text:
Influence of age and 17β-estradiol on kisspeptin, neurokinin B, and prodynorphin gene expression in the arcuate-median eminence of female rhesus macaques
Dominique H. Eghlidi, Gwendolen E. Haley, Nigel C. Noriega, Steven G. Kohama, Henryk F. Urbanski
The neuropeptides kisspeptin, neurokinin B, and dynorphin A (collectively abbreviated as KNDy) are, respectively, encoded by KiSS-1, NKB, and PDYN and are coexpressed by neurons of the hypothalamic arcuate nucleus (ARC). Here, using quantitative real-time PCR, we examined age-related changes in the expression of genes encoding KNDy and associated receptors G protein-coupled receptor 54 (encoded by GPR54), neurokinin 3 receptor (encoded by NK3), and κ-opioid receptor (encoded by KOR), in the female rhesus macaque ARC-median eminence (ARC-ME). Expression of KiSS-1 and NKB was highly elevated in old perimenopausal compared with young or middle-aged premenopausal animals. To test whether these age-related changes could be attributed to perimenopausal loss of sex steroids, we then examined KNDy, GPR54, NK3, and KOR expression changes in response to ovariectomy (OVX) and exposure to 17β-estradiol (E2). Short-term (7 months) OVX (with or without 1 month of estrogen replacement) failed to modulate the expression of any of the KNDy-related genes. In contrast, long-term (∼4 yr) OVX significantly increased KiSS-1 and NKB expression, and this was reversed by E2 administration. Finally, we examined the expression of KNDy-related genes in young adult females during the early follicular, late follicular, or midluteal phases of their menstrual cycle but found no difference. Together, the results suggest that short-term alterations in circulating E2 levels, such as those occurring during the menstrual cycle, may have little effect on the ARC-ME expression of KNDy and associated receptors. Nevertheless, they clearly demonstrate that loss of ovarian steroid negative feedback that occurs during perimenopause plays a major role in modulating the activity of KNDy circuits of the aging primate ARC-ME.
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preproenkephalin Medicine & Life Sciences 96%
Kisspeptins Medicine & Life Sciences 94%
Median Eminence Medicine & Life Sciences 83%
Arcuate Nucleus of Hypothalamus Medicine & Life Sciences 81%
Macaca mulatta Medicine & Life Sciences 68%
Estradiol Medicine & Life Sciences 57%
Gene Expression Medicine & Life Sciences 39%
Menstrual Cycle Medicine & Life Sciences 27%
Eghlidi, D. H., Haley, G. E., Noriega, N. C., Kohama, S. G., & Urbanski, H. F. (2010). Influence of age and 17β-estradiol on kisspeptin, neurokinin B, and prodynorphin gene expression in the arcuate-median eminence of female rhesus macaques. Endocrinology, 151(8), 3783-3794. https://doi.org/10.1210/en.2010-0198
Influence of age and 17β-estradiol on kisspeptin, neurokinin B, and prodynorphin gene expression in the arcuate-median eminence of female rhesus macaques. / Eghlidi, Dominique H.; Haley, Gwendolen E.; Noriega, Nigel C. et al.
Eghlidi, DH, Haley, GE, Noriega, NC, Kohama, SG & Urbanski, HF 2010, 'Influence of age and 17β-estradiol on kisspeptin, neurokinin B, and prodynorphin gene expression in the arcuate-median eminence of female rhesus macaques', Endocrinology, vol. 151, no. 8, pp. 3783-3794. https://doi.org/10.1210/en.2010-0198
Eghlidi DH, Haley GE, Noriega NC, Kohama SG, Urbanski HF. Influence of age and 17β-estradiol on kisspeptin, neurokinin B, and prodynorphin gene expression in the arcuate-median eminence of female rhesus macaques. Endocrinology. 2010 Aug;151(8):3783-3794. doi: 10.1210/en.2010-0198
Eghlidi, Dominique H. ; Haley, Gwendolen E. ; Noriega, Nigel C. et al. / Influence of age and 17β-estradiol on kisspeptin, neurokinin B, and prodynorphin gene expression in the arcuate-median eminence of female rhesus macaques. In: Endocrinology. 2010 ; Vol. 151, No. 8. pp. 3783-3794.
@article{ae6b5af223b14403b94743deaee84efc,
title = "Influence of age and 17β-estradiol on kisspeptin, neurokinin B, and prodynorphin gene expression in the arcuate-median eminence of female rhesus macaques",
abstract = "The neuropeptides kisspeptin, neurokinin B, and dynorphin A (collectively abbreviated as KNDy) are, respectively, encoded by KiSS-1, NKB, and PDYN and are coexpressed by neurons of the hypothalamic arcuate nucleus (ARC). Here, using quantitative real-time PCR, we examined age-related changes in the expression of genes encoding KNDy and associated receptors G protein-coupled receptor 54 (encoded by GPR54), neurokinin 3 receptor (encoded by NK3), and κ-opioid receptor (encoded by KOR), in the female rhesus macaque ARC-median eminence (ARC-ME). Expression of KiSS-1 and NKB was highly elevated in old perimenopausal compared with young or middle-aged premenopausal animals. To test whether these age-related changes could be attributed to perimenopausal loss of sex steroids, we then examined KNDy, GPR54, NK3, and KOR expression changes in response to ovariectomy (OVX) and exposure to 17β-estradiol (E2). Short-term (7 months) OVX (with or without 1 month of estrogen replacement) failed to modulate the expression of any of the KNDy-related genes. In contrast, long-term (∼4 yr) OVX significantly increased KiSS-1 and NKB expression, and this was reversed by E2 administration. Finally, we examined the expression of KNDy-related genes in young adult females during the early follicular, late follicular, or midluteal phases of their menstrual cycle but found no difference. Together, the results suggest that short-term alterations in circulating E2 levels, such as those occurring during the menstrual cycle, may have little effect on the ARC-ME expression of KNDy and associated receptors. Nevertheless, they clearly demonstrate that loss of ovarian steroid negative feedback that occurs during perimenopause plays a major role in modulating the activity of KNDy circuits of the aging primate ARC-ME.",
author = "Eghlidi, {Dominique H.} and Haley, {Gwendolen E.} and Noriega, {Nigel C.} and Kohama, {Steven G.} and Urbanski, {Henryk F.}",
T1 - Influence of age and 17β-estradiol on kisspeptin, neurokinin B, and prodynorphin gene expression in the arcuate-median eminence of female rhesus macaques
AU - Eghlidi, Dominique H.
AU - Haley, Gwendolen E.
AU - Noriega, Nigel C.
AU - Kohama, Steven G.
AU - Urbanski, Henryk F.
N2 - The neuropeptides kisspeptin, neurokinin B, and dynorphin A (collectively abbreviated as KNDy) are, respectively, encoded by KiSS-1, NKB, and PDYN and are coexpressed by neurons of the hypothalamic arcuate nucleus (ARC). Here, using quantitative real-time PCR, we examined age-related changes in the expression of genes encoding KNDy and associated receptors G protein-coupled receptor 54 (encoded by GPR54), neurokinin 3 receptor (encoded by NK3), and κ-opioid receptor (encoded by KOR), in the female rhesus macaque ARC-median eminence (ARC-ME). Expression of KiSS-1 and NKB was highly elevated in old perimenopausal compared with young or middle-aged premenopausal animals. To test whether these age-related changes could be attributed to perimenopausal loss of sex steroids, we then examined KNDy, GPR54, NK3, and KOR expression changes in response to ovariectomy (OVX) and exposure to 17β-estradiol (E2). Short-term (7 months) OVX (with or without 1 month of estrogen replacement) failed to modulate the expression of any of the KNDy-related genes. In contrast, long-term (∼4 yr) OVX significantly increased KiSS-1 and NKB expression, and this was reversed by E2 administration. Finally, we examined the expression of KNDy-related genes in young adult females during the early follicular, late follicular, or midluteal phases of their menstrual cycle but found no difference. Together, the results suggest that short-term alterations in circulating E2 levels, such as those occurring during the menstrual cycle, may have little effect on the ARC-ME expression of KNDy and associated receptors. Nevertheless, they clearly demonstrate that loss of ovarian steroid negative feedback that occurs during perimenopause plays a major role in modulating the activity of KNDy circuits of the aging primate ARC-ME.
AB - The neuropeptides kisspeptin, neurokinin B, and dynorphin A (collectively abbreviated as KNDy) are, respectively, encoded by KiSS-1, NKB, and PDYN and are coexpressed by neurons of the hypothalamic arcuate nucleus (ARC). Here, using quantitative real-time PCR, we examined age-related changes in the expression of genes encoding KNDy and associated receptors G protein-coupled receptor 54 (encoded by GPR54), neurokinin 3 receptor (encoded by NK3), and κ-opioid receptor (encoded by KOR), in the female rhesus macaque ARC-median eminence (ARC-ME). Expression of KiSS-1 and NKB was highly elevated in old perimenopausal compared with young or middle-aged premenopausal animals. To test whether these age-related changes could be attributed to perimenopausal loss of sex steroids, we then examined KNDy, GPR54, NK3, and KOR expression changes in response to ovariectomy (OVX) and exposure to 17β-estradiol (E2). Short-term (7 months) OVX (with or without 1 month of estrogen replacement) failed to modulate the expression of any of the KNDy-related genes. In contrast, long-term (∼4 yr) OVX significantly increased KiSS-1 and NKB expression, and this was reversed by E2 administration. Finally, we examined the expression of KNDy-related genes in young adult females during the early follicular, late follicular, or midluteal phases of their menstrual cycle but found no difference. Together, the results suggest that short-term alterations in circulating E2 levels, such as those occurring during the menstrual cycle, may have little effect on the ARC-ME expression of KNDy and associated receptors. Nevertheless, they clearly demonstrate that loss of ovarian steroid negative feedback that occurs during perimenopause plays a major role in modulating the activity of KNDy circuits of the aging primate ARC-ME.


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\section{Introduction}

Consider a wireless cellular system with $K$ user terminals (UTs)
per cell where all users share the same bandwidth and
Base Stations (BSs) are arranged on a uniform grid on a line (see Fig.~\ref{Fig_infinite_linear_cellular}).
This model was pioneered by Wyner in
\cite{Wyner94} under a very simplified channel gain assumption,
where the path gain to the closest BS is 1, the
path gain to adjacent BSs is $\alpha$ and it is zero elsewhere.
Wyner considered optimal joint processing of all base stations.
Later, Shamai and Wyner \cite{Shamai97} considered a similar model
with frequency flat fading and more conventional decoding schemes,
ranging from the standard separated base station processing to some forms
of {\em limited cooperation}. A very large literature followed and extended these
works in various ways (see for example \cite{Somekh97,Sanderovich07}).

In this paper we focus on the uplink of a {\em conventional system},
such that each BS decodes only the users in its own cell and treats inter-cell interference
as noise. We extend the model in two directions: 1) we consider realistic propagation channels
determined by a position-dependent path loss,
and by a slowly time-varying frequency-selective fading channel;
2) we compare the {\em optimal} isolated cell delay-limited scheme \cite{Hanly98}
with the Proportional Fair Scheduling (PFS) scheme \cite{Bender00,Tse02}.

In a delay-limited scheme each user transmits at a fixed rate in
each fading block, and the system uses power control in order to
cope with the time-varying channel conditions \cite{Hanly98}. A
delay-limited system achieves ``hard fairness'' (HF), in the sense
that each user transmits at its own desired rate, independently of
the fading channel realization. On the other hand, generally
higher throughput can be achieved by relaxing the fixed rate per slot constraint.
Under variable rate allocation, the sum throughput is maximized by
letting only the user with the best channel transmit in each slot \cite{Tse98}.
However, if users are affected by different distance-dependent path losses
that change on a time-scale much slower than the small scale fading,
this strategy may result in a very unfair resource allocation.
In this case, PFS achieves a desirable tradeoff, by maximizing the geometric
mean of the long-term average throughputs of the users \cite{Tse02}.

HF and PFS have been compared in terms of system throughput versus $E_b/N_0$ in
\cite{Caire07} for the single-cell case.  This comparison is indeed relevant:
HF models how voice-based systems work today
and how Quality-of-Service guaranteed systems  will work in the future
(each user makes a rate request and the system struggles to accommodate it).
In contrast, PFS is being implemented in the so-called EV-DO 3rd
generation systems \cite{Bender00} in order to take advantage of delay-tolerant
data traffic.  Hence, a meaningful question is: {\em what system capacity loss is to be expected by imposing hard-fairness?}
In this paper we address this question by extending the results of \cite{Caire07}
to the multicell case with conventional decoding (i.e., without joint processing of the BSs).

\section{System Model}

Each cell experiences interference from the signals transmitted by UTs
in other cells.  Frequency-selectivity is modeled by considering $M$ parallel
frequency-flat subchannels.  Roughly speaking, we may identify
$M$ with the number of fading coherence bandwidths in the system
signal bandwidth \cite{Tse05,Biglieri98,Proakis01}.
The received signal at BS $n$ in subchannel $m$
is given by
\begin{equation}
r^m (n) = \sum_{j=-\infty}^{\infty} \sum_{k=1}^K h_k^m(n,j)
x_k^m(j) + z^m(n)
\end{equation}
where $h_k^m(n,j)$ denotes the $m$-th subchannel gain from user
$k$ at cell $j$ to cell $n$, and $x_{k}^m(j)$ denotes the signal
of $m$-th subchannel transmitted by user $k$ in cell $j$, and
$z^m(n)$ is an additive white Gaussian noise with variance $N_0$.
The channel (power) gain is given by $g_k^m(n,j)=|h_k^m(n,j)|^2$
and the transmit power of a user is given by
$\mbox{\bb E}[|x_k^m(j)|^2]=E_k^m(j)$.

We model the channel gain as the product of two terms, $g_k^m(n,j)=s_k(n,j) f_k^m(n,j)$,
where $s_k(n,j)$ denotes a frequency-flat path gain that depends on
the distance between the BS and the UT, and $f_k^m(n,j)$ is a
``small-scale'' fading term that depends on local scattering environment around
user terminal \cite{Proakis01}.
These two components are mutually independent as they are due to
different propagation effects.
Path loss takes the expression $s_k(n,j) = d_k(n,j)^{-\alpha}$,
where $d_k(n,j)$ denotes the the distance from base station $n$ to
user $k$ in cell $j$ and $\alpha$ is the path loss exponent. We
assume that UTs are not located in a forbidden region at distance less than
$\delta$ from the BS so that the path loss does not diverge.
When the users are uniformly distributed in each cell, the cdf of
$s\equiv s_k(n,n)$ is given by
\begin{equation}
F_s(x) = \left\{ \begin{array}{ll}
  0, & x<r^{-\alpha} \\
  1-\frac{x^{-\frac{1}{\alpha}}-\delta}{r-\delta}, & r^{-\alpha} \leq x < \delta^{-\alpha} \\
  1, & x\geq \delta^{-\alpha}
\end{array} \right.  \label{pathloss_cdf}
\end{equation}
where the cell radius is $r$ and the minimum distance between BSs is $D = 2r$.
The distance $d_k(n,j)$, $n\neq j$, is given by $d_k(n,j) = |n-j|D - d_k(j,j)$
or $d_k(n,j) = |n-j|D + d_k(j,j),$ depending on the location of
user $k$ in cell $j$ as shown in Fig.~\ref{Fig_pathloss_model}.
Consequently, the path loss $s_k(n,j)$ can be expressed as
\begin{equation} s_k(n,j)=\theta_k(n,j)
\left(|n-j|D-s_k(j,j)^{-\frac{1}{\alpha}}\right)^{-\alpha} +
(1-\theta_k(n,j))
\left(|n-j|D+s_k(j,j)^{-\frac{1}{\alpha}}\right)^{-\alpha},
\label{s_k(n,j)}
\end{equation}
where $\theta_k(n,j)$ takes on values 0 or 1 with equal probability since
the UTs are distributed uniformly in each cell.
We assume that the path losses change in time on a very slow scale,
and can be considered as random, but constant, over the whole duration
of transmission. In contrast, the small-scale fading changes relatively rapidly, even
for moderately mobile users \cite{Tse05}. We assume Rayleigh block-fading, changing in an
ergodic stationary manner from block to block, i.i.d., on the $M$ subchannels,
$F_{f}(x)=1-e^{-x}$.

As in \cite{Shamai97}, we assume that all users send independently generated Gaussian random codes.
Let $R_k^m(n)$ denote the rate per symbol allocated by user $k$ in cell $n$ on
subchannel $m$. When outer-cell interference is akin Gaussian noise,
the uplink {\em capacity} region of cell $n$ for fixed channel gains is given by the set of inequalities
\begin{equation}
\sum_{k\in S} R_k^m (n) \leq \log\left(1+\frac{\sum_{k\in S}
g_k^m(n,n) E_k^m(n)}{N_0+I^m(n)} \right) \label{mac_capacity}
\end{equation}
for all $S \subseteq \{1,2,\cdots,K \}$, where the interference at cell $n$ on subchannel $m$ is given by
\begin{equation}
I^m(n) = \sum_{j\neq n}\sum_{k=1}^K g_k^m(n,j) E_k^m(j).
\end{equation}
The capacity region of the $M$ parallel channel case can be
achieved by splitting each user information into $M$ parallel
streams and sending the independent codewords over parallel
channels. The aggregate rate of user $k$ at cell $n$ is given by
\begin{eqnarray}
R_k(n) = \sum_{m=1}^M R_k^m(n),~~~k=1,2,\cdots,K. \label{R_k}
\end{eqnarray}

\subsection{Delay-Limited Systems}

In a delay-limited system, the rates $R_k(n)$ are fixed \emph{a
priori}, and the system allocates the transmit energies in order
such that the rate $K$-tuple $(R_1(n),\cdots,R_K(n))$ is
inside the achievable region in each fading block \cite{Hanly98, Caire07}.
We define the system $E_b/N_0$ under a coding strategy that
supports user rates $(R_1(n),\cdots,R_K(n))$ as
\begin{eqnarray}
\left( \frac{E_b}{N_0} \right)_{\rm sys} = \frac{1}{N_0 \Gamma}
\sum_{k=1}^K \sum_{m=1}^M E_k^m(n), \label{Eb_N0_def}
\end{eqnarray}
where the total number of bits per cell, $\Gamma$, is given by
$\Gamma=\sum_{k=1}^K R_k(n)$. \footnote{Note that we can omit the
cell index $n$ in defining $(E_b/N_0)_{\rm sys}$ as each cell is
symmetric under the assumptions of the infinite linear cellular
model and the symmetric channel distributions.} The system
spectral efficiency $\textsf{C}$ is given by $\textsf{C} =
\frac{\Gamma}{M}$ and it is expressed in bits per second per hertz
(bit/s/Hz) or, equivalently, in bits per dimension.

For given user rates $(R_1(n),\cdots,R_K(n))$, we allocate the
partial rates $R_k^m(n)$ under the constraints (\ref{R_k}) in
order to minimize $(E_b/N_0)_{\rm sys}$. As a subproblem for this
optimization problem, we first consider the $m$-th subchannel
energy allocation assuming that the partial rates and the
interference $I^m(n)$ are given. Thanks to the fact that the
received energy region is a contra-polymatrid \cite{Hanly98}, the
optimal energy allocation is given explicitly as
\begin{eqnarray}
E_{\pi_k^m(n)}^m(n) = \frac{N_0+I^m(n)}{g_{\pi_k^m(n)}^m(n,n)}
\left[\exp\left(\sum_{i\leq k} R_{\pi_i^m(n)}^m(n) \right) -
\exp\left( \sum_{i<k} R_{\pi_i^m(n)}^m(n) \right) \right],
\label{opt_energy}
\end{eqnarray}
where $\pi^m(n)$ is the permutation of $\{1,2,\cdots,K\}$ that
sorts the channel gains in increasing order, i.e.,
$g_{\pi_1^m(n)}^m(n,n) \leq \cdots \leq g_{\pi_K^m(n)}^m(n,n)$ and
the decoding order at base station $n$ is given by $\pi_K^m(n)$
(decoded first), $\pi_{K-1}^m(n), \cdots, \pi_1^m(n)$ (decoded
last).

Since the energy allocation in (\ref{opt_energy}) is the minimum
sum energy allocation for given partial rates, it remains to
minimize the total energy by optimizing the partial rates under the constraints
(\ref{R_k}). Inserting (\ref{opt_energy}) into (\ref{Eb_N0_def}),
we obtain the optimization problem for minimum system $E_b/N_0$
\begin{eqnarray}
\min \frac{1}{N_0 \Gamma} \sum_{k=1}^K \sum_{m=1}^M
\frac{N_0+I^m(n)}{g_{\pi_k^m(n)}^m(n,n)}
\left[\exp\left(\sum_{i\leq k} R_{\pi_i^m(n)}^m(n) \right) -
\exp\left( \sum_{i<k} R_{\pi_i^m(n)}^m(n) \right) \right]
\label{opt_prob}
\end{eqnarray}
under the constraints (\ref{R_k}). The interference experienced by
base station $n$ on subchannel $m$ takes on the expression
\begin{equation}
I^m(n) = \sum_{j\neq n}\sum_{k=1}^K g_{\pi_k^m(n)}^m(n,j)
\frac{N_0+I^m(j)}{g_{\pi_k^m(j)}^m(j,j)}
\left[\exp\left(\sum_{i\leq k} R_{\pi_i^m(j)}^m(j) \right) -
\exp\left( \sum_{i<k} R_{\pi_i^m(j)}^m(j) \right) \right]
\label{interference}
\end{equation}
because the partial rate allocation is performed locally at each
cell. An operating point $(\left  (\frac{E_b}{N_0} \right )_{\rm sys}, \textsf{C})$ on
the power/spectral efficiency plane is a function of both the
signaling strategy and the individual user rates as well as of
the channel gain joint distribution.

\subsection{Delay-Tolerant Systems}

In a delay-tolerant system, the user rates can be adapted
according to their instantaneous channel condition to achieve
higher throughput at the cost of increasing delay. We consider the
constant power allocation and let $\rho=E_{tot}/N_0$ denote the
transmit SNR in each slot. Note that the water-filling power
allocation tends to the constant power allocation as SNR
increases. We also assume that the channel gains are independent
but not necessarily identically distributed across the users, and
symmetrically distributed across the subchannels, that is, for any
permutation $\pi$ of $\{1,2,\cdots,M\}$, the joint cumulative
distribution function of channel gains satisfy
$F(g_k^1(n,j),\cdots,g_k^M(n,j))=F(g_k^{\pi_1}(n,j),\cdots,g_k^{\pi_M}(n,j))$
for all $k$, $n$, $j$. This means that no subchannel is
statistically worse or better than any other.

PFS allocates slots fairly among users in the case of a near-far situation
\cite{Tse02}. The PFS algorithm serves user $k$ on subchannel
$m$ in cell $n$ if $\hat{k}_m(n) = k$, where
\begin{eqnarray}
\hat{k}_m(n) = \arg \max_{k'=1,\cdots,K} \frac{1}{T_{k'}(n)} \log
\left(1+ \frac{\rho g_{k'}^m(n,n)}{1+\rho \sum_{j\neq n}
g_{\hat{k}_m(j)}^m(n,j)} \right). \label{PFS}
\end{eqnarray}
$\hat{k}_m(j)$ denotes the index of the user selected for
transmission on $m$-th subchanel in cell $j$ by the PFS scheduling
and $T_k(n)$ denotes the long-term average user throughput of user
$k$ in cell $n$.

\section{Delay-Limited Systems for a Large Number of Users}

The optimization problem (\ref{opt_prob}) for fixed inter-cell
interference is convex, but it does not yield a closed form
solution. In order to gain insight into the problem we investigate
the asymptotic case for $K \rightarrow \infty$, for which a closed
form solution exists. We make the following assumptions:

\newcounter{steps}
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  \setlength{\leftmargin}{8.6mm}
  \setlength{\labelwidth}{6.0mm}
  \setlength{\labelsep}{2.7mm}
 }
 \item $M$ is fixed while $K$ becomes arbitrarily large.
 \item As $K\rightarrow \infty$, the empirical distributions of
 $s_k(n,n)$, $\{ f_k^m(n,j) : m = 1,\ldots,M\}$ and of $\theta_k(n,j)$
converge almost surely to given deterministic cdfs,
$F_s(x)$, $F_f(y_1,\cdots,y_M)$, and $F_{\theta}(\phi)$,
respectively.
The cdf $F_f(y_1,\cdots,y_M)$ is symmetric, with identical marginal cdfs $F_f(y)$.
The cdf $F_\theta(\phi)$ has two mass points at 0 and $1$, with equal probability mass $1/2$.

 \item For a given system throughput $\Gamma$, the user individual
rates are given by $R_k(n)=\frac{\Gamma}{K}\nu_k(n)$, where
$\nu_k(n)$ is the rate allocation factor for user $k$ in cell $n$.
As $K\rightarrow \infty$, the empirical rate distribution
converges almost surely to a given deterministic cdf
$F_{\nu}(\mu)$ with mean 1 and support in $[a,b]$, where $0\leq
a\leq b<\infty$ are constants independent of $K$.

 \item The rate allocation factors are fixed \emph{a priori},
independently of the realization of the channel gains. Therefore,
the empirical joint distribution of
$\{(s_k(n,n),f_k^1(n,j),\cdots,f_k^M(n,j),$
$\theta_k(n,j),\nu_k(n) ):k=1,\cdots,K\}$ converges
to the product cdf $F_s(x)F_f(y_1,\cdots,y_M)F_{\theta}(\phi)F_{\nu}(\mu)$.\footnote{
We remark here that this assumption reflects the delay-limited nature
of the problem: the user rates are fixed \emph{a priori} and
independently of the channel realization.}
\end{list}

\subsection{Asymptotic Performance}

In the single cell case, the minimum $(E_b/N_0)_{\rm sys}$ for
given $\textsf{C}$ is given by \cite{Caire07}
\begin{equation}
\left( \frac{E_b}{N_0} \right)_{\rm sys}^{\rm SC} = \log(2) \int
2^{\textsf{C} G_M(x)} \frac{dG_M(x)}{x}.
\label{Eb_N0_SC}
\end{equation}
where $G_M(x)$ denotes the cdf of $s \max\{f^1,\ldots,f^M\}$,
where $s \sim F_s(x)$ and $(f^1,\ldots,f^M) \sim
F_f(y_1,\ldots,y_M)$.  For the infinite linear array cellular
model, we establish the following result.

\vspace{5pt}
\begin{theorem} \label{Theorem1}
Under the assumptions A1, A2, A3, and A4, as $K\rightarrow \infty$
the minimum $(E_b/N_0)_{\rm sys}$ for given system spectral
efficiency $\textsf{C}$ in an infinite linear array cellular model
is given by
\begin{equation}
\left( \frac{E_b}{N_0} \right)_{\rm sys}^{\rm MC} = \frac{\log(2)
\int 2^{\textsf{C} G_M(x)}
\frac{d G_M(x)}{x}} {1-\textsf{C} \log(2) \iint
2^{\textsf{C} G_M(xy)} \Phi(x) \frac{dF_s(x)
dH_M(y)}{xy} }. \label{Eb_N0_MC}
\end{equation}
where $H_M(y)$ denotes the cdf of $\max\{f^1,\ldots,f^M\}$ where
$(f^1,\ldots,f^M) \sim F_f(y_1,\ldots,y_M)$, and where the
function $\Phi(s)$ is given by
\begin{equation}
\Phi(x) =
D^{-\alpha}\left(\zeta\left(\alpha,1-\frac{x^{-1/\alpha}}{D}\right)
+\zeta\left(\alpha,1+\frac{x^{-1/\alpha}}{D}\right) \right),
\label{Psi_func}
\end{equation}
where $\zeta(a,x)$ is the Riemann zeta function
\cite{Gradshteyn00}.
The minimum $(E_b/N_0)_{\rm sys}$ is achieved
by letting each user transmit on its best subchannel only, and by
using superposition coding and successive decoding on each
subchannel.
\end{theorem}
\begin{proof}
See Appendix A.
\end{proof}
\vspace{5pt}

If we compare (\ref{Eb_N0_MC}) with (\ref{Eb_N0_SC}), we can
observe three facts: First, $(E_b/N_0)_{\rm sys}^{\rm MC}$ is
higher than $(E_b/N_0)_{\rm sys}^{\rm SC}$ for all $\textsf{C}$.
So, for the multiple cell case, the higher $(E_b/N_0)_{\rm sys}$
is required to achieve the same system spectral efficiency
$\textsf{C}$ due to intercell interference. Second,
$(E_b/N_0)_{\rm sys}^{\rm MC}$ converges to $(E_b/N_0)_{\rm
sys}^{\rm SC}$, as $\textsf{C}\rightarrow 0$. For low spectral
efficiency, low transmit energy causes negligible interference to
other cells, which effectively turns the multiple cell case into
the single cell (interference-free) case. Third, there exists a
spectral efficiency limit in the multiple cell case because
$(E_b/N_0)_{\rm sys}^{\rm SC}$ tends to infinity as $\textsf{C}
\rightarrow \textsf{C}_0$, where $\textsf{C}_0$ is the root of the
denominator of the right hand side in (\ref{Eb_N0_MC})
\begin{equation}
1-\textsf{C}_0 \log(2) \iint 2^{\textsf{C}_0 G_{M}(xy)}
\Phi(x) \frac{dF_s(x) dH_M(y)}{xy}=0. \label{C_0}
\end{equation}
Any spectral efficiency can be supported in the single cell case
as long as the transmit energy is high. However, in the multiple
cell case, high transmit energy to support high spectral
efficiency may cause significant interference to other cells, thus
demanding more transmit energy to combat interference from other
cells. Eventually, additional energy to combat interference
becomes tremendously large even in a finite spectral efficiency.
So, the multiple cell case is an \emph{interference-limited}
system due to this fundamental spectral efficiency limit.

Fig.~\ref{Fig_capacity_HFS} shows the spectral efficiency achieved
by the delay-limited systems for infinite number of users and we
can confirm the three properties described above. The
spectral efficiency limits for $M=10$ and $M=20$ are
$\textsf{C}_0=4.2$ bits/s/Hz and $\textsf{C}_0=4.73$ bits/s/Hz,
respectively.

\subsection{Asymptotic Results for Simplified Multiple Cell Model}

A well-known approach to the computation of system
capacity of a conventional multi-cell
system consists of modeling the inter-cell interference power as
proportional to the total transmit power in each cell
\cite{Viterbi95}. In this section we validate this approach and
compute explicitly the proportionality constant. Let the
interference level experienced by each cell be given by
\begin{equation}
I^m = \beta \sum_{k=1}^K E_k^m. \label{simple_interference}
\end{equation}
where we drop the cell index $n$ for simplicity. In
(\ref{simple_interference}), the ratio of interference and total
transmit power is given by $\beta$. The capacity region of cell
$n$ for cell-site optimal joint decoding is given by
\begin{equation}
\sum_{k\in S} R_k^m \leq \log\left(1+\frac{\sum_{k\in S} g_k^m
E_k^m}{N_0+I^m} \right) \label{simple_mac_capacity}
\end{equation}
for all $S \subseteq \{1,2,\cdots,K \}$. For given $I^m$, the
minimum total energy supporting a given rate ${\bf
R}^m=(R_1^m,\cdots,R_K^m)$ with gains ${\bf
g}^m=(g_1^m,\cdots,g_K^m)$ is given by
\begin{eqnarray}
E_{\pi_k^m}^m = \frac{N_0+I^m}{g_{\pi_k^m}^m}
\left[\exp\left(\sum_{i\leq k} R_{\pi_i^m}^m \right) - \exp\left(
\sum_{i< k} R_{\pi_i^m}^m \right) \right],
\label{simple_opt_energy}
\end{eqnarray}
where, as before, $\pi^m$ is sorting permutation of the channel gains in increasing order.
We have:

\vspace{5pt}
\begin{theorem} \label{Theorem2}
Under the assumptions A1, A2, A3, and A4, as $K\rightarrow \infty$
the minimum $(E_b/N_0)_{\rm sys}$ for given system spectral
efficiency $\textsf{C}$ in a simple multiple cell model is given by
\begin{equation}
\left( \frac{E_b}{N_0} \right)_{\rm sys}^{\rm MC} = \frac{\log(2)
\int 2^{\textsf{C} G_{M}(x)}
\frac{dG_M(x)}{x}} {1- \beta \textsf{C} \log(2) \int
2^{\textsf{C} G_M(x)}\frac{dG_M(x)}{x}}
\label{simple_Eb_N0_MC}
\end{equation}
The minimum $(E_b/N_0)_{\rm sys}$ is achieved by letting each user
transmit on its best subchannel only, and by using superposition
coding and successive decoding on each subchannel.
\end{theorem}
\begin{proof}
See Appendix C.
\end{proof}
\vspace{5pt}

By equating (\ref{Eb_N0_MC}) and (\ref{simple_Eb_N0_MC}), we can
solve for $\beta$ and obtain
\begin{equation}
\beta = \frac{\iint 2^{\textsf{C} G_M(xy)} \Phi(x)
\frac{dF_s(x) dH_M(y)}{xy}} {\int 2^{\textsf{C}
G_M(x)}\frac{dG_M(x)}{x}}.
\label{eq_beta}
\end{equation}
Since $\zeta(a,1-x)+\zeta(a,1+x)$ is an increasing function of $x$
and $0 \leq s^{-1/\alpha}/D \leq 1/2$, we have
\begin{equation}
2D^{-\alpha} \zeta(\alpha,1)\leq \beta \leq D^{-\alpha}
(\zeta(\alpha,1/2)+\zeta(\alpha,3/2)). \label{beta_bound}
\end{equation}
For example, if $\alpha=2$, then $3.29D^{-2} \leq \beta \leq 5.87
D^{-2}$. This implies that there exists $\beta$ in $[2D^{-\alpha}
\zeta(\alpha,1), D^{-\alpha}
(\zeta(\alpha,1/2)+\zeta(\alpha,3/2))]$ such that the simple
multiple cell model yields a valid result as in the infinite
linear array cellular model. Fig.~\ref{Fig_beta} shows $\beta$
in (\ref{eq_beta}) and its upper and lower bounds in
(\ref{beta_bound}) for $M=10$, path loss exponent $\alpha=2$, the
cell size $D=2$, and the forbidden region $\delta=0.01$. We
observe that $\beta$ changes very little with respect to the spectral
efficiency \textsf{C}. Hence, assuming $\beta$ constant as commonly done
in simplified multicell analysis \cite{Viterbi95} yields good approximations.

We can rewrite the minimum $(E_b/N_0)_{\rm sys}^{\rm MC}$ in terms
of $(E_b/N_0)_{\rm sys}^{\rm SC}$
\begin{equation}
\left( \frac{E_b}{N_0} \right)_{\rm sys}^{\rm MC} = \frac{\left(
\frac{E_b}{N_0} \right)_{\rm sys}^{\rm SC}} {1-\beta \textsf{C}
\left( \frac{E_b}{N_0} \right)_{\rm sys}^{\rm SC}}.
\end{equation}
So, the performance degradation due to the multiple cell
interference is up to 3 dB as long as
\begin{equation}
\left( \frac{E_b}{N_0} \right)_{\rm sys}^{\rm SC} \leq
\frac{1}{2\beta \textsf{C}}. \label{noise_limited_condition}
\end{equation}
In the case when the interference dominates the noise, the
performance degradation is more than 3 dB and the $(E_b/N_0)_{\rm
sys}^{\rm MC}$ tends to infinity as $\textsf{C} \rightarrow
\textsf{C}_0$. We may summarize the conditions that determine the
performance of the multiple cell case
\begin{eqnarray}
\left\{ \begin{array}{cl}
 \left( \frac{E_b}{N_0} \right)_{\rm sys}^{\rm SC} \leq
\frac{1}{2\beta \textsf{C}},&\mbox{noise dominated region} \\
 \frac{1}{2\beta \textsf{C}} < \left( \frac{E_b}{N_0} \right)_{\rm sys}^{\rm SC}
< \frac{1}{\beta \textsf{C}},&\mbox{interference dominated region} \\
 \left( \frac{E_b}{N_0} \right)_{\rm sys}^{\rm SC} \geq \frac{1}{\beta \textsf{C}},&\mbox{forbidden region}
\end{array} \right. \label{condition_for_mc}
\end{eqnarray}
For $\textsf{C}=2.8$ bit/s/Hz, $\left( \frac{E_b}{N_0}
\right)_{\rm sys}^{\rm SC}=-7.967$ dB and $\frac{1}{2\beta
\textsf{C}}=-7.896$ dB when $\beta\simeq 1.1$ (see
Fig.~\ref{Fig_beta}). According to (\ref{condition_for_mc}), the
system may be regarded as operated in the noise-dominated region
because $\left( \frac{E_b}{N_0} \right)_{\rm sys}^{\rm
SC}<\frac{1}{2\beta \textsf{C}}$. We can verify this by checking
$\left( \frac{E_b}{N_0} \right)_{\rm sys}^{\rm MC}=-5.024$ dB in
Fig.~\ref{Fig_capacity_HFS}. So, as long as the target spectral
efficiency is less than 2.8 bit/s/Hz, additionally required energy
in the multiple cell is no more than 3 dB compared to the single
cell.\footnote{We remark here that the values of $E_b/N_0$ should
be considered on a relative scale: a horizontal dB shift of all
these curves can be obtained simply by rescaling the cell size $D$
and by introducing a path-loss normalization factor. However, the
relevant information here is captured by the relative values
(differences in dB) and by the spectral efficiency.} It should
also be noticed that practical systems nowadays achieve spectral
efficiencies of about 1 bit/s/Hz. It follows that, in practice,
there is still a significant gap before the interference-limited
nature of the multicell system becomes significant. Therefore,
implementing some clever low complexity separated cell-site
processing may not be a bad idea from the practical system
engineering viewpoint. For example, using successive decoding at
each BS, as the system analyzed in this paper, represents already
a remarkable step-forward with respect to orthogonal or
semi-orthogonal conventional techniques such as TDMA, FDMA, CDMA,
OFDM or a mixture thereof.

\subsection{Partial Reuse Transmission}

In order to alleviate the interference limited nature of the multi-cell delay-limited system,
we introduce a partial reuse transmission scheme. We notice here that in the traditional Wyner cellular model
non-adjacent do not interfere. Trivially, a reuse factor of 2 (even-odd cells)
yields a non-interference limited system \cite{Shamai97}. With a more realistic distance-dependent path loss model
as in our case, this is no longer true and reuse factor optimization becomes more delicate.

Without loss of generality we can set the cell size $D=2$ (generalization is trivially obtained by cell re-scaling).
Fig.~\ref{Fig_partial_reuse} shows the system model for
partial reuse transmission. We classify cells into even and odd groups, according to their index parity.
We also divide each cell into an inner and outer regions, where the inner region radius is $\delta \leq r_0 \leq 1$.
The users located in the inner zone transmit in each slot. Users located in the
outer zone transmit signals alternately, at even slot times if they belong to an even cell, or odd slot times if they belong to an odd cell.  Consequently, the activity duty cycle of the users located in the outer zone
is 0.5.

From (\ref{pathloss_cdf}),  the cdf of the path loss when users
are uniformly distributed in $(a, b)$ is given by
\begin{equation}
F_s^{(a,b)}(x) = \left\{ \begin{array}{ll}
  0, & x<b^{-\alpha} \\
  1-\frac{x^{-\frac{1}{\alpha}}-a}{b-a}, & b^{-\alpha} \leq x < a^{-\alpha} \\
  1, & x\geq a^{-\alpha}
\end{array}. \right.  \label{pathloss_cdf_(a,b)}
\end{equation}
Due to the symmetry, it suffices to focus only on the phase where
even cells are fully active. In this case, the active users in odd
cells are uniformly distributed in $(\delta, r_0)$ while the users
in even cells are uniformly distributed in $(\delta, 1)$. Letting
$K_0 = K$ denote the number of active users in even cells, the
number of active users in odd cells is
$K_1=\frac{r_0-\delta}{1-\delta}K$. Similarly, we set the number
of bits transmitted in each odd cell is
$\Gamma_1=\frac{r_0-\delta}{1-\delta}\Gamma_0$ when the number of
bits transmitted in each even cell is $\Gamma_0$. Based on these,
the spectral efficiency in this partial reuse system is
\begin{equation}
\textsf{C} =
\frac{\Gamma_0+\Gamma_1}{2M}=\frac{\Gamma_0}{2M}\frac{1+r_0-2\delta}{1-\delta}.
\label{C_partial}
\end{equation}
Also, the $(E_b/N_0)_{\rm sys}$ is defined by
\begin{equation}
\left(\frac{E_b}{N_0}\right)_{\rm sys} = \frac{E_{\rm
tot}^0+E_{\rm tot}^1}{N_0 (\Gamma_0+\Gamma_1)} = \frac{E_{\rm
tot}^0+E_{\rm tot}^1}{2N_0 M \textsf{C}} ,
\label{Eb_N0_partial_def}
\end{equation}
where $E_{\rm tot}^0$ and $E_{\rm tot}^1$ denote the total
transmitted energy in each even cell and each odd cell,
respectively. We have:

\vspace{5pt}
\begin{theorem}  \label{Theorem_HF_Partial}
Under the assumptions A1, A2, A3, and A4, as $K\rightarrow
\infty$, the minimum $(E_b/N_0)_{\rm sys}$ for given system
spectral efficiency $\textsf{C}$ for the partial reuse system in
an infinite linear array cellular model is given by
\begin{eqnarray}
\left( \frac{E_b}{N_0} \right)_{\rm sys}^{\rm MC-Partial} =
\frac{1}{2}
\frac{1+A_{01}-A_{11}}{1-A_{00}-A_{11}-A_{01}A_{10}+A_{01}A_{11}}
\frac{2(1-\delta)}{1+r_0-2\delta} \nonumber \\ \cdot \log(2) \int
2^{\frac{2\textsf{C}(1-\delta)}{1+r_0-2\delta}
G_M(x)}
\frac{dG_M(x)}{x}~~ \nonumber \\
+\frac{1}{2}
\frac{1+A_{10}-A_{00}}{1-A_{00}-A_{11}-A_{01}A_{10}+A_{01}A_{11}}
\frac{2(r_0-\delta)}{1+r_0-2\delta} \nonumber \\ \cdot \log(2) \int
2^{\frac{2\textsf{C}(r_0-\delta)}{1+r_0-2\delta}
G_M^{(\delta,r_0)}(x)}
\frac{dG_M^{(\delta,r_0)}(x)}{x}~.
\label{E_b/N_0_HF_Partial}
\end{eqnarray}
Here $A_{ij}$'s are given by \alpheqn
\begin{eqnarray}
A_{00} = \log(2) \frac{2\textsf{C}(1-\delta)}{1+r_0-2\delta} \iint
2^{\frac{2\textsf{C}(1-\delta)}{1+r_0-2\delta}
G_M(xy)} \Phi_0(x)
\frac{dF_s(x) dH_M(y)}{xy}~  \label{eq30a} \\
A_{01} = \log(2) \frac{2\textsf{C}(r_0-\delta)}{1+r_0-2\delta} \iint
2^{\frac{2\textsf{C}(r_0-\delta)}{1+r_0-2\delta}
G_M^{(\delta,r_0)}(xy)} \Phi_1(x)
\frac{dF_s^{(\delta,r_0)}(x) dH_M(y)}{xy}  \label{eq30b} \\
A_{10} = \log(2) \frac{2\textsf{C}(1-\delta)}{1+r_0-2\delta} \iint
2^{\frac{2\textsf{C}(1-\delta)}{1+r_0-2\delta}
G_M(xy)} \Phi_1(x)
\frac{dF_s(x) dH_M(y)}{xy}~  \label{eq30c} \\
A_{11} = \log(2) \frac{2\textsf{C}(r_0-\delta)}{1+r_0-2\delta} \iint
2^{\frac{2\textsf{C}(r_0-\delta)}{1+r_0-2\delta}
G_M^{(\delta,r_0)}(xy)} \Phi_0(x)
\frac{dF_s^{(\delta,r_0)}(x) dH_M(y)}{xy} \label{eq30d}
\end{eqnarray} \reseteqn
where \alpheqn
\begin{eqnarray}
\Phi_0(x) =
4^{-\alpha}\left(\zeta\left(\alpha,1-\frac{x^{-1/\alpha}}{4}\right)
+\zeta\left(\alpha,1+\frac{x^{-1/\alpha}}{4}\right) \right)~~
 \\
\Phi_1(x) =
4^{-\alpha}\left(\zeta\left(\alpha,\frac{1}{2}-\frac{x^{-1/\alpha}}{4}\right)
+\zeta\left(\alpha,\frac{1}{2}+\frac{x^{-1/\alpha}}{4}\right)
\right).
\end{eqnarray} \reseteqn
\end{theorem}
\vspace{5pt}
\begin{proof}
See Appendix D.
\end{proof}
\vspace{5pt}

In the expressions above, $G_M^{(\delta,r_0)}(x)$ denotes the cdf
of $s \max\{f^1,\ldots,f^M\}$ when $s \sim F_s^{(\delta,r_0)}(x)$
and $(f^1,\ldots,f^M) \sim F_f(y_1,\ldots,y_M)$. It can be readily
checked that the case of $r_0=1$ coincides with the full reuse,
that we have analyzed separately in Theorem \ref{Theorem1}. If
$r_0=\delta$, then half of cells are silent to reduce
interference, which corresponds to the reuse factor of 2, also
referred to as intercell time division transmission system. We
also notice that the performance of frequency reuse, i.e.,
partitioning the $M$ frequency subchannels into two subsets and
allocate them to even and odd cells, cannot outperform the
``time'' reuse studied in Theorem \ref{Theorem_HF_Partial} since
the frequency diversity in each cell would be reduced. Of course,
the reuse parameter $r_0$ may be optimized in order to minimize
the required $(E_b/N_0)_{\rm sys}$ for given spectral efficiency
\textsf{C}.

Fig.~\ref{Fig_EbN0_partial_r_0} depicts $(E_b/N_0)_{\rm sys}^{\rm
MC-Partial}$ in (\ref{E_b/N_0_HF_Partial}) versus $r_0$ for the
partial reuse transmission system. There is an optimal $r_0 \in
(\delta , 1]$ that minimizes $(E_b/N_0)_{\rm sys}^{\rm
MC-Partial}$ for given \textsf{C}. At $\textsf{C}=4$, the
$(E_b/N_0)_{\rm sys}$ difference between the full transmission and
the optimal partial reuse transmission is about 7 dB, that is
quite significant. This shows that large gains can be realized by
careful optimization of the partial reuse factor.

Fig.~\ref{Fig_EbN0_partial_opt} shows the optimal partial reuse
transmission scheme compared with the full transmission schemes in
the single and multiple cell cases and reuse 2 scheme
in the multiple cell case.
We notice here that the gain due to partial reuse comes at the expenses of relaxing slightly the
HF constraint: users in the outer zone of each cell is half of their requested rate,
since they are served with duty cycle 0.5. For this reason, in the regime of small $\textsf{C}$ the multicell system with partial reuse
outperforms the single-cell HF system.

\section{Performance of Delay-Tolerant Systems}

A delay-tolerant system can achieve high spectral efficiency at
the cost of loose delay requirement.
In a distance-dependent path-loss scenario as considered here, it can be shown that PFS  serves
user at the peak of its own small-scale fading, i.e., the path-loss takes no role in the channel allocation
\cite{Caire07}. In this section, we investigate the performance of PFS in our multi-cell setting.

\subsection{Spectral Efficiency Bounds}

According to Theorem 1 in \cite{Caire07}, in the case of the
single cell, the spectral efficiency \textsf{C} as a function of
$\left( \frac{E_b}{N_0} \right)_{\rm sys}$ is given implicitly by
\begin{eqnarray}
\textsf{C}=\mbox{\bb E} \left[ \log_2(1+\rho s  \max_k f_k )\right] \nonumber \\
\left( \frac{E_b}{N_0} \right)_{\rm sys}^{\rm SC-PFS} =
\frac{\rho}{\textsf{C}}~~~~~~~~~~~~~~~~~~~~~~~~~~~
\end{eqnarray}
where $s \sim F_s(x)$ given by
(\ref{pathloss_cdf}) and $f_k$ is distributed as the frequency
selective block fading of user $k$.
Similarly, we have:

\vspace{5pt}
\begin{theorem}
For any given $K$, the spectral efficiency and the system
$E_b/N_0$ achieved by the PFS are given by
\begin{eqnarray}
\textsf{C}=\mbox{\bb E} \left[ \log_2\left(1+ \frac{\rho s  \max_{k=1,2,\cdots,K} f_k}{1+\rho \sum_{j\neq n} g_{\hat{k}(j)}(n,j)} \right)\right] \nonumber \\
\left( \frac{E_b}{N_0} \right)_{\rm sys}^{\rm MC-PFS} =
\frac{\rho}{\textsf{C}}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\label{Eb_N0_PFS}
\end{eqnarray}
where $\hat{k}(j)$ is the index of the user selected for transmission in cell $j$.
\end{theorem}
\vspace{5pt}
\begin{proof}
Because of the symmetry of the small-scale fading distribution,
the average throughput for each subchannel is identical. So, it is
sufficient to focus on a single subchannel and we drop the
subchannel index $m$. Since each user in the same cell gets the
same interference $I=\sum_{j\neq n} g_{\hat{k}(j)}(n,j)$ in
(\ref{PFS}), the PFS scheduling decision in the multiple cell is
the same with that in the single cell if the noise is replaced
with $1+\rho I$. From Theorem 1 in \cite{Caire07}, the result
follows straightforwardly.
\end{proof}
\vspace{5pt}

From (\ref{s_k(n,j)}), $g_{\hat{k}(j)}(n,j)$ can be expressed as
\begin{equation}
g_{\hat{k}(j)}(n,j) \sim \left( \theta (|n-j|D-U)^{-\alpha} +
(1-\theta) (|n-j|D+U)^{-\alpha} \right) f(n,j),
\label{interf_dist}
\end{equation}
where $U$ is a uniform random variable ranging from $\delta$ to
$r$, $\theta$ is a binary random variable taking 0 or 1 with equal
probability, and $f(n,j)$ is unit-mean exponentially distributed.

We can derive some bounds for the spectral efficiency as shown in
Appendix E. The lower bound follows from Jensen's inequality and is given by
\begin{eqnarray}
\textsf{C} \geq \mbox{\bb E} \left[ \log_2\left(1+ \frac{\rho s  \max_k
f_k}{1+\rho I_0} \right)\right], \label{PFS_lower_bound}
\end{eqnarray}
where the average interference is given by
\begin{eqnarray}
I_0=\mbox{\bb E}\left[ \sum_{j\neq n} g_{\hat{k}(j)}(n,j) \right]~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \nonumber \\
= \left\{
\begin{array}{ll}
 \frac{1}{D(D/2-\delta)} \left(2-\frac{D}{\delta}+\pi\cot\left(\frac{\pi\delta}{D} \right) \right), & \mbox{if }\alpha=2 \\
 \frac{D^{-\alpha+1}}{(\alpha-1)(D/2-\delta)}\left(2^{\alpha-1}
+\zeta\left(\alpha-1,1+\frac{\delta}{D}\right)-\zeta\left(\alpha-1,1-\frac{\delta}{D}\right)\right)
,& \mbox{if }\alpha\neq 2.
\end{array} \right.  \label{PFS_avg_interference}
\end{eqnarray}
A simple upper bound is derived by considering the interference
only from the two nearest cells
\begin{eqnarray}
\textsf{C} &\leq& \mbox{\bb E} \left[ \log_2\left(1+ \rho s \max_k f_k +\rho
I_0^{(2)} \right)\right]
\nonumber \\
&& - \mbox{\bb E} \left[ \log_2 \left( 1+\rho
(g_{\hat{k}(n-1)}(n,n-1)+g_{\hat{k}(n+1)}(n,n+1)) \right) \right].
\label{PFS_upper_bound}
\end{eqnarray}
where the average interference is given by
\begin{equation}
I_0^{(2)}=\frac{D^{-\alpha+1}}{(\alpha-1)(D/2-\delta)} \left(
2^{\alpha-1}-\left(\frac{2}{3}\right)^{\alpha-1}+\left(1+\frac{\delta}{D}\right)^{-\alpha+1}-\left(1-\frac{\delta}{D}\right)^{-\alpha+1}\right).
\end{equation}

The proportional fairness system is also interference limited for
any given finite $K$, because, as SNR increases, the
spectral efficiency converges to a finite value
\begin{equation}
\textsf{C}_0=\lim_{\rho \rightarrow \infty} \textsf{C} = \mbox{\bb E} \left[
\log_2\left( \frac{ s  \max_k f_k}{\sum_{j\neq n}
g_{\hat{k}(j)}(n,j)} \right)\right]. \label{PFS_cap_limit}
\end{equation}
However, under mild conditions on $f_k$ (e.g., they are independent and identically distributed (i.i.d.)
exponential random variables), for $K \rightarrow \infty$ we have \cite{Sharif05}
\begin{equation}
\Pr(|\max_k f_k - \log K|< \log\log K) \geq 1-O\left(
\frac{1}{\log K} \right).  \label{mud_gain}
\end{equation}
Consequently, the spectral efficiency limit $\textsf{C}_0$ is
order of $O(\log\log K)$. Notice that the spectral efficiency of the
delay-limited system converges to a finite value $\textsf{C}_0$ in
(\ref{C_0}) as $E_b/N_0\rightarrow \infty$, also in the case of $K \rightarrow \infty$.
We conclude that the PFS delay-tolerant system has merit in terms of spectral efficiency
limit because its spectral efficiency increases without bound as the number of users tends to
infinity (another manifestation of the ubiquitous {\em multiuser diversity} principle).

Fig.~\ref{Fig_capacity_PFS} shows the performance of PFS, the
lower bound (\ref{PFS_lower_bound}), and the upper bound
(\ref{PFS_upper_bound}). The lower bound is quite tight and the
upper bound is close to the simulation result at low SNR regime.
Compared to Fig.~\ref{Fig_capacity_HFS}, we can observe that PFS
generally outperforms the delay-limited system with infinite number of users
in the regime of low to moderate SNR and yields similar throughput at high SNR.

\section{Conclusions}

We provided a closed-form analysis of the system spectral efficiency vs. the system $E_b/N_0$
for two types of systems in a multi-cell ``Wyner-like'' cellular scenario, under the assumption of
conventional single-BS processing. On one hand, we have a hard-fairness system
where users transmit at fixed instantaneous rates in each slot and the system allocates power and makes use of
successive interference cancellation decoding at each BS in order to minimize the required power to
accommodate the user rate requests. On the other hand, we have a delay-tolerant
system with variable instantaneous rate allocation in order to maximize the long-term system throughput
subject to the proportional fairness constraint.
Beyond the pleasant analytical details, the main outcomes of this work are:
1) We validated analytically the popular simplified multi-cell model that treats inter-cell interference power as
proportional to the total cell power, evaluating the proportionality factor $\beta$ and showing that it is indeed close
to a constant almost independent of spectral efficiency;
2) We showed that significant gains can be obtained by optimizing the partial reuse factor, letting users in the outer region of cells
to transmit with duty-cycle 0.5;
3) We showed that PFS yields significant gains at low spectral efficiency, while for a finite number of users and high SNR
the two systems are quite comparable. Hence, there is no silver bullet associated with PFS,
and only a moderate increase in throughput over a hard-fairness system can be expected by exploiting the delay-tolerant nature of data traffic;  4) However, as $K$ increases, PSF yields an unbounded spectral efficiency while the HF
system becomes interference limited.
This is a manifestation of the multiuser diversity in a multi-cell environment.

\begin{center}
\textsc{Appendix A}

\textsc{Proof of Theorem \ref{Theorem1}}
\end{center}

The proof is based on the proof of Theorem 2 in \cite{Caire07}.
Let $R_k^m(n)=\frac{1}{K}\Gamma \nu_k^m (n)$, for all $k$ and $m$,
be the partial rate allocation. We rewrite (\ref{opt_prob}) for
given interference $I^m(n)$ to minimize
\begin{eqnarray}
\frac{1}{N_0 \Gamma} \sum_{k=1}^K \sum_{m=1}^M
\frac{N_0+I^m(n)}{g_{\pi_k^m(n)}^m(n)}
\exp\left(\frac{\Gamma}{K}\sum_{i<k} \nu_{\pi_i^m(n)}^m(n) \right)
\left(\exp\left( \frac{\Gamma}{K} \nu_{\pi_k^m(n)}^m(n) \right) -
1 \right) \label{minimize_energy}
\end{eqnarray}
subject to $\sum_{m=1}^M \nu_k^m(n) = \nu_k(n)$, $k=1,2,\cdots,K$,
and to the non-negative constraints $\nu_k^m(n)\geq 0$ for all $k$
and $m$. Since $\exp(x)=1+x+o(x)$ for small $x$, the objective
function for large $K$ can be written as
\begin{eqnarray}
\frac{1}{N_0} \frac{1}{K} \sum_{k=1}^K \sum_{m=1}^M (N_0+I^m(n))
\frac{\nu_{\pi_k^m(n)}^m(n)}{g_{\pi_k^m(n)}^m(n)}
\exp\left(\frac{\Gamma}{K}\sum_{i<k} \nu_{\pi_i^m(n)}^m(n) \right)
\label{minimize_energy2}
\end{eqnarray}
and its associated Lagrangian function is
\begin{eqnarray}
\mathcal{L}=\frac{1}{N_0} \frac{1}{K} \sum_{k=1}^K \sum_{m=1}^M
\frac{\nu_{\pi_k^m(n)}^m(n)}{d_{\pi_k^m(n)}^m(n)}
\exp\left(\frac{\Gamma}{K}\sum_{i<k} \nu_{\pi_i^m(n)}^m(n) \right)
- \sum_{k=1}^K \lambda_k^{(K)}\left( \sum_{m=1}^M \nu_k^m-\nu_k
\right),
\end{eqnarray}
where the channel SINR (signal to interference plus noise ratio)
$d_{\pi_k^m(n)}^m(n)$ is given by $d_{\pi_k^m(n)}^m(n)\equiv
g_{\pi_k^m(n)}^m(n)/(N_0+I^m(n))$. We have an optimization problem
for each $K$ and $\lambda_k^{(K)}$ denotes the $k$th Lagrangian
multiplier of the problem. By differentiating with respect to
$\nu_k^m$, and by letting $\pi_i^m=k$, i.e., user $k$ is ranked in
the $i$th position on subchannel $m$, the Kuhn-Tucker condition
becomes
\begin{eqnarray}
\frac{1}{d_{k}^m(n)}\exp\left(\frac{\Gamma}{K}\sum_{l<i}\nu_{\pi_l^m(n)}^m(n)\right)
+\frac{\Gamma}{K}\sum_{l>i}
\frac{\nu_{\pi_l^m(n)}^m(n)}{d_{\pi_l^m(n)}^m(n)}
\exp\left(\frac{\Gamma}{K}\sum_{l'<l} \nu_{\pi_{l'}^m(n)}^m(n)
\right) \geq \lambda_k^{(K)}, \label{KKT_condition}
\end{eqnarray}
where we have multiplied both sides by $KN_0$ and have replaced
$KN_0\lambda_k^{(K)}$ by $\lambda_k^{(K)}$. At this point, we notice that
(\ref{KKT_condition}) coincides with
the Kuhn-Tucker condition for the single cell case (see proof of Theorem 2 in \cite{Caire07})
provided that SNR \cite{Caire07} is replaced by
the SINR $g_{\pi_k^m(n)}^m(n)/(N_0+I^m(n))$.
Consequently, the asymptotically optimal rate allocation for large
$K$ is given by \cite{Caire07}
\begin{equation}
\nu_k^m(n) = \left\{ \begin{array}{cc}
  \nu_k(n), & \mbox{for}~m=m_k \\
  0, & \mbox{for}~m\neq m_k
\end{array} \right.  \label{optimal_nu}
\end{equation}
where $m_k=\arg\max_l \{g_k^l(n,n)/(N_0+I^m(n))\}$ denotes the
index of the subchannel for which user $k$ has {\em maximum SINR}.
As a byproduct, we have that allocating each user to its own
best  SINR channel is asymptotically optimal for large $K$ in the multiple cell case.
Since the subchannel distribution is symmetric and
users are distributed uniformly in each cell, the interference
$I^m(n)$ converges to the same non-random limit for all $m$ and $n$ as $K \rightarrow \infty$.
If the interference is the same, then the optimal rate allocation for the multiple
cell consists of allocating each user on its best SNR channel, i.e.,
$m_k=\arg\max_l \{g_k^l(n,n)\}$.

Now we show that allocating each user on its subchannel with best channel
gain  makes indeed the interference $I^m(n)$ constant with $m$ (constant with the cell index $n$ follows immediately form the symmetry of the system). From (\ref{interference}), for large $K$, the
interference power is expressed as
\begin{eqnarray}
I^m(n) = \sum_{j\neq n} (N_0+I^m(j)) \frac{\Gamma}{K} \sum_{k=1}^K
\frac{g_{\pi_k^m(n)}^m(n,j)
\nu_{\pi_k^m(j)}^m(j)}{g_{\pi_k^m(j)}^m(j,j)}
 \exp\left(\frac{\Gamma}{K}\sum_{i<k}
\nu_{\pi_i^m(j)}^m(j) \right) + o(1/K). \label{inteference_exp}
\end{eqnarray}
As we adopt an optimal successive decoding scheme in each BS,
$\nu_k^m(j)$ takes the form in (\ref{optimal_nu}). Suppose that $g_k^m(j,j)$,
given by $s_k(j,j)=x$ and $f_k^m(j,j)=y$, is ranked in the $i$th
position by the permutation $\pi^m(j)$. By using
Lemma~\ref{lemma1} in Appendix B, we get
\begin{equation}
\exp\left(\frac{\Gamma}{K}\sum_{i<k} \nu_{\pi_i^m(j)}^m(j) \right)
\rightarrow \exp\left(\frac{\Gamma}{M} G_M(xy)
\right)
\end{equation}
as $K\rightarrow \infty$.
From (\ref{s_k(n,j)}), averaging with respect to $\theta_k(n,j)$, we get
\begin{equation}
\mbox{\bb E} \left[ s_k(n,j) | s_k(j,j) \right] = \frac{1}{2} \left(
\left(|n-j|D-s_k(j,j)^{-\frac{1}{\alpha}}\right)^{-\alpha} +
\left(|n-j|D+s_k(j,j)^{-\frac{1}{\alpha}}\right)^{-\alpha} \right)
\end{equation}
According to Lemma~\ref{lemma2} in Appendix B, the asymptotic
interference in (\ref{inteference_exp}) for large $K$ becomes
\begin{eqnarray}
I^m(n) = \sum_{j\neq n} (N_0+I^m(j)) \frac{\Gamma}{M} \iint
e^{\frac{\Gamma}{M} G_M(xy)} \Phi(x,|n-j|D)
\frac{dF_s(x)dH_M(y)}{xy},~ \label{interference_relation}
\end{eqnarray}
where $\Phi(x,y)\equiv \frac{1}{2} \left(
\left(y-x^{-\frac{1}{\alpha}}\right)^{-\alpha} +
\left(y+x^{-\frac{1}{\alpha}}\right)^{-\alpha}\right)$. Note that
the interference relation in (\ref{interference_relation}) is
symmetric with respect to $I^m(n)$ for all $m$ and $n$. Therefore,
as $K\rightarrow \infty$, the interference at each base station
converges to the same value, $I^m(n)\rightarrow I_0$, for all $m$
and $n$. As $I^m(n)\rightarrow I_0$, we have
\begin{eqnarray}
I_0 = (N_0+I_0) \frac{\Gamma}{M} \iint e^{\frac{\Gamma}{M}
G_M(xy) } \Phi(x)
\frac{dF_s(x)dH_M(y)}{xy},
\label{interference_relation2}
\end{eqnarray}
where $\Phi(x) \equiv 2 \sum_{j=1}^{\infty} \Phi(x,jD)$. Solving
(\ref{interference_relation2}) with respect to $I_0$, we obtain
the asymptotic interference under the single-cell optimal
successive decoding strategy. In addition, by Lemma~\ref{lemma1}
in Appendix B, allocating each user on its own best channel makes
the minimum $(E_b/N_0)_{\rm sys}$ in (\ref{minimize_energy2})
converges to the following
\begin{equation}
\left( \frac{E_b}{N_0} \right)_{\rm sys} \rightarrow
\frac{N_0+I_0}{N_0} \int_0^{\infty} \frac{1}{x} \exp\left(
\frac{\Gamma}{M} G_M(x) \right)dG_M(x)
\label{EbN0_int}
\end{equation}
as $K\rightarrow \infty$ (again, this can be easily shown based on
the proof of Theorem 2 in \cite{Caire07}). Inserting the
expression of $I_0$ from (\ref{interference_relation2}) into
(\ref{EbN0_int}) and expressing $\Gamma$ in bits, we eventually
arrive at the desired result.

\begin{center}
\textsc{Appendix B}

\textsc{Lemmas}
\end{center}

\begin{lemma} \cite{Caire07} \label{lemma1}
Let $\textsl{A}$ be an interval, and $\textsl{A}(m)$ denote the
set
\begin{equation}
\textsl{A}(m)=\left
\{k|s_k(n,n)\max\{f_k^1(n,n),\cdots,f_k^M(n,n)\}\in \textsl{A},
m_k=m \right \},
\end{equation}
where $m_k\equiv \arg\max_l \{f_k^l(n,n)\}$. Also, let $g(x)$
denote a continuous measurable function in $x\in A$. Under
Assumptions A2, A3, and A4, we get
\begin{equation}
\frac{1}{K}\sum_{k\in \textsl{A}(m)} g(s_k(n,n) f_k^m(n,n))  \nu_k
\rightarrow \frac{1}{M}\int_{x\in\textsl{A}} g(x)
dG_M(x)
\end{equation}
with probability 1, as $K\rightarrow \infty$ and $M$ is fixed.
\end{lemma}

\begin{lemma} \label{lemma2}
Let $\textsl{A}$ be an interval, and $\textsl{A}(m)$ denote the
set
\begin{equation}
\textsl{A}(m)=\left
\{k|s_k(j,j)\max\{f_k^1(j,j),\cdots,f_k^M(j,j)\}\in \textsl{A},
m_k=m \right \},
\end{equation}
where $m_k\equiv \arg\max_l \{f_k^l(j,j)\}$. Also, let
$g(x,\phi)$ denote a continuous measurable function in $x\in A$
and $\phi \geq 0$. Under Assumptions A2, A3, and A4, we get
\begin{equation} \label{bubu}
\frac{1}{K}\sum_{k\in \textsl{A}(m)}
\frac{g(s_k(j,j),\theta_k(n,j)) f_k^m(n,j) \nu_k}{s_k(j,j)
f_k^m(j,j)} \rightarrow \frac{1}{M}\iint_{xy\in\textsl{A}}
\frac{\mbox{\bb E}_{\theta}[g(x,\theta)]}{xy} dF_s(x)dH_M(y)
\end{equation}
with probability 1, as $K\rightarrow \infty$ and $M$ is fixed.
In (\ref{bubu}), $\mbox{\bb E}_\theta[\cdot]$ denotes expectation with respect to
$\theta \sim F_\theta(\phi)$.
\end{lemma}
\begin{proof}
It follows from the convergence of the empirical cdfs that, as
$K\rightarrow \infty$, we have
\begin{eqnarray}
\frac{1}{K}\sum_{k\in \textsl{A}(m)}
\frac{g(s_k(j,j),\theta_k(n,j)) f_k^m(n,j) \nu_k}{s_k(j,j)
f_k^m(j,j)}~~~~~~~~~~~~~~~~~~~~~~~~~~ \nonumber \\ =
\frac{|\textsl{A}(m)|}{K} \frac{1}{|\textsl{A}(m)|} \sum_{k\in
\textsl{A}(m)} \frac{g(s_k(j,j),\theta_k(n,j))
f_k^m(n,j) \nu_k}{s_k(j,j) f_k^m(j,j)}~~~~~~~~~~ \nonumber \\
\rightarrow \mbox{\bb P}(k\in \textsl{A}(m))
\mbox{\bb E}\left[\frac{g(s_k(j,j),\theta_k(n,j)) f_k^m(n,j) \nu_k}{s_k(j,j)
f_k^m(j,j)} | k\in \textsl{A}(m) \right].
\end{eqnarray}
We have
\begin{eqnarray}
\mbox{\bb P}(k\in \textsl{A}(m)) &=&
\mbox{\bb P}(s_k(j,j)\max\{f_k^1(j,j),\cdots,f_k^M(j,j)\}\in \textsl{A}|
m_k=m)\mbox{\bb P}(m_k=m) \nonumber \\ &=&\frac{1}{M}\int_{\textsl{A}}
dG_M(x). \label{prob1}
\end{eqnarray}
where $G_M(x)$ is the limit empirical cdf of $s_k(j,j)\max\{f_k^1(j,j),\cdots,f_k^M(j,j)\}$, which exists a.s..
By Assumptions A3 and A4, we also have
\begin{eqnarray}
\mbox{\bb E}\left[\left . \frac{g(s_k(j,j),\theta_k(n,j)) f_k^m(n,j) \nu_k}{s_k(j,j)
f_k^m(j,j)} \right | k\in \textsl{A}(m) \right]~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \nonumber \\
= \mbox{\bb E}\left[\left . \frac{g(s_k(j,j),\theta_k(n,j))}{s_k(j,j) f_k^m(j,j)} \right |
k\in \textsl{A}(m) \right] \mbox{\bb E}\left[ f_k^m(n,j) \nu_k | k\in
\textsl{A}(m) \right]~~~~~~~~~~~~~~~~~~~~~~~~~~ \nonumber \\
= \mbox{\bb E}\left[\frac{g(s_k(j,j),\theta_k(n,j))}{s_k(j,j)
\max\{f_k^1(j,j),\cdots,f_k^M(j,j)\}} | s_k(j,j)
\max\{f_k^1(j,j),\cdots,f_k^M(j,j)\} \in \textsl{A} \right]
\nonumber \\
=\frac{1}{\int_{\textsl{A}} dG_M(x)} \iint_{xy\in
\textsl{A}} \frac{\mbox{\bb E}_{\theta}[g(x,\theta)]}{xy}
dF_s(x)dH_M(y)~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\label{prob2}
\end{eqnarray}
where $H_M(y)$ denotes the limit empirical cdf of $\max\{f_k^1(j,j),\cdots,f_k^M(j,j)\}$.
By using (\ref{prob1}) and (\ref{prob2}) the result follows.
\end{proof}

\begin{center}
\textsc{Appendix C}

\textsc{Proof of Theorem \ref{Theorem2}}
\end{center}

We can write the minimization of $(E_b/N_0)_{\rm sys}$ for given
interference $I^m$ as minimize
\begin{eqnarray}
\frac{N_0+I^m}{N_0 \Gamma} \sum_{k=1}^K \sum_{m=1}^M
\frac{1}{g_{\pi_k^m}^m} \exp\left(\frac{\Gamma}{K}\sum_{i<k}
\nu_{\pi_i^m}^m \right) \left(\exp\left( \frac{\Gamma}{K}
\nu_{\pi_k^m}^m \right) - 1 \right) \label{simple_minimize_energy}
\end{eqnarray}
subject to $\sum_{m=1}^M \nu_k^m = \nu_k$, $k=1,2,\cdots,K$, and
to the non-negative constraints $\nu_k^m\geq 0$ for all $k$. As
done in Appendix A, the solution to the problem
(\ref{simple_minimize_energy}) for given interference level $I^m$
is the same with the solution of Theorem 2 in \cite{Caire07}.
Consequently, the asymptotically optimal rate allocation for large
$K$ is given by (\ref{optimal_nu}). In addition, by
Lemma~\ref{lemma1} in Appendix B, for given $I^m$, the minimum
$(E_b/N_0)_{\rm sys}$ converges to the following
\begin{equation}
\left( \frac{E_b}{N_0} \right)_{\rm sys} \rightarrow
\frac{N_0+I^m}{N_0} \int \frac{1}{x} \exp\left( \frac{\Gamma}{M}
G_M(x) \right)dG_M(x)
\label{Simple_EbN0_int}
\end{equation}
Inserting (\ref{simple_opt_energy}) into
(\ref{simple_interference}) and applying Lemma~\ref{lemma1} in
Appendix B once again, we get
\begin{eqnarray}
I^m = \beta(N_0+I^m) \int \frac{1}{x} \exp\left(\frac{\Gamma}{M}
G_M(x)\right) dG_M(x)
\end{eqnarray}
as $K\rightarrow \infty$. Since this equation holds for all $m$,
$I^m$ converges to the same value $I_0$ independent of $m$.
Consequently, the minimum $(E_b/N_0)_{\rm sys}$ converges to the
following
\begin{equation}
\left( \frac{E_b}{N_0} \right)_{\rm sys} \rightarrow \frac{ \int
\frac{1}{x} e^{\frac{\Gamma}{M} G_M(x)}
dG_M(x)} {1- \beta \frac{\Gamma}{M} \int \frac{1}{x}
e^{\frac{\Gamma}{M} G_M(x)} dG_M(x)}
\label{Simple_EbN0_int2}
\end{equation}
as $K\rightarrow \infty$. Expressing $\Gamma$ in bits yields the
desired result.

\begin{center}
\textsc{Appendix D}

\textsc{Proof of Theorem \ref{Theorem_HF_Partial}}
\end{center}

The proof is very similar to the proof of Theorem \ref{Theorem1}.
Therefore, we give only a sketch and leave out trivial details for
the sake of space limitation. From (\ref{C_partial}), the numbers
of bits transmitted in each even cell and each odd cell are
$\Gamma_0=\frac{2M\textsf{C}(1-\delta)}{1+r_0-2\delta}$ and
$\Gamma_1=\frac{2M\textsf{C}(r_0-\delta)}{1+r_0-2\delta}$. From
(\ref{inteference_exp}), the interference experienced by cell $n$
is given by
\begin{eqnarray}
I^m(n) = \sum_{j\neq n,~{\rm even}~j} (N_0+I^m(j))
\frac{\Gamma_0}{K_0} \sum_{k=1}^{K_0} \frac{g_{\pi_k^m(n)}^m(n,j)
\nu_{\pi_k^m(j)}^m(j)}{g_{\pi_k^m(j)}^m(j,j)}
 \exp\left(\frac{\Gamma_0}{K_0}\sum_{i<k}
\nu_{\pi_i^m(j)}^m(j) \right)~~~~~~~~  \nonumber \\
+ \sum_{j\neq n,~{\rm odd}~j} (N_0+I^m(j)) \frac{\Gamma_1}{K_1}
\sum_{k=1}^{K_1} \frac{g_{\pi_k^m(n)}^m(n,j)
\nu_{\pi_k^m(j)}^m(j)}{g_{\pi_k^m(j)}^m(j,j)}
 \exp\left(\frac{\Gamma_1}{K_1}\sum_{i<k}
\nu_{\pi_i^m(j)}^m(j) \right)  + o(1/K)
\end{eqnarray}
The asymptotically optimal rate allocation is to allocate each
user on its best SNR channel as done in Theorem~\ref{Theorem1}.
According to Lemma~\ref{lemma2} in Appendix B, as $K\rightarrow
\infty$, the asymptotic interference converges to
\begin{eqnarray}
& I^m(n) = \sum_{j\neq n,~{\rm even}~j} (N_0+I^m(j))
\frac{\Gamma_0}{M} \iint e^{\frac{\Gamma_0}{M}
G_M(xy)} \Phi(x,2|n-j|)
\frac{dF_s(x)dH_M(y)}{xy} & \nonumber \\
& +
\sum_{j\neq n,~{\rm odd}~j} (N_0+I^m(j)) \frac{\Gamma_1}{M} \iint
e^{\frac{\Gamma_1}{M} G_M^{(\delta,r_0)}(xy)}
\Phi(x,2|n-j|)
\frac{dF_s^{(\delta,r_0)}(x) dH_M(y)}{xy}, &
\label{int_partial}
\end{eqnarray}
where $\Phi(x,y)\equiv \frac{1}{2} \left(
\left(y-x^{-\frac{1}{\alpha}}\right)^{-\alpha} +
\left(y+x^{-\frac{1}{\alpha}}\right)^{-\alpha}\right)$. Due to its
symmetry, the interference power at each even cell converges to
the same limit $I_0$, and the interference power to all odd cells
converges to the same limit $I_1$. Now we define $\Phi_0(x)$ and
$\Phi_1(x)$ to be $\Phi_0(x)\equiv 2\sum_{j=2,4,\cdots}
\Phi(x,2j)$ and $\Phi_1(x)\equiv 2\sum_{j=1,3,\cdots} \Phi(x,2j)$.
Then, we can rearrange (\ref{int_partial}) to get
\begin{eqnarray}
I_0&=&(N_0+I_0)A_{00}+(N_0+I_1)A_{01} \nonumber \\
I_1&=&(N_0+I_0)A_{10}+(N_0+I_1)A_{11}, \label{I_0,I_1_eq}
\end{eqnarray}
where $A_{00},A_{01},A_{10},A_{11}$ are given by (\ref{eq30a}) --
(\ref{eq30d}), respectively. Solving for $I_0,I_1$ in terms of the
$A_{i,j}$'s and expressing the limiting total energy in each even
and odd cell using Lemma 1 in Appendix B, we finally get the
desired result.

\begin{center}
\textsc{Appendix E}

\textsc{Derivation of lower and upper bounds for PFS}
\end{center}

We derive the lower bound using Jensen's inequality. The
average interference can be obtained by
\begin{eqnarray}
I_0 = 2 \sum_{j=1}^{\infty}
\mbox{\bb E}\left[\frac{1}{2}(jD-U)^{-\alpha}+\frac{1}{2}(jD+U)^{-\alpha}
\right]~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\nonumber \\
= \frac{D^{-\alpha+1}}{(\alpha-1)(D/2-\delta)} \left(2^{\alpha-1}+
\sum_{j=1}^{\infty} \left( \frac{1}{(j+\delta/D)^{\alpha-1}}
-\frac{1}{(j-\delta/D)^{\alpha-1}} \right) \right),~~~~~
\end{eqnarray}
where we used $D=2r$. From the identity \cite{Gradshteyn00}
\begin{equation}
\sum_{j=1}^{\infty} \frac{1}{j^2-x^2}=\frac{1}{2x^2}(1-\pi x
\cot(\pi x))
\end{equation}
for $\alpha=2$, then the average interference is given by
(\ref{PFS_avg_interference}). Because $\log(1+x/(1+y))$ is convex
with respect to $y$, Jensen's inequality yields the lower bound in
(\ref{PFS_lower_bound}).

For the upper bound, we only consider the interference from two
nearest cells
\begin{eqnarray}
C &\leq& \mbox{\bb E} \left[ \log_2\left(1+ \frac{\rho s  \max_k f_k}{1+\rho
(g_{\hat{k}(n-1)}(n,n-1)+g_{\hat{k}(n+1)}(n,n+1)) }
\right)\right] \nonumber \\
 &\leq& \mbox{\bb E} \left[ \log_2\left(1+ \rho s
\max_k f_k +\rho I_0^{(2)} \right)\right] \nonumber \\
&& - \mbox{\bb E} \left[ \log_2 \left( 1+\rho
(g_{\hat{k}(n-1)}(n,n-1)+g_{\hat{k}(n+1)}(n,n+1)) \right) \right],
\end{eqnarray}
where the last inequality follows from Jensen's inequality and the
average interference from the two nearest cells is given by
\begin{eqnarray}
I_0^{(2)} = \mbox{\bb E}[g_{\hat{k}(n-1)}(n,n-1)+g_{\hat{k}(n+1)}(n,n+1))]~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\nonumber \\
= \frac{D^{-\alpha+1}}{(\alpha-1)(D/2-\delta)} \left(
2^{\alpha-1}-\left(\frac{2}{3}\right)^{\alpha-1}+\left(1+\frac{\delta}{D}\right)^{-\alpha+1}-\left(1-\frac{\delta}{D}\right)^{-\alpha+1}\right).
\end{eqnarray}

\newpage

redpajama_set_name: RedPajamaArXiv


Tonight I feel like a five year old child on Christmas Eve. It is 8.30pm and I am in bed!
(I know, Right?!) Dinner, laundry, lunches, homework – check! Electric blanket on, the last of my glass of wine on my bedside table, son fast asleep and mom - headphones in and laptop where it belongs… on lap! Haha!
Thursday evenings are always festive in my house (or my mom's) – as we rotate weekly dinners. One week her place, one mine and so it goes. My significant other will stand firm in the fact that our weekly rotation has very little to do with the food and a whole lot more to do with the conversation and wine. He is not entirely wrong… haha! Either way, Thursday night is always looked forward to.
As 'Phuza' is the traditional Zulu term for drink, it follows that 'Phuza Thursday' is the act of drinking on a Thursday night. This has become a burgeoning tradition among South African youth, with Thursday nights almost replacing Friday nights as occasions for drinking.
My mother and I have mastered the art of "Phuza Thursday" (respectfully and moderately of course) – well in my opinion anyway… lol. In fact I think most mature people have mastered the art of partying with intelligence.
It is quite simply really, start early – end early. Wake up and enjoy the next day! Boom! – Yip!
What were we all thinking when we were in our late teens and twenties?! Why on God's green earth did we obliterate ourselves until the point of hearing the 5am birds singing?
Old people are smart I tell you – SMART! lol!
Isn't it funny how life goes full circle?
• When you were young you hated bathing – now it's a treat.
now you can't get into it early enough.
• When you were a teen you knew everything. Now you listen a whole lot more.
I literally couldn't shower fast enough this evening so that I could climb into bed and bore you all with the random thoughts that waft through my mind. Hehe!
Children have all the uninhibited enthusiasm that so many of us lack as adults and old folk have all the experience of life and its (not always so golden) platters. If we could all learn how to combine the best of those two puzzle pieces, we would see some amazing results.
In fact, I would go so far as to say that the world's most influential creatives and 'other' - have never been conformists and in many ways have been 'child like' in their approach to things – whether it be their thinking patterns, emotions, tempers, lifestyles etc.
They have, to my mind – mastered the art of "life in reverse". Living, with the experience and "savvy" of an adult - but retaining the raw, limitless, doubt free spirit of a child.
Watch children. Learn from them – they are FAR wiser than adults. You will naturally combine what you see with what you know ;) Truestory.


Classic method, zero dosage spumante, 18 months in steel tank, from ponca (marl and sandstone) soils of the terraced Romain vineyard. 80% Chardonnay, 20% Pinot Blanc.
"Ponca", marl and sandstone flysch, is what makes the Colli Orientali's wines unique. At the Rodaro winery, meticulous work in their terraced vineyards and the willingness to experiment have given rise to many unique and different wines, with this spumante being the latest addition to Paolo Rodaro's production.
Harvested in September at 130meters elevation from his Romain terraced vineyard, this spumante is made with the classic (champanoise) method. Very fine mousse, with yeastiness and notes of toasted nuts and bread, mineral driven and clean as can be expected from a zero dosage Brut Nature.


text:
We need a national citizenship drive
By Howard Husock
Jul 12, 2019 | 2:39 PM
File photo (Getty Images)
Now that President Trump has dropped his push to include a question about citizenship on the 2020 Census, Democrats will be tempted to claim victory and move on. Instead, both parties should promote a long-term solution to the issues raised by there being so many non-citizens counted in the drawing of congressional districts: encourage legal immigrants to become naturalized American citizens. Doing so would require both an endorsement of the meaning and value of citizenship but also a lowering of existing monetary and logistical barriers to naturalization.
It may well be true that including the citizenship question on the Census form would have discouraged immigrants, intimidated by the president's rhetoric and stepped-up deportation threats, not to respond and thus to go uncounted. But that does not mean the status quo should not be changed. Democrats should acknowledge that, just as the structure of the Senate currently favors Republicans, to, too, does the way in which the Constitution requires that congressional districts be determined favors Democrats.
Because representation is determined not by the number of potential voters but, as per the Constitution, a count of all residents, no matter their legal status, parts of the country where Democrats predominate tend to benefit. According to an Axios analysis, there are some 50 Democratic districts in which the foreign-born population tops 40%. There are only 10 such Republican districts. Counting residents who cannot vote may swell the number of Democratic seats — but it's not good for democracy when residents have no voice in who represents them.
The best way to rectify the problem is a national naturalization drive. But doing so means making it easier for legal immigrants to become citizens. That does not mean that we should make the 10-question citizenship test easier nor make it available in languages other than English. Just as we did during the wave of early 20th century immigration, we should want new Americans to know the common language, in part for their own benefit.
But it's worth asking why undertaking the naturalization process should require, as it currently does, a $730 fee. This is no small amount for a working class newcomer. The process should not be without any cost — we tend to value more that for which we pay — but $100 or even less seems reasonable.
[More Opinion] They just killed my son again: Eric Garner's mother Gwen Carr says the Department of Justice has failed her family »
Just as important, steps must be taken to make the naturalization process go more quickly. The Immigration and Naturalization Service reports that, 2017, a total of 986,851 applications for citizenship were submitted but only 707,265 persons were naturalized. The figures were similar for 2016: 972,151 applications but only 753,060 approved. About 80,000 applications were denied each year — leaving many others just in limbo. We can do better, and we have. In 2008, perhaps reflecting the legacy of the pro-immigration George W. Bush Administration, more than 1 million persons became naturalized citizens.
There are political reasons for Democrats to resist a citizenship campaign. But it's the right thing for America and the right next step to take.
Husock is vice president for research and publications at the Manhattan Institute.


Back in the 1920s and early '30s, Fats Waller sold a lot of songs for next to nothing. Much to his consternation, some of these melodies turned up later as hugely successful hits with other peoples' names attached. He adamantly insisted that he'd written both "On the Sunny Side of the Street" and "I Can't Give You Anything But Love," gorgeous tunes that do seem to be cut from the same fabric as "Honeysuckle Rose," "My Fate Is in Your Hands," and "Keepin' Out of Mischief Now." Bearing this in mind, Waller's magnificently cynical duet with Una Mae Carlisle is not only one of the most skillfully humorous records he ever made, but might also have served as a sort of artistic vengeance upon Jimmy McHugh and Dorothy Fields. In any case, Una Mae has a wonderfully sultry voice and Fats is remarkably funny. "The Darktown Strutters' Ball" features a non-electric taxi cab horn and a very hot band, goaded by Waller's shouting and piano plunking. "Swinga-Dilla Street" demonstrates his mature bouncing Hammond organ style. "At Twilight" was composed by Thomas and Anita Waller. It is romantic, soft, and lovely, like something from Duke Ellington. "Black Maria," that cheerful ode to a paddy wagon, swings hard. Rosy and idyllic, "Mighty Fine" is most memorable for its interesting punch line: "Exuberance is the spontaneity of life." Waller's band had already recorded two excellent versions of "The Moon Is Low" in 1939. The first 1940 take is a nice hasty jam, but nothing could top what they'd done with it a few months earlier. Unissued until the 1970s, "The Moon Is Low, Pt. 2" uses the Hammond organ in place of piano. Fats slips into in a strange key at one point, which is probably why this take was rejected. "Fat and Greasy" is a remake of a nasty tune recorded in 1935 by Waller's big band. This version actually includes the verse, for what it's worth. It's kind of strange to hear Fats making fun of obesity. "You Run Your Mouth" is famous for its prickly assertiveness. This is much better than the sloppy, apparently inebriated second take. Fats makes all kinds of weird noises with his lips during "Eep, Ipe, Wanna Piece of Pie," a very funny song for children or twisted adults. "My Mommie Sent Me to the Store" is even better, with a hilarious vocal exchange between the bandmembers and their fearless leader. Does the word "rye" refer to bread or liquids? Only mother knows for sure. "Dry Bones," cast in a minor mode, is wonderfully spooky and expertly timed. It's much better than anyone else's version of this old song. Deep in the throes of his own theater of the absurd, Fats invokes "fine neck bones and rice" with almost religious fervor.


text:
UNC Asheville Knocks Liberty Out Of Big South Volleyball Tourney
November 15, 2005 Asheville, N.C. RSS
UNC Asheville used a balanced attack to advance past Liberty, 3-0 in the quarterfinals of the 2005 Big South Conference Volleyball Tournament 3-0 (30-20, 30-22, 30-23) to end the Lady Flames' season.
The third-seeded Bulldogs advance to the semifinals of the Big South Conference Tournament at Coastal Carolina this weekend. Asheville plays second-seeded Coastal Carolina Saturday afternoon at 5 p.m.
The Bulldogs (24-6) had six different players with at least six kills. Junior Elif Unlu led Asheville with 10 kills, nine digs and five blocks. Sophomore Ashley Wrightenberry finished with 10 kills and eight blocks.
Wrightenberry hit .727 in the match as she had her eight kills in 11 attempts. Senior Allison Lambert added eight kills and two blocks. Talia Ogle and Rose Butler contributed seven kills each. Setter Amanda Wiles had six kills and 35 assists. The defense was led by Ogle with 16 digs. Krysti Havens contributed
12 digs.
Liberty (9-22) was led by Janell Migeot with 11 kills and seven digs. Mary Alice Pike finished with nine kills and seven digs.
The win was Asheville's sixth straight and avenged a 3-0 defeat to Liberty in last year's Big South Conference Tournament quarterfinals. The Bulldogs have swept their last five opponents and gave won 16 consecutive games.
Asheville started out quickly as it jumped out to a 10-5 lead and never led the Lady Flames get closer than three points the rest of the game. Liberty made 15 attack errors in the opening game as UNCA won 30-20.
In the second game, Asheville's blocking took over as the Bulldogs recorded 10 blocks in the game. UNCA used a 5-0 run midway through the second game and rolled to a 30-22 win.
The third game was closer before Asheville took charge with a 6-1 run to build a 20-15 lead. Liberty could get no closer than three points. The last point of the match came when Butler and Wrightenberry blocked a Liberty attack.
The loss brings the Lady Flames' 2005 season to an end.


Source: Info in the About Flury: Journey Of The Snowman was from the press kit from the publicity team.
Source: Info in the About Jack: The Tale Of Frost was from the press kit from the publicity team.
Source: Info in the About Claus: Legend Of The Fat Man was from the press kit from the publicity team.
One winner wins e-copies of all 3 books in the Claus series so far: Claus, Jack and Flury. Each individual blogger also can hold a giveaway for an e-copy of the winner's choice from Tony Bertauski's books.
For a chance to win enter the rafflecopter below.


text:
About Carnegie
Change country, current country: Sweden
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Our advisory
Strong performance in the face of changed market conditions for the first half of 2022
SCA, Vaisala and OX2 are the winners of Carnegie Sustainability Award 2022
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Force of the forest, impact through measurements and renewable energy to combat environmental changes are in focus for the winners of Carnegie Sustainability Award 2022.
"There are few things that will impact investor decisions more than how effectively a company can navigate the new green transition. The demand for companies with ESG integrated in the business model are enormous", says Lena Österberg, Head of Sustainability, Research & Strategy at Carnegie. "Both from a risk perspective as well as an business opportunity perspective", she continues.
Since 2019 Carnegie has awarded companies for their sustainability work with the purpose to put the spotlight on sustainability metrics that create shareholder value and present good examples to inspire companies and investors. The winners in 2022 are all companies associated with environmental changes and energy solutions, two of the most important topics in the market of today.
The winners of 2022
Large Cap: SCA
SCA is a Swedish forestry company and Europe's largest private forest owner with 2.6m hectares of forest land in Sweden, covering some 6% of the Swedish land area. The company's forests bind carbon equivalent to 10% of Sweden's CO2 emissions. SCA has a vertically integrated value chain with production facilities relatively close to its forest holdings, which enables the supply of sustainably sourced wood raw materials and energy. In 2021, 100% of its forests were FSC and PEFC certified. The company is CO2 positive and has set ambitious targets to further increase its positive contribution, but also has targets for zero waste, a fossil-free value chain, and to increase wind power production on its land to 11TWh.
Small Cap: Vaisala
With progressing climate change, Vaisala's products will be critical to monitor more extreme weather and environmental changes. The company has been a sustainability forerunner and early invented and implemented energy-saving improvements that only recently have become a norm. The company's products have for decades been built for extended endurance and minimum waste, and helping its customers use resources more efficiently. The company is acknowledged for its strong company culture and high ethical standards.
Best newcomer : OX2
Carnegie also highlights strong sustainably work among new companies on the stock exchange in the hopes that aspiring companies will, at an early stage, include sustainability in their long-term strategy. This year's winner in the 'Best newcomer' category, OX2, set out to be a fully sustainable business from start in 2004. The company is a leading pure-play developer of wind and solar assets, having developed and constructed more than 2.5GW of onshore wind capacity. The political and public pressure to develop renewables to combat climate change has never been stronger, and a vast and fast deployment is necessary to deliver on the EU's 2050 CO2 neutrality target. Here, skilled developers such as OX2 have a critical role to play so that we can successfully manage the transition.
About the award:
Carnegie established Carnegie Sustainability Award in 2019. The winners in three categories are chosen from among the 400 listed companies, approximately 95 percent of the stock market value in the Nordics, analysed by Carnegie and the purpose is to highlight good examples of sustainable companies and inspire others. The award is presented at Carnegie Sustainability seminar 31 of May 2022.
Previous winners:
Large-cap:
2021 Ørsted
2020 Neste
2019 Nibe
Small-cap:
2021Camurus
2020 Scatec Solar
2019 Thule
Best Newcomer:
2021 Pexip
2020 K2A.
2019 NCAB
© Copyright Carnegie Investment Bank AB 2020. All rights reserved.


Through her unusual approach, Barbara Wildenboer creates art that's often complex and intriguing. When she started as an artist, she was focused on combining media with photography, but as of recently she has shifted to creating sculptures out of old books. With this original concept, Wildenboer's work shows us a completely new approach to sculpturing using many different techniques. In items which would look old and boring to one person, Wildenboer finds a new opportunity to make people aware of problems such as environmental issues and endangerment of plants, animals and life in general.
By turning books into sculptures, Wildenboer gives these old books new artistic meaning. From being a work of literature, books become a sculpture. They are still perceived visually of course, but the way they are perceived has changed. These books may not be read anymore, but now they are being examined and watched – just like a sculpture. She cuts them apart, almost destroys them, and then she brings them back again in a new form. It could be described as an artistic reincarnation. One form dies, so that another one can be born. There are two aspects that give the viewer an insight into the previous function of the book. The first is the book title printed on the hard cover and the second one are the pages ripped into nerve-like shapes which present the core of the book that doesn't exist anymore – the text.
As opposite to the low-detail artworks, Barbara's Rythmical Arrangement of Vortices is a different story. The details are strong and numerous, but at the same time the simplicity is kept. The clock mechanism is shown through various smaller pieces, which together form a single art piece. Every part is similar in its round shape, but each one can be presented as an art-piece for itself without losing the point. In other works (Sublimation), she uses wood and hand-blown glass to create unique and simple tear sculptures. Both pieces are similar in the way they are created from a number of smaller pieces, but the difference is not in the way they are presented, but how they are constructed. Through the hand- blown glass technique tears create a simple, yet deep impression. On the other hand, through paper sculpturing, the round paper sculptures are more complex and they leaves a highly organized and functional impression. Exactly like a clock mechanism.
The round paper sculptures are complex and they leave a highly organized and functional impression.
Barbara Wildenboer's art is simple yet complex. Because of that, many viewers might be puzzled when they lay their eyes on her work for the first time. The confusion will most likely be followed by an intrigue. That is why the art-piece titles play a vital role in bringing Barbara's work to life. Similar to the book titles, they give people a small, but vital insight into understanding the book – an insight with which the viewers are free to do as they please.
She is represented by The Cat Street Gallery in Hong Kong.
Barbara Wildenboer lives and works in Pretoria, South Africa.


Is the Gold Rock Casino a Golden Opportunity to Win at Slots?
What's the welcome like if you wash up on this rock? Gold Rock Casino hits you with a busy homepage that looks the part and has a cool welcome offer to treat you to as well.
But let's dive deeper to find out why people are flocking to Gold Rock Casino. Will you be joining them?
A welcome bonus we can all get behind!
This one delivers a huge 200% on the first deposit you make when you sign up to the casino. That could be worth up to $500 if you max out the bonus, so make the most of this deal. Be sure to read the terms and conditions too though. There are wagering requirements for the deal, and you'll need to adhere to those.
You may have spotted the Rival logo on the homepage of this site. That means the site is powered by Rival, and that means all their games are available to play here. Once you have downloaded the casino software, you're good to get underway with a collection of sensational games to try.
We've revealed there are lots of Rival games here already, but what does that mean? Well, you can choose from slot games with either three or five reels in play. You can also sample some of their best i-Slots – and these are unique to Rival too, so you'll get the lot if you sign up to Gold Rock Casino now. They tend to pack in lots of special features and bonuses, so watch out for those.
The design of the Gold Rock Casino is very good indeed and is smooth and slick. Some designs go over the top and end up being too busy to work, but that isn't the case with this one. Make sure you make the most of this casino, along with all its Rival slots and games, because this is one you will not want to miss.
With the best welcome deal in town, followed by the chance to grab other bonuses and promotions as well, you will feel right at home from the moment you download their software. Are you ready to get involved and make the most of your time at Gold Rock?


Award-winning actress Angel Locsin is set to return on primetime TV via ABS-CBN's upcoming family drama series "The Legal Wife," where she will star with Jericho Rosales, Maja Salvador, and new Kapamilya star JC de Vera.
Angel last starred in the hit fantasy series in 2010 "Imortal" opposite John Lloyd Cruz.
Meanwhile, "The Legal Wife" is a milestone TV project for JC as it is his first ever teleserye in the Kapamilya network.
ABS-CBN's newest family drama is a TV masterpiece of award-winning directors Rory Quintos and Dado Lumibao, and part of the continuing celebration of 60 years of Philippine television.


You will only see the public (readable) repos here until you login.
Get access to an account. If you want (need) one, send me an email at joachim<...something ...>giaever.org.
NOTE! Regards to a new server and a new setup you will have to alter all of your existing ./git/config to use the user git.giaever.org@git.giaever.org instead of giaever@git.giaever.org.
Regards to the change in software where we now support https and ssh with git, and every repository is managed on-site and not back-end through XML-RPC anymore. BY NOW: Use it as you would have used GitHub, and if you're stuck - search for it on Google. Happy go lucky!
Dont worry! I will add them to the new site. Just be patient.
Do I need keys now?
Complete the key generation and remember where your key is stored. Its normally stored in ~/.ssh/id_rsa, but for me who has keys on many different hosts normally gives each key a unique name, i.e ~/.ssh/id_git_giaever_org.
Ensure the permission to the config-file is correct: chmod 0600 ~/.ssh/config.
Copy the printed content and paste that text where you add your key (login required) by pressing CTRL+v. Please give the key a good description as you may have several keys on different computers.
Hi there, You've successfully authenticated, but Gogs does not provide shell access.
If this is unexpected, please log in with password and setup Gogs under another user.
Clone your repo, to your local disk (note the extra /git-directory): $ git clone "repo url: either ssh or https>"
If you havent set your username as global, you should set them for this repo. Use --global flag after config to set it globally, for every repo.
git config user.name "My name"
$ git commit -am "Added files to project"
You need to be approved to get an GIT account here, and there is no such thing as a registration form, so dont bother look for it.
Well. First of all it "cost money" (if you need several private repos), and if you have the knowlegde; why pay for it? :) Since I'm (and probably the users of this GIT) mostly can't share our code, its better to have our own solution which we can adapt to our needs.


Using just a few ingredients you probably have on hand, you can make your own natural, homemade deodorant in no time! This homemade deodorant recipe is healthier and budget friendly!
If you are like me, you probably hate chemicals! They practically took over all of the aspects of our lives, we eat and breathe chemicals. It's probably impossible to leave chemicals free but I like to try and limit using those as much as possible. I make my own natural cleaning products (you can find the list here), I make my own compost instead of buying it and I stopped using softener in my laundry in favor of vinegar and now I thought I should give this homemade deodorant recipe a try.
I must admit, I was a bit skeptical at first mainly because I didn't want to use any binding agent in my deodorant (like Cocoa or Mango butter), other than coconut oil so I expected it to simply crumble. But to my surprise it all "glued" together nicely and stayed like that even when I applied to skin. I did have to rub my armpits a little after applying the deodorant, just to make sure the baking soda would stay put but that's just a teeny tiny detail that didn't bother me.
I love the homemade deodorant because it doesn't clog the pores and doesn't block the sweat glands as the commercial version does which is thought by some to be a possible cause of breast cancer. The deodorant kept it's scent the whole day and I didn't need to re apply. I used one of my favorite essential oils, Purification (a special blend of Lemongrass, Rosemary, Melaleuca, Myrtle, Citronella and Lavandin).
Purification is specifically designed to neutralize odors, making it perfect for this project!
Make sure baking soda is very fine. If not, run it through a grinder first.
Mix well with cornstarch or arrowroot powder.
Pour 2 tablespoons of coconut oil over the dry ingredients and mix very well with a fork.
Add the rest of the coconut oil and mix well. You should now get a paste like texture.
With a teaspoon, transfer the mixture to an old deodorant stick or to a small jar.