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The latest in news, events, and technical articles
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fourth ruler of the Northern Song dynasty.
How ‘three Chinas’ – Emperor Renzong’s, Xi Jinping’s Communist ‘Party-State,’ and little democratic Taiwan - each in their way grappled or are grappling with these two epidemics a thousand years apart makes for some instructive contrasts.
“Outlaws of the Marsh” tells how, when a plague struck the land in 1049 AD, and people all over China snowed the imperial court under with petitions for help, the scholar-officials of Renzong’s government labored to find ways to “save the people.” A thousand years later, when doctors in Wuhan warned that a dangerous new SARS-like virus was jumping from person to person, they found no such sympathetic ear from their government. Instead, the regime’s censorship and inaction helped the virus break free into the rest of China and the world. Meanwhile, liberal democratic Taiwan began testing travelers from Wuhan weeks before Xi Jinping's party-state had even admitted to a problem in that city. Taiwan drew on widespread civic participation and trust in its people to beat back a Chinese cyber-disinformation campaign that tried to subvert the country's elections on January 11. Now the country is using the same strengths to help beat back the coronavirus more effectively than nearly anywhere in the world.
Is it intolerably naive to think that the communist political system is more prone to spawning epidemics than the liberal democratic? True, all governments face the same dilemmas when confronted by the emerging threat of an infectious disease outbreak, including the temptation to lie and delay action. These choices can be summarized as a “Pandemic Game.” Not all political regimes have the same incentives to succumb to temptation, though. In particular, there is that irrepressible need of the communists to claim they are entirely correct about everything. I suggest that, given varying incentives, a communist regime is more likely to succumb to the temptation to lie and gamble with the health of the world than a liberal democratic - or than Emperor Renzong's enlightened scholar-officials for that matter.
“Men dozed off at noon midst gay birds and flowers”
The Song dynasty at its zenith was by many accounts a time when, according to a poem by the scholar Shao Yaofu, quoted at the beginning of “Outlaws of the Marsh,”.
Poetic extravagance, admittedly, but historians concur on the basics. Despite the incursion of powerful tribes like the Khitans, Tanguts and Jurchens, “At the height of its prosperity the Song was one of the most humane, cultured and intellectual societies in Chinese history, and perhaps in all of world history,” says Dieter Kuhn in “The Age of Confucian Rule: The Song Transformation of China.”
Emperor Renzong was “one of the most able and humane rulers in Chinese history. Under him the Song government is generally believed to have come closer than ever before to reaching the Confucian ideal of just government,” says Encylopaedia Britannica. He was “merciful, tolerant, modest, lenient, and frugal,” according to the records of the official History of Song, adds Wikipedia.
In the first twenty-seven years of Renzong’s long reign, says “Outlaws of the Marsh”:
“all went well. Grain harvests were large; the people were happy at their work; no one kept articles lost by others on the road; doors were left unlocked at night.”
But then:
[the capital city], published the officially approved prescriptions and spent his own money on medicines in an attempt to save the people. But to no avail. The plague grew worse. All the high civil and military officials conferred. Then they gathered in the Hall of the Water Clock and waited for daybreak, when court would be held, so that they could appeal to the emperor.”
At court the next morning, Premier Zhao Zhe advances respectfully and tells Renzong:
“‘The plague is raging unabated in the capital. Victims among the soldiers and the people are many. We hope Your Majesty, in a forgiving and benevolent spirit, will reduce prison sentences and cut taxes, and pray to Heaven that the people be relieved.”
Deputy Premier Fan Zhongyan suggests the emperor summon the Divine Teacher of the Taoists to the capital, to conduct a great prayer service in the imperial park, for in this way, “the people will be saved.”
“Emperor Ren Zong approved Fan's proposal.”
It is tempting to saddle up and join Marshall Hong and his cavalry on their dash 600 miles southward from the capital, Kaifeng, to Jiangxi province, and to witness Hong’s
misadventures on Dragon and Tiger Mountain, which comprise the next pages in the exuberant tale of “Outlaws of the Marsh.” (Incidentally, Marshall Hong’s route south would have taken him near or through Wuhan, the incubator for the worldwide coronavirus plague of 2020.)
But let’s stay with Renzong and his worried ministers. How far can one trust the novel’s account of their efforts to stem the epidemic?
One probably shouldn’t use a “Robin Hood” adventure story stitched together out of folk narratives some three or four hundred years after the event as an authoritative historical source. Still, it is comforting that, in outline, “Outlaws of the Marsh” does rest on a skeleton of historical facts. For example, it places the epidemic in the 27th year of Renzong’s reign, or AD 1049, a date confirmed by William McNeill in “Plagues and Peoples,” a study of infectious diseases in world history. (An appendix to McNeill’s book lists some 291 epidemics somewhere in China between 243 BC and 1911 AD, one every 7-8 years, on average.)
More important, perhaps, is what the "Outlaws of the Marsh" reveals indirectly, in passing references and unstated assumptions, not about this or that policy detail, but rather about what were likely widely shared attitudes about the right political relationship between rulers and ruled.
The most telling, if easily overlooked, detail in the novel's account of the plague is this:
“The imperial court was snowed under with petitions for relief from every district and prefecture.”
That is, local people and governments in counties, prefectures, and provinces were not afraid to report the bad news to the imperial court, appear to have had no reason to shut up and stay quiet. Instead, the stout fellows expect, and forthrightly, loudly, petition for help from their rulers.
They are not mistaken in this expectation. The sense of obligation, the sympathy with which the rulers react, and try to “save the people” is apparent. The high officials confer anxiously all through the night; they try first one thing and then another; they grope forward by trial and error. There is no boasting, no effort to shift the blame, no presumption of omniscience.
Renzong’s government was ignorant of the modern microbial theory of infectious diseases, and hence of the preventive measures used today based on that knowledge, like ‘social distancing’ or vaccines. But, within the understanding of the times, its efforts are impressive nonetheless. It publicizes approved medicines and helps defray their cost. It cancels all taxes, in line with the hefty coronavirus social relief and stimulus packages that governments are rushing through today. Prisoner amnesties, unusual elsewhere, were common in the legal system of Imperial China, especially from the Han through the Song dynasties, a useful proof of the benevolence of a ruler who rightly enjoyed the favor of Heaven. Opinions will differ on the efficacy of public prayers as a method to combat the plague. But they must have been a potent reminder that, absolute autocrat though he be, the Emperor could not act arbitrarily or unjustly, but always within the moral order of Heaven.
The description of the humane, pragmatic spirit animating Renzong’s government in “Outlaws of the Marsh” accords quite well with the account of Song politics at this time in F.W. Mote’s great “Imperial China 900-1800.”
The Northern Song (960-1127 AD), was a great age in China’s intellectual history, says Mote. There was the consolidation of a scholar-official governing class selected more impartially and from a more diverse social background than ever before, through the system of civil service examinations. Some of the leading statesmen of Renzong’s reign were ‘new men,’ from relatively humble origins, who championed morally idealistic Confucian precepts and reforms at the court. Alongside the revival of Confucianism, with its supreme value on ‘ren’ - ‘humaneness’ or ‘humanity’ - Song political thinkers also valued ‘gongli sixiang,’ ‘utilitarian thought’ or ‘beneficial effectiveness.’ Reformers in the 1040s sought a practical focus on social and political problems. They tried to improve the quality of the government, for example by changing the content of the civil service examinations “from demonstrating skills in writing correct and ingenious poetry to skills in thinking about practical statecraft,” by rewarding the able and weeding out the incompetent, and by leaving more resources in the hands of the local governments.
Empire of Lies
Into how different a moral universe we fall in turning from Renzong and the enlightened scholar-officials of the Northern Song to Xi Jinping’s Communist Party-State.
Here, the officials closest to the ground - the Wuhan doctors who in December last year were treating an escalating number of patients with some severe new respiratory ailment – found no sympathetic ear for their warnings. Here, instead, the regime’s inaction and censorship helped the virus break free into the rest of China and the world, as detailed accounts in Axios, the Wall Street Journal, the New York Times, Caixin Global, National Review and other outlets show.
On December 27, doctors in Wuhan began to hear back from Chinese genomics testing companies that patient samples from local hospitals pointed to a dangerous SARS-like coronavirus. The doctors were also independently concluding that the virus was spreading from person to person. Dr. Ai Fen, head of the emergency department at Wuhan Central information reported this terrifying news to the hospital and the regional branch of the China CDC. But when a colleague, Dr. Li Wenliang, posted it to a social media group for doctors on December 30, and the information leaked to the public, the party-state jumped into action. Dr. Ai was reprimanded for “spreading rumors,” causing panic, and “damaging the stability” of Wuhan. The Wuhan Public Security Bureau hauled in Dr. Li and other doctors to recant their mistakes while arresting several members of the general public for passing on the truth that a deadly virus was afoot in Wuhan.
It was not until January 23, 2020, that the central authorities put Wuhan and three other cities under lockdown, almost four weeks after the first results from the genomics testing companies. These weeks in January are when some 150 million Chinese travel around the country to their ancestral homes for the Chinese New Year. Cell phone data suggest some 7 million people left Wuhan in these weeks, as shown in this engaging graphic in the New York Times, many of them carrying the coronavirus and helping to seed new outbreaks all over China. Thousands also flew abroad from Wuhan. As many as 15,000 flew to Bangkok, the most popular destination, where the first international coronavirus case appeared in mid-January. A recent study suggests that earlier action to curb the Wuhan outbreak could have reduced the reported number of infections in China at the end of February by up to 95%.
Playing the Blame Game
So why did Xi Jinping’s party-state, with its ambition to direct every aspect of Chinese life, fail to act sooner, setting off this great pandemic? Was it due to the dysfunction of a specific level or region within the party-state? Or is the communist system itself to blame?
On the first question, Tom Greer at The Scholar’s Stage warned towards the end of January, that:
“There is a lot of commentary out there blaming various parts of the Chinese governing system for the epidemic. Some place personal blame on Chairman Xi, others on the provincial party standing committee of Hubei, or on the mayor of Wuhan. Frankly, we still do not have enough information to make that call. We likely will not have a detailed picture of who knew what when, and who directed whom to do what, for several months still. Those who try to convince you otherwise are rumor-mongering. “
One wonders if now, two months later, there is enough data for the specialist China-watchers to make that call. The New York Times seems to think so: an informative but ultimately unpersuasive article in the paper on March 29, “China Created a Fail-Safe System to Track Contagions. It Failed,” puts the blame firmly on the heads of the Wuhan city and Hubei provincial party-state functionaries.
Thus, after China’s humiliating loss of face due to its earlier cover-up and inaction during the 2003 SARS epidemic, the country put in place a “world-class” infectious disease reporting system that was “fast, thorough and…immune to meddling.” Hospitals would input patient data into a computer and instantly notify the central health authorities in Beijing. A model of transparency! But, if the system had a flaw, it was in not accounting for the incentives that would face the all-too-human players in a real crisis. Thus, with a new SARS-like coronavirus spreading between persons in Wuhan, hospital administrators chose not to use the system, taking the bad news directly to the local party-state functionaries, on whose approval, presumably, their futures depend. You’re the boss, you decide!
According to the Times article, it was the city and provincial party-state officials, who, afraid of “upsetting Beijing,” covered up the facts from the central authority. But this is implausible at several levels. The fact that the local bosses led a cover-up from the public does not imply that they also hid the bad news from their bosses at the center. Wouldn’t such a subterfuge be a rash, career-ending act, given the severe reinforcement of centralized, top-down Communist Party-State control under Xi Jinping?
Further, there is both circumstantial and direct evidence that the center was well aware of the real situation in Wuhan from early on.
Thus, for example, China made its first report to the World Health Organization on December 31, according to its obligations under the International Health Regulations (IHR) of 2005. These Regulations were strengthened in 2005 to preclude just the sort of cover-up and inaction that China had perpetrated during the 2003 SARS crisis. They require each country’s National IHR Focal Point to report within 24 hours of assessing public health information “of all events which may constitute a public health emergency of international concern within its territory.” By this point, the center already knew of the evidence pointing to a dangerous SARS-like coronavirus and the likelihood of person-to-person transmission, if for no other reason than because, as the Times article notes, the center was aware of the social media outcry over the information leaked by Dr. Li Wenliang on December 30. In all likelihood, that information had also gone up the internal party-state channels.
Yet China’s December 31 communication with the WHO refers blandly to only a “pneumonia of unknown cause,” with “no evidence of significant human-to-human transmission.” A second communication to the WHO on January 12 reports a “novel coronavirus,” but continues to maintain “no clear evidence of human-to-human transmission.” China repeatedly refused offers of investigative help from the WHO and the US CDC. Only on February 14 did it disclose that 1700 health care workers were infected. Little wonder that a professor of international law can claim that China has – again! –breached its obligations under the IHR, and that under international law it is liable for the vast human and economic damage the coronavirus pandemic is wreaking on the world. A decision packed with such danger for China’s international standing could only have been made at a high central level.
But the most direct evidence that the highest levels of the party-state were responsible for the cover-up is simply that Xi Jinping tells us so. The party journal Qiushi (“Seeking Truth”) has released a speech by Xi Jinping to the politburo on January 7, in which he takes command and gives instructions on how to handle the crisis. That is more than two weeks before the lockdown on Wuhan and other cities.
Those were weeks when the virus was spreading fast in Wuhan, even as the Wuhan Health Commission insisted there were no new cases. That was when the Wuhan and Hubei party-state organizations held their lavish annual meetings in the city without ever mentioning the epidemic, culminating in a ‘pot luck’ banquet to celebrate the Chinese New Year for some 40,000 families. That was when infected people were streaming out of Wuhan to other parts of China and other countries; when China was continuing to mislead the WHO. An article in the Straits Times records the bewilderment among many China experts: why would Xi want to associate himself and the top party leadership with all that?
Playing the Pandemic Game: Communist Party-State versus Liberal-Democracy
Let’s put aside that particular mystery, conclude that there is enough evidence to implicate all levels of the Chinese party-state in the initial cover-up and failure to act, and return to our original question. Why would the party-state want to pursue this disastrous course for the first month of the crisis anyway? Is the communist system inherently more liable to spawning pandemics?
Tom Greer, in the article already cited, says that:
“I'm also unconvinced that the Communist system itself deserves special blame for the epidemic. The truth is that the cycle of denial, political games, and over-reaction that has marked this virus' spread fit a historical pattern that democracies have often fallen victim to (consider the events surrounding San Fransisco's brush with the bubonic plague in 1900). Down playing news of a novel disease only to pivot to an extreme, coercive response when public panic begins is common ... human pride, not something unique to Communist politics, drove those decisions.”
I agree with a lot of this. All governments do face the same strategic dilemma when confronted by an emerging infectious disease threat, including a temptation to lie and delay action. But I want to respectfully push back against the proposition that a communist one-party state has no greater political incentive to succumb to that temptation than a liberal democracy.
A World Bank paper analyzing lessons from the SARS epidemic summarizes the dilemma this way:
“Especially in the early stages, there may be considerable uncertainty whether a disease outbreak will develop into a more serious epidemic. Sometimes a rash of illness from a known or unknown source will raise concerns but then just peters out and is forgotten. Given that the outbreak might simply fizzle out of its own accord, there is an incentive for the authorities to “wait and see”. This incentive will be particularly strong when a premature announcement could lead to severe economic losses because of panic, excessive preventive actions by the public (for example flight from affected areas), declines in foreign tourism and severe and often unwarranted trade restrictions on the country’s exports imposed by other governments…
There could also be large political costs for a government that is seen to “cry wolf” and cause losses “unnecessarily”. In 1976 an outbreak of swine flu in the US raised concerns about a potential human pandemic influenza. Instead of downplaying or concealing the threat the authorities “did the right thing” and launched an aggressive public communication and preventive immunization program to counter it. Unfortunately this effort turned into a political disaster for the administration when a pandemic failed to occur and, instead, the vaccination program itself led to medical complications, some deaths and a “litigation nightmare”.
Alongside these putative benefits from delaying risk communications, there are also, of course, major potential costs. An obvious one is the increase in epidemiological risk from delays in launching public health measures and in calling on international partners for technical and financial help to counter the outbreak. This risk may be profound for new diseases such as SARS or a new pandemic influenza virus.”
We can pose this dilemma as a two-stage “pandemic game” between the government and nature. In the first stage, faced with an emerging but uncertain infectious disease threat, the government has two choices. It can cover-up and wait, or tell the truth and take prompt preventive action. At a second stage, nature acts: the infectious disease threat either just fizzles out (a false alarm), or it ramps up into an epidemic. The following two-by-two matrix outlines the four possible outcomes. Each outcome cell contains likely features of that scenario and some examples.
Note that communist and liberal democratic regimes seem to have different “revealed preferences,” at least at an impressionistic level. China consistently prefers “cover-up and wait,” as in SARS and the present coronavirus crisis. Liberal democracies like the U.S. are more willing to take a chance on truth-telling, as in the swine flu epidemics of 1976 and 2009.
The Pandemic Game matrix shows there are two ways for a government to “get it right.” Box (1), when the government covers-up and the threat fizzles out, is the best, “zero cost” outcome for any political system: there is no death, no economic disruption, and no political cost. In Box (4), in a real epidemic, the government’s early action limits death and economic damage, winning credit for its farsightedness both at home and abroad. In both these cases, there is no apparent reason why the costs or benefits should differ between a communist party-state and a liberal democracy.
The crucial difference across regimes, though, is how they handle “getting it wrong.”
Under the communist system, the Party cannot admit to getting anything wrong. Its claim to absolute political power rests on an assertion of its superior scientific wisdom and its technocratic ability to foresee and solve any problem whatsoever. Hence the endless, intolerable boastful lying in the propaganda of communist regimes. In “The Coronavirus is a Disease of Chinese Autocracy,” the political scientist Minxin Pei says:
“It should be no surprise that history is repeating itself in China. To maintain its authority, the Communist Party of China must keep the public convinced that everything is going according to plan” (my emphasis.)
Consider Box 2, where a government tells the truth and takes preventive action, but there is no epidemic. In these situations, the announcement of a potential threat itself can cause a certain amount of panic, spontaneous social distancing, and economic damage. The political cost to the government in Box (2) will be higher under a communist than under a liberal democratic regime. Under liberal democracy, no one expects the government to be right all or most of the time. The government may even get some credit for “trying to do the right thing.” If it loses the next election, there will be another a few years later. For a communist government, the public ridicule of “getting it wrong while trying to do the right thing” would be intolerable. Even with full control over the media, it would be hard for a communist government to cover-up its own highly visible campaign against an epidemic that did not happen.
The relative political costs are reversed under Box (3), in which there is a government cover-up, followed by a full-blown epidemic. Under a liberal democratic regime, a free press, separation of powers, and competition among political parties would quickly expose the original cover-up. The shame of having gambled with the lives of millions would fling the ruling party from power and keep it from recovery for long after. A communist government, on the other hand, can do the very things the Chinese party-state is doing now. It has all the tools of repression and control to cover-up the original cover-up. “East Asia has always been at war with the coronavirus,” as it were.
It can blank out any information about the cover-up from social media and suppress any dissenting voices. Dr. Ai Fen, the Wuhan doctor who first raised the alarm, gave a magazine interview exposing the cover-up, and has now “disappeared,” presumably taken into custody. It can launch a highly militarized, heavy-handed campaign to curb the pandemic, with brutal disregard for the costs to the public, such as getting welded into your home. It can launch an enormous campaign of lies and disinformation, underplaying the continuing death toll in China by several orders of magnitude, and overplaying the success of its pandemic control strategy. It can spin conspiracy theories about the origin of the virus – anywhere but China! – and extol the wisdom of Xi Jinping, The People’s Leader in the global fight against the pandemic.
The invaluable “China Uncensored” sifts through the disinformation here:
In “The Theater of State Power,” Andrew Batson, an economic researcher in Beijing, says of the party-state’s virus crackdown:
…
The leaders of the Chinese party-state believes their distinctive version of socialism is superior, and that this superiority consists of an ability to exercise state power more forcefully and effectively than other governments…Whatever finally happens with the outbreak, the one thing that the propaganda narrative will not allow is a full debate over the costs and benefits of the government’s response. The only possible answer is that only state power could solve the problem, and it did.”
Returning to the Pandemic Game, I hope to have shown that, compared to liberal democracy, the communist system always has more reason to “Cover-up and Wait,” rather than “Tell the Truth and Act Early.” The qualification is essential. I have not shown that a communist system will always choose the cover-up strategy. But it is always more likely to go down that path and gamble with the world’s health than is a liberal democracy and is deserving of special blame for that reason.
It is perhaps impossible to make a meaningful comparison on this score with Renzong’s government, so vastly greater are the knowledge and state capacity available in our times. Still, it is hard to imagine that a government as steeped in traditional Confucian values of humanity and piety as Renzong’s would want deliberately to go down so impious and cynical a path.
Another China is Possible: The Democratic Taiwan Model
To squash news of the virus in Wuhan was not the only reprehensible act of the Chinese regime in December and early January. It was also pushing a powerful cyber-disinformation campaign to undermine the January 11 elections in Taiwan and defeat President Tsai Ing-wen of the Democratic Progressive Party.
Tsai Ing-wen won by a record majority of 8.2 million. China’s cyber-assault failed abjectly, beaten by Taiwanese digital initiatives based on civic participation and trust in the people. Taiwan has now deployed these tools to devise one of the most effective coronavirus campaigns anywhere.
In “Taiwan’s Disinformation Solution,” Jacob Mchangama and Jacob Parello-Plesner say that:
“Arguably the most critical Taiwanese response to disinformation has been civic tech initiatives that harness the digital power of the people… According to [Digital Minister Audrey] Tang, immunizing democracies against disinformation from below requires trusting citizens and civil society rather than viewing them as a fickle mob ready to believe whatever outrageous rumors are being spread by the enemies of democracy. In short, when it comes to countering disinformation, citizens of democracies should be treated as a resource, not a liability.
Inside government, Tang’s approach has helped cut response time on disinformation down to two hours or less. Moreover, cooperation with civil society organizations such as g0v (pronounced “gov-zero”) has allowed Tang’s Anti-Troll Army to collect and analyze reams of data and carefully target its response in order to optimize efficiency and reach... [Such tools] can help stop the spread of disinformation in the otherwise fertile ground of end-to-end encrypted messaging services, but without resorting to surveillance, censorship, or draconian measures such as Internet shut downs.”
Taiwan, only two hours flying time from Wuhan, is one of the countries most at risk from the coronavirus. Half a million Taiwanese work in China and a million mainlanders visit Taiwan in a typical year. Yet it had experienced only 339 confirmed cases and five deaths as of April 2, in a population of 24 million. Taiwan imposed travel restrictions early – it began testing inbound travelers from Wuhan at the start of January, weeks before China had even admitted to a problem in that city! It has successfully limited local community transmission, in part with “top-down” integrated databases of health and travel information to identify cases and classify risks, but also with “bottom-up” online tools to provide transparency and foster civic participation.
In “How Civic Technology Can Help Stop a Pandemic,” Jaron Lanier and E.Glen Weyl, detail how:
“.”
Lanier and Weyl conclude:
.”
The Mandate of Heaven has descended from Emperor Renzong not to Xi Jinping's Party-State but to plucky little liberal democratic Taiwan. It is a sight that, I feel sure, will gladden the heart of that “merciful, tolerant, modest, lenient, and frugal” Emperor, as he looks down from Heaven.
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TITLE: Is there always a prime number between $p_n^2$ and $p_{n+1}^2$?
QUESTION [12 upvotes]: The following table indicates that there is a prime number p between the square of two consecutive primes.
$$
\displaystyle
\begin{array}{rrrr}
\text{n} & p_n^2 & p_{n+1}^2 & \text{p} \\
\hline
1 & 4 & 9 & 7 \\
2 & 9 & 25 & 23 \\
3 & 25 & 49 & 47 \\
4 & 49 & 121 & 113 \\
5 & 121 & 169 & 167 \\
6 & 169 & 289 & 283 \\
7 & 289 & 361 & 359 \\
8 & 361 & 529 & 523 \\
9 & 529 & 841 & 839 \\
10 & 841 & 961 & 953
\end{array}
$$
Can anyone prove that for each natural number $n$ there is always a prime number $p$, such that $p_n^2<p<p_{n+1}^2$ ?
REPLY [0 votes]: This question is Brocard's conjecture. See https://en.wikipedia.org/wiki/Brocard%27s_conjecture.
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."
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In a 1990 cover story for The Nation, Contributing Editor Kai Bird called Jimmy Carter "the very model of an ex-president." He described his work on human rights, education, preventive health care, and conflict resolution as a "return to the populist warpath, telling people what he perceives to be the hard truths on the larger issues."
Bird noted that his take on Carter wasn't altogether too common: "…he was never a liberal as defined by the party's traditional liberal constituency groups."
Yet more than 25 years later, Carter has become the moral standard-bearer for the progressive Democratic flank. As Patrick Doherty's recent Tompaine.com blog "
Want.
The.
Watch the news out of House Speaker Dennis Hastert's office today. It may well be the site of the best the debate about the continued funding of the U.S. occupation of Iraq.
Anti-war activists plan to visit the Illinois Republican's office this afternoon and to begin reading aloud the names of U.S. soldiers and Iraqis killed in the war. They say they won't stop until Hastert meets with them to discuss the $67 billion "supplemental" military spending bill that is scheduled for a House vote late today.
They want Hastert to agree to oppose the White House's request for the additional money top fund wars in Iraq and Afghanistan..
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P.O.T.G. BULLETIN (Movie):
Will Ferrell stars as long-term Congressman Cam Brady who commits a major public gaffe before an upcoming election, in Warner Bros.' “The Campaign,” a mud-slinging, back-stabbing, home-wrecking comedy.
When Brady's scandal exposed his vulnerability,.
After four consecutive terms with no opposition, Cam Brady has embraced his lifestyle as a career Congressman with a great sense of ease and entitlement…and every expectation of sliding into a fifth. Says Ferrell, “Cam’s a pretty lazy politician. He’s been touted as a possible vice presidential candidate, which shows how high his aspirations go, but that’s only because he imagines the job as a lot of ribbon-cutting, fancy balls, celebrity perks and kicking back. He’s also morally corrupt.”
Moreover, Ferrell adds, “He’s an expert at saying nothing, with that super-polished way politicians have in responding to questions with statements like, ‘Thank you very much for your concern,’ or ‘I appreciate your carving out 15 minutes of your day to come down here to speak about the problems we all face,’ and then not actually providing an answer. It was so much fun to adopt those speech patterns.”
When in doubt, Cam employs the guaranteed crowd-pleaser “Support our troops!,” with the hope that the ensuing applause will drown out any inconvenient follow-ups.
But even with such an undistinguished record, Cam might have easily ridden the wave of public indifference into another term if he hadn’t gotten sloppy. “He leaves a salacious message on what he thinks is his mistress’s voicemail and it turns out to be the home of a very respectable family with young children,” director Jay Roach reveals. “Suddenly it’s a huge story. His poll numbers plummet.”
“The Campaign” also lampoons one of Roach’s favorite PR tools: the ubiquitous catch phrase. Says the director, “People are always reaching for catchy, meme ideas to carry the essence of who they are; loaded but largely meaningless phrases for the short-attention-span public, that we all seem to fall for, time and again. I’d love to be in the brainstorming meetings for some of these and see how they come up with a winner. For Cam Brady, we went with ‘America, Jesus, Freedom.’ Amplify and repeat. Because these are the words he believes Americans want to hear. It seems that candidates can’t get anywhere now without talking about freedom as if they invented the notion, and they have to paint themselves as the most patriotic of Americans—certainly more patriotic than their opponents, who they’d like us to believe are in league with terrorists.”
Notes Ferrell, “Cam’s big slogan isn’t really a slogan. It’s not even a sentence. It’s just words, like his other battle cry, ‘Cam Brady in 0-12,’ which doesn’t even make sense, numerically, but sounds powerful and decisive.”
Opening across the Philippines on Aug. 29, “The Campaign” is distributed in the Philippines by Warner Bros. Pictures, a Warner Bros. Entertainment Company.
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Preferences - (VCOrganizer Version 8 Preview)
To enter the Global Settings, click the "Settings" (
To enter the individual friend's Settings, open your friend chat screen (from the Friends section), and select "Settings" from the menu.
These are some of the major settings you can configure
Connection SettingsConnection - Server Host, Port or Service Name. By default, it's Google Talk Server.
Compress Connection Data
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\begin{document}
\title{On projective varieties of
dimension $n+k$ covered by $k$-spaces}
\author[E.\ Mezzetti, O.\ Tommasi] {E.\ Mezzetti $^1$, O.\ Tommasi}
\address{Dipartimento di Scienze
Matematiche, Universit\`a di Trieste, Via Valerio 12/1, 34127 Trieste, Italy}
\email{mezzette@univ.trieste.it}
\address{Department of Mathematics, University of Nijmegen, Toernooiveld,
6525 ED Nijmegen, The Netherlands}
\email{tommasi@sci.kun.nl}
\date{}
\subjclass{ Primary 14N20, 14J30; Secondary 53A20, 14D05, 14M05}
\keywords{Focal scheme, second fundamental form, Gauss map, tangent
space to Grassmannian,
ruled threefolds}
\thanks{$^1$ Supported by University of Trieste (fondi 60\%), MURST,
project \lq\lq Geometria sulle variet\`a algebriche'' and by INdAM,
project \lq\lq Cohomological
properties of projective schemes''}
\headsep=40pt
\footskip=45pt
\headheight 20pt
\begin{abstract} We study families of linear spaces in
projective space
whose union is a proper subvariety $X$ of the expected dimension.
We establish relations between configurations of focal points and existence
or non-existence of a fixed tangent space to $X$ along a
general element of the family. We apply our results to the
classification of ruled $3$-dimensional varieties. \end{abstract}
\maketitle
\section*{Introduction}
Since the publication of \cite{GH79} there has been a renewal of
interest in the study of differential geometric properties of
algebraic varieties.
The bases of this study are to be found in classical
works, such as several papers by C. Segre (particularly
\cite{S:sup} and \cite{S:fuochi}).
There, topics such as the second fundamental
form of projective varieties, varieties with degenerate Gauss mapping and
in general varieties ruled by linear subspaces are introduced and
discussed.
Recently, contributions on these topics have been given by
Akivis, Goldberg, Landsberg, Rogora
(\cite{AG},\cite{L99},\cite{AGL},\cite{R}). These papers highlight
the importance of the study of the {\em focal scheme}.
The foci are
a classical tool for families of linear spaces (see \cite{S:fuochi}).
In modern algebraic geometry it has been reformulated by means of the
focal diagram in the paper of Ci\-li\-ber\-to and Ser\-ne\-si (\cite{CS})
and has been applied to the study of congruences of lines
(\cite{ABT},\cite{Arr},\cite{DePoi}).
In this paper we will
deal with families of linear spaces that generate proper subvarieties of the
expected dimension in the
projective space. For instance, let us
consider a family
$B$ of $k$-spaces in the projective space $\bb{P}^N$,
the variety $X$ ruled by $B$, and assume $\dim B=n$, $\dim X= n+k<N$.
Then, we will take into consideration
the relationship between the existence and
the properties of the focal scheme on a general space of $B$,
and the existence of spaces of dimension $\leq n+k$ tangent to $X$ along
a general space of $B$. A complete description of this
relationship for a family of lines will be given in the following theorem:
\begin{thm}\label{thm:C}
Let $B\subset\mathbb{G}(1,N)$ be a family of lines in $\mathbb{P}^N$
of dimension $n,\ n\leq N-2$. Suppose that the union of
the lines belonging to $B$ is an algebraic variety $X$ of dimension $n+1$.
Then, for all $k$ in the range $0\leq k\leq n$, the following are
equivalent:\\
(i) the focal locus on the general element $r\in B$
has length $k$;\\
(ii) X has a fixed tangent $\mathbb{P}^{k+1}$ along every general
$r\in B$. \end{thm}
There is an analogue
to Theorem \ref{thm:C} for varieties with degenerate
Gauss mapping:
\begin{thm}\label{thm:B}
Let $B$ be a family of linear subspaces of $\mathbb{P}^N$ of dimension
$k$, and denote by $n$ the dimension of $B$. Suppose that the union of
the $k$-planes of the family $B$ is an algebraic variety
$X\subset\mathbb{P}^N$ of dimension $n+k<N$. Then the following are
equivalent:\\
(i) the tangent space to $X$ is constant along general
elements of $B$;\\
(ii) for all $\Lambda$ belonging to an open set
of $B$ the focal subvariety of $B$ is a hypersurface of $\Lambda$ of
degree $n$; otherwise all points of $\Lambda$ are focal.
\end{thm}
We will apply Theorem \ref{thm:C} and Theorem \ref{thm:B} to the
study of ruled varieties of dimension 3.
Our results comprise and
complete what is shown in previous papers, such as
\cite{GH79}, \cite{R}, \cite{AGL}. It should be noted, however, that
the result in
\cite{GH79} about varieties with degenerate Gauss mapping
is not precisely stated, and that
\cite{R} considers only necessary conditions and not sufficient ones.
We will give the classification of
threefolds with a tangent 2-plane constant along lines in Theorem \ref{thm:c1},
and that of threefolds with
degenerate Gauss mapping in Theorem \ref{thm:c2}.
\begin{thm}\label{thm:c1}
Let $B$ be a surface in the Grassmannian $\mathbb{G}(1,N)$,
with $N\geq4$.
Suppose that the union of the lines belonging to $B$ is an algebraic
variety $X$
of dimension 3, and that the Gauss image of $X$ has
dimension 3. Then, along a general line of $B$ there is a fixed tangent
2-plane not contained in $X$ if and only if $X$ is one of
the following:\begin{enumerate}
\item a union of lines, all tangent to a surface $S\subset\bb{P}^N$,
whose direction at the tangency point
is not in general a conjugate direction for the second fundamental
form of $S$;
\item the union of a one-dimensional family of
2-dimensional cones,
whose vertices sweep a curve.
\end{enumerate}\end{thm}
\begin{thm}\label{thm:c2}
Let $X$ be a variety of dimension 3 with degenerate Gauss mapping.
Then, one of the following holds
\begin{enumerate}
\item\label{G2} the Gauss image of $X$ has dimension 2, and $X$ is one of
the following:
\begin{enumerate}
\item a union of lines bitangent to a surface;
\item there are two surfaces such that $X$ is a union of lines tangent to
both;
\item a union of lines tangent to a surface, and meeting a fixed curve;
\item the union of asymptotic tangent lines of a surface;
\item the join of two curves;
\item the variety of secant lines of a curve;
\item a band (see Definition \ref{band});
\item the cone over a surface, with a point as vertex.
\end{enumerate}
\item\label{G1} the Gauss image of $X$ has dimension 1,
and $X$ is built up by a composite construction of cones
and varieties ruled by osculating spaces over some curve. \end{enumerate}
All these cases are possible, and each of them always
represents a class of varieties with degenerate Gauss mapping.
\end{thm}
The plan of the paper is as follows: In Section
\ref{sec:foci}
we introduce the notion of foci for a family of linear
spaces and
we give the interpretation of foci in terms of tangent
spaces to the Grassmannian.
In Section \ref{dim3} we prove the two classification theorems for
ruled varieties of dimension 3. We prove moreover that all
surfaces $S$ appearing in
cases (a)-(d) of Theorem \ref{thm:c1} are not general, but must satisfy
the condition that the osculating space to $S$ at a general point
has dimension at most 4.
In Section \ref{s:rette}
we prove Theorem
\ref{thm:C} and establish
the properties of the focal locus in the case of varieties
ruled by lines. In the last section we consider varieties
ruled by subspaces of higher dimension.
We prove by means of an example
that Theorem \ref{thm:C} cannot be extended to a family
of subspaces of dimension $\geq2$.
However, Theorem \ref{thm:B} shows that
a description is still possible for
varieties with degenerate Gauss mapping. This result has
already
been proved by Akivis and Goldberg in \cite{AG} with differential
geometry techniques; we give now an algebraic proof of it.
\section*{Notation}
We will study projective algebraic varieties over the complex field
or, more generally,
over an
algebraically closed field $\mathbb{K}$ with
$char\ \mathbb{K} =0$.\\
$V$ will denote a linear space of dimension $N+1$ over $\mathbb{K}$,
and $\mathbb{P}^N=\mathbb{P}(V)$ the projectivization of $V$. Analogously,
$\bb{A}^{N+1}=\bb{A}(V)$
will denote the affine space associated to
$V$.\\ If $\Lambda\subset \bb{P}^N$ is a projective linear subspace,
$\hat{\Lambda}\subset V$ will denote the linear subspace associated
to $\Lambda$ such that $\Lambda=\bb{P}(\hat{\Lambda})$.\\
$[v] \in \mathbb{P}(V)$ will denote the point of $\bb{P}^N$ corresponding
to the equivalence class of $v\in V\smallsetminus\{0\}$.\\
$T_xX$ will denote the Zariski tangent space to the variety $X$ at
its point $x$,
while we will denote by $\mathbb{T}_xX\subset\bb{P}^N$ the embedded
tangent space to $X$ at $x$.\\
$G(h,V)$ will denote the Grassmannian variety of linear subspaces of
dimension $h$ in $V$. $\mathbb{G}(k,N)$ will denote the Grassmannian
of projective
subspaces of dimension $k$ of $\bb{P}^N$. We will use
the same symbol to denote the points of the Grassmannian and the
corresponding linear subspaces.
\section{Focal diagram}\label{sec:foci}
Let $B\subset\bb{G}(k,N)$ be a family of dimension $n$ of $k$-spaces
in $\bb{P}^N$.
Denote by $B'$ a desingularization of $B$ and by $\mathcal{I}$
the incidence correspondence of $B'$, with the natural projections
$$\begin{array}{ccccc} \;B'&\xleftarrow[\mspace{50mu}]{p_1}&B'
\times\bb{P}^N&\xrightarrow[\mspace{50mu}]{p_2}&\bb{P}^N\\[-6pt]
&&\cup &&\\[-6pt]
B&\xleftarrow[\mspace{50mu}]{g}&\mathcal{I}&\xrightarrow[\mspace{50mu}]{f}&\;
\mathbb{P}^N.\end{array}$$
In what follows, we will restrict ourselves to families $B$ such that
the image of $f$ (i.e. the union of the lines belonging to $B$)
is a variety $X$ of dimension $n+k$. This is the same as assuming
the general fibre of $f: \mathcal{I}\longrightarrow X$ to be finite.
\begin{Def}
A point $x\in X$ is said to be a {\em fundamental point} of the family
$B$ if the fibre $f^{-1}(x)$ has positive dimension.\\
This condition defines a closed subset of $X$ called the {\em
fundamental locus} $\Phi$ of $B$.
\end{Def}
On the basis of this set-up, we can construct a
commutative diagram of exact sequences, called the {\em focal diagram} of $B$:
$$\begin{CD}
@. @. 0 @.\\
@. @. @VVV\\
@. @. \left( p_1^*(\mathcal{T}_{B'})\right)\,_{|\mathcal{I}}@>\chi>>
\mathcal{N}_{\mathcal{I}|B' \times \mathbb{P}^N}\\
@. @. @VVV @|\\
0 @>>> \mathcal{T}_{\mathcal{I}} @>>> \mathcal{T}_{B' \times
\mathbb{P}^N}{}_{|\mathcal{I}} @>>> \mathcal{N}_{\mathcal{I}|B'
\times \mathbb{P}^N} @>>> 0\\
@. @VdfVV @VVV\\
@. f^*(\mathcal{T}_{\mathbb{P}^N}) @=
\left( p_2^*(\mathcal{T}_{\mathbb{P}^N})\right)\,_{|\mathcal{I}}@.\\
@. @. @VVV\\
@. @. \;0.
\end{CD}$$
The focal diagram is built up by crossing
the exact sequence
defining the normal sheaf to $\mathcal{I}$ inside $B'\times\bb{P}^N$
with the sequence (restricted to $\mathcal{I}$) expressing the
tangent sheaf of the product variety $B'\times\bb{P}^N$ as a product
of tangent sheaves.
\begin{Def}
The map denoted by $\chi$ in the focal diagram is
called the {\em characteristic map} of the family $B$.\\
For every $\Lambda\in B_{ns}$ the restriction of $\chi$ to $g^{-1}(\Lambda)$
is called the {\em characteristic map of $B$ relative to $\Lambda$};
it lies in the following diagram:
$$\begin{CD}
\chi(\Lambda):\ @. T_\Lambda B'
\otimes \mathcal{O}_\Lambda @>>> \mathcal{N}_{\Lambda|\mathbb{P}^N}\\
\scriptstyle @. \wr@| \wr@|\\
\textstyle @. \mathcal{O}^{m}_\Lambda @>>> \mathcal{O}^{N-k}_\Lambda (1). \end{CD}$$
\end{Def}
\begin{Def}
The condition
$$\rank{\chi} (\Lambda,x) < \min \{\rank(\left( p_1^*(\mathcal{T}_{B'})
\right)\,_{|\mathcal{I}}), \rank(\mathcal{N}_{\mathcal{I}|B' \times
\mathbb{P}^N})\}$$
defines a closed subscheme $V(\chi)\subset\mathcal{I}$ which will
be called the \emph{subscheme of first order foci} (or, simply, the
{\em focal subscheme}) of the family $B$. Analogously, $F = f
(V(\chi))$ is called the \emph{locus of first order foci}, or the
{\em focal locus} of $B$ in $\mathbb{P}^N$.
\end{Def}
By the commutative property of the focal diagram, the focal
locus has a double interpretation. Indeed, the kernel of $\chi$
and the kernel of $df$ must coincide (as subsheaves of
$\mathcal{T}_{B
\times \mathbb{P}^N}{}_{|\mathcal{I}})$). Then the focal locus
is the ramification locus of $f$. As a consequence, the fundamental
locus is contained in the focal locus. These considerations can be rephrased
by the following proposition.
\begin{prop}\label{equiv}
The following
are equivalent:\\
1. the rank of $\chi$ is maximal;\\
2. the rank of $df$ is maximal;\\
3. $V(\chi)$ is a closed proper subscheme of $\mathcal{I}$. \end{prop}
We have assumed that the union of the $k$-spaces belonging to $B$
is a variety $X$ of dimension $n+k$. By Proposition \ref{equiv},
this implies that a general
point on a general space of $B$ is not a focus. Nevertheless,
some particular spaces of $B$ can be contained in the focal locus:
they are called {\em focal spaces}. \vskip 6pt
The characteristic map is
closely connected with the structure of the tangent space to
the Grassmannian variety as a space of homomorphisms (see \cite{H92}).
Let $B$ be a subvariety of $\bb{G}(k,N)$. We can identify this Grassmannian
with the Grassmannian of linear subspaces of dimension $k+1$ of
$V$, $G(k+1,V)$. Then, by associating to each $\Lambda\in B$ the affine
cone $\hat{\Lambda}\subset\bb{A}(V)= \bb{A}^{N+1}$,
we can construct a new incidence correspondence
$\mathcal{I}'\subset B'
\times\bb{A}^{N+1}$ and projections $f',g'$:
$$\begin{array}{ccccc} \;B'&\xleftarrow[\mspace{50mu}]{q_1}&B'
\times\bb{A}^{N+1}
&\xrightarrow[\mspace{50mu}]{q_2}&\bb{A}^{N+1}\\[-6pt] &&
\mspace{-18mu}\cup&&\\[-6pt]
B&\xleftarrow[\mspace{50mu}]{g'}&\mspace{-18mu}\mathcal{I'}\ &
\xrightarrow[\mspace{50mu}]{f'}&\;
\mathbb{A}^{N+1}.\end{array}$$
Considering
$B$ as a family of subspaces in $\bb{A}^{N+1}$ yields an affine
version of the focal diagram:
$$\begin{CD}
@. @. 0 @.\\
@. @. @VVV\\
@. @. \left( q_1^*(\mathcal{T}_{B'})\right)\,_{|\mathcal{I}'}@>\chi'>>
\mathcal{N}_{\mathcal{I}'|B' \times \mathbb{A}^{N+1}}\\
@. @. @VVV @|\\
0 @>>> \mathcal{T}_{\mathcal{I}'} @>>> \mathcal{T}_{B' \times
\mathbb{A}^{N+1}}{}_{|\mathcal{I}'} @>>> \mathcal{N}_{\mathcal{I}'|B' \times
\mathbb{A}^{N+1}} @>>> 0\\
@. @Vdf'VV @VVV\\
@. {f'}^*(\mathcal{T}_{\mathbb{A}^{N+1}}) @=
\left( q_2^*(\mathcal{T}_{ \mathbb{A}^{N+1}})\right)\,_{|\mathcal{I}'}@.\\
@. @. @VVV\\
@. @. \;0.
\end{CD}$$
As in the projective case, we can define the characteristic
map $\chi'$ relative to $\Lambda$, a non-singular element of $B$,
$$\chi'(\Lambda):\ T_\Lambda B\otimes\mathcal{O}_{\hat{\Lambda}}
\longrightarrow \mathcal{N}_{\hat{\Lambda}|\bb{A}^{N+1}}.$$
If we compare the definition of the focal diagram and
the characterization of $T_\Lambda B$ as a space of homomorphisms,
we easily get the following proposition.
\begin{prop}\label{prop:h&f}
Let $\Lambda$ be a non-singular point of $B\subset\bb{G}(k,N)=G(k+1,V)$.
Let us consider $T_\Lambda B$ as a linear subspace of $T_\Lambda\bb{G}(k,N)
\cong\Hom(\hat{\Lambda},V/\hat{\Lambda})$. Then the
characteristic map $\chi'$ relative to $\Lambda$, considered
as a morphism of vector bundles, for all $v\in \hat{\Lambda}$ associates to
$\eta\in T_\Lambda B$ the normal vector $\eta (v)$.
\end{prop}
The projectivization of the usual characteristic map $\chi$
coincides with that of the affine version $\chi'$, so we have:
\begin{cor}\label{cor:h&f}
Let $\Lambda$ be a non-singular point of $B\subset\bb{G}(k,N)$.
Let us consider $T_\Lambda B$ as a linear subspace of
$T_\Lambda\bb{G}(k,N)\cong\Hom(\hat{\Lambda},V/\hat{\Lambda})$.
Then the projectivization of the characteristic map $\chi$ relative
to $\Lambda$, considered as a morphism of vector bundles, for all
$p\in\Lambda$ associates to $[\eta]\in \bb{P}(T_\Lambda B)$
the point $[\eta (v)]\in\bb{P}(V/\hat{\Lambda})$, where $v\in V$ is such that
$[v]=p$. \end{cor}
This corollary yields an interpretation of
focal points, which is particularly clear in the case of a family of lines.
\begin{rem}\label{fuochimult}
Consider a variety $B\subset\bb{G}(1,N)$ and a general line
$r\in B$. Then, the foci on $r$ are the points $p=[v]$ such
that $v\in\ker\eta$
for a non-trivial $\eta\in T_rB$. Since under our hypotheses
the rank of the general $\eta\in T_rB$ is 2, the existence
of focal points depends on the existence of rank 1
homomorphisms in $T_rB$.
Focal points with multiplicity represent a special case. A point
$p=[v]\in r$ is a focal point of multiplicity $\geq2$ if and only
if there exist two linearly independent tangent vectors
$\eta_1,\eta_2\in T_rB$ verifying
$$\begin{array}{ll} \eta_1(v)=0,&Im\ (\eta_1)\neq0,\\
\eta_2(v)\in Im\ (\eta_1),& Im\ (\eta_2)\neq Im\ (\eta_1),\end{array}$$
since the condition on multiplicity is that the composition of the
characteristic
map relative to $r$ with the natural map $V/\hat{r}
\longrightarrow (V/\hat{r})/Im\ (\eta_1)$ has not maximal rank.
Iteration of this construction provides the characterization
for focal points of higher multiplicity.
\end{rem}
By means of the Pl\"ucker embedding, we can consider
the embedded tangent
space to
$B\subset\bb{G}(1,N)$ at a point $\Lambda$. In the case of lines,
there is a connection
between the existence of focal points on $r\in B$
and the existence of a line in $\bb{T}_rB\cap\bb{G}(1,N)$.
\begin{prop}
Le $B\subset\bb{G}(1,N)$ be a family of lines. Let $r$
be a general element of $B$.
Suppose $r$ is not focal: then, there is a bijection
between the focal points on $r$ and
the lines in the intersection of the Grassmannian
$\bb{G}(1,N)$ with $\bb{T}_rB$ (embedded in $\bb{P}(\bigwedge^2V)$). \end{prop}
\proof
We know that $[v]\in r$ is a focal point if and only if $v\in\ker\eta$,
where $\eta\in T_rB$ has rank 1. With a simple computation,
it is possible to prove that if a homomorphism $\eta\in T_r\bb{G}(1,N)$ has
rank 1 then the pencil of lines passing through $\bb{P}(\ker\eta)$ and
lying in $Im\ (\eta) \oplus r$ is a line contained in the
intersection of $\bb{T}_rB$ with the Grassmannian.
The converse is also true:
if there is a line in the intersection, then we can find a
homomorphism $\eta$ of rank 1 and hence a focal point.
\qed
\section{Varieties of dimension 3}\label{dim3}
We will apply the study of the focal locus to the specific
problem
of classifying ruled varieties of dimension 3 with
degenerate tangential properties. More precisely, we will consider:
\begin{enumerate}
\item\label{a} varieties ruled by lines with a
constant tangent 2-plane along
any line of the ruling;
\item varieties ruled by lines with a constant tangent space of dimension 3
along every line;
\item varieties ruled by planes with a constant tangent space of dimension
3 along every plane.
\end{enumerate}
Note that the last two cases yield the classification
of varieties of
dimension 3 with degenerate Gauss mapping. In the proofs we will
use also some results to be proved in Sections 3 and 4.
The classical references for our approach to classification
are the works of C. Segre. Particularly, a classical proof of
the classification of case \ref{a} can be found in \cite{S:fuochi}.
The classification of varieties of dimension 3 with degenerate
Gauss mapping has
already been presented recently in \cite{R} and \cite{AGL}.
In both papers, the classification is based on the study of
the focal scheme
of the family of fibres of the Gauss map, but there
is no distinction between
strict focal locus and (total) focal locus
(see Definition \ref{d:strict}). In \cite{R} the
classification is outlined
without
a study of the second fundamental form of focal surfaces.
Therefore, there is no description of how to construct a
variety with degenerate Gauss mapping. In \cite{AGL}
one of the cases (that of bands) is not
completely solved.
\vskip 12 pt
In what follows, the concept of {\em conjugate directions}
for the second fundamental form will naturally arise. We will denote
by $$II_y: T_yY\otimes T_yY\longrightarrow N_yY$$
the {\em second fundamental form} of a variety $Y$ at a non-singular point
$y$ (for the definition, see \cite{H92}). It is a symmetric bilinear
form, so it can be
interpreted as a linear system of quadrics
$|II_y|$ in
$\bb{P}(T_yY)$. The dimension of the linear system is linked with the
dimension of the second osculating space to $Y$ in $y$,
$T^{(2)}_yY$, by the relation
$$\dim|II_y|=\dim T^{(2)}_yY-\dim T_yY-1.$$
\begin{Def}
Let $Y\subset\bb{P}^N$ be a variety, and $y$ be a non-singular point of it.
Then two tangent vectors $v,w\in T_yY$ are said to represent {\em
conjugate directions}
at $y$ if $II_y(v,w)=0$. This means that the
points $[v],[w]$ are conjugate with respect to all quadrics in
$|II_y|$.\\ If there is a
selfconjugate tangent vector, its direction is
called an {\em asymptotic direction} at $y$.
\end{Def}
The existence of conjugate directions at every non-singular point is
not a general fact. It is related to the dimension of the second
osculating space to the variety at the general point. We are
interested in the study of conjugate directions for surfaces.
For general surfaces at general points the dimension of the second
osculating space is 5. In this case, at a general point there
are no conjugate directions. Conjugate directions exist only if the
dimension of the second osculating space is $\leq4$.
If the dimension is 3, every direction possesses a conjugate
direction. It is well known
that in $\bb{P}^N$, $N\geq4$, this property holds only for
developable surfaces, i.e. cones and varieties
swept out by tangent lines to a curve.
\begin{Def}[\cite{S:sup}]
A surface is called a $\varPhi$ surface if and only
if the dimension of its second osculating space at the general point is 4.
\end{Def}
\begin{prop}
For a surface $S\subset\bb{P}^N$, $N\geq5$, the following properties are
equivalent:
\\(i) $S$ is a $\varPhi$ surface;
\\(ii) at a general point of $S$ there is exactly one couple of conjugate
directions, or one asymptotic direction;
\\(iii) the union of the tangent planes to $S$ is a
variety of dimension $4$
with tangent space fixed along those planes.
\end{prop}
\proof The equivalence of the first two properties is a
consequence of the fact that a linear system of quadrics in $\bb{P}^1$
admits exactly one couple of conjugate points if and only if its
dimension is 1.
Let now $S\subset\bb{P}^N$ be a surface: let us denote by $V$ the
closure in $\bb{P}^N$ of the union of the tangent planes to $S$ at
its non-singular points. Then the dimension of $V$ is 4 if and only if
$S$ is not a developable surface or a plane. Let us consider the
osculating space $\bb{T}^2_pS$ to $S$ at a general point $p$, embedded
in $\bb{P}^N$. Using a local parametric representation of $S$, it is
easy to show the following equality:
$$\bb{T}^2_pS=\overline{\bigcup_{q\in\bb{T}_pS\cap V_{ns}}\bb{T}_qV}.$$
This implies that the tangent space to $V$ is constant along planes
if and only if the dimension of the osculating space to $S$ equals the
dimension of $V$.
Hence, the equivalence of (i) and (iii) is established. \qed
\begin{rem}
The general situation for the union of tangent planes
to a surface is that the fibres of the Gauss map are 1-dimensional.
\end{rem}
When we have a $\varPhi$ surface $S$, we can always construct an
irreducible family $\Sigma$ of dimension 2, whose elements are
lines tangent to $S$, such that for each general point $p\in S$
there is exactly one line of $\Sigma$ tangent to $S$ at $p$, and
moreover its tangent direction at $p$ is conjugate to some (other)
tangent direction.
In this
case, we will say that the lines of $\Sigma$ {\em admit a conjugate
direction} on $S$. If at
the general point of $S$ the
two conjugate directions coincide, i.e. there is an asymptotic
direction, then the lines of $\Sigma$ are called
{\em asymptotic lines} on $S$.
\vskip 12pt
Let us consider case \ref{a} first. In this case,
we have a 3-dimensional variety $X$ which is covered by the lines
belonging to a surface $B$ in the
Grassmannian $\bb{G}(1,N)$, such that $X$ has a constant tangent
plane along a general line of $B$.
A classification of these families is provided by
Theorem \ref{thm:c1} (see also \cite{M}).
\vskip 6pt
\par{\em Proof of Theorem \ref{thm:c1}. }
We can apply Theorem \ref{thm:C} to the family $B$.
The existence of the tangent plane implies
then that on the general line belonging to $B$
there exists one focal point
(with multiplicity 1).
This focus cannot be a
fixed point $p$.
In this case there would be
a $2$-dimensional subfamily of lines of $B$ passing through $p$, and $p$
would be a focal point of multiplicity 2, which is not allowed.
Then, considering the closure of the union of the focal points on
such lines (the {\em strict} focal locus, in the terminology to be
introduced in \S 3), we get two possibilities: we can obtain a surface
$S$, or a curve $C$. In the former case, the first part of the claim
follows from Theorem \ref{thm:A}. The exception considered in our
statement is necessary in order to exclude varieties with degenerate
Gauss mapping, as we will see later. In the latter case, $C$ lies
in the fundamental locus of $B$, which yields the second part of the
claim.
By a direct calculation, we can
check that the union of tangent
lines to a surface $S$ has a constant tangent plane along a
general line $r$. This plane is the tangent plane to $S$ at the
point of tangency of $r$. Analogously, the fixed tangent plane along the
lines of a cone is contained in the tangent space to the union of cones. \qed
\begin{rem}
In the hypotheses of Theorem \ref{thm:c1} we have excluded the (trivial)
case of varieties $X$ ruled by planes. In this case, the Fano variety of
lines has dimension $>2$, but it is always
possible to find a subvariety $B$ of it, with $\dim B=2$, such that
the lines of $B$ cover $X$. There are two ways of
constructing $B$. We can choose a unisecant curve $C$ to the family of
planes and consider for every plane the pencil of lines
with center the corresponding point of $C$. The points of $C$ are fundamental
points of $B$, and in general they are not singular points of $X$.
Note that a general ruled surface in $\bb{G}(1,N)$ gives an example of
this situation. Otherwise, inside every plane we can fix a curve
(varying algebraically with the plane) and consider the family of its
tangent lines. Also in this case the focal points are not in general
singular for $X$. In fact, in both cases the focal points have no real
geometric meaning for $X$. \end{rem}
We will prove now Theorem \ref{thm:c2}, giving the classification of
varieties of dimension 3 with degenerate Gauss
mapping.
\vskip 6pt
\par{\em Proof of Theorem \ref{thm:c2}. }
Part \ref{G1} is well known and classical. We give
here a simple proof based on the analysis of foci.
Let us suppose that $X\subset\bb{P}^N$ is a 3-dimensional
variety with Gauss map whose
fibres have dimension 2. Let us consider the
family $B\subset\bb{G}(2,N)$ of the fibres of the Gauss map of $X$.
Then, by Theorem
\ref{thm:B}, there is a focal line on every general plane of $B$.
If there is a fundamental line $L$, $X$ must be a cone over a curve,
with vertex $L$. Otherwise, the focal locus is a ruled
surface $S$, and, by Theorem \ref{thm:A}, every plane of $B$ is tangent to
$S$ along a line of its ruling. Hence $S$ is a surface with
degenerate Gauss mapping, so $S$ is a cone or the tangent
developable to a curve. In the first case, $X$ is a
cone, with a point as vertex, over the tangent developable to a curve.
In the second case, $X$ is the union of osculating planes to a curve.
We will prove now part 1.
For more details see also \cite{tesi}.
Let $B\subset\bb{G}(1,N)$ be the
family of fibres of the Gauss map of $X$. By
\ref{thm:C}, on a general line of $B$ there are two foci (counting
multiplicity). Then, we will consider the number of distinct focal
points on a general line belonging to $B$, the number (1 or 2) and the
dimension of the irreducible components of the strict focal locus
(see Definition \ref{d:strict}), i.e. the variety obtained as closure
of the union of focal points on non-focal lines of $B$. This is a
general procedure, which will be extended to varieties of higher
dimension in \S \ref{s:rette}.
If the focal points on a general line
of $B$ are distinct, Theorem \ref{thm:A} gives us the classification
of all possible cases, as arranged in Table~1.
\begin{table}
\begin{tabular}{|m{4cm} | m{3cm}| m{3cm}|}
\hline\hline
foci on a general line & strict focal
locus & description\\ \hline\hline
two distinct points&$\mspace{-9mu}$\begin{tabular}{m{3cm}|m{3cm}|}
each point sweeps a
surface&$\mspace{-9mu}$\begin{tabular}{m{3cm}}{{union of lines
bitangent to a surface}}
\\\hline{{union of lines tangent to two surfaces}}\\\end{tabular}\\ \hline
a point sweeps a surface, the other sweeps a curve &
union of lines tangent to a surface and meeting a curve\\\hline
each point sweeps a curve&$\mspace{-9mu}$\begin{tabular}{m{3cm}}secant
variety of a curve \\\hline join of two curves\\\end{tabular}\end{tabular}\\
\hline
\end{tabular}
\caption{Two distinct foci}
\end{table}
If on a general line of $B$ there is one focal point
with multiplicity 2, we need more information.
That can be provided considering the interpretation, given in
Section \ref{sec:foci}, of the
characteristic map of $B$
relative to a general $r\in B$ as describing
the subspace $T_rB\subset T_r\bb{G}(1,N)\cong\Hom(\hat{r},V/\hat{r})$.
Using it,
we will prove now that, if the strict focal locus is a surface $S$, then a
general line of $B$ represents an asymptotic direction of $S$, i.e. a
selfconjugate direction with respect
to the second fundamental form of $S$.
Let $r$ be a general fibre of the Gauss map and $F=[v]$ be the
(double) focus on $r$:
then by Remark \ref{fuochimult} there exist two linearly independent
tangent vectors $\eta_1$,
$\eta_2$ in
$T_rB$ such that $\eta_1(r)=0$ in $V/\hat r$ and $\eta_2(v)\in
Im\ (\eta_1)$. Let $\{b_1(t)\}$
be an arc of smooth curve in $B$, parametrized by an open disc
containing the origin, with
$b_1(0)=r$ and
$b_1'(0)=\eta_1$ and let
$\{c_1(t)\}$ be a lifting of
$\{b_1(t)\}$ through $F$, i.e. any regular curve in $X$ such that
$c_1(t)\in b_1(t)$ for all
$t$ and $c_1(0)=F$. Then $r$ is the tangent line to the curve
$\{c_1(t)\}$ at $F$. In
particular, the curve
$C$ ($\subset S$) generated by the unique focus of $b_1(t)$ as $t$
varies in the disc
is such a lifting. If
$\{g_1(t)\}$ is another lifting of $\{b_1(t)\}$ but with $g_1(0)\neq
F$, then the tangent
vector
$g_1'(0)$ is not parallel to $r$ so, together with $r$, it generates
the tangent plane at
$g_1(0)$ to the ruled surface
$Y$, union of the lines $b_1(t)$. Since dim $Im\ (\eta_1)=1$, this
plane is constant along
$r$, so
it coincides with the osculating plane to the curve $C$ at $F$.
Let now $\{b_2(t)\}$ be a regular curve in $B$ such that $b_2(0)=r$
and $b_2'(0)=\eta_2$: if
$\{d_2(t)\}$ is a lifting of its
through $F$, then its tangent line
at $F$ is contained in the
osculating plane to $C$ at $F$. In particular, we can choose as
lifting the curve $D$ of the
foci of the lines $b_2(t)$. Because of the generality assumptions,
the tangent plane to $S$
at $F$ is generated by the tangent lines to $C$ and $D$, so it is the
osculating plane to
$C$. We have thus proved that through a general point $F$ of $S$ there
is a curve $C$ whose
osculating plane at $F$ coincides with the tangent plane to $S$. The
tangent line to $C$,
which is a general line of $B$, is therefore an asymptotic tangent
line of $S$: this proves
our claim.
\begin{table}
\begin{tabular}{|m{4cm} | m{3cm}| m{3cm}|}
\hline\hline
foci on a general line & strict focal locus & description\\
\hline\hline
a point with multiplicity 2&$\mspace{-9mu}$\begin{tabular}{m{3cm}|m{3cm}|}
the focal point sweeps a surface & union of asymptotic lines\\\hline the focal
point sweeps a curve &band\\\hline
the focal point is fixed & cone\end{tabular} \\
\hline
\end{tabular}
\caption{One double focus}
\end{table}
If the strict
focal locus is a curve $C$, then we we will show that $X$ is not just
a union of
cones, as in the
case in which the focal point has multiplicity 1, but a
union of planes tangent to $C$. We proceed as in the previous case:
let $r\in B$ be a general
line and $F=[v]$ be its focus. Since $F$ is a fundamental point for
the family $B$, there is a
curve $Z$ in the Grassmannian, passing through $r$, which represents
the lines of $B$
through $F$. It is easy to show that every lifting of $Z$ through $F$
has $r$ as tangent line
at $F$, so $\eta_1$, the tangent vector to $Z$ at $r$, is such that
$\eta_1(v)=0$. But $F$
is a focus with multiplicity two and dim $T_rB=2$, so it follows that
every regular curve
contained in $B$, passing through $r$ but with tangent vector
$\eta_2$ different from
$\eta_1$, is such that $\eta_2(v)$ belongs to the image of $\eta_1$.
The focal curve $C$ can
be interpreted as a lifting of such a curve: let $w$ be its tangent
vector at $F$, then the
plane generated by $r$ and $w$ contains also the tangent line to any
lifting of $Z$ at its
intersection point with $r$. Let $\phi(t)$ be a local parametrization
of such a lifting,
with $\phi(0)=P\in r$,
then we have that $\phi'(0)$
lies in the plane generated by $w$ and the direction of $r$.
By repeated derivations,
we get that all derivatives $\phi^{(k)}(0)$ belong to this plane,
hence the whole curve is
contained in it. Therefore every lifting of $Z$ is a plane curve,
which proves that the lines
of $B$ passing through $F$ form a pencil, contained in the plane
generated by $r$ and by the
tangent line to $C$ at $F$.
So $B$ is a ruled surface. In this
case, $X$ is called a
3-dimensional band.
The precise definition is the
following (see \cite{AG}):
\begin{Def}\label{band}
A variety $X\subset \mathbb{P}^N$ is said to be a
\textit{3-dimensional band} if
there exist two distinct curves $C,D\subset X$, not belonging both
to the same $\mathbb{P}^3$, and a birational equivalence $\psi: C
\longrightarrow D$, such that $X$ is the closure of the union of the
planes lying in the image of the morphism: $$\begin{array}{lccc}f:
&C_0 &\longrightarrow &\mathbb{G} (2,N)\\&p &\longrightarrow
&\la\mathbb{T}_pC,
\psi(p)\ra,\end{array}$$ where $C_0$ is a non-singular open
subset of $C$ contained in the domain of definition of $\psi$.\end{Def}
Table~2 describes all varieties ruled by a 2-dimensional family
of lines with a focal point of multiplicity 2 on the general line.
\qed
\vskip 12pt
By a direct calculation, it is possible to find out that every variety
obtained in Theorem \ref{thm:c1} is a variety with degenerate Gauss
mapping.
The interesting point
is that, whereas any curve can be obtained as the focal
curve of a $3$-dimensional variety with degenerate Gauss mapping, the
focal surfaces must verify some special conditions. For instance, it
is not a general fact for a surface that there exists a family of
dimension $2$ of bitangent lines.
\begin{thm}
Let $X\subset\bb{P}^N$ be a variety of dimension 3
with Gauss image of dimension 2, satisfying one
of the conditions (a)-(d) in Theorem \ref{thm:c2}. \\
Suppose that the strict focal
locus of the family $B$ of the fibres of the Gauss map of $X$
has an irreducible component $S$ of dimension 2.
Then $S$ is either a developable surface or a $\varPhi$ surface. Moreover the
lines of
$B$ are tangent to $S$ and they are either asymptotic
tangent lines or they admit a conjugate direction.
\end{thm}
\proof
If $X$ is as in (d), then the Theorem is clearly true. So we
assume that on a general fibre
of the Gauss map there are two distinct foci. Let $F_1\in S$ be general: it is
a focus on a non-focal line $r$, which contains also a second focus $F_2$. So
there exist two tangent vectors $\eta_1, \eta_2\in T_rB$, such that, for all
regular curves
$\{b_i(t)\}\subset B$,
$i=1,2$, with $b_i(0)=r$ and $b_i'(0)=\eta_i$, every lifting through
$F_i$ has $r$ as tangent
line at $F_i$. As a lifting of $\{b_1(t)\}$, we can choose a curve
$C_1\subset S$, with local
parametrization $\{c_1(t)\}$, such that $c_1(t)$ is a focus of the
line $b_1(t)$ for all $t$.
Note that $Im\ (\eta_1)$, which is 1-dimensional, is
generated by the tangent
vector $x'(0)$ for all choice of a lifting $x(t)$ of $b_1(t)$ with
$x(0)\neq F_1$. Hence, as
in the proof of Theorem \ref{thm:c2}, one proves that $x'(0)$
belongs to the osculating
plane to the curve $C_1$.
Assume now that $X$ satisfies conditions (a) or (b).
Then, the previous construction
can be repeated
for the second focus $F_2$ on $r$
relatively to the focal surface $S'$ to which it belongs,
which coincides with $S$
in case (a) or is the second component of the strict focal locus of
$X$ in case (b). This
gives a second curve $C_2\subset S'$ passing through $F_2$ and
with $\bb T _{F_1}C_1=r=\bb T _{F_2}C_2$. Let now.
$D_2$ be the curve generated by the second focus of the lines $b_1(t)$, and
similarly $D_1$ be the curve generated by the second focus of the lines
$b_2(t)$. Note that
$C_1\neq D_1$ and
$C_2\neq D_2$. We can choose
a local parametrization for $S$ of the form
$\psi(t,s)$, where $\psi(0,0)=F_1$,
$\psi(t,0)$ and $\psi(0,s)$ are local parametrizations of
respectively $C_1$ and $D_1$.
By considering the other focus, we get a parametrization $\phi(t,s)$
of the second surface $S'$ near $F_2$
such that
$\phi(t,0)$ and $\phi(0,s)$ are local parametrizations of
respectively $D_2$ and $C_2$.
By comparing the tangent vectors, we get: $\psi_t=\phi_s$,
$\phi_t\in\langle\psi_t,\psi_{tt}\rangle$,
$\psi_s\in\langle\phi_s,\phi_{ss}\rangle$, and also
$\psi_{tt}\in\langle\phi_t, \phi_s \rangle$,
$\phi_{ss}\in\langle\psi_t,\psi_s\rangle$.
So we can deduce that $\psi_{ts}\in\langle\psi_t,\psi_s\rangle$. Hence the
pair of vectors $(\psi_t, \psi_s)$ annihilates the second fundamental form of
$S$, and they represent conjugate directions.
If we are in case (c), $F_2$ is a fundamental point for the family $B$, so
there are infinitely many lines of $B$ through $F_2$. Each of them contains
also a second focus, describing a curve $E$. In this case we can find a local
parametrization of $S$, $\psi(t,s)$, centred at $F_1$
and such that
$\psi(t,0)$ describes $C_1$ and
$\psi(0,s)$ describes $E$. Note that $\psi_t(0,s)$ is the direction of the
line of the ruling passing through $\psi(0,s)$, and $\psi_{ts}$ is tangent at
$F_1$ to the cone of vertex $F_2$ on the curve $E$, therefore it is contained
in the tangent plane to this cone along $r$. But this plane is generated by
$\psi_t$ and $\psi_s$, so it coincides with the tangent plane to $S$ at the
point $F_1$. We conclude then as in the previous case.
\qed
\vskip 12pt
We will close this section with a remark on the second fundamental
form. It is known
(\cite{GH79}) that the second fundamental form of the varieties with degenerate
Gauss mapping has non-empty singular locus. In
particular, this singular locus is a point in the case of varieties
of dimension
3 with Gauss image of dimension 2. Assume that $X$ is such
a variety, which is not a hypersurface. Then there is a connection between the
properties of the second fundamental form and the configuration of
focal points on the general fibre of the Gauss map of $X$. Indeed, if $X$ is
a variety with distinct focal points of multiplicity 1, then the
dimension of the second osculating space is 5 and the second fundamental form
is a pencil of conics with a point both as base and as singular
locus. If $X$ has one focal point of multiplicity 2 on the general line and
is not a cone, then the dimension of the second osculating
space is also 5, but the pencil of conics of the second fundamental form has
a line as
base locus. In the case of cones over
a surface, the dimension of the second osculating space is 6 (in general),
so the second fundamental form is a net of conics and the base locus can
only be a point, coinciding with the singular point.
\
\section{Varieties covered by lines}\label{s:rette}
Let $B\subset\mathbb{G}(1,N)$ be a family of lines in $\mathbb{P}^N$ of
dimension $n\leq N-2$. Suppose that the union of the lines belonging to $B$
is an algebraic variety $X$ of dimension $n+1$. When this
condition holds, we do not expect in general cases to find any focal point.
In particular, a general line of $B$ cannot be focal. This
allows us to consider the length of the focal locus on the general $r\in B$.
It turns out that such length has a geometric interpretation
in terms of fixed tangent spaces along $r$. Theorem \ref{thm:C} states that
the length of the focal locus on $r\in B$ is $k$ if and only if
$X$ possesses a fixed tangent space of dimension $k+1$ along a
general line $r$.
We will prove it now.
\vskip 12 pt
\par {\em Proof of Theorem \ref{thm:C}.}
We can suppose without loss of generality that $X$ is a hypersurface,
i.e. $N=n+2$.
Indeed, if $X$ is not a hypersurface, we can project it to $\mathbb{P}^{n+2}$,
and a general projection will not affect either its
tangential properties or its focal ones.
Let $r$ be a general point of $B$. Suppose that on $r$ there are exactly
$k$ focal points, counting multiplicity. They are
the points where the characteristic map relative to $r$,
$$\begin{CD}\chi(r):\ @. T_rB\otimes\mathcal{O}_r@>>>
\mathcal{N}_{r|\bb{P}^N}\\ @. \wr@| \wr@|\\
@.\mathcal{O}^n_r@.\mathcal{O}_r(1)^{n+1}
\end{CD}$$
has not maximal rank. If we choose projective coordinates
$x_0,x_1$ on $r$, by means of the natural identification
given above, we can represent $\chi(r)$ by an $n\times
(n+1)$ matrix $$A=\begin{pmatrix}l_{1,1}&\dots&l_{1,n}\\
\vdots&\cdots &\vdots
\\l_{n+1,1}&\ldots&l_{n+1,n}\end{pmatrix},$$ whose
entries $l_{i,j}$ are linear forms in $x_0,x_1$. Let us
consider the minors (with sign)
of $A$ of maximal order, $$\phi_i=(-1)^{i+1}\det
\begin{pmatrix}l_{1,1}&\dots&l_{1,n}\\ \vdots&\cdots &\vdots
\\\widehat{l_{i,1}}&\dots&\widehat{l_{i,n}}\\ \vdots&\cdots &
\vdots \\l_{n+1,1}&\ldots&l_{n+1,n}\end{pmatrix},\ i=1,\dots,n+1.$$
The existence of $k$ focal points implies that $\phi_1,\dots,
\phi_{n+1}$ have a common factor $F$ of degree $k$. So we have
the relations $\phi_i=F \psi_i$, where $\psi_1,\dots,\psi_{n+1}$ are suitable
polynomials of degree $n-k$ in $x_0,x_1$. We are interested
in finding vectors tangent to $X$ at every point of $r$.
This means we seek
normal vectors of coordinates $(v_1,\dots,v_{n+1})$
belonging to the image of $\chi(r)$ in every point of $r$. This can
be expressed by
the condition
$$\det\begin{pmatrix}v_1&l_{1,1}&\dots&l_{1,n}\\
\vdots&\vdots&\cdots &\vdots \\v_{n+1}&l_{n+1,1}&
\ldots&l_{n+1,n}\end{pmatrix}=0,$$
or, equivalently, \begin{equation*}\tag{*}v_1\psi_1+
\dots+v_{n+1}\psi_{n+1}=0.\end{equation*} As there are
$n-k+1$ monomials of degree $n-k$ in $x_0,x_1$, equation
(*) is equivalent to a system of $n-k+1$ homogeneous linear
equations
in the indeterminates $v_1,\dots,v_{n+1}$. So there are at
least $k$ linearly independent solutions. Denote by $V'$
a linear subspace of dimension $k$ of the space of solutions. If we identify
$V/\hat{r}$ with a subspace $W$ complementary of $\hat{r}$,
the vectors of $V'\subset V/\hat{r}$ are tangent to $X$ at every point of
$r$. Then $\mathbb{P}(V')$ is a tangent subspace of
dimension $k+1$ contained in the tangent space to $X$ at every point
of $r$. That
proves implication (i) $\Rightarrow$ (ii).
Let $r$ be a general point of $B$. Now we will prove that
if there is a constant tangent space of dimension $k+1$ along
$r$ then there are $k$ focal points on $r$ (counting multiplicity).
As in the previous part, we will denote by $A=(l_{i,j})$ the matrix
representing $\chi(r)$. What we want to show is that the minors
$\phi_1,\dots,\phi_{n+1}$ have a common factor of degree $k$.
We know that condition $$\det\begin{pmatrix}v_1&l_{1,1}&\dots&l_{1,n}\\
\vdots&\vdots& \cdots &\vdots
\\v_{n+1}&l_{n+1,1}&\ldots&l_{n+1,n}\end{pmatrix}=0$$
is satisfied for every $v=(v_1,\dots,v_{n+1})$
belonging to a normal subspace of dimension $k$.
We can assume without loss of generality that this normal subspace is
$$V'=\la(\overbrace{0,\dots,0}^{n-k+1},1,\overbrace{0,
\dots,0}^{k-1}),( \overbrace{0,\dots,0}^{n-k+2},1,\overbrace{0,\dots,0}^{k-2}),
\dots,(\overbrace{0 ,\dots,0}^n,1)\ra.$$ This is the
same as supposing that the last $k$ minors of order $n$ of $A$ are 0, i.e.
$$\phi_{n-k+2}=\dots=\phi_{n+1}=0.$$
In the following, we will denote by $A^{j_1,\dots,j_h}_{i_1,\dots,i_h}$
the determinant of the square submatrix of the $j_1,\dots,j_h$-th rows
and the $i_1,\dots,i_h$-th columns of $A$.
Let us consider the remaining forms $\phi_{1},\dots,\phi_{n-k+1}$.
Being minors of the matrix $A$, they satisfy the following
homogeneous system of degree 1
$$\sist
l_{1,1} \phi_1 + l_{2,1} \phi_2 + \dots + l_{n-k+1,1}
\phi_{n-k+1} = 0\\ \hdotsfor{1}\\
l_{1,n} \phi_1 + l_{2,n} \phi_2 + \dots + l_{n-k+1,n} \phi_{n-k+1}
= 0.\sistt$$ Fix two equations of the system above by
choosing two
indices $1\leq i_1<i_2\leq n$.
We can multiply the first equation by $l_{n-k+1,i_2}$,
the second one by $l_{n-k+1,i_1}$ and subtract: we
get a homogeneous relation
among $\phi_1,\dots,\phi_{n-k}$ with coefficients of degree 2.
Considering every possible choice of $i_1,i_2$,
we obtain the homogeneous system
$$\sist
A^{1,n-k}_{i_1,i_2}\phi_1+\dots+A^{n-k,n-k+1}_{i_1,i_2}\phi_{n-k}=0\\
1\leq i_1<i_2\leq n.\sistt
$$
In an analogous way we can find homogeneous relations with coefficients
of every degree between 2 and $n-k$, involving less and less minors.
For the highest degree we have a system of $\binom{n}{k}$
equations in 2 minors. For $\phi_1,\phi_2$, for instance, we have
$$\sist
A^{1,3,4,\dots,n-k+1}_{i_1,\dots,i_{n-k}}\phi_1+
A^{2,3,\dots,n-k+1}_{i_1,\dots,i
_{n-k}}\phi_2=0\\
0\leq i_1<i_2<\dots<i_{n-k}\leq n.\sistt
$$
As the relations belonging to this system cannot all be trivial,
we get that $\phi_1$ and $\phi_2$ must have a common factor of
degree $\geq k$. Moreover, it is possible to prove that
$\phi_1,\dots,\phi_{n-k}$ have a common factor of degree $k$ .
In fact, consider the system of relations (of degree $n-k-1$)
among 3 minors, say $\phi_1,\phi_2,\phi_3$:
$$\sist A^{1,4,\dots,n-k+1}_{i_1,\dots,i_{n-k-1}}\phi_1+A^{2,4,
\dots,n-k+1}_{i_1, \dots,i_{n-k-1}}\phi_2+A^{3,4,\dots,n-k+1}_{i_1,
\dots,i_{n-k-1}}\phi_3=0\\ 0\leq i_1<i_2<\dots<i_{n-k-1}\leq n.\sistt
$$
Denote by $F$ a common factor of degree $k$ of $\phi_1,\phi_2$.
Suppose $F\nmid \phi_3$: then there is a factor $G$ of $F$ such
that $G$ divides $A^{3,4,\dots,n-k}_{i_1,\dots,i_{n-k-2}}$ for
any choice of $i_1,\dots,i_{n-k-2}$. This means that $G$ divides
both $A^{1,3,4,\dots,n-k}_{i_1,\dots,i_{n-k-1}}$ and
$A^{2,3,\dots,n-k}_{i_1,\dots,i_{n-k-1}}$. Then $\phi_1$ and $\phi_2$
have a common factor of degree $\geq k+\deg G$, and we can check
whether this new polynomial of higher degree and $\phi_3$
have a common factor of degree $k$ or not. If the answer is negative,
we can iterate the construction until we find the factor we look for,
after deleting all common factors of
$A^{1,3,4,\dots,n-k}_{i_1,\dots,i_{n-k-1}}$ and $A^{2,3,
\dots,n-k}_{i_1,\dots,i_{n-k-1}}$.
\qed
\vskip 12pt
\begin{rem}
If there are more than $n$ focal points on a line $r\in B$,
then $r$ is a focal line.
\end{rem}
Theorem \ref{thm:C} allows us to give a rough description
of the focal locus of a variety $X$ ruled by an $n$-dimensional family of
lines, once we
know the dimension of the constant tangent space along a general
line.
\begin{Def}\label{d:strict}
Let $B\subset\bb{G}(1,N)$ be a subvariety of the Grassmannian,
such that its general element is not focal. Let $k$ be the
degree of the focal locus on a general element $r\in B$.
Let us denote by $U\subset B$ the open set of the lines on which the
focal locus
is a proper subscheme of length $k$. Then the closure in
$\bb{P}^N$ of the union of the focal loci on the lines of $U$ is
called the {\em
strict focal locus} of $B$.\end{Def}
\begin{rem} For $B\subset\bb{G}(1,N)$, $\dim B=n$, if $n=k$,
then $U$ is the open set of non-focal lines, and the strict focal
locus is simply the closure in $\bb{P}^N$ of the union of
focal points on the non-focal lines of $B$.\end{rem}
The study of the strict focal locus enables us to formulate a
pattern of classification of varieties of dimension $n+1$ ruled
by an $n$-dimensional family of lines. First of all, any such variety is
characterized by the number and the multiplicity of the distinct
focal points on a general line. Then we can study the strict focal locus
of $X$, and, in particular, the number of components and their
dimensions.\\
The maximal possible dimension for a component of the strict
focal locus is $n$. If there is a component of dimension $<n$,
then through every point of it there pass
infinitely many lines
of $B$. So
this component must be contained in the fundamental locus of the
family $B$. If there are components of dimension $n$ of the focal locus,
then every line of $B$ is tangent to them. This is a particular
case of a property of the focal locus that holds for varieties ruled by
linear subspaces of dimension $\geq 1$ too. So we will prove
it in the general case.
\begin{thm}\label{thm:A}
Let $B\subset\bb{G}(k,N)$ be a family of $k$-spaces in $\bb{P}^N$
of dimension $n\leq N-k$. Suppose that the union of the $k$-planes
belonging to $B$ is a
variety $X$ of dimension $k+n$, and that the focal
locus has codimension 1 in $X$.\\ Then every general subspace
$\Lambda$ belonging to
$B$ is tangent to $F$ at all the focal points on
$\Lambda$ that are not fundamental points.
\end{thm}
\proof
Let us consider $\mathcal{I}$, the desingularization of the
incidence correspondence of $B$, and the natural projections $f,g$
$$\begin{CD}\mathcal{I}@>f>>\mathbb{P}^N\\
@VgVV\\
B.\end{CD}$$
Let $p$ be a general point of $g^{-1}(\Lambda)$, belonging
to the focal subvariety $V(\chi)\subset\mathcal{I}$.
By definition, the differential
of $f$ in $p$,
$$d_pf: T_p\mathcal{I} \longrightarrow T_{f(p)} \bb{P}^N$$
has a non-trivial kernel.\\
Let us consider the restriction of $f$ to the focal subvariety,
$$\begin{array}{lccc}f: & \mathcal{I}& \xrightarrow{\ \ \ \ \ \ \
}&\,\;\mathbb{P}^N\\[-3pt]
&\scriptstyle{\cup} &&\scriptstyle{\cup} \\
&V (\chi)& \xrightarrow{\ \ \ \ \ \ \ }& F.\end{array}$$
As we
supposed $char\ \bb{K}=0$, the algebraic geometry analogue of
Sard's Theorem holds (\cite [p. 176]{H92}). So
for a general $p\in
V( \chi)$ the homomorphism
$d_pf|_{V (\chi)}: T_pV(\chi) \longrightarrow T_{f (p)}F$ is
surjective.
Since not all focal points are fundamental points and $f$ has
finite-dimensional fibres, $\dim V(\chi)=\dim F$. Thus $d_pf|_{V(\chi)}$
is an isomorphism.\\ Now consider again the differential of $f$
in a general point $p$ of $V(\chi)\cap g^{-1}(\Lambda)$, $$d_pf:
T_p\mathcal{I} \rightarrow T_{f (p)} \mathbb{P}^N.$$
We know that $d_pf$ has a non-trivial kernel. We already know as
well that its image contains $T_{f (p)}F$. Since $V (\chi)$ is a
codimension 1 subvariety of $\mathcal{I}$, $T_pV (\chi)$ is a linear
subspace of codimension 1 in $T_p{\mathcal{I}}$. Hence
$d_pf (T_p\mathcal{I}) = T_pF$. As $g^{-1} (
\Lambda) \subset \mathbb{T}_p\mathcal{I}$, we
have $\Lambda = d_pf (g^{-1} (\Lambda) \subset \mathbb{T}_{f (p)}F$.
\qed
\vskip 12pt
Coming back to varieties with constant tangent space along lines,
assume that on a general line $r\in B$ there are focal
points which are not fundamental points. Then the strict focal
locus has a
component $Y$ of pure codimension 1 in $X$, and every line in $U$ is
tangent to $Y$ at its focal, non-fundamental points.
In Theorem \ref{thm:C} we can consider the two extremal cases:
namely, $k=0$ and $k=n$. In the first case, the theorem implies that
a variety ruled by lines has no focal point on a general line if and only
if the only fixed tangent space along a general line is the line
itself. In the second case, we obtain a characterization of the varieties
whose degenerate Gauss map has 1-dimensional fibres, i.e.
varieties of dimension $n+1$ with tangent space constant along lines.
\begin{cor}\label{cor:C}
Let $B\subset\mathbb{G}(1,N)$ be a family of lines in $\mathbb{P}^N$
of dimension $n,\ n\leq N-2$. Suppose that the union of the lines
belonging to
$B$ is an algebraic variety $X$ of dimension $n+1$.
Then
the following are equivalent:\\
(i) the focal locus on the general element $r\in B$ consists of $n$ points
(counting multiplicity);\\
(ii) the tangent space to $X$ is constant along the lines of $B$. \end{cor}
\begin{rem}
If $X$ is not ruled by linear subspaces of dimension $\geq2$,
then condition (i) implies that $B$ is the family of the fibres of
the Gauss map.
If $X$ possesses a higher dimensional ruling, then the
fibres of the Gauss map may have dimension greater than 1.\end{rem}
\section{Varieties ruled by linear subspaces}\label{gen}
In this section we try to find out whether the results established in the
previous section may be extended to
varieties ruled by linear subspaces of dimension $>1$.
In particular we expect that in the case of a
family of linear subspaces of dimension $k$,
the existence of constant tangent
spaces gives a focal hypersurface on the general $k$-space.
This is true for varieties with degenerate Gauss mapping,
for which
Theorem \ref{thm:B} yields a straightforward generalization
of Corollary \ref{cor:C}.
\vskip 12pt
\par{\em Proof of Theorem \ref{thm:B}.}
Let $X\subset\mathbb{P}^N$ be a projective variety of dimension $n+k$,
with Gauss map whose fibres have dimension $k$. We want to prove
that condition (ii) holds for the family $B\subset\mathbb{G}(k,N)$ of
fibres of the Gauss map of $X$. Let $\Lambda$ be a general element
of $B$. Let us consider the characteristic map of $B$ relative to
$\Lambda$,
$$\begin{CD}
\chi(\Lambda):\ @. T_\Lambda B \otimes \mathcal{O}_\Lambda @>>>
\mathcal{N}_{\Lambda|\mathbb{P}^N}\\
@. \wr@| \wr@|\\
@. \mathcal{O}^{n}_\Lambda @>>> \mathcal{O}^{N-k}_\Lambda (1).\end{CD}$$
$\chi(\Lambda)$ is represented by an $n\times(N-k)$ matrix,
whose entries are linear forms
on $\Lambda$. The columns of this matrix evaluated in a point
$p\in\Lambda$ can be regarded as vectors $L_1 (p),\dots,L_{n}(p)$
in $V/\hat{\Lambda}$. Let us denote by $\Pi$ the fixed tangent space
to $X$ along $\Lambda$. Then the image of $\chi(\Lambda)$ in any
point $p\in\Lambda$ is contained in $\hat{\Pi}/\hat{\Lambda}$, which
is a fixed subspace of $V/\hat{\Lambda}$ of dimension $n$. If we
consider the coordinates of the normal vectors $L_1 (p),\dots,L_{n}(p)$
in $\hat{\Pi}/\hat{\Lambda}$, we get a matrix $(m_{ij})_{i,j=1,\dots,n}$.
Then the condition defining the focal locus on $\Lambda$ is
$$\det(m_{ij}) = 0,$$
which in general cases gives a hypersurface on $\Lambda$
of degree $n$, even if it is possible in special cases that
all $\Lambda$ is focal.
Suppose now that $B\subset\mathbb{G}(n,k)$ satisfies condition
(ii). If we fix a general $\Lambda\in B$, the focal variety on $\Lambda$
is a hypersurface of degree $n$. Then on the general line $r\subset\Lambda$
there are $n$ (not necessarily distinct) focal points, which
are the points where the morphism
$$\lambda:\ T_\Lambda B\otimes\mathcal{O}_r
\longrightarrow(\mathcal{N}_{\Lambda|\mathbb{P}^N})|_r,$$
given by the restriction of the characteristic
map, has not maximal rank. We can adapt to $\lambda$ the procedure
applied in the proof of
the implication (ii) $\Rightarrow$ (i) of Theorem
\ref{thm:C}. In this way, we find that there is a fixed subspace
$W(r)$ of dimension
$n$ contained in the image of the characteristic map
in any point of $r$. Now we choose a general point $p$ in $\Lambda$;
particularly, $p$
is non-focal and smooth. We restrict to an affine open
set $U_0\subset\mathbb{P}^N$ and consider a system of
affine coordinates on $U_0$
such that $p$ is the point $(0,\dots,0)$. $\Lambda_0=\Lambda\cap U_0$
is a linear space of dimension $k$. We can fix $k$ lines
$r_1,\dots,r_k$ through $p$\ spanning $\Lambda_0$, such that
for all $j$ on $r_j$ there are $n$ focal points (considered with
multiplicity). On any $r_j$ there is a fixed tangent subspace,
spanned by $r_j$ and $W(r_j)$. So the tangent space to $X$ in $p$ does
contain all lines $r_1,\dots,r_k$ (spanning $\Lambda_0$) and
all linear subspaces $W(r_1),\dots,W(r_k)$, which implies that all the spaces
$W(r_j)$ must coincide for dimensional reasons. In this way we
have found a fixed linear space $W$ of dimension $n$, such that in any
smooth point of $\Lambda_0$ the tangent space to $X$ is spanned by
$\Lambda_0$ and $W$. \qed
\vskip 12pt
In the general case of varieties ruled by lines, it was possible
to find non-focal lines on which there were more than
the general number of focal points. Under the hypotheses of
Theorem \ref{thm:B}, the
open set of subspaces on which the focal locus has
degree $k$ coincides with the set of non-focal subspaces.
So Theorem \ref{thm:B}
allows us to describe possible characterizations
of the strict focal locus for varieties with degenerate Gauss mapping.
In this case, the strict focal locus is defined
as the closure in $\bb{P}^N$ of the union of the focal points on
non-focal subspaces.
For varieties with degenerate Gauss mapping, the focal locus
is contained in the singular locus of the variety.
The converse is not true in general.
\begin{thm}
Let $X$ be a variety with degenerate Gauss mapping, and denote by
$B$ the family of fibres of the Gauss map of $X$. Then,
the focal points of $B$ are singular points of $X$.
\end{thm}
\proof Let us recall that the focal points are the
ramification points of the projection $f: \mathcal{I}\longrightarrow X$
from the desingularization of the incidence correspondence of $B$
to $X$. As the degree of $f$ is 1, either
the focal points are points where $f$ is not
finite, or they are necessarily non-normal points of $X$. In the
former case, they are fundamental points of $B$;
in the latter, they are a fortiori singular points of $X$. Since
through a fundamental
point there pass at least two different fibres of the Gauss map,
also the fundamental points are always singular.
\qed
\begin{rem}
This theorem can be extended to every variety $X$ ruled by a
family $B$, such that the projection from the desingularized
incidence corrispondence to $X$ has degree 1. In this case all
focal points not belonging to the fundamental locus are non-normal points,
but nothing can be said about fundamental points. \end{rem}
Theorem \ref{thm:B} could suggest that
also a more general equivalence holds true, i.e. that, given
a variety $X$ of dimension $n+k$ covered by a family $B$ of $k$-spaces with
$\dim B=n$, $X$ possesses a constant tangent space of dimension
$k+h$ along a general $\Lambda\in B$ if and only if the focal locus of $B$
on a general $\Lambda\in B$ is a hypersurface of degree $h$.
Unfortunately, there are counterexamples of this equivalence even for the
first possible non-trivial case, that is for varieties
ruled by a family of planes with focal lines. Observe that this case is the
simplest possible not covered by Theorem \ref{thm:C} or Theorem
\ref{thm:B} either.
\begin{ese}We will give two examples of varieties of dimension
4 ruled by a $2$-dimensional family $B$ of planes, with a focal line
on the general $\Lambda\in B$. We will see that the tangential
properties along the planes of the ruling are not the same in the two cases.
Let us consider a variety $Y$ of dimension 3 ruled by lines,
with a fixed tangent plane along the general line
of the ruling, but no higher dimensional constant tangent space. Then
the family of
tangent planes has a focal line on the general element,
and this line is precisely the line of the ruling of $Y$. In this case it is
possible to prove that the union of the family of
tangent planes is a variety $X$ of dimension 4 with a fixed tangent
$\bb{P}^3$ along every plane. So, for the variety $X$ the relationship between
the dimension of the fixed tangent space along
the planes of the ruling and the degree of the focal locus holds.
Now, let $Z$ be a variety of dimension 3
ruled by lines, with constant tangent
space along the lines of the ruling. Denote by $B$ the 2-dimensional
family of such lines, i.e. (in
general) the family of the fibres of the
Gauss map. Then we can choose a family $C\subset\bb{G}(2,N)$ of
planes such that, for every line
$r$ in $B$, there is a plane in $C$ containing
$r$ and lying in the constant tangent space to $Z$ along $r$. On a
general plane $\Pi$ in
$C$ the line $r$ of $B$ such that $r\subset\Pi$ is a
focal line. Assume that the union of the planes of $C$ is a variety
$X$ of dimension
4. It is possible to prove that along a general line in
$\Pi$ there is a constant $\bb{P}^3$ tangent to $X$, but that this $\bb{P}^3$
depends on the chosen line, so that there is no constant
$\bb{P}^3$ tangent to $X$ along $\Pi$. This example shows
therefore that the relationship
previously proposed is not always valid.
\end{ese}
Concluding, all we know in general cases is that if a
variety $X$ of dimension $n+k$, ruled by an
$n$-dimensional family of $k$-spaces, possesses
a fixed space of dimension $k+h$ tangent
along a general $k$-space, then the focal locus
on the general $k$-space of the ruling must contain a hypersurface of
degree $\geq h$. If
we know the degree of the focal locus, we
only know the maximal dimension of a space tangent to $X$ along the
general line lying in a
space of the ruling, which can vary with the choice of the line.
| 23,986
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Peterbilt – one of the most famous brands of trucks around the world. These huge powerful cars trimmed with the chromeplated details – the real kings of roads. The history of this automobile building enterprise originates in 1915. Then the small company on production of passenger cars, buses and trucks was founded by brothers Frank and William.
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Despite the time of troubles of "a great depression", demand for trucks I didn't fall, and under the auspices of two serious companies Fageol Motors Co I continued to let out new models of trucks.
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In 1981 the market was entered by the updated version of model 352 – Peterbilt 362 with the cabin located over the engine. The car exterior was changed, and a cabin equipped with a continuous windshield. 1988 brought one more version of model 352 – Peterbilt 372 Winnebago. The design of this model differed in aerodynamic forms, and the cabin, as well as at the previous representatives, was directly over the car motor.
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3 Trends Impacting Small and Medium Businesses Now
Small and medium-sized business owners have experienced year-over-year improvement of business results, and they’re increasingly optimistic about the current economic environment. Marketing channels that will see the biggest investment include email, social media, display, mobile and search.
In short, marketing is now front and center for small businesses, who are considering what they can do to increase their momentum. We’ll see business owners continue to focus on three core areas that will shape the marketing ecosystem as a result:
- Social media.
The majority of small businesses now have some kind of social media presence, whether on Facebook, Twitter or other emerging platforms and they’re becoming more mature in their use of these platforms. We often see small business, small businesses need to be engaged in constant conversation with their customers, responding to inquiries, soliciting feedback and creating the perception that the business is accessible and interested.
Small businesses. Small business must educate themselves on the rudiments of social media measurement to get a basic grasp of whether awareness, likes and site visits correlate with business goals, especially sales.
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Digital channels such as email, social media, SMS (texts) and POS are all conduits for ad retargeting and nurturing your customer relationships. Each time a potential or return customer encounters your brand, incremental data is collected, which small and medium businesses proprietor of supermarkets with several stores around the region can use a combination of digital and physical communications to manage local customer relationships.
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Since You’ve Been Gone
Posted by Kendra on October 30, 2014
Mary Jennifer Payne is the author of Finding Jade, Book One in the Daughters of Light Series, the YA novel Since You’ve Been Gone, and several YA graphic novels. Her writing has been published in journals, anthologies, and magazines in Canada and abroad. She teaches special education with the Toronto District School Board and lives in Toronto.
| 42,259
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TITLE: Let $f$ be a polynomial function on the entire complex plane such that $() ≠ 0$ for $z$ such that $|z|< 1$ ; then which of the following is true?
QUESTION [1 upvotes]: Problem: Let $~f~$ be a polynomial function on the entire complex plane such that $~() ≠ 0~$ for $~z~$ such that $~|z|< 1~$ ; then $$\dfrac {1}{2}~∫_{|z|=1} \dfrac{f'()}{()} $$
$1. ~~$can take any integer values
$2. ~~$can take any value
$3. ~~$is zero
$4. ~~$is equal to degree of $f$.
My thoughts: Argument theorem says, if $f(z)$ be analytic inside and on a simple closed curve C except for a finite number of poles inside C. Suppose that $f(z)\ne0$ on C. If N and P are, respectively the number of zeros and poles of $f(z)$ inside C, counting multiplicities, then $$\dfrac {1}{2}~\oint_C \dfrac{f'()}{()} =N-P~.$$
We know that every polynomial of degree $n$ has exactly $n$ zeros and has no pole. Therefore in this case $~N=n~($degree of the polynomial function$)~,~P=0~.$ Therefore $$\dfrac {1}{2}~∫_{|z|=1} \dfrac{f'()}{()} =\text{degree of the polynomial function}~f$$ So here option $4$ is true but answer given here is option $3$. Please help.
REPLY [1 votes]: One direct prof. We can assume that the leading coefficient is $1$. Since $f(z)$ is a polynomial with degree $n$ on the entire complex plan, $f(z)= \prod_{k=1}^{n} \left(z-z_k \right)$ where $z_k, k = \overline{1,n}$. Then:
$$ f'(z) = \sum_{k=1}^{n}\prod_{j\neq k}(z-z_j) \Rightarrow \frac{f'(z)}{f(z)} = \sum_{k=1}^{n} \frac{1}{z-z_k}, z \neq z_k$$
Now, combine with the fact that none of $z_i \notin \overline{B}(0,1)$ by Cauchy's integral theorem:
$$\dfrac {1}{2}~\oint_C \dfrac{f'()}{()} = \frac{1}{2\pi i} \sum_{k=1}^{n} \oint_{\vert z \vert = 1} \frac{\mathrm{d}z}{z-z_k} = 0$$
| 24,587
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$1,650 3 bed 616 La Costa St, Minneola FL, 34715
$1,650 3 bed
616 La Costa St, 34715
Commute to Downtown Clermont
616 La Costa St has a Walk Score of 21 out of 100. This location is a Car-Dependent neighborhood so almost all errands require a car.
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Minimum Security deposit of $1,650.00
Rent includes Pool care and Lawn care
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Directions: From R27 and R50, North on R27 about 3 miles. Turn right at Southern Breeze Dr (Quail Valley). Turn right onto LaCosta (4th street). House is near end of street on left.
(RLNE2121171)
Pets allowed:
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A location in Minneola
Almost all errands require a car.
Minimal bike infrastructure.
Explore how far you can travel by car, bus, bike and foot from 616 La Costa St.
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Está vd. en Home / Video / Viaje Virtual / Gressoney-Saint-Jean and Gressoney-la-Trinité
Located in the upper Valle del Lys and surrounded by peaks over 3,000 metres in height, Gressoney-Saint-Jean has been an important tourist venue since the early twentieth century. The population preserves the local dialect and the traditions. At the bottom of the valley is Gressoney-la-Trinité.
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Umber. 28,000 bottles made.
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\begin{document}
\title[ Finite Dynamical Systems, Linear Automata, and Finite Fields]{ Finite Dynamical Systems, Linear Automata, and Finite Fields}
\author
{O. MORENO}
\address{ Department of Mathematics and Computer Science\\
University of Puerto Rico at Rio Piedras\\
Rio Piedras, PR 00931, moreno@uprr.pr}
\author{D. BOLLMAN}
\address{Department of Mathematics\\
University of Puerto Rico at Mayaguez\\
Mayaguez, PR 00681-9018\\
bollman@cs.uprm.edu }
\author{M.A. AVI\~{N}O-DIAZ}
\address{Department of Mathematics\\
University of Puerto Rico at Cayey\\
Cayey,PR 00777\\
mavino@cayey.upr.edu}
\subjclass{11T06, 37B10, 92B05}
\keywords {finite dynamical system, finite field, linearized
polynomial, linear automata, biomathematics}
\maketitle
\begin{abstract}
We establish a connection between finite fields and finite
dynamical systems. We show how this connection can be used to shed light
on some problems in finite dynamical systems and in particular, in
linear systems.
\end{abstract}
\medskip
\medskip
\section{ Introduction}
There is a natural correspondence between the set $GF(p^r)$ and
the set ${{\bf Z}_p}^r$ of $r$-tuples over ${\bf Z}_p,$ p prime.
Furthermore, $GF(p^r)$ is a vector space over $GF(p)={\bf Z}_p$
and linear transformations over ${{\bf Z}_p}^r$ correspond to
linearized polynomials over $GF(p^r).$ In this ongoing work, we
use these facts to study some problems in finite dynamical systems
and linear automata.
\par
In Section 2, we study finite dynamical systems and how our approach
can be used in the classification problem. In Section 3, we study
linearized polynomials and how they can be applied to a problem in
linear finite state machines, which in turn arises from a problem in
crystallographic FFTs.
\section{Finite dynamical systems}
Finite dynamical systems are important in applications to
computational molecular biology. They are useful in microarrays of
genes in order to find the best model that fits a given data, the
so called ``reverse engineering problem.'' (See
http://industry.ebi.ac.uk/~brazama/Genenets.) Laubenbacher and
Pareigis [2] define a finite dynamical system as a function
$f:k^n\rightarrow k^n,$ constructed by the following data:
1. $k=\{0,1\}$
2. a finite graph $F$ on $n$ vertices.
3. a family of ``local" update functions $f_a:k^n\rightarrow k^n$,
one for each vertex $a\in F$, which changes only the coordinate
corresponding to $a$, and computes the binary state of vertex $a$.
These functions are local in the sense that they only depend on
those variables which are connected to $a\in F$.
4. an ``update schedule" $\pi$, which specifies an order on the
vertices of $F$, represented by a permutation $\pi\in S_n$.
The function $f$ is then constructed by composing the local
functions according to the update schedule $\pi$, that is
$$f=f_{\pi (n)} \circ \cdots \circ f_{\pi (1)}: k^n\rightarrow
k^n.$$
In [3], Laubenbacher and Pareigis called the above function
a permutation sequential dynamical system, and extended
the definition in
different directions. In particular, they take $k$ as an
arbitrary set. For our purposes it is convenient to consider
$k={\bf Z}_m=\{0,1,\cdots,m-1\}$,
\medskip
\par\noindent
Definition 2.1. A finite dynamical system (FDS) is a pair $(V,f)$
where $V$ is the {\it set} of vectors over a finite field and
$f:V\rightarrow V.$
\medskip
\par\noindent
Definition 2.2. The state diagram of a FDS $(V,f)$ is the digraph
whose vertices are members of $V$ and whose edges are the set of
all $(x,f(x)),$ where $x\in V.$
\medskip
\par\noindent
Remark: Note how for an FDS $({{\bf Z}_p}^r,f),$ the function can
be viewed naturally as an $r$-tuple $(f_1,f_2,\cdots,f_r)$ of
functions $f_i:{\bf Z}_p\rightarrow {\bf Z}_p,$ $i\le i \le r.$ It
is also important to note that, using Langrange interpolation, any
function from a finite field to itself can be realized as a
polynomial [4]. Hence each of the $f_i$ can be regarded as a
polynomial over ${\bf Z}_p.$ A similar remark applies to a FDS of
the form $(GF(p^n),f).$
\medskip
\par\noindent
Definition 2.3. Two FDSs are isomorphic if their state diagrams
are isomorphic.
\medskip
\par
Two FDSs are isomorphic if and only if their state diagram are
isomorphic as digraphs. A \textit{limit cycle} is simply a directed
cycle in the state diagram ${\mathcal S}_f.$ A loop is a limit cycle
consisting of a single vertex and in the case that it occurs, it is
a fixed point of the FDS $f.$ We denote by ${\mathcal L}_f$ the
subdigraph of ${\mathcal S}_f $ induced by all the arcs of the limit
cycles.
\medskip
\par
Definition 2.4. Let $f:V\rightarrow V$ be a FDS with state diagram
${\mathcal S}_f$ and with subdigraph ${\mathcal L}_f$ of limit
cycles. Then $x\in k$ is a vertex in ${\mathcal L}_f$ if and only if
there exists a positive integer $m$ such that
$f^{m}(x)=x$. The minimum $m$ such that $f^{m}(x)=x$ for all $
x\in {\mathcal L}_f$ is called the \textit{order} of the system
$f$, denoted by Order$(f)$. (See [2])
\medskip
\par\noindent
Directed paths in ${\mathcal S}_f$ correspond to iterations of $f$
on the element at the beginning of the path. Since the set ${{\bf
Z}_p}^r$ is finite, any directed path must eventually enter a limit
cycle. Thus each connected component of ${\mathcal S}_f$ consist of
one limit cycle, together with \textit{transients}, that is,
directed paths having no repeated vertices and ending in a vertex
that is part of limit cycle.
\medskip\medskip
\medskip
\par
One of the main problems in FDSs is their classification. Loosely speaking,
this is the problem of determining of two arbitrarily given FDSs whether or
not they are isomorphic. One of our goals
in this work is to facilitate the solution of the classification problem
though the association given in
\medskip
\par\noindent
Theorem 2.5. For any fixed basis $\alpha_1,\cdots,\alpha_r$ of
$GF(p^r)$ there is a natural one-one correspondence between the
FDSs over $GF(p^r)$ and those over ${{\bf Z}_p}^r.$
\medskip
\par\noindent
{\it Proof.} There is a natural correspondence between the sets
${{\bf Z}_p}^r$ and $GF(p^r)$, namely,
$(x_1,x_2,\cdots,x_r)\leftrightarrow x_1\alpha_1+\cdots
x_r\alpha_r.$ Now given $f:{{\bf Z}_p}^r \rightarrow {{\bf
Z}_p}^r,$ define $L:GF(p^r)\rightarrow GF(p^r)$ such that for each
$(x_1,x_2,\cdots,x_r)$ in ${{\bf Z}_p}^r,$ $L(x)=f_1(x_1)\alpha_1
+ \cdots f_r(x_r)\alpha_r,$ where $x=x_1\alpha_1 + \cdots +
x_r\alpha_r$ and $f=(f_1,\cdots,f_r).$ Conversely, given
$L:GF(p^r)\rightarrow GF(p^r),$ if $L(x_1\alpha_1+\cdots
x_r\alpha_r)=y_1\alpha_1+\cdots y_r\alpha_r,$ there corresponds a
function $f=(f_1,f_2,\cdots,f_r)$ such that
$f(x_1,\cdots,x_r)=(y_1,\cdots,y_r),$ where $f_i(x_i)=y_i$ for
each $i=1,2,\cdots r.$ Since this correspondence is onto and the
two sets are finite with the same number of elements, it is also
one-one.
\medskip
\par\noindent
Corollary 2.6. If $S_1=({{\bf Z}_p}^r,f)$ and $S_2=({{\bf
Z}_p}^r,f')$ are FDSs and $f$ corresponds to $L(x)$ with respect
to the basis $\alpha_1,\cdots,\alpha_r$ and $f'$ corresponds also
to $L(x),$ but with respect to another basis, then $S_1$ and $S_2$
are isomorphic.
\medskip
\par
This latter corollary says that our approach is quite useful for the
classification problem. On the other hand,
\medskip
\par\noindent
Theorem 2.7. For any fixed basis $\alpha_1,\cdots,\alpha_r$ of
$GF(p^r),$ there is a natural correspondence between the FDSs over
$(GF(p^r))^n$ and those over $({\bf Z}_p)^{rn}.$
\medskip
\par
In other words, it is redundant to study both types of these FDSs,
but each is important given the classification problem.
\section{Linear FDSs and linearized polynomials.}
A linear finite dynamical
system or a linear (autonomous) finite state machine is a FDS
$({{\bf Z}_p}^r,f)$ in which $f$ is a linear transformation on
${{\bf Z}_p}^r$ regarded as a vector space over ${\bf Z}_p.$ We
shall see in this case that there is a useful correspondence between
linear FDSs and linearized polynomials.
\par
The correspondence $x\rightarrow x^{p^i},$ $i=0,1,\cdots,p^{r-1},$
gives the Galois automorphisms of $GF(p^r).$ A linearized
polynomial $L(x)$ is a polynomial generated by these
automorphisms. In other words, $L(x)=\sum_{i=0}^{r-1}A_ix^{p^i},$
where $A_i \in GF(p^r.)$ We note that if $y,z \in GF(p^r)$ and
$\lambda \in GF(p),$ then $L(x+y)=L(x)+L(y)$ and $L\lambda x) =
\lambda L(x).$ Thus, $L(x)$ is a linear function on $GF(p^r)$
regarded as a vector space over $GF(p).$ Furthermore the
correspondence between $GF(p^r)$ and ${{\bf Z}_p}^r$ given in
Theorem 1 is an isomorphism as a vector space over ${\bf Z}_p$ .
Since there are $(p^r)^r$ linearized polynomials, this coincides
with all the linear functions on ${{\bf Z}_p}^r.$ It is easy to
see that if $f:{{\bf Z}_p}^r \rightarrow {{\bf Z}_p}^r$ is a FDS
associated to the linearized polynomial
$L(x)=\sum_{i=0}^{r-1}A_ix^{p^i},$ then ker$f$ is the set of all
roots of $L(x)$. So, $f$ is invertible if and only if the only
root of $L$ in $GF(p^r)$ is $0$.
\par
Given a linear autonomous machines $S=({{\bf Z}_p}^r,F),$ if $f$
is a nonsingular linear transformation, i.e., an invertible matrix
over ${\bf Z}_p,$ then the state space ${{\bf Z}_p}^r$ decomposes
into disjoint ``orbits'' or ``cycles.'' Based on the above
observation, the same also holds for machines $(GF(p^r),L)$ and it
is interesting to note how properties of linearized polynomials
determine this orbit structure, much in the same way as the
properties of the elementary divisors of $f$ determine the orbit
structure of $({\bf Z}_p^r,f)$ in the classical theory. However,
our motivation for studying linearized polynomials stems from a
more general problem which arises in crystallographic FFTs [5].
Let us briefly describe this problem.
\par
Crystallographic data can introduce structured symmetries in the
inputs of a multidimensional discrete Fourier transform, which
in term introduce symmetries into the outputs. In order to avoid
redundant calculations, it is of interest to exploit these
symmetries. Assuming that symmetries are given by an $n\times n$
matrix $S$ over ${\bf Z}_p,$ for prime $p$ edge length, we can
reduce the complexity of the FFT by determining a matrix $M$ with
$MS=SM$ and $M^tS=SM^t$ (where $M^t$ denotes the transpose of $M$)
that minimizes the number of ``$MS$-orbits.'' A vector $x\in
Z_p^n$ belongs to an {\it MS-orbit} of length $k$ if and only if
$M^kx=S^ix$ for some $i.$ The cases $n=2$ and $n=3$ are of
particular interest.
\par
For $n=2,$ for example, i.e., $F:{\bf Z}_p\times {\bf Z}_p
\rightarrow {\bf Z}_p\times {\bf Z}_p,$ this corresponds to the
study of linearized polynomials $F(x)=Ax^p+Bx,$ where $A,B\in
GF(p^2),$ where $F$ is an invertible map. The first question is,
when is $F(x)$ invertible in $GF(p^2)?$
\medskip
\par\noindent
Lemma 3.1. $F(x)$ is invertible if $A^{p+1}\ne B^{p+1}.$
\medskip
\par\noindent
{\it Proof.} $F(x)$ is invertible if and only if it is one-one,
i.e., if $ker\;F=0.$
In other words, if $F(x)=0$ has only $x=0$ as a solution over $GF(p^2).$
But $x\ne 0$ and $Ax^p+Bx=0$ imply that
$X^{p-1}=-{{B}\over{A}}$ and so raising both sides to the power $p+1,$
we obtain $x^{p^2-1}={{B^{p+1}}\over{A^{p+1}}}$ and $A^{p+1}=B^{p+1}.$
\medskip
\par
In the remainder of this section we consider the class ${\mathcal
L}_p$ of linearized polynomials $L(x)=\sum_{i=0}^{r-1}A_ix^{p^i}$
where $A_i\in GF(p).$ These types of polynomials have important
properties which we outline below.
\medskip
\par\noindent
Property I. If $L(x),L'(x)\in {\mathcal L}_p$ then
$L(L'(x))=L'(L(x)).$ (Note that this means that the corresponding
matrices commute).
\medskip
\par\noindent
Property II. Given $L(x)\in {\mathcal L}_p,$ the class of $L'(x)$
satisfying Property I is precisely ${\mathcal L}_p.$
\medskip
\par\noindent
Property III. Using a normal basis, the matrix corresponding to
$L(x)\in {\mathcal L}_p$ is symmetric. Consequently, the transpose
matrix also commutes.
\medskip
\par\noindent
Definition 3.2. If $L(x)=\sum_{i=0}^{r-1}A_ix^{p^i}$ is a
linearized polynomial, then its {\it associate} is
$l(x)=\sum_{i=0}^{r-1}A_iX^i.$
\medskip
\par\noindent
Definition 3.3. $L^i(x)=L(x)$ and $L^{n+1}(x)=L(L^n(x)).$ That is,
$L^n(x)$ is the $n$-fold composition of $L$ with itself.
\medskip
\par\noindent
Property IV. $L^n(x)=x$ modulo $x^{p^r}=x$ if and only if
$(l(x))^n=1$ modulo $x^r=1.$
\section{ Systems over ${\bf Z}_{p^n}$}
In this section we study FDS over ${\bf Z}_{p^n}$, that is systems
$(V,f)$ where $f:{{\bf Z}_{p^n}}^r\rightarrow {{\bf Z}_{p^n}}^r$
and $V$ is the ${\bf Z}_{p^n}$-module ${{\bf Z}_{p^n}}^r$. We use
Definitions 2.2, 2.3, and 2.4,
but now we are working in the
${\bf Z}_{p^n}$-module $V$. Therefore a linear FDS is an
endomorphism of the ${\bf Z}_{p^n}$-module $V$.
Let $g:{{\bf Z}_p}^n\rightarrow {\bf Z}_{p^n}$ be a bijection.
Then the product function $g^r:{{\bf Z}_p}^{nr}\rightarrow {\bf
Z}_{p^n}^r$ given by $g (a_1,a_2,\cdots
,a_r)=(g(a_1),g(a_2),\cdots ,g(a_r))$ is a bijection too.
\medskip
\par\noindent
Proposition 4.1.
Let $f:{{\bf Z}_{p^n}}^r\rightarrow {{\bf Z}_{p^n}}^r$ be a FDS. Let
$\overline f:{{\bf Z}_p}^{nr}\rightarrow {{\bf Z}_p}^{nr}$
be the FDS such that
$g^r\circ \overline f= f \circ g^r$. Then
$\overline f$ and $f$ have state diagrams isomorphic.
\medskip
\par\noindent
{\it Proof.}
Since $g^r$ is a bijection, there exists
$(g^r)^{-1}=g^{-r}$. Then the system
$f_1=g^{-r}\circ f \circ
g^r$ has the same state diagram to $f$. In fact, set $x_1=g^{-r}(x)$ and $y_1=g^{-r}(y)=(g^{-r}\circ f \circ
g^r)(x_1)$. Now, suppose $(x,y=f(x))$ is an edge in the state
diagram of $f$. Then $(x_1, y_1)$ is an edge in the state diagram of $f_1$.
On the other hand, $g^r\circ f_1= f \circ
g^r$. So, $\overline f=f_1$ and our claim holds.
\medskip
\par\noindent
\medskip
Definition 4.2. With the notation above, if $f$ is a linear FDS
over ${\bf Z}_{p^n}$ then the system $f$ will be called the linear
system associated to $\overline f$ by the bijection $g$.
\medskip
\par\noindent
\medskip
We use the the linear system associated to a non-linear FDS
$\overline f$ to describe its state diagram and its order.
\medskip\par\noindent
Example
Let $f:({\bf Z}_{2^3})^2\rightarrow ({\bf Z}_{2^3})^2$ be a linear system, given by
$f(a,b)=(5b,a+2b)$. For any bijection $g:({\bf Z}_2)^3\rightarrow
{\bf Z}_{2^3}$ we have an induced systems $\overline f$ with the
same state diagram of the system $f$. Now using the method given
in [1], we find the Order of $f$ and the Order of $\overline f$
for any bijection $g$. The matrix
$$A=\left(\begin{array}{cc}
0 & 5 \\
1 & 2
\end{array}\right)\equiv \left(\begin{array}{cc}
0& 1 \\
1 & 0
\end{array}\right) \ (\hbox{ mod $2$ } )$$ has minimal polynomial
\[m(x)=(x-1)^2\] and the Order of $A$ modulo $2$ is
$e=2$. Since \[A^2=\left(\begin{array}{cc}
5 & 2 \\
2 & 1
\end{array}\right) (\hbox{ mod $2^3$ } ),\] the largest positive integer
$\beta$ such that $A^2\equiv I \ (\hbox{ mod } 2^\beta)$ is $\beta
=1$. Then $A^2$ has Order $4$ and $A$ has Order $8$ modulo $2^3$.
\section{ Future Work}
We will exploit the ideas presented here to seek a polynomial
solution to the reverse engineering problem. We also make use of
our theory to develop an efficient algorithm to determine, given a
matrix $S$ of symmetries, a matrix $M$ that minimizes the number of
$MS$-orbits in the precomputation phase of crystallographic FFTs.
\smallskip
\section*{Acknowledgments}
The work of the first two authors
was supported in part by the NSF grants CISE-MI and NIH/NIGMS S06GM08103.
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Manchester United and Manchester City play out a dull 0-0 derby draw
MANCHESTER, England -- Three thoughts on a dull 0-0 draw between Manchester United and Manchester City at Old Trafford on Sunday.
1. The dullest derby ever?
There have been 170 Manchester derbies spread over 121 years. There is a strong case for arguing the best, whether Manchester United's 4-3 win in 2009 or Manchester City's 6-1 triumph in 2011, are among the most recent. Sunday's game ranked at the other end of the spectrum. The spectacular has been replaced by the subdued, the extraordinary by the very ordinary.
The 0-0 draw only briefly threatened to rise from mediocrity in the closing minutes, when Anthony Martial's gorgeous pass was first volleyed on to the bar by Jesse Lingard, and then, with a rather more direct approach, Marouane Fellaini found Chris Smalling with a flick-on and the defender's shot was tipped past the post by Joe Hart.
That apart, it was forgettable fare played by two cautious teams. Manuel Pellegrini, often an advocate of attacking football, set the tone when he submitted his team sheet and inserted Fernando, an extra defensive midfielder. By the time Martin Demichelis came off the bench to replace Yaya Toure, he had three.
Louis van Gaal, who relishes such tactical battles, is entitled to argue that his substitutions of Lingard and Fellaini almost conjured a winner, but familiar criticisms are valid. United were ponderous in possession, particularly in a dreadful first half. They scarcely played with the ambition of potential champions and swathes of the game went by with very little happening.
The result means United retain their excellent defensive record at Old Trafford, where they have conceded only one league goal, and City a two-point advantage over their neighbours. Sadly, studying the league table could be as enjoyable as watching the match.
There were penalty appeals, with Ander Herrera aghast that Mark Clattenburg did not deem Raheem Sterling's challenge on him a foul early in the second half. At either end, Yaya Toure and Smalling almost headed in corners, but should anyone purchase a DVD of the game, they may want to fast forward much of the first 80 minutes.
2. Martial looks lively on the left
United may not have secured much silverware since Sir Alex Ferguson's retirement, but they did parade one award winner before kickoff. Martial was presented with his trophy as the Premier League's player of the month for September, and while October's award may be bound for Jamie Vardy, the sense is that Martial will not prove a one-off winner.
In any case, "second-month syndrome" has brought a second position for Martial: September's striker has become October's winger. Van Gaal believes Martial can play in any of four positions: on either flank, as a striker or as a No. 10. Much of his brief stint with Monaco was spent on the left.
If Martial reprised an earlier role at Everton in part to stymie Seamus Coleman, here his immediate opponent was the less attacking Bacary Sagna. Van Gaal's deployment of the world's most expensive teenager could not be explained by his defensive duties alone.
Every United lineup can feel contrived as Van Gaal tries to find a way to accommodate Wayne Rooney, who entered his third decade of life much as he left his second, with the sort of performance that will only prompt further criticism of the United untouchable. Rooney had his head stapled after a collision with Vincent Kompany. He bled for the United cause but it rarely appeared he would score a record 12th derby goal.
Rooney was once their dynamic teenager. Now Martial, the £36 million man, plays that role. A slaloming solo run brought the crowd to life, and it seems as though it is Martial's task to electrify Old Trafford. He offers the acceleration in a three-speed attack, with Rooney and then Juan Mata operating in rather lower gears.
Martial's lithe changes of direction give him a capacity to skip away from opponents. Even those with a turn of pace themselves can be resorted to bringing him down and both Fernandinho and Kompany were cautioned for fouls on the 19-year-old. His creativity was also apparent, as he almost provided Lingard with a winner.
All in all, Martial's derby debut showcased his promise, but it would have been intriguing to see what he could have done if unleashed as the main striker.
3. Bony is no Aguero
At such exalted levels, players can be damned by comparison. Wilfried Bony is not Sergio Aguero, and suffers because of it. So did City as they lacked their premier striker -- United can be grateful his deputy is not in the same class.
The Argentinian has proved rather too elusive, as his tally of six goals in his last four derbies suggests. Bony does not offer such slippery speed, compensating with brute force. But with Daley Blind benched, United opted for bruisers of their own at the back in the sizeable figures of Phil Jones and Smalling.
As Bony struggled at Old Trafford, the threat once again came from De Bruyne, whose talents extend to offering menace in a variety of roles. But even the Belgian was betrayed by his final ball and was shackled well by Marcos Rojo; Pellegrini opted against moving him to the middle in place of Bony again when the ineffective Raheem Sterling was replaced instead. The Ivorian eventually departed for Kelechi Iheanacho in the closing minutes, meaning his City tally now stands at four goals in 22 games.
There is a case for selecting Bony against Crystal Palace in the Capital One Cup on Wednesday to try to spur him into form, but considering what a damp squib his City career has been thus far, some supporters would rather see the opportunity go to the talented teenager Iheanacho. Either way, Aguero's monthlong absence with a hamstring injury cannot end soon enough for.
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Toro y Moi’s most recent album, released on Jan. 18, radiates just what we all need: Outer Peace. A chillwave genius and a talented graphic designer, Toro y Moi (also known as Chazwick Bradley Bundick) has incorporated his experimental brilliance in the 10 track LP. The 2019 album follows after almost a decade since the start of his professional music career when he released his debut album Causers of This in 2010.
Since the official album drop, Outer Peace has been reviewed and praised by Pitchfork and NPR, among many others. In his interview with NPR and its special First Listen program, Moi described how he incorporates technology in Outer Peace.
“Technology is allowing people to become creative at home and become almost like entrepreneurs just from their desks,” he said. “I felt like that’s who I really wanted to connect with [on this album] — the people that are grinding behind the computer in a creative way.”
The iconic album cover is equally as stunning as the music: An image of Toro y Moi on an unconventional orange yoga ball chair, facing his brightly lit musical equipment before a vibrant backdrop reminds us of the otherworldly experience that accompanies the art of soundscapes in the present age of technology.
The opening track, ”Fading,” is a combination of techno, house, pop, and basically every chime and bell one may find on the electronic drum kits of a music software mixer. The overlap of echo effects, coupled with all kinds of rhythmic sequences, makes it a song you would add to your carefully curated, celestial-themed playlist. After the last line of the song, “everything is fading, fading, fading,” Toro y Moi jumps right into “Ordinary Pleasure,” a Daft Punk inspired tune that reminds me of “Mirage” from his 2017 Boo Boo album but at a faster tempo. The groove continues in “Laws of the Universe,” where his subtle scat singing joins in on a vintage synth keyboard solo.
By the time you get to “Miss Me,” featuring R&B singer ABRA, or “darkwaveduchess” as she calls herself on Instagram, the mood changes to one of sombre contemplation, a feeling I would imagine one might experience while floating alone in outer space. At the same time “New House” touches on themes with repetitive lyrics carrying the weight of heavier life realities: “I want a brand new house / Something I can not buy, something I can afford.”
While Toro y Moi explores questions of belonging, or the lack thereof, he confidently claims ownership of his self-driven creativity, especially in “Freelance.” And, of course, you could not forget the eye-catching music video that goes with it: Toro y Moi, the freelancer himself, bobs his head while a film camera zooms in and out, his music production studio now the central subject of a photography shoot. In fact, “Freelance” is only one of many other hit music video masterpieces. “Girl Like You” and “Say That,” my personal favorites, are like short, experimental films, showcasing breathtakingly visual aestheticism that speaks for the artistry of Toro’s imaginative potential.
In another interview with Complex, Toro y Moi revealed that his favorite track from Outer Peace is “Who I Am,” a song with both an uplifting beat and a rather poignant message about identity crisis: “Add an accent to your sound / Now I don’t know who I am.” Expressing his current preference for short songs that fit a general mood, Toro y Moi explains that the album fits into “playlist culture,” with each song claiming their own segments under a common “futuristic vibe.” The immediacy of the short but meaningful melodies creates one cohesive picture that leaves a strong impression on the listener.
The last two tracks of the album feature WET and Instupendo in “Monte Carlo” and “50-50,” respectively. “Monte Carlo,” while also very futuristic, draws inspiration from the 1987 Monte Carlo, a vintage, analog car model that Toro y Moi reimagines. And finally, “50-50” ends in a trance-like, lo-fi beat with subtle hi-hats that make it all the more dynamic. The way the song eases its way into silence almost seems to resolve the existential questions that he incorporates in each song.
By the end Outer Peace makes you feel as though you yourself have attained inner peace, ironically. It takes you back to your quiet room, where a dim light hovers over your computer and your keyboard, a space that you wire yourself up in. Outer Peace is the perfect album for the introversion that exists within all of us at the end of a challenging day.
Please note All comments are eligible for publication in The News-Letter.
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My commission project with my friends army, the Night Wolves, continues. It's a pretty good deal really. He converts/assembles his minis, then gives them to me for painting.
Today I finished one of his Rune Priests. His painting requests were relatively simple: He wanted the outstretched hand to glow as though it were shooting lightning, the hair to look like a graying redhead, and the spear to look badass. Here's what I came up with:
It isn't as noticeable in the pictures, but I also tried my hand at an electrical effect on the spear to go along with the lightning motif.
The Thousand Son on the base was its own little accomplishment. the 1k Son & Rune Priest use a pretty similar palette, so I tried to lighten the prone marine's armor, & did a little OSL green to differentiate it & draw a little attention to it.
~Muninn
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TITLE: Show that the maximum no. of bonds possible in a chain of $n$ elements, such that an element only bonds with another element once, is $n^2$
QUESTION [3 upvotes]: Show that the maximum no. of bonds possible in a chain of $n$ elements, such that an element only bonds with another element once, is $n^2$. What this means will be clear by a representation;
As you can see if $n=1$, the max no. of chains will be $1^2$, if it's $2$ then the max no. becomes $2^2$ and so on. I want to show that the maximum no. of chains possible will be $n^2$.
Also for example, $☆—◌$ & $◌—☆$ are treated differently because it is >Star-bonds-with-circle and the latter is >Circle-bonds-with-star.
And also before anyone says there are $n$ elements and a element can bond with $n$ elements (including itself), and so it should be $n^2$. I want to see that this is possible, in a semi-rigorous way. What if it isn't possible, how do we prove that?
Another subset question that I came up with this, is to find how many different combinations are possible with $n$ elements so as to give $n^2$ chains. Maybe this one's for later, but still intriguing. I don't know a whole lot of terminology, symbols nor representations used in Graph theory, feel free to edit it in.
REPLY [2 votes]: One way to describe such chains is to rephrase the problem as follows. We have a directed graph on $n$ vertices (such as $\{☆,□,◌\}$) in which each of the $n^2$ possible directed edges appears once. Here's an example of the graph for $n=3$:
A chain of length $n^2$ is a walk in this graph which visits each of the $n^2$ edges (the arrows) once. This is known as an Eulerian walk. For directed graphs, we know that whenever the number of edges going into a vertex equals the number of edges going out, an Eulerian walk (a) exists, and (b) always returns to the starting vertex at the end. (You may have noticed that in each of your chains, the first and last symbols are the same.)
So we know that there is a solution. What next?
By the BEST theorem, the number of such Eulerian circuits is equal to $$t(G)\prod_{v \in V} (\deg(v)-1)!$$ where $\deg(v)$ is the number of edges going out of a vertex $v$, and $t(G)$ is the number of arborescences of $G$ rooted at some vertex. Arborescences in general are complicated, but in the case of this graph, they are the same as trees on $n$ vertices, and there are $n^{n-2}$ of these by Cayley's theorem. Putting these together, we get that the number of Eulerian circuits is
$$
n^{n-2} (n-1)!^n
$$
although we should probably also multiply by $n^2$ to account for the fact that we can start an Eulerian circuit in any of $n^2$ places and get a chain. Simplifying, this gives us a final answer of $n!^n$ possible chains.
| 103,035
|
CMAT 2020 Results Declared at cmat.nta.nic.in, Check Final Merit List and Official Answer Key here
The National Testing Agency (NTA) announced the results of the Common Management Admission Test (CMAT) on 4th February 2020. The CMAT 2020 was conducted on 28th January in 275 test centres across India.
Candidates can download their individual scorecard and final from the official website of NTA. Over 63 thousand candidates out of 74486 registered applicants appeared for the exam.
Gopaljee Jha secured the 1st rank in the merit list by securing 100.00 NTA Score or percentile. He is the only candidate achieving the perfect percentile in CMAT 2020.
The topper amongst female candidates is Kalisetty Udaya Sandhya Lakshmi Devi with 99.993 NTA Score or percentile. Overall seven applicants obtained 99.99+ percentile.
Steps to Download CMAT 2020 Scorecard
Step- 1: Visit the official website of NTA CMAT- cmat.nta.nic.in
Step-2: Click on the ‘View Result/ Scorecard’ tab.
Step-3: Enter your CMAT Application Number, Date of Birth and the Security pin.
Step-4: Click on the ‘Submit’ button. The page will display your scorecard.
Step-5: Download the result and take a print out of the CMAT Scorecard.
Validity of CMAT Scorecard
The CMAT 2020 scorecard will be valid for one academic year, till 31st December 2020.
CMAT 2020 Official Answer Key
The NTA has also released final answer key for CMAT along with the results and merit list. Applicants can download the official answer key from the following link or by visiting the official website of NTA CMAT.
CMAT 2020 Final Answer Key Download
How to check CMAT 2020 All India Rank?
The NTA has also published the final merit list containing the names, sectional percentile and overall NTA score. If 2 or more candidates secure the same marks, the candidate obtaining higher marks in the following order will attain a higher rank.
The sectional order to determine the rank will be comparing scores in the Quantitative Techniques & Data Interpretation,
Logical Reasoning, Language Comprehension and General Awareness Candidates in order. Applicants with lesser incorrect responses and candidates older in age will also gain an advantage in the tie break.
Find the link to download the final merit list to check All India Rank in CMAT 2020 below.
CMAT 2020 Toppers
Gopaljee Jha from Jharkhand secured a perfect 100 percentile. He is only candidate this year to do so. Kalisetty Udaya Sandhya from Telangana topped amongst the female candidates with 99.993 percentile.
Delhi candidate Fahad Nizam got the second rank with an overall percentile of 99.998. Tanuj Daga, a candidate from Madhya Pradesh, obtained 99.886 percentile to top amongst the PwD candidates.
Stay connected with fellow students on PaGaLGuY for CMAT 2020 Exam
Top 10 CMAT candidates with Overall marks and Percentile
CMAT 2020 Highlights
● The National Testing Agency was the conducting body for the CMAT for the second time.
● More than 1000 management institutes will accept the CMAT 2020 Scorecard.
● The CMAT exam was conducted on 28th January in 104 cities from 9:30 am to 12:30 pm.
● The CMAT 2020 was conducted in 245 test centres across India.
● The difficulty level of the CMAT paper was moderate.
| 287,327
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by Sean Orr | The great vindication? Study finds number of neighbourhood’s social housing units actually on the rise. ”. This has been my entire point all along. Yes, change is happening, but unless you’re a wide-eyed anarcho-syndicalist who is unable or refuses to see nuances in capitalism – that projects like 60 West Cordova and independent restaurants like L’Abattoir (who in turn hire grumpy bloggers like me) can be sensitive to demographic concerns – then the debate is purely, uselessly academic.
So how do the “Marxists” respond? Well, they first held a meeting (rough video facsimile at the top), wherein they decided that the City lied! City skews numbers to hide loss of low income housing – 430 units lost in the last year.
The city report, which is in fact only a slideshow presentation for a public relations campaign, suggests that the number of low-income singles housing in the downtown core has increased…However, Ward’s Tyee article fails to take into account that the same slideshow also states that out of the 4,482 low-income SRO units in 2012, only 24% of the units rent at welfare rate… The problem is that the city’s SRA By-law literally does not count these units as losses in the affordable housing stock, regardless of what price they climb to.
Ok. Fair enough. But what does a restaurant have to do with that? Raise the bloody welfare rates now and the point is moot.
But how do the self-styled class warriors respond? They steal the little sandwich-board sign from out front of Save On Meats. Because that’s how the Russian Revolution started, am I right?
Related: saying you will hire from within the community sure sounds good, but it doesn’t always work: Businesses strain to retain Downtown Eastside workers. “Just like the neighbourhood residents, they’re figuring out how to succeed in their own way.”
Meanwhile, did you know that a middling pizzeria is responsible for the wholesale displacement of Quebecois squeegee kids and patchouli-covered trust-fund hippies on Commercial Drive? Yup. Pizzeria allegedly vandalized by Vancouver anti-gentrification group. Would it be too predictable of me to now form an Anti “Anti Gentrification Front” Front?
And check out these fucking gentrifiers: Introducing the Brewed Awakening feature tap at Pat’s Pub & Brewhouse.
Sorry, but Canada was never the No. 1 place to live. I don’t know if it was on purpose, but I love how the first word there is sorry. So Canadian!
Bonus: Victoria’s Johnson Street Bridge: a thoroughfare of song.
| 284,361
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\begin{document}
\title{Lie structure of associative algebras containing matrix subalgebras}
\author{Alexander Baranov}
\address{Department of Mathematics, University of Leicester, Leicester, LE1
7RH, UK}
\email{ab155@le.ac.uk}
\begin{abstract}
We prove that the commutator subalgebra of the associative algebra
containing a matrix subalgebra is perfect.
\end{abstract}
\maketitle
\global\long\def\bbR{\mathbb{R}}
\global\long\def\ccR{\mathcal{R}}
\begin{singlespace}
\global\long\def\bbF{\mathbb{F}}
\end{singlespace}
\global\long\def\ccF{\mathcal{F}}
\global\long\def\ccL{\mathcal{L}}
\global\long\def\ccP{\mathcal{P}}
\global\long\def\ccB{\mathcal{B}}
\global\long\def\ccS{\mathcal{S}}
\begin{singlespace}
\global\long\def\dlim{\operatorname{\underrightarrow{{\rm lim}}}}
\global\long\def\ker{\operatorname{\rm ker}}
\global\long\def\Im{\operatorname{\rm Im}}
\global\long\def\End{\operatorname{\rm End}}
\global\long\def\dim{\operatorname{\rm dim}}
\global\long\def\core{\operatorname{\rm core}}
\end{singlespace}
\global\long\def\skew{\operatorname{\rm skew}}
\global\long\def\Soc{\operatorname{\rm Soc}}
\global\long\def\Range{\operatorname{\rm Range}}
\global\long\def\rank{\operatorname{\rm rank}}
\global\long\def\ad{\operatorname{\rm ad}}
\global\long\def\gl{\operatorname{\rm gl}}
\global\long\def\sl{\operatorname{\rm sl}}
\global\long\def\sp{\operatorname{\rm sp}}
\global\long\def\so{\operatorname{\rm so}}
\global\long\def\u{\operatorname{\rm u^{*}}}
\global\long\def\su{\operatorname{\rm su^{*}}}
\global\long\def\rad{\operatorname{\rm rad}}
\section{Introduction}
The ground field $\bbF$ is algebraically closed of characteristic
$p\ge0$. Throughout the paper, $A$ is a (non-unital) associative
algebra over $\bbF$ containing a non-zero semisimple finite dimensional
subalgebra $S$. Recall that $A$ becomes a Lie algebra $A^{(-)}$
under the Lie bracket $[x,y]=xy-yx$. We denote by $A^{(1)}=[A^{(-)},A^{(-)}]$
its derived subalgebra.
Recall that a Lie algebra $L$ is said to be \emph{perfect} if $[L,L]=L$.
We say that an associative algebra $A$ is\emph{ $k$-perfect} ($k\ge1)$
if $A$ has no proper ideals of codimension $\le k$. Let $S$ be
a finite dimensional semisimple algebra over $F$. Then it is the
direct sum of matrix ideals, i.e. $S=M_{n_{1}}(\bbF)\oplus\dots\oplus M_{n_{s}}(\bbF)$.
Thus, $S$ is $k$-perfect if and only if $n_{i}>\sqrt{k}$ for all
$i=1,\dots,s$.
One of our main results is the following theorem.
\begin{thm}
\label{thm:main} Let $A$ be an associative algebra over $\bbF$
containing a non-zero semisimple finite dimensional subalgebra $S$.
Suppose $A$ is generated by $S$ as an ideal and $S$ is either $1$-perfect
with $p\ne2$ or $4$-perfect with $p=2$. Then
(1) $A^{(1)}$ is perfect;
(2) $A=A^{(1)}A^{(1)}+A^{(1)}$;
(3) $A^{(1)}$ is generated by $S^{(1)}$ as an ideal;
(4) $A^{(1)}$ is $\Gamma$-graded where $\Gamma$ consists of the
roots of $S^{(1)}$ and the weights of the natural and conatural $S^{(1)}$-modules.
\end{thm}
As a corollary, we get the following result for the finite dimensional
algebras, which is a generalization of \cite[Corollary 6.4]{Bav=000026Zal}
(where the case of $4$-perfect algebras in characteristic zero was
proved).
\begin{thm}
\label{thm:fd} Let $A$ be a finite dimensional algebra over $\bbF.$
Suppose that $A$ is either $4$-perfect or $1$-perfect with $p\neq2$.
Then $A^{(1)}$ is perfect and $A=A^{(1)}A^{(1)}+A^{(1)}$.
\end{thm}
As an application, we are going to show that the Lie algebras $A^{(1)}$
which appear in Theorem \ref{thm:main} are actually \emph{root-graded}.
The following definition a slight generalization of Berman and Moody's
definition of root graded Lie algebras in \cite{berman1992lie}.
\begin{defn}
\begin{singlespace}
\noindent Let $\Delta$ be a root system and let $\Gamma$ be a finite
set of integral weights of $\Delta$ containing $\Delta$ and $\{0\}$.
A Lie algebra $L$ over a field $\bbF$ of characteristic zero is
said to be \emph{$(\Gamma,\mathfrak{g})$-graded }(or simply\emph{
$\Gamma$-graded}) if
\noindent $(\Gamma1)$ $L$ contains as a subalgebra a finite-dimensional
split semisimple Lie algebra
\[
\mathit{\mathrm{\mathfrak{g}=\mathfrak{h}}\oplus\mathrm{\underset{\alpha\in\mathit{\text{\ensuremath{\Delta}}}}{\bigoplus}}}\mathit{\mathrm{\mathfrak{g}}}_{\alpha},
\]
whose root system is $\Delta$ relative to a split Cartan subalgebra
$\text{\ensuremath{\mathfrak{h}}}=\mathfrak{g}_{0}$;
\noindent $(\Gamma2)$ $L=\underset{\alpha\in\Gamma}{\bigoplus}L_{\alpha}$
where $L_{\alpha}=\left\{ x\in\mathrm{\mathit{L}\mid\left[\mathit{h,x}\right]=\mathit{\alpha\left(h\right)x\text{ for all }h\in\mathrm{\mathit{\mathfrak{h}}}}}\right\} $;
\noindent $(\Gamma3)$ $L_{0}=\underset{\alpha,-\alpha\in\Gamma\setminus\{0\}}{\overset{}{\sum}}\left[L_{\alpha},L_{-\alpha}\right]$.
\end{singlespace}
\end{defn}
\begin{thm}
\label{thm:rg} Let $A$ and $S$ be as in Theorem \ref{thm:main}.
Let $Q_{1},\dots,Q_{t}$ be the simple components of the Lie algebra
$S^{(1)}$. Then the Lie algebra $A^{(1)}$ is $\Gamma$-graded where
$\Gamma$ consists of the roots of $S^{(1)}$ and all weights of the
form $\lambda_{i}+\lambda_{j}$ where $1\le i<j\le t$ and $\lambda_{i}$
is either zero or one of the weights of the natural or conatural $Q_{i}$-modules.
\end{thm}
\section{$S$-decomposition of $A$}
Throughout the paper, $A$ is a (non-unital) associative algebra over
$\bbF$ containing a non-zero semisimple finite dimensional subalgebra
$S$. We denote by $\hat{A}$ the unital algebra obtained by adjoining
an identity element $\mathbf{1}=\mathbf{1}_{\hat{A}}$ to $A$, i.e.
$\hat{A}=A\oplus\bbF\mathbf{1}$ as a vector space and $A$ is an
ideal of $\hat{A}$ of codimension 1. Similarly, we denote by $\hat{S}$
the unital semisimple subalgebra $S\oplus\bbF\mathbf{1}$ of $\hat{A}$.
We denote by $M_{n}$ the algebra of $n\times n$ matrices over $\bbF$.
Let $\{S_{i}:i\in I\}$ be the set of the simple components of $S$.
Then the set of the simple components of $\hat{S}$ is $\{S_{i}:i\in\hat{I}\}$
where $\hat{I}=I\cup\{0\}$ and $S_{0}\cong\bbF$. Let $\mathbf{1}_{S_{i}}$
be the identity element of $S_{i}$, $i\in\hat{I}$. Then $S_{0}=\bbF\mathbf{1}_{S_{0}}$,
$\mathbf{1}=\sum_{i\in\hat{I}}\mathbf{1}_{S_{i}}$ and $\{\mathbf{1}_{S_{i}}\mid i\in\hat{I}\}$
is the complete set of the primitive central idempotents of $\hat{S}$.
We identify each $S_{i}$ with the matrix algebra $M_{n_{i}}$ (so
$\dim S_{i}=n_{i}^{2}$). Let $V_{i}$ be the natural left $S_{i}$-module.
Then the dual space $V_{i}^{*}$ is the natural right $S_{i}$-module.
We identify the space $V_{i}$ (resp. $V_{i}^{*}$) with the column
(resp. row) space $\bbF^{n_{i}}$.
Let $M$ be a (not necessarily unital) $S$-bimodule. Set $\mathbf{1}m=m\mathbf{1}=m$
for all $m\in M$. Then $M$ is a unital $\hat{S}$-bimodule or equivalently,
unital left $\mathcal{S}$-module, where
\[
\mathcal{S}=\hat{S}\otimes\hat{S}^{op}=\underset{i,j\in\hat{I}}{\bigoplus}S_{i}\otimes S_{j}^{op}
\]
is the\emph{ enveloping algebra }of $S$. Put $\mathcal{S}_{ij}=S_{i}\otimes S_{j}^{op}\cong M_{n_{i}}(\bbF)\otimes M_{n_{j}}(\bbF)\cong M_{n_{i}n_{j}}(\bbF)$
for all $i,j\in\hat{I}$. Then each $\mathcal{S}_{ij}$ is a simple
component of $\mathcal{S}$ and $\mathcal{S}=\bigoplus_{i,j\in\hat{I}}\mathcal{S}_{ij}$
is semisimple. Thus, $M$ is the direct sum (possibly infinite) of
simple left $\mathcal{S}$-modules, or equivalently, $\hat{S}$-bimodules.
Put $V_{ij}=V_{i}\otimes V_{j}^{*}$ for all $i,j\in\hat{I}$. We
identify each $V_{ij}$ with the space of all $n_{i}\times n_{j}$
matrices over $\bbF$. Then $V_{ij}$ is the natural left $\mathcal{S}_{ij}$-module
(resp. $S_{i}$-$S_{j}$-bimodule). Hence, as a left $\mathcal{S}$-module,
$M$ is the direct sum of copies of $V_{ij}$, $i,j\in\hat{I}$. By
collecting together the isomorphic copies, we obtain the following.
\begin{lem}
\label{lem:M bimodule} Let $M$ be an $S$-bimodule. Then, as an
$\hat{S}$-bimodule,
\[
M\cong\bigoplus_{i,j\in\hat{I}}V_{ij}\otimes\Lambda_{M}(i,j)
\]
for some vector spaces $\Lambda_{M}(i,j)$.
\end{lem}
\begin{rem}
Recall that $S_{0}\cong\bbF$ is 1-dimensional and $\hat{S}=S_{0}\oplus S$.
Therefore, $V_{00}$ is a 1-dimensional $S_{0}$-bimodule, $M_{0}:=V_{00}\otimes\Lambda_{M}(0,0)$
is a\emph{ trivial} $S$-sub-bimodule of $M$ (i.e. $SM_{0}=M_{0}S=0$)
and $M=(SM+MS)\oplus M_{0}$.
\end{rem}
Recall that $S$ is a subalgebra of $A$, so $A$ is an $S$-bimodule.
Hence, by Lemma \ref{lem:M bimodule}, there is an $\hat{S}$-bimodule
isomorphism
\begin{equation}
\theta:A\rightarrow\bigoplus_{i,j\in\hat{I}}V_{ij}\otimes\Lambda_{A}(i,j),\label{eq:A to Wij.^(i,j)}
\end{equation}
where $\hat{I}=I\cup\{0\}$, $\Lambda_{A}(i,j)$ are vector spaces
and $V_{ij}$ is the space of all $n_{i}\times n_{j}$ matrices over
$\bbF$. Hence, the space
\[
\mathbf{V}=\underset{i,j\in\hat{I}}{\bigoplus}V_{ij}
\]
is an associative algebra with respect to the standard matrix multiplication
(with $V_{ij}V_{st}=0$ unless $j=s$). We denote by $\{e_{st}^{ij}\mid1\leq s\leq n_{i},\ 1\le t\leq n_{j}\}$
(or simply $e_{st}$) the standard basis of $V_{ij}$ consisting of
matrix units. For example, $\{e_{11},e_{12},\ldots,e_{1n_{j}}\}$
is the basis of $V_{0n_{j}}=V_{0}\otimes V_{j}^{*}$. We will identify
$A$ with $\theta(A)$.
\selectlanguage{english}
\selectlanguage{british}
Consider any two simple $\ccS$-submodules (or equivalently, $\hat{S}$-sub-bimodules)
$W_{ij}=V_{ij}\otimes\lambda_{ij}$ and $W_{st}=V_{st}\otimes\lambda_{st}$
of $A$. Then the product $W_{ij}W_{st}=W_{ij}S_{j}S_{s}W_{st}=0$
unless $j=s$. If $j=s$, then
\[
W_{ij}W_{jt}=W_{ij}S_{j}W_{jt}=(V_{ij}\otimes\lambda_{ij})S_{j}(V_{jt}\otimes\lambda_{jt})
\]
which is a homomorphic image of the $\mathcal{S}_{it}$-module
\[
V_{ij}\otimes_{S_{j}}V_{jt}\cong V_{i}\otimes V_{j}^{*}\otimes_{S_{j}}V_{j}\otimes V_{t}^{*}\cong V_{i}\otimes V_{t}^{*}=V_{it}.
\]
Therefore,
\begin{equation}
(V_{ij}\otimes\lambda_{ij})(V_{jt}\otimes\lambda_{jt})=V_{it}\otimes\lambda_{it}\label{eq:vtimesv}
\end{equation}
for some $\lambda_{it}\in\Lambda_{A}(i,t)$ (defined up to a scalar
multiple). Note that $e_{11}$ exists in every $V_{ij}$ for $i,j\in\hat{I}$
and the product $(e_{11}\otimes\lambda_{ij})(e_{11}\otimes\lambda_{jt})$
must be equal to a scalar multiple of $e_{11}\otimes\lambda_{it}$.
Thus, by rescaling $\lambda_{it}$ if necessary, we can assume that
there is a binary product $(\lambda_{ij},\lambda_{jt})\mapsto\lambda_{ij}\lambda_{jt}$
such that
\[
(e_{11}\otimes\lambda_{ij})(e_{11}\otimes\lambda_{jt})=e_{11}\otimes\lambda_{ij}\lambda_{jt}
\]
for all $\lambda_{ij}\in\Lambda_{A}(i,j)$ and $\lambda_{jt}\in\Lambda_{A}(j,t)$.
It is easy to see that this binary product is bilinear and associative
(since the product of $V$'s in (\ref{eq:vtimesv}) is associative).
We get the following.
\begin{lem}
\label{lem:^(i,j)} There is a multiplication structure on the space
$\Lambda_{A}:=\bigoplus_{i,j\in\hat{I}}\Lambda_{A}(i,j)$ satisfying
the following conditions:
(i) $\Lambda_{A}(i,j)\Lambda_{A}(s,t)=0$, if $j\neq s$;
(ii) $\Lambda_{A}(i,j)\Lambda_{A}(j,t)\subseteq\Lambda_{A}(i,t)$;
(iii) $\Lambda_{A}$ is an associative algebra with respect to this
multiplication;
(iv) $S_{i}=V_{ii}\otimes\mathbf{1}_{i}$ ($i\in I$) where $\mathbf{1}_{i}$
is the identity element of the subalgebra $\Lambda_{A}(i,i)$;
(v) $S=\bigoplus_{i\in I}S_{i}=\bigoplus_{i\in I}V_{ii}\otimes\mathbf{1}_{i}$.
\end{lem}
Since both $\mathbf{V}$ and $\Lambda_{A}$ are associative algebras,
their tensor product $\mathbf{V}\otimes\Lambda_{A}$ is also an associative
algebra. It is easy to see that $\bigoplus_{i,j\in\hat{I}}V_{ij}\otimes\Lambda_{A}(i,j)$
is a subalgebra of $\mathbf{V}\otimes\Lambda_{A}$ with the following
product: for all $X\otimes\lambda\in V_{ij}\otimes\Lambda_{A}(i,j)$
and $Y\otimes\mu\in V_{st}\otimes\Lambda_{A}(s,t)$,
\[
(X\otimes\lambda)(Y\otimes\mu)=XY\otimes\lambda\mu
\]
where $XY$ is the matrix multiplication (with $XY=0$ if $j\neq s$)
and $\lambda\mu$ is the multiplication in $\Lambda_{A}$. Moreover,
this subalgebra is isomorphic to $A$. Thus we get the following.
\begin{prop}
\label{prop:Vij.^(i,j ) is an algebra} $\theta$ is an isomorphism
of associative algebras.
\end{prop}
\begin{rem}
Suppose that $A$ is finite dimensional and $S$ in Proposition \ref{prop:Vij.^(i,j ) is an algebra}
is a Levi (i.e. maximal semisimple) subalgebra of $A$. Then $\Lambda_{A}/\rad\Lambda_{A}\cong\bigoplus_{i\in I}\bbF\mathbf{1}_{i}$.
In particular, $\Lambda_{A}$ is a basic algebra. Moreover, it is
not difficult to see that $\Lambda_{A}$ is Morita equivalent to $A$.
\end{rem}
We will identify the algebra $A$ with its image $\theta(A)=\bigoplus_{i,j\in\hat{I}}V_{ij}\otimes\Lambda_{A}(i,j)$.
\begin{defn}
\label{def:S-decomposition} We say that $A=\bigoplus_{i,j\in\hat{I}}V_{ij}\otimes\Lambda_{A}(i,j)$
is the \emph{$S$-decomposition} of $A$.
\end{defn}
Let $A=\bigoplus_{i,j\in\hat{I}}V_{ij}\otimes\Lambda_{A}(i,j)$ be
the $S$-decomposition of $A$ and let $B$ be a subalgebra of $A$
such that $SB+BS\subseteq B$. Then $B$ is an $S$-sub-bimodule of
$A$ and
\begin{equation}
B=\bigoplus_{i,j\in\hat{I}}V_{ij}\otimes\Lambda_{B}(i,j)\label{eq:Bdec}
\end{equation}
where $\Lambda_{B}=\bigoplus_{i,j\in\hat{I}}\Lambda_{B}(i,j)$ is
a subspace of $\Lambda_{A}=\bigoplus_{i,j\in\hat{I}}(i,j)$. We say
that (\ref{eq:Bdec}) is the \emph{$S$-decomposition} of $B$. Using
Proposition \ref{prop:Vij.^(i,j ) is an algebra}, we obtain the
following.
\begin{prop}
\label{prop:^B sub ^A} Let $B$ be a subalgebra of $A$ such that
$SB+BS\subseteq B$. Then
(i) $\Lambda_{B}$ is a subalgebra of $\Lambda_{A}$.
(ii) If $B$ is an ideal of $A$, then $\Lambda_{B}$ is an ideal
of $\Lambda_{A}$.
\end{prop}
\begin{defn}
\label{def:B is S-mod} We say that a subalgebra $B$ of $A$ is $S$\emph{-modgenerated}
if $SB+BS\subseteq B$ and $B$ is generated (as an algebra) by non-trivial
simple $S$-sub-bimodules of $B$.
\end{defn}
Let $B$ be a subalgebra of $A$ such that $SB+BS\subseteq B$ and
let $B_{S}$ be the subalgebra of $B$ generated by non-trivial simple
$S$-sub-bimodules of $B$. Then $B_{S}$ is a subalgebra of $A$
with $SB_{S}+B_{S}S\subseteq B_{S}$. Let $B_{S}=\bigoplus_{i,j\in\hat{I}}V_{ij}\otimes\Lambda_{B_{S}}(i,j)$
be the $S$-decomposition of $B_{S}$. Since $B_{S}$ is a subalgebra
of $B$, we have $\Lambda_{B_{S}}(i,j)\subseteq\Lambda_{B}(i,j)$
for all $i.j\in\hat{I}$. On the other hand, $B_{S}$ is generated
by all $V_{ij}\otimes\Lambda_{B}(i,j)$ with $(i,j)\neq(0,0)$, so
$\Lambda_{B_{S}}(i,j)=\Lambda_{B}(i,j)$ for all $(i,j)\ne(0,0)$.
We have the following proposition.
\begin{prop}
\label{B_S} (i) $\Lambda_{B_{S}}(0,0)=\sum_{i\in I}\Lambda_{B_{S}}(0,i)\Lambda_{B_{S}}(i,0)$.
(ii) $\Lambda_{B_{S}}(0,0)=\sum_{i\in I}\Lambda_{B}(0,i)\Lambda_{B}(i,0)$.
(iii) $B_{S}$ is an ideal of $B$.
\end{prop}
\begin{proof}
(i) Let $B'_{S}=\bigoplus_{(i,j)\neq(0,0)}V_{ij}\otimes\Lambda_{B_{S}}(i,j)$.
Then $B_{S}$ is generated by $B'_{S}$. Choose any product $\Pi=V_{i_{1}j_{1}}\otimes\Lambda_{B_{S}}(i_{1},j_{1})\ldots V_{i_{k}j_{k}}\otimes\Lambda_{B_{S}}(i_{k},j_{k})$,
where $k\geq2$, in $B_{S}$. Suppose that $\Pi$ is not in $B_{S}$.
Then by Proposition \ref{prop:Vij.^(i,j ) is an algebra}, $j_{1}=i_{2}$,
$j_{2}=i_{3}$, $\ldots$, $j_{k-1}=i_{k}$ and $i_{1}=j_{k}=0$,
so $j_{1}\neq0$. Therefore, by using Proposition \ref{prop:Vij.^(i,j ) is an algebra},
we obtain
\begin{eqnarray*}
\Pi & = & (V_{0j_{1}}\otimes\Lambda_{B_{S}}(0,j_{1}))((V_{j_{1}j_{2}}\otimes\Lambda_{B_{S}}(j_{1},j_{2})\ldots(V_{j_{k-1}0}\otimes\Lambda_{B_{S}}(j_{k-1},0))\\
& \subseteq & (V_{0j_{1}}\otimes\Lambda_{B_{S}}(0,j_{1})(V_{j_{1}0}\otimes\Lambda_{B_{S}}(j_{1},0)\subseteq V_{00}\otimes\Lambda_{B_{S}}(0,j_{1})\Lambda_{B_{S}}(j_{1},0).
\end{eqnarray*}
Hence, $\Lambda_{B_{S}}(0,0)=\sum_{i\in I}\Lambda_{B_{S}}(0,i)\Lambda_{B_{S}}(i,0)$,
as required.
(ii) Follows from (i) since $\Lambda_{B_{S}}(i,j)=\Lambda_{B}(i,j)$
for all $(i,j)\ne(0,0)$.
(iii) We need to show that $\Lambda_{B}\Lambda_{B_{S}}\subseteq\Lambda_{B_{S}}$
(the case $\Lambda_{B_{S}}\Lambda_{B}\subseteq\Lambda_{B_{S}}$ is
similar). Since $\Lambda_{B_{S}}(i,j)=\Lambda_{B}(i,j)$ for all $(i,j)\ne(0,0)$
and $\Lambda_{B_{S}}$ is an algebra, we have $\Lambda_{B}(i,j)\Lambda_{B_{S}}\subseteq\Lambda_{B_{S}}$
for all $(i,j)\neq(0,0)$. It remain to note that $\Lambda_{B}(0,0)\Lambda_{B_{S}}\subseteq\Lambda_{B_{S}}$.
Indeed, for all $i\in I$ we have
\[
\Lambda_{B}(0,0)\Lambda_{B_{S}}(0,i)\subseteq\Lambda_{B}(0,0)\Lambda_{B}(0,i)\subseteq\Lambda_{B}(0,i)=\Lambda_{B_{S}}(0,i).
\]
Therefore, $\Lambda_{B_{S}}$ is an ideal of $\Lambda_{B}$.
\end{proof}
\begin{prop}
\label{prop:(i) =00003D (ii) =00003D (iii)} The following are equivalent
(i) $A$ is $S$-modgenerated.
(ii) $\Lambda_{A}(0,0)=\sum_{i\in I}\Lambda_{A}(0,i)\Lambda_{A}(i,0)$.
(iii) The ideal of $A$ generated by $S$ coincides with $A$.
\end{prop}
\begin{proof}
Let $A_{S}$ be the subalgebra of $A$ generated by non-trivial irreducible
$S$-sub-bimodules of $A$.
$(i)\Rightarrow(ii)$: Suppose that $A$ is $S$-modgenerated. Then
$A=A_{S}$, so $(ii)$ follows from Proposition \ref{B_S}(ii).
$(ii)\Rightarrow(i)$: Suppose $(ii)$ holds. Then by Proposition
\ref{B_S}(iii),
\[
\Lambda_{A_{S}}(0,0)=\sum_{i\in I}\Lambda_{A}(0,i)\Lambda_{A}(i,0)=\Lambda_{A}(0,0).
\]
Therefore, $A_{S}=A$, as required.
$(ii)\Leftrightarrow(iii)$: Note that the ideal $T$ of $A$ generated
by $S$ contains $A_{S}$. On the other hand, by Proposition \ref{B_S}(i),
$A_{S}$ is an ideal of $A$ containing $S,$ so $T\subseteq A_{S}$.
Therefore, $T=A_{S}$, as required.
\end{proof}
\begin{defn}
We say that $A$ is \emph{$M_{k}$-modgenerated} if $A$ is $S$-modgenerated
with $S$ isomorphic to the matrix algebra $M_{k}(\bbF)$.
\end{defn}
\begin{prop}
Suppose that $A$ is $S$-modgenerated and $S$ is $1$-perfect (resp.
$4$-perfect). Then $A$ is $M_{2}$-modgenerated (resp. $M_{3}$-modgenerated).
\end{prop}
\begin{proof}
Suppose that $S$ is $1$-perfect (the case of $4$-perfect $S$ is
similar). Let $\{S_{i}\mid i\in I\}$ be the set of the simple components
of $S$. Since $S$ is $1$-perfect, each $S_{i}$ contains a subalgebra
$P_{i}\cong M_{2}(\bbF)$. Fix any subalgebra $P$ of $S$ such that
$P\cong M_{2}(\bbF)$ and the projection of $P$ onto $S_{i}$ is
$P_{i}$. Then $S$ is generated by $P$ as an ideal. Since $A$ is
$S$-modgenerated, by Proposition \ref{prop:(i) =00003D (ii) =00003D (iii)},
$A$ is generated by $S$ as an ideal. Therefore, $A$ is generated
by $P$ as an ideal, so $A$ is $M_{2}$-modgenerated.
\end{proof}
\begin{prop}
\label{levi} Let $A$ be a perfect finite dimensional associative
algebra and let $S$ be a Levi subalgebra of $A$. Then $A$ is $S$-modgenerated.
\end{prop}
\begin{proof}
Let $T$ be an ideal of $A$ generated by $S$. By Proposition \ref{prop:(i) =00003D (ii) =00003D (iii)},
we need to show that $A=T$. Since $S$ is Levi subalgebra of $A$
and $S\subseteq T$, $A/T$ is nilpotent. As $A$ is perfect, $A/T=0$,
so $T=A$, as required.
\end{proof}
\section{Lie Structure of $A$}
Throughout this section, $A$ is an associative algebra, $S$ is a
finite dimensional semisimple subalgebra of $A$ such that $A$ is
$S$-modgenerated, $A=\bigoplus_{i,j\in\hat{I}}V_{ij}\otimes\Lambda_{A}(i,j)$
is the $S$-decomposition for $A$, $\{S_{i}\mid i\in I\}$ is the
set of the simple components of $S$, each $S_{i}$ is identified
with $M_{n_{i}}(\bbF)$, $B$ is a subalgebra of $A$ such that $BS+SB\subseteq B$,
$B_{S}$ is the subalgebra of $B$ generated by all non-trivial irreducible
$S$-sub-bimodules of $B$, $A=\bigoplus_{i,j\in\hat{I}}V_{ij}\otimes\Lambda_{A}(i,j)$
(resp. $B_{S}=\bigoplus_{i,j\in\hat{I}}V_{ij}\otimes\Lambda_{B_{S}}(i,j)$)
is the $S$-decomposition of $A$ (resp. $B_{S}$), $\{e_{st}^{ij}\mid1\leq s\leq n_{i},\ 1\le t\leq n_{j}\}$
(or simply $e_{st}$) is the standard basis of $V_{ij}$ consisting
of the matrix units and $S=\bigoplus_{i\in I}V_{ii}\otimes\mathbf{1}_{i}$.
In this section we suppose that $S$ is $1$-perfect if $p\ne2$ or
$4$-perfect if $p=2$. In particular, $n_{i}\ge2$ for all $i\in I$
and $n_{0}=1$.
We define $V'_{i,j}$ ($i,j\in\hat{I}$) as follows:
\begin{equation}
V'_{ij}=\begin{cases}
\{X\in V_{ii}\mid tr(X)=0\}\qquad, & \text{ if }i=j\\
V_{ij}, & \text{ if }i\neq j
\end{cases}.\label{eq:W'ij}
\end{equation}
Throughout this section, $L$ is the Lie algebra of $A$ generated
by all $V'_{ij}\otimes\Lambda_{A}(i,j)$ and $Q:=[S,S]=\bigoplus_{i\in I}Q_{i}\subseteq L$
where $Q_{i}=V'_{ii}\otimes\mathbf{1}_{i}$ are ideals of the Lie
algebra $Q$. Note that each $Q_{i}\cong sl_{n_{i}}(\bbF$) is simple
if $p=0$ or $p\not|n_{i}$. In the case $p|n_{i}$, the algebra $Q_{i}$
is \emph{quasisimple}, i.e. $Q_{i}$ is perfect and $Q_{i}/Z(Q_{i})$
is simple (here we use that $n_{i}\ge3$ if $p=2$).
\begin{prop}
\label{prop:L is perfect} $L$ is perfect.
\end{prop}
\begin{proof}
We need to show that $L\subseteq[L,L]$. It is enough to prove that
$V'_{ij}\otimes\Lambda_{A}(i,j)\subseteq[L,L]$ for all $(i,j)\neq(0,0)$.
Let $V'_{ij}\otimes\lambda\in V'_{ij}\otimes\Lambda_{A}(i,j)$. Suppose
that $i=j$. Since $Q\subseteq L$ and $V'_{ii}\cong Q_{i}$ as a
$Q_{i}$-module, we have
\begin{eqnarray*}
[L,L] & \supseteq & [V'_{ii}\otimes\mathbf{1}_{i},V'_{ii}\otimes\lambda]=[Q_{i},V'_{ii}\otimes\lambda]=V'_{ii}\otimes\lambda.
\end{eqnarray*}
Suppose now that $i\neq j$. Since $(i,j)\neq(0,0)$, at least one
of the indices, say $i$, is non-zero. Since $V'_{ij}$ is isomorphic
to the direct sum of $n_{j}$ copies of $V_{i}$ as $Q_{i}$-module,
we have
\begin{eqnarray*}
[L,L] & \supseteq & [V'_{ii}\otimes\mathbf{1}_{i},V'_{ij}\otimes\lambda]=[Q_{i},V'_{ij}\otimes\lambda]=V'_{ij}\otimes\lambda,
\end{eqnarray*}
as required. Therefore, $L$ is perfect.
\end{proof}
\begin{lem}
\label{lem:Vii * ^(i,i) } $[V_{ii}\otimes\Lambda_{A}(i,i),V_{ii}\otimes\Lambda_{A}(i,i)]\subseteq L$
for all $i\in I$.
\end{lem}
\begin{proof}
Denote by \textbf{$\mathbf{\varepsilon}_{i}$} the identity matrix
in $V_{ii}$. Let\noun{ }$X\otimes\lambda,Y\otimes\mu\in V_{ii}\otimes\Lambda_{A}(i,i)$.
We wish to show that $[X\otimes\lambda,Y\otimes\mu]\in L$. Recall
that $[L,L]=L$. We have two cases depending on whether $p$ divides
$n_{i}$ or not.
Suppose first that $p$ does not divide $n_{i}$. In this case $\varepsilon_{i}\not\in L$
and $V_{ii}=V'_{ii}\oplus\bbF\varepsilon_{i}$. Let $X',Y'\in V'_{ii}$.
By linearity, it suffices to show that $[X'\otimes\lambda,Y'\otimes\mu]$,
$[X'\otimes\lambda,\varepsilon_{i}\otimes\mu]$ and $[\varepsilon_{i}\otimes\lambda,\varepsilon_{i}\otimes\mu]$
are in $L$. We have $[X'\otimes\lambda,Y'\otimes\mu]\in[L,L]=L$
and $[X'\otimes\lambda,\varepsilon_{i}\otimes\mu]=X'\otimes\lambda\mu-X'\otimes\mu\lambda\in L$,
as required. Note that $e_{rq}\in V'_{ii}$ for all $r\neq q$, so
$[e_{rq}\otimes\lambda,e_{qr}\otimes\mu]\in[L,L]=L$. Hence, we have
the following system of commutators in $L$:
\[
\begin{array}{ccc}
[e_{12}\otimes\lambda,e_{21}\otimes\mu] & = & e_{11}\otimes\lambda\mu-e_{22}\otimes\mu\lambda\in L,\\
{}[e_{23}\otimes\lambda,e_{32}\otimes\mu] & = & e_{22}\otimes\lambda\mu-e_{33}\otimes\mu\lambda\in L,\\
\vdots & \vdots & \vdots\\
{}[e_{n-1,n}\otimes\lambda,e_{n,n-1}\otimes\mu] & = & e_{n-1,n-1}\otimes\lambda\mu-e_{nn}\otimes\mu\lambda\in L,\\
{}[e_{n1}\otimes\lambda,e_{1n}\otimes\mu] & = & e_{nn}\otimes\lambda\mu-e_{11}\otimes\mu\lambda\in L
\end{array}
\]
where $n=n_{i}$. Since the sum of all expressions on the right equals
to $\varepsilon_{i}\otimes\lambda\mu-\varepsilon_{i}\otimes\mu\lambda=[\varepsilon_{i}\otimes\lambda,\varepsilon_{i}\otimes\mu]$
we have $[\varepsilon_{i}\otimes\lambda,\varepsilon_{i}\otimes\mu]\in L$,
as required.
Now, suppose that $p$ divides $n_{i}$. In this case we have $\varepsilon_{i}\in V'_{ii}$
and $V_{ii}=V'_{ii}+\bbF e_{11}$. Let $X',Y'\in V'_{ii}$. By linearity,
it suffices to show that $[X'\otimes\lambda,Y'\otimes\mu]$, $[X'\otimes\lambda,e_{11}\otimes\mu]$
and $[e_{11}\otimes\lambda,e_{11}\otimes\mu]$ are in $L$. As before,
$[X'\otimes\lambda,Y'\otimes\mu]\in[L,L]=L$ and $[X'\otimes\lambda,e_{11}\otimes\mu]=X'e_{11}\otimes\lambda\mu-e_{11}X'\otimes\mu\lambda$.
Put $X'e_{11}=X_{1}+\beta e_{11}$ and $e_{11}X'=X_{2}+\beta e_{11}$,
where $X_{1},X_{2}\in V'_{ii}$. Then
\[
[X'\otimes\lambda,e_{11}\otimes\mu]=X_{1}\otimes\lambda\mu+\beta e_{11}\otimes\lambda\mu-X_{2}\otimes\mu\lambda-\beta e_{11}\otimes\mu\lambda.
\]
Note that $X_{1}\otimes\lambda\mu,X_{2}\otimes\mu\lambda\in L$. Hence,
it remains to show that $e_{11}\otimes\lambda\mu-e_{11}\otimes\mu\lambda\in L$.
Recall that $e_{rq}\in V'_{ii}$ for all $r\neq q$. Hence,
\[
[e_{rq}\otimes\lambda,e_{qr}\otimes\mu]\in[V'_{ii}\otimes\Lambda_{A}(i,i),V'\otimes\Lambda_{A}(i,i)]\subseteq[L,L]=L.
\]
Hence, we have the following system of commutators
\[
\begin{array}{ccc}
[e_{12}\otimes\mu,e_{21}\otimes\lambda] & = & e_{11}\otimes\mu\lambda-e_{22}\otimes\lambda\mu\in L,\\
{}[e_{13}\otimes\mu,e_{31}\otimes\lambda] & = & e_{11}\otimes\mu\lambda-e_{33}\otimes\lambda\mu\in L,\\
\vdots & \vdots & \vdots\\
{}[e_{1,n}\otimes\mu,e_{n,1}\otimes\lambda] & = & e_{11}\otimes\mu\lambda-e_{nn}\otimes\lambda\mu\in L,
\end{array}
\]
where $n=n_{i}$. The sum of all these commutators $\stackrel[k=2]{n}{\sum}[e_{1k}\otimes\mu,e_{k1}\otimes\lambda]\in L$.
Since $p|n_{i}$, we have $ne_{11}=0$, so this sum is
\[
\stackrel[k=2]{n}{\sum}(e_{11}\otimes\mu\lambda-e_{kk}\otimes\lambda\mu)=-e_{11}\otimes\mu\lambda+e_{11}\otimes\lambda\mu-\varepsilon_{i}\otimes\lambda\mu\in L.
\]
Hence, $e_{11}\otimes\lambda\mu-e_{11}\otimes\mu\lambda\in L$, as
required.
\end{proof}
\begin{lem}
\label{lem:=00005BL,Vii * ^A(i,i)=00005D sub L} $[L,V_{ii}\otimes\Lambda_{A}(i,i)]\subseteq L$
for all $i\in I$.
\end{lem}
\begin{proof}
Note that
\[
[L,V_{ii}\otimes\Lambda_{A}(i,i)]\subseteq[A,V_{ii}\otimes\Lambda_{A}(i,i)]=\bigoplus_{(i,j)\neq(0,0)}[V_{ij}\otimes\Lambda_{A}(i,j),V_{ii}\otimes\Lambda_{A}(i,i)]].
\]
Hence, we have two cases depending on the values of $i$ and $j$.
If $i=j$, then by Lemma \ref{lem:Vii * ^(i,i) }, $[V_{ii}\otimes\Lambda_{A}(i,i),V_{ii}\otimes\Lambda_{A}(i,i)]\subseteq L$.
Suppose that $i\neq j$. Then by using (\ref{eq:W'ij}) we obtain
\[
[V_{ij}\otimes\Lambda_{A}(i,j),V_{ii}\otimes\Lambda_{A}(i,i)]]\subseteq V_{ij}\otimes\Lambda_{A}(i,j)=V'_{ij}\otimes\Lambda_{A}(i,j)\subseteq L.
\]
Thus, $[L,V_{ii}\otimes\Lambda_{A}(i,i)]\subseteq L$.
\end{proof}
\begin{lem}
\label{lem:V00 * ^(0,0) } $[V_{00}\otimes\Lambda_{A}(0,0),V_{00}\otimes\Lambda_{A}(0,0)]\subseteq L$.
\end{lem}
\begin{proof}
Let $X\otimes\lambda,Y\otimes\mu\in V_{00}\otimes\Lambda_{A}(0,0)$.
By Proposition \ref{prop:(i) =00003D (ii) =00003D (iii)}, we have
\begin{equation}
\Lambda_{A}(0,0)=\sum_{i\in I}\Lambda_{A}(0,i)\Lambda_{A}(i,0).\label{eq:W00-1}
\end{equation}
Hence,
\begin{eqnarray*}
V_{00}\otimes\Lambda_{A}(0,0) & = & V_{00}\otimes\sum_{i\in I}\Lambda_{A}(0,i)\Lambda_{A}(i,0)\\
& = & \sum_{i\in I}(V_{0i}\otimes\Lambda_{A}(0,i))(V_{i0}\otimes\Lambda_{A}(i,0).
\end{eqnarray*}
Thus, $X\otimes\lambda=\sum_{i\in I}(X_{0i}\otimes\lambda_{0i})(X_{i0}\otimes\lambda_{i0})$
and $Y\otimes\mu=\sum_{i\in I}(Y_{0i}\otimes\mu_{0i})(Y_{i0}\otimes\mu_{i0})$,
where $X_{0i}\otimes\lambda_{0i},Y_{0i}\otimes\mu_{0i}\in V_{0i}\otimes\Lambda_{A}(0,i)$
and $X_{i0}\otimes\lambda_{i0},Y_{i0}\otimes\mu_{i0}\in V_{i0}\otimes\Lambda_{A}(i,0)$.
By \noun{(\ref{eq:W'ij}), }we have $V_{ij}\otimes\Lambda_{A}(i,j)=V'_{ij}\otimes\Lambda_{A}(i,j)\subseteq L$
for all $i\neq j$, so
\[
X_{0i}\otimes\lambda_{0i},X_{i0}\otimes\lambda_{0i},Y_{0i}\otimes\mu_{0i},Y_{i0}\otimes\mu_{i0}\in L.
\]
We wish to show that $[X\otimes\lambda,Y\otimes\mu]\in L$. Note that
\begin{eqnarray*}
[X\otimes\lambda,Y\otimes\mu] & = & \underset{(i,j)\neq(0,0)}{\sum}[(X_{0i}\otimes\lambda_{0i})(X_{i0}\otimes\lambda_{i0}),(Y_{0j}\otimes\mu_{0j})(Y_{j0}\otimes\mu_{j0})]\\
& = & \underset{(i,j)\neq(0,0)}{\sum}[X_{0i}X_{i0}\otimes\lambda_{0i}\lambda_{i0},Y_{0j}Y_{j0}\otimes\mu_{0j}\mu_{j0}],
\end{eqnarray*}
so without loose of generality it is enough to show that
\[
[X_{01}X_{10}\otimes\lambda_{01}\lambda_{10},Y_{01}Y_{10}\otimes\mu_{01}\mu_{10}]\in L.
\]
Put $x_{0}=X_{01}X_{10}\otimes\lambda_{01}\lambda_{10}$ and $y_{0}=Y_{01}Y_{10}\otimes\mu_{01}\mu_{10}$.
Then we need to show that $[x_{0},y_{0}]\in L$. Note that $x_{0},y_{0}\in V_{00}\otimes\Lambda_{A}(0,0)$.
Let
\[
x_{L}=[X_{01}\otimes\lambda_{01},X_{10}\otimes\lambda_{10}]=X_{01}X_{10}\otimes\lambda_{01}\lambda_{10}-X_{10}X_{01}\otimes\lambda_{10}\lambda_{01}=x_{0}-x_{1},
\]
where $x_{1}=X_{10}X_{01}\otimes\lambda_{10}\lambda_{01}$ and
\[
y_{L}=[Y_{01}\otimes\mu_{01},Y_{10}\otimes\mu_{10}]=Y_{01}Y_{10}\otimes\mu_{01}\mu_{10}-Y_{10}Y_{01}-\mu_{10}\mu_{01}=y_{0}-y_{1},
\]
where $y_{1}=Y_{10}Y_{01}\otimes\mu_{10}\mu_{01}$. Then $x_{L},y_{L}\in[L,L]=L$,
and $x_{1},y_{1}\in V_{11}\otimes\Lambda_{A}(1,1)$. Since $x_{0}=x_{L}+x_{1}$
and $y_{0}=y_{L}+y_{1}$, we have
\begin{eqnarray}
[x_{0},y_{0}] & = & [x_{L}+x_{1},y_{L}+y_{1}]\nonumber \\
& = & [x_{L},y_{L}]+[x_{L},y_{1}]+[x_{1},y_{L}]+[x_{1},y_{1}].\label{eq:=00005Bx0,y0=00005D-1}
\end{eqnarray}
To show that $[x_{0},y_{0}]\in L$, it suffices to show that each
term on the right side of (\ref{eq:=00005Bx0,y0=00005D-1}) is in
$L$. We have $[x_{L},y_{L}]\in[L,L]=L$. As $i\in I$, by Lemma \ref{lem:Vii * ^(i,i) },
$[x_{1},y_{1}]\in L$. Moreover, by Lemma \ref{lem:=00005BL,Vii * ^A(i,i)=00005D sub L},
$[L,V_{ii}\otimes\Lambda_{A}(i,i)]\subseteq L$, so $[x_{L},y_{1}],[x_{1},y_{L}]\in L$.
Therefore, $[x_{0},y_{0}]\in L$. Thus, $[X\otimes\lambda,Y\otimes\mu]\in L$
for all $i$, $j$, $s$ and $t$, as required.
\end{proof}
By combining Lemma \ref{lem:Vii * ^(i,i) } and Lemma \ref{lem:V00 * ^(0,0) },
we obtain the following.
\begin{prop}
\label{prop:Vii * ^(i,i) for all i in I hat} $[V_{ii}\otimes\Lambda_{A}(i,i),V_{ii}\otimes\Lambda_{A}(i,i)]\subseteq L$
for all $i\in\hat{I}$.
\end{prop}
\begin{thm}
\label{thm:L=00003D=00005BA,A=00005D is perfect} $A^{(1)}$ is perfect
and $A=A^{(1)}A^{(1)}+A^{(1)}$.
\end{thm}
\begin{proof}
We identify $A$ with $\bigoplus_{i,j\in\hat{I}}V_{ij}\otimes\Lambda_{A}(i,j)$.
By Lemma \ref{prop:Vij.^(i,j ) is an algebra}, $S=\bigoplus_{i\in I}S_{i}=\bigoplus_{i\in I}V_{ii}\otimes\mathbf{1}_{i}$
is a Levi subalgebra of $A$. As above we fix a matrix realization
$M_{n_{i}}(\bbF)$ for each simple component $S_{i}$ of $S$. Let
$L$ be a Lie subalgebra of $A$ generated by all $V'_{i,j}\otimes\Lambda_{A}(i,j)$
for $(i,j)\neq(0,0)$, where $V'_{ij}$ is defined in (\ref{eq:W'ij}).
Then by Proposition \ref{prop:L is perfect}, $L$ is perfect and
$Q=[S,S]=\bigoplus_{i\in I}V'_{ii}\otimes\mathbf{1}_{i}\subseteq L$.
We wish to show that $L=A^{(1)}$. Since $L$ is a perfect Lie subalgebra
of $A^{(-)}$, we have $L=L^{(1)}\subseteq A^{(1)}$. It remains to
show that $A^{(1)}\subseteq L$. Let $X\otimes\lambda\in V_{ij}\otimes\Lambda_{A}(i,j)$
and $Y\otimes\mu\in V_{st}\otimes\Lambda_{A}(s,t)$, where $i,j,s,t\in\hat{I}$.
We wish to show that
\begin{equation}
[X\otimes\lambda,Y\otimes\mu]=XY\otimes\lambda\mu-YX\otimes\mu\lambda\in L.\label{eq:=00005BX,Y=00005D=00003DXY-YX}
\end{equation}
If $i\neq t$ and $j\neq s$, then obviously we have $[X\otimes\lambda,Y\otimes\mu]=0\in L$.
Suppose that $j=s$. First, consider the case when $i\neq t$. Then
$YX\otimes\mu\lambda=0$. By using the formula (\ref{eq:W'ij}) we
obtain $XY\otimes\lambda\mu\in V_{it}\otimes\Lambda_{A}(i,t)=V'_{it}\otimes\Lambda_{A}(i,t)\subseteq L$.
Therefore, $[X\otimes\lambda,Y\otimes\mu]=XY\otimes\lambda\mu\in L$.
Now, suppose that $i=t$. If $i\neq j$, then by formula (\ref{eq:W'ij}),
$V_{ij}\otimes\Lambda_{A}(i,j)=V'_{ij}\otimes\Lambda_{A}(i,j)\subseteq L$
for all $i\neq j$. Hence,
\[
[X\otimes\lambda,Y\otimes\mu]\in[V'_{ij}\otimes\Lambda_{A}(i,j),V'_{ji}\otimes\Lambda_{A}(j,i)]\subseteq[L,L]=L.
\]
Suppose that $i=j$. Then by Proposition \ref{prop:Vii * ^(i,i) for all i in I hat},
$[V_{ii}\otimes\Lambda_{A}(i,i),V_{ii}\otimes\Lambda_{A}(i,i)]\in L$
for all $i\in\hat{I}$. Hence, $[X\otimes\lambda,Y\otimes\mu]\in L$
for all $i$, $j$, $s$ and $t$. Therefore, $A^{(1)}=L$ is perfect,
as required.
It remains to show that $A=LL+L$. Note that $V'_{ij}\otimes\Lambda_{A}(i,j)\subseteq L\subseteq A$
for all $(i,j)\neq(0,0)$. Suppose that $j\neq0$. Then for all $\lambda\in\Lambda_{A}(i,j)$
we have $(e_{11}\otimes\lambda)(e_{12}\otimes\mathbf{1}_{i})\in(V'_{ij}\otimes\lambda)(V'_{jj}\otimes\mathbf{1}_{j})\subseteq LQ\subseteq LL$.
Therefore, $V_{ij}\otimes\Lambda_{A}(i,j)\subseteq LL+L$ for all
$(i,j)\neq(0,0)$. Now, assume that $j=0$. Then either $i\neq0$
or $i=0$. The case when $i\neq0$ is similar to above. Suppose that
$i=0$. Then by using Proposition \ref{prop:(i) =00003D (ii) =00003D (iii)}
and the formula (\ref{eq:W'ij}) we obtain
\begin{eqnarray*}
V_{00}\otimes\Lambda_{A}(0,0) & = & V_{00}\otimes\sum_{i\in I}\Lambda_{A}(0,i)\Lambda_{A}(i,0)\\
& = & \sum_{i\in I}(V_{0i}\otimes\Lambda_{A}(0,i))(V_{i0}\otimes\Lambda_{A}(i,0)\\
& = & \sum_{i\in I}(V'_{0i}\otimes\Lambda_{A}(0,i))(V'_{i0}\otimes\Lambda_{A}(i,0)\subseteq LL.
\end{eqnarray*}
Therefore, $A=LL+L$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:main}]
This follows from Proposition \ref{prop:(i) =00003D (ii) =00003D (iii)}
and Theorem \ref{thm:L=00003D=00005BA,A=00005D is perfect}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:fd}]
Let $S$ be a Levi subsalgebra of $A$. Then by Proposition \ref{levi},
$A$ is $S$-modgenerated. Since $S\cong A/\rad(A)$, the algebra
$S$ is $k$-perfect whenever $A$ is $k$-perfect. Therefore the
result follows from Theorem \ref{thm:L=00003D=00005BA,A=00005D is perfect}.
\end{proof}
| 4,513
|
Includes: Double Bimini top, black (1.25" frame with SST quick release), Vertical Bow curtain, Side curtains, Aft curtain
Includes: (4) Mounted Anodes
Includes: Reinforced nose cone (rough water), Full aluminum skin, Fuel tank increases to 37 gallons, Max weight capacity changes to 2,679 lbs and 14-15 persons
Includes: 27" center tube (full), Reinforced nose cone (rough water), Full-length keel, Strakes, lifting (REQUIRED, 27" tubes, recommend 90HP and Up), Full aluminum skin, Fuel tank increases to 37 gallons, Max weight capacity changes to 2,923 lbs and 15-16 persons
Includes: Ski tow bar, stainless steel, V3 Action Package, In-floor storage (center)
Overall Length with ladder (ft/in): 25' 9"
Pontoon Length (ft/in): 24' 0"
Pontoon Diameter (.080 gauge) (in): 25"
Beam (ft/in): 8' 6"
Weight (2 tubes/3 tubes) (lbs): 2,705/3,400
Max Weight Capacity (2 tubes/3 tubes) (lbs): 2,314/2,679
Max Weight (persons): 12-16
Max Person Capacity (2 tubes/3 tubes) (persons): 12-13/14-15
Min HP: 9.9
Max HP: 200
Max HP (2 tubes/3 tubes): 150/200
Fuel Capacity (2 tubes/V3 Action Package) (gals): 28/37
Draft Up (in): 12"
Draft Down (in): 21"
Bridge Clearance (bimini and stern lights down) (in): 62"
FEATURES
CONSOLE/INSTRUMENTATION
Console with lit rocker switches, 3 drink holders, 12V power outlet, cubby, lockable glove box, storage with door below and space for 7" electronics (replaces cubby) (starboard)
Windscreen, removable
Tachometer and trim gauges, analog
Speedometer, GPS driven
Steering wheel, premium, aluminum polished with comfort grip
Tilt steering
Fuel gauge
Horn
FLOORING
Decking, marine-grade 3/4" 7-ply pressure treated plywood
SEATING
Seat, helm, highback recliner with slider and swivel (1)
Seat, chaise lounge chair, pillow top, extended (starboard, bow)
Seat, chaise lounge chair, pillow top, extended (port, bow)
Seat, L chaise lounge chair, pillow top (port, stern)
CANVAS
Bimini top, 10' black (1.25" frame with SST quick release)
Cover, mooring, black snapless
Transhield transportation cover, reusable
STORAGE/CONVENIENCE
Storage compartments (below all lounge chairs)
Storage compartment, large with sunpad (port, stern)
Storage compartment, trash can
Drink holders (7 including table)
INTERIOR
Gates, (4) (center bow, port across from helm, starboard in front of helm, starboard stern)
Lanyard, ignition stop switch
Gate floor stops (dealer installed)
Light, navigation
Light, courtesy
EXTERIOR
Keel, full length, extruded (3/8" thick)
Cross members, aluminum extruded
Full-length torsion control riser bracket with AM/FM, bluetooth capability, USB audio and charging, auxiliary and 4 speakers
Table, wood grain with 2 molded drink holders (dealer installed)
Tube accent graphics (matches fence color).
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New log Burner – Tips for use
Much excitement here this week. We’ve got a new log burner. The mess and dust has been horrendous but it is so worth it.
If you haven’t had a log burner there are a few things that you need to know to really get the most out of that metal box in the fireplace that I thought would be worth sharing;.
Finally just enjoy it! The log burner for us is not our only source of heating we have the central heating too. Make sure you shop around for the best quotes on the central Heating to find your best option.
| 161,158
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{\bf Problem.} Compute
\[e^{2 \pi i/13} + e^{4 \pi i/13} + e^{6 \pi i/13} + \dots + e^{24 \pi i/13}.\]
{\bf Level.} Level 3
{\bf Type.} Precalculus
{\bf Solution.} Let $\omega = e^{2 \pi i/13}.$ Then from the formula for a geometric sequence,
\begin{align*}
e^{2 \pi i/13} + e^{4 \pi i/13} + e^{6 \pi i/13} + \dots + e^{24 \pi i/13} &= \omega + \omega^2 + \omega^3 + \dots + \omega^{12} \\
&= \omega (1 + \omega + \omega^2 + \dots + \omega^{11}) \\
&= \omega \cdot \frac{1 - \omega^{12}}{1 - \omega} \\
&= \frac{\omega - \omega^{13}}{1 - \omega}.
\end{align*}Since $\omega^{13} = (e^{2 \pi i/13})^{13} = e^{2 \pi i} = 1,$
\[\frac{\omega - \omega^{13}}{1 - \omega} = \frac{\omega - 1}{1 - \omega} = \boxed{-1}.\]
| 17,030
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\begin{document}
\markboth{\scriptsize{EACEHL}}{\scriptsize{EACEHL}}
\title{BRASCAMP-LIEB INEQUALITIES FOR NON-COMMUTATIVE INTEGRATION}
\author{\vspace{5pt} Eric A. Carlen$^1$ and
Elliott H. Lieb$^{2}$ \\
\vspace{5pt}\small{$1.$ Department of Mathematics, Hill Center,}\\[-6pt]
\small{Rutgers University,
110 Frelinghuysen Road
Piscataway NJ 08854-8019 USA}\\
\vspace{5pt}\small{$2.$ Departments of Mathematics and Physics, Jadwin
Hall,} \\[-6pt]
\small{Princeton University, P.~O.~Box 708, Princeton, NJ
08544}\\
}
\date{\version}
\maketitle
\footnotetext
[1]{Work partially
supported by U.S. National Science Foundation
grant DMS 06-00037. }
\footnotetext
[2]{Work partially
supported by U.S. National Science Foundation
grant PHY 06-52854.\\
\copyright\, 2008 by the authors. This paper may be reproduced, in its
entirety, for non-commercial purposes.}
\def\mn{{\bf M}_n}
\def\hn{{\bf H}_n}
\def\hnp{{\bf H}_n^+}
\def\hmnp{{\bf H}_{mn}^+}
\def\h{{\cal H}}
\def\A{{\mathfrak A}}
\def\B{{\mathfrak B}}
\def\CC{{\mathfrak C}}
\def\dd{{\rm d}}
\def\tr{{\rm Tr}}
\begin{abstract}
We formulate a non-commutative analog of the Brascamp-Lieb inequality, and prove it in several
concrete settings.
\end{abstract}
\medskip
\leftline{\footnotesize{\qquad Mathematics subject classification numbers: 47C15, 15A45, 26D15}}
\leftline{\footnotesize{\qquad Key Words: inequalities, traces, non-commutative integration}}
\section{Introduction} \label{intro}
\subsection{Young's inequality in
the context of ordinary Lebesgue integration}
In this paper, we shall extend the class of generalized Young's
inequalities known as Brascamp-Lieb inequalities (B-L inequalities)
to an operator algebra setting entailing non-commutative integration.
\if false We begin by specifying precisely what we mean by a generalized
Young's inequality in the context of ordinary Lebesgue integration. \fi
The original Young's inequality \cite{Y12} states that for non
negative measurable functions $f_1$, $f_2$ and $f_3$ on $\R$, and
$1\le p_1,p_2,p_3 \le \infty$, with $1/p_1 + 1/p_2 + 1/p_3 = 2$,
\begin{equation}\label{basicy}
\int_{\R^2}f_1(x)f_2(x-y)f_3(y)\dd x\dd y \le \left(\int_\R f_1^{p_1}(t)\dd t\right)^{1/p_1}
\left(\int_\R f_2^{p_2}(t)\dd t\right)^{1/p_2}\left(\int_\R f_3^{p_3}(t)\dd t\right)^{1/p_3}\ .
\end{equation}
Thus, it provides an estimate of the integral of a product of
functions in terms of a product of $L^p$ norms of these functions. The
crucial difference with a H\"older type inequality is that the
integrals on the right are integrals over only $\R$, while the
integrals on the left are integrals over $\R^2$, and none of the three
factors in the product on the left -- $f(x)$, $g(x-y)$ or $h(y)$ --
are integrable (to any power) on $\R^2$.
To frame the inequality in terms that are more amenable to the
generalizations considered here, define the maps $\phi_j: \R^2 \to
\R$, $j=1,2,3$, by
$$\phi_1(x,y) = x\qquad \phi_2(x,y) = x-y\qquad{\rm and}\qquad \phi_3(x,y) = y\ .$$
Then (\ref{basicy}) can be rewritten as
\begin{equation}\label{basicy2}
\int_{\R^2}\left(\prod_{j=1}^3 f_j\circ \phi_j\right) \dd^2 x \le \prod_{j=1}^3 \left(\int_\R f_j^{p_j}(t)\dd t\right)^{1/p_j}\ .
\end{equation}
There is now no particular reason to limit ourselves to products of only three functions, or to integrals over $\R^2$ and $\R$, or even any Euclidean space for that matter:
\begin{defi} Given measure spaces $(\Omega, \mathcal S , \mu)$ and
$(M_j, \mathcal M_j , \nu_j)$, $j=1,\dots,N$, not necessarily
distinct, together with measurable functions $\phi_j:\Omega \to M_j$
and numbers $p_1,\dots,p_N$ with $1\le p_j \le \infty$, $1 \le j\le
N$, we say that {\em a B-L inequality holds for
$\{\phi_1,\dots,\phi_N\}$ and $\{p_1,\dots,p_N\}$} in case there
is a finite constant $C$ such that
\begin{equation}\label{gy}
\int_\Omega \prod_{j=1}^N f_j\circ \phi_j {\rm d}\mu \le C \prod_{j=1}^N\|f_j\|_{L^{p_j}(\nu_j)}\
\end{equation}
\end{defi}
\noindent{holds} whenever each $f_j$ is non negative and measurable on $M_j$, $j=1,\dots,N$.
There are by now many examples. One of the oldest is the original {\em
discrete Young's inequality}. In the current notation, this
concerns the case in which $\Omega = \Z^2$ equipped with counting
measure, $N =3$, and each $M_j$ is $\Z$, equipped with counting
measure. Then with
$$\phi_1(m,n) = m\qquad \phi_2(m,n) = m-n\qquad{\rm and} \qquad\phi_3(m,n) = n\ ,$$
(\ref{basicy2}) holds for any three non-negative functions $f_j:\Z \to
\R_+$ under the same conditions on the $p_j$ as in the continuous
case; i.e., $1/p_1 + 1/p_2 + 1/p_3 = 2$. There is a significant
difference: In the discrete case, the constant $C=1$ is sharp, and
there is equality if and only if one of the $f_j$ is identically zero,
or else $f_1$ vanishes except at some $m_0$, $f_3$ vanishes except at
some $n_0$, and $f_2$ vanishes except at $m_0-n_0$. The inequality
itself is due to Young \cite{Y12}, while the statement about cases of
equality is proved in \cite{HLP}, where the authors also consider
extensions to more than three functions.
In the continuous case, a much wider generalization to more than three
functions was made by B-L in \cite{BL}, where the sharp constant in Young's
inequality -- which is strictly less than $1$ unless $p_1 = p_2 =1$ --
was obtained, with a proof that the only non-negative functions yielding
equality are certain Gaussian functions. (This best constant was also
obtained at the same time by Beckner \cite{Beckner}, for three functions.)
These inequalities generalize from $\R$ to $\R^n$. The complete
generalization to the case in which the $M_j$ are all Euclidean
spaces, but of different dimension, and the $\phi_j$ are linear
transformations from $\R^n$ to $M_j$, was proved by Lieb \cite{L90}.
Again, the maximizers are Gaussians. Another proof of this
generalized version, together with a reverse form, was obtained by
Barthe \cite{B}, who also provided a detailed analysis of the cases of
equality in the original B-L inequality from \cite{BL}. The cases of
equality in the higher dimensional generalization from \cite{L90} were
analyzed in detail in \cite{BCCT1,BCCT2}.
Examples in which $\Omega$ is the sphere $S^{N-1}$ or the permutation
group ${\cal S}^N$ were proved in \cite{CLL1,CLL2}, and the above
definition of B-L inequalities in arbitrary measure spaces is taken
from \cite{CC}, where a duality between B-L inequalities and
subadditivity of entropy inequalities is proved.
\subsection{A generalized Young's inequality in
the context of non commutative integration}
In non commutative integration theory, as developed by Irving Segal
\cite{S53,S56,S65}, the basic framework is a triple $(\h,\A,\lambda)$
where $\h$ is a Hilbert space, $\A$ is a $W^*$ algebra (a von Neumann
algebra) of operators on $\h$, and $\lambda$ is a positive linear
functional on the finite rank operators in $\A$. In Segal's picture,
the algebra $\A$ corresponds to the algebra of bounded measurable
functions, and applying the linear positive linear functional
$\lambda$ to a positive operator corresponds to taking the integral of
a positive function. That is,
$$A \mapsto \lambda(A)\qquad{\rm corresponds\ to}\qquad
f \mapsto \int_M f\dd \nu\ .$$
Such a triple $(\h,\A,\lambda)$ is called a {\em non commutative
integration space}. Certain natural regularity properties must be
imposed on $\lambda$ if one is to get a well-behaved non-commutative
integration theory, but we shall not go into these here as the
examples that we consider are all based on the case in which $\lambda$
is the {\em trace} on operators on $\h$, or some closely related
functional, for which discussion of these extra conditions would be a
digression.
In this operator algebra setting, there are natural non-commutative
analogs of the usual $L^p$ spaces: If $A$ is a finite rank operator in
$\A$, and $1\le q < \infty$, define
$$\|A\|_{q,\lambda} = \left(\lambda(A^*A)^{q/2}\right)^{1/q}\ .$$
This defines a norm (under appropriate conditions on $\lambda$ that
are obvious for the trace), and the completion of the space of finite
rank operator in $\A$ under this norm defines a non-commutative $L^p$
space. (The completion may contain unbounded operators ``affiliated''
to $\A$.) For more on the general theory of non-commutative
integration, see the early papers \cite{Di53,S53,S65,St59} and the
more recent work in \cite{FK,H,K2,N74}.
To frame an analog of (\ref{gy}) in an operator algebra setting, we
replace the measure spaces by non commutative integration spaces:
$$
(\Omega, \mathcal S , \mu) \longrightarrow (\h, \A,\lambda) \qquad{\rm
and}\qquad (M_j, \mathcal M_j , \nu_j) \longrightarrow (\h_j,\A_j,
\lambda_j)\qquad j =1,\dots,N \ .$$
The right hand side of (\ref{gy})
has an obvious generalization to the operator algebra setting in terms
of the non-commutative $L_p$ norms introduced above.
As for the left hand side of (\ref{gy}), regard $f_j \mapsto f_j\circ
\phi_j$ as a $W^*$ algebra homomorphism (which, restricted to the
$W^*$ algebra $L^\infty(M_j)$, it is), and suppose we are given $W^*$
homomorphisms
$$\phi_j : \A_j \to \A\ .$$
Then each $\phi_j(A_j)$ belongs to $\A$, however in the
non-commutative case, the product of the $\phi_j(A_j)$ depends on
their order in the product, and need not be self-adjoint even -- let
alone positive -- even if each of the $A_j$ are positive.
Therefore, let us return to the left side of (\ref{gy}), and suppose
that each $f_j$ is strictly positive. Then defining
$$h_j = \ln(f_j) \qquad{\rm so \ that}\qquad f_j\circ \phi_j = e^{h\circ \phi_j}\ ,$$
we can then rewrite (\ref{gy}) as
\begin{equation}\label{gy2}
\int_\Omega \exp\left( \sum_{j=1}^N h_j\circ \phi_j\right) {\rm d}\mu
\le C \prod_{j=1}^N\| e^{h_j}\|_{L^{p_j}(\nu_j)}\ ,
\end{equation}
We can now formulate our operator algebra analog of (\ref{gy}):
\begin{defi}\label{ncgy}
Given non commutative integration spaces $(\h,\A,\lambda)$ and $(\h_j,\A_j,\lambda_j)$, $j=1,\dots,N$,
together with $W^*$ algebra homomorphisms $\phi_j:\A_j\to \A$, $j=1,\dots,N$, and indices
$1\le p_j\le \infty$, $j=1,\dots,N$, a {\em non-commutative B-L inequality holds
for $\{\phi_1,\dots,\phi_N\}$ and $\{p_1,\dots,p_N\}$} if there is a finite constant $C$
so that
\begin{equation}\label{intprod3}
\boxed{ \ \lambda\left(\exp\left[ \sum_{j=1}^N \phi_j(H_j)\right]\right) \le C
\prod_{j=1}^N\left(\lambda_j\exp\left[ p_j H_j \right]\right)^{1/p_j} \ }
\end{equation}
whenever $H_j$ is self-adjoint in $\A_j$, $j = 1,\dots,N$.
\end{defi}
In this paper, we are concerned with determining the indices and the
best constant $C$ for which such an inequality holds, and shall focus
on two examples: The first concerns {\em operators on tensor products
of Hilbert spaces}, and the second concerns {\em Clifford algebras}.
\subsection{A generalized Young's inequality for tensor products}
\begin{exam}\label{tp}
Let $\h_j$, $j=1,\dots,n$ be separable Hilbert spaces, and let
Let ${\cal K}$
denote the tensor product
$${\cal K} = \h_1\otimes \cdots \otimes \h_n\ .$$
Define $\A$ to be $\B({\cal K})$, the algebra of bounded operators on
${\cal K}$, and define $\lambda$ to be $\tr$, the trace $\tr$ on
${\cal K}$, so that $(\h,\A,\lambda) = ({\cal K},\B({\cal K}),\tr)$.
For any non empty subset $J$ of $\{1,\dots,n\}$, let ${\cal K}_J$
denote the tensor product
$${\cal K}_J = \bigotimes_{j\in J}\h_j\ .$$ Define
$\A_J$ to be $\B({\cal K}_J)$, the algebra of bounded operators on ${\cal K}_J$, and define
$\lambda_J$ be $\tr_J$, the trace on ${\cal K}_J$, so that $(\h_J,\A_J,\lambda_J) = ({\cal K}_J,\B({\cal K}_J),\tr_J)$.
There are natural homomorphisms $\phi_J$ embedding the $2^n-1$
algebras $\A_J$ into $\A$. For instance, if $J = \{1,2\}$,
\begin{equation}\label{embed1}
\phi_{\{1,2\}}(A_1\otimes A_2) = A_1\otimes A_2\otimes I_{\h_3}\otimes \cdots \otimes I_{\h_N}\ ,
\end{equation}
and is extended linearly.
It is obvious that in case $J\cap K = \emptyset$ and $J\cup K
=\{1,\dots,n\}$, then for all $H_J \in \A_J$ and $H_K \in \A_K$,
\begin{equation}\label{gt1}
\tr\left(e^{H_J+H_K}\right) = \tr_J\left(e^{H_J}\right)\tr_K\left(e^{H_K}\right)\ ,
\end{equation}
but things are more interesting when $J\cap K\ne \emptyset$ and $J$
and $K$ are both proper subsets of $\{1,\dots,n\}$. If $H_J$ and
$H_K$ do not commute, which is the typical situation for $J\cap K\ne
\emptyset$, one can estimate the left hand side of (\ref{gt1}) by first
applying the Golden--Thompson inequality \cite{Go,T}, which says that
for self-adjoint operators $H_J$ and $H_K$, $$\tr\left(e^{H_J+H_K}\right)
\le \tr\left(e^{H_J}e^{H_K}\right) \ .$$ One might then apply H\"older's
inequality -- but if $J$ and $K$ are proper subsets of $\{1,\dots,n\}$,
this will yield a finite bound if and only if all of the Hilbert
spaces whose indices are not included in {\em both} $J$ and $K$ are
finite dimensional. Even then, the bound depends on the dimension in an
unpleasant way. The non-commutative B-L Inequalities provided by the
next theorem do not have this defect.
\begin{thm}\label{simpletp} Let $J_1,\dots, J_N$ be $N$ non empty
subsets of $\{1,\dots,n\}$ For each $i \in \{1,\dots,n\}$, let
$p(i)$ denote the number of the sets $J_1,\dots,J_N$ that contain
$i$, and let $p$ denote the minimum of the $p(i)$. Then, for
self-adjoint operators $H_j$ on ${\cal K}_{J_j}$, $j=1,\dots,N$,
\begin{equation}\label{tpgy}
\boxed{\ \tr \left(\exp\left[ \sum_{j=1}^N \phi_{J_j}(H_j)\right]\right) \le
\prod_{j=1}^N\left(\tr_{J_j}\ e^{ q H_j }\right)^{1/q} \ }
\end{equation}
for $q=p$ (and hence all $1 \le q \le p$), while for all $q>p$, it is
possible for the left hand side to be infinite, while the right hand
side is finite.
\end{thm}
Note that in Theorem~\ref{simpletp}, the constant $C$ in Definition
(\ref{ncgy}) is $1$. The fact that the constant $C=1$ is best
possible, and that the inequality cannot hold for $q>p =
\min\{p(1),\dots,p(N)\}$ is easy to see by considering the case that
each $\h_j$ has finite dimension $d_j$, and $H_j=0$ for each $j$. Then
$$\tr \left(\exp\left[ \sum_{j=1}^N \phi_{J_j}(H_j)\right]\right) = \prod_{j=1}^n d_j \qquad{\rm and}\qquad
\prod_{j=1}^N \left(\tr_{J_j} e^{ q H_j }\right)^{1/q} = \prod_{j=1}^N \prod_{k\in J_j} d_k^{1/q} =
\prod_{j=1}^n d_j^{p(j)/q}\ .$$
We will prove the inequality (\ref{tpgy}) for $q=p$ in Section~\ref{SSA}.
As an example, consider the case in which $n=6$, $N =3$ and
$$J_1 = \{1,2,3\}\quad J_2=\{3,4,5\} \qquad{\rm and}\quad J_3 = \{5,6,1\}\ .$$
Here, $p=1$, and hence
\begin{equation}\label{intprod4c}
\tr \left(\exp\left[ \sum_{j=1}^3 \phi_{J_j}(H_j)\right]\right) \le
\prod_{j=1}^3\left(\tr_{J_j}\ e^{ H_j }\right) \ .
\end{equation}
The inequality (\ref{intprod4c}) can obviously be extended to larger
tensor products, and has an interesting statistical mechanical
interpretation as a bound on the {\em partition function} of a
collection of interacting spins in terms of a product of partition
functions of simple constituent sub-systems.
To estimate the left side of (\ref{intprod4c}) without using
Theorem~\ref{simpletp}, one might use the Golden-Thompson inequality
and then Schwarz's inequality to write
$$\tr \left(\exp\left[ \sum_{j=1}^3 \phi_{J_j}(H_j)\right]\right) \le
\tr \left( e^{ \phi_1(H_1)+ \phi_3(H_3)} e^{\phi_2(H_2)} \right) \le
\left(\tr\ e^{2[\phi_1(H_1)+ \phi_3(H_3)]}\right)^{1/2}\left(\tr\
e^{2\phi_2(H_2)}\right)^{1/2}\ .$$ While the $L^2$ norms are an
improvement over the $L^1$ norms in (\ref{intprod4c}), the traces are
now over the entire tensor product space. Thus, for example,
$$\left(\tr\ e^{2\phi_2(H_2)}\right)^{1/2} = (d_1d_2d_6)^{1/2}\left(\tr_{J_2}\ e^{2 H_2 }\right)^{1/2}$$
where $d_j$ is the dimension of Hilbert space $\h_j$. This dimension dependence may be unfavorable if any of the dimensions is large.
\end{exam}
\subsection{A generalized Young's inequality in Clifford algebras}
Our next example concerns Clifford algebras, which as Segal emphasized
\cite{S56}, allow one to represent Fermion Fock space as an $L^2$
space -- albeit a non-commutative $L^2$ space, but still with many of
the advantages of having a Hilbert space represented as a function
space, as in the usual Schr\"odinger representation in quantum
mechanics.
In the finite dimensional setting, with $n$ degrees of freedom, one
starts with $n$ operators $Q_1,\dots,Q_n$ on some Hilbert space $\h$
that satisfy the {\em canonical anticommutation relations}
$$Q_iQ_j + Q_jQ_i = 2\delta_{i,j} I\ .$$
One can concretely construct such operators acting on $\h =
(\C^2)^{\otimes n}$, the $n$--fold tensor product of $\C^2$ with
itself; see \cite{BW}. The Clifford algebra $\CC$ is the operator
algebra on $\h$ that is generated by $Q_1,\dots,Q_n$.
The Clifford algebra $\CC$ itself is $2^n$ dimensional. In fact, let
$\alpha = (\alpha_1,\dots,\alpha_n)$ be a {\em Fermionic multi-index},
which means that each $\alpha_j$ is either $0$ or $1$. Then define
\begin{equation}\label{multii}
Q^\alpha = Q_1^{\alpha_1}Q_2^{\alpha_2}\cdots Q_n^{\alpha_n}\ .
\end{equation}
it is easy to see that the $2^n$ operators $Q^\alpha$ are a basis for
the Clifford algebra, so that any operator $A$ in $\CC$ has a unique
expression
$$A = \sum_{\alpha}x_\alpha Q^\alpha\ .$$
The linear functional $\tau$ on $\CC$ is defined by
\begin{equation}\label{taudef}
\tau\left(\sum_{\alpha}x_\alpha Q^\alpha\right) = x_{(0,\dots,0)}\ .
\end{equation}
That is, $\tau$ acting on $A$ picks off the coefficient of the
identity in $A= \sum_{\alpha} x_\alpha Q^\alpha$. It turns out that
when the Clifford algebra is constructed in the way described here, as
an algebra operators on the $2^n$ dimensional space $\h$, $\tau$ is
nothing other than the normalized trace:
$$\tau(A) = \frac{1}{2^n}\tr_\h(A)\ .$$
Hence $\tau$ is a positive linear functional, and $((\C^2)^{\otimes
n},\CC, \tau)$ is a non commutative integration space in the sense
of Segal.
Clifford algebras have infinitely many subalgebras that are also
Clifford algebras of lower dimension. This is in contrast to the
setting described in Example~\ref{tp}, in which the only natural
subalgebras are the $2^n-1$ subalgebras corresponding to the $2^n-1 $
non empty subsets of the index set $\{1,\dots,n\}$.
To describe these subalgebras, let ${\cal J}$ be the {\em canonical
injection} of $\R^n$ into $\CC$, which is given by
\begin{equation}\label{caninj}{\cal J}((x_1,\dots,x_n)) = \sum_{j=1}^n x_jQ_j\ .
\end{equation}
If $x$ and $y$ are any two vectors in $\R^n$, it is easy to see from
the canonical anticommutation relations that
$$ ({\cal J}(x))({\cal J}(y)) = 2(x\cdot y) I\ .$$
Hence if $V$ is {\em any} $m$ dimensional subspace of $\R^n$, and
$\{u_1,\dots,u_m\}$ is {\em any} orthonormal basis for $V$, the $m$
operators
$${\cal J}(u_1), \dots, {\cal J}(u_m)$$
again satisfy the canonical anticommutation relations, and generate a
subalgebra of $\CC$ that is denoted by $\CC(V)$, and referred to as
{\em the Clifford algebra over $V$}. In the same vein, it is
convenient to refer to $\CC$ itself as the {\em Clifford algebra over
$\R^n$}. Obviously, $\CC(V)$ is naturally isomorphic to
$\CC(\R^m)$, and for $A \in \CC(V)$ one may compute $\tau(A)$ using
either the normalized trace $\tau$ inherited from $\CC$, or the
normalized trace $\tau_V$ induced by the identification of $\CC(V)$
with $\CC(\R^m)$.
As Segal emphasized, $((\C^2)^{\otimes n},\CC,\tau)$ is in many way a
non-commutative analog of the Gaussian measure space
$(\R^n,\gamma(x)\dd x)$ where
\begin{equation}\label{gauss}
\gamma(x) = \frac{1}{(2\pi)^{n/2}}e^{-|x|^2/2}\ .
\end{equation}
In fact, just as orthogonality implies independence in
$(\R^n,\gamma(x)\dd x)$, if $V$ and $W$ are two orthogonal subspaces
of $\R^n$, and if $A\in \CC(V)$ and $B\in \CC(W)$, then
$$\tau(AB) = \tau(A)\tau(B)\ .$$
The results we prove here reenforce this analogy. We are now ready to
introduce our next example:
\begin{exam}\label{cliff}
For some $n>1$, let $\A$ be the Clifford algebra over $\R^n$ with
its usual inner product, and let $\A$ be equipped with its unique
tracial state $\tau$, which is simply the normalized trace.
For each $j=1,\dots,N$, let $V_j$ be a subspace on $\R^n$, and let
$\A_j$ be $\CC(V_j)$, the Clifford algebra over $V_j$ with the inner
product $V_j$ inherits from $\R^n$. Let $\A_j$ be equipped with its
unique tracial state $\tau_j$. The natural embedding of $V_j$ into
$\R^n$ induces a homomorphism of $\A_j$ into $\A$, and we define
this to be $\phi_j$. In this setting, we shall prove
\begin{thm}\label{cy} Let $V_1,\dots,V_N$ be $N$ subspaces of
$\R^n$, and let $\A_j$ be the Clifford algebra over $V_j$ with the
inner product $V_j$ inherits from $\R^n$, and let $\A_j$ be
equipped with its unique tracial state $\tau_j$. Let $\phi_j$ be
the natural homomorphism of $\A_j$ into $\A$ induced by the
natural embedding of $V_j$ into $\R^n$. Then
\begin{equation}\label{intprod5b}
\boxed{\
\tau \left(\exp\left[ \sum_{j=1}^N \phi_j(H_j)\right]\right) \le
\prod_{j=1}^N\left(\tau_j\ e^{ p_j H_j }\right)^{1/p_j} \ }
\end{equation}
for all self-adjoint operators $H_j\in \A_j$ if and only if
\begin{equation}\label{fjc}
\sum_{j=1}^N\frac{1}{p_j}P_j \le I_{\R^n}\ .
\end{equation}
where
$P_j$ is the orthogonal projection onto
$V_j$ in $\R^n$.
\end{thm}
In the special case in which ${\rm dim}(V_j) = 1$ for each $j$,
(\ref{intprod5b}) reduces to an interesting inequality for the
hyperbolic cosine. Indeed, let $u_j$ be one of the two unit vectors in
$V_j$.
Then, with $u_j\otimes u_j$ denoting the orthogonal projection onto the span of $u_j$, (\ref{fjc})
reduces to
\begin{equation}\label{fjc1d}
\sum_{j=1}^N\frac{1}{p_j}u_j\otimes u_j \le I_{\R^n}\ .
\end{equation}
The greater simplification, however, is that in this case, the space
of self-adjoint operators in each $\A_j$ is just two dimensional, and
with ${\cal J}$ denoting the canonical injection defined in
(\ref{caninj}),
$$H_j = a_jI + b_j{\cal J}(u_j)$$
for some real numbers $a_j$ and $b_j$. Then
$$\sum_{j=1}^N H_j = \left(\sum_{j=1}^N a_j\right)I + {\cal J}\left(\sum_{j=1}^N b_ju_j\right)\ .$$
This operator has exactly two eigenvalues,
$$\left(\sum_{j=1}^N a_j\right) \pm \left|\sum_{j=1}^N b_ju_j\right|\ $$
with equal multiplicities.
Likewise, $p_jH_j$ has exactly two eigenvalues $p_ja_j \pm p_jb_j$
with equal multiplicities. Hence, in this case, (\ref{intprod5b})
reduces to
\begin{equation}\label{cosh1}
\cosh\left(\left|{\textstyle \sum_{j=1}^N b_ju_j}\right|\right) \le \prod_{j=1}^N\left(\cosh(p_jb_j)\right)^{1/p_j}\quad{\rm for\ all}\quad (b_1,\dots,b_N) \in \R^N\ ,
\end{equation}
which, according to the theorem, must hold whenever (\ref{fjc1d}) is
satisfied. (The $a_j$'s make the same contribution to both sides, and
may be cancelled away.) Taking the logarithm of both sides, this can
be rewritten as
\begin{equation}\label{cosh2}
\ln\cosh\left(\left|{\textstyle \sum_{j=1}^N b_ju_j}\right|\right) \le \sum_{j=1}^N\frac{1}{p_j}
\ln\cosh(p_jb_j) \quad{\rm for\ all}\quad (b_1,\dots,b_N) \in \R^N\ ,
\end{equation}
and this inequality must hold whenever the unit vectors
$\{u_1,\dots,u_N\}$ and the positive numbers $\{p_1,\dots,p_N\}$ are
such that (\ref{fjc1d}) is satisfied.
Later on, we shall give an elementary proof of this inequality, and
hence an elementary proof of Theorem~\ref{cy} when each $V_j$ is one
dimensional. Our proof of the other cases is less than elementary, and
even our elementary proof of (\ref{cosh2}) is less than direct.
\end{exam}
\section{Subadditivty of Entropy and Generalized Young's Inequalities}\label{subadd}
In the examples we have introduced in the previous section, the
positive linear functionals $\lambda$ under consideration are either
traces or normalized traces. Throughout this section, we assume that
our non commutative integration spaces $(\h,\A,\lambda)$ are based on
{\em tracial} positive linear functionals $\lambda$. That is, we
require that for all $A,B \in \A$,
$$\lambda(AB) = \lambda(BA)\ .$$
In such a non commutative integration space $(\h,\A,\lambda)$, a {\em
probability density} is a non negative element $\rho$ of $\A$ such
that $\lambda(\rho) =1$. Indeed, the tracial property of $\lambda$
ensures that
$$\lambda(\rho A) = \lambda(A\rho) = \lambda(\rho^{1/2}A\rho^{1/2})$$
so that $A \mapsto \lambda(\rho A)$ is a positive linear functional that is $1$ on the identity.
Now suppose we have $N$ non-commutative integration spaces
$(\h_j,\A_j,\lambda_j)$ and $W^*$ homomorphism $\phi_j: \A_j \to \A$.
Then these homomorphisms induce maps from the space of probability
densities on $\A$ to the spaces of probability densities on the
$\A_j$:
For any probability density $\rho$ on $(\A, \lambda)$, let $\rho_j$ be
the probability density on $(\A_j, \lambda_j)$ by
$$\lambda_j(\rho_j A) = \lambda(\rho \phi_j(A))$$
for all $A \in \A_j$.
For example, in the setting of Example~\ref{tp}, $\rho_{J_j}$ is just
the partial trace of $\rho$ over $\bigotimes_{k\in J_j^c}\h_{k}$
leaving an operator on $\bigotimes_{k\in J_j}\h_{k}$. In the Clifford
algebra setting of Example~\ref{cliff}, $\rho_j$ is simply the
orthogonal projection of $\rho$ in $L^2(\CC,\tau)$ onto $\CC(V_j)$,
which is also known as the conditional expectation \cite{Um1} of
$\rho$ given $\CC(V_j)$.
In this section, we are concerned with the relations between the {\em entropies} of $\rho$ and the $\rho_1,\dots,\rho_N$.
The entropy of a probability density $\rho$, $S(\rho)$, is defined by
$$S(\rho) = -\lambda(\rho \ln \rho)\ .$$
Evidently, the entropy functional is concave on the set of probability densities.
\begin{defi}\label{gsa}
Given tracial non-commutative integration spaces $(\h,\A,\lambda)$
and $(\h_j,\A_j,\lambda_j)$, $j=1,\dots,N$, together with $W^*$
algebra homomorphisms $\phi_j:\A_j\to \A$, $j=1,\dots,N$, and
numbers $1\le p_j\le \infty$, $j=1,\dots,N$, a {\em generalized
subadditivity of entropy inequality} holds if there is a finite
constant $C$ so that
\begin{equation}\label{genen}
\boxed{\
\sum_{j=1}^N \frac{1}{p_j} S(\rho_j) \ge S(\rho)- \ln C\ }
\end{equation}
for all probability densities $\rho$ in $\A$.
\end{defi}
It turns out that for tracial non-commutative integration spaces,
generalized subadditivity of entropy inequalities and B-L inequalities
are dual to one another, just as they are in the commutative case
\cite{CC}, so that if one holds, so does the other, with the same
values of $p_1,\dots,p_N$ and $C$. The following is in fact a direct
non-commutative analog of the main theorem of \cite{CC}.
\begin{thm}\label{equiv} Let
$(\h,\A,\lambda)$ and $(\h_j\A_j,\lambda_j)$, $j=1,\dots,N$, be
tracial non-commutative integration spaces. Let $\phi_j:\A_j\to \A$,
$j=1,\dots,N$ be $W^*$ algebra homomorphisms. Then for any numbers
$1\le p_j\le \infty$, $j=1,\dots,N$, and any finite constant $C$,
the generalized subadditivity of entropy inequality (\ref{genen}) is
true for all probability densities $\rho$ on $\A$ if and only if the
non-commutative B-L inequality (\ref{intprod3}) is true for all
self-adjoint $H_j\in \A_j$, $j=1,\dots,N$, with the same
$p_1,\dots,p_N$ and the same $C$.
\end{thm}
As a consequence of Theorem~\ref{equiv}, one strategy for proving a non-commutative
B-L inequality
is to prove the corresponding generalized subadditivity of entropy inequality. We shall see in our examples that this is an effective strategy; indeed, this is how we prove Theorems~\ref{simpletp}
and \ref{cy}.
In the current tracial context, the proof of Theorem~\ref{equiv} is a
direct adaptation of the proof of the corresponding result in the
context of Lebesgue integration given in \cite{CC}. It turns on a
well--known formula for the Legendre transform of the entropy. For
completeness, we give this formula in Lemma~\ref{leg} below. Before
stating the lemma, it is convenient to extend the definition of $S$ to
all of $\A_{\rm sa}$, the subspace of self-adjoint elements of $\A$,
as follows:
\begin{equation}\label{hdef}
S(A) =
\begin{cases}
-\lambda(A \ln A) & \text{if $A\ge 0$ and $\lambda(A) = 1$,}
\\
-\infty & \text{otherwise.}
\end{cases}
\end{equation}
\begin{lm}\label{leg} Let $\A$ be $\B(\h)$, the algebra of bounded operators on a separable Hilbert space $\h$.
Let $\lambda$ denote either the trace $\tr$ on $\h$, or, if $\h$ is
finite dimensional, the normalized trace $\tau$. Then for all $A\in
\A_{\rm sa}$,
\begin{equation}\label{forone}
-S(A) = \sup_{H \in \A_{\rm sa}}\left\{
\lambda(AH) - \ln\left(\lambda\left(e^H \right)\right)\right\} \ .
\end{equation}
The supremum is an attained maximum if and only if $A$ is a strictly
positive probability density, in which case it is attained at $H$ if
and only if $H = \ln A + cI$ for some $c\in \R$. Consequently, for
all $H\in \A_{\rm sa}$,
\begin{equation}\label{fortwo}
\ln\left(\lambda\left(e^H\right)\right) = \sup_{A \in \A_{\rm sa}}\left\{ \lambda(AH)+ S(A)\right\} \ .
\end{equation}
The supremum is a maximum at all points of the domain of $
\ln\left(\lambda\left(e^H\right)\right)$, in which case it is attained
only at the single point $A = e^H/(\lambda(e^H))$.
\end{lm}
\noindent{\bf Proof:} We consider first the case that $\lambda = \tr$,
and $\h$ has finite dimension $d$. To see that the supremum is
$\infty$ unless $0 \le A \le I$, let $c$ be any real number, and let
$u$ be any unit vector. Then let $H$ be $c$ times the orthogonal
projection onto $u$. For this choice of $H$,
$$\lambda(AH) - \ln\left(\lambda\left(e^H \right)\right) = c\langle u,Au\rangle - \ln (e^c + (d-1))\ .$$
If $\langle u,Au\rangle<0$, this tends to $\infty$ as $c$ tends to $-\infty$. If
$\langle u,Au\rangle>1$, this tends to $\infty$ as $c$ tends to $\infty$. Hence we need only consider $0 \le A \le I$.
Next, taking $H = cI$, $c\in \R$,
$$\lambda(AH) - \ln\left(\lambda\left(e^H \right)\right) = c\lambda(A) - c - \ln (d)\ .$$
Unless $\lambda(A) = 1$, this tends to $\infty$ as $c$ tends to $\infty$. Hence we need only consider
the case that $A$ is a density matrix $\rho$.
Let $\rho$ be any density matrix on $\h$ and let $H$ be any
self-adjoint operator such that $\tr(e^H) < \infty$. Then define the
density matrix $\sigma$ by
$$\sigma = \frac{e^H}{\tr(e^H)}\ .$$
Then, by the positivity of the relative entropy,
$$\tr( \rho\ln \rho - \rho \ln \sigma) \ge 0$$
with equality if and only if $\sigma = \rho$. But by the definition of
$\sigma$, this reduces to
$$\tr (\rho \ln \rho) \ge \tr(\rho H) - \ln\left(\tr\left(e^H\right)\right)\ ,$$
with equality if and only if $H = \ln \rho$. From here, there rest is
very simple, including the treatment of the normalized trace.. \lanbox
\medskip
Petz \cite{P88} has shown how to extend Lemma~\ref{leg} to a much more
general context, and his result can be used to extend the validity
Theorem ~\ref{equiv} beyond the tracial case. However, since the
examples in which we prove the generalized subadditivity inequality
here are tracial, we shall not go into this.
\medskip
\noindent{\bf Proof of Theorem~\ref{equiv}:}
Suppose first that the non-commutative B-L inequality (\ref{intprod3})
holds. Then, for any probability density $\rho$ in $\A$, and any
self-adjoint $H_j \in \A_j$, $j=1,\dots,N$, apply (\ref{forone}) with
$A = \rho$ and $H = \sum_{j=1}^N\phi_j(H_j)$ to obtain
\begin{eqnarray}
-S(\rho) &\ge& \lambda\left(\rho\left[\sum_{j=1}^N\phi_j(H_j)\right]\right) -
\ln\left[\lambda \left( \exp\left[\sum_{j=1}^N\phi_j(H_j)\right]\right)\right]\nonumber\\
&=&\sum_{j=1}^N\lambda_j(\rho_j H_j) - \ln\left[\lambda \left( \exp\left[\sum_{j=1}^N\phi_j(H_j)\right]\right)\right]\nonumber\\
&\ge&\sum_{j=1}^N\lambda_j(\rho_j H_j) - \ln\left[C\prod_{j=1}^N
\lambda_j \left( e^{p_jH_j}\right)^{1/p_j}\right]\nonumber\\
&=&\sum_{j=1}^N\frac{1}{p_j}\left[\lambda_j(\rho_j [p_jH_j]) - \ln\left(
\lambda_j \left( e^{[p_jH_j]}\right)\right)\right] - \ln C\ .\nonumber\\
\end{eqnarray}
The first inequality here is (\ref{forone}), and the second is
the non-commutative B-L inequality (\ref{intprod3}).
Now choosing $p_jH_j$ to maximize $\lambda_j(\rho_j [p_jH_j]) -
\ln\left( \lambda_j \left( e^{[p_jH_j]}\right)\right)$, we get from
(\ref{forone}) once more that
$$\lambda_j(\rho_j [p_jH_j]) - \ln\left(
\lambda_j \left( e^{[p_jH_j]}\right)\right) = -S(\rho_j) = \lambda_j(\rho_j\ln \rho_j)\ .$$
Thus, we have proved (\ref{genen}) with the same $p_1,\dots,p_N$ and $C$ that we had in
(\ref{intprod3}).
Next, suppose that (\ref{genen}) is true. We shall show that in this
case, the non-commutative B-L inequality (\ref{intprod3}) holds with
the same $p_1,\dots,p_N$ and $C$. To do this, let the self-sadjoint
operators $H_1,\dots,H_N$ be given, and define
$$\rho = \left[\lambda \left( \exp\left[\sum_{j=1}^N\phi_j(H_j)\right]\right)\right]^{-1}
\exp\left[\sum_{j=1}^N\phi_j(H_j)\right]\ .$$
Then by Lemma~\ref{leg},
\begin{eqnarray}\label{back}
\ln \left[\lambda \left( \exp\left[\sum_{j=1}^N\phi_j(H_j)\right]\right)\right] &=&
\lambda\left(\rho \left[\sum_{j=1}^N\phi_j(H_j)\right]\right) + S(\rho)\nonumber\\
&=& \sum_{j=1}^N\lambda_j \left[\rho_j H_j\right] + S(\rho)\nonumber\\
&\le& \sum_{j=1}^N \frac{1}{p_j}\left[\lambda_j \left[\rho_j (p_jH_j)\right] + S(\rho_j)\right] + \ln C \nonumber\\
&\le& \sum_{j=1}^N \frac{1}{p_j}\ln \left[\lambda_j \left(\exp( p_jH_j)\right) \right] + \ln C \nonumber\\
\end{eqnarray}
The first inequality is the generalized subadditivity of entropy
inequality (\ref{genen}), and the second is (\ref{fortwo}).
Exponentiating both sides of (\ref{back}), we obtain (\ref{intprod3})
with the same $p_1,\dots,p_N$ and $C$ that we had in (\ref{genen}).
\lanbox \medskip
\section{Proof of the generalized subadditivity of entropy inequality for tensor products of Hilbert spaces}\label{SSA}
The crucial tool that we use in this section is the {\em strong
subadditivity of the entropy} \cite{LR}, which we now recall in a
formulation that is suited to our purposes.
Suppose, as in Example~\ref{tp}, that we are given $n$ separable
Hilbert spaces $\h_1,\dots,\h_n$. As before, let ${\cal K}$ denote
their tensor product, and for any non empty subset $J$ of
$\{1,\dots,n\}$, let ${\cal K}_J$ denote $\bigotimes_{j\in J}\h_j$.
For a density matrix $\rho$ on ${\cal K}$, and any non empty subset
$J$ of $\{1,\dots,n\}$, define $\rho_J = \tr_{J^c}\rho$ to be the
density matrix on ${\cal K}_J$ induced by the natural injection of
$\B({\cal K}_J)$ into $\B({\cal K})$. As noted above, $\rho_J$ is
nothing other than the partial trace of $\rho$ over the complementary
product of Hilbert spaces, $\bigotimes_{j\notin J}\h_j$.
The strong subadditivity of entropy is expressed by the inequality
stating that for all nonempty $J,K \subset \{1,\dots,n\}$,
\begin{equation}\label{ssa}
S(\rho_{J}) + S(\rho_{K}) \ge S(\rho_{J\cup K}) + S(\rho_{J\cap K})\ .
\end{equation}
In case $\J\cap K = \emptyset$, it reduce to the
ordinary subadditivity of the entropy, which is the elementary inequality
\begin{equation}\label{rsa}
S(\rho_J) + S(\rho_K) \ge S(\rho_{J\cup K})\qquad{\rm for}\quad J\cap K = \emptyset\ .
\end{equation}
Combining these, we have
\begin{eqnarray}
S(\rho_{\{1,2\}}) + S(\rho_{\{2,3\}})+ S(\rho_{\{3,1\}}) &\ge& S(\rho_{\{1,2,3\}})+ S(\rho_{\{2\}}) + S(\rho_{\{1,3\}})\nonumber\\
&\ge &
2S(\rho_{\{1,2,3\}})\ ,\nonumber\\
\end{eqnarray}
where the first inequality is the strong subadditivity (\ref{ssa}) and
the second is the ordinary subadditivity (\ref{rsa}). Thus, for $n=3$
and $J_1= \{1,2\}$, $J_2= \{2,3\}$ and $J_3= \{3,1\}$, we obtain
$$\frac{1}{2}\sum_{j=1}^N S(\rho_{J_j}) \ge S(\rho)\ .$$
In this example, each index $i \in \{1,1,3\}$ belonged to exactly two
of the set $J_1$, $J_2$ and $J_3$, and this is the source of the facto
of $1/2$ in the inequality. The same procedure leads to the following
result:
\begin{thm}\label{stpen} Let $J_1,\dots, J_N$ be $N$ non empty subsets of $\{1,\dots,n\}$ For each $i \in \{1,\dots,n\}$, let $p(i)$
denote the number of the sets $J_1,\dots,J_N$ that contain $i$, and
let $p$ denote the minimum of the $p(i)$. Then
\begin{equation}\label{sentp}
\frac{1}{p}\sum_{j=1}^N S(\rho_{J_j}) \ge S(\rho)
\end{equation}
for all density matrices $\rho$ on ${\cal K} = \h_1\otimes \cdots\otimes \h_n$.
\end{thm}
\noindent{\bf Proof:} Simply use strong subadditivty to combine
overlapping sets to produce as many ``complete'' sets as possible, as
in the example above. Clearly, there can be no more than $p$ of these.
If $p(i) > p$ for some indices $i$, there will be ``left over''
partial sets. The entropy is always non negative, and therefore,
discarding the corresponding entropies gives us $\sum_{j=1}^N
S(\rho_{J_j}) \ge pS(\rho)$, and hence the inequality. \lanbox
It is now a very simple matter to prove Theorem~\ref{simpletp}:
\noindent{\bf Proof of Theorem~\ref{simpletp}:} By the remarks made
after the statement of the theorem, all that remains to be proved is
the inequality (\ref{tpgy}) for $q=p$. However, this follows directly
from Theorem~\ref{equiv} and Theorem~\ref{stpen}. \lanbox
\section{On the generalized Young's inequality with a Gaussian reference measure}\label{gaussY}
Before turning to the proof of our non-commutative B-L inequality in
Clifford algebras, we discuss the commutative case in which the
reference measures is Gaussian. We do this here for two reasons:
First, as noted, a Clifford algebra $\CC$ with its normalized trace
$\tau$ is a non commutative analog of a Gaussian measure space. This
analogy is strong enough that we shall be able to pattern our analysis
in the Clifford algebra case on an analysis of the Gaussian case.
Second, the Gaussian inequality is of interest in itself, and seems
not to have been fully studied before. Suppose that $V_1,\dots,V_N$
are $N$ non zero subspaces of $\R^n$, and for each $j$, define $\phi_j
= P_j$ to be the orthogonal projection of $\R^n$ onto $V_j$. Equip
$\R^n$ and equip each $V_j$ with Lebesgue measure. Then the problem of
determining for which sets of indices $\{p_1,\dots,p_N\}$ there exists
a finite constant $C$ so that (\ref{gy}) holds for all non-negative
measurable functions $f_j$ on $V_j$, $j=1,\dots,N$ is highly non
trivial, and has only recently been fully solved \cite{BCCT1,BCCT2}.
Moreover, determining the value of the best constant $C$ for those
choices of $\{p_1,\dots,p_N\}$ is still a challenging finite
dimensional variational problem for which there is no general explicit
solution.
In contrast, suppose we are given a non-degenerate Gaussian measure on
$\R^n$. It will be convenient to take the covariance matrix of the
Gaussian to define the inner product, so that the Gaussian becomes a
unit Gaussian. For each positive integer $m$, define $\gamma_m(x) =
(2\pi)^{-m/2} e^{-|x|^2/2}$ on $\R^m$. Then equipping $\R^n$ with the
measure $\gamma_n(x)\dd x$ and equipping each $V_j$ with the
$\gamma_{d_j}(x)\dd x$, $d_j$ being the dimension of $V_j$, it turns
out that there is a {\em very simple necessary and sufficient
condition} on the indices $\{p_1,\dots,p_N\}$ for the constant $C$
to be finite, and better yet, {\em the best constant $C$ is always $1$
whenever it is finite}:
\begin{thm}\label{gaussbl} Let $V_1,\dots,V_N$ be $N$ non zero subspaces of $\R^n$,
and for each $j$, and let $d_j$ denote the dimension of $V_j$.
Define $P_j$ to be the orthogonal projection of $\R^n$ onto $V_j$.
Given the numbers $p_j$, $1\le p_j < \infty$ for $j = 1,\dots,N$,
there exists a finite constant $C$ such that
\begin{equation}\label{blg}
\int_{\R^n} \prod_{j=1}^N f_j\circ P_j(x) \gamma_n(x){\rm d}x \le C
\prod_{j=1}^N\left(\int_{V_j}f_j^{p_j}(y)\gamma_{d_j}(y){\rm d}y\right)^{1/p_j}\
\end{equation}
holds for all non-negative $f_j$ on $V_j$, $j=1,\dots,N$, if and only if
\begin{equation}\label{fja}
\sum_{j=1}^N \frac{1}{p_j} P_j \leq {\rm Id}_{\R^n}
\end{equation}
and in this case, $C=1$.
\end{thm}
\medskip
We hasten to point out that this theorem is partially known. In the
special case that each of the subspaces $V_j$ is one dimensional,
Barthe and Cordero-Erausquin \cite{BC}, have the {\em sufficiency} of
the condition (\ref{fja}) which reduces to
\begin{equation}\label{geo}\sum_{j=1}^N \frac{1}{p_j}u_j\otimes u_j = {\rm Id}_{\R^n}
\end{equation}
with each $u_j$ being a unit vector spanning $V_j$. They did this as
an intermediate step in a short proof of the {\em Lebesgue measure}
version of the B-L inequality under the condition (\ref{geo}) -- the
so-called {\em geometric case}. Perhaps because their main focus was
the Lebesgue measure case, in which (\ref{geo}) is not a necessary
condition for finiteness of the constant $C$, they did not address the
necessity of this condition in the Gaussian case.
Indeed, the inequality (\ref{blg}) is equivalent to its Lebesgue
measure analog, which is known to hold with the constant $C=1$ under
the condition (\ref{fja}) \cite{BCCT1,BCCT2}. To see this, define
$g_1,\dots,g_N$ by
$$g_j(y) = f_j(y)(\gamma_j(y))^{1/d_j}\qquad j = 1,\dots,N\ .$$
As noted in \cite{BC}, this change of variable allows one to pass back
and forth between the Gaussian and Lebesgue measure version of the B-L
inequality -- under the condition (\ref{fjc}).
Nonetheless, it is worthwhile to give a proof of Theorem~\ref{gaussbl}
here for two reasons: First, it may be surprising that the condition
(\ref{fjc}) is necessary for the inequality to hold with any finite
constant at all. Second, the proof we will give of sufficiency of the
condition (\ref{fjc}) serves as a model for the proof of the
corresponding theorem in the Clifford algebra case that we consider in
the next section.
In proving Theorem~\ref{gaussbl}, our first step is to pass to the
problem of proving a generalized subadditivity inequality. Because
the commutative version of Theorem~\ref{equiv} has been proved in
\cite{CC}, Theorem~\ref{gaussSA} theorem below on subbadditivity of
entropy with respect to a Gaussian reference measure is equivalent to
Theorem~\ref{gaussbl}. Hence, it suffices to prove one of the other.
Before stating and proving the subadditivty theorem, we first recall
that for any probability density $\rho$ on $(\R^m,\dd \gamma_m)$, the
entropy of $\rho$, is defined by
$$S(\rho) = -\int_{\R^m}\rho(y)\ln\rho(y) \gamma_m(y)\dd y\ .$$
Note that the relative entropy of $\rho(y)\gamma_m(y)\dd y$ to
$\gamma_m(y)\dd y$ is $-S(\rho)$; in the convention employed here, the
entropy $S$ is concave, and the relative entropy is convex.
\begin{thm}\label{gaussSA} Let $V_1,\dots,V_N$ be $N$ non zero
subspaces of $\R^n$, and for each $j$, and let $d_j$ denote the
dimension of $V_j$. Define $P_j$ to be the orthogonal projection of
$\R^n$ onto $V_j$. For any probability density $\rho$ on $(\R^n,\dd
\gamma_n)$, let $\rho_{V_j}$ denote the marginal density on
$(V_j,\dd \gamma_{d_j})$. Then, given the numbers $p_j$, $1\le p_j
< \infty$ for $j = 1,\dots,N$, there exists a finite constant $C$
such that
\begin{equation}\label{sag}
\sum_{j=1}^N\frac{1}{p_j}S(\rho_{V_j}) \ge S(\rho) - \ln(C)
\end{equation}
holds for all probability densities $\rho$ on
$(\R^n,\dd \gamma_n)$, if an only if
\begin{equation}\label{fj}
\sum_{j=1}^N \frac{1}{p_j} P_j \leq I
\end{equation}
and in this case, $\ln(C)=0$.
\end{thm}
We first prove necessity of the condition (\ref{fj}):
\begin{lm} The condition (\ref{fj}) in Theorem~\ref{gaussSA} is necessary.
\end{lm}
\noindent{\bf Proof:} It suffices to consider densities of the form
$$\rho(x) = \exp(b\cdot x - |b|^2/2)\ ,$$
for $b\in \R^n$.
Then
$$\rho_{V_j}(x) = \exp(P_jb\cdot y - |P_jb|^2/2)\ ,$$
and we compute:
$$S(\rho) = -\frac{|b|^2}{2}\qquad{\rm and}\qquad S(\rho_{V_j}) = -\frac{|P_jb|^2}{2}\ .$$
Thus
$$\sum_{j=1}^N\frac{1}{p_j}S(\rho_{V_j}) - S(\rho) = b\cdot \left[ Id_{\R^n} - \sum_{j=1}^N\frac{1}{p_j}P_j\right]b\ ,$$
and evidently this is bounded below if and only if (\ref{fj}) is satisfied. \lanbox
\subsection{Proof of sufficiency}
The sufficiency of the condition (\ref{fj}) will be proved using an
interpolation between an arbitrary density $\rho$ and the uniform
density that is provided by the Mehler semigroup. (Indeed, Barthe and
Coredero-Erausquin used the Mehler semigroup in their work \cite{BC}
mentioned above, but in a direct proof of the Gaussian B-L inequality
inspired by the heat-flow method introduced in \cite{CLL1}. The heat
flow approach to prove subadditivity inequalities was developed in
\cite{BCM} and \cite{CC}.)
The Mehler semigroup is the strongly continuous semigroup of
positivity preserving contractions on $L^2(\R^n ,\gamma_n(x){\rm d}x)$
whose generator $-{\cal N}$ is given by the Dirichlet form
\begin{equation}\label{gener}
{\cal E}(f,g) = \int_{\R^n} \nabla f^*(x)\cdot \nabla g(x)\gamma_n(x){\rm d}x
\end{equation}
through $\langle f, {\cal N} g \rangle_{L^2(\gamma_n)} = {\cal E}(f,g)$,
where $f^*$ is the complex conjugate of $f$.
Integrating by parts, one finds
$${\cal N} = -(\Delta - x\cdot\nabla)\ ,$$
The eigenvalues of ${\cal N}$ are the non-negative integers, and the
eigenfunctions are the Hermite polynomials. (In certain physical
contexts, the eigenvalues count occupancy of quantum state and ${\cal
N}$ is called the {\em Boson number operator}.)
There is a simple explicit formula for the $e^{-t{\cal N}}$:
\begin{equation}\label{mfor}
e^{-t{\cal N}}f(x) = \int_{\R^n} f\left(e^{-t}x + \sqrt{1 - e^{2t}}y\right)\gamma_n(y)\dd y\ ,
\end{equation}
which is easily checked.
Since evidently ${\cal N}1 = 0$, and $e^{-t{\cal N}}$ is self-adjoint,
it also preserves integrals against $\gamma_n(x){\rm d}x$, and hence,
if $\rho$ is any probability density, so is each $\rho_t := e^{-t{\cal
N}}$. As one sees from (\ref{mfor}),
\begin{equation}\label{conver}
\lim_{t\to\infty}e^{-t{\cal N}}\rho(x) = 1\ ,
\end{equation}
the uniform probability density on $(\R^n, \gamma_n(x){\rm d}x)$, and
thus the Mehler semigroup provides us with an interpolation between
any probability density $\rho$ and the uniform density $1$.
This interpolation is well-behaved with respect to the operation of
taking marginals: Consider any probability density $\rho$ on
$(\R^n,\gamma_n(x){\rm d}x)$, and any $m$ dimensional subspace $V$ of
$\R^n$. Let $\rho_V$ be the marginal density of $\rho$ as in
Theorem~\ref{gaussSA}. Then of course, we may regard $\rho_V$ as a
probability density on $(\R^n,\gamma_n(x){\rm d}x)$ that is constant
along directions in $V^\perp$. (Simply compose $\rho_V$ with $P_V$.)
Interpreted this way, so that both $\rho$ and $\rho_V$ are functions
on $\R^N$,
\begin{equation}\label{compo}
\left(e^{-t{\cal N}}\rho\right)_V = e^{-t{\cal N}} \left(\rho_V\right)\ .
\end{equation}
That is, taking marginals commutes with the action of the Mehler semigroup.
The next point to note is that the entropy is monotone increasing
along this interpolation: Differentiating, with $\rho_t = e^{-t{\cal
N}}\rho$,
$$\frac{{\rm d}}{{\rm d}t} S(\rho_t) = -\int_{\R^n} \ln(\rho_t)(\Delta - x\cdot \nabla)\rho_t \gamma_n \dd x =
\int_{\R^n}\nabla \ln \rho_t \cdot \nabla \rho_t \gamma_n \dd x= {\cal
E}(\ln \rho_t, \rho_t)\ .$$ For any smooth density $\rho$, $ {\cal
E}(\ln \rho, \rho) = \int_{\R^n}\nabla \ln \rho \cdot \nabla \rho
\gamma_n \dd x = \int_{\R^n}|\nabla \ln \rho |^2 \rho \gamma_n \dd x
$, and hence $S(\rho_t) $ is strictly increasing for all $t$.
Moreover, since $(x,t) \mapsto |x|^2/t$ is jointly convex on
$\R^n\times \R_+$, $\rho\mapsto {\cal E}(\ln \rho, \rho)$ has a unique
extension as a convex functional the set of all probability densities
on $(\R^n, \gamma_n(x){\rm d}x)$.
\begin{defi}[Entropy Production]\label{eprodg}
The
{\it entropy production} functional is the convex functional $D(\rho)$ on probability
densities on $(\R^n, \gamma_n(x){\rm d}x)$ given by
\begin{equation}\label{epfo}
D(\rho) = \int_{\R^n} \ln \rho(x) {\cal N}\rho(x) \gamma_n(x) {\rm d}x = {\cal E}(\ln \rho, \rho)\ .
\end{equation}
\end{defi}
With this definition,
$$\frac{{\rm d}}{{\rm d}t} S(e^{-t{\cal N}}\rho) = D(e^{-t{\cal N}}\rho)\ .$$
Now because of (\ref{compo}), for any subspace $V$ of $\R^n$,
$$\frac{{\rm d}}{{\rm d}t} S([e^{-t{\cal N}}\rho]_V) = D([e^{-t{\cal N}}\rho]_V)\ .$$
Now, since $[e^{-t{\cal N}}\rho]_V$ is constant along directions
orthogonal to $V$, the derivatives in those directions that figure in
$D([e^{-t{\cal N}}\rho]_V)$ are irrelevant; we need only take
derivatives along directions in $V$. This consideration leads to the
definitions of the {\em restricted number operator}, and the {\em
restricted entropy production}:
Given an $m$ dimensional subspace $V$ of $\R^n$,
let $P_V$ be the orthogonal projection onto $V$. The restricted number operator ${\cal N}_V$
is the self-adjoint operator on
$L^2(\R^n ,\gamma_n(x){\rm d}x)$ defined through
\begin{equation}\label{renum}
\langle f, {\cal N}_V g \rangle_{L^2(\gamma_n)} = \int_{\R^n} \nabla f^*(x)\cdot P_V\nabla g(x)\gamma_n(x){\rm d}x\ ,
\end{equation}
and the {\em restricted entropy production functional} $D_V(\rho)$ is the convex functional
given by
\begin{equation}\label{repdef}
D_V(\rho) = \int_{\R^n} \left({\cal N}_V \ln \rho(x) \right)\rho(x) \gamma_n(x) {\rm d}x \ .
\end{equation}
With this definition, $D(\rho_V) = D_V(\rho_V)$, however, there is a
crucial difference between $D_V(\rho)$ and $D(\rho_V)$:
\begin{lm}\label{gaussmon} For any smooth probability density $\rho$ on
$(\R^n,\gamma_n(x){\rm d}x)$, and any non $m$ dimensional subspace
$V$ of $\R^n$, let $\rho_V$ be the corresponding marginal density
regarded as a probability density on $(\R^n,\gamma_n(x){\rm d}x)$.
Then
\begin{equation}\label{monpro}
D(\rho_V) \le D_V(\rho)\ .
\end{equation}
\end{lm}
\noindent{\bf Proof:}
Regard $\rho_{V}$
as a function on $\R^n$ (by composing it with $P_V$).
Assume that $\rho$ is smooth and bounded above and below
by strictly positive numbers. Notice that since $\rho_V$ is constant
constant along directions in $V^\perp$,
$${\cal N}\ln \rho_V = {\cal N}_V\ln \rho_V\ ,$$
and hence
Then, integrating by parts, and using the definition of $\rho_{V}$
and the Schwarz inequality, we obtain:
$$D(\rho_V)= \int_{\R^n} \left[{\cal N}_V\ln \rho_{V}(x) \right] \rho_{V} (x)
\gamma_{n}(x) \dd x = \int_{\R^n} \left[{\cal N}_V\ln \rho_{V} (x)\right] \rho (x)
\gamma_{n}(x) \dd x\ ,$$
where we have used the definition of $\rho_V$ to replace the second $\rho_V$ be $\rho$ itself.
Then, by the definition of ${\cal N}_V$, and the Schwarz inequality,
\begin{eqnarray}\label{sw}
D(\rho_V)
&=& \int_{\R^n}\left(\nabla \ln \rho_{V} (x)\right)\cdot P_{V}\nabla \rho(x)
\gamma_{n} \dd x\nonumber\\
&=& \int_{\R^n}\left(\nabla \ln \rho_{V} (x)\right)\cdot P_{V}\left(\nabla \ln \rho(x)\right)
\rho(x) \gamma_{n}(x) \dd x\nonumber\\
&\le& \left(\int_{\R^n}\left|\nabla \ln \rho_{V} (x)\right|^2 \rho(x)
\gamma_{n}(x) \dd x\right)^{1/2}\left(\int_{\R^n}\left|P_V\nabla
\ln \rho(x) \right|^2\rho
\gamma_{n} \dd x\right)^{1/2}\nonumber\\
\end{eqnarray}
In the first factor in the last line, we may replace $\rho$ by $\rho_V$ since
$\left|\nabla \ln \rho_{V} (x)\right|^2$ depends on $x$ only through $P_Vx$. Hence
this factor is
simply $\sqrt{D(\rho_V)}$, and the second factor is $\sqrt{D_V(\rho)}$.
\lanbox
The proof we have just given is patterned on the proof of an
analogous result in the Lebesgue measure case in \cite{CC}, which in
turn is based on similar arguments in \cite{C} and \cite{BCM}. It is
somewhat more complicated to adapt the argument to the Clifford
algebra setting, but this is what we shall do in the next section.
We are now ready to prove the sufficiency of condition (\ref{fj}):
\begin{lm}\label{gausssuf}The condition (\ref{fj}) in Theorem~\ref{gaussSA} is sufficient.
\end{lm}
\noindent{\bf Proof:} For a probability density $\rho$ on $(\R^n,\dd
\gamma_n)$ $S(\rho)> -\infty$, it is easy to see that
$$\lim_{t\to\infty}S(e^{-t{\cal N}}\rho) = S(1) = 0$$
and hence, $\lim_{t\to\infty}S(e^{-t{\cal N}}(\rho_{V_j})) = 0$ for
each $j=1,\dots,N$. Therefore, it suffices to show that
$$a(t) := \left[\sum_{j=1}^N\frac{1}{p_j}S(e^{-t{\cal N}} \rho_{V_j}) - S(e^{-t{\cal N}} \rho) \right]$$
is monotone decreasing in $t$.
Differentiating, and using (\ref{compo}), and then Lemma~\ref{gaussmon},
\begin{eqnarray}
\frac{{\rm d}}{{\rm d}t}a(t) &=& \left[\sum_{j=1}^N\frac{1}{p_j}D((e^{-t{\cal N}} \rho)_{V_j}) - D(e^{-t{\cal N}} \rho) \right]\nonumber\\
&\le & \left[\sum_{j=1}^N\frac{1}{p_j}D_{V_j}(e^{-t{\cal N}} \rho) - D(e^{-t{\cal N}} \rho) \right]\nonumber\\
\end{eqnarray}
Now note that by (\ref{repdef}), whenever (\ref{fj}) is satisfied,
$$\sum_{j=1}^N\frac{1}{p_j}D_{V_j}(\sigma) \le D( \sigma)$$
for any smooth density $\sigma$. Hence the derivative of $\alpha(t)$
is negative for all $t>0$. \lanbox
\section{Generalized subadditivity of the entropy in Clifford algebras}
In this section we shall prove
\begin{thm}\label{cye} Let $V_1,\dots,V_N$ be $N$ subspaces of $\R^n$,
and let $\A_j$ be the Clifford algebra over $V_j$ with the inner
product $V_j$ inherits from $\R^n$, and let $\A_j$ be equipped with
its unique tracial state $\tau_j$. For any probability density
$\rho\in \A$, let $\rho_{V_j}$ be the induced probability density in
$\A_j$. Let $S(\rho) = \tau(\rho\ln \rho)$ and $S(\rho_{V_j}) =
\tau_j(\rho_{V_j}\ln \rho_{V_j})$
Then
\begin{equation}\label{clifs1}
\sum_{j=1}^N\frac{1}{p_j} S(\rho_{V_j}) \ge S(\rho)
\end{equation}
for all probability densities $\rho\in \A$ if and only if
\begin{equation}\label{fjc2}
\sum_{j=1}^N\frac{1}{p_j}P_j \le I_{\R^n}\ .
\end{equation}
where
$P_j$ is the orthogonal projection onto
$V_j$ in $\R^n$.
\end{thm}
Granted this result, we have:
\noindent{\bf Proof of Theorem~\ref{cy}:} Theorem~\ref{equiv} and Theorem~\ref{cye}
together prove Theorem~\ref{cy}. \lanbox
We shall now prove
Theorem~\ref{cye}.
As before, we begin by proving the necessity of (\ref{fjc2}).
\begin{lm}\label{clnec}
The condition (\ref{fjc2}) in Theorem~\ref{cye} is necessary.
\end{lm}
Proof: For any vector $a = (a_1,\dots,a_n)\in \R^n$, define
$$\rho_a = I + \sum_{j=1}^n a_jQ_j = I + a\cdot Q\ .$$
Then $\rho_a$ is a probability density if and only if $|a| \le 1$.
Indeed, $\rho_a$ has only two eigenvalues, $1\pm |a|$, with equal
multiplicity.
Then $(\rho_a)_{V_j} = I + (P_ja)\cdot Q$, and so $(\rho_a)_{V_j}$ has
only two eigenvalues, $1\pm |P_j a|$, with equal multiplicity.
Therefore,
\begin{equation}\label{ents1}
S(\rho_a) = -\psi(|a|)\qquad{\rm and}\qquad S((\rho_a)_{V_j}) = -\psi(|P_ja|) \ .
\end{equation}
where $\psi(x)$ is the convex function defined by
\begin{equation}\label{psidef}
\psi(x) =
\begin{cases}
\frac{1}{2}\left[ (1+x)\ln(1+x) + (1-x)\ln(1-x)\right] & \text{if $|x| \le 1$.}
\\
\infty & \text{otherwise,}
\end{cases}
\end{equation}
Thus, for (\ref{clifs1}) to hold for each $\rho_a$, $|a|\le 1$, it must be the case that
\begin{equation}\label{ents2}
\sum_{j=1}^N\frac{1}{p_j}\psi(|P_ja|) \le \psi(|a|)\qquad {\rm for\ all\ } a\ {\rm with} \ |a| \le 1\ .
\end{equation}
Then since $\psi(x) = x^2 + {\cal O}(x^4)$, replacing $a$ by $ta$, $0<
t < 1$, we see that (\ref{fjc2}) must hold. \lanbox
Because of (\ref{ents1}), once we have proved Theorem~\ref{cye}, we
will have a proof of (\ref{ents2}). However, it is of interest to
have a direct proof of this inequality.
\begin{prop}\label{elem}
The inequality (\ref{ents2}) holds whenever (\ref{fjc2}) is satisfied.
\end{prop}
\noindent{\bf Proof:} An easy calculation of derivatives shows that
$$\psi'(x) = {\rm arctanh}(x)\qquad {\rm and}\qquad \psi''(x) = \frac{1}{1-x^2}$$
for $|x| < 1$.
Now fix any $a$ with $|a| < 1$. Then, for $t>0$, define
$$\eta(t) = \psi(e^{-t}|a|) - \sum_{j=1}^N\frac{1}{p_j}\psi(e^{-t}|P_ja|) \ .$$
We have to show that $\eta(t)>0$ for all $t>0$. Since evidently $\lim_{t\to\infty}\eta(t) =0$,
it suffices to show that $\eta'(t) < 0$ for all $t>0$.
Differentiating, we find
$$\eta'(t) = -e^{-t}\left[ |a|\ {\rm arctanh}(e^{-t}|a|) - \sum_{j=1}^N\frac{1}{p_j}
|P_j a|\ {\rm arctanh}(e^{-t}|P_ja|)\right] := e^{-t}\theta(t)\ .$$
Hence, it suffices to show that $\theta(t)\ge 0$ for all $t>0$. Since once again,
$\lim_{t\to\infty}\theta(t) =0$,
it suffices to show that $\theta'(t) < 0$ for all $t>0$.
Differentiating, we find
$$\theta'(t) = -e^{-t}\left[\frac{ |a|^2}{1-e^{-2t}|a|^2}- \sum_{j=1}^N \frac{1}{p_j}
\frac{|P_j a|^2}{1-e^{-2t}|P_ja|^2}\right] \ .$$ Multiplying through
by $e^{-t}$, and absorbing a factor of $e^{-t}$ into $a$, it suffices
to show that
\begin{equation}\label{rat1}
\frac{ |a|^2}{1-|a|^2}\ge \sum_{j=1}^N \frac{1}{p_j}
\frac{|P_j a|^2}{1-|P_ja|^2}
\end{equation}
for all $|a|\le 1$. However, since $|a| \ge |P_ja|$,
$$\frac{|P_j a|^2}{1-|a|^2} \ge \frac{|P_j a|^2}{1-|P_ja|^2}\ ,$$
and thus (\ref{rat1}) follows from (\ref{fjc2}). \lanbox
We are now in a position to give an elementary proof of
Theorem~\ref{cy} in the special case that each $V_j$ is one
dimensional. As explained in Example~\ref{cliff}, it suffices in this
case to prove the following:
\begin{prop}\label{onedimc} Suppose $\{u_1,\dots,u_N\}$ is any set of $N$ unit vectors in $\R^n$, and
$\{p_1,\dots,p_N\}$ is any set of $N$ positive numbers such that
\begin{equation}\label{fjfj}
\sum_{j=1}^N c_j u_j \otimes u_j =I_{\R^n}\ .
\end{equation}
Then for any $b = (b_1,\dots,b_N)\ in \R^N$,
\begin{equation}\label{cosh2b}
\ln\cosh\left(\left|{\textstyle \sum_{j=1}^N b_ju_j}\right|\right) \le \sum_{j=1}^N\frac{1}{p_j}
\ln\cosh(p_jb_j) \ .
\end{equation}
\end{prop}
\noindent{\bf Proof:} Let $\psi^*(x)$ denote the function $\psi^*(x) =
\ln \cosh(x)$, $x\in \R$. The notation is meant to indicate the well
known fact, easily checked, that $\psi^*$ is the Legendre transform of
the function $\psi$ defined in (\ref{psidef}).
Now, given a set of $N$ orthogonal projections $\{P_1,\dots,P_N\}$
satisfying (\ref{fjc2}), we may make any choice of a unit vector $u_j$
from the range of $P_j$, and then the $N$ unit vectors
$\{u_1,\dots,u_N\}$ will satisfy (\ref{fjfj}). Conversely, given any
set of $N$ unit vectors $\{u_1,\dots,u_N\}$ that satisfy (\ref{fjfj}),
we may take $P_j = u_j\otimes u_j$, and then (\ref{fjc2}) is
satisfied. Hence, we suppose we are given a a set of $N$ orthogonal
projections $\{P_1,\dots,P_N\}$ satisfying (\ref{fjc2}), and for each
$j$, $u_j$ is a unit vector in the range of $P_j$.
Then for any $b \in \R^n$,
\begin{eqnarray}
\psi^*\left(\left|{\textstyle \sum_{j=1}^N b_ju_j}\right|\right)
&=& \sup_{a\in \R^n}\left\{ a \cdot \sum_{j=1}^N b_ju_j - \psi(|a|)\right\}\nonumber\\
&=& \sup_{|a| \le 1}\left\{ \sum_{j=1}^N P_ja \cdot b_ju_j - \psi(|a|)\right\}\nonumber\\
&\le& \sup_{|a| \le 1}\left\{ \sum_{j=1}^N P_ja \cdot b_ju_j - \sum_{j=1}^N\frac{1}{p_j}\psi(|P_ja|)\right\}\nonumber\\
&\le& \sup_{|a| \le 1}\left\{ \sum_{j=1}^N |P_ja| | b_j| - \sum_{j=1}^N\frac{1}{p_j}\psi(|P_ja|)\right\}\nonumber\\
&=& \sup_{|a| \le 1}\left\{ \sum_{j=1}^N\frac{1}{p_j} \bigl[ |P_ja| p_j| b_j| -\psi(|P_ja|)\bigr]\right\}\nonumber\\
\end{eqnarray}
where the first inequality is from (\ref{ents2}), and the second is from Schwarz. Then, by the definition of the Legendre transform, for any $a$,
$$\psi^*(p_jb_j) \ge |P_ja| ( p_j| b_j|) -\psi(|P_ja|)\ ,$$
we obtain
$$\psi^*\left(\left|{\textstyle \sum_{j=1}^N b_ju_j}\right|\right) \le \sum_{j=1}^N\frac{1}{p_j}
\psi^*(p_jb_j)\ ,$$
which is (\ref{cosh2b}). \lanbox
We now prove Theorem~\ref{cye} in full generality. This gives another
proof of the last two propositions, but by less elementary means. The
proof will follow the basic pattern of the proof of
Theorem~\ref{gaussSA}, and use the Clifford algebra analog of the
Mehler semigroup. This is the so-called Clifford--Mehler semigroup,
about which we now recall a few relevant facts.
\subsection{About the Clifford--Mehler semigroup}
There is also a differential calculus in the Clifford algebra. Let
$Q_1,\dots,Q_n$ be any set of $n$ generators for the Clifford algebra
$\CC$ over $\R^n$. For $A\in \CC$, define
$$\nabla_i(A) = \frac{1}{2}\left[Q_iA - \Gamma(A)Q_i\right]\ ,$$
where $\Gamma$ is the {\em grading operator} on $\CC$: That is, using the notation in (\ref{multii}),
$$\Gamma(Q^{\alpha}) = (-1)^{|\alpha|}Q^\alpha\ .$$
One computes that $\nabla_i(Q^\alpha) = 0$ is $\alpha(i) = 0$, and
otherwise, $\nabla_i(Q^\alpha) = 0$ is what one gets by anti-commuting
the factor of $Q_i$ through to the left, and then removing it. In this
sense it is like a differentiation operator, and what is more, it is a
skew derivation on $\CC$, which means that for all and $A$ and $B$ in
$\CC$, $\nabla_j(AB) = \nabla_j(A) B + \Gamma(A)\nabla_j(B)$.
The Clifford algebra analog of the Gaussian energy integral
(\ref{gener}) is given by
\begin{equation}\label{generc}
{\cal E}(A,B) =\tau\left( \sum_{j=1}^n \nabla_j A^*\nabla_j B\right)\ ,
\end{equation}
for all $A,B\in \CC$. This is the {\em Clifford Dirichlet form} studied by Gross.
Then, the Fermionic number operator, also denoted ${\cal N}$, is defined by
$$ {\cal E}(A,B) = \tau(A^* {\cal N}(B))\ .$$
It is easy to see that the spectrum of ${\cal N}$ consists of the non
negative integers $\{0,1,\dots,n\}$ and that
\begin{equation}\label{deg}
{\cal N}Q^\alpha = |\alpha|Q^\alpha\ .
\end{equation}
The Clifford Mehler semigroup is then given by $e^{-t{\cal N}}$. It
is clear from this definition, (\ref{taudef}) and (\ref{deg}) that for
any $A\in \CC$, $\lim_{t\to\infty}e^{-t{\cal N}}(A) = \tau(A)I$. Thus
for any probability density $\rho$ in $\CC$,
$$t\mapsto \rho_t = e^{-t{\cal N}}(\rho)$$
provides an interpolation between $\rho$ and $I$, and each $\rho_t$ is
a probability density. This corresponds exactly to the Mehler
semigroup interpolation that was used to prove Theorem~\ref{gaussSA},
and we shall use it here in the same way, though some additional
complications shall arise.
${\cal N}$ does not depend on the choice of the set of generators
$Q_1,\dots,Q_n$. Indeed, if $\{u_1,\dots,u_n\}$ is any orthonormal
basis of $\R^n$, and we define $\widetilde Q_j = u_j\cdot Q\qquad j
=1,\dots,n$, then the Clifford Dirichlet form that one obtains using
this basis to define the derivatives is the same as the original.
In particular, given an $m$ dimensional subspace $V$ of $\R^n$, we may
choose $\{u_1,\dots,u_n\}$ so that $\{u_1,\dots,u_m\}$ is an
orthonormal basis for $V$, and then the first $m$ generators will be a
set of generators for $\CC_V$. We then define the {\em reduced
Clifford Dirichlet form} ${\cal E}_V$ by
\begin{equation}\label{generc2}
{\cal E}_V(A,B) =\tau\left( \sum_{i,j=1}^n \nabla_i A^*[P_V]_{i,j} \nabla_j B\right)\ ,
\end{equation}
where $[P_V]_{i,j}$ is the $i,j$th entry of the $n\times n$ matrix for $P_V$.
The restricted number operator ${\cal N}_V$ is then the self-adjoint operator on $L^2(\CC)$
given by
$\tau(A^* {\cal N}_V(B)) = {\cal E}_V(A,B)$.
Now, for any probability density $\rho$ in $\CC$ let $\rho_V$ be the
corresponding marginal density regarded as an operator in $\CC$ by
identifying it with $\phi_V(\rho_V)$, where $\phi_V$ is the canonical
embedding of $\CC(V)$ into $\CC(\R^n)$. Then it is an easy consequence
of the definitions that
\begin{equation}\label{clcond}
\left(e^{-t{\cal N}}\rho\right)_V = e^{-t{\cal N}} \left(\rho_V\right) = e^{-t{\cal N}_V} \left(\rho_V\right)\ .
\end{equation}
Also, under the condition (\ref{fjc2}), it is easy to see that
\begin{equation}\label{fjc3}
\sum_{j=1}^N\frac{1}{p_j}{\cal N}_{V_j} \le {\cal N}\ .
\end{equation}
Finally, we introduce {\it entropy production} $D(\rho)$: With $\rho_t := e^{-t{\cal N}}\rho$
,we differentiate and find
$$\frac{{\rm d}}{{\rm d}t} S\left(\rho_t\right) =
\tau\left(\ln(\rho_t){\cal N}(\rho_t)\right) = {\cal E}(\ln(\rho_t),\rho_t)\ .$$
\begin{defi}[Entropy Production] The {\it entropy production}
functional at a probability density $\rho$ is the functional defined
by
\begin{equation}\label{epformb}
D(\rho) = \tau\left(\ln(\rho){\cal N}(\rho)\right) = {\cal E}(\ln(\rho),\rho)\ .
\end{equation}
Given an $m$ dimensional subspace $V$ of $\R^n$, the {\it restricted
entropy production} functional at a probability density $\rho$ is
the functional defined by
\begin{equation}\label{epform}
D_V(\rho) = \tau\left(\ln(\rho){\cal N}_V(\rho)\right) = {\cal E}_V(\ln(\rho),\rho)\ .
\end{equation}
\end{defi}
The following lemma is the basis of our proof of the sufficiency of
(\ref{fjc2}). In the course of proving it, we shall see that both
$D(\rho)$ and $D_V(\rho)$ are convex functionals, which is somewhat
less obvious than in the Gaussian case.
\begin{lm}\label{clmon} For any any probability density $\rho$ in $\CC(\R^n)$, and any
$m$ dimensional subspace $V$ of $\R^n$, let $\rho_V$ be the
corresponding marginal probability density regarded as a probability
density in $\CC(\R^n)$. Then
$$D(\rho_V) \le D_V(\rho)\ .$$
\end{lm}
\noindent{\bf Proof:} We choose an orthonormal basis
$\{u_1,\dots,u_n\}$ for $\R^n$ such that $\{u_1,\dots,u_m\}$ is an
orthonormal basis for $V$. Without loss of generality, we may suppose
that $\{u_1,\dots,u_n\}$ is the standard basis so that
$\{Q_1,\dots,Q_m\}$ is a set of generators for $\CC(V)$. Then,
\begin{equation}\label{generc3}
{\cal E}(A,B) =\tau\left( \sum_{j=1}^n \nabla_j A^*\nabla_j B\right)\qquad{\rm and}\qquad
{\cal E}_V(A,B) =\tau\left( \sum_{j=1}^m \nabla_j A^*\nabla_j B\right)\ .
\end{equation}
It will be convenient to define
${\cal N}_j = \nabla_j^*\nabla_j\qquad j = 1,\dots,n$.
Then we have
\begin{equation}\label{gen44}
{\cal N} = \sum_{j=1}^n{\cal N}_j \qquad{\rm and}\qquad
{\cal N}_V = \sum_{j=1}^m {\cal N}_j \ ,
\end{equation}
and so
\begin{equation}\label{pro3}
D_V(\rho) = \sum_{j=1}^m \tau\left( \ln \rho, {\cal N}_j \rho\right)\ .
\end{equation}
Since
${\displaystyle {\cal N}_j Q^{\alpha} =
\begin{cases}
Q^{\alpha} & \text{if $\alpha(j) =1$,}
\\
0 & \text{$\alpha(j)=0$,}
\end{cases}
}$, each ${\cal N}_j$ is an orthogonal projection, and so (\ref{pro3})
can be rewritten as
\begin{equation}\label{pro3b}
D_V(\rho) = \sum_{j=1}^m \tau\left( {\cal N}_j(\ln \rho), {\cal N}_j \rho\right)\ .
\end{equation}
To proceed, we use a formula of Gross \cite{G75} for ${\cal N}_j f(A)$
where $A\in \CC(\R^n)$, and $f$ is a continuous function. To write
down Gross's formula, first write $A = B + Q_jC$ where both $B$ and
$C$ are linear combinations of the $Q^\alpha$ with $\alpha(j) =0$.
Then define $\widehat A = B - Q_jC$. Notice that if $\rho$ is a
probability density, then $\widehat\rho$ is again a probability
density. Gross's formula is
$${\cal N}_j f(A) = \frac{1}{2}\left[ f(A) - f(\widehat A\ )\right]\ .$$
To prove this formula, notice that there is a unitary operator $U$
such that $\widehat A = UAU^*$. (If the dimension $n$ is odd, one can
take $U$ to be the product, in some order, of all of the $Q_k$ for
$k\ne j$; if the dimension is even, one can add another generator.)
Therefore,
$$\widehat{f(A)} = U f(A) U^* = f(UAU^*) = f(\widehat A\ )\ .$$
Using this together with the fact that for any $A\in \A$, ${\cal N}_jA = (1/2)[A - \widehat{A}]$,
we obtain Gross's formula, which we now apply as follows:
\begin{eqnarray}
\tau\left( {\cal N}_j(\ln \rho) {\cal N}_j \rho\right) &=&
\frac{1}{4}\tau\left( \left[ \ln(\rho) - \ln(\widehat\rho)\right] \left[ \rho - \widehat\rho\right] \right)\nonumber\\
&=&
\frac{1}{4}\tau\left( \ln(\rho)\left[ \rho - \widehat\rho\right] \right) + \frac{1}{4}\tau\left( \ln(\widehat\rho)\left[ \widehat\rho - \rho\right] \right)\nonumber\\
&=& \frac{1}{4} H[\rho|\widehat\rho\ ] + \frac{1}{4} H[\widehat\rho\ |\rho] \nonumber\\
\end{eqnarray}
where $H[\rho|\sigma] = \tau \rho(\ ln \rho - \ln \sigma)$ is the {\em
relative entropy} of $\rho$ with respect to $\sigma$. As is well
known, $(\rho,\sigma) \mapsto H(\rho|\sigma)$ is jointly convex, and
hence
$$\rho \mapsto \tau\left( (\ln \rho){\cal N}_j \rho\right) $$
is convex. Furthermore, by the fundamental monotonicity property of
the relative entropy under conditional expectations \cite{U2},
$$H(\rho_V|\sigma_V) \le H(\rho|\sigma)$$
for any two probability densities $\rho$ and $\sigma$. It follows that
$ \tau\left( (\ln \rho_V) {\cal N}_j \rho_V\right) \le \tau\left( (\ln
\rho) {\cal N}_j \rho\right)$. Summing on $j$ from $1$ to $m$, we
find
$$D(\rho_V) = D_V(\rho_V) = \sum_{j=1}^m\tau\left( (\ln \rho_V) {\cal N}_j \rho_V\right) \le
\sum_{j=1}^m \tau\left( (\ln \rho) {\cal N}_j \rho\right) = D_V(\rho)\
.$$ \lanbox
\subsection{Proof of the sufficiency}
\begin{lm}\label{clsuf}The condition (\ref{fj}) in Theorem~\ref{gaussSA} is sufficient.
\end{lm}
\noindent{\bf Proof:} For a probability density $\rho$ in $\CC(\R^n)$
it is easy to see that
$$\lim_{t\to\infty}S(e^{-t{\cal N}}\rho) = S(1) = 0$$
and hence,
$\lim_{t\to\infty}S(e^{-t{\cal N}}(\rho_{V_j})) = 0$ for each $j=1,\dots,N$.
Therefore,
it suffices to show that
$$a(t) := \left[\sum_{j=1}^N\frac{1}{p_j}S(e^{-t{\cal N}} \rho_{V_j}) - S(e^{-t{\cal N}} \rho) \right]$$
is monotone decreasing in $t$.
Differentiating, and using (\ref{clcond}), and then Lemma~\ref{clmon},
\begin{eqnarray}
\frac{{\rm d}}{{\rm d}t}a(t) &=& \left[\sum_{j=1}^N\frac{1}{p_j}D((e^{-t{\cal N}} \rho)_{V_j}) - D(e^{-t{\cal N}} \rho) \right]\nonumber\\
&\le & \left[\sum_{j=1}^N\frac{1}{p_j}D_{V_j}(e^{-t{\cal N}} \rho) - D(e^{-t{\cal N}} \rho) \right]\nonumber\\
\end{eqnarray}
Now note that by (\ref{repdef}), whenever (\ref{fj}) is satisfied,
${\displaystyle \sum_{j=1}^N\frac{1}{p_j}D_{V_j}(\sigma) \le D(
\sigma)}$ for any smooth density $\sigma$. Hence the derivative of
$\alpha(t)$ is negative for all $t>0$. \lanbox
Notice that the proof is almost identical, symbol for symbol, with
that of the corresponding proof in the Gaussian case. The main
difference of course is that the proof of the main lemma,
Lemma~\ref{clmon}, is considerably more intricate than that of its
Gaussian counterpart.
\noindent{\bf Proof of Theorem~\ref{cye}:} This now follows immediately from Lemma~\ref{clnec} and \ref{clsuf}. \lanbox
\bigskip
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Slashdot Log In
Wachowski Brothers and the Speed Racer Movie
Posted by kdawson on Mon Aug 27, 2007 04:15 AM
from the camera-has-guards dept.
from the camera-has-guards dept.
Steven Weintraub writes "Susan Sarandon talks about the Wachowski Brothers Speed Racer movie and confirms the revolutionary way the brothers are making the film — the entire frame will be in focus like a cartoon."
Related Stories
Firehose:Speed Racer movie - everything will be in focus! by Anonymous Coward
This discussion has been archived. No new comments can be posted.
Wachowski Brothers and the Speed Racer Movie | Log In/Create an Account | Top | 333 comments (Spill at 50!) | Index Only | Search Discussion
The Fine Print: The following comments are owned by whoever posted them. We are not responsible for them in any way.
The "Revolutionary New Camera" (Score:5, Informative)
()
Re:Go Speed Racer Go! (Score:5, Funny)
Focus is a tool (Score:5, Insightful)
I hope they don't spend a lot of money/effort on this "feature", the way they did on the game-quality 3D graphics of the Burly Brawl (ref: Matrix 2).
Re:Focus is a tool (Score:5, Insightful)
Re:Focus is a tool (Score:5, Funny)
(Last Journal: Friday November 09, @04:36PM)
But perhaps they couldn't. Perhaps what you perceive as a choice between 10 better ways on their part is a choice, but at the same time the choice they made is the only choice they could have made. You choose to think otherwise, but do you really have a choice in thinking you have a choice, did you choose to have a choice, or did you decide anyway?
The Wachowski brothers made the choices they made because they were the only choices they could have made. (Continued ad-nausium until the exciting car chase in the middle of the film. To be continued after car chase.)
Re:Focus is a tool (Score:5, Insightful)
()
I can ask dozen different questions, each with a simple answer, but that most people will fumble at. Not because it is difficult to execute the conclusion, but that the conclusion is non-obvious from the offset. Only once it is presented to all the answers, including those to which you would find 'better' become 'obvious'.
Hindsight is 20/20.
Making a graphic novel into a movie sounds easy. The average shmuck(by your own logic, I suppose that would include you), might say "Pffft...The story was already written down and framed, how could they screw THAT up?"
But only once you realize that you have 2 hours of film, a certain budget, actors with certain demands and a market with certain thirsts does the enormity of the task become apparent. How would you convety something that takes 2 hours to -read- into 2 hours of action? And how do you pull it off without boring the snot out of people or resorting to the cheap trick of keeping the silly camera moving too goddamn fast to make out the shortcomings of the choreography(I'm looking at you Transformers and Borne Supremacy).
I happened to like V for Vendetta.
I loved the first Matrix movie, the second one was meh and the third one was crap in my opinion. They shouldn't have been done. But given the massive plot hole-ridden concept the original was based on, I guess they sorta painted themselves in a corner.
But besides all that, I will ask a simple question: how do you make a boiled egg stand straight up on a table without using any materials except the egg and the solid table(no tablecloths, salt, etc...
The solution is simple. But can you think of it?
The answer(in reverse, right to left):
To prove my point, after reading the answer(if you could), the solution becomes far more obvious then it was from the offset.
The big problem is sometimes the average shmuck thinks of himself too highly to probe deeper then a superficial holier then thou, self gratifying way a la Simpsons ComicBookGuy.
Re:Focus is a tool (Score:5, Funny)
Re:Focus is a tool (Score:5, Interesting)
There are other ways than depth of field to emphasize an object, but its not easy even in stills photography. In movies i'd guess its going to be very hard to get the right "look" consistently. Good luck to them.
Re:Focus is a tool (Score:5, Informative)
()
Re:Focus is a tool (Score:4, Informative)
A pinhole camera has infinite depth of field. Of course it has some other problems, diffraction, sensitivity, etc.
If you have enough light, fast film, and shoot with a tight aperture, you can get very wide depth of field. Just two or three "layers" would be enough for effectively infinite depth of field even at film resolution. However compositing the layers would be a bit of a chore. For a feature length film, the compositing process would need to be automatic, perhaps assisted with something like a scanning laser rangefinder.
Martin
Re:Focus is a tool (Score:5, Funny)
I prefer to have the subject circled with a big red arrow pointing at it.
Re:Focus is a tool (Score:4, Interesting)
()
Re:Focus is a tool (Score:4, Informative)
()
Re:Good, another movie I don't need to watch (Score:4, Funny)
Re:Good, another movie I don't need to watch (Score:5, Funny)
( | Last Journal: Monday November 20 2006, @09:27AM)
Re:Good, another movie I don't need to watch (Score:4, Insightful)
But that was the problem - the first one was completely fresh and different (for mainstream audiences not into anime and extreme martial arts) - the sequels were obliged to follow broadly the same style, but by the time they came out, bullet time, wire-work Kung-Fu and "extreme" fight scenes had become cliched. Have you noticed how tame the bank lobby shootout scene looks today, compared with the first time you saw it? The long delay (probably not helped by the death of two cast members and the post-9/11 hiatus for any film in which things got blowed up) didn't help.
Its not as if the plot of the sequels was any sillier than the first movie (the whole humans as power sources thing - holy thermodynamics Batman!) just that the first film was such compellingly brilliant eye candy that your brain's services were not required, and we never worried about why someone punching you in VR should give you a fat lip in reality. By "Reloaded" we'd seen it all before (with freeze frame, commentary and white rabbits too, thanks to the original's role in popularizing DVD) and were starting to worry about plot holes.
...plus the first film had the "advantage" that it came out fairly close to Star Wars Episode one, and benefitted from rather favorable comparisons... (NB: I still think that Universal should have gambled and released "Serenity" head-to-head against "Revenge of the Sith" - then they'd have been a story, and people love to root for the little guy).
Re:Good, another movie I don't need to watch (Score:5, Interesting)
()
Reloaded was bad because it was utterly bereft of a plot. It was like a bad japanese RPG - they kept going to the Oracle to get quests.
Re:Good, another movie I don't need to watch (Score:5, Interesting)
( | Last Journal: Thursday September 09 2004, @11:35AM)
"In 1948, behavioral psychologist B.F. Skinner published an article in the Journal of Experimental Psychology, in which he describes his pigeons exhibiting what appeared to be superstitious behavior. One pigeon was making turns in its cage, another would swing its head in a pendulum motion, while others also displayed a variety of other behaviors.."
Re:Good, another movie I don't need to watch (Score:5, Informative)
Re:Good, another movie I don't need to watch (Score:5, Funny)
(Last Journal: Sunday October 07, @01:01AM)
Also, I am smoking Camel Turkish Silver. Don't see the relevance, but I'm happy to answer you.
Stop the hating (Score:5, Interesting)
( | Last Journal: Tuesday July 03, @11:44PM)
So yes, studios still very much listen to these guys, and they should.
The major flaw with the Matrix sequels was the script, which had too much exposition. V For Vendetta proved they could take a lengthy graphic novel that is heavy on exposition, and not overload their movie with it. And from AICN's script review of Speed Racer, it will be a movie that focuses primarily on intense action sequences.
In case anyone forgot, Matrix Reloaded, horrid exposition and all, still happens to feature perhaps the most insane freeway sequence in the history of film. The State of California wouldn't let them film it on any of their highways because they said the script for that sequence was unfilmable, and it was guaranteed to kill people in the process.
I'd wager that any real student or lover of film is still very much interested in how these guys will continue to innovate in later movies, even if their previous films have flaws. In fact, I don't think I've ever seen a perfect film. Even my absolute favorites still have glaring flaws.
So the cameras are on loan from unseen-U library? (Score:4, Funny)
Great... (Score:5, Funny)
Wow. (Score:4, Funny)
()
How can I patent this?
Re:Wow. (Score:5, Interesting)
How can I patent this?
What's revolutionary is they shoot every scene with several cameras at the same time (or several times with the same camera), using different focus planes each time to cover the entire depth range.
Then they assemble them post-production and boost the saturation, for that very special cartoony-colors, always-in-focus look... otherwise known as how the photos of throw-away consumer cameras look like.
Yea, all the wasted effort... keep in mind the movie took at least twice longer to shoot because they had to use blue screens even for a scene with nothing special in it (only to assist the post-production assembling of the planes).
Newfangled Oldfangled? (Score:5, Informative)
Re:Newfangled Oldfangled? (Score:5, Informative)
()
First, like you say, go farther away and use a tele-lens to pull the foreground to the wanted size. This has the side-effect that, as you say, the background becomes bigger and appears closer to the foreground. (because what matters is the *relative* distance, having the actors 5 meters away and the explosion 50 meters away means the actors are 10 times closer. Having the actors 50 meters away and the explosion 100 meters away means the explosion is only twice as far away, so if you compensate by zooming until the actors are same size on screen, the end-result is a explosion that is visually 5 times larger than in the first case)
Second, use a smaller aperture. With an infinitely small aperture, you get everything in focus, with a small aperture you get a very large focused area.
hmm... (Score:4, Interesting)
Surely they will follow much of the original Speed Racer construction formula and have lots of close-up shots, re-used footage and the same 4 panels of background speeding by as Speed and Racer X do their thing.
If the story villains don't have polygonal moustaches than I'm not going.
Hmm (Score:5, Insightful)
Deep Focus? (Score:5, Interesting)
()
So it would appear that they're making some differences with color, etc., but yeah - I'd like to see a still or two at least.
Re:Deep Focus? (Score:5, Insightful)
Re:Deep Focus? (Score:5, Informative)
()
Deep focus will still give you a depth of field, you just play around with everything in the frame to ensure it's within the hyperfocal distance of the lens.
With this new one, they're taking it one step further - if two things need to appear in the frame, but it's not possible to have them both in focus, they'll be filmed separately and stitched together so absolutely everything is sharp and crisp...
Great. (Score:3, Funny)
Re:Story this time? (Score:5, Insightful)
Re:Brothers? (Score:5, Informative)
1.) Kym Barret (The Matrix,Reloaded,Revolutions) will be doing the costume design.
2.) John Gaeta (The Matrix, inventor of Bullet Time..) is the visual effects supervisor.
3.) Owen Patterson (The Matrix, etc) is the production designer.
4.) Peter Fernandez (The original American voice of Speed Racer) will have an appearance in the film.
Re:Brothers? (Score:5, Insightful)
(Last Journal: Friday November 09, @04:36PM)
He's saying they're both the same thing because they both involve multiple still cameras. This, of course, means that the field of special effects has had no innovations whatsoever since the end of the 19th century, when motion pictures were invented. Anyone who thought Birth of a Nation, Citizen Kane, 2001, Star Wars, Blade Runner, The Matrix, et al, were in any way different to anything produced before them clearly was just imagining it because some of the technology they used had something in common with technology that had previously been invented.
3d too I hope. (Score:5, Interesting)
()
This is a first for Hollywood! (Score:5, Funny)
A film where the script, the acting and now the image are all flat and two-dimensional !
Woo-hoo! Next they'll invent super-xylem vision, so they can all be wooden as well!
Pretty light on details (Score:5, Insightful)
()
Personally I couldn't glean almost anything useful from the article.
Ironic (Score:4, Interesting)
()
It's ironic that they would choose this movie to highlight such an effect. As a cartoon watcher in the 1970s, I noticed that Speed Racer was one of the few that would on occasion actually use out-of-focus backgrounds in some scenes.
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| 206,352
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TITLE: $V(R)=Af(R\cdot B)$, where A and B are constant, prove that curl V is perpendicular to both A and B
QUESTION [2 upvotes]: If $V(R)$ can be expressed as $V(R)=Af(R\cdot B)$, where $A$ and $B$ are constant, prove that curl $V$ is perpendicular to both $A$ and $B$.
REPLY [1 votes]: Writing out $\nabla \times V$, we obtain
$\nabla \times V = (V_{z,y} - V_{y, z})\mathbf i + (V_{x,z} - V_{z, x})\mathbf j + (V_{y,x} - V_{x, y})\mathbf k, \tag{1}$
Where the comma notation is used for partial derivatives, i.e. $V_{z, y} = \frac{\partial V_z}{\partial y}$ etc., and $\mathbf i, \mathbf j,\mathbf k$ are the usual basis vectors in $\Bbb R^3$. Let's evaluate $V_{z, y}$:
$V_{z, y} = A_z \frac{\partial f(R \cdot B)}{\partial y} = A_z B_y f'(R \cdot B), \tag{2}$
where
$\frac{\partial f(R \cdot B)}{\partial y} = f'(R \cdot B) B_y \tag{3}$
by the chain rule and the fact that $R \cdot B = xB_x + yB_y + zB_z$. Likewise
$V_{y, z} = A_y \frac{\partial f(R \cdot B)}{\partial z} = A_y B_z f'(R \cdot B), \tag{4}$
so that
$V_{z, y} - V_{y, z} = f'(R \cdot B)(A_zB_y - A_yB_z), \tag{5}$
which is exactly the $\mathbf i$-component of $B \times A$. Performing similar operations on the $\mathbf j$- and $\mathbf k$-components of $\nabla \times V$, one sees that
$\nabla \times V = f'(R \cdot B)(B \times A); \tag{6}$
this formula immediately implies that $\nabla \times V$ is perpedicular to both $A$ and $B$.
Hope this helps. Cheers,
and as always,
Fiat Lux!!!
| 218,078
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Dear Parents and Guardians,
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| 85,904
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\section{Order of Automorphism Group of Cyclic Group}
Tags: Automorphism Groups, Cyclic Groups
\begin{theorem}
Let $C_n$ denote the [[Definition:Cyclic Group|cyclic group]] of [[Definition:Order of Group|order]] $n$.
Let $\Aut {C_n}$ denote the [[Definition:Automorphism Group|automorphism group]] of $C_n$.
Then:
:$\order {\Aut {C_n} } = \map \phi n$
where:
:$\order {\, \cdot \,}$ denotes the [[Definition:Order of Group|order]] of a [[Definition:Group|group]]
:$\map \phi n$ denotes the [[Definition:Euler Phi Function|Euler $\phi$ function]].
\end{theorem}
\begin{proof}
Let $g$ be a [[Definition:Generator of Cyclic Group|generator]] of $C_n$.
Let $\varphi$ be an [[Definition:Group Automorphism|automorphism]] on $C_n$.
By [[Homomorphic Image of Cyclic Group is Cyclic Group]], $\map \varphi g$ is a [[Definition:Generator of Cyclic Group|generator]] of $C_n$.
By [[Homomorphism of Generated Group]], $\varphi$ is uniquely determined by $\map \varphi g$.
By [[Finite Cyclic Group has Euler Phi Generators]], there are $\map \phi n$ possible values for $\map \varphi g$.
Therefore there are $\map \phi n$ [[Definition:Group Automorphism|automorphisms]] on $C_n$:
:$\order {\Aut {C_n} } = \map \phi n$
{{qed}}
\end{proof}
| 128,881
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TITLE: Lie algebra with cyclic structure constants
QUESTION [1 upvotes]: Given a Lie algebra (finite-dimensional, over a field) and a basis, denote the structure constants in the usual way: $[e_i,e_j]=\sum_kc_{ij}^ke_k$. We say that the structure constants are cyclic if $c_{ij}^k=c_{jk}^i$ for all $i,j,k$ (note that this depends on the choice of basis).
The Lie algebra being given, the existence of a basis with cyclic structure constant is a nontrivial condition as it can easily be checked not to exist in a non-abelian 2-dimensional Lie algebra.
Which Lie algebras admit a basis with cyclic structure constants?
REPLY [5 votes]: Endow your algebra with the symmetric bilinear form making $\{e_i\}$ an orthornormal basis (the Gram matrix is the identity matrix). Since $c_{ij}^k=([e_i,e_j],e_k)$, the cyclicity condition is equivalent to the invariance of the form:
$$
([e_i,e_j],e_k)=(e_i,[e_j,e_k]).
$$
Thus a finite-dimensional Lie algebra admits a basis with cyclic structure constants if and only if it admits an invariant symmetric bilinear form with Gram matrix the identity matrix. If the base field is algebraically closed not of characteristic 2, the latter condition simplifies to "admits a nondegenerate invariant symmetric bilinear form".
| 141,909
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The murder of counter-protestor Heather Hayer at the Charlottesville neo-Nazi march.
The shootings at an Orlando night club, a Las Vegas country music concert, churches in Charleston and in Texas, a Thousand Oaks bar, schools in Parkland, Florida and across the country,
The specter of domestic terrorism has been threatening the nation with increasing frequency, driven by a surge of attacks motivated by right-wing ideologies. Yet the response of the Trump administration — never shy about fear-mongering over a supposed invasion at the southern border by people that they consider international terrorists — is to disband the Homeland Security Department’s group of intelligence analysts focused on domestic terrorism, according to an article in The Daily Beast.
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The decision goes beyond the counter-intuitive to land decisively in the realm of idiotic dereliction of duty given the rise in overall terror incidents in the United States over the past 15 years, the vast majority of which were perpetrated by right-wing extremists.
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According to The Daily Beast, many current and former DHS officials are worried about the decision, saying that it has resulted in a significant drop in the number of analytic reports produced by DHS about domestic terrorism.
“’It’s especially problematic given the growth in right-wing extremism and domestic terrorism we are seeing in the U.S. and abroad,’ one former intelligence official told The Daily Beast.”
The move to eliminate the domestic terrorism focus of the DHS’s Office of Intelligence and Analysis (I&A), as the analysts’ department was called, stems from a reorganization of the department by the Trump administration’s new I&A chief, David Glawe, who last year reassigned to other projects the staff responsible for sharing information with state and local law enforcement to help them protect their communities from domestic terror threats.
“We’ve noticed I&A has significantly reduced their production on homegrown violent extremism and domestic terrorism while those remain among the most serious terrorism threats to the homeland,” one DHS official said
When contacted by The Daily Beast to respond to the reports of the shift in focus away from the all-too-real domestic threats, Glawe issued this statement:
.”
Another unnamed senior DHS official also tried to dispute the idea that the agency’s focus had been deliberately shifted.
.”
Local officials who had relied upon reports received from DHS/I&A tell another story. Sgt. Mike Abdeen of the Los Angeles County Sheriff’s Department noted that the previously frequent communications with the DHS had shriveled down to practically nothing in recent months.
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.”
While former officials who have led the Office of Intelligence and Analysis believe that, if anything, the department should be ramping up its anti-domestic terrorism operations, current officials argue that any responsibility for preventing domestic terrorism belongs to the FBI.
A former DHS official who concentrated on violent extremism, Nate Snyder, believes that the DHS reorganization belies the Trump administration message that it takes domestic terrorism,” said Snyder. “You can’t have it both ways.”
Could it be that the Trump administration — which has been accused of catering to the large number of white nationalists in the president’s core base — is loathe to be seen as working against the violent purveyors of the same message that Trump tweets out every day? Perhaps it’s time for America to realize that this administration is not working in our own best interests.
Follow Vinnie Longobardo on Twitter.
Original reporting by Betsy Woodruff at The Daily Beast.
| 67,651
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Peterson's algorithm (or Peterson's solution) is a concurrent programming algorithm for mutual exclusion that allows two or more processes to share a single-use resource without conflict, using only shared memory for communication. It was formulated by Gary L. Peterson in 1981. While Peterson's original formulation worked with only two processes, the algorithm can be generalized for more than two.
The algorithm uses two variables: <code>flag</code> and <code>turn</code>. A <code>flag[n]</code> value of <code>true</code> indicates that the process <code>n</code> wants to enter the critical section. Entrance to the critical section is granted for process P0 if P1 does not want to enter its critical section or if P1 has given priority to P0 by setting <code>turn</code> to <code>0</code>.
The algorithm satisfies the three essential criteria to solve the critical-section problem. The while condition works even with preemption.
Since <code>turn</code> can take on one of two values, it can be replaced by a single bit, meaning that the algorithm requires only three bits of memory.
P0 and P1 can never be in the critical section at the same time: If P0 is in its critical section, then <code>flag[0]</code> is true. In addition, either <code>flag[1]</code> is <code>false</code> (meaning that P1 has left its critical section), or <code>turn</code> is <code>0</code> (meaning that P1 is just now trying to enter the critical section, but graciously waiting), or P1 is at label <code>P1_gate</code> (trying to enter its critical section, after setting <code>flag[1]</code> to <code>true</code> but before setting <code>turn</code> to <code>0</code> and busy waiting). So if both processes are in their critical sections, then we conclude that the state must satisfy <code>flag[0]</code> and <code>flag[1]</code> and <code>turn = 0</code> and <code>turn = 1</code>. No state can satisfy both <code>turn = 0</code> and <code>turn = 1</code>, so there can be no state where both processes are in their critical sections.
(This recounts an argument that is made rigorous in.)
Progress is defined as the following: if no process is executing in its critical section and some processes wish to enter their critical sections, then only those processes that are not executing in their remainder sections can participate in making the decision as to which process will enter its critical section next. Note that for a process or thread, the remainder sections are parts of the code that are not related to the critical section. This selection cannot be postponed indefinitely. Instead of a boolean flag, it requires an integer variable per process, stored in a single-writer/multiple-reader (SWMR) atomic register, and N − 1 additional variables in similar registers. The registers can be represented in pseudocode as arrays:
level : array of N integers
last_to_enter : array of N − 1 integers
The level variables take on values up to N − 1, each representing a distinct "waiting room" before the critical section. Processes advance from one room to the next, finishing in room N − 1, which is the critical section. Specifically, to acquire a lock, process i executes
i ← ProcessNo
for ℓ from 0 to N − 1 exclusive
level[i] ← ℓ
last_to_enter[ℓ] ← i
while last_to_enter[ℓ] = i and there exists k ≠ i, such that level[k] ≥ ℓ
wait
To release the lock upon exiting the critical section, process i sets level[i] to −1.
That this algorithm achieves mutual exclusion can be proven as follows. Process i exits the inner loop when there is either no process with a higher level than level[i], so the next waiting room is free; or, when i ≠ last_to_enter[ℓ], so another process joined its waiting room. At level zero, then, even if all N processes were to enter waiting room zero at the same time, no more than N − 1 will proceed to the next room, the final one finding itself the last to enter the room. Similarly, at the next level, N − 2 will proceed, etc., until at the final level, only one process is allowed to leave the waiting room and enter the critical section, giving mutual exclusion.
The algorithm can also be shown to be starvation-free, meaning that all processes that enter the loop eventually exit it (assuming that they don't stay in the critical section indefinitely). The proof proceeds by induction from N − 1 downward. A process at N − 1 is in the critical section and by assumption will exit it. At all lower levels ℓ, it is impossible for a process i to wait forever, since either another process j will enter the waiting room, setting last_to_enter[ℓ] ← j and "liberating" i; or this never happens, but then all processes j that are also in the waiting rooms must be at higher levels and by the inductive hypothesis, they will eventually finish the loop and reset their levels, so that for all k ≠ i, level[k] < ℓ and i again exits the loop.
Starvation freedom is in fact the highest liveness guarantee that the algorithm gives; unlike the two-process Peterson algorithm, the filter algorithm does not guarantee bounded waiting.
When working at the hardware level, Peterson's algorithm is typically not needed to achieve atomic access.
Some processors have special instructions, like test-and-set or compare-and-swap, which, by locking the memory bus, can be used to provide mutual exclusion in SMP systems.
Most modern CPUs reorder memory accesses to improve execution efficiency (see memory ordering for types of reordering allowed). Such processors invariably give some way to force ordering in a stream of memory accesses, typically through a memory barrier instruction. Implementation of Peterson's and related algorithms on processors that reorder memory accesses generally requires use of such operations to work correctly to keep sequential operations from happening in an incorrect order. Note that reordering of memory accesses can happen even on processors that don't reorder instructions (such as the PowerPC processor in the Xbox 360).
Most such CPUs also have some sort of guaranteed atomic operation, such as <code>XCHG</code> on x86 processors and load-link/store-conditional on Alpha, MIPS, PowerPC, and other architectures. These instructions are intended to provide a way to build synchronization primitives more efficiently than can be done with pure shared memory approaches.
| 135,148
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About
The Student Success Centre aims to provide the resources and support for equipping and empowering you to realize your academic potential. Here you can find helpful resources for the nitty-gritty of academic work, opportunities to connect with fellow peers, and opportunities to explore and discover how God has shaped and gifted you for lives of service in His kingdom.
Learn more about the services and support we offer by browsing through our webpage. If you have any questions, please do not hesitate to contact us.
| 247,797
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On a scale of 1-10, I think most people would be a 10. I mean, I don’t know anyone who weighs less than 10. Maybe Steven Tyler.
Funny thoughts on weight
7.21.2011
An ant can lift up to 50 times its own weight, or, for comparison, at least 5 runway models.
1.20.2011
Pregnant pause: The silence that follows when a woman who is not expecting is asked about her pregnancy.
1.13.2011
The universe is constantly expanding. Who are my waistline and I to argue?
8.12.2010
One of the funniest things in the world is a big dog dragging a scrawny person behind it.
6.2.2010
How much do I have to eat to become too big to fail?
9.14.2009
Obscure Medical Fact #2: Pound for pound, humans weigh as much as horses.
| 389,327
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[Resolved] Sitewide & Group Forums
Hi there,
I’ve been using group forums on my site for a while.
I installed sitewide forums as well, I wanted to set up a forum space for some archived topics and import a phpBB forum, as well. Was going to just call these “archived-forums” or something to keep them distinct.
I don’t think I want to move forward this way anymore.
My question is, if I uninstall bbPress from my plugins, will it take my already-functioning and adopted group forums with it?
In other words, if I remove bbPress, will my group forums still be around and functional?
You must be logged in to reply to this topic.
| 384,189
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\begin{document}
\title[Alternating sign multibump solutions]{Alternating sign multibump solutions of nonlinear elliptic equations in
expanding tubular domains}
\author{Nils Ackermann}
\address{Instituto de Matem\'{a}ticas, Universidad Nacional Aut\'{o}noma de M\'{e}xico,
Circuito Exterior, C.U., 04510 M\'{e}xico D.F., Mexico.\smallskip}
\author{M\'{o}nica Clapp}
\address{Instituto de Matem\'{a}ticas, Universidad Nacional Aut\'{o}noma de M\'{e}xico,
Circuito Exterior, C.U., 04510 M\'{e}xico D.F., Mexico.\smallskip}
\author{Filomena Pacella}
\address{Dipartimento di Matematica, Universit\`{a} "La Sapienza" di Roma, P.le A. Moro
2, 00185 Roma, Italia}
\thanks{This research was partially supported by CONACYT grant 129847 and
PAPIIT-DGAPA-UNAM grants IN101209 and IN106612 (Mexico), and by exchange funds
of the Universit\`{a} \textquotedblleft La Sapienza\textquotedblright\ di Roma (Italy).}
\date{October, 2012}
\begin{abstract}
Let $\Gamma$ denote a smooth simple curve in $\mathbb{R}^{N}$, $N\geq2$,
possibly with boundary. Let $\Omega_{R}$ be the open normal tubular
neighborhood of radius $1$ of the expanded curve $R\Gamma:=\{Rx\mid x\in
\Gamma\smallsetminus\partial\Gamma\}$. Consider the superlinear problem
$-\Delta u+\lambda u=f(u)$ on the domains $\Omega_{R}$, as $R\rightarrow
\infty$, with homogeneous Dirichlet boundary condition. We prove the existence
of multibump solutions with bumps lined up along $R\Gamma$ with alternating
signs. The function $f$ is superlinear at $0$ and at $\infty$, but it is not
assumed to be odd.
If the boundary of the curve is nonempty our results give examples of
contractible domains in which the problem has multiple sign changing solutions.
\end{abstract}
\maketitle
\section{Introduction}
\label{sec:introduction}
Let $\gamma\in C^{3}([0,1],\mathbb{R}^{N})$, $N\geq2$, be a curve without
self-intersections except possibly for $\gamma(0)=\gamma(1)$. In this case we
also assume that $\dot{\gamma}(0)=\dot{\gamma}(1)$. For $R>0$ define
\begin{equation}
\Omega_{R}:=\text{int}{\bigcup_{t\in\lbrack0,1]}}\{R\gamma(t)+v\mid
v\in\mathbb{R}^{N},\ \lvert v\rvert<1,\ \dot{\gamma}(t)\cdot v=0\},
\label{eq:72}
\end{equation}
where int$(X)$ denotes the interior of $X$ in $\mathbb{R}^{N}$. Thus, for $R$
large enough, $\Omega_{R}$ is the tubular neighborhood of radius $1$ of the
$1$-dimensional submanifold $\Gamma_{R}$ of $\mathbb{R}^{N}$ defined as
\begin{equation*}
\Gamma_{R}:=
\begin{cases}
\{R\gamma(t)\mid t\in\lbrack0,1]\}, & \qquad\text{if }\gamma(0)=\gamma(1),\\
\{R\gamma(t)\mid t\in(0,1)\}, & \qquad\text{if }\gamma(0)\neq\gamma(1).
\end{cases}
\end{equation*}
We are interested in finding solutions to the problem
\begin{equation}
\left\{
\begin{array}
[c]{ll}
-\Delta u+\lambda u=f(u) & \text{in }\Omega_{R},\\
u=0 & \text{on }\partial\Omega_{R},
\end{array}
\right. \label{eq:1}
\end{equation}
for $R$ large enough.
Let $\lambda_{1,1}$ be the first eigenvalue of the Laplace operator $-\Delta$
in the unit ball in $\mathbb{R}^{N-1}$ with Dirichlet boundary conditions. Set
$p_{S}:=\infty$ if $N=1,2$ and $p_{S}:=(N+2)/(N-2)$ if $N\geq3$. We make the
following assumptions:
\begin{enumerate}
\item[(H1) ] $\lambda>-\lambda_{1,1}$.
\item[(H2) ] $f\in C^{1}(\mathbb{R})\cap C^{3}(\mathbb{R}\backslash\{0\})$.
\item[(H3)] There are $C>0$ and $p_{1},p_{2}\in(1,p_{S})$ such that $p_{1}\leq
p_{2}$ and
\begin{equation*}
\lvert f^{(k)}(u)\rvert\leq C(\lvert u\rvert^{p_{1}-k}+\lvert u\rvert
^{p_{2}-k})
\end{equation*}
for $k\in\{0,1,2,3\}$ and $u\neq0$.
\item[(H4)] $f(u)u>0$ for all $u\neq0$.
\end{enumerate}
Note that
\begin{equation}
f(0)=f^{\prime}(0)=0. \label{eq:50}
\end{equation}
For example, the standard nonlinearity $f(u):=\lvert u\rvert^{p-1}u$ satisfies
(H1)-(H4) if $p\in(1,p_{S})$.
We write a point in $\mathbb{R}^{N}$ as $(\xi,\eta)$, with $\xi\in\mathbb{R}$
and $\eta\in\mathbb{R}^{N-1}$, and denote the cylinder in $\mathbb{R}^{N}$ of
radius $1$ around the $\xi$-axis by
\begin{equation*}
\mathbb{L}:=\{(\xi,\eta)\in\mathbb{R}^{N}\mid\lvert\eta\rvert<1\}.
\end{equation*}
Locally, $\mathbb{L}$ is the limit domain of $\Omega_{R}$ as $R\rightarrow
\infty$. So we consider the limit problem
\begin{equation}
\left\{
\begin{array}
[c]{l}
-\Delta u+\lambda u=f(u),\\
u\in H_{0}^{1}(\mathbb{L)}.
\end{array}
\right. \label{eq:2}
\end{equation}
By Lemma~\ref{lem:schroed-invertible} below, the operator $-\Delta+\lambda$
with Dirichlet boundary conditions in $L^{2}(\mathbb{L})$ has a positive
spectrum. If $f$ satisfies an Ambrosetti-Rabinowitz type condition the
mountain pass theorem, together with the translation invariance in the $\xi
$-direction and concentration compactness, yields a positive and a negative
solution to \eqref{eq:2}, having minimal energy in their respective cones. We
add the following assumption:
\begin{enumerate}
\item[(H5)] Problem \eqref{eq:2} has a positive solution $U^{+}$ and a
negative solution $U^{-}$ which are nondegenerate, in the sense that the
solution space of the linearized problem
\begin{equation*}
-\Delta u+\lambda u=f^{\prime}(U^{\pm})u,\qquad u\in H_{0}^{1}(\mathbb{L}),
\end{equation*}
has dimension one.
\end{enumerate}
Note that the solution space of the linearized problem\ must have at least
dimension one, due to the invariance under translations. Hypothesis (H5)
requires that these are the only elements in the kernel of the linearization.
This condition is not easy to check, even for the standard nonlinearity
$f(u):=u^{p}.$ For this $f,$ Dancer showed in \cite{MR1962054} that (H5) holds true
either for $\lambda=0$ and almost every $p\in(1,p_{S}),$ or for almost every
$\lambda\in(0,\infty)$ and every $p\in(1,p_{S}).$
By \cite[Theorem~1.2]{MR1395408} the solutions $U^{\pm}$ are radially symmetric in
$\eta$ and decreasing in $\lvert\eta\rvert$. Moreover, by \cite[Theorem~6.2]
{MR1039342}, after a translation in the $\xi$-direction, we may assume that they are
also even in $\xi$ and decreasing in $\left\vert \xi\right\vert $. It follows
that they have a unique extremal point at $0$. We extend $U^{\pm}$ to all of
$\mathbb{R}^{N}$ by setting them as $0$ outside of $\mathbb{L}$.
For each $x\in\Gamma_{R}$ we choose a linear isometry $A_{x}$ which maps the
tangent space of $\Gamma_{R}$ at $x$ onto $\mathbb{R}\times\{0\}$ and its
orthogonal complement onto $\{0\}\times\mathbb{R}^{N-1}$, and we define
\begin{equation}
U_{x,R}^{\pm}(y):=U^{\pm}(A_{x}(y-x))\qquad\text{for all }y\in\mathbb{R}^{N}.
\label{eq:3}
\end{equation}
Since $U^{\pm}$ is radially symmetric in $\xi$ and in $\eta$, the function
$U_{x,R}^{\pm}$ is independent of the choice of $A_{x}$.
The parametrization $\gamma$ induces an orientation on $\Gamma_{R}$ which
allows to give an order to every finite set of points in $\Gamma_{R}.$ We
shall say that $(x_{1},\ldots,x_{n})\in\left( \Gamma_{R}\right) ^{n}$ is an
$n$-\emph{chain} in $\Gamma_{R}$ if there exist $0\leq t_{1}<t_{2}
<\cdots<t_{n}<1$ such that
\begin{equation}
x_{i}=R\gamma(t_{i})\qquad\text{for }i=1,2,\ldots,n. \label{eq:33}
\end{equation}
If $\gamma(0)=\gamma(1)$ a circular shift $(x_{i},\ldots,x_{n},x_{1}
,\ldots,x_{i-1})$ of an $n$-chain will also be called an $n$-chain. We shall
prove the following results.
\begin{theorem}
\label{thm:no-boundary} Assume that $\gamma(0)=\gamma(1)$. Suppose also that
\emph{(H1)-(H5)} hold. For each $k\in\mathbb{N}$ there exists $R_{k}>0$ such
that for every $R\geq R_{k}$ there are a $2k$-chain $(x_{R,1},x_{R,2}
,\ldots,x_{R,2k})\in\left( \Gamma_{R}\right) ^{2k}$ and a solution $u_{R}$
of \eqref{eq:1} such that
\begin{equation}
u_{R}=\sum_{i=1}^{k}(U_{x_{R,2i-1},R}^{+}+U_{x_{R,2i},R}^{-})+o(1)
\label{eq:4}
\end{equation}
in $H^{1}(\mathbb{R}^{N})$ as $R\rightarrow\infty$. Moreover, $\lvert
x_{R,i}-x_{R,j}\rvert\rightarrow\infty$ as $R\rightarrow\infty$, if $i\neq j$.
\end{theorem}
\begin{theorem}
\label{thm:with-boundary} Assume that $\gamma(0)\neq\gamma(1).$ Suppose also
that \emph{(H1)-(H5)} hold. For each $n\in\mathbb{N}$, $n\geq2$, there exists
$R_{n}>0$ such that for every $R\geq R_{n}$ there are an $n$-chain
$(x_{R,1},x_{R,2},\ldots,x_{R,n})\in\left( \Gamma_{R}\right) ^{n}$ and a
solution $u_{R}$ of \eqref{eq:1} such that
\begin{equation}
u_{R}=\sum_{i=1}^{k}(U_{x_{R,2i-1},R}^{+}+U_{x_{R,2i},R}^{-})+(n-2k)U_{x_{R,n}
,R}^{+}+o(1) \label{eq:5}
\end{equation}
in $H^{1}(\mathbb{R}^{N})$ as $R\rightarrow\infty$, where $k$ is the largest
integer smaller than or equal to $n/2.$ Moreover, as $R\rightarrow\infty$,
$\lvert x_{R,i}-x_{R,j}\rvert\rightarrow\infty$ if $i\neq j$, and
\emph{dist}$(x_{R,i},\partial\Gamma_{R})\rightarrow\infty$ for all $i $.
\end{theorem}
All solutions constructed in Theorems~\ref{thm:no-boundary} and
\ref{thm:with-boundary} change sign. If $\gamma$ is a closed curve these
solutions have an even number of bumps with alternating signs along the curve,
whereas in the open-end case $\gamma(0)\neq\gamma(1)$ the number of
alternating bumps may be even or odd. Note that the term $(n-2k)$ in
Theorem~\ref{thm:with-boundary} is $0$ if $n$ is even, and it is $1$ if $n$ is
odd. In the first case we have a positive bump at one end and a negative bump
at the other end of the domain, and in the second case we have positive bumps
at both ends. Of course, applying Theorem~\ref{thm:with-boundary} with $f(u)$
replaced by $-f(-u)$ and then multiplying the obtained multibump solution by
$-1$, we obtain a solution with negative bumps at both ends, as well.
Observe that in the open-end case the domains $\Omega_{R}$ are contractible,
and they are even convex if $\Gamma$ is a segment. This means that to get
multiplicity of sign changing solutions neither topological nor particular
geometrical assumptions are needed. This stands in contrast with the case of
positive solutions where it has been conjectured that for some power-type
nonlinearities only one positive solution exists in any convex
domain~\cite{MR949628}, as it does in a ball. Of course this difference between
multiplicity of positive and sign changing solutions can be easily understood
by looking at odd nonlinearities. In fact, if $f$ is odd (for example, if
$f(u)=\lvert u\rvert^{p-1}u$, $p\in(1,p_{S})$) it is well known that
infinitely many sign changing solutions exist in any bounded domain. Our
results \emph{do not assume that} $f$ \emph{is odd}, therefore multiplicity of
sign changing solutions is not so obvious. In fact, if $f$ is not odd only few
multiplicity results are available, see e.g. \cite{MR2037264, MR1627654}.
Dancer exhibited positive solutions with multiple bumps for \textquotedblleft
dumbbell shaped domains\textquotedblright\ \cite{MR949628,MR1072904}. Sign changing
solutions may also be constructed in domains of this type. On the other hand,
if $\Gamma$ is a segment, Theorem~\ref{thm:with-boundary} yields examples of
\emph{convex} domains in which problem \eqref{eq:1} has at least $k$ nodal
solutions with up to $k+1$ peaks, for any given $k$, without assuming that $f$
is odd. We believe this is the first result of this type.
As in other similar problems, the procedure to prove
Theorems~\ref{thm:no-boundary} and \ref{thm:with-boundary} is to consider
approximate solutions to problem \eqref{eq:1} and then show that near them a
true solution exists. So, to start, we need to make a good guess as to what
the approximate solutions should be. The geometry of our expanding domains
suggests looking at functions of the form
\begin{equation*}
U_{x_{R,1},R}^{+}+U_{x_{R,2},R}^{-}+U_{x_{R,3},R}^{+}+U_{x_{R,4},R}^{-}+\cdots
\end{equation*}
for finitely many points $x_{R,1},x_{R,2},x_{R,3},x_{R,4},...$, ordered along
the curve, whose number is even if the curve is closed. Then some estimates
are needed to show that these are indeed good approximate solutions and to
compute the order of the approximation. To prove the existence of a true
solution near them we follow a well-known Lyapunov-Schmidt reduction
procedure, which relies on the contraction mapping principle. This requires
again careful estimates on the approximate solutions and their linearization.
Finally, a critical point of the reduced problem is obtained by a
minimization. Here the crucial role is played by the fact that the interaction
between a positive and a negative bump increases the value of the energy
functional. This explains why the bumps should be placed along the tube with
alternating signs and why the number of bumps must be even in the closed tube
case (Theorem~\ref{thm:no-boundary}). In the open-end case
(Theorem~\ref{thm:with-boundary}) the energy also increases as a bump
approaches an end of the tube. Therefore, in both cases, a solution to the
reduced problem is obtained by minimizing the energy.
It is harder to prove similar results when $\Gamma$ is a higher dimensional
manifold, instead of a curve. For positive solutions some results were
obtained by Dancer and Yan \cite{MR1886955} when $\Gamma$ is the boundary of a convex
domain. Positive multibump solutions in a tubular neighborhood of an expanding
compact manifold have been constructed in \cite{aclapa-2}. The problem of
constructing sign changing solutions in such domains is more subtle and
requires minimax arguments.
The outline of the paper is as follows: In section~\ref{sec:prel-estim-1} we
have collected some tools, and results about the linear problem.
Section~\ref{sec:energy-estimates} contains the essential energy estimates,
while in section~\ref{sec:finite-dimens-reduct} we describe the finite
dimensional reduction and prove our main results.
\begin{acknowledgement}
Filomena Pacella wishes to thank the Mathematics Institute at UNAM and
Nils Ackermann and M\'{o}nica Clapp wish to thank the Mathematics Department
of the Universit\`{a} \textquotedblleft La Sapienza\textquotedblright\ di Roma
for their kind and warm hospitality.
\end{acknowledgement}
\section{Preliminaries}
\label{sec:prel-estim-1}
\subsection{Algebraic and geometric tools}
\label{sec:algebr-geom-tools}We start with some elementary lemmas which will
be used later to estimate the interactions.
\begin{lemma}
\label{lem:interaction-exponential} Suppose that $\mu_{k}>\bar{\mu}\geq0$ for
$k=1,2,3$. Then there is $C>0$ such that the inequalities
\begin{equation}
\int_{\mathbb{R}^{N}}\mathrm{e}^{-\mu_{1}\lvert x-x_{1}\rvert}\mathrm{e}
^{-\mu_{2}\lvert x-x_{2}\rvert}\,\mathrm{d}x\leq C\mathrm{e}^{-\bar{\mu}\lvert
x_{1}-x_{2}\rvert} \label{eq:19}
\end{equation}
and
\begin{equation}
\int_{\mathbb{R}^{N}}\mathrm{e}^{-\mu_{1}\lvert x-x_{1}\rvert}\mathrm{e}
^{-\mu_{2}\lvert x-x_{2}\rvert}\mathrm{e}^{-\mu_{3}\lvert x-x_{3}\rvert
}\,\mathrm{d}x\leq C\exp\biggl (-\bar{\mu}\min_{x\in\mathbb{R}^{N}}\sum
_{k=1}^{3}\lvert x-x_{k}\rvert\biggr ) \label{eq:20}
\end{equation}
hold true for all $x_{1},x_{2},x_{3}\in\mathbb{R}^{N}$.
\end{lemma}
\begin{proof}
Since $\bar{\mu}\lvert x_{1}-x_{2}\rvert+(\mu_{2}-\bar{\mu})\lvert
x-x_{2}\rvert\leq\bar{\mu}(\lvert x-x_{1}\rvert+\lvert x-x_{2}\rvert)+(\mu
_{2}-\bar{\mu})\lvert x-x_{2}\rvert\leq\mu_{1}\lvert x-x_{1}\rvert+\mu
_{2}\lvert x-x_{2}\rvert,$ we have that
\begin{equation*}
\int_{\mathbb{R}^{N}}\mathrm{e}^{-\mu_{1}\lvert x-x_{1}\rvert}\mathrm{e}
^{-\mu_{2}\lvert x-x_{2}\rvert}\,\mathrm{d}x\leq\int_{\mathbb{R}^{N}
}\mathrm{e}^{-\bar{\mu}\lvert x_{1}-x_{2}\rvert}\mathrm{e}^{-(\mu_{2}-\bar
{\mu})\lvert x-x_{2}\rvert}\,\mathrm{d}x=C\mathrm{e}^{-\bar{\mu}\lvert
x_{1}-x_{2}\rvert},
\end{equation*}
as claimed. The proof of the other inequality is similar.
\end{proof}
\begin{lemma}
\label{lem:splitting-f} There exists $\alpha\in(1/2,1]$ with the following
property: for any given $\widetilde{C}_{1}\geq1$ and $n\in\mathbb{N}$ there is
a constant $\widetilde{C}_{2}=\widetilde{C}_{2}(\alpha,n,\widetilde{C}_{1})>0$
such that the inequalities
\begin{equation}
\biggl \lvert f\biggl(\sum_{i=1}^{n}u_{i}\biggr)-\sum_{i=1}^{n}f(u_{i}
)\biggr \rvert\leq\widetilde{C}_{2}\sum_{i<j}\lvert u_{i}u_{j}\rvert^{\alpha},
\label{eq:21}
\end{equation}
\begin{equation}
\biggl \lvert F\biggl(\sum_{i=1}^{n}u_{i}\biggr)-\sum_{i=1}^{n}F(u_{i}
)-\sum_{i\neq j}f(u_{i})u_{j}\biggr \rvert\leq\widetilde{C}_{2}\biggl(\sum
_{i<j}\lvert u_{i}u_{j}\rvert^{2\alpha}+\sum_{i<j<k}\lvert u_{i}u_{j}
u_{k}\rvert^{2/3}\biggr), \label{eq:22}
\end{equation}
hold true for all $u_{1},u_{2},\dots,u_{n}\in\mathbb{R}$ with $\lvert
u_{i}\rvert\leq\widetilde{C}_{1}$.
\end{lemma}
\begin{proof}
Observe that (H3) implies that there is a constant $C>0$ such that
\begin{equation}
\left\vert f^{(k)}(u)\right\vert \leq C\left\vert u\right\vert ^{p_{1}
-k}\qquad\text{if }\left\vert u\right\vert \leq\widetilde{C}_{1},\ u\neq0.
\label{eq:29}
\end{equation}
Set $\alpha:=\min\{(p_{1}+1)/4,1\}\in(1/2,1]$. It is tedious but elementary to
prove that the inequalities
\begin{equation}
\left\vert f(u+v)-f(u)-f(v)\right\vert \leq C\left\vert uv\right\vert
^{\alpha} \label{eq:9}
\end{equation}
and
\begin{equation}
\left\vert F(u+v)-F(u)-F(v)-f(u)v-f(v)u\right\vert \leq C\left\vert
uv\right\vert ^{2\alpha} \label{eq:11}
\end{equation}
hold true for some constant $C>0$, if $\left\vert u\right\vert ,\left\vert
v\right\vert \leq\widetilde{C}_{1}$. These are inequalities \eqref{eq:21} and
\eqref{eq:22} for $n=2$. For $n>2$ inequalities \eqref{eq:21} and
\eqref{eq:22} follow easily by induction on $n$.
\end{proof}
The right-hand side of inequality \eqref{eq:22} indicates that we will need to
consider triple interactions. The following lemma will be useful to estimate them.
\begin{lemma}
\label{lem:triangle-geometry} Consider a triangle in $\mathbb{R}^{N}$ with
vertices $x_{1},x_{2},x_{3}\in\mathbb{R}^{N}$ and side lengths $w\leq v\leq
u$. Denote $s:=\min_{x\in\mathbb{R}^{N}}\sum_{k=1}^{3}\lvert x-x_{k}\rvert$.
Then the following statements are true:
\begin{enumerate}
\item[(a)] If one of the interior angles is larger than or equal to $2\pi/3$,
then $s=v+w$.
\item[(b)] In any case, $s\geq(w+v+u)/2$.
\end{enumerate}
\end{lemma}
\begin{proof}
The following facts from triangle geometry may be found in \cite{MR1573157}. The
minimum $s$ is achieved at a unique point $x_{0}$ in $\mathbb{R}^{N}$. In case
(a) that point is the vertex of the triangle with the largest interior angle,
so the claim follows immediately.
To prove (b) observe that adding up the inequalities $\lvert x_{i}-x_{0}
\rvert+\lvert x_{j}-x_{0}\rvert\geq\lvert x_{i}-x_{j}\rvert$, $i\neq j$,
yields
\begin{equation*}
2s=2\sum_{k=1}^{3}\lvert x_{0}-x_{k}\rvert\geq w+v+u\qquad\forall
x\in\mathbb{R}^{N},
\end{equation*}
as claimed.
\end{proof}
\begin{lemma}
\label{lem:est-intersection-balls} For $n\in\mathbb{N}$ there is a constant
$C=C(n)$ such that if $x_{1},x_{2}\in\mathbb{R}^{n}$ satisfy $\lvert
x_{1}-x_{2}\rvert<1$ and if $r\in\lbrack1,\lvert x_{1}-x_{2}\rvert+1]$ then
\begin{align}
\text{\emph{vol}}_{n}\left( B_{r}(x_{2})\smallsetminus B_{1}(x_{1})\right)
& \leq C\left( \left\vert x_{1}-x_{2}\right\vert +r-1\right) ,
\label{eq:53}\\begin{equation*}1ex]
\sup_{x\in\partial B_{r}(x_{2})}\text{\emph{dist}}(x,\partial B_{1}(x_{1}))
& \leq\left\vert x_{1}-x_{2}\right\vert +r-1,\label{eq:35}\\
\sup_{x\in\partial B_{1}(x_{1})}\text{\emph{dist}}(x,\partial B_{r}(x_{2}))
& \leq\left\vert x_{1}-x_{2}\right\vert +r-1. \label{eq:42}
\end{align}
Here $\emph{vol}_{n}$ denotes the Lebesgue measure in $\mathbb{R}^{n}$ and
$B_{r}(x):=\{y\in\mathbb{R}^{n}\mid\lvert y-x\rvert<r\}$.
\end{lemma}
\begin{proof}
Let $\omega_{k}$ denote the volume of the unit ball in $\mathbb{R}^{k}$. Set
$d:=\left\vert x_{1}-x_{2}\right\vert $. Without loss of generality we may
suppose that $x_{1}=0$ and $x_{2}=(d,0,\dots,0)$. Set $B_{1}:=B_{1}(0).$ Since
$B_{r}(x_{2})\smallsetminus B_{1}\subset(B_{1}(x_{2})\smallsetminus B_{1}
)\cup(B_{r}(x_{2})\smallsetminus B_{1}(x_{2}))$ and $r\in\lbrack1,2], $ we
have that
\begin{align*}
\text{vol}_{n}(B_{r}(x_{2})\smallsetminus B_{1}) & \leq\text{vol}_{n}
(B_{1}(x_{2})\smallsetminus B_{1})+\omega_{n}(r^{n}-1)\\
& \leq\text{vol}_{n}(B_{1}\smallsetminus B_{1}(x_{2}))+\omega_{n}
(2^{n}-1)(r-1).
\end{align*}
Write $x=(t,y)\in\mathbb{R}^{n}$ with $t\in\mathbb{R}$ and $y\in
\mathbb{R}^{n-1}.$ By symmetry considerations,
\begin{equation*}
\text{vol}_{n}(B_{1}\smallsetminus B_{1}(x_{2}))=\text{vol}_{n}\{x\in
B_{1}\mid\left\vert t\right\vert \leq d/2\}\leq\omega_{n-1}d.
\end{equation*}
Together with the previous inequality, this proves \eqref{eq:53}. An obvious
geometric argument proves \eqref{eq:35} and \eqref{eq:42}.
\end{proof}
\subsection{Analysis of linear operators and the limit problem}
\label{sec:analys-line-oper}Next we will show that $-\Delta+\lambda$ satisfies
the strong maximum principle on $\mathbb{L}$ and $\Omega_{R}$ for $R$ large if
$\lambda>-\lambda_{1,1}$.
For $r>0$ let $\lambda_{1,r}$ denote the smallest Dirichlet eigenvalue of
$-\Delta$ in the open ball $B_{r}^{N-1}:=\{\eta\in\mathbb{R}^{N-1}\mid
\lvert\eta\rvert<r\}$ of radius $r$ in $\mathbb{R}^{N-1}$, and let
$\vartheta_{1,r}$ be the positive eigenfunction corresponding to
$\lambda_{1,r}$, normalized by $\lVert\vartheta_{1,r}\rVert_{L^{2}}=1$. The
following result is well known.
\begin{lemma}
\label{lem:schroed-invertible} If $\lambda_{1}(\mathbb{L})$ denotes the bottom
of the spectrum of $-\Delta$ in $L^{2}(\mathbb{L})$ with Dirichlet boundary
conditions, then $\lambda_{1}(\mathbb{L})=\lambda_{1,1}$.
\end{lemma}
Next, we construct a positive superharmonic function for $-\Delta+\lambda$ in
$\Omega_{R}$ for $R$ large. This allows to estimate the bottom of the spectrum
of $-\Delta$ in $L^{2}(\Omega_{R})$ from below and provides a maximum
principle for $-\Delta+\lambda.$
As before, we write a point in $\mathbb{R}^{N}$ as $(\xi,\eta)$, where $\xi
\in\mathbb{R}$ and $\eta\in\mathbb{R}^{N-1}.$
\begin{lemma}
\label{lem:existence-positive-supersolution} If $\lambda>-\lambda_{1,1},$
there exists a superharmonic function for $-\Delta+\lambda$ in $C^{2}
(\mathbb{L})\cap C(\overline{\mathbb{L}})$ which is positive on $\overline
{\mathbb{L}}$. If $R$ is large enough then there exists a superharmonic
function for $-\Delta+\lambda$ in $C^{2}(\Omega_{R})\cap C(\overline
{\Omega_{R}})$ which is positive on $\overline{\Omega_{R}}$.
\end{lemma}
\begin{proof}
We fix $r>1$ close enough to $1$ so that $\lambda_{1,r}+\lambda>0$. Then
$W(\xi,\eta):=\vartheta_{1,r}(\eta)$ satisfies
\begin{equation*}
(-\Delta+\lambda)W=(\lambda_{1,r}+\lambda)W>0\text{ in }\mathbb{L}\text{
\ \ and \ \ }\min_{\overline{\mathbb{L}}}W>0.
\end{equation*}
This proves the first assertion.
To prove the second one note first that, for $R\geq1$ large enough, the set
$\Omega_{R,r}:=\{x\in\mathbb{R}^{N}\mid$dist$(x,\Gamma_{R})<r\}$ is a tubular
neighborhood of $\Gamma_{R}$. Since $\vartheta_{1,r}$ is radial, we may write
$\vartheta_{1,r}(\eta)=\vartheta_{1,r}(\left\vert \eta\right\vert )$ and
define
\begin{equation*}
W(x):=\vartheta_{1,r}(\text{dist}(x,\Gamma_{R}))\text{\quad for }x\in
\Omega_{R,r}.
\end{equation*}
Clearly, $\min\limits_{\overline{\Omega_{R}}}W>0$ for $R$ large enough. We
claim that
\begin{equation}
W\in C^{2}(\Omega_{R})\cap C(\overline{\Omega_{R}}) \label{eq:99}
\end{equation}
and
\begin{equation}
\min_{\Omega_{R}}\left( {(-\Delta+\lambda)W}\right) >0 \label{eq:100}
\end{equation}
for $R$ large enough. To prove this claims we fix $y_{0}\in\Omega_{R}$ and we
define locally, around $y_{0},$ a diffeomorphism from $\Omega_{R}$ to the unit
normal bundle of $\Gamma_{R}$ as follows: after a change of coordinates we may
assume that $0\in\Gamma_{R}$ and that dist$(y_{0},\Gamma_{R})=\left\vert
y_{0}\right\vert .$ We may also assume that the tangent space to $\Gamma_{R}$
at $0$ is $\mathbb{R}\times\{0\}.$ Then, $y_{0}\in\{0\}\times\mathbb{R}
^{N-1}.$ Let $\tau\colon(-\varepsilon,\varepsilon)\rightarrow\mathbb{R}^{N}$
be a parametrization by arc length of $\Gamma$ such that $\tau(0)=0$ and
$\tau^{\prime}(0)=(1,0).$ For $\xi\in(-R\varepsilon,R\varepsilon)$ and
$\eta\in\mathbb{R}^{N-1},$ set $\tau_{R}(\xi):=R\tau(\frac{\xi}{R})$ and let
$h_{R}(\xi,\eta)$ be the orthogonal projection of $(0,\eta)$ onto the space
$\tau^{\prime}(\xi)^{\perp}=\{x\in\mathbb{R}^{N}:x\cdot\tau^{\prime}(\frac
{\xi}{R})=0\}.$ Now define
\begin{equation*}
\Phi_{R}(\xi,\eta):=\tau_{R}\left( \xi\right) +\frac{\left\vert
\eta\right\vert }{\left\vert h_{R}(\xi,\eta)\right\vert }h_{R}(\xi,\eta).
\end{equation*}
Note that $\Phi_{R}(0,\eta)=(0,\eta)$. Moreover,
\begin{equation}
\mathrm{D}\Phi_{R}(0,\eta)=
\begin{pmatrix}
1-\frac{1}{R}\left[ (0,\eta)\cdot\tau^{\prime\prime}(0)\right] & 0\\
0 & I_{N-1}
\end{pmatrix}
. \label{eq:87}
\end{equation}
Therefore, $\Phi_{R}$ is a $C^{2}$-diffeomorphism between neighborhoods of
$\overline{\{0\}\times B_{1}^{N-1}}$ for $R$ large enough. Note that, since
$h_{R}(\xi,\eta)$ is orthogonal to $\Gamma_{R}$ at $\tau_{R}(\xi),$
\begin{equation*}
\text{dist}(\Phi_{R}(\xi,\eta),\Gamma_{R})=\left\vert \Phi_{R}(\xi,\eta
)-\frac{1}{R}\tau(\xi)\right\vert =\left\vert \eta\right\vert .
\end{equation*}
This implies that
\begin{equation*}
W(\Phi_{R}(\xi,\eta))=\vartheta_{1,r}(\left\vert \eta\right\vert ).
\end{equation*}
So, since $\Phi_{R}$ is a local $C^{2}$-diffeomorphism at $y_{0}$, this
identity proves \eqref{eq:99}.
To prove \eqref{eq:100} it is enough to show that
\begin{equation}
(-\Delta+\lambda)W(0,\eta)\geq C>0 \label{eq:88}
\end{equation}
for $\eta\in B_{1}^{N-1}$ and large $R$, where $C$ is independent of $y_{0}$
and $R$. A straightforward computation shows that
\begin{equation*}
(-\Delta+\lambda)W(0,\eta)=\left( {\lambda_{1,r}+\lambda}\right)
\vartheta_{1,r}(\eta)+O(\left\vert \mathrm{D}^{2}\Phi_{R}(0,\eta)\right\vert
),
\end{equation*}
independently of $y_{0}$, and that that
\begin{equation}
\mathrm{D}^{2}\Phi_{R}(0,\eta)\rightarrow0\qquad\text{as $R\rightarrow\infty$,
independently of $y_{0}$ and $\eta$.} \label{eq:102}
\end{equation}
Since $\vartheta_{1,r}$ is positive and continuous on $\overline{B_{1}^{N-1}}$
we may set
\begin{equation*}
C:=\frac{\lambda_{1,r}+\lambda}{2}\min_{\left\vert \eta\right\vert \leq
1}\vartheta_{1,r}(\eta)>0
\end{equation*}
and obtain \eqref{eq:88}.
\end{proof}
\begin{corollary}
\label{cor:spectrum-positive-omega-r} If $\lambda_{1}(\Omega_{R})$ denotes the
bottom of the spectrum of $-\Delta$ in $L^{2}(\Omega_{R})$ with Dirichlet
boundary conditions, then
\begin{equation*}
\liminf_{R\rightarrow\infty}\lambda_{1}(\Omega_{R})\geq\lambda_{1,1}.
\end{equation*}
\end{corollary}
\begin{proof}
A standard argument, using Lemma~\ref{lem:existence-positive-supersolution},
proves this claim.
\end{proof}
The following fact will play a crucial role to obtain asymptotic estimates for
the energy functional and its gradient.
\begin{corollary}
\label{rem:maximum-principle} If $\lambda>-\lambda_{1,1}$ the operator
$-\Delta+\lambda$ satisfies the strong maximum principle in any subdomain of
$\mathbb{L}$ and in any subdomain of $\Omega_{R}$ for $R$ large enough.
\end{corollary}
\begin{proof}
This follows from Lemma~\ref{lem:existence-positive-supersolution} and
\cite[Theorem~1]{MR0271508}.
\end{proof}
We shall also need the following decay estimates for the solutions $U^{\pm}$
to the limit problem \eqref{eq:2}. They follow immediately from
\cite[Proposition~4.2]{MR1039342}.
\begin{lemma}
\label{decayU} There are constants $C_{1},C_{2}>0$ such that
\begin{equation*}
C_{1}\mathrm{e}^{-\mu\lvert\xi\rvert}\vartheta_{1,1}(\eta)\leq\lvert U^{\pm
}(\xi,\eta)\rvert\leq C_{2}\mathrm{e}^{-\mu\lvert\xi\rvert}\vartheta
_{1,1}(\eta)\qquad\text{for all }(\xi,\eta)\in\mathbb{L}
\end{equation*}
where $\mu:=\sqrt{\lambda+\lambda_{1,1}}.$
\end{lemma}
\section{Asymptotics of the energy and its gradient}
\label{sec:energy-estimates}
We assume from now on that $\lambda>-\lambda_{1,1}.$ Let $\mathbb{L}
_{s}:=\{(\xi,\eta)\in\mathbb{R}^{1}\times\mathbb{R}^{N-1}:\lvert\eta
\rvert<s\}.$ We fix $r_{0}>1$ such that $\lambda_{1,r_{0}}+\lambda>0$ and, for
$R>0$, $x\in\Gamma_{R}$ and $s\in\lbrack1,r_{0}],$ we set
\begin{equation*}
\mathbb{L}_{s,x}:=\{x+A_{x}^{-1}(z):z\in\mathbb{L}_{s}\}
\end{equation*}
with $A_{x}$ as in (\ref{eq:3})$.$ Note that the first eigenvalue of $-\Delta$
in $H_{0}^{1}(\Omega_{R}\cap\mathbb{L}_{s,x})$ satisfies $\lambda_{1}
(\Omega_{R}\cap\mathbb{L}_{s,x})+\lambda>0$ for large $R$, because $\Omega
_{R}\cap\mathbb{L}_{s,x}$ is an open bounded subset of $\mathbb{L}_{r_{0},x}$.
We write $V_{x,s,R}^{\pm}$ for the unique solution to the problem
\begin{equation}
\left\{
\begin{array}
[c]{ll}
-\Delta u+\lambda u=f(U_{x,R}^{\pm}) & \text{in }\Omega_{R}\cap\mathbb{L}
_{s,x},\\
u=0 & \text{on }\partial\left( \Omega_{R}\cap\mathbb{L}_{s,x}\right) ,
\end{array}
\right. \label{V}
\end{equation}
with $U_{x,R}^{\pm}$ as in (\ref{eq:3}). By the maximum principle and
assuption (H4), $V_{x,s,R}^{+}$ is positive and $V_{x,s,R}^{-}$ is negative
for large $R.$ We extend $V_{x,s,R}^{\pm}$ to all of $\mathbb{R}^{N}$ by
defining it as $0$ outside of $\Omega_{R}\cap\mathbb{L}_{s,x}$.\ When $s=1$ we
omit it from the notation and write $\mathbb{L}_{x},$ $V_{x,R}$ instead of
$\mathbb{L}_{1,x},$ $V_{x,1,R}.$
\subsection{The closed tube case}
In this subsection we assume that $\gamma(0)=\gamma(1).$ The following decay
estimates hold true.
\begin{lemma}
\label{lem:decay-V}For each $s\in\lbrack1,r_{0})$ there are positive constants
$c_{3},c_{4}$ and $R_{0},$ independent of $x\in\Gamma_{R},$ such that all
quantities
\begin{equation*}
\left\vert U_{x,R}^{\pm}(y)\right\vert ,\quad\left\vert \nabla U_{x,R}^{\pm
}(y)\right\vert ,\quad\left\vert V_{x,s,R}^{\pm}(y)\right\vert ,\quad
\left\vert \nabla V_{x,s,R}^{\pm}(y)\right\vert ,
\end{equation*}
are bounded by $c_{3}e^{-c_{4}\left\vert y-x\right\vert }$ for all $R\geq
R_{0}$ and almost all $y\in\mathbb{R}^{N}.$ Moreover,
\begin{equation*}
\left\vert D^{2}U_{x,R}^{\pm}(y)\right\vert \text{\quad and\quad}\left\vert
D^{2}V_{x,s,R}^{\pm}(y)\right\vert
\end{equation*}
are bounded uniformly in $\mathbb{L}_{x}$\ and $\Omega_{R}\cap\mathbb{L}
_{s,x}$\ respectively, independently of $R\geq R_{0}.$
\end{lemma}
\begin{proof}
Lemma \ref{decayU}, together with standard regularity estimates, yields the
estimates for $U_{x,R}^{\pm}$ and its derivatives.\newline To prove the
estimates for $V_{x,s,R}^{\pm}$ we assume without loss of generality that
$x=0$ and that $\mathbb{R}\times\{0\}$ is the tangent space to $\Gamma_{R}$ at
$0.$ Then there exists $\tilde{c}_{s}>0$ such that $\vartheta_{1,r_{0}}
(\eta)\geq\tilde{c}_{s}$ for all $\eta\in B_{s}^{N-1}, $ where $\vartheta
_{1,r_{0}}$ is the positive first Dirichlet eigenfunction of $-\Delta$ in the
ball of radius $r_{0}$ (as in the beginning of subsection
\ref{sec:analys-line-oper}). We write $y\in\mathbb{L}_{s}$ as $(\xi,\eta)$
with $\xi\in\mathbb{R}$ and $\eta\in B_{s}^{N-1}, $ and set
\begin{equation*}
W(y):=e^{-\nu\left\vert \xi\right\vert }\vartheta_{1,r_{0}}(\eta)
\end{equation*}
where $\nu$ is a small positive constant, independent of $R,$ which will be
fixed next. A straightforward computation gives
\begin{align*}
-\Delta W(y)+\lambda W(y) & =\left( \frac{\left( N-1\right) \nu
}{\left\vert \xi\right\vert }-\nu^{2}+\lambda_{1,r_{0}}+\lambda\right) W(y)\\
& >\left( \lambda_{1,r_{0}}+\lambda-\nu^{2}\right) \tilde{c}_{s}
e^{-\nu\left\vert \xi\right\vert }.
\end{align*}
Since $\lambda_{1,r_{0}}+\lambda>0$ we have that $\lambda_{1,r_{0}}
+\lambda-\nu^{2}>0$ if $\nu$ is small enough. On the other hand, assumption
(H3) on $f$ together with Lemma \ref{decayU} yield that
\begin{equation*}
f(U_{x,R}^{+})\leq b_{1}e^{-\mu p_{1}\left\vert \xi\right\vert },
\end{equation*}
for some large enough $b_{1}>0$. Since $V_{x,s,R}^{+}$ satisfies (\ref{V}) the
maximum principle implies that $V_{x,s,R}^{+}\leq b_{2}W$ with $b_{2}
:=b_{1}\tilde{c}_{s}^{-1}\left( \lambda_{1,r_{0}}+\lambda-\nu^{2}\right)
^{-1}.$ This gives the exponential bound on $V_{x,s,R}^{+}.$ Similarly for
$V_{x,s,R}^{-}.$ Regularity estimates, using the results in \cite{MR1156467}, yield
the estimates for its derivatives.
\end{proof}
Set
\begin{equation*}
F(u):=\int_{0}^{u}f(s)\,\mathrm{d}s\qquad\text{if }u\in\mathbb{R}\text{.}
\end{equation*}
Then, by (H3),
\begin{equation}
\lvert F(u)\rvert\leq C(\lvert u\rvert^{p_{1}+1}+\lvert u\rvert^{p_{2}
+1})\qquad\text{for all }u\in\mathbb{R}\text{.} \label{eq:49}
\end{equation}
\begin{lemma}
\label{lem:u-v-est}For $s\in\lbrack1,r_{0})$ and $p\in(0,\infty)$ the
asymptotic estimates
\begin{align}
\int_{\mathbb{R}^{N}}\lvert V_{x,s,R}^{\pm}-U_{x,R}^{\pm}\rvert^{p} &
=O(R^{-\min\{p,1\}}),\label{eq:13}\\
\int_{\mathbb{R}^{N}}\lvert\nabla V_{x,s,R}^{\pm}-\nabla U_{x,R}^{\pm}
\rvert^{2} & =O(R^{-1}),\label{eq:12}\\
\int_{\mathbb{R}^{N}}\lvert F(V_{x,s,R}^{\pm})-F(U_{x,R}^{\pm})\rvert &
=O(R^{-1}),\label{eq:14}\\
\int_{\mathbb{R}^{N}}\lvert f(V_{x,s,R}^{\pm})-f(U_{x,R}^{\pm})\rvert^{p} &
=O(R^{-\min\{p,1\}}), \label{eq:15}
\end{align}
hold true as $R\rightarrow\infty,$ independently of $x\in\Gamma_{R}.$
\end{lemma}
\begin{proof}
Let $x$ be a point on $\Gamma.$ After translation and rotation we may assume
that $x=0$ and that $\mathbb{R}\times\{0\}$ is the tangent space to $\Gamma$
at $0.$ Since $\Gamma$ is compact there exist $\delta,\rho>0$, independent of
$x$, and a $C^{3}$-function $h:(-\rho,\rho)\rightarrow B_{\delta}^{N-1}$ such
that
\begin{equation*}
\Gamma\cap\left( (-\rho,\rho)\times B_{\delta}^{N-1}\right) =\{(\xi
,h(\xi))\mid\xi\in(-\rho,\rho)\},
\end{equation*}
and the derivatives of $h$ up to the order $3$ are bounded independently of
$\xi\in(-\rho,\rho)$ and $x\in\Gamma.$ Setting $h_{R}(\xi):=Rh(\xi/R)$ we have
that
\begin{equation*}
\widetilde{\Gamma}_{R}:=\Gamma_{R}\cap\left( (-\rho R,\rho R)\times B_{\delta
R}^{N-1}\right) =\{(\xi,h_{R}(\xi))\mid\xi\in(-\rho R,\rho R)\}.
\end{equation*}
An easy argument using Taylor's theorem and geometric considerations shows
that there is a constant $C,$ independent of $x$, such that
\begin{equation}
\left\vert h_{R}(\xi)\right\vert \leq\frac{C\xi^{2}}{R},\text{\quad}\left\vert
h_{R}^{\prime}(\xi)\right\vert \leq\frac{C\left\vert \xi\right\vert }
{R}\text{\quad and\quad}\left\vert y-h_{R}(\xi)\right\vert \leq1+\frac
{C(1+\xi^{2})}{R^{2}} \label{eq:55}
\end{equation}
for all $\xi\in(-\rho R+1,\rho R-1)$ and $y\in\mathbb{R}^{N-1}$ with
$(\xi,y)\in\Omega_{R}$. It follows that
\begin{equation}
\{\xi\}\times B_{1}^{N-1}(h_{R}(\xi))\subset\left[ \{\xi\}\times
\mathbb{R}^{N-1}\right] \cap{\Omega}_{R}\subset\{\xi\}\times
B_{1+C(1+\left\vert \xi\right\vert ^{2})/R^{2}}^{N-1}(h_{R}(\xi))
\label{eq:56}
\end{equation}
for all $\xi\in(-\rho R+1,\rho R-1)$ and $R$ large enough. Consider the set
\begin{equation*}
Q_{R}:=(-R^{1/4},R^{1/4})\times B_{s}^{N-1}\subset\mathbb{L}_{s}.
\end{equation*}
We express $\mathbb{R}^{N}$ as the union of the sets
\begin{equation}
\mathbb{R}^{N}\smallsetminus Q_{R},\hspace{0.25in}Q_{R}\cap\left( \Omega
_{R}\smallsetminus\mathbb{L}\right) ,\hspace{0.25in}Q_{R}\cap\left(
\mathbb{L}\smallsetminus\Omega_{R}\right) ,\hspace{0.25in}Q_{R}\cap
\mathbb{L}\cap\Omega_{R}. \label{eq:63}
\end{equation}
We will show that the estimates \eqref{eq:13}, \eqref{eq:12}, \eqref{eq:14},
\eqref{eq:15}, hold true for the integrals over each one of these sets. Note
that the integrals over $Q_{R}\smallsetminus\left( \mathbb{L}\cup\Omega
_{R}\right) $ are zero.
\begin{claim}
\label{c1}Estimate \eqref{eq:13} holds true for the integral over
$\mathbb{R}^{N}\smallsetminus Q_{R}.$
\end{claim}
By Lemma~\ref{lem:decay-V} there are positive constants $\widetilde{C}
_{1},\widetilde{C}_{2}$ such that
\begin{equation}
\lvert V_{x,s,R}^{\pm}(\xi,\eta)-U_{x,R}^{\pm}(\xi,\eta)\rvert^{p}
\leq\widetilde{C}_{1}\mathrm{e}^{-\widetilde{C}_{2}(\left\vert \xi\right\vert
+\left\vert \eta\right\vert )} \label{eq:59}
\end{equation}
for all $(\xi,\eta)\in\mathbb{R}\times\mathbb{R}^{N-1}$. This immediately
yields Claim~\ref{c1}.
\begin{claim}
\label{c2}Estimate \eqref{eq:13} holds true for the integral over $Q_{R}
\cap\left( \Omega_{R}\smallsetminus\mathbb{L}\right) .$
\end{claim}
By \eqref{eq:59}, \eqref{eq:56}, Lemma~\ref{lem:est-intersection-balls} and
\eqref{eq:55} it holds that
\begin{align*}
& \int_{Q_{R}\cap\left( \Omega_{R}\smallsetminus\mathbb{L}\right) }\lvert
V_{x,s,R}^{\pm}-U_{x,R}^{\pm}\rvert^{p}\text{ }\\
& \leq\widetilde{C}_{1}\int_{-R^{1/4}}^{R^{1/4}}\mathrm{e}^{-\widetilde
{C}_{2}\left\vert \xi\right\vert }\text{vol}_{N-1}\left( B_{1+C(1+\left\vert
\xi\right\vert ^{2})/R^{2}}^{N-1}(h_{R}(\xi))\smallsetminus B_{1}
^{N-1}(0)\right) \mathrm{d}\xi\\
& \leq C\int_{-R^{1/4}}^{R^{1/4}}\mathrm{e}^{-\widetilde{C}_{2}\left\vert
\xi\right\vert }(\left\vert h_{R}(\xi)\right\vert +C(1+\xi^{2})/R^{2})\text{
}\mathrm{d}\xi\\
& \leq\frac{C}{R}\int_{-\infty}^{\infty}\mathrm{e}^{-\widetilde{C}
_{2}\left\vert \xi\right\vert }(1+\xi^{2})\text{ }\mathrm{d}\xi=O(R^{-1})
\end{align*}
as $R\rightarrow\infty.$
\begin{claim}
\label{c3}Estimate \eqref{eq:13} holds true for the integral over $Q_{R}
\cap\left( \mathbb{L}\smallsetminus\Omega_{R}\right) .$
\end{claim}
The proof is similar to that of Claim 2, using this time the first inclusion
in \eqref{eq:56}.
\begin{claim}
\label{c4}Estimate \eqref{eq:13} holds true for the integral over $Q_{R}
\cap\mathbb{L}\cap\Omega_{R}.$
\end{claim}
Set $D_{R}:=Q_{R}\cap\mathbb{L}\cap\Omega_{R}.$ First we prove that, for some
suitable constant $C$ independent of $x$ and $R$,
\begin{equation}
\left\vert V_{x,s,R}^{\pm}(\xi,\eta)-U_{x,R}^{\pm}(\xi,\eta)\right\vert \leq
C\mathrm{e}^{-C_{4}\left\vert \xi\right\vert }\frac{1+\xi^{2}}{R}
\label{eq:64}
\end{equation}
for all $(\xi,\eta)\in\partial D_{R}$. Let $(\xi,\eta)\in\partial D_{R}.$ If
$(\xi,\eta)\in\partial\mathbb{L}$, Lemma~\ref{lem:est-intersection-balls},
together with \eqref{eq:56}, and \eqref{eq:55}, yields
\begin{equation}
\text{dist}((\xi,\eta),\partial\Omega_{R})\leq\left\vert h_{R}(\xi)\right\vert
+C\frac{1+\xi^{2}}{R^{2}}\leq C\frac{1+\xi^{2}}{R}. \label{eq:85}
\end{equation}
Similarly, if $(\xi,\eta)\in\partial\Omega_{R}$ then
\begin{equation*}
\text{dist}((\xi,\eta),\partial\mathbb{L})\leq C\frac{1+\xi^{2}}{R}.
\end{equation*}
Since $U_{x,R}^{\pm}$ vanishes on $\partial\mathbb{L}$ and $V_{x,s,R}^{\pm}$
vanishes on $\partial\Omega_{R}$, the estimates in Lemma~\ref{lem:decay-V}
yield inequality \eqref{eq:64}. Next we set $W(y):=e^{-\nu\left\vert
\xi\right\vert }\vartheta_{1,r_{0}}(\eta)$ with $\nu\in(0,C_{4})$ as in the
proof of Lemma~\ref{lem:decay-V}. By \eqref{eq:64} there exists $C>0$ such
that
\begin{equation*}
\left\vert V_{x,s,R}^{\pm}(\xi,\eta)-U_{x,R}^{\pm}(\xi,\eta)\right\vert
\leq\frac{C}{R}W(\xi,\eta)
\end{equation*}
for all $(\xi,\eta)\in\partial D_{R}$. Since $V_{x,R}^{\pm}-U_{x,R}^{\pm}$ is
harmonic for $-\Delta+\lambda$ in $D_{R}$ the maximum principle implies that
\begin{equation*}
\left\vert V_{x,s,R}^{\pm}(\xi,\eta)-U_{x,R}^{\pm}(\xi,\eta)\right\vert
\leq\frac{C}{R}W(\xi,\eta)=\frac{C}{R}e^{-\nu\left\vert \xi\right\vert
}\vartheta_{1,r_{0}}(\eta)
\end{equation*}
for all $(\xi,\eta)\in D_{R}$, with $C$ independent of $x$ and $R$.
Therefore,
\begin{equation}
\int_{D_{R}}\left\vert V_{x,s,R}^{\pm}-U_{x,R}^{\pm}\right\vert ^{p}=O(R^{-p})
\label{eq:67}
\end{equation}
as $R\rightarrow\infty$. This proves Claim \ref{c4}.
\begin{claim}
\label{c5}Estimate \eqref{eq:12} holds true for the integrals over
$\mathbb{R}^{N}\smallsetminus Q_{R},$ $Q_{R}\cap\left( \Omega_{R}
\smallsetminus\mathbb{L}\right) $ and $Q_{R}\cap\left( \mathbb{L}
\smallsetminus\Omega_{R}\right) .$
\end{claim}
The same arguments as in the proofs of Claims \ref{c1}, \ref{c2} and \ref{c3}
yield this claim.
\begin{claim}
\label{c6}Estimate \eqref{eq:12} holds true for the integral over $Q_{R}
\cap\mathbb{L}\cap\Omega_{R}.$
\end{claim}
Set $D_{R}:=Q_{R}\cap\mathbb{L}\cap\Omega_{R}.$ The functions $U_{x,R}^{\pm}$
and $V_{x,R}^{\pm}$ can be extended to $C^{2}$-functions in neighborhoods of
$\mathbb{L}$ and $\Omega_{R}$, respectively. Denote by $Y_{R}$ the difference
of these extensions on a neighborhood of $\overline{D_{R}}$. Note that $D_{R}$
has Lipschitz boundary if $R$ is large enough. Hence we can apply the
Gauss-Green theorem (see e.g.\ \cite[Theorem~5.8.2]{MR1014685} and the remark
following it) and obtain that
\begin{equation}
\int_{D_{R}}\left\vert \nabla Y_{R}\right\vert ^{2}=\int_{\partial D_{R}}
Y_{R}\,n_{R}(x)\cdot\nabla Y_{R}\,\mathrm{d}H_{N-1}(x)-\lambda\int_{D_{R}
}Y_{R}^{2} \label{eq:105}
\end{equation}
Here $n_{R}(x)$ denotes the measure theoretic exterior normal to $\partial
D_{R}$ at $x$, and $H_{N-1}$ denotes $(N-1)$-dimensional Hausdorff measure. By
Lemma~\ref{lem:decay-V}, $\nabla Y_{R}$ is bounded uniformly and independently
of $R$. Hence \eqref{eq:105} and \eqref{eq:64} imply
\begin{align*}
\int_{D_{R}}\left\vert \nabla Y_{R}\right\vert ^{2} & \leq\frac{C}{R}\left(
\int_{\partial D_{R}}\mathrm{e}^{-C_{4}\left\vert \xi\right\vert
}(1+\left\vert \xi\right\vert ^{2})\,\mathrm{d}H_{N-1}(x)+\int_{D_{R}
}\mathrm{e}^{-C_{4}\left\vert \xi\right\vert }(1+\left\vert \xi\right\vert
^{2})\,\mathrm{d}\xi\,\mathrm{d}\eta\right) \\
& =O(R^{-1}).
\end{align*}
This proves Claim \ref{c6}.
\begin{claim}
\label{c7}Estimates \eqref{eq:14} and \eqref{eq:15} hold true.
\end{claim}
These estimates follow easily from \eqref{eq:13} since $U_{x,R}^{\pm}$ and
$V_{x,R}^{\pm}$ are bounded uniformly as $R\rightarrow\infty$ and $F$ and $f$
are continuously differentiable.
\end{proof}
The energy functional for the Dirichlet problem $-\Delta u+\lambda u=f(u)$ in
a domain $\Omega\subseteq\mathbb{R}^{N}$ is given by
\begin{equation*}
J_{\Omega}(u):=\frac{1}{2}\int_{\Omega}(\lvert\nabla u\rvert^{2}+\lambda
u^{2})-\int_{\Omega}F(u),\qquad u\in H_{0}^{1}(\Omega).
\end{equation*}
By (H2), (H3) and \eqref{eq:49} $J_{\Omega}$ is well defined and twice
continuously differentiable on $H_{0}^{1}(\Omega)$, with $\mathrm{D}
^{2}J_{\Omega}$ globally H\"{o}lder continuous on bounded subsets of
$H_{0}^{1}(\Omega)$.
\begin{lemma}
\label{lem:nearness-u-v} The estimates
\begin{align}
\sup_{x\in\Gamma_{R}}\lVert V_{x,R}^{\pm}-U_{x,R}^{\pm}\rVert_{H^{1}
(\mathbb{R}^{N})} & =O(R^{-1/2}),\label{eq:16}\\
\sup_{x\in\Gamma_{R}}\lvert J_{\Omega_{R}}(V_{x,R}^{\pm})-J_{\mathbb{L}
}(U^{\pm})\rvert & =O(R^{-1}),\label{eq:17}\\
\sup_{x\in\Gamma_{R}}\lVert\nabla J_{\Omega_{R}}(V_{x,R}^{\pm})\rVert
_{H_{0}^{1}(\Omega_{R})} & =O(R^{-1/2}), \label{eq:18}
\end{align}
hold true as $R\rightarrow\infty$.
\end{lemma}
\begin{proof}
Estimates \eqref{eq:16} and \eqref{eq:17} follow immediately from Lemma
\ref{lem:u-v-est}. To prove the third one we choose $s\in(1,r_{0})$ and a
cut-off function $\chi\in C^{\infty}(\mathbb{R}^{N-1})$ with $\chi(\eta)=1$ if
$\left\vert \eta\right\vert \leq1$ and $\chi(\eta)=0$ if $\left\vert
\eta\right\vert \geq s.$ Fix $R$ and $x\in\Gamma_{R}.$ Assuming that $x=0$ and
that $\mathbb{R}\times\{0\}$ is the tangent space to $\Gamma_{R}$ at $0,$ we
write $v\in H_{0}^{1}(\Omega_{R})$ as $v=v_{1}+v_{2}$ where $v_{1}(\xi
,\eta):=\chi(\eta)v(\xi,\eta).$ Then $v_{1}\in H_{0}^{1}(\Omega_{R}
\cap\mathbb{L}_{s,x}),$ supp$(v_{2})\subset\Omega_{R}\smallsetminus
\mathbb{L}_{x}$ and there exists a constant $c_{s}, $ independent of $R$ and
$x,$ such that $\left\Vert v_{1}\right\Vert _{H^{1}(\mathbb{R}^{N})}\leq
c_{s}\left\Vert v\right\Vert _{H^{1}(\mathbb{R}^{N})}$ for all $v\in H_{0}
^{1}(\Omega_{R}).$ From the definition of $V_{x,s,R}^{\pm}$ and Lemma
\ref{lem:u-v-est} we obtain
\begin{align*}
\left\vert DJ_{R}(V_{x,R}^{\pm})v\right\vert & =\left\vert DJ_{R}
(V_{x,R}^{\pm})v_{1}\right\vert \\
& \leq\left\vert DJ_{R}(V_{x,s,R}^{\pm})v_{1}\right\vert +\left\vert
DJ_{R}(V_{x,R}^{\pm})v_{1}-DJ_{R}(V_{x,s,R}^{\pm})v_{1}\right\vert \\
& \leq\left\vert \int_{\mathbb{R}^{N}}\left( f(U_{x,R}^{\pm})-f(V_{x,s,R}
^{\pm})\right) v_{1}\right\vert +O(R^{-1/2})\left\Vert v_{1}\right\Vert
_{H^{1}(\mathbb{R}^{N})}\\
& \leq O(R^{-1/2})\left\Vert v\right\Vert _{H^{1}(\mathbb{R}^{N})},
\end{align*}
as claimed.
\end{proof}
For $m=1,2$ we consider functions $g_{m}\colon\mathbb{R}^{+}\rightarrow
\mathbb{R}^{+}$ (to be fixed later) satisfying
\begin{align}
g_{2} & <g_{1}, & & \label{eq:24}\\
g_{m}(R) & \rightarrow\infty & \text{as }R & \rightarrow\infty,\text{ for
}m=1,2,\label{eq:10}\\
g_{m}(R) & =o(R) & \text{as }R & \rightarrow\infty,\text{ for }m=1,2.
\label{eq:25}
\end{align}
Let $D_{m,R}$ be the set of points $(x_{1},x_{2},\dots,x_{n})$ in $(\Gamma
_{R})^{n}$ such that there exist $i,j\in\{1,2,\dots,n\}$ with $i\neq j$ and
$\lvert x_{i}-x_{j}\rvert\leq g_{m}(R)$, and let
\begin{equation}
\mathcal{U}_{m,R}:=\{(x_{1},x_{2},\dots,x_{n})\in(\Gamma_{R})^{n}
\smallsetminus D_{m,R}\mid(x_{1},x_{2},\dots,x_{n})\text{ is an $n $-chain}\},
\label{Uclosed}
\end{equation}
see \eqref{eq:33} for the definition of an $n$-chain. Then $\mathcal{U}_{1,R}
$ and $\mathcal{U}_{2,R}$ are open subsets of $(\Gamma_{R})^{n}$ such that
$\overline{\mathcal{U}_{1,R}}\subset\mathcal{U}_{2,R}$. For $i,j\in
\{1,2,\dots,n\}\ $we set
\begin{equation*}
d_{n}(i,j):=\min\{\left\vert i-j\right\vert ,\left\vert i-j+n\right\vert
,\left\vert i-j-n\right\vert \}.
\end{equation*}
$d_{n}(i,j)$ is the distance from $i$ to the set of integers which are
congruent to $j$ mod $n.$
\begin{lemma}
\label{lem:chains}For $R$ large enough and every $(x_{1},x_{2},\dots,x_{n}
)\in\overline{\mathcal{U}_{1,R}}$ we have that
\begin{equation}
s(R):=\min_{x\in\mathbb{R}^{N}}\left( \lvert x-x_{i}\rvert+\lvert
x-x_{j}\rvert+\lvert x-x_{\ell}\rvert\right) \geq2g_{1}(R) \label{sR}
\end{equation}
for any $i,j,\ell\in\{1,2,\dots,n\},$ and
\begin{equation}
\lvert x_{i}-x_{j}\rvert\geq\frac{4}{3}g_{1}(R)\text{\qquad if }d_{n}
(i,j)\geq2. \label{d>2}
\end{equation}
\end{lemma}
\begin{proof}
Since $\Gamma$ is compact there exists $\varrho>0$ with the following properties:
\begin{enumerate}
\item[(i)] If $x,y\in\Gamma$ and $0<\left\vert x-y\right\vert <2\varrho$ then
there exists a connected component $\mathcal{C}$ of $\Gamma\smallsetminus
\{x,y\}$ such that $\left\vert x-z\right\vert +\left\vert z-y\right\vert
\leq\frac{3} {2}\left\vert x-y\right\vert $ for every $z\in\mathcal{C}$.
\item[(ii)] If $x,y,z$ are three different points in $\Gamma$, $\left\vert
x-y\right\vert <2\varrho$ and $\left\vert z-y\right\vert <2\varrho$ then one
of the angles of the triangle with vertices $x,y,z$ is larger that $2\pi/3.$
\end{enumerate}
Fix $R$ large enough so that $\frac{g_{1}(R)}{R}<\varrho.$
Let $(x_{1},x_{2},\dots,x_{n})\in\overline{\mathcal{U}_{1,R}}.$ If
\begin{equation*}
\max\left( \lvert x_{j}-x_{i}\rvert+\lvert x_{\ell}-x_{j}\rvert+\lvert
x_{i}-x_{\ell}\rvert\right) <2\varrho R,
\end{equation*}
the points $\frac{x_{i}}{R}
,\frac{x_{j}}{R},\frac{x_{\ell}}{R}\in\Gamma$ satisfy the hypothesis of (ii)
and, therefore, the triangle with vertices $x_{i},x_{j},x_{\ell}$ has an angle
which is larger that $2\pi/3.$ It follows from Lemma
\ref{lem:triangle-geometry} that $s(R)\geq2g_{1}(R).$ If, on the other hand,
$\lvert x_{j}-x_{i}\rvert\geq2\varrho R$ then $\lvert x_{j}-x_{i}\rvert
\geq2g_{1}(R),$ and Lemma \ref{lem:triangle-geometry} implies that
$s(R)\geq2g_{1}(R).$ This proves (\ref{sR}).
To prove (\ref{d>2}) we argue by contradiction. Assume there are $(x_{1}
,x_{2},\dots,x_{n})\in\overline{\mathcal{U}_{1,R}}$ and\ $i,j\in
\{1,2,\dots,n\}$ such that $d_{n}(i,j)\geq2$ and $\lvert x_{i}-x_{j}
\rvert<\frac{4}{3}g_{1}(R).$ Then $0<\left\vert \frac{x_{i}}{R}-\frac{x_{j}
}{R}\right\vert <2\varrho$. Since $d_{n}(i,j)\geq2$ there is a point $x_{\ell}
$ in the $n$-chain, which lies between $x_{i}$ and $x_{j},$ such that
$\frac{x_{\ell}}{R}$ belongs to the connected component of $\Gamma
\smallsetminus\{\frac{x_{i}}{R},\frac{x_{j}}{R}\}$ to which the conclusion of
(i) applies. Then, $2g_{1}(R)\leq\lvert x_{i}-x_{\ell}\rvert+\lvert x_{\ell
}-x_{j}\rvert\leq\frac{3}{2}\lvert x_{i}-x_{j}\rvert,$ a contradiction.
\end{proof}
In the rest of this subsection we assume that $n=2k.$ For $X=(x_{1}
,x_{2},\dots,x_{n})\in\mathcal{U}_{2,R}$ we define $\varphi_{R}\colon
\mathcal{U}_{2,R}\rightarrow H_{0}^{1}(\Omega_{R})$ by
\begin{equation}
\varphi_{R}(X):=
{\sum_{i=1}^{k}}
(V_{x_{2i-1},R}^{+}+V_{x_{2i},R}^{-}). \label{immersion}
\end{equation}
For fixed $X=(x_{1},x_{2},\dots,x_{n})$ it will be convenient to write
\begin{equation}
\overline{U}_{i}:=\left\{
\begin{array}
[c]{ll}
U_{x_{i},R}^{+} & \text{if }i\text{ is odd,}\\
U_{x_{i},R}^{-} & \text{if }i\text{ is even,}
\end{array}
\right. \text{\qquad}\overline{V}_{i}:=\left\{
\begin{array}
[c]{ll}
V_{x_{i},R}^{+} & \text{if }i\text{ is odd,}\\
V_{x_{i},R}^{-} & \text{if }i\text{ is even.}
\end{array}
\right. \label{bar}
\end{equation}
Then
\begin{equation*}
\varphi_{R}(X)=
{\sum_{i=1}^{n}}
\overline{V}_{i}.
\end{equation*}
\begin{proposition}
\label{prop:lyapunov-gradient-estimate} Let $\alpha$ be as in
\emph{Lemma~\ref{lem:splitting-f}} and fix $\alpha^{\prime}\in(1/2,\alpha)$.
Then
\begin{equation*}
\sup_{X\in\mathcal{U}_{2,R}}\lVert\nabla J_{\Omega_{R}}(\varphi_{R}
(X))\rVert_{H_{0}^{1}(\Omega_{R})}=O(\mathrm{e}^{-\alpha^{\prime}\mu g_{2}
(R)})+O(R^{-1/2})
\end{equation*}
as $R\rightarrow\infty$.
\end{proposition}
\begin{proof}
Fix $X=(x_{1},x_{2},\dots,x_{n})\in\mathcal{U}_{2,R}.$ If $v\in H_{0}
^{1}(\Omega_{R})$ satisfies $\left\Vert v\right\Vert _{H_{0}^{1}(\Omega_{R}
)}=1$ then, using Lemmas \ref{lem:nearness-u-v}, \ref{lem:splitting-f},
\ref{lem:u-v-est}, \ref{lem:decay-V} and \ref{lem:interaction-exponential}\ we
obtain
\begin{multline*}
\left\vert DJ_{\Omega_{R}}(\varphi_{R}(X))\left[ v\right]
\right\vert \\
\begin{aligned}
&=\left\vert {\sum_{i=1}^{n}}
DJ_{\Omega_{R}}(\varphi_{R}(\overline{V}_{i}))\left[ v\right]
+\int _{\Omega_{R}}\left( {\sum_{i=1}^{n}}
f(\overline{V}_{i})-f\left( {\sum_{i=1}^{n}}
\overline{V}_{i}\right) \right) v\right\vert \\
& \leq {\sum_{i=1}^{n}} \lVert\nabla
J_{\Omega_{R}}(\overline{V}_{i})\rVert_{H_{0}^{1}(\Omega_{R}
)}+\left( \int_{\Omega_{R}}\left\vert {\sum_{i=1}^{n}}
f(\overline{V}_{i})-f\left( {\sum_{i=1}^{n}}
\overline{V}_{i}\right) \right\vert ^{2}\right) ^{1/2}\\
& \leq O(R^{-1/2})+C {\sum_{i<j}} \left(
\int_{\Omega_{R}}\lvert\overline{V}_{i}\overline{V}_{j}\rvert
^{2\alpha}\right) ^{1/2}\\
& =O(R^{-1/2})+C {\sum_{i<j}} \left(
\int_{\Omega_{R}}\lvert\overline{U}_{i}\overline{U}_{j}\rvert
^{2\alpha}\right) ^{1/2}\\
& =O(R^{-1/2})+O(e^{-\alpha^{\prime}\mu g_{2}(R)}).
\end{aligned}
\end{multline*}
These estimates are independent of the choice of $X.$
\end{proof}
Recall that $n=2k$ and set
\begin{equation*}
E_{n}:=k\left[ J_{\mathbb{L}}(U^{+})+J_{\mathbb{L}}(U^{-})\right] .
\end{equation*}
\begin{proposition}
\label{prop:jr-high-on-boundary} There exists $\beta>0$ such that
\begin{equation*}
\inf_{X\in\partial\mathcal{U}_{1,R}}J_{\Omega_{R}}(\varphi_{R}(X))\geq
E_{n}+\beta\mathrm{e}^{-\mu g_{1}(R)}+o(\mathrm{e}^{-\mu g_{1}(R)}
)+O(R^{-2/3})
\end{equation*}
as $R\rightarrow\infty$.
\end{proposition}
\begin{proof}
If $X=(x_{1},x_{2},\dots,x_{n})\in\partial\mathcal{U}_{1,R}$ there are
$i_{0},j_{0}\in\{1,2,\dots,n\}$ such that $\lvert x_{i_{0}}-x_{j_{0}}
\rvert=g_{1}(R).$ By Lemma \ref{lem:chains}, $d_{n}(i_{0},j_{0})=1$ for $R$
large enough. Then, assumption (H4) implies that
\begin{equation}
f(\overline{U}_{i_{0}})\overline{U}_{j_{0}}\leq0. \label{eq:117}
\end{equation}
On the other hand, it follows from Lemma \ref{decayU} and property
\eqref{eq:25} that there exist $r,\varepsilon>0$ such that $\left\vert
f(\overline{U}_{i_{0}})\right\vert \geq\varepsilon$ and $\left\vert
\overline{U}_{j_{0}}\right\vert \geq C_{1}\mathrm{e}^{-\mu g_{1}(R)}$ in
$B_{r}(x_{i_{0}})$ for $R$ large enough, independently of the choice of
$X\in\partial\mathcal{U}_{1,R}$. Hence
\begin{equation}
\frac{1}{2}\int_{\mathbb{R}^{N}}\left\vert f(\overline{U}_{i_{0}})\overline
{U}_{j_{0}}\right\vert \,\mathrm{d}x\geq\beta\mathrm{e}^{-\mu g_{1}(R)}
\label{eq:23}
\end{equation}
for some $\beta>0$ and large $R$. Moreover, Lemmas \ref{lem:chains},
\ref{lem:interaction-exponential} and \ref{decayU} yield
\begin{equation}
\int_{\mathbb{R}^{N}}\left\vert f(\overline{U}_{i})\overline{U}_{j}\right\vert
\,\mathrm{d}x=o(\mathrm{e}^{-\mu g_{1}(R)})\qquad\text{if }d_{n}(i,j)\geq2,
\label{eq:27}
\end{equation}
as $R\rightarrow\infty$.
Since $\overline{U}_{i}$ and $\overline{V}_{i}$ are uniformly bounded, using
Lemma~\ref{lem:splitting-f}, estimate \eqref{eq:13}, and Lemmas
\ref{lem:chains} and \ref{lem:interaction-exponential}, we obtain
\begin{align}
& \left\vert \int_{\Omega_{R}}\left[ F(
{\sum_{i}}
\overline{V}_{i})-
{\sum_{i}}
F(\overline{V}_{i})\right] -
{\sum_{i\neq j}}
\int_{\Omega_{R}}f(\overline{V}_{i})\overline{V}_{j}\right\vert \label{eq:107}
\\
& \leq C
{\sum_{i<j}}
\int_{\Omega_{R}}\left\vert \overline{V}_{i}\overline{V}_{j}\right\vert
^{2\alpha}+C
{\sum_{i<j<k}}
\int_{\Omega_{R}}\left\vert \overline{V}_{i}\overline{V}_{j}\overline{V}
_{k}\right\vert ^{2/3}\nonumber\\
& =C
{\sum_{i<j}}
\int_{\Omega_{R}}\left\vert \overline{U}_{i}\overline{U}_{j}\right\vert
^{2\alpha}+C
{\sum_{i<j<k}}
\int_{\Omega_{R}}\left\vert \overline{U}_{i}\overline{U}_{j}\overline{U}
_{k}\right\vert ^{2/3}+O(R^{-2/3})\nonumber\\
& =o(\mathrm{e}^{-\mu g_{1}(R)})+O(R^{-2/3}).\nonumber
\end{align}
Therefore, using estimates \eqref{eq:17}, \eqref{eq:13}, \eqref{eq:117},
\eqref{eq:23} and \eqref{eq:27} we conclude that
\begin{align*}
J_{\Omega_{R}}(\varphi_{R}(X))= &
{\sum_{i=1}^{n}}
J_{\Omega_{R}}(\overline{V}_{i})+\frac{1}{2}
{\sum_{i\neq j}}
\int_{\Omega_{R}}\left( \nabla\overline{V}_{i}\cdot\nabla\overline{V}
_{j}+\lambda\overline{V}_{i}\overline{V}_{j}\right) \,\mathrm{d}x\\
& -\int_{\Omega_{R}}\left[ F(
{\sum_{i}}
\overline{V}_{i})-
{\sum_{i}}
F(\overline{V}_{i})\right] \,\mathrm{d}x\\
= & E_{n}+\frac{1}{2}
{\sum_{i\neq j}}
\int_{\Omega_{R}}f(\overline{U}_{i})\overline{V}_{j}\,\mathrm{d}x-
{\sum_{i\neq j}}
\int_{\Omega_{R}}f(\overline{V}_{i})\overline{V}_{j}\,\mathrm{d}x\\
& +o(\mathrm{e}^{-\mu g_{1}(R)})+O(R^{-2/3})\\
= & E_{n}-\frac{1}{2}
{\sum_{i\neq j}}
\int_{\Omega_{R}}f(\overline{U}_{i})\overline{U}_{j}\,\mathrm{d}
x+o(\mathrm{e}^{-\mu g_{1}(R)})+O(R^{-2/3})\\
\geq & E_{n}+\frac{1}{2}\int_{\mathbb{R}^{N}}\left\vert f(\overline{U}
_{i_{0}})\overline{U}_{j_{0}}\right\vert \,\mathrm{d}x-\frac{1}{2}
{\sum_{d_{n}(i,j)\geq2}}
\int_{\Omega_{R}}\left\vert f(\overline{U}_{i})\overline{U}_{j}\right\vert
\,\mathrm{d}x\\
& +o(\mathrm{e}^{-\mu g_{1}(R)})+O(R^{-2/3})\\
\geq & E_{n}+\beta\mathrm{e}^{-\mu g_{1}(R)}+o(\mathrm{e}^{-\mu g_{1}
(R)})+O(R^{-2/3}).
\end{align*}
This asymptotic estimate is independent of $X$.
\end{proof}
\begin{proposition}
\label{prop:jr-low-in-interior} The estimate
\begin{equation*}
\inf_{X\in\mathcal{U}_{1,R}}J_{\Omega_{R}}(\varphi_{R}(X))\leq E_{n}
+o(\mathrm{e}^{-\mu g_{1}(R)})+O(R^{-2/3})
\end{equation*}
holds true as $R\rightarrow\infty$.
\end{proposition}
\begin{proof}
Fix $0<t_{1}<t_{2}<\dots<t_{n}<1$ and set $x_{R,i}:=R\gamma(t_{i})\in
\Gamma_{R}.$ By \eqref{eq:25}, $X_{R}:=(x_{R,1},x_{R,2},\dots,x_{R,n}
)\in\mathcal{U}_{1,R}$ for large $R$. As in the proof of
Proposition~\ref{prop:jr-high-on-boundary} we obtain
\begin{equation*}
J_{\Omega_{R}}(\varphi_{R}(X_{R}))=E_{n}-\frac{1}{2}
{\sum_{i\neq j}}
\int_{\mathbb{R}^{N}}f(\overline{U}_{i})\overline{U}_{j}\,\mathrm{d}
x+o(\mathrm{e}^{-\mu g_{1}(R)})+O(R^{-2/3}).
\end{equation*}
Choose $\varepsilon\in(0,\mu)$ and $\delta\in(0,{\min_{i\neq j}}\left\vert
\gamma(t_{i})-\gamma(t_{j})\right\vert )$. Then $\left\vert x_{R,i}
-x_{R,j}\right\vert \geq\delta R$ if $i\neq j$.
Lemma~\ref{lem:interaction-exponential} and Lemma \ref{decayU} imply that
\begin{equation*}
\int_{\mathbb{R}^{N}}f(\overline{U}_{i})\overline{U}_{j}\,\mathrm{d}
x=O(\mathrm{e}^{-(\mu-\varepsilon)\delta R})=o(\mathrm{e}^{-\mu g_{1}
(R)})\qquad\text{if }i\neq j,
\end{equation*}
and our claim follows.
\end{proof}
\subsection{The open-end tube case}
We now suppose that $\gamma(0)\neq\gamma(1).$ In this case we need also to
estimate the effect of the ends of the tubular domain on $V_{x,R}^{\pm}$. We
start by comparing the solutions $U^{\pm}$ to the limit problem in
$\mathbb{L}$\ with their projections onto a finite cylinder
\begin{equation*}
\mathbb{L}_{a,b}:=(-a,b)\times B_{1}^{N-1},\text{\qquad}a,b>0.
\end{equation*}
Let $\widetilde{U}_{a,b}^{\pm}$ be the unique solution of
\begin{equation}
\left\{
\begin{array}
[c]{ll}
-\Delta u+\lambda u=f(U^{\pm}) & \text{in }\mathbb{L}_{a,b}\text{,}\\
u=0 & \text{on }\partial\mathbb{L}_{a,b}\text{.}
\end{array}
\right. \label{Utilde}
\end{equation}
Again, we consider $\widetilde{U}_{a,b}^{\pm}$ to be defined in $\mathbb{R}
^{N}$.
\begin{lemma}
\label{lem:tu-u-est-open}The inequalities
\begin{equation}
0\leq\widetilde{U}_{a,b}^{+}(\xi,\eta)\leq U^{+}(\xi,\eta),\qquad U^{-}
(\xi,\eta)\leq\widetilde{U}_{a,b}^{-}(\xi,\eta)\leq0, \label{eq:108}
\end{equation}
and
\begin{equation}
\lvert U^{\pm}(\xi,\eta)-\widetilde{U}_{a,b}^{\pm}(\xi,\eta)\rvert\leq
C_{2}\vartheta_{1,1}(\eta)\left( \mathrm{e}^{-\mu(a+\lvert\xi+a\rvert
)}+\mathrm{e}^{-\mu(b+\lvert\xi-b\rvert)}\right) \label{eq:43}
\end{equation}
hold true for all $(\xi,\eta)\in\mathbb{L}$, where $C_{2}$ is the same
constant as in \emph{Lemma \ref{decayU}}. Moreover, there are $C_{5},C_{6}>0$
such that
\begin{equation}
C_{5}\mathrm{e}^{-\mu\min\{a,b\}}\leq\lVert U^{\pm}-\widetilde{U}_{a,b}^{\pm
}\rVert_{H_{0}^{1}(\mathbb{L})}\leq C_{6}\mathrm{e}^{-\mu\min\{a,b\}}.
\label{eq:44}
\end{equation}
\end{lemma}
\begin{proof}
Note that $U^{\pm}$ and $\widetilde{U}_{a,b}^{\pm}$ are in $C^{2}
(\mathbb{L}_{a,b})\cap C(\overline{\mathbb{L}_{a,b}})$. Set $Y_{a,b}:=U^{\pm
}-\widetilde{U}_{a,b}^{\pm}.$ We claim that the inequalities
\begin{align}
& C_{1}\vartheta_{1,1}(\eta)\max\{\mathrm{e}^{-\mu(a+\lvert\xi+a\rvert
)},\mathrm{e}^{-\mu(b+\lvert\xi-b\rvert)}\}\leq Y_{a,b}\label{Y}\\
& \qquad\qquad\leq C_{2}\vartheta_{1,1}(\eta)(\mathrm{e}^{-\mu(a+\lvert
\xi+a\rvert)}+\mathrm{e}^{-\mu(b+\lvert\xi-b\rvert)})\nonumber
\end{align}
hold true for all $(\xi,\eta)\in\mathbb{L}$, where $C_{1}$ and $C_{2}$ are the
constants in Lemma \ref{decayU}. This is trivially true in $\mathbb{L}
\smallsetminus\mathbb{L}_{a,b}.$ For $(\xi,\eta)\in\mathbb{L}_{a,b}$ it
follows from the maximum principle, because the equalities
\begin{align*}
(-\Delta+\lambda)\vartheta_{1,1}(\eta)\mathrm{e}^{-\mu(a+\lvert\xi+a\rvert)}
& =0,\\
(-\Delta+\lambda)\vartheta_{1,1}(\eta)\mathrm{e}^{-\mu(b+\lvert\xi-b\rvert)}
& =0,\\
(-\Delta+\lambda)Y_{a,b} & =0,
\end{align*}
hold true in $\mathbb{L}_{a,b}$. Inequalities \eqref{eq:108} and \eqref{eq:43}
are now a consequence of \eqref{Y} and the maximum principle.
Next we prove \eqref{eq:44}. A straightforward computation using \eqref{Y}
yields
\begin{equation}
\left\Vert Y_{a,b}\right\Vert _{L^{2}(\mathbb{L})}=O(\mathrm{e}^{-\mu
\min\{a,b\}}) \label{eq:40}
\end{equation}
as $a,b\rightarrow\infty$. A standard regularity argument,
cf. \cite[Theorem~9.12]{MR737190}, yields
\begin{equation*}
\left\Vert \nabla Y_{a,b}\right\Vert _{L^{2}(\mathbb{L})}=O(\mathrm{e}
^{-\mu\min\{a,b\}}).
\end{equation*}
which, together with \eqref{eq:40}, this gives the inequality in the
right-hand side of \eqref{eq:44}. To prove the other inequality it is enough
to show that
\begin{equation}
\left\Vert Y_{a,b}\right\Vert _{L^{2}(\mathbb{L})}\geq C\mathrm{e}^{-\mu
\min\{a,b\}}, \label{eq:41}
\end{equation}
where $C$ is independent of $a$ and $b$. Note that
\begin{equation*}
a+\left\vert \xi+a\right\vert \leq b+\left\vert \xi-b\right\vert \quad\text{if
and only if}\quad\xi\leq b-a.
\end{equation*}
It follows that
\begin{align}
& \int_{\mathbb{R}}\max\{\mathrm{e}^{-\mu(a+\lvert\xi+a\rvert)}
,\mathrm{e}^{-\mu(b+\lvert\xi-b\rvert)}\}\,\mathrm{d}\xi\label{eq:109}\\
& =\frac{1}{\mu}(\mathrm{e}^{-2\mu a}-\mathrm{e}^{-2\mu(a+b)}+\mathrm{e}
^{-2\mu b})\nonumber\\
& \geq\frac{1}{\mu}\max\{\mathrm{e}{^{-2\mu a},\mathrm{e}^{-2\mu b}\}}
=\frac{1}{\mu}\mathrm{e}^{-2\mu\min\{a,b\}},\nonumber
\end{align}
which together with \eqref{Y} yields \eqref{eq:41}. The proof is complete.
\end{proof}
Next we compare $V_{x,R}^{\pm}$ with the function
\begin{equation*}
W_{x,R}^{\pm}(y):=\widetilde{U}_{\lvert x-R\gamma(0)\rvert,\lvert
x-R\gamma(1)\rvert}^{\pm}(A_{x}(y-x)),\qquad y\in\mathbb{R}^{N},
\end{equation*}
with $A_{x}$ as in \eqref{eq:3} and $\widetilde{U}_{a,b}^{\pm}$ as in
\eqref{Utilde}. Thus, the support of $W_{x,R}^{\pm}$ is contained in a copy of
the finite cylinder $\mathbb{L}_{\lvert x-R\gamma(0)\rvert,\lvert
x-R\gamma(1)\rvert},$ obtained by translating $0$ to $x$ and identifying
$\mathbb{R}\times\{0\}$ with the tangent space to $\Gamma_{R}$ at $x.$
\begin{lemma}
\label{lem:w-v-est-open}For $s\in\lbrack1,r_{0})$ and $p\in(0,\infty)$ the
asymptotic estimates
\begin{align}
\int_{\mathbb{R}^{N}}\lvert V_{x,s,R}^{\pm}-W_{x,R}^{\pm}\rvert^{p}
\,\mathrm{d}y & =O(R^{-\min\{p,1\}}),\label{eq:36}\\
\int_{\mathbb{R}^{N}}\lvert\nabla V_{x,s,R}^{\pm}-\nabla W_{x,R}^{\pm}
\rvert^{2}\,\mathrm{d}y & =O(R^{-1}),\label{eq:37}\\
\int_{\mathbb{R}^{N}}\lvert F(V_{x,s,R}^{\pm})-F(W_{x,R}^{\pm})\rvert
\,\mathrm{d}y & =O(R^{-1}),\label{eq:38}\\
\int_{\mathbb{R}^{N}}\lvert f(V_{x,s,R}^{\pm})-f(W_{x,R}^{\pm})\rvert
^{p}\,\mathrm{d}y & =O(R^{-\min\{p,1\}}), \label{eq:39}
\end{align}
hold true as $R\rightarrow\infty,$ independently of $x\in\Gamma_{R}.$
\end{lemma}
\begin{proof}
Let $x_{R}\in\Gamma_{R}.$ If $x_{R}$ is far from the boundary the proof is
similar to that of Lemma~\ref{lem:u-v-est}, but if $x_{R}$ is close to the
boundary the proof requires some new geometric considerations. More precisely,
we consider two cases:
a) $\left\vert x_{R}-R\gamma(0)\right\vert \geq2R^{1/4}$ and $\left\vert
x_{R}-R\gamma(1)\right\vert \geq2R^{1/4}.$ Then the proof is the same as that
of Lemma~\ref{lem:u-v-est}.
b) Either $\left\vert x_{R}-R\gamma(0)\right\vert <2R^{1/4}$ or $\left\vert
x_{R}-R\gamma(1)\right\vert <2R^{1/4}$. Since both cases are similar, we only
consider the case
\begin{equation}
b_{R}:=\left\vert x_{R}-R\gamma(1)\right\vert <2R^{1/4}. \label{eq:65}
\end{equation}
For each $R$ we fix a coordinate system by identifying $x_{R}$ with $0$ and
the tangent space to $\Gamma_{R}$ at $x_{R}$ with $\mathbb{R}\times\{0\},$
preserving the orientation. In this coordinate system we consider the infinite
cylinder $\mathbb{L}$ and the finite cylinders
\begin{equation*}
\begin{array}
[c]{l}
\mathbb{L}_{R}:=\mathbb{L}_{\lvert x_{R}-R\gamma(0)\rvert,\lvert x_{R}
-R\gamma(1)\rvert},\medskip\\
Q_{R}:=(-R^{1/3},R^{1/3})\times B_{s}^{N-1}(0),
\end{array}
\end{equation*}
and we write $R\gamma(1)=(\xi_{R},\eta_{R}).$ Note that, since $\frac{x_{R}
}{R}\rightarrow\gamma(1)$ as $R\rightarrow\infty,$ the end of $\Omega_{R}$
which contains $R\gamma(0)$ lies outside of $Q_{R}$ for $R$ large enough.
We may assume that $\gamma$ is defined in some interval $(0,1+\varepsilon), $
$\varepsilon>0,$ and write $\widetilde{\Gamma}_{R}:=\{R\gamma(t)\mid
t\in\lbrack0,1+\varepsilon)\}$ and $\widetilde{\Omega}_{R}$ for its tubular
neighborhood of radius $1.$ Then $\widetilde{\Gamma}_{R}\cap Q_{R}$ is
contained in the graph of a $C^{3}$-function $h_{R}:(-R^{1/3},R^{1/3}
)\rightarrow\mathbb{R}^{N-1}$ for large $R$. As before, inequalities
\eqref{eq:55}\ hold for $h_{R}.$ Since $0\leq\xi_{R}\leq b_{R}$ and $b_{R}
^{2}-\xi_{R}^{2}=\eta_{R}^{2}=h_{R}(\xi_{R})^{2}$ we obtain
\begin{equation}
\left\vert b_{R}-\xi_{R}\right\vert =\frac{h_{R}(\xi_{R})^{2}}{\left\vert
b_{R}+\xi_{R}\right\vert }\leq\frac{C\xi_{R}}{2R^{2}}\leq C\frac{b_{R}}{R^{2}
}. \label{eq:73}
\end{equation}
Next, we express $\mathbb{R}^{N}$ as the union of the sets
\begin{equation}
\begin{array}
[c]{l}
D_{R}^{1}:=\mathbb{R}^{N}\smallsetminus Q_{R},\smallskip\\
D_{R}^{2}:=Q_{R}\cap\left[ \left( \Omega_{R}\cup\mathbb{L}_{R}\right)
\smallsetminus(\widetilde{\Omega}_{R}\cap\mathbb{L})\right] ,\smallskip\\
D_{R}^{3}:=Q_{R}\cap\left[ (\Omega_{R}\cap\left( \mathbb{L}\smallsetminus
\mathbb{L}_{R}\right) )\cup((\widetilde{\Omega}_{R}\smallsetminus\Omega
_{R})\cap\mathbb{L})\right] ,\smallskip\\
D_{R}^{4}:=Q_{R}\cap\mathbb{L}_{R}\cap\Omega_{R},
\end{array}
\label{partition}
\end{equation}
and we show that the estimate \eqref{eq:36} holds true for the integral over
each one of these sets. Note that $D_{R}^{2}\subset Q_{R}\cap\lbrack
(\widetilde{\Omega}_{R}\cup\mathbb{L})\smallsetminus(\widetilde{\Omega}
_{R}\cap\mathbb{L})].$ Thus, the arguments for $D_{R}^{1}$ and $D_{R}^{2}$ are
the same as those\ given to prove Claims 1-3 in Lemma \ref{lem:u-v-est}. To
prove estimate \eqref{eq:36} over $D_{R}^{3},$ first observe that the angle
$\alpha_{R}$ between $\{b_{R}\}\times\mathbb{R}^{N-1}$ and the end of
$\Omega_{R}$ which contains $R\gamma(1)$ is the same as the angle between the
tangent space to $\Gamma_{R}$ at $x_{R},$ which we have identified with
$\mathbb{R}\times\{0\},$ and the tangent space to $\widetilde{\Gamma}_{R}$ at
$R\gamma(1).$ Therefore, using \eqref{eq:55} we obtain that
\begin{equation}
\tan\alpha_{R}=\left\vert h_{R}^{\prime}(\xi_{R})\right\vert \leq C\frac
{b_{R}}{R}. \label{eq:81}
\end{equation}
Since diam$(B_{1}^{N-1})=2$ it follows that
\begin{align}
D_{R}^{3} & \subset[\xi_{R}-2\tan\alpha_{R},\,b_{R}+2\tan\alpha_{R}]\times
B_{1}^{N-1}\label{eq:79}\\
& \subset[b_{R}-s_{R},b_{R}+s_{R}]\times B_{1}^{N-1},\nonumber
\end{align}
where $s_{R}\geq0$ satisfies
\begin{equation}
s_{R}\leq C\left( \frac{b_{R}}{R^{2}}+\frac{b_{R}}{R}\right) \leq
C\frac{b_{R}}{R}. \label{eq:80}
\end{equation}
Here we have used \eqref{eq:73} and \eqref{eq:81}. Therefore, using Lemma
\ref{lem:decay-V}\ we conclude that
\begin{align}
\int_{D_{R}^{3}}\lvert V_{x,s,R}^{\pm}-W_{x,R}^{\pm}\rvert^{p}\,\mathrm{d}y
& \leq C\int_{b_{R}-s_{R}}^{b_{R}+s_{R}}\mathrm{e}^{-pC_{4}\xi}
\,\mathrm{d}\xi\label{eq:83}\\
& =C\mathrm{e}^{-pC_{4}b_{R}}\sinh(pC_{4}s_{R})\nonumber\\
& \leq C\mathrm{e}^{-pC_{4}b_{R}}\frac{b_{R}}{R}=O(R^{-1}).\nonumber
\end{align}
for $R$ large enough. To prove estimate \eqref{eq:36} over $D_{R}^{4},$ we
start by estimating $\lvert V_{x,s,R}^{\pm}-W_{x,R}^{\pm}\rvert$ on $\partial
D_{R}^{4}.$ If $(\xi,\eta)\in\partial D_{R}^{4}\cap\partial\mathbb{L}$ then,
as in \eqref{eq:85}, we have that
\begin{equation*}
\text{dist}((\xi,\eta),\partial\widetilde{\Omega}_{R})\leq C\frac{1+\xi^{2}
}{R}.
\end{equation*}
Similarly, if $(\xi,\eta)\in\partial D_{R}^{4}\cap\partial\widetilde{\Omega
}_{R}$ then
\begin{equation*}
\text{dist}((\xi,\eta),\partial\mathbb{L})\leq C\frac{1+\xi^{2}}{R}.
\end{equation*}
Moreover, if $(\xi,\eta)\in\partial D_{R}^{4}\cap\partial\Omega_{R}
\cap\widetilde{\Omega}_{R}$ then
\begin{equation*}
\text{dist}((\xi,\eta),\partial\mathbb{L}_{R}\cap\mathbb{L})\leq2s_{R}\leq
C\frac{b_{R}}{R}\leq C\frac{\xi+s_{R}}{R}\leq C\left( \frac{\xi}{R}
+\frac{b_{R}}{R^{2}}\right) \leq C\frac{1+\xi^{2}}{R}.
\end{equation*}
Similarly, if $(\xi,\eta)\in\partial D_{R}^{4}\cap\partial\mathbb{L}_{R}
\cap\mathbb{L}$ then
\begin{equation*}
\text{dist}((\xi,\eta),\partial\Omega_{R}\cap\widetilde{\Omega}_{R})\leq
C\frac{1+\xi^{2}}{R}.
\end{equation*}
Since $V_{x,s,R}^{\pm}=0$ in $\mathbb{R}^{N}\smallsetminus\Omega_{R}$ and
$W_{x,R}^{\pm}=0$ in $\mathbb{R}^{N}\smallsetminus\mathbb{L}_{R},$ arguing as
in the proof of Claim 4 of Lemma \ref{lem:u-v-est}, we conclude that
\begin{equation}
\lvert V_{x,s,R}^{\pm}-W_{x,R}^{\pm}\rvert\leq C\mathrm{e}^{-C_{4}\left\vert
\xi\right\vert }\frac{1+\xi^{2}}{R}\qquad\text{on $\partial$}D_{R}^{4}\text{,}
\label{eq:84}
\end{equation}
and that
\begin{equation}
\int_{D_{R}^{4}}\lvert V_{x,s,R}^{\pm}-W_{x,R}^{\pm}\rvert^{p}\,\mathrm{d}
y=O(R^{-p}). \label{eq:82}
\end{equation}
This finishes the proof of \eqref{eq:36}.
The proof of \eqref{eq:37} is analogous to that of \eqref{eq:12}, using the
partition \eqref{partition}. Equations \eqref{eq:38} and \eqref{eq:39} follow
from \eqref{eq:36} as in the proof of Lemma~\ref{lem:u-v-est}.
\end{proof}
Again, we consider functions $g_{m}\colon\mathbb{R}^{+}\rightarrow
\mathbb{R}^{+}$ (to be fixed later) satisfying \eqref{eq:24}-\eqref{eq:25},
but this time we define $D_{m,R}$ as the set of points $(x_{1},x_{2}
,\dots,x_{n})$ in $(\Gamma_{R})^{n}$ such that either there exist
$i,j\in\{1,2,\dots,n\}$ with $i\neq j$ and $\lvert x_{i}-x_{j}\rvert\leq
g_{m}(R)$, or there exists $i\in\{1,2,\dots,n\}$ with $2$dist$(x_{i}
,\partial\Gamma_{R})\leq g_{m}(R).$ Then we define
\begin{equation}
\mathcal{U}_{m,R}:=\{(x_{1},x_{2},\dots,x_{n})\in(\Gamma_{R})^{n}
\smallsetminus D_{m,R}\mid(x_{1},x_{2},\dots,x_{n})\text{ is an $n $-chain}\}.
\label{Uopen}
\end{equation}
\begin{lemma}
\label{lem:nearness-w-v-open}The estimates
\begin{align}
\sup_{x\in\Gamma_{R}}\left\Vert V_{x,R}^{\pm}-W_{x,R}^{\pm}\right\Vert
_{H_{0}^{1}(\mathbb{R}^{N})} & =O(R^{-1/2}),\label{eq:28}\\
\sup_{x\in\Gamma_{R}}\left\vert J_{\Omega_{R}}(V_{x,R}^{\pm})-J_{\mathbb{L}
}(W_{x,R}^{\pm})\right\vert & =O(R^{-1}),\label{eq:45}\\
\sup_{\substack{x\in\Gamma_{R} \\\text{\emph{dist}}(x,\partial\Gamma_{R})\geq
g_{2}(R)/2}}\left\Vert \nabla J_{\Omega_{R}}(V_{x,R}^{\pm})\right\Vert
_{H_{0}^{1}(\Omega_{R})} & =O(R^{-1/2})+O(\mathrm{e}^{-\min\{p_{1},2\}\mu
g_{2}(R)/2}) \label{eq:46}
\end{align}
hold true as $R\rightarrow\infty$.
\end{lemma}
\begin{proof}
Estimates \eqref{eq:28} and \eqref{eq:45} follow immediately from
Lemma~\ref{lem:w-v-est-open}. To prove \eqref{eq:46} we first observe that
$|tU^{\pm}+(1-t)\widetilde{U}_{a,b}^{\pm}|\leq\left\vert U^{\pm}\right\vert $
for every $t\in\lbrack0,1]$ by \eqref{eq:108}. Moreover, (H3) implies that
$\left\vert f^{\prime}(u)\right\vert \leq C\left\vert u\right\vert ^{p_{1}-1}
$ for some constant $C$ which depends only on an upper bound for $\left\vert
u\right\vert $. Therefore Lemma~\ref{decayU} and inequality \eqref{eq:43}
imply
\begin{align*}
\int_{\mathbb{L}}|f(U^{\pm})-f(\widetilde{U}_{a,b}^{\pm})|^{2}\,\mathrm{d}x
& \leq\int_{\mathbb{L}}\left( \int_{0}^{1}|f^{\prime}(tU^{\pm}
)+(1-t)\widetilde{U}_{a,b}^{\pm}|\,\mathrm{d}t\right) ^{2}|U^{\pm}
-\widetilde{U}_{a,b}^{\pm}|^{2}\,\mathrm{d}x\\
& \leq C\int_{\mathbb{R}}\mathrm{e}^{-2(p_{1}-1)\mu}\mathrm{e}^{-2\mu
(a+\lvert\xi+a\rvert)}+\mathrm{e}^{-2\mu(b+\lvert\xi-b\rvert)}\,\mathrm{d}
\xi\\
& =O(\mathrm{e}^{-2\min\{p_{1},2\}\mu\min\{a,b\}}),
\end{align*}
as $a,b\rightarrow\infty$. Therefore,
\begin{align*}
\left\Vert f(U_{x,R}^{\pm})-f(V_{x,R}^{\pm})\right\Vert _{L^{2}} &
\leq\left\Vert f(U_{x,R}^{\pm})-f(W_{x,R}^{\pm})\right\Vert _{L^{2}
}+\left\Vert f(W_{x,R}^{\pm})-f(V_{x,R}^{\pm})\right\Vert _{L^{2}}\\
& \leq O(\mathrm{e}^{-\min\{p_{1},2\}\mu g_{2}(R)/2})+O(R^{-1/2}),
\end{align*}
as $R\rightarrow\infty$. Arguing as in the proof of \eqref{eq:18},\ using this
estimate, we obtain \eqref{eq:46}.
\end{proof}
Define $\varphi_{R}\colon\mathcal{U}_{2,R}\rightarrow H_{0}^{1}(\Omega_{R})$
by
\begin{equation}
\varphi_{R}(X):=
{\sum_{i=1}^{k}}
(V_{x_{2i-1},R}^{+}+V_{x_{2i},R}^{-})+(n-2k)V_{x_{n},R}^{+},\qquad
X=(x_{1},x_{2},\dots,x_{n}), \label{phiopen}
\end{equation}
where $k$ is the largest integer smaller than or equal to $\frac{n}{2}.$ This
time we do not require that $n$ is even.
Next we show that the statements of Propositions
\ref{prop:lyapunov-gradient-estimate}--\ref{prop:jr-low-in-interior} are also
true for these new data. We set $\overline{U}_{i}$ and $\overline{V}_{i}$ as
in (\ref{bar}). Similarly, we set
\begin{equation*}
\overline{W}_{i}:=\left\{
\begin{array}
[c]{ll}
W_{x_{i},R}^{+} & \text{if }i\text{ is odd,}\\
W_{x_{i},R}^{-} & \text{if }i\text{ is even.}
\end{array}
\right.
\end{equation*}
\begin{proposition}
\label{prop:lyapunov-gradient-estimate-open}Let $\alpha$ be as in
\emph{Lemma~\ref{lem:splitting-f}} and fix $\alpha^{\prime}\in(1/2,\min
\{\alpha,p_{1}/2,1\})$. Then
\begin{equation*}
\sup_{X\in\mathcal{U}_{2,R}}\lVert\nabla J_{\Omega_{R}}(\varphi_{R}
(X))\rVert_{H_{0}^{1}(\Omega_{R})}=O(\mathrm{e}^{-\alpha^{\prime}\mu g_{2}
(R)})+O(R^{-1/2})
\end{equation*}
as $R\rightarrow\infty$.
\end{proposition}
\begin{proof}
The proof is completely analogous to that of
Proposition~\ref{prop:lyapunov-gradient-estimate}, using this time Lemmas
\ref{lem:nearness-w-v-open}\ and \ref{lem:w-v-est-open}.
\end{proof}
Set
\begin{equation*}
E_{n}:=k\left[ J_{\mathbb{L}}(U^{+})+J_{\mathbb{L}}(U^{-})\right]
+(n-2k)J_{\mathbb{L}}(U^{+}).
\end{equation*}
\begin{proposition}
\label{prop:jr-high-on-boundary-open}There exists $\beta>0$ such that
\begin{equation*}
\inf_{X\in\partial\mathcal{U}_{1,R}}J_{\Omega_{R}}(\varphi_{R}(X))\geq
E_{n}+\beta\mathrm{e}^{-\mu g_{1}(R)}+o(\mathrm{e}^{-\mu g_{1}(R)}
)+O(R^{-2/3})
\end{equation*}
as $R\rightarrow\infty$.
\end{proposition}
\begin{proof}
Let $X=(x_{1},x_{2},\dots,x_{n})\in\partial\mathcal{U}_{1,R}.$ The proof is
similar to that of Proposition \ref{prop:jr-high-on-boundary} except that now
we must replace $\overline{U}_{i}$ by $\overline{W}_{i}.$ So, in order to
arrive to the conclusion, we need the following estimates:
\begin{align}
J_{x_{i}+A_{x_{i}}^{-1}\mathbb{L}}(\overline{W}_{i}) & \geq J_{x_{i}
+A_{x_{i}}^{-1}\mathbb{L}}(\overline{U}_{i})+C\mathrm{e}^{-2\mu\text{dist}
(x_{i},\partial\Gamma_{R})},\label{ener}\\
\int_{\mathbb{R}^{N}}f(\overline{W}_{i})\overline{W}_{j} & =\int
_{\mathbb{R}^{N}}f(\overline{U}_{i})\overline{U}_{j}+o(\mathrm{e}^{-\mu
g_{1}(R)}). \label{inter}
\end{align}
Let us prove the first one. After an appropriate change of coordinates
$\overline{U}_{i}$ becomes $U^{\pm}$\ and $\overline{W}_{i}$ becomes
$\widetilde{U}_{a,b}^{\pm}.$ Recall that $J_{\mathbb{L}}^{\prime}(U^{\pm}
)=0$\ and observe that $\left\vert f^{\prime}(U^{\pm}(\xi,\eta))\right\vert
\leq C\mathrm{e}^{-(p_{1}-1)\mu\left\vert \xi\right\vert }$ due to condition
(H3) and Lemma~\ref{decayU}. So using Lemma \ref{lem:tu-u-est-open}\ we obtain
\begin{align*}
J_{\mathbb{L}}(\widetilde{U}_{a,b}^{\pm}) & =J_{\mathbb{L}}(U^{\pm}
)+\frac{1}{2}J_{\mathbb{L}}^{\prime\prime}(U^{\pm})[\widetilde{U}_{a,b}^{\pm
}-U^{\pm},\widetilde{U}_{a,b}^{\pm}-U^{\pm}]+o(\lVert U^{\pm}-\widetilde
{U}_{a,b}^{\pm}\rVert_{H_{0}^{1}(\mathbb{L})}^{2})\\
& \geq J_{\mathbb{L}}(U^{\pm})+\frac{1}{2}\lVert U^{\pm}-\widetilde{U}
_{a,b}^{\pm}\rVert_{H_{0}^{1}(\mathbb{L})}^{2}+o(\lVert U^{\pm}-\widetilde
{U}_{a,b}^{\pm}\rVert_{H_{0}^{1}(\mathbb{L})}^{2})\\
& \geq J_{\mathbb{L}}(U^{\pm})+C\mathrm{e}^{-2\mu\min\{a,b\}}
\end{align*}
for $R$ large enough. This proves (\ref{ener}).
To prove the second estimate it suffices to show that
\begin{align}
\int_{\mathbb{R}^{N}}({f(\overline{W}_{i})-f(\overline{U}_{i})}\overline
{W}_{j} & =o(\mathrm{e}^{-\mu g_{1}(R)})\label{eq:112}\\
\int_{\mathbb{R}^{N}}f(\overline{U}_{i})(\overline{W}_{j}-\overline{U}_{j})
& =o(\mathrm{e}^{-\mu g_{1}(R)}) \label{eq:113}
\end{align}
as $R\rightarrow\infty$. Since the proof of both estimates is similar, we only
prove \eqref{eq:112}. After a change of coordinates we may assume that
$x_{i}=0$ and that the tangent space to $\Gamma_{R}$ at $x_{i}$ is
$\mathbb{R}\times\{0\}$. Then we set $a:=\left\vert R\gamma(0)\right\vert $
and $b:=\left\vert R\gamma(1)\right\vert .$ We may assume without loss of
generality that $a\leq b.$ Since $\left\vert \overline{W}_{j}(x)\right\vert
\leq C\mathrm{e}^{-\mu\left\vert x-x_{j}\right\vert }$ by \eqref{eq:108} and
Lemma~\ref{decayU}, the proof of \eqref{eq:112}\ reduces to showing that
\begin{equation}
\int_{\mathbb{L}}\left\vert f(U^{\pm}(x))-f(\widetilde{U}_{a,b}^{\pm
}(x))\right\vert \mathrm{e}^{-\mu\left\vert x-x_{j}\right\vert }
\mathrm{d}x=o(\mathrm{e}^{-\mu g_{1}(R)}) \label{eq:115}
\end{equation}
as $R\rightarrow\infty$. We distinguish two cases: If $\left\vert
x_{j}\right\vert \geq2g_{1}(R),$ using condition (H3),
Lemma~\ref{lem:interaction-exponential} and \eqref{eq:10} we obtain
\begin{align*}
\int_{\mathbb{L}}\left\vert f(U^{\pm}(x))-f(\widetilde{U}_{a,b}^{\pm
}(x))\right\vert \mathrm{e}^{-\mu\left\vert x-x_{j}\right\vert }\mathrm{d}x
& \leq C\int_{\mathbb{L}}\mathrm{e}^{-p_{1}\mu\left\vert x\right\vert
}\mathrm{e}^{-\mu\left\vert x-x_{j}\right\vert }\mathrm{d}x\\
& \leq C\mathrm{e}^{-\mu\left\vert x_{j}\right\vert }\leq C\mathrm{e}^{-2\mu
g_{1}(R)}=o(\mathrm{e}^{-\mu g_{1}(R)})
\end{align*}
as $R\rightarrow\infty$. On the other hand, if $\left\vert x_{j}\right\vert
\leq2g_{1}(R)$ we write $x_{j}=(\xi_{j},\eta_{j})$ and use the Lipschitz
continuity of $f$ on bounded sets and \eqref{eq:43} to obtain
\begin{align*}
& \int_{\mathbb{L}}\left\vert f(U^{\pm}(x))-f(\widetilde{U}_{a,b}^{\pm
}(x))\right\vert \mathrm{e}^{-\mu\left\vert x-x_{j}\right\vert }\mathrm{d}x\\
& \leq C\int_{\mathbb{R}}\left( \mathrm{e}^{-\mu(a+\lvert\xi+a\rvert
)}+\mathrm{e}^{-\mu(b+\lvert\xi-b\rvert)}\right) \mathrm{e}^{-\mu\left\vert
\xi-\xi_{j}\right\vert }\mathrm{d}\xi\\
& \leq C\left( \mathrm{e}^{-\mu(a+\lvert\xi_{j}+a\rvert)}+\mathrm{e}
^{-\mu(\lvert\xi_{j}-b\rvert)}\right) \\
& =C\mathrm{e}^{-\mu(a+\lvert\xi_{j}+a\rvert)}+o(\mathrm{e}^{-\mu g_{1}(R)})
\end{align*}
The last equality follows from $\left\vert x_{j}\right\vert \leq2g_{1}(R),$
$b:=\left\vert R\gamma(1)\right\vert $ and \eqref{eq:25}. Now, if $j>i$ we
have that $a\geq\frac{3}{2}g_{1}(R)(1+o(1))$, and if $j<i$ we have that
$\xi_{j}+a\geq\frac{3}{2}g_{1}(R)(1+o(1))$ as $R\rightarrow\infty$. So in both
cases $\mathrm{e}^{-\mu(a+\lvert\xi_{j}+a\rvert)}=o(\mathrm{e}^{-\mu g_{1}
(R)})$. This proves \eqref{eq:115} and, hence, \eqref{eq:112}.
Now we may argue as in Proposition \ref{prop:jr-high-on-boundary}. The
analogue of \eqref{eq:107} with ${\overline{U}_{i}}$ replaced by
${\overline{W}_{i}}$ is obtained in a similar way. Therefore, using estimates
\eqref{ener}, \eqref{inter}\ and \eqref{d>2} we conclude that
\begin{align*}
J_{\Omega_{R}}(\varphi_{R}(X))= &
{\sum_{i=1}^{n}}
J_{\Omega_{R}}(\overline{V}_{i})+\frac{1}{2}
{\sum_{i\neq j}}
\int_{\Omega_{R}}f(\overline{U}_{i})\overline{V}_{j}\,\mathrm{d}x-
{\sum_{i\neq j}}
\int_{\Omega_{R}}f(\overline{V}_{i})\overline{V}_{j}\,\mathrm{d}x\\
& +o(\mathrm{e}^{-\mu g_{1}(R)})+O(R^{-2/3})\\
= &
{\sum_{i=1}^{n}}
J_{x_{i}+A_{x_{i}}^{-1}\mathbb{L}}(\overline{W}_{i})+\frac{1}{2}
{\sum_{i\neq j}}
\int_{\Omega_{R}}f(\overline{U}_{i})\overline{W}_{j}\,\mathrm{d}x-
{\sum_{i\neq j}}
\int_{\Omega_{R}}f(\overline{W}_{i})\overline{W}_{j}\,\mathrm{d}x\\
& +o(\mathrm{e}^{-\mu g_{1}(R)})+O(R^{-2/3})\\
\geq & E_{n}+C
{\sum_{i=1}^{n}}
\mathrm{e}^{-2\mu\text{dist}(x_{i},\partial\Gamma_{R})}+\frac{1}{2}
{\sum_{\left\vert i-j\right\vert =1}}
\int_{\Omega_{R}}\left\vert f(\overline{U}_{i})\overline{U}_{j}\right\vert
\,\mathrm{d}x\\
& +o(\mathrm{e}^{-\mu g_{1}(R)})+O(R^{-2/3}).
\end{align*}
Since $X\in\partial\mathcal{U}_{1,R},$ either dist$(x_{1},\partial\Gamma
_{R})=g_{1}(R)/2$ or dist$(x_{n},\partial\Gamma_{R})=g_{1}(R)/2$ or
$\left\vert x_{i+1}-x_{i}\right\vert =g_{1}(R)$ for some $i=1,...,n-1.$ In any
case, our claim follows.
\end{proof}
\begin{proposition}
\label{prop:jr-low-in-interior-open}The estimate
\begin{equation*}
\inf_{X\in\mathcal{U}_{1,R}}J_{\Omega_{R}}(\varphi_{R}(X))\leq E_{n}
+o(\mathrm{e}^{-\mu g_{1}(R)})+O(R^{-2/3})
\end{equation*}
holds true as $R\rightarrow\infty$.
\end{proposition}
\begin{proof}
The proof is similar to that of Proposition \ref{prop:jr-low-in-interior},
this time taking into account that dist$(x_{i},\partial\Gamma_{R})\geq CR$ for
some $C>0$ and every $R$ and $i.$
\end{proof}
\section{Proof of the Main Results}
\label{sec:finite-dimens-reduct}
\subsection{The Finite Dimensional Reduction}
\label{sec:finite-dimens-reduct-1}
Let $\mathcal{U}_{2,R}$ and $\varphi_{R}:\mathcal{U}_{2,R}\rightarrow
H_{0}^{1}(\Omega_{R})$ be as in (\ref{Uclosed}) and (\ref{immersion}) when
$\Gamma$ is a closed curve and as in (\ref{Uopen}) and (\ref{phiopen}) if
$\gamma(0)\neq\gamma(1).$ Set
\begin{equation*}
\Sigma_{R}:=\varphi_{R}(\mathcal{U}_{2,R}).
\end{equation*}
\begin{lemma}
\label{lem:phi-immersion} $\Sigma_{R}$ is a finite dimensional $C^{2}
$-submanifold of $H_{0}^{1}(\Omega_{R})$.
\end{lemma}
\begin{proof}
It is easy to see that the map $\varphi_{R}$ is a $C^{2}$-immersion. If
$\partial\Gamma\neq\emptyset$ or $n\leq2$ then $\varphi_{R}$ is injective, and
hence $\Sigma_{R}$ is a submanifold of $H_{0}^{1}(\Omega_{R})$. On the other
hand, if $\partial\Gamma=\emptyset$ and $n\geq4$ then $\varphi_{R}$ is not
injective: two points in $\mathcal{U}_{2,R}$ have the same image under
$\varphi_{R}$ if and only if one of them is obtained from the other after a
finite number of shifts of the form $x_{i}\mapsto x_{i+2}$. Since the group of
permutations acts freely on $\mathcal{U}_{2,R},$ $\Sigma_{R}$ is a submanifold
of $H_{0}^{1}(\Omega_{R})$ also in this case.
\end{proof}
We shall reduce the problem of finding a critical point of $J_{\Omega_{R}}$ to
that of finding a critical point of a function $G_{R}:\Sigma_{R}
\rightarrow\mathbb{R}$, which will be defined below.
For $u\in\Sigma_{R}$ we denote by $T_{u}\Sigma_{R}$ the tangent space to
$\Sigma_{R}$ at $u$, by $T_{u}^{\perp}\Sigma_{R}$ its orthogonal complement in
$H_{0}^{1}(\Omega_{R})$ and by $P_{u,R}^{\perp}:H_{0}^{1}(\Omega
_{R})\rightarrow T_{u}^{\perp}\Sigma_{R}$ the orthogonal projection. We
consider $\mathrm{D}^{2}J_{\Omega_{R}}(u)$ as the derivative of the gradient
vector field $\nabla J_{\Omega_{R}}:H_{0}^{1}(\Omega_{R})\rightarrow H_{0}
^{1}(\Omega_{R})$ at $u,$ and define
\begin{equation*}
L_{u,R}:=P_{u,R}^{\perp}\mathrm{D}^{2}J_{\Omega_{R}}(u)|_{T_{u}^{\perp}
\Sigma_{R}}:T_{u}^{\perp}\Sigma_{R}\rightarrow T_{u}^{\perp}\Sigma_{R}.
\end{equation*}
We write $\mathcal{L}(T_{u}^{\perp}\Sigma_{R})$ for the space of bounded
linear operators from $T_{u}^{\perp}\Sigma_{R}$ into itself.
\begin{lemma}
\label{prop:lyapunov-inverse-estimate} If $R$ is large enough and $u\in
\Sigma_{R}$, then $L_{u,R}$ is invertible in $\mathcal{L}(T_{u}^{\perp}
\Sigma_{R})$ and
\begin{equation*}
\limsup_{R\rightarrow\infty}\sup_{u\in\Sigma_{R}}\left\Vert L_{u,R}
^{-1}\right\Vert _{\mathcal{L}(T_{u}^{\perp}\Sigma_{R})}<\infty.
\end{equation*}
\end{lemma}
\begin{proof}
The proof of this fact is standard, see for example Lemma~3.8(v) in \cite{MR2216902}.
\end{proof}
\begin{lemma}
\label{lem:existence-reduction} There exist $r_{0}>0$ and $R_{1}\geq1$ such
that for $R\geq R_{1}$ and for every $u\in\Sigma_{R}$ there is a unique
$v_{u}\in u+T_{u}^{\perp}\Sigma_{R}$ which satisfies $\lVert u-v_{u}
\rVert_{H_{0}^{1}(\Omega_{R})}<r_{0}$ and $P_{u,R}^{\perp}\nabla J_{\Omega
_{R}}(v_{u})=0$. The estimates
\begin{equation}
\lVert u-v_{u}\rVert_{H_{0}^{1}(\Omega_{R})}=O(\lVert\nabla J_{\Omega_{R}
}(u)\rVert_{H_{0}^{1}(\Omega_{R})}) \label{eq:30}
\end{equation}
and
\begin{equation}
\lvert J_{\Omega_{R}}(u)-J_{\Omega_{R}}(v_{u})\rvert=O(\lVert\nabla
J_{\Omega_{R}}(u)\rVert_{H_{0}^{1}(\Omega_{R})}^{2}) \label{eq:31}
\end{equation}
hold true as $R\rightarrow\infty$, independently of $u\in\Sigma_{R}$.
Moreover, the operator
\begin{equation*}
P_{u,R}^{\perp}\mathrm{D}^{2}J_{\Omega_{R}}(v_{u})|_{T_{u}^{\perp}\Sigma_{R}
}:T_{u}^{\perp}\Sigma_{R}\rightarrow T_{u}^{\perp}\Sigma_{R}
\end{equation*}
is invertible in $\mathcal{L}(T_{u}^{\perp}\Sigma_{R})$.
\end{lemma}
\begin{proof}
Along this proof $B_{r}Z$ will denote the open ball of radius $r$ centered at
$0$ in a normed space $Z$, and $\overline{B}_{r}Z$ will denote its closure.
By Lemma~\ref{prop:lyapunov-inverse-estimate} we may fix $M\geq1$ satisfying
\begin{equation}
M>\limsup_{R\rightarrow\infty}\sup_{u\in\Sigma_{R}}\left\Vert L_{u,R}
^{-1}\right\Vert _{\mathcal{L}(T_{u}^{\perp}\Sigma_{R})}. \label{eq:119}
\end{equation}
Clearly,
\begin{equation*}
C_{0}:=\limsup_{R\rightarrow\infty}\sup_{u\in\Sigma_{R}}\left\Vert
u\right\Vert _{H_{0}^{1}(\Omega_{R})}<\infty.
\end{equation*}
Condition~(H3) yields
\begin{equation}
\limsup_{R\rightarrow\infty}\left\Vert J_{\Omega_{R}}\right\Vert
_{C^{2,\bar{\alpha}}(B_{2C_{0}}H_{0}^{1}(\Omega_{R}))}<\infty\label{eq:8}
\end{equation}
for some $\bar{\alpha}\in(0,1]$.
By Lemma~\ref{prop:lyapunov-inverse-estimate} and \eqref{eq:8} there is
$r_{0}>0$ such that for $R$ large enough
\begin{equation}
\left\Vert \mathrm{D}^{2}J_{\Omega_{R}}(u)-\mathrm{D}^{2}J_{\Omega_{R}
}(v)\right\Vert _{\mathcal{L}(H_{0}^{1}(\Omega_{R}))}\leq\frac{1}{2M}
\label{eq:32}
\end{equation}
and $P_{u,R}^{\perp}\mathrm{D}^{2}J_{\Omega_{R}}(v)|_{T_{u}^{\perp}\Sigma_{R}
}$ is invertible in $\mathcal{L}(T_{u}^{\perp}\Sigma_{R}),$ for every
$u\in\Sigma_{R}$ and $v\in H_{0}^{1}(\Omega_{R})$ with $\left\Vert
u-v\right\Vert _{H_{0}^{1}(\Omega_{R})}\leq r_{0}$. Moreover, for $R$ large
enough,
\begin{equation}
\sup_{u\in\Sigma_{R}}\lVert\nabla J_{\Omega_{R}}(u)\rVert_{H_{0}^{1}
(\Omega_{R})}\leq\frac{r_{0}}{2M}. \label{eq:7}
\end{equation}
because of Propositions \ref{prop:lyapunov-gradient-estimate} and
\ref{prop:lyapunov-gradient-estimate-open}. Fix $u\in\Sigma_{R}$ and define
$g\colon T_{u}^{\perp}\Sigma_{R}\rightarrow T_{u}^{\perp}\Sigma_{R}$ by
\begin{equation*}
g(w):=w-L_{u,R}^{-1}P_{u,R}^{\perp}\nabla J_{\Omega_{R}}(u+w).
\end{equation*}
If $w\in\overline{B}_{r_{0}}T_{u}^{\perp}\Sigma_{R}$ it follows from
\eqref{eq:32} and \eqref{eq:7} that
\begin{align}
\left\Vert g(w)\right\Vert & \leq M\left\Vert \mathrm{D}^{2}J_{\Omega_{R}
}(u)w-\nabla J_{\Omega_{R}}(u+w)\right\Vert \nonumber\\
& =M\left\Vert -\nabla J_{\Omega_{R}}(u)-\int_{0}^{1}(\mathrm{D}^{2}
J_{\Omega_{R}}(u+tw)-\mathrm{D}^{2}J_{\Omega_{R}}(u))w\text{ \textrm{d}
}t\right\Vert \label{eq:6}\\
& \leq M\left( \left\Vert \nabla J_{\Omega_{R}}(u)\right\Vert +\frac
{\left\Vert w\right\Vert }{2M}\right) \leq r_{0}.\nonumber
\end{align}
Hence, $g$ maps $\overline{B}_{r_{0}}T_{u}^{\perp}\Sigma_{R}$ into itself.
Moreover, by \eqref{eq:119} and \eqref{eq:32} we have
\begin{equation*}
\left\Vert g^{\prime}(w)\right\Vert \leq\left\Vert L_{u,R}^{-1}\right\Vert
\left\Vert \mathrm{D}^{2}J_{\Omega_{R}}(u)-\mathrm{D}^{2}J_{\Omega_{R}
}(u+w)\right\Vert \leq\frac{1}{2}.
\end{equation*}
Therefore, $g$ is a contraction on $\overline{B}_{r_{0}}T_{u}^{\perp}
\Sigma_{R}$ and by Banach's fixed point theorem $g$ has a unique fixed point
$w_{u}\in\overline{B}_{r_{0}}T_{u}^{\perp}\Sigma_{R}$. Thus, $v_{u}:=u+w_{u}$
is then the only zero of $P_{u,R}^{\perp}\nabla J_{\Omega_{R}}$ in
$u+\overline{B}_{r_{0}}T_{u}^{\perp}\Sigma_{R}.$
Inequality \eqref{eq:6} with $w:=w_{u}=g(w_{u})$ yields $\left\Vert
w_{u}\right\Vert \leq2M\left\Vert \nabla J_{\Omega_{R}}(u)\right\Vert $ and
hence \eqref{eq:30}. Moreover, since $\mathrm{D}J_{\Omega_{R}}(v_{u}
)[w_{u}]=0,$
\begin{align}
& \lvert J_{\Omega_{R}}(u)-J_{\Omega_{R}}(v_{u})\rvert\label{eq:34}\\
& \leq\left\vert \mathrm{D}J_{\Omega_{R}}(v_{u})[w_{u}]\right\vert +\int
_{0}^{t}(1-t)\left\vert \mathrm{D}^{2}J_{\Omega_{R}}(u+(1-t)w_{u})\left[
w_{u},w_{u}\right] \right\vert \text{ \textrm{d}}t\nonumber\\
& \leq C\left\Vert w_{u}\right\Vert ^{2}\nonumber
\end{align}
for some constant $C$ independent of $R$ and $u$. Now \eqref{eq:30} and
\eqref{eq:34} imply \eqref{eq:31}. Finally, if $R$ is large enough,
\eqref{eq:30} implies the strict inequality $\lVert u-v_{u}\rVert_{H_{0}
^{1}(\Omega_{R})}<r_{0}$, as stated in the lemma.
\end{proof}
We now fix $r_{0}$ and $R_{1}$ as in Lemma~\ref{lem:existence-reduction}. If
$R\geq R_{1}$ we define $G_{R}\colon\Sigma_{R}\rightarrow\mathbb{R}$ by
\begin{equation*}
G_{R}(u):=J_{\Omega_{R}}(v_{u}).
\end{equation*}
where $v_{u}$ is given by Lemma~\ref{lem:existence-reduction}.
\begin{proposition}
\label{lem:reduced-problem} For $R\geq R_{1}$ the map $G_{R}$ is in
$C^{1}(\Sigma_{R},\mathbb{R})$. If $u\in\Sigma_{R}$ is a critical point of
$G_{R}$ then $v_{u}$ is a critical point of $J_{\Omega_{R}}$.
\end{proposition}
\begin{proof}
The map $u\mapsto v_{u}$ is a cross section of the normal disc bundle of
radius $r_{0}$ over $\Sigma_{R},$ so its image $\tilde{\Sigma}_{R}
:=\{v_{u}:u\in\Sigma_{R}\}$ is a submanifold which is transversal to the
fibres, that is, $H_{0}^{1}(\Omega)=T_{v_{u}}\tilde{\Sigma}_{R}\oplus
T_{u}^{\perp}\Sigma_{R}.$ The map $\psi_{R}:\Sigma_{R}\rightarrow\tilde
{\Sigma}_{R}$ given by $\psi_{R}(u):=v_{u}$ is a $C^{1}$-diffeomorphism.
Therefore $G_{R}$ is of class $C^{1}$ and, since $\mathrm{D}G_{R}
(u)=\mathrm{D}J_{\Omega_{R}}(v_{u})\circ\mathrm{D}\psi_{R}(u),$ we have that
$\mathrm{D}J_{\Omega_{R}}(v_{u})w=0$ for every $w\in T_{v_{u}}\tilde{\Sigma
}_{R}$ if $u$ is a critical point of $G_{R}.$ But $v_{u}$ was chosen so that
$\mathrm{D}J_{\Omega_{R}}(v_{u})z=0$ for every $z\in T_{u}^{\perp}\Sigma_{R}.$
Hence, $v_{u}$ is a critical point of $J_{\Omega_{R}}$ if $u$ is a critical
point of $G_{R}.$
\end{proof}
\subsection{The proof of Theorems \ref{thm:no-boundary} and
\ref{thm:with-boundary}}
\label{sec:find-locat-crit}
From Propositions \ref{prop:lyapunov-gradient-estimate},
\ref{prop:jr-high-on-boundary} and \ref{prop:jr-low-in-interior} if
$\gamma(0)=\gamma(1)$, or from Propositions
\ref{prop:lyapunov-gradient-estimate-open},
\ref{prop:jr-high-on-boundary-open} and \ref{prop:jr-low-in-interior-open} if
$\gamma(0)\neq\gamma(1)$, and estimate \eqref{eq:31}, we obtain
\begin{align*}
\min_{X\in\partial\mathcal{U}_{1,R}}G_{R}(\varphi_{R}(X))\geq & E_{n}
+\beta\mathrm{e}^{-\mu g_{1}(R)}+o(\mathrm{e}^{-\mu g_{1}(R)})\\
& +O(R^{-2/3})+O(\mathrm{e}^{-2\alpha^{\prime}\mu g_{2}(R)})\\
\min_{X\in\overline{\mathcal{U}_{1,R}}}G_{R}(\varphi_{R}(X))\leq &
E_{n}+o(\mathrm{e}^{-\mu g_{1}(R)})+O(R^{-2/3})+O(\mathrm{e}^{-2\alpha
^{\prime}\mu g_{2}(R)}).
\end{align*}
We set
\begin{equation*}
g_{1}(R):=\frac{1}{2\mu}\log R\qquad\text{and}\qquad g_{2}(R):=\left(
\frac{1}{2}+\frac{1}{4\alpha^{\prime}}\right) g_{1}(R).
\end{equation*}
Since $\alpha^{\prime}>1/2,$ these functions satisfy \eqref{eq:24},
\eqref{eq:10}, and \eqref{eq:25}. Note that
\begin{equation*}
R^{-2/3}=o(\mathrm{e}^{-\mu g_{1}(R)})\qquad\text{and}\qquad\mathrm{e}
^{-2\alpha^{\prime}\mu g_{2}(R)}=o(\mathrm{e}^{-\mu g_{1}(R)}).
\end{equation*}
Therefore,
\begin{equation*}
\min G_{R}(\varphi_{R}(\overline{\mathcal{U}_{1,R}}))<\min G_{R}(\varphi
_{R}(\partial\mathcal{U}_{1,R}))
\end{equation*}
if $R$ is large enough. It follows that $G_{R}$ has a local minimum
$w_{R}:=\varphi_{R}(X_{R})$ in $\varphi_{R}(\mathcal{U}_{1,R})\subset
\Sigma_{R}$. Hence, by Lemma~\ref{lem:reduced-problem}, $u_{R}:=v_{w_{R}}$ is
a critical point of $J_{\Omega_{R}}$.
Moreover, by \eqref{eq:30}, we have that $u_{R}=\varphi_{R}(X_{R})+o(1)$ in
$H_{0}^{1}(\Omega_{R})$ as $R\rightarrow\infty$. This, together with estimates
\eqref{eq:16}, \eqref{eq:28} and \eqref{eq:44}, yields \eqref{eq:4} and \eqref{eq:5}.
Finally, \eqref{eq:10} implies that $\lvert x_{R,i}-x_{R,j}\rvert
\rightarrow\infty$ if $i\neq j$ and that dist$(x_{R,i},\partial\Gamma
_{R})\rightarrow\infty$ for all $i$, as $R\rightarrow\infty$. The proofs of
Theorems \ref{thm:no-boundary} and \ref{thm:with-boundary} are complete.
\ \hfill$\square$
\bibliographystyle{amsplain-abbrv}
\bibliography{aclapa-1}
\end{document}
| 193,305
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TITLE: Final Objects of a (Coslice?) Category
QUESTION [2 upvotes]: I'm reading Aluffi's Algebra: Chapter 0 and in example 5.6 / exercise 5.5, he asks what the terminal object/objects (he refers to it as singular in the example but in plural in the exercise) are the in following category. He also remarks that its a supremely uninteresting one.
The category is defined as such:
Given an equivalence relation $\sim$ on set $A$,
Objects of the category are morphisms $A \overset{\varphi}{\longrightarrow} Z$ from the set $A$ to any set $Z$ such that $\alpha \sim \alpha' \Rightarrow \varphi(\alpha) = \varphi(\alpha')$ (I think this is an example of a coslice category).
Morphisms between objects $(\varphi_1,Z_1) \rightarrow (\varphi_2,Z_2)$ are defined as morphisms $\sigma$ such that the following commutes (can't draw a commutative diagram with diagonal arrows in mathjax so imagine $A$ being the bottom of a triangle with the other two vertices being $Z_1$ and $Z_2$) :
$$ A \overset{\varphi_1}{\longrightarrow} Ζ_1 \overset{\sigma}{\longrightarrow} Z_2 \overset{\varphi_2}{\longleftarrow} A $$
So far I've figured that both the morphism $(1_A, A)$ (that is the identity morphism from $A$ to $A$) and the morphisms of type $(const_Α, S)$ (where $S$ is any singleton set and $const_A$ is the constant function from $A$ to that set) appear to be final objects in this category.
The unique morphisms from any $(\varphi, Z)$ to $(1_A , A)$ are the reverse functions $\varphi^{-1}$ while the unique morphism for any from any $(\varphi, Z)$ to $(const_Α, S)$ are the corresponding constant functions from $Z$ to the singleton $S$.
Is this correct? Any help is greatly appreciated!
REPLY [3 votes]: There are some problems with your answer.
First of all, a terminal object is unique up to (unique) isomorphism. Hence you cannot have two non isomorphic terminal objects.
You are right that the singleton set with the constant map is a terminal object. (And every terminal object is a singleton set).
However, $(id_A,A)$ is not a terminal object. In fact, it is not an object at all unless the equivalence relation $\sim$ is the identity. Remember that an object of your category is a map $\varphi:A\rightarrow Z$ such that $\varphi(a)=\varphi(a')$ if $a\sim a'$. The identity satisfies this condition if and only if $\sim$ is the equality.
But even in that case, it is not the terminal object. You said, the unique function form $(\varphi,Z)$ to $(id_A,A)$ is the reverse function $\varphi^{-1}$. But it does not exist if $\varphi$ is not bijective. Can you show that there is no map from $(const,\{*\})$ to $(id_A,A)$ ?
But for this category, this is the initial object that is very interesting. Can you see what it is ? Hint : it is a usual construction given a set equipped with an equivalence relation.
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TITLE: Problem in quantum information theory
QUESTION [1 upvotes]: I am reading the following paper: on quantum information theory
Does anybody understand the estimate at the bottom of page $7$
$$\left\lVert \rho-\rho_n \right\rVert \le 2 \left\lVert (id_A-P_n)\otimes id_R \rho (P_n \otimes id_R)\right\rVert_1+ 2 r_n\ ?$$
I have troubles understanding what happened in this step.
Can anybody explain this one to me?
REPLY [1 votes]: Here is one estimate; it doesn't go exactly as in the paper, but it achieves the same. Since $r_n\to0$, for $n$ big enough one may assume that $1-r_n\geq1/2$. One has
\begin{align}
\|\rho-\rho_n\|_1
&\leq\|\rho-\rho(P_n\otimes I_R)\|_1+\|\rho(P_n\otimes I_R)-(P_n\otimes I_r)\rho(P_n\otimes I_R)\|_1+\\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \|(P_n\otimes I_R)\rho(P_n\otimes I_R)-(1-r_n)^{-1}(P_n\otimes I_r)\rho(P_n\otimes I_R)\|_1\\ \ \\
&=\|\rho[(I_A-P_n)\otimes I_R)]\|_1+\|((I_A-P_n)\otimes I_R)\rho(P_n\otimes I_R)\|_1\\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\|(P_n\otimes I_R)\rho(P_n\otimes I_R)\|_1\,(1-(1-r_n)^{-1})\\ \ \\
&\leq 2\|\rho[(I_A-P_n)\otimes I_R)]\|_1+2r_n.
\end{align}
The estimates used at the end come from the inequality $\|AB\|_1\leq\|A\|\,\|B\|_1$, and the fact that $\|P_n\otimes I_R\|=1$, $\|\rho\|_1=1$. The condition $1-r_n\geq1/2$ gives $(1-(1-r_n)^{-1})\leq 2r_n$.
Finally, if $Q$ is a projection, write $Q\rho=V|Q\rho|$ the polar decomposition, and then (using Cauchy-Schwarz, that $VV^*\leq I$ and that $\rho\geq0$)
$$
\|Q\rho\|_1=\operatorname{Tr}(V^*Q\rho)=\operatorname{Tr}(V^*Q\rho^{1/2}\rho^{1/2})
\leq \operatorname{Tr}(\rho^{1/2}QVV^*Q\rho^{1/2})^{1/2}\,\operatorname{Tr}(\rho)^{1/2}
\leq\operatorname{Tr}(Q\rho)^{1/2}.
$$
So
$$
\|\rho[(I_A-P_n)\otimes I_R)]\|_1\leq\sqrt{\operatorname{Tr}(\rho[(I_A-P_n)\otimes I_R)])}=\sqrt{r_n}.
$$
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Taking some of the ‘wild’ out of the West in saloons of Virginia City
July 14, 2005
VIRGINIA CITY – In Hollywood’s history books, old-time saloons are often peopled exclusively with rowdy sots, shifty bartenders and freelance felons knocking back shots of high-proof whiskey while waiting for the opportunity to kill someone.
While great for building bad-guy narratives and for character-actors seeking work, it just begins to tell the whole story.
In her book “Boomtown Saloons: Archaeology and History in Virginia City,” Dr. Kelly Dixon of the University of Montana challenges the movie-set burlesque to a quick-draw contest, conjuring up visions of fine crystal stemware and brandy snifters in place of shattered shot glasses, trombones instead of player pianos and good quality food instead of whatever you managed to shoot that day.
“There was definitely an element of the so-called ‘Wild West’ in the saloons,” says Dixon. “But there really was a much more complex, hyper-cosmopolitan aspect to these establishments.”
“Not everybody went to the saloon looking to fight.”
Through archaeological excavations of four 19th Century Virginia City saloons, Dixon has come up with what she believes is a more authentic western saloon experience.
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At its peak, Virginia City had a population of 20,000 and a rumored 100 different saloons to quench its thirst.
“These were people coming from all over the world with their customs, their religions, their food, their superstitions,” says Dixon. “They all converged in boomtowns like Virginia City. The American west represented a fusion of everyone on earth.”
For archaeologists like Dixon, Virginia City is the Pompeii of the West.
“There is so much there, buried underground,” she says.
Some of which, like the bones found at the Boston Saloon were the smoking gun, not of random pistol play, but of the fine cuts of meat available at the black-owned business.
In 1866, the Territorial Enterprise reported on the shooting of a white man named “Frenchy” at the Boston.
It was an accident, but the shooting made headlines.
Dixon sees the incident another way. Just one year after the Civil War, here were the signs of an integrated bar.
“There’s no evidence that blacks were allowed in a place like the Piper’s Opera House saloon,” says Dixon. “But it does show a bigger picture – a shared heritage.”
Dr. Dixon will be reading from and signing copies of her book tonight at 6:30 p.m. at Fourth Ward School Museum in Virginia City.
— Contact reporter Peter Thompson at pthompson@nevadaappeal.com or 881-1215.
Booksigning
What: Dr. Kelly Dixon reads from and signs “Boomtown Saloons: Archaeology and History in Virginia City”
Where: Fourth Ward School Museum at the south end of C Street in Virginia City
When: 6:30 tonight
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Turkmens of Iraq: The Third Ethnic Component of Iraq
Turkmens of Iraq: The Third Ethnic Component of Iraq
I. Introduction
Turkmens are the third largest ethnic group in Iraq after Arabs and Kurds. Today, the Iraqi Turkmen population is estimated to be at around 2.5-3 million, taking into account all available estimates of towns and townships that they live in. They may constitute around 10% of the population, however previous sources provided estimates such as 14%-16% of the Iraqi population.[1] Turkmens speak the Turkmen language which is one of the Turkish languages similar to the Azerbaijani language (Azeri Turkish). Turkmens of Iraq have been settled in Iraq before the Ottoman Empire in 1516 A.D. In general terms, 60% Iraqi Turkmens are Sunni whereas 40% of them are Shia Muslims. In addition, there are Christian Turkmens living in Kirkuk (Gala Kawar).
Turkmens in the modern Iraqi history have been undergoing decades of assimilation campaigns in Iraq –often, in a more brutal fashion than the one carried out against Kurds.
II. Origin of Turkmens
Most people link the presence of Turkmens in Iraq to Ottoman Empire. However, Turkmens have been in Iraq much earlier than the Ottomans. There is a research indicating that the Sumerians who moved from central Asia in 5000 B.C. and settled around the river Tigris, were originally ancestors of Turkmens. This argument is based on some similarities found between Sumerian and Turkish language, as there are around 600 words similar to each other in both languages. There have also been similarities such as the shape of their skulls between Mongolian Turks and Sumerians.
The historian Abbas Al Azwai in his book “Iraqi History between Two Occupations”, writes that Turkmens came from Turkmenistan, lived in Khurasan (Iran) and migrated from there to various parts of the world. The Russian historian, Bartold, in his book “History of Turk in Central Asia” claimed that they were three groups of nations: Akaz, Al Karlok, Al Tokaz or Alguz. They lived in the land extending from the Kazar Sea to China border and formed two biggest Empires in the history of Seljuk and Ottoman Empires. They moved to Iraq during different periods of Islamic invasion or in more specific terms during the caliphate of Omar Al Kattab. Others claim that 2,000 Turkmen fighters came to Iraq during the Ammuyiat period, subsequently more people came during the Abbasid period. They formed six Turkmen countries over 900 years. They lived in harmony with Arabs and other people in their lands.
According to the Encyclopedia Britannica, the name Turkmen is a synonym of Oguz, which includes all Turkish population that lives in the southwestern Central Asia. This includes the Turkish population of Turkey, Republic of Azerbaijan, Azerbaijan of Iran, Turkmenistan and other countries (i.e., Afghanistan, Iraq, Syria, Greece, Cyprus, Bulgaria, Yugoslavia and other European countries). In the Turkish language, men/man means powerful, magnificent, pure or original; so Turkmen in Turkish means a pure/original Turk. Also, others stated that Turkmen means Turkish man or Turkish fighter.
Turkmens of Iraq have established several Turkmen states in Iraq, such as Seljuk Empire (1118-1194), Dynasty of Mosul Atabekians (1127-1233), Dynasty of Erbil Atabekians (1144-1209), Turkmen Dynasty in Kirkuk (1230), State of Kara Koyunlu (1411-1470) and State of Ak Koyunlu (1470-1508).
To summarize, Turkmens are the descendants of those Oguz tribes who originally came from Central Asia. As stated by the El-Maroozi, the Oguz were divided into two main branches. The first branch settled in the cities; the second branches which settled in plain grasslands, and the Oguz tribes who converted to Islam were called Turkmen.
Turkmens did not leave their ancestral lands in one massive migration. However, they departed their land in successive waves over a long period and eventually settled in the Middle East.[2]
III. Population of Turkmens
Some sources generously estimate that Turkmens consist of up to 16% of the Iraqi population, and others estimate 10% of the Iraqi population. If the current Iraqi population is 35 million, this means that Turkmen population would be more than 3 million (see Figure 1).
There is no accurate census about the number of Turkmens in Iraq. In recent years, there are only estimates for various reasons: Turkmens have been subject to systematic assimilation and deliberate displacement over decades for ethnic cleansing. For instance, they had been displaced from their lands where they had been living for hundreds of years.
During Saddam’s ruling period, between 1970 and 2003, many Turkmen people were forced to change their identity and ethnicity and to become Arab. Turkmens had no right to build up or own a land in Kirkuk unless they changed their identity and they became Arab. In addition, Turkmen citizens were forced by the official channels (The Ministry of Planning) and were being paid as little as 500 Iraqi dinars (equal to ¼ USA dollars) to do so. They were allowed to sell their lands, but not to buy.
Arabs were given free grants and lands to come to live in Kirkuk in order to change the demographic nature of the city.
After Saddam’s ruling period in 2003, the situation of Turkmens has not been better off, with many feeling that the post-Saddam period is worse under Kurdish control; when Kurds took control of Kirkuk, all the government buildings, empty houses as well as the military campuses, were turned into houses for Kurdish families which were also brought illegally to change the demographics of the city. This was done in a very speedy way.
There were thousands of disputed lands and assets confiscated from Turkmen citizens during the Ba’ath regime, which have not been returned back to their original owners, in Tal Afar, Erbil, Mosul, Kirkuk, Yayachi, Tassain, Turklan, Taza, Tuzkhormatu and many other Turkmen towns and villages.
IV. Language of Turkmens
The main spoken language in Turkmen Eli (Turkmen homeland) is a Turkmen dialect. This is a part of the Western Turkish language group that also includes Turkish spoken in Turkey, Cyprus, the Balkans, Iranian Azerbaijan (south) and Republic of Azerbaijan (north), northern Syria, Iran, Turkmenistan and southern Turkistan (northern Afghanistan). The Turkmen language, with its various accents, is closer to the Turkish spoken language in both Azerbaijan (Republic of Azerbaijan, Azerbaijan of Iran) and Urfa in southeastern Turkey rather than the Turkmen language in the Republic of Turkmenistan.[3]
Spoken language is the foundation of Iraqi Turkmen culture, folk literature, group identity, ethnic consciousness and world outlook. The spoken mother tongue is naturally passed on to new generations and this, naturally, creates a strong bond uniting the Turkish-speaking peoples of Iraq. However, there is a group of Turkmen called Christian Turkmen of Kirkuk Castle “Kala Gaweri”, which has, for centuries, lived in Kirkuk among Turkmens. They have their own script, bible and mourning songs. However, all these activities are practiced in the Turkmen language.
Unfortunately, compulsory education in Arabic has led to the weakening and deterioration of the spoken Turkish from generation to generation. In fact, the older generation with no formal education speaks relatively pure and more correct Turkish.
Formal written Turkish is the second major source of the Turkish language in Iraq. Local dialects have not found their place in written literature. Turkmens have adopted formal Anatolian Turkish as the written language. Up until the 16th Century, the literary works of Turkmens were written in Azeri dialect, but from the second half of that century onwards, the written literature of Turkmens came under influence of the rising Ottoman language, a western Turkish dialect. However, after the First World War with the separation of Turkey, Turkmens have continued with their preference of Anatolian Turkish by using Arabic letters.
V. Geographical Location of Turkmens
For centuries, Turkmen territories were considered as a buffer zone separating Arabs from Kurds. Cultural, social, religious, economic and political factors have considerably influenced the relations and distribution of the population of Turkmens in the area. The Turkmens of Iraq are mainly merchants, manual labourers and professionals.
Turkmens are concentrated mainly in the northern and central regions of Iraq. This is a diagonal strip of land stretching from Telfar at the north Syrian border to Diayla in the middle part of Iraq.[4]
In this region, there are several major cities and some smaller districts, where Turkmens are living. These are mainly four provinces of Iraq; Erbil, Kirkuk, Saladin and Diyala. In addition to these, historically Turkmen-populated areas have encompassed Telfar; the towns and villages around Mosul such as Al Rashadiya, Shierkan, Nabi Yunis; villages of Shabak around Erbil, Kupery, Kirkuk city, Tassain, Tazakurmatoo, Dakook (Taook), Bashir, Tuzkhormatu, Amerli, Bastamali, Yengaja, Brawachilli, Karanaz, Shasewan, and many other villages around Amerli, Kifri, Karatappa, Karaghan (Jalwalaa), Sharaban (Magdadia), Kizilarbatt (Alsayadia), Kanakeen, Mandeli, Kazania.
An estimate of ¼ million Turkmen lives in the capital city Baghdad too; in Ragiba Katoon, Al Fathal, Al Aathamia, Alsalyiagh, Kanbar Ali, Hay Adan, Zyuna, Hay Oor, Alsahab city and in other places of Al karagh part of Baghdad.
There are Turkmens in other middle and southern part of Iraq as well, from Albayat, Alkarghol, Alsalahi, Al Amerli, and Alatragchji, settled in Babal, Al Messan, Karbala and Basra.
Turkmens, are known as a community greatly attached to their national consciousness, tradition and religion.
VI. Turkmen Families and their Social Life
There are some similarities between Turkmen society and Arabic and also Kurdish societies.
In the rural areas, it is mostly tribal, where people and head of families will be loyal to their head of the tribe. You find out that the family will be proud of their origins and they are using their tribal titles and families’ name, –some of the names may come from their grandfathers or great grandfathers’ names. There are some positive aspects for such allegiances, where the families will be competing for various economic, social and political affairs. However, there are some negative aspects such that they have to be adherent to their cultural rules such as paying a ransom or using these tribes as a means of gaining political or economic gain. Unfortunately, this issue has been recently used and also during the Ba’ath era to gain political seats in parliament.
Turkmen people living in urban parts and cities are more loyal to their families than to their tribes. In recent days, this issue may have contributed to Turkmens’ losing in the elections and failure of a fairer representation.
Turkmens appear to be weak in such gains, as they have been divided amongst themselves, between, religious groups, some being Sunni and others Shia, and some being secular against being Islamic, while others being with the extremist versus moderate groups. Some are nationalist and others are loyal to Turkey. All above issues have caused significant damages to the whole of Turkmen society in the recent political climate of the new Iraqi political system.
Another reason that Turkmen have been less represented is that Turkmens have lived in harmony with other groups such as Arabs, Kurds and Christians, Shabaks, Yazidis in Mosul province and in the past with Jewish community. Turkmen people accepted the intermarriage between themselves and other communities which may have led to further underrepresentation as well.
VII. Turkmens Living in Harmony with Other Iraqi Ethnic Groups
There are, until today, Christian Turkmens who live in Kirkuk old Castle (Kirkuk Kala). Turkmens have lived in harmony with Jewish people, especially before 1948, as since then, most of them have left Iraq for Israel. Turkmens were and are easy to mix with and live in harmony with Arabs and Kurds, through intermarriages happening between Sunni and Shia as well as with other ethnic groups.
The old religions of the Turkmen were Al Shamaniya, Judaism, Buddhism and Zaradishet, but Turkmens converted to Islam after Islamic forces’ conquest of central.[5]
The majority of Turkmens are Muslims and threy are divided into two Muslim faiths: Shiite and Sunni. In addition, there are about 30,000 Christian “catholic” Turks living in Iraq. They are called the Kala Gaweri.
These two Turkmen Muslim and Christian sects helped Turkmens to be more dynamic in the Iraqi society. They facilitated mixed marriages with Arabs and Kurds; therefore, there are a large number of Arab tribes who have originated from Turkmen, such as Albayati. There is no difference at all between the Sunni and Shiite Turkmens regarding the dialogue, language or culture. Intermarriage between the Shiite and Sunni Turkmen is very common. However, some Kurdish militias are trying to utilize various methods to divide the Turkmen community by carrying out a policy of divide and conquer.
VIII. Turkmens’ Contributions for Building up Iraqi Society
There are many well-announced Turkmen scholars who contributed to various fields of education, culture, academia, military and medical for building up Iraqi civilization in the recent history of Iraq.
Professor Mustafa Jawad, Dr. Ihsan Aldogramachi , who was the ambassador of UNICEF, lived in Turkey and refused to be Turkish President. Dr. Salam Al Dogramachi was one of the leading professors in pediatric oncology and hematology in Iraq for many years.
Turkmens gave birth to two well-known poets, like Fazooly Al Baghdadi who has been buried in the Imam Hussain’s Mosque in Karbala.
Dr. Sinnan Saeed was one of the first PhD scholars in media studies, who first put the corner stone of Iraqi media in Baghdad University in 1975.
Dr. Ibrahim Al Dakookly wrote the first letter in Al Aalam in Arab world, 1972, he produced weekly newspaper, and first news journal about role of Media in Arab world.
IX. Modern History of the Iraqi Turkmens
Many considered the maintenance of Iraq’s territorial integrity as a critical issue due to the knowledge of the country’s enormous ethnic and religious diversity. One must also take into account the aspirations of these groups and the problems they are facing now. For better understanding, this historical period will be divided into four stages.[6]
1. Post-Ottoman Empire (1924-1958)
Over the twentieth century, and until now, Turkmens have been subject to many atrocities and programed aggressions, starting with the massacre of 1924 in Kirkuk, to Kwar Baghi events in 1946, and the massacre of 1959.
Under the Iraqi constitution of 1925, both Turkmens and Kurds had the right to use their own languages in schools, government offices and press.
It is stated in the Royal Constitution, which was valid until 1958, that the Iraqi State consisted of Arabs, Kurds, Turkmens and other minorities.
According to Article 14 of the same constitution, Turks, like other minorities, were also entitled to receive an education in their own language and to be in charge of their own educational institutions. In fact, until the proclamation of the republic, various constitutional amendments did not cause ethnic or political discrimination.
The military coup of 1958, that toppled the monarchy, brought a glimpse of hope for Turkmens at first when they heard radio announcements by coup leader General Abdul-Kerim Qasim and his deputy General Abdul-Salam Arif that Iraq was made up of three main ethnic groups: Arab, Kurd and Turkmen. Turkmens interpreted these statements as the end of the suppression. However, happy days did not last long.
2. Post-Monarchy (1958-1970)
As a result of the general amnesty, once Kurdish leader Mullah Mustafa Barzani returned from the Soviet Union, he started negotiations for an autonomous Kurdish region. This has increased the tension in the region and, as the result of this incitement, for the first time in history, clashes between Turkmens and Kurds took place with heavy casualties. When the new regime decided to steer a policy independent of other influential Arab states, the Communist Party and Kurds gained favour with the political ascendancy, and, soon afterwards, Turkmens in Kirkuk were attacked on the false pretext that they helped the Mosul resistance.
On 14th July, 1959, Kirkuk was put under curfew and its population slaughtered by Communists and Kurds. When 25 innocent Turkmen civilians were killed and 130 people were injured in day light in streets of Kirkuk, this was known to be one of the city’s most brutal moments in history.
This massacre was totally disregarded by the world that turned a blind eye to it. It was only after this massacre that the Communist Kurds became so bold as to ask for the inclusion of Kirkuk in their autonomous region under negotiations. Attempts by the Iraqi government to restrict the operations of foreign oil companies and its threats towards Kuwait’s oil put it at loggerheads with other Arab countries and Great Britain.
The ensuing era of General Abdul- Salam Arif (1963-1967) was one of the best periods for the Turkmens in Iraq. Turkmens were allowed to operate cultural associations and schools, publish magazines and newspapers in the Latin characters of Turkish, and get some posts in government. They demonstrated excellently that as citizens of Iraq, they could work for their country and live in cooperation with other Iraqis.
3. Arabization Period (1970-2003)
Then, the Ba’ath party rule, commencing in 1968, opened one of the darkest chapters in Turkmen history. The Ba’ath party forced people to sign petitions asking for the closure of Turkish language schools, and to appoint Arab administrators in Turkmen areas. Boycotts by Turkmens were suppressed in a bloody means.
Many Turkmen traders and professionals were captured and imprisoned. In early 1970, Mr. Mohammad Salah, who was the Head of Kirkuk Trade Union was the first Iraqi executed by Ba’ath rulers together with many Turkmen intellects and human rights activists.
In 1971, the Artist Hussain Ali Damerchi was killed along with many students, teachers, and professionals after peaceful demonstration, as the Turkmen speaking schools were abolished and all Turkmen rights were cancelled after only a year of having been issued.
In November 1979, four of most influential Turkmen people were captured: Dr. Najidat Kojak, Professor in Engineering College of Baghdad University; Abdullah Abdul Al Rahman, who was a retired general, who was the chair of Turkmen Brotherhood Club; Professor Raza Damerchi, the Chief Director of Forests, in Iraqi Agriculture Ministry and the well-known trade man, Adaal Sherif. They were subject of worst physical abuse and torture and later in January 1980, they were killed without even charging them with any criminal charges or court proceedings.
The 1980s saw the execution of countless Turkmen leaders and elders who were, often falsely, accused of spying for Turkey or Iran. During the Iran-Iraq war, dozens of Turkmen villages were totally bulldozed to the ground. Many young Turkmen people (from the Shia community) were captured, they disappeared from Telfar, Kirkuk, Tasseen, Bashir, Dakook, Tuzkhormatu, Tazakurmatoo, Amerli, Quratappa, Kifiri, Kanakeen, Mandeli, Kazania, Baghdad. Some were accused of being part of Islamic movements and of being loyal to Iran and others accused of being loyal to Turkish government.
Mr. Aziz Alsamanji in his book published in 1999 in London, “The political history of Turkmen of Iraq”, a list of 283 Turkmen people were executed by Saddam’s regime between 1980-1990.[7] Furthermore, he published another list of 75 Turkmens who were killed by shooting in the uprising of 28th of March 1991. All of those people were professionals, university students and other served in the military services. He documented a further list of 103 Turkmens who were imprisoned, and another 13 people who disappeared and never returned to the families.
Mofak Salman wrote in his book, that the Turkmen Cultural Directorate that was set up by the government to bring Turkmens under strict control was not working according to the government plans.[8] Therefore, the Iraqi government started a new strategy to replace all Turkmen teachers with Arab teachers; they also sent all Turkmen teachers to non-Turkmen areas. An all-out assimilation campaign against Turkmens was unleashed. Young Turkmen people holding university degrees were given jobs in non-Turkmen areas. Arabs were encouraged to settle in Turkmen areas with rewards of 15,000 Iraqi Dinars to each person. Those Arabs who bought farmlands were offered an extra reward ranging between 7,000 and 10,000 Dinars (approximately $30,000), and the lands confiscated from Turkmens under various pretexts, were given to Arabs.
Young Arab men were encouraged to marry Turkmen girls with offers of 10,000 Iraqi Dinars. All this was designed to change the demographic balance of the Turkmen-dominated region, with its capital city Kirkuk.
This was followed by government decrees that changed Kirkuk’s name to that of Al-Tamim and also changed its administrative borders, taking other Turkmen towns like Tuzkhormatu and Kifri from Kirkuk to other provinces.
Subsequently, the Ba’ath government banned the use of the Turkmen language in public. Religious leaders who did not speak Arabic, were forced to deliver sermons in Arabic, and when they failed to, they were executed.
While the Islamic Union of Iraqi Turkmens, in their well-documented book, published in detail, the name of 432 Turkmen people, who were executed and assassinated by Saddam’s regime between 1979-1991.[9]
The Chief of Iraqi Revolution, said to the retired General Abdul Hussain Mula Ibrahim originally from Tuzkhormatu, when he read his execution order, that he should be hanged and killed twice, once for being Turkmen and second time for being Shia. However, Abdul Hussain could not tolerate the brutality and passed away from the torture.
Turkmens have been severely intimidated into silence, and they have been waiting helplessly, not knowing what to do. Here, I would like to mention the 1987 national census in Iraq, as it is relevant to a number of ethnic groups. In this census, Turkmens were openly threatened to declare themselves as either Arabs or Kurds. If they declared themselves as Turkmens, they would be deported to Saudi.
As a result of Erbil events in 31th August 1996, many Turkmens were captured, and on 2nd September 1996, 25 Turkmen citizens were executed.[10]
The decomposition of the Iraqi Turkmens was an Iraqi policy inherited from one government to the subsequent one. The aim was to remove Turkmens from the oil-rich northern region and to disperse them to the south of Iraq.
4. Targeting Turkmens after 2003 (Kurdization)
After 2003, Shiite Turkmens have been a target of systematic terror attacks in various ways, although the attack seems mainly on Shiite Turkmens, however Sunni Turkmens also had their own share as people are mixed together, living next to each other and married to each other. All Turkmen areas indiscriminately had many attacks from Telfar, Erbil, Mosul, Kirkuk, especially Tassin area, Bashir, Taza, Tuzkhormatu, and Amerli.[11] These are some example of atrocities but not the exhausted list of all the attacks.
4.1. Kirkuk
From 2003 onwards, the Iraqi Turkmens have continued to be subjected to targeted campaigns of intimidation, assimilation, kidnapping, threatening and land confiscation practices, which have resulted in wide-scale emigration. Moreover, Turkmen political actors are often targeted based on their ethnicity, religion and political opinion. In 2011, e.g., the headquarters of the Iraqi Turkmen Front in Kirkuk were completely demolished by explosives. Many university students, scholars, lectures were attacked and killed.
Many Turkmen doctors and professionals were target of killing and kidnapping, almost all of them received letters asking them to leave or pay a ransom. It is estimated that Turkmens paid more than 50 million US dollars until today many medical colleagues left as result of such indiscriminate threats.[12] Indeed many young doctors and university graduates left to other parts of Iraq if not to Turkey or elsewhere in the world.
A report from Iraqi Turkmen Doctors Association reported that Turkmen medical sector specifically, were a target for abductions, kidnapping and assassinations in Kirkuk to drain the city from their minds and intellect. They listed 46, most of whom were Turkmen doctors from Kirkuk alone, who were kidnapped and ransomed for $10,000-50,000 for their release; some of whom were killed and others left the city for good.[13]
In Kirkuk, 95% of the terror attacks targeted Turkmens, Turkmen neighborhoods left no protection despite the heavily presences of security forces which are protecting non-Turkmen neighbourhoods like Kurdish residential areas, and this is exactly what is happening and happened in Tuzkurmatu town.
4.2. Telfar
On 9th of September 2004 and 5th September 2005, Telfar was attacked by tanks, helicopters, soldiers, leaving 1,350 dead people and 2,650 injured, including many children, women and elderlies. During this period more than 48,000 families were displaced from Telfar.
While Telfar was a site of daily attacks of car bombs, kidnapping, killing by various methods, on 9th of July 2009 two suicide bombers killed themselves in the middle of the town, killing more than 34 people and injured hundreds with many houses and belongings were destroyed.
4.3. Tazakhormatu
It is located 20 km south of Kirkuk and it had its own share from terror attacks, on 20th of July 2009, a large explosion of a trailer in the middle of busy market similar to Amerli attack, killed 82 persons and injured 228 people and many shops, and more than 80 houses were collapsed.
4.4. Tuzkhormatu
Countless Turkmen people from Tuzkhormatu were killed and targeted by various terrorist attacks, from kidnapping, road side bombs, car bombs, suicide bombers, head hunting and targeted explosions of their houses and neighborhoods. Explosion of Mosques, worship places like Hussinyia, even nurseries and primary schools and high schools were targets, killing innocent children and people regardless. All these attacks were mainly in the streets of Turkmen neighbourhood.
In January 2013, a suicide bomber exploded himself in the middle of gathering of funeral, killed more than 42 people and injured more than 70 people.
13th of June another deadly suicide bomber attack on peaceful demonstration in Tuzkurmatu killed the Iraqi Turkmen Front Vice president Ali Hashim Mukhtar Oglu with other 13 TurkmenS prominent people and injured more than 30 people.
July 2013, a massive car bomb exploded in 5 a.m. while people sleeping in their beds, in a Turkmen neighbourhood, killed 12 people, children, elderly and young people regardless and more than 20 houses were destroyed with more than 50 people who got injured.
More than 1,500 Turkmen people killed in Tuz, and more than 1,000 houses were destroyed and more than thousands of families were forced to leave their homeland and to move to the south especially to Karbala and Baghdad as they were being fearful of their lives.
It was reported that, between January 2013 and August 2013; “Three hundred attacks took place in the province of Kirkuk”, with “Two hundred seventy attacks” were in Salah al-Din, mainly in Tuzkhurmatu.
4.5. Amerli
A small district located 20 km south of Tuzkhurmatu, which is 80 km south of Kirkuk city. In July 2007, Amerli was subject to a deadly trailer bomb explosion in the middle of a busy market where 160 civilians were killed, more than 300 people were wounded and more than 100 were destroyed. The attack left behind many widowed, orphaned and disabled children and adults. Since then many young people and professionals were targets for deliberate killing on their way to work between Amerli, Tikirit and Kirkuk .
X. Recent Atrocities against Turkmens by Islamic State of Iraq and Sham (ISIS)
Moreover, the recent rapid rise of the ISIS in Iraq has left the state in chaos and its minorities extremely vulnerable, of which in particular Turkmens and Assyrians, as they do not have their own security forces. Reportedly, on 15th June 2014, ISIS fighters took over Telfar, which is mostly populated by Turkmens. In total, 100 people were killed and 200.000 people are estimated to have fled Telfar according to Human Rights Watch.[14] ISIS forces kidnapped at least 40 Shiite Turkmens and ordered 950 Shiite Turkmen families to leave the villages of Guba and Shireekhan. Many more than 100 Turkmen families were forced from Al Rashidyia village, and other Turkmen villages around Mosul (UN Report, 2014). In another report, an estimate of 350,000 Turkmen people from Telfar were displaced.
Bashir a district located at southwest of Kirkuk city, is one of the Shiite Turkmen villages, which were destroyed when Saddam forced their habitants to leave, confiscated their lands, killed many youth and imprisoned others. After 2003, many orders from central government were dismissed and local Arab tribes who took over Bashir lands refused to leave.
On 12th and 13th of June 2014, ISIS terrorists attacked Bashir civilians, kidnapping, killing, abducting, raping children and women. Mosques and worship places were destroyed, 59 people, including three children and women were killed. Little girls and young women were raped and then killed and their corpses were hung from the lamp posts. Around 1000 families fled from Bashir.
Macro Babille; the United Nations children’s fund representative in Iraq, said; that “ISIS militants have massacred 700 Turkmen civilians, including women, children and the elderly, in a northern Iraqi village, Bashir between July 11 and 12”.
Brawachilli and many other villages around Amerli were attacked, people were killed indiscriminately including, sick, children, women and elderly, some people managed to escape their villages by leaving behind the most vulnerable to be abused and killed. Their houses, mosques, lands and livestock were destroyed.
Amerli was under siege for 80 days (17/06/2014-31/08/14) under harsh inhumane conditions, with food, water and electricity supplies were cut off. 20,000 people were under daily attacks. More than 50 children, along of 10 new born babies lost their lives in one day as result of lack of milk and nutrition, more than 100 people were, perished as result of the siege and daily attacks.
Michael Knights is a Boston-based Lafer fellow of the Washington Institute, in his article (Iraq’s City of Orphans), urged US government to rescue Amerli people and argued why the international communities ignored thousands more Iraqi communities of Amerli.
Michael stated in his article that saving Iraqi Turkmens is a Win-Win-Win. A U.S.-backed effort to save besieged Iraqi Turkmens in the Tuzkhurmatu district could bring Baghdad, the Kurds, and Turkey into a joint fight against the ongoing jihadist offensive.[15]
Ayatullah Ali Sistani, Shiite Muslims’ most influential scholar, said through his representative, Abd al-Mehdi al-Karbalai that ”.[16]
The people of Amerli, with their determination, defeated ISIS by defending themselves with the help of Iraqi coalitions from various groups and Kurdish Peshermega as well as with the help of international communities from the USA, and other Western countries.
The American help to save Amerli came very late, which left some doubts in the minds of many Turkmens as to the intentions of America in supporting Turkmens in their struggle.
Amerli was the first town in Iraq stood bravely against the atrocities of ISIS. As Mr. Knight predicted that Iraqis with its various fractions including Kurds came together to defend and fight against the evil ISIS. However, the Turkish government was no longer interested in helping Turkmens any more for reasons out of scope of this review.
Reports from Turkmen Front, human right office in London, June 2014 and from Beladi Strategic Center, said that “the human misery and severe human suffering continue in the Turkmen towns and cities after the invasion of the ISIS, especially in Telfar, Shirghan, Qura Quyan, Bashir, Tazakurmatu, Tuzkurmatu, as well as Al Bayat villages and towns around Tuzkurmatu in Salahadin, other villages and towns around Mosul, Al Sa’dia district and other villages in Diyala province”.[17]
The above communities have suffered from killings via car bombs, suicide bombers, road side bomb explosions as well explosions of their houses by various means prior to the above events. Many loved ones have been killed, and hundreds disabled, leaving behind numerous widows, and orphans, let alone the daily mental anguish and fear of what will happen next. However, these atrocities have not affected Turkmens alone, but later others; Christians and Yazidi ethnic minorities, Shabak and Kurd Shia Faili were targeted as well.[18]
XI. Conclusion
The Iraqi Turkmens have suffered immensely from intimidation by the central government, Kurds and extrajudicial militia groups for religious and ethnic reasons, ‘Arabization’ assimilation policies during the Saddam Hussein regime, and the “Kurdization” policy after 2003.
Since 2003, various groups, including extrajudicial militia, have targeted Turkmens via car bombings, assassinations, kidnappings, arbitrary arrests, harassment and torture, with impunity and torture.
The most recent events by ISIS attacks on Turkmens did not come as a surprise; it is an extension of decades of assimilations, intimidations by Arabs and Kurds, in order to forcefully displace them from their motherland where they had been living for centuries.
The official combination of the assimilation policy and the decomposition policy was successfully played out for years, but assimilation and decomposition would not have been enough to erase or eradicate Turkmens and the languages of the Iraqi Turkmens.
There is no evident sign that official Turkish positions on the Iraqi Turkmens will change. However, changes to improve the present conditions and to solve the Iraqi Turkmen problems within the Iraqi sovereignty may be timely, especially with a view to the Kurdish Spring in northern Iraq. The Kurdish Spring encouraged by the United States, the European Union, Russia and others needs to be emulated by the Iraqi Turkmens who cannot afford to be isolated from northern Iraq geographically or politically, and, least of all, economically.
It is very clear that various policy makers of Western countries calling to divide Iraq into three states of: Arab Shias in the south, Kurds in the north and Sunnis in the west, ignoring the human rights and existence of the original Turkmen people in this land and other ethnic minorities. However, Turkmens should not be seen as a danger to Iraqi sovereignty but as a credit to strengthen Iraqi stability and as a part of the big mosaic of Iraqi unity. The Iraqi government must guarantee human rights for all citizens, regardless of ethnicity, and reach an agreement about these issues, with representation from Kurds, Turkmens and Arabs. They should present a regional security plan in which the Turkmens are given a role to play. The second stipulation concerning Turkmens is that the world should be reminded of Turkmens’ presence in Iraq. Baghdad should be aware of this presence, and it should be noted that providing certain rights and guarantees to Turkmens would contribute to the ending of the division of the country.
Above all, Turkmen people themselves need to wake up, with its various fractions; Shia, Sunni, Muslim and non-Muslim should come together and extend their hands to each other in order to help their wounded and shattered people with many, orphaned children and women who were left defenseless and who were struggling day by day. To take them back to their own home land.
Please cite this publication as follows:
Albayati, Z. J. & Albayati, E. (February, 2015), “Turkmens of Iraq: The Third Ethnic Component of Iraq”, Vol. IV, Issue 2, pp.6-28, Centre for Policy and Research on Turkey (ResearchTurkey), London, Research Turkey. ()
XII. References
FOAB Newsletter: Sistani urges support for Amerli; August 22nd 2014.
Political history of Turkmen of Iraq; Aziz Samanji; 1993; London,UK.
Report of Beladi center for Strategic studies and Research; Department of Human Rights, Baghdad, Iraq; August 2014.
Report of Turkmen front party; Atrocities that Iraqi Turkmen subjected to by ISIS; July 2014.
Report of Iraq Turkmen Doctors Association; Iraq-Kirkuk;
Saving Iraqi Turkmens Is a Win-Win-Win (PolicyWatch 2285);
Turkmen of Iraq; Mofak Salman; 2007, Dublin; Ireland.
Turkmen Martyrs, by Islamic Iraqi Turkmen; 1999; London, UK.
Unrepresented Nations and Peoples Organization in cooperation with Iraqi Turkmen Front; UNPO Alternative report; July 2014.
UN Report, 2014.
Available Websites:
XIII. Endnotes
[1] Mofak Salman, Turkmen of Iraq, 2007, Dublin, Ireland.
[2].
.
[4].
[5] Mofak Salman, Turkmen of Iraq, 2007, Dublin, Ireland.
.
[7] Aziz Samanji, Political history of Turkmen of Iraq, 1993, London, UK.
[8] Mofak Salman, Turkmen of Iraq, 2007, Dublin. Ireland.
[9] Turkmen Martyrs, by Islamic Iraqi Turkmen, 1999, London, UK.
[10]
[11].
Report of Iraq Turkmen Doctors Association, Iraq-Kirkuk;
[12]Report of Turkmen front party, Atrocities that Iraqi Turkmen subjected to by ISIS, July 2014.
[13]Report of Iraq Turkmen Doctors Association, Iraq-Kirkuk;
[14]Unrepresented Nations and Peoples Organization in cooperation with Iraqi Turkmen Front; UNPO Alternative report; July 2014.
[15] See
Saving Iraqi Turkmens Is a Win-Win-Win (Knights | PolicyWatch 2285);
[16] Sistani urges support for Amerli, FOAB Newsletter, August, 2014;
[17] See Atrocities that Iraqi Turkmen subjected to by ISIS, Report of Turkmen front party, July 2014.
UNPO Alternative Report, Unrepresented Nations and Peoples Organization in cooperation with Iraqi Turkmen Front, July 2014.
Report of Beladi center for Strategic studies and Research, Department of Human Rights, Baghdad, Iraq, August 2014.
[18] UNPO Alternative Report, Unrepresented Nations and Peoples Organization in cooperation with Iraqi Turkmen Front, July 2014.
TÜRKMENLER kan ve gözyaşının tarihidir.Osmanlı mirasıdır selçuklu torunlarıdır Atatük çok mücadele etmesine rağmen bizdne koparılan bir parçadır.Bugün ırak pkk ışıd arasnda kalmış sürülen ölen bezdirilen sahipsiz bir halk olmuşlardır.Birlik ve beraberlik dliyorum Büyük Amirli Direnişi ,Beşir zaferiyle Daha aydınlık günler gelecek Türkiye çok daha fazla sahip çıkmalı Türkmenler 1957 nüfus sayımında % 9 olarak belirlenmiştir. Günümüz ırak nüfusuna oranı 3 milyondur.Irak meclisinde 10 Türkmen vekil bulunmaktadır.
Interesting topic and questions I think that more indneeedpnt reporting would be good and is key to covering any international conflict or any issue, national or international. A vast majority of U.S. journalists in Iraq are/were embedded reporters (to my understanding), and while I wouldn’t be the one to volunteer leaving the protection that the military offers in a place of war and conflict, in order to tell the whole story, we need journalists who are brave enough to. In 2005, The New York Times also uncovered some information that revealed that many T.V. news stations were airing or not significantly editing propoganda packages sent by the government in 2003, and that would be something that would need to change, too. A move toward more indneeedpnt reporting and more Americans reading news sources that don’t contain AP reports would not only be good for coverage and American democracy, but it would revitalize the field of journalism and create more jobs for journalists if people weren’t satisfied merely with what the AP or other syndicated sources are handing out.
32 gün Türkmen belgeseli
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Dian Shepperson Mills' 'Commentary' in BioNews 493 (2/2/2009) 'Why fertility patients should consider what they eat before resorting to more invasive treatment' directs us to observational studies on the links between diet and (in)fertility as well as pregnancy outcomes. Since the original observations were made in the 1980s there has been an increasing understanding of the importance of optimal male and female nutrition as an adjunct to fertility treatments, not an alternative.
A large number of factors affecting fertility in women can be affected by, if not attributed to, lifestyle issues. Obesity in women has been clearly linked to ovulatory disorders, poorer pregnancy outcomes, and an increased likelihood of miscarriage. In an IVF setting, obese patients generally demonstrate a resistance to ovarian stimulation and require larger doses of gonadotrophins to effect the same response as patients in the healthy weight range (1). Many studies have demonstrated that even moderate weight loss can be sufficient to stimulate the resumption of spontaneous ovulation. However, ovulatory disorders are not purely the domain of the obese. Some studies have demonstrated that disorders of ovulation can also be linked to increased consumption of energy from trans-fats (when compared with carbohydrates or unsaturated fats). A mere extra two per cent of daily energy derived from dietary trans-fats (instead of unsaturated fats) may be enough to double an individual's risk of infertility from ovulatory disorder (2). On the flipside, the likelihood of ovulatory disorders occurring appears to be reduced by multivitamin intake - perhaps due to the protective effects of antioxidants (3).
Antioxidants also play in important role in oocyte (egg cell) quality. The normal mature oocyte is suspended in metaphase II (MII) [a stage in gamete development] where it stays until fertilisation occurs and metaphase is resumed. An excess of reactive oxygen species (ROS) due to insufficient antioxidant activity can adversely affect the supporting granulosa and luteal cells and result in reduced gonadotrophin action, DNA damage and may prevent the resumption of metaphase (4).
Men wishing to father children would also do well to watch their nutrient intake. Excess ROS production in men has been linked to lipid peroxidation of the sperm membranes and DNA fragmentation. Membrane damage is associated with a reduced ability of the sperm to bind to the oocyte, whilst DNA fragmentation is strongly associated with poor embryo development and poor pregnancy outcomes (5).
In addition, optimal nutrition during gametogenesis, pregnancy and after birth is now recognised as a determinant of future childhood and adult health status and susceptibility to disease (6,7,8). More intriguing still is the suggestion that environmental factors (grandpaternal nutritional status during mid childhood) can be linked to the mortality risk ratio (in their grandsons) two generations later (9). One mechanism by which this may occur is through changes in gene expression without changes in DNA sequence. Such 'epigenetic' changes may result from environmental factors such as folate status and in some situations possibly the IVF process itself (10). The UK committee on Toxicology concluded in 2008 (11) that 'There is reasonable evidence that epigenetic changes associated with environmental exposures during development can result in adverse effects'. We should clearly include 'environmental exposure' to nutritional deficiencies in this context. Nutritional intervention periconceptually, throughout pregnancy and in early childhood should be a priority for health care systems around the world. In Australia there is a particular need for this in relation to the burden of chronic disease in the indigenous Aboriginal community (12).
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You can end your current Flexcar plan and return your car at any time by visiting your account page. When you return your Flexcar vehicle, just follow the steps below:
- Prior to dropping the car off at the location:
- Remove all your personal belongings from the car
- Be sure your Flexcar is returned in the same condition in which it was received, clear of any dirt, stains, trash, or odors
- Failure to do so will result in a reconditioning fee. If your car requires reconditioning, you will be notified and charged a fee to clean and/or repair the car to return to subscription condition. This cost is dependent upon the amount of reconditioning required.
- Fill up the car with a full tank of gas
- Cars returned without a full tank of gas (>90% of tank full) will be charged for the fuel required to top up the tank. We use AAA published values to determine the rate a customer is charged per gallon. In addition, a $5 flat surcharge is added to that cost to reimburse for Flexcar handling and processing fees incurred.
- Once you arrive at the location:
- Leave the keys in the glovebox
- In your Flexcar app, open the Upcoming Return screen
- Review the checklist
- When ready, lock the car
Before ending your Flexcar plan, you should know:
- Your Flexcar plan will be set to close by the end of the current plan week.
- We are unable to provide a partial or full refund of any weekly fee of the current week.
- Any additional drivers on your account will also be closed by ending your Flexcar plan.
- If you have a remaining balance on your account, it will be charged or credited to your payment option on file when your Flexcar plan ends.
- Tolls, charges, and violations will be charged to your payment option on file once they are received.
- You may be subject to additional fees if the vehicle is not returned in the same condition in which it was received.
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Karen Cruz
Karen Cruz was born in San Juan, Puerto Rico. She went to college to study elementary education. She is passionate about providing excellent customer service and educating our clients regarding the legal process effectively alongside our attorneys. Her goal is to provide the best support and understanding when it comes to legal service in our community. Karen has excellent interpersonal and communication skills, is very comprehensive and loves helping others. It is extremely important for her to excel on her professional skills and functions as that will only further add to the benefits our clients and their families receive from our help. She is dedicated to her faith, family and work. Karen also enjoys in her leisure time traveling and spending a sunny at the beach..
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Fifth Principle Task Force
Approved by the UUA Board of Trustees, October 21, 2007
The Unitarian Universalist Association (UUA) Board of Trustees voted to create a Fifth Principle Task Force per the attached charge:
Charge: The Fifth Principle Task Force will present two or more recommendations on the future configuration and content of General Assembly (GA). Areas to be examined include but are not limited to:
- off-site participation in GA, including voting
- reconfiguration of GA content to include current pre-GA functions, including leadership development and continuing education for laypersons and religious professionals
- GA frequency
- GA duration
Methods: The Task Force will meet the UUA requirements for openness.
Prior Related Task Forces: The Task Force will review the reports of the Economic Accessibility and GA Technology Task Forces.
Budget: The Task Force budget will provide for two face-to-face meetings per year, one of which should be held at GA.
Membership: Five persons appointed by the UUA Board of Trustees and a Board liaison. The Director of the General Assembly and Conference Services Office will serve as staff liaison. The President and Moderator shall serve as ex-officio members of the Task Force. All members must be very familiar with the role of General Assembly in the democratic processes of the Association. To be considered for appointment, fill out the Committee on Committees application form.
Staff Support: Staff support for the Task Force will be provided by the General Assembly and Conference Services Office.
Timeline: The Task Force will begin its work prior to GA 2008 and provide a final report to the UUA Board of Trustees by April 2010.
For more information contact coc @ uua.org.
Last updated on Wednesday, July 1, 2009.
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Top: Arts: Music: Styles: J: Jazz: News_and_Media
For the means of dissemination of information about jazz as opposed to the music itself. Includes timely coverage by mass media such as magazines, e-zines, newspapers, radio, and television, as well as non-recurring works such as books and movies.
No category description found
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Wildwood Lakes Campsite
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May 17, 2017
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Fifth round day, where the ‘big guns’ enter the competition to do battle with the minnows; a day of great excitement and shocks galore...
Er, wait a minute, I’ll stop you there.
Without trying to sound wise after the event, I glanced down through the fixtures the other day and thought to myself it was entirely conceivable that every single one of the games – particularly those involving senior clubs – could go entirely as anticipated with the favourites for each tie going through. Lo and behold, that’s exactly what happened.
Setting aside Warrenpoint Town (more on them later) beating Cliftonville on penalties a couple of season ago and, by and large, shock results don’t tend to happen in the Irish Cup, certainly not on a regular basis, and there’s a very good reason for that.
Unlike in the FA Cup in England, clubs in Northern Ireland treat the country’s premier knockout with the reverence it deserves - even though the famous old trophy doesn’t even have a sponsor at present.
My second biggest pet hate of the ‘Sky’ era in English football – after people who purport to ‘follow’ a particular club simply because it’s fashionable despite their interaction not stretching beyond watching Jeff Stelling on Soccer Saturday – is the manner in which the importance of the FA Cup has been diluted to the point where it is viewed as little more than a distraction.
I find it extraordinary that many top-flight clubs in England shun the cup in favour of their more pressing priority of finishing fourth from bottom of the Premier League.
Not so in Northern Ireland, thankfully. A quick glance at the teamlists from the weekend and you don’t see Irish League clubs making nine or 10 changes from the previous week’s line-up.
Instead, Irish Cup fifth round day is a day in which Irish Premiership clubs try to field their STRONGEST possible team because they realise the benefits a cup run can bring in terms of prestige and raising both profile and revenue.
Ironically, the closest thing to an Irish Cup upset at the weekend came on our own doorstep as Ballymena United laboured to an uninspiring victory over a Warrenpoint Town side who were simply brilliant for most of Saturday’s tie.
Thank goodness Glenn Ferguson didn’t take the view with Alan Davidson that ‘we’ve an important league game at Glentoran on Tuesday night’ and leave the fit-again midfielder out of his squad, otherwise the Sky Blues would be facing the prospect of a nerve-jangling replay.
Davidson’s cracking late strike dug United out of a massive hole in a game in which the better team certainly lost.
Warrenpoint fully deserved the standing ovation they received from Ballymena’s supporters as they left the pitch.
I’ve often felt that the gesture of applauding an opposing team off is a largely insincere one, and more of a reflection as to the crowd’s opinion of their own team’s performance, but this reception was heartfelt.
Such are the fine margins involved that instead of facing potential Irish Cup humiliation, Ballymena squeaked through to set up a mouth-watering and money-spinning sixth round tie against Coleraine – although Health and Safety will undoubtedly dictate numbers once again, as per Boxing Day.
Do you think Glenn Ferguson and Oran Kearney will ‘rest’ players that day? Not on your life!
* Follow Ballymena Times Sports Editor Stephen Alexander on Twitter (@Stephen_Bmena).
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TITLE: Direct sum and basis for a vector space
QUESTION [0 upvotes]: I was reading Axler "Linear Algebra Done Right" and he defines the concept of direct sum as following:
Suppose $ U_{1}, U_{2}, \dots , U_{m} $ are subspaces of a vector space $ V. $ The sum $ U_{1} + U_{2} + \dots + U_{m} $ is called the direct sum if each element of $ U_{1} + U_{2} + \dots + U_{m} $ can be written in only one way as a sum $ u_{1} + u_{2} + \dots + u_{m} $ where each $ u_{i} $ is in $ U_{i}. $ So if you pick $ m $ vectors $ u_{i} $ each in $ U_{i}, $ then those vectors form a basis for $ V, $ is it true to conclude that way?
REPLY [2 votes]: If for each $1 \leq i \leq m$ you choose a basis $\mathcal{B}_i = (u^i_1, \dotsc, u^i_{n_i})$ of $U_i$ then you get that
$$
\mathcal{B} = (u^1_1, \dotsc, u^1_{n_1}, u^2_1, \dotsc, u^2_{n_2}, \dotsc, u^m_1, \dotsc, u^m_{n_m})
$$
is a basis of $V$.
It is a generating set because each $v \in V$ can be written as $v = \sum_{i=1}^m u_i$ with $u_i \in U_i$ and then each $u_i$ can be written as a linear combination $u_i = \sum_{j=1}^{n_i} \lambda^i_j u^i_j$, resulting in
$$
v
= \sum_{i=1}^m u_i
= \sum_{i=1}^m \sum_{j=1}^{n_i} \lambda^i_j u^i_j.
$$
If on the other hand
$$
0 = \sum_{i=1}^m \underbrace{\sum_{j=1}^{n_i} \lambda^i_j u^i_j}_{\in U_i}
$$
then by the uniqueness it follows that $\sum_{j=1}^{n_i} \lambda^i_j u^i_j = 0$ for each $i$, so by the linear independence of $\mathcal{B}_i$ it follows that $\lambda^i_j = 0$ for all $i,j$. So $\mathcal{B}$ is also linear independent.
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\begin{document}
\title[The simplest cohomological invariants for vertex algebras]
{The simplest cohomological invariants for vertex algebras}
\author{A. Zuevsky}
\address{Institute of Mathematics \\ Czech Academy of Sciences\\ \\ Zitna 25, 11567 \\ Prague\\ Czech Republic}
\email{zuevsky@yahoo.com}
\begin{abstract}
For the double complex structure of grading-restricted vertex algebra cohomology defined in \cite{Huang},
we introduce a multiplication of elements of double complex spaces.
We show that the orthogonality and bi-grading conditions applied on double complex spaces, provide in relation
among mappings and actions of co-boundary operators.
Thus, we endow the double complex spaces with structure of bi-graded differential algebra.
We then introduce the simples cohomology classes for a grading-restricted vertex algebra, and
show their independence on the choice of mappings from double complex spaces.
We prove that its cohomology class does not depend on mappings representing of
the double complex spaces.
Finally, we show that the orthogonality relations together with the bi-grading condition
bring about generators and commutation relations for a continual Lie algebra.
AMS Classification: 53C12, 57R20, 17B69
\end{abstract}
\keywords{Vertex algebras, cohomological invariants, cohomology classes}
\vskip12pt
\maketitle
\section{Introduction: $\overline{W}$-valued rational functions}
\label{valued}
In \cite{Huang} the cohomology theory for a grading-restricted vertex algebra \cite{K} (see Appendix \ref{grading})
was introduced.
The definition of double complex spaces and co-boundary operators,
uses an interpretation of vertex algebras in terms of rational functions constructed from
matrix elements \cite{H2} for a grading-restricted vertex algebra.
The notion of composability (see Section \ref{composable}) of double complex space elements with a number of vertex operators, is essentially
involved in the formulation.
Then the cohomology of such complexes defines in the standard way a cohomology of a grading-restricted vertex
algebras.
It is an important problem to study possible cohomological classes for vertex algebras.
In this paper we do the first steps to discover simplest cohomological
invariants associated to the setup described above.
For that purpose we first endow the double complex spaces with natural product, derive a counterpart of Leibniz formula
for the action of co-boundary operators.
Then we introduce the notion of a cohomological class for a vertex algebra.
The orthogonality condition of double complex space is then defined.
We show that the orthogonality being applied to the double complex spaces leads to relations among mappings and actions
of co-boundary operators.
The simplest non-vanishing cohomology classes for a grading-restricted vertex algebra is then derived.
We show that such classes are independent of the choice of elements of the double complex spaces.
Finally, we discuss occurring relations of a vertex algebra double complex relations with
a continual Lie algebra \cite{saver}.
For further applications of material introduced in this paper, we would mention
the natural question of searching for more general cohomological invariants for a grading-restricted vertex algebra.
Concerning possible applications, one can use the cohomological classes we derive to compute
higher cohomologies of grading-restricted vertex algebras.
Let $V$ be a grading-restricted
vertex algebra, and $W$ a a grading-restricted generalized $V$-module (see Appendix \ref{grading}).
One defines the configuration space
\cite{Huang}:
\[
F_{n}\C=\{(z_{1}, \dots, z_{n})\in \C^{n}\;|\; z_{i}\ne z_{j}, i\ne j\},
\]
for $n\in \Z_{+}$.
\begin{definition}
A $\overline{W}$-valued rational function $\F$ in $(z_{1}, \dots, z_{n})$
with the only possible poles at
$z_{i}=z_{j}$, $i\ne j$,
is a map
\begin{eqnarray*}
\F: F_{n}\C &\to& \overline{W},
\\
(z_{1}, \dots, z_{n}) &\mapsto& \F(z_{1}, \dots, z_{n}),
\end{eqnarray*}
such that for any $w'\in W'$,
\begin{equation}
\label{def}
\langle w', \F(z_{1}, \dots, z_{n}) \rangle,
\end{equation}
is a rational function in $(z_{1}, \dots, z_{n})$
with the only possible poles at
$z_{i}=z_{j}$, $i\ne j$.
Such map is called in what fallows $\overline{W}$-valued rational function in
$(z_{1}, \dots, z_{n})$ with possible other poles.
Denote the space of all $\overline{W}$-valued rational functions in
$(z_{1}, \dots, z_{n})$ by $\overline{W}_{z_{1}, \dots, z_{n}}$.
\end{definition}
Namely,
if a meromorphic function $f(z_{1}, \dots, z_{n})$ on a region in $\C^{n}$
can be analytically extended to a rational function in $(z_{1}, \dots, z_{n})$,
then the notation
$R(f(z_{1}, \dots, z_{n}))$,
is used to denote such rational function.
Note that the set of a grading-restricted vertex algebra elements $(v_1, \ldots, v_n)$ associated with
corresponding $(z_1, \ldots, z_n)$ play the role of non-commutative parameters for a function $\F$ in \eqref{def}.
Let us introduce the definition of a $\W_{z_1, \ldots, z_n}$-space:
\begin{definition}
\label{wspace}
We define the space $\W_{z_1, \dots, z_n}$ of
$\overline{W}_{z_{1}, \dots, z_{n}}$-valued rational forms $\Phi$
with each vertex algebra element entry $v_i$, $1 \le i \le n$
of a quasi-conformal grading-restricted vertex algebra $V$ tensored with power $\wt(v_i)$-differential of
corresponding formal parameter $z_i$, i.e.,
\begin{eqnarray}
\label{bomba}
&&
\Phi \left(v_{1}, z_1; \ldots;
v_{n}, z_n\right)
\nn
&& \qquad
= \F \left(dz_1^{{\rm \wt}(v_1)} \otimes v_{1}, z_1; \ldots;
dz_n^{{\rm \wt}(v_n)} \otimes v_{n}, z_n\right) \in \W_{z_1, \dots, z_n}.
\end{eqnarray}
where $\F \in \overline{W}_{z_1, \dots, z_n}$.
\end{definition}
\begin{definition}
One defines an action of $S_{n}$ on the space $\hom(V^{\otimes n},
\W
_{z_{1}, \dots, z_{n}})$ of linear maps from
$V^{\otimes n}$ to $
W
_{z_{1}, \dots, z_{n}}$ by
\begin{equation}
\label{sigmaction}
\sigma(\Phi)(v_{1}\otimes \cdots\otimes v_{n})(z_{1}, \dots, z_{n}) ,
=\Phi(v_{\sigma(1)}\otimes \cdots\otimes v_{\sigma(n)})(z_{\sigma(1)}, \dots, z_{\sigma(n)}),
\end{equation}
for $\sigma\in S_{n}$ and $v_{1}, \dots, v_{n}\in V$, $\Phi \in \W_{z_{1}, \dots, z_{n}}$.
We will use the notation $\sigma_{i_{1}, \dots, i_{n}}\in S_{n}$, to denote the
the permutation given by $\sigma_{i_{1}, \dots, i_{n}}(j)=i_{j}$,
for $j=1, \dots, n$.
\end{definition}
\begin{definition}
\label{wspace}
For $n\in \Z_{+}$,
a linear map
\[
\F(v_{1}, z_{1}; \ldots ; v_{n}, z_{n})
= V^{\otimes n}\to
\W_{z_{1}, \dots, z_{n}},
\]
is said to have
the $L_V(-1)$-derivative property if
\begin{equation}
\label{lder1}
(i) \qquad
\partial_{z_{i}} \F (v_{1}, z_{1}; \ldots ; v_{n}, z_{n})
=
\F(v_{1}, z_{1}; \ldots; L_{V}(-1)v_{i}, z_i; \ldots ; v_{n}, z_{n}),
\end{equation}
for $i=1, \dots, n$, $(v_{1}, \dots, v_{n}) \in V$, $w'\in W$,
and
\begin{eqnarray}
\label{lder2}
(ii) \qquad \sum\limits_{i=1}^n\partial_{z_{i}}
\F(v_{1}, z_{1}; \ldots ; v_{n}, z_{n})=
L_{W}(-1).\F(v_{1}, z_{1}; \ldots ; v_{n}, z_{n}),
\end{eqnarray}
with some action $"."$ of $L_{W}(-1)$ on $\F(v_{1}, z_{1}; \ldots ; v_{n}, z_{n})$.
\end{definition}
\begin{definition}
A linear map
\[
\F: V^{\otimes n} \to \W_{z_{1}, \dots, z_{n}}
\]
has the $L_W{(0)}$-conjugation property if for $(v_{1}, \dots, v_{n}) \in V$,
$(z_{1}, \dots, z_{n})\in F_{n}\C$, and $z\in \C^{\times}$, such that
$(zz_{1}, \dots, zz_{n})\in F_{n}\C$,
\begin{eqnarray}
\label{loconj}
z^{L_{W}(0)}
\F \left(v_{1}, z_1; \ldots; v_{n}, z_{n} \right)
=
\F\left(z^{ L_V{(0)} } v_{1}, zz_{1}; \ldots ; z^{L_V{(0)} } v_{n}, zz_{n}\right).
\end{eqnarray}
\end{definition}
\subsection{E-elements}
For $w\in W$, the $\overline{W}$-valued function
$E^{(n)}_{W}(v_{1}\otimes \cdots\otimes v_{n}; w)$ is
given by
$$
E^{(n)}_{W}(v_{1}\otimes \cdots\otimes v_{n}; w)(z_{1}, \dots, z_{n})
=E(Y_{W}(v_{1}, z_{1})\cdots Y_{W}(v_{n}, z_{n})w),
$$
where an element $E(.)\in \overline{W}$ is
given by
\[
\langle w',E(.)\rangle =R(\langle w', . \rangle),
\]
and $R(.)$ denotes the rationalization in the sense of \cite{Huang}.
Namely,
if a meromorphic function $f(z_{1}, \dots, z_{n})$ on a region in $\C^{n}$
can be analytically extended to a rational function in $(z_{1}, \dots, z_{n})$,
then the notation $R(f(z_{1}, \dots, z_{n}))$ is used to denote such rational function.
One defines
\[
E^{W; (n)}_{WV}(w; v_{1}\otimes \cdots\otimes v_{n})
=E^{(n)}_{W}(v_{1}\otimes \cdots\otimes v_{n}; w),
\]
where
$E^{W; (n)}_{WV}(w; v_{1}\otimes \cdots\otimes v_{n})$ is
an element of $\overline{W}_{z_{1}, \dots, z_{n}}$.
One defines
\[
\Phi\circ \left(E^{(l_{1})}_{V;\;\one}\otimes \cdots \otimes E^{(l_{n})}_{V;\;\one}\right):
V^{\otimes m+n}\to
\overline{W}_{z_{1}, \dots, z_{m+n}},
\]
by
\begin{eqnarray*}
\lefteqn{(\Phi\circ (E^{(l_{1})}_{V;\;\one}\otimes \cdots \otimes
E^{(l_{n})}_{V;\;\one}))(v_{1}\otimes \cdots \otimes v_{m+n-1})}\nn
&&=E(\Phi(E^{(l_{1})}_{V; \one}(v_{1}\otimes \cdots \otimes v_{l_{1}})\otimes \cdots
E^{(l_{n})}_{V; \one}
(v_{l_{1}+\cdots +l_{n-1}+1}\otimes \cdots
\otimes v_{l_{1}+\cdots +l_{n-1}+l_{n}}))),
\end{eqnarray*}
and
\[
E^{(m)}_{W}\circ_{m+1}\Phi: V^{\otimes m+n}\to
\overline{W}_{z_{1}, \dots,
z_{m+n-1}},
\]
is given by
\begin{eqnarray*}
\lefteqn{
(E^{(m)}_{W}\circ_{m+1}\Phi)(v_{1}\otimes \cdots \otimes v_{m+n})
}\nn
&&
=E(E^{(m)}_{W}(v_{1}\otimes \cdots\otimes v_{m};
\Phi(v_{m+1}\otimes \cdots\otimes v_{m+n}))).
\end{eqnarray*}
Finally,
\[
E^{W; (m)}_{WV}\circ_{0}\Phi: V^{\otimes m+n}\to
\overline{W}_{z_{1}, \dots,
z_{m+n-1}},
\]
is defined by
\begin{eqnarray*}
(E^{W; (m)}_{WV}\circ_{0}\Phi)(v_{1}\otimes \cdots \otimes v_{m+n})
=E(E^{W; (m)}_{WV}(\Phi(v_{1}\otimes \cdots\otimes v_{n})
; v_{n+1}\otimes \cdots\otimes v_{n+m})).
\end{eqnarray*}
In the case that $l_{1}=\cdots=l_{i-1}=l_{i+1}=1$ and $l_{i}=m-n-1$, for some $1 \le i \le n$,
we will use $\Phi\circ_{i} E^{(l_{i})}_{V;\;\one}$ to
denote $\Phi\circ (E^{(l_{1})}_{V;\;\one}\otimes \cdots
\otimes E^{(l_{n})}_{V;\;\one})$.
\subsection{Maps composable with vertex operators}
\label{composable}
Since $\overline{W}$-valued rational functions above are valued in $\overline{W}$,
for $z\in \C^{\times}$, $u, v\in V$, $w\in W$, $Y_{V}(u, z)v\in \overline{V}$,
and $Y_{W}(u, z)v\in \overline{W}$,
one might not be able to compose in general a linear map from
a tensor power of $V$ to $\overline{W}_{z_{1}, \dots, z_{n}}$ with
vertex operators.
Thus in \cite{Huang} they consider linear maps
from tensor powers of $V$ to $\overline{W}_{z_{1}, \dots, z_{n}}$
such that these maps can be composed with vertex operators in the sense mentioned above.
\begin{definition}
\label{mapco}
For a $V$-module $W=\coprod_{n\in \C}W_{(n)}$ and $m\in \C$,
let $P_{m}: \overline{W}\to W_{(m)}$ be
the projection from $\overline{W}$ to $W_{(m)}$.
Let $\Phi: V^{\otimes n}\to
\overline{W}_{z_{1}, \dots, z_{n}}$ be a linear map. For $m\in \N$,
$\Phi$ is said \cite{Huang} to be composable with $m$ vertex operators if
the following conditions are satisfied:
\begin{enumerate}
\item Let $l_{1}, \dots, l_{n}\in \Z_+$ such that $l_{1}+\cdots +l_{n}=m+n$,
$v_{1}, \dots, v_{m+n}\in V$ and $w'\in W'$. Set
\begin{eqnarray*}\label{psi-i}
\Psi_{i}
&
=
&
E^{(l_{i})}_{V}(v_{k_1}
\otimes
\cdots \otimes v_{k_i}
; \one_{V})
(z_{k_1},
\dots,
z_{k_i}
),
\end{eqnarray*}
where ${k_1}={l_{1}+\cdots +l_{i-1}+1}$, ..., $v_{k_i}={l_{1}+\cdots +l_{i-1}+l_{i}}$,
for $i=1, \dots, n$. Then there exist positive integers $N^n_m(v_{i}, v_{j})$
depending only on $v_{i}$ and $v_{j}$ for $i, j=1, \dots, k$, $i\ne j$ such that the series
\[
\sum_{r_{1}, \dots, r_{n}\in \Z}\langle w',
(\Phi(P_{r_{1}}\Psi_{1}\otimes \dots\otimes
P_{r_n} \Psi_{n}))(\zeta_{1}, \dots,
\zeta_{n})\rangle,
\]
is absolutely convergent when
$|z_{l_{1}+\cdots +l_{i-1}+p}-\zeta_{i}|
+ |z_{l_{1}+\cdots +l_{j-1}+q}-\zeta_{i}|< |\zeta_{i}
-\zeta_{j}|$,
for $i,j=1, \dots, k$, $i\ne j$ and for $p=1,
\dots, l_i$ and $q=1, \dots, l_j$.
The sum must be analytically extended to a
rational function
in $(z_{1}, \dots, z_{m+n})$,
independent of $(\zeta_{1}, \dots, \zeta_{n})$,
with the only possible poles at
$z_{i}=z_{j}$, of order less than or equal to
$N^n_m(v_{i}, v_{j})$, for $i,j=1, \dots, k$, $i\ne j$.
\item For $v_{1}, \dots, v_{m+n}\in V$, there exist
positive integers $N^n_m(v_{i}, v_{j})$, depending only on $v_{i}$ and
$v_{j}$, for $i, j=1, \dots, k$, $i\ne j$, such that for $w'\in W'$, and
${\bf v}_{n,m}=(v_{1+m}\otimes \cdots\otimes v_{n+m})$, ${\bf z}_{n,m}=(z_{1+m}, \dots, z_{n+m})$, such that
\[
\sum_{q\in \C}\langle w',
(E^{(m)}_{W}(v_{1}\otimes \cdots\otimes v_{m};
P_{q}((\Phi({\bf v}_{n,m}))({\bf z}_{n,m})))\rangle,
\]
is absolutely convergent when $z_{i}\ne z_{j}$, $i\ne j$
$|z_{i}|>|z_{k}|>0$ for $i=1, \dots, m$, and
$k=m+1, \dots, m+n$, and the sum can be analytically extended to a
rational function
in $(z_{1}, \dots, z_{m+n})$ with the only possible poles at
$z_{i}=z_{j}$, of orders less than or equal to
$N^n_m(v_{i}, v_{j})$, for $i, j=1, \dots, k$, $i\ne j$,.
\end{enumerate}
\end{definition}
In \cite{Huang} one finds:
\begin{proposition}
\label{}
The subspace of $\hom(V^{\otimes n},
\W
_{z_{1}, \dots, z_{n}})$ consisting of linear maps
having
the $L(-1)$-derivative property, having the $L(0)$-conjugation property
or being composable with $m$ vertex operators is invariant under the
action of $S_{n}$.
\end{proposition}
\section{Chain complexes and cohomologies}
\label{complexes}
Let us recall the definition of shuffles \cite{Huang}.
\begin{definition}
For $l \in \N$ and $1\le s \le l-1$, let $J_{l; s}$ be the set of elements of
$S_{l}$ which preserve the order of the first $s$ numbers and the order of the last
$l-s$ numbers, i.e.,
\[
J_{l, s}=\{\sigma\in S_{l}\;|\;\sigma(1)<\cdots <\sigma(s),\;
\sigma(s+1)<\cdots <\sigma(l)\}.
\]
The elements of $J_{l; s}$ are called shuffles. Let $J_{l; s}^{-1}=\{\sigma\;|\;
\sigma\in J_{l; s}\}$.
\end{definition}
Now we introduce the notion of a $C^n_m(V, \W)$-space:
\begin{definition}
\label{kakashka}
Let $V$ be a vertex operator algebra and $W$ a $V$-module.
For $n\in \Z_{+}$, let $ {C}_{0}^{n}(V, \W)$ be the vector space of all
linear maps from $V^{\otimes n}$ to $\W_{z_{1}, \dots, z_{n}}$
satisfying the $L(-1)$-derivative property and the $L(0)$-conjugation property.
For $m$, $n\in \Z_{+}$,
let $ {C}_{m}^{n}(V, \W)$ be the vector spaces of all
linear maps from $V^{\otimes n}$ to $\W_{z_{1}, \dots, z_{n}}$
composable with $m$ vertex operators, and satisfying the $L(-1)$-derivative
property, the $L(0)$-conjugation property, and such that
\begin{equation}
\label{shushu}
\sum_{\sigma\in J_{l; s}^{-1}}(-1)^{|\sigma|}
\sigma\left(\Phi(v_{\sigma(1)}\otimes \cdots \otimes v_{\sigma(l)})\right)=0.
\end{equation}
\end{definition}
Using a generalization of the construciton of the vertex algebra bundle and coordinate-free formulation of
vertex operators in \cite{BZF} for the case of $\W$-valued forms, we obtain
following
\begin{lemma}
\label{popa}
that an element \eqref{bomba} of $C^n_m(V, \W)$ is invariant with respect the group
${\rm Aut}_{z_1, \ldots, z_n}\Oo^{(n)}$ of $n$-dimensional independent changes of formal parameters
\[
(z_1, \ldots, z_n) \mapsto (\rho_1(z_1, \ldots, z_n), \ldots, \rho_n(z_1, \ldots, z_n)).
\]
\hfill $\square$
\end{lemma}
We also find in \cite{Huang}
\begin{proposition}
Let $ {C}_{m}^{0}(V, \W)=\W$. Then we have
$C_{m}^{n}(V, \W)\subset {C}_{m-1}^{n}(V, \W)$,
for $m\in \Z_{+}$.
\end{proposition}
In \cite{Huang} the co-boundary operator for the double complex spaces ${C}_{m}^{n}(V, \W)$ was introduced:
\begin{equation}
\label{hatdelta}
{\delta}^{n}_{m}: {C}_{m}^{n}(V, \W)
\to {C}_{m-1}^{n+1}(V, \W).
\end{equation}
For $\Phi \in {C}_{m}^{n}(V, \W)$, it is given by
\begin{equation}
\label{marsha}
{\delta}^{n}_{m}(\Phi)
=E^{(1)}_{W}\circ_{2} \Phi
+\sum_{i=1}^{n}(-1)^{i}\Phi\circ_{i} E^{(2)}_{V; \one}
+(-1)^{n+1}
\sigma_{n+1, 1, \dots, n}(E^{(1)}_{W}\circ_{2}
\Phi),
\end{equation}
where $\circ_i$ is defined in Subsection \ref{valued}.
Explicitly,
for $v_{1}, \dots, v_{n+1}\in V$, $w'\in W'$
and $(z_{1}, \dots, z_{n+1})\in
F_{n+1}\C$,
\begin{eqnarray*}
\lefteqn{\langle w', (( {\delta}^{n}_{m}(\Phi))(v_{1}\otimes \cdots\otimes v_{n+1}))
(z_{1}, \dots, z_{n+1})\rangle}\nn
&&=R(\langle w', Y_{W}(v_{1}, z_{1})(\Phi(v_{2}\otimes \cdots\otimes v_{n+1}))
(z_{2}, \dots, z_{n+1})\rangle)\nn
&&\quad +\sum_{i=1}^{n}(-1)^{i}R(\langle w',
(\Phi(v_{1}\otimes \cdots \otimes v_{i-1} \otimes
Y_{V}(v_{i}, z_{i}-z_{i+1})v_{i+1}\nn
&&\quad\quad\quad\quad\quad\quad\quad\quad\quad
\quad\quad\otimes \cdots \otimes v_{n+1}))
(z_{1}, \dots, z_{i-1}, z_{i+1}, \dots, z_{n+1})\rangle)\nn
&&\quad + (-1)^{n+1}R(\langle w', Y_{W}(v_{n+1}, z_{n+1})
(\Phi(v_{1}\otimes \cdots \otimes v_{n}))(z_{1}, \dots, z_{n})\rangle).
\end{eqnarray*}
In the case $n=2$, there is a subspace
of $ {C}_{0}^{2}(V, \W)$
containing $ {C}_{m}^{2}(V, \W)$ for all $m\in \Z_{+}$ such that
$ {\delta}^{2}_{m}$ is still defined on this subspace.
Let $ {C}_{\frac{1}{2}}^{2}(V, \W)$ be the subspace of $ {C}_{0}^{2}(V, \W)$
consisting of elements $\Phi$ such that for $v_{1}, v_{2}, v_{3}\in V$, $w'\in W'$,
\begin{eqnarray*}
&& \sum_{r\in \C}\big(\langle w', E^{(1)}_{W}(v_{1};
P_{r}((\Phi(v_{2}\otimes v_{3}))(z_{2}-\zeta, z_{3}-\zeta)))(z_{1}, \zeta)\rangle\nn
&&\quad+\langle w', (\Phi(v_{1}\otimes P_{r}((E^{(2)}_{V}(v_{2}\otimes v_{3}; \one))
(z_{2}-\zeta, z_{3}-\zeta))))
(z_{1}, \zeta)\rangle\big),
\end{eqnarray*}
and
\begin{eqnarray*}
\lefteqn{\sum_{r\in \C}\big(\langle w',
(\Phi(P_{r}((E^{(2)}_{V}(v_{1}\otimes v_{2}; \one))(z_{1}-\zeta, z_{2}-\zeta))
\otimes v_{3}))
(\zeta, z_{3})\rangle}\nn
&&\quad +\langle w',
E^{W; (1)}_{WV}(P_{r}((\Phi(v_{1}\otimes v_{2}))(z_{1}-\zeta, z_{2}-\zeta));
v_{3}))(\zeta, z_{3})\rangle\big)
\end{eqnarray*}
are absolutely convergent in the regions $|z_{1}-\zeta|>|z_{2}-\zeta|, |z_{2}-\zeta|>0$ and
$|\zeta-z_{3}|>|z_{1}-\zeta|, |z_{2}-\zeta|>0$, respectively,
and can be analytically extended to
rational functions in $z_{1}$ and $z_{2}$ with the only possible poles at
$z_{1}, z_{2}=0$ and $z_{1}=z_{2}$.
It is clear that
${C}_{m}^{2}(V, \W)\subset {C}_{\frac{1}{2}}^{2}(V, \W)$
for $m\in \Z_{+}$.
The co-boundary operator
\begin{equation}
\label{halfdelta}
{\delta}^{2}_{\frac{1}{2}}: {C}_{\frac{1}{2}}^{2}(V, \W)
\to {C}_{0}^{3}(V, \W),
\end{equation}
is defined in \cite{Huang} by
\begin{eqnarray}
\label{halfdelta1}
&& {\delta}^{2}_{\frac{1}{2}}(\Phi)
= E^{(1)}_{W} \circ_2 \Phi
+ \sum\limits_{i=1}^2 (-1)^i E^{(2)}_{V, \one_V} \circ_i \Phi
+ E^{W; (1)}_{WV}\circ_2\Phi,
\nn
&&\langle w', (({\delta}^{2}_{\frac{1}{2}}(\Phi))
(v_{1}\otimes v_{2} \otimes v_{3}))(z_{1}, z_{2}, z_{3})\rangle
\nn
&&
=R(\langle w', (E^{(1)}_{W}(v_{1};
\Phi(v_{2}\otimes v_{3}))(z_{1}, z_{2}, z_{3})\rangle\nn
&&
\quad \quad
+\langle w', (\Phi(v_{1}\otimes E^{(2)}_{V}(v_{2}\otimes v_{3}; \one)))
(z_{1}, z_{2}, z_{3})\rangle)
\nn
&&\quad
-R(\langle w',
(\Phi(E^{(2)}_{V}(v_{1}\otimes v_{2}; \one))
\otimes v_{3}))(z_{1}, z_{2}, z_{3})\rangle
\nn
&&\quad \quad
+\langle w',
(E^{W; (1)}_{WV}(\Phi(v_{1}\otimes v_{2}); v_{3}))
(z_{1}, z_{2}, z_{3})\rangle)
\end{eqnarray}
for $w'\in W'$,
$\Phi\in {C}_{\frac{1}{2}}^{2}(V, \W)$,
$v_{1}, v_{2}, v_{3}\in V$ and $(z_{1}, z_{2}, z_{3})\in F_{3}\C$.
Consider the short sequence of the double complex spaces
\begin{equation}
\label{shortseq}
0\longrightarrow C_{3}^{0}(V, \W)
\stackrel{\delta_{3}^{0}}{\longrightarrow}
C_{2}^{1}(V, \W)
\stackrel{\delta_{2}^{1}}{\longrightarrow}C_{\frac{1}{2}}^{2}(V, \W)
\stackrel{\delta_{\frac{1}{2}}^{2}}{\longrightarrow}
C_{0}^{3}(V, \W)\longrightarrow 0,
\end{equation}
of (\ref{hatdelta}).
The first and last arrows are trivial embeddings and projections.
In \cite{Huang} we find:
\begin{proposition}
\label{delta-square}
For $n\in \N$ and $m\in \Z_{+}+1$, the co-boundary operators \eqref{marsha} and \eqref{halfdelta1} satisfy
the chain complex conditions, i.e.,
\[
{\delta}^{n+1}_{m-1}\circ {\delta}^{n}_{m}=0,
\]
\[
{\delta}^{2}_{\frac{1}{2}}\circ {\delta}^{1}_{2}=0.
\]
\end{proposition}
Since
\[
{\delta}_{2}^{1}( {C}_{2}^{1}(V, \W))\subset
{C}_{1}^{2}(V, \W)\subset
{C}_{\frac{1}{2}}^{2}(V, \W),
\]
the second formula follows from the first one, and
\[
{\delta}^{2}_{\frac{1}{2}}\circ {\delta}^{1}_{2}
= {\delta}^{2}_{1}\circ {\delta}^{1}_{2}
=0.
\]
Using the double complexes (\ref{hatdelta}) and (\ref{halfdelta}),
for $m\in \Z_{+}$ and $n\in \N$, one introduces in \cite{Huang}
the $n$-th cohomology $H^{n}_{m}(V, W)$ of a grading-restricted vertex algebra $V$
with coefficient in $W$, and composable with $m$ vertex operators
to be
\[
H_{m}^{n}(V, \W)=\ker \delta^{n}_{m}/\mbox{\rm im}\; \delta^{n-1}_{m+1},
\]
\[
H^{2}_{\frac{1}{2}}(V, \W)
=\ker \delta^{2}_{\frac{1}{2}}/\mbox{\rm im}\; \delta_{2}^{1}.
\]
\section{The $\epsilon$-product of $C^n_m(V, \W)$-spaces}
\label{productc}
In this section we introduce
definition of the $\epsilon$-product of
double complex spaces $C^n_m(V, \W)$ with the image in another double complex space coherent with respect
to the original differential \eqref{hatdelta}, and satisfying the symmetry \eqref{shushu},
$L_V(0)$-conjugation \eqref{loconj}, and $L_V(-1)$-derivative \eqref{lder1} properties
and derive an analogue of Leibniz formula.
\subsection{Motivation and geometrical interpretation}
The structure of $C^n_m(V, \W)$-spaces
is quite complicated and it is difficult to introduce algebraically a product
of its elements.
In order to define an appropriate product of two $C^n_m(V, \W)$-spaces
we first have to interpret
them geometrically.
Basically, a $C^n_m(V, \W)$-space
must be associated with a certain model space, the algebraic $\W$-language should be
transferred to a geometrical one, two model spaces should be "connected" appropriately, and,
finally, a product should be
defined.
For two
$\W_{x_1, \ldots, x_k}$- and
$\W_{y_{1}, \ldots, y_{n}}$-spaces we first associate formal complex parameters
in the sets
$(x_1, \ldots, x_k)$ and $(y_{1}, \ldots, y_n)$
to parameters of two auxiliary
spaces.
Then we describe a geometric procedure to
form a resulting model space
by combining two original model spaces.
Formal parameters of
$\W_{z_1, \ldots, z_{k+n}}$ should be then identified with
parameters of the resulting space.
Note that
according to our assumption, $(x_1, \ldots, x_k) \in F_k\C$, and $(y_{1}, \ldots, y_{n}) \in F_{n}\C$.
As it follows from the definition of the configuration space $F_n\C$ in Subsection \ref{valued},
in the case of coincidence of two
formal parameters they are excluded from $F_n\C$.
In general, it may happen that some number $r$ of formal parameters of $\W_{x_1, \ldots, x_k}$ coincide with some
$r$ formal parameters of $\W_{y_{1}, \ldots, y_{n}}$ on the whole $\C$ (or on a domain of definition).
Then,
we exclude one formal parameter from each coinciding pair.
We require that the set of formal parameters
\begin{equation}
\label{zsto}
(z_1, \ldots, z_{k+n-r})= ( \ldots, {x}_{i_1}, \ldots, {x}_{i_r}, \ldots ;
\ldots, \widehat {y}_{j_1}, \ldots, \widehat{y}_{j_r}, \ldots ),
\end{equation}
where $\; \widehat{.} \; $ denotes the exclusion of corresponding formal parameter for
$x_{i_l}=y_{j_l}$, $1 \le l \le r$,
for the resulting model space
would belong to $F_{k+n-r}\C$.
We denote this operation of formal parameters exclusion by
$\widehat{R}\;\F(x_1, \ldots, x_k; y_{1}, \ldots, y_{n}; \epsilon)$.
Now we formulate the definition of the $\epsilon$-product of two $C^n_m(V, \W)$-spaces:
\begin{definition}
For
$\F(v_1, x_1; \ldots; v_{k}, x_k) \in C^{k}_{m}(V, \W)$, and
$\F(v'_{1}, y_{1}; \ldots; v'_{n}, y_{n}) \in C_{m'}^{n}(V, \W)$
the product
\begin{eqnarray}
\label{gendef}
&& \F(v_1, x_1; \ldots; v_{k}, x_k) \cdot_\epsilon \F(v'_{1}, y_{1}; \ldots; v'_{n}, y_{n})
\nn
&&
\qquad \qquad \mapsto
\widehat{R} \; \F\left( v_1, x_1; \ldots; v_{k}, x_k; v'_{1}, y_{1}; \ldots; v'_{n}, y_{n}; \epsilon\right),
\end{eqnarray}
is a $\W_{
z_1, \ldots, z_{k+n-r}
}$-valued rational
form
\begin{eqnarray}
\label{Z2n_pt_epsss}
&& \langle w', \widehat{R} \; \F (v_1, x_1; \ldots; v_{k}, x_k; v'_{1}, y_{1}; \ldots; v'_{n}, y_{n}; \epsilon) \rangle
\nn
& & \quad =
\sum_{u\in V }
\langle w', Y^{W}_{WV}\left(
\F (v_{1}, x_{1}; \ldots; v_{k}, x_{k}), \zeta_1\right)\; u \rangle
\nn
& &
\quad
\langle w', Y^{W}_{WV}\left(
\F(v'_{1}, y_{1}; \ldots; {v'}_{i_1}, \widehat{y}_{i_1};
\ldots; \ldots; {v'}_{j_r}, \widehat{y}_{j_r}; \ldots; v'_{n}, y_{n})
, \zeta_{2}\right) \; \overline{u} \rangle,
\end{eqnarray}
via \eqref{def},
parametrized by
$\zeta_1$, $\zeta_2 \in \C$, and we exclude all monomials $(x_{i_l} - y_{j_l})$, $1 \le l \le r$, from
\eqref{Z2n_pt_epsss}.
The sum
is taken over any $V_{l}$-basis $\{u\}$,
where $\overline{u}$ is the dual of $u$ with respect to a non-degenerate bilinear form
$\langle .\ , . \rangle_\lambda$, \eqref{eq: inv bil form} over $V$, (see Appendix \ref{grading}).
\end{definition}
\begin{remark}
Due to the symmetry of the geometrical interpretation describe above, we could exclude
from the set $(x_1, \ldots, x_k)$ in \eqref{Z2n_pt_epsss}
$r$
formal parameters which belong
to coinciding pairs resulting to the same
definition of the $\epsilon$-product.
\end{remark}
By the standard reasoning \cite{FHL, Zhu},
\eqref{Z2n_pt_epsss} does not depend on the choice of a basis of $u \in V_l$, $l \in \Z$.
In the case when multiplied forms $\F$ do not contain $V$-elements, i.e., for $\Phi$, $\Psi \in \W$,
\eqref{Z2n_pt_epsss} defines the product $\Phi \cdot_\epsilon \Psi$ associated to
a
rational function:
\begin{eqnarray}
\label{Z2_part}
{\mathcal R(\epsilon)}= \sum_{l \in \Z } \epsilon^l
\sum_{u\in V_l }
\langle w', Y^{W}_{WV}\left(
\Phi, \zeta_1\right) \; u \rangle
\langle w', Y^{W}_{WV}\left(
\Psi,
\zeta_2 \right) \; \overline{u} \rangle,
\end{eqnarray}
which defines $\F(\epsilon) \in \W$ via $\mathcal R(\epsilon)=\langle w', \F(\epsilon)\rangle$.
\subsection{Convergence and properties of of the $\epsilon$-product}
In order to prove convergence of a product of elements of two spaces
$\W_{x_1, \ldots, x_k}$
and $\W_{y_1, \ldots, y_n}$ of rational $\W$-valued forms,
we have to use a geometrical interpretation \cite{H2, Y}.
Recall that a $\W_{z_1, \ldots, z_n}$-space is defined by means of matrix elements of the form \eqref{def}.
For a vertex algebra $V$, this corresponds \cite{FHL} to a matrix element of a number of $V$-vertex operators
with formal parameters identified with local coordinates on a Riemann sphere.
Geometrically, each space $\W_{z_1, \ldots, z_n}$ can be also associated to a Riemann sphere
with a few marked points,
and local coordinates
vanishing at these points
\cite{H2}.
An extra point
can be associated to a center of an annulus used in order
to sew the sphere with another sphere.
The product \eqref{Z2n_pt_epsss} has then a geometric interpretation.
The resulting model space would also be associated to a Riemann sphere formed as a result of sewing procedure.
In Appendix \ref{sphere} we describe explicitly the geometrical procedure of sewing of two spheres \cite{Y}.
Let us identify (as in \cite{H2, Y, Zhu, TUY, FMS, BZF}) two sets $(x_1, \ldots, x_k)$ and $(y_1, \ldots, y_n)$ of
complex formal parameters,
with local
coordinates of two sets of points on the first and the second Riemann spheres correspondingly.
Identify complex parameters $\zeta_1$, $\zeta_2$ of \eqref{Z2n_pt_epsss} with coordinates \eqref{disk} of
the annuluses \eqref{zhopki}.
After identification of annuluses $\mathcal A_a$ and $\mathcal A_{\overline{a}}$,
$r$ coinciding coordinates may occur. This takes into account case of coinciding formal parameters.
In this way, we construct the map \eqref{gendef}.
As we
see in \eqref{Z2n_pt_epsss},
the product is defined
by a sum of products of matrix elements \cite{FHL} associated to each of two spheres.
Such sum is supposed to describe a $\W$-valued rational differential form defined
on a sphere formed
as a result of geometrical sewing \cite{Y} of two initial spheres.
Since two initial spaces $\W_{x_1, \ldots, x_k}$ and $\W_{y_1, \ldots, y_n}$
are defined through rational-valued forms
expressed by
matrix elements of the form \eqref{def}.
We then arrive at
the resulting product defines a
$\W_{z_1, \ldots, z_{k+n-r}
}$-valued rational
form by means of an absolute convergent matrix element on the resulting sphere.
The complex sewing parameter, parameterizing the module space of sewin spheres, parametrizes also the product of
$\W$-spaces.
Next, we formulate
\begin{definition}
\label{sprod}
We define the
action of an element $\sigma \in S_{k+n-r}$ on the product of
$\F (v_{1}, x_{1}; \ldots; v_{k}, x_{k}) \in \W_{x_1, \ldots, x_k}$, and
$\F (v'_{1}, y_{1}; \ldots; v'_{n}, y_{n}) \in \W_{y_1, \ldots, y_n}$, as
\begin{eqnarray}
\label{Z2n_pt_epsss}
&& \langle w', \sigma(\widehat{R}\;\F) (v_{1}, x_{1}; \ldots; v_{k}, x_{k}; v'_{1}, y_{1}; \ldots; v'_{n}, y_{n};
\epsilon) \rangle
\nn
&&
\qquad =\langle w', \F (\widetilde{v}_{\sigma(1)}, z_{\sigma(1)}; \ldots; \widetilde{v}_{\sigma(k+n-r)}, z_{\sigma(k+n-r)};
\epsilon) \rangle
\nn
& & \qquad =
\sum_{u\in V }
\langle w', Y^{W}_{WV}\left(
\F (\widetilde{v}_{\sigma(1)}, z_{\sigma(1)}; \ldots; \widetilde{v}_{\sigma(k)}, z_{\sigma(k)}), \zeta_1\right)\; u \rangle
\nn
& &
\qquad
\langle w', Y^{W}_{WV}\left(
\F
(\widetilde{v}_{\sigma(k+1)}, z_{\sigma(k+1)}; \ldots; \widetilde{v}_{\sigma(k+n-r)}, z_{\sigma(k+n-r)}) , \zeta_{2}\right) \; \overline{u} \rangle,
\end{eqnarray}
where by $(\widetilde{v}_{\sigma(1)}, \ldots, \widetilde{v}_{\sigma(k+n-r)})$ we denote a permutation of
\begin{equation}
\label{notari}
(\widetilde{v}_{1}, \ldots, \widetilde{v}_{k+n-r})
=(v_1, \ldots; v_k; \ldots, \widehat{v}'_{j_1}, \ldots, \widehat{v}'_{j_r}, \ldots ).
\end{equation}
\end{definition}
Let $t$ be the number of common vertex operators the mappings
$\F(v_{1}, x_{1}$; $\ldots$; $v_{k}, x_{k}) \in C^{k}_{m}(V, \W)$ and
$\F(v'_{1}, y_{1}; \ldots; v'_{n}, y_{n}) \in C^{n}_{m'}(V, \W)$,
are composable with.
The rational
form corresponding to the
$\epsilon$-product $\widehat{R} \F\left(v_1, x_1; \ldots; v_{k}, x_k; v'_{1}, y_{1}; \ldots; v'_{n}, y_{n}
; \epsilon\right)$ converges in $\epsilon$, and
satisfies \eqref{shushu}, $L_V(0)$-conjugation \eqref{loconj} and
$L_V(-1)$-derivative \eqref{lder1} properties.
Using Definition \ref{kakashka} of $C^n_m(V, \W)$-space and Definition \ref{mapco} of mappsings composable with
vertex operators,
we then have
\begin{proposition}
\label{tolsto}
For $\F(v_1, x_1; \ldots; v_{k}, x_k) \in C_{m}^{k}(V, \W)$ and
$\F(v'_{1}, y_{1}; \ldots; v'_{n}, y_{n})\in C_{m'}^{n}(V, \W)$,
the product $\widehat{R} \F\left(v_1, x_1; \ldots; v_{k}, x_k; v'_{1}, y_{1}; \ldots; v'_{n}, y_{n}
; \epsilon\right)$ \eqref{Z2n_pt_epsss}
belongs to the space $C^{k+n-r
}_{m+m'-t
}(V, \W)$, i.e.,
\begin{equation}
\label{toporno}
\cdot_\epsilon : C^{k}_{m}(V,\W) \times C_{m'}^{n}(V, \W) \to C_{m+m'-t
}^{k+n-r}(V, \W).
\end{equation}
\hfill $\square$
\end{proposition}
\begin{remark}
Note that due to \eqref{wprop}, in
Definition
\eqref{Z2n_pt_epsss}
it is assumed that
$\F (v_{1}, x_{1} $; $ \ldots $ ; $ v_{k}, x_{k})$ and $\F(v'_{1}, y_1; \ldots; v'_{n}, y_{n})$
are composable with the $V$-module $W$ vertex operators
$Y_W(u, -\zeta_1)$ and $Y_W(\overline{u}, -\zeta_2)$ correspondingly.
The product \eqref{Z2n_pt_epsss} is actually defined by a sum of
products of matrix elements of ordinary $V$-module $W$ vertex operators
acting on $\W
$-elements.
The elements
$u \in V$ and $\overline{u} \in V'$
are connected by \eqref{dubay}, and $\zeta_1$, $\zeta_2$ are related by \eqref{pinch}.
The form
of the product defined above is natural in terms of the theory of chacaters for vertex operator algebras
\cite{TUY, FMS, Zhu}.
\end{remark}
\begin{remark}
For purposes of construction of cohomological invariant, we do not exclude in this paper
the case of $r$ pais of common formal parameters $x_i=y_j$, $1 \le i \le k$, $1 \le j \le n$, for
$\F(v_1, x_1; \ldots; v_{k}, x_k) \in C_{m}^{k}(V, \W)$ and
$\F(v'_{1}, y_{1}; \ldots; v'_{n}, y_{n})\in C_{m'}^{n}(V, \W)$ in Proposition \ref{}.
Such formal parameter pairs are excluded from the right hand side of the map \eqref{toporno}.
\end{remark}
We then have
two corollaries:
\begin{corollary}
For the spaces $\W_{x_1, \ldots, x_k}$ and $\W_{y_1, \ldots, y_n}$
with the product \eqref{Z2n_pt_epsss} $\F \in \W_{z_1, \ldots, z_{k+n-r}
}$,
the subspace of $\hom(V^{\otimes n},
{\W}_{z_1, \ldots, z_{k+n-r}}$
consisting of linear maps
having the $L_W(-1)$-derivative property, having the $L_V(0)$-conjugation property
or being composable with $m$ vertex operators is invariant under the
action of $S_{k+n-r}$.
\end{corollary}
\begin{corollary}
\label{functionformprop}
For a fixed set $(v_1, \ldots v_k; v_{k+1}, \ldots, v_{k+n}) \in V$ of vertex algebra elements, and
fixed $k+n$, and $m+m'$,
the $\epsilon$-product
$\F(v_1, z_1; \ldots; v_k, z_k; v_{k+1}, z_{k+1}; \ldots $ ; $ v_{k+n-r}, y_{k+n-r}; \epsilon)$,
\[
\cdot_{\epsilon}: C^{k}_m(V, \W) \times C^{n}_{m'}(V, \W) \rightarrow C^{k+n-r}_{m+m'-t}(V, \W),
\]
of the spaces $C^{k}_{m}(V, \W)$ and $C^{n}_{m'}(V, \W)$,
for all choices
of $k$, $n$, $m$, $m'\ge 0$,
is the same element of $C^{k+n-r}_{m+m'-t}(V, \W)$
for all possible $k \ge 0$.
\hfill $\square$
\end{corollary}
By Lemma \ref{popa}, elements of the space $C^{k+n-r}_{m+m'-t}$ resulting from the $\epsilon$-product are
invariant with respect to changes of formal parameters of the group
${\rm Aut}_{z_1, \ldots, z_{k+n-r}}\Oo^{(k+n-r)}$.
We then have
\begin{definition}
For fixed sets $(v_1, \ldots, v_k)$, $(v'_1, \ldots, v'_n) \in V$,
$(x_1, \ldots, x_k)\in \C$, $(y_1, \ldots, y_n$) $\in \C$,
we call the set of all $\W_{x_1, \ldots, x_{k} ; y_1, \ldots, y_{n}}$-valued rational forms
$\widehat{R} \F(
v_1, x_1; \ldots; v_k, x_{k} $ ; $ v'_1, y_1; \ldots; v'_{n}, y_{n};
\epsilon)$ defined by \eqref{Z2n_pt_epsss}
with the parameter $\epsilon$ exhausting all possible values,
the complete product of the spaces $\W_{x_1, \ldots, x_k}$ and $\W_{y_{1}, \ldots, y_n}$.
\end{definition}
\subsection{Coboundary operator acting on the product space}
In Proposition \ref{tolsto} we proved that the product \eqref{Z2n_pt_epsss} of elements
$\F_1 \C_{m}^{k}(V, \W)$ and $\F_2 \in C_{m'}^{n}(V, \W)$ belongs to $C^{k+n-r}_{m+m'-t}(V, \W)$.
Thus, the product admits the action ot the differential operator $\delta^{k+n-r}_{m+m'-t}$ defined in
\eqref{hatdelta} where $r$ is the number of common formal parameters, and $t$ the number of commpon
composable vertex
operators for $\F_1$ and $\F_2$.
The co-boundary operator \eqref{hatdelta}
possesses a variation of Leibniz law with respect to the product
\eqref{Z2n_pt_epsss}.
We then have
\begin{proposition}
\label{tosya}
For $\F(v_{1}, x_{1}; \ldots; v_{k}, x_{k}) \in C_{m}^{k}(V, \W)$
and
$\F(v'_{1}, y_{1}; \ldots; v'_{n}, y_{n}) \in C_{m'}^{n}(V, \W)$,
the action of $\delta_{m + m'-t}^{k + n-r}$ on their product \eqref{Z2n_pt_epsss} is given by
\begin{eqnarray}
\label{leibniz}
&& \delta_{m + m'-t}^{k + n-r} \left( \F (v_{1}, x_{1}; \ldots; v_{k}, x_{k})
\cdot_{\epsilon} \F (v'_{1}, y_{1}; \ldots; v'_{n}, y_{n}) \right)
\nn
&&
\qquad =
\left( \delta^{k}_{m} \F (\widetilde{v}_{1}, z_{1}; \ldots; \widetilde{v}_{k}, z_{k}) \right)
\cdot_{\epsilon} \F (\widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+n}, z_{k+n-r})
\nn
&&
\; + (-1)^k
\F (\widetilde{v}_{1}, z_{1}; \ldots; \widetilde{v}_{k}, z_{k}) \cdot_{\epsilon} \left( \delta^{n-r}_{m'-t}
\F(\widetilde{v}_{1}, z_{k+1}; \ldots;
\widetilde{v}_{k+n-r}, z_{k+n-r}) \right),
\nn
&&
\end{eqnarray}
where we use the notation as in \eqref{zsto} and \eqref{notari}.
\end{proposition}
Appendix \ref{duda} contains the proof of this Proposition.
\begin{remark}
Checking
\eqref{hatdelta} we see that an extra arbitrary vertex algebra element $v_{n+1} \in V$, as well as corresponding
extra arbitrary formal parameter $z_{n+1}$ appear as a result of the action of $\delta^{n}_m$ on
$\F \in C^n_m(V, \W)$ mapping it to $C^{n+1}_{m-1}(V, \W)$.
In application to the $\epsilon$-product \eqref{Z2n_pt_epsss} these extra arbitrary elements are involved in the
definition of the action of $\delta_{m + m'-t}^{k + n-r}$ on
$\F (v_{1}, x_{1}; \ldots; v_{k}, x_{k})
\cdot_{\epsilon} \F (v'_{1}, y_{1}; \ldots; v'_{n}, y_{n})$.
\end{remark}
Note that both sides of \eqref{leibniz} belong to the space
$C_{m + m'-t + 1}^{n + n' -r +1}(V, W)$.
The co-boundary operators $\delta^n_m$ and $\delta^{n'}_{m'}$
in \eqref{leibniz} do not include the number of common vertex algebra elements
(and formal parameters), neither the number of common vertex operators corresponding mappings composable with.
The dependence on common vertex algebra elements, parameters, and composable vertex operators is taken into
account in mappings multiplying the action of co-boundary operators on $\Phi$.
Finally, we have the following
\begin{corollary}
The multiplication \eqref{Z2n_pt_epsss} extends the chain-cochain
complex
structure of Proposition \ref{delta-square} to all products $C^k_m(V, \W) \times C^{n}_{m'}(V, \W)$,
$k$, $n \ge0$, $m$, $m' \ge0$.
\hfill $\qed$
\end{corollary}
\begin{corollary}
The product \eqref{Z2n_pt_epsss} and the product operator \eqref{hatdelta}
endow the space $C^k_m(V, \W)$ $\times$ $ C^n_m(V, \W)$, $k$, $n \ge0$, $m$, $m' \ge0$,
with the structure of a
bi-graded differential algebra $\mathcal G(V, \W, \cdot_\epsilon, \delta^{k+n-r}_{m+m'-t})$.
\hfill $\qed$
\end{corollary}
For elements of the spaces $C^2_{ex}(V, \W)$
we have the following
\begin{corollary}
The product of elements of the spaces $C^{2}_{ex} (V, \W)$ and $C^n_{m} (V, \W)$ is given by
\eqref{Z2n_pt_epsss},
\begin{equation}
\label{pupa3}
\cdot_\epsilon: C^{2}_{ex} (V, \W) \times C^n_{m} (V, \W) \to C^{n+2-r}_{m} (V, \W),
\end{equation}
and, in particular,
\[
\cdot_\epsilon: C^{2}_{ex} (V, \W) \times C^{2}_{ex} (V, \W) \to C^{4-r}_{0} (V, \W).
\]
\hfill $\square$
\end{corollary}
\subsection{The commutator}
Let us consider the mappings
$\Phi(v_1, z_1 $ ; $ \ldots $; $ v_{n}, z_k)$ $\in$ $C_{m}^{k}(V, \W)$, and
$\Psi (v_{k+1}, z_{k+1};
\ldots;
v_{k+n}, z_{k+n}) \in C_{m'}^{n}(V, \W)$,
with have $r$ common vertex algebra elements (and, correspondingly, $r$ formal variables), and
$t$ common vertex operators mappings $\Phi$ and $\Psi$ are composable with.
Note that when applying the co-boundary operators \eqref{marsha} and \eqref{halfdelta1} to a map
$\Phi(v_1, z_1;
\ldots;
v_n, z_n)
\in C^n_m(V, \W)$,
\[
\delta^n_m: \Phi(v_1, z_1;
\ldots;
v_n,z_n)
\to
\Phi(v'_1, z'_1;
\ldots;
v'_{n+1}, z'_{n+1})
\in C^{n+1}_{m-1}(V, \W),
\]
one does not necessary assume that we keep
the same set of vertex algebra elements/formal parameters and
vertex operators composable with for $\delta^n_m \Phi$,
though it might happen that some of them could be common with $\Phi$.
Let us define an extra product (related to the $\epsilon$-product)
the product of $\Phi$ and $\Psi$,
\begin{eqnarray}
&&
\Phi \cdot \Psi: V^{\otimes(k +n-r)} \to \W_{z_1, \ldots, z_{k+ n-r}}, \;
\\
\label{product}
&&
\Phi \cdot \Psi = \left[\Phi,_{\cdot \epsilon} \Psi\right]= \Phi \cdot_\epsilon \Psi- \Psi \cdot_\epsilon \Phi,
\end{eqnarray}
where brackets denote ordinary commutator in $\W_{z_1, \ldots, z_{k+ n-r}}$.
Due to the properties of the maps $\Phi\in C_{m}^{k}(V, \W)$ and
$\Psi\in C_{m'}^{n}(V, \W)$, the map $(\Phi \cdot_\epsilon \Psi)$
belongs to the space $C_{m + m'- t }^{k +n-r}(V, \W)$.
For $k=n$ and
\[
\Psi (v_{n+1}, z_{n+1};
\ldots;
v_{ 2n}, z_{2n})
=
\Phi(v_{1}, z_1;
\ldots;
v_{ n}, z_n),
\]
we obtain from \eqref{product} and \eqref{Z2n_pt_epsss} that
\begin{eqnarray}
\label{fifi}
\Phi(v_{1}, z_1;
\ldots;
v_{ n}, z_n)
\cdot
\Phi(v_{1}, z_1;
\ldots;
v_{ n}, z_n)
=0.
\end{eqnarray}
\section{The invariants}
\label{invariants}
In this section we provide the main result of the paper by deriving
the simplest cohomological invariants associated to the short double complex \eqref{halfdelta}
for a grading-restricted vertex algebra.
Let us give first some further definitions.
In this section we skip the dependence on vertex algebra elements and formal parameters in notations for
elements of $C_{n}^{m}(V, \W)$.
\begin{definition}
In analogy with differential forms, we call a map
$\Phi \in C_{m}^{n}(V, \W)$ closed if
\[
\delta^{n}_{m} \Phi=0.
\]
For $m \ge 1$, we call it exact if there exists $\Psi \in C_{m-1}^{n+1}(V, \W)$
such that
\[
\Psi=\delta^{n}_{m} \Phi.
\]
\end{definition}
\begin{definition}
For $\Phi \in {C}^{n}_{m}(V, \W)$ we call the cohomology class of mappings
$\left[ \Phi \right]$ the set of all closed forms that differs from $\Phi$ by an
exact mapping, i.e., for $\chi \in {C}^{n-1}_{m+1}$,
\[
\left[ \Phi \right]= \Phi + \delta^{n-1}_{m+1} \chi,
\]
(we assume that both parts of the last formula belongs to the same space ${C}^{n}_{m}(V, \W)$).
\end{definition}
Under a natural extra condition, the short double complex \eqref{shortseq} allows us to establish relations
among elements of double complex spaces.
In particular, we require that for a pair of double complex spaces $C^{n_1}_{k_1}(V, \W)$ and
$C^{n_2}_{k_2}(V, \W)$ there exist subspaces
${C'}^{n_1}_{k_1}(V, \W) \subset C^{n_1}_{k_1}(V, \W)$ and
${C'}^{n_2}_{k_2}(V, \W) \subset C^{n_2}_{k_2}(V, \W)$ such that for $\Phi_1 \in {C'}^{n_1}_{k_1}(V, \W)$ and
$\Phi_2 \in C'^{n_2}_{k_2}(V, \W)$,
\begin{equation}
\label{ortho}
\Phi_1 \cdot \delta^{n_2}_{k_2} \Phi_2=0,
\end{equation}
namely,
$\Phi_1$ supposed to be orthogonal to $\delta^{n_2}_{k_2} \Phi_2$
(i.e., commutative with respect to the product \eqref{product}).
We call this {\it the orthogonality condition} for
mappings and actions of co-boundary operators for a double complex.
It is easy to see that
the assumption to belong to the same double complex space for both sides of the equations following from the orthogonality
condition applies the bi-grading condition on double complex spaces.
Note that in the case of differential forms considered on a smooth manifold,
the Frobenius theorem for a distribution provides the orthogonality condition.
In this Section we derive algebraic relations occurring from the orthogonality condition on
the short double complex \eqref{shortseq}.
We formulate
\begin{proposition}
The orthogonality condition for the short double complex sequence \eqref{shortseq} determines the cohomological classes:
\begin{equation}
\label{stupor}
\left[\left(\delta^{1}_{2} \Phi \right)\cdot \Phi \right], \;
\left[\left(\delta^{0}_{3} \chi \right)\cdot \chi \right], \;
\left[\left(\delta^{1}_{t} \alpha \right)\cdot \alpha \right],
\end{equation}
for $0 \le t \le 2$, with
non-vanishing $\left(\delta^{1}_{2} \Phi \right)\cdot \Phi$,
$\left(\delta^{0}_{3} \chi \right)\cdot \chi$, and
$\left(\delta^{1}_{t} \alpha \right)\cdot \alpha$.
These classes are independent on the choice of $\Phi \in C^{1}_{2}(V, \W)$, $\chi \in C^{0}_{3}(V, \W)$, and
$\alpha \in C^1_t(V, \W)$.
\end{proposition}
\begin{remark}
A cohomology class with vanishing $\left(\delta^{1}_{2} \Phi \right)\cdot \Phi \cdot \alpha$ is given by
$\left[ \left(\delta^{1}_{2} \Phi \right)\cdot \Phi \cdot \alpha \right]$.
\end{remark}
\begin{proof}
Let us consider two maps $\chi \in C^{0}_{3}(V, \W)$, $\Phi \in C^{1}_{2}(V, \W)$.
We require them to be orthogonal, i.e.,
\begin{equation}
\label{isxco}
\Phi \cdot \delta^{0}_{3} \chi=0.
\end{equation}
Thus, there exists $\alpha \in C^{n}_{m}(V, \W)$, such that
\begin{equation}
\label{uravnenie}
\delta^{0}_{3} \chi= \Phi \cdot \alpha,
\end{equation}
and
$1=1 + n -r$, $2=2+m-t$, i.e., $n=r$, which leads to $r=1$; $m=t$, $0\le t \le 2$, i.e.,
$\alpha \in C^{1}_{t}(V, \W)$.
All other orthogonality conditions for the short sequence \eqref{shortseq} does not allow relations of the form
\eqref{uravnenie}.
Consider now \eqref{isxco}.
We obtain, using \eqref{leibniz}
\[
\delta^{2-r'}_{4-t'} (\Phi \cdot \delta^{0}_{3} \chi)=
\left(\delta^{1}_{2} \Phi\right) \cdot \delta^{0}_{3} \chi + \Phi \cdot \delta^{1}_{2} \delta^{0}_{3} \chi=
\left(\delta^{1}_{2} \Phi\right) \cdot \delta^{0}_{3} \chi= \left(\delta^{1}_{2} \Phi\right) \cdot \Phi \cdot
\alpha.
\]
Thus
\[
0=\delta^{3-r'}_{3-t'} \delta^{2-r'}_{4-t'} (\Phi \cdot \delta^{0}_{3} \chi)=
\delta^{3-r'}_{3-t'} \left( \left(\delta^{1}_{2} \Phi\right) \cdot \Phi \cdot
\alpha. \right),
\]
and $\left(\left(\delta^{1}_{2} \Phi\right) \cdot \Phi \cdot \alpha\right) $ is closed.
At the same time,
from \eqref{isxco}
it follows that
\[
0=\delta^{1}_{2} \Phi \cdot \delta^{0}_{3} \chi- \Phi \cdot\delta^{1}_{2}\delta^{0}_{3} \chi
= \left( \Phi\cdot \delta^{0}_{3} \chi \right).
\]
Thus
\[
\delta^{1}_{2} \Phi \cdot \delta^{0}_{3} \chi= \delta^{1}_{2} \Phi \cdot \Phi \cdot \alpha =0.
\]
Consider \eqref{uravnenie}.
Acting by $\delta^{1}_{2}$ and substituting back we obtain
\[
0= \delta^{1}_{2} \delta^{0}_{3} \chi= \delta^{1}_{2}(\Phi \cdot \alpha)=
\delta^{1}_{2}(\Phi) \cdot \alpha - \Phi \cdot \delta^{1}_{t} \alpha.
\]
thus
\[
\delta^{1}_{2}(\Phi) \cdot \alpha = \Phi \cdot \delta^{1}_{t} \alpha.
\]
The last equality trivializes on applying $\delta^{3}_{t+1}$ to both sides.
Let us show now the non-vanishing property of $\left(\left(\delta^{1}_{2} \Phi \right)\cdot \Phi\right)$.
Indeed, suppose $\left(\delta^{1}_{2} \Phi \right)\cdot \Phi=0$. Then there exists $\gamma \in C^{n}_{m}(V, \W)$,
such that $\delta^{1}_{2} \Phi =\gamma \cdot \Phi$. Both sides of the last equality should belong to the same double complex
space but one can see that it is not possible.
Thus, $\left(\delta^{1}_{2} \Phi \right)\cdot \Phi$ is non-vanishing.
One proves in the same way that $\left(\delta^{0}_{3} \chi \right)\cdot \chi$ and
$\left(\delta^{1}_{t} \alpha \right)\cdot \alpha$ do not vanish too.
Now let us show that $\left[\left(\delta^{1}_{2} \Phi \right)\cdot \Phi \right]$
is invariant, i.e., it does not depend on the choice of $\Phi \in C^1_2(V, \W)$.
Substitute $\Phi$ by
$\left(\Phi + \eta\right)\in C^{1}_{2}(V, \W)$.
We have
\begin{eqnarray}
\label{pokaz}
\nonumber
\left(\delta^{1}_{2} \left( \Phi + \eta \right) \right) \cdot \left( \Phi + \eta \right) &=&
\left(\delta^{1}_{2} \Phi\right) \cdot \Phi
+ \left( \left(\delta^{1}_{2} \Phi \right)\cdot \eta
- \Phi \cdot \delta^{1}_{2} \eta \right)
\nn
&+& \left( \Phi \cdot \delta^{1}_{2} \eta +
\delta^{1}_{2} \eta \cdot \Phi \right)
+
\left(\delta^{1}_{2} \eta \right) \cdot \eta.
\end{eqnarray}
Since
\[
\left( \Phi \cdot \delta^{1}_{2} \eta +
\left(\delta^{1}_{2} \eta\right) \cdot \Phi \right)=
\Phi \delta^{1}_{2} \eta - (\delta^{1}_{2} \eta) \Phi
+ \left(\delta^{1}_{2} \eta\right) \Phi - \Phi \; \delta^{1}_{2} \eta=0,
\]
then \eqref{pokaz} represents the same cohomology class
$\left[ \left(\delta^{1}_{2} \Phi \right) \cdot \Phi \cdot \alpha \right]$.
The same folds for $\left[\left(\delta^{0}_{3} \chi \right)\cdot \chi \right]$, and
$\left[\left(\delta^{1}_{t} \alpha \right)\cdot \alpha \right]$.
\end{proof}
\begin{remark}
Due to Proposition \ref{}, all chahomological classes are invariant with respect to correponding group
${\rm Aut}_{z_1, \ldots, z_n}\Oo^{(n)}$
changes of formal parameters.
\end{remark}
The orthogonality condition for a double complex sequence \eqref{shortseq},
together with the action of co-boundary operators
\eqref{hatdelta} and \eqref{halfdelta}, and the multiplication formulas \eqref{product}--\eqref{leibniz},
define a differential bi-graded algebra depending on vertex algebra elements and formal parameters.
In particular, for the short sequence \eqref{shortseq}, we obtain in this way the generators and commutation relations
for a continual Lie algebra $\mathcal G(V)$ (a generalization of ordinary Lie algebras with continual
space of roots, c.f. \cite{saver})
with the continual root space represented by a grading-restricted vertex algebra $V$.
\begin{lemma}
For the short sequence \eqref{shortseq} we get a continual Lie algebra $\mathcal G(V)$
with generators
\begin{equation}
\label{generators}
\left\{\Phi(v_1),\; \chi, \;\alpha(v_2),\; \delta^1_2 \Phi(v_1),\; \delta^0_3 \chi,\; \delta^1_t \alpha (v_2),
0 \le t \le 2
\right\},
\end{equation}
and commutation relations for a continual Lie algebra $\mathcal G(V)$
\begin{eqnarray}
\label{comid}
\; \;
\Phi\cdot \delta^1_t \alpha&=&\alpha \cdot \delta^1_2 \Phi \ne 0,
\nn
\delta^{0}_{3} \chi&=& \Phi \cdot \alpha,
\end{eqnarray}
with all other relations being trivial.
The sum of cohomological classes \eqref{stupor} provides an invariant of $\mathcal G(V)$.
\end{lemma}
\begin{proof}
Recall that $\Phi(v_1)(z_1) \in C^1_2(V, \W)$, $\chi \in C^0_3(V, \W)$, $\alpha \in C^1_t(V, \W)$, $0 \le t \le 2$.
One easily checks the commutation relations coming from the orthogonality and bi-grading conditions.
Further applications of \eqref{hatdelta}, \eqref{halfdelta}, and \eqref{ortho} to \eqref{shortseq}
lead to trivial results.
$\Phi \cdot \delta^1_t \alpha\ne 0$ is proven by contradiction.
It is easy to check Jacobi identities for \eqref{generators} and \eqref{comid}.
With a redefinition
\begin{eqnarray}
\label{redifinition}
H&=&\delta^0_3\chi,
\nn
H^*&=&\chi,
\nn
X_+(v_1)&=&\Phi(v_1),
\nn
X_-(v_2)&=&\alpha(v_2),
\nn
Y_+(v_1)&=&\delta^1_2\Phi(v_1),
\nn
Y_-(v_2)&=&\delta^1_t\alpha(v_2),
\end{eqnarray}
the commutation relations \eqref{comid} become:
\begin{eqnarray*}
\left[X_+(v_1), X_-(v_2) \right]&=&H,
\nn
\left[X_+(v_1), Y_-(v_1) \right] &=& \left[X_-(v_2), Y_+(v_1) \right],
\end{eqnarray*}
i.e., the orthogonality condition brings about a representation of an affinization \cite{K} of continual counterpart
of the Lie algebra $sl_2$.
Vertex algebra elements in \eqref{redifinition} play the role of
roots belonging to continual non-commutative root space given by a vertex algebra $V$.
\end{proof}
\section*{Acknowledgments}
The author would like to thank
Y.-Z. Huang, H. V. L\^e, and P. Somberg
for related discussions.
Research of the author was supported by the GACR project 18-00496S and RVO: 67985840.
\section{Appendix: Grading-restricted vertex algebras and their modules}
\label{grading}
In this section, following \cite{Huang} we recall basic properties of
grading-restricted vertex algebras and their grading-restricted generalized
modules, useful for our purposes in later sections.
We work over the base field $\C$ of complex numbers.
A vertex algebra
$(V,Y_V,\mathbf{1})$, cf. \cite{K}, consists of a $\Z$-graded complex vector space
\[
V =
\bigoplus_{n\in\Z}\,V_{(n)}, \quad \dim V_{(n)}<\infty\,\, \mbox{for each}\,\, n\in \Z,
\]
and linear map
\[
Y_V:V\rightarrow {\rm End \;}(V)[[z,z^{-1}]],
\]
for a formal parameter $z$ and a
distinguished vector $\mathbf{1_V}\in V$.
The evaluation of $Y_V$ on $v\in V$ is the vertex operator
\[
Y_V(v)\equiv Y_V(v,z) = \sum_{n\in\Z}v(n)z^{-n-1},
\]
with components
\[
(Y_{V}(v))_{n}=v(n)\in {\rm End \;}(V),
\]
where $Y_V(v,z)\mathbf{1} = v+O(z)$.
Now we describe further restrictions \cite{Huang}, defining a grading-restricted vertex algebra:
\noindent
\begin{enumerate}
\item
\noindent
{Grading-restriction condition}:
$V_{(n)}$ is finite dimensional for all $n\in \Z$, and $V_{(n)}=0$ for $n\ll 0$.
\item { Lower-truncation condition}:
For $u, v\in V$, $Y_{V}(u, z)v$ contains only finitely many negative
power terms, that is, $Y_{V}(u, z)v\in V((z))$ (the space of formal
Laurent series in $z$ with coefficients in $V$).
\item { Identity property}:
Let $\one_{V}$ be the identity operator on $V$. Then
\[
Y_{V}(\mathbf{1}_V, z)={\rm Id}_{V}.
\]
\item { Creation property}: For $u\in V$, $Y_{V}(u, z)\mathbf{1}_V\in V[[z]]$
and
\[
\lim_{z\to 0}Y_{V}(u, z)\mathbf{1}_V=u.
\]
\item { Duality}: For $u_{1}, u_{2}, v\in V$,
$v'\in V'=\coprod_{n\in \mathbb{Z}}V_{(n)}^{*}$ ($V_{(n)}^{*}$ denotes
the dual vector space to $V_{(n)}$ and $\langle\,. ,. \rangle$ the evaluation
pairing $V'\otimes V\to \C$), the series
$\langle v', Y_{V}(u_{2}, z_{2})Y_{V}(u_{1}, z_{1})v\rangle$, and
$\langle v', Y_{V}(Y_{V}(u_{1}, z_{1}-z_{2})u_{2}, z_{2})v\rangle$,
are absolutely convergent
in the regions $|z_{1}|>|z_{2}|>0$, $|z_{2}|>|z_{1}|>0$,
$|z_{2}|>|z_{1}-z_{2}|>0$, respectively, to a common rational function
in $z_{1}$ and $z_{2}$ with the only possible poles at $z_{1}=0=z_{2}$ and
$z_{1}=z_{2}$.
One assumes the existence of Virasoro vector $\omega\in V$:
its vertex operator
$Y(\omega, z)=\sum_{n\in\Z}L(n)z^{-n-2}$
is determined by Virasoro operators $L(n): V\to V$ fulfilling
(notice that with abuse of notation we denote $L_V(n)=L(n)$)
\[
[L(m), L(n)]=(m-n)L(m+n)+\frac{c}{12}(m^{3}-m)\delta_{m+b, 0}{\rm Id_V},
\]
($c$ is called the {\it central charge} of $V$).
The grading operator is given by $L(0)u=nu,\quad u\in V_{(n)}$,
($n$ is called the weight of $u$ and denoted by $\wt(u)$).
\item { $L_V(0)$-bracket formula}: Let $L_{V}(0): V\to V$
be defined by $L_{V}(0)v=nv$ for $v\in V_{(n)}$. Then
\[
[L_{V}(0), Y_{V}(v, z)]=Y_{V}(L_{V}(0)v, z)+z\frac{d}{dz}Y_{V}(v, z),
\]
for $v\in V$.
\item { $L_V(-1)$-derivative property}:
Let $L_{V}(-1): V\to V$ be the operator
given by
\[
L_{V}(-1)v=\res_{z}z^{-2}Y_{V}(v, z)\one=Y_{(-2)}(v)\one,
\]
for $v\in V$. Then for $v\in V$,
\begin{equation*}
\label{derprop}
\frac{d}{dz}Y_{V}(u, z)=Y_{V}(L_{V}(-1)u, z)=[L_{V}(-1), Y_{V}(u, z)].
\end{equation*}
\end{enumerate}
Correspondingly, a {grading-restricted generalized $V$-module} is a vector space
$W$ equipped with a vertex operator map
\[
Y_{W}: V\otimes W \to W[[z, z^{-1}]],
\]
\begin{eqnarray*}
u\otimes w&\mapsto & Y_{W}(u, w)\equiv Y_{W}(u, z)w=\sum_{n\in \Z}(Y_{W})_{n}(u,w)z^{-n-1},
\end{eqnarray*}
and linear operators $L_{W}(0)$ and $L_{W}(-1)$ on $W$ satisfying conditions similar as in the
definition for a grading-restricted vertex algebra. In particular,
\begin{enumerate}
\item {Grading-restriction condition}:
The vector space $W$ is $\mathbb C$-graded, that is,
$W=\coprod_{\alpha\in \mathbb{C}}W_{(\alpha)}$, such that
$W_{(\alpha)}=0$ when the real part of $\alpha$ is sufficiently negative.
\item { Lower-truncation condition}:
For $u\in V$ and $w\in W$, $Y_{W}(u, z)w$ contains only finitely many negative
power terms, that is, $Y_{W}(u, z)w\in W((z))$.
\item { Identity property}:
Let ${\rm Id}_{W}$ be the identity operator on $W$,
$Y_{W}(\mathbf{1}, z)={\rm Id}_{W}$.
\item { Duality}: For $u_{1}$, $u_{2}\in V$, $w\in W$,
$w'\in W'=\coprod_{n\in \mathbb{Z}}W_{(n)}^{*}$ ($W'$ is
the dual $V$-module to $W$), the series
\begin{eqnarray}
\label{porosyataw}
&&
\langle w', Y_{W}(u_{1}, z_{1})Y_{W}(u_{2}, z_{2})w\rangle,
\nn
&&
\langle w', Y_{W}(u_{2}, z_{2})Y_{W}(u_{1}, z_{1})w\rangle,
\nn
&&
\langle w', Y_{W}(Y_{V}(u_{1}, z_{1}-z_{2})u_{2}, z_{2})w\rangle,
\end{eqnarray}
are absolutely convergent
in the regions $|z_{1}|>|z_{2}|>0$, $|z_{2}|>|z_{1}|>0$,
$|z_{2}|>|z_{1}-z_{2}|>0$, respectively, to a common rational function
in $z_{1}$ and $z_{2}$ with the only possible poles at $z_{1}=0=z_{2}$ and
$z_{1}=z_{2}$.
The locality
\[
Y_W(v_1, z_1) Y_W(v_2, z_2) \sim Y_W(v_2, z_2) Y_W(v_1, z_1),
\]
and associativity
\[
Y_W(v_1, z_1) Y_W(v_2, z_2) \sim Y_W(Y_V v_1, z_1- z_2) v_2, z_2),
\]
properties for the vertex operators in a $V$-module $W$ follow from the
Jacobi identity \cite{K}.
\item { $L_{W}(0)$-bracket formula}: For $v\in V$,
\[
[L_{W}(0), Y_{W}(v, z)]=Y_{W}(L(0)v, z)+z\frac{d}{dz}Y_{W}(v, z).
\]
\item { $L_W(0)$-grading property}: For $w\in W_{(\alpha)}$, there exists
$N\in \Z_{+}$ such that $(L_{W}(0)-\alpha)^{N}w=0$.
\item { $L_W(-1)$-derivative property}: For $v\in V$,
\[
\frac{d}{dz}Y_{W}(u, z)=Y_{W}(L_{V}(-1)u, z)=[L_{W}(-1), Y_{W}(u, z)].
\]
\end{enumerate}
For $v\in V$, and $w \in W$, the intertwining operator
\begin{eqnarray}
\label{interop}
&& Y_{WV}^{W}: V\to W,
\nn
&&
v \mapsto Y_{WV}^{W}(w, z) v,
\end{eqnarray}
is defined by
\begin{eqnarray}
\label{wprop}
Y_{WV}^{W}(w, z) v= e^{zL_W(-1)} Y_{W}(v, -z) w.
\end{eqnarray}
\subsection{Non-degenerate invariant bilinear form on $V$}
\label{liza}
The subalgebra
\[
\{L_V(-1),L_V(0),L_V(1)\}\cong SL(2,\mathbb{C}),
\]
associated with M\"{o}bius transformations on
$z$ naturally acts on $V$, (cf., e.g. \cite{K}).
In particular,
\begin{equation}
\gamma_{\lambda}=\left(
\begin{array}{cc}
0 & \lambda\\
-\lambda & 0\\
\end{array}
\right)
:z\mapsto w=-\frac{\lambda^{2}}{z},
\label{eq: gam_lam}
\end{equation}
is generated by
\[
T_{\lambda }= \exp\left(\lambda L_V{(-1)}\right)
\; \exp\left({\lambda}^{-1}L_V(1)\right) \; \exp\left(\lambda L_V(-1)\right),
\]
where
\begin{equation}
T_{\lambda }Y(u,z)T_{\lambda }^{-1}=
Y\left(\exp \left(-\frac{z}{\lambda^{2}}L_V(1)\right)
\left(-\frac{z}{\lambda}\right)^{-2L_V(0)}u,-\frac{\lambda^{2}}{z}\right). \label{eq: Y_U}
\end{equation}
In our considerations (cf. Appendix \ref{sphere}) of Riemann sphere
sewing, we use in particular,
the M\"{o}bius map
\[
z\mapsto z'= \epsilon/z,
\]
associated with the sewing condition \eqref{pinch} with
\begin{equation}
\lambda=-\xi\epsilon^{\frac{1}{2}},
\label{eq:lamb_eps}
\end{equation}
with $\xi\in\{\pm \sqrt{-1}\}$.
The adjoint vertex operator \cite{K, FHL}
is defined by
\begin{equation}
Y^{\dagger }(u,z)=\sum_{n\in \Z}u^{\dagger }(n)z^{-n-1}= T_{\lambda}Y(u,z)T_{\lambda}^{-1}. \label{eq: adj op}
\end{equation}
A bilinear form $\langle . , . \rangle_{\lambda}$ on $V$ is
invariant if for all $a,b,u\in V$,
if
\begin{equation}
\langle Y(u,z)a,b\rangle_{\lambda} =
\langle a,Y^{\dagger }(u,z)b\rangle_{\lambda},
\label{eq: inv bil form}
\end{equation}
i.e.
\[
\langle u(n)a,b\rangle_{\lambda} =
\langle a,u^{\dagger }(n)b\rangle_{\lambda}.
\]
Thus it follows that
\begin{equation}
\label{dubay}
\langle L_V(0)a,b\rangle_{\lambda} =\langle a,L_V(0)b\rangle_{\lambda},
\end{equation}
so that
\begin{equation}
\label{condip}
\langle a,b\rangle_{\lambda} =0,
\end{equation}
if $wt(a)\not=wt(b)$ for homogeneous $a,b$.
One also finds
\[
\langle a,b\rangle_{\lambda} = \langle b,a \rangle_{\lambda}.
\]
The form
$\langle . , .\rangle_{\lambda}$ is unique up to normalization if $L_V(1)V_{1}=V_{0}$.
Given any $V$ basis $\{ u^{\alpha}\}$ we define the
dual $V$ basis $\{ \overline{u}^{\beta}\}$ where
\[
\langle u^{\alpha} ,\overline{u}^{\beta}\rangle_{\lambda}=\delta^{\alpha\beta}.
\]
\section{Appendix:
A sphere formed from sewing of two spheres}
\label{sphere}
The matrix element for a number of vertex operators of a vertex algebra is usually associated \cite{FHL, FMS, TUY}
with a vertex algebra character on a sphere. We extrapolate this notion to the case of $\W_{z_1, \ldots, z_n}$ spaces.
In Section \ref{productc} we explained that a space $\W_{z_1, \ldots, z_n}$ can be associated with a Riemann sphere with marked points,
while the product of two such spaces is then associated with a sewing of such two spheres with a number
of marked
points
and extra points with local coordinates identified with formal parameters of $\W_{x_1, \ldots, x_k}$ and $\W_{y_1, \ldots, y_n}$.
In order to supply an appropriate geometric construction for the product,
we use the $\epsilon$-sewing procedure (described in this Appendix) for two initial spheres to obtain a matrix element associated with \eqref{gendef}.
\begin{remark}
In addition to the $\epsilon$-sewing procedure of two initial spheres, one can alternatively use
the self-sewing procedure \cite{Y} for the sphere to get, at first, the torus, and then by sending parameters
to appropriate limit by shrinking genus to zero. As a result, one obtains again the sphere but with a
different parameterization. In the case of spheres, such a procedure
consideration of the product of $\W$-spaces so we focus in this paper on the $\epsilon$-formalizm only.
\end{remark}
In our particular case of $\W$-values rational functions obtained from matrix elements \eqref{def}
two initial
auxiliary
spaces we take Riemann spheres $\Sigma^{(0)}_a$, $a=1$, $2$, and the resulting
space is formed by
the sphere $\Sigma^{(0)}$ obtained by the procedure of sewing $\Sigma^{(0)}_a$.
The formal parameters $(x_1, \ldots, x_k)$ and $(y_{1}, \ldots, y_n)$ are identified with
local coordinates of $k$ and $n$ points on two initial spheres $\Sigma^{(0)}_a$, $a=1$, $2$ correspondingly.
In the $\epsilon$ sewing procedure, some $r$ points
among
$(p_1, \ldots, p_k)$
may coincide with
points
among $(p'_{1}, \ldots, p'_n)$
when we identify the annuluses \eqref{zhopki}.
This corresponds to the singular case of coincidence of $r$ formal parameters.
Consider the sphere formed by sewing together two initial spheres in the sewing scheme referred to
as the $\epsilon$-formalism in \cite{Y}.
Let $\Sigma_a^{(0)}$,
$a=1$, $2$
be
to initial spheres.
Introduce a complex sewing
parameter $\epsilon$ where
\[
|\epsilon |\leq r_{1}r_{2},
\]
Consider $k$ distinct points on $p_i \in \Sigma_{1}^{(0)}$, $i=1, \ldots, k$,
with local coordinates $(x_1, \ldots, x_{k}) \in F_{k}\C$,
and distinct points $p_j \in \Sigma_{2}^{(0)}$, $j=1, \ldots, n$,
with local coordinates $(y_{1},\ldots ,y_{n})\in F_{n}\C$,
with
\[
\left\vert x_{i}\right\vert
\geq |\epsilon |/r_{2},
\]
\[
\left\vert y_{i}\right\vert \geq |\epsilon |/r_{1}.
\]
Choose a local coordinate $z_{a}\in \mathbb{C}$
on $\Sigma^{(0)}_a$ in the
neighborhood of points $p_{a}\in\Sigma^{(0)}_a$, $a=1$, $2$.
Consider the closed disks
\[
\left\vert \zeta_{a} \right\vert \leq r_{a},
\]
and excise the disk
\begin{equation}
\label{disk}
\{
\zeta_{a}, \; \left\vert \zeta_{a}\right\vert \leq |\epsilon |/r_{\overline{a}}\}\subset
\Sigma^{(0)}_a,
\end{equation}
to form a punctured sphere
\begin{equation*}
\widehat{\Sigma}^{(0)}_a=\Sigma^{(0)}_a \backslash \{\zeta_{a},\left\vert
\zeta_{a}\right\vert \leq |\epsilon |/r_{\overline{a}}\}.
\end{equation*}
We use the convention
\begin{equation}
\overline{1}=2,\quad \overline{2}=1.
\label{bardef}
\end{equation}
Define the annulus
\begin{equation}
\label{zhopki}
\mathcal{A}_{a}=\left\{\zeta_{a},|\epsilon |/r_{\overline{a}}\leq \left\vert
\zeta_{a}\right\vert \leq r_{a}\right\}\subset \widehat{\Sigma}^{(0)}_a,
\end{equation}
and identify $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ as a single region
$\mathcal{A}=\mathcal{A}_{1}\simeq \mathcal{A}_{2}$ via the sewing relation
\begin{equation}
\zeta_{1}\zeta_{2}=\epsilon. \label{pinch}
\end{equation}
In this way we obtain a genus zero compact Riemann surface
\[
\Sigma^{(0)}=\left\{ \widehat {\Sigma}^{(0)}_1
\backslash \mathcal{A}_{1} \right\}
\cup \left\{\widehat{\Sigma}^{(0)}_2 \backslash
\mathcal{A}_{2}\right\}\cup \mathcal{A}.
\]
This sphere form a suitable geometrical model for the construction of a product of $\W$-valued rational forms
in Section \ref{productc}.
\section{Appendix: proof of Proposition \ref{tosya}}
\label{duda}
\begin{proof}
For a vertex operator $Y_{V, W}(v,z)$ let us introduce a notation $\omega_{V, W}=Y_{V, W}(v,z)\; dz^{{\rm wt} v}$.
Let us use notations \eqref{zsto} and \eqref{notari}.
According to \eqref{hatdelta}, the action of
$\delta_{m + m'-t}^{k + n-r}$ on $\widehat{R} \F(
v_1, x_1; \ldots; v_k, x_k; v'_1, y_1; \ldots; v'_k, y_n
; \epsilon)$
is given by
\begin{eqnarray*}
&&
\langle w',
\delta_{m + m'-t}^{k + n-r} \widehat{R} \; \F(
v_1, x_1; \ldots; v_k, x_k; v'_1, y_1; \ldots; v'_n, y_n
; \epsilon) \rangle
\nn
&& \quad =
\langle w', \sum_{i=1}^{k
}(-1)^{i} \;
\widehat{R} \; \F ( \widetilde{v}_1, z_1; \ldots; \widetilde{v}_{i-1}, z_{i-1}; \; \omega_V (\widetilde{v}_i, z_i
- z_{i+1})
\widetilde{v}_{i+1}, z_{i+1}; \; \widetilde{v}_{i+2}, z_{i+2};
\nn
&& \qquad \qquad \qquad
\ldots; \widetilde{v}_k, z_k; \widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+n}, z_{k+n}; \epsilon ) \rangle
\end{eqnarray*}
\begin{eqnarray*}
&& \qquad +
\sum_{i=1}^{n-r}(-1)^{i} \; \langle w',
\F \left( \widetilde{v}_1, z_1; \ldots; \widetilde{v}_k, z_k;
\widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+i-1}, z_{k+i-1}; \right.
\nn
&&\qquad \qquad \qquad
\omega_V \left(\widetilde{v}_{k+i}, z_{k+i}
- z_{k+i+1} ) \;
\widetilde{v}_{k+i+1}, z_{k+i+1}; \right.
\nn
&&
\left.
\qquad \qquad \qquad \qquad \qquad \qquad
\widetilde{v}_{k+i+2}, z_{k+i+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}; \epsilon \right) \rangle
\end{eqnarray*}
\begin{eqnarray*}
&& \qquad + \langle w',
\omega_W \left(\widetilde{v}_1, z_1
\right) \; \F (\widetilde{v}_2, z_2
; \ldots; \widetilde{v}_{k}, z_k; \widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}; \epsilon
) \rangle
\end{eqnarray*}
\begin{eqnarray*}
& &\qquad + \langle w, (-1)^{k+n+1-r}
\omega_W(\widetilde{v}_{k+n-r+1}, z_{k+n-r+1}
)
\;
\nn
&&
\qquad \qquad \qquad \F(\widetilde{v}_1, z_1
; \ldots; \widetilde{v}_k, z_k; \widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}; \epsilon )
\rangle
\end{eqnarray*}
\begin{eqnarray*}
&& \quad =
\sum\limits_{u\in V}
\langle w', \sum_{i=1}^{k}(-1)^{i} \; Y^W_{VW}(
\F ( \widetilde{v}_1, z_1; \ldots; \widetilde{v}_{i-1}, z_{i-1}; \; \omega_V (\widetilde{v}_i, z_i
- z_{i+1})
\widetilde{v}_{i+1}, z_{i+1}; \;
\nn
&& \qquad \qquad \qquad
\widetilde{v}_{i+2}, z_{i+2}; \ldots; \widetilde{v}_k, z_k), \zeta_1) u \rangle
\nn
&&
\qquad \qquad \qquad \qquad \qquad \qquad
\langle w', Y^W_{VW}( \F(\widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}), \zeta_2)
\overline{u} \rangle
\end{eqnarray*}
\begin{eqnarray*}
&& \qquad +
\sum\limits_{u\in V} \sum_{i=1}^{n-r}(-1)^{i} \; \langle w', Y^W_{VW}(
\F \left( \widetilde{v}_1, z_1; \ldots; \widetilde{v}_k, z_k), \zeta_1) u \rangle \right.
\nn
&&
\qquad \qquad \qquad
\langle w',
Y^W_{VW}( \F(\widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+i-1}, z_{k+i-1};
\nn
&&
\qquad \qquad \qquad \qquad \omega_V ( \widetilde{v}_i, z_{k+i}
- z_{k+i+1}) \;
\widetilde{v}_{k+i+1}, z_{k+i+1}; \widetilde{v}_{k+i+2}, z_{k+i+2};
\nn
&&
\qquad \qquad \qquad \qquad \qquad \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r} ), \zeta_2) \overline{u} \rangle
\end{eqnarray*}
\begin{eqnarray*}
&& \qquad + \sum\limits_{u\in V} \langle w', Y^W_{VW}(
\omega_W \left(\widetilde{v}_1, z_1 \right) \; \F (\widetilde{v}_2, z_2 ; \ldots; \widetilde{v}_{k}, z_k), \zeta_1) u \rangle
\nn
&&
\qquad \qquad \langle w', Y^W_{VW}( \F( \widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r} ), \zeta_2) \overline{u} \rangle
\end{eqnarray*}
\begin{eqnarray*}
&&
\qquad + \sum\limits_{u\in V} \langle w', Y^W_{VW}( (-1)^{k+1}
\omega_W \left(\widetilde{v}_{k+1}, z_{k+1}
\right) \; \F (\widetilde{v}_1, z_1 ; \ldots; \widetilde{v}_{k}, z_k), \zeta_1) u\rangle
\nn
&& \qquad \qquad \qquad
\langle w', Y^W_{VW}(
\F( \widetilde{v}_{k+2}, z_{k+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}), \zeta_2) \overline{u} \rangle
\end{eqnarray*}
\begin{eqnarray*}
&&
\qquad -
\sum\limits_{u\in V} \langle w', (-1)^{k+1} \langle w', Y^W_{VW}(
\omega_W \left(\widetilde{v}_{k+1}, z_{k+1}
\right) \; \F (\widetilde{v}_1, z_1 ; \ldots; \widetilde{v}_{k}, z_k), \zeta_1) u \rangle
\nn
&&
\qquad \qquad \qquad
\langle w', Y^W_{VW}( \F(\widetilde{v}_{k+2}, z_{k+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}), \zeta_2) \overline{u}
\rangle
\nn
&&
\qquad + \sum\limits_{u\in V} \langle w', Y^W_{VW}(
\F(\widetilde{v}_1, z_1; \ldots; \widetilde{v}_k, z_k), \zeta_1) u\rangle
\nn
&&
\qquad \qquad
\langle w', Y^W_{VW}(
\omega_W(\widetilde{v}_{k+n-r+1}, z_{k+n-r+1})\;
\nn
&&
\qquad \qquad \qquad \qquad \F(\widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r} ), \zeta_2) \overline{u}\rangle
\nn
&& \qquad
- \sum\limits_{u\in V} \langle w', Y^W_{VW}(
\F(\widetilde{v}_1, z_1; \ldots; \widetilde{v}_k, z_k), \zeta_1) \rangle
\nn
&&
\qquad \qquad \langle w', Y^W_{VW}(
\omega_W(\widetilde{v}_{k+n-r+1}, z_{k+n-r+1})
\nn
&&
\qquad \qquad \qquad \qquad \F( \widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r} ), \zeta_2) \rangle
\end{eqnarray*}
\begin{eqnarray*}
&& \quad =
\sum\limits_{u\in V}
\langle w', \; Y^W_{VW} (
\delta^k_m\F ( \widetilde{v}_1, z_1; \ldots; \widetilde{v}_k, z_k), \zeta_1 ) u \rangle
\nn
&&
\qquad \qquad \qquad \qquad
\langle w', Y^W_{VW}( \F(\widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}), \zeta_2)
\overline{u} \rangle
\nn
&& \qquad + (-1)^k
\sum\limits_{u\in V} \langle w', Y^W_{VW}(
\F ( \widetilde{v}_1, z_1; \ldots; \widetilde{v}_k, z_k), \zeta_1) u \rangle
\nn
&&
\qquad \qquad \qquad
\langle w', Y^W_{VW}( \delta^{n-r}_{m'-t}
\F(\widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r} ), \zeta_2 ) \overline{u} \rangle
\end{eqnarray*}
\begin{eqnarray*}
&& \quad =
\langle w',
\delta^k_m\F ( \widetilde{v}_1, z_1; \ldots; \widetilde{v}_k, z_k) \cdot_\epsilon
\langle w', \F(\widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}) \rangle
\nn
&& \qquad + (-1)^k
\langle w',
\F ( \widetilde{v}_1, z_1; \ldots; \widetilde{v}_k, z_k) \cdot_{\epsilon}
\delta^{n-r}_{m'-t} \F(\widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r} ) \rangle,
\end{eqnarray*}
since,
\begin{eqnarray*}
&& \sum\limits_{u\in V} \langle w', (-1)^{k+1} Y^W_{VW}(
\omega_W \left(\widetilde{v}_{k+1}, z_{k+1}
\right) \; \F (\widetilde{v}_1, z_1 ; \ldots; \widetilde{v}_{k}, z_k), \zeta_1) u \rangle
\nn
&&
\qquad \qquad \langle w', Y^W_{VW}( \F(\widetilde{v}_{k+2}, z_{k+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}), \zeta_2) \overline{u}
\rangle
\end{eqnarray*}
\begin{eqnarray*}
&&
=\sum\limits_{u\in V} \langle w', (-1)^{k+1} e^{\zeta_1 L_W{(-1)}} Y_W(u, -\zeta_1) \;
\omega_W \left(\widetilde{v}_{k+1}, z_{k+1}
\right) \; \F (\widetilde{v}_1, z_1 ; \ldots; \widetilde{v}_{k}, z_k) \rangle
\nn
&&
\qquad \qquad \langle w', Y^W_{VW}( \F(\widetilde{v}_{k+2}, z_{k+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}), \zeta_2) \overline{u}
\rangle
\end{eqnarray*}
\begin{eqnarray*}
&&
=\sum\limits_{u\in V} \langle w', (-1)^{k+1} e^{\zeta_1 L_W{(-1)}} \omega_W \left(\widetilde{v}_{k+1}, z_{k+1} \right)
Y_W(u, -\zeta_1) \;
\; \F (\widetilde{v}_1, z_1 ; \ldots; \widetilde{v}_{k}, z_k) \rangle
\nn
&&
\qquad \qquad \langle w', Y^W_{VW}( \F(\widetilde{v}_{k+2}, z_{k+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}), \zeta_2) \overline{u}
\rangle
\end{eqnarray*}
\begin{eqnarray*}
\nn
&&
=
\sum\limits_{u\in V} \langle w', (-1)^{k+1} \;
\omega_W \left(\widetilde{v}_{k+1}, z_{k+1} +\zeta_1 \right)\; e^{\zeta_1 L_W{(-1)}}
Y_W(u, -\zeta_1) \;
\; \F (\widetilde{v}_1, z_1 ; \ldots; \widetilde{v}_{k}, z_k) \rangle
\nn
&&
\qquad \qquad \langle w', Y^W_{VW}( \F(\widetilde{v}_{k+2}, z_{k+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}), \zeta_2) \overline{u}
\rangle
\end{eqnarray*}
\begin{eqnarray*}
&&
=\sum\limits_{v\in V}
\sum\limits_{u\in V}
\langle v', (-1)^{k+1} \; \omega_W \left(\widetilde{v}_{k+1}, z_{k+1}+\zeta_1 \right) w \rangle
\nn
&&
\qquad \qquad \langle w', e^{\zeta_1 L_W{(-1)}}
Y_W(u, -\zeta_1) \;
\; \F (\widetilde{v}_1, z_1 ; \ldots; \widetilde{v}_{k}, z_k) \rangle
\nn
&&
\qquad \qquad \langle w', Y^W_{VW}( \F(\widetilde{v}_{k+2}, z_{k+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}), \zeta_2) \overline{u}
\rangle
\end{eqnarray*}
\begin{eqnarray*}
&&
=
\sum\limits_{u\in V}
\langle w', e^{\zeta_1 L_W{(-1)}}
Y_W(u, -\zeta_1) \;
\; \F (\widetilde{v}_1, z_1 ; \ldots; \widetilde{v}_{k}, z_k) \rangle
\nn
&&
\qquad \qquad \sum\limits_{v\in V} \langle v', (-1)^{k+1} \; \omega_W \left(\widetilde{v}_{k+1}, z_{k+1}+\zeta_1 \right) w \rangle
\nn
&&
\langle w', Y^W_{VW}( \F(\widetilde{v}_{k+2}, z_{k+2}; \ldots;
\nn
&&
\qquad \qquad \qquad \qquad \widetilde{v}_{k+n-r}, z_{k+n-r}), \zeta_2) \overline{u}
\rangle
\end{eqnarray*}
\begin{eqnarray*}
&&
=
\sum\limits_{u\in V}
\langle w',
Y^{W}_{VW}(
\F (\widetilde{v}_1, z_1 ; \ldots; \widetilde{v}_{k}, z_k) , \zeta_1) u \; \rangle
\nn
&&
\qquad \qquad
\langle w', (-1)^{k+1} \; \omega_W \left(\widetilde{v}_{k+1}, z_{k+1}+\zeta_1 \right)
\;
\nn
&&
\qquad \qquad Y^W_{VW}( \F(\widetilde{v}_{k+2}, z_{k+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}), \zeta_2) \overline{u}
\rangle
\end{eqnarray*}
\begin{eqnarray*}
&&
=
\sum\limits_{u\in V}
\langle w',
Y^{W}_{VW}(
\F (\widetilde{v}_1, z_1 ; \ldots; \widetilde{v}_{k}, z_k) , \zeta_1) u \; \rangle
\nn
&&
\qquad \qquad
\langle w', (-1)^{k+1} \; \omega_W \left(\widetilde{v}_{k+1}, z_{k+1}+\zeta_1 \right)
\nn
&&
\qquad \qquad
\; e^{\zeta_2 L_W{(-1)}} Y_W(\overline{u}, -\zeta_2) \; \F(\widetilde{v}_{k+2}, z_{k+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r})
\rangle
\end{eqnarray*}
\begin{eqnarray*}
&&
=
\sum\limits_{u\in V}
\langle w',
Y^{W}_{VW}(
\F (\widetilde{v}_1, z_1 ; \ldots; \widetilde{v}_{k}, z_k) , \zeta_1) u \; \rangle
\nn
&&
\qquad
\langle w', (-1)^{k+1} \;
\; e^{\zeta_2 L_W{(-1)}} \; Y_W(\overline{u}, -\zeta_2)
\;
\omega_W \left(\widetilde{v}_{k+1}, z_{k+1}+\zeta_1-\zeta_2 \right)
\nn
&&
\qquad \qquad \; \F(\widetilde{v}_{k+2}, z_{k+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r})
\rangle
\end{eqnarray*}
\begin{eqnarray*}
&&
\qquad \qquad
= \sum\limits_{u\in V} \langle w', Y^W_{VW}(
\F(\widetilde{v}_1, z_1; \ldots; \widetilde{v}_k, z_k), \zeta_1) u \rangle
\nn
&&
\qquad \qquad \langle w', Y^W_{VW}(
\omega_W(\widetilde{v}_{k+1}, z_{k+1}) \; \F( \widetilde{v}_{k+2}, z_{k+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r} ), \zeta_2) \overline{u} \rangle,
\end{eqnarray*}
due to locality \eqref{porosyataw} of vertex opertors, and arbitrarness of $\widetilde{v}_{k+1}\in V$ and $z_{k+1}$,
we can always put
\[
\omega_W \left(\widetilde{v}_{k+1}, z_{k+1}+\zeta_1-\zeta_2 \right) =\omega_W(\widetilde{v}_{k+2}, z_{k+2}),
\]
for $\widetilde{v}_{k+1}=\widetilde{v}_{k+2}$, $z_{k+2}= z_{k+1}+\zeta_2-\zeta_1$.
\end{proof}
| 123,435
|
Every reasons Austria is the best place to go wine tasting in the world.
1. Roll out the Red (or White) Carpet
The glory of Vienna makes itself known on your first taxi ride through the center of town. Few cities can equal its architectural grandeur. The city is stuffed with impressive facades, and dotted by palaces. Never tasted wine in a palace? You're in for a treat. If you happen to be in town for one of the city's major wine events, you'll get the chance to sip wines in rooms from which a large chunk of Europe was ruled until the 19th Century. The bi-annual VieVinum is the largest of these events, and is held in the massive Hofburg, the imperial palace of the Hapsburg empire. With a glass of something good in your hand, you'll feel positively regal. Then when you've had enough, go for a concert in one of the world's acoustically perfect concert halls, and a late dinner at a Michelin-starred restaurant.
2. Wine Bars... Everywhere
They exist all over Austria, but the institution of the heuriger finds its most perfect expression in Vienna, where literally hundreds of these convivial outlets for drinking, dining, and talking fill the city. The culture of the heuriger can be traced back to the Empress Maria Theresa and her son, Emperor Joseph II, who passed laws allowing any farmer to make and sell their own wines (and certain prepared foods) without taxation from the empire. Almost instantly a new industry arose. Now you can find several hundred heurigen throughout Vienna, each offering wine by the glass or by the flask, a bit to eat, and a convivial, casual family atmosphere often featuring live music.
3. Downtown Vineyards
The laws that gave rise to heurigen and embedded wine deeply in the culture of Austria also preserved one of the wonders of the wine world. Vienna boasts something no other major capital city in the world can: an entire wine region within the city limits. For centuries, vineyards have dotted the hills surrounding downtown Vienna, coexisting alongside posh real estate that everyone from Mozart to Freud inhabited at one time or another. Thanks to the economic viability of the heuriger (which, because wines aren't bottled or shipped anywhere, allows a farmer to profit from a small parcel of grapes) these vineyards were never sold to make way for more houses, and still produce wine. Chief among their products is a wine named Gemischter Satz, a white blend of sometimes up to twenty different grape varieties that remains one of the most unique (and tasty) white wines of the world.
4. 50 Minutes to Wine Country
Even though wine country is walking distance from the center of Vienna, plenty more awaits an easy (and pleasurable) hour's drive from the capital. Hop onto the well-maintained autobahns and speed (within the limit, of course) your way to one of Austria's several unique wine countries. Find your way to the Thermenregion where Cistercians settled centuries ago, because, in part, it was a good place to grow their noble Burgundian grape varieties. Wind your way along the Danube and up a stunning river valley into the Kremstal, home to picturesque villages full of crisp white wines. Or make your way south to Burgenland, on the shores of the Neusiedlersee, where intense red wines co-exist with sweet whites that can hold their own among the greatest sweet wines of the world.
5. The Wachau
And then there is the Wachau, which alone recommends Austria as a destination for the wine lover. Most wine growing regions of the world are beautiful, but a very few rise to the label of spectacular. The Wachau, with its steep slopes of terraced vineyards, and small terra-cotta roofed villages on the banks of the Danube certainly qualifies for superlatives of many kinds. Spending a day hiking (or biking!) through the vineyards with their panoramic views, tasting world-class Riesling and Grüner Veltliner, and then eating a lovely dinner in small restaurants that feel like an extension of a family's kitchen is an experience that every wine lover should have.
6. The Elegance of Zalto
Scenery, and even wine quality aside, tasting wine at wineries in Austria is better than most any other region in the world simply because chances are whatever you're tasting will be poured for you in a glass made by Zalto. No matter what you believe about whether a wine glass affects the way a wine tastes (I'm among the unbelievers) there's no denying that tasting wine from Zalto stems is among the most pleasurable ways to go. These ultra-thin hand-blown crystal stems are a downright pleasure to hold and drink from, thanks to fantastic design and craftsmanship. These delicate works of art cost $50 or more each, and therefore aren't practical for wine lovers of modest means, but they're one of the most refined aspects of tasting wine in Austria.
7, 8, and 9: Riesling, Grüner Veltliner and Blaufränkisch
Oh yeah. The wines. They're really good. Really, really good. Most wine lovers have heard of Grüner Veltliner at this point, the spicy, crisp, and fruity white wine that largely put Austria on the world's radar, with the help of evangelical sommeliers all over the globe. The grape finds varied expressions in Austria's many wine regions, but hits the highest notes in the Wachau, the Kremstal, and the Kamptal regions. These three regions also grow world-class (mostly dry) crystalline citrus and stone Rieslings that can age for decades. The much more obscure red Blaufränkisch grape is quickly becoming Austria's third claim to fame. In the hands of many of the Burgunland region's talented winemakers, the bright berry, earth-tinged Blaufränkisch has begun to achieve heights of expression reserved for only the world's noblest grape varieties.
10. Austrian Hospitality
Having traveled to many wine regions in the world, I can say that Austria ranks among the highest for gracious hospitality. Certainly when I travel as a journalist, I am received well by those who expect me, but I also visit plenty of restaurants, hotels, and even a few wineries to whom I am just another English-speaking tourist. Ive found the Austrians to be a warm, welcoming lot. Quick to celebrate, quicker to smile, and quite eager to provide a great experience to their guests. This masterful hospitality I believe traces its roots back to the same cultural origins as the heurigen. For centuries, a good portion of Austria's population has been welcoming strangers as well as neighbors into their homes to eat and drink, and it really shows.
I've now been to Austria twice, and I'm already plotting my return. If you haven't yet been, I highly recommend it.
Great post Alder, I couldn't agree more with all 10 on your list! THE best wine travel experience for me yet, only if the US World Cup games began before midnight the 10 days in Austria would have been perfect. Oh seriously no complaints...on and those trays of cream puffs they served at dinner one night...trays full! Can not wait to return soon too - Ali
Great post Alder. I will take up your suggestion at the first opportunity.
Loved the Rieslings from Styria. Wish I could find them in the South Bay. Our wine bar experience in Innsbruck was awful due to the fact that everyone smoked. How could they enjoy the wine.
Guess we will have to travel to the outdoor wine bars.
| 188,347
|
CHARLESTON - A Cape Girardeau man was found guilty for his role in the armed robbery of a Miner business Tuesday by a Mississippi County jury.
Darrin Meuir, 24, was found guilty of the class A felony of first degree robbery and an unclassified felony of armed criminal action in connection with the Sept. 13, 2001, robbery of the Keller Truck Stop in Miner.
The robbery occurred around 3:30 a.m. at the truck stop. According to reports, Christopher Shipman had entered the business wearing a gas mask as a disguise while being armed with what appeared to be a black revolver. At gunpoint, Shipman demanded money for cashier Lois Lutes and escaped from the truck stop with approximately $600 and a cash drawer. Shipman got into a car that was driven by Meuir.
The report stated that Meuir drove Shipman to the robbery and was keeping watch while armed with a fully loaded 30-06 rifle. Both defendants then drove out of Miner and into Sikeston, where they were pursued by officers from the Miner Police Department. The car was eventually stopped by police and money from the robbery was located in the car. Both suspects were arrested at that time.
During the trial, Lutes testified that after she surrendered the cash drawer and money to Shipman, she immediately called the police. Miner officer Anthony Moody said he heard the call go out that a robbery had occurred and shortly thereafter, he spotted a car exiting onto Interstate 55 from Highway HH.
Moody said he got behind the car and was able to determine with his spotlight that the passenger had long blonde hair and matched the description of the robber. He followed the suspect vehicle back through Miner and into Sikeston, where it was eventually pulled over by Miner Police Chief Roger Moore with assistance from the Sikeston Department of Public Safety.
Donald Massey, also an officer with the Miner Police Department, said that after investigating the crime scene, he had found footprints leading to and from Keller's from a secluded spot on the shoulder of Highway H. He said he observed and photographed tire marks indicating that the getaway car was parked behind a tall privacy fence outside the view from Keller's Truck Stop.
Moore testified that after the suspect vehicle was stopped, he identified the passenger as Shipman and the driver as Meuir. He said Meuir was armed with a loaded 30-06 scoped rifle at the time of the stop. He also said a further search of the car revealed the cash drawer from Keller's, the money and Lutes' register card that was also taken in the robbery. The gas mask and gun used by Shipman in the robbery were also recovered.
Although Shipman later confessed that he committed the robbery, Meuir made no statements to the police, nor did he testify at trial.
Mississippi County Assistant Prosecuting Attorney Gregory B. Spencer said the state prosecuted Meuir on the theory of accomplice liability. In Missouri, a person who aids or encourages another in the commission of a crime is just as guilty as the person committing the crime. "Without the assistance of Mr. Meuir," Spencer said, "Chris Shipman never would have been able to commit this robbery in the manner that he did."
Spencer said that without the hard work of the Miner Police Department, the men might have escaped. "And with the brave testimony of Lois Lutes, these dangerous guys will not breathe free air for hopefully a long time."
In November 2001, Shipman pleaded guilty to the robbery instead of going to trial. Even though he had no prior convictions, Shipman will still have to serve at least 85 percent of the 10-year sentence handed to him by Judge David Dolan before being eligible for parole. Robbery in the first degree is defined by the Missouri Legislature as a "dangerous offense."
Since Meuir has prior felony convictions, Dolan found him to be a prior and persistent felony offender. As a result, the jury did not decide the issue of punishment in his case. Dolan has sent sentencing for Oct. 8 in Mississippi County, after a pre-sentence investigation is completed by the Department of Probation and Parole.
Spencer prosecuted the case on behalf of Scott County, even though he is now employed in Mississippi County.
| 350,425
|
The office of the Sheriff is well under way. Heck, I might even get it finished today. I got some thick balsa, cut a few small holes for windows and a big one for the door. I superglued the bits together and supported the joints with pins as before, using a few more pins to give bars to the windows. Then I got some pieces of plasticard tube (round and rectangular). I cut two lengths of the rectangular tube to the width of the Sheriff's Office, one for the bottom of the bars and one for the top. I cut a load of 5mm long bits of 4mm tube to serve as mounting points, then cut lots of 2.4mm tube to provide the bars themselves. I glued every bar in place save the end one, which I left loose. It provided the hinge for the door, you see. The details of the door's construction speak for themselves, I fancy. I cut small bits and bit bits for this and that, and cut a couple of bits of rectangular tube to fill the large gaps above and below the door proper. There's a gap for food to be passed through, but no lock.
Really, there's very little left to do. I need a desk, chair and rack of rifles for the Sheriff, and a door, roof and sign for the whole structure. Then I can stick some paint on it. It's been very easy to assemble, this wee thingy. Enjoy the pics, folks!
| 153,760
|
TITLE: Integral $\int_{0}^{1} x^2 e^{-2\pi i k x}\,dx$
QUESTION [0 upvotes]: I am trying to figure out the following integral
$$\int_{0}^{1} x^2 e^{-2\pi i k x}\,dx$$
but cannot get the correct result which should be
$$\frac{-1}{2\pi i k} + \frac{1}{2(\pi k)^2}$$
Here is what I have:
$$\int_{0}^{1} x^2 e^{-2\pi i k x}\,dx$$
Integration by parts:
$$f'(x) = e^{-2\pi i k x}, f(x) = \frac{e^{-2\pi i k x}}{-2\pi i k},$$
$$g(x) = x^2, g'(x) = 2x$$
$$\left[\frac{e^{-2\pi i k x}}{-2\pi i k} \cdot x^2\right]_0^1 - \int_{0}^{1} 2x \cdot \frac{e^{-2\pi i k x}}{-2\pi i k}\,dx$$
$$\left[\frac{e^{-2\pi i k}}{-2\pi i k}- \frac{1}{-2\pi i k}\right] + \frac{1}{\pi i k} \int_{0}^{1} e^{-2\pi i k x} \cdot x\,dx$$
Using integration by parts again
$$f'(x) = e^{-2\pi i k x}, f(x) = \frac{e^{-2\pi i k x}}{-2\pi i k},$$
$$g(x) = x, g'(x) = 1$$
$$\left[\frac{e^{-2\pi i k}}{-2\pi i k}+ \frac{1}{2\pi i k}\right] + \frac{1}{\pi i k} \left( \left[\frac{e^{-2\pi i k x}}{-2\pi i k} \cdot x\right]_0^1 - \int_{0}^{1} \frac{e^{-2\pi i k x}}{-2\pi i k}\cdot 1\,dx \right)$$
$$\left[\frac{e^{-2\pi i k}}{-2\pi i k}+ \frac{1}{2\pi i k}\right] + \frac{1}{\pi i k} \left( \frac{e^{-2\pi i k}}{-2\pi i k} + \frac{1}{2\pi i k} \left[\frac{e^{-2\pi i k}}{-2\pi i k} + \frac{1}{2\pi i k} \right] \right)$$
$$\frac{e^{-2\pi i k}}{-2\pi i k}+ \frac{1}{2\pi i k} + \frac{1}{\pi i k} \left(\frac{e^{-2\pi i k}}{-2\pi i k} - \frac{e^{-2\pi i k}}{4\pi^2 i^2 k^2} + \frac{1}{4\pi^2 i^2 k^2}\right)$$
using $e^{-2 \pi i k} = 1$ and $i^2 = -1$
$$\frac{1}{-2\pi i k} + \frac{1}{2\pi i k} + \frac{1}{\pi i k} \left(\frac{1}{-2 \pi i k} + \frac{1}{4\pi^2k^2} - \frac{1}{4\pi^2k^2} \right) $$
$$\frac{1}{2\pi^2 k^2}$$
I hope I did not make too many notational mistakes and included enough steps.
REPLY [2 votes]: Here is one way to integrate
\begin{align}
\int_{0}^{1} x^2 e^{-2\pi i k x}dx
=&\ \frac{d^2}{da^2}\left(\int_0^1 e^{-ax}dx\right)_{a=2\pi ik}
=\frac{d^2}{da^2}\left(\frac{1-e^{-a}}a\right)_{a=2\pi ik}\\
=&\ \left[\frac{2(1-e^{-a})}{a^3} -e^{-a}\left(\frac2{a^2}+\frac1a\right)\right]_{a=2\pi ik}=\frac{1}{2\pi^2k^2}- \frac{1}{2\pi i k}
\end{align}
where $e^{-2\pi i k }=1 $ is recognized.
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Dyson - V6 Animal Cord-Free Stick Vacuum - Iron
This item is no longer available in new condition.
See similar items below.
Description
Features
Keeps carpets and hard floors looking great
Care for surfaces of all types with flexible cleaning options.
Bagless technology
Saves time and energy with no bags to buy or replace.
Rechargeable battery
Provides up to 20 minutes of run time per full charge.
Helps you keep your home clean
With a 120V motor, you can achieve reliable suction.
Crevice tool
Allows you to clean in tight spaces. Combination dusting brush/upholstery tool helps you clean your furniture thoroughly.
0.1-gallon dust cup
Holds an amount of dirt, so you can empty it less.
Motorized brush for tough tasks
Lifts stubborn pet hair and ground-in dirt from surfaces while powerful suction carries it away.
Lightweight design
Helps you transport the vacuum from room to room and floor to floor.
100W of suction power
Offer reliable suction to thoroughly clean your carpets, office, vehicle and more.
What's Included
- Dyson V6 Animal
- Crevice tool, motorized brush, combination accessory tool, mini motorized tool, docking station
Key Specs
- Cleaning Path Width9.8 inches
- Vacuum TypeStick vacuums
- Bin Capacity0.1 gallons
- Product Weight4.7 pounds
- BaglessYes
- Compatible Floor TypeAll floors
- Corded/CordlessCordless
- Filter TypeOther
- Multi SurfaceYes
- Battery Charge Time210 minutes
- Attachments IncludedCrevice tool, motorized brush, combination accessory tool, mini motorized tool, docking station
- Maximum Runtime20 minutes
- Crevice ToolYes
General
- Product NameV6 Animal Cord-Free Stick Vacuum
- BrandDyson
- Model Number216829-01
- ColorIron
- Color CategoryGray
Dimension
- Product Height47.8 inches
- Product Length8.2 inches
- Product Width9.8 inches
Performance
- Adjustable SuctionYes
Feature
- BaglessYes
- Filter TypeOther
- Voltage120 volts
Warranty
- Manufacturer's Warranty - Parts2 years
- Manufacturer's Warranty - Labor2 years
Other
- UPC885609008349
Customer reviews
Rating 4.6 out of 5 stars with 2452 reviews(2,452 customer reviews)
Rating by feature
- Value4.0
Rating 4.0 out of 5 stars
- Quality4.3
Rating 4.3 out of 5 stars
- Ease of Use4.7
Rating 4.7 out of 5 stars
to a friend
Rated 5 out of 5 stars
Excellent purchase.||Posted .
We are living full time in our motor home and needed something compact and powerful. I didn't realize how much I would like this vacuum. This is excellent for this or any purpose.
This review is from Dyson - V6 Animal Cord-Free Stick Vacuum - PurpleI would recommend this to a friend
Rated 2 out of 5 stars
Disappointed||Posted .Owned for 1 year when reviewed.
June ‘21 was 2 yrs, and Nov. ‘21 it’s failing me. Will run for 2-3 minutes then shuts off. Place it back in the charging cradle, and instantly remove and it will run another 2-3 minutes. Not sure what is wrong with it, but there must be a reason Dyson only has a 2 yr warranty for cordless vacuums unlike the 5 yr on corded ones.
This review is from Dyson - V6 Animal Cord-Free Stick Vacuum - PurpleNo, I would not recommend this to a friend
Rated 5 out of 5 stars
Outstanding Light Weight Vac||Posted .
The Dyson is mounted in convenient recharging station and is exceptionally easy to use!
This review is from Dyson - V6 Animal Cord-Free Stick Vacuum - PurpleI would recommend this to a friend
Rated 3 out of 5 stars
Disappointed that Dyson could be so petty.Posted .
This is not my first Dyson but it may be my last. I have an animal upright, and when I got tired of dragging it to the carport to clean the dog hair out of the car, I went shopping for a handheld. I bought a V6 trigger for about $140 and was proud of the sales price. It worked great in the car, and other places around the house that the animal seemed too big for. I thought about the stairs but wished I had the stick and powerhead that the cordless vacuum had. I called Dyson and they said that they don't sell those parts for the trigger. I would need to buy a cordless vacuum. The V6 animal went on sale so I bought it. It came with the same accessories that the V6 trigger came with except for the stick and the powerhead. The vacuums look exactly alike. But the stick and powerhead fits only one. The attachments are altered just enough that they are not interchangeable. For a company that sells vacuums that are hundreds of dollars, you can afford to have interchangeable parts!No, I would not recommend this to a friend
Brand response from DysonProductExpertPosted .
Hi there, this is Krystal with Dyson. Thank you for taking the time to leave a review. The Dyson V6 Stick vacuums are interchangeable in that you can use it as a stick vacuum or handheld. However, if you decide to purchase just a handheld, like the V6 Trigger, it is just a handheld as advertised.
We would value the opportunity to address any questions or concerns you may have. Feel free to contact us at askdysonUS@dyson.com and we will be happy to help.
Best Wishes,
Krystal
Dyson Customer Support
Rated 5 out of 5 stars
How do I love thee? Let me count the ways . . .||Posted .. I mean, how good can a vacuum really be to justify paying that price for it? Then I saw the commercial for the Dyson cordless V6 Animal vacuums -- I admit I was intrigued! I still didn't believe they could really be that great, but when they went on sale on Black Friday 2016 I finally decided to purchase one. I got it from Best Buy knowing I could bring it back if it was just mediocre as I expected it to be. How very wrong I was! This vacuum is worth every penny! I've recently retired and am cleaning my house from top to bottom, and this vacuum has made it so EASY! It's lightweight, and easy to keep with me no matter what kind of tight space I'm working in -- the attachments are lightweight and remove and attach easily, and the attachments reach virtually ANYWHERE! The vacuum is weighted for a woman -- so I can easily maneuver it anywhere I need to, just as they showed on the demonstration videos! I have mild arthritis in my shoulder and arm and hand, but it doesn't hurt when I use the Dyson. No more lifting a heavy vacuum in tight spaces, and not being able to maneuver it where I need it to go without hurting myself. I vacuumed under my bed, and the vacuum just laid flat and I could vacuum all the way under it easily and quickly! It's so lightweight and balanced in such a way that vacuuming just seems sort of intuitive, like it "knows" what I want it to do, and it never disappoints! It removes dust and dirt from corners and tight spaces quickly and easily! This vacuum is really NOT comparable to any other vacuum on the market. Its far above and beyond a normal vacuum cleaner. As for the battery life, it's worked perfectly for me! I have a very small house, and only work in a small area at one time, so I just plug it back into its holder when I'm not using it and it charges right back up. The charging wall holder is just amazing and so convenient!! The one con would be that the debris cannister is quite small -- we have three very large dogs, so I end up with a LOT of hair in the cannister every time I vacuum, but it's really simple to empty it out, so it hasn't been a problem. I'm just so VERY glad I bought this vacuum. I never thought I could be so enthusiastic about a vacuum cleaner, but I love my Dyson!!!
This review is from Dyson - V6 Animal Cord-Free Stick Vacuum - PurpleI would recommend this to a friend
Rated 3 out of 5 stars
I have mixed feelings about this one, here is why||Posted .
I received this as a present. And I was excited to get such a great gift! Normally I would not buy expensive stuff like this for myself, but cleaning the house needs to get done. I thought at first, after seeing the reviews, that the negatives of battery time and small dust cup maybe were just having expectations that were too high. But after using it for a short while, I find those two things to be really frustrating. Why? I have a modest townhouse, with tile/wood floors downstairs. This area needs to be vacuumed each day, and we have a dog with long fur, which makes it even more necessary. My battery vac, which I've bought two of in two years had just broken down again, a motor problem, just out of warranty. And I would use that each night, and I'd fill the dust cup, empty it, and do it again the next day. But with the Dyson, I have to empty the cup 3x for the same cleaning area and conditions. And I find myself draining the battery by the time I'm done - the other battery vac I had, didn't have either issue. Now, I think I'm getting more dirt up, and certainly more dust, it is evident by the amount of stuff filling up the cup. But, while getting the dust up is great, it is hard to get the unit clean. I can remove the dust cup, I can remove the cylindrical filter, but I cannot really clean the mesh/screen by removing and washing it. This makes no sense at all. And I've noticed an accumulation of dust inside of the tube, which is a bit concerning thinking about this long term. Most amazing to me, was when I turned it on, I smelled that dog smell thing. I have hepa filters around the house, and have always had hepa vacs that don't emit that smell, well for the last 10 years anyway. For a pet vac, I found out it doesn't have a hepa filter; come on, really? A pet vac without a hepa filter, and for $300 on a black friday special? The dust builds up on the inside rubber seal, which is also not easy to clean as it stays attached to the area you cannot remove and wash. Then without a hepa filter, and a dirty seal you cannot easily clean, it is making for a long term frustration trying to keep it clean. So, now I'm thinking about the minimalist battery issue, the small dust cup issue, and the lack of a hepa filter, and I'm wondering if it is really worth the money. The problem is that battery vacs are mostly not good. I have had several and they were not powerful - the Dyson is really powerful, while it lasts. It is balanced and not heavy, and even has a way to stand it against the wall so it doesn't fall. All of this is really good, no doubt. But do the negatives outweigh the positives? Normally I use the heavy duty plug in vac a couple of times a week, and use a battery vac each night in between, and for quick pickups. The plug in vac, which you've seen on infomercials, is amazing. But it is big, heavy, and a pain to unravel the cord, plug in, fight with the cord while vacuuming, then wrap it all up - it just adds time to an already short day. The real answer is what Dyson has come out with, a longer lasting battery vac, with a hepa filter. But that is hundreds more. Is it worth it? Is it worth keeping something that on a great sale costs $300? Will this last? I know the suction goes down with the dust in the area that is hard to clean (they say to run the dust brush over it to get the dust out of there, but that doesn't work so well). Good thing Best Buy has a good return policy and I'll give it awhile to see if in fact it is worth keeping. Maybe the other units will come down in price once these are sold out. Dyson has amazing products, no doubt. And this is better than any other battery vac on the market today, because it does really outshine the performance of other battery vacs. But, in the back of my mind is how much this cost and I expected those concerns I've had (and others have had) were not really big deals - but they are a concern. Time will tell, but I thought I should mention these things so you see if you really want to invest the money for something that is already expensive, and if you are, maybe spending a bit more would be the better way to go. I'm not cleaning nut or anything, just so you know. But it seems I've added to my chore list now, cleaning the vac every couple of nights, because of the dust build up and the lack of being able to easily clean it out.
This review is from Dyson - V6 Animal Cord-Free Stick Vacuum - PurpleI would recommend this to a friend
Brand response from DysonProductExpertPosted .
Hi Stevefromtheburbs. Thank you for providing your review on the V6 Animal. We are pleased to hear that your are satisfied with the machines performance. As for cleaning the machine, we do recommend that it is cleaned occasionally to keep the machine working as intended. We do recommend using a can of compressed air to blow out any debris that may build up in the shroud area of the cyclone as we do not recommend getting the cyclone wet.
We would value the opportunity to learn more about your experience, please contact us at askdysonUS@dyson.com and we will be happy to help.
Best Wishes,
Krystal
Dyson Customer Service
Rated 5 out of 5 stars
Great vacuum and its cordless||Posted .
We were tired of pushing around heavy vacuums so we decided to get a lightweight vacuum. We considered the Shark corded and cordless, but the Dyson passed our in-store tests. We have 2/3 tile floors and 1/3 carpeted floors at our house. It really meets all of our requirements, except for runtime. We were not going to pay an additional $150 for the V8 with its double battery, so that is all the complaining I can give this topic for the V6. Its only other downside is that you have to keep the trigger depressed for it to run, and that gets tiring after 5 minutes; a lock or a palm switch would be easier. It lasts about 20 minutes, or on MAX only 6 minutes. It took a few days with charge cycles and runtime to figure out that we never really need MAX. Either floor type worked fine at the regular setting and we could see everything swirling around the canister while we cleaned, so we knew it was doing its job. So one or two rooms at a time and back to the charger. Because of its small shape a that it is so lightweight it gets right under sofas, beds, and tables; no more missed areas. It is fun to use still after a month. And in dustbuster mode, it kicks butt; I never had anything that could clean the cars so easily. Only the shop vac had more suction, but now I don't have to wind up 50' of extension cord. Because of the ease of use and low weight, we vacuum sofas and seats which were previously avoided due to size and weight of a shop vac. Now the seats are maintained cleaner than before.
This review is from Dyson - V6 Animal Cord-Free Stick Vacuum - PurpleI would recommend this to a friend
Rated 5 out of 5 stars
Amazing and powerful yet small||Posted .
Read a bunch of reviews and a lot of people had mostly only positive things to say. I've gifted two dysons in the past but hadn't bought one for myself, so when I saw that this was on sale I figured why not give it a shot. Plus it is a good size for my living space and Is cordless. This vacuum is very powerful for being as small as it is. Has plenty of handy attachments, it's light and picked up everything - the tiniest of dust particles - my old vacuum couldn't. Dual power levels means you can run at max suction power for a shorter amount of time or on the lower setting for much longer.. I can't wait to see how this works out over the next few years but for now I would recommend this to anyone that is looking for a light cordless handheld vacuum for an apartment or smaller living space.
This review is from Dyson - V6 Animal Cord-Free Stick Vacuum - PurpleI would recommend this to a friend
Q: QuestionI bought my dyson v6 3 months ago and now there is a flashing red light, doesn’t charge and won’t start. What sound I do next?
Asked by Madeline.
- A:Answer I have the same issue with the V7. My filters haven't had the chance to get dirty. In fact, the filters are still bright white. Do I attempt to return it to the store or attempt to contact Dyson per the email that a Dyson rep posted? Battery issues seem to be an ongoing issue with more than one model. Hopefully, this is resolved easily and painlessly.
Answered by Lance
Q: QuestionDoes this exhaust toward the users face at the rear of the top of the vacuum? Is there a chance that it will blow in the face of the user?
Asked by Flower56.
- A:Answer It absolutely blows directly in you face. Only part I hate
Answered by Yeap
Q: QuestionDoes it come with the Multi-angle brush, Mini Soft dusting brush and Stiff bristle brush as described on Dyson.com?
Asked by Mike.
- A:Answer Comes with the attachments in pic!
Answered by Spark
Q: Questiondo I have to hold a button/trigger for it to workdo I have to hold a button/trigger for it to work
Asked by Wohlf.
- A:Answer Yes, the button or trigger needs to be held in, but it's really not that big of a deal. I think it's good because you only get about 20 minutes of battery life and you can easily turn off when moving locations to save battery. The trigger is really sensitive too so you barely think you are holding the button in.
Answered by KaChank
Q: Questionis this for carpeting too?
Asked by Anonymous.
- A:Answer Yes! I use it on my area rugs/carpet and on my hardwood and tile floors. You can go from floor to floor without having to change anything.
Answered by GigiRM
Q: QuestionHow long does it work for on one charge? I saw the new version lasts longer. Is this the new version? Does it last longer?
Asked by Sol.
- A:Answer Still about 20 minutes
Answered by Journey
Q: QuestionDoes it have an extra rechargeable battery or can you buy one?
Asked by Senior66.
- A:Answer Hi Senior66. It's Vince with Dyson. A spare battery assembly is available from Dyson directly. You can reach out to get more info via email at askdysonus@dyson.com. Best wishes, Vince P.
Answered by DysonProductExpert
Q: QuestionWhat is the difference between Dyson - V6 Animal Bagless Cordless Stick Vacuum - Nickel/Purple model #: 210692-01 and the Dyson - V6 Bagless Cordless Stick Vacuum - White/Iron Model #:209472-01?
Asked by Eason.
- A:Answer Hello! This is Miella from Dyson. The V6 Animal comes with more tools specifically designed for homes with pets (like the mini motorized tool and mini soft dusting brush) and the direct-drive cleanerhead, which is 75% more powerful than the standard motorized cleanerhead. The V6 comes with the standard combination tool and crevice tool (the Animal comes with these as well), and the standard motorized cleanerhead. Hope this helps! Miella Dyson Customer Service.
Answered by DysonProductExpert
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The seemingly unstoppable Jodi Picoult delivers another heart-wrenching page-turner in Sing You Home, a stirring exploration of same-sex couples’ reproductive rights. Fast-paced and three-dimensional, the novel does justice to this pivotal civil rights issue, and Picoult again proves herself the queen of heartfelt social statement.
Forty-year-old music therapist Zoe Baxter and her husband, Max, have tried to have a child for nine years. When their fifth in-vitro fertilization attempt ends in a stillbirth, Max files for divorce, unwilling to try fathering a child again.
Backsliding into alcoholism, Max moves in with his brother, Reid, and sister-in-law, Liddy, who are also struggling with infertility. Confidence at rock bottom, Max comes under the influence of the charismatic, ultraconservative Pastor Clive at Reid’s evangelical church. Meanwhile, Zoe develops a close friendship with high school guidance counselor Vanessa Shaw and, to her own surprise, falls in love with her.
Zoe and Vanessa marry, and when they discuss the possibility of parenting, Zoe remembers that three frozen embryos remain from her last round of IVF with Max. When Zoe asks Max for consent to obtain them, a heated court battle erupts in which Max tries to prevent the “pre-born children” from being brought into Zoe and Vanessa’s “sinful” household. Coached by Pastor Clive and a media-drunk attorney, Max wants Reid and Liddy to be awarded the embryos instead.
Told from the perspectives of Zoe, Max and Vanessa, the story takes beautiful shape as Zoe’s loving but troubled relationship with Max falls apart and her tender one with Vanessa begins. Included with the book, a CD of songs performed by “Zoe” (with lyrics by Picoult) adds further dimension to the novel. The born-again Max sometimes verges on cartoonish, but his complicated relationship with his sister-in-law and his memories of marriage to Zoe pull his character back from the brink. At the same time, Picoult’s deft weaving of past and present gives Zoe and Vanessa engrossing depth from start to finish, and readers will be hard-pressed to put the book down before that finish comes. Thoroughly satisfying, Sing You Home truly sings.
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\section{Proof of the Main Theorem}
\label{sec:proof}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
Recall that we assume, see~Subsection~\ref{sec:convention}, that we have chosen an admissible asymptotically regular embedding $\imath:\Sigma\rightarrow T^*M$. From the assumptions $H_{k+1}(\Sigma)\not=0$ and $H_{k+1}(\ls)=H_{k+2}(\ls)=0$, we are able to construct linking sets $A$ and $B$ in the loop space, cf.~Lemma~\ref{lem:link}. We estimate $\A$ on $A$ and $B$ in Proposition~\ref{prop:estimates}. This gives rise to estimates for the penalized functionals $\A_\epsilon$ and this in turn gives the existence of critical points, cf.~Proposition~\ref{propo:crit}. Under the assumption of flat ends, Proposition~\ref{prop:pen} produces a critical point of $\A$, by taking the limit $\epsilon\rightarrow 0$. The critical point of $\A$ corresponds to a closed characteristic on $\Sigma$.
\end{proof}
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The queen of ‘glamour’ Madhuri Dixit turns 48 on 15th May, 2015. Her companions from the film industry greeted her a prosperous Happy Birthday via Twitter.
Dixit debuted 30 years ago in Bollywood with ‘Abodh’, by Hiten Nag. She became widespread with ‘Tezaab’, released in 1988, featuring Anil Kapoor. The doors of success and stardom were opened for her by 1990.
Once came a time when every director wanted to sign Madhuri Dixit (now Madhuri Dixit Nene), she represented lead cast in movies from 1990-2006 such as Ram Lakhan, Parinda, Dil, Saajan, Beta, Khalnayak, Hum Aapke Hain Kaun..!, Devdas and Dil Toh Pagal Hai. Indian government gave her the prestigious ‘Padma Shri’ in 2008.
Gulaab Gang – Madhuri Dixit
To have a family, she resigned Bollywood for almost 9-10 years and settled in America. She has two children, 1-1 son & daughter, Ryaan Nene and Arin Nene. She came back through ‘Dedh Ishqiya’ & ‘Gulab Gang’, both released in 2014. She is an undisputed rare dancer. Nowadays, she judges a dance reality show ‘Jhalak Dikhla Jaa’ with two more individuals.
Read the tweets by her companions and friends from the film industry:-
Wishing my favourite forever @MadhuriDixit a very Happy Birthday pic.twitter.com/zMUjbastXK
— Riteish Deshmukh (@Riteishd) May 15, 2015
Riteish Deshmukh, Twitter.
Happy bday to one of my true inspirations, the ever gorgeous, who danced her way through our hearts n still does @MadhuriDixit ma’am 🙂 — Yami Gautam (@yamigautam) May 15, 2015
Yami Gautum, Twitter.
Happy birthdayyy to the dancing goddess of our country @MadhuriDixit !!! wish you all the happiness in the world ma’am!!! ❤️🎂
— ShraddhaVINNIEKapoor (@ShraddhaKapoor) May 15, 2015
Shraddha Kapoor, Twitter.
Happpy bday @MadhuriDixit ..hav a great one mam — Karan Wahi (@karan009wahi) May 15, 2015
Karan Wahi, Twitter.
Hope you liked the news story, share it with your friends via buttons given below. Stay tuned for more Bollywood updates.
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TITLE: Common Galois extension over $\mathbb Q $
QUESTION [4 upvotes]: Suppose $L'$ is a fixed cyclic galois extension over $\mathbb {Q} $ of degree $4$.Now we know that there exists also a degree $k$ extension $L$ over $L'$ but the extension $(L/\mathbb{Q})$ may not be the galois extension. So my question is, can we always find such cyclic extension $(L/L')$ of degree $k$ such that $(L/\mathbb Q) $ is also a cyclic galois extension of degree $4k$?
REPLY [7 votes]: If $k$ is odd, then yes. If $L'/\mathbb{Q}$ is a cyclic extension of degree $4$, choose an extension $M/\mathbb{Q}$ that is cyclic of degree $k$. Then the compositum $L'M/\mathbb{Q}$ will have ${\rm Gal}(L'M/\mathbb{Q}) \cong {\rm Gal}(L'/\mathbb{Q}) \times {\rm Gal}(M/\mathbb{Q})$ which will be cyclic of order $4k$.
If $k$ is even, the answer is no. In particular, if $L'/\mathbb{Q}$ is a cyclic extension that is not totally real (like $L' = \mathbb{Q}(\zeta_{5})$), then it is not possible to find a degree $8$ extension $L/\mathbb{Q}$ with ${\rm Gal}(L/\mathbb{Q}) \cong \mathbb{Z}/8\mathbb{Z}$. If such an extension were to exist and $c \in {\rm Gal}(L/\mathbb{Q})$ is complex conjugation, then $c^{2} = 1$ and this implies that $\langle c \rangle = {\rm Gal}(L/L')$, which forces $c|_{L'} = 1$ and hence $L'$ must be totally real.
It should be possible to give necessary and sufficient conditions on when it is possible to find such an extension $L$. It is well-known that a quadratic extension $K/\mathbb{Q}$ can be embedded in a $\mathbb{Z}/4\mathbb{Z}$ extension if and only if $K$ is totally real and the odd primes that ramify in $K$ are all $\equiv 1 \pmod{4}$. There is an extensive literature on Galois embedding problems. For a short intro, see Section 3.4 of Klüners and Malle's paper "A database for field extensions of the rationals". For much more detail, read Chapter 4 of Malle and Matzat's "Inverse Galois Theory".
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When ever you start getting serious with any form of discipline, the higher the level the more expensive the equipment can become.
PC gaming in particular can send you on the endless search for the ultimate setup. With technology constantly improving its easy to blow money of the newest hottest accessory for your gaming arsenal.
This is where companies like Logitech come in, releasing high performance PC gaming accessories at very reasonable price points.
We recently got hands on with there latest release –
the Logitech G502 HERO High Performance Gaming Mouse
HERO is actually a acronym meaning – High Efficiency Rated Optical
This is the key feature that differentiates it from its predecessor The Proteus Spectrum.
The sensor goes all the way up to 16,000 dpi which is fast becoming the new benchmark.
It has the same layout as all the other mice in the G502 range which is a good thing, as the layout has made these some of the highest sold gaming mice on the market.. This will sample the colours from your monitor and match them with the RGB lighting on the mouse itself.
One other change that I really liked is the thinner lead that less prone to kinking, something that my Proteus Spectrum suffered from.
Tech Specs
Sensor: HERO™
Resolution: 100 – 16,000 dpi
Software: Logitech G HUB
Lighting: RGB
RRP: $149 (AUD)
Overview-
This isn’t a huge step up from previous models, but does show how Logitech continues to release affordable state of the art gaming mice.
This would definitely be a good buy if you are looking at your first high performance Gaming Mouse.
Score – 4 / 5
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Defined-contribution plans such as a 401(k), 403(b), 457(b) plan or profit-sharing plans are a self-directed way to save for retirement: you get to decide how much you want to put into the plan using tax-deferred dollars, and you get to choose from a selection of investments (or participate in company profits) as offered by the plan. Once you retire, however, the IRS has some say in how much and when you get to spend this money.
If you plan to use your 401(k) or defined-contribution plan to fund your income needs during retirement, then you will want to be aware of the IRS tax rules that tell you when you must take the money out. If you choose to roll some or all of your 401(k) in an IRA, then you should be aware of the RDM rules for IRAs.
RMD RULES FOR DEFINED-CONTRIBUTION PLANS
The amount of income you are required to withdraw from your defined-contribution plan is called your Required Minimum Distribution or RMD. The rules set forth by the Internal Revenue Service (IRS) for defined-contribution plans are slightly different than the rules for IRA plans. One of the most notable differences is that if you have more than one 401(k) or defined contribution plan – such as from multiple jobs or places of employment – you have to make each RMD withdrawal separately, for each plan. The exception to this is if you have more than one 403(b) tax-sheltered annuity.
For 401(k), profit-sharing, 403(b), or other defined contribution plans, you have until the LATER of the following two scenarios:
April 1 of the year following the calendar year in which you reach age 70½.
OR
The year in which you retire (if allowed by your plan).
Please note: for profit-sharing plans, if you are a 5% owner, you must start your RMDs according to the rules for IRAS – by APRIL 1st of the year following the year you turn 70½. The IRS specifies April 1 of the year following your 70½ birthday, but after this first withdrawal, subsequent withdrawals must be made by December 31st.
Q AND A FOR DEFINED-CONTRIBUTION PLANS
How much do I have to take out for my RMD? It is your age and the account value that determine the amount you must withdraw. Another difference between an IRA and a defined contribution plan is that with these kinds of plans, your plan sponsor or administrator should calculate the amount of your RMD for you.
Can you take out more than the RMD amount? Yes. Any amount you take out will be considered as part of your taxable income for the year. If you take out more than your RMD one year, however, you cannot apply the excess withdrawal amount to the following year.
Can I take out more than one withdrawal during the year to meet my RMD? Yes, once you turn 70½, you can take multiple withdrawals throughout the year as long as you reach your total RMD requirement by the specified time.
What is the penalty for NOT taking out your RMD? You may have to pay a 50 percent excise tax on the amount you failed to withdraw, in addition to the taxes owed. This penalty is also charged if you fail to take out enough money.
As part of a comprehensive retirement planning service, many financial advisors help their clients to plan for their RMD. To find out the most advantageous way to maximize the money in your 401(k) or defined-contribution plan, feel free to contact us and one of our financial experts will get back to you.
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TITLE: Meaning of historical fluxion notation
QUESTION [6 upvotes]: I've noticed that in 18th century English books on calculus writers would say that 'the fluxion of $ax$ is $a\dot{x}$' and 'the fluxion of $x^n$ is $n x^{n-1} \dot{x}$'. What does this extra '$\dot{x}$' at the end of the formulas for fluxions (derivatives) signify?
P.S. Apologies if historical questions are not allowed here.
REPLY [6 votes]: In the physics applications that Newton was interested in, his functions were mostly functions of time. Since he was usually differentiating with respect to time, it was OK that his dot notation, unlike Leibniz's notation, didn't indicate what he was differentiating with respect to. Newton used the symbol $o$ to indicate a fixed infinitesimal interval of time. So in Newton's notation $\dot{x}$ would be Leibniz's $dx/dt$, while $\dot{x}o$ would be $dx$ --- an infinitesimal difference, called a "moment."
Newton also used a notational shortcut that caused confusion. He used a convention that when it was clear from context that he was talking about a moment, then $\dot{x}$ would implicitly mean $\dot{x}o$, i.e., the $o$ could be left out because it was too cumbersome to write it all the time. Because of this, English mathematicians began to muddy the notational waters by not distinguishing the two notions. See Boyer, p. 201, and p. 114 for the definition of "moment."
So when someone using Newton's notation says that the fluxion of $x^n$ is $nx^{n-1}\dot{x}$, what they mean could be two different but equivalent things in Leibniz notation. Either:
(1) $d(x^n)=nx^{n-1}dx$ (where $dx$ is the same as $\dot{x}o$, and the $o$ is implicit); or
(2) $d(x^n)/dt=nx^{n-1}dx/dt$
In 17th- and 18th-century mathematics, the difference between (1) and (2) is purely a matter of dividing both sides of the equation by $dt$. (There was no notion, as in modern NSA, that a derivative is the standard part of the ratio of infinitesimals, which is not quite the same thing as the ratio of infinitesimals.)
Boyer, The History of the Calculus and its Conceptual Development. https://archive.org/details/TheHistoryOfTheCalculusAndItsConceptualDevelopment
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Personal Statement: I'm a writer, songwriter, poet, puppeteer, and artist. A lot of my works can be found on my website, BANANA PATCH FANTASY PRODUCTIONS ().
Although I would appreciate an opportunity to be self-supporting with my many projects, the purpose of my website is to define my individuality and share it with others who are appreciative of it.
Art formsMusic, Film/Video, Performance Art, Multidisciplinary, Dance
Type of artistComposer, Director, Performer, Painter/Sculptor, Writer
The Fund for Women
Artists
3739 Balboa Street #181
San Francisco, CA 94121
Phone: (415) 751-2202
info@womenarts.org
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2006-04-26 / Briefly Noted
Two-Session Bereavement Workshop at Putnam Hospital Center in May
For anyone who is grieving the loss of a loved one, Putnam Hospital Center is offering a free two-session bereavement workshop in May. "Bereavement: Exploring and Understanding Your Feelings" will be held on Tuesdays, May 16th and 23rd, from 7:00 to 8:30 pm in the ground floor cafe classrooms at the
The full version of this story is available only to subscribers. Subscribe now to access to full versions of news stories in our archive of over 12,000 stories going back to 1999.
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TITLE: Why doesn't the generating function for Fibonacci match its characteristic polynomial?
QUESTION [2 upvotes]: For Fibonacci numbers I usually see generating function $\frac{1}{1-x-x^2}$ or $\frac{x}{1-x-x^2}$ depending on initial terms.
But the denominator, $1-x-x^2$, seems different from the usual characteristic polynomial $x^2 - x - 1 = 0$. It doesn't appear to be a simple multiplication by $-1$ either since the $x$ term is negative in both cases.
I'm asking this simpler question because I am getting strange and unexpected results over here.
REPLY [4 votes]: If you write down the recurrence relation for Fibonacci numbers and the consequence it has on the generating function, you will immediately notice that $1-x-x^2$ is just the characteristic polynomial of the Fibonacci sequence with its coefficients written in the opposite order, i.e. $x^2\,p\left(\frac{1}{x}\right)$ with $p(x)$ being $x^2-x-1$.
Addendum: such "reversal" process has an interesting side-effect: the radius of convergence of the generating function is exactly the reciprocal of the modulus of the largest root of the characteristic polynomial.
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Hawaii Occupational Safety and Health
Initial approval date: Dec. 28, 1973
Program certification date: April 26, 1978
Final approval date: April 30, 1984
Final approval voluntarily withdrawn: Sept. 21, 2012
Every year, OSHA evaluates each of the 27 State Plan states and territories. Today, we’re looking at the federal agency’s review of Hawaii.
Hawaii Occupational Safety and Health suffered a setback a few years ago when it wasn’t able to maintain enough staff or conduct enough inspections. As a result, the state and federal agencies entered into an agreement where federal OSHA would take over some of the state’s duties while HIOSH makes internal improvements.
In OSHA’s fiscal year 2013 report on HIOSH, the federal agency observes Hawaii has made improvements and regained much of its enforcement authority. However, HIOSH needs to improve its implementation of policies and regulations, and the hiring and retaining of qualified personnel remains a problem for the state agency.
Of the 16 recommendations OSHA made in 2012, the state completed six, including:
- Improve response time of complaints to within two work days
- Assign inspections for compliance staff in such a way to maximize resources
- Ensure health inspectors conduct appropriate sampling during inspections
In addition to the 10 recommendations from 2012 being carried over, federal OSHA issued six new recommendations in its FY 2013 report. Those new recommendations include:
- Ensure abatement verification of serious, willful and repeat violations
- Follow procedures in investigations of discrimination complaints
- Develop and document methods used to target high-hazard industries for inspections
Percent of HIOSH inspections where serious, willful or repeat violations were issued:
FY 2011 – 56%
FY 2012 – 58%
FY 2013 – 81%
Three-year State Plan national rate – 57%
FY 2011 – 89%
FY 2012 – 52%
FY 2013 – 75%
Three-year State Plan national rate – 54%
FY 2011 – $1,118
FY 2012 – $1,278
FY 2013 – $1,305
Three-year national average – $2,245
The opinions expressed in "On Safety" do not necessarily reflect those of the National Safety Council or affiliated local Chapters.
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\subsection{\titlecap{centralized-distributed trade-off: an analytical insight}}\label{sec:cont-time-single-int-trade-off}
\done{
For \tcb{a single integrator model in continuous time,
the \tradeoff can be explicitly quantified by utilizing~\cref{prop:subopt-gain} to compute the feedback gains and
rewriting the objective function as the product}
\begin{equation}\label{eq:cont-time-single-int-trade-off-mult}
\var = \underbrace{f(n)}_{\tilde{J}_{\textrm{latency}}(n)} \cdot \ \underbrace{\sum_{j=2}^N \tilde{C}_{j}^*(n)}_{\tilde{J}_{\textrm{network}}(n)}
\end{equation}
where
$ \sigma_{I}^{2}(\tilde{\gpos}_j^*) = \tilde{C}_{j}^*(n)\taun $
and $ \tilde{C}_{j}^*(n) $ only depends on $ n $
and can be computed \tcb{exactly; see Appendix~\ref{app:cont-time-single-int-suboptimal-variance-computation}.}
{This holds because
the suboptimal eigenvalues have expression $ \tilde{\gpos}_j^* = \tilde{c}_j^*(n)\opteig $
; cf.~\cref{prop:subopt-gain}.}
Such a decomposition can be interpreted as a
decoupling of \tcb{the impact of network ($ \tilde{c}_j^*(n) $) and latency ($ \opteig $) effects on the control design.}
By inspection, it can be seen that $ \tilde{J}_{\textrm{network}}(n) $ is a decreasing function of $n$
and that $ \tilde{J}_\textrm{latency}(n) $ is determined by $ f(n) $.
Furthermore, for sublinear rates $ f(\cdot) $,
the steady-state variance can be also written according to~\eqref{eq:trade-off},
\begin{equation}\label{eq:cont-time-single-int-trade-off}
\var = \underbrace{f(n)\cdot\sum_{j=2}^N\left(\tilde{C}_{j}^*(n)-C^*\right)}_{\tcb{J_{\textrm{network}}(n)}} +
\underbrace{(N-1)C^*f(n)}_{\tcb{J_{\textrm{latency}}(n)}}
\end{equation}
where $ \varx{\opteig}{I} = C^*\taun $ is the optimal variance \tcb{according to~\eqref{eq:cont-time-single-int-steady-state-variance} and~\cref{lem:optimal-variance-explicit}.}
Indeed,
the summation decreases with superlinear rate,
so that $ J_{\textrm{network}}(n) $ is a decreasing sequence.
The terms in $ J_{\textrm{network}}(n) $,
each associated with a decoupled subsystem~\eqref{eq:cont-time-single-int-subsystem},
illustrate benefits of communication:
as $ n $ increases, the eigenvalues of $ K $ have more degrees of freedom
and can squeeze more tightly about $ \opteig $,
reducing performance gaps between subsystems and theoretical optimum.
We note that $ J_{\textrm{network}}(n) $ vanishes for the fully connected architecture.
{Even though analytical expressions could not be obtained for minimum-variance designs,
the curves in~\autoref{fig:opt-var} exhibit trade-offs which are consistent with the above analysis.}
\iffalse
\mjmargin{I feel that statements in this remark should be sharpened. Alternatively, you can remove it. One thing to keep in mind is that $J_{\textrm{network}}$ would not vanish even with a fully connected topology if you penalize control effort in the objective function.}
\canOmit{
\begin{rem}
Even though the trade-off could not be written analytically
for the minimum-variance control,
this seems to be the case.
In particular, numerical tests yield the network-latency split structure $ c^*(n)\opteig $
also for the optimal eigenvalues.
\autoref{fig:trade-off} shows the costs
defined in~\eqref{eq:cont-time-single-int-trade-off} for
the solution of~\eqref{eq:cont-time-single-int-variance-minimization}.
\end{rem}
}
\fi
}
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Guest Post by David Roche – David Roche and his wife Megan are the coaches at Some Work, All Play, where they work with some of the top mountain, ultra, and trail runners in the world. SWAP team members have won many national championships and qualified for dozens of national teams, with a unifying trait of loving the trails and the mountains.
Adrian Ballinger and Cory Richards are now two weeks into their Everest No Filter 2.0 expedition, an attempt to climb Mt. Everest without supplemental oxygen. If Cory reaches the top, it will be his second Everest summit without oxygen. Adrian is in pursuit of his first summit without oxygen to go with his six previous Everest summits achieved with supplemental oxygen.
The lack of supplemental oxygen changes the game in high altitude mountaineering. At sea level, the effective oxygen concentration is 20.9%. At 23,000 feet, which they reached on May 4th and 5th, the concentration is 8.7%, and at the Death Zone above 8000 meters, that drops below 8%. In other words, there is around 60% less oxygen at the highest elevation they’ve reached on the mountain.
To compensate, they have to breathe faster, and the body naturally releases EPO to increase red blood cell production. This hypoxia is an added stress when Adrian and Cory have stopped climbing and enter a recovery cycle as well. The lack of oxygen slows down recovery since the body is working hard just to try to achieve homeostasis. Their bodies are stressed to the limit with activity but then can’t get back to the sea level baseline due to the ongoing hypoxic conditions. So up above 20,000 feet, it’s all about balancing stress with rest, keeping in mind that the body may never fully adapt and that it’s essential to get down off the high slopes of the mountains before the ongoing hypoxia takes too large a toll.
Their Strava files are a treasure trove of insights into how they’re performing on the climb. For instance, there are spikes in the data when grade-adjusted pace is faster for short periods of time, likely due to advantageous terrain. At certain points, their heart rate data at certain grade-adjusted paces is remarkable. On May 4, for example, they have a stretch at 7-minutes-per-mile (grade-adjusted pace) with sub-120 heart rate at 23,000 feet of elevation.
In other words, even with 60% less oxygen than at sea level, they are moving around 3-hour marathon pace for a quick spurt.
While that is not a sustainable push, these impressive spikes in the data indicate that they are not overly stressed and that they are adapting well. The varying duration of climbs shows that they are effectively using rest days to maximize output on certain key efforts. The largest day to date was a 12-mile, 5,000-vertical-foot effort on April 30 where they spent more than 90 minutes above zone 2 heart rate, nearly triple any other day. While that might not seem extreme at lower elevations, up above 20,000 feet, those efforts likely feel like their hearts are beating out of their chests and they are breathing out of a paper bag. Notably, they took two rest days before and after that key effort, letting their bodies adapt to the difficult stress of high performance on top of the world.
As they continue their climb to the summit, the pace should slow for clear reasons–there will be almost no oxygen and the terrain is intense. In addition, their heart rate curves will probably flatten out–they won’t be hitting high max heart rates, but they won’t have the respite of their heart rates going much lower during rest either. Their bodies are always working, whether in the tent or on the mountain. As the climb progresses and Adrian and Cory get closer to their ultimate objective, here are some key performance indicators you can look for in their activities:
1. Whether there are still peaks in the data with faster grade-adjusted paces.
2. Whether their heart rates stay under control.
3.The offset between moving time and elapsed time, which could serve as a proxy for brutality of the terrain and the number of rest breaks they have to take.
The toughest days still lay ahead.
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SPOILERS. MAYBE?
My choice is obviously obvious. For me, I'm going to choose Clementine from the new amazingly fantastic The Walking Dead game by Telltale. I've never ever cared for a character as I do for her. Maybe it's because she's a child? Maybe because she's so damn adorably innocent? Maybe because the worlds gone to shit (in the game) and she's a little angel? I put that girl before anything in the game and will destroy planets if I have to if it means protecting her.
Spoilers ahead!
Whenever I do something that she sees as wrong
I feel terrible when she looks sad or scared. Exposing her to messed up shit kills me but I still make sure she's always safe. I only stopped HER when they were having dinner in episode 2. You know why dinner tasted so good.
So answer my question buddy/guy/friend.
Edit: So Crisis Core: Final Fantasy VII which is an amazing game. I completely forgot that Zack Fair, the protagonist, is actually the first character I really ever cared about especially aftertalked about FF7 and I was reminded about
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Dwyane Wade: Heat stars 'not looking elsewhere' before 2014 free agency
The coming of 2013 Media Day unofficially marks the start of the countdown to 2014 free agency, but Dwyane Wade doesn't get the sense that the Heat's "Big Three" is looking to go its separate ways just yet.
Wade, LeBron James and Chris Bosh can all become unrestricted free agents next summer, and James, the back-to-back NBA MVP, has been noncommittal when discussing his future. That "leaving open all options" approach isn't exactly keeping Wade awake at nights.
"You have concern when you feel people want to go elsewhere," Wade said, according to the Sun-Sentinel. "I don't think nobody is looking to go elsewhere."
On Monday, James again ducked questions about his future.
"Being a leader of this team, I owe it to this organization, to my teammates to not talk about it," he said, according to USA Today Sports.
Although Wade made it clear that James will need to make his own free-agency decision, he said James' focus was in the right place as the Heat look to become the first team to three-peat since the 2000-2002 Lakers.
"We all know inside our locker room that LeBron is committed to this team," Wade said, according to the Associated Press.
James could become the first player to win the NBA MVP award in three consecutive seasons since Larry Bird in 1984, 1985 and 1986.
“I got better," James said, according to Fox Sports Florida. "I’m a better basketball player than I was last year in every aspect.’’Wade, James and Bosh all possess the ability to opt in with the Heat for both the 2014-15 and 2015-16 seasons.
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If you go to 6 studios and have the postcard signed, you will be entered in a raffle and eligible for $50 towards a glass piece from one of the 19 artists.No need to RSVP. Thanks to those who have already responded to me. At Margie’s Glass, there will be a raffle for a fused glass piece. You don’t have to be present for the drawing to win.
Project on Site: You can also make a free trinket that will be fired after the Studio Tour and Sale. The trinket can be picked up Sunday, November 17th from 1 to 3 PM. Please feel free to forward this email to your friends.
NOTE: Sorry for the inconvenience but there is a possible road closure at 185th and Sandy Blvd. To get to my house if you are coming off of Hwy 84 at 181st continue past Sa ndy Blvd on 181st which becomes Airport Way. At the first stoplight turn right onto Riverside Parkway, turn left at the first stop sign onto 185th, then right onto Marine Drive. Make a right onto Interlachen Lane. My driveway is the first driveway on the left.
Hope to see you here,
Margie Rieff
Artist Statement
Think of curves, nature, gardens, and color. I enjoy making whimsical, imaginative pieces where I feel like I am five years old. My journey with glass has been fun, exceptional, and mind bending.
Working under Teri Neville, I gained the skills to experiment with different techniques, and, in spite of mistakes, the pieces always turned out beautifully.
Over the past decade, I have become a fused glass, multi-day workshop junkie taught by many different talented glass artists from around the country. Of note is Ann Cavanaugh who taught painting thick landscapes with frit (small pieces of glass) and powder.
Living in the Northwest, the hot bed of glass, I explored functional items, fusing, slumping, sand painting, hot glass manipulation, roll ups, air brushing, painting with enamels, penning, shading, and sgraffito (drawing in glass powder).
I am often surprised when I finish a glass art piece, put it on the mantle, and see that subconsciously I have used my mother’s colors. My mother, a professionally trained artist, crafted collages from her etchings and prints similar to the depth of fused glass.
My glass has been exhibited at:
- Infusion Gallery in Troutdale, Lorang Gallery in Cascade Locks
- Ryan Gallery in Lincoln City
- A juried Gresham Arts Show: Chromatics.
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Friends of Lord Howe Island Newsletter No.17 Summer 2006 Calling names for 2006 weed trips Several of the winter 2006 weed weeks are full already. If you haven’t booked yet and are interested in a trip please contact Ian Hutton as soon as possible as we need to finalise numbers. Phone 02 6563 2447 or Email: [email protected] Dates available June 10-17 June 25-July 2 July 22 to 29 Aug 19 to 26 Aug 26 to Sept 3 Friends President Des Thompson elected to the new Board on Lord Howe Island In the recent LHI Board elections held in February 2006, Friends of LHI President Des Thompson stood for nomination, and was successful in being elected as one of the four Island representatives on the seven person Board. Participation in valuable contribution to the weed problem on Lord Howe Island... Groups working to remove Asparagus fern from Transit Hill and the settlement area. Other Island members elected are Barney Nichols, Stan Fenton and Gary Crombie. Mainland member Helen Clements stood down after years on the Board, and Jeff Angel from the Total Environment Centre was appointed a community environment representative. Des hopes to see continuing care of the environment as one of the key focuses on the Island. OAM awarded to Ian Hutton In the Australia Day honours list for 2006, Friends of LHI vice president Ian Hutton was awarded the OAM for services to conservation and tourism on Lord Howe Island. During the weeding weeks the afternoons are spent exploring the island, discovering the flora, fauna, geology, marine life in the company of good friends. The conservation contribution was mainly for the promotion of the awareness of weeds as the biggest environmental threat to the Island; and along with Rymill Abell, the development of the Bush Regeneration tours to Lord Howe Island. 1 Presidents Report Congratulations to Ian Hutton for his award of OAM announced on Australia Day. I have talked to Ian and he is very appreciative of the opportunities the Island has given him and the support from all those involved in the Friends of LHI. I put my name in the ring in the recent LHI Board election and was successful in being elected by the community. I see some very important issues facing the Island in the coming years and will do my best to ensure the Island community views are put forward as strongly as possible. I believe for the Island to flourish it must have the community participating in management decisions as they are the people who know and treasure the Island and their way of life. $21,335.91 in our regular account. We have just pledged $5,000 to support a permanent display on weeds at the Lord Howe Island Museum. Hopefully this will be completed for members’ inspection during the weeding trips this year. The funding for this has been made possible by your generosity in supporting the Friends. Your continued support is appreciated in supporting weed eradication into the future. All donations over $2 to the FLHI are tax deductible; we welcome these on their own or with subscription renewals to P.O Box 155, Lord Howe Island, 2898. I will also be committed to the environment of Lord Howe Island, particularly the imperative of continuing the fight against weeds. I also hope to be able to encourage more local participation in the Friends of LHI. Good work continues on the weed front at Lord Howe Island, with the LHI Board’s weed teams now covering large areas on the ground removing the worst of our weeds. Des Thompson Treasurers Report Here we are again at the end of summer. As the heat fades and autumn approaches my thoughts turn to Lord Howe. Our winter escape to paradise and the satisfaction that comes with helping to preserve it. You may have noticed the additional trip at the start of Spring- this is an exciting addition to the line up of weeding trips and an opportunity to witness a different phase of the islands bird life. We have been a little delayed this year in getting membership renewals out. Indeed some of you may have just received yours. Many thanks to all those who have pre-empted this and sent in their cheque. Please remember to let us know if any of your contact details change. Don’t forget to take advantage of our multiyear renewals- Buy 4 years of membership and get the fifth year free! Our accounts are currently healthy with $3,994.02 in our gift fund account and Steve Gale Friends to fund weed display at Lord Howe Island Museum At the February meeting of the Friends of Lord Howe Island, the committee agreed to fund to $5000 for a display at the LHI museum on weeds. Weeds are the biggest environmental threat to the World Heritage values of Lord Howe Island. In the past ten years great advances have been made in dealing with the weed problem – the volunteer Friends of Lord Howe Island was set up, weed mapping has been done, a weed strategy has been drawn up, major grants have been available for employment of local and offshore workers to deal with weeds in a systematic way and a quarantine strategy has been completed for the Island. This is a start but the effort will need to continue for ten to twenty years. To outline the successes plus the task ahead, a display at the museum is seen as a tool to educate and enthuse the residents and visitors to the situation. The weeds display will be incorporated into a major section of the museum dealing with alien species and their effects on the Island, 2 LHI Board to commence Owl culling LHI Board Manager Nick Rigby has advised that a program will commence to cull Masked Owls from Lord Howe Island. Nick also said “I would rather see an eradication program in place, but in the short term the Board will cull Owls.” Masked owls are an introduced species that are having an impact on the birds of Lord Howe Island. They are known to take Woodhens, White terns, Black Noddies, Black-winged petrels, Providence Petrels and probably other bird species on the Island as well. Seabirds are vulnerable to predation by owls, because they are used to breeding on this and other islands that are free of predators and thus they have no instinctive behaviour to avoid predation. White terns are particularly vulnerable because of their white colour, and are easily observed at night when owls hunt. Black-winged petrels are vulnerable because their habit is to come into breeding colonies at night and sit on the surface, outside their burrows. Black-winged Petrel attacked by Owl. Owls could also be implicated in the demise of breeding populations of Kermadec Petrels and White-bellied Storm Petrels on the main Island in the 1920s. This is often attributed to rats arriving on the Island, but the introduced Owls may be also involved. Masked Owl Tyto novaehollandiae Masked Owls were brought to the Island by the NSW government in the early 1920’s to combat the rats that were accidentally introduced in 1918. Most of the Owls were Masked Owls from Tasmania (Tyto novaehollandiae ssp castanops) but colour variations amongst the Lord Howe Island birds indicate some from mainland NSW or Victoria were also shipped over. While the Owls did and still do eat rats, they also prey upon birds including Woodhens, White terns, Black noddies, Black-winged petrels, Providence Petrels and probably other bird species on the Island as well. Ornithologist Dr Ben Miller lived on the Island in 1978 and 1979 and was convinced that the Owls were a major predator on Woodhens, and recommended that Owls be removed from the Island. For many years throughout the 1980’s LHI Board staff did shoot the Owls. Masked Owls are not and never have been part of the natural ecosystem of Lord Howe Island and should be regarded as feral pests, just the same as cats, goats, pigs and rats. While the Owls may be a protected species on mainland Australia the birds are they are killing on Lord Howe Island are listed as vulnerable or endangered. LHI Board to set up laboratory to test and monitor ground water. The LHI Board has put out expressions of interest from suitably qualified Island persons to set up a water quality test system in the Island Research laboratory. This will be to test for Biochemical Oxygen demand, suspended solids, pH and presence of coliform and E-coli. It will also involve setting up a computer generated database to record test results. Barrie Rogers, the LHI Board Technical Services Manager said “This work will identify problems with the Island’s groundwater and go a long way towards developing a groundwater strategy.” 3 Formosan Lily Most Friends of LHI would be aware of the presence of Formosan lily on the Island. The LHI Board has committed funds to research into the weed and ways to control it. TROUBLE BLOOMS IN PARADISE A beautiful invader is threatening one of Australia's most precious and beloved natural wonders - the World Heritage-listed Lord Howe Island, off the coast of NSW. A Taiwanese native bulb, Formosa lily or tiger lily (Lilium formosanum), has spread across the island from the coastal dunes to the high mountain slopes, raising the spectre of previous invaders like the rat which had catastrophic impacts on the unique Lord Howe ecology. "Since establishment it has spread across the island, colonizing virtually every ecosystem and habitat from the beach, to the forest, to the upper slopes of Mt Gower and Mt Lidgbird," she says. "It has become well established over large areas both in the open and in the forest, displacing native plants." The lily has also naturalised along the east coast of mainland Australia and could emerge as a major problem there too. "Although Formosa lily is perennial, during the winter it dies back to an underground bulb whose size varies with the age of the individual. On Lord Howe, aboveground growth begins in July and plants produce a single shoot between then and mid summer. "Around January, this shoot produces multiple large white flowers tinged with purple, and each of these gives rise to a large capsule containing many hundreds of tiny seeds which are easily scattered by the wind." Lord Howe faces a number of threats from invasive plants including Crofton weed, bitou bush and cherry guava, and in 2004, the Lord Howe Island Board secured a NSW Environmental Trust Grant to tackle some of the island's environmental weeds. Susie Warner's PhD project is one component of this work. Formosan lily Lilium formosanum Lying 700 km north-east of Sydney, Lord Howe has a global reputation for its spectacular native flora - including its famous cloud forest - and the fact that over 40 per cent of its 241 native plants are found nowhere else on earth. However, this unique assemblage of plants and animals that has arisen from both Australasian and Pacific island ancestry over millions of years is under threat from invasive plants. Over 300 introduced plant species have been recorded on the island - exceeding the number of native species. The Formosa lily is an emerging problem, says Weeds CRC doctoral researcher Susie Warner. The bulb is native to Taiwan but has been widely cultivated as an ornamental around the world. The first herbarium record for the plant on Lord Howe was in the early 1970s and is another of Australia's now-notorious "garden escapes". "My research is examining key aspects of the population biology of Formosa lily, including its germination requirements, soil seed-banks, seed production and habitat requirements. "I'm also exploring the possible impacts the lily is having on native Lord Howe plant species with the aim of providing some recommendations about how best to control it." The challenge is enormous. As a bulb, the lily hides most of its mass safely underground where it is hard to get at with physical or chemical control - successfully enabling it to compete with native plants for nutrients and water. Its tiny, light seeds mean it will be almost impossible to prevent it from spreading. More information: Susie Warner, Weeds CRC and University of New England Phone: 02 6773 5213 Email: [email protected] 4 Report on August 6 to 13 2005 weed trip Elizabeth Brown was leader on this trip and we had a mix of regulars including Gwenda Lister, Peter and Joan Brown plus Ian and Margaret Parsonson. We also had quite a few first timers, including Judith and Bruce Tressider. Bruce said he and his wife were not fanatical weeders, but had heard of the program and just had to come along to see what it was all about. At this time of the year the Lignum vitae tree Sophora howinsula is in flower and it put on a good display for the group. Highlights of the week were the Boat Harbour walk, North Bay AND we got a round island cruise in. Report on August 28 - September 4 trip. Bill McDonald from Queensland Herbarium led this trip and quite a few Queenslanders did join him including Keith and Jo Weir, Mike and Elizabeth Russell, and Jim and Jennifer Peat. Also farmer John Zyla from Goulburn on his 8th trip! We also had two international visitors – John Millett and Maureen Young from New Zealand who were most impressed by the Island and the weeding program. August 6 to 13 2005 group Sue Stevens enjoyed the week immensely and wrote: Little did I imagine, when I gazed down at this magic Island from Air New Zealand 32 years ago that I would one day be getting up close and personal digging asparagus fern under a beautiful rainforest canopy with a bunch of dedicated Lord Howe Island Lovers whose dream is fulfilled with so much fun and hard work. The sandwiching of work, environmental adventures, food and bonhomie is a perfect mix for each and every day. The group had to tackle a particularly difficult block on Transit Hill- one that had a fairly open canopy and hence a lot of Smilax vine to crawl through. Still they came to the task and completely cleared the block. Aug/September 2005 group The group started work on the ground asparagus above Pinetrees. On Wednesday, as a break we had a group at Gai Wilsons removing climbing asparagus, while another group tackled a Madeira vine outbreak in Stevens Reserve sighted by the previous group in August. This outbreak proved bigger than first seen and this group felt compelled to tackle this, so the remainder of the week was taken up removing many bags of Madeira vine, leaves, tubers and roots. September group removing Madeira vine Lignum vitae in bloom 5 Kew palm study released In past issues of the FLHI Newsletter we have reported on the research into Howea palms by botanists from Kew gardens in England. Their findings have just been published in the prestigious Nature Magazine February issue. Following is the press release. Evolutionary experts have found two of the best examples yet for cases of sympatric speciation, which is when two species diverge from a single ancestor without geographical isolation. This issue has been a bone of contention among evolutionists: many suspect that such speciation is possible, but it has been fiendishly difficult to prove. Botanist Vincent Savolainen and his colleagues offer convincing evidence in the form of two sister species of palm tree on Lord Howe Island, a remote outpost almost 600 kilometres off the eastern coast of Australia. As they report in a study published online by Nature, genetic studies of the two species show that they are indeed sisters, and diverged much more recently than the island's creation. This shows that the two species have always lived side by side, making their speciation almost certainly sympatric, the authors explain. Species generally diverge when they become reproductively isolated - usually through geographical isolation (giving rise to allopatric speciation). But here the two species seem to have diverged after they began flowering at different times of year, probably as a result of differing soil 25th Anniversary program of Woodhen recovery As reported in FLHI newsletter 16, the Island museum held celebrations in November 2005 for the 25th anniversary of the first captive bred Woodhen hatching in the Woodhen recovery program. Key people involved in the recovery program attended the celebrations. John Disney was Curator of birds with the Australian Museum in the 1970’s and commenced studies over a ten year period into the surviving Woodhens on the top of Mount Gower. He was joined by Dr Peter Fullagar who was then with CSIRO Division of Wildlife. The pair made treks at least once a year throughout the 1970’s to the top of Mount Gower where they camped and surveyed the Woodhen population across the whole summit of the mountain. The camp on the summit of Mount Gower Their studies recommended that a full time ornithologist be appointed to the Island for two year to further study the Woodhen and make conservation recommendations. Bill Baker and Dave Springate on LHI Big drought on Lord Howe Island The Island is facing severe water shortage, and many rainforest plants are wilting and have leaves dying. In summer 2002 we reported in the FLHI newsletter No 8 of a drought affecting the plants; this current drought seems even more severe on the forest. The average rainfall for January to March on Lord Howe Island is 403 mm and for 2006 the total so far has been just 98 mm Dr Ben Miller was the NSW NPWS Ornithologist at the time and he lived on the Island in 1978 and 1979. Together with local rangers Paul Beaumont and Bruce Thompson, Ben Miller continued monitoring the summit Woodhens plus others in more remote areas of the Island. His recommendations included ridding the Island of feral cats and pigs, a task done by local residents and rangers in 1979. Then a captive breeding program was established on the Island, taking three pairs of Woodhen off the top of Mount Gower and relocating them in a special compound constructed in Stevens Reserve for this purpose of captive breeding. New Zealand aviculturists Glenn Fraser was employed to oversee the project, and he successfully raised 6 93 Woodhen chicks at the centre, for release around the Island. Today there are approximately 300 Woodhens on the Island. Function at the LHI Museum for the Woodhen None of this could have taken place without the funding of $250,000, which was raised by the Foundation for National Parks and Wildlife. The Foundation continues its commitment to conservation on Lord Howe Island today – funding such projects as rodent cost benefit analysis, and research into Currawongs. The Foundation also funded a new display for the museum on the Woodhen story. The Woodhen recovery program still ranks as one of the most successful endangered bird recovery programs of the world, and demonstrates that captive breeding is a solution to recovery of bird endangered bird populations. John Disney and Peter Fullagar were overjoyed to see Woodhens across the lowlands where they had not seen them during their ten years study in the 1970’s. The LHI Museum published a book on the Woodhen, with funding by the LHI Board. The book costs $12 and is available from the Island museum – Friends of LHI can order through FLHI PO Box 155 Lord Howe Island NSW 2898. New ranger for Lord Howe Island. In February LHI Board ranger Sean Thompson transferred to the DEC near Myall Lakes. The new ranger is Meg Lorang, who commenced work on 27 February for a one year term. Meg has spent the last two years as a DEC ranger in the Tenterfield area in the NSW Northern Tablelands. Prior to that Meg had worked in the western NSW based in Bourke. Meg also has five years experience as an Environmental Officer for local and State government. Meg has an arts degree majoring in Geography from Sydney University, a Graduate Diploma of Environmental Science from Murdoch University and a Master of Wildlife Management from Macquarie University. She also has a boating license, SCUBA diving certification and a firearms license. Meg will be based in the Board’s Environmental Unit and will be involved in a range of projects including threatened species recovery, noxious weed, pest species eradication and community consultation and education. We welcome Meg to LHI and look forward to working with her on the winter weeding program. Children from the LHI Central School made a great display to celebrate the Woodhen storythere were posters, dioramas, poems and stories which provided great interest for tourists through summer. 7 Unusual Bird Sightings November 2005 to February 2006 Black-winged stilt at North Bay On 11th November 2005, two Black-winged stilts turned up at North Bay. In a strong southerly gale on 13th one moved off the Island, and the other stayed around the Old Settlement creek until 15 December before disappearing. The last record of this species at Lord Howe Island was in October 1888 when a pair was collected. The Black-winged stilt occurs over all of Australia, New Guinea and New Zealand at inland swamps, lagoons and saline lakes, where they are often found in large numbers. Some regularly breed in swamps near Sydney. On 10th and 11th November and again 19, 20, 21December a single Caspian Tern was on the Moseley Park swamp opposite the airstrip. The Caspian Tern is easily recognised by its size - much larger than the Sooty tern; white body, light grey upper wings, a shaggy black cap and a large red bill. It is a widespread species around tropics and subtropics of the world. It is known from all parts of Australia, coastal as well as inland, but is seldom common and usually seen in single pairs or alone. A vagrant to LHI, with the first record only in 2000; now one is seen most years about the same time. Caspian Tern Black-winged Stilt An immature Kelp Gull was on the Island 22nd at North Bay, and found dead at Old Settlement Beach on 23rd. A rare bird for Lord Howe Island, with just four records here. -------------------------------------------------------2006 weed trip booking form. Friends of Lord Howe Island newsletter compiled by Ian Hutton. Friends of LHI PO Box 155 Lord Howe Island NSW 2898 Email [email protected] -----------------------------------------------------------Please book me on the Lord Howe Island bush regeneration tour Included is my deposit of $200 per person Please send form and deposit cheque to: Lord Howe Island Nature Tours PO Box 157 Lord Howe Island NSW 2898 Mr/Mrs/Ms/Miss______________________ ------------------------------------------------------ ____________________________________ Full price $1775 ex Sydney or Brisbane, twin share (Sole rooms subject to availability - $1930) Deposit $ 200 (cheque made to PINETREES) Balance $1575 (twin share) or $1730 (sole use) 45 days prior to trip. ----------------------------------------------------------- ____________________________________ Address ____________________________________ Phone _____________________ Email _______________________ QANTAS Frequent Flyer number ________________________ I prefer to go on dates: Caspian Tern 8
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"Ma Bulle", la princesse petit pois
En temps qu'artiste invité au salon Art to Play 2012 à Nantes, j'ai dû participer à ce concours nommé "Ma Bulle", où chacun devait réaliser un dessin au format A4 dans lequel un espace blanc serait réservé pour un personnage, un visage.
Ce dessin tire son inspiration du conte "La princesse petit pois", avec une "petite" variante. J'ai eu cette idée en pensant à Vanessa (qui pose ici), dont le dos fragile lui cause bien des soucis, mais qui est à la fois capable de s'endormir à peu près n'importe où dans des postures terriblement inconfortables.
L'image en entier: Princesse petit pois
Le lien pour voter sur facebook: Lien
"No Doll" painting complete
C'est un projet ancien, dont l'idée m'a traversé l'esprit pour la première fois en 2005. Le concept a muri au travers d'un nombre conséquent d'études et de croquis. Mon amie Jo est venue à mon studio bordeaux au printemps 2010 pour poser.
J'ai donné mon premier coup de pinceau le 1er janvier 2011, à 0h00. C'était un réveillon que j'avais passé seul, exalté. J'ai terminé cette toile le 31 décembre, autour de 21h00, y ai posé ma signature. Un an de travail, pas toujours assidu, et le voici, perché sur son cheval Larmes de Joie au Jardin des Géantes, Sous la Rouille.<<
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\begin{document}
\maketitle
\begin{abstract}
We show that the irregular connection on $\bbG_m$ constructed by
Frenkel-Gross (\cite{FG}) and the one constructed by
Heinloth-Ng\^{o}-Yun (\cite{HNY}) are the same, which confirms
Conjecture 2.16 of \cite{HNY}.
\end{abstract}
\bigskip
\bigskip
\centerline{\sc Contents}
\medskip
\noindent \ \ \ \ \ \ Introduction\\
\S~1. \ Kac-Moody chiral algebras\\
\S~2. \ Hecke eigensheaves\\
\S~3. \ Some geometry of the local Hitchin map\\
\S~4. \ Endomorphisms of some vacuum modules\\
\S~5. \ Proof of a conjecture in \cite{HNY}\\
\phantom{aa} \ \ References\\
\bigskip
\medskip
\medskip
\section*{Introduction}
We show that the irregular connection on $\bbG_m$ constructed by
Frenkel-Gross (\cite{FG}) and the one constructed by
Heinloth-Ng\^{o}-Yun (\cite{HNY}) are the same, which confirms
Conjecture 2.16 of \cite{HNY}.
The proof is simple, modulo the big machinery of quantization of
Hitchin's integrable systems as developed by Beilinson-Drinfeld
(\cite{BD}). The key idea is as follows: instead of directly showing that the two connections constructed in \cite{FG} and \cite{HNY} are isomorphic, we go the opposite direction by showing that their corresponding automorphic sheaves under the Langlands correspondence are isomorphic. More precisely,
Let $\calE$ be the irregular
connection on $\bbG_m$ as constructed by Frenkel-Gross. It admits a
natural oper form. We apply (a variant of) Beilinson-Drinfeld's machinery to
produce an automorphic D-module on the corresponding moduli space of
$G$-bundles, with $\calE$ its Hecke eigenvalue. We show that this
automorphic D-module is equivariant with respect to the unipotent
group $I(1)/I(2)$ (see \cite{HNY} or below for the notation) against a
non-degenerate additive character $\Psi$. By the uniqueness of such
D-modules on the moduli space, the automorphic
D-module constructed using Beilinson-Drinfeld's machinery is the
same as the automorphic D-module explicitly constructed by
Heinloth-Ng\^{o}-Yun. Since the irregular connection on $\bbG_m$
constructed in \cite{HNY} is by definition the Hecke-eigenvalue of
this automorphic D-module, it must be the same as $\calE$. We emphasize here that the simple nature of the proof is due to the existence of the ``Galois-to-automorphic" direction for the Langlands
correspondence over $\bbC$. Similarly, the existence of such direction over finite fields (which remains conjectural) would imply the unicity of the Kloosterman sheaves as conjectured in \cite{HNY}.
Let us brief describe the main new ingredients needed in the proof of the conjecture. Let $X$ be a smooth projective curve over $\bbC$ and let $\calG$ be a smooth affine group scheme over $X$. Let $\Bun_\calG$ denote the moduli stack of $\calG$-torsors on $X$.
Assume that there is some dense open subset $U\subset X$ such that $\calG|_U=G\times U$ for some reductive group $G$ over $\bbC$. Then there is the usual Hitchin map
\[h^{cl}: T^*\Bun_\calG\to \on{Hitch}(U),\]
where $\on{Hitch}(U)$ is the Hitchin base for $U$ (which is not finite dimensional if $U$ is affine, see \eqref{Hitchin base}). The map factors through a finite dimensional closed subscheme $\on{Hitch}(X)_\calG\subset \on{Hitch}(U)$.
The new difficulty is that, unlike the case when $\calG=G\times X$ is constant, in general $h^{cl}$ is not a completely integrable system, and in particular $\bbC[\on{Hitch}(X)_{\calG}]$ does not provide a maximal Poisson commuting subalgebra of the ring of regular functions on $T^*\Bun_\calG$. For every $x\in X\setminus U$, let $K_x:=\calG(\calO_x)$ and let $P_x$ be the normalizer of $K_x$ in $G(F_x)$, where $\calO_x$ denotes the complete local ring at $x$ and $F_x$ denotes the fractional field of $\calO_x$ (in the note, $\calG(\calO_x)$ will be a Moy-Prasad subgroup so $P_x$ will be a parahoric subgroup of $G(F_x)$). Then $Q_x:=P_x/K_x$ acts on $\Bun_\calG$ by translations and therefore there is the moment map $\mu_x: T^*\Bun_\calG\to \frakq_x^*$ where $\frakq_x$ is the Lie algebra of $Q_x:=P_x/K_x$. Then one has the enhanced Hitchin map
\[h^{cl}\times\prod_x \mu_x: T^*\Bun_\calG\to \on{Hitch}(X)_\calG\times \prod_x \frakq_x^*.\]
Our observation is that in some nice cases (e.g. $K_x$ is the pro-unipotent radical of a pararhoric or $K_x$ is a subgroup of $G(F_x)$ considered in \cite{RY}, see \S~\ref{loc Hitchin}), one can obtain a maximal Possion commuting subalgebra of the ring of regular functions on $T^*\Bun_\calG$ from this enhanced system. Then we can quantize this system by (a generalization of) Beilinson-Drinfeld's machinery to obtain an affine scheme $\Spec A$ and Hecke eigensheaves parameterized by it. A new feature is that points on $\Spec A$ do not parameterize opers in general, but by construction there is a map from $\Spec A$ to the space of opers. E.g., in the case considered in \S~\ref{Proof}, $\Spec A$ paramterizes linear forms on $\Lie(I(1)/I(2))$.
Let us summarize the contents of the note.
We first review of the main results of
\cite{BD} in \S~\ref{KMCA}-\S~\ref{recon}. We take the opportunity to describe
a slightly generalized version of \cite{BD}
in order to deal with the level structures (in particular see Corollary \ref{variant}). These results are probably known to experts but seem not to exist in literature yet.
In \S~\ref{loc Hitchin}, we study some geometry of the local Hitchin map by relating the invariant theory of simple Lie algebras over local fields with the invariant theory of Vinberg's theta groups. In particular, we determine the images of the dual of some subalgebras of a loop algebra under this map. Then we quantize these results in \S~\ref{endo alg} to get some information of the endomorphism algebras of some modules of an affine Kac-Moody algebra at the critical level. These two sections contain results more than needed in the proof of the main theorem. It may be useful to relate the subsequent works \cite{Y} and \cite{C}.
The main result is proved in \S~\ref{Proof}.
\medskip\noindent\bf Notations. \rm
In the note, $X$ will be a smooth projective curve over $\bbC$. For
every closed point $x\in X$, let $\calO_x$ denote the complete local
ring of $X$ at $x$ and let $F_x$ denote its fractional field. Let
$D_x=\Spec \calO_x$ and $D_x^\times=\Spec F_x$. For a scheme $T$ of finite type over $\calO_x$, we denote by $T(\calO_x)$ the jet space $T$. For an affine scheme $T$ of finite type over $F_x$, we denote by $T(F_x)$ the ind-scheme of the loop space of $T$.
In the note, unless otherwise specified, $\calG$ is a
(fiberwise) connected smooth affine group scheme over $X$ such that
$G=\calG\otimes \bbC(X)$ is reductive. We denote by $\Bun_\calG$ the stack of $\calG$-bundles on $X$. This is a smooth algebraic stack over $\bbC$.
In the note, a presheaf means a covariant functor from the category of commutative $\bbC$-algebras to the category of sets.
Let $\calF\to\calF'$ be a morphism of presheaves and let $u:\Spec R\to \calF'$ be morphism. We use $\calF\otimes R$, or $\calF_R$ to denote $\calF\times_{\calF'}\Spec R$ if no confusion is likely to arise.
For
an affine (ind-)scheme $T$, we denote by $\on{Fun} T$ or $\bbC[T]$ the
(pro-)algebra of regular functions on $T$.
\medskip\noindent\bf Acknowledgement. \rm The author thanks the referee for careful reading and critical questioning of the early version of the note. The author also thanks T.-H. Chen and M. Kamgarpour for very useful comments. The work is partially supported by NSF grant DMS-1001280/1313894 and DMS-1303296/1535464 and the AMS Centennial Fellowship.
\section{Kac-Moody chiral algebras}\label{KMCA}
In the following two sections, we review the main results of the work \cite{BD}, with some generalizations in order to deal with level structures.
\subsection{Beilinson-Drinfeld Grassmannians}
We refer to \cite[\S~3]{Z2} for a general introduction.
For $x\in X$, let $\Gr_{\calG,x}$ denote the
affine Grassmannian of $\calG$ at $x$. This is a presheaf that assigns to every $\bbC$-algebra $R$ the pairs
$(\calF,\beta_x)$, where $\calF$ is a $\calG$-torsor on $X\otimes R$ and
$\beta_x$ is a trivialization of $\calF$ away from the graph $\Gamma_x$ of $x$. Then $\Gr_{\calG,x}$ is represented by an ind-scheme over $\bbC$, ind of finite type, and we can write
$$\Gr_{\calG,x}\simeq G(F_x)/K_x,$$ where $K_x=\calG(\calO_x)$, regarded as a pro-algebraic group over $\bbC$.
By forgetting $\beta_x$, we get a map
$$u_x: \Gr_{\calG,x}\to \Bun_\calG.$$
By allowing multiple points and allowing points to move, one obtain the Ran version of the Beilinson-Drinfeld Grassmannian $\Gr_{\calG, \Ran(X)}$, which is a presheaf\footnote{Note that it is not a sheaf.} that assigns to every $\bbC$-algebra $R$, the triples $(\{x_i\}, \calF,\beta)$, where $\{x_i\}\subset X(R)$ is a finite non-empty subset, $\calF$ is a $\calG$-torsor on $X\otimes R$, and $\beta$ is a trivialization of $\calF$ away from the union of the graphs $\cup_i \Gamma_{x_i}$ of $\{x_i\}$. There is a natural projection
\[q: \Gr_{\calG,\Ran(X)}\to \Ran(X),\]
to the Ran space $\Ran(X)$ of $X$, which assigns to every $\bbC$-algebra $R$, the finite non-empty subsets of $X(R)$. If $U\subset X$ is an open subset, we have the Ran space $\Ran(U)$ of $U$, which is an open sub-presheaf of $\Ran(X)$. Let $\Gr_{\calG,\Ran(U)}:=\Gr_{\calG,\Ran(X)}\times_{\Ran(X)}\Ran(U)$ denote the base change. There is the (ad\`elic version) uniformization map
$$u_{\Ran}: \Gr_{\calG,\Ran(U)}\to \Bun_\calG,$$
by forgetting $(\{x_i\},\beta)$, and a unit section
\[e_\Ran:\Ran(U)\to \Gr_{\calG,\Ran(U)}\]
given by the canonical trivialization of the trivial $\calG$-torsor. In the sequel, if $\calG$ and $U$ are clear from the context, we write $\Ran(U)$ by $\Ran$ and $\Gr_{\calG,\Ran(U)}$ by $\Gr_\Ran$ for brevity.
A salient feature of the Beilinson-Drinfeld Grassmannian is its factorizable property (cf. \cite[\S~3.10.16]{BD1}). First, the Ran space $\Ran=\Ran(U)$ has a semi-group structure given by union of points
\[\on{union}: \Ran\times \Ran\to \Ran,\quad (\{x_i\},\{x_j\})\mapsto \{x_i,x_j\}.\]
Let $(\Ran\times \Ran)_{disj}$ denote the open sub presheaf consisting of those $\{x_i\}$ and $\{x_j\}$ with $\{x_i\}\cap \{x_j\}=\emptyset$. Then the factorization property amounts to a canonical isomorphism
\[ \Gr_{\Ran}\times_{\Ran}(\Ran\times\Ran)_{disj}\simeq \Gr_{\Ran}\times\Gr_{\Ran}|_{(\Ran\times\Ran)_{disj}},\]
compatible with the unit section $e:\Ran\to\Gr_\Ran$ and satisfying a natural cocycle condition over $(\Ran\times\Ran\times \Ran)_{disj}$.
Next, we recall a factorizable line bundle on $\Gr_\Ran$.
Let $\omega_{\Bun_\calG}$ be the canonical sheaf of
$\Bun_\calG$. This is in fact a line bundle that can be described as follows. Let $\calE$ denote the universal $\calG$-torsor on $X\times\Bun_\calG$ and $\ad\calE=\calE\times^\calG \Lie\calG$ be the adjoint bundle.
Then $\omega_{\Bun_\calG}$ is isomorphic to the determinant line bundle $\det(\ad\calE)$ for $\ad\calE$. Its fiber over the trivial bundle is $\det(\Lie \calG)$. Therefore, we can normalize $\omega_{\Bun_\calG}$ to a rigidified line bundle as
\[\omega_{\Bun_\calG}^{\sharp}=\omega_{\Bun_{\calG}} \otimes \det(\Lie \calG)^{-1}.\]
Let $\calL_{2c,x}=u_x^*\omega_{\Bun_\calG}^\sharp\in \Pic(\Gr_{\calG,x})$, and let $\calL_{2c,\Ran}=u_{\Ran}^*\omega_{\Bun_\calG}^\sharp$.
\begin{lem}\label{fl}
The line bundle $\calL_{2c,\Ran}$ admits a natural factorizable structure, compatible with the factorizable structure of $\Gr_\Ran$.
\end{lem}
\begin{proof}
Let $\calG'=\GL(\Lie \calG)$. This is an inner form of $\GL_n$ on $X$. Let $\calL_{\det}$ denote the determinant line bundle on $\Bun_{\calG'}$ as well as its pullback to $\Gr_{\calG',\Ran(X)}$. Then $\calL_{2c}$ is just the pullback of $\calL_{\det}$ along the natural morphism $\Ad:\Gr_{\calG,\Ran(X)}\to\Gr_{\calG',\Ran(X)}$, sending a $\calG$-torsor to its adjoint bundle. Note that $\ad$ is compatible with the natural factorizable structures on $\Gr_{\calG,\Ran(X)}$ and on $\Gr_{\calG',\Ran(X)}$. Therefore, it is enough to show that $\calL_{\det}$ admits a natural (graded) factorizable structure, which is well-known (e.g. see \cite[Remark 3.2.8]{Z2}).
\end{proof}
\subsection{Kac-Moody chiral algebras}\label{KM chiral}
Let $F=\bbC(X)$. We assume that $\calG$ is a smooth, affine fiberwise connected group scheme over $X$ with $G:=\calG\otimes F$ simple and simply-connected.
Let us briefly discuss the Kac-Moody chiral algebra attached to $\calG$, generalizing the usual Kac-Moody chiral algebra for the constant group scheme over $X$.
Let $q: \Gr_{\calG}:=X\times_{\Ran(X)}\Gr_{\calG,\Ran(X)}\to X$ be the Beilinson-Drinfeld Grassmannian over $X$, and $e:X\to\Gr_{\calG}$ the unit section given by the trivial
$\calG$-torsor. Let $\calL_{2c}$ be the pullback of the line bundle $\calL_{2c,\Ran}$ to
$\Gr_{\calG}$. Let $\delta_e$ be the $\calL_{2c}^{1/2}$-twisted delta right $D$-module along the unit section\footnote{Recall that given a line bundle $\calL$ on an algebraic variety $X$, it makes sense to talk about $\calL^\la$-twisted $D$-modules on $X$ for any $\la\in\bbC$.}. Let us define
\[\calV ac_X:=q_*(\delta_e),\]
the push-forward of $\delta_e$ as a quasi-coherent sheaf.
\begin{lem}\label{flat}
The sheaf
$\calV ac_X$ is flat as an $\calO_X$-module.
\end{lem}
\begin{proof}First note that $\pi:\Gr_\calG\to X$ is formally smooth. Indeed, let $\calL^+\calG$ (resp. $\calL\calG$) be the global jet (resp. loop) group of $\calG$, which classifies $(x,\gamma)$, where $x\in X$ and $\gamma: D_x\to \calG$ (resp. $\gamma: D_x^\times\to \calG$) is a section. Then $\calL^+\calG$ and $\calL\calG$ are formally smooth. As $\Gr_\calG=\calL\calG/\calL^+\calG$, it is formally smooth.
Then one can argue as \cite[\S~7.11.8]{BD} to show that the tangent space $T_e\Gr_\calG$ of $\Gr_\calG$ along $e:X\to \Gr_\calG$ is a flat $\calO_X$-module, and $\delta_e$ admits a filtration whose associated graded is $\on{Sym}T_e\Gr_\calG$. Therefore, $\calV ac_X$ is flat.
\end{proof}
We refer to \cite{BD1} for the generalities of chiral algebras.
\begin{lem}\label{chiral}
$\calV ac_X$ has a natural chiral algebra structure over $X$.
\end{lem}
\begin{proof}For each finite non-empty set $I$, we have $\Gr_{\calG,X^I}\to X^I$ and therefore the similarly defined $\calV ac_{X^I}$.
Using the dictionary between chiral algebras and factorization algebras \cite[Chap. 2, Sect. 4]{BD1}, the lemma is equivalent to saying that there is a natural factorization structure on the collection $\{\calV ac_{X^I}\otimes\omega_{X^I}^{-1}\}$. This follows from Lemma \ref{fl}.
\end{proof}
We call $\calV ac_X$ the Kac-Moody chiral algebra associated to $\calG$.
\subsection{Vacuum modules}\label{vac mod}
By flatness, the fiber of $\calV ac_X$ over a point $x\in X$ is
\[\Vac_x:=\Gamma(\Gr_{\calG,x},\delta_e|_{\Gr_{\calG,x}}).\]
We need a more representation theoretical description of $\Vac_x$ in some cases.
For the purpose, we need to fix a few notations.
We write $\frakg_x=\Lie \calG\otimes F_x$, regarded as an infinite dimensional Lie algebra over $\bbC$. Let $\hat{\frakg}_x$ be the ``level one" Kac-Moody central extension of $\frakg_x$ (i.e., the completion of the derived subalgebra of one of those in \cite[Theorem 7.4, 8.3]{Ka}). We fix an Iwahori subgroup $I\subset G(F_x)$, and let $\Lambda_i, i=0,\ldots,\ell$ be the fundamental weights of $\hat{\frakg}_x$ (with respect to $I$).
Let $\rho_{\on{aff}}=\sum \Lambda_i$. For a standard parahoric subgroup $P\supset I$ of $G(F_x)$, let $\rho_P$ denote the half sum of affine roots in $P/I$. As before, we write $K_x=\calG(\calO_x)$, and let $\frakk_x$ denote its Lie algebra. Its pre-image in $\hat{\frakg}_{x}$ is denoted by $\hat{\frakk}_{x}$.
\begin{lem}\label{vacuum I}
Assume that $K_x=P$ is a standard parahoric subgroup. Then as a representation of $\hat{\frakg}_x$,
\[\Vac_x=\on{Ind}_{\hat{\frakk}_x}^{\hat{\frakg}_x} (\bbC_{-\rho_{\on{aff}}+\rho_P}),\]
where $\bbC_{-\rho_{\on{aff}}+\rho_P}$ is the $1$-dimensional representation of $\hat{\frakk}_x$ given by the character $-\rho_{\on{aff}}+\rho_P$.
\end{lem}
\begin{proof}The the lemma follows from the calculation of \cite[\S~4.1]{Z}. Note that the line bundle $\calL_{2c,x}$ here corresponds to $\calL_{2c}^{-1}$ in \emph{loc. cit.}.
\end{proof}
As explained in \cite[\S~4.1]{Z}, the restriction of $-\rho_{\on{aff}}+\rho_P$ to the center of $\hat{\frakg}_x$ is always given by multiplication by $-h^\vee$, the dual Coxeter number of (the split form of) $G$. Therefore, it is customary to consider $\Vac_x$ as a module of the critical extension $\hat{\frakg}_{c,x}$ of $\frakg_x$, i.e. $(-h^\vee)$-multiple of the ``level one" central extension $\hat{\frakg}_x$, so that the center of $\hat{\frakg}_{c,x}$ acts on $\Vac_x$ by identity. In the sequel, we adapt this latter point of view. Let $\mathbf{1}_x$ (or $\mathbf{1}$ for simplicity) denote the central element in $\hat{\frakg}_{c,x}$ that acts on $\Vac_x$ by identity.
When $\Hom(K_x,\bbG_m)=1$, e.g. $K_x$ is a maximal parahoric or a pro-unipotent group, there is another expression of $\Vac_x$, which might justify the notation. Namely, in this case, there is a unique splitting of the Lie algebra $\hat{\frakk}_x\simeq \bbC\mathbf{1}\oplus \frakk_x$ coming from the splitting at the group level. Then
\begin{equation}\label{vacuum II}
\Vac_x=\on{Ind}_{\bbC\mathbf{1}\oplus \frakk_x}^{\hat{\frakg}_{c,x}} (\bbC),
\end{equation}
where $\mathbf{1}$ acts on $\bbC$ by identity and $\frakk_x$ acts trivially. That is, $\Vac_x$ is a ``vacuum" module.
\begin{rmk}\label{diff splitting}
However, if $\Hom(K_x,\bbG_m)$ is non-trivial, there is no ``natural" splitting of $\hat{\frakk}_x$. For example, if $K_x=I$ is the Iwahori subgroup, by embedding $I$ into different maximal parahorics one obtains \emph{different} splittings of $\hat{\frakk}_x$, and if one writes $\Vac_x$ as the form $\on{Ind}_{\bbC\mathbf{1}\oplus \frakk_x}^{\hat{\frakg}_{c,x}} (\bbC_\chi)$ using these splittings, $\frakk_x$ will act on $\bbC_\chi$ via different characters $\chi$.
\end{rmk}
\quash{
It will be useful to have another expression of $\Vac_x$,
Recall that the line bundle $\calL_\crit$ is canonically trivialized over the unit section of $\Gr_{\calG}$ and therefore,
$\hat{\frakk}_{x}$ canonically splits as $\frakk_x\oplus\bbC\bf{1}$ as Lie aglebras.
Therefore, we can write
\[\Vac_x=\Ind_{\frakk_x+\bbC\bf{1}}^{\hat{\frakg}_{c,x}}(\bbC_\chi),\]
where $\bbC_\chi$ is 1-dimensional, on which $\mathbf{1}$ acts as identity and $\frakk_x$ acts via certain character $\chi$.
\quash{
Note that $G=\calG\otimes F$ is automatically quasi-split. We choose a pinning $(G,B,T,e)$ of $G$, and a pinning $(G_0,B_0,T_,e_0)$ of its split form over $\bbC$, we fix once and for all an isomorphism
\begin{equation}
(G,B,T,e)\otimes_F\overline F\simeq (G_0,B_0,T_0,e_0)\otimes_\bbC \overline{F}
\end{equation}
It induces a map
\begin{equation}
\psi:\Gal(\overline F/F)\to \Aut(G_0,B_0,T_0,e_0)\simeq \on{Out}(G_0)
\end{equation} such that the natural action of $\ga\in\Gal(\overline{F}/F)$ on the left hand side corresponds to the action $\psi(\ga)\otimes\ga$ on the right hand side.
It also defines an apartment $\calA$ together with a special point $v_0\in \calA$ of $G\otimes F_x$ for every $x\in X$.
$$\calA=\Hom_{F_x}(\bbG_m,T_{F_x})\otimes\bbR\simeq \Hom(\bbG_m,T_0)^{\Gal(\overline F_x/F_x)}\otimes\bbR,$$
}
Finally, if $K_x$ is pro-unipotent, then $\chi$ is trivial.
\begin{rmk}The module $\Vac_x$ is not always isomorphic to $\Ind_{\Lie
K_x+\bbC\bf{1}}^{\hat{\frakg}_{\crit,x}}(\on{triv})$, due to the
twist by $\calL_{\crit}$. For example, if $K_x$ is an Iwahori
subgroup,
\[\Vac_x=\Ind_{\Lie
K_x+\bbC\bf{1}}^{\hat{\frakg}_{\crit,x}}(\bbC_{-\rho}),\] is the
Verma module of highest weight $-\rho$ ($-\rho$ is anti-dominant
w.r.t. the chosen $K_x$).
\end{rmk}
}
\subsection{The center}For any chiral algebra $\calA$ over a curve, one can associate the
algebra of its endomorphisms, denoted by $\calE nd(\calA)$. As
a sheaf of $\calO_X$-modules on $X$, it is defined as
\[\calE nd(\calA)=\calH om_\calA(\calA,\calA),\]
where $\calH om_{\calA}$ is taken in the category of chiral $\calA$-modules. If $\calA$ is $\calO_X$-flat, which is the case we will be considering, it is quasi-coherent.
Obviously, $\calE nd(\calA)$ is an algebra by composition. Less
obviously, there is a natural chiral algebra structure on $\calE
nd(\calA)\otimes\omega_X$ which is compatible with the algebra
structure. In fact, the natural map $\calE nd(\calA)\otimes\omega_X\to \calA$ identifies $\calE nd(\calA)\otimes\omega_X$ with the center $\frakz(\calA)$ of $\calA$.
Therefore, $\calE nd(\calA)$ is a commutative
$\calD_X$-algebra. There is a natural
injective mapping $\calE nd(\calA)_x\to \End(\calA_x)$ which is not
necessarily an isomorphism in general, where $\End(\calA_x)$ is the
endomorphism algebra of $\calA_x$ as a chiral $\calA$-module. However,
this is an isomorphism if there is some open neighborhood $U$
containing $x$ such that $\calA|_U$ is constructed from a vertex
algebra (in the sense of \cite[Chap. 19]{FB}). We refer to \cite{N} for details of the above discussion. We apply the above discussion to $\calE nd(\calV ac_X)$. In particular, if $\calG$ is unramified at $x$, $\calE nd(\calV ac_X)_x\simeq \End(\Vac_x)$.
To continue, we assume that $G=\calG\otimes F$ is split, i.e. it is the base change of simple complex Lie group $G_0$ from $\bbC$ to $F$. The reason we make this assumption is that the relevant theory for twisted affine algebras has not been fully developed.
We denote by $\frakg$ the Lie
algebra of $G$ and ${^L}\frakg$ the Langlands dual Lie algebra, equipped with a Borel subalgebra ${^L}\frakb$. Following the usual convention, we consider ${^L}\frakg$ and ${^L}\frakb$ as finite dimensional Lie algebras over $\bbC$ (while $\frakg$ is a Lie algebra over $F$).
Recall the definition of opers (cf. \cite[\S~3]{BD}). Let $\on{Op}_{{^L}\frakg}(D_x)$ (resp.
$\on{Op}_{{^L}\frakg}(D^\times_x)$) be the scheme (resp. ind-scheme) of
${^L}\frakg$-opers on the disc $D_x$ (resp. punctured disc $D^\times_x$). Let $A_{{^L}\frakg}(D_x)$ (resp. $A_{{^L}\frakg}(D_x^\times)$) denote the ring of its regular functions.
Let $\hat{U}_{c}(\frakg_x)$ denote the completed universal enveloping algebra of $\hat{\frakg}_{c,x}$, and let $\frakZ_x$ denote the center of $\hat{U}_{c}(\frakg_x)$. Then the Feigin-Frenkel
isomorphism (\cite[\S~3.7.13]{BD}, \cite{F}) asserts that there is a canonical isomorphism
\begin{equation}\label{FeFr}
\varphi_x:A_{{^L}\frakg}(D^\times_x)\simeq \frakZ_x,
\end{equation}
satisfying certain compatibility conditions (one of which will be reviewed in \S~\ref{endo alg}).
Then we obtain
\[A_{{^L}\frakg}(D^\times_x)\simeq\frakZ_x\to\End(\on{Vac}_x).\]
If $\calG$ is unramified at $x$, this map factors as
\[A_{^{L}\frakg}(D_x^\times)\twoheadrightarrow A_{^{L}\frakg}(D_x)\simeq\frakz_x:=\End(\Vac_x).\]
Let $U\subset X$ be an open subscheme such that $\calG|_U\simeq G_0\times U$. Let $\on{Op}_{{^L}\frakg}|_U$ be the $\calD_U$-scheme over $U$, whose
fiber over $x\in U$ is the scheme of ${^L}\frakg$-opers on $D_x$.
Then by the above generality, the Feigin-Frenkel
isomorphism gives rise to
\[\Spec \calE nd(\calV ac_U)\simeq \on{Op}_{{^L}\frakg}|_U.\]
Recall that for a commutative $\calD_U$-algebra $\calB$, one can take
the algebra of its horizontal sections $H_\nabla(U,\calB)$ (or
so-called conformal blocks) \cite[\S~2.6]{BD}, which is usually a
topological commutative algebra. For example,
\[\Spec
H_{\nabla}(U,\on{Op}_{{^L}\frakg})=\on{Op}_{{^L}\frakg}(U)\] is the
ind-scheme of ${^L}\frakg$-opers on $U$ (\cite[\S~3.3]{BD}).
As the map
$$H_{\nabla}(U,\calE nd(\calV ac_U))\to H_{\nabla}(X,\calE nd(\calV
ac_X))$$ is surjective, we have a closed embedding
\[\Spec H_{\nabla}(X,\calE nd(\calV ac_X))\to\on{Op}_{{^L}\frakg}(U).\]
Let $\Op_{{^L}\frakg}(X)_\calG$ denote the image of this closed
embedding. This is a subscheme (rather than an ind-scheme) of
$\on{Op}_{{^L}\frakg}(U)$.
We recall the characterization $\on{Op}_{{^L}\frakg}(X)_{\calG}$.
\begin{lem}\label{support}
Let $X\setminus U=\{x_1,\ldots,x_n\}$.
Assume that the support of $\Vac_{x_i}$ (as an
$\frakZ_{x_i}$-module) is
$Z_{x_i}\subset\on{Op}_{{^L}\frakg}(D^\times_{x_i})$, i.e.
$\on{Fun}(Z_{x_i})=\on{Im} (A_{{^L}\frakg}(D_{x_i}^\times)\to
\End(\Vac_{x_i}))$. Then $$\on{Op}_{{^L}\frakg}(X)_{\calG}\simeq
\on{Op}_{{^L}\frakg}(U)\times_{\prod_i\on{Op}_{{^L}\frakg}(D^\times_{x_i})}\prod
Z_{x_i}.$$
\end{lem}
\begin{proof}Note that $\calV ac_X$ is flat, and therefore by Lemma \ref{flat}, $\calE nd(\calV ac_X)_{x_i}\to \calE nd(\Vac_{x_i})$ is injective. Recall that $\Spec \calE nd(\calV ac_X)_{x_i}$ is identified with the scheme of horizontal sections of $\calE nd(\calV ac_X)$ on the disc $D_{x_i}$. On the other hand, the ind-scheme of horizontal sections of $\calE nd(\calV ac_X)$ on $D_{x_i}^\times$ is just $\on{Op}_{{^L}\frakg}(D_{x_i}^\times)$. Therefore $A_{{^L}\frakg}(D_{x_i}^\times)\to
\calE nd(\calV ac_X)_{x_i}$ is surjective.
Therefore,
$$\calE nd(\calV ac_X)_{x_i}=\on{Im} (A_{{^L}\frakg}(D_{x_i}^\times)\to
\End(\Vac_{x_i})).$$ Then
the lemma follows from the last paragraph of \cite[\S~2.4.12]{BD1} applied to the commutative chiral algebra $\calE nd(\calV ac_X)$.
\end{proof}
In \S~\ref{endo alg}, we will give an explicit description of $Z_{x_i}$ in some cases.
\section{Hecke eigensheaves}\label{recon}
\subsection{Square root of $\omega_{\Bun_\calG}^\sharp$}
In order to construct the Hecke eigensheaves, it is important to have a square root $\omega_{\Bun_\calG}^{1/2}$ of $\omega_{\Bun_\calG}^\sharp$ as a line bundle. In the case considered in \S~\ref{Proof}, the existence of such a square root is very easy (see Remark \ref{sqr5}). But in general, this is subtle. The case when $\calG$ is a constant semisimple group scheme is discussed in \cite[\S~4]{BD}. Here, we describe a result for the case when $\calG$ is non-constant, but $G=\calG\otimes\bbC(X)$ is simple and simply-connected. The readers can skip this subsection.
We write $F=\bbC(X)$. We make the following assumptions on $\calG$.
\begin{enumerate}
\item[(i)] $G=\calG\otimes F$ is absolutely simple and simply-connected;
\item[(ii)] for every $x\in X$, either $\calG|_{\calO_x}$ is a parahoric group scheme or the fiber of $\calG$ at $x$ is a unipotent group.
\end{enumerate}
Recall the line bundle $\calL_{2c,x}=u_x^*\omega_{\Bun_\calG}^\sharp$.
\begin{prop}\label{square root}
Assumptions are as above. Then a square root of $\omega_{\Bun_{\calG}}^\sharp$ exists as a line bundle if and only if for every $x$ such that $\calG|_{\calO_x}$ is a parahoric group, the line bundle $\calL_{2c,x}$ admits a square root as a line bundle. In addition, such a square root is unique.
\end{prop}
We denote the unique square root, rigidified at the trivial $\calG$-torsor, by $\omega_{\Bun_\calG}^{1/2}$.
The ``only if" part is trivial. Note that there are examples of parahorics of $G(F_x)$ such that $\calL_{2c,x}$ does not admit a square root (cf. \cite[Remark 6.1]{Z}). The ``if" part and the uniqueness is based on the following proposition.
\begin{prop}Let $U\subset X$ be an open subset such that for every $x\in X\setminus U$, the fiber of $\calG$ at $x$ is unipotent. Then
the pullback functor $u_{\Ran}^*$ induces an equivalence of categories between rigidified line bundles on $\Bun_\calG$ and on $\Gr_{\calG,\Ran(U)}$.
\end{prop}
This statement is a slight variant/generalization of \cite[4.3.14]{BD} or \cite[4.9.1]{BD1}.
It allows us to reduce the problem to construct a square root of $\calL_{2c,\Ran}$. Then using the factorization property, we can work \'{e}tale locally around a point. We omit the details (but see \cite[\S~4.2]{Z2}) since we do not really use Proposition \ref{square root} in \S~\ref{Proof}.
\subsection{Construction of Hecke eigensheaves}
We further assume that $G:=\calG\otimes F=G_0\otimes_{\bbC} F$, where $G_0$ is a simple, simply-connected complex Lie group.
Now we assume that $\Bun_\calG$ is ``good" in the sense of
Beilinson-Drinfeld, i.e.
$$\dim T^*\Bun_\calG=2\dim\Bun_\calG.$$
In this case one can construct the D-module of the sheaf of
critically twisted (a.k.a. $\omega_{\Bun_\calG}^{1/2}$-twisted)
differential operators on the smooth site $(\Bun_\calG)_{sm}$ of
$\Bun_\calG$, denoted by $\calD'$. Let $D'=(\underline{\End}
\calD')^{op}$ be the sheaf of endomorphisms of $\calD'$ as a twisted
D-module. Then $D'$ is a sheaf of associative algebras on
$(\Bun_\calG)_{sm}$ and $D'\simeq (D')^{op}$. For more details, we
refer to \cite[\S~1]{BD}. \quash{We just mention that if $U\subset \Bun_\calG$ is a (smooth) open subscheme, then $\calD'|_U\simeq \calD_U\otimes \omega_{\Bun_\calG}^{1/2}|_U$, where $\calD_U$ is the usual sheaf of differential operators on $U$.}
Let $\Bun_{\calG,x}$ be the scheme classifying pairs
$(\calF,\beta)$, where $\calF$ is a $\calG$-torsor on $X$ and
$\beta$ is a trivialization of $\calF$ on $D_x=\Spec \calO_x$. It
admits a $(\hat{\frakg}_{c,x},K_x)$-action, and
$\Bun_{\calG,x}/K_x\simeq \Bun_\calG$. Now applying the standard
localization construction to the Harish-Chandra module $\on{Vac}_x$
(cf. \cite[\S~1]{BD}) gives rise to
\[\on{Loc}(\on{Vac}_x)\simeq\calD'\]
as critically twisted $D$-modules on $\Bun_\calG$. It induces a natural ring homomorphism
\[h_x: \frakZ_x\to \End(\Vac_x)\to \Gamma(\Bun_\calG, D').\]
In fact, the mappings $h_x$ can be organized into a horizontal morphism $h$
of $\calD_X$-algebras over $X$ (we refer to \cite[\S~2.6]{BD} for
the generalities of $\calD_X$-algebras and \cite[\S~2.8]{BD} for the horizontality of $h$),
\begin{equation}\label{hglob}
h: \calE nd(\calV ac_X)\to\Gamma(\Bun_\calG,D')\otimes\calO_X.
\end{equation}
\quash{Namely, by varying $x$, we get a scheme $\Bun_{\calG, X}$ over $X$ classifying a $\calG$-torsor $\calF$, a point $x\in X$ and a trivialization of $\calF$ along the graph $\Gamma_x$ of $x$. According to \cite[\S~2.8]{BD}, this is a crystal of schemes over $X$, and the natural map $\Bun_{\calG,X}\to \Bun_\calG\times X$ by forgetting the trivialization is
Indeed, the construction of $h$ is as in \cite[\S~2.8]{BD}, where the horizontality of $h|_U$ is also proved. But if there is an algebra homomorphism of $\calD_X$-algebras that is horizontal over some open subscheme $U\subset X$, then this homomorphism is horizontal over the whole curve.}
By taking the horizontal sections, one obtains a mapping
\begin{equation}\label{hhorizontal}
h_{\nabla}:H_{\nabla}(X,\calE nd(\calV
ac_X))\to\Gamma(\Bun_{\calG},D').
\end{equation}
Therefore, \eqref{hhorizontal} can be rewrite as a mapping
\begin{equation}\label{hhorizontal1}h_{\nabla}:\on{Fun}\on{Op}_{{^L}\frakg}(U)\twoheadrightarrow\on{Fun}\on{Op}_{{^L}\frakg}(X)_{\calG}\to
\Gamma(\Bun_{\calG},D').\end{equation}
The mapping \eqref{hhorizontal1} is a quantization of a classical
Hitchin system. Namely, as explained in \cite[\S
3.1.13]{BD} (or see \S~\ref{endo alg} for a review), there is a natural filtration on the algebra $\on{Fun}\on{Op}_{{^L}\frakg}(U)$ whose
associated graded is the algebra of functions on the classical
Hitchin space
\begin{equation}\label{Hitchin base}
\on{Hitch}(U)=\Gamma(U,\frakc^*\times^{\bbG_m}\omega_U^\times).
\end{equation}
Here, using the fact that $\calG$ is unramified over $U$, we regard
$$\frakc^*=\frakg^*/\!\!/G=\Spec F[\frakg^*]^G$$ as a scheme over $U$.
On the other hand, there is a natural
filtration on $\Gamma(\Bun_\calG,D')$ coming from the order of the
differential operators. Then \eqref{hhorizontal1} is strictly
compatible with the filtration and the associated graded map gives
rise to the classical Hitchin map
\[h^{cl}: T^*\Bun_\calG\to \on{Hitch}(U).\]
Its image is the closed
subscheme $\on{Hitch}(X)_\calG\subset\on{Hitch}(U)$ whose algebra of
functions is the associated graded of $\on{Fun}\on{Op}_{{^L}\frakg}(X)_\calG$.
Let ${^L}G$ be the Langlands dual group of $G$, which is of adjoint type. The following theorem summarizes the main result of \cite{BD}.
\begin{thm}\label{BDmain}
Assume that the line bundle $\omega_{\Bun_\calG}^{1/2}$ exists.
Let $\chi\in
\on{Op}_{{^L}\frakg}(X)_{\calG}\subset\on{Op}_{{^L}\frakg}(U)$ be a
closed point, which gives rise to a ${^L}\frakg$-oper $\calE$ on
$U$. Let
$$\varphi_\chi:\on{Fun}\on{Op}_{{^L}\frakg}(X)_{\calG}\to\bbC$$ be the
corresponding homomorphism of $\bbC$-algebras. Then
\[\Aut_{\calE}:=(\calD'\otimes^L_{\on{Fun}\on{Op}_{{^L}\frakg}(X)_{\calG},\varphi_\chi}\bbC)\otimes\omega_{\Bun_\calG}^{-1/2}\]
is a Hecke-eigensheaf on $\Bun_\calG$ with respect to $\calE$
(regarded as a ${^L}G$-local system).
\end{thm}
\begin{rmk}The statement of the about theorem is weaker than the
main theorem in \cite{BD}, where $\calG$ is the
constant group scheme (the unramified case). In this case Beilinson-Drinfeld proved that
$\on{Op}_{{^L}\frakg}(X)_{\calG}=\on{Op}_{{^L}\frakg}(X)$ is the
space of ${^L}\frakg$-opers on $X$, that
$\on{Fun}\on{Op}_{{^L}\frakg}(X)\simeq\Gamma(\Bun_G,D')$, and that $\calD'$ is flat over $\on{Fun}\on{Op}_{{^L}\frakg}(X)$. It then follows that in the definition of $\Aut_\calE$, the derived tensor product can be replaced by the underived tensor product, and that $\Aut_\calE$ is a non-zero
holonomic D-module.
The proof of these assertions relies on the fact that the
classical Hitchin map is a completely integrable system. However, for general level structures, we do not know whether $\calD'$ is always flat over $\on{Fun}\on{Op}_{{^L}\frakg}(X)_{\calG}$. In addition,
the automorphic D-modules constructed as above will not be holonomic, as $\on{Fun}\on{Op}_{{^L}\frakg}(X)_{\calG}\subset\Gamma(\Bun_\calG,D')$ might be too small. For example, this will be the case for the group scheme $\calG=\calG(0,2)$ introduced in \S~\ref{Proof}.
\end{rmk}
Sometimes, we can remedy the situation by replacing $\on{Fun}\on{Op}_{{^L}\frakg}(X)_\calG$ by a larger commutative subalgebra in $\Gamma(\Bun_\calG,D')$. To state our result, we first give a brief review of the local ingredient needed in the proof of Theorem \ref{BDmain}. Let $x\in U\subset X$, so $\calG$ is unramified at $x$. Then $\Gr_{\calG,x}$ is the usual affine Grassmannian $\Gr_{G_0,x}$ of $G_0$ at $x$. Let $\on{Sph}_x=\on{D}\mbox{-mod}(\Gr_{G_0,x})^{G_0(\calO_x)}$ be the Satake category of $G_0(\calO_x)$-equivariant D-modules on $\Gr_{G_0,x}$, and let $\calS_x: \on{Rep}({^L}G)\to \on{Sph}_x$ denote the geometric Satake equivalence. As explained in \cite[\S~7.8]{BD}, there is a convolution action $\star$ of $\on{Sph}_x$ on the derived category of Harish-Chandra $(\hat{\frakg}_{c,x},G_0(\calO_x))$-modules.
The local Hecke eigen property of $\Vac_x$ (see \cite[\S~5.5]{BD}) asserts that for every $V\in \on{Rep}({^L}G)$ there is
a canonical isomorphism of Harish-Chandra $(\hat{\frakg}_{c,x},K_x)$-modules,
\begin{equation}\label{local Hecke}
\calS_x(V)\star \Vac_x\simeq (\calE_x)_V \otimes_{\frakz_x} \Vac_x,
\end{equation}
where $\calE_x$ is the fiber over $x\in D_x$ of the universal ${^L}G$-torsor on $\on{Op}_{{^L}\frakg}(D_x)\hat{\times}D_x$, and $(\calE_x)_V$ is its twist by $V$, regarded as a finite projective $\frakz_x\simeq A_{{^L}\frakg}(D_x)$ module. In particular, the isomorphism commutes with the actions of $\frakz_x$ on both sides.
The localization functor intertwines the action of $\on{Sph}_x$ on the derived category of $(\hat{\frakg}_{c,x},G_0(\calO_x))$-modules and on the derived category of D-modules on $\Bun_\calG$. Then by letting $x$ move in $U$, one obtains a canonical isomorphism
\begin{equation}\label{global Hecke}
\calS_U(V)\star \calD'\simeq \calE_V\otimes_{A_{{^L}\frakg}(U)} \calD',
\end{equation}
of D-modules on $\Bun_{\calG}\times X$, compatible with the actions of $A_{{^L}\frakg}(U)$ on both sides. Here $\calS_U$ is the family version of $\calS_x$ over $U$.
Now let $y\in X\setminus U$, and let $A_y\subset \End(\Vac_y)$ be a commutative subalgebra.
We regard $\Vac_x\otimes \Vac_y$ as the representation of $\hat{\frakg}_{c,x,y}:=\hat{\frakg}_{c,x}\oplus\hat{\frakg}_{c,y}/(\mathbf{1}_x-\mathbf{1}_y)$. Then by tensoring with $\Vac_y$, \eqref{local Hecke} induces a canonical isomorphism of Harish-Chandra $(\hat{\frakg}_{c,x,y},K_x\times K_y)$-modules,
\[
\calS_x(V)\star \Vac_x\otimes \Vac_y\simeq (\calE_x)_V \otimes_{\frakz_x} \Vac_x\otimes \Vac_y,
\]
compatible with the $A_y\otimes \frakz_x$-module structures. By the (two-points version) localization functor and allowing $x$ to move in $U$ (but fixing $y$), we see that the isomorphism \eqref{global Hecke} is compatible with the $A_y\otimes A_{{^L}\frakg}(U)$-module structures on both side.
Now for every $y\in X\setminus U$, let $A_y$ be a commutative subalgebra $\End(\Vac_y)$ containing the image of $\frakZ_y\to \End(\Vac_y)$. Then the map in \eqref{hhorizontal1} can be upgraded to a map
$$A_{{^L}\frakg}(U)\otimes \prod_y A_y\to \Gamma(\Bun_\calG, D').$$
Let $A$ denote its image. Then $A$ is a commutative subalgebra of $\Gamma(\Bun_\calG, D')$ containing $\on{Fun}\on{Op}_{{^L}\frakg}(X)_\calG$. Let $\pi: \Spec A\to \Op_{{^L}\frakg}(X)_\calG$ denote the corresponding map of schemes induced by this inclusion.
\begin{cor}\label{variant}
Assume that $\calD'$ is $A$-flat.
Then for every $\chi\in \Spec A$, the corresponding D-module
$$\Aut_\calE:=\omega_{\Bun_\calG}^{-1/2}\otimes (\calD'\otimes_{A,\varphi_\chi}\bbC)$$ is a Hecke eigensheaf with eigenvalue $\calE$, where $\varphi_\chi:A\to\bbC$ is the homomorphism corresponding to $\chi$, and $\calE$ is the ${^L}G$-local system on $U$ corresponding to the oper $\pi(\chi)\in \on{Op}_{{^L}\frakg}(X)_{\calG}\subset \on{Op}_{{^L}\frakg}(U)$.
\end{cor}
\begin{proof}As explained as above, the isomorphism \eqref{global Hecke} is compatible with the $A$-module structure on both sides. Then $\omega_{\Bun_\calG}^{-1/2}\otimes(\calD'\otimes^L_{A,\varphi}\bbC)$ is a Hecke eigensheaf with eigenvalue $\calE$. Finally $\calD'\otimes^L_{A,\varphi}\bbC=\calD'\otimes_{A,\varphi}\bbC$ by the flatness, giving the corollary.
\end{proof}
\quash{
We need to make use of the following. Assume we have a homomorphism $\calG_1\to \calG_2$ of two models of $G$, which is an isomorphism over $U\subset X$, and for every $x\in X\setminus U$, $(K_1)_x\subset (K_2)_x$ is a closed normal subgroup. Then $K=\prod_x (K_2)_x/(K_1)_x$ acts on $\Bun_{\calG_1}$ and the quotient is $\Bun_{\calG_2}$.
The D-module $\calD_{\Bun_{\calG_2}}$ is (weakly) $K$-equivariant.
\begin{lem}
The map $h_\nabla: \Op_{{^L}\frakg}(X)_{\calG_1}\to \Gamma(\Bun_{\calG_1},D')$ factors through $h_\nabla:\Op_{^{L}\frakg}(X)_{\calG_1}\to \Gamma(\Bun_{\calG_1},D')^K$.
\end{lem}
\begin{proof}
Note that the natural map
\[\frakZ_x\to \End \Vac_x\simeq \Vac_x^{K_x}\]
factors through $\frakZ_x\to \Vac_x^{N_{G(F_x)}(K_x)}\subset \Vac_x^{K_x}$.
\[\widehat{U}_{\crit}(\frakg_x)^{\calG(F_x)}\to (U(\frakg_x)/U(\frakg_x)\frakk_x)^{K_x}.\]
We consider the following more in general situation.
Let $K_1\subset K_2$ be a closed normal subgroup and $K_2/K_1$ is an affine algebraic group. Let $S$ be a scheme, equipped with a Harish-Chandra action $(\frakg, K_2)$. Let
\end{proof}
}
\section{Some geometry of the local Hitchin map}\label{loc Hitchin}
This section is purely local. We fix a point $x\in X$ on the curve. To simplify the notation, we denote the local field $F_x$ by $F$, and its ring of integers by $\calO$, its maximal ideal $\frakm$. Likewise, we use $D$ (resp. $D^\times$) to denote the disc (resp. the punctured disc) around $x$. After choosing a uniformizer $t\in\frakm$, we have $F\simeq \bbC((t))$.
For a line bundle $\calL$ on $D=\Spec\calO$, let $\calL(m)$ denote $\calL(m\cdot x)$ where $x\in D$ is the closed point.
\subsection{The space of invariant polynomials}
Let $\frakg$ be a simple Lie algebra over a field $k$ of characteristic zero, and let $G$ (resp. $G_\ad$ denote the corresponding simply-connected (resp. adjoint) algebraic group. Let
$$\chi:\frakg^*\to\frakc^*$$
denote the usual Chevalley map, where $\frakc^*:=\frakg/\!\! /G=\Spec k[\frakg^*]^G$ is the GIT quotient of $\frakg^*$ under the adjoint action of $G$.
For our purpose, we need to recall some finer structures on $\frakc^*$.
Let ${^L}\frakg$ be the dual Lie algebra over $k$, and let $\frakc_{{^L}\frakg}= k[{^L}\frakg]^{{^L}G}$ denote its space of characteristic polynomials. Let $\on{Out}({^L}\frakg)$ be the group of outer automorphisms of ${^L}\frakg$.
Then the Lie algebra $\frakg$ gives a homomorphism $\psi: \Gal(\overline k/k)\to \on{Out}(\frakg_{\overline k})=\on{Out}({^L}\frakg)$ up to conjugacy.
Then we have the canonical isomorphism
\[\frakc^*\simeq (\frakc_{{^L}\frakg}\otimes\overline k)^{\Gal(\overline k/k)}.\]
Now we give another description of $\frakc_{{^L}\frakg}$ with the action of $\on{Out}({^L}\frakg)$.
Let $V_{{^L}\frakg}$ be the Lie algebra of the universal unipotent regular centralizer of ${^L}\frakg$, equipped with a canonical $\bbG_m$-action.
Let us recall its definition. Namely, let $J_{{^L}\frakg}\to \frakc_{{^L}\frakg}$ denote the universal centralizer group scheme \`{a} la Ng\^{o} \cite[\S~2]{Ng}. Then
$$V_{{^L}\frakg}: =\Lie J_{{^L}\frakg}|_0,$$
where $0\in \frakc_{{^L}\frakg}$ corresponds to the regular nilpotent conjugacy class.
By definition, for a regular nilpotent element $e$, there is a canonical $\bbG_m$-equivariant isomorphism $V_{{^L}\frakg}\simeq {^L}\frakg^e$, and
after fixing a principal $\fraks\frakl_2$-triple $\{e,h,f\}$ of ${^L}\frakg$, there are the isomorphisms
\begin{equation}\label{flat coordinate}
V_{{^L}\frakg}\simeq f+{^L}\frakg^e\simeq \frakc_{{^L}\frakg}.
\end{equation}
This isomorphism is independent of the choice of the $\fraks\frakl_2$-triple, and therefore endows $\frakc_{{^L}\frakg}$ with a vector space structure.
In addition, if we fix a pinning $({^L}\frakg,{^L}\frakb,{^L}\frakt,e)$, $\Aut({^L}\frakg,{^L}\frakb,{^L}\frakt,e)$ will act on ${^L}\frakg^e$ by linear transformations. By transport of structure, we obtain an action of $\on{Out}({^L}\frakg)$ on $V_{{^L}\frakg}$, which is independent of the choice of the pinning and commutes with the $\bbG_m$-action. In addition, the isomorphism \eqref{flat coordinate} is
$\on{Out}({^L}\frakg)$-equivariant. If we define a graded $k$-vector space as
\[{^L}V_\frakg:=(V_{{^L}\frakg}\otimes \overline k)^{\Gal(\overline k/k)},\]
we get a canonical $\bbG_m$-equivariant isomorphism
\begin{equation}\label{flat str}
{^L}V_\frakg \simeq \frakc^*,
\end{equation}
giving $\frakc^*$ a graded vector space structure.
\subsection{The local Hitchin map}
Now we apply the above discussion to $k=F$ so $\frakg$ be a simple Lie algebra over $F$. Then $V_{{^L}\frakg}$ has a natural $\bbG_m$-stable $\calO$-structure.
and therefore ${^L}V_{\frakg}$ admits a $\bbG_m$-stable $\calO$-lattice
$${^L}V_{\frakg,\calO}:=(V_{{^L}\frakg}\otimes \overline{\calO})^{\Gal(\overline{F}/F)}.$$
Similarly there is a canonical $\bbG_m$-stable integral structure on $\frakc^*$, denoted by $\frakc^*_{\calO}$. The isomorphism \eqref{flat str} is compatible with the integral structures.
Let
\[{^L}V_{\frakg,\calO}=\bigoplus_i {^L}V_{\frakg,d_i}\]
denote the grading under the action of $\bbG_m$. We arrange the weights appearing in the above decomposition into a non-decreasing sequence $\{d_1,\ldots,d_\ell\}$ such that $d_i$ appears in the sequence $\dim {^L}(V_{\frakg,d_i}\otimes F)$ times. These positive numbers are sometimes called the (fundamental) degrees of $\frakg$. By \eqref{flat str}, for a $1$-dimensional vector space over $F$ (or an invertible $\calO$-module) $\calL$,
\[\frakc^*\times^{\bbG_m}\calL^\times\simeq \bigoplus_i \calL^{d_i}\otimes {^L}V_{\frakg,d_i}.\]
Now we regard the Chevalley map $\chi$ as a map of $\bbC$-indschemes.
\quash{We identify $\frakg^*$ with $\frakg\otimes_F \omega_F$ as $k$-indschemes via the residue of the Killing form
$$\frakg\times (\frakg\otimes\omega_F)\stackrel{B}{\longrightarrow} \omega_F\stackrel{\Res_F}{\longrightarrow} \bbC.$$ } It induces the local Hitchin map of ind-schemes
\[h^{cl}_{x}:\frakg^*\otimes\omega_F\to \frakc^*\times^{\bbG_m}\omega_F^\times=\bigoplus_i \omega_F^{d_i}\otimes {^L}V_{\frakg,d_i}.\]
We will be interested in the image of certain $\calO$-lattices of $\frakg^*$ under the local Hitchin map.
\subsection{Local Hitchin maps for Moy-Prasad subgroups}
Let $P\subset G(F)$ be a parahoric subgroup. It admits a filtration by normal subgroups (the Moy-Prasad filtration)
\[P=P(0)\rhd P(1)\rhd P(2)\rhd\cdots\]
where $L_P:=P(0)/P(1)$ is the Levi quotient of $P$. Let
\[\frakg\supset \frakp=\frakp(0)\supset \frakp(1)\supset\frakp(2)\supset\cdots,\]
denote the corresponding filtration on Lie algebras. Then the exponential map induces isomorphisms
\[\frakp(i)/\frakp(i+1)\simeq P(i)/P(i+1), \quad i\geq 1.\]
We denote $\frakp(1)/\frakp(2)\simeq P(1)/P(2)$ by $V_P$.
\quash{Recall its definition. Let $x$ be a point in the building of $\frakg$ such that the Lie algebra of the stabilizer of $x$ is $\frakp$. Then one defines $\frakp_{x,r}$ be the Lie algebra generated by $\frakt$ and $\frakg_{\al}$}
There is an integer $m$ such that $\frakm\frakp(i)=\frakp(i+m)$. Therefore, we can extend the above filtration of $\frakp$ to a filtration of $\frakg$ by $\calO$-lattices given by
\[\frakp(i)=\frakm^{-N}\frakp(i+Nm), \quad N\gg 0.\]
Note that for $i<0$, $\frakp(i)$ is not a subalgebra. Let $\frakp(i)^\vee$ be the $\calO$-dual of $\frakp(i)$ in $\frakg^*$, and let $\frakp(i)^\perp=\frakp(i)^\vee\otimes_\calO\omega_\calO$. We have the chain of $\calO$-submodules in $\frakg^*\otimes\omega_F$
\[\cdots\subset \frakp(0)^\perp\subset \frakp(1)^\perp \subset\frakp(2)^\perp\cdots.\]
\begin{prop}\label{size of image}
The following diagram is commutative
\[\begin{CD}
\frakp(n)^\perp @>>>\frakg^*\otimes\omega_F\\
@VVV@VV h^{cl}_{x}V\\
\bigoplus_i \omega_{\calO}^{d_i}((d_i-\lceil\frac{d_i(1-n)}{m}\rceil)) \otimes {^L}V_{\frakg,d_i}@>>>\bigoplus_i \omega_F^{d_i}\otimes {^L}V_{\frakg,d_i}.
\end{CD}\]
\end{prop}
\begin{rmk}
For $n\geq 1$, one shall regard this proposition as a classical limit of the main theorem of \cite{CK}.
\end{rmk}
\begin{proof}
The proof is a simple application of \cite[Theorem 4.1]{RY} in our set-up, which generalizes Kac's work (cf. \cite[Ch. 8]{Ka}) on the realization of (twisted) affine algebras via loop algebras. We will give a relatively self-contained account of Reeder-Yu's result. However, we shall warn the readers from the beginning that our notations are different from Reeder-Yu's.
We choose a pinning $(\frakg,\frakb,\frakt, e)$ of $\frakg$. Let $(\frakg_0,\frakb_0,\frakt_0,e_0)$ be a split form of $\frakg$ together with a pinning defined over $\bbC$. We fix once and for all an isomorphism
\begin{equation}\label{pinning}
(\frakg,\frakb,\frakt,e)\otimes_F\overline{F}\simeq (\frakg_0,\frakb_0,\frakt_0,e_0)\otimes_\bbC\overline{F},
\end{equation}
where $\overline F$ is an algebraic closure of $F$.
It induces a map
\begin{equation}
\psi:\Gal(\overline F/F)\to \Aut(\frakg_0,\frakb_0,\frakt_0,e_0)\simeq \on{Out}(\frakg_0)
\end{equation} such that the natural action of $\ga\in\Gal(\overline{F}/F)$ on the left hand side corresponds to the action $\psi(\ga)\otimes\ga$ on the right hand side.
These data determine:
\begin{enumerate}
\item an apartment $\calA$ corresponding to $\frakt$ in the building of $\frakg$;
\item a special point $x_0\in\calA$ corresponding to the parahoric algebra $(\frakg_0\otimes\overline\calO)^{\Gal(\overline{F}/F)}$;
\quash{\item Using this point, we identify $\calA$ with the vector space $\xcoch(T_0)^{\Gal(\overline{F}/F)}\otimes\bbR$, where $T_0$ is the torus in $G_0$ whose Lie algebra is $\frakt_0$;
An alcove $C\subset\calA$ whose closure contains the point $x_0$ and that is contained in the Weyl chamber in $\xcoch(T_0)\otimes\bbR$ determined by $\frakb_0$.}
\item a set of simple affine roots $\{\al_0,\ldots,\al_\ell\}$ such that $\al_1(x_0)=\cdots=\al_\ell(x_0)=0$ and that the vector parts of $\al_i, i=1,\ldots,\ell$ form the set of simple roots of $\frakg$ with respect to $\frakb$.
\end{enumerate}
Explicitly, we can identify $\calA$ with the $\bbR$-span of the $\Gal(\overline F/F)$-invariant subspace of the cocharacter group $\xcoch(T_{0,\ad})=\Hom_k(\bbG_m,T_{0,\ad})$, where $T_{0,\ad}$ is the adjoint torus with Lie algebra $\frakt_0$. We can also identify $x_0$ with the origin of $\xcoch(T_{0,\ad})^{\Gal(\overline F/F)}$.
If $\frakg=\frakg_0\otimes_{\bbC}F$ and if \eqref{pinning} arises as the base change (i.e. the split case), then $\{\al_1,\ldots,\al_\ell\}$ are the set of simple roots of $\frakg_0$ with respect to $\frakb_0$ (regarded as affine functions on $\calA$), and $\al_0=1-\theta$, where $\theta$ is the highest root.
Let $\{a_i\}$ be Kac's labels of the affine Dynkin diagram of $\frakg$ (cf. \cite[Ch. 4]{Ka} TABLE Aff 1,2), and $r=1,2,3$ be the order of the image of $\psi:\Gal(\overline F/F)\to \Aut(\frakg_0,\frakb_0,\frakt_0,e_0)$. We regard affine roots as affine functions on $\calA$. Under the (usual) normalization
\[r\sum a_i\al_i=1.\]
Let $C$ be the (closed) alcove in $\calA$ defined by $\{x\in\calA\mid \al_i(x)\geq 1\}$. Clearly, we can prove the proposition just for standard parahorics, i.e. those corresponding to facets of $C$. For such a parahoric $P$, let $x_P$ denote the barycenter of the corresponding facet. We can write the affine coordinates with respect to $\{\al_0,\ldots,\al_\ell\}$ (a.k.a. the Kac coordinates) of $x_P$ as
\[(\frac{s_0}{m},\ldots,\frac{s_\ell}{m}),\]
where $s_i\in\{0,1\}$ and $m=r\sum a_is_i$.
Note that $m(x_P-x_0)$ pairing with any root of $\frakg$ is an integer and therefore defines a cocharacter
\[\eta_P: \bbG_m\to T_\ad,\]
where $T_\ad\subset G_\ad$ is the torus whose Lie algebra is $\frakt$.
The isomorphism \eqref{pinning} induces an isomorphism $\Hom_F(\bbG_m,T_\ad)\simeq \Hom_k(\bbG_m,T_{0,\ad})^{\Gal(\overline F/F)}$, and therefore
we may regard $\eta_P$ as a cocharacter $\bbG_m\to T_0$.
Let $E/F$ be the (unique) Galois extension of degree $m$ inside $\overline F$. Then $\psi$ factors through $\Gal(E/F)\to\Aut(\frakg_0,\frakb_0,\frakt_0,e_0)$, and the isomorphism \eqref{pinning} descends to $E$. Following Kac, we define a torsion automorphism of $\frakg_0$ as
\[\theta_P: \Gal(E/F)\to\Aut(\frakg_0),\quad \theta_P(\ga)=\psi(\ga) \Ad(\eta_P(\zeta)^{-1}),\]
where $\zeta\in\mu_m$ corresponds to $\ga$ under the canonical identification $\Gal(E/F)=\mu_m$.
We can write
\[\frakg_0=\bigoplus_{i\in \bbZ/m\bbZ} \frakg_0(i),\]
where $\frakg_0(i)$ is the weight space of $\theta$ corresponding to $i\in \bbZ/m\bbZ=\mu_m^*$. Explicitly, $\frakg_0(i)$ is the $\zeta^{-i}$-eigenspace of $\theta_P(\ga)$.
We endow $\frakg_0\otimes E$ with an action of $\Gal(E/F)$ given by $\theta_P(\ga)\otimes\ga$ for $\ga\in\Gal(E/F)$. Explicitly, let $u$ be a uniformizer of $E$ such that $t=u^m$ is a uniformizer of $F$. Then $\zeta=\ga(u)/u$ so $\ga$ acts on $\frakg_0\otimes u^i$ as $\psi(\ga)\eta_P(\zeta^{-1})\otimes \zeta^i$.
In particular, for a generator $\ga$ of $\Gal(E/F)$,
$(\frakg_0\otimes E)^{\theta_P(\ga)\times\ga}$ is the $u$-adic completion of $\bigoplus_{i\in \bbZ} \frakg_0(i)\otimes u^i$.
Note that as $F$-automorphisms of $\frakg_0\otimes E$,
\[\psi(\ga)\times\ga= \Ad(\eta_P(u^{-1}))(\theta_P(\ga)\times \ga) \Ad(\eta_P(u)).\]
Combining with \eqref{pinning}, we obtain the isomorphism constructed by Kac (\cite[Theorem 8.5]{Ka})
\begin{equation}\label{Kac isom}
K:\frakg=(\frakg\otimes_FE)^{\ga}\simeq (\frakg_0\otimes_k E)^{\psi(\ga)\times\ga}\stackrel{\Ad(\eta_P(u))}{\simeq} (\frakg_0\otimes_k E)^{\theta_P(\ga)\times\ga},
\end{equation}
for any choice $\ga$ of the generator of $\Gal(E/F)$. By abuse of notation, we also use $K$ to denote the induced embedding $\frakg\simeq (\frakg_0\otimes_k E)^{\theta_P(\ga)\times\ga}\to \frakg_0\otimes E$.
Let us have a more detailed analysis of the isomorphism $K$. First,
$$K(\frakm^n\frakt)=\frakt_0^{\theta_P(\ga)}\otimes \frakm^n=\sum_{j\geq n} \frakt_0^{\psi(\ga)}\otimes u^{mj}.$$
Next, let $\al$ be an affine root of $\frakg$ and let $\frakg_\al$ denote the corresponding affine root space (which is one-dimensional over $\bbC$). Let $\{b_i\}$ be the set of roots of $\frakg_0$ that restrict to the linear part $\dot{\al}$ of $\al$, and $\{\frakg_{0,b_i}\}$ be the corresponding root spaces. Then
\[K(\frakg_\al)= \sum_i \frakg_{0,b_i} \otimes u^{\langle\eta_P,b_i\rangle+m\al(x_0)}\subset \frakg_0(j)\otimes u^j\]
for $j=\langle\eta_P,b_i\rangle+m\al(x_0)$.
Note that we can also rewrite this number as
\[m\dot{\al}(x_P-x_0)+m\al(x_0)=m\al(x_P).\]
Now recall that by the definition of the Moy-Prasad filtration, $\frakp(n)$ is the subalgebra of $\frakg$ generated by $\frakm^{\left\lfloor \frac{n}{m}\right\rfloor}\frakt$, and all the affine root subspaces $\frakg_{\al}$ such that $\al(x_P)\geq \frac{n}{m}$. Therefore, $K$ restricts to an isomorphism
\[K:\frakp(n)\simeq (\frakg_0\otimes_k \frakm_E^n)^{\theta_P(\ga)\times\ga}=\prod_{j\geq n} \frakg_0(j)\otimes u^j,\]
which is \cite[Theorem 4.1]{RY}
Note that by replacing $\frakg$ by $\frakg^*$ in the above paragraphs, we obtain an isomorphism $K^*: \frakg^*\simeq (\frakg^*_0\otimes E)^{\theta_P(\ga)\times \ga}$
and therefore we obtain
\[DK:\frakg^*\otimes \omega_F\simeq (\frakg^*_0\otimes_k \omega_E)^{\theta_P(\ga)\times\ga}\subset \frakg^*_0\otimes_k \omega_E, \quad \xi(t)\frac{dt}{t}\mapsto mK^*(\xi(t))\frac{du}{u}.\]
We need two properties of $DK$: (i) the following diagram is commutative,
\begin{equation}\label{aux2}
\begin{CD}
\frakg^*\otimes \omega_F@>DK>>\frakg^*_0\otimes \omega_E\\
@VVV@VVV\\
\frakc^*\times^{\bbG_m}\omega^\times_F@>>>\frakc^*_0\times^{\bbG_m}\omega^\times_E.
\end{CD}
\end{equation}
(ii) $K\otimes DK$ commutes with the pairings induced by the residue up to the scalar $m$. I.e.,
for $X\in \frakg, \ \xi\frac{dt}{t}\in \frakg\otimes\omega_F$,
\begin{equation}\label{inv res}
\Res_F (\xi,X)\frac{dt}{t}= m\Res_E (K^*(\xi),K(X))\frac{du}{u}=m\Res_E(DK(\xi\frac{du}{u}),K(X)).
\end{equation}
Since under the residue pairings, the orthogonal complement of $(\frakg_0\otimes \frakm_E^n)$ is $(\frakg^*_0\otimes \frakm_E^{-n}\omega_{\calO_E})$, and the orthogonal complement of $\frakp(n)$ is $\frakp(n)^\perp$,
we obtain from \eqref{inv res} that
\begin{equation}\label{aux1}
\frakp(n)^\perp=DK^{-1}(\frakg^*\otimes\omega_F\cap \frakm_E^{-n}(\frakg^*_0\otimes\omega_{\calO_E})).
\end{equation}
Note that under the map $\frakg_0\otimes\omega_E\to \frakc_0\otimes \omega_E$, the closed subscheme
$\frakm_E^{-n}(\frakg^*_0\otimes\omega_{\calO_E})$ maps surjectively to $\bigoplus \omega_{\calO_E}^{d_i}(nd_i)$. This is clear for $n=0$ and therefore for all $n$.
On the other hand,
\begin{lem} Inside $\omega^d_{E}$,
\[ \omega_{\calO_E}^{d}(n)\cap \omega^d_F= \omega_{\calO_F}^{d}(\lceil\frac{(d-n)}{m}\rceil).\]
\end{lem}
\begin{proof}Let $f(t)(\frac{dt}{t})^d\in \omega_F^d$, then $f(u^m)m^d(\frac{du}{u})^d\in \omega_{\calO_E}^d(n)$ if and only if
$$m\on{ord}_t(f)-d\geq -n.$$ So $\on{ord}_t(f)\geq \lceil\frac{(d-n)}{m}\rceil$.
\end{proof}
Now the proposition follows easily from \eqref{aux2} \eqref{aux1} and the above lemma.
\end{proof}
\begin{rmk}\label{dual}
When studying harmonic analysis on $\frakg$, it is customary to regard the (local) Hitchin map as a map $\frakg\otimes \omega_F\to \frakc\times^{\bbG_m}\omega_F$, and it's natural to describe the image of $\frakp(n)$ under this map. But this reduces the study the image of $\frakp(n)^\perp$ under $h_x^{cl}$ by the following reason.
Note that the isomorphism $\frakg^*\simeq \frakg$ induced by the Killing form intertwines $K$ and $K^*$. Therefore,
\begin{equation*}
\frakp(n)^\perp= \frakp(1-n)\otimes_{\calO}\omega_\calO(1)=\frakp(1-n-m)\otimes_{\calO}\omega_{\calO}.
\end{equation*}
\end{rmk}
\medskip
We consider a few well-known special cases of Proposition \ref{size of image}.
If $n=0$, it says that the local Hitchin map restricts to a map
\[\frakp(0)^\perp\to \bigoplus_{i}\omega^{d_i}_{\calO}(\left\lfloor\frac{(m-1)d_i}{m} \right\rfloor)\otimes {^L}V_{\frakg,d_i}\]
In particular,
\begin{itemize}
\item if $m=1$, so $\frakp\simeq \frakg_0\otimes_k \calO$ is hyperspecial, then we have the unramified local Hitchin map
$$\frakp(0)^\perp\to \bigoplus_{i}\omega_{\calO}^{d_i}\otimes {^L}V_{\frakg,d_i}=\frakc^*_{\calO}\times^{\bbG_m}\omega^\times_{\calO}.$$
\item If $m$ is the (twisted) Coxeter number of $\frakg$, then $\frakp$ must be an Iwahori subalgebra and the local Hitchin map restricts to the map
$$\frakp(0)^\perp\to \bigoplus_{i}\omega_{\calO}^{d_i}(d_i-1)\otimes {^L}V_{\frakg,d_i}.$$
\quash{
\item If $m=2$, then $\frakp$ is the so-called Gross parahoric and we have
$$\frakp(0)^\perp \to \bigoplus_{i}\omega_{\calO}^{d_i}(\left\lfloor\frac{d_i}{2}\right\rfloor)\otimes V_{\frakg,d_i}.$$}
\end{itemize}
It's known that in both cases the maps are surjective.
If $n=1$, it says that the local Hitchin map restricts to a map
\begin{equation}\label{image p(1)}
\frakp(1)^\perp\to\bigoplus_{i}\omega^{d_i}_{\calO}(d_i)\otimes V_{\frakg,d_i}\simeq \frakc_{\calO}\times^{\bbG_m}\omega_{\calO}(1)^\times=: \on{Hitch}(D)_{\on{RS}},
\end{equation}
which is known to be surjective if $\frakp$ is an Iwahori subalgebra (so $\frakp(1)=[\frakp(0),\frakp(0)]$). Since for a general parahoric subalgebra $\frakp$, $\frakp(1)^\perp$ contains the orthogonal complement of the pro-nilpotent radical of some Iwahori subalgebra, we see that
\begin{cor}
The map \eqref{image p(1)} is surjective for every $\frakp$.
\end{cor}
\medskip
Now we consider the special case when $\theta_P:\mu_m\to\Aut(\frakg_0)$ corresponding to the parahoric $P$ constructed above is principal (or called $N$-regular as in \cite{P}). This means that $\theta_P$ is
conjugated by an inner automorphism to
\[\theta'_m=\psi \rho: \mu_m\to \Aut(\frakg_0),\]
where $\rho$ as usual denotes the half sum of positive roots of $\frakg_0$ (with respect to $\frakb_0$). In this case, there exists a regular nilpotent element $e'$ of $\frakg_0$ (maybe different from $e_0$ we fixed before) in $\frakg_0(1)$.
\quash{According to Kac (\cite[Theorem 8.6]{Ka}), $\theta_P$ and $\theta'$ are conjugate if and only if the Kac coordinate of $x_P$ can be transferred to $(\frac{r+1-h_{\psi}}{m},\frac{1}{m},\ldots,\frac{1}{m})$ under the Iwahori-Weyl group of $G(F)$. }
Clearly, $m=1$ (so that $\frakp$ is hyperspecial) is the case. In addition, it is proved in \cite[Corollary 5.1]{RY} that if $m$ is a regular elliptic number of $\frakg_0$, then $\theta_P$ is principle. In particular, if $P=I$ is an Iwahori subgroup, then $\theta_I$ is principal.
Assume that $\theta_P$ is principal. Let us denote
$$\on{Hitch}(D)_{\frakp(n)}=\bigoplus_i\omega_{\calO}^{d_i}(d_i-\lceil\frac{d_i(1-n)}{m}\rceil)\otimes {^L}V_{\frakg,d_i}\subset \on{Hitch}(D^\times):=\frakc^*\times^{\bbG_m}\omega_F^\times.$$
\begin{prop}\label{surj}
Assume that $\theta_P$ is principle. Then the map
$$\frakp(2)^\perp\to\on{Hitch}(D)_{\frakp(2)}$$ is surjective. In particular, the map $\on{Fun}\on{Hitch}(D)_{\frakp(2)}\to \on{Fun}\frakp(2)^\perp$ is injective.
\end{prop}
\begin{rmk}There are only a few parahorics $\frakp$ of $\frakg$ such that $\theta_P$ is principal. It will be interesting to describe the precise image of $\frakp(n)^\perp$ under the local Hitchin map for a general parahoric $\frakp$ and general $n$. We are informed that D. Baraglia and M. Kamgarpour have made some progress in this direction.
\end{rmk}
\begin{proof}
It is convenient to identify $\frakg_0^*$ with $\frakg_0$ using the Killing form. We can choose a principal $\fraks\frakl_2$-triple $\{e',h',f'\}$ of $\frakg_0$ where $e'\in\frakg_0(1), h'\in \frakg_0(0), f'\in \frakg_0(-1)$.
Then the Kostant section
\[\kappa:\frakc_0\times^{\bbG_m}\omega^\times_E\to \frakg_0\otimes \omega_E\]
associated to $\{e',h',f'\}$ maps the element
$$(t^{-\lfloor\frac{d_1}{m}\rfloor}c_1(t)(\frac{dt}{t})^{d_1}\otimes v_1,\ldots,t^{-\lfloor\frac{d_\ell}{m}\rfloor}c_\ell(t)(\frac{dt}{t})^{d_\ell}\otimes v_\ell)\in \on{Hitch}(D)_{\frakp(2)}$$ to
\begin{equation}\label{section}
(f'+ \sum_i u^{-d_i-m\lfloor\frac{d_i}{m}\rfloor}c_i(u^m)p_i) du,
\end{equation}
where $p_i'\in \frakg_0^{e'}$ such that $\chi(f'+p_i)=v_i$. Let us regard $h'$ as a cocharacter $h':\bbG_m\to G_0$. Then after conjugating by $\Ad(h'(u))$, the element \eqref{section} becomes
\[(u^{-1}f'+ \sum_i u^{d_i-m\lfloor\frac{d_i}{m}\rfloor-1}c_i(u^m)p_i)\frac{du}{u}.\]
As $p_i\in \frakg_0(d_i-1)$, the pre-image under $DK$ of the above element belongs to $\frakp(2)^\perp$. This implies the proposition.
\end{proof}
There is the following map
\begin{equation}\label{loc res}
\on{Hitch}(D)_{\frakp(2)}\to \bigoplus_{d_i\mid m} \omega^{d_i}_\calO(d_i+\frac{d_i}{m})/\omega^{d_i}_\calO(d_i+\frac{d_i}{m}-1)\otimes {^L}V_{\frakg,d_i},
\end{equation}
which fits into the commutative diagram
\[\begin{CD}
\frakp(2)^\perp @>>> \frakg^*_0(-1)\otimes\omega_{\calO_E}(2)|_0=\frakp(2)^\perp/\frakp(1)^\perp\\
@VVV@VVV\\
\on{Hitch}(D)_{\frakp(2)}@>>> \bigoplus_{d_i\mid m} \omega^{d_i}_\calO(d_i+\frac{d_i}{m})/\omega^{d_i}_\calO(d_i+\frac{d_i}{m}-1)\otimes {^L}V_{\frakg,d_i}.
\end{CD}\]
In addition, the right vertical map factors through
\begin{equation}\label{vinberg quot}
\frakg^*_0(-1)\otimes \omega_{\calO_E}(2)|_0\to (\frakg^*_0(-1)/\!\!/L_P)\times^{\bbG_m}\omega_{\calO_E}(2)|_0\to \bigoplus_{d_i\mid m} \omega^{d_i}_\calO(d_i+\frac{d_i}{m})/\omega^{d_i}_\calO(d_i+\frac{d_i}{m}-1)\otimes {^L}V_{\frakg,d_i}.
\end{equation}
Since $\theta_P$ is principal, the map
$$(f'+\frakg_0^{e'})\cap \frakg_0(-1)=\{f+ \sum_{m\mid d_i} c_ip_i\}\to \frakg_0(-1)/\!\!/L_P$$ is an isomorphism by \cite[Theorem 3.5]{P}. Therefore the last map in \eqref{vinberg quot} is an isomorphism.
Note that the pairing $\Res_F: \frakp(2)^\perp\times \frakp(1)\to\bbC$ identifies $\frakg^*_0(-1)\otimes\omega_{\calO_E}(2)|_0$ with the dual $V_P^*$ of $\frakg_0(1)=V_P$, and therefore we naturally
identify $(\frakg^*_0(-1)/\!\!/L_P)\times^{\bbG_m}\omega_{\calO_E}(2)|_0$ with $V^*_P/\!\!/L_P$.
We therefore obtain
\begin{prop}\label{new residue}
Assume that $\theta_P$ is principal. Then
there is the following commutative diagram
\[\begin{CD}
\frakp(2)^\perp @>>> V_P^*\\
@VVV@VVV\\
\on{Hitch}(D)_{\frakp(2)}@>>> V_P^*/\!\!/L_P,
\end{CD}\]
with all arrows surjective.
\end{prop}
\begin{rmk}\label{first residue}
We will not make use of this remark. One should compare the above proposition with the following commutative diagram, which holds for any $\frakp$,
\[\begin{CD}
\frakp(1)^\perp @>>> ( \Lie L_P)^*\simeq \frakp(1)^\perp/\frakp(0)^\perp\\
@VVV@VVV\\
\on{Hitch}(D)_{\on{RS}}@>>> (\frakc^*_{\calO}\times^{\bbG_m}\omega_{\calO}(1)^{\times})|_0.
\end{CD}\]
Note that the bottom arrow in the diagram is the classical limit of the residue map studied in \cite{BD,FGa}.
\end{rmk}
\section{Endomorphisms of some vacuum modules}\label{endo alg}
We continue to fix a point $x\in X$, and for simplicity write $F_x$ by $F$, etc.
\subsection{Endomorphisms of some vacuum modules}Let us quantize the picture discussed in the previous section.
As in \S~\ref{vac mod}, we restrict to the case when $\frakg$ is a split Lie algebra. Without loss of generality we can assume that $\frakg=\frakg_0\otimes_kF$.
Also we denote by
$\hat{\frakg}_c$ the \emph{critical} central extension of $\frakg$ as in \S~\ref{vac mod}, and let $\hat{U}_c(\frakg)$ be the corresponding completed universal enveloping algebra.
Let $P\subset G(F)$ be a parahoric subgroup and $P=P(0)\rhd P(1)\rhd P(2)\rhd\cdots$ be the corresponding Moy-Prasad filtration. As $P(n)$ for $n>0$ is a pro-unipotent group, we have a natural splitting $\widehat{\frakp(n)}=\frakp(n)\oplus \bbC\mathbf{1}$, where $\widehat{\frakp(n)}$ denotes the preimage of $\frakp(n)$ in $\hat{\frakg}_c$.
Now for $n>0$, we define
\begin{equation}\label{vacp}
\Vac_{\frakp(n)}= \on{Ind}_{\frakp(n)\oplus \bbC\mathbf{1}}^{\hat{\frakg}_c}(\bbC),
\end{equation}
where $\frakp(n)$ acts trivially on $\bbC$ and $\mathbf{1}$ acts as the identity.
Let $\frakZ$ be the center of $\hat{U}_c(\frakg)$ endowed with the natural filtration. By \cite[Theorem 3.7.8]{BD}, the natural map
\begin{equation}\label{classical center}
\on{gr}\frakZ\to \on{Fun}\on{Hitch}(D^\times)
\end{equation}
is an isomorphism.
We write
\[\frakZ\to \frakZ_{\frakp(n)}\hookrightarrow \End \Vac_{\frakp(n)}\simeq \Vac_{\frakp(n)}^{P(n)}\subset \Vac_{\frakp(n)}.\]
All the maps are strictly compatible with the filtrations so that taking the associated graded gives
\[\on{Fun}\on{Hitch}(D^\times)\twoheadrightarrow \on{Fun}\on{Hitch}(D)_{\frakp(i)}\subset \on{Fun}(\frakp(i)^\perp).\]
\quash{
\medskip
\noindent\bf Conjecture. \rm The natural map
\[\frakZ\to \Vac_{\frakp(i)}^{P(0)}\]
is surjective.
\medskip
\begin{prop}
The conjecture holds if $G$ is split over $F$.
\end{prop}
\begin{proof}
The usual filtration on $\hat{U}_{\crit}(\frakg)$ induces the filtrations on $\frakZ$ and on $\Vac_{\frakp(i)}^{P}$, and it is enough to show that the composition
\[\on{gr}\frakZ\to \on{gr}(\Vac_{\frakp(i)}^{P})\hookrightarrow (\on{gr}\Vac_{\frakp(i)})^{P}\]
is surjective.
Recall by \cite[Theorem 3.7.8]{BD}, the natural injective map
$\on{gr}\frakZ \to \on{Fun}\on{Hitch}(D^\times)$ is an isomorphism. So it is enough to show that
\begin{prop}
The natural map $\on{Fun}\on{Hitch}(D^\times)\to \on{Sym}(L\frakg/\frakp(i))^{P}$ is surjective.
\end{prop}
\begin{proof}Using the residue pairing and by choosing a uniformizer $t\in F$, we identify $ \on{Sym}(L\frakg/\frakp(i))$ as the algebra of functions on $\frakp(-i)$. Then we need to show that ever $P$ invariant function on $\frakp(-i)$ can be extended to a $G(F)$-invariant function on $\frakg((t))$.
To prove the proposition, we use a construction by V. Kac to relates finite order automorphisms of a complex simple Lie algebra $\frakh$ to parahoric subalgebras in the (twisted) loop algebra $L\frakg$ of $\frakh$. We following the exposition of \cite{GLRY}
So let $\frakh$ be a simple Lie algebra over $\bbC$ and let $\theta$ be a finite order automorphism of $\frakh$. Assume that the order of $\theta$ is $m$ and let $\zeta$ be an $m$th root of unit. We write
\[\frakh=\bigoplus_{i\in \bbZ/m\bbZ} \frakh(i),\]
where $\frakh(i)$ is the eigenspace of $\theta$ with eigenvalue $\zeta^i$. More canonically, let $\theta: \mu_m\to \Aut(\frakh)$ be an injective homomorphism. Then we can decompose
\[\frakh=\bigoplus_{i\in \bbZ/m}\frakh(i),\]
where $\bbZ/m$ is regard as the group of characters of $\mu_m$.
We fix a Cartan sub algebra $\frakt\subset\frakh$. Let $R\subset \frakt^*$ be the set of roots. We fix $\Delta\subset R$ a set of simple roots.
By the classification of finite order automorphisms of $\frakg$ by V. Kac, we have
\[\Aut(R,\Delta)\to \Aut(\frakg)_{\on{tor}}\]
we can write
\[\theta= \on{Int}(g)\sigma,\]
where $\on{Int}(g)\in G_\ad$ is an inner automorphism, $\sigma$ is an outer automorphism of order $e=1,2$ or $3$. In addition, one can choose $g$ as
\[g= \sum_{i=1}^{\ell} s_i\omega_i(\zeta),\]
where $\omega_i:\bbG_m\to T_\ad$ are the fundamental coweights, $s_i\in\bbZ_{\geq 0} ,i=0,\ldots,\ell$ such that
\[\sum a_is_i=m,\]
where $a_i$ are the Kac-labelling of the affine Dynkin diagram. We write $x=\frac{1}{m}\sum s_i\omega_i \in \xcoch(T)_{\bbQ}$.
\begin{thm}
Let $L\frakh=\frakh\otimes\bbC((u))$, endowed with an action of $\theta$, where $\theta$ acts on $\frakg$ as before, and on $u$ as $\theta(u)=\zeta^{-1}(u)$. Then $(L\frakg)^\theta$ is a (twisted) loop algebra and $(L^+\frakg)^\theta$ is a parahoric subalgebra corresponding to $x$. In addition, the natural filtration by the order of $u$ induces the Moy-Prasad
\end{thm}
The first statement is due to V. Kac. The last two statement
Recall the following map constructed in \cite[\S~2.4.2]{BD}.
\[T_{cl}: \frakh((u)) \to \frakh((s))[[u]],\quad T_{cl}\varphi=\sum_k \frac{(D^k\varphi)(s)}{k!}u^k.\]
More precisely, $T_{cl}$ as a projective system of morphisms of ind-schemes over $\bbC$,
\[T_{cl,N}: \frakh((u))\to \frakh((s))[u]/u^{N+1}, \quad T_{cl,N}\varphi=\sum_{k=0}^N \frac{(D^k\varphi)(s)}{k!}u^k\]
A direct calculation shows that $T_{cl,N}$ is $\theta$-equivariant.
Let $f: \frakh[[u]]\to \bbC$ be a regular function, coming from $\frakh[[u]]\to \frakh[u]/u^N\to \bbC$. Then by base change
Therefore, we obtain
\end{proof}
\end{proof}
\begin{cor}The natural map
$$\sigma: \on{gr}(\Vac_{\frakp(i)}^P)\to (\on{gr}(\Vac_{\frakp(i)}))^P$$ is an isomorphism.
\end{cor}
Let $\frakZ_{\frakp(i)}$ be the support of $\Vac_{\frakp(i)}$. Then
}
As
\[\Vac_{\frakp(2)}=\on{Ind}^{\hat{\frakg}_{c}}_{\frakp(2)+\bbC\mathbf{1}}\bbC=\on{Ind}^{\hat{\frakg}_{c}}_{\frakp(1)+\bbC\mathbf{1}}U(V_P),\]
we obtain an injective algebra homomorphism
\[U(V_P)\simeq \End_{\frakp(1)}U(V_P)\subset \End(\Vac_{\frakp(2)}).\]
\begin{prop}\label{endo diag}
Assume that $\theta_P$ is principal. Then there is the following natural commutative diagram
\[\begin{CD}
U(V_P)^{L_P}@>>> \frakZ_{\frakp(2)}\\
@VVV@VVV\\
U(V_P)@>>> \End(\Vac_{\frakp(2)})
\end{CD}\]
In fact, $\frakZ_{\frakp(2)}\cap U(V_P)=U(V_P)^{L_P}$.
\end{prop}
\begin{proof}Let $W$ be the intersection of $\frakZ_{\frakp(2)}$ and $U(V_P)$ inside $\End(\Vac_{\frakp(2)})$. Under the following natural maps,
\[\End(\Vac_{\frakp(2)})\simeq \Vac_{\frakp(2)}^{\frakp(2)}\subset \Vac_{\frakp(2)},\]
the subalgebra $\frakZ_{\frakp(2)}\subset \End(\Vac_{\frakp(2)})$ is in fact contained in $\Vac_{\frakp(2)}^P\subset \Vac_{\frakp(2)}$, and $U(V_P)$ is just the subspace in $\Vac_{\frakp(2)}$ generated from the vacuum vector by $\frakp(1)$.
Therefore $W$ is contained in $U(V_P)^{L_P}=U(V_P)\cap \Vac_{\frakp(2)}^P$.
The filtrations on all the above spaces are induced from the natural filtration on $\hat{U}_{c}(\frakg)$. Therefore $\on{gr}W$ is the intersection of $\on{gr}\frakZ_{\frakp(2)}=\on{Fun}\on{Hitch}(D)_{\frakp(2)}$ and $\on{gr}U(V_P)=\on{Fun}(V_P^*)$ inside $\on{gr}\Vac_{\frakp(2)}=\on{Fun} \frakp(2)^\perp$. Therefore by Proposition \ref{new residue}, we have inclusions
\[(\on{Fun} V_P^*)^{L_P}\subset\on{gr}W\subset \on{gr}(U(V_P)^{L_P})\subset (\on{Fun} V_P^*)^{L_P}.\]
Clearly, the composition of these inclusions is just the identity map.
Therefore $\on{gr}W=\on{gr}(U(V_P)^{L_P})$. Together with $W\subset U(V_P)^{L_P}$, it implies that $W=U(V_P)^{L_P}$.
\end{proof}
\begin{rmk}
(i) In the case when $P$ is an Iwahori subgroup, T.-H. Chen (private communication) proved this proposition by a different (and more direct) argument. He then in turn deduced Proposition \ref{new residue} from it in this case.
(ii) It is easy to see that the subalgebras $U(V_P)$ and $\frakZ_{\frakp(2)}$ in $\End(\Vac_{\frakp(2)})$ commute with each other, and therefore induces a map
\[U(V_P)\otimes_{U(V_P)^{L_P}} \frakZ_{\frakp(2)}\to \End(\Vac_{\frakp(2)}).\]
This is probably an isomorphism.
\end{rmk}
\subsection{Endomorphism algebras and opers}
Recall that we denote by ${^L}\frakg$ the Langlands dual Lie algebra of $\frakg$, equipped with a Borel subalgebra ${^L}\frakb$. As we assume that $\frakg$ is split, ${^L}V_\frakg=V_{{^L}\frakg}$ and we will following \cite{BD} to use the later notation in this subsection.
Recall that there is a natural $V_{{^L}\frakg}\otimes\omega_F$-torsor structure on $\on{Op}_{{^L}\frakg}(D^\times)$ (cf. \cite[3.1.9]{BD}), which
induces a natural filtration on $A_{{^L}\frakg}(D^\times)$ (cf. \cite[\S~3.1.13]{BD}) with a canonical isomorphism
$$\on{gr}A_{{^L}\frakg}(D^\times)\simeq \on{Fun} V_{{^L}\frakg}\otimes\omega_F\simeq \on{Fun}\frakc_{{^L}\frakg}\times^{\bbG_m}\omega_F^\times\simeq \on{Fun}\frakc^*\times^{\bbG_m}\omega_F^\times=\on{Fun}\on{Hitch}(D^\times).$$
One of the properties of the Feigin-Frenkel isomorphism \eqref{FeFr} is that it is
compatible with the filtration such that the associated graded gives \eqref{classical center}. Then we may regard $\Spec \frakZ_{\frakp(n)}=: \on{Op}_{{^L}\frakg}(D)_{\frakp(n)}$ as a closed subscheme of $\on{Op}_{{^L}\frakg}(D^\times)$, which we now describe for $n=2$.
Let $\Op_{^{L}\frakg}(D)$ be the scheme (of infinite type) of
${^L}\frakg$-opers on $D$. As before, it has a natural $V_{{^L}\frakg}\otimes\omega_{\calO}$-torsor structure. We have the natural inclusions of pro-unipotent groups
\[V_{{^L}\frakg}\otimes\omega_{\calO}\subset\bigoplus \omega_{\calO}^{d_i}(d_i+\left\lfloor\frac{d_i}{m}\right\rfloor)\otimes V_{{^L}\frakg,d_i}.\]
Then
\begin{lem} The scheme
$\on{Op}_{{^L}\frakg}(D)_{\frakp(2)}$ is the $(\bigoplus_i \omega_{\calO}^{d_i}(d_i+\left\lfloor\frac{d_i}{m}\right\rfloor)\otimes V_{{^L}\frakg,d_i})$-torsor induced from the $V_{{^L}\frakg}\otimes\omega_{\calO}$-torsor $\Op_{^{L}\frakg}(D)$.
\end{lem}
\begin{proof}
Let us temporarily denote by $\on{Op}_{{^L}\frakg}(D)'_{\frakp(2)}$ this induced torsor. Since $\on{Op}_{{^L}\frakg}(D^\times)$ is also induced from $\on{Op}_{{^L}\frakg}(D)$ via $V_{{^L}\frakg}\otimes\omega_{\calO}\subset V_{{^L}\frakg}\otimes\omega_F$, $\on{Op}_{{^L}\frakg}(D)'_{\frakp(2)}$ is a closed subscheme of $\on{Op}_{{^L}\frakg}(D^\times)$ and the filtration on $\on{Fun}\on{Op}_{{^L}\frakg}(D)'_{\frakp(2)}$ is the quotient filtration on $A_{{^L}\frakg}(D^\times)$. Therefore, it is enough to show that the associated graded of $\on{Fun}\on{Op}_{{^L}\frakg}(D)'_{\frakp(2)}$ and of $\on{Fun}\on{Op}_{{^L}\frakg}(D)_{\frakp(2)}$ coincide in $\on{Hitch}(D^\times)$, which is clear.
\end{proof}
More explicitly, we can describe $\on{Op}_{{^L}\frakg}(D)_{\frakp(2)}$ as follows. We fix a uniformizer $t\in F$, and a principal $\fraks\frakl_2$-triple $\{e,h,f\}$ of ${^L}\frakg$ with $e\in{^L}\frakb$, then it is well-known (e.g. \cite[\S~3.5.6]{BD} or \cite[\S~4.2.4]{F}) that $\on{Op}_{{^L}\frakg}(D^\times)$ can be identified with
\begin{equation}\label{oper space}
\{\nabla=d+(f+u)dt \mid u\in {^L}\frakb(F)\}/{^L}U(F)\simeq \{\nabla=d+(f+v)dt\mid v\in V_{{^L}\frakg}\otimes F\},
\end{equation}
where ${^L}U$ is the unipotent radical of the ${^L}B$.
Then $\on{Op}_{{^L}\frakg}(D)_{\frakp(2)}$ is the subspace of operators as above such that
\begin{equation}\label{irr oper}
v\in \bigoplus_i \omega^{d_i}_{\calO}(d_i+\left\lfloor \frac{d_i}{m}\right\rfloor)\otimes V_{{^L}\frakg,d_i}.
\end{equation}
\begin{rmk}
There is the notion of the slope of a ${^L}\frakg$-local system on the punctured disc $D^\times$ (e.g. \cite[\S~5]{FG} or \cite[\S~2]{CK}). It is not hard to see that $\on{Op}_{{^L}\frakg}(D)_{\frakp(2)}$ is the (reduced) subscheme of $\on{Op}_{{^L}\frakg}(D^\times)$ such that the underlying local system has slope $\leq \frac{1}{m}$. We do not use this fact in the sequel.
\end{rmk}
\section{Proof of a conjecture in \cite{HNY}}\label{Proof} Now we specialize the group scheme $\calG$ over $X=\bbP^1$. Let $G_0$ be a
simple, simply-connected complex Lie group, of rank $\ell$. Let us
fix $B_0\subset G_0$ a Borel subgroup and $B_0^{\opp}$ an opposite Borel
subgroup. The unipotent radical of $B_0$ (resp. $B_0^{\opp}$) is denoted by
$U_0$ (resp. $U_0^{\opp}$). Following \cite{HNY}, we denote by $\calG(0,1)$
the group scheme on $\bbP^1$ obtained from the dilatation of the constant group scheme
$G_0\times\bbP^1$ along $B_0^{\opp}\times\{0\}\subset G_0\times\{0\}$ and along
$U_0\times\{\infty\}\subset G_0\times\{\infty\}$. Explicitly,
$$\calG(0,1)(\calO_0)=\on{ev}_0^{-1}(B_0^{\opp}),\quad \calG(0,1)(\calO_\infty)=\on{ev}_\infty^{-1}(U_0),$$ where $\on{ev}_0: G_0(\calO_0)\to G_0$ (resp. $\on{ev}_\infty:G_0(\calO_\infty)\to G_0$) is the evaluation map.
Following \emph{loc.
cit.}, we denote $I(1)=\calG(0,1)(\calO_\infty)$.
Let $\calG(0,2)\to\calG(0,1)$ be the dilatation of $\calG(0,1)$ at $\infty$, such
that it is an isomorphism away from $\infty$, and that at $\infty$ it induces
$$\calG(0,2)(\calO_\infty)=I(2):=[I(1),I(1)]\subset I(1)=\calG(0,1)(\calO_\infty).$$
To simplify notations, in the sequel, $\calG(0,1)$ is denoted by $\calG'$ and $\calG(0,2)$ is denoted by $\calG$.
Note that we can apply all the results in \S~\ref{loc Hitchin} to the standard Iwahori subgroup $I=I(0)\subset G(F_\infty)$. Following notations in that section, let
$V=V_I=I(1)/I(2)$, which is isomorphic to
$\prod_{i=0}^{\ell}U_{\al_i}$, where $\al_i$ are simple affine roots,
and $U_{\al_i}$ are the corresponding affine root groups. We identify $V$ with its Lie algebra via the exponential map as explained before. Let $T=L_I= I(0)/I(1)$, which is the Cartan torus acting on $V$ by conjugation.
\quash{
Let us choose for
each $\al_i$ an isomorphism $\Psi_i:U_{\al_i}\simeq\bbG_a$. Then we
obtain a well-defined morphism
\[\Psi: I(1)\to I(1)/I(2)\simeq \prod_{i=0}^{\ell}U_{\al_i}\simeq\prod\bbG_a\stackrel{\on{sum}}{\to}\bbG_a.\]
Let $I_\Psi:=\ker\Psi\subset I(1)$.}
As explained in \emph{loc. cit.}, there is an open substack $*\subset \Bun_{\calG'}$ corresponding to trivializable $\calG'$-torsors. Its pre-image in
$\Bun_{\calG}$ is isomorphic to $V$, denoted by $\mathring{\Bun}_\calG$.
\begin{lem}\label{goodness}
The stack $\Bun_{\calG}$ is good in the sense of \cite[\S~1.1.1]{BD}.
\end{lem}
\begin{proof}Since $\Bun_{\calG}$ is a principal bundle
over $\Bun_{\calG'}$ under the group $V$,
it is enough to show that $\Bun_{\calG'}$ is good. But this follows from the fact that $\Bun_{\calG'}$ has a stratification by
elements in the affine Weyl group of $G$ such that the stratum
corresponding to $w$ has codimension $\ell(w)$ and the stabilizer
group has dimension $\ell(w)$.
Indeed, by \cite[Proposition 1]{HNY},
\[\Bun_{\calG'}=\calG'_{out}\backslash \Gr_{\calG',0},\]
where $\calG'_{out}=\calG'(\bbP^1\setminus\{0\})$ is the ind-group representing sections of $\calG'$ over $\bbP^1\setminus\{0\}$.
By \cite[Corollary 3]{HNY}, $\dim\Bun_{\calG'}=0$.
Note that $\Gr_{\calG',0}$ is just the usual affine flag variety for $G_0(F_0)$. By \cite[Theorem 7]{Fa}, for every $w$ in the affine Weyl group $W_{\on{aff}}$, the double quotient
\[C_w:=\calG'_{out}\backslash \calG'_{out}\ \cdot\ w\ \cdot\ \calG'(\calO_0)/\calG'(\calO_0)\]
is represented by an Artin stack, locally closed in $\Bun_{\calG'}$. In addition, $\dim C_w=-\ell(w)$. This proves the claimed fact.
\end{proof}
\begin{rmk}\label{sqr5}
Again, according to \cite[Theorem 7]{Fa}, the closure $\overline{C}_s$ of $C_s$ for a simple reflection $s$ is a divisor on $\Bun_{\calG'}$. Together with the calculation in \cite[\S~4.1]{Z}, we see twice of the sum of all these divisors gives the line bundle $\omega_{\Bun_{\calG'}}^{-1}$. In particular, the square root $\omega_{\Bun_{\calG'}}^{1/2}$ exists. Note that the pullback of $\omega_{\Bun_{\calG'}}^{1/2}$ is isomorphic to $\omega_{\Bun_{\calG}}^{1/2}$, and therefore $\omega_{\Bun_{\calG}}^{1/2}$ is canonically trivialized over $\mathring{\Bun}_\calG$ .
\end{rmk}
\quash{Let $S_w$ denote the preimage in $\Bun_{\calG(0,\Psi)}$ of
$C_w\subset \Bun_{\calG(0,1)}$. Then
$S_1\simeq\bbA^1$, and
\begin{lem}
For a simple reflection $s$, $S_1\cup
S_s\simeq\bbP^1$. In particular, any regular function on
$\Bun_{\calG(0,\Psi)}$ is constant.
\end{lem}
\begin{proof}
Indeed, let $i_s:\SL_2\to G(F_0)$ denote the $\SL_2$ for the simple affine root corresponding to $s$. It induces an embedding $\bbP^1\to \Gr_{\calG(0,1),0}$, still denoted by $i_s$. As $\bbP^1\times \calG(0,1)_{out}\to \Gr_{\calG(0,1),0}$ is an open immersion, we see it projects to an open immersion $\bbP^1\to \Bun_{\calG(0,1)}$.
\end{proof}}
Now we consider the Hitchin map
\[h^{cl}: T^*\Bun_\calG\to \on{Hitch}(X)_{\calG}\subset \on{Hitch}(U).\]
Applying results in \S~\ref{loc Hitchin} and the similar argument of Lemma \ref{support}, it is easy to calculate $\on{Hitch}(X)_{\calG}$ in this case
\[\bigoplus_ {i<\ell} \Gamma(X,\omega_X^{d_i}((d_i-1)\cdot 0+ d_i\cdot \infty)\otimes V_{\frakg_0,d_i}\bigoplus \Gamma(X,\omega_X^{d_\ell}((d_\ell-1)\cdot 0+ (d_\ell+1)\cdot \infty)\otimes V_{\frakg_0,d_\ell},\]
which is isomorphic to $\bbA^1$.
Let $\mu: T^*\Bun_{\calG}\to V^*$ be the moment map for the action of $V$ on $\Bun_\calG$.
\begin{lem}\label{flat moment}
The moment map $\mu: T^*\Bun_\calG\to V^*$ is flat.
\end{lem}
\begin{proof}Consider the Hamiltonian reduction $\mu^{-1}(0)/V\simeq T^*\Bun_{\calG'}$. By the proof of Lemma \ref{goodness}, $\Bun_{\calG'}$ is good, so $T^*\Bun_{\calG'}$ is of dimension zero. This implies that $\dim \mu^{-1}(0)=\dim V$. Therefore, $\mu$ is flat.
\end{proof}
We also need a global version of
Proposition \ref{new residue} in this case.
\begin{lem}\label{global new residue}
There is the following commutative diagram, with all arrows surjective.
\[\begin{CD}
T^*\Bun_\calG @>\mu>> V^*\\
@VVV@VVV\\
\on{Hitch}(X)_{\calG}@>>> V^*/\!\! /T.
\end{CD}\]
In addition, the bottom arrow is an isomorphism.
\end{lem}
\begin{proof}
By restricting to the formal neighbourhood of $\infty$, the global Hitchin map embeds into the local Hitchin map, and therefore Proposition \ref{new residue} gives the commutative diagram. To see the bottom arrow is an isomorphism, it is enough to observe that in our special case the composition
$$\on{Hitch}(X)_\calG\to \on{Hitch}(D)_{\frakp(2)}\to \bigoplus_{d_i\mid m} \omega^{d_i}_\calO(d_i+\frac{d_i}{m})/\omega^{d_i}_\calO(d_i+\frac{d_i}{m}-1)\otimes{^L}V_{\frakg,d_i}= \omega^{d_\ell}_\calO(d_\ell+1)/\omega^{d_\ell}_\calO(d_\ell)$$ is an isomorphism.
\end{proof}
\begin{rmk}\label{general vinberg}
The above discussions can be generalized. Namely, given any simple, simply-connected group $G$ over $\bbC((t))$ and a parahoric subgroup $P\subset G(\bbC((t)))$, one can construct the corresponding group scheme $\calG=\calG(0,2)_P$ over $\bbP^1$, unramified over $\bbP^1\setminus\{0,\infty\}$, and $\calG(\calO_0)\simeq P^{\on{opp}}$ and $\calG(\calO_\infty)= P(2)$, generalizing $\calG(0,2)$. Then Lemma \ref{goodness} and \ref{flat moment} generalize in this case. In addition, if $\theta_P$ is principal, Lemma \ref{global new residue} generalizes as well, i.e. we have
\[\begin{CD}
T^*\Bun_\calG @>\mu>> V_P^*\\
@VVV@VVV\\
\on{Hitch}(X)_{\calG}@>>> V_P^*/\!\! /L_P.
\end{CD}\]
To prove this, applying Proposition \ref{size of image}, we conclude that $\on{Hitch}(X)_\calG\to V^*_P/\!\!/L_P$ is a closed embedding. But $T^*\Bun_\calG\to V^*_P\to V^*_P/\!\!/L_P$ is clearly surjective, so $\on{Hitch}(X)_\calG\simeq V^*_P/\!\!/L_P$ if $\theta_P$ is principal.
\end{rmk}
Now we quantize the above picture.
Let us describe $\on{Op}_{{^L}\frakg}(X)_{\calG}$ in this
case.
At $0\in\bbP^1$, $K_0=\calG(\calO_0)=I^{\opp}:=\on{ev}_0^{-1}(B^{\opp})$. We use the inclusion $K_0\subset G_0(\calO_0)$ to split $\hat{\frakk}_0\simeq \frakk_0\oplus\bbC\mathbf{1}$ (see Remark \ref{diff splitting} for the subtlety). Then Lemma \ref{vacuum I} specializes to
\[\on{Vac}_0=\Ind_{\Lie I^{\opp} + \bbC \bf{1}}^{\hat{\frakg}_{c,0}}(\bbC_{\rho}),\]
which is just the Verma module $\bbM^{\opp}_{\rho}$ for $I^{\opp}$.
Here $\rho$ is half sum of roots of $B$, so is \emph{anti-dominant} w.r.t. $B^{\opp}$. It is known
(\cite[Corollary 13.3.2]{FGa} or \cite[Theorem 9.5.3]{F}) that
$\on{Fun}\on{Op}_{{^L}\frakg}(D^\times_0)\to\End (\bbM^{\opp}_{\rho})$
induces an isomorphism
\[\on{Fun}\on{Op}_{{^L}\frakg}(D_0)_{\varpi(0)}\simeq\End(\bbM^{\opp}_{\rho}),\]
where $\on{Op}_{{^L}\frakg}(D_0)_{\varpi(0)}$ is the scheme of
${^L}\frakg$ opers on $D_0$ with regular singularities and
residue $\varpi(0)$. In fact, in \emph{loc. cit.}, this is proved for the Verma module $\bbM_{-\rho}$ of $I$. But
if we choose an element $\tilde{w}_0\in G(\bbC)$, conjugation by which switches $B$ and $B^{\opp}$, and define the new action of $\hat{\frakg}_{c,0}$ on $\bbM_{-\rho}$ by $X\cdot v=\Ad_{\tilde{w}_0}(X) v$, then the resulting module
$\bbM_{-\rho}^{\tilde{w}_0}$ is isomorphic to $\bbM_{\rho}^{\opp}$.
Therefore the central supports of $\bbM_{-\rho}$ and $\bbM_{\rho}^{\opp}$ coincide.
We refer to \cite[\S~2]{FGa} for the precise definition of $\on{Op}_{{^L}\frakg}(D_0)_{\varpi(0)}$ as a moduli scheme. Here, we describe this space in concrete terms following the style as in \S~\ref{endo alg}. Let
$\{e,h,f\}$ be a principal $\fraks\frakl_2$-triple as with with $e\in {^L}\frakb$. Then
after choosing a
uniformizer $t$ of the disc $D_0$,
$\on{Op}_{{^L}\frakg}(D_0)_{\varpi(0)}$ is the space of operators
\begin{equation}\label{zero residue}
\{\partial_t+\frac{f}{t}+{^L}\frakg^e\otimes \calO_0\}.
\end{equation}
Indeed, it is known from \cite[\S~9.1]{F} that the space of opers with regularities can be identified with the space of operators of the form
$$\{\partial_t+\frac{1}{t}(f+ {^L}\frakb\otimes\calO_0)\}/{^L}U(\calO_0)\simeq \{\partial_t+\frac{1}{t}(f+{^L}\frakg^e\otimes \calO_0)\}.$$ The condition of the residue cuts out the subspace \eqref{zero residue}. It is also explained in \emph{loc. cit.} that after a gauge transformation, \eqref{zero residue} embeds into \eqref{oper space} as a closed subscheme.
\quash{
Then
after choosing a
uniformizer $z$ of the disc $D_0$,
$\on{Op}_{{^L}\frakg}(D_0)_{\varpi(0)}$ is the space of operators of
the form
\[\partial_z+\frac{f}{z}+{^L}\frakb[[z]].\]
up to ${^L}U(\calO_0)$-gauge equivalence. Indeed, the space of opers with regular singularities is the space of operators of the form $\partial_z+\frac{f}{z}+\frac{1}{z}{^L}\frakb[[z]]$ up to ${^L}U(\calO_0)$-gauge equivalence. The condition that it residue is $\varpi(0)$
Equivalently, we can describe it as follows
Let us complete $f$ to an $\fraks\frakl_2$-triple
$\{e,\gamma,f\}$ with $e\in {^L}\frakn$, and $[\gamma,e]=e$. Let ${^L}\frakg^e=\oplus_{i=1}^{\ell}{^L}\frakg^e_i$ be the
centralizer of $e$ in ${^L}\frakg$, decomposed
according to the
principal grading by $\gamma$, and let $d_i$ be the corresponding degree of ${^L}\frakg^e_i$ (so $\{d_i\}$ are the exponents of ${^L}\frakg$). Then
after choosing a
uniformizer $z$ of the disc $D_0$,
$\on{Op}_{{^L}\frakg}(D_0)_{\varpi(0)}$ is the space of operators of
the form
\[\partial_z+f+\gamma+\sum_{i=1}^{\ell} z^{-d_i}({^L}\frakg^e_i)[[z]].\]
I.e. $\on{Op}_{{^L}\frakg}(D_0)_{\varpi(0)}$ is a torsor under the infinite dimensional affine space $\sum_{i=1}^{\ell} z^{-d_i}({^L}\frakg^e_i)[[z]]$.
}
At $\infty\in\bbP^1$, $K_\infty=\calG(0,2)(\calO_\infty)=I(2)$, so
\[\on{Vac}_\infty=\Vac_{\frakp(2)},\]
as introduced in \S~\ref{endo alg}.
As in this case $\frakp$ is the Iwahori subalgebra so $m=h$ is the Coxeter number, we also write $\on{Op}_{{^L}\frakg}(D)_{\frakp(2)}$ as defined in \S~\ref{endo alg} by
$\on{Op}_{{^L}\frakg}(D_\infty)_{1/h}$.
Choose a uniformizer $t\in F_\infty$. Recall from \eqref{oper space} and \eqref{irr oper} that
$\on{Op}_{{^L}\frakg}(D_\infty)_{1/h}\subset \on{Op}_{{^L}\frakg}(D^\times_\infty)$ is the space of operators of
the form
\[\partial_t+f+\sum_{i<\ell} t^{-d_i}V_{{^L}\frakg,d_i}\otimes \calO_\infty+t^{-d_\ell-1}V_{{^L}\frakg,d_\ell}\otimes \calO_\infty.\]
After a gauge transformation, this space is identified with
\[\{\partial_t+\frac{f}{t}+\frac{1}{t}{^L}\frakb\otimes\calO_\infty+\frac{1}{t^2}{^L}\frakg_\theta\otimes\calO_\infty\}/{^L}U(\calO_\infty),\]
where ${^L}\frakg_\theta$ is the root subspace for the highest root $\theta$ of ${^L}\frakg$ (with respect to ${^L}\frakb$).
Therefore, by Lemma \ref{support}, $\on{Op}_{{^L}\frakg}(X)_{\calG}$ is isomorphic
to
\[\on{Op}_{{^L}\frakg}(X)_{(0,\varpi(0)),(\infty,1/h)}:=\on{Op}_{{^L}\frakg}(D_\infty)_{1/h}\times_{\on{Op}_{{^L}\frakg}(D_\infty^\times)}\on{Op}_{{^L}\frakg}(\bbG_m)\times_{\on{Op}_{{^L}\frakg}(D_0^\times)}\on{Op}_{{^L}\frakg}(D_0)_{\varpi(0)}.\]
As observed in \cite[\S~5]{FG},
\begin{lem}There is a (non-canonical) isomorphism
$\on{Op}_{{^L}\frakg}(X)_{(0,\varpi(0)),(\infty,1/h)}\simeq\bbA^1$.
\end{lem}
\begin{proof}
Indeed, any ${^L}B$-torsor on $\bbG_m$ is trivial. If we fix the $\fraks\frakl_2$-triple $\{e,h,f\}$ as above,
then after choosing a coordinate $z$ on $\bbG_m$, the space of opers on $\bbG_m$ is the space of operators of the form
\[\nabla=\partial_z+ \frac{f}{z} + v , \]
where $v(z)\in {^L}\frakg^e[z,z^{-1}]$. The condition at $0$ implies that $v(z)\in {^L}\frakg^e[z]$. At the $\infty$, the local coordinate is $t=1/z$ so such an operator becomes
\[\nabla=\partial_t+\frac{f}{t} + \frac{1}{t^2}v(\frac{1}{t}).\]
The condition at $\infty$ then implies that $v\in {^L}\frakg_\theta$ is constant. Choosing a root vector $e_\theta\in {^L}\frakg_\theta$,
Then the space $\on{Op}_{{^L}\frakg}(X)_{(0,\varpi(0)),(\infty,1/h)}$ is the space of opers of the
form
\[\nabla=\partial_z+ \frac{f}{z}+a e_{\theta},\]
with $a\in\bbC$.
So it is isomorphic to $\Spec \bbC[a]$.
\end{proof}
Now combining Proposition \ref{endo diag} and Lemma \ref{global new residue} we obtain
\begin{lem}
The following diagram is commutative.
\[\begin{CD}
U(V)^T@>\simeq>> \on{Fun}\on{Op}_{{^L}\frakg}(X)_{(0,\varpi(0)),(\infty,1/h)}\\
@VVV@VVh_\nabla V\\
U(V)@>>> \Gamma(\Bun_{\calG},D').
\end{CD}\]
\end{lem}
Now we can apply Corollary \ref{variant}, where $A$ is replaced by $U(V)$ in the current situation.
Given $\chi\in \Spec U(V)$, corresponding to the character $\varphi_\chi: U(V)\to \bbC$, the point $\pi(\chi)$ on $\on{Op}_{{^L}\frakg}(X)_{(0,\varpi(0)),(\infty,1/h)}$ then can be represented by the trivial ${^L}B$-torsor on $\bbG_m$ with the connection given by
\begin{equation}\label{Kl}
\nabla=\partial_z+ \frac{f}{z}+ \varphi_\chi(g)e_\theta,
\end{equation}
where $g\in U(V)^T$ is the generator of this algebra corresponding to $a$ under the isomorphism $U(V)^T\simeq \on{Fun}\on{Op}_{{^L}\frakg}(X)_{(0,\varpi(0)),(\infty,1/h)}$, and
\[\Aut_\calE=\omega_{\Bun_\calG}^{-1/2}\otimes (\calD'\otimes_{U(V),\varphi_\chi}\bbC)\]
is a Hecke-eigensheaf for this connection $\calE$. Note that since $\mu^{-1}(0)$ is a Lagrangian, $\Aut_\calE$ is holonomic.
We specialize to the case when the character $\varphi_\chi:U(V)\to \bbC$ is induced by an additive character $\Psi: V\to \bbG_a$. We use $\calL_\Psi$ to denote the pullback via $\Psi$ of the exponential D-module on $\bbG_a$, which is a rank one local system on $V$.
Since the map $U(V)\to \Gamma(\Bun_{\calG},D')$ geometrically comes from the action of $V$ on $\Bun_{\calG}$, $\Aut_\calE$ is in fact $(V,\calL_\Psi)$-equivariant on $\Bun_{\calG}$. Let $\mathring{\Bun}_{\calG}\subset\Bun_{\calG}$ be the open substack introduced before Lemma \ref{goodness}, on which $V$ acts simply-transitively. Since $\omega_{\Bun_\calG}^{-1/2}$ is canonically trivialized over this open substack (Remark \ref{sqr5}), the restriction of $\Aut_\calE$ to $\mathring{\Bun}_{\calG}$ is just $\calL_\Psi$.
We further specialize to the case when $\Psi$ is generic (see \cite[\S~1.3]{HNY} for the meaning). Then as argued in \cite[Lemma 2.3]{HNY} (which works in the D-module sitting without change), any $(V,\calL_\Psi)$-equivariant holonomic D-module supported on $\Bun_\calG\setminus\mathring{\Bun}_{\calG}$ is zero (in fact, holonomicity is unnecessary for this statement). Therefore,
$\Aut_\calE$ is isomorphic to the immediate (in fact, clean) extension of $\Aut_\calE|_{\mathring{\Bun}_{\calG}}$. As the Kloosterman D-module constructed in \cite{HNY} is the eigenvalue of $\Aut_\calE$, it coincides with the connection \eqref{Kl}. On the other hand, $\Psi$ is generic if and only if the induced character $\varphi_\chi: U(V)\to\bbC$ takes non-zero value on $g$. Therefore the connection \eqref{Kl} for generic $\Psi$ also coincides with the one constructed in \cite{FG}. We are done.
\begin{rmk}Note that even $\Psi$ is non-generic, $\Aut_\calE$ is still a Hecke eigensheaf, but for a tame local system on $\bbG_m$ (since in this case $\varphi_\chi(g)=0$). However, in this case it is not easy to express $\Aut_\calE$ as an extension of the local system on $\mathring{\Bun}_\calG$ by \emph{sheaf theoretical} operations.
\end{rmk}
| 67,152
|
TITLE: Measurability of the angular limit function
QUESTION [1 upvotes]: Let $\mathbb T$ be the unit circle and suppose that $f\in L^1(\mathbb T)$ is real-valued. Then its Poisson integral $F=P[f]$
is real-valued, too. Let
$$Osz[f](e^{i\theta}):=\limsup_{z\to e^{i\theta}\atop z\in S_\alpha(\theta)} F(z)-
\liminf_{z\to e^{i\theta}\atop z\in S_\alpha(\theta)} F(z)$$
be the oscillation of $F$ in the cone ${S_\alpha(\theta):=\{z\in \mathbb D: |\arg(1-e^{-i\theta}z)|<\alpha\}}$,
$0<\alpha<\pi/2$.
By The Hardy-Littlewood maximality theorem, $Osz[f]$ is well-defined and finite a.e. (for details, one may see the book "Bounded analytic functions" by J. B.Garnett). Why $Osz[f]$ is measurable? Note that, in general, the supremum over an uncountable family of measurable functions is not measurable, in general.
REPLY [0 votes]: May be one could proceed as follows:
It obviously suffices to prove that
$$V:=\{\theta\in\mathbb R: \hspace{-3mm} \limsup_{z\to e^{i\theta},~z\in S_\alpha(\theta)} F(z)>\eta\}$$ and
$$\{\theta\in\mathbb R:\hspace{-3mm} \liminf_{z\to e^{i\theta}~ z\in S_\alpha(\theta)} F(z)<\eta\}$$
are open sets in $\mathbb R$ for each $\eta\in \mathbb R$. Note that
there is a constant $k_\alpha>1$ such that
for every $z=re^{it}\in S_\alpha(\theta)$, we have $\mu:=|t-\theta|<k_\alpha(1-|r|)$. Hence,
if $\theta\in V$ and $F(re^{it})>\eta$ for some $r\geq r_0$, we choose $\theta'$ so close to $\theta$ that
$$|t-\theta'|\leq |t-\theta|+|\theta-\theta'|<\mu+\delta<k_\alpha(1-|r|).$$
Hence $re^{it}\in S_\alpha(\theta')$ and so
$$\limsup_{z\to e^{i\theta},~z\in S_\alpha(\theta')} F(z)\geq F(re^{it})>\eta.$$
We conclude that ${\theta'\in V}$.
| 73,423
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Geostrom: Gerard Butler disaster movie is a disaster at the box office, expected to lose Rs 650 crore
According to a new report, Gerard Butler’s new disaster movie, Geostorm, is expected to lose over $100 million (Rs 650 crore) at the box office.hollywood Updated: Oct 25, 2017 13:32 IST
The new Gerard Butler disaster movie Geostorm is expected to lose Warner Bros over $100 million (Rs 650 crore), a new report says.
According to the Wrap, the film, which was produced on a $120 million budget has so far collected $66 million worldwide. Box office analysts told the Wrap that the break even point for the film is between $300 and $350 million. Warner also cut down on the film’s marketing budget after it was delayed by almost 20 months due to expensive reshoots. The final film attracted toxic reviews, sitting at 13% on the review aggregator website Rotten Tomatoes.
“There really wasn’t a lot of advertising for this movie, and that’s a sign that WB was willing to cut their losses,” said Exhibitor Relations analyst Jeff Bock.
“By the end of its run,” the report says “analysts expect that the film will be lucky to make $200 million.”
This is Warner Bros’ second major flop of the year, after Guy Ritchie’s King Arthur: Legend of the Sword lost over $100 million. But the studio’s last four releases - Wonder Woman, Dunkirk, Annabelle: Creation and It, have grossed a combined $2.3 billion worldwide.
Follow @htshowbiz for more
| 121,424
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\begin{document}
\dedicatory{To Jon F. Carlson on his 80th birthday.}
\begin{abstract}
We develop a support theory for elementary supergroup schemes, over a field of positive characteristic $p\ge 3$, starting with a definition of a $\pi$-point generalising cyclic shifted subgroups of Carlson for elementary abelian groups and $\pi$-points of Friedlander and Pevtsova for finite group schemes. These are defined in terms of maps from the graded algebra $k[t,\tau]/(t^p-\tau^2)$, where $t$ has even degree and $\tau$ has odd degree. The strength of the theory is demonstrated by classifying the parity change invariant localising subcategories of the stable module category of an elementary supergroup scheme.
\end{abstract}
\keywords{complete intersection, $\pi$-point, support, stratification of module category, elementary supergroup scheme}
\subjclass[2020]{18G65 (primary); 18G80, 13E10, 16W55, 16T05}
\date{August 6, 2020}
\maketitle
\setcounter{tocdepth}{1}
\tableofcontents
\section*{Introduction}
Carlson~\cite{Carlson:1983a} introduced two notions of variety for a finitely generated module over an elementary abelian $p$-group. One, the rank variety, is based on restrictions to cyclic shifted subgroups, while the other is a cohomological support variety. This theory was generalised to infinitely generated modules by Benson, Carlson and Rickard~\cite{Benson/Carlson/Rickard:1996a} by using cyclic shifted subgroups defined over extension fields where enough generic points exist.
The notion of rank variety was put in the more general context of a finite group scheme $G$ over a field $k$ by Friedlander and Pevtsova \cite{Friedlander/Pevtsova:2007a}, through the theory of $\pi$-points. A $\pi$-point is a flat algebra homomorphism from $K[t]/(t^p)$ to $KG_K$, where $K$ is an extension field of $k$. There is an equivalence relation on $\pi$-points, and in the case of an elementary abelian group there is exactly one shifted subgroup in each equivalence class over a large enough field.
In a parallel development a theory of support varieties based on cohomology, and applicable in a rather broad context, was developed in \cite{Benson/Iyengar/Krause:2009a, Benson/Iyengar/Krause:2011a}. Combining those ideas with the theory of $\pi$-points eventually led to a classification of the localising, and colocalising, subcategories of the stable module category of a finite group scheme by the current authors~\cite{Benson/Iyengar/Krause/Pevtsova:2018a}.
In~\cite{Benson/Iyengar/Krause/Pevtsova:bikp5} we began a program to extend all these results to the world of supergroup schemes. In that work we identified a family of elementary supergroups schemes and proved that the projectivity of modules over a unipotent supergroup scheme can be detected by its restrictions to the elementary ones, possibly defined over extensions fields. This is in analogy with Chouinard's theorem for finite groups.
In this paper we develop a theory of $\pi$-points for the elementary supergroup schemes $\mcE$ introduced in~\cite{Benson/Iyengar/Krause/Pevtsova:bikp5}, and classify the localising subcategories of its stable module category. This feeds into the proof of a similar classification for finite unipotent supergroup schemes, presented in \cite{Benson/Iyengar/Krause/Pevtsova:bikp9}. It transpires that rather than flat maps from $K[t]/(t^p)$, we have to consider the $K$-algebra
\[
\test_K=K[t,\tau]/(\tau^2-t^p)
\]
where $t$ is in even degree and $\tau$ is in odd degree, and maps of finite flat dimension
\[
\alpha\colon\test_K\to K\mcE_K \,.
\]
The basic new result in our work is that $\pi$-points detect projectivity: a $k\mcE$-module $M$ is projective if, and only if, for each $\pi$-point $\alpha$ as above the restriction of the $K\mcE_K$-module $K\otimes_kM$ to $\test_K$ has finite flat dimension; a corresponding statement involving $\Hom_k(K,M)$, rather than $K\otimes_kM$, also holds.
From the point of view of commutative algebra, the group algebra $k\mcE$ is a complete intersection, and for such rings Avramov~\cite{Avramov:1989a} has developed a theory of support sets for modules, as yet another extension of Carlson's work. Our proof of the detection theorem for $\mcE$ goes by relating $\pi$-points to support sets over $k\mcE$. However, we need a version of the theory that applies also to infinite dimensional modules. This is presented in the Appendix.
With the detection theorem on hand, we put an equivalence relation on
$\pi$-points, analogous to the one
in~\cite{Friedlander/Pevtsova:2007a}, and exhibit an explicit set of
representatives of these equivalence classes, up to linear multiples,
analogous to the cyclic shifted subgroup approach in
\cite{Benson/Carlson/Rickard:1996a,Carlson:1983a}. These form a
projective space over $k$, and give rise to various notions of support
for modules over $k\mcE$. This allows us to classify the localising
subcategories of the stable category of $k\mcE$-modules. For a finite
dimensional $k\mcE$-module $M$ one also has a rank variety, in exact analogy with Carlson's theory of rank varieties for elementary abelian groups. This variety is determined by a rank condition on an explicitly defined matrix, and in particular it is a closed subset.
\subsection*{Outline of the paper}
The basic definitions concerning supergroup schemes, including the structure of the elementary ones, is recalled in \intref{Section}{se:prelim}.
In \intref{Section}{se:supportsets} we record the desired statements concerning support sets of modules over $k\mcE$, by specialising results for general complete intersections established in \intref{Appendix}{se:appendix}. The link to $\pi$-points, and a proof of the detection theorem discussed above, is presented in \intref{Section}{se:pi-points}. The proofs again require quite substantial input from the homological theory of complete intersection rings, and also basic facts about the representation theory of the algebra $\test$ recalled in \intref{Section}{se:test}. From this point on, the narrative unfolds in the expected way: Rank varieties for finite dimensional modules are introduced in \intref{Section}{se:rank}, leading to an explicit method for computing them. The equivalence relation on $\pi$-points is discussed in \intref{Section}{se:pi-and-cohomology}. \intref{Section}{se:support-and-cosupport} brings in the cohomological notions of support and cosupport, culminating with the classification results.
\begin{ack}
It is a pleasure to acknowledge the support provided by the American
Institute of Mathematics in San Jose, California, through their
``Research in Squares" program. We also acknowledge the National Science
Foundation under Grant No.\ DMS-1440140 which supported three of the
authors (DB, SBI, JP) while they were in residence at the Mathematical Sciences
Research Institute in Berkeley, California, during the Spring 2018
semester. Finally, two of the authors (DB, JP) are grateful for hospitality
provided by City, University of London.
SBI was partly supported by NSF grants DMS-1700985 and DMS-2001368.
JP was partly supported by NSF grants DMS-1501146,
DMS-1901854, and a Brian and Tiffinie Pang faculty fellowship.
\end{ack}
\section{Elementary supergroup schemes}
\label{se:prelim}
Throughout $k$ will be a field of positive characteristic $p\ge 3$. A \emph{superalgebra} will mean a $\bbZ/2$-graded algebra, and a \emph{graded} module over such an algebra will be assumed to be a $\bbZ/2$-graded left module. The category of graded modules over a superalgebra $A$ is denoted $\Mod(A)$; these are allowed to be infinitely generated. The full subcategory of finitely generated ones is denoted $\mod(A)$. When $M$ is a graded $A$-module, $\Pi M$ denotes the module with the zero and one components swapped. For $m\in M$, the action of an element $a$ on $\Pi m$ is given by
\[
a\cdot \Pi m \colonequals (-1)^{|a|}\Pi(am) \quad\text{in $\Pi M$.}
\]
A superalgebra $A$ is \emph{commutative} if $yx=(-1)^{|x||y|}xy$ for all $x,y$ in $A$.
An \emph{affine supergroup scheme} over $k$ is a covariant functor from commutative superalgebras to groups, whose underlying functor to sets is representable. If $G$ is a supergroup scheme its \emph{coordinate ring} $k[G]$ is the representing object. By applying Yoneda's lemma to the group multiplication and inverse maps, it is a commutative Hopf superalgebra. This gives a contravariant equivalence of categories between affine supergroup schemes and commutative Hopf superalgebras.
A \emph{finite} supergroup scheme $G$ is an affine supergroup scheme
whose coordinate ring is finite dimensional. In this case, the dual
$kG=\Hom_k(k[G],k)$ is a finite dimensional cocommutative Hopf
superalgebra called the \emph{group ring} of $G$. This gives a
covariant equivalence of categories between finite supergroup schemes
and finite dimensional cocommutative Hopf superalgebras. We denote by
$\StMod(kG)$ the \emph{stable module category} which is obtained from
$\Mod(kG)$ by annihilating all projective modules. Note that this
carries the structure of a triangulated category since $kG$ is a
self-injective algebra. We write $\stmod(kG)$ for the full subcategory
of finite dimensional modules.
Following~\cite{Benson/Iyengar/Krause/Pevtsova:bikp5} we say that a finite supergroup scheme $E$ over $k$ is \emph{elementary} if it is isomorphic to a quotient of $E_{m,n}^-\times(\bbZ/p)^r$, where $E_{m,n}^-$ is the Witt elementary, explicitly described in~\cite[Definition~3.3]{Benson/Iyengar/Krause/Pevtsova:bikp5}. The main theorem of that paper states that if $G$ is a unipotent finite supergroup scheme over $k$ then a $kG$-module $M$ is projective if and only if, for all extension fields $K$, and all elementary sub-supergroup schemes $\mcE$ of $G_K$, the module $M_K=K\otimes_kM$ is a projective $K\mcE$-module. It also gives a similar condition for the nilpotence of an element of cohomology.
We will be concerned \emph{only} with the algebra structure of $kE$
and the existence of a comultiplicaton; the explicit formula for the latter plays no role. Nevertheless, here is a brief description of the supergroup schemes $E_{m,n}^-$ and their finite quotients.
Let $\bbG_{a(n)}$ be the $n$th Frobenius kernel of the additive group scheme $\bbG_a$. We denote by $\bbG_a^-$ the supergroup scheme with the group algebra $\Lambda^*(\sigma) \cong k[\sigma]/\sigma^2$ with generator $\sigma$ in odd degree and primitive. Let $W_m$ be the affine group scheme of Witt vectors of length $m$, and let $W_{m,n}$ be the $n$th Frobenius kernel of $W_m$. Hence, $W_{m,n}$ is a finite connected group scheme of height $n$. Then the finite group scheme $E_{m,n}$ is defined as a quotient
\[
E_{m,n} : = W_{m,n}/W_{m-1,n-1}\,.
\]
The \emph{Witt elementary} super group $E_{m,n}^-$ is a finite supergroup scheme determined uniquely by the following two properties:
\begin{enumerate}
\item It fits into an extension $1 \to E_{m,n} \to E_{m,n}^- \to \bbG_a^- \to 1$,
\item $kE_{m,n}^- \cong k[s_1,\dots,s_{n-1},s_n,\si]/(s_1^p, \dots,s_{n-1}^p,s_n^{p^m},\sigma^2-s_n^{p})$.
\end{enumerate}
See \cite[\S 8]{Benson/Iyengar/Krause/Pevtsova:bikp5} for details.
The quotients of $E_{m,n}^-\times(\bbZ/p)^r$ can be completely classified using the theory of Dieudonn\'e modules and, up to isomorphism, fall into one of the following classes:
\begin{enumerate}[{\quad\rm(i)}]
\item
$\bbG_{a(n)} \times (\bbZ/p)^{\times s}$ with $n,s\ge 0$,
\item
$\bbG_{a(n)} \times \bbG_a^- \times (\bbZ/p)^{\times s}$ with $n,s\ge 0$,
\item
$E_{m,n}^- \times (\bbZ/p)^{\times s}$ with $m\ge 1$, $n\ge 2$, $s\ge 0$, or
\item
$E_{m,n,\mu}^-\times (\bbZ/p)^{\times s}$ with $m,n\ge 1$, $0\ne\mu\in k$ and $s\ge 0$.
\end{enumerate}
The last family involves supergroups $E^-_{m,n,\mu}$ which are
quotients of $E_{m+1, n+1}^-$ but have the same group algebra structure
as $E_{m+1, n}^-$. In the present work we are only concerned with the
algebra structure of $kE$, so we do not distinguish between cases
(iii) and (iv). Therefore, ignoring the comultiplication, the group
algebra of interest is one of the following:
\[
kE \cong
\begin{cases} k[s_1,\dots,s_n]/(s_1^p,\dots,s_n^p) \\
k[s_1,\dots,s_n,\sigma]/(s_1^p,\dots,s_{n}^p, \si^2) \\
k[s_1,\dots,s_n,\sigma]/(s_1^p,\dots,s_{n-1}^p,s_n^{p^{m}},s_n^p-\si^2) \quad \text{with $m\ge 2$.}
\end{cases}
\]
where the $s_i$ have degree $0$ and $\sigma$ has degree $1$. The first case occurs when the supergroup scheme is a group scheme. For these the representation theory has been analysed in detail in \cite{Benson/Carlson/Rickard:1996a, Benson/Iyengar/Krause:2011b, Carlson:1983a, Friedlander/Pevtsova:2007a}.
Our goal is to write down suitable analogues of the main theorems of
these papers for elementary supergroup schemes of the second and
third form. Among these, the third case is the one that presents and
most challenges and is the focus of the bulk of this work. The second case is discussed in the last \intref{Section}{se:others}.
Henceforth $k$ will be, as before, a field of positive characteristic $p\ge 3$ and $\mcE$ the supergroup scheme with group algebra
\begin{equation}
\label{eq:kE}
k\mcE\colonequals \frac{k[s_1,\dots,s_n, \sigma]}{(s_1^p, \dots,s_{n-1}^p, s_n^{p^m}, s_n^p-\sigma^2)}
\end{equation}
with $|s_i|=0$ and $|\sigma|=1$.
\section{Support sets}
\label{se:supportsets}
In this section we describe supports sets of $k\mcE$-modules, following the general theory for complete intersections developed in the Appendix.
We write $\bbA^n(k)$ for the $n$-tuples of elements, $
(a_1,\dots,a_n)$, of $k$, and $\bbP^{n-1}(k)$ for the non-zero
elements of $\bbA^n(k)$ modulo scalar multiplication. The image of
$(a_1,\dots,a_n)$ in $\bbP^{n-1}(k)$ is denoted $[a_1,\dots,a_n]$.
For a singly or doubly graded ring $R$ we write $\Proj R$ for the set
of homogeneous prime ideals other than the maximal ideal of non-zero
degree elements, topologised with the Zariski topology.
\begin{construction}
\label{con:powerseries}
Consider the ring of formal power series
\[
P \colonequals \pos{\bss, \sigma} \colonequals \pos{s_1,\dots,s_n, \sigma}
\]
in indeterminates $\bss, \sigma$. This is a noetherian local ring and $\fm\colonequals (\bss,\sigma)$ is its maximal ideal. In the ring $P$ consider the ideal
\begin{equation}
\label{eq:mI}
I=(s_1^p,\dots,s_{n-1}^{p}, s_n^{p^m}, s_n^p - \sigma^2)\,.
\end{equation}
As $k$-algebras one has $k\mcE = P/I$. Moreover $s_1^p,\dots,s_{n-1}^{p}, s_n^{p^m}, s_n^p - \sigma^2$ is a regular sequence in $P$. This can be verified directly from the definition, recalled in \intref{Appendix}{se:appendix}. Alternatively, one observes that this is a sequence of length $n+1$ in the local ring $P$ that is regular of Krull dimension $n+1$, and that the quotient ring, namely $k\mcE$, has Krull dimension zero; see \cite[Theorem~2.1.2]{Bruns/Herzog:1998a}.
The upshot is that $k\mcE$ is complete intersection. In particular the constructions and results presented in \intref{Appendix}{se:appendix} apply to $k\mcE$. One remark is in order before we can march on.
\end{construction}
\begin{remark}
\label{re:projectives-ungraded}
Recall that $k\mcE$ is a superalgebra. A graded $k\mcE$-module $M$ is projective if and only if it is projective when viewed as an (ungraded) $k\mcE$-module.
One way to verify this is to note that $k\mcE$ is an artinian local ring, and hence projective modules (graded or not) are free. It is clear that being free does not depend on the grading.
\end{remark}
\begin{definition}
Fix $\bsa \colonequals [a_1,\dots,a_{n+1}]$ in $\bbP^{n}(k)$ and consider the hypersurface $P_{\bsa}\colonequals P/(h_{\bsa})$, defined by the polynomial
\begin{equation}
\label{eq:hypersurface}
h_{\bsa} (\bss,\sigma)\colonequals a_1s_1^p + \cdots a_{n-1}s_{n-1}^p + a_ns_n^{p^m} + a_{n+1}(s_n^p-\sigma^2)\,.
\end{equation}
Since $h_{\bsa}(\bss,\sigma)$ is in $I$ there is a surjection $\beta_{\bsa}\colon P_{\bsa} \to k\mcE$. Given any $k\mcE$-module $M$ write $\beta^*_{\bsa}(M)$ for its restriction to $P_{\bsa}$, and set
\begin{equation}
\label{eq:Av-variety}
\hysupp_\mcE(M)\colonequals \{\bsa\in \bbP^{n}(k) \mid \fdim \beta_{\bsa}^*(M) =\infty \}\,.
\end{equation}
This is the \emph{support set} of $M$ defined through hypersurfaces, following Avramov~\cite{Avramov:1989a}.
\end{definition}
\begin{definition}
\label{de:base-change}
Given a $k\mcE$-module $M$ and an extension field $K$ of $k$ set
\[
M_K \colonequals K\otimes_k M \quad\text{and}\quad M^K \colonequals \Hom_k(K,M)
\]
viewed as $K\mcE$-modules in the natural way. The tensor product and Hom are taken in the graded category, with $K$ in degree $0$. When $K$ is a finite extension of $k$, the $K\mcE$-modules $M_K$ and $M^K$ are isomorphic.
Observe that $M_K$ and $M^K$ can also be realised as
\[
M_K = K\mcE\otimes_{k\mcE} M \quad\text{and} \quad M^K = \Hom_{k\mcE}(K\mcE,M)\,.
\]
This reconciles the notation here with that introduced in \intref{Definition}{de:extending-fields}.
\end{definition}
\begin{theorem}
\label{th:support-sets}
Let $M$ be a $k\mcE$-module. The following conditions are equivalent:
\begin{enumerate}[\quad\rm(1)]
\item
$M$ is projective;
\item
$\hysupp_{\mcE_K}(M_K)=\varnothing$ for any field extension $K$ of $k$;
\item
$\hysupp_{\mcE_K}(M^K)=\varnothing$ for any field extension $K$ of $k$.
\end{enumerate}
In \emph{(2)} and \emph{(3)} it suffices to take $K=k$ if $\rank_kM$ is finite and $k$ is algebraically closed. In any case it suffices to take for $K$ an algebraically closed extension of $k$ of transcendence degree at least $n$.
\end{theorem}
\begin{proof}
Observe that $M$ is projective if and only if it has finite projective dimension; this holds, for example, because $k\mcE$ is finite dimensional and commutative. Thus putting together \intref{Theorem}{th:Av} and \intref{Theorem}{th:appendix} yields the desired result.
\end{proof}
\subsection*{Cohomology of $k\mcE$}
The cohomology ring of $k\mcE$ was recorded in \cite[Theorems~5.10 and~5.11]{Benson/Iyengar/Krause/Pevtsova:bikp5}. It takes the form
\begin{equation}
\label{eq:Ecohomology}
H^{*,*}(\mcE,k)= \Lambda(u_1,\dots,u_n) \otimes_k k[x_1,\dots,x_n,\zeta]
\end{equation}
where the degrees are $|u_i|=(1,0)$, and $|x_i|=(2,0)$, $|\zeta|=(1,1)$. Here, the first degree is cohomological and the second comes from the $\bbZ/2$-grading on $k\mcE$. The numbering is chosen so that $x_i$ and $u_i$ restrict to zero on the subalgebra generated by $s_j$ if $i\ne j$, and $\zeta$ has non-zero restriction to the subalgebra generated by $\sigma$.
The nil radical of $H^{*,*}(\mcE,k)$ is generated by $ u_1,\dots,u_n$. The quotient modulo this ideal is
\[
H^{*,*}(\mcE,k)/\sqrt{0} = k[\bsx,\zeta] \quad\text{where}\quad \bsx=x_1,\dots,x_n\,.
\]
This ring has the property that the internal degree of a non-zero element is congruent to its cohomological degree modulo two.
It follows that if we take the $\Proj$ of this ring as a doubly graded ring or as a singly graded ring by ignoring the internal degree, we get the same homogeneous prime ideals, that is to say, the natural map is a homeomorphism:
\[
\Proj H^{*,*}(E,k) \xra{\ \cong\ } \Proj H^{*}(E,k)\,.
\]
Moreover the inclusion $k[\bsx,\zeta^2]\subset k[\bsx,\zeta]$ induces an isomorphism
\[
\Proj k[\bsx,\zeta]\xra{\ \cong \ } \Proj k[\bsx,\zeta^2]\,.
\]
Thus $\Proj H^{*,*}(\mcE,k)$ identifies with the standard projective space.
\subsection*{Cohomology of $P_{\bsa}$}
Fix $\bsa=[a_1,\dots,a_{n+1}] \in \bbP^n(k)$. The cohomology of the hypersurface $P_{\bsa}$ is
\begin{equation}
\label{eq:Pacohomology}
\Ext^*_{P_{\bsa}}(k,k) =
\begin{cases}
\Lambda(v_1,\dots,v_n)\otimes_k k[\eta] & \text{if $a_{n+1}\ne 0$} \\
\Lambda(v_1,\dots,v_n)\otimes_k k[y] \otimes \Lambda(\eta) & \text{if $a_{n+1}=0$}
\end{cases}
\end{equation}
where the $v_i$ and $\eta$ have cohomological degree one, and $y$ is of degree two; see, for example, ~\cite[Theorem~5]{Sjodin:1976a}. The map $\beta_{\bsa}\colon P_{\bsa}\to k\mcE$ induces a map of $k$-algebras
\[
\beta^*_{\bsa}\colonequals \Ext^*_{\beta_{\bsa}}(k,k)\colon \Ext_{\mcE}^{*}(k,k) \lra \Ext^{*}_{P_{\bsa}}(k,k)\,.
\]
In the notation of \eqref{eq:Ecohomology} and \eqref{eq:Pacohomology}, and for $1\le i\le n$, this map is given by
\begin{align*}
&\beta^*_{\bsa}(u_i) = v_i \\
&\beta^*_{\bsa}(x_i) =
\begin{cases}
\displaystyle{\frac{a_i}{a_{n+1}}} \eta^2 & \text{when $a_{n+1}\ne 0$} \\
a_i y & \text{when $a_{n+1}=0$}
\end{cases} \\
&\beta^*_{\bsa}(\zeta) = \eta
\end{align*}
Let $\fp(\bsa)$ denote the radical of $\Ker(\beta^*_{\bsa})$; it is a prime ideal because the target modulo its radical is a domain. A direct computation reveals that
\begin{equation}
\label{eq:beta-prime}
\fp(\bsa) =
\begin{cases}
\left(a_{n+1}x_i - a_i \zeta^2\mid 1\le i \le n\right) +(u_1,\dots,u_n) & \text{if $a_{n+1}\ne 0$}\\
(\zeta)+ \left(a_ix_j - a_j x_i\mid 1\le i < j\le n\right) + (u_1,\dots,u_n) & \text{if $a_{n+1}= 0$.}
\end{cases}
\end{equation}
So one gets a map
\[
\bbP^{n}(k) \lra \Proj H^*(\mcE,k) \quad\text{where}\quad \bsa \mapsto \sqrt{\Ker(\beta^*_{\bsa})}\,.
\]
When $k$ is algebraically closed, this map is an isomorphism onto the closed points, and through this we can identify $\hysupp_{\mcE}(M)$ with a subset of $\Proj H^*(\mcE,k)$.
\section{The algebra $\test$}
\label{se:test}
In this section we discuss modules over the superalgebra
\begin{equation}
\label{eq:fT}
\test \colonequals \frac{k[t,\tau]}{(t^p-\tau^2)}
\end{equation}
where $t$ is of even degree and $\tau$ is of odd degree. This algebra is commutative but not graded-commutative since the odd degree element $\tau$ does not square to zero. As a ring $\test$ is a noetherian domain of Krull dimension one.
We are interested in the representation theory of $\test$, and this is governed by the maximal Cohen--Macaulay modules. We will be interested also in the infinitely generated analogues of the MCM modules, namely the \emph{G-projective modules}, where ``G" stands for ``Gorenstein". In what follows, we speak of ``G-projectives" rather than MCM modules, for consistency.
\subsection*{G-projective modules}
Recall that the ring $\test$ is Gorenstein of Krull dimension one. A module $M$ is G-projective if and only if $\Ext_{\test}^1(M,\test)=0$; see \cite[\S4]{Christensen:2000a}. for details, including the definition of G-projectives. In particular, the first syzygy module of any $\test$-module is G-projective. Moreover, the finitely generated G-projective modules are precisely the MCM; \cite[\S2.1]{Bruns/Herzog:1998a} and also \cite[Theorems~3.3.7 and 3.3.10(d)]{Bruns/Herzog:1998a}. Since there are only finitely many (isomorphism classes of) indecomposable finitely generated $\test$-modules, each G-projective module is a direct sum of finitely generated modules; see Beligiannis~\cite[Theorem~4.20]{Beligiannis:2011a}.
Let $\GProj(\test)$ be the full subcategory of $\Mod(\test)$ consisting of G-projective modules. It is a Frobenius exact category, with projectives (and hence injectives) the projective $\test$-modules. We write $\uGProj(\test)$ for its stable category, viewed as a triangulated category. One has an approximation functor $\Mod(\test)\to\uGProj(\test)$, which induces a right adjoint to the inclusion $\uGProj(\test)\subseteq \underline{\Mod}(\test)$ into the category of all $A$-modules modulo projectives. Thus an $\test$-module has finite projective dimension if and only if its image in $\uGProj(\test)$ under the approximation functor is zero; this is why we care about the latter category. We write $\uGproj(\test)$ for the subcategory $\uGProj(\test)$ consisting of finitely generated modules.
Ignoring grading, the classification of the indecomposable finitely generated G-projective $\test$-modules appears in Jacobinski~\cite{Jacobinski:1967a}, and is incorporated in the ADE classification, for $\test$ is an $A_{p-1}$ singularity; see \cite[Proposition~5.11]{Yoshino:1990a}. Besides $A$ itself, these are the ideals $M_i\colonequals (t^{i},\tau)$ for $1\le i\le (p-1)/2$; observe that $M_{p-i}\cong M_i$ as ungraded modules.
Taking grading into account, the $\{M_{i}\}_{i=1}^{p-1}$ are all the isomorphism classes of graded, non-projective, G-projective modules, up to shift. Thus $M_i$ has generators $\alpha$ and $\beta$ of degrees zero and one, respectively, and satisfy $\tau\alpha=t^i\beta$, $t^{p-i}\alpha=\tau\beta$. The module $\Omega(M_i)$ looks similar, but with $\alpha$ in degree one and $\beta$ in degree zero, so $\Omega(M_i)\cong M_{p-i}$. The minimal free resolution of $M_i$ is the complex
\begin{equation}
\label{eq:resol}
\cdots \to
\test \oplus \Pi\test \xrightarrow{\left(\begin{smallmatrix} \tau & t^i \\ -t^{p-i} &
-\tau\end{smallmatrix}\right)} \Pi\test\oplus \test \xrightarrow{\left(\begin{smallmatrix} \tau & t^i \\
-t^{p-i} & -\tau \end{smallmatrix}\right)} \test\oplus \Pi\test \to 0\,.
\end{equation}
Here $\Pi\test$ is a free $\test$-module on a single odd degree generator.
The Auslander--Reiten quiver for ungraded finitely generated G-projective $\test$-modules is
\[
\xymatrix{[\test] \ar@<.7ex>[r] & [M_1] \ar@<.7ex>[l]\ar@<.7ex>[r] & [M_2]
\ar@<.7ex>[l] \ar@<.7ex>[r] & \cdots \ar@<.7ex>[l] \ar@<.7ex>[r]&
*[]{\, [M_{\frac{(p-1)}{2}}]\qquad\quad }\ar@(ur,dr)[] \ar@<.7ex>[l] }
\]
See \cite[(5.12)]{Yoshino:1990a}. For graded modules, this unfolds to give
\[
\xymatrix@=6mm
{[\test]\ar@<.7ex>[r] & [M_1] \ar@<.7ex>[l] \ar@<.7ex>[r] & \cdots \ar@<.7ex>[l] \ar@<.7ex>[r] &
[M_{\frac{(p-1)}{2}}] \ar@<.7ex>[l]\ar@<.7ex>[r] & [M_{\frac{(p+1)}{2}}]
\ar@<.7ex>[l] \ar@<.7ex>[r] & \cdots \ar@<.7ex>[l] \ar@<.7ex>[r] &
[M_{p-1}] \ar@<.7ex>[l] \ar@<.7ex>[r] & [\Pi\test] \ar@<.7ex>[l]}
\]
Ignoring the projectives, this is the same as the Auslander--Reiten quiver for ungraded $k[t]/(t^p)$-modules; cf.~ \intref{Proposition}{pr:detect}.
\subsection*{Tensor structures}
The $k$-algebra $\test$ has a canonical structure of a cocommutative Hopf superalgebra over $k$, where the elements $\tau$ and $t$ are primitives. Its counit, or augmentation, is the map of $k$-algebras
\[
\test\lra k\quad\text{where $t,\tau\mapsto 0 $.}
\]
The tensor product of any $\test$-module with a projective is projective, and hence the same is true of modules of finite projective dimension. Thus $\uGProj(\test)$ is a tensor triangulated category and the approximation functor is a tensor functor.
The ideal $(\tau)$ in $\test$ is primitively generated, so the quotient $k[t]/(t^p)$ inherits a Hopf structure compatible with the surjection $\test\to k[t]/(t^p)$. So $\StMod(k[t]/(t^p))$, the stable category of ungraded $k[t]/(t^p)$-modules is also tensor triangulated, and restriction followed by the approximation functor induces a tensor functor
\begin{equation}
\label{eq:stable-TT}
\StMod(k[t]/(t^p)) \lra \uGProj(\test)\,.
\end{equation}
The result below is well-known and easy to verify.
\begin{proposition}
\label{pr:detect}
The functor \eqref{eq:stable-TT} is an equivalence of categories; it restricts to an equivalence $\stmod(k[t]/(t^p))\simeq \uGproj(\test)$. \qed
\end{proposition}
Next we present a criterion for detecting when certain $A$-modules
have finite flat dimension. This is used in \intref{Section}{se:rank}
to construct rank varieties for modules over elementary supergroup
schemes. It is also with this application in mind that we switch to
speaking of flat dimension of a module, rather than its projective
dimension. Observe that one is finite if and only if the other is
finite, for $\dim A=1$.
\begin{definition}
\label{de:maximal-rank}
Any $k$-linear map $f\colon V\to V$ of vector spaces with $f^2=0$ satisfies $\Im(f)\subseteq \Ker(f)$; when equality holds we say $f$ has \emph{maximal image}. When $\rank_kM$ is finite the preceding inclusion yields inequalities
\[
\rank_k f \colonequals \rank_k\Im(f) \le \frac{\rank_k V}2 \le \rank_k\Ker(f)\,.
\]
Evidently $f$ has maximal image precisely when $\rank_kf = (\rank_kV)/2$. The result below is a super analogue of \cite{Avramov/Iyengar:2018a}. \end{definition}
\begin{theorem}
\label{th:rank}
Let $M$ be a graded $\test$-module on which $t$ (equivalently, $\tau$) is nilpotent. Then $M$ has finite flat dimension
if and only if the square zero map
\begin{equation}
\label{eq:2x2}
\begin{pmatrix} \tau & t \\ -t^{p-1} & -\tau \end{pmatrix} \colon M\oplus M \lra M\oplus M
\end{equation}
has maximal image. If $\rank_kM$ is finite, this happens if and only if the rank of the map above is equal to $\rank_kM$; otherwise the rank is strictly less.
\end{theorem}
\begin{proof}
Given that $\tau^2=t^p$ in $\test$, when $t$ or $\tau$ is nilpotent on $M$, both are nilpotent and then $M$ is $(\tau,t)$-torsion. Observe that $M$ has finite flat dimension if and only if it has finite injective dimension, and since $\dim \test=1$, the latter condition is equivalent to the injective dimension of $M$ being at most one; this follows, for example, from \intref{Lemma}{le:pd-test}, for $A$ is a hypersurface.
Since $M$ is $(\tau,t)$-torsion, the minimal injective resolution of
$M$ consists only of copies of the injective hull of $k$. It follows that $M$ has injective dimension at most one if and only if $\Ext^2_{\test}(k,M)=0$. Using the resolution \eqref{eq:resol} with $i=1$ for $J_1=k$, we see that $\Ext^2_{\test}(k,M)$ is isomorphic to the middle homology in the sequence
\[
\Pi M \oplus M
\xrightarrow{\left(\begin{smallmatrix}\tau & t^i \\ -t^{p-i} & -\tau \end{smallmatrix}\right)}
M \oplus \Pi M
\xrightarrow{\left(\begin{smallmatrix}\tau & t^i \\ -t^{p-i} & -\tau \end{smallmatrix}\right)}
\Pi M \oplus M.
\]
This justifies the first claim in the statement of the theorem. It also follows that when $\rank_kM$ is finite, $\Ext^2_{\test}(k,M)=0$ if and only if the rank of the matrix \eqref{eq:2x2} on $M \oplus M$ is equal to $\rank_kM$, and otherwise the rank is strictly less.
\end{proof}
\subsection*{Cohomology}
The ring $\test$ is a hypersurface and its cohomology ring
\[
H^{*,*}(\test,k)\colonequals \Ext^{*,*}_{\test}(k,k)
\]
is well-known---see \cite[Theorem~5]{Sjodin:1976a}---and easy to compute using the minimal resolution of $k$ given in \eqref{eq:resol}, for $i=1$. Namely, there is an isomorphism of superalgebras
\begin{equation}
\label{eq:Acohomology}
H^{*,*}(\test,k) \cong \Lambda(u) \otimes_k k[\eta]
\end{equation}
with $|u|=(1,0)$ and $|\eta|=(1,1)$.
\section{$\pi$-points}
\label{se:pi-points}
As before $k$ is a field of positive characteristic $p\ge 3$. We refocus on elementary supergroup schemes $\mcE$ over $k$ whose group algebra is of the form \eqref{eq:kE}. In this section we introduce $\pi$-points for $\mcE$, based on maps from the $k$-algebra $\test$ discussed in \intref{Section}{se:test}. The main result is \intref{Theorem}{th:factorisation} that relates support defined via $\pi$-points to support sets introduced in \intref{Section}{se:supportsets}. As a consequence, we deduce that $\pi$-points detect projectivity; see \intref{Theorem}{th:detection}.
\begin{definition}
\label{defn:pi-point}
A \emph{$\pi$-point} of $\mcE$ consists of an extension field $K$ of $k$ together with a map of graded $K$-algebras
\[
\alpha \colon \test_K \to K\mcE_K \quad\text{where $\test_K\colonequals K\otimes_k\test$},
\]
such that $K\mcE_K$ has finite flat dimension as an $\test_K$-module via $\alpha$. Since $\alpha$ is a map of $K$-algebras and $\rank_K K\mcE_K$ is finite, the latter is a finitely generated $\test_K$-module. In particular its flat dimension is the same as its projective dimension.
Of particular interest are maps $\alpha_{\bsa}\colon \test_K \to K\mcE_K$ defined by the assignment
\begin{align}
\label{eq:alpha}
\begin{split}
\alpha_{\bsa}(t) &= a_1 s_1 + \cdots + a_{n-1}s_{n-1} + a_n s_n^{p^{m-1}} + a_{n+1}^2s_n\,, \\
\alpha_{\bsa}(\tau)&=a_{n+1}^p\sigma \\
\end{split}
\end{align}
for $\bsa\colonequals (a_1,\dots,a_{n+1})\in \bbA^{n+1}(K)$. As we shall see in \intref{Proposition}{pr:reps} such a map is a $\pi$-point, that is to say, has finite flat dimension, if and only if $\bsa\ne 0$.
\end{definition}
Our first task is to describe criteria that detect when a given map of $k$-algebras
\[
\alpha\colon \test \to k\mcE
\]
has finite flat dimension.
\begin{construction}
\label{con:factorisation}
We keep the notation from \intref{Construction}{con:powerseries}. In particular, $k\mcE = P/I$ where $P=\pos{\bss,\sigma}$ is the power series over $k$ in indeterminates $\bss = s_1,\dots,s_n$ and $\sigma$, and $I$ is the ideal generated by the regular sequence
\[
s_1^p,\dots,s_{n-1}^p, s_n^{p^m}, s_n^p-\sigma^2\,.
\]
It is helpful also to consider the following ideal and the $k$-algebra it determines
\begin{equation}
\label{eq:J}
J\colonequals (s_1^{p},\dots, s_{n-1}^p, s_n^{p^m}) \subset \pos{\bss} \quad\text{and}\quad S \colonequals \pos{\bss}/J\,.
\end{equation}
Observe that $S$ is a $k$-subalgebra of $k\mcE$ and the latter is free as an $S$-module, with basis $\{1,\sigma\}$. A map of $k$-algebras $\alpha \colon \test \to k\mcE$ is thus determined by polynomials $f(\bss), g(\bss)$ in $k[\bss]$, uniquely defined only modulo $J$, such that
\begin{equation}
\label{eq:fg-condition}
f(\bss)^p \equiv g(\bss)^2 s_n^p \quad \text{modulo} \quad J
\end{equation}
where $\alpha(t)= f(\bss)$ and $\alpha(\tau)= g(\bss)\sigma$. Set
\[
P_{\alpha} \colonequals \frac{\pos{\bss,\sigma}}{(f(\bss)^p - g(\bss)^2\sigma^2)}\,.
\]
Observe that the element $(f(\bss)^p - g(\bss)^2\sigma^2)$ is in the ideal $I$ defining $k\mcE$; see \eqref{eq:fg-condition}. Thus $\alpha$ can be factored as
\begin{equation}
\label{eq:factor-alpha}
\test \xra{\ \dot\alpha\ } P_{\alpha} \xra{\ \bar{\alpha}\ } k\mcE
\end{equation}
where $\dot\alpha$ maps $(t,\tau)$ to $(f(\bss),g(\bss)\sigma)$ and $\bar\alpha$ is the canonical surjection.
\end{construction}
The result below contains criteria for detecting when a given map of $k$-algebras from $\test$ to $k\mcE$ is a $\pi$-point.
\begin{theorem}
\label{th:factorisation}
In the notation of \intref{Construction}{con:factorisation} assume that
$\alpha$ does not factor through the surjection $\test\to k$. Then the
map $\dot\alpha$ has finite flat dimension, and for any $k\mcE$-module
$M$ the module $\alpha^*(M)$ has finite flat dimension if and only if
${\bar\alpha}^*(M)$ has finite flat dimension.
\end{theorem}
\begin{proof}
As $\alpha$ does not factor through the surjection $\test\to k$, at
least one $f(\bss)$ or $g(\bss)$ is non-zero in $k\mcE$. This implies that $f(\bss)^p-g(\bss)^2\sigma^2$ is non-zero in $\pos{\bss,\sigma}$.
Consider the map of $k$-algebras
\[
\tilde\alpha \colon \pos{t,\tau} \lra \pos{\bss,\sigma} \quad\text{where}\quad t\mapsto f(\bss) \quad \text{and}\quad \tau\mapsto g(\bss)\sigma\,.
\]
This is a lift of the map $\dot\alpha$, meaning that the following square commutes
\[
\begin{tikzcd}
\pos{t,\tau} \arrow[r, "{\tilde\alpha}"] \arrow[d,twoheadrightarrow] & \pos{\bss,\sigma} \arrow[d,twoheadrightarrow] \\
\test \arrow[r,"{\dot\alpha}"] & P_\alpha
\end{tikzcd}
\]
Here the vertical maps are the canonical surjections. We claim that this diagram of $k$-algebras is a Tor-independent fibre square:
\[
\Tor^{\pos {t,\tau}}_i(\pos{\bss,\sigma},\test) =
\begin{cases}
P_\alpha & \text{for $i=0$}\\
0 & \text{for $i\ne 0$.}
\end{cases}
\]
Indeed, the complex $0\to \pos{t,\tau}\xra{t^p-\tau^2} \pos{t,\tau}\to 0$ is a free resolution of $\test$ over $\pos{t,\tau}$, so the Tor-modules in question are the homologies of the complex
\[
\begin{tikzcd}
0 \arrow[r] & \pos{\bss,\sigma} \arrow[rr,"{f(\bss)^p-g(\bss)^2\sigma^2}"] && \pos{\bss,\sigma} \arrow[r] & 0\,.
\end{tikzcd}
\]
In degree zero the homology is evidently $P_\alpha$. Since $f(\bss)^p-g(\bss)^2\sigma^2$ is non-zero the complex above has no homology in other degrees. This justifies the claim.
The ring $\pos{t,\tau}$ is regular, of Krull dimension two, so any map out of it, in particular $\tilde\alpha$, has finite flat dimension. And since the square above is a Tor-independent fibre square, we deduce that the flat dimension of $\dot\alpha$ is finite as well.
As to the second part of the statement: Since $\dot\alpha$ has finite flat dimension, when ${\bar{\alpha}}^*(M)$ has finite flat dimension, so does the $\test$-module $\alpha^*(M)={\dot\alpha}^*({\bar{\alpha}}^*(M))$. The proof of the converse exploits the following statement about the bounded derived category, $\dbcat{P_\alpha}$, of the hypersurface $P_\alpha$:
\[
k \in \Thick(F) \quad \text{for $F\in\dbcat{P_\alpha}$ not perfect.}
\]
See \cite[Corollary~6.9]{Takahashi:2010a}. We will need this result for the $P_\alpha$-complex
\[
F\colonequals P_\alpha\lotimes_{\test} k \,,
\]
where $P_\alpha$ acts through the left-hand factor of the tensor product. Observe that this $P_\alpha$-complex is in $\dbcat{P_\alpha}$ for one has isomorphisms
\begin{align*}
P_\alpha \lotimes_{\test} k
&\cong ( \pos{\bss,\sigma} \lotimes_{\pos{t,\tau}} \test ) \lotimes_{\test} k \\
&\cong \pos{\bss,\sigma} \lotimes_{\test} k
\end{align*}
in $\dbcat{ \pos{\bss,\sigma}}$. Moreover $F$ is not perfect because
\[
k\lotimes_{P_\alpha} F = k\lotimes_{P_\alpha} (P_\alpha \lotimes_{\test} k) \cong k\lotimes_{\test} k
\]
which has non-zero homology in all non-negative degrees.
Now suppose that $\alpha^*(M)$ has finite flat dimension, so $\Tor^{\test}_i(M,k)=0$ for $i\gg 0$. Since $\dot\alpha$ is flat, associativity of tensor products yields isomorphisms
\[
M\lotimes_{\test} k \cong M \lotimes_{P_{\alpha}} (P_\alpha \lotimes_{\test} k) \cong M\lotimes_{P_{\alpha}} F\,.
\]
We deduce that
\[
\Tor^{P_{\alpha}}_i(M,F)=0 \quad \text{for $i\gg 0$.}
\]
As noted above, the $P_\alpha$-module $k$ is in $\Thick(F)$ in $\dcat{P_\alpha}$. It follows that
\[
\Tor^{P_{\alpha}}_i(M,k)=0 \quad \text{for $i\gg 0$}
\]
as well. Since $M$ has finite Loewy length, we conclude that the flat dimension of $\bar{\alpha}^*(M)$ is finite, as desired; see \intref{Lemma}{le:pd-test}.
\end{proof}
\begin{corollary}
\label{co:factorisation}
For any map of $k$-algebras $\alpha\colon A\to k\mcE$ the following conditions are equivalent.
\begin{enumerate}[\quad\rm(1)]
\item
$\alpha$ is a $\pi$-point;
\item
$\dot\alpha$ and $\bar{\alpha}$ are of finite flat dimension;
\item
$\bar{\alpha}$ is complete intersection of codimension $n$;
\item
$f(\bss)^p - g(\bss)^2\sigma^2$ is not in $\fm I$.
\end{enumerate}
\end{corollary}
\begin{proof}
(1)$\Leftrightarrow$(2) In either case, since the flat dimension of the $\test$-module $k$ is infinite, $\alpha$ cannot factor through $k$. Thus the desired equivalence holds by \intref{Theorem}{th:factorisation} applied to $M=k\mcE$.
\smallskip
(2)$\Rightarrow$(4) If $f(\bss)^p - g(\bss)^2\sigma^2$ is in $\fm I$, the flat dimension of $\bar{\alpha}$ is infinite; this is by a result of Shamash~\cite[Theorem~1]{Shamash:1969a}. See also \cite[Theorem~2.1(3)]{Avramov/Iyengar:2018a}.
\smallskip
(3)$\Leftrightarrow$(4) The ideal $I$ is evidently a complete intersection and minimally generated by $n+1$ elements. Thus the hypothesis in (3) is equivalent to the condition that the element $f(\bss)^p - g(\bss)^2\sigma^2$ can be extended to a minimal generating set for the ideal $I$; that is to say, it is not in $\fm I$.
\smallskip
(4)$\Rightarrow$(1) Any complete intersection map has finite flat dimension; this applies in particular to $\bar{\alpha}$. By the equivalence of (4) and (3), one has that $f(\bss)^p - g(\bss)^2\sigma^2$ is not in $\fm I$, and in particular, $\alpha$ does not factor through $k$. As already discussed above, this implies $\dot\alpha$ has finite flat dimension.
\end{proof}
\begin{remark}
In the factorisation \eqref{eq:factor-alpha} of a $\pi$-point, the map $\dot\alpha$ need not be flat. Indeed, it is easy to verify that that the map is flat if and only if the polynomials $f(\bss),g(\bss)$ can be chosen without non-trivial common factors; equivalently, that they form a regular sequence in the ring $\pos{\bss,\sigma}$.
Here is an example where this property does not hold: Fix an integer $m\ge 3$, an element $a\in k$ such that $a^2\ne 1$, and consider the $\pi$-point
\[
\alpha\colon \test \lra \frac{\pos{s,\sigma}}{(s^{p^m},s^p-\sigma^2)} \quad
\text{where} \quad t\mapsto s^{p^{m-1}} \quad \text{and}\quad \tau \mapsto a s^{n}
\]
for $n=(p^m-p)/2$. That this is indeed a $\pi$-point can be easily checked using criterion (4) in \intref{Corollary}{co:factorisation}.
On the other hand, for the ``standard" $\pi$-points $\alpha_{\bsa}$ described in \eqref{eq:alpha}, the maps $\dot\alpha_{\bsa}$ are flat, because $(f(\bss),g(\bss))=1$. This is noteworthy because, for our purposes, any $\pi$ point is equivalent to a standard one; see \intref{Proposition}{pr:reps}.
\end{remark}
The following lemma is intended for use in the proof of \intref{Proposition}{pr:reps}.
\begin{lemma}
\label{le:reps}
Let $f(\bss),g(\bss)$ be polynomials in $k[\bss]$ satisfying equation \eqref{eq:fg-condition}. Then there exist uniquely defined elements $b_1,\dots,b_{n+1}$ in $k$ such that
\[
f(\bss)^p - g(\bss)^2\sigma^2 \equiv b_{1}s_1^{p} + \dots + b_{n-1} s_{n-1}^p + b_n s_n^{p^m} + b_{n+1} (s_n^p-\sigma^2)
\]
modulo the ideal $\fm I$ in $k[\bss,\sigma]$, where $\fm$ and $I$ are as in \eqref{eq:mI}.
\end{lemma}
\begin{proof}
With $J$ as in \eqref{eq:J}, the action of $k[\bss]$ on $J/{\bss}J$ factors through the quotient $k[\bss]/(\bss)\cong k$, so $J/{\bss}J$ is a $k$-vector space with basis the residue classes of elements
\[
s_1^{p},\dots, s_{n-1}^p, s_n^{p^m}\,.
\]
Thus each element of $J$ is equivalent modulo ${\bss}J$ to an element of the form
\[
b_1 s_1^{p} +\cdots + b_{n-1}s_{n-1}^p + b_n s_n^{p^m}
\]
for uniquely defined elements $b_1,\dots,b_n$ in $k^n$. This applies in particular to the element $f(\bss)^p - g(\bss)^2s_n^p $, which is in $J$ by our hypothesis. Observe that in the ring $k[\bss,\sigma]$ the extended ideal ${\bss}Jk[\bss, \sigma]$ is contained in $\fm I$, for $\bss\subset \fm$ and $J\subset I$. Therefore in the ring $k[\bss,\sigma]$ we get that
\begin{align*}
f(\bss)^p - g(\bss)^2\sigma^2
&= f(\bss)^p - g(\bss)^2s_n^p + g(\bss)^2(s_n^p- \sigma^2 ) \\
&\equiv b_1 s_1^{p} + \cdots + b_{n-1} s_{n-1}^p +b_n s_n^{p^m} + g(\bss)^2(s_n^p- \sigma^2) \\
&\equiv b_1 s_1^{p} + \cdots + b_{n-1} s_{n-1}^p +b_n s_n^{p^m} + g(0)^2(s_n^p- \sigma^2)
\end{align*}
where the equivalence is modulo $\fm I$; for the last step observe that ${\bss}(s_n^p-\sigma^2)\subset \fm I$. Setting $b_{n+1}= g(0)^2$ completes the proof.
\end{proof}
\begin{proposition}
\label{pr:reps}
Assume $k$ is closed under taking $p$th roots and square roots. Let $\alpha\colon \test\to k\mcE_k$ be a map of $k$-algebras and $(b_1,\dots,b_{n+1})\in \bbA^{n+1}(k)$ associated to $\alpha$ by \intref{Lemma}{le:reps}. Set $a_i = b_i^{1/p}$ for $1\le i\le n$ and $a_{n+1} = b_{n+1}^{1/2p}$, and $\alpha_{\bsa}\colon \test\to k\mcE$ the morphism of $k$-algebras defined in \eqref{eq:alpha}. The following statements hold.
\begin{enumerate}[\quad\rm(1)]
\item
The map $\alpha$ is a $\pi$-point if and only if $(a_1,\dots,a_{n+1})\ne 0$.
\item
When $\alpha$ is a $\pi$-point, and $M$ is a $k\mcE$-module, the following are equivalent:
\begin{enumerate}[\quad\rm(a)]
\item
$\alpha^*(M)$ has finite flat dimension;
\item
$\alpha_{\bsa}^*(M)$ has finite flat dimension.
\end{enumerate}
\end{enumerate}
\end{proposition}
\begin{proof}
\intref{Lemma}{le:reps} implies that $(b_1,\dots,b_{n+1})\ne 0$ holds if and only if $f(\bss)^p-g(\bss)^2\sigma^2$ is not in $\fm I$. Thus (1) is a consequence of \intref{Corollary}{co:factorisation}.
Let $(f_{\bsa}(\bss),g_{\bsa}(\bss))$ be the polynomials defining the map $\alpha_{\bsa}$; thus
\begin{align*}
f_{\bsa}(\bss) &= a_1 s_1 + a_{n-1}s_{n-1} +a_n s_n^{p^{m-1}} + \bsa^2_ns_n \\
g_{\bsa}(\bss) &= a_{n+1}^p\,.
\end{align*}
Therefore by the choice of the $a_i$ and \intref{Lemma}{le:reps} one has that
\[
f(\bss)^p-g(\bss)^2\sigma^2 \equiv f_{\bsa}(\bss)^p-g_{\bsa}(\bss)^2\sigma^2 \pmod {\fm I}\,.
\]
The desired result is thus a special case of \cite[Theorem~2.1(2)]{Avramov/Iyengar:2018a}.
\end{proof}
Next we record the following consequence of the preceding results; this is the main outcome of this section, as far as the sequel is concerned.
\begin{theorem}
\label{th:detection}
Let $M$ be a $k\mcE$-module. The following conditions are equivalent:
\begin{enumerate}[\quad\rm(1)]
\item
$M$ is projective;
\item
$\alpha^*(M_K)$ has finite flat dimension for every $\pi$-point $\alpha\colon \test_K\to K\mcE_K$;
\item
$\alpha^*(M^K)$ has finite flat dimension for every $\pi$-point $\alpha\colon \test_K\to K\mcE_K$.
\end{enumerate}
\end{theorem}
\begin{proof}
For any extension field $K$ of $k$ and
$\bsa = (a_1,\dots,a_{n+1}) \in \bbA^{n+1}(K)$, and for the
$\pi$-point $\alpha_{\bsa}\colon \test_K\to K\mcE$ described by
\eqref{eq:alpha}, one has that $\alpha_{\bsa}^*(M_K)$, respectively
$\alpha_{\bsa}^*(M^K)$, has finite flat dimension if and only if
$(a_1^p,\dots,a_n^p,a_{n+1}^{2p})$ is in $\hysupp_{\mcE_K}(M_K)$,
respectively, in $\hysupp_{\mcE_K}(M^K)$. This claim is a direct
consequence of \intref{Proposition}{pr:reps}(2). Thus the desired
equivalence follows from \intref{Theorem}{th:support-sets}.
\end{proof}
\section{The rank variety}
\label{se:rank}
In this section, we restrict our attention to finitely generated $k\mcE$-modules, and describe the analogue of Carlson's rank variety~\cite{Carlson:1983a}, relating it to support sets introduced in \intref{Section}{se:supportsets}. The field $k$, always of positive characteristic $p\ge 3$, will be algebraically closed.
\begin{definition}
For a point $\bsa = (a_1,\dots,a_{n+1})$ in $\bbA^{n+1}(k)$ let $\alpha_{\bsa} \colon \test\to k\mcE$ be the $\pi$-point defined in \eqref{eq:alpha}.
Let $M$ be a finitely generated $k\mcE$-module and consider the square
zero map
\begin{equation}
\label{eq:Mmatrix}
\begin{pmatrix}
\alpha_{\bsa}(\tau) & \alpha_{\bsa}(t) \\
-\alpha_{\bsa}(t)^{p-1} & -\alpha_{\bsa}(\tau)
\end{pmatrix} \colon M\oplus M \lra M\oplus M\,.
\end{equation}
The rank of this map is at most that of $M$; see \intref{Theorem}{th:rank}. The \emph{rank variety} of $M$, denoted $V^r_\mcE(M)$, is the subset of $\bbA^{n+1}(k)$ consisting those $\bsa$ for which the map in \eqref{eq:Mmatrix} fails to have maximal rank, in the sense of \intref{Definition}{de:maximal-rank}. In particular, $0$ is in the rank variety of any $M$.
\end{definition}
\begin{remark}
In contrast with support sets the rank varieties are introduced as subsets of affine space rather than projective space. Observe that swapping the diagonal entries in the matrix \eqref{eq:Mmatrix} yields the matrix corresponding to the point $(a_1,\dots, -a_{n+1})$. In other words, the rank variety on $M$ is invariant under the $\bbZ/2$ action on the last coordinate that sends $a_{n+1}$ to $-a_{n+1}$. The intervention of this automorphism is not surprising. It plays a role in Deligne's work on Tannakian categories~\cite[Section~8]{Deligne:1990a}, where it is the generator for the fundamental group of the category of super vector spaces. In his description~\cite{Deligne:2002a} of symmetric tensor abelian categories of moderate growth in characteristic zero, this central involution is an essential ingredient.
There is more: Consider the action of $k^\times$ on $\bbA^{n+1}(k)$ given by
\begin{equation}
\label{eq:k-action}
\lambda(a_1,\dots,a_{n+1})\colonequals (\lambda^2 a_1,\dots, \lambda^2 a_n, \lambda a_{n+1})
\end{equation}
and the map $F\colon \bbA^{n+1}(k)\to \bbP^{n}(k)$ where
\[
(a_1,\dots,a_{n+1}) \mapsto [a_1^p,\dots, a_n^p,a_{n+1}^{2p}]\,.
\]
Observe that $F(\lambda \bsa) = \lambda^{2p} F(\bsa)$. This observation is used in the proof of the result below, in which part (1) compares the rank variety of $M$ to its support set~\eqref{eq:Av-variety}.
\end{remark}
\begin{theorem}
\label{th:carlson}
Let $M$ be a finitely generated $k\mcE$-module.
\begin{enumerate}[\quad\rm(1)]
\item
The rank variety $V^r_\mcE(M)$ is a closed subset of $\bbA^{n+1}(k)$, homogeneous for the action of $k^\times$ on $\bbA^{n+1}$ defined in \eqref{eq:k-action}.
\item
A point $\bsa\in \bbA^{n+1}(k)\setminus\{0\}$ is in $V^r_{\mcE}(M)$ if and only if $F(\bsa)$ is in $\hysupp_{\mcE}(M)$.
\item
The $k\mcE$-module $M$ is projective if and only if $V^r_\mcE(M)=\{0\}$.
\end{enumerate}
\end{theorem}
\begin{proof}
The first part of (2) is immediate from the description of the two
varieties and \intref{Proposition}{pr:reps}(2).
(1) The condition for a point $\bsa$ to be in $V^r_\mcE(M)$ is given by the vanishing of the minors of the matrix \eqref{eq:Mmatrix} of size equal to $\rank_kM$. These are polynomial relations in the coordinates, so $V^r_{\mcE}(M)$ is closed under the Zariski topology on $\bbA^{n+1}(k)$.
Fix $\bsa$ in $V^r_{\mcE}(M)\setminus\{0\}$, so that $F(\bsa)$ is in $\hysupp_{\mcE}(M)$, by (2). We deduce that the point $F(\lambda\bsa) = \lambda^{2p}F(\bsa)$ is in $\hysupp_{\mcE}(M)$ as well. Thus $\lambda\bsa$ is in $V^r_{\mcE}(M)$, again by (2). This justifies the assertion that the rank variety is homogeneous.
(3) The map $F$ is onto because the field $k$ is algebraically
closed. Thus (3) is immediate from (2) and the fact that $\hysupp_{\mcE}(M)=\varnothing$ if and only if $M$ is projective; see \intref{Theorem}{th:support-sets}.
\end{proof}
\begin{remark}
The matrix \eqref{eq:Mmatrix} corresponding to $\lambda \bsa$ is
\[
\begin{pmatrix}
\lambda^p\alpha_{\bsa}(\tau) & \lambda^2\alpha_{\bsa}(t) \\
-\lambda^{2p-2}\alpha_{\bsa}(t)^{p-1} & -\lambda^p\alpha_{\bsa}(\tau)
\end{pmatrix}
\]
It follows from \intref{Theorem}{th:carlson}(2) that this matrix has
maximal rank if and only if the one in \eqref{eq:Mmatrix} does. Thus
the subset $V^r_{\mcE}(M)$ of $\bbA^{n+1}(k)$ is homogenous for the
$k^{\times}$-action in \eqref{eq:k-action}. A direct proof of this
observation seems complicated.
\end{remark}
\section{$\pi$-points and cohomology}
\label{se:pi-and-cohomology}
In this section we set up the equivalence relation on $\pi$-points for $k\mcE$ and the bijection between their equivalence classes and points in the weighted projective space $\Proj H^{*,*}(\mcE,k)$. The development is modelled on the one for group schemes due to Friedlander and Pevtsova~\cite{Friedlander/Pevtsova:2007a}.
Given a $\pi$-point $\alpha\colon \test_K\to K\mcE$ the restriction functor
\[
\alpha^*\colon \Mod(K\mcE_K) \to \Mod(\test_K)
\]
takes projective modules to modules of finite flat dimension. Composed with the approximation functor, it induces an exact functor
\begin{equation}
\label{eq:alpha*}
\alpha^*\colon \StMod(K\mcE_K)\lra \uGProj(\test_K).
\end{equation}
This need not be compatible with the tensor structures, for $\alpha$ need not be a morphism of Hopf algebras.
\begin{definition}
If $\alpha\colon \test_K \to K\mcE_K$ is a $\pi$-point of $\mcE$, we write $H^{*,*}(\alpha)$ for the composition of homomorphism of $k$-algebras
\[
H^{*,*}(\mcE,k) = \Ext^{*,*}_{k\mcE}(k,k) \xrightarrow{K \otimes_k -}
\Ext^{*,*}_{K\mcE_K}(K,K) \xrightarrow{\alpha^*} \Ext^{*,*}_{\test_K}(K,K)
\]
The cohomology of the algebra $\test$ was described in \eqref{eq:Acohomology}; modulo its radical it is a domain. Thus the radical of the kernel of $H^{*,*}(\alpha)$ is a prime ideal, denoted $\fp(\alpha)$.
\end{definition}
\begin{example}
Fix $\bsa \in \bbA^{n+1}(k)$ and let $\alpha_{\bsa}$ be the $\pi$-point described in \eqref{eq:alpha}. In the notation of \eqref{eq:Ecohomology} and \eqref{eq:Acohomology}, the map induced in cohomology by $\alpha_{\bsa}$ is given by
\begin{alignat*}{4}
&\alpha^*_{\bsa}(u_i) = a_i v& \quad &\text{for $1\le i\le n-1$}& \quad\text{and}\quad & \alpha^*_{\bsa}(u_n) = a_{n+1}^2 v \\
&\alpha^*_{\bsa}(x_i) = a_i^p \eta^2& \quad &\text{for $1\le i\le n$}&\quad\text{and}\quad & \alpha^*_{\bsa}(\zeta) = a_{n+1}^p \eta.
\end{alignat*}
Therefore the associated point is $\Proj H^*(\mcE,k)$ is
\begin{equation}
\label{eq:alpha-prime}
\fp(\alpha_{\bsa}) =
\begin{cases}
\left(a^{2p}_{n+1}x_i - a^p_i \zeta^2\mid 1\le i \le n\right)+(u_1,\cdots,u_n) & \text{if $a_{n+1}\ne 0$}\\
(\zeta)+ \left(a^p_i x_j -a^p_j x_i\mid 1\le i < j\le n\right)+(u_1,\cdots,u_n) & \text{if $a_{n+1}= 0$.}
\end{cases}
\end{equation}
Compare this with \eqref{eq:beta-prime}. In particular, when $k$ is closed under taking square roots and $p$th roots (for example, if $k$ is algebraically closed) each rational point in $\Proj H^*(\mcE,k)$ occurs as $\fp(\alpha)$, for some $\pi$-point $\alpha$. Here is the general statement.
\end{example}
\begin{proposition}
Given a point $\fp$ in $\Proj H^{*,*}(\mcE,k)$, there exists a field extension $K$ of $k$ and a $\pi$-point $\alpha\colon \test_K\to K\mcE_K$ of the
form~\eqref{eq:alpha} such that $\fp(\alpha)=\fp$.
\end{proposition}
\begin{proof}
Choose a perfect field extension $K$ of $k$ containing the function field for $\fp$, that is, the degree $(0,0)$ elements of the graded field of fractions of $H^{*,*}(\mcE,k)/\fp$. Then there is a closed point $\fm\in \Proj H^{*,*}(\mcE_K,K)$ lying over $\fp$. In coordinates, this is given by $[b_1,\cdots,b_{n+1}]$ with not all the $b_i$ equal to zero. Setting $a_i=b_i^{1/p}$, the map $\alpha_{\bsa}$ defined by \eqref{eq:alpha} has $\fp(\alpha_{\bsa})=\fp$.
\end{proof}
The following definition is the analogue of~\cite[2.1]{Friedlander/Pevtsova:2007a}.
\begin{definition}
We say that $\pi$-points $\alpha\colon \test_K\to K\mcE_K$ and $\beta\colon \test_L\to L\mcE_L$ are \emph{equivalent}, and write $\alpha\sim\beta$, if for each finitely generated $k\mcE$-module $M$, the $\test_K$-module $\alpha^*(M_K)$ has finite flat dimension if and only if the $\test_L$-module $\beta^*(M_L)$ does.
\end{definition}
Other ways of expressing this equivalence relation are given in \intref{Theorem}{th:equiv}. In preparation for this, we recall the definition of Carlson's modules. For any non-zero element $\xi\in H^{2n,0}(\mcE,k)$ let $L_\xi$ be the kernel of a representative cocycle $\hat \xi\colon \Omega^{2n}(k) \to k$. Thus we have an exact sequence of $k\mcE$-modules
\begin{equation}
\label{eq:Carlson}
0 \to L_\xi \to \Omega^{2n}(k) \xrightarrow{\hat\xi} k \to 0
\end{equation}
and a corresponding triangle $L_\xi \to \Omega^{2n}(k) \to k\to$ in $\StMod(k\mcE)$. The following result is analogous to \cite[Proposition~2.3]{Friedlander/Pevtsova:2005a} and \cite[Proposition~2.9]{Friedlander/Pevtsova:2007a}.
\begin{proposition}
\label{pr:Lxi}
Let $\alpha\colon \test_K\to K\mcE_K$ be a $\pi$-point. An element $\xi\in H^{2n,0}(\mcE,k)$ is not in $\fp(\alpha)$ if and only if $\alpha^*(L_\xi)_K$ has finite flat dimension. If $\alpha$, $\beta$ are $\pi$-points with $\fp(\alpha)\ne \fp(\beta)$ then $\alpha$ and $\beta$ are inequivalent.
\end{proposition}
\begin{proof}
Applying $(-)_K$ to the exact triangle defining $L_\xi$ yields a triangle
\[
(L_\xi)_K \lra \Omega^{2n}(K) \lra K\lra
\]
in $\StMod(K\mcE_K)$. Apply $\alpha^*$ to this triangle, and keep in mind ~\eqref{eq:alpha*}, to get a triangle
\[
\alpha^*(L_\xi)_K \lra \Omega^{2n}(\alpha^*(K)) \lra \alpha^*(K)\lra
\]
in $\uGProj(\test)$. Now $H^{2n,0}(\test,K)$ is two dimensional, but its image in $H^{*,*}(\test,K)$ modulo $\sqrt{0}$ is one dimensional. Elements outside $\sqrt{0}$ induce an isomorphism from $\Omega^{2n}(\alpha^*(K))$ to $\alpha^*(K)$ in $\uGProj(\test)$. So $\alpha^*(L_\xi)_K$ has finite flat dimension if and only if it is zero in $\uGProj(\test)$, which happens if and only if $\alpha^*(\xi)$ is non-zero in $H^{2n,0}(\test,K)$ modulo $\sqrt{0}$, and this happens if and only if $\xi\not\in\fp(\alpha)$.
If $\fp(\alpha)\ne \fp(\beta)$, choose $\xi\in H^{2n,0}(\mcE,k)$ in $\fp(\beta)$ but not in $\fp(\alpha)$. Then $\beta^*(L_\xi)_L$ has infinite flat dimension while $\alpha^*(L_\xi)_K$ has finite flat dimension. Since $L_\xi$ is finitely generated, it follows that $\alpha$ and $\beta$ are inequivalent.
\end{proof}
\begin{theorem}
\label{th:equiv}
Let $\mcE$ be an elementary supergroup scheme. Let $\alpha\colon \test_K\to K\mcE_K$, $\beta\colon \test_L\to L\mcE_L$ be $\pi$-points of $\mcE$. Then the following are equivalent:
\begin{enumerate}[\quad\rm(1)]
\item
$\alpha$ and $\beta$ are equivalent.
\item
For all $k\mcE$-modules $M$, the $\test_K$-module $\alpha^*(M_K)$ has finite flat dimension if and only if the $\test_L$-module $\beta^*(M_L)$ has finite flat dimension.
\item
For all $k\mcE$-modules $M$, the $\test_K$-module $\alpha^*(M^K)$ has finite flat dimension if and only if the $\test_L$-module $\beta^*(M^L)$ has finite flat dimension.
\item
$\fp(\alpha)=\fp(\beta)$.
\end{enumerate}
Thus the map sending $\alpha$ to $\fp(\alpha)$ induces a bijection between the set of equivalence classes of $\pi$-points and the set $\Proj H^{*,*}(G,k)$.
\end{theorem}
\begin{proof}
It is clear that (2) and (3), even limited to finitely generated
modules, imply (1). \intref{Proposition}{pr:Lxi} shows that (1)
implies (4). In the rest of the proof we suppose that (4) holds, and
verify that (2) and (3) do. By extending fields if necessary, we may assume that $k$ is algebraically closed, and that $L=k=K$.
We can also assume that $\alpha=\alpha_{\bsa}$ and $\beta=\alpha_{\bsb}$, as in \eqref{eq:alpha}, for some $\bsa,\bsb\in \bbA^{n+1}(k)$.
Indeed, \intref{Proposition}{pr:reps} yields that $\alpha$ and $\beta$ are equivalent as $\pi$-points to $\alpha_{\bsa}$ and $\alpha_{\bsb}$, respectively, for some $\bsa,\bsb$. Moreover $\fp(\alpha) = \fp(\alpha_{\bsa})$ and $\fp(\beta)=\fp(\alpha_{\bsb})$, for we already know (1)$\Rightarrow$(4). This justifies the assumption.
It is clear from \eqref{eq:alpha-prime} that $\fp(\alpha_{\bsa})=\fp(\alpha_{\bsb})$ implies
\[
(a_1^p,\dots,a_n^p,a_{n+1}^{2p})= (b_1^p,\dots,b_n^p,b_{n+1}^{2p})\,.
\]
Then, in the notation of \eqref{eq:factor-alpha} one gets that $\bar{\alpha}_{\bsa} = \bar{\alpha}_{\bsb}$, and so \intref{Theorem}{th:factorisation} implies that (2) and (3) hold.
\end{proof}
\section{Support and cosupport}
\label{se:support-and-cosupport}
In this section we define the $\pi$-support and $\pi$-cosupport of a $k\mcE$-module $M$, and the cohomological support and cosupport,
and prove that these two notions of support, and of cosupport, coincide. Once this is done \cite{Benson/Iyengar/Krause/Pevtsova:2017a} provides a path to the classification of the localising subcategories of $\StMod k\mcE$.
\begin{definition}
Let $M$ be a $k\mcE$-module. The \emph{$\pi$-support} of $M$, denoted $\pisupp_\mcE(M)$, is the subset of $\Proj H^*(\mcE,k)$ consisting of the primes $\fp(\alpha)$ where $\alpha\colon\test_K\to K\mcE_K$ is a $\pi$-point such that the flat dimension of $\alpha^*(M_K)$ is infinite. Replacing
$\alpha^*(M_K)$ by $\alpha^*(M^K)$ gives the \emph{$\pi$-cosupport} of $M$, denoted $\picosupp_\mcE(M)$.
\end{definition}
\begin{remark}
For a finitely generated $k\mcE$-module $M$, we have
\[
\pisupp_\mcE(M)=\picosupp_\mcE(M)\,.
\]
This is usually not true for infinitely generated modules; see \cite[Example~4.7]{Benson/Iyengar/Krause/Pevtsova:2017a}.
\end{remark}
The result below is now only a reformulation of \intref{Theorem}{th:detection}.
\begin{theorem}
\label{th:pi-proj}
Let $M$ be a $k\mcE$-module. The following conditions are equivalent.
\begin{enumerate}[\quad\rm(1)]
\item $M$ is projective,
\item $\pisupp_\mcE(M)=\varnothing$,
\item $\picosupp_\mcE(M)=\varnothing$. \qed
\end{enumerate}
\end{theorem}
The following proposition exhibits the effect of changing the Hopf
structure, in order to prove formulas for the support of tensor product
and Hom of two representations.
\begin{proposition}
\label{pr:superFP}
Let $M$ be a $k$-vector space and let $\beta, \gamma,\delta,\ve \colon M\to M$ be commuting $k$-linear maps such that
\[
\gamma^{p}=0= \delta^{p^m} \quad \text{and}\quad \delta^p = \ve^2
\]
for some integer $m\ge 1$. Then the map
\[
\begin{pmatrix}
\ve & \delta \\
-\delta^{p-1} & -\ve
\end{pmatrix} \colon M\oplus M \lra M \oplus M
\]
is square zero. It has maximal image if and only if the map
\[
\begin{pmatrix}
\ve & \delta+ \beta \gamma \\
-(\delta+ \beta \gamma)^{p-1} & -\ve
\end{pmatrix}
\col M\oplus M \lra M \oplus M
\]
does.
\end{proposition}
\begin{proof}
We managed, not without some difficulty, to adapt the proof of~\cite[2.2]{Friedlander/Pevtsova:2005a} to this context. Here is an alternative approach, suggested by \cite[\S5.4]{Avramov/Iyengar:2018a} that makes it clear that what we seek is a variation on \intref{Proposition}{pr:reps}. Akin to the proof of \cite[5.6]{Avramov/Iyengar:2018a} we consider the $k$-algebra
\[
R \colonequals \frac{\pos{b,c,d,e}}{(c^{p}, d^{p^m}, d^p-e^2)}
\]
and maps of $k$-algebras
\[
\begin{tikzcd}
\displaystyle{\frac{k[t,\tau]}{(t^p-\tau^2)}} \arrow[r,yshift=0.7ex,"{\alpha_1}"] \arrow[r,yshift=-0.7ex,swap,"{\alpha_2}"]
& R
\end{tikzcd} \quad\text{where} \quad
\begin{aligned}
&\alpha_1(t) = d & \text{ and } \alpha_1(\tau) = e \\
&\alpha_2(t) = d + bc & \text{ and } \alpha_2(\tau) = e.
\end{aligned}
\]
Given \intref{Theorem}{th:rank}, the desired result is then that $\alpha_1^*(M)$ has finite flat dimension if and only if $\alpha_2^*(M)$ does. In the notation of \intref{Theorem}{th:factorisation} the pertinent hypersurfaces in $\pos{b,c,d,e}$ are defined by polynomials
\[
d^p -e^2 \quad\text{and} \quad (d+bc)^p - e^2\,.
\]
Evidently these polynomials are congruent modulo $(b,c,d,e)(c^p,d^{p^m},d^p-e^2)$. This justifies the second equivalence below:
\begin{align*}
\alpha_1^*(M) \text{ has finite flat dimension }
& \iff \bar{\alpha}_1^*(M) \text{ has finite flat dimension } \\
& \iff \bar{\alpha}_2^*(M) \text{ has finite flat dimension } \\
&\iff \alpha_2^*(M) \text{ has finite flat dimension }
\end{align*}
The first and the last one are by \intref{Theorem}{th:factorisation}.
\end{proof}
The result below is the analogue of~\cite[Lemma~3.9]{Friedlander/Pevtsova:2005a} and ~\cite[Lemma~4.3]{Benson/Iyengar/Krause/Pevtsova:2017a}.
\begin{lemma}
\label{le:tensor-hom-fpd}
Let $\alpha\colon \test\to K\mcE_K$ be a $\pi$-point and let $M,N$ be $K\mcE_K$-modules. The following conditions are equivalent:
\begin{enumerate}[\quad\rm(1)]
\item $\alpha^*(M\otimes_K N)$ has finite flat dimension.
\item $\alpha^*(\Hom_K(M,N))$ has finite flat dimension.
\item $\alpha^*(M)$ or $\alpha^*(N)$ has finite flat dimension.
\end{enumerate}
\end{lemma}
\begin{proof}
If $\alpha$ were a homomorphism of Hopf algebras, then the induced restriction functor $\alpha^*$ from \eqref{eq:alpha*} is a tensor functor, so combining it with \intref{Proposition}{pr:detect} one gets tensor functors
\[
\StMod(K\mcE_K)\xrightarrow{\alpha^*} \uGProj(\test) \simeq \StMod(K[t]/(t^p)).
\]
In $\StMod(K[t]/(t^p))$, the tensor product or Hom of two modules is projective (i.e., zero) if and only if one of them is projective, which proves the lemma in this case.
The general case is tackled as in the proof of \cite[3.9]{Friedlander/Pevtsova:2005a}: By \intref{Proposition}{pr:reps}, we may assume that $\alpha$ has the
form~\eqref{eq:alpha}. We may therefore change the Hopf algebra structure on $K\mcE_K$ so that $\alpha$ is a homomorphism of Hopf algebras,
by making all the $s_i$ and $\sigma$ primitive:
\[
\Delta(s_i)=s_i\otimes 1 + 1 \otimes s_i \quad\text{and}\quad
\Delta(\sigma)=\sigma\otimes 1 + 1 \otimes \sigma\,.
\]
For any element $x$ of the augmentation ideal $I$ of $K\mcE_K$ this changes $\Delta(x)$ by an element of $I \otimes I$.
Now consider the action on $(M \otimes_K N) \oplus (M \otimes_K N)$ of the matrix
\[
\begin{pmatrix} \tau & t \\
-t^{p-1} & -\tau
\end{pmatrix}
\]
via $\alpha$. The effect of changing from the old to the new diagonal
on this action is a sequence of changes where $t\otimes 1 + 1 \otimes
t$ is replaced by $t\otimes 1 + 1 \otimes t + u \otimes v$ with
$u^p=0=v^p$. Applying \intref{Proposition}{pr:superFP}, we see that
this does not affect the maximal image property. So by \intref{Theorem}{th:rank}, it does not change whether $M \otimes_K N$ has finite projective
dimension. The argument for $\Hom_K(M,N)$ is similar.
\end{proof}
\begin{theorem}
\label{th:pi-tensor-hom}
Let $M$ and $N$ be $k\mcE$-modules. Then the following holds.
\begin{enumerate}[\quad\rm(1)]
\item
$\pisupp_\mcE(M\otimes_k N) = \pisupp_\mcE(M)\cap \pisupp_\mcE(N)$.
\item
$\picosupp_\mcE(\Hom_k(M,N)) =
\pisupp_\mcE(M)\cap \picosupp_\mcE(N)$.
\end{enumerate}
\end{theorem}
\begin{proof}
For (i), we use the isomorphism
\[
M_K \otimes_K N_K \cong (M \otimes_k N)_K.
\]
For (ii), we use the isomorphism
\begin{equation*}
\Hom_k(M,N)^K \cong \Hom_K(M_K,N^K).
\end{equation*}
In both cases, we then use \intref{Lemma}{le:tensor-hom-fpd}.
\end{proof}
Next we introduce cohomological support and cosupport, following \cite{Benson/Iyengar/Krause:2009a,Benson/Iyengar/Krause:2012b}. Recall that the stable category $\StMod(k\mcE)$ has two gradings: an internal one with shift $\Pi$ and a cohomological one with shift $\Omega^{-1}$, and a natural isomorphism $\Pi\Omega\cong\Omega\Pi$. Its
centre $Z\StMod(k\mcE)$ is doubly graded, and $Z^{i,j}\StMod(k\mcE)$ consists of those natural transformations $\gamma$ from the identity to $\Omega^{-i}\Pi^{j}$ which satisfy $\gamma\Omega=-\Omega\gamma$ and $\gamma\Pi=-\Pi\gamma$. This is a doubly graded commutative ring, in the sense that if $x\in Z^{i,j}$ and $y\in Z^{i',j'}$ then
\[
yx = (-1)^{(i+j)(i'+j')}xy.
\]
The category $\StMod(k\mcE)$ is $H^{*,*}(\mcE,k)$-linear, in the sense of \cite{Benson/Iyengar/Krause:2009a}. Namely, there is a homomorphism of doubly graded $k$-algebras $H^{*,*}(\mcE,k) \to Z\StMod(k\mcE)$, given in the obvious way. For each $\fp\in\Proj H^{*,*}(\mcE,k)$ there is a local cohomology functor
\[
\Gamma_\fp\colon\StMod(k\mcE)\to\StMod(k\mcE)
\]
and a local homology functor
\[
\Lambda_\fp\colon\StMod(k\mcE)\to\StMod(k\mcE)
\]
defined in terms of this action; see \cite{Benson/Iyengar/Krause:2009a,Benson/Iyengar/Krause:2012b}.
\begin{definition}
The \emph{support} and \emph{cosupport} of a $k\mcE$-module $M$ are the subsets
\begin{align*}
\supp_\mcE(M)&\colonequals \{\fp\in\Proj H^{*,*}(\mcE,k) \mid \Gamma_\fp(M)\ne 0 \} \\
\cosupp_\mcE(M)& \colonequals \{\fp\in\Proj H^{*,*}(\mcE,k) \mid \Lambda_\fp(M) \ne 0 \}.
\end{align*}
See \cite[\S5]{Benson/Iyengar/Krause:2009a} and \cite[\S4]{Benson/Iyengar/Krause:2012b}.
\end{definition}
\begin{proposition}
\label{pr:pi-Gamma-p}
We have $\pisupp(\Gamma_\fp(k))=\{\fp\}$.
\end{proposition}
\begin{proof}
This statement is independent from the Hopf structure of $k\mcE$,
since the local cohomology functors $\Gamma_\fq$ do not change when
the Hopf structure is changed; see
\cite[Corollary~3.3]{Benson/Iyengar/Krause:2011b}.
Thus we can use a Hopf structure with the property that the $\pi$-points of the form \eqref{eq:alpha} are Hopf maps $\alpha\colon\test\to K\mcE_K$.
If $N$ is a finite dimensional $k\mcE$-module then consider the commutative diagram of finite maps
\[
\xymatrix{H^{*,*}(K\mcE_K,K) \ar[r]^{\alpha^*}\ar[d]_{-\otimes N} &
H^{*,*}(\test,K)\ar[d]^{-\otimes\alpha^*(N)} \\
\Ext^{*,*}_{K\mcE_K}(N,N) \ar[r]^(0.44){\alpha^*} &
\Ext^{*,*}_{\test}(\alpha^*(N),\alpha^*(N)).}
\]
If $\fp(\alpha)$ is in the $\pi$-support of $N$ then the kernel of the top map followed by the right hand map is not contained in $\fp(\alpha)$. So the kernel of the left hand map is not contained in $\fp(\alpha)$ and hence $\fp(\alpha)\in\supp(N)$. Thus we have $\pisupp(N)\subseteq\supp(N)$.
For a subset $\fa\subseteq H^{*,*}(\mcE,k)$ let $\mcV(\fa)$ denote the
set primes $\fp\in\Proj H^{*,*}(\mcE,k)$ such that $\fa\subseteq\fp$.
Since the
module $\Gamma_\fp(k)$ is a filtered colimit of finite dimensional
modules whose support is contained in $\mcV(\fp)$, it follows that
$\pisupp(\Gamma_\fp(k))\subseteq\mcV(\fp)$.
Now consider the Carlson module $L_\xi$ for an element
$\xi\in H^{2n,0}(\mcE,k)$, see \eqref{eq:Carlson}.
\intref{Proposition}{pr:Lxi} shows that $\pisupp(L_\xi)=\supp(L_\xi)$
equals the set of primes containing $\xi$. By
\intref{Theorem}{th:pi-tensor-hom}, it follows that if we have a
sequence of such elements, $\xi_1,\dots,\xi_m$ then
\[
\pisupp(L_{\xi_1}\otimes \dots\otimes L_{\xi_m}) = \mcV(\xi_1)\cap\dots\cap \mcV(\xi_m)=\mcV(\xi_1,\dots,\xi_m),
\]
which is also equal to $\supp(L_{\xi_1}\otimes \cdots \otimes L_{\xi_m})$. So there are finite dimensional modules with any given closed subset
for both its $\pi$-support and its support. If $N$ is such a module whose support is properly contained in $\mcV(\fp)$ then $\Gamma_\fp(k)\otimes N =
\Gamma_\fp(N)$ is projective, and so again using \intref{Theorem}{th:pi-tensor-hom},
we have
\[
\pisupp(\Gamma_\fp(k))\cap\pisupp(N)=\varnothing\,.
\]
This shows that $\pisupp(\Gamma_\fp(k))\subseteq\{\fp\}$. Since $\Gamma_\fp(k)$ is not projective, it now follows from \intref{Theorem}{th:pi-proj}
that $\pisupp(\Gamma_\fp(k))=\{\fp\}$.
\end{proof}
\begin{theorem}
\label{th:pi-equals-H}
Let $\mcE$ be an elementary supergroup scheme and $M$ a $k\mcE$-module. Then $\cosupp_\mcE(M)=\picosupp_\mcE(M)$ and $\supp_\mcE(M)=\pisupp_\mcE(M)$.
\end{theorem}
\begin{proof}
Using \intref{Proposition}{pr:pi-Gamma-p}, \intref{Theorem}{th:pi-proj} and \intref{Theorem}{th:pi-tensor-hom}, the proof is exactly the same as
that of \cite[Theorem~6.1]{Benson/Iyengar/Krause/Pevtsova:2017a}.
\end{proof}
\begin{corollary}
For all $k\mcE$-modules $M$ and $N$ we have
\begin{enumerate}[\quad\rm(1)]
\item $\supp_\mcE(M \otimes_k N) = \supp_\mcE(M)\cap\supp_\mcE(N)$,
\item $\cosupp_\mcE(\Hom_k(M,N)) = \supp_\mcE(M) \cap\cosupp_\mcE(N)$.
\end{enumerate}
\end{corollary}
\begin{proof}
This follows from \intref{Theorem}{th:pi-equals-H} and \intref{Theorem}{th:pi-tensor-hom}.
\end{proof}
Given these expressions for the (cohomological) support and cosupport one can mimic the proof of \cite[Theorem~7.1]{Benson/Iyengar/Krause/Pevtsova:2017a} to deduce:
\begin{theorem}
\label{th:localising}
Let $\mcE$ be an elementary supergroup scheme over $k$. The stable module category $\StMod(k\mcE)$ is stratified as a $\bbZ/2$-graded triangulated category by the natural action of the cohomology ring $H^{*,*}(\mcE,k)$. Therefore the assignment
\[
\bfC \mapsto \bigcup_{M\in\bfC}\supp_\mcE(M)
\]
gives a one to one correspondence between the localising subcategories of $\StMod(k\mcE)$ invariant under the parity change operator $\Pi$ and the subsets of $\Proj H^{*,*}(\mcE,k)$. \qed
\end{theorem}
From the preceding result, and following the by now well-trodden path
discovered by Neeman~\cite{Neeman:1992a} one obtains a classification
of thick subcategories of $\stmod k\mcE$; see also the proof of \cite[Theorem~7.3]{Benson/Iyengar/Krause/Pevtsova:2017a}.
\begin{corollary}
\label{co:thick}
Let $\mcE$ be an elementary supergroup scheme over $k$. There is a one to one correspondence between thick subcategories of
$\stmod(k\mcE)$ invariant under $\Pi$ and the specialisation closed subsets of $\Proj H^{*,*}(\mcE,k)$. \qed
\end{corollary}
\section{The remaining cases}
\label{se:others}
The gist of this section is that the results in
\intref{Section}{se:supportsets} up to \intref{Section}{se:support-and-cosupport} carry over to the other elementary supergroup schemes recalled in \intref{Section}{se:prelim}. We begin with the supergroup scheme $\mcE$ with group algebra
\begin{equation}
\label{se:kE-exterior}
k\mcE \colonequals k[s_1,\dots,s_n,\sigma]/(s_1^p,\dots,s_n^p,\sigma^2)
\end{equation}
with $|s_i|=0$ and $|\sigma|=1$. As an algebra $k\mcE$ is the tensor product of the group algebra of an elementary abelian $p$-group of rank $n$, and an exterior algebra on one generator, $\sigma$. This is again a complete intersection so the results of \intref{Appendix}{se:appendix} apply. In particular, there is a notion of support sets for $k\mcE$-modules, which are subsets of $\bbP^{n}(k)$, and these detect projectivity. The theory of $\pi$-points also carries over, and \intref{Theorem}{th:detection} carries over to these algebras. The only difference is that the representatives of $\pi$-points are given by maps $\alpha_{\bsa}\colon \test_K\to K\mcE_K$, where $K$ is a field extension of $k$, and
\begin{align}
\label{eq:alpha-exterior}
\begin{split}
\alpha_{\bsa}(t) &= a_1 s_1 + \cdots + a_n s_n\,, \\
\alpha_{\bsa}(\tau)&=a_{n+1}\sigma \\
\end{split}
\end{align}
for $\bsa\in \bbA^{n+1}(K)$; cf.~ \eqref{eq:alpha}. This has the
consequence that the rank variety of a finitely generated
$k\mcE$-module $M$, defined only when $k$ is algebraically closed,
consists of those points $\bsa\in \bbA^{n+1}(k)$ for which the square zero map
\begin{equation}
\label{eq:Mmatrix-exterior}
\begin{pmatrix}
\alpha_{\bsa}(\tau) & \alpha_{\bsa}(t) \\
-\alpha_{\bsa}(t) & -\alpha_{\bsa}(\tau)
\end{pmatrix} \colon M\oplus M \lra M\oplus M
\end{equation}
has rank at most $\rank_kM$. It is an affine cone in $\bbA^n(k)$;
what is more, it is invariant under the $\bbZ/2$-action that negates
the last coordinate: $a_{n+1}\mapsto -a_{n+1}$, because $M$ is
$\bbZ/2$-graded. The map $V^r_{\mcE}(M)\to \hysupp_{\mcE}(M)$ that
comes up in the proof of \intref{Theorem}{th:carlson} is equivariant
with respect to the usual Frobenius map.
Finally we come to elementary abelian $p$-groups. The observation that tackles this case applies more generally to any finite group scheme $\mcE$, viewed as a supergroup scheme. For such an $\mcE$ and any map of $k$-algebras $\alpha\colon \test\to k\mcE$, one has $\alpha(\tau)=0$ for degree reasons, so there is an induced map of $k$-algebras
\[
\bar{\alpha} \colon \test/(\tau) \lra k\mcE\,.
\]
Observe that $\test/(\tau)\cong k[t]/(t^p)$, so $\bar{\alpha}$ is
a candidate for a $\pi$-point of $\mcE$, in the sense of \cite{Friedlander/Pevtsova:2007a}.
\begin{lemma}
Let $\mcE$ be any finite group scheme. A map of $k$-algebras $\alpha\colon \test \to k\mcE$ has finite flat dimension if and only if the induced map $\bar{\alpha}\colon \test/(\tau) \to k\mcE$ is flat.
\end{lemma}
\begin{proof}
Since $\tau$ is not a zero divisor in $\test$, the canonical surjection $\ve\colon \test\to \test/(t)$ has flat dimension one. Thus if $\bar{\alpha}$ is flat, $\alpha$ has finite flat dimension.
Assume the flat dimension of $\alpha$ is finite, that is to say, the
flat dimension of $k\mcE$ viewed as a module over $\test$ via $\alpha$
is finite. The action of $\test$ on $k\mcE$ factors through
$\test/(\tau)\cong k[t]/(t^p)$, and as a $k[t]/(t^p)$-module $k\mcE$
is a direct sum of copies of the cyclic modules $k[t]/(t^i)$, for
$1\le i\le p$. Evidently as an $\test$-module $k[t]/(t^i)$ has finite
flat dimension only when $i=p$; see \eqref{eq:resol}. Thus the
hypothesis on $\alpha$ implies that $k\mcE$ is a direct sum of copies
of $k[t]/(t^p)$, and hence $\bar{\alpha}$ is flat, as claimed.
\end{proof}
Given the preceding result it is clear that $\pi$-points for elementary abelian $p$-groups defined via $\test$ have the same properties as the ones introduced by Carlson~\cite{Carlson:1983a}, so nothing more needs to be said about this situation.
\appendix
\section{Complete intersections}
\label{se:appendix}
In this section we describe support sets of modules over complete intersections, following Avramov~\cite{Avramov:1989a}, with a view towards detecting finite flat dimension of modules over complete intersections. With this in mind we have chosen to work with complete intersection rings that are quotients of power series rings, and modules of finite Loewy length. This simplifies the exposition at various points. This utilitarian approach is also why we do not develop the material beyond \intref{Theorem}{th:appendix}, which can be taken as a starting point for a $\pi$-point approach to arbitrary modules over complete intersections.
Throughout this appendix $k$ will be field; there are no restrictions on the characteristic. Set $P\colonequals \pos {\bsx}$, the power ring over $k$ in indeterminates $\bsx\colonequals x_1,\dots,x_c$. In particular $P$ is a noetherian local ring, with maximal ideal $(\bsx)$. It is to ensure these properties that we work with the ring of formal power series. An alternative would be to take the localisation of the polynomial ring over $(\bsx)$, localised at $(\bsx)$.
Let $\bsf\colonequals f_1,\dots,f_c$ be elements in $P$ such that
\begin{enumerate}
\item
$(\bsf)$ is in $(\bsx)^2$, and
\item
$f_i$ is not a zero-divisor in $P/(f_1,\dots, f_{i-1})$ for $1\le i\le i$.
\end{enumerate}
Condition (2) states that the $\bsf$ is a regular sequence in $P$, and hence the ring
\[
R\colonequals P/(f_1,\dots,f_c)
\]
is a complete intersection local ring, of codimension $c$ in $P$. Let $\fm$ denote the maximal ideal $(\bsx)R$ of $R$.
\subsection*{Generic hypersurfaces}
For each $\bsa \colonequals [a_1,\dots,a_{c}]$ in $\bbP^{c-1}(k)$ set
\[
h_{\bsa} (\bsx)\colonequals a_1f_1 + \cdots + a_cf_c \quad \text{and}\quad P_{\bsa}\colonequals P/(h_{\bsa}(\bsx))\,.
\]
While $h_{\bsa}(\bsx)$ depends on a representative for $\bsa$, the ideal it generates does not, so there is no ambiguity in the notation $P_{\bsa}$. Since $h_{\bsa}(\bsx)$ is in $(\bsf)$ there is a surjection
\[
\beta_{\bsa}\colon P_{\bsa} \lra R\,.
\]
Let $M$ be an $R$-module and $\beta^*_{\bsa}(M)$ its restriction to $P_{\bsa}$. We shall be interested in the \emph{support set} of $M$, defined to be:
\begin{equation}
\hysupp_R(M)\colonequals \{\bsa\in \bbP^{c-1}(k) \mid \fdim \beta_{\bsa}^*(M) =\infty \}
\end{equation}
The language is borrowed from \cite[\S3]{Avramov/Iyengar:2018a},
whilst the result below is from~\cite[Corollaries 3.11,
3.12]{Avramov:1989a}. In these sources, projective dimension, rather
than flat dimension, is used, but this makes no difference, for one is finite
if and only if the other is finite, and the dimensions coincide when $M$ is finitely
generated, because finitely generated flat modules are projective.
\begin{theorem}
\label{th:Av}
Assume $k$ is algebraically closed and that $M$ is a finitely generated $R$-module. The subset $\hysupp_{R}(M)\subseteq \bbP^{c-1}(k)$ is closed in the Zariski topology. One has $\hysupp_{R}(M)=\varnothing$ if and only if $M$ has finite flat dimension. \qed
\end{theorem}
We need an extension of the preceding result that applies also to infinitely generated modules. For simplicity, we consider only $R$-modules $M$ of \emph{finite Loewy length}, meaning $\fm^s M =0$ for some $s\gg 0$. Such modules form a Serre subcategory of $\Mod R$, though not a localising subcategory. The main case of interest is when $R$ has codimension $n$, equivalently, of Krull dimension zero, in which case each $R$-module has this property. However the proof of \intref{Theorem}{th:appendix} below involves an induction on the codimension of $R$, and then it becomes convenient to work with modules of finite Loewy length.
The result however requires only that the module is $\fm$-torsion, that is to say that each element is annihilated by a power of $\fm$.
\begin{lemma}
\label{le:pd-test}
When $M$ is $\fm$-torsion the following conditions are equivalent:
\begin{enumerate}[\quad\rm(a)]
\item
$\fdim_RM < \infty$
\item
$\Tor^R_i(k,M)=0$ for $i\gg 0$;
\item
$\Ext_R^i(k,M)=0$ for $i\gg 0$.
\end{enumerate}
\end{lemma}
\begin{proof}
Any complete intersection ring is Gorenstein, and when such a ring has finite Krull dimension---for example, if it is a quotient of $P$---a module has finite flat dimension if and only if it has finite injective dimension; see \cite[(3.3.4)]{Christensen:2000a}. Given this observation and the hypothesis that $M$ is $\fm$-torsion, the equivalence of the stated conditions follows from \cite[Propositions 5.3.F, 5.3.I]{Avramov/Foxby:1991a}.
\end{proof}
For what follows we need to consider support sets defined over
extension fields.
\begin{definition}\label{de:extending-fields}
Let $K$ be an extension field of $k$. We set $P_K\colonequals \pos[K]{\bsx}$ and consider $P$ as a subring of $P_K$ in the obvious way. For any quotient ring $A$ of $P$, set $A_K\colonequals P_K\otimes_PA$; this is then a quotient ring of $P_K$. It is easy to verify that the extension $P\to P_K$ is flat, and hence so is the extension $A\to A_K$.
In particular, $R_K$ is the quotient of $P_K$ by the ideal generated
by $f_1,\dots,f_c$, viewed as elements of $P_K$. Hence $R_K$ is a
complete intersection of codimension $c$ in $P_K$, with maximal ideal
$\fm R_K$. For an $R$-module $M$ set
\[
M_K \colonequals R_K\otimes_R M \qquad\text{and}\qquad M^K\colonequals \Hom_{R}(R_K,M)
\]
viewed as $R_K$-modules.
\end{definition}
\begin{remark}\label{re:extending-fields}
The modules $M_K$ and $M^K$ have finite Loewy length provided that
$M$ has finite Loewy length. This is clear since $\fm^s M=0$ implies
$\fm^s M_K=0 = \fm^s M^K$.
\end{remark}
\begin{lemma}
\label{le:ext-tor}
When $M$ is an $R$-module of finite Loewy length, for each $i$ one has
\begin{align*}
\Tor^{R_K}_i(K,M_K) &\cong K\otimes_k \Tor^R_i(k,M) \\
\Ext_{R_K}^i(K,M^K) &\cong \Hom_k(K,\Ext_R^i(k,M))\,.
\end{align*}
\end{lemma}
\begin{proof}
We verify the isomorphism involving Ext; the first one is a tad easier to verify for $R_K$ is flat over $R$. By hypothesis, there exists a positive integer $s$ such that $\fm^s M=0$. Set $A\colonequals R/\fm^s$; then $A_K \cong R_K/\fm^s R_K$. Observe that $A$ is finite dimensional over $k$; given this, it is easy to verify that $A_K$ is projective as an $A$-module. This justifies the second isomorphism in $\dcat{R_K}$ below
\[
\RHom_R(R_K,M) \cong \RHom_A(A_K,M)\cong \Hom_A(A_K,M)\,.
\]
The first one is adjunction. We conclude that $\Hom_R(R_K,M) \cong \RHom_R(R_K,M)$. This justifies the first isomorphism below
\begin{align*}
\Ext^i_{R_K}(K,M^K)
&\cong \Ext^i_{R_K}(K,\RHom_R(R_K,M)) \\
& \cong \Ext^i_R(K,M) \\
& \cong \Hom_k(K,\Ext^i_R(k,M))\,.
\end{align*}
The second one is standard adjunction and the last one is just standard.
\end{proof}
We also need the following remarks concerning cohomology operators; we focus on codimension two for this is all that is needed in the sequel.
See \cite{Avramov/Buchweitz:2000a} for details.
\begin{remark}
\label{re:codim2}
Let $P\to Q\to R$ be a factorisation of the surjection $P\to R$ such that the kernel of the map $Q\to R$ can be defined by a $Q$-regular sequence, say $g_1,g_2$. Set $\fn \colonequals (\bsx) Q$; this is a maximal ideal of $Q$ lying over the maximal ideal $\fm$ of $R$. Set $J\colonequals (g_1,g_2)$. There is a natural embedding of $k$-vector spaces
\[
\Hom_k(J /{\fn J}, k) \hookrightarrow \Ext^2_R(k,k)\,.
\]
The residue classes of $g_1,g_2$ are a basis for the $k$-vector space $J/\fq J$; let $\chi_1,\chi_2$ denote the image in $\Ext^2_R(k,k)$ of the dual basis. These are the cohomology operators constructed by Gulliksen~\cite{Gulliksen:1974a} and Eisenbud~\cite{Eisenbud:1980a} associated to $Q\to R$. They lie in the center of the $k$-algebra $\Ext_R(k,k)$.
Fix a point $\bsb\colonequals (b_1,b_2)$ in $k^2\setminus \{0\}$ and
set $Q_{\bsb}\colonequals Q/(b_1g_1+b_2g_2)$. Then one has a
surjection $Q_{\bsb}\to R$, with kernel generated by
$J/(b_1g_1+b_2g_2)$; it is easy to check that this ideal can be generated any element $c_1g_1+c_2g_2$ such that $(c_1,c_2)$ is not a scalar multiple of $(b_1,b_2)$; no such element is a zero divisor. The surjection
\[
\frac J{\fn J} \twoheadrightarrow \frac J{(\fn J + b_1g_1+b_2g_2)}
\]
yields an embedding
\[
\Hom_k(\frac J{(\fn J + b_1g_1+b_2g_2)},k) \hookrightarrow \Hom_k(\frac J {\fn J}, k) \hookrightarrow \Ext^2_R(k,k)\,.
\]
From this it follows that the cohomology operator corresponding to the surjection $Q_{\bsb}\to R$ is $b_2\chi_1 - b_1\chi_2$.
For any $R$-module $M$, the standard change of rings spectral sequence associated to $Q_{\bsb}\to R$ yields an exact sequence
\begin{equation}
\label{eq:chi-les}
\cdots \to \Ext^{i+1}_{Q_{\bsb}}(k,M) \to \Ext^{i}_{R}(k,M) \xra{b_2\chi_1 - b_1\chi_2} \Ext^{i+2}_{R}(k,M) \to \Ext^{i+2}_{Q_{\bsb}}(k,M)\to \cdots
\end{equation}
\end{remark}
The result below extends part of \intref{Theorem}{th:Av}.
\begin{theorem}
\label{th:appendix}
Let $R$ be as above and $M$ an $R$-module of finite Loewy length. The following conditions are equivalent:
\begin{enumerate}[\quad\rm(1)]
\item
$\fdim_RM$ is finite;
\item
$\hysupp_{R_K}(M_K)=\varnothing$ for any field extension $K$ of $k$;
\item
$\hysupp_{R_K}(M^K)=\varnothing$ for any field extension $K$ of $k$.
\end{enumerate}
Moreover, in \emph{(2)} and \emph{(3)} it suffices to take for $K$ an algebraically closed extension of $k$
of transcendence degree at least $c-1$.
\end{theorem}
\begin{proof}
The proof combines ideas from \cite{Avramov:1989a} and \cite{Benson/Carlson/Rickard:1996a}. \intref{Remark}{re:extending-fields} and \intref{Remark}{re:codim2} will be used implicitly in what follows.
\medskip
(1)$\Rightarrow$(2) For any $\bsa$ in $\bbP^{c-1}(k)$ the map $P_{\bsa}\to R$ is a complete intersection, of codimension $c-1$; in particular $\fdim_{P_{\bsa}}R$ is finite. Thus $\fdim_RM$ finite implies $\fdim_{P_{\bsa}}M$ is finite as well, and hence $\hysupp_R(M)=\varnothing$. It remains to observe that if $\fdim_RM$ is finite, then so is $\fdim_{R_K}M_K$; for example, by combining \intref{Lemma}{le:pd-test} and \intref{Lemma}{le:ext-tor}.
\medskip
(1)$\Rightarrow$(3) This can be proved akin to the previous implication.
\medskip
(3)$\Rightarrow$(1)
We argue by induction on $c$. The result is a tautology when $c=1$, for then $\hysupp_R(M)=\varnothing$ is equivalent to $\fdim_RM<\infty$, by definition of the support set. We can thus assume $c\ge 2$, and that the desired conclusion holds for complete intersections of codimension $c-1$. Set $Q\colonequals P/(f_3,\dots,f_{c})$, so that one has a surjection $Q\to R$.
Since $R=Q/(f_1,f_2)$ and $f_1,f_2$ is a $Q$-regular sequence, corresponding to the surjection $Q\to R$ one has a subspace $k\chi_1 + k\chi_2$ of $\Ext^2_R(k,k)$; see \intref{Remark}{re:codim2}. Fix a field extension $K$ of $k$, and elements $b_1,b_2$ in $K^2$, not both zero. Set $Q_{\bsb}\colonequals Q_K/(b_1f_{1}+b_2f_{2})$. The map $Q_K\to R_K$ factors as $Q_K\to Q_{\bsb} \to R_K$.
We wish to apply the induction hypothesis to $Q_{\bsb}$. This ring is the quotient of $P_K$ by the ideal generated by $b_1f_1+b_2f_2,f_3,\dots, f_c$, and the latter is a regular sequence in $P_K$. Moreover for any field extension $L$ of $K$, any $L$-linear combination of this sequence is an $L$-linear combination of $f_1,\dots,f_c$. Keeping in mind that $\Hom_{R_K}(R_L,M^K)\cong M^L$ as $R_L$-modules, it is now a routine verification that
\[
\hysupp_{(Q_{\bsb})_L}(M^L)= \varnothing
\]
as a subset of $L^{c-1}$. The upshot is that $\fdim_{Q_{\bsb}}(M^K)$ is finite, by the induction hypothesis. Since the Krull dimension of $Q_{\bsb}$ is at most $n$, it follows that
\[
\fdim_{Q_{\bsb}}M^K\le n\,.
\]
Fix $i\ge n$ and consider the following snippet
\[
\Ext^{i+1}_{Q_{\bsb}}(K,M^K) \to \Ext^{i}_{R_K}(K,M^K) \xra{\ b_2\chi_1 - b_1\chi_2\ } \Ext^{i+2}_{R_K}(K,M^K) \to \Ext^{i+2}_{Q_{\bsb}}(K,M^K)
\]
of the exact sequence from \eqref{eq:chi-les} associated to $Q_{\bsb}\to R_K$. Thus the choice of $i$, the exact sequence above, and \intref{Lemma}{le:ext-tor} imply that one has an isomorphism
\[
\Hom_k(K, \Ext^{i}_R(k,M)) \xra[\cong]{\ b_2\chi_1 - b_1\chi_2\ } \Hom_k(K, \Ext^{i+2}_R(k,M))\,.
\]
Since $(b_1,b_2)$ in $K^2\setminus \{0\}$ was arbitrary, \cite[Lemma~5.1]{\bikp:2017a} implies $\Ext^{i}_R(k,M)=0$. Since this holds for each $i\ge n$ we deduce that $\fdim_RM\le i$, by \intref{Lemma}{le:pd-test}.
\medskip
(2)$\Rightarrow$(1) can be proved along the lines of (3)$\Rightarrow$(1). Instead of \cite[Lemma~5.1]{Benson/Iyengar/Krause/Pevtsova:2017a}, one applies its analogue for tensor products \cite[Theorem~5.2]{Benson/Carlson/Rickard:1996a}. These results also make it clear that it suffices to consider algebraic closed extension fields $K$ of transcendence degree $c-1$.
\end{proof}
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I totally stole the title of today's column from Abbie Hoffman. Then again, he promoted people stealing things from him, so I guess it's okay.
(Yes, Abbie Hoffman. My cultural references are very contemporary. Try to keep up.)
Off to burn my bra,
Brutalism
1 comment:
I recommended it to my doctor but I think he was a little confused as to why I was recommending it and he also seemed a little confused as to why I had randomly shown up to his house. Specifically in his kitchen.
| 60,464
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\begin{document}
\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
\def\bq{\begin{quote}}
\def\eq{\end{quote}}
\section{Introduction}
In the Duistermaat-Heckman ($DH$) localization \cite{DE} the integral of the exponential of the hamiltonian of a torus action on a compact symplectic manifold is computed exactly by the saddle-point method
as a sum on the fixed points of the torus action. The $DH$ localization has a cohomological interpretation \cite{A} according to which
the exponential of a closed form is deformed by adding a coboundary that does not change the integral of the closed form. The coboundary is rescaled by a large factor
and the modified integral is evaluated exactly by the saddle point method. Witten localization \cite{W1} extended the $DH$ idea to certain supersymmetric functional integrals that are viewed as cohomology classes for which the differential is a twisted supercharge. Nekrasov \cite{N} reproduced the Seiberg-Witten solution \cite{W3} for the prepotential of $ \cal{N}$ $=2$ $SUSY$ $YM$
by localizing the partition function, i.e. the "cohomology of $1$", whose logarithm is the prepotential,
on a sum of finite dimensional integrals over the moduli space of instantons \cite{N}.
There is a torus action, $U(1)^{N-1} \times U(1) \times U(1)$, on this moduli space compactified by a non-commutative deformation of space-time \cite{N}.
$U(1)^{N-1}$ is the unbroken gauge group of $ \cal{N}$ $=2$ $SUSY$ $YM$ in the phase spontaneously broken by the condensation of the eigenvalues of a complex Higgs field and $U(1) \times U(1)$ is the rotational symmetry in the non-commutative background.
As a consequence Nekrasov further reduced the partition function to a sum on the fixed points of the aforementioned torus action on the instantons moduli by means of the $DH$ localization \cite{N}.
Nekrasov exact localization on a sum of saddle points has the interesting consequence that the beta function for the Wilsonian coupling constant
of $ \cal{N}$ $=2$ $SUSY$ $YM$ is one-loop exact, a result already known \cite{AA}. The Wilsonian coupling occurs in the Wilsonian normalization of the action as opposed to the canonical coupling that occurs by rescaling the kinetic term in such a way that is independent on the coupling. The two definitions of the coupling have different beta functions in general \cite{AA}.
In the pure $SU(N)$ $YM$ theory there is no natural cohomology because of the lack of supersymmetry.
However, we may wonder as to whether a different kind of localization holds, perhaps linked to the large-$N$ limit.
It is clear that such localization may exist only for special observables, since this is already the case in $SUSY$ gauge theories.
The aim of this paper is to show that a new kind of localization holds in the large-$N$ limit of pure $YM$.
In pure $YM$ in absence of a natural cohomology we developed new localization techniques based on homology theory and on a new holomorphic version of the loop equation for special Wilson loops \cite{MB}, called twistor Wilson loops for geometrical reasons that we do not discuss here,
whose v.e.v. is invariant at quantum level for deformations that are vanishing boundaries in homology (i.e. backtracking arcs). In fact twistor Wilson loops are trivial in the large-$N$ limit and in the technical sense of being invariant for the addition of backtracking arcs they are in the "homology of $1$ ".
However, the localization of the holomorphic loop equation by homology admits a simpler interpretation
directly in terms of the functional integral, as localization on the fixed points of a semigroup contracting the functional measure and leaving invariant
the v.e.v. of twistor Wilson loops. This new localization has many analogies with Nekrasov localization on the fixed points of the torus action
on the moduli of (non-commutative) instantons.
Thus it is this simpler theory that we describe in the following section.
As a consequence the large-$N$ Wilsonian beta function of $YM$, in certain new variables of anti-selfdual ($ASD$) type that are defined through a non-$SUSY$ version \cite{MB} of the Nicolai map
\cite{Nic} of $ \cal{N}$ $=1$ $SUSY$ $YM$, that was worked out by De Alfaro-Fubini-Furlan-Veneziano in the eighties \cite{V}, is one-loop exact as in $SUSY$ gauge theories.
\section{Localization in pure large-$N$ $YM$ theory on the fixed points of a semigroup of contractions}
We define twistor Wilson loops in the $YM$ theory
with gauge group $U(N)$ on $R^2 \times R^2 _{\theta}$ with complex coordinates $(z=x_0+i x_1, \bar z=x_0-i x_1, u=x_2+i x_3, \bar u=x_2-i x_3)$
and non-commutative parameter $\theta$, satisfying $ [\partial_u, \partial_{\bar u}]=\theta^{-1} 1$, (see \cite{MA,AG} for a review of non-commutative gauge theories)
as follows:
\bea
Tr \Psi_{\lambda}(L_{ww})=Tr P \exp i \int_{L_{ww}}(A_z+\lambda D_u) dz+(A_{\bar z}+ \lambda^{-1}D_{\bar u}) d \bar z ,
\eea
where $D_u=\partial_u+i A_u$ is the covariant derivative along the non-commutative direction $u$. The plane $(z, \bar z)$ is instead commutative. The loop, $L_{ww}$, starts and ends at the marked point, $w$. The trace in Eq.(2.1)
is over the tensor product of the $U(N)$ Lie algebra and of the infinite dimensional Fock space that defines
the Hilbert space representation of the non-commutative plane $(u, \bar u)$ \cite{MA,AG}. The limit of infinite non-commutativity in the plane $(u, \bar u)$ is understood, being equivalent to the large-$N$ limit of the commutative gauge theory \cite{MA,AG}. Therefore non-commutativity is for us just a mean to define the large-$N$ limit as well as it is for Nekrasov just a mean to compactify the moduli space of instantons.
It easy to prove that the v.e.v. of the twistor Wilson loops is independent on the parameter $\lambda$:
\bea
<\frac{1}{N}Tr\Psi_{\lambda}(L_{ww})>=<\frac{1}{N}Tr\Psi_{1}(L_{ww})> .
\eea
The proof is obtained changing variables, rescaling functional derivatives in the usual definition of the functional integral
of the non-commutative $YM$ theory \cite{MA,AG}. The formal non-commutative integration measure is invariant under such rescaling because of the
pairwise cancellation of the powers of $\lambda$ and $\lambda^{-1}$. The non-commutative $YM$ action, proportional to
$Tr( -i[ D_{\alpha}, D_{\beta}]-\theta^{-1}_{\alpha \beta})^2$, is invariant because of rotational invariance in the non-commutative plane. The only possibly dangerous terms couple the non-commutative
parameter to the commutator $[ D_{u}, D_{\bar u}]$, while all the other mixed terms are zero in our case. But the commutator is invariant under $\lambda$-rescaling.
In fact the twistor Wilson loops are trivially $1$ at large-$\theta$ to all orders in the 't Hooft coupling constant $g$:
\bea
\lim_{\theta \rightarrow \infty} <\frac{1}{N}Tr\Psi_{\lambda}(L_{ww})>=1 .
\eea
We do not give a diagrammatic proof of the triviality in this paper,
but we show that indeed triviality holds to the lowest non-trivial order in perturbation theory.
We have in the Feynman gauge in the large-$\theta$ limit:
\bea
<\int_{L_{ww}}(A_z+\lambda D_u) dz+(A_{\bar z}+ \lambda^{-1}D_{\bar u}) d \bar z \int_{L_{ww}}(A_z+\lambda D_u) dz+(A_{\bar z}+ \lambda^{-1}D_{\bar u}) d \bar z> \nonumber \\
= 2 \int_{L_{ww}} dz\int_{L_{ww}}d \bar z (<A_z A_{\bar z}>+i^2 <A_u A_{\bar u}>)
=0 .
\eea
We use the $\lambda$-independence to prove that the v.e.v. of twistor Wilson loops is localized on the fixed points of the semigroup
rescaling $\lambda$.
It is convenient to choose our twistor Wilson loops in the adjoint representation and to use the fact that in the large-$N$ limit
their v.e.v. factorizes in the product of the v.e.v. of the fundamental representation and of its conjugate.
Then, for the factor in the fundamental representation, localization proceeds as follows.
We write the $YM$ partition function by means of a non-$SUSY$ analogue \cite{MB} of the Nicolai map \cite{Nic} of $\cal{N}$ $=1$ $SUSY$ $YM$ theory \cite{V},
introducing in the functional integral the appropriate resolution of identity:
\bea
1= \int \delta(F^{-}_{\alpha \beta}-\mu^{-}_{\alpha \beta}) \delta\mu^{-}_{\alpha \beta} ,
\eea
\bea
Z=\int \exp(-\frac{N 8 \pi^2 }{g^2} Q-\frac{N}{4g^2} \sum_{\alpha \neq \beta} \int Tr_f(\mu^{-2}_{\alpha \beta}) d^4x)
\delta(F^{-}_{\alpha \beta}-\mu^{-}_{\alpha \beta}) \delta\mu^{-}_{\alpha \beta} \delta A_{\alpha} .
\eea
$Q$ is the second Chern class (the topological charge) and $\mu^{-}_{\alpha \beta}$
is a field of $ASD$ type. The equations of $ASD$ type in the resolution of identity,
$F_{01}-F_{23}=\mu^-_{01} ,
F_{02}-F_{31}=\mu^-_{02} ,
F_{03}-F_{12}=\mu^-_{03} $,
can be rewritten in the form of a Hitchin system (taking into account the central extension that occurs in the non-commutative case), $
-i F_A+[D,\bar D] -\theta^{-1}1=\mu^0=\frac{1}{2}\mu^-_{01} ,
-i\partial_{A} \bar D= n=\frac{1}{4}(\mu^-_{02}+i\mu^-_{03}) ,
-i\bar \partial_A D=\bar n=\frac{1}{4}(\mu^-_{02}-i\mu^-_{03}) $,
or equivalently in terms of the non-hermitian
connection whose holonomy is computed by the twistor Wilson loop with parameter $\rho$,
$B_{\rho}=A+\rho D+ \rho^{-1} \bar D=(A_z+ \rho D_u) dz+(A_{\bar z}+ \rho^{-1} D_{\bar u}) d \bar z$:
$-i F_{B_{\rho}} -\theta^{-1}1= \mu_{\rho}=\mu^0+\rho^{-1} n- \rho \bar n ,
-i\partial_{A} \bar D= n ,
-i\bar \partial_A D=\bar n$.
The resolution of identity in the functional integral
then reads:
\bea
1=\int \delta n \delta \bar n \int_{C_{\rho}} \delta \mu_{\rho} \delta(-i F_{B_{\rho}} - \mu_{\rho}-\theta^{-1}1) \delta(-i\partial_{A} \bar D- n) \delta(-i\bar \partial_A D- \bar n) ,
\eea
where the measure, $\delta \mu_{\rho}$, along the path, $C_{\rho}$, is over the non-hermitian path with fixed $n$ and $\bar n$ and varying $\mu^0$. The resolution
of identity is independent, as $\rho$ varies, on the complex path of integration $C_{\rho}$.
Let us consider the v.e.v. of twistor Wilson loops:
\bea
\int \delta n \delta \bar n \int_{C_{\rho}}\delta \mu_{\rho} \exp(-\frac{N 8 \pi^2 }{g^2} Q-\frac{N 4}{g^2} \int Tr_f( \mu^{0})^2 +4Tr_f(n \bar n) d^4x) \nonumber\\
Tr_f P \exp i \int_{L_{ww}}(A_z+\lambda D_u) dz+(A_{\bar z}+ \lambda^{-1}D_{\bar u}) d \bar z \nonumber \\
\delta(-i F_{B_{\rho}} - \mu_{\rho}-\theta^{-1}1) \delta(-i\partial_{A} \bar D- n) \delta(-i\bar \partial_A D- \bar n)
\delta A \delta \bar A \delta D \delta \bar D
\eea
and let us change variables in the functional integral rescaling the non-commutative covariant derivatives:
\bea
\int \delta n \delta \bar n \int_{C_{\rho}}\delta \mu_{\rho} \exp(-\frac{N 8 \pi^2 }{g^2} Q-\frac{N4}{g^2} \int Tr_f(\mu^{0})^2+4Tr_f(n \bar n) d^4x)\nonumber \\
Tr_f P \exp i \int_{L_{ww}}(A_z+D'_u) dz+(A_{\bar z}+ D'_{\bar u}) d \bar z \nonumber \\
\delta( -i F_A+ [D',\bar D']-\theta^{-1}1-\mu^0
-i\frac{\lambda}{\rho} \partial_{A} \bar D' +i \frac{\rho}{\lambda} \bar \partial_A D' -\rho^{-1} n+\rho \bar n)\nonumber \\
\delta(-i \lambda \partial_{A} \bar D'-n) \delta(-i \lambda ^{-1}\bar \partial_A D'- \bar n)
\delta A \delta \bar A \delta D' \delta \bar D' .
\eea
Taking the limit $\lambda \rightarrow 0$ inside the functional integral, the last line implies localization on $ n=0$ and
$\bar \partial_A D' =0$. The $\delta n$ integral is performed by means of the delta function.
The independence on the path $C_{\rho}$ in the neighborhood of $\rho=0$, that we denote, choosing $\rho=\lambda$, $C_{0^+}$, implies that the $\delta \bar n$ integral decouples and that
$\partial_{A} \bar D'=0$ as well.
Indeed on $C_{0^+}$ the argument of the remaining delta function contains the combination of a hermitian $-i F_A+ [D',\bar D']-\theta^{-1}1-\mu^0$
and an anti-hermitian $-i \partial_{A} \bar D' +i \bar \partial_A D'$ part, whose sum can be zero only if the two terms are zero separately, using implicitly the constraint
from the conjugate representation. Therefore $-i \partial_{A} \bar D' +i \bar \partial_A D'=0$ on
$C_{0^+}$ and because $\bar \partial_A D' =0$ also $\partial_{A} \bar D'=0$.
We notice that the localized density has a holomorphic ambiguity,
since we can represent the same measure using a different density making holomorphic transformations without spoiling the localization: $
\delta \mu_{0^{+}}= \frac{\delta \mu_{0^{+}}} {\delta \mu'_{0^{+}}} \delta \mu'_{0^{+}} $.
This holomorphic ambiguity (and the associated holomorphic anomaly) can be resolved only through the more refined theory
of the homological localization of the loop equation \cite{MB1} that will not be discussed here. This theory requires that $\mu'$ be chosen in the holomorphic gauge, $B_{\bar z}=0$.
The final result for the localized effective measure is:
\bea
\big[ \int_{C_{0^{+}}} \delta \mu'_{0^{+}} \frac{\delta \mu_{0^{+}} } { \delta \mu'_{0^{+}} }
\exp(-\frac{N 8 \pi^2 }{g^2} Q-\frac{N}{4g^2} \sum_{\alpha \neq \beta} \int Tr_f(\mu^{-2}_{\alpha \beta}) d^4x)
\delta(F^{-}_{\alpha \beta}-\mu^{-}_{\alpha \beta}) \big]_{n=\bar n=0} \delta A_{\alpha} ,
\eea
where we have reintroduced the covariant notation.
A delicate point arises about the meaning of the residual complex functional integral at the fixed points.
We would like to reduce it to something more manageable. In the $SUSY$ case the original functional integral is reduced to a sum over finite dimensional integrals over instantons
by cohomological localization. Of course this is not possible in the pure $YM$ case. But the basic idea in the $YM$ theory is to reduce the resolution of identity due to the non-$SUSY$ Nicolai map in Eq.(2.5) to finite dimensional integrals
on a dense set in function space in a neighborhood of the fixed points by interpreting it as hyper-Kahler reduction on a lattice of surface operators:
\bea
1= \prod_p \int d n_p d\bar n_p \int_{C} d\mu_p \delta(-i F_{B} -\sum_{p} \mu_p \delta^{(2)}(z-z_{p_{(u, \bar u)}}) -\theta^{-1}1) \nonumber \\
\delta(-i\partial_{A} \bar D-\sum_{p} n_p \delta^{(2)} (z-z_{p_{(u, \bar u)}}) )
\delta(-i\bar \partial_A D- \sum_{p} \bar n_p \delta^{(2)} (z-z_{p_{(u, \bar u)}}))
\eea
Codimension-two singularities of this kind were introduced many years ago in \cite{MB2,MB3} in the pure $YM$ theory as an "elliptic fibration of parabolic bundles" for the purpose of getting control over the large-$N$ limit of the pure $YM$ theory exploiting the integrability of the Hitchin fibration.
Later, in \cite{W2}, they were introduced in the $\cal{N}$ $=4$ $SUSY$ $YM$ theory for the study of the geometric Langlands correspondence, under the name of "surface operators", and this is now the name universally used in the physics literature. In fact they have been studied in the mathematics literature at classical level in \cite{KM} as singular instantons.
The localization on the fixed points in Eq.(2.10) implies $n_p=\bar n_p=0$.
We want to understand the residual $d\mu_p$ integration. The hyper-Kahler reduction on surface operators on non-commutative space-time is not immediately understood
because of the non-commutativity. But on a non-commutative torus, for (large) rational values of $2 \pi \theta$ in units of the torus area, we can use the Morita equivalence (see \cite{AG} for a review) of the non-commutative $U(N)$ $YM$
theory to a commutative one with larger $U(N')$ gauge group and some units of a background 't Hooft flux. The large-$N$ limit must now be taken via rational sequences because of Morita duality, but this is not restrictive for twistor Wilson loops because of their triviality.
If the $(z, \bar z )$ plane is compactified to a sphere the standard hyper-Kahler reduction has a topological interpretation as defining a representation of the fundamental group of the punctured sphere, $\prod_p M_p = 1$, via the holonomy representation of the connection $B$, where $M_p= P \exp i \int_{L_p} B_z dz$ is the holonomy in the holomorphic gauge, $B_{\bar z}=0$, along vanishing cycles, $L_p$, encircling the punctures, $p$. In the holomorphic gauge $B$ is a holomorphic connection with regular singularities.
The complexification of the global gauge group acts on the holonomy at one point, $p_1$, by the adjoint action in such a way that $M_{p_1}$can be put in canonical form.
$M_{p_1}$ can be diagonalized if it has distinct eigenvalues, while in general it can be put in Jordan form.
In the large-$N$ limit it is possible to restrict the integration measure $d\mu_p$ to orbits whose holonomies have fixed eigenvalues since this restriction
implies an error of subleading order in $\frac{1}{N}$.
In addition by translational invariance the conjugacy class of the orbits at all the points $p$ must be the same.
Finally the global $SU(N)$ gauge group of the $YM$ theory (and not only the $U(1)^{N-1}$ torus as in the Nekrasov case) must fix $M_{p_1}$, i.e. $g M_{p_1} g^{-1}=M_{p_1}$, since otherwise $M_{p_1}$ would break spontaneously the global gauge symmetry. Therefore $M_{p_1}$
must be central and thus must be in $Z_N$. But then all the orbits collapse to a point and there are no moduli at the fixed points.
However, in any neighborhood of the fixed points of the global gauge group, defined deforming infinitesimally the eigenvalues, the orbits are non-trivial and moduli there exist.
Thus if we first compute the effective measure in a neighborhood of the fixed points and then we sit on the fixed points
the induced measure will contain the powers of the Pauli-Villars regulator due to the moduli.
This has an analogue in the localization of the $\cal{N}$ $=2$ $SUSY$ $YM$ partition function, where
generically instantons have moduli (this is essential to get the correct beta function in that case too), but the instantons at the fixed points of
the torus action have not. Thus at the fixed points the contour integral over $\mu_{0^{+}}$ collapses to a discrete sum over sectors with $Z_N$ holonomy.
Now the integration on the gauge connection in Eq.(2.10) can be explicitly performed in the Feynman gauge to obtain:
\bea
\sum_{Z_N} [\exp(-\frac{N 8 \pi^2 }{g_W^2} Q-\frac{N}{4g_W^2} \sum_{\alpha \neq \beta} \int Tr_f(\mu^{-2}_{\alpha \beta}) d^4x)\nonumber \\
Det^{-\frac{1}{2}}(-\Delta_A \delta_{\alpha \beta} -i ad_{ \mu^-_{\alpha \beta}} ) Det(-\Delta_A)
(\frac{\Lambda}{2 \pi})^{n_b} Det^{\frac{1}{2}} \omega \frac{\delta \mu_{0^{+}} } { \delta \mu'_{0^{+}} } \times c.c. \big] _{n=\bar n=0}.
\eea
The complex conjugate factor arises by the conjugate representation. According to Eq.(2.3) the holonomy of twistor Wilson loops at the fixed points is trivial, because of the cancellation of $Z_N$ factors
between the fundamental and the conjugate representations.
The connection, $A$, denotes the solution of the equation
$[F^-_{\alpha \beta}- \sum_{p} \mu ^-_{\alpha \beta}(p) \delta^2(z-z_{p_{(u, \bar u)}}) =0]_{n=\bar n=0} $ in each $Z_N$ sector.
$Det^{\frac{1}{2}} \omega$ is the contribution of the $n_b$ zero modes due to the moduli and $\Lambda$ the corresponding Pauli-Villars regulator.
Important technical issues are the control over the ultraviolet quadratic divergence of the classical action on surface operators by means of the partial non-commutative Eguchi-Kawai reduction \cite{MB, MA}
and the point splitting regularization of the loop expansion of the functional determinants in the background of the lattice of surface operators \cite{MB1}.
The beta function for the Wilsonian coupling of the large-$N$ $YM$ theory in the $ASD$ variables is exactly one-loop
and coincides with the result of one-loop perturbation theory.
However, the one-loop result for $\beta_0= \frac{1}{(4\pi)^2} \frac{11}{3}$ is obtained as the sum $\frac{1}{(4\pi)^2} (\frac{5}{3}+2)$.
The first term is the total contribution of the determinant due to the non-$SUSY$ Nicolai map and of the Faddeev-Popov determinant, and gives rise to the multiplicative renormalization factor, $Z^{-1}=1-\frac{1}{(4\pi)^2} (\frac{10}{3}) g_W^2 \log(\frac{\Lambda}{\mu})$, in the $ASD$ variables. This contribution to the beta function occurs generically around any translational invariant background of surface operators.
The second term is due to the zero modes and depends on the relative normalization between the classical action in the reduced Eguchi-Kawai theory, $\frac{2(2 \pi)^2}{2 g_W^2} k( N-k) +c.c.$, and the real dimension of the moduli space, $2k(N-k)$, of the adjoint orbit, $g_p \lambda g_p^{-1}$, in an infinitesimal neighborhood of a surface operator with $Z_N$ holonomy and of its complex conjugate.
At the same time, in the regularization scheme of the holomorphic loop equation \cite{MB},
by rescaling the fields in canonical form the following relation is obtained between the canonical and the Wilsonian coupling constant:
\bea
\frac{1}{2 g^2_W}
= \frac{1}{2 g^2_c}+ \frac{4}{(4 \pi)^2} \log g_c + \frac{1}{(4 \pi)^2} \log Z .
\eea
Differentiating this relation it follows the canonical beta function \cite{MB} that has a structure of $NSVZ$ type and reproduces the first two universal perturbative coefficients:
\bea
\frac{\partial g_c}{\partial \log \Lambda}=\frac{-\beta_0 g_c ^3 +
\frac{1}{(4 \pi)^2} g_c ^3 \frac{\partial \log Z}{\partial \log \Lambda} }{1- \frac{4}{(4 \pi)^2} g_c^2 } .
\eea
\acknowledgments{We thank the organizers of the Lattice 2010 conference, and in particular Guido Martinelli and Giancarlo Rossi,
for the fruitful atmosphere, for the interesting discussions and for inviting us to give this talk.}
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How living and led by example. For my family, this didn’t mean living off the grid, homeschooling and cultivating organic kale. It meant prioritizing family time, freedom and self-actualization over the material trappings associated with “success”. It meant forgoing career opportunities in the big city in favor of working flexible hours in our small town. It meant canoe trips, bagged lunches and backyard games instead of amusement parks, restaurants and cable TV.
My only semblance of teenage rebellion was briefly straying from my family’s simple living values. There was a time when I would have proudly told you that making my fortune was motivating and important to me. It wasn’t until I came upon a fork in the road that I was forced to ponder these values for myself.
In my third year at Queen’s University in 2011, I was invited to meet with a partner from a multi-billion dollar private equity fund that had donated one of my scholarships. Over lunch, I learned that his firm recruited math and physics students and I was offered an internship on the spot. At the time, this seemed like the golden ticket to wealth I had craved. But something gave me pause. After mulling over the offer, I ended up choosing a different path for many reasons, ultimately turning my back on a conventional career.
I had almost forgotten about this anecdote until the past year. I finally had a small taste of the financial success I had once coveted and, admittedly, I found it addictive. For a time, I again lost touch with the value of simple living. My orientation towards triathlon also shifted. I’ve always been pragmatic about racing and the business side of the sport, but I caught myself cynically eyeing up opportunities on the basis of earnings alone.
One of the many lessons of the past year is that money can’t sustain passion and enthusiasm and it sure as hell isn’t what inspired me to pursue professional triathlon.
My Fourth Year Pro Triathlon Budget
My Third Year Pro Triathlon Budget
My Second Year Pro Triathlon Budget
My Rookie Year Pro Triathlon Budget
2018 Retrospective
2018 was the best year of my life by almost any measure. I won five races, nailed my IRONMAN debut, and made more money than the past two years combined. I also finally closed on my first house, wrapped up my overambitious home reno (well mostly) and my boyfriend moved in with me. Things were going swimmingly.
However, as much as I enjoyed the ride, that success was also a double-edged sword. For all the excitement of the past year, there were also some tough lessons which caught me off guard. In my soul-baring essay, “Growing Pains,” I wrote about my struggles with anxiety and perfectionism, exacerbated by the newfound pressure, expectation, overcommitment and scrutiny that I experienced. Over the off-season, I was almost as miserable and reclusive as my years in university. I even questioned whether the brutality of IRONMAN training had somehow rewired my brain and predisposed me to anxiety.
Earning more money was also unexpectedly stressful in its own way. First world problems, anyone? I spent more time than ever managing sponsor contracts, bringing my bookkeeping up to snuff, investing and filing tax returns in multiple countries, all by myself. I was adulting hard.
I also debated whether I should continue the annual tradition of publishing my finances following a point raised in an interview. The question was whether I would risk alienating or annoying some readers now that my income was more than double the Canadian median. An ironic fact about triathlon is that your average amateur handily out-earns the vast majority of pro triathletes (see USAT’s membership demographics). After all, triathlon is the new golf! On the other hand, some of my readers are struggling millennials or penniless athletes chasing Olympic dreams. Perspectives will vary.
Let’s also recall that I barely broke even in my rookie pro season after years as an amateur. So my average hourly rate in triathlon still makes Walmart’s compensation package look positively lavish!
Wherever you stand, my intent with these posts is to uphold my reputation for transparency, to inform and dispel misconceptions, and maybe even to inspire.
2018 Budget
Revenue, expenses and earnings before taxes (EBT) from triathlon over my five years as a professional are summarized below. Throughout this post, I’ll give dollar figures in both Canadian and US dollars (converted at the average annual exchange rate) since the majority of my revenue and expenses are in US dollars though I live in Canada.
A detailed breakdown of my 2018 triathlon budget is shown below. Like previous budgets, this one strictly includes triathlon-related revenue and expenses. Non-triathlon income and expenses are discussed later.
Revenue
Prize Money
I made more in prize money in 2018 than any previous season despite racing less often. Other than a flat tire in Texas and an off day in Monterrey, I managed to consistently execute some of my best performances. Prize money amounts below are in US dollars and include any amounts subject to foreign tax withholding*.
- 1st IRONMAN Chattanooga: $15,000*
- 1st IRONMAN Mont Tremblant: $12,000
- 1st IRONMAN 70.3 Eagleman: $6,000*
- 1st IRONMAN 70.3 Victoria: $5,000
- 1st IRONMAN 70.3 Taiwan: $3,000*
- 7th IRONMAN 70.3 Monterrey: $750
- 22nd IRONMAN 70.3 Texas: $0
- Total: US$41,750 / CA$54,100
Sponsorship
Cash sponsorship accounted for the greatest share of my income in 2018 at CA$64,000, up over 60% from 2017 and about 440% from 2016. I saw growth in both performance bonuses and base salaries. More reliable fixed income allowed me to focus on top performances at fewer races.
I’ve written a good deal about sponsorship in previous editions of my budget and I have only a little to add this year. Generally, sponsorship compensation can take several forms, listed below. Only the first two categories are directly accounted for in my budget. I also had strong returns in the equity stake I acquired as compensation from a sponsor, making it one of my most lucrative deals.
- base salary/stipend
- performance bonus
- expense allowance (e.g. covering travel)
- equity
- product
- exposure & other services
I view sponsorship with the same mentality behind a diversified investment portfolio. I aim to have contracts featuring compensation from a balanced mix of these categories. I also seek a combination of endemic (triathlon industry) and non-endemic (non-triathlon) sponsors. This balanced portfolio hedges against fluctuations in both my performance and the state of the triathlon and broader endurance sports industries.
I set a cap of 8 to 10 sponsors, which is the maximum I can realistically maintain without some unacceptable compromises. As a result, I’ve become very up front about what I can offer and what compensation I expect in return. It’s well worth my time to carefully research and vet prospective sponsors. Only about one in ten opportunities actually pan out, so I’ve learned to say no thanks and to accept rejection.
Financial compensation is only one piece of the puzzle. A respectful rapport, reasonable expectations, and a long-term commitment are more important to me than the money on the table.
That said, I generally don’t accept product-only deals. I would rather purchase my ideal products without any concessions or strings attached. I’m astounded by what some athletes—pros and age groupers alike—are willing to do for a little free product. In my opinion, this does the industry a disservice and inundates social media with low quality advertising. I do my best to disclose sponsored content and to only partner with companies that I truly respect.
I’ve continued to work with an agent, Claire Duncan. I enjoy managing the bulk of my contracts, though Claire has been very helpful with negotiations and introductions. Some doors simply aren’t open to a self-represented athlete, particularly larger corporate opportunities. It can also be challenging to protect my interests in periodic negotiations with sponsors with whom I maintain a close and friendly rapport. Claire, on the other hand, can play hardball. In many ways, an agent helps free up my mental bandwidth.
Non-Triathlon Income
For the fourth consecutive year, most of my income came from triathlon. I drew on some investment income over the first part of the year to fund the remainder of my home renovation. I also received what will likely be my last paycheck from from the environmental science consulting firm that I worked for since 2011. I’m very grateful for this engaging work that afforded me the flexibility to pursue pro triathlon.
Expenses
Here’s a breakdown of how I spent money in 2018, rounded to the nearest thousand dollars and guesstimated in some cases. Note that despite paying tax in three countries, my income tax rate was relatively low due to a large Registered Retirement Savings Plan (RRSP) contribution.
- Savings: $37,000
- Living expenses: $26,000
- Income tax: $20,000
- Triathlon expenses: $18,000
- Home renovations: $15,000
- Home furnishings: $10,000
- Entertainment & vacations: $3,000
- Miscellaneous: $2,000
- Total: ~CA$131,000 / ~US$101,000
Triathlon Expenses
Triathlon expenses appear to be higher than previous years, though in reality this increase reflected more efficient use of tax deductions and better bookkeeping. For the first time, I claimed a fraction (~10%) of qualifying home expenses since I effectively run my business out of my house.
Travel expenses were half of the previous year due to a shorter race schedule and the generosity of Giles Atkinson, one of my earliest supporters. Earlier in my career, I would scrimp on travel wherever possible. This led to some
traumatic character building experiences (avoid the $30/night hotels!). Statistics suggest that the amount I’m willing to pay to avoid a red-eye flight goes up by $100 every year. I’m getting too old for that shit!
Living Expenses
In every edition of my budget, I’ve made reference to my thriftiness and financial restraint (“pathological cheapness” in the words of my dad). Years of scrimping and saving, including returning to live with my parents for three years after university, set me up to buy my first house—an 1890s fixer upper in downtown Guelph. After a nightmarish year of delays and renovations, I finally moved in in early 2018.
Last year, I noticed a shift in my spending habits. Something about having more cash on hand and dealing with unfamiliarly large sums during renovations began to erode my financial discipline. I didn’t exactly trade my Prius for a Lambo, and I still can’t stomach buying $9 loaves of artisanal peasant bread like my parents, but my expenses definitely began to creep up. For example, online shopping for home improvement and decor became a guilty pleasure. To combat this creeping consumerism and stay true to my minimalist roots, I took small steps like challenging myself not to buy any new clothes for the entire year.
Even so, my living expenses remained well within my means. I questioned whether I should have bought a fancier house, but decided that I was more comfortable with my modest mortgage (~CA$180,000) and the flexibility and peace of mind that affords. The common practice of taking on the largest possible mortgage and living house poor because real estate is lauded as a “good investment” feels like collective insanity fueled by the promise of growth from a bygone era.
My average monthly living expenses are itemized below, excluding major planned expenditures such as home renovations, furniture and vacations.
My boyfriend of four years, James, moved in with me last year before renovations were even complete. James just graduated from the University of Guelph, therefore we had to come up with an equitable financial arrangement to address both our income disparity and the fact that our home is technically my asset. Many of my millennial friends find themselves in similar financial situations with their partners. Our arrangement is that James contributes about CA$500 per month towards shared expenses plus a portion of our grocery and internet bills. This is fair market rent for a bedroom in a student rooming house and roughly one quarter fair market rent on a home like ours. If and when James has the capital and the inclination to put money towards the mortgage or upkeep, we’ll come up with a new arrangement.
Saving & Investing
I’m not exactly passionate about investing, despite my penchant for financial planning. I grudgingly stay informed by following several blogs and subreddits. I aim for a relatively hands-off investing strategy and resist reacting to market movement.
I learned some valuable lessons about investing over the past couple years. I got my start as a teenager investing exclusively in exchange-traded funds and index funds, a proven—if a little boring—approach. After a decade of solid returns during a prolonged bull market, I decided to get fancier. In my first foray into stocks, I tripled my investment in cannabis stocks within a year, timing the market with near perfection. Emboldened by what was mostly dumb luck, I hunted for the next big thing and found cryptocurrency. You can guess how this ended up! Mercifully, I limited my “fun” risk-taking picks to a tiny fraction of my portfolio, making this a relatively cheap but effective lesson in market dynamics and principles of investing.
My current asset allocation is roughly 35% Canadian ETFs, 34% US ETFs, 30% developed and emerging international ETFs, and 1% cryptocurrency (excluding my equity in a sponsor). I don’t currently hold any fixed-income securities, which some may consider a bold or foolish move. My rationale is that I’m still in my twenties and I don’t intend to touch these savings for decades, so I have plenty of time to recoup losses. I also maintain a healthy balance outside of investments in the form of cash, accounts receivable and eye-wateringly expensive triathlon equipment, which could cover living expenses for at least a year or allow me to buy the dip in the event of a recession.
Note: The registered accounts discussed here only apply to Canadians, though other countries have similar tax shelters.
I shifted from saving mostly within my Tax Free Savings Account (TFSA) to my Registered Retirement Savings Plan (RRSP, similar to the US 401k) this year. Without getting into the weeds, it tends to be more tax efficient to contribute to a RRSP than a TFSA when your income is high relative to other years. I maxed out my RRSP contribution room which lowered my average Canadian income tax rate by about 12%, saving me over $12,000 in tax. Going forward, I’ll aim to max out my annual contributions in both of these tax-sheltered accounts.
Concluding Thoughts
There’s an adage that athletes love to cite, something to the effect of “you learn more from failure than from success.” This platitude may offer some consolation in the face of disappointment, but it’s never resonated with me. 2018 was the most successful year of my life, yet it also carried some of the most humbling lessons and greatest personal and professional development. I was forced to contemplate my own definition of success. That’s still a work in progress, but I’m increasingly convinced that the success I care about most can’t be measured in dollars or wins or followers or magazine covers. I’m reacquainting myself with my motivations in triathlon, beyond the unpredictable external gratification and even more fickle financial rewards.
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We all loved the duo of Samwise Gamgee and Frodo Baggins from Lord of the Rings. After all, they single-handedly saved Middle Earth from the evils of Sauron. But, it had been a long time since we heard his beautiful speech from The Two Towers. So, Sean Astin decided to give the fans a trip down memory lane.
Sean Astin re-reads the amazing Two Towers Speech
Sam’s speech from The Two Towers was a very hopeful speech. And he delivered again, this time amid the Coronavirus quarantine, to give his fans a brief moment of nostalgia and happiness. Here’s Sean Astin reading the speech out loud on Twitter.
By popular demand! More @SeanAstin reading from #LOTR. Sam’s Osgiliath speech from The Two Towers. #JRRTolkien pic.twitter.com/SbfLBKV7tm
— Linda Iroff (@LindaofNote) April 11, 2020
It’s interesting to think that the speech wasn’t a part of the original script. Astin confirmed that Peter Jackson later added it as reassuring words for viewers after the 9/11 attacks in New York City.
The speech is everything we want to hear right now
Sam’s amazing Two Towers speech to Frodo Baggins is perfect for today’s circumstances too. The speech is as follows: the world, Mr. Frodo. And it’s worth fighting for.
This Two Towers speech is something fans still hold close to their heart. And they let Sean Astin know in the replies too, for instance:
Not gonna lie, shed a tear listening to that. Eighteen years, and, unshockingly, he's still the same old Samwise!
— Caleb (@JCBagginsTTOA) April 12, 2020
Is it weird this popped up on my feed just as I was watching the scene? pic.twitter.com/87apaSAzFv
— enjoying life … (@inlovewithfx) April 12, 2020
It’s not easy to live in quarantine for months, in fear of an invisible disease. But, these words from Sean Astin can go a long way in comforting people in this time of need.
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\begin{document}
\title[LG-systems and $\mathcal{D}^b(X)$ of toric Fano manifolds with small $\rho(X)$]{On Landau-Ginzburg systems and $\mathcal{D}^b(X)$ of various toric Fano manifolds with small picard group}
\address{School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 6997801, Israel}
\email{yochayjerby@post.tau.ac.il}
\date{\today}
\author{Yochay Jerby}
\begin{abstract} For a toric Fano manifold $X$ denote by $Crit(X) \subset (\mathbb{C}^{\ast})^n$ the solution scheme of the Landau-Ginzburg system of equations of $X$. Examples of toric Fano manifolds with $rk(Pic(X)) \leq 3$ which admit full strongly exceptional collections of line bundles were recently found by various authors. For these examples we construct a map $E : Crit(X) \rightarrow Pic(X)$ whose image $\mathcal{E}=\left \{ E(z) \vert z \in Crit(X) \right \}$ is a full strongly exceptional collection satisfying the M-aligned property. That is, under this map, the groups $Hom(E(z),E(w))$ for $z,w \in Crit(X)$ are naturally related to the structure of the monodromy group acting on $Crit(X)$.
\end{abstract}
\maketitle
\section{Introduction and Summary of Main Results}
\label{s:intro}
\hspace{-0.6cm} Let $X$ be a smooth algebraic manifold and let $\mathcal{D}^b(X)$ be the bounded derived category of coherent sheaves on $X$, see \cite{GM,T}. A fundamental question in the study of $\mathcal{D}^b(X)$ is the question of existence of full strongly exceptional collections $\mathcal{E} = \left \{ E_1,...,E_N \right \} \subset \mathcal{D}^b(X)$. Such collections satisfy the property that the adjoint functors $$\begin{array}{ccc} R Hom_X(T, -) : \mathcal{D}^b(X) \rightarrow \mathcal{D}^b(A_\mathcal{E}) & ; & - \otimes^L_{A_{\mathcal{E}}} T : \mathcal{D}^b(A_{\mathcal{E}}) \rightarrow \mathcal{D}^b(X) \end{array} $$ are equivalences of categories where $T:=\bigoplus_{i=1}^NE_i$ and $A_{\mathcal{E}}=End(T)$ is the corresponding endomorphism ring. The first example of such a collection is $$\mathcal{E} = \left \{ \mathcal{O}, \mathcal{O}(1),...,\mathcal{O}(s) \right \} \subset Pic(\mathbb{P}^s)$$ found by Beilinson in \cite{B}. When $X$ is a toric manifold one further asks the more refined question of whether $\mathcal{D}^b(X)$ admits an exceptional collection whose elements are line bundles $\mathcal{E} \subset Pic(X)$, rather than general elements of $\mathcal{D}^b(X)$?
\hspace{-0.6cm} Let $X$ be a $s$-dimensional toric Fano manifold given by a Fano polytope $\Delta$ and and let $\Delta^{\circ}$ be the polar polytope of $\Delta$. Let $f_X = \sum_{n \in \Delta^{\circ} \cap \mathbb{Z}^n} z^n \in \mathbb{C}[z_1^{\pm},...,z_s^{\pm}]$ be the Landau-Ginzburg potential associated to $X$, see \cite{Ba,FOOO,OT}. Recall that the Landau-Ginzburg system of equations is given by $$ z_i \frac{\partial}{\partial z_i } f_X(z_1,...,z_s)=0 \hspace{0.5cm} \textrm{ for } \hspace{0.25cm} i=1,...,s $$ and denote by $Crit(X) \subset (\mathbb{C}^{\ast})^s$ the corresponding solution scheme.
\hspace{-0.6cm} In \cite{J} we defined an exceptional map to be a map of the form
$E : Crit(X) \rightarrow Pic(X)$ such that $\mathcal{E}_E(X) := E(Crit(X)) \subset Pic(X)$ is a full strongly exceptional collection of line bundles and introduced natural exceptional maps for the five Del-Pezzo surfaces and various three dimensional toric Fano manifolds. Our main observation in \cite{J} was that, in the considered cases, the defined exceptional maps exhibit a non-trivial property to which we refer as the M-aligned property. In short, M-aligned means that the algebraic structure of the spaces $Hom(E(z),E(w))$ for $z,w \in Crit(X)$ could be further realted to the structure of the geometric monodromy group acting on $Crit(X)$.
\hspace{-0.6cm} In general, not much is known on the question of which toric Fano manifolds admit full strongly exceptional collections of line bundles in $Pic(X)$. And, with full generality, the question seems out of reach with current techniques. However, full strongly exceptional collections for specific classes of toric Fano manifolds with small $\rho(X) = rank(Pic(X))$ were recently found by various authors, see \cite{CMR3,CMR4,DLM}. Our aim in this work is to introduce exceptional maps $E : Crit(X) \rightarrow Pic(X) $ for these classes and study the $M$-aligned property for these maps.
\hspace{-0.6cm} Defining a map of the form $E: Crit(X) \rightarrow Pic(X)$ requires the association of integral invariants to the elements of the solution scheme $Crit(X)$. The Landau-Ginzburg potential $f_X$ is an element of the space $$ L(\Delta^{\circ}) : = \left \{ \sum_{n \in \Delta^{\circ} \cap \mathbb{Z}^n} u_n z^n \vert u_n \in \mathbb{C}^{\ast} \right \} \subset \mathbb{C}[z_1^{\pm},...,z_n^{\pm}]$$ Similarly, for $f_u(z) = \sum_n u_n z^n \in L(\Delta^{\circ})$ define the corresponding Landau-Ginzburg system of equations $$ z_i \frac{\partial}{\partial z_i } f_u(z_1,...,z_n) =0 \hspace{0.5cm} \textrm{ for } \hspace{0.25cm} i=1,...,n $$ and denote by $Crit(X; f_u ) \subset (\mathbb{C}^{\ast})^n$ the solution scheme. Our first main observation, is that the arguments of the solution scheme $Crit(X; f_u)$ converge to roots of unity as $Log \vert u \vert \rightarrow \pm \infty$. We view these collections of roots of unity as giving an "asymptotic" generalization of the roots of unity arising directly in the case of projective space.
\hspace{-0.6cm} Denote by $ \Delta(k)$ the set of $k$-dimensional facets of $\Delta$. Note that $ \Delta( n-1) \simeq \Delta^{\circ}(0)$ and denote by $F_n \in \Delta(n-1)$ the facet corresponding to $n \in \Delta^{\circ} (0)$. Denote by $V_X(F_n)$ the $T$-orbit closure corresponding to the facet $F_n$. In particular, for a solution $z \in Crit(X ; f_u) $ define the $T$-invariant $\mathbb{R}$-divisor $$ D(z) := \sum_{n \in \Delta^{\circ} \cap \mathbb{Z}^n} Arg(z^n) \cdot V_X(F_n) \in Div_T(X) \otimes \mathbb{R} $$ In the cases considered, $Pic(X)$ admits a natural set of generators. In particular, denote by $\left [D(z) \right ] \in Pic(X)$ the integral part of $D(z)$ via the map $Div_T(X) \otimes \mathbb{R} \rightarrow Pic(X) \otimes \mathbb{R}$. In the first part of the paper we show:
\bigskip
\hspace{-0.6cm} \bf Theorem A: \rm Let $X$ be a toric Fano manifold of one of the following classes:
\bigskip
(a) $X = \mathbb{P}(\mathcal{O}_{\mathbb{P}^s}\oplus \left (\oplus_{i=1}^r\mathcal{O}_{\mathbb{P}^s}(a_i) \right ) )$, a projective bundle with $\sum_{i=1}^r a_i \leq s$.
\bigskip
(b) $X = Bl_B (\mathbb{P}^{n-r} \times \mathbb{P}^r)$, where $B=\mathbb{P}^{n-r-1} \times \mathbb{P}^{r-1}$.
\bigskip
(c) $X=Bl_B(\mathbb{P}(\mathcal{O}_{\mathbb{P}^{n-1}} \oplus \mathcal{O}_{\mathbb{P}^{n-1}}(b)))$, where $B=\mathbb{P}^{n-2}$ and $b<n-1$.
\bigskip
\hspace{-0.6cm} Then $X$ admits an exceptional map $E : Crit(X ; f_t) \rightarrow Pic(X)$ given by $z \mapsto \left [ D(z) \right ]$ for $0<<t$ big enough.
\bigskip
\hspace{-0.6cm} Note that for (a) one has $\rho(X)=rk(Pic(X)) =2$ while for (b) and (c) one has $\rho(X) =3$. In fact, due to Kleindschmit's classification theorem any toric manifold with $\rho(X)=2$ is of class (a), see \cite{K}. The exceptional collections arising for (a) coincide with those found by Costa and Miro-Roig in \cite{CMR3}. The exceptional collections arsing for classes (b) and (c) coincide with those found by Costa and Miro-Roig in \cite{CMR4}
and by Lason, Michalek and Dey in \cite{DLM}, respectively. Both of which are special cases of a more general construction for a wider class of toric manifolds with $\rho(X) =3$ due to \cite{LM}. Let us illustrate the above with the following example:
\bigskip
\hspace{-0.6cm} \bf Example \rm (projective line): For $X= \mathbb{P}^s$ the Landau-Ginzburg potential is given by $f(z_1,...,z_s)=z_1+...+z_s +\frac{1}{z_1 \cdot ... \cdot z_s}$ and the corresponding system
of equations is $$ z_i \frac{\partial}{\partial z_i } f_X(z_1,...,z_s)=z_i - \frac{1}{z_1 \cdot ... \cdot z_s} =0 \hspace{0.5cm} \textrm{ for } \hspace{0.25cm} i=1,...,s $$ The solution scheme $Crit(\mathbb{P}^s) \subset (\mathbb{C}^{\ast})^s$ is given by
$z_k= ( e^{\frac{2 \pi ki}{s+1}},...,e^{\frac{2 \pi k i}{s+1}}) $ for $k=0,...,s$, via direct computation. We have $Div_T(X) = \bigoplus_{i=1}^r V_X(F_i) \cdot \mathbb{Z}$ where $V_X(F_i)$ is the projective hyperplane defined by $z_i=0$. The exceptional map is hence given by $$E(z_k) = \left [\frac{k}{s+1} \cdot V_X(F_1) + ... + \frac{k}{s+1} \cdot V_X(F_s) \right ] = k \cdot H$$ where $H \in Pic(X)$ is the positive generator.
\bigskip
\hspace{-0.6cm} A full strongly exceptional collection of line bundles $\mathcal{E} \subset Pic(X) $ on a toric manifold $X$ is associated with a quiver $Q_{\mathcal{E}}$ whose vertices are the elements of $\mathcal{E}$ and edges connecting two elements $E_1,E_2 \in \mathcal{E}$ correspond to the $T$-invariant divisors appearing in the natural splitting $$ Hom(E_1,E_2) \simeq \bigoplus_{D \in Div_T(X)} Hom_D(E_1,E_2)$$ The main feature of the exceptional maps, defined in \cite{J}, is the $M$-aligned property, which relates the quivers $Q_{\mathcal{E}}$ to monodromies acting on the solution scheme $Crit(X; f_t)$. The second part of the paper is devoted for the description of the $M$-aligned for the exceptional maps introduced for the classes (a)-(c) in Theorem A. For instance, for a projective bundle $X = \mathbb{P}( \mathcal{O} \oplus ( \oplus_{i=1}^r \mathcal{O}(a_i)))$ the corresponding potential function is given by $$ f_t(z,w) :=1+ \sum_{i=1}^s z_i + \sum_{i=1}^r w_i + e^{t} \cdot \frac{w_1^{a_1} \cdot ... \cdot w_r^{a_r}}{z_1 \cdot ... \cdot z_s} + \frac{1}{w_1 \cdot ... \cdot w_r}$$ for a $T$-divisor $$D:= \sum_{i=1}^s n_i \cdot V_X(e_i) + \sum_{i=s+1}^{s+r} m_i \cdot V_X(e_i)+ n_0 \cdot V_X \left ( \sum_{i=s+1}^r a_i e_i - \sum_{i=1}^s e_i \right) + m_0 \cdot V_X \left (-\sum_{i=s+1}^r e_i \right ) $$ we associate following loop of Laurent polynomials in $L(\Delta^{\circ})$: $$ \gamma_D^{\theta}(z,w) :=1+ \sum_{i=1}^s e^{2 \pi i n_i \theta} z_i + \sum_{i=1}^r e^{2 \pi i m_i \theta} w_i + e^{t+2 \pi i n_0 \theta} \cdot \frac{w_1^{a_1} \cdot ... \cdot w_r^{a_r}}{z_1 \cdot ... \cdot z_s} +e^{2 \pi i m_0 \theta} \frac{1}{w_1 \cdot ... \cdot w_r}$$ for $ \theta \in [0,1]$. Note that $\gamma^0_D = \gamma^1_D$. In particular, following the solution schemes $Crit(X ; \gamma_D^{\theta})$ along $\theta \in [0,1]$ one gets a map $M_D : Crit(X ; f_t) \rightarrow Crit(X ; f_t)$, to which we refer as the monodromy map corresponding to the divisor $D$. We show:
\bigskip
\hspace{-0.6cm} \bf Theorem B \rm ($M$-aligned property for projective bundles): Let $z_1,z_2 \in Crit(X ; f_t)$ be two elements and $D \in Div_T(X)$ be a $T$-divisor. Then $$ Hom_D(E(z_1),E(z_2)) \neq 0 \Rightarrow M_D(z_1) = z_2$$ where $E : Crit(X ; f_t) \rightarrow Pic(X)$ is the exceptional map of Theorem A.
\bigskip
\hspace{-0.6cm} We refer to this property as $M$-algined to indicate that the $Hom$-spaces between elements of the collection $\mathcal{E}$ are algined with the corresponding monodromy actions on $Crit(X ; f_t)$. In section 4 we prove Theorem B and further consider the $M$-aligned property for the classes (b) and (c) of Theorem A, as well.
\bigskip
\hspace{-0.6cm} The rest of the work is organized as follows: In section 2 we recall relevant facts on toric Fano manifolds and their derived categories of coherent sheaves and recall the definition of the exceptional collections for the classes (a)-(c). In section 3 we study the Landau-Ginzburg system for classes (a)-(c) and define the exceptional map. In section 4 we show the M-aligned property. In section 5 we discuss concluding remarks and relations to further topics of mirror symmetry.
\section{Relevant Facts on Toric Fano Manifolds}
\label{s:Rfotfm}
\hspace{-0.6cm} Let $N \simeq \mathbb{Z}^n$ be a lattice and let $M = N^{\vee}=Hom(N, \mathbb{Z})$ be the dual lattice. Denote by $N_{\mathbb{R}} = N \otimes \mathbb{R}$ and $M_{\mathbb{R}}=M \otimes \mathbb{R}$ the corresponding vector spaces. Let $ \Delta \subset M_{\mathbb{R}}$ be an integral polytope and let $$ \Delta^{\circ} = \left \{ n \mid (m,n) \geq -1 \textrm{ for every } m \in \Delta \right \} \subset N_{\mathbb{R}}$$ be the \emph{polar} polytope of $\Delta$. The polytope $\Delta \subset M_{\mathbb{R}}$ is said to be \emph{reflexive} if $0 \in \Delta$ and $\Delta^{\circ} \subset N_{\mathbb{R}}$ is integral. A reflexive polytope $\Delta$ is said to be \emph{Fano} if every facet of $\Delta^{\circ}$ is the convex hall of a basis of $M$.
\hspace{-0.6cm} To an integral polytope $\Delta \subset M_{\mathbb{R}}$ associate the space $$ L(\Delta) = \bigoplus_{m \in \Delta \cap M} \mathbb{C} m $$ of Laurent polynomials whose Newton polytope is $\Delta$. Denote by $i_{\Delta} : (\mathbb{C}^{\ast})^n \rightarrow \mathbb{P}(L(\Delta)^{\vee})$ the embedding given by $ z \mapsto [z^m \mid m \in \Delta \cap M] $. The \emph{toric variety} $X_{\Delta} \subset \mathbb{P}(L(\Delta)^{\vee})$ corresponding to the polytope $\Delta \subset M_{\mathbb{R}}$ is defined
to be the compactification of the image $i_{\Delta}((\mathbb{C}^{\ast})^n) \subset \mathbb{P}(L(\Delta)^{\vee})$. A toric variety $X_{\Delta}$ is said to be Fano if its anticanonical class $-K_X$ is Cartier and ample. In \cite{Ba2} Batyrev shows that $X_{\Delta}$ is a Fano variety if $\Delta$ is reflexive and, in this case, the embedding
$i_{\Delta}$ is the anti-canonical embedding. The Fano variety $X_{\Delta}$ is smooth if and only if $\Delta^{\circ}$ is a Fano polytope.
\hspace{-0.6cm} Denote by $\Delta(k)$ the set of $k$-dimensional faces of $\Delta$ and denote by $ V_X(F) \subset X$ the orbit closure of the orbit corresponding to the facet $F \in \Delta(k)$ in $X$, see \cite{F,O}. In particular, consider the group of toric divisors $$ Div_T(X) := \bigoplus_{F \in \Delta(n-1)} \mathbb{Z} \cdot V_X(F)$$ Assuming $X$ is smooth the group $Pic(X)$ is described in terms of the short exact sequence $$ 0 \rightarrow M \rightarrow
Div_T(X) \rightarrow Pic(X) \rightarrow 0 $$ where the map on the left hand side is given by $ m \rightarrow \sum_F \left < m, n_F \right > \cdot V_X(F) $ where $n_F \in \mathbb{N}_{\mathbb{R}}$ is the unit normal to the hyperplane spanned by the facet $F \in \Delta(n-1)$. In particular, note that $$ \rho(X)= rank \left ( Pic (X) \right ) = \vert \Delta(n-1) \vert -n $$ Moreover, when $\Delta$ is reflexive one has $\Delta^{\circ}(0)=\left \{ n_F \vert F \in \Delta(n-1) \right \} \subset N_{\mathbb{R}}$. We thus sometimes denote $V_X(n_F)$ for the $T$-invariant divisor $V_X(F)$. We denote by $Div_T^+(X) $ the semi-group of all toric divisors $ \sum_{F} m_F \cdot V_X(F)$ with $0 \leq m_F$ for any $F \in \Delta(n-1)$.
\hspace{-0.6cm} Let $X$ be a smooth projective variety and let $\mathcal{D}^b(X)$ be the derived category of bounded complexes of
coherent sheaves of $\mathcal{O}_X$-modules, see \cite{GM,T}. For a finite dimensional algebra $A$ denote by $\mathcal{D}^b(A)$ the derived category of bounded complexes of finite dimensional right modules over $A$. Given an object $T \in \mathcal{D}^b(X)$ denote by $A_T=Hom(T,T)$ the corresponding endomorphism algebra.
\bigskip
\hspace{-0.6cm} \bf Definition 2.1: \rm An object $T \in \mathcal{D}^b(X)$ is called a \emph{tilting object} if the corresponding adjoint functors $$\begin{array}{ccc} R Hom_X(T, -) : \mathcal{D}^b(X) \rightarrow \mathcal{D}^b(A_T) & ; & - \otimes^L_{A_T} T : \mathcal{D}^b(A_T) \rightarrow \mathcal{D}^b(X) \end{array} $$ are equivalences of categories. A locally free tilting object is called a tilting bundle.
\bigskip
\hspace{-0.6cm} An object $ E \in \mathcal{D}^b(X)$ is said to be \emph{exceptional} if $Hom(E,E)=\mathbb{C}$ and $Ext^i(E,E)=0$ for $0<i$. We have:
\bigskip
\hspace{-0.6cm} \bf Definition 2.2: \rm An ordered collection $ \mathcal{E} = \left \{ E_1,...,E_N \right \} \subset \mathcal{D}^b(X)$ is said to be an \emph{exceptional collection} if each $E_j$ is exceptional and $Ext^i(E_k,E_j) =0$ for $j<k \textrm{ and } 0 \leq i $. An exceptional collection is said to be \emph{strongly exceptional} if also $Ext^i(E_j,E_k)=0$ for $j \leq k$ and $0<i$. A strongly exceptional collection is called \emph{full} if its elements generate $\mathcal{D}^b(X)$ as a triangulated category.
\bigskip
\hspace{-0.6cm} The importance of full strongly exceptional collections in tilting theory is due to the following properties, see \cite{Bo,K}:
\bigskip
- If $ \mathcal{E}$ is a full strongly exceptional collection then $T = \bigoplus_{i=1}^N E_i$ is a tilting object.
\bigskip
- If $T = \bigoplus_{i=1}^N E_i$ is a tilting object and $ \mathcal{E} \subset Pic(X) $ then $\mathcal{E}$ can be ordered as a full
\hspace{0.2cm} strongly exceptional collection of line bundles.
\bigskip
\hspace{-0.6cm} Let us note the following examples of toric Fano manifolds, which would be considered in the continuation:
\bigskip
\hspace{-0.6cm} \bf Example 2.3 \rm (Projective bundles): By a result of Kleinschmidt's \cite{Kl} the class of toric manifolds with $\rho(X) = rk(Pic(X))=2$ consists of the projective bundles $$ X_a=\mathbb{P} \left (\mathcal{O}_{\mathbb{P}^s} \oplus \bigoplus_{i=1}^r \mathcal{O}_{\mathbb{P}^s}(a_i) \right ) \hspace{0.5cm} \textrm{ with } \hspace{0.25cm} 0 \leq a_1 \leq... \leq a_r$$ see also \cite{CoLS}. Set $a_0 =0$. Consider the lattice $N = \mathbb{Z}^{s+r}$ and let $v_1,...,v_s$ be the standard basis elements of $\mathbb{Z}^s$ and $e_1,...,e_r$ be the standard basis elements of $\mathbb{Z}^r$. Set $v_0 =- \sum_{i=1}^s u_i+ \sum_{i=1}^s a_i e_i $ and $ e_0 =- \sum_{i=1}^r e_i$. Let $\Delta_a^{\circ} \subset N_{\mathbb{R}}$ be the polytope whose vertex set is given by $$\Delta_a^{\circ}(0) = \left \{ v_0,...,v_s,e_0,...,e_r \right \}$$ It is straightforward to verify that $\Delta^{\circ}_a$ is a Fano polytope if and only if $\sum_{i=1}^r a_i \leq s$. In particular, in this case $X_a \simeq X_{\Delta_a}$, see \cite{CoLS}. One has $$ Pic(X_a) = \xi \cdot \mathbb{Z} \oplus \pi^{\ast}H \mathbb{Z}$$ where $\xi$ is the class of the tautological bundle and $\pi^{\ast}H $ is the pullback of the generator $H$ of $Pic(\mathbb{P}^s) \simeq H \cdot \mathbb{Z}$ under the projection $\pi : X_a \rightarrow \mathbb{P}^s$. Note that the following holds $$ \begin{array}{ccc} [V_X(v_i)] = \pi^{\ast}H & ; & [V_X(e_j)]=\xi - a_i \cdot \pi^{\ast} H \end{array}$$ for $0 \leq i \leq s $ and $ 0 \leq j \leq r$. In \cite{CMR3} Costa and Mir$\acute{\textrm{o}}$-Roig show that the collection of line bundles $\mathcal{E} = \left \{E_{kl} \right \}_{k=0,l=0}^{s,r} \subset Pic(X)$ where $$ E_{kl} := k \cdot \pi^{\ast} H + l \cdot \xi \hspace{0.5cm} \textrm{ for } \hspace{0.25cm} 0 \leq k \leq s \hspace{0.1cm} , \hspace{0.1cm} 0 \leq l \leq r $$ admits the structure of a full strongly exceptional collection.
\bigskip
\hspace{-0.6cm} \bf Example 2.4: \rm For $r \leq n $ let $ X_{n,r}=Bl_B(\mathbb{P}^{n-r} \times \mathbb{P}^r)$ be the blow up of $\mathbb{P}^{n-r} \times \mathbb{P}^r$ along the multi-linear codimension two subspace $B = \mathbb{P}^{n-r-1} \times \mathbb{P}^{r-1}$. Let $e_1,...,e_n$ be the standard basis of $M_{\mathbb{R}}$. The vertices of the polar polytope are given by $ \Delta^{\circ}(0) = \left \{ e_1,...,e_n, v_1 , v_2, v_3 \right \}$ where $$ \begin{array}{ccccc} v_1 = -\sum_{i=1}^{n-r} e_i & ; & v_2 = -\sum_{i=n-r+1}^n e_i & ; & v_3 = -\sum_{i=1}^n e_i \end{array} $$ In particular, $$ Pic(X_{n,r}) = U \cdot \mathbb{Z} \oplus V \cdot \mathbb{Z} \oplus E \cdot \mathbb{Z}$$ with $$\begin{array}{ccccc} U:=[V_X(e_i)]=[ V_X(v_1) + V_X(v_3) ] & ; & V:=[V_X(e_j)]=[V_X(v_2) +V_X(v_3)] & ; & E: = [V_X(v_3)] \end{array} $$ where $1 \leq i \leq n-r$ and $n-r+1 \leq j \leq n$. Set $$ E_{kl}:= kU+lV \hspace{0.5cm} \textrm{ for } \hspace{0.25cm} 0 \leq k \leq n-r \textrm{ , } 0 \leq l \leq r$$ and $$ F_{m,n} :=mU +n V - E \hspace{0.5cm} \textrm{ for } \hspace{0.25cm} 1 \leq m \leq n-r \textrm{ , } 1 \leq n \leq r$$ In \cite{CMR4} Costa and Miro-Roig show that the following collection $$ \mathcal{E}_{n,r} = \left \{E_{kl} \right \}_{k,l=0}^{n-r,r} \cup \left \{ F_{m,n} \right \}_{m,n=1}^{n-r,r} \subset Pic(X_{n,r})$$ admits the structure of a full strongly exceptional collection.
\bigskip
\hspace{-0.6cm} \bf Example 2.5: \rm For $b < n-1$ let $X_{n,b} = Bl_B ( \mathbb{P}( \mathcal{O}_{\mathbb{P}^{n-1}} \oplus \mathcal{O}_{\mathbb{P}^{n-1}}(b)))$ the blow-up of the projective bundle $ \mathbb{P}( \mathcal{O}_{\mathbb{P}^{n-1}} \oplus \mathcal{O}_{\mathbb{P}^{n-1}}(b))$ along $B \simeq \mathbb{P}^{n-2}$. The vertices of the polar polytope are given by $ \Delta^{\circ}(0) = \left \{ e_1,...,e_n, u_1 , u_2, u_3 \right \}$ where $$ \begin{array}{ccccc} u_1 = -e_n & ; & u_2 = - \sum_{i=1}^{n-1} e_i- be_n & ; & u_3 = -\sum_{i=1}^{n-1} e_i - (b+1)e_n \end{array} $$ We have $$ Pic(X) \simeq V \cdot \mathbb{Z} \oplus Y \cdot \mathbb{Z} \oplus T \cdot \mathbb{Z}$$ where $V=[V_X(e_i)]$ for $i=1,...,n-1$, $Y=[V_X(u_3)]$ and $T=[V_X(u_1)]$. For $0 \leq k \leq n-1$, $ 0 \leq l \leq 1 $ and $1 \leq m \leq n-1$ set $$ \begin{array}{ccc} E_{kl} :=k V + l(Y+T+bV) & ; & F_m := (m+b) V + T \end{array} $$ Michalek, Lason and Dey show in \cite{DLM} that $$ \mathcal{E}_{n,b} := \left \{E_{kl} \right \}_{k=0,l=0}^{n-1,1} \cup \left \{F_m \right \}_{m=1}^{n-1} \subset Pic(X) $$ admits the structure of a full strongly exceptional collection.
\bigskip
\hspace{-0.6cm} Let us conclude this section by recalling a few facts about the Frobenius toric map and Frobenius splitting:
\bigskip
\hspace{-0.6cm} \bf Remark 2.6 \rm (Exceptional collections and Frobenius splitting): Let $ l \in \mathbb{N}$ be an integer and let $F_l : (\mathbb{C}^{\ast})^s \rightarrow (\mathbb{C}^{\ast})^s$ be the $l$-th power map given by $(x_1,...,x_s) \mapsto (x_1^l,...,x_s^l)$. The extension of this map to the whole toric manifold $X$, which we also denote by $F_l : X \rightarrow X $, is called the $l$-th toric Frobenius mapping. In \cite{T} Thomoson described the following formula for the Frobenius splitting of the trivial bundle $$ (F_l)_{\ast} \mathcal{O} = \bigoplus_{D \in Pic(X)} \mathcal{O}(D)^{m(D)} $$ where $m(D)$ is the number of points in the cube $Div_T(X)/l \cdot Div_T(X)$ representing the class $-lD \in Pic(X)$, see also \cite{A}. Let $\widetilde{F}_l : Div_T(X)/ l \cdot Div_T(X) \rightarrow Pic(X)$ be the function given by $ D \mapsto [D/l]$. In particular, in the limit as $l \rightarrow \infty$ we get a map $\widetilde{F} : \mathbb{T}^{r} \rightarrow Pic(X)$, where $r = rk(Div_T(X))$, see \cite{B}. Let $\mathcal{B}_X \subset Pic(X)$ be the image of the map $\widetilde{F}$. The exceptional collection of Example 2.3-4 satisfy $\mathcal{E}_X = \mathcal{B}_X$ while the exceptional collection of Example 4.5 satisfies $\mathcal{E}_X \subset \mathcal{B}_X$.
\hspace{-0.6cm} Let $\mathcal{B}_X \subset Pic(X)$ be the image set of the map $\widetilde{F}$. In general, $\chi(X) \leq \vert \mathcal{B}_X \vert$. Hence, the number of elements in $\mathcal{B}_X$ exceeds the expected number of elements of an exceptional collection. On the one hand, many results from recent years seem to indicate an ilusive relationship between the set $\mathcal{B}_X$ and full strongly exceptional collections of line bundles, see \cite{Bo3,CMR2,CMR4,LM}. On the other hand, Michalek, Lason and Dey found examples of toric Fano manifolds for which the set $\mathcal{B}_X$ admits no subset which is a full strongly exceptional collection, see \cite{LM}. An explanation for the distinction between the two cases is yet unclear.
\section{Variations of the LG-system and exceptional maps}
\label{s:LGM}
\hspace{-0.6cm} Let $X$ be a $n$-dimensional toric Fano manifold given by a Fano polytope $\Delta \subset M_{\mathbb{R}}$ and let $\Delta^{\circ} \subset N_{\mathbb{R}}$ be the corresponding polar polytope. Set $$L(\Delta^{\circ}):= \left \{ \sum_{n \in \Delta^{\circ} \cap \mathbb{Z}^n} u_n z^n \vert u_n \in \mathbb{C}^{\ast} \right \} \subset \mathbb{C}[z_1^{\pm},...,z_n^{\pm}]$$ We refer to $$ z_i \frac{\partial}{\partial z_i } f_u(z_1,...,z_n)=0 \hspace{0.5cm} \textrm{ for } \hspace{0.25cm} i=1,...,n $$ as the LG-system of equations associated to an element $ f_u(z) = \sum_{n \in \Delta^{\circ} \cap \mathbb{Z}^n} u_n z^n $ and denote by $Crit(X; f_u) \subset (\mathbb{C}^{\ast})^n$ the corresponding solution scheme. We refer to the element $f_X(z) = \sum_{n \in \Delta^{\circ} \cap \mathbb{Z}^n} z^n$ as the LG-potential of $X$.
\bigskip
\hspace{-0.6cm} \bf Definition 3.1: \rm A map $E : Crit(X; f_u) \rightarrow Pic(X)$ is an exceptional map if the image $E(Crit(X; f_u)) \subset Pic(X)$ is a full strongly exceptional collection.
\bigskip
\hspace{-0.6cm} Our main observation in this section is that roots of unity arise when considering the asymptotics of $Crit(X ; f_u)$ as $Log \vert u \vert \rightarrow \pm \infty$. We view these roots of unity as a generalization of the roots of unity arising in $Crit(X)$ in the case of projective space $X= \mathbb{P}^n$.
\hspace{-0.6cm} Let $Arg: (\mathbb{C}^{\ast})^n \rightarrow \mathbb{T}^n$ be the map given by $ (r_1 e^{2 \pi i \theta_1},..., r_n e^{2 \pi i \theta_n}) \mapsto (\theta_1 ,...,\theta_n )$. For any $z \in Crit(X ; f_u) $ define the $\mathbb{R}$-divisor $$ D_u(z) := \sum_{n \in \Delta^{\circ}(0)} Arg(z^n) \cdot V_X(F_n) \in Div_T(X) \otimes \mathbb{R}$$ Assume $L_1,...,L_{\rho} \in Pic(X)$ is a given basis. Let $D = \sum_n a_n \cdot V_X(F_n) \in Div_T(X) \otimes \mathbb{R}$ be an $\mathbb{R}$-divisor and denote by $ [D] := \sum_{i=1}^{\rho} b_i \cdot L_i$ the corresponding element of $Pic(X) \otimes \mathbb{R}$. We denote by $$ [D]_{\mathbb{Z}} := \sum_{i=1}^{\rho} [b_i]_{\mathbb{Z}} \cdot L_i \in Pic(X)$$ the line bundle obtained by replacing the coefficients of $[D]$ with their integer part. Define the map $E_u : Crit(X ; f_u) \rightarrow Pic(X)$ given by $z \mapsto [D_u(z)]_{\mathbb{Z}} \in Pic(X)$. In the continuation of this section we show how maps of the form $E_u$ give rise to exceptional maps for the toric Fano manifolds of examples 2.3-2.5 ((a)-(c) of the introduction).
\bigskip
\hspace{-0.6cm} \bf Remark 3.1 \rm (Geometric viewpoint): Let $\left \{f_u \right \}_{u \in \mathbb{C}^{\ast}} \subset L(\Delta^{\circ})$ be a $1$-parametric family of Laurent polynomials. Consider the Riemann surface $$ C: = \left \{ \left ( z ,u \right ) \vert z \in Crit(X; f_u) \right \} \subset (\mathbb{C}^{\ast})^n \times \mathbb{C}^{\ast}$$ Denote by $\pi : C \rightarrow \mathbb{C}^{\ast}$ the projection on the second factor which expresses $C$ as an algebraic fibration over $\mathbb{C}^{\ast}$ of rank $N=\chi(X)$. Denote by $C(u) = \pi^{-1}(u)$ for $u \in \mathbb{C}^{\ast}$. A graphic illustration of $C$ together with the curves $C(t)$ for $0 \leq t$ for the Hirzebruch surface $X= \mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(1))$ is as follows:
\begin{center}
$$ \begin{tikzpicture}[scale=.25]
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(-5,3) .. controls +(-60:1) and +(-120:2) .. (6,3)
(8,3) ellipse (2cm and 1cm)
(10,3) .. controls +(-60:1) and +(-120:1) .. (2,-10)
(-0,-10) ellipse (2cm and 1cm)
(-2,-10).. controls +(-60:1) and +(-120:1) .. (-9,3);
\node (a) at (5,-3) {};
\node (b) at (-4,-4.5){};
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\node (d) at (4.5,-1){} ;
\fill[black] (a) circle (10pt);
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(a) edge [->,bend right=10,line width=1.05pt] (a1)
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(d) edge [->,bend left=10,line width=1.05pt](c1);
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$$
\end{center}
\hspace{-0.6cm} An amusing analogy can be drawn between the resulting dynamics and the cue game of "pool". Indeed, consider the Riemann surface $C$ as a "pool table", the cusps of the surface as the "pockets", and the set $C(0) \simeq Crit(X)$ as an initial set of "balls". In this analogy the dynamics of $C(t)$ describes the path in which the balls approach the various "pockets" of the table as $ t \rightarrow \pm \infty $.
\bigskip
\subsection{Exceptional map for projective bundles} For the projective bundle $X_a$ the Landau-Ginzburg potential is given by $$ f(z,w)=1+\sum_{i=1}^s z_i+\sum_{i=1}^r w_i + \frac{w_1^{a_1} \cdot ... \cdot w_r^{a_r}}{z_1 \cdot ... \cdot z_s } + \frac{1}{w_1 \cdot ... \cdot w_r } \in L(\Delta_a^{\circ})$$ We consider the $1$-parametric family of Laurent polynomials $$ f_u(z,w):=1+\sum_{i=1}^s z_i+\sum_{i=1}^{r} w_i + e^{u} \cdot \frac{w_1^{a_1} \cdot ... \cdot w_r^{a_r}}{z_1 \cdot ... \cdot z_s } + \frac{1}{w_1 \cdot ... \cdot w_r } \in L(\Delta_a^{\circ})$$ for $u \in \mathbb{C}$.
\hspace{-0.6cm} Let $A(V):=Arg(V) \subset \mathbb{T}^n$ be the image of the algebraic subvariety $V \subset (\mathbb{C}^{\ast})^n $ under the argument map. Such sets are known as \emph{co-amoebas}, see \cite{PT}. For $1 \leq i \leq s$ and $1 \leq j \leq r$ consider the following sub-varieties of $(\mathbb{C}^{\ast})^{s+r}$: $$ \begin{array}{ccc} V_i^u = \left \{ z_i - e^u \frac{ \prod_{i=1}^r w_i^{a_i} }{\prod_{i=1}^s z_i }= 0 \right \} & ; & W_j^u = \left \{ w_i +a_i e^u \frac{ \prod_{i=1}^r w_i^{a_i} }{\prod_{i=1}^s z_i}-\frac{1}{ \prod_{i=1}^r w_i} = 0 \right \} \end{array} $$ For the co-amoeba one has $$ A(Crit(X ; f_u) ) \subset \left (\bigcap_{i=1}^s A(V_i^u) \right ) \cap \left ( \bigcap_{i=1}^r A(W_i^u) \right ) \subset \mathbb{T}^{s+r} $$ Let $( \theta_1,...,\theta_s, \delta_1,...,\delta_r)$ be coordinates on $\mathbb{T}^{s+r}$. We have, via straight-forward computation:
\bigskip
\hspace{-0.6cm} \bf Lemma 3.1.1: \rm For $1 \leq i \leq s$ and $1 \leq j \leq r$:
\bigskip
(1) $lim_{t \rightarrow -\infty} A(V_i^t)= \left \{ \theta_i + \sum_{i=1}^s \theta_i - \sum_{j=1}^r a_j \delta_j =0 \right \} \subset \mathbb{T}^{s+r}$
\bigskip
(2) $ lim_{t \rightarrow -\infty} A(W_j^t)= \left \{ \delta_j + \sum_{j=1}^r \delta_j=0 \right \} \subset \mathbb{T}^{s+r}$
\bigskip
\hspace{-0.6cm} Let $ \Theta : (\mathbb{C}^{\ast})^{s+r} \rightarrow (\mathbb{T}^{\ast})^2$ be the map given by $$ (z_1,...,z_s,w_1,...,w_r) \mapsto Arg \left (
\frac{ \prod_{i=1}^r w^{a_i}}{ \prod_{i=1}^s z_i} , \frac{1}{\prod_{i=1}^r w_i } \right ) $$ We have:
\bigskip
\hspace{-0.6cm} \bf Proposition 3.1.2: \rm $$ lim_{ t \rightarrow -\infty} \left ( \Theta(Crit(X;f_t)) \right ) = \left \{ \left ( \frac{l \sum_{i=1}^r a_i}{(s+1)(r+1)} + \frac{k}{s+1} ,\frac {l}{r+1} \right ) \right \}_{k=0,l=0}^{s,r} \subset \mathbb{T}^2 $$
\bigskip
\hspace{-0.6cm} \bf Proof: \rm Set $A_i = lim_{t \rightarrow - \infty} A(V_i^t)$ and $B_j=lim_{t \rightarrow -\infty} A(W_j^t)$ for $1 \leq i \leq s $ and $1 \leq j \leq r$. If $(\theta , \delta) \in \bigcap_{j=1}^r B_j$ then $\delta : = \delta_1= ... = \delta_r$ and $ (r+1) \delta =0 $ in $\mathbb{T}$. If $$( \theta, \delta) \in \left (\bigcap_{i=1}^s A_i \right ) \cap \left ( \bigcap_{j=1}^r B_j \right )$$ then $ \theta=\theta_1 =... = \theta_s$ and $ \delta= \frac{l}{r+1} $ for some $0 \leq l \leq r$. As $(s+1) \theta - \sum_{j=1}^r a_j \delta$ we get $ \theta = \sum_{j=1}^r \frac{a_j l}{(s+1)(r+1)} + \frac{k}{s+1} $ for $1 \leq k \leq s$. As there are exactly $(r+1)(s+1)$ such elements $(\theta, \delta)$ we get $lim_{t \rightarrow -\infty} A(Crit(X ; f_t)) = ( \bigcap_{i=1}^s A_i ) \cap ( \bigcap_{j=1}^r B_j )$. $\square$
\bigskip
\hspace{-0.6cm} Let $i : (\mathbb{C}^{\ast})^{r+s} \rightarrow (\mathbb{C}^{\ast})^{r+s+2}$ be the map given by $$(z_1,...,z_s,w_1,...,w_r) \mapsto \left ( z_1,...,z_s,w_1,...,w_r, \frac{\prod_{i=1}^r w_i^{a_i}}{\prod_{i=1}^s z_i} , \frac{1}{\prod_{i=1}^r w_i} \right ) $$ We have:
\bigskip
\hspace{-0.6cm} \bf Corollary 3.1.4 \rm The map $E: Crit(X_a; f_t) \rightarrow Pic(X)$ given by $E(z) = [D(z)]_{\mathbb{Z}}$ is an exceptional map when $ t \rightarrow - \infty$.
\bigskip
\hspace{-0.6cm} \bf Proof: \rm Set $\Theta_{k,l}=\left ( \theta_{k,l}(a), ..., \theta_{k,l}(a), \rho_{l},...,\rho_{l}, \theta_{k,l}(a), \rho_{l} \right ) \in \mathbb{T}^{r+s+2}$ where $ \rho_l = \frac{l}{r}$ and $\theta_{k,l}(a) = \sum_{j=1}^r \frac{a_j l}{(s+1)(r+1)} + \frac{k}{s+1} $. We have: $$ [D(\Theta_{k,l})]_{\mathbb{Z}}= \left [ \sum_{i=0}^s \theta_{k,l}(a) \cdot V_X(v_i) + \sum_{j=0}^r \rho_l \cdot V_X(e_i) \right ]_{\mathbb{Z}}
= $$ $$= \left [ \left ( \frac{l \sum a_i}{r+1} + k \right ) \cdot \pi^{\ast} H + \sum_{i=0}^r \frac{ l}{r+1} \left ( \xi - a_i \cdot \pi^{\ast}H \right ) \right ]= k \cdot \pi^{\ast} H + l \cdot \xi $$ $\square$
\bigskip
\hspace{-0.6cm} Note that in the definition of the exceptional map we considered the limit $t \rightarrow - \infty$ . It is interesting to ask whether the limit $t \rightarrow \infty$ can also be interpreted in terms of the exceptional map $E$. Denote by $\Theta_{\pm}(X) :=lim_{t \rightarrow \pm} \left ( \Theta(Crit(X ; f_t)) \right )$. Consider the following example:
\bigskip
\hspace{-0.6cm} \bf Example 3.1.5 \rm (The Hirzebruch surface): Let $X = \mathbb{P}( \mathcal{O} \oplus \mathcal{O}(1))$ be the Hirzebruch surface. Recall that $$ X = \left \{ ([z_0:z_1:z_2],[\lambda_0: \lambda_1] ) \vert \lambda_0 z_0 + \lambda_1 z_1 =0 \right \} \subset \mathbb{P}^2 \times \mathbb{P}^1$$ Denote by $ p : X \rightarrow \mathbb{P}^2$ and $\pi : X \rightarrow \mathbb{P}^1$ the projection to the first and second factor, respectively. Note that $p$ expresses $X$ as the blow up of $\mathbb{P}^2$ at the point $[0:0:1] \in \mathbb{P}^2$ and $\pi$ is the fibration map. The group $Pic(X)$ is described, in turn, in the following two ways $$ Pic(X) \simeq p^{\ast} H_{\mathbb{P}^2} \cdot \mathbb{Z} \oplus E \cdot \mathbb{Z} \simeq \pi^{\ast} H_{\mathbb{P}^1} \cdot \mathbb{Z} \oplus \xi \cdot \mathbb{Z}$$ Where $E$ is class of the the line bundle whose first Chern class $c_1(E) \in H^2(X ; \mathbb{Z})$ is the Poincare dual of the exceptional divisor and $\xi$ is the class of the tautological bundle of $\pi$. The exceptional collection is expressed in these bases by $$ \mathcal{E}_X = \left \{ 0, p^{\ast} H_{ \mathbb{P}^2}-E , 2 p^{\ast} H_{\mathbb{P}^2} - E , p^{\ast} H_{\mathbb{P}^2} \right \} = \left \{ 0 , \pi^{\ast} H_{\mathbb{P}^1} , \pi^{\ast} H_{\mathbb{P}^1}+ \xi, \xi \right \} $$ Let us note that we have $p_{\ast} \left \{ 0, p^{\ast} H_{ \mathbb{P}^2} , 2 p^{\ast} H_{\mathbb{P}^2} - E \right \}= \left \{ 0, H_{\mathbb{P}^2}, 2H_{\mathbb{P}^2} \right \} = \mathcal{E}_{\mathbb{P}^2}$, while we think of the additional element $p^{\ast} H- E$ as "added by the blow up".
\hspace{-0.6cm} On the other hand, direct computation gives $\Theta_+(X) = \mu(3) \cup \mu(1)$ where $\mu(n)=\left \{ e^{\frac{2 \pi k i}{n}} \vert k=0,...,n-1 \right \} \subset \mathbb{T} $ is the set of $n$-roots of unity for $n \in \mathbb{N}$. (see illustration in Remark 3.4). For $(k,l) \in \mathbb{Z}/2 \mathbb{Z} \oplus \mathbb{Z}/ 2 \mathbb{Z}$ let $\gamma_{kl}(t)= (z_{kl}(t),w_{kl}(t)) \in (\mathbb{C}^{\ast})^{s+r}$ be
the smooth curve defined by the condition $(z_{kl}(t) , w_{kl}(t)) \in Crit(X ; f_t)$ for $ t \in \mathbb{R}$ and $E(z_{kl}(0),w_{kl}(0)) = E_{kl}$ . Define the map $I^+: Crit(X) \rightarrow \Theta_+(X)$ by $$I^+(z_{kl},w_{kl}) :=lim_{t \rightarrow \infty}( \Theta (z_{kl}(t) ,w_{kl}(t)))$$ By direct computation $$ \begin{array}{cccccccc} I^+((z_{00},w_{00})) = \rho_3^0 & ; & I^+((z_{01},w_{01})) = \rho_3^1 & ; & I^+((z_{11},w_{11})) = \rho_3^2 & ; & I^+((z_{10},w_{10})) = 1 \end{array} $$ where $ \rho = e^{\frac{ 2 \pi i }{3} } \in \mu(3)$. Simlarly, define the map $I^-: Crit(X) \rightarrow \Theta_-(X)$,
on the other hand, taking $t \rightarrow - \infty$ in the limit. Note that this is the way we defined the exceptional map $E$ in the first place. We thus view the map $I: \Theta_-(X) \rightarrow \Theta_+(X)$ given by $I = I^+ \circ (I^-)^{-1}$ as a "geometric interpolation" between the bundle description of $\mathcal{E}_X$ and the blow up description of $\mathcal{E}_X$.
\bigskip
\subsection{Exceptional map for the class (b)} The Landau-Ginzburg potential for $X_{n,r}$ is given by $$ f(z,w) = \sum_{i=1}^{n-r} z_i + \sum_{i=1}^r w_i + \frac{1}{\prod_{i=1}^{n-r} z_i} + \frac{1}{\prod_{i=1}^r w_i} + \frac{1}{\prod_{i=1}^{n-r} z_i \cdot \prod_{i=1}^r w_i } \in L (\Delta_{n,r}^{\circ})$$ Instead, for $t \in \mathbb{R}$ we consider the $1$-parametric family $$f_{t} (z,w) = \sum_{i=1}^{n-r} e^{-t} z_i + \sum_{i=1}^r e^{-t} w_i + \frac{1}{\prod_{i=1}^{n-r} z_i} + \frac{1}{\prod_{i=1}^r w_i} + \frac{1}{\prod_{i=1}^{n-r} z_i \cdot \prod_{i=1}^r w_i } \in L (\Delta_{n,r}^{\circ})$$ The corresponding system of equations is $$ \begin{array}{ccc} e^{-t} z_i - \frac{1}{\prod_{i=1}^{n-r} z_i}- \frac{1}{\prod_{i=1}^{n-r} z_i \cdot \prod_{i=1}^r w_i } =0 & ; &
e^{-t} w_j - \frac{1}{\prod_{i=1}^{r} w_i}- \frac{1}{\prod_{i=1}^{n-r} z_i \cdot \prod_{i=1}^r w_i } =0 \end{array}$$ for $1 \leq i \leq n-r$ and $1 \leq j \leq r$. One can see that $ z_1 =...= z_{n-r}$ and $ w_1 = ... = w_r $, and we set $z=z_i$ and $w= w_j$.
In particular, the equations turn to be $$ \left \{ \begin{array}{c} e^{-t} z - \frac{1}{z^{n-r}}- \frac{1}{z^{n-r} w^r } =0 \\
e^{-t} w - \frac{1}{w^r}- \frac{1}{z^{n-r} w^r } =0 \end{array} \right. $$ Let $\Theta: (\mathbb{C}^{\ast})^n \rightarrow \mathbb{T}^2$ given by $(z_1,...,z_{n-r},w_1,...,w_r) \mapsto Arg(z_1,w_1)$. We have:
\bigskip
\hspace{-0.6cm} \bf Proposition 3.2.1: \rm The elements of $lim_{t \rightarrow \infty} \Theta(Crit(X_{n,r}; f_t)) \subset \mathbb{T}^{2}$ are of the following two classes $$ \Theta_{k,l}= \left (k \cdot \rho_{n-r+1}, l \cdot \rho_{r+1} \right ) \hspace{0.5cm} \textrm{ for } \hspace{0.25cm} 0 \leq k \leq n-r, 0 \leq l \leq r$$ and $$ \Theta'_{m,n} = (\rho_{2(n-r)} \cdot m \cdot \rho_{n-r} , \rho_{2r} \cdot n \cdot \rho_{r} ) \hspace{0.5cm} \textrm{ for } \hspace{0.25cm} 0 \leq m \leq n-r-1, 0 \leq n \leq r-1$$
\hspace{-0.6cm} \bf Proof: \rm Geometrically, the elements of $Crit(X_{n,r},f_t) \subset (\mathbb{C}^{\ast})^2$ are given as the intersection of the two curves $$\begin{array}{ccc} C_1 := \left \{ e^{-t} z - \frac{1}{z^{n-r}} - \frac{1}{z^{n-r} \cdot w^r} =0 \right \} \subset (\mathbb{C}^{\ast})^2 & ; & C_2 := \left \{ e^{-t} w - \frac{1}{w^{r}} - \frac{1}{z^{n-r} \cdot w^r} =0 \right \} \subset (\mathbb{C}^{\ast})^2 \end{array}$$ Let $Log_r \vert \cdot \vert : (\mathbb{C}^2) \rightarrow \mathbb{R}^2$ and $Arg: (\mathbb{C}^{\ast})^2 \rightarrow \mathbb{T}^2$ be the maps given by $$ \begin{array}{ccc} (r_1 e^{2 \pi i \theta_1} , r_2 e^{2 \pi i \theta_2}) \mapsto\left ( \frac{log(r_1)}{log(r)} , \frac{log (r_2)}{log(r)} \right ) & ; & (r_1 e^{2 \pi i \theta_1} , r_2 e^{2 \pi i \theta_2}) \mapsto (\theta_1 ,\theta_2 ) \end{array}$$ respectively. Let $T(C_i) := Log \vert C_1 \vert \subset \mathbb{R}^2 $ and $A(C_i) := Arg(C_i)$ be the amobea and co-amobea of $C_i$ for $i=1,2$. The corresponding tropical curves $\widetilde{T}(C_i) := lim_{r \rightarrow \infty} Log_r \vert C_i \vert$ are given as follows: $$ T(C_1) = \left \{ \begin{array}{c} x_1 =\frac{t}{n-r+1} \textrm{ , } \\ x_2 \leq 0 \end{array} \right \} \cup \left \{\begin{array}{c} x_2 = 0 \textrm{ , } \\ x_1 \leq \frac{t}{n-r+1} \end{array} \right \} \cup \left \{ \begin{array}{c} (n-r+1) x_1+ r x_2 = t \textrm{ , } \\ x_2 \leq 0 \end{array}
\right \} $$
$$ T(C_2) = \left \{ \begin{array}{c} x_2 =\frac{t}{r+1} \textrm{ , } \\ x_2 \leq 0 \end{array} \right \} \cup \left \{\begin{array}{c} x_1 = 0 \textrm{ , } \\ x_2 \leq \frac{t}{r+1} \end{array} \right \} \cup \left \{ \begin{array}{c} (n-r) x_1+ (r+1) x_2 = t \textrm{ , } \\ x_1 \leq 0 \end{array}
\right \} $$ These two tropical curves, which are the spines of the corresponding amoeba, intersect at two points $$ \widetilde{T}(C_1 ) \cap \widetilde{T}(C_2) = \left \{ ( 0,0) , \left ( \frac{t}{n-r+1}, \frac{t}{r+1} \right ) \right \} \subset \mathbb{R}^2$$ The corresponding co-tropical curves $\widetilde{A}(C_i) := \partial A(C_i)$ are given by: $$ A(C_1) = \left \{ \theta_1 =\frac{k}{n-r+1} \right \}_{k=0}^{n-r} \cup \left \{ r \theta_2 =\frac{1}{2}+k \right \}_{k=0}^{r-1} \cup \left \{ (n-r+1) \theta_1+ r \theta_2 = k \right \}_{k=0}^{r-1} $$ $$ A(C_2) = \left \{ \theta_2 =\frac{k}{r+1} \right \}_{k=0}^{r+1} \cup \left \{ (n-r) \theta_1 =\frac{1}{2}+k \right \}_{k=0}^{n-r-1} \cup \left \{ (n-r) \theta_1+ (r+1) \theta_2 = k \right \}_{k=0}^{n-r-1} $$ Each component of the co-tropical curve corresponds to one of the tentecals of the tropical curve. Thus, for the intersection point of the tropical curves $$\left ( \frac{t}{n-r+1}, \frac{t}{r+1} \right ) \in \widetilde{T}(C_1) \cap \widetilde{T}(C_2)$$ the corresponding intersections for the co-tropical curves are $$ \left \{ (\theta_1, \theta_2) \vert \theta_1 = \frac{k_1}{n-r+1} \textrm{ and } \theta_2 = \frac{k_2}{r+1} \right \}_{k_1=0,k_2=0}^{n-r,r} \subset \mathbb{T}^2 $$ For the intersection point of the tropical curves $ \left ( 0,0 \right ) \in \widetilde{T}(C_1) \cap \widetilde{T}(C_2)$ the corresponding intersections for the co-tropical curves are $$ \left \{ (\theta_1, \theta_2) \vert \theta_1 = \frac{1}{2(n-r)} +\frac{k_1}{n-r} \textrm{ and } \theta_2 = \frac{1}{2r} +\frac{k_2}{r} \right \}_{k_1=0,k_2=0}^{n-r-1,r-1} \subset \mathbb{T}^2 $$ $\square$
\bigskip
\hspace{-0.6cm} We have:
\bigskip
\hspace{-0.6cm} \bf Corollary 3.2.2: \rm The map $E: Crit(X_{n,r} ; f_t) \rightarrow Pic(X)$ given by $E(z) =[D(z)]_{\mathbb{Z}}$ is an exceptional map when $ t \rightarrow \infty$.
\bigskip
\hspace{-0.6cm} \bf Proof: \rm We have $$ [D(\Theta_{k,l})]_{\mathbb{Z}} = \left [ \frac{ (n-r) k}{n-r+1} \cdot U+ \frac{r \cdot l}{r+1} \cdot V + \right. $$ $$ \left.+\frac{k}{n-r+1} \left (U-E \right ) + \frac{l}{r+1} \left ( V-E \right ) + \left ( \frac{k}{n-r+1} + \frac{l}{r+1} \right ) E \right ]_{\mathbb{Z}} = k \cdot U + l \cdot V$$ and
$$ [D (\Theta'_{m,n})]_{\mathbb{Z}} = \left [\left ( \frac{1}{2} +m \right ) \cdot U+ \left ( \frac{1}{2} + n \right ) \cdot V + \frac{1}{2} \left (U-E \right ) + \frac{1}{2} \left ( V-E \right ) \right ]_{\mathbb{Z}} = (1+m) \cdot U +(1+n) \cdot V - E $$ $\square$
\bigskip
\subsection{Exceptional map for the class (c)} The Landau-Ginzburg potential for $X_{n,b}$ is given by $$ f(z,w) = \sum_{i=1}^{n-1} z_i +e^{-t} w + \frac{1}{\prod_{i=1}^{n-1} z_i \cdot w^b} + \frac{1}{\prod_{i=1}^{n-1} z_i \cdot w^{b+1}} + \frac{1}{ w } \in L (\Delta_{n,b}^{\circ})$$ Instead, for $t \in \mathbb{R}$ we consider the $1$-parametric family$$ f_t(z,w) = \sum_{i=1}^{n-1} e^{-t} z_i + e^{-t} w + \frac{1}{\prod_{i=1}^{n-1} z_i \cdot w^b} -\frac{i}{\prod_{i=1}^{n-1} z_i \cdot w^{b+1}} + \frac{1}{ w } \in L (\Delta_{n,b}^{\circ})$$ The corresponding system of equations is $$ \begin{array}{ccc} e^{-t} z_i - \frac{1}{\prod_{i=1}^{n-1} z_i \cdot w^b}+\frac{i}{\prod_{i=1}^{n-1} z_i \cdot w^{b+1} } =0 & ; &
e^{-t} w - \frac{b}{\prod_{i=1}^{n-1} z_i \cdot w^b}+ \frac{(b+1)i}{\prod_{i=1}^{n-1} z_i \cdot w^{b+1} } - \frac{1}{w} =0 \end{array}$$ for $1 \leq i \leq n-1$. One can see that $ z_1 =...= z_{n-1}$ and we set $z=z_i$.
In particular, the equations turn to be $$ \left \{ \begin{array}{c} e^{-t} z - \frac{1}{z^{n-1} \cdot w^b}+\frac{i}{z^{n-1} w^{b+1} } =0 \\ e^{-t} w - \frac{b}{z^{n-1} \cdot w^b}+ \frac{(b+1)i}{z^{n-1} w^{b+1} }- \frac{1}{w} =0 \end{array} \right. $$ Let $\Theta: (\mathbb{C}^{\ast})^n \rightarrow \mathbb{T}^2$ given by $(z_1,...,z_{n-1},w) \mapsto Arg(z_1,w)$. We have:
\bigskip
\hspace{-0.6cm} \bf Proposition 3.3.1: \rm The elements of $lim_{t \rightarrow \infty} \Theta(Crit(X_{n,b}; f_t)) \subset \mathbb{T}^{2}$ are of the following two classes $$ \Theta_{k,l} = \left (( b \cdot l \cdot \rho_{2n}) \cdot (k \cdot \rho_n), \frac{l}{2} \right ) \hspace{0.5cm} \textrm{ for } \hspace{0.25cm} 0 \leq k \leq n-1, 0 \leq l \leq 1$$ and $$ \Theta'_{m} = \left ( (3(b+1) \cdot \rho_{4(n-1)} ) \cdot (m \cdot \rho_{n-1}) , \frac{1}{4} \right ) \hspace{0.5cm} \textrm{ for } \hspace{0.25cm} 0 \leq m \leq n-2$$
\bigskip
\hspace{-0.6cm} \bf Proof: \rm As $ t \rightarrow \infty $ the solutions converge to the union of the solutions of the two systems $$ \left \{ \begin{array}{c} - \frac{1}{z^{n-1} \cdot w^b}+\frac{i}{z^{n-1} w^{b+1} } =0 \\ - \frac{b}{z^{n-1} \cdot w^b}+ \frac{(b+1)i}{z^{n-1} w^{b+1} }- \frac{1}{w} =0 \end{array} \right. \hspace{0.25cm} \textrm{ and } \hspace{0.25cm} \left \{ \begin{array}{c} e^{-t} z - \frac{1}{z^{n-1} \cdot w^b} =0 \\ e^{-t} w - \frac{b}{z^{n-1} \cdot w^b}- \frac{1}{w} =0 \end{array} \right. $$ The solutions of the first system are given by direct computation by $$ \left \{ (\rho_{4(n-1)}^{3(b+1)} \cdot \rho_{n-1}^m, i) \vert m=0,...,n-2 \right \} $$ For the second equation we have $$ e^{-t} w -be^{-t} z - \frac{1}{w} = 0 $$ solving this quadratic equation gives $$ w_{\pm} = \frac{ be^{-t} z \pm \sqrt{b z e^{-2t} + 4 e^{-t} } }{ 2e^{-t}} \rightarrow \pm e^{\frac{t}{2}} $$ Substituting in the first equation gives $ z^n w^b e^{-t} = 1 $. For $w_+$ we have $$ z= e^{\frac{(2-b)t}{2n}} \rho_n^k \hspace{0.5cm} \textrm{ for } \hspace{0.25cm} k=0,...,n-1$$ For $w_-$ we have $$ z= e^{\frac{(2-b)t}{2n}}\rho^b_{2n} \cdot \rho_n^k \hspace{0.5cm} \textrm{ for } \hspace{0.25cm} k=0,...,n-1$$ which gives the required result. $\square$
\bigskip
\hspace{-0.6cm} We have:
\bigskip
\hspace{-0.6cm} \bf Corollary 3.3.2: \rm The map $E: Crit(X_{n,b} ; f_t) \rightarrow Pic(X)$ given by $E(z) =[D(z)]_{\mathbb{Z}}$ is an exceptional map when $ t \rightarrow \infty$.
\bigskip
\hspace{-0.6cm} \bf Proof: \rm For $0 \leq k \leq n-1$ we have $$ [D(\Theta_{k,0})]_{\mathbb{Z}}=\left [ \frac{(n-1)k}{n} \cdot V + \frac{k}{n} \cdot (V-Y) + \left ( \frac{k}{n} + \frac{3}{4} \right ) Y \right]_{\mathbb{Z}} = k \cdot V $$ and $$ [D(\Theta_{k,1})]_{\mathbb{Z}} =\left [ \left ( \frac{(n-1)b}{2n}+\frac{(n-1)k}{n} \right ) \cdot V + \frac{1}{2}(Y+T+bV) + \left (\frac{b}{2n}+\frac{k}{n} \right) \cdot (V-Y) \right.$$ $$ \left. +\left ( \frac{b}{2n} + \frac{k}{n} + \frac{1}{2}+\frac{3}{4} \right ) Y + \frac{1}{2}T \right ]_{\mathbb{Z}} = k \cdot V + (Y+T+bV)$$ and $$ [D( \Theta'_{m,n})]_{\mathbb{Z}} = \left [
\left ( \frac{3b+1}{4}+m \right ) \cdot V + \frac{1}{4} (Y+T+bV) + \frac{1}{4} (V-Y) \right ]_{\mathbb{Z}} = (m+1) \cdot V + T $$ for $0 \leq m \leq n-2$. $\square$
\bigskip
\hspace{-0.6cm} Let us conclude this section with the following remark:
\bigskip
\hspace{-0.6cm} \bf Remark 3.3.3 \rm (Exceptional maps and the Frobenius mapping): Let $\widetilde{F} : \mathbb{T}^{r} \rightarrow Pic(X)$, where $r = rk(Div_T(X))$, be the Frobenius map described in Remark 2.6. Consider the map $i : (\mathbb{C}^{\ast})^n \rightarrow (\mathbb{C}^{\ast})^r$ given by $z \mapsto (z^{n_1},...,z^{n_r})$ where $\Delta^{\circ}(0)= \left \{ n_1,...,n_r \right \}$. Note that by definition of the Frobenius map, we can equivalently write the exceptional maps as $E(z) = \widetilde{F}(Arg(i(z)))$ for $z \in Crit(X ; f_t)$ for $0<<t$ big enough. In particular, we geometrically interpret the torus $\mathbb{T}^r$ (which in the Frobenius setting was defined as the limit of $Div_T(X)/l \cdot Div_T(X)$ as $l \rightarrow \infty$) as the argument torus of the maximal algebraic tori of $\mathbb{P}(L(\Delta^{\circ})^{\vee})$.
\section{Monodromies and the Endomorphism Ring}
\label{s:LGM}
\hspace{-0.6cm} Given a full strongly exceptional collection $\mathcal{E} = \left \{ E_i \right \}_{i=1}^N \subset Pic(X)$ one is interested in the structure of its endomorphism algebra $$A_{\mathcal{E}}= End \left ( \bigoplus_{i=1}^N E_i \right )= \bigoplus_{i,j=0}^N Hom(E_i,E_j)= \bigoplus_{i,j=0}^N H^0(X ; E_j \otimes E_i^{-1})$$ Our aim in this section is to illustrate how this algebra is naturally reflected in the monodromy group action of the Landau-Ginzburg system and the Frobenius stratification, for the classes of toric Fano manifolds (a)-(c), considered above.
\hspace{-0.6cm} Recall that a \emph{quiver with relations} $\widetilde{Q}=(Q,R)$ is a directed graph $Q$ with a two sided ideal $R$ in the path algebra $\mathbb{C}Q$ of $Q$, see \cite{DW}. In particular, a quiver with relations $\widetilde{Q}$ determines the
associative algebra $A_{\widetilde{Q}}=\mathbb{C}Q/R$, called the path algebra of $\widetilde{Q}$. In general, a collection of elements $\mathcal{C} \subset \mathcal{D}^b(X)$ and a basis $ B \subset A_{\mathcal{C}}:= End \left ( \bigoplus_{E \in \mathcal{C}} E \right )$
determine a quiver with relations
$\widetilde{Q}(\mathcal{C},B)$ whose vertex set is $\mathcal{C}$ such that $A_{\mathcal{C}} \simeq A_{Q(\mathcal{C},B)}$, see \cite{K}.
\hspace{-0.6cm} Let $Gr(X)$ be the Grassmaniann of subspaces $W \subset \mathbb{P}(L(\Delta^{\circ})^{\vee})$ of $codim(W)=n$. Denote by $Crit(X ; W) := X^{\circ} \cap W$ for $W \in Gr(X)$. Note the natural map $ W : L(\Delta^{\circ}) \rightarrow Gr(X)$ given by $$ f \mapsto \bigcap_{i=1}^n \left \{ z_i \frac{\partial}{\partial z_i} f =0 \right \} $$ In particular $Crit(X ; W(f)) = Crit(X ; f)$. Let $R_X \subset Gr(X)$ be the hypersurface of all $W \in Gr(X)$ such that $Crit(X;W)$ is non-reduced. Whenever $Crit(X; W(f_u))$ is reduced, one obtains, via standard analytic continuation, a monodromy map of the following form $$ M : \pi_1(Gr(X)
\setminus R_X, W(f_u)) \rightarrow Aut(Crit(X; W(f_u)))$$ On the other hand, as the line bundles in the exceptional collections $ \mathcal{E}$ are all $T$-equivariant the $Hom$-spaces between the elements of the collection admit a decomposition $$ Hom(E_i, E_j) = \bigoplus_{D \in Div_T(X)} Hom_D(E_i,E_j)$$ For each of the classes (a)-(c) we define a natural map $\Gamma : Div_T(X) \rightarrow \pi_1(Gr(X) \setminus R_X , f_u)$ with the property $$ Hom_D(E(z_i),E(z_j)) \neq 0 \Rightarrow M(\Gamma_D)(z_i)=z_j $$ We thus define $$ Hom_{mon} (z_i , z_j) := \bigoplus_{\left \{ D \vert M(\Gamma_D)(z_i)=z_j \right \}} D \cdot \mathbb{Z} $$ Our aim in this section is to show how exceptional maps $E : Crit(X ; f_u ) \rightarrow Pic(X)$ of section 3 satisfy $$Hom(E(z), E(w) ) \subset Hom_{mon} (z,w) \hspace{0.5cm} \textrm{ for any } \hspace{0.25cm} z,w \in Crit(X ; f_u)$$
\bigskip
\hspace{-0.6cm} \bf Remark 4.1: \rm We refer to this as the $M$-aligned property to note that the exceptional maps in section 3 are defined such that the algebraic $Hom$-functor is aligned with the geometric monodromies of the Landau-Ginzburg system. Note that this property is non-trivial as, in particular, it implies that even for an individual $T$-divisor $D \in Div_T(X)$, the corresponding monodromy $M(\Gamma_D)$ is aligned for all elements of the collection at once.
\subsection{M-algined property for projective bundles} Let $X= \mathbb{P} \left (\mathcal{O}_{\mathbb{P}^s} \oplus \bigoplus_{i=1}^r \mathcal{O}_{\mathbb{P}^s}(a_i) \right ) $ be a projective Fano bundle. Note that, in our case $$Div_T(X) = \left ( \bigoplus_{i=1}^s \mathbb{Z} \cdot V_X(v_i) \right )
\bigoplus \left ( \bigoplus_{i=0}^r \mathbb{Z} \cdot V_X(e_i) \right )$$ Let $L_{kl}=k \cdot \pi^{\ast} H + l \cdot \xi \in Pic(X)$ be any element. Then $$ H^0 (X ; L_{kl}) \simeq \left \{ \sum_{i=0}^s n_i V_X(v_i) + \sum_{i=0}^r m_i V_X(e_i) \bigg \vert \vert m \vert = l \textrm{ and } \vert n \vert = k+ \sum_{i=0}^r m_i a_i \right \} \subset Div^+_T(X)$$
\hspace{-0.6cm} In particular, the algebra $A_{\mathcal{E}}$ comes with the basis $\left \{ V(v_0),...,V(v_s),V(e_0),...,V(e_r) \right \}$. We denote the resulting quiver by $Q_s(a_0,...,a_r)$. For example, the quiver $Q_3(0,1,2)$ for $X=\mathbb{P}\left ( \mathcal{O}_{\mathbb{P}^3} \oplus \mathcal{O}_{\mathbb{P}^3}(1) \oplus \mathcal{O}_{\mathbb{P}^3}(2) \right )$ is the following
$$
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=1.9cm,main node/.style={font=\sffamily\bfseries \small}]
\node (1) {$E_{00}$};
\node (4) [right of=1] {$E_{10}$};
\node (7) [right of=4] {$E_{20}$};
\node (10) [right of=7] {$E_{30}$};
\node (2) [below of=4] {$E_{01}$};
\node (5) [right of=2] {$E_{11}$};
\node (8) [right of=5] {$E_{21}$};
\node (11) [right of=8] {$E_{31}$};
\node (3) [below of=5] {$E_{02}$};
\node (6) [right of=3] {$E_{12}$};
\node (9) [right of=6] {$E_{22}$};
\node (12) [right of=9] {$E_{32}$};
\path[every node/.style={font=\sffamily\small}]
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(11) edge node{} (12)
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edge [green] node{} (6);
\end{tikzpicture}
$$ For a divisor $D= \sum_{i=0}^s n_i V_X(v_i) + \sum_{i=0}^r m_i V_X(e_i) \in Div_T(X) $ and $u \in \mathbb{C}$ consider the loop $$ \gamma^u_{D}(\theta) := \sum_{i=1}^s e^{2 \pi i n_i \theta } z_i + \sum_{i=1}^{r} e^{2 \pi i m_i \theta} w_i + e^{u} \cdot e^{2 \pi i n_0 \theta }
\frac{ \prod_{i=1}^r w_i^{a_i}}{ \prod_{i=1}^s z_s} + \frac{e^{2 \pi i m_0 \theta} }{\prod_{i=1}^r w_i } $$ For $ \theta \in [0,1)$. Define $ \Gamma_D : = lim_{t \rightarrow -\infty} [\gamma_D^t] \in \pi_1(L(\Delta^{\circ}) \setminus R_X , f_X) $ and set $\widetilde{M}_D := M(\Gamma_D) \in Aut(Crit(X;f_t))$. Express the solution scheme as $$Crit(X;f_t) = \left \{ (z_{kl},w_{kl} ) \right \}_{k=0,l=0}^{s,r} \simeq \mathbb{Z} / (r+1) \mathbb{Z} \oplus \mathbb{Z} / (s+1) \mathbb{Z}$$ where $E((z_{kl},w_{kl}))=E_{kl}$. We have:
\bigskip
\hspace{-0.6cm} \bf Theorem 4.2 \rm (M-aligned property (a)): For $(k,l) \in \mathbb{Z}/(s+1) \mathbb{Z} \oplus \mathbb{Z} / (r+1) \mathbb{Z} \simeq Crit(X)$ the monodromy action satisfies:
\bigskip
(a) $\widetilde{M}_{V(v_j)}(k,l)=(k+1,l)$ for $j=0,...,s$.
\bigskip
(b) $\widetilde{M}_{V(e_j)}(k,l)=(k-a_j,l+1)$ for $j=0,...,r$.
\bigskip
\hspace{-0.6cm} \bf Proof : \rm For a divisor $D \in Div_T(X)$ and $ \theta \in [0,1)$ Set $$ \begin{array}{ccc} V^{u, \theta}_{D,i} := \left \{ e^{2 \pi i n_i \theta} z_i - e^u e^{ 2 \pi i n_0 \theta} \frac{ \prod_{i=1}^r w_i^{a_i}}{\prod_{i=1}^s z_i }=0 \right \} & ; & W^{u , \theta}_{D,j} := \left \{ e^{2 \pi i m_j}w_j +a_i e^u e^{2 \pi i n_0 \theta} \frac{ \prod_{i=1}^r w_i^{a_i}}{ \prod_{i=1}^s z_i} - \frac{e^{ 2 \pi i m_0}}{\prod_{i=1}^r w_i } = 0 \right \} \end{array}$$ where $ 1 \leq i \leq s$, $1 \leq j \leq r$ and $u \in \mathbb{C}$. Let $( \theta_1,...,\theta_s, \delta_1,...,\delta_r)$ be coordinates on $\mathbb{T}^{s+r}$. It is clear that:
\bigskip
- $A_{D,i}^{ t, \theta}:= lim_{ t \rightarrow -\infty} A(V^{t,\theta}_{D,i}) = \left \{ \theta_i + \sum_{i=1}^s \theta_i - \sum_{j=1}^r a_j \delta_j +(n_i-n_0) \theta=0 \right \} \subset \mathbb{T}^{s+r}$
\bigskip
- $B_{D,j}^{t , \theta}:= lim_{t \rightarrow -\infty} A(W_{D,j}^{t, \theta})= \left \{ \delta_j + \sum_{j=1}^r \delta_j + (m_j-m_0) \theta =0 \right \} \subset \mathbb{T}^{s+r}$
\bigskip
\hspace{-0.6cm} For $D = V(v_0) $ we have $ (\theta,\delta) \in \bigcap_{j=1}^r B_{D,j}^{t,\theta}$ then $ \delta: = \delta_1 = ... = \delta_r$ and $ (r+1) \delta = 0$ hence $ \delta = \frac{l}{r+1}$ for some $0 \leq l \leq r$. Assume further that $ (\theta,\delta) \in (\bigcap_{i=1}^s A_{D,i}^{t, \theta}) \cap( \bigcap_{j=1}^r B_{D,j}^{t,\theta})$ then $ \widetilde{ \theta} = \theta_1 = ... = \theta_s$ and $ (s+1) \widetilde{ \theta} - \sum_{j=1}^r \frac{ a_j l }{r+1} - \theta= 0$. Hence, $\widetilde{\theta} = \frac{k}{s+1} + \frac{ l \sum_{j=1}^r a_j}{(s+1)(r+1)} + \frac{\theta}{(s+1)} $ for some $0 \leq k \leq s$.
\hspace{-0.6cm} For $D= V(v_i)$ if $ (\theta,\delta) \in (\bigcap_{i=1}^s A_{D,i}^{t, \theta}) \cap( \bigcap_{j=1}^r B_{D,j}^{t,\theta})$ then $ \widetilde{ \theta} = \theta_1 = ...= \hat{\theta_i}=... = \theta_s$ and $ \theta_i = \widetilde{ \theta} - \theta$ and again $ (s+1) \widetilde{ \theta} - \sum_{j=1}^r \frac{ a_j l }{r+1} - \theta= 0$. Hence, $\widetilde{\theta} = \frac{k}{s+1} + \frac{ l \sum_{j=1}^r a_j}{(s+1)(r+1)} + \frac{\theta}{(s+1)} $ for some $0 \leq k \leq s$
\hspace{-0.6cm} For $D=V(e_0)$ we have $ (\theta,\delta) \in \bigcap_{j=1}^r B_{D,j}^{t,\theta}$ then $ \delta: = \delta_1 = ... = \delta_r$ and $ (r+1) \delta = \theta $ hence $ \delta = \frac{l+ \theta}{r+1}$ for some $0 \leq l \leq r$. Assume further that $ (\theta,\delta) \in (\bigcap_{i=1}^s A_{D,i}^{t, \theta}) \cap( \bigcap_{j=1}^r B_{D,j}^{t,\theta})$ then $ \widetilde{ \theta} = \theta_1 = ... = \theta_s$ and $ (s+1) \widetilde{ \theta} - \sum_{j=1}^r \frac{ a_j (l+ \theta) }{r+1} = 0$. Hence, $\widetilde{\theta} = \frac{k}{s+1} + \frac{ (l+ \theta) \sum_{j=1}^r a_j}{(s+1)(r+1)} $ for some $ 0 \leq k \leq s$.
\hspace{-0.6cm} For $D=V(e_j)$ we have $ (\theta,\delta) \in \bigcap_{j=1}^r B_{D,j}^{t,\theta}$ then $ \delta: = \delta_1 = ... = \hat{ \delta_j} =...= \delta_r$ and $\delta_j= \delta - \theta$ hence $(r+1) \delta = \theta $ and $ \delta = \frac{l+ \theta}{r+1}$ for some $0 \leq l \leq r$. Assume $ (\theta,\delta) \in (\bigcap_{i=1}^s A_{D,i}^{t, \theta}) \cap( \bigcap_{j=1}^r B_{D,j}^{t,\theta})$ then $ \widetilde{ \theta} = \theta_1 = ... = \theta_s$ and $ (s+1) \widetilde{ \theta} - \sum_{j=1}^r \frac{ a_j (l+ \theta) }{r+1} + a_j \theta= 0$. Hence, $\widetilde{\theta} = \frac{k-a_j \theta}{s+1} + \frac{ (l+ \theta) \sum_{j=1}^r a_j}{(s+1)(r+1)} $ for some $ 0 \leq k \leq s$. $ \square$
\bigskip
\hspace{-0.6cm} For instance, consider the following example:
\bigskip
\hspace{-0.6cm} \bf Example 4.3 \rm (monodromies for $X=\mathbb{P}\left ( \mathcal{O}_{\mathbb{P}^3} \oplus \mathcal{O}_{\mathbb{P}^3}(1) \oplus \mathcal{O}_{\mathbb{P}^3}(2) \right )$): The following diagram outlines the corresponding monodromies
on $\mathbb{T}^2$:
$$
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\node (c12) at (4,-1.1){};
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\node (d12) at (-4,-1.1){};
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\node (b2) at (1,-3.664){};
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\node (d2) at (-3,-3.664){};
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\node (c22) at (4,-3.764){};
\node (c23) at (4,-3.364){};
\node (c24) at (4,-3.564){};
\node (d21) at (-4,-3.964){};
\node (d22) at (-4,-3.764){};
\node (d23) at (-4,-3.364){};
\node (d24) at (-4,-3.564){};
\node (a3) at (-1,-6.328){};
\node (b3) at (1,-6.328){};
\node (c3) at (3,-6.328){};
\node (d3) at (-3,-6.328){};
\node (c31) at (4,-6.628){};
\node (c32) at (4,-6.428){};
\node (c33) at (4,-6.028){};
\node (c34) at (4,-6.228){};
\node (d31) at (-4,-6.628){};
\node (d32) at (-4,-6.428){};
\node (d33) at (-4,-6.028){};
\node (d34) at (-4,-6.228){};
\node (a4) at (-1,-8){};
\node (b4) at (1,-8){};
\node (c4) at (3,-8){};
\node (d4) at (-3,-8){};
\node (aa1) at (-3.3,-0.6){$z_{00}$};
\node (bb1) at (-1.3,-0.6){$z_{10}$};
\node (cc1) at (0.7,-0.6){$z_{20}$};
\node (dd1) at (2.7,-0.6){$z_{30}$};
\node (aa2) at (-1.3,-3.264){$z_{01}$};
\node (bb2) at (0.7,-3.264){$z_{11}$};
\node (cc2) at (2.7,-3.264){$z_{21}$};
\node (dd2) at (-3.3,-3.264){$z_{31}$};
\node (aa3) at (-1.3,-5.828){$z_{32}$};
\node (bb3) at (0.7,-5.828){$z_{02}$};
\node (cc3) at (2.7,-5.828){$z_{12}$};
\node (dd3) at (-3.3,-5.828){$z_{22}$};
\fill[black] (a1) circle (4.5pt);
\fill[black] (b1) circle (4.5pt);
\fill[black] (c1) circle (4.5pt);
\fill[black] (d1) circle (4.5pt);
\fill[black] (a2) circle (4.5pt);
\fill[black] (b2) circle (4.5pt);
\fill[black] (c2) circle (4.5pt);
\fill[black] (d2) circle (4.5pt);
\fill[black] (a3) circle (4.5pt);
\fill[black] (b3) circle (4.5pt);
\fill[black] (c3) circle (4.5pt);
\fill[black] (d3) circle (4.5pt);
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(a44) edge node{} (a2)
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(c44) edge node{} (b2)
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(e44) edge node{} (c2)
(f44) edge node{} (c3)
(g44) edge node{} (d2)
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(a05) edge [thick,green] node{} (a1)
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(c05) edge [thick,green] node{} (c1)
(d05) edge [thick,green] node{} (d1)
(a0) edge [thick,red] node{} (a1)
(b0) edge [thick,red] node{} (b1)
(c0) edge [thick,red] node{} (c1)
(d0) edge [thick,red] node{} (d1)
(a1) edge [thick,bend right =30,blue] node{} (b1)
edge [thick,bend right =10,blue] node{} (b1)
edge [thick,bend left =30,blue] node{} (b1)
edge [thick,bend left =10,blue] node{} (b1)
edge [thick, red] node{} (a2)
edge [thick,green] node{} (d2)
edge node{} (a00)
(b1) edge [thick,bend right=30,blue] node{} (c1)
edge [thick,bend right =10,blue] node{} (c1)
edge [thick,bend left =30,blue] node{} (c1)
edge [thick,bend left =10,blue] node{} (c1)
edge [thick,red] node{} (b2)
edge [thick,green] node{} (a2)
edge node{} (b00)
(c1) edge [thick,bend right=0,blue] node{} (c11)
edge [thick,bend right=0,blue] node{} (c12)
edge [thick,bend left=0,blue] node{} (c13)
edge [thick,bend left=0,blue] node{} (c14)
edge [thick,red] node{} (c2)
edge [thick,green] node{} (b2)
edge node{} (c00)
(d11) edge [thick,bend right=0,blue] node{} (d1)
(d12) edge [thick,bend right=0,blue] node{} (d1)
(d13) edge [thick,bend left=0,blue] node{} (d1)
(d14) edge [thick,bend left=0,blue] node{} (d1)
(d1) edge [thick,bend right=30,blue] node{} (a1)
edge [thick,bend right =10,blue] node{} (a1)
edge [thick,bend left =30,blue] node{} (a1)
edge [thick,bend left =10,blue] node{} (a1)
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edge [thick,green] node{} (d6)
edge node{} (d00)
(c6) edge [thick,green] node{} (c2)
(a2) edge [thick,bend right=30,blue] node{} (b2)
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edge [thick,bend left =10,blue] node{} (b2)
edge [thick,red] node{} (a3)
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edge node{} (a01)
(b2) edge [thick,bend right =30,blue] node{} (c2)
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edge [thick,bend left =30,blue] node{} (c2)
edge [thick,bend left =10,blue] node{} (c2)
edge [thick,red] node{} (b3)
edge [thick,green] node{} (a3)
edge node{} (b01)
(c2)edge [thick,bend right=0,blue] node{} (c21)
edge [thick,bend right=0,blue] node{} (c22)
edge [thick,bend left=0,blue] node{} (c23)
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edge node{} (d01)
(a3) edge [thick,bend right=30,blue] node{} (b3)
edge [thick,bend right =10,blue] node{} (b3)
edge [thick,bend left =30,blue] node{} (b3)
edge [thick,bend left =10,blue] node{} (b3)
edge [thick,red] node{} (a4)
edge node{} (b1)
edge [thick,green] node{} (d55)
(b3) edge [thick,bend right=30,blue] node{} (c3)
edge [thick,bend right =10,blue] node{} (c3)
edge [thick,bend left =30,blue] node{} (c3)
edge [thick,bend left =10,blue] node{} (c3)
edge [thick,red] node{} (b4)
edge node{} (c1)
edge [thick,green] node{} (a55)
(c3) edge [thick,bend right=0,blue] node{} (c31)
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edge [thick,red] node{} (c4)
edge node{} (c7)
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(d31) edge [thick,bend right=0,blue] node{} (d3)
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(d34) edge [thick,bend left=0,blue] node{} (d3)
(d3) edge [thick,bend right = 30,blue] node{} (a3)
edge [thick,bend right =10,blue] node{} (a3)
edge [thick,bend left =30,blue] node{} (a3)
edge [thick,bend left =10,blue] node{} (a3)
edge [thick,red] node{} (d4)
edge node{} (a1)
edge [thick,green] node{} (d8)
(c5) edge [thick,green] node{} (c3)
(d7) edge node{} (d1);
\end{tikzpicture}$$ Blue lines describe the monodromy action of $v_0,v_1,v_2,v_3$ (which are, in practice, all linear in the horizontal direction), black lines describe the action of $e_0$ while red and green lines describe the action of $e_1,e_2$ respectively.
\bigskip
\hspace{-0.6cm} For a divisor $D \in Div_T(X)$ set $$ \begin{array}{ccc} \vert D \vert_1 :=\sum_{i=0}^s n_i - \sum_{i=0}^r a_i m_i & ; & \vert D \vert_2 = \sum_{i=0}^r m_i \end{array}$$ Set $$ Div^+(k,l) := \left \{ D \vert 0 < k+\vert D \vert_1 \leq s \textrm{ and } 0 < l + \vert D \vert_2 \leq r \right \} \subset Div^+_T(X)$$ For two solutions $(k_1,l_1),(k_2,l_2) \in \mathbb{Z}/(s+1) \mathbb{Z} \oplus \mathbb{Z}/ (r+1) \mathbb{Z}$ we define $$ Hom_{mon}((k_1,l_1),(k_2,l_2) ) := \bigoplus_{D \in M((k_1,l_1),(k_2,l_2)) } \widetilde{M}_D \cdot \mathbb{Z}$$ where $$ M((k_1,l_1),(k_2,l_2)) := \left \{ D \vert \widetilde{M}_D(k_1,l_1) = (k_2,l_2) \textrm{ and } D \in Div^+(k_1,l_1) \right \} $$ Let us note that in terms of these "hom"-functors the M-aligned property could be formulated as follows:
\bigskip
\hspace{-0.6cm} \bf Corollary 4.4 \rm (M-Aligned property (a) II): For any two solutions $(k_1,l_1),(k_2,l_2) \in Crit(X)$ the following holds $$Hom(E_{k_1l_1}, E_{k_2l_2}) \simeq Hom_{mon}((k_1,l_1),(k_2,l_2))$$ Furthermore, the composition map $$ Hom(E_{k_1l_1},E_{k_2l_2}) \otimes Hom(E_{k_2l_2},E_{k_3l_3}) \rightarrow Hom(E_{k_1l_1},E_{k_3l_3})$$ is induced by the map $$Mon((k_1,l_1),(k_2,l_2)) \times Mon((k_2,l_2),(k_3,l_3)) \rightarrow Mon((k_1,l_1),(k_3,l_3)) $$ given by $(D_1,D_2) \mapsto D_1+D_2$.
\bigskip
\hspace{-0.6cm} Motivated by mirror symmetry it is interesting to ask to which extent such "artificially defined" hom-functors between solutions of the asymptotic LG-equations can be related to known geometric structures on the $A$-side, we refer the reader to section 5 for a discussion.
\subsection{M-aligned property for the class (b)} For $r \leq n $ let $ X_{n,r}=Bl_B(\mathbb{P}^{n-r} \times \mathbb{P}^r)$ be the blow up of $\mathbb{P}^{n-r} \times \mathbb{P}^r$ along the multi-linear codimension two subspace $B = \mathbb{P}^{n-r-1} \times \mathbb{P}^{r-1}$. One has $$Div_T(X) = \left ( \bigoplus_{i=1}^n \mathbb{Z} \cdot V_X(e_i) \right )
\bigoplus \left ( \bigoplus_{i=1}^3 \mathbb{Z} \cdot V_X(v_i) \right )$$ Let $L_{nml}=n \cdot V_X(v_1) +m \cdot V_X(v_2) + l \cdot V_X(v_3) \in Pic(X)$ be any element. Then a $T$-divisor satisfies $D \in H^0 (X ; L_{nml})$ if and only if $D \in Div_T^+(X)$ is of the form $$ D = \sum_{i=1}^{n-r}n'_i V_X(e_i) + \sum_{n-r+1}^n m'_i V_X(e_i) + n'' V_X(v_1) + m'' V_X (v_2) + l'' V_X(v_3)$$ where $$ \begin{array}{ccccc} n'' = ( n- \vert n' \vert) & ; & m''=(m-\vert m' \vert) & ; & l'' = (l+ \vert n' \vert + \vert m' \vert +n +m) \end{array}$$
\hspace{-0.6cm} In particular, $\left \{ V(e_1),...,V(e_n),V(v_1),V(v_2),V(v_3) \right \}$ is a basis for the algebra $A_{\mathcal{E}}$. We denote the resulting quiver by $Q(n,r)$. For example, for $X=Bl_B (\mathbb{P}^2 \times \mathbb{P}^2)$ the quiver $Q(4,2)$ is the following
$$
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=1.9cm ]
\node (1) {$E_{20}$};
\node (4) [left of=1] {$E_{10}$};
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\node (5) [left of=2] {$E_{11}$};
\node (8) [left of=5] {$E_{01}$};
\node (3) [below of=5] {$E_{22}$};
\node (6) [left of=3] {$E_{12}$};
\node (9) [left of=6] {$E_{02}$};
\node (10) [right of=3] {$F_{11}$};
\node (11) [right of=10] {$F_{12}$};
\node (12) [below of=10] {$F_{22}$};
\node (13) [left of=12] {$F_{12}$};
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(7) edge [bend right=20,blue] node {} (4)
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\end{tikzpicture}
$$
\hspace{-0.6cm} Where two of the blue lines stand for the elements $V_X(e_1), V_X(e_2)$, two of the black lines for $V_X(e_3),V_X(e_4)$, green lines for $V_X(v_1)$, yellow for $V_X(v_2)$ and red for $V_X(v_3)$. The additional blue and black lines stand for $V_X(v_1) + V_X(v_3)$ and $V_X(v_2) +V_X(v_3)$ respectively.
\hspace{-0.6cm} On the other hand, on the Landau-Ginzburg side, the base point is given by the system of equations $$ W^t_X : = \left \{ \begin{array}{c} e^{-t} z_i - \frac{1}{ \prod_{i=1}^{n-r} z_i } - \frac{1}{\prod_{i=1}^{n-r} z_i \cdot \prod_{i=1}^r w_i } =0 \\ e^{-t} w_j - \frac{1}{ \prod_{i=1}^{r} w_i } - \frac{1}{\prod_{i=1}^{n-r} z_i \cdot \prod_{i=1}^r w_i } =0 \end{array} \right \}_{i=1,j=1}^{n-r,r} $$ For $ 1 \leq i \leq n-r$, $n-r+1 \leq j \leq n$ and $ \theta \in [0,1)$ we define the following loops in $Gr(X) \setminus R_X$:
$$ W^t_{V_X(e_i)}( \theta) : = \left \{ \begin{array}{c} e^{2 \pi i \delta(i,i') \theta} z_{i'} - \frac{1}{ \prod_{i=1}^{n-r} z_i } - \frac{1}{\prod_{i=1}^{n-r} z_i \cdot \prod_{i=1}^r w_i } =0 \\ e^{-t} w_j - \frac{1}{ \prod_{i=1}^{r} w_i } - \frac{1}{\prod_{i'=1}^{n-r} z_i \cdot \prod_{i=1}^r w_i } =0 \end{array} \right \}_{i'=1,j=1}^{n-r,r} $$
$$ W^t_{V_X(e_j)}( \theta) : = \left \{ \begin{array}{c} e^{-t} z_i - \frac{1}{ \prod_{i=1}^{n-r} z_i } - \frac{1}{\prod_{i=1}^{n-r} z_i \cdot \prod_{i=1}^r w_i } =0 \\ e^{2 \pi i \delta(j,j') \theta} w_{j'} - \frac{1}{ \prod_{i=1}^{r} w_i } - \frac{1}{\prod_{i=1}^{n-r} z_i \cdot \prod_{i=1}^r w_i } =0 \end{array} \right \}_{i=1,j'=1}^{n-r,r} $$
$$ \widetilde{W}^t_{V_X(e_i)-V_X(v_3)}( \theta) : = \left \{ \begin{array}{c} e^{2 \pi i \delta(i,i') \theta} z_{i'} - \frac{1}{ \prod_{i=1}^{n-r} z_i } - \frac{1}{\prod_{i=1}^{n-r} z_i \cdot \prod_{i=1}^r w_i } =0 \\ e^{-t} w_j - \frac{1}{ \prod_{i=1}^{r} w_i } - \frac{e^{-2 \pi i \theta}}{\prod_{i'=1}^{n-r} z_i \cdot \prod_{i=1}^r w_i } =0 \end{array} \right \}_{i'=1,j=1}^{n-r,r} $$
$$ \widetilde{W}^t_{V_X(e_j)-V_X(v_3)}( \theta) : = \left \{ \begin{array}{c} e^{-t} z_i - \frac{1}{ \prod_{i=1}^{n-r} z_i } - \frac{e^{-2 \pi i \theta}}{\prod_{i=1}^{n-r} z_i \cdot \prod_{i=1}^r w_i } =0 \\ e^{2 \pi i \delta(j,j') \theta} w_{j'} - \frac{1}{ \prod_{i=1}^{r} w_i } - \frac{1}{\prod_{i'=1}^{n-r} z_i \cdot \prod_{i=1}^r w_i } =0 \end{array} \right \}_{i=1,j'=1}^{n-r,r} $$
$$ \widetilde{W}^t_{V_X(v_3)}( \theta) : = \left \{ \begin{array}{c} z_i - \frac{1}{ \prod_{i=1}^{n-r} z_i } - \frac{1}{\prod_{i=1}^{n-r} z_i \cdot \prod_{i=1}^r w_i } =0 \\ e^{-t} w_j - \frac{1}{ \prod_{i=1}^{r} w_i } - \frac{e^{2 \pi i \theta}}{\prod_{i'=1}^{n-r} z_i \cdot \prod_{i=1}^r w_i } =0 \end{array} \right \}_{i=1,j=1}^{n-r,r} $$
$$ \widetilde{W'}^t_{V_X(v_3)}( \theta) : = \left \{ \begin{array}{c} e^{-t} z_{i'} - \frac{1}{ \prod_{i=1}^{n-r} z_i } - \frac{e^{-2 \pi i \theta}}{\prod_{i=1}^{n-r} z_i \cdot \prod_{i=1}^r w_i } =0 \\ w_j - \frac{1}{ \prod_{i=1}^{r} w_i } - \frac{1}{\prod_{i'=1}^{n-r} z_i \cdot \prod_{i=1}^r w_i } =0 \end{array} \right \}_{i=1,j=1}^{n-r,r} $$ Set also $$ U^t(\theta) : = \left \{ \begin{array}{c} e^{-t} z_i - \frac{1}{ \prod_{i=1}^{n-r} z_i } - \frac{1}{\prod_{i=1}^{n-r} z_i \cdot \prod_{i=1}^r w_i } =0 \\ e^{-t (1- \theta)} w_j - \frac{1}{ \prod_{i=1}^{r} w_i } - \frac{1}{\prod_{i=1}^{n-r} z_i \cdot \prod_{i=1}^r w_i } =0 \end{array} \right \}_{i=1,j=1}^{n-r,r} $$ $$ V^t( \theta) : = \left \{ \begin{array}{c} e^{-t (1- \theta)} z_i - \frac{1}{ \prod_{i=1}^{n-r} z_i } - \frac{1}{\prod_{i=1}^{n-r} z_i \cdot \prod_{i=1}^r w_i } =0 \\ e^{-t} w_j - \frac{1}{ \prod_{i=1}^{r} w_i } - \frac{1}{\prod_{i=1}^{n-r} z_i \cdot \prod_{i=1}^r w_i } =0 \end{array} \right \}_{i=1,j=1}^{n-r,r}$$
\hspace{-0.6cm} Set $$\begin{array}{cccc} \Gamma_{V_X(e_i)}: = [(U^t)^{-1} \circ W^t_{V_X(e_i)} \circ U^t] & ; & \Gamma_{V_X(e_j)}:=[(V^t)^{-1} \circ W^t_{V_X(e_j)} \circ V^t] & ; \end{array} $$
$$\begin{array}{cccc} \Gamma_{V_X(v_1)}:= [(U^t)^{-1} \circ \widetilde{W}^t_{V_X(e_i)-V_X(v_3)} \circ U^t] & ; & \Gamma_{V_X(v_2)} :=
[(V^t)^{-1} \circ \widetilde{W}^t_{V_X(e_j)-V_X(v_3)} \circ V^t] & ; \end{array} $$
$$\begin{array}{ccc} \Gamma_{V_X(v_3)}:= [(U^t)^{-1} \circ \widetilde{W}^t_{V_X(v_3)} \circ U^t] & ; & \Gamma'_{V_X(v_3)} :=
[(V^t)^{-1} \circ \widetilde{W'}^t_{V_X(v_3)} \circ V^t] \end{array} $$ Let $$ Crit(X_{n,r} ; W^t_X) = \left \{ (z_k,w_l) \right \}_{k=0,l=0}^{n-r,r} \bigcup \left \{ (z'_m,w'_n) \right \}_{m=0,n=0}^{n-r-1,r-1} $$ be as in Definition 3.2.1. The following example illustrates the $M$-aligned property for the class (b):
\bigskip
\hspace{-0.6cm} \bf Example 4.5 \rm (monodromies for $X= Bl_B(\mathbb{P}^2 \times \mathbb{P}^2)$ with $B= \mathbb{P}^1 \times \mathbb{P}^1$): The exceptional map of Definition 3.2.1 is defined in terms of the solutions of the LG-system for $$ f_t(z,w) = e^{-t} z_1+e^{-t}z_2+e^{-t}w_1+e^{-t}w_2 + \frac{1}{z_1z_2} + \frac{1}{w_1 w_2 } + \frac{1}{z_1z_2w_1w_2} $$ for $0<<t$. The following is a graphical description of the $z_1$ and $w_1$ coordinates of the solution scheme $Crit(X ; W_X^t)$:
$$ \begin{array}{ccc} \begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=3.5cm,main node/.style={circle,draw,font=\sffamily\bfseries \small}, scale=1.4]
\draw [->,thick] (0,-2)--(0,2.5) node (yaxis) [above] {$w_1(Crit(X; W^t))$}
|- (-2.5,0)--(2.5,0) node (xaxis) [right] {};
\node (a) at (0, 2.08669){};
\node (b) at (1.56015, -1.10404){};
\node (c) at (-1.56015, -1.10404){};
\node (d) at (1.84216, -1.03819){};
\node (e) at (-1.84216, -1.03819){};
\node (f) at (0.133691, 1.87899){};
\node (g) at (-0.133691, 1.87899){};
\node (h) at (1.66754, -0.770428){};
\node (h1) at (-1.66754, -0.770428){};
\node (h2) at (0.982687, 0.0606989){};
\node (h3) at (-0.982687, 0.0606989){};
\node (h4) at (1.0074, -0.0703753){};
\node (h5) at (-1.0074, -0.0703753){};
\fill[black] (a) circle (2.5pt);
\fill[black] (b) circle (2.5pt);
\fill[black] (c) circle (2.5pt);
\fill[black] (d) circle (2.5pt);
\fill[black] (e) circle (2.5pt);
\fill[black] (f) circle (2.5pt);
\fill[black] (g) circle (2.5pt);
\fill[black] (h) circle (2.5pt);
\fill[black] (h1) circle (2.5pt);
\fill[black] (h2) circle (2.5pt);
\fill[black] (h3) circle (2.5pt);
\fill[black] (h4) circle (2.5pt);
\fill[black] (h5) circle (2.5pt);
\node (a1) at (0.25, 2.28669){$E_{10}$};
\node (b1) at (1.56015, -1.40404){$E_{21}$};
\node (c1) at (-1.56015, -1.40404){$E_{02}$};
\node (d1) at (2.14216, -1.03819){$E_{11}$};
\node (e1) at (-2.14216, -1.03819){$E_{12}$};
\node (f1) at (0.433691, 1.87899){$E_{20}$};
\node (g1) at (-0.433691, 1.87899){$E_{00}$};
\node (h1) at (1.86754, -0.540428){$E_{01}$};
\node (h11) at (-1.86754, -0.540428){$E_{22}$};
\node (h21) at (1.182687, 0.3606989){$F_{21}$};
\node (h31) at (-1.182687, 0.3606989){$F_{12}$};
\node (h41) at (1.3074, -0.2703753){$F_{11}$};
\node (h51) at (-1.3074, -0.2703753){$F_{22}$};
\end{tikzpicture}
& &
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=3.5cm,main node/.style={circle,draw,font=\sffamily\bfseries \small}, scale=1.4]
\draw [->,thick] (0,-2)--(0,2.5) node (yaxis) [above] {$z_1(Crit(X;W^t_t))$}
|- (-2.5,0)--(2.5,0) node (xaxis) [right] {};
\node (a) at (0, 2.08669){};
\node (b) at (1.56015, -1.10404){};
\node (c) at (-1.56015, -1.10404){};
\node (d) at (1.84216, -1.03819){};
\node (e) at (-1.84216, -1.03819){};
\node (f) at (0.133691, 1.87899){};
\node (g) at (-0.133691, 1.87899){};
\node (h) at (1.66754, -0.770428){};
\node (h1) at (-1.66754, -0.770428){};
\node (h2) at (0.982687, 0.0606989){};
\node (h3) at (-0.982687, 0.0606989){};
\node (h4) at (1.0074, -0.0703753){};
\node (h5) at (-1.0074, -0.0703753){};
\fill[black] (a) circle (2.5pt);
\fill[black] (b) circle (2.5pt);
\fill[black] (c) circle (2.5pt);
\fill[black] (d) circle (2.5pt);
\fill[black] (e) circle (2.5pt);
\fill[black] (f) circle (2.5pt);
\fill[black] (g) circle (2.5pt);
\fill[black] (h) circle (2.5pt);
\fill[black] (h1) circle (2.5pt);
\fill[black] (h2) circle (2.5pt);
\fill[black] (h3) circle (2.5pt);
\fill[black] (h4) circle (2.5pt);
\fill[black] (h5) circle (2.5pt);
\node (a1) at (0.25, 2.28669){$E_{01}$};
\node (b1) at (1.56015, -1.40404){$E_{12}$};
\node (c1) at (-1.56015, -1.40404){$E_{20}$};
\node (d1) at (2.14216, -1.03819){$E_{11}$};
\node (e1) at (-2.14216, -1.03819){$E_{21}$};
\node (f1) at (0.433691, 1.87899){$E_{02}$};
\node (g1) at (-0.433691, 1.87899){$E_{00}$};
\node (h1) at (1.86754, -0.540428){$E_{10}$};
\node (h11) at (-1.86754, -0.540428){$E_{22}$};
\node (h21) at (1.182687, 0.3606989){$F_{12}$};
\node (h31) at (-1.182687, 0.3606989){$F_{21}$};
\node (h41) at (1.3074, -0.2703753){$F_{11}$};
\node (h51) at (-1.3074, -0.2703753){$F_{22}$};
\end{tikzpicture}
\end{array}
$$ The action of $U^t(\theta)$ and $V^t(\theta)$ for $[0,1)$ is described as follows:
$$ \begin{array}{ccc} \begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=3.5cm,main node/.style={circle,draw,font=\sffamily\bfseries \small}, scale=1.4]
\draw [->,thick] (0,-2)--(0,2.5) node (yaxis) [above] {$w_1(U^t( \theta))$}
|- (-2.5,0)--(2.5,0) node (xaxis) [right] {};
\node (a) at (0, 2.08669){};
\node (b) at (1.56015, -1.10404){};
\node (c) at (-1.56015, -1.10404){};
\node (d) at (1.84216, -1.03819){};
\node (e) at (-1.84216, -1.03819){};
\node (f) at (0.133691, 1.87899){};
\node (g) at (-0.133691, 1.87899){};
\node (h) at (1.66754, -0.770428){};
\node (h1) at (-1.66754, -0.770428){};
\node (h2) at (0.982687, 0.0606989){};
\node (h3) at (-0.982687, 0.0606989){};
\node (h4) at (1.0074, -0.0703753){};
\node (h5) at (-1.0074, -0.0703753){};
\node (a1) at (0, 2.41225){};
\node (b1) at (1.34988, -1.50026){};
\node (c1) at (-1.34988, -1.50026){};
\node (d1) at (0.797981, 0.294386){};
\node (e1) at (-0.797981, 0.294386){};
\node (f1) at (0.579635, 1.76278){};
\node (g1) at (-0.579635, 1.76278){};
\node (h01) at (2.16423, -1.20049){};
\node (h11) at (-2.16423, -1.20049){};
\node (h21) at (-0.748239, -0.343339){};
\node (h31) at (0.748239, -0.343339){};
\node (h41) at (-1.98059, -0.219197){};
\node (h51) at (1.98059, -0.219197){};
\node (a11) at (0.3, 2.41225){$E_{10}$};
\node (b11) at (1.54988, -1.80026){$E_{21}$};
\node (c11) at (-1.54988, -1.80026){$E_{02}$};
\node (d11) at (0.797981, 0.594386){$F_{10}$};
\node (e11) at (-0.797981, 0.594386){$F_{12}$};
\node (f11) at (0.979635, 1.76278){$E_{20}$};
\node (g11) at (-0.979635, 1.76278){$E_{00}$};
\node (h011) at (2.46423, -1.40049){$E_{11}$};
\node (h111) at (-2.46423, -1.40049){$E_{12}$};
\node (h211) at (-0.348239, -0.643339){$F_{22}$};
\node (h311) at (0.348239, -0.643339){$F_{11}$};
\node (h411) at (-2.38059, -0.219197){$E_{22}$};
\node (h511) at (2.38059, -0.219197){$E_{01}$};
\fill[black] (a) circle (1.5pt);
\fill[black] (b) circle (1.5pt);
\fill[black] (c) circle (1.5pt);
\fill[black] (d) circle (1.5pt);
\fill[black] (e) circle (1.5pt);
\fill[black] (f) circle (1.5pt);
\fill[black] (g) circle (1.5pt);
\fill[black] (h) circle (1.5pt);
\fill[black] (h1) circle (1.5pt);
\fill[black] (h2) circle (1.5pt);
\fill[black] (h3) circle (1.5pt);
\fill[black] (h4) circle (1.5pt);
\fill[black] (h5) circle (1.5pt);
\fill[black] (a1) circle (2.5pt);
\fill[black] (b1) circle (2.5pt);
\fill[black] (c1) circle (2.5pt);
\fill[black] (d1) circle (2.5pt);
\fill[black] (e1) circle (2.5pt);
\fill[black] (f1) circle (2.5pt);
\fill[black] (g1) circle (2.5pt);
\fill[black] (h01) circle (2.5pt);
\fill[black] (h11) circle (2.5pt);
\fill[black] (h21) circle (2.5pt);
\fill[black] (h31) circle (2.5pt);
\fill[black] (h41) circle (2.5pt);
\fill[black] (h51) circle (2.5pt);
\path[every node/.style={font=\sffamily\small}]
(a) edge (a1)
(b) edge (b1)
(c) edge (c1)
(d) edge (h01)
(e) edge (h11)
(f) edge (f1)
(g) edge (g1)
(h) edge (h51)
(h1) edge (h41)
(h2) edge (d1)
(h3) edge (e1)
(h4) edge (h31)
(h5) edge (h21);
\end{tikzpicture}
& &
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=3.5cm,main node/.style={circle,draw,font=\sffamily\bfseries \small}, scale=1.4]
\draw [->,thick] (0,-2)--(0,2.5) node (yaxis) [above] {$z_1(V^t(\theta))$}
|- (-2.5,0)--(2.5,0) node (xaxis) [right] {};
\node (a) at (0, 2.08669){};
\node (b) at (1.56015, -1.10404){};
\node (c) at (-1.56015, -1.10404){};
\node (d) at (1.84216, -1.03819){};
\node (e) at (-1.84216, -1.03819){};
\node (f) at (0.133691, 1.87899){};
\node (g) at (-0.133691, 1.87899){};
\node (h) at (1.66754, -0.770428){};
\node (h1) at (-1.66754, -0.770428){};
\node (h2) at (0.982687, 0.0606989){};
\node (h3) at (-0.982687, 0.0606989){};
\node (h4) at (1.0074, -0.0703753){};
\node (h5) at (-1.0074, -0.0703753){};
\node (a1) at (0, 2.41225){};
\node (b1) at (1.34988, -1.50026){};
\node (c1) at (-1.34988, -1.50026){};
\node (d1) at (0.797981, 0.294386){};
\node (e1) at (-0.797981, 0.294386){};
\node (f1) at (0.579635, 1.76278){};
\node (g1) at (-0.579635, 1.76278){};
\node (h01) at (2.16423, -1.20049){};
\node (h11) at (-2.16423, -1.20049){};
\node (h21) at (-0.748239, -0.343339){};
\node (h31) at (0.748239, -0.343339){};
\node (h41) at (-1.98059, -0.219197){};
\node (h51) at (1.98059, -0.219197){};
\node (a11) at (0.3, 2.41225){$E_{01}$};
\node (b11) at (1.54988, -1.80026){$E_{12}$};
\node (c11) at (-1.54988, -1.80026){$E_{20}$};
\node (d11) at (0.797981, 0.594386){$F_{01}$};
\node (e11) at (-0.797981, 0.594386){$F_{21}$};
\node (f11) at (0.979635, 1.76278){$E_{02}$};
\node (g11) at (-0.979635, 1.76278){$E_{00}$};
\node (h011) at (2.46423, -1.40049){$E_{11}$};
\node (h111) at (-2.46423, -1.40049){$E_{21}$};
\node (h211) at (-0.348239, -0.643339){$F_{22}$};
\node (h311) at (0.348239, -0.643339){$F_{11}$};
\node (h411) at (-2.38059, -0.219197){$E_{22}$};
\node (h511) at (2.38059, -0.219197){$E_{10}$};
\fill[black] (a) circle (1.5pt);
\fill[black] (b) circle (1.5pt);
\fill[black] (c) circle (1.5pt);
\fill[black] (d) circle (1.5pt);
\fill[black] (e) circle (1.5pt);
\fill[black] (f) circle (1.5pt);
\fill[black] (g) circle (1.5pt);
\fill[black] (h) circle (1.5pt);
\fill[black] (h1) circle (1.5pt);
\fill[black] (h2) circle (1.5pt);
\fill[black] (h3) circle (1.5pt);
\fill[black] (h4) circle (1.5pt);
\fill[black] (h5) circle (1.5pt);
\fill[black] (a1) circle (2.5pt);
\fill[black] (b1) circle (2.5pt);
\fill[black] (c1) circle (2.5pt);
\fill[black] (d1) circle (2.5pt);
\fill[black] (e1) circle (2.5pt);
\fill[black] (f1) circle (2.5pt);
\fill[black] (g1) circle (2.5pt);
\fill[black] (h01) circle (2.5pt);
\fill[black] (h11) circle (2.5pt);
\fill[black] (h21) circle (2.5pt);
\fill[black] (h31) circle (2.5pt);
\fill[black] (h41) circle (2.5pt);
\fill[black] (h51) circle (2.5pt);
\path[every node/.style={font=\sffamily\small}]
(a) edge (a1)
(b) edge (b1)
(c) edge (c1)
(d) edge (h01)
(e) edge (h11)
(f) edge (f1)
(g) edge (g1)
(h) edge (h51)
(h1) edge (h41)
(h2) edge (d1)
(h3) edge (e1)
(h4) edge (h31)
(h5) edge (h21);
\end{tikzpicture}
\end{array}
$$
\hspace{-0.6cm} The following is a graphical description of the corresponding monodromy actions:
$$ \begin{array}{ccc} \begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=3.5cm,main node/.style={circle,draw,font=\sffamily\bfseries \small}, scale=1.4]
\draw [->,thick] (0,-2)--(0,2.5) node {}(yaxis) [above]
|- (-2.5,0)--(2.5,0) node {}(xaxis) [right] {} ;
\node (a1) at (0, 2.41225){};
\node (b1) at (1.34988, -1.50026){};
\node (c1) at (-1.34988, -1.50026){};
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\node (b11) at (1.54988, -1.80026){$E_{21}$};
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\node (d11) at (0.797981, 0.594386){$F_{21}$};
\node (e11) at (-0.797981, 0.594386){$F_{12}$};
\node (f11) at (0.979635, 1.76278){$E_{20}$};
\node (g11) at (-0.979635, 1.76278){$E_{00}$};
\node (h011) at (2.46423, -1.40049){$E_{11}$};
\node (h111) at (-2.46423, -1.40049){$E_{12}$};
\node (h211) at (-0.348239, -0.643339){$F_{22}$};
\node (h311) at (0.348239, -0.643339){$F_{11}$};
\node (h411) at (-2.38059, -0.219197){$E_{22}$};
\node (h511) at (2.38059, -0.219197){$E_{01}$};
\fill[black] (a1) circle (2.5pt);
\fill[black] (b1) circle (2.5pt);
\fill[black] (c1) circle (2.5pt);
\fill[black] (d1) circle (2.5pt);
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\fill[black] (f1) circle (2.5pt);
\fill[black] (g1) circle (2.5pt);
\fill[black] (h01) circle (2.5pt);
\fill[black] (h11) circle (2.5pt);
\fill[black] (h21) circle (2.5pt);
\fill[black] (h31) circle (2.5pt);
\fill[black] (h41) circle (2.5pt);
\fill[black] (h51) circle (2.5pt);
\path[every node/.style={font=\sffamily\small}]
(a1) edge [blue,->,bend left] (f1)
(b1) edge [blue,->,bend left] (h31)
(c1) edge [blue,bend left] (h11)
edge [green] (e1)
(d1) edge [blue,->,bend left] (h51)
edge [red] (b1)
(e1) edge [blue,->,bend left] (h21)
edge [red] (h11)
(f1) edge [blue,->,bend left] (g1)
(g1) edge [blue,->,bend left] (a1)
(h01) edge [blue,->,bend left] (b1)
edge [green] (d1)
(h11) edge [blue,bend left] (h41)
edge [green] (h21)
(h21) edge [blue,->,bend left] (c1)
edge (h41)
(h31) edge [blue,->,bend left] (d1)
edge [red] (h01)
(h41) edge [blue,->,bend left] (e1)
(h51) edge [blue,->,bend left] (h01)
edge [green] (h31);
\end{tikzpicture}
& &
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=3.5cm,main node/.style={circle,draw,font=\sffamily\bfseries \small}, scale=1.4]
\draw [->,thick] (0,-2)--(0,2.5) node {} (yaxis) [above]
|- (-2.5,0)--(2.5,0) node {} (xaxis) [right] {};
\node (a1) at (0, 2.41225){};
\node (b1) at (1.34988, -1.50026){};
\node (c1) at (-1.34988, -1.50026){};
\node (d1) at (0.797981, 0.294386){};
\node (e1) at (-0.797981, 0.294386){};
\node (f1) at (0.579635, 1.76278){};
\node (g1) at (-0.579635, 1.76278){};
\node (h01) at (2.16423, -1.20049){};
\node (h11) at (-2.16423, -1.20049){};
\node (h21) at (-0.748239, -0.343339){};
\node (h31) at (0.748239, -0.343339){};
\node (h41) at (-1.98059, -0.219197){};
\node (h51) at (1.98059, -0.219197){};
\node (a11) at (0.3, 2.41225){$E_{01}$};
\node (b11) at (1.54988, -1.80026){$E_{12}$};
\node (c11) at (-1.54988, -1.80026){$E_{20}$};
\node (d11) at (0.797981, 0.594386){$F_{12}$};
\node (e11) at (-0.797981, 0.594386){$F_{21}$};
\node (f11) at (0.979635, 1.76278){$E_{02}$};
\node (g11) at (-0.979635, 1.76278){$E_{00}$};
\node (h011) at (2.46423, -1.40049){$E_{11}$};
\node (h111) at (-2.46423, -1.40049){$E_{21}$};
\node (h211) at (-0.348239, -0.643339){$F_{22}$};
\node (h311) at (0.348239, -0.643339){$F_{11}$};
\node (h411) at (-2.38059, -0.219197){$E_{22}$};
\node (h511) at (2.38059, -0.219197){$E_{10}$};
\fill[black] (a1) circle (2.5pt);
\fill[black] (b1) circle (2.5pt);
\fill[black] (c1) circle (2.5pt);
\fill[black] (d1) circle (2.5pt);
\fill[black] (e1) circle (2.5pt);
\fill[black] (f1) circle (2.5pt);
\fill[black] (g1) circle (2.5pt);
\fill[black] (h01) circle (2.5pt);
\fill[black] (h11) circle (2.5pt);
\fill[black] (h21) circle (2.5pt);
\fill[black] (h31) circle (2.5pt);
\fill[black] (h41) circle (2.5pt);
\fill[black] (h51) circle (2.5pt);
\path[every node/.style={font=\sffamily\small}]
(a1) edge [->,bend left] (f1)
(b1) edge [->,bend left] (h31)
(c1) edge [bend left] (h11)
edge [yellow] (e1)
(d1) edge [->,bend left] (h51)
edge [red] (b1)
(e1) edge [->,bend left] (h21)
edge [red] (h11)
(f1) edge [->,bend left] (g1)
(g1) edge [->,bend left] (a1)
(h01) edge [->,bend left] (b1)
edge [yellow] (d1)
(h11) edge [bend left] (h41)
edge [yellow] (h21)
(h21) edge [->,bend left] (c1)
edge [red] (h41)
(h31) edge [->,bend left] (d1)
edge [red] (h01)
(h41) edge [->,bend left] (e1)
(h51) edge [->,bend left] (h01)
edge [yellow] (h31);
\end{tikzpicture}
\end{array}
$$
\hspace{-0.6cm} Blue lines represent the monodromies of $[\widetilde{W}^t_{V_X(e_i)}]$ for $i=1,...,n-r$, black lines $[\widetilde{W}^t_{V_X(e_j)}]$ for $j=n-r+1,...,n$, green lines represent the monodromies
of $[\widetilde{W}^t_{V_X(v_1)}]$, yellow lines $[\widetilde{W}^t_{V_X(v_2)}]$ and red lines the monodromies of $[\widetilde{W}^t_{V_X(v_3)}]=[\widetilde{W}'^t_{V_X(v_3)}]$. In particular, one has
\bigskip
(1) $M( \Gamma_{V_X(e_i)}) (z_k,w_l) = (z_{k+1},w_l) $ and $M(\Gamma_{V_X(e_i)} ) (z'_m,w'_n)= (z'_{m+1},w'_n)$.
\bigskip
(2) $M( \Gamma_{V_X(e_j)}) (z_k,w_l) = (z_k,w_{l+1}) $ and $M(\Gamma_{V_X(e_j)} ) (z'_m,w'_n)= (z'_m,w'_{n+1})$.
\bigskip
(3) $M(\Gamma_{V_X(v_1)}) (z_k,w_l) = (z'_{k+1},w'_l)$.
\bigskip
(4) $M(\Gamma_{V_X(v_2)}) (z_k,w_l) = (z'_k,w'_{l+1})$.
\bigskip
(5) $M(\Gamma_{V_X(v_3)}) (z'_k,w'_l) = M(\Gamma'_{V_X(v_3)})(z'_k,w'_l)=(z_k,w_l)$.
\bigskip
\hspace{-0.6cm} Relations (1)-(5) establish the $M$-aligned property in this case.
\subsection{M-aligned property for the class (c)} For $b < n-1 $ let $ X_{n,b}=Bl_B(
\mathbb{P}(\mathcal{O}_{\mathbb{P}^{n-1}}) \oplus \mathcal{O}_{\mathbb{P}^{n-1}}(b))$ be the blow up of the projective bundle $\mathbb{P}(\mathcal{O}_{\mathbb{P}^{n-1}} \oplus \mathcal{O}_{\mathbb{P}^{n-1}}(b))$ along the codimension two subspace $B = \mathbb{P}^{n-2}$. One has $$Div_T(X) = \left ( \bigoplus_{i=1}^n \mathbb{Z} \cdot V_X(e_i) \right )
\bigoplus \left ( \bigoplus_{i=1}^3 \mathbb{Z} \cdot V_X(u_i) \right )$$ Let $L_{nml}=n \cdot V +m T + l \cdot Y \in Pic(X)$ be any element. Then a $T$-divisor satisfies $D \in H^0 (X ; L_{nml})$ if and only if $D \in Div_T^+(X)$ is of the form $$ D = \sum_{i=1}^{n-1} n'_i V_X(e_i) + m' V_X(e_n) + n'' V_X(u_1) + m'' V_X (u_2) + l'' V_X(u_3)$$ where $$ \begin{array}{ccccc} n = ( \vert n' \vert + m' +m'' \cdot b) & ; & m=(m''-m'+l'') & ; & l = (n''+m'') \end{array}$$
\hspace{-0.6cm} In particular, $\left \{ V(e_1),...,V(e_n),V(u_1),V(u_2),V(u_3) \right \}$ is a basis for the algebra $A_{\mathcal{E}}$. We denote the resulting quiver by $Q(n,r)$. For example, for $X=Bl_B (\mathbb{P}^2 \times \mathbb{P}^2)$ the quiver $Q(4,2)$ is the following
$$
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=1.9cm]
\node (1) {$E_{01}$};
\node (4) [right of=1] {$E_{11}$};
\node (7) [right of=4] {$E_{21}$};
\node (10) [right of=7] {$E_{31}$};
\node (13) [right of=10] {$E_{31}$};
\node (2) [below of=4] {$E_{00}$};
\node (5) [right of=2] {$E_{10}$};
\node (8) [right of=5] {$E_{20}$};
\node (11) [right of=8] {$E_{30}$};
\node (14) [right of=11] {$E_{30}$};
\node (3) [above of =4] {$F_1$};
\node (6) [above of =7] {$F_2$};
\node (9) [above of=10] {$F_3$};
\node (12) [above of=13] {$F_4$};
\path[every node/.style={font=\sffamily\small}]
(1) edge [bend right=30,blue] node {} (4)
edge [bend right =10,blue] node {} (4)
edge [bend left=30, blue] node {} (4)
edge [bend left = 10,blue] node {} (4)
edge [green] node {} (3)
(2) edge [bend right=30,blue] node {} (5)
edge [bend right = 10,blue] node {} (5)
edge [bend left=30, blue] node {} (5)
edge [bend left = 10, blue] node{} (5)
edge node{} (1)
(3) edge [bend right=30, blue] node {} (6)
edge [bend right = 10,blue] node {} (6)
edge [bend left=30, blue] node {} (6)
edge [bend left = 10,blue] node{} (6)
edge [yellow] node {} (4)
(4) edge [bend right=30, blue] node {} (7)
edge [bend right = 10,blue] node {} (7)
edge [bend left=30, blue] node {} (7)
edge [bend left = 10, blue] node{} (7)
edge [green] node {}(6)
(5) edge [bend right=30, blue] node {} (8)
edge [bend right = 10,blue] node {} (8)
edge [bend left=30, blue] node {} (8)
edge [bend left =10, blue] node{} (8)
edge [red] node {}(3)
edge node{} (4)
(6) edge [bend right, blue] node {} (9)
edge [bend right =10,blue] node {} (9)
edge [bend left=30, blue] node {} (9)
edge [bend left=10,blue] node{} (9)
edge [yellow] node{} (7)
(7) edge [green] node {} (9)
edge [bend right=30, blue] node {} (10)
edge [bend right=10,blue] node {} (10)
edge [bend left=30, blue] node {} (10)
edge [bend left=10,blue] node{} (10)
(8) edge [red] node {}(6)
edge [bend right=30, blue] node {} (11)
edge [bend left=30,blue] node {} (11)
edge [bend left=10, blue] node {} (11)
edge [bend right=10,blue] node{} (11)
edge node{} (7)
(9) edge [bend right, blue] node {} (12)
edge [bend right =10,blue] node {} (12)
edge [bend left=30, blue] node {} (12)
edge [bend left=10,blue] node{} (12)
edge [yellow] node{} (10)
(10) edge [green] node {} (12)
edge [bend right=30, blue] node {} (13)
edge [bend right=10,blue] node {} (13)
edge [bend left=30, blue] node {} (13)
edge [bend left=10,blue] node{} (13)
(11) edge [red] node {}(9)
edge [bend right=30, blue] node {} (14)
edge [bend left=30,blue] node {} (14)
edge [bend left=10, blue] node {} (14)
edge [bend right=10,blue] node{} (14)
edge node{} (10)
(12) edge [yellow] node{} (13)
(14) edge node{} (13)
edge [red] node {} (12);
\end{tikzpicture}
$$ For a divisor $D \in Div_T(X) $ as above and $u \in \mathbb{C}$ consider the loop $$ \gamma^u_{D}(\theta) := \sum_{i=1}^{n-1} e^{-u+2 \pi i n'_i \theta } z_i + e^{2 \pi i m' \theta} w + \frac{e^{2 \pi i n'' \theta }}{ \prod_{i=1}^{n-1} z_i \cdot w^b} + \frac{e^{2 \pi i m'' \theta+ \pi i} }{\prod_{i=1}^{n-1} z_i \cdot w^{b+1} } + \frac{e^{2 \pi i l'' \theta}}{w} $$ For $ \theta \in [0,1)$. Define $ \Gamma_D : = [\gamma_D^t]$ for $0<<t$ and set $\widetilde{M}_D := M(\Gamma_D) \in Aut(Crit(X_{n,b} ; f_t))$. Let $$Crit(X) = \left \{ (z_{k},w_{l} ) \right \}_{k=0,l=0}^{n-1,1} \bigcup \left \{ (z'_m,w'_m) \right \}_{m=1}^{n-1} \subset (\mathbb{C}^{\ast})^{n-1} \times \mathbb{C}^{\ast} $$ be as in Proposition 3.3.1. The following example illustrates the $M$-aligned property for the class (c):
\bigskip
\hspace{-0.6cm} \bf Example 4.6 \rm (monodromies for $X= Bl_B(\mathcal{O}_{\mathbb{P}^4} \oplus \mathcal{O}_{\mathbb{P}^4} )$ with $B= \mathbb{P}^3$): The exceptional map of Definition 3.3.2 is defined in terms of the solutions of the LG-system for $$ f_t(z,w) = e^{-t} z_1+e^{-t}z_2+e^{-t}z_3+e^{-t}z_4+e^{-t} w + \frac{1}{z_1z_2 z_3 z_4} + \frac{i}{z_1 z_2 z_3 z_4 w } + \frac{1}{w} $$ for $0<<t$. The following is a graphical description of the $z_i$-coordinate of the solution scheme $Crit(X ; f_t)$:
$$ \begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=3.5cm,main node/.style={circle,draw,font=\sffamily\bfseries \small}, scale=10.4]
\draw [->,thick] (0,-0.4)--(0,0.4) node (yaxis) [above] {$z(Crit(X; f_t))$}
|- (-0.5,0)--(0.5,0) node (xaxis) [right] {};
\node (a) at (0.19662, -0.0331365){};
\node (b) at (-0.19662, -0.0331365){};
\node (c) at (0.0309771, -0.198729){};
\node (d) at (-0.0309771, -0.198729){};
\node (e) at (0.0905276, 0.175303){};
\node (f) at (-0.0905276, 0.175303){};
\node (g) at (0.267681, -0.262253){};
\node (h) at (-0.267681, -0.262253){};
\node (h1) at (0.267703, 0.273023){};
\node (h2) at (-0.267703, 0.273023){};
\node (h3) at (0.177486, -0.0926983){};
\node (h4) at (-0.177486, -0.0926983){};
\node (h5) at (0.140639, 0.13849){};
\node (h6) at (-0.140639, 0.13849){};
\fill[black] (a) circle (0.5pt);
\fill[black] (b) circle (0.5pt);
\fill[black] (c) circle (0.5pt);
\fill[black] (d) circle (0.5pt);
\fill[black] (e) circle (0.5pt);
\fill[black] (f) circle (0.5pt);
\fill[black] (g) circle (0.5pt);
\fill[black] (h) circle (0.5pt);
\fill[black] (h1) circle (0.5pt);
\fill[black] (h2) circle (0.5pt);
\fill[black] (h3) circle (0.5pt);
\fill[black] (h4) circle (0.5pt);
\fill[black] (h5) circle (0.5pt);
\fill[black] (h6) circle (0.5pt);
\node (a1) at (0.24662, -0.0371365){$E_{00}$};
\node (b1) at (-0.24662, -0.0371365){$E_{21}$};
\node (c1) at (0.0709771, -0.228729){$E_{10}$};
\node (d1) at (-0.0709771, -0.228729){$E_{11}$};
\node (e1) at (0.1305276, 0.215303){$E_{40}$};
\node (f1) at (-0.1305276, 0.215303){$E_{31}$};
\node (g1) at (0.317681, -0.312253){$F_1$};
\node (h01) at (-0.317681, -0.312253){$F_2$};
\node (h11) at (0.317703, 0.313023){$F_4$};
\node (h12) at (-0.317703, 0.313023){$F_3$};
\node (h13) at (0.217486, -0.1326983){$E_{01}$};
\node (h14) at (-0.217486, -0.1326983){$E_{20}$};
\node (h15) at (0.200639, 0.17849){$E_{41}$};
\node (h16) at (-0.200639, 0.17849){$E_{30}$ };
\end{tikzpicture}$$
\hspace{-0.6cm} The corresponding monodromies are described as follows:
$$ \begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=3.5cm,main node/.style={circle,draw,font=\sffamily\bfseries \small}, scale=10.4]
\draw [->,thick] (0,-0.4)--(0,0.4) node (yaxis) [above] {$\textrm{ action of } \widetilde{M}_{V(e_i)} \textrm{ for } i=1,...,n-1$}
|- (-0.5,0)--(0.5,0) node (xaxis) [right] {};
\node (a) at (0.19662, -0.0331365){};
\node (b) at (-0.19662, -0.0331365){};
\node (c) at (0.0309771, -0.198729){};
\node (d) at (-0.0309771, -0.198729){};
\node (e) at (0.0905276, 0.175303){};
\node (f) at (-0.0905276, 0.175303){};
\node (g) at (0.267681, -0.262253){};
\node (h) at (-0.267681, -0.262253){};
\node (h1) at (0.267703, 0.273023){};
\node (h2) at (-0.267703, 0.273023){};
\node (h3) at (0.177486, -0.0926983){};
\node (h4) at (-0.177486, -0.0926983){};
\node (h5) at (0.140639, 0.13849){};
\node (h6) at (-0.140639, 0.13849){};
\fill[black] (a) circle (0.5pt);
\fill[black] (b) circle (0.5pt);
\fill[black] (c) circle (0.5pt);
\fill[black] (d) circle (0.5pt);
\fill[black] (e) circle (0.5pt);
\fill[black] (f) circle (0.5pt);
\fill[black] (g) circle (0.5pt);
\fill[black] (h) circle (0.5pt);
\fill[black] (h1) circle (0.5pt);
\fill[black] (h2) circle (0.5pt);
\fill[black] (h3) circle (0.5pt);
\fill[black] (h4) circle (0.5pt);
\fill[black] (h5) circle (0.5pt);
\fill[black] (h6) circle (0.5pt);
\node (a1) at (0.24662, -0.0371365){$E_{00}$};
\node (b1) at (-0.24662, -0.0371365){$E_{21}$};
\node (c1) at (0.0709771, -0.228729){$E_{10}$};
\node (d1) at (-0.0709771, -0.228729){$E_{11}$};
\node (e1) at (0.1305276, 0.215303){$E_{40}$};
\node (f1) at (-0.1305276, 0.215303){$E_{31}$};
\node (g1) at (0.317681, -0.312253){$F_1$};
\node (h01) at (-0.317681, -0.312253){$F_2$};
\node (h11) at (0.317703, 0.313023){$F_4$};
\node (h12) at (-0.317703, 0.313023){$F_3$};
\node (h13) at (0.217486, -0.1326983){$E_{01}$};
\node (h14) at (-0.217486, -0.1326983){$E_{20}$};
\node (h15) at (0.200639, 0.17849){$E_{41}$};
\node (h16) at (-0.200639, 0.17849){$E_{30}$ };
\path[every node/.style={font=\sffamily\small}]
(h) edge [blue, ->,bend left] (h2)
(h2) edge [blue,->,bend left] (h1)
(h1) edge [blue,->,bend left] (g)
(g) edge [blue,->,bend left] (h)
(b) [blue] edge (f)
(f) [blue] edge (h5)
(h5) [blue] edge (h3)
(h3) [blue] edge (d)
(d) [blue] edge (b)
(h4) [blue] edge (h6)
(h6) [blue] edge (e)
(e) [blue] edge (a)
(a) [blue] edge (c)
(c) [blue] edge (h4);
\end{tikzpicture}$$
$$ \begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=3.5cm,main node/.style={circle,draw,font=\sffamily\bfseries \small}, scale=10.4]
\draw [->,thick] (0,-0.4)--(0,0.4) node (yaxis) [above] {$\textrm{ action of } \widetilde{M}_{V(u_2)}$}
|- (-0.5,0)--(0.5,0) node (xaxis) [right] {};
\node (a) at (0.19662, -0.0331365){};
\node (b) at (-0.19662, -0.0331365){};
\node (c) at (0.0309771, -0.198729){};
\node (d) at (-0.0309771, -0.198729){};
\node (e) at (0.0905276, 0.175303){};
\node (f) at (-0.0905276, 0.175303){};
\node (g) at (0.267681, -0.262253){};
\node (h) at (-0.267681, -0.262253){};
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\bigskip
(1) $\widetilde{M}_{V(e_i)}(w_k,w_l)=(z_{k+1},w_l)$ and $ \widetilde{M}_{V(e_i)}(z'_m,w'_m) = (z'_{m+1},w'_m)$ for $i=0,...,n-1$.
\bigskip
(2) $\widetilde{M}_{V(e_n)}(z_k,w_0)=(z_{k-b},w_{1})$ for $b+1 \leq k \leq n-1$.
\bigskip
(3) $\widetilde{M}_{V(u_1)}(z_k,w_0)=(z'_{k-b},w'_k)$ for $b+1 \leq k \leq n-1$.
\bigskip
(4) $\widetilde{M}_{V(u_2)}(z_k,w_1)=(z'_{k+1},w'_{k+1})$ for $0 \leq k \leq n-2$.
\bigskip
(5) $\widetilde{M}_{V(u_3)}(z'_k,w'_k)=(z_k,w_1)$ for $1 \leq k \leq n-1$.
\bigskip
\hspace{-0.6cm} Relations (1)-(5) establish the $M$-aligned property in this case.
\section{Discussion and Concluding Remarks}
\label{s:LGM}
\bigskip
\hspace{-0.6cm} We would like to conclude with the following remarks and questions:
\bigskip
\hspace{-0.6cm} \bf (a) Monodromies and Lagrangian submanifolds: \rm A leading source of interest for the study of the structure of $\mathcal{D}^b(X)$, in recent years, has been their role in the famous homological mirror symmetry conjecture due to Kontsevich, see \cite{K}. For a toric Fano manifold, the analog statement of homological mirror symmetry relates the structure of $\mathcal{D}^b(X)$ to the structure of the Fukaya-Seidel cateogry $Fuk(\widetilde{Y}^{\circ})$, where $\widetilde{Y}^{\circ}$ is a disingularization of a hyperplane section $Y^{\circ}$ of $X^{\circ}$, see \cite{AKO,Ko2,S}. In this setting, homological mirror symmetry was proven by Abouzaid in \cite{Ab1,Ab2}. On the other hand, the Fukaya-Seidel category is known to admit exceptional collections of Lagrangian spheres, due Seidel's Lefschetz thimble construction, see for instance \cite{S2}. Thus, via homological mirror symmetry these collections translate to exceptional collections of objects in $\mathcal{D}^b(X)$. However, it seems a non-trivial question to discern, in terms of the Fukay-Seidel category itself, which such collections of vanishing spheres (if any) in $Fuk(\widetilde{Y}^{\circ})$ actually correspond to collections of line bundles in $\mathcal{D}^b(X)$, under the mirror map. Examples could be found in \cite{AKO}. It is thus interesting to pose the following question:
\bigskip
\hspace{-0.6cm} \bf Question: \rm Is it possible to naturally associate Lagrangian submanifolds $L(z) \subset \widetilde{Y}^{\circ}$ to solutions of the Landau-Ginzburg system $z \in Crit(X)$ with the property $$HF(L(z),L(w)) \simeq Hom_{mon}(z,w) \hspace{0.5cm} \textrm{ for} \hspace{0.25cm} z,w \in Crit(X)$$ where $HF$ stands for Lagrangian Floer homology?
\bigskip
\hspace{-0.6cm} (b) \bf Further toric Fano manifolds: \rm It is interesting to ask to which extent the methods of this work could be generalized to further classes of torc Fano manifolds? In particular, the main question seems to be in which cases there exists an asymptotic deformation $f_t$ of the Landau-Ginzburg potential $f_X$ such that the corresponding map $E : Crit(X ; f_t) \rightarrow Pic(X)$, defined in section 3, is exceptional for $0<<t$.
\hspace{-0.6cm} Regrading the deformation procedure, let us note that the zero set $V(f)= \left \{f=0 \right \} \subset (\mathbb{C}^{\ast})^n$ of an element $f \in L(\Delta^{\circ})$
is an affine Calabi-Yau hypersurface. In \cite{Ba2} Batyrev introduced $\mathcal{M}(\Delta^{\circ})$ the toric moduli of such affine Calabi-Yau hyper-surfaces which is a $\rho(X)$-dimensional singular toric variety obtained as the quotient of $L(\Delta^{\circ})$ by appropriate equivalence relations. In \cite{Ba2} Batyrev further shows that $PH^{n-1}(V(f)) \simeq Jac(f)$, where $Jac(f)$ is the function ring of the solution scheme $Crit(X ; f) \subset (\mathbb{C}^{\ast})^n$.
\hspace{-0.6cm} In this sense our approach could be viewed as a suggesting that in the toric Fano case homological data about the structure of $\mathcal{D}^b(X)$, could, in fact, be extracted from the local behavior around the boundary of the B-model moduli, which in our case is $\mathcal{M}(\Delta^{\circ})$, rather than the Fukaya category appearing in the general homological mirror symmetry conjecture, whose structure is typically much harder to analyze.
\bigskip
\hspace{-0.6cm} \bf Acknowledgements: \rm The author would like to thank Leonid Polterovich for his ongoing support and interest in this project. The author would also like to thank Denis Auroux for valuable comments and suggestions on previous versions of this work. This research has been partially supported by the European
Research Council Advanced grant 338809 and AdG 669655.
| 53,731
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Theina and India, to an economically-stable southeast Asia. But because of the Myanmar authorities’ historical inability to govern the northern and eastern part of the country bordering India, Bangladesh and southern China, a vast area has remained lawless, threatening to undermine the region as a whole economically. The killing of Muslims by chauvinist Buddhists, with apparent support from the Buddhist Sangha, could have fatal consequences for years to come for those who are not involved in these insane activities but merely reside in the neighborhood.
According to available media reports, on Oct. 21, a violent mob led by a group of Buddhist fanatics, killed resident Muslims and burned down their homes in Rakhine state, located in western Myanmar. A Rakhine government spokesman put the death toll at 112, but within hours state media had revised it to 67 killed, 95 wounded and nearly 3,000 houses destroyed from Oct. 21 to Oct. 25. All those killed were Arakanese Muslims, called Rohingyas. According to Iran’s Press TV, Myanmar army forces provided the Buddhists with containers of petrol to set ablaze the houses of Muslim villagers, forcing them out of their homes. This latest violence against the Muslims by Myanmar’s Buddhists has also drawn the attention of the United Nations. On Oct. 25, Secretary-General Ban Ki-moon urged the authorities in Myanmar to take action to bring an end to the lawlessness currently affecting the state, with the UN chief’s spokesperson describing the outbreak of communal violence in five townships in the state’s north as “deeply troubling.”
Earlier this year on June 4, Buddhist residents in western Burma killed at least nine Muslims as sectarian tension worsens in the region, police say. Reports. The riots continued throughout the month of June.
The New York-based Human Rights Watch released satellite images of the “near total destruction” of a once-thriving coastal community that was reduced to ashes around Kyaukpyu, an industrial zone important to Chinese energy interests.
Chain in the Buddhist Fanatics
Persecution of the Arakanese Muslims by the Buddhists has been a pattern in Myanmar’s national fabric for decades. The heart of the problem lies with the Buddhist-majority government of Myanmar, which refuses to recognize the Rohingyas and has classified them as illegal migrants despite the fact that they are said to be Muslim descendants of Persian, Turkish, Bengali and Pushtun origin who migrated to Myanmar as early as in the 8th century. As a result, Myanmar’s estimated 800,000 Rohingyas have remained officially stateless, and are considered outsiders and not one of country’s 135 officially-recognized ethnic groups. Extremely disturbing is the fact that this persecution of Muslims has now gone beyond the Arakan Hills, and Muslims are now under the Buddhist gun throughout Myanmar.
The Buddhist-led anti-Muslim viciousness took this violent turn in 1978, when more than 200,000 Rohingyas were forced to flee to Bangladesh following targeted Myanmar army operations against them. The army’s aim, as Yangon explained at the time, was to scrutinize individuals living in Rakhine State and designate them as citizens or foreigners in accordance with the law and take action against those who were identified as foreigners by labeling them illegal.
At the time, a report issued by the London-based Amnesty International pointed out that the military campaign directly targeted civilians and resulted in widespread human rights abuses, including extrajudicial killings, rape, destruction of mosques and other religious persecutions. Most of those refugees were later repatriated; but in 1991, more than 250,000 Rohingyas again fled their homes to Bangladesh, citing religious persecution at the hands of the military junta and the Buddhist fanatics, who had joined the organized pogrom.
The persecution of Muslims in Myanmar has now gone beyond the vicinity of the Arakan Hills. Two incidents of grenade attacks on Muslim mosques occurred on Oct. 28 in two different townships in Karen state, which is near Thailand. A bomb exploded at Ye-Nan-Si-Gone mosque of Kawkareik and the Jameh mosque in Kyone-Doe was attacked by grenades, according to local Muslim residents.
Recent intelligence reports indicate that the Democratic Karen Buddhist Army (DKBA), a breakaway group of former Buddhist soldiers and officers of the predominantly Karen Christian-led Karen National Liberation Army (KNLA), has been involved in the destruction of mosques and forced relocation of Muslim villagers. DKBA soldiers have allegedly tried to force Muslims to worship Buddhist monks and have put up Buddhist altars. Restrictions have also been placed on Muslims to force them to become vegetarian. Both the DKBA and the State Peace and Development Council (SPDC) have forced Muslims in Karen State to do menial labor for them on a regular basis, reports claim.
Where Is Suu Kyi and her Democratic Fervor?
Also disturbing is the fact that neither the military authorities nor the civilian political groups have shown any inclination to openly oppose the marauders or come to the aid of their Muslim victims. Myanmar’s military junta has time and again sided with the anti-Muslim fanatics while maintaining its own complex political equations with the Buddhist monks intact. The behind-the-back shaking of hands between the fanatic Buddhists and the military junta on the persecution of Muslims has not gone altogether unnoticed. On the contrary, the fanatics want their backing by the military to be known.
In September, about six weeks before the latest violent attacks, hundreds of Buddhist monks marched in Yangon, calling for the deportation of Rohingya Muslims from Myanmar. Voice of America cited analysts saying the country’s Buddhist chauvinism was shaped by authorities’ attempts to form a “national identity.” The monks were supporting a suggestion by President Thein Sein, who made a public statement saying the Muslim minority, numbering close to a million, should be segregated and deported.
It is evident that the hard-headed military junta, seeking support of the Buddhist fanatics for its own political longevity, is silently backing the persecution of Muslims. And the same could be said for Myanmar’s Nobel Peace Prize winner, Aung Sun Suu Kyi. Though she lived for decades in the international limelight as a victim at the hands of military dictators, Suu Kyi has been noticeably silent. A blue-blooded Burman like Thein Sein, she traveled out of Myanmar to accept her Nobel Peace Prize last July at a time when the Arakanese Muslims’ homes were burnt and people were killed by those whom Suu Kyi represents. (The peace credentials conferred by the prestigious Nobel Prize have already been called into question, of course, by US President Barack Obama, who is directing the unconstitutional and extrajudicial killing of individuals in foreign lands by unmanned aerial vehicles known as drones and labeling all the dead as terrorists.)
Speaking at the International Labor Organization’s (ILO) annual conference last June in Geneva at the height of the June riots, Suu Kyi said that “fear of illegal immigration” lay at the heart of the violence between ethnic Arakanese and the stateless Rohingya minority. .
Suu Kyi has tried to make the issue part of her political campaign by saying the problem lies in the fact that the junta is not ruling the country effectively. But did she visit the victims? No, she did not. And that is because her political campaign is based on projecting a common cause between ethnic Rakhines and Burmans under a common “Buddhist Burmese” identity. And to attain this merger, ostensibly a political necessity, she has little to do with what is happening to the Rohingya Muslims. The nature of this common cause is out there in the open for all to see as the military junta, as well as senior opposition leaders from Suu Kyi’s own party—including Tin Oo, Nyan Win and Win Tin - have together ganged up against the Rohingya Muslims.
The Jihadi Threat
Naypyidaw must notice that its “business as usual” approach in dealing with its Muslim population may end up costing it a lot. Besides the UN, the 57-nation Organization of Islamic Conference (OIC) and a slew of human rights groups have spoken out against anti-Muslim activities within Myanmar. Turkey has taken a serious note of these developments.
Turkish Prime Minister Erdogan’s wife, Emine Erdogan, and daughter Sumeyye Erdogan visited Myanmar to get accurate information about the situation. Mrs. Erdogan has expressed her horror and dismay at the anti-Muslim pogrom, calling it “unbelievable” and “shocking.” The Turkish news weekly, Journal of Turkish Weekly, said: “What’s going on in the region is just an ethnic cleansing, which has been common throughout the 20th century in Burma against Muslims. The majority of the Buddhist population of the country see themselves as a powerful nation and recognize themselves the ‘right’ to attack and chase off the Muslim residents, who were seen as ‘subordinates’ and ‘aliens’ in the country.”
Such reports will not help the Myanmar junta or the civilian democratic forces under Nobel laureate Aung Sun Suu Kyi. And greater threats are lurking around the corner. Muslims around the world, even before the “Arab Awakening,” had begun to take note of the persecution of Muslims. Some of these Muslims have turned toward armed militancy to deal with their persecutors. Their militant and violent acts have been financially supported by the oil and gas-rich Gulf Arabs, Saudi Arabia and Qatar, in particular. With the tacit approval of colonial European powers and the United States, these militants have recently turned Libya and Syria upside down and have also moved toward Islamicizing Egypt, Tunisia and Pakistan, among other nations.
This is a threat that not only extends its ominous shadow over Myanmar, but over a large area where efforts are being made by the major nations of the region to ensure peace and stability. Those efforts focus on accelerating the growth of their physical economies. To say the least, it would be naïve for the Myanmar authorities, and the political figures of that country who have sided with the Buddhist fanatics because of their own prejudices to gain domestic political mileage, to believe that the uprooted Rohingyas, driven away from their homes through the use of violence, will take it lying down and not reach out for support from militant Islamic forces elsewhere.
There are media reports that reveal a militant Rohingya group, the Rohingya Solidarity Organization (RSO), has set up a network in the Cox’s Bazaar district of southern Bangladesh. Afghan Taliban instructors were also seen in some of the RSO camps along the Bangladesh-Myanmar border, while some RSO rebels were reported to be undergoing training in the Afghan province of Khost along with Hizb-e-Islami fighters. One CNN video, reportedly secured from al-Qaeda’s archives in Afghanistan in August 2002, shows Muslim allies from “Burma” receiving training in Afghanistan.
In addition, it has been reported that following the atrocities committed in the Arakan Hills, Prince Khalid bin Sultan bin Abdul Aziz Al Saud, commander of the Saudi Arabian Military, visited Dhaka and recommended waging a military action against Burma.
In 1998, the RSO and Arakan Rohingya Islamic Front (ARIF) combined to found the Rohingya National Council (RNC). The Rohingya National Army (RNA) was also established as the RNC’s armed wing; and the Arakan Rohingya National Organization (ARNO) began to organize all the different Rohingya insurgents into one group. As Wikileaks revealed at the time, ARNO had an estimated strength of about 200 militants, of whom about 170 were equipped with a variety of arms. Ninety members of ARNO were selected to attend a guerrilla warfare course, a variety of explosives courses and heavy-weapons courses held in Libya and Afghanistan in August 2001.
In January 2012, six months before the June riots, a Bangladeshi newspaper reported a merger of different Rohingya militant groups in Bangladesh. According to the news, the RSO, Arakan Movement, Arakan People’s Freedom Party and ARNO had decided to form an alliance and work together.
These are danger signals for the Myanmar authorities to take note of. If the Myanmar leaders do not wake up to these warnings, the region could turn into another hotbed for jihadis posing threats not only to Naypyidaw/Yangon, but also to Dhaka and India’s troubled northeast.
The author is South Asian Analyst at Executive Intelligence Review
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\section{Extensions}\label{extensions}
To extend the results regarding LOBC to more general equations, we must first guarantee that the global existence result of \cite{barles2004generalized} applies to the equations considered. In fact, that the result applies to our model equation, \eqref{modeleq}, will follow as a particular case. For convenience, we restate part of what was mentioned in the introduction. Namely, consider
\begin{equation*}
u_t - F(D^2 u) = f(Du) \textrm{ in } \Omega \times (0,T),
\end{equation*}
where the nonlinearities are as follows: $F:S(n) \rightarrow \mathbb{R}$ is uniformly elliptic, i.e.,
\begin{equation*}
\Mcalm(X - Y) \leq F(X) - F(Y) \leq \Mcalp (X - Y) \quad \textrm{for all } X,Y \in S(n),
\end{equation*}
and vanishes at zero. In particular, this implies that
\begin{equation}\label{extremalbounds}
\Mcalm(X) \leq F(X) \leq \Mcalp(X)\quad \textrm{for all} \ X\in S(n).
\end{equation}
The gradient nonlinearity $f:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfies $f(\xi) \geq |\xi|^2 h(|\xi|)$ for all $\xi\in \mathbb{R}^n$, where $h:\mathbb{R}\rightarrow\mathbb{R}$ satisfies the growth condition \eqref{growthh} and
\begin{align}
& h=h(s) \textrm{ is positive nondecreasing for } s>0,\label{hhyp1}\\
& s\mapsto s^2h(s) \textrm{ is convex},\\
& h(yz) \leq C(h(y) + h(z)) \textrm{ for large } y,z>0 \textrm{ and some } C>0.\label{hhyp3}
\end{align}
The last condition implies that $h$ grows more slowly than any positive power. Examples which satisfy the conditions above are $h(s) = (\log s)^p$ and $h(s) = (\log s)^p (\log \log s)^q$, for large $s$ and $p,q>0$.
\subsection{Comparison, existence and uniqueness}\label{comparisonsubsec}
We look to verify hypotheses (H1) and (H2) needed for the Strong Comparison Principle, Theorem \ref{SCR}. We begin by setting
\begin{equation}
G(x,r,\xi,X) = G(\xi,X) = -F(X) + f(\xi),
\end{equation}
where $\xi\in\mathbb{R}^n$, $X\in S(n)$, and the nonlinearities $F, f$ (and consequently, $h$) are as above. Note that this is compatible with the exchange of sub- and supersolutions mentioned in Remark \ref{signchange}, since $\tilde{F}:S(n) \rightarrow \mathbb{R}$ given by
\begin{equation*}
\tilde{F}(X) = - F(-X), \quad \textrm{ for all } X\in S(n)
\end{equation*}
is uniformly elliptic and vanishes at zero if $F$ does.
We proceed to check (i) through (iii) of property (P) for $h_1(s)=s^2 h(s)$, with $h$ as above:
\begin{enumerate}[(i)]
\item By the growth condition \eqref{growthh}, we have
\begin{equation*}
\int_1^\infty \frac{s}{h_1(s)}\,ds = \int_1^\infty \frac{s}{s^2h(s)} = \int_1^\infty \frac{1}{sh(s)}\,ds < \infty.
\end{equation*}
\item Since $h$ is nondecreasing,
\begin{equation*}
L\mapsto L^2s^2 h(Ls) - CL^2s^2h(s) = L^2s^2 (h(Ls) - Ch(s))
\end{equation*}
is increasing for all $s>0$ and $L\geq 1$.
\item Since $L,s>0$ will be taken large, it is equivalent to show that, for fixed $C,\tilde{C}>0$ and $\epsilon>0$,
\begin{align*}
& h(Ls) - Ch(s) \geq \epsilon,\\
& \epsilon > \nicefrac{\tilde{C}}{Ls}.
\end{align*}
It follows from the growth condition \eqref{growthh} on $h$ that $h(s)\rightarrow\infty$ as $s\rightarrow\infty$. Hence, fixing $s>0$ and taking large enough $L=L(s)>1$, we get $h(Ls) \geq \epsilon + Ch(s).$ The second inequality above comes from choosing $L$ large as well.
\end{enumerate}
This shows \eqref{uniparaboleq} satisfies (H1). Now on to (H2). The second matrix inequality in (H2) implies $X\leq Y + o(1)$. Hence, $\mu X\leq Y + o(1)$ for any $0<\mu<1$, and from the uniform ellipticity and the definition of the Pucci operator, this gives
\begin{equation*}
F(\mu X) - F(Y) \leq \Mcalp(\mu X - Y) \leq o(1).
\end{equation*}
For the contribution of the gradient term to the estimate of (H2), if suffices to have $h_1$ above (i.e., $s\mapsto s^2h(s)$) be locally Lipschitz and satisfy the following, as noted in Example 1 of \cite{barles2004generalized}:
\begin{enumerate}[(H1)]
\setcounter{enumi}{2}
\item For all $C>0$, there exists a sequence $0<\mu_\epsilon<1$ defined for $0<\epsilon\leq 1$ such that $\mu_\epsilon\rightarrow 1$ as $\epsilon\rightarrow 0$ and such that for all large $r>0$ large enough and $0<\epsilon$ small enough, we have:
\begin{equation*}
C\epsilon r \sup_{0\leq \tau \leq r(1+C\epsilon)} |h_1'(\tau)| \leq (1-\mu_\epsilon) \inf_{\tau\geq r(1-C(1-\mu_\epsilon))} (h_1'(\tau)\tau - h_1(\tau)).
\end{equation*}
\end{enumerate}
Given the properties of $h$ above, a lengthy but straightforward computation shows that to verify (H3) it suffices to choose $\mu_\epsilon$ such that $\epsilon^{-1}(1-\mu_\epsilon)\rightarrow +\infty$ as $\epsilon\rightarrow 0$.
\subsection{Loss of boundary conditions}\label{subsLOBCextens}
What follows is an extension of Theorem \ref{mainonball} that includes a more general gradient term, with a suitable growth condition. Consider
\begin{equation}\label{gengradtermeq}
u_t - \Mcalm(D^2 u) = g(|Du|) \ \textrm{ in } \ B_1(0) \times (0,T),
\end{equation}
where $g:\mathbb{R}\to\mathbb{R}$, $g$ is convex increasing for $s\geq 0$, and $g(0) = 0$. Note that \eqref{gengradtermeq} has no ``$(x,t)$-dependence'', so that Lemma \ref{infconvpreserves} applies directly. Lemma \ref{radiallemma} applies as well if $u_0$ is radially symmetric, given that $g=g(|Du|)$.
On the other hand, Proposition \ref{appeqlemma} requires a slight adaptation. In the case that $w(\hat{x},\hat{t}) < u_{\epsilon, \kappa}(\hat{x},\hat{t})$, where $(\hat{x},\hat{t})$ is a point of second-order differentiability of the regularized function $w$, we must take the regularization parameter $\delta = \delta(g)$ small enough so that
\begin{align*}
w_t(\hat{x},\hat{t}) - \Mcalm(D^2w(\hat{x},\hat{t})) - g(|Dw(\hat{x},\hat{t})|) \geq{}& - \frac{K}{\kappa^\frac{1}{2}} + \frac{\lambda}{\delta} - \frac{\Lambda(n-1)}{\epsilon} - g\left(\frac{K}{\epsilon^\frac{1}{2}}\right)\\
\geq{}& 0
\end{align*}
to get that $w$ is a supersolution.
For the statement of the following Lemma, we define
\begin{equation*}
a(s) = \sup_{y>0} \frac{g^{-1}(ys)}{g^{-1}(y)}, \quad s\geq 0,
\end{equation*}
and recall the definition of the convex conjugate,
\begin{equation*}
g^*(s) = \sup\,\{\,ys - g(y)\,|\, y \in \mathbb{R}\,\}.
\end{equation*}
\begin{lemma}\label{gengradtermlemma}
Let $u_0\in C(\overline{B_1(0)})$ be a radial function, and $g$ as described above. Assume also that $g$ is such that \eqref{gengradtermeq} satisfies \emph{(H1)} and \emph{(H2)} of Section \ref{comparison}. If
\begin{equation}\label{growthg*}
\int_1^\infty \frac{g^*(La(s))}{s^2} \,ds < \infty
\end{equation}
for all $L>0$, then there exist positive constants $\delta$ and $M$, depending only on $\lambda, \Lambda, n$ and $g$, such that, if
\begin{equation}\label{intconditiongengrad}
\int_\delta^{1-\delta} u_0(r) - \frac{1}{2}\|u_0\|_\infty \,dr > M,
\end{equation}
then the solution $u$ of \eqref{gengradtermeq}, \eqref{boundarydata}, \eqref{initialdata} with $\Omega = B_1(0)$ and initial data $u_0$ has LOBC at some finite time $T=T(u_0)$.
\end{lemma}
\textit{Proof:} Aside from using all the auxiliary results leading to Theorem \ref{mainonball}, the proof follows that of Theorem 5.2 in \cite{souplet2002gradient}. We repeat most of the argument for convenience. Once more, we proceed by contradiction, assuming $u$ is a solution which satisfies \eqref{boundarydata} in the classical sense. We consider $\varphi_1$ as previously defined, and again denote by $w$ the function obtained by regularizing the radial part of the solution $u$ of \eqref{gengradtermeq} for $\Omega=B_1(0)$. This function now satisfies, for arbitrary $\epsilon>0$ and $0< t_0 < t_1<T$,
\begin{equation*}
\tilde{\rho} w_t \geq (\rho w')' + \tilde{\rho}g(|w'|) \quad \textrm{for a.e. } r\in(\epsilon,1-\epsilon), t\in (t_0,t_1).
\end{equation*}
From the definition of $a$, setting $y = g(|w'(r,t)|)\varphi_1(r,t)$ for ${r\in(\epsilon, 1-\epsilon)}$, ${ t_0<t<t_1}$, we have
\begin{equation*}
a(\nicefrac{1}{\varphi_1}) g^{-1}(g(|w'|)\varphi_1) \geq g^{-1}(\nicefrac{y}{\varphi_1}) = |w'|.
\end{equation*}
Hence,
\begin{equation}\label{g-a-ineq}
g(|w'|)\varphi_1 \geq g\left(\frac{|w'|}{a(\nicefrac{1}{\varphi_1})}\right).
\end{equation}
Let $L>0$ to be chosen later. By the definition of the convex conjugate, we have
\begin{equation*}
L|w'| = \frac{|w'|}{a(\nicefrac{1}{\varphi_1})}La(\nicefrac{1}{\varphi_1}) \leq g\left(\frac{|w'|}{a(\nicefrac{1}{\varphi_1})}\right) + g^*(La(\nicefrac{1}{\varphi_1}))
\end{equation*}
for a.e.~$r\in(\epsilon, 1-\epsilon), t_0<t<t_1$ (we have omitted the arguments for simplicity). This is an instance of Fenchel's inequality, analogous to that of Hölder's inequality in the proof of Theorem \ref{mainonball}.
Using \eqref{g-a-ineq}, we have
\begin{equation*}
L\int_{\epsilon}^{1-\epsilon} |w'| \tilde{\rho} \,dr \leq \int_{\epsilon}^{1-\epsilon} g(|w'|)\varphi_1 \tilde{\rho} \,dr + \int_{\epsilon}^{1-\epsilon} g^*(La(\nicefrac{1}{\varphi_1})) \tilde{\rho} \,dr.
\end{equation*}
That the second integral is finite follows from \eqref{growthg*}. It can also be proven that it is bounded by a constant C that does not depend on $\epsilon$, as in Lemma \ref{eigenlemma}.
Arguing as in the beginning of the proof of Theorem \ref{mainonball}, we obtain
\begin{equation*}
\dot{z}(t) + \lambda_1 z(t) \geq I
\end{equation*}
where $z$ is defined exactly as before, but now
\begin{align*}
I :={}& \int_{\epsilon}^{1 - \epsilon} g(|w'|) \varphi_1 \tilde{\rho} \,dr\\
\geq{}& L \int_{\epsilon}^{1 - \epsilon} |w'|\, \tilde{\rho} \,dr - \int_{\epsilon}^{1-\epsilon} g^*(La(\nicefrac{1}{\varphi_1})) \tilde{\rho} \,dr\\
\geq{}& L \int_{\epsilon}^{1 - \epsilon} |w'|\, \tilde{\rho} \,dr - C\\
\geq{}& L \left(\int_{\epsilon}^{1 - \epsilon} |w|\, \tilde{\rho} \,dr - w(1-\epsilon, t) \right) - C,
\end{align*}
where the last inequality is follows by applying Lemma \ref{weighedpoincare}. Setting $L$ sufficiently large, in terms of $\|\varphi_1\|_\infty$ and $\lambda_1$ (both of which are independent of $\epsilon$), and noting that $w(1-\epsilon, t)=o(1)$ as $\epsilon\to 0$, as before, we obtain
\begin{equation}\label{EDOgengrad}
\dot{z}(t) \geq z(t) - C \quad \textrm{for a.e. } t\in(t_0,t_1),
\end{equation}
which we can integrate to get
\begin{equation}\label{EDOgengradint}
z(t) \geq (z(t_0) - C)e^{t - t_0} \quad \textrm{for all } t\in(t_0,t_1).
\end{equation}
To conclude, note that \eqref{EDOgengrad} does not blowup in finite time, as does \eqref{blowupODE}. We will find a contradiction in the form of a bound, derived from \eqref{EDOgengradint}, which we can easily violate by choosing the appropriate $u_0$. Also, since our aim is to prove nonexistence beyond some finite time, we may assume $T>0$ is large to achieve this contradiction. By choosing the time-regularization parameter small as well, the difference $t_1-t_0$ can be made large as well, say $t_1-t_0\geq\nicefrac{T}{2}$.
As in \eqref{estimzinit}, we have
\begin{equation}\label{estimminitzext}
z(t_0) \geq \int_{\delta}^{1-\delta} u_0(r) \hat{\varphi}(r) \hat{\rho}(r)\,dr + o(1), \quad \textrm{as }\epsilon,\, t_0\to 0.
\end{equation}
On the other hand, by evaluating \eqref{EDOgengradint} at $t=t_1$,
\begin{align*}
z(t_0) \leq{}& e^{-(t_1-t_0)} z(t) + C\\
\leq{}& e^{-\frac{T}{2}}\int_{\epsilon}^{1-\epsilon} w(r,t)\varphi_1(r)\tilde{\rho} \,dr + C.
\end{align*}
Recalling the bounds for $\varphi_1,\,\tilde{\rho}$, that $\hat{\varphi},\,\hat{\rho}$ are bounded by below by a positive constant in $[\delta, 1-\delta]$, and that $\|w\|_\infty\leq\|u_0\|_\infty$, we take $T$ sufficiently large so that
\begin{equation*}
z(t_0) \leq \frac{1}{2} \int_{\delta}^{1-\delta} \|u_0\|_\infty\hat{\varphi}\hat{\rho}\,dr + C.
\end{equation*}
Combining both estimates for $z(t_0)$, we have
\begin{equation*}
\int_\delta^{1-\delta} (u_0 - \frac{1}{2}\|u_0\|_\infty) \hat{\varphi}(r)\hat{\rho}(r) \,dr < C,
\end{equation*}
where $C=C(\lambda, \Lambda, n, g)$. This bound is readily violated by choosing a suitably large $u_0$, and as before, it is equivalent to the one in the statement of the lemma since $\hat{\varphi}$ and $\hat{\rho}$ are uniformly bounded from below in $[\delta, 1-\delta]$. \hfill\qed
Finally, we proceed with the proof of the extension mentioned in the introduction.
\noindent\textit{Proof of Theorem \ref{uniparabolthm}:} First, define $g(s) = s^2 h(s)$, where $h$ satisfies {\eqref{hhyp1}-\eqref{hhyp3}} and the growth condition \eqref{growthh}. It is proven in Lemma 5.3 and the Completion of Theorem 2.2 in \cite{souplet2002gradient} that $g$ so defined satisfies the hypothesis of Lemma \ref{gengradtermlemma}, including \eqref{growthg*}. Furthermore, using \eqref{extremalbounds}, if $u$ is a solution of \eqref{uniparaboleq}, formally we have that
\begin{align*}
u_t - \Mcalm(D^2 u) \geq u_t - F(D^2 u) = f(Du) \geq |Du|^2 h(|Du|) ={}& g(|Du|)\\
&{}\textrm{ in } \Omega \times (0,T),
\end{align*}
and this is readily checked using test functions. Hence, $u$ is a supersolution of \eqref{gengradtermeq} in $\Omega\times (0,T)$. Using the interior sphere condition, without loss of generality we may assume that $B_1(0)\subset \Omega$ with $B_1(0)$ tangent to $\partial \Omega$ at some point $x_0\in\partial \Omega \cap \partial B_1(0)$ (This is equivalent to repeating the constructions of Theorems \ref{mainonball} and \ref{gengradtermlemma} on a ball of arbitrary radius and performing translation.) Arguing as in the proof of Corollary \ref{gendomain}, we conclude that $u$ is a supersolution of \eqref{gengradtermeq} in $B_1(0)\times (0,T)$. Let $\tilde{u}_0 \in C(\overline{B_1(0)})$ nonnegative, radially symmetric and satisfies \eqref{intconditiongengrad}, and consider the solution $\tilde{u}$ of \eqref{gengradtermeq} with initial data $\tilde{u}_0$ and homogeneous boundary data. By Lemma \ref{gengradtermlemma}, $\tilde{u}$ has LOBC in finite time $T'>0$, and since it is radially symmetric we may conclude that LOBC occurs at $x_0\in B_1(0)$. Now define
\begin{equation*}
u_0(x) =\left\{\begin{array}{rl}
\tilde{u}_0(x), &\quad x\in B_1(0)\\[6pt]
0, &\quad x\in \overline{\Omega}\backslash B_1(0).
\end{array}\right.
\end{equation*}
By comparison, we have that the solution $u$ of \eqref{uniparaboleq}-\eqref{boundarydata}-\eqref{initialdata} with initial data $u_0$ satisfies $u(x_0,T')\geq\tilde{u}(x_0,T')>0$, hence $u$ has LOBC.\hfill\qed
\begin{remark}
The more general nonlinearities do not admit a rescaling argument like the one given in the proof of Corollary \ref{gendomain}. Furthermore, \eqref{uniparaboleq} may no longer have radial symmetry. The preceding argument is in some sense simpler, but it does not provide a condition one can check for any given initial data like the one in Corollary \ref{gendomainbis}.
\end{remark}
\begin{remark}
The typical example for the nonlinearity $h$ in Theorem \ref{uniparabolthm} is $h(s) = (\log s)^q$ for $q>0$ and large $s$. In this case, the growth condition \eqref{growthh} forces that $q>1$. This is consistent with what is known to be a more precise condition for preventing GBU in the case of the viscous Hamilton-Jacobi equation: for
\begin{equation*}
u_t - \Delta u = f(u,\nabla u) \quad \textrm{ in } \Omega\times (0,T),
\end{equation*}
GBU does not occur if
\begin{equation*}
|f(u,\nabla u)| \leq C(u)(1 + |\nabla u|^2)h(|\nabla u|)
\end{equation*}
where $C(u)$ is locally bounded, and $h$ is positive nondecreasing and satisfies
\begin{equation*}
\int_1^\infty \frac{1}{sh(s)} = \infty.
\end{equation*}
See \cite{quittner2007superlinear}, Ch. IV, and the references therein.
\end{remark}
\begin{remark}\label{rmkextSphereUni}
The proof of Theorem \ref{uniparabolthm} uses only the uniform interior sphere condition. Both interior and exterior sphere conditions are assumed to establish a connection between Theorems \ref{localexthm} and \ref{uniparabolthm}. Specifically, to have both results applicable in the same situation. See also Remarks \ref{intSphereLocal}.
\end{remark}
| 67,597
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Evocative and the antithesis of Instagram...the work of Adam Harriden
Evocative and the antithesis of Instagram...the work of Adam Harriden
Photographer Adam Harriden eschews both the latest in high-tech cameras and the immediacy of Instagram in favour of the Polaroid camera.
Australian photographer Adam Harridan loves working with Polaroid because of its capacity to capture feeling and memory in a way the razor sharp digital photography we see around us everyday can't. Pictured above: evoking distant memories of carefree holidays and youth, perhaps, tinged with a sadness that what's gone won't come again..Two Lovers
Polaroid cameras were the height of cool in the '70s and '80s, and memory of someone saying 'smile' and a picture of the occasion being passed round a few minutes later makes you wonder they haven't been updated, given that for all the photos we have on our phones, hardly any of them ever get printed..
For photographer Adam Harriden, Polaroid is an artform and he uses Polaroid film because it creates a particular look and atmosphere - much as Kodachrome film had its particular intensity of colour. 'For me we live in era were technology makes everything so easy,' he says. 'Polaroid for me offers the opposite - a challenge to create something with faults, cracks, blurs, a chance to get it wrong and yet create something so unique and beautiful. And what's so wrong with something being difficult? Polaroid is difficult at every stage and to me that's art.'
Polaroid pictures did fade over time, given the chemicals that processed the image before your eyes kept on working. Harriden's pictures will not fade away, and obviously he's developed his own techniques using the medium of Polaroid. 'It is a very technical process and it has its limitations: you can't shoot into light, you can't shoot in the dark, you can't be to far away or to close, each shot needs to be transferred in the dark so I've had to make my own portable dark room that I set up on the beach or wherever I am.
'The film - which isn't on sale to the public any more - is very expensive, and there's a real craft and skill to get it right - in the beginning I was working through trial and error and it still means a lot to me when I get it right and achieve a great shot.'
The charm of Polaroid film lies in its imperfections, says Harriden, who thinks advanced digital cameras have been detrimental to photography as an artform: 'How do photographers today differentiate themselves when all you need is a good camera, the right location, and your done? Everyone's doing the same thing these days and perfection gets very boring fast.
'So to me, the more cracks, blurs, and pixellations the better.'
Harriden, who uses a 1968 103 Polaroid LAND camera, can print his images in any size up to 1m square and he thinks his images look best when in large format. 'The bigger I print the work the more amazing it looks because the little imperfections and individual traits all come out, so to speak.'
As to how eco his work is, he thinks he's doing ok. 'I try to use recycled paper for my prints and eco friendly inks. And our frames are all sustainable Tasmanian oak.'
You can order images online and Harriden ships worldwide.
To see him at work, see this video
| 179,409
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17 Answers
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Working Hardly on OC with hope that it is Legit and will pay for my hard work and effort.
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| 135,724
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Danny Barton
Benjamin Bosse High Class of 1967
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| 59,636
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TITLE: Determining whether equality $ \|T v\| = \|T\| \cdot \|v\|$ is possible
QUESTION [2 upvotes]: As an exercise, I'm supposed to determine whether for the operators $A_\lambda:g(t)\mapsto \sqrt\lambda g(\lambda t)$ on $C[0,1],\lambda\in (0,1)$ and $T_f:g\mapsto g(t)f(t)$ on $L^2$ it is possible to have equality $ \|T v\| = \|T\| \cdot \|v\|$ for some $v\neq 0$, where $T$ is each of the operators above? I'm pretty clueles here.
I found $\|A_{\lambda}\|=\lambda$ and $\|T_f\|=\|f\|_\infty$.
$$\|A_\lambda f\|^2=\lambda\int_0^1f(\lambda t)^2dt=\lambda \int _0^\lambda f(t)^2=\lambda\int _0^1f(t)^2\chi_{[0,\lambda]}^2\leq \lambda \|f\|\|\chi_{[0,\lambda]}\|=\lambda ^2\|f\|$$ so taking $\|f\|=1$ and taking the supremum gives just $\lambda$.
REPLY [2 votes]: Your sequence of equalities should have been
$$
\|A_\lambda f\|^2=\lambda\int_0^1f(\lambda t)^2dt= \int _0^\lambda f(t)^2=\int _0^1f(t)^2\chi_{[0,\lambda]}^2\leq \|f\|\|\chi_{[0,\lambda]}\|=\lambda \|f\|.
$$
Now, I don't know what inequality you are using, but I cannot make sense of it. And you get $\|f\|$ instead of $\|f\|^2$, which is a bad sign.
The norm of $A_\lambda$ is $1$. Because
$$
\|A_\lambda f\|^2=\lambda\int_0^1f(\lambda t)^2dt= \int _0^\lambda f(t)^2
\leq \int _0^1 f(t)^2=\|f\|^2,
$$
so $\|A_\lambda\|\leq1$. Now let $h\in C[0,1]$ be supported in $[0,\lambda]$. Then
$$
\|A_\lambda h\|^2=\int_0^1\,\lambda\,|h(\lambda t)|^2=\int_0^\lambda |h(t)|^2\,dt=\int_0^1|h(t)|^2\,dt=\|h\|^2.
$$
In particular, $\|A_\lambda h\|=\|A_\lambda\|\,\|h\|$.
For $T_f$, if $\|T_fg\|=\|f\|_\infty\,\|g\|$, this equality (squared) is
$$
\int_0^1 |f(t)|^2\,|g(t)|^2\,dt=\int_0^1\|f|_\infty^2\,|g(t)|^2\,dt,
$$
so
$$
0=\int_0^1(\|f\|_\infty^2-|f(t)|^2)\,|g(t)|^2\,dt.
$$
Since the integrand is non-negative, the only way the integral can be zero is if $$
(\|f\|_\infty^2-|f(t)|^2)\,|g(t)|^2=0\ \ \ \text{a.e.}
$$
So we need $g$ to be zero on all those points where $|f(t)|<\|f\|_\infty$. This could easily fail, if $|f(t)|<\|f\|_\infty$ a.e., then only $g=0$ would satisfy the equalty. At the other end of the spectrum if $f$ is constant, the equality holds for any $g$.
| 170,861
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The spiritual path is lonely
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04.03.2017 at 05:41 Bashicage:
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| 185,985
|
SoMe: Instagram Stories – a change in Social Media
This is an unusual article for my lifestyle blog but today, I would like to introduce you to a new category: Social Media. As some of you might know, I am really a big fan of Social Media and I see it as a perfect outlet for a person or company to reach out to […]
| 196,511
|
10:00am:
The occasion begins in the Temple of Damerika, Goddess of earth and motherhood, who the Queen regularly worshipped. The Queen lies in her coffin, looking stunning and very peaceful in her gown. Whoever laid her out did a fabulous job, she could almost be sleeping. Flowers of the kingdom cover the coffin and were laid carefully all around.
10:34am:
For the ceremony, only the Royal family, the priestesses of Damerika and a small number of dignitaries are admitted. Oh, and little me of course. The priestesses light about a billion candles and sing some chants to lighten the mood or something. The High Priestess blesses the Queen's body and spirit and then she and the King commence the 'returning to the earth' rights in the ancient language of Abevorn, ensuring Queen Janna will be welcomed into Damerika's realm.
1:03pm:
The ceremony took about three hours to complete. It was all very worshipful and sombre and, frankly, a little dull. If I hadn't had my puzzle beads and if the family of Lady Wolfrich hadn't turned up in those dreadfully unfashionable wolfskins they're so fond of, I really not sure how I would have kept myself entertained.
1:28pm:
The coffin is now being covered in black silk. It will be born to the Royal Carriage by the Queen's Brothers, Duke Tomithy and Duke Sengir, her brother-in-law, Lord Magnus, and his fellow Lord-Councillors, Lord Hawkcroft, Lady Wolfrich and the Duchess Summer-Mae. Ordinarily, Prince Mikael would have been one of the coffin-bearers, but with the foolish boy off lost somewhere, the Duchess Summer-Mae insisted on taking his place. Her pale yellow dress accompanied by an ebony headscarf is somewhat at odds with the monotone black worn by everyone else, but she is such a determined free spirit and does look rather stylish, so I say we should forgive her.
1:31pm:
The King takes his place in the carriage ahead of Queen Janna's coffin. Curtains are drawn across the windows, presumably so the public cannot see their regent weep, even though I'm sure everyone could sympathise with tears at a time like this. Though for the womenfolk and more trendsetting menfolk, I recommend applying some sort of make-up waterproofing before turning on the waterworks. Both tar and pitch possess admirable waterproofing properties, while also serving as a more than adequate form of mascara.
1:34pm:
The funeral procession gets underway. I shall be riding alongside the escort of Royal Guard. The guard look exceptionally spiffy in the black version of their uniforms. Their stoic and admirable devotion to duty means I have so far managed to pinch three bottoms without even a flinch, let alone a reprimand.
1:48pm:
The procession moves through the City slowly, starting down Temple Street then turning north onto Market Street to begin a loop of the key areas of the Royal City and to give people a chance to pay their respects. The route has been marked with flags to inform the otherwise ignorant populace where they should wait.
2:13pm:
The crowds are eerily silent with many holding lit candles. A few peasants wave as we pass. Others stare on with tears staining their cheeks. I too find myself getting emotional, but manage to sniff it back before it blots my notes.
2:50pm:
We travel from Market Street in the heart of the trade district, along Silver Street in the jewellery quarter and onto Spice Lane, which is rich with the amazing smells that it gave the street its name. Moving on past the premises of Isabel Taurmond, perfumier to Royalty, we smell the special scent created in memory of the Queen. Its delicate grace notes are clear and sweet over the less delectable smell of fish wafting in from the Southern Docks.
3:14pm:
The procession moves slowly and when we enter the flower markets of Blossom Boulevard, we find locals have strewn blossoms over the road like a carpet.
3:23pm:
People are still holding candles and generally looking glum and we turn onto Summer Boulevard, passing the cafes and bakeries, before finally making our way via Angel Row to head to the Dockmaster's Gate and an emotional exit from the City.
3:34pm:
Oh. My. Gods. And. Or. Goddesses.
Can it be?
Honest to goodness, I do believe it is - riding in through the Dockmaster's Gate, accompanied by grizzled riders whose faces I do not recognise - it's... Prince Mikael!
He looks changed from the lanky youth we all know and have mild affection for. His skin is red and rough and covered in dirt and scratches. His clothes are equally worn and ragged - grossly unbecoming of the heir to the throne. What on earth has he been up to these past few weeks beyond saving up a heart attack for our dear Royal Costumier?
The funeral procession halts and Prince Mikael's uncouth party stops before it. The Prince himself dismounts so quickly he almost falls, but his fall becomes a stumble that carries him over to the Royal Carriage. There he drops to knees. A grubby hand is raised to touch the glass protecting the Queen's coffin from the world outside.
It looks as if Prince Mikael is about to speak, but the words catch in his throat. He hangs his head instead, his body visibly trembling.
I look to the doors of the carriage and their covered windows, half-expecting the King to step out to comfort his son, but they remain closed. A member of the public attempts to do what the King would not: a girl of no more than sixteen years of age breaks from the crowd and rushes over to the Prince.
She doesn't reach him.
A member of the Royal Guard urges his horse forward and clatters the girl to the ground with a strike from his shield. Shouts of protest go up from the crowd and the Royal Guard lining their street raise swords to dissuade further dissent.
But now the Prince is back on his feet, wiping at his eyes with the back of a forearm. All eyes are upon him as stops by the door of the carriage. Words are exchanged with those inside, but I don't hear them. All I see is the Prince's face turning from sadness to anger, then he storms over to his horse, re-mounts and sends it galloping on toward the Royal Castle, his companions in tow.
Wow.
I don't think any of us were expecting that.
Except maybe the clairvoyants. And the soothsayers. And the oracles. But if they were expecting it, they certainly didn't tell the rest of us.
The procession is moving again. I don't know if the hush now descended upon the crowd is due to respect for the departed or shock at what they've just seen.
4:15pm:
The streets were lined ten deep or more on the streets of the City in their stunning quiet vigil, but the sheer mass of people outside the gates is staggering. It's a mournful sea of faces. As we travel on, the crowd spontaneously begin to sing one of the old songs that the Queen always loved so much. That's the moment I may have blotted my notes a little.
Conclusion:
Well, that's it, darlings! I will be sending regular updates on the Queen's final journey back to her birthplace, so keep reading the Herald and, more importantly, keeping reading me!
By Katar Bristicus
| 131,220
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Severn Trent De Nora Strengthens Global Sales and Support Network for BALPURE® and MARINER OMNIPURE® Product Lines
FORT WASHINGTON, Pa. – boat building market. Recently Turkey has been receiving remarkable.
About Severn Trent De Nora, LLC
For over 35 years, Severn Trent De Nora, LLC () has been designing, manufacturing and offering innovative electrolytic disinfection treatment solutions to the marine and offshore markets. With a proven history and a track record of delivering economical and reliable disinfection solutions, Severn Trent De Nora sets a new standard in ballast water treatment with the Type-Approved BALPURE® system.
Severn Trent De Nora is a joint venture between Severn Trent Services and Industrie De Nora. Severn Trent Services is a business of Severn Trent PLC (), a leading FTSE100 company.
About Delmar Marine
Delmar Marine was founded in 2004 and represents well-known global manufacturers covering a wide range of equipment to service the Turkish shipping and yachting industry. Delmar Marine employs qualified naval architects and marine engineers with a permanent staff presently consisting of 30 persons. The company's main office is located in Istanbul, Turkey, with branches in Izmýr/Turkiye and Rotterdam/Holland.
| 18,609
|
So many pixels have been killed over the 84 Lumber Super Bowl spot, I’m reluctant to add too many more. But I do think this ad is important, and worthy of note, because it does a lot of important things very well. It may also signal (or at least reflect) a shift in the way “popular advertising” is created and consumed.
Also, it’s hard to ignore some parallels between The Journey Begins and another seminal Super Bowl ad, one that shows up on so many “Best Ever” lists As a reminder, here are the 2 spots:
The similarities go beyond the numbers 8 and 4.
They don’t “sell”
Much of the criticism around 1984 centered on its obscurity. What are they selling? Why should I buy it? What are the Reasons to Believe? As a (then) smaller, challenger brand, Apple realized that it mattered more to stake out a place in the public consciousness, and the national conversation, then it did moving units. They trusted that would come as a result and we know how the story ended. But despite the fact that this ad is now hailed as history-making, there were many at the time who saw it as a monumental waste of time and money. Just like, today, some experts and others are saying that 84L wasted $5 Million, just like that.
They have value (and Values) beyond immediate ROI.
It cost 84L a lot of money to run this spot. A lot. But that paid for itself, countless times over, in the earned media it got them in the run-up and aftermath of the Super Bowl. They may not have sold one additional plank of wood because of it. Apple may not have been able to attribute a single sale to 1984. Of course today, earned media and buzz is what it is all about. But Give 84L credit for understanding that they’d get a lot more mileage about a serious and sober story about opportunity than from another hipster irony-fest.
They’re cinematic.
Much in the way Ridley Scott and Chiat-Day’s 1984 reflected and even presaged the video storytelling techniques of its time, so too does The Journey. It’s rapid cutting, untethered camera and changing POV (from Rob Shapiro, Creative Director and his team at Brunner) reads mostly like an indie film or a doc-wth-heart. Not to say that other spots during SB LI didn’t have artistic merit, but this one reflects the creativity of the moment unlike the others.
They proudly declare Creativity Matters.
Can we talk a minute about how stunningly well cast and acted this spot is? My goodness, if you don’t choke back a tear when the mother first realizes the wall might end their journey or when the girl hands her the flag made from found objects, you may not actually have a beating heart. Just as Apple could have run another product-feature-heavy hard sell, 84L could have made their recruitment spot all sunny, glowing “Morning in America” fare. Neither creative team was content with that. We know what it did for Apple. It will be interesting to see where it takes 84L
| 350,745
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TITLE: Equation with a definite integral - can I differentiate it?
QUESTION [2 upvotes]: I have an equation like this:
$$te^{t} = \int\nolimits_0^t e^\tau u(\tau)d\tau$$
I don't really know how to solve it.. Would it be possible to differentiate both sides of the equation? If so, how can I do it - is it like differentiate one side and then the other or is it more complex? And if it is possible what conditions need to be satisfied in order to do it?
REPLY [4 votes]: If we differentiate the left hand side we get $e^t+t e^t$, if we differentiate the right hand side we get $e^t u(t)$ by the fundamental theorem of calculus. Then we see
$$u(t)=t+1$$
An we are done. I think we need to demand that $u(t)$ is continuous to get uniqueness. In general $u(t)=t+1$ almost everywhere.
| 119,101
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