_{SL}p_{SR}p_{LR} + \frac{2}{(\theta_s+1)(\theta_s+2)} (1-p_{SL}) p_{SL}^2 p_{LR}^2 \\
a_{22} & = \frac{{\theta_s}^2}{(\theta_s +1)(\theta_s +2)} p_{LR} + \frac{\theta_s}{(\theta_s +1)(\theta_s+2)}(3p_{LR} - 3 p_{SR} p_{SR}) + \\
&\quad + \frac{2}{(\theta_s+2)(\theta_s+1)} p_{LR}(1-p_{SL})(1+p_{SL}-4p_{SL}p_{SR} + 2p_{SL}^2) \\
a_{23} & = \frac{\theta_s^2}{(\theta_s +1)(\theta_s +2)}(1-p_{LR}) + \\
& \quad + \frac{\theta_s}{(\theta_s+1)(\theta_s+2)} (1-p_{LR})(1-p_{SL}^2 + 1- p_{SL}p_{SR}+ 1- p_{SL}p_{SR}) +\\
& \quad + \frac{2}{(\theta_s+1)(\theta_s+2)} (1-p_{SL})(1-p_{LR})(1+p_{SL}- 2p_{SL}p_{SR}) \\
a_{31} & =  \frac{2\theta_s}{(\theta_s+1)(\theta_s+2)(\theta_s+3)} p_{SL}^2 p_{SR}^2  \\
a_{32} & = \frac{ 4 \theta_s^2}{(\theta_s+1)(\theta_s+2)(\theta_s+3)} p_{SL}p_{SR}  + \\
& \quad + \frac{2\theta_s}{(\theta_s+1)(\theta_s+2)(\theta_s+3)}(p_{SR}p_{SL}(1-p_{SR}p_{SL})) + \\
& \quad + \frac{4 \theta_s}{(\theta_s+1)(\theta_s+2)(\theta_s+3)} p_{SL}p_{SR}(2-p_{SL}-p_{SR})+ \\
& \quad + \frac{3!}{(\theta_s+1)(\theta_s+2)(\theta_s+3)} (4 (1-p_{SL})(p_{SL}(1-p_{SR})p_{SR}))\\
a_{33} & =  \frac{ \theta_s^3 }{(\theta_s + 1)(\theta_s + 2)(\theta_s +3)} + \\
& \quad + \frac{\theta_s^2}{(\theta_s + 1)(\theta_s + 2)(\theta_s + 3)} (4(1 - p_{SL}p_{SR}) + (1 - p_{SL}p_{SL}) + (1 - p_{SR}p_{SR})) + \\
          & \quad + \frac{4\theta_s}{(\theta_s + 1)(\theta_s + 2)(\theta_s + 3)}((1 -p_{SR})(1-p_{SL})(1-p_{SR})+ (1 - p_{SR})(1 -p_{SR})p_{SL}) + \\
& \quad + \frac{4\theta_s}{(\theta_s + 1)(\theta_s + 2)(\theta_s + 3)}( 2(1-p_{SR})(1-p_{SL})p_{SR} + (1-p_{SL})(1-p_{SL})p_{SR})+\\
&\quad  + \frac{4\theta_s}{(\theta_s + 1)(\theta_s + 2)(\theta_s + 3)} ( 2(1-p_{SR})p_{SL}(1-p_{SL}) + (1-p_{SL})(1-p_{SL})(1-p_{SR})) + \\
& \quad + \frac{\theta_s}{(\theta_s + 1)(\theta_s + 2)(\theta_s + 3)}(2(1 - p_{SL}p_{SR})(1 - p_{SL}p_{SR}) +  (1 - p_{SL}p_{SL})(1 - p_{SR}p_{SR}) ) + \\
& \quad +  \frac{6}{(\theta_s + 1)(\theta_s + 2)(\theta_s + 3)}((1 - p_{SL})(1 - p_{SR})(1 +  p_{SL} + p_{SR} - 3p_{SL}p_{SR}))
\end{align*}

and matrix $$B= (b_{ij})_{1\leq i,j \leq 3}$$ given by

\begin{align*}
b_{11} & =  \frac{\theta_s}{\theta_s+1} p_{LS}^2 p_{SR}^2 \\
b_{12} & =\frac{2\theta_s}{\theta_s+1} p_{SR} p_{LS}(1-p_{LS}p_{SR}) + \frac{4}{\theta_s+1} p_{LS}(1-p_{LS})p_{SR} (1-p_{SR}) \\
b_{13} & =  \frac{\theta_s}{\theta_s+1} (1-p_{LS}p_{SR})^2 + \\
& \quad + \frac{1}{\theta_s+1} (1-p_{LS})(1-p_{SR})(2p_{LS}(1-p_{SR})+2(1-p_{LS})p_{SR} +(1-p_{LS})(1-p_{SR})) \\
b_{21} & = \frac{\theta_s}{(\theta_s+1)(\theta_s+2)}p_{LS}^2p_{SR}^2 \\
b_{22} & =  \frac{\theta_s^2}{(\theta_s+1)(\theta_s+2)} p_{LS} p_{SR} + \\
& \quad + \frac{\theta_s}{(\theta_s+1)(\theta_s+2)}(p_{LS}p_{SR} (1-p_{LS}p_{SR})  +  (2-p_{SR} -p_{LS})p_{SR}p_{LS} + p_{SR} p_{LS}(1-p_{LS}p_{SR})) + \\
& \quad + \frac{8}{(\theta_s+1)(\theta_s+2)} p_{LS}(1-p_{LS})p_{SR} (1-p_{SR})\\
b_{23} & =  \frac{\theta_s^2}{(\theta_s+1)(\theta_s+2)}(1-p_{SR}p_{LS}) + \\
& \quad + \frac{\theta_s}{(\theta_s+1)(\theta_s+2)}((1-p_{LS})^2(1-p_{SR}) + 2p_{LS}(1-p_{LS})(1-p_{SR})) + \\
& \quad + \frac{\theta_s}{(\theta_s+1)(\theta_s+2)}( p_{SR}(1-p_{LS})^2 + (1-p_{LS}p_{SR})^2 + (1-p_{SR})^2(1-p_{LS})) + \\
& \quad + \frac{\theta_s}{(\theta_s+1)(\theta_s+2)}(  2 p_{SR}(1-p_{SR})(1-p_{LS})+ p_{LS}(1-p_{SR})^2) \\
& \quad + \frac{2}{(\theta_s+1)(\theta_s+2)} (1-p_{LS})(1-p_{SR})(2p_{LS}(1-p_{SR}))\\
& \quad + \frac{2}{(\theta_s+1)(\theta_s+2)}(2(1-p_{LS})p_{SR} +(1-p_{LS})(1-p_{SR}))\\
b_{31} & = a_{31} \qquad \textrm{ with } p_{LS} \textrm{ instead of } p_{SL}  \\
b_{32} & = a_{32} \qquad \textrm{ with } p_{LS} \textrm{ instead of } p_{SL}  \\
b_{33} & = a_{33} \qquad \textrm{ with } p_{LS} \textrm{ instead of } p_{SL}
\end{align*}

To see the above equations, consider for example in geometry (a) the term $a_{21}$:
 In this case at time $T/2$ there are two pairs, which are heterozygous in both
loci and exactly one of the pairs is linked. If the two pairs have two different ancestors
 with respect to the selected locus neither a $SL$-recombination nor a $SR$-recombination must happen for the unlinked pair, nor
 a $LR$-recombination event to the linked pair. Therefore the probability to stay linked also at the beginning of the sweep is
 $\frac{\theta_s}{(\theta_s+1)(\theta_s+2)} p_{SL}p_{SR}p_{LR}$ using Ewens sampling
 formula.
 If the two pairs have a single ancestor, the linked pair has to change backgrounds,
i.e. a $SL$-recombination event has to take place. Therefore we obtain in this case the probability $\frac{2}{(\theta_s+1)(\theta_s+2)} (1-p_{SL}) p_{SL}^2 p_{LR}^2$.
 The sum of these two probabilities gives $a_{21}$.
The other terms can be explained in an analogous manner.\\

\medskip

\noindent$\textrm{\textbf{ Step 5}}$ : Collecting all together

\medskip

We have

$$(\widehat{\mathcal{X}_0}, \widehat{\mathcal{Y}_0}, \widehat{\mathcal{Z}_0})^{T} =E \cdot  F \cdot C\cdot A \cdot E^{-1} (\mathcal{X}_T, \mathcal{Y}_T, \mathcal{Z}_T)^{T}$$
for geometry (a) and

$$(\widehat{\mathcal{X}_0}, \widehat{\mathcal{Y}_0},\widehat{ \mathcal{Z}_0})^{T} = E \cdot F \cdot C\cdot B \cdot E^{-1}
 (\mathcal{X}_T, \mathcal{Y}_T, \mathcal{Z}_T)^{T} $$
for geometry (b).

With this we can compute $\widehat{\sigma^2_D} = \widehat{\mathcal{Z}_0}/ \widehat{\mathcal{X}_0}$ (recalling
 Equation (\ref{sigma})).

A calculation with Mathematica gives Equations (\ref{big1})-(\ref{big4}), if terms of order $\theta_s^2$ and $1/n$ are ignored.
 $\square$

\subsubsection*{Acknowledgement}
I am grateful to Peter Pfaffelhuber for many fruitful discussions.
 Many thanks to Greg Ewing for providing and helping me with msms and to Franz Baumdicker and Joachim Hermisson for helpful comments on the manuscript.
I acknowledge support from the DFG Forschergruppe 1078 "Natural
selection in structured populations".

redpajama_set_name: RedPajamaArXiv


I was somewhat incredulous that we were flying all the way to the other side of the world to adopt a child, since we had many irons in the fire in the United States. Adopting a baby from our own hometown would have been just fine with us. But God apparently wanted us to go to Russia, because none of these other things worked out.
A driver and an interpreter picked us up at the airport and dropped us at our hotel an hour later They would be back tomorrow to take us to a smaller airport for our flight to Izhevsk.
Although we just wanted to go to sleep, as it was about 2am our time, they say when you fly overseas you must stay awake until bedtime in your new time zone or you will have a much harder time acclimating. So we headed to Red Square, the Kremlin and St. Basil's Cathedral.
It was spectacular, but my memory is that I just wanted to lie down somewhere. We stayed awake until about 4pm and then couldn't stave off sleep any longer. We woke at 2am and slowly got ready for the trip to Izhevsk.
It was a 2 hour small plane ride strait east of Moscow, right before hitting the Ural Mountains.
The flight was fine. We were in a TY 134 like a small DC-9. Izhevsk is wooded with some fields. The grassland looks like Oklahoma, but the soil is pitch black. The airport looked like Ponca city's – small, but there were a good number of jets.
There is Matt in the lower left corner.
An interpreter met us and got us to our hotel. It is set in an old apartment, but the entry is an added-on building they renovated.
Our hotel, or motel, or whatever it was is behind us.
There are tons of these 8-10 story apartment blocks. They are awful and no one mows the grass. To us it looks like the slums, but we walk around and feel perfectly safe.
We have been working on speaking Russian. The locals appreciate it and light up when we speak it. We've been into town on two trips. The first was to a church to the west of Udmurt University.
Allison got the priest to let us take photos.
I lit a candle for the boys.
Wow. It is a green church. What a use of color!
Women had to wear head coverings so I used my sweater. Nice.
Next trip we went to the town square or rather town center. We got there late and had to leave because I thought the sun was setting – oops – it goes down about midnight. A very nice town center. Don't know if we get to see Alexander today. We have to be interviewed by the Minister of Education at 2pm. It is all very "go with the flow." Lots of waiting.
The next day a driver/interpreter picked us up at our small motel. First we had to go do some paperwork at the agency.
Ok…Looks like the beginning of a scary movie. These are the hallways to the agency office. It did get better.
Then our guide drove us (like a maniac I might add) 45 minutes to a smaller town north of Izhevsk called Votkinsk. We were literally out in the middle of rural Russia far from any known reality, completely dependent on our driver. What a deal. What an experience. What the heck are we doing?
We pull up to a very rundown building and are told to go through "that" door. Ok, whatever you say.
We sort of tentatively walk in and someone points to another door. We walk in that door and are greeted by the director, the interpreter, and the social worker. We are all crowded in a tiny office and they proceed to ask us questions and interview us. Then we ask a few questions.
Finally they bring Alexander, 15 months old, to us. He is a darling little boy. We play with him for maybe 30 minutes in the directors office. We ask a few more questions and then it's time to go.
I sit in the back seat and cover my eyes as this guy drives. Matt is in the front seat. (I feel sorry for him.) We didn't have time to talk. Cars are honking constantly, inches from each other, going 70 miles an hour, swerving in, out and around. We are each thinking separately about what we just experienced and praying that we make it back to our motel alive.
Our driver let us out and said he would pick us up at 8am to go back to the orphanage for a second visit. Oh, ok. Well, that is good. We can see him again. When we finally got to talk, we both were a little unsure. It was all very intense. The orphanage was dark and depressing. We didn't talk much about it though because we knew we were going back tomorrow and we would get a better idea and then really talk.
Because of your unfailing love, I can enter your house; with deepest awe I will worship at your temple. Lead me in the right path O Lord… Tell me clearly what to do and show me which way to turn. Psalm 5:7-8.
The next morning we head back to Votkinsk. Another hair-raising drive. This time they take us to another room with a few toys and then bring Alexander to us. We spend about 30 minutes with him. Again he is a precious boy, but I can sense that both of us feel there is something not right about this.
We had spoken to our agency back in Texas the night before and Doris could tell we were unsure. She told us to go ahead and tell them yes and take our time to think about it and then we could, as she said, "deny the referrel" later, but if we said no now, it would be difficult to change our minds later. Ok. We are just going with the flow here.
So we tell the orphanage that we are interested and our agency would work out the details. We get back into the car for the final (thank you God) 45-minute trip back to Izhevsk. Both Matt and I are in a strange space. Again, we can't talk now, but must sit with our thoughts alone during the drive back. Probably a good thing.
By the time we get back we both say to each other "this is not right." We aren't sure why. We just know we are not meant to adopt this boy. It is breaking my heart to leave this child here, but we've just got to be obedient to the leaning of the Holy Spirit.
We have prayed deeply and we believe You, Father, have not called us to bring Alexander home with us. We absolutely believe if we sense Your will like we do, that his (Alexander's) forever family is soon coming. We must be obedient, even if it is difficult. We will continue to pray for him. Please bless him Jesus.
Paperchase – What documents are necessary to adopt?


Music Producer, Best Seller Author, Yoga & Reiki Master, Certified Health & Wellness Educator, and International DJ, Bibba Pacheco holds one of the most extraordinary careers that an artist of to the electronic music segment can achieve. Her exceptional talent comes with a brilliant academic background: Bachelor of Social Communication, Advertising/Marketing, Degree in Film/ Cinematography/ VideoProduction/USA, Science of Psychology Coaching/Harvard University/USA, Yoga Instructor/Chopra University/USA, Music Theory/Classical Piano and Electronic Music Production – AIMEC/BR. This explosive confluence of predicates certainly would be enough for the artist to develop worldwide prominence. Bibba Pacheco has performed with grace all over the world. And the exhibition of her success has made the brand *DJBIBBAPACHECO known for its contemporary elegance, often suggested to events of the major national and international brands. The musician has performed at all clubs and music festivals around Brazil and played for over 100 million viewers as a guest of Globo, the 2nd world's largest TV network. These are the attributes that guided her tours in Europe, Latin America and United States, followed by her gigs at the Winter Music Conference, the largest conference of DJs in the world.
Being a national and international reference is a natural consequence for who turned music a way of living. In tuned with musical trends, Bibba Pacheco writes, composes, produces, sings her tracks, commercialized worldwide. The soundtrack of her life is guided by a fundamental principle: social responsibility, part of her music profits are donated to charities.
Citizen of the world, she has lived in Sydney, London, Miami and Barcelona, always improving her studies and works in music, yoga, psychology, health and well-being, the jetsetter deepens herself in every culture of the countries she visits, in order to prepare an exclusive set for each show. Bibba Pacheco captivates crowds through her positive energy, melodic uplifting productions, that makes her appearances remarkable on her successful path.
Produtora Musical, Autora Best Seller do livro "O Poder da Alegria", Mestre de Yoga/Reiki, Educadora Certificada de Bem Estar & Saúde, e DJ Internacional, Bibba Pacheco tem uma das mais extraordinárias carreiras que uma artista ligada ao segmento da música eletrônica pode alcançar. Seu talento excepcional vem de sua formação acadêmica brilhante: Bacharel em Comunicação Social/Publicidade e Propaganda, pós-graduada em Cinematography/Film/EUA, com diplomas em Psychology Coach pela Harvard University/EUA, Mestre de Yoga pela Chopra University/EUA, Teoria da Música/Piano Clássico e Produção de Música Eletrônica – AIMEC. Essa explosiva confluência de predicados certamente já seria suficiente para que a produtora musical ganhasse destaque mundial.
A exposição do seu sucesso tornou a marca *DJBIBBAPACHECO uma categoria conhecida por ser sinônimo de elegância contemporânea, sendo frequentemente sugerida para eventos de grandes marcas nacionais e internacionais. A bela já se apresentou em todos os clubs espalhados de norte a sul do país, com DJs como David Guetta, Tiesto e para 100 milhões de telespectadores como convidada da Rede Globo. Realizou turnês pela Europa, América Latina, e tocou na Winter Music Conference nos Estados Unidos, maior conferência de DJs do mundo.
Ser referência nacional e internacional é, em se tratando de Bibba Pacheco, consequência natural para quem fez da música sua razão de ser. Antenadíssima com as tendências musicais em sua seara de atuação, a produtora musical escreve e canta suas tracks, reunidas em compilações/CDs comercializados mundialmente. A trilha sonora de sua vida é embalada por um princípio essencial: o da responsabilidade social, tanto que parte da sua renda é revertida para entidades carentes.
Cidadã do mundo, com o conhecimento de quem já morou em Sydney, Londres, Miami e Barcelona, sempre aprimorando seus estudos e trabalhos sobre música, yoga, psicologia, comportamento humano, saúde e bem-estar, a jetsetter busca mergulhar na cultura de cada país que visita, além de preparar um set exclusivo para cada show. Bibba Pacheco cativa multidões através da sua energia positiva e produções melódicas, tornando cada aparição marcante em sua trajetória de sucesso.
Musician, Writer, Yoga and Reiki Master, Health/Wellness Educator, and International DJ, Bibba Pacheco holds one of the most extraordinary careers that an artist connected to the segment of electronic music can achieve. Her exceptional talent comes with a brilliant academic background: Bachelor of Social Communication/Advertising, post-graduated in Film/Cinematography/USA, with degrees in Psychology Coach/Harvard University/USA, Yoga Instructor/Chopra University/USA, Music Theory/Classical Piano and Electronic Music Production – AIMEC.
This explosive confluence of predicates certainly would be enough for the artist to develop worldwide prominence. Bibba Pacheco has performed with grace all over the globe. And the exhibition of her success has made the brand *DJBIBBAPACHECO a category known for its contemporary elegance, often suggested to events of major national and international brands.
The musician has performed at all the super clubs from north to south of Brazil and played for over 100 million viewers as a guest of Globo, the 2nd world´s largest TV network. These are the attributes that guided her tours in Europe, Latin America and United States, followed by her gigs at the Winter Music Conference, the largest conference of DJs in the world.
Being a national and international reference is natural consequence for who turned music a way of living. In tuned with musical trends, Bibba Pacheco writes, composes, produces, sings her tracks, commercialized worldwide. The soundtrack of her life is guided by a fundamental principle: social responsibility, part of the proceeds of her music are donated to charities.
Citizen of the world, with the knowledge of who has lived in Sydney, London, Miami and Barcelona, always improving her studies and works in music, yoga, psychology, health and well-being, the jetsetter deepens herself in every culture of countries she visits, in order to prepare an exclusive set for each show. Bibba Pacheco captivates crowds through her positive energy and melodic productions, that makes her appearances remarkable on her successful path.


Location The hotel is situated in the business district, close to White Marsh Mall. Guests are also able to visit the Baltimore Aquarium, Oriole Park at Camden Yards and the M&T Bank Stadium, each of which is roughly 20 km away from the hotel. Regional attractions also include Baltimore Museum of Art and Fort McHenry. The American Visionary Art Museum is about 12 km away. Baltimore/Washington International Thurgood Marshall Airport is approximately 25 km away from the hotel. Facilities The air-conditioned hotel comprises a total of 131 rooms. Individual storeys are accessible via the lift or stairs. Amenities include a cloakroom and a safe. Wireless internet access in public areas allows guests to stay connected. Culinary facilities include a breakfast room. Additional facilities at the accommodation include a newspaper stand. Those arriving in their own vehicles can leave them in the car park of the establishment. Available services and facilities include room service and a laundry service. Rooms Each of the rooms is appointed with air conditioning, central heating and a bathroom. The carpeted rooms feature a double bed or a king-size bed. There is also a desk. A fridge, a microwave and a tea/coffee station ensure a comfortable stay. An ironing set is also available for travellers' convenience. Internet access, a telephone, a TV, a radio and WiFi add to the comfort of the holiday. Some of the rooms are adapted for wheelchair users. The bathrooms are fitted with a shower and a bathtub. A hairdryer is also available. The hotel offers family rooms and non-smoking rooms. Sports/Entertainment The accommodation offers attractions including sport and entertainment opportunities. The establishment features an outdoor pool. Sun loungers provide a great place to sunbathe. A range of options are available, including a gym, a spa and a solarium. Meals Bed and breakfast is bookable. Options include breakfast. Payment The following credit cards are accepted at the hotel: American Express, VISA and MasterCard.
The hotel is situated in the business district, close to White Marsh Mall. Guests are also able to visit the Baltimore Aquarium, Oriole Park at Camden Yards and the M&T Bank Stadium, each of which is roughly 20 km away from the hotel. Regional attractions also include Baltimore Museum of Art and Fort McHenry. The American Visionary Art Museum is about 12 km away. Baltimore/Washington International Thurgood Marshall Airport is approximately 25 km away from the hotel.


text:
Petanque World Championships Single in Nice
©FIPJP
Petanque World Championships Single Women and Men poster
The City of Nice will host the first ever Petanque World Championships Single Women and Men from January 22 to 25, 2015, a great première.
A great première in the history of the Petanque: from January 23 to 25, 2015, women and men players from 50 nations from 5 continents will compete for the ultimate title of World Champion Single in Nice at the Municipal Boulodromes Gé Gallaratto and Henri Bernard.
Nice loves La Petanque
Since 2008, Christian Estrosi, MP-Mayor of Nice, President of the Nice Côte d'Azur Metrople, has worked to make of Nice a leading destination for hosting major sporting events, generating activity and jobs. This strategy continues to bear fruit, since, after the success of the 1st International Single Trophy Henri Bernard (named after the former President), the International Federation of Pétanque and Provençal Game decided to create the World Championships Single Women and Men in Nice.
Nice, which has 76 bowling associations, over 5,000 licensees and 72 bowls pitches distributed in the neighborhoods of the city, has become one of the capitals of Petanque.
Port de Nice ©DR
Christian Estrosi's commitment
The city of Nice deserved more than any other, this new competition. First, because the President Henri Bernard is a child of Nice and what better tribute than creating a new competition in his home, and moreover, at the boulodrome bearing his name. But also, for the involvement of the Municipality and its MP-Mayor Christian Estrosi, who is supporting all sports and especially the sports of bowling.
Thus, at his initiative, in 2015 will take place the European Masters Games, which are sort of the "Veterans" Olympics, which will bring together 27 disciplines including the Pétanque.
La Petanque
In 2007, La Petanque celebrated its 100 years. With the values of the Provençal Game: atmosphere, team spirit, emotion and pleasure, the Petanque set out to conquer the world. Today, the Petanque is the most popular bowls sport with some 600,000 licensees in 94 national federations on 5 continents and millions of fans who are playing for fun.
The FIPJP (International Federation of Pétanque and Provençal Game) is a worldwide umbrella organization of all the Petanque and Provençal game licensees – two nearby sports that have a common modern history.
Port Lympia
Next, the Olympics
Through the work of FIPJP, the Petanque takes part in events like the World Games, the Asian Games, the Mediterranean Games, the Pacific Games. In 1986, the Petanque has been recognized by the International Olympic Committee (IOC) and the FIPJP is a founding member of the World Confederation of Boules Sports (CMSB), where she is striving, with other bowling sports, to achieve the admission of the Petanque in the Olympic program.
Petanque World Championships Single Women and Men
Nice – January 23 to 25, 2015
Random draw
On January 15, 2015, Christian Estrosi, MP-Mayor of Nice, President of the Nice Côte d'Azur Metrople, presented the competition at the Villa Massena. He was in company of Claude Azema, the President of the Fédération Internationale de Pétanque et de Jeu Provençal (FIPJP) and Stéphane Pintus, President of the Organization Committee. After offering them a set of 3 boules engraved with the shield of arms of the City of Nice, Christian Estrosi then drew lots to select the first table of parties.
Claude Azema, Christian Estrosi & Stéphane Pintus
fipjp-cdm.com
Click on pictures to enlarge – ©YesICannes.com – All rights reserved
Stéphane Pintus
Claude Azema
Christian Estrosi
Christian Estrosi offering a set of 3 boules
Set of boules with arms of Nice
Petanque World Championships Single in Nice was last modified: January 19th, 2015 by tamel
Posted in: Sports
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redpajama_set_name: RedPajamaCommonCrawl


text:
NDC promises 3 million jobs in Mahama's second term
Sun, 20 Nov 2016 Source: 3 News
The National Democratic Congress is promising to invest a whooping GHC13 billion in 184 projects across the country if given the chance for another four-year term.
Deputy Minister of Communication, Ato Sarpong, who announced this, explained that the investment would create 3 million jobs within the four-year period.
Addressing a Job Summit in Kumasi, he said young people would have to upgrade their skills and knowledge to be positioned to grab the job opportunities.
The job summit provided a platform for participants to know the existing job opportunities in the country and what the NDC government intends to do in the next four years to create more jobs.
Mr Sarpong said the government would create the 3 million jobs through investment in infrastructure and the revival of strategic state-owned enterprises.
"The government has planned to invest GHc 13 billion to create 3 million jobs through the construction of roads, schools, hospitals, and domestic revival of strategic state owned enterprises".
Mr. Sarpong noted that the job opportunities created by the government would be for those who position themselves to be gainfully employed.
"The government will create the jobs but it is only those who are prepared and have upgraded their skills and knowledge are the ones who are going to benefit from the job opportunities," he pointed out.
Chief of Staff, Julius Debrah advised the youth to have faith in themselves and capitalise on their potentials.
He said though the government is committed to providing jobs to the citizenry, the youth must adopt entrepreneurial skills to enable them own businesses.
A former deputy minister for Western Region, Emilia Arthur, highlighted opportunities under the Youth Enterprise Support (YES).
She asked the unemployed youth to take advantage of the programme to get jobs and further upgrade their skills.
Meanwhile, the chief of staff and his team interacted with Zongo chiefs and imams at Abuakwa in the Atwima Nwabiagya South constituency.
Julius Debrah called on the constituents to vote and retain President Mahama to promote the welfare of Zongo communities.
The Krontihene of Abuakwa Zongo, Alhaji Nuhu Alhassan Dickleburgh assured the chief of staff of their votes.
Julius Debrah introduced the NDC parliamentary candidate for Atwima Nwabiagya South constituency, Nana Asare Bediako, to the people.
Source: 3 News


text:
Computing explanations for interactive constraint-based systems
Name: PapadopoulosA_PhD ...
Description: Full text available
Papadopoulos, Alexandre
Copyright: © 2011, Alexandre Papadopoulos
Related: http://library.ucc.ie/record=b2028173~S0
Citation: Papadopoulos, A. 2011. Computing explanations for interactive constraint-based systems. PhD Thesis, University College Cork.
Supervisor(s): O'Sullivan, Barry
Constraint programming has emerged as a successful paradigm for modelling combinatorial problems arising from practical situations. In many of those situations, we are not provided with an immutable set of constraints. Instead, a user will modify his requirements, in an interactive fashion, until he is satisfied with a solution. Examples of such applications include, amongst others, model-based diagnosis, expert systems, product configurators. The system he interacts with must be able to assist him by showing the consequences of his requirements. Explanations are the ideal tool for providing this assistance. However, existing notions of explanations fail to provide sufficient information. We define new forms of explanations that aim to be more informative. Even if explanation generation is a very hard task, in the applications we consider, we must manage to provide a satisfactory level of interactivity and, therefore, we cannot afford long computational times. We introduce the concept of representative sets of relaxations, a compact set of relaxations that shows the user at least one way to satisfy each of his requirements and at least one way to relax them, and present an algorithm that efficiently computes such sets. We introduce the concept of most soluble relaxations, maximising the number of products they allow. We present algorithms to compute such relaxations in times compatible with interactivity, achieving this by indifferently making use of different types of compiled representations. We propose to generalise the concept of prime implicates to constraint problems with the concept of domain consequences, and suggest to generate them as a compilation strategy. This sets a new approach in compilation, and allows to address explanation-related queries in an efficient way. We define ordered automata to compactly represent large sets of domain consequences, in an orthogonal way from existing compilation techniques that represent large sets of solutions.
Computer Science - Doctoral Theses [63]
Except where otherwise noted, this item's license is described as © 2011, Alexandre Papadopoulos