text:
\section{Introduction}\label{SectionOne}
The need for ubiquitous coverage and the increasing demand for high data rate services keep constant pressure on the mobile network infrastructure. There has been intense research to improve the system spectral efficiency and coverage, and a significant part of this effort focused on developing and optimizing the multiple access techniques. One such promising technique that been recently proposed is the low density spreading (LDS), which intelligently manages the multiple access interference to offer efficient and low complexity multiuser detection (MUD)~\cite{Tse_LDS}. It was shown that the low density spreading multiple access (LDSMA) enables near optimal MUD using belief propagation with affordable complexity~\cite{Tse_LDS,YoshidaTanaka_SSCDMA,RaymondSaad_SSCDMA,GuoWang_SSCDMA}. The LDSMA concept has been extended to multicarrier communication to cope with the multipath channel effect~\cite{FerThesis}. In multicarrier-LDSMA (MC-LDSMA), each symbol is spread over a small number of the available subcarriers, and each subcarrier is shared by more than one user. The main focus in the literature of LDS, was on the evaluation of the asymptotic behaviour of the belief propagation algorithm and optimization of the system's bit error rate (BER) performance~\cite{Tse_LDS,YoshidaTanaka_SSCDMA,RaymondSaad_SSCDMA,GuoWang_SSCDMA}.

Given the limited spectrum availability, radio resource allocation is essential for optimizing the performance of mobile communication systems. Most of existing work on radio resource allocation for uplink scenario can be divided into two categories. The first category deals with resource allocation for generic multiple access channel (MAC), where only the users' power constraints are considered~\cite{Verdu_ISI,Tse_Polymatroid,WeiYu_Jour}. In the second category, specific multiple access schemes such as orthogonal frequency division multiple access (OFDMA) and time division multiple access are considered. Comparing to the generic MAC, in the specific multiple access schemes, extra constraints need to be satisfied in the resource allocation problem. In general, these constraints make the allocation problem non-linear and non-convex, which motivates suboptimal solutions, and can be resolved by analysing the optimality conditions~\cite{NgSung_SPA_UL,HuangBerry_SPA_UL,EURASIP_OFDMA} or based on the game theory approach~\cite{YPW_Auction,HJL_Nash_J,Noh_Game}. Although, there has been a lot of work on the radio resource allocation, the existing schemes cannot be directly applied to the MC-LDSMA system under question.

The LDS structure imposes constraints on the number of users that share the same subcarrier and the number of subcarriers used for spreading each symbol, which make the radio resource allocation problem more challenging. We have proposed a heuristic algorithm for radio resource allocation without considering the spreading to simplify the problem~\cite{MyPaper_VTC11}. In this paper, we consider the radio resource allocation for MC-LDSMA by taking into account the spreading, which significantly change the structure of the problem. We will consider the radio resource allocation for two cases; single-user and multiuser. The contributions of this paper can be summarized as follows. i)~A single-user optimal power allocation scheme that consists of maximum ratio transmission (MRT) and water-filling is proposed. ii)~An optimal subcarrier partitioning scheme for the proposed power allocation is derived. iii)~Assuming continuous frequency selective channels, an optimal power allocation for MC-LDSMA is derived under certain relaxed conditions. iv)~Using the single-user power allocation algorithm and the insights gained from the multiuser optimal solution, two suboptimal subcarrier and power allocation algorithms are proposed. v)~The effects of subcarrier loading, effective spreading factor and power allocation are investigated through Monte Carlo simulations.\\
The rest of the paper is organized as follows. In Section~\ref{SectionTwo}, background information on the optimal power allocation for frequency selective MAC and the MC-LDSMA system model are presented. The optimal power allocation and subcarriers partitioning schemes for the single-user case are introduced in Section~\ref{SectionThree}. In Section~\ref{SectionFour}, the optimality conditions of radio resource allocation for the multiuser case in addition to suboptimal algorithms are provided. In Section~\ref{SectionFive}, we evaluate the performance of the proposed algorithms in single-user and multiuser scenarios using Monte Carlo simulations. Finally, concluding remarks are drawn in~Section~\ref{SectionSix}. Table~\ref{table_ListSyms} summarizes the frequently used mathematical symbols in this paper.
\section{Background and System Model}\label{SectionTwo}
\begin{table}[!t]
\caption{List of Symbols.}
\centering
\begin{tabular}{c||l}
\hline
Symbol & Description\\
\hline
$\mathcal{K}$ & Set of users\\

$K$ & Total number of users\\

$\mathcal{N}$ & Set of subcarriers\\

$N$ & Total number of subcarriers\\

$\mathcal{N}_k$ & Set of subcarriers allocated to user $k$\\

$N_k$ & Number of subcarriers allocated to user $k$\\

$\mathcal{M}_k$ & Set of subcarriers' subsets/groups of user $k$\\

$M_k$ & Number of symbols/subcarriers' subsets of user $k$\\

$P_k$ & Maximum power of user $k$\\

$w_k$ & Weight assigned to user $k$\\

$R_k$ & Rate of user $k$\\

$\lambda_k$ & Lagrange multiplier of the $k$th user's power constraint\\

$\mathcal{I}_k$ & Set of users that interfere with the $k$th user\\

$q$ & Interference from users not cancelled yet\\

$\sigma^2$ & Noise power spectral density\\

$\pi(.)$ & Permutation\\

$d_c$ & Number of users at each subcarrier in LDS\\

$d_v$ & Effective spreading factor in LDS\\

$\mathcal{D}_{k,m}$ & Set of subcarriers to spread symbol $m$ of user $k$ in LDS\\

$R_{k,m}$ & $k$th user rate on the $m$th subset of subcarriers in LDS\\

$u_{k,m}$ & $k$th user utility on the $m$th subset of subcarriers in LDS\\

$u_{k,n}$ & $k$th user utility on the $n$th subcarrier\\

$u_k(q,f)$ & $k$th user utility on the frequency $f$ with interference $q$\\

$\mathcal{B}$ & Set of users allocated the same weight/priority\\

$\bar{w}_l$ & Weight assigned to the $l$th group of users $(\mathcal{B}_l)$\\

$\mathcal{S}_f$ \& $\mathcal{S}_n$ & Set of active users at frequency $f$ \& subcarrier $n$\\

$\mathcal{P}_k(f)$ \& $p_{k,n}$ & $k$th user power at frequency $f$ \& subcarrier $n$\\

$x_k(f)$/$x_{k,n}$ & Frequency/subcarrier allocation index for user $k$\\

$\alpha_k(f)$ \& $\alpha_{k,n}$ & $k$th user channel response at frequency $f$ \& subcarrier $n$\\

$h_k(f)$ \& $h_{k,n}$ & $k$th user channel gain at frequency $f$ \& subcarrier $n$\\
\hline
\end{tabular}
\label{table_ListSyms}
\end{table}
The uplink scenario can be represented by a multiple access channel, where a set of users $\mathcal{K}=\{1, \dots, K\}$ transmit to a single base-station in the presence of additive Gaussian noise over frequency selective channels. As the channel frequency responses are not flat, it is expected that dynamic power allocation will improve the system performance. Considering perfect channel knowledge at the transmitter, the goal is to allocate the optimal transmit power for each user that maximizes the weighted sum-rate. The weights are used to prioritize the users as a fairness mechanism. The users are subject to individual maximum power constraints, $\mathbf{P}=[P_1, \dots, P_K]$. The capacity region of frequency selective MAC is given by\vspace{-0.1cm}
\begin{multline}\label{Eq_MAC_CapArea}
\hspace{-0.5cm}\bigcup_{\mathcal{P}\in\mathcal{F}}\Bigg\{\textbf{R}:\hspace{-0.1cm}\sum_{k\in \mathcal{S}} R_k\leq\hspace{-0.15cm}\int_{-W/2}^{W/2}\log \left(1+\frac{1}{\sigma^2}\displaystyle\sum_{k\in \mathcal{S}} \mathcal{P}_k(f) h_k(f)\right)df,\\
\forall \mathcal{S}\subset \mathcal{K}\Bigg\},
\end{multline}
where $\mathcal{S}$ is any subset of the total set of the users $\mathcal{K}$, and $\mathcal{F}$ is the set of feasible power allocation policies that satisfy the power constraint
\begin{equation}
\mathcal{F}\equiv \left\{\mathcal{P}:\int_{-W/2}^{W/2} \mathcal{P}_k(f)df\leq P_k, \forall k\right\}.
\end{equation}
Here, $W$ is the total bandwidth and $\mathcal{P}$ is a power allocation policy such that $\mathcal{P}_k(f)$ is the power user $k$ allocates at frequency $f$, and $h_k(f)=|\alpha_k(f)|^2$, where $\alpha_k(f)$ is the $k$th user channel frequency response at frequency $f$ and $\sigma^2$ is the noise power spectral density. This capacity area has $K!$ vertices in the positive quadrant, and each vertex is achievable by a successive decoding using one of the possible $K!$ distinct permutations of the set $\mathcal{K}$. The vertex $\mathbf{R}_\pi$, which corresponds to permutation $\pi(.)$, is given by
\begin{equation}
R_{\pi(k)}=\int_{-W/2}^{W/2} \log\Bigg( 1+\frac{\mathcal{P}_{\pi(k)}(f) h_{\pi(k)}(f)}{\sigma^2+\displaystyle\sum_{i=k+1}^{K}\mathcal{P}_{\pi(i)}(f) h_{\pi(i)}(f)}\Bigg)df.
\end{equation}
The problem of finding the optimum power allocation can be formulated as follows.\\
\textbf{Problem MAC}: \textit{Weighted sum-rate maximization for frequency selective MAC}
\begin{equation}\label{Eq_MACfs_Obj}
\max_{\mathcal{P}}\quad \sum_{k\in \mathcal{K}} w_{\pi(k)}R_{\pi(k)},
\end{equation}
subject to
\begin{equation}\label{Eq_MACfs_PowerConst}
\int_{-W/2}^{W/2} \mathcal{P}_k(f)df\leq P_k,\quad \forall k\in \mathcal{K},
\end{equation}
where $w_k$ is the weight associated with user $k$. By exploiting the Polymatroid properties of the capacity region~\eqref{Eq_MAC_CapArea} and using the Lagrange multipliers, Tse and Hanly~\cite{Tse_Polymatroid} have shown that optimal power allocation can be carried out by optimization at each frequency, and the optimal solution for a given $\boldsymbol{\lambda}=[\lambda_1, \dots, \lambda_K]$ is given by
\begin{multline}
\sum_{k\in\mathcal{K}} w_{\pi(k)} \log\left( 1+\frac{\mathcal{P}^\star_{\pi(k)}(f) h_{\pi(k)}(f)}{\sigma^2+\displaystyle\sum_{i=k+1}^{K}\mathcal{P}^\star_{\pi(i)}(f) h_{\pi(i)}(f)}\right)\\-\sum_{k\in\mathcal{K}} \lambda_k \mathcal{P}^\star_k(f)= \int_0^\infty u^\star(q,f)dq,
\end{multline}
where
\begin{equation}\label{Eq_UtilityMax}
u^\star(q,f)=\Big[\max_{k\in\mathcal{K}}u_k(q,f)\Big]^+,
\end{equation}
\begin{equation}
u_k(q,f)=\frac{w_k}{\sigma^2+q}-\frac{\lambda_k}{ h_k(f)}.
\end{equation}
Here, $u_k(q,f)$ is the utility of user $k$ on frequency $f$, and $q$ is the residual interference due to the users that are not yet cancelled. $\lambda_k$ is the Lagrange multiplier associated with the power constraint of user $k$. The optimal power allocation can be obtained via a greedy algorithm, which is summarized as follows~\cite{MyPaper_VTC11}. Let $\sigma^2+q$ represents the current ``interference level'' due to the noise and received power of users not yet cancelled. Starting with received power $q=0$, the optimal solution can be obtained by adding a user, at each interference level, that leads to the largest increase in the objective function. The decoding order that achieves the optimal solution is in increasing order of the users' weights $w_{\pi(1)}\leq w_{\pi(2)}\dots \leq w_{\pi(K)}$. It is worth mentioning that the solution is only valid for continuous channel gain or when the number of discrete channel gains reaches infinity~\cite{Tse_Polymatroid}.

For MAC, users' total transmit power $(P_k)$ is the only constraint in the optimization problem. For special multiple access schemes, an additional constraint is imposed on the optimization problem to control the number of users that share the same frequency. Let $\mathcal{S}_f$ be the set of active users at frequency $f$, $\mathcal{S}_f=\{k: \mathcal{P}_k(f)>0\}$. As a special multiple access scheme, in MC-LDSMA the condition, $\vert \mathcal{S}_f\vert\leq d_c$, must be satisfied. On the other hand, in OFDMA $\vert \mathcal{S}_f\vert\in\{0, 1\}$ must be satisfied. As will be shown in Section~IV, adding such a constraint to the optimization problem makes the problem non-convex. However, it will be proven that MC-LDSMA can be a special case of the generic MAC with certain conditions on the users' weights, and the power allocation algorithm for MAC can be used to find the optimal power allocation for MC-LDSMA. Therefore, the constraint will be fulfilled without explicitly imposing it in the optimization problem.

\begin{figure}[!t]
\centering
\includegraphics[width=8cm]{Figure_1}
\caption{Uplink MC-LDSMA block diagram.}
\label{Figure1_LDSbd}
\end{figure}

Now, the system model of a single-cell MC-LDSMA will be presented, which can also be found in~\cite{MyPaper_IWCMC}. The conceptual block diagram of an uplink MC-LDSMA system is depicted in Fig.~\ref{Figure1_LDSbd}. The total frequency band is divided into a set of subcarriers $\mathcal{N}=\{1, \dots, N\}$. Let $\textbf{a}_k$ be a data vector of user $k$ consisting of $M_k$ modulated data symbols and denoted as $\textbf{a}_k = [a_{k,1}, a_{k,2}, \dots, a_{k,M_k}]^T$. The signature matrix $\textbf{S}_k$ assigned for the $k$th user consists of $M_k$ LDS signatures, i.e. $\textbf{S}_k = [\textbf{s}_{k,1}, \textbf{s}_{k,2} \dots, \textbf{s}_{k,M_k}]$. Each LDS signature, $\textbf{s}_{k,m}\in\mathbb{C}^{N\times1}$, is a sparse vector consisting of $N$ chips among which only $d_v$ chips have non-zero values, where $d_v$ is the effective spreading factor. Each data symbol $a_{k,m}$ will be spread using the $m$th spreading sequence. Let $\textbf{c}_k = [c_{k,1}, c_{k,2}, \dots, c_{k,N}]^T$ denotes the chips vector belonging to user $k$ after the spreading process, which is given by $\textbf{c}_k = \textbf{S}_k \textbf{a}_k$.

At the receiver side, users' signals that are using the same subcarrier will be superimposed. However, the number of users interfere at each subcarrier $(d_c)$ is much less than the total number of users, i.e. $d_c\ll K$. After performing OFDM demodulation operation at the receiver, the received signal at subcarrier $n$ is given by
\begin{equation}\label{Eq_LDS_RecSign}
y_n = \sum_{k\in\mathcal{S}_n} \alpha_{k,n}c_{k,n} + v_n,
\end{equation}
where $v_n$ is the additive white Gaussian noise (AWGN) and $\mathcal{S}_n$ is the set of users that use subcarrier $n$. This signal is passed to LDS MUD to separate users' symbols using close to optimum chip-level iterative MUD based on message passing algorithm. More details about the LDS receiver can be found in~\cite{Tse_LDS}.
\section{Single-User Radio Resource Allocation}\label{SectionThree}
In this section, the single-user power allocation with LDS will be considered. The power allocation for LDS systems is more challenging compared with the non-spreading case, which has a well-known water-filling solution. Assuming that a set of subcarriers, $\mathcal{N}_k=\{1, \dots, N_k\}$, are allocated to a user, the user will partition this set of subcarriers into $M_k$ subsets/groups indexed by $\mathcal{M}_k=\{1, \dots, M_k\}$. Each subset of subcarriers will be used to spread one symbol, and the total number of transmitted symbols will be $M_k=N_k/d_v$. Let the $m$th subset of subcarriers denoted by $\mathcal{D}_{k,m}$, where $|\mathcal{D}_{k,m}|=d_v$, is used to spread the $m$th symbol, then the rate achieved on that subset is given by
\begin{equation}\label{Eq_SymRate}
R_{k,m}=\log{\left(1+\left(\sum_{n\in\mathcal{D}_{k,m}} \sqrt{p_{k,n}g_{k,n}}\right)^2\right)},\quad\forall m\in\mathcal{M}_k,
\end{equation}
where $g_{k,n}=\frac{h_{k,n}}{\sigma^2W/N}$. Note that~\eqref{Eq_LDS_RecSign} represents the received signal from all the users in each subcarrier before the LDS detector (belief propagation algorithm). It is worth mentioning that in multiple access techniques that don't use spreading, such as OFDMA, the information/symbols on each subcarrier are independent, hence, the rate achieved on a subset of subcarriers is obtained by a sum of the rate achieved on each subcarrier. However, this is not the case for spreading techniques, like MC-LDSMA. With spreading the same information (modulation symbol) is repeated on a set of subcarriers. Thus, the rate achieved on a set of subcarriers will be a function of the effective signal to noise ratio (SNR) on the subcarrier. Furthermore, in the optimization problem, we consider the effective SNR on the subcarriers that are used by each user/symbol after the LDS detector, which is represented in~\eqref{Eq_SymRate}.

The single-user power allocation problem for LDS can be split into two parts: Firstly, what is the optimal subcarriers partitioning that maximizes the user rate? Secondly, for a given subcarriers partitioning, what is the optimal power allocation? These two issues will be addressed in the sequel.
\subsection{Power Allocation}
For a given subcarriers partitioning scheme, the power allocation problem can be formulated as
\begin{equation}\label{Eq_SU_MaxObj}
\max_{p_{k,n}} \sum_{m\in\mathcal{M}_k}\log{\left(1+\left(\sum_{n\in\mathcal{D}_{k,m}} \sqrt{p_{k,n} g_{k,n}}\right)^2\right)},
\end{equation}
subject to
\begin{equation}
\sum_{m\in\mathcal{M}_k}\sum_{n\in\mathcal{D}_{k,m}} p_{k,n} = P_k,\quad \text{and}\quad p_{k,n}\geq 0.
\end{equation}
To find the optimal power allocation, first the power allocation within each symbol is considered then the power allocation over the symbols is optimized. Let $\bar{p}_{k,m}=\sum_{n\in\mathcal{D}_{k,m}}p_{k,n}$ be the power allocated to symbol $m$. The optimal power allocation of the symbol power, $\bar{p}_{k,m}$, over the symbol's subcarriers, $\mathcal{D}_{k,m}$, is given by maximum ratio transmission (MRT)
\begin{equation}\label{Eq_MRTPowerAllocation}
p_{k,n}=\frac{g_{k,n}}{\sum_{n\in\mathcal{D}_{k,m}} g_{k,n}}\ \bar{p}_{k,m},\quad \forall n\in\mathcal{D}_{k,m}.
\end{equation}
The MRT is optimum in the sense it maximizes the received SNR of the symbol~\cite[Chapter 7]{WC_GS_Book}. Consequently, by substituting~\eqref{Eq_MRTPowerAllocation} in the objective function~\eqref{Eq_SU_MaxObj}, the optimization problem will be
\begin{equation}\label{Eq_SymbolsRate}
\max_{\bar{p}_{k,m}} \sum_{m\in\mathcal{M}_k}\log{(1+ \bar{p}_{k,m}\bar{g}_{k,m})},
\end{equation}
subject to
\begin{equation}\label{Eq_SymPowerConst}
\sum_{m\in\mathcal{M}_k} \bar{p}_{k,m} = P_k,\quad \text{and}\quad \bar{p}_{k,m}\geq 0,
\end{equation}
where
\begin{equation}\label{Eq_EffecSymGain}
\bar{g}_{k,m}=\displaystyle\sum_{n\in\mathcal{D}_{k,m}}g_{k,n},
\end{equation}
which can be solved by single-user water-filling algorithm. The water-filling solution can be expressed as follows
\begin{equation}\label{Eq_SymbolWaterFilling}
\bar{p}^\star_{k,m}=\left[\frac{1}{\lambda_k}-\frac{1}{\bar{g}_{k,m}}\right]^+,
\end{equation}
where $[x]^+=\max (0,x)$ and $\lambda_k$ is the Lagrange multiplier that should satisfy the power constraint in~\eqref{Eq_SymPowerConst}. Hence, the single-user power allocation for MC-LDSMA will be conducted in two steps: the first step is water-filling to find the power for each symbol; the second step is MRT of the symbol power over the symbol's subcarriers. This power allocation algorithm will be referred to as MRT-WF.
\subsection{Subcarriers Partitioning}
In the MC-LDSMA technique, the allocated subcarriers to a user, $\mathcal{N}_k$, should be partitioned into $M_k$ subsets, each of which will be used to transmit one symbol. In the previous section, the optimal power allocation scheme for a given subcarriers partitioning has been derived. In this section, we investigate how to optimize the partitioning of available subcarriers. In conventional LDS codes, the subcarriers are partitioned randomly. The random partitioning has shown satisfactory BER performance~\cite{GuoWang_SSCDMA,FerThesis}. However, the partitioning scheme has not yet been investigated from the user rate optimization perspective. Here, the subcarriers partitioning will be studied under the objective of user rate maximization. The problem of partitioning the subcarriers to maximize the user rate can be formulated as follows
\begin{equation}\label{Eq_SU_PartOptiProb}
\max_{\mathcal{D}_{k,m}\in\mathcal{N}_k} \sum_{m\in\mathcal{M}_k}R_{k,m}(\mathcal{D}_{k,m}),
\end{equation}
subject to
\begin{equation}
\mathcal{D}_{k,m} \cap\mathcal{D}_{k,j}=\emptyset,\quad \forall m\neq j,\ m, j\in\mathcal{M}_k,
\end{equation}
\begin{equation}\label{Eq_SU_CardConst}
|\mathcal{D}_{k,m}|=d_v, \quad\forall m\in\mathcal{M}_k.
\end{equation}
This is a combinatorial optimization problem with a large search space. The number of possible LDS codes to be generated from the $N_k$ subcarriers for a specific spreading factor $d_v$ is given by
\begin{equation}
C_{LDS}=\frac{1}{2}{2d_v\choose d_v}\sum_{i=0}^{M_k-3} {N_k-1-i d_v\choose d_v-1}.
\end{equation}
In fact, $C_{LDS}$ represents the cardinality of the feasible search space of problem~(\ref{Eq_SU_PartOptiProb}~-~\ref{Eq_SU_CardConst}). In order to see how large is the search space, let us consider $N_k=32$ and $d_v=2$. In this case, the number possible LDS codes will be $C_{LDS}=1.92\times 10^{17}$. Apparently, brute-force search is infeasible for practical systems, and partitioning schemes with low complexity need to be sought. Even though the subcarriers partitioning problem is non-convex, there is an underlying Schur-convex structure, which can be utilized to solve the problem.

Let $\mathbf{g}_k=[\bar{g}_{k,1}, \dots, \bar{g}_{k,M_k}]$ represents the vector of symbols' gains for a given subcarrier partitioning scheme. Using the MRT-WF allocation and by substituting the optimal power~\eqref{Eq_SymbolWaterFilling} in the user rate~\eqref{Eq_SymbolsRate}, the user rate can be formulated as a function of the symbol gains as follows
\begin{equation}\label{Eq_SymbolsRate_2}
R_k(\mathbf{g}_k)= \sum_{m\in\mathcal{M}_k}\log{\left(\frac{\bar{g}_{k,m}}{\lambda_k}\right)},
\end{equation}
where $\frac{1}{\lambda_k}$ is the water-level and it is given by
\begin{equation}\label{Eq_WaterLevel_1}
\frac{1}{\lambda_k}=\frac{1}{M_k}\left(P_k+\sum_{m\in\mathcal{M}_k}\frac{1}{\bar{g}_{k,m}}\right).
\end{equation}
Based on this new formulation of the user rate, the following theorem can be introduced to link between the majorization of the symbol gains and the user rate with MRT-WF power allocation.
\begin{thm}\label{ThrmRateSchur}
The user rate $R_k(\mathbf{g}_k)$ is Schur-convex function of the symbol gains vector $\mathbf{g}_k$, and
\begin{equation}
\mathbf{g}_k\succ\hat{\mathbf{g}}_k\quad \Rightarrow\quad R_k(\mathbf{g}_k)\geq R_k(\hat{\mathbf{g}}_k),
\end{equation}
if $P_k<M_k^3-M_k^2-M_k$, where $\mathbf{x}\succ\mathbf{y}$ means $\mathbf{x}$ majorize $\mathbf{y}$.
\end{thm}
\begin{IEEEproof}
See Appendix~\ref{App_RateScurProof}
\end{IEEEproof}
Theorem~\ref{ThrmRateSchur} shows that the user rate~$R_k(\mathbf{g}_k)$ will be increased by majorizing the symbols' gains vector $\mathbf{g}_k$.
After showing the effect of symbols gains majorization on the user rate, the subcarrier partitioning can thus be derived based on this finding.
\begin{lem}\label{Lem_SC_Partitiong}
The optimal subcarrier partitioning that maximizes the user rate with MRT-WF power allocation is given by
\begin{equation}\label{Eq_GreedyScPartition}
\hspace{-0.18cm}\mathcal{D}^\star_{k,m}=\{g_{k,\pi(\hat{m})}, g_{k,\pi(\hat{m}+1)}, \dots,\ g_{k,\pi(\hat{m}+d_v-1)}\},\ \forall m\in\mathcal{M}_k,
\end{equation}
where $\hat{m} = (m-1)d_v+1$, and $\pi$ represents a permutation of the subcarrier gains in a descending order such that $g_{k,\pi(1)}\geq g_{k,\pi(2)}\geq\dots \geq g_{k,\pi(N_k)}$.
\end{lem}
\begin{IEEEproof}
See Appendix~\ref{App_SymMMVproof}
\end{IEEEproof}
In the proposed scheme, the subcarriers are sorted in descending order then the first best $d_v$ subcarriers are combined to create one symbol, the second best $d_v$ for another symbol and so on. This partitioning scheme results in symbols' gains vector $(\mathbf{g}_k)$ that majorize any other partitioning scheme. The proposed scheme will be compared with brute-force search in Section~V. For comparison purposes, another partitioning scheme will be included, which attempts to generate the least majorized vector (LMV) by making the symbols gains close to each other as much as possible. In this scheme, the subcarriers are sorted in descending order $g_{k,\pi(1)}\geq g_{k,\pi(2)}\geq\dots \geq g_{k,\pi(N_k)}$. Then, the subcarriers that have the best channel gains are combined with the subcarriers that have the least gains to create one LDS symbol. The LMV partitioning scheme can be defined as
\begin{equation}\label{Eq_PartitionSceme1}
\mathcal{D}_{k,m}=\left\{ \begin{array}{ll}
\{g_{k,\pi(m)},\ g_{k,\pi(N_k-m+1)}\}, & \text{for}\ d_v=2,\\
\\

\{g_{k,\pi(m)},\ g_{k,\pi(N_k-2m+1)}, & \\

g_{k,\pi(N_k-2m+2)}\}, & \text{for}\ d_v=3.
\end{array} \right.
\end{equation}
\section{Multiuser Radio Resource Allocation}\label{SectionFour}
In this section, the radio resource allocation for the MC-LDSMA in the multiuser case will be considered. Firstly, the subcarrier and power allocation problem for MC-LDSMA is formulated. The optimality conditions of power allocation for MC-LDSMA under relaxed conditions will be provided. This will shed light on the derivation of suboptimal subcarrier and power allocation algorithms.
\subsection{Problem Formulation}
To formulate the subcarrier and power allocation problem for MC-LDSMA system, let $x_{k,n}$ be the channel allocation index such that $x_{k,n}=1$ if subcarrier $n$ is allocated to user $k$ and $x_{k,n}=0$ otherwise. Consequently, for a given subcarrier partitioning, the weighted sum-rate maximization problem for MC-LDSMA system can be formulated as follows.\\
\textbf{Problem LDS}: \textit{Weighted sum-rate maximization for MC-LDSMA}
\begin{equation}\label{Eq_LDS_Obj}
\max_{x_{k,n},p_{k,n}} \sum_{k\in\mathcal{K}}w_{\pi(k)}\sum_{m\in\mathcal{M}_{\pi(k)}}R_{\pi(k),m},
\end{equation}
where
\begin{equation}
R_{\pi(k),m}=\log{\left(1+\left(\sum_{n\in\mathcal{D}_{\pi(k),m}}\sqrt{snr_{\pi(k),n}}\right)^2\right)},
\end{equation}
\begin{equation}
snr_{\pi(k),n}= \frac{p_{\pi(k),n}h_{\pi(k),n}}{\sigma^2W/N+\displaystyle\sum_{j=k+1}^K x_{\pi(j),n}p_{\pi(j),n}h_{\pi(j),n}},
\end{equation}
subject to
\begin{equation}
\sum_{m\in\mathcal{M}_k}\sum_{n\in\mathcal{D}_{k,m}} p_{k,n} \leq P_k, \quad \forall k\in\mathcal{K},
\end{equation}
\begin{equation}
p_{k,n}\geq 0, \quad \forall k\in\mathcal{K},\hspace{0.2cm} n\in\mathcal{N},
\end{equation}
\begin{equation}\label{Eq_UsersPerSubc}
\sum_{k\in\mathcal{K}} x_{k,n} \leq d_c, \quad \forall n\in\mathcal{N},
\end{equation}
\begin{equation}\label{Eq_Binar_Q}
x_{k,n}\in \{0,1\}, \quad\forall k\in\mathcal{K},\hspace{0.2cm} n\in\mathcal{N}.
\end{equation}
The constraints in~\eqref{Eq_UsersPerSubc} and~\eqref{Eq_Binar_Q} provide control on the number of users per subcarrier, which are referred to as the loading constraints. Note that there is an implicit spreading constraint, i.e. $|\mathcal{D}_{k,m}|>1$. The loading and spreading constraints distinguish the MC-LDSMA system from generic MAC. The \emph{Problem LDS} is non-convex for two reasons. Firstly, the binary constraint in~\eqref{Eq_Binar_Q} is a non-convex set. Secondly, the interference term in the objective function~\eqref{Eq_LDS_Obj} makes it a non-convex function. Unlike the case in OFDMA, the constraint in \eqref{Eq_Binar_Q} cannot be relaxed to take any real value in the interval $[0, 1]$ to make the problem tractable~\cite{NgSung_SPA_UL,HuangBerry_SPA_UL,EURASIP_OFDMA}. In OFDMA, when the constraint is relaxed the users are still orthogonal to each other. If the binary constraint is relaxed in LDS, all the users may interfere at each subcarrier, which violate the main concept of LDS that only $d_c$ users are allowed to interfere with each other at the same subcarrier. Furthermore, it can result in a situation when all the users transmit on all the subcarriers, leading to a fully connected graph rather than a low density one.
\subsection{Optimality Conditions for MC-LDSMA}
In this section, the optimality conditions for radio resource allocation in MC-LDSMA under relaxed conditions will be presented. The aim is to address the problem at a more fundamental level by finding the relationship between MC-LDSMA and the optimal multiple access scheme. Then the insights gained from the optimality conditions will be used to design suboptimal algorithms for subcarrier and power allocation in the next section. The spreading constraint is relaxed to find the solution to the problem that satisfies the other conditions, specifically, the loading constraints. Here, it will be shown that \emph{Problem LDS} (\ref{Eq_LDS_Obj}~-~\ref{Eq_Binar_Q}) can be solved by solving \emph{Problem MAC} (\ref{Eq_MACfs_Obj}~and~\ref{Eq_MACfs_PowerConst}) under specific conditions. To relax the spreading constraint, let $d_v=1$, and by allowing the channel gain to be continuous, the following relaxed problem is obtained.

\noindent \textbf{Problem Relaxed LDS:}
\begin{equation}\label{Eq_ReLDS_Obj}
\max_{x_k(f),\mathcal{P}_k(f)} \sum_{k\in\mathcal{K}}w_{\pi(k)}x_{\pi(k)}(f)R_{\pi(k)},
\end{equation}
where
\begin{multline}
R_{\pi(k)}=\\ \int_{-W/2}^{W/2} \log{\Bigg(1+\frac{\mathcal{P}_{\pi(k)}(f)h_{\pi(k)}(f)}{\sigma^2+\displaystyle\sum_{i=k+1}^K x_{\pi(i)}(f)\mathcal{P}_{\pi(i)}(f)h_{\pi(i)}(f)}\Bigg)df},
\end{multline}
subject to
\begin{equation}
\int_{-W/2}^{W/2} \mathcal{P}_k(f)df\leq P_k, \quad \forall k\in\mathcal{K},
\end{equation}
\begin{equation}
\mathcal{P}_k(f)\geq 0, \quad \forall k\in\mathcal{K},
\end{equation}
\begin{equation}\label{Eq_UsersPerSubcR}
\sum_{k\in\mathcal{K}} x_k(f) \leq d_c,
\end{equation}
\begin{equation}\label{Eq_Binar_QR}
x_k(f)\in \{0,1\}, \quad\forall k\in\mathcal{K}.
\end{equation}
In this new formulation, the problem is still non-convex due to the binary constraint~\eqref{Eq_Binar_QR}. However, this relaxation will allow us to satisfy the loading constraints~\eqref{Eq_UsersPerSubcR} and~\eqref{Eq_Binar_QR} without explicitly imposing them in the optimization problem as follows.
\begin{lem}\label{Lemma1}
The solution of \emph{Problem Relaxed LDS} (\ref{Eq_ReLDS_Obj}~-~\ref{Eq_Binar_QR}) can be obtained by solving \emph{Problem MAC} (\ref{Eq_MACfs_Obj}~and~\ref{Eq_MACfs_PowerConst}) with users grouped into $d_{c}$ groups and the users in each group, $\mathcal{B}_l, l=1, \dots, d_{c}$, are assigned the same weight
\begin{equation}\label{Eq_UsersGrouping}
w_k= \bar w_l,\quad \forall k\in\mathcal{B}_l,\quad l=1, \dots, d_{c}.
\end{equation}
\end{lem}
\begin{proof}
We have to prove that at each frequency, only one user from each group may be chosen for all the interference levels. With users grouping~\eqref{Eq_UsersGrouping}, the maximization problem in~\eqref{Eq_UtilityMax} can be divided into number of sub-problems, each for one group $\mathcal{B}_l$
\begin{equation}
u^*(q,f)=\max_l\Big(\max_{k\in\mathcal{B}_l}\;u_k(q,f)\Big).
\end{equation}
In each group of users $\mathcal{B}_l$, the term $\frac{\bar w_{l}}{\sigma^2+q}$ will be the same for all the users in the group and the user with the smallest $\frac{\lambda_k}{h_k(f)}$ will be dominant for each frequency~$f$

\begin{equation}
u_k(q,f)=\frac{\bar w_l}{\sigma^2+q}-\frac{\lambda_k}{h_k(f)},\quad \forall k\in\mathcal{B}_l,
\end{equation}
\begin{multline}
u_k(q,f) > u_j(q,f),\quad\text{for}\quad q\in[0,\infty),\quad if\\ \quad\frac{\lambda_k}{h_k(f)}<\frac{\lambda_j}{h_k(f)},\quad \forall j\neq k,\quad k, j\in\mathcal{B}_l.\nonumber
\end{multline}
Hence, for each frequency, only one user from each group will be selected, which is given by
\begin{equation}
k_l^*(f)=\arg\min_{k\in\mathcal{B}_l}\;\frac{\lambda_k}{h_k(f)}.
\end{equation}
Consequently, no more than $d_c$ users will share each frequency~$f$.
\end{proof}
The objective is that, by grouping the users into groups and allocating the same weight for the users in the same group, the users sharing each frequency will not exceed the number of groups. The complexity of the optimal solution lies in the computation of the Lagrange multipliers, which are computed by an iterative algorithm to satisfy the power constraints~\cite{Tse_Polymatroid}. Consequently, the complexity for solving Problem~MAC is still too high for practical cellular systems. Furthermore, partitioning of the available bandwidth in an optimal fashion is difficult to achieve practically. However, the optimal algorithm and Lemma~$1$ shed light on the structure of the optimal solution, which will be useful in designing suboptimal algorithms.
\subsection{Suboptimal Algorithms}
As discussed in the previous section, finding the optimal solution to \emph{Problem LDS} (\ref{Eq_LDS_Obj}~-~\ref{Eq_Binar_Q}) is computationally intensive and impractical. Therefore, suboptimal subcarrier and power allocation algorithms with low complexity are presented here. We propose two suboptimal solutions that differ in the stage where the spreading is considered. In the first algorithm, the effect of spreading is taken into account from the outset, and in the following steps we consider:
\begin{itemize}
\item Each user will partition the available subcarriers, $\mathcal{N}$, into $M$ subsets using the proposed partitioning scheme~\eqref{Eq_PartitionSceme1}.
\item Each user performs MRT-WF,~\eqref{Eq_MRTPowerAllocation}~and~\eqref{Eq_SymbolWaterFilling}, as single-user power allocation strategy.
\item The subcarriers are allocated on set basis. In other words, at each iteration of the algorithm, $d_v$ subcarriers are allocated to one user.
\end{itemize}
In the second algorithm, no spreading is assumed at the start and
\begin{itemize}
\item The subcarriers will not be partitioned at the first stage.
\item Single-user water-filling power allocation is used for each user.
\item The subcarriers are allocated individually, i.e., at each iteration of the algorithm one subcarrier is allocated to one user.
\item After the subcarrier and power allocation is finished, the subcarriers are partitioned using~\eqref{Eq_PartitionSceme1} and the spreading effect is taken into account for rate calculation.
\end{itemize}
As the first algorithm is based on MRT-WF, it is referred to it as MUMRT, where MU stands for multiuser. The second algorithm is referred to as MUWF, since it is based on water-filling power allocation.

For the MUMRT algorithm, let us define the utility of the $k$th user on the $m$th set of subcarriers, $\mathcal{D}_{k,m}$, as follows\\
\begin{multline}\label{Eq_UtilityMUMRT}
u_{k,m}=\\w_k \log{\left(1+\left(\sum_{n\in\mathcal{D}_{k,m}}\sqrt{\frac{h_{k,n}p_{k,n}}{\displaystyle\sum_{i\in\mathcal{I}_k}x_{i,n}h_{i,n}p_{i,n}+\sigma^2W/N}}\right)^2\right)},
\end{multline}
where $\mathcal{I}_k$ is the set of users that interfere on the $k$th user, and based on the optimal successive decoding, it is given by $\mathcal{I}_k=\{i:w_i>w_k\}$. In other words, each user will experience interference only from the users with higher weights. The suboptimal subcarrier and power allocation algorithm consists of two phases. In the first phase, the utilities of every user on each set of subcarriers are calculated. To this end, the MRT-WF single-user power allocation algorithm and partitioning scheme developed in the previous section are used. In the second phase, a set of subcarriers is allocated to one user based on the utilities calculated in the first phase. The algorithm iteratively allocates a set of subcarriers in each iteration. Analogous to the optimal algorithm, the allocated subcarriers are no longer available to the users from the same group.

Starting with zero interference, each user will calculate the utility on every set of subcarriers. Based on the calculated utilities, one user will be allocated one set of subcarriers. The interference is updated according to the subcarrier allocation and the users' utilities are recalculated. The algorithm iterate until all the users have zero-utility on all the unallocated subcarriers.

Two different subcarrier allocation criteria are considered for the second phase of the algorithm. The first one is allocating the set of subcarriers to the user that has the maximum utility value, $u_{k,m}$, and we refer to this subcarrier allocation criterion as SA$1$. This criterion is inspired by the fact that in the optimal algorithm, maximizing the main objective is decoupled into maximizing over every channel state. The second criterion, which is referred to as SA$2$, is to allocate a set of subcarriers to the user that achieve the maximum increase in the objective function~\eqref{Eq_LDS_Obj}
\begin{equation}
k^\star=\arg\max_k w_k (R_k-R_k^a),
\end{equation}
where $R_k^a$ is the rate of user $k$ using the subcarriers that already allocated to that user $(\mathcal{N}_k)$, and $R_k$ is the rate on all the subcarriers $(\mathcal{N}_k\cup \mathcal{N}_k^u)$ calculated at the power allocation phase.

The second algorithm, MUWF, has the same structure as the first algorithm but with different utility function which is defined at each subcarrier instead of set of subcarriers as follows
\begin{equation}\label{Eq_UtilityMUWF}
u_{k,n}=w_k \log{\left(1+\frac{h_{k,n}p_{k,n}}{\displaystyle\sum_{i\in\mathcal{I}_k}x_{i,n}h_{i,n}p_{i,n}+\sigma^2W/N}\right)}.
\end{equation}
Also, single-user water-filling is used for power allocation and one subcarrier is allocated in each iteration. After allocating all the subcarriers to the users, each user partitions the allocated subcarriers using the proposed partitioning scheme~\eqref{Eq_GreedyScPartition}, and the MRT-WF power allocation is implemented on the allocated subcarriers.
\begin{algorithm}[!t]
\small
\caption{\small Iterative Subcarrier and Power Allocation (MUMRT)}
\label{ALG_MUWF1}
\begin{algorithmic}[1]
\STATE \textbf{Users Grouping:} Set $w_k=\bar{w}_l,\quad\forall k\in\mathcal{B}_l,\quad l=1, \dots, d_c$.
\STATE \textbf{Initialization:} Put $\mathbf{J}=\textbf{0}$, $\mathcal{N}_k=\emptyset$ and $\mathcal{N}_k^u=\mathcal{N},\hspace{0.3cm}\forall k\in\mathcal{K}$.
\REPEAT
\STATE \textbf{Subcarriers Partitioning:} Partition the subcarriers using~\eqref{Eq_GreedyScPartition}:\\
$\mathcal{N}_k\rightarrow\mathcal{M}_k,\ \text{and}\ \mathcal{N}_k^u\rightarrow\mathcal{M}_k^u,\quad \forall k\in\mathcal{K}$.
\STATE \textbf{Power Allocation:} Considering the interference from other users, $\textbf{J}$, each user performs MRT-WF over $(\mathcal{N}_k\cup \mathcal{N}_k^u)$.
\STATE \textbf{Subcarrier Allocation (SA):} Calculate $u_{k,m}$ using~\eqref{Eq_UtilityMUMRT}$\ \forall k$:\\
\textbf{Option 1 (SA1):} Choose the pair such that:\\ $(k^\star,m^\star)=\arg\displaystyle\max_{k\in\mathcal{K},m\in\mathcal{M}_k^u} u_{k,m}$.\\
\textbf{Option 2 (SA2):} Perform power allocation for all $k\in\mathcal{K},\ n\in\mathcal{N}_k$ and calculate $R^a_k$. Choose the pair $(k^\star,m^\star)$ such that:\\
$k^\star=\arg\displaystyle\max_k w_k (R_k-R_k^a)$ and $m^\star=\arg\displaystyle\max_{m\in\mathcal{M}_{k^\star}^u} u_{k^\star,m}.$
\STATE Allocate the subcarriers subset $\mathcal{D}_{k^\star, m^\star}$ to user $k^\star$:
\\Set $x_{k^\star,n}=1,\quad\mathcal{N}_{k^\star}=\mathcal{N}_{k^\star}\cup \{n\},\quad \forall n\in\mathcal{D}_{k^\star, m^\star}$.
\STATE Remove the allocated subcarriers $(\mathcal{D}_{k^\star, m^\star})$ from the available subcarriers of the users' group $(\mathcal{B}_l)$ that includes $k^\star$:\\
$\mathcal{N}_k^u=\mathcal{N}_k^u\setminus{n},\quad\forall n\in\mathcal{D}_{k^\star, m^\star},\quad\forall k\in\mathcal{B}_l$.
\STATE Update the interference matrix elements $j_{k,n}\;\forall k\in\mathcal{I}_{k^\star}^c$,\\ $n\in\mathcal{D}_{k^\star, m^\star}$, where $\mathcal{I}_{k}^c=\mathcal{K}\setminus(\mathcal{I}_{k}\cup \{k\})$.
\UNTIL{$u_{k,m}=0,\quad\forall m\in\mathcal{M}_k^u$} or $\mathcal{M}_k^u=\emptyset,$ $\quad\forall k\in\mathcal{K}$.
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[!t]
\small
\caption{\small Iterative Subcarrier and Power Allocation (MUWF)}
\label{ALG_MUWF2}
\begin{algorithmic}[1]
\STATE \textbf{Users Grouping:} Set $w_k= \bar{w}_l,\quad \forall k\in\mathcal{B}_l,\quad l=1, \dots, d_c$.
\STATE \textbf{Initialization:} Put $\mathbf{J}=\textbf{0}$, $\mathcal{N}_k=\emptyset$ and $\mathcal{N}_k^u=\mathcal{N},\hspace{0.3cm}\forall k\in\mathcal{K}$.
\REPEAT
\STATE \textbf{Power Allocation:} Considering the interference from other users, $\textbf{J}$, each user performs water-filling over $\mathcal{N}_k\cup \mathcal{N}_k^u$.
\STATE \textbf{Subcarrier Allocation (SA):} Calculate $u_{k,n}$ using~\eqref{Eq_UtilityMUWF}$\ \forall k$:\\
\textbf{Option 1 (SA1):} Choose the pair $(k^\star,n^\star)$ such that:\\ $(k^\star,n^\star)=\arg\displaystyle\max_{k\in\mathcal{K},m\in\mathcal{N}_k^u} u_{k,n}$.\\
\textbf{Option 2 (SA2):} Perform power allocation for all $k\in\mathcal{K},\ n\in\mathcal{N}_k$ and calculate $R^a_k$. Choose the pair $(k^\star,n^\star)$ such that:\\
$k^\star=\arg\displaystyle\max_k w_k (R_k-R_k^a)$ and $n^\star=\arg\displaystyle\max_{m\in\mathcal{N}_{k^\star}^u} u_{k^\star,n}.$
\STATE Allocate subcarrier $n^\star$ to user $k^\star$:
\\Set $x_{k^\star,n^\star}=1, \quad \text{and} \quad\mathcal{N}_{k^\star}=\mathcal{N}_{k^\star}\cup \{n^\star\}$.
\STATE Remove the subcarrier $n^\star$ from the available subcarriers of the users' group $(\mathcal{B}_l)$ that includes $k^\star$:
$\mathcal{N}_k^u\hspace{-0.05cm}=\mathcal{N}_k^u\hspace{-0.02cm}\setminus\{n^\star\},\ \forall k\in\mathcal{B}_l$.
\STATE Update the interference matrix elements $j_{k,n^\star}\;\forall k\in\mathcal{I}_{k^\star}^c$,
where $\mathcal{I}_{k}^c=\mathcal{K}\setminus(\mathcal{I}_{k}\cup \{k\})$.
\UNTIL{$u_{k,n}=0,\quad\forall n\in\mathcal{N}_k^u$} or $\mathcal{N}_k^u=\emptyset,$ $\quad\forall k\in\mathcal{K}$.
\STATE \textbf{Final power allocation}: Partition the subcarriers using~\eqref{Eq_GreedyScPartition} and implement MRT-WF for each user on the subcarriers allocated to it.
\end{algorithmic}
\end{algorithm}

The proposed subcarrier and power allocation algorithms, MUMRT and MUWF, are summarized in Algorithm~\ref{ALG_MUWF1}~and~\ref{ALG_MUWF2}, respectively. In Algorithm~\ref{ALG_MUWF1}~and~\ref{ALG_MUWF2}, the notation $\mathcal{N}_k^u$ represents the set of subcarriers that are not allocated to user $k$ or any user in its group. $\mathcal{M}_k^u$ is the set of symbols after subcarriers partitioning of $\mathcal{N}_k^u$. The interference matrix, $\mathbf{J}\in\mathcal{R}_{+}^{K\times N}$, indicates the interference every user will experience on each subcarrier from other users, where $j_{k,n}=\sum_{i\in\mathcal{I}_k}x_{i,n}h_{i,n}p_{i,n}$. It is worth mentioning that the algorithms have a deterministic number of iterations to allocate all the subcarriers for a given $d_c$ and $d_v$. The first algorithm, as it allocates $d_v$ subcarriers in each iteration, requires $N d_c/d_v$ maximum number of iterations to allocate all the available subcarriers. On the other hand, by allocating one subcarrier in each iteration, the second algorithm requires $N d_c$ maximum number of iterations.
\section{Numerical Results}\label{SectionFive}
\begin{figure}[!t]
\centering
\includegraphics[width=8.5cm]{Figure_2}
\caption{Spectral efficiency comparison for different subcarrier partitioning schemes (with MRT-WF and $d_v=2$).}
\label{fig_PartitionMRTRes_dv2}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=8.5cm]{Figure_3}
\caption{Spectral efficiency comparison for different subcarrier partitioning schemes (with MRT-WF and $d_v=3$).}
\label{fig_PartitionMRTRes_dv3}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=8.5cm]{Figure_4}
\caption{Spectral efficiency comparison for the two algorithms with different subcarrier allocation criteria ($d_c=6$ and $d_v=2$).}
\label{fig_SE_MRTvsWF_SA1vsSA2}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=8.5cm]{Figure_5}
\caption{Outage probability comparison for the two algorithms with different subcarrier allocation criteria ($d_c=6$ and $d_v=2$).}
\label{fig_OutageZero_MRTvsWF_SA1vsSA2}
\end{figure}

In this section, the performance of the proposed algorithms is evaluated through Monte Carlo simulation. A single hexagonal cell with $1$~km inradius is considered, and the users' locations are randomly generated and uniformly distributed over the cell. The maximum transmit power of each user is $23$~dBm and the system bandwidth is $5$~MHz consisting of $30$~subcarriers. The link gain between the base-station and a user is given as the product of path loss and fast fading effects. ITU pedestrian~B channel model~\cite{IMT_Guideline} is adopted for fast fading generation and the simplified model~\cite{WC_GS_Book} for the path loss. The noise power spectral density is assumed to be $-173$~dBm/Hz. Spectral efficiency and outage probability are used as the performance evaluation metrics. Taking into account fairness among users, the outage probability is defined as the number of users with zero rates divided by total number of users. To impose fairness in the system, users are grouped into $d_c$ groups based on their location from the base-station, and the group of users that are furthest from the base-station are assigned higher weight comparing to users that are closer to the base-station.

Starting with the single-user case, Figs.~\ref{fig_PartitionMRTRes_dv2} and~\ref{fig_PartitionMRTRes_dv3} show the user spectral efficiency for different subcarriers partitioning schemes with $d_v=2$ and~$3$, respectively. Due to the complexity of brute-force search, small numbers of subcarriers ($N$) have been considered in this case. As it can be observed from the figures, the proposed partitioning scheme achieves the same spectral efficiency as the brute-force search and significantly outperforms the random and LMV schemes. This provide a numerical proof on the optimality of the proposed scheme. The random and LMV schemes only achieve about $86\%$ and $73\%$ of the optimal solution, respectively, which indicates the sub-optimality of these schemes. Furthermore, the very poor performance of the LMV partitioning, demonstrate the importance of vector majorization in generating the symbols gains, where it shows that a less majorized gain vector will produce very low user rate.

\begin{figure}[!t]
\centering
\includegraphics[width=8.5cm]{Figure_6}
\caption{Spectral efficiency comparison for different subcarrier loading $(d_c)$ (with MUMRT-SA1 and $d_v=2$).}
\label{fig_SE_MUMRT_SA1_diffDc}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=8.5cm]{Figure_7}
\caption{Outage probability comparison for different subcarrier loading $(d_c)$ (with MUMRT-SA1 and $d_v=2$).}
\label{fig_OutageZero_MUMRT_SA1_diffDc}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=8.5cm]{Figure_8}
\caption{Spectral efficiency comparison for different effective spreading factors (with MUMRT-SA$1$).}
\label{fig_Spectral_Eff_2}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=8.5cm]{Figure_9}
\caption{Spectral efficiency comparison between MC-LDSMA (with MUMRT-SA$1$ and $d_v=2$) and OFDMA systems.}
\label{fig_Spectral_Eff_3}
\end{figure}

Figs.~\ref{fig_SE_MRTvsWF_SA1vsSA2}~and~\ref{fig_OutageZero_MRTvsWF_SA1vsSA2} present the spectral efficiency and outage probability, respectively, in the multiuser case, for the two proposed algorithms together with the static subcarrier and power allocation case. The figures show the difference between the two algorithms (MUMRT and MUWF) and the two subcarrier allocation criteria (SA$1$ and SA$2$). In the static case, the subcarriers are equally distributed among the users, and the power of each user is equally allocated on its subcarriers. It is evident from Fig.~\ref{fig_SE_MRTvsWF_SA1vsSA2} that both algorithms significantly improve the system spectral efficiency comparing to the static subcarrier and power allocation. Also, as the figure shows, the MUMRT algorithm achieves higher spectral efficiency compared with the MUWF algorithm. As shown in Fig.~\ref{fig_OutageZero_MRTvsWF_SA1vsSA2}, MUWF algorithm achieves less outage probability thanks to the capability of allocating individual subcarriers to the users. Furthermore, it can be observed from Fig.~\ref{fig_SE_MRTvsWF_SA1vsSA2} that the two subcarrier allocation criteria achieve almost the same spectral efficiency. The SA$1$ criterion has a marginal gain compared with SA$2$ in MUMRT algorithm. However, as it can be seen from Fig.~\ref{fig_OutageZero_MRTvsWF_SA1vsSA2}, SA$2$ provides better fairness among users compared to SA$1$ by achieving less outage probability. Consequently, it can be concluded that SA$1$ is more greedy in the resource allocation in comparison to SA$2$.

To demonstrate the effect of subcarrier loading, $d_c$, Figs.~\ref{fig_SE_MUMRT_SA1_diffDc}~and~\ref{fig_OutageZero_MUMRT_SA1_diffDc} show the spectral efficiency and outage probability, respectively, for different subcarrier loading values. It can be seen from the figures that increasing the subcarrier loading improves the system performance in terms of spectral efficiency and outage probability at the cost of increased receiver complexity. Increasing the subcarrier loading will allow subcarriers to be reused by more users, which will increase the spectral efficiency. Also, the number of users' groups will be increased, resulting in improved fairness to the system, which consequently improves the outage probability. Furthermore, it can be observed that the required subcarrier loading increases as the total number of users increase to maintain a desired system performance. As an example, when the total number of users is $50$ and the subcarrier loading are $2$ and $3$, the outage probabilities will be $54\%$ and $36\%$, respectively. Therefore, the subcarrier loading needs to be adapted to the system load (total number of users) to balance between the system performance and complexity.

In order to quantify the effect of the effective spreading factor $d_v$ on the system performance, Fig.~\ref{fig_Spectral_Eff_2} presents the spectral efficiency for $d_v=2$ and $d_v=3$. As is evident from the figure, increasing the effective spreading factor will decrease the spectral efficiency. Also, the subcarrier loading has more impact for higher spreading factors. This result may give indication that it is more spectral efficient to not spread and use an orthogonal scheme. The following discussion will clarify this issue by comparing between MC-LDSMA and orthogonal transmission.

\begin{figure}[!t]
\centering
\includegraphics[width=8.5cm]{Figure_10}
\caption{Outage probability comparison between MC-LDSMA (with MUMRT-SA$1$ and $d_v=2$) and OFDMA systems.}
\label{fig_Outage_Pro_3}
\end{figure}
From system level perspective, spreading has advantage and disadvantage. The advantage of spreading is to allow the users share the subcarriers at the same time, which will increase the system capacity comparing to orthogonal transmission. On the other hand, spreading reduces the user's rate~\cite{Massey_ITspread}. Thus, it is essential to investigate if the gain of allowing the users to share the subcarriers is sufficient to compensate the loss due to spreading. To this end, as a non-orthogonal transmission, MC-LDSMA will be compared with OFDMA system, as an orthogonal transmission technique. For OFDMA system, the subcarrier and power allocation algorithm developed in~\cite{EURASIP_OFDMA} will be used.

Figs.~\ref{fig_Spectral_Eff_3}~and~\ref{fig_Outage_Pro_3} show the spectral efficiency and outage probability, respectively, for MC-LDSMA and OFDMA with different subcarrier loading $(d_c)$. For the OFDMA system, the $(d_c)$ value is just for the weight calculation to keep the comparison fair and it does not affect the subcarrier loading as it is always equal to $1$. As is clear from Fig.~\ref{fig_Spectral_Eff_3}, for very small number of users ($5$ and $10$), OFDMA slightly outperforms MC-LDSMA. However, for high number of users, MC-LDSMA significantly outperforms OFDMA. This due the fact that when the total number of users is small there is no need for reusing the subcarriers, and the available subcarriers are enough to support all the users.

On the other hand, when the total number of users is high, the subcarrier sharing is essential to support more users and achieve higher spectral efficiency. Furthermore, as can be observed from the figure, using higher fairness by increasing the number of users groups, results in a marginal reduction in the OFDMA spectral efficiency. This is because more fairness will be imposed and the subcarrier loading is not changed. In contrast, in MC-LDSMA, the increase in fairness is combined with an increase in the subcarrier loading, which improves the system performance. Also, as Fig.~\ref{fig_Outage_Pro_3} shows, MC-LDSMA is fairer comparing to OFDMA system, and the resulting outage probability is much less than OFDMA. As an example, when the total number of users is $50$, OFDMA has outage probability about $43\%$, while for MC-LDSMA $21\%,\ 6.3\% \text{ and } 2.3\%$ can be achieved with $d_c=4, 6 \text{ and } 8$, respectively.

From the results shown in Figs.~\ref{fig_Spectral_Eff_3} and~\ref{fig_Outage_Pro_3}, it can be concluded that low density spreading with the proposed radio resource allocation algorithms can significantly increase the system performance in terms of spectral efficiency and outage probability compared with orthogonal transmission techniques. These gains can be translated into an increase in the number of supported users within the cell and higher users' data rates.
\section{Conclusion}\label{SectionSix}
In this paper, we considered the problem of radio resource allocation for single-cell MC-LDSMA system in single-user and multiuser scenarios. For single-user radio resource allocation, an optimal power allocation has been derived, which consists of two steps: water-filling over the symbols and maximum ratio transmission over the subcarriers of each symbol. Also, we have shown that the optimal subcarriers partitioning scheme is the one that gives the most majorized symbols' gain vector. Numerical analysis revealed that the proposed partitioning scheme can significantly improve the user rate comparing to the conventional random partitioning. For multiuser radio resource allocation in MC-LDSMA, an analysis for the optimality conditions is provided and an optimal power allocation for a relaxed problem has been derived for continuous frequency selective channel. Using the structure of the optimal solution and the single-user power allocation, two suboptimal subcarrier and power allocation algorithms are proposed. The main difference between the two algorithms is the spreading effect: the first algorithm (MUMRT) takes into account the spreading from the onset of the allocation process and the subcarriers are allocated on subsets basis, whereas the second algorithm (MUWF) assumes no spreading and the subcarriers are allocated individually. The performance of the proposed algorithms have been evaluated by simulations, and results show that the proposed algorithms significantly improve the system spectral efficiency comparing to static subcarrier and power allocation. MUMRT algorithm has shown better performance comparing to the other one, and one subcarrier allocation (SA$1$) criterion has shown more greedy nature comparing to the other one (SA$2$). It is shown that there is a trade-off between the complexity (subcarrier loading) and the system performance, and the subcarrier loading can be chosen accordingly based on the system loading. Furthermore, we found that increasing the spreading factor to more than $2$ will decrease the system performance. Finally, a comparison between MC-LDSMA and OFDMA systems has been carried out and shown that MC-LDSMA can improve the system performance in terms of spectral efficiency and outage probability comparing to OFDMA.

\appendices
\section{Proof of Theorem~\ref{ThrmRateSchur}}\label{App_RateScurProof}
\begin{IEEEproof}
The proof of the theorem makes use of the properties of the elementary symmetric polynomials by using them as bridge to link between symbols gains majorization and the user rate. As the relation between the elementary symmetric polynomials and vector majorization is known, the only thing required is to show the connection between the user rate and the elementary symmetric polynomials. To this end, the user rate will be reformulated as a function of the elementary symmetric polynomials. The water-level in~\eqref{Eq_WaterLevel_1} can be rewritten as
\begin{equation}
\frac{1}{\lambda_k}=\frac{1}{M_k}\left(P_k+\frac{\displaystyle\sum_{m=1}^{M_k}\prod_{i=1\backslash m}^{M_k} g_{k,i}}{\displaystyle\prod_{m=1}^{M_k} g_{k,m}}\right),
\end{equation}
which can be further simplified using elementary symmetric polynomials as follows
\begin{equation}
\frac{1}{\lambda_k}=\frac{S_{M_k}(\mathbf{g}_k)P_k+S_{M_k-1}(\mathbf{g}_k)}{M_k S_{M_k}(\mathbf{g}_k)},
\end{equation}
where $S_i(.)$ is the $i$th elementary symmetric polynomial and given by~\cite{SymmetricFunBook}
\begin{equation}
S_{M_k}(\mathbf{g}_k)=\prod_{m=1}^{M_k} g_{k,m},
\end{equation}
\begin{equation}
S_{M_k-1}(\mathbf{g}_k)=\sum_{m=1}^{M_k}\prod_{i=1\backslash m}^{M_k} g_{k,i}.
\end{equation}
Consequently, the user rate~\eqref{Eq_SymbolsRate_2} will be
\begin{eqnarray}\label{Eq_SymbolsRate_3}
R_k(\mathbf{g}_k)&=& \sum_{m\in\mathcal{M}_k}\log{\left(\frac{\bar{g}_{k,m}}{\lambda_k}\right)}=\log{\left(\prod_{m=1}^{M_k}\frac{\bar{g}_{k,m}}{\lambda_k}\right)}\nonumber\\
&=& \log{\left(\frac{1}{\lambda_k^{M_k}}S_{M_k}(\mathbf{g}_k)\right)}\nonumber\\
&=& M_k\log{\left(\frac{1}{\lambda_k}\right)}+\log{\left(S_{M_k}(\mathbf{g}_k)\right)}\nonumber\\
&=& M_k\log{\left(\frac{S_{M_k}(\mathbf{g}_k)P_k+S_{M_k-1}(\mathbf{g}_k)}{M_k S_{M_k}(\mathbf{g}_k)}\right)}\nonumber\\
&&+\log{\left(S_{M_k}(\mathbf{g}_k)\right)}\nonumber\\
&=& M_k\log{\Big(S_{M_k}(\mathbf{g}_k)P_k+S_{M_k-1}(\mathbf{g}_k)\Big)}\nonumber\\
&&-(M_k-1)\log{\left(S_{M_k}(\mathbf{g}_k)\right)}-M_k\log{(M_k)}.\nonumber\\
\end{eqnarray}
Both $S_{M_k}(\mathbf{g}_k)$ and $S_{M_k-1}(\mathbf{g}_k)$ are Schur-concave functions~\cite[Chapter~3]{Major_Marshall_Book}, and will be decreased by majorizing $\mathbf{g}_k$.
Now, the change in the user rate with respect to $S_{M_k}(\mathbf{g}_k)$ and $S_{M_k-1}(\mathbf{g}_k)$ can be found through the partial derivative of the user rate as follows
\begin{equation}\label{Eq_RateDerivative1}
\frac{\partial R_k(\mathbf{g}_k)}{\partial S_{M_k}}=\frac{S_{M_k}(\mathbf{g}_k)P_k-(M_k-1)S_{M_k-1}(\mathbf{g}_k)}{S^2_{M_k}(\mathbf{g}_k)P_k+S_{M_k}(\mathbf{g}_k)S_{M_k-1}(\mathbf{g}_k)},
\end{equation}
\begin{equation}\label{Eq_RateDerivative2}
\frac{\partial R_k(\mathbf{g}_k)}{\partial S_{M_k-1}}=\frac{M_k}{S^2_{M_k}(\mathbf{g}_k)P_k+S_{M_k}(\mathbf{g}_k)S_{M_k-1}(\mathbf{g}_k)}.
\end{equation}
Consequently, the user rate will be a decreasing function of majorizing $\mathbf{g}_k$ if the total change with respect to $S_{M_k}(\mathbf{g}_k)$ and $S_{M_k-1}(\mathbf{g}_k)$ is negative, i.e.
\begin{equation}
\frac{\partial R_k(\mathbf{g}_k)}{\partial S_{M_k}}+\frac{\partial R_k(\mathbf{g}_k)}{\partial S_{M_k-1}}<0,
\end{equation}
which is satisfied if the following condition holds
\begin{equation}
P_k<(M_k-1)\frac{S_{M_k-1}(\mathbf{g}_k)}{S_{M_k}(\mathbf{g}_k)}-M_k.
\end{equation}
In fact, $\frac{S_{M_k-1}(\mathbf{g}_k)}{S_{M_k}(\mathbf{g}_k)}$ is a Schur-convex function of $\mathbf{g}_k$~\cite[Chapter~3]{Major_Marshall_Book}, hence, the minimum its achieved when all the elements in $\mathbf{g}_k$ are equal, and the upper bound on the user power will be
\begin{equation}\label{Eq_PowerBound}
P_k<(M_k-1)\frac{\frac{M_k}{M_k^{M_k-1}}}{\frac{1}{M_k^{M_k}}}-M_k\quad \Longrightarrow\quad P_k<M_k^3-M_k^2-M_k.
\end{equation}
This implies that if the user total power $(P_k)$ is less than the bound in \eqref{Eq_PowerBound}, the user rate will be decreased with the increase in $S_{M_k}(\mathbf{g}_k)$ and $S_{M_k-1}(\mathbf{g}_k)$. Consequently, the user rate~\eqref{Eq_SymbolsRate_2} will be increased by majorizing $\mathbf{g}_k$.
\end{IEEEproof}
\section{Proof of Lemma~\ref{Lem_SC_Partitiong}}\label{App_SymMMVproof}
\begin{IEEEproof}
The optimality proof of this scheme can be done by showing that the proposed partitioning scheme will give symbols gains vector that majorize any other partitioning scheme. Vectors majorization is defined as follows.
\begin{defn}\label{Def_Majorization}
\textbf{(Majorization of vectors~\cite[Chapter~1]{Major_Marshall_Book}):} For two vectors $\mathbf{x}, \mathbf{y}\in\mathcal{R}^{N_k}$ with descending order such that $x_1\geq x_2\geq\dots \geq x_{N_k}\geq 0$ and $y_1\geq y_2\geq\dots \geq y_{N_k}\geq 0$, $\mathbf{x}$ is said to be majorize $\mathbf{y}$ and written as $\mathbf{x}\succ\mathbf{y}$ if
\begin{displaymath}
\sum_{i=1}^n x_i \geq \sum_{i=1}^n y_i,\quad n=1,\dots, N_k-1\quad \text{and}\quad \sum_{i=1}^{N_k} x_i = \sum_{i=1}^{N_k} y_i.
\end{displaymath}
\end{defn}
Let $\pi$ represents the permutation of the subcarriers gains in descending order and $\rho$ is any other permutation. Based on these permutations it is clear that
\begin{equation}
\sum_{i=1}^m x_i \geq \sum_{i=1}^m y_i,\quad m=1,\dots, M_k-1\quad \text{and}\quad \sum_{i=1}^{M_k} x_i= \sum_{i=1}^{M_k} y_i,
\end{equation}
where $x_i=\{g_{k,\pi(i)}, g_{k,\pi(i+1)}, \dots,\ g_{k,\pi(i+d_v-1)}\}$ and $y_i=\{g_{k,\rho(i)}, g_{k,\rho(i+1)}, \dots,\ g_{k,\rho(i+d_v-1)}\}$. Consequently, vector $\mathbf{x}$ is majorizing vector $\mathbf{y}$ according to Definition~\ref{Def_Majorization}.
\end{IEEEproof}

\ifCLASSOPTIONcaptionsoff
\newpage
\fi

\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


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text:
Endexx Appoints Chyna Whyne to its Jamaican Board
World-Renowned Alexander Technique, Yoga and Stiletto Expert Joins Endexx Advisory Board to Propel the Company's CBD Brand Internationally
CAVE CREEK, AZ, Aug. 02, 2018 (GLOBE NEWSWIRE) — Endexx Corporation (OTC PINK: EDXC), an international provider of innovative phytonutrient-based food and nutritional products, is pleased to announce the engagement of Chyna Whyne http://walkinstilettos.com/chyna-whyne to its Board of Directors of Go Green Global Enterprises' operation in Jamaica. Chyna Whyne, who has been titled "stiletto guru" by multiple international magazines, has trained thousands of women on how to walk for hours in high heels, comfortably and effortlessly, without suffering neck, leg, shoulder, or back problems. Her Walking in Stilettos™ program, which is now accredited by the University of Delaware and certified by Dr. Brown, utilizes the Alexander Technique education process to improve participants' posture, balance, movement, and coordination.
With the engagement, Ms. Whyne will help bring awareness and educate the use of CBD to a broader female audience as well as help propel the Endexx line of CBD products internationally. As a collaboration between the two powerhouses, Ms. Whyne and Endexx are currently collaborating on a new health and beauty line of products which include "Stiletto Mist", a CBD mist spray aimed at reducing pain and inflammation of tired and aching feet, ankles and legs, associated with women wearing high heels.
"We are truly excited about Chyna Whyne joining our board. As an expert in the art of walking in stilettos, Ms. Whyne is a perfect addition to our expanding team and we look forward to unveiling our latest product which will target the female demographic to bring comfort and confidence to women who love wearing heels," said Endexx Chief Executive Officer Todd Davis. Mr. Davis added, "The world market for women wearing high heels is immense, and Endexx has a therapeutic and healthy CBD infused spray for all occasions."
Ms. Whyne will be presenting a seminar on 'Mastering the Art of Wearing High Heels' at the World Congress in Loyola University, Chicago, and on August 2nd, taking part as a presenter in the Alexander Teachers' Healing in Diversity panel.
The Alexander Technique teaches women how to present themselves with elegance and confidence in professional, social, and business environments and about the importance of good posture, balance, and coordination, which has been credited for relieving neck, back, and shoulder pains.
About Endexx
Endexx, with its collaborative partners and consultants, develops and distributes two consumable product lines derived from industrial hemp, which is organic and naturally rich in phyto-cannabinoids under the brand name CBD Unlimited. Phyto-Bites® is its CBD-infused soft chews for dogs and cats. The dog treats are formulated to promote health and support the reduction of separation anxiety, pain and inflammation. The company also has two technology products and services that launched in 2014 – the m3hub and the AutoSpense™. The AutoSpense™ is a commercial-grade inventory control and dispensing device that provides up-to-the-minute accounting details and ensures both product and patient security. By automating the dispensing process, AutoSpense™ increases productivity and reduces costs for pharmacy and retailers, stopping shrink and theft, while enhancing customer experience and reducing transaction time for customers. Websites include: www.cbdunlimited.com, www.endexx.com and www.phytobites.com.
This press release may contain "forward-looking statements" within the meaning of the Private Securities Litigation Reform Act of 1995, such as statements relating to financial results and plans for future development activities and are thus prospective. Forward-looking statements include all statements that are not statements of historical fact regarding intent, belief or current expectations of the company, its directors or its officers. Investors are cautioned that any such forward-looking statements are not guarantees of future performance and involve risks and uncertainties, many of which are beyond the company's ability to control. Actual results may differ materially from those projected in the forward-looking statements. Among the factors that could cause actual results to differ materially from those indicated in the forward-looking statements are risks and uncertainties associated with the company's business and finances in general, including the ability to continue and manage its growth, competition, global economic conditions and other factors discussed in detail in the Company's periodic filings with the Securities and Exchange Commission. The company undertakes no obligation to update any forward-looking statements.
For further investor and media information, contact:
Todd Davis
endexx@endexx.com
What is CBD Microdosing?
Endexx Announces Global CBD License & Distribution Deal With Cannabis Infusion Brand The Mad Hatter Coffee & Tea Co.
Endexx Initiates Adaptive Clinical Study With Proprietary CBD Capsule Formulation To Address Neuropathic Pain

redpajama_set_name: RedPajamaCommonCrawl


Training teachers in the new science standards should have been a budget priority, some educators said.
I attended the six days of study and deliberation by this panel, and was impressed with the depth and breadth of its expertise, and with its thoroughness and thoughtfulness.
Implementation statewide of the Next Generation Science Standards – usually referred to by its acronym NGSS – is expected to take three to five years. But there is much that local education agencies – which include school districts, county offices of education and charter schools – can do now.
Some districts have indicated they want to start professional development and learning about the new science standards right away. Others will want to take longer, waiting until the California Department of Education (CDE) develops a curriculum framework, and until more is known about what kind of statewide science assessment -– the tests students will take that are aligned with the new standards – will be adopted and when.
The CDE is charged with reviewing and endorsing instructional materials aligned with the new science standards. But recent changes in California law now allow districts and charters to choose any instructional materials they want to use, and to conduct their own review of those materials to ensure they are aligned with the state's standards.
Another significant development is that by this July all districts and charter schools in California must have a Local Control and Accountability Plan (LCAP) that addresses how they will meet the eight new "state priorities" spelled out in the state's new school financing law. One of those priorities (#2) is the implementation of the state's academic standards, including the new science standards.
Local school boards will need to include plans and annual goals in their LCAPs for phased progress in implementing the new science standards, and will need to allocate resources to support the implementation, such as professional development and increased instructional time.
Districts and charter schools can use funding they receive from the Local Control Funding Formula and other sources to do so. In addition, the one-time $1.25 billion in the state budget for implementing the Common Core standards in English language arts and math for 2013-14 and 2014-15 can also be used for moving forward on the new science standards.
Including an implementation plan for the Next Generation Science Standards in the Local Control and Accountability Plan due by July 1, 2014 will assure parents and other stakeholders that any given district or charter school is making science education a priority. This is especially important in kindergarten through 8th grade, where cuts to science education instructional time and resources have been the deepest over the last decade. It will also ensure that parents and the community can monitor the progress of a district or charter school in implementing the science standards, thus ensuring all students have access to a quality science education as part of a broad course of study, and that all students are ready for new tests aligned with the new science standards when they become available in a few years.
How can individuals or organizations advocate for implementation and funding for the Next Generation Science Standards in 2014-15? They can send letters to their local district's elected board and superintendent, call and ask when the public hearings will be held and attend those hearings, ask how parent groups will be meaningfully engaged, and encourage like-minded science education stakeholders and champions to do the same.
After many years of reduced budgets, school districts are faced with many competing priorities for their funds for 2014-15. Many will have very tough decisions to make in meeting the requirements and expectations of the new school financing law, implementing multiple new academic standards, preparing for a new state assessment in math and English language arts and balancing the requests of their many stakeholders (parent groups, teacher bargaining units, community and business leaders).
That said, with the adoption of the Next Generation Science Standards, California is poised to become a national leader in K-12 science education. All of our K-12 students can become among the best educated in the country in science and engineering.
But this can only happen with quality implementation of California's new science standards across all the state's districts, schools and charter schools – for the benefit of all of California's students.
Trish Williams is a member of the California State Board of Education, and has been a Board liaison lead for implementation of the Next Generation Science Standards since 2011. She was formerly executive director of EdSource.
If the State of California wants teachers to implement NGSS why is their conference for a fee?If I want to go, I have to pay for it myself, take personal leave days, and the sub comes out of my account. With all that you wrote, how do teachers get their districts to pay for getting PD in order to implement NGSS?
I would call the NGSS a science version of the common core notion, but science education has typically set its own individual course, starting with the ambitious idea of Project 2061 of taking the long view of a national view of what should be learned in science — the number in the title is a date, and is the next time we will be able to view Halley's comet.
There is a problem with the website. I posted in the "judge denies motion" article, but it says I posted on this one, but there are no posts excepting this statement.
We started our transition process in Maryland last June. As a result, I started a blog to share the process we are using and obtain additional input. No sense we should do this alone.
Forgive this question. I am not a science teacher and feel very ignorant about these standards. But….I thought we were going to have national standards under the aegis of CCSS?
It's also hard for me to believe there are very many parents out there eager to read these standards and check to see that their second grader is getting everything stipulated on a state list.
The CCSS are for English Language Arts and Math only. The science standards are the "Next Generation Science Standards" which are different.


text:
package ca.wescook.homesteadcompanion;

import ca.wescook.homesteadcompanion.proxy.CommonProxy;
import net.minecraftforge.fml.common.Mod;
import net.minecraftforge.fml.common.Mod.EventHandler;
import net.minecraftforge.fml.common.SidedProxy;
import net.minecraftforge.fml.common.event.FMLInitializationEvent;
import net.minecraftforge.fml.common.event.FMLPostInitializationEvent;
import net.minecraftforge.fml.common.event.FMLPreInitializationEvent;

import static ca.wescook.homesteadcompanion.HomesteadCompanion.MODID;
import static ca.wescook.homesteadcompanion.HomesteadCompanion.MODNAME;

@Mod(modid = MODID, name = MODNAME, version = "@VERSION@")
public class HomesteadCompanion {
	public static final String MODID = "homesteadcompanion";
	public static final String MODNAME = "Homestead Companion";

	// Create instance of mod
	@Mod.Instance
	public static HomesteadCompanion instance;

	// Create instance of proxy
	// This will vary depending on if the client or server is running
	@SidedProxy(clientSide="ca.wescook.homesteadcompanion.proxy.ClientProxy", serverSide="ca.wescook.homesteadcompanion.proxy.ServerProxy")
	static private CommonProxy proxy;

	// Events
	@EventHandler
	public void preInit(FMLPreInitializationEvent event) {
		HomesteadCompanion.proxy.preInit(event);
	}

	@EventHandler
	public void init(FMLInitializationEvent event) {
		HomesteadCompanion.proxy.init(event);
	}

	@EventHandler
	public void postInit(FMLPostInitializationEvent event) {
		HomesteadCompanion.proxy.postInit(event);
	}
}

redpajama_set_name: RedPajamaGithub


text:
Куева де лос Серес () насеље је у Мексику у савезној држави Оахака у општини Ваутепек. Насеље се налази на надморској висини од 1548 м.

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Према подацима из 2010. године у насељу је живело 431 становника.

Хронологија

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 Савезне државе Мексика

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Насеља у општини Ваутепек (Оахака)
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redpajama_set_name: RedPajamaWikipedia


text:
Nashville schools board takes first look at its 2020-21 school year budget
Jason Gonzales
Nashville public schools officials project more than $20 million in required increases next fiscal year on textbooks and salaries.
The money is necessary spending even before the school board has laid out its funding priorities and doesn't include millions more for other required cost increases in the 2020-21 school year.
it's unclear just how much the district will need to spend overall, but the high costs could leave little for priorities set by the Metro Nashville Public Schools board. That doesn't mean the school board, however, won't bring forward a budget that includes what board Chair Anna Sheperd called during a Friday budget retreat an "aspirational" funding amount.
"We weren't brave enough to bring (Council) a fully-funded budget last year because, quite frankly, they all would have a stroke," Shepherd said.
This year, though, she said she wants the board to show Metro Council and Mayor John Cooper what "fully funded" means. The district plans to release its budget during the board's February 11 meeting, according to Chief Operating Officer Chris Henson.
Required spending this year includes ensuring a January 3% salary increase for employees is made long-term, Henson said. The money used by the Mayor's Office for the salary increase was one-time funding.
Other needs include funds to bolster teaching for the district's continually growing English-learning student population, to sustain pre-Kindergarten classes and for projected water rate cost increases, according to Henson.
And the district will also need to send money to charter schools for enrollment growth in that sector.
When those required items are squared away in the budget, a survey from the board indicated they will try to prioritize employee compensation.
But any increase for employees could also prove costly, with a recent study by the district's human resources staff found that to bring mid-career teacher pay to about $64,000 a year — comparable to Nashville's median income — the city would have to pay more than $100 million more a year.
And the survey also shows the board would like to prioritize spending on social-emotional learning, as well as more instructional materials and textbooks for teachers.
Want to read more stories like this? A subscription to one of our Tennessee publications gets you unlimited access to all the latest news and the ability to tap into stories, photos and videos from throughout the USA TODAY Network's 109 local sites.
Reach Jason Gonzales at jagonzales@tennessean.com and on Twitter @ByJasonGonzales.


The 2k, will it ever end?
Making our way to Lodz (line though L, sorry), my counter sprung to 0 again. After taking the picture, a ritual with ever 500km and now being the fourth time, I realized that it doesn't reset itself. Other digits, ones I never paid attention too, represent the total number, wheras the one I focused at … More The 2k, will it ever end?
Leave a comment The 2k, will it ever end?


"It should be formally announced sometime probably next week that we will have 100 percent of the caliphate", Mr. Trump told his audience of worldwide leaders in brief remarks.
"We will not avert our eyes from a regime that chants death to America and threatens genocide against the Jewish people", he continued, to applause, mostly from the Republican side.
The president told representatives of a 79-member, US -led coalition fighting IS that the militants held a tiny percentage of the vast territory they claimed as their "caliphate". "I have also accelerated our negotiations to reach a political settlement in Afghanistan".
Several of Trump's fellow Republicans strongly disagreed with his plans to withdraw 2,000 USA troops from Syria on the grounds that militants no longer pose a threat. "They have the ability of coming back together if we don't", General Joseph Votel, head of the USA military's Central Command, told a Senate Armed Services Committee hearing hours before the president's public address.
He said that the United States "mission is unwavering, but we need your help to accomplish it, just as we've had over the past months and years".
The State of the Union is a yearly presidential address where the USA leader touts his accomplishments and outlines his vision for the future of the country.
Representatives from 79 countries gathered at the Pentagon near Washington, DC for an all-day meeting on defeating Islamic State and other enemies of the US-led coalition in the Middle East. President Donald Trump and State Secretary Mike Pompeo addressed the meeting.
"I want to applaud the outstanding men and women of the United States military, incredible people, for their remarkable skill and precision throughout this entire campaign", the president told a gathering of leaders and top officials from countries who signed on to the coalition. "We have to protect other things that we have".
On Tuesday, the Senate passed a non-binding resolution condemning Trump's plans to withdraw troops from those two conflicts, and slapped new sanctions on the Syrian government.
As is tradition, Trump's national televised speech was followed by a Democratic response, this time given by Stacey Abrams, who narrowly lost her bid to become the first African-American governor of a southern USA state.


Very economic commercial premise in Patraix district (Valencia),it is located in calle Cuenca, a busy street. The general condition is ready to enter, with the exception of the bathroom that must be remodeled.There is a possibility to make a garret of 5x2x1 of high. With an area of 22m2 and clear space. It is perfect to set a bussines, to use as a studio or as a storage. Ask for further information.


One of our favourite burger joints in Dubai never fails to impress with its over-the-top dishes, but now we're getting a whole new batch of grub to get, fittingly, bananas about.
New York-inspired venue Black Tap has just launched a brand-new menu, featuring mega new editions to its burger, CrazyShakes and taco menu.
First off, foodies can dig into six new burgers including "The Gulf Burger" featuring Black Tap's prime Greater Omaha USA beef burger stacked with arugula, zaatar, marinated halloumi cheese, pesto mayo, sun dried tomatoes, avocado and parmesan cheese. How do they pack all that in two buns?
There's also the "Truffle Burger" a wagyu steak patty topped with black truffle mayo, the "Filet au Poivre Burger", which boasts a tenderloin steak piled high with peppercorn sauce, frizzled onions and swiss cheese or blue cheese.
The final three include the "Pizza Burger", "Chili Cheese Burger" and "Reuben Burger." Add them up and Black Tap's gourmet burger count now stands at 17. Fabulous.
As for the CrazyShakes, customers can get their hands on "The Crazy Banana," stacked with a Nutella-frosted rim, topped with crushed pretzels, a fried banana, white Kinder Bueno chocolate, whipped cream, chocolate chips and chocolate caramel drizzle.
The other newbie includes "The New Yorker," a strawberry-flavoured shake featuring a vanilla-frosted rim with crushed Graham crackers, topped with a strawberry cheesecake slice, whipped cream, strawberry sauce drizzle and a cherry on top.
Fan of tacos? There's two new offerings to try, including the Carne Asada with marinated steak, pico de gallo, caramelised onions and roasted tomato salsa, while the Shrimp taco, which is topped with Mexican hot sauce and served with cabbage, avocado and corn also looks very appealing.
Breakfast is also now available at Black Tap's JBR branch between 9am and 2pm.
You can chomp on all the new additions at three of Black Tap's venues across Dubai, including Rixos Premium JBR, Jumeirah Al Naseem and the InterContinental Dubai Festival City.
Make sure you arrive with an empty stomach!


I like this because it includes a strong two point  linear cantor set attractor with a rotation that looks like it should be symmetric, but as your look at the details, the internals are off.  Yet, the transformations are so simple the actual symmetries reveal themselves from inspection.
This one reminded me of a fractal version of the tic marks on a ruler. Â It's up there with the minimal spiral for one of my favorite illustrations of the self similarity of fractals. Â As they become more complex, it becomes more and more difficult to make out the relationships between the different parts. Â This one being more "one" dimensional seems to make it easier to take in.
This one feels like it is constructed out of lines rather than individual points, which seem to make the relationships less mind bending, even though the definitions only differ by a few numbers.  The moment it really sunk in that every individual point in the circles I was looking at really really was circular patterns all the way down.  I'd been looking at and computing images of various fractals for over 20 years at that point, but had never made that particular connection before.  I had to see the parts as an exact copy of the whole instead of  each one being subtly different.
This image, while not very visually interesting has a lot of accidental symmetry.
These guys use four transformations with simple rotation parameters. All I changed is the scaling factor of the two central transformations and their centers. It shows how the nature and feeling of the image can change dramatically with just a few little parameter changes.
This has an almost Mayan feel getting ready for the upcoming apocalypse.


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Former Pakistani child bride shares story with students
Amin starts foundation to help victims of forced marriages
Posted Thursday, January 5, 2017 1:00 pm
By Brian Stieglitz
When Pakistani-American Naila Amin, 27, of Baldwin, stood in front of Mepham High School's Leadership and Voices of the Past students, she said she was nervous. Amin has shared her story many times before, but Dec. 20 was the first time she told it to students who were the age she was when it happened.
Rina Sarfraz, a Mepham student, had been following Amin's story, and reached out online to ask if she would speak to a high school class. Amin wanted to keep spreading awareness through her story, and her nerves eased when she saw how curious and engaged the students were during her presentation. She said that her goal is to ensure that no woman is alone, like she was, if her childhood is cut short by a forced marriage.
Amin came to America at age 4, from a remote village in the Attock tribal region of Pakistan. She traveled with her pregnant mother and three siblings to be with her father, a Merchant Marine stationed in New York City. They lived in Astoria, Queens, where Amin attended kindergarten.
When she was 8, her family visited Pakistan, where, in keeping with Muslim traditions, Amin was affianced to her first cousin, Tariq. He was 13 years older. She officially signed her marriage contract, or Nikah, to be Tariq's wife before she entered high school. However, at 14, she started dating a boy named Edy. Her parents called it adultery when they found out. Amin said this led to a physical fight and her first attempt to run away from home.
She stayed with Edy and his family, who brought her to a social worker at the school she attended in Queens. After seeing Amin's bruises, the social worker called Child Protective Services. She became a ward of the state, which meant that the New York government promised to protect her. "That's what saved my life," she said.
According to Amin, she was lost in the foster care system. They could not find a Muslim foster home or even provide halal food, which is required under Muslim law. "Nobody knew what to do with me," she said, referring to the foster care process in which her CPS provider moved her to new homes. Eventually it occurred to Amin that there were no international or Muslim children at any of these places. She wanted to practice her faith and belong to a Muslim community, so she ran away from one of the homes and returned to her parents' house. They then flew to Pakistan together and started living with Tariq.
Amin ran away from her husband's home in Pakistan. When her family brought her back, Tariq and her father beat her. Amin's mother screamed for them to stop, but they never did, she said.
"We need to implement more programs to help our citizens [overseas]," Amin said. "God knows how many women are trapped."
One afternoon in 2005, Amin escaped and was running through the streets of Islamabad. She did not know where to go and, out of desperation, called an uncle for help. She begged him to call Child Protective Services in the U.S. because she feared her family would kill her to protect their honor, a Pakistani tradition when a child brings shame to family. Instead, Amin's uncle brought her home to Tariq, who accused him of eloping with his bride. This led to a fight between the two men, and out of anger, the uncle finally called CPS.
"If I was not a ward of the state, I was not coming back," Amin said, noting that the pull of New York state and its foster care system was strong enough to get her home. Her mother came back to America, too. When they landed at Kennedy Airport, several police officers, social workers and government officials greeted Amin and took her to a psychiatric hospital to be evaluated for post-traumatic stress disorder.
Eventually she moved in with her mother in Baldwin and formed a stable relationship with her. "She cooked me breakfast this morning," she told the Herald. Today Amin continues to live with her mother, has a boyfriend and a 10-year-old son named Aayan.
She recently interned at the Ayaan Hirsi Ali Foundation, an organization that works to protect women's rights and end honor violence. She aspires to be a social worker and is studying toward her associate's degree at Nassau Community College. She will be the first female in her family to graduate from college.
Amin said that when a young woman is being forced to marry and speaks out about it, "As a social worker, I can't intervene unless there is abuse." She said that she hopes to change that with the Naila Amin Foundation, which she said will provide a home for victims of forced marriages. In October she received a letter from the IRS, approving her organization as a nonprofit. In November Amin received the Susan B. Anthony Award from the New York City chapter of the National Organization for Women for her dedication to her cause.
"I really do believe in this dream of mine," she said. "Once I find this safe space, this shelter, I could reach out to legislators and senators and make people see that this is really an issue that is happening here."
Amin said she hopes New Jersey Bill A3091, which would bar those under age 18 from marrying or entering a civil union regardless of parental consent, can raise more awareness in New York. In November the bill advanced in the state's Assembly Judiciary Committee with 64 passing votes, and is now pending in the State Senate.
Amin plans to give more presentations at high schools and colleges. She found the experience at Mepham rewarding, and was surprised by the number of questions she was asked and the sincere interest of the students. "It was like looking back in time to see a younger version of me in the students," she said. "I needed that."
For more information or to donate to Amin's cause, visit her Gofundme page at www.gofundme.com/NailaAminNaf.
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text:
Едуард Штраус (, 14 лютого 1835 — 28 грудня 1916) — австрійський композитор і диригент. Один з трьох синів відомого композитора Йогана Штрауса.

Біографія

Народився 14 лютого 1835 року у Відні. Батько, Йоганн Штраус (батько), практично не займався вихованням своїх дітей, оскільки постійно гастролював по Європі. Але, єдине за чим він дуже суворо стежив і того ж вимагав від своєї дружини Ганни, це щоб діти не вигадували музику і не вчилися грати на скрипці. Але, старший брат Йоганн, незважаючи на заборони батька потайки вчився грати на скрипці і заохотив до цього і Едуарда.

Батько бачив в ньому дипломата, але Едуард цілком присвятив себе музиці.

Після смерті батька, на сімейній раді в 1853 році було вирішено, що оскільки брат Йоганн не може розірватися між гастролями, виступами в Відню і письменництвом, то другий брат Йозеф повинен підміняти брата і диригувати батьківським оркестром. Коли старший брат Йоганн став пропадати по півроку за кордоном Йозефу довелося дуже багато працювати і оскільки він був дуже слабкого здоров'я, то врешті-решт перевтомлювався. І тоді на допомогу прийшов Едуард. 5 лютого 1861 року в Залі Софії грали три оркестри, кожним з яких керував один із трьох братів Штраусів.

Відтоді на афішах писали просто «Штраус» і публіка не знала кого з братів вона слухатиме.

Едуард як і брати диригував, грав на скрипці і вигадував вальси. У Едуарда не було скільки-небудь помітного творчого дарування, проте він вигадав близько двох сотень танцювальних п'єс. Вважався найкращим інтерпретатором творів свого батька і братів. Едуард був надзвичайно красивий і тому дуже подобався публіці. Він диригував Капелою Штрауса впродовж трьох десятиліть, аж до її розпуску в 1901 році.

У 1906 він опублікував спогади — дуже суб'єктивну, мстиву і недостовірну хроніку сімейства Штраусів.

Він концертував до глибокої старості, але писати вальси у нього виходило все гірше і гірше. Тоді, оголосивши, що епоха віденського вальсу закінчена, він спалив цілу скриню рукописів батька і братів в печах для випалення цегли. Власник цегляної майстерні благав Едуарда цього не робити, але він був непохитний.

Помер Едуард 28 грудня 1916 року. Його син Йоганн Штраус (1866—1939) намагався воскресити сімейну традицію і гастролював по Європі перед Першою світовою війною.

Штраус-фестивалі

Щороку в Європі проходить «Штраус-фестивалі» присвячений творчості Штраусів. Іспанія, Італія, Австрія, Португалія, Франція, Німеччина — насолоджуються прекрасними вальсами, польками, маршами, галопами, увертюрами і аріями з оперет відомої династії композиторів і диригентів 19 століття. — Штраусів.

Творчість
Написав близько 150 творів, в яких видно прагнення Едуарда наслідувати традиції свого батька і старшого брата.
 «Шлях відкритий», швидка полька
 «Імператорський вальс»
 «Уранішні листки», вальс
 «Грім і блискавка», полька
 «Троянди півдня», вальс

Див. також
 Йоганн Штраус (батько)
 Йоганн Штраус (син)
 Йозеф Штраус

Посилання
 ШТРАУС
 Энциклопедический словарь Ф. А. Брокгауза и И. А. Ефрона
 «Казки віденського…»

Примітки

Персоналії за алфавітом
Австрійські композитори


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