text:
Home | addiction guide | polydrug use
What Can Happen if You Mix Ecstasy with Methamphetamines?
Ecstasy (also known as MDMA) and methamphetamine (also known as meth) are both illicit stimulant drugs that are common in the nightclub or "rave" scene. This has led to them being classified as "club drugs." People at parties or dances use them to get a burst of energy, feel powerful, and enhance the experiences of dancing, lights, and being close to people.
Clubs and raves are notorious for being scenes of recreational drug use. Intoxicants like ecstasy are often readily available at these parties, and engaging in some drug abuse is often expected. It's also very common for club drugs to be combined to enhance their effects or create new experiences of intoxication.
While meth is thought of as pure stimulant, producing intense energy and a feeling of invincibility, ecstasy is a more complex drug. While it enhances energy levels, it also produces feelings of ecstatic happiness and love, and promotes intense empathy with other people. It has psychoactive effects, making bright, colored lights and different textures more pleasing to the user. It's also known for causing dehydration and raising body temperature. Combined with another stimulant, this can become dangerous.
Worried about your ecstasy or meth use? Take our free and confidential addiction assessment today.
Health Effects and Overdose
Unfortunately, ecstasy users may find themselves taking both the MDMA and meth at the same time whether they know it or not. Ecstasy tablets may come with simple characters on them, but there's no way for the average person to determine if the pill is pure MDMA or if it's been mixed with another drug to make it cheaper to produce. Some people may pass off tablets of entirely different drugs as ecstasy, leaving the user at the mercy of the unpredictable effects of the real substance. One chemist stated that the ecstasy pills received by her lab for testing tend to only be 5-10 percent pure.
If you are taking ecstasy and meth at the same time, the combined stimulant effects can result in serious overheating and dehydration. Raves and dance parties can be hot and, of course, involve vigorous physical activity. People have died from heat stroke and severe dehydration at these events, typically due to the use of multiple drugs. Body temperature can increase to 106 degrees or more in these conditions, which can result in organ damage or failure.
Stimulant overdose symptoms include:
Chest pain or tightness
Profuse sweating and chills
Racing pulse
Irregular breathing or shortness of breath
Muscle jerking or rigidity
Intense anxiety
Agitation or irritability
Hypervigilance
Mixing any stimulants will increase the effects exponentially, along with the risk of overdose. Increased blood pressure is one of the most dangerous side effects as it increases the risk of stroke and heart attack.
Over time, constant stimulant intake can wear down the blood vessel walls and weaken the heart muscle, risking a rupture or sudden heart failure.
It's common knowledge among the club drug scenes that taking breaks, keeping cool, and drinking plenty of water are essential for safety during ecstasy use. However, ecstasy and many other club drugs can reduce inhibitions and impulse control, leading to the use of other stimulants and a failure to exercise safety precautions. At the same time, you may not be getting ecstasy when you think you are. The only way to avoid mixing ecstasy and meth for sure is to either have a drug testing lab available or avoid taking the drug altogether.
Are There Dangers of Taking OTC Medications With Drugs or Alcohol?
Dangers of Mixing Adderall with Other Drugs
What Happens if You Use Drugs or Alcohol while Using Nicotine Patches?
The Dangers of Mixing Painkillers & Benzodiazepine Anxiety Medication
What Can Happen if You Take Ecstasy with Other Club Drugs?
What Can Happen if You Take Cocaine with Alcohol?
What Issues Are There from Combining Cough Medicine & Antihistamines?


In the case that an individual faces a slip and fall accident on snow or ice they might be entitled to file a lawsuit against the responsible party. In fact, snow and ic accidents are some of the most common types of slip and fall accidents. The law requires that property owners maintain responsibility for their premises by shoveling any present snow or ice. In the case that a property owner fails to do so and an individual is injured as a result, the property owner might be held liable for his or her negligent behavior. At Queller, Fisher, Washor, Fuchs & Kool our premises liability attorneys have experience defending the rights of those who have been unnecessarily injured in slip-and-fall accidents. Our attorneys have secured a number of favorable verdicts in slip and fall cases, amongst which there is one for $1.5 million. Our experienced attorneys will do their very best to ensure that you receive your maximum deserved compensation.
Commercial, residential and public property owners can be held liable for accumulations of snow or ice on their property in the case that the fail to shovel, plot or salt their property. According to New York City law property owners are required to clear any accumulations of snow or ice on their property within a reasonable time period following a storm. Typically, property owners are required to remove any accumulation on their property within four hours of a storm. In the case that a property owner fails to follow through on their duty and as a result an individual is injured, the property owner can be held liable for his or her negligence.
Most commonly, accidents occur in New York City on private property or on sidewalks, In the case that the accident occurs on private property, the landowner or the tenant might be held liable for their negligence. Alternatively, if the accident occurs on a sidewalk, the property owner adjacent to the sidewalk might be held liable for his negligent behavior.
Establishing liability or fault depends on issue of "notice". This issue surmises a conjecture as to whether enough time had passed between the occurrence of the storm and the accident for a reasonable person to have taken notice of and removed the snow or ice. In order to prove "notice" or that the defendant knew or should have known of the dangerous condition prior to your accident, there are numerous issues that must be carefully investigated by your attorney. Additionally, the discovery phase of your case needs to be carried out properly if a settlement is to be recovered.
One such issue is when the poor weather occurred. Weather reports can often be obtained to determine how much and what type of precipitation (rain, sleet, snow) fell on the day of your accident and on the days before your accident. The records should be obtained from the weather station closest to the accident location, and a certified copy of such records may be needed for admissibility purposes in the event that your case goes to trial. Of course, if the condition that caused your accident still exists, photographs should be taken whenever possible. Also, an investigation should be done to determine the identity of any eyewitnesses to the conditions that were present at the time of your accident. Such witnesses may be helpful at trial, or in laying the foundation for settling your case through deposition testimony regarding the dangerous conditions.
The identity of any and all employees or agents of the defendant who had opportunity to observe and/or become aware of the dangerous conditions.
The identity of any persons whose job responsibilities included monitoring, inspecting, or cleaning the accident location, including but not limited to the responsibility of clearing the relevant area of snow and ice.
Any schedules that were in effect and any practice and procedure that the property owner or other defendant followed for inspecting or cleaning the property.
If the defendant claims they actually did take steps to clear the sidewalk, questions need to be answered regarding the manner in which it is claimed that the sidewalk was cleared. For example, what equipment was allegedly used to clear the sidewalk? Was a snowplow or other machinery used? Were manual methods to clear the sidewalk such as use of snow shovels or salting of the sidewalk employed? If so who did this and when?
The aforementioned issues are amongst the myriad of questions your attorney at Queller, Fisher, Washor, Fuchs & Kool will investigate. Lawsuits involving slip and fall accidents on ice normally deal with transient conditions. A transient condition is one, which may have existed temporarily but vanishes soon after. The property owner or tenant may defend your lawsuit by claiming that the dangerous or defective condition never existed in the first place. Alternatively the defendant may argue that even if the condition did exist, they did not know about it in time so that in the exercise of reasonable care they could have done anything about it prior to the time your accident happened. Additionally, the defendant might also claim that the victim's injuries are unrelated to the accident or that the accident did not even occur to being with. Companies and individual's attempting to avoid liability will attempt to put forth a myriad of reasons. Consequently, it is imperative that you retain a qualified and experienced attorney in the case that you have been seriously injured as a result of a slip or fall on snow or ice.
However, the sustaining of an injury is not sufficient grounds for a lawsuit, it must also be demonstrated that the injury resulted in economic or non-economic damages. Economic damages pertain to any medical costs incurred due to the accident that are not covered by insurance where as non economic damages refer to any emotion stress or pain and suffering caused by the accident.
Individuals injured due to a slip and fall accident on snow or ice may have the option to recover compensation through a premises liability lawsuit. Amongst the compensable damages are economic damages such as lost wages or medical bills. Lost wages refer to any loss of income or damage to future earning potential, which needs to be compensated. Additionally, victims may also receive compensation for medical bills, procedures and medications resulting from the injury. In the case that an injury proves to be fatal, the family of the victim may be able to recoup compensation through a wrongful death lawsuit. In a wrongful death lawsuit, families of victims may be able to recoup compensation for damages such as loss of consortium, lost wages and damage to future earning potential, funeral costs and pain and suffering.


text:
<?php
namespace Clockwork\DataSource;

use Clockwork\DataSource\DataSource;
use Clockwork\Request\Log;
use Clockwork\Request\Request;
use Clockwork\Request\Timeline;
use Illuminate\Foundation\Application;
use Symfony\Component\HttpFoundation\Response;

/**
 * Data source for Laravel 4 framework, provides application log, timeline, request and response information
 */
class LaravelDataSource extends DataSource
{
	/**
	 * Laravel application from which the data is retrieved
	 */
	protected $app;

	/**
	 * Laravel response from which the data is retrieved
	 */
	protected $response;

	/**
	 * Log data structure
	 */
	protected $log;

	/**
	 * Timeline data structure
	 */
	protected $timeline;

	/**
	 * Create a new data source, takes Laravel application instance as an argument
	 */
	public function __construct(Application $app)
	{
		$this->app = $app;
	}

	/**
	 * Adds request method, uri, controller, headers, response status, timeline data and log entries to the request
	 */
	public function resolve(Request $request)
	{
		$request->method         = $this->getRequestMethod();
		$request->uri            = $this->getRequestUri();
		$request->controller     = $this->getController();
		$request->headers        = $this->getRequestHeaders();
		$request->responseStatus = $this->getResponseStatus();
		$request->routes         = $this->getRoutes();

		$request->timelineData = $this->getTimeline()->finalize($request->time);
		$request->log          = array_merge($request->log, $this->getLog()->toArray());

		return $request;
	}

	/**
	 * Set a custom response instance
	 */
	public function setResponse(Response $response)
	{
		$this->response = $response;
	}

	/**
	 * Return the log data structure (creates default Log instance if none set)
	 */
	public function getLog()
	{
		if (!$this->log)
			$this->log = new Log();

		return $this->log;
	}

	/**
	 * Set a custom log data structure
	 */
	public function setLog(Log $log)
	{
		$this->log = $log;
	}

	/**
	 * Return the timeline data structure (creates default Timeline instance if none set)
	 */
	public function getTimeline()
	{
		if (!$this->timeline)
			$this->timeline = new Timeline();

		return $this->timeline;
	}

	/**
	 * Set a custom timeline data-structure
	 */
	public function setTimeline(Timeline $timeline)
	{
		$this->timeline = $timeline;
	}

	/**
	 * Hook up callbacks for various Laravel events, providing information for timeline and log entries
	 */
	public function listenToEvents()
	{
		$timeline = $this->getTimeline();

		$timeline->startEvent('total', 'Total execution time.', 'start');

		$timeline->startEvent('initialisation', 'Application initialisation.', 'start');

		$this->app->booting(function() use($timeline)
		{
			$timeline->endEvent('initialisation');
			$timeline->startEvent('boot', 'Framework booting.');
			$timeline->startEvent('run', 'Framework running.');
		});

		$this->app->booted(function() use($timeline)
		{
			$timeline->endEvent('boot');
		});

		$this->app->shutdown(function() use($timeline)
		{
			$timeline->endEvent('run');
		});

		$this->app->before(function() use($timeline)
		{
			$timeline->startEvent('dispatch', 'Router dispatch.');
		});

		$this->app->after(function() use($timeline)
		{
			$timeline->endEvent('dispatch');
		});

		$this->app['events']->listen('clockwork.controller.start', function() use($timeline)
		{
			$timeline->startEvent('controller', 'Controller running.');
		});
		$this->app['events']->listen('clockwork.controller.end', function() use($timeline)
		{
			$timeline->endEvent('controller');
		});

		$log = $this->getLog();

		$this->app['events']->listen('illuminate.log', function($level, $message) use($log)
		{
			switch ($level) {
				case 'debug': $level = Log::DEBUG; break;
				case 'info': $level = Log::INFO; break;
				case 'notice': $level = Log::NOTICE; break;
				case 'warning': $level = Log::WARNING; break;
				case 'error': $level = Log::ERROR; break;
				case 'critical': $level = Log::ERROR; break;
				case 'alert': $level = Log::WARNING; break;
				case 'emergency': $level = Log::ERROR; break;
				default: $level = Log::INFO;
			}

			$log->log($message, $level);
		});
	}

	/**
	 * Return a textual representation of current route's controller
	 */
	protected function getController()
	{
		$router = $this->app['router'];

		if (method_exists($router, 'getCurrentRoute')) { // Laravel 4.0
			$controller = $router->getCurrentRoute()->getAction();
		} else { // Laravel 4.1
			$controller = $router->current()->getActionName();
		}

		if ($controller instanceof Closure) {
			$controller = 'anonymous function';
		} elseif (is_object($controller)) {
			$controller = 'instance of ' . get_class($controller);
		} else if (is_array($controller) && count($controller) == 2) {
			if (is_object($controller[0]))
				$controller = get_class($controller[0]) . '->' . $controller[1];
			else
				$controller = $controller[0] . '::' . $controller[1];
		} else if (!is_string($controller)) {
			$controller = null;
		}

		return $controller;
	}

	/**
	 * Return request headers
	 */
	protected function getRequestHeaders()
	{
		return $this->app['request']->headers->all();
	}

	/**
	 * Return request method
	 */
	protected function getRequestMethod()
	{
		return $this->app['request']->getMethod();
	}

	/**
	 * Return request URI
	 */
	protected function getRequestUri()
	{
		return $this->app['request']->getRequestUri();
	}

	/**
	 * Return response status code
	 */
	protected function getResponseStatus()
	{
		return $this->response->getStatusCode();
	}

	/**
	 * Return array of application routes
	 */
	protected function getRoutes()
	{
		$router = $this->app['router'];
		$routesData = array();

		if (method_exists($router, 'getCurrentRoute')) { // Laravel 4.0
			$routes = $router->getRoutes()->all();

			foreach ($routes as $name => $route) {
				$routesData[] = array(
					'method' => implode(', ', $route->getMethods()),
					'uri'    => $route->getPath(),
					'name'   => $name,
					'action' => $route->getAction() ?: 'anonymous function',
					'before' => implode(', ', $route->getBeforeFilters()),
					'after'  => implode(', ', $route->getAfterFilters()),
				);
			}
		} else { // Laravel 4.1
			$routes = $router->getRoutes();

			foreach ($routes as $route) {
				$routesData[] = array(
					'method' => implode(', ', $route->methods()),
					'uri'    => $route->uri(),
					'name'   => $route->getName(),
					'action' => $route->getActionName() ?: 'anonymous function',
					'before' => implode(', ', array_keys($route->beforeFilters())),
					'after'  => implode(', ', array_keys($route->afterFilters())),
				);
			}
		}

		return $routesData;
	}
}

redpajama_set_name: RedPajamaGithub


This pack will be available as 1 zip file which contains 30 JPG Images about Ombre Gold Paint Brush Stroke Watercolor Digital Papers. Each paper was created with in 12 x 12 inches (30.48 cm) and has 300 dpi resolution for the best quality printing. This pack is not seamless.
- Measurement: 12" x 12"
Thank you for your visit. Please feel free to contact me with any questions.


text:
Owner of landmark South Jersey restaurant damaged in fire vows to reopen
Updated Jan 30, 2019; Posted Aug 27, 2018
By Bill Duhart | For NJ.com
A fire in a restaurant in the Mullica Hill historic district of Harrison caused extensive damage to the kitchen area and apartments above the more than century-old building, but did not cause any injuries, the township and business owner said.
The blaze at the blueplate restaurant was reported by an alarm system at 11:59 p.m. Sunday. It was brought under control 40 minutes later, according to fire Chief Matt Cardile of the Harmony Fire Company.
Two residents in the apartment over the restaurant escaped the fire without injury, with a pet cat and rabbit, Cardile said.
A fire at blueplate restaurant in Mullica Hill may keep it closed for up to six weeks.
The fire appears to have started near an exterior light and spread into the building, restaurant owner James Malaby said. Cardile said one of the residents sprayed water from a hose and may have helped prevent a larger blaze.
The blueplate is located in the historic district in the Mullica Hill section of Harrison. The 68 acres here are listed on the National Registry of Historic Places, and features dozens of antique shops.
It also features ghost stories.
"Apparently, we have ghosts," said Malaby, who had the Jersey Unique Minds Paranormal group sweep his 4,200-square-foot building a few years ago with electronic equipment to detect paranormal activity ."We've had stories at different times of year from customers about a gentlemen ghost we have here. It's fun to talk about."
A fire Sunday at the blueplate in Mullica Hill may close the restaurant for up to six weeks.
Malaby said the fire may keep his restaurant -- known for popular Farm to Fork American fare -- closed for up to six weeks. The wooden, four-square colonial building has housed a restaurant for nearly 70 years. Malaby has owned the blueplate for 14 years. Previous restaurants here were the Hilltop Inn and DiVinci's.
Bill Duhart may be reached at bduhart@njadvancemedia.com. Follow him on Twitter @bduhart. Find NJ.com on Facebook.
Have a tip? Tell us. nj.com/tips

redpajama_set_name: RedPajamaCommonCrawl


text:
Adobe Scoops Up Figma in $20 Billion Deal
By Kehl Bayern / September 15, 2022
It looks like Adobe's empire is expanding just a little bit more.
MacBook Pro screen showing Adobe application icons. Photo by Mikaela Shannon
Well, that depends on whether or not you consider $USD 20 billion a little bit but, to hear Adobe tell the tale, they're getting quite a bargain with their purchase of design platform Figma.
As The Verge points out, the purchase caused quite a stir in the design community as Figma itself has largely offered a compelling alternative to Adobe's suite of products. From a business standpoint, however, it makes perfect sense. After all, if you can't beat them (enough), join them or, in this case, buy them.
"Together, Adobe and Figma will reimagine the future of creativity and productivity, accelerate creativity on the web, advance product design and inspire global communities of creators, designers and developers…The combined company will have a massive, fast-growing market opportunity and capabilities to drive significant value for customers, shareholders and the industry," The Verge quotes Adobe on the deal.
For the immediate future, though, it looks like everything is going to remain as-is at Figma and that probably suits their users just fine. Even so, the eventual merger of the two platforms will come sometime down the road. The Verge points out that Figma's major strength – a web-based platform – is one key area where Adobe and its competitors have had a hard time in a comment that perhaps gives us even more insight into the huge sum paid for the relatively young company. And let's not act like Figma won't benefit from Adobe vast well of resources and expertise in fields outside of Figma's core offerings. Whether it all works out or not, we'll be sure to let you know.
Do you use Figma? What about Adobe products? What do you think of Figma's new owners? Let us know in the comments.
Also, check out our other photography news at this link.
[The Verge]
Getty Images Sues Stable Diffusion Over Alleged Content Scraping
Sell Your Prints on Flickr with Latest Update
Monetize Your Photos on Twitter with New Coins Feature
Like Everything Else These Days, Kodak Film Prices Set to Rise in 2023


text:
\section{Introduction}

Device-to-device (D2D) communications enables direct communications between two user
devices without traversing the base station (BS) or core network, and is a promising way to achieve the high throughput goal of 5th generation (5G) cellular networks \cite{doppler2009device,Andrews.D2D,zhang2013exploring,Andreev.JSAC}. The typical use-cases of D2D communications include
cellular offloading, content distribution, and relaying, \emph{etc.} \cite{Survey.D2D}, where content delivery service has attracted considerable attention recently, since it accounts for the majority of the explosive increasing traffic load.

Motivated by the observation that a large amount of content delivery requests are asynchronous but redundant, i.e., the same content is requested repeatedly at different times, caching has long been studied as a technique to improve performance of wired networks. Due to the rapid reduction in cost of storage device, caching at the wireless edge is also recognized as a promising way for delivering popular contents nowadays, which can improve the network throughput, energy efficiency and the quality of user experience (QoE) \cite{wang2014cache,Procach14,Ali13,Dong,Higgins12,LHui14,Chen15}. However, different from wired networks, the performance of wireless networks is fundamentally limited by the interference, which inevitably limits the throughput gain from local caching.

To take the advantage of the storage device at smart phones, cache-enabled D2D communications has been proposed recently, which can offload the content delivery traffic and hence boost the network throughput significantly \cite{Golrezaei.TWC,JMY.JSAC}. Since only the users in proximity communicate to each other, the interference in D2D networks is strong, which needs to be carefully controlled.
In an early work of studying cache-enabled D2D communications, the D2D users are divided into clusters. Then, the intra-cluster interference among D2D links is managed by using time division multiple access (TDMA), while the inter-cluster interference between D2D links is simply treated as noise \cite{Golrezaei.TWC}. In \cite{JMY.JSAC}, only the D2D link from one of the four adjacent clusters is allowed to be active at the same time-frequency resource block, in order to avoid strong inter-cluster interference among adjacent clusters.
In \cite{IA.D2D}, interference alignment was employed to  mitigate the interference  among D2D links, but only three D2D links were coordinated within each cluster, and the interference among clusters was again treated as noise.
In \cite{CD2D.15,GC13.Relay,C2D.Relay}, cooperative relay techniques were proposed to mitigate the interference between cellular
and D2D links, which however can not manage the interference among the D2D links.

It is well known that if several transmitters have the required data for some users, they can jointly transmit to the users without generating interference. In fact, if contents have been locally cached, cooperative transmission without data exchange among transmitters becomes possible, which can transform interference into spatial multiplexing gain. Based on such an interesting observation, a BS cooperative transmission strategy was proposed in \cite{Lau.Tran13} by exploiting the caches at BSs, where precoding and cache control were optimized to guarantee the QoE of users. Inspired by this work, a natural question is: can we apply cooperative transmission in D2D communications with caching?

Fortunately, cooperative transmission is possible in practice due to the following reasons. (i) In D2D communications, the D2D transmitter (DT) has been proposed to assist other users in additional to transmitting data to its destined D2D receiver (DR), e.g., with cooperative relay \cite{GC13.Relay}. The users have the incentive to do this if their own QoE can be improved or their costs can be compensated by some other rewards \cite{incentive}. (ii) To facilitate cooperative transmission, the global channel state information (CSI) is required to compute the precoding matrix. The CSI among D2D links can be obtained at DTs and the BS through channel probing and feedback \cite{CSI}. Then, the precoding vectors can be computed at the BS and sent to the cooperative DTs via multicast. (iii) The synchronization among cooperative DTs is more easier to be implemented than that in Ad-hoc networks, because it can be realized with the assist of the BS \cite{Andrews.D2D}. Besides, the synchronization can also be realized at users by using the methods proposed in \cite{SYN.13}.

In this paper, we propose an {opportunistic cooperation strategy} for cache-enabled D2D communications to manage the interference among D2D links. Different from the BS cooperative transmission strategy \cite{Lau.Tran13}, the cooperation strategy for D2D communications needs to be optimized in a different way. Considering that D2D communications is applicable to users in proximity, we divide the D2D users into virtual clusters. To maximize the opportunity of cooperative transmission via D2D links, we take both redundant caching and diversity caching into account in the users among the clusters, which differs from \cite{Lau.Tran13} where all BSs cache the
same files. When some users have cached the files requested
by other users called DRs, these users act as DTs to
jointly transmit the requested files to the DRs.
Because only some D2D links can employ cooperative transmission, we assign different frequency bands to cooperative and non-cooperative links to avoid mutual interference. To maximize the average network throughput without compromising the experience of non-cooperative users, we jointly optimize the cluster size and bandwidth allocation under the minimal average user data rate constraint.

The contributions of this paper are summarized as follows:
\begin{itemize}
	\item We propose an opportunistic cooperation strategy to manage the interference among D2D links, which improve the network throughput remakably.
	\item We jointly optimize the cluster size and bandwidth allocation and obtain the closed-form expression of optimal bandwidth allocation factor.
\end{itemize}

The rest of the paper is organized as follows. Section \ref{sec:system model} presents the system model. Section \ref{sec:optimiazation} introduces the cooperation strategy, derives the average network throughput and average user data rate, and jointly optimizes bandwidth allocation and cluster size. Section \ref{sec:simulation} provides numerical and simulation results. Section \ref{sec:conclusion} concludes the paper.

\section{System Model}
\label{sec:system model}

Consider a cellular network, where $M$ single-antenna users are uniformly located in a square hotspot within a macro cell,
where the area is with side length of $D_c$ as shown in Fig. \ref{fig.ici-d2d}. Each user is willing to store $N$ files in its local cache and can act as a helper to share files. When a helper conveys a file in local cache via D2D link to a DR requesting the file, the helper becomes a DT. The BS is aware of the cached files at each user and coordinates the D2D communications.

\subsection{Content Popularity}

We consider a static content catalog including $N^f$ files that the users may request, where the files are indexed in a descending order of popularity, e.g., the 1st file
is the most popular file. The probability that the $i$th file is requested by a user is assumed to follow a Zipf distribution,
\begin{equation}
	\label{equ.P_N_f}
	P_{N^f}(i)=i^{-\beta}/\sum_{j=1}^{N^f}j^{-\beta},
\end{equation}
where $\sum_{i=1}^{N_f}P_{N^f}(i)=1$, and the parameter $\beta$ reflects skewness of the popularity distribution, with large $\beta$ meaning that a few files are requested by the majority of users \cite{Zipf99}.

\begin{figure}[!hb]
	\centering

	\includegraphics[width=0.45\textwidth]{D2D_MODEL}\\
	\caption{Cluster division model, ``UE" means user equipment.}\label{fig.ici-d2d}
\end{figure}
\subsection{Communication Protocol}
D2D links can be established among users in proximity. A widely used communication protocol for D2D communications is that two user equipments (UEs) can communicate if their distance is smaller than a given distance \cite{Golrezaei.TWC,Kumar.C}.
To restrict the D2D link distance and make the analysis tractable, the square hotspot area is divided into $B$ smaller square areas called clusters, where the side length of each cluster is $D=D_c/\sqrt{B}$. Only the users within the same cluster can establish D2D link. For mathematical simplicity, we assume that the number of users per cluster is $K=M/B$ and each user is assumed to transmit with the same power $P$ as in \cite{JMY.JSAC}.

We consider overlay inband D2D \cite{Survey.D2D}, and assume that a fixed bandwidth of $W$ is assigned to the D2D links.

\section{Opportunistic Cooperation Strategy}
\label{sec:optimiazation}
In this section, we first introduce a caching policy to provide high opportunity for cache-enabled cooperative D2D transmission. Then, we propose an opportunistic cooperative transmission policy. Finally, we optimize two key parameters in the strategy to maximize the network throughput.

\subsection{Caching Policy}
To maximize the probability that a user can fetch files through D2D links, the users within a cluster should cache different files. To maximize the probability of cooperative transmission among DTs in different clusters, the files cached at the users of each cluster should be the same.
This suggest that the caching policy needs to balance the
diversity of content with the redundancy of the replicas of popular
contents. To this end, we consider the following caching policy.

According to the user cache size $N$, all files are divided into $K_0=N^f/N$ groups. The $k$th \emph{file group} $\mathcal{G}_k$ consists of the $(k-1)N+1$th to the $kN$th files where $1 \leq k\leq K_0$, e.g., the 1st file group $\mathcal{G}_1$ contains the most popular $N$ files. Then, the probability that a user requests a file within the $k$th \emph{file group} $\mathcal{G}_k$ can be obtained as
\begin{equation}
	\label{equ.P_k}
	P_k= \sum_{i=(k-1)N+1}^{kN}P_{N^f}(i) = \frac{\sum_{i=(k-1)N+1}^{kN}i^{-\beta}}{\sum_{j=1}^{N^f}j^{-\beta}} .
\end{equation}

In every cluster, the $k$th user caches the $k$th file group $\mathcal{G}_k$. Then, every user in each cluster caches different files, i.e., diversity caching is achieved within each cluster, and the most popular $KN$ files are cached in every cluster, i.e., redundant caching is achieved among clusters. Because each cluster contains $K$ users, the file groups with indices exceeding $K$ (i.e.,  $\mathcal{G}_k$, $k>K$ ) are not cached at users.

When $K=K_0$, all the $N^f$ files can be cached at the users in each cluster and all user requests can be served via D2D links, therefore it is not necessary to assign more than $K_0$ users to each cluster. For this reason, we assume $K \leq	K_0$.

In practice, the files can be proactively downloaded by the operator from the  BS to the cache at each user via multicast during off-peak times according to the  user demand statistics.

\subsection{Opportunistic Cooperative Transmission Policy}
According to whether a user can find the requested file in its local cluster, we can classify the users into two types.

{\bf D2D users:} If the file requested by a user is cached at any UE in the cluster it belongs to (called \emph{local cluster of the user}), then the user can directly obtain the file with D2D communication. Such a user is referred to as a D2D user. Besides, if the file requested by a user is in its local cache, it can retrieve the file immediately with zero delay, but we ignore this case for analysis simplicity as in \cite{JMY.JSAC}.

{\bf Cellular users:} If the file requested by a user is not cached in the UEs within its local cluster, the user fetches the file from the BS and becomes a regular cellular user. The number of cellular users is denoted as $N^b$.

\begin{figure}[!h]
	\centering

	\includegraphics[width=0.4\textwidth]{Coop}\\
	\caption{Illustration of the opportunistic cooperaton strategy. Catalog size $N_f=20$, $B=4$ clusters in the hotspot, $K=3$ users in each cluster and each user caches $N=5$ files. All $20$ files are divided into $4$ groups according to content popularity, e.g., $\mathcal{G}_1 = \{1,2,3,4,5\}$.}\label{fig.Coop}
\end{figure}

For easy understanding, we introduce the strategy with the help of an example.

\subsubsection{Cooperative D2D Users}
If there exists at least one user in a cluster requesting the files in $\mathcal{G}_k$, then we say that the cluster \emph{hits the $k$th file group}. In Fig. \ref{fig.Coop}, the users in the first cluster respectively request the files in $\mathcal{G}_1$, $\mathcal{G}_2$ and $\mathcal{G}_4$, and hence the first cluster hits the  $\{1,2,4\}$th file groups.

If every cluster hits the same file group $\mathcal{G}_k$, the $k$th user in each cluster who caches the file group $\mathcal{G}_k$ can act as a DT, and all DTs in these clusters  cooperatively transmit files to the DRs requesting the files in $\mathcal{G}_k$.\footnote{Though further improvement is possible by allowing cooperation among less than $B$ clusters (called partial cooperation), we only consider the full cooperation among all clusters for mathematical tractability. The impact of partial cooperation is shown via simulation in Section \ref{sec:simulation}.} Those DRs are referred to as {\bf cooperative D2D users} (\emph{Coop users} for short), whose number is denoted as $N^c$.

In Fig. \ref{fig.Coop}, every cluster hits the $1$st file group. Hence, the $1$st users  in all the four clusters who cache the files in $\mathcal{G}_1$ can act as DTs to cooperatively transmit files with indices $\{4,2,1,5\}$ respectively to the $3$rd user in cluster $1$, the $3$rd user in cluster $2$, the $2$nd user in cluster $3$, and the $3$rd user in cluster $4$, as shown in Fig. \ref{fig.Coop_transmission}.

\begin{figure}[!htp]
	\centering

	\includegraphics[width=0.4\textwidth]{Coop_transmission}\\
	\caption{Illustration for cooperative transmission from multiple DTs to DRs.}\label{fig.Coop_transmission}
\end{figure}
The remaining users except the cellular and Coop users are {\bf non-cooperative D2D users} (\emph{N-Coop users} for short), whose number is $N^n= M-N^b-N^c$.

\subsubsection{Interference Control}
Due to the random locations of the DTs in proximity, the interference in the network needs to be carefully controlled even with the cooperative transmission.

{\bf Inter-type interference:} To avoid the mutual interference between \emph{Coop users} and \emph{N-Coop users}, we assign $\eta W$ for Coop users and the remaining bandwidth $(1-\eta) W$ for N-Coop users, where $\eta$ is the bandwidth allocation factor and $0 \leq \eta < 1$. A large value of $\eta$ means that more bandwidth is allocated to Coop users.

{\bf Intra-cluster interference:} Considering that the users within each cluster can not cooperate due to caching different files, we randomly select one Coop D2D link and one N-Coop D2D link respectively in each cluster to transmit at the same time to avoid intra-cluster interference as in \cite{Golrezaei.TWC,JMY.JSAC}.

{\bf Inter-cluster interference:} There is no inter-cluster interference among Coop users owing to the joint transmission from multiple DTs, but there exists inter-cluster interference between N-Coop users, which is regarded as noise.

\subsubsection{Operation Modes}

Due to the opportunistic nature of establishing the cooperative D2D links, the network may operate in the following two modes.
\begin{itemize}
	\item In \emph{Mode} $0$, there does not exist any file group hit by every cluster, i.e., all D2D users are N-Coop users. Then, all the DTs transmit independently, and the bandwidth $W$ is assigned to the N-Coop users, i.e.,
	$\eta=0$.
	\item In \emph{Mode} $1$, there exist file groups hit by every cluster, i.e., there exist Coop users. Then, $0<\eta<1$.
\end{itemize}

To become the Coop users, the users are not necessary to request the \emph{same file}, but to request the \emph{files in the same group}. Hence the cooperative probability, i.e., the  probability that the network operates in \emph{Mode 1}, is high, which increases with the number of users in each cluster $K$.
To see this, we derive  the cooperative probability $P^c$ as follows.

A cluster hits the $k$th file group if at least one of the $K$ users in the cluster requests a file in $\mathcal{G}_k$, whose probability is denoted as $P^h_k$. It is the complement of the probability that no user requests any file in the $k$th file group, which is $(1-P_k)^K$. Then, from (\ref{equ.P_k}), $P^h_k$ can be obtained as
\begin{equation}
	\label{equ.P^h_k}
	P^h_k=1-(1-P_k)^K
\end{equation}
which  increases with  $K$, because $0 \leq P_k \leq 1$.

Cooperative probability is the probability that there exists at least one file group hit by all the $B$ clusters. It is the complement of the probability that there is no file group hit by all the $B$ clusters, and hence can be derived
as
\begin{equation}
	\label{equ.P^c}
	P^c=1-\prod_{k=1}^{K}(1-(P^h_k)^B),
\end{equation}
where  $1-(P^h_k)^B$ is the probability that the number of clusters hitting the $k$th file group is less than $B$, which decreases with the growth of $K$ since $B=M/K$. Therefore, for a given value of $M$, $P^c$ is an increasing function of $K$.

\subsubsection{Key Parameters}
Since only one Coop D2D link per cluster is allowed to be active each time,  $B$ users out of all Coop users can be scheduled simultaneously in \emph{Mode} 1. Therefore, the number of {\bf active Coop users} is $N^a=B$ in \emph{Mode} $1$, and is $N^a=0$ in \emph{Mode} 0.\footnote{For the considered strategy, the number of active Coop users is less than the number of Coop users, i.e., $N^a \le N^c$.} With the cooperative probability, the average number of active Coop users can be obtained as
\begin{equation}
	\label{equ.E_N^a}
	\bar{N^a}= BP^c+0(1-P^c)=BP^c.
\end{equation}

$\bar{N^a}$ characterizes how many \emph{interference-free D2D links} can transmit at the same time-frequency resources in average, which can reflect  the multiplexing gain. In general, the number of interference-free D2D links demonstrates the same trend with the network throughput, as to be verified in Section \ref{sec:simulation}. Hence, a large value of $\bar{N^a}$ implies a high network throughput. When the number of users per cluster $K$ is large, the cooperative probability is high, but the number of active Coop users is small since $N^a=B$ and $B=M/K$. This suggests that there is a tradeoff between two counter-running effects: a small value of $K$ leads to more active Coop users if the system operates in Mode 1; a large value of $K$ yields high cooperative probability. In other words, to maximize the network throughput, the cluster size should be optimized, which is reflected by the number of users per cluster $K$ since the number of users in the hotspot $M$ is given.

Due to the multiplexing gain and interference-free transmission, the average data rate of Coop users usually exceeds that of N-Coop users. As a result, the overall network throughput will be reduced if we simply assign identical bandwidth to these two types of D2D users. Owing to the same reason, simply allocating all the bandwidth to Coop users can maximize the network throughput, but no N-Coop users can be served. This indicates that the bandwidth allocation factor $\eta$ should be optimized to maximize the throughput of the network under the constraint on the data rate of each user to avoid unfairness.

\subsection{Optimization of Cluster Size and Bandwidth Allocation}
In this subsection, we jointly optimize the bandwidth allocation factor $\eta$ and cluster size $K$ to maximize the average network throughput under a constraint that the average user data rate is larger than a given value, $\mu$ (Mbps).
Because we assume overlay D2D communications, only D2D users are considered in the network throughput.

In the sequel, we first derive the average network throughput achieved by all D2D users. Then, we derive the average data rate for each of Coop and N-Coop users. Finally, we find the optimal cluster size $K$ and bandwidth allocation factor $\eta$.

\subsubsection{Average Network Throughput}
Recall that only one Coop D2D link (if any) and one N-Coop D2D link are scheduled per cluster in each time. Then,
the average throughput of the network operating in \emph{Mode} 0 can be obtained as follows,
\begin{equation}
	\label{equ.bar_R_0}
	\bar{R_0} = \mathbb{E}\{W\sum_{i=1}^{B}R_i^n\} \overset{(a)}{=} WB\bar{R_i^n},
\end{equation}
where the expectation is taken over small scale channel fading and user location, $(a)$ comes from the fact that all users are randomly located and transmit with equal power, and ${R_i^n}$ and $\bar{R_i^n}$ are the instantaneous and average data rate per unit bandwidth per second  of the N-Coop link in the $i$th cluster,  respectively.

Analogically, the average throughput of the network operating in \emph{Mode} 1 can be obtained as
\begin{equation}
	\label{equ.bar_R_1}
	\begin{split}
		\bar{R_1} &= \mathbb{E}\{ \eta W\sum_{i=1}^{B}R_i^c\ + (1-\eta)W\sum_{i=1}^{B}R_i^n\} \\
		&= WB(\eta \bar{R_i^c}+(1-\eta) \bar{R_i^n} ),
	\end{split}
\end{equation}
where ${R_i^c}$ and $\bar{R_i^c}$ are the instantaneous and average data rate per unit bandwidth per second of the Coop link in the $i$th cluster,  respectively.

Further considering the cooperative probability $P^c$ in (\ref{equ.P^c}), the average throughput of the network is
\begin{equation}
\label{equ.bar_R}
\begin{split}
\bar{R} &= P^c\bar{R_1} + (1-P^c)\bar{R_0}\\
& = WB(P^c\eta \bar{R_i^c}+(1-P^c\eta)\bar{R_i^n}).
\end{split}
\end{equation}

\vspace{2mm}\begin{proposition}
	\label{p:1}
	The average data rate per unit bandwidth per second of the N-Coop link in the $i$th cluster is
	\begin{equation}
		\label{equ.bar_R_i^n3}
		\begin{split}
			\bar{R_i^n} = \log_2(Q_1(\alpha))-\log_2(Q_2(\alpha))-3,
		\end{split}
	\end{equation}
	where $Q_1(\alpha)\triangleq\int_{0}^{\sqrt{2}}r^{-\alpha}g(r)dr +8\int_{0}^{\sqrt{5}}r^{-\alpha}f(r)dr$,
	$Q_2(\alpha)\triangleq\int_{0}^{\sqrt{5}}r^{-\alpha}f(r)dr$, and $f(r)$ and $g(r)$ are in closed-form expression defined in Appendix \ref{a:1}.
\end{proposition}
\begin{IEEEproof}
	See Appendix \ref{a:1}.
\end{IEEEproof}\vspace{2mm}

In Proposition \ref{p:1}, $Q_1(\alpha)$ and $Q_2(\alpha)$ are easy to be computed numerically.
We can see that $\bar{R_i^n}$ only depends on the path loss exponent $\alpha$.

\vspace{2mm}\begin{proposition}
	\label{p:2}
	The average data rate per unit bandwidth per second of the Coop link in the $i$th cluster is

	\begin{equation}
		\label{equ.bar_R_i^c2}
		\begin{split}
			\bar{R_i^c} =\log_2(1+\frac{PD^{-\alpha}}{B\sigma^2}Q_1(\alpha)).
		\end{split}
	\end{equation}
\end{proposition}
\begin{IEEEproof}
	See Appendix \ref{a:2}.
\end{IEEEproof}\vspace{2mm}

By substituting  (\ref{equ.bar_R_i^n3}) and (\ref{equ.bar_R_i^c2}) into (\ref{equ.bar_R}), the average network throughput can be obtained as
\begin{equation}
	\label{equ.bar_R2}
	\begin{split}
		\bar{R} &= WBP^c\eta ( \log_2(Q_1(\alpha))-\log_2(Q_2(\alpha))-3)\\ &+WB(1-P^c\eta)\log_2(1+\frac{PD^{-\alpha}}{B\sigma^2}Q_1(\alpha).
	\end{split}
\end{equation}
\subsubsection{Average User Data Rate}

Since only one N-Coop user and one Coop user (if exists) are active in a cluster each time, with
round robin scheduling, the average data rates of N-Coop and Coop users can be
respectively obtained  from (\ref{equ.bar_R_i^n3}) and (\ref{equ.bar_R_i^c2}) as follows
\begin{align}
	\bar{R^n_u}   & =W(1-\eta) \mathbb{E}\{\frac{B R_i^n}{N^n} \}\nonumber \overset{(a)}{\approx} \frac{WB(1-\eta)\bar{R_i^n}}{\bar{N^n}} \nonumber \\&= \frac{WB(1-\eta)}{\bar{N^n}}\log_2(1+\frac{PD^{-\alpha}}{B\sigma^2}Q_1(\alpha)),  \nonumber\\
	\bar{R^c_u}  & =W\eta \mathbb{E}\{\frac{B R_i^c}{N^c} \}\nonumber  \overset{(b)}{\approx} \frac{WB\eta \bar{R_i^c}}{\bar{N^c}} \nonumber \\& = \frac{WB\eta(\log_2(Q_1(\alpha))-\log_2(Q_2(\alpha))-3) }{\bar{N^c}},\label{equ.bar_R^c_n}
\end{align}
where (a) and (b) come from the fact that $R_i^n$ and $N^n$ are independent random variables and the same to $R_i^c$ and $N^c$, thus $\mathbb{E}\{R_i^n/N^n\} = \mathbb{E}\{R_i^n\}\mathbb{E}\{1/N^n\} \approx
\mathbb{E}\{R_i^n\}/\mathbb{E}\{N^n\} = \bar{R_i^n}/\bar{N^n}$ according to (\ref{equ.1_approx}) and $\mathbb{E}\{R_i^c/N^c\} \approx \bar{R_i^c}/\bar{N^c}$ analogically. $\bar{N^c}=\mathbb{E}\{N^c\}$ and $\bar{N^n}=\mathbb{E}\{N^n\}$ are the average numbers of Coop users and N-Coop users,  respectively.

\vspace{2mm}\begin{proposition}
	\label{p:3}
	The average number of Coop users is

	\begin{equation}
		\label{equ.bar_N_c}
		\begin{split}
			\bar{N^c}

			&= \sum_{\Phi_\mathcal{N}}\prod_{i=1}^{B}\frac{K!\prod_{k=1}^{K_0} (P_k)^{n_{ik}}}{\prod_{j=1}^{K_0}n_{ij}!} \sum_{k=1}^{K}\sum_{i=1}^{B}\zeta(k) n_{ik},
		\end{split}
	\end{equation}
which can be approximated as
	\begin{equation}
		\label{equ.bar_N_c2}
		\begin{split}
			&\bar{N^c} \approx \bar{N^c_1} +\bar{N^c_2},
		\end{split}
	\end{equation}
	where $\zeta(k)$, $n_{ik}$, $\bar{N^c_1}$ and $\bar{N^c_1}$ are defined in Appendix \ref{a:3}.
\end{proposition}
\begin{IEEEproof}
	See Appendix \ref{a:3}.
\end{IEEEproof}\vspace{2mm}

Though we can use similar way to derive the average number of N-Coop users $\bar{N}^n$ as for  $\bar{N}^c$, the resulting expression is complicated.
Considering that $N^n+N^c+N^b=M$, we can obtain $\bar{N}^n$ by deriving the average number of cellular users $\bar{N^b}=\mathbb{E}\{N^b\}$.
Since all
requests follow a Zipf distribution independently, the number of users that can not fetch files via D2D is a random variable following a Binomial distribution and $N^b \sim B(M,1-\sum_{k=1}^{K}P_k)$. Therefore, $\bar{N^b}=M(1-\sum_{k=1}^{K}P_k)$. Then, the average number of N-Coop users is
\begin{equation}
	\label{equ.bar_N_n}
	\bar{N^n} = M - \bar{N^c} - \bar{N^b} .
\end{equation}

With the average number of Coop and N-Coop users $\bar{N^c}$ and $\bar{N^n}$, we can obtain the corresponding average user data rate  $\bar{R}^c_u$ and $\bar{R}^n_u$ using \eqref{equ.bar_R^c_n}.

\subsubsection{Joint Optimization of $\eta$ and $K$}
The bandwidth allocation factor and cluster size that maximize the average network throughput under the constraint of average user data rate can be optimized from the following problem
\begin{equation}
	\label{equ.opt2}
	\begin{aligned}
		&\max_{\eta,K}\,\, &&\bar{R} \\
		&s.t.\quad
		&&\bar{R}^c_u\geq \mu, ~~ \bar{R}^n_u \geq \mu,\\
		& && 0< \eta \leq 1, ~~ KB=M.
	\end{aligned}
\end{equation}

Since the number of users per cluster $K$ is an integer, we can find the joint optimal solution by first finding optimal $\eta$ for any given $K$ and then enumerating $K$ until the value of $\bar{R}$ computed by \eqref{equ.bar_R2} achieves the maximum under the two constraints.

By taking the derivative of $\bar{R}$  in (\ref{equ.bar_R}) with respect to $\eta$, we have $\frac{\partial \bar{R}}{\partial \bar{\eta}} = WBP^c(\bar{R_i^c}-\bar{R_i^n})$. The constraints $\bar{R}^c_u\geq \mu$ and $\bar{R}^n_u \geq \mu$ can be respectively rewritten as $\eta \leq \frac{WB\bar{R^n_i}-\mu\bar{N^n}}{WB\bar{R^n_i}}$ and $\eta \geq \frac{\bar{N^c}\mu}{WB\bar{R^c_i}}$ according to \eqref{equ.bar_R^c_n}. Therefore, if $\bar{R^c_i} \geq \bar{R^n_i}$, $\bar{R}$ is an increasing function of $\eta$, then the optimal solution of problem \eqref{equ.opt2} for any given $K$ is $\eta^*_K=\frac{WB\bar{R^n_i}-\mu\bar{N^n}}{WB\bar{R^n_i}}$; otherwise, $\bar{R}$ is a decreasing function of $\eta$, and $\eta^*_K=  \frac{\bar{N^c}\mu}{WB\bar{R^c_i}}$.

\vspace{2mm}\begin{proposition} \label{p:4}
	If $\bar{I_i} \geq B\sigma^2$, then $\bar{R^c_i} \geq \bar{R^n_i}$.
\end{proposition}

\begin{IEEEproof}
	See Appendix \ref{a:4}.
\end{IEEEproof}\vspace{2mm}

Since the power of interference among N-Coop users $\bar{I_i}$ defined in \eqref{equ.SINR_R^n} is much larger than noise variance $\sigma^2$ in D2D communications and $B$ is finite, the condition $\bar{I_i}\geq B\sigma^2$ in Proposition
\ref{p:4} is easy to be satisfied. Consequently, the optimal value of $\eta$ for a given $K$ is
\begin{equation}
	\label{equ.opt_eat}
	\eta^*_K = 1 - \frac{\mu\bar{N^n}}{WB\bar{R^n_i}},
\end{equation}
which only depends on the average data rate of N-Coop link ${\bar R^n_i}$ and the number of N-Coop users $\bar{N^n}$.

The joint optimal solution $K$ and $\eta$ can be found by one-dimensional searching and $\eta^*$ is with closed-form expression, and is hence with low complexity. Because $K^*$ and $\eta^*$ depend on $W$, $\mu$, $B$, $M$, $N_f$, $\beta$ and $N$, which are usually fixed for a long time, they are unnecessary to be updated frequently.

\section{Simulation and Numerical Results}

\label{sec:simulation} In this section, we validate the analysis and evaluate the performance of the proposed opportunistic cooperation strategy by simulation and numerical results.

In the simulation, we consider a square hotspot area with the side length $D_c=100$ m, where $M=180$ users are randomly located. Such a setting reflects relatively high user density as in \cite{3GPP.density}, where more than one user is located within an area of $10\times10$ $m^2$.
The
path-loss model is $37.6+36.8\log_{10}(r)$ \cite{JMY.JSAC}. Each user is with transmit power $P=23$ dBm. $W=20$ MHz, and $\sigma^2=-100$ dBm.
The file catalog size $N^f=300$, and each user is willing to cache $N=10$ files. The parameter of Zipf distribution is $\beta =0 \sim 1$. The user data rate constraint is
$\mu = 1 \sim 2$ Mbps.
This setup is used in the sequel unless otherwise specified.

\subsection{Impact of Cluster Size and Number of Total Users}

In Fig. \ref{fig.2}, we provide numerical results of the average number of active Coop users $\bar{N^a}$ obtained from \eqref{equ.E_N^a} and simulation results for the average sum rate of the active Coop users versus the number of users per cluster $K$. It is shown from Fig. \ref{fig.2}(a) that $\bar{N^a}$ increases with $\beta$. This is because with large $\beta$, a few popular files are requested by majority of users, which leads to high cooperative probability $P^c$. With the increase of $K$, $\bar{N^a}$ first increases and then decreases. It is shown from Fig. \ref{fig.2}(b) that with the growth of $K$, the average sum rate of active Coop users exhibits the same trends with $\bar{N^a}$, which agrees to the analysis in section III.A.

\begin{figure}[!htb]
	\centering

	\includegraphics[width=0.5\textwidth]{G_C_D2D_1_1}\\
	\caption{$\bar{N^a}$ and average sum rate of active Coop users versus $K$.}\label{fig.2}
\end{figure}
\begin{figure}[!b]
	\centering

	\includegraphics[width=0.5\textwidth]{G_C_D2D_2}\\
	\caption{$P^c$ of (a) full cooperative, and (b) partial cooperation.}\label{fig.3}

\end{figure}

In Fig. \ref{fig.3}, we simulate the cooperative probability $P^c$. It is shown from Fig. \ref{fig.3}(a) that $P^c$
increases with $K$, which agrees with the analysis after (\ref{equ.P^c}). Moreover, the cooperative probability is high although  the full cooperation is allowed only when all clusters hit the same file group, especially when $\beta$ is not small, say $\beta > 0.4$.
In Fig. \ref{fig.3}(b), we show the impact of partial cooperation by changing the minimal number of clusters allowed to cooperate.\footnote{We set such a minimal number because when we allow fewer clusters to cooperate, the multiplexing gain will reduce despite that $P_c$ will be higher, and then the resulting throughput will reduce.  When this minimal number of clusters is six, it becomes the full cooperation strategy.} As expected, $P^c$ can be improved if we allow partial cooperation among clusters, but not significant. Therefore, the throughput gain from partial cooperation over full cooperation is marginal as to be illustrated later.

In Fig. \ref{fig.1a}, we provide numerical results of the optimal cluster size $K^*$ and the resulting average number of active Coop users $\bar{N^a}$ for different number of total users in the hotspot area $M$. When $M=1000$, there are $10$ users in an area of $10 \times10$ $m^2$, corresponding to a very high traffic load \cite{3GPP.density}. It is shown from Fig. \ref{fig.1a}(a) that $K^*$ decreases with $\beta$ as expected. With the growth of $M$, $K^*$ first increases  and then approaches a constant that equals to $K_0=N^f/N$. This is because when $K=K_0$, all files in the catalog can be cached at the users in each cluster. Assigning more than $K_0$ users to each cluster can not increase the number of Coop users. It is shown from Fig. \ref{fig.1a}(b) that  with the growth of $M$, the average number of active Coop users monotonously increases when $\beta$ is large but first increases and then decreases when $\beta$ is small. This implies that if the user density is large but $\beta$ is small, the proposed opportunistic cooperation strategy may be not useful.

\begin{figure}[!htb]
	\centering

	\includegraphics[width=0.5\textwidth]{G_C_D2D_1_2}\\
	\caption{$K^*$ and $\bar{N^a}$ versus the number of total users in the hotpot $M$.}\label{fig.1a}
\end{figure}

\subsection{Accuracy of the Approximations}
In Fig. \ref{fig.4}, we evaluate the accuracy of all the approximations used in deriving the throughput and the number of users by simulation and numerical results.
In Fig.
\ref{fig.4}(a), the numerical results of $\bar{R^c_i}$ and $\bar{R^n_i}$ are respectively obtained from (\ref{equ.bar_R_i^n3}) and (\ref{equ.bar_R_i^c2}) by changing the path loss exponent $\alpha$ from $2$ to $4$. We can see that with the growth of $\alpha$, the average data rate per unit bandwidth per
second of the Coop link decreases but that for the N-Coop link increases. This is because both signal and interference power decrease when $\alpha$ increases, but the interference power decreases more rapidly due to larger distance of interference link than that of signal link.
In Fig. \ref{fig.4}(b), the numerical results of the number of Coop users $\bar{N^c}$ and the number of N-Coop users $\bar{N^n}$ are  respectively obtained from \eqref{equ.bar_N_c2} and \eqref{equ.bar_N_n}. As expected, when $K$ increases, the number of Coop users increases due to the decrease of the number of clusters. However, the number of N-Coop users first decreases and then increases slowly with $K$. This is because with the growth of $K$, more files can be cached in each cluster and thus the total number of D2D users increases, and the number of N-Coop users changes as in \eqref{equ.bar_N_n}.
We can see that the approximations are accurate, except the average data rate
per unit bandwidth per second of the Coop link when $\alpha$ is large, which comes from the first order approximation in \eqref{equ.bar_R_i^n2}.

\begin{figure}[!htb]
	\centering

	\includegraphics[width=0.5\textwidth]{G_C_D2D_3}\\
	\caption{Average data rate
per unit bandwidth per second and number of users.  ``Num.'': Numerical results, ``Sim.'': Simulation results.}\label{fig.4}
\end{figure}

\subsection{Optimal Bandwidth Allocation and Network Throughput}

In Fig. \ref{fig.5}(a), we present the optimal solution of problem  (\ref{equ.opt2}) $\eta^*$ versus the Zipf parameter $\beta$. We can see that $\eta^*$ increases with $\beta$. This is because the number of interference-free links, $\bar{N^a}$, increases with $\beta$, and then allocating more bandwidth to Coop users can increase the network throughput. $\eta^*$
decreases as $\mu$ increases, because more bandwidth is needed for N-Coop users to support higher user data rate.

In Fig. \ref{fig.5}(b), we provide simulation results for the maximal network throughput. In the legends, ``$\eta=0$" refers to the strategy in  \cite{Golrezaei.TWC} (without interference coordination) and ``TDMA" is the strategy in \cite{JMY.JSAC} (without strong inter-cluster
interference), which serve as the baselines for comparison.  ``$\eta=0.5, K=30$" refers a cooperation strategy without optimizing $K$ and $\eta$, which allocates equal bandwidth to Coop users and N-Coop users. ``$\eta^*, K=30$" refers to a cooperation strategy without optimizing $K$ but only optimizing $\eta$. We can see that optimizing the bandwidth allocation becomes necessary when $\beta >0.5$, while optimizing the cluster size is always necessary but the gain from optimization grows with $\beta$. With $K^*$ and $\eta^*$, the throughput gain over the baseline for $\beta=1$ is  $400\% \sim 500\%$, which demonstrates that the proposed opportunistic cooperation strategy can boost the network throughput remarkably. Even when $\beta=0$, where file popularity follows a uniform distribution, the throughput gain is still $60\% \sim 80\%$. We also demonstrate the performance of ``Partial-Coop", i.e., allowing cooperation among less than $B$ clusters. It is shown that the performance can be improved by allowing partial cooperation, but the gain is marginal.

\begin{figure}[!htb]
	\centering

	\includegraphics[width=0.5\textwidth]{G_C_D2D_4}\\
	\caption{$\eta^*$ and maximal average network throughput versus $\beta$. }\label{fig.5}
\end{figure}

\section{Conclusions}
\label{sec:conclusion} In this paper, we proposed an opportunistic cooperation strategy for cache-enabled D2D communications. We jointly optimized the cluster size and the bandwidth allocated to Coop and N-Coop users to maximize the network throughput with minimal user data rate constraint. Simulation results showed that the proposed strategy can boost the throughput even when the content popularity follows a uniform distribution, and the gain over existing strategies is remarkable when the popularity distribution is more skewed.

\appendices
\numberwithin{equation}{section}

\section{Proof of Proposition \ref{p:1}}
\label{a:1}
Without cooperation, the closest DT in the same cluster of its corresponding DR delivers the requested file to the DR, which treats the inter-cluster interference as noise when decoding the desired signal. The signal to interference plus noise ratio (SINR) at the DR of the active N-Coop link in the $i$th cluster can be expressed as
\begin{equation}
	\label{equ.SINR_R^n}
	\gamma_i^n = \frac{P|h_{ii}|^2r_{ii}^{-\alpha}}{I_i+\sigma^2},
\end{equation}
where $P$ is the transmit power, $\sigma^2$ is the variance of white Gaussian noise, $I_i = P\sum_{j=1,j \neq i}^{B}r_{ij}^{-\alpha}|h_{ij}|^2$
is the total power of inter-cluster interference, $h_{ij}$ and $r_{ij}$ are respectively the channel coefficient and distance between the DT  and the DR, $\alpha$ is the path loss exponent, and both the interference channel coefficient $h_{ij}$ $(i\neq j)$ and the desired channel coefficient $h_{ii}$ follow a complex Gaussian distribution with zero mean and unit variance.

Due to the short distance between D2D links, the D2D network is interference-limited and hence the noise can be ignored, i.e., $I_i \gg \sigma^2$.

Then,  from \eqref{equ.SINR_R^n}
the data rate per unit bandwidth per second for the N-Coop link in the $i$th cluster is $R_i^n = \log_2(1+\frac{P|h_{ii}|^2r_{ii}^{-\alpha}}{I_i})$.

Considering that $|h_{ij}|^2$ follows an Exponential distribution, which is a special case of the Gamma
distribution, the interference power $I_i$ (which is a sum of random variables following a Gamma distribution)  can be approximated as a Gamma distribution
\cite{Gamma.2011}. Further consider that for a Gamma distributed random variable $X$ with parameters $k$ and $\theta$,
$\mathbb{E}\{\ln(X)\}=\psi(k)+\ln(\theta)$, where $\psi(k)$ is the Digamma function \cite{abramowitz1964}. Then, the average  data rate per unit bandwidth per second  of the N-Coop link can be obtained according to Proposition 9 in \cite{Gamma.2011} as
\begin{equation}
	\label{equ.E_h_R_i^n}
	\mathbb{E}_{\mathbf{h}}\{R_i^n\} \approx \log_2(1+\frac{Pr_{ii}^{-\alpha}}{\bar{I_i}}),
\end{equation}
where $\mathbb{E}_{\mathbf{h}}\{\cdot\}$ represents the expectation taken over small scale channel fading, $\bar{I_i} = P\sum_{j=1,j \neq i}^{B}r_{ij}^{-\alpha} $ is the average total power of the inter-cluster interference.

Since channel fading and user location are distributed independently, the average data rate per unit bandwidth per second  of the N-Coop link taken over both channel fading and user location
can be obtained as
\begin{equation}
	\label{equ.bar_R_i^n}
	\begin{split}
		&\bar{R_i^n} =\mathbb{E}_{\textbf{p}} \{ \log_2(1+\frac{Pr_{ii}^{-\alpha}}{\bar{I_i}}) \},
	\end{split}
\end{equation}
where $\mathbb{E}_{\mathbf{p}}\{\cdot\}$ denotes the expectation taken over user location.

Because the joint probability density function (pdf) of the distances among D2D users is hard to obtain, we introduce the first order approximation to derive
the expression of $\bar{R_i^n}$. Specifically, for a random variable $X$, the expectation of a function of $X$, $\varphi( X)$, can be approximated as
\cite{approx}
\begin{equation}
	\label{equ.1_approx}
	\begin{split}
		\mathbb{E}\{\varphi( X)\}& = \mathbb{E}\{\varphi(\mu_x +  X - \mu_x )\}  \\
		&\approx \mathbb{E}\{\varphi(\mu_x) + \varphi'(\mu_x)( X-\mu_x)\} \\
		&= \varphi(\mu_x),
	\end{split}
\end{equation}
where $\mu_x = \mathbb{E}\{ X\}$ and the approximation is accurate when the variance of $X$ is small. With this approximation, $\bar{R_i^n}$ in (\ref{equ.bar_R_i^n}) can be approximated as
\begin{equation}
	\label{equ.bar_R_i^n2}
	\bar{R_i^n} \approx \log_2(\mathbb{E}_{\textbf{p}}\{Pr_{ii}^{-\alpha}+\bar{I_i}\}) - \log_2(\mathbb{E}_{\textbf{p}}\{\bar{I_i}\}).
\end{equation}
The pdf of the signal link distance $r_{ii}$ can be obtained from \cite{PDF}
by variable substitution $r=r_{ii}/D$ as
\begin{eqnarray}
	\label{equ.g_r_ii}
	g(r)=\frac{1}{D}
	\begin{cases}
		2r(r^2-4r+\pi), &0\leq r<1 \\
		8r\epsilon-2r(r^2+2) \\+4r(\arcsin(\frac{1}{r})\\-\arccos(\frac{1}{r})), &1\leq r<\sqrt{2}\\
	\end{cases}.
\end{eqnarray}

To simplify the analysis, the interference link distance $r_{ij}$ is assumed to have the same distribution $f(r)$, where $r=r_{ij}/D$. We can derive the pdf of the interference link distance $r_{ij}$ as follows.
\begin{figure}[!t]
	\centering

	\includegraphics[width=0.3\textwidth]{Appendix}\\
	\caption{The distance of users in two adjacent cluster}\label{fig.6}
\end{figure}

Denote the position of the DR in the $i$th cluster as $(x_i,y_i)$ and the position of the DT in the $j$th cluster as $(x_j,y_j)$, as illustrated in Fig. \ref{fig.6}. The link distance between them normalized by the cluster side $D$ can be expressed as $d_{ij} = \frac{\sqrt{(x_i-x_j)^2 + (y_i-y_j)^2}}{D} =\sqrt{(\Delta x)^2 + (\Delta y)^2}$, where $\Delta x = \frac{x_i-x_j}{D}$ and $\Delta y = \frac{y_i-y_j}{D}$. The pdf of $|\Delta y|$ can be obtained according to \cite{PDF} as
\begin{eqnarray}
	\label{equ.dy.pdf}
	p_{\Delta y} (v)=
	\begin{cases}
		{2(1-v)}, &0 \leq v \leq 1 \\
		0, &\text{otherwise}\\
	\end{cases}.
\end{eqnarray}
Analogically, the pdf of $|\Delta x|$ can be obtained as
\begin{eqnarray}
	\label{equ.dx.pdf}
	p_{\Delta x} (u)=
	\begin{cases}
		1-|1-u|, &0 \leq u \leq 2 \\
		0, &\text{otherwise}\\
	\end{cases}.
\end{eqnarray}

Then, the cumulative probability distribution function (cdf) of $d_{ij}$ is
\begin{equation}
	\label{equ.r.cdf}
	\begin{split}
		F_d(r) &= \mathbb{P} \{d_{ij} \leq r\} =  \mathbb{P}  \{\sqrt{|\Delta x|^2 + |\Delta y|^2} \leq r \} \\
		&= \iint du dv p_{\Delta x,\Delta y}(u,v) \\
		&\stackrel{(a)}{=} \iint du dv p_{\Delta x} (u) p_{\Delta y} (v),
	\end{split}
\end{equation}
where (a) comes from the fact that $|\Delta x|$ and $|\Delta y|$ are independent random variables.

When $0 \leq r \leq 1$, from \eqref{equ.dx.pdf} and \eqref{equ.dy.pdf}, we can obtain $p_{\Delta x,\Delta y}(u,v) = 2u(1-v)$, where $0 \leq u,v \leq 1$. Then, the cdf in  \eqref{equ.r.cdf} can be derived as
\begin{equation}
	\label{equ.r.cdf.1}
	\begin{split}
		F_d(r) &=  \iint du dv p_{\Delta x,\Delta y}(u,v) \\
		&= \int_{0}^{r}dv \int_{0}^{\sqrt{r^2-v^2}}2u(1-v) du \\
		&= \frac{2}{3}r^3 - \frac{1}{4}r^4,
	\end{split}
\end{equation}

When $1 \leq r \leq \sqrt{2}$, we have
\begin{eqnarray}
	\nonumber
	\label{equ.dxdy.pdf2}
	p_{\Delta x,\Delta y}(u,v)=
	\begin{cases}
		2u(1-v), &0 \leq u \leq 1, 0 \leq v \leq 1 \\
		2(2-u)(1-v), &1 \leq u \leq 2, 0 \leq v \leq 1\\
		0, &\text{otherwise}\\
	\end{cases},
\end{eqnarray}
and the corresponding cdf can be derived as
\begin{equation}
	\label{equ.r.cdf.2}
	\begin{split}
		&F_d(r)= \\
		&\int_{0}^{\sqrt{r^2-1}}dv \left( \int_{0}^{1} 2u(1-v) du +\int_{1}^{\sqrt{r^2-v^2}}2(2-u)(1-v) du \right)\\
		&+\int_{\sqrt{r^2-1}}^{1}dv \int_{0}^{\sqrt{r^2-v^2}}2u(1-v) du\\
		&= \frac{5}{4} + \epsilon \left(  -2r^2+(1+r^2)\epsilon + \frac{2}{3}\epsilon^2 -\frac{1}{2}\epsilon^2   \right) \\
		& + r^2 \left( 2 \sin^{-1}(\frac{3}{r}) + \frac{1}{2} - \frac{4}{3}r \right) ,
	\end{split}
\end{equation}
where $\epsilon \triangleq \sqrt{r^2-1}$.

Analogically, when $\sqrt{2} \leq r \leq 2 $,  the cdf can be derived as
\begin{equation}
	\label{equ.r.cdf.3}
	\begin{split}
		F_d(r) &=  \int_{0}^{1}dv   \int_{0}^{1} 2u(1-v) du\\
		& +\int_{0}^{1}dv  \int_{1}^{\sqrt{r^2-v^2}}2(2-u)(1-v) du \\
		&= -\frac{11}{12} + 2\epsilon - \frac{r^2}{2} + 2r^2\sin^{-1}(\frac{1}{r}) + \frac{4}{3}(\epsilon^3-r^3).
	\end{split}
\end{equation}

When $ 2  \leq r \leq \sqrt{5} $, the cdf can be derived as
\begin{equation}
	\label{equ.r.cdf.4}
	\begin{split}
		F_d(r) &= 1- \int_{\sqrt{r^2-4}}^{1}dv\int_{\sqrt{r^2-v^2}}^{2} 2u(1-v) du \\
		&= -\frac{45}{12} - \frac{r^2}{2} - \frac{r^4}{2} -2r^2 \left(   \sin^{-1}(\frac{\xi}{r}) - \sin^{-1}(\frac{1}{r}) \right)\\
		& + r^2\xi + 2\epsilon -\frac{1}{3} \xi^3 + \frac{4}{3} \epsilon^3 +\frac{1}{4}\xi,
	\end{split}
\end{equation}
where  $\xi \triangleq \sqrt{r^2-4}$.

Finally, by combining \eqref{equ.r.cdf.1} \eqref{equ.r.cdf.2} \eqref{equ.r.cdf.3} and \eqref{equ.r.cdf.4}, and considering the pdf $f(r) = \frac{dF(r)}{dr}$, the pdf of the interference link distance $r_{ij}$ is
\begin{eqnarray}
	\label{equ.f_r_ij}
	f(r)=\frac{1}{D}
	\begin{cases}
		2r^2-r^3, &0\leq r<1 \\
		2r-4r^2+2r^3\\
		-2r\epsilon+\frac{2r}{\epsilon}\\
		-\frac{2r^3}{\epsilon}+4r\arcsin(\frac{\epsilon}{r}), &1\leq r<\sqrt{2}\\
		4r\epsilon+4r\arcsin(\frac{1}{r})\\
		-r-4r^2, &\sqrt{2}\leq r<2\\
		-5r-r^3+4r\epsilon\\
		-4r\arcsin(\frac{\xi}{r})-\arcsin(\frac{1}{r})\\
		-\frac{4r}{\xi}+r\xi+\frac{r^3}{\xi}, &2\leq r<\sqrt{5}\\
	\end{cases}
\end{eqnarray}
where $r = r_{ij}/D$, $\epsilon \triangleq \sqrt{r^2-1}$, and $\xi \triangleq \sqrt{r^2-4}$.

Since the interference generated by DTs far away from the DR can be ignored due to pathloss, we only consider dominant interference generated from the nearest eight clusters around the $i$th cluster as shown in Fig. \ref{fig.ici-d2d}. Then, by substituting the pdf of interference and signal link distance into \eqref{equ.bar_R_i^n2}, we can obtain the average data rate per unit bandwidth per second of the N-Coop link in the $i$th cluster as
\begin{equation}
	\label{a:equ.bar_R_i^n3}
	\begin{split}
		\bar{R_i^n} &\approx \log_2\left(\int_{0}^{\sqrt{2}D}Pr_{ii}^{-\alpha}g(\frac{r_{ii}}{D})dr_{ii} \right.\\ &\left.+8\int_{0}^{\sqrt{5}D}Pr_{ij}^{-\alpha}f(\frac{r_{ij}}{D})dr_{ij}\right)\\
		& - \log_2\left(8\int_{0}^{\sqrt{5}D}Pr_{ij}^{-\alpha}f(\frac{r_{ij}}{D})dr_{ij} \right) \\
		&= \log_2\left(\int_{0}^{\sqrt{2}}Pr^{-\alpha}g(r)dr +8\int_{0}^{\sqrt{5}}Pr^{-\alpha}f(r)dr\right) \\
		&- \log_2\left(8\int_{0}^{\sqrt{5}}Pr^{-\alpha}f(r)dr\right) \\
		&= \log_2(Q_1(\alpha))-\log_2(Q_2(\alpha))-3,
	\end{split}
\end{equation}
This proves Proposition \ref{p:1}.

\section{Proof of Proposition \ref{p:2}}
\label{a:2}
In \emph{Mode 1}, the cooperative DTs jointly  transmit the requested files to the Coop users with zero-forcing beamforming, which is of low complexity and hence practical. Then, the SINR at the DR of the active Coop link in the $i$th cluster can be expressed as
\begin{equation}
	\label{equ.SINR_R^c}
	\begin{split}
	\gamma_i^c &= \frac{P\| \textbf{h}_{i}\|^2\delta_i}{\sigma^2} \\
	&\approx  \frac{P\sum_{j=1}^{B}r_{ij}^{-\alpha}| h_{ij}|^2}{B\sigma^2} ,
	\end{split}
\end{equation}
where $\textbf{h}_{i}=[\sqrt{r_{i1}^{-\alpha}}h_{i1},\sqrt{r_{i2}^{-\alpha}}h_{i2},...,\sqrt{r_{iB}^{-\alpha}}h_{iB}]$ is the composite channel vector
between all DTs and the DR, $0\le \delta_i \le 1$, and a larger value of $\delta_i$ indicates a better orthogonality
between $\textbf{h}_{i}$ and $\textbf{h}_{j}$ for $i \ne j$. The approximation comes from the fact $\delta_i \approx (BN^t-B+1)/{B}={1}/{B}$ \cite{ZQ.TVT13}, where $N^t$ is the number of antennas per DT and $N^t=1$ in this paper. This approximation is accurate when the variance of $\delta_i$ is small.

Using the same approximation as deriving (\ref{equ.E_h_R_i^n}), the average data rate per unit bandwidth per second of the Coop link  is obtained as
\begin{equation}
	\label{equ.bar_R_i^c}
	\begin{split}
		&\bar{R_i^c} \approx \mathbb{E}_{\textbf{p}} \{ \log_2(1+\frac{P\sum_{j=1}^{B}r_{ij}^{-\alpha}}{B\sigma^2}) \}.
	\end{split}
\end{equation}

By applying the first-order approximation in (\ref{equ.1_approx}), using (\ref{equ.g_r_ii}) and (\ref{equ.f_r_ij}), and only considering dominant signal, we can obtain the average data rate per unit bandwidth per second for the Coop link as
\begin{equation}
\label{equ.bar_R_i^c2_app}
\begin{split}
\bar{R_i^c} &\approx \log_2\left(B\sigma^2 +8\int_{0}^{\sqrt{5}D}Pr_{ij}^{-\alpha}f(\frac{r_{ij}}{D})dr_{ij} \right.\\
&\left. + \int_{0}^{\sqrt{2}D}Pr_{ii}^{-\alpha}g(\frac{r_{ii}}{D})dr_{ii} \right) - \log_2\left( B \sigma^2 \right) \\
&=\log_2(1+\frac{PD^{-\alpha}}{B\sigma^2}Q_1(\alpha)).
\end{split}
\end{equation}
This proves Proposition \ref{p:2}.

\section{Proof of Proposition \ref{p:3}}
\label{a:3}
Denote the number of users in the $i$th cluster who request files in $\mathcal{G}_k$ as $n_{ik}$,  $1 \leq k \leq K_0$, $1
\leq i \leq B$.

Since the users request files independently, the probability that the combination of the numbers of users in the $i$th cluster who request files in each file group is $\{n_{i1},n_{i2},...,n_{i{K_0}}\} \triangleq \mathcal{N}_{i}$
can be derived as
\begin{equation}
	\label{equ.P_N_i}
	\begin{split}
		{p_{\mathcal{N}_{i}}}& = \prod_{m=1}^{K_0}C_{K-\sum_{j=1}^{m-1}n_{ij}}^{n_{im}}\prod_{k=1}^{K_0} (P_k)^{n_{ik}} \\
		&\overset{(a)}{=} \frac{K!\prod_{k=1}^{K_0} (P_k)^{n_{ik}}}{\prod_{j=1}^{K_0}n_{ij}!},
	\end{split}
\end{equation}
where $(a)$ comes from $C_{n}^{m}C_{n-m}^{k}=\frac{n!}{m!k!(n-m-k)!}$.

Only when all $B$ clusters hit a file group $\mathcal{G}_k$ and $k \leq K$ (i.e., $n_{ik}>0$  is satisfied for $k \leq K$  and any $i$, $1 \leq i \leq
B$), the users requesting the files within $\mathcal{G}_k$ are Coop users, and we call $\mathcal{G}_k$ a \emph{hit file group}.

The number of Coop users for
all hit file groups can be obtained as
\begin{equation}
	\label{equ.N_c}
	\begin{split}
		&{N^c} = \sum_{k=1}^{K}\sum_{i=1}^{B}\zeta(k) n_{ik},
	\end{split}
\end{equation}
where $\zeta(k)=\lceil \sum_{i=1}^{B} u(n_{ik})-B+1) \rceil ^+$ ($k\leq K$) indicates that whether $\mathcal{G}_k$ is a \emph{hit file group}, the function $u(x)=1$ when
$x>0$, otherwise $u(x)=0$, and $\lceil \Lambda \rceil ^+= \max(\Lambda,0)$.

Denote $\mathcal{N}=\{\mathcal{N}_1,\mathcal{N}_2,...,\mathcal{N}_B\}$, and
$\Phi_\mathcal{N}=\{\mathcal{N}|n_{ik}\geq 0,\sum_{k=1}^{K_0}n_{ik}=K\}$ represents a set of all possible combinations of $\mathcal{N}$. Then, by taking average of $N^c$ in
(\ref{equ.N_c}) over $\Phi_\mathcal{N}$, we can derive the average number of Coop users as
\begin{equation}
	\label{equ.bar_N_c_app}
	\begin{split}
		\bar{N^c}

		&=  \sum_{\Phi_\mathcal{N}}\prod_{i=1}^{B}{p_{\mathcal{N}_{i}}}{N^c}  \\
		&= \sum_{\Phi_\mathcal{N}}\prod_{i=1}^{B}\frac{K!\prod_{k=1}^{K_0} (P_k)^{n_{ik}}}{\prod_{j=1}^{K_0}n_{ij}!} \sum_{k=1}^{K}\sum_{i=1}^{B}\zeta(k) n_{ik} .
	\end{split}
\end{equation}
The number of Coop users can be computed accurately using (\ref{equ.bar_N_c_app}). However, the cardinality of $\Phi_\mathcal{N}$ is $K_0^{(K-1)B}$, and hence the computational complexity
exponentially increases with $K$. For example, when $K=K_0=15$ and $B=9$, we obtain $|\Phi_\mathcal{N}| = 1.35 \times10^{140}$.

In the sequel, we seek an alternative solution.

Noticing that the probability  that multiple \emph{hit file groups} exist simultaneously decreases with the growth of the number of {hit file groups}, we
only consider the case where only one or two hit file groups exist to approximate the average number of Coop users as follows. This approximation is accurate when the number of clusters is large.

The probability that a \emph{hit file group} only contains $\mathcal{G}_k$ can be obtained from (\ref{equ.P^h_k}) as $\prod_{j=1,j\neq k}^{K}(1-{(P^h_j)}^B){(P^h_k)}^{B}$. As a result, the probability of $n_{ik}=m$ ($1 \leq m \leq K$) when a
\emph{hit file group} only contains $\mathcal{G}_k$  is $C_{K}^{m}(P_{k})^m(1-P_{k})^{(K-m)}/P^h_k$. Then, the average number of Coop
users in the cases where only one hit file group exists can be derived as
\begin{equation}
	\label{N_c_appr_1}
	\begin{split}
		\bar{N^c_1} &= \sum_{k=1}^{K} \prod_{j=1,j\neq k}^{K} (1-{(P^h_j)}^B){(P^h_k)}^{B}\cdot\\ &B\sum_{m=1}^{K}\frac{C_{K}^{m}(P_{k})^m(1-P_{k})^{(K-m)}}{P^h_k}m\\
		&= \sum_{k=1}^{K} \prod_{j=1,j\neq k}^{K}(1-{(P^h_j)}^B){(P^h_k)}^{B-1}BP_kK.
	\end{split}
\end{equation}

The probability that there only exist two \emph{hit file groups}, $\mathcal{G}_{k_1}$ and $\mathcal{G}_{k_2}$, can be obtained from (\ref{equ.P^h_k}) as $\prod_{j=1,j\neq k_1,k_2}^{K}(1-{(P^h_j)}^B)({P^h_{k_1}})^{B}({P^h_{k_2}})^B$. Consequently, the
probability of $n_{ik_1}=m_1$ and $n_{ik_2}=m_2$ (where $1 \leq m_1 \leq K$, $1 \leq m_2 \leq K$ and $2 \leq m_1+m_2=m \leq K$) when the \emph{hit file groups} only contain $\mathcal{G}_{k_1}$ and $\mathcal{G}_{k_2}$  is $C_{m}^{m_1}C_{m-m_1}^{m_2}  \frac{{(p_{k_1})}^{m_1}{(p_{k_2})}^{m_2}}{P^h_{k_1}P^h_{k_2}}$. Then, the average number of Coop
users in the case where only two hit file groups exist can be derived as
\begin{equation}
	\label{N_c_appr_2}
	\begin{split}
		\bar{N^c_2} &=\sum_{\Phi_{k_1k_2}} \prod_{j=1,j\neq k_1,k_2}^{K}\left(1-{(P^h_j)}^B\right)({P^h_{k_1}})^{B}({P^h_{k_2}})^B \cdot\\
		&B  \sum_{m=2}^{K}\sum_{\Phi_{m_1m_2}} C_{m}^{m_1}C_{m-m_1}^{m_2}  \frac{{(p_{k_1})}^{m_1}{(p_{k_2})}^{m_2}}{P^h_{k_1}P^h_{k_2}} m\\
		&=\sum_{\Phi_{k_1k_2}}\prod_{k=1,k\neq i,j}^{K}\left(1-{(P^h_k)}^B\right)({P^h_i}{P^h_j})^{B-1} \cdot \\
		&B  \sum_{m=2}^{K}\sum_{\Phi_{m_1m_2}}\frac{mm!}{m_1!m_2!}{(p_{k_1})}^{m_1}{(p_{k_2})}^{m_2},
	\end{split}
\end{equation}
where $\Phi_{k_1k_2} = \{k_1,k_2|0\leq k_1,k_2\leq K,i\neq j\}$, $\Phi_{m_1m_2}=\{m_1,m_2|m_1+m_2=m\}$, whose  cardinality are  $K^2$ and $m^2$ ($m \leq K$), respectively.

By combining \eqref{N_c_appr_1} and \eqref{N_c_appr_2}, the average number of Coop users is approximated as
\begin{equation}
	\label{equ.bar_N_c2_app}
	\begin{split}
		&\bar{N^c} \approx \bar{N^c_1} +\bar{N^c_2},
	\end{split}
\end{equation}
which can be obtained much easier than \eqref{equ.bar_N_c_app}.

This proves Proposition \ref{p:3}.

\section{Proof of Proposition \ref{p:4}}
\label{a:4}

By simply subtracting $\bar{R^n_i}$ from $\bar{R^c_i}$ and considering $\bar{I}_i \gg \sigma^2$,
we can obtain that
\begin{equation}
	\label{equ.difference}
	\begin{split}
		\bar{R^c_i}-\bar{R^n_i} &= \mathbb{E}\{\log_2(1+\frac{P\sum_{j=1}^{B}r_{ij}^{-\alpha}}{B\sigma^2})\} \\ &-\mathbb{E}\{\log_2(1+\frac{Pr_{ii}^{-\alpha}}{\bar{I_i}})\} \nonumber \\
		& =\mathbb{E}\{\log_2(\frac{\bar{I_i}}{B\sigma^2})\}. \nonumber\\
	\end{split}
\end{equation}
When $\log_2(\frac{\bar{I_i}}{B\sigma^2})\geq 0$ (i.e., $\bar{I_i}\geq B\sigma^2$), $\bar{R^c_i}-\bar{R^n_i}\geq 0$.

This proves Proposition \ref{p:4}.

\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


text:
Natural and Mathematical Sciences
Measurement of the top quark Yukawa coupling from t¯t kinematic distributions in the dilepton final state in proton-proton collisions at √s = 13 TeV
dc.contributor.author Sirunyan, A. M.
dc.contributor.author Işıldak, Bora
dc.identifier.issn 2470-0010 en_US
dc.identifier.uri https://journals.aps.org/prd/abstract/10.1103/PhysRevD.102.092013
dc.description.abstract A measurement of the Higgs boson Yukawa coupling to the top quark is presented using proton-proton collision data at root s = 13 TeV, corresponding to an integrated luminosity of 137 fb(-1), recorded with the CMS detector. The coupling strength with respect to the standard model value, Y-t, is determined from kinematic distributions in t (t) over bar final states containing ee, mu mu, or e mu pairs. Variations of the Yukawa coupling strength lead to modified distributions for t (t) over bar production. In particular, the distributions of the t (t) over bar mass of the a system and the rapidity difference of the top quark and antiquark are sensitive to the value of Y-t. The measurement yields a best fit value of Y-t = 1.16(-0.35)(+0.24), bounding Y-t < 1.54 at a 95% confidence level. en_US
dc.description.sponsorship BMBWF and FWF (Austria); FNRS and FWO (Belgium); CNPq, CAPES, FAPERJ, FAPERGS, and FAPESP (Brazil); MES (Bulgaria); CERN; CAS, MoST, and NSFC (China); COLCIENCIAS (Colombia); MSES and CSF (Croatia); RIF (Cyprus); SENESCYT (Ecuador); MoER, ERC IUT, PUT and ERDF (Estonia); Academy of Finland, MEC, and HIP (Finland); CEA and CNRS/IN2P3 (France); BMBF, DFG, and HGF (Germany); GSRT (Greece); NKFIA (Hungary); DAE and DST (India); IPM (Iran); SFI (Ireland); INFN (Italy); MSIP and NRF (Republic of Korea); MES (Latvia); LAS (Lithuania); MOE and UM (Malaysia); BUAP, CINVESTAV, CONACYT, LNS, SEP, and UASLP-FAI (Mexico); MOS (Montenegro); MBIE (New Zealand); PAEC (Pakistan); MSHE and NSC (Poland); FCT (Portugal); JINR (Dubna); MON, RosAtom, RAS, RFBR, and NRC KI (Russia); MESTD (Serbia); SEIDI, CPAN, PCTI, and FEDER (Spain); MOSTR (Sri Lanka); Swiss Funding Agencies (Switzerland); MST (Taipei); ThEPCenter, IPST, STAR, and NSTDA (Thailand); TUBITAK and TAEK (Turkey); NASU (Ukraine); STFC (United Kingdom); DOE and NSF (USA). Individuals have received support from the Marie-Curie program and the European Research Council andHorizon 2020Grants, ContractNo. 675440, No. 752730, and No. 765710 (European Union); the Leventis Foundation; the A. P. Sloan Foundation; the Alexander von Humboldt Foundation; the Belgian Federal Science Policy Office; the Fonds pour la Formation a la Recherche dans l'Industrie et dans l'Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWTBelgium); the F. R. S.-FNRS and FWO (Belgium) under the "Excellence of Science-EOS"-be.h Project No. 30820817; the Beijing Municipal Science and Technology Commission, No. Z191100007219010; the Ministry of Education, Youth and Sports (MEYS) of the Czech Republic; the Deutsche Forschungsgemeinschaft (DFG) under Germany's Excellence Strategy-EXC 2121 "Quantum Universe"-390833306; the Lendulet ("Momentum") Program and the Janos Bolyai Research Scholarship of the Hungarian Academy of Sciences, the New National Excellence Program UNKP, the NKFIA research Grants No. 123842, No. 123959, No. 124845, No. 124850, No. 125105, No. 128713, No. 128786, and No. 129058 (Hungary); the Council of Science and Industrial Research, India; the HOMING PLUS program of the Foundation for Polish Science, cofinanced from European Union, Regional Development Fund, the Mobility Plus program of the Ministry of Science and Higher Education, the National Science Center (Poland), Contracts Harmonia 2014/14/M/ST2/00428, Opus 2014/13/B/ST2/02543, 2014/15/B/ST2/03998, and 2015/19/B/ST2/02861, Sonata-bis 2012/07/E/ST2/01406; the National Priorities Research Program by Qatar National Research Fund; the Ministry of Science and Higher Education, Project No. 02.a03.21. 0005 (Russia); the Tomsk Polytechnic University Competitiveness Enhancement Program; the Programa Estatal de Fomento de la Investigacion Cientifica y Tecnica de Excelencia Maria de Maeztu, grantMDM-20150509 and the Programa Severo Ochoa del Principado de Asturias; the Thalis and Aristeia programs cofinanced by EU-ESF and the Greek NSRF; the Rachadapisek Sompot Fund for Postdoctoral Fellowship, Chulalongkorn University and the ChulalongkornAcademic into Its 2ndCenturyProject Advancement Project (Thailand); the Graduate Research Fellowship Program of the National Science Foundation, Grant No. DGE-141911; the Kavli Foundation; the Nvidia Corporation; the SuperMicro Corporation; the Welch Foundation, Contract No. C-1845; and the Weston Havens Foundation (USA).
dc.publisher American Physical Society en_US
dc.relation.ispartof Physical Review D
dc.title Measurement of the top quark Yukawa coupling from t¯t kinematic distributions in the dilepton final state in proton-proton collisions at √s = 13 TeV en_US
dc.description.version Publisher version en_US
dc.contributor.authorID (ORCID 0000-0002-0283-5234 & YÖK ID 124605) Işıldak, Bora
dc.creator The CMS Collaboration
dc.identifier.volume 102 en_US
dc.identifier.doi 10.1103/PhysRevD.102.092013 en_US
Measurement of the top quark ...


The final round of the Midweek MTB Madness at Lee Quarry, Bacup on Thursday 23rd July.
Well done to Andrew Lindley, just pipped at the finish line into third place. Andy also came third overall in the series.
Mark McPhillips came ninth after riding some interesting lines to the amusement of the marshals. He also picked up a spot prize worth £50.
Anthony Jevons was 17th in the enthusiasts class in his second XC race.
Maggie was second in the women's category, and first in the series.
There has been an excellent turnout by Stockport Clarion riders for all 5 rounds. Looking forward to next year. Big thanks to the organisers.


...I think it's perfectly acceptable for folk to have welcomed the appointment, I think it's perfectly acceptable that folk gave him a chance. I did that myself, but his record has been very poor so it's perfectly acceptable to highlight that now.
Few more days now. Bye.
I am always ready to give anyone a chance to prove themselves at the club, it isn't fawning enthusiasm, you just hope when someone is appointed or a player is signed that it works out for themselves and the club.
The biggest issue with Walsh as I see it is that he either hasn't asserted himself sufficiently as DOF and allowed muddled decision making, or, if he has asserted his opinion, that it has proven to be clearly wrong in the mix and quality of signings made.
Us mere mortals on GOT are just giving an opinion with no consequences if it's right or wrong. Perhaps Elstone who takes the biggest non-footballing wage out of the club, and Moshiri and Kenwright, as the effective custodians, are responsible for the outcome of hiring a man with no experience in the DOF role, and evidently no clear understanding of it either.
We talk about people in black and white terms, absolute success or failure, but the reality is very many shades of grey. Walsh isn't fully culpable, of course not, but he is an integral part of a failing system and a cog that we can do without. I'll welcome Brands as a person with experience in the role we are filling, you hope someone will get it right eventually and his credentials seem reasonably good to me.
We talk about people in black and white terms, but the reality is very made shades of grey. Walsh isn't fully culpable, of course not, but he is an integral part of a failing system and a cog that we can do without. I'll welcome Brands as a person with experience in the role we are filling, you hope someone will get it right eventually and his credentials seem reasonably good to me.
I'm just glad I knew nothing about Walsh, and I know even less about Brands.
Ignorance is bliss. I can't lose.
I have no issue with my comment. I still feel a DOF is essential in a league where, good or bad, clubs are turning over managers. Having a stable presence is essential.
I think all of his signings at Leicester the season after they won the league flopped too. So not just with us. Time to go I'd say.
..falling off the page again.
I'm just reminding people to think twice before they hype up Marcel " Never heard of him till a few weeks ago " Brands.
Exactly, some lads here talk as if they work with him for decades.
..if he's worst than Walsh we'll be in dire straights. The reality is it's difficult to anticipate what a new DoF will be like, but it's not difficult to assess what the current one is like.
You`d really have to go some to be worse than Walsh though.
Can't believe he is still here. He is just as culpable as Koeman for the squandering of 200m, which has left us with a squad that is not only unbalanced but short in quality. I have raised this question before, how many clubs have had a successful DOF/Manager combination?
United in their period of dominance didn't have a DOF. They had an outstanding manager and a very astute chief executive in David Gill. If we were to replace elstrone with a decent chief executive, would there be any need for a DOF?
I think it's just a case of Everton being Everton, and exhausting all possible poor options before settling on the correct one.
He's wasted the Stones and Lukaku cash. Now we have no quality players left to sell and a team of over payed players.


text:
Marathon to miejscowość (ang. town) w Kanadzie, w prowincji Ontario, w dystrykcie Thunder Bay.

Powierzchnia Marathon to 170,48 km².
Według danych spisu powszechnego z roku 2001 Marathon liczy 4416 mieszkańców (25,90 os./km²).

Dystrykt Thunder Bay

redpajama_set_name: RedPajamaWikipedia


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Posted: July 10, 2019 02:24 PM CDT
Updated: July 10, 2019 02:24 PM CDT
People cheer as members of the US Women's National Soccer Team travel down the "Canyon of Heroes" in a ticker tape parade on July 10, 2019, in New York City. [ + - ]
Members of the United States Women's National Soccer Team are honored at a ceremony at City Hall on July 10, 2019, in New York City. The honor followed a ticker tape parade up lower Manhattan's "Canyon of Heroes" to celebrate their gold medal victory in the 2019 Women's World Cup in France. [ + - ]
Soccer fans line the parade route as members of the United States Women's National Soccer Team are honored at a ceremony at City Hall on July 10, 2019, in New York City. [ + - ]
Megan Rapinoe and members of the United States Women's National Soccer Team are honored at a ceremony at City Hall on July 10, 2019, in New York City. [ + - ]
(L to R) President of the USA Soccer Federation Carlos Cordeiro, New York Mayor Bill de Blasio, his Wife Chirlane McCray, Megan Rapinoe, Allie Long, and Alex Morgan celebrate during the U.S. Women's National Soccer Team Victory Parade and City Hall Ceremony on July 10, 2019, in New York City. [ + - ]
The USA Women's National Soccer Team celebrate during a Victory Ticker Tape Parade down the Canyon of Heroes on July 10, 2019, in New York City. [ + - ]
Julie Ertz celebrates during the U.S. Women's National Soccer Team Victory Parade and City Hall Ceremony on July 10, 2019, in New York City. [ + - ]
Fans celebrate during a Victory Ticker Tape Parade for the U.S. Women's National Soccer Team down the Canyon of Heroes on July 10, 2019, in New York City. [ + - ]
Alex Morgan, Ali Krieger and Megan Rapinoe celebrate during a Victory Ticker Tape Parade for the U.S. Women's National Soccer Team down the Canyon of Heroes on July 10, 2019, in New York City. [ + - ]
People cheer as members of the US Women's National Soccer Team travel down the "Canyon of Heroes" in a ticker tape parade on July 10, 2019, in New York City. The team defeated the Netherlands 2-0 Sunday in France to win the 2019 Women's World Cup. [ + - ]
Fans celebrate during a Victory Ticker Tape Parade for the U.S. Women's National Soccer Team down the Canyon of Heroes on July 10, 2019, in New York City. The USA defeated the Netherlands on Sunday to win the 2019 FIFA Women's World Cup in France. [ + - ]
A fan looks on during the U.S. Women's National Soccer Team Victory Parade and City Hall Ceremony on July 10, 2019, in New York City. [ + - ]
Megan Rapinoe (C) and Ashlyn Harris (R) celebrate while riding on a float during the U.S. Women's National Soccer Team Victory Parade through the Canyon of Heroes on July 10, 2019, in New York City. The team defeated the Netherlands 2-0 Sunday in France to win the 2019 Women's World Cup. [ + - ]
The U.S. Women's National Team was honored Wednesday in New York City with a ticker tape parade and ceremony following their World Cup victory over the Netherlands on Sunday in France.
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In 'Million Dollar Listing' Lessons to Improve Your Online Advertising, I explained how important it is to "stage" your offer so it matches exactly what the market wants. In this article, I'll explain how to do that ("stage" your offer) using the information readily available in Google AdWords.
Once you dig into the data, you'll find that you actually know much more about your prospect than you probably thought you did. The key to this process is to first assess all the data you have, and then create the most compelling offer for each of your prospect segments.
The most important and most revealing source of data is the keyword phrase typed into Google. The keyword tells you what your prospect is looking for. If the keyword is "how to run barefoot" then that person is most likely looking for information in the form of an article or video. If the keyword is "barefoot running clinic" then that person is interested in taking a class to learn how to run barefoot.
See the difference? If you're selling a barefoot running clinic, then the keyword "how to run barefoot" is probably not going to work because the offer is not in line with what the person is looking for. Use the keyword phrase to figure out exactly what your prospect is looking for. Then match your ad to the intent of the keyword phrase.
In a lot of markets, the time of day when your prospect is searching online plays a big role. For example, I tend to search for a restaurant for lunch around 11am. Clearly, an ad with a special lunch offer will work better around 11am than at 5pm.
What else do you know about your prospect based on the day or time of day of the search? Can you tailor your offer differently for workdays versus weekends? Does your prospect research at night and make purchases during office hours? If you aren't sure, then dig into your historical data to see when your ads perform the best and when they perform the worst.
The physical location of the searcher obviously gives you valuable insight. One of my private clients analyzed their entire database of US customers to determine their top performing zip codes. And based on this information, we now bid differently and tailor the ad messages accordingly.
How could you change your ad copy if you knew where the prospect was searching? Would you offer a different product based on location? In AdWords, you DO know where the person is located and you CAN bid and/or change your offer to improve response.
The device now needs to be a factor in your advertising decisions. When someone searches "sushi" on their smartphone, then what are they really looking for? They are looking for a sushi restaurant. If it's 11am, then they are probably looking for a sushi lunch special. And if they are on the Upper West Side of NYC, then they are looking for a sushi lunch special in the Upper West Side. See how much info you already know just from someone searching "sushi" on their mobile device?
The final piece of data is only available if the person previously visited your website. This is called an Audience in Google AdWords and you can use this to further refine your ad copy. Let's use the same example above.
Your prospect is on a mobile device searching for "sushi" at 11am on the Upper West Side of NYC. Now with Audience data, it's possible to know whether or not this person already visited your website. In fact, you may have an Audience only for previous customers and you could use that information to create an ad only for existing customers.
As you can see, when you combine ALL 5 of these data points, you start to get a very clear picture about who you're targeting with your ads. Next, it's up to you to create the perfect ad that will outperform all of your competitors.


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Now reading: 5 Signs You Are in a Toxic Relationship
5 Signs You Are in a Toxic Relationship
Being in a relationship can be a blessing or a curse. You either feel inspired daily, loved incredibly and whole-heartedly supported, or you feel like you're constantly defending yourself, being broken down and fighting more than laughing. Here are five signs you are in a toxic relationship. 1. The Controller No one should be telling you when you're allowed to do something or what you're not allowed to do. Don't put a leash on anyone, and don't let anyone put one on you. 2. The Debbie Downer We all know that negativity is a total downer, but it can be hard to avoid. When it comes to relationships, it's important to keep each other positive, happy and full of light. if you are bringing your partner down, or feel like they're sucking the life out of you with their bad vibes, then you're allowing the relationship to be associated with cynicism. 3. The Liar If you notice your partner is always coming up with white lies to their friends and family, you can bet they're doing it to you as well. And if you're finding yourself doing this, become aware of it and make an effort to stop. No one likes a liar. It can lead to big lies that will certainly affect the relationship. 4. The Perfectionist There's always more weight to lose, more dishes to clean, more projects to complete, more fake smiles to shine. When you are a perfectionist, you expect your significant other to live up to that standard, and that is a surefire way to set them up for failure. Watch for the signs to find out sooner or later if you're dating a perfectionist as well. 5. The Cheater Cheating doesn't need much of an explanation. If you're doing it or your significant other is, there is a huge lack of respect. Do you feel like you are in a toxic relationship? Source: Elite Daily Do you follow us on Instagram? [caption id="attachment_108834" align="alignnone" width="100"] @BodyRockTV[/caption]
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