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When 'Akbar the Great' begged an Indian Queen to spare his life: Congress scraps the glorious history of Kiran Kumari, a Symbol of Bravery and Honour
Kiran Kumari, the Queen of Bikaner, is one Legend who grounded and humiliated the Mughal Emperor Akbar, who tried to violate the modest of women under the guise of anonymity in Meena Bazar
Agnima Sharma, Jaipur
Congress, over the years, has been scrapping the glorious history of Indian royals such as Kiran Kumari, the queen of Bikaner who made a mark in Indian history for her act of bravery. Her story should be known by one and all as she is one woman whose story speaks of woman valour, self-respect, dynamism and empowerment. Once upon a time, she single-handedly grounded the Mughal emperor, Akbar, under her feet who begged her to spare his life. In fact, India has many such inspiring stories of Indian kings and queens which are still to be known by its people. However, Congress seems busy glorifying the acts of a few of its leaders, says a veteran leader Bajrang Singh Shekhawat from Bikaner.
Shekhawat in Bikaner represents many social organisations and feels proud to be born in India which is the land of warriors.
However, presently he feels equally pained at the way the stories of brave soldiers are slowly losing their name under Congress rule.
“We remember this brave queen on every festival and occasion as it was because of her brave act that Meena Bazaar being organised in Delhi was stopped,” he says adding that he wants her story to symbolise woman power.
The story of Kiran Devi recently went viral soon after BJP state president Madanlal Saini on the occasion of Maharana Pratap Jayanti, quoted Akbar as a characterless man in Jaipur.
Saini claimed that Mughal emperor Akbar once attempted to misbehave with Bikaner queen in the women-run market, Meena Bazaar in Delhi. However, the Bikaner queen, Kiran Devi, saw through his ill intentions and pushed him to the ground and pointed a knife to his chest. She asked who he was and Akbar told her that the emperor of Hindustan was lying on her feet begging for his life. Meena Bazaar was closed since that day, said Saini.
He also said that comparing Maharana Pratap with the Mughal emperor is like making a mockery of history.
Saini further claimed that the misdeeds of Akbar have been recorded in history and he used to organise Meena Bazaar with the same purpose.
Akbar “was never great” and was visiting Meena Bazaar just to “exploit” women, said Saini. Adding that “world is well aware of the fact that Akbar organised ‘Meena Bazaars’ a market meant only for women. Surprisingly, men were not allowed to visit the market. However, Akbar visited them hiding his identity to commit ‘dushkarm’,” claimed Saini.
Meanwhile, another resident of Bikaner Lokendra Singh claims that the story of Queen Kiran Kumari should stand for woman empowerment which stand strong in historical pages of India. She was one brave lady who single-handedly dragged the Mughal emperor under her feet and brought Meena Bazaar to an end. Let the world know her story, he says.
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On a surprising verse from Lil Wayne‘s Dedication 4 mixtape, Nicki Minaj rapped a line referencing Mitt Romney over Kanye West and G.O.O.D. Music‘s “Mercy” instrumental. During the verse Nicki Minaj reveals that she will vote for Mitt Romney in the upcoming election and says that she is a republican.
“I’m a republican voting for Mitt Romney, you lazy b*tches are f*cking up the economy”. –Nicki Minaj
Continue reading to listen to Nicki Minaj‘s Mitt Romney endorsing verse.
Nicki Minaj‘s full “Mercy” freestyle with Lil Wayne which the clip above was taken from can be heard here.
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Q:
Размер картинки, зависящий от размера родителя
Делаю адаптивную вестку и столкнулся с проблемой.
В некоторых постах есть картинки. Надо сделать так, чтоб при изменении родительского элемента менялись размеры и самой картинки. Вот такая задача. Желательно это сделать без JavaScript и кросбраузерно (кроме IE6).
A:
Хороший вопрос. Здесь нужно использовать процентные величины. Например родитель #main, потомок обычная картинка img.
Тогда можем поработать с css:
#main { width: 100%; height: 150px; }
#main img { width: 40%; height: 100%; }
img возьмёт величины своего родителя, и относительно их будет вычислять свои размеры. Вообще, в таких ситуациях не желательно использовать js, а процентные величины. У меня может быть отключен js.
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Quneitra Governorate clashes (2012–14)
The 2012–14 Quneitra Governorate clashes began in early November 2012, when the Syrian Army began engaging with rebels in several towns and villages of the Quneitra Governorate. The clashes quickly intensified and spilled into the UN-supervised neutral demilitarized zone between Syrian controlled territory and the Israeli-occupied Golan Heights.
The fighting came to international attention when on March 2013, Syrian rebels took hostage 21 Filipino UN personnel, who had been a part of the UN Disengagement Observer Force in the neutral buffer zone between Syria and Israel. According to UN official they were taken hostage near Observation Post 58, which had sustained damage and was evacuated the previous weekend, following heavy combat in close proximity at Al Jamla. The UN personnel were later released with Jordanian mediation.
Israel has been briefly involved in the fighting in several incidents, such as on 11 November 2012, when mortar shells from Syria landed near an Israeli military outpost in the Israeli-occupied Golan Heights, responding by firing "warning shots" into Syria. This accounted for the first direct cross-border incidents between the two countries since the Yom Kippur War nearly forty years prior. Other occasions of short cross-border fire exchanges followed in early 2013 and on March 2014, with several wounded Israeli soldiers reported in each incident and one Israeli civilian killed.
The clashes in the Governorate were eclipsed by the 2014 Quneitra offensive, launched by rebels on late August 2014, resulting in take-over of much of the governorate by mid-September.
Background
Quneitra Governorate came into its modern shape in 1946, with Syrian independence from French Mandatory rule. The borders of the Governorate were extended during the 1948 War with Israel, as Syria occupied parts of the Jordan and Huleh Valleys, adjacent to the Sea of Galilee. During the Six-Day War in 1967, Israel captured the Western Golan Heights from Syria, effectively reduced Syrian-controlled Quneitra part to one third of its size. After a failed attempt to recapture the region in the Yom Kippur War of 1973, Syria and Israel have remained in a shaky truce with a United Nations-monitored demilitarized zone (DMZ) separating the countries. Many countries have condemned Israel's occupation of the Golan Heights, especially their unilateral annexation of the area in 1981 and subsequent settlement construction.
The border remained quiet for nearly four decades until the outbreak of the Arab Spring. During the 2011 Israeli border demonstrations, Palestinian protesters approached the border and were subsequently fired upon by Israeli forces. Four demonstrators were killed and dozens were injured. Additionally, Israeli soldiers were injured when protesters attempted to cross into the Druze town of Majdal Shams located on the Israeli-occupied side of the ceasefire line. As the Syrian civil war progressed, border clashes began to escalate, with spillover conflicts in Lebanon and Turkey prompting fears of an escalation to a wider regional conflict.
There were some concerns of civil unrest on the Israeli side of the border as well, particularly among the Golan Druze. The Druze population of the Israeli-occupied portion of Golan Heights numbers around 20,000 individuals, with majority of them still holding Syrian citizenship. Prior to the war, the Golan Druze were overwhelmingly in support of the government of Bashar al-Assad, as his government has long been staunchly supportive of their interests and opposition to Israeli rule. Many of them were able to conduct business across the border in Syria as a result of agreements between the Syrian and Israeli governments. As the civil war deepened, however, a minority of Golan Druze began to voice opposition to the Assad government. According to local sources, perhaps 22 individuals had crossed the border into Syria to fight for the rebels by late September 2012. Public support for the Syrian government nevertheless remains high, while rumours of pro-Assad spies intimidate potential dissenters fearful of being banned from cross-border trade.
Involved actors in Quneitra region
Baathist Syria and allies
The Katiba Ahrar Haḍr (Battalion of the Free Men of Haḍr), referring to the Druze village of Haḍr in the Jabal al-Sheikh region, formed on 28 January 2013 in response to disillusionment with government policies of conscription into the Syrian Army as well as apparent extortionist practices on the part of the People's Committees set up to coordinate the activities of Druze militias with the Syrian Army. In the group's formation video, the battalion declares affiliation with the FSA-banner Military Council of the Quneitra and Golan region.
Syrian opposition
Brigadier General Abdul-Ilah al-Bashir, the leader of the Quneitra Military Council, a Syrian rebel coalition affiliated with the Free Syrian Army, was appointed the Chief-of-Staff of the FSA's Supreme Military Council (SMC) on 16 February 2014. A rebel offensive in May 2014 to capture the town of Qahtaniya and the Riwadi and Hamidyya checkpoints involved the Syrian Revolutionaries Front (SRF), Saraya al-Jihad, Bait al-Maqdis Group, Ahl Assalaf Youth, Jund al-Rahman Brigades and Mujahideen al-Sham Movement.
The Furqan Brigade (al Quneitra) is a branch of the Damascus-based Islamist Furqan Brigade, operating in the southwest of Syria near the Israeli-occupied Golan Heights The Islamic Front is a coalition of several rebel groups, including Ahrar al Sham, which is linked to al Qaeda.
The Syrian Revolutionaries Front (SRF) is an alliance formed in December 2013 by the Free Syrian Army as a response to the merger of Islamist Syrian rebels into the Islamic Front. It assisted the Islamic Front and Al Nusrah in 'liberating' ar-Rawadia and Humaydia in the Quneitra countryside in September 2014.
Mujahedeen
In August 2014, it was reported that Jabhat al-Nusra, along with Fallujah-Houran Brigade, Syria Revolutionaries Front, Saraya al-Jihad, Bayt al-Maqdis and Ahrar al-Sham, began a battle called "the real promise" to seize control of the devastated city of Quneitra and the crossing connecting it with the Golan Heights. The source was surprised by the participation of Islamic factions in the battle, since they were absent for months from the fighting in the southern region, especially in Daraa, before simultaneously deciding to participate alongside the Syria Revolutionaries Front, its most bitter enemy.
Others
Other rebel factions in Quneitra included: Ansar al-Huda; Fursan al-Ababil; Jabhat Ansar al-Islam; Youth of Sunnah Brigade, and probably other local groups.
Timeline
2012
On 18 July 2012, the Free Syrian Army captured Jubata al-Khashab and started operations against the Syrian Army from it.
On 2 November 2012, Syrian Army tanks crossed into the UN-administered demilitarized zone and clashed with rebels near the village of Beer Ajam. During the clashes, stray bullets hit an Israeli patrol in the area. Israel responded on 5 November by filing a complaint with the United Nations Security Council, claiming Syria violated the 1974 Agreement on Disengagement signed following the Yom Kippur War. The terms of the 1974 armistice prevent the Syrian military from conducting operations within the DMZ. Sources within the Israeli military cite these restrictions as a potential reason why the armed opposition drew the Syrian Army into combat in the area.
By 10 November, at least 30 Syrian rebels and soldiers had been killed in the fighting, according to the Syrian Observatory for Human Rights. Clashes were reportedly ongoing in and around the villages of Bariqa, Beer Ajam and al-Hersh. Two days later, Syrian Government forces began shelling opposition positions in the village of Bariqa near the border with Israel. A foreign journalist reported seeing the fighting from the Israeli side of the border, with government forces driving rebels toward the border with heavy artillery. About thirty minutes later, a shell from Syria landed near Tel Hazeka in the Golan Heights. Israel retaliated by shelling Syrian government positions with Merkava tanks, resulting in "direct hits" on the sources of the fire. According to Israeli Army Radio, the Assad government requested that Israel stop firing, though it was not clear if the Israeli shelling caused any casualties.
By 13 November, a force of at least 200 rebels had captured the Syrian-side villages of Beer Ajam and Bariqa. Rebels were reportedly in control of the areas to the north and south of Quneitra. The following day, Israeli defense minister Ehud Barak claimed that rebels were in control of most of the villages on the eastern slopes of the Golan Heights, and that the Syrian Army had been unable to enter them.
On 28 December, SOHR reported that six rebels, including a commander, and five government soldiers were killed in combat in the villages of Ruwayhinah and Zubaydah. The following day, SOHR reported that two rebels died of their wounds incurred in earlier combat with government forces in the area.
2013
On 6 March 21 UN personnel were taken hostage by Syrian rebels in the neutral zone. They were later released with Jordanian mediation.
On 24 March, the IDF fired a guided missile at a Syrian machine gun nest after Israeli troops were shot at twice in the Golan Heights. No Israeli soldiers were hurt in the shooting, during which army vehicles were hit.
On 7 May, 18 rebel fighters were reported killed in heavy fighting in the province.
On 21 May, there was an exchange of fire between Syria and Israel in the Golan Heights. An Israeli vehicle was hit by Syrian fire with the Israeli's retaliating and destroying the source of the attack.
Early on 6 June, rebels attacked and temporarily captured a Golan border crossing. However, the same day, government forces counter-attacked with tanks and armoured personnel carriers, recapturing the crossing. Al Jazeera's Sue Turton, reporting from the Golan Heights, said that this marked a significant point in the crisis. Rebels also attacked a military checkpoint in the largely destroyed and abandoned city of Quneitra. A shell landed in an UN base nearby as a result of Government-Rebel fire-exchanges. An Austrian defense ministry official confirmed to the Associated Press that rebel troops captured the crossing point and that UN forces have withdrawn from the area.
On 16 July, rebel fighters retreated from the al-Qahtaniya village because of heavy bombardment by the army on the village after violent clashes.
2014
On 31 January 2014, rebels captured al-Susiyah city and 5 other villages. A week later, rebels managed to stop an army convoy that was heading to Al-Dwaieh village.
On 16 February, Abdel al-Ilah al-Bachir, chief of the FSA Military Council in Quneitra, was appointed chief of staff of the Free Syrian Army.
On 18 February, the Army launched a surprise offensive during the morning using tanks and air strikes against the villages of al-Hajjeh, al-Dawayeh al-Kubra, al-Sughra, Bir Ajam and al-Buraika in the central and southern parts of al-Quneitra. 4 days later, the Army and National Defense Force captured the areas of Rasm al-Hour and Rasm al Sayd, south of the town of Quneitra. SOHR confirmed troops were on the offensive, adding that the air force was taking part in the attack. Two days later, rebels managed to 'infiltrate' the Abu-Dhiab and Khalil tank platoons in Tal-Al-Jabieh area, seizing various weapons and two tanks. It was also reported that about 40 regular army soldiers were killed and a number of others captured. Al-Arabiya reported this as a major advance of FSA in Quneitra governorate. On 26 February, the Army dispatched reinforcements to the province, after recent gains by rebels there. It was also reported that rebels declared their control of over 80% of the Golan countryside through coordination committees. They also announced a new offensive against the bases in Quneitra, especially Brigade 61.
On 1 March, two rockets were fired on an Israeli post on Mount Hermon, in what is widely believed to be a retaliation for an Israeli airstrike on a Hezbollah target near the Lebanese-Syrian border. On 18 March, an Israeli jeep traveling on the Golan Heights near the Syrian border came under attack, when an explosive device was detonated in its vicinity. One soldier was seriously wounded. Another three soldiers sustained light-to-moderate injuries. IDF 155 Artillery battery returned fire across the border following the incident and shot several shells on a Syrian outpost. Israel responded by carrying out multiple airstrikes against Syrian targets, including a military headquarters, artillery batteries and a training base of the Syrian army. The Syrian army reported that the Israeli airstrikes killed one Syrian soldier and wounded seven. On 28 March, Israeli soldiers opened fire on two gunmen seen attempting to sabotage the border fence with Syria on the Golan Heights. IDF said both armed suspects were struck by the gunfire.
On 7 April, rebels announced the start of a new offensive in the Tell Ahmar area to capture two strategic hills. They managed to advance while damaging a tank according to the SOHR. The rebels captured one of the strategic hills later that day. Rebels also captured the village of Tulul al-Humur after besieging it for weeks.
On 8 May, rebels launched a military operation against al-Qahtania, al-Hamedia, Quneitra Crossing and the al-Rawadi checkpoint in Quneitra province.
On 23 June, the IDF launched several airstrikes targeting government troops in retaliation for an attack the day before that killed an Israeli teenager from the Arab village of Arraba. Four soldiers were killed and nine injured during the strikes. On 15 July 2014, the IDF bombarded the city of Quneitra, killing at least 4 persons. It also bombarded the Brigade 90 base.
Aftermath
On 27 August, rebels took control of the Quneitra Crossing between Syria and the Israeli-occupied Golan Heights. At least 20 soldiers and 14 rebels were killed during the battle. Fighting in the area continued in towns northeast of the crossing, while the IDF shelled two Syrian army positions in retiliation of six mortar shells that fell in the Israeli-occupied Golan Heights and the wounding of an Israeli officer. The Al-Nusra Front, Ahrar ash-Sham and other rebel groups (including moderate groups) participated in the fighting. The next day, fighters from the Nusra front captured 44-45 U.N. peacekeepers and surrounded 75 others, resulting in a gun fight that lasted over 7 hours. A group of 35 U.N. soldiers were successfully escorted out of the UN encampment in Breiqa by their colleagues. Rebels tried to breach the Rwihana U.N. encampment, but the attack was repelled by the U.N. defenders with support from the Syrian Army. The remaining 40 peacekeepers were eventually evacuated during the night of 29 August, after a ceasefire was established.
Reactions
Iran
Observers in the Arab world have been warning for years about growing evidence of Iranian expansionism. Tehran has invested huge resources in making Syria a Shiite state. Dr. Shimon Shapira, a retired brigadier general of the Israel Defense Forces (IDF), has written a paper unambiguously titled "Iran's Plans to Take Over Syria," which emphasizes comments made by Mehdi Taaib, the head of Ayatollah Ali Khamenei's think tank, that Syria is "35th district of Iran and it has greater strategic importance for Iran than Khuzestan [an Arab-populated district inside Iran]." Iran is also recruiting Shiite forces from various countries for fighting in Syria. As Syria disintegrates into a patchwork of areas, Iran is said to aim to have a network of militias in place inside Syria to protect its vital interests, regardless of what happens to Assad. Qasem Suleimani, the commander of the Quds Force of the Revolutionary Guard Corps, is reported to have prepared an operational plan named after him based upon the establishment of a 150,000-man force for Syria, the majority of whom will come from Iran, Iraq, and a smaller number from Hizbullah and the Gulf states. He has been the spearhead of Iranian military activism in the Middle East. In January 2012, he declared that the Islamic Republic controlled "one way or another" Iraq and South Lebanon. In March 2014, an article in the Guardian estimated the numbers of Shia fighters in Syria range between 8,000 and 15,000. The Kurdish head of Iraq's parliamentary security and defence committee said in March 2015 there were around 30,000 Iranians fighting ISIL in Iraq (and Syria?). The same month, it was reported elsewhere that Iran's Islamic Revolutionary Guard Corps has mobilized roughly 70,000 fighters in Syria since the war began to prop up Assad's army which had been reduced to less than half through casualties, desertions and draft dodging and that there are between 5,000-10,000 Hezbollah fighters in Syria. This would be around 10% of the estimated 70,000 well trained soldiers of Hezbollah. Refugee Afghan Shiite jihadists living in Iran are said to have provided the largest supply of non-Arab foreigners to the Syrian battlefield. Meanwhile, a U.S. intelligence official in February 2015 estimated around 20,000 foreign fighters had joined jihadist organizations in Syria and Iraq.
However the Fair Observer site, while noting that the Revolutionary Guards are administratively embedded in the most critical Syrian government agencies reflecting a commissar-type pattern, states their "battle management strategy in Syria appears to be premised on a light Iranian footprint on the ground and the use of proxy militias as force multipliers to manage the battle space. This juncture of the Syrian war is characterized by the Revolutionary Guard managing the battle space by liaising across hundreds of Shiite, Alawite and Ismaili (and some allied Christian) militias characterized by loyalty ultimately to the Revolutionary Guard which facilitates their funding either directly or through surrogates such as the Syrian government. The Washington Institute produced in 2015 a lengthy and detailed analysis of Iran and Hezbollah's strategy in developing Syrian militias which alleges that along with Hezbollah, refugee Iraqi Shiite jihadists have formed the core Iranian proxy units sent to Syria, and that attached advisors from Hezbollah and IRGC influenced the groups' religious and ideological development. This has altered the conflict narrative from the Assad line emphasizing a secular "fight against terrorists". Hezbollah has assumed a lead role in the February 2015 offensive in the south. Many Shia militiamen from Iraq who had been fighting alongside Assad's forces started to return last autumn to neighboring Iraq to assist in the pushback against the Sunni militants of Islamic State, also known as ISIL who seized a swath of western and northern Iraq in the summer. Top Iranian commanders from Iran's Revolutionary Guards, including the commander of its Quds Force Qasem Soleimani, who were frequently in Syria to help direct military tactics are now more often in Iraq, say Baghdad-based Western diplomats. The Iranian quest to achieve regional dominance through its proxies has thus been aided by the return from Syria of experienced Iraqi fighters whose organizations have thus expanded from having bases in Syria alone to also having ones in Iraq.
Osnet reports that the Syrian Government/Hezbollah led drive towards Quneitra via Tel Harra has a number of unique features. For the first time in the Syrian theatre, the forces are commanded in full by an Iranian operations room. Its goal reaches beyond ensuring the survival of Syria. Rather they aim to make Quneitra the seat of their forward command and bring Iranian Revolutionary Guards (IRGC) within sight and firing range of Israeli military forces. At a 7 May 2013 meeting with Iranian foreign minister Ali Akbar Salehi, Syrian President Bashar Al-Assad announced, "The Golan will become a front of resistance." Tel al-Hara, at 3500 feet the tallest peak in the Golan range, and overlooking Israel's outposts, is important because it was formerly a Syrian fortress with tens of square kilometers of bunkers, funnels and defensive positions. Also perched there were advanced Russian radar stations, which kept track of Israel military and air force movements across the border. These stations were connected to the Middle East intelligence networks of the IRGC and kept Tehran abreast of Israeli military movements and deployments." Iran's aims in deploying in the Golan Heights is to deter Israel from acting against its nuclear program, defend Syria as part of the resistance axis, and establish an active front for anti-Israel terror attacks in the Golan and even to liberate the Israeli-occupied Golan. Hezbollah's strategy could also be termed defensive in that they fear the possibility that Israel will close in on it from Mount Dov to the west, using the Al-Nusra Front and moderate opposition forces to lay siege to southern Lebanon, thus causing difficulties for the organization's activities there. In addition, and perhaps more importantly, Hezbollah and Assad fear that Israel is carving out a path to Damascus via Quneitra and Daraa — one that will allow it easy access to the Syrian capital in the event of war. Hezbollah and Iran have asked the Palestinian resistance movements, such as Hamas's Al-Qassam Brigades and the many factions of Palestinians living in refugee camps in Lebanon and Syria, to join their front against Israel.
Israel
With the breakdown of the status quo on the Golan Heights front, which had been quiet since the Separation of Forces Agreement of 1974, Israel's eyes have been anxiously focused on developments in Quneitra. Amos Yadlin, a former senior Israeli intelligence officer and now a member of the opposition Zionist Union party, told the Wall Street Journal: "There is no doubt that Hezbollah and Iran are the major threat to Israel, much more than the radical Sunni Islamists, who are also an enemy." A prominent Middle East researcher stated that Israel established contacts with members of the Syrian opposition abroad during the second half of 2012 and the idea of a buffer zone emerged during secret meetings in Amman in early 2013.
Importantly, Israel explicitly threatened to fire on any Syrian government armor which enters the demilitarized zone in Quneitra, which would be a breach of the May 1974 disengagement agreement between the two countries. Sedqi al-Maqet, a Syrian Druze, pro-Assad activist who lives in the Israeli occupied Golan was interdicted in February 2014 after reporting online he had witnessed meetings between Israeli armed forces in the Golan and what he termed terrorists active in the Syrian-controlled sector of the Golan.
Casualties
Syrian Arab Army and allies
53 soldiers and officials killed on 1 February 2013 by rebels.
24 Syrian soldiers killed and 7 injured by Israeli military.
5 Arab Tawhid Party militants killed in the town of Arna in Mount Hermon on 4 November 2013.
Syrian rebels
At least 220 rebels killed: Claims by Hezbollah and the Iranian News agency FARS No significant events took place on either date and neither figure is confirmed by any credible verifiable source.
20 killed on 27.03.14 (Hezbollah claim)
200+ killed 5 May 2014
Civilians
11+ civilians killed (including 1 foreign):
2+ Syrian civilians killed in Syrian gov-t bombings
8 Syrian civilians killed by Israeli Army bombardment.
1 Israeli civilian killed and three injured in spillover.
See also
2012 Syrian–Turkish border clashes
Cities and towns during the Syrian Civil War
2014 Daraa offensive
2015 Southern Syria offensive
2018 Southern Syria offensive
References
Category:Military operations of the Syrian Civil War in 2012
Category:Military operations of the Syrian Civil War in 2013
Category:Military operations of the Syrian Civil War in 2014
Category:Quneitra Governorate in the Syrian Civil War
Category:Military operations of the Syrian Civil War involving the Syrian government
Category:Military operations of the Syrian Civil War involving the al-Nusra Front
Category:Military operations of the Syrian Civil War involving the Free Syrian Army
Category:Israeli involvement in the Syrian Civil War
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In 1978 Italy implemented Law Number 180, the reform law that blocked all new admissions to public mental hospitals. After 40 years without mental hospitals, we aim at understanding the consequences of the Italian reform in terms of mental health care facility and staff availability. We compared the organization of the Italian mental health system with that of countries belonging to the Group of 7 (G7) major advanced economies. Italy has nearly 8 psychiatrists, 20 nurses, 2 social workers and less than 3 psychologists per 100,000 population, while for example in France there were 22 psychiatrists, in Japan 102 nurses, in the United States 18 social workers, and in Canada and France more than 45 psychologists per 100,000 population. In terms of inpatient facilities, no beds in mental hospitals were available in Italy, while in the other G7 countries mental hospital beds ranged from 8 in the United Kingdom to 204 in Japan per 100 000 population. In Italy there were fewer beds for acute care in general hospitals but more beds in community residential facilities than in the other G7 countries. Service use data showed variability in the provision of mental health care throughout the country. Soon after the implementation of the Italian reform the absolute number of compulsory admissions progressively declined, from more than 20,000 in 1978 to less than 9000 in 2015. Alongside the progressive decline of psychiatric beds imposed by Law 180, the age-adjusted suicide rate remained stable, ranging from 7·1/100,000 population in 1978 to 6·3/100,000 population in 2012. The population of psychiatric patients placed in Italian forensic psychiatric hospitals progressively declined. During the last 40 years without mental hospitals, Italy has seen a progressive consolidation of a community-based system of mental health care. We highlighted, however, reasons for concern, including a decreasing staffing level, a potential use of community residential facilities as long-stay residential services, a still too high variability in service provision across the country, and lack of national data on physical restraints. At a national level, the resources allocated to mental health care are lower in Italy than in other high-income countries.
A radical change in the organization of mental health care occurred in Italy in 1978 as a consequence of the implementation of the Italian Law Number 180, the reform law that marked the transition from a hospital-based system of care to a model of community mental health care (Box 1) [1–8]. Law 180 blocked all new admissions to public mental hospitals, with immediate effect (i.e. from 1978), as well as readmissions, 2 years later. Consequently, the psychiatric hospital population (78,538 individuals in 1978) dropped by 53% between 1978 and 1987, further declined to 7704 in 1998, and the final dismantling was completed by year 2000 [9, 10].
After 40 years of community mental health care, here we provide an overview of the mental health system in Italy, with emphasis on understanding the consequences of the Italian reform in terms of mental health care facility and staff availability. Using available data taken from both international and national sources (Box 2) [11–15], we compared the organization of the Italian mental health system with that of countries belonging to the Group of 7 (G7) major advanced economies. Additionally, we described trends in compulsory admissions and suicide rates in Italy in the 40 years after the implementation of Law 180.
Box 1. Summary of the main characteristics of the 1978 Italian psychiatric reform
The main principle of Law 180 is that patients with mental disorders have the right to be treated the same way as patients with other diseases, which means the following:
Acute mental health conditions have to be managed in psychiatric wards located in general hospitals. These wards cannot exceed 15 beds.
Treatments should be provided on a voluntary basis, with compulsory admissions reserved for the following specific circumstances: (1) an emergency intervention is needed; (2) the patient refuses treatment; (3) alternative community treatment is impossible.
Compulsory admissions need to be formally authorized by the Mayor and can only be undertaken in general hospital psychiatric wards.
New community-based services were to be established to provide mental health care to the population of a given catchment area.
Gradual closure of public mental hospitals by blocking all new admissions.
Box 2. Data source
We used the Organisation for Economic Co-operation and Development (OECD) database to gather information on demographic and economic indicators, psychiatric bed availability and age-standardised suicide rates for Italy and the other G7 countries [11].
The WHO Global Health Observatory [12] and the WHO Mental Health ATLAS-2014 repository [13] were used to extract data on inpatient and outpatient resources for mental health care (both public and private) in Italy and in the other G7 countries. WHO definitions of mental health staff, inpatient and outpatient facilities were used. For inpatient facilities, the following WHO categories were used: mental hospitals (public and private non-profit and for-profit specialized hospital-based facilities that provide inpatient care and long-stay residential services for people with mental disorders), psychiatric wards in general hospitals (public and private non-profit and for-profit psychiatric units usually located within general hospitals that provide inpatient care for the management of acute mental disorders), community residential facilities (public and private non-profit and for-profit non-hospital, community-based mental health facilities that provide overnight residence for people with mental disorders).
From the recently implemented Italian national mental health information system data on the availability and use of mental health facilities (both public and private) were gathered for the year 2015 [14]. The following information was extracted for each Italian region: treated prevalence of any mental disorders (number of individuals with at least one contact with psychiatric services during 2015/10,000 population); treated incidence of any mental disorders (number of individuals with a first ever contact with psychiatric services during 2015/10,000 population); rate of individuals under the care of mental health outpatient facilities (per 10,000 population); rate of individuals under the care of day treatment facilities (per 10,000 population); admissions to community residential facilities (per 10,000 population); admissions to psychiatric wards of general hospitals (per 10,000 population); rate of compulsory admissions (per 10,000 population); proportion of outpatients visits within 30 day after hospital discharge.
As additional source of information, we used the Italian Central Institute of Statistics (ISTAT) data to describe the total number of compulsory admissions and the proportion of all psychiatric admissions that were compulsory from 1978 onwards [15]. Data released from the Commission on psychiatric forensic facilities were used to compute the number of psychiatric patients placed in forensic psychiatric hospitals from 1978 onwards [16].
Italy is the fourth most populous European state after Germany, France and the United Kingdom. It hosts a growing proportion of foreign population, which is approaching 10%, as in Germany (Table 1). In 2014, the number of healthy life years at birth was estimated at 83 years, similar to Japan and higher than the other G7 countries. The unemployment rate in 2016 was close to 12%, with a gross domestic product much lower than the other G7 countries. In 2011, the proportion of government expenditures on mental health was half than Germany or France (Table 1).
Table 1
Demographic and economic indicators for Italy and the other countries belonging to the Group of 7 (G7) major advanced economies (OECD data)
Canada
France
Germany
Italy
Japan
UK
USA
Year
Population (million persons)
35.54
64.06
80.89
60.44
127.51
63.65
318.85
2014
Foreign population (% of population)
NA
NA
9.29
8.11
1.62
7.70
6.96
2013
Healthy life expectancy at birth (years)
NA
82.40
81.20
83.20
83.70
81.40
78.80
2014
Unemployment rate (% of labour force)
6.99
10.05
4.10
11.68
3.11
4.80
4.86
2016
Gross domestic product (total, US dollars/capita)
44,025
41,489
48,839
38,146
41,534
42,651
57,325
2016
Government expenditures on mental health (% of total expenditure on health)
7.20
12.91
11.00
5.00
4.94
NA
NA
2011
OECD Organisation for Economic Co-operation and Development
NA not available
Italy, in comparison with the other G7 countries, has fewer human resources for mental health care (Table 2). According to WHO ATLAS-2014, there were nearly 8 psychiatrists, 20 nurses, 2 social workers and less than 3 psychologists per 100,000 population, while for example in France there were 22 psychiatrists, in Japan 102 nurses, in the United States 18 social workers, and in Canada and France more than 45 psychologists per 100,000 population (Table 2).
Table 2
Staff availability and resources for mental health care in Italy and in the other G7 countries
Canada
France
Germany
Italy
Japan
UK
USA
Staffa
Psychiatrists working in mental health sector (per 100,000)
12.61
22.35
15.23
7.83
10.1
14.63
12.40
Nurses working in mental health sector (per 100 000)
65.0
86.21
56.06
19.28
102.55
67.35
3.07
Social workers working in mental health sector (per 100 000)
NA
3.83
NA
1.93
6.06
1.99
17.93
Psychologists working in mental health sector (per 100 000)
46.56
47.9
NA
2.58
3.99
12.83
29.03
Inpatient facilitiesb
Beds for mental health in general hospitals (per 100 000)
NA
22.72
41.08
10.95
73.12
50.63
14.36
Beds in community residential facilities (per 100 000)
NA
NA
NA
46.41
16.23
2.28
22.29
Beds in mental hospitals (per 100 000)
31.38
71.81
47.62
0
204.4
7.99
19.44
Outpatient facilitiesb
Mental health outpatient facilities (per 100,000)
NA
5.75
30.32
1.43
2.31
4.94
1.95
Day treatment facilities (per 100,000)
NA
3.50
0.61
1.34
1.05
2.88
NA
NA not available
aFrom WHO Global Health Observatory (GHO)
bFrom WHO ATLAS
In terms of inpatient facilities, no beds in public mental hospitals were available in Italy, as required by Law 180, while in the other G7 countries mental hospital beds showed high variability, ranging from 8 in the United Kingdom to 204 in Japan per 100,000 population. In Italy there were fewer beds for acute care in general hospitals than in the other G7 countries, with Japan having more than 70 beds in general hospitals and Italy around 10/100,000 population. However, In Italy the rate of beds in community residential facilities was higher than in other countries where this information was available (Table 2).
Soon after the implementation of the Italian reform the absolute number of compulsory admissions progressively declined, from more than 20,000 in 1978 to less than 9000 in 2015. Similarly, the proportion of psychiatric admissions that were compulsory progressively declined from 1978 to 2005, and remained stable thereafter, accounting for less than 5% of all psychiatric admissions (Fig. 1).
Fig. 1
Compulsory psychiatric admissions in Italy, 1978–2015 (ISTAT data)
Figure 2 describes the age-adjusted suicide rate in Italy from 1978 onwards, alongside the progressive decline of psychiatric beds imposed by Law 180. In 1978 there were 7.1 suicides per 100,000 population, while in 2012 there were 6.3 suicides per 100,000 population, with the highest rate in 1985 (8.8 suicides per 100,000) and the lowest in 2006 (5.6/100,000). Lack of a clear relationship between psychiatric bed availability and suicides was also suggested by Fig. 3, where psychiatric beds for the G7 countries are reported alongside the national rate of suicides. In Japan the rate of suicide was the highest among the G7 countries, despite more than 250 psychiatric beds per 100,000 population, while in the United States there were high suicide rates with relatively few psychiatric beds. The United Kingdom showed a situation similar to Italy, with few beds and relatively low suicide rates.
Fig. 2
Availability of acute-care psychiatric beds in Italy and age-standardised suicide rates from 1978 to 2012 (OECD data, 2004 and 2005 are missing)
Unfortunately, there is little epidemiological data on the population of psychiatric patients placed in forensic psychiatric hospitals from 1978 onwards. In 1980, the population comprised 1424 people, in 1987 there were 977 people and in 2012 there were 1264 people [16]. In 2016, after the phasing out of forensic psychiatric hospitals, there were 541 individuals placed in newly developed residential facilities providing intensive mental health care to socially dangerous individuals with mental disorders [16]. Additionally, there were 201 individuals with mental disorders placed in psychiatric units in prison, yielding an overall number of 742 people for the year 2016 [16].
In Table 3 service use data for mental health care in Italy is presented for the year 2015. Substantial variability in the provision of mental health care can be observed throughout the country. For example, the treated prevalence of mental disorders, a proxy indicator of the coverage capacity of community psychiatric services, ranged from 205 individuals per 10,000 population in Emilia Romagna (north of Italy) to 108 in Basilicata (south). Similar differences were observed for the treated incidence of mental disorders, although a north to south gradient was not observed, being higher in Liguria and Friuli (north of Italy) and lower in Lombardy (north), Tuscany (north), Umbria (centre), Marche (centre) and Basilicata (south). On average, in Italy there were 150 individuals per 10,000 population under the care of mental health outpatient facilities, with wide regional differences, and 6 individuals per 10,000 population under the care of day treatment facilities (Table 3).
Table 3
Treated prevalence and treated incidence of mental disorders, and service use data for mental health care in Italian regions, year 2015 (Italian national mental health information system data)
Italian region (north to south)
Rate per 10,000 population
Percentage
Treated prevalence of mental disorders
Treated incidence of mental disorders (first-ever cases)
Individuals under the care of mental health outpatient facilities
Individuals under the care of day treatment facilities
Admissions to community residential facilities
Admissions to psychiatric wards
Compulsory admissions
Outpatient visits within 30 days after hospital discharge
Piemonte
163.68
67.64
164.13
8.79
5.34
24.20
1.37
52.1
Valle D’Aosta
NA
NA
NA
NA
NA
30.17
2.99
NA
Lombardia
176.65
46.52
172.54
6.70
5.71
23.78
0.96
55.3
Bolzano
NA
NA
NA
NA
NA
40.19
0.22
NA
Trento
165.96
57.54
177.27
4.77
3.70
19.26
0.95
86.7
Veneto
143.40
67.99
142.29
13.98
4.56
28.69
0.98
34.9
Friuli Venezia Giulia
116.52
120.20
113.86
10.45
2.95
6.34
0.43
69.9
Liguria
175.33
131.32
176.26
6.66
10.35
37.46
1.19
41.7
Emilia Romagna
205.82
79.38
206.25
5.54
14.27
26.78
2.64
56.9
Toscana
110.49
39.01
110.66
3.07
3.62
23.23
1.16
49.3
Umbria
164.89
48.03
184.12
3.74
9.79
10.48
1.94
30.1
Marche
158.94
44.05
168.84
4.35
8.84
24.25
5.68
49.7
Lazio
138.60
76.39
131.45
5.59
10.13
17.31
1.46
NA
Abruzzo
142.41
66.83
141.82
4.37
3.88
24.50
1.49
32.1
Molise
165.10
71.77
167.24
2.06
5.05
22.00
1.61
63.6
Campania
139.39
54.05
153.06
3.47
1.56
9.06
1.90
57.8
Puglia
167.58
79.69
159.83
4.27
6.25
17.35
2.07
47.3
Basilicata
107.63
47.98
135.73
2.53
5.94
18.75
0.72
NA
Calabria
161.34
105.85
222.09
0.30
0.37
16.38
2.10
50.0
Sicilia
186.33
90.27
196.19
3.70
4.99
28.40
3.08
42.5
Sardegna
NA
NA
NA
NA
NA
19.48
2.33
NA
Italy
159.40
68.13
153.87
5.91
6.10
21.87
1.73
49.4
NA not available
In terms of bed use, there were slightly more than 20 admissions to general hospital beds per 10,000 population, with substantial variability in terms of proportion of patients with an outpatient visit within 30 days after discharge, ranging from nearly 90% in Trento (north) to less than 35% in Veneto (north), 32% in Abruzzo (centre) and 30% in Umbria (centre). The rate of compulsory admissions was 1.73/10,000 population, ranging from 5.68 in Marche (centre) to 0.22 in Bolzano (north) and 0.43 in Friuli (north). On average, there were 6 admissions to community residential facilities per 10,000 population, with substantial variability. Interestingly, the average length of stay in these facilities was higher than 750 days, ranging from 30 days in Campania (south) to 2 269 days (more than 6 years) in Veneto (north).
It has often been emphasised the closing of mental hospitals as the main objective of the Italian reform, while its first and main aim is that individuals with mental disorders are treated the same way as individuals with other diseases. Implementing this principle has determined a shift in the role and focus of psychiatry, from custody and coercion to treatment and care. All the practical changes to the Italian mental health system have been a consequence of this paradigm shift: the total dismantling of old asylums, the development of psychiatric wards in general hospitals and the implementation of a community-based system of mental health care.
Compulsory admissions and suicides
A hard indicator of the shift from custody to care is a progressive decline in compulsory admissions, both in terms of absolute numbers and in terms of proportion of psychiatric admissions that were involuntary. In other countries different trends have been observed. In the United Kingdom, for example, the number of uses of the Mental Health Act has been rising, with the highest ever year-on-year rise (10%) to 58,400 detentions in 2014/15 [17]. More than half of admissions to psychiatric hospitals in England are now involuntary, the highest rate recorded since the 1983 Mental Health Act, with wide local variations [18].
A decreasing availability of psychiatric beds has been suggested as one explanation for the rise in compulsory admissions [19]. On similar grounds, in the United States a decreasing availability of psychiatric beds has been suggested as one explanation for the rise in suicide rates [20–23]. The natural experiment offered by the Italian reform would suggest that a direct and linear relationship between psychiatric bed availability and these public health indicators should not be expected. Despite a dramatic decrease in acute-care hospital beds, compulsory admissions decreased and suicide rates remained stable. Data from other G7 countries would reinforce this point, as there are countries with high rates of both beds and suicides, countries with low rates of beds and suicides, and countries with diverging rates. Of course we acknowledge that a wide variety of social, economic, health, mental health and context variables may significantly affect such indicators, and therefore no causal inference can be derived from these descriptive data. However, for the same reasons we argue that increasing the number of psychiatric beds may hardly be considered an evidence-based public health measure to decrease the rates of suicides and the rates of involuntary admissions.
Data available on individuals placed in forensic facilities from 1978 onwards suggests that the phasing out of mental hospitals has not determined an increase of this population, which has declined. Unfortunately, no data are available on the true prevalence of mental disorders in people placed in Italian prisons. A study conducted in one prison found a prevalence of 19.3% of one or more diagnostic and statistical manual of mental disorders, fourth edition, axis I current mental disorders (excluding substance misuse) [24], which seems in line with international estimates [25].
Mental health workers
Fewer human resources were available in Italy than in other high-income countries. WHO ATLAS showed that the median number of mental health workers per 100,000 population vary from below 1/100,000 population in low-income countries to over 50 in high-income countries [26]. In Italy there were 33 workers per 100,000, which is below the median of 43.5/100,000 population in Europe and below the median of 52.3/100,000 population in high-income countries. The global median is 9/100,000 population, or less than one mental health worker for every 10,000 people [26]. Although it may be argued that the Italian experience suggests that human resources are not as important as system organisation, it is nevertheless true that staff availability is associated with the capacity of providing mental health care which, in turn, affects the coverage for severe mental disorders, which is one of the main targets mentioned by the WHO action plan [27]. Related to this, Italy has the lowest gross domestic product among the G7 countries, with the lowest proportion of government expenditures on mental health. Looking ahead, this may represent a key challenge for the sustainability of the Italian mental health care system, and for the quality of health care provided by mental health services.
Community residential facilities
In Italy we recorded more beds in community residential facilities as compared with other high-income countries. These are non-hospital, community-based facilities that provide overnight residence for people with mental disorders. Usually these facilities serve individuals with relatively stable mental disorders who require rehabilitation interventions. In Italy both public and private non-profit and for-profit facilities are available. A challenging issue is that a length of stay exceeding 2 years on average, and reaching 6 years in some Italian regions, may suggest that these facilities, rather than focusing on rehabilitation, provide inpatient care and long-stay residential services. This was also suggested by the PROGRES survey, which showed that patients in residential facilities were mostly males, with low education, and with a disability pension in the majority of cases Almost half of the sample surveyed was totally inactive, not even assisting with their facility’s daily activities. Extremely low resident turnover emerged as one of the most relevant problems [28–30]. Looking ahead, we argue that the mission and operational definition of residential facilities should be reconsidered, perhaps recognising that for many long-term, disabled patients, these facilities currently represent ‘‘homes for life’’ rather than rehabilitation sites. In this perspective, we recognise some ambiguity in their role, being focus on rehabilitation and care but also on some degree of protection, with a risk of gently switching back to custody as main mission.
Variability in service provision
In terms of regional differences, we highlighted a marked variation in service provision for different areas of the country, especially between the more wealthy areas of northern and central Italy and the poorer regions of the south. It was particularly worrying to note a marked variation in the proportion of discharged patients seen within a month, which is an indicator of continuity of care between hospital and the community, an aspect that is usually considered quite strong in the Italian mental health care system. Not only wide differences were observed in different areas of the country, but the average percentage of 49% is well below the average for European countries and for high-income countries, which is 81 and 76%, respectively [13]. Looking ahead, we argue that continuity of mental health care should receive more attention by policy makers and team leaders who have planning and clinical responsibilities, taking advantage of the recently implemented Italian national mental health information system that may play a key role in monitoring this indicator and in providing data to check if poor continuity of care is associated with other facility-related variables, for example the mental health staffing level [31].
Limitations
The description of the Italian reform presented here has several limitations. A first problem is that national statistics describing health systems may have some imprecisions that cannot be quantified. However, WHO and OECD data are based on operational definitions to decrease ambiguity and to guide towards a common interpretation. WHO ATLAS, for example, has a glossary of terms to precisely characterise facilities, workers, and all the service use data that were collected [13]. A second issue is that national statistics do not capture the type and quality of care provided by Italian mental health facilities. However, at the end of the 1990s, two consecutive nationwide projects gathered an unprecedented amount of data about residential care and acute inpatient care [28–30, 32]. On the whole, the data collected highlighted several critical issues, such as a large regional variability in the availability of residential and acute inpatient beds, a delay between symptom onset and first contact with psychiatric services, and a substantial proportion of patients that seem not to receive fully adequate care [28, 29]. Other studies conducted on large, representative numbers of patients in treatment showed that the quality of mental health care may often be of limited quality [33–35].
Overall, during the last 40 years without mental hospitals, Italy has seen a progressive consolidation of a community-based system of mental health care. The Italian experience would suggest that the number of psychiatric beds may not represent a key factor for public health indicators such as rates of suicides, involuntary admissions, and people placed in forensic facilities. We highlighted, however, reasons for concern, including a decreasing staffing level, a potential use of community residential facilities as long-stay residential services, and lack of community alternatives to acute inpatient admissions. Action is therefore required to reverse these trends. At a national level, the resources allocated to mental health care are lower in Italy than in other high-income countries. Consequently, apart from notable exceptions, the organization of services has remained very similar to that implemented 40 years ago. This does not consider the fact that the Italian society has been profoundly changing and the needs of special populations, for example the elderly and adolescents, as well as the needs of new populations, such as economic migrants, asylum seekers and refugees [36], are not receiving enough consideration in current service planning and delivery [37]. Additionally, very few evidence-based specific interventions and treatment modalities, such as early intervention teams for first-episode psychosis, for example, have been implemented. Italy needs to improve what is called ‘translational epidemiology’ in psychiatry [38].
Policy makers and clinical team leaders, with the involvement of a variety of stakeholders and the wider society, should be able to generate a new and innovative vision for the future of mental health care, motivating all the actors involved to work together, as a team, towards new achievements, aiming for continuous improvement and continuous reinforcement of treatment and care as main mission.
Authors’ contributions
CB and BS conceptualised the study. CB was responsible for data curation. CB wrote the first draft and CB, DP and BS reviewed and edited subsequent versions of the manuscript. All authors read and approved the final manuscript.
Authors’ information
CB coordinates the activities of the WHO Collaborating Centre for Research and Training in Mental Health and Service Evaluation at the University of Verona, Italy, and conducts research in the field of evidence-based global mental health. DP is resident in psychiatry and Ph.D. student at the University of Verona. BS is a psychiatrist who has made, and continues to make, a major contribution to global mental health, by his sustained and vigorous efforts for the advancement of psychiatry and mental health around the world, and particularly for those who are least able to speak for themselves. BS had the opportunity to work with Franco Basaglia, an Italian psychiatrist who gave impetus to the development and implementation of the Italian reform.
Consent for publication
Ethics approval and consent to participate
Funding
This study was performed without any external support.
Publisher’s Note
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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
WHO Collaborating Centre for Research and Training in Mental Health and Service Evaluation, Department of Neuroscience, Biomedicine and Movement Sciences, Section of Psychiatry, University of Verona, Policlinico GB Rossi, Piazzale Scuro, 10, 37134 Verona, Italy
Amaddeo F, Barbui C, Tansella M. State of psychiatry in Italy 35 years after psychiatric reform. A critical appraisal of national and local data. Int Rev Psychiatry. 2012;24:314–20.View ArticlePubMedGoogle Scholar
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Q:
HBase schema design in storing query log
Recently, I'm working on make a solution for storing user's search log/query log into a HBase table.
Let's simple the raw Query log:
query timestamp req_cookie req_ip ...
Data access patterns:
scan through all querys within a time range.
scan through all search history with a specified query
I came up with the following row-key design:
<query>_<timestamp>
But the query may be very long or in different encoding, put query directly into the rowkey seems unwise.
I'm looking for help in optimizing this schema, anybody handling this scenario before?
A:
1- You can do a full table scan with a timerange. In case you need realtime responses you have to maintain a reverse row-key table <timestamp>_<query> (plan your region splitting policy carefully first).
Be warned that sequential row key prefixes will get some of your
regions very hot if you have a lot of concurrence, so it would be wise
to buffer writes to that table. Additionally, if you get more writes than a single region can handle you're going to implement some sort of sharding prefix (i.e modulo of the timestamp), although this will make your
retrievals a lot more complex (you'll have to merge the results of
multiple scans).
2- Hash the query string in a way that you always have a fixed-length row key without having to care about encoding (MD5 maybe?)
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On the go and no time to finish that story right now? Your News is the place for you to save content to read later from any device. Register with us and content you save will appear here so you can access them to read later.
Dom 'Furious' George: So many years, so many killed
Dom is Jamie Mackay's reluctant sidekick and long-suffering verbal punching bag. When he's not being abused at work he's being abused at home by his wife and kids. His cheery disposition is further enhanced by the fact that he has to get up at 4 am every day to host The Country Early Edition on Radio Sport. (5-6 am Tue-Sat). Tune in...if you dare.
Whether you were working the land, attending dawn services or simply relaxing on Anzac Day, I'm sure most of you stopped at least once and contemplated the reason for the public holiday.
The day has become increasingly important for our fledgling little nation as we, like all countries, reflect on our sense of identity. For me, it was genuinely relaxing - watching war documentaries and movies on Netflix. And no, for the younger, more savvy readers - I'm married; Netflix and chill sadly means actually putting your feet up in front of the telly.
However, watching the news coverage of the day (including The Country correspondent David Broome and his 12-year-old offspring engaging in their own verbal offensive against a group of peace protesters) and hearing our politicians speak of the commemorations, it struck me how much of the Gallipoli campaign has become about dates and numbers.
So many years, so many decades, so many killed, so many wounded. History, as some assume, is not the study of dates and times; it's about gathering information from primary sources and trying to understand why particular events occurred and perhaps even detecting certain patterns in human behaviour that may prove useful for future generations.
Naturally dates are important, as are numbers, but they're not the imperative of history.
We seem fixated on comparing the number of dead in one battle to the number of dead in others, even pitting war casualties against airplane and earthquake casualties, for example, although to what end I'm unsure. That approach to me seems utterly pointless.
In fact, simply reeling off a few numbers without any sort of context is a sure way to dull the point that's trying to be made, rather than engaging people with the lessons to be learned from events such as Gallipoli. Understanding why it happened, what actually took place and what affect it had, both immediate and long-term, is of infinitely more use to us than a body count.
A couple of years ago Jamie Mackay and I decided we would like to host an ANZAC Day show on Newstalk ZB. Permission was granted from those who grant permissions for such things and The Digger's Breakfast became a reality.
We divvied up the interviews between us - he took the Prime Minister and our Australian Correspondent Chris Russell, among others, while I was to call on the expertise of historians Professor Tom Brooking from Otago University and the Curator of the Toitu Settlers Museum in Dunedin Sean Brosnahan.
Sadly, the shoelace tying the Auckland studio to Dunedin must have snapped in the night and we were beset by that old worn-out phrase used by broadcasters when things go tits-up - "technical difficulties". Unfortunately not all the interviews made it to air but I decided I'd resurface them this year on The Country Early Edition. And listening to them again, I once again appreciated the stories that can be gleaned from the likes of old documents and photographs, along with the first-hand accounts of those present.
Professor Brooking told the story of Alexander Aitken and his violin; an inspiring tale about one of New Zealand's greatest mathematicians, who would go on to work with the team that cracked the Enigma code, and whose cabin mate won the instrument en route to Gallipoli. As he couldn't play it, he gave it to Aitken who entertained the troops virtually every night of the campaign to take their minds off what they'd endured that day and what was surely to come the next. Such was Aitken's proficiency, he would eventually play with the London Philharmonic Orchestra and also pen one of the most detailed and important accounts of New Zealand's WWI effort, 'From Gallipoli to the Somme'.
Sean Brosnahan spoke of many diary entries from the Peninsula that were unusually positive. Unlike the Western Front, dozens of soldiers told of the physical beauty of the environment, especially the spectacular sunsets over the Aegean Sea. For the more educated members of the Anzac forces, the Classical nature of their locale was certainly not lost on them either; after all, Troy was just down the road.
But perhaps the most unassuming story I've come across recently has nothing to do with bullets and bayonets but relates directly to our Anzac commemorations. As Ron Palenski explains in his 2010 book, 'On This Day in NZ', poppies are traditionally used to mark Armistice Day, the date on which the First World War officially ended. But back in 1921 the shipment of poppies from France was delayed, thwarting the RSA's plans to promote the wearing of poppies on November 11.
They made the pragmatic decision to hold them over until the New Year and dish them out on April 25, Anzac Day. Now there's a story.
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The Post
When it comes to a perfectly calibrated film going experience, "The Post" more than delivers. Steven Spielberg's swift and stirring retelling of how the Washington Post published the Pentagon Papers and how it changed contemporary journalism in the process has both intellectual and emotional heft.
It has a felicitous feel for its period -- mostly 1971 -- and the political contentiousness of its time: Nixon is in the White House and the Vietnam War plods on while the streets are filled with protestors in love beads. It's a time when everyone smokes cigarettes, covert communications are made on pay phones and the sleek caftans worn by Katherine Graham (an impeccable Meryl Streep), the Post publisher, reek of the things Elizabeth Taylor wore in her films.
But beyond the fashions, what Spielberg and screenwriters Liz Hannah and Josh Singer do extraordinarily well is pinpoint the moment when the American post war bubble burst -- America was losing a war and didn't know how to cope with it. It is the moment when chaos seeped into all corners of the culture, even Graham's elegant townhouse, where she is seen giving dinner parties to such politicos as Robert McNamara, the former Secretary of Defense who is a prominent player in the events found in the Pentagon Papers.
Graham had became the publisher of the Post after her husband's suicide eight years before. To keep the paper solvent, she's agreed to letting ownership go public; the only caveat being that in the first week of the offering, prospective buyers can opt out if there's some sort of extraordinary event. It will never happen, Graham is told, by one of her confidantes.
But such an extraordinary thing happens when the New York Times publishes a front page expose about how America became involved and persisted in Vietnam, despite the fact that it was a losing proposition. On the same day the Times report breaks, the Post leads with a story about Tricia Nixon's wedding -- a turn of events that made Post editor Ben Bradlee (Tom Hanks) envious of its New York rival. But when the Nixon administration prosecutes the Times for publishing top secret material, the Post picks up the battle, but only if Graham agrees.
Spielberg tells this story with a terse urgency, yet there's nothing showy about the direction, which has the feel of a good HBO film. What may be most satisfying is how sober and observant it is, and how well it builds to its emotional catharsis. It also helps to have a cast like this one. Everyone, from Hanks and Streep to Sarah Paulson, Bob Odenkirk, Tracy Letts, Bradley Whitford, Bruce Greenwood, Carrie Coon and Matthew Rhys is letter-perfect. Especially funny is Michael Stuhlbarg's foppish portrayal of Bradlee's counterpart at the Times, Abe Rosenthal.
It's great to see Tom Hanks in a role that allows his acting to breathe. Far more volatile than Jason Robards, who won an Oscar playing the previous onscreen Bradlee in "All the President's Men," Hanks appears to be having fun playing this now legendary inside-the-beltway personality. (To learn more about the real-life Bradlee, catch the HBO documentary "The Newspaperman: The Life and Times of Ben Bradlee" currently airing.)
Streep skillfully embodies Graham's cultured demeanor, but also digs into her ambivalent personality, which gives her character (and the story) a tension. How she deals with the great changes occurring around her and comes into her own in the process gives the film its emotional pull.
It also underscores the subtext of feminist empowerment that is skillfully woven into the narrative, most pertinently in a missive that Tony Bradlee (Paulson) tells her husband in explaining the gravity of Graham's decision. That Streep makes her moment of choice resonate so emotionally is (again) testament to her greatness as a film actress.
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"The Post" recounts the simultaneous crossroads faced by both the now-nationally important newspaper and neophyte owner Kay Graham. It's also the first example of head-on cinematic resistance to Trumpism. Heroism, then, all around.
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When I think of Answers, I still think of it as the simple question and answers website it once was. The fact is, though, it has changed quite a bit in recent years, with the company expanding to become a platform with a suite of sites that provide those analytics and solutions to other verticals and websites.
The idea is to get content from Answers to sites all around the Web, in order to to help them boost brand engagement, retail sales and customer satisfaction. And the company is moving forward with that plan by acquiring customer experience analytics firm ForeSee, it was announced on Monday.
Terms of the deal were not disclosed, but it was revealed that the deal was funded via a debt financing transaction led by SunTrust Robinson Humphrey and Silicon Valley Bank in Answers Corporation, and equity investments in AFCV Holdings LLC from existing investors, including Summit Partners, TA Associates, AFCV founders and management.
TechCrunch previously reported that the deal was worth over $200 million, and that the debt financing was worth more than $300 million. Answers CEO David Karandish has also confirmed these figures to VatorNews.
As per the deal, ForeSee will retain total autonomy," a ForeSee spokesperson told me. That means that the company will be keeping all of its current employees, as well as staying in the same offices in Ann Arbor.
Founded in 2001, ForeSee works with some of the world's biggest companies in order to help them figure out how the customer experience on their websites, call centers, brick-and-mortar locations and mobile apps impacts loyalty sales.
The company's research allows it to go deeper than simply asking customers what they like.
For example, research about top holiday retailers can show how a good or bad store and customer experience will predict how likely people are to shop more, return to that store and recommend it to others.
By acquiring ForeSee, Answers will now be able provide its customers with a full suite of solutions that span the customer life cycle, from optimizing customer acquisition to analyzing the customer experience to predicting future customer behavior.
This is a mutually benefitial acquisition: for ForeSee, its customer will have now have access to Answers' product offerings, including information from the company's other sites, as well as the other acquisitions that it has made in the last 18 months, ResellerRatings, Webcollage and Easy2.
For Answers, it means that its content will now appear on sites for half of the Fortune 500, and 82 of the top 100 retail sites, Karandish said.
What really brought the two companies together, though, was a shared vision and approach.
"We are two tech driven companies, a shared vision: the ability to help consumers, brands and retailrs to make good decisions," he told me. "And, on the culture side, we are two Midwestern based companies, with Answers based in St. Louis and ForeSee in Ann Arbor. I'd say we both have a certain Mid-Western approach."
Being purchased by Answers was a great opportunity for the company, ForeSee CEO Larry Freed wrote in a blog post on Monday.
"Joining with Answers also allows us to be part of a full lifecycle solution, from optimizing customer acquisition to analyzing the customer experience to predicting future customer behavior," he said.
"It’s a partnership that makes a lot of sense for our business and for Answers and the suite of companies in its portfolio."
Founded in 1998 as GuruNet, and originally located in Jerusalem, the now St. Louis-based Answers, which calls itself “the world's top destination for trusted Q&A content,” operates Answers.com, WikiAnswers, ReferenceAnswers, VideoAnswers, and five international language Q&A communities in Spanish, French, Italian, German and Tagalog.
Answers, which was purchased by AFCV Holdings in February 2011 for $127 million in cash. Its platform currently contains more than 17 billion answers, with a user base of 160 million. The site estimates that it gets 10,000 new registered users every hour.
In the next 12 to 18 months, Karandish says that he wants Answers to have the "credible threat" of an IPO.
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Movement Disorders Program
You are here
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Texas Medical Center
Make a Difference
The Promise Campaign will expand our care to more children, and offer comfort to the people who love them.
Seeing patients with impairments of body movement and control
The Pediatric Movement Disorders Program at Texas Children’s Hospital sees patients with impairments of body movement and control. Each patient receives a thorough evaluation to diagnose the specific type of movement disorder and develop an individualized plan of treatment.
The Pediatric Movement Disorders Program is a multidisciplinary effort of Neurology and Developmental Neuroscience in collaboration with the departments of Physical Medicine, Neurosurgery and multiple others.
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Cumulus-oocyte interactions and programmed cell death in bovine embryos produced in vitro.
This study investigated the interactions between cumulus cells (CCs) and oocytes and programmed cell death in bovine cumulus-oocyte complexes (COCs) with different morphological characteristics. DNA fragmentation was assessed in CCs at 0 and 24 h of maturation, as well as parthenogenetic developmental competence on the 9th day post-activation, blastocyst quality and BCL-2 and BAX transcript levels in matured CCs. Most immature oocytes in the COC-A group (full cumulus and several compact layers) were in the initial germinal vesicle (iGV) stage, exhibiting minimal or no DNA damage. In contrast, after follicle removal, the COCB (partial cumulus and one or two cell layers) and C (expanded cumulus) groups presented in more advanced GV stages and exhibited DNA fragmentation. After maturation, significant increases in fragmented nuclei were noted in COCC and COCB groups. Embryos resulting from the COC-A developed more rapidly and had increased competence compared to embryos resulting from groups COCB and COCC. The COCB group exhibited the highest BAX protein levels and a reduced BCL-2/BAX protein ratio. The results show a negative correlation between nuclear fragmentation and embryonic development potential in COCs with different morphologies. In addition, a low BCL-2/BAX protein ratio might be associated with an increase in nuclear fragmentation in CCs.
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Police Blotter: Recent crime in Berkeley
Selected calls for service reported in Berkeley from Dec. 16-23, via CrimeMapping.com.
Selected calls for service reported in Berkeley from Dec. 16-23, via CrimeMapping.com.
This is a partial list of recent crimes in Berkeley based on information from the Berkeley Police Department and the UC Berkeley Police Department, unless otherwise noted. [Note to readers: We're trying out a new format, and will be experimenting in the coming weeks to get you more information, so please provide feedback in the comments section below if you notice anything new or have ideas for more coverage.] See past Crime Blotters here.
The following represents just a sampling of calls, many of which were selected by the Berkeley Police Department. From Dec. 16-23, there were 23 burglary reports, 42 auto break-in or theft reports, and six stolen vehicle reports within the city limits, according to CrimeMapping.com. Nine robberies and seven assaults were reported.
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Indian cricket going in circles
At the best of times, the Indian cricket fan’s hair is at the gravest risk: the team flutters from the sublime to the ludicrous and back so quickly that you often get the urge to tear it off.
You can imagine the plight of the hapless hair when the team is caught in a particularly perplexing maze; after the rout in England and the hiding in Australia, it is clearly lost in one of those intricate lanes.
Not too surprisingly, India have embarked on a fanciful experiment: pompously called ‘the rotation policy’, in this the three openers in the squad take turns at missing a game so that the youngsters in the middle order get more opportunities.
On the surface, it seems like a smart, even laudable, move: after all, the future lies in the hands, wrists and willows of the youth brigade. But one wonders why this masterstroke didn’t strike them during the Tests, when the seniors couldn’t make mincemeat of the ball.
Sehwag, Gambhir, Dravid, Laxman, Dhoni and, to some extent, even Tendulkar looked like they needed a break; yet, none of them was rested and India’s famed batting lineup looked like a pale shadow of itself. If nothing, fresh legs would have forced the Aussies to draw up a new plan for the new batsman.
Rohit Sharma was itching to have a go; yet, he was kept on the sidelines, match after match. Worse, he was left on sharp tenterhooks in the run-up to each Test, with no clear indication if he was in or out. The pressure and anxious wait must have taken a toll on him. And it’s showing in the One-dayers.
One of India’s finer talents might wither away even before he can blossom, let alone bloom. Rohit, thankfully, has already experienced the vicissitudes of life and won’t give in without a wholesome fight. He knows that runs are his only ally in tough times and he will have to tap them for survival.
Virat Kohli too teetered towards the edge, primarily because of the fear of failure; luckily, the team management gave him one last chance, that too at treacherous Perth, and he passed his test of fire. One can only wonder if Rohit too would have already crossed his big hurdle, if he had been drafted in at the right time.
The team seems to have learnt its lesson. However, one suspects it’s not as simple as it seems: it’s now an open secret that Sehwag and Dhoni don’t see eye to eye. Was he a victim of this ego game in the opening encounter? Is that why Gambhir was dropped in the second match and the bogey of rotation policy raised?
When it comes to India, you can never say. Anyway, such an important policy shift cannot be left at the whims and fancies of the captain or the touring think-tank; it has to be at the behest of the selection panel, and that too only if it is convinced that it’s the only way forward into the future.
Otherwise, the aim always has to be to field the best possible combination: the best players, the in-form batsmen and bowlers must be preferred, unless there are injury worries. Or indeed, if somebody is showing clear signs of fatigue or jadedness.
Sadly, these are not the best of times for Indian cricket. Which means your hair is not safe.
DISCLAIMER : Views expressed above are the author's own.
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Author
Bobilli Vijay Kumar is TOI's National Sports Editor. He writes a weekly column, mainly on cricket; he has many interests, though, going way beyond sport itself: you can't keep him out of a movie hall or away from Floyd or Doors. He loves to cook and show off his French; he can play a tune or two on the guitar too. His current fancy, though, is salsa.
Bobilli Vijay Kumar is TOI's National Sports Editor. He writes a weekly column, mainly on cricket; he has many interests, though, going way beyond sport its. . .
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This proposal requests funds to purchase a robot to set up crystallization experiments for the Structural Biology Resource Center at The Rockefeller University. The robot will aid researchers in obtaining diffraction quality crystals of biological macromolecules involved in bacterial transcription and its regulation, bacterial two-component histidine kinase signaling, bacterial pathogenesis, hepatitis C virus replication, and eukaryotic cell division. Obtaining crystals is often the rate-limiting step in X-ray structure determination. The robot will greatly increase the efficiency of crystallization trials by reducing the amount of time and sample required. This, in turn, will increase the success rate of NIH-funded research, and will increase utilization of the NIH-funded X-ray equipment already in the Resource Center. Due to the smaller quantities of materials required for the robot as opposed to manual setup, the robot will also allow extensive crystallization trials for projects that would otherwise not be possible. In addition, crystallization experiments will have a smaller impact on the environment by generating much less waste. The crystallization robot requested is the Screenmaker96+8 from Innovadyne - $145000. This robot can perform three distinct processes needed to set up crystallization experiments. Step 1: transfer of reservoir solutions from the sparse matrix stocks to the 96 positions in the crystallization tray;Step 2: deposition of nano-liter volumes of this reservoir onto the drop position and deposition of the protein nano-liter volume onto the same spot;Step 3: to refine the condition by making custom grids by varying two components in the lead condition to create new opportunities for improved crystal growth. Another very important feature of the Screenmaker96+8 is that the time required to place the drops (both the precipitant solution and the protein) is significantly less than other crystallization robots, thereby reducing dehydration time by at least half. The crystallization robot will be housed in the newly created Structural Biology Resource Center, supported by The Rockefeller University under the umbrella of the Research Support office. The Center will supervise the robot's use, train users and maintain the instrument in top working condition, and will be responsible for fair, relevant and efficient utilization of the robot. Additional parts needed: computer $2450, installation $4000, two nest chiller system $10450, for sensitive samples and extended warranty $32000. (Prices quoted here do not reflect 10% academic discount applied to total requested in budget, see price quote in Equipment.doc). PUBLIC HEALTH RELEVANCE: Determining the three-dimensional structure of biological macromolecules is key in understanding their function and in opening new venues for designing drugs and cures for human diseases. This can be accomplished through the technique known as X-ray crystallography, where the rate-limiting step is often the crystallization of the target molecule, a menial process that is labor intensive and time consuming. This proposal requests a robot that performs the steps involved in setting up crystallization trials of biologically important molecules, decreasing the labor 10 fold and decreasing the generation of waste while increasing the efficiency and scope of the experimental projects.
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1. Field of the Invention
The present invention relates to a resurfacing tool and more particularly pertains to reshaping and resurfacing edges of windshield wiper blades, squeegees, dustpans, and the like with a resurfacing tool.
2. Description of the Prior Art
The use of windshield blade cleaners is known in the prior art. More specifically, windshield blade cleaners heretofore devised and utilized for the purpose of cleaning windshield blades are known to consist basically of familiar, expected and obvious structural configurations, notwithstanding the myriad of designs encompassed by the crowded prior art which have been developed for the fulfillment of countless objectives and requirements.
By way of example, U.S. Pat. No. 3,708,924 to Prunchak discloses a windshield blade reconditioner.
U.S. Pat. No. 5,426,895 to Siciliano et al. discloses a windshield wiper blade cleaner.
U.S. Pat. No. Des. 322,922 to Swanson et al. discloses the ornamental design for a ski edge sharpener.
U.S. Pat. No. 4,617,765 to Weiler discloses a wiper blade edger.
U.S. Pat. No. 3,886,657 to Fabian discloses a windshield wiper sharpener.
U.S. Pat. No. 5,381,629 to Salvail discloses a portable sharpener.
While these devices fulfill their respective, particular objective and requirements, the aforementioned patents do not describe a resurfacing tool for reshaping and resurfacing edges of windshield wiper blades, squeegees, dustpans, and the like.
In this respect, the resurfacing tool according to the present invention substantially departs from the conventional concepts and designs of the prior art, and in doing so provides an apparatus primarily developed for the purpose of reshaping and resurfacing edges of windshield wiper blades, squeegees, dustpans, and the like.
Therefore, it can be appreciated that there exists a continuing need for new and improved resurfacing tool which can be used for reshaping and resurfacing edges of windshield wiper blades, squeegees, dustpans, and the like. In this regard, the present invention substantially fulfills this need.
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Q:
NSSortDescriptor get key from core data in Swift 4
I am new to IOs developement and core Data.
In my application I want to use NSSortDescriptor to sort Data before fetching it to my tableview.
In my core data I have one table "TBL_Products" with two fields "product_name" and "Quantity"
The code below work perfectly fine
let MySortDescriptor = NSSortDescriptor(key: "product_name" , ascending: true)
But what I hate about it is I am hard coding the key name so the app will crash if the column name has been changed.
Is there is a way do to something like this (the code below not working):
let MySortDescriptor = NSSortDescriptor(key: TBL_Products.product_name.descriptor , ascending: true)
A:
Use can use #keyPath directive:
NSSortDescriptor(key: #keyPath(TBL_Products.product_name), ascending: true)
The compiler replaces that with a string containing the property name.
It is also useful in Core Data predicates, e.g.
NSPredicate(format: "%K == %@", #keyPath(TBL_Products.product_name), "someProduct")
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The effect of papulacandin B on (1----3)-beta-D-glucan synthetases. A possible relationship between inhibition and enzyme conformation.
The antibiotic, papulacandin B, inhibited growth or (1----3)-beta-D-glucan synthetase (or both) in the fungi Saccharomyces cerevisiae, Hansenula anomala, Neurospora crassa, Cryptococcus laurentii, Schizophyllum commune and Wangiella dermatitidis. No effect was observed on Achlya ambisexualis. There was no apparent correlation between the inhibition of growth and that of the synthetase. With most of the fungal extracts, the inhibition of glucan synthetase by papulacandin B became less pronounced as the substrate (UDP-glucose) concentration was decreased. At very low levels of UDP-glucose, with the enzymes from S. cerevisiae and W. dermatitidis, the antibiotic stimulated the activity of glucan synthetase. As further studied with the W. dermatitidis enzyme, those low concentrations of UDP-glucose corresponded to a sigmoidal portion of the rate vs. substrate curve. The sigmoid segment of the curve extended to higher concentrations of UDP-glucose as the temperature was increased. Concomitantly, the range of substrate concentrations at which papulacandin B stimulated the reaction or was noninhibitory was broadened. It is tentatively concluded that glucan synthetase may exist in more than one interconvertible form. The stimulatory effect of papulacandin B is possibly due to preferential binding to the active form of the enzyme. The equilibrium between these forms could be shifted by structural changes in the membrane in which the enzyme is embedded. The lack of correlation between the effects of papulacandin B in whole cells and in extracts is discussed in terms of the variations in membrane structure in the two situations.
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A current ELD mandate waiver which postponed the measure for trucks carrying produce and other ag products ends March 18.
In a letter submitted Feb. 20, the United Fresh Produce Association, Western Growers, the National Potato Council, the U.S Apple Association and more than 20 other produce groups said a combination of factors have driven up transportation costs.
“With the electronic logging device (ELD) mandate, driver shortages, and other issues, there have been considerable increases in transportation costs for fresh produce causing devastating effects on our industry,” the letter said. “We are hearing from many of our members across multiple commodities and sectors throughout the country that shippers are having to pay two or three times, occasionally more, the normal rate for transporting their product.”
ELD concerns
The letter said feedback from producers and trucking operations indicates many ELDs on the market are not able to accommodate the agricultural exemption that is provided under the hours-of-service regulations. Under the agricultural exemption, hours-of-service regulations do not apply to the transportation of agricultural commodities operating within a 150-air mile radius of a pick-up.
“We believe that this extension would provide a reasonable period of time for FMCSA to work with the technology providers in developing a program to verify that the ELDs on the market can perform the tasks that the rule mandates and allow trucks hauling agricultural commodities to fully utilize the 150-mile exemption,” according to the letter.
The coalition is asking the agency to consider hours-of-service modifications to accommodate the realities of loading and unloading fresh produce.
“The unpredictability of loading and unloading times as it relates to fresh fruits and vegetables can significantly detract from the on-duty hours drivers are allowed in a day,” according to the letter, which notes that two-to four-hour delays at loading are not uncommon.
“We encourage FMCSA to consider flexibility under either the ELD rule or the hours-of-service rule for truck drivers who are idling, waiting or traveling small distances reflective of negotiating a congested terminal to be considered in an exempt status,” according to the letter. “We do not believe that this type of activity is as demanding as over-the-road driving and therefore should not contribute to maximum driving times.”
Allow packing facilities, cold storages and other locations to be considered as a “source” location under the hours-of-service regulation.
Allow the agricultural exemption’s 150-air-mile radius to begin at the final pick up point for multi-point pickups. Drivers make multiple pick-ups from small packinghouses or cold storage facilities to fill their load before continuing to final destinations. “We would encourage the 150 air-mile radius to begin at the location of the last pick-up point so as not to disrupt current supply chains and accommodate the operational efficiencies organically created by the marketplace over the last 100-plus years,” according to the letter.
Clearly define that empty trucks are covered under the agricultural exemption. According to the letter, agricultural exemptions should be clearly defined to include unladen trucks as eligible if they are traveling to a facility exclusively to pick up an order.
The deal is expected to close in the first quarter of this year, according to a news release.
“Mann Packing’s strength in the vegetable category, one of the fastest-growing fresh food segments, will provide us with synergies, enhancing our ability to better serve our combined customers and address consumers’ needs for healthier products,” Mohammad Abu-Ghazaleh, Del Monte chairman and CEO, said in the release. “This acquisition is a significant step toward our goal to be the world’s leading supplier of healthful, wholesome and nutritious fresh and prepared food and beverages for consumers.”
Mann Packing sales in 2017 were $535 million.
“Everyone at Mann is excited with this development,” Lorri Koster, chairman and CEO of Mann Packing, said in the release. “We share Del Monte’s values and commitment of providing fresh, high-quality produce based foods that are nutritious and delicious. Both our companies have been successful in their own right with their superior quality, service and value to our customers and consumers in all channels throughout North America. This will only be enhanced by combining the business expertise and skills of two of the industry’s premiere organizations.”
A system patented by Walmart aims to address one of the top drawbacks for would-be online shoppers: the desire to pick their own produce.
The “Fresh Online Experience,” a process Walmart outlined in a patent published Dec. 28, would allow consumers to remotely approve or reject specific produce items prepared for online orders. The service could be used for other fresh items as well.
When placing an order, consumers could select which items to confirm. Once two-dimensional or three-dimensional photos of the produce have been sent, the consumer has a set amount of time to approve or reject the items.
Walmart explained its rationale for the system — for which fulfillment could be manual or automated — in the background section of patent.
“A customer when visiting a retail store can inspect and choose produce that seems to look like the highest quality,” the company stated in the document. “However, a customer who orders the same item from a retail store website for grocery pickup and/or delivery has to rely on the store associate to choose the actual item to be delivered. They may be dissatisfied with the result.
“It is desirable for the customer to be able to request images of the item in the retail store, so that the customer can be satisfied with their online purchase,” Walmart stated.
The company has patented numerous other ideas over the years that have not been deployed. E-commerce, however, has been a major area of growth for Walmart, and inability to inspect produce and other fresh item is one of the most cited reasons people give for not grocery shopping online.
The declaration lets farmers and ranchers in those areas seek support, including emergency loans, from the Farm Service Agency, according to a news release.
“I thank U.S. Secretary of Agriculture Sonny Perdue for taking action to support Florida’s farmers and ranchers still picking up the pieces from Hurricane Irma,” Florida agriculture commissioner Adam Putnam said in the release Oct. 13. “Our preliminary estimates peg the total damage at more than $2.5 billion, but it’s important to recognize that the damage is still unfolding.
“Today’s disaster declaration provides much needed support, and I will continue working with (Florida Gov. Rick Scott) and our leaders in Washington to get Florida agriculture the relief it needs to rebuild,” Putnam said.
The Packer’s Midwest Produce Expo is back in Kansas City, Mo., Aug. 14-16 for its sixth annual edition, and this year the show focuses on a huge generation of consumers who are changing food retailing: millennials.
“Millennial Mindset is our theme this year to get the produce industry better information about reaching this important group of consumers,” said The Packer Publisher Shannon Shuman
The Packer Editor Greg Johnson and Produce Retailer Editor Pamela Riemenschneider will present a millennial version of their Fresh Trends Quiz Show, which uses Fresh Trends 2017 data to show attendees how to better market to millennials, and it uses real-time audience answers in the presentation.
“Pamela and I have some surprises for our attendees,” Johnson said. “For instance, younger consumers tend to buy fruits and vegetables less often than other age groups according to our Fresh Trends survey, but they’re even with or above the other groups on tropical fruits. We’ll explain why and what retailers can do with that information.”
Keynote speaker Matt Beaudreau has given keynote presentations to many corporate groups, and he uses proprietary data on millennials to show how to better market to and employ them.
The Millennial Mindset education program starts with a presentation on e-commerce and grocery delivery by Erick Taylor, president and CEO of Pyramid Foods, a Rogersville, Mo., retail chain, which operates 52 stores under several banners in Missouri, Oklahoma and Kansas.
Then, Garland Perkins, U.S. retail solutions with The Oppenheimer Group, Vancouver, British Columbia, will analyze millennials using her personal experience and professional experiences.
The 5-hour expo is Aug. 15 at the headquarter hotel, the Sheraton Kansas City Hotel at Crown Center.
Homeowners with citrus trees in their yards can apply online to have a vial of tiny parasitic wasps mailed to them, that can then be released onto citrus trees.
To defend the state’s citrus crop from an industry-crippling infection, scientists with the Florida Department of Agriculture & Consumer Services are fighting pest with parasites.
Florida residents can apply online to the department for tiny parasitic wasps called tamarixia that hunt the Asian citrus psyllid, an invasive insect that spreads the fatal disease “citrus greening.”
The psyllid carries the infection, which plugs the plant’s phloem, starves the tree and causes fruit to drop prematurely. Tamarixia feed on the pest and lay eggs inside young psyllids, killing them and, hopefully, the bacteria that cause the disease, said biological scientist Gloria Lotz.
At a mass-rearing lab in Gainesville, one of a few throughout the state, Lotz and fellow researchers supply over 1 million tamarixia every year to commercial citrus growers and now, Florida residents who want to protect their backyard citrus trees.
The tamarixia release program is one of several tools researchers and growers use to slow greening’s spread, including pesticides to kill the disease-causing bacteria and hydroponic systems to keep infected plants healthy.
But there’s no single solution to a complex problem like citrus greening. It’s infected nearly 100 percent of the state’s mature citrus trees, said Steve Futch, a citrus agent at the UF/IFAS Citrus Research and Education Center in Lake Alfred.
Biocontrol methods usually operate as a “series of waves,” he said; when there are fewer pests, the parasite that hunts them starts to decline, too.
The chances of eradicating the psyllid and the infection with tamarixia are slim, he said — but it should work well in smaller, urban environments, where wasps can fly between citrus trees on different properties.
The citrus industry employs nearly 76,000 growers, truckers, pickers, and packers who face job loss if crop production continues to decline. But Futch said despite the bleak prognosis, Florida’s staple crop will survive—though it may be a bit smaller.
“There will always be a citrus industry in Florida,” he said. “It will be different in the future than it is today and in the past.”
Citrus tree owners can apply here to have a small vial of the tiny wasps sent to their home: http://bit.ly/2vfcI5V
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Q:
Uncaught TypeError: undefined is not a function when using a jQuery plugin in Magento
I am working on a Magento Extension. Magento has a known issue when you try to use jQuery in Magento because Magento uses the Prototype library.
The work around for it is to put your jQuery code in no conflict mode like this...
jQuery.noConflict();
Once I did this, it resolved 90% of my problems i was having with JavaScript errors. However I still have 3 major problems with JavaScript right now and I believe they are related...
Uncaught TypeError: undefined is not a function
http://www.codedevelopr.com/screenshots/2014/2014-08-23_16-14-37.png
When I view the line numbers that it is reporting these errors from I see this...
Line 1168 jQuery(".acc-wizard").accwizard({ now this accwizard() is a function that is loaded from a jQuery plugin file. I have verified the file is loaded and it is loaded after my jQuery and after I set the no conflict mode for jQuery. I am not sure if something in the plugin file needs to be changed as well to work with the no conflict mode or why it is saying it is undefine?
Same situation with simplecolorpicker() on line 1180.
I have uploaded the file that holds the accwizard() jQuery Plugin, it is about 14kb in size and can be found here http://www.codedevelopr.com/screenshots/2014/acc-wizard-bs-3.js I figured it is a little to large to post that code here.
Can anyone help me to get these errors resolved? It seems any jQuery plugin I try to use results in this undefined error above?
UPDATE
So I have been experimenting with a lot of things with no luck...that is until I tried loading all my JS files inline in my actual template page that my extension uses...once I do that, it all works with none of these errors.
This is frustrating though as I much prefer to have separate JS file for my JS...instead of loading 3 JS files I even tried putting all 3 into 1 file and loading that 1 file but I still get the JS errors...now when I copy that 1 file that had all 3 files combined and put it directly in the template file...everything works perfectly! This makes no sense to me, please help?
A:
you may need to use a closure
;(function($){
// your code
})(jQuery);
Move your code inside the closure and use $ instead of jQuery
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---
abstract: 'We prove conjecture due to Erickson-Semenoff-Zarembo and Drukker-Gross which relates supersymmetric circular Wilson loop operators in the $\CalN=4$ supersymmetric Yang-Mills theory with a Gaussian matrix model. We also compute the partition function and give a new matrix model formula for the expectation value of a supersymmetric circular Wilson loop operator for the pure $\CalN=2$ and the $\CalN=2^*$ supersymmetric Yang-Mills theory on a four-sphere. A four-dimensional $\CalN=2$ superconformal gauge theory is treated similarly.'
address: 'Physics Department, Princeton University, Princeton NJ 08544'
author:
- Vasily Pestun
bibliography:
- 'lib.bib'
date: 'December, 2007'
title: 'Localization of gauge theory on a four-sphere and supersymmetric Wilson loops '
---
[^1]
ITEP-TH-41/07\
PUTP-2248
Introduction
============
Topological gauge theory can be obtained by a twist of $\CalN=2$ supersymmetric Yang-Mills theory [@Witten:1988ze]. The path integral of the twisted theory localizes to the moduli space of instantons and computes the Donaldson-Witten invariants of four-manifolds [@Witten:1988ze; @MR1094734; @MR710056].
In a flat space the twisting does not change the Lagrangian. In [@Nekrasov:2002qd] Nekrasov used a $\U(1)^2$ subgroup of the $\SO(4)$ Lorentz symmetry on $\BR^4$ to define a $\U(1)^2$-equivariant version of the topological partition function, or, equivalently, the partition function of the $\CalN=2$ supersymmetric gauge theory in the $\Omega$-deformed background [@Nekrasov:2003rj]. The integral over moduli space of instantons $\CalM_{inst}$ localizes at the fixed point set of a group which acts on $\CalM_{inst}$ by Lorentz rotations of the space-time and gauge transformations at infinity. The partition function ${Z_{\text{inst}}}(\ve_1,\ve_2,a)$ depends on the parameters $(\ve_1,\ve_2)$, which generate $\U(1)^2$ Lorentz rotations, and the parameter $a \in
\g$, which generates gauge transformations at infinity. By $\g$ we denote the Lie algebra of the gauge group. This partition function is finite because the $\Omega$-background effectively confines the dynamics to a finite volume $V_{\text{eff}} = \frac {1}{\ve_1 \ve_2}$. In the limit of vanishing $\Omega$-deformation ($\ve_1,\ve_2 \to 0$) the effective volume $V_{\text{eff}}$ diverges as well as the free energy $F=-\log {Z_{\text{inst}}}$. But the specific free energy $F/V_{\text{eff}}$ has a well-defined limit, which actually coincides with Seiberg-Witten low-energy effective prepotenial $\CalF(a)$ of the $\CalN=2$ supersymmetric Yang-Mills theory [@Seiberg:1994aj; @Seiberg:1994rs]. In this way instanton counting gives a derivation of Seiberg-Witten prepotential from the first principles.
In this paper we consider another interesting situation where an analytical computation of the partition function is possible. We consider the $\CalN=2$, the $\CalN=2^{*}$ and the $\CalN=4$ Yang-Mills theory on a four-sphere $S^4$ equipped with the standard round metric.[^2]
There are no zero modes for the gauge fields, because the first cohomology group of $S^4$ is trivial. There are no zero modes for the fermions. This follows from the fact that the Laplacian operator on a compact space is semipositve and the formula $\Dslash^2 = \Delta + \frac R 4$, where by $\Dslash$ we denote the Dirac operator, by $\Delta$ the Laplacian, and by $R$ the scalar curvature, which is positive on $S^4$. There are no zero modes for the scalar fields, because there is a mass term in the Lagrangian proportional to the scalar curvature.
Observing that there are no zero modes at all, we can try to integrate over all fields in the path integral and to compute the full partition function of the theory. In addition, we would like to compute expectation values of certain interesting observables.
In this paper we are mostly interested in the observable defined by the supersymmetric circular Wilson loop operator (see Fig. \[fig:Wilson-loop\]) $$\begin{aligned}
\label{eq:Wilson-loop-defined}
W_R(C) = \tr_{R} \Pexp \oint_{C} (A_{\mu} dx^{\mu} + i \Phi_{0}^{E} ds).
\end{aligned}$$
Here $R$ is a representation of the gauge group, $\Pexp$ is the path-ordered exponent, $C$ is a circular loop located at the equator of $S^4$, $A_{\mu}$ is the gauge field and $i \Phi_{{0}}^{E}$ is one of the scalar fields of the $\CalN=2$ vector multiplet. We reserve notation $\Phi_{{0}}^E$ for the scalar field in a theory obtained by dimensional reduction of a theory in Euclidean signature. Our conventions are that all fields take values in the real Lie algebra of the gauge group. For example, if the gauge group is $\U(N)$, then all fields can be represented by antihermitian matrices. The covariant derivative is $D_{\mu} = \p_{\mu} + A_{\mu}$ and the field strength is $F_{\mu \nu} = [D_{\mu}, D_{\nu}]$.
{width="4cm"} \[fig:Wilson-loop\]
In [@Erickson:2000af] Erickson, Semenoff and Zarembo conjectured that the expectation value $ \langle W_R(C) \rangle$ of the Wilson loop operator in the four-dimensional $\CalN=4$ $\SU(N)$ gauge theory in the large $N$ limit can be exactly computed by summing all rainbow diagrams in Feynman gauge. The combinatorics of rainbow diagrams can be represented by a Gaussian matrix model. In [@Erickson:2000af] the conjecture was tested at one-loop level in gauge theory. In [@Drukker:2000rr] Drukker and Gross conjectured that the exact relation to the Gaussian matrix model holds for any $N$ and argued that the expectation value of the Wilson loop operator can be computed by a matrix model. However, Drukker-Gross argument does not prove that this matrix model is Gaussian.
In the context of the $AdS/CFT$ correspondence [@Maldacena:1997re; @Witten:1998qj; @Gubser:1998bc] the conjecture was relevant for many works; see for example [@Maldacena:1998im; @Berenstein:1998ij; @Rey:1998ik; @Drukker:1999zq; @Semenoff:2001xp; @Zarembo:2002an; @Zarembo:2002ph; @Tseytlin:2002tr; @Bianchi:2002gz; @Semenoff:2002kk; @Pestun:2002mr; @Drukker:2005kx; @Semenoff:2006am; @Drukker:2006zk; @Drukker:2007yx; @Yamaguchi:2007ps; @Okuyama:2006jc; @Okuyama:2006ir; @Gomis:2006im; @Tai:2006bt; @Giombi:2006de; @Chen:2006iu; @Hartnoll:2006ib; @Drukker:2006ga; @Drukker:2007dw; @Chu:2007pb] and references there in. But there has been no direct gauge theory derivation of the conjecture beyond the two-loop level [@Plefka:2001bu; @Arutyunov:2001hs] and some attempts to evaluate the first instanton corrections [@Bianchi:2002gz].
In this paper, we prove the Erickson-Semenoff-Zarembo/Drukker-Gross conjecture for the $\CalN=4$ supersymmetric Yang-Mills theory formulated for an arbitrary gauge group. Let $r$ be the radius of $S^4$. The conjecture states that $$\label{eq:main-result-op}
\langle W_{R}(C) \rangle_{\text{$\CalN=4$ on $S^4$}} = \frac { \int_{\g} [da] \, e^{-\frac { 4 \pi^2 r^2} {g^{2}_{YM}}(a,a)} \tr_R e^{2 \pi r i a} }
{ \int_{\g} [da] \, e^{-\frac { 4 \pi^2 r^2} {g^{2}_{YM}} (a,a)} }.$$
The finite dimensional integrals in this formula are taken over the Lie algebra $\g$ of the gauge group, $a$ denotes an element of $\g$. By $(a,a)$ for $a \in \g$ we denote an invariant positive definite quadratic form on $\g$. Our convention is that the kinetic term in the gauge theory is normalized as $\frac {1} {4 g_{YM}^2} \int d^4 x \sqrt{g} (F_{\mu \nu},F^{\mu \nu})$. The formula (\[eq:main-result-op\]) can be rewritten in terms of the integral over the Cartan subalgebra of $\g$ with insertion of the usual Weyl measure $\Delta(a) = \prod_{\alpha \in \text{roots of $\g$}} \alpha \cdot a$.
We also get a new formula for the $\langle W_R(C) \rangle$ in the $\CalN=2$ and the $\CalN=2^*$ supersymmetric Yang-Mills theory. As in the $\CalN=4$ case, the result can be written in terms of a matrix model. However, this matrix model is much more complicated than a Gaussian matrix model. We derive this matrix model action up to all orders in perturbation theory. Then we argue what is the non-perturbative contribution of all instanton/anti-instanton corrections.
Our main result is $$\label{eq:main-result}
\boxed{ Z^{\CalN}_{S^4} \langle W_R(C) \rangle_{\CalN} = \frac {1} {\vol(G)} \int_{\g} [da] \, e^{-\frac { 4 \pi^2 r^2} {g^{2}_{YM}} (a,a) } Z^{\CalN}_{\text{1-loop}}(ia)|{Z_{\text{inst}}}^{\CalN}(r^{-1},r^{-1},ia)|^2 \tr_R e^{2\pi r i a}}.$$
Here $Z^{\CalN}_{S^4}$ is the partition function of the $\CalN=2$, the $\CalN=2^*$ or the $\CalN=4$ supersymmetric Yang-Mills theory on $S^4$, defined by the path integral over all fields in the theory, and $\langle W_R(C) \rangle_{\CalN}$ is the expectation value of $W_R(C)$ in the corresponding theory. In particular, if we take $R$ to be the trivial one-dimensional representation, the formula says that the partition function $Z_{S^4}^{\CalN}$ is computed by the following finite-dimensional integral: $$\label{eq:main-result2}
Z^{\CalN}_{S^4} = \frac {1} {\vol (G)}\int [da] e^{-\frac { 4 \pi^2 r^2} {g^{2}_{YM}} (a,a) } Z^{\CalN}_{\text{1-loop}}(ia)|{Z_{\text{inst}}}^{\CalN}(r^{-1},r^{-1},ia)|^2.$$
In other words, we show that the Wilson loop observable is compatible with the localization of the path integral to the finite dimensional integral and that $$\boxed{ \langle W_{R}(C) \rangle_{\text{4d theory}} = \langle \tr_R e^{2 \pi r i a} \rangle_{\text{matrix model}}},$$ where the matrix model measure $\langle \dots \rangle_{\text{matrix model}}$ is given by the integrand in .
The factor $Z_{\text{1-loop}}(ia)$ is a certain infinite dimensional product, which appears as a determinant in the localization computation. It can be expressed in terms of a product of Barnes $G$-functions [@adamchik-2003]. In the most general $\CalN=2^*$ case, the factor $Z_{\text{1-loop}}(ia)$ is given by the formula . The $\CalN=2$ and $\CalN=4$ cases can be obtained by taking respectively limits $m=\infty$ and $m=0$, where $m$ is the hypermultiplet mass in the $\CalN=2^{*}$ theory. For the $\CalN=4$ theory we get $Z_{\text{1-loop}}=1$.
The factor ${Z_{\text{inst}}}(\ve_1,\ve_2,ia)$ is Nekrasov’s partition function [@Nekrasov:2003rj] of point instantons in the equivariant theory on $\BR^4$. In the $\CalN=2^*$ case it is given by the formula (\[eq:Z-inst-N-star\]). In the limit $m=\infty$, one gets the $\CalN=2$ case (\[eq:Z-inst\]), in the limit $m=0$ one gets the $\CalN=4$ case. In the $\CalN=4$ case, the instanton partition function (\[eq:Z-N-4-eta\]) does not depend on $a$. Therefore in the $\CalN=4$ case, instantons do not contribute to the expectation value $\langle
W_R(C) \rangle$.
Our claim about vanishing of instanton corrections for the $\CalN=4$ theory contradicts to the results of [@Bianchi:2002gz], where the first instanton correction for the $\SU(2)$ gauge group was found to be non-zero. In [@Bianchi:2002gz] the authors introduced a certain cut-off on the instanton moduli space, which is not compatible with the relevant supersymmetry of the theory and the Wilson loop operator. Our instanton calculation is based on Nekrasov’s partition function on $\BR^4$. This partition function is regularized by a certain non-commutative deformation of $\BR^4$ compatible with the relevant supersymmetry. Though we do not write down explicitly the non-commutative deformation of the theory on $S^4$, we assume that such deformation can be well defined. We also assume that in a small neighbourhood of the North or the South pole of $S^4$ this non-commutative deformation agrees with the non-commutative deformation used by Nekrasov [@Nekrasov:2002qd] on $\BR^4$.
Since both ${Z_{\text{inst}}}(\ve_1,\ve_2,ia)$ and its complex conjugate enter the formula, this means that we count both instantons and anti-instantons. The formula is similar to Ooguri-Strominger-Vafa relation between the black hole entropy and the topological string partition function [@Ooguri:2004zv; @Beasley:2006us] $$\label{eq:OSV}
Z_{BH} \propto |Z_{top}|^2.$$
Actually the localization computation is compatible with more general observables than a single Wilson loop in representation $R$ inserted on the equator . Let us fix two opposite points on the $S^4$ and call them the North and the South poles. Then we can consider a class of Wilson loops placed on circles of arbitrary radius such that they all have a common center at the North pole, and such that they all can be transformed to each other by a composition of a dilation in the North-South direction and by an anti-self-dual rotation in the $\SU(2)_L$ left subgroup of the $\SO(4)$ subgroup of the $S^4$ isometry group which fixes the North pole. However, for Wilson loops of not maximal size, we need to change the relative coefficient between the gauge and the scalar field terms in (\[eq:Wilson-loop-defined\]). Let $C_\theta$ be a circle located at an arbitrary polar angle $\theta$ measured from the North pole (at the equator $\sin \theta = 1$). Then we consider $$\begin{aligned}
\label{eq:Wilson-loop-gen-defined}
W_R(C_\theta) = \tr_{R} \Pexp \oint_{C_\theta} (A_{\mu} dx^{\mu} + \frac {1
} {\sin \theta} (i\Phi_{0}^{E} + \Phi_9 \cos \theta) ds ),
\end{aligned}$$ where $ \Phi_0^E$ and $\Phi_9$ are the scalar fields of the $\CalN=2$ vector multiplet.
Equivalently this can be rewritten as $$\label{eq:Wilson-loop-gen-defined-eq}
W_R(C_\theta) = \tr_{R} \Pexp \oint_{C_\theta} (A_{\mu} dx^{\mu} + (i \Phi_{0}^{E} + \Phi_9 \cos \theta) r d\alpha).$$ where $\alpha \in [0, 2\pi)$ is an angular coordinate on the circle $C$. Formally, as the size of the circle vanishes ($\theta \to 0$) we get a “holomorphic” observable $W_R(C_{\theta \to 0}) = \tr_{R} \exp 2\pi r \Phi(\text{N})$ where $\Phi(\text{N})$ is the complex scalar field $i\Phi_0^E + \Phi_9$ evaluated at the North pole. In the opposite limit ($\theta \to \pi$) we get an “anti-holomoprhic” observable $W_R(C_{\theta \to \pi}) = \tr_{R} \exp 2\pi r \bar \Phi(\text{S})$, where $\bar \Phi({S})$ is the conjugated scalar field $-i \Phi_0^E + \Phi_9$ evaluated at the South pole. However, in the actual computation of the path integral we will always assume a finite size of $C$, so that the operator $W_R(C)$ is well defined.
Then for an arbitrary set $\{W_{R_1}(C_{\theta_1}),\dots,
W_{R_n}(C_{\theta_n})\}$ of Wilson loops in the class we described above we obtain $$\boxed{ \langle W_{R_1}(C_{\theta_1} ) \dots W_{R_n}(C_{\theta_n}) \rangle_{\text{4d theory}} = \langle \tr_{R_1} e^{2 \pi r i a} \dots
\tr_{R_n} e^{2 \pi r i a} \rangle_{\text{matrix model}}}.$$
The Drukker-Gross argument only applies to the case of a single circle which can be related to a straight line on $\BR^4$ by a conformal transformation, but in the present approach we can consider several circles simultaneously.
So far we described the class of observables which we can compute in the massive $\CalN=2^{*}$ theory. All these observables are invariant under the same operator $Q$ generated by a conformal Killing spinor on $S^4$ of constant norm. This operator $Q$ is a fermionic symmetry at quantum level.
Now we describe more general classes of circular Wilson loops which can be solved in $\CalN=4$ theory. Thanks to the conformal symmetry of the $\CalN=4$ theory there is a whole family of operators $\{Q(t)\}$ where $t$ runs from $0$ to $\infty$, which we can use for the localization computation. The case $t=1$ corresponds to the conformal Killing spinor of constant norm and to the observables which we study in the $\CalN=2^{*}$ theory. However, for a general $t$ in the $\CalN=4$ theory we can take $$\begin{aligned}
\label{eq:Wilson-loop-gen-defined-more-general}
W_R(C_\theta,t) = \tr_{R} \Pexp \oint_{C_\theta} \left(A_{\mu} dx^{\mu} + \frac {1
} {t \sin \theta} \left( (\cos^2 \frac \theta 2+t^2 \sin^2 \frac \theta 2) i\Phi_{0}^{E} + \Phi_9
( \cos^2 \frac \theta 2 - t^2 \sin^2 \frac \theta 2 )\right) ds \right).
\end{aligned}$$ At $t \sin \frac \theta 2 = \cos \frac \theta 2$ we get the Wilson loop (\[eq:Wilson-loop-defined\]) with the same relative coefficient $1$ between $A_{\mu}$ and $i \Phi_{0}^E$ but of arbitrary size. The $\CalN=4$ theory with insertion of the operator $W_{R}(C_{\theta},t)$ still localizes to the Gaussian matrix model.
The idea underlying localization is that in some situations the integral is exactly equal to its semiclassical approximation. For example, the Duistermaat-Heckman formula says [@MR674406] $$\int_M \frac {\omega^{n}}{ (2 \pi)^n n!} e^{i H(\phi)} = i^n \sum_{p \in F}
\frac {e^{i H(\phi)}} {\prod \alpha_i^{p}(\phi)},$$ where $(M,\omega)$ is a symplectic manifold, and $H: M \to g^*$ is a moment map[^3] for a Hamiltonian action of $G=\U(1)^k$ on $M$. The Duistermaat-Heckman formula is a particular case of a more general Atiyah-Bott-Berline-Vergne localization formula [@MR721448; @MR685019]. Let an abelian group $G$ act on a compact manifold $M$. We consider the complex of $G$-equivariant differential forms on $M$ valued in functions on $\g$ with the differential $Q = d - \phi^{a} i_{a}$. The differential squares to a symmetry transformation $Q^2 = - \phi^{a}
\CalL_{v^{a}}$. Here $\CalL_{v^{a}}$ represents the action of $G$ on $M$. Hence $Q^2$ annihilates $G$-invariant objects. Then for any $Q$-closed form $\alpha$, Atiyah-Bott-Berline-Vergne localization formula is $$\int_{M} \alpha = \int_{F} \frac {{i^*_F \alpha}}{e(N_{F})},$$ where $F \overset{i}{\hookrightarrow} M$ is the $G$-fixed point set, and $e(N_{F})$ is the equivariant Euler class of the normal bundle of $F$ in $M$. When $F$ is a discrete set of points, the equivariant Euler class $e(N_F)$ at each point $f \in F$ is simply the determinant of the representation in which $\g$ acts on the tangent bundle of $M$ at a point $f$.
Localization can be argued in the following way [@Witten:1988ze; @Witten:1991zz]. Let $Q$ be a fermionic symmetry of a theory. Let $Q^2 = \CalL_{\phi}$ be some bosonic symmetry. Let $S$ be a $Q$-invariant action, so that $QS = 0$. Consider a functional $V$ which is invariant under $\CalL_{\phi}$, so that $Q^2 V = 0$. Deformation of the action by a $Q$-exact term $QV$ can be written as a total derivative and does not change the integral up to boundary contributions $$\frac {d} {d t} \int e^{S + t Q V } = \int \{Q,V\} e^{S + t Q V} = \int \{Q,
V e^{S + t Q V}\} = 0.$$ As $t \to \infty$, the one-loop approximation at the critical set of $QV$ becomes exact. Then for a sufficiently nice $V$, the integral is computed by evaluating $S$ at critical points of $QV$ and the corresponding one-loop determinant.
We apply this strategy to the $\CalN=2$, the $\CalN=2^*$ and the $\CalN=4$ supersymmetric Yang-Mills gauge theories on $S^4$ and show that the path integral is localized to the constant modes of the scalar field $\Phi_0$ with all other fields vanishing. In this way we also compute exactly the expectation value of the circular supersymmetric Wilson loop operator .
*Remark.* Most of the presented arguments in this work should apply to an $\CalN=2$ theory with an arbitrary matter content. For a technical reasons related to the regularization issues, we limit our discussion to the $\CalN=2$ theory with a single $\CalN=2$ massive hypermultiplet in the adjoint representation, also known as the $\CalN=2^{*}$. By taking the limit of vanishing or infinite mass we can respectively recover the $\CalN=4$ or the $\CalN=2$ theory.
Still we will give in (\[eq:Z-1-loop-any-matter\]) a formula for the factor $Z_{\text{1-loop}}$ for an $\CalN=2$ gauge theory with a massless hypermultiplet in such representation that the theory is conformal. Perhaps, one could check our result by the traditional Feynman diagram computations directly in the gauge theory. To simplify comparison, we will give an explicit expansion in $g_{YM}$ up to the sixth order of the expectation value of the Wilson loop operator for the $\CalN=2$ theory with the gauge group $\SU(2)$ and 4 hypermultiplets in the fundamental representation (see (\[se:first-corrections\])) $$\langle e^{2\pi n a} \rangle_{\text{matrix model}} = 1 + \frac 3 {2 \cdot 2^2} n^2 g_{YM}^2 + \frac 5 {8\cdot 2^4} n^4 g_{YM}^4
+ \frac {7} {48 \cdot 2^6} n^6 g_{YM}^6 - \frac{35 \cdot 12 \cdot \zeta(3)} {2^4 (4\pi)^2} n^2 g_{YM}^6 + O(g_{YM}^8),$$ In this formula $a \in \BR$ is a coordinate on the Cartan algebra $\h$ of $\g$. By an integer $n \in \h^{*}$ we denote a weight. For example, if the Wilson loop is taken in the spin-$j$ representation, where $j$ is a half-integer, the weights are $\{-2j,-2j+2, \dots, 2j\}$. Hence we get $\langle W_j(C) \rangle=\langle \sum_{m=-j}^{j} e^{4\pi m a} \rangle_{MM}$.
We shall note that the first difference between the $\CalN=2$ superconformal theory and the $\CalN=4$ theory appears at the order $g^{6}_{YM}$, up to which the Feynman diagrams in the $\CalN=4$ theory were computed in [@Plefka:2001bu; @Arutyunov:2001hs]. Therefore a direct computation of Feynman diagrams in the $\CalN=2$ theory up to this order seems to be possible and would be a non-trivial test of our results.
Some unusual features in this work are: (i) the theory localizes not on a counting problem, but on a nontrivial matrix model, (ii) there is a one-loop factor involving an index theorem for transversally elliptic operators [@MR0482866; @MR0341538].
In section 2 we give details about the $\CalN=2$, the $\CalN=2^*$ and $\CalN=4$ SYM theories on a four-sphere $S^4$. In section 3 we make a localization argument to compute the partition function for these theories. Section 4 explains the computation of the one-loop determinant [@MR0341538; @MR0482866], or, mathematically speaking, of the equivariant Euler class of the infinite-dimensional normal bundle in the localization formula. In section 5 we consider instanton corrections.
There are some open questions and immediate directions in which one can proceed:
1. [ One can consider more general supersymmetric Wilson loops like studied in [@Drukker:2007qr; @Drukker:2007yx; @Drukker:2007dw] and try to prove the conjectural relations of those with matrix models or two-dimensional super Yang-Mills theory. Perhaps it will be also possible to extend the analysis of those more general loops to (superconformal) $\CalN=2$ theories like it is done in the present work. ]{}
2. [ Using localisation, one can try to solve exactly for an expectation value of a circular supersymmetric ’t Hooft-Wilson operator (this is a generalization of Wilson loop in which the loop carries both electric and magnetic charges) [@Kapustin:2006hi; @Kapustin:2005py; @Kapustin:2006pk]. The expectation values of such operators should transform in the right way under the $S$-duality transformation which replaces the coupling constant by its inverse and the gauge group $G$ by its Langlands dual $^LG$. Perhaps this could tell us more on the four-dimensional gauge theory and geometric Langlands [@Kapustin:2006pk] where ’t Hooft-Wilson loops play the key role.]{}
3. [It would be interesting to find more precise relation between our formulas, and Ooguri-Strominger-Vafa [@Ooguri:2004zv] conjecture (\[eq:OSV\]). There could be a four-dimensional analogue of the $tt^{*}$-fusion [@Cecotti:1991me]. ]{}
I would like to thank E. Witten and N. Nekrasov for many stimulating discussions, important comments and suggestions. I thank M. Atiyah, N. Berkovits, A. Dymarsky, D. Gaiotto, S. Gukov, J. Maldacena, I. Klebanov, H. Nakajima, A. Neitzke, I. Singer, M. Rocek, K. Zarembo for interesting discussions and remarks. Part of this research was done during my visit to Physics Department of Harvard University in January 2007. The work was supported in part by Federal Agency of Atomic Energy of Russia, grant RFBR 07-02-00645, grant for support of scientific schools NSh-8004.2006.2 and grant of the National Science Foundation PHY-0243680. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of these funding agencies.
Fields, action and symmetries {#se:fields-action-symmetries}
=============================
To write down the action of the $\CalN=4$ SYM on $S^4$, we use dimensional reduction of the $\CalN=1$ SYM [@Brink:1977bc] on $\BR^{9,1}$. By $G$ we denote the gauge group. By $A_{M}$ with $M = 0, \dots, 9$ we denote the components of the gauge field in ten dimensions, where we take the Minkowski metric $ds^2 =
- dx_0^2 + dx_1^2 + \dots + dx_9^2$. When we write formulas in Euclidean signature so that the metric is $ds^2 = dx_0^2 + dx_1^2 + \dots + dx_9^2$, we use notation $A_0^E$ for the zero component of the gauge field.
By $\Psi$ we denote a sixteen real component ten-dimensional Majorana-Weyl fermion valued in the adjoint representation of $G$. (In Euclidean signature $\Psi$ is not real, but its complex conjugate does not appear in the theory.) The ten-dimensional action $S= \int d^{10} x \CalL$ with the Lagrangian $$\begin{aligned}
\CalL = \frac {1} {2 g^2_{YM}} \lb \frac 1 2 F_{MN} F^{MN} - \Psi \Gamma^{M}
D_{M} \Psi \rb\end{aligned}$$ is invariant under the supersymmetry transformations $$\begin{aligned}
& \delta_{\ve} A_{M} = \ve \Gamma_{M} \Psi \\
& \delta_{\ve} \Psi = \frac 1 2 F_{MN} \Gamma^{MN} \ve.\end{aligned}$$ Here $\ve$ is a constant Majorana-Weyl spinor parameterizing the supersymmetry transformations in ten dimensions. (See appendix \[sec:Octonionic-gamma-matrices\] for our conventions on the algebra of gamma-matrices.)
We do not write explicitly the color and spinor indices. We also assume that in all bilinear terms the color indices are contracted using some invariant positive definite bilinear form (Killing form) on the Lie algebra $\g$ of the gauge group. Sometimes we denote this Killing form by $(\cdot,\cdot)$. In Euclidean signature we integrate over fields which all take value in the real Lie algebra of the gauge group. For example, for the $\U(N)$ gauge group all fields are represented by the antihermitian matrices, and we can define the Killing from on $\g$ as $(a,b) = -\tr_{\text{F}} a b$, where $\tr_{\text{F}}$ is the trace in the fundamental representation.
We take $(x_1,\dots,x_4)$ to be the coordinates along the four-dimensional space-time, and we make dimensional reduction in the remaining directions: $0,5,\dots 8, 9$. Note that the four-dimensional space-time has Euclidean signature.
Now we describe the symmetries of the four-dimensional theory if we start from Minkowski signature in ten dimensions. Note that we make dimensional reduction along the time-like coordinate $x_0$. Therefore we get the wrong sign for the kinetic term for the scalar field $\Phi_0$, where $\Phi_0$ denotes the $0$-th component of the gauge field $A_M$ after dimensional reduction. To make sure that the path integral is well defined and convergent, in this case in the path integral for the four-dimensional theory we integrate over imaginary $\Phi_0$. Actually this means that the path integral is the same as in the Euclidean signature with all bosonic fields taken real.
The ten-dimensional $\Spin(9,1)$ Lorentz symmetry group is broken to $\Spin(4) \times \Spin(5,1)^R$, where the first factor is the four-dimensional Lorentz group acting on $(x_1,
\dots, x_4)$ and the second factor is the R-symmetry group acting on $(x_5,
\dots, x_9, x_0)$. It is convenient to split the four-dimensional Lorentz group as $\Spin(4) =
SU(2)_L \times SU(2)_R$, and brake the $\Spin(5,1)^R$-symmetry group into $\Spin(4)^R \times \SO(1,1)^R=\SU(2)_L^R \times \SU(2)_R^R \times \SO(1,1)^R$. The components of the ten-dimensional gauge field, which become scalars after the dimensional reduction are denoted by $\Phi_A$ with $A=0,5,\dots,9$. Let us write the bosonic fields and the symmetry groups under which they transform: $$\overbrace{A_{1}, \dots A_{4}}^{ SU(2)_L \times SU(2)_R } \, \,
\overbrace{\Phi_5, \dots, \Phi_8}^{{ \SU(2)_L^R \times \SU(2)_R^R }} \, \,
\overbrace{\Phi_9,\Phi_0}^{{\SO(1,1)^R}}.$$
Using a certain Majorana-Weyl representation of the Clifford algebra $\Cl(9,1)$ (see appendix \[sec:Octonionic-gamma-matrices\] for our conventions), we write $\Psi$ in terms of four four-dimensional chiral spinors as $$\Psi = \begin{pmatrix}
\psi^L \\ \chi^R \\ \psi^R \\ \chi^L
\end{pmatrix}.$$
Each of these spinors ($\psi^L,\chi^R,\psi^R,\chi^L$) has four real components. Their transformation properties are summarized in the table:
------- ------------ ------------ ------------ -------------- -------------- --------------
$\ve$ $\Psi$ $\SU(2)_L$ $\SU(2)_R$ $\SU(2)_L^R$ $\SU(2)_R^R$ $\SO(1,1)^R$
$ * $ $\psi^{L}$ $1/2$ 0 $1/2$ 0 $+$
$ 0 $ $\chi^{R}$ $0$ $1/2$ 0 $1/2$ $+$
$ * $ $\psi^{R}$ $0$ $1/2$ $1/2$ 0 $-$
$ 0 $ $\chi^{L}$ $1/2$ 0 0 $1/2$ $-$
------- ------------ ------------ ------------ -------------- -------------- --------------
Let the spinor $\ve$ be the parameter of the supersymmetry transformations. We restrict the $\CalN=4$ supersymmetry algebra to the $\CalN=2$ subalgebra by taking $\ve$ in the $+1$-eigenspace of the operator $\Gamma^{5678}$. Such spinor $\ve$ has the structure $$\ve=
\begin{pmatrix}
* \\ 0 \\ * \\ 0
\end{pmatrix},$$ transforms in the spin-$\frac 1 2$ representation of the $\SU(2)^R_L$ and in the trivial representation of the $\SU(2)^R_R$.
With respect to the supersymmetry transformation generated by such $\ve$, the $\CalN=4$ gauge multiplet splits in two parts
- [ $(A_1\dots A_4,\Phi_9,\Phi_0,\, \psi^L, \psi^{R})$ is the $\CalN=2$ vector multiplet ]{}
- [ $(\Phi_5 \dots \Phi_8,\, \chi^L, \chi^R)$ is the $\CalN=2$ hypermultiplet]{}.
So far we considered dimensional reduction from $\BR^{9,1}$ to the flat space $\BR^4$. Now we would like to put the theory on a four-sphere $S^4$. We denote by $A_{\mu}$ with $\mu = 1,\dots, 4$ the four-dimensional gauge field and by $\Phi_{A}$ with $A =
0,5,\dots,9$ the four-dimensional scalar fields. The only required modification of the action is a coupling of the scalar fields to the scalar curvature of space-time. Namely, the kinetic term must be changed as $(\partial \Phi)^2 \to
(\partial \Phi)^2 + \frac R 6 \Phi^2$, where $R$ is the scalar curvature. One way to see why this is the natural kinetic term for the scalar fields is to use the argument of the conformal invariance. Namely, one can check that $\int d^{4} x \, \sqrt{g} ( (\p \Phi)^2 + \frac R 6 \Phi^2 )$ is invariant under Weyl transformations of the metric $g_{\mu \nu} \to e^{2\Omega}g_{\mu \nu}$ and scalar fields $\Phi \to e^{-\Omega} \Phi$. Then the action on $S^4$ of the $\CalN=4$ SYM is $$\label{eq:SYM_S^4_N=4}
S_{\CalN=4} = \frac 1 {2 g_{YM}^2} \int_{S^4} \sqrt{g} d^4 x \left(\frac 1 2 F_{MN} F^{MN}
- \Psi \gamma^{M} D_{M} \Psi + \frac 2 {r^2} \Phi^A \Phi_A \right),$$ where we used the fact that the scalar curvature of a $d$-sphere $S^d$ of radius $r$ is $\frac {d(d-1)} {r^2}$.
The action is invariant under the $\CalN=4$ superconformal transformations $$\begin{aligned}
\label{eq:SYM_S^4_N=4-trans}
& \delta_{\ve} A_{M} = \ve \Gamma_{M} \Psi \\
& \delta_{\ve} \Psi = \frac 1 2 F_{MN} \Gamma^{MN} \ve + \frac 1 2 \Gamma_{\mu A}
\Phi^A \nabla^{\mu} \ve,\end{aligned}$$ where $\ve$ is a conformal Killing spinor solving the equations $$\begin{aligned}
\label{eq:conformal-Killing}
& \nabla_{\mu} \ve = \Gamma_{\mu} \tilde \ve \\
& \nabla_{\mu} \tilde \ve = - \frac {1} {4 r^2} \Gamma_{\mu} \ve.\end{aligned}$$ (See e.g. [@math0202008v1] for a review on conformal Killing spinors, and for the explicit solution of these equations on $S^4$ see appendix \[se:Killing-spinors\].) To get intution about the meaning of $\ve$ and $\tilde \ve$ we can take the flat space limit $r \to \infty$. In this limit $\tilde \ve$ becomes covariantly constant spinor $\tilde \ve = \hat \ve_{c}$, while $\ve$ becomes a spinor with at most linear dependence on flat coordinates $x^{\mu}$ on $\BR^4$: $\ve = \hat \ve_s + x^{\mu} \Gamma_{\mu} \hat \ve_{c}$. By $\hat \ve_{s}$ and $\hat \ve_{c}$ we denote some constant spinors. Then $\hat \ve_{s}$ generates supersymmetry transformations, while $\hat \ve_{c}$ generates special superconformal symmetry transformations.
The superconformal algebra closes only on-shell. Let $\delta_{\ve}^2$ be the square of the fermionic transformation $\delta_{\ve}$ generated by a spinor $\ve$. After some algebra (see appendix \[se:off-shell susy\]) we obtain $$\begin{aligned}
\label{eq:delta2-closed-main-text}
& \delta_{\ve}^2 A_{\mu} = - (\ve \Gamma^{\nu} \ve) F_{\nu \mu} - [(\ve \Gamma^{B}\ve) \Phi_B, D_{\mu}] \\
& \delta_{\ve}^2 \Phi_A = - (\ve \Gamma^{\nu} \ve) D_{\nu} \Phi_{A} - [(\ve
\Gamma^{B} \ve )\Phi_B, \Phi_A] +
2 (\tilde \ve \Gamma_{AB} \ve) \Phi^{B} - 2(\ve \tilde \ve) \Phi_A \\
& \delta_{\ve}^{2} \Psi = -(\ve \Gamma^{\nu} \ve) D_{\nu} \Psi - [(\ve \Gamma^{B}
\ve )\Phi_B, \Psi] - \frac 1 2 (\tilde \ve \Gamma_{\mu \nu} \ve) \Gamma^{\mu
\nu} \Psi + \frac 1 2 (\tilde \ve \Gamma_{AB} \ve) \Gamma^{AB} \Psi
-3(\tilde \ve \ve) \Psi + \text{eom}[\Psi].
\end{aligned}$$ Here the term denoted by $\text{eom}[\Psi]$ is proportional to the Dirac equation of motion for fermions $\Psi$ $$\label{eq:opsi}
\text{eom}[\Psi] = \frac 1 2 (\ve \Gamma_N \ve) \tilde \Gamma^{N} \Dslash \Psi - (\ve \Dslash \Psi) \ve.$$
The square of the supersymmetry transformation can be written as $$\label{eq:square-susy}
\delta^{2}_{\ve} = -\CalL_{v} - R - \Omega.$$
The first term is the gauge covariant Lie derivative $\CalL_{v}$ in the direction of the vector field $$\label{eq:v-in-terms-eps}
v^M = \ve \gamma^{M} \ve.$$ For example, $\CalL_{v}$ acts on scalar fields as follows: $\CalL_v {\Phi_A} = v^M D_{M} \Phi=
v^{\mu} D_{\mu} \Phi_A + v^{B} [\Phi_{B}, \Phi]$. Here $D_{\mu}$ is the usual covariant derivative $D_{\mu} = \p_{\mu} + A_{\mu}$
To explain what the gauge covariant Lie derivative means geometrically, first we consider the situation when the gauge bundle, say $E$, is trivial. We fix some flat background connection $A_{\mu}^{(0)}$ and choose a gauge such that $A_{\mu}^{(0)} = 0$. For any connection $A$ on $E$ we define $\tilde A = A - A^{(0)}$. The field $\tilde A$ transforms as a one-form valued in the adjoint representation of $E$. The path integral over $A$ is equivalent to the path integral over $\tilde A$. Then we can write the gauge covariant Lie derivative $\CalL_{v}$ as follows $$\CalL_{v} = L_{v} + G_{\Phi}.$$ Here $L_{v}$ is a usual Lie derivative in the direction of the vector field $v^{\mu}$. The action of $L_{v}$ on the gauge bundle is defined by the background connection $A^{(0)}$ which we set to zero. The second term $G_{\Phi}$ is the gauge transformation generated by the adjoint valued scalar field $\Phi$ where $$\label{eq:Phi-def}
\Phi = v^{M} \tilde A_{M}.$$ The gauge transformation $G_{\Phi}$ acts on the matter and the gauge fields in the usual way: $G_{\Phi} \Phi_{A} = [\Phi,\Phi_A]$, $G_{\Phi} \cdot A_{\mu} = [\Phi,D_{\mu}] = -D_{\mu} \Phi $.
The term denoted by $R$ in is a $\Spin(5,1)^R$-symmetry transformation. It acts on scalar fields as $(R \cdot \Phi)_{A} = R_{AB}
\Phi^{B}$, and on fermions as $R \cdot \Psi = \frac 1 4 R_{AB} \Gamma^{AB}
\Psi$, where $R_{AB} = 2 \ve \tilde \Gamma_{AB} \tilde \ve$. When $\ve$ and $\tilde \ve$ are restricted to the $\CalN=2$ subspace of $\CalN=4$ algebra, ($\Gamma^{5678} \ve = \ve$ and $\Gamma^{5678} \tilde \ve = \tilde
\ve$), the matrix $R_{AB}$ with $A,B=5,\dots, 8$ is an anti-self-dual (left) generator of $\SO(4)^{R}$ rotations. In other words, when we restrict $\ve$ to the $\CalN=2$ subalgebra of the $\CalN=4$ algebra, the $\SO(4)^{R}$ $R$-symmetry group restricts to its $\SU(2)^R_L$ subgroup. The fermionic fields of the $\CalN=2$ vector multiplet (we call them $\psi$) transform in the trivial representation of $R$, while the fermionic fields of the $\CalN=2$ hypermultiplet (we call them $\chi$) transform in the spin-$\frac 1 2$ representation of $R$.
Finally, the term denoted by $\Omega$ in generates a local dilatation with the parameter $2(\ve \tilde
\ve)$, under which the gauge fields do not transform, the scalar fields transform with weight $1$, and the fermions transform with weight $\frac 3 2$. (In other words, if we make Weyl transformation $g_{\mu \nu} \to e^{2\Omega} g_{\mu \nu}$, we should scale the fields as $A_{\mu} \to A_{\mu},
\Phi \to e^{-\Omega} \Phi, \Psi \to e^{-\frac 3 2 \Omega} \Psi$ to keep the action invariant.)
Classically, it is easy to restrict the fields and the symmetries of the $\CalN=4$ SYM to the pure $\CalN=2$ SYM: one can discard all fields of the $\CalN=2$ hypermultiplet and restrict $\ve$ by the condition $\Gamma^{5678} \ve = \ve$. The resulting action is invariant under $\CalN=2$ superconformal symmetry. On quantum level the pure $\CalN=2$ SYM is not conformally invariant. We will be able to give a precise definition of the quantum $\CalN=2$ theory on $S^4$, considering it as the $\CalN=4$ theory softly broken by giving a mass term to the hypermultiplet, which we will send to the infinity in the end.
If we start from Minkowski signature in the ten dimensional theory, then classically the supersymmetry groups for the $\CalN=4$, the $\CalN=2$, and the $\CalN=2^*$ Yang-Mills theories on $S^4$ are the following.
In the $\CalN=2$ case, $\ve$ is a Dirac spinor on $S^4$. The equation has 16 linearly independent solutions, which correspond to the fermionic generators of the $\CalN=2$ superconformal algebra. Intuitively, $8$ generators out of these 16 correspond to $8$ charges of $\CalN=2$ supersymmetry algebra on $\BR^4$, and the other $8$ correspond to the remaining generators of $\CalN=2$ superconformal algebra. The full $\CalN=2$ superconformal group on $S^4$ is $\SL(1|2, \BH)$.[^4] Its bosonic subgroup is $\SL(1,\BH)\times \SL(2,\BH)\times \SO(1,1)$. The first factor $\SL(1,\BH) \simeq \SU(2)$ generates the $R$-symmetry $\SU(2)^R_L$ transformations. The second factor $\SL(2,\BH) \simeq \SU^{*}(4,\BC) \simeq \Spin(5,1)$ generates conformal transformations of $S^4$. The third factor $\SO(1,1)^R$ generates the $\SO(1,1)^R$ symmetry transformations. The fermionic generators of $\SL(1,2|\BH)$ transform in the $\mathbf{2+2'}$ of the $\SL(2,\BH)$, where $\mathbf{2}$ denotes the fundamental representation of $\SL(2,\BH)$ of quaternionic dimension two. This representation can be identified with the fundamental representation $\mathbf{4}$ of $\SU^{*}(4)$ of complex dimension four, or with chiral (Weyl) spinor representation of the conformal group $\Spin(5,1)$. The other representatfion $\mathbf{2'}$ corresponds to the other chiral spinor represenation of $\Spin(5,1)$ of the opposite chirality.
In the $\CalN=4$ case we do not impose the chirality condition on $\ve$. Hence a sixteen component Majorana-Weyl spinor $\ve$ of $\Spin(9,1)$ reduces to a pair of the four-dimensional Dirac spinors $(\ve_{\psi}, \ve_{\chi})$, where $\ve_{\psi}$ and $\ve_{\chi}$ are elements of the $+1$ and $-1$ eigenspaces of the chirality operator $\Gamma^{5678}$ respectively. Each of the Dirac spinors $\ve_{\psi}$ and $\ve_{\chi}$ independently satisfies the conformal Killing spinor equation (\[eq:conformal-Killing\]) because the operators $\Gamma_{\mu}$ do not mix the $+1$ and $-1$ eigenspaces of $\Gamma^{5678}$. Then we get 16+16 = 32 linearly independent conformal Killing spinors. Each of these spinors corresponds to a generator of the $\CalN=4$ superconformal symmetry. One can check that the full $\CalN=4$ superconformal group on $S^4$ is $\PSL(2|2, \BH)$.
To describe the $\CalN=2^{*}$ theory on $S^4$, which is obtained by giving mass to the hypermultiplet, we need some more details on Killing spinors on $S^4$. Because mass terms break conformal invariance, we should expect the $\CalN=2^{*}$ theory to be invariant only under $8$ out of $16$ fermionic symmetries of the $\CalN=2$ superconformal group $\SL(1,2|\BH)$. In other words, we should impose some additional restrictions on $\ve$. Let us desribe this theory in more details.
First we explicitly give a general solution for the conformal spinor Killing equation on $S^4$. Let $x^{\mu}$ be the stereographic coordinates on $S^4$. The origin corresponds to the North pole, the infinity corresponds to the South pole. If $r$ is the radius of $S^4$, then the metric has the form $$\label{eq:metric-S4-stereo}
g_{\mu \nu} = \delta_{\mu \nu} e^{2\Omega},
\quad \text{where} \quad e^{2 \Omega} := \frac 1 { (1 + \frac {x^2} {4 r^2})^2 }.$$ We use the vielbein $e_{\mu}^{i} = \delta^{i}_{\mu} e^{\Omega}$ where $\delta^{i}_{\mu}$ is the Kronecker delta, the index $\mu=1,\dots,4$ is the space-time index, the index $i=1,\dots,4$ enumerates vielbein elements. The solution of the conformal Killing equation is (see appendix \[se:Killing-spinors\]) $$\begin{aligned}
\label{eq:Killing-solution-in-S^4-stereo}
\ve = \frac 1 {\sqrt{1+ \frac {x^2} {4 r^2}}} (\hat \ve_{s} + x^{i} \Gamma_{i} \hat \ve_{c}) \\
\tilde \ve = \frac 1 {\sqrt{1+ \frac {x^2} {4 r^2}}} (\hat \ve_{c} - \frac
{x^{i} \Gamma_{i} }{4 r^2} \hat \ve_{s}),\end{aligned}$$ where $\hat \ve_{s}$ and $\hat \ve_{c}$ are Dirac spinor valued constants.
Classically, the action of $\CalN=2$ SYM on $\BR^4$ with a massless hypermultiplet is invariant under the $\CalN=2$ superconformal group, which has 16 fermionic generators. Turning on non-zero mass of the hypermultiplet breaks 8 superconformal fermionic symmetries, but preserves the other 8 fermionic symmetries which generate the $\CalN=2$ supersymmetry. These 8 charges are known to be preserved on quantum level [@Seiberg:1994rs]. The $\CalN=2$ supersymmetry algebra closes to the scale preserving transformations: the translations on $\BR^4$. These scale preserving transformations are symmetries of the massive theory as well.
Following the same logic, we would like to find a subgroup, which will be called $\CalS$, of the $\CalN=2$ superconformal group on $S^4$ with the following properties. The supergroup $\CalS \subset \SL(1|2,\BH)$ contains $8$ fermionic generators, the bosonic transformations of $\CalS$ are the scale preserving transformations and are compatible with mass terms for the hypermultiplet. The group $\CalS$ is what we call the $\CalN=2$ supersymmetry group on $S^4$.
The conformal group of $S^4$ is $\SO(5,1)$. The scale preserving subgroup of the $\SO(5,1)$ is the $\SO(5)$ isometry group of $S^4$. We require that the space-time bosonic part of $\CalS$ is a subgroup of this $\SO(5)$. This means that for any conformal Killing spinor $\ve$ that generates a fermionic transformation of $\CalS$, the dilatation parameter $(\tilde \ve \ve)$ in the $\delta^2_{\ve}$ vanishes.
For a general $\ve$ in the $\CalN=2$ superconformal group, the transformation $\delta_{\ve}^2$ contains $\SO(1,1)^{R}$ generator. Since the $\SO(1,1)^R$ symmetry is broken explicitly by hypermultiplet mass terms, and since it is broken on quantum level in the usual $\CalN=2$ theory in the flat space[^5], we require that $\CalS$ contains no $\SO(1,1)^{R}$ transformations. In other words, the conformal Killing spinors $\ve$ which generate transformations of $\CalS$ are restricted by the condition that the $\SO(1,1)^R$ generator in $\delta_{\ve}^2$ vanishes. By equation this means $\tilde \ve
\Gamma^{09} \ve = 0$.
Using the explicit solution we rewrite the equation $(\tilde \ve
\ve) = (\tilde \ve \Gamma^{09} \ve) = 0$ in terms of $\hat \ve_{s}$ and $\hat
\ve_{c}$ $$\label{eq:good-Killing-spinors}
\begin{aligned}
\hat \ve_{s} \hat \ve_{c} = \hat \ve_{s} \Gamma^{09} \hat \ve_{c} = 0 \\
\hat \ve_{c} \Gamma^{\mu} \hat \ve_{c} - \frac 1 {4 r^2} \hat \ve_{s}
\Gamma^{\mu} \hat \ve_{s} =0.
\end{aligned}$$ To solve the second equation, we take chiral $\hat \ve_{s}$ and $\hat
\ve_{c}$ with respect to the four-dimensional chirality operator $\Gamma^{1234}$. Since the operators $\Gamma^{\mu}$ reverse the four-dimensional chirality, both terms in the second equation vanish automatically. There are two interesting cases: (i) the chirality of $\hat \ve_{s}$ and $\hat \ve_{c}$ is opposite, (ii) the chirality of $\hat \ve_{s}$ and $\hat \ve_{c}$ is the same. The main focus of this work is on the second case.
1\. In the first case we can assume that $$\ve_{s}^{L} = 0, \quad \hat \ve_{c}^{R} = 0.$$ Here by $\ve_{s}^{L}$ and $\ve_{s}^{R}$ we denote left/right four-dimensional chiral components. They are respectively defined as the $-1/+1$ eigenspaces of the chirality operator $\Gamma^{1234}$. In this case the first equation in (\[eq:good-Killing-spinors\]) is also automatically satisfied. Moreover, the spinors $\ve$ and $\tilde \ve$ also have opposite chirality over the whole $S^4$. Hence we have $8$ generators, say $\hat \ve_{s}^{R}$ and $\hat \ve_{c}^{L}$, which anticommute to pure gauge transformations generated by the scalar field $\Phi := (\ve \Gamma^{A} \ve) \Phi_{A}$. The $\delta_{\ve}$-closed observables are the gauge invariant functions of $\Phi$ and their descendants. One could try to interpret such $\delta_{\ve}$ as a cohomological BRST operator $Q$ and to relate in this way the physical $\CalN=2$ gauge theory on $S^4$ with the topological Donaldson-Witten theory. That does not work, because in the present case the conformal Killing spinor $\ve$, generated by such $\hat \ve_{s}$ and $\hat \ve_{c}$ necessary vanishes somewhere on $S^4$. Of course, in the twisted theory [@Witten:1988ze; @Vafa:1994tf] the problem does not arise, since $\ve$ is a scalar and can be set to be a non-zero constant everywhere. However, our goal is to treat the non-twisted theory. Moreover, the circular Wilson loop operator $W_R(C)$ is not closed under such $\delta_{\ve}$. Thus we turn to the second case.
2\. The spinors $\hat \ve_{s}$ and $\hat \ve_{c}$ have the same chirality, say left, and the first equation restricts them to be orthogonal $$\hat \ve_{s}^{R} = 0, \quad \hat \ve_{c}^{R} = 0, \quad (\hat \ve_{s}^L \hat
\ve_{c}^L) =0.$$ The Killing vector field $v^{\mu} = \ve \Gamma^{\mu} \ve$, associated with the $\de_{\ve}^2$, generates an anti-self-dual (left) rotation of $S^4$ around the North pole. In addition, $\de_{\ve}^2$ generates a $\SU(2)_{L}^R$-symmetry transformation and a gauge symmetry transformation. The spinor $\ve$ is chiral only at the North and the South poles of $S^4$, but not at any other point. At the North pole $\ve$ is left, at the South pole $\ve $ is right. We can find circular Wilson loop operators of the form which are invariant under such $\delta_{\ve}$. Conversely, for any given circular Wilson loop $W_R(C)$ of the form we can find a suitable conformal Killing spinor $\de_{\ve}$ which annihilates $W_R(C)$. (The North pole is picked up at the center of the $W_R(C)$.) If the spinors $\hat \ve_{s}$ and $\hat \ve_{c}$ are both non zero, then $\ve$ is a nowhere vanishing spinor on $S^4$. We can use such $\de_{\ve}$ to relate the physical $\CalN=2$ gauge theory on $S^4$ to a somewhat unusual equivariant topological theory, and apply localization methods developed for topological theories [@Witten:1988ze; @MR1094734] to solve for $\langle W_R(C) \rangle$. The relation has the simplest form if the norm of $\ve$ is constant.
Before proceeding to this equivariant topological theory, we would like to finish our description of the supersymmetry group $\CalS$ of the $\CalN=2^{*}$ theory on $S^4$. First we find the maximal set of linearly independent conformal Killing spinors $\{\ve^i\}$ that simultaneously satisfy the equations $$\label{eq:no-dilat-cond}
\ve^{(i} \tilde \ve^{j)} =
\ve^{(i} \Gamma^{09} \tilde \ve^{j)} = 0,$$ and then we find what superconformal group is generated by this set. One can show that the equivalent way to formulate the conformal Killing spinor equation for the spinors in the $+1$ space of the chirality operator $\Gamma^{5678}$ is the following $$\label{eq:Killing-spinor1}
D_{\mu} \ve = \frac 1 {2 r} \Gamma_{\mu} \Lambda \ve,$$ where $\Lambda$ is a generator of $SU(2)^{R}_L$-symmetry. For example, if we start from the ten-dimensional Minkowski signature we can take $\Lambda=\Gamma^{0}\Gamma_{ij}$ where $5 \leq i < j \leq 8$. If we start from the ten-dimensional Euclidean signature we can take $\Lambda=-i\Gamma^{0} \Gamma_{ij}$ where $5 \leq i < j \leq 8$. Equivalently, $\Lambda$ is a real antisymmetric matrix, which acts in the $+1$ eigenspace of $\Gamma^{5678}$, satisfies $\Lambda^2 = -1$ and commutes with $\Gamma^{m}$ for $m=1,\dots,4,0,9$. The equation has 8 linearly independent solutions. Let $V_{\Lambda}$ be the vector space that they span. Then the space of solutions of the conformal Killing spinor equations is $V_{\Lambda} \oplus V_{-\Lambda}$, where we take $\tilde \ve = \frac {1} {2r} \Lambda \ve$.
The spinors in the space $V_{\Lambda}$ satisfy our requirement , because $\Lambda$ is antisymmetric and commutes with $\Gamma^{{9}}$. The generators $\{\delta_{\ve} | \ve \in V_{\Lambda}\}$ anticommute to generators of $\Spin(5)
\times \SO(2)^{R}$, where $\Spin(5)$ rotates $S^4$, and $\SO(2)^{R}$ is a subgroup of the $\SU(2)^R_L$-symmetry group. This $\SO(2)^R$ subgroup is generated by $\Lambda$. The space $V_{\Lambda}$ transforms in the fundamental representation of $\Sp(4) \simeq \Spin(5)$. We conclude that restricting the fermionic generators to the space $V_{\Lambda}$ of breaks the full $\CalN=2$ superconformal group $\SL(1|2,\BH)$ to the supergroup $\OSp(2|4)$, where the choice of the $\SU(2)^{R}_L$ generator $\Lambda$ defines the embedding of the $\SO(2)_{R}$ in the $\SU(2)^{R}_L$.
Besides the spaces $V_{\Lambda}$, obtained as solutions of , we can find other half-dimensional fermionic subspaces of the $\CalN=2$ superconformal group satisfying . These spaces can be obtained by $SO(1,1)_{R}$ twisting of $V_{\Lambda}$. Indeed, if the spinors $\ve$ and $\tilde \ve$ satisfy , then so do the spinors $\ve' =
e^{\frac 1 2 \beta \Gamma^{{0}{9}}} \ve $ and $\tilde \ve' = e^{-\frac 1 2
\beta \Gamma^{{0}{9}} \tilde \ve}$, where $\Gamma^{{0}{9}}$ generates $\SO(1,1)_{R'}$, and $\beta$ is a parameter of the twisting. The $\SO(1,1)_{R}$ twisted space $V_{\Lambda, \beta}$ is equivalently a space of solutions to the twisted Killing equation $$\label{eq:Killing-spinor11}
D_{\mu} \ve = \frac 1 {2 r} \Gamma_{\mu} e^{ -\beta \Gamma^{{0}{9}}} \Lambda \ve.$$
We summarize, that restriction to the half-dimensional fermionic subspace by equation breaks the $\CalN=2$ superconformal group $\SL(1|2,\BH)$ down to $\OSp(2|4)$. The choice of $\OSp(2|4)$ is defined by the generator of $\SU(2)_{R}$ symmetry $\Lambda$, and the generator of $\SO(1,1)_{R'}$ symmetry $\beta$.
If we require that the Wilson loop operator is closed with respect to $\delta_{\ve}$ with $\ve \in V_{\Lambda,\beta}$, then the parameter $\beta$ is related to the radius of the Wilson loop. In the ten-dimensional Minkowski conventions, the Wilson loop operator has the form $$\label{eq:eqLoop}
\begin{aligned}
W_R(\rho) = \tr_{R} \Pexp \oint_{C} ((A_{\mu} \frac {dx^{\mu}}{ds} +
\Phi_{0}) ds).
\end{aligned}$$ Let the circular contour $C$ be $(x^{1},x^{2}, x^{3}, x^{4}) = t (\cos \alpha,
\sin \alpha, 0, 0)$ in the stereographic coordinates. Here $t = 2 r \tan \frac
{\theta_0} 2$ for the Wilson loop located at the polar angle $\theta_0$. The combination $v^{M} A_{M} = v^{\mu} A_{\mu} + v^{A} \phi_{A}$ is annihilated by $\delta_{\ve}$, since $(\ve \Gamma^{M} \ve) (\psi \Gamma_M {\ve})$ vanishes because of the triality identity . Then the Wilson loop is $\delta_{\ve}$-closed if $(v^{\mu}, v^{{9}},
v^{{0}}) = (\frac {dx^{\mu}}{ds}, 0, 1)$. Using $\Gamma^{{0}}=1$ and the explicit form for $\ve$ we get $$\hat \ve_{c} = \frac 1 {t} \Gamma_{12} \hat \ve_{s}.$$ To satisfy we must have $$\hat \ve_{c} = \frac 1 {2 r} e^{-\beta \Gamma^{{0}{9}}} \Lambda \hat \ve_{s}.$$ Let chirality of $\hat \ve_{s}, \hat \ve_{c}$ be positive at $x=0$. Then $
\beta = \log \frac {t} {2r}$, and $(\Lambda - \Gamma_{12}) \hat \ve_{s}
= 0$. This equation has a non-zero solution for $\hat \ve_{s}$ only when $\det (\Lambda - \Gamma_{12})=0$. That determines $\Lambda$ uniquely up to a sign. In other words, the choice of the position of the Wilson loop on $S^4$ determines the way the $\SU(2)_R$ symmetry group breaks to $\SO(2)$, and the size of the Wilson loop determines the $\SO(1,1)$ twist parameter $\beta$. For the Wilson loop located at the equator $t=2 r$.
A very nice property of the conformal Killing spinor $\ve$ generating $\OSp(2|4)$ is that it has a constant norm over $S^4$, similarly to a supersymmetry transformation on flat space. Since $\OSp(2|4)$ has $8$ fermionic generators, contains only scale preserving transformations, and it is generated by spinors of constant norm on $S^4$, we call it $\CalN=2$ supersymmetry on $S^4$. So we have found that $\CalS = \OSp(2|4)$.
Now we show that it is possible to add a mass term for the hypermultiplet fields and preserve the $\OSp(2|4)$ symmetry. From now we will assume that the Wilson loop is located at the equator, so that $\ve$ has a constant norm. To generate such mass term in four dimensions we use Scherk-Schwarz reduction of ten-dimensional $\CalN=1$ SYM. Namely, we turn on a Wilson line in the $\SU(2)^{R}_{R}$ symmetry group along the coordinate $x_0$. The $\CalN=2$ vector multiplet fields $A_{\mu}, \Phi_0,
\Phi_9, \Psi$ are not charged under $\SU(2)^R_{R}$, therefore their kinetic terms are not changed. The hypermultiplet fields $\chi$ and $\Phi_{i}$ with $i =
5,\dots,8$ transform in the spin-$\frac 1 2$ representation under $\SU(2)^R_R$. Explicitly it means that we should replace $D_0 \Phi_{i}$ by $D_{0} \Phi_{i} + M_{ij} \Phi_j$, and $D_0 \chi$ by $D_0 \chi + \frac 1 4 M_{ij} \Gamma_{ij} \chi$, where an antisymmetric $4\times
4 $ matrix $M_{ij}$ with $i,j =5,\dots, 8$ is a generator of the $\SU(2)^{R}_{R}$ symmetry. Since $F_{0i}$ is replaced by $[\Phi_0, \Phi_i] + M_{ij} \Phi_j$, the $F_{0i} F^{0i}$ term in the action generates mass for the scalars of the hypermultiplet.
On the flat space, the resulting action is still invariant under the usual $\CalN=2$ supersymmetry. However, on $S^4$ we need to be more careful with the $\ve$-derivative terms in the supersymmetry transformations. Let us explicitly compute variation of the Scherk-Schwarz deformed $\CalN=4$ theory on $S^4$. We use the conformal Killing spinor $\ve$ in the $\CalN=2$ superconformal subsector, i.e. $\Gamma^{5678} \ve = \ve$. Then $\ve$ is not charged under $SU(2)_{R}^{R}$, so $D_0 \ve = 0$. Variation of by gives us (we write variation of the Lagrangian up to total derivative terms since they vanish after integration over the compact space $S^4$) $$\begin{gathered}
\delta_{\ve}( \frac 1 2 F_{MN} F^{MN} - \Psi \Gamma^{M} D_{M} \Psi + \frac 2 {r^2}
\Phi_{A} \Phi^{A}) = \\ = 2 D_{M} (\ve \Gamma_{N} \Psi) F^{MN} + 2 \Psi
\Gamma^{M} D_{M} ( \frac 1 2 F_{PQ} \Gamma^{PQ} \ve -2 \Phi_{A} \tilde
\Gamma^{A} \tilde \ve) + \frac 4 {r^2} (\ve \Gamma^{A} \psi) \Phi_{A} = \\= -
2 (\ve \Gamma_{N} \Psi) D_{M} F^{MN} + \Psi D_{M} F_{PQ} \Gamma^{M}
\Gamma^{PQ} \ve + \Psi \Gamma^{M} \Gamma^{PQ} F_{PQ} D_{M} \ve - 4 \Psi
\Gamma^{M} \tilde \Gamma^{A} \tilde \ve D_{M} \Phi_{A} + \\+ \frac 1 {r^2}
\Psi \Gamma^{\mu} \tilde \Gamma^{A} \Phi_{A} \Gamma_{\mu} \ve + \frac 4 {r^2}
(\ve \Gamma^{A} \Psi) \Phi_A = \dots\end{gathered}$$ Using $$\label{eq:GammaMPQ}
\Gamma^M \Gamma^{PQ} = \frac 1 3 (\Gamma^{M} \Gamma^{PQ} + \Gamma^{P} \Gamma^{QM} + \Gamma^{M} \Gamma^{PQ})
+ 2 g^{M[P} \Gamma^{Q]}$$ and the Bianchi identity, we see that the first term cancels the second, and that the last two terms cancel each other. Then $$\begin{gathered}
\dots = \Psi \Gamma^{\mu} \Gamma^{PQ} \Gamma_{\mu} \tilde \ve F_{PQ} - 4\Psi
\Gamma^{M} \tilde \Gamma^{A} \tilde \ve D_{M} \Phi_{A} = 4 \Psi \tilde
\Gamma^{MA} \tilde \ve F_{MA} - 4 \Psi \Gamma^{M} \tilde \Gamma^{A} \tilde \ve
D_{M} \Phi_{A}\end{gathered}$$ where we use the index conventions $M,N,P,Q = 0,\dots,9$, $\mu=1,\dots,4$, $A
=5,\dots,9,0$. In the absence of Scherk-Schwarz deformation we have $F_{MA} =
D_{M} \Phi_{A}$ for all $M=0,\dots, 9$ and $A =5,\dots,9,0$, hence the two terms cancel. After the deformation, we have $F_{0i} = D_{0} \Phi_i$, but $F_{i0} =
-D_{0} \Phi_{i} = -[\Phi_0, \Phi_i] - M_{ij} \Phi_{j} = D_i \Phi_0 - M_{ij}
\Phi_j$. Therefore, the naively Scherk-Schwarz deformed $\CalN=4$ theory on $S^4$ is not invariant under arbitrary $\CalN=2$ superconformal transformation: $$\delta_{\ve}( \frac 1 2 F_{MN} F^{MN} - \Psi \Gamma^{M} D_{M} \Psi + \frac 2 {r^2} \Phi_{A} \Phi^{A}) = - 4 \Psi \Gamma^{i} \tilde \Gamma^{0} \tilde \ve M_{ij} \Phi_{j}.$$ This is the natural consequence of adding mass terms to the Lagrangian. Nevertheless, we can add some other terms to the action in such a way to make the action invariant under the $\OSp(2|4)$ subgroup of $\CalN=2$ superconformal group on $S^4$. We use the fact that $\ve$ generating a transformation in the $\OSp(2|4)$ subgroup satisfies the conformal Killing equation with $\tilde \ve = \frac {1}
{2r} \Lambda \ve$, where $\Lambda$ is a generator of $\SU(2)_{L}^R$-group normalized as $\Lambda^2 = -1$. Let us take $\Lambda = \frac 1 4 \Gamma_{kl}
R_{kl}$ where $R_{kl}$ is an anti-self-dual matrix normalized as $R_{kl}R^{kl} =
4$, where $k,l = 5, \dots, 8$. Then we get $$\begin{aligned}
\delta_{\ve}( \frac 1 2 F_{MN} F^{MN} - \Psi \Gamma^{M} D_{M} \Psi + \frac 2 {r^2} \Phi_{A} \Phi^{A}) = \frac 1 {2r} \Psi \Gamma^{0} \Gamma^{i} \Gamma^{kl} \ve R_{kl} M_{ij} \Phi_{j} = \\
= \frac {1} {2r} (\Psi \Gamma^{i} \ve) R_{ki} M_{kj} \Phi_{j} = \frac {1}
{2r} (\delta_{\ve} \Phi^{i}) (R_{ki} M_{kj}) \Phi_j
\end{aligned}$$ Hence, the addition of $\frac {-1} {4r} (R_{ki} M_{kj}) \Phi^{i} \Phi^{j}$ term to the Scherk-Schwarz deformed action on $S^4$ makes the action invariant under the $\OSp(2|4)$.
Let us summarize. The action $$\begin{aligned}
S_{\CalN=2^{*}} = \frac {1} {2 g_{YM}^2} \int d^4 x \sqrt{g} \lb \frac 1 2
F_{MN} F^{MN} - \Psi \Gamma^{M} D_{M} \Psi + \frac 2 {r^2} \Phi_{A} \Phi^{A} -
\frac {1} {4r} (R_{ki} M_{kj}) \Phi^{i} \Phi^{j} \rb,\end{aligned}$$ where $D_{0}\Phi^{i} = [\Phi_0, \cdot] + M_{ij} \Phi^{j}$ and $D_{0} \Psi =
[\Phi_0, \Psi] + \frac 1 4 \Gamma^{ij} M_{ij} \Psi$, is invariant under the $\OSp(2|4)$ transformations, generated by conformal Killing spinors solving $D_{\mu} \ve = \frac {1} {8r} \Gamma_{\mu} \Gamma^{0kl} R_{kl} \ve$ with $\ve$ restricted to $\CalN=2$ subspace $\Gamma^{5678} \ve = \ve$.
Since $\delta^2_{\ve}$ generates a covariant Lie derivative along the vector field $-v^{M} = -\ve \Gamma^M \ve$, in particular it is contributed by the gauge transformation along the $0$-th direction. After we turned on mass for the hypermultiplet by Scherk-Schwarz mechanism, $\delta^2_{\ve}$ gets new contributions on the hypermultiplet $$\begin{aligned}
\delta^2_{\ve} \Phi_i= \delta^2_{\ve, \text{M=0}} \Phi_i - v^{0} M_{ij} \Phi_j \\
\delta^2_{\ve} \chi = \delta^2_{\ve, \text{M=0}} \chi -\frac 1 4 v^{0}
M_{ij} \Gamma^{ij} \chi.
\end{aligned}$$
So far we computed $\delta^2_{\ve}$ on-shell. To use the localization method we need an off-shell closed formulation of the fermionic symmetry of the theory. The pure $\CalN=2$ SYM can be easily closed by means of three auxiliary scalar fields, but it is well known that the off-shell closure of $\CalN=2$ hypermultiplet is impossible with a finite number of auxiliary fields. For our purposes we do not need to close off-shell the whole $\OSp(2|4)$ symmetry group. Since the localization computation uses only one fermionic generator $Q_{\ve}$, it is enough to close off-shell only the symmetry generated by this $\ve$.
To close off-shell the relevant supersymmetry of the $\CalN=4$ theory on $S^4$ we make the dimensional reduction of Berkovits method [@Berkovits:1993zz] used for the ten-dimensional $\CalN=1$ SYM, see also [@Evans:1994cb; @Baulieu:2007ew]. The number of auxiliary fields compensates the difference between the number of fermionic and bosonic off-shell degrees of freedom modulo gauge transformations. In the $\CalN=4$ case we add $16 - (10-1)
= 7 $ auxiliary fields $K_i$ with free quadratic action and modify the superconformal transformations to $$\begin{aligned}
\label{eq:off-shell-susy}
& \delta_{\ve} A_{M} = \Psi \Gamma_{M} \ve \\
& \delta_{\ve} \Psi = \frac 1 2 \gamma^{MN} F_{MN} + \frac 1 2 \gamma^{\mu A} \phi_{A} D_{\mu} \ve + K^i \nu_i \\
& \delta_{\ve} K_i = - \nu_i \gamma^{M} D_{M} \Psi,
\end{aligned}$$ where spinors $\nu_i$ with $i=1, \dots, 7$ are required to satisfy $$\begin{aligned}
\label{eq:nu1-relations1}
&\ve \Gamma^M \nu_i = 0 \\
\label{eq:nu1-relations2}
&\frac 1 2 (\ve \Gamma_N \ve) \tilde \Gamma^{N}_{\alpha \beta} =
\nu^i_{\alpha} \nu^i_{\beta} + \ve_{\alpha} \ve_{\beta} \\
\label{eq:nu1-relations3}
&\nu_i \Gamma^M \nu_j = \delta_{ij} \ve \Gamma^M \ve.\end{aligned}$$ For any non-zero Majorana-Weyl spinor $\ve$ of $\Spin(9,1)$ there exist seven linearly independent spinors $\nu_{i}$, which satisfy these equations[^6] [@Berkovits:1993zz]. They are determined up to an $\SO(7)$ transformations. The equation ensures closure on $A_M$, the equation ensures closure on $\Psi$, and the equations and ensure closure on $K$ $$\label{eq:delta2K}
\delta_{\ve}^2 K_i =
- (\ve \gamma^{M} \ve) D_{M} K^{i} - (\nu_{[i} \gamma^{M} D_{M} \nu_{j]}) K^{j}
- 4(\tilde \ve \ve) K_{i}.$$
If $E_K$ is an $\SO(7) \otimes \mathrm{ad} (G) $ vector bundle over $S^4$ whose sections correspond to the auxiliary fields $K_i$, then can be interpreted as a covariant Lie derivative action along the vector field $v^{\mu}$, or in other words as a lift of the $L_{v}$ action on $S^4$ to the action on the vector bundle $E_{K} \to S^4$. A conformal Killing spinor $\ve$ generating a transformation of the $\OSp(2|4)$ subgroup can be represented in the following form (see appendix \[se:Killing-spinors\] for details) $$\ve (x) = \exp \lb \frac \theta 2 n_{i }(x) \Gamma^{i} \Gamma^{9} \rb \hat \ve_{s},$$ where $x^i$ are the stereographic coordinates on $S^4$, $n_{i}$ is the unit vector in the direction of the vector field $v_{i} =
\frac 1 r x^{i} \omega_{i j} $. We use the conformal Killing spinor $\ve(x)$ such that $(\ve(x), \ve(x)) = 1$ and $\Gamma^{9} \hat \ve_{s}
= \hat \ve_{s}$. The matrix $\omega_{ij}$ is the anti-self-dual generator of $\SU(2)_L \subset \SO(4)$ rotation around the North pole in $\delta^2_{\ve}$. We see that the conformal Killing spinor $\ve(x)$ at an arbitrary point $x$ is obtained by $\Spin(5)$ rotation $\exp (\frac \theta 2 n_{i}(x) \Gamma^{i}
\Gamma^9)$ of its value at the origin $\ve(0) = \hat \ve_s$.
For the closure of $\CalN=4$ symmetry we need seven spinors $\nu_i$ which satisfy -. Following [@Berkovits:1993zz], at the origin we can take $\hat \nu_i = \Gamma^{i8} \hat \ve_{s}$ for $i =
1\dots 7$, and then transform $\hat \nu_i$ to an arbitrary point on $S^4$ as $$\label{eq:nu-defined}
\nu_i(x) = \exp (\frac \theta 2 n_{i}(x) \Gamma^{i8}) \hat \ve_{s}.$$
Finally, we conclude that the action $$\begin{aligned}
\label{eq:actionNstar-offshell}
S_{\CalN=2^{*}} = \frac {1} {2 g_{YM}^2} \int d^4 x \sqrt{g} \lb \frac 1 2
F_{MN} F^{MN} - \Psi \Gamma^{M} D_{M} \Psi + \frac 2 {r^2} \Phi_{A} \Phi^{A} -
\frac {1} {4r} (R_{ki} M_{kj}) \Phi^{i} \Phi^{j} - K_i K_i\rb,\end{aligned}$$ is invariant under the off-shell supersymmetry $Q_{\ve}$ given by with $\nu_i$ defined by . Though we will not need this fact, we remark that it is possible to simultaneously close four fermionic symmetries generating the $\OSp(2|2)$ subgroup of $\OSp(2|4)$. The space-time part of this $\OSp(2|2)$ subgroup consists of anti-self-dual rotations around the North pole on $S^4$.
Localization
============
As explained in the introduction, to localize the theory we deform the action by a $Q$-exact term $$\begin{aligned}
\label{eq:deformation}
S \to S + t Q V.\end{aligned}$$ Since we use $Q$ which squares to a symmetry of the theory, and since the action and the Wilson loop observable are $Q$-closed, we can use the localization argument. For $Q^2$-invariant $V$, the deformation (\[eq:deformation\]) does not change the expectation value of $Q$-closed observables. Hence, when we send $t$ to infinity, the theory localizes to some set $F$ of critical points of $QV$, over which we will integrate in the end. The measure in the integral over $F$ comes from the restriction of the action $S$ to $F$ and the determinant of the kinetic term of $QV$ which counts fluctuations in the normal directions to $F$.
To ensure convergence of the four-dimensional path integral, we compute it for a theory obtained by dimensional reduction from a theory in ten-dimensional Euclidean signature. To technically simplify the description of the symmetries in the previous section, we used ten-dimensional Minkowski signature. We can keep Minkowski metric $g_{MN}$ and Minkowski gamma-matrices $\Gamma_{M}$ and still get the same partition function as in Euclidean signature by making Wick rotation of the $\Phi_0$ field. In other words, the path integral, computed with Minkowski metric $g_{MN}$ but with $\Phi_0$ substituted by $i\Phi_0^E$ where $\Phi_0^E$ is real, is convergent and is equal to the Euclidean path integral. We also integrate over imaginary contour for the auxilary fields $K_i$, so that $K_i = iK_i^E$, where $K_i^E$ is real.
For localization computation we will take the following functional $$\begin{aligned}
\label{eq:V-definition}
V = (\Psi, \overline{Q\Psi}).\end{aligned}$$ Then the bosonic part of the $QV$-term is a positive definite functional $$\begin{aligned}
S^{Q}|_{bos} = (Q\Psi, \overline{Q\Psi}).\end{aligned}$$
Explicitly we have $$\begin{aligned}
\label{eq:QPsi}
Q \Psi = \frac 1 2 F_{MN} \Gamma^{MN} \ve + \frac 1 2 \Phi_{A} \Gamma^{\mu A} \nabla_{\mu} \ve + K^{i} \nu_{i} \\
\overline{ Q \Psi} = \frac 1 2 F_{MN} \tilde \Gamma^{MN} \ve + \frac 1 2
\Phi^{A} \tilde \Gamma^{\mu A} \nabla_{\mu} \ve - K^{i} \nu_i,
\end{aligned}$$ where $\tilde \Gamma^{0} = -\Gamma^{0}, \tilde \Gamma^{M} = \Gamma^{M}$ for $M=1,\dots,9$, and $\Gamma^{MN} = \tilde \Gamma^{[M} \Gamma^{N]}, \tilde
\Gamma^{MN} = \Gamma^{[M} \tilde \Gamma^{N]}$.
Before proceeding to technical details of the computation, let us explicitly define the conformal Killing spinor $\ve$ which we will use, and find the vector field $v^{M} = \ve \Gamma^{M} \ve$ generated by the corresponding $\delta_{\ve}^2$. We take $\ve$ in the form (\[eq:Killing-solution-in-S\^4-stereo\]), where $\hat \ve_{s}$ is any spinor such that
1. [The chirality operator $\Gamma^{5678}$ acts on $\hat \ve_{s}$ by $1$]{}
2. [The chirality operator $\Gamma^{1234}$ acts on $\hat \ve_{s}$ it by $-1$]{}
3. [$\hat \ve_{s} \hat \ve_{s} = 1$]{}
The first condition means that $\ve$ generates transformation inside the $\CalN=2$ superconformal subgroup of $\CalN=4$ superconformal group. The second condition ensures that $\ve$ is a four-dimensional left chiral spinor on the North pole of $S^4$. The third condition is a conventional normalization. In our conventions for the gamma-matrices (appendix \[sec:Octonionic-gamma-matrices\]) we can take $\hat \ve_{s} = (1,0,\dots, 0)^{t}$. Let the Wilson loop be located at the equator and invariant under anti-self-dual rotations in the $\SO(4)$ group of rotations around the North pole. To be concrete, let the Wilson loop be placed in the $(x_1, x_2)$ plane. Then we take $\hat \ve_{c} = \frac {1} {2r} \Gamma^{12} \hat \ve_{s}$. The conformal Killing spinor $\ve$ defined by such $\hat \ve_{s}$ and $\hat \ve_{c}$ has a constant unit norm over the whole four-sphere ($(\ve \ve) = 1$). At the North pole the spinor $\ve$ is purely left, at the South pole the spinor $\ve$ is purely right.
Now we compute the components of the vector field $v^{M} = \ve \Gamma^{M} \ve$. If we assume ten-dimensional Minkowski signature, then we get $$\begin{aligned}
& v_{t} = \sin \theta \\
& v^0 = 1 \\
& v^{9} = \cos \theta \\
& v^{i} = 0 \quad \text{for} \quad i=5,\dots,8,
\end{aligned}$$ where $\theta$ is the polar angle on $S^4$ such that the Wilson loop is placed at $\theta = \frac \pi 2$, the North pole is at $\theta = 0$, and the South pole is at $\theta = \pi$. The four-dimensional space-time component $v_{t}$ of $v^{M}$ has length $\sin \theta$ and is directed along the orbits of the $\U(1) \subset \SU(2)_L \subset \SO(4)$ group which rotates the $(x_1, x_2)$ plane. If we switch to the ten-dimensional Euclidean signature, then $v^{0}=i$ while the other componens are the same as in Minkowski signature.
To simplify $S^{Q}|_{bos}$ we use the Bianchi identity for $F_{MN}$, the gamma-matrices algebra and integration by parts. The principal contribution to $S^{Q}|_{bos}$ is the curvature term $$S_{FF} = \frac 1 4 (\ve \tilde \Gamma^{N} \Gamma^{M} \tilde \Gamma^{P} \Gamma^{Q} \ve) F^{MN} F^{PQ}$$ The $F_{MN} K_i$ cross-terms vanish because $\nu_i \Gamma^{0} \Gamma^{M} \ve
= \nu_i \Gamma^{M} \ve = 0$. Then we have a simple contribution from auxiliary $KK$-term $$S_{KK} = -K_i K^i.$$ In the flat space limit, $r \to \infty$ the spinor $\ve$ is covariantly constant $\nabla_{\mu} \ve =0$. Therefore, in the flat space we simply have $S^{Q}|_{bos} = S_{FF} + S_{KK}$. Up to the total derivatives and $\nabla_{\mu} \ve$-terms, using the Bianchi identity and the gamma-matrices algebra, we can see that $S_{FF}$ is equivalent to the usual Yang-Mills action $\frac 1 2 F^{MN} F_{MN}$. When the space is curved and $\nabla_{\mu} \ve \neq 0$ we shall make more careful computation. Using we get $$S_{FF} = \frac 1 2 F^{MN} F_{MN} + \frac 1 4 \ve \tilde \Gamma^{N} \Gamma^{M} \tilde \Gamma^{P} \Gamma^{Q} \ve
\frac 1 3 \lb F_{MN} F_{PQ} + F_{PN} F_{QM} + F_{QN} F_{MP} \rb.$$
To simplify the last term, first we break the indices into two groups: $M,N,P,Q=(1,\dots,4,9,0)$ and $M,N,P,Q = (5, \dots, 8)$ describing respectively the fields of the vector and hyper multiplet. Using $\Gamma^{5678}\ve=\ve$ we can see that the nonvanishing terms have only zero, two or four of indices in the hypermultiplet range $(5,\dots, 8)$. We call the resulting terms as vector-vector, vector-hyper and hyper-hyper respectively. First we consider vector-vector terms. For vector-vector terms we split indices to the gauge field part $(1,\dots,4)$ and to the scalar part $(0,9)$. The nonvanishing gauge field terms all have different values of $M,N,P,Q$. Then their contribution is simplified to $$\begin{gathered}
\frac 1 4 \cdot \frac 1 3 \cdot 24 \cdot \ve \Gamma^{1234} \ve (F^{21} F^{34}
+ F^{31}F^{42}+F^{41}F^{23}) = - \frac 1 2 \ve \Gamma^{1234} \ve (F,*F) =
-\frac 1 2 \cos \theta (F, *F),\end{gathered}$$ where $*F$ is the Hodge dual of $F$. All terms in which one of the indices is $0$ vanish because $\Gamma^{MPQ}$ is antisymmetric matrix, hence $\ve \Gamma^{0} \Gamma^{MPQ} \ve = 0$. Then the remaining vector-vector terms have $D_{\mu}\Phi_9 F$ structure. Integrating by parts and using Bianchi identity we get $$- \frac 1 3 D_{\mu} (\ve \Gamma^9 \Gamma^{\mu \nu \rho} \ve) \Phi_9 F_{\nu \rho} + \text{cyclic}(\mu \nu \rho)
= 4 (\tilde \ve \Gamma^9 \Gamma^{ \mu \nu} \ve) \Phi_9 F_{\mu \nu}.$$
Doing similar algebra we get the contribution to the vector-hyper mixing terms in $S_{FF}$ $$-8 \tilde \ve \Gamma^9 \Gamma^{ij} \ve \Phi_i [\Phi_9, \Phi_j] - 6 \tilde \ve \Gamma^{\mu} \Gamma^{ij}
\ve \Phi_i D_{\mu} \Phi_j$$ We sum up all contributions to $S_{FF}$ and obtain $$S_{FF} = \frac 1 2 F_{MN} F^{MN} - \frac 1 2 \cos \theta F_{\mu \nu} (*F)^{\mu \nu} +
4 (\tilde \ve \Gamma^9 \Gamma^{ \mu \nu} \ve) \Phi_9 F_{\mu \nu} -8 \tilde \ve \Gamma^9 \Gamma^{ij} \ve \Phi_i [\Phi_9, \Phi_j] - 6 \tilde \ve \Gamma^{\mu} \Gamma^{ij}
\Phi_i D_{\mu} \Phi_j.$$
Next we consider the cross-terms between $\Phi^A$ and $F_{MN}$ in $S^{Q}|_{bos}$ $$S_{F\Phi} = -\tilde \ve \Gamma^{A} \tilde \Gamma^{M} \Gamma^{N} \ve \Phi_A
F_{MN} - \tilde \ve \tilde \Gamma^{A} \Gamma^{M} \tilde \Gamma^{N} \ve \Phi_A
F_{MN}.$$ We consider separately the cases when the index $A$ is in the set $\{0,9\}$ and in the set $\{5,\dots,8\}$. The terms with index $A=0$ all vanish because $\tilde \Gamma^{0} = - \Gamma^0$ and because $\tilde \ve \Gamma^{M} \ve =0$ for our choice of $\ve$ in $\OSp(2|4)$. Next we take index $A=9$. The only nonvanishing terms are $$- 2 \tilde \ve \Gamma^{9} \Gamma^{\mu \nu} \ve \Phi_9 F_{\mu \nu} - 2 \tilde \ve
\Gamma^{9} \Gamma^{ij} \ve \Phi_9 [\Phi_i, \Phi_j],$$ where $\mu, \nu = 1, \dots, 4$ and $i,j = 5, \dots, 8$. Finally, we consider the case when the index $A$ is in the hypermultiplet range $5, \dots 8$. The result is $$4 \tilde \ve \Gamma^{\mu } \Gamma^{ij} \ve \Phi_i D_{\mu} \Phi_j + 4 \tilde \ve
\Gamma^{9 } \Gamma^{ij} \ve \Phi_i [\Phi_9, \Phi_j].$$ Then $$S_{F\Phi} = - 2 \tilde \ve \Gamma^{9} \Gamma^{\mu \nu} \ve \Phi_9 F_{\mu \nu} +
4 \tilde \ve \Gamma^{\mu } \Gamma^{ij} \ve \Phi_i D_{\mu} \Phi_j + 6 \tilde \ve
\Gamma^{9 } \Gamma^{ij} \ve \Phi_i [\Phi_9, \Phi_j].$$ The $\Phi\Phi$ term is easy $$S_{\Phi\Phi} = 4 \Phi^{A} \Phi^{B} \tilde \ve \Gamma^{A} \tilde \Gamma^{B}
\tilde \ve = 4 \tilde \ve \tilde \ve \Phi^A \Phi_A.$$ Finally, we need the $\Phi K$ cross-term. Only $\Phi_0$ contributes $$S_{\Phi K} = 2 K_i \Phi_0 \nu_i \tilde \Gamma^{0} \tilde \ve - 2 K_i \Phi_0
\nu_i \Gamma^{0} \tilde \ve = - 4 K_i \Phi_0 \nu_i \tilde \ve.$$ The total result is $$\begin{gathered}
S^{Q}|_{bos} = S_{FF} + S_{F\Phi} + S_{\Phi \Phi} + S_{\Phi K} + S_{KK} = \\
\frac 1 2 F_{MN} F^{MN} - \frac 1 2 \cos \theta F_{\mu \nu} (*F)^{\mu \nu} + 2
\tilde \ve \Gamma^9 \Gamma^{ \mu \nu} \ve \Phi_9 F_{\mu \nu} -2 \tilde \ve
\Gamma^9 \Gamma^{ij} \ve \Phi_i [\Phi_9, \Phi_j] - 2 \tilde \ve \Gamma^{\mu}
\Gamma^{ij} \ve \Phi_i D_{\mu} \Phi_j + \\
+ 4 (\tilde \ve \tilde \ve) \Phi_A \Phi^A - 4 K_i \Phi_0 \nu_i \tilde \ve -
K_i K^i\end{gathered}$$ The next step in the localization procedure is to find the critical points of the $S^{Q}|_{bos}$. Our strategy will be to represent $S^{Q}|_{bos}$ as a sum of semipositive terms (full squares) and find the field configurations which ensure vanishing all of them.
First we combine the four-dimensional curvature terms together with the $\Phi_9$-mixing terms $$\begin{gathered}
\label{eq:Q-exact-curvature}
\frac 1 2 F^{\mu \nu} F_{\mu \nu} -\frac 1 2 \cos \theta F^{\mu \nu} (*F)_{\mu
\nu} + 2 \tilde \ve \Gamma^9 \Gamma^{ \mu \nu} \ve \Phi_9 F_{\mu \nu} + 4
(\tilde \ve \tilde \ve) \Phi_9 \Phi^9 = \\ = \cos^2 \frac \theta 2 ( F^-_{\mu
\nu} + w_{\mu \nu}^- \Phi_9)^2 + \sin^2 \frac \theta 2 (F^{+}_{\mu \nu} +
w_{\mu \nu}^{+} \Phi_9)^2.\end{gathered}$$ where $$\begin{aligned}
w_{\mu \nu}^- = \frac 1 {\cos^{2} \frac \theta 2 } \tilde \ve^{L} \Gamma^{9} \Gamma_{\mu \nu} \ve^{L} \\
w_{\mu \nu}^+ = \frac 1 {\sin^{2} \frac \theta 2 } \tilde \ve^{R} \Gamma^{9}
\Gamma_{\mu \nu} \ve^{R}.
\end{aligned}$$
Next we make a full square with the terms $$D_m \Phi_{i} D^m \Phi^i -2 \tilde \ve \Gamma^9 \Gamma^{ij} \ve \Phi_i [\Phi_9,
\Phi_j] - 2 \tilde \ve \Gamma^{\mu} \Gamma^{ij} \ve \Phi_i D_{\mu} \Phi_j = (D_m
\Phi_j - \tilde \ve \Gamma_m \Gamma_{ij} \ve \Phi^i)^2 - \Phi^i \Phi_i (\tilde
\ve \tilde \ve) (\ve \ve).$$ Finally we absorb the mixing term $K_i \Phi_0$ as follows $$- 4 (\tilde \ve \tilde \ve) \Phi_0 \Phi_0 - 4 \Phi_0 K_{i} (\nu^{i} \tilde
\ve) - K_i K^i = -(K_i + 2 \Phi_0 (\nu_i \tilde \ve))^2.$$ We use the following relations through out the computation $$(\ve \ve) = 1, \quad (\ve^L \ve^L) = \cos^2 \frac \theta 2, \quad (\ve^R
\ve^R) = \sin^2 \frac \theta 2, \quad (\tilde \ve \tilde \ve) = \frac 1 {4 r^2},
\quad w^{-}_{\mu \nu} w^{-\mu \nu} = w^{+}_{\mu \nu} w^{+\mu \nu} = \frac 1
{r^2}.$$
The final result is $$S^{Q}|_{bos} = S^{Q}_{vect,bos} + S^{Q}_{hyper,bos}.$$ Here $$\label{eq:N-2local}
S^{Q}_{vect,bos} = \cos^2 \frac \theta 2 ( F^-_{\mu \nu} + w_{\mu \nu}^- \Phi_9)^2 +
\sin^2 \frac \theta 2 (F^{+}_{\mu \nu} + w_{\mu \nu}^{+} \Phi_9)^2 + (D_{\mu} \Phi_a)^2 + \frac 1 2 [\Phi_a, \Phi_b] [\Phi^a, \Phi^b] + (K_i^E + w_i \Phi_0^E )^2$$ where the indices $a,b=0,9$ run over the scalars of the vector multiplet, the index $i=5,6,7$ runs over the three auxiliary fields for the vector multiplet, and $w_i = 2(\nu_i \tilde \ve)$ has norm $w_i w^i = \frac 1 {r^2}$.
At this moment we also switched to the fields $\Phi_0^E, K_i^E$ which are related to the original fields in Minkowski signature as $\Phi_0 = i\Phi_0^E, K_i = iK_i^E$. Equivalently, we could make the computation in the Euclidean signature from the very beginning keeping all fields real. In this case some imaginary coefficients would appear in the supersymmetry transformations: we would write down $i$ in front of the fields $K_i$ and would replace the $\Gamma^{0}$ matrix by $i \Gamma^{0}$.
One could worry then that such supersymmetry transformations spoil the reality conditions on the fields. However, our localization computation is not affected. The Lagrangian and the theory is still invarint under such transformations if we understand the action as an analytically continued functional to the space of complexified fields. The path integral is understood as an integral of a holomorphic functional of fields over a certain real half-dimensional “contour of integration” in the complexified space of fields. Strictly speaking, the bar in the formula (\[eq:QPsi\]) for $\overline{Q\Psi}$ literally means complex conjugation only if we assume that we use that contour of integration which we described before: all fields are real except $\Phi_0$ and $K_i$ which are imaginary. For a general contour of integration in the path integral we just use the functional $V$ (\[eq:V-definition\]) where $\overline{Q\Psi}$ is *defined* by the second line of (\[eq:QPsi\]). This means that the functional $V$ holomorphically depends on all complexified fields. The bosonic part of $QV$ is positive definite after restriction to the correct contour of integration.
From any point of view, we should stress that $\delta_\ve$ squares to a complexified gauge transformation, whose scalar generator is $i\Phi_0^E + \cos \theta \Phi_9+\sin \theta A_{v}$, where $\Phi_0^E$, $\Phi_9^E$ and $A_{v}$ take value in the real Lie algebra of the gauge group, and where $A_{v}$ is the component of the gauge field in the direction of the vector field $v^{\mu}$. The theory is similar to the Donaldson theory near the North pole where this generator becomes $i \Phi_0^E + \Phi_9$, and anti-Donaldson theory near the South pole where this generator becomes $i \Phi_0^E - \Phi_9$.
The hypermultiplet contribution is $$S^{Q}_{hyper,bos} = (D_0 \Phi_i)^2 + (D_m \Phi_j - f_{m i j}\Phi^i)^2 + \frac 1
2 [\Phi_i, \Phi_j][\Phi^i,\Phi^j] + \frac 3 {4 r^2} \Phi^i \Phi_i + K^E_I K^E_I,$$ where $m=1,\dots,5$, $i=5, \dots, 8$, $I=1, \dots 4$ and $f_{m i j} = \tilde \ve
\Gamma_m \Gamma_{ij} \ve$. We see that with our choice of the “integration contour” in the space of complexified fields (all fields are real except $K_i, K_I$ and $\Phi_0$ which are pure imaginary), all terms in the action $S^{Q}|_{bos}$ are semi-positive definite. Therefore, in the limit $t \to \infty$ we need to care in the path integral only about the locus at which all squares vanish and small fluctuations in the normal directions.
For the hypermultiplet action we get a simple “vanishing theorem”: because of the quadratic term $\frac 3 {4 r^2} \Phi^i \Phi_i$, the functional $S^{Q}_{hyper, bos}$ vanishes iff all fields $\Phi_i$ vanish.
Next consider zeroes of the term $S^{Q}_{vect,bos}$. The term $(D_{\mu} \Phi_9)^2$ ensures that the field $\Phi_9$ must be covariantly constant. Away from the North and the South poles and requiring that the curvature terms vanish, we get the equations $$F_{\mu \nu} = -w_{\mu \nu} \Phi_9$$ where $w_{\mu \nu} = w^{-}_{\mu \nu} +
w^{+}_{\mu \nu}$. The curvature $F_{\mu \nu}$ satisfies Bianchi identity, hence we must have $$\label{eq:vanishing-th}
d_{[\lambda} w_{\mu \nu]} \Phi_9 = 0.$$ It is easy to check that away from the North and the South poles, $d_{[\lambda}
w_{\mu \nu]}$ does not vanish, hence $\Phi_9$ and $F_{\mu\nu}$ must vanish. The kinetic term $(D_\mu \Phi^E_0)^2$ ensures that $\Phi_0^E$ is covariantly constant. Since $F_{\mu \nu} =0$ we can assume that the gauge field vanish, then $\Phi_0^E$ is a constant field over $S^4$. We call this constant $a_E$ and conclude, that up to a gauge transformation, at smooth configurations we must have $$\label{eq:good-configurations}
S^{Q}_{bos} = 0 \Rightarrow
\begin{cases}
A_{\mu} = 0 \quad \mu = 1, \dots 4 \\
\Phi_{i} = 0 \quad i = 5, \dots, 9 \\
\Phi_0^E = a_E \quad \text{constant over $S^4$} \\
K_i^E = -w_i a_E \\
K_I = 0
\end{cases}.$$ This is the key step in the localization procedure and in the proof of the Erickson-Semenoff-Zarembo/Drukker-Gross conjecture about circular Wilson loop operators. The infinite-dimensional path integral localizes to the finite dimensional locus , and the integral over $a_E \in \g$ is the resulting matrix model.
Let us evaluate the $S_{YM}$ action at . The nonvanishing terms are only $$S_{YM}[a] = \frac 1 { 2 g_{YM}^2} \int d^4 x \sqrt {g} \lb
\frac 2 {r^2} (\Phi^E_0)^2 + (K_i^E)^2 \rb = \frac 1 { 2 g_{YM}^2} \vol(S^4)
\frac 3 {r^2} a_E^2 = \frac {4 \pi^2 r^2 }{g^2_{YM}} a_E^2$$ where we used $w_i w^i = \frac 1 {r^2}$ and the volume of the four-sphere $\frac 8 3 \pi^2 r^4$. We obtained precisely the Drukker-Gross matrix model.
Let us check that the coefficient is correct. Recall, that the original action has the following propagators in Feynman gauge on $\BR^4$ $$\begin{aligned}
\langle A_{\mu} (x) A_{\nu}(x') \rangle = \frac {g_{YM}^2 } {4 \pi^2} \frac {g_{\mu \nu}}{ (x-x')^2} \\
\langle \Phi^E_0 (x) \Phi^E_0 (x') \rangle = \frac {g_{YM}^2} {4 \pi^2} \frac
{1}{ (x-x')^2}.\end{aligned}$$ Hence, the correlator functions which appear in the perturbative expansion of the Wilson loop operator, have the structure $$\langle A_{\mu}(\alpha) \dot x^{\mu} \, A_{\nu}(\alpha') \dot x^{\nu} +
i \Phi^E_0(\alpha) i \Phi^E_0(\alpha') \rangle = -\frac {g_{YM}^2} {4 \pi^2 r^2 }
\frac {\cos(\alpha - \alpha')-1} {4 \sin^2 \frac {\alpha - \alpha'}{2}} = -\frac
{g_{YM}^2} {8 \pi^2 r^2},$$ where $\alpha$ denotes an angular coordinate on the loop. That was the original motivation for Erickson-Semenoff-Zarembo conjecture [@Erickson:2000af]. We see that the first order perturbation theory agrees with the matrix model action derived by localization. The power of the localization computation is that it actually proves the relation between the field theory and the matrix model in all orders in perturbation theory. It is also capable of taking into account instanton effects, which we describe shortly after computing the fluctuation determinant near the locus and confirming the exact solution.
We remark that for the $\CalN = 2^*$ theory, the same argument about zeroes of $S^{Q}|_{bos}$ holds. To ensure that all terms are positive definite, we take the mass parameter $M_{ij}$ in the Scherk-Schwarz reduction to be pure imaginary antisymmetric self-dual matrix. Then the action of the mass deformed $\CalN=2^{*}$ theory at configurations reduces to the same matrix model action. However, as we will see shortly, when the mass parameter $M_{ij}$ is non zero, the matrix model measure for the $\CalN=2^*$ theory is corrected by a non-trivial determinant.
Determinant factor
==================
Gauge-fixing complex
--------------------
Because of the infinite-dimensional gauge symmetry of the action we need to work with the gauge-fixed theory. We use the Faddeev-Popov ghost fields and introduce the following BRST like complex with the differential $\delta$: $$\begin{aligned}
\delta X &= -[c,X] & \delta c &= -a_0 -\frac 1 2 [c,c] & \delta \tilde c & = b&
\delta \tilde a_0 & = \tilde c_0& \delta b_0 & = c_0 \\
& &\delta a_0 &= 0 & \delta b & = [a_0, \tilde c]& \delta \tilde c_0 & =
[a_0,\bar a_0]& \delta c_0& = [a_0,b_0].
\end{aligned}$$ Here $X$ stands for all physical and auxiliary fields entering . All other fields are the gauge-fixing fields. By $[c,X]$ we denote a gauge transformation with a parameter $c$ of any field $X$. (For the gauge fields $A_{\mu}$ we have $\delta A_{\mu} =
-[c,\nabla_{\mu}]$. The gauge transformation of $\Phi = v^{M} A_{M}$ is $\delta
\Phi = [v^{\mu} D_{\mu} + v^{A} \Phi_{A},c]=\ad(\Phi) c+L_{v}c$, where $\ad(\Phi) c$ is the pointwise adjoint action of $\Phi$ on $c$ involving no differential operators). The fields $c$ and $\tilde c$ are the usual Faddeev-Popov ghost and anti-ghost. The bosonic field $b$ is the standard Lagrange multiplier used in $R_{\xi}$-gauge, where the gauge fixing is done by adding terms like $(b,i d^* A + \frac \xi 2 b)$ and $(\tilde c, d^* \nabla_{A} c)$ to the action. The fields $c$ and $\tilde c$ actually have zero modes. To treat them systematically we add constant fields $c_0,\tilde c_0,a_0,\tilde a_0,b_0$ to the gauge-fixing complex. The field $a_0$ is interpreted as a ghost field for the ghost $c$. The fields $a_0, \tilde a_0, b_0$ are bosonic, and the fields $c_0, \tilde c_0$ are fermionic. The operator $\delta$ squares to the gauge transformation by the constant bosonic field $a_0$ $$\delta^2 \cdot = [a_0, \cdot].$$ The gauge invariant action and observable are $\delta$-closed $$\delta S_{YM}[X] = 0,$$ therefore their correlation functions are not changed when we add the $\delta$-exact gauge-fixing term.
When we combine the gauge-fixing terms with the physical action, we will see that the convergence of the path integral requires the imaginary contour of integration for the constant field $a_0$. This field $a_0$ later will be identified with the zero mode of the physical field $\Phi_0$ which is integrated over imaginary contour. To have consistent notations we set $a_0 = i a_0^E$ and assume that $a_0^E$ is integrated over the real contour.
The $\delta$-exact term $$\begin{gathered}
\label{eq:gauge-fix-term}
S^{\delta}_{g.f.} = \delta( (\tilde c, i d^*A + \frac {\xi_1} 2 b + i b_0) - (c,
\tilde a_0 - \frac {\xi_2} 2 a_0) ) = \\= (b, i d^*A + \frac {\xi_1} 2 b + i b_0) - (\tilde c, id^* \nabla_A c + ic_0 + \frac {\xi_1} 2 [a_0,\tilde c]) + (-ia_0^E + \frac 1 2 [c,c], \tilde a_0- \frac {\xi_2} 2 ia_0^E) + (c, i\tilde c_0)\end{gathered}$$ properly fixes the gauge.
Assuming that all bosonic fields are real, the bosonic part of gauge-fixed action has strictly positive definite quadratic term for all fields and ghosts at $\xi_1, \xi_2 >0$.
By general arguments the partition function does not depend on the parameters $\xi_1, \xi_2$ in the $\delta$-exact term. Let us fix $\xi_1=0$ and demonstrate explicitly independece on $\xi_2$ and equivalence with the standard gauge-fixing procedure. First we do Gaussian integral integral over $a_0^E$ and get $$(ia_0^E + \frac 1 2 [c,c],i \tilde a_0- \frac {\xi_2} 2 ia_0^E) \to +\frac {1} {2 \xi_2} (\tilde a_0 - \frac {\xi_2} {4} [c,c])^2.$$ Then we do Gaussian integral over $\tilde a_0$ and the above term goes away completely. The determinant coming from the Gaussian integral over $\tilde a_0$ is inverse to the determinant coming from the Gaussian integral over $a_0$. Then we integrate the zero mode of $b$ against $b_0$. Then integral over non-zero modes of $b$ gives Dirac delta-functional inserted at the gauge-fixing hypersurface $d^* A=0$. The remaining terms are $$(\tilde c, id^* \nabla_A c) + i(\tilde c, c_0) + i (c,\tilde c_0).$$ We can integrate out $c_0$ with the zero mode of $\tilde c$, and $\tilde c_0$ with the zero mode of $c$. Then we are left with the integral over $c$ and $\tilde c$ with the zero modes projected out and the gauge-fixing term $$(\tilde c, id^* \nabla_A c).$$ This reproduces the usual Faddeev-Popov determinant $\det'(d^{*} \nabla_A)$ which we need to insert into the path integral for the partition function after restricting to the gauge-fixing hypersurface $d^* A = 0$. The symbol $'$ means that the determinant is computed on the space without the zero modes.
We summarize the gauge fixing procedure by the formula $$\begin{gathered}
\label{eq:Z_{phys}}
Z = \frac {1} {\vol(\CalG,g_{YM})} \int [DX] e^{-S_{YM}[X]} = \frac {1} {\vol(\CalG)} \int
[DX] e^{-S_{YM}[X]}
\int_{g \in \CalG'} [Dg] \, \delta_{Dirac}( d^*A^{g}) \det{}^{'}( d^* \nabla_{A} ) =\\
=\frac {\vol(\CalG',g_{YM})} {\vol(\CalG,g_{YM})} \int [DX Db' Dc' D\tilde c'] e^{-S_{YM}[X] - \int_{S^4} \sqrt{g} \, d^4 x (i(b,d^*A) - (\tilde c, i d^* \nabla_A c))} =\\
= \frac {1} {\vol{(G,g_{YM})}} \int [DX \, Db \, Db_0 \, Dc \, Dc_0 \, D\tilde c \,
D\tilde c_0 \, Da_0 \, D\tilde a_0] e^{-S_{YM}[X] - S_{g.f.}^{\delta}[X,ghosts]},\end{gathered}$$ where $\CalG'=\CalG/G$ is the coset of the group of gauge transformations by constant gauge transformations. We shall note that in our conventions for the gauge theory Lagrangian $\frac{1}{4 g_{YM}^2} (F,F)$, where $F = dA + A \wedge A$, we need to take the volume of the group of gauge transformations with respect to the measure which is rescaled by a power of the coupling constant $g_{YM}$. In other words, we take $\vol(G,g_{YM}) = g_{YM}^{dim G} \vol(G)$, where $\vol(G)$ is the volume of the gauge group computed with respect to the Haar measure induced by the coupling constant independent Killing form $(,)$ on the Lie algebra.
Supersymmetry complex
---------------------
To compute the path integral, it is convenient to bring the supersymmetry transformations to a cohomological form by a change of variables. (This change of variables involves no Jacobian, one can think about it as a change of notations.) We use the fact that conformal Killing spinor $\ve$ in has constant unit norm at any point on $S^4$. Then the set of sixteen spinors consisting of $\{\Gamma^{M} \ve\}$ for $M=1,\dots,9$ and $\{\nu_i\}$ for $i = 1,\dots, 7$ form an orthonormal basis for the space of $\Spin(9,1)$ Majorana-Weyl spinors reduced on $S^4$. We expand $\Psi$ over this basis $$\Psi = \sum_{M=1}^{9} \Psi_M \Gamma^M \ve + \sum_{i=1}^{7} \Upsilon_i \nu^i.$$ In new notations $(\Psi_M, \Upsilon_i)$, the supersymmetry transformations take the following form:
$$\begin{aligned}
&\begin{cases}
{s}A_{M} = \Psi_M \\
{s}\Psi_{M} = -(L_v + R +M +G_{\Phi}) A_{M}
\end{cases} \\
&\begin{cases} {s}\Upsilon_i = H^i \\
{s}H^i = -(L_v + R + M + G_{\Phi}) \Upsilon_i,
\end{cases}
\end{aligned}$$
where $$H^i \equiv K^{i} + w_i \Phi_0 + s_i(A_M).$$ Now $s$ denotes $\delta_{\ve}$ to distinguish it from the differential $\delta$ of the Faddeev-Popov complex. By $L_v$ we denote the Lie derivative in the direction of the vector field $v^{\mu}$, $R$ denotes the $R$-symmetry transformation in $\SU^{R}_{L}$, $M$ denotes the mass-term induced transformation by $M_{ij}$ in $\SU^{R}_{R}$, and $G_{\Phi}$ denotes the gauge transformation by $\Phi$. The functions $s_i(A_M)$ with $i=1,\dots, 7$ are the “equations” of the equivariant theory $$s_i(A_M) = \frac 1 2 F_{MN} \nu_i \Gamma^{MN} \ve + \frac 1 2 \Phi_A \nu_i \Gamma^{\mu A} \nabla_{\mu} \ve \quad \text{for} \quad M,N = 1, \dots, 9 \quad A =5,\dots,9.$$
Even shorter, we can write the supersymmetry complex like $$\begin{aligned}
& s X = X'\\
& s X' = [\phi + \ve, X],
\end{aligned}$$ and $s \phi = 0$, where we denoted $\phi = - \Phi$, $[\phi,X] = -G_{\Phi} X$ and $[\ve,X]= -(L_v +
R + M) X $.
All fields except $\Phi$ (\[eq:Phi-def\]) are grouped in $s$-doublets $(X, X')$, where the fields $X$ and $X'$ have opposite statistics. We can think about fields $X$ as coordinates on some infinite-dimensional supermanifold $\CalM$, on which group $\CalG$ acts. The fields $X'$ can be interpreted as de Rham differentials $X'
\equiv d X$, if we identify the operator $s$ with the differential in the Cartan model of $\CalG$-equivariant cohomology on $\CalM$ $$s = d + \phi^a i_{v^a}$$ where $\phi^a$ are the coordinates on the Lie algebra $\g$ of the group $\CalG$ with respect to some basis $\{e_a\}$, and $i_{v^a}$ is the contraction with a vector field $v^a$ representing action of $e_{a}$ on $\CalM$. The differential $s$ squares to the Lie derivative $\CalL_{\phi}$. In the present case, the group $\CalG$ is a semi-direct product $$\label{eq:group_G}
\CalG=\CalG_{gauge} \ltimes \U(1)$$ of the infinite-dimensional group of gauge transformations $\CalG_{gauge}$ and the $\U(1)$ subgroup of the $\OSp(2|4)$ symmetry group generated by the conformal Killing spinor $\ve$.
In the path integral for the partition function $Z_{phys}$, we integrate $s$-equivariantly closed form $e^S$ over $\CalM$ and then over $\phi$. See [@Vafa:1994tf; @Labastida:1997vq; @Kapustin:2006pk] for twisted $\CalN=4$ SYM related theories which have similar cohomological structure, and [@Witten:1992xu] where similar integration over the parameter of the equivariant cohomology is performed.
The combined $Q$-complex
------------------------
So far we constructed separately the gauge-fixing complex with the differential $\delta$ and the supersymmetry complex with the differential $s$: $$\begin{aligned}
\delta a_0 &= 0 & \delta X &= -[c,X] & \delta c &= -a_0 -\frac 1 2 [c,c] & \delta \tilde c & = b& \delta \tilde a_0 & = \tilde c_0& \delta b_0 & = c_0 \\
& & \delta X' &= -[c, X'] & \delta \phi &= -[c + \ve, \phi] & \delta b & = [a_0, \tilde c]& \delta \tilde c_0 & = [a_0,\tilde a_0]& \delta c_0& = [a_0,b_0] \\
s a_0 &= 0 & s X &= X' & s c &= \phi & s \tilde c & = 0& s\tilde a_0 & = 0& s b_0 & = 0 \\
& & s X' &= [\phi+\ve, X] & s \phi &= 0 & s b & = [\ve,\tilde c]& s \tilde c_0 &
= 0 & s c_0& = 0.
\end{aligned}$$ Here we summarize the anticommutators for $\delta$ and $s$: $$\begin{aligned}
\{\delta,\delta\} X^{(\prime)} &= [a_0, X^{(\prime)}] & \{\delta,\delta\} (ghost) &= [a_0, ghost] \\
\{s,s\} X^{(\prime)} & = [\phi+\ve,X^{(\prime)}] & \{s,s\} (ghost) &= 0 \\
\{s,\delta\} X^{(\prime)} & = -[\phi,X^{(\prime)}] & \{s,\delta\} (ghost) &= [\ve, ghost].
\end{aligned}$$ In this formula $X^{(')}$ stands for all physical and auxilary fields $X$ and $X'$, and $ghost$ stands for any field of the BRST gauge fixing complex.
Now we combine the operators $\delta$ and $s$ and define a fermionic operator $Q$: $$Q = s+\delta.$$ Then we get $$\begin{aligned}
Q X &= X' -[c,X] & Q c &= \phi -a_0 -\frac 1 2 [c,c] & Q \tilde c & = b& Q \tilde a_0 & = \tilde c_0& Q b_0 & = c_0 \\
Q X' &= [\phi+\ve, X] - [c, X'] &Q \phi &= -[c, \phi + \ve ] & Q b &=[a_0+\ve,\tilde c] & Q \tilde c_0 & = [a_0,\tilde c_0] & Q c_0 & = [a_0, b_0] \\
Q a_0 &= 0,
\end{aligned}$$ This means that $Q$ satisfies on all fields $$Q^2 \cdot = [a_0 + \ve, \cdot] .$$
In other words, $Q$ squares to a constant gauge transformation generated by $a_0$ and the $\U(1)$ anti-self-dual Lorentz rotation around the North pole generated by $\ve$.
Now, since $sS_{phys} = 0$ and $\delta
S_{phys} = 0$ we have $$Q S_{phys}= 0.$$ We would like to make sure that the gauge-fixing term is also $Q$-closed so that we could use the localization argument.
We will take the following $Q$-exact gauge-fixing term: $$\begin{gathered}
S_{g.f.}^{Q} = (\delta + s) ((\tilde c, i d^*A + \frac {\xi_1} 2 b +
i b_0) - (c, \tilde a_0 - \frac {\xi_2} 2 a_0) )
= S_{g.f}^{\delta} - (\tilde c, s(i d^*A + \frac {\xi_1} 2 b + i b_0)) - (\phi, \tilde a_0) =\\
= S_{g.f}^{\delta} - (\tilde c, d^*\psi + \frac {\xi_1} 2 [\ve, \tilde c]) - (\phi, \tilde a_0 - \frac {\xi_2} 2 a_0)\end{gathered}$$ The replacement of $S_{g.f.}^{\delta}$ by $S_{g.f.}^{Q}$ does not change the partition function $Z_{phys}$ . We can easily see this at $\xi_1 = 0$. Integrating over $a_0$ we get $$(ia_0^E + \frac 1 2 [c,c] - \phi, \tilde a_0 - \frac {\xi_2} {2} ia_0^E) \to \frac {1}{2 \xi_2}\lb -\frac {\xi_2} {2}
( \frac 1 2 [c,c] - \phi) + i \tilde a_0\rb^2.$$ After we integrate over $\tilde a_0$ the above term goes away completely. The determinants for the Gaussian integrals over $a_0$ and $\tilde a_0$ cancel. Then we are left with the following gauge-fixing terms $$i(b, d^*A + b_0) - i(\tilde c, d^* \nabla c + c_0) + i (c, \tilde c) - (\tilde c, d^{*} \psi),$$ where $\psi$ is the fermionic one-form which is the superpartner of the gauge field $A$. Then we note that the term $(\tilde c, d^{*} \psi)$ does not change the fermionic determinant arising from the integral over $c, \tilde c, c_0$ and $\tilde c_0$. The reason is that all modes of $c$ are coupled to $\tilde c$ by this quadratic action $$i(\bar c, d^* \nabla c + c_0) + i(c, \tilde c),$$ and that there are no other terms in the gauge-fixed action which contain modes of $c$. In other words, if treat the term $(\tilde c, d^{*} \psi)$ as a perturbation to the usual gauge fixed action, all diagrams with it vanish because $\tilde c$ can be connected by a propagator only to $c$, but there are no other terms which generate vertices with $c$.
In other words we did the following. The action of the theory gauge-fixed in the standard way (\[eq:Z\_[phys]{}\]) is $\delta$-closed, but not $Q$-closed. We make the action $Q$-closed by adding such terms to it which do not change the path integral. The fact that the partition function does not change can be also shown by making a change of variables which has trivial Jacobian.
We conclude that the total gauge-fixed action $$\tilde S_{phys} = S_{phys} + S'_{g.f.}$$ is $Q$-closed $$Q \tilde S_{phys} = 0,$$ and that the partition function defined by the path integral over all fields and ghosts with the action $\tilde S_{phys}$ is equal to the standard partition function with the usual gauge-fixing (\[eq:Z\_[phys]{}\]).
It is possible to write the operator $Q$ in the canonical form; namely $Q$ is the equivariant differential in the Cartan model for the $\tilde G = G \ltimes \U(1)$ cohomology generated by $a_0$ and $\ve$ on the space of all other fields over which we integrate in the path integral . The multiplets $(\tilde c, b)$, $(\tilde a_0, \tilde c_0)$ and $(b_0 , c_0)$ are already in the canonical form. To bring the transformations of $(X,X')$ and $(c,\phi)$ to the canonical form we make a change of variables $$\begin{aligned}
\tilde X' = X' - [c,X] \\
\tilde \phi = \phi - a_0 - \frac 1 2 [c,c].
\end{aligned}$$ Such change of variables has trivial Jacobian and does not change the path integral. In terms of new fields, the $Q$-complex is canonical: all fields are grouped in doublets $(Field, Field')$, while $Q$ acts as $$\begin{aligned}
\label{eq:Q_complex_long}
Q (Field) &= (Field') \\
Q (Field') &= [a_0 + \ve, Field] .
\end{aligned}$$ Moreover, $Qa_0 = Q \ve = 0$.
Now recall Atiyah-Bott-Berline-Vergne localization formula for the integrals of the equivariantly closed differential forms [@MR721448; @MR685019] $$\label{eq:ABBV}
\int_{\CalM} \alpha = \int_{F \subset \CalM} \frac {i^*_{F} \alpha} { e(\CalN)}.$$ The numerator corresponds to the physical action evaluated at the critical locus of the $tQV$ term. The equivariant Euler class of the normal bundle in the denominator is just a determinant, coming from the Gaussian integral using quadratic part of $tQV$ in the normal directions $\CalN$. We will argue that this determinant can be expressed as a product of weights for the group action on $\CalN$ defined by . The basic difference with the usual localization formula is that the manifold $\CalM$ in our problem is not a usual manifold, but an (infinite-dimensional) supermanifold. Hence, the equivariant Euler class must be understood in a super-formalism [@lavaud-equiv; @lavaud-superpf]. In our case it is just a super-determinant. If we split the normal bundle to the bosonic and the fermionic subspaces, the resulting determinant is the product of weights on the bosonic subspace divided by the product of weights on the fermionic subspace.
Before making gauge-fixing procedure we argued previously that the theory localizes to the zero modes of the field $\Phi_0$. The localization argument for the gauge-fixed theory remains the same, except that now we can identify the zero mode of the field $\Phi_0$ with $a_0$. Indeed, if we first integrate over $\tilde a_0$ using gauge fixing terms at $\xi_2 = 0$ $$(ia_0^E + \frac 1 2 [c,c] - i \phi^E, \tilde a_0),$$ we get the constraint that the zero mode of $\phi^E$ is equal to $a_0^E$.
Computation of the derminant by the index theory of transversally elliptic operators
------------------------------------------------------------------------------------
We write the linearization of the $Q$-complex in the form $$\begin{aligned}
\label{eq:eqX}
Q X_0 &= X_0' & Q X_1 &= X_1' \\
Q X_0' &= R_0 X_0 & Q X_1' &= R_1 X_1
\end{aligned}$$ where all bosonic and fermionic fields in the first line of are denoted as $X_0$ and $X_1$ respectively, and their $Q$-differentials are denoted as $X_0'$ and $X_1'$. So $X_0, X_1'$ are bosonic, and $X_0', X_1$ are fermionic fields.
The quadratic part of the functional $V$ is $$\begin{aligned}
V^{(2)} = \left(
\begin{array}{c}
X_0' \\ X_1
\end{array}
\right)^{t} \left(
\begin{array}{cc}
D_{00} & D_{01} \\
D_{10} & D_{11} \\
\end{array}
\right) \left(
\begin{array}{c}
X_0 \\
X_1' \\
\end{array}
\right),\end{aligned}$$ where $D_{00}, D_{01}, D_{10}, D_{11}$ are some differential operators. Then we have $$QV^{(2)} = (X_{bos}, K_{bos} X_{bos}) + (X_{ferm}, K_{ferm} X_{ferm}),$$ where the kinetic operators $K_{bos}, K_{ferm}$ are expressed in terms of $D_{00}, D_{01}, D_{10}, D_{11}$ and $R_0,R_1$ in a certain way. The Gaussian integral gives $$\label{eq:Gauss-Z-1-loop}
Z_{\text{1-loop}} = \left ( \frac { \det K_{bos}}{\det K_{ferm}}
\right)^{-\frac 1 2}.$$ Let $E_0$ and $E_1$ denote the vector bundles whose sections can be identified with fields $X_0,X_1$. Some linear algebra shows that this ratio of the determinants depends only on the representation structure $R$ on the kernel and cokernel spaces of the operator $D_{10}: \Gamma(E_0) \to
\Gamma(E_1)$. Namely we have $$\frac { \det K_{bos}} { \det K_{ferm}} = \frac { \det_{\ker D_{10}} R} { \det_{\coker D_{10}} R}.$$
The operator $D_{10}$ in our problem is not an ordinary elliptic operator, but a transversally elliptic operator with respect to the $U(1)$ rotation of $S^4$.
This means the following. Let $E_0$ and $E_1$ be vector bundles over a manifold $X$ and $D: \Gamma(E_0) \to \Gamma(E_1)$ be a differential operator. (In our problem $X = S^{4}$.) Let a compact Lie group $\tilde G$ act on $X$ such that its action preserves all structures. Let $\pi: T^*X \to X$ be the cotangent bundle of $X$. Then pullback $\pi^*E_i$ is a bundle over $T^*X$. By definition, a symbol of the differential operator $D: \Gamma(E_0) \to
\Gamma(E_1)$ is a vector bundle homomorphism $\sigma(D): \pi^* E_0 \to \pi^*
E_1$, such that in local coordinates $x_i$, the symbol is defined by replacing all partial derivatives in the highest order component of $D$ by momenta, so that $\frac {\p} {\p
x^i} \to i p_i$, and then taking $p_i$ to be coordinates on fibers of $T^*X$. The operator $D$ is called elliptic if its symbol $\sigma(D)$ is invertible on $T^*X \setminus 0$, where $0$ denotes the zero section. The kernel and cokernel of an elliptic operator are finite dimensional vector spaces. Using the Atiyah-Singer index theory [@Atiyah-Singer:1968-ind1; @Atiyah-Segal:1968-ind2; @Atiyah-Singer:1968-ind3; @MR0190950; @MR0212836; @MR0232406] one can find a formal difference of representations in which $\tilde G$ acts on these spaces, as we will see in a moment. However, we will see as well that the operator $D_{10}$ is not elliptic, so the ordinary Atiyah-Singer index theory does not apply. There is a generalization of Atiyah-Singer index theory for operators which are elliptic only in directions transverse to the $\tilde G$-orbits [@MR0482866; @MR0341538]. Such operators are called transversally elliptic. In other words, for any point $x \in X$ we consider the subspace $T_{\tilde G}^*X_x$ of the $T^*X_x$, which consists of elements which are orthogonal to the $\tilde G$-orbit through $x$. We have $$T_{\tilde G}^*X_{x} = \{ p \in T^{*}X_x \quad\mathrm{such \quad that} \quad p \cdot v(\tilde g) = 0 \quad \forall \tilde g \in \mathrm{Lie}(\tilde G) \},$$ where $v(\tilde G)$ denotes a vector field on $X$ generated by an element $\tilde g $ of the Lie algebra of $\tilde G$. Then the family of the vector spaces $T^*_{\tilde G} X$ over $X$ is defined as the union of $T_{\tilde G}^*X_{x}$ for all $x \in X$. The notion of a family of vector spaces over some base is similar to the notion of a vector bundle, except that dimension of fibers can jump. The operator $D$ is called transversally elliptic if its symbol $\sigma(D)$ is invertible on $T^*_{\tilde G} X \setminus 0$. Computing the symbol of $D_{10}$, we will see explicitly in (\[eq:symbol-explicit\]) that $D_{10}$ is not an elliptic operator, but a transversally elliptic one. The kernel and the cokernel of such an operator are not generally finite dimensional vector spaces, but if we decompose them into irreducible representations, then each irreducible representation appears with a finite multiplicity [@MR0482866; @MR0341538]. So we have $$\begin{aligned}
\ker D_{10} = \oplus_{\alpha} m_{\alpha}^{(0)} R_{\alpha} \\
\coker D_{10} = \oplus_{\alpha} m_{\alpha}^{(1)} R_{\alpha},
\end{aligned}$$ where $\alpha$ runs over irreducible representations of $\tilde G$, and $m_{\alpha}$ denotes the multiplicity of the irreducible representation $R_{\alpha}$. Then $$\label{eq:defKK}
\frac { \det K_{bos}} { \det K_{ferm}} = \prod_{\alpha} (\det R_{\alpha})^{m^{(0)}_{\alpha} - m^{(1)}_{\alpha}}.$$
Thus we need to know only the difference of multiplicities $m^{(0)}_{\alpha}$ and $m^{(1)}_{\alpha}$ of irreducible representations into which the kernel and cokernel of $D_{10}$ can be decomposed. To find this difference we use Atiyah-Singer index theory [@MR0482866; @MR0341538] for transversally elliptic operators, which generalizes the usual theory [@Atiyah-Singer:1968-ind1; @Atiyah-Segal:1968-ind2; @Atiyah-Singer:1968-ind3; @MR0190950; @MR0212836; @MR0232406]. In our problem, $R_{\alpha}$ is an irreducible representation of the group $\tilde G =\U(1)\times G$. We also denote this $\U(1)$ group by $H$, so that $\tilde G = H \times G$. The relevant representations of $G$ are those in which the physical fields transform (we will consider only the adjoint representation), but all representations of $H=\U(1)$ arise. Let $q \in \BC, |q|=1$ denote an element of $\U(1)$. Irreducible representations of $\U(1)$ are labeled by integers $n$, so that the character of representation $n$ is $q^n$. The $\U(1)$-equivariant index of $D_{10}$ is defined as $$\ind (D_{10}) = \tr_{\ker D_{10}} R(q) - \tr_{\coker D_{10}} R(q) =
\sum_{n} ( m^{(0)}_n - m^{(1)}_n) q^{n}.$$ Hence, if we compute the equivariant index of $D_{10}$ as a series in $q$, we will know $m^{(0)}_n - m^{(1)}_n$ and will be able to evaluate .
To compute the index of $D_{10}$, first we need to describe the bundles $E_0,
E_1$ and the symbol of the operator $D_{10}: \Gamma(E_0) \to \Gamma(E_1)$. The collective notation $X_0, X_0', X_1, X_1'$ corresponds to the original fields in the following way $$\begin{aligned}
X_0 &= (A_{M}, \tilde a_0, b_0) & X_1 &= (\Upsilon_i, c, \tilde c) \\
X'_1 &= (\tilde \Psi_{M}, \tilde c_0, c_0) & X_1' &= (\tilde H_i, \tilde \phi,
b).
\end{aligned}$$
The space of all fields decomposes in a way compatible with $Q$-action into direct sum of two subspaces: the fields of vector multiplet and hypermultiplet. The vector subspace also includes fields of the gauge fixing complex. The vector subspace consists of $$X_0^{vect} = (\Phi_9, A_M, \tilde a_0, b_0) \quad \text{for } M=1,\dots,4 \quad
X_1^{vect} = (\Upsilon_i, c, \tilde c ) \quad \text{for } i=5, \dots, 7$$ and their $Q$-superpartners. The hyper subspace consists of $$X_0^{hyper} = (A_M) \quad \text{for } M=5,\dots,8 \quad
X_1^{hyper} = (\Upsilon_i ) \quad \text{for } i =1, \dots, 4$$ and their $Q$-superpartners. The operator $D_{10}$ does not mix the vector and hyper subspaces. So the vector bundles split as $E_0 = E_0^{vect} \oplus
E_{0}^{hyper}$, and $E_1 = E_1^{vect} \oplus E_1^{hyper}$, as well as the operator $D_{10} =D_{10}^{vect} + D_{10}^{hyper}$, where $D_{10}^{vect}:
\Gamma(E_0^{vect}) \to \Gamma(E_1^{vect})$ and $D_{10}^{hyper}: \Gamma(E_0^{hyper})
\to \Gamma(E_1^{hyper})$.
First we consider the index of $D_{10}^{vect}$. The constant fields $(\tilde a_0,
b_0)$ are in the kernel of $D_{10}^{vect}$ and have zero $\U(1)$ weights, hence their contribution to the index is $2$. The remaining fields, denoted by $X_0^{vect'}$, are identified with sections of bundle $(T^* \oplus \CalE) \otimes
\ad E$, where $T^*$ is the cotangent bundle, and $\CalE$ is the rank one trivial bundles over $S^4$. The fields $X_1^{vect'}$ are identified with sections of $(\CalE^{3} \oplus \CalE^2) \otimes \ad E$, where $\CalE^3$ is the rank three trivial bundle of auxiliary scalar fields, and $\CalE^2$ is the rank two trivial bundle of the gauge fixing fields $c$ and $\tilde c$. Because of the difference due to $(\tilde a_0, b_0)$ contribution we have $$\label{eq:difference-d-dprime}
\ind(D_{10}^{vect}) = \ind'(D_{10}^{vect}) + 2.$$ Now we compute the symbol of the operator $D_{10}^{vect}$. The relevant terms are $$V^{(2)} = (\tilde c, d^{*} A) + (c, \nabla_{\mu} \CalL_{v} A_{\mu}) + (\Upsilon_i, (*F_{0i}) - F_{0i} \cos \theta
+ \nabla_{i} \Phi_9 \sin \theta),$$ where index $i$ runs over vielbein elements on $S^4$.
We chose a vielbein in such a way that $i=1$ is the direction of the $\U(1)$ vector field, and $i=2,3,4$ are the remaining orthogonal directions. The term $(c,
\nabla_{\mu} \CalL_{v} A_{\mu})$ comes from the term $(\psi_{\mu}, \CalL_{v}
A_{\mu})$ and the relation $\psi_{\mu} = \tilde \psi_{\mu} - \nabla_{\mu} c$. Then the symbol $\sigma(D_{10}^{vect}): \pi^* E_{0}^{vect} \to \pi^{*}E_1^{vect}$, where $\pi$ denotes the projection of the cotangent bundle $\pi: T^*X \to X$, is represented by the following matrix
$$\label{eq:symbol-explicit}
\begin{pmatrix}
c \\
\tilde c \\
\Upsilon_1 \\
\Upsilon_2 \\
\Upsilon_3 \\
\end{pmatrix}
\leftarrow
\begin{pmatrix}
{ c_{\theta} }p^2 & { s_{\theta} }\vect{p}^2 & -{ s_{\theta} }p_2 p_1 & -{ s_{\theta} }p_3 p_1 & -{ s_{\theta} }p_4 p_1 \\
0 & p_1 & p_2 & p_3 & p_4 \\
{ s_{\theta} }p_2 & -{ c_{\theta} }p_2 & { c_{\theta} }p_1 & -p_4 & p_3 \\
{ s_{\theta} }p_3 & -{ c_{\theta} }p_3 & p_4 & { c_{\theta} }p_1 & -p_2 \\
{ s_{\theta} }p_4 & -{ c_{\theta} }p_4 & -p_3 & p_2 & { c_{\theta} }p_1
\end{pmatrix}
\begin{pmatrix}
\Phi_9 \\ A_1 \\ A_2 \\ A_3 \\ A_4
\end{pmatrix}.$$
Here $p_{i}$ for $i=1,\dots,4$ denotes coordinates on fibers of $T^*X$, $\vect{p}=(p_2,p_3,p_4)$ denotes coordinate on fibers of $T^*_H X$, and ${ c_{\theta} }\equiv \cos \theta$, ${ s_{\theta} }\equiv \sin \theta$. In other words, $\vect{p}$ is a momentum orthogonal to the direction of the $\U(1)$ vector field on $S^4$. After a change of coordinates on fibers of bundles $E_0 \to T^*X$ and $E_1 \to T^*X$ $$\begin{aligned}
c \to c + { s_{\theta} }p_0 \tilde c \\
\Phi_9 \to { c_{\theta} }\Phi_9 + { s_{\theta} }A_1 \\
A_1 \to - { s_{\theta} }\Phi_9 + { c_{\theta} }A_1,
\end{aligned}$$ the matrix of the symbol of $D_{10}^{vect}$ takes the form $$\label{eq:symbol-simplified-nearly}
\begin{pmatrix}
p^2 & 0 & 0 & 0 & 0 \\
{ s_{\theta} }p_1 & { c_{\theta} }p_1 & p_2 & p_3 & p_4 \\
0 & -p_2 & { c_{\theta} }p_2 & -p_4 & p_3 \\
0 & -p_3 & p_3 & { c_{\theta} }p_2 & -p_2 \\
0 & -p_4 & -p_3 & p_2 & { c_{\theta} }p_2 .
\end{pmatrix}.$$ The term ${ s_{\theta} }p_1$ in the first column of the second line can be also removed by subtracting the first line multiplied by ${ s_{\theta} }p_1 /p^2$. Then the notrivial part of the symbol is represented by the following $4 \times 4 $ matrix $$\label{eq:symbol-simplified}
\sigma=\begin{pmatrix}
{ c_{\theta} }p_1 & p_2 & p_3 & p_4 \\
-p_2 & { c_{\theta} }p_2 & -p_4 & p_3 \\
-p_3 & p_3 & { c_{\theta} }p_2 & -p_2 \\
-p_4 & -p_3 & p_2 & { c_{\theta} }p_2 .
\end{pmatrix}.$$ The determinant of this matrix is $(\cos^2\theta p_1^2 + \vect{p}^2)^2$. First of all, we see that the symbol is not elliptic at the equator of $S^4$, since if $\cos \theta = 0$ we can take $(p_1 \neq 0, \vect{p}=0)$ and the determinant will vanish. But the symbol is transversally elliptic with respect to the $H=\U(1)$ group, since its determinant is always non-zero whenever $\vect{p} \neq 0$. Indeed, to check if the symbol is transversally elliptic, we need to consider only non-zero momenta orthogonal to the $\U(1)$ orbits. In our notations that means $p_1=0, \vect{p} \neq 0$.
In a neighborhood of the North pole (${ c_{\theta} }=1$) the symbol is equivalent to the elliptic symbol of the standard anti-self-dual complex $(d,d^{-})$ $$\label{eq:asd-complex}
\Omega^{0} \overset{d}{\to} \Omega^{1} \overset{d^{-}}{\to} \Omega^{2-},$$ while in a neighborhood of the South pole (${ c_{\theta} }= -1$), the symbol is equivalent to the elliptic symbol of the standard self-dual complex $(d,d^{+})$ $$\label{eq:sd-complex}
\Omega^{0} \overset{d}{\to} \Omega^{1} \overset{d^{+}}{\to} \Omega^{2+}.$$ Intuitively one can see that from the structure of the $QV$-action .
In the elliptic case, one could use Atiyah-Bott formula [@MR0212836; @MR0232406] to compute the index as a sum of local contributions from $H$-fixed points on $X$. In the transversally elliptic case the situation is more complicated. By definition, the index is a sum of characters of irreducible representations. We have $$\label{eq:index-u-1-general}
\ind (D) = \sum_{n=-\infty}^{\infty} a_n q^n,$$ where $a_n = m_n^{(0)} - m_n^{(1)}$ is a difference of multiplicities in which irreducible representation $n$ appears in the kernel and cokernel of $D$. In the elliptic case, only a finite number of $a_n$ does not vanish, so that the index is a finite polynomial in $q$ and $q^{-1}$. This also means that the index is a regular function on the group $H=\U(1)$. In the transversally elliptic case, the series (\[eq:index-u-1-general\]) can be infinite, so that index is generally not a regular function. However, Atiyah and Singer showed [@MR0482866; @MR0341538] that in the tranversally elliptic case, all coefficients $a_n$ are finite, and that the index is well defined as a distribution (a generalized function) on the group.
For example, consider the zero operator acting on functions on a circle $X=S^{1}$, so $D: C^{\infty}(S^1) \to 0$. This is a transversally elliptic operator with respect to the canonical $\U(1)$ action on $S^1$. The kernel of the zero operator is the space of all functions on $S^1$, the cokernel is zero. Then $m_n^{(0)} = 1, m_{n}^{(1)}=0$ for all $n$, so the index is $\sum_{n=-\infty}^{\infty}{q^n}$, which is the Dirac delta-function supported at $q=1$.
The equivariant index theory can be generalized to the transversally elliptic case [@MR0482866; @MR0341538; @MR1369411; @MR1288997]. The idea is that we can cut a $H$-manifold $X$ into small neighborhoods of $H$-fixed points and the remaining subspace $Y \subset X$ on which $H$ acts freely. By definition, at each $H$-fixed point the symbol of transversally elliptic operator is actually elliptic, so the ordinary equivariant index theory applies. Since $H$ acts freely on $Y$, we can consider the quotient $Y/H$. A $H$-transversally elliptic operator on $Y$ gives us an elliptic operator on $Y/H$. Then we can combine the representation theory of $G$ and the usual index theory on the quotient $Y/H$ to find the index of transversally elliptic operator on $Y$ [@MR0482866].
Let $R(H)$ be the space of regular functions on $H$ (the space of finite polynomials in $q$ and $q^{-1}$). Let $\CalD'(H)$ denote the space of distributions (generalized functions) on $H$ (not necessarily finite series in $q$ and $q^{-1}$). The space of distributions $\CalD'(H)$ is a module over the space of regular functions $R(H)$, since there is a well defined term by term multiplication of series in $q$ and $q^{-1}$ by finite polynomials in $q$ and $q^{-1}$. Some singular generalized functions such as the Dirac delta-function $\sum_{n=-\infty}^{\infty} q^{n}$ can be annihilated by non-zero regular functions. For example, Dirac delta-function $\sum_{n=-\infty}^{\infty} q^{n} \in \CalD'(H)$ vanishes after multiplication to $(1-q)$. Such elements of $\CalD'(H)$ which can be annihilated by non-zero regular functions in $R(H)$ are called torsion elements.
To find the index of transversally elliptic operator up to a distribution supported at $q=1$ (a torsion element of $\CalD'(H)$), we can use the usual Atiyah-Bott formula [@MR0232406; @MR0212836; @MR0190950] (see appendix ). This formula gives a contribution to the index from each fixed point as a rational function of $q$. This function is generally singular at $q=1$. For example, if $H=\U(1)$ acts on $\BC$ as $z \to q z$, then the Atiyah-Bott formula for the index of the $\bar \p$-operator at the fixed point $z=0$ gives $$\label{eq:Daulbealt-example}
\ind ( \bar \p)|_0 = \frac {1} {1 - q^{-1}}.$$
To get a distribution associated with this rational function, we need to expand it in series in $q$ and $q^{-1}$. Of course, the result is not unique, but different expansions differ only by a distribtuion supported at $q=1$. For $H=\U(1)$, there are two basic ways, or regularizations, which fix the singular part [@MR0482866]. The regularization $[f(q)]_+$ is defined by taking expansion at $q=0$. This gives us a series infinite in positive powers of $q$. The regularization $[f(q)]_-$ is defined by taking expansion at $q=\infty$. This gives us a series infinite in negative powers of $q$. These two regularizations differ by a distribution supported at $q=1$. For example, for the $\bar \p$-operator we get as the difference the Dirac delta-function $[ (1-q^{-1})^{-1}]_+ - [(1-q^{-1})^{-1}]_- = - \sum_{n=-\infty}^{n=\infty} q^{n}$.
Let $X=\BC^n$ be a $H=\U(1)$ module with positive weights $m_1, \dots, m_n$, so that $\U(1)$ acts as $z_i \to q^{m_i} z_i$, and let $Y=\{0\}$ be the $H$-fixed point set. Let $v$ be the vector field generated by the $\U(1)$ action on $X$. Let $\sigma(D)$ be an elliptic symbol defined on $T^* X |_Y$, i.e. defined on the fiber of the cotangent bundle to $X$ at the origin. Atiyah showed [@MR0482866] that we can use the vector field $v$ in two different ways, called $[\cdot ]_{+}$ and $[\cdot]_{-}$, to construct a transversally elliptic symbol $\tilde \sigma=[\sigma]_{\pm}$ on the whole space $T_{H}^* X$ such that $\tilde \sigma$ is an isomorphism outside of the origin $Y$. (If $(x,p)$ are coordinates on $T^*X$, then, loosely speaking, we take $\tilde\sigma(x,p) = \sigma(0,p \pm v)$. See appendix \[se:trans-elliptic\] for more precise details). Then the index of the transversally elliptic symbol $\tilde \sigma$ is well defined as a distribution on $H$. Moreover, if $\ind(\sigma)$ is a rational function of $q$ associated at the fixed point $Y$ to the elliptic symbol $\sigma$ by Atiyah-Bott formula, then $$\label{eq:atiyha-trans-elliptic}
\ind([\sigma]_\pm) = [\ind(\sigma)]_{\pm}.$$
We apply this procedure to our problem. Namely, we use the vector field generated by the $H=\U(1)$-action on $X=S^4$ to trivialize the symbol $\sigma(D_{10}^{vect})$ everywhere on $T^*_H
X$ except at the North and the South pole. Then the index is equal to the sum of contributions from the fixed points, where each contribution is expanded in positive or negative powers of $q$ according to the (\[eq:atiyha-trans-elliptic\]). More concretely, we trivialize the transversally elliptic symbol $\sigma = \sigma(D_{10}^{vect})$ everywhere outside the North and the South poles on $T^*_H X$ by replacing ${ c_{\theta} }p_1$ by ${ c_{\theta} }p_1 + v$ on the diagonal in (\[eq:symbol-simplified\]), where $v = \sin \theta$. In other words, we deform the operator by adding the Lie derivative in the direction of the vector field $v$. The resulting symbol $$\label{eq:symbol-simplified-new}
\tilde \sigma=
\begin{pmatrix}
{ c_{\theta} }p_1 + { s_{\theta} }& p_2 & p_3 & p_4 \\
-p_2 & { c_{\theta} }p_1 + { s_{\theta} }& -p_4 & p_3 \\
-p_3 & p_4 & { c_{\theta} }p_1 +{ s_{\theta} }& -p_2 \\
-p_4 & -p_3 & p_2 & { c_{\theta} }p_1 + { s_{\theta} }\end{pmatrix}.$$ has determinant $(\vect{p}^2 + ({ c_{\theta} }p_1 + { s_{\theta} })^2)^2$ which is non-zero everywhere outside the North and the South poles at $T^{*}_{H} X$. (To check this, take $p_1 = 0$ and ${ s_{\theta} }> 0$.) The index of $\tilde \sigma$ is equal to the index of $\sigma$, since $\tilde \sigma$ is a continous deformation of $\sigma$. On the other hand, since $\tilde \sigma$ is an isomorphism outside of the North and the South pole, to get the index of $\tilde \sigma$ we sum up contributions from the North and the South pole. At the North pole $\cos \theta = 1$. Therefore, in a small neighborhood of the North pole, the transversally elliptic symbol $\tilde \sigma$ coincides with the symbol associated to the elliptic symbol $\tilde \sigma_{\theta =0}$ by the $[\cdot ]_{+}$ regularization. At the South pole $\cos \theta = -1$. Therefore, in a small neighborhood of the South pole, the transversally elliptic symbol $\tilde \sigma$ coincides with the symbol associated to the elliptic symbol $\tilde \sigma_{\theta =\pi}$ by the $[ \cdot ]_{-}$ regularization.
Finally we obtain $$\ind'(D_{10}^{vect}) = \left [ \ind (d,d^{-}) |_{\theta=0} \right ]_+ +
\left [ \ind (d,d^{+}) |_{\theta=\pi} \right ]_- .$$
One could probably also derive this result following the procedure in [@MR792703], where the index theorem for the Dirac operator was obtained using the deformation $\Gamma^{\mu} D_{\mu} \to \Gamma^{\mu}D_{\mu}
+ t \Gamma^{\mu} v_{\mu}$.
Let $z_1, z_2$ be complex coordinates in a small neighboorhod of the South pole, such that the $\U(1)$ action is $z_1 \to q z_1, z_2 \to q z_2$. With respect to this action the complexified self-dual complex is isomorphic to the Dolbeault $\bar \p$-complex twisted by the bundle $\CalO \oplus \Lambda^2 T^*_{1,0}$. Using the fact that the index of $\bar \p$ operator is $(1-q^{-1})^{-2}$, we get $$\ind'(D_{10}^{vect}) = \left [ - \frac {1+q^2} {(1-q)^2}\right]_{+} +
\left [ - \frac {1+q^2} {(1-q)^2}\right]_{-},$$ where $[f(q)]_\pm$ respectively means to take expansion of $f(q)$ in positive or negative powers of $q$. In our conventions $E_0$ corresponds to the middle term of the standard (anti)-self dual complex (\[eq:sd-complex\]), therefore we get an extra minus sign.
Finally, $$\begin{gathered}
\ind (D^{vect}_{10}) = 2+\ind'(D_{10}^{vect}) = \\= 2 -(1+q^2)(1+2q+3q^2+\dots) - (1+q^{-2})(1+2q^{-1} + 3q^{-2} + \dots) = \\ = -\sum_{n=-\infty}^{\infty} | 2n | q^{n}.\end{gathered}$$
Note that in the computation of the index for the vector multiplet, the chirality of the complex coincides with the chirality of the $\U(1)$ rotation near each of the fixed points.
Now we proceed to the hypermultiplet contribution to the index. The computation is similar to the vector multiplet. The transversally elliptic operator $D_{10}^{hyper}: \Gamma(E_0^{hyper}) \to \Gamma(E_1^{hyper})$ can be trivialized everywhere over $T^*_G X$ except fixed points, where it is isomorphic to the self-dual complex at the North pole, or anti-self-dual complex at the South pole. For the hypermultiplet the chirality of the complex is opposite to the chirality of the $\U(1)$ rotation near each of the fixed points. Then, using that the index of the twisted Dolbeault operator is $(1+q
q^{-1})/((1-q)(1-q^{-1}))$, we get $$\ind_q (D^{hyper}_{10}) = \left [ - \frac {2} {(1-q)(1-q^{-1})}\right]_{+} +
\left [ - \frac {2} {(1-q)(1-q^{-1})}\right]_{-},$$ which results in $$\ind_q (D^{hyper}_{10}) = +\sum_{n=-\infty}^{\infty} |2n| q^{-n}.$$
So far we considered the massless hypermultiplet. In this case its contribution to the index exactly cancels the vector multiplet. Hence, the determinant factor in the $\CalN=4$ theory is trivial. This finishes the proof that the Erickson-Semenoff-Zarembo matrix model is exact in all orders of perturbation theory.
In the $\CalN=2^{*}$ case the situation is more interesting. Now the hypermultiplet is massive. In the transformations the action of $R$ is contributed by the $\SU(2)^R_R$ generator $M_{ij}$. We normalize it as $M_{ij}
M^{ij} = 4 m^2$. The hypermultiplet fields transform in the spin-$\frac 1 2$ representation of $\SU(2)^R_R$. Therefore, in the massive case the index is multiplied by the spin-$\frac 1 2$ character relative to the massless case: $\frac 1 2 (e^{im}
+ e^{-im})$. Hence all $\U(1)$-eigenspaces split into half-dimensional subspaces with eigenvalues shifted by $\pm m$.
Finally, all fields transform in the adjoint representation of gauge group. Making a constant gauge transformation we can assume that the generator $a_0$ is in the Cartan subalgebra of the Lie algebra $\g$ of the gauge group. Then non-zero eigenvalues of $a_0$ in the adjoint representation are $\{\alpha\cdot a_0 \}$, where $\alpha$ runs over all roots of $\g$. Hence, combining all contributions to the index, we obtain for the $\CalN=2^*$ theory $$\left( \frac {\det K_{bos}} { \det K_{ferm}} \right)_{\CalN=2^*} = \prod_{\text{roots }\alpha}
\prod_{n=-\infty}^{\infty} \left [ \frac { (\alpha\cdot a_0 + n \ve + m)(\alpha \cdot a_0
+ n \ve - m)} {( \alpha \cdot a_0 + n\ve)^2 } \right ]^{|n|}.$$ Here we denote $\ve = r^{-1}$. The term $n\ve$ comes from a weight $n$ representation of the $\U(1)$, the term $\alpha \cdot a_0$ is an eigenvalue of $a_0$ acting on the eigensubspace of the adjoint representation corresponding to root $\alpha$.
We argued before that to ensure convergence of the path integral the mass parameter and the scalar field $\Phi_0$ should be taken imaginary if we work with ten-dimensional Minkowski signature. The parameter $a_0$ is also imaginary since it is identified with the zero mode of $\Phi_0$. Let us denote $m = i m_E, a_0 = i a_E \equiv i a^E_0$. Then, recalling (\[eq:Gauss-Z-1-loop\]) we get $$\label{eq:Z-1-loop-N=2}
Z_{\text{1-loop}}^{\CalN=2^*}(ia_E) = \prod_{\text{roots }\alpha} \prod_{n=1}^{\infty} \left [
\frac { ( (\alpha \cdot a_E)^2 + \ve^2 n^2)^2 } { ((\alpha \cdot a_E + m_E)^2 + \ve^2 n^2)
(( \alpha \cdot a_E - m_E)^2 + \ve^2 n^2 )} \right ]^{\frac n 2}.$$ This product requires some regularization which we explain in a moment.
Recall the product formula for the Barnes $G$-function (see e.g. [@adamchik-2003]) $$G(1+z) = (2\pi)^{z/2} e^{-((1+ \gamma z^2) +z)/2} \prod_{n=1}^{\infty} \left(1 + \frac z n \right)^n e^{-z +
\genfrac{}{}{}{1}{ z^2}{2n}},$$ where $\gamma$ is the Euler constant. Then we introduce a function $H(z) = G(1+z) G(1-z)$ and obtain $$H(z) = e^{-(1+\gamma) z^2} \prod_{n=1}^{\infty} \left(1 - \frac {z^2}{n^2}\right)^{n} \prod_{n=1}^{\infty}
e^{\genfrac{}{}{}{1}{z^2} {n}}.$$ Using this relation we obtain formally $$\begin{gathered}
\label{eq:Z-Barnes-hyper}
Z^{\CalN=2^*}_{\text{1-loop}}(ia_E) =
\exp\left( \frac {m_E^2}{ \ve^2} \left(
(1+\gamma) - \sum_{n=1}^{\infty} \frac {1} {n} \right)\right) \times \\
\times \prod_{\text{roots }\alpha} \frac { H \left(i\alpha \cdot a_E/ \ve\right)}
{ \left[ H\left((i \alpha \cdot a_E+ i m_E) / \ve \right) H\left((i\alpha \cdot a_E-i m_E) / \ve \right) \right]^{1/2}}. \end{gathered}$$ The first factor $\exp(\dots)$ is divergent, but it does not depend on $a_E$. Therefore it cancels when we compute expectation value of the operators which localize to functions of $a_E$, such as the circular supersymmetric Wilson loop operator. Therefore we can remove this factor from the partition function. The resulting product of the $G$-functions is a well defined analytic function of $a_E$.
Our result is consistent with the renormalization properties of the gauge theory. To check that the $\beta$-function comes out right, we need asymptotic expansion of the $G$-function at large $z$ $$\log G(1+z) = \frac 1 {12} - \log A + \frac {z} 2 \log 2 \pi + \left( \frac {z^2} 2 - \frac 1 {12} \right) \log z
- \frac 3 4 {z^2} + \sum_{k=1}^{\infty} \frac {B_{2k+2}} {4k(k+1)z^{2k}},$$ where $A$ is a constant and $B_{n}$ are Bernoulli numbers. Then $$\label{eq:asymptotic-Barnes}
\frac 1 2 \left(\log G(1+iz_E) + \log G(1-iz_E)\right) =
\frac 1 {12} - \log A + \left(-\frac {z_E^2} {2} - \frac {1} {12}\right) \log z_E + \frac 3 4 z_E^2 + \dots$$
If we take a limit of very large mass of the hypermultiplet, we expect to get the minimal $\CalN=2$ theory at the energy scales much lower then the mass of the hypermultiplet. At large $m$, we expand the denominator in , corresponding to the hypermultiplet contribution to $Z_{\text{1-loop}}$, and get $$Z_{\text{1-loop}}^{hyper} = const(m_E) + \left(const + \log \frac {m_E} {\ve} \sum_{\alpha}
\frac {(\alpha \cdot a_E)^2}{\ve^2}\right) + \CalO(\frac 1 {m^2}).$$ The important dependence on $a_E$ can be combined with the classical Gaussian action in the matrix model $$\label{eq:renorm-g}
\frac {4 \pi^2 r^2 }{g_{YM}^2} (a_E,a_E) \to \left( \frac {4 \pi^2 r^2 }{g_{YM}^2} - \frac {C_2}{\ve^2} \log {\frac {m_E}\ve} \right) (a_E,a_E),$$ where $C_2$ denotes the proportionality constant of the second Casimir $\tr_{\mathrm{Ad}} T_{a} T_{b} = C_{2} \delta_a \delta_b$. We can write that as $$\label{eq:running}
\frac 1 {\tilde g^2_{YM}} = \frac 1 {g^2_{YM}} - \frac{C_2}{4\pi^2} \log \frac {m_E} \ve$$ where $\tilde g^2_{YM}$ has a simple meaning of the renormalized coupling constant. In other words, the bare microscopical constant $g^2_{YM}$ is defined at the UV scale $m_E$ and higher (in that region it does not run because of restored $\CalN=4$ supersymmetry). At scales less than $m_E$, the coupling constant runs by beta-function of pure $\CalN=2$ theory. Recall that the one-loop beta function for a gauge theory with $N_{f}$ Dirac fermions and $N_{s}$ complex scalars in adjoint representation is $$\frac {\p g(\mu)}{\p \log \mu} = \beta(g) = -\frac {C_2 g^3}{(4\pi)^2}\lb \frac {11}{3} - \frac 4 3 N_{f} - \frac 1 3 N_{c} \rb.$$ Taking $N_{f} = N_{s} = 1$ for a pure $\CalN=2$ theory we get precisely the relation , which says that $\tilde g^2_{YM}$ is the running coupling constant at the IR scale $\ve=r^{-1}$, which is the lowest scale for the theory on $S^4$ of radius $r$. This is also the scale of the Wilson loop operator, since it is placed on the equator.
We can check that the resulting integral over $a_E$ is always convergent as long as the bare coupling constant $g_{YM}^2$ is positive, in other words as long as the original action is positive definite. First of all, the Barnes function $G(1+z)$ does not have poles or zeroes on the imaginary contour $\Re z = 0$ over which we integrate. To see that the integral also behaves nicely at infinity we use the asymptotic expansion .
In the pure $\CalN=2$ case the leading term in the exponent comes from the numerator of $Z_{\text{1-loop}}$ and is equal to $-\frac 1 2 z_E^2 \log z_E$. This is a negative function which grows in absolute value faster than any other terms including the renormalized quadratic term even if $\tilde g^2_{YM}$ formally becomes negative.
In the $\CalN=2^*$ case we need to take asymptotic expansion at large $z_E$ of both the numerator and denominator of to check convergence at infinity. The leading terms $(\alpha \cdot a_E)^2 \log
(\alpha \cdot a_E)$ cancel, and the next order term is proportional to $m^2_E \log
(\alpha \cdot a_E)$. This does not spoil the convergence insured by the Gaussian classical factor $\exp(-\frac{4\pi^2 r^2}{g^2_{YM}} (a_E,a_E))$.
To summarize, in the pure $\CalN=2$ theory we need to insert the factor $$\label{eq:Z-1-loop-N-2}
Z_{\text{1-loop}}^{\CalN=2} = \prod_{\text{roots }\alpha} H \left( {i\alpha \cdot a_E}/ {\ve}\right),$$ under the integral in the matrix model and to substitute $g_{YM}$ by the renormalized coupling constant $\tilde g_{YM}$ in the Gaussian classical action.
When we set $m=0$ we get the $\CalN=4$ theory. The numerator coming from the vector multiplet exactly cancels the denominator coming from the hypermultiplet in the formula (\[eq:Z-Barnes-hyper\]) and we get $$\label{eq:Z-1-loop-N-4}
Z_{\text{1-loop}}^{\CalN=4} = 1.$$
We shall note that most of the above computations are generalized easily for the $\CalN=2$ theory with a massless hypermultiplet taken in an arbitrary representation. Let us denote this represenation by $W$. Analogously to the case of the adjoint representation, one can get a formula $$\label{eq:Z-1-loop-any-matter}
Z_{1-loop}^{\CalN=2, W}(ia_E) = \frac{ \prod_{\alpha \in \text{weights}(\text{Ad})} H(i\alpha \cdot a_E /\ve) }
{ \prod_{w \in \text{weights}(W)} H(iw\cdot a_E/\ve) }.$$ Strictly speaking, this formula is valid in the situations when the infinite product of weights for the vector multiplet and hypermultiplet is proportional to the product of Barnes $G$-functions with the same divergent factor. That happens for such representations $W$ when $\sum_{\alpha} (\alpha \cdot a)^2 = \sum_{w} (w \cdot a)^2$ for any $a \in \g$. This is actually the condition of vanishing $\beta$-function for the $\CalN=2$ theory with a hypermultiplet in representation $W$. Therefore we claim that the formula (\[eq:Z-1-loop-any-matter\]) literally holds for all $\CalN=2$ superconformal theories. In a general $\CalN=2$ case, the one-loop determinant requires regularization similarly to what we did for the pure $\CalN=2$ theory.
It would be interesting to combine the factor $Z_{1-loop}$ with the partition function of instanton corrections $|Z_{inst}|^2$ in an arbitrary $\CalN=2$ superconformal case, integrate over $a_E$ and check predictions of the $S$-duality for these theories (see e.g.[@Argyres:2007cn; @Kapustin:2006hi]).
Example
-------
Before turning to the instanton corrections, let us give a simplest example of a non-trival prediction of the formula (\[eq:Z-1-loop-any-matter\]), which perhaps can be checked using the traditional methods of the perturbation theory.
Take the $\CalN=2$ theory with with the $\SU(2)$ gauge group and 4 hypermultiplets in the fundamental representation. We choose coordinate $a$ on the Cartan subalgebra of the real Lie algebra of the gauge group $\SU(2)$, such that an element $a$ is represented by an anti-hermitian matrix $diag(ia,-ia)$. Let the invariant bilinear form on the Lie algebra be minus the trace in the fundamental representation, and let the kinetic term of the Yang-Mills action be normalized as $\frac {1} {4 g_{YM^2}} \int d^4 x \sqrt{g} (F_{\mu \nu},F^{\mu \nu})$. The weights $w$ in the spin-$j$ representation run from $-2j$ to $2j$. In the adjoint representaton ($j=1$) we have $\{\alpha \cdot a \} = \{-2a,0,2a\}$. In the fundamental representation ($j=\frac 1 2$) we have $\{w \cdot a \} = \{a,-a\}$. We also have $(a,a) = 2a^2$. The matrix model for the expectation value of the Wilson loop in the spin-$j$ representation is $$\langle \tr_j \Pexp (\int Adx + i\Phi_0 ds) \rangle = Z^{-1} \int_{-\infty}^{\infty} da e^{ - \frac {8 \pi^2} {g_{YM}^2} a^2 } (2a)^2 \frac { H(2ia) H(-2ia)} { (H(ia) H(-ia))^4} ( \sum_{m=-j}^{j} e^{4\pi m a} ),$$ where $Z$ is a constant independent of the inserted Wilson loop operator. The extra factor $(2a)^2$ is the usual Vandermonde determinant appearing when we switch to the integral over the Cartan subalgebra from the integral over the whole Lie algebra. At the weak coupling $g_{YM} \to 0$ we can evaluate this integral as a series in $g_{YM}$. For the Barnes G-function we use Taylor series expansion at small $z$ $$\log G(1+z) = \frac 1 2 (\log (2 \pi) -1) z - (1+ \gamma) \frac {z^2} {2} + \sum_{n=3}^{\infty} (-1)^{n-1} \zeta(n-1)
\frac{z^n} {n}.$$ After some algebra one gets the following perturbative result for the expectation value of $e^{2\pi n a}$ in the matrix model (we write here $g=g_{YM}$) $$\label{se:first-corrections}
\langle e^{2\pi n a} \rangle = 1 + \frac 3 {2 \cdot 2^2} n^2 g^2 + \frac 5 {8\cdot 2^4} n^4 g^4
+ \frac {7} {48 \cdot 2^6} n^6 g^6 + \frac{35} {2^4 (4\pi)^2} t_2 n^2 g^6 + O(g^8),$$ where $t_2$ is the coefficient coming from the expansion of the Barnes $G$-function. It is expressed in terms of Riemann zeta-function $$t_2 = -12 \zeta(3).$$ To get this result we expanded the determinant factor in powers of $a$: $$\log \lb \frac { H(2ia) H(-2ia)} { (H(ia) H(-ia))^4} \rb = -8 \sum_{k=2}^{\infty} \frac {\zeta(2k-1)} {k} (2^{2k-2} -1)
(-1)^k a^{2k} =: \sum_{k=2}^{\infty} t_{k} a^{2k}.$$ Then for a Gaussian measure $\int da \, e^{-\frac {1} {2 \sigma^2} a^2 }$ with $\sigma^2 = \frac {g^2_{YM}}{16 \pi^2}$ we have
$$\left \langle a^2 \exp \left (\sum t_{k} a^{2k} \right) e^{qa} \right \rangle_{\text{gauss}} =
\left (\frac{\p} {\p q} \right)^2 \exp\left( \sum t_k
\left( \frac {\p} {\p q} \right)^k \right) e^{\frac{1}{2} q^2 \sigma^2}.$$
The perturbative result for the $\CalN=4$ $\SU(2)$ theory is given by the same formula but with $t_k = 0$: $$\langle e^{qa} \rangle_{\CalN=4} = (1 + \sigma^2 q^2) \exp ( \frac 1 2 \sigma^2 q^2 )=1+\frac 3 2 ( \sigma q)^2 + \frac 5 8
(\sigma q)^4 + \frac {7} {48} ( \sigma q)^6 + O((\sigma q)^8).$$
Taking $q = 2 \pi n$ and $\sigma = \frac{g_{YM}}{4 \pi}$ we get the result (\[se:first-corrections\]) for the $\CalN=4$ theory with $t_2=0$. For a superconformal $\CalN=2$ theory the Gaussian matrix model action is corrected by the terms $t_{k} a^{2k}$. The first correction is quartic $t_2 a^4$, and at the lowest order it gives the result (\[se:first-corrections\]) for the $\SU(2)$ theory with 4 hypermultiplets in the fundamental representation.
The first difference for $\langle W_{R} (C) \rangle$ between the $\CalN=2$ $\SU(2)$ gauge theory with 4 fundamental hypermultiplets and the $\CalN=4$ $\SU(2)$ gauge theory appears at the order $g_{YM}^6$. This is the order of the two-loop level Feynman diagram computations which have been done in the gauge theory for the $\CalN=4$ case [@Plefka:2001bu; @Arutyunov:2001hs].
In the matrix model it is very easy to get the higher terms in the expansion over $g_{YM}$. On the other hand, the complexity of the Feynman diagram computations done directly in the gauge theory grows enormously with the number of loops.
Now we will argue that we can improve the matrix model by taking into account all instanton corrections of the theory, so that the result becomes non-perturbatively exact.
Instanton corrections
=====================
When we argued by that the theory localizes to the trivial gauge field configurations, we used the fact that $d_{[\lambda}w_{\mu
\nu]}$ does not vanish everywhere except at the North and the South poles and we assumed smooth gauge fields. Dropping the smoothness condition, we can only say that the gauge field strength must vanish everywhere away from the North and South poles. If we allow field configurations like Dirac-delta function, then the gauge field strength can be supported at the poles and still be consistent with vanishing $tQV$-term. From we see that $F^{+}$ might be non zero at the North pole, where $\sin^2 \frac \theta 2$ vanish, while $F^{-}$ might be non zero at the South pole, where $\cos^2 \frac \theta 2$ vanish. Thus, if we allow non-smooth gauge fields in the path integral, we should count configurations with point anti-instantons ($F^{-}=0$) localized at the North pole, and point instantons ($F^{+}=0$) localized at the South pole. The $Q$-complex on $S^4$ in our problem in a neighborhood of the South/the North pole coincides with the $Q$-complex of the topological ($F^{+}=0$)/anti-topological ($F^-=0$) gauge theory on $\BR^4$ in the $\Omega$-background studied by Nekrasov [@Nekrasov:2002qd]. There the moduli space of solutions to $F^{+}=0$ modulo gauge transformations was taken equivariantly under the $\U(1)^2$ action on $\BR^4 \simeq \BC^2$ by $z_{1} \to
e^{i\ve_1} z_1, z_{2} \to e^{i \ve_2} z_2$, and gauge transformations at infinity with generator $a \in \g$. Making the correspondence between the theory on $S^4$ in a local neighborhood of the North pole and the theory on $\BR^4$ we should take $\ve_{1} = \ve_{2} = r^{-1}$, since for the problem on $S^4$, the chirality of the equations at the North pole coincides with the chirality of the generator of the Lorentz rotations $d_{[\mu} v_{\nu]}$. The same applies to the South pole: the chirality of the equations is reversed as well as the chirality of the generator of the Lorentz rotations.
In this section we consider only the case of the $\U(N)$ gauge group. We use the following conventions. The solutions of the equations $F^{+}=0$ are called instantons. The solutions of the equations $F^{-} = 0$ are called anti-instantons.
We define the instanton charge as the second Chern class[^7] $$k = c_2 = - \frac {1} {8 \pi^2} \int F \wedge F,$$ and modify the action by the $\theta$-term $$S_{YM} \to S_{YM} + \frac {i \theta} {8 \pi^2} \int F \wedge F.$$ At $F^{+}=0$ we have $ \sqrt{g} F_{\mu \nu}F^{\mu \nu} d^4 x = 2 F \wedge *F = -2 F \wedge F$. Then the Yang-Mills action of instanon of charge $k$ is $$S_{YM}(k) = \frac 1 {4 g_{YM}^2} \int \sqrt{g} d^4 x F_{\mu \nu} F^{\mu \nu} + \frac {i \theta} {8\pi^2} \int
F \wedge F = \lb \frac {4\pi^2} { g_{YM}^2} - {i \theta} \rb k.$$ Its contribution to the partition function is proportional to $$\exp (-S_{YM}(k)) = \exp \lb 2 \pi i \tau k \rb = {q}^{k},$$ where we introduced the complexified coupling constant $$\tau = \frac {2 \pi i} {g_{YM}^2 } + \frac {\theta}{ 2\pi},$$ and the expansion parameter $${q}=\exp(2\pi i\tau).$$ (The expansion parameter $q$ in this section should not be confused with the formal generator of the $\U(1)$ group used to compute the index of the transversally elliptic operator in the previous section).
Near the South pole the theory on $S^4$ looks like topological theory with the equations $F^{+}=0$, so that only point instantons contribute. Near the North pole the situation is opposite: the equations are replaced by $F^{-}=0$, therefore we need to count anti-instantons. The generating function of anti-instantons is the same as the generating function of instantons with replacement of the expansion parameter $q$ by its complex conjugate $\bar q$.
For the $\U(N)$ gauge group the explicit formula for the equivariant instanton partition function on $\BR^4$ is [@Nekrasov:2002qd; @Nekrasov:2003rj; @Flume:2002az; @Bruzzo:2002xf; @nakajima-inst; @nakajima-lectures] $$\label{eq:Z-inst}
Z_{\text{inst}}^{\CalN=2}(\ve_1,\ve_2,a) = \sum_{\vect{Y}} \frac {{{q}}^{|Y|}} {\prod_{\alpha, \beta=1}^{N}
n^{\vect{Y}}_{\alpha, \beta} (\ve_1,\ve_2, \vect{a})},$$ where we sum over an ordered set of $N$ Young diagrams $\{Y_\alpha\}$ with $\alpha = 1\dots N$. By $|\vect{Y}|$ we denote the total size of all diagrams in a set $|\vect{Y}|=\sum{|Y_{\alpha}|}$. The total size is equal to the instanton number. The factor $n_{\alpha, \beta}^{\vect{Y}}(\ve_1, \ve_2, \vect{a})$ denotes the equivariant Euler class of the tangent space to the instanton moduli space at the fixed point labeled by $\vect{Y}$. It is given by $$\begin{gathered}
n^{\vect{Y}}_{\alpha, \beta}(\ve_1, \ve_2, \vect{a}) = \prod_{s \in
Y_{\alpha}} ( - h_{Y_{\beta}}(s) \ve_1 + (v_{Y_{\alpha}}(s)+1)\ve_2 +
a_{\beta} - a_{\alpha}) \times \\ \times \prod_{t \in Y_{\beta}}
((h_{Y_{\alpha}}(t) + 1 ) \ve_1 - v_{Y_{\beta}}(t) \ve_2 + a_{\beta} -
a_{\alpha}).\end{gathered}$$ (We assume that an element $a$ in the Cartan subalgebra of $\mathfrak{u}(N)$ is represented by a diagonal matrix $(ia_{1}, \dots, ia_{N})$.) Here $s$ and $t$ run over squares of Young diagrams $Y_{\alpha}$ and $Y_{\beta}$. Let $Y$ is a Young diagram $\nu_{1} \geq \nu_{2} \dots \geq \nu_{\nu_{1}'}$, where $\nu_{i}$ is the length of the $i$-th column, $\nu_{j}'$ is the length of the $j$-th row. If a square $s=(i,j)$ is located at the $i$-th column and the $j$-th row then $v_{Y}(s) = \nu_{i}(Y) - j$ and $h_{Y}(s)=\nu_{j}'(Y) - i$. In other words, $v_{Y}(s)$ and $h_{Y}(s)$ is respectively the vertical and horizontal distance from the square $s$ to the edge of the diagram $Y$. We can rewrite the product in the denominator of as $$\prod_{\alpha, \beta=1}^{N}
n^{\vect{Y}}_{\alpha, \beta} (\ve_1,\ve_2, \vect{a}) = \prod_{\alpha,\beta=1}^{N} \prod_{s \in Y_{\alpha}}
E_{\alpha\beta}(s)
(\ve_1 + \ve_2 - E_{\alpha\beta}(s)),$$ where $$E_{\alpha\beta} (s)= (-h_{Y_{\beta}}(s) \ve_1 + (v_{Y_{\alpha}}(s) + 1)\ve_2 + a_{\beta} - a_{\alpha}).$$ We will give a few simplest examples of evaluation of this formula. First consider $\U(1)$ case. Then we sum over all Young diagrams of one color. At one instanton level $k=1$, there is only one diagram $Y=(1)$. Then $E_{11}=\ve_2$, so that $$Z^{\CalN=2}_{k=1}(\ve_1,\ve_2,a) = \frac {1}{ \ve_2 \ve_1}.$$ At two instanton level $k=2$, there are two diagrams $Y=(2,0)$ and $Y=(1,1)$. Their contribution is $$Z^{\CalN=2}_{k=2}(\ve_1,\ve_2,a_1) = \frac{1}{(2\ve_2)(\ve_1 - \ve_2)(\ve_2)(\ve_1)} +
\frac {1} {(-\ve_1 + \ve_2)(2 \ve_1) (\ve_2) (\ve_1)} = \frac {1}{2(\ve_1 \ve_2)^2}$$ At three instanton level $k=3$, there are three diagrams $Y=(3,0)$, $Y=(2,1)$ and $Y=(1,1,1)$. Their contribution is $$\begin{gathered}
Z^{\CalN=2}_{k=3}(a,\ve_1,\ve_2) = \frac {1} { (\ve_2)(\ve_1) (2 \ve_2) (\ve_1 - \ve_2) (3\ve_2) (\ve_1 - 2 \ve_2)} + \\
+ \frac {1} { (\ve_2)(\ve_1) (2\ve_2 -\ve_1) (2 \ve_1 - \ve_2) (\ve_2) (\ve_1)} +\frac {1} { (\ve_2) (\ve_1) (\ve_2 - \ve_1)(2\ve_1) (\ve_2 - 2\ve_1) (3\ve_2)} = \\
= \frac {1} {6 (\ve_1 \ve_2)^3}\end{gathered}$$ At an arbitrary instanton level $k$, the sum of all Young diagrams of order $k$ simplifies to $$Z^{\CalN=2}_{k}(\ve_1,\ve_2,a) = \frac {1} { k! (\ve_1 \ve_2)^k},$$ hence $$Z^{{\CalN=2}}_{\U(1)}(\ve_1,\ve_2,a) = \sum_{k=1}^{\infty} \frac {{q}^k}{k! (\ve_1\ve_2)^k} =
\exp\left(\frac {{q}}{\ve_1 \ve_2}\right).$$ Now we consider a few instantons for the $\U(2)$ gauge group. At one instanton there are two colored Young diagrams $((1),0)$ and $(0,(1))$ contributing $$\begin{gathered}
Z^{\CalN=2}_{k=1}(\ve_1,\ve_2,a_1,a_2) = \frac {1} { \ve_1 \ve_2 (a_2 - a_1 +
\ve_1 + \ve_2)(a_1 - a_2) }
+ \frac {1} { (a_1 - a_2 + \ve_1 + \ve_2)(a_2 - a_1) \ve_1 \ve_2} = \\
=\frac {2} {\ve_1 \ve_2 ( (\ve_1+\ve_2)^2 - a^2)},\end{gathered}$$ where we denoted $ a=a_2 - a_1$. As the instanton number grows, its contribution becomes more and more complicated rational function of $a_i$. For example, at $k=2$ we get (we set $a=ia_E$, where $a_E$ is real) $$Z^{\CalN=2}_{k=2}(\ve_1,\ve_2,ia_E) = \frac{ (2a_E^2 + 8 \ve_1^2 + 8 \ve_2^2 + 17 \ve_1 \ve_2) }
{ ((\ve_1+2 \ve_2)^2 + a_E^2)((2\ve_1+ \ve_2)^2 + a_E^2)((\ve_1 + \ve_2)^2 + a_E^2)\ve_1^2 \ve_2^2 }.$$ Generally, instanton contributions are certain rational functions of $a_i$ and $\ve_i$. Contrary to the case $\ve_{1}=-\ve_2 = \hbar$, which is often taken in the literature to simplify the instanton partition function [@Nekrasov:2002qd; @Nekrasov:2003rj; @nakajima-lectures], in our problem we get the same signs: $\ve_{1} = \ve_{2} = \frac {1} {r}$. Looking at the examples above, one can note an important property of the instanton contributions at $\ve_{1} = \ve_{2}$; they do not have poles at the integration contour for $a_{i}$. Recall that in the matrix integral we integrate over imaginary $a=ia_E$, while $\ve_1$ and $\ve_2$ is real. Generally, the denominator contains factors $n_1 \ve_1 + n_2 \ve_2 + a$, where $n_1$ and $n_2$ are some numbers. There is a pole at the integration contour only if $n_1 \ve_1
+ n_2 \ve_2=0$. Though it happens regularly at $\ve_1 = -\ve_2$, it never happens at $\ve_1 = \ve_2$. (This fact was checked explicitly up to $k=5$ instantons for $\U(2)$ gauge group and actually one can show it in general.[^8]) Therefore the integrand in is a smooth function everywhere at the integration domain and it also decreases rapidly at infinity. Thus the integral is convergent and well defined.
In the $\CalN=2^*$ case, each instanton contribution is multiplied by a new factor. This factor is equal to the product of the same weights as in the denominator, but shifted by the hypermultiplet mass. From [@Nekrasov:2002qd; @Nekrasov:2003rj; @Bruzzo:2002xf] we get that the instanton partition function is given by $$\label{eq:Z-inst-N-star}
Z^{\CalN=2^{*}}_{\text{inst}}(\ve_1, \ve_2,\tilde m,a) = \sum_{\vect{Y}} {q}^{|\vect{Y}|} \prod_{\alpha,\beta=1}^{N} \prod_{s \in Y_{\alpha}}
\frac { (E_{\alpha \beta}(s) -\tilde m )(\ve_1 + \ve_2 - E_{\alpha \beta}(s) -\tilde m)} { E_{\alpha \beta}(s)(\ve_1 + \ve_2 - E_{\alpha \beta}(s)) }.$$ where $\tilde m$ is the equivariant parameter used to introduce hypermultiplet mass in the Nekrasov’s theory in the $\Omega$-background [@Nekrasov:2002qd] related to the mass of hypermultiplet used in the present work as $\tilde m = m + (\ve_1 + \ve_2)/2$ (see [@Okuda:2010] for details)[^9]. For example, $$\begin{gathered}
Z^{\CalN=2^*}_{k=1}(\ve_1,\ve_2, \tilde m, a_1,a_2) = \frac { (\ve_1- \tilde m) (\ve_2-\tilde m) (a_2 -
a_1 + \ve_1 + \ve_2- \tilde m)(a_1 - a_2- \tilde m) } { \ve_1 \ve_2
(a_2 - a_1 + \ve_1 + \ve_2)(a_1 - a_2) } + \\
+ \frac { (a_1 - a_2 + \ve_1 + \ve_2- \tilde m)(a_2 - a_1- \tilde m) (\ve_1-\tilde m) (\ve_2-\tilde m) } { (a_1 - a_2 + \ve_1 + \ve_2)(a_2 - a_1) \ve_1 \ve_2} = \\
=\frac{ 2(\tilde m-\ve_2)(\tilde m-\ve_1)( \tilde m^2-a^2- \tilde m(\ve_1 + \ve_2)+(\ve_1+\ve_2)^2) } {
((\ve_1 + \ve_2)^2 - a^2) \ve_1 \ve_2 }\end{gathered}$$ The integrand is still a smooth function on the whole integration domain and decreases sufficiently fast at infinity.
Hence, we conclude that the matrix integral, with all instanton corrections included, is well defined in the $\CalN=2$, the $\CalN=2^*$ and the $\CalN=4$ cases, and that it gives the exact partition function of these theories on $S^4$. The expectation value of a supersymmetric circular Wilson operator on $S^4$ in an arbitrary representation is equal to the expectation value of the operator $\tr_{R} e^{2\pi i r a}$ in this matrix model.
For a generic $m$, the one-loop determinant factor $Z_{\text{1-loop}}$ and the instanton factor $Z_{\text{inst}}$ are nontrivial. However, for $m=0$, when $\CalN=4$ symmetry, is recovered $Z_{\text{1-loop}} = 1$ as well as $Z_{\text{inst}} = 1$ [@Okuda:2010]. We conclude that in the $\CalN=4$ theory there are no instanton corrections, and the the Gaussian matrix model conjecture (\[eq:main-result-op\]) is exact.
Another point in the parameter space of the $\CalN=2^{*}$ theory, the point $\tilde m = 0$ is also interesting. It is easy to evaluate $Z_{\text{1-loop}}$ and $Z_{\text{inst}}$. The numerator and denominator cancel each other in each of the fixed point instanton contribution to $Z_{\text{inst}}$, hence in the $\U(N)$ theory $$\label{eq:Z-N-4-inst}
Z_{\U(N), \text{inst}}^{\tilde m = 0 } = \sum_{\vect{Y}} {q}^{|\vect{Y}|} = \prod_{k=1}^{\infty} \frac 1 {(1-{q}^k)^N}$$ is the generating function for the number of $N$-colored partitions.
Using the definition of the Dedekind eta-function $\eta(\tau) = {q}^{1/24}\prod_{k=1}^{\infty}(1-{q}^k)$ we can write $$\label{eq:Z-N-4-eta}
Z_{\U(N), \text{inst}}^{\tilde m = 0} = \lb \frac {1} {{q}^{-1/24} \eta(\tau)} \rb^{N}.$$
At $\tilde m = 0$ most of the factors in the infinite product (\[eq:Z-1-loop-N-2\]) cancel each other, and we are left with $$Z_{\U(N), \text{1-loop}}^{\tilde m = 0}(ia_E) = \prod_{\text{roots }\alpha} \frac{1} { |(\alpha \cdot a_E)|}$$ We see that the 1-loop contribution at $\tilde m = 0$ gives exactly the inverse Weyl measure in the reduction of the integral over $\g$ to the Cartan algebra of $\g$. Therefore, the total partition function at $\tilde m = 0$ for $\U(N)$ theory is given by (we set $\ve = \frac 1 r = 1$) $$Z_{\U(N)}^{\tilde m = 0} = |Z_{\U(N), \text{inst}}^{\tilde m = 0 }|^2 \int d^N a \, e^{- \frac { 4 \pi^2 }{g_{YM}^2} a_E^2} = \lb \frac {1} { ( {q}\bar {q})^{-1/24} \eta(\tau) \bar \eta(\tau) \sqrt{ 2 \tau_2}} \rb^{N}$$
This function does not transform well under $S$-duality $\tau \to -1/\tau$. However, it is possible to add to the theory $c$-number gravitational curvature terms which shift the action by a constant [@Vafa:1994tf], for example we can add the following $R^2$-term: $$S_{YM} \to S_{YM} - 2 \pi \tau_2 \frac {1} {24} \frac {N} {32 \pi^2} \int_{S^4} R_{\mu \nu \rho \lambda}R^{\mu \nu \rho \lambda}.$$ Such $R^2$ terms generally appear as gravitational corrections to an effective action on a brane in string theory [@Bachas:1999um]. This $R^2$ term cancels the extra factor ${q}^{-1/24}$ in the partition function, and after such correction we get $$\label{eq:z-final-1}
Z^{\tilde m = 0 }_{\U(N), \text{$R^2$ background}} = \frac {1} { ( \eta(\tau) \bar \eta (\tau) \sqrt{2 \tau_2})^{N}}.$$ So far we discussed instanton corrections only to the partition function. Now we consider corrections to the Wilson loop operator. One can show that the Wilson loop $W(C)$ which we consider is in the same $\delta_{\ve}$ cohomology class as the operator $\tr_{R} \exp (\frac {2\pi} {\ve} \Phi)$ inserted at the North pole, where $\Phi = i \Phi_0^E + \Phi_9$. Instanton corrections to the operator $\exp ( \beta \Phi)$ in the $\CalN=2$ equivariant theory on $\BR^4$ for a given asymptotic of $\Phi$ at infinity were computed in [@Losev:2003py; @Flume:2004rp; @nakajima-lectures; @Nekrasov:2003rj]. Using these results, one can actually see that if $\beta = \frac {2 \pi n}{\ve}$ where $n$ is integer, there are no instanton corrections to the operator $\tr_{R} \exp(\beta \Phi)$. In other words, the operator $\tr_R \exp( \beta \Phi)$ in the field theory is replaced simply by the operator $\tr_R \exp(2 \pi i r a)$ in the matrix model.
This is exactly the case of Wilson loop operator which we consider. In other words, even after taking into account the instanton corrections, we still conclude that the Wilson loop operator $W(C)$ corresponds to the operator $\tr_R \exp(2\pi i r a)$ in the matrix model. However, the expectation value of $W(C)$ in a generic $\CalN=2$ theory receives corrections because the measure in the matrix integral (\[eq:main-result\]) is corrected by the insertion of the instanton factor $|Z_{\text{inst}}(ia, \ve,\ve)|^2$.
Clifford algebra {#sec:Octonionic-gamma-matrices}
================
We use the following conventions to denote symmetrized and antisymmetrized tensors: $$\label{eq:symmetr-antisymmetr}
\begin{aligned}
a_{[i} b_{j]}= \frac 1 2 (a_{i} b_{j} - a_{j} b_{i}) \\
a_{\{i} b_{j\}}= \frac 1 2 (a_{i} b_{j} + a_{j} b_{i}),
\end{aligned}$$ where $a$ and $b$ are any indexed variables.
Let us summarize here our conventions on gamma-matrices in ten dimensions. We start with Minkowski metric $ds^2 = -dx_0^2 + dx_1^2 + \dots dx_9^2$. Capital letters from the middle of the Latin alphabet normally are used to denote ten-dimensional space-time indices $M,N,P,Q = 0,\dots, 9$. Let $\gamma^{M}$ for $M=0,\dots,9$ be $32 \times 32$ matrices representing the Clifford algebra $\Cl(9,1)$. They satisfy the standard anticommutation relations $$\label{eq:Gamma-anti-comm}
\gamma^{\{M} \gamma^{N\}} = g^{MN},$$ where $g^{MN}$ is the metric. The corresponding representation of $\Spin(9,1)$ has rank 32 and can be decomposed into irreducible spin representations $\CalS^+$ and $\CalS^-$ of rank 16. The chirality operator $$\gamma^{11} = \gamma^{1}\gamma^{2}\dots \gamma^{9} \gamma^{0}$$ acts on $\CalS^{+}$ and $\CalS^{-}$ as multiplication by $1$ and $-1$, respectively. The gamma-matrices $\Gamma^{M}$ reverse chirality, so $\Gamma^{M}: \CalS^{\pm} \to \CalS^{\mp}$. We can write $\gamma^{M}$ in the block form $$\label{eq:gamma-ident}
\gamma^{M} = \begin{pmatrix}
0 & \tilde \Gamma^{M} \\
\Gamma^{M} & 0 \\
\end{pmatrix},$$ assuming that we write the rank 32 spin representation of $\Spin(9,1)$ as $$\begin{pmatrix}
\CalS^{+} \\
\CalS^{-}
\end{pmatrix}.$$ Let $\Gamma^{M}$ and $\tilde \Gamma^{M}$ be the chiral “half” gamma-matrices appearing in (\[eq:gamma-ident\]). Then $$\label{eq:gamma-anti-comm}
\tilde \Gamma^{\{M} \Gamma^{N\}} = g^{MN}, \quad
\Gamma^{\{M} \tilde \Gamma^{N\}} = g^{MN}.$$ We define $\gamma^{MN}, \Gamma^{MN}$ and $\tilde \Gamma^{MN}$ as follows $$\label{eq:gamma-commutator}
\gamma^{MN} = \gamma^{[M} \gamma^{N]} = \begin{pmatrix}
\tilde \Gamma^{[M} \Gamma^{N]} & 0 \\
0 & \Gamma^{[M} \tilde \Gamma^{N]} \\
\end{pmatrix} =:
\begin{pmatrix}
\Gamma^{MN} & 0\\
0 & \tilde \Gamma^{MN} \\
\end{pmatrix}.$$ Using anticommutation relations we get $$\label{eq:gamma-tri}
\Gamma^{M} \Gamma^{PQ} = 4 g^{M[P} \Gamma^{Q]} + \tilde \Gamma^{PQ} \Gamma^{M}.$$ For computations in the four-dimensional theory, we will often need to split the ten-dimensional space-time indices into two groups. The first group contains four-dimensional space-time indices in the range $1,\dots,4$, which we denote by Greek latter in the middle of the alphabet $\mu, \nu, \lambda, \rho$. The second group contains the indices for the normal directions, running over $5,\dots,9,0$, which we denote by capital letters from the beginning of the Latin alphabet $A,B,C,D$. As usual, the repeated index means summation over it. Then we have the following identities $$\label{eq:gamma-tri-spec}
\begin{aligned}
& \Gamma_{\mu A} \tilde \Gamma^{\mu} = - 4 \tilde \Gamma_{A} \\
& \Gamma^{\mu} \Gamma_{\nu \rho} \tilde \Gamma_{\mu} = 0 \\
& \Gamma^{\mu} \Gamma_{\nu A} \tilde \Gamma_{\mu} = 2 \tilde \Gamma_{\nu A} \\
& \Gamma^{\mu} \Gamma_{A B} \tilde \Gamma_{\mu} = 4 \tilde \Gamma_{AB}
\end{aligned}$$ We choose matrices $\Gamma_{M}$ and $\tilde \Gamma^{M}$ to be symmetric: $$(\Gamma^{M})^{T} = \Gamma_{M} \quad (\tilde \Gamma^{M})^{T} = \tilde \Gamma^{M}.$$ Then we get $(\Gamma^{MN})^{T} = - \tilde \Gamma^{MN}$, so the representations $\CalS^{+}$ and $\CalS^{-}$ are dual to each other.
There is a very important “triality identity” which appears in the computations involving ten-dimensional supersymmetry: $$\label{eq:triality}
(\Gamma_{M})_{ \alpha_1 \{\alpha_2} (\Gamma^{M})_{\alpha_3 \alpha_4\}} = 0,$$ where $\alpha_1,\alpha_2,\alpha_3,\alpha_4=1,\dots,16$ are the matrix indices of $\Gamma^{M}$.
All gamma-matrices relations above are valid both for Minkowski and Euclidean signature.
The difference between gamma-matrices for Minkowski and Euclidean signature is the following. In Minkowski signature we choose $\Gamma^{M}$ to be real. In Euclidean signature we use the following matrices $\{i\Gamma^{0}, \Gamma^{1}, \dots, \Gamma^{9}\}$. Therefore all Euclidean gamma-matrices are real except $\Gamma^{0}$, which is imaginary. In Euclidean signagure the representation $\CalS^{+}$ and $\CalS^{-}$ are unitary. Since in Euclidean signature they are also dual to each other, we conclude that in Euclidean signature $\CalS^{+}$ and $\CalS^{-}$ are complex conjugate representations.
It is convenient to use octonions to explicitly write down $\Gamma^{M}$. In Minkowski signature we choose $$\begin{aligned}
&\Gamma^{i} = \left(
\begin{array}{cc}
0 & E_{i}^{T} \\
E_{i} & 0 \\
\end{array}
\right), \quad i=1\dots 7 \\
&\Gamma^9 = \left(
\begin{array}{cc}
1_{8\times8} & 0 \\
0 & -1_{8\times 8} \\
\end{array}
\right), \\
&\Gamma^0 = \left(
\begin{array}{cc}
1_{8\times8} & 0 \\
0 & 1_{8\times 8} \\
\end{array}
\right),
\end{aligned}$$ where $E_{i}$ for $i = 1 \dots 8$ are $8\times8$ matrices representing left multiplication of the octonions.
Let $e_{i}$ with $i=1\dots 8$ be the generators of the octonion algebra with the octonionic structure constants $c^{k}_{ij}$ defined by the multiplication table $e_{i} \cdot e_{j} = c^{k}_{ij}
e_{k}$. Then $(E_i)^{k}_{j} = c^{k}_{ij}$. The element $e_1$ is the identity. To be concrete, we define the multiplication table by specifying the triples which have cyclic multiplication table: $ (234),(256),(357),(458),(836),(647),(728)$ (e.g. $e_2 e_3 = e_4$, etc.). Then one can check that $E_{i}$ have the following form $$\begin{aligned}
& E_{\mu} = \left(
\begin{array}{cc}
J_{\mu} & 0 \\
0 & \bar{J_\mu} \\
\end{array} \right), \quad \mu=1\dots 4 \\
& E_{A} = \left(
\begin{array}{cc}
0 & -J_{A}^{T}\\
J_{A} & 0 \\
\end{array}
\right), \quad A = 5 \dots 8,
\end{aligned}$$ where $J_{\mu}$ for $\mu = 1 \dots 4$ are the $4 \times 4$ matrices representing generators of quaternion algebra by the left action, while $\bar{J_{\mu}}$ are the $4\times 4$ matrices representing generators of quaternion algebra by the right action. Concretely we obtain $$(J_{1},J_{2},J_{3},J_{4}) =\tiny \left( \left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right),
\left(
\begin{array}{cccc}
0 & -1& 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 \\
0 & 0 & 1& 0 \\
\end{array}
\right),
\left(
\begin{array}{cccc}
0 & 0 & -1& 0 \\
0 & 0 & 0& 1 \\
1 & 0 & 0 & 0 \\
0 & -1& 0 & 0 \\
\end{array}
\right),
\left(
\begin{array}{cccc}
0 & 0 & 0 & -1\\
0 & 0 & -1& 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
\end{array}
\right) \right),$$ with the relations $$J_{i} J_{j} = \ve_{ijk} J_{k}, \quad i,j,k=2 \dots 4,$$ and $$(\bar {J_{1}},\bar {J_{2}},\bar {J_{3}},\bar {J_{4}}) =\tiny \left( \left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right),
\left(
\begin{array}{cccc}
0 & -1& 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 &-1& 0 \\
\end{array}
\right),
\left(
\begin{array}{cccc}
0 & 0 & -1& 0 \\
0 & 0 & 0& -1\\
1 & 0 & 0 & 0 \\
0 & 1& 0 & 0 \\
\end{array}
\right),
\left(
\begin{array}{cccc}
0 & 0 & 0 & -1\\
0 & 0 & 1& 0 \\
0 & -1& 0 & 0 \\
1 & 0 & 0 & 0 \\
\end{array}
\right) \right),$$ with the relations $$\bar J_{i} \bar J_{j} = -\ve_{ijk} \bar J_{k}, \quad i,j,k=2 \dots 4.$$ Similarly, $$({J_{5}},{J_{6}},{J_{7}},{J_{8}}) =\tiny \left( \left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1 \\
\end{array}
\right),
\left(
\begin{array}{cccc}
0 & 1& 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 &-1& 0 \\
\end{array}
\right),
\left(
\begin{array}{cccc}
0 & 0 & 1& 0 \\
0 & 0 & 0& -1\\
1 & 0 & 0 & 0 \\
0 & 1& 0 & 0 \\
\end{array}
\right),
\left(
\begin{array}{cccc}
0 & 0 & 0 & 1\\
0 & 0 & 1& 0 \\
0 & -1& 0 & 0 \\
1 & 0 & 0 & 0 \\
\end{array}
\right) \right).$$
We choose orientation in the $(1\dots 4)$-plane and the $(5\dots 8)$-plane by saying that $1234$ and $5678$ are the the positive cycles.
Then the matrices $\Gamma^{\mu \nu}$ for $\mu,\nu=1 \dots 4$ and $\Gamma^{i j}$ for $i,j=5 \dots 8$ have the following block decomposition: $$\begin{aligned}
\Gamma_{\mu \nu} = \left(
\begin{array}{cc}
E_{[\mu}^{T} E_{\nu]} & 0 \\
0 & E_{[\mu} E_{\nu]}^{T}\\
\end{array}
\right) = \left(
\begin{array}{cccc}
J_{\mu \nu}^{-} & 0 & 0 & 0 \\
0 & \bar J_{\mu \nu}^{+} & 0 & 0 \\
0 & 0 & -J_{\mu \nu}^{+} & 0 \\
0 & 0 & 0 & -\bar J_{\mu \nu}^{-} \\
\end{array}
\right), \\
\Gamma_{i j} = \left(
\begin{array}{cc}
E_{[i}^{T} E_{i]} & 0 \\
0 & E_{[i} E_{i]}^{T}\\
\end{array}
\right) = \left(
\begin{array}{cccc}
- \bar J_{ i j}^{-} & 0 & 0 & 0 \\
0 & - J_{ij}^{+} & 0 & 0 \\
0 & 0 & -\bar J_{ij}^{-} & 0 \\
0 & 0 & 0 & - J_{ij}^{+} \\
\end{array}
\right), \\
\end{aligned}$$ where the $\pm$-superscript denotes the self-dual and anti-self-dual tensors; $J_{12} = J_1^{T} J_2 = J_2$, etc.
Then we define the four-dimensional chirality operator acting in tangent directions to the four-dimensional space-time : $${\Gamma^{(\overline{14})}}=\Gamma_1 \Gamma_2 \Gamma_3 \Gamma_4.$$ It is represented by the matrix $${\Gamma^{(\overline{14})}}= \left(
\begin{array}{cccc}
1_{4\times 4} & 0 & 0 & 0 \\
0 & -1_{4\times 4} & 0 & 0 \\
0 & 0 & -1_{4\times 4} & 0 \\
0 & 0 & 0 & 1_{4\times 4} \\
\end{array}
\right).$$
Similarly, we define the four-dimensional chirality operator $${\Gamma^{(\overline{58})}}=\Gamma_5 \Gamma_6 \Gamma_7 \Gamma_8,$$ acting in four normal directions $M=5\dots 8$. It is represented by the matrix $${\Gamma^{(\overline{58})}}= \left(
\begin{array}{cccc}
1_{4\times 4} & 0 & 0 & 0 \\
0 & -1_{4\times 4} & 0 & 0 \\
0 & 0 & 1_{4\times 4} & 0 \\
0 & 0 & 0 & -1_{4\times 4} \\
\end{array}
\right).$$
Finally, we define the eight-dimensional chirality operator $$\Gamma^{9} = {\Gamma^{(\overline{14})}}{\Gamma^{(\overline{58})}}.$$ It is represented by the matrix $$\Gamma^{9} = \left(
\begin{array}{cccc}
1_{4\times 4} & 0 & 0 & 0 \\
0 & 1_{4\times 4} & 0 & 0 \\
0 & 0 & -1_{4\times 4} & 0 \\
0 & 0 & 0 & -1_{4\times 4} \\
\end{array}
\right).$$
The representation $\bf{16} = \CalS^{+}$ (a sixteen component Majorana-Weyl fermion of $\Spin(9,1)$) then splits as $\bf{16} = \bf{8} + \bf{8'}$ with respect to the $\Spin(8) \subset \Spin(9,1)$ acting in the directions $M=1,\dots, 8$. Then we brake $\Spin(8)$ as $\Spin(8) \hookleftarrow \Spin(4) \times \Spin(4)^R$, where the group $\Spin(4)$ acts in the directions $M=1, \dots, 4$, while the group $\Spin(4)^R$ acts in the directions $M = 5, \dots, 8$. We write the $\Spin(4)$ as $\Spin(4) = \SU(2)_L \times \SU(2)_R$ and the $\Spin(4)^R$ as $\Spin(4)^R = \SU(2)_L^R \times \SU(2)_R^R$. With respect to these $\SU(2)$-subgroups, the representation $\bf{16} = \CalS^{+}$ of $\Spin(9,1)$ transforms as $${\bf 16 = (2,1,2,1) + (1,2,1,2) + (1,2,2,1) + (2,1,1,2)}.$$
As we mentioned before, the only difference between the gamma-matrices in the Euclidean and Minkowski case is that we multiply the matrix $\Gamma^{0}$ by $i \equiv \sqrt{-1}$, so the Euclidean gamma-matrices are: $$\begin{aligned}
&\Gamma^{M} = \left(
\begin{array}{cc}
0 & E_{M}^{T} \\
E_{M} & 0 \\
\end{array}
\right), \quad M=1\dots 7 \\
&\Gamma^9 = \left(
\begin{array}{cc}
1_{8\times8} & 0 \\
0 & -1_{8\times 8} \\
\end{array}
\right), \\
&\Gamma^0 = \left(
\begin{array}{cc}
i1_{8\times8} & 0 \\
0 & i1_{8\times 8} \\
\end{array}
\right).
\end{aligned}$$
Conformal killing spinors on $S^4$ \[se:Killing-spinors\]
=========================================================
The explicit form of the Killing spinor on $S^4$ depends on the vielbein. For solution in spherical coordinates see [@Lu:1998nu]. In stereographic coordinates the solution has simpler form and is easily related to the flat limit.
Pick up a point on $S^4$, call it the North pole, and call the opposite point the South pole. Let $x^{\mu}$ be the stereographic coordinates on $S^4$ in the neighborhood of the North pole. The metric has the following form $$\label{eq:metric-S4-stereo-app}
g_{\mu \nu} = \delta_{\mu \nu} e^{2\Omega},
\quad \text{where} \quad e^{2 \Omega} := \frac 1 { (1 + \frac {x^2} {4 r^2})^2 }.$$ By $\theta$ we denote the polar angle in spherical coordinates measure from the North pole. In other words, $\theta=0$ is the North pole, $\theta=\frac {\pi} {2}$ is the equator, and $\theta = \pi$ is the South pole. We have $|x| = 2 r \tan \frac \theta 2$ and $e^{\Omega} = \cos^2 \frac \theta 2$. Fix the vielbein[^10] $e^{\hat \mu}_{\lambda} = \delta^{\hat \mu}_{\lambda} e^{\Omega}$. The spin connection $\omega^{\hat \mu}_{\hat \nu \lambda}$ induced by the Levi-Civita connection can be computed using the Weyl transformation of the flat metric $\delta_{\mu \nu} \mapsto e^{2 \Omega} \delta_{\mu \nu}$. Under such transformation $\omega^{\hat \mu}_{\hat \nu \mu} \mapsto \omega^{\hat \mu}_{\hat
\nu \lambda} + (e^{\hat \mu}_{\lambda} e^{\nu}_{\hat \nu} \Omega_\nu - e_{\hat
\nu \lambda} e^{\hat \mu \nu} \Omega_{\nu})$. Since in the flat case $\omega^{\hat \mu}_{\hat \nu \lambda} = 0$, we get $$\omega^{\hat \mu}_{\hat \nu \lambda} = (e^{\hat \mu}_{\lambda} e^{\nu}_{\hat \nu} \Omega_\nu - e_{\hat \nu \lambda} e^{\hat \mu \nu} \Omega_{\nu}),$$ where $\Omega_{\nu} := \p_{\nu} \Omega$.
The conformal Killing spinor equation takes the explicit form $$\label{eq:Killing-equation-ster}
\begin{aligned}
&(\p_{\lambda} + \frac 1 4 \omega_{\hat \mu \hat \nu \lambda} \Gamma^{\hat \mu \hat \nu} ) \ve = \Gamma_{\lambda} \tilde \ve\\
&(\p_{\lambda} + \frac 1 4 \omega_{\hat \mu \hat \nu \lambda} \Gamma^{\hat
\mu \hat \nu} ) \tilde \ve = - \frac 1 {4 r^2} \Gamma_{\lambda} \ve;
\end{aligned}$$ At the flat limit $r = \infty$ the equations simplify as $\p_{\lambda} \ve =
\Gamma_{\lambda} \tilde \ve$ and $\p_{\lambda} \tilde \ve = 0$; hence the flat space solution is $$\begin{aligned}
\label{eq:conformal_susy_in_R^4-sol}
&\ve = \hat \ve_{s} + x^{\hat \mu} \Gamma_{\hat \mu} \hat{\ve_c} \\\
&\tilde \ve = \hat \ve_c,
\end{aligned}$$ where $\hat \ve_{s}, \hat \ve_{c}$ are constant spinors on $\BR^4$. The spinor $\hat \ve_{s}$ generates usual supersymmetry transformations, the spinor $\hat \ve_c$ generates special superconformal transformations.
For an arbitrary $r$ the solution is $$\begin{aligned}
\label{eq:Killing-solution-in-S^4-stereo-app}
\ve = \frac 1 {\sqrt{1+ \frac {x^2} {4 r^2}}} (\hat \ve_{s} + x^{\hat \mu} \Gamma_{\hat \mu} \hat \ve_{c}) \\
\tilde \ve = \frac 1 {\sqrt{1+ \frac {x^2} {4 r^2}}} (\hat \ve_{c} - \frac
{x^{\hat \mu} \Gamma_{\hat \mu} }{4 r^2} \hat \ve_{s}),\end{aligned}$$ where $\hat \ve_{s}$ and $\hat \ve_{c}$ are arbitrary spinor parameters.
Consider the case when $\ve$ is the conformal Killing spinors generating a transformation of an $\OSp(2|4)$ subgroup. We take chiral $\hat \ve_{s}$ and $\hat \ve_{c}$, such that ${\Gamma^{9}}\hat \ve_s = \hat \ve_s$ and ${\Gamma^{9}}\hat
\ve_{c} = \hat \ve_{c}$, so $$\begin{aligned}
\label{eq:Killing-solution-in-S^4-stereo2}
\ve = \frac 1 {\sqrt{1+ \frac {x^2} {4 r^2}}} (\hat \ve_{s} + x^{\hat \mu}
\Gamma_{\hat \mu} {\Gamma^{9}}\hat \ve_{c}).\end{aligned}$$ Moreover, for such spinor $\ve$ we have $\hat \ve_{c} = \frac 1 {2r} \frac 1 4
\omega_{\hat \mu \hat \nu} \Gamma^{\hat \mu \hat \nu} \hat \ve_{s}$, where $\omega_{\hat \mu \hat \nu}$ is an anti self-dual generator of $SO(4)$ normalized $\omega_{\hat \mu \hat \nu} \omega^{\hat \mu \hat \nu} = 4$.
This means that $\delta_{\ve}$ squares to a rotation around the North pole generated by $\omega$. Then $(\hat \ve_{c}, \hat \ve_{c}) = \frac 1 {4r^2} (\hat \ve_{s}, \hat \ve_{s})$, and thus $(\ve,\ve)$ is constant over $S^4$.
Take $(\hat \ve_{s}, \hat \ve_{s}) =1$. Then we get the vector field $v_{\hat \nu} = \ve \Gamma_{\hat
\nu} \ve = 2 \hat \ve_{s} \Gamma_{\hat \nu} \Gamma_{\hat \mu} x^{\hat \mu}
\hat \ve^{c} = 2 \hat \ve_{s} \Gamma_{\hat \nu} \Gamma_{\hat \mu} x^{\hat \mu}
\frac 1 {2r} \frac 1 4 \omega_{\hat \rho \hat \lambda}\Gamma^{\hat \rho \hat
\lambda} \hat \ve_{s} = \frac 1 r x^{\hat \mu} \omega_{\hat \mu \hat \nu}
(\hat \ve_{s} \hat \ve_{s}) = \frac 1 r x^{\hat \mu} \omega_{\hat \mu \hat \nu}
$. Using this identity we can rewrite conformal Killing spinor $\ve\equiv\ve(x)$ as $$\begin{aligned}
\label{eq:Killing-solution-in-S^4-stereo3}
\ve(x) = \frac 1 {\sqrt{1+ \frac {x^2} {4 r^2}}} (\hat \ve_{s} + \frac 1 {2r}
\frac 1 4 x^{\hat \mu} \Gamma_{\hat \mu} \omega_{\hat \rho \hat \lambda}
\Gamma^{\hat \rho \hat \lambda} {\Gamma^{9}}\hat \ve_{s}) =
\frac 1 {\sqrt{1+ \frac {x^2} {4 r^2}}} (\hat \ve_{s} + \frac {1} {2r} x^{\hat \rho} \Gamma^{\hat \lambda} \omega_{\hat \rho \hat \lambda} {\Gamma^{9}}\hat \ve_{s}) = \\
=\frac 1 {\sqrt{1+ \frac {x^2} {4 r^2}}} (\hat \ve_{s} + \frac 1 2 v_{\hat
\lambda} \Gamma^{\hat \lambda} {\Gamma^{9}}\hat \ve_{s}) = \frac 1 {\sqrt{1+ \frac
{x^2} {4 r^2}}} (\hat \ve_{s} + \frac {|x|}{2r} n_{\hat \lambda}(x)
\Gamma^{\hat \lambda} {\Gamma^{9}}\hat \ve_{s}) = \\= \lb \cos \frac \theta 2 + \sin
\frac \theta 2 (n_{\hat \lambda}(x) \Gamma^{\hat \lambda} {\Gamma^{9}})\rb \hat
\ve_{s} = \exp \lb \frac \theta 2 n_{\hat \lambda}(x) \Gamma^{\hat \lambda}
{\Gamma^{9}}\rb \hat \ve_{s},\end{aligned}$$ where $n_{\hat \lambda}$ is the unit vector in the direction of the vector field $v_{\hat \lambda}$. The aim of these manipulations was to represent the spinor $\ve(x)$ at an arbitrary point $x$ by an explicit $\Spin(5)$ rotation $R(x) =
\exp \frac \theta 2 (n_{\hat \lambda}(x) \Gamma^{\hat \lambda} {\Gamma^{9}})$ of its value at the origin $\ve(0) = \hat \ve_s$.
Off-shell supersymmetry\[se:off-shell susy\]
============================================
Let $\delta_{\ve}$ be the supersymmetry transformation generated by a Killing spinor $\ve$. Then the square of $\delta_{\ve}$ is computed as follows $$\delta_{\ve}^2 A_{M} =\delta_{\ve} (\ve \Gamma_{M} \Psi) = \ve \Gamma_{M} ( \frac 1 2 \Gamma^{PQ} \ve F_{PQ} + \frac 1 2 \Gamma^{\mu A} \Phi_{A} D_{\mu} \ve ).$$ Since $$\ve \Gamma_{M} \Gamma_{PQ} \ve = \ve \Gamma_{PQ}^T \Gamma_M \ve = -\ve \tilde
\Gamma_{PQ} \Gamma_{M} \ve = \frac 1 2 \ve (\Gamma_{M} \Gamma_{PQ} - \tilde
\Gamma_{PQ} \Gamma_{M}) = 2 g_{M[P} \ve \Gamma_{Q]} \ve ,$$ the first term for $\delta_{\ve}^2 A_{M}$ gives $ -\ve \Gamma^{N} \ve F_{NM}$. The second term is $$\frac 1 2 \ve \Gamma_{M} \Gamma^{\mu A } \Phi_{A} D_{\mu} \ve = -2 \ve
\Gamma_M \tilde \Gamma_{A} \ve \Phi^{A}.$$ Then $$\label{eq:delta2onA}
\delta_{\ve}^2 A_{M} = -(\ve \Gamma^{N} \ve) F_{NM} -
2 \ve \Gamma_{M} \tilde \Gamma_{A} \ve \Phi^{A}.$$ Restricting the index $m$ to the range of $\mu$ or $A$ we get respectively $$\begin{aligned}
& \delta_{\ve}^2 A_{\mu} = - v^{\nu} F_{\nu \mu} - [v^{B} \Phi_B, D_{\mu}] \\
& \delta_{\ve}^2 \Phi_A = - v^{\nu} D_{\nu} \Phi_{A} - [v^{B} \Phi_B, \Phi_A] - 2
\ve \tilde \Gamma_{AB} \tilde \ve \Phi^{B} - 2\ve \tilde \ve \Phi_A,
\end{aligned}$$ where we introduced the vector field $v$ $$\label{eq:v-in-terms-eps-2}
v^\mu \equiv \ve \Gamma^{\mu} \ve, \quad v^{A} \equiv \ve \Gamma^{A} \ve.$$ Therefore $$\delta_{\ve}^2 = -L_{v} - G_{v^M A_{M}} - R - \Omega.$$ Here $L_{v}$ is the Lie derivative in the direction of the vector field $v^{\mu}$. The transformation $G_{v^{M}A_{M}}$ is the gauge transformation generated by the parameter $v^{M}A_{M}$. On matter fields $G$ acts as $G_{u}
\cdot \Phi \equiv [u, \Phi]$, on gauge fields $G$ acts as $G_{u} \cdot A_{\mu} =
- D_{\mu} u$. The transformation $R$ is the rotation of the scalar fields $(R
\cdot \Phi)_{A} = R_{AB} \Phi^{B}$ with the generator $R_{AB} = 2 \ve \tilde
\Gamma_{AB} \tilde \ve$. Finally, the transformation $\Omega$ is the dilation transformation with the parameter $2(\ve \tilde \ve)$.
The $\delta_{\ve}^2$ acts on the fermions as follows $$\begin{gathered}
\label{eq:delta-fermion}
\delta_{\ve}^2 \Psi = D_{M} (\ve \Gamma_N \Psi) \Gamma^{MN} \ve +
\frac 1 2 \Gamma^{\mu A} (\ve \Gamma_{A} \Psi) D_{\mu} \ve = \\
= (\ve \Gamma_N D_{M} \Psi) \Gamma^{MN} \ve + ((D_{\mu} \ve) \Gamma_{N} \Psi)
\Gamma^{\mu N} \ve + \frac 1 2 \Gamma^{\mu A} (\ve \Gamma_{A} \Psi) D_{\mu}
\ve.\end{gathered}$$ From the “triality identity” we have $\Gamma_{N \alpha_2 (\alpha_1
}\Gamma^{N}_{\alpha_3) \xi } = -\frac 1 2 \Gamma^{N}_{\alpha_2 \xi} \Gamma_{N
\alpha_1 \alpha_3}.$ Then the first term gives $$\begin{gathered}
\label{eq:delta-fermion-on-shell}
(\ve \Gamma_N D_{M} \Psi) (\Gamma^{MN} \ve)_{\alpha_4} =
(\ve \Gamma_N D_{M} \Psi) ((\tilde \Gamma^{M} \Gamma^{N} \ve)_{\alpha_4} - g^{MN}\ve_{\alpha_4}) = \\
= \ve^{\alpha_1} \Gamma_{N \alpha_1 \alpha_2 } D_{M} \Psi^{\alpha_2} \tilde
\Gamma^{M}_{\alpha_4 \xi} \Gamma^{N}_{\xi \alpha_3} \ve^{\alpha_3}
- (\ve \Gamma^{N} D_{N} \Psi) \ve_{\alpha_4}=\\
=-\frac 1 2 (\ve^{\alpha_1} \Gamma_{N \alpha_1 \alpha_3} \ve^{\alpha_3})
(\tilde \Gamma^{M}_{\alpha_4 \xi} \Gamma^{N}_{\alpha_2 \xi} D_{M} \Psi^{\alpha_2}) - (\ve \Gamma^{N} D_{N} \Psi) \ve_{\alpha_4}=\\
= -\frac 1 2 (\ve \Gamma_N \ve) (\tilde \Gamma^{M} \Gamma^{N} D_{M}
\Psi)_{\alpha_4}
-(\ve \Gamma^{N} D_{N} \Psi) \ve_{\alpha_4} = \\
= -\frac 1 2 (\ve \Gamma_N \ve) ( - \tilde \Gamma^{N} \Gamma^{M} D_{M} \Psi +
2 D_N \Psi)_{\alpha_4} - (\ve \Gamma^N D_{N} \Psi) \ve_{\alpha_4} = \\
= \frac 1 2 (\ve \Gamma_N \ve) \tilde \Gamma^{N} (\Dslash \Psi)_{\alpha_4} -
(\ve \Gamma^{N} \ve) (D_{N} \Psi)_{\alpha_4} - (\ve \Dslash \Psi)
\ve_{\alpha_4}.\end{gathered}$$ The first and the third term in the last line vanish on-shell. When we add auxiliary fields, they will cancel the first and the third term explicitly. Then we get $$\delta_{\ve}^{2} \Psi = -(\ve \Gamma^{N} \ve) D_{N} \Psi
+ ( \Psi \Gamma_{N} D_{\mu} \ve) \Gamma^{\mu N} \ve +
\frac 1 2 \Gamma^{\mu A} (\ve \Gamma_{A} \Psi) D_{\mu} \ve + \text{eom}[\Psi], \\$$ where $\text{eom}[\Psi]$ stands for the terms proportional to the Dirac equation of motion for $\Psi$. Then we rewrite the last two terms as follows $$\begin{gathered}
\label{eq:begin-trans}
(\Psi \Gamma_N \Gamma_{\mu} \tilde \ve) \Gamma^{\mu N} \ve + \frac 1 2
\Gamma^{\mu A} (\ve \Gamma_{A} \Psi) \Gamma_{\mu}
\tilde \ve = \\
= (\Psi \Gamma_N \Gamma_{\mu} \tilde \ve) (\tilde \Gamma^{\mu} \Gamma^{N} - g^{\mu
N}) \ve - 2 (\ve \Gamma_{A} \Psi) \tilde \Gamma^{A} \tilde \ve = (\Psi
\Gamma_N \Gamma_\mu \tilde \ve) \tilde \Gamma^{\mu} \Gamma^{N} \ve
- 4(\Psi \tilde \ve) \ve - 2 (\ve \Gamma_{A} \Psi) \tilde \Gamma^{A} \tilde \ve \\
\overset{triality}{=} -(\tilde \ve \tilde \Gamma_{\mu} \Gamma_N \ve) \tilde
\Gamma^{\mu} \Gamma^{N} \Psi -(\ve \Gamma_N \Psi) \tilde \Gamma^{\mu}
\Gamma^{N} \Gamma_{\mu} \tilde \ve
-4 (\Psi \tilde \ve) \ve - 2 (\ve \Gamma_{A} \Psi) \tilde \Gamma^{A} \tilde \ve = \\
= -(\tilde \ve \tilde \Gamma_{\mu} \Gamma_\nu \ve) \tilde \Gamma^{\mu} \Gamma^{\nu}
\Psi - (\tilde \ve \tilde \Gamma_{\mu} \Gamma_A \ve) \tilde \Gamma^{\mu} \Gamma^{A}
\Psi + 2(\ve \Gamma_\nu \Psi) \Gamma^{\nu} \tilde \ve + 4 (\ve \Gamma_{A} \Psi)
\tilde \Gamma^{A}
\tilde \ve - 4 (\Psi \tilde \ve) \ve - 2 (\ve \Gamma_{A} \Psi) \tilde \Gamma^{A} \tilde \ve =\\
-(\tilde \ve \Gamma_{\mu \nu} \ve) \Gamma^{\mu \nu} \ve - 4 (\ve \tilde \ve)
\Psi\ -(\tilde \ve \Gamma_{\mu A} \ve) \Gamma^{\mu A} \ve + 2( \ve
\Gamma_{\nu} \Psi) \tilde \Gamma^{\nu} \tilde \ve +
2(\ve \Gamma_{A} \Psi) \tilde \Gamma^{A} \tilde \ve - 4( \Psi \tilde \ve) \ve =\\
= -\frac 1 2 (\tilde \ve \Gamma_{\mu \nu} \ve ) \Gamma^{\mu \nu} \Psi - \frac 1
2 (\tilde \ve \Gamma_{\mu \nu} \ve) \Gamma^{\mu \nu} \Psi -4 (\ve \tilde \ve)
\Psi - (\tilde \ve \Gamma_{\mu A} \ve) \Gamma^{\mu A} \ve - \frac 1 2 (\tilde
\ve \Gamma_{AB} \ve) \Gamma^{AB} \Psi + \\ + \frac 1 2 (\tilde \ve \Gamma_{AB} \ve)
\Gamma^{AB} \Psi +
2 (\ve \Gamma_{N} \Psi) \tilde \Gamma^{N} \tilde \ve - 4(\Psi \tilde \ve) \ve = \\
= \left ( -\frac 1 2 (\tilde \ve \Gamma_{\mu \nu} \ve) \Gamma^{\mu \nu} \Psi +
\frac 1 2 (\tilde \ve \Gamma_{AB} \ve) \Gamma^{AB} \Psi \right)+\\ + \left (
-\frac 1 2 (\tilde \ve \Gamma_{MN} \ve) \Gamma^{MN} \Psi - 4( \ve \tilde
\ve)\Psi - 4(\Psi \tilde \ve) \ve + 2 (\ve \Gamma_{N} \Psi) \tilde
\Gamma^{N} \tilde \ve \right)\end{gathered}$$ The first term is a part of the Lie derivative along the vector field $v^{\mu} =
(\ve \Gamma^{\mu} \ve)$ acting on $\Psi$. The second term correspond to the rotations of the scalar fields $\Phi^{A}$ by the generator $R_{AB}$ and the properly induced action on the fermions.
In the $\CalN=4$ case we use Fierz identity for $\Gamma^{MN}_{\alpha_1 \alpha_2}
\Gamma_{MN \, \alpha_3 \alpha_4}$ in the last line of to see that all term in the second pair of parentheses are canceled except for $-3(\ve \tilde \ve)\Psi$, so that $$\label{eq:delta4fermions}
\delta_{\ve}^{2} \Psi = -(\ve \Gamma^{N} \ve) D_{N} \Psi -
\frac 1 2 (\tilde \ve \Gamma_{\mu \nu} \ve) \Gamma^{\mu \nu} \Psi
- \frac 1 2 (\ve \tilde \Gamma_{AB} \tilde \ve) \Gamma^{AB} \Psi -
3(\tilde \ve \ve) \Psi + \text{eom}[\Psi].$$
To achieve off-shell closure in the $\CalN=4$ case we add seven auxiliary fields $K_i$ with $i=1, \dots, 7$ and modify the transformations as $$\begin{aligned}
& \delta_{\ve} \Psi = \frac 1 2 \Gamma^{MN} F_{MN} + \frac 1 2 \Gamma^{\mu A} \Phi_{A} D_{\mu} \ve + K^i \nu_i \\
& \delta_{\ve} K_i = -\nu_i \Gamma^{M} D_{M} \Psi.
\end{aligned}$$ Here we introduced seven spinors $\nu_i$. They depend on choice of the conformal Killing spinor $\ve$ and are required to satisfy the following relations: $$\begin{aligned}
\label{eq:nu-relations1}
&\ve \Gamma^M \nu_i = 0 \\
\label{eq:nu-relations2}
&\frac 1 2 (\ve \Gamma_N \ve) \tilde \Gamma^{N}_{\alpha \beta} =
\nu^i_{\alpha} \nu^i_{\beta} + \ve_{\alpha} \ve_{\beta} \\
\label{eq:nu-relations3}
&\nu_i \Gamma^M \nu_j = \delta_{ij} \ve \Gamma_M \ve.\end{aligned}$$ The equation ensures closure on $A_M$, the equation ensures closure on $\Psi$.
The new term in the transformations for $\Psi$ modifies the last line of as $$\delta_{\ve} (K^i \nu_i ) = -(\nu_i \Dslash \Psi) \nu_i.$$ Then the terms in $\delta_{\ve}^2
\Psi$ which were not taken into an account in are $$-(\nu_i \Dslash \Psi) \nu_i + \frac 1 2 (\ve \Gamma_N \ve) \tilde \Gamma^{N} \Dslash \Psi -
(\ve \Dslash \Psi) \ve.$$ This expression is identically zero because of (\[eq:nu-relations2\]). Hence, after inclusion of the auxiliary fields $K_i$, the formula for $\delta_{\ve}^2 \Psi$ is valid off-shell.
For the transformation $\delta_{\ve}^2 K_i$ we get $$\delta_{\ve}^2 K_i = -\nu_i \Gamma^{M} [ (\ve \Gamma_M \Psi), \Psi ]
- \nu_i \Gamma^{M} D_{M} (\frac 1 2 \Gamma^{PQ} F_{PQ} \ve + \frac 1 2 \Gamma^{\mu A}
\Phi_A D_{\mu} \ve + K^{i} \nu_i).$$
Using the gamma matrix “triality identity” the first term is transformed to $\frac 1 2 (\nu_i \Gamma^M \ve) [(\Psi, \Gamma^{M} \Psi)]$, which vanishes because of . The second term with derivative acting on $F$ is equal by Bianchi identity to $(\nu_i \Gamma_N \ve) D_{M} F^{MN}$ and vanishes because of . Then we use to simplify the remaining terms $$\begin{gathered}
\delta_{\ve}^2 K_i = - \frac 1 2 \nu_i \Gamma^{\mu} \Gamma^{PQ} \Gamma_{\mu} \tilde
\ve F_{PQ} - \frac 1 2 (\nu_i \Gamma^{M} \Gamma_{\mu A} \Gamma^{\mu} \tilde
\ve) D_{M} \Phi_A - \frac 1 2 (-\frac 1 {4 r^2}) \Phi_A \nu_i \Gamma^{\nu}
\Gamma^{\mu A}
\Gamma_{\mu} \Gamma_{\nu} \ve -\\
- \nu_i \Gamma^{M} (D_{M} K^{j}) \nu_j- (\nu_i \Gamma^{\mu} D_{\mu} \nu_j)
K^{j} = -\frac 1 2 (4) \nu_i \tilde \Gamma^{M B} \tilde \ve D_{M} \Phi_B
-\frac 1 2 (-4) \nu_i \tilde \Gamma^{M B} \tilde \ve D_{M} \Phi_B
+ (\frac 2 {r^2}) \nu_i \Gamma^{A} \ve \Phi_A + \\
- (\nu_i \Gamma^{M} \nu_j) D_{M} K^{j} - (\nu_i \Gamma^{\mu} D_{\mu} \nu_j)
K^{j} =
- (\ve \Gamma^{M} \ve) D_{M} K^{j} - (\nu_i \Gamma^{M} D_{M} \nu_j) K^{j} =\\
= - (\ve \Gamma^{M} \ve) D_{M} K^{i} - (\nu_{[i} \Gamma^{\mu} D_{\mu}
\nu_{j]}) K^{j} - 4(\tilde \ve \ve) K_{i}.\end{gathered}$$ To get the last line we use the differential of , i.e. $\nu_{(i} \Dslash \nu_{j)} = 4 (\ve \tilde \ve) \delta_{ij}$.
Now we consider separately the case of pure $\CalN=2$ Yang-Mills. First we rewrite the last terms in (\[eq:begin-trans\]) as follows (here $d$ is the dimension of uncompactified theory) $$\begin{gathered}
\label{eq:remain}
(\tilde \ve \Gamma_{M N} \ve) \Gamma^{MN} \Psi = (\tilde \ve \tilde
\Gamma_{M} \Gamma_N \ve) \Gamma^{MN} \Psi = (\tilde \ve \tilde \Gamma_{M}
\Gamma_N \ve)
\tilde \Gamma^{M} \Gamma^{N} \Psi - d(\tilde \ve \ve) \Psi \overset{triality}{=} \\
- (\ve \Gamma_N \Psi) \tilde \Gamma^{M} \Gamma^{N} \tilde \Gamma_{M} \tilde
\ve - (\Psi \Gamma_N \tilde \Gamma_M \tilde \ve) \tilde \Gamma^{M} \Gamma^{N}
\ve - d(\tilde \ve \ve) \Psi = (d-2) (\ve \Gamma_N \Psi) \tilde \Gamma^{N}
\tilde \ve -(\Psi \Gamma_N \tilde \Gamma_M \tilde \ve) \tilde \Gamma^{M}
\Gamma^{N} \ve - d(\tilde \ve \ve) \Psi.\end{gathered}$$
For the pure $\CalN=2$ theory in four-dimensions we take $d=6$ and get $$\begin{gathered}
\label{eq:tolambda}
\left ( -\frac 1 2 (\tilde \ve \Gamma_{MN} \ve) \Gamma^{MN} \Psi - 4( \ve
\tilde \ve)\Psi - 4(\Psi \tilde \ve) \ve + 2 (\ve \Gamma_{N} \Psi) \tilde
\Gamma^{N}
\tilde \ve \right) = \\
-\frac 1 2 \left ( 4 (\ve \Gamma_N \Psi) \tilde \Gamma^{N} \tilde \ve -(\Psi
\Gamma_N \tilde \Gamma_M \tilde \ve) \tilde \Gamma^{M} \Gamma^{N} \ve -
6(\tilde \ve \ve) \Psi \right ) -\\ - 4( \ve \tilde \ve)\Psi - 4(\Psi \tilde
\ve) \ve + 2 (\ve \Gamma_{N} \Psi) \tilde \Gamma^{N} \tilde \ve = \frac 1 2
(\Psi \Gamma_N \tilde \Gamma_M \tilde \ve) \tilde \Gamma^{M} \Gamma^{N} \ve
-(\ve \tilde \ve) \Psi - 4(\Psi \tilde \ve) \ve = \\ = \frac 1 2
(\Psi(-\Gamma_M \tilde \Gamma_{N} + 2 g_{MN}) \tilde \ve) \tilde \Gamma^{M}
\Gamma^N \ve -(\ve \tilde \ve) \Psi - 4(\Psi \tilde \ve) \ve = \\ = -\frac 1 2
(\Psi \Gamma_M \tilde \Gamma_{N} \tilde \ve) \tilde \Gamma^{M} \Gamma^{N} + 6
(\Psi \tilde \ve) \ve -(\ve \tilde \ve) \Psi - 4(\Psi \tilde \ve) \ve = -\frac
1 2 (\Psi \Gamma_M \tilde \Gamma_{N} \tilde \ve) \tilde \Gamma^{M} \Gamma^{N}
\ve + 2 (\Psi \tilde \ve) \ve -(\ve \tilde \ve) \Psi.\end{gathered}$$ We express the first term in terms of the triplet of matrices $\Lambda^{i}$, which are defined as a set of three antisymmetric matrices such that $$\begin{aligned}
\label{eq:Lambda-relations}
&\Lambda^{i}_{\alpha_1 \alpha_3} \Lambda^{j}_{\alpha_2 \alpha_3} =
\epsilon^{ijk} \Lambda^{k}_{\alpha_1 \alpha_2} + \delta^{ij} 1_{\alpha_1 \alpha_2},
\quad i,j,k = 1,\dots, 3.\\
&[\Lambda_i, \Gamma^{M}] = 0 \\
&\label{eq:Lambda-ident}
\frac 1 2 \Gamma^{M}_{\alpha_1 \alpha_2 } \tilde \Gamma_{M \, \alpha_3 \alpha_4 } =
\delta_{\alpha_2 (\alpha_1 } \delta_{\alpha_3) \alpha_4} -
\Lambda^i_{ \alpha_2 (\alpha_1 } \Lambda^i_{\alpha_3) \alpha_4}.\end{aligned}$$
Then we get $$(\Psi \Gamma_M \tilde \Gamma_{N} \tilde \ve) \tilde \Gamma^{M} \Gamma^{N} \ve =
4 (\Psi \tilde \ve) \ve + 4 (\ve \tilde \ve)\Psi +
4 (\ve \Lambda^{i} \tilde \ve) \Lambda^i \Psi,$$ and finally the equation turns into $$-2 (\Psi \tilde \ve) \ve - 2 (\ve \tilde \ve)\Psi -
2(\ve \Lambda^{i} \tilde \ve) \Lambda^i \Psi +
2 (\Psi \tilde \ve) \ve -(\ve \tilde \ve) \Psi =
- 2(\ve \Lambda^{i} \tilde \ve) \Lambda^i \Psi - 3(\tilde \ve \ve) \Psi.$$ That completes simplification of $\delta_{\ve}^{2}$ acting on fermions $$\label{eq:delta2fermions}
\delta_{\ve}^{2} \Psi = -(\ve \Gamma^{N} \ve) D_{N} \Psi -
\frac 1 2 (\tilde \ve \Gamma_{\mu \nu} \ve) \Gamma^{\mu \nu} \Psi
- \frac 1 2 (\ve \tilde \Gamma_{AB} \tilde \ve) \Gamma^{AB} \Psi -
2(\ve \Lambda^{i} \tilde \ve) \Lambda^i \Psi - 3(\tilde \ve \ve) \Psi.$$ It has the structure $$\delta_{\ve}^{2} \Psi = -L_{v} \Psi - G_{v^{N} A_{N}} \Psi - R \Psi - R' \Psi - \Omega \Psi,$$ where the notations for the generators are the same as in the bosonic case. The only new generator here is $R'$, corresponding to the term $\delta_{\ve}^2 \Psi = -
2(\ve \Lambda^{i} \tilde \ve) \Lambda^i \Psi$. It generates an $\SU(2)_L$ R-symmetry transformation of $\CalN=2$ which acts trivially on the bosonic fields of the theory, and as $\Psi \mapsto e^{r_i \Lambda_i} \Psi$ on fermionic fields.
To achieve off-shell closure in $\CalN=2$ case we add a triplet of auxiliary fields $K_i$ and modify the transformations as $$\begin{aligned}
& \delta_{\ve} \Psi = \frac 1 2 \Gamma^{MN} F_{MN} + \frac 1 2 \Gamma^{\mu A} \Phi_{A} D_{\mu} \ve + K^i \Lambda_i \ve \\
& \delta_{\ve} K_i = \ve \Lambda_i \Gamma^{M} D_{M} \Psi,
\end{aligned}$$ The new term in the transformations for $\Psi$ modifies the last line of as $$\delta_{\ve}(K^i \Lambda_i \ve) = (\ve \Lambda_i \Dslash \Psi) \Lambda_i \ve.$$ Then the terms in $\delta_{\ve}^2
\Psi$ which were not taken into an account in are $$(\ve \Lambda_i \Dslash \Psi) \Lambda_i \ve + \frac 1 2 (\ve \Gamma_N \ve) \tilde \Gamma^{N} \Dslash \Psi -
(\ve \Dslash \Psi) \ve.$$ This expression is identically zero because of the relation . Hence, after inclusion of the auxiliary fields $K_i$, the formula for $\delta_{\ve}^2 \Psi$ is valid off-shell.
*Remark.* The second equation follows from the first equation and the third equation as follows. Let $$M_{\alpha \beta} = \nu^i_{\alpha} \nu^i_{\beta} + \ve_{\alpha} \ve_{\beta}.$$ We want to show that $M_{\alpha \beta} = \frac 1 2 v_N \tilde \Gamma^{N}_{\alpha
\beta}$, that is the matrix $M_{\alpha \beta}$ can be expanded over the matrices $\tilde \Gamma^{N}_{\alpha \beta}$ with the coefficients $\frac 1 2
v_N$. Fix the positive definite metric on the space $\BR^{16 \times 16}$ of $16
\times 16$ matrices as $(M,M): = M_{\alpha \beta} M_{\alpha \beta}$. Since $\tilde \Gamma^N = \Gamma_N$ and $\Gamma^{\alpha \beta}_M \tilde
\Gamma^N_{\alpha \beta} = 16 \delta^N_M$, the set of 10 matrices $\frac 1 4
\Gamma_N$ is orthonormal in $\BR^{16 \times 16}$. Complete this set to the basis of $\BR^{16 \times 16}$. Then the coefficient $m_N$ of $\frac 1 4
\Gamma_N$ in the expansion of $M$ over this basis is given by the scalar product $$m_N = (M, \frac 1 4 \Gamma_N) = \frac 1 4(\nu^i \Gamma_N \nu^i + \ve \Gamma_N
\ve)= 2 v_N .$$ Therefore we have $M = 2 v_N (\frac 1 4 \Gamma_N) + (\dots)$, where $(\dots)$ stands for possible other terms in the expansion over the completion of the set $\{\frac 1 4 \Gamma_N\}$ to the basis of $\BR^{16 \times
16}$. To prove that all other terms vanish, compare the norm of $M$ $$(M,M) = (\ve \ve)(\ve \ve) + (v_i v_j)(v_iv_j) = (\ve\ve) + \delta_{ij}
(\ve\ve) \delta_{ij} (\ve\ve) = 8 (\ve\ve)(\ve \ve)$$ with the $\sum_{N} m_N^2$ $$\sum_{N} m_N^2 = 4 v_N v_N = 4 (\ve \Gamma_N \ve) (\ve \tilde \Gamma^N \ve) =
4 ( ( \ve \Gamma_N \ve) (\ve \Gamma^N \ve) + 2 (\ve \ve) (\ve \ve)) = 8 (\ve
\ve) (\ve \ve).$$ Since the norms are the same, $(M,M) = \sum_{N} m_N^2$, and the metric is positive definite, we conclude that all other coefficients vanish.
Index of transversally elliptic operators \[se:trans-elliptic\]
===============================================================
Here we collect some facts about indices of transversally elliptic operators mostly following Atiyah’s book [@MR0482866]. See also [@MR0341538].
Let $\dots \to E^{i} \overset{D_i}{\to} E^{i+1} \to \dots $ be an elliptic complex of vector bundles over a manifold $X$. Let a Lie group $G$ act on $X$ and bundles $E^i$. This means that for any transformation $g: X \to X$, which sends a point $x \in X$ to $g(x)$, we are given a vector bundle homomorphisms $\gamma^i: g^* E^i \to E^i$. Then we have natural linear maps $\hat \gamma^i: \Gamma(E^i) \to \Gamma(E^i)$ defined by $\hat \gamma^i = \gamma^i \circ g^*$. On any section $s(x) \in \Gamma(E^i)$ the map $\hat \gamma^i$ acts by the formula $(\hat \gamma s)(x) = \gamma_x s
(g(x))$. We assume that $\hat \gamma$ commutes with the differential operators $D_{i}$ of the complex $E$. Then $\hat \gamma$ descends to a well-defined action on the cohomology groups $H^{i}(E)$.
The $G$-equivariant index is defined as $$\label{eq:equiv-index-def}
\ind_g(E) = \sum_{i} (-1)^i \tr_{H^i} \hat \gamma^i.$$
In the case when the set of $G$-fixed points is discrete and the action of $G$ is nice in a neighborhood of each of the fixed point, the Atiyah-Bott fixed point formula says [@MR0232406; @MR0212836; @MR0190950] $$\label{eq:Atiyah-Bott-formula}
\ind_g(E) = \sum_{x \in \text{fixed point set } } \frac {\sum (-1)^i \tr \gamma_x^i} { |\det (1 - dg(x))|}.$$
This formula can be easily argued in the following way (see [@Goodman:1985bw] for a derivation using supersymmetric quantum mechanics). For an illustration we consider the case when the complex $E$ consists of two vector bundles $E^0 \overset{D}{\to} E^1$, and we assume that the bundles are equipped with a hermitian $G$-invariant metric. Let $D: \Gamma(E^0) \to \Gamma(E^1)$ be the differential. Then we consider the Laplacian $\Delta = D D^* + D^*D$. The zero modes of the Laplacian are identified with the cohomology groups of $E$, which are in this case: $H^{0}(E) = \ker D$ and $H^1 (E) = \coker D$. Hence, the index can be computed as $$\ind_{g} (E) = \lim_{t \to \infty} \str_{\Gamma(E)} \hat \gamma e^{-t
\Delta}.$$ Here the supertrace for operators acting on $\Gamma(E)$ is defined assuming even parity on $\Gamma(E^0)$ and odd parity on $\Gamma(E^1)$. However, the expression under the limit sign actually does not depend on $t$ because $[\Delta,\hat \gamma]=0$. Taking the limit $t \to 0$ we get supertrace of $\hat
\gamma$. The trace can be easily taken in the coordinate representation. By definition, the operator $\hat \gamma$ has kernel $\hat \gamma(x,y) = \gamma_x \delta
(g(x) - y)$ if we write $(\hat \gamma s)(x) = \int_{X} \hat \gamma(x,y) s(y)$. Here $\delta(x)$ is the Dirac delta-function. Taking the trace we get Atiyah-Bott result $$\begin{gathered}
\label{eq:index-derivation}
\ind_{g} (E) = \lim_{t \to 0} \str_{\Gamma(E)} \hat \gamma e^{-t \Delta} = \int dx \str_{E_x} \hat \gamma(x,x) = \int dx \str_{E_x} \gamma_{x} \delta (g(x) - x) = \\
= \sum_{g(x) = x} \frac { \str_{E_{x}} \gamma_x } { | \det (1 - dg(x))| }.\end{gathered}$$
Let $X$ be a complex manifold of dimension $n$. Consider the complex of $(0,p)$-forms with the differential $\bar \p$. Let $G= \U(1)$ acts on $X$ holomorphically. In a neighborhood of a fixed point we can choose such coordinates $z^1, \dots, z^n$ that an element $g \in G$ acts by $z^i \to q_i
z^i$. If $z^i$ transforms in a $\U(1)$ representation $m_i \in \BZ$, and we parameterize $\U(1)$ by a unit circle $\{|q|=1, q \in \BC\}$, then $q_i =
q^{m_i}$. One-forms $f_{\bar i}$ transform as $f_{\bar i}
\to \bar q_i^{-1} f_{\bar i}$. Since $|q|=1$ we have $f_{\bar i} \to q_i
f_{\bar i}$. Computing the supertrace for the numerator on external powers of the anti-holomorphic subspace of the fiber of the cotangent bundle at the origin, we get $\str_{\Omega^{0,\bullet}} q = \prod_{i=1}^{n} (1 - q_i)$. The denominator is $\prod_{i=1}^{n} (1-q_i)(1-q_i^{-1})$. Then contribution of a fixed point with weights $\{q_1,\dots, q_n\}$ to the index of $\bar \p$ is $$\ind_q (\bar \p)|_0 = \frac {1} {\prod_{i=1}^{n} (1 - q_i^{-1})}.$$
Let $\pi: T^*X \to X$ be the cotangent bundle. Then $\pi^*E_i$ are the bundles over $T^*X$. The symbol of the differential operator $D: \Gamma(E_0) \to
\Gamma(E_1)$ is a vector bundle homomorphism $\sigma(D): \pi^* E_0 \to \pi^*
E_1$. In local coordinates $x_i$ it is defined by replacing all partial derivatives in the highest order component of $D$ by momenta, so $\frac {\p} {\p
x^i} \to i p_i$, and then taking $p_i$ to be coordinates on fibers of $T^*X$. Let the family of the vector spaces $T^*_{G} X$ be a union of vector spaces $T^{*}_G X_x$ over all points $x\in X$, where $T^{*}_G X_x$ denotes a subspace of $T^* X$ transversal to the $G$-orbit through $x$. The operator $D$ is transversally elliptic if its symbol $\sigma(D)$ is invertible on $T^*_G X \setminus 0$, where $0$ denotes the zero section.
We need a few notions of $K$-theory [@MR0224083]. Let ${\mathrm{Vect}}(X)$ be the set of isomorphism classes of vector bundles on $X$. It is an abelian semigroup where addition is defined as the direct sum of vector bundles. For any abelian semigroup $A$ we can associate an abelian group $K(A)$ by taking all equivalence classes of pairs $(a,b) \sim (a+c,b+c)$, where $a,b,c
\in A$. Taking ${\mathrm{Vect}}(X)$ as $A$ we define the $K$-theory group $K(X)$. Its elements are pairs of isomorphism classes of vector bundles $(E_0,E_1)$ over $X$ up to the equivalence relation $(E_1,E_1) \sim (E_0 \oplus H, E_1 \oplus H)$ for all vector bundles $H$ over $X$. If $X$ is a space with a basepoint $x_0$, then we define $\tilde K(X)$ as a kernel of the map $i^*: K(X) \to K(x_0)$ where $i:
x_0 \to X$ is the inclusion map. Next we define relative $K$-theory group $K(X,Y)$ for a compact pair of spaces $(X,Y)$. Let $X/Y$ be the space obtained by considering all points in $Y$ to be equivalent and taking this equivalence class as a basepoint. Then $K(X,Y)$ is defined as $\tilde K(X/Y)$. Equivalently, $K(X,Y)$ consists of pairs of vector bundles $(E_0,E_1)$ over $X$ such that $E_0$ is isomorphic to $E_1$ over $Y$, and considered up to the equivalence relation $(E_0,E_1) \sim (E_0 \oplus H, E_1 \oplus H)$ for all vector bundles $H$ over $X$. For a non-compact space, such as a total space of vector bundle $V \to X$, we define $K(V)$ as $\tilde K (X^V)$, where $X^V$ is a one-point compactification of $V$, or equivalently $B(V)/S(V)$, where $B(V)$ and $S(V)$ is respectively a unit ball and unit sphere on $V$.
If a group $G$ acts on $X$ we can consider the set of isomorphism classes of $G$-vector bundles over $X$. It is an abelian semi-group, to which we associate an abelian group $K_G(X)$. All constructions above can be done in $G$-equivariant fashion.
Since the symbol of a transversally elliptic operator is an isomorphism $\sigma(D):
\pi^* E \to \pi^* F $ of vector bundles over $T^*_{G} X$ outside of zero section, by definition it represents an element of $K_{G}(T^*_G X)$. One can show that the index of transversally elliptic operator does not depend on continuous deformations of it symbol, hence it depends only on the homotopy type of the symbol. The index vanishes for a symbol which is induced by an isomorphism of vector bundles $E$ and $F$. Therefore the index of $D$ depends only on an element of $K_{G}(T^*_G X)$ which represents symbol $\sigma(D)$.
The equivariant index was defined for any group element $g$ as an alternating sum of traces of $g$ in representations $R^{i}$ in which $G$ acts on the cohomology groups $H^i$ of the complex $E$. One can show that for transversally elliptic operators the representations $R^i$ can be decomposed into a direct sum of irreducible representations where each irreducible representation enters with a finite multiplicity. In the elliptic case the number of irreducible representations which appear is finite since cohomology groups $H^i$ have finite dimensions. Let $\chi_\alpha$ be a character for each irreducible representation $\alpha$. Then the index of transversally elliptic operator is $\sum_{\alpha} m_{\alpha} \chi_{\alpha}$ where $m_{\alpha}$ are finite integer multiplicities. Thus the index can be regarded as a distribution on $G$, so that the multiplicities $m_{\alpha}$ are coefficients in its Fourier series expansion. Let $\CalD'(G)$ be the space of distributions on $G$.
Consider an example. Let $X$ be a circle $S^1$ on which group $G=\U(1)$ acts in a natural way. Let $E_0$ be the trivial rank one bundle $\CalE$ over $S^1$, and $E_1$ be the zero bundle. Let $D: \Gamma(\CalE) \to 0$ be the zero operator. Then the cohomology group $H^0$ is the space of all functions on a circle, and $H^1$ vanishes. Functions on a circle can be decomposed into Fourier modes labeled by integers, so that each mode corresponds to an irreducible representation of $\U(1)$. If $q=e^{i\alpha}$ for $\alpha \in [0,2\pi)$ denotes an element of $\U(1)$, then we obtain the index $$\ind 0 = \sum_{-\infty}^{\infty} q^{n} = \sum_{-\infty}^{\infty} e^{in\alpha}
= 2 \pi \delta (\alpha).$$
We see that the index is not a smooth function on $\U(1)$, but a distribution – the Dirac delta-function.
We learned that the index is a map from $K$-theory group of $T^*_{G} X$ to distributions on $G$ $$\ind: K_{G} (T^*_G X ) \to \CalD'(G).$$ Moreover, the index is a group homomorphism with respect to the abelian group structure on $K_{G}(T^*_G X)$ and the addition operation on $\CalD'(G)$. The abelian groups $\CalD'(G)$ and $K_{G}(T^*_G X)$ are modules over the character ring $R(G)$. Indeed, $K_{G}(pt) = R(G)$ since elements of $R(G)$ are formal linear combinations of irreducible representations of $G$, and $K_{G}(X)$ has a module structure over $K_{G}(pt)$, since we can take tensor products of vector bundles representing $K_{G}(X)$ with trivial vector bundles representing $K_{G}(pt)$. The module $\CalD'(G)$ has a torsion submodule. For example, the Dirac delta-function on a circle supported at $q=1$ is a torsion element of $\CalD'(\U(1))$, because it is annihilated by $q-1$. One can show that the support of the index is a subset of points $g \in G$ for which $X^{g} \neq
\varnothing$, where $X^{g} \subset X$ is the $g$-fixed set. If $G$ acts freely on $X$ then the index is supported at the identity of $G$, hence it is a pure torsion element.
From now we consider the case $G=\U(1)$. We can find torsion free part of the index if we know it as a function on a generic group element $g \neq \Id$. If $X^g$ consists of non-degenerate points, then we can repeat the argument used in the elliptic case and obtain the formula . In the elliptic case, separate contributions from fixed points are not well defined at $q=1$, but the total sum is well defined, since the index is a finite polynomial in $q$ and $q^{-1}$. In the transversally elliptic case, if we add contributions of fixed points formally defined by the formula , we will obtain correctly only the torsion free part of the index. In other words, we will obtain the index up to a singular distribution supported at $q=1$.
To fix the torsion part, we should find a way in which we associate distributions to rational functions given by the formula . This procedure is explained in details in [@MR0482866]. For example, the contribution to the index of $\bar \p$ operator from the origin of $\BC$ as a rational function is $$\ind_q(\bar \p)|_0 = \frac {1} {1-q^{-1}}.$$ There are two basic ways to associate a distribution to it, which we call expansions in positive or negative powers of $q$: $$\begin{aligned}
&\left [ \frac 1 {1-q^{-1}} \right]_+ = - \frac {q} {1-q} = - \sum_{n=1}^{\infty} q^n \\
&\left [ \frac 1 {1-q^{-1}} \right]_- = - \sum_{n=0}^{\infty} q^{-n}.\end{aligned}$$ These two regularizations differ by a torsion element (a distribution supported at $q=1$): $$\left [ \frac 1 {1-q^{-1}} \right]_+ - \left [ \frac 1 {1-q^{-1}} \right]_- =
- \sum_{n=-\infty}^{\infty} q^n = - 2\pi i \delta (q-1).$$
The decomposition of $K_{G} (T^*_G X) $ to the torsion part and the torsion free part can be described by the exact sequence $$\label{eq:exact-sequence}
0 \to K_G(T^*_G(X \setminus Y)) \to K_G (T^*_G X) \to K_G (T^*X |_Y) \to 0,$$ where $Y$ is the fixed point set in $X$. Since $G$ acts freely on $X \setminus
Y$, the image of $K_{G}(T^*_G(X \setminus Y))$ under the index homomorphism is a torsion submodule of $\CalD'(G)$. The last term of the sequence is the torsion free quotient determined completely by the fixed point set $Y$. Using a vector field $v$ on $X$ generated by action of $G$, it is possible to construct two homomorphisms $$\theta^{\pm}: K_G (T^*X|_Y) \to K_G (T^*_G X),$$ where $\pm$ signs correspond to a choice of the direction of the vector field. First, given a symbol $\sigma: \pi^*E_0 \to \pi^*E_1$, representing an element of $K_G(T^*X|_Y)$, we extend it to an open neighborhood $U$ of $Y$. It is an isomorphism outside of the zero section. Second, we define a symbol $\tilde
\sigma$, restricting symbol $\sigma$ to fibers of $T^*_G X$ shifted in the direction of the vector field $v$ $$\tilde \sigma(x,p) = \sigma (x, p + v e^{-p^2}),$$ where $(x,p)$ are local coordinates on $T^*X$ in a neighborhood of $Y$. Outside of $Y$ the symbol $\tilde \sigma$ is an isomorphism for all points on fibers of $T^*_GX$ (not only outside of zero section). In other words, $\tilde \sigma$ is an isomorphism everywhere in the neighborhood $U$ outside of the fixed point set $Y$. Hence $\tilde \sigma$ represents an element of $K_G(T^*_G U)$. Since $U$ is open in $X$, using the natural homomorphism $K_G( T^*_GU) \to K_G (T^*_G X)$ we get an element of $K_G (T^*_G X)$.
Applying this construction to the space $X=\BC^n$ on which $\U(1)$ acts with positive weights $m_1,\dots, m_n$, and taking generator of $K(T^* \BC^n|_0)$ associated with $\bar \p$ operator, we get its images under $\theta^{\pm}$ in $K_G(T^*_G \BC^n)$. A direct computation shows that $$\ind \theta^{\pm} [\bar \p] = \left [ \frac 1 { \prod(1-q^{-m_i})}
\right]_{\pm} .$$
Now assume that using the vector field $v$ it is possible to trivialize a transversally elliptic operator everywhere on $T^*_GX$ outside of the fixed point set $Y$, and that in a neighborhood of the fixed point set the trivialization is isomorphic to just described with some choice of $\pm$ signs for each fixed point. Then the index is computed by summing contributions from the set of fixed points, where each contribution is regularized by an expansion in positive or negative powers of $q$, according to the choice of sign for the $\theta$ homomorphism.
For example in this way we get the $\U(1)$ index of the following operator on $\CP^{1}$: $$\label{eq:cp1-trans-ellipt}
\ind ( f(\theta) \bar \p + (1-f(\theta)) \p) =
\left[ \frac {1} {1-q^{-1}} \right]_+ + \left[ \frac {1} {1-q^{-1}} \right]_-.$$ Here $\theta$ denotes the polar angle on $\CP^1$ measured from the North pole, and $f(\theta) = \cos^2 (\theta/2)$, so that the operator is approximately $\bar \p $ at the North pole and $\p$ at the South pole. It fails to be elliptic at the equator, but it is transversally elliptic with respect to the canonical $\U(1)$ action on $\CP^1$ whose fixed points are the North and South poles.
[^1]: On leave of absence from ITEP, Moscow, 117259, Russia.
[^2]: What we call $\CalN=2$ supersymmetry on $S^4$ is explained in section \[se:fields-action-symmetries\]. It would be interesting to extend the analysis to more general backgrounds [@Karlhede:1988ax].
[^3]: In other words, $ i_{\phi} \omega = d H(\phi)$ for any $\phi \in g$, where $i_{\phi}$ is a contraction with a vector field generated by $\phi$.
[^4]: By $\SL(n,\BH)$ we mean group of general linear transformation $\GL(n, \BH)$ over quaternions factored by $\BR^{*}$, so that the real dimension of $\SL(n,\BH)$ is $4n^2-1$.
[^5]: See e.g. [@Seiberg:1994aj; @Seiberg:1994rs] keeping in mind that if we start from the Euclidean signature in ten dimensions, the $\SO(1,1)^R$ group is replaced by the usual $U(1)^{R}$ symmetry of $\CalN=2$ theory.
[^6]: The author thanks N.Berkovits for communications.
[^7]: For $U(N)$ bundles we have the total Chern class $c = \det (1 + \frac {iF} { 2\pi}) = \prod (1 +x_i) = c_0 + c_1 + \dots$, where $F$ is the curvature which takes value in the Lie algebra of the gauge group, $x_i$ are the Chern roots, and $c_k$ is polynomial of degree $k$ in $x_i$. We have $c_2 = \sum_{i < j} x_i x_j = \frac 1 2 (\sum x_i)^2 - \frac 1 2 \sum x_i^2$. If $c_1 = \sum x_i$ vanishes, we get $c_2 = -\frac 1 2 \int \tr
\frac {iF}{2\pi} \wedge \frac {iF}{2\pi}= \frac 1 {8\pi^2} \int \tr F \wedge F = -\frac 1 {8 \pi^2} \int (F,\wedge F)$, where the trace is taken in the fundamental representation. The parentheses $(a,b)=-\tr a b$ denote the positive definite bilinear form on the Lie algebra which is assumed in the most of the formulas.
[^8]: The author thanks H. Nakajima for a discussion.
[^9]: The correction $\tilde m = m + (\ve_1 + \ve_2)/2$ appeared in the v2 of this preprint, while in v1 it was erroneously assumed that $\tilde m = m$. The author thanks Takuya Okuda for pointing out this issue.
[^10]: In this section we use the indices $\hat \mu ,\hat
\nu=1,\dots,4 $ to enumerate the vielbein basis elements, that is $e^{\hat \mu
}_{\lambda} e^{\hat \nu}_{\nu} = \delta^{\hat \mu \hat \nu}$ where $\delta^{\hat \mu \hat \nu}$ is the four-dimensional Kronecker symbol. Then $\Gamma^{\hat \mu}$ are the four-dimensional gamma-matrices normalized as $\Gamma^{(\hat \mu} \Gamma^{\hat \nu)}= \delta^{\hat \mu \hat \nu}$,
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Q:
can't find out the period of my signal
I've a noisy signal, which is the sound of motor with a constant speed, so the sound "should" be periodic, I know that there is a way to use the autocorrelation function to get the period,I did it, but I can't figure out the period. Any idea how to do that bellow the signal and the result of the autocorrelation :
the signal
the autocorrelation result :
A:
Here's an attempt to do what you need in scilab.
The top plot shows some data that I synthesized. The second plot shows the auto-correlation of the raw data (zoomed in around the central peak of the auto-correlation). The red circles show the peaks found using this find_peaks function.
The final plot shows the difference between all the peak locations. This will be an estimate of the period. Because you're not really guaranteed that the underlying period will be an integer number of samples, you should probably find the mean of these values.
In this case, the "true" period is 1/f0 = 11.191996, and taking diffs = diff(peaks); and then mean(diffs(10:173)) yields 11.195122.
Code below.
N = 1000;
f0 = 0.0893495634;
phi = rand(1,1,'uniform')*2*%pi;
sigma = 0.5;
x = sin(2*%pi*[0:N-1]*f0 + phi) + sigma*rand(1,N,'normal');
XC = xcorr(x);
clf
subplot(311)
plot(x);
subplot(312)
plot(XC);
peaks=peak_detect(XC,0);
plot(peaks,XC(peaks),'ro')
a = get('current_axes');
a.data_bounds=[950 1050 -500 800];
subplot(313)
plot(diff(peaks));
A:
it seems that are you applying some window function in your signal (autocorrelation Plot seems windowed) !
To do its work, split your signal in a constant framed data overlaped or not, apply autocorrelation function:
$$y(k)=\sum_{n=0}^{N-1}x(k) * x(n+k)$$
At the end find the peak position and congrats you found the period !
How do it in matlab here
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Xavier Tondo
Xavier Tondo Volpini (5 November 1978 in Valls, Spain – 23 May 2011 in Monachil, Spain) was a Spanish professional road racing cyclist who specialized in mountain stages of bicycle races.
Death
Tondo was killed after being apparently trapped between his own garage door and car, and crushed by the door, while preparing his bicycle for a training ride. Teammate Beñat Intxausti was with him at the time of the accident.
To commemorate Xavier Tondo the 100%Tondo sportive is held yearly, starting in Sant Joan les Fonts and finishing in Vallter 2000.
Drugs
In February 2011, it was reported that a pro cyclist had tipped off police about a doping ring. Tondo was later identified as that cyclist.
According to the Spanish newspaper El País, Tondo received an email in December 2010, which offered several doping products, including EPO, human growth hormone, Nandrolone, and Clenbuterol, all at low prices. Tondo gave the email to the police.
Palmarès
2002
1st Stage 7 Tour of Qinghai Lake
2005
1st Overall Volta ao Alentejo
1st Stage 4
1st Stage 3 Vuelta a Asturias
2007
1st Overall Troféu Joaquim Agostinho
1st Prologue
1st Overall Volta a Portugal
2008
1st Subida al Naranco
2009
1st Stage 5 Tour de San Luis
8th Overall Volta a Catalunya
2010
1st Stage 6 Paris-Nice
2nd Overall Volta a Catalunya
1st Stage 3
5th Overall Vuelta a España
2011
1st Stage 4 Tour de San Luis (ITT)
1st Overall Vuelta a Castilla y León
5th Overall Tour of the Basque Country
6th Overall Volta a Catalunya
See also
Professionals who died during training and other cycling related deaths
References
External links
100% Tondo commemorative sportive
Category:1978 births
Category:2011 deaths
Category:People from Valls
Category:Spanish male cyclists
Category:Accidental deaths in Spain
Category:Vuelta a España cyclists
Category:Giro d'Italia cyclists
Category:Volta a Portugal winners
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Raleigh native Chesson Hadley gets big win in Boise Open
(Metro Golf Magazines File Photo)Chesson Hadley fired a 6-under 65 in the final round of the Albertsons Boise Open to move closer to fully exempt status on the PGA Tour.
BOISE, Idaho (AP) — Chesson Hadley rallied to win the Albertsons Boise Open on Sunday to put himself in position for fully exempt PGA Tour status halfway through the four-event Web.com Tour Finals.
Hadley shot a 6-under 65 at Hillcrest Country Club — birdieing Nos. 15-17 and closing with a par — for a one-stroke victory over Ted Potter Jr. (67) and Jonathan Randolph (69).
His PGA Tour card already secured with a ninth-place finish on the Web.com Tour's regular-season money list, Hadley earned $180,000 to overtake Brice Garnett for the No. 1 spot on the combined regular-season and Finals money list in a bid for fully exempt PGA Tour status. He also tops the Finals money list. If he takes either spot, he will also earn an exemption into The Players Championship.
"I proved to myself that I'm good enough to play on tour," said Hadley, 2014 Puerto Rico Open winner on the PGA Tour. "When you lose your card, you start thinking, and you get to some pretty dark places. It's so awesome to be going back there. I'm kind of fired up about getting that No. 1 spot."
Hadley finished at 16-under 268 for his second Web.com Tour victory of the season and fourth overall. He began the round five strokes behind leader Tyler Duncan.
"I know I came out of nowhere. No cameras showed up on me until I was on 16," Hadley said. "Obviously, every win is such a great feeling. The course was not playing easy, and I just kind of hung in there."
Hadley, Potter and the other top-25 finishers on the Web.com regular-season money list have earned PGA Tour cards. They are competing against each other for tour priority, with regular-season earnings counting in their totals.
The others — Nos. 26-75 on the Web.com regular-season money list, Nos. 126-200 in the PGA Tour's FedEx Cup standings, and non-members with enough money to have placed in the top 200 in the FedEx Cup had they been eligible — are playing for an additional 25 cards.
Randolph wrapped up a card with his $88,000 check.
"Obviously, that was the goal at the start of the week," Randolph said. "I know that's where I belong, and I proved it to myself, being able to come out here and get it done this week."
Alex Cejka also clinched a card. He was fourth at 14 under after a 69.
Duncan had a 74 to tie for fifth at 12 under. He also locked up a tour card.
Six-time PGA Tour winner Hunter Mahan had a 74 to tie for 61st at 3 under. He has made $21,320 in the two events after finishing 182nd in the FedEx Cup standings.
Copyright 2017 The Associated Press. All rights reserved. This material may not be published, broadcast, rewritten or redistributed.
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Direct coupling of solid phase microextraction with electrospray ionization mass spectrometry: A Case study for detection of ketamine in urine.
Electrospray ionization mass spectrometry (ESI-MS) is a commonly used technique for analysis of various samples. Solid phase microextraction (SPME) is a simple and efficient technique that combines both sampling and sample preparation into one consolidated step, preconcentrating extracted analytes for ultra-sensitive analysis. Historically, SPME has been coupled with chromatography-based techniques for sample separation prior to analysis, however more recently, the chromatographic step has been omitted, with the SPME device directly coupled with the mass spectrometer. In this study, direct coupling of SPME with ESI-MS was developed, and extensively validated to quantitate ketamine from human urine, employing a practical experimental workflow and no extensive hardware modification to the equipment. Among the different fibers evaluated, SPME device coated with C18/benzenesulfonic acid particles was selected for the analysis due to its good selectivity and signal response. Different approaches, including desorption spray, dripping, desorption ESI and nano-ESI were attempted for elution and ionization of the analytes extracted using the SPME fibers. The results showed that the desorption spray and nano-ESI methods offered better signal response and signal duration than the others that were evaluated. The analytical performance of the SPME-nano-ESI-MS setup was excellent, including limit of detection (LOD) of 0.027 ng/mL, limit of quantitation (LOQ) of 0.1 ng/mL, linear range of 0.1-500.0 ng/mL (R2 = 0.9995) and recoveries of 90.8-109.4% with RSD 3.4-10.6% for three validation points at 4.0, 40.0 and 400.0 ng/mL, far better than the performance of conventional methods. The results herein presented, demonstrated that the direct coupling of SPME fibers with ESI-MS-based systems allowed for the simple and ultra-sensitive determination of analytes from raw samples such as human urine.
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Q:
DNN theme not loaded after migration
I'm absolutely new to DNN world, and I have to migrate a bunch of websites from a web server to another.
Following my expectations and some "guide" on the web, i did:
Exported SqlServer databases from old server
Imported all databases in new server
Copied the whole c:\inetput\vhosts directory from old server to the new one
Created manually the vhosts entries in IIS to host the websites (setting the vhost on the httpdocs dir and converting to application the subfolder "portal"
After some problem with app pool, user permissions, database user configurations etc. i reached to get websites running.
But what it happens is that the websites seems to load the default "theme" instead of the one that was using in production server. What did I forgot?
A:
There is likely an Error that is being thrown with the current "skin" so you'll need to get into the Event Viewer (under the admin page) if you can get logged in, or into the EventLog table in the database to see what errors are being thrown.
select top 50 * From eventlog order by logcreatedate desc
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Sunny's World Cup go-slow
The first World Cup match was an occasion tarnished by the most bizarre of innings from Sunil Gavaskar
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Q:
Jquery : Function to autoscroll the page on load
I need a jquery function that will scroll the page (from where the user currently is) by 100 pixels. Please note, this function does not scroll from the top, it scrolls from the current position.
so far i have :
<script type="text/javascript">
$(document).ready(function() {
$(window).scrollTop($(window).scrollTop()+100);
});
</script>
It doesn't do anything.
A:
Thanks to @HirstoYankov for partly helping.
Below is the code that fixed it:
<script type="text/javascript">
$(window).load(function () {
$(window).scrollTop($(window).scrollTop()+1);
});
</script>
Its important to use $(window).load and not $(document).ready
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5 Ways I Overcame My Tech Distractions (With More Tech)
October 23, 2017
As I was typing this line, my phone buzzed to let me know I’d been mentioned on Twitter. When I returned to type these words, the thought struck me that I had better check if my editor had replied to my email.
With flurries of app notifications, 24/7 emails, and the mere thought of our smartphones, staying focused with multiple screens is no easy task, whether we are interrupted by other apps, or we’ve simply decided to check Facebook in the middle of writing an email.
A digital detox might minimize these interruptions, but as a technology writer, I was convinced there were tech-based means to manage my attention.
Luckily, productivity coach Craig Jarrow, who founded the blog Time Management Ninja, agreed with me. “Technology can be a time-waster — but it can also be our biggest time-saver,” he says. Here’s how I experimented with tech to reclaim my focus and eliminate tech distractions…with more tech:
1. Blocking Notifications
The chime of a smartphone notification is enough to distract me from most tasks — and I’m not alone. According to researchers, simply hearing a notification distracted people and reduced their performance nearly as much as if they had stopped to respond to the alert.
What’s more, the average person checks their phone about 60 times a day. “When we interrupt our current task , it involves [more brainpower] to re-uptake the task, which impacts productivity and focus,” says Lee Hadlington, a senior lecturer at De Montfort University and author of Cybercognition: Brain, Behaviour and the Digital World.
But blocking notifications from all apps had me checking my phone more — a common consequence, it turns out. “Turning off all notifications can result in as many self-interruptions as having them on,” says Hadlington. “Our smartphones are our conduit to what’s going on in the world. If your alerts are off, you might be equally distracted wondering what could have been happening.”
“Instead, pare down the number of apps that get to send notifications,” Jarrow says. Think again: does that shopping app need to send alerts about new collections?
2. Blocking the Internet and Social Media
Even without the Pavlovian buzz of notifications, having the internet at my fingertips often inspires me to head down the rabbit hole of cute animal videos and endless newsfeed scrolling.
“The thing about our engagement with technology is that the thrill of reading your newsfeed or performing a search doesn’t dissipate over time,” Hadlington says.
So I decided to go cold-turkey. I downloaded a program to block all social sites, and another that would block the internet entirely. Being unable to access Facebook during periods of work felt good — the act of trying to load Facebook and failing seemed to slowly erode my habit of checking my newsfeed at the first twinge of boredom.
However, while the programs are great for project work that eschews the web, my job involves Twitter, and for that matter, the internet. For an everyday solution, “use your programs in full-screen mode to ground you in the activity you’re doing,” says Jarrow. There are also writing processor programs designed to do just that.
3. Testing my Bluetooth Button
I had high hopes for Saent, an app and Bluetooth-connected button that promised to maximize my productivity. The trick is that you link a physical cue — pressing the sleek, highly tactile button — with the start of a distraction-free work period on your computer that’s followed by a timed break. It’s based on the Pomodoro technique of time management, but the systematic division of projects into time chunks might not suit all working styles.
“It depends on the industry and individual differences whether this method can work. Some people can’t block out all distractions [during a focus period],” says Hadlington.
On some days, I could focus much longer than others. On those days, I didn’t want a reminder that my focus session was up.
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Equipment Dimensions & Weight
Length (cm/inches)
263 inches / 668 cm
Width (cm/inches)
188 inches / 478 cm
Height (cm/inches)
93 inches / 237 cm
Manufacturer reserves the right to change or alter specifications at any time. Precor protects its product by vigorously enforcing its patent, trademark, copyright and other applicable intellectual property rights in the USA and in other countries. Precor is a registered trademark of Precor Incorporated.
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Background {#Sec1}
==========
Trauma is a major public health problem in India as a result of accelerated urbanization and industrialization \[[@CR1]\]. The increase in trauma in recent years has led to a greater incidence of fractures treated with internal fixation, with tibia fractures being reported as one of the most common and complicated fractures \[[@CR1], [@CR2]\]. Tibia fractures that are treated with surgery are at risk of serious and debilitating infections \[[@CR3]\]. Previous research has suggested that the incidence of infections following internal fixation is higher in low and middle income countries (LMICs), as operating rooms are often not sterile and may contain microbes responsible for wound infection \[[@CR4]--[@CR6]\].
Surgical site infection (SSI) following internal fracture fixation poses large socioeconomic and quality of life implications for the patient \[[@CR3], [@CR7]\]. There is an increased risk within LMICs due to the disproportionately large number of trauma incidents, particularly attributed to motor vehicle accidents, coupled with the aforementioned risk factors of poor sterilization of orthopaedic wards and long times between fracture and surgery \[[@CR4]--[@CR6]\].
The primary objectives of the current study were to determine the incidence of infection within one year of surgery and to describe the distribution of infections by location (superficial, deep) and time of diagnosis (early, delayed, late) for open and closed tibia fracture patients in India. Secondary objectives were to: 1) describe the symptoms, management, and treatment outcomes of infections, 2) explore the effect of fracture type, hospital type, time to surgery, and planned duration of antibiotics on the incidence of infection, 3) compare the proportion of fractures healed at 12 months in patients with and without infections, and 4) evaluate health-related quality of life over 12 months in patients with and without infections.
Methods {#Sec2}
=======
Study overview {#Sec3}
--------------
We conducted a multi-center, prospective, observational cohort study to investigate the incidence of infections within one year for open and closed tibia fracture patients who were treated with internal fixation. The method of internal fixation was left to the discretion of the attending surgeon. After obtaining informed consent, baseline and surgical data were recorded. Approval was obtained from the Institutional Review Board (Aurora, Ontario) and each hospital's local Ethics Committee prior to commencing study activities.
Participant identification and eligibility criteria {#Sec4}
---------------------------------------------------
Patients who presented to one of the participating hospitals with a tibia fracture treated with internal fixation were screened for study eligibility. The inclusion criteria were: 1) Men and women who are 18 years of age or older. 2) Open or closed tibia fracture (AO 41, 42, and 43) treated by internal fixation (plate or nail) or by external fixation with planned conversion to plate or nail. 3) Ability to understand the content of the subject information/informed consent form and to be willing to participate in the clinical investigation. 4) Provided written informed consent.
The exclusion criteria were: 1) Previous wound infection or osteomyelitis at the same limb (according to subject history). 2) Patients who plan to undergo conversion surgery at a different hospital. 3) Previous fracture with retained hardware in injured extremity that will interfere with implant fixation. 4) Immunological deficiency disease. 5) Tumor related fractures. 6) Any severe systemic disease: class V-VI of the American Society of Anesthesiologists (ASA) physical status classification \[[@CR8]\]. 6) Recent history of substance abuse that would preclude reliable assessment. 7) Patient is a prisoner. 8) Participation in any other medical device or medicinal product study within the previous month that could influence the results of the present study. Reasons for ineligibility were documented.
Data collection {#Sec5}
---------------
After providing informed consent, baseline information was documented and participants underwent a haematology analysis (Leucocyte count, CRP level, and ESR level) and radiographs (AP, lateral) before surgery. Details regarding the surgical procedure, including antibiotic prophylaxis, were documented. Post-operatively participants underwent a haematology analysis and x-rays.
Participants with any symptoms of surgical site infections underwent further investigations including radiological assessment, hematological analysis and bacteriological culture and antibiogram whenever possible to determine whether infection was present. If infection was diagnosed, infection management including administered antibiotics, wound care, surgical intervention performed, and infection treatment outcome were recorded.
Participants attended clinic visits at 3 months, 6 months, and 12 months post-surgery, or were contacted by telephone to collect as much information as possible if unable to attend a follow-up visit. At each visit, patients were assessed for infections and fracture healing. Haematology and radiographs were taken as standard of care. Antibiotic use was documented. Participants also completed the EuroQol-5 Dimensions (EQ-5D). The EQ-5D is a standardized instrument for use as a measure of health outcome, primarily designed for self-completion. At the 12 month visit, any planned revision surgeries were also documented.
Confirmation of eligibility and review of infections {#Sec6}
----------------------------------------------------
An independent Adjudication Committee comprised of three orthopaedic trauma surgeons confirmed the eligibility for cases in which patient eligibility was in doubt. They also reviewed reported infections to confirm the presence of infection and classify the infection as a superficial incisional surgical site infection (SSI) or a deep incisional SSI using CDC criteria \[[@CR9]\]. They also confirmed the timing of infection as early (onset of symptoms within 2 weeks of injury), delayed (onset of symptoms 2--10 weeks after injury) or, late (onset of symptoms more than 10 weeks after injury).
Data analysis {#Sec7}
-------------
We summarize participant characteristics using descriptive statistics expressed as means and standard deviations for continuous variables) or counts and percentages for categorical variables. For analysis of primary outcomes, the incidence of infection within one year of the internal fixation surgery was reported as a proportion. A Fisher's exact test was used to determine if the incidence of infection and infection type (superficial versus deep) differed across fracture types (open versus closed fractures). Infections were classified by timing of onset and a Chi-square test was used to compare the incidence of early versus delayed versus late onset infections across fracture types.
For analysis of secondary outcomes, infection symptoms and management in open and closed fracture patients were summarized using descriptive statistics. Infection outcomes and fracture healing at 12 months for open and closed fracture patients are reported as proportions. A Chi-square test was used to compare if the incidence of infections differed by hospital type (public versus private versus combination), as well as by fixation technique used and Gustilo classification for open fractures. Fisher's exact tests were used to determine if the incidence of infection differed across timing of surgery (within 6 h of injury versus greater than 6 h from injury), and by fixation device material (stainless steel versus titanium). Fracture healing status at 12 months in patients with an infection versus those without was also compared using a Fischer's exact test. The EQ-5D scores are reported as means and standard deviations for participants with and without infections. Time to healing in patients with and without infections was explored using a *t*-test, and EQ-5D scores in patients with and without infections were compared using two-way repeated measures ANOVA. Level of significance was determined as *p* \< 0.05. Data analysis was done using Statistical Analysis Software (SAS, v9.2, Cary, North Carolina, United States).
Results {#Sec8}
=======
Of the 899 patients screened for participation, 800 met the inclusion criteria and provided informed consent (Fig. [1](#Fig1){ref-type="fig"}). The Adjudication Committee deemed 13 participants to be ineligible. 787 participants were included in the analyses and 768 participants completed the 12 month follow-up.Fig. 1Participant Flow Diagram
Demographics and fracture characteristics {#Sec9}
-----------------------------------------
The mean age of the study participants was 40.1 ± 14.0 years and the majority were male (79.8%) (Table [1](#Tab1){ref-type="table"}). The majority of participants included in this study were healthy (88.9% had no comorbidities) and non-smokers (95.6%). Less than 20% of participants had insurance. Most participants had completed secondary school (33.8%), junior college (14.2%), or university (36.0%). The most common mechanism of injury was motor vehicle accidents (71.4%) (Table [2](#Tab2){ref-type="table"}). Less than one-third of participants (28.5%) had additional injuries or fractures. The majority of participants had closed fractures (625 participants, 79.4%), with 162 participants (20.6%) having open fractures.Table 1Participant DemographicsCharacteristicN (%) *N* = 787Age (mean ± SD)40.1 ± 14.0Sex Female159 (20.2%) Male628 (79.8%)Ethnicity Indian787 (100.0%)Education None52 (6.6%) Primary school74 (9.4%) Secondary school266 (33.8%) Junior college112 (14.2%) University283 (36.0%)Insurance None650 (82.6%) Governmental27 (3.4%) Private110 (14.0%)Smoker No752 (95.6%) Yes22 (2.8%) Former13 (1.7%)Co-morbidity None700 (88.9%) Yes^a^87 (11.1%) High Blood Pressure54 (62.1%) Heart Disease16 (18.4%) Osteoporosis5 (5.7%) Osteoarthritis or5 (5.7%) Degenerative Arthritis Lung Disease4 (4.6%) Stomach Disease or3 (3.4%) Ulcer Kidney Disease2 (2.3%) Liver Disease2 (2.3%) Hepatitis B2 (2.3%) Epilepsy2 (2.3%) Blood Disorder or1 (1.1%) Anemia Osteopenia1 (1.1%) Cancer1 (1.1%) Cervical Tubercular1 (1.1%) Lymphadenitis Fever Since 2 Days1 (1.1%) Cough1 (1.1%) Hyperlipidemia1 (1.1%) Unspecified1 (1.1%)Diabetic No734 (93.3%) Yes -- Insulin-dependent24 (3.0%) Yes -- Insulin-independent29 (3.7%)^a^Does not equal to 100% due to patients having multiple comorbidities Table 2Injury and Fracture CharacteristicsCharacteristicN (%) *N* = 787Mechanism of injury Motor vehicle accident562 (71.4%) Fall186 (23.6%) Struck30 (3.8%) Other3 (0.4%) Twisting3 (0.4%) Sports2 (0.3%) Stress Fracture1 (0.1%)Work-related injury70 (8.9%)Additional fractures / injuries224 (28.5%)AO -- Müller Fracture Classification 41-Proximal255 (32.4%) 42-Diaphyseal337 (42.8%) 43-Distal95 (12.1%) 44-Malleolar100 (12.7%)Open fracture162 (20.6%) Gustilo classification I67 (41.4%) II44 (27.2%) IIIA28 (17.3%) IIIB21 (13.0%) IIIC1 (0.6%)Closed fracture625 (79.4%) Tscherne classification 0276 (44.2%) 1251 (40.2%) 282 (13.1%) 316 (2.6%)Totals may not add due to missing data
Surgical management and peri-operative care {#Sec10}
-------------------------------------------
The mean time from injury to surgery was 73 ± 107.5 h (Table [3](#Tab3){ref-type="table"}). The time to surgery was 64 ± 108.0 h for open fractures and 76 ± 108.0 h for closed fractures. The majority of fractures were stabilized with a plate (43.2%) or a reamed intramedullary nail (41.3%). The majority of implants were from local (Indian) manufacturers (86.0%) versus global manufacturers (14.0%).Table 3Surgical and Peri-Operative ManagementOpen *N* = 162 N (%)Closed *N* = 625 N (%)Total *N* = 787 N (%)Duration time from injury to surgery (hours) (Mean ± SD)64.0 ± 108.076.0 ± 108.073.1 ± 107.5Duration time from hospital admission to surgery (hours) (Mean ± SD)55.1 ± 78.059.0 ± 78.058.1 ± 77.3Previous irrigation and debridement57 (35.2%)1 (0.2%)58 (6.5%)Duration of definitive fixation surgery (hours) (Mean ± SD)2.12 ± 1.01.53 ± 1.01.57 ± 1.0Method of fixation Plate32 (19.8%)308 (49.3%)340 (43.2%) Reamed intramedullary nail102 (63.0%)223 (35.7%)325 (41.3%) Screw7 (4.3%)62 (9.9%)69 (8.8%) Unreamed intramedullary nail13 (8.0%)16 (2.6%)29 (3.7%) Screw and wire4 (2.5%)5 (0.8%)9 (1.1%) Wire0 (0.0%)5 (0.8%)5 (0.6%) Intramedullary nail (unspecified)3 (1.9%)1 (0.2%)4 (0.5%) Plate and wire0 (0.0%)2 (0.3%)2 (0.3%) Plate and reamed nail0 (0.0%)1 (0.2%)1 (0.1%) External fixator0 (0.0%)1 (0.2%)1 (0.1%)Implant manufacturer Local147 (90.7%)530 (84.8%)677 (86.0%) Global15 (9.3%)95 (15.2%)110 (14.0%)Additional surgical procedures performed43 (26.5%)89 (14.2%)132 (16.8%)Antibiotics administered during surgery149 (92.0%)576 (92.2%)725 (92.1%)Participant returned to operating room for repeat debridement/irrigation9 (5.6%)1 (0.2%)10 (1.3%)Antibiotics received post-operative157 (97.5%)605 (97.0%)762 (96.8%)Planned duration of post-op antibiotics9.1 days ± 5.08.3 days ± 5.08.5 days ± 5.0Drains used post-op110 (68.3%)354 (56.7%)464 (59.11%)Wound manually cleaned152 (94.4%)415 (66.5%)567 (72.3%)Totals may not add due to missing data
Almost all participants with an open fracture had an irrigation and debridement (92.0%). Approximately one third (35.2%) of open fractures had an irrigation and debridement prior to their surgery for definitive fixation. Very few open fracture participants (5.6%) returned to the operating room for subsequent irrigation and debridement (Table [3](#Tab3){ref-type="table"}). Post-operative drains were used in 68.3% of open fracture patients and 56.7% of closed fracture patients.
Almost all participants received antibiotics during surgery (92.1%) and after surgery (96.8%). The antibiotics were prescribed for an average of 9.1 ± 5.0 days in open fractures and 8.3 ± 5.0 days in closed fractures. The vast majority of patients received Cephalosporin (757 patients). The majority of open fracture participants (94.4%) had their wound cleaned manually post-operatively and two thirds of closed fracture participants (66.5%) had their wound cleaned manually (Table [3](#Tab3){ref-type="table"}). Proximal fractures were most commonly treated with plating (87.8%), while diaphyseal fractures and distal fractures were most commonly treated with reamed intramedullary nailing (81.3%, and 49.5% respectively) (Table [4](#Tab4){ref-type="table"}).Table 4Method of Fixation for AO Fracture TypesAO ClassificationTotal *N* = 787 N (%)Proximal (*N* = 254)Diaphyseal (*N* = 337)Distal (*N* = 95)Malleolar (*N* = 100)Method of fixation Plate223 (87.8%)36 (10.7%)39 (41.0%)42 (42.0%)340 (43.2%) Reamed intramedullary nail2 (0.8%)274 (81.3%)47 (49.5%)2 (2.0%)325 (41.3%) Screw27 (10.6%)1 (0.3%)2 (2.1%)39 (39.0%)69 (8.8%) Unreamed intramedullary nail1 (0.4%)22 (6.5%)3 (3.2%)3 (3.0%)29 (3.7%) Screw and wire0 (0.0%)0 (0.0%)0 (0.0%)9 (9.0%)9 (1.1%) Wire0 (0.0%)0 (0.0%)2 (2.1%)3 (3.0%)5 (0.6%) Intramedullary nail (unspecified)0 (0.0%)3 (0.9%)1 (1.1%)0 (0.0%)4 (0.5%) Plate and wire0 (0.0%)0 (0.0%)0 (0.0%)2 (2.0%)2 (0.3%) Plate and reamed nail0 (0.0%)1 (0.3%)0 (0.0%)0 (0.0%)1 (0.1%) External fixator1 (0.4%)0 (0.0%)0 (0.0%)0 (0.0%)1 (0.1%)Totals may not add due to missing data
Incidence of infection {#Sec11}
----------------------
The incidence of infection within 12 months of surgery was 2.9% (23 participants). The incidence of infection was higher in open fractures (8.0%) (13 infections) as compared to closed fractures (1.6%) (10 infections) (*p* \< 0.0001) (Table [5](#Tab5){ref-type="table"}). Of the 13 infections in open fractures, 1 occurred in a Gustilo Type I fracture, 3 in Type II fractures, and 9 in Type III fractures (*p* = 0.0002) (Table [5](#Tab5){ref-type="table"}). There were 7 deep infections (5 in open fractures and 2 in closed fractures) and 16 superficial infections (8 in open fractures and 8 in closed fractures) (*p* = 0.2362). There were 5 early infections, 7 delayed infections, and 11 late infections (*p* = 0.1675). Infections were seen within patients treated with plating (11 infections), reamed intramedullary nailing (7 infections), unreamed intramedullary nailing (3 infections), screw (1 infection), and unspecified intramedullary nailing (1 infection) (*p* = 0.1110). There was no significant difference between infection rate and implant material used (stainless steel versus titanium, *p* = 0.9643) (Table [5](#Tab5){ref-type="table"}).Table 5Characteristics of Infected and Non-Infected FracturesInfection N(%) *N* = 23No Infection N(%) *N* = 764Total N(%) *N* = 787*p* valueFracture type\<0.0001 Open13 (56.5%)149 (19.5%)162 (20.6%) Closed10 (43.5%)615 (80.5%)625 (79.4%)Open fracture Gustilo classification (*N* = 162)0.0002 I1 (7.7%)66 (44.3%)67 (41.4%) II3 (23.1%)41 (27.5%)44 (27.2%) IIIA2 (15.4%)26 (17.4%)28 (17.3%) IIIB7 (53.8%)14 (9.4%)21 (13.0%) IIIC0 (0.0%)1 (0.7%)1 (0.6%)Closed Fracture Tscherne classification (*N* = 625)0.0588 01 (10.0%)275 (44.7%)276 (44.2%) 15 (50.0%)246 (40.0%)251 (40.2%) 23 (30.0%)79 (12.8%)82 (13.1%) 31 (10.0%)15 (2.4%)16 (2.6%)Hospital Type0.9938 Private20 (87.0%)662 (86.6%)682 (86.7%) Public2 (8.7%)65 (8.5%)67 (8.5%) Combination1 (4.3%)37 (4.8%)38 (4.8%)Surgical Delay \>6 h0.6921 Yes21 (91.3%)670 (87.8%)691 (87.9%) No2 (8.7%)93 (12.2%)95 (12.1%)Method of fixation0.1110 Plate11 (47.8%)329 (43.1%)340 (43.2%) Reamed intramedullary nail7 (30.4%)318 (41.6%)325 (41.3%) Screw1 (4.3%)68 (8.9%)69 (8.8%) Unreamed intramedullary nail3 (13.0%)26 (3.4%)29 (3.7%) Screw and wire0 (0.0%)9 (1.2%)9 (1.1%) Wire0 (0.0%)5 (0.7%)5 (0.6%) Intramedullary nail (unspecified)1 (4.3%)3 (0.4%)4 (0.5%) Plate and wire0 (0.0%)2 (0.3%)2 (0.3%) Plate and reamed nail0 (0.0%)1 (0.1%)1 (0.1%) External fixator0 (0.0%)1 (0.1%)1 (0.1%)Implant Material0.9643 Stainless steel17 (73.9%)567 (74.2%)584 (74.2%) Titanium6 (26.1%)196 (25.7%)202 (25.7%)Fracture healed radiographically by 12 months13 (56.5%)674 (88.2%)687 (87.3%)\<0.0001Radiographic healing time in days (Mean ± SD)223.0 ± 102.6149.2 ± 72.0150.6 ± 73.30.0234Totals may not add due to missing data
Infection incidence was 2.9% at both private hospitals and public or combination hospitals (*p* = 0.9938) (Table [5](#Tab5){ref-type="table"}). There was no difference in the incidence of infection between participants who had surgery within 6 h of their injury compared to greater than 6 h of their injury (*p* = 0.6921) (Table [5](#Tab5){ref-type="table"}).
Fracture healing and health-related quality of life {#Sec12}
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The mean time to fracture healing was 171.5 ± 88.4 days for open fractures and the majority (83.5%) had healed by 12 months. The mean fracture healing time for closed fractures was 145.5 ± 68.2 days and 91.0% had healed by 12 months. Fractures that were infected took 223.0 ± 102.6 days to radiographically heal, whereas fractures that were not infected took 149.0 ± 72.0 days to heal (*p* = 0.0234) (Table [5](#Tab5){ref-type="table"}). Approximately half (56.5%) of the infected fractures were radiographically healed at 12 months, compared to almost all non-infected fractures (88.2%) (*p* \< 0.0001) (Table [5](#Tab5){ref-type="table"}). Participants who had an open fracture and an infection had the lowest EQ-5D scores at 6 months and 12 months (*p* \< 0.0001) and their scores did not return to baseline at 12 months (Fig. [2](#Fig2){ref-type="fig"}).Fig. 2EQ-5D over time. Open-No Infection coincides with Closed-No Infection. Mean EQ-5D was used as a measure of health outcome over 12 months post-surgery
Infection characteristics and management {#Sec13}
----------------------------------------
Symptoms of infections included purulent drainage (62.5%), wound healing disturbance (54.2%), erythema (37.5%), and local pain (33.3%) (Table [6](#Tab6){ref-type="table"}). Bacterial cultures (52.2%) were taken in approximately half of the participants with infections and positive bacterial cultures were found in 44.4% of the cultures taken. The most common classification of organisms isolated was aerobic gram positive (69.2%) and aerobic gram negative (15.4%). The most commonly isolated organism was staphylocococcus aureaus (84.6%).Table 6Infection Symptoms and ManagementOpen N(%)\
*N* = 13Closed N(%)\
*N* = 10Total N(%)\
*N* = 23Symptoms present Purulent drainage7 (53.8%)7 (70.0%)14 (60.9%) Wound healing disturbance5 (38.4%)7 (70.0%)12 (52.2%) Persisting/increasing local pain6 (46.2%)1 (10.0%)7 (30.4%) Erythema3 (23.1%)5 (50.0%)8 (34.8%) Edema1 (7.7%)2 (20.0%)3 (13.0%) Fever2 (15.4%)0 (0.0%)2 (8.3%) Pain1 (7.7%)0 (0.0%)1 (4.2%) Maggots1 (7.7%)0 (0.0%)1 (4.2%)Bacteria culture found positive5 (71.4%)5 (83.3%)10 (76.9%)Class of organisms isolated Aerobic Gram positive4 (80.0%)3 (60.0%)7 (70.0%) Aerobic Gram negative1 (20.0%)0 (0.0%)1 (10.0%) Anaerobic Gram positive0 (0.0%)1 (20.0%)1 (10.0%) Anaerobic Gram negative0 (0.0%)1 (20.0%)1 (10.0%)Organisms collected *Staphylococcus aureus*4 (80.0%)5 (83.3%)9 (81.8%) *Pseudomonas* spp.1 (20.0%)0 (0.0%)1 (9.1%) *Enterobacter Serus*0 (0.0%)1 (16.7%)1 (9.1%)Infection treated by Antibiotics only7 (53.8%)6 (60.0%)13 (56.5%) Antibiotics and Surgery6 (46.2%)2 (20.0%)8 (34.8%) Surgery only0 (0.0%)2 (20.0%)2 (8.7%)Drainage used4 (30.8%)5 (50.0%)9 (39.1%)Wound manually cleaned12 (92.3%)10 (100.0%)22 (95.6%)Totals may not add due to missing data
The majority of the infections were treated with antibiotics only (53.8%) or with antibiotics and surgery (38.5%). Cephalosporin was the most commonly used antibiotic. Drainage was used in 30.8% of participants with infections and open fractures and 50.0% of participants with closed fractures and infections, and the wound was manually cleaned in 96.2% of participants. At 12 months, 4 infections had completely resolved without persistent drainage and recovery was still in progress for 19 infections.
Discussion {#Sec14}
==========
This study found a low incidence of infections following surgical management of tibia fractures in a cohort of 787 participants in India. The incidence of infection for closed and open fractures was 1.6% and 8.0%, respectively. These incidences are similar to those seen in the largest investigation of tibia shaft fractures in developed countries, which were 1.9% in closed fractures and 8.8% in open fractures \[[@CR10]\]. This result is surprising, as many study participants had injuries resulting from motor vehicle accidents (71.4%), experienced long times to surgery (64 h for open and 76 for closed fractures), and did not have health insurance (\<20% had insurance). The typical time between injury and surgery in developed countries for open fractures has been reported to be 9.8 h, which is drastically shorter than the mean time to surgery of 64 h for open fractures observed within this study \[[@CR11]\]. Contributing factors may include patients living in rural areas of India unable to travel to a hospital in an appropriate time, as well as patient overcrowding in a hospital. Overcrowding can result in delayed treatment, long patient waiting time and stay, overburdened working staff, and poor patient outcomes \[[@CR12]\]. Despite these factors, one possible explanation for the low incidence of infection seen in our study is that participants received prophylactic antibiotics for a mean of 8.3 days in closed fractures and 9.1 days in open fractures, which is much longer than North American standard practice \[[@CR13]--[@CR15]\]. The typical length of prophylactic antibiotic use for tibia fractures in the literature ranges from 1--5 days, demonstrating an extended length of prophylactic antibiotic use seen within our cohort \[[@CR15], [@CR16]\]. However, this is specific to open fractures in India without evidence of use in closed fractures. The widespread and prolonged use of prophylactic antibiotics within this study may be a contributor to the low incidence of infection given that prophylactic antibiotic use has been suggested to reduce the risk of infection after internal tibial fracture fixation by 29% \[[@CR17]\].
Although the incidence of infection seen within our study is relatively low, infection management may not have been optimal. The large number of participants with an unresolved infection at 12 months post-fracture (10 open and 9 closed fractures) suggests that once an infection was present there was difficultly in managing it. Current guidelines outline drainage, debridement, and specific antibiotic prescription as the hallmark treatment regimen for SSI; however infection management within our study was seen to primarily consist of antibiotic alone (53.8%) \[[@CR18], [@CR19]\]. The prolonged length of infection duration within our study may be a result of the limited use of surgical intervention to address infection, as only 38.5% of infections were treated surgically despite surgical debridement being a core component of SSI treatment in current guideline recommendations \[[@CR20]\].
Our EQ-5D results suggest that reducing the incidence of infection is important in increasing participant recovery and quality of life parameters following a tibia fracture, whether it be open or closed, as infection resulted in longer time required to heal and significantly decreased health related quality of life measures. These results align with the literature, which shows that the occurrence of infection significantly decreases participant quality of life when compared to individuals who avoid surgical site infection \[[@CR21]\].
This study is strengthened by its prospective design. The study had clearly defined eligibility criteria prior to study initiation to ensure that all included participants were an appropriate representation of the target population. Additional strength was gained through the large sample size and use of multi-centre recruitment. The study also was able to capture details regarding current clinical practices in India, as standardized treatment methods and antibiotic regimens were not provided for the study. Attending surgeons treated patients as they would in typical clinical practice, and eligibility criteria did not remove patients based on clinical factors such as prolonged delay between injury and treatment. This was important to ensure that results were an honest depiction of tibial fracture patients seen in India. The study is limited by the low number of events seen within the cohort, as only 23 infections were seen across all participants. This may be a result of the large proportion of closed fractures within the study, as they are generally at low risk of infection. The low number of infections decreases the power of our statistical analysis, as the sample size of infections is small. Another potential limitation of this study is that the hospitals that participated may not be representative of the average Indian hospital, as they were large and modern facilities with experience in clinical research. An additional limitation arises due to the inclusion of all types of tibial fractures, making the results more difficult to be used for drawing conclusions for specific tibial fracture types. Furthermore, the study results are specific to India, and cannot appropriately be generalized to other patient populations. However, while these infection rates are specific to India, they align with previously published results of tibia fracture infection after internal fixation within LMICs \[[@CR16]\]. Lastly, there exist several risk factors for surgical site infections that were not investigated outside of smoking and diabetes. Future studies should aim to explore other risk factors and determine if there is a relationship between incidences of infection and fracture type/location.
This is the first investigation to our knowledge to provide a thorough overview of the incidence of infection, prognostic factors, prophylactic antibiotic use, infection management options, and patient quality of life for patients who undergo internal fixation of a tibia fracture in India. This investigation allows for the calculation of required subjects for additional studies in this area of research. Although the incidence of infection within this study is similar to that observed in North America, participants in this study received prophylactic antibiotics for considerably longer than North American standard practice.
Conclusion {#Sec15}
==========
The incidence of infection within this study is similar to that seen in developed countries within the current literature. The duration of prophylactic antibiotic use in India was much longer than typical North American regimens. Future research should aim to identify the best practices for management of infection and for prophylactic antibiotic use to ensure a strict treatment algorithm is established for the management of soft tissue and fracture morphology while avoiding unnecessary overuse.
ASA
: Society of anesthesiologists
EQ-5D
: EurQol-5 Dimensions
LMIC
: Low and middle income countries
SSI
: Surgical site infections
We would like to acknowledge the INFINITI Study Investigators who include Prakash Doshi, Dr. Pritesh Shantaram Khardlkar, and Dr. Lokesh Naik G. (Dr. Balabhai Nanavati Hospital), Vijay H. Upadhyay, Mahipatsinh Parmar, and Brijesh Patel (Hi Tech Hospital), Kiran Doshi, Suryakant Bharamgunde, and Arjun Patil (Kolhapur Institute of Orthopedics and Trauma), Chetan Pradhan and Nalini Gawande (Sancheti Institute for Orthopedics and Rehabilitation), Sunil Kulkarni, Santosh Janyavar, and Dr. Gaurav Gujarathi (Orthopedic Hospital and Post Graduate Institute of Swasthiyog Pratishthan), Sunil M. Shahane and Dr. Rajeev Reddy(Dr. R. N. Cooper Hospital), Mandeep Dhillon, Sarvdeep Dhatt, and Jujhar Singh (Post Graduate Institute of Medical Education and Research), Rajeev Naik, Dr.Shivappa Teli, and Dr.Sailesh.C.J (Dr. B. R. Ambedkar Medical College Hospital), Hitesh Gopalan (Malankara Orthodox Syrian Church Medical College Hospital) and Vijay Kakatkar and Yogesh Dhole (Jai Hospital Pvt. Ltd.).
Funding {#FPar1}
=======
This study was funded by the AO Foundation via the TK Trauma Network.
Availability of data and materials {#FPar2}
==================================
All data generated or analysed during this study are included in this published article. EQ-5D data is only a summary and the raw data is unavailable.
Authors' contributions {#FPar3}
======================
All authors contributed to the study design, study conducted, and the drafting of the manuscript. PD, HG, CP, and SK enrolled patients into the study and SS and MB provided study oversight. All authors read and approved the final manuscript. Corresponding author is MB.
Competing interests {#FPar4}
===================
The authors declare that they have no competing interests.
Consent for publication {#FPar5}
=======================
Not applicable.
Ethics approval and consent to participate {#FPar6}
==========================================
Ethics approval from the Institutional Review Board (IRB Services) and consent to participate was submitted and approved prior to enrollment. The following ethics committees approved the study in India: Nanavati Hospital Ethics Committee, Cooper Hospital Ethics Committee, Ambedkar Hospital Ethics Committee, Sujlam Independent Ethics Committee, Pawar Medical College, Hospital and Research Centre Ethics Committee, Sancheti Institutional Review Board, Malankara Orthodox Syrian Church Medical College Institutional Ethics Committee, Swasthiyog Pratishthan Ethics Committee, and Postgraduate Institute of Medical Education and Research Ethics Committee.
Publisher's Note {#FPar7}
================
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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The GAD approach as an alternative to create viable housing environments
Abstract:
Paper presented at the XXXIII IAHS World Congress on Housing, 27-30 September 2005,"Transforming Housing Environments through Design", University of Pretoria.
ABSTRACT: Housing is a basic need after food. Generally, low-income people living in developing countries are still barred access to legal housing process. Whereas in developed countries both women and men participate in housing provision, in developing countries it is in fact still male “business”. Although lately many concepts and methods were developed to involve women participation in it, yet, they are still subordinate; having limited access both to housing finance (such as housing credit) or being involved in the planning of housing process. Although they are more at home, taking care of children and spend a lot of time in the neighbourhood of their homes. They are the most affected if the access to infrastructure and utilities (such as water and the way to obtain) is insufficient. Due to their small role in creating their housing environments, they are not able to develop themselves, adversely affecting members of the household in improving a better and healthy life. Some housing projects in developing countries show how low income communities achieved to build their housing as desired, for example participation in a community based organisation. Such projects seem to be successful. However, are such successes enjoyed by the whole family? To create an appropriate housing environment the participation of both genders are required.
This paper will analyse the GAD (Gender and Development) Approach as an alternative to create viable housing environments. GAD approach focuses on the participation of both sexes. Their equity is a fundamental issue. The approach is not the current one (since mid 1990s). However, the transfer of this approach in housing environments is still low. The GAD approach will be tried to be implicated in some housing environment projects experiences which are supposed to be successful. Several case studies in Indonesia are described in this paper. The result will show; whether by means of the GAD approach a viable housing environment can be created both for women and men.
Description:
Authors of papers in the proceedings and CD-ROM ceded copyright to the IAHS and UP. Authors furthermore declare that papers are their original work, not previously published and take responsibility for copyrighted excerpts from other works, included in their papers with due acknowledgment in the written manuscript. Furthermore, that papers describe genuine research or review work, contain no defamatory or unlawful statements and do not infringe the rights of others. The IAHS and UP may assign any or all of its rights and obligations under this agreement.
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The production of bis(hydroxyalkyl) esters of benzenedicarboxylic acids such as bis(2-hydroxyethyl) terephthalate has become of significant commercial importance in recent years because these diesters can be polymerized to form linear super polyesters. These polyesters such as polyethylene terephthalate are widely used in textiles, tire cord, and the like.
The present invention has developed from the investigation of new polymeric compositions derived from bis(2-hydroxyethyl) terephthalate which is now a inexpensive and readily available commercial product. It was deemed desirable to endeavor to introduce the excellent properties of bis(2-hydroxyethyl) terephthalate into polymeric compositions which could then be cured to high molecular weight, three-dimensional resinous structures that are substantially infusible and insoluble.
It is thus an object of the present invention to provide novel curable compositions based on bis(2-hydroxyethyl) terephthalate.
It is another object of this invention to provide methods for curing thermoplastic bis(2-hydroxyethyl) terephthalate polymers in the form of thermoset shaped articles and coatings that have resistance to all ordinary solvents and are not adversely affected by ambient conditions of heat and light.
A further object of this invention resides in the provision of block polymers of bis(2-hydroxyethyl) terephthalate polyether derivatives.
Other objects and advantages of the present invention will become apparent from the following description and examples.
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Our channels
E3 2015: Shenmue III Kickstarter Campaign Launched
Yu Suzuki has launched a Kickstarter campaign for Shenmue III. Yes, you heard that right. Shenmue III. You can access the campaign here, and check out a trailer above. We’ll update the post when more details become available.
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New Delhi: Union minister Ravi Shankar Prasad on Tuesday dared the Congress to come clean on the VVIP chopper deal with AgustaWestland deal after an Italian court observed that it was “proven” that illegal money had made their way to Indian officials.
“Now that bribe-givers have been convicted, what should happen to the bribe-takers? Will Mr. (former defence minister A.K.) Antony publicly give a statement on this? Will he accept that his partymen are involved in the scam?” he said.
Senior Congress leader Antony was defence minister when the VVIP helicopter deal with Italy-based firm AgustaWestland was being finalised and sealed in February 2010 during the United Progressive Alliance government.
In a 225-page judgment, a judge at the Milan Court of Appeals found that bribe was paid by the firm to Indian officials to get the $530 million contract for the supply of 12 AW101 choppers.
The Indian government, however, cancelled the deal in 2013 when a controversy over the deal emerged with the arrest of Agusta’s parent organisation Finmeccanica CEO Giuseppe Orsi by Italian authorities.
For all the latest National News, download NewsX App
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Contrary to popular belief, not all certified organic and natural makeup is created using the same ingredients or with the same quality. When looking for a brand that won’t be as harsh with inorganic chemicals and ingredients such as talc, everyone needs to know the signs of what makes a truly certified organic makeup that sources only the best for their customers. In particular, certified organic beauty products that are truly organic are going to be certified with an official label on their packaging and product website.
Certified organic beauty means the creators have to use organic ingredients in their beauty products as well as being carefully regulated by certain organizations that find them organic and safe for use. Here are some things to look out for when shopping for organic beauty products:
Anywhere on the packaging or the makeup itself it says that it’s paraben, PCB, or phthalate free. These are harsh chemicals used to create both the makeup and packaging. For example, if the beauty company is using PCBs in their plastic, that chemical can leech into the makeup and go into the skin.
Most makeup and beauty product are regulated through the USDA, but there are some natural ingredients that aren’t regulated by the USDA that everyone should be aware of. Things such as plant-derived ingredients and essential oils are not listed under the USDA but are much healthier for you and your skin than chemicals. Certified organic makeup and beauty products are going to have a certified USDA label or similar organic certification label, too.
There are other associations throughout the world that are promoting organic beauty products and all of them are going to have a label placed on certified products in order to show which products are truly organic.
But why is there so much buzz about certified beauty and makeup? The ingredients used to create these beauty products are naturally made and that means they will be gentler on the skin without losing their “pop” of color. Most makeup these days uses chemicals in order to keep things fresher and brighter, but these chemicals build up in the skin over time and can cause rashes, allergic reactions, and health problems. Also, some women live a vegan or gluten-free lifestyle, and don’t want to compromise on their makeup, so they too have options when it comes to eyeshadow, lipstick, and foundation.
The current market for certified organic makeup is booming and there are many options when it comes to living a sustainable and healthy lifestyle. However, everyone should also know what to look for when shopping for truly certified organic beauty products, so they don’t have to worry about any traces of chemicals in the makeup they put on their skin.
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The present invention relates to programming and control systems. More particularly, this invention relates to flowchart-based programming and control systems that include active debugging objects.
Programming and control systems are generally used for controlling processes that involve devices such as relays, solenoids, motors, valves, switches, and other electrical and electromechanical devices. The processes that are controlled include machining, drilling, welding, spraying paint, mixing materials, assembling parts, handling materials, and other similar processes.
Conventional programming and control systems generally employed ladder diagrams and relay ladder logic (RLL) to control the operation of the devices that are associated with the processes. In practice, however, programmers tend to use a flowchart to initially define the operation of the devices in the process. Then, the programmers manually translated the flowchart into the ladder diagrams. The programmers employ the flowcharts as a first step because the flowcharts emulate human thought processes whereas the ladder diagrams do not. Subsequently, developers created programming and control systems that control the process directly from the flowchart logic. One flowchart-based system is disclosed in xe2x80x9cContinuous Flowchart, Improved Data Format and Debugging System For Programming and Operation of Machinesxe2x80x9d, U.S. Pat. No. 4,852,047, which is hereby incorporated by reference.
Flowcharts generally include action blocks that represent an operation or action based on current input and output data. Action blocks generally have one entry point that is usually located at the top and one exit point that is usually located at the bottom. A branching or decision block is a diamond-shaped block that represents a branch in the control path based on the results of a decision. Decision blocks generally have one entry point that is usually located at the top and two exit points that are usually located at the side and the bottom. Using combinations of the action and decision blocks, a programmer creates a flowchart that controls one or more devices that are associated with a process.
Programming and control systems generally provide an operating mode and a debugging mode. In the debugging mode, the programmer monitors the flowchart object code as it is executed. In some systems, a currently executing logic block is highlighted in real time while it is executing. Multiple watch windows allow a user to view different flowcharts or different parts of the same flowchart while it is executing. The debugging mode may also include a step-by-step executing mode. In other words, the developer uses a mouse or a keyboard to trigger one logic block to be executed at a time. After each logic block is executed, the developer checks the state of control variables and device operation to determine whether a desired response or logic state is achieved. The debugging process often takes a long time and increases the cost of the project.
A machine programming and control system according to the present invention includes a computer with a processor, memory, and a display and a device associated with a process. A flowcharting module executed by the computer generates and edits a flowchart that contains action and decision blocks that define logic for operating the device to further the process. The flowcharting module allows active debugging objects to be added to the flowchart.
In other features, the active debugging objects provide information relating to debugging of flowchart code and/or debugging of a portion of the process that is related to the flowchart code. The active debugging objects may include an icon that is added to the flowchart adjacent to related flowchart code to visually identify the active debugging object.
In still other features, the active debugging object provides at least one of audio, a movie clip, a link to a website and textual information. The active debugging object includes at least one of audio, a movie clip, a link to a web site and text demonstrating the desired operation of the process. The active debugging object includes at least one of audio, a movie clip, a link to a website and text describing correct values for control variables at a first logical point in the flowchart.
Further areas of applicability of the present invention will become apparent from the detailed description provided hereinafter. It should be understood that the detailed description and specific examples, while indicating the preferred embodiment of the invention, are intended for purposes of illustration only and are not intended to limit the scope of the invention.
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Q:
What determines the efficiency of electron production in photosynthetic bacteria?
Is there a specific gene involved, perhaps? Would one be able to genetically engineer a bacterium to oxidize water and generate electrons quicker? I am speaking about this biological problem in terms of an application to solar cells.
Edit: I am not inquiring to the existence of a gene responsible for photosynthesis, I am inquiring as to why some exoelectrogenic bacteria can produce electrons in electron transport more efficiently than others. Is there a reason for this that can be traced to a certain gene that makes certain exoelectrogens superior to others?
A:
You may or may not consider this an actual answer to your entire question, but it's interesting nonetheless.
A physicist friend of mine did some work recently modelling the quantum dynamics of photosynthetic complexes, and their coupling constants for passing energy throughout the photosystem. His results showed that, (and I believe this is in agreement with the literature), that photosynthetic systems are actually about as efficient as it is possible to be (not only this, but they are the most efficient energetic systems known to man I believe - over something like ~90% efficient).
If you begin altering positions of chromophores and so on, the photosystems exhibit remarkable robustness and you get a more or less logistic decay in efficiency (i.e. removing one or 2 chromophores results in marginal reductions in efficiency, more still in considerably more severe debilitation, until ultimately it's efficiency is markedly lessened, but further manipulation has a plateau in efficacy reduction).
Much of the work that's been done on the quantum biology of photosystems has concerned the FMO Complex (Fenna-Matthews-Olson) which is a simplified model system from photosynthetic algae - you might find more detailed answers to your queries by reading up in this area.
As for discrepancies seen in photosynthetic effectiveness bought about genetically, there are theories that suggest that the organisms are actually protecting themselves. E.g. to avoid over-production of ROS species. I can't offer you much in the way of a well evidenced case for this though, it's just a conversation I had with a post-doc at work recently, who completed his PhD on photosynthetic cyanobacteria.
EDIT Here's one of the papers about optimality in the FMO:
http://aip.scitation.org/doi/pdf/10.1063/1.4930110
A:
@Joe Healey's answer is great, but I would like to expand on the subject, having done a thorough literature review on the subject before.
Before we do that, I would like to clear up some misunderstandings that you seem to have about the electron transfer.
Just like @stords said, electrons aren't generated during photosynthesis. Where it comes from is from the oxidation of water mediated by the reaction center. The reaction center is part of the photosystem, which contains all the light-dependent parts of the reactions. The oxidation is performed by the oxygen-evolving complex. This process of oxidation isn't very well understood. However, most photosynthetic bacteria contain similar complexes. Note that green sulfur bacteria doesn't use H2O for oxidation, but instead H2S.
Reaction center
Now, in the light-dependent reactions, we have what's known as a light-harvesting complex, which contains many pigments that absorb light and get excited. The pigments then transfer the energy to the reaction center, which also gets the electrons from the oxidation. So when the energy from the antenna reaches the reaction center, the energy excites the electron to a higher energy level. This higher energy electron is transferred to the electron transport chain, where it reduces various electron acceptors, and while doing so, using the energy released to pump electrons to make a proton gradient. This proton gradient is used to make ATP. Even in bacteria, the process is very similar. The exact details can be read here
Excitation efficiency
Now, answering your question, what really determines the efficiency of the electron transfer is actually the energy transfer occurring in the light-harvesting complex (LHC). Now, different organisms have different mechanisms, but all scientists agree that there is some quantum mechanics behind this. In the following discussion, I will be referring to the dynamics of the FMO complex, the antenna complex found the green sulfur bacteria, and commonly used to study the quantum mechanical interactions of energy transfer.
Now when we have this transfer of energy from one pigment molecule (chlorophyll in plants, bacteriochlorophyll a in green sullfur bacteria), we call it an exciton. This exciton is a quantum mechanical state. Now, in quantum mechanics, there is a concept called superposition. Basically, superposition means that a single quantum mechanical state is composed of multiple states. In the case of the light-harvesting complex, when we say that the exciton is in a superposition, what this means is that the energy can travel in all possible pathways from a pigment to reaction center, and when the best pathway is found, the interactions in the complex causes the state to collapse into the best state. This is how the transfer of energy occurs in less than 1 ns!
This dynamics is very complicated, and very surprising, since this quantum mechanical dynamics is occurring in an open system. Typically, it is very hard to observe quantum mechanical dynamics because of something known as decoherence. This is when a mixed (superposition) state collapses into one state. However, interestingly, the complex uses this decoherence to collapse into the best path for the exciton to reach the reaction center.
This field is very interesting. If you have any questions, please leave a comment. I am a little rusty on this subject, but I am very interested in this subject. If you want to see more examples of quantum mechanics in biology, look up "quantum biology". Quantum biology is a new field studying the intersection of quantum mechanics and biology.
References:
https://en.wikipedia.org/wiki/Light-dependent_reactions
http://rsif.royalsocietypublishing.org/content/11/92/20130901
https://en.wikipedia.org/wiki/Fenna-Matthews-Olson_complex
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Donate
Please click on the button below to donate to SHARE or WHEEL via PayPal. SHARE is a 501c(3) non-profit registered in the state of Washington and all donations are tax-deductible. WHEEL shares SHARE's 501c(3).
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Tagged: Criminal Assault
CHARLOTTE, N.C. -- Kurt Busch was cleared Wednesday to get back in his race car and attempt to rebuild a career that was halted two days before the Daytona 500 when NASCAR suspended him for allegedly assaulting his ex-girlfriend.
A Little Rock woman said Tuesday that Jermain Taylor threatened to fatally shoot her 5-, 3-, and 1-year-old children, and that she knocked the gun away after the cursing middleweight boxing champion put the weapon to her husband's head.
DOVER, Del. -- A friend who counselled NASCAR driver Kurt Busch's ex-girlfriend amid the couple's breakup last year testified Tuesday that Patricia Driscoll initially was brokenhearted but later set out for revenge.
By Josh Cornfield
TRENTON, N.J. -- A woman who said Friday that Bill Cosby had drugged her and sexually assaulted her in 1979 also accused another famous man of attempted sexual assault: sportscaster Marv Albert, who pleaded guilty to assault and battery the day after her surprise testimony against him.
LAS VEGAS -- Rap music figure Marion "Suge" Knight had a Las Vegas traffic warrant reduced to a parking ticket on Thursday, but was returned in custody to the Clark County jail to await a Monday extradition hearing in a California robbery case.
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Nidal Hasan, a Muslim, killed a bunch of people at Ft. Hood. Clearly, this calls for the angriest members of the right wing to compete to say the most paleoracist anti-Muslim thing. Today: Pat Robertson vs. Tunku Varadarajan.
So you are dealing with not a religion. You're dealing with a political system. And I think we should treat it as such and treat its adherences as such as we would members of the Communist Party or members of some fascist group.
[We] must ask whether we are confronting a new phenomenon of violent rage, one we might dub—disconcertingly—"Going Muslim." This phrase would describe the turn of events where a seemingly integrated Muslim-American—a friendly donut vendor in New York, say, or an officer in the U.S. Army at Fort Hood—discards his apparent integration into American society and elects to vindicate his religion in an act of messianic violence against his fellow Americans.
Not the friendly donut vendor!
The difference between "going postal," in the conventional sense, and "going Muslim," in the sense that I suggest, is that there would not necessarily be a psychological "snapping" point in the case of the imminently violent Muslim; instead, there could be a calculated discarding of camouflage—the camouflage of integration—in an act of revelatory catharsis.
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Q:
Finite, abelian, yet "fugitive" orthogonal subgroups
Update July 29, 2013.
I have still not found a good textbook for this topic, if you point one to me I will be grateful :) I have accepted BS's answer anyway, since their explanation was useful to me and gave me a good starting point for further research. I ended up finding a very helpful resource for this topic: the online notes of the course Introduction to Topological Groups, by Dikran Dikranjan, University of Udine. I recommend these notes to anyone interested on this topic, or on the Pontryagin-Van Kampen duality. The content of these notes has been partially published as An elementary approach to Haar integration and Pontryagin duality in locally compact abelian groups, Dikranjan and Stoyanov, Topology and its Applications, Volume 158, issue 15, 2011), p. 1942-1961, DOI: 10.1016/j.topol.2011.06.037, MR: 2825348, Elsevier Science.
A popular concept in quantum computation, used extensively to design algorithms for finite-abelian-groups, are the so-called orthogonal subgroups
Let $G=\mathbb{Z}_{d_1}\times\ldots\times\mathbb{Z}_{d_m}$ be a finite abelian group, the orthogonal subgroup $H^{\perp}$ of $H$ a subgroup of $G$ is defined as:
$$H^\perp:=\lbrace g\in H : \chi_g(h)=1 \quad\text{for all } h \in H\rbrace,$$
where $\chi_g$ are the characters of $G$:
$$ \chi_g(h) = \exp{\left(2\pi \sum_{i=1}^{m} g_i h_i/d_i \right)} \quad \text{for all } \quad g, h \in G $$
Given two subgroups $H$ and $K$, basic Character Theory allows one to quickly derive
\begin{matrix}
(1)\ H^{\perp\,\perp} = H & (2)\ |H^{\perp}| = |G|/|H| \\
(3)\ H\subset K \iff K^{\perp}\subset H^{\perp} & (4)\ (H\cap K)^{\perp} = \langle H^{\perp} , K^{\perp} \rangle
\end{matrix}
Question.
This structure is extensively used in some important quantum algorithms and appears in quite a bunch of relatively-recent research papers. Yet, and though it looks pretty basic, I can not find some standard textbook where this is defined and that includes proofs of propositions (1-4). I would like to find such a reference since I often discuss these concepts with people not fluent with Character or Group theory. Also, I would like to know if the name "orthogonal-subgroup" is used by mathematicians.
A:
These results belong to what is called duality in finite abelian groups, a theory that has been generalized by Pontryagin and others in the 30's to locally compact abelian groups.
Another keyword here is "Discrete Fourier Transform", although it is mainly used for cyclic groups (of order $2^N$ for FFT) or $GF(2)$ vector spaces (analysis of boolean functions).
It is also a chapter in the representation theory of finite groups, founded by Frobenius in the end of 19th century, namely the case of finite abelian groups, where the irreducible characters are all homomorphisms to $S^1$. This case was already known to Gauss and Dirichlet.
By the way, the orthogonal you define is more naturally seen as a subgroup of the dual group $\hat{G}=Hom(G,S^1)$, which in (multiplicative) duality with $G$ via the evaluation $G\times \hat{G}\to S^1$.
It is isomorphic to $G$, but not naturally so.
I would bet that almost any book on either Pontryagin duality or representation theory of finite groups has the results you want to cite, but I have only been able to find this link to an online encyclopedy, which easily implies your claims, since 3 and 4 follow quickly from 1 and 2.
Hope this helps.
A:
In today's updates you ask specifically about published textbook treatments of duality for finite abelian groups, including in particular a proof of (4). That may be found in B. Huppert's books,
Endliche Gruppen. I (1967), §V.6 "Charaktere abelscher Gruppen" (pp. 487-490);
Character theory of finite groups (1998), §5 "Characters of abelian groups" (pp. 58-65).
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Minuscule 2813
Minuscule 2813 (in the Gregory-Aland numbering), is a Greek minuscule manuscript of the New Testament, on 151 parchment leaves (11.9 cm by 9.5 cm). Dated paleographically to the 13th century.
Description
The codex contains Luke and John with some lacunae. The text is written in one column per page, in 19 lines per page. It contains a miniature before Gospel of John. It is rubbed. The manuscript was bound with John preceding Luke. The leaves are numbered and according to the these numbers Luke preceded John before bounding.
Kurt Aland the Greek text of the codex did not place in any Category.
It was not examined by the Claremont Profile Method.
Currently the codex is in private hands.
See also
List of New Testament minuscules (2001–)
Textual criticism
References
External links
Images of manuscript 2813 at the CSNTM
Category:Greek New Testament minuscules
Category:13th-century biblical manuscripts
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Q:
Display recurring dates from SQL Database using PHP
I'm accessing a database created by another web company to retrieve event information for my current client. My client enters info for events and notes whether the date is recurring or not. I'm trying to display all the recurring dates. So far I have been able to get everything to display, regular dates as well as recurring.
The tables are laid out as follows:
Events
Events_Recurring
Here is part of the Events table BIGGER PICTURE
This is what the Events_Recurring table looks like
When the client checks it as recurring, the events_recurring table creates a row with the Event ID and other information like what day of the week or month the event is recurring on.
I'm just not sure how to display multiples of that certain ID that is recurring. I have a start date, and end date I can access, as well as what day of the week it is recurring on.
So for example: If this event reoccured every thursday. and I knew it started on Jan 1st and ended Jan 31st, can I run through that and spit out 4 different events all with the date of every Thursday in January?
Here is the full code I am working with, it's a little messy while trying to figure this out. I'm checking for the recurrence towards the bottom
// Access external database
$events_db = new wpdb(TOP SECRET CREDENTIALS HERE);
$events_db->show_errors();
if ($events_db) :
// Query Events Database
$events = $events_db->get_results(
"
SELECT ID, RequestDateStart, RequestDateEnd, Ministry, RequestTimeStart, EventName, CoordinatorName, EventDescription, Location
FROM gc_events
WHERE PrivateEvent = 0
AND Ministry = 15
AND date(RequestDateStart)>=date(NOW())
ORDER BY RequestDateStart
"
);
// Create the event data that will be displayed
foreach ($events as $event) :
// Store Event ID in a variable
$masterID = $event->ID;
echo '<div class="col-12">';
echo '<strong>ID:</strong> ' . $event->ID . '<br /><strong>Event Name:</strong> ' . $event->EventName . '<br /><strong>Leader:</strong> ' . $event->CoordinatorName . '<br /><strong>Date:</strong> ' . date('l, F j',strtotime($event->RequestDateStart)) . '<br /><strong>Start Time:</strong> ' . date('g:i a',strtotime($event->RequestTimeStart));
// CHECK IF RECURRING
$recurring_events = $events_db->get_results(
"
SELECT gc_event_id, period, day
FROM gc_event_recurring
WHERE gc_event_id = '$masterID'
"
);
foreach ($recurring_events as $recurring_event) :
if ($recurring_event->period === 'week') {
echo '<div class="col-12"><strong>↑ WEEKLY</strong><br />';
echo $recurring_event->day;
echo '</div>';
}
endforeach;
echo '</div>';
endforeach;
endif;
The result I am getting right now (with recurring events) is
Event: Weekly Prayer
Date: Feb 1, 2013
The result I would like is
Event: Weekly Prayer
Date: Feb 1, 2013
Event: Weekly Prayer
Date: Feb 8, 2013
Event: Weekly Prayer
Date: Feb 15, 2013
Event: Weekly Prayer
Date: Feb 22, 2013
This would be if the start date was Feb 1st and end date was Feb 28th.
A:
A word of advice.
Although designing a database to store the 'description' of the repetition pattern is a very clean approach from a design point-of-view, you may get a lot of problems down the way.
I've done a project with a similar approach a while ago (I will look up the database design and add that to my answer) and, although I was able to reproduce the exact date/times of the recurring events, you will run into problems in the following situations; most originate from this:
the recurring events describe the repetition pattern, so the actual (individual) events are no physical records in your database
If the customer decides to add a new event, how will you check if it overlaps with any existing event? You'll have to calculate all 'events', based on the repetition pattern.
If the customer decides that the scheduled time for an event needs to be changed, how will you have this change apply to all future events and not for events that are in the past (you'll have to duplicate the original event, modify its end-date, and set the duplicated event with a new start-date)
If the customer decides he wants to remove a single day from the repetition pattern (e.g. a single event has ben canceled), you will also have to split the original event into two separate repetitions, or have a 'canceled/blocked' dates/times table
If people need to 'book' for specific events, you won't be able to attach them to a 'real' event-record, because the individual events because they are not physically present in the database. e.g. to check if a single event can be re-scheduled or canceled, you'll need to do this from code as the database cannot make use of foreign-key constraints to automatically update related reservations
Regarding performance; because individual events are not physically stored, they will have to be calculated every time you want to show them. Consider having 1000 recurring events in the database and try to show a 'calendar' of week 23 two years from now. You'll have to analyze all recurring-events patterns and calculate all events that they produce!
All depends of course on the actual usage of your system, but I wanted to warn you for problems we've run into.
Here's the schema for the 'schedules' table (contains recurring events pattern);
CREATE TABLE IF NOT EXISTS `schedules` (
`id` int(11) NOT NULL auto_increment,
`date_start` date NOT NULL,
`time_start` time NOT NULL,
`time_end` time NOT NULL,
`recur_until` date default NULL COMMENT 'end date when recurrence stops',
`recur_freq` varchar(30) default NULL COMMENT 'null, "secondly", "minutely", "hourly", "daily", "weekly", "monthly", "yearly"',
`recur_interval` smallint(5) unsigned default NULL COMMENT 'e.g. 1 for each day/week, 2 for every other day/week',
`recur_byday` smallint(5) unsigned default NULL COMMENT 'BITWISE; monday = 1, sunday = 64',
PRIMARY KEY (`id`)
) ENGINE=InnoDB DEFAULT CHARSET=utf8 ROW_FORMAT=DYNAMIC;
How to circumvent the problems described
Fully describing a solution to these problems won't probably be suitable here, but here's some things to consider;
Storing a recurring event as described on itself is not bad practice. It perfectly describes when, and how often, an event should take place. However, the lack of physical records for the actual events is what causes the problem.
When creating or modifying a recurring event, calculate all resulting events and store them as physical records. These records can be queried, 'reservations' can be attached to them and you'll be able to make use of database features, like foreign-key-constraints to handle them properly.
When storing the individual events as described in 1., make sure you're keeping a reference to the 'schedule' that they belong to. If (for example) the customer wants to change the time of a recurring event, you'll be able to update all related (individual) events.
Keep in mind that in situation 2, you'll probably only want to update future events, so the 'recurring event' will still need to be 'split' in two to achieve that. In which case 'future' events need to be attached to the new 'recurring event', old events stay attached to the existing 'recurring event'
invest time in your database/software design, properly investigate if the design will 'work' for the thing you're trying to achieve. Test it, try things and if they don't work, don't hesitate to 'throw it away', often it's easier to start from scratch than try to 'fix' things. A proper design will take time and may take several 'redesigns' to get it right, but it will save you time and money in the end.
Hope this helps, good luck!
A:
foreach ($events as $event) :
// Store Event ID in a variable
$masterID = $event->ID;
echo '<div class="col-12">';
echo '<strong>ID:</strong> ' . $event->ID . '<br /><strong>Event Name:</strong> ' . $event->EventName . '<br /><strong>Leader:</strong> ' . $event->CoordinatorName . '<br /><strong>Date:</strong> ' . date('l, F j',strtotime($event->RequestDateStart)) . '<br /><strong>Start Time:</strong> ' . date('g:i a',strtotime($event->RequestTimeStart));
// CHECK IF RECURRING
$recurring_events = $events_db->get_results(
"
SELECT gc_event_id, period, day
FROM gc_event_recurring
WHERE gc_event_id = '$masterID'
"
);
foreach ($recurring_events as $recurring_event) :
if ($recurring_event->period == 'week') {
$StartDate = strtotime($event->RequestDateStart);
$EndDate = strtotime($event->RequestDateEnd);
$TotalDays = round(($EndDate-$StartDate)/(60*60*24*7));
for($i = 0 ;$i<($TotalDays-1);$i++)
{
$StartDate += (60*60*24*7);
echo '<div class="col-12">';
echo '<strong>ID:</strong> ' . $event->ID . '<br /><strong>Event Name:</strong> ' . $event->EventName . '<br /><strong>Leader:</strong> ' . $event->CoordinatorName . '<br /><strong>Date:</strong> ' . date('l, F j',$StartDate) . '<br /><strong>Start Time:</strong> ' . date('g:i a',strtotime($event->RequestTimeStart));
}
}
endforeach;
echo '</div>';
endforeach;
try this and tell me if it works
| null |
minipile
|
NaturalLanguage
|
mit
| null |
---
abstract: 'In this article we review classical and recent results in anomalous diffusion and provide mechanisms useful for the study of the fundamentals of certain processes, mainly in condensed matter physics, chemistry and biology. Emphasis will be given to some methods applied in the analysis and characterization of diffusive regimes through the memory function, the mixing condition (or irreversibility), and ergodicity. Those methods can be used in the study of small-scale systems, ranging in size from single-molecule to particle clusters and including among others polymers, proteins, ion channels and biological cells, whose diffusive properties have received much attention lately.'
author:
- 'Fernando A. Oliveira'
- 'Rogelma M. S. Ferreira'
- 'Luciano C. Lapas'
- 'Mendeli H. Vainstein'
bibliography:
- 'referencias.bib'
title: 'Anomalous diffusion: A basic mechanism for the evolution of inhomogeneous systems'
---
Introduction
============
General concepts
----------------
Diffusion is a basic transport process involved in the evolution of many non-equilibrium systems towards equilibrium [@Vainstein06; @Shlesinger93; @Metzler99; @Metzler00; @Morgado02; @Metzler04; @Sancho04; @Costa03; @Lapas08; @Weron10; @Thiel13; @Mckinley18; @Flekkoy17]. By diffusion, particles (or molecules) spread from regions of high concentration to those of low concentration leading, via a gradual mixing, to a situation in which they become evenly dispersed. Diffusion is of fundamental importance in many disciplines; for example, in growth phenomena, the Edwards-Wilkinson equation is given by a diffusion equation plus a noise [@Barabasi95]. In cell biology it constitutes a main form of transport for amino acids and other nutrients within cells [@Murray2002].
For more than two hundred years [@Vainstein06], diffusion has been a widely studied phenomena in natural science due to it’s large number of applications. Initially, the experiments carried out by Robert Brown [@Brown28; @Brown28a] called the attention to the random trajectories of small particles of polen and also of inorganic matter. This irregular motion, later named Brownian motion, can be modeled by a random walk in which the mean square displacement is given by Einstein’s relation [@Vainstein06] $$\label{eq.sqr_disp}
\langle (\Delta r )^2 \rangle = 2dDt,$$ where $\Delta r$ is the displacement of the Brownian particle in a given time interval $t$, $d$ is the spatial dimension, and $D$ the diffusion coefficient. Whereas a single Brownian particle trajectory is chaotic, averaging over many trajectories reveals a regular behavior.
The purpose of this article is to review recent efforts aiming at formulating a theory for anomalous diffusion processes– i.e., those where the mean square displacement does not follow Equation (\[eq.sqr\_disp\]) which can be applied to many different situations from theoretical physics to biology.
In the next section, we introduce the pioneering works on diffusion, and then call attention to the existence of anomalous diffusion. We discuss the main methods to treat anomalous diffusion and concentrate our efforts on the discussion of the generalized Langevin formalism.
The tale of the three giants
----------------------------
At the birth of the gravitational theory Isaac Newton mentioned that he build up his theory based on the previous works of giants. Also, one cannot talk about diffusion without underscoring the works of Albert Einstein, Marian Smoluchowski, and Paul Langevin. At the dawn of the last century, the atomic theory was not widely accepted by the physical community. Einstein believed that the motion observed by Brown was due to the collisions of molecules such as proposed by Boltzmann in his famous equation. However, Boltzmann’s equation was difficult to solve and therefore he proposed a simpler analysis combining the kinetic theory of molecules with the Fick’s law, from which he obtained the diffusion equation $$\label{diff}
\frac{\partial \rho(x,t)}{\partial t}=D\, \nabla^2\rho(x,t),$$ where $\rho(x,t)$ is the density of particles at position $x$ and time $t$. The solution of the above equation yields $\rho(x,t)$ and $$\lim _{t\rightarrow \infty }\langle x^{2}(t)\rangle =\int x^2(t)\rho(x,t) dx= 2Dt,
\label{X2}$$ where $\langle x(t) \rangle =0$, assuming the symmetry $ \rho(-x,t) = \rho(x,t)$. For simplicity, we have considered the motion one-dimensional and set all particles at the origin at time $t=0$. Generalization to two and three dimensions is straightforward. He then considered the molecules as single non-interacting spherical particles with radius $a$ and mass $m$, and subject to a friction $\gamma$ when moving in the liquid [@Vainstein06; @Einstein1905; @Einstein56] .
Finally, he deduced the famous Einstein-Stokes relation [@Einstein56] for the diffusion constant $$D=\frac{RT}{6\pi N_a a \eta}=\frac{RT}{mN_a\gamma}=\frac{RT\mu}{N_a},
\label{D}$$ where $N_a$ is Avogadro’s number, fundamental for atomic theory, but unknown at the time, $R$ is the gas constant, $\mu$ the mobility, $\eta$ the viscosity, and $T$ the temperature.
The scientific community became very excited and, in the years following Einstein’s papers, some dedicated experiments helped to verify Equation (\[X2\]). Diffusion constants were measured and it became possible to estimate Avogadro’s number from different experiments. Moreover, it was possible to estimate the radius of molecules with few hundreds of atoms. Einstein was successful in demonstrating that atoms and molecules were not mere illusions as critics used to suggest. Finally, the theory of Brownian motion started to set a firm ground. For instance, it became possible to associate diffusion with conductivity in the case of a gas of charged particles. Supposing each has the same charge $e$ and is subjected to a time dependent electric field $\vec{E}=\vec{{E}}(\omega) \exp(-i\omega t)$, the conductivity $\tilde{\sigma}(\omega)$ can be defined by $\vec{J}(\omega)=\tilde{\sigma}(\omega)\vec{E}(\omega)$, where $\vec{J}(\omega)$ is the current. Now, it was possible to relate the diffusivity $\tilde{D}(w)$ with $\tilde{\sigma}(\omega)$ by the relation [@Vainstein06; @Dyre00; @Oliveira05] $$\tilde{\sigma}(\omega)=\frac{ne^2}{k_BT}\widetilde{D}(\omega),
\label{Domega}$$ where is $n$ the carrier density. From that it was obvious that connections between a diversity of response functions could be obtained.
Two major achievements in the theory of stochastic motion were due to Smoluchowski. As expressed by Novak [*et al.*]{} [@Nowak17], “One was the Smoluchowski equation describing the motion of a diffusive particle in an external force field, known in the Western literature as the Fokker-Planck equation [@Risken89; @Salinas01]" $$\label{FP}
\begin{split}
\frac{\partial P(v,t,v_0)}{\partial t}=- \frac{\partial}{\partial v} &\left[ A(v)P(v,t,v_0) \right] \\
&+ \frac{1}{2} \frac{\partial^2}{\partial v^2} \left[ B(v)P(v,t,v_0) \right] ,
\end{split}$$ where $$A(v)=\frac{1}{\delta t}\int_{-\infty}^{\infty}(v'-v)P(v',\delta t,v)\,dv'$$ and $$B(v)=\frac{1}{\delta t}\int_{-\infty}^{\infty}(v'-v)^2P(v',\delta t,v)\,dv'.$$ Here $P(v', \delta t,v)$ is the transition probability between two states with different velocities. The second one, a fundamental cornerstone of molecular physical chemistry and of cellular biochemistry [@Nowak17; @Gadomski18] is “Smoluchowski’s theory of diffusion limited coagulation of two colloidal particles." Unfortunately, due to his premature death, the Nobel prize was not awarded to Smoluchowski. However, the scientific community pays tribute to him [@Smoluchowski17].
The last of the three giants was Langevin, who considered Newton’s second law of motion for a particle as [@Langevin08] $$m\frac{dv(t)}{dt}=-m\gamma v(t)+f(t),
\label{L}$$ dividing the environment’s (thermal bath) influence into two parts: a slow dissipative force, $-m\gamma v$, with time scale $\tau= \gamma^{-1}$, and a fast random force $f(t)$, which changes in a time scale $\Delta t \ll \tau $, subject to the conditions $$\label{fmed}
\langle f(t)\rangle =0,$$ $$\label{fv}
\langle f(t)v(0)\rangle =0,$$ and $$\langle f(t)f(t')\rangle= \Lambda \,\delta(t-t').$$ If we solve Equation (\[L\]) and, using the equipartition theorem, impose $\langle v^2(t \rightarrow \infty) \rangle= \langle v^2\rangle_{eq}=k_BT/m$, where $k_B=R/N_a$ is the Boltzmann constant, we obtain $\Lambda =2m \gamma k_BT $ and write $$\langle f(t)f(t')\rangle=2m \gamma k_BT\delta(t-t').
\label{FDT0}$$ This last equation establishes a relation between the fluctuation and the dissipation in the system reconnecting the useful, although artificial, separation of the two forces. This relation has been named the fluctuation-dissipation theorem (FDT) and is one the most important theorems of statistical physics. Equations (\[fmed\]), (\[fv\]), and (\[FDT0\]) yield the velocity-velocity correlation function that reads [@Reichl98] $$\label{Cv1}
C_v(t)= \langle v(t+t')v(t')\rangle=(k_BT/m) \exp(-\gamma t),$$ the mean square displacement $$\label{X22}
\langle x^2(t\gg \tau) \rangle= \int_0^tdt'\int_0^t dt''\langle v(t')v(t'')\rangle =2Dt,$$ and $$\label{Kubo}
D=\int_0^{\infty}C_v(t)dt,$$ known as the Kubo formula. This bring us back to Einstein’s results for diffusion, Equation (\[D\]).
The simplification introduced by the Langevin formalism makes it easy to carry out analytical calculations and computer simulations. Consequently, the Langevin equation, and its generalization (Sec. 3), has been applied successfully to the study of many different systems such as chain dynamics [@Toussaint04; @Oliveira94; @Oliveira96; @Oliveira98a; @Maroja01; @Dias05], liquids[@Rahman62; @Yulmetyev03], ratchets [@Bao03a; @Bao06], and synchronization [@Longa96; @Ciesla01]. However, its major importance was to relate fluctuation with dissipation.
The Fokker Planck equation (\[FP\]) with the Langevin choices becomes $$\label{FP2}
\frac{\partial P(v,t)}{\partial t}=\gamma\frac{\partial}{\partial v}\left[ v P(v,t)\right] +\gamma\frac{k_BT}{m} \frac{\partial^2}{\partial v^2}\left[ P(v,t)\right],$$ known as the Ornstein-Uhlenbeck equation. It obviously yields the same result as that of Einstein and Langevin. This completes the tale of the three giants. Their work established the bases of non-equilibrium statistical mechanics opening a new field in research for the next decades. For example, it was demonstrated that hydrodynamics could be obtained from the Boltzmann equation in particular situations [@Huang87], so that the physics community could appreciate the work of yet another giant.
Breakdown of the normal diffusive regime
========================================
The different facets of the anomaly: subdiffusion and superdiffusion
--------------------------------------------------------------------
Inspection of different supposedly diffusive processes such as enhanced diffusion in the intracellular medium [@Holek09], cell migration in monolayers [@Palmieri15], Levy flight search on a polymeric DNA [@Lomholt05], or the Brownian motion in an inhomogeneous medium [@Durang15] reveals that the previous framework is not always fulfilled. Instead, in these and other similar examples the mean square displacement deviates from the linear temporal evolution. One of the most common anomalous behaviours is given by $$\label{X2anomalous}
\lim_{t \rightarrow \infty} \left\langle r^2(t)\right\rangle \sim t^\alpha,$$ where $\alpha \neq 1$ is a real positive number [@Metzler00; @Morgado02; @Metzler04; @Morgado04].
{width="90.00000%"}
The origin of this discrepancy is the tacit assumption made in the derivation of (\[eq.sqr\_disp\]) that the Brownian particle moves in an infinite structureless medium acting as a heat bath. This assumption is generally incorrect when the Brownian motion takes place in a complex medium, as is the case of the previously mentioned examples. We illustrate in Fig. (\[fig:X2anomalous\]) the mean square displacement $\langle x^ 2(t) \rangle$ as a function of $t$ for three distinct Brownian motions in one dimension. From the upper curve downwards, we have $\alpha=1.5$ (superdiffusion), $\alpha=1$ (normal diffusion), and $\alpha=0.5$ (subdiffusion).
The recent interest in the study of complex systems has led to an increased focus on anomalous diffusion [@Metzler00; @Morgado02; @Metzler04; @Sancho04; @Yulmetyev03; @Holek09; @Durang15], which has been described in biological systems [@Holek09; @Palmieri15; @Lomholt05], protostellar birth [@Vaytet18], complex fluids [@Mason95; @Grmela97; @Bakk02; @Sehnem2014; @Sehnem15; @cabreira18], electronic transportation [@deBrito95; @Monte00; @Monte02; @Kumakura05; @Borges06; @Novak05], porous media and infiltration [@Filipovitch16; @Reis16; @Reis18], drug delivery [@GomesFilho16; @Ignacio17; @Gun17; @Soares17], fractal structures and networks [@Mandelbrot82; @Stauffer95; @Cristea14; @Balankin17; @Balankin18], and in water’s anomalous behavior [@Barbosa11; @Bertolazzo15; @Silva15], phase transitions in synchronizing oscillators [@Ciesla01; @Longa96; @Bier16; @Pinto16; @Pinto17], to name a few.
There are many formalisms that describe anomalous diffusion, ranging from thermodynamics [@Perez-Madrid04a; @Rubi88; @kusmierz18], and fractional derivatives [@Metzler00; @Metzler04] to generalized Langevin equations (GLE) [@Hanggi90; @Morgado07; @Lapas07]. The goal of this short review is to call attention to relevant research within the GLE framework and to some fundamental theorems in statistical mechanics.
The generalized Langevin equation approach
==========================================
Non-Markovian processes
-----------------------
The Langevin formalism within its classical description has some restrictions:
1. It has a relaxation time $\tau=\gamma^{-1}$ , and an unspecified $\Delta t \ll \tau $, while a diffusive process in a real system can present several time scales;
2. For short times $t < \Delta t$, predictions are unrealistic; for example, the derivative of $C_v(t)$ is zero [@Lee83] at $t=0$, while in Langevin’s formalism $C_v(t)=\exp(-\gamma | t| )$, exhibiting a discontinuity.
3. If no external field is applied it cannot predict anomalous diffusion.
Up to now we have used time and ensemble averages without distinction, i.e. we implicitly used the Boltzmann Ergodic hypothesis (EH). The ensemble average for a variable $B(t)$ is defined by $$\label{Bens}
\langle B(t) \rangle= \int \exp{(- E/(k_BT))}\Omega(E,B)B(t)dE,$$where $\Omega(E,B)$ is the number of states for a given energy $E$. For the time average we have $$\overline{B(t)}=\frac{1}{\tau_0}\int_{-\tau_0/2}^{\tau_0/2}B(t+t')dt'.
\label{time}$$ For times $\tau_0 \gg \tau$, the Ergodic Hypothesis (EH) reads $$\label{EH}
\overline{B(t)} = \langle B(t) \rangle,$$which means that the system should be able to reach every accessible state in configuration space given enough time. This is expected to be true for equilibrated macroscopic systems and also for systems that suffer small perturbations close to equilibrium.
In this section, we generalize Langevin‘s equation and study some of its consequence for diffusion. The first correction to the FDT was done by Nyquist [@Nyquist28] who formulated a quantum version of the FDT. Later on Mori [@Mori65; @Mori65] and Kubo [@Kubo74] used a projection operator method to obtain the equation of motion, the Generalized Langevin equation (GLE), $$\label{GLE}
\frac{d A(t)}{d t}=-\int _{0}^{t}\Pi (t-t')A(t')\,d t'+F(t),$$ for a dynamical operator $A(t)$, where $\Pi (t)$ is a non-Markovian memory and $F(t)$ is a random variable subject to
1. $\overline{ F(t) } =0$,
2. $\overline{ F(t)A(0)} =0 $, and,
3. the Kubo fluctuation-dissipation theorem (FDT) [@Kubo91; @Kubo66] $$\label{fdt}
C_F(t-t')=\overline{ F(t)F(t')} = \langle A^2 \rangle_{eq} \, \Pi(t-t').$$
In this way, it is clear that time translational invariance holds in the Kubo formalism. An alternative to the projection operators, the recurrence relation method, was derived by Lee [@Lee83; @Lee82; @Lee83a; @Lee84].
It is easy to show that Equation (\[GLE\]) can give rise both to normal and to anomalous diffusion [@Vainstein06]. Let us consider two limiting case examples: first, when $\Pi(t)=2\gamma \delta(t)$, we recover the normal Langevin equation (\[L\]) with normal diffusion ($\alpha=1$); second, for a constant memory $\Pi(t)=K$, Equation (\[GLE\]) becomes $$\frac{d^2x}{dt^2}=-Kx +F(t),
\label{harm_osc}$$ where $$\label{x}
x(t)=\int_0^t A(t')\,d t'.$$ Equation (\[harm\_osc\]) is the equation of motion for a harmonic oscillator with zero diffusion constant ($D=0$) and with exponent $\alpha=0$. Consequently, we may have different classes of diffusion for distinct memories.
Now, we can study the asymptotic behavior of Equation (\[x\]) or of its second moment, Equation (\[X2anomalous\]), to characterize the type of diffusion presented by the system for any memory $\Pi(t)$. Much information about the system’s relaxation properties can be obtained by studying the correlation function $$\label{CA}
C_A(t,t')=\overline{ A(t)A(t')}= C_A(t-t').$$ The existence of stationary states warrants time-translation invariance so that the the two-time correlation function becomes a function only of the difference between two times, such as in the Kubo FDT above. We shall return to this point later. The main equation we are interested in is Equation (\[eq.sqr\_disp\]), where for anomalous diffusion, $D$ should be replaced by $D(t)$ now defined by $D(t)= \int_0^{t}C_A(t')dt'$ then $$\label{Kubo2}
\lim_{t \rightarrow \infty}D(t)=\lim_{t \rightarrow \infty} \int_0^{t}C_A(t')dt'=\lim_{z \rightarrow 0} \widetilde{C}_A(z).$$ From here onwards, the tilde over the function stands for the Laplace transform. For the last equality, we use the final-value theorem for Laplace transforms [@Gluskin03], which states that if $f(t)$ is bounded on $(0,\infty )$ and $\lim_{t \rightarrow \infty} f ( t ) $ has a finite limit, then $ \lim _{t\rightarrow \infty }f(t) = \lim_{z\rightarrow 0}{z \widetilde{f}(z)}$. Also note that the Laplace transform of the integral of a function is the Laplace transform of the function divided by $z$, then $\widetilde{D}(z)=\widetilde{C}_A(z)/z$. Using $z \propto 1/t$, we obtain the asymptotic behavior of $D(t)$. In order to do that, we multiply Eq. ( \[GLE\]) by $A(0)$, take the average and use the conditions $(1)$ and $(2)$ above for the noise to obtain the self-consistent equation $$\label{self_consistent}
\frac{d R(t)}{d t}=-\int_0^t \Pi(t-t')R(t')\,d t',$$ where for simplicity we have defined $$\label{cor}
R(t)=\frac{C_A(t)}{C_A(0)}.$$ Note that from Equation (\[GLE\]), we need to average from a large number of stochastic trajectories, while to obtain Eq. (\[cor\]), it is only necessary to solve a single equation, i.e. Eq. (\[self\_consistent\]). Further insight can be gained by analyzing the Laplace transformed version of Equation (\[self\_consistent\]) $$\label{laplace_R}
\widetilde{R}(z)=\frac{1}{z+\widetilde{\Pi }(z)}.$$ From here, it is clear that the knowledge of $\widetilde{\Pi}(z)$ in the limit $z\to 0$ completely defines the asymptotic dynamics. For instance, if $\lim_{z \rightarrow 0} \widetilde{\Pi}(z) \propto z^\mu$ then Equation (\[Kubo2\]) becomes $$\label{diff_beta}
\lim_{t \rightarrow \infty}D(t) \propto t^\beta$$ and consequently [@Morgado02] $$\label{alphaeq}
\alpha=\beta+1,$$ where $$\label{expbeta}
\beta=
\begin{cases}
\mu , &\text{ if~~ } -1<\mu <1\\
1, &\text{ if~~ } \mu \geq 1,\\
\end{cases}$$ We see that we have a cutoff limit for the exponent $\beta$ and, therefore, also for $\alpha$.
Due to the existence of correlations in the GLE that arise through hydrodynamical interactions [@Reichl98], it has been proposed [@Morgado02; @Vainstein05; @Morgado04] to establish a connection between the random force $F(t)$ and the noise density of states $\rho (\omega )$ of the surrounding media, modeled as a thermal bath of harmonic oscillators [@Morgado02; @Hanggi90] of the form $$\label{Noise}
F(t)= \int C (\omega )\cos [\omega t+\phi(\omega)]d \omega,$$ where $0 < \phi < \pi$ are random phases. Now, using the Kubo FDT, eq. (\[fdt\]), and time averaging over the cosines, we obtain the memory as [@Morgado02; @Costa03] $$\label{memory}
\Pi (t)=\int \rho (\omega )\cos (\omega t)d \omega,$$ which is an even function independent of the noise distribution. Here, $\rho(\omega)=C^2 (\omega )\langle A^2 \rangle_{eq}/2$.
The consequences of considering a colored noise given by a generalization of the Debye spectrum $$\label{noise_dos}
\rho (\omega )=
\begin{cases}
\frac{2\gamma }{ \pi } \left( \frac{ \omega }{\omega_s} \right )^\nu, & \text{ if } \omega<\omega_s\\
0, &\text{ otherwise},
\end{cases}$$ with $\omega_s$ as a Debye cutoff frequency were analyzed in detail in [@Vainstein06a]. The reason for the choice of this functional form for the noise density of states is that it was previously shown in [@Morgado02; @Costa03] that if $\widetilde{\Pi}(z) \propto z^\mu$ as $z\rightarrow 0$, then the same restriction as in Equation (\[expbeta\]), with $\nu=\mu$, applies and the diffusion exponent [@Morgado02] is given by Equation (\[alphaeq\]).
Later, this problem was revisited by Ferreira *et al.* [@Ferreira12] in which a generalized version of Equation (\[X2anomalous\]) was considered, namely $$\label{X2anomalous2}
\lim_{t \rightarrow \infty} \left\langle r^2(t)\right\rangle
\sim t^{\alpha}(ln(t))^{ \pm n}.$$
Most authors [@Metzler00; @Morgado02; @Metzler04; @Morgado04] have reported the cases of anomalous diffusion where, $n=0$ and $\alpha \neq 1 $. However, some authors such as for example, Srokowksi [@Srokowski00; @Srokowski13] reports situations were for $t \rightarrow \infty $, the dispersion behaves as $$\langle x^2(t) \rangle \propto t/\ln(t),$$i.e., a weak subdiffusive behavior for which we can say that $\alpha = 1^-$. In this way Ferreira *et al.* [@Ferreira12] generalizes the concept of $\alpha$, to associate with Eq. (\[X2anomalous2\]), the $ \alpha^{\pm}$ exponents, which arise analogously to the critical exponents of a phase transition [@kadanoff67; @kadanoff00; @Kenna06]. For example, in magnetic systems with temperatures $T$ close to the transition temperature $T_c$, the specific heat at zero field, $H=0$, exhibits the power law behavior $C_{H=0} \propto|T-T_c|^{-\alpha}$, where $\alpha$ is the critical exponent. However, for the two-dimensional Ising model [@kadanoff67] the critical exponent can be considered $\alpha = 0^{+}$, since the specific heat behaves logarithmically, $ C_{H=0} \propto \ln{|T-T_c|}$, instead. Logarithmic corrections [@Kenna06] to scaling have also been applied to the diluted Ising model in two dimensions in [@Kenna08]. This generalized nomenclature is pertinent because there are many possible combinations of both logarithmic and power law behaviors. This result highlights the existence of different types of diffusion.
In this way, for the density of states (\[noise\_dos\]), the generalized $\alpha$ becomes $$\label{diff_exponent}
\alpha=
\begin{cases}
2, &\text{ if~~ } \nu>1\\
2^-, &\text{ if~~ } \nu=1\\
1+\nu, &\text{ if~~ } -1<\nu<1.
\end{cases}$$ In Equation (\[noise\_dos\]), we choose the constant such that for normal diffusion $\widetilde{\Gamma}(z=0)=\gamma$. The exponent for Ballistic diffusion (BD), $\alpha=2$, is the maximum for diffusion in the absence of an external field. The slow ballistic motion $\alpha=2^{-}$ has properties that differ markedly from the ballistic case, see sections (3.3 and 3.4).
Non-exponential relaxation
--------------------------
Besides the importance of the asymptotic behavior, the study of the correlation $R(t)$ for finite times is also obviously significant, and there exists a vast literature describing non-exponential behavior of correlation functions in systems ranging from plasmas to hydrated proteins [@Rubi04; @Santamaria-Holek04; @Vainstein03a; @Santos00; @Benmouna01; @Peyrard01; @Colaiori01; @Ferreira91; @Bouchaud91], since the pioneering works of Rudolph Kohlrausch [@Kohlrausch54] who described charge relaxation in Leyden jars using stretched exponentials, $ R(t) \approx \exp{[-(t/\tau )^{\beta} ]} $ with $0 < \beta < 1$, and his son Friedrich Kohlrausch [@Kohlrausch63], who observed two universal behaviors: the stretched exponential and the power law. Since many features are shared among such systems and those that present anomalous diffusion [@Vainstein05; @Lapas15], it is natural that similar methods of analysis can be applied to both. For example, from Equation (\[self\_consistent\]), $\frac{dR(t)}{dt}$ must be zero at $t=0$, which is at odds with the result $R(t)=\exp(-\gamma \vert t\vert )$ of the memoryless Langevin equation. Nevertheless, we know that the exponential can be a reasonable approximation in some cases: Vainstein [*et al.*]{} [@Vainstein06a] have presented a large diversity of correlation functions that can be obtained from Equation (\[self\_consistent\]) once $\Pi(t)$ is known. Since, from Equation (\[memory\]), $\Pi(t)$ is an even function then we can write $$\label{memory2}
\Pi (t)=\sum_{n=0}^{\infty}b_n t^{2n}.$$ From Equation (\[self\_consistent\]), they proved that $R(t)$ must also be an even function, therefore $$\label{R2}
R(t)=\sum_{n=0}^{\infty}a_n t^{2n},$$ with $a_0 = R(0) = 1$. We insert Equations (\[memory2\]) and (\[R2\]) into Equation (\[self\_consistent\]) to obtain the recurrence relation [@Vainstein06a] $$a_n=-\frac{2\gamma \omega_s}{\pi(2n)!}\sum_{l=0}^{n-1}\frac{(-1)^l[2(n-1-l)]!\, \omega_s^{2l}}{(2l+1+\nu)} a_{n-1-l},$$ which shows the richness and complexity of behavior that can arise from a non-Markovian model. The above defined convergent power series represents a large class of functions, including the Mittag-Leffler function [@Mittag-Leffler05] which behaves as a stretched exponential for short times and as an inverse power law in the long time scale. Note that even for the simplest case of normal diffusion, $\nu=0$, $R(t)$ is not an exponential since at the origin its derivative is zero; however, for broad-band noise $\omega_s \gg \gamma $, i.e., in the limit of white noise it approaches the exponential $R(t)= \exp(-\gamma t)$, for times larger than $\tau_s= \omega_s^{-1}$.
{width="50.00000%"}
In Fig. (\[fig:correlation\_anomalous\]), from [@Vainstein06a], we display the rich behavior of the correlation function $R(t)$ for the case of normal diffusion. Here, $\gamma=1$ and $\omega_s=2 $ and $20$, for curves $a$ and $b$, respectively. Curve $c$ is the plot of $\exp(-\gamma t)$. In the inset, we highlight that although curve $b$ and the exponential approach one another for long times, for short times they differ appreciably. Also plotted is $\cos(\omega_0 t)$, with $\omega_0 = \sqrt{\Pi(0)}$.
Basic theorems of statistical mechanics
---------------------------------------
### The decay towards an equilibrium state
One of the most important aspects of dynamics is to observe the asymptotic behavior of a system or how it approaches equilibrium (or not). Observe that a direct solution for Equation (\[GLE\]) is $$A(t)=A(0)R(t)+\int_0^tR\left(t-t'\right)F(t)dt\label{eq:sol_gle}.$$ Given the initial states $A(0)$, it is possible to average over many trajectories to obtain the temporal evolution of the moments $\overline{ A^{n}(t)}$, with $n=1,2,\ldots$. The first moment arises directly from an average of Equation (\[eq:sol\_gle\]), $$\overline{ A(t)}=\overline{ A(0)} R(t)\text{.}\label{p_ave}$$ Taking the square of Equation (\[eq:sol\_gle\]) and averaging, we obtain $$\overline{ A^2}=\overline{ A^2}+R^2(t)\left[\overline{ A^2(0)}-\langle A^{2}\rangle_{eq}\right]\text{.}\label{p2_ave}$$ The skewness is defined as a measure of the degree of asymmetry of the distribution of $A(t)$, and is given by [@Lapas07] $$\label{ske}
\zeta(t)=\left[\frac{\sigma_A(0)}{\sigma_A(t)} \right]\zeta(0)R^3(t),$$ where $\sigma_A(t)=\overline{ A^2(t)}-\overline{A(t)}^2$. We also obtain the non-Gaussian factor [@Lapas07] $$\label{NG}
\eta(t)=\left[\frac{\overline{ A^2(0)}}{\overline{ A^2(t)}} \right]\eta(0)R^4(t).$$ Consequently, we see that $R(t)$ completely determines these averages. One can note that if the system is originally at equilibrium $\overline{A(0)}=0$ and $\overline{ A^2(0)} =\left\langle A^2\right\rangle_{eq}$, then the system remains in equilibrium. If the system is not in equilibrium, then $$\lim_{t \rightarrow \infty }R(t)=0
\label{MC}$$ drives the system towards equilibrium. The condition stated in Equation (\[MC\]) is called the mixing condition (MC) and is a fundamental concept in statistical mechanics, which asserts that after a long time the system reaches equilibrium and forgets all initial conditions.
Note that parity is conserved as well: if the initial skewness is null, $\zeta(0)=0$ in Eq. (\[ske\]), it will remain null during the whole evolution; the same holds for the nongaussian factor.
### The Kinchin theorem and ergodicity
The Khinchin theorem (KT) [@Lapas08; @Khinchin49] states that if the the MC holds, then ergodicity holds. As shown below, anomalous diffusion is a good field for testing ergodicity breaking [@Lapas08; @Weron10; @Lapas07]. For a situation where the MC is violated, we have $$\kappa= \lim_{t \rightarrow \infty }R(t)=\lim_{z \rightarrow 0}z\widetilde{R}(z)=\lim_{z \rightarrow 0}\left[1+\frac{\widetilde{\Pi}(z)}{z} \right]^{-1}\neq 0,
\label{MCV}$$ where $\kappa$ is the nonergodic factor [@Costa03; @Lapas08; @Bao06; @Bao05a]. For example for $\widetilde{\Pi}(z\rightarrow 0)\propto bz$ , $ \alpha=2$, and $$\kappa = \frac{1}{1+b}.
\label{MCV2}$$ I.e., the MC is violated in the ballistic motion, and consequently ergodicity is violated, but not the KT. Note that for $\alpha=2^{-}$, where $R(t \rightarrow \infty) \rightarrow 1/\ln(t) \rightarrow 0$, the MC is not violated. In this way the MC is satisfied for all diffusive processes in the range $0<\alpha<2^{-}$ [@Ferreira12].
It is interesting to observe that Equation (\[MCV\]) for long time behavior is equivalent to the condition $$\Lambda=\int U(\vec{r})\,d\vec{r}\to \infty,$$ for systems with long range interactions [@Silvestre; @Campa09], where $U(\vec{r})$ is the potential between the particles and the integration is performed over all space.
### Gaussianization
{width="30.00000%"} {width="30.00000%"} {width="30.00000%"}
As an illustration of the analytical results, we numerically integrated the GLE, Equation (\[GLE\]), to approximate the particle velocity distribution function, using Equations (\[memory\]) and (\[noise\_dos\]) to generate the memory for $\nu=-0.5$, $0$, and $0.5$, which by Equation (\[diff\_exponent\]) give $\alpha = 0.5$, $1$, and $1.5$, respectively. The results are exhibited in Fig. (\[fig:hist\]), from [@Lapas08], where we show the probability distribution functions as function of the momentum $p$. From left to right, we have subdiffusion ($\nu=-0.5$), normal diffusion ($\nu=0$), and superdiffusion ($\nu=0.5$) for the values $a=\frac{2\gamma}{\pi}=0.25$. A value $\omega_s=0.5$ was used for normal and superdiffusion. In the case of subdiffusion, a broader noise $\omega_s=2$ is needed for it to arrive at the stationary state. It is expected that $R(t\rightarrow\infty)=0$ in all cases, and that the EH will be valid even for the subdiffusion (superdiffusion). It should be noted that despite large fluctuations in the time average, there is a good agreement between the ensemble and time distributions, in agreement with Eqs. (\[p\_ave\]) to (\[MC\]), indicating the validity of the EH. In all cases, the distributions converge to the expected Maxwell-Boltzmann distribution, in accordance with analytical results [@Lapas07].
{width="80.00000%"}
In Fig. (\[fig:ngaussian\] we show the evolution of the nongaussian factor. We see that for the Ballistic diffusion (BD), it does not reach a null value, but it evolves towards it. Note that even for a situation where $\kappa \neq 0$, the nongaussian factor will be very small and the probability of it being non-zero in simulations after a long time is very small. For example for $\kappa=0.1$, there will a factor of $10^{-4}$ in relation (\[NG\]).
{width="80.00000%"}
In Fig. (\[fig:ekt\]), from [@Lapas07], we show the evolution of the kinetic temperature of the system by taking $A^2(t)$, with $A=P$, the momentum of the particles. In this case we should have $ \langle A^2 \rangle_{eq} -\langle A \rangle^2_{eq} \propto T$ [@Lapas07] and the temperature evolution can be obtained from Equation (\[p2\_ave\]). We consider both normal and ballistic diffusion (BD) in a reservoir characterized by $T=1.0$ with initial high and low temperatures $T_0 = 1.5$ and $T_0 = 0.5$, respectively. For normal diffusion (dashed curve), we have $R(t) = \exp{(-\gamma t)}$, with $\gamma = 10^{-3}$. For BD, $R(t)$ is calculated numerically in [@Lapas07] with $\gamma = 1$. Since BD’s relaxation is slow, we take a “friction” $\gamma$ a thousand times larger than that of normal diffusion for comparison. As expected, in the case of normal diffusion the system’s temperature always relaxes to that of the reservoir, while for BD the temperature approaches that of the reservoir without reaching it [@Lapas07; @Lapas15]. Figure (\[fig:ngaussian\]) displays the normalized non-Gaussian factor, Equation (\[NG\]), as a function of time [@Lapas07] for the cases in the previous figure, with the same convention for the labeling. For normal diffusion, the system’s probability distribution evolves towards a Gaussian, which is not the case for BD. In the latter case, $R(t)$ oscillates around the value predicted by Equation (\[MCV2\]), even for long times, In both cases, the initial probability distribution function was the Laplace distribution, with $\langle A^2(0)\rangle=1$ and $\langle A(0)\rangle=0$.
### The fluctuation-dissipation theorem
Figures (\[fig:hist\])–(\[fig:ngaussian\]) are just illustrations of the results that can be obtained analytically from Equations (\[MC\]) to (\[EH\]). Lapas [*et al.*]{} [@Lapas08; @Lapas07] have shown that the KT is valid for all forms of diffusion, and that the ballistic diffusion violates the EH, but not the KT.
Although it is expected that a system in contact with a heat reservoir will be driven to equilibrium by fluctuations, Figs.(\[fig:ekt\]) and (\[fig:ngaussian\]) show that it is not always the case and in some far from equilibrium situation it may not happen. The concept of “far from equilibrium” is itself sometimes misleading, since it depends not only on the initial conditions, but also on the possible trajectories the system may follow [@Costa03; @Rubi04; @Santamaria-Holek04]. It was a known fact that the FDT can be violated in many slow relaxation processes [@Parisi97; @Vainstein06; @Dybiec12], however, it was a surprise that it could occur in a GLE without disorder [@cugliandolo94] or an external field [@Costa03; @Dyre00; @Vainstein06]. Finally, Costa [*et al.*]{} [@Costa03] have called attention to the fact that if after a long time the fluctuations are not enough to drive the system to equilibrium then the fluctuation-dissipation theorem is violated. For the dissipation relation to be fulfilled, the MC condition, Equation (\[MC\]), must be valid.
Beyond the basics... and more basic
===================================
In the last sections we have discussed some basic results for anomalous diffusion under the point of view of the formalism of the generalized Langevin equation which yield two main features: simplicity and exact results. Obviously, this approach does not exhaust the subject, since diffusion is a basic phenomenon in physics it is a starting point to many different formalisms which we shall briefly discuss.
Fractional Fokker-Planck equation
---------------------------------
We have seen that, in principle, all kinds of anomalous diffusion can be described by the GLE formalism, which is itself well established from the Mori method. Since normal diffusion can be studied both from Langevin equations and from Fokker-Planck equations, we would expect to obtain a generalized Fokker-Planck formalism for anomalous diffusion. Indeed, fractal formulations of the Fokker-Planck equation (FFPE) have been widely used in the literature in the last decades, in which the evolution of the probability distribution function $P(x,t)$ reads [@Metzler99; @Metzler00; @Metzler04] $$\label{FFPE}
\frac{\partial^\alpha P(x,t)}{\partial t^\alpha}=\left[ \frac{1}{m\gamma_\alpha}\frac{\partial U(x)}{\partial x}+D_\alpha \frac{\partial^2}{\partial x^2}\right]P(x,t),$$ where on the left-hand-side a fractional Rieman-Liouville time-derivative is defined as [@oldham74] $$\frac{\partial^\alpha P(x,t)}{\partial t^\alpha}=\frac{1}{\Gamma(1-\alpha)}\frac{\partial}{\partial t}\int_0^t\frac{P(x,s)}{(t-s)^\alpha}ds,$$ with $0 < \alpha < 1$. Note that the definition of fractional derivatives is not unique [@oldham74; @kilbas06], with a variety of possibilities, physically (almost) totally unexplored. We notice here that the nonlocal character of the fractional Fokker-Planck equation is similar to the memory kernel in the GLE. On the right hand side, $\gamma_\alpha$ is a generalized friction, $U(x)$ is the potential, while $D_\alpha$ is generalized Einstein-Stokes relation $$D_\alpha= \frac{k_BT}{m\gamma_\alpha}.$$The major result from Equation (\[FFPE\]) for a force-free diffusion is the asymptotic solution $$\lim_{t \rightarrow \infty}\langle x^2(t) \rangle = \frac{2D_\alpha}{\Gamma(1+\alpha)} t^\alpha.$$ Again, for $\alpha \neq 1$, we reach an anomalous diffusion regime [@Shlesinger93; @Metzler00; @Metzler04; @Klafter96].
Interface growth
-----------------
Models for interface growth generally consider the random deposition of particles that diffuse to a surface and, as such, have been studied with Langevin equations and modified diffusion equations. Since diffusion with subsequent deposition is ubiquitous, rough surfaces at the interface of two media are very common in nature [@Barabasi95; @Edwards82; @Kardar86; @Hansen00; @Cordeiro01; @Schmittbuhl06] and the description of interface evolution is a very interesting problem in statistical physics. The major objective here is to describe the temporal evolution of the height $h(\vec{x},t)$ of the interface between two substrates [@Barabasi95; @Edwards82], where $\vec{x}$ is the $d$ dimensional position vector and $t$ is the time. We outline the evolution of $h$ in Fig. (\[fig:h\]). In (a), we show a real forest fire propagation, a very complex situation. However, we can focus on the interface between the burnt and unburned regions. In (b), we provide a snapshot at a fixed time $t$ of the height $h(\vec{x},t)$ for a microscopic growth process in which the green medium is penetrating the blue one with arbitrary units. These types of dynamics in $d+1$ dimensional space are easy to understand, but not so simple to solve analytically. Experiments can be done for $d=1,2,3$; however, they present hard theoretical problems for any $d$.
The two main quantities of interest are the average height $$\label{hmedio}
\langle h(t) \rangle =\frac{1}{V}\int h(\vec{x},t) d\vec{x},$$ where $V$ is the sample volume, and the standard deviation $$w^2(t)=\langle h^2(t) \rangle - \langle h(t) \rangle^2,
\label{wt}$$ often called the surface width $w(t)$, or roughness. Not surprisingly, this height fluctuation has a lot of information about the physical processes governing the system. The evolution of $w(t)$ observed through experiments, computer simulations, and a few analytical results gives us some general features of growth dynamics.
Starting with a flat surface, $w(0)=0$, the evolution exhibits four distinct regions: (a) for a very short period $0<t<t_0$, during which correlations are negligible, the process is a random deposition $w(t) \sim t^{1/2}$; (b) for $ t_0 < t < t_\times$ we have $ w(t) \sim t^{\beta}$. Here the $t_\times$ follows a power law of the form $t_\times \sim L^{z}$, where $L$ is the size of the sample, $z$ is the dynamic exponent, and $\beta$ the growth exponent; (c) there is a transition region for $ t \sim t_\times$; (d) finally, for $t \gg t_\times$, the dynamic equilibrium leads to surface width saturation, $w_s$, which also follows a power law $ w_s \sim L^{\alpha}$, where $\alpha$ is the roughness exponent. The crossing of the curves $ w(t) \sim t^{\beta}$ and $w \sim w_s$ yields the universal relation $$\label{universal}
z=\frac{\alpha}{\beta}.$$ It should be pointed out that these exponents are not related with those of the previous section.
To obtain $w(t)$, we need to know $h(\vec{x},t)$ and there are two major theoretical ways to attack this problem: 1. Continuous growth equations; 2. Discrete growth models. Fig. (\[fig:h\]a) is an example of the first, while Fig. (\[fig:h\]b) is an example of the second.
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### Equation of motion and symmetries
The growth, in general, is due to the number of particles per unit of time $G(\vec{x},t)$ arriving on the surface at the position $\vec{x}$ and time $t$. The particle flux is not uniform since the particles are deposited at random positions [@Barabasi95]. Therefore, the evolution of $h(\vec{x},t)$ can be described by $$\label{dht}
\frac{\partial h(\vec{x},t)}{\partial t}=G=F+\chi(\vec{x},t),$$ where the first term $F$ is the average number of particles arriving at site $\vec{x}$. The second term, $\chi(\vec{x},t)$, reflects the random fluctuations and satisfies $$\langle \chi({\vec{x},t})\rangle =0,\text{ and }$$ $$\langle \chi({\vec{x},t})\chi({\vec{x}\,',t'}) \rangle =2D_g\delta(\vec{x}-\vec{x}\,')\delta(t-t'),$$ where, $D_g$ measures the degree of growth randomness. The deterministic flux $F$ must satisfy certain symmetry requirements, such as invariance under translation of position, $\vec{x} \rightarrow \vec{x}+\vec{x_0}$, height, $h \rightarrow h +h_0$ and time $t \rightarrow t +t_0$. To satisfy these conditions, it must depend only on derivatives, which, again by the symmetries $\vec{x} \rightarrow -\vec{x} $ and $h \rightarrow - h $, must be of even order [@Barabasi95; @Edwards82]. Besides this, considering that symmetrically possible terms such as $ \nabla^{2n} h(\vec{x},t)$, for $n=2,3,\dotsc$, are irrelevant in the long wavelength limit, since they go to zero faster than $ \nabla^{2} h(\vec{x},t)$, we obtain, at the lowest-order in the derivatives [@Edwards82], $$\label{EWE}
\frac{\partial h(\vec{x},t)}{\partial t}= \nu \nabla^2 h(\vec{x},t)+\chi(\vec{x},t),$$ known as Edwards-Wilkinson equation (EW). Note that it is basically a diffusion equation, as (\[diff\]), plus a noise, where we have a surface tension $\nu$ associated with the Laplacian smoothening mechanism. The random deposition model with surface relaxation is in the same universality class as the EW model [@Horowitz01], i.e., they share the same exponents.
Because there are, however, a large class of growth phenomena which are not described by the EW equation, new formulations become necessary. The recently proposed Arcetri models [@Henkel15] allow the study of more rigid interfaces than the EW model and still allow for exact solutions in any dimension $d$. Depending on the initial conditions, both growing interfaces and particle motion on the lattice can be modeled. Another classical model was proposed by Kardar [*et al.*]{} [@Kardar86], inspired by the stochastic Burgers equation. They observed that lateral growth could be added to this equation via the nonlinear term of the Burgers equation so that Equation (\[EWE\]) becomes
$$\label{KPZ}
\dfrac{\partial h(\vec{x},t)}{\partial t}=\nu \nabla^2 h(\vec{x},t) +\dfrac{\lambda}{2}[\vec{\nabla}h(\vec{x},t)]^2 + \chi(\vec{x},t).$$
Since its formulation, the Kardar-Parisi-Zhang equation (KPZ) has been a prime model in the description of growth dynamics. The nonlinear term includes a new constant $\lambda$ associated with the tilt mechanism and breaks down the symmetry $h \rightarrow - h $. Consequently, the universality class of KPZ is different from that of EW.
Equation (\[KPZ\]) is the simplest nonlinear equation that can describe a large number of growth processes [@Barabasi95; @Kardar86]. However, the apparent simplicity of this equation is misleading, since the nonlinear gradient term combined with the noise makes it one of the toughest problems in modern mathematical physics [@Hairer13; @Sasamoto10]. On the other hand, its complexity is compensated for its generality. It is connected to a large number of stochastic processes, such as the direct polymer model [@Kardar85], the weakly asymmetric simple exclusion process [@Bertine97], the totally asymmetric exclusion process [@Spitzer70], direct d-mer diffusion [@Odor10], fire propagation [@Mylles01; @Mylles03; @Merikoski03], atomic deposition [@Csahok92], evolution of bacterial colonies [@Ben-Jacob94; @Matsushita90], turbulent liquid-crystals [@Takeuchi11; @Takeuchi12; @Takeuchi13], polymer deposition in semiconductors[@Almeida17], and etching [@Mello01; @Reis03; @Reis04; @Reis05; @Oliveira08; @Tang10; @Xun12; @Rodrigues15; @Mello15; @Alves16; @Carrasco16; @Carrasco18].
This problem in non-equilibrium statistical physics is analogous to the Ising model for equilibrium statistical physics, which is used as a basic model for understanding a large class of phenomena. The search for exact solutions to this equation in the $d+1$ dimensional space has resulted in important contributions to mathematics; however, to date, they have been obtained only for specific situations and are limited to $1+1$ dimensions [@Kardar86; @Sasamoto10] .
### Scaling invariance
In this small subsection we limit ourselves to scaling symmetries only. However, in interface growth, the shape of single-time and two-time responses and height correlation functions can be derived from the assumption of a local time-dependent scale-invariance [@Henkel17]. For the KPZ equation, a systematic test of aging scaling was performed, showing the scaling relations for the two-time spatio-temporal autocorrelator and for the time-integrated response function [@Reis04; @Henkel12; @Kelling17a].
The scaling invariance can be investigated through the transformation [@Barabasi95] $\vec{x} \rightarrow b \vec{x}$, $h \rightarrow b^\alpha$ and $t \rightarrow t^z$ which yields, for the KPZ equation, (\[KPZ\]), $$\label{nu}
\nu \rightarrow b^{z-2} \nu,$$ $$\label{Dg}
D_g \rightarrow b^{z-d-2\alpha} D_g,$$ and $$\label{lambda}
\lambda \rightarrow b^{\alpha+z-2} \lambda$$Scaling invariance demands that the exponents in Equations (\[nu\],\[Dg\], \[lambda\]) must be zero. However, a simple inspection shows that they are inconsistent. The renormalization group approach of Kardar [*el al.*]{} [@Kardar86], uses the nonlinear term as a perturbation, and their result shows that only Equation (\[lambda\]) remains invariant, yielding the famous Galilean invariance $$\alpha+z=2.
\label{GI}$$ Equations (\[nu\]) and (\[Dg\]) are corrected by the renormalization, and the final result yields $\alpha=1/2$, $\beta=1/3 $, and $z=3/2$, for $1+1$ dimensions. Since the KPZ renormalization approach is valid only for $1+1$ dimensions, questions about the validity of the Galilean invariance [@Wio10b; @Wio17] for $d>1$ and the existence of an upper critical dimension for KPZ [@Francesca01; @Schwartz12] have been raised.
For $d>1$, the numerical simulation of the KPZ equation is not an easy task [@Wio10b; @Wio17; @lam98; @Xu06; @HalpinHealy15; @Torres17], and the use of cellular automata models [@Mello01; @Reis03; @Reis04; @Reis05; @Oliveira08; @Tang10; @Xun12; @Rodrigues15; @Mello15; @Alves16; @Carrasco16; @Carrasco18; @Kelling17; @Predota96; @Chua05; @Buceta14] has become increasingly common for growth simulations. Polynuclear growth (PNG), is a typical example of a discrete model that has received a lot of attention, and the outstanding works of Prähofer and Spohn [@Prahofer00] and Johansson [@Johansson00] drive the way to the exact solution of the distributions of the heights fluctuations $f(h,t)$ in the KPZ equation for $1+1$ dimensions [@Sasamoto10]. By construction, $h\rightarrow h-\langle h\rangle $, then $f(h,t)$ has zero mean, so its skewness and kurtosis are the most important quantities to observe [@Takeuchi13; @Oliveira13; @Alves13; @Almeida14; @Halpin-Healy95]. In addition, Langevin equations for growth models have been discussed by some authors [@Haselwandter06; @Haselwandter08; @Silveira12]. Several works have been done in the weakly asymmetric simple exclusion process [@Bertine97], the totally asymmetric exclusion process [@Spitzer70; @Alcaraz99], and the direct d-mer diffusion model [@Odor10]: for a review see [@HalpinHealy15; @Halpin-Healy95; @Meakin93; @Krug97]. More experimentally measured exponents for growing interfaces in four universality classes (KPZ, quenched KPZ, EW and Arcetri) can be found in [@Henkel15] and references therein.
### Cellular automata growth models
{width="100.00000%"}
Cellular automata are described by simple rules, which allow us to inquire about relevant properties of a complex dynamical system. For a given growth model, the first question to be answered is if the model belongs to the same universality class as KPZ. In this context, we expose here the etching model [@Mello01], which has attracted considerable attention in recent years [@Reis03; @Reis04; @Reis05; @Oliveira08; @Tang10; @Xun12; @Rodrigues15; @Mello15; @Alves16; @Carrasco16; @Carrasco18]. These studies suggest a close relation between the etching model and KPZ. For example, for $1+1$ dimensions Alves [*et al.*]{} [@Alves16] have proven that $\alpha=1/2$ exactly. Unfortunately, their method does not allow to obtain $\beta$ or $z$.
The etching model describes the mechanism of an acid eroding a surface. The detailed description of the model can be found in [@Mello01]) For simplicity, let us consider a site $i$ in a hypercube of side $L$ and volume $V=L^d$ in space $\Psi$ of dimension $d$, and look at one of its nearest neighbors, $j$. If $h_{j} > h_i$, then it becomes equal to $h_i$. We then define the etching model following the steps:
1. At time t we randomly choose a site $i\in{V}$.
2. If $h_{j}(t) > h_i(t)$, do $h_{j}(t+\Delta t)=h_i(t)$.
3. Do $h_i(t+\Delta t)=h_i(t)-1.$
In Fig. (\[fig:etching\]), we show the mechanism of the etching model in one dimension: (a) step 1, we randomly select a site at time $t$, here $i=2$ shown as green in the figure; step 2, the site $i=2$ interacts with its neighbors $j=1,3$. Then in Fig. (\[fig:etching\]2) $h_1(t+\Delta t)=h_2(t)$, meaning that it is strongly affected, and $h_3(t+\Delta t)=h_3(t)$, meaning it is not affected; (c) step 3, $h_2(t+\Delta t)=h_2(t)-1$. We describe here a process of strong interactions between the site and its neighbors.
Using these rules and averaging over $N_e$ numerical experiments, it is possible to obtain the exponents $\alpha$, $\beta$, and $z$ [@Mello01; @Reis03; @Reis04; @Reis05; @Oliveira08; @Tang10; @Xun12; @Rodrigues15; @Mello15; @Alves16; @Carrasco16; @Carrasco18]. Finally, the upper critical limit for the etching model (KPZ) was recently discussed by Rodrigues [*et al.*]{} [@Rodrigues15], where it was shown numerically that there is no upper critical dimension for the etching model up to $6+1$ dimensions. Thus, we have established a lower limit for the KPZ critical dimension, i.e., if it exists then $d_c>6$, in agreement with other authors [@Odor10]. They have shown that the Galilean invariance (\[GI\]) remains valid for $d \leq 6$ as well. These problems, however, still lack an exact solution.
Reaction-diffusion processes
-----------------------------
One could not complete a work on diffusion without a brief discussion of reaction-diffusion processes [@Alcaraz99; @ben00; @abad02; @shapoval18]. Exactly solvable reaction-diffusion models consist largely of single species reactions in one dimension, e.g., variations of the coalescence process, $A + A \rightarrow A + S $ [@doering89; @krebs95; @simon95] and the annihilation process $A + A \rightarrow S + S $ [@krebs95; @simon95], where $A$ and $S$ denote occupied and empty sites, respectively. These simple reactions display a wide range of behavior characteristic of non-equilibrium kinetics, such as self organization, pattern formation, and kinetic phase transitions. Interval methods have provided many exact solutions for one-dimensional coalescence and annihilation models. The method of empty intervals, applicable to coalescence models, requires solution of an infinite hierarchy of differential difference equations for the probabilities $E_n$ of finding $n$ consecutive lattice sites simultaneously empty. For annihilation models, the method of parity intervals similarly requires determination of $G_n$, the probability of $n$ consecutive lattice sites containing an even number of particles [@abad02]. In the continuous limit, such models can be described exactly in terms of the probability $E(t,x)$ of finding an empty interval of size $x$ at time $t$. Under fairly weak conditions, the equation of motion for $E(t,x)$ is a diffusion equation, albeit with the unusual boundary condition $E(t,0)=1$. In several cases, the exact and fluctuation-dominated behaviour in $d=1$ has been seen experimentally, since the 1990s. This is one of the rare cases where theoretical statistical mechanics can be compared with experiments and also shows that simple mean-field schemes are not enough. For an introduction see [@ben00], and for up to date references see [@shapoval18].
Another important new development in diffusion concerns stochastic resets, as they were introduced by Evans and S.N. Majumdar (see [@evans11; @evans14; @durang14] and references therein). This leads to important modifications of the stationary state and raises the question how to consider the status of the general theorem described in the present review.
Conclusions
===========
In this short review we address diffusion, as a modern and important topic of statistical physics, with broad applications [@Metzler00; @Morgado02; @Metzler04; @Costa03; @Lapas08; @Ciesla01; @Holek09]. We emphasize the study of systems with memory, for which a generalized Langevin equation applies and describe how the diffusion exponent is obtained from the memory [@Morgado02; @Ferreira12]. We highlight the properties of response functions [@Lapas07; @Vainstein06] for the processes of anomalous relaxation [@Vainstein06; @Lapas15], Gaussianization and ergodicity [@Lapas08; @Lapas07]. Moreover, we have established a hierarchy: the mixing condition is stronger than the ergodic hypothesis, which is itself stronger than the fluctuation-dissipation theorem. It is important to call attention again to Costa [@Costa03], where it was observed that in the ballistic diffusion the fluctuation-dissipation theorem fails. Indeed, for the ballistic motion which, on average, behaves as a Newtonian particle with constant velocity, the fluctuations are not enough to bring the system to equilibrium, i.e., the dissipation, which for large times decays as $1/t$, does not balance the fluctuations. This work points out that the violation of the mixing condition breaks down ergodicity, as required by the Khinchin theorem and the fluctuation dissipation theorem.
In conclusion, since the mechanism of diffusion is present in most nonequilibrium processes, diffusion is an exhaustive subject, and we have only called attention to some of its aspects. Consequently, we apologize to the authors of important works not mentioned here , such as aging [@cugliandolo94; @Reis04; @Henkel12; @Kelling17a; @hodge95] for example.
Conflict of Interest Statement {#conflict-of-interest-statement .unnumbered}
==============================
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Author Contributions {#author-contributions .unnumbered}
====================
FAO suggested the work, and wrote some sections; RMSF was responsible for the section on non-Markovian processes and selected the figures; LCL wrote the ergodicity and decay towards equilibrium sections; MHV collaborated on the section on the generalized Langevin equation and reedited the text.
Acknowledgments {#acknowledgments .unnumbered}
===============
This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, CNPq and FAPDF. MHV was supported by a Senior fellowship (88881.119772/2016-01) from Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) at MIT. MHV would like to thank Prof. Jeff Gore for the hospitality at Physics of Living Systems, MIT.
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13th Asian Continental – a two-nation battle
by Sagar Shah
4/19/2014 – The Open Section has 64 players, including 32 GMs and 11 IMs, and a 2398 average rating. There are 32 prizes, with the winner receiving US $6000. The Women's section has 32 players and a decent rating average of 2141, with the first prize being $3000. The event is essentially a powerhouse confrontation of the two big chess nations: India and China. Three rounds have been played.
Everyone uses ChessBase, from the World Champion to the amateur next door. It is the program of choice for anyone who loves the game and wants to know more about it. Start your personal success story with ChessBase 14 and enjoy your chess even more.
Everyone uses ChessBase, from the World Champion to the amateur next door. It is the program of choice for anyone who loves the game and wants to know more about it. Start your personal success story with ChessBase 14 and enjoy your chess even more.
Suppose you are a budding talent from Asia and your dream is to attain
the absolute top chess honour of becoming the World Chess Champion. How
can this dream be converted into reality? Here is the route.
You take part in the Asian Continental Chess Championships that have
started in UAE, specifically in Sharjah, from the 16-26th April 2014.
Finish amongst top 5 players of the tournament and qualify for the
World Cup that is to be held in Baku, Azerbaijan in 2015.
Win the World cup or finish as a runner up and get qualified for the
Candidates 2016.
Win the Candidates 2016 tournament.
Challenge the World Champion (most probably Magnus Carlsen who might
have reached 3000 ELO by then!)
Beat the World Champion and become the undisputed World Chess Champion!
Quite a long process, I agree, but this gives an opportunity to every single
player to dream big! You can definitely understand the importance Asian
Continental Chess Championships hold. The Asian Continental Chess Championships
both for Open and Women category are being held from 16th-26th April in
the city of Sharjah, UAE.
The biggest incentive for players apart from becoming the Asian champion
and winning good prize money is the qualification to World Cup 2015. The
top five players from the Asian Continental Open category and the winner
of Asian Continental women will qualify for the World Cup 2015 in Baku,
Azerbaijan.
The Sharjah Cricket Stadium, currently holding
a very big event of Indian Premier League (IPL)
But let us focus our attention on chess and the Asian Continental Championship.
The Open Section is extremely strong with 64 players and an average rating
of 2398. There are 32 GMs and 11 IMs in the field. It is a nine round event,
with a single round every day. The first prize is US $6000, and there are
in all 32 prizes. So half the players will go back home with some prize
money, and the last place also has a decent amount of $600.
This tournament is more like two powerhouses of Asian Chess, India and
China, fighting out against each other for the top honours. In the top ten
seeds of the tournament there are four Chinese and four Indian players!
I have compared one Chinese player with one Indian, just to give you an
idea as to how well matched they are!
Bu Xiangzhi (2699) and Krishnan Sasikiran (2680)
Bu Xiangzhi (2699) is the top seed of the event. Though he has not been
playing very actively recently he still remains a huge force to reckon.
Bu was at one point the youngest GM in the world when he achieved the feat
at 13 years 10 months and 13 days.
While Bu’s experience at top level chess has been quite extensive,
the Indian GM Krishnan Sasikiran too needs no introduction. Sasikiran is
one of the most talented and hard-working players in Indian chess. He has
not only locked horns with the best but also beaten world class players
like Aronian, Anand, Ponomariov, Adams etc. He has already qualified for
the World Cup 2015 on the basis of becoming National Champion in December
2013.
Yu Yangyi (2667) and Parimarjan Negi (2640)
Parimarjan Negi, born in 1992, is considered by many as the brightest
talent in India currently. He became a GM at the tender age for 13 years
three months and 13 days. He was the champion of this event in 2012, so
he knows what is like to be the best in Asia. Recently he got admitted into
the world famous Stanford University, which is a huge feat for a chess player
in general.
Yu Yangyi born in 1994 was not as quick as Negi to achieve his GM title.
However he more than compensated it by his recent achievements. He became
the World Junior Champion in 2013, and if we go by the names of the past
World Junior champions like Anand, Aronian, Kasparov etc. we realize that
this kid has a very bright future ahead of him. Just a month ago he won
the very strong Chinese Championships. The World Junior 2013 title has qualified
Yu for the World Cup 2015.
Ni Hua (2654) and Surya Shekhar Ganguly (2631)
Touted as huge talents, both these players were expected to break into
the 2700 mark with ease. While Ni Hua who was at one point at a high with
a rating of 2724, he has now lost almost 70 rating points. Ganguly on the
other hand, rendered invaluable services to Vishy Anand, which helped the
latter to become World Champion or retain his title. Both these players
must have come to the tournament with grim determination to perform to their
true potential and become the Asian Champion.
Wei Yi (2629) and Abhijeet Gupta (2630)
While age is not the common factor amongst these two players, their strength
is. Almost equally rated both these players have a lot of achievements to
their credit. Wei Yi has the amazing distinction of become the youngest
player in the history of chess to break in 2600, at the age of 14 years
and four months. The ease with which he beat Nepomniachtchi and Shirov in
World Cup 2013 just shows his immense talent. Even Garry Kasparov has made
a note of him as a rising star.
On the other hand Abhijeet Gupta has a penchant for winning tournaments.
He became the Junior World Champion in 2008, and has won many prestigious
events, like the Prasvanath International Open, the Commonwealth Chess Championships,
the Al Ain Classic, the Philadelphia Open – and the list continues.
Just to make my task easier I can direct you to his
newly launched website to know more about his achievements.
To the not so observant viewer, this might seem like an India-China match.
However there are two more spots left in the top ten seeds and they are
taken up by players from Uzbekistan and Vietnam.
Rustam Kasimdzhanov is the second seed of the event with a rating of 2693.
He already won the World Cup in 2004 and become the FIDE World Chess Champion.
He worked for many years as Vishy Anand’s second and helped him in
his bid to become the World Champion. The list of top ten seeds is completed
by Nguyen Ngoc Truongson (2621) who is one of the best players from Vietnam
after Le Quang Liem.
The Indians have sent a very strong contingent with all of their four young
talents in the form of Vidit Gujrathi (2601, top left), B Adhiban (2609,
top right), Lalith Babu (2594, bottom left) and S.P. Sethuraman (2578, bottom
right) taking part in this year’s Asian Continental. These four youngsters
are marching ahead as a unit when it comes to their ratings, and all of
them are a grave threat to the top ten seeds of the tournament.
The Women's Championship
With the top five Chinese women Hou Yifan, Xie Jun, Zhao Xue, Ju Wenjun
and Ruan Lufei, and the top two Indian women Koneru Humpy and Dronavalli
Harika, not taking part in this year’s Asian Continental women championship,
the strength of the tournament is quite diminished. Yet it boasts of a decent
rating average of 2141, with 32 players in the field. The first prize is
US $3000.
Chinese WGM Tan Zhongyi, rated 2488, is the
top seed in the Women’s Section
Asian Continental Blitz Championship
On 19th April after the 3rd round, a blitz championship will be held, with
a first prize of US $1500. The time control of the event is 3 minutes +
2 sec increment. GM Pentala Harikrishna from India, with an ELO of 2726
the fourth highest rated player in Asia after Anand, Wang Hao and Wesley
So, will take part in the Asian Continental Blitz Championship. He will
be the hot favourite to win the title.
Sagar ShahSagar Shah is an International Master from India with two GM norms. He is also a chartered accountant and would like to become the first CA+GM of India. He loves to cover chess tournaments, as that helps him understand and improve at the game he loves so much. He is the co-founder of the ChessBase India website.
See also
4/23/2017 – In conjunction with the Kasparov Chess Foundation's 15th Anniversary Celebration, the Kasparov Chess Foundation Asia-Pacific proposed five activities as its contribution and four of them came together with a tour of four countries in Asia with the common theme of Chess in Education. Here is an illustrated report by Peter Long
See also
2/22/2017 – We recently reported that the "golden couple" GM Zhang Zhong and WGM Li Ruofan, Singapore's top players, had been "banned" from participating in the the forthcoming Asian Zonal 3.3, because they showed "insufficient involvement in the local chess life." The Association of Chess Professionals protested, and now the Singapore Chess Federation has replied with the following letter.
Video
Tired of spending hours and hours on the boring theory of your favourite opening? Then here is your solution, play an Anti-Sicilian with 3.Bb5 against 2...d6 or 2...Nc6, and 3.d3 against 2...e6. In 60 minutes you will get a crash course in how to avoid mainstream theory and in understanding the ideas of this Anti-Sicilian setup. After these 60 minutes you should be able to survive the Sicilian for a long time, without being bothered by new developments found by engine x supported by an x-core machine. Now that it finally comes down to understanding, let's play chess!
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Q:
Cannot deploy to Azure using git
I am using the Xamarin studio on mac to develop a ASP.NET MVC Razor application. Everything is working locally, but when I push to azure, it fails to deploy. Here is the output:
remote: Running deployment command...
remote: Handling .NET Web Application deployment.
remote: ..........
remote: Installing 'Microsoft.AspNet.WebPages 3.2.0'.
remote: Successfully installed 'Microsoft.AspNet.WebPages 3.2.0'.
remote: Installing 'Microsoft.Data.OData 5.6.2'.
remote: Installing 'Microsoft.AspNet.Mvc 5.2.0'.
remote: Successfully installed 'Microsoft.AspNet.Mvc 5.2.0'.
remote: Installing 'Microsoft.Web.Infrastructure 1.0.0.0'.
remote: Successfully installed 'Microsoft.Web.Infrastructure 1.0.0.0'.
remote: Installing 'Microsoft.Data.Edm 5.6.2'.
remote: Installing 'System.Spatial 5.6.2'.
remote: Successfully installed 'Microsoft.Data.OData 5.6.2'.
remote: Successfully installed 'System.Spatial 5.6.2'.
remote: Successfully installed 'Microsoft.Data.Edm 5.6.2'.
remote: Installing 'Microsoft.Data.Services.Client 5.6.2'.
remote: Successfully installed 'Microsoft.Data.Services.Client 5.6.2'.
remote: ......................
remote: Installing 'Newtonsoft.Json 5.0.8'.
remote: Successfully installed 'Newtonsoft.Json 5.0.8'.
remote: Installing 'WindowsAzure.Storage 4.3.0'.
remote: Successfully installed 'WindowsAzure.Storage 4.3.0'.
remote: Installing 'Microsoft.AspNet.Razor 3.2.0'.
remote: Installing 'Microsoft.WindowsAzure.ConfigurationManager 1.8.0.0'.
remote: Successfully installed 'Microsoft.AspNet.Razor 3.2.0'.
remote: Successfully installed 'Microsoft.WindowsAzure.ConfigurationManager 1.8.0.0'.
remote: ..
remote: WeddingSite -> D:\home\site\repository\WeddingSite\bin\WeddingSite.dll
remote: D:\home\site\repository\WeddingSite\WeddingSite.csproj : error MSB4057: The target "pipelinePreDeployCopyAllFilesToOneFolder" does not exist in the project.
remote: Failed exitCode=1, command="D:\Windows\Microsoft.NET\Framework\v4.0.30319\MSBuild.exe" "D:\home\site\repository\WeddingSite\WeddingSite.csproj" /nologo /verbosity:m /t:Build /t:pipelinePreDeployCopyAllFilesToOneFolder /p:_PackageTempDir="C:\DWASFiles\Sites\#1raouf-wedding\Temp\5314389e-4e5d-40f3-a7ce-0493eae3f70c";AutoParameterizationWebConfigConnectionStrings=false;Configuration=Release /p:SolutionDir="D:\home\site\repository\.\\"
remote: An error has occurred during web site deployment.
remote:
remote: Error - Changes committed to remote repository but deployment to website failed.
To https://[email protected]:443/raouf-wedding.git
* [new branch] master -> master
I cannot figure out the issue.
A:
I had the same problem, here are steps I used to fix it:
1. Edit .csproj file and add following in tag, it should be the first tag in the file.
<Import Project="$(VSToolsPath)\WebApplications\Microsoft.WebApplication.targets" Condition="'$(VSToolsPath)' != ''" />
<Import Project="$(MSBuildExtensionsPath32)\Microsoft\VisualStudio\v12.0\WebApplications\Microsoft.WebApplication.targets" Condition="true" />
You need to make sure that for (MSBuildExtensionsPath32) you are pointing to the correct path, in my case I needed to use v12.0 but if you have earlier version of VS then 2013 it might e.g.: be v11.
And also in web.config I added this section:
<configuration>
<system.webServer>
<validation validateIntegratedModeConfiguration="false"/>
</system.webServer>
</configuration>
I also changed pipeline mode in azure to Classic.
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TRANSFER TRACKER STATUS: Rumor
Bundesliga side Borussia Monchengladbach have beaten out Turkish giants Besiktas and English promotion hopefuls Leeds United for the services of Canada national team forward Cyle Larin, according to a report from The Sun (UK).
The 22-year-old Larin has spent his three-year pro career entirely with Orlando City SC, scoring 43 goals in 87 league appearances, but this fall had indicated his desire for a transfer to Europe.
That was perhaps no shock after the July arrival of striker Dom Dwyer and reported interest from abroad in Larin in the summer window.
A move to Germany would be a bit of a surprise, however, after reports of a potential switch to join Besiktas and Canada teammate Atiba Hutchinson appeared to have legs.
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Q:
Kendoui DropDownList on grid with Ajax and paramter (ASP.NET MVC)
I have a KendoUI grid that uses a DropDown List. So each element of the grid has a dropdown list. The DropDownList is defined in a partial view.
@(Html.Kendo().DropDownList()
.Name("positions")
.DataValueField("EmpId")
.DataTextField("EmpName")
.DataSource(source =>
{
source.Read(read =>
{
read.Action("_AjaxGetEmps", "Emp", new { Empid = <empid of currently selected grid row> });
}).ServerFiltering(true);
})
)
What do I put where is? What I'm trying to do is reference a field from the grid of the currently selected row. Each row of the grid can have different values in the drop down and I need to pass the value into the AjaxGetEmps method. I'm using ASP.NET MVC with the Razor view engine.
A:
You have to pass the Empid parameter via the Data method instead of giving directly the parameter like this :
@(Html.Kendo().DropDownList()
.Name("positions")
.DataValueField("EmpId")
.DataTextField("EmpName")
.DataSource(source =>
{
source.Read(read =>
{
read.Action("_AjaxGetEmps", "Emp")
.Data("getCurrentEmpid"); // this links to a javascript function
// which will get the current emp id
}).ServerFiltering(true);
})
)
and the javascript function should be implemented like this :
function getCurrentEmpid() {
var grid = $("#idGrid").data("kendoGrid"); // where "idGrid" is the id of your grid
return {
Empid: grid.dataItem(grid.select()).Empid
}
}
Here grid.select() returns the selected line in your grid and grid.dataItem(row) get the data associated with this row. So here Empid should be the id of your model class.
Note also that if you have the flag GridSelectionMode to Multiple you will have to loop threw the grid.select() array...
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The medium budget film is steadily climbing the ladder of success at the box office and has collected over Rs.25 crore in its opening weekend.
New Delhi: Director Shoojit Sircar's slice-of-life family entertainer “Piku” has got a thumbs up not just from the film fraternity and critics, but also movie buffs. The medium budget film is steadily climbing the ladder of success at the box office and has collected over Rs.25 crore in its opening weekend.
The film, which released on 8 May across 1,300 screens in India and in around 200 screens globally, features megastar Amitabh Bachchan, Deepika Padukone and Irrfan Khan.
“Piku” raked in Rs.5.32 crore on Friday, and witnessed a phenomenal growth via positive word-of-mouth on Saturday with Rs.8.70 crore and Rs.11.20 crore on Sunday -- taking its combined total to Rs.25.22 crore in its opening weekend, according to trade analyst Taran Adarsh.
“Take a bow, Team #Piku... Just look at the REMARKABLE growth in the biz of #Piku. Proves yet again CONTENT is KING," Adarsh tweeted on Monday.
The viewership of the film -- which has ‘Motion Se Hi Emotion' as its tagline -- is only poised towards growth, say other industry members.
“The response has been superb till now," Deepak Sharma, chief operating officer, PVR Pictures Ltd, told IANS, and added that the movie went houseful over the weekend.
"It took a drastic jump in collection due to the word-of-mouth publicity since Friday. It's too early to decide the future of 'Piku', but I think it will have a long run in cinemas," Sharma added.
Devang Sampat, business head - Strategy, Cinepolis, said the film's "appealing content and word of mouth is sure to attract and sustain the momentum of the weekend".
"It is sure to sustain the footfalls for the weekdays as well. It has registered over 60 percent occupancy on an average over the weekend in all Cinepolis theatres," Sampat added.
Served with dollops of comedy and emotions, "Piku", which stresses on the camaraderie between a father (Amitabh) and daughter (Deepika), has received rave reviews. Even in theatres, it has spread loud cheers and applause by people of all age groups.
As far as the future of "Piku" on the box office is concerned, it will be interesting to see how well the film will do in upcoming weeks where it is likely to face stiff competition from Ranbir Kapoor and Anushka Sharma-starrer period crime drama film "Bombay Velvet", which releases on Friday.
In the week thereafter, there is the much-awaited “Tanu Weds Manu Returns”.
IANS
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Ultra Pill Splitter
Split pills cleanly in half with the strong stainless steel blade of this convenient Pill Splitter. A pill grip holds the pill firmly in place while cutting. This clear Ultra Splitter features opaque blue grips for easy handling and a self-retracting blade guard for safety.
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1988 United States Senate election in Connecticut
The 1988 United States Senate election in Connecticut took place on November 8, 1988. Incumbent Republican U.S. Senator Lowell Weicker ran for re-election to a fourth term, but was defeated by Democrat Joe Lieberman, the Connecticut Attorney General and eventual 2000 nominee for Vice President of the United States who would remain in office until 2013, when he retired.
Major candidates
Democratic
Joe Lieberman, Connecticut Attorney General
Republican
Lowell P. Weicker Jr., incumbent U.S. Senator since 1971
Results
See also
United States Senate elections, 1988
References
Connecticut
1988
United States Senate
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Q:
Flask/SQLAlchemy Relationships
So I was reading about database relationships and I seem to be puzzled by this
below is the code of Miguel Grinberg in his blog
from app import db
class User(db.Model):
id = db.Column(db.Integer, primary_key=True)
nickname = db.Column(db.String(64), index=True, unique=True)
email = db.Column(db.String(120), index=True, unique=True)
posts = db.relationship('Post', backref='author', lazy='dynamic')
def __repr__(self):
return '<User %r>' % (self.nickname)
class Post(db.Model):
id = db.Column(db.Integer, primary_key = True)
body = db.Column(db.String(140))
timestamp = db.Column(db.DateTime)
user_id = db.Column(db.Integer, db.ForeignKey('user.id'))
def __repr__(self):
return '<Post %r>' % (self.body)
correct me if my understanding is wrong
posts = db.relationship('Post', backref='author', lazy='dynamic')
the posts attribute is having a relationship to the Post model but what does the posts attribute become? a field? what can it contain?
and what does backref mean? it's being defined as author but there is no author attribute in neither Models.
I looked into the docs and well as any newb would have it. I still don't quite get it.
so TL:DR
what does posts become?
what does backref do?
A:
posts is an list that you can use as abstraction in SQLAlchemy to access. SQLAlchemy will translate it to the appropriate SQL query when you use it
(it indicates a has-many relationship)
backref - hints SQLAlchemy that the property in post to indicate that it actually belongs to a specific User. In this case it is a mistake (look at the comments at the bottom of the blog) and should be user
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Actions
New Orleans wins NBA draft lottery, right to pick No. 1
New Orleans has bucked the odds and won the NBA draft lottery, and now will have the first chance at choosing Zion Williamson next month.
The Pelicans won the lottery for the first time since 2012, when they selected Anthony Davis. And the lottery win comes after a season when Davis wanted a trade — something that might not seem so appealing to him now, not with Williamson likely coming to New Orleans.
Memphis will choose second, New York third and the Los Angeles Lakers will pick fourth.
Williamson says he has never been to New Orleans. That might soon change.
“It’s just another positive event for us,” new Pelicans general manager David Griffin said.
The Pelicans were the biggest winners — and the Grizzlies and the Lakers had reason to celebrate as well. They all moved up, much to the chagrin of teams like New York, Cleveland and Phoenix. The Suns, Cavs and Knicks all had the best chance of winning the lottery, and neither even got as much as a No. 2 pick.
Williamson will have an excellent chance at seeing his name added to the list of other No. 1 picks who entered the league with much fanfare in the lottery era, players like LeBron James, Tim Duncan, Shaquille O’Neal, David Robinson and Allen Iverson.
Cleveland will pick fifth, Phoenix sixth, Chicago seventh, Atlanta eighth, Washington ninth, Atlanta again at 10th, Minnesota 11th, Charlotte 12th, Miami 13th and Boston 14th. That pick was conveyed to the Celtics by Sacramento as part of an earlier trade.
UCF center Tacko Fall, the 7-foot-6 draft hopeful, played against Williamson in the second round of the NCAA tournament — a 77-76 Duke victory that went down to the very last moment, a game where Williamson scored 32 points. Fall was asked Tuesday who he would take with the No. 1 pick in this draft, and he did not hesitate for even a second before answering.
“Zion,” Fall said. “He’s a once-in-a-generation player. I’d seen him on TV a lot, but when you play against this kid, you just see it. He’s different.”
Williamson was watching the lottery result unveiling from a stageside seat, along with Murray State’s Ja Morant, Texas Tech’s Jarrett Culver, Indiana’s Romeo Langford and more. Williamson was one of three now-former Duke starters at the lottery — R.J. Barrett and Cam Reddish were among the invited players as well.
“For three of us to be here, it’s crazy,” Williamson told NBA TV before the event started. “I don’t think you usually see anything like that.”
Copyright 2019 The Associated Press. All rights reserved. This material may not be published, broadcast, rewritten or redistributed.
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The number of qualified controllers at the Westbury, LI, center that monitors New York’s air traffic is down to 158 — far fewer than the 205 to 270 authorized in the Federal Aviation Administration budget, the controllers’ union says.
There are 58 trainees but that’s not enough, says Dean Iacopelli, president of the National Air Traffic Controllers Association.
“We have a slew of people whose sum total of . . . experience is, maybe they were on an airplane once,” he said.
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Comprising of 61ha* (151 acres*) of land and located in the Shire of Donnybrook – Balingup.
The property is located in the aesthetically attractive Preston River valley within easy commuting distance of Bunbury and approximately 30 km from Donnybrook, Collie and Balingup.
The locality of Noggerup comprises a small-town site with a very small population and generally supports lifestyle/weekender occ...
Opportunities abound on this former Plant / Nursery Farm which is currently approved for and being utilised for Light Industrial use. You will need to look past the sorted piles of scrap and the layers of dust to see the massive potential of this Nursery / Industrial property. At settlement the seller will present the property in a clean functioning state. Potential to develop a vegetable or market garden on unused land?
Located 8 kms from Boyup Brook and 23 Kms from Bridgetown this is an amazing property with approx. 37 Acres of well-established saltbush and pasture. 30,000 salt bush plants are established to provide high protein feed for sheep
Abundant water from 3 Large dams and 76 acres of natural bush with Grass trees, Jarrah, Marri, and wandoo. With abundant wildlife and wildflowers in spring and summer.
Pasture area is well fenced with 1.8 meter posts with 9 plain wires plus I electric.
There is an established 24 m x 10 m shed with 2 large entrances and a rainwater Tank also a weekender shed still...
This well known and respected nursery is being offered for sale freehold with at extensive plant and equipment list and approximately 8000m2 of hothouse growing space.
The hothouses are fully irrigated by overhead sprinklers and have easy access for loading and unloading seedling trays. The owners of 30 years have designed this enterprise for ease of operating and management , the water in the 3 million gallon dam is almost entirely fed from run off from the hothouse rooves through a system of 150mm pipes. The total land area is 3.7ha (approx 9 acres) and includes a 3 bedroom home for...
This is a Fantastic opportunity for an astute investor in the hospitality realm or simply people dreaming of starting their own lifestyle business in one of the most beautiful places in Australia.
Own a lifestyle accommodation business with an already extremely strong presence, visibility and reputation in all the right accommodation and social media sites across the internet - with an enviable list of top rated testimonials and traveler reviews... Even a "certificate of Excellence" from TripAdvisor! We invite you to check them out yourself.
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Forecast universe prolongation of wanton oil and other liquids in 2017 and 2018 was revised somewhat downward in a Jun book of EIA’s Short-Term Energy Outlook (STEO), that was released after a May 25 proclamation by a Organization of a Petroleum Exporting Countries (OPEC) of an prolongation to prolongation cuts that were creatively set to finish this month.
Image credit: U.S. Energy Information Administration
OPEC’s wanton oil prolongation aim will sojourn during 32.5 million barrels per day (b/d) by a finish of a initial entertain of 2018. Given a extended prolongation cuts, EIA now forecasts OPEC members’ wanton oil prolongation to normal 32.3 million b/d in 2017 and 32.8 million b/d in 2018, down 0.2 million b/d and 0.4 million b/d, respectively, from a prior STEO. Total OPEC glass fuels prolongation is also approaching to be reduce than formerly forecast. However, stability prolongation expansion in many non-OPEC countries is approaching to assuage a gait of tellurian glass fuels register draws in 2017. EIA expects a tiny register build in 2018.
Inventory draws approaching in a second and third buliding of 2017 advise a probability of some increases in wanton oil prices over a entrance months. However, since U.S. parsimonious oil prolongation is comparatively manageable to changes in oil prices compared with offshore production, and even given an estimated six-month loiter between a change in oil prices and satisfied production, aloft wanton oil prices in mid-2017 have a intensity to lift U.S. supply in 2018.
Image credit: U.S. Energy Information Administration
The largest tellurian register boost in a foresee occurs in a second entertain of 2018, when Brazilian and OPEC prolongation are approaching to boost by 570,000 b/d and 220,000 b/d, respectively. Supply expansion in 2018 could minister to downward vigour in oil prices as early as late 2017. EIA’s STEO foresee assumes OPEC cuts will be extended over Mar 2018 though that non-compliance will start to grow late in 2017 and boost in a second half of 2018. Although this foresee reflects a arrogance of augmenting non-compliance with a second production-cut prolongation in 2018, any prolongation provides some support for wanton oil prices, even if usually temporarily, that would partially equivalent downward cost vigour from flourishing inventories.
The Jun STEO forecasts a 2017 normal mark cost for Brent wanton oil of $53/barrel (b), with prices augmenting to $56/b in 2018. Average West Texas Intermediate (WTI) prices are foresee to be $2/b reduce than Brent prices in both 2017 and 2018. As always, all oil cost forecasts are theme to substantial uncertainty. For example, EIA’s foresee for a normal WTI cost in Sep 2017 is $51/b, though research of options trade suggests marketplace expectations operation from $39/b to $64/b during a 95% certainty interval.
EIA expects U.S. wanton oil prolongation to boost by 2018, averaging 9.3 million b/d in 2017 and 10.0 million b/d in 2018. The 2018 STEO foresee exceeds a prior record U.S. prolongation turn of 9.6 million b/d set in 1970. Growth in U.S. prolongation of wanton oil and hydrocarbon gas liquids has been a largest writer to a 820,000 b/d of non-OPEC liquids supply expansion from Jan by May 2017. Continued increases in drilling activity in U.S. shale basins, quite in Texas, support prolongation increases via a forecast.
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You thought it was funny that Arma 3 now has kart racing , didn't you? Well pull yourself together, because the April Fools gag is now an actual feature, and it's managed to raise a million Czech koruna for the country's Red Cross. That's nearly US$50 grand.
Everyone wins, because it means Bohemia Interactive has road tested its unique approach to DLC, which provides access to all new content free-of-charge with “a few reasonable restrictions” that you can pay remove. The model is designed to avoid splitting the playerbase between the DLC have and have-nots, and it seems to have worked out well.
"Alongside helping us to evaluate Arma 3's new approach to DLC, with the backing of our community, Arma 3 Karts has managed to raise money for a great cause,” Arma 3 creative director Jay Crowe said . “A big thanks to our players, both for their support and their valuable feedback, which we'll use to refine our implementation. Overall, we're positive about the preliminary results and responses, which seems to benefit Arma 3 players and help sustain Arma 3 development."
Arma 3 Karts is still on sale for EU$1.49 / US$1.99 / £1.19, but can also be purchased in a bundle alongside the forthcoming Helicopters and Marksmen DLC.
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SMSFs Still On Track To Meet Retirement Goals
SMSFs in or transitioning to retirement remain appropriately positioned to reach their retirement goals, but more savings in super are required now due to a weaker investment outlook, a new report from the SMSF Association and Accurium has revealed.
The database represents SMSF households that are in or phasing into retirement.
The sixth volume of the report revealed due to a weaker investment outlook that indicates returns will be lower for longer, SMSF retirees will need more savings to achieve their retirement goals.
The amount needed for a 65-year-old SMSF couple to afford a comfortable retirement, as defined by the Association of Superannuation Funds of Australia (ASFA), increased by 17 per cent from $702,000 to $824,000.
Over time, the quality and cost of someone’s desired lifestyle increases, as reflected in the ASFA budget standards, meaning SMSFs need more money today in order to afford the same desired lifestyle, the report said.
However, 66 per cent of SMSF trustees can remain confident – with an 80 per cent probability – that they are well placed to live comfortably in retirement on $60,063 a year.
At a higher spending level of $70,000 a year, 50 per cent of SMSFs can be confident of achieving this goal, while 39 per cent can be very confident of doing so.
The report also found the median desired spending level in retirement for an SMSF couple is $78,800, up from $75,000 last year.
However, 24 per cent of SMSF couples aim to spend over $100,000 a year in retirement.
At this aspirational level of $100,000 a year spending, 28 per cent of SMSFs can be confident in achieving this goal, while 20 per cent can be very confident.
The report said the median balance for a two-member SMSF at retirement rose to $1.137 million for the 2016 financial year, compared with $1.124 million in the previous year.
SMSF median balances at retirement have risen 17 per cent since 2010.
“SMSF balances over the year to 30 June 2016 increased only modestly, only 1.2 per cent, and that was based on an imputed investment return of 1 per cent,” Accurium general manager Douglas McBirnie told a media briefing in Sydney today.
“So future returns were reflective of a weaker year in investment markets and those returns are before tax but after expenses.
“That all being said, the balances of SMSFs are still pretty healthy and SMSFs are in a good place to meet a comfortable retirement lifestyle.”
SMSF Association head of technical Peter Hogan added that the idea of the research was to paint a picture of how well placed SMSFs are in meeting their retirement goals.
“It does highlight some challenging times for SMSFs, but also for the superannuation industry generally,” Hogan noted.
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The present invention relates to charging devices for battery operated, cordless tools, appliances and the like. Normally such devices are connected to a single tool or appliance for recharging via an AC power source. In other instances the tool or appliance carries its own recharging device together with a suitable cord for connecting it to a power source.
In today's household where a growing dependency on power tools and appliances is clearly in evidence, there is a need for storing such appliances in an orderly fashion, usually in a storage closet or in a kitchen, workshop area, garage, basement or the like. Since in many cases such tools and appliances are hung on a rack-like device or stored in an array of compartments or trays, the need for recharging such devices in an orderly and simple manner is desirable, especially if all of such appliances and tools can be charged at the same time by a common means or system in which selected tools or appliances can be removed for use without affecting the charging operation of the remaining tools and appliances and, further, wherein additional tools and appliances can be added into the charging system without changing that system.
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376 N.W.2d 893 (1985)
Dan LUNDY, Individually, and on Behalf of all others similarly situated, Appellant,
v.
IOWA DEPARTMENT OF HUMAN SERVICES, Appellee.
No. 85-208.
Supreme Court of Iowa.
November 13, 1985.
*894 Dennis Groenenboom and Martin Ozga, Des Moines, for appellant.
Thomas J. Miller, Atty. Gen., and Candy Morgan, Asst. Atty. Gen., for appellee.
Considered by REYNOLDSON, C.J., and UHLENHOPP, McCORMICK, SCHULTZ and WOLLE, JJ.
McCORMICK, Justice
This appeal involves a challenge to agency rulemaking procedures. The federal government made funds available to states to administer a work registration and job search program for food stamp recipients. Petitioner Dan Lundy, an Iowa food stamp recipient, filed a petition for judicial review for himself and as a purported class action alleging that respondent Iowa Department of Human Services violated rulemaking requirements of the Iowa Administrative Procedures Act (IAPA) in entering and implementing a work registration and job search program in Iowa. The department filed a motion to dismiss the petition which the district court sustained. We reverse the district court.
The question is whether the district court's ruling can be sustained on the grounds either that petitioner is not a person "who is aggrieved or adversely affected by any final agency action" or has not "exhausted all adequate administrative remedies." Only a person or party who meets both these requirements is entitled to judicial review under the IAPA. See Iowa Code§ 17A.19(1) (1985). Because we are reviewing a district court ruling sustaining a motion to dismiss, we view the allegations of the petition in their light most favorable to petitioner, resolve doubts in his favor, and will uphold the ruling only if he could not establish his right to judicial review under any state of facts provable under the allegations of the petition. See Curtis v. Board of Supervisors of Clinton County, 270 N.W.2d 447, 448 (Iowa 1978). At issue is petitioner's right of access to district court, not the merits of his allegations.
I. Whether petitioner is not a person aggrieved or adversely affected by any final agency action. The challenged agency action in the present case is rulemaking. Petitioner alleged the department violated the procedural requirements of section 17A.4 in adopting the challenged rules. Under the allegations of the petition the department action in adopting the rules was complete. Thus we take it as true that the petition attacks final agency action. We believe we must also take it as true that petitioner was aggrieved or adversely affected.
Petitioner alleged he is a food stamp recipient subject to the invalid rules. He alleged that the department exercised an option to enter the mandatory work registration and job search program. Because of insufficient funds the department could not implement the program for all food stamp recipients. As a result the department adopted a policy of requiring the first 2,461 food stamp recipients each month whose social security numbers ended with *895 certain digits to seek work through the Iowa Department of Job Service. Petitioner alleged the departmental decisions in entering and implementing the program constituted rulemaking which is invalid because of noncompliance with the public participation and other procedural requirements of section 17A.4.
To show he was aggrieved or adversely affected by the agency action, petitioner must demonstrate a specific, personal and legal interest in the subject matter and a special and injurious effect on such interest. See City of Des Moines v. Public Employment Relations Board, 275 N.W.2d 753, 759 (Iowa 1979). Obviously petitioner's allegations showed he had an interest in the agency action that distinguished him from members of the community at large. Id. Moreover, the fact he is subject to the allegedly invalid rules demonstrates the requisite injurious effect. Id.
We note that a person or party challenging agency rulemaking procedures under section 17A.4 is not required to show personal prejudice. See Iowa Bankers Association v. Iowa Credit Union Department, 335 N.W.2d 439, 447 (Iowa 1983). We do not address petitioner's right to maintain a class action. His motion for certification of the case as a class action was not ruled on and the issue is not before us. See Iowa R.Civ.P. 42.2.
We hold that the district court ruling cannot be upheld on the ground that petitioner is not a person aggrieved or adversely affected by final agency action.
II. Whether petitioner has not exhausted all adequate administrative remedies. Failure of an agency to comply substantially with the procedural requirements of section 17A.4 makes the resulting rule invalid. Section 17A.4(3) provides:
No rule adopted after July 1, 1975, is valid unless adopted in substantial compliance with the above requirements of this section. However, a rule shall be conclusively presumed to have been made in compliance with all of the above procedural requirements of this section if it has not been invalidated on the grounds of noncompliance in a proceeding commenced within two years after its effective date.
The General Assembly obviously contemplated that the rule would be attacked in a "proceeding" initiated for that purpose. The issue here is whether such proceeding must be brought before the agency or whether it may be brought in district court as in the present case.
This court has entertained at least two cases in which a challenge to rulemaking was initiated by judicial review proceedings in district court. Parole revocation rulemaking was challenged directly in Airhart v. Iowa Department of Social Services, 248 N.W.2d 83 (Iowa 1976), and credit department rulemaking was challenged directly in Iowa Bankers Association v. Iowa Credit Union Department, 335 N.W.2d 439 (Iowa 1983). The exhaustion issue, however, was not raised or addressed in those cases.
Before an exhaustion requirement can be imposed, two conditions must be met: "An administrative remedy must exist for the claimed wrong, and the statutes must expressly or impliedly require that remedy to be exhausted before resort to the courts." Rowen v. LeMars Mutual Insurance Company, 230 N.W.2d 905, 909 (Iowa 1975). In the present case, the department contends petitioner was required to exhaust either his right to a contested case proceeding, his right to petition for the adoption of rules, or his right to petition for a declaratory ruling.
It is true that petitioner would be entitled to litigate the validity of the rules in a contested case if the rules were applied to him. Nevertheless section 17A.4(3) plainly allows an attack on the validity of rules based on rulemaking infractions without waiting for a contested case to arise. See Iowa Bankers Association, 335 N.W.2d at 444 ("We believe the legislature intended to make a judicial remedy available to any person or party who can show the requisite injury.") Our determination of this issue is unaffected by the department's *896 out-of-the-record assertion that petitioner is a party in a contested case involving application to him of the challenged rules. Questions concerning the effect of that proceeding are not implicated in the exhaustion argument. We find that petitioner was not barred from challenging the agency action by his right to attack the rules in a contested case if they were applied to him.
Nor do we perceive an exhaustion requirement in petitioner's right to petition for rulemaking under section 17A.7. He was not interested in advocating promulgation, amendment or repeal of rules as substantive matters. This distinguishes the present case from Schmitt v. Iowa Department of Social Services, 263 N.W.2d 739 (Iowa 1978). Instead petitioner sought only to challenge the validity of the rules on procedural grounds. While a petition for repeal of the rules might have been a way to present the issue to the department, section 17A.7 is not designed as a means for challenging the validity of rules. A petition for repeal assumes a valid rule is in place. It also assumes the agency admits its challenged policies and practices constitute rules, which in this case it does not. We find no legislative intention that a petition for rulemaking be brought as a condition precedent to a judicial review action challenging the validity of rules on procedural grounds.
We reach the same conclusion regarding the declaratory ruling provision of section 17A.9. That provision requires agencies to "provide by rule for the filing and prompt disposition of petitions for declaratory rulings as to the applicability of any statutory provision, rule or other written statement of law or policy, decision or order of the agency...." Petitioner attacks the validity of the alleged rules on procedural grounds, not their applicability in a particular state of facts. The present situation is thus distinguishable from that in City of Des Moines v. Des Moines Police Bargaining Unit Association, 360 N.W.2d 729 (Iowa 1985).
We find no basis for holding that the legislature either provided an adequate administrative remedy or intended that an administrative remedy be exhausted before a judicial review proceeding can be initiated in district court to challenge procedural irregularities in agency rulemaking pursuant to section 17A.4(3). Instead, we believe the purposes of section 17A.4(3) are furthered by permitting the challenge to be commenced by a judicial review petition in district court. This court has noted:
The purpose of section 17A.4(3) is to enforce strict compliance with statutory rule-making procedures, in view of the tendency by some administrators to skirt the requirements. [citation] The provision effects the general IAPA purposes of increasing public accountability of agencies, fostering public participation in rule-making, and assuring agency adherence to a uniform minimum procedure. [citation] We are instructed to broadly construe the IAPA in furtherance of these purposes. [citation]
Iowa Bankers Association, 335 N.W.2d at 447.
Section 17A.4(3) is derived from the Revised Model State Administrative Procedure Act. See Revised Model Administrative Procedure Act § 3(c), 14 U.L.A. 387 (1980). At least one commentator on the model act believes the challenge under the provision is to be made in court. See 1 F. Cooper, State Administrative Law 206-07 (1965). Professor Bonfield obviously interprets the Iowa provision in the same way:
[T]he two year statute of limitation bars invalidation of an agency rule because of procedural failures only as to those proceedings that have not been commenced in an appropriate court within two years of the rule's effective date; the actual invalidation may finally occur several years later when those proceedings that have been timely filed have been finally terminated. ... Consequently, an agency whose rule has been held void in any Iowa District Court proceeding (or Federal District Court proceeding) commenced within two years from the time the rule was issued must take action if it wishes *897 to save the rule; it must appeal the lower court decision.
A. Bonfield, The Iowa Administrative Procedures Act: Background, Construction, Applicability, Public Access to Agency Law, The Rulemaking Process, 60 Iowa L.Rev. 731, 874-75 (1975).
We find no merit in the department's exhaustion argument.
We hold that the district court erred in sustaining the department's motion to dismiss.
REVERSED.
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No luck for the Swiss on the last day of the qualification phase at the Laser Worlds. The light wind specialist Guillaume Girod SUI could not defend his place in the Goldfleet despite of the slack wind conditions, dropping on rank 62. He will continue the Championship in the Silverfleet as do Christian Steiger SUI (69th) and Christoph Bottoni SUI (72nd). With this situation it is clear, that no Swiss Laser will be at the 2012 Olympics in Weymouth/London. At the (big breeze) Hyères Worldcup last week, Christoph Bottoni SUI - with the best result of his career - missed the nations' objective set by Swiss Olympic by a very small margin. To be noted, that contrary to other Olympic classes, the Laser Worldcups have practically always a top nations participation and in no other class, the 12 nations' barrier requested by Swiss Olympic is so demanding. The results.
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New Hampshire voters are famously spoiled. If any given person only gets to talk to a candidate two times before primary day, it's considered an insult, a snub. Most people would tire of the never-ending exposure to pandering primary candidates, showing up at their diners or Chick-fil-As or sports bars for folksy conversations with "everyday folks." This isn't even counting the bombardment of television ads, set to commence soon, as candidates "seriously considering the possibility of running for president," as Jeb Bush puts it, make it official. But they love it up here. One attendee at an event this weekend was handing out business cards. The occupation? "Registered New Hampshire Voter."
The official candidates -- Rand Paul, Marco Rubio, and Ted Cruz -- gathered at Nashua, New Hampshire's Crowne Plaza hotel this Friday and Saturday for the First in the Nation Republican Leadership Summit. The summit also featured unofficial candidates who are almost certainly running but can't quite say so yet for legal/fundraising reasons: Scott Walker, Chris Christie, Jeb Bush, Bobby Jindal, Rick Perry, Carly Fiorina, and Mike Huckabee, to name a few. You also had your for-pretend candidates, like Donald Trump and Lindsey Graham. Beneath them are the YOLO candidates, like former Virginia Gov. Jim Gilmore, former New York Gov. George Pataki, and former U.N. ambassador John Bolton. All here. Former Maryland Gov. Bob Ehrlich occupies whatever tier is beneath Gilmore and Pataki, and he was here, too. At the very bottom of the pile is this guy. Here.
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New Hampshire Republicans are not the same as Iowa Republicans. They're blunt and talk fast and are in no way intimidated to ask a governor or senator or ambassador whatever's on their mind, in a shrewd, distrusting what's your angle here? sort of way. There's a libertarian bent to the state's conservative politics and, by and large, no one wants to waste any precious time talking about, say, gay marriage.
This is not to say that the conservative die-hards in New Hampshire don't have their own quirks. The ones in attendance at the FITN summit, at least, care a lot about ISIS establishing beachheads on the continent and the scourge of "illegals." If you were wondering last fall why New Hampshire GOP Senate candidate Scott Brown kept going on about ISIS fighters crossing the southern border to infect Americans with Ebola, a few conversations at the Crowne Plaza in Nashua would make it quite clear. There's a paranoid style to New Hampshire politics.
***
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"He's trying to destroy the country," a woman in black-and-white shoes and a Diane Keaton-style tie/vest combination, whispered to me during the speech from former U.N. Ambassador John Bolton. She was referring to the current Democratic president -- or "Democrat president," in the parlance of right-wing conferences. Bolton had just said that the "principle responsibility of the President of the United States is to protect the country," and for this woman, and presumably everyone else in the room, Barack Obama has performed poorly on that score.
Bizarrely enough, I'm more familiar with Bolton's talking points than any other candidate's, and the man presents a terrifying view of the world. Everyone is trying to bomb us and our president won't bomb them, and Bolton can't stand this, and he's going to make America care about his particular brand of foreign policy and not all these wishy-washy foofoo issues like education or health care. ISIS is coming for us, Russia is coming for us, China, you name it. And oh, this Iran deal? "The most serious act of appeasement in American history." Bolton likes to close with a dose of levity, about how he has a special understanding of Hillary Clinton since he was a year apart from Bill and Hillary at Yale Law School. Hillary was a "radical" then, and she's a radical now. According to Bolton, the way people are in graduate school is pretty much the way they are the rest of their lives, and Hillary Clinton is every bit the leftist that Elizabeth Warren and Barack Obama are. Scared yet?
"I read an article in the Investor's Business Daily that scared the crap out of me," an older man began his question to Bolton. Turns out ISIS is setting up training camps in New Mexico, or maybe it's Mexico-Mexico, and they're pouring across the border. (The article in question cites a report from Judicial Watch, a right-wing paranoiac crank website, saying ISIS has "spotters" in New Mexico to aid in deadly terrorist crossings.) This news didn't surprise John Bolton at all. John Bolton is never surprised. John Bolton has seen some things.
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"I've been reading some things about ISIL," another older man says. He's been reading the Internet and you just won't believe the stuff he's finding. ISIS is trying to establish a Caliphate and, according to the Koran, once the Caliphate is established, all Muslims are required to pledge allegiance to it and kill all non-participants. Why aren't we stopping Islam? John Bolton is of course aware of this, and not surprised by it, and to him the fact that Barack Obama won't even name the enemy gives the enemy enormous power.
But these two questioners were warm-up acts for the Greatest Question Ever. It came from the woman sitting next to me, the same one who had whispered to me earlier that Obama is "trying to destroy the country." She didn't mean this metaphorically. She had evidence -- proof so ironclad that it could only have come from a weird multicolor-font chain email that old people forward to each other.
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"What do you think about the War on Terror within?" She asks? Obama has been "filling the government with Muslims. He is filling the State Department with the Muslim Brotherhood." It gets worse. This woman has also heard that Barack Obama is "amassing tanks and artillery weapons" to stage a revolution "in the streets." He may have several four- and five-star generals who are going along with this. And then there's the--
"The answer is no," John Bolton cut her off, laughing nervously. She had managed to surprise John Bolton, and that's an accomplishment.
***
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"Well, you're probably nicer to me than the conservative media," Mike Huckabee told me, after I was introduced to him as a reporter from a liberal news outlet.
I had been watching him on Fox News from the hotel lounge, announcing that he announce his presidential decision in early May. Right after the segment aired, Mike Huckabee walked through the lounge on his way to something else. Like a moron, I pointed at the TV and said to him "YOU WERE JUST ON THE TV," as though bewildered that someone can climb out of a television set and into real life.
Can't say I've ever been very nice to Mike Huckabee, but it's true that he's about to go to war with conservative media. Small-government fiscal conservatives have never liked Huckabee. For all outspoken social conservatism and a foreign policy informed by fundamentalist interpretation of Revelation, he's never been much of a budget-cutter. (He'll claim that a lot of this had to do with the Democratic legislature he dealt with when he was governor of Arkansas.) And in the past few days, he's been the (unofficial) presidential candidate coming out hardest against Chris Christie's plan to cut Social Security.
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He's not framing it in liberal terms -- about how means-testing Social Security would welfarize the program and erode its support -- and he looked sort of confused when I tried to explain that argument. But Huckabee has a problem, as a vast majority of those affected almost certainly will, with people paying into Social Security for decades and then not getting back what's owed them, whether those cuts come in the form of means-testing, a raise in the retirement age, or linking cost-of-living increases to chained CPI. And since he's a professional politician who understands his audience, he's spinning it another way: the government is stealing the money of the people. You lent the government your money, and they're not letting you have it back. It's Big Government Theft. That'll do.
Huckabee is good with people. So is Chris Christie, in his own, unorthodox style. It's an attribute he must have great strength in, since he's preparing a campaign centered on yelling at people about cutting Social Security and Medicare.
Christie came to the summit to continue selling his plan. Sharp cuts to social insurance programs are an extension of the overall Christie message of tough love, or a belief that in America's heart of heart lurks a masochist. Christie said he is starting his run (Unofficially! Per the lawyers! ) with a 12-point proposal on cutting entitlement spending because he wants to run a campaign based on "strength, clarity, and hard truths."
"Leadership" is the other word Christie loves to employ. To Christie, "leadership" means the willingness to put forth these "hard truths," like how Social Security needs a good whooping. It's a testament to his skills that he's even trying something like this. But in order for it to be successful he'll need to convince people that what he's saying, and the policies he's pushing, are the only possible correct ones. That he's telling the truth, and anyone who disagrees with him is a liar, and his plan is literally the only way to shore up Social Security, Medicare and Medicaid for the long haul. If anyone can do this, it's Christie. But it's far from a sure thing that anyone can do this.
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Another plank of Christie's big plan is to reform Social Security disability benefits. There are too many working-age people out there falsely claiming disability and we need to get them back in the workforce. Grr! During his Q&A segment, a woman took issue with this part of the plan. She told him that she has a 24-year-old son with Asperger's syndrome and he can't keep a job, so he really needs disability benefits. Christie assures her that her son's case is legitimate and he has no intention of harming him
The woman, having secured a pledge from Christie to protect her son's bennies, then asked Christie what he's going to do about all the illegals immigrants coming to take our jobs. Near the end of his answer Christie acknowledged, in another one of his hard truths, that the 11 or 12 million undocumented immigrants in the country cannot be relied upon to all "self-deport." (Poor Mitt Romney.) Was this the part where humanity's truthiest truth-teller was going to go all-in for amnesty? Not quite. He merely suggested that leaders of both parties are going to have to come together to find a solution for that. The reason it hasn't happened already, according to Christie, is that there's been "no leadership from the White House." You might think that pushing with its political might a bipartisan comprehensive immigration reform bill would count as some kind of leadership on this issue, but apparently not. That's just another hard truth from Chris Christie.
***
Independent Journal Review is a conservative viral news, politics and culture site that gets something like ten trillion visitors a day. Isn't that neat? The outlet was a sponsor of the summit and held party at a nearby restaurant, in collaboration with Facebook. It was open bar and complemented by a sensational spread of fancy cheeses and little steak nibbly things (I am not a food writer) and fresh, raw oysters on the half shell. Free swag all over the place. Internet money is the best money.
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I walked out of the bathroom and there was Marco Rubio sitting at a big table all to himself, on a laptop, as though cramming a college paper about British Romanticism. He was fielding questions on Facebook. Rubio, 43, is considered the youthful, hip candidate in the field, the '90s rap fan who really gets millennial culture. He's all about these edgy, disruptive new technologies that allow politicians to respond to questions people submit on a website.
"Facebook has gotten older, but Instagram has gotten younger," Rubio said to an Instagram representative after the Q&A. He quickly corrected himself. "I mean, Twitter has gotten older, but Instagram has gotten younger." Exactly.
After Rubio, it was time for millennials' real presidential standard-bearer, John Bolton, to do his Facebook Q&A. Scott Brown was also at this party for some reason.
***
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Ted Cruz shows no intention of dropping anytime soon his joke about how he would shut down the IRS and put all of its employees on the southern border, though now he includes a disclaimer to "the media" that he's being tongue-in-cheek. Eliminating the IRS is part of his tax reform plan, which is to institute a flat tax where returns can be filed on a standard-sized postcard. (Which agency collects and enforces the flat tax postcards in our post-IRS utopia? Details, details, blah blah blah. We're no fun, in the media.) Cruz would also "repeal every word" of Obamacare and "repeal every word" of Common Core. He makes it easy to remember, at least. You can fit Ted Cruz's policy platform on a standard-size postcard.
Cruz, the final speaker of the event on Saturday, was only able to take a couple of questions. The Crowne Plaza, in an aggressive pursuit of billings, booked the ballroom where the conference was taking place for a wedding reception at 5:00. Cruz finished speaking at 4:00. The tight booking made for some interesting scenes as conference hangers-on mingled with the newlyweds in their formalwear. Wisconsin Gov. Scott Walker was speaking at a small dinner event for select guests at the other end of the hotel, but he took a respite from the dinner to congratulate the bridge and groom outside the bathroom.
A few hours later, I walked out of my hotel with a few other reporters, and there was Ted Cruz getting out of his car. He looked exhausted, so naturally we harassed him with more questions. I asked him about Chris Christie's Social Security plan and its many, many hard truths. He wouldn't endorse the plan itself, of course, but he did lay out his broad outlines for Social Security reform. He supports gradually increasing the retirement age, and "having Social Security benefits grow to match inflation, rather than having growth exceed inflation." (This is the idea behind linking cost-of-living increases to the slower-growing "chained CPI" measure of inflation.)
Just as the questions wrapped up, with Cruz's aides tugging on him to walk away from reporters and get inside the sanctuary of his hotel room -- you get the sense that this is a common occurrence -- Cruz turned back and said to me that he hasn't given up on winning me over. (More tugging from the aides.) It turns out that Sen. Ted Cruz is familiar with Salon's coverage of Ted Cruz, which is largely negative. Though he didn't say so explicitly, Ted Cruz clearly understands that his fate in the New Hampshire GOP primary rests on his ability to woo Salon. The sooner they all realize this, the better. We take this gatekeeper role seriously and look forward to further engagement with him and our 18 other candidate-friends from the weekend. Good luck to all.
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Signet ring cell adenocarcinoma of prostate.
Primary signet ring cell adenocarcinoma of the prostate is a rare malignancy with a total of 13 cases reported to date in the English literature. We report a very unusual case of signet ring adenocarcinoma of the prostate occurring in a patient who presented initially with irritative voiding symptoms and a bladder mass. Results of immunohistochemical, flow cytometric, and cytogenetic analyses of the tumor are presented.
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Q:
Avoiding boilerplate code due to equivalent constructors
I have an ADT as follows:
Prelude> data Bond = FixedRateBond Float Float | FloatingRateBond Float Float
I want to do an operation on every value constructors of this ADT as follows:
Prelude> let foo :: Bond -> Float
Prelude| foo (FixedRateBond a b) = a + b
Prelude| foo (FloatingRateBond a b) = a + b
As you can see I have code duplication here; for every value I have a + b. I will have more value constructors so this is going to be repeated even more. To me this is code smell, but I don't know how I would refactor it to eliminate the duplicated code. Is there a functional way to avoid this repeated code? This is a trivial example as I have stripped down the real problem to bare essentials to explain the problem.
A:
You're correct. This is a code smell, and it's actually a very common modelling mistake. All you need to do is just factor the rate-type out. E.g.,
data RateType = Fixed | Floating
data Bond = Bond RateType Float Float
Then you'll have
foo :: Bond -> Float
foo (Bond _ a b) = a + b
atop of other benefits like RateType now actually being a type, which you can have Enum and Bounded instances for.
Basically, the rule of thumb here is: if you have multiple constructors implementing the same thing, there must be an enum asking to be factored out.
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901
What is the p'th term of 241978908, 241978909, 241978910, 241978911, 241978912, 241978913?
p + 241978907
What is the k'th term of 5733, 10865, 16017, 21189, 26381?
10*k**2 + 5102*k + 621
What is the m'th term of 1840, 3633, 5810, 8563, 12084, 16565?
32*m**3 + 1569*m + 239
What is the k'th term of 196339, 785343, 1767015, 3141355, 4908363, 7068039?
196334*k**2 + 2*k + 3
What is the p'th term of -5128807, -5128804, -5128801, -5128798, -5128795?
3*p - 5128810
What is the n'th term of -899, -2668, -6171, -12278, -21859, -35784?
-145*n**3 + 3*n**2 - 763*n + 6
What is the f'th term of 790, 3307, 7512, 13405, 20986?
844*f**2 - 15*f - 39
What is the h'th term of 65817750, 65817751, 65817752, 65817753, 65817754?
h + 65817749
What is the t'th term of -18111, -61256, -133163, -233826, -363239, -521396?
t**3 - 14387*t**2 + 9*t - 3734
What is the k'th term of 7556, 9461, 11350, 13217, 15056, 16861, 18626, 20345?
-k**3 - 2*k**2 + 1918*k + 5641
What is the o'th term of -50542, -201720, -453562, -806068, -1259238, -1813072?
-50332*o**2 - 182*o - 28
What is the c'th term of 427031, 425690, 424349, 423008, 421667?
-1341*c + 428372
What is the o'th term of -35133, -70346, -105559, -140772, -175985?
-35213*o + 80
What is the t'th term of -155041653, -155041644, -155041631, -155041614?
2*t**2 + 3*t - 155041658
What is the f'th term of -524207, -524248, -524285, -524318, -524347, -524372, -524393?
2*f**2 - 47*f - 524162
What is the j'th term of 178, 365, 492, 547, 518?
-2*j**3 - 18*j**2 + 255*j - 57
What is the v'th term of -1514965, -1515003, -1515041, -1515079, -1515117, -1515155?
-38*v - 1514927
What is the y'th term of -532, -2151, -4898, -8773, -13776, -19907?
-564*y**2 + 73*y - 41
What is the p'th term of -329, -2605, -8653, -20315, -39433, -67849, -107405?
-307*p**3 - 44*p**2 + 5*p + 17
What is the u'th term of -819, 130, 1065, 1980, 2869, 3726, 4545, 5320?
-u**3 - u**2 + 959*u - 1776
What is the w'th term of 44086319, 88172639, 132258959?
44086320*w - 1
What is the a'th term of 11315086, 11315081, 11315076, 11315071?
-5*a + 11315091
What is the d'th term of -14328, -14029, -13730, -13431?
299*d - 14627
What is the t'th term of 3392212, 3392208, 3392204?
-4*t + 3392216
What is the x'th term of -14273, -27623, -40973?
-13350*x - 923
What is the b'th term of 1114, 1051, 946, 799, 610?
-21*b**2 + 1135
What is the j'th term of 226700453, 453400906, 680101359, 906801812, 1133502265?
226700453*j
What is the i'th term of 116838, 232486, 348134, 463782, 579430, 695078?
115648*i + 1190
What is the a'th term of 293349, 293379, 293409, 293439?
30*a + 293319
What is the w'th term of 93987275, 187974551, 281961829, 375949109, 469936391, 563923675, 657910961?
w**2 + 93987273*w + 1
What is the v'th term of 1336649, 1336697, 1336745, 1336793, 1336841, 1336889?
48*v + 1336601
What is the m'th term of -400923, -801876, -1202829, -1603782?
-400953*m + 30
What is the v'th term of -70447560, -70447562, -70447566, -70447572?
-v**2 + v - 70447560
What is the y'th term of 849441, 849440, 849439, 849438, 849437?
-y + 849442
What is the h'th term of 3531, 3607, 3749, 3957, 4231, 4571, 4977?
33*h**2 - 23*h + 3521
What is the q'th term of -3165, -708, 1753, 4218, 6687, 9160, 11637?
2*q**2 + 2451*q - 5618
What is the n'th term of 440581, 440564, 440547, 440530, 440513?
-17*n + 440598
What is the p'th term of -393645, -786226, -1178807, -1571388, -1963969, -2356550?
-392581*p - 1064
What is the g'th term of 143524, 287441, 431364, 575293, 719228?
3*g**2 + 143908*g - 387
What is the o'th term of 8642, 34777, 78384, 139457, 217990, 313977, 427412?
-o**3 + 8742*o**2 - 84*o - 15
What is the w'th term of -1265793, -1265780, -1265765, -1265748, -1265729?
w**2 + 10*w - 1265804
What is the n'th term of 5257116, 10514249, 15771382?
5257133*n - 17
What is the o'th term of 49717484, 99434965, 149152446, 198869927, 248587408?
49717481*o + 3
What is the d'th term of -99, -373, -887, -1683, -2803?
-7*d**3 - 78*d**2 + 9*d - 23
What is the b'th term of 904, -882, -2668, -4454?
-1786*b + 2690
What is the w'th term of 9718, 37533, 83448, 147463, 229578, 329793, 448108?
9050*w**2 + 665*w + 3
What is the u'th term of 1907124, 7628634, 17164484, 30514674, 47679204?
1907170*u**2 - 46
What is the l'th term of 1685038, 3370066, 5055094, 6740122, 8425150?
1685028*l + 10
What is the i'th term of -16251, -47693, -94325, -156147?
-7595*i**2 - 8657*i + 1
What is the a'th term of -1325, 2714, 13703, 35116, 70427?
579*a**3 + a**2 - 17*a - 1888
What is the o'th term of 7944318, 15888639, 23832960, 31777281, 39721602, 47665923?
7944321*o - 3
What is the r'th term of 686, 2946, 6656, 11786, 18306, 26186?
-5*r**3 + 755*r**2 + 30*r - 94
What is the m'th term of 2401, 1512, 505, -680, -2103, -3824?
-10*m**3 + m**2 - 822*m + 3232
What is the o'th term of -17886840, -17886843, -17886846?
-3*o - 17886837
What is the u'th term of 4304205, 8608424, 12912643, 17216862?
4304219*u - 14
What is the x'th term of -8190, -32746, -73724, -131166, -205114?
-7*x**3 - 8169*x**2 - 14
What is the b'th term of -131456, -130717, -129978?
739*b - 132195
What is the w'th term of 99, 632, 1727, 3540, 6227, 9944?
26*w**3 + 125*w**2 - 24*w - 28
What is the o'th term of 15484, 31436, 47388?
15952*o - 468
What is the t'th term of -85, -59, -189, -475, -917?
-78*t**2 + 260*t - 267
What is the o'th term of -2106742, -2106661, -2106578, -2106493, -2106406?
o**2 + 78*o - 2106821
What is the s'th term of -5382, -5222, -5034, -4824, -4598?
-s**3 + 20*s**2 + 107*s - 5508
What is the p'th term of -1521892, -1521915, -1521954, -1522009?
-8*p**2 + p - 1521885
What is the r'th term of -488834, -488815, -488784, -488741, -488686, -488619, -488540?
6*r**2 + r - 488841
What is the w'th term of 956490, 3825910, 8608274, 15303588, 23911858, 34433090?
w**3 + 956466*w**2 + 15*w + 8
What is the y'th term of -29539, -30403, -31267?
-864*y - 28675
What is the q'th term of -956, -1298, -1654, -2030, -2432, -2866, -3338, -3854?
-q**3 - q**2 - 332*q - 622
What is the c'th term of -2375557, -4751415, -7127271, -9503125, -11878977?
c**2 - 2375861*c + 303
What is the m'th term of -3108705, -3108697, -3108689, -3108681?
8*m - 3108713
What is the c'th term of 8193, 14798, 21403, 28008, 34613, 41218?
6605*c + 1588
What is the a'th term of -22776, -46384, -69994, -93606, -117220, -140836, -164454?
-a**2 - 23605*a + 830
What is the u'th term of 6254, 5975, 5692, 5405?
-2*u**2 - 273*u + 6529
What is the c'th term of -13451, -22153, -36651, -56945?
-2898*c**2 - 8*c - 10545
What is the j'th term of 1853, 1789, 1681, 1529, 1333, 1093?
-22*j**2 + 2*j + 1873
What is the w'th term of -49510, -99169, -148828, -198487, -248146, -297805?
-49659*w + 149
What is the h'th term of 3681830, 3681823, 3681816, 3681809, 3681802, 3681795?
-7*h + 3681837
What is the v'th term of 220092, 220103, 220114, 220125?
11*v + 220081
What is the i'th term of -402411000, -804821998, -1207232996, -1609643994, -2012054992?
-402410998*i - 2
What is the h'th term of -29671815, -29671816, -29671817, -29671818, -29671819?
-h - 29671814
What is the p'th term of -1, 38, -7, -136, -349, -646, -1027?
-42*p**2 + 165*p - 124
What is the f'th term of 85560, 171073, 256558, 342015, 427444, 512845, 598218?
-14*f**2 + 85555*f + 19
What is the d'th term of 401, 154, -223, -700, -1247?
5*d**3 - 95*d**2 + 3*d + 488
What is the y'th term of 92742, 92255, 91444, 90315, 88874?
y**3 - 168*y**2 + 10*y + 92899
What is the s'th term of -144847133, -144847145, -144847165, -144847193, -144847229?
-4*s**2 - 144847129
What is the r'th term of 2193, 7366, 15507, 26616?
1484*r**2 + 721*r - 12
What is the m'th term of -173, -765, -2267, -5117, -9753, -16613?
-73*m**3 - 17*m**2 - 30*m - 53
What is the b'th term of -18271222, -18271219, -18271216?
3*b - 18271225
What is the x'th term of 1411, 5696, 12823, 22792, 35603, 51256, 69751?
1421*x**2 + 22*x - 32
What is the c'th term of -2470481, -9881894, -22234249, -39527546, -61761785, -88936966, -121053089?
-2470471*c**2 - 10
What is the k'th term of -414282, -828563, -1242844?
-414281*k - 1
What is the b'th term of -3471585, -13886348, -31244287, -55545402, -86789693, -124977160?
-3471588*b**2 + b + 2
What is the n'th term of -4621871, -4622080, -4622289?
-209*n - 4621662
What is the f'th t
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mit
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1. Field of the Invention
Embodiments of the present invention relate generally to network communications and more specifically to a system and method for intelligently load balancing and failing over network traffic using a hash engine.
2. Description of the Related Art
Performance and reliability are key requirements for modern computer networks. When a new network connection is initiated on a computing device that includes a plurality of network interface cards (“NICs”), the operating system typically selects a NIC for that connection without regard to the utilization or error rate of each NIC. One disadvantage of such environments is that new connections are often assigned to a NIC that is overloaded and/or unreliable, leading to reduced network performance for the new connection and possibly for other existing connections on the selected NIC.
Some prior art solutions attempt to avoid assigning new connections to overloaded and/or unreliable NICs by maintaining a sophisticated data structure containing “connection state.” Analyzing this data structure allows a network device driver to determine which NICs are overloaded and/or unreliable. However, the structure of this connection state does not lend itself to efficiently identifying which NIC has been assigned to each connection or to efficiently redistributing connections from an overloaded or unreliable NIC to a fully functional NIC. Additionally, maintaining and analyzing this connection state is computationally expensive, which can degrade computational and network performance for the computing device.
As the foregoing illustrates, what is needed in the art is a more efficient technique for distributing and redistributing network connections across NICs in a computing device.
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Nikolas Cruz's psychosis ended in a bloody massacre not only because of the stunning incompetence of the Broward County Sheriff's Department. It was also the result of liberal insanity working exactly as it was intended to.
School and law enforcement officials knew Cruz was a ticking time bomb. They did nothing because of a deliberate, willful, bragged-about policy to end the "school-to-prison pipeline." This is the feature part of the story, not the bug part.
If Cruz had taken out full-page ads in the local newspapers, he could not have demonstrated more clearly that he was a dangerous psychotic. He assaulted students, cursed out teachers, kicked in classroom doors, started fist fights, threw chairs, threatened to kill other students, mutilated small animals, pulled a rifle on his mother, drank gasoline and cut himself, among other "red flags."
Over and over again, students at Marjory Stoneman Douglas High School reported Cruz's terrifying behavior to school administrators, including Kelvin Greenleaf, "security specialist," and Peter Mahmood, head of JROTC.
At least three students showed school administrators Cruz's near-constant messages threatening to kill them—e.g., "I am going to enjoy seeing you down on the grass," "Im going to watch ypu bleed," "iam going to shoot you dead”—including one that came with a photo of Cruz’s guns. They warned school authorities that he was bringing weapons to school. They filed written reports.
Threatening to kill someone is a felony. In addition to locking Cruz away for a while, having a felony record would have prevented him from purchasing a gun.
All the school had to do was risk Cruz not going to college, and depriving Yale University of a Latino class member, by reporting a few of his felonies—and there would have been no mass shooting.
But Cruz was never arrested. He wasn't referred to law enforcement. He wasn't even expelled.
Instead, Cruz was just moved around from school to school—six transfers in three years. But he was always sent back to Marjory Stoneman Douglas High School, in order to mainstream him, so that he could get a good job someday!
The moronic idea behind the "school-to-prison pipeline" is that the only reason so many "black and brown bodies" are in prison is because they were disciplined in high school, diminishing their opportunities. End the discipline and ... problem solved!
It's like "The Wizard of Oz" in reverse. The Wizard told the Scarecrow: You don't need an education, you just need a diploma! The school-to-prison pipeline idiocy tells students: You don't need to behave in high school, you just need to leave with no criminal record!
Of course, killjoys will say that removing the consequences of bad behavior only encourages more bad behavior. But that's not the view of Learned Professionals, who took summer courses at Michigan State Ed School.
In a stroke of genius, they realized that the only problem criminals have is that people keep lists of their criminal activities. It's the list that prevents them from getting into M.I.T. and designing space stations on Mars. Where they will cure cancer.
This primitive, stone-age thinking was made official Broward County policy in a Nov. 5, 2013, agreement titled "Collaborative Agreement on School Discipline."
The first "whereas" clause of the agreement states that "the use of arrests and referrals to the criminal justice system may decrease a student's chance of graduation, entering higher education, joining the military and getting a job."
Get it? It's the arrest—not the behavior that led to the arrest—that reduces a student's chance at a successful life. (For example, just look at how much the district's refusal to arrest Nikolas Cruz helped him!) [See also Eliminating The School To Prison Pipeline, by Robert W. Runcie, Broward County School Department, 2016 PDF]
The agreement's third "whereas” clause specifically cites "students of color" as victims of the old, racist policy of treating criminal behavior criminally.
Say, in the middle of a drive to cut back on the arrest or expulsion of "students of color," how do you suppose the school dealt with a kid named "Nikolas Cruz"? Might there be some connection between his Hispanic last name and the school's abject refusal to do anything about Cruz's repeated criminal behavior?
Just a few months ago, the superintendent of Broward County Public Schools, Robert W. Runcie, was actually bragging about how student arrests had plummeted under his bold leadership.
When he took over in 2011, the district had "the highest number of school-related arrests in the state." But today, he boasted, Broward has "one of the lowest rates of arrest in the state." By the simple expedient of ignoring criminal behavior, student arrests had declined by a whopping 78 percent.
FOOTBALL COACH: "When I took over this team a year ago, we were last in the league in pass defense. Today, we no longer keep that statistic!"
When it comes to spectacular crimes, it's usually hard to say how it could have been prevented. But in this case, we have a paper trail. In the pursuit of a demented ideology, specific people agreed not to report, arrest or prosecute dangerous students like Nikolas Cruz.
These were the parties to the Nov. 5, 2013, agreement that ensured Cruz would be out on the street with full access to firearms:
Robert W. Runcie (pictured right) Superintendent of Schools
Peter M. Weinstein, Chief Judge of the 17th Judicial Circuit
Michael J. Satz, State Attorney
Howard Finkelstein, Public Defender
Scott Israel, Broward County Sheriff
Franklin Adderley, Chief of the Fort Lauderdale Police Department
Wansley Walters, Secretary of the Florida Department of Juvenile Justice
Marsha Ellison, President of the Fort Lauderdale Branch of the NAACP and Chair of the Juvenile Justice Advisory Board
Nikolas Cruz may be crazy , but the parties to that agreement are crazy, too. They decided to make high school students their guinea pigs for an experiment based on a noxious ideology. The blood of 17 people is on their hands.
COPYRIGHT 2017 ANN COULTER
DISTRIBUTED BY ANDREWS MCMEEL SYNDICATION
Ann Coulter is the legal correspondent for Human Events and is the author of TWELVE New York Times bestsellers—collect them here.
Her book, ¡Adios America! The Left’s Plan To Turn Our Country Into A Third World Hell Hole, was released on June 1, 2015. Her latest book is IN TRUMP WE TRUST: E Pluribus Awesome
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Q:
Swift4 - Error Domain=NSCocoaErrorDomain Code=4865
func DoLogin(_ email:String, _ password:String)
{
struct user : Decodable {
let userid: Int
let sfname: String
let slname: String
let email: String
let sid: Int
}
let url = URL(string: ".....")!
var request = URLRequest(url: url)
request.setValue("application/x-www-form-urlencoded", forHTTPHeaderField: "Content-Type")
request.httpMethod = "POST"
let postString = "email=" + email + "&password=" + password + ""
request.httpBody = postString.data(using: .utf8)
let task = URLSession.shared.dataTask(with: request) { data, response, error in
guard let data = data, error == nil else { // check for fundamental networking error
print(error!)
return
}
if let httpStatus = response as? HTTPURLResponse, httpStatus.statusCode != 200 { // check for http errors
print("statusCode should be 200, but is \(httpStatus.statusCode)")
print(response!)
}
let responseString = String(data: data, encoding: .utf8)
print(responseString!)
do {
let myStruct = try JSONDecoder().decode(user.self, from: data)
print(myStruct)
} catch let error as NSError {
print(error)
}
}
task.resume()
}
So the aim is to save the JSON response into the 'user' class so i can use the variables to insert the data into an sql database. The problem I'm currently getting is the Error message...
"Error Domain=NSCocoaErrorDomain Code=4865 "No value associated with key userid ("userid")." UserInfo={NSCodingPath=(
), NSDebugDescription=No value associated with key userid ("userid").}"
I feel the problem is that the HTTP response is returning the data back in an array form which is then not being able to be decoded (HTTP response listed below which is the responseString which I've been using for testing purposes)
{"user":{"userid":2,"sfname":"John","slname":"Doe","email":"[email protected]","sid":123456}}
Heres the PHP that is used to return the data.
public function getUserByEmail($email)
{
$stmt = $this->conn->prepare("SELECT userid, sfname, slname, email, sid FROM students WHERE email = ?");
$stmt->bind_param("s", $email);
$stmt->execute();
$stmt->bind_result($userid, $sfname, $slname, $email, $sid);
$stmt->fetch();
$user = array();
$user['userid'] = $userid;
$user['sfname'] = $sfname;
$user['slname'] = $slname;
$user['email'] = $email;
$user['sid'] = $sid;
return $user;
}
Thanks in advance :D
A:
As you already mention yourself the structure of your JSON doesn't match the structure of your user struct.
You could do two things: Try to figure out why the returned JSON is wrapped in another JSON object or you create a wrapper struct which matches the structure of the received JSON.
The second approach should result in somethin like this:
struct UserWrapper: Decodable {
let user:user
}
Then when your create your user from the JSON simply to this
let wrapper = try JSONDecoder().decode(UserWrapper.self, from: data)
myStruct = wrapper.user
By the way: I would recommend you to read a Swift style guide. By convention function names should start with lowercases and class/struct names with uppercases.
Also named parameters in functions are a very cool thing to document your code.
Something like
func doLogin(userWithMail email: String, andPassword password: String) {...}
// ... other stuff
// call
doLogin(userWithMail: "[email protected]", andPassword: "1234567")
could be more readable. Just in case you have to share your codebase in the future. ;D
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Learn. Succeed. Have Fun!
Experiential Learning is the foundation of a broad-based Missouri S&T education.
The Student Design and Experiential Learning Center (SDLEC) serves as the business incubator and support center for fourteen multi-disciplinary student-managed design teams.
The SDELC is housed in the Kummer Student Design Center, a new facility that provides design team members with advanced computer design labs and software, a complete manufacturing and testing center, business offices and logistical assets, along with the technical, marketing, communication and fundraising support necessary to prepare students for successful careers even before graduation.
Team membership is open to, and encouraged of, S&T students of all academic majors, not just engineering. Team-based learning blends traditional classroom instruction with the critical “outside-the-box” thinking necessary to be successful in a fast-paced development project.
Design team participants enjoy:
24-7 facility access
Strong faculty, staff, business and community support
Networking opportunities with employers and university competitors
Specialized training
Global experience
JOIN A DESIGN TEAM TODAY!
Have fun and make new friends while gaining valuable career experience.
Learn the development process that organizations use to bring products to market.
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Lichen striatus: clinical and laboratory features of 115 children.
To analyze the clinical features, response to treatment, and follow-up of lichen striatus and any associated symptoms or disease, we designed a retrospective study involving 115 affected children at the Pediatric Dermatology Unit of the Department of Dermatology of the University of Bologna, Bologna, Italy. Between January 1989 and January 2000 we diagnosed lichen striatus in 37 boys and 78 girls (mean age 4 years 5 months). We studied their family history and the season of onset, morphology, distribution, extent, duration, histopathology, and treatment of their lichen striatus. We found that family history was negative in all our patients except for two pairs of siblings. The majority of children had the disease in the cold seasons; precipitating factors were found in only five cases. The most frequently involved sites were the limbs, with no substantial difference between upper and lower limb involvement. When lichen striatus was located on the trunk and face, it always followed Blaschko lines; in seven children the bands on the limbs appeared to be along the axial lines of Sherrington. In 70 cases, lichen striatus was associated with atopy. The mean duration of the disease was 6 months and relapses were observed in five children, and in one instance the disease had a prolonged course. Only a few case study series of lichen striatus in children have been reported and ours is the largest to date. The etiology of lichen striatus remains unknown in the majority of our patients. The confirmed association with atopy observed in our patients may be a predisposing factor. It has generally been accepted that lichen striatus follows the lines of Blaschko, and this distribution is a sign of both a topographic and a pathogenetic concept. In patients where lichen striatus is along axial lines, a locus minoris resistentiae, we suppose that this distribution may only be an illusory phenomenon in instances in which the trigger factor prefers this route, consisting of several successive Blaschko lines, but appearing as a single band.
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Synthesis, spectroscopic characterization, antimicrobial, DNA binding and oxidative-induced DNA cleavage activities: new oxovanadium(IV) complexes of 2-(2-hydroxybenzylideneamino)isoindoline-1,3-dione.
A mononuclear complex [(phen)VO(bid)]SO(4)·3H(2)O (1), (phen=1,10-phenanthroline) and a binuclear [(VObid)(2)]·H(2)O (2), with Hbid [(E)-2-(2-hydroxybenzylideneamino)isoindoline-1,3-dione] were prepared and characterized by elemental analysis, IR, mass, UV-Vis spectral studies, ESR, molar conductance and thermogravimetric analysis. The DNA-binding properties of the complexes 1 and 2 were investigated by UV-spectroscopy, fluorescence spectroscopy and viscosity measurements. The superoxide dismutase-like activity of the complexes has been determined. The spectral results suggest that the complexes 1 and 2 can bind to DNA by intercalation. The cleavage properties of these complexes with super coiled pUC19 have been studied using the gel electrophoresis method, where both complexes 1 and 2 displayed chemical nuclease activity in the absence and presence of H(2)O(2)via an oxidative mechanism. All the complexes inhibit the growth of both Gram positive and Gram negative bacteria to competent level. Minimum inhibitory concentration (MIC) was determined by microtiter method.
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A powerful jet from a supermassive black hole is blasting a nearby galaxy in the system known as 3C 321, according to new results from NASA. This galactic violence, never seen before, could have a profound effect on any planets in the path of the jet and trigger a burst of star formation in the wake of its destruction.
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Facilitated antigen presentation and its inhibition by blocking IgG antibodies depends on IgE repertoire complexity.
The antibody repertoires of allergic subjects are characterized by the presence of allergen-specific IgE antibodies. We have previously shown that the composition of the IgE repertoire is critical for allergen-mediated activation of human effector cells. Activation of CD4(+) T cells in allergic subjects is highly potentiated by the process of facilitated antigen presentation (FAP), in which allergen in complex with IgE is taken up by B cells through the low-affinity IgE receptor CD23 and presented to T cells. We sought to investigate the influence of IgE repertoire complexity on the formation of IgE/allergen/CD23 complexes on B cells and subsequent T-cell activation. Using defined allergen-specific recombinant IgE and IgG antibodies, we investigated the influence of individual IgE affinity, IgE clonality, specific IgE concentration, and the ratio between IgE specificities on IgE/allergen/CD23 complex formation in vitro. Although IgE affinity is an important factor, IgE clonality seems to be governing complex formation, especially with medium- and low-affinity IgE antibodies. We demonstrate that differences in allergen-specific IgE affinity correlate with the efficiency of subsequent T-cell activation. In addition, we show that the complexity of an IgE repertoire also affects the ability of allergen-specific IgG antibodies to block FAP. The composition of allergen-specific IgE repertoires in individual patients, especially with respect to IgE clonality, might play an important role in the manifestation of allergic disease not only for the immediate allergic reaction through activation of basophils and mast cells but also for the exacerbation of allergic inflammation through recurring activation of allergen-specific T cells by FAP.
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44% of Voters Believe Repealing Any Part of Obamacare Is a Good Start
51% of voters oppose the individual mandate
Forty-four percent of registered voters want to get rid of Obamacare and believe that repealing any part of it is a good way to start, according to a Morning Consult poll.
The group polled 4,577 registered voters in November and December of last year and asked them various questions about the Affordable Care Act.
There were 39 percent of registered voters who disagreed with repealing any part of Obamacare, and 16 percent were unsure.
Survey responders were asked whether they favored or opposed the individual mandate, which requires that individuals purchase health insurance or pay a penalty to the IRS.
Thirty-three percent of respondents strongly opposed the mandate and 18 percent somewhat opposed it. Only 35 percent of respondents either strongly or somewhat favored the mandate, and 15 percent were unsure.
In addition to repealing the individual mandate, many were supportive of repealing Obamacare's Independent Payment Advisory Board (IPAB). According to the American Medical Association, some support IPAB because it's meant to protect health care costs from political pressure. However, they support its repeal because of its authority and lack of flexibility.
According to the survey, 43 percent agree that the individual mandate and IPAB are bad policies, and repealing both would be a big step in repealing Obamacare. Only 20 percent disagreed with this, and 37 percent were unsure.
"With last year's repeal of the individual mandate, Republicans in Congress have the momentum to further undo the damage Obamacare has inflicted on the health care system," said Taxpayers Protection Alliance president David Williams. "Next on the list should be repealing IPAB, one of the worst parts of Obamacare. IPAB allows unelected bureaucrats to cut Medicare without congressional approval and inserts government between Americans and their doctors."
"Giving 15 unelected officials the authority to make cuts to Medicare without oversight is undemocratic and unacceptable," said Williams. "Fortunately, President Trump's voters share this view. In fact, 60 percent of voters who supported the president in 2016 believe that repealing IPAB is the next [best] step in repealing the ACA in 2018." Williams continued, "Republicans in Congress should do what they were elected to do and repeal Obamacare. If that's too difficult, IPAB repeal should be next on their list. Their voters will reward them."
Ali MeyerEmail Ali | Full Bio | RSSAli Meyer is a staff writer with the Washington Free Beacon covering economic issues that expose government waste, fraud, and abuse. Prior to the Free Beacon, she was a multimedia reporter with CNSNews.com where her work appeared on outlets such as Drudge Report and Fox News. She also interned with the Heritage Foundation and Pacific Research Institute. Her Twitter handle is @DJAliMeyer, and her email address is [email protected].
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MATHEMATICS, SIMULATION AND MODELLING
SystemModeler
25 June 2012
Wolfram
SystemModeler, a high-fidelity modelling environment that uses versatile symbolic components and computation to drive design efficiency, has been released. Integrating with the Wolfram technology platform, it enables modelling, many types of analysis and reporting, resulting in a fully agile design optimisation loop.
‘I'm excited to see technologies as diverse as linguistic querying, computable documents, symbolic computations and the world's most advanced statistical analysis start to combine with SystemModeler into a single, integrated design workflow,’ said Conrad Wolfram, director of Strategic & International Development at Wolfram. ‘We're late to systems engineering, but right at the start of the new era of integrated design optimisation.’
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Darrynane Cottages Bodmin Moor CornwallDog Friendly Self-Catering Cottages in St Breward. Nestling in a sheltered valley on the edge of Bodmin Moor. The cottage and the garden is fully fenced to aid keeping your pet within the property grounds. Wheelchair access is available through the french windows.
Pet Friendly Self-Catering Cottages around Flushing. Waterside holiday home with no expense spared. stunning water views. A sea-side terrace and 2 lounges and a wood-burner make this the perfect year-round destination.
Dog Friendly Self-Catering Cottages. Stunning development of twenty-eight stylish and contemporary holiday villas is set within the St Merryn Holiday Village near Padstow. There are several nearby beaches which allow dogs year round, the nearest of which is Constantine Bay, approximately 3 miles away.
Dog Friendly Self-catering in St. Mawes Cornwall. Located close to lovely walks around the estuary and coastal footpath with outstanding scenery to take in. Chyandour is just one of the many cottages that Creekside Cottages manage. They have a good selection of properties in Falmouth, Mylor, Flushing and Restronguet. Sleeps 6.
Dog Friendly Self-catering accommodation 1bed to 3 bed. We are in the countryside near St Austell, and close to The Eden Project. We provide on-site personal kennels should you wish to visit an attraction that does not allow dogs, you can leave your dog in safety.
3 bed Static Caravan sleeps 6. Located on the Killigarth holiday park. You will have the full amenities of the holiday park. From Killigarth Manor there is a coastal walk to Polperro passing two small beaches with rockpools. Dolphins have been seen off the bay. Visit Smugglers Rest before your return.
1 bed Self-catering accommodation St Austell. Delightful Shepherds Huts that are dog friendly. Each hut has an enclosed and secure garden area, a kennel should you wish to visit a non dog friendly attraction. We are in the countryside near St Austell, and close to The Eden Project.
Four Self-catering cottages in Liskeard. Situated in an area of outstanding natural beauty. Sage cottage has immediate access to 6 acres of grounds with wonderful views on all sides. St Neot is just 1 mile away with pub & Village shop.
Bed & Breakfast - welcomes pets. Situated in the village of Trewoon just outside St. Austell town. Offering a spacious luxury accommodation with en-suite rooms. 1.9 miles from the centre of St Austell. The Eden Project, The Lost Gardens of Heligan and more attractions.
Self-Catering Cottages Bude. Holiday properties stand in 14 acres of peaceful woodland, perfect for dogs and owners. An easy 1 mile stroll along the stunning valley footpath, through the meadows and woodland, leads to the beach.
Self-Catering Cottages. Converted barn offering light and airy open plan living. It has a traditional country kitchen, incorporating a spacious dining area, leading to the living room with open fire. UK's leading dog friendly, self catering holiday accommodation, with access to around 100 acres of unspoilt woodland and open pasture. 3 bed, sleeps 5,
2 bed static Caravan - Sleeps 6. dog friendly. Farmpark is ideally situated between the famous fishing and holiday resort of Looe and the picturesque Polperro. A small well kept dog friendly holiday park
Dog Friendly Hotel Torpoint. Four legged friends are very welcome - providing they bring along their well behaved owners of course!! The South West Coast Path is just a few moments’ walk away, with its beautiful beaches and rugged coastline.
Pet Friendly Self-Catering in St Ives Bay Cornwall. The large expanse of beach a few minutes walk from the cottages, unrestricted and available to dogs throughout the year Beach in walking distance | WIFI dogs allowed.
Pet Friendly B and B in Lostwithiel. Elegant and grand Victorian home located in the picturesque and historic town of Lostwithiel, known as the antiques capital of Cornwall. Built in 1880 it retains a wealth of original features, including fireplaces in most of the large rooms.
Dog Friendly Self-Catering Cottages in St Breward. Nestling in a sheltered valley on the edge of Bodmin Moor. The cottage and the garden is fully fenced to aid keeping your pet within the property grounds. Wheelchair access is available through the french windows.
Rose in vale Country House Hotel St. Agnes CornwallDogs Welcome. Stunning Georgian Country House Hotel on north Cornish coast. Relaxed atmosphere, fantastic food and no children under 12. Rose in Vale in Cornwall is a dog friendly hotel and we are pleased to welcome dogs.Tel: 01872 552202 | - ref: 622
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Gene Demby
Before coming to NPR, he served as the managing editor for Huffington Post's BlackVoices following its launch. He later covered politics.
Prior to that role he spent six years in various positions at The New York Times. While working for the Times in 2007, he started a blog about race, culture, politics and media called PostBourgie, which won the 2009 Black Weblog Award for Best News/Politics Site.
Demby is an avid runner, mainly because he wants to stay alive long enough to finally see the Sixers and Eagles win championships in their respective sports. You can follow him on Twitter at @GeeDee215.
The death of Muhammad Ali — one of the world's greatest boxers — has come with a wave of tributes and memorials.We've been taken back to his most triumphant fights and were reminded of just how handsome he was. (I mean, did we ever reallyforget?)
Ahead of our forthcoming podcast, I've been heads-down in some reading and interviews about the way we talk about, well, white people. Whiteness has always been a central dynamic of American cultural and political life, though we don't tend to talk about it as such.
It's been only a year and a half since the social protest movement around police violence commonly referred to as Black Lives Matter emerged as a major political force.
Much of this movement's momentum-building and organizing happened on Twitter, and a fascinating new study by media scholars Charlton McIlwain, Deen Freelon and Meredith Clark mapped out how it happened and who drove.
This summer, football players at Northwestern University came very close to successfully forming a union — not to demand that they be paid, but to demand better scholarships and safety protocols. Had their bid succeeded, it might have changed college athletics — and, indeed, higher education — in some fundamental ways.
A few years ago, a good friend and I were walking near downtown Philadelphia, not far from my old elementary school, Thomas C. Durham, on 16th and Lombard. The school was built on the edge of a black neighborhood in South Philly in the early 1900s, and its design earned it a spot on the National Register of Historic Places when I was in the third grade. I nudged my friend to take a quick detour with me.
Despite the fiery, complicated past of the 6200 block of Osage Avenue in West Philadelphia, Gerald Renfrow is bullish on its future.
He's one to know; he has lived here forever. His parents bought one of the bigger houses on the corner of 62nd and Osage Avenue and he grew up there. When it was time for him to buy his own home, he landed just up the block and raised his own kids there.
Talk to some of the folks who lived through the bombing of 62nd and Osage Avenue in West Philadelphia 30 years ago, and you'll notice that they refer to the event by its full date. May 13, 1985.
That's how Gerald Renfrow refers to it when we talk about the inferno. His house is about 30 yards from the compound on which the bomb was dropped — practically ground zero. He'd been living there since long before the bombing, and now he's the block captain, trying to hold on to the home where he grew up and raised his own family.
New York City's public school system is vast, with more than a million students spread across thousands of schools. And like the city itself, it's remarkably diverse — about 15 percent Asian, just under 30 percent black, about 40 percent Latino, and about 15 percent white, with all sorts of finer shadings of ethnicity, nationality and language in that mix.
Even before the unrest in Ferguson, Mo., or the Eric Garner incident in New York City last summer, Charles Ramsey, Philadelphia's police commissioner, called on the federal government to look into how the officers in his department used force, and how their use of force might contribute to the department's often strained relationship with the city's residents.
Updated on Feb. 4 at 12:30 p.m. ET: The board of directors for the Howard University Middle School of Mathematics and Science issued a statement on the dismissal of three social studies teachers, indicating that the school is governed by an independent nonprofit organization and regulated by the D.C. Charter School Board. Its also confirms that three teachers resigned from the university effective Jan. 27. From the statement:
By now, you've surely seen Jonathan Chait's sprawling takedown of what he describes as a dangerous resurgence of political correctness in the 21st century. In his telling, a "PC culture" that flourished on college campuses in the '90s is back, stronger than ever thanks to Twitter and social media, and it's been crippling political discourse — and maybe even democracy itself.
It's Halloween — a time for Frankenstein monsters and vampires and werewolves. But many of us have our own monsters from different cultures, and When we threw out a call to our readers asking what ghost stories and folktales they grew up with in their own traditions, we got back stories of creatures stalking the shadows of Latin American hallways and vengeful demons from South Asia with backwards feet. (And that's before we get to the were-hyenas and the infernal bathroom stalls.) Below are some of the best we've found or that were told to us from Code Switch readers.
Editor's Note: In an earlier version of this story, we had two videos of encounters with the police. They contained graphic language and violence, so we've removed them from the story. If you still want to see them, we've included links.
There's a common argument around Muslim extremism that calls for moderate Muslims to denounce and condemn radical adherents of Islam. Many folks push back on that idea by pointing out that Islam isn't a monolith, that there are well north of a billion Muslims in the world, and that it's wrong to conflate the small number of dangerous radicals with everyone who belongs to the faith.
Those very tensions are playing out right now in the Somali immigrant communities of Minneapolis and St. Paul.
Over the past week, Adrian Peterson, the Minnesota Vikings' all-world running back and one of the NFL's biggest stars, has become the face of corporal punishment in America. Peterson turned himself in to police over the weekend on charges of child abuse after he allegedly hit his son with a switch that left welts on his body.
Over the past week, much of the nation's attention has been trained on the town of Ferguson, Mo., following an incident there in which a police officer shot an unarmed black teenager named Michael Brown. Like similar stories, the Michael Brown shooting has become a flashpoint for conversations about race and policing, and there have been heated, chaotic showdowns between the police there and protesters.
Here's some of what's been written about the shooting and the reaction to it in the week since.
Over at NewsOne, Donovan X. Ramsey contrasted two approaches President Obama has taken with black audiences: 1) the finger-wagging, pull-up-your-pants approach that he often takes with African-Americans, like the graduates at all-male Morehouse College ("We've got no time for excuses ... nobody is going to give you anything you haven't earned"), and 2) the laudatory tone he took with young African leaders who traveled to D.C. this week for the Africa Summit.
We have a default template for the way we process mass shootings. We scour through every available scrap of the perpetrators' interior lives – Facebook postings, YouTube videos, interviews with former roommates — to try to find out what drove them to kill. The sites of the massacres become a kind of shorthand: Columbine, Sandy Hook, Fort Hood. We conduct protracted, unsatisfying conversations about gun rights, and about mental illness, and about how we have to make sure that they never happen again.
If it seems like we talk about housing a lot on Code Switch, it's because we do. But the fact is it's really hard to talk about all the ways race correlates to different outcomes — in health or education, say— without talking about where people live. Take household wealth, for example: The major reason whites have so much more of it is because of how much likelier they are not just to own homes, but to own homes in places where that property might appreciate in value.
The Tewaaraton Award is college lacrosse's equivalent of the Heisman Trophy, given to the best player in the country each year. The award takes its name from the Mohawk word for lacrosse, as a way to honor the sport's Native American origins. The bronze trophy depicts a Mohawk man with a lacrosse stick, surging forward.
We play for each other, for our fans, and for our families — not Donald Sterling.
That was the general message that players for the Los Angeles Clippers reiterated, off-mic, when the Sterling fiasco blew up over the weekend. They were being buffeted by questions about how, exactly, they might respond to allegations that Sterling, the team owner, had been recorded saying that he did not want black people to attend his team's games. Would they boycott? Would they be focused enough to be able to play?
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2014 SkyPilot Wingman highline – Photos
2014 SkyPilot Wingman highline – Photos
SkyPilot Mountain near Squamish was one of the first major mountain objects we decided to take on when first starting into mountaineering. Claiming several lives over the past few years, SkyPilot is a real mountain with real consequences.
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It is required to, or at least expedient to, subject lens blanks made from phototropic glass compositions to a heat treatment, after initial forming thereof, to develop the phototropic characteristics or properties of the glass compositions from which the lens blanks are formed. Such a heat treatment is sometimes performed at a temperature above the softening point temperature of the glass of the lens blanks and, therefore, the lens blanks can sag or otherwise become misshaped if not suitably supported during said heat treatment thereof. For example, it may be required, or at least expedient to, subject lens blanks made from glass compositions such as disclosed in U.S. Pat. Nos. 3,197,296; 3,208,860, and 3,325,299, issued July 27, 1965, Sept. 28, 1965, and June 13, 1967, respectively, to heat treatments, at a temperature above the softening point temperature of the glass of the lens blanks, for the purpose of developing or inducing the phototropic characteristics or properties of said compositions and lens blanks made from such compositions. In such event said lens blanks should be suitably supported to prevent sagging or deformation thereof. The aforesaid supporting of said lens blanks has heretofore been performed by disposing and clamping the flat surfaces of each lens blank between the flat surfaces of a pair of flat disks of a suitable material and then subjecting such blank to the necessary heat treatment. This however imparts to said surfaces of the treated lens blank any roughness of said surfaces of said disks regardless of the minuteness of such roughness, and the treated lens blank, or an eyeglass lens formed, therefrom must, therefore, be ground and polished to remove said roughness imparted thereto during the heat treatment thereof.
It is economically desirable that the making of non-prescription sunglass lenses be performed at as low of a cost as possible and, to such end, it has been suggested that lens blanks can be sagged into suitably shaped mold carriers to form the contour or curvature desired for the sunglass lenses since such lenses are not prescription lenses requiring precise shaping by precise grinding thereof. It has then been further suggested that the lens blanks could be sagged into suitably shaped mold cavities to attain the desired eyeglass lens shapes during the aforesaid heat treatment and another economic advantage thereby attained. However, the surfaces of said mold cavities impart roughness to the convex surfaces of the shaped lenses in a manner similar to that in which roughness is imparted to the lens blanks by the surfaces of said disks during said heat treatment of such blanks and, therefore, said convex surfaces of the lens shaped by sagging of lens blanks into mold cavities would also have to be ground and polished to remove said roughness imparted thereto.
In view of the above it is an object of the present invention to provide a method of heat treating a plurality of lens blanks at a temperature above the softening point temperature of the glass of such blanks and without distortion of such lens blanks and, while such plurality of lens blanks are continued to be heated, to sag part of each of the blanks into a shape having a curvature desired for an eyeglass lens.
It is another object of the present invention to economically heat treat and sag lens blanks, made from a phototropic glass composition having a softening point temperature below the temperature required or desired for heat treatment of such blanks to develop the phototropic characteristics or properties of the lens blanks, into shapes having curved contours suitable for eyeglass lenses, the heat treatment and sagging of the lens blanks being performed on a plurality of said blanks at the same time.
Other objects and characteristic features will become apparent as the description proceeds.
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Q:
Equation for Comparing Autos
I am buying a used vehicle. I want an equation that will allow me to compare two vehicle's value that are identical in every way except price and mileage. For example:
Car A: $20395 cost and 83400 miles
Car B: $17090 cost and 10500 miles
Car C: $24000 cost and 39000 miles
Miles: lower miles = good
Cost: higher cost = bad
What is a ratio/equation that can produce a number to compare these two variables across vehicles?
A:
Decide on a number of miles that represents a wore-out car and call that number as w.
Then
(miles / w) * price = index value .
But large differences in price become relatively insignificant at small mileages.
New edit:
Or instead
index value = (w - miles) / price .
That's actually miles-remaining-per-dollar. But that doesn't hold for negative values.
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Antoni Mroczkowski
Antoni Mroczkowski was a Polish ace pilot in the Imperial Russian Air Force during the World War I with 5 confirmed kills.
Biography
In August 1914 Antoni Mroczkowski served in the Imperial Russian Army. In 1915 he completed pilot training in Sevastopol. From 1915 to 1917 he flew in the 24th Air corps. He scored his first victory on an Albatros near Tuchyn in Volhynia. He was wounded two times, and was also shot down in error by Russian artillery. In 1917 he was promoted captain.
In Odessa Mroczkowski joined a Polish Air Force unit and from 1919 he served in an escadrille attached to the 10th Infantry Division. He worked as instructor in a flying school in Warsaw, then he was a test pilot in Centralne Warsztaty Lotnicze. In 1920 he was assigned to the 19th Fighter Escadrille. In 1921 Morczkowski was given an indefinite leave of absence. Later he returned to his profession as test pilot in Plage i Laśkiewicz in Lublin. He was fired for participation in a strike.
In 1939 Mroczkowski arrived in France then he reached Great Britain. In the UK he flew multi-engine airplanes. Due to his age he was transferred to the ground service.
Mroczkowski came back in Poland in 1946. During his career he flew over 8000 hours on 85 different aircraft.
Antoni Mroczkowski died in Warsaw on 26 December 1970.
Awards
Virtuti Militari, Silver Cross
Order of Polonia Restituta, Knight's Cross
Cross of Independence
Medal of the 10th Anniversary of People's Poland
References
Further reading
Jerzy Jędrzejewski. Polscy piloci doświadczalni. Biblioteka Historyczna Instytutu Lotnictwa 2014 r.
External links
Biografia Antoniego Mroczkowskiego
Wzmianka o asach myśliwskich okresu I wojny światowej wraz z Antonim Mroczkowskim
Category:1970 deaths
Category:1896 births
Category:Russian aviators
Category:Polish aviators
Category:Russian military personnel of World War I
Category:Imperial Russian Air Force personnel
Category:Russian World War I flying aces
Category:Recipients of the Silver Cross of the Virtuti Militari
Category:Recipients of the Order of Polonia Restituta
Category:Recipients of the Cross of Independence
Category:Test pilots
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Some of Microsoft's older operating systems will soon take another step towards the end of their lifecycles.
Businesses using Windows XP will need to upgrade to Service Pack 3 (SP3) of the operating system by July in order to continue receiving support from Microsoft.
Although XP is currently in its extended support phase, which runs until April 2014, users will need to be using a supported service pack to be eligible for this. The cutoff for XP SP2 support is 13 July 2010, while support for XP SP1 ended in October 2006. XP users will need to install SP3 through the Windows Update online service. Those not using XP SP2 will need to install this before downloading SP3.
Read more of "XP and Windows 2000: Time running out on support" at Silicon.com.
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[Acute carotid thrombosis in penetrating traumatism: is revascularization necessary in the patient without central neurological deficit?].
Carotid lesions require priority in both evaluation and treatment due to their high morbidity and mortality. Controversy about therapeutic behavior in these patients with or without central neurological deficit is still under in discussion. To present a patient with acute carotid thrombosis due to a shotgun wound and discuss its therapeutic behavior. Hospital de Urgencias in Córdoba city. A 15-year-old male patient is presented with a "point-blank" shotgun wound in the soft parts of the left cervical region, and a left carotid thrombosis with no central neurological deficit. Wound toilette and carotid revascularization by means of resection and venous by-pass with external carotid ligature was performed. The procedure was finished by delaging for plastic reconstruction of the cervical injury. Carotid postoperative angiographic control showed good permeability with no carotid flow alteration. Penetrating carotid injuries should be resolved, if technically possible, with revascularization of the carotid sector. This procedure has to be aborted if the patient is in coma or the lesion is difficult to repair, in such a case ligature should be carried out.
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Q:
UITableViewCell is displayed incorrectly the first time
I'm toggling between different cell backgrounds (white and lightgray) and font properties (bold and normal) with the following code after cell creation or reuse:
- (UITableViewCell *)tableView:(UITableView *)tableView
cellForRowAtIndexPath:(NSIndexPath *)indexPath {
...
UIView* cellBackgroundView = [[[UIView alloc] initWithFrame:CGRectZero] autorelease];
UIFont *font;
if ([[userModel suspended] boolValue]) {
[cellBackgroundView setBackgroundColor:[UIColor lightGrayColor]];
font = [UIFont italicSystemFontOfSize:[[[cell textLabel] font] pointSize]];
} else {
[cellBackgroundView setBackgroundColor:[UIColor whiteColor]];
font = [UIFont boldSystemFontOfSize:[[[cell textLabel] font] pointSize]];
}
[cell setBackgroundView:cellBackgroundView];
[[cell textLabel] setFont:font];
[[cell textLabel] setBackgroundColor:[UIColor clearColor]];
[[cell detailTextLabel] setBackgroundColor:[UIColor clearColor]];
[[cell textLabel] setText:[NSString stringWithFormat:@"%@, %@",
[userModel familyName], [userModel givenName]]];
[[cell detailTextLabel] setText:[userModel userName]];
return cell;
}
The problem is that if a cell that needs to be lightgray and italic is on the first set of cells displayed after loading, its background appears lightgray (correctly) but its font is normal (wrong).
If I scroll down and have the cell redisplayed, then it displays as expected.
Thanks,
Jorge
A:
In the line
font = [UIFont italicSystemFontOfSize:[[[cell textLabel] font] pointSize]]
you are assuming that the cell textLabel already exists and has a correct font. I would NSLog the font just before that call. Also why not just specify the fontsize here:
font = [UIFont italicSystemFontOfSize:12.0]
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ENEL GREEN POWER BECOMES BRAZIL S TOP SOLAR PLAYER WITH 553 MW OF NEW PV CAPACITY IN LANDMARK TENDER AWARD
Published on Monday, 31 August 2015
EGP won more solar capacity than any other participant in Brazils Leilão de Reserva public tender.
The company has been awarded the right to sign 20-year energy supply contracts with the 103 MW Horizonte MP, 158 MW Lapa and 292 MW Nova Olinda photovoltaic projects
EGP will be investing a total of approx. 600 million US dollars in the construction of the plants
Rome and Rio de Janeiro, August 31st, 2015 EGP has been awarded the right to sign 20-year energy supply contracts in Brazil for a total of 553 MW with its three new solar photovoltaic projects Horizonte MP (103 MW), Lapa (158 MW) and Nova Olinda (292 MW) following the Leilão de Reserva public tender. The addition of this award to EGPs existing operational presence in the country means that the company is now the top player across the entire solar industry of Brazil in terms of installed capacity and project portfolio. The outcome of this tender, which awarded more capacity to EGP than any other participants, adds to the 11 MW of Fontes Solar I and II, Brazils largest photovoltaic complex currently in operation, and to a further 254 MW awarded to the company in November 2014 for the construction of the Ituverava solar project.
We are extremely pleased about this landmark win, thanks to which we became the number one company in the Brazilian solar industry said EGP CEO Francesco Venturini. The 818 MW of total capacity won in all photovoltaic auctions launched in Brazil so far, the 700 MW won in wind power auctions and 102 MW won in hydropower auctions in the country since 2010 all offer further validation of our Latin America growth strategy, the strength of which is based on our focus on technologies that are approaching or are at grid parity, developing and delivering innovative solutions and extracting synergies with other Enel Group companies in the region.
EGP will be investing a total of approximately 600 million US dollars in the construction of the three new solar facilities that will be completed and enter operation by 2017, in line with the growth targets set out in the companys current business plan, which provides for the construction of 7.1 GW of additional capacity worldwide by 2019.
The three projects, which will be built in areas which enjoy high levels of solar radiation, will generate approximately 1.2 TWh of green energy, making a significant contribution to meeting Brazils need for new power generation. The 20-year supply contracts awarded to EGP provide for the sale of specified volumes of energy generated by the plants to the Brazilian Chamber of Commercialisation of Electric Energy (CCEE Camara de Comercializaçao da Energia Eletrica).
Horizonte MP will be built in Tabocas do Brejo Velho in the State of Bahia, which is located in Brazils north-east. Once up and running, the plant will generate annually about 223 GWh of renewable energy while avoiding the emission of around 67,000 tonnes of CO2 into the atmosphere. The plant will be located next to EGPs Ituverava solar project, which was awarded to the company in November 2014s Leilão de Reserva tender, meaning that EGP will be able to pool resources both during the construction of Horizonte MP and the operation of the two projects, which will share the same connection infrastructure.
The Lapa project will be built in Bom Jesus da Lapa in the State of Bahia. Once up and running, the plant will generate about 340 GWh each year while avoiding the emission of approximately 102,000 tonnes of CO2 into the atmosphere.
Nova Olinda will be constructed in Ribeira do Piaui in the State of Piaui. The plant will generate about 604 GWh per year once fully operational, avoiding the emission of around 181,000 tonnes of CO2 into the atmosphere in the process.
Enel Green Power is the Enel Group company fully dedicated to the international development and management of renewable energy sources, with operations in Europe, the Americas and Africa. With a generation capacity equal to approximately 32 billion kWh in 2014 from water, sun, wind and the Earths heat enough to meet the energy needs of more than 11 million households Enel Green Power is a world leader in the sector thanks to its well-balanced generation mix that provides generation volumes well over the sector average. The company has an installed capacity of more than 9,900 MW from a mix of sources including wind, solar, hydropower, geothermal and biomass. The company has about 740 plants operating in 15 countries.
All Enel Green Power press releases are also available in smartphone and tablet versions. You can download the Enel Mobile app at Apple Store and Google Play
Enel S.p.A. provides for the dissemination to the public of regulated information by using SDIR NIS, managed by BIt Market Services, a London Stock Exchange Group's company, with registered office at Milan, Piazza degli Affari, 6. For the storage of regulated information made available to the public, Enel S.p.A. has adhered, as from July 1st, 2015 to the authorized mechanism denominated “NIS-Storage”, available at the address www.emarketstorage.com, managed by the above mentioned BIt Market Services S.p.A. and authorized by CONSOB with the resolution No. 19067 of November 19th, 2014. From May 19th 2014 to June 30th 2015 Enel S.p.A. used the authorized mechanism for the storage of regulated information denominated “1Info”, available at the address www.1info.it, managed by Computershare S.p.A. with registered office in Milan and authorized by CONSOB with resolution No. 18852 of April 9th, 2014
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Q:
Spectrum of a field
Let's $F$ be a field. What is $\operatorname{Spec}(F)$? I know that $\operatorname{Spec}(R)$ for ring $R$ is the set of prime ideals of $R$. But field doesn't have any non-trivial ideals.
Thanks a lot!
A:
As you say $\mathrm{Spec}(R)$ is defined to be the set of all prime ideals of $R$. If $R$ is a field, the only proper ideal is $0$ hence you get $\mathrm{Spec}(F) = \{0\}$.
It gets more interesting if your space is a ring that is not a field, like for example $R = \mathbb Z$. Then you can endow it with the following topology: each closed set in the space corresponds to an ideal $J$ of $R$, defined as $C(J) = \{ p \mid p \text{ a prime ideal of } R \text{ such that } J \subset p\}$.
Now what does $\mathrm{Spec}(\mathbb Z)$ endowed with this topology look like? Well, first of all, the points in our space correspond to prime ideals and since $\mathbb Z$ is a principal ideal domain, each point looks like $p\mathbb Z$. Note that the zero ideal $\{0\}$ is prime if and only if the ring is an integral domain, so in this case, zero is also a point in our space.
Next we want to know what closed sets look like. For this, let's stick a prime ideal into $C(\cdot)$ and see what comes out: $C(p\mathbb Z) = \{ p \mathbb Z \}$ which means, every singleton set in our space is closed.
Now what are the closed sets corresponding to non-prime ideals? Well, $n$ has only a finite number of prime divisors and each point in $C(n\mathbb Z)$ corresponds to a divisor of $n$: $C(n\mathbb Z) = \{ p \mathbb Z \mid p \mathbb Z \text{ a prime ideal containing } n \mathbb Z \}$.
Now we know that all closed sets are finite.
Edit (I apologise for the blunder kindly pointed out by Rene and t.b.)
You need to be careful about what open sets, i.e. complements of closed sets look like. You can easily trick yourself into believing that since a set is closed if and only if it's finite, $\mathrm{Spec}(\mathbb Z)$ has the cofinite topology. But this is false. To see this note that if we indeed had the cofinite topology, $\mathrm{Spec}(\mathbb Z) \setminus \{\langle 0 \rangle \}$ would be open. But for this to be true, $\langle 0 \rangle$ would have to be closed which means that we would have to have an ideal $I$ in $\mathbb Z$ such that the only prime ideal it is contained in is $\langle 0 \rangle$. But this implies that $I = \langle 0 \rangle$ which implies that $I$ is contained in every prime ideal. Hence there is no ideal $I$ such that $C(I) = \{ \langle 0 \rangle \}$.
As pointed out in Rene's answer, it boils down to all open sets contain zero since the complement of a closed set, $C(n\mathbb Z)^c$, is all prime ideals contained in $n \mathbb Z$ which always includes zero since we're in an integral domain so that the zero ideal is prime.
A:
$Spec(Z)$ does not have the cofinite topology. The non-empty
open sets are precisely the cofinite sets that contain the zero ideal.
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Earl Honaman
Earl M. Honaman (April 13, 1904 – March 17, 1982) was a suffragan bishop of the Episcopal Diocese of Central Pennsylvania, serving from 1956 to 1969.
Personal life
Honaman was a native of Lancaster, Pennsylvania and was the son of Walter K. and Ada R. Honaman. He received his Bachelor of Arts degree from Franklin and Marshall College in 1922 where he was a member of Phi Beta Kappa and Sigma Pi fraternity. He graduated cum laude from the Philadelphia Divinity School in 1928 and was ordained that same year. He was married to Mary Shenk Honaman and they had two sons, Frederich and Walter.
Military Chaplain
Honaman was a chaplain in the United States Army Reserve from 1934 until 1940. After the start of World War II, he became a chaplain in the United States Army, with the rank of lieutenant colonel from 1941 until 1945. His army experience as Division Chaplain of the 28th Infantry Division included the famous Battle of the Bulge. During the war he received the Bronze Star. From 1946 to 1950 he served as chaplain in the Pennsylvania National Guard. He again served as a chaplain for the United States Army from 1950 until 1951.
In recognition of his distinguished service to Church and country, the Doctor of Divinity degree, honoris causa, was conferred upon Bishop Honaman by both the Philadelphia Divinity School and Franklin and Marshall College.
Civilian Ministry
Except for the years when he was in the military, Rev. Honaman spent his entire ministry in his native diocese, serving, in turn, as vicar, as rector, and as archdeacon. He was Rector of St. John's Church in York and Rector of St. John's Church in Carlisle. He was elected Suffragan Bishop of the Diocese of Central Pennsylvania in the fall of 1955,.
References
External links
Retired Episcopal Bishop Honaman Dies
Emerald of Sigma Pi
Category:1904 births
Category:1982 deaths
Category:Bishops of the Episcopal Church in the United States of America
Category:American military personnel of World War II
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M.Sc. Ricardo Martins
About M.Sc. Ricardo Martins
I am a deeply passionated about improving the way people attend their emotional and cognitive needs.
I'm happy to...
Go fora coffee
Show youmy company
Contributecontent
Host eventsin my area
I am also a design and marketing teacher at Federal University, member of International Institute of Information Design, Communication Research Institute, and also have Human Change Management Book of Knowledge Certification.
M.Sc. Ricardo’s interest in Service Design
My main interest on service design is related to implementation issues. According to some reports (IBM, Gartner Group, McKinsey, Forrester Research) 70% of change initiatives fail in organizations. The designers also face particular problems during the implementation process. Maybe because designers favor the planning stage and generation of ideas, but devote little energy in the execution process. My research is targeted to fill the gap between design idea and realizing its offerings.
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Assessing Fukushima-Derived Radiocesium in Migratory Pacific Predators.
The 2011 release of Fukushima-derived radionuclides into the Pacific Ocean made migratory sharks, teleosts, and marine mammals a source of speculation and anxiety regarding radiocesium (134+137Cs) contamination, despite a lack of actual radiocesium measurements for these taxa. We measured radiocesium in a diverse suite of large predators from the North Pacific Ocean and report no detectable (i.e., ≥ 0.1 Bq kg-1 dry wt) Fukushima-derived 134Cs in all samples, except in one olive ridley sea turtle (Lepidochelys olivacea) with trace levels (0.1 Bq kg-1). Levels of 137Cs varied within and across taxa, but were generally consistent with pre-Fukushima levels and were lower than naturally occurring 40K by one to one to two orders of magnitude. Predator size had a weaker effect on 137Cs and 40K levels than tissue lipid content. Predator stable isotope values (δ13C and δ15N) were used to infer recent migration patterns, and showed that predators in the central, eastern, and western Pacific should not be assumed to accumulate detectable levels of radiocesium a priori. Nondetection of 134Cs and low levels of 137Cs in diverse marine megafauna far from Fukushima confirms negligible increases in radiocesium, with levels comparable to those prior to the release from Fukushima. Reported levels can inform recently developed models of cesium transport and bioaccumulation in marine species.
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Time-Limited, Exclusive Holiday Deal!
Ever Wanted To Get Bigger, Stronger and Leaner But Didn’t Know Where To Start?
The Ultimate Blueprint is a Complete System!
N
Video Modules
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All the information you need in one place. You can watch on your phone anywhere or listen while working out or during your daily commute.
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You can get the essential info to fast-track your results or dig in deep to understand all the nitty-gritty details of the program.
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– Intro
– Workouts
– Key Points
– Recap of the Blueprint
Blueprint Loading Patterns
– Intro to Loading Patterns
– Getting Started with Loading Patterns
– German Loading Pattern #1
– German Loading Pattern #2
– Russian Strength Program Generator
– The 10% Solution
– One Workout a Week to a New Peak
– Two Workouts a Week to a New Peak
Ideas for Famine Phase
– DIET IDEAS FAMINE PHASES
– TRAINING IDEAS FOR FAMINE PHASE
Ideas for Feast Phase
– TRAINING IDEAS – FEAST PHASE
– DIET IDEAS – FEAST PHASE
Re-Comp Training
– TRI-PHASIC ENERGY SYSTEM TRAINING: Background
– RAISING ABSOLUTE STRENGTH: What’s Old Is New Again
– THE RUSSIAN STEP UP
– THE BENT ARM PULLOVER
– TNT TRAINING: The Finer Points
– Guidance On Cardio – What Type, Where, When & How
– HOW TO RECOVER LIKE YOU’RE ON STEROIDS
– TNT DIET
The Blueprint and Adaptogens
– TALKING TO THE CHEMIST
– THE 4 PHYSIOLOGICAL STATES
– MOVING IN AND OUT OF EACH STATE
– RUSSIAN ROULET? NYET! HOW ADAPTOGENS WERE REALLY USED IN SPORT
– FALSE TRANSFERENCE
– ANABOLICS vs. ADAPTOGENS: MECHANISM OF ACTION
– SIGNS IN CYBERSPACE: CLUES
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More Tips and Tricks
– TIMING OF ECDY INTAKE
– ONE OTHER, VERY IMPORTANT POINT
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There is nothing else to buy! The training and diet principles WORK all by themselves, period. If you decide to enhance the program with scientifically sound, “Blueprint Tested” supplements then you’ll see that those natural bodybuilding supplements can expand your success.
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We are so confident that you will be happy about The Ultimate Blueprint that we offer a 30 day guarantee.
If within 30 days of your purchase, you are not 100% satisfied with the value of the training, I will give you your money back.
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My bodyfat is literally melting away. Today was the 2-3 rep range for the feast phase. I surprised myself yet again and am proud to say I destroyed all my previous PR’s.”
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From The Blueprint Collective forums
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“I have been a subscriber to the Blue Print Bulletin for over a year now.
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“I am still trying to wrap my head around the progress I have made just from the famine period to present.
My bodyfat is literally melting away. Today was the 2-3 rep range for the feast phase. I surprised myself yet again and am proud to say I destroyed all my previous PR’s.”
Savabrat
from The Blueprint Collective forums
“My life has been forever changed. Most other sources of bodybuilding information regurgitate the same basic knowledge. But what I found in the blueprints and the bulletin was different from everything else out there. In the blueprints, Rob Regish spills the beans and provides much of what he’s learned from years and years of training.
Every time I think that I’ve scoured the Internet to find all the beneficial supplements, the Blueprint bulletin comes out and blows my mind. In the bulletin, Rob Regish usually provides several overviews of the latest, as well as the old-school, supplements that one could consider. But he’s not a bro, and tells his readers which ones they should not take as well as those that they should take. He’s not even selling these supplements. In fact, many of his best recommendations are inexpensive, easy-to-find items.”
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Blueprint Believer, Louisville, KY
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Q. I currently own a previous version of the Blueprint or Meteoric. Can I get a discount for the Ultimate Blueprint?
A. Thank you, we appreciate your loyalty! If you’ve read it this far, you most likely love what you got and are looking for more. You should have received a discount code. If you don’t have one, please contact us so we can review your account.
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Patient and public involvement in emergency care research.
Patients participate in emergency care research and are the intended beneficiaries of research findings. The public provide substantial funding for research through taxation and charitable donations. If we do research to benefit patients and the public are funding the research, then patients and the public should be involved in the planning, prioritisation, design, conduct and oversight of research, yet patient and public involvement (or more simply, public involvement, since patients are also members of the public) has only recently developed in emergency care research. In this article, we describe what public involvement is and how it can help emergency care research. We use the development of a pioneering public involvement group in emergency care, the Sheffield Emergency Care Forum, to provide insights into the potential and challenges of public involvement in emergency care research.
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Q:
Mongodb findOne object in array by id
Im trying to find a specific entry to my database document by the users id and the by the selected field and item. I would like the items object to be returned. This is the document structure:
{
"_id": ObjectId("58edfea4b27fd0547375eeb4"),
"user_id": ObjectId("58d2dd4c8207c28149dbc748"),
"calories": 2000,
"date": 20170312,
"snacks": [ ],
"dinner": [
{
"nutrients": {
"protein": "11.6",
"carbs": "29.4",
"fat": "7.9"
},
"servings": "75",
"calories": 750,
"name": "Meat feast stone baked pizza"
},
{
"nutrients": {
"protein": "6.8",
"carbs": "54",
"fat": "30.6"
},
"servings": "25",
"calories": 550,
"name": "Mc Coy's Cheddar and onion"
}],
"lunch": [],
"breakfast": [],
}
What I have done so far is get the users id, date and then the selected meal and item to search for. What im getting returned is the entire meal array however I only want the food items object returned.
user_food.findOne({user_id : req.session.user_id, date: today},{'dinner': 'Meat feast stone baked pizza'},function(err, item){
if(err){
console.log("something went wrong: " + err);
return res.status(500).send(err);
}
else{
console.log(item);
return res.status(200).send(item);
}
});
What im getting returned is this:
"dinner": [
{
"nutrients": {
"protein": "11.6",
"carbs": "29.4",
"fat": "7.9"
},
"servings": "75",
"calories": 750,
"name": "Meat feast stone baked pizza"
},
{
"nutrients": {
"protein": "6.8",
"carbs": "54",
"fat": "30.6"
},
"servings": "25",
"calories": 550,
"name": "Mc Coy's Cheddar and onion"
}]
what I want is simply:
{
"nutrients": {
"protein": "11.6",
"carbs": "29.4",
"fat": "7.9"
},
"servings": "75",
"calories": 750,
"name": "Meat feast stone baked pizza"
}
A:
Try this:
user_food.
findOne({
user_id : req.session.user_id,
date: today,
'dinner.name': 'Meat feast stone baked pizza'
},{
'dinner.$' : 1
},function(err, item){
....
});
dinner.$ will only return the dinner items which match the criteria, i.e where dinner.name : Meat feast stone baked pizza.
Read about $(positional) operator to know more about limiting the contents of array from query result and returning element matching the query document.
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Q:
Drawing grid lines across the image uisng opencv python
Using OpenCV python, I want to make a grid when I switch on my camera. Can you guys help me with a logic or code.
Please find the image link below for better understanding.
Camera switched on and pointed to a floor
Grid Lines are split across the whole image
A:
Here is the solution for my question guys. Make use of it.
import matplotlib.pyplot as plt
import matplotlib.ticker as plticker
try:
from PIL import Image
except ImportError:
import Image
# Open image file
image = Image.open('bird.jpg')
my_dpi=200.
# Set up figure
fig=plt.figure(figsize=(float(image.size[0])/my_dpi,float(image.size[1])/my_dpi),dpi=my_dpi)
ax=fig.add_subplot(111)
# Remove whitespace from around the image
fig.subplots_adjust(left=0,right=1,bottom=0,top=1)
# Set the gridding interval: here we use the major tick interval
myInterval=300.
loc = plticker.MultipleLocator(base=myInterval)
ax.xaxis.set_major_locator(loc)
ax.yaxis.set_major_locator(loc)
# Add the grid
ax.grid(which='major', axis='both', linestyle='-', color='g')
# Add the image
ax.imshow(image)
# Find number of gridsquares in x and y direction
nx=abs(int(float(ax.get_xlim()[1]-ax.get_xlim()[0])/float(myInterval)))
ny=abs(int(float(ax.get_ylim()[1]-ax.get_ylim()[0])/float(myInterval)))
# Save the figure
fig.savefig('birdgrid_without_Label.jpg')
A:
def draw_grid(img, line_color=(0, 255, 0), thickness=1, type_=_cv2.LINE_AA, pxstep=50):
'''(ndarray, 3-tuple, int, int) -> void
draw gridlines on img
line_color:
BGR representation of colour
thickness:
line thickness
type:
8, 4 or cv2.LINE_AA
pxstep:
grid line frequency in pixels
'''
x = pxstep
y = pxstep
while x < img.shape[1]:
_cv2.line(img, (x, 0), (x, img.shape[0]), color=line_color, lineType=type_, thickness=thickness)
x += pxstep
while y < img.shape[0]:
_cv2.line(img, (0, y), (img.shape[1], y), color=line_color, lineType=type_, thickness=thickness)
y += pxstep
A:
You can draw lines on the input image using the cv2.line() function. So depending on where you want to draw the lines, your basic code will look like:
img = cv2.imread(r"path\to\img")
cv2.line(img, (start_x, start_y), (end_x, end_y), (255, 0, 0), 1, 1)
To get the dimensions of the image, you can use img.shape which will return (height, width).
To draw a vertical line through the center for example, your code would look like:
cv2.line(img, (int(img.shape[1]/2), 0),(int(img.shape[1]/2), img.shape[0]), (255, 0, 0), 1, 1)
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Use of TLS parameters to model anisotropic displacements in macromolecular refinement.
An essential step in macromolecular refinement is the selection of model parameters which give as good a description of the experimental data as possible while retaining a realistic data-to-parameter ratio. This is particularly true of the choice of atomic displacement parameters, where the move from individual isotropic to individual anisotropic refinement involves a sixfold increase in the number of required displacement parameters. The number of refinement parameters can be reduced by using collective variables rather than independent atomic variables and one of the simplest examples of this is the TLS parameterization for describing the translation, libration and screw-rotation displacements of a pseudo-rigid body. This article describes the implementation of the TLS parameterization in the macromolecular refinement program REFMAC. Derivatives of the residual with respect to the TLS parameters are expanded in terms of the derivatives with respect to individual anisotropic U values, which in turn are calculated using a fast Fourier transform technique. TLS refinement is therefore fast and can be used routinely. Examples of TLS refinement are given for glyceraldehyde-3-phosphate dehydrogenase (GAPDH) and a transcription activator GerE, for both of which there is data to only 2.0 A, so that individual anisotropic refinement is not feasible. GAPDH has been refined with between one and four TLS groups in the asymmetric unit and GerE with six TLS groups. In both cases, inclusion of TLS parameters gives improved refinement statistics and in particular an improvement in R and free R values of several percent. Furthermore, GAPDH and GerE have two and six molecules in the asymmetric unit, respectively, and in each case the displacement parameters differ significantly between molecules. These differences are well accounted for by the TLS parameterization, leaving residual local displacements which are very similar between molecules and to which NCS restraints can be applied.
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Search form
Lely Juno 100 Automatic Feed Pusher
The Lely Juno moves along the feeding alley automatically, following the feed fence. While operating, the machine pushes feed toward the feed fence without disturbing the cows. Because the Lely Juno is a stand-alone machine, barn modifications are seldom required; it can be used in almost any type of barn.
Every farm is different, and every feeding alley is different. That’s why, three years after the introduction of the Lely Juno 150, a second Juno family member has been introduced: the Lely Juno 100.
The Juno 100 is a more compact, more financially attractive model. Because of its smaller diameter, the Juno 100 is well-suited for barns with a smaller feeding alley.
For both models, the charging station serves at the point of departure and arrival for each feeding round; it is installed at a suitable location in the feeding alley. Because of the various built-in sensors, the Lely Juno can drive diverse routes.
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Virginia Democratic-controlled legislature advanced the red flag bill that allows authorities to confiscate guns from people accused of being a threat to themselves and others.
The Senate Bill SB240 passed alongside three other gun control bills Senate Bill 35, Senate Bill 69, and Senate Bill 70. The bills authorize localities to impose firearm bans from public spaces during some events, imposes the limit of one handgun purchase per month, and enforce universal background checks for all firearm transfers respectively.
Another Democratic-sponsored bill bans indoor shooting ranges at offices with more than 50 employees. Yet another Democratic bill HB 162 offers immunity to gun-free zones if a person sustains firearm-related injuries in such places.
We are Texalorians... Weapons are our religion.... This is the way, y'all... Show your Texalorian Pride with this kick ass shirt!
The democrats ruled out any amendments that the Republicans suggested and proceeded to approve the bills. Republican members protested the passing of the bills saying that it amounted to an assault of civil liberties. The Senate Bill SB240 will go through a vote where it is expected to pass.
The democratic legislature also killed 11 Republican bills within two hours with some bills being kicked out in less than 10 minutes.
Republican-sponsored HB 1470 and HB 1471 introduced by Christopher Head from Botetourt sought to create a process for compliance for a landowner whose property spans different localities with different gun laws.
Other bills sought to allow the carrying of firearms in places of worship. Religious leaders had made these requests in order to ensure the security of their places of worship. However, the democratic legislature ignores their requests and trashed the bills. Another Republican-sponsored bill would allow citizens to carry without a concealed gun without a permit.
The sane gun debate in Virginia ended when the democrats funded by rich anti-gun billionaires such as Michael Bloomberg took over the house after over two decades of failure.
Meanwhile, Lt. Gov. Dan Patrick is pushing universal background checks in our state, and if left unchecked, it will not be long until we see bills like this getting traction in Austin. Texas is embarrassingly ranked 29th for gun rights, and with the help of the political elites in Austin, we will surely rank among New York, California, and Illinois if left to their own devices. Please join our fight today, and help us restore Texas' place as the standard for the U.S.
Join LSGR - https://www.lsgr.live/join
Sign our petition - https://www.lsgr.live/petition
Send a postcard to Dan Patrick - https://www.lsgr.live/store/Postcard-to-Lt-Gov-Patrick-p158378407
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Incidence and consequences of neonatal alloimmune thrombocytopenia: a systematic review.
Neonatal alloimmune thrombocytopenia (NAIT) is a potentially devastating disease that may lead to intracranial hemorrhage in the fetus or neonate, often with death or major neurologic damage. There are no routine screening programs for NAIT, preventive measures are taken only in a subsequent pregnancy. To estimate the population incidence of NAIT and its consequences, we conducted a review of the literature. Our results may aid in the design of a screening program. An electronic literature search included Medline, Embase, Cochrane database and references of retrieved articles. Eligible for inclusion were all prospective studies aimed at diagnosing NAIT in a general, nonselected newborn population, with sufficient information on platelet count at birth, bleeding complications, and treatment. Titles and abstracts were reviewed, followed by review of full text publications. Studies were independently assessed by 2 reviewers for methodologic quality. Disagreements were resolved by consensus, including a third reviewer. From the initial 768 studies, 21 remained for full text analysis, 6 of which met the inclusion criteria. In total, 59,425 newborns were screened, with severe thrombocytopenia in 89 cases (0.15%). NAIT was diagnosed in 24 of these 89 newborns (27%). In 6 (25%) of these cases, an intracranial hemorrhage was found, all likely of antenatal origin. NAIT is among the most important causes of neonatal thrombocytopenia. Intracranial hemorrhage due to NAIT occurs in 10 per 100 000 neonates, commonly before birth. Screening for NAIT might be effective but should be done antenatally.
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Woman sneaks out with puppy under her shirt
A couple walked into a Murfreesboro pet store, and they allegedly walked out with a Pekingese puppy underneath a woman's shirt.
The couple entered Animal City Pet Store. The girl put the 12-week-old puppy under her shirt, pet store workers say.
"She picked up the puppy, was holding him for a little bit, walked down the aisle and lifted up her shirt," said Amanda Kiviniemi of Animal City. "As soon as that puppy was under there, she walked out."
The puppy is valued at $700. Store workers said the theft is about a lot more than that.
"It's one thing to shoplift a thing, which is terrible in and of itself, but to shoplift a living being that needs care and has people who worry about it just makes that a different thing entirely," said Kiviniemi.
"I guess if you're trying to steal a puppy, that would be the only place to put it, but at the same time why? Why do that? That jeopardizes the puppy," said Mallory Saladino.
The theft happened on Wednesday.
A police officer who saw media reports about it recognized the woman as the person who lives next door to him.
The puppy was returned to the store last night. Workers said they will press charges.
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Vegetable and fruit intake and pancreatic cancer in a population-based case-control study in the San Francisco bay area.
Pancreatic cancer is one of the most devastating and rapidly fatal cancers, yet little is known about the primary cause and prevention of this disease. We conducted a population-based case-control study to investigate the association between vegetables and fruits and pancreatic cancer. Between 1995 and 1999, 532 cases and 1,701 age- and sex-matched controls completed direct interviews using a semiquantitative food-frequency questionnaire. No proxy interviews were conducted. We observed inverse associations between consumption of total and specific vegetables and fruits and the risk of pancreatic cancer. The odds ratio and 95% confidence interval for the highest versus the lowest quartile of total vegetable intake was 0.45 (0.32-0.62), trend P < 0.0001; and for total fruits and fruit juice was 0.72 (0.54-0.98), trend P = 0.06. Odds ratios and 95% confidence intervals for the highest versus the lowest quartile of specific vegetables and fruits were: 0.63 (0.47-0.83) for dark leafy vegetables, 0.76 (0.56-1.0) for cruciferous vegetables, 0.59 (0.43-0.81) for yellow vegetables, 0.56 (0.41-0.76) for carrots, 0.51 (0.38-0.70) for beans, 0.46 (0.33-0.63) for onions and garlic, and 0.78 (0.58-1.0) for citrus fruits and juice. Compared with less than five servings per day of total vegetables and fruits combined, the risk of pancreatic cancer was 0.49 (0.36-0.68) for more than nine servings per day. These results suggest that increasing vegetable and fruit consumption, already recommended for the prevention of several other chronic diseases, may impart some protection against developing pancreatic cancer.
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