text:
Give Goods
Government will regret shelving plans to introduce minimum price for alcohol
In his introduction to the Government's 2012 Alcohol Strategy, the Prime Minister was clear: When it comes to the UK's problems with alcohol, he wrote, a real effort must be made to 'get to grips with the root cause of the problem. And that means coming down hard on cheap alcohol'.
'So', wrote Mr Cameron, 'we are going to introduce a minimum unit price. For the first time it will be illegal for shops to sell alcohol for less than this set price per unit'.
'Of course' he continued, 'I know the proposals in this strategy won't be universally popular. But the responsibility of being in government isn't always about doing the popular thing. It's about doing the right thing. Binge drinking is a serious problem. And I make no excuses for clamping down on it'.
Each of these statements is completely correct. But yesterday (17th July), and sixteen months after Mr Cameron's words were published, it was announced that plans to introduce Minimum Unit Price (or MUP) have been shelved.
Addaction firmly believes that this is the wrong decision and one that the Government should and will regret making. Make no mistake, MUP is not the only thing they should be doing, but cheap alcohol causes immense harm and puts huge pressure on our police forces and our hospitals. And quite simply, it is a killer.
That is why, like many others working in the field of public health and alcohol treatment, we are extremely disappointed that a wealth of good evidence (showing how such a measure would save thousands of lives) has been shamefully ignored.
Since its inception, Addaction has worked as part of the Alcohol Responsibility Deal - a coalition of retailers, NGOs, public health providers and the drinks industry which (as its name suggests) has worked to create a more responsible attitude towards alcohol in this country. Central to this aim has been changing the ways in which alcohol is sold.
Of course, there have been many opposing views among members of the Responsibility Deal, and one of the main arguments against MUP has been that it would punish responsible drinkers. Addaction strongly disagrees with this assumption, as the evidence shows that people drinking within recommended guidelines will see no real difference in the amount they spend. Last year, the Prime Minister also agreed with us, writing 'If the minimum price is around 40p a unit, it won't affect the price of a pint in a pub. In fact, pubs may benefit by making the cheap alternatives in supermarkets more expensive'.
Yesterday, many of our colleagues understandably withdrew from the Responsibility Deal, stating that 'it is perfectly clear that MUP has fallen victim to a concerted and shameful campaign of lobbying by sections of the drinks industry who are putting profits before health and public safety'. Addaction agrees that this message is the one that the government is sending, as do others – including the Conservative MP Sarah Wollaston, who accused politicians of 'roll[ing] over to big alcohol lobbyists'.
This cannot be the end of the matter. That's why Addaction will retain its place in the Responsibility Deal to ensure our sector is represented, and so that discussions with government around MUP are quickly resumed. And we also urge councils in England and Wales to quickly instate proposals (such as those currently being considered by Cheshire East) to introduce minimum unit pricing at a local level.
David Cameron was right in his introduction to the government's 2012 strategy. Then, he could see that alcohol is a public health issue, not a business issue. And while it is right that the drinks industry should take far more responsibility when it comes to the problems alcohol causes in the UK, this can only complement decisive legislation from government.
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text:
It's official: Oak Ridge National Laboratory has fastest computer
Oak Ridge's Summit supercomputer, designed to reclaim the title "world's fastest," has officially done so.
The designation came at the ISC High Performance conference in Frankfurt, Germany, according to a news release from Oak Ridge National Laboratory. Eight times more powerful than its predecessor, Titan, the Summit supercomputer reclaims the title from China's Sunway TaihuLight.
"With Summit, researchers will be able to simulate and explore complex phenomena and obtain results in disciplines ranging from quantum materials and chemistry, advanced fission and fusion energy, to bioenergy and foundational biosciences, faster and in greater detail," ORNL Director Thomas Zacharia said in the announcement. "In addition to traditional modeling and simulation, Summit will also serve as an artificial intelligence and deep learning behemoth, capable of analyzing massive amounts of data and automating critical steps of the discovery process."
Summit, built by IBM, is the third computer at Oak Ridge to be ranked number one. In November 2009 and June 2010 the Jaguar system held the title, and Summit's predecessor Titan Cray X was the world's fastest at its November 2012 debut.
China and Switzerland quickly caught up, and Titan dropped to fourth place in the world last year.
During a benchmark test Summit reached 122.3 petaflops, or 122,300 trillion calculations per second, according to the news release. It may be capable of 200 petaflops, eight times Titan's top speed.
Summit has 4,608 servers, each with two 22-core IBM Power9 processors and six NVIDIA Tesla V100 graphics processing unit accelerators, ORNL announced. It runs a standard Linux operating system from Red Hat in Red Hat Enterprise Linux.
It's seen as a big step toward developing the first exascale computer, capable of a billion billion double precision floating point operations per second. Summit has already done some exascale calculations, making more than 1.8 quintillion calculations in a single second in bioenergy and human health research, Zacharia said at its debut.
"This is the first time anyone has broken the exascale barrier," he said then. "Today's Summit also gives us confidence we can deliver on a fully capable exascale computing resource by the year 2021."
State and federal elected officials hailed Summit at its activation and this announcement, including U.S. Sen. Lamar Alexander, R-Tenn.
"Supercomputers cost money, a lot of it," Alexander said in a news release. "And this also would not have happened unless bipartisan majorities in Congress had for several years appropriated record levels of taxpayer dollars to make our country first in the world in supercomputing. President Trump has signed two of those appropriations bills and should include supercomputing as part of his 'America First' agenda. This is one more example of how federal government sponsored research has been a crucial part of making it possible for the United States to produce 24 percent of the world's wealth for just 5 percent of the world's people."
Alexander chairs the Senate subcommittee on Energy and Water appropriations, which funds supercomputing programs.
Summit reportedly cost upward of $200 million. It will be made available to researchers in the U.S. Department of Energy's Innovative and Novel Computational Impact on Theory and Experiment user program in January 2019, the news release said.
Researchers have been preparing to use Summit for three years. According to the news release, future users and projects include:
Salman Habib of Argonne National Laboratory, whose team is modeling the large-scale structure and distribution of matter over the 13-billion-year lifespan of the universe;
Dmytro Bykov of ORNL, whose team aims to describe the electronic structure of large molecular systems using quantum chemistry techniques, with targeted applications that include pharmacology and nanotechnology;
Abhishek Singharoy of Arizona State University, whose team is investigating the mechanics of a biological motor called ATP synthase in all-atom detail, a study which may aid the design of bioinspired clean energy technology;
Gaute Hagen of ORNL, whose team is calculating the forces within atomic nuclei to study phenomena such as neutrinoless double-beta decay, a hypothesized form of radioactive decay; and
Joe Oefelein of Georgia Tech, whose team is carrying out combustion simulations that closely match engine operating conditions to inform the design of fuel-efficient, low-emission engines.
Source: Knoxville News Sentinel, by Jim Gaines

redpajama_set_name: RedPajamaCommonCrawl


Tempo was designed by Naoto Fukasawa for Magis. The wall clock convinces with its minimalist form speaking like nearly all of the products designed by the popular Japanese designer. The battery-powered wall clock is manufactured out of an on-off injection method ABS-plastic. It is available in four variants: pointer and frame in brown, black, grey and orange (the dial is constantly white). The Tempo wall clock from Magis was presented on the Milan fair in the year 2010 for the first time and it is produced since 2011.


text:
10 Facts about Girl Scout Cookies you might not know
Date: January 28, 2016Author: AbFabSkyLife 1 Comment
It's that time of year again. It's a new year, and to paraphrase the 1989 cult classic Troop Beverly Hills, "It's cookie time!" They've been deep fried at state fairs and incorporated into a Sprinkles Ice Cream seasonal flavor. Everybody, it seems, has their favorite flavor. They're sold in offices, in front of grocery stores, by the girls themselves, often with the support of parents, grandparents, and friends. Like them or not, they're a part of the cultural landscape every midwinter.
"This cookie season is a landmark one for our iconic Girl Scout Cookie Program," said Anna Maria Chávez, CEO of Girl Scouts of the USA. "In responding to an important consumer trend, we've brought back our delicious gluten-free cookie varieties, allowing even more cookie customers to indulge while supporting girls and their communities. And with Digital Cookie 2.0, Girl Scouts are able to further explore STEM (science, technology, engineering, and math), while still engaging in more traditional, including door-to-door, sales and giving back to their communities."
However, as an organization that keeps evolving (a Girl Scout organization in Western Washington recently refused a $100,000 donation from a donor that stipulated the funds not be used in support of transgender scouts), there are new changes to the cookie program every year, so examiner.com has listed some fun facts about Girl Scout Cookies you may not be familiar with.
To find cookies in your area, visit girlscoutcookies.org.
The tradition is almost a century old Photo courtesy of Girl Scouts, used with permission
The tradition is almost a century old
The sale of cookies to finance troop activities and philanthropy began as early as 1917 in Muskogee, Oklahoma. The cookies were home baked, and the girls sold them in their high school cafeteria. Girl Scouts were selling cookies as a fundraiser less than five years after the organization began in the United States.
Cookie sales teach five essential skills Photo courtesy of Girl Scouts, used with permission
Cookie sales teach five essential skills
In addition to raising money for their local troop, Girl Scouts learn five essential skills from the sale of cookies: Goal Setting, Decision Making, Money Management, People Skills, and Business Ethics. Buying a box of Girl Scout cookies helps prepare girls with the confidence, drive, and experience that they'll need to succeed in whatever their chosen field may be.
You can donate cookies Photo courtesy of Girl Scouts, used with permission
You can donate cookies
Whether purchasing directly from your Girl Scout Troop or via their digital sales program Digital Cookie 2.0, newly upgraded this year, purchased boxes can be donated back to the Troop for them to donate as they see fit. Girl Scout Cookies often go to shelters, food pantries, and emergency relief organizations.
There's a whole weekend dedicated to Girl Scout Cookies Photo courtesy of Girl Scouts, used with permission
There's a whole weekend dedicated to Girl Scout Cookies
National Girl Scout Cookie Weekend is a new national holiday for Girl Scout Cookie fans and will take place February 26-28, 2016. This year, the weekend will honor girls who use their cookie revenue to benefit their communities, such as girls who fundraised to purchase a bulletproof vest for a female police officer to others who created care packages for patients at a children's cancer center.
Cookie sales have gone digital Photo courtesy of Girl Scouts, used with permission
Cookie sales have gone digital
Using Digital Cookie 2.0, girls can track their sales, learn about running a business, set sales goals, and explore ways to invest their earnings in their communities. Over 90% of councils are using the Digital Cookie platform. Customers can use the Girl Scout Cookie Finder app to find girls selling cookies, or log on to girlscoutcookies.org. In 2015 350,000 orders for 2.5 million boxes of cookies were placed using Digital Cookie.
Girl Scout Cookies have a hashtag Photo courtesy of Girl Scouts, used with permission
Girl Scout Cookies have a hashtag
Girls, parents, and customers can use the hashtag #CookieBoss to find stories about cookie sales, goal achievements, and community projects supported by cookie sales on social media.
There are gluten-free cookies Photo courtesy of Girl Scouts, used with permission
There are gluten-free cookies
Two gluten-free Girl Scout cookie varieties were introduced last year, and they're returning this year due to popular demand. The cookies are produced in a dedicated gluten-free facility and are NSF-certified to be gluten-free. The two flavors are Gluten-Free Trios, made with peanut butter, chocolate chips, butter, and gluten-free whole grain oats; and Gluten-Free Toffee-tastic, buttery cookies with golden toffee bits.
Girl Scout Cookies inspire girls to futures in STEM Photo courtesy of Girl Scouts, used with permission
Girl Scout Cookies inspire girls to futures in STEM
Girl Scouts of the USA held the first-ever Girls' STEM Summit at the Consumer Electronics Show (CES) this year, where attendees discussed a future for girls in STEM and addressed gender barriers that start during childhood, in order to compel the next generation of women in STEM.
Cookie names vary regionally Photo courtesy of Girl Scouts, used with permission
Cookie names vary regionally
In some parts of the country, they're called Samoas, in others, they're called Caramel deLites. Some buys may know them as Trefoils, while others may have grown up with Shortbread. Same with Peanut Butter Patties and Tagalongs. Different bakeries are the reason for the name variations, but the cookies are very similar despite the difference. A full listing of all the Girl Scout Cookies available this year can be found here.
Thin Mints are the most popular cookie Photo courtesy of Girl Scouts, used with permission
Thin Mints are the most popular cookie
Thin Mints are the most popular cookie sold, and not surprisingly the most popular cookie sold through Digital Cookie last year. Perennially popular Samoas gave Thin Mints a run for their money, but the Gluten-Free Trios were also among the top-selling cookies last year.
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Constructed of 1½" square tubing, upper and lower pegs available in 5"length. Length opens to 52½" and closes to 3". Width opens to 57' and closes 24".
Constructed of 1½" square tubing, upper and lower pegs available in 7.5"length. Length opens to 52½" and closes to 3". Width opens to 57' and closes 24".
Measures 32"L x 25.5" w x 33"H. Made of 2" square tubing. Maximum load capaicity is 600 lbs.
Storage Bin Specifications: Measures 48" long X 24" wide X 12" deep.


text:
John Tardy – Vocals
Trevor Peres – Guitar
Kenny Andrews – Guitar
Terry Butler – Bass
Donald Tardy – Drums
Upon switching their name from Xecutioner to OBITUARY, the career of one of the most successful and influential death metal bands began. Hailing from Florida and featuring brothers John Tardy (Vocals), Donald Tardy (Drums), Trevor Peres (Guitar), Allen West (Guitar), and Daniel Tucker (Bass), the band signed to Roadracer Records, a now defunct division of Roadrunner, for the recording of their debut album, the immense and immeasurably heavy Slowly We Rot (1989). The album was engineered by the legendary Scott Burns at Morrisound Studio, which would come to be the most sought after facility for production of albums during the rise of death metal in the 1990s. Unlike much death metal preceding it, the album had a sludgy feel and integrated devastatingly slow passages along with obliterating overtures that reached far beyond any point of mayhem that metal had yet to reach. The result was a carnal pleasure for doom, death and thrash fans alike coupling the adrenaline of a speedball with the slow, degrading measures of a sewer at dusk. Like them or not, Obituary were unlike anything anyone had heard before.
Slowly We Rot was chaotic, bass heavy mix of manic guitar solos and crashing drums. The album's defining feature, however, was vocalist John Tardy's disarmingly horrific, gargling style, that created guttural chasms of dread which, though often mimicked, have been achieved by no one else. Drastically shifting tempos and Tardy's unique vocal style became the band's trademarks, which clearly distinguished them from the rest of the burgeoning Florida scene. Nowhere are these two features more apparent than on the prophetic title track of their debut. The fact that OBITUARY refrained from printing lyric sheets with their albums led to rumors that the band hadn't written any lyrics at all. Even so, any true fans have every grunt, growl and howling shriek committed to memory as if they were blistering utterances from God. What does not exist cannot be remembered, and an OBITUARY show is testimony to the re-creation of what one's ears couldn't believe in the first place. The ironically titled live album Dead worked quite well for the quintet, dividing the listeners conscience yet leaving much to the imagination. While such differing sensations once seemed incongruous, OBITUARY have proven their ability to merge unlikely dichotomies, from their slow-as-hell-yet-fast-as-fuck style to the non-evil, homegrown approach to what would largely become the satanized, bastardized, make-up wearing movement known as death metal.
The maturation of the musicians into songwriters taking more visionary and complex forms would soon be heard world wide as OBITUARY took metal by storm in 1990. Despite their youth upon release of their sophomore offering, Cause of Death embodied the confident swagger of the most fearsome pack-leading hound. From the insidious growls of John's vocals to the barrage of Donald Tardy's thunderously-metered explosions of double bass, Cause of Death was the intention and method as promised by the early threat of Slowly We Rot; for OBITUARY, death was just the beginning. Accordingly, the title track alone ("Cause of Death") would be heard, regurgitated, manipulated, complimented and collapsed—but never duplicated—on third and forth generation death metal albums for years to come. Lovecraftian imagery and aural morbidity aside, even a deaf man found fear when confronted by the formidable visage of guitarist Peres; entering Frank Watkins, the hulking henchman of a bassist from South Florida, finally provided long-needed and powerful rhythm stability to the line up. However, the grinding of the axes would not be complete until the return of Xecutioner veteran Allen West, who, along with Peres and Tardy, crafted the foundation for most of OBITUARY's most primordial and historic moments. Attack now whole, OBITUARY had given birth—sight, sound and feel—to a true horror greater than metal had ever known.
The paradox herein lies that Obituary was anything but a summation to and end, but more an exploratory journey into the infinite dehumanization of all that is known, as confronted brazenly by their best selling release yet, The End Complete and later followed by the cynical and dark expedition of World Demise. Reunited with songwriter West, the band was conjoined like quintuplets sharing life and a name. Though finality was possibly inferred by these titles, OBITUARY was anything but finished. Ironically, the images conjured by songs such as "Don't Care", "Platonic Disease" and "World Demise" seemingly foretold of the millennium as can now be seen daily, displayed plainly across the screens of CNN and reality TV programs world wide; not bad for a bunch of rednecks from Florida with Budweiser dreams and bongwater nightmares.
After the release of Back from the Dead (1997), OBITUARY disbanded to focus on individual projects until reforming in 2003 to release the highly acclaimed Frozen in Time (2005) full-length. Longtime producer Scott Burns contributed to the album after 11 years since co-producing World Demise. The long awaited return brought forth the fan-favorite opening track "Redneck Stomp" and other long-lasting, tour classics such as "On The Floor" and "Slow Death." Frozen in Time would be the last album the band released through Roadrunner Records as the band went on to sign with Candlelight Records for the following two releases, Xecutioner's Return (2007) and Darkest Day (2009).
Fast forward to 2014, when after a five year gap between albums, OBITUARY successfully crowdfunded their new album and subsequently partnered with Relapse Records for it's release. Reinvigorated with a new lineup featuring the additions of legendary bassist Terry Butler (Death, Massacre, Six Feet Under) and lead guitarist Ken Andrews. The result was Inked In Blood (2014) which immediately took the world by storm. The next couple years revolved around an extensive world tour schedule that took them to sold-out venues across the globe including tours with Cannibal Corpse, Carcass and Exodus and multiple mainstage festival appearances including 7000 Tons of Metal, Maryland Deathfest, Hellfest, Obscene Extreme, Full Terror Assault, Neurotic Deathfest, Graspop Metal Meeting, and the band's own inaugural Florida Metal Fest.
To end 2016, OBITUARY released a surprise, live EP, Ten Thousand Ways to Die, featuring two brand new songs plus twelve classics and fan favorites recorded across eleven different cities during the Inked In Blood world tour. The EP left die-hard fans craving more, unbeknownst that it was just the beginning.
Now in 2017, OBITUARY return with their self-titled, 10th studio album, further cementing their legacy as one of the most important metal bands of all time! Picking up where Inked In Blood left off, OBITUARY show no signs of slowing down as they continue to reign as Kings of death metal. Recorded at their home studio in Tampa, FL, Obituary is a 10 track tour-de-force of bone-pulverising death metal that is as heavy, uncompromising and infectious as anything they've released in their historic, nearly 30-year career!


text:
\section{Introduction}

\subsection{Models for in-vivo virus dynamics}
A number of population dynamics models have been proposed in order to describe the HIV in-vivo dynamics \cite{Perelson:Nelson:1999,Nowak:May:2000}. Although these models have distinct features, since they attempt  to incorporate different aspects of the interaction between the virus and the  immune system, many of them share a common long-term behaviour, evolving towards an isolated equilibrium state \cite{Nowak:May:2000}.

The basic model for the HIV in-vivo dynamics is given by a three-by-three, first-order system of ordinary differential equations (ODEs)---\cite{Nowak:Bangham:1996,Bonhoeffer:May:Nowak:1997,Nowak:May:2000}:
\begin{equation}
 \left\{
\begin{array}{rcl}
 \dot{x}&=&\lambda-dx-\beta xv,\\
\dot{y}&=&\beta x v - ay,\\
\dot{v}&=& ky-uv.\\
\end{array}
\right.
\label{nb:basic}
\end{equation}
In this model, $x$ denotes the uninfected cells, $y$ the infected cells and $v$ the free virus particles. The average lifetime of an infected cell is $1/a$, while the average lifetime of a virus particle is $1/u$. The total number of virus particles  produced by an infected cell is $k/a$. Healthy cells are infected at a rate $\beta xv$.  New CD4+ T cells are produced, in the thymus, at a rate $\lambda$, and die at a rate $dx$.

System \eqref{nb:basic} has two equilibrium points:
\begin{enumerate}
 \item The disease free equilibrium:  $x^*=\lambda/d$, $y^*=v^*=0$;
\item The endemic equilibrium: $x^*=au/\beta k$, $y^*=(\beta\lambda k - dau)/\beta a k$, $v^*=(\beta\lambda k - dau)/\beta a u$.
\end{enumerate}
The long term dynamics of System~\eqref{nb:basic} can be entirely described in terms of the dimensionless
 parameter
\begin{equation}
 R_0=\frac{\beta\lambda k}{dau},
\label{r0:def}
\end{equation}
also known as the \textit{basic reproductive ratio}.

If $R_0\le 1$, the disease free equilibrium is a global attractor, and the infection cannot persist. If $R_0>1$, the endemic equilibrium becomes a global attractor, and the infections persists indefinitely. This has been first observed numerically ~\cite{Nowak:Bangham:1996,Bonhoeffer:May:Nowak:1997,Nowak:May:2000}. Mathematical proofs of these global stability characteristics were given by \citeasnoun{Li:Muldowney:1995}, using Hirsch's theory of competitive differential systems---see \citeasnoun{Smith:1995}---and, more recently  by \citeasnoun{Leenheer:Smith:2003} and \citeasnoun{Korobeinikov:2004} using a Lyapunov function approach.

Given the notable ability of the HIV to escape from immune response, there is interest in studying models that account for a more detailed immune response as, for instance, the role of cytotoxic T lymphocytes (CTLs). An example is the following four-by-four system of ODEs \cite{Nowak:Bangham:1996}:
\begin{equation}
 \left\{
\begin{array}{rcl}
 \dot{x}&=&\lambda-dx-\beta xv,\\
\dot{y}&=&\beta x v - ay - pyz,\\
\dot{v}&=& ky-uv,\\
\dot{z}&=&cyz-bz.
\end{array}
\right.
\label{nb:immune}
\end{equation}
System \eqref{nb:immune} extends \eqref{nb:basic} by introducing the $z$ variable, that denotes the CTL response. Infected cells are killed at a rate $pyz$, while antigen stimulation produces CTL cells at a rate $cyz$. In the absence of such a stimulation, CTL cells decay at a rate $bz$.

In the same vein, the high mutation rate of HIV naturally leads  to the study of the interplay between immune response and virus diversity for a number of different strains. The immune response produces a selection pressure on these different strains of the virus, as discussed in \citeasnoun{Nowak:Bangham:1996}, when studying and  numerically analysing a $(3n+1)$-by-$(3n+1)$ first-order ODE system of the form:
\begin{equation}
 \left\{
\begin{array}{rclr}
  \dot{x}&=&\lambda-dx-x\sum_{i=1}^n\beta_i v_i,& \\
\dot{y_i}&=&\beta_i x v_i - a_iy_i - p_iy_iz_i,& i=1,\ldots,n,\\
\dot{v_i}&=& k_iy_i-u_iv_i, & i=1,\ldots,n,\\
\dot{z_i}&=&c_iy_iz_i-b_iz_i, & i=1,\ldots,n.
\end{array}
\right.
\label{nb:antigenic}
\end{equation}
System \eqref{nb:antigenic} is a slightly generalised form of the system studied by \citeasnoun{Nowak:Bangham:1996}, where  the more restricted case $a_i=a$, $p_i=p$, $u_i=u$, $c_i=c$, and $b_i=b$ was addressed.

In all these models, there is the question as whether the  long term dynamics approaches an equilibrium or, more generally, an attractor, and how this might depend on the initial condition. There is compelling numerical evidence---cf. \cite{Nowak:Bangham:1996,Bonhoeffer:May:Nowak:1997,Nowak:May:2000}---that Systems \eqref{nb:immune} and \eqref{nb:antigenic} are globally asymptotically stable. However, no mathematical proof of this fact seems to be available. In System~\eqref{nb:antigenic} there is also the question of determining the antigenic diversity at  the equilibrium.

In this work, we study the stability characteristics of the models given by \eqref{nb:antigenic} following a Lyapunov approach. The Lyapunov functional used here has been used before by \citeasnoun{Korobeinikov:Wake:1999}, in the global analysis of three-dimensional predator-prey systems, and by \citeasnoun{Korobeinikov:2004} and \citeasnoun{Korobeinikov:2004b} in the global analysis of various virus models. More precisely, by using an appropriate linear combination of this Lyapunov functional,  we are able to show global asymptotic stability results for System \eqref{nb:antigenic}, and hence to \eqref{nb:immune}.

The plan for this article goes as follows: We close this introductory section with further biological background and motivations. In Section~\ref{prelims} we address some preliminary issues such as choice of dimensionless variables and parameter reductions.
We study the global stability characteristics of model~\eqref{nb:immune}.
In Section~\ref{antimodel}, we study the equilibria and global stability of model~\eqref{nb:antigenic} under the assumption of unique fitnesses of the strains. In this case,  we determine the possible equilibria of \eqref{nb:antigenic}, and show than there are $2^{n-1}(n+2)$ equilibrium points. We also show that system is globally asymptotic stable, and we determine what is the global attractor in the nonnegative orthant of $\mathbb{R}^{3n+1}$. As a byproduct, we characterise the attained diversity and show that it is monotonically increasing with the strength of the immune response. Some additional results for the case of nonunique fitnesses are also presented.
We conclude in Section~\ref{conclusions} with a discussion of some of the implications
of our results.

\subsection{Biological Background and Motivation}

The models studied by equation~\ref{nb:antigenic} have many potential biological applications. Most notably, to within-host infections connected to
cytotoxic T lymphocytes with antigenic variation including, but not restricted to, HIV infection.
A better understanding of how the within-host HIV, interacts with immune cells seems
to be a key factor in the development of effective long-term therapies or possibly preventive vaccines
for deadly diseases such as the acquired immunodeficiency syndrome \cite{Nowak:May:2000}.
Mathematical modeling of the underlying biological mechanisms and a good understanding of the theoretical implications of such models is crucial in this process.
Indeed, it helps clarifying and testing assumptions, finding the smallest number of determining factors to explain the biological phenomena, and analysing the experimental results \cite{AB2003}.
Furthermore, modeling has already impacted on research at molecular level \cite{Nowak:May:2000} and  important
results have been obtained in modeling the virus dynamics for several infections, such as the
HIV \cite{Nowak:Bangham:1996,PKD1993,PNML1996}, hepatitis B \cite{MRB1991}, hepatitis C \cite{NLDG1998}, and influenza \cite{BR1994a}.

In the particular case of the HIV infection, the dynamics of the within-host infection goes as follows: First, the HIV enters a T cell.
Being a retrovirus, once the HIV is
inside the T~cell, it makes a DNA copy of its viral RNA. For this process it requires the reverse transcriptase (RT) enzyme. The DNA of the virus is then inserted in the T-cell's DNA. The latter in turn will produce viral particles that can bud off the T~cell to infect other ones.  Before one such viral particle leaves
the infected cell, it must be equipped with {\em protease}, which is an enzyme used to cleave a long
protein chain. Without protease the virus particle is uncapable of infecting other T cells.

One of the key characteristics of HIV is its extensive genetic variability. In fact, the HIV seems
to be changing continuously in the course of each infection and typically the virus strain that initiates the patient's infection differs from the one found a year ore more after the infection. In this respect, the introduction of the different strains in the model
is crucial for it to be realistic.

In general terms, one can also say that Model (\ref{nb:antigenic}) is similar in spirit to other models such as food-chain models. The latter have attracted substantial interest by a number of authors. See for example~\cite{roysolimano,kooibsb2001,kooibsb1998} and references therein.
However, the presence of more general quadratic terms or of logistic terms on the
right hand side of the different ``strains" leads to a potentially richer dynamics than
the globally stable present in Model \ref{nb:antigenic}. See for example \cite{kooibsb2001} for a bifurcation analysis of certain food chain models.

\section{Preliminaries} \label{prelims}

\subsection{Parameter reduction}

As already noticed in the introduction, a more restricted form of \eqref{nb:antigenic} with a number of parameters being strain-independent has been studied by \citeasnoun{Nowak:Bangham:1996}. It turns out that some parameters in \eqref{nb:antigenic} can indeed be taken to be strain-independent, as we now show.

We start by noting that if $k_i=0$, then $v_i$ decays exponentially with rate $u_i$. Also, if $p_i=0$, then the dynamics of $z_i$ does not impinges on the rest of the system. Thus, without loss of generality, we can assume that $k_i,p_i\not=0$, $i=1,\ldots,n$. In this case, following \citeasnoun{Pastore:2005}, we rescale the $v_i$s and $\beta_i$s. In addition, we also rescale the $z_i$s. More precisely, the change of variables
\begin{equation}
 v_i\mapsto \frac{k_i}{k}v_i, \quad
z_i\mapsto \frac{p}{p_i}z_i
\quad\text{and}\quad
\beta_i \mapsto \frac{k}{k_i}\beta_i
\label{red:change}
\end{equation}
takes \eqref{nb:antigenic} into
\begin{equation}
 \left\{
\begin{array}{rclr}
\dot{x}&=&\lambda-dx-x\sum_{i=1}^n\beta_i v_i,& \\
\dot{y_i}&=&\beta_i x v_i - a_iy_i - py_iz_i,& i=1,\ldots,n,\\
\dot{v_i}&=& ky_i-u_iv_i, & i=1,\ldots,n,\\
\dot{z_i}&=&c_iy_iz_i-b_iz_i, & i=1,\ldots,n.
\end{array}
\right.
\label{our:antigenic}
\end{equation}
Intuitively, the change of variables \eqref{red:change} reflects that only the ratio $\beta_i/k_i$ turns out to be important, and that this can be already taken into account in the $\beta_i$s, provided we rescale the $v_i$s. Moreover, it also shows that the precise value $p_i$ does not matter, as long as it is nonzero.

\subsection{Dimensionless constants}

In \citeasnoun{Nowak:Bangham:1996}, it was already  observed that, in addition to $R_0$, the quantity $cy^*/b$ is also important in determining the global equilibria.  In a more precise fashion, \citeasnoun{Nowak:May:2000} define
\[
 R_{I}=1+\frac{\beta b k}{c d u},
\]
which they term the basic reproductive ratio in the presence of immune response.
\
However, we follow \citeasnoun{Pastore:2005}, and find  more convenient to write
\[
 R_{I}=1+\frac{R_0}{I_0},
\]
where
\[
 I_0=\frac{c\lambda}{ab}.
\]
An alternative dimensionless constant is the CTL reproduction number given by
\[
 P_0=\frac{I_0(R_0-1)}{R_0}
\]
Although only two constants among $R_0$, $I_0$ and $P_0$ are independent, and are sufficient to describe the regimes of \eqref{our:antigenic}, we have chosen to use both, three constants, as some conditions are better characterised by $P_0$, while much of the algebra in the Lyapunov functional derivatives is better handled by expressing them in terms of $R_0$ and $I0$. Thus, we shall use the strain dependent constants:
\begin{equation}
 R_0^i=\frac{\beta_i\lambda k}{da_iu_i},\quad I_0^i=\frac{c_i\lambda}{a_ib_i}\quad
\text{and}\quad P_0^i=\frac{I_0^i(R_0^i-1)}{R_0^i}.
\end{equation}
\subsection{Strain sets}

In order to deal with plethora of equilibria that arises in System~\eqref{our:antigenic}, we shall now define some notation for some special set of strain indices. This will allows us to deal conveniently with the combinatorial structure of the equilibria.

Without loss of generality, we shall assume that the strains are indexed by increasing order of $R_0^i$.

Let $\mathcal{N}=\{1,2,\ldots,n\}$. Then, we define the set of the strong responders as
\[
 \mathcal{S}=\{i \in\mathcal{N} \,| \,P_0^i>1\}.
\]

\begin{defi}
 We shall say that the set $\mathcal{S}$ of strong responders is consistent, if
\[
 \mathcal{S}=\{1,\ldots,m\}, \quad 1\leq m\leq n.
\]
This is certainly the cases, if the $I_0^i$ satisfy $I_0^{i}\geq I_0^{i+1}$. In particular, this holds if $I_0^i=I_0$ as in the model studied by \citeasnoun{Nowak:Bangham:1996}.
\end{defi}
Given a set of indices $\mathcal{I}$, we define
\[
 \rho_0^\mathcal{I}=\sum_{i\in\mathcal{I}}\frac{R_0^i}{I_0^i}.
\]
Two important definitions are given below:
\begin{defi}
We shall say that $\mathcal{I}\subset\mathcal{S}$ is an antigenic set, if
\begin{equation}
 R_0^i\geq 1 + \rho_0^\mathcal{I},\quad i\in\mathcal{I}
\label{ineq:cond:1}
\end{equation}
holds. In addition, if
\begin{equation}
 R_0^i\leq 1 + \rho_0^\mathcal{I},\, i\not\in\mathcal{I}.\label{ineq:cond:2}
\end{equation}
also holds, we shall say that $\mathcal{I}$ is a stable antigenic set.

Notice that, if $\mathcal{S}\not=\emptyset$, we have that $\mathcal{I}=\{1\}$ is an antigenic set. Let $l$ be the largest integer for which the set $\mathcal{J}=\{1,\ldots,l\}$ is an antigenic set. Then we shall say that $\mathcal{J}$ is the maximal antigenic set.
\end{defi}
Two important facts about the maximal and stable antigenic sets are collected below:
\begin{lem}
\label{lem:sets}
Assume that $\mathcal{S}\not=\emptyset$, and that the strain basic reproductive numbers are distinct.
\begin{enumerate}
\item
If a stable antigenic set exists, then it  is also the  maximal antigenic set. In particular, stable antigenic sets are unique.
 \item If $\mathcal{N}$ is the maximal antigenic set, then it is a stable antigenic set.
\end{enumerate}
\end{lem}
\begin{proof}
\begin{enumerate}

\item Assume that $\mathcal{I}$ is a stable antigenic set and let  $i_l=\max\mathcal{I}$. If $1\leq k<l$, then $i_k\in\mathcal{I}$. Indeed, by the increasing ordering and \eqref{ineq:cond:1}, we have that
\[
 R_0^{i_k}> R_0^{i_l}\geq 1+\rho_0^\mathcal{I}.
\]
But this contradicts \eqref{ineq:cond:2}, thus we must have $i_k\in\mathcal{I}$.
Now assume that $\mathcal{I}'=\{i_1,\ldots,i_{l+1}\}$ is also an antigenic set. Then we must have
\[
 R_0^{l+1}\geq 1 + \rho_0^{\mathcal{I}'}= 1 + \rho_0^\mathcal{I}+\frac{R_0^{l+1}}{I_0^{l+1}}\geq 1+\rho^\mathcal{I}.
\]
But this again contradicts \eqref{ineq:cond:2} and, hence, that $\mathcal{I}$ is stable antigenic. Therefore, it is maximal.
\item This follows since, in this case, \eqref{ineq:cond:2} cannot be violated.
\end{enumerate}
\end{proof}

\section{The model with antigenic variation}\label{antimodel}

In this section we shall study the stability of of \eqref{our:antigenic} in the non-negative orthant of $\mathbb{R}^{3n+1}$ which we shall denote by $\O$. The positive orthant will be denoted by $\O^+$.

We observe that the planes $z_i=0$ and that $\O$
are positive invariant sets for \eqref{nb:antigenic}, since the field points inwards.

The equilibria and stability characteristics of \eqref{our:antigenic} depend  significantly whether the $R_0^i$s are distinct or not. In \S 3.1 and \S 3.2, we describe the equilibria and study their stability in the case of unique fitnesses, i.e., we assume that if $R_0^i=R_0^j$, then $i=j$. In this case, with the adopted order, we have  that
\[
 R_0^{i}>R_0^{i+1},\quad i=1,\ldots,n-1.
\]
Additional remarks when the fitnesses are not unique can be found in Section 3.3.

\subsection{Equilibria}

 Let $\mathcal{N}=\{1,2,\ldots,n\}$.  It turns out that the equilibria of \eqref{nb:antigenic} can be conveniently indexed by $(j,\mathcal{J})$, where $\mathcal{J}\subseteq\mathcal{N}$, and either $j=0$ or $j\not\in\mathcal{J}$. The corresponding equilibrium point will be denoted by $X_{j,\mathcal{J}}$.

Using this notation, we have

\begin{lem}
\label{lem:eq}
System \eqref{our:antigenic} has $2^{n-1}(2+n)$ equilibrium points which can be written as
\[
 X_{j,\mathcal{J}} = \left(\frac{\lambda}{d}Q_{j,\mathcal{J}}^x,\frac{\lambda}{a_1}Q_{j,\mathcal{J}}^{y_1},\ldots,\frac{\lambda}{a_n}Q_{j,\mathcal{J}}^{y_n}
,\frac{d}{\beta_1}Q_{j,\mathcal{J}}^{v_1},\ldots,\frac{d}{\beta_n}Q_{j,\mathcal{J}}^{v_n},\frac{a_1}{p}Q_{j,\mathcal{J}}^{z_1},
\ldots,\frac{a_n}{p}Q_{j,\mathcal{J}}^{z_n}\right),
\]
where
\begin{enumerate}
 \item
\[
Q_{0,\emptyset}^x=1,\quad\text{and}\quad Q_{0,\emptyset}^{y_i}=Q_{0,\emptyset}^{v_i}=Q_{0,\emptyset}^{z_i}=0.
\]
\item If $j$ is  such that $1\leq j \leq n$, then we have
\[
Q_{j,\emptyset}^x=\frac{1}{R_0^j},\quad Q_{j,\emptyset}^{y_j}=1-\frac{1}{R_0^j},\quad
Q_{j,\emptyset}^{v_j}=R_0^j-1\quad\text{and}\quad Q_{j,\emptyset}^{z_j}=0.
\]
and
 \[
Q_{j,\emptyset}^{y_i}=Q_{j,\emptyset}^{v_i}=Q_{j,\emptyset}^{z_i}=0,\quad  i=1,\ldots,n,\quad i\not=j.
\]
\item Given $\mathcal{J}\subseteq \mathcal{N}$, we have
\[
 Q_{0,\mathcal{J}}^x=\frac{1}{1+\rho_0^{\cj}}
\]
and
\[
 Q_{0,\mathcal{J}}^{y_i}=\frac{1}{I_0^i},\quad Q_{0,\mathcal{J}}^{v_i}=\frac{R_0^i}{I_0^i},\quad Q_{0,\mathcal{J}}^{z_i}=\frac{R_0^i}{1+\rho_0^{\cj}}-1, \quad i\in\mathcal{J};
\]
also
\[
 Q_{0,\mathcal{J}}^{y_i}= Q_{0,\mathcal{J}}^{v_i}= Q_{0,\mathcal{J}}^{z_i}=0,\quad i\not\in\mathcal{J}.
\]
\item Given a proper subset $\mathcal{J}\subset \mathcal{N}$, and  $1\leq j'\leq n, j'\not\in \mathcal{J}$, we have that
\[
 Q_{j',\mathcal{J}}^x=\frac{1}{R_0^{j'}},\quad Q_{j',\mathcal{J}}^{y_{j'}}=1-\frac{1}{R_0^{j'}}-\frac{\rho_0^{\cj}}{R_0^{j'}},\quad
Q_{j',\mathcal{J}}^{v_{j'}}=R_0^{j'}-1-\rho_0^{\cj},\quad Q_{j',\mathcal{J}}^{z_{j'}}=0;
\]

for $i\in \mathcal{J}$, we have
\[
 Q_{j',\mathcal{J}}^{y_i}=\frac{1}{I_0^i},\quad Q_{j',\mathcal{J}}^{v_i}=\frac{R_0^i}{I_0^i}, \quad
Q_{j',\mathcal{J}}^{z_i}=\frac{R_0^i}{R_0^{j'}}-1.
\]
For $i\not\in \mathcal{J}$, and $i\not=j'$, we have
\[
 Q_{j',\mathcal{J}}^{y_i}=Q_{j',\mathcal{J}}^{v_i}=Q_{j',\mathcal{J}}^{z_i}=0.
\]
\end{enumerate}
\end{lem}

\begin{proof}
 The first equilibrium is trivial. The second type of equilibria is obtained by choosing an index $j$ such that $z_j=0$, but $y_j\not=0$. We can choose only one such $j$, since this determines $x$. For the other indices $i$, we set $y_i=v_i=z_i=0$. The first equation then determines $v_j$. The third type is obtained by choosing a set $\mathcal{J}$ of indices, such that, for $i\in\mathcal{J}$, we have $z_i\not=0$. This readily determines $y_i$ and $v_i$. For $i\not\in\mathcal{J}$, we have $y_i=v_i=z_i=0$. The first equation, then, determines $x$. Finally, the last equilibria is found by having a set of indices $\mathcal{J}$, as in the equilibrium of the third type, and then choosing an index $j'\not\in\mathcal{J}$ as in the second equilibrium. Again, only one such $j'$ can be chosen.
\end{proof}

\subsection{Stability analysis}

We are now ready to study the global stability of the equilibria of System~\ref{our:antigenic}. Surprisingly, although there is a large number of equilibria, only four of them will be globally stable. In what follows, unless otherwise is said, we shall assume that that $R_0^{i}>R_0^{i+1}$, for $i=1,\ldots,n-1$, and that the set of strong responders is consistent.

\begin{thm}
 For system \eqref{our:antigenic}, defined on $\O$, and  with initial condition at its interior, there is always a globally asymptotically stable equilibrium
given as follows:
\begin{enumerate}
 \item $X_{0,\emptyset}$, if $R_0^n\leq 1$;
\item $X_{1,\emptyset}$, if $1<R_0^1$, and $P_0^1\leq1$.
\item If $P_0^1>1$, let $\mathcal{J}$ be the maximal antigenic set. Then
\begin{enumerate}
\item If $\mathcal{J}$ is a stable antigenic set, then the equilibrium $X_{0,\mathcal{J}}$ is globally asymptotically stable.
\item  Otherwise, let $j'$ be the smallest integer such that
$j'\not\in\mathcal{J}$, which exists by virtue of Lemma~\ref{lem:sets}. Then the equilibrium $X_{j',\mathcal{J}}$ is globally asymptotically stable.
\end{enumerate}
\end{enumerate}

\label{thm:mutation}
\end{thm}

\begin{proof}[Proof of Theorem~\ref{thm:mutation}]

Following \citeasnoun{Korobeinikov:2004}, we shall use the following Lyapunov function:
\begin{align*}
 V(x,\mathbf{y},\mathbf{v},\mathbf{z})&=x-x^*\ln(x/x^*) + \sum_{i=1}^n\left(y_i-y_i^*\ln(y_i/y_i^*)\right) +\\
& \qquad + \sum_{i=1}^nC_i\left(v_i-v_i^*\ln(v_i/v_i^*)\right)
+ p\sum_{i=1}^n\frac{1}{c_i}\left(z_i-z_i^*\ln(z_i/z^*)\right),
\end{align*}
where $C_i$ will be a constant to be specified later on.

Then, using the uniform notation of the the equilibria of \eqref{our:antigenic}, that is set  in Lemma~\ref{lem:eq}, see \S3.1, we have that
\begin{align}
 \dot{V}=& \frac{d}{dt}V(x(t),\mathbf{y}(t),\mathbf{v}(t),\mathbf{z}(t))\nonumber\\
&\lambda\left[ +  Q_{j,\mathcal{J}}^x+ \sum_{i=1}^nQ_{j,\mathcal{J}}^{y_i} + \frac{d}{\lambda}\sum_{i=1}^nC_i\frac{u_i}{\beta_i}Q_{j,\mathcal{J}}^{v_i}+ \sum_{i=1}^n\frac{Q_{j,\mathcal{J}}^{z_i}}{I_0^i}\right]-\left[dx+\frac{\lambda^2 Q_{j,\mathcal{J}}^x}{dx}\right]-\nonumber\\
&-\lambda\sum_{i=1}^n\frac{\beta_i}{a_i}Q_{j,\mathcal{J}}^{y_i}\frac{xv_i}{y_i} - dk\sum_{i=1}^n\frac{C_i}{\beta_i}Q_{j,\mathcal{J}}^{v_i}\frac{y_i}{v_i}
 + \sum_{i=1}^ny_i\left[kC_i-a_i-a_iQ_{j,\mathcal{J}}^{z_i}\right] +\label{master}\\
& \frac{\lambda}{d}Q_{j,\mathcal{J}}^x\sum_{i=1}^nv_i\beta_i -\sum_{i=1}^nC_iu_iv_i
 + p\lambda\sum_{i=1}^na_iz_i\left[Q_{j,\mathcal{J}}^{y_i}-\frac{1}{I_0^i}\right].\nonumber
\end{align}

For the first two equilibria, we consider the Lyapunov function \eqref{master}, with $C_i=a_i/k$.
Then, on using the structure of equilibria of \eqref{our:antigenic}, we may write $\dot{V}$ as follows:
\begin{align*}
 \dot{V}=& \lambda\left[1+Q^x_{j,\mathcal{J}} + \sum_{i=1}^nQ^{y_i}_{j,\mathcal{J}}+\sum_{i=1}^n\frac{Q^{v_i}_{j,\mathcal{J}}}{R_0^i}+\sum_{i=1}^n\frac{Q^{z_i}_{j,\mathcal{J}}}{I_0^i}\right]-\left[dx+\frac{\lambda^2 Q^x_{j,\mathcal{J}}}{dx}\right]-\\
&\quad -\lambda\sum_{i=1}^n\frac{\beta_i}{a_i} Q^{y_i}_{j,\mathcal{J}}\frac{xv_i}{y_i}-
d\sum_{i=1}^n\frac{a_i}{\beta_i}Q^{v_i}_{j,\mathcal{J}}\frac{y_i}{v_i}-\sum_{i=1}^na_iQ^{z_i}_{j,\mathcal{J}}y_i +\\
&\quad +
\frac{\lambda}{d}\sum_{i=1}^n\beta_iv_i\left[Q^x_{j,\mathcal{J}}-\frac{1}{R_0^i}\right]+p\lambda\sum_{i=1}^n\frac{z_i}{a_i}\left[Q^{y_i}_{j,\mathcal{J}}-\frac{1}{I_0^i}\right].
\end{align*}
For $X_{0,\emptyset}$, we find, using Lemma~\ref{lem:eq}, that
\[
 \dot{V}=2\lambda-\left[dx+\frac{\lambda^2}{dx}\right]+\frac{\lambda}{d}\sum_{i=1}^n\beta_iv_i\left[1-\frac{1}{R_0^i}\right]-p\lambda\sum_{i=1}^n\frac{z_i}{a_iI_0^i}.
\]
Since $R_0^i\leq1$, for $i=1,\dots,n$, and
\[
 dx+\frac{\lambda^2}{dx}\geq 2\lambda,
\]
we have that $\dot{V}<0$ in $\O^+$. Hence, that $X_{0,\emptyset}$ is globally asymptotically stable in this case.

Now, suppose that $1<R_0^1$, and that $P_0^1\leq1$.
In this case, Lemma~\ref{lem:eq} yields that
\begin{align*}
 \dot{V}=& \lambda\left[3\left(-\frac{1}{R_0^1}\right)+\frac{2}{R_0^1} \right]-\left[dx+\frac{\lambda^2}{R_0^1dx}\right]-
\frac{\lambda}{a_1}\beta_1 \left(1-\frac{1}{R_0^1}\right)\frac{xv_1}{y_1}-
\frac{a_1d}{\beta_1}(R_0^1-1)\frac{y_1}{v_1} +\\
&\quad +
\frac{\lambda}{d}\sum_{i=1}^n\beta_iv_i\left[\frac{1}{R_0^1}-\frac{1}{R_0^i}\right]+p\lambda\frac{z_1}{a_1}\left[1-\frac{1}{R_0^1}-\frac{1}{I_0^1}\right]-p\lambda\sum_{i=2}^n\frac{z_i}{a_iI_0^i}.
\end{align*}
The last term in $\dot{V}$ is clearly negative. We also observe that, since $R_0^{i}<R_0^{1}$, for $1 < i\leq n$, we have that
\[
\sum_{i=1}^n\beta_iv_i\left[\frac{1}{R_0^1}-\frac{1}{R_0^i}\right]<0.
\]
Also, since $P_0^1\leq1$, then we have that $R_0^1 \leq 1+ R_0^1/I_0^1$. Therefore, the last three terms in the expression for $\dot{V}$ are negative.

For the remaining terms, let us write
\[
 \frac{\lambda^2}{R_0^1}=\left(\frac{\lambda}{R_0^1}\right)^2 + \left(1-\frac{1}{R_0^1}\right)\frac{\lambda^2}{R_0^1}.
\]
Then we have that
\[
 dx+\left(\frac{\lambda^2}{R_0^1dx}\right)^2\geq 2\frac{\lambda}{R_0^1},
\]
and that
\[
.\frac{\lambda^2}{R_0^1dx}\left(1-\frac{1}{R_0^1}\right) +\lambda\frac{\beta_1}{a_1}\left(1-\frac{1}{R_0^1}\right)\frac{xv_1}{y_1}+\frac{da_1}{\beta_1}(R_0^1-1)\frac{y_1}{v_1}\geq 3\lambda\left(1-\frac{1}{R_0^1}\right).
\]
Thus $\dot{V}<0$ in $\O^+$, and hence we have that $X_{1,\emptyset}$ is a globally asymptotically stable equilibrium.

Finally, if $P_0^1>1$, then let $\mathcal{J}$ be the maximal antigenic set. First, we  assume that $\mathcal{J}$ is stable antigenic and show that $X_{0,\mathcal{J}}$ is globally asymptotically stable. In this case, we use the Lyapunov function \eqref{master}, with $C_i=x^*\beta_i/u_i$.

Using Lemma~\ref{lem:eq}, this can be further recast as $\dot{V}=\dot{V_1}+\dot{V_2}$, where
\begin{align*}
 \dot{V_1}=&\lambda\left[3 - \frac{1}{1+\rho_0^{\cj}}\right]-\left[dx+\frac{\lambda^2 }{dx\left(1+\rho_0^{\cj}\right)}\right]
- \lambda\sum_{i\in\mathcal{J}}\frac{\beta_i}{I_0^ia_i} \frac{xv_i}{y_i}
-\frac{\lambda k}{1+\rho_0^{\cj}}\sum_{i\in\mathcal{J}} \frac{R_0^i}{u_iI_0^i}\frac{y_i}{v_i} ;\\
\dot{V_2}=&\sum_{i\not\in\mathcal{J}}a_iy_i\left[\frac{R_0^i}{1+\rho_0^{\cj}}-1\right] -p\lambda\sum_{i\not\in\mathcal{J}}\frac{z_i}{a_iI_0^i}.
\end{align*}
We treat $\dot{V_2}$ first. The last term is clearly negative. Also, since for $i\not\in\mathcal{J}$, we have that
\[
 \frac{R_0^i}{1+\rho_0^{\cj}}<1\quad\text{and thus that}\quad \sum_{i\not\in\mathcal{J}}y_i\left[\frac{R_0^i}{1+\rho_0^{\cj}}-1\right]<0.
\]
Therefore, $\dot{V_2}<0$, when $\mathcal{J}\not=\mathcal{N}$.

Let
\[
 \eta=\frac{\rho_0^{\cj}}{1+\rho_0^{\cj}}\quad\text{and}\quad
\eta_i=\frac{\frac{R_0^i}{I_0^i}}{1+\rho_0^{\cj}},\quad i\in\mathcal{J}.
\]
Then, we may write $\dot{V_1}$ as
\begin{align*}
 \dot{V_1}&=\lambda\left[3 - \frac{1}{1+\frac{\rho_0^{\cj}}{I_0}}\right]-\left[dx+\frac{ \lambda^2}{dx\left(1+\rho_0^{\cj}\right)^2}\right]
-\\
&\qquad - \sum_{i\in\mathcal{J}}\frac{\lambda^2\eta_i}{dx\left(1+\rho_0^{\cj}\right)}
- \lambda\sum_{i\in\mathcal{J}}\frac{\beta_i}{I_0^ia_i} \frac{xv_i}{y_i}
-\frac{\lambda k}{1+\rho_0^{\cj}}\sum_{i\in\mathcal{J}} \frac{R_0^i}{u_iI_0^i}\frac{y_i}{v_i}
\end{align*}
For each $i\in\mathcal{J}$, we have
\[
 -\frac{\lambda^2\frac{R_0^i}{I_0^i}}{dx\left(1+\rho_0^{\cj}\right)^2}
- \lambda\frac{\beta_i}{I_0^ia_i} \frac{xv_i}{y_i}
-\frac{\lambda k}{1+\rho_0^{\cj}}\frac{R_0^i}{u_iI_0^i}\frac{y_i}{v_i}\leq
-3\lambda\left(\frac{\frac{R_0^i}{I_0^i}}{1+\rho_0^{\cj}}\right),
\]

and that
\[
 dx+\frac{\lambda^2}{dx\left(1+\rho_0^{\cj}\right)^2}>2\lambda\frac{1}{1+\rho_0^{\cj}}.
\]
After combining these estimates and summing for $i\in\mathcal{J}$, we get that $\dot{V_1}\leq0$
and thus we have the result, If $\mathcal{J}$ is a proper subset of $\mathcal{N}$. In the case that $\mathcal{J}=\mathcal{N}$, we have $\dot{V}\leq0$, with equality ocurring only when
\[
 x=\frac{\lambda}{d}Q_{0,\mathcal{J}}^x\quad\text{and}\quad \frac{v_i}{y_i}=\frac{k_i}{u_i}.
\]
Inasmuch this plane is not invariant by the corresponding flow---other than the point $X_{0,\mathcal{J}}$---we have global stability as a consequence of LaSalle's theorem ~\cite{LaSalle:1964}.
For the fourth point, we use a mix of the two Lyapunov functions above, namely:
 \begin{align*}
 V(x,\mathbf{y},\mathbf{v},\mathbf{z})=&x-x^*\ln(x/x^*) + \sum_{i=1}^n\left(y_i-y_i^*\ln(y_i/y_i^*)\right) + \\
&\qquad +x^*\sum_{\sti{i=1}{i\not=j'}}^n\frac{\beta_i}{u_i}\left(v_i-v_i^*\ln(v_i/v_i^*)\right)
 + p\sum_{i=1}^n\frac{1}{c_i}\left(z_i-z_i^*\ln(z_i/z^*)\right)+\\
&\qquad + \frac{a_{j'}}{k}\left(v_{j'}-v_{j'}^*\ln(v_{j'}/v_{j'}^*)\right).
\end{align*}

Computing $\dot{V}$ and using the uniform notation, we find that
\begin{align*}
 \dot{V}=&\lambda\left[1 + Q_{j',\mathcal{J}}^x+\sum_{i=1}^nQ_{j',\mathcal{J}}^{y_i} + Q_{j',\mathcal{J}}^x\sum_{\sti{i=1}{i\not=j'}}^nQ_{j',\mathcal{J}}^{v_i} + \sum_{i=1}^n \frac{Q_{j',\mathcal{J}}^{z_i}}{I_0^i} + \frac{Q^{v_{j'}}_{j',\mathcal{J}}}{R_0^{j'}} \right]-\left[dx+\frac{\lambda^2 Q_{j',\mathcal{J}}^x}{dx}\right]-\\
&\qquad - \lambda\sum_{i=1}^n\frac{\beta_i Q_{j',\mathcal{J}}^{y_i}}{a_i} \frac{xv_i}{y_i}
-\lambda kQ_{j',\mathcal{J}}^x\sum_{\sti{i=1}{i\not=j'}}^n\frac{Q_{j',\mathcal{J}}^{v_i}}{u_i}\frac{y_i}{v_i} + \sum_{\sti{i=1}{i\not=j'}}^ny_i\left[\frac{\beta_i \lambda k}{du_i}Q_{j',\mathcal{J}}^x-a_i-a_iQ_{j',\mathcal{J}}^{z_i}\right]+\\ &\qquad +p\lambda\sum_{i=1}^n\frac{z_i}{a_i}\left[Q_{j',\mathcal{J}}^{y_i}-\frac{1}{I_0^i}\right] -
\frac{da_{j'}}{\beta_{j'}}Q^{v_{j'}}_{j',\mathcal{J}}\frac{y_{j'}}{v_{j'}} - a_{j'}Q^{z_{j'}}_{j',\mathcal{J}}y_{j'} +
\frac{\lambda\beta_{j'}}{d}v_{j'}\left[Q^x_{j',\mathcal{J}}-\frac{1}{R_0^{j'}}\right]
\end{align*}

On using Lemma~\ref{lem:eq}, and that
\[
1=\left(1-\frac{1}{R_0^{j'}}-\frac{\rho_0^{\cj}}{R_0^{j'}}\right)+\frac{1}{R_0^{j'}}+\frac{\rho_0^{\cj}}{R_0^{j'}},
\]
where each term in the sum is positive, but smaller than one, we rewrite it as $ \dot{V}=\dot{V_1}+\dot{V_2} + \dot{V_3} + \dot{V_4}$, where
\begin{align*}
\dot{V_1}&=\left(1-\frac{1}{R_0^{j'}}-\frac{\rho_0^{\cj}}{R_0^{j'}}\right)\left[3\lambda-\frac{\lambda^2}{dxR_0^{j'}}-\frac{\lambda\beta_{j'}}{a_{j'}}\frac{xv_{j'}}{y_{j'}}-\frac{da_{j'}R_0^{j'}}{\beta_{j'}}\frac{y_{j'}}{v_{j'}}\right],\\
\dot{V_2}&=\frac{2\lambda}{R_0^{j'}}-\left[dx+\frac{\lambda^2}{dx(R_0^{j'})^2}\right],\\
\dot{V_3}&=3\lambda\frac{\rho_0^{\cj}}{R_0^{j'}}-\frac{\lambda^2\rho_0^{\cj}}{dx(R_0^{j'})^2}-\lambda\sum_{i\in\mathcal{J}}\frac{\beta_i}{a_iI_0^i}\frac{xv_i}{y_i}-\frac{\lambda k}{R_0^{j'}}\sum_{i\in\mathcal{J}}\frac{R_0^i}{u_iI_0^i}\frac{y_i}{v_i},\\
\dot{V_4}&=\frac{p\lambda z_{j'}}{a_{j'}}\left(1-\frac{1}{R_0^{j'}}-\frac{\rho_0^{\cj}}{R_0^{j'}}-\frac{1}{I_0^{j'}}\right).
\end{align*}
The terms $\dot{V_1}$, $\dot{V_2}$ and $\dot{V_3}$ can be treated similarly as in the previous equilibria and are all nonpositive in the interior.

As for $\dot{V_4}$, first we observe that
\[
 1-\frac{1}{R_0^{j'}}-\frac{\rho_0^{\cj}}{R_0^{j'}}-\frac{1}{I_0^{j'}}=\frac{1}{R_0^{j'}}\left(R_0^{j'}-1-\rho_0^{\cj}-\frac{R_0^{j'}}{I_0^{j'}}\right)
\]
if $j'\not\in\mathcal{S}$, then we that
\[
 R_0^{j'}-1-\rho_0^{\cj}-\frac{R_0^{j'}}{I_0^{j'}}<R_0^{j'}-1-\frac{R_0^{j'}}{I_0^{j'}}\leq0.
\]
If $j'\in\mathcal{S}$, then let $\mathcal{J}'=\mathcal{J}\cup\{j'\}$. Then we have that
\[
 1-\frac{1}{R_0^{j'}}-\frac{\rho_0^{\cj}}{R_0^{j'}}-\frac{1}{I_0^{j'}}=\frac{1}{R_0^{j'}}\left(R_0^{j'}-1 -\rho_0^{\mathcal{J}'}\right).
\]
Since $\mathcal{J}$ is the maximal antigenic set, we must have that
\[
 R_0^{j'}-1 -\rho_0^{\mathcal{J}'}\leq0.
\]
\end{proof}
\subsection{Nonunique Fitness}

Given the non-generic nature of this case,  we shall only briefly discuss the stability when some of the strains have the same fitness, i.e., there exists at lest one  index set $\Gamma$, such that $R_0^i=R_0^j$, for $i,j\in\Gamma$. Notice that, in this case, we have non-isolated equilibria.

We start by observing that the computation with the Lyapunov function for the equilibrium $X_{0,\emptyset}$ does not depend on the uniqueness of fitness. Hence we have
\begin{cor}
 If $R_0^i\leq1$, for $i=1,\ldots,n$, then $X_{0,\emptyset}$ is a globally asymptotically stable equilibrium.
\end{cor}
The case $I_0^i=I_0$, $1<R_0^1$ and $P_0^1<1$ can also be partially treated:
\begin{prop}
\label{prop:ls}
 Let $\Gamma$ be the set of indices $i\in\mathcal{N}$ such that $R_0^i=R_0^n$. Let $E_\Gamma$ be the set satisfying
\[
x^*=\frac{\lambda}{dR_0^n},\quad y_j=v_j=0,\quad j\not\in\Gamma,\quad \sum_{j\in\Gamma}\beta_jv_j=\frac{dx^*-\lambda}{x^*}, \quad v_j\geq0,
\]
\[
v_i=\frac{k}{u}y_i,\quad\text{and}\quad z_i=0, \quad i\in\mathcal{N}.
\]
If $\mathbf{x}(t,\mathbf{x}_0)$ is a solution of \eqref{nb:antigenic}, with initial condition $\mathbf{x}_0$, then
\[
 \mathbf{x}(t)\to E_\Gamma,\quad\text{as}\quad t\to\infty.
\]
\end{prop}

\begin{proof}
Let us denote the omega set of $\mathbf{x}(t,\mathbf{x}_0)$ by $\Omega(\mathbf{x_0})$. As shown in \cite{Pastore:2005}, the solutions to \eqref{nb:antigenic} are bounded in $\mathbb{R}_+^{3n+1}$.
Hence, $\Omega(\mathbf{x_0})$ is compact.

Using the same Lyapunov function for $X_{1,\emptyset}$ as in Section 3.2, we find that
\begin{align*}
 \dot{V}=& \lambda\left[3-\frac{1}{R_0^1} \right]-\left[dx+\frac{\lambda^2}{R_0^1dx}\right]-
\frac{\lambda}{a}\beta_1 \left(1-\frac{1}{R_0^1}\right)\frac{xv_1}{y_1}-
\frac{ad}{\beta_1}(R_0^1-1)\frac{y_1}{v_1} +\\
&\quad +
\frac{\lambda}{d}\sum_{i\not\in\Gamma}\beta_iv_i\left[\frac{1}{R_0^1}-\frac{1}{R_0^i}\right]+\frac{p\lambda}{a}\sum_{i=1}^nz_i\left[1-\frac{1}{R_0^i}-\frac{1}{I_0}\right].
\end{align*}
The same calculations in section 3.2 shows that $\dot{V}\leq0$. However, notice that $\dot{V}=0$ in $E_\Gamma$. If $s\in\mathbb{R}^N$ and $S\subset\mathbb{R}^N$ is closed, then let
\[
 d(s,S)=\min_{s'\in S}\|s-s'\|.
\]
LaSalle's invariant principle then yields that,
\[
 \lim_{t\to\infty}d(\mathbf{x}(t),E_\Gamma)=0.
\]

\end{proof}

\begin{cor}
 If $R_0^1>R_0^i$ for $i=2,\ldots,n$, then $X_{1,\emptyset}$ is a globally asymptotically stable equilibrium, when $1<R_0^1\leq1+R_0^1/I_0$.
\end{cor}
We have performed numerical calculations of \eqref{nb:antigenic}, using a high order Runge-Kutta method, which suggest that  in the case treated by proposition~1, a solution of System~\eqref{nb:antigenic} will converge to a unique equilibrium point in $E_\Gamma$, that depends  only on the initial condition.

Finally, when the viable set of strains is not the full antigenic variation, we have
\begin{prop}
 If $I_0^i=I_0$, $P_0^1>1$. Assume that  $\mathcal{J}\not=\mathcal{I}$ is a stable antigenic set, then we have that $X_{0,\mathcal{J}}$ is a globally asymptotic stable equilibrium.
\end{prop}

\begin{proof}
 In this case, the estimate  $\dot{V}_1\leq0$ remains valid. Moreover, if $\mathcal{J}\not=\mathcal{I}$, then we must have $\dot{V}_2<0$, which yields the result.
\end{proof}

\section{Conclusions} \label{conclusions}
In this work, we have performed a thorough study of System~\eqref{nb:antigenic}.
As a preliminary result, we have shown that, when the both the virus production rate and the CTL interaction rate are nonzero for all strains, then System~\eqref{nb:antigenic} is dynamically equivalent to System~\eqref{our:antigenic}, which has strain-independent virus production  and CTL interaction rates. In particular, the precise nonzero values of the CTL interaction rates are completely irrelevant to the dynamical behaviour of the system. This seems to suggest that a more refined model which is able to capture this difference is needed.

We have also identified all the $2^{n-1}(2+n)$ equilibria of \eqref{our:antigenic} in Lemma~\ref{lem:eq}. When $n$ is large this can be quite a large number. Nevertheless, under the hypothesis of unique fitness, we were able to show that only four of them are dynamically relevant. More precisely, we assume that the virus basic reproduction rates, $R_0^i$ are distinct and that the CTL reproduction numbers, $P_0^i$ have the same ordering as the $R_0^i$. This last condition is automatically satisfied if $I_0^i=I_0$ for all $i$. In this case, Theorem~\ref{thm:mutation} shows that, if the largest reproduction number, $R_0^1$, is smaller than one, the the disease-free equilibrium---$X_{0,\emptyset}$ in the notation of Lemma~\ref{lem:eq}---is globally asymptotic stable. In this case, no strain is viable and the infection dies out. On the other hand, if $R_0^1>1$, but $P_0^1<1$, then only the first strain survives, and the infection persists. If $P_0^1>1$, then we have that the two outcomes are possible: either a (unique) stable antigenic set exists and then $X_{0,\mathcal{J}}$ is globally asymptotic stable. In this case, the set $\mathcal{J}$ determines the antigenic diversity.   In other words, a strong immune response generates a larger antigenic variation. Alternatively, there exits  a pair $(j',\mathcal{J})$ such that the point $X_{j',\mathcal{J}}$ is globally asymptotic stable. In this case, the strain with the weakest fitness  will not actually trigger the CTL response at all in the long run.
In the case of absence of antigenic variation, i.e. System~\eqref{nb:immune}, then only the first outcome is possible. We were unable to interpret in a biological sense the combinatorial conditions of existence of a stable antigenic set, and we believe that this should be addressed in the future.
The results presented in Theorem~\ref{thm:mutation} show rigorously some of the inferences that have already been made in \citeasnoun{Nowak:Bangham:1996} based on extensive simulations of System~\eqref{nb:antigenic}.

We have also shown some results for very special  cases in which the $R_0^i$s are not distinct. In these cases, the equilibria is not isolated and this complicates the matters further. Also, we have not addressed that case when the set of strong responders is not consistent, and this might also merit further study in the future.

\bibliographystyle{harvard}

redpajama_set_name: RedPajamaArXiv


text:
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Trump calls for revoking New York Times, Washington Post Pulitzers for Mueller coverage
March 28, 2019 politico Leave a Comment
white house President Donald Trump on Friday called for the New York Times and the Washington Post to have their Pulitzer prizes rescinded for their coverage of the special counsel's Russia investigation. "So funny that The New York Times & The Washington Post got a Pulitzer Prize for their coverage (100% NEGATIVE and FAKE!) of Collusion with Russia - And there was No Collusion! So, they were either duped or corrupt? In any event, their prizes should be taken away by the Committee!" Trump tweeted. Story Continued Below Trump has lashed out at the media in recent days after a synopsis of special counsel Robert Mueller's report was released by Attorney General William Barr. The full report has yet to be released. The newspapers received a joint Pulitzer Prize last year for their coverage of the Mueller investigation and the connections between Russia's interference in the 2016 elections and Trump campaign members and aides.Trump has calledthe New York Times the … [Read more...] about Trump calls for revoking New York Times, Washington Post Pulitzers for Mueller coverage
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Trump Voters Trust Infowars as Much as the New York Times and Washington Post, Poll Shows
February 14, 2019 newsweek Leave a Comment
U.S. Politics President Donald Trump has long attacked the news media—and The Washington Post and The New York Times have been no exception. And it appears his supporters aren't exactly fans of those outlets, either. A new poll from YouGov/The Economist released this week showed folks who voted for Trump in 2016 trust Infowars—the infamous conspiracy site headed by Alex Jones—about as much as the Post or Times. The survey asked respondents, "How trustworthy do you rate the following news organizations?" before listing off a number of outlets. Fifteen percent of Trump voters found Infowars trustworthy to some degree—4 percent "very trustworthy" and 11 percent "trustworthy." Fifteen percent of Trump voters deemed The New York Times trustworthy to some degree—5 percent "very trustworthy" and 10 percent "trustworthy"—while 15 percent viewed The Washington Post trustworthy to some degree (5 percent "very trustworthy" and … [Read more...] about Trump Voters Trust Infowars as Much as the New York Times and Washington Post, Poll Shows
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Hailing the New York Times: Trump loves his hometown paper
August 17, 2018 chron Leave a Comment
David Nakamura, The Washington Post Published 1:58 pm CDT, Friday, August 17, 2018 SOUTHAMPTON, N.Y. - As has been noted before, President Donald Trump attacks the "failing New York Times" and other legacy news outlets even though he appears to secretly crave their endorsement, considering he came up in a pre-internet era of print behemoths. Another sign of this came Friday on Air Force One as the president traveled to a Republican fundraiser here, a day after he declared on Twitter that "THE FAKE NEWS MEDIA IS THE OPPOSITION PARTY." Midway through the flight, a presidential aide entered the press corps cabin in the rear of the jet holding a newspaper clipping from the Times. She handed it to Tom Brenner, a Times freelancer who had traveled with Trump on a trip to Fort Drum, New York, on Monday. The clip included a photo of Trump boarding the presidential jet with a cloud of smoke in the background from a ceremonial cannon shot during Trump's visit. Trump, the aide … [Read more...] about Hailing the New York Times: Trump loves his hometown paper
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The joy of traveling alone: A New York Times travel writer describes why she loves exploring solo
August 8, 2018 sfgate Leave a Comment
Bella DePaulo, The Washington Post Published 11:07 am PDT, Wednesday, August 8, 2018 Photo: Viking Press. Image 1of/1 CaptionClose Image 1 of 1 "Alone Time: Four Seasons, Four Cities, and the Pleasures of Solitude" by Stephanie Rosenbloom. "Alone Time: Four Seasons, Four Cities, and the Pleasures of Solitude" by Stephanie Rosenbloom. Photo: Viking Press. The joy of traveling alone: A New York Times travel writer describes why she loves exploring solo 1 / 1 Back to Gallery As a travel writer for the New York Times, Stephanie Rosenbloom has scored many plum assignments. One of them - spending five days in Paris, on her own - stayed with her long after she returned. To find out why her senses pricked up there - and why she was able to delight in the smallest things there - she returned to Paris and also traveled … [Read more...] about The joy of traveling alone: A New York Times travel writer describes why she loves exploring solo
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New York Times publisher and Trump clash over president's threats against journalism
July 29, 2018 startribune Leave a Comment
BRIDGEWATER, N.J. – President Donald Trump and the publisher of The New York Times, A.G. Sulzberger, engaged in a fierce public clash Sunday over Trump's threats against journalism, after Sulzberger said the president misrepresented a private meeting and Trump accused The Times and other papers of putting lives at risk with irresponsible reporting. Trump said on Twitter that he and Sulzberger had discussed "the vast amounts of Fake News being put out by the media & how that Fake News has morphed into phrase, 'Enemy of the People.' Sad!" In a five-paragraph statement issued two hours after the tweet, Sulzberger said he had accepted Trump's invitation for the July 20 meeting mainly to raise his concerns about the president's "deeply troubling anti-press rhetoric." "I told the president directly that I thought that his language was not just divisive but increasingly dangerous," said Sulzberger, who became … [Read more...] about New York Times publisher and Trump clash over president's threats against journalism
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text:
8.5/10 Gamespot 7/10 IGN 80% Metacritic
Min System Requirements
OS: Windows® XP Service Pack 2 or later
Processor: 2.0 Ghz or higher (single core)
Graphics: DirectX®9 compliant video card or greater, 256 MB video RAM or higher, Shader Model 1.1 or higher (Laptop versions of these chipsets may work but are not supported. Updates to your video and sound card drivers may be required)
Take to the skies in the World's favourite flight simulator!
The multi award winning Microsoft Flight Simulator X lands on Steam for the first time. Take off from anywhere in the world, flying some of the world's most iconic aircraft to any one of 24,000 destinations. Microsoft Flight Simulator X Steam Edition has updated multiplayer and Windows 8.1 support.
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FSX Steam Edition offers players a connected world where they can choose who they want to be, from air-traffic controller to pilot or co-pilot. Racing mode allows you to compete against friends with four types of racing, including Red Bull Air Race courses, the unlimited Reno National Championship course, as well as cross country, competition sailplane courses and fictional courses like the Hoop and Jet Canyon. Test your skills with three different levels of difficulty, from simple pylon racing to racing highly challenging courses in a variety of weather conditions.
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text:
/* ###
 * IP: GHIDRA
 *
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *
 *      http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */
package ghidra.dbg.model;

import java.util.Deque;
import java.util.LinkedList;
import java.util.concurrent.CompletableFuture;

import ghidra.dbg.DebuggerModelFactory;
import ghidra.dbg.DebuggerObjectModel;
import ghidra.dbg.util.ConfigurableFactory.FactoryDescription;

@FactoryDescription(brief = "Mocked Client", htmlDetails = TestDebuggerModelFactory.FAKE_DETAILS)
public class TestDebuggerModelFactory implements DebuggerModelFactory {
	public static final String FAKE_DETAILS = "A 'connection' to a fake debugger";
	public static final String FAKE_DETAILS_HTML =
		"<html><b>Description:</b> A 'connection' to a fake debugger";
	public static final String FAKE_OPTION_NAME = "Test String";
	public static final String FAKE_DEFAULT = "Default test string";

	protected final Deque<CompletableFuture<DebuggerObjectModel>> buildQueue =
		new LinkedList<>();

	@FactoryOption(FAKE_OPTION_NAME)
	public final Property<String> testStringOption =
		Property.fromAccessors(String.class, this::getTestString, this::setTestString);

	private String testString = FAKE_DEFAULT;

	public TestDebuggerModelFactory() {
	}

	public String getTestString() {
		return testString;
	}

	public void setTestString(String testString) {
		this.testString = testString;
	}

	@Override
	public CompletableFuture<? extends DebuggerObjectModel> build() {
		CompletableFuture<DebuggerObjectModel> future = new CompletableFuture<>();
		buildQueue.offer(future);
		return future;
	}

	public CompletableFuture<DebuggerObjectModel> pollBuild() {
		return buildQueue.poll();
	}
}

redpajama_set_name: RedPajamaGithub


text:
Employees with disabilities of Mediaset in Spain 2013-2015, by job category
Number of employees with disabilities of Mediaset in Spain from 2013 to 2015, by job category
by Statista Research Department, last edited Feb 1, 2016
This statistic shows the number of employees with disabilities of Mediaset in Spain from 2013 to 2015, by job category. The average number of employees who had a disability within this media group amounted to six during the measured period.
TV channels in Sweden 2018, by audience share
Number of TV households in Sweden 2008-2017
Ranking of TV channels in Sweden 2018, by daily reach
Average daily TV viewing time in Sweden 2008-2018
Statistics on "Television in Sweden"
Public service television
Advertising & commercial TV
Number of TV households in Sweden from 2008 to 2017 (in millions)Number of TV households in Sweden 2008-2017
Revenues from television services in Sweden from 2009 to 2017 (in million SEK)Revenues from television services in Sweden 2009-2017
Revenues from television services in Sweden from 2009 to 2017, by distribution platform (in million SEK)Revenues from television services in Sweden 2009-2017, by platform
Ranking of selected TV companies in Sweden in 2017, by revenue (in million SEK)Ranking of TV companies in Sweden 2017, by revenue
Ranking of TV channel families in Sweden in 2017, by audience shareRanking of TV channel families in Sweden 2017, by audience share
Average daily TV reach in Sweden from 2008 to 2018Average daily TV reach in Sweden 2008-2018
Average daily television viewing time in Sweden from 2008 to 2018 (in minutes)Average daily TV viewing time in Sweden 2008-2018
Ranking of television channels in Sweden in 2018, by audience shareTV channels in Sweden 2018, by audience share
Share of individuals with access to TV sets at home in Sweden in 2018, by gender and age Share of individuals with access to TV sets at home in Sweden 2018, by gender and age
Do you have access to a TV set?Access to a TV set among children in Sweden 2016, by age group
Does your child have access to TV sets?Small children's access to TV sets in Sweden 2016, by age group
Number of subscriptions on television services in Sweden from 2007 to 2017, by distribution platform (in 1,000s)Number of subscriptions on television services in Sweden 2007-2017, by platform
Share of TV households receiving TV via antenna in Sweden in 2018, by gender and age Share of TV households receiving TV via antenna in Sweden 2018, by gender and age
Share of TV households receiving TV via broadband/cable in Sweden in 2018, by gender and age TV households receiving TV via broadband/cable in Sweden 2018, by gender and age
Share of TV households receiving TV via satellite in Sweden in 2018, by gender and age TV households receiving TV via satellite in Sweden 2018, by gender and age
Share of TV households with HD services in Sweden in 2016, by platformShare of TV households with HD services in Sweden 2016, by platform
Revenue of Sveriges Television AB from 2010 to 2018 (in billion SEK)Revenue of Sveriges Television AB 2010-2018
Profit of Sveriges Television AB from 2007 to 2017 (in million SEK)Profit of Sveriges Television AB 2007-2017
Number of employees of Sveriges Television AB from 2007 to 2017Number of employees of Sveriges Television AB 2007-2017
Audience share of the public service television of Sveriges Televsion AB channels in Sweden from 2007 to 2017Audience share of the public service television channels of SVT in Sweden 2007-2017
Air time of channels of Sveriges Television AB in 2017 and 2018 (in hours)Air time of SVT channels 2017-2018
Net revenue of the Swedish Educational Broadcasting Company (UR) from 2015 to 2018 (in million SEK)Net revenue of the Swedish Educational Broadcasting Company (UR) 2015-2018
Number of employees of the Swedish Educational Broadcasting Company (UR) from 2013 to 2017Number of employees of the Swedish Educational Broadcasting Company (UR) 2013-2017
Air time of Swedish Educational Broadcasting Company's (UR) TV telecasts for children and youths in 2018 (in hours)Air time of Swedish Educational Broadcasting Company's TV telecasts 2018
Television advertising expenditures in Sweden from 2011 to 2017 (in million SEK)Television advertising expenditures in Sweden 2011-2017
Revenues of selected commercial television broadcasters in Sweden in 2017 (in million SEK)Revenues of selected commercial television broadcasters in Sweden 2017
Profit or loss of selected commercial television broadcasters in Sweden in 2017 (in million SEK)Profit of selected commercial television broadcasters in Sweden 2017
Audience share of the Swedish commercial television channel TV3 from 2008 to 2018Audience share of TV3 in Sweden 2008-2018
Audience share of the Swedish commercial television channel Kanal 5 from 2008 to 2018Audience share of Kanal 5 in Sweden 2008-2018
Share of individuals taking note of TV advertising in Sweden from 2007 to 2017Share of individuals taking note of TV advertising in Sweden 2007-2017
Do you have a positive or a negative perception of advertising in TV?User evaluation of advertising in TV in Sweden 2007-2017
Average daily television viewing time in Sweden from 2008 to 2018, by gender (in minutes)Average daily TV viewing time in Sweden 2008-2018, by gender
Average daily television viewing time in Sweden from 2014 to 2018, by age group (in minutes)Average daily TV viewing time in Sweden 2014-2018, by age
Average daily television viewing time in Sweden in 2017 and 2018, by days of the week (in minutes)Average daily TV viewing time in Sweden 2017-2018, by days of the week
Share of individuals who have watched television daily in Sweden from 2009 to 2018Daily TV consumption in Sweden 2009-2018
Share of individuals who have watched television daily in Sweden in 2018, by age groupDaily TV consumption in Sweden 2018, by age
Share of individuals watching traditional TV weekly in Sweden in 2018, by ageShare of individuals watching traditional TV weekly in Sweden 2018, by age
Average daily TV reach in Sweden in 2017 and 2018, by age and genderAverage daily TV reach in Sweden 2017-2018, by age and gender
Ranking of TV channels in Sweden in 2018, by daily reachRanking of TV channels in Sweden 2018, by daily reach
Share of individuals watching online television services per day in Sweden from 2011 to 2018Share of individuals watching online TV per day in Sweden 2011-2018
Share of individuals watching online television services per day in Sweden from 2011 to 2018, by genderWatching online TV per day in Sweden 2011-2018, by gender
Share of individuals watching online television per day in Sweden from 2011 to 2018, by age groupWatching online TV per day in Sweden 2011-2018, by age group
Could you tell me to what extent you watch TV via the Internet?Frequency of TV viewing via internet in Sweden 2018
Average viewing time of online TV in Sweden daily in 2017 and 2018, by network (in hours)Daily viewing time of online TV in Sweden 2017-2018, by network
Distribution of viewing time of online TV in Sweden in 2018, by channel familiesDistribution of viewing time of online TV in Sweden 2018, by channel families
Average daily viewing time of the Swedish VOD service SVT Play from 2016 to 2018 (in hours)Daily viewing time of the Swedish VOD service SVT Play 2016-2018
Average daily viewing time of the Swedish VOD service of TV4 from 2016 to 2018 (in hours)Daily viewing time of TV4 Play in Sweden 2016-2018
Mediaset employment in Spain in 2017, by company
Percentage of employees of Mediaset España 2013-2017, by type of contract
Mean age of Mediaset employees in Spain 2013-2018
Employees of Mediaset España in 2017, by age and gender
Mediaset España net advertising revenues 2013-2017
Share of female employees of the media group Atresmedia in Spain 2013-2018
Mediaset TV programs ranked by the number of social network comments in Spain in 2015
Share of broadcast rights providers of Mediaset content in Spain in 2018, by region
Average number of online TV users in Spain in 2015, by media group
Television and radio net revenue of the media group Atresmedia in Spain 2014-2018
Television and radio gross operating income of Atresmedia in Spain 2014-2018
Goods and services' expenditure of Atresmedia in Spain 2013-2018
Subtitled programs broadcasted by Mediaset in Spain in 2018, by content type
Prisa employment worldwide in 2014, by country and gender
Personnel costs of selected regional radio and TV corporations Spain 2015
Number of incidents in the television content of Atresmedia in Spain 2009-2018
Mediaset Group
RAI (Radiotelevisione Italiana)
Mediaset: annual report 2017
Mediaset: 2017 Full Year Results Presentation
Mediaset: relazione finanziaria semestrale 2016
Mediaset: bilancio 2015
Number of TV households in Sweden from 2008 to 2017 (in millions)
Revenues from television services in Sweden from 2009 to 2017 (in million SEK)
Revenues from television services in Sweden from 2009 to 2017, by distribution platform (in million SEK)
Ranking of selected TV companies in Sweden in 2017, by revenue (in million SEK)
Ranking of TV channel families in Sweden in 2017, by audience share
Average daily TV reach in Sweden from 2008 to 2018
Average daily television viewing time in Sweden from 2008 to 2018 (in minutes)
Ranking of television channels in Sweden in 2018, by audience share
Ranking of TV programs in Sweden in 2018, by number of viewers (in 1,000s)
Share of individuals with access to TV sets at home in Sweden in 2018, by gender and age
Do you have access to a TV set?
Does your child have access to TV sets?
Number of subscriptions on television services in Sweden from 2007 to 2017, by distribution platform (in 1,000s)
Share of TV households receiving TV via antenna in Sweden in 2018, by gender and age
Share of TV households receiving TV via broadband/cable in Sweden in 2018, by gender and age
Share of TV households receiving TV via satellite in Sweden in 2018, by gender and age
Share of TV households with HD services in Sweden in 2016, by platform
Number of television channels available and established in Sweden as of December 2018, by genre
Revenue of Sveriges Television AB from 2010 to 2018 (in billion SEK)
Profit of Sveriges Television AB from 2007 to 2017 (in million SEK)
Number of employees of Sveriges Television AB from 2007 to 2017
Audience share of the public service television of Sveriges Televsion AB channels in Sweden from 2007 to 2017
Air time of channels of Sveriges Television AB in 2017 and 2018 (in hours)
Net revenue of the Swedish Educational Broadcasting Company (UR) from 2015 to 2018 (in million SEK)
Number of employees of the Swedish Educational Broadcasting Company (UR) from 2013 to 2017
Air time of Swedish Educational Broadcasting Company's (UR) TV telecasts for children and youths in 2018 (in hours)
Air time of Swedish Educational Broadcasting Company's (UR) telecasts in 2017, by topic (in hours)
Television advertising expenditures in Sweden from 2011 to 2017 (in million SEK)
Revenues of selected commercial television broadcasters in Sweden in 2017 (in million SEK)
Profit or loss of selected commercial television broadcasters in Sweden in 2017 (in million SEK)
Audience share of the Swedish commercial television channel TV3 from 2008 to 2018
Audience share of the Swedish commercial television channel Kanal 5 from 2008 to 2018
Share of individuals taking note of TV advertising in Sweden from 2007 to 2017
Do you have a positive or a negative perception of advertising in TV?
Average daily television viewing time in Sweden from 2008 to 2018, by gender (in minutes)
Average daily television viewing time in Sweden from 2014 to 2018, by age group (in minutes)
Average daily television viewing time in Sweden in 2017 and 2018, by days of the week (in minutes)
Share of individuals who have watched television daily in Sweden from 2009 to 2018
Share of individuals who have watched television daily in Sweden in 2018, by age group
Share of individuals watching traditional TV weekly in Sweden in 2018, by age
Average daily TV reach in Sweden in 2017 and 2018, by age and gender
Ranking of TV channels in Sweden in 2018, by daily reach
Could you tell me to what extent you watch television on a TV set?
Share of individuals watching online television services per day in Sweden from 2011 to 2018
Share of individuals watching online television services per day in Sweden from 2011 to 2018, by gender
Share of individuals watching online television per day in Sweden from 2011 to 2018, by age group
Could you tell me to what extent you watch TV via the Internet?
Average viewing time of online TV in Sweden daily in 2017 and 2018, by network (in hours)
Distribution of viewing time of online TV in Sweden in 2018, by channel families
Average daily viewing time of the Swedish VOD service SVT Play from 2016 to 2018 (in hours)
Average daily viewing time of the Swedish VOD service of TV4 from 2016 to 2018 (in hours)
Average daily viewing time of the VOD service of Discovery Networks Sweden from 2016 to 2018 (in hours)
Average number of employees of Mediaset España in 2017, by company
Percentage distribution of employees of Mediaset España from 2013 to 2017, by type of contract
Average age of employees in Mediaset in Spain from 2013 to 2018
Number of employees of Mediaset España n 2017, by age and gender
Net advertising revenues of Mediaset España from 2013 to 2017 (in million euros)
Female employment of Atresmedia as a percentage of the total number of employees of the media group in Spain from 2013 to 2018
Most popular Mediaset TV programs based on the number of social network comments in Spain in 2015
Percentage distribution of the broadcast rights providers of Mediaset programs in Spain in 2018, by region
Average number of online television users in Spain in 2015, by media group (in millions)
Net revenue of the media group Atresmedia generated by television and radio from 2014 to 2018 (in millions euros)
Gross operating income of Atresmedia generated by television and radio between 2014 and 2018 (in million euros)
Expenditure on goods and services of the media group Atresmedia from 2013 to 2018 (in million euros)
Total number of subtitled programs broadcasted by Mediaset TV channels in Spain in 2018, by content type
Number of employees of Prisa group worldwide in 2014, by country and gender
Personnel costs in the regional radio and TV corporations of Andalusia, Galicia and the Basque Country in 2015 (in million euros)
Average number of incidents recorded by the Spanish media group Atresmedia during the production, recording and broadcasting of television content from 2009 to 2018


Due to the fact that card-not-present credit card transactions traditionally carry a higher potential for fraud, the credit card industry uses two tools to help E-Commerce merchants determine the legitimacy of any transaction. These tools are the Address Verification Service (AVS) and Card Verification Value (CVV2) codes. These codes are returned from the card issuing banks when an authorization is obtained.
Navigate to Security Settings Fraud Screening. This is where you can change your filtering options for the transactions that you process. You can choose to reject transactions that are processed through the system based on various responses that come back from the bank processor. You can also choose to reject transactions based on the response that comes from the bank based on the CVV2 number the customer provided with their credit card.
Click on the check box next to each response you would like to reject.
Recommended Minimum Settings: We recommend that you reject at least the N responses for both the AVS and CVV2 checks. Fraudulent transactions will frequently have an N response for either the AVS or CVV2 check, so by rejecting any sale where this value is returned will provide you some protection against fraudulent transactions. You may elect to reject other responses based upon your business's needs; merchants providing goods or services in higher-risk markets or selling high-ticket items may need to be more cautious and thus be more restrictive in their AVS and CVV2 settings.
Click the Submit button at the bottom of the screen to save your changes.
X: Both address and ZIP code match.
AVS and CVV2 response codes are only sent by the card issuing bank when a charge is authorized. When the gateway receives the response from the card issuing bank, the AVS and CVV2 codes returned are checked against the codes you have elected to reject. If the AVS or CVV2 code matches a response you are rejecting, the gateway will fail the transaction and issue an immediate reversal. For further explanation of authorizations and how they affect the card balance, see authorizations vs. settlements.


Ease joint pain and stiffness, from age or injury A combination of glucosamine sulfate and chondroitin sulfate with additional nutritional cofactors, which together help maintain joint health and mobility. These nutrients are especially important during the ageing process, as the 'rebuilding' process slows down, when levels of these nutrients can decline and result in joint problems.
Glucosamine, chondroitin and sulfur are the building blocks for replacing damaged and worn away joint cartilage. A combination is thought to be more effective than Glucosamine alone. Assists with overall joint mobility. Gluten, Wheat, Dairy, Soy and Preservative Free.


text:
> servo v. : WATCH OVER, OBSERVE

>Latin.

Servo provides a simple interface for exposing and publishing application metrics in Java.  The primary goals are:

* **Leverage JMX**: JMX is the standard monitoring interface for Java and can be queried by many existing tools.
* **Keep It Simple**: It should be trivial to expose metrics and publish metrics without having to write lots of code such as [MBean interfaces](http://docs.oracle.com/javase/tutorial/jmx/mbeans/standard.html).
* **Flexible Publishing**: Once metrics are exposed, it should be easy to regularly poll the metrics and make them available for internal reporting systems, logs, and services like [Amazon CloudWatch](http://aws.amazon.com/cloudwatch/).

This has already been implemented inside of Netflix and most of our applications currently use it.

# Project Details

### Build Status

| Branch | Status |
| ------ | ------ |
| master | [![Build Status](https://netflixoss.ci.cloudbees.com/job/servo-master/badge/icon)](https://netflixoss.ci.cloudbees.com/job/servo-master/) |
| pull-requests | [![Build Status](https://netflixoss.ci.cloudbees.com/job/servo-master/badge/icon)](https://netflixoss.ci.cloudbees.com/job/servo-pull-requests/) |

### Versioning

Servo is released with a 0.X.Y version because it has not yet reached full API stability.

Given a version number MAJOR.MINOR.PATCH, increment the:

* MINOR version when there are binary incompatible changes, and
* PATCH version when new functionality or bug fixes are backwards compatible.

### Documentation

 * [GitHub Wiki](https://github.com/Netflix/servo/wiki)
 * [Javadoc](http://netflix.github.io/servo/current/servo-core/docs/javadoc/)

### Communication

* Google Group: [Netflix Atlas](https://groups.google.com/forum/#!forum/netflix-atlas)
* For bugs, feedback, questions and discussion please use [GitHub Issues](https://github.com/Netflix/servo/issues).
* If you want to help contribute to the project, see [CONTRIBUTING.md](https://github.com/Netflix/servo/blob/master/CONTRIBUTING.md) for details.

## Project Usage

### Build

To build the Servo project:

```
$ git clone https://github.com/Netflix/servo.git
$ cd servo
$ ./gradlew build
```

More details can be found on the [Getting Started](https://github.com/Netflix/servo/wiki/Getting-Started) page of the wiki.

### Binaries

Binaries and dependency information can be found at [Maven Central](http://search.maven.org/#search%7Cga%7C1%7Ccom.netflix.servo).

Maven Example:

```
<dependency>
    <groupId>com.netflix.servo</groupId>
    <artifactId>servo-core</artifactId>
    <version>0.8.0</version>
</dependency>
```

Ivy Example:

```
<dependency org="com.netflix.servo" name="servo-core" rev="0.8.0" />
```

## License

Copyright 2014 Netflix, Inc.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at:

http://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.


text:
EV maker Rivian prices IPO above range to raise over $11.9 billion
Echo Wang and Anirban Sen
November 9, 2021, 7:20 PM ·2 min read
By Echo Wang and Anirban Sen
NEW YORK (Reuters) - Amazon-backed electric vehicle (EV) maker Rivian sold shares in its initial public offering (IPO) at $78 per share, above its target range, to raise more than $11.93 billion, the company said on Tuesday.
The flotation ranks among the top 10 IPOs of all time in the United States, according to data from Dealogic.
Rivian priced an upsized IPO of 153 million shares at $78 per share, the company said. On Friday, Rivian had boosted its price range to $72 to $74 per share, up from an earlier range of $57 to $62.
The mammoth IPO follows a highly successful investor roadshow last week, with Wall Street's biggest institutional investors betting on Rivian to be the next big player in a sector dominated by Tesla.
The EV market is witnessing a surge in demand globally, as consumers are growing more environmentally conscious and betting on eco-friendly vehicles. Tesla produced a record number of cars in its latest quarter and is building factories all over the world.
Founded in 2009, Rivian has been investing heavily to ramp up production. It is doubling down on production of its upscale all-electric R1T pickup truck, which was launched in September.
The EV maker had about 55,400 R1T and R1S pre-orders in the United States and Canada as of Oct. 31, according to a recent regulatory filing.
Rivian said it had started deliveries of the R1T and had recorded revenue for the three months ended Sept. 30. The net loss for that period, however, is estimated to increase due to higher production costs.
Amazon, which recently disclosed it holds an approximately 20% stake in Rivian, has ordered 100,000 of Rivian's electric delivery vans as part of the e-commerce company's broader effort to cut its carbon footprint.
Rivian will list its shares on the Nasdaq under the symbol "RIVN" on Wednesday.
Morgan Stanley, Goldman Sachs and J.P. Morgan are the lead underwriters for the offering.
(Reporting by Echo Wang in New York and Anirban Sen in Bengaluru; additional reporting by Nishit Jogi; Editing by Chris Reese and Sherry Jacob-Phillips)
Why Unilever Stock Rose as Much as 10% Today
A day after a sharp drop, shares of the European consumer products and food maker rose. Only it wasn't a simple rebound -- Unilever did an about-face.
3 Warren Buffett dividend stocks yielding as high as 4.6%
Still earning peanuts in your savings account? These 3 income stocks might help.