text:
\section{Introduction}
Interference coordination has re-emerged as a fundamental challenge
for next-generation cellular wireless systems. Traditional
macrocellular deployments are likely to be supplemented with smaller
femtocells and relays, with mixtures of restricted and open access,
often deployed in an ad hoc manner \cite{ChaAndG:08,LopezVRZ:09}.
Such deployments may create much stronger and highly variable (in
time and space) interference conditions than those experienced in
current macrocellular networks, and traditional cellular power and
rate control may not be adequate \cite{FemtoForum:10}. To address
this challenge, a key focus of the current 3GPP LTE-Advanced
standardization efforts is on the design of an interference
coordination framework for such unplanned cellular deployments of
base-stations with widely different transmission
powers~\cite{3GPPICIC}. Current release of the LTE
specification~\cite{3GPP36.300} provides simple methods for inter-cell
interference coordination (ICIC), see e.g.,~\cite{fodor_2009}. Along
with the design of mechanisms for ICIC, algorithms to exploit such
mechanisms are an active area of research as well, see for
example,~\cite{kang_2008},~\cite{sundaresan_2008}.

Mathematically, interference coordination is a complex distributed
optimization problem involving scheduling decisions at the
transmitters of multiple interfering links.
In this work, we consider a general \emph{linear mixing}
interference model where the scheduling decisions
in each link are represented as vector (e.g., transmission
powers on different sub-bands in frequency, beamforming weights) and
the interference on each link is a linear
combination of the scheduling vectors on the other links.
Associated with each link at a given time is a \emph{utility function} which
describes the benefit to a link as a function of the scheduling
vector from the serving transmitter and interference from the other
transmitters.  The linear mixing model is extremely general
and can apply to a large class of interference models and objectives.
Computing maximum weighted matching for queue
stability~\cite{TassiulasE:92} and maximization of sum utility of
average rates for fairness~\cite{stolyar_2005} are special cases of
this formulation.

In the past few years, algorithms for special cases of the above
problem have been extensively studied. In many cases, algorithms
with provable desired properties have been obtained -- for example,
there is a rich literature on distributed power control methods to
achieve a desired SINR for each link~\cite{chiang_2008} and to
maximize a certain class of utility functions of
SINRs~\cite{huang_2006}, approximation algorithms for maximum weight
matching for combinatorial interference model were obtained
in~\cite{sharma_2007, gupta_2007}, efficient methods to compute
optimal beamforming vector for multiuser downlink~\cite{TanCS:2010}
and maximizing sum utility of SINRs on uplink~\cite{tan_isit},
stabilizing policies for collision sense multiple access (CSMA) type
of models based on simulated annealing were derived
in~\cite{rajagopalan_2009}. More generally, heuristic algorithms
have been constructed to solve certain specific problems
approximately in, for example,~\cite{StolyarVis:2009,madan_2010}.
While these algorithms perform well in practice in spite of no
provable guarantees, the insights and approximations used to obtain
these algorithms are very specific to the problem under
consideration.

In this paper, we make the following contributions:

\noindent\emph{BP Framework:} We consider a belief propagation (BP)
framework for a very
general wireless scheduling and interference coordination
optimization problem. The underlying optimization problem is posed
as a problem of estimating marginals of a joint probability
distribution. BP provides a systematic and general approach to
obtain distributed algorithms; it can be used with arbitrary
nonlinear utility functions and scheduling vectors sets, which
enable the algorithm to be applied a range of complex scheduling
problems including power control, subband scheduling and distributed
beamforming.
Also, while we do not obtain any theoretical guarantees, in
practice a few iterations of BP generate a good operating point.
Thus, it typically has faster convergence than gradient based
algorithms (e.g.,10s of iterations in~\cite{tan_isit}) or simulated
annealing~\cite{Hajek:88}.

\noindent \emph{Approximation Algorithms:} It is well known that
implementing BP for distributed optimization problems entails high
computational complexity and communication overhead.
Exploiting the linear mixing interference model and applying
Gaussian  and first-order approximations
similar to~\cite{BouCaire:02,MontanariT:06,GuoW:06,BayatiM:10arxiv,Rangan:10arXiv},
we develop an approximate BP method that has low complexity,
distributed implementation and minimal messaging.
Along many links, messages can
be carried in small payloads and can be broadcast without separate
unicast transmissions, which is particularly crucial for wireless
systems.  Moreover, the resulting algorithm has a natural interpretation
has a ``soft" RTS / CTS type handshaking.  The approximate BP algorithm
is also similar to the recent approximate message passing (AMP) algorithm
in \cite{DonohoMM:09arxiv,BayatiM:10arxiv} and this connection may be useful
for further analysis.

\noindent \emph{Numerical Results:} Through simulations for
femtocell deployments, we demonstrate that approximate BP provides
good performance for sub-band and power allocation to maximize
utilities of rates, sub-band and power allocation to maximize a
weighted sum of rates, and beamforming optimization to maximize
utilities of average rates. Also, although the BP algorithm requires
multiple exchange of messages before each scheduling decision, our
simulations indicate good performance with only two rounds of
messaging. Thus, the approximate BP approach is a promising paradigm
for new emerging cellular deployments with large interference
variations in time and space compared to current predominantly
macro-only deployments.

\subsection{Previous Work}

Algorithms for ICIC in LTE macrocells have been considered in a
large number of works in both the uplink and downlink
\cite{RahmanYW:09,FodorKRRSM:09,XiangLH:07}. These works are
generally based on adaptive subband scheduling and
fractional frequency reuse (FFR) methods
\cite{HanPLAJ:08,StolyarVis:2009} and exploit statistics over large
numbers of mobiles per macrocell.
Interference mitigation in femtocells has focussed on similar techniques as well
as frequency planning, power control~\cite{chandra_2008}, or
semi-static resource allocation~\cite{kang_2008,humblet_2009,sundaresan_2008}.
As we will demonstrate
in the simulations, the BP methods presented here can also be used
for adaptive subband scheduling as a special case of the linear mixing interference
model.
Also, much of the ICIC work has considered slowly varying allocations
that don't change over few 10s to 100s of milliseconds.
Due to the low messaging overhead, it is possible that approximate BP
can also be used for more dynamic interference
management in femtocell deployments, where there is high variability
in load and interference from one timeslot to another.

For scheduling based on more dynamic traffic statistics such as
queue lengths and head-of-line delays, variants of maximum weight
scheduling can be used~\cite{Tassiulas:98},\cite{shakkottai_2001}.
Unfortunately, computing a maximum weight schedule is generally
NP-hard, and much work has thus focused on approximate algorithms.
In addition to the works mentioned above,
the works \cite{Tassiulas:98} and \cite{GiacconePS:03} proposed
randomized linear complexity (but centralized) algorithms, and
\cite{JiangWal:08,RajagopalanSS:09,NiSrikant:09} present simple distributed
algorithms for combinatorial interference models. Greedy maximal
weight matching for such interference models has been considered
in~\cite{DimakisW:06,JooLinShroff:09,LeconteNS:09}.

However, many of the above works apply a hard constraint
interference models, where neighboring links cannot transmit
simultaneously. Cellular systems in contrast permit multiple
interfering links to transmit simultaneously and then use rate
control to adapt to the resulting signal-to-interference and noise
ratio (SINR). Thus, the degradation in rate with interference is
gradual, and are difficult to capture in the combinatorial
interference model. In contrast, ``soft" interference effects can be
easily modeled in the BP utility framework.

However, we note that there is an important theoretical connection
between the methods in \cite{JiangWal:08,RajagopalanSS:09} and
the BP method considered here. As we will discuss below, BP arrives
at the scheduling decision by estimating the marginals of a certain
joint probability distribution function given in \eqref{eq:probF}.
The CSMA-type methods in \cite{JiangWal:08,RajagopalanSS:09}
can be seen as a simulated annealing
(SA) method for selecting a random scheduling vector from precisely
the same distribution in the context of a constraint combinatorial
interference model. SA can be seen as an asymptotically
exact but slow method \cite{Hajek:88} for solving the optimization problem.
In constrast, BP is approximate, but potentially faster.

General overviews of BP can be found in a number of works including
\cite{Pearl:88,WainwrightJ:08}. In the context of wireless
scheduling, theoretical guarantees have been obtained for on-off
channels and the combinatorial contention graph
model~\cite{SanghaviMW:07, BayatiSS:08}. The methods here can be
seen as a generalization of these methods to soft interference
models with larger class of scheduling vectors.

\section{Problem Formulation} \label{sec:prob}

We consider a wireless scheduling problem with $n$ links.
The transmitter of each link $j=1,\ldots,n$, denoted TX $j$,
must select some \emph{scheduling vector}
$\mathbf{x}_j \in {\cal X} \subseteq {\mathbb{R}}^{n_x}$, which contains
$n_x$ parameters related to link $i$.
Examples of these parameters will be given below.
The selection of the scheduling vectors results in
an \emph{interference vector} $\mathbf{z}_i \in {\mathbb{R}}^{n_z}$
at the receiver of each link $i$, denoted RX $i$.
The interference is assumed to be
a linear function of the scheduling vectors of the other links,
\begin{equation} \label{eq:zidef}
    \mathbf{z}_i = \sum_{j=1}^n \mathbf{A}_{ij}\mathbf{x}_j
\end{equation}
for some matrices $\mathbf{A}_{ij} \in {\mathbb{R}}^{n_z \times n_x}$.  We assume that
$\mathbf{A}_{ii} = 0$, so that link $i$ does not interfere with itself.
We will let $\mathbf{x}$ and $\mathbf{z}$ be the column vectors with entries
$\mathbf{x}_j$ and $\mathbf{z}_i$,
\begin{eqnarray*}
    \mathbf{x} &=& \left[ \mathbf{x}_1' \cdots \mathbf{x}_n'\right]' \in {\mathbb{R}}^{n_xn} \\
    \mathbf{z} &=& \left[ \mathbf{z}_1' \cdots \mathbf{z}_n'\right]' \in {\mathbb{R}}^{n_zn},
\end{eqnarray*}
and write $\mathbf{z} = \mathbf{A} \mathbf{x}$ where $\mathbf{A}$ is the block matrix with
entries $\mathbf{A}_{ij}$. We call $\mathbf{A}$ the \emph{interference matrix}.
Also, we let $\mathbf{A}_i$ denote the $i$th column of the $\mathbf{A}$ so that
$\mathbf{z}_i = \mathbf{A}_i\mathbf{x}$.

Associated with each link $i$, is some utility function
$f_i(\mathbf{x}_i,\mathbf{z}_i)$ of the scheduling vector $\mathbf{x}_i$
and interference vector $\mathbf{z}_i$.  The scheduling problem is to
maximize the overall utility
\begin{equation} \label{eq:opt}
    \max_{\mathbf{x}} F(\mathbf{x}), \ \ \ F(\mathbf{x}) = \sum_{i=1}^n f_i(\mathbf{x}_i, \mathbf{z}_i).
\end{equation}
We will sometimes call the optimization problem \eqref{eq:opt},
an optimization with \emph{linear mixing} to stress the linear dependence
of the interference on the transmit vectors.

\section{Linear Mixing Utility Examples} \label{sec:ex}
The linear mixing formulation above is extremely general
and can incorporate a large class of utility functions and
interference models.

A general treatment of utility functions for wireless scheduling
can be found in \cite{KelleyMT:98,ShakkottaiS:07}.
In our simulations, we will consider utility maximization for both
static and time-varying problems.
For static optimization, the scheduling vectors $\mathbf{x}_j$
are selected once for a long time period
and the utility function is typically of the form
\begin{equation} \label{eq:utilStatic}
    f_i(\mathbf{x}_i,\mathbf{z}_i) = U_i(R_i(\mathbf{x}_i,\mathbf{z}_i)),
\end{equation}
where $R_i(\mathbf{x}_i,\mathbf{z}_i)$ is the long-term rate as a function of the
TX vector $\mathbf{x}_i$ and interference $\mathbf{z}_i$ and $U_i(R)$ is the utility
as a function of the rate.
The problem formulation above can incorporate any of the
common utility functions including: $U_i(R) = R$ which
results in a sum rate optimization; $U_i(R) = \log(R)$ which is the proportional
fair metric and $U_i(R) = -\beta R^{-\beta}$ for some $\beta > 0$ called
an $\beta$-fair utility.  Penalties can also be added if there is a cost
associated with the selection of the TX vector $\mathbf{x}_j$ such as power.

To accommodate time-varying channels and traffic loads,
many cellular systems enable fast dynamic scheduling in time slots
in the order of 1 to 2 ms.  For these systems, the utility maximization can be
re-run in each time slot.  One common approach is that
in each time slot $t=0,1,2\ldots$,
the scheduler uses a utility of the form
\begin{equation} \label{eq:wtUtil}
   f_i(t,\mathbf{x}_i,\mathbf{z}_i) = w_i(t)R_i(t,\mathbf{x}_i,\mathbf{z}_i), \ \ \
\end{equation}
where $w_i(t)$ is a time-varying weight given by the marginal utility
\begin{equation} \label{eq:wtMargUtil}
   w_i(t) = \frac{\partial U_i(\overline{R}_i(t))}{\partial R},
\end{equation} and $\overline{R}_i(t)$ is exponentially weighted average rate updated
as \begin{equation} \label{eq:rateFilt}
    \overline{R}_i(t+1) = (1-\alpha)\overline{R}_i(t) + \alpha
    R_i(t,\widehat{\mathbf{x}}_i(t),\widehat{\mathbf{z}}_i(t)),
\end{equation}
where $\widehat{\mathbf{x}}_i(t)$ is the TX vector and  $ \widehat{\mathbf{z}}_i(t)$ is
the interference for link $i$ at time $t$. Any maxima of the
optimization \eqref{eq:opt} with the weighted utility
\eqref{eq:wtUtil} is called a \emph{maximal weight matching}. A
well-known result of stochastic approximation~\cite{stolyar_2005} is
that if $\alpha \rightarrow 0$, and the scheduler performs the maximum
weight matching with the marginal utilities \eqref{eq:wtMargUtil},
then for a large class of processes, the resulting average rates
will maximize the total utility $\sum_i U_i(\overline{R}_i(t))$.

The above utilities are designed for infinite backlog queues.
For delay sensitive traffic, one can take the weights $w_i(t)$ to be the queue length
or head-of-line delay.  Maximal weight matching performed with these weights
generally results in so-called throughput optimal performance \cite{TassiulasE:92}.
These results also apply to multihop networks with the so-called backpressure
weights.

In addition to incorporating general utilities, an appealing feature of the
linear mixing framework is that a large class of interference models
can also be considered, including, for example:
\begin{itemize}
\item \emph{Flat fading with power control:}  In this case, $x_j$ is a scalar representing
the transmit power, and $A_{ij}$ is the gain from TX $j$ to RX $i$, so that $z_i$ is the
total interference at RX $i$.  The rate, $R_i(x_i,z_i)$ can then be described
as a function of the SINR $g_ix_i/z_i$, where $g_i$ is the channel gain along
link $i$.  Arbitrary SINR to rate mappings may be used.
Note that a special case of on-off channels where $x_j$ is zero or a maximum
transmit power can be used.

\item \emph{Multiple subbands:}  The above example is easily extended to the
case of multiple subbands.  As described in the Introduction, subband scheduling
is one of the key motivating features of LTE, but the optimization is difficult.
To handle multiple subbands, we simply let $\mathbf{x}_j$ and $\mathbf{z}_i$ be the vectors
of transmit  and interference powers in each subband
and $\mathbf{A}_{ij}$ be a diagonal matrix with channel gains in each subband.

\item \emph{Beamforming and linear precoding:}  The linear mixing formulation
can also incorporate problems with transmit beamforming or linear
precoding.  For example, suppose a link has
$N$ transmit antennas and one receive antenna.
If each transmitter TX $j$ uses a beamforming vector $\mathbf{b}_j \in {\mathbb{C}}^N$,
and $\mathbf{g}_{ij} \in {\mathbb{C}}^N$ is the channel from TX $j$ to RX $i$,
the interference at RX $i$ is given by
\[
    z_i = \sum_{j \neq i} \mathbf{g}_{ij}'\mathbf{b}_j\mathbf{b}_j'\mathbf{g}_{ij},
\]
which is linear in the rank one matrices $\mathbf{b}_j\mathbf{b}_j'$.
Hence, if we let $\mathbf{x}_j \in {\mathbb{C}}^{N^2}$ be the column vector with entries
of the matrix $\mathbf{b}_j\mathbf{b}_j'$, the interference $z_i$ can be represented
as a linear combination of the vectors $\mathbf{x}_j$.
The idea can also be generalized to precoding matrices with
multiple transmit streams.
\end{itemize}

\section{Belief Propagation} \label{sec:BpAlgo}

\subsection{Standard BP} \label{sec:stdBP}
We begin by briefly reviewing how we would apply standard BP
 to the optimization \eqref{eq:opt}.
Let $u > 0$ and define the probability distribution \begin{equation}
\label{eq:probF}
    p(\mathbf{x}) = \frac{1}{Z} \exp(uF(\mathbf{x})) = \frac{1}{Z} \prod_{i=1}^n
        \exp(uf_i(\mathbf{x}_i,\mathbf{z}_i)),
\end{equation} where $Z$ is a normalization constant called the partition
function (it is a function of $u$).  BP can be seen as a method to
estimate the marginal distributions of the distribution $p(\mathbf{x})$
with respect to the variables $\mathbf{x}_j$. From these marginals, one
can compute the marginal expectations $\widehat{\mathbf{x}}_j = \mathbf{E}\left( \mathbf{x}_j
\right)$. A standard result of large deviations~\cite{DemboZ:98}
 is that as $u \rightarrow
\infty$,
 under suitable conditions, $p(\mathbf{x})$ concentrates around the
maxima of $F(\mathbf{x})$ and
\[
    \lim_{u \rightarrow \infty} \widehat{\mathbf{x}} = \mathop{\mathrm{arg\,max}}_{\mathbf{x}} F(\mathbf{x}).
\]
So, if we can estimate the marginal expectations of the probability
distribution (\ref{eq:probF}) for large $u$, we can recover a good
estimate for the maximization of (\ref{eq:opt}).

To compute the marginal distributions, BP associates with the
interference matrix $\mathbf{A}$ a bipartite graph $G=(V,E)$ called the
\emph{factor} or \emph{Tanner} graph.  The vertices $V$ consists of
$n$ \emph{transmitter nodes} associated with the transmitters TX
$j$, and $n$ \emph{receiver nodes} associated with the receivers RX
$i$. There is an edge $(i,j) \in E$ if and only if $i=j$ or
$\mathbf{A}_{ij}$ is non-zero -- that is TX $j$ has some influence on the
interference or signal at RX $i$. We let $N_{\rm rx}(i)$ and $N_{\rm tx}(j)$ be
the neighbors sets of the nodes RX $i$ and TX $j$ in graph $G$,
respectively.

With this graph, BP iteratively passes \emph{beliefs} along the
edges of the graph that represent estimates of the marginal
distributions of $p(\mathbf{x})$ with respect to the variables $\mathbf{x}_j$.
In the context of the wireless scheduling problem, we can interpret
the iterations as \emph{rounds}, where computations are first
performed at the receivers and then at the transmitters. We index
the round by $t$, and let $p_{i \rightarrow j}(t,\cdot): {\cal X} \mapsto{\mathbb{R}}$
denote the belief message from RX $i$ to TX $j$ in the receiver half
of the round. The reverse belief message from TX $j$ to RX $i$ is
denoted $p_{i \leftarrow j}(t,\cdot): {\cal X} \mapsto{\mathbb{R}}$. $p_{i \leftarrow
j}(t,\mathbf{x}_j)$ and $p_{i \leftarrow j}(t,\mathbf{x}_j)$ denote the values of the
beliefs at $\mathbf{x}_j$. After some fixed number of rounds, the
algorithm is stopped and a final scheduling decision, meaning a
selection of the TX vectors $\mathbf{x}_j$, is made by the transmitters.
The steps for BP are as follows:

\begin{enumerate}
\item \emph{Initialization:}  Set $t=0$ and for all $(i,j) \in E$, let
$p_{i \leftarrow j}(t,\mathbf{x}_j)$ be some initial distribution on $\mathbf{x}_j$.
This distribution could be, for example, the uniform distribution on
the set ${\cal X}$.
\item \emph{RX node update:}  In the RX half of the round, each RX $i$
sends a belief message to the transmitters TX $j$ with $j \in
N_{\rm rx}(i)$ given by \begin{equation} \label{eq:outBP}
    p_{i \rightarrow j}(t,\mathbf{x}_j) =
        \mathbf{E}\left[ \exp(uf_i(\mathbf{x}_i,\mathbf{z}_i)) \mid \mathbf{x}_j \right],
\end{equation} where $\mathbf{z}_i = \mathbf{A}_i \mathbf{x}$ as in (\ref{eq:zidef}) and the
expectation is over independent $\mathbf{x}_k \sim p_{i \leftarrow k}(t,\mathbf{x}_k)$,
$\forall k\in N_{\rm rx}(i)$.
\item \emph{TX node update:}  In the TX half of the round, each TX $j$
sends a belief message back to the receivers RX $i$ with $i \in
N_{\rm tx}(j)$ given by \begin{equation} \label{eq:inBP}
    p_{i \leftarrow j}(t+1,\mathbf{x}_j) = \frac{1}{Z}
    \prod_{\ell \in N_{\rm tx}(j) \neq i} p_{\ell \rightarrow j}(t,\mathbf{x}_j),
\end{equation} where $Z$ is a normalization constant and the product is over
all $\ell \in N_{\rm tx}(j)$ with $\ell \neq i$. The iteration number is
incremented, $t=t+1$, and we return to step 2 until a sufficient
number of rounds have been performed.
\item \emph{Final solution:}  The final estimate for the marginal
distribution of $\mathbf{x}_j$ is given by \begin{equation} \label{eq:xhatBP}
    p_j(t+1,\mathbf{x}_j) = \frac{1}{Z} \prod_{i \in N_{\rm tx}(j)} p_{i \rightarrow j}(t,\mathbf{x}_j).
\end{equation} The scheduling vector can be selected as the maximum of this
marginal distribution.
\end{enumerate}

In the case when the graph $G$ has no cycles, it can be shown that
the $p_j(t,\mathbf{x}_j)$ converges to the true marginal distribution of
$p(\mathbf{x})$ in (\ref{eq:probF}) with respect to the variable $\mathbf{x}_j$.
However, for general $G$, BP is approximate.  A complete treatment
of BP is beyond the scope of this work -- the reader is referred to
the references above.

Implementation of the above BP algorithm in wireless networks is
challenging due to two reasons:
\begin{itemize}
\item \emph{High computational complexity:} The
expectation in \eqref{eq:outBP} requires integration over all the
variables $\mathbf{x}_r$ with $r \in N_{\rm rx}(i)$ and $r \neq j$. This
computation grows exponentially in $|N_{\rm rx}(i)|$, which is the number
of transmitters interfering with RX $i$.  If this set is large, the
computation is prohibitive.
\item \emph{High messaging overhead:} Passing the beliefs requires
\emph{unicast} messages between each RX and each TX (which are
neighbors as per graph $G$) as opposed to single broadcast message.
Also, in each round $t$, the messages comprise of the beliefs $p_{i
\rightarrow j}(t,\mathbf{x}_j)$ and $p_{i \leftarrow j}(t,\mathbf{x}_j)$ for all values $\mathbf{x}_j
\in {\cal X}$. If ${\cal X}$ is large, the messaging overhead may be
significant, and it grows with the number of transmitters
interfering at a receiver.
\end{itemize}
\subsection{Gaussian Approximation} \label{sec:gaussBP}

We first consider the simplification of the RX node update
\eqref{eq:outBP}. We describe the simplification in log domain. Let
$\Delta_{i \rightarrow j}(t,\cdot)$ and $\Delta_{i \leftarrow j}(t,\cdot)$ be
\emph{log likelihood functions}, meaning any functions such that
\begin{subequations} \label{eq:DeltaLR}
\begin{eqnarray}
    \Delta_{i \rightarrow j}(t,\mathbf{x}_j) &:=& \frac{1}{u}  \log \left[
        p_{i \rightarrow j}(t,\mathbf{x}_j) \right] + \mbox{const}, \label{eq:DeltaRight} \\
    \Delta_{i \leftarrow j}(t,\mathbf{x}_j) &:=& \frac{1}{u}  \log \left[
        p_{i \leftarrow j}(t,\mathbf{x}_j) \right] + \mbox{const}. \label{eq:DeltaLeft}
\end{eqnarray}
\end{subequations}
where the constants do not depend on $\mathbf{x}_j$ (although they may
depend on $t$ and the indices $i$ and $j$). Observe that we can
recover the probabilities from the log likelihoods by the relation
\begin{subequations} \label{eq:probDeltaLR}
\begin{eqnarray}
    p_{i \rightarrow j}(t,\mathbf{x}_j) &\propto& \exp\left[ u \Delta_{i \rightarrow j}(t,\mathbf{x}_j) \right]
        \label{eq:probDeltaR} \\
    p_{i \leftarrow j}(t,\mathbf{x}_j) &\propto& \exp\left[ u \Delta_{i \leftarrow j}(t,\mathbf{x}_j) \right].
        \label{eq:probDeltaL}
\end{eqnarray}
\end{subequations}

We now consider the simplification of the RX node update
\eqref{eq:outBP} under two cases:  when $i=j$ and when $i \neq j$.
We begin with the case when $i=j$. The expectation in
(\ref{eq:outBP}) is to be evaluated with $\mathbf{z}_i$ given by
(\ref{eq:zidef}) with the variables $\mathbf{x}_j$ being independent and
$\mathbf{x}_j \sim p_{i \leftarrow j}(t,\mathbf{x}_j)$. Let $\widehat{\mathbf{x}}_{i \leftarrow j}(t)$ and
$\mathbf{Q}^x_{i \leftarrow j}(t)$ be the mean and $1/u$ times the variance of
the distribution $p_{i \leftarrow j}(t,\cdot)$. Then, under the simplifying
assumption that the summation in (\ref{eq:zidef}) consists of a
large number of independent terms, we can apply the Central Limit
Theorem and approximate the distribution of $\mathbf{z}_i$ with \begin{equation}
\label{eq:ziGauss}
    \mathbf{z}_i = {\cal N}(\widehat{\mathbf{s}}_{ii}(t), \mathbf{Q}^s_{ii}(t)/u),
\end{equation} where
\begin{subequations} \label{eq:supdate}
\begin{eqnarray}
    \widehat{\mathbf{s}}_{ii}(t) &=& \sum_{j \in N_{\rm rx}(i)} \mathbf{A}_{ij} \widehat{\mathbf{x}}_{i \leftarrow j}(t) \\
    \mathbf{Q}^s_{ii}(t) &=& \sum_{j \in N_{\rm rx}(i)} \mathbf{A}_{ij} \mathbf{Q}^x_{i\leftarrow j}(t)\mathbf{A}_{ij}',
\end{eqnarray}
\end{subequations}
which have the interpretation as a mean and variance of the
interference at RX $i$. Applying the Gaussian approximation
\eqref{eq:ziGauss} to the expectation (\ref{eq:outBP}) shows that we
can write $\Delta_{i \rightarrow i}(\cdot)$ in \eqref{eq:DeltaRight} as \begin{equation}
\label{eq:DeltaiiRZ}
    \Delta_{i \rightarrow i}(t,\mathbf{x}_i) \approx \frac{1}{u} \log \left[
        Z_{i0}(\mathbf{x}_i,\widehat{\mathbf{s}}_{ii}(t), \mathbf{Q}^s_{ii}(t)) \right],
\end{equation} where $Z_{i0}(\cdot)$ is the partition function \begin{equation}
\label{eq:Zi0}
    Z_{i0}(\mathbf{x}_i,\widehat{\mathbf{s}}_{ii},\mathbf{Q}^s_{ii}) = \int
        \exp\left[ uL_{i0}(\mathbf{x}_i,\mathbf{z}_i, \widehat{\mathbf{s}}_{ii},\mathbf{Q}^s_{ii})\right] d\mathbf{z}_i,
\end{equation} and \begin{equation} \label{eq:Li0}
    L_{i0}(\mathbf{x}_i,\mathbf{z}_i, \widehat{\mathbf{s}}_{ii},\mathbf{Q}^s_{ii}) = f_i(\mathbf{x}_i,\mathbf{z}_i)
    - \frac{1}{2} (\mathbf{z}_i - \widehat{\mathbf{s}}_{ii})'\mathbf{Q}^{-s}_{ii}(\mathbf{z}_i - \widehat{\mathbf{s}}_i)
\end{equation} and $\mathbf{Q}_{ii}^{-s}$ is the matrix inverse of $\mathbf{Q}_{ii}^s$.

Next consider the case $i \neq j$. A similar argument as above shows
that, conditional on $\mathbf{x}_j$, the distribution of $\mathbf{z}_i$  can be
approximated by the Gaussian \begin{equation} \label{eq:zijGauss}
    \mathbf{z}_i = {\cal N}(\widehat{\mathbf{s}}_{ij}(t), \mathbf{Q}^s_{ij}(t)/u),
\end{equation} where
\begin{subequations} \label{eq:supdateij}
\begin{eqnarray}
    \widehat{\mathbf{s}}_{ij}(t) &=& \sum_{r \in N_{\rm rx}(i) \neq j} \mathbf{A}_{ir} \widehat{\mathbf{x}}_{i \leftarrow r}(t)
        + \mathbf{A}_{ij}\mathbf{x}_j \nonumber \\
        &=& \widehat{\mathbf{s}}_{ii}(t) + \mathbf{A}_{ij}(\mathbf{x}_j-\widehat{\mathbf{x}}_{i \leftarrow j}(t)) \\
    \mathbf{Q}^s_{ij}(t) &=& \sum_{r \in N_{\rm rx}(i) \neq j} \mathbf{A}_{ir} \mathbf{Q}^x_{i\leftarrow r}(t)\mathbf{A}_{ir}', \nonumber \\
        &=& \mathbf{Q}^s_{ii}(t) - \mathbf{A}_{ij} \mathbf{Q}^x_{i\leftarrow j}(t)\mathbf{A}_{ij}'.
\end{eqnarray}
\end{subequations}
Then, applying the Gaussian approximation (\ref{eq:zijGauss}) along
with \eqref{eq:DeltaLeft} to the expectation (\ref{eq:outBP}) shows
that we can write $\Delta_{i \rightarrow j}(\cdot)$ in \eqref{eq:DeltaRight}
as \begin{equation} \label{eq:DeltaijRZ}
    \Delta_{i \rightarrow j}(t,\mathbf{x}_j)
    \approx \frac{1}{u}\log\left[ Z_i(\Delta_{i \leftarrow i}(t,\cdot),\widehat{\mathbf{s}}_{ij}(t), \mathbf{Q}^s_{ij}(t))
        \right],
\end{equation} where
\[
    Z_{i}(\Delta(\cdot),\widehat{\mathbf{s}}_{ij},\mathbf{Q}^s_{ij}) = \int
        \exp\left[ uL_{i}(\mathbf{x}_i,\mathbf{z}_i, \widehat{\mathbf{s}}_{ij},\mathbf{Q}^s_{ij})\right] d\mathbf{x}_i d\mathbf{z}_i,
\]
and \begin{equation} \label{eq:Li}
    L_{i}(\mathbf{x}_i,\mathbf{z}_i, \widehat{\mathbf{s}}_{ij},\mathbf{Q}^s_{ij}) = L_{i0}(\mathbf{x}_i,\mathbf{z}_i, \widehat{\mathbf{s}}_{ij},\mathbf{Q}^s_{ij}) + \Delta(\mathbf{x}_i).
\end{equation}

Then, the RX and TX node update steps of the BP algorithm with
Gaussian approximation of the interference at the RX are:
\begin{itemize}
\item \emph{RX node update:}
The above equations are used to simplify the standard BP algorithm
as follows. Each RX $i$ receives mean and variances $\widehat{\mathbf{x}}_{i \leftarrow
j}(t)$ and $\mathbf{Q}^x_{i \leftarrow j}(t)$ of the vectors from the
transmitters TX $j$ with $j \in N_{\rm rx}(i)$. RX $i$ also receives the
function $\Delta_{i \leftarrow i}(t,\cdot)$ from its serving transmitter,
TX $i$. RX $i$ then computes the interference means and variances in
\eqref{eq:supdate} and \eqref{eq:supdateij}. Then, for each TX $j$
it sends back the log likelihoods $\Delta_{i \rightarrow j}(t,\cdot)$ by
evaluating the log partition functions \eqref{eq:DeltaiiRZ} and
\eqref{eq:DeltaijRZ}.

\item \emph{TX node update:}
It can be verified that converting the update \eqref{eq:inBP} to log
domain yields \begin{equation} \label{eq:DeltaijLSum}
    \Delta_{i \leftarrow j}(t+1,\mathbf{x}_j) = \sum_{\ell \in N_{\rm tx}(j) \neq i}
        \Delta_{\ell \rightarrow j}(t,\mathbf{x}_j).
\end{equation} Each TX $j$ can first computes the log likelihoods $\Delta_{i
\leftarrow j}(t+1,\mathbf{x}_j)$ from the log likelihoods $\Delta_{\ell \rightarrow
j}(t,\mathbf{x}_j)$ from the receivers RX $\ell$ with $\ell \in N_{\rm tx}(j)$,
$\ell \neq i$. Then, using the log likelihoods $\Delta_{i \leftarrow
j}(t+1,\mathbf{x}_j)$ , TX $j$ computes the mean and variance $\widehat{\mathbf{x}}_{i
\leftarrow j}(t+1)$ and $\mathbf{Q}^x_{i \leftarrow j}(t+1)$ from the probability
distribution $p_{i \leftarrow j}(t+1,\mathbf{x}_j)$ in \eqref{eq:probDeltaL}.
Then, TX $j$ sends messages to the receivers as described in the RX
node update above.
\end{itemize}
After a sufficient number of rounds, TX $j$ can compute the final
log likelihood \begin{equation} \label{eq:DeltajLSum}
    \Delta_{j}(t+1,\mathbf{x}_j) = \sum_{\ell \in N_{\rm tx}(j)} \Delta_{\ell \rightarrow j}(t,\mathbf{x}_j),
\end{equation} and then compute the final scheduling vector as the one which
maximizes the above function.

We have therefore simplified the standard BP algorithm by
eliminating the exponential complexity of the RX update
\eqref{eq:outBP}, and replaced this computation with a Gaussian
approximation.

\subsection{First Order Approximations} \label{sec:firstOrderApprox}
Unfortunately, the Gaussian approximation above does not
significantly reduce the messaging overhead.  Each RX and TX must
still send separate unicast messages to every TX or RX in its
neighbor set.  Also, although the TX must only send a mean and a
covariance matrix $\widehat{\mathbf{x}}_{i \leftarrow j}(t)$ and $\mathbf{Q}^x_{i \leftarrow j}(t)$,
the receivers must send the entire log likelihood functions
$\Delta_{i \rightarrow j}(t,\mathbf{x}_j)$.

The messaging overhead can be reduced via selective use of first
order approximations as follows: Divide the edges $(i,j) \in E$ with
$i\neq j$
 into two sets -- \emph{weak} and \emph{strong} -- depending on whether
 $\mathbf{A}_{ij}$ is small or large.
Along a strong edge $(i,j)$, RX $i$ and TX $j$ exchange the full
unicast messages described above. However, for the weak edges, the
messages can be replaced with a first order approximation described
below. The precise classification rule between weak and strong edges
is an algorithm parameter that can be used to trade off complexity
and accuracy.

To describe the first order approximation along the weak edges,
suppose $\mathbf{A}_{ij}$ is small for some edge $(i,j)$. Let
$\hat{\mathbf{x}}_j(t)$ be the mean value of the distribution
corresponding to the log likelihood $\Delta_j(t,\cdot)$ in
\eqref{eq:DeltajLSum}. Now consider the log likelihood $\Delta_{i
\rightarrow j}(t,\mathbf{x}_j)$ in \eqref{eq:DeltaijRZ}. Applying
\eqref{eq:supdateij}
\begin{eqnarray}
    \lefteqn{ \Delta_{i \rightarrow j}(t,\mathbf{x}_j)
    \approx \frac{1}{u}\log\left[ Z_i(\Delta_{i \leftarrow i}(t,\cdot),\widehat{\mathbf{s}}_{ij}(t), \mathbf{Q}^s_{ij}(t))
        \right]}, \nonumber \\
        &\stackrel{(a)}{\approx}&
        \frac{1}{u}\log\left[ Z_i(\Delta_{i \leftarrow i}(t,\cdot),\right. \nonumber\\
        & &\left. \widehat{\mathbf{s}}_{ii}(t)+
            \mathbf{A}_{ij}(\mathbf{x}_j-\widehat{\mathbf{x}}_{j}(t)), \mathbf{Q}^s_{ii}(t)) \right], \nonumber \\
       &\stackrel{(b)}{\approx}&
            \mathbf{u}'_{ij}(t) \mathbf{x}_j  + \mbox{ const}     \label{eq:DelijApprox}
\end{eqnarray}
where (a) follows from \eqref{eq:supdateij} and the
approximation that $\mathbf{Q}^s_{ij}(t) \approx \mathbf{Q}^s_{ii}(t)$ and
$\widehat{\mathbf{x}}_{i \leftarrow j}(t) \approx \widehat{\mathbf{x}}_j(t)$ when $\mathbf{A}_{ij}$ is
small; and (b) follows from taking a Taylor's approximation with
\begin{equation} \label{eq:uij}
    \mathbf{u}_{ij}(t) = \mathbf{A}'_{ij}D'_{i1}(t)-
            \mathbf{A}_{ij}'D_{i2}(t)\mathbf{A}_{ij}\widehat{\mathbf{x}}_{j}(t),
\end{equation}
where for $r=1,2$, $D_{ir}(t)$, are the derivatives
\begin{equation} \label{eq:logZDeriv}
    D_{ir}(t) := \frac{1}{u}\frac{\partial^r}{\partial \widehat{\mathbf{s}}^r}
        \log \left[Z_i(\Delta_{i \leftarrow i}(\cdot),\widehat{\mathbf{s}}_{ii}(t), \mathbf{Q}^s_{ii}(t)) \right].
\end{equation}
The constant in \eqref{eq:DelijApprox} is independent of
$\mathbf{x}_j$. Using standard properties of the cumulant function
\cite{WainwrightJ:08}, it can be shown that the derivative is given
by
\begin{equation} \label{eq:derivUtil}
    D_{ir}(t) = \mathbf{E}\left[ \left.
    \frac{\partial^r}{\partial \mathbf{z}_i^r} f_i(\mathbf{x}_i,\mathbf{z}_i)
    \right| \widehat{\mathbf{s}}_{ii}(t),\mathbf{Q}^s_{ii}(t), \Delta_{i \leftarrow i}(t,\cdot) \right],
\end{equation}
where the expectation is with respect to the conditional distribution
\begin{eqnarray}
    \lefteqn{ p_{\mathbf{x},\mathbf{z}|\widehat{\mathbf{s}},\mathbf{Q}^s}(\mathbf{x}_i,\mathbf{z}_i\mid
    \widehat{\mathbf{s}}_{ii}(t),\mathbf{Q}^s_{ii}(t)) }     \nonumber \\
     &:=& \frac{1}{Z_i}\exp\left[
        uL_i(\mathbf{x}_i,\mathbf{z}_i, \widehat{\mathbf{s}}_{ii}(t),\mathbf{Q}^s_{ii}(t)\right],
        \label{eq:pxz}
\end{eqnarray}
and $L_i(\cdot)$ is defined in \eqref{eq:Li} and
\eqref{eq:Li0}. The dependence on $\Delta_{i \leftarrow i}(\cdot)$ in \eqref{eq:derivUtil}
is implicit in \eqref{eq:Li}.
Note that the derivatives $D_{ir}(t)$ in
\eqref{eq:derivUtil} can be interpreted as a sensitivity of the
expected utility $f_i(\mathbf{x}_i,\mathbf{z}_i)$ to changes in the interference
$\mathbf{z}_i$.

The computation of $\mathbf{x}_{i \leftarrow j}(t)$ can also be simplified as
follows: Recall that $\widehat{\mathbf{x}}_{i \leftarrow j}(t)$ and $\widehat{\mathbf{x}}_j(t)$ are
the expectation with respect to the likelihood functions $\Delta_{i
\leftarrow j}(t,\cdot)$ in \eqref{eq:DeltaijLSum} and $\Delta_j(t,\cdot)$
in \eqref{eq:DeltajLSum}, respectively. Therefore, we can write them
as
\begin{subequations}
\begin{eqnarray}
    \widehat{\mathbf{x}}_{i \leftarrow j}(t) &=& \mathbf{E}\left[ \mathbf{x}_j | \Delta_{i \leftarrow j}(t,\cdot) \right]
        \label{eq:xijDel} \\
    \widehat{\mathbf{x}}_{j}(t) &=& \mathbf{E}\left[ \mathbf{x}_j | \Delta_{j}(t,\cdot) \right],  \label{eq:xjDel}
\end{eqnarray}
\end{subequations}
where we have used the notation $\mathbf{E}( g(\mathbf{x}) | \Delta(\mathbf{x}) )$ to
denote the expectation of $g(\mathbf{x})$ with respect to the probability
distribution
\[
    p_\Delta(\mathbf{x}) \propto \exp\left[u \Delta(\mathbf{x}) \right].
\]
It is easy to check that for small perturbations of $\epsilon(\mathbf{x})$
of $\Delta(\mathbf{x})$, we have the first order approximation \begin{eqnarray}
    \lefteqn{ \mathbf{E}(g(\mathbf{x})|\Delta + \epsilon) \approx
        \mathbf{E}(g(\mathbf{x})|\Delta)   } \nonumber \\
    &+& u\left[ \mathbf{E}(g(\mathbf{x})\epsilon(\mathbf{x})|\Delta) -
    \mathbf{E}(g(\mathbf{x})|\Delta)\mathbf{E}(\epsilon(\mathbf{x})|\Delta)\right]. \label{eq:EdelApprox}
\end{eqnarray}
Therefore,
\begin{eqnarray}
    \lefteqn{ \widehat{\mathbf{x}}_{i \leftarrow j}(t+1) \stackrel{(a)}{=}
        \mathbf{E}\left[ \mathbf{x}_j | \Delta_j(t+1,\cdot) - \Delta_{i \rightarrow j}(t,\cdot)\right]
        } \nonumber \\
    &\stackrel{(b)}{\approx}&
        \mathbf{E}(\mathbf{x}_j|\Delta_j(t+1,\cdot)) -
        u\mathbf{E}(\mathbf{x}_j\Delta_{i \rightarrow j}(t,\mathbf{x}_j)|\Delta_j(t+1,\cdot)) \nonumber \\
    &-& u\mathbf{E}(\mathbf{x}_j|\Delta_j(t,\cdot))
     \mathbf{E}(\Delta_{i \rightarrow j}(t,\mathbf{x}_j)|\Delta_j(t+1,\cdot)) \nonumber \\
    &\stackrel{(c)}{\approx}&  \widehat{\mathbf{x}}_j(t+1) - \mathbf{Q}^x_j(t+1)\mathbf{u}_{ij}(t)
    \label{eq:xijApprox}
\end{eqnarray}
where (a) follows from \eqref{eq:xijDel} along with
\eqref{eq:DeltaijLSum} and \eqref{eq:DeltajLSum}; (b) follows from
\eqref{eq:EdelApprox}; and (c) follows from \eqref{eq:xjDel} and
\eqref{eq:DelijApprox}.

We can use the above relations to define the following algorithm:
\begin{enumerate}
\item \emph{Initialization:}  Set $t=0$.  Each TX $j$
broadcasts an initial scheduling vector $\widehat{\mathbf{x}}_j(t)$ and variance
$\mathbf{Q}^x_j(t)$. These can be based on the mean and variance of
$\mathbf{x}_j$ over the set ${\cal X}$. The receivers RX $i$ sets $\widehat{\mathbf{x}}_{i
\leftarrow j}(t) = \mathbf{x}_j(t)$ and $\mathbf{Q}^x_{i\leftarrow j}(t)=\mathbf{Q}^x_j(t)$ for all
$j$, and $\Delta_{i \leftarrow i}(t,\mathbf{x}_i) = 0$ for all $\mathbf{x}_i$.

\item \emph{RX node update:}  In the RX half of the round,
each RX $i$ first computes the interference means and variances
$\widehat{\mathbf{s}}_{ii}(t)$ and $\mathbf{Q}^s_{ii}(t)$ in \eqref{eq:supdate} and
$\widehat{\mathbf{s}}_{ij}(t)$ and $\mathbf{Q}^s_{ij}(t)$ in \eqref{eq:supdateij}.
Then, RX $i$ computes the log likelihood function $\Delta_{i \rightarrow
i}(t,\mathbf{x}_i)$ in \eqref{eq:DeltaiiRZ} and sends it as a unicast
message to its serving transmitter TX $i$. Also, for each strong
edge $(i,j)$, RX $i$ computes the log likelihood functions
$\Delta_{i \rightarrow j}(t,\mathbf{x}_j)$ in \eqref{eq:DeltaijRZ} and sends it as
a unicast message to the interfering TX $j$.  For the weak edges, RX
$i$ simply computes the sensitivity $D_{i1}(t)$ in
\eqref{eq:derivUtil} and broadcasts it to all other transmitters TX
$j$. The receiver also computes $\mathbf{u}_{ij}(t)$ in \eqref{eq:uij} and
stores it for the next round.

\item \emph{TX node update:}  In the TX half of the round, each transmitter TX
$j$ would have received the log likelihoods $\Delta_{i \rightarrow
j}(t,\mathbf{x}_j)$ from the receivers RX $i$, along the edges $(i,j)$
that were strong.  For any weak edge $(i,j)$, TX $j$ can
approximately compute the log likelihood from the sensitivity
$D_{i1}(t)$ using \eqref{eq:DelijApprox}. TX $j$ can then compute
the log likelihoods $\Delta_{i \leftarrow j}(t+1,\mathbf{x}_j)$ from
\eqref{eq:DeltaijLSum} for all $i \in N_{\rm tx}(j)$ and the log
likelihood $\Delta_j(t+1,\mathbf{x}_j)$ in \eqref{eq:DeltajLSum}. TX $j$
sends the receiver RX $j$ that it is serving the entire log
likelihood $\Delta_{j \leftarrow j}(t+1,\mathbf{x}_j)$. For the receivers RX $i$
such that $(i,j)$ is a strong edge, TX $j$ computes the mean
$\widehat{\mathbf{x}}_{i \leftarrow j}(t+1)$ and variance $\mathbf{Q}_{i \leftarrow j}^x(t+1)$ from
the log likelihood $\Delta_{i \leftarrow j}(t+1,\mathbf{x}_j)$ and sends it as a
unicast message to RX $i$. Each TX $j$ also computes the mean and
variance $\widehat{\mathbf{x}}_j(t+1)$ and $\mathbf{Q}^x_j(t+1)$ from the log
likelihood $\Delta_j(t+1,\mathbf{x}_j)$ and broadcasts the quantities to
the other receivers.  Any receiver RX $i$ that is along a weak edge
$(i,j)$ can then approximately compute $\widehat{\mathbf{x}}_{i \leftarrow j}(t)$ from
\eqref{eq:xijApprox}.

The round number is incremented, $t=t+1$, and we return to Step 2
until a fixed number of rounds have been performed.

\item \emph{Final solution:}  After the final round,
each transmitter TX $j$ takes the scheduling vectors to be the
vector $\mathbf{x}_j$ that maximizes the log likelihood
$\Delta_j(t,\mathbf{x}_j)$.
\end{enumerate}

\begin{figure}
\begin{center}

  \includegraphics[height=2in]{figures/msgFlow.pdf}
\end{center}
\caption{One round of the BP messages along the weak edges
interpreted as a ``soft" RTS / CTS mechanism.
}
\label{fig:bpMsgFlow}
\end{figure}

The message flow along the weak edges has an appealing
interpretation. Consider Fig.\ \ref{fig:bpMsgFlow} where a
transmitter TX1 attempts to send data to a receiver RX1.  The
receiver RX1 experiences interference from a second transmitter TX2,
while the transmitter TX1 causes interference onto a victim receiver
RX3.  Fig.\ \ref{fig:bpMsgFlow} shows the messages along the weak
edges in one round of the BP algorithm to coordinate the
interference. The transmitters TX1 and TX2 will broadcast the mean
and variance $\widehat{\mathbf{x}}_j(t)$ and $\mathbf{Q}^x_j(t)$ of their intended
transmit vectors. These transmissions can be interpreted as ``soft"
request to sends (RTS).  They are soft since the intended transmit
vectors are signaled by a distribution.  Based on the transmit
vector distribution from the interfering TX2, the
 receiver RX1 replies to the serving TX1 with an estimate of the interference
 $\widehat{\mathbf{s}}_{11}(t)$ and $\mathbf{Q}^s_{11}(t)$ which can be interpreted
as a ``soft" channel
quality indication (CQI).  As victim receivers, RX1 and RX3 also compute
the sensitivities to the interference level by the derivatives $D_{ir}(t)$.
These values can be interpreted as soft clear to send (CTS) indications,
since instead of a binary go/no go type CTS, they signal a soft cost
on changes in the interference from other transmitters.
\subsection{Communication and Computation Complexity}
We summarize the complexity of one round of the three variations of
BP in Tables~\ref{tab:comp_compl} and~\ref{tab:comm_compl}.
Specifically, we focus on RX $i$ and TX $j$ and consider how the
complexity grows with ${\cal X}$, $N_{\rm rx}(i)$ and $N_{\rm tx}(j)$.

\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|}
 \hline   Method & RX $i$ & TX $j$ \\
\hline Exact & $O\left(\left|{\cal X}\right|^{|N_{\rm rx}(i)|}\right)$&  $O\left(|{\cal X}||N_{\rm tx}(j)|\right)$ \\
\hline Gaussian Approx. & $O\left( |N_{\rm rx}(i)| \right)$ & $O\left(|{\cal X}||N_{\rm tx}(j)|\right)$\\
\hline First Order &$O\left(|N_{\rm rx}(i)| \right)$ & $O\left(|N_{\rm tx}(j)| \right)$\\
\hline
\end{tabular}
\end{center}
\caption{Computational Complexity}
\label{tab:comp_compl}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|}
 \hline Method & RX $i$ & TX $j$ \\
\hline Exact & $O\left(|{\cal X}||N_{\rm rx}(i)|\right)$& $O\left(|{\cal X}||N_{\rm tx}(j)|\right)$\\
\hline Gaussian Approx. & $O\left(|{\cal X}||N_{\rm rx}(i)|\right)$&  $O\left(|N_{\rm tx}(j)|\right)$\\
\hline First Order & O(1) & O(1)\\
\hline
\end{tabular}
\end{center}
\caption{Communication Complexity}
\label{tab:comm_compl}
\end{table}

\section{Numerical Simulation} \label{sec:sim}

The BP algorithm was simulated on a
a simplified version of an industry standard
model for LTE femtocell evaluation in \cite{FemtoForum:10}.
The simulation parameters are shown in Table \ref{tbl:simParam}.
The network consists of a 3 x 3 grid of 10m x 10m apartments
with active links in 5 of the 9 apartments.
Each link consists of one femto BS transmitting to one femto mobile
(called a UE, or user equipment, in 3GPP terminology).
Due to restricted association, UEs connect to the femto BS in their
apartment even if it is not the BS with the minimum path loss.
As mentioned in the introduction, this scenario exposes many links
to strong interference, and thus presents a good test scenario for
advanced interference coordination algorithms.

\begin{table}
\begin{center}
\begin{tabular}{|p{1in}|p{2in}|}
\hline
Parameter & Value \\ \hline
Network topology & $3 \times 3$ apartment model, with active links in 5 of the 9 apartments.
\\ \hline
Carrier freq & 2 GHz \\ \hline
Bandwidth & 5 MHz \\ \hline
Wall loss & 0 or 10 dB \\ \hline
Lognormal shadowing & 10 dB std.\ dev.\ \\ \hline
Path loss & $38.46 + 20\log_{10}(R) + 0.7R$ dB, $R$ distance in meters. \\ \hline
Femto BS TX power & 0 dBm \\ \hline
Femto UE noise figure & 4 dB \\ \hline
\end{tabular}
\end{center}
\caption{Simulation parameters.}
\label{tbl:simParam}
\end{table}

In the first simulation, we considered a time-varying simulation
with a simple on-off model where, in each time slot,
each link either transmits at the max power or is completely off.
As described in Section \ref{sec:ex}, for the time-varying problem,
the utility maximization was rerun in each
time slot with the weighted sum rate utility \eqref{eq:wtUtil} with weights
\eqref{eq:wtMargUtil}.  We used the proportional fair utility $U_i(R) = \log(R)$.
We generated independent flat fades on the links in each slot,
and took a filter time constant in \eqref{eq:rateFilt} was $\alpha = 0.1$.
For each random realization, or ``drop", of the femto network, we ran the simulation
over 100 time slots and measured the average rate.
The wall loss in the femto model was 10 dB.

\begin{figure}
\begin{center}

  \includegraphics[height=2.5in]{figures/dynSim_wall0.pdf}
  \includegraphics[height=2.5in]{figures/dynSimUtil_wall10.pdf}
\end{center}
\caption{Downlink femtocell simulation with an on-off channel model
and time-varying flat fading.
The top panel shows the rate CDFs across all links and drops
using various optimization algorithms for weight matching.
The bottom panel is the CDF of system utility (represented
as the harmonic mean rate) across different drops.}
\label{fig:dynSim}
\end{figure}

The top panel of Fig.\ \ref{fig:dynSim} plots
the cumulative distribution function (CDF)
of the time-averaged rates for 5 links and 100 drops
comparing various optimization methods for computing the maximum weighted
matching optimization \eqref{eq:opt}.
The curve ``reuse 1" is the case when all links transmit at max power.
Two cases of BP
are simulated:  (i) 4 rounds of BP in each time slot
using the Gaussian approximation in Section
\ref{sec:gaussBP} but no first order approximations; and
(ii) using only two rounds of BP
with both the Gaussian approximations and first order approximations on all interfering links less than 0 dB below the serving link.
We see that even the linear approximated BP with 2 rounds does significantly better
than reuse 1, and is not that far from the 4 round BP.
The gap between BP and reuse 1 particularly large at low rates.
For example, for the 20\% worst links,
BP offers almost a factor of 5 improvement in rate over reuse 1.

Also plotted is the rate CDF with optimal matching in each time slot
based on an exhaustive centralized search.
BP performs reasonably close to this curve, although there is still an
obvious gap, again at low rates.

The bottom panel of Fig.\ \ref{fig:dynSim} shows the CDF
of the total system utility over the 100 drops.  Since the optimization used
a log utility, the optimization is equivalent to maximizing the harmonic
mean rate over the 5 links.  We see in this plot that the approximate BP
algorithms are close to optimal and significantly better than reuse 1.

As a second simulation, we considered the same problem but with a
static single optimization and $K=4$ subbands.  Random fading was
generated in each subband, so the simulation would be applicable
in the case with frequency selective fading with coherence bandwidth
roughly equal to the subband bandwidth.  The optimization was over
 scheduling vectors  $\mathbf{x}_j \in {\mathbb{C}}^K$ with each component being on or off,
 so there are $2^K-1=15$ possible non-zero scheduling vectors in each link.
Optimal subband allocation is a well-known
challenging but important problem for OFDMA systems like LTE.

\begin{figure}
\begin{center}

  \includegraphics[height=2.5in]{figures/subTest_wall10.pdf}
\end{center}
\caption{Subband static optimization.
Performance comparison for downlink femtocell rates with
various optimizations for 4 independently faded subbands.
}
\label{fig:subSim}
\end{figure}

Under these assumptions, Fig.\ \ref{fig:subSim} shows the CDF
of the rates under various optimization methods.
Similar to the dynamic single subband case, we see that BP
provides significant gains over simple reuse 1, even when we use
linear approximations and only four rounds.  Also, for most links,
BP achieves a rate reasonably close to the optimal subband allocation found by
exhaustive search over all subband allocations over all rates.
However, for the lowest rate links, there is a significant gap
between BP and optimal.  Thus, even though BP outperforms reuse 1
significantly in this regime, there is significant room for improvement.

The BP method can also be applied to beamforming (BF) problems.
As a simple simulation, we consider again the femtocell deployment
with transmit beamforming with two antennas with half wavelength spacing
spacing and one receive antenna.  We neglect scattering so the channel
appears as a linear phase across the two antennas on all links.
For the transmit vectors $\mathbf{x}_j$, we optimize beamforming angles
over 10 angles uniformly spaced between 0 and $\pi$.
A performance comparison of the rate CDFs under various optimization
algorithms is shown in Fig.~\ref{fig:bfSim}.
In this case, we took a wall loss of 0 dB.
The curve labeled ``opt serving link only" is the case when the
BF vector is chosen to maximize the signal strength from the serving
link only without regard to interference.  This simple method provides
the baseline.  We see that BP provides some gains over serving link
optimization.  For example, the median rate with BP is approximately 50\%
higher than using optimal BF on the serving link only.
Also, BP appears to be reasonably close to the optimal BF selection
based on exhaustive search.

\begin{figure}
\begin{center}

  \includegraphics[height=2.5in]{figures/bfTest_wall00.pdf}
\end{center}
\caption{Beamforming optimization.
Downlink femtocell rates for
various optimizations with 2 TX antenna beamforming.
Channel models assume no scattering and beamforming optimization
is performed over linear phase beamforming vectors.
}
\label{fig:bfSim}
\end{figure}

\section{Conclusions} \label{sec:conclusions}

We formulated a general wireless scheduling and interference
coordination problem as an optimization problem with linear mixing
utilities. This was cast in a BP framework where the goal is to
compute the marginal distributions of a joint probability function.
Using Gaussian and linear approximations, we obtained a distributed
interference coordination algorithms with low overhead. The
algorithm has a natural interpretation as a soft RTS/CTS scheme.
Numerical simulations demonstrated that the resulting algorithm is
close to an optimal scheme for dynamic and static sub-band
optimization as well as for beamforming coordination across cells.
Moreover, the results show that the algorithm computes a good
operating point in just two to four iterations making it very
attractive to be used in practical wireless cellular systems. In the
future, we plan to explore connections between our work and the AMP
framework in~\cite{MontanariT:06} to obtain performance bounds for
large random networks.

\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


text:
Q: Is the phrase "these police" or "those police" grammatically incorrect? Grammarly keeps trying to change it to "this police." I am talking about multiple police. What am I supposed to say in this case?

A: It's a bit hard to say without an example sentence, but most likely you want to simply say "the police". This phrasing is imprecise in terms of number or location.

redpajama_set_name: RedPajamaStackExchange


Wilhelm von Jülich-Berg-Ravensberg was born 9 January 1455 to Gerhard von Jülich-Berg (c1416-1475) and Sophie von Sachsen-Lauenburg (c1432-1473) and died 6 September 1511 of unspecified causes. He married Elisabeth von Nassau-Saarbrücken (?-c1478) 1472 JL . He married Sibylle von Brandenburg (1467-1524) 25 July 1481 JL . Notable ancestors include Henry II of England (1133-1189), William I of England (1027-1087), Charlemagne (747-814), Hugh Capet (c940-996), Alfred the Great (849-899). Ancestors are from France, the United Kingdom, Germany, Belgium, the Czech Republic, the Netherlands, Hungary, Spain, Russia, Ukraine, Belarus, Italy, Turkey, the Byzantine Empire, Sweden.


Location, Location, Location ~ Just a few blocks to downtown restaurants, beaches, schools and the Cocoa Beach Country Club & Golf Course, this nearly a 1/4 acre corner lot, 3BR 2BA, 1, 638 sq ft under air, deep water canal property is a well built concrete block cul de sac home mostly in it's original as is condition waiting for your designer touch. Original owner occupied home has always been well maintained with beautiful terrazzo floors under carpet, updated seawall, updated new electrical panel, A/C unit and sprinkler system. Home boasts an eat in kitchen, large family room with a fireplace, laundry/pantry room and a large bonus storage room attached. Bright and sunny home with views of the canal the minute you step in the door. Easy to show ~ must see ~ priced right ~ won't last !


text:
Engineers taking more time to assess Webb's behavior in space before continuing deployments
STORY WRITTEN FOR CBS NEWS & USED WITH PERMISSION
Northrop Grumman teams work with the sunshield on the James Webb Space Telescope during ground testing. Credit: Northrop Grumman
Engineers activating the James Webb Space Telescope decided Sunday to hold off tightening up the observatory's critical sunshade to allow more time to check out the performance of its power systems and overall behavior now that several major deployments are complete.
"Nothing we can learn from simulations on the ground is as good as analyzing the observatory when it's up and running," Bill Ochs, the Webb project manager, said in a NASA blog post Sunday. "Now is the time … to learn everything we can about its baseline operations. Then we will take the next steps."
NASA is not providing "live" coverage of Webb's deployments and has not held a media briefing since the telescope's launch on Christmas Day. But the latest blog post said engineers wanted to more thoroughly characterize the telescope's performance now that it's finally in space while making sure motors needed for sunshade tensioning are at the "optimal" temperatures before proceeding.
No technical details were provided.
Since launch on Christmas Day, Webb has successfully fine-tuned its trajectory with two precision thruster firings, deployed its critical solar panel, unlimbered the high-gain antenna it will use to relay science data back to Earth and extended a "momentum flap" to counteract the destabilizing pressure of the solar wind.
It's also elevated its primary mirror and science instruments by about four feet to further isolate them from the heat generated by on-board electronics and other systems.
On Friday, two telescoping booms extended to either side of two pallets, pulling out and unfolding Webb's tennis court-size sunshade to kick off one of the most complex procedures in the observatory's initial activation.
Made up of five hair-thin Kapton layers, the sunshade is critical to Webb's goal of capturing faint light from the first stars and galaxies to light up in the wake of the Big Bang birth of the cosmos nearly 14 billion years ago.
To register that ancient radiation, now stretched out into the infrared by the expansion of space itself, Webb must be chilled to within 50 degrees of absolute zero, or nearly 400 degrees below zero Fahrenheit. The light- and heat-blocking shield needed to do that, which was folded up for launch like a skydiver's parachute, is now in the process of being extracted.
The two pallets holding the sunshade were deployed and locked in place Tuesday, one on either side of Webb's 21.3-foot primary mirror. On Thursday, protective covers were commanded to roll off each pallet, exposing the still-folded sunshade membranes to space.
The actual deployment began Friday when the two telescoping booms at right angles to the pallets began extending, one at a time, slowly pulling out both sides of the shield and unfolding the membranes in the process.
That work started later than expected to give engineers time to confirm 107 membrane retention devices, used to hold the folded layers in place during launch, had worked as required.
They did, and with both booms extended to give the sunshade its iconic kite-like shape, all five layers must now be pulled taut using motor-driven cables running through scores of pulleys. Tensioning is required to produce a gap between each layer, providing space for excess heat to migrate outward to the sides.
Because the boom extension work took longer than expected, mission managers put tensioning on hold Saturday to give the team a chance to catch its collective breath. Another delay was ordered Sunday, in part to make sure the motors needed for the shade's full extraction were at the required temperatures.
"We've spent 20 years on the ground with Webb, designing, developing and testing," said Mike Menzel, Webb's lead systems engineer. "We've had a week to see how the observatory actually behaves in space. It's not uncommon to learn certain characteristics of your spacecraft once you're in flight. That's what we're doing right now.
"So far, the major deployments we've executed have gone about as smoothly as we could have hoped for. But we want to take our time and understand everything we can about the observatory before moving forward."
Source: Space
The post Engineers taking more time to assess Webb's behavior in space before continuing deployments appeared first on CableFreeTV.
Twitter bans Rep. Marjorie Taylor-Greene's personal account over COVID-19 misinformation
SpaceX's first launch of 2022 will deploy more Starlink internet satellites

redpajama_set_name: RedPajamaCommonCrawl


Chen, G. and Luo, Y. (2016). "A BIM and Ontology-based Intelligent Application Framework", Proceedings of the 2016 IEEE Advanced Information Management, Communicates, Electronic and Automation Control Conference, Xi'an, China.
Wu, W. and Luo, Y. (2016). "Pedagogy and assessment of student learning in BIM and sustainable design and construction", Journal of Information Technology in Construction, Special issue: 9th AiC BIM Academic Symposium & Job Task Analysis Review Conference, Vol. 21, pg. 218-232, http://www.itcon.org/2016/15.
Luo, Y., Wu, W., and Bagdasarov, Z. (2016). "Students' Perceptions of Tablet-Enhanced Learning Environment on the Development of Communication and Critical Thinking Skills: An Interdisciplinary Study", Proceedings of 123rd ASEE Annual Conference & Exposition, New Orleans, LA.
Luo, Y. and Wu, W. (2015). "Sustainable Design with BIM Facilitation in Project-based Learning", Proceedings of the 2015 International Conference on Sustainable Design, Engineering and Construction, Chicago, IL.
Wu, W. and Luo, Y. (2015). "Investigating the Synergies of Sustainability and Building Information Modeling through Collaborative Project-based Learning", Proceedings of 122nd ASEE Annual Conference & Exposition, Seattle, WA.
Luo, Y. (2014). "A Collaborative Learning Community on Service-learning in Energy Conservation and Efficiency", Proceedings of the Annual Conference of ASCE Construction Research Congress: Construction in a Global Network, Atlanta, GA.
Luo, Y. (2013). "A Learning Community on Service-learning in Energy Conservation and Efficiency", Poster presentation at the 16th Annual Continuums of Service Conference, Portland, OR.
Luo, Y. and Becker, G. (2013). "Green Buildings to Sustain California's Central Valley", Proceedings of the 5th International Conference on Construction Engineering and Project Management, Garden Grove, CA.
Crask, L., Luo, Y., and Dyson, A. (2011). "Eco-Village: A Humane Solution to the Homeless", Proceedings of the International Conference on Sustainable Design & Construction.
Luo, Y. (2011). "Green the Campus through Integrated Teaching and Research", Proceedings of the International Conference on Sustainable Design & Construction.
Luo, Y. Crask, L. Dyson, A., Zoghi, M., and Hyatt, B. (2011). "The Eco-Village Experience at California State University, Fresno: An Integrated Approach to Service-Learning", Metropolitan Universities Journal, 21(3), 16-26.
Harmsen, F., Hunsaker, D., Water, P., and Luo, Y. (2008). "Mitigation and Adaptation Strategies for Climate Change in Fresno California", An Assessment Report to the Fresno City Council, CSU Fresno.
Luo, Y. (2008). " Decision Support for Prefabrication Strategy Selection on Building Systems", Ph.D. Dissertation, The Pennsylvania State University.
Luo, Y., Riley, D.R., Horman, M.J., and Kremer, G. (2007). "Optimizing Prefabrication Strategy Selection", Proceedings of the Annual Conference of ASCE Construction Research Congress: A Global Community.
Luo, Y., Riley, D.R., Horman, M.J., and Kremer, G. (2006). "Decision Support Methodology for Prefabrication Decisions on Green Building Projects", Proceedings of the CIB W092 Symposium on Sustainability and Value Through Construction Procurement, Manchester, UK.
Luo, Y. and Riley, D.R. (2005). "First Cost Savings through Design for Deconstruction Strategies", Proceedings of The Building Material Reuse and Recycling: Deconstruction '05 International Conference, Atlanta, Georgia.
Luo, Y., Riley, D.R., and Horman, M.J. (2005). "Lean Principles for Prefabrication in Green Design-build (GDB) Projects", Proceedings of the 13th Annual Conference of IGLC, Sydney, Australia.
Keelor, D.C., Luo, Y., Earls, C.J., and Yulismana, Y. (2004). "Service Load Effective Compression Flange Width in Composite Designs Involving Steel Stringers with FRP Decks", Journal of Composites for Construction, ASCE, Vol.8 No. 4, Reston, Virginia, pp. 289-297.
Luo, Y., Keelor, D.C., and Earls, C.J. (2002). "Field Monitoring of the Boyer Bridge Fiber Reinforced Polymer Deck Installation", Proceedings of the International Bridge Conference, Pittsburgh, Pennsylvania, paper IBC-02-52, CD-ROM.


Luxury motor yacht Mary is a stylish 68 feet Ferretti, which combines comfort and style along with exceptional performance. She is based in the wonderful Greek island of Rhodes (Dodecanese) and she is available for private charters in the Greek islands of Dodecanese (Symi, Halki, Patmos, Kos, Leros etc).
Motor Yacht Mary offers a generous beam which one will usually meet in larger yachts. Sunbathe or enjoy afternoon barbeques under the shade of the bimini on the spacious sun deck. The table on the aft deck offers an ideal environment for breakfast and brunch while appreciating the timeless simplicity and beauty of the seascape view. Unwind with your favorite DVD or CD in the full width Salon, furnished with rich cream leather and fully air conditioned. The raised dining area features a briarwood dining table, comfortably seating eight. Gourmet meals, tailor made to personal taste, will be served from the adjacent galley.
MY Mary has the most up to date navigation and communication equipment, assuring your safety and ease at all times. The carefully chosen crew will attend your every wish and pamper you and your guests. The crew have their own separate quarters and deck access, guaranteeing complete privacy for you and your guests.
The full beam Master cabin features a queen-size bed, video and audio entertainment, writing desk and en-suite facilities. Up to six additional guests can be accommodated in the VIP cabin (also with a queen-size bed and en-suite facilities) and two Twin cabins. All cabins are fully air conditioned.
Click here to download the yacht's e-brochure.


text:
Stephen Fowler Wilson (September 4, 1821 – March 30, 1897) was an American lawyer, politician and judge from Pennsylvania who served as a Republican member of the U.S. House of Representatives for Pennsylvania's 18th congressional district from 1865 to 1869.  He also served as a member of the Pennsylvania State Senate for the 1st district from 1863 to 1865.

Early life and education
Wilson was born in Columbia Township, Pennsylvania the youngest of seven children born to George and Jane Wilson. He studied law under the Honorable James Lowrey and in 1845 was admitted to the Tioga County bar. He practiced law for almost 25 years.  He was originally a Democrat but switched to a Republican when that party was founded.  He held several local offices.

Career
He was a member of the Pennsylvania State Senate for the 1st district from 1863 to 1865 and served in one session after he had been elected a Representative to Congress.  He was a delegate to the 1864 Republican National Convention.

Wilson was elected as a Republican to the Thirty-ninth and Fortieth Congresses.  He was appointed additional judge of the fourth judicial district of Pennsylvania in 1871 to fill a vacancy.  He was elected additional judge and served ten years.  He was appointed associate justice of the supreme court of the Territory of New Mexico by President Chester A. Arthur on October 16, 1884.  He was president judge of the fourth judicial district of Pennsylvania from 1887 to 1889.  He resumed the practice of law in Wellsboro, Pennsylvania, where he died in 1897.  Interment in Wellsboro Cemetery in Wellsboro, Pennsylvania.

Notes

Sources

The Political Graveyard

|-

1821 births
1897 deaths
19th-century American politicians
New Mexico Territory judges
Pennsylvania lawyers
Pennsylvania state court judges
Republican Party Pennsylvania state senators
People from Bradford County, Pennsylvania
Republican Party members of the United States House of Representatives from Pennsylvania
19th-century American judges
19th-century American lawyers

redpajama_set_name: RedPajamaWikipedia


text:
Liqui Moly Asia Pacific's organisational structure updates
With effect from 1 January 2022, Mr Brendan Ang has been appointed as the new Chief Executive Officer / Managing Director of Liqui Moly...
Liqui Moly Asia Pacific to implement first DigiThree next-level secured office solutions
Amsterdam, Singapore , December 7, 2021 – DigiThree Labs, a leader in metaverse-ready, and blockchain-based enterprise solutions, and...
Office Closure Notice
To Our Valued Customers, Please be informed that our office will be closed on Christmas Eve (24/12/21) and New Year Eve (31/12/21)....
© 2020 by Liqui Moly Asia Pacific.
Your Competent Partner for Automotive/Industrial Lubricants, Additives, Chemical Fluids and Care Products.


Floor sawing is an effective way to cut through horizontal reinforced concrete or granite without the need for heavy machinery. Thanks to the precision of using a large spinning circular blade, it's ideal for roadworks and other jobs that require clean lines cut through concrete up to half a metre thick.
We've provided specialist concrete removal services for over 20 years, building up a long list of satisfied clients as we go. Whatever the job, give us a call and we'll send round one of our specialists to assess the work needed. We'll always provide a competitive quote and make sure the work is completed as quickly as possible.
Floor sawing uses a large, spinning, diamond-tipped circular blade to cut a precise line through concrete or asphalt up to half a metre deep. They're mainly used for road works, creating trenches for cables or pipes, or for removing large sections of concrete flooring which can then be broken up with heavier machinery.
We use both interior and exterior floor sawing machines, depending on the location of the contract. Our interior saws are powered by electric motors, which ensure no fumes are given off; ideal for enclosed spaces, whereas exterior saws use diesel engines to spin the blade.
Despite the size of the saws, the amount of dust and mess left behind is minimal and we aim to minimise any disturbance during the work. If noise is an issue, let us know and we'll ensure the work is finished as quickly and as quietly as possible.
Need to remove a large section of concrete floor?


text:
Princess Victoria Style»Handbags Lookbook
Accessories Hats
Crown Princess Victoria of Sweden and Prince Daniel of Sweden arrive to attend the Royal Wedding of Prince William to Catherine Middleton at Westminster Abbey on April 29, 2011 in London, England. The marriage of the second in line to the British throne is to be led by the Archbishop of Canterbury and will be attended by 1900 guests, including foreign Royal family members and heads of state. Thousands of well-wishers from around the world have also flocked to London to witness the spectacle and pageantry of the Royal Wedding.
(April 28, 2011 - Source: Chris Jackson/Getty Images Europe) see more angles »
Princess Victoria Envelope Clutch
Princess Victoria paired her royal Elie Saab dress with a matching envelope clutch.
Browse: Princess Victoria | Envelope Clutch
Show Princess Victoria Looks With:


The Mountain Between Us is a romantic-drama from executive producers Fred Berger (La La Land) and Becki Cross Trujillo (Daredevil, The Lookout). The film stars Kate Winslet and Idris Elba as two strangers whose lives become entwined with one another after a plane crash leaves both of them in the remote wilderness of a snowy landscape with their only hope for survival: each other.
Ben Bass (Idris Elba) and Alex Martin (Kate Winslet) may have not known each other before that fateful plane crash flown by pilot Walter (Beau Bridges), but after crash-landing in a remote area, the two individuals (and a puppy which miraculously survives) become dependent on one another to brave the harsh environment. With no one else to turn to in the desolate snow, Ben and Alex begin to trek across the mountains together as they journey towards their unknown destination. As they come to know each other, a friendship and romance begins to blossom.
A huge part of the reason why this film works well in any measure is because of the chemistry and craft of both Idris Elba and Kate Winslet. Both deliver worthwhile performances which certainly elevate the somewhat unexceptional and conventional storytelling. Winslet and Elba deliver as the lead performers.
The film does have interesting stunts and CGI work in a few scenes. Given that this is primarily a romantic drama, it was a bit of a surprise to see some good work done for the action-oriented scenes (like the plane crash and some of the trekking across the outdoors landscape). It works well for the genre.
The film has some noteworthy productions aspects. The music score composed by Ramin Djawadi (Game of Thrones, Iron Man) is serviceable in the film and fits the mood. It's not amongst the best scores of the year, by any measure, but it's a decent score befitting the film.
The cinematography by Mandy Walker (Hidden Figures, Australia) is one of the strengths: this is a beautifully photographed film with vast outdoor landscapes. The production design by Patrice Vermette (Arrival, Sicario) is effective with regards to presenting the outdoors environments as well. Costumes are designed by Renee Ehrlich Kalfus (Hidden Figures, Chocolat).
Based on the novel by Charles Martin, The Mountain Between Us screenplay was written by Chris Weitz (About a Boy, Rogue One: A Star Wars Story). The story is unexceptional in bringing surprises or deeply woven depth to the characters and story. However, it does a acceptable job of crafting a modestly enjoyable romantic-drama storyline with a strong backdrop.
Directed by Hany Abu-Assad (Paradise Now, Omar), The Mountain Between Us is a lightly entertaining and engaging film. While some of the scenes feel a bit more preposterous than others (for example, the dog in the film often seems barely bothered by being outdoors in the cold), the overall effect of the filmmaking is a enjoyable (if occasionally silly) ride. The film manages to excel primarily through the romance between Winslet and Elba's characters. The effort is a worthwhile one without ever crossing the barrier between mere entertainment and dramatic art.
The Mountain Between Us arrives on Blu-ray with a 1080p high definition presentation. Presented in the original 2.39:1 widescreen aspect ratio, the film has a generally strong presentation with a decent 25 mbps encode. The film has strong colors and the clarity is generally satisfactory overall. While the film has some soft elements and some (very minor) banding in a few moments, it's still a worthy presentation. It could have benefited some from having a stronger encode rate but it's a acceptable presentation which will leave most viewers satisfied.
The audio is presented in DTS-HD Master Audio 5.1 surround. This is a generally strong lossless audio presentation for the film. The surrounds aren't as heavily utilized as one might hope but given the film's mostly romantic-drama genre it isn't surprising the surround mixing is a little limited in scope. The clarity and depth is good (and that is one of the most important parts). Dialogue is clear and easy to understand.
Optional English SDH, Spanish, and French subtitles are included. Optional English descriptive audio in 5.1 surround sound and Spanish/French 5.1 Dolby Digital audio is also included on the release.
Please Note: This release is a combo pack release including a Blu-ray + DVD + Digital HD.
Love and Survival: Creating Chemistry (HD, 13 min.) is a featurette about the film's setting/cold of the outdoors, production, story, and how the cast worked together on the film. This featurette includes behind the scenes footage and interviews with the cast and crew.
Mountain Between Them: Shooting in Isolation (HD, 10 min.) is about shooting outdoors in the cold snowy area, where they did most of the production. Though the production climate was certainly cold, the crew found ways to keep spirits lifted together. Features interviews with production team members and the director.
The Wilds: Survival Stunts (HD, 6 min.) is about the stunts of the film (especially the scene with the airplane crashing and how they made that sequence).
Photo Gallery with several high quality production photograph stills taken during production of the film.
Sneak Peak: Other trailers for recent films by Fox.
The Mountain Between Us is a modestly entertaining romantic-drama featuring excellent performances from Kate Winslet and Idris Elba. The film is a decent audience-pleaser to experience on a rainy afternoon. It's not high-art but it fits the bill for a decent drama that entertains more than it falters.


Applications are invited from suitably qualified applicants for Timecon Kenya Limited Job Vacancy 2018 Interested applicant(s) ought to follow the details and procedures on this page to secure this opportunity.
Timecon Kenya Limited is a Business Software Company established with the goal of providing solutions to businesses of all sizes such as start-ups as well as growing and established organizations so as to improve their business processes.
NB: On successful completion of the internship program, the candidate may be considered for employment.
Interested candidates are requested to submit their application through our careers page on our website or directly send your Curriculum Vitae by 23rd November 2018.


With only a handful of legislative days remaining in 2017, many Americans are waiting with bated breath to find out when (or if) President Trump's proposed tax reform plan will change the tax landscape. As you already know, any changes to tax policy may mean you'll need to revisit certain parts of your estate plan to ensure everything continues to effectively protect your family.
The Trump administration and Congressional Republicans have been busy with many issues, but, as of now, there have not been any significant legislative enactments. Proposals ranging from tax cuts or reform, immigration reform, healthcare reform, and more still loom large.
If Trump's tax plan pushes through as it currently reads, it will create a system with three tax brackets rather than seven. It would also completely eliminate estate and generation-skipping transfer taxes. Mortgage interest deductions would get an overhaul, and most itemized deductions would disappear. The stated goal of the President and Congress is a simplification of the tax rules, but like all prior tax laws, there will still be complexity.
Yes, these are all very substantial changes. And in addition to the uncertainty about how these changes would impact you as an individual, it also still remains to be seen whether this proposal will become law or not. So we find ourselves with two primary considerations: The tax proposal would mean drastic changes, and no one is certain when, or even if, it will go into effect.
Is your estate plan in standby mode?
This means the best course of action is to hold tight until the outcome becomes clear, right? Actually, this attitude could cost you. Now is not the time to wait for news from the government before taking action, even though uncertainty about tax and health care policies may leave you thinking that doing nothing right now is appropriate.
But, the "wait and see" approach can spell disaster for your family if you die or become incapacitated without adequate, up-to-date planning in place — an obsolete estate plan or a plan that doesn't match your current estate planning goals and needs won't serve you no matter what happens in the White House, but an up-to-date one always will. That's why the process of reviewing and revising your wills, trusts, and other estate planning documents now is as important as ever.
Regardless of what the future holds, you should always focus on what you can control. Although changes to tax rates, deductions, exemptions, and so on may impact how much tax must be paid in any given year and the future of the death tax is uncertain, there are no credible proposals that do away with taxes altogether. You must have some plan in place to ensure your financial legacy avoids as much tax as possible, while also achieving all of your other estate planning goals and protecting your family's wealth.
Am I 100% confident that I've named the best possible trustees, executors, agents, and guardians for children?
Am I 100% confident that my children, grandchildren, other relatives, charities, or others designated to receive my estate are the same beneficiaries I would name today?
Do my trust asset schedules accurately list all of my financial accounts, insurance policies, retirement plans, rental or vacation properties, and other assets?
If you aren't entirely comfortable with your answers to the questions above, then the best response to "How do I move forward in uncertain times?" is to get your planning 100% up-to-date by giving us a call so we can ensure it meets your current goals and needs.


The video work History Now consists of two recordings running synchronously. One is from 1993 and shows the artist's mother with her two sons standing on the quay in front of the refugee camp "Flotel Europa". They are making a video letter to the mother's parents who've stayed behind in the midst of the Bosnian war. Juxtaposed on the screen is a recording of the artist's mother today watching and commenting on the 1993 footage. As viewers, we watch today a past along with its protagonist and experience the collapse of time that memory engenders. Tomic's method gives us not only access to the past but also insight into how we understand and connect to it now.


Mathletics-4- Schools is presenting a Micro Mathletics workshop on Wednesday 8th May 2019, from 5 to 7pm.
Mathletics 4 Schools is a unique co-curricular programme for primary schools which combines maths with athletics in a fun way. As an introduction to our Mathletics programme we are offering a workshop which will include instruction on a number of 10 Minute Mathletics games. These games can be incorporated into your everyday maths class and will offer a different more hands on approach to problem solving. The key goal is to improve the overall attitude towards maths at a young age.
Niamh Lawlor of Púca Puppets will take teachers through the basics of puppetry performance for classroom contexts over two sessions, a total of 5 hours of Continuing Professional Development Training. Niamh has extensive experience both as professional performer and in schools, creches and other contexts.
This seminar is aimed at teachers of SPHE and will address ways of educating young people about personal safety, protection and abuse. It will provide an opportunity for participants to familiarise themselves with the Chid Protection Procedures for Primary and Post-Primary Schools 2017 and the PDST Personal Safety lessons developed for Junior Cycle. Full substitution. Booking via sphe.ie only.


text:
Hong Kong cardinal warns priests to 'watch your language…
Hong Kong cardinal warns priests to 'watch your language' in homilies
01 Sep 2020 0 shares 1 views
Published 01 Sep 2020
Washington, D.C. Newsroom, Sep 1, 2020 / 02:10 pm (CNA).- The leader of the Diocese of Hong Kong has written to all the clergy telling them to avoid politics in homilies and admonishing them for "offensive" and "provocative" preaching.
The letter, obtained by CNA, was sent from Cardinal John Ton Hon, the former Bishop of Hong Kong and current administrator of the diocese, on Friday. It came amid ongoing crackdowns by local authorities on pro-democracy demonstrations, after the implementation of the new National Security Law in July.
"The homily is not meant to convey the preacher's personal views (such as his own own view on a social or political issue) but God's message," the letter reminds priests.
"Our homilies must not lose touch with our daily life and the concrete situation of society," the cardinal wrote. "A reference to or brief analysis of current social issues would often be helpful and sometimes even necessary for a homily" with the aim of "encouraging the faithful… to bear Christian witness in social life and in social transformation," he added.
"However, slandering and offensive expressions insinuating or instigating hatred and social disorder are unchristian and inappropriate for the liturgy."
Although the letter does not explicitly reference the law and ongoing political situation, the cardinal warned that priests and deacons "should be well aware that our faithful are all the time listening to what we say and watching what we do."
"Therefore, we have to be prudent and attentive to what we say in our homilies, sermons and speeches."
Cardinal Hon has previously spoken in support of the National Security Law, which came into force July 1, and insisted it would have "no effect" on the local Church's religious freedom. Since the law came into force, Catholic journalists, political activists, and businessmen have been arrested on charges of sedition.
"A church and its sanctuary are sacred places where the faithful come to encounter God," Cardinal Hon wrote Friday. "In a critical time like today, our faithful are hoping to hear something comforting, constructive and encouraging from the preachers during the liturgy."
"Through preaching, we can help ease their mind. This could not be achieved using abusive or provoking expressions."
"Let us in particular watch our language," he said.
Hon's letter to clergy met with opposition in some quarters of the local Church. One priest told CNA it "went down like a bucket of cold sick" with some local Catholics.
The priest, who asked not to be named because of concerns he could be prosecuted under the National Security Law, told CNA on Tuesday that many local Catholics were dismayed by Cardinal Hon's actions.
"The youth of the Church is for democracy, they simply are," he told CNA. "They are looking for leadership, and I doubt you would find any Catholic under 35 here who is not angry and does not see the chancery as siding with the people tear-gassing them in the streets."
"The Church is for human dignity, for human rights. To pretend like we can sing a lullaby for a homily in these times is absurd."
Another local priest, who also asked not to be named, told CNA that the cardinal's letter was meant to have a "chilling effect" on priests speaking about the political situation.
"The government see the presence of very many Catholics in the demonstrations. They hear the words of [Cardinal] Zen and [Cardinal] Bo and they see the Church can be somehow supportive of civil freedoms."
"The cardinal [Hon] does not want the Church to become a target, I think, and he wants people to lay off attacking the government in homilies – he said himself, people are always watching."
The priest said it was important to note that, although many in the Church, especially young Catholics, were sympathetic to the pro-democracy movement, they did not represent the whole diocese.
"Certainly there are Catholics here – including some priests – who very much pro-the government, even pro-the mainland. There is a line between supporting the freedom of the Church, and even social justice, and making the diocese a nakedly political actor."
Headed as a "Fraternal Admonition," Hon's letter recalled the liturgical context of the homily which, he said, should focus on God. He also insisted that clergy respect public health measures in the wake of the coronavirus pandemic and to lead by example.
"In solidarity with our local community and for the sake of the common good, let us take the lead in complying with anti-pandemic measures enforced by the government," Hon wrote.
The letter to clergy was released on the same day Ton intervened to stop a Catholic group affiliated with the diocese from running a prayer for democratic freedoms in Hong Kong in local newspapers.
The Justice and Peace Commission of Hong Kong Catholic Diocese, whose membership is appointed by the bishop, had organized a fundraising campaign to place a prayer in local newspapers which included the sentence "As the city of Hong Kong is under threats of abusive control, we pray for your mercy."
On Saturday, the commission announced it was cancelling the plan to publish the prayer after intervention from Hon.
"Although the diocese supports the kind deeds of prayer for Hong Kong, it does not support the method of fundraising and the content of the prayer to be published by this commission," said the commission in a statement.
Both the letter to clergy and the cancellation of the newspaper prayer follow an Aug. 11 instruction from the diocesan episcopal delegate for education, who wrote to the leaders of Hong Kong's Catholic schools telling them to explain the provisions of the new National Security Law to students, and to encourage patriotic "values" at school.
As part of helping students understand the new law's provisions, teachers are to "foster the correct values on [students'] national identity" and to respect Chinese national symbols including the flag and national anthem, the letter said.
The delegate's letter also called for schools to put in place structures for evaluating "materials, assignments, examination papers and books" used by teachers to prevent "unilateral promotion of political messages, positions or views."
Cardinal Zen: Jimmy Lai arrest is part of campaign of 'political intimidation'
CNA 10 Dec 2020
Washington, D.C. Newsroom, Dec 10, 2020 / 11:00 am (CNA).- Cardinal Joseph Zen Ze-kiun has warned that the recent arrest of Jimmy..
Cardinal Zen: Church is losing 'credibility' to evangelize China
CNA 21 Sep 2020
Hong Kong cardinal: priests must 'watch our language' (CNA)
Catholic Culture 02 Sep 2020


text:
Verwendung von Cookies/
What are cookie files?
Our website www.2n.cz use cookie files. Cookie files are data files which are stored in the devices of individual users (i.e. into your devices) during the visit of our website. Our website thus remembers e.g. settings and operations of individual users; therefore the user does not have to enter the settings which have been already made repeatedly during his/her next visit.
Cookie files enable our websites to distinguish individual users without the necessity of their direct identification. They make their use more comfortable for the user without a risk of misuse of personal data. Cookie files cannot be used to identify the user and/or his/her logging data.
By using our websites, you consent to the use of cookie files.
What kind of cookie files do we use?
Our websites use "long term" cookie files (persistent cookies) and "short term" cookie files (session cookies). Long term cookies remain in the device for a certain time, or also without time limitation also after the browser has been closed. Short term cookies are temporary and remain stored in the browser until the user closes it.
For which purposes do we use cookie files?
These cookie files are necessary for the correct function of our websites. They enable the user to move on our websites and to use their functions. These cookie files do not gather any information on the user which might be used for marketing or recording the sites visited by him/her on the Internet. They are usually intended for the given session and expire when the user terminates his/her visit of such websites. This category of cookie files cannot be banned.
Functional cookie files
These cookie files enable our websites to remember choices and preferences of the user in order not to have to enter them again in the future (such as the user's name, language and region in which the user is located, etc.). The purpose of this is to offer improved and individualized functions for the user. This category of cookie files cannot be banned.
Analytical cookie files
Analytical, i.e. performance, cookies help us to identify and calculate the number of visitors (users) and to find out how they move on our websites when they use them. They gather information such as, the used Internet browsers and operation systems, number of visits on websites, average duration of visit on the websites or information from which site the user clicked on our website. They enable us to identify general patterns of using our website, but not a manner of using it by a particular user. These cookie files usually remain in the user's computer until they are deleted.
Advertising cookie files
These cookie files record visits of our websites by the user, visited sites and links on which the user clicked. They gather information on the user's habits when browsing websites and store the information that the user visited our website. Our company (including third parties authorised by our company) may make use of these information on order to increase the relevance of our websites, contents and displayed advertisements given the specific interests of the user. They are also used to limit the frequency of displayed advertisements and to measure the effectiveness of advertising campaigns. These cookie files usually remain in the user's computer until they are deleted.
Do we use cookie files of third parties?
Besides our own cookie files (cookie files of the first party), we also use cookie files of third parties in certain cases, which enables us to better understand and to improve the effectiveness of our advertisements. These cookies are provided by third parties, and 2N TELEKOMUNIKACE a.s. has no access to data reading or data recording contained in them. We use following tools and services: Google Tag Manager, Google Universal Analytics to analyse the total number of visitors and Google AdWords, including remarketing/retargeting and surveys of demography and user's interests. Tools or applications which are interlinked with providers of services of social media such as, Facebook, Twitter, Google+, LinkedIn or YouTube may also appear on our websites in order to improve the Internet surfing and to simplify the content sharing. We also use Zopim chat for the communication with a visitor and Lucky Orange heat maps.
How can I check or delete cookie files?
Most Internet browsers allow the automatic storing of cookie files. Browser settings may be changed, the use of cookie files may be limited or blocked.
Google Chrome: support.google.com
Mozilla Firefox: support.mozilla.org
Safari: support.apple.com
Internet Explorer: windows.microsoft.com
Opera: help.opera.com


When I think of IIM Udaipur and what makes it different a lot of things come to mind that makes my college different from every other b-school of the country. However, when I thought a little deeper, I came to realize that it is not the beauty of Udaipur or IIM Udaipur. It is not the humongous campus of the college or the lakes that are being built in the campus. It is not the night walks that I take under the starry sky. It is not the beautiful green mountains that I see every morning from my room's window. It is not the academic rigor or the sacrosanct deadlines. It is not the focus on academic excellence or entrepreneurship. It is not how lively the class becomes during discussions. So if none of these amazing things makes IIM Udaipur different then what does? The answer to this is so simple and intuitive that you will not hear it from any of the students here because in our two years journey it gets ingrained in our very nature, and the answer is the people of IIM Udaipur. Yes it is the people of the entire community of IIM Udaipur that makes it different. In the last year that I have spent on campus, there is not a single person that I have come across and thought that he/she is ordinary. Every student/ faculty/ staff member has something so unique in them that the learning opportunity at IIM Udaipur never ends. Every interaction, every communication, every mail that you read, every person you come in contact with you always feel that you can learn something from them. I think this is what makes IIM Udaipur different. It is not the place but the kind of people that come to IIM Udaipur. It is the culture of IIM Udaipur. It is the connect that students have not only with the college and fellow students but also with the alums, with the faculty and staff, with the local community and with nature. Coming to IIM Udaipur was definitely one of the best decisions of my life. It is not only a place which is different from others, but it has also helped me see what is it that is different and unique in me that I can offer the world. A journey at IIM Udaipur is genuinely a Transformational Journey.
Which Group of Aditya Birla do I want to Work with?
If I get an opportunity, I would want to work for Aditya Birla Capital Limited. Being a finance enthusiast seeking career opportunities in the same Aditya Birla Capital is a dream company to work for. It is one of the largest financial services players in India with presence in a wide range of sectors such as asset management, life insurance, private equity, corporate lending, wealth management, etc. It is like a paradise for a finance enthusiast with opportunity to get exposure to almost all the financial sectors which would add tremendously to anyone's knowledge who joins the company. In short, if you are a finance enthusiast than Aditya Birla Capital is the place to be.


Our Little Miracles: Paci Free!
Our house is now a paci-free zone! Right before the boys' second nap on Friday I told them that this would be the last time they got to have their pacifier, and to enjoy it. When they got up from their nap, they kept it for a little bit, but then Phillip got them. First, he was playing with Hayden and Hayden handed it to him and he put it up. Then, he asked Carson to bring his paci to him, and he did. When we put the boys down for bed, they (mainly Hayden) cried off and on for 30 minutes or so - much shorter than I thought. However, Carson woke up around 3:00 crying and of course, woke Hayden up. We were up for a little bit with them, but then put them back down. Phillip says Carson cried for over an hour (I somehow fell asleep). The next day, they had 2 one-hour naps (they usually have 2 two-hour naps). It didn't take them too long to get to sleep Saturday night (being so tired probably helped), and they slept all night! I was SO happy. On Sunday, they had a 3 hour nap (somewhat typical because they miss their first nap), and then slept all night again.
All in all, they are doing good. They still fuss when you put them down to sleep, but not for too long - only a couple of minutes. I didn't really know what to expect, but from what I hear, they are doing really good. I just hope I found all of the pacifiers we have in the house and they don't find one and we have to start over!
I talked Phillip into taking the boys' pictures without pacifiers while they were sleeping. Carson cooperated and stayed asleep, but Hayden woke up for the picture (but quickly went back to sleep). They are such good boys!
I just love to check your blog site and watch these boys grow up!
Keep it up for all of us that don't get to see you very often. Even if I only live in Pocahontas.