text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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Reece is a given name and surname that derives from the Welsh name Rhys. Notable people with the name include:
Surname
Alan Reece (1927–2012), English engineer and entrepreneur
Alex Reece, British musician
Alicia Reece (born 1971), American politician
Andy Reece (born 1962), English footballer
Angel Katherine Reece, better known as Hailey Hatred (born 1983), American professional wrestler
B. Carroll Reece (1889–1961), American politician
Barbara Massey Reece (born 1942), American politician
Beasley Reece (born 1954), American football player
Bessie Reece
Bob Reece (born 1951), American baseball player
Brian Reece (1913–1962), English actor
Byron Herbert Reece (1917–1958), American author
Caley Reece (born 1979), Australian kickboxer
Carlton Reece (1915–1963), Guyanese cricketer
Carmen Reece, British singer, songwriter, musician and producer
Charlie Reece (born 1989), English footballer
Christopher Reece (born 1959), American musician
Cleo Reece, activist and filmmaker
Courtenay Reece (1899–1984), Barbadian first-class cricketer and cricket umpire
Damon Reece (born 1967), English drummer
Danny Reece (born 1955), American football player
Dave Reece (born 1948), American ice hockeyplayer
David Reece, American singer
Dizzy Reece (born 1931), Jamaican-born hard bop jazz trumpeter
Don Reece (1919–1992), American football player
Donald James Reece (born 1934), Emeritus Archbishop of the Roman Catholic Archdiocese of Kingston in Jamaica
E Reece, American rapper-songwriter, actor, and model
Eric Reece (1909–1999), Premier of Tasmania
Erik Reece, American writer
Eryn Reece, American bartender
Florence Reece (1900–1986), American social activist, poet, and folksong writer
Gabrielle Reece (born 1970), American volleyball player and fashion model
Geoff Reece (born 1952), American football player
Gerald Reece (1897–1985), British writer and colonial administrator
Gil Reece (1942–2003), Welsh international footballer
Gordon Reece (1929–2001), British journalist and television producer
Jack Reece (1927–1966), New Zealand cricketer
James Reece (disambiguation)
Jane Reece (born 1944), American geneticist
Jason Reece (born 1971), American drummer and vocalist
Jimmy Reece (1929–1958), American racecar driver
John Reece (born 1957), British businessman
Kensley Reece (born 1945), former Barbadian cyclist
Lewis Reece (born 1991), Welsh rugby league footballer
Louise Goff Reece (1898–1970), American politician
Luis Reece (born 1990), English cricketer
Marcel Reece (born 1985), American football player
Marilyn Jorgenson Reece (1926–2004), American civil engineer
Maynard Reece (1920–2020), American artist
Nicholas Reece (born 1974), Australian politician and policy activist
Paul Reece (born 1968), English footballer
Richard Reece (born 1939), numismatist and academic
Sevu Reece (born 1997), Fiji-born, New Zealand rugby union player
Shakera Reece (born 1988), Barbadian sprinter
Skeena Reece (born 1974), Canadian First Nations artist
Spencer Reece, poet and presbyter
Stephanie Reece (born 1970), American tennis player
Travis Reece (born 1975), American football player
William Reece (1856–1930), mayor of Christchurch, New Zealand
William Lewis Reece (born 1959), American serial killer
Given name
Reece Beckles-Richards (born 1995), Antigua and Barbudan international footballer
Reece Beekman (born 2001), American college basketball player
Reece Bellotti (born 1990), English boxer
Reece Blayney (born 1985), Australian rugby league footballer
Reece Boughton (born 1995), English rugby union player
Reece Brown (footballer, born 1991), English footballer
Reece Brown (footballer, born 1996), English footballer
Reece Burke (born 1996), English footballer
Reece Caira (born 1993), Australian footballer
Reece Caudle (1888–1955), American politician
Reece Chapman-Smith (born 1998), English rugby league footballer
Reece Cole (born 1998), English footballer
Reece Conca (born 1992), Australian rules footballer
Reece Connolly (born 1992), English footballer
Reece Crowther (born 1988), Australian footballer
Reece Deakin (born 1996), Welsh footballer
Reece Dinsdale (born 1959), English actor and director
Reece Donaldson (born 1994), Scottish footballer
Reece Douglas (born 1994), English actor
Reece Fielding (born 1998), English footballer
Reece Flanagan (born 1994), English footballer
Reece Gaines (born 1981), American basketball player and coach
Reece Gaskell (born 2000s), English footballer
Reece Gray (born 1992), English footballer
Reece Grego-Cox (born 1996), English footballer
Reece Hales (born 1995), English footballer
Reece Hall-Johnson (born 1995), English footballer
Reece Hands (born 1993), English footballer
Reece Hewat (born 1995), South African-born Australian rugby union player
Reece Hodge (born 1994), Australian rugby union player
Reece Hoffman (born 2001), Australian rugby league footballer
Reece Humphrey (born 1986), American freestyle wrestler
Reece Hussain (born 1995), English cricketer
Reece Hutchinson (born 2000), English footballer
Reece James (disambiguation)
Reece James (footballer, born 1993), English footballer
Reece James (footballer, born 1999), English footballer
Reece Jones (disambiguation), several people
Reece Jones (artist) (born 1976), British artist
Reece Jones (footballer) (born 1992), Welsh international footballer
Reece Jones (geographer) (born 1976), American political geographer
Reece Kelly, Irish cricketer
Reece Kershaw, Commissioner of the Australian Federal Police
Reece Lyne (born 1992), English rugby league footballer
Reece Lyon (born 2000), Scottish footballer
Reece Marshall (born 1994), rugby union player
Reece Mastin, Australian singer; see Reece Mastin
Reece McAlear (born 2002), Scottish footballer
Reece McFadden (born 1995), Scottish boxer
Reece McGinley (born 2000), Northern Irish footballer
Reece Mitchell (born 1995), English footballer
Reece Morrison (born 1945), American football player
Reece Noi (born 1988), British actor and writer
Reece Oxford (born 1998), English footballer
Reece Papuni (born 1987), light heavyweight boxer
Reece Prescod (born 1996), British sprinter
Reece Ritchie (born 1986), English actor
Reece Robinson (born 1987), Lebanon international rugby league footballer
Reece Robinson (darts player) (born 1992), English darts player
Reece Robson (born 1998), Australian rugby league footballer
Reece Scarlett (born 1993), Canadian ice hockey player
Reece Shearsmith (born 1969), British comedian
Reece Shipley (1921–1998), American country musician
Reece Simmonds (born 1980), Australian rugby league footballer
Reece Styche (born 1989), English footballer
Reece Joseph Staunton (born 2001), footballer
Reece Thompson (born 1988), Canadian actor
Reece Thompson (footballer) (born 1993), English footballer
Reece Tollenaere (born 1977), Australian footballer
Reece Topley (born 1994), English international cricketer)
Reece Ushijima (born 2003), Japanese-American racing driver
Reece Wabara (born 1991), English businessman and footballer
Reece Waldock, Australian public servant
Reece Webb-Foster (born 1998), English footballer
Reece Whitby (born 1964), Australian politician
Reece Whitley (born 2000), American competitive swimmer
Reece Willcox (born 1994), Canadian ice hockey player
Reece Williams (born 1985), Australian rugby league footballer and referee
Reece Williams (cricketer) (born 1988), South African cricketer
Reece Willison (born 1999), Scottish footballer
Reece Wilson (born 1996), Scottish downhill mountain biker
Reece Young (born 1979), New Zealand Test cricketer
Other names
Uncle Reece, stage name of Maurice Hicks Jr. (born 1984), American musician
See also
Rees (surname)
Reese (given name)
References | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,482 |
Antônio José Fernandes Barroso, mais conhecido como Tuninho Barroso (Rio de Janeiro, 27 de maio de 1956), é um treinador de futebol brasileiro que esteve a frente da Seleção Brasileira Sub-20 na Copa do Mundo FIFA Sub-20 de 1997.
Carreira
Toninho treinou várias equipes menores no Brasil e, em 1997, foi nomeado treinador da Equipe Nacional de Futebol Sub-20 do Brasil, e permaneceu até 1999. Entretanto, ele treinou o por cinco jogos em 1998 e também em outra oportunidades enquanto permaneceu no clube. Em 2008, assumiu o cargo de treinador da .
Títulos
Nova Iguaçu
Campeonato Carioca - Série B1: 2005
Ligações externas
Naturais da cidade do Rio de Janeiro
Treinadores de futebol do Rio de Janeiro
Treinadores do São Cristóvão de Futebol e Regatas
Treinadores da Seleção Brasileira de Futebol Sub-20
Treinadores de futebol do Clube de Regatas do Flamengo
Treinadores do Botafogo Futebol Clube (João Pessoa)
Treinadores do Campinense Clube
Treinadores do Serrano Foot Ball Club
Treinadores do Veranópolis Esporte Clube Recreativo e Cultural
Treinadores do Nova Iguaçu Futebol Clube
Treinadores do Club de Regatas Vasco da Gama
Treinadores do Duque de Caxias Futebol Clube | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,452 |
if [[ -z "$RAY_HASH" ]]; then
echo "RAY_HASH env var should be provided"
exit 1
fi
if [[ -z "$RAY_VERSION" ]]; then
echo "RAY_VERSION env var should be provided"
exit 1
fi
download_wheel() {
WHEEL_URL=$1
OUTPUT_FILE=${WHEEL_URL##*/}
if [ "${OVERWRITE-}" == "1" ] || [ ! -f "${OUTPUT_FILE}" ]; then
wget "${WHEEL_URL}"
fi
}
# Linux.
echo "Downloading Ray core Linux wheels"
download_wheel "https://s3-us-west-2.amazonaws.com/ray-wheels/releases/$RAY_VERSION/$RAY_HASH/ray-$RAY_VERSION-cp36-cp36m-manylinux2014_x86_64.whl"
download_wheel "https://s3-us-west-2.amazonaws.com/ray-wheels/releases/$RAY_VERSION/$RAY_HASH/ray-$RAY_VERSION-cp37-cp37m-manylinux2014_x86_64.whl"
download_wheel "https://s3-us-west-2.amazonaws.com/ray-wheels/releases/$RAY_VERSION/$RAY_HASH/ray-$RAY_VERSION-cp38-cp38-manylinux2014_x86_64.whl"
download_wheel "https://s3-us-west-2.amazonaws.com/ray-wheels/releases/$RAY_VERSION/$RAY_HASH/ray-$RAY_VERSION-cp39-cp39-manylinux2014_x86_64.whl"
download_wheel "https://s3-us-west-2.amazonaws.com/ray-wheels/releases/$RAY_VERSION/$RAY_HASH/ray-$RAY_VERSION-cp310-cp310-manylinux2014_x86_64.whl"
# macOS.
echo "Downloading Ray core MacOS wheels"
download_wheel "https://s3-us-west-2.amazonaws.com/ray-wheels/releases/$RAY_VERSION/$RAY_HASH/ray-$RAY_VERSION-cp36-cp36m-macosx_10_15_intel.whl"
download_wheel "https://s3-us-west-2.amazonaws.com/ray-wheels/releases/$RAY_VERSION/$RAY_HASH/ray-$RAY_VERSION-cp37-cp37m-macosx_10_15_intel.whl"
download_wheel "https://s3-us-west-2.amazonaws.com/ray-wheels/releases/$RAY_VERSION/$RAY_HASH/ray-$RAY_VERSION-cp38-cp38-macosx_10_15_x86_64.whl"
download_wheel "https://s3-us-west-2.amazonaws.com/ray-wheels/releases/$RAY_VERSION/$RAY_HASH/ray-$RAY_VERSION-cp39-cp39-macosx_10_15_x86_64.whl"
download_wheel "https://s3-us-west-2.amazonaws.com/ray-wheels/releases/$RAY_VERSION/$RAY_HASH/ray-$RAY_VERSION-cp310-cp310-macosx_10_15_universal2.whl"
# Windows.
echo "Downloading Ray core Windows wheels"
download_wheel "https://ray-wheels.s3-us-west-2.amazonaws.com/releases/$RAY_VERSION/$RAY_HASH/ray-$RAY_VERSION-cp37-cp37m-win_amd64.whl"
download_wheel "https://ray-wheels.s3-us-west-2.amazonaws.com/releases/$RAY_VERSION/$RAY_HASH/ray-$RAY_VERSION-cp38-cp38-win_amd64.whl"
download_wheel "https://ray-wheels.s3-us-west-2.amazonaws.com/releases/$RAY_VERSION/$RAY_HASH/ray-$RAY_VERSION-cp39-cp39-win_amd64.whl"
# download_wheel "https://ray-wheels.s3-us-west-2.amazonaws.com/releases/$RAY_VERSION/$RAY_HASH/ray-$RAY_VERSION-cp310-cp310-win_amd64.whl"
# Linux CPP.
echo "Downloading Ray CPP Linux wheels"
download_wheel "https://s3-us-west-2.amazonaws.com/ray-wheels/releases/$RAY_VERSION/$RAY_HASH/ray_cpp-$RAY_VERSION-cp36-cp36m-manylinux2014_x86_64.whl"
download_wheel "https://s3-us-west-2.amazonaws.com/ray-wheels/releases/$RAY_VERSION/$RAY_HASH/ray_cpp-$RAY_VERSION-cp37-cp37m-manylinux2014_x86_64.whl"
download_wheel "https://s3-us-west-2.amazonaws.com/ray-wheels/releases/$RAY_VERSION/$RAY_HASH/ray_cpp-$RAY_VERSION-cp38-cp38-manylinux2014_x86_64.whl"
download_wheel "https://s3-us-west-2.amazonaws.com/ray-wheels/releases/$RAY_VERSION/$RAY_HASH/ray_cpp-$RAY_VERSION-cp39-cp39-manylinux2014_x86_64.whl"
download_wheel "https://s3-us-west-2.amazonaws.com/ray-wheels/releases/$RAY_VERSION/$RAY_HASH/ray_cpp-$RAY_VERSION-cp310-cp310-manylinux2014_x86_64.whl"
# macOS CPP.
echo "Downloading Ray CPP MacOS wheels"
download_wheel "https://s3-us-west-2.amazonaws.com/ray-wheels/releases/$RAY_VERSION/$RAY_HASH/ray_cpp-$RAY_VERSION-cp36-cp36m-macosx_10_15_intel.whl"
download_wheel "https://s3-us-west-2.amazonaws.com/ray-wheels/releases/$RAY_VERSION/$RAY_HASH/ray_cpp-$RAY_VERSION-cp37-cp37m-macosx_10_15_intel.whl"
download_wheel "https://s3-us-west-2.amazonaws.com/ray-wheels/releases/$RAY_VERSION/$RAY_HASH/ray_cpp-$RAY_VERSION-cp38-cp38-macosx_10_15_x86_64.whl"
download_wheel "https://s3-us-west-2.amazonaws.com/ray-wheels/releases/$RAY_VERSION/$RAY_HASH/ray_cpp-$RAY_VERSION-cp39-cp39-macosx_10_15_x86_64.whl"
download_wheel "https://s3-us-west-2.amazonaws.com/ray-wheels/releases/$RAY_VERSION/$RAY_HASH/ray_cpp-$RAY_VERSION-cp310-cp310-macosx_10_15_universal2.whl"
# Windows CPP.
echo "Downloading Ray CPP Windows wheels"
download_wheel "https://ray-wheels.s3-us-west-2.amazonaws.com/releases/$RAY_VERSION/$RAY_HASH/ray_cpp-$RAY_VERSION-cp37-cp37m-win_amd64.whl"
download_wheel "https://ray-wheels.s3-us-west-2.amazonaws.com/releases/$RAY_VERSION/$RAY_HASH/ray_cpp-$RAY_VERSION-cp38-cp38-win_amd64.whl"
download_wheel "https://ray-wheels.s3-us-west-2.amazonaws.com/releases/$RAY_VERSION/$RAY_HASH/ray_cpp-$RAY_VERSION-cp39-cp39-win_amd64.whl"
download_wheel "https://ray-wheels.s3-us-west-2.amazonaws.com/releases/$RAY_VERSION/$RAY_HASH/ray_cpp-$RAY_VERSION-cp310-cp310-win_amd64.whl"
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,444 |
Draft of Chapter X, "Shattered Dreams"
King, Martin Luther, Jr.
July 1, 1962 to March 31, 1963
How do we determine conjectured information? ?
Atlanta, Ga.
Martin Luther King, Jr. - Career in Ministry
Martin Luther King, Jr. - Education
King blends thoughts on unmet expectations from preachers Frederick Meek, Leslie Weatherhead, Howard Thurman, and J. Wallace Hamilton in this sermon. He writes that African Americans have "long dreamed of freedom," and asserts, "Moreover, through our suffering in this oppressive prison and our non-violent struggle to get out of it, we may give the kind of spiritual dynamic to western civilization that it so desperately needs to survive." As in the version of this sermon he preached at Dexter in 1959, King elaborates on the apostle Paul's unfulfilled desire to travel to Spain.1 He asks the congregation to learn from Paul's "unwanted and unfortunate circumstance…in developing the capacity to accept the finite disappointment and yet cling to the infinite hope."
"When I take my journey into Spain, I will come unto you." Romans 15:24.
Our sermon today brings us face to face with one of the most agonizing problems of human experience. Very few, if any, of us are able to see all of our hopes fulfilled. So many of the hopes and promises of our mortal days are unrealized. Each of us, like Shubert, begins composing a symphony that is never finished.2 There is much truth in George Frederick Watts' imaginative portrayal of Hope in his picture entitled Hope. He depicts Hope as seated atop our planet, but her head is sadly bowed and her fingers are plucking one unbroken harp string.3 Who has not had to face the agony of blasted hopes and shattered dreams?
If we turn back to the life of the Apostle Paul, we find a very potent example of this problem of disappointed hopes.4 In his letter to the Christians at Rome Paul wrote: "When I take my journey into Spain, I will come unto you." It was one of Paul's greatest hopes to go to Spain, the edge of the then known world, where he could further spread the Christian gospel. And on his way to Spain, he planned to visit that valiant group of Christians in Rome, the capital city of the world. He looked forward to the day that he would have personal fellowship with those people whom he greeted in his letter as "Christians in the household of Caesar."5 The more he thought about it the more his heart exuded with joy. All of his attention now would be turned toward the preparation of carrying the gospel to the city of Rome with its many gods, and to Spain, the end of the then known world.6
But notice what happened to this noble dream and this glowing hope that gripped Paul's life. He never got to Rome in the sense that he had hoped. He went there only as a prisoner and not as a free man. He spent his days in that ancient city in a little prison cell, held captive because of his daring faith in Jesus Christ. Neither was Paul able to walk the dusty roads of Spain, nor see its curvacious slopes, nor watch its busy coast life, because he died a martyr's death in Rome.7 The story of Paul's life was the tragic story of a shattered dream and a blasted hope.
Life is full of this experience. There is hardly anyone here this morning who has not set out for some distant Spain, some momentous goal, some glorious realization, only to find that we had to settle for much less. We were never able to walk as free men through the streets of our Rome. Instead we were forced to live our lives in a little confining cell which circumstance had built around us.8 Life seems to have a fatal flaw, and history seems to have an irrational and unpredictable streak. Ultimately we all die not having received what was promised. Our dreams are constantly tossed and blown by staggering winds of disappointment.9
Mahatma Gandhi, after long years of struggle for independence, dreamed of a united India, only to see that dream trampled over by a bloody religious war between the Hindus and the Moslems which led to the division of India and Pakistan.10 Woodrow Wilson dreamed of a league of nations, but he died with the dream shattered. The Negro slaves of America longed for freedom with all their passionate endeavors, but many died without receiving it. Jesus, prayed in the garden of Gethsemane that the cup might pass, but he had to drink it to the last bitter dregs.11 The Apostle Paul prayed fervently for the "thorn" to be removed from his flesh, but he went to his grave with this desire unfulfilled.12 Shattered dreams! Blasted hopes! This is life.
What does one do under such circumstances? This is a central question, for we must determine how to live in a world where our highest hopes are not fulfilled.
It is quite possible for one to seek to deal with this problem by distilling all his frustrations into a core of bitterness and resentment of spirit. The persons who follow this path develop a hardness of attitude and a coldness of heart. They develop a bitter hatred for life itself. In fact, hate becomes the dominant force in their lives. They hate God, they hate the people around them, and they hate themselves. Since they can't corner God or life, they take out their vengeance on other people. If they are married they are extremely cruel to their mate. If they have children, they treat them in the most inhuman manner. When they are not beating them, they are screaming at them; and when they are not screaming at them, they are cursing them. In short, they are mean.13 They love nobody and they demand no love. They trust no one and do not expect anyone to trust them. They find fault in everything and everybody. They always complain. You have seen people like this. They are cruel, vindictive and merciless.14
The terrible thing about this approach is that it poisons the soul and scars the personality. It does more harm to the person who harbours it than to anyone else. Many physical ailments are touched off by bitter resentment. Medical science has revealed that many cases of arthritis, gastric ulcer and asthma are caused by the long continuance of emotional poison in the mind. They are often psychosomatic, that is to say, they show in the body, but they are caused in the mind. There can be no doubt that resentment is a harmful reaction to disappointment and capable of setting up actual physical illness.15
Another possible reaction to the experience of blasted hopes is for the individuals to withdraw completely into themselves. They become absolute introverts. They allow no one to come into their lives and they refuse to go out to others. Such persons give up in the struggle of life. They lose the zest for living. They attempt to escape the disappointments of life by lifting their minds to a transcendent realm of cold indifference. Detachment is the word that may describe them. They are too unconcerned to love and they are too passionless to hate. They are too detached to be selfish and too lifeless to be unselfish. They are too indifferent to experience moments of joy and they are too cold to experience moments of sorrow.16 In short, such people are neither dead nor alive; they merely exist. Their eyes behold the beauties of nature, and yet they do not see them. Their ears are subjected to the majestic sounds of great music, and yet they do not hear it. Their hands gently touch a charming little baby, and yet they do not feel him. There is nothing of the aliveness of life left in them; there is only the dull motion of bare existence. Their disappointed hope leads them to a crippling cynicism. With Omar Khayyam they would affirm:
"The Worldly Hope men set their
Hearts upon turns to ashes—or it
prospers; and anon,
Like Snow upon the Desert's dusty Face
Lighting a little Hour or two-is gone."17
One can very easily see the danger of this reaction. It is, at bottom, based on an attempt to escape life. Psychiatrists tell us the more individuals attempt to engage in these escapes from reality the thinner and thinner their personalities become until ultimately they split. This is one of the causal sources of the schiophrenic personality.
Another way that people respond to life's disappointments is to adopt a philosophy of fatalism. This is the idea that whatever happens must happen, and that all events are determined by necessity. Fatalism implies something foreordained and inescapable. The people who subscribe to The this philosophy follow a course of absolute resignation. They resign themselves to what they consider their fate. They see themselves as little more than helpless orphans thrown out in the terrifying immensities of space. Since they believe that man has no freedom, they seek neither to deliberate nor to make decisions. They wait passively for external forces to deliberate and decide for them. They never actively seek to change their circumstances, since they believe that all circumstances, like the Greek tragedies, are controlled by irresistible and foreordained forces. Often the fatalists are very religious people who see God as the determiner and controller of destiny. Everything, they feel, is God's will, however evil it happens to be. This view is expressed in the verse of one of our Christian hymns:
"Though dark my path and sad my lot,
Let me be still and murmur not,
But breathe the prayer divinely taught,
Thy will be done."18
So the fatalists go through life with the conviction that freedom is a myth. They end up with a paralyzing determinism, saying that we are
"But helpless Pieces of the Game He plays
Upon this chequer-board of Night and Days,
and that we need not trouble our minds about the future—"Who knows?" Nor about the past, for
"The Moving Finger writes, and having writ
moves on… Neither tears not wit can
cancel out a line of it."19
For one to sink in the quicksands of this type of fatalism is both intellectually and psychologically stiffling. Since freedom is a part of the essence of man, the fatalist, in his denial of freedom, becomes a puppet and not a person. He is right in his conviction that there is no absolute freedom, and that freedom always operates within the framework of predestined structure. Thus a man is free to go north from Atlanta to Washington or South from Atlanta to Miami. But he is not free to go north to Miami or South to Washington. Freedom is always within destiny. But there is freedom. We are both free and destined. Freedom is the act of deliberating, deciding and responding within our destined nature. Even if destiny prevents our going to some attractive Spain, there still remains in us the capacity to take this disappointment, to {answer} it, to make our individual response to it, to stand up to it and do something with it. Fatalism doesn't see this. It leaves the individual stymied and helplessly inadequate for life.
But even more, fatalism is based on a terrible conception of God. It sees everything that happens, evil and good alike, as the will of God. Any healthy religion will rise above the idea that God wills evil. It is true that God has to permit evil in order to preserve the freedom of man. But this does not mean that he causes it. That which is willed is intended, and the idea that God intends for a child to be born blind, or that God gives cancer to this person and inflicts insanity upon another is rank heresy. Such a false idea makes God into a devil rather than a loving Father. So fatalism is a tragic and dangerous way to deal with the problem of unfulfilled dreams.20
What, then, is the answer? We must accept our unwanted and unfortunate circumstance and yet cling to a radiant hope. The answer lies in developing the capacity to accept the finite disappointment and yet cling to the infinite hope. In speaking of acceptance, I do not mean the grim, bitter acceptance of those who are fatalistic. I mean the kind of acceptance that Jeremiah achieved as expressed in the words, "this is my grief and I must bear it."21
This means sitting down and honestly confronting your shattered dream. Don't follow the escapist method of trying "to put it out of your mind." This will lead to repression which is always psychologically injurious. Place it at the forefront of your mind and stare daringly at it. Then ask yourself, "how can I transform this liability into an asset?" "How can I, confined in some narrow Roman cell, unable to reach life's Rome {Spain}, transform this cell from a dungeon of shame to a haven of redemptive suffering." Almost anything that happens to us can be woven into the purposes of God. It may lengthen our cords of sympathy. It may break our self-centered pride. Even the cross, which was willed by wicked men, was woven by God into the redemption of the world.
Many of the world's most influential characters have transformed their thorns into a crown. Charles Darwin was almost always physically ill. Robert Louis Stevenson was inflicted with tuberculosis. Helen Keller was blind and deaf.22 But they did not respond to these conditions with bitter resentment and grim fatalism. Rather they stood up to life, and, through the exercise of a dynamic will, transformed a negative into a positive.23 [George Frideric] Handel confronted the most difficult and trying circumstances in his life. Says his biographer: "His health and his fortunes had reached the lowest ebb. His right side had become paralyzed, and his money was all gone. His creditors seized him and threatened him with imprisonment. For a brief time he was tempted to give up the fight—but then he rebounded again to compose the greatest of his inspirations, the epic "Messiah." So, the "Hallelujah Chorus" was born, not in a desired Spain, but in a narrow cell of undesirable circumstances.
Wanting Spain and getting a narrow cell in a Roman prison, how familiar an experience that is! But to take the Roman prison, the broken, the left-over of a disappointed expectation, and make of it an opportunity to serve God's purpose, how much less familiar that is!24 Yet, powerful living has always involved such a victoryover one's own soul and one's situation.
We as a people have long dreamed of freedom, but we are still confined to an oppressive prison of segregation and discrimination.25 Must we respond to this disappointed hope with bitterness and cynicism? Certainly not, for this will only distort and poison our personality. Must we conclude that the existence of segregation is a part of the will of God, and thereby resign ourselves to the fate of oppression. Of course not, for such a course would be blasphemy, because it attributes to God something that should be attributed to the devil. Moreover, to accept passively an unjust system is to cooperate with that system; thereby the oppressed become as evil as the oppressor. Our most fruitful course of action will be to stand up with a courageous determination, moving on non-violently amid obstacles and setbacks, facing disappointments and yet clinging to the hope. It will be this determination and final refusal to be stopped that will eventually open the door of fulfillment. While still in the prison of segregation we must ask, "How can I turn this liability into an asset?" It is possible that, recognizing the necessity of suffering, we can make of it a virtue. To suffer in a righteous cause is to grow to our humanity's full stature.26 If only to save ourselves from bitterness, we need the vision to see the ordeals of this generation as the opportunity to transfigure ourselves and American society. Moreover, through our suffering in this oppressive prison and our non-violent struggle to get out of it, we may give the kind of spiritual dynamic to western civilization that it so desperately needs to survive.
Of course some of us will die having not received the promise of freedom. But we must continue to move on. On the one hand we must accept the finite disappointment, but in spite of this we must maintain the infinite hope. This is the only way that we will be able to live without the fatigue of bitterness and the drain of resentment.
This was the secret of the survival of our slave foreparents. Slavery was a low, dirty, inhuman business. When the slaves were taken from Africa, they were cut off from their family ties, and chained to ships like beasts. There is nothing more tragic than to cut a person off from his family, his language, and his roots.27 In many instances, during the days of slavery, husbands were cut off from wives and children were separated from parents. The women were often forced to satisfy the biological urges of the master himself, and the slave husband was powerless to intervene.28 Yet, in spite of these inexpressible cruelties, our foreparents continued to live and develop. Even though they could expect nothing the next morning but the long rows of cotton, the sweltering heat and the rawhide whip of the overseer, they continued to dream of a better day.29 They accepted the fact of slavery and yet clung to the hope of freedom. Their hope continued even amid a seemingly hopeless situation. They took the pessimism of life and filtered it in their own souls and fashioned it into a creative optimism that gave them strength to carry on. With their bottomless vitality they continually transformed the darkness of frustration into the light of hope. They had the "courage to be."30
When I first flew from New York to London, it was in the days of the propellor type aircraft. The flight over took 9 ½ hours. (The jets can make the flight in six hours.) On returning to the States from London I discovered that the flying time would be twelve hours and a half. This confused me for the moment. I knew that the distance returning to New York was the same as the distance from New York to London. Why this difference of three hours, I asked myself. Soon the pilot walked through the plane to greet the passengers. As soon as he got to me I raised the question of the difference in flight time. His answer was simple and to the point. "You must understand something about the winds," he said. "When we leave New York," 'he continued, "the winds are in our favor; we have a strong tail wind. When we return to New York from London, the winds are against us; we have a strong headwind." And then he said, "don't worry though, these four engines are fully capable of battling the winds, and even though it takes three hours longer we will get to New York." Well, life is like this. There are times when the winds are in our favor—moments of joy, moments of great triumph, moments of fulfillment. But there are times when the winds are against us, times when strong head winds of disappointment and sorrow beat unrelentingly upon our lives.31 We must decide whether we will allow the winds to overwhelm us or whether we will journey across life's mighty Atlantic with our inner spiritual engines equipped to go in spite of the winds. This refusal to be stopped, this "courage to be," this determination to go on living "inspite of," is the God in man. He who has made this discovery knows that no burden can overwhelm him and no wind of adversity can blow his hope away. He can stand anything that can happen to him.
Certainly the Apostle Paul had this type [of] "courage to be." His life was a continual round of disappointments. He started out for Spain and ended up in a Roman prison. He wanted to go to Bithynia but ended up in Troas.32 Everywhere he turned he faced broken plans. He was jailed, mobbed beaten and shipwrecked in his gallant program of preaching spreading the gospel of Christ. But he did not allow these conditions to overwhelm him.33 "I have learned," he said, "in whatsoever state I am, therewith to be content."34 Paul did not mean that he had learned to be complacent. There is nothing in the life of Paul which could characterize him as a complacent man. [Edward] Gibbon in his Decline and Fall of the Roman Empire says, "Paul has done more to promote the idea of freedom and liberty than any man who set foot on western soil." This does not sound like a complacent man. So Paul is not saying that he had learned to dwell in a valley of stagnant complacency. Neither is he saying that he had learned to resign himself to some tragic fate. Paul meant that he had learned to stand up amid the disappointment of life without despairing. He had discovered the distinction between a tranquil soul and the outward accidents of circumstance. He had learned to live from within instead of from without.35
The person who makes this magnificent discovery will, like Paul, be the recipient of true peace. Indeed, he will possess that peace which passeth all understanding.36 The peace which the world understands is that which comes with the removal of the burden or the pain. It is a peace which only comes on beautiful summer days, when the skies are clear and the sun shines in all of its scintilating beauty. It is a peace that comes when the pocketbook is filled and the body has no aches or pains. It is a peace that can only come by reaching the Spain of one's hope and staying out of the filthy jail. But this is not true peace. Real peace is something inward, a tranquility of soul amid terrors of trouble. It is inner calm amid the howland rage of outer storm. True peace is like a hurricane. Around its circomference rages howling and jostling winds of destruction, while at its center all is serenely quiet. This is why true peace passeth all understanding. It is easy to understand how one can have peace when everything is going right, and one is "up and in." But it is difficult to understand how one can have unruffled tranquility when he is "down and out," when the burden still lies heavy upon one's shoulders, when the pain still throbs annoyingly in one's body, when the prison cell still surrounds one with unbearable agony, and when the disappointment is inescapably real. True peace is peace amid story, tranquility amid disaster. It is a calm that exceeds all description and all explanation.37
Peace was Jesus' chief legacy. He said, "peace I leave with you, my peace I give unto you."38 This peace is there for us to iner inherit if we will only accept it through faith. Paul at Philippi, body beaten and bloody, incarcerated in a dark and desolate dungeon, feet chained and spirit tired, could joyously sing the songs of Zion at midnight.39 The early Christians, with the fierce faces of hungry lions standing before them and the excruciating pain of the chopping block only a step away, could face these pending disasters rejoicing that they had been dem deemed worthy to suffer disgrace for the sake of Christ. The Negro slaves, standing tiredly in the sizzling heat with the whip lashes freshly etched on their backs could sing triumphantly, "By and by I'm gwin to lay down this heavy load."40 This was peace amid storm.41
In the final analysis our ability to deal creatively with shattered dreams and blasted hopes will be determined by the extent of our faith in God. A genuine faith will imbue us with the conviction that there is a God beyond time and a Life beyond Life. Thus we know that we are not alone in any circumstance, however dismal and catastrophic it may be. God dwells with us in life's confining and oppressive cells. And even if we die there having not received the earthly promise, he will walk with us down that mysterious road called death, and lead us at last to that indescribable city that he has prepared for us. Let us never feel that God's creative power is exhausted by this earthly life, and his majestic love is locked within the limited walls of time and space. This would be a strongly irrational universe if God did not bring about an ultimate wedding of virtue and fulfillment. This would be an absurdly absurdly meaningless universe if death turned out to a blind alley leading the human race into a state of nothingness. God, through Christ has taken the sting from death, and it no longer has dominion over us.42 This earthly life is merely an embryonic prelude to a new awakening, and death is an open door that leads us into life eternal.
With this faith we can accept nobly what cannot be changed, and face disappointments and sorrow with an inner poise. We will have the power to absorb the most excruciating pain without losing our sense of hope. We will then know that in life and death, God will take care of us.43
"Be not dismayed, what-ere betide,
God will take care of you.
Beneath his wings of live abide,
"Thro' days of toil when heart doth fail,
When dangers fierce your path assail,
"God will take care of you, through every day,
"O're all the way.
He will take care of you,
God will take care of you."44
{But helpless pieces of the game he plays upon this chequer Board of Nights and days.}45
1. King, Unfulfilled Hopes, Sermon Delivered at Dexter Avenue Baptist Church, 5 April 1959, pp. 359-367 in this volume. King annotated the chapter titled "Shattered Dreams" in his personal copy of J. Wallace Hamilton's Horns and Halos in Human Nature (pp. 25-34).
2. Franz Schubert (1797-1828), an Austrian composer, completed only two movements of his eighth symphony.
3. British sculptor and painter George Frederick Watts produced "Hope" in 1886; George A. Buttrick, Sermons Preached in a University Church (New York: Abingdon Press, 1959), p. 110: "Thus George Frederick Watts depicts Hope herself as seated atop our planet indeed, but with head forlornly bowed and her fingers plucking one unbroken harp string."
4. The phrase "If we turn back to the life of the Apostle Paul" was replaced by "In Paul's letter to the Roman Christians" in the published version (King, Strength to Love, p. 78).
5. The preceding two sentences were altered in the published version: "On his return he wished to have personal fellowship with that valiant group of Roman Christians" (p. 78).
6. In the published version: "His preparations now centered in carrying the gospel to the capital city of Rome and to Spain at the distant fringe of the empire" (p. 78). Meek, "Strength in Adversity": "Paul had high hopes of going to Spain, the edge of the then known world, that he might take there his word about the Christian Gospel. And on the way he planned to visit the Christian folk in Rome, the capital city of the world. Paul wanted to see that little valiant group of Christians, folk whom he saluted in his letter as 'Christians in the household of Caesar.' The more he thought about his planned journey, the more his heart was warmed by it. Imagine, Rome with its many gods and with its great power, subject to the Christian Gospel."
7. Meek, "Strength in Adversity": "Paul did get to Rome, but he went as a prisoner and not as a freeman. Paul lived in Rome at the expense of the Roman government in a prison cell, held captive because of his faith. And Paul never saw the mountains and the plains and the coast life of Spain, because he died a martyr's death before the hope of his mission could ever be fulfilled."
8. Meek, "Strength in Adversity": "How many of us in one way or another have dreamed our dreams of going to Spain, of fulfilling some far reaching hope, of doing valiantly for a great cause. But we never reached the Spain of our dreams. We had to settle for a far shorter journey. We were never able to wander freely about the streets of our Rome. Instead, we looked out through the little windows of some confining cell which the circumstances of life had built around us."
9. The preceding two sentences were replaced in the published version: "Like Abraham, we too sojourn in the land of promise, but so often we do not become 'heirs with him of the same promise.' Always our reach exceeds our grasp" (p. 79).
10. The Muslim state of Pakistan was founded as a provision of the Indian Independence Act of 1947. Immediately following independence, border disputes and religious conflicts erupted between India and Pakistan, killing hundreds of thousands and displacing millions. The preceding sentence was altered in the published version: "After struggling for years to achieve independence, Mahatma Gandhi witnessed a bloody religious war between the Hindus and the Moslems, and the subsequent division of India and Pakistan shattered his heart's desire for a united nation" (p. 79).
11. Cf. Matthew 26:39.
12. Cf. 2 Corinthians 12:7-10; Howard Thurman, Deep River, pp. 34-35: "Jesus, in the garden of Gethsemane, prayed that the cup might pass, but he had to drink it to the last bitter dregs. The Apostle Paul prayed for the 'thorn' to be taken from his flesh, but he had to carry the thorn to his grave." In the published version the phrase "he went to his grave with this desire unfulfilled" was replaced with "the pain and annoyance continued to the end of his days" (p. 79).
13. Thurman, Deep River, p. 37: "It is quite possible to become obsessed with the idea of making everything and everybody atone for one's predicament. All one's frustrations may be distilled into a core of bitterness and disillusionment that expresses itself in a hardness of attitude and a total mercilessness—in short, one may become mean. You have seen people like that. They seem to have a demoniacal grudge against life." King also paraphrased this passage in a handwritten note for a sermon; see King, Notes on Deep River by Howard Thurman, October 1960.
14. The preceding twelve sentences were altered in the published version: "Because he cannot corner God or life, he releases his pent-up vindictiveness in hostility toward other people. He may be extremely cruel to his mate and inhuman to his children. In short, meanness becomes his dominating characteristic. He loves no one and requires love from no one. He trusts no one and does not expect others to trust him. He finds fault in everything and everybody, and he continually complains" (p. 79). King's notes on Deep River also included these words: "They trust no one and have no interest in doing so…For them life is essentially evil, and they are essentially vengeful…They have nothing to lose because they have lost everything" (King, Notes on Deep River, October 1960).
15. The preceding four sentences were altered in the published version: "Medical science reveals that such physical ailments as arthritis, gastric ulcer, and asthma have on occasion been encouraged by bitter resentments. Psychosomatic medicine, dealing with bodily sicknesses which come from mental illnesses, shows how deep resentment may result in physical deterioration" (p. 80). Leslie Weatherhead, "The Nature of Christ's Temptations," in The Key Next Door (New York: Abingdon Press, 1960), pp. 50-51: "I must speak very carefully here, but it is known that arthritis is sometimes touched off by resentment—it would be foolish to overlook the fact that some illnesses are what we now delight to call psychosomatic. That is to say, they show in the body, but they are caused or touched off in the mind, and arthritis is one of them. Asthma is another; the gastric ulcer is another…There can be no doubt that resentment, long harbored in the mind, is a faulty reaction to grief and capable of setting up actual physical illness."
16. The repetition of the phrase "they are" in the preceding three sentences was omitted in the published version (p. 80).
17. Khayyám, Rubáiyát, XVI.
18. King quotes Charlotte Elliott's hymn, "My God and Father! While I Stray" (1834).
19. King quotes from Khayyám, Rubáiyát, LXIX and LXXI. The final line in the published version of this sermon read: "nor all your Piety nor Wit / Shall lure it back to cancel half a Line, / Nor all your Tears wash out a Word of it"(p. 81).
20. The phrase "as are bitterness and withdrawal" was inserted at the end of this sentence in the published version (p. 82).
21. Cf. Jeremiah 10:19.
22. Fosdick, On Being a Real Person, p. 6: &;ldquo;Charles Darwin, as he himself said, 'almost continually unwell'; Robert Louis Stevenson, with his tuberculosis; Helen Keller, blind and deaf."
23. The preceding six sentences were condensed in the published version: "Many of the world's most influential personalities have exchanged their thorns for crowns. Charles Darwin, suffering from a recurrent physical illness; Robert Louis Stevenson, plagued with tuberculosis; and Helen Keller, inflicted with blindness and deafness, responded not with bitterness or fatalism, but rather by the exercise of a dynamic will transformed negative circumstances into positive assets" (p. 82).
24. The preceding two sentences were condensed in the published version: "How familiar is the experience of longing for Spain and settling for a Roman prison, and how less familiar the transforming of the broken remains of a disappointed expectation into opportunities to serve God's purpose!" (p. 83).
25. The word "Negroes" replaced "as a people" in the published version (p. 83).
26. The preceding two sentences were condensed in the published version: "By recognizing the necessity of suffering in a righteous cause, we may possibly achieve our humanity's full stature" (p. 83).
27. Thurman, Deep River, p. 35: "But it must be intimately remembered that slavery was a dirty, sordid, inhuman business. When the slaves were taken from their homeland, the primary social unit was destroyed, and all immediate tribal and family ties were ruthlessly broken… There is no more hapless victim than one who is cut off from family, from language, from one's roots."
28. Thurman, Deep River, p. 36: "In instance after instance, husbands were sold from wives, children were separated from parents; a complete and withering attack was made on the sanctity of the home and the family. Added to all this, the slave women were constantly at the mercy of the lust and rapacity of the master himself, while the slave husband or father was powerless to intervene."
29. Thurman, Deep River, p. 35: "For the slave, freedom was not on the horizon; there stretched ahead the long road down which there marched in interminable lines only the rows of cotton, the sizzling heat, the riding overseer with his rawhide whip, the auction block where families were torn asunder, the barking of the bloodhounds."
30. King invokes the title to Paul Tillich's book The Courage to Be.
31. The preceding three sentences were altered in the published version: "At times in our lives the tail winds of joy, triumph, and fulfillment favor us, and at times the head winds of disappointment, sorrow, and tragedy beat unrelentingly against us" (p. 84).
32. Cf. Acts 16:7-9.
33. In the published version, the preceding five sentences were altered: "On every side were broken plans and shattered dreams. Planning to visit Spain, he was consigned to a Roman prison. Hoping to go to Bithynia, he was sidetracked to Troas. His gallant mission for Christ was measured 'in journeyings often, in perils of waters, in perils of robbers, in perils by mine own countrymen, in perils by the heathen, in perils in the city, in perils in the wilderness, in perils in the sea, in perils among false brethren.' Did he permit these conditions to master him?" (pp. 84-85).
34. Philippians 4:11.
35. The previous six sentences were altered in the published version: "Does this sound like complacency? Nor did he learn resignation to inscrutable fate. By discovering the distinction between spiritual tranquillity and the outward accidents of circumstance, Paul learned to stand tall and without despairing amid the disappointments of life" (p. 85).
36. Cf. Philippians 4:7.
37. The preceding paragraph was altered in the published version: "Each of us who makes this magnificent discovery will, like Paul, be a recipient of that true peace 'which passeth all understanding.' Peace as the world commonly understands it comes when the summer sky is clear and the sun shines in scintillating beauty, when the pocketbook is full, when the mind and body are free of ache and pain, and when the shores of Spain have been reached. But this is not true peace. The peace of which Paul spoke is a calmness of soul amid terrors of trouble, inner tranquillity amid the howl and rage of outer storm, the serene quiet at the center of a hurricane amid the howling and jostling winds. We readily understand the meaning of peace when everything is going right and when one is 'up and in,' but we are baffled when Paul speaks of that true peace which comes when a man is 'down and out,' when burdens lie heavy upon his shoulders, when pain throbs annoyingly in his body, when he is confined by the stone walls of a prison cell, and when disappointment is inescapably real. True peace, a calm that exceeds all description and all explanation, is peace amid storm and tranquillity amid disaster" (p. 85).
38. John 14:27.
39. Cf. Acts 16:22-25.
40. King quotes from the spiritual "Bye and Bye."
41. In the published version: "These are living examples of peace that passeth all understanding" (p. 85).
42. Cf. 1 Corinthians 15:55-57.
43. This sentence was omitted and the previous sentence continued "for we know, as Paul testified, in life or in death, in Spain or Rome, 'that all things work together for good to them that love God, to them who are the called according to his purpose'" (p. 86).
44. King quotes Civilla D. Martin's hymn "God Will Take Care of You" (1904).
45. Khayyám, Rubáiyát, LXIX.
MLKP, MBU, Martin Luther King, Jr., Papers, 1954-1968, Howard Gotlieb Archival Research Center, Boston University, Boston, Mass.
The Papers of Martin Luther King, Jr. Volume VI: Advocate of the Social Gospel, September 1948 – March 1963
Clayborne Carson, Susan Carson, Susan Englander, Troy Jackson, and Gerald L. Smith, eds. | {
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ACCEPTED
#### According to
Index Fungorum
#### Published in
null
#### Original name
Tremella rosea Höhn.
### Remarks
null | {
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Parks and Neighborhoods committee recap
Yesterday was our last committee meeting before our annual 6-week plunge into the budget, so we had a longer agenda than usual. Councilmembers Godden and Conlin joined me at the table this week. You can see the Seattle Channel video of the whole meeting here.
We voted to approve acquisition of property at 6311 California Avenue to double the size of Morgan Junction Park.
After Parks Superintendent Christopher Williams gave us his Superintendent's Report, we looked at the following items. All the legislation that we voted on will be taken up by Full Council on September 24. That's the same Full Council meeting that will receive the Mayor's Budget and that includes a scheduled vote on the Arena MOU, so it's likely to be a busy one.
We voted to adopt Resolution 31384, which approves the composition of the Major Institution Citizens Advisory Committee (CAC) for Swedish Medical Center's Cherry Hill Campus. Formed as part of the Major Institutions and Schools program, CACs are designed to provide a way for neighbors of Seattle's Hospitals, Universities and Colleges to be directly involved in the development plans for those institutions to assure that neighborhood concerns are considered when those plans are made. All their meetings are open to the public. As you'll see if you visit the link, the committee is comprised of 12 members, 5 of whom are considered "near neighbors." There are also three alternates. The total applicant pool was 29.
We've had Parks Administrative office in the RDA Building at 800 Maynard South since 1998. Our lease was slated to expire in November, so after a review to ensure that the office space was still needed and within our price range, City staff negotiated a new seven-year lease that keeps the office rental flat for three years, and then escalates by inflation or 3%, whichever is less. We voted to approve Council Bill 117562, which authorizes the Director of Finance and Administrative Services to execute the lease agreement between the City and Fu Quan, LLC., the property owner. Here's a presentation with a few highlights of the RDA Building Lease.
We held C.B. 117587 which relates to jurisdiction and funding for the 14th Avenue NW park in Ballard without discussion or a vote, pending some further investigation into the best way for Parks and SDOT (Seattle Department of Transportation) to collaboratively manage jurisdiction over the proposed park boulevard.
We voted to approve C.B. 117586, which authorizes acquisition of land that will expand West Seattle's Morgan Junction park under the 2008 Parks and Green Space Levy. The Morgan Junction residential urban village had been identified as significantly underserved with open space, and this park, after it is developed, will double the size of Morgan Junction Park.
Finally, we voted to approve C.B. 117579, which transfers jurisdiction of the Sound Way property in the West Duwamish Greenbelt from the Department of Parks and Recreation to Seattle Public Utilities for maintenance, repair and operation of existing drainage facilities.
Posted: September 21st, 2012 under Neighborhoods, Parks, Planning and Land Use, Random | {
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} | 3,286 |
Reading: Kid Cudi Revealed He Recorded 11 Songs In The Span Of A Week
Gigi Lamayne so in love with Burna Boy
Msaki announces her final gig guide
ChatGPT puts Google, Meta and other brands under huge pressure
Kid Cudi Revealed He Recorded 11 Songs In The Span Of A Week
It seems some new Kid Cudi music is underway.
This past week, the rapper took to Twitter to announce that he had recorded six new songs in the span of three days.
A few days later, he shared that he had recorded a total of 11 songs throughout the week.
"My goal was to do a whole new album worth of songs this week," he said. "12 was my goal. Might get 14."
11 songs in 5 days. My goal was to do a whole new album worth of songs this week. 12 was my goal. Might get 14 😶🌫️
— The Chosen One (@KiDCuDi) December 2, 2022
Back in September upon the release of his 10th studio album, Entergalactic, Cudi revealed in an interview with Apple Music 1's Zane Lowe that he didn't have any desire to make any more albums.
"I have other things I wanna do," he said. "I do not see me never making music. […] But as far as getting in the studio and working on an album and then going and touring it? I just don't have it in me."
Last month, he revealed that he still had one more album left in his contract, though fans shouldn't expect it within the next year.
TAGGED: Cudi, Kid, recorded, revealed, Songs, Span, week | {
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JAHEIM HEADLEY DELIGHTED TO GET ON SCORESHEET
Max Barton
@htafcacademy
Jaheim Headley on target in Under-19s' latest fixture
- Jaheim Headley pleased to be on the scoresheet
- Fullback
- Under-19s drew 2-2 with Middlesbrough
Huddersfield Town's Under-19s fullback Jaheim Headley was delighted to score in his side's 2-2 draw with Middlesbrough at Rockcliffe Park on Friday afternoon.
The 19-year-old's 56th minute strike got Town back on level terms having trailed 2-1 at half time.
Headley showed good attacking intent throughout the game by troubling the Middlesbrough defence down the left side.
The Young Terrier admitted that Middlesbrough's XI, which included two First Team players, proved a tough test.
"The first half we were decent and the second half we kicked on well after they found ways to stop us playing our game.
"The first half especially they tried to stop us from playing out, they made it really difficult for us to play, but we were able to overcome that."
Despite any tactics to nullify the Town attack, Headley was able to find the back of the net following a winding run down the left. The fullback was pleased with his attacking performance.
"It's always a nice thing to score a goal so I'm really happy with that.
"I'm always looking to try and get into goalscoring positions as much as possible.
"I'm constantly working on improving my end product whether that's a cross or a shot."
The youngster said that attacking intent is important to the way the team plays and something he knows that Worthy likes to see from him.
"Worthy does encourage me to get forward, he encourages everyone to get forward to be honest.
"It's important to get forward for the way that we play, we look to win the ball back from the opposing winger and then get past him.
"It's something we've been working on to open up spaces for other players on the pitch and then overload them when we're in the box."
Headley hopes to acclimatise to the physical demands of playing fullback whilst increasing his number of attacking returns.
"It's very physically demanding playing fullback, there's a lot of running forward and running back to get back into position.
"I want to improve my end product and to understand the game even more, but I do think I'm seeing improvements in that area already."
Middlesbrough vs Huddersfield Town on 29 Jan 21
Jaheim Headley
REPORT: WORKSOP TOWN 2-1 YOUNG TERRIERS
Young Terriers narrowly beaten in second pre-season friendly
MICHAEL ACQUAH: WE NEED TO PLAY WITH MORE FIGHT
Michael Acquah reviews narrow Blackpool defeat
REPORT: WALSALL 3-0 TOWN U19s
Young Terriers beaten on the road
CARLOS PREVIEWS READING
Town Head Coach discusses upcoming trip to Reading FC
BRODIE SPENCER 'BUZZING' TO GET NAME ON SCORESHEET
Defender on 8-1 win over Salford City
PAT JONES PREVIEWS B TEAM vs SALFORD
Winger opens up to HTTV about missing start of season through injury
TOFFS: IT'S DISAPPOINTING
Left back reviews Swansea City draw | {
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Campiglossa helveola este o specie de muște din genul Campiglossa, familia Tephritidae. A fost descrisă pentru prima dată de Ito în anul 1984. Conform Catalogue of Life specia Campiglossa helveola nu are subspecii cunoscute.
Referințe
Campiglossa | {
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(function (app) {
'use strict';
applitude.register('mvc', {
Models: {}
});
applitude.register('mixinTest', {
mixins: 'mvc'
});
applitude.register('overrideTest', {
Models: {
override: true
},
mixins: 'mvc'
});
describe('mixinTest', function () {
it('should add mixins', function () {
expect(app.mixinTest.Models).toBeDefined();
});
});
describe('mvc plugin', function () {
it('should exist', function () {
expect(app.mvc).toBeDefined();
});
});
describe('overrideTest', function () {
it('should let module override mixins', function () {
expect(app.overrideTest.Models.override).toBeDefined();
});
});
}(applitude));
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,892 |
Q: How can I find the file name in a directory? For example, in the path C:\Users\Username\filename.txt, how would I print filename.txt?
Here is an example of a code:
>>> x = input("Enter file path: ")
>>> Enter file path: C:\Users\Username\filename.txt
# Now print filename.txt because that is the name of the file in the variable x
A: Split the path by \ and get the last element (which is supposed to be the filename):
print(x.split('\\')[-1]) # ==> filename.txt
A: The best way to do this intelligently is with the os module.
import os
Path = 'C:\Users\Username\filename.txt'
f_path, f_name = os.path.split(Path)
>f_name
'filename.txt'
A: Try this:
x = input("Enter file path: ")
with open(x, 'r') as f:
print(f.read())
A: method 1, you need to know the length of the filename:
path = "C:\Users\Username\filename.txt"
file = path[-12:] # ===> filename.txt
method 2, using os.path.split:
import os
file = os.path.split(path)[1]
method 3, using str.split. You need to make sure the path is seperated by '/':
file = path.split('/')[-1]
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,509 |
Q: Mean squared error vs. mean squared prediction error What is the semantic difference between Mean Squared Error (MSE) and Mean Squared Prediction Error (MSPE)?
A: The difference is not the mathematical expression, but rather what you are measuring.
Mean squared error measures the expected squared distance between an estimator and the true underlying parameter:
$$\text{MSE}(\hat{\theta}) = E\left[(\hat{\theta} - \theta)^2\right].$$
It is thus a measurement of the quality of an estimator.
The mean squared prediction error measures the expected squared distance between what your predictor predicts for a specific value and what the true value is:
$$\text{MSPE}(L) = E\left[\sum_{i=1}^n\left(g(x_i) - \widehat{g}(x_i)\right)^2\right].$$
It is thus a measurement of the quality of a predictor.
The most important thing to understand is the difference between a predictor and an estimator. An example of an estimator would be taking the average height a sample of people to estimate the average height of a population. An example of a predictor is to average the height of an individual's two parents to guess his specific height. They are thus solving two very different problems.
A: There is a correction to the second equation about:
$MSPE(L)=\mathbb{E}\Big[\Big(g(X))-\widehat{g}(X)\Big)^{2}\Big]$;
where $X$ is a random variable.
It is important to remember that when we are working with MSPE or MSEP (I usually use the last expression) we are dealing with random variables. We want to predict an unobserved random variable $X$ using an estimator which is also a random variable (usually constructed with a sample data). There, we have a great difference with the most used expression MSE. In that case, we are dealing with a population parameter $\theta$ that it is a constant and the estimator is again a random variable.
In a more realistic scenario, we are dealing with conditional expectation for the MSPE because if we want to predict we need to measure the quality of our estimator based on the information used in the sample data. So, our definition of MSPE would be:
$MSPE(L)=\mathbb{E}\Big[\Big(g(X))-\widehat{g}(X)\Big)^{2}\Big|\;\mathcal{G}\; \Big]$; where $\mathcal{G}$ is a $\sigma$-algebra and $\widehat{g}(X)$ is $\mathcal{G}$-measurable. We can say that $X$ is $\mathcal{F}$-measurable in the measurable space $(\Omega, \mathcal{F})$ and $g$ is a borel-measurable function, so by Doob-Dynkin $g(X)$ is also $\mathcal{F}$-measurable.
A: Typically, MSE involves only training data. The error here refers to how far the observed training response data is from the fitted response data (based on a model fit on the training data itself).
On the other hand, MSPE typically involves a testing set that was not part of the model training. The error here refers to how far the predicted testing data (predicted based on a model already fit on the training data) is from the observed testing data.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,426 |
Q: PHP generate PDF and edit I would like to ask
Is it possible to edit the PDF file? For example, insert a new pages between pages , like add page between page 17 and 19.
Which library can i use?
A: one way to do this is to use ghostscript to split the pdf into individual page images and then you can add additional pages and rearrange pages as needed with tcpdf or similar pdf library.
A: You can generate pdf easily using FPDF library http://www.fpdf.org/. But you can not edit them via this library. You can use FPDI library mentioned here: PDF Editing in PHP?.
A: Here is my view:
It is easy to edit Hypertext rather than PDF it self. I use TCPDF to convert HTML to PDF.
Take a look at this. It might be helpful : TCPDF_import is not bringing in an existing file
Thanks, Let us know if you find a better solution.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,288 |
\section{Introduction}
For $A\in\mathbb{R}^{m\times n}$, consider the linear inequality
system\begin{subequations}\label{eq:O_p_vN_pair}
\begin{equation}
A^{T}y>0,\label{eq:perceptron-ineq}
\end{equation}
and its alternative
\begin{equation}
Ax=0,\mathbf{1}^{T}x=1\mbox{ and }x\geq0,\label{eq:von-Neumann-ineq}
\end{equation}
\end{subequations}where $\mathbf{1}$ stands for the vector of all
ones. More generally, for a closed convex cone $K\subset\mathbb{R}^{n}$
with interior, consider the conic system \begin{subequations}\label{eq:G_p_vN_pair}
\begin{equation}
A^{T}y\in\mbox{\rm int}(K^{*}),\label{eq:perceptron-ineq-1}
\end{equation}
and its alternative
\begin{equation}
Ax=0,\bar{u}^{T}x=1\mbox{ and }x\in K.\label{eq:von-Neumann-ineq-1}
\end{equation}
\end{subequations}where $K^{*}:=\{z:z^{T}x\geq0\mbox{ for all }x\in K\}$
is the (positive) dual cone of $K$ and $\bar{u}$ is some point in
$\mbox{\rm int}(K^{*})$. It is easy to see that \eqref{eq:O_p_vN_pair} is
the particular case of \eqref{eq:G_p_vN_pair} when $K=\mathbb{R}_{+}^{n}$
and $\bar{u}=\mathbf{1}$. An easy variant of Farkas's Lemma shows
that exactly one of \eqref{eq:perceptron-ineq} and \eqref{eq:von-Neumann-ineq}
is feasible. A similar result is easily seen to hold for \eqref{eq:G_p_vN_pair}.
The perceptron algorithm is a simple iterative algorithm that finds
a solution to system \eqref{eq:perceptron-ineq} if it is feasible.
The von Neumann algorithm is a simple iterative algorithm that finds
an approximate solution to \eqref{eq:von-Neumann-ineq} if it is feasible,
and it can give a solution $y$ to \eqref{eq:perceptron-ineq} if
\eqref{eq:von-Neumann-ineq} turns out to be infeasible.
We recall some of the history of the perceptron and von Neumann algorithms.
The perceptron algorithm was introduced in \cite{Rosenblatt58} for
solving classification problems in machine learning. The von Neumann
algorithm was privately communicated by von Neumann to Dantzig in
the late 1940s, and later studied by Dantzig \cite{Dantzig92,Dantzig92_conv}.
Block \cite{Block62} and Novikoff \cite{Novikoff62} showed that
when \eqref{eq:perceptron-ineq} is feasible, the perceptron algorithm
finds a solution to \eqref{eq:perceptron-ineq} after at most $1/[\rho(A)]^{2}$
iterations, where $\rho(A)$, a condition number defined in \cite{CheungCucker01_MP},
is defined by
\[
\rho(A):=\left|\max_{\|y\|_{2}=1}\min_{j=1,\dots,n}\frac{a_{j}^{T}y}{\|a_{j}\|_{2}}\right|.
\]
When \eqref{eq:perceptron-ineq} is feasible, $\rho(A)$ is precisely
the width of the feasibility cone $\{y:A^{T}y\geq0\}$ as defined
in \cite{FreundVera99}. This condition number traces its roots to
\cite{Renegar_SIOPT95} (see also \cite{Renegar_MP95,PenaRenegar00}).
Epelman and Freund \cite{EpelmanFreund00} showed that the von Neumann
algorithm either computes an $\epsilon$-solution to \eqref{eq:von-Neumann-ineq}
in $O(\frac{1}{\rho(A)^{2}}\log(\frac{1}{\epsilon}))$ iterations
when \eqref{eq:von-Neumann-ineq} is feasible, or finds a solution
to the alternative system \eqref{eq:perceptron-ineq} in $O(1/\rho(A)^{2})$
iterations if \eqref{eq:perceptron-ineq} is feasible. They also treated
the generalized pair \eqref{eq:G_p_vN_pair}. (See also \cite{EpelmanFreund02}.)
Consider the problem
\begin{eqnarray}
& \min_{x\in\mathbb{R}^{n}} & \frac{1}{2}\|Ax\|^{2}\label{eq:primal-QP}\\
& \mbox{s.t.} & \bar{u}^{T}x=1\nonumber \\
& & x\in K.\nonumber
\end{eqnarray}
It is clear that \eqref{eq:von-Neumann-ineq-1} is feasible if and
only if the objective value of \eqref{eq:primal-QP} is zero. Alternatives
for solving \eqref{eq:G_p_vN_pair} include the interior point algorithm
and the ellipsoid algorithm. Both the interior point and ellipsoid
methods are sophisticated algorithms that give a better complexity
bound, but require significant computational effort to perform each
iteration.
For the case where $K=\mathbb{R}_{+}^{n}$ and $\bar{u}=\mathbf{1}$,
an active set quadratic programming (QP) algorithm can also be an
alternative. An active set QP algorithm can easily solve \eqref{eq:O_p_vN_pair}
if $m$ and $n$ are small, and the subproblems in each iteration
are easily solved when $m$ is small. For larger problems, an active
set QP algorithm is considered to be as efficient as the simplex method
in practice. An advantage of the active set QP method is that the
minimum of \eqref{eq:primal-QP} can be attained in finitely many
iterations. If the active set QP algorithm is used to find a feasible
solution to \eqref{eq:perceptron-ineq}, the algorithm can terminate
before the minimizer is found.
The perceptron and von Neumann algorithms are a first-order methods
for solving \eqref{eq:O_p_vN_pair} in that the computational effort
in each iteration is small, but one would need much more iterations
than a more sophisticated algorithm like the interior point method
or the ellipsoid method. For large scale problems, a first-order method
may be the only reasonable approach. Since the von Neumann algorithm
uses only matrix vector multiplications and do not solve linear systems,
it is also useful for sparse problems.
Problem \eqref{eq:von-Neumann-ineq} is a particular case of the problem
of finding whether the convex hulls of two sets of points overlap.
More precisely, for $A\in\mathbb{R}^{m\times n_{1}}$ and $B\in\mathbb{R}^{m\times n_{2}}$,
consider
\begin{eqnarray}
& \min_{x\in\mathbb{R}^{n_{1}},z\in\mathbb{R}^{n_{2}}} & \frac{1}{2}\|Ax-Bz\|^{2}\label{eq:von-Neumann-gen}\\
& \mbox{s.t.} & \mathbf{1}^{T}x=1\nonumber \\
& & \mathbf{1}^{T}z=1\nonumber \\
& & x,z\geq0.\nonumber
\end{eqnarray}
The sets are the vectors spanned by the columns of $A$ and $B$ respectively.
When $B$ is the zero vector of size $m\times1$, then \eqref{eq:von-Neumann-gen}
reduces to to \eqref{eq:primal-QP}. When the convex hull of these
two sets of points do not overlap, the classification problem is the
problem of finding a good separating hyperplane between these two
sets. Research in the classification problem has gone on to handle
the misclassifications of some of the points \cite{Scheinberg06}.
There are other accelerations of the perceptron and von Neumann algorithms
in the literature. A smoothed perceptron von Neumann algorithm was
studied in \cite{SoheiliPena12}, who were in turn motivated by the
smoothing techniques in \cite{Nesterov05_SIOPT}. A randomized rescaled
version of the perceptron algorithm was proposed in \cite{DunaganVempala06_MP}
that terminates in $O(m\log(\frac{1}{\rho(A)}))$ with high probability.
The randomized algorithm was extended to more general conic systems
in \cite{BelloniFreundVempala09_MOR}.
Another well-known algorithm for solving feasibility problems is the
method of alternating projections. The idea of using a QP as an intermediate
step to accelerate the method of alternating projections was studied
by the author for general feasibility problems in \cite{cut_Pang12,SHDQP,improved_SIP},
though the idea had been studied for particular cases in \cite{Pierra84,BausCombKruk06}
(intersection of an affine space and a halfspace), \cite{G-P98,G-P01}
(under smoothness conditions) and \cite{Fukushima82} (for the convex
inequality problem). For more information, we refer to the references
in the papers mentioned earlier, and highlight \cite{BB96_survey,EsRa11}
as well as \cite[Subsubsection 4.5.4]{BZ05} for more information
about the theory and method of alternating projections. It is natural
to ask whether a QP can accelerate algorithms for solving \eqref{eq:O_p_vN_pair}
and \eqref{eq:G_p_vN_pair}.
\subsection{Contributions of this paper}
We make use of the fact that it is relatively easy to project a point
onto the convex hull of a small number of points using an active set
QP algorithm to generalize the von Neumann algorithm. When the size
of the set that defines the convex hull to be projected on equals
two, then our algorithm becomes the setting of the von Neumann algorithm.
The size of this set can be chosen to be as large as one can reasonably
can to increase efficiency, as long as each iteration is still manageable.
(See lines 7-10 of Algorithm \ref{alg:enhanced-vN}, Remark \ref{rem:vN-particular-case}
and the subsequent discussion.)
For the case of \eqref{eq:O_p_vN_pair}, the generalized algorithm
(Algorithm \ref{alg:enhanced-vN}) is a variant of an active set QP
algorithm, and it converges to a point in finitely many iterations
to find a $y$ satisfying \eqref{eq:perceptron-ineq} or an $x$ satisfying
\eqref{eq:von-Neumann-ineq}, whichever one is feasible.
For the case of \eqref{eq:G_p_vN_pair}, Theorem \ref{thm:fin-conv-1}
proves that Algorithm \ref{alg:enhanced-vN} converges to a point
in finitely many iterations if $0\in\mbox{\rm int}(S)$, where $S=\mbox{\rm int}\{Ap:\bar{u}^{T}p=1,p\in K\}$,
and all previously identified points are kept. For the case when $m=2$
and $0$ is on the boundary of $S$, we show that the convergence
of $\{\|y_{i}\|\}_{i}$ in Algorithm \ref{alg:enhanced-vN} to zero
is linear with rate at worst $1/\sqrt{2}$ in Theorem \ref{thm:lin-conv-best},
and further analyze the behavior of Algorithm \ref{alg:enhanced-vN}
in Section \ref{sec:More-analysis} by appealing to epigraphs and
the monotonicity of subdifferentials.
\subsection{Notation}
We list down some common notation used in this paper, which are rather
standard material in convex analysis \cite{Rockafellar70}. Let $C\subset\mathbb{R}^{n}$
be a set.
\begin{lyxlist}{00.00.0000}
\item [{$\mbox{\rm aff}(C)$}] The affine hull of the set $C$.
\item [{$\mbox{\rm conv}(C)$}] The convex hull of the set $C$.
\end{lyxlist}
If $C$ is a closed convex set, we have the following notation.
\begin{lyxlist}{00.00.0000}
\item [{$T_{C}(x)$}] The tangent cone of the set $C$ at $x$.
\item [{$\partial C$}] The boundary of $C$.
\end{lyxlist}
For a convex function $f:\mathbb{R}\to\mathbb{R}$, we have the following
notation.
\begin{lyxlist}{00.00.0000}
\item [{$\partial f(x)$}] The subdifferential of $f$ at $x$, and $\partial f(x)$
is a subset of $\mathbb{R}$.
\end{lyxlist}
\section{Algorithm}
In this section, we propose Algorithm \ref{alg:enhanced-vN} for solving
\eqref{eq:G_p_vN_pair}, and describe the algorithmic issues incrementally.
We start by describing Algorithm \ref{alg:enhanced-vN}.
\begin{algorithm}
\label{alg:enhanced-vN}(Algorithm for system \eqref{eq:G_p_vN_pair})
For $A\in\mathbb{R}^{m\times n}$ and a closed convex cone $K\subset\mathbb{R}^{n}$
with interior, this algorithm finds either a $y\in\mathbb{R}^{m}$
satisfying \eqref{eq:perceptron-ineq-1} or an $x\in\mathbb{R}^{n}$
satisfying \eqref{eq:von-Neumann-ineq-1}.
01 $\quad$Set $x_{0}=p_{0}\in\{x:\bar{u}^{T}x=1,x\in K\}$, $y_{0}=Ax_{0}$
and $i=0$
02 $\quad$Loop
03 $\quad$$\qquad$Find some $p_{i+1}$ such that $p_{i+1}\in K$,
$\bar{u}^{T}p_{i+1}=1$ and $p_{i+1}^{T}Ay_{i}\leq0$.
04 $\quad$$\qquad$$\qquad$If $p^{T}A^{T}y_{i}>0$ for all $p\in K$
such that $\bar{u}^{T}p=1$,
05$\quad$$\qquad$$\qquad$$\qquad$then $A^{T}y_{i}\in\mbox{\rm int}(K^{*})$,
solving \eqref{eq:perceptron-ineq-1}, so we exit.
06 $\quad$$\qquad$\textbf{(Distance reduction)}
07 $\quad$$\qquad$Let $C_{i+1}=\{Ac_{i+1,1},\dots,Ac_{i+1,k_{i+1}}\}$
be a finite subset of
08$\quad$$\qquad$$\qquad$$\mbox{\rm conv}\{Ap_{0},Ap_{1},\dots,Ap_{i+1}\}$.
09 $\quad$$\qquad$Let $y_{i+1}:=P_{\scriptsize\mbox{\rm conv}(\tilde{C})}(0)=\sum_{j=1}^{k_{i+1}}\lambda_{j}^{(i+1)}Ac_{i+1,j}$,
where $\tilde{C}\subset C_{i+1}$,
10$\quad$$\qquad$$\qquad$$\sum_{j=1}^{k_{i+1}}\lambda_{j}^{(i+1)}=1$
and $\lambda_{j}^{(i+1)}\geq0$ for all $j\in\{1,\dots,k_{i+1}\}$.
11 $\quad$$\qquad$Let $x_{i+1}=\sum_{j=1}^{k_{i+1}}\lambda_{j}^{(i+1)}c_{i+1,j}$,
and $i\leftarrow i+1$.
12 $\quad$$\qquad$Perform an \textbf{aggregation step }to reduce
the size of $C_{i}$.
13 $\quad$until $\|y_{i}\|=\|Ax_{i}\|$ small.\end{algorithm}
\begin{rem}
(Choice of $p_{i+1}$) The $p_{i+1}$ in line 3 is typically chosen
by solving the conic section optimization problem
\begin{eqnarray}
p_{i+1}= & \arg\min_{p} & p^{T}A^{T}y_{i}\label{eq:new-p-i-1-formula}\\
& \mbox{s.t.} & \bar{u}^{T}p=1\nonumber \\
& & p\in K.\nonumber
\end{eqnarray}
In the case where $\bar{u}=\mathbf{1}$ and $K=\mathbb{R}_{+}^{n}$,
the vector $p_{i+1}$ is easily seen to be the elementary vector $e_{j}$,
where $j$ corresponds to the coordinate of $A^{T}y_{i}$ with the
minimum value. In the case where $K$ is the semidefinite cone $\mathcal{S}_{+}^{k\times k}$
and $\bar{u}$ is the identity matrix in $\mathbb{R}^{k\times k}$,
the optimization objective is now $\langle p,A^{T}y_{i}\rangle$,
where $A^{T}$ is the adjoint of the operator $A:\mathcal{S}_{+}^{k\times k}\to\mathbb{R}^{m}$
and $\langle,\rangle$ corresponds to the trace inner product. A minimizer
of \eqref{eq:new-p-i-1-formula} is easily obtained once the eigenvalue
factorization of $A^{T}y_{i}$ is obtained. If $K$ is the direct
sum of sets of the form $\mathbb{R}_{+}^{n}$ and semidefinite cones,
the problem \eqref{eq:new-p-i-1-formula} can still be easily solved.
More elaboration is given in \cite{EpelmanFreund00} for example.
\end{rem}
We show that Algorithm \ref{alg:enhanced-vN} is an enhancement to
the von Neumann algorithm.
\begin{rem}
\label{rem:vN-particular-case}(von Neumann algorithm) The generalized
von Neumann algorithm to solve \eqref{eq:G_p_vN_pair} in \cite{EpelmanFreund00}
is a particular case of Algorithm \ref{alg:enhanced-vN} when the
$p_{i+1}$ in line 3 is chosen by \eqref{eq:new-p-i-1-formula} and,
most importantly, the set $C_{i+1}$ in line 7 is chosen to be $\{y_{i},Ap_{i+1}\}$.
Choices of $p_{i+1}$ different from \eqref{eq:new-p-i-1-formula}
were explored. See for example \cite{SoheiliPena13}. The classical
von Neumann algorithm is the particular case when $K=\mathbb{R}_{+}^{n}$,
\textbf{$\bar{u}=\mathbf{1}$}, $p_{0}=\frac{1}{n}\mathbf{1}$. When
$K$ is a product of semidefinite cones and cones of the type $\mathbb{R}_{+}^{k}$,
the optimization problem \eqref{eq:new-p-i-1-formula} is easy to
solve. (See for example \cite{EpelmanFreund00}.)
\end{rem}
The key generalization over the von Neumann algorithm in Algorithm
\ref{alg:enhanced-vN} is lines 7 to 10. The set $C_{i}$ is typically
taken to be of size 2, but we notice that it is still easy to project
onto the convex hull of a small number of points using an active set
quadratic programming (QP) algorithm. An active set QP algorithm is
considered to be as efficient as the simplex method in practice, and
we will describe the active set QP algorithm in Algorithm \ref{alg:active-QP}
later.
\begin{rem}
(The $\tilde{C}$ in line 9) One would ideally choose $\tilde{C}=C_{i+1}$
in line 9 of Algorithm \ref{alg:enhanced-vN}, but it may take prohibitively
many iterations in order for the active set QP algorithm to find $P_{\scriptsize\mbox{\rm conv}(C_{i+1})}(0)$.
The active set QP algorithm finds $P_{\scriptsize\mbox{\rm conv}(\tilde{C})}(0)$
for different active sets $\tilde{C}$, ensuring a reduction in $d(0,\mbox{\rm conv}(\tilde{C}))$
at each iteration. (See Proposition \ref{prop:Facts-active-QP}, in
particular (4) and (5).) If the active set QP algorithm is expected
to take too many iterations before completion, then one can stop earlier
and find a new element to add to $C_{i}$ in line 7 of Algorithm \ref{alg:enhanced-vN}
instead.
\end{rem}
A useful property is the following.
\begin{rem}
\label{rem:finiteness-certificate}(Feasibility certificate) Suppose
\[
|\{\lambda_{j}^{(i)}:\lambda_{j}^{(i)}>0\}|=m+1,
\]
(i.e., there are $m+1$ positive terms in $\{\lambda_{j}^{(i)}\}_{j=1}^{k_{i}}$)
at line 12 of Algorithm \ref{alg:enhanced-vN}. The set $\{Ac_{i,j}:\lambda_{j}^{(i)}>0\}$
then contains $m+1$ points. If in addition, the affine space $\mbox{\rm aff}\{Ac_{i,j}:\lambda_{j}^{(i)}>0\}$
equals $\mathbb{R}^{m}$, then $0$ lies in $\mbox{\rm int}\,\mbox{\rm conv}\{Ac_{i,j}:\lambda_{j}^{(i)}>0\}$.
This condition can be used as an effective certificate of the feasibility
of \eqref{eq:von-Neumann-ineq-1}. Similar ideas, referred to as bracketing,
were proposed in \cite{Dantzig92}.
\end{rem}
\subsection{\label{sub:agg_strat}Aggregation strategies}
In line 12 of Algorithm \ref{alg:enhanced-vN}, we provided an option
of performing an aggregation step to reduce the size of the set $C_{i}$
so that each iteration can be performed in a reasonable amount of
effort. We now show how this aggregation can be done. Recall that
$C_{i}=\{Ac_{i,1},\dots,Ac_{i,k_{i}}\}$. The iterate $y_{i}$ can
be written as
\begin{equation}
y_{i}=\sum_{j=1}^{k_{i}}\lambda_{j}^{(i)}Ac_{i,j},\mbox{ where }\sum_{j=1}^{k_{i}}\lambda_{j}^{(i)}=1\mbox{ and }\lambda_{j}^{(i)}\geq0\mbox{ for all }j\in\{1,\dots,k_{i}\}.\label{eq:y-lambda-c}
\end{equation}
In other words, $y_{i}$ is a convex combination of some of the elements
in $C_{i}$. A logical first step to reduce the size of $C_{i}$ is
to discard indices $j$ such that $\lambda_{j}^{(i)}=0$. By Caratheodory's
theorem, the size of the set $C_{i}\subset\mathbb{R}^{m}$ can be
reduced to be at most $m+1$. If $|C_{i}|=m+1$, then it means that
$0$ lies in $\mbox{\rm int}\,\mbox{\rm conv}(C_{i})$, which ends our algorithm (see
Remark \ref{rem:finiteness-certificate}). If $|C_{i}|$ is still
not small enough so that the iterations of Algorithm \ref{alg:enhanced-vN}
can be easily performed, then we can aggregate to reduce the size
of $C_{i}$ by 1, as described below.
\begin{rem}
\label{rem:agg_strat}(Aggregation procedure 1) If the set $C_{i}=\{Ac_{i,1},\dots,Ac_{i,k_{i}}\}$,
the point $y_{i}$ and the vector $\lambda^{(i)}$ satisfy \eqref{eq:y-lambda-c}
and that $\lambda_{j}^{(i)}>0$ for all $j\in\{1,\dots,k_{i}\}$,
we can reduce the size of the active set $C_{i}$ by one using the
following procedure.
\begin{itemize}
\item Find 2 elements of $C_{i}$, say $\tilde{c}_{k_{i}-1}$ and $\tilde{c}_{k_{i}}$,
using one of the following strategies
\begin{itemize}
\item The elements $\tilde{c}_{k_{i}-1}$ and $\tilde{c}_{k_{i}}$ are the
oldest elements not to have been aggregated.
\item The coefficients $\lambda_{k_{i}-1}^{(i)}$ and $\lambda_{k_{i}}^{(i)}$
are the smallest.
\item The coefficients $\lambda_{k_{i}-1}^{(i)}$ and $\lambda_{k_{i}}^{(i)}$
are the largest.
\end{itemize}
\item Set $\tilde{c}_{k_{i}-1}\leftarrow\frac{\lambda_{k_{i}-1}^{(i)}}{\lambda_{k_{i}-1}^{(i)}+\lambda_{k_{i}}^{(i)}}\tilde{c}_{k_{i}-1}+\frac{\lambda_{k_{i}}^{(i)}}{\lambda_{k_{i}-1}^{(i)}+\lambda_{k_{i}}^{(i)}}\tilde{c}_{k_{i}}$,
$\lambda_{k_{i-1}}^{(i)}\leftarrow\lambda_{k_{i-1}}^{(i)}+\lambda_{k_{i}}^{(i)}$,
$\lambda_{k_{i}}^{(i)}\leftarrow0$, and $C_{i}\leftarrow C_{i}\backslash\{\tilde{c}_{k_{i}}\}$.
\end{itemize}
\end{rem}
In order to reduce the set $C_{i}$ to a manageable size, one can
perform as many iterations of the procedure in Remark \ref{rem:agg_strat}
as needed to drop more points, or to amend the procedure to drop more
points per iteration.
Consider points $p_{i+1}$ obtained by the optimization procedure
\eqref{eq:new-p-i-1-formula}. The point $Ap_{i+1}$ minimizes $\{y_{i}^{T}v:v=Ap,\bar{u}^{T}p=1,p\in K\}$.
Hence $Ap_{i+1}$ lies on the boundary of the set $S:=\{Ap:\bar{u}^{T}p=1,p\in K\}$.
If we aggregate by taking some weighted average of some of the points
in $\{Ap_{0},\dots,Ap_{i+1}\}$, the new points obtained can lie in
the relative interior of $S$. It may be desirable to keep as many
points on the boundary of $S$ as possible, and we describe a second
aggregation procedure.
\begin{rem}
\label{rem:agg_strat_2}(Aggregation procedure 2) For the setting
in Remark \ref{rem:agg_strat}, we can consider an alternative aggregation
strategy.
\begin{itemize}
\item If no vector had been obtained by an aggregation process (i.e., if
there were no vectors of the form $\tilde{c}_{k_{i}-1}$ produced
at the end of Remark \ref{rem:agg_strat}),
\begin{itemize}
\item Perform the procedure in Remark \ref{rem:agg_strat}, but store the
aggregated vector (i.e., the vector $\tilde{c}_{k_{i}-1}$) in $\tilde{c}_{1}$
instead after some change of indices.
\end{itemize}
\item else
\begin{itemize}
\item Choose a vector, say $\tilde{c}_{k_{i}}$, by some criterion (for
example, the ones similar to Remark \ref{rem:agg_strat}), and aggregate
with the steps $\tilde{c}_{1}\leftarrow\frac{\lambda_{1}^{(i)}}{\lambda_{1}^{(i)}+\lambda_{k_{i}}^{(i)}}\tilde{c}_{1}+\frac{\lambda_{k_{i}}^{(i)}}{\lambda_{k_{i}-1}^{(i)}+\lambda_{k_{i}}^{(i)}}\tilde{c}_{k_{i}}$,
$\lambda_{1}^{(i)}\leftarrow\lambda_{1}^{(i)}+\lambda_{k_{i}}^{(i)}$,
$\lambda_{k_{i}}^{(i)}\leftarrow0$, and $C_{i}\leftarrow C_{i}\backslash\{\tilde{c}_{k_{i}}\}$.
\end{itemize}
\end{itemize}
\end{rem}
\subsection{Primal active set quadratic programming}
We now discuss the primal active set quadratic programming algorithm
for performing the projection $y_{i+1}=P_{\scriptsize\mbox{\rm conv}(C_{i+1})}(0)$
in line 9 of Algorithm \ref{alg:enhanced-vN}.
\begin{algorithm}
\label{alg:active-QP}(Active set QP algorithm for $y=P_{\scriptsize\mbox{\rm conv}(C)}(0)$)
Let $C=\{c_{1},\dots,c_{k}\}$ be a finite set, as in the setting
of line 9 in Algorithm \ref{alg:enhanced-vN}. We give the full details
of how to evaluate $y=P_{\scriptsize\mbox{\rm conv}(C)}(0)$, as well to find
the multipliers $\lambda\in\mathbb{R}^{k}$ such that $y=\sum_{j=1}^{k}\lambda_{j}c_{j}$,
$\sum_{j=1}^{k}\lambda_{j}=1$.
01$\quad$Find $j_{0}^{*}\in\{1,\dots,k\}$.
02$\quad$Set $\lambda^{(0)}=e_{j_{0}^{*}}$, $J_{0}=\{j_{0}^{*}\}$,
$\tilde{C}_{0}=\{c_{j_{0}^{*}}\}$ and $\tilde{y}_{0}=c_{j_{0}^{*}}$.
03$\quad$Set $i=1$.
04$\quad$Begin outer loop \textbf{{[}Find entering index{]}}
05$\quad$$\quad$Find an index $j_{i}^{*}$ such that $\tilde{y}_{i-1}^{T}c_{j_{i}^{*}}<\tilde{y}_{i-1}^{T}c$
for all (or for any) $c\in\tilde{C}_{i-1}$.
06$\quad$$\quad$$\quad$If such $j_{i}^{*}$ does not exist, then
$\tilde{y}_{i-1}=P_{\scriptsize\mbox{\rm conv}(C)}(0)$, and we end.
07$\quad$$\quad$Set $\tilde{C}=\tilde{C}_{i-1}$, $\tilde{J}=J_{i-1}$,
$\tilde{y}=\tilde{y}_{i-1}$, $\tilde{\lambda}=\lambda^{(i-1)}$.
08$\quad$$\quad$Begin inner loop \textbf{{[}Tries to add $j_{i}^{*}$
to active set{]}}
09$\quad$$\quad$$\quad$Find $y^{\prime}=P_{\scriptsize\mbox{\rm aff}(\tilde{C}\cup\{c_{j_{i}^{*}}\})}(0)$,
and write $y^{\prime}=[\sum_{j\in\tilde{J}}\lambda_{j}^{\prime}c_{j}]+\lambda_{j_{i}^{*}}^{\prime}c_{j_{i}^{*}}$
10$\quad$$\quad$$\quad$$\quad$for some $\lambda^{\prime}\in\mathbb{R}^{k}$
such that $\mathbf{1}^{T}\lambda^{\prime}=1$ and $\lambda_{j}^{\prime}=0$
if $j\notin[\tilde{J}\cup\{j_{i}^{*}\}]$.
11$\quad$$\quad$$\quad$Find the largest $\bar{t}\in(0,1]$ such
that $\tilde{\lambda}+\bar{t}[\lambda^{\prime}-\tilde{\lambda}]\geq0$
12$\quad$$\quad$$\quad$If $\bar{t}=1$, then \textbf{{[}Found better
active set{]}}
13$\quad$$\quad$$\quad$$\quad$Set $\tilde{J}\leftarrow\tilde{J}\cup\{j_{i}^{*}\}$,
$J_{i}\leftarrow\{j\in\tilde{J}:\lambda_{j}^{(i)}>0\}$, $\tilde{C}_{i}\leftarrow\{c_{j}\}_{j\in J_{i}}$.
14$\quad$$\quad$$\quad$$\quad$Set $\lambda^{(i)}=\lambda^{\prime}$,
$\tilde{y}_{i}=y^{\prime}$ and exit inner loop
15$\quad$$\quad$$\quad$else\textbf{ {[}Remove non-active vertex{]}}
16$\quad$$\quad$$\quad$$\quad$At least one component of $\tilde{\lambda}+\bar{t}[\lambda^{\prime}-\tilde{\lambda}]$
equals zero, say $\tilde{j}$.
17$\quad$$\quad$$\quad$$\quad$Set $\tilde{y}\leftarrow\tilde{y}+\bar{t}[y^{\prime}-\tilde{y}]$,
$\tilde{\lambda}\leftarrow\tilde{\lambda}+\bar{t}[\lambda^{\prime}-\tilde{\lambda}]$
and $\tilde{J}\leftarrow\tilde{J}\backslash\{\tilde{j}\}$.
18$\quad$$\quad$$\quad$end
19$\quad$$\quad$$i\leftarrow i+1$
20$\quad$$\quad$end inner loop
21$\quad$end outer loop
\end{algorithm}
Some explanation of Algorithm \ref{alg:active-QP} is in order. The
following proposition collects some facts, some of which are analogues
of well known facts of active set quadratic programming algorithms.
\begin{prop}
\label{prop:Facts-active-QP}(Facts of Algorithm \ref{alg:active-QP})
For Algorithm \ref{alg:active-QP}, we have the following facts. At
each iteration $i$,
\begin{enumerate}
\item $\lambda^{(i)}\in\mathbb{R}_{+}^{n}$, $\mathbf{1}^{T}\lambda^{(i)}=1$,
and $\tilde{y}_{i}=\sum_{j=1}^{k}\lambda_{j}^{(i)}c_{j}$.
\item $\lambda_{j}^{(i)}>0$ if and only if $j\in J_{i}$.
\item $\tilde{y}_{i}=P_{\scriptsize\mbox{\rm conv}(\tilde{C}_{i})}(0)$ and $\tilde{y}_{i}=\sum_{j=1}^{k}\lambda_{j}^{(i)}c_{j}=\sum_{j\in\tilde{J}_{i}}\lambda_{j}^{(i)}c_{j}$.
\item $\{j_{i+1}^{*}\}\subset J_{i+1}\subset J_{i}\cup\{j_{i+1}^{*}\}$.
\item {[}Improvement{]} $\|\tilde{y}_{i+1}\|<\|\tilde{y}_{i}\|$ unless
$\tilde{y}_{i}=P_{\scriptsize\mbox{\rm conv}(C)}(0)$. \\
(Note that $\|\tilde{y}_{i}\|=d(0,\mbox{\rm conv}(\tilde{C}_{i}))$.)
\item {[}Finite convergence{]} There is some $i^{*}$ such that $\tilde{y}_{i^{*}}=P_{\scriptsize\mbox{\rm conv}(C)}(0)$.
\end{enumerate}
\end{prop}
For the formula $\tilde{y}_{i-1}^{T}c_{j_{i}}<\tilde{y}_{i-1}^{T}c$
for all $c\in\tilde{C}_{i-1}$ in line 5 of Algorithm \ref{alg:active-QP},
we note that since $\tilde{y}_{i-1}=P_{\tilde{C}_{i-1}}(0)$,
\begin{equation}
\tilde{y}_{i-1}^{T}c_{j_{1}}=\tilde{y}_{i-1}^{T}c_{j_{2}}\mbox{ for all }j_{1},j_{2}\in J_{i-1}.\label{eq:y-c-any-index}
\end{equation}
Thus the ``for all'' there is equivalent to ``for any''.
Algorithm \ref{alg:active-QP} works in the following manner. Property
(1) is clear. In view of property (2), $J_{i}$ is also the active
set. At each iteration, the pair $(\tilde{y}_{i},J_{i})$ satisfies
properties (2) and (3). Take any $\tilde{c}\in\tilde{C}_{i}$. If
$\tilde{y}_{i}^{T}c\geq\tilde{y}_{i}^{T}\tilde{c}$ for all $c\in C$,
then in view of \eqref{eq:y-c-any-index}, we can deduce (with a bit
of effort) that $\tilde{y}_{i}=P_{\scriptsize\mbox{\rm conv}(C)}(0)$. Otherwise,
we can find an index $j_{i+1}^{*}$ in line 5. For the next pair $(\tilde{y}_{i+1},J_{i+1})$,
we find $\tilde{y}_{i+1}=P_{\scriptsize\mbox{\rm conv}(\tilde{C}_{i}\cup\{c_{j_{i+1}^{*}}\})}(0)$,
but $\tilde{C}_{i}\cup\{c_{j_{i+1}^{*}}\}$ is not necessarily the
active set satisfying property (2). The removal of elements not in
the active set (through checking the sign of $\lambda$ in the inner
loop) gives us $J_{i+1}$ satisfying properties (2) and (4). Since
\[
\|\tilde{y}_{i+1}\|=d\big(0,\mbox{\rm conv}(\tilde{C}_{i}\cup\{c_{j_{i+1}^{*}}\})\big)<d\big(0,\mbox{\rm conv}(\tilde{C}_{i})\big)=\|\tilde{y}_{i}\|,
\]
property (5) is satisfied. Property (6) follows from property (5)
and the fact that the active set $J_{i}$ can take on only finitely
many possibilities.
\begin{rem}
(von Neumann algorithm via Algorithm \ref{alg:active-QP}) We show
how we can use Algorithm \ref{alg:active-QP} directly to solve the
system \eqref{eq:O_p_vN_pair} through \eqref{eq:primal-QP}. Let
$C=\{a_{1},\dots,a_{n}\}$ be the columns of $A$. It is clear that
$0$ lies in the convex hull of $C$ if and only if \eqref{eq:von-Neumann-ineq}
has a solution. The difference between using Algorithm \ref{alg:enhanced-vN}
for \eqref{eq:O_p_vN_pair} and using Algorithm \ref{alg:active-QP}
directly are
\begin{enumerate}
\item Algorithm \ref{alg:enhanced-vN} can stop when it finds a $y$ such
that $A^{T}y>0$, whereas Algorithm \ref{alg:active-QP} as stated
would stop only at $P_{\scriptsize\mbox{\rm conv}(C)}(0)$.
\item Algorithm \ref{alg:enhanced-vN} allows for aggregation to reduce
the size of $C$ but not Algorithm \ref{alg:active-QP}.
\end{enumerate}
If no aggregation is performed in Algorithm \ref{alg:enhanced-vN},
then finite convergence follows from the fact that the active set
can take on finitely many possibilities and (5) of Proposition \ref{prop:Facts-active-QP}.
\end{rem}
We remark on our choice of the QP algorithm.
\begin{rem}
(Choice of QP algorithm) A QP can be solved by an interior point method
or by a dual active set QP algorithm \cite{Goldfarb_Idnani}. We believe
that our choice of a QP algorithm is most appropriate because of the
following consequence of Proposition \ref{prop:Facts-active-QP}(5):
If we expect that we still need many iterations to solve the QP, we
can abort the QP solver halfway and the iterate $\tilde{y}_{i}$ obtained
so far would be closer to $0$ than what we started with. We might
not be able to get such an improved iterate if other QP solvers were
used.
\end{rem}
We remark on how one can speed up the implementation of Algorithm
\ref{alg:active-QP}.
\begin{rem}
\label{rem:accelerate-primal-QP}(On projecting onto affine spaces
in Algorithm \ref{alg:active-QP}) Recall that in line 9 of Algorithm
\ref{alg:active-QP}, we need an algorithm to find $y^{\prime}=P_{\scriptsize\mbox{\rm aff}(C)}(0)$,
where $C$ is a set of points $\{c_{1},\dots,c_{k}\}$. Finding this
projection is equivalent to finding $\gamma\in\mathbb{R}^{k-1}$ such
that
\[
y^{\prime}=c_{k}+\sum_{j=1}^{k-1}\gamma_{j}[c_{j}-c_{k}]\mbox{ and }y^{\prime}\perp[c_{j}-c_{k}]\mbox{ for all }j\in\{1,\dots,k-1\}.
\]
Let $\tilde{A}\in\mathbb{R}^{m\times(k-1)}$ be such that the $j$th
column is $[c_{j}-c_{k}]$ have the QR factorization $A=QR$. Then
one can easily figure that $\gamma=-R^{-1}Q^{T}c_{k}$. The bottleneck
in implementing Algorithm \ref{alg:active-QP} is thus to calculate
the QR factorization of matrices of the form $\tilde{A}$. One need
not calculate these QR factorizations from scratch, and can update
these QR factorizations whenever new columns are added or removed
using Given's rotations or Householder reflections. We refer the reader
to \cite{NW06} and the references therein for more details.
\end{rem}
More intuition is given in Figure \ref{fig:sample-runs-QP-alg},
where we show a sample run of Algorithm \ref{alg:active-QP} (or Algorithm
\ref{alg:enhanced-vN}) and the von Neumann algorithm. For this example
where $A\in\mathbb{R}^{2\times3}$, the von Neumann Algorithm takes
many iterations before it can certify the infeasibility of the QP,
while Algorithm \ref{alg:active-QP} finds the projection of $0$
onto $\mbox{\rm conv}\{a_{1},a_{2},a_{3}\}$ in two steps.
\begin{figure}[!h]
\begin{tabular}{|c|c|c|}
\hline
\multicolumn{3}{|l|}{von Neumann Algorithm}\tabularnewline
\hline
\hline
\includegraphics[scale=0.2]{vNm_01} & \includegraphics[scale=0.2]{vNm-02} & \includegraphics[scale=0.2]{vNm_03}\tabularnewline
\hline
\multicolumn{3}{c}{}\tabularnewline
\hline
\multicolumn{3}{|l|}{Algorithm \ref{alg:enhanced-vN}}\tabularnewline
\hline
\hline
\includegraphics[scale=0.2]{vNm_01} & \includegraphics[scale=0.2]{vNm-02} & \includegraphics[scale=0.2]{AQP_03}\tabularnewline
\hline
\end{tabular}
\caption{\label{fig:sample-runs-QP-alg}The diagram on the top shows iterations
of the von Neumann algorithm, while the diagram on the bottom shows
iterations of Algorithm \ref{alg:enhanced-vN}. To find $x_{2}$ in
Algorithm \ref{alg:enhanced-vN}, the point $a_{3}$ is identified
in line 3. The algorithm projects onto $\mbox{\rm conv}\{a_{1},a_{2},a_{3}\}$
by moving from $x_{1}$ onto the line segment $[a_{2},a_{3}]$, then
moving along the line segment to get $x_{2}$. }
\end{figure}
We generalize Algorithm \ref{alg:enhanced-vN} to handle \eqref{eq:von-Neumann-gen}
in Algorithm \ref{alg:extended-primal-QP} below. A similar approach
was attempted in \cite{Rujan93} using the dual active set QP algorithm
\cite{Goldfarb_Idnani}. A dual quadratic programming algorithm is
considered to be better for general QP problems because there is no
need to find a feasible starting point. Since a feasible point to
\eqref{eq:von-Neumann-gen} is readily available, the primal approach
is not disadvantaged. More importantly, we feel that the primal QP
approach is better for \eqref{eq:von-Neumann-gen} because it works
with vectors in $\mathbb{R}^{m}$, whereas the dual approach works
with vectors in $\mathbb{R}^{n_{1}+n_{2}}$, and we expect $m\ll n_{1}+n_{2}$.
It is unclear whether this generalization is original or not, but
we feel that it is worthwhile to make a connection. Algorithm \ref{alg:extended-primal-QP}
can be pieced from the general structure of a primal active set QP
algorithm, and is similar to Algorithm \ref{alg:enhanced-vN}. Furthermore,
we will not elaborate Algorithm \ref{alg:extended-primal-QP}, nor
will the rest of this paper depend on Algorithm \ref{alg:extended-primal-QP},
so we shall be brief.
\begin{algorithm}
\label{alg:extended-primal-QP}(Active set QP algorithm for \eqref{eq:von-Neumann-gen})
For $A\in\mathbb{R}^{m\times n_{1}}$ and $B\in\mathbb{R}^{m\times n_{2}}$,
this algorithm finds either a feasible pair $(x,z)$ of \eqref{eq:von-Neumann-gen}
satisfying $Ax=Bz$, or a $y\in\mathbb{R}^{m}$ such that
\begin{equation}
\min_{j\in\{1,\dots,n_{1}\}}a_{j}^{T}y>\max_{k\in\{1,\dots,n_{2}\}}b_{k}^{T}y,\label{eq:separating-hyperplane}
\end{equation}
where $\{a_{j}\}_{j=1}^{n_{1}}$ are the columns of $A$ and $\{b_{k}\}_{k=1}^{n_{2}}$
are the columns of $B$.
Choose $j\in\{1,\dots,n_{1}\}$, $k\in\{1,\dots,n_{2}\}$.
Set $J=\{j\}$, $K=\{k\}$ $x_{0}=e_{j}$, $y_{0}=Ax_{0}-Bz_{0}$
and $i=0$
Loop
$\quad$Find either some $j^{*}\in\{1,\dots,n_{1}\}$ such that $a_{j^{*}}^{T}y_{i}\leq\beta:=\max_{k\in K}b_{k}^{T}y_{i}$,
$\quad$$\quad$or some $k^{*}\in\{1,\dots,n_{2}\}$ such that $b_{k^{*}}^{T}y_{i}\geq\alpha:=\min_{j\in J}a_{j}^{T}y_{i}$.
$\quad$If no such $j^{*}$ or $k^{*}$ exists, then $y_{i}$ solves
\eqref{eq:separating-hyperplane}, and we exit.
$\quad$\textbf{(Distance reduction loop)}
$\quad$Loop
$\quad$$\quad$If $j^{*}$ was found earlier
$\quad$$\quad$$\quad$Find closest points between
$\quad$$\quad$$\quad$$\quad$$S_{1}:=\mbox{\rm aff}(\{a_{j}:j\in J\cup\{j^{*}\}\})$
and $S_{2}:=\mbox{\rm aff}(\{b_{k}:k\in K\})$,
$\quad$$\quad$$\quad$$\quad$$\quad$say $s_{1}\in S_{1}$ and
$s_{2}\in S_{2}$.
$\quad$$\quad$$\quad$$\quad$Write $s_{1}$ as $Ad_{1}$ and $s_{2}$
as $Bd_{2}$.
$\quad$$\quad$$\quad$Let
\begin{eqnarray*}
t_{1} & = & \min\left\{ \frac{(x_{i})_{j}}{-(d_{1}-x_{i})_{j}}:j\in J,(d_{1}-x_{i})_{j}<0\right\} \\
t_{2} & = & \min\left\{ \frac{(z_{i})_{k}}{-(d_{2}-z_{i})_{k}}:k\in K,(d_{2}-z_{i})_{k}<0\right\} \\
t & = & \min\{t_{1},t_{2},1\}
\end{eqnarray*}
$\quad$$\quad$$\quad$If $t=1$, then take $x_{i+1}\leftarrow d_{1}$
and $z_{i+1}\leftarrow d_{2}$,
$\quad$$\quad$$\quad$$\quad$set $J\leftarrow J\cup\{j^{*}\}$
and exit loop.
$\quad$$\quad$$\quad$If $t\in(0,1)$, then $x_{i}\leftarrow x_{i}+t(d_{1}-x_{i})$
and $z_{i}\leftarrow z_{i}+t(d_{2}-z_{i})$,
$\quad$$\quad$$\quad$$\quad$and drop the appropriate element in
either $j^{\prime}\in J$ or $k^{\prime}\in K$
$\quad$$\quad$$\quad$$\quad$such that $[x_{i}+t(d_{1}-x_{i})]_{j^{\prime}}=0$
or $[z_{i}+t(d_{2}-z_{i})]_{k^{\prime}}=0$.
$\quad$$\quad$end if
$\quad$$\quad$(The case where $k^{*}$ was found instead is similar)
$\quad$end loop.
$\quad$Set $y_{i}=Ax_{i}-Bz_{i}$
until $\|y_{i}\|$ small.
\end{algorithm}
\section{\label{sec:basic-analysis}Analysis of Algorithm \ref{alg:enhanced-vN}}
In this section, we prove some results of Algorithm \ref{alg:enhanced-vN}.
Theorem \ref{thm:fin-conv-1} gives conditions under which Algorithm
\ref{alg:enhanced-vN} terminates in finitely many iterations when
$0\in\mbox{\rm int}(S)$, where
\begin{equation}
S:=\{Ap:\bar{u}^{T}p=1,p\in K\}.\label{eq:set-S}
\end{equation}
We treat the case when $0$ lies in the boundary of $S$ and $S\subset\mathbb{R}^{2}$
in Subsection \ref{sub:bdry-case}, and show in Theorem \ref{thm:lin-conv-best}
that in such a case, we can expect linear convergence of $\{\|y_{i}\|\}_{i}$
to zero with a rate of at worst $1/\sqrt{2}$.
We recall the convergence rates of the generalized von Neumann algorithm
for \eqref{eq:G_p_vN_pair}.
\begin{rem}
(Convergence results from \cite{EpelmanFreund00}) The $\{\|y_{i}\|\}_{i}$
for the generalized von Neumann Algorithm for \eqref{eq:G_p_vN_pair}
is shown to be at worst linear when $0\in\mbox{\rm int}(S)$, and at worst
sublinear with rate $O(\frac{1}{\sqrt{i}})$ when $0\in\partial S$
in \cite{EpelmanFreund00}. The second result can also be traced back
to \cite{Dantzig92_conv}. These results can be extended by copying
the proofs almost word for word for Algorithm \ref{alg:enhanced-vN}
as long as in line 8, $Ap_{i+1}$ and $y_{i}$ are contained in $\mbox{\rm conv}(\tilde{C})$.
In this paper, we shall concentrate on how we can get better rates
than those in \cite{EpelmanFreund00} when $\tilde{C}=C_{i+1}$.
\end{rem}
We recall a easy result. A proof can be found in \cite{FreundVera99}
for example.
\begin{prop}
(Compactness of $S$) Suppose $\bar{u}\in\mbox{\rm int}(K^{*})$. Then the
set $\{p:\bar{u}^{T}p=1,p\in K\}$ is compact. The set $S$ in \eqref{eq:set-S}
is compact as well.
\end{prop}
Our first result is the finite convergence of Algorithm \ref{alg:enhanced-vN}
if $0\in\mbox{\rm int}(S)$.
\begin{thm}
\label{thm:fin-conv-1}(Finite convergence) Suppose Algorithm \ref{alg:enhanced-vN}
is used to solve \eqref{eq:G_p_vN_pair} for which \eqref{eq:von-Neumann-ineq-1}
is feasible. If $0\in\mbox{\rm int}(S)$, where $S$ is as defined in \eqref{eq:set-S},
and the choices $p_{i+1}$ in line 3 and $C_{i+1}$ in line 7 are
chosen by \eqref{eq:new-p-i-1-formula} and
\[
C_{i+1}=\{Ap_{0},\dots,Ap_{i+1}\}
\]
for all iterations $i$, then Algorithm \ref{alg:enhanced-vN} converges
in finitely many iterations.\end{thm}
\begin{proof}
Seeking a contradiction, suppose Algorithm \ref{alg:enhanced-vN}
runs indefinitely. Since $0\in\mbox{\rm int}(S)$, let $\delta>0$ be such
that $\mathbb{B}(0,\delta)\subset S$, and let $M:=\max_{s\in S}\|s\|$.
Recall that $y_{i}=Ap_{i}$ for all $i$. We use induction to prove
that
\begin{equation}
\angle[Ap_{j}]0[Ap_{k}]\geq\sin^{-1}(\delta/M)\mbox{ for all }0<j<k,\label{eq:fin-conv-indn-hyp}
\end{equation}
which leads to a contradiction because of the compactness of the unit
ball in $\mathbb{R}^{m}$.
Suppose \eqref{eq:fin-conv-indn-hyp} is true for all $k\leq i$.
We show that \eqref{eq:fin-conv-indn-hyp} is true for $k=i+1$. Since
$y_{i}$ equals $P_{C_{i}}(0)$, where $C_{i}=\{Ap_{0},\dots,Ap_{i}\}$,
we have
\begin{equation}
y_{i}^{T}Ap_{j}>0\mbox{ for all }j\in\{0,\dots,i\}.\label{eq:fin-conv-step-1}
\end{equation}
The point $p_{i+1}$ is chosen so that $Ap_{i+1}$ is a minimizer
of $\min\{[\frac{y_{i}}{\|y_{i}\|}]^{T}s:s\in S\}$. Note that since
$\mathbb{B}(0,\delta)\subset S$, $[\frac{y_{i}}{\|y_{i}\|}]^{T}Ap_{i+1}\leq-\delta$,
and $\|Ap_{i+1}\|\leq M$, we have
\begin{equation}
\left[\frac{y_{i}}{\|y_{i}\|}\right]^{T}\frac{Ap_{i+1}}{\|Ap_{i+1}\|}\leq-\frac{\delta}{M}.\label{eq:fin-conv-step-2}
\end{equation}
Note that \eqref{eq:fin-conv-step-2} implies that $\angle[Ap_{i+1}]0v\geq\sin^{-1}(\delta/M)$
for all $v$ such that $y_{i}^{T}v\geq0$. Combining \eqref{eq:fin-conv-step-1}
and the induction hypothesis, we see that \eqref{eq:fin-conv-indn-hyp}
is true for $k=i+1$. This ends the proof of our result.
\end{proof}
\subsection{\label{sub:bdry-case}When $0\in\partial S$}
We now treat the case when $0$ lies in the boundary of $S$ (as defined
in \eqref{eq:set-S}) and $A\in\mathbb{R}^{2\times n}$ (i.e., $m=2$).
This setting implies $S\subset\mathbb{R}^{2}$, which in turn allows
for a detailed analysis.
If $p_{i+1}$ is chosen by \eqref{eq:new-p-i-1-formula}, then $Ap_{i+1}$
is a minimizer of $\min\{y_{i}^{T}s:s\in S\}$. Since $0\in\partial S$,
we look at $T_{S}(0)$, the tangent cone of $S$ at $0$, and the
case when $\dim(T_{S}(0))$ equals two. If $\dim(T_{S}(0))$ equals
to one instead, then $\dim(S)$ equals one, in which case $S$ is
a line segment. Once the end points of the line segment are identified,
we know all that we need about the set $S$.
In view of the above discussions, we simplify Algorithm \ref{alg:enhanced-vN}
to the particular setting of interest where $m=2$ and $0\in\partial S$.
\begin{algorithm}
\label{alg:simple-enh-vN}(Algorithm for system \eqref{eq:G_p_vN_pair})
For a compact convex set $S\subset\mathbb{R}^{2}$ containing $0$
on its boundary, this algorithm tries to find a sequence of iterates
$y\in\mathbb{R}^{2}$ converging to $0$.
01$\quad$Set $y_{0}$ in the boundary of $S$ and $i=0$
02$\quad$Loop
03$\quad$$\qquad$Find $s_{i+1}$ such that $s_{i+1}$ is a minimizer
of $\min_{s\in S}\, y_{i}^{T}s.$
04$\quad$$\qquad$Let $C_{i+1}=\{y_{0},s_{1},\dots,s_{i+1}\}$,
$y_{i+1}:=P_{\scriptsize\mbox{\rm conv}(C_{i+1})}(0)$ and $i\leftarrow i+1$
05$\quad$until $\|y_{i}\|$ small.
\end{algorithm}
Note that lines 3-5 of Algorithm \ref{alg:enhanced-vN} accommodate
for the cases when $0\in\mbox{\rm int}(S)$, $0\in\partial S$ and $0\notin S$.
Since we only wish to study the case where $0\in\partial S$, we took
out the corresponding lines in Algorithm \ref{alg:simple-enh-vN}.
We also enforced that $y_{0}$ lies on the boundary of $S$ to simplify
our analysis.
In line 4 of Algorithm \ref{alg:simple-enh-vN}, we do not try to
reduce the size of $C_{i+1}$. The next result shows that since $m=2$,
there is no need to reduce the size of $C_{i+1}$. For 2 points $\alpha,\beta\in\mathbb{R}^{2}$,
we let $(\alpha,\beta)$ denote the set
\[
(\alpha,\beta)=\{t\alpha+(1-t)\beta:t\in(0,1)\}.
\]
\begin{thm}
\label{thm:bisec-enhanced-vN}(Bisection behavior of Algorithm \ref{alg:simple-enh-vN})
In Algorithm \ref{alg:simple-enh-vN}, suppose $y_{0}\neq0$ and $\dim(S)=2$.
Then for each $i\geq0$,
\begin{enumerate}
\item Either $s_{1}=0$ or $y_{1}\in(y_{0},s_{1})$.
\item Suppose $s_{i}\neq0$, $y_{i}\neq0$, and that $y_{i}\in(\bar{c}_{i},s_{i})$
for some $\bar{c}_{i}\in C_{i-1}$. \\
Then either $y_{i+1}\in(\bar{c}_{i},s_{i+1})$, $y_{i+1}\in(s_{i},s_{i+1})$,
or $s_{i+1}=0$.
\end{enumerate}
\end{thm}
\begin{figure}[!h]
\includegraphics[scale=0.5]{set_S}\caption{\label{fig:illustrate-bisect}Illustration of Theorem \ref{thm:bisec-enhanced-vN}.}
\end{figure}
\begin{proof}
If at any point $s_{i+1}=0$, then $y_{i+1}=P_{C_{i+1}}(0)=0$, resulting
in the termination of Algorithm \ref{alg:simple-enh-vN}. We shall
rule this case out to simplify our proof.
We first prove (1). For $i=0$, $y_{1}=P_{\scriptsize\mbox{\rm conv}\{y_{0},s_{1}\}}(0)$.
Since $s_{1}$ is chosen to be a minimizer of $\min_{s\in S}y_{0}^{T}s$,
we have $y_{0}^{T}s_{1}\leq y_{0}^{T}0=0$. Since $s_{1}\neq0$ and
$y_{0}\neq0$, this means that $\angle y_{0}0s_{1}\geq\pi/2$, which
implies that $\angle0y_{0}s_{1}<\pi/2$ and $\angle0s_{1}y_{0}<\pi/2$.
So $y_{1}=P_{\scriptsize\mbox{\rm conv}\{y_{0},s_{1}\}}(0)$ must lie in the
set $(y_{0},s_{1})$.
\textbf{}The projection of $0$ onto the polyhedron $\mbox{\rm conv}(C_{i})$
must land on a face of the polyhedron. If such a face is 2-dimensional,
then this means that $0$ lies in the (relative) interior of $\mbox{\rm conv}(C_{i})$,
but this would imply that $0$ lies in the interior of $S$ as $\mbox{\rm int}\,\mbox{\rm conv}(C_{i})\subset\mbox{\rm int}(S)$.
If the face if 0-dimensional, this means that $y_{i}=P_{\scriptsize\mbox{\rm conv}(C_{i})}(0)$
is a point in $C_{i}$. The other possibility is that the face is
1-dimensional, which corresponds to $y_{i}=P_{\scriptsize\mbox{\rm conv}(C_{i})}(0)$
being in $(c_{1},c_{2})$, where $c_{1},c_{2}$ are distinct elements
in $C_{i}$.
We prove (2) by induction. Statement (1) shows that the base case
holds. Suppose our claim is true for $i=i^{*}$. Then $y_{i^{*}}=(\bar{c}_{i^{*}},s_{i^{*}})$
for some $\bar{c}_{i^{*}}\in C_{i^{*}}$. Now, $y_{i^{*}}=P_{\scriptsize\mbox{\rm conv} C_{i^{*}}}(0)$
implies that
\begin{equation}
y_{i^{*}}^{T}c\geq y_{i^{*}}^{T}y_{i^{*}}>0\mbox{ for all }c\in\mbox{\rm conv}(C_{i^{*}}).\label{eq:y-c-geq-0-conv-C}
\end{equation}
\textbf{Claim 1: $y_{i^{*}+1}$ cannot be a point in $C_{i^{*}+1}$. }
We take a look at $y_{i^{*}+1}=P_{\scriptsize\mbox{\rm conv}(C_{i^{*}+1})}(0)$.
If $y_{i^{*}+1}$ is some point in $C_{i^{*}+1}$, then the possibilities
are that $y_{i^{*}+1}\in C_{i^{*}}$ or $y_{i^{*}+1}=s_{i^{*}+1}$.
If $y_{i^{*}+1}\in C_{i^{*}}$, then note that $\mbox{\rm conv}(C_{i^{*}})\subset\mbox{\rm conv}(C_{i^{*}+1})$,
so $y_{i^{*}}=P_{\scriptsize\mbox{\rm conv}(C_{i^{*}})}$ must be a point in
$C_{i^{*}}$ as well, but this is ruled out by the induction hypothesis.
We now rule out $y_{i^{*}+1}=s_{i^{*}+1}$. Now $s_{i^{*}+1}\in\mbox{\rm conv}(C_{i^{*}+1})$
and $y_{i^{*}}\in\mbox{\rm conv}(C_{i^{*}})\subset\mbox{\rm conv}(C_{i^{*}+1})$. Recall
that $s_{i^{*}+1}$ is chosen to be a minimizer of $\min_{s\in S}y_{i^{*}}^{T}s$,
so
\begin{equation}
y_{i^{*}}^{T}s_{i^{*}+1}\leq y_{i^{*}}^{T}0=0,\label{eq:y-s-1-geq-0}
\end{equation}
or $\angle y_{i^{*}}0s_{i^{*}+1}\geq\pi/2$. Since $y_{i^{*}}\neq0$
and $s_{i^{*}+1}\neq0$, we have $\angle y_{i^{*}}s_{i^{*}+1}0<\pi/2$
and $\angle s_{i^{*}+1}y_{i^{*}}0<\pi/2$, so
\[
d(0,\{s_{i^{*}+1},y_{i^{*}}\})>d(0,\mbox{\rm conv}\{s_{i^{*}+1},y_{i^{*}}\})\geq d(0,\mbox{\rm conv}(C_{i^{*}+1})).
\]
Thus $y_{i^{*}+1}$ cannot be a point in $C_{i^{*}+1}$.
\textbf{Claim 2: If $y_{i^{*}}\in(\bar{c}_{i^{*}},s_{i^{*}})$, then
either $y_{i^{*}+1}\in(\bar{c}_{i^{*}},s_{i^{*}+1})$ or $y_{i^{*}+1}\in(s_{i^{*}},s_{i^{*}+1})$ }
If $y_{i^{*}+1}$ lies in some line segment $(c_{1},c_{2})$, where
$c_{1}$ and $c_{2}$ are distinct elements in $C_{i^{*}}$, then
\[
d(0,C_{i^{*}})=d(0,y_{i^{*}})>d\big(0,(s_{i^{*}+1},y_{i^{*}})\big)\geq d(0,C_{i^{*}+1})=d(0,(c_{1},c_{2}))\geq d(0,C_{i^{*}}),
\]
which is absurd. Thus $y_{i^{*}+1}$ must lie in the segment $(c,s_{i^{*}+1})$
for some $c\in C_{i^{*}}$. We need to prove that $c$ can only be
either $\bar{c}_{i^{*}}$ or $s_{i^{*}}$.
Since $y_{i^{*}}\in(\bar{c}_{i^{*}},s_{i*})$ and $\bar{c}_{i^{*}}$
and $s_{i^{*}}$ both lie on the boundary of $S$, the line $\mbox{\rm aff}(\{\bar{c}_{i^{*}},s_{i^{*}}\})$
is a supporting hyperplane of $\mbox{\rm conv}(C_{i^{*}})$ at $y_{i^{*}}$.
Take any $c\in C_{i^{*}}\backslash\{\bar{c}_{i^{*}},s_{i^{*}}\}$.
Since $S\subset\mathbb{R}^{2}$, we make use of \eqref{eq:y-s-1-geq-0}
and \eqref{eq:y-c-geq-0-conv-C} to see that the line segment $[c,s_{i^{*}+1}]$
has to intersect somewhere in the line segment $[\bar{c}_{i^{*}},s_{i^{*}}]$.
By working out the possibilities in $\mathbb{R}^{2}$, we see that
\[
\min\big(d(0,[\bar{c}_{i^{*}},s_{i^{*}+1}]),d(0,[s_{i^{*}},s_{i^{*}+1}])\big)\leq d(0,[c,s_{i^{*}+1}]).
\]
Thus $y_{i^{*}+1}$ has to be in $(\bar{c}_{i^{*}},s_{i^{*}+1})$
or $(s_{i^{*}},s_{i^{*}+1})$.
\end{proof}
The consequence of Theorem \ref{thm:bisec-enhanced-vN} is that when
$\dim(S)=2$, there is no need to revisit dropped boundary points
of $S$ in the active set QP algorithm to project onto the convex
hull of an increasing set of points $C_{i}$.
Another way to interpret Theorem \ref{thm:bisec-enhanced-vN} is as
follows. The boundary of $S$ is homeomorphic to the sphere $\{x\in\mathbb{R}^{2},|x|=1\}$.
Algorithm \ref{alg:simple-enh-vN} is a bisection strategy. In iteration
$i$ when $y_{i}\in(\bar{c}_{i},s_{i})$ as in the notation of Theorem
\ref{thm:bisec-enhanced-vN}, Algorithm \ref{alg:simple-enh-vN} identifies
that $0$ lies on the path along the boundary from $\bar{c}_{i}$
to $s_{i}$. After the next iteration, either $y_{i+1}\in(\bar{c}_{i},s_{i+1})$
or $y_{i+1}\in(s_{i+1},s_{i})$. This means that Algorithm \ref{alg:simple-enh-vN}
has found that $0$ lies along the path along the boundary of $S$
from $s_{i+1}$ to either $\bar{c}_{i}$ or $s_{i}$. Notice that
even if $0$ were very close to $\bar{c}_{i}$ (or $s_{i}$ instead)
for example, the next point $s_{i+1}$ depends only on the geometry
of $S$ and not on the position of $y_{i}$. We shall see in Proposition
\ref{prop:any-sequence} that the ratio between $\|s_{i+1}-\bar{c}_{i}\|$
and $\|s_{i+1}-s_{i}\|$ can be arbitrarily large or small.
As a consequence of Theorem \ref{thm:bisec-enhanced-vN}, we prove
that the convergence of $\{\|y_{i}\|\}_{i}$ to zero is at least linear.
\begin{thm}
\label{thm:lin-conv-best}(Linear convergence of $\{\|y_{i}\|\}_{i}$
in Algorithm \ref{alg:simple-enh-vN}) The convergence of $\{\|y_{i}\|\}_{i}$
in Algorithm \ref{alg:simple-enh-vN} to zero is at worst linear with
constant $1/\sqrt{2}$. \end{thm}
\begin{proof}
Let $w_{i}=\|\bar{c}_{i}-s_{i}\|$. We assume that the convergence
of $\{\|y_{i}\|\}_{i}$ to zero is not finite. We deal with the easier
case first.
\textbf{Case 1: There is some $K>0$ such that if $i^{*}$ is large
enough, $i>i^{*}$ and $\frac{w_{i}}{w_{i^{*}}}\leq2^{-(i-i^{*})/2}$,
then $\|y_{i}\|\leq Kw_{i}\leq2^{-(i-i^{*})/2}Kw_{i^{*}}$.}
Recall $S$ is a convex set in $\mathbb{R}^{2}$ and $0$ lies in
the path from $\bar{c}_{i}$ to $s_{i}$ along $\partial S$. If $\frac{w_{i}}{w_{i^{*}}}\leq2^{-(i-i^{*})/2}$,
we must have $w_{i}\to0$, so $\bar{c}_{i}\to0$ and $s_{i}\to0$.
The limit $\lim_{i\to\infty}\angle\bar{c}_{i}0s_{i}$ exists as $\{\angle\bar{c}_{i}0s_{i}\}_{i}$
is nondecreasing and equals
\[
\lim_{i\to\infty}\angle\bar{c}_{i}0s_{i}=\max\{\angle v_{1}0v_{2}:v_{1},v_{2}\in T_{S}(0)\},
\]
which is finite. Let the limit above be $\theta>0$. We can use elementary
geometry to figure that $\|y_{i}\|$, which is also $d(0,\mbox{\rm aff}(\{\bar{c}_{i},s_{i}\}))$,
equals
\[
\|y_{i}\|=\frac{w_{i}\sin\angle\bar{c}_{i}s_{i}0\sin\angle s_{i}\bar{c}_{i}0}{\sin\angle\bar{c}_{i}0s_{i}},
\]
Moreover,
\begin{eqnarray*}
\limsup_{i\to\infty}\frac{\sin\angle\bar{c}_{i}s_{i}0\sin\angle s_{i}\bar{c}_{i}0}{\sin\angle\bar{c}_{i}0s_{i}} & \leq & \lim_{i\to\infty}\frac{[\sin(\frac{1}{2}[\pi-\angle\bar{c}_{i}0s_{i}])]^{2}}{\sin\angle\bar{c}_{i}0s_{i}}\\
& = & \begin{cases}
\frac{[\sin(\frac{1}{2}[\pi-\theta])]^{2}}{\sin\theta} & \mbox{ if }\theta<\pi\\
0 & \mbox{ if }\theta=\pi.
\end{cases}
\end{eqnarray*}
There is some $K>0$ such that if $i^{*}$ is large enough and $i>i^{*}$,
then $\|y_{i}\|\leq Kw_{i}$. The remaining inequality is easy, and
this ends our proof for case 1.
Let $q_{i}$ be a point such that $\mbox{\rm aff}(\{s_{i},q_{i}\})$ is a supporting
hyperplane of $S$ at $s_{i}$, and $q_{i}$ lies on the same side
of $\mbox{\rm aff}(\{\bar{c}_{i},s_{i}\})$ as $0$. Similarly, let $\bar{q}_{i}$
be a point such that $\mbox{\rm aff}(\{\bar{c}_{i},\bar{q}_{i}\})$ is a supporting
hyperplane of $S$ at $\bar{c}_{i}$ and $\bar{q}_{i}$ lies on the
same side of $\mbox{\rm aff}(\{\bar{c}_{i},s_{i}\})$ as $0$. See Figure \ref{fig:pf-lin-conv}.
\textbf{Claim 1:} \textbf{If $\angle\bar{c}_{i}s_{i}q_{i}$ and $\angle s_{i}\bar{c}_{i}\bar{q}_{i}$
are acute, then both $\angle\bar{c}_{i+1}s_{i+1}q_{i+1}$ and $\angle s_{i+1}\bar{c}_{i+1}\bar{q}_{i+1}$}
\textbf{are acute.}
Without loss of generality, assume that $\bar{c}_{i+1}=\bar{c}_{i}$.
(The other possibility of $\bar{c}_{i+1}=s_{i}$ is similar.) One
can see from Figure \ref{fig:pf-lin-conv} that $\bar{q}_{i+1}$ can
be taken to be $\bar{q}_{i}$. The $q_{i+1}$ is also easy to choose.
One can see that $\angle\bar{c}_{i+1}s_{i+1}q_{i+1}$ and $\angle s_{i+1}\bar{c}_{i+1}\bar{q}_{i+1}$
are both acute as claimed.
\begin{figure}
\begin{tabular}{|c|c|}
\hline
\includegraphics[scale=0.4]{pf01} & \includegraphics[scale=0.4]{pf2}\tabularnewline
\hline
\end{tabular}
\caption{\label{fig:pf-lin-conv}The diagram on the left is that of the proof
of Claim 1 in Theorem \ref{thm:lin-conv-best}, while the diagram
on the right is that of the proof of Claim 3 in the same theorem.}
\end{figure}
\textbf{Claim 2: For $i$ large enough, both $\angle\bar{c}_{i}s_{i}q_{i}$
and $\angle s_{i}\bar{c}_{i}\bar{q}_{i}$ are acute.}
Note that $\bar{c}_{1}=y_{0}$. One can easily see that $\angle y_{0}s_{1}q_{1}$
is acute. However, the angle $\angle s_{1}y_{0}\bar{q}_{1}$ is not
necessarily acute. If every point on the line segment $[y_{0},0]$
is on the boundary of $S$, then we can choose $\bar{q}_{1}$ so that
$\angle s_{1}y_{0}\bar{q}_{1}$ is acute.
Consider the following statement:
\begin{itemize}
\item [(*)] Unless every point on the line segment $[y_{0},0]$ is on the
boundary of $S$ (which was already treated in the previous paragraph),
eventually $\bar{c}_{i}\neq y_{0}$ for all $i$ large enough.
\end{itemize}
We now show that ({*}) implies our claim at hand. Suppose ({*}) is
true. Let $i^{*}$ be the smallest $i$ such that $\bar{c}_{i}\neq y_{0}$.
This would mean that $\bar{c}_{i^{*}}=s_{i^{*}-1}$. The angle $\angle s_{i^{*}}\bar{c}_{i^{*}}\bar{q}_{i^{*}}$
can be checked to be acute, and so would $\angle\bar{c}_{i^{*}}s_{i^{*}}q_{i^{*}}$.
We now prove ({*}) by contradiction. Suppose $\bar{c}_{i}=y_{0}$
for all $i$. The points $\{s_{i}\}_{i}$ trace a path along $\partial S$
getting closer to $0$. Let $s^{*}:=\lim_{i\to\infty}s_{i}$ and
\[
y^{*}:=\lim_{i\to\infty}\frac{P_{\scriptsize\mbox{\rm conv}(\{y_{0},s_{i}\})}(0)}{\|P_{\scriptsize\mbox{\rm conv}(\{y_{0},s_{i}\})}(0)\|}.
\]
If $s^{*}\neq0$, we can see that $y^{*}=\frac{P_{\scriptsize\mbox{\rm conv}(\{y_{0},s^{*}\})}(0)}{\|P_{\scriptsize\mbox{\rm conv}(\{y_{0},s^{*}\})}(0)\|}$.
If $s^{*}=0$, then $y^{*}$ is the vector perpendicular to $\mbox{\rm aff}(\{y_{0},0\})$
such that $s_{1}^{T}y^{*}>0$. Since all points in the line segment
$(y_{0},0)$ lie in $\mbox{\rm int}(S)$, all points in the line segment $(y_{0},s^{*})$
also lie in $\mbox{\rm int}(S)$. Any minimizer of $\min\{s^{T}y^{*}:s\in S\}$
lies on the path along the boundary of $S$ between $s^{*}$ and $y_{0}$.
So if $s_{i}$ were sufficiently close to $s^{*}$, $s_{i+1}$ would
be forced to be on the boundary of $S$ between $s^{*}$ and $y_{0}$
as well. This contradicts the assumption that $s^{*}=\lim_{i\to\infty}s_{i}$,
ending the proof of the claim.
Let $A_{i}$ be the area of $S\cap H_{i}$, where $H_{i}$ is the
halfspace with boundary $\mbox{\rm aff}(\{\bar{c}_{i},s_{i}\})$ containing
$0$. See Figure \ref{fig:pf-lin-conv}.
\textbf{Claim 3: $2A_{i+1}\leq A_{i}$ if $i$ is large enough so
that claim 2 holds}
Let the triangle $\mbox{\rm conv}(\{\bar{c}_{i},s_{i},s_{i+1}\})$ be $T_{i}$.
See Figure \ref{fig:pf-lin-conv}. If $i$ is large enough so that
claim 2 holds, then the set $S\cap H_{i}$ is bounded by four lines:
the line $\mbox{\rm aff}(\{\bar{c}_{i},s_{i}\})$, the line parallel to $\mbox{\rm aff}(\{\bar{c}_{i},s_{i}\})$
through $s_{i+1}$, and the lines perpendicular to $\mbox{\rm aff}(\{\bar{c}_{i},s_{i}\})$
through $\bar{c}_{i}$ and $s_{i}$. The rectangle formed, which we
call $R_{i}$, has twice the area of $T_{i}$. It is clear that $[S\cap H_{i+1}]\cup T_{i}\subset S\cap H_{i}\subset R_{i}$,
which implies $A_{i+1}+\mbox{area}(T_{i})\leq A_{i}$. Also, $S\cap H_{i+1}\subset R_{i}\backslash T_{i}$,
which implies $A_{i+1}\leq\mbox{area}(T_{i})$. Thus $2A_{i+1}\leq A_{i}$,
which is the conclusion we seek.
We now consider the second case.
\textbf{Case 2: If $i^{*}$ is large enough, $i>i^{*}$ and $\frac{w_{i}}{w_{i^{*}}}\geq2^{-(i-i^{*})/2}$,
then $\|y_{i}\|\leq\frac{2A_{i}}{w_{i}}\leq\frac{2^{-i+i^{*}+1}A_{i^{*}}}{2^{-(i-i^{*})/2}w_{i^{*}}}=2^{-(i-i^{*})/2}\frac{2A_{i^{*}}}{w_{i^{*}}}$.}
It is clear from elementary geometry that
\[
\|y_{i}\|w_{i}\leq d(s_{i+1},\mbox{\rm aff}(\{\bar{c}_{i},s_{i}\}))w_{i}=\mbox{area}(R_{i})=2\mbox{area}(T_{i})\leq2A_{i},
\]
or in other words $\|y_{i}\|\leq\frac{2A_{i}}{w_{i}}$. Claim 3 implies
that $A_{i}\leq2^{-i+i^{*}}A_{i^{*}}$ if $i>i^{*}$ and $i^{*}$
is large enough. This ends the proof of our result for case 2.
Putting together the two cases gives us the result at hand.
\end{proof}
\section{\label{sec:More-analysis}More on Algorithm \ref{alg:simple-enh-vN}}
In this section, we continue from the developments in Section \ref{sec:basic-analysis}
and elaborate on the behavior of Algorithm \ref{alg:simple-enh-vN}
by using an epigraphical and subdifferential analysis.
When $\dim(S)=2$, the intersection of $\mathbb{B}(0,\delta)\cap S$
is, up to a rotation, the intersection of a compact convex set and
the epigraph of some convex function, say $f$. This is described
in Figure \ref{fig:epiLipschitzian}. \marginpar{Remember to put this figure and the subsequent figure to latex file
to make the latex pictures render properly.}(The set $S$ is said to be epi-Lipschitzian \cite{Rockafellar79_epiL}
at $0$ in the sense of variational analysis. For more information,
see \cite{Cla83,Mor06,RW98} for example.)
\begin{figure}[!h]
\scalebox{0.45}[0.45]{\input{rot.pstex_t}}
\caption{\label{fig:epiLipschitzian} The set $S$ is epi-Lipschitzian at $0$.
Therefore, we can rotate the set $S$ so that locally at $0$, it
is the epigraph of some convex function $f$. We will need to use
the subdifferential mapping $\partial f(\cdot)$ for our later analysis.}
\end{figure}
We look at the graph of subdifferential $\partial f:\mathbb{R}\rightrightarrows\mathbb{R}$,
where ``$\rightrightarrows$'' signifies that $\partial f(\cdot)$
is a set-valued map, or in other words, $\partial f(x)$ is in general
a subset of $\mathbb{R}$. Since $f(\cdot)$ is convex, it is well-known
that the subdifferential mapping $\partial f(\cdot)$ is \emph{monotone},
i.e., if $v_{1}\in\partial f(x_{1})$, $v_{2}\in\partial f(x_{2})$
and $x_{1}\leq x_{2}$, then $v_{1}\leq v_{2}$. In view of monotonicity,
the points of discontinuity of $\partial f(\cdot)$ on an interval
is of measure zero and the function $\partial f(\cdot)$ is integrable,
i.e.,
\[
\int_{\alpha}^{\beta}\partial f(x)dx=f(\beta)-f(\alpha).
\]
We now state an algorithm expressed in terms of $f$ and $\partial f$,
and show its relationship with Algorithm \ref{alg:enhanced-vN}.
\begin{algorithm}
\label{alg:fn-bracketing}(A bracketing algorithm) For $a_{0},b_{0}>0$,
let $f:[-a_{0},b_{0}]\to\mathbb{R}$ be a convex function with a minimizer
at $0$. We want to find a minimizer of $f(\cdot)$ with the following
steps.
01$\quad$Start with $i=0$
02$\quad$Loop
03$\quad$Find a point in $[\partial f]^{-1}(\frac{f(b_{i})-f(-a_{i})}{a_{i}+b_{i}})$,
say $c_{i}$, which lies in the interval $[-a_{i},b_{i}]$.
04$\quad$$\quad$If $c_{i}<0$, then $a_{i+1}\leftarrow-c_{i}$ and
$b_{i+1}\leftarrow b_{i}$.
05$\quad$$\quad$If $c_{i}>0$, then $a_{i+1}\leftarrow a_{i}$ and
$b_{i+1}\leftarrow c_{i}$.
06$\quad$$\quad$If $a_{i+1}+b_{i+1}$ is sufficiently small or $c_{i}=0$,
then end algorithm.
07$\quad$$\quad$$i\leftarrow i+1$
08$\quad$end loop
\end{algorithm}
At each step of Algorithm \ref{alg:fn-bracketing}, we find $a_{i}$
and $b_{i}$ such that $0\in(-a_{i},b_{i})$. Each iteration improves
either the left or right end point.
In line 3 of Algorithm \ref{alg:simple-enh-vN}, we find a minimizer
of $\min_{s\in S}y_{i}^{T}s$, where $y_{i}$ is the projection of
$0$ onto $C_{i}$. In line 3 of Algorithm \ref{alg:fn-bracketing},
we find a minimizer of $x\mapsto f(x)-[\frac{f(b_{i})-f(-a_{i})}{a_{i}+b_{i}}]^{T}x$
by finding a point $c_{i}$ such that $\frac{f(b_{i})-f(-a_{i})}{a_{i}+b_{i}}\in\partial f(c_{i})$.
It is clear to see that line 3 of both algorithms are equivalent.
The following result shows the basic convergence of Algorithm \ref{alg:fn-bracketing}.
\begin{thm}
\label{thm:interval-analysis}(Basic convergence of Algorithm \ref{alg:fn-bracketing})
Let $\bar{a}$ and $\bar{b}$ be two positive numbers, and $f:[-\bar{a},\bar{b}]\to\mathbb{R}$.
Suppose $a^{\prime}\in[0,\bar{a}]$ and $b^{\prime}\in[0,\bar{b}]$
are such that
\[
f(x)\begin{cases}
=0 & \mbox{if }x\in[-a^{\prime},b^{\prime}]\\
>0 & \mbox{otherwise}.
\end{cases}
\]
Then the iterates $\{a_{i}\}_{i}$ and $\{b_{i}\}_{i}$ of Algorithm
\ref{alg:fn-bracketing} are such that $\{a_{i}\}_{i}$ and $\{b_{i}\}_{i}$
are non-increasing sequences such that for each $i$, either $a_{i+1}<a_{i}$
or $b_{i+1}<b_{i}$. Furthermore, one of these possibilities happen
\begin{enumerate}
\item Algorithm \ref{alg:fn-bracketing} finds a point in $[\partial f]^{-1}(0)$.
(i.e., a minimizer of $f$ is found.)
\item $\lim_{i\to\infty}a_{i}=a^{\prime}$ and $\lim_{i\to\infty}b_{i}=b^{\prime}$.
\end{enumerate}
\end{thm}
\begin{proof}
Assume that Algorithm \ref{alg:fn-bracketing} does not encounter
a point in $[\partial f]^{-1}(0)$. We try to show that only case
(2) can happen.
When $a^{\prime}=\bar{a}$ and $b^{\prime}=\bar{b}$, then $\partial f(x)=\{0\}$
for all $x\in(-\bar{a},\bar{b})$, so case (1) must happen. When $a^{\prime}<\bar{a}$
and $b^{\prime}=\bar{b}$, Algorithm \ref{alg:fn-bracketing} applied
to $f(\cdot)$ gives equivalent iterates as Algorithm \ref{alg:fn-bracketing}
applied to $f(-\cdot)$, where the $a$'s and $b$'s swap roles, reducing
to the case where $a^{\prime}=\bar{a}$ and $b^{\prime}<\bar{b}$.
We look at two cases from here onwards.
\textbf{Case A: $a^{\prime}=\bar{a}$ and $b^{\prime}<\bar{b}$.}
It is obvious that $\lim_{i\to\infty}a_{i}=\bar{a}=a^{\prime}$. If
case (1) is not encountered, then $b_{i}>b^{\prime}$ for all $i$.
We prove that $\lim_{i\to\infty}b_{i}=b^{\prime}$. Let $\partial f(b_{i})=[s_{i,3},s_{i,4}]$
for all $i$. Now,
\[
s_{i}^{\prime}:=\frac{1}{\bar{a}+b_{i}}\int_{-\bar{a}}^{b_{i}}\partial f(x)dx\leq\frac{1}{\bar{a}+b_{i}}\int_{0}^{b_{i}}s_{i,3}dx\leq\frac{\bar{b}s_{i,3}}{\bar{a}+\bar{b}}.
\]
It is clear to see that $s_{i}^{\prime}\in(0,\frac{\bar{b}}{\bar{a}+\bar{b}}s_{i,3})$.
Thus $b_{i+1}=[\partial f]^{-1}(s_{i}^{\prime})$ would be such that
$b_{i+1}<b_{i}$. Since $\partial f(b_{i+1})=[s_{i+1,3},s_{i+1,4}]$,
we see that $s_{i+1,3}\leq\frac{\bar{b}}{\bar{a}+\bar{b}}s_{i,3}$,
which implies $\lim_{i\to\infty}s_{i+1,3}=0$. Thus $\lim_{i\to\infty}b_{i}=b^{\prime}$
and we are done.
\textbf{Case B: $a^{\prime}<\bar{a}$ and $b^{\prime}<\bar{b}$.}
If case (1) is not encountered, then $a_{i}>a^{\prime}$ and $b_{i}>b^{\prime}$
for all $i$. We prove that case (2) must hold.
Consider $s_{i}^{\prime}=\frac{1}{a_{i}+b_{i}}\int_{-a_{i}}^{b_{i}}\partial f(x)dx$.
Let
\[
\partial f(-a_{i})=[s_{i,1},s_{i,2}]\mbox{ and }\partial f(b_{i})=[s_{i,3},s_{i,4}].
\]
By the monotonicity of $\partial f(\cdot)$, we have $s_{i,1}\leq s_{i,2}\leq0\leq s_{i,3}\leq s_{i,4}$.
We have $s_{i}^{\prime}\in[s_{i,2},s_{i,3}]$, and $s_{i}^{\prime}$
equals $s_{i,2}$ only if $\partial f(x)=\{s_{i,2}\}$ for all $x\in(-a_{i},b_{i})$.
This cannot happen as $a_{i}>a^{\prime}$ would ensure that $s_{i,2}<0$,
and $b_{i}>b^{\prime}$ would then imply $0\notin\partial f(0)$,
which is a contradiction. We can also argue that $s_{i}^{\prime}$
cannot be $s_{i,3}$. Thus $s_{i}^{\prime}\in(s_{i,2},s_{i,3})$.
By the workings of Algorithm \ref{alg:fn-bracketing}, we either have
$-a_{i+1}\in[\partial f]^{-1}(s_{i}^{\prime})$ or $b_{i+1}\in[\partial f]^{-1}(s_{i}^{\prime})$,
which will mean that either $a_{i+1}<a_{i}$ or $b_{i+1}<b_{i}$.
Thus the sequences $\{a_{i}\}_{i}$ and $\{b_{i}\}_{i}$ are nonincreasing,
and for each $i$, either $a_{i+1}<a_{i}$ or $b_{i+1}<b_{i}$.
Let $b^{*}:=\lim_{i\to\infty}b_{i}$ and $a^{*}:=\lim_{i\to\infty}a_{i}$.
It is clear that $b^{*}\geq b^{\prime}$ and $a^{*}\geq a^{\prime}$.
We prove that $b^{\prime}=b^{*}$ and $a^{\prime}=a^{*}$. Let $\partial f(a^{*})=[s_{1}^{*},s_{2}^{*}]$
and $\partial f(b^{*})=[s_{3}^{*},s_{4}^{*}]$. It is clear that $s_{1}^{*}\leq s_{2}^{*}\leq0\leq s_{3}^{*}\leq s_{4}^{*}$.
If $a^{*}>a^{\prime}$, then $s_{2}^{*}<0$. Otherwise $b^{*}>b^{\prime}$
gives $s_{3}^{*}>0$. In either case, we have $s_{1}^{*}\leq s_{2}^{*}<s_{3}^{*}\leq s_{4}^{*}$.
Now,
\[
\lim_{i\to\infty}\frac{1}{a_{i}+b_{i}}\int_{-a_{i}}^{b_{i}}\partial f(x)dx=\frac{1}{a^{*}+b^{*}}\int_{-a^{*}}^{b^{*}}\partial f(x)dx.
\]
Since either $a^{*}>a^{\prime}\geq0$ or $b^{*}>b^{\prime}\geq0$,
we have $a^{*}+b^{*}>0$, so the limit above is well defined. Let
this limit be $s^{*}$. It is clear that $s^{*}\in[s_{2}^{*},s_{3}^{*}]$.
\textbf{Claim: If $s^{*}\in\{s_{2}^{*},s_{3}^{*}\}$, then $a^{*}=a^{\prime}$
and $b^{*}=b^{\prime}$.}
Consider the case when $s^{*}=s_{2}^{*}$. We must have
\begin{equation}
\partial f(x)=\{s_{2}^{*}\}\mbox{ for all }x\in(-a^{*},b^{*}).\label{eq:partial-f-x-s-2-1}
\end{equation}
If $b^{*}>0$, then the inequality $s_{2}^{*}\leq0$ and $\partial f(x)\in[0,\infty)$
for all $x\in(0,b^{*})$ forces $s_{2}^{*}=0$, which gives $b^{*}\leq b^{\prime}$,
and in turn $b^{*}=b^{\prime}$. We are left with showing that $a^{*}=a^{\prime}$.
Seeking a contradiction, suppose $a^{*}=a^{\prime}$. Recall that
this implies $s_{2}^{*}<0$. Since $\partial f(x)\in[0,\infty)$ for
all $x\in(0,b^{*})$, \eqref{eq:partial-f-x-s-2-1} implies $b^{*}=0$.
So $\partial f(x)=\{s_{2}^{*}\}$ for all $x\in(-a^{*},0)$. Let $\gamma>0$
be such that
\[
\int_{-a^{*}}^{\gamma}\partial f(x)dx=a^{*}s_{2}^{*}+\int_{0}^{\gamma}\partial f(x)dx<0.
\]
The local Lipschitz continuity of $f(\cdot)$ at $0$ implies that
$\partial f(\cdot)$ is locally bounded at $0$, so such a $\gamma$
must exist. If $b_{i}<\gamma$, then $\int_{-a_{i}}^{b_{i}}\partial f(x)dx\leq\int_{-a^{*}}^{\gamma}\partial f(x)dx<0$,
so $[\partial f]^{-1}(\frac{1}{a_{i}+b_{i}}\int_{-a_{i}}^{b_{i}}\partial f(x)dx)<0$.
This means that only $a_{i}$ would decrease and $b_{i}$ would remain
constant, contradicting the fact that $b^{*}=\lim_{i\to\infty}b_{i}=0$.
This ends the proof of our claim when $s^{*}=s_{2}^{*}$. The case
when $s^{*}=s_{3}^{*}$ is similar. This ends the proof of our claim.
Recalling the situation before our claim, we have $s^{*}\in(s_{2}^{*},s_{3}^{*})$.
This means that either $-a_{i+1}\in(-a^{*},b^{*})$ or $b_{i+1}\in(-a^{*},b^{*})$
for $i$ large enough, which contradicts the definition of $a^{*}$
and $b^{*}$. So $b^{\prime}=b^{*}$ and $a^{\prime}=a^{*}$ as needed.
\end{proof}
Theorem \ref{thm:interval-analysis} shows that if $0\notin\partial f(-a_{i})$
and $0\notin\partial f(b_{i})$ for all $i$, the only situation when
the iterates $\{a_{i}\}_{i}$ and $\{b_{i}\}_{i}$ of Algorithm \ref{alg:fn-bracketing}
do not both converge to zero is when both $a^{\prime}=\lim_{i\to\infty}a_{i}$
and $b^{\prime}=\lim_{i\to\infty}b_{i}$ are such that $-a^{\prime}$
and $b^{\prime}$ minimize $f(\cdot)$, and $0\in[-a^{\prime},b^{\prime}]$.
When $0\in\partial f(-a_{i})$ or $0\in\partial f(b_{i})$ for some
iterate $a_{i}>0$ or $b_{i}>0$, we can assume without loss of generality
that $0\in\partial f(-a_{i})$. Algorithm \ref{alg:fn-bracketing}
would continue with the iterates $a_{i}$ staying put, and $\{b_{i}\}_{i}$
strictly decreasing to a minimizer of $f(\cdot)$. The point $0$
would lie in $[-a^{\prime},b^{\prime}]$.
The observation in the last paragraph shows the following behavior
of Algorithm \ref{alg:simple-enh-vN}: When there is a nontrivial
line segment on $\partial S$ such that $0$ lies somewhere on the
line segment, the cluster points of the iterates $\{s_{i}\}_{i}$
of Algorithm \ref{alg:simple-enh-vN} will land on the line segment.
Furthermore, $0$ lies in the convex hull of the cluster points of
$\{s_{i}\}_{i}$.
There is no fixed behavior of the iterates $\{a_{i}\}_{i}$ and $\{b_{i}\}_{i}$
of Algorithm \ref{alg:fn-bracketing}, as the following result shows.
\begin{prop}
\label{prop:any-sequence}(Arbitrary decrease in width) Let $\bar{a}$
and $\bar{b}$ be two positive numbers, and $f:[-\bar{a},\bar{b}]\to\mathbb{R}$.
Let the nonincreasing, nonnegative sequences $\{a_{i}\}_{i}$ and
$\{b_{i}\}_{i}$ be such that
\begin{enumerate}
\item $a_{0}=\bar{a}$ and $b_{0}=\bar{b}$, and
\item For each $i$, either $a_{i+1}<a_{i}$ and $b_{i+1}=b_{i}$, or $a_{i+1}=a_{i}$
and $b_{i+1}<b_{i}$.
\end{enumerate}
We can choose a proper convex function $f:[-\bar{a},\bar{b}]\to\mathbb{R}$
such that Algorithm \ref{alg:fn-bracketing} generates the iterates
$\{a_{i}\}_{i}$ and $\{b_{i}\}_{i}$. \end{prop}
\begin{proof}
Let $a^{\prime}:=\lim_{i\to\infty}a_{i}$ and $b^{\prime}:=\lim_{i\to\infty}b_{i}$.
Define $f(\cdot)$ to be zero on $[-a^{\prime},b^{\prime}]$. We now
define $f(\cdot)$ on the rest of $[-\bar{a},\bar{b}]$. Construct
the sequences of nonincreasing positive numbers $\{\alpha_{i}\}_{i}$,
$\{\beta_{i}\}_{i}$ and $\{\gamma_{i}\}_{i}$ satisfying the following
rules:
\begin{enumerate}
\item [(A)]If $a_{i+1}<a_{i}$ and $b_{i+1}=b_{i}$, then \textrm{$[a_{i+1}+b_{i}]\gamma_{i}+[a_{i}-a_{i+1}][-\alpha_{i}]<[a_{i}+b_{i}][-\gamma_{i}]$}
\item [(B)]If $a_{i+1}=a_{i}$ and $b_{i+1}<b_{i}$, then \textrm{$[a_{i}+b_{i+1}][-\gamma_{i}]+[b_{i}-b_{i+1}][\beta_{i}]>[a_{i}+b_{i}][\gamma_{i}]$}
\item [(C)]$\alpha_{i+1}=\beta_{i+1}=\gamma_{i}$ and $\gamma_{i+1}\leq\gamma_{i}$
for all $i\geq0$.
\item [(D)]$\alpha_{0}=\beta_{0}=1$.
\end{enumerate}
We can construct the sequences inductively with $\alpha_{0}$ and
$\beta_{0}$ defined through (D), $\gamma_{i}$ defined by $\alpha_{i}$
and $\beta_{i}$ through (A) and (B), and $\alpha_{i+1}$ and $\beta_{i+1}$
defined by $\gamma_{i}$ through (C). Define $\partial f(\cdot)$
by
\[
\partial f(x):=\begin{cases}
\{-\alpha_{i}\} & \mbox{ if }x\in(-a_{i},-a_{i+1})\\
\{\beta_{i}\} & \mbox{ if }x\in(b_{i+1},b_{i}).
\end{cases}
\]
The function $f(\cdot)$ can be inferred from $f(x)=\int_{0}^{x}\partial f(x)dx$
since the monotone function $\partial f(\cdot)$ is integrable. We
now verify that Algorithm \ref{alg:fn-bracketing} applied to $f(\cdot)$
generates the sequence $\{a_{i}\}_{i}$ and $\{b_{i}\}_{i}$. We first
look at the case where $a_{i+1}<a_{i}$ and $b_{i+1}=b_{i}$. Here,
\begin{eqnarray*}
\int_{-a_{i}}^{b_{i}}\partial f(x)dx & = & \int_{-a_{i+1}}^{b_{i}}\partial f(x)dx+\int_{-a_{i}}^{-a_{i+1}}\partial f(x)dx\\
& \leq & [a_{i+1}+b_{i}]\gamma_{i}+[a_{i}-a_{i+1}][-\alpha_{i}]\\
& < & [a_{i}+b_{i}][-\gamma_{i}]\\
\Rightarrow\frac{1}{a_{i}+b_{i}}\int_{-a_{i}}^{b_{i}}\partial f(x)dx & < & -\gamma_{i}.
\end{eqnarray*}
It is clear that $\frac{1}{a_{i}+b_{i}}\int_{-a_{i}}^{b_{i}}\partial f(x)dx>-\alpha_{i}$.
Since condition (C) implies that $\partial f(x)\in[-\gamma_{i},\gamma_{i}]$
for all $x\in(-a_{i+1},b_{i+1})$, this implies that $[\partial f]^{-1}(\frac{1}{a_{i}+b_{i}}\int_{-a_{i}}^{b_{i}}\partial f(x)dx)=-a_{i+1}$.
This means that from the end points $-a_{i}$ and $b_{i}$ at iteration
$i$, the next endpoints are indeed $-a_{i+1}$ and $b_{i}$ as claimed.
The case when $a_{i+1}=a_{i}$ and $b_{i+1}<b_{i}$ is similar.
\end{proof}
Even though Proposition \ref{prop:any-sequence} shows that the width
of the intervals can decrease at any rate in Algorithm \ref{alg:fn-bracketing},
the proof of case 2 in Theorem \ref{thm:lin-conv-best} shows that
$\{\|y_{i}\|\}_{i}$ converges quickly.
We give conditions such that the width of the intervals in Algorithm
\ref{alg:fn-bracketing} decreases at a linear rate.
\begin{thm}
\label{thm:lin-conv-1}(Bracketing in Algorithm \ref{alg:fn-bracketing})
Let $f:\mathbb{R}\to\mathbb{R}$ be a convex function such that $f(0)=0$
and $f(\cdot)$ is differentiable at $0$ with $f^{\prime}(0)=0$
in Algorithm \ref{alg:fn-bracketing}. Suppose further that $\partial f(\cdot)$
has left derivative $f_{-}^{\prime\prime}(0)$ and right derivative
$f_{+}^{\prime\prime}(0)$ which are formally defined as
\begin{equation}
f_{-}^{\prime\prime}(0):=\lim_{t\searrow0,v\in\partial f(-t)}\frac{1}{-t}[v-f^{\prime}(0)]\mbox{ and }f_{+}^{\prime\prime}(0):=\lim_{t\searrow0,v\in\partial f(t)}\frac{1}{t}[v-f^{\prime}(0)].\label{eq:LR-derv-subdiff}
\end{equation}
In view of the convexity of $f$ (i.e. monotonicity of $\partial f(\cdot)$),
we have $f_{-}^{\prime\prime}(0)\geq0$ and $f_{+}^{\prime\prime}(0)\geq0$.
Suppose $f_{-}^{\prime\prime}(0)>0$ and $f_{+}^{\prime\prime}(0)>0$.
Then the width of the interval $[-a_{i},b_{i}]$, easily seen to be
$a_{i}+b_{i}$, decreases at a linear rate.
\end{thm}
The formulas $f_{-}^{\prime\prime}(0)$ and $f_{+}^{\prime\prime}(0)$
are defined by \eqref{eq:LR-derv-subdiff} instead of
\[
f_{-}^{\prime\prime}(0)=\lim_{t\searrow0}\frac{1}{-t}[f^{\prime}(-t)-f^{\prime}(0)]\mbox{ and }f_{+}^{\prime\prime}(0):=\lim_{t\searrow0}\frac{1}{t}[f^{\prime}(t)-f^{\prime}(0)],
\]
because \eqref{eq:LR-derv-subdiff} does not require the differentiability
of $f(\cdot)$ in a neighborhood of $0$ and is more general. We now
prove Theorem \ref{thm:lin-conv-1}.
\begin{proof}
In view of the existence of the limits in \eqref{eq:LR-derv-subdiff},
for any constants $g_{l,l}$ and $g_{l,u}$ such that $0<g_{l,l}<f_{-}^{\prime\prime}(0)<g_{l,u}$,
we can find $\delta>0$ such that if $x\in[-\delta,0)$, then $\partial f(x)\subset[-|x|g_{l,u},-|x|g_{l,l}]$.
Similarly, for any constants $g_{r,l}$ and $g_{r,u}$ such that $0<g_{r,l}<f_{+}^{\prime\prime}(0)<g_{r,u}$,
we can reduce $\delta>0$ if necessary so that if $x\in(0,\delta]$,
then $\partial f(x)\subset[xg_{r,l},xg_{r,u}]$. We shall also assume
that
\begin{equation}
\frac{g_{l,u}}{g_{l,l}}<R\mbox{ and }\frac{g_{r,u}}{g_{r,l}}<R\mbox{, where }R\mbox{ can be made arbitrarily close to }1.\label{eq:ratio-R-formula}
\end{equation}
After one iteration, the interval $[-a_{i},b_{i}]$ becomes either
$[c_{i},b_{i}]$ or $[-a_{i},c_{i}]$, depending on the sign of $c_{i}$.
We now try to find upper and lower bounds on $c_{i}$. Suppose $a_{i}\in(0,\delta]$
and $b_{i}\in(0,\delta]$. Then the graph of $\partial f(\cdot)$
on $[-\delta,\delta]$ is bounded from below by the piecewise linear
function $h_{L}:[-\delta,\delta]\to\mathbb{R}$ and the from above
by the piecewise linear function $h_{U}:[-\delta,\delta]\to\mathbb{R}$
(See Figure \ref{fig:wedge}) defined respectively by
\[
h_{L}(x):=\begin{cases}
g_{l,u}x & \mbox{ if }x\in[-\delta,0]\\
g_{r,l}x & \mbox{ if }x\in[0,\delta]
\end{cases}\mbox{ and }h_{U}(x):=\begin{cases}
g_{l,l}x & \mbox{ if }x\in[-\delta,0]\\
g_{r,u}x & \mbox{ if }x\in[0,\delta].
\end{cases}
\]
\begin{figure}[!h]
\scalebox{0.5}[0.5]{\input{wedge.pstex_t}}\caption{\label{fig:wedge}Illustration of how $h_{L}(\cdot)$ and $h_{U}(\cdot)$
compare to $\partial f(\cdot)$. }
\end{figure}
We now estimate an upper bound on $c$. An upper bound on $\frac{1}{a_{i}+b_{i}}[f(b_{i})-f(-a_{i})]$,
which equals $\frac{1}{a_{i}+b_{i}}\int_{-a_{i}}^{b_{i}}f^{\prime}(x)dx$
since $f^{\prime}(\cdot)$ is integrable, is $v_{U}:=\frac{1}{a_{i}+b_{i}}\int_{-a_{i}}^{b_{i}}h_{U}(x)dx$.
Since $h_{L}(\cdot)\leq\partial f(\cdot)$, $c_{i}\leq h_{L}^{-1}(v_{U})$.
We now proceed to calculate these values.
We can calculate that $v_{U}:=\frac{1}{a_{i}+b_{i}}[-\frac{1}{2}g_{l,l}a_{i}^{2}+\frac{1}{2}g_{r,u}b_{i}^{2}]$.
We are interested in the upper bound of $c_{i}$ in the case when
the next interval is $[-a_{i},c_{i}]$, so we only consider the case
where $v_{U}>0$. In this case, $h_{L}^{-1}(v_{U})=\frac{1}{g_{r,l}[a_{i}+b_{i}]}[-\frac{1}{2}g_{l,l}a_{i}^{2}+\frac{1}{2}g_{r,u}b_{i}^{2}]$.
The width of the interval $[-a_{i},c_{i}]$ divided by the width of
$[a_{i},b_{i}]$ is estimated as follows.
\begin{eqnarray*}
\frac{a_{i}+c_{i}}{a_{i}+b_{i}} & \leq & \frac{a_{i}+h_{L}^{-1}(v_{U})}{a_{i}+b_{i}}\\
& = & \frac{2g_{r,l}[a_{i}+b_{i}]a_{i}+[-g_{l,l}a_{i}^{2}+g_{r,u}b_{i}^{2}]}{2g_{r,l}[a_{i}+b_{i}]^{2}}\\
& = & \frac{1}{2}+\frac{[g_{r,l}-g_{l,l}]a_{i}^{2}+[g_{r,u}-g_{r,l}]b_{i}^{2}}{2g_{r,l}[a_{i}+b_{i}]^{2}}\\
& \leq & \frac{1}{2}+\frac{[1-\frac{g_{l,l}}{g_{r,l}}]a_{i}^{2}+[R-1]b_{i}^{2}}{2[a_{i}+b_{i}]^{2}}\\
& \leq & \frac{1}{2}+\frac{1}{2}\underbrace{\left(\left[1-\frac{g_{l,l}}{g_{r,l}}\right]+[R-1]\right)}_{(*)},
\end{eqnarray*}
where the ratio $R$ is as defined in \eqref{eq:ratio-R-formula}.
The term $[R-1]$ can be made arbitrarily close to zero. The term
$\frac{g_{l,l}}{g_{r,l}}$ can be made arbitrarily close to $t:=f_{-}^{\prime\prime}(0)/f_{+}^{\prime\prime}(0)$.
In other words, $[1-\frac{g_{l,l}}{g_{r,l}}]$ can be arbitrarily
close to $[1-t]$. If $[1-t]<0$, then with proper choices of $g_{l,l}$,
$g_{r,l}$, $g_{l,u}$ and $g_{r,u}$, we can make $(*)$ negative,
in which case the ratio $\frac{a_{i}+c_{i}}{a_{i}+b_{i}}$ is less
than $1/2$. If $[1-t]\geq0$, we still have $[1-t]<1$, so with the
proper choice of constants, we can ensure that $\frac{a_{i}+c_{i}}{a_{i}+b_{i}}\leq\frac{3}{4}+\frac{1}{4}[1-t]$,
which still ensures that the reduction of the width of the intervals
is still linear.
The calculations for finding a lower bound on $c_{i}$ is similar.
The lower bound is of interest when the next interval is $[c_{i},b_{i}]$,
and that $c_{i}<0$. Thus $c_{i}<h_{U}^{-1}(v_{L})$, where $v_{L}:=\frac{1}{a_{i}+b_{i}}[-\frac{1}{2}g_{l,u}a_{i}^{2}+\frac{1}{2}g_{r,l}b_{i}^{2}]$.
So
\begin{eqnarray*}
\frac{-c_{i}+b_{i}}{a_{i}+b_{i}} & \leq & \frac{-h_{U}^{-1}(v_{L})+b_{i}}{a_{i}+b_{i}}\\
& = & \frac{1}{2}+\frac{[g_{l,u}-g_{l,l}]a_{i}^{2}+[g_{l,l}-g_{r,l}]b_{i}^{2}}{2g_{l,l}[a_{i}+b_{i}]^{2}}\\
& \leq & \frac{1}{2}+\frac{[R-1]a_{i}^{2}+[1-\frac{g_{r,l}}{g_{l,l}}]b_{i}^{2}}{2[a_{i}+b_{i}]^{2}}\\
& \leq & \frac{1}{2}+\frac{1}{2}\left([R-1]+\left[1-\frac{g_{r,l}}{g_{l,l}}\right]\right).
\end{eqnarray*}
Once again, the ratio is $\frac{g_{r,l}}{g_{l,l}}$ can be chosen
arbitrarily close to $1/t$, where $t$ was as defined earlier. If
$[1-\frac{1}{t}]<0$, we will have $\frac{-c_{i}+b_{i}}{a_{i}+b_{i}}<\frac{1}{2}$
eventually. If $[1-\frac{1}{t}]>0$, we still have $[1-\frac{1}{t}]<1$,
in which case we can ensure that $\frac{a_{i}+c_{i}}{a_{i}+b_{i}}\leq\frac{3}{4}+\frac{1}{4}[1-\frac{1}{t}]$.
No matter the case, we have a linear rate of convergence of the width
of the intervals to zero.\end{proof}
\begin{cor}
\label{cor:lin-conv-2}(Linear convergence of Algorithm \ref{alg:fn-bracketing})
With the additional assumptions in Theorem \ref{thm:lin-conv-1},
the iterates of Algorithm \ref{alg:fn-bracketing} are such that the
sequence
\begin{equation}
\big\{ d\big((0,0),\{(-a_{i},f(-a_{i})),(b_{i},f(b_{i}))\}\big)\big\}_{i},\label{eq:min-formula}
\end{equation}
where the distance in $\mathbb{R}^{2}$ is measured by the 2-norm,
is bounded above by a linearly convergence sequence. The corresponding
sequence $\{\|y_{i}\|\}_{i}$ in Algorithm \ref{alg:simple-enh-vN}
is bounded by a linearly convergent sequence. \end{cor}
\begin{proof}
By Theorem \ref{thm:lin-conv-1}, the width of the intervals $[a_{i},b_{i}]$
converges linearly to zero. Hence $\{\min(a_{i},b_{i})\}_{i}$ is
bounded by a linearly convergent sequence. Note that $f(\cdot)$,
being convex, is locally Lipschitz at $0$ with some constant $L$,
so $f(-a_{i})\leq La_{i}$ and $f(b_{i})\leq Lb_{i}$. We can thus
easily obtain the first conclusion.
The points $(0,0)$, $(-a_{i},f(-a_{i}))$ and $(b_{i},f(b_{i}))$
are points in the epigraph of $f(\cdot)$, and the formula in \eqref{eq:min-formula}
is an upper bound on the distance from $(0,0)$ to the line segment
connecting the points $(-a_{i},f(-a_{i}))$ and $(b_{i},f(b_{i}))$.
Hence the second statement is clear.
\end{proof}
The assumptions of Theorem \ref{thm:lin-conv-1} correspond to a second
order property on the boundary of $S$ at $0$. With added structure,
Algorithm \ref{alg:fn-bracketing} and \ref{alg:simple-enh-vN} can
converge faster. For example, if $S$ is polyhedral and Algorithm
\ref{alg:simple-enh-vN} chooses the extreme points, we have finite
convergence of $y_{i}$ to $0$ because there are only finitely many
extreme points for a polyhedron.
\begin{rem}
\label{rem:difficulties}(Difficulties in extending to $m>2$) For
much of this section and the last, we analyzed the case where $m=2$
in Algorithm \ref{alg:enhanced-vN}. We expect the extension to $m>2$
to be difficult, and the following are some of the reasons.
\begin{enumerate}
\item We made a connection to monotonicity of $\partial f(\cdot)$ here
and proved our results using single variable analysis. These need
to be extended to higher dimensions for $m>2$.
\item Proposition \ref{thm:bisec-enhanced-vN} cannot be easily extended
to the higher dimensional case. It is not necessarily true that for
the higher dimensional case, the projection will be on a face that
is of codimension 1.
\item For the 2 dimensional case, we see that $\mbox{\rm aff}(\{\bar{c}_{i},s_{i}\})\cap S$
is equal to $[\bar{c}_{i},s_{i}]$ if $0\notin[\bar{c}_{i},s_{i}]$.
One can see that if $S$ is a sphere in $\mathbb{R}^{3}$, for any
3 points $a$, $b$ and $c$ on $\partial S$, we do not have $\mbox{\rm aff}(\{a,b,c\})\cap S=\mbox{\rm conv}(\{a,b,c\})$.
\item The projection onto the convex hull of two points is easy, and we
can write down an analytic formula to help in our analysis. However,
it is difficult to write down such a formula for the projection onto
the convex hull of 3 or more points in higher dimensions, even if
this projection can be solved quite effectively using the methods
discussed earlier.
\end{enumerate}
\end{rem}
\section{Numerical experiments}
We perform some numerical experiments to show that Algorithm \ref{alg:enhanced-vN}
is more effective for some problem instances.
We generate our random matrices $A\in\mathbb{R}^{30\times80000}$
using the following code segment in Matlab:\texttt{
\begin{eqnarray}
& & \mbox{A=rand(30,80000)-ones(30,80000)*0.315;}\label{eq:generate-A}\\
& & \mbox{for i=1:80000}\nonumber \\
& & \quad\mbox{A(:,i)=A(:,i)/norm(A(:,i));}\nonumber \\
& & \mbox{end}\nonumber
\end{eqnarray}
}Through our experiments, we found that this choice of parameters
generate problem instances for which either \eqref{eq:von-Neumann-ineq}
is feasible (in which case the von Neumann algorithm cannot converge
finitely), or \eqref{eq:perceptron-ineq} is feasible but the von
Neumann algorithm typically takes many iterations, sometimes more
than 2000 iterations, before it terminates.
\subsection{Numerical experiment 1: Comparison against von Neumann algorithm
when $A^{T}y>0$ feasible}
We ran experiments for 491 different matrices $A\in\mathbb{R}^{30\times80000}$
generated by \eqref{eq:generate-A} such that \eqref{eq:perceptron-ineq}
holds (i.e., $0$ does not lie in the convex hull of the elements
generated by the columns of $A$). We calculated the number of iterations
needed for the von Neumann algorithm to find a $y$ satisfying \eqref{eq:perceptron-ineq},
and for Algorithm \ref{alg:enhanced-vN} with various limits on the
size of the active set (See Subsection \ref{sub:agg_strat}) to do
the same. The aggregation strategy is the one in Remark \ref{rem:agg_strat_2},
where we aggregate the oldest element(s) that have not been aggregated.
We set a limit of 2000 for the number of iterations.
We first look at the results obtained from the conducting experiments
on 491 different matrices $A\in\mathbb{R}^{30\times80000}$. We look
at Table \ref{tab:Alg-QP-vs-vNm} for a comparison of the number of
iterations needed by Algorithm \ref{alg:enhanced-vN} to find a $y$
such that $A^{T}y>0$ versus the number of iterations needed by the
von Neumann algorithm. We shall use the following convention in our
diagrams and tables in this section:
\begin{defn}
\label{def:A-N-notation-numerical}Let $A_{N}$ denote Algorithm \ref{alg:enhanced-vN}
where the size of the set $C_{i}$ is bounded above by $N-1$ after
line 12 is performed. In other words, we aggregate according to Remark
\ref{rem:agg_strat_2} when the size of the set $C_{i}$ equals $N$.
\end{defn}
\begin{table}[!h]
\begin{tabular}{|c|r|r|r|r|r|r|r|r|}
\hline
\multicolumn{9}{|c|}{Comparing iteration counts of Algorithm \ref{alg:enhanced-vN} against
von Neumann Algorithm}\tabularnewline
\hline
$N$ & \multicolumn{2}{r|}{$A_{2}>A_{N}$} & \multicolumn{2}{r|}{$A_{2}<A_{N}$} & \multicolumn{2}{r|}{$A_{2}=A_{N}\leq2000$} & \multicolumn{2}{r|}{$A_{2}>2000$ and $A_{N}>2000$ }\tabularnewline
\hline
\hline
5 & 63 & 12.9\% & 260 & 53.0\% & 0 & 0\% & 168 & 34.2\%\tabularnewline
\hline
10 & 170 & 34.6\% & 152 & 31.0\% & 4 & 0.8\% & 165 & 33.6\%\tabularnewline
\hline
15 & 342 & 69.7\% & 18 & 3.7\% & 2 & 0.4\% & 129 & 26.3\%\tabularnewline
\hline
20 & 409 & 83.3\% & 1 & 0.2\% & 0 & 0\% & 81 & 16.5\%\tabularnewline
\hline
25 & 453 & 92.3\% & 0 & 0\% & 0 & 0\% & 38 & 7.7\%\tabularnewline
\hline
31 & 491 & 100\% & 0 & 0\% & 0 & 0\% & 0 & 0\%\tabularnewline
\hline
\end{tabular}\caption{\label{tab:Alg-QP-vs-vNm}Refer to the definition of $A_{N}$ in Definition
\ref{def:A-N-notation-numerical}. This table compares the number
of times in 491 experiments where the number of iterations to find
a $y$ s.t. $A^{T}y>0$ for the von Neumann algorithm (or $A_{2}$)
uses is greater than/ less than/ equal to that of $A_{N}$. The last
column represents the number of times both $A_{2}$ and $A_{N}$ reach
their limit of 2001 iterations.}
\end{table}
Recall that $A_{2}$ refers to the von Neumann algorithm (see Remark
\ref{rem:vN-particular-case}). It can be seen that the von Neumann
Algorithm has consistently used fewer iterations than $A_{5}$, and
it is quite competitive with $A_{10}$. As we increase the maximum
size of the active set, the number of iterations needed gets better
compared to the von Neumann algorithm $A_{2}$.
\begin{figure}[!h]
\includegraphics[scale=0.5]{analyze_data}\includegraphics[scale=0.5]{manage_data}
\caption{\label{fig:exp-histogram}Number of iterations needed for Algorithm
\ref{alg:enhanced-vN} with various parameters to find a $y$ such
that $A^{T}y>0$. In the graph on the right, each vertical line corresponds
to a particular experiment. In the diagram on the left, the number
of iterations needed for all experiments are obtained and sorted.}
\end{figure}
We explain the diagrams in Figure \ref{fig:exp-histogram}, and we
first look at the diagram on the left. In our 491 experiments where
there is a $y$ such that $A^{T}y>0$, we found that overall, the
von Neumann algorithm $A_{2}$ uses fewer iterations to find the $y$
such that $A^{T}y>0$ than $A_{5}$. As we increase the tolerance
of the size of the set $C_{i}$ before we aggregate, the number of
iterations needed to find this $y$ decreases.
In the diagram on the right of Figure \ref{fig:exp-histogram}, we
sort the experiments so that each vertical line corresponds to a particular
experiment. We see that the von Neumann algorithm usually takes more
iterations than Algorithm \ref{alg:enhanced-vN} than $A_{15}$, though
we notice a few rare instances of when $A_{15}$ takes more iterations
than the von Neumann algorithm. We observe the general pattern that
the larger the tolerance before aggregating $C_{i}$, the fewer iterations
it takes for Algorithm \ref{alg:enhanced-vN}. In fact, when there
is no aggregation, Algorithm \ref{alg:enhanced-vN} takes less than
80 iterations to decide whether \eqref{eq:perceptron-ineq} or \eqref{eq:von-Neumann-ineq}
is feasible.
We now look at a particular anomalous experiment, and explain diagrams
in Figure \ref{fig:exp-data-vNm-QP}. For this particular experiment,
we plot the norm $\|y_{i}\|$ with respect to the iteration $i$.
This example is unusual because the von Neumann algorithm takes fewer
iterations than $A_{20}$. The plots are drawn for each iteration
till we have found $y_{i}$ such that $A^{T}y_{i}>0$. Even though
the von Neumann algorithm takes fewer iterations to get a $y_{i}$
such that $A^{T}y_{i}>0$, the norms of $\|y_{i}\|$ decrease much
slower than all versions of Algorithm \ref{alg:enhanced-vN}.
\begin{figure}[!h]
\includegraphics[scale=0.5]{normplot01}
\includegraphics[scale=0.5]{Adataplot01}\includegraphics[scale=0.5]{Adataplot02}
\caption{\label{fig:exp-data-vNm-QP}The diagrams show an anomalous example
where the von Neumann algorithm takes fewer iterations to find a $y$
such that $A^{T}y>0$ than $A_{20}$.}
\end{figure}
We now explain the bottom diagrams in Figure \ref{fig:exp-data-vNm-QP}.
At each iteration $i$, we calculate
\[
q_{1,i}:=\max_{j\in\{1,\dots,n\}}-a_{j}^{T}y_{i},
\]
just like in solving \eqref{eq:new-p-i-1-formula}. If this quantity
is negative, then $A^{T}y_{i}>0$, and we end. The bottom left diagram
shows that, other than a general downward trend, there is no clear
pattern in the dependence of $q_{1,i}$ on $i$. In the bottom right
diagram, we calculate
\[
q_{2,i}:=\min_{k\in\{1,\dots,i\}}[\max_{j\in\{1,\dots,n\}}-a_{j}^{T}y_{i}].
\]
We observe that in general, the larger the limit the size of the active
set, the faster $q_{2,i}$ (and hence $q_{1,i}$) decreases.
\subsection{A note on number of iterations and time}
We have only discussed the performance of the algorithms we test in
terms of iteration counts instead of the time taken. For our experiments
so far, we plot the time taken per iteration versus the number of
iterations for our implementation of Algorithm \ref{alg:enhanced-vN}
as well as our implementation of the von Neumann algorithm in Matlab.
These are shown in Figure \ref{fig:time-per-iter}. The time taken
per iteration for Algorithm \ref{alg:enhanced-vN} is seen to be between
0.0255 seconds to 0.0290 seconds regardless of the size of $C_{i}$.
The time taken per iteration for the von Neumann algorithm is seen
to be between 0.0043 seconds to 0.0047 seconds. Since the running
time of the algorithms and the iteration numbers differ only up to
a constant factor that is implementation dependent, we shall analyze
our algorithms only in terms of the number of iterations. Moreover,
it may be possible to improve this ratio in favor of Algorithm \ref{alg:enhanced-vN}
if the accelerations in Remark \ref{rem:accelerate-primal-QP} are
carried out, especially when $m$ is large.
\begin{figure}[!h]
\includegraphics[scale=0.5]{time_per_iter_evN}\includegraphics[scale=0.5]{time_per_iter_vN}
\caption{\label{fig:time-per-iter}The diagram on the left shows the time taken
per iteration for $A_{5}$, $A_{10}$, $A_{15}$, $A_{20}$, $A_{25}$
and $A_{31}$. It can be seen that the size of the set $C_{i}$ does
not affect the time per iteration. The diagram on the right shows
the time taken per iteration for our implementation of the von Neumann
algorithm. }
\end{figure}
\subsection{\label{sub:agg_strats}Numerical experiment 2: Aggregation strategies}
Table \ref{tab:test_high_agg} below compares the running times of
353 experiments for the case when \eqref{eq:perceptron-ineq} is feasible.
Rows 1-3 look at the number of iterations it takes to find a certificate
vector $y$, while rows 4-6 look at the norms $\|Ax_{i}\|=\|y_{i}\|$
of the iterates at the $80$th iteration for the various aggregation
methods. For the test on the norms $\|Ax_{i}\|$, the undecided column
denotes the number of times at least one algorithm has found a $y$
such that $A^{T}y>0$ in $80$ iterations. The experiments suggest
that the best aggregation method is to aggregate the oldest elements
that have not been aggregated. There might be other factors that we
have not identified which determine the performance of an aggregation
strategy. There could also be better aggregation strategies other
than the ones we have tried.
\begin{table}[h]
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
& 353 runs for \eqref{eq:perceptron-ineq} feasible & \multicolumn{5}{c|}{Best aggregation method}\tabularnewline
\hline
& & (1) & (2) & (3) & Ties & Undecided\tabularnewline
\hline
\hline
1 & No. iters for $A_{25}$ to find $y$ solving \eqref{eq:perceptron-ineq} & 178 & 39 & 102 & 34 & 0\tabularnewline
\hline
2 & No. iters for $A_{20}$ to find $y$ solving \eqref{eq:perceptron-ineq} & 173 & 11 & 110 & 59 & 0\tabularnewline
\hline
3 & No. iters for $A_{15}$ to find $y$ solving \eqref{eq:perceptron-ineq} & 155 & 10 & 101 & 87 & 0\tabularnewline
\hline
4 & $\|Ax_{i}\|$ at $i=80$ for $A_{25}$ & 62 & 70 & 90 & 0 & 131\tabularnewline
\hline
5 & $\|Ax_{i}\|$ at $i=80$ for $A_{20}$ & 142 & 67 & 96 & 0 & 48\tabularnewline
\hline
6 & $\|Ax_{i}\|$ at $i=80$ for $A_{15}$ & 240 & 37 & 64 & 0 & 12\tabularnewline
\hline
\hline
& 126 runs for \eqref{eq:von-Neumann-ineq} feasible & & & & & \tabularnewline
\hline
\hline
7 & $\|Ax_{i}\|$ at $i=400$ for $A_{15}$ & 114 & 10 & 0 & 0 & 2\tabularnewline
\hline
8 & $\|Ax_{i}\|$ at $i=400$ for $A_{10}$ & 121 & 5 & 0 & 0 & 0\tabularnewline
\hline
9 & $\|Ax_{i}\|$ at $i=400$ for $A_{5}$ & 123 & 2 & 1 & 0 & 0\tabularnewline
\hline
\end{tabular}
\caption{\label{tab:test_high_agg}Experiments on which aggregation strategy
is best among those presented in Remark \ref{rem:agg_strat_2}. The
strategies in the three columns are: (1) oldest aggregated, (2) lowest
coefficient aggregated, and (3) highest coefficient aggregated, as
according to the description in Remark \ref{rem:agg_strat_2}, which
relies on Remark \ref{rem:agg_strat}. For rows 1-6, the 353 experiments
are for when \eqref{eq:perceptron-ineq} is feasible. For rows 7-9,
the 126 experiments are for when \eqref{eq:von-Neumann-ineq} is feasible.
We refer to Subsection \ref{sub:agg_strats} for more details.}
\end{table}
Table \ref{tab:test_high_agg} also compares the running times of
126 experiments for which \eqref{eq:von-Neumann-ineq} is feasible.
When \eqref{eq:von-Neumann-ineq} is feasible, we want to find iterates
$x$ such that $\|Ax\|$ is small. To evaluate the performance of
the aggregation strategies, we look at how the norms of the values
$\|Ax_{i}\|$ vary with the iteration count $i$ for different strategies.
For the test on the norms $\|Ax_{i}\|$, the undecided column denotes
the number of times numerical errors resulting from $\|Ax_{i}\|$
were encountered for at least one algorithm in $400$ iterations.
It is quite clear that the strategy of aggregating the oldest point
obtained is the best strategy among our experiments.
\begin{figure}
\includegraphics[scale=0.5]{vN_run}\caption{Plot of an experiment for which \eqref{eq:von-Neumann-ineq} is feasible.
The black dotted plot is for $A_{2}$ (the von Neumann algorithm).
The red plots are for the three versions of aggregation strategies
in Remark \ref{rem:agg_strat_2} for $A_{5}$. The blue and magenta
plots are for $A_{10}$ and $A_{15}$ respectively. }
\end{figure}
\section{Conclusion}
We introduced an improvement of the distance reduction step in the
generalized von Neumann algorithm in Algorithm \ref{alg:enhanced-vN}
by projecting onto the convex hull of a set of points $C_{i}$ using
a primal active set QP algorithm. The size of $C_{i}$, $|C_{i}|$,
can be chosen to be as large as possible, as long as each iteration
is manageable. If $|C_{i}|$ is increased, the cost of each iteration
increases, but we expect better iterates when \eqref{eq:von-Neumann-ineq-1}
is feasible. This is verified by our numerical experiments. When \eqref{eq:perceptron-ineq-1}
is feasible, we can find a solution to \eqref{eq:perceptron-ineq-1}
if $|C_{i}|$ is relatively large. But interestingly, if $|C_{i}|$
is small but bigger than $2$, the performance can be poorer than
von Neumann's algorithm on average.
On the theoretical side, Theorem \ref{thm:fin-conv-1} studies the
behavior of Algorithm \ref{alg:enhanced-vN} when $0\in\mbox{\rm int}(S)$
and the rest of Sections \ref{sec:basic-analysis} and \ref{sec:More-analysis}
study the behavior of Algorithm \ref{alg:enhanced-vN} when $0\in\partial S$
and $A\in\mathbb{R}^{2\times n}$. A natural follow up question that
better models how Algorithm \ref{alg:enhanced-vN} can be used in
practice is to study what happens when $|C_{i}|$ is of moderate size.
It appears hard to prove such results because there is no easy formula
for the projection onto $\mbox{\rm conv}(C_{i})$ when $|C_{i}|>2$. Remark
\ref{rem:difficulties} also shows the difficulties for extending
our results to the case when $0\in\partial S$ and $A\in\mathbb{R}^{m\times n}$
for $m>2$. Nevertheless, the results here can give an idea of what
can be expected to be true in higher dimensions.
\begin{acknowledgement*}
We thank Marina Epelman for organizing Freundfest honoring Robert
M. Freund's 60th birthday, where Javier Pe\~{n}a talked on how Rob
Freund's contributions in the perceptron and von Neumann algorithms
influenced his recent work. We also thank Javier Pe\~{n}a for further
conversations.
\end{acknowledgement*}
\bibliographystyle{amsalpha}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,428 |
Q: Getting selectedrow in RowCommand i have a gridview as:
<asp:GridView ID="gdvOpinions" runat="server" Width="100%" CellPadding="4"
ForeColor="#333333" GridLines="None" Height="100px" Visible="False"
AutoGenerateColumns="False" onrowcommand="gdvOpinions_RowCommand" >
<AlternatingRowStyle BackColor="White" ForeColor="#284775" />
<Columns>
<asp:ButtonField ButtonType="Button" CommandName="Confirm" Text="تائید" />
<asp:ButtonField ButtonType="Button" CommandName="Delete" Text="حذف" />
<asp:BoundField DataField="ID" HeaderText="ID" />
<asp:BoundField DataField="مقاله" HeaderText="مقاله" >
<ControlStyle Width="150px" />
</asp:BoundField>
<asp:BoundField DataField="نظر کاربر" HeaderText="نظر کاربر" >
<ControlStyle Width="250px" />
</asp:BoundField>
</Columns>
<EditRowStyle BackColor="#999999" />
<FooterStyle BackColor="#5D7B9D" Font-Bold="True" ForeColor="White" />
<HeaderStyle BackColor="#5D7B9D" Font-Bold="True" ForeColor="White" />
<PagerStyle BackColor="#284775" ForeColor="White" HorizontalAlign="Center" />
<RowStyle BackColor="#F7F6F3" ForeColor="#333333" />
<SelectedRowStyle BackColor="#E2DED6" Font-Bold="True" ForeColor="#333333" />
<SortedAscendingCellStyle BackColor="#E9E7E2" />
<SortedAscendingHeaderStyle BackColor="#506C8C" />
<SortedDescendingCellStyle BackColor="#FFFDF8" />
<SortedDescendingHeaderStyle BackColor="#6F8DAE" />
</asp:GridView>
As you see, i have two buttonfield for every row, but when i click one of these buttons and want to get their row in
gdvOpinions_RowCommand
the
gdvOpinions.SelectedRow
returns null;
how should i find which row is selected?
A: <asp:GridView ID="GridView1" runat="server" AutoGenerateColumns = "false"
OnRowCommand = "OnRowCommand">
<Columns>
<asp:ButtonField CommandName = "ButtonField" DataTextField = "CustomerID"
ButtonType = "Button"/>
</Columns>
</asp:GridView>
protected void OnRowCommand(object sender, GridViewCommandEventArgs e)
{
int index = Convert.ToInt32(e.CommandArgument);
GridViewRow gvRow = GridView1.Rows[index];
}
A: From what I understand out of your question, if you are trying to get the row for which the button was clicked, then in that case this might be the right thing to do :
GridViewRow gRow = (GridViewRow)((Control)e.CommandSource).NamingContainer;
Now here, this is for a control such as a Button or ImageButton. You might have to improvise on this for ButtonField.
You can also change the ButtonField to a simple TemplateField with a Button.
Hope this helps.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,807 |
{"url":"https:\/\/gosamples.dev\/case-insensitive-string-comparison\/","text":"## \ud83d\ude4c Case-insensitive string comparison in Go\n\n##### May 25, 2021\nintroduction strings\n\nIs there a case-insensitive string comparison function in Go? Of course! Although the name does not seem to indicate it, strings.EqualFold deals with it:\n\npackage main\n\nimport (\n\"fmt\"\n\"strings\"\n)\n\nfunc main() {\nfoo1 := \"foo\"\nfoo2 := \"FOO\"\nfmt.Println(strings.EqualFold(foo1, foo2))\n}\n\n\nYou may ask now why we can\u2019t convert both strings to upper or lowercase and, in this way, compare if they are case-insensitive equal. Of course, it works, but not for any case and any language. For example, in Greek, there are 3 forms of sigma letter:\n\ng1 := \"\u03c2\" \/\/ a final lowercase sigma\ng2 := \"\u03a3\" \/\/ a capital sigma\ng3 := \"\u03c3\" \/\/ a non-final sigma\n\nfmt.Println(strings.ToLower(g1))\nfmt.Println(strings.ToLower(g2))\nfmt.Println(strings.ToLower(g3))\n\nfmt.Println(strings.EqualFold(g1, g2))\nfmt.Println(strings.EqualFold(g1, g3))\nfmt.Println(strings.EqualFold(g2, g3))\n\n\nOutput:\n\n\u03c2\n\u03c3\n\u03c3\ntrue\ntrue\ntrue\n\nConverting them to lowercase doesn\u2019t give the same form, but a comparison using strings.EqualFold informs that they are equal. This is because strings.EqualFold uses case folding (now it\u2019s clear why the function is named EqualFold) method which respects the case rules of different languages, so it always should be preffered method of case-insensitive comparison.\n\n#### \u2712\ufe0f Write to a CSV file in Go\n\n##### Learn how to write data to a CSV or TSV file\nintroduction file\n\n#### \ud83c\udfc1 Check if a string starts with a substring in Go\n\n##### Learn how to use strings.HasPrefix() function\nintroduction strings\n\n#### \u270d\ufe0f Write to a file in Go\n\n##### Learn how to write any data to a file\nintroduction file","date":"2021-06-14 14:05:44","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.26829805970191956, \"perplexity\": 5980.148086850694}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623487612537.23\/warc\/CC-MAIN-20210614135913-20210614165913-00492.warc.gz\"}"} | null | null |
/*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.gbif.occurrence.hive.udf;
import org.apache.hadoop.io.Text;
import org.junit.jupiter.api.Test;
import static org.junit.jupiter.api.Assertions.assertEquals;
import static org.junit.jupiter.api.Assertions.assertNull;
public class ToLocalISO8601UDFTest {
@Test
public void testLocalDateFormatter() {
ToLocalISO8601UDF function = new ToLocalISO8601UDF();
assertNull(function.evaluate(new Text()));
assertNull(function.evaluate(new Text("")));
assertEquals("2020-04-02T16:48:54", function.evaluate(new Text("1585846134000")).toString());
assertEquals("2020-04-02T16:48:54.001", function.evaluate(new Text("1585846134001")).toString());
}
@Test
public void testUtcDateFormatter() {
ToISO8601UDF function = new ToISO8601UDF();
assertNull(function.evaluate(new Text()));
assertNull(function.evaluate(new Text("")));
assertEquals("2020-04-02T16:48:54Z", function.evaluate(new Text("1585846134000")).toString());
assertEquals("2020-04-02T16:48:54.001Z", function.evaluate(new Text("1585846134001")).toString());
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,199 |
Tillandsia 'Kacey' es un cultivar híbrido del género Tillandsia perteneciente a la familia Bromeliaceae.
Es un híbrido creado con las especies Tillandsia bulbosa × Tillandsia butzii.
Referencias
BSI BCR Entry for 'Kacey'
Híbridos de Tillandsia | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,575 |
If you're not growing, you're shrinking.
I am a firm believer in this truth, and it's part of what fuels me every day to learn as much as possible.
This morning, as I listened to Malcom Gladwell's "Outliers"–the supremely talented author reads the book himself–I was struck by his mention of the three elements that add up to work fulfillment.
They are autonomy, complexity, and a relationship between work and reward.
The second piece of that equation, complexity, is what I enjoy most about the ongoing assignment I have with the Chicago chapter of the Urban Land Institute.
Since September, I have had five opportunities to tackle what is (for me, anyway) the complex task of distilling more than an hour's worth of ULI discussion into a cohesive 1,000- to 1,200-word summary. You can read my reports, including the one from last Thursday's session, here.
Having covered government bodies for the better part of two decades, I have a finely tuned ear to what a variety of people say, and how it all fits into the larger web of all that is said. But because ULI talks encompass topics with which I have little familiarity, the process presents an entirely new challenge.
Although ULI gives me two days to file a summary, I usually file my report within six hours. The rapid turn-around isn't as noble as it may seem–I am eager to flesh out the session in writing before the impression of what I just learned fades to gray.
This entry was posted in Uncategorized and tagged complexity, growth, Malcolm Gladwell, Outliers, Urban Land Institute by Matt Baron. Bookmark the permalink. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,149 |
Q: XMLHttpRequest for sending files is not sending the data to the server I want to upload files from a angularjs 2 app to my python server application.
the formData look perfect before sending it to the server. Containing name and file data.
On the other end is a Python API waiting to accept the file data. But i don't get the data. Debugger PDB get activated as the send is done by angularjs 2 app but i am missing the file data.
What could i be doing wrong?
upload.service.ts
import {Injectable} from 'angular2/core';
import {Http, Headers} from 'angular2/http';
@Injectable()
export class UploadService {
constructor(private http: Http) { }
uploader(files: any) {
let xhr = new XMLHttpRequest();
const formData = new FormData();
for(var i = 0; i < files.length; i++){
formData.append(files[i].name, files[i]);
}
var headers = new Headers();
headers.append('Content-Type', 'multipart/form-data');
xhr.onreadystatechange = function () {
if (xhr.readyState === 4) {
if (xhr.status === 200) {
resolve(JSON.parse(xhr.response))
} else {
reject(xhr.response)
}
}
}
xhr.open('POST','http://192.168.1.205:5000/zino', true)
xhr.send(formData)
}
}
python flask API debugger
As you see, the request.values are empty :(
<Request 'http://192.168.1.205:5000/zino' [POST]>
(Pdb) request.values
CombinedMultiDict([ImmutableMultiDict([]), ImmutableMultiDict([])])
A: This is a working example for RC angular2
import {Injectable} from '@angular/core';
import {Http, Headers, RequestOptions} from '@angular/http';
@Injectable()
export class UploadService {
constructor(private http: Http) { }
uploader(files: any) {
let xhr = new XMLHttpRequest()
const formData = new FormData()
for(var i = 0; i < files.length; i++){
formData.append(files[i].name, files[i])
}
xhr.onreadystatechange = function () {
if (xhr.readyState === 4) {
if (xhr.status === 200) {
//resolve(JSON.parse(xhr.response))
} else {
//reject(xhr.response)
}
}
}
xhr.open('POST','/server/upload-api', true)
xhr.setRequestHeader('X-Requested-With', 'XMLHttpRequest')
xhr.send(formData)
}
}
On the python side you find the files in
(Pdb) req.httprequest.files
ImmutableDict({'IMGP0269.jpg': <FileStorage: u'IMGP0269.jpg' ('image/jpeg')>, 'IMGP0270.jpg': <FileStorage: u'IMGP0270.jpg' ('image/jpeg')>})
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,523 |
|}
|}
De N457 is een gewestweg in België tussen Leupegem (N60) Schorisse (N454).
De weg heeft een lengte van ongeveer 6 kilometer. De gehele weg bestaat uit twee rijstroken voor beide rijrichtingen samen.
Sinds 2018 wordt de weg heraangelegd met fietspaden.
Plaatsen langs de N457
Etikhove
Maarke-Kerkem
Schorisse
457
Weg in Oost-Vlaanderen | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 381 |
\section*{Introduction}
A (deformation) quantization of a symplectic manifold is a noncommutative deformation of the structure sheaf which is, in a certain sense, compatible with
the symplectic structure.
We know that every symplectic manifold admits quantizations (cf. ~\cite{DWL1}; ~\cite{DWL2}; ~\cite{Fe}), but in general
the quantization is neither unique nor canonically constructed.
Therefore, construction of quantizations will be one of the important subjects in this theory.
In \cite{BZBi},
D. Ben-Zvi and I. Biswas provided a canonical construction
of a quantization on (the total space of) the cotangent bundle minus the zero section on a given Riemann surface $C$ equipped with a projective structure.
Here, recall that a projective structure on $C$ is an atlas of coordinate charts defining $C$ whose transition functions may be expressed as Mobi\"{u}s transformations.
Projective structures
have a very major role to play in understanding the framework of uniformization theorem of Riemann surfaces
and have mutually equivalent objects, including $\mathrm{PGL}_2$-opers and projective connections, etc..
Every Riemann surface $C$ admits a projective structure, and the space of all projective structures on $C$ forms an affine space for the space $H^0 (C, \Omega_{C}^{\otimes 2})$ of quadratic differentials on $C$.
The idea behind the construction of D. Ben-Zvi and I. Biswas
is that the canonical construction of a quantization on the complex projective line $\mathbb{P}^1_\mathbb{C}$, which is invariant under the action of $\mathrm{PGL}_2(\mathbb{C})$ (= the group of Mobi\"{u}s transformations on $\mathbb{P}^1_\mathbb{C}$), may extend naturally to any Riemann surface once we choose a projective structure.
On algebraic curves in positive characteristic, there are analogous objects of projective structures, called {\it Frobenius-projective structures}.
The notion of a Frobenius-projective structure was
introduced by Y. Hoshi (cf. ~\cite{Hos}, \S\,2, Definition 2.1) as a certain collection of locally defined \'{e}tale maps on a prescribed curve to the projective line.
Just as in the complex case, any smooth curve in positive characteristic admits such a structure.
Also, Frobenius-projective structures are equivalent to, e.g., {\it dormant} $\mathrm{PGL}_2$-opers and projective connections {\it having a full set of solutions}.
That is to say, given a connected smooth curve $X$ in characteristic $p>2$ and a theta characteristic $\mathbb{L} := (\mathcal{L}, \psi_\mathcal{L} : \mathcal{L}^{\otimes 2} \stackrel{\sim}{\migi} \Omega_X)$ (cf. \S\,\ref{S8}), we obtain
the following diagram consisting of bijective correspondences in parallel with
the classical result on Riemann surfaces:
\begin{align}
\vcenter{\xymatrix@C=16pt@R=16pt{
& \mathfrak{P} \mathfrak{S}_X^\mathrm{F} \ar[ldd] \ar[rdd]&
\\
& & \\
\mathfrak{O} \mathfrak{p}^{^\mathrm{Zzz...}}_{\mathrm{PGL}_2, X} \ar[rr]_{\sim} \ar[uur]^{\rotatebox{40}{{\tiny $\sim$}}}& & \mathfrak{P} \mathfrak{C}^{2, \mathrm{full}}_{X, \mathbb{L}}, \ar[uul]_{\rotatebox{-40}{{\tiny $\sim$}}} \ar[ll]
}}
\end{align}
where
\begin{itemize}
\item[] \hspace{-3mm} $\mathfrak{P} \mathfrak{S}_X^\mathrm{F}$ := the set of Frobenius-projective structures on $X$ (cf. (\ref{Efllo2}));
\vspace{1mm}
\item[] \hspace{-3mm} $\mathfrak{O} \mathfrak{p}^{^\mathrm{Zzz...}}_{\mathrm{PGL}_2, X}$ := the set of isomorphism classes of dormant $\mathrm{PGL}_2$-opers on $X$ (cf. (\ref{E69207}));
\vspace{1mm}
\item[] \hspace{-3mm} $\mathfrak{P} \mathfrak{C}^{2, \mathrm{full}}_{X, \mathbb{L}}$ := the set of projective connections for $\mathbb{L}$
having a full set of solutions (cf. (\ref{E41180})).
\end{itemize}
(We also discuss, in the present paper, certain intermidiate objects equivalent to them, called {\it dormant $(\mathrm{SL}_2, \mathbb{L})$-opers}.)
The purpose of the present paper is to prove an analogous assertion of D. Ben-Zvi and I. Biswas, i.e., a canonical construction of a quantization by means of the curve $X$ together with a choice among such additional structures.
In ~\cite{BK1}, ~\cite{Kon}, and ~\cite{Y}, it has been shown that the theory of quantizations can be made to work in the algebraic setting.
Also, we can find, in ~\cite{BeKa1} (and ~\cite{BeKa2}), the study of a special class of quantizations on symplectic algebraic varieties in positive characteristic, called {\it Frobenius-constant quantizations}.
They are quantizations with large center in some suitable sense, and has a cohomological classification given in the point of view of formal geometry.
Let $\mathbb{A} (\Omega_X)^\times$ denote the complement of the zero section in (the total space of) the cotangent bundle of $X$; it admits
a symplectic structure $\check{\omega}^\mathrm{can}$ defined as one half of the Liouville symplectic form.
Thus, it makes sense to speak of a (Frobenius-constant) quantization on the symplectic variety $(\mathbb{A} (\Omega_X)^\times, \check{\omega}^\mathrm{can})$.
Denote (cf. (\ref{Er33})) by
\begin{align}
\mathfrak{Q}^{\mathrm{FC}}_{(\mathbb{A} (\Omega_X)^\times, \check{\omega}^\mathrm{can})}
\end{align}
the set of Frobenius-constant quantizations on
$(\mathbb{A} (\Omega_X)^\times, \check{\omega}^\mathrm{can})$.
Then, the main result of the present paper (cf. Theorem \ref{T0090}) provides a canonical {\it injective} assignment from
a Frobenius-projective structure (or equivalently, a dormant indigenous bundle, or a projective connection having a full set of solutions) to a Frobenius-constant quantization on $(\mathbb{A} (\Omega_X), \check{\omega}^\mathrm{can})$, as displayed below:
\begin{align}
\hspace{-5mm}\vcenter{\xymatrix@C=16pt@R=16pt{
& \mathfrak{P} \mathfrak{S}_X^\mathrm{F} \ar[ldd] \ar[rdd]&
\\
& & \\
\mathfrak{O} \mathfrak{p}^{^\mathrm{Zzz...}}_{\mathrm{PGL}_2, X} \ar[rr]_{\sim} \ar[uur]^{\rotatebox{40}{{\tiny $\sim$}}}& & \mathfrak{P} \mathfrak{C}^{2, \mathrm{full}}_{X, \mathbb{L}}, \ar[uul]_{\rotatebox{-40}{{\tiny $\sim$}}} \ar[ll]
}}
\hspace{15mm} \scalebox{2}{$\rightsquigarrow$} \hspace{15mm} \mathfrak{Q}^\mathrm{FC}_{(\mathbb{A}(\Omega_X)^\times, \check{\omega}^\mathrm{can})}.
\end{align}
In particular, we can think of $\mathfrak{P} \mathfrak{S}^\mathrm{F}_X$ as a subset of $\mathfrak{Q}^\mathrm{FC}_{(\mathbb{A}(\Omega_X)^\times, \check{\omega}^\mathrm{can})}$ via this assignment, and hence, give a lower bound of the number of Frobenius-constant quantizations on $(\mathbb{A} (\Omega_X)^\times, \check{\omega}^\mathrm{can})$ by applying the result in ~\cite{Wak}.
The present paper is organized as follows.
The first section contains the necessary definitions
and conventions used in our discussion, including a symplectic structure, a differential operator, and a theta characteristic.
In the second section, we recall the notion of a Frobenius-constant quantization and discuss some related topics.
For instance, it is observed that Frobenius-constant quantizations are functorial with respect to pull-back via \'{e}tale morphisms (cf. \S\,\ref{S10}) and have a descent property via finite Galois coverings (cf. \S\,\ref{S41}).
In the third section, we discuss various bijective correspondences between
Frobenius-projective structures on a curve and some equivalent objects, i.e., dormant $\mathrm{PGL}_2$-opers, dormant $(\mathrm{SL}_2, \mathbb{L})$-opers, and projective connections with a full set of solutions.
Some results mentioned in that section have been essentially obtained in other literatures, e.g., ~\cite{Hos} (which gives $\mathfrak{P} \mathfrak{S}^\mathrm{F}_X \stackrel{\sim}{\migi} \mathfrak{O} \mathfrak{p}^{^\mathrm{Zzz...}}_{\mathrm{PGL}_2, X}$) and ~\cite{BD2} together with ~\cite{Kat2}
(which gives $\mathfrak{O} \mathfrak{p}^{^\mathrm{Zzz...}}_{(\mathrm{SL}_n, \mathbb{L}), X}\stackrel{\sim}{\migi} \mathfrak{P} \mathfrak{C}^{n, \mathrm{full}}_{X, \mathbb{L}}$).
Also, in ~\cite{Wak2}, we can find generalizations of these correspondences to a family of pointed stable curves.
But, unfortunately, many of them seem not to be standard and are unavoidable when we complete the proof of the main theorem, so we decided to review them here and the contents became nearly self-contained.
The fourth section is devoted to state and prove the main theorem.
As carried out in ~\cite{BZBi}, we first construct, by means of a Frobenius projective structure, a Frobenius-constant quantization on the complement of the zero section in the total space of $\mathcal{L}$.
This quantization turns out to be invariant under the natural involution, and hence, descends to a Frobenius-constant quantization on $(\mathbb{A} (\Omega_X)^\times, \check{\omega}^\mathrm{can})$.
Moreover, the injectivity of this assignment is proved by examining the behavior of the noncommutative multiplication in each quantization.
In the final section,
we discuss (cf. Theorem \ref{T009h1gh}) a higher-dimensional variant of our main theorem, which may be thought of as a positive characteristic analogue of a result in ~\cite{Bi}.
\vspace{5mm}
\hspace{-4mm}{\bf Acknowledgement} \leavevmode\\
\ \ \
The author would like to thank all algebraic curves equipped with a Frobenius-projective structure, who live in the world of mathematics, for their useful comments and heartfelt encouragement. The author was partially supported by the Grant-in-Aid for Scientific Research (KAKENHI No. 18K13385).
\vspace{10mm}
\section{Preliminaries} \vspace{3mm}
In this section, we prepare the notation and conventions used in the present paper.
Throughout the present paper, let us fix an odd prime $p$ and an algebraically closed field $k$ of characteristic $p$.
Unless otherwise stated, all schemes and morphisms of schemes are implicitly assumed to be over $k$, and products of schemes are taken over $k$.
We use the word {\it variety} (resp., {\it curve}) to mean a finite type integral scheme over $k$ (resp., a finite type integral scheme over $k$ of dimension $1$).
For each positive integer $n$, we shall write
$\mathbb{A}^n$ (resp., $\mathbb{P}^n$) for the affine space (resp., the projective space) over $k$ of dimension $n$.
Also, write
$\mathbb{A}^{n \times} := \mathbb{A}^n \setminus \{0\}$.
\vspace{5mm}
\subsection{Vector bundles} \label{S7}
\leavevmode\\ \vspace{-4mm}
Let
$S$ be a smooth variety of dimension $n \geq 0$.
Given a vector bundle $\mathcal{F}$ on $S$ (i.e., a locally free coherent sheaf on $S$),
we denote by $\mathcal{F}^\vee$ its dual sheaf, i.e., $\mathcal{F}^\vee := \mathcal{H} om_{\mathcal{O}_X} (\mathcal{F}, \mathcal{O}_X)$.
Let $\mathbb{A} (\mathcal{F})$ and $\mathbb{P}(\mathcal{F})$ denote the relative affine and projective spaces respectively associated with $\mathcal{F}$, i.e.,
\begin{align} \label{E04}
\mbA (\mathcal{F}) := \mathcal{S} pec (S^\bullet (\mathcal{F}^\vee)), \ \ \ \mathbb{P} (\mathcal{F}) := \mathcal{P} roj (S^\bullet (\mathcal{F}^\vee)),
\end{align}
where $S^\bullet (\mathcal{F}^\vee) \ \left(:= \bigoplus_{i \geq 0}S^i (\mathcal{F}^\vee)\right)$ denotes the symmetric algebra over $\mathcal{O}_S$ associated with $\mathcal{F}^\vee$.
Also, write
\begin{align}
\mbA (\mathcal{F})^\times
\end{align}
for the complement of the zero section $S \rightarrow \mathbb{A} (\mathcal{F})$ in $\mbA (\mathcal{F})$, which admits a natural projection
\begin{align} \label{E06}
\pi_\mathcal{F} : \mbA (\mathcal{F})^\times \rightarrow \mathbb{P} (\mathcal{F})
\end{align}
over $S$.
We shall write $\Omega_{S}$ for the sheaf of $1$-forms (in other words, the cotangent bundle) on
$S$ relative to $k$ and $\mathcal{T}_S$ for its dual.
By the smoothness assumption on $S$, both $\Omega_S$ and $\mathcal{T}_S$ turn out to be vector bundles
of rank $n$.
We write
$d : \mathcal{O}_S \rightarrow \Omega_S$
for the universal derivation.
Moreover, denote by $\omega_{S}$ the canonical line bundle of $S$ (relative to $k$), which is canonically isomorphic to the determinant line bundle $\mathrm{det}(\Omega_S) := \bigwedge^n \Omega_S$ of $\Omega_S$.
\vspace{5mm}
\subsection{Symplectic structures} \label{S700}
\leavevmode\\ \vspace{-4mm}
Recall that a {\bf symplectic structure} on $S$ is a nondegenerate closed $2$-form $\omega \in \Gamma (S, \bigwedge^2\Omega_{S})$.
Here, we say that $\omega$ is {\it nondegenerate} if the morphism $\Omega_{S} \rightarrow \mathcal{T}_{S} \ \left(= \Omega_{S}^\vee \right)$ induced naturally by $\omega$ is an isomorphism.
A {\bf symplectic variety (over $k$)} is a pair $(S, \omega)$ consisting of a smooth variety $S$ and a symplectic structure $\omega$ on it.
An {\bf isomorphism} $(S, \omega) \stackrel{\sim}{\migi} (S', \omega')$ between symplectic varieties is an isomorphism $S \stackrel{\sim}{\migi} S'$ preserving the respective symplectic structures.
As is well-known, the variety $\mbA(\Omega_S)$ (i.e., the total space of the cotangent bundle of $S$) has a canonical symplectic structure
\begin{align} \label{W104}
\omega^\mathrm{can}_S\in \Gamma (\mbA (\Omega_S), {\bigwedge}^2 \Omega_{\mbA (\Omega_S)}).
\end{align}
often called the {\it Liouville symplectic form}.
If there is no fear of causing confusion, we write $\omega^\mathrm{can}$ instead of $\omega^\mathrm{can}_S$ for simplicity.
If $q_1, \cdots, q_{n}$ are local coordinates in $S$ and $q^\vee_1, \cdots, q^\vee_{n}$ denote the dual coordinates in $\mbA (\Omega_{S})$, then $\omega^{\mathrm{can}}$ may be expressed locally as
$\omega^{\mathrm{can}} = \sum_{i=1}^{n} dq^\vee_i \wedge dq_i$.
By abuse of notation, we also use the notation $\omega^\mathrm{can}$ to denote the restriction of $\omega^\mathrm{can}$ to the open subscheme $\mbA(\Omega_S)^\times \ \left(\subseteq \mbA (\Omega_S) \right)$.
Also, for each $c \in k^\times$, $c \cdot \omega^\mathrm{can}$ forms a symplectic structure.
In particular, by letting $\check{\omega}^\mathrm{can} := \frac{1}{2}\cdot \omega^\mathrm{can}$,
we have symplectic varieties
\begin{align} \label{Errgt8}
(\mbA(\Omega_S), \check{\omega}^\mathrm{can}), \hspace{5mm}
(\mbA(\Omega_S)^\times, \check{\omega}^\mathrm{can}).
\end{align}
\vspace{5mm}
\subsection{Differential operators} \label{S83}
\leavevmode\\ \vspace{-4mm}
We shall recall the notion of a differential operator.
Let $\mathcal{L}_i$ ($i=1,2$) be line bundles on $S$.
By a {\it differential operator} from $\mathcal{L}_1$ to $\mathcal{L}_2$, we mean a $k$-linear morphism $D :\mathcal{L}_1 \rightarrow \mathcal{L}_2$
locally expressed, after fixing identifications $\mathcal{L}_1 \cong \mathcal{L}_2 \cong \mathcal{O}_S$ and a local coordinate system $\vec{x}:= (x_1, \cdots, x_n)$ in $S$, as
\begin{align} \label{E442}
D
: v \mapsto
D(v) = \sum_{\alpha \in \mathbb{Z}_{\geq 0}^n} a_\alpha \cdot \partial^\alpha_{\vec{x}} (v)
\end{align}
by means of some local sections $a_\alpha \in \mathcal{O}_S$ with $a_\alpha =0$ for almost all $\alpha$,
where
for each $\alpha := (\alpha_1, \cdots, \alpha_n) \in \mathbb{Z}_{\geq 0}^n$,
we write
$\partial^\alpha_{\vec{x}}(v) := \frac{\partial^{|\alpha|}(v)}{\partial x_1^{\alpha_1} \cdots \partial x_n^{\alpha_n}}$ ($|\alpha| := \alpha_1 + \cdots + \alpha_n$).
If $a_\alpha=0$ for any $\alpha$ with $|\alpha| \geq p$,
then $j_\mathrm{max} := \mathrm{max}\left\{ |\alpha| \, | \, a_\alpha \neq 0 \right\} \ (<p)$ is well-defined (i.e., depend only on $D$, not on the choice of the local expression (\ref{E442})).
In this situation,
we say that $D$ is {\it of order $j_\mathrm{max}$}. (We say that $D$ is {\it of order $-\infty$} if $D=0$.)
Given a nonnegative integer $j$ with $j<p$, we denote by
\begin{align}
\mathcal{D} \textit{iff}^{\,\leq j}_{\mathcal{L}_1, \mathcal{L}_2}
\end{align}
the Zariski sheaf on $S$ consisting of locally defined differential operators from $\mathcal{L}_1$ to $\mathcal{L}_2$ of order $\leq j$; it is a subsheaf of the sheaf $\mathcal{H} om_k (\mathcal{L}_1, \mathcal{L}_2)$ of locally defined $k$-linear morphisms $\mathcal{L}_1 \rightarrow \mathcal{L}_2$.
In the case where $\mathcal{L}_1 = \mathcal{L}_2 =\mathcal{O}_X$, we write
\begin{align}
\mathcal{D}_X^{\leq i} := \mathcal{D} \textit{iff}^{\,\leq j}_{\mathcal{O}_X, \mathcal{O}_X}.
\end{align}
Note that $\mathcal{D} \textit{iff}^{\,\leq j}_{\mathcal{L}_1, \mathcal{L}_2}$
admits two different structures of $\mathcal{O}_S$-module --- one as given by left multiplication (where we denote this $\mathcal{O}_S$-module by
${^l \mathcal{D}} \textit{iff}^{\,\leq j}_{\mathcal{L}_1, \mathcal{L}_2}$),
and the other given by right multiplication (where we denote this $\mathcal{O}_S$-module by ${^r \mathcal{D}} \textit{iff}^{\,\leq j}_{\mathcal{L}_1, \mathcal{L}_2}$) ---.
Given an $\mathcal{O}_X$-module $\mathcal{F}$, we equip the tensor product $\mathcal{F} \otimes {^l \mathcal{D}}_{X}^{\leq j}$ (resp., ${^r \mathcal{D}}_X^{\leq j} \otimes \mathcal{F}$) with an $\mathcal{O}_X$-module structure arising from
${^r \mathcal{D}}_X^{\leq j}$ (resp., ${^l \mathcal{D}}_X^{\leq j}$).
Then,
the composition with the $k$-linear morphism $\mathcal{L}_2 \otimes \mathcal{D}_X^{\leq j} \rightarrow \mathcal{L}_2$ given by
$v \otimes D \mapsto v \otimes D(1)$ yields
an identification
\begin{align} \label{Edr56}
\mathcal{H} om_{\mathcal{O}_X} (\mathcal{L}_1, \mathcal{L}_2 \otimes \mathcal{D}_X^{\leq j}) \stackrel{\sim}{\migi} \mathcal{D} {\it iff}_{\mathcal{L}_1, \mathcal{L}_2}^{\leq j}.
\end{align}
Moreover, the assignment $D = \sum_{\alpha \in \mathbb{Z}_{\geq 0}^n} a_\alpha \cdot \partial^\alpha_{\vec{x}} \mapsto \sum_{|\alpha| =j} a_\alpha \cdot \partial^\alpha_{\vec{x}}$ gives a well-defined isomorphism of $\mathcal{O}_S$-modules
\begin{align}
{ \mathcal{D}} \textit{iff}^{\,\leq j}_{\mathcal{L}_1, \mathcal{L}_2}/{\mathcal{D}} \textit{iff}^{\,\leq (j-1)}_{\mathcal{L}_1, \mathcal{L}_2}
\stackrel{\sim}{\migi} \mathcal{H} om_{\mathcal{O}_S} (\mathcal{L}_1, \mathcal{L}_2 \otimes S^j (\mathcal{T}_{S})),
\end{align}
where $S^j(\mathcal{T}_{S})$ denotes the $j$-th component of the symmetric power of $\mathcal{T}_S$.
Denote by $\Sigma$ the composite
\begin{align}
\Sigma : \mathcal{D} \textit{iff}^{\,\leq j}_{\mathcal{L}_1, \mathcal{L}_2} \twoheadrightarrow
\mathcal{D} \textit{iff}^{\,\leq j}_{\mathcal{L}_1, \mathcal{L}_2}/\mathcal{D} \textit{iff}^{\,\leq (j-1)}_{\mathcal{L}_1, \mathcal{L}_2} \stackrel{\sim}{\migi} \mathcal{H} om_{\mathcal{O}_S} (\mathcal{L}_1, \mathcal{L}_2 \otimes S^j (\mathcal{T}_{S})).
\end{align}
For each local section $D \in \mathcal{D} \textit{iff}^{\,\leq j}_{\mathcal{L}_1, \mathcal{L}_2}$, we refer to $\Sigma (D)$ as the {\it principal symbol} of $D$.
Next, let us write
\begin{align}
S^{(1)}
\end{align}
for the Frobenius twist of $S$ over $k$ (i.e., the base-change of $S$ via the absolute Frobenius morphism of $k$)
and
\begin{align}
F_{S/k}: S \rightarrow S^{(1)}
\end{align}
for the relative Frobenius morphism of $S$ over $k$.
To simplify the notation,
we regard each $\mathcal{O}_{S^{(1)}}$-module (resp., $\mathcal{O}_S$-module)
as a sheaf on $S$ (resp., on $S^{(1)}$) via the underlying homeomorphism of $F_{S/k}$.
Notice that each differential operator $D : \mathcal{L}_1 \rightarrow \mathcal{L}_2$ of order $j$ may be
considered as an {\it $\mathcal{O}_{S^{(1)}}$-linear} morphism
$F_{S/k*}(\mathcal{L}_1) \rightarrow F_{S/k*}(\mathcal{L}_2)$
via the underlying homeomorphism of $F_{S/k}$.
It follows that the kernel $\mathrm{Ker}(D)$ forms an $\mathcal{O}_{S^{(1)}}$-submodule of $F_{S/k*}(\mathcal{L}_1)$.
\vspace{3mm}
\begin{defi} \label{D01fgh} \leavevmode\\
\ \ \
We shall say that {\bf $D$ has a full set of solutions} if $\mathrm{Ker}(D)$ is a vector bundle on $S^{(1)}$ of rank $j$.
\end{defi}\vspace{1mm}
\vspace{5mm}
\subsection{Theta characteristics} \label{S8}
\leavevmode\\ \vspace{-4mm}
By a {\bf theta characteristic} on $S$, we mean
a pair
\begin{align}
\mathbb{L} := (\mathcal{L}, \psi_\mathcal{L})
\end{align}
consisting of a line bundle $\mathcal{L}$ on $S$ and an isomorphism $\psi_\mathcal{L} : \mathcal{L}^{\otimes (n+1)} \stackrel{\sim}{\migi} \omega_S$ between line bundles.
As is well-known,
any smooth curve always admits a theta characteristic.
\vspace{3mm}
\begin{exa} \label{E5d10}
\leavevmode\\
\ \ \
We shall observe that there exists a canonical theta characteristic on the projective space $\mathbb{P}^{n} = \mathrm{Proj}(k[x_0, x_1, \cdots, x_n])$.
Let
\begin{align} \label{Effgh}
\eta_0 : \mathcal{O}_{\mathbb{P}^n}(-1) \rightarrow \mathcal{O}_{\mathbb{P}^n}^{\oplus (n+1)}
\end{align}
be the $\mathcal{O}_{\mathbb{P}^n}$-linear injection given by $w \mapsto \sum_{i=0}^n w x_i\cdot e_i$ for each local section $w \in \mathcal{O}_{\mathbb{P}^n}(-1)$, where $(e_0, \cdots, e_n)$ is a canonical basis of $\mathcal{O}_{\mathbb{P}^n}^{\oplus (n+1)}$.
The composite
\begin{align}
\mathcal{O}_{\mathbb{P}^n}(-1) \xrightarrow{\eta_0} \mathcal{O}_{\mathbb{P}^n}^{\oplus (n+1)} \xrightarrow{d^{\oplus (n+1)}} \Omega_{\mathbb{P}^n} \otimes \mathcal{O}_{\mathbb{P}^n}^{\oplus (n+1)} \twoheadrightarrow \Omega_{\mathbb{P}^n} \otimes (\mathcal{O}_{\mathbb{P}^n}^{\oplus (n+1)}/\mathrm{Im}(\eta_0)),
\end{align}
which is verified to be $\mathcal{O}_{\mathbb{P}^n}$-linear, induces an isomorphism of $\mathcal{O}_{\mathbb{P}^n}$-modules
\begin{align} \label{E5568}
\mathcal{O}_{\mathbb{P}^n}(-1) \otimes (\mathcal{O}_{\mathbb{P}^n}^{\oplus (n+1)}/\mathrm{Im}(\eta_0))^\vee \stackrel{\sim}{\migi} \Omega_{\mathbb{P}^n}.
\end{align}
Moreover,
we have a composite isomorphism
\begin{align} \label{G3345}
\mathcal{O}_{\mathbb{P}^n} \stackrel{\sim}{\migi} \mathrm{det} (\mathcal{O}_{\mathbb{P}^n}^{\oplus (n+1)})
\stackrel{\sim}{\migi} \mathcal{O}_{\mathbb{P}^n}(-1)\otimes \mathrm{det} (\mathcal{O}_{\mathbb{P}^n}^{\oplus (n+1)}/\mathrm{Im}(\eta_0)),
\end{align}
where the first isomorphism is given by $1 \mapsto e_0 \wedge \cdots \wedge e_n$ and the second isomorphism arises from
the short exact sequence
\begin{align}
0 \xrightarrow{} \mathcal{O}_{\mathbb{P}^n}(-1) \xrightarrow{\eta_0} \mathcal{O}_{\mathbb{P}^n}^{\oplus (n+1)} \xrightarrow{} \mathcal{O}_{\mathbb{P}^n}^{\oplus (n+1)}/\mathrm{Im}(\eta_0) \xrightarrow{} 0.
\end{align}
Denote by $\psi_0$ the composite isomorphism
\begin{align}
\psi_0 : \mathcal{O}_{\mathbb{P}^n}(-1)^{\otimes (n+1)} \ &\left(=\mathcal{O}_{\mathbb{P}^n}(-n) \otimes \mathcal{O}_{\mathbb{P}^n}(-1) \right) \\
& \, \stackrel{\sim}{\migi} \mathcal{O}_{\mathbb{P}^n}(-n) \otimes \mathrm{det}(\mathcal{O}_{\mathbb{P}^n}^{\oplus (n+1)}/\mathrm{Im}(\eta_0))^\vee \notag \\
& \, \stackrel{\sim}{\migi} \mathrm{det}(\mathcal{O}_{\mathbb{P}^n}(-1) \otimes (\mathcal{O}_{\mathbb{P}^n}^{\oplus (n+1)}/\mathrm{Im}(\eta_0))^\vee) \notag \\
& \, \stackrel{\sim}{\migi} \left(\mathrm{det}(\Omega_{\mathbb{P}^n}) = \right) \ \omega_S, \notag
\end{align}
where the first isomorphism follows from (\ref{G3345}) and the third isomorphism follows from (\ref{E5568}).
Thus, we have obtained a theta characteristic
\begin{align} \label{E09076}
\mathbb{L}_0 := (\mathcal{O}_{\mathbb{P}^n}(-1), \psi_0)
\end{align}
on $\mathbb{P}^n$.
\end{exa}
\vspace{3mm}
\vspace{10mm}
\section{Frobenius-constant quantizations} \vspace{3mm}
In this section, we recall the notion of a Frobenius-constant (= FC) quantization on a given symplectic variety and discuss some related topics.
\vspace{5mm}
\subsection{Quentizations} \label{S4}
\leavevmode\\ \vspace{-4mm}
Let $(S, \omega_S)$ be a symplectic variety.
The nondegenerate pairing $\mathcal{T}_{S} \otimes_{\mathcal{O}_S} \mathcal{T}_{S} \rightarrow \mathcal{O}_S$ given by $\omega_S$
becomes a pairing
$\omega^{-1}_S : \Omega_{S} \otimes_{\mathcal{O}_S} \Omega_{S} \rightarrow \mathcal{O}_S$ via $\Omega_S \stackrel{\sim}{\migi} \mathcal{T}_S$ induced by $\omega_S$.
Thus, we obtain a skew-symmetric $k$-bilinear map
\begin{align}
\{-, - \}_{\omega} : \mathcal{O}_S \times \mathcal{O}_S \rightarrow \mathcal{O}_S
\notag \end{align}
defined by
$\{ f, g \}_{\omega} := \omega^{-1}_S(df, dg)$.
One verifies from the closedness of $\omega_S$ that $\{-, -\}_\omega$ defines a Poisson bracket in the usual sense.
Here, let $k[[\hslash]]$ denote the ring of formal power series in the variable $\hslash$ over $k$, and write $\mathcal{O}_S [[\hslash ]] := \varprojlim_{j \geq 1} \mathcal{O}_S [\hslash]/(\hslash^j)$.
In this article, we shall define
a {\bf quantization} on $(S, \omega_S)$ to be
a sheaf of (noncommutative) flat $k[[\hslash ]]$-algebras $\mathcal{O}^\hslash_S$ on $S$
such that $\mathcal{O}^\hslash_S = \mathcal{O}_S [[\hslash ]]$ (as an equality of sheaves of $k[[\hslash]]$-modules)
and
the
commutator in $\mathcal{O}^\hslash_S$ is equal to $\hslash \cdot \{-,-\}_\omega$ mod $\hslash^2 \cdot \mathcal{O}_S^\hslash$.
Moreover,
a {\bf Frobenius-constant quantization} (or, an {\bf FC quantization}, for short) on $(S, \omega_S)$ (cf. ~\cite{BeKa1}, Definition 3.3; \cite{BeKa2}, Definitions 1.1 and 1.4) is
a quantizaton $\mathcal{O}^\hslash_S$ on $(S, \omega_S)$ such that
the image of the natural inclusion
$\mathcal{O}_{S^{(1)}} [[\hslash]]\hookrightarrow \mathcal{O}_S[[\hslash ]] \ \left(= \mathcal{O}^\hslash_S\right)$ coincides with
the center $Z(\mathcal{O}^\hslash_S)$ of $\mathcal{O}_S^\hslash$.
We shall write
\begin{align} \label{Er33}
\mathfrak{Q}_{(S, \omega_S)}^{\mathrm{FC}}
\end{align}
for the set of FC quantizations on $(S, \omega_S)$.
\vspace{5mm}
\subsection{Pull-back of FC quantizations} \label{S10}
\leavevmode\\ \vspace{-4mm}
If we are given an FC quantization on a prescribed symplectic variety,
then it induces an FC quantization on each open subvariety via restriction.
More generally,
we can construct the pull-back of an FC quantization via an {\it \'{e}tale} morphism, as follows.
Let
$(T, \omega_T)$ be another symplectic variety
and $f : T \rightarrow S$ an \'{e}tale morphism
with $f^*(\omega_S) = \omega_T$.
The \'{e}taleness of $f$ implies that
the commutative square diagram
\begin{align} \label{E073}
\xymatrix{
T \ar[r]^{f} \ar[d]_{F_{T/k}} & \ar[d]^{F_{S/k}} S
\\
T^{(1)} \ar[r]_{f^{(1)}} & S^{(1)}
}
\end{align}
is cartesian, where $f^{(1)}$ denotes the base-change of $f$ via the absolute Frobenius morphism of $k$.
Given an FC quantization $\mathcal{O}_{S}^\hslash$
on $(S, \omega_S)$,
we set
\begin{align}
f^* (\mathcal{O}_{S}^\hslash) := \varprojlim_{j >0} \left(\mathcal{O}_{T^{(1)}}\otimes_{f^{-1}(\mathcal{O}_{S^{(1)}})} f^{-1}(\mathcal{O}_{S}^\hslash/(\hslash^j) \right).
\end{align}
Then, since
$Z (f^* (\mathcal{O}_{S}^\hslash)) \cong \mathcal{O}_{T^{(1)}} \otimes_{f^{-1}(\mathcal{O}_{S^{(1)}})} \varprojlim_{j>0}\left(Z (\mathcal{O}_S^\hslash)/(\hslash^j) \right)$,
the sheaf $f^* (\mathcal{O}_{S}^\hslash)$ specifies an FC quantization on $(T, \omega_T)$.
We shall refer to $f^* (\mathcal{O}_{S}^\hslash)$ as the {\bf pull-back} of $\mathcal{O}_S^\hslash$ (via $f$).
\vspace{5mm}
\subsection{Equivariant FC quantizations} \label{S41}
\leavevmode\\ \vspace{-4mm}
Let $(S, \omega_S)$ be as above
and $G$ a finite group acting freely on $(S, \omega_S)$, i.e., $S$ is equipped with a free $G$-action preserving $\omega_S$.
A {\bf Frobenius $G$-constant quantization} (or, a {\bf $G$-FC quantization}) on $(S, \omega_{S})$ (cf. ~\cite{BeKa1}, Definition 5.5) is an FC quantization
$\mathcal{O}_S^\hslash$ on $(S, \omega_S)$ compatible, in the natural sense, with the $G$-action on $S$.
Denote by
\begin{align}
\mathfrak{Q}^{G\text{-}\mathrm{FC}}_{(S, \omega_S)}
\end{align}
the set of
$G$-FC quantizations on $(S, \omega_S)$.
We obtain the natural forgetting map
\begin{align} \label{W400}
\mathfrak{Q}^{G\text{-}\mathrm{FC}}_{(S, \omega_S)} \longrightarrow \mathfrak{Q}^{\mathrm{FC}}_{(S, \omega_S)}.
\end{align}
Furthermore, let $(T, \omega_{T})$ be the quotient of $(S, \omega_S)$ by the $G$-action.
The quotient morphism $f : S \rightarrow T$ is a Galois \'{e}tale covering with Galois group $G$ and
$f^*(\omega_{T}) = \omega_S$.
Hence, pulling-back via $\pi$ induces a map of sets
\begin{align} \label{W1004}
\mathfrak{Q}_{(T, \omega_T)}^{\mathrm{FC}} \longrightarrow \mathfrak{Q}_{(S, \omega_S)}^{\mathrm{FC}}.
\end{align}
If $\mathcal{O}^\hslash_{T}$ is an FC quantization on $(T, \omega_{T})$,
then the pull-back $f^*(\mathcal{O}^\hslash_T)$ has naturally a structure of $G$-FC quantization
since the $G$-actions on $S$ and $S^{(1)}$
are compatible via $F_{S/k}$.
Conversely, let $\mathcal{O}^\hslash_S$ be a $G$-FC quantization on $(S, \omega_S)$.
Then, the sheaf $f_*(\mathcal{O}^\hslash_S)^G$ of $G$-invariant sections of
$f_*(\mathcal{O}^\hslash_S)$
specifies an FC quantization on $(T, \omega_T)$.
One verifies immediately that the assignments
$\mathcal{O}_T^\hslash \mapsto f^*(\mathcal{O}_T^\hslash)$ and $\mathcal{O}_S^\hslash \mapsto f_*(\mathcal{O}_S^\hslash)^G$
give
a bijection correspondence
\begin{align} \label{W1005}
\mathfrak{Q}^{\mathrm{FC}}_{(T, \omega_T)} \stackrel{\sim}{\longrightarrow}\mathfrak{Q}^{G\text{-}\mathrm{FC}}_{(S, \omega_S)}
\end{align}
making the following diagram commute:
\begin{align}
\vcenter{\xymatrix{
\mathfrak{Q}^{\mathrm{FC}}_{(T, \omega_T)} \ar[rr]_{\sim}^{(\ref{W1005})} \ar[rd]_{(\ref{W1004})} & & \mathfrak{Q}^{G\text{-}\mathrm{FC}}_{(S, \omega_S)} \ar[ld]^{(\ref{W400})}
\\
& \mathfrak{Q}^{\mathrm{FC}}_{(T, \omega_T)}. &
}}
\end{align}
\vspace{5mm}
\subsection{Formal Weyl algebras} \label{S15}
\leavevmode\\ \vspace{-4mm}
In this subsection, we recall a canonical (Frobenius-constant) quantization on the affine space $\mathbb{A}^{2n} := \mathrm{Spec} (k[x_1, \cdots, x_n, y_1, \cdots, y_n])$ ($n>0$)
equipped with the symplectic structure
\begin{align}
\omega^\mathrm{Weyl} := \sum_{i=1}^n d y_i \wedge d x_i.
\end{align}
Here, notice that the Poisson bracket $\{ -, -\}_{\omega}$ associated with $\omega^\mathrm{Weyl}$ is given by
$\{ f, g\}_{\omega} = \sum_{i=1}^n \frac{\partial f}{\partial y_i} \cdot \frac{\partial g}{\partial x_i} - \frac{\partial f}{\partial x_i} \cdot \frac{\partial g}{\partial y_i}$ (for any local sections $f, g \in \mathcal{O}_{\mathbb{A}^{2n}}$).
For each commutative ring $R$ over $k$, we define $W^{2n}_R$ to be the (noncommutative) $R[[\hslash ]]$-algebra $W^{2n} := R[x_1, \cdots, x_n, y_1, \cdots, y_n][[\hslash ]]$ equipped with the multiplication ``$*$" given by
\begin{align}
f * g := \sum_{\alpha \in \mathbb{Z}_{\geq 0}^n} \frac{\hslash^{|\alpha|}}{\alpha !} \cdot \partial^\alpha_{\vec{y}}(f) \cdot \partial^\alpha_{\vec{x}}(g)
\end{align}
for any $f, g \in R[x_1, \cdots, x_n, y_1, \cdots, y_n]$.
Hence, the $R[[\hslash]]$-algebra $W^{2n}_R$ is generated by the elements $x_1, \cdots, x_n, y_1, \cdots, y_n$ subject to relations
\begin{align}
[x_i, x_j] = [y_i, y_j] =0, \hspace{5mm} [y_i, x_j] = \delta_{ij} \cdot \hslash
\end{align}
for all $0 < i, j \leq n$.
Since the center of this algebra coincides with $R[x^p_1, \cdots x_n^p, y^p_1, \cdots, y_n^p][[\hslash ]]$, $W^{2n}_R$ may be thought of as an $R[x^p_1, \cdots x_n^p, y^p_1, \cdots, y_n^p][[\hslash ]]$-algebra.
Here, write $\mathrm{Sp}_{2n}$ for the symplectic group over $k$ of rank $n$, i.e.,
\begin{align}
\mathrm{Sp}_{2n}(R) = \left\{ A \in \mathrm{GL}_{2n}(R) \, | \, {^t A} J_{2n} A = J_{2n}\right\}
\end{align}
for each commutative ring $R$ over $k$, where $J_{2n}:= \begin{pmatrix} O & E \\ -E & O\end{pmatrix}$ ($E$ denotes the unit matrix of size $n$).
Each $A \in \mathrm{Sp}_{2n}(R)$ yields an automorphism $\eta_A$ of $W^{2n}_R$ given by $(x_1, \cdots, x_n, y_1, \cdots, y_n) \mapsto (x_1, \cdots, x_n, y_1, \cdots, y_n){^t A}$.
If $\mathrm{Aut}(W^{2n}_R)$ denotes the automorphism group of the $R[[\hslash ]]$-algebra $W^{2n}_R$, then the assignment $A \mapsto \eta_A$ determines an injective homomorphism
\begin{align} \label{E4522}
\mathrm{Sp}_{2n}(R) \rightarrow \mathrm{Aut}(W^{2n}_R).
\end{align}
Notice that $W^{2n}_k$ gives rise to
an $\mathcal{O}_{(\mathbb{A}^{2n})^{(1)}}[[\hslash]]$-algebra
\begin{align} \label{Efgklo}
\mathcal{W}^{2n}_k,
\end{align}
which specifies
an FC quantization on $(\mathbb{A}^{2n}, \omega^\mathrm{Weyl})$, as well as on $(\mathbb{A}^{2n\times}, \omega^\mathrm{Weyl})$ via restriction.
The variety $\mathbb{A}^{2n\times}$ admits
a free action of $\mu_2 := \{ \pm1\}$ such that the automorphism corresponding to $-1 \in \mu_2$ is given by $(x, y) \mapsto (-x, -y)$.
This action preserves $\omega^\mathrm{Weyl}$, and we obtain the quotient symplectic variety
\begin{align}
(\mathbb{A}^{2n \times}_{/\mu_2}, \omega_{/\mu_2}^\mathrm{Weyl}).
\end{align}
One verifies that $\mathcal{W}^{2n}_k$ is a $\mu_2$-FC quantization, which descends to a FC quantization
\begin{align} \label{E4889}
\mathcal{W}^{2n}_{/\mu_2}
\end{align}
on $(\mathbb{A}^{2n \times}_{/\mu_2}, \omega_{/\mu_2}^\mathrm{Weyl})$.
(In our discussion, we will use this quantization only in the case $n=1$.)
\vspace{10mm}
\section{Frobenius-projective structures and related objects} \vspace{3mm}
In this section, we review a positive characteristic analogue of a complex projective structure, called a Frobenius-projective structure.
Also, we
discuss various bijective correspondences between Frobenius-projective structures on a curve
and some equivalent objects, i.e., dormant $\mathrm{PGL}_2$-opers, dormant $(\mathrm{SL}_2, \mathbb{L})$-opers, and projective connections with a full set of solutions.
\vspace{5mm}
\subsection{Frobenius-projective structures} \label{S1}
\leavevmode\\ \vspace{-4mm}
Let $n$ be a positive integer and denote by
$\mathrm{PGL}_{n+1}$ the projective linear group over $k$ of rank $n+1$, which
is naturally identified with the automorphism group of $\mathbb{P}^n$.
Given each algebraic group $G$ over $k$ and a smooth variety $S$, we denote by
$G_{S}$ the sheaf of groups on $S$ represented by $G$.
Write
\begin{align}
G_{S}^\mathrm{F} := F^{-1}_{S/k} (G_{S^{(1)}}) \ \left(\subseteq G_{S} \right).
\end{align}
Also, we shall write
\begin{align}
\mathcal{P}^{\text{\'{e}t}}
\end{align}
for the sheaf of sets on $S$ that assigns, to each open subscheme $U$ of $S$, the set of \'{e}tale morphisms $\phi : U \rightarrow \mathbb{P}^n$ from $U$ to $\mathbb{P}^n$.
Each local section $\phi$ of $\mathcal{P}^{\text{\'{e}t}}$ may be regarded, by taking its graph, as a local section $\Gamma_\phi : U \rightarrow U \times \mathbb{P}^n$ of the trivial $\mathbb{P}^n$-bundle $U \times \mathbb{P}^n \xrightarrow{\mathrm{pr}_1} U$.
The sheaf $\mathcal{P}^{\text{\'{e}t}}$ has a $(\mathrm{PGL}_{n})_S^\mathrm{F}$-action described as follows.
Let $U$ be an open subscheme, $\phi : U \rightarrow \mathbb{P}^n$ an element of $\mathcal{P}^{\text{\'{e}t}}(U)$, and
$A$ an element of $(\mathrm{PGL}_{n})_S^\mathrm{F}(U) \ \left(\subseteq \mathrm{PGL}_n (U) \right)$.
Then, one verifies immediately that the composite
\begin{align}
A (\phi) : U \xrightarrow{\Gamma_\phi} U \times \mathbb{P}^n \xrightarrow{A} U \times \mathbb{P}^n \xrightarrow{\mathrm{pr}_2} \mathbb{P}^n
\end{align}
specifies an element of $\mathcal{P}^{\text{\'{e}t}}(U)$.
The assignment $(A, \phi) \mapsto A (\phi)$ defines a $(\mathrm{PGL}_{n})_S^\mathrm{F}$-action on $\mathcal{P}^{\text{\'{e}t}}$.
\vspace{3mm}
\begin{defi}[cf. ~\cite{Hos}, \S\,2, Definition 2.1 for the case $n=1$] \label{D01} \leavevmode\\
\ \ \ We shall say that a subsheaf $\mathcal{S}^\heartsuit$ of $\mathcal{P}^{\text{\'{e}t}}$ is a {\bf Frobenius-projective structure (of level $1$)} on $S$ if $\mathcal{S}^\heartsuit$ is closed under the $(\mathrm{PGL}_{n})_S^\mathrm{F}$-action on $\mathcal{P}^{\text{\'{e}t}}$, and moreover, forms a $(\mathrm{PGL}_{n})_S^\mathrm{F}$-torsor on $S$ with respect to the resulting $(\mathrm{PGL}_{n})_S^\mathrm{F}$-action on $\mathcal{S}^\heartsuit$.
\end{defi}\vspace{1mm}
\vspace{3mm}
We shall write
\begin{align} \label{Efllo2}
\mathfrak{P} \mathfrak{S}^\mathrm{F}_S
\end{align}
for the set of Frobenius-projective structures on $S$.
\vspace{5mm}
\subsection{Dormant indigenous bundles} \label{S1}
\leavevmode\\ \vspace{-4mm}
In what follows, let us fix a smooth curve $X$.
Recall from, e.g., ~\cite{Wak}, \S\,2, Definition 2.1 (i), that an {\bf indigenous bundle} (or, a $\mathrm{PGL}_2$-oper) on $X$ is
a triple $\mathcal{E}^\spadesuit :=(\mathcal{E}, \nabla_\mathcal{E}, \sigma_\mathcal{E})$ consisting of
a flat $\mathbb{P}^1$-bundle $(\mathcal{E}, \nabla_\mathcal{E})$ on $X$
(i.e., a pair of a $\mathbb{P}^1$-bundle $\mathcal{E}$ on $X$ and a connection $\nabla_\mathcal{E}$ on $\mathcal{E}$)
and a global section $\sigma_\mathcal{E} : X \rightarrow \mathcal{E}$ such that the Kodaira-Spencer map $\mathcal{T}_{X/k} \rightarrow \sigma_\mathcal{E}^*(\mathcal{T}_{\mathcal{E}/X})$ associated to $\sigma_\mathcal{E}$
is nowhere vanishing.
(We omit to describe in detail the definition of an indigenous bundle because it will not be necessary for our discussion.)
We shall say that an indigenous bundle $\mathcal{E}^\spadesuit := (\mathcal{E}, \nabla_\mathcal{E}, \sigma_\mathcal{E})$ is {\bf dormant} if the connection $\nabla_\mathcal{E}$ has vanishing $p$-curvature (cf. ~\cite{Wak}, \S\,3, Definition 3.1).
Write
\begin{align} \label{E69207}
\mathfrak{I} \mathfrak{B}_X \left( \text{resp.,} \ \mathfrak{I} \mathfrak{B}_X^{^\mathrm{Zzz...}}\right)
\end{align}
for the set of isomorphism classes of indigenous bundles (resp., dormant indigenous bundles) on $X$.
Then,
there exists a canonical bijection of sets
\begin{align} \label{W101}
\mathfrak{P} \mathfrak{S}^\mathrm{F}_X \stackrel{\sim}{\migi} \mathfrak{I} \mathfrak{B}_X^{^\mathrm{Zzz...}}.
\end{align}
(~\cite{Hos}, \S\,3, Proposition 3.11 in the case where $X$ is proper).
In fact, let $\mathcal{S}^\heartsuit$ be a Frobenius-projective structure on $X$.
The $\mathrm{PGL}_2$-torsor over $X^{(1)}$ corresponding to $\mathcal{S}^\heartsuit$ via
the underlying homeomorphism of $F_{X/k}$ specifies
a $\mathbb{P}^1$-bundle
over $X^{(1)}$.
The pull-back $\mathcal{E}_\mathcal{S}$ of this $\mathbb{P}^1$-bundle over $X$
admits naturally a connection $\nabla^\mathrm{can}_{\mathcal{E}_\mathcal{S}}$ with vanishing $p$-curvature (cf. ~\cite{Kat}, \S\,5, Theorem 5.1).
Moreover, the local sections $U \rightarrow U \times \mathbb{P}^1$ (for various open subschemes $U$ of $X$) classified by sections in $\mathcal{S}^\heartsuit$ may be glued together to obtain a well-defined global section $\sigma_{\mathcal{E}_\mathcal{S}} : X \rightarrow \mathcal{E}_\mathcal{S}$. It follows from the definition of a Frobenius-projective structure that
the resulting triple
\begin{align} \label{W60}
\mathcal{E}_\mathcal{S}^\spadesuit := (\mathcal{E}_\mathcal{S}, \nabla^\mathrm{can}_{\mathcal{E}_\mathcal{S}}, \sigma_{\mathcal{E}_\mathcal{S}})
\end{align}
specifies a dormant indigenous bundle on $X$.
The resulting assignment $\mathcal{S}^\heartsuit \mapsto \mathcal{E}_\mathcal{S}^\spadesuit$ gives the bijection (\ref{W101}).
\vspace{3mm}
\begin{rema} \label{R111}
\leavevmode\\
\ \ \
If $X$ is proper, then we know an explicit formula for computing the number of dormant indigenous bundles on $X$, as proved in
~\cite{Wak}, Theorem A.
In particular, there exists at least one dormant indigenous bundle on any (not necessarily proper) smooth curve as it has a smooth compactification.
\end{rema}
\vspace{5mm}
\subsection{Dormant indigenous vector bundles} \label{S1}
\leavevmode\\ \vspace{-4mm}
In this subsection, we describe indigenous bundles and their higher-rank generalizations in terms of vector bundles.
Let $n$ be a positive integer with $n<p$.
Here, recall that, for each vector bundle $\mathcal{F}$ on $X$ of rank $n$,
a {\it connection} on $\mathcal{F}$ means a $k$-linear morphism $\nabla_\mathcal{F} : \mathcal{F} \rightarrow \Omega_{X}\otimes \mathcal{F}$ satisfying that $\nabla_\mathcal{F} (a \cdot v) = da \otimes v + a \cdot \nabla_\mathcal{F} (v)$ for any local sections $a \in \mathcal{O}_X$ and $v \in \mathcal{F}$.
Given such a connection $\nabla_\mathcal{F}$, we have a connection $\nabla_{\mathrm{det}(\mathcal{F})}$ on the determinant bundle $\mathrm{det}(\mathcal{F})$ induced by $\nabla_\mathcal{F}$, i.e., given by $\nabla_{\mathrm{det}(\mathcal{F})}(a_{1} \wedge \cdots \wedge a_{n}) = \sum_{i=1}^n a_1 \wedge \cdots \wedge \nabla_\mathcal{F} (a_i) \wedge \cdots \wedge a_n$, where $n:=\mathrm{rk}(\mathcal{F})$.
Recall (cf. ~\cite{BD2}, \S\,2.1) that a {\bf $\mathrm{GL}_n$-oper} on $X$ is a collection of data
\begin{align}
(\mathcal{F}, \nabla_\mathcal{F}, \mathcal{F}^\bullet)
\end{align}
consisting of a rank $n$ vector bundle $\mathcal{F}$ on $X$, a connection $\nabla_\mathcal{F}$
on $\mathcal{F}$, and an $n$-step decreasing filtration $\mathcal{F}^\bullet := \{ \mathcal{F}^j \}_{j=0}^n$ on $\mathcal{F}$ satisfying the following conditions
\begin{itemize}
\item[$\bullet$]
Each $\mathcal{F}^j$ is a subbundle of $\mathcal{F}$ such that
$\mathcal{F}^0 = \mathcal{F}$, $\mathcal{F}^n = \mathcal{F}$, and $\mathrm{gr}^j_\mathcal{F} := \mathcal{F}^j/\mathcal{F}^{j+1}$ ($0\leq j \leq n-1$) is a line bundle;
\item[$\bullet$]
$\nabla_\mathcal{F} (\mathcal{F}^j) \subseteq \Omega_X \otimes \mathcal{F}^{j-1}$ ($1 \leq j \leq n-1$) and the morphism
$\mathrm{KS}^j_{\mathcal{F}^\bullet} :
\mathrm{gr}^j_\mathcal{F} \rightarrow \Omega_X \otimes \mathrm{gr}^{j-1}_\mathcal{F}$
induced by $\nabla_\mathcal{F}$ (which is verified to be $\mathcal{O}_X$-linear) is an isomorphism.
\end{itemize}
In particular,
a $\mathrm{GL}_2$-oper
on $X$ is determined by a triple $(\mathcal{F}, \nabla_\mathcal{F}, \mathcal{N}_\mathcal{F})$ consisting of a pair $(\mathcal{F}, \nabla_\mathcal{F})$ as above (with $n=2$) and a line subbundle $\mathcal{N}_\mathcal{F}$ of $\mathcal{F}$ such that
the $\mathcal{O}_X$-linear composite
\begin{align} \label{W80}
\mathrm{KS}_{\mathcal{N}_\mathcal{F}} : \mathcal{N}_\mathcal{F} \hookrightarrow \mathcal{F} \xrightarrow{\nabla_\mathcal{F}} \Omega_{X}\otimes \mathcal{F} \twoheadrightarrow \Omega_X \otimes (\mathcal{F}/\mathcal{N}_\mathcal{F})
\end{align}
(called the {\it Kodaira-Spencer map} associated with $\mathcal{N}_\mathcal{F}$) is an isomorphism.
Next, let us fix a theta characteristic $\mathbb{L} := (\mathcal{L}, \psi_\mathcal{L})$ (cf. \S\,\ref{S8}) of $X$.
\vspace{3mm}
\begin{defi}[cf. ~\cite{Wak}, \S\,2, Definition 2.3 (ii), in the case $n=2$] \label{D10128} \leavevmode\\
\ \ \
An {\bf $(\mathrm{SL}_n, \mathbb{L})$-oper} on $X$
is a collection of data
\begin{align}
\mathcal{F}^\diamondsuit :=(\mathcal{F}, \nabla_\mathcal{F}, \mathcal{F}^\bullet, \eta_\mathcal{F}),
\end{align}
where $(\mathcal{F}, \nabla_\mathcal{F}, \mathcal{F}^\bullet)$ is a $\mathrm{GL}_n$-oper on $X$ and $\eta_\mathcal{F}$ denotes an isomorphism $\mathcal{L}^{\otimes (n-1)} \stackrel{\sim}{\migi} \mathcal{F}^{n-1}$
such that
the connection $\nabla_{\mathrm{det}(\mathcal{F})}$ on $\mathrm{det}(\mathcal{F})$
coincides with $d : \mathcal{O}_X \rightarrow \Omega_X$ via the composite isomorphism
\begin{align} \label{Effgh3}
\delta_{\mathcal{F}^\diamondsuit} : \mathrm{det}(\mathcal{F})\left(= \bigotimes_{j=0}^{n-1} \mathrm{gr}^j_\mathcal{F} \right) &\xrightarrow{\otimes_j \kappa_j} \bigotimes_{j=0}^{n-1} \Omega_X^{\otimes (j-n+1)} \otimes \mathcal{F}^{n-1} \\
& \xrightarrow{\sim} \Omega_{X}^{\otimes \frac{-n(n-1)}{2}} \otimes (\mathcal{F}^{n-1})^{\otimes n} \notag \\
& \xrightarrow{\sim} \Omega_{X}^{\otimes \frac{-n(n-1)}{2}} \otimes \mathcal{L}^{\otimes n (n-1)}\notag \\
& \xrightarrow{\sim} \mathcal{O}_X, \notag
\end{align}
where the third isomorphism arises from $\eta_\mathcal{F}$, the fourth isomorphism arises from $\psi_\mathcal{F}$, and
each $\kappa_j$ ($j=0, \cdots, n-1$) in the first isomorphism denotes the composite isomorphism
\begin{align}
\mathrm{gr}^j_\mathcal{F} \stackrel{\sim}{\migi} \Omega^{\otimes (-1)}_X \otimes \mathrm{gr}^{j+1}_\mathcal{F} \stackrel{\sim}{\migi} \cdots \stackrel{\sim}{\migi} \Omega_{X}^{\otimes (j-n+1)} \otimes \mathrm{gr}^{n-1}_\mathcal{F} \ \left(=\Omega_{X}^{\otimes (j-n+1)} \otimes \mathcal{F}^{n-1} \right)
\end{align}
each of whose constituent arises from $\mathrm{KS}^{(-)}_{\mathcal{F}^\bullet}$.
In a natural manner, we can define the notion of an isomorphism between $(\mathrm{SL}_n, \mathbb{L})$-opers.
Also, we shall say that an $(\mathrm{SL}_n, \mathbb{L})$-oper is {\bf dormant} if it has vanishing $p$-curvature.
\end{defi}\vspace{1mm}
\vspace{3mm}
We denote by
\begin{align}
\mathfrak{O} \mathfrak{p}_{(\mathrm{SL}_n, \mathbb{L}), X} \left(\text{resp.,} \ \mathfrak{O} \mathfrak{p}_{(\mathrm{SL}_n, \mathbb{L}), X}^{^\mathrm{Zzz...}}\right)
\end{align}
the set of isomorphism classes of $(\mathrm{SL}_n, \mathbb{L})$-opers (resp., dormant $(\mathrm{SL}_n, \mathbb{L})$-opers) on $X$.
According to ~\cite{Wak}, \S\,2, Proposition 2.4, there exists a canonical bijection
\begin{align} \label{E02f34}
\mathfrak{I} \mathfrak{B}_{X} \stackrel{\sim}{\migi} \mathfrak{O} \mathfrak{p}_{(\mathrm{SL}_n, \mathbb{L}), X},
\end{align}
which restricts to a bijection
\begin{align} \label{E02234}
\mathfrak{I} \mathfrak{B}_{X}^{^\mathrm{Zzz...}} \stackrel{\sim}{\migi} \mathfrak{O} \mathfrak{p}_{(\mathrm{SL}_n, \mathbb{L}), X}^{^\mathrm{Zzz...}}.
\end{align}
Let $\mathcal{E}^\spadesuit := (\mathcal{E}, \nabla_\mathcal{E}, \sigma_\mathcal{E})$ be an indigenous bundle on $X$ and $\mathcal{F}^\diamondsuit := (\mathcal{F}, \nabla_\mathcal{F}, \mathcal{N}_\mathcal{F}, \eta_\mathcal{F})$ denote the $(\mathrm{SL}_2, \mathbb{L})$-oper corresponding to $\mathcal{E}^\spadesuit$.
Then, $(\mathcal{E}, \nabla_\mathcal{E})$ may be obtained from $(\mathcal{F}, \nabla_\mathcal{F})$ via projectivization, and $\eta_\mathcal{F} : \mathcal{L} \stackrel{\sim}{\migi} \mathcal{N}_\mathcal{F} \ \left(\subseteq \mathcal{F} \right)$ induces a morphism
\begin{align} \label{Ega3t}
\eta^\mbA_\mathcal{F} : \mbA (\mathcal{L})^\times \rightarrow \mbA (\mathcal{F})^\times
\end{align}
over $X$
such that the following square diagram
\begin{align} \label{Ddd01}
\vcenter{\xymatrix@C=36pt@R=36pt{
\mbA (\mathcal{L})^\times \ar[r]^{\eta^\mbA_\mathcal{F}} \ar[d]
& \mbA (\mathcal{F})^\times \ar[d]^{\pi_\mathcal{F}}
\\
X \ar[r]_-{\sigma_\mathcal{E}} & \mathcal{E} \ \left(= \mathbb{P} (\mathcal{F}) \right)
}}
\end{align}
is commutative and cartesian.
\vspace{3mm}
\begin{exa} \label{E510d}
\leavevmode\\
\ \ \
Let $\mathcal{F}^\diamondsuit := (\mathcal{F}, \nabla_\mathcal{F}, \mathcal{N}_\mathcal{F}, \eta_\mathcal{F})$ be an
$(\mathrm{SL}_2, \mathbb{L})$-oper
on $X$.
The $(n-1)$-th symmetric power $\mathcal{F}_{\mathrm{SL}_n} := S^{n-1}(\mathcal{F})$ of $\mathcal{F}$ is a rank $n$ vector bundle.
For each $j=0, 1, \cdots, n$, the image $\mathcal{F}^j_{\mathrm{SL}_n}$
of the natural morphism $\mathcal{N}_\mathcal{F}^{\otimes j} \otimes S^{n-1-j}(\mathcal{F})\rightarrow S^{n-1}(\mathcal{F})$ is
a rank $(n-j)$ subbundle of $\mathcal{F}_{\mathrm{SL}_n}$.
The collection $\mathcal{F}_{\mathrm{SL}_n}^\bullet := \{ \mathcal{F}_{\mathrm{SL}_n}^j \}_{j=0}^n$ forms an $n$-step decreasing filtration on $\mathcal{F}_{\mathrm{SL}_n}$.
Let $\eta_{\mathcal{F}, \mathrm{SL}_n} : \mathcal{L}^{\otimes (n-1)} \stackrel{\sim}{\migi} \mathcal{F}_{\mathrm{SL}_n}^{n-1}$ be the composite isomorphism of $\eta_\mathcal{F}^{\otimes (n-1)} : \mathcal{L}^{\otimes (n-1)} \stackrel{\sim}{\migi} \mathcal{N}_\mathcal{F}^{\otimes (n-1)}$ with the natural isomorphism $\mathcal{N}^{\otimes (n-1)}_{\mathcal{F}} \stackrel{\sim}{\migi} \mathcal{F}_{\mathrm{SL}_n}^{n-1}$.
Also, let $\nabla_{\mathcal{F}, \mathrm{SL}_n}$ be the connection on $\mathcal{F}_{\mathrm{SL}_n}$ induced naturally by $\nabla_\mathcal{F}$.
Then, one verifies immediately that the collection of data
\begin{align}
\mathcal{F}^\diamondsuit_{\mathrm{SL}_n} := (\mathcal{F}_{\mathrm{SL}_n}, \nabla_{\mathcal{F}, \mathrm{SL}_n}, \mathcal{F}^\bullet_{\mathrm{SL}_n}, \eta_{\mathcal{F}, \mathrm{SL}_n})
\end{align}
forms an $(\mathrm{SL}_n, \mathbb{L})$-oper on $X$.
If, moreover, $\mathcal{F}^\diamondsuit$ is dormant, then the resulting $(\mathrm{SL}_n, \mathbb{L})$-oper $\mathcal{F}^\diamondsuit_{\mathrm{SL}_n}$ is dormant.
Thus, the assignment $\mathcal{F}^\diamondsuit \mapsto \mathcal{F}^\diamondsuit_{\mathrm{SL}_n}$ defines a map of sets
\begin{align} \label{Erty7}
\mathfrak{O} \mathfrak{p}_{(\mathrm{SL}_2, \mathbb{L}), X} \rightarrow \mathfrak{O} \mathfrak{p}_{(\mathrm{SL}_n, \mathbb{L}), X}
\end{align}
which restricts to a map $\mathfrak{O} \mathfrak{p}_{(\mathrm{SL}_2, \mathbb{L}), X}^{^\mathrm{Zzz...}} \rightarrow \mathfrak{O} \mathfrak{p}_{(\mathrm{SL}_n, \mathbb{L}), X}^{^\mathrm{Zzz...}}$.
\end{exa}
\vspace{5mm}
\subsection{Projective connections} \label{S63}
\leavevmode\\ \vspace{-4mm}
In this subsection, we discuss higher-order projective connections.
Let $\mathbb{L}$ be as above and let $D : \mathcal{L}^{\otimes (-n+1)} \rightarrow \mathcal{L}^{\otimes (n+1)}$ be an $n$-th order differential operator
(i.e., an element of $\mathcal{D} {\it iff}^{\leq n}_{ \mathcal{L}^{\otimes (-n+1)}, \mathcal{L}^{\otimes (n+1)}}$) satisfying the equality $\Sigma (D)=1$ (cf. \S\,\ref{S83}) under the identification
\begin{align}
\mathcal{H} om_{\mathcal{O}_X}(\mathcal{L}^{\otimes (-n+1)}, \mathcal{L}^{\otimes (n+1)}\otimes S^n(\mathcal{T}_X)) \stackrel{\sim}{\migi} \mathcal{H} om_{\mathcal{O}_X} (\mathcal{L}^{\otimes (-n+1)}, \mathcal{L}^{\otimes (-n+1)}) \stackrel{\sim}{\migi} \mathcal{O}_X
\end{align}
induced by $\psi_\mathcal{L}$.
Denote by ${^t D}$ the transpose of $D$, which is a differential operator
$\Omega_X \otimes (\mathcal{L}^{\otimes (n+1)})^\vee \rightarrow \Omega_X \otimes (\mathcal{L}^{\otimes (-n+1)})^\vee)$.
If $D$ is locally expressed as $D = \sum_{i=0}^n a_i \cdot \partial^i$ (for a local generator $\partial$ of $\mathcal{T}_X$), then
${^t D} = \sum_{i=0}^n (-\partial)^i \cdot a_i$.
Since $\psi_\mathcal{L}$ allows us to consider $\Omega_X \otimes (\mathcal{L}^{\otimes (n+1)})^\vee$ and $\Omega_X \otimes (\mathcal{L}^{\otimes (-n+1)})^\vee)$ as $\mathcal{L}^{\otimes (-n+1)}$ and $\mathcal{L}^{\otimes (n+1)}$ respectively, $D$ may be thought of as a differential operator in $\mathcal{D} {\it iff}^{\leq n}_{ \mathcal{L}^{\otimes (-n+1)}, \mathcal{L}^{\otimes (n+1)}}$.
In particular, it makes sense to speak of the operator $D' := \frac{1}{2} \cdot (D -(-1)^{n} \cdot {^t D})$.
Moreover, by the equality $\Sigma (D) =1$, the operator $D'$ turns out to be of order $n-1$.
We refer to the principal symbol $\Sigma_\mathrm{sub}(D) := \Sigma (D')$ of $D'$ as the {\it subprincipal symbol} of $D$.
\vspace{3mm}
\begin{defi} \label{D01ff} \leavevmode\\
\ \ \
An {\bf $n$-th order projective connection} for $\mathbb{L}$ is
an $n$-th order differential operator $D^\clubsuit : \mathcal{L}^{\otimes (-n+1)} \rightarrow \mathcal{L}^{\otimes (n+1)}$ with $\Sigma (D^\clubsuit) =1$ and $\Sigma_\mathrm{sub}(D^\clubsuit) =0$.
For simplicity, we refer to each second order projective connection as a {\bf projective connection}.
\end{defi}\vspace{1mm}
\vspace{3mm}
We shall write
\begin{align} \label{E41180}
\mathfrak{P} \mathfrak{C}^n_{X, \mathbb{L}} \ \left(\text{resp.,} \ \mathfrak{P} \mathfrak{C}_{X, \mathbb{L}}^{n, \mathrm{full}} \right)
\end{align}
for
the set of $n$-th order projective connections for $\mathbb{L}$ (resp., the set of $n$-th order projective connections for $\mathbb{L}$ having a full set of solutions).
\vspace{3mm}
\begin{prop}[cf. ~\cite{BD2}, \S\,2.1 and \S\,2.8; ~\cite{Kat2}, Proposition 6.0.5] \label{L100} \leavevmode\\
\ \ \
There exists a canonical bijection
\begin{align} \label{F001}
\mathfrak{O} \mathfrak{p}_{(\mathrm{SL}_n, \mathbb{L}), X} \stackrel{\sim}{\migi} \mathfrak{P} \mathfrak{C}^n_{X, \mathbb{L}}
\end{align}
restricting to a bijection
\begin{align} \label{F002}
\mathfrak{O} \mathfrak{p}_{(\mathrm{SL}_n, \mathbb{L}), X}^{^\mathrm{Zzz...}} \stackrel{\sim}{\migi} \mathfrak{P} \mathfrak{C}^{n, \mathrm{full}}_{X, \mathbb{L}}.
\end{align}
\end{prop}
\begin{proof}
First, we shall construct the bijection (\ref{F001}).
Let $\mathcal{F}^{\diamondsuit}:= (\mathcal{F}, \nabla_\mathcal{F}, \mathcal{F}^\bullet, \eta_\mathcal{F})$ be an $(\mathrm{SL}_n, \mathbb{L})$-oper on $X$.
The connection $\nabla_\mathcal{F}$
induces, inductively on $i \leq n$,
an $\mathcal{O}_X$-linear morphism
$\nabla_\mathcal{F}^{\mathcal{D}, i} : \mathcal{D}_X^{\leq i} \otimes \mathcal{F} \rightarrow \mathcal{F}$
determined by the condition that $\nabla^{\mathcal{D}, 0}_\mathcal{F} = \mathrm{id}_\mathcal{F}$ and $\nabla_\mathcal{F}^{\mathcal{D}, i}(\partial^i \otimes v) = \langle \partial, \nabla_\mathcal{F}(\nabla_\mathcal{F}^{\mathcal{D}, i-1} (\partial^{i-1} \otimes v))\rangle$ for any local generator $\partial \in \mathcal{T}_X$ and any local section $v \in \mathcal{F}$.
By the definition of a $\mathrm{GL}_n$-oper, we see (by induction on $i$) that
the morphism $\nabla_\mathcal{F}^{\mathcal{D}, i}$ for $i \leq n-1$ restricts to an isomorphism $\mathcal{D}_{X}^{\leq i} \otimes \mathcal{F}^{n-1} \rightarrow \mathcal{F}^{n-i-1}$ and hence $\nabla_\mathcal{F}^{\mathcal{D}, n}$ is surjective.
The composite $(\nabla^{\mathcal{D}, n-1}_{\mathcal{F}})^{-1} \circ \nabla^{\mathcal{D}, n}_{\mathcal{F}}$, regarded as an $\mathcal{O}_X$-linear morphism $\mathcal{D}_X^{\leq n} \otimes \mathcal{L}^{\otimes (n-1)} \rightarrow \mathcal{D}_X^{\leq (n-1)} \otimes \mathcal{L}^{\otimes (n-1)}$ via $\eta_\mathcal{F}$,
determines a split surjection of the following short exact sequence:
\begin{align} \label{Eddfg}
0 \longrightarrow \mathcal{D}^{\leq (n-1)}_X \otimes \mathcal{L}^{\otimes (n-1)} \longrightarrow \mathcal{D}^{\leq n}_X \otimes \mathcal{L}^{\otimes (n-1)} \longrightarrow (\mathcal{D}^{\leq n}_X/\mathcal{D}^{\leq (n-1)}_X) \otimes \mathcal{L}^{\otimes (n-1)} \longrightarrow 0.
\end{align}
Let us consider the composite $\mathcal{L}^{\otimes (-n-1)} \rightarrow \mathcal{D}_X^{\leq n} \otimes \mathcal{L}^{\otimes (n-1)}$ of the corresponding split injection
$\left(\mathcal{T}_X^{\otimes n} \otimes \mathcal{L}^{\otimes (n-1)}= \right) \ (\mathcal{D}^{\leq n}_X/\mathcal{D}^{\leq (n-1)}_X) \otimes \mathcal{L}^{\otimes (n-1)} \hookrightarrow \mathcal{D}_X^{\leq n} \otimes \mathcal{L}^{\otimes (n-1)}$
and the isomorphism $\mathcal{L}^{\otimes (-n-1)} \stackrel{\sim}{\migi} \mathcal{T}_X^{\otimes n} \otimes \mathcal{L}^{\otimes (n-1)}$ induced naturally by $\psi_\mathcal{L}$;
it corresponds to an $\mathcal{O}_X$-linear morphism $\mathcal{L}^{\otimes (-n+1)} \rightarrow \mathcal{L}^{\otimes (n+1)} \otimes \mathcal{D}_X^{\leq n}$, or equivalently, an $n$-th differential operator $D^\clubsuit_{\mathcal{F}} : \mathcal{L}^{\otimes (-n+1)} \rightarrow \mathcal{L}^{\otimes (n+1)}$ (cf. (\ref{Edr56})). One verifies immediately that $\Sigma (D^\clubsuit_{\mathcal{F}}) =1$, and moreover, (by taking account of the fact that $(\mathrm{det}(\mathcal{F}), \nabla_{\mathrm{det}(\mathcal{F})}) \cong (\mathcal{O}, d)$) that $\Sigma_\mathrm{sub}(D^\clubsuit_{\mathcal{F}}) =0$.
Thus, $D^\clubsuit_{\mathcal{F}}$ specifies a projective connection for $\mathbb{L}$.
Thus, the assignment $\mathcal{F}^\diamondsuit \mapsto D^\clubsuit_{\mathcal{F}}$ defines a map of sets $\mathfrak{O} \mathfrak{p}_{(\mathrm{SL}_n, \mathbb{L}), X} \rightarrow \mathfrak{P} \mathfrak{C}^n_{X, \mathbb{L}}$.
Conversely, let $D^\clubsuit$ be a projective connection belonging to $\mathfrak{P} \mathfrak{C}^n_{X, \mathbb{L}}$.
For each $i=0, \cdots, n$, we shall write
$\mathcal{F}_D^i := \mathcal{D}_{X}^{\leq (n-i-1)}\otimes \mathcal{L}^{\otimes (n-1)}$ and $\mathcal{F}_D^\bullet := \{ \mathcal{F}_D^i \}_{i=0}^n$.
The operator $D^\clubsuit$ may be thought of as an $\mathcal{O}_X$-linear morphism $\mathcal{L}^{\otimes (-n+1)} \rightarrow \mathcal{L}^{\otimes (n+1)} \otimes \mathcal{D}_{X}^{\leq n}$ (via (\ref{Edr56})), or equivalently, $\left((\mathcal{D}_X^{\leq n}/\mathcal{D}_X^{\leq (n-1)}) \otimes \mathcal{L}^{\otimes (n-1)}= \right) \mathcal{L}^{\otimes (-n-1)} \rightarrow \mathcal{D}_X^{\leq n} \otimes \mathcal{L}^{\otimes (n-1)}$.
It specifies a split injection of (\ref{Eddfg}),
where we shall write $\nabla'$ for the corresponding split surjection $\mathcal{D}_X^{\leq n} \otimes \mathcal{L}^{\otimes (n-1)} \twoheadrightarrow \mathcal{D}_X^{\leq (n-1)} \otimes \mathcal{L}^{\otimes (n-1)} \ \left(= \mathcal{F}^0_D \right)$.
Then, there exists a unique connection $\nabla_D$ on $\mathcal{F}_D^0 \ \left(\subseteq \mathcal{D}_X^{\leq n}\otimes \mathcal{L}^{\otimes (n-1)} \right)$ determined by the condition that
$\langle \partial, \nabla_D (\partial^i \otimes v)\rangle = \nabla' (\partial^{i+1} \otimes v)$ ($i= 0, \cdots, n-1$) for any local generator $\partial \in \mathcal{T}_X$ and any local section $v \in \mathcal{L}^{\otimes (n-1)}$.
If $\eta_D$ denotes the natural isomorphism $\mathcal{L}^{\otimes (n-1)} \stackrel{\sim}{\migi} \mathcal{F}_D^{n-1}$, then (because of the assumption that $\Sigma (D) =1$ and $\Sigma_\mathrm{sub}(D)=0$) the collection
$\mathcal{F}_D^\diamondsuit := (\mathcal{F}^0_D, \nabla_D, \mathcal{F}_D^\bullet, \eta_D)$ forms an $(\mathrm{SL}_n, \mathbb{L})$-oper on $X$.
One verifies that the assignment $D^\clubsuit \mapsto \mathcal{F}_D^\diamondsuit$ turns out to be the inverse of the map $\mathfrak{O} \mathfrak{p}_{(\mathrm{SL}_n, \mathbb{L}), X} \rightarrow \mathfrak{P} \mathfrak{C}^n_{X, \mathbb{L}}$ obtained above, which completes the former assertion.
Next, we shall consider the latter assertion.
Let us take a projective connection $D^\clubsuit$ belonging to $\mathfrak{P} \mathfrak{C}^n_{X, \mathbb{L}}$, and denote by $\mathcal{F}^\diamondsuit := (\mathcal{F}, \nabla_\mathcal{F}, \mathcal{F}^\bullet, \eta_\mathcal{F})$ the corresponding $(\mathrm{SL}_n, \mathbb{L})$-oper constructed by the above steps.
If $D^\clubsuit$ may be expressed (after choosing a local identification $\mathcal{L} \cong \mathcal{O}_X$) locally as $D^\clubsuit = \partial^n + q_1 \partial^{n-1}+ \cdots + q_{n-1}\partial + q_n$ (for a local generator $\partial \in \mathcal{T}_X$ and local sections $q_1, \cdots, q_n \in \mathcal{O}_X$),
then the dual connection $\nabla_\mathcal{F}^\vee$ of $\nabla_\mathcal{F}$
may be expressed locally (with respect to a suitable local basis) as
\begin{align}
\nabla_D^\vee = \partial - \begin{pmatrix} -q_1 & -q_2& -q_3& \cdots & -q_{n-1}& - q_n \\ 1& 0 & 0& \cdots & 0 & \\ 0 & 1 & 0& \cdots & 0 & 0 \\ 0 & 0 &1& \cdots & 0& 0 \\ \vdots & \vdots & \vdots & \ddots& \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 1 & 0 \end{pmatrix}.
\end{align}
Then, $y \mapsto {^t(}\partial^{n-1}(y), \cdots, \partial (y), y)$
gives a bijective correspondence between the solutions of the differential equation $D^\clubsuit(y) =0$ and the horizontal (with respect to $\nabla_\mathcal{F}^\vee$) local sections of $\mathcal{F}^\vee$.
This implies that $D^\clubsuit$ has a full set of solutions if and only if the connection $\nabla_\mathcal{F}^\vee$, as well as $\nabla_\mathcal{F}$, has vanishing $p$-curvature.
Consequently, the bijection $\mathfrak{O} \mathfrak{p}_{(\mathrm{SL}_n, \mathbb{L}), X} \stackrel{\sim}{\migi} \mathfrak{P} \mathfrak{C}^n_{X, \mathbb{L}}$ restricts to a bijection $\mathfrak{O} \mathfrak{p}^{^\mathrm{Zzz...}}_{(\mathrm{SL}_n, \mathbb{L}), X} \stackrel{\sim}{\migi} \mathfrak{P} \mathfrak{C}^{n, \mathrm{full}}_{X, \mathbb{L}}$, as desired.
\end{proof}
\vspace{3mm}
Thus, we have obtained various maps of sets, as displayed below:
\begin{align} \label{E46090}
\vcenter{\xymatrix{
\mathfrak{P} \mathfrak{S}_X^\mathrm{F} \ar[r]_-\sim^-{(\ref{W101})} & \mathfrak{I} \mathfrak{B}^{^\mathrm{Zzz...}}_X \ar[r]_-\sim^-{(\ref{E02234})} \ar[d]^{\mathrm{incl.}} & \mathfrak{O} \mathfrak{p}^{^\mathrm{Zzz...}}_{(\mathrm{SL}_2, \mathbb{L}), X} \ar[r]^-{(\ref{F002})}_-\sim \ar[d]^{\mathrm{incl.}} & \mathfrak{P} \mathfrak{C}^{2, \mathrm{full}}_{X, \mathbb{L}} \ar[d]^{\mathrm{incl.}}\\
& \mathfrak{I} \mathfrak{B}_X \ar[r]^-\sim_-{(\ref{E02f34})} & \mathfrak{O} \mathfrak{p}_{(\mathrm{SL}_2, \mathbb{L}), X} \ar[r]^-\sim_-{(\ref{F001})} & \mathfrak{P} \mathfrak{C}^2_{X, \mathbb{L}},
}}
\end{align}
where
all the vertical arrows are natural inclusions.
Moreover, there is a map for the set of projective connections (resp., having a full set of solutions) to the set of $n$-th order projective connections (resp., having a full set of solutions)
\begin{align} \label{E4289}
\xi^{2 \rightarrow n} : \mathfrak{P} \mathfrak{C}^2_{X, \mathbb{L}} \rightarrow \mathfrak{P} \mathfrak{C}^n_{X, \mathbb{L}} \ \left(\text{resp.,} \ \xi^{2 \rightarrow n} : \mathfrak{P} \mathfrak{C}^{2, \mathrm{full}}_{X, \mathbb{L}} \rightarrow \mathfrak{P} \mathfrak{C}^{n, \mathrm{full}}_{X, \mathbb{L}} \right)
\end{align}
constructed in such a way that the square diagram
\begin{align}
\vcenter{\xymatrix{
\mathfrak{O} \mathfrak{p}_{(\mathrm{SL}_2, \mathbb{L}), X} \ar[r]^{(\ref{Erty7})} \ar[d]^\wr_{(\ref{F001})} & \mathfrak{O} \mathfrak{p}_{(\mathrm{SL}_n, \mathbb{L}), X} \ar[d]_\wr^{(\ref{F001})}
\\
\mathfrak{P} \mathfrak{C}^2_{X, \mathbb{L}} \ar[r]_{\xi^{2 \rightarrow n}} & \mathfrak{P} \mathfrak{C}^n_{X, \mathbb{L}}
}}
\
\left(\text{resp.,} \vcenter{\xymatrix{\mathfrak{O} \mathfrak{p}_{(\mathrm{SL}_2, \mathbb{L}), X} \ar[r]^{(\ref{Erty7})} \ar[d]^\wr_{(\ref{F001})} & \mathfrak{O} \mathfrak{p}_{(\mathrm{SL}_n, \mathbb{L}), X} \ar[d]_\wr^{(\ref{F001})}
\\
\mathfrak{P} \mathfrak{C}^2_{X, \mathbb{L}} \ar[r]_{\xi^{2 \rightarrow n}} & \mathfrak{P} \mathfrak{C}^n_{X, \mathbb{L}}
}} \right)
\end{align}
is commutative.
\vspace{5mm}
\subsection{Case of the projective line} \label{S1}
\leavevmode\\ \vspace{-4mm}
In this subsection, we shall consider the case where
$X= \mathbb{P}^1 \ \left(= \mathrm{Proj}(k [x, y]) \right)$ equipped with the theta characteristic $\mathbb{L}_0$ (cf. (\ref{E09076})).
We will observe that, in this case, there is a typical example of a Frobenius-projective structure (resp., a dormant indigenous bundle; resp., a dormant $(\mathrm{SL}_n, \mathbb{L}_0)$-oper; resp., a projective connection for $\mathbb{L}_0$ having a full set of solutions) on $\mathbb{P}^1$, which will be denoted by $\mathcal{S}_0^\heartsuit$ (resp., $\mathcal{E}^\spadesuit_0$; resp., $\mathcal{F}^{\diamondsuit}_0$; resp., $D_0^\clubsuit$).
First, we define
\begin{align}
\mathcal{S}_0^\heartsuit
\end{align}
to be the subsheaf of $\mathcal{P}^{\text{\'{e}t}}$ (for $X = \mathbb{P}^1$) which, to any open subscheme $U$ of $\mathbb{P}^1$, assigns the set
\begin{align}
S_0^\heartsuit(U) :=
\{ A (\mathrm{op}_U) \in \mathcal{P}^{\text{\'{e}t}}(U) \, | \, A \in (\mathrm{PGL}_{2})_{\mathbb{P}^1}^\mathrm{F} (U) \},
\end{align}
where $\mathrm{op}_U$ denotes the natural open immersion $U \hookrightarrow \mathbb{P}^1$.
Then, $\mathcal{S}_0^\heartsuit$ forms a trivial $(\mathrm{PGL}_2)_{\mathbb{P}^1}^\mathrm{F}$-torsor, and hence, specifies a Frobenius-projective structure on $\mathbb{P}^1$.
Next, we shall write $\mathcal{E}_0 := \mathbb{P}^1 \times \mathbb{P}^1$, which defines
the trivial $\mathbb{P}^1$-bundle on $\mathbb{P}^1$ by
regarding
the first projection $\mathrm{pr}_1 : \mathcal{E}_0 \rightarrow \mathbb{P}^1$
as its structure morphism.
Write $\nabla_0$ for the trivial connection on this trivial $\mathbb{P}^1$-bundle; it is clear that $\nabla_0$ has vanishing $p$-curvature.
The Kodaira-Spencer map (with respect to $\nabla_0$) of the diagonal embedding $\sigma_0 : \mathbb{P}^1 \rightarrow \mathbb{P}^1 \times \mathbb{P}^1 \ \left(= \mathcal{E}_0 \right)$ is nowhere vanishing.
Thus, the triple
\begin{align} \label{E469}
\mathcal{E}^\spadesuit_0 := (\mathcal{E}_0, \nabla_0, \sigma_0)
\end{align}
forms a dormant indigenous bundle on $\mathbb{P}^1$.
Moreover, recall the injection $\eta_0: \mathcal{O}_{\mathbb{P}^1}(-1) \hookrightarrow \mathcal{O}_{\mathbb{P}^1}^{\oplus 2}$ introduced in Example \ref{E5d10}, (\ref{Effgh}) (of the case $n=1$), which we shall
identify with the resulting isomorphism from $\mathcal{O}_{\mathbb{P}^1}(-1)$ onto its image.
Then, the collection
\begin{align}
\mathcal{F}^\diamondsuit_0 := (\mathcal{O}_{\mathbb{P}^1}^{\oplus 2}, d^{\oplus 2}, \mathcal{O}_{\mathbb{P}^1}(-1), \eta_{0})
\end{align}
forms a dormant $(\mathrm{SL}_2, \mathbb{L}_{0})$-oper on $\mathbb{P}^1$.
Finally,
let us consider the second order differential operator $D_{0, u}^\clubsuit : \mathcal{O}_{\mathbb{P}^1}(1)|_U \rightarrow \mathcal{O}_{\mathbb{P}^1}(-3)|_U$ (resp., $D_{0, t}^\clubsuit : \mathcal{O}_{\mathbb{P}^1}(1)|_T \rightarrow \mathcal{O}_{\mathbb{P}^1}(-3)|_T$) on the open subscheme $U := \mathrm{Spec}(k[u])$ (resp., $T := \mathrm{Spec}(k[t])$) of $\mathbb{P}^1$, where $u := x/y$ (resp., $t:= y/x$), given by
$f (u) \cdot y \mapsto \frac{\partial^2 f}{\partial u^2}(u) \cdot y^{-3}$
(resp., $g (t) \cdot x \mapsto \frac{\partial^2 g}{\partial t^2}(t) \cdot x^{-3}$).
Then,
$D_{0, u}^\clubsuit$ and $D_{0, t}^\clubsuit$ may be glued together to obtain a globally defined differential operator
\begin{align}
D_0^\clubsuit : \mathcal{O}_{\mathbb{P}^1}(-1)^\vee \ \left(=\mathcal{O}_{\mathbb{P}^1}(1) \right) \rightarrow \mathcal{O}_{\mathbb{P}^1}(-1)^{\otimes 3} \ \left(= \mathcal{O}_{\mathbb{P}^1}(-3) \right)
\end{align}
forming a projective connection for $\mathbb{L}_0$.
\vspace{3mm}
\begin{prop} \label{P205} \leavevmode\\
\ \ \
All the sets $\mathfrak{P} \mathfrak{S}_{\mathbb{P}^1}^\mathrm{F}$, $\mathfrak{I} \mathfrak{B}_{\mathbb{P}^1}$, $\mathfrak{I} \mathfrak{B}_{\mathbb{P}^1}^{^\mathrm{Zzz...}}$, $\mathfrak{O} \mathfrak{p}_{(\mathrm{SL}_2, \mathbb{L}_0), \mathbb{P}^1}$, $\mathfrak{O} \mathfrak{p}_{(\mathrm{SL}_2, \mathbb{L}_0), \mathbb{P}^1}^{^\mathrm{Zzz...}}$, $\mathfrak{P} \mathfrak{C}_{\mathbb{P}^1, \mathbb{L}_0}^2$, and $\mathfrak{P} \mathfrak{C}_{\mathbb{P}^1, \mathbb{L}_0}^{2, \mathrm{full}}$ are singletons respectively.
That is to say,
\begin{align}
\mathfrak{P} \mathfrak{S}_{\mathbb{P}^1}^\mathrm{F} &= \left\{\mathcal{S}^\heartsuit_0 \right\}, \\
\mathfrak{I} \mathfrak{B}_{\mathbb{P}^1}^{^\mathrm{Zzz...}} & =\mathfrak{I} \mathfrak{B}_{\mathbb{P}^1} = \left\{ \mathcal{E}^{\spadesuit}_0\right\}, \notag\\
\mathfrak{O} \mathfrak{p}_{(\mathrm{SL}_2, \mathbb{L}_0), \mathbb{P}^1} & = \mathfrak{O} \mathfrak{p}_{(\mathrm{SL}_2, \mathbb{L}_0), \mathbb{P}^1}^{^\mathrm{Zzz...}} = \left\{ \mathcal{F}^{\diamondsuit}_0\right\}, \notag \\
\mathfrak{P} \mathfrak{C}^2_{\mathbb{P}^1, \mathbb{L}_0} & = \mathfrak{P} \mathfrak{C}^{2, \mathrm{full}}_{\mathbb{P}^1, \mathbb{L}_0} = \left\{ D^\clubsuit_0 \right\}. \notag
\end{align}
\end{prop}
\begin{proof}
By the various bijections in (\ref{E46090}),
it suffices to prove that $\mathfrak{P} \mathfrak{C}^2_{\mathbb{P}^1, \mathbb{L}_0}$ contains at most one element.
As discussed in the proof of Proposition \ref{L100},
each element of $\mathfrak{P} \mathfrak{C}^2_{\mathbb{P}^1, \mathbb{L}_0}$ corresponds to
a splitting of
the short exact sequence
\begin{align} \label{E009ju}
0 \longrightarrow \mathcal{D}_{\mathbb{P}^1}^{\leq 1} \otimes \mathcal{O}_{\mathbb{P}^1}(-1) \longrightarrow
\mathcal{D}_{\mathbb{P}^1}^{\leq 2} \otimes \mathcal{O}_{\mathbb{P}^1}(-1) \longrightarrow
(\mathcal{D}_{\mathbb{P}^1}^{\leq 2} /\mathcal{D}_{\mathbb{P}^1}^{\leq 1}) \otimes \mathcal{O}_{\mathbb{P}^1}(-1) \longrightarrow 0.
\end{align}
Here, observe that
$(\mathcal{D}_{\mathbb{P}^1}^{\leq 2} /\mathcal{D}_{\mathbb{P}^1}^{\leq 1}) \otimes \mathcal{O}_{\mathbb{P}^1}(-1) \cong \mathcal{O}_{\mathbb{P}^1}(3)$ and $\mathcal{D}_{\mathbb{P}^1}^{\leq 1} \otimes \mathcal{O}_{\mathbb{P}^1}(-1)$ is an extension of $\mathcal{O}_{\mathbb{P}^1}(1)$ by $\mathcal{O}_{\mathbb{P}^1}(-1)$.
Hence, $\mathrm{Hom}_{\mathcal{O}_{\mathbb{P}^1}}((\mathcal{D}_{\mathbb{P}^1}^{\leq 2} /\mathcal{D}_{\mathbb{P}^1}^{\leq 1}) \otimes \mathcal{O}_{\mathbb{P}^1}(-1), \mathcal{D}_{\mathbb{P}^1}^{\leq 1} \otimes \mathcal{O}_{\mathbb{P}^1}(-1)) =0$, which implies that
there is no splitting of (\ref{E009ju}) but the splitting corresponding to $D_0^\clubsuit$.
This completes the proof of the assertion.
\end{proof}
\vspace{3mm}
\begin{rema} \label{R001}
\leavevmode\\
\ \ \
By an argument similar to the argument in the proof of Proposition \ref{P205},
we can prove that both $\mathfrak{P} \mathfrak{C}_{\mathbb{P}^1, \mathbb{L}_0}^n$ and $\mathfrak{P} \mathfrak{C}_{\mathbb{P}^1, \mathbb{L}_0}^{n, \mathrm{full}}$ (for any $n \geq 2$) are singletons.
In particular, the unique element can be obtained as the image $\xi^{2 \rightarrow n} (D_0^\clubsuit)$ of $D_0^\clubsuit$ via $\xi^{2 \rightarrow n}$ (cf. (\ref{E4289})).
On the open subscheme
$U := \mathrm{Spec}(k[u])$ (resp., $T:= \mathrm{Spec}(k[t])$) as before,
the operator $\xi^{2 \rightarrow n} (D_0^\clubsuit) |_{U} : \mathcal{O}_{\mathbb{P}^1} (n-1)|_U \rightarrow \mathcal{O}_{\mathbb{P}^1}(-n-1)|_U$ (resp., $\xi^{2 \rightarrow n} (D_0^\clubsuit) |_{T} : \mathcal{O}_{\mathbb{P}^1} (n-1)|_T \rightarrow \mathcal{O}_{\mathbb{P}^1}(-n-1)|_T$) may be expressed as
\begin{align}
f (u) \cdot y^{n-1} \rightarrow \frac{\partial^n f}{\partial u^n} (u) \cdot y^{-n-1} \ \left(\text{resp.,} \ g (t) \cdot x^{n-1} \rightarrow \frac{\partial^n g}{\partial t^n} (t) \cdot x^{-n-1} \right).
\end{align}
\end{rema}
\vspace{10mm}
\section{The main theorem} \vspace{3mm}
The fourth section is devoted to state and prove the main theorem of the present paper.
\vspace{5mm}
\subsection{Statement of the main theorem} \label{S1g1}
\leavevmode\\ \vspace{-4mm}
Let us fix a smooth curve $X$, a theta characteristic $\mathbb{L} := (\mathcal{L}, \psi_\mathcal{L})$ on $X$.
The morphism
\begin{align} \label{W200}
\psi^\mbA_{\mathcal{L}} : \mbA(\mathcal{L})^\times \rightarrow \mbA(\Omega_X)^\times
\end{align}
over $X$ between algebraic surfaces defined by $\psi^\mbA_{\mathcal{L}} (v) = \psi_\mathcal{L} (v \otimes v)$ (for each local section $v \in \mathcal{L}$)
is a Galois
double covering whose Galois group is isomorphic to $\mu_2 = \{ \pm 1 \}$.
The automorphism of $\mbA(\mathcal{L})^\times$ determined by $(-1) \in \mu_2$ is
given by $v \mapsto - v$.
Let us write
\begin{align} \label{Erww23}
\check{\omega}_{\mathbb{L}} : = (\psi_{\mathcal{L}}^\mbA)^*\left(\check{\omega}^{\mathrm{can}}\right)
\in \Gamma (\mbA(\mathcal{L})^\times, \bigwedge^2 \Omega_{\mbA(\mathcal{L})^\times})
\end{align}
(cf. (\ref{W104})), which
specifies a symplectic structure on $\mbA(\mathcal{L})^\times$.
In particular, the pair
\begin{align}
(\mbA(\mathcal{L})^\times, \check{\omega}_{\mathbb{L}}).
\end{align}
forms a symplectic variety equipped with a $\mu_2$-action.
Then, the main result of the present paper is as follows.
\vspace{3mm}
\begin{thm} \label{T0090}\leavevmode\\
\ \ \
There exists a canonical construction of a $\mu_2$-FC quantization on $(\mbA (\mathcal{L})^\times, \check{\omega}_\mathbb{L})$ by means of a Frobenius-projective structure (or equivalently, a dormant indigenous bundle, a dormant $(\mathrm{SL}_2, \mathbb{L})$-oper, or a projective connection for $\mathbb{L}$ having a full set of solutions) on $X$.
The resulting map of sets
\begin{align}
\bigstar_{X, \mathbb{L}} : \mathfrak{P} \mathfrak{S}^\mathrm{F}_X \rightarrow \mathfrak{Q}^{\mu_2 \text{-}\mathrm{FC}}_{(\mbA(\mathcal{L})^\times, \check{\omega}_\mathbb{L})}
\end{align}
is injective and the composite injection
\begin{align}
\bigstar_X : \mathfrak{P} \mathfrak{S}^\mathrm{F}_X \hookrightarrow \mathfrak{Q}^\mathrm{FC}_{(\mbA(\Omega_X)^\times, \check{\omega}^\mathrm{can})}
\end{align}
of this map and the natural bijection $\mathfrak{Q}^{\mu_2 \text{-}\mathrm{FC}}_{(\mbA(\mathcal{L})^\times, \check{\omega}_\mathbb{L})} \stackrel{\sim}{\migi} \mathfrak{Q}^{\mathrm{FC}}_{(\mbA(\Omega_X)^\times, \check{\omega}^\mathrm{can})}$ (i.e., the inverse of (\ref{W1005})) does not depend on the choice of the theta characteristic $\mathbb{L}$.
\end{thm}
\vspace{3mm}
In
the rest of this section, we shall will prove the above theorem.
\vspace{5mm}
\subsection{Step I: Local construction} \label{S11}
\leavevmode\\ \vspace{-4mm}
Let
$\mathcal{S}^\heartsuit$
be a Frobenius-projective structure on $X$, and denote by $\mathcal{E}^\spadesuit := (\mathcal{E}, \nabla_\mathcal{E}, \sigma_\mathcal{E})$ and $\mathcal{F}^\diamondsuit := (\mathcal{F}, \nabla_\mathcal{F}, \mathcal{N}_\mathcal{F}, \eta_\mathcal{F})$ the corresponding indigenous bundle and $(\mathrm{SL}_2, \mathbb{L})$-oper respectively.
In this first step, we construct FC quantizations on various open subschemes of $(\mbA (\mathcal{L})^\times, \check{\omega}_\mathbb{L})$.
Since $\nabla_\mathcal{F}$ has vanishing $p$-curvature,
$(\mathcal{F}, \nabla_\mathcal{F})$ is locally trivial.
More precisely,
there exists a collection
\begin{align} \label{E004}
\{ (U_\alpha, \gamma_\alpha ) \}_{\alpha \in I}
\end{align}
of pairs $(U_\alpha, \gamma_\alpha)$ (indexed by a set $I$), where
$\{ U_\alpha \}_{\alpha \in I}$ is an open covering of $X$ and
$\gamma_\alpha$ (for each $\alpha \in I$) denotes an $\mathcal{O}_{U_\alpha}$-linear isomorphism $\mathcal{F} |_{U_\alpha} \stackrel{\sim}{\migi} \mathcal{O}_{U_\alpha}^{\oplus 2}$
inducing, via taking determinants, the isomorphism $\delta_{\mathcal{F}^\diamondsuit} |_{U_\alpha}$ (cf. (\ref{Effgh3})).
Each isomorphism $\gamma_\alpha$ induces an isomorphism
\begin{align}
\gamma^\mathbb{A}_{\alpha} : \mbA (\mathcal{F} |_{U_\alpha})^\times \stackrel{\sim}{\migi} \left(\mbA (\mathcal{O}_{U_\alpha}^{\oplus 2})^\times\stackrel{\sim}{\migi} \right) \ U_\alpha \times \mathbb{A}^{2 \times} \end{align}
over $U_\alpha$, where $\mathbb{A}^{2 \times} := \mathrm{Spec}(k[x, y])$ and the isomorphism in the parenthesis is given by $x \mapsto (1, 0)$, $y \mapsto (0, 1)$.
Moreover,
denote by $\Phi_\alpha$ the composite
\begin{align} \label{Ert52}
\Phi_\alpha : \mbA (\mathcal{L} |_{U_\alpha})^\times \xrightarrow{\eta_\mathcal{F}^\mbA |_{U_\alpha}} \mbA (\mathcal{F} |_{U_\alpha})^\times \xrightarrow{\gamma_{\alpha, \mbA}}
U_\alpha \times \mathbb{A}^{2 \times} \xrightarrow{\mathrm{pr}_2} \mathbb{A}^{2 \times},
\end{align}
which is compatible with the respective $\mu_2$-actions on $\mbA (\mathcal{L} |_{U_\alpha})^\times$ and $\mathbb{A}^{2 \times}$.
The morphism $\mbA (\Omega_{U_\alpha})^\times \stackrel{\sim}{\migi} \mathbb{A}^{2 \times}_{/\mu_2}$ induced via quotient does not depend on the choice of $\mathbb{L}$.
\vspace{3mm}
\begin{lemma} \label{Lff1}\leavevmode\\
\ \ \
The morphism $\Phi_\alpha$ is \'{e}tale and satisfies the equality $\Phi_\alpha^*(\omega^{\mathrm{Weyl}}) = \check{\omega}_\mathbb{L} |_{U_\alpha}$.
\end{lemma}
\begin{proof}
To begin with, we shall consider the case where $X =U_\alpha = \mathbb{P}^1 \ (= \mathrm{Proj}(k [x, y]))$, $\mathbb{L} = \mathbb{L}_0$, $\mathcal{F}^\diamondsuit = \mathcal{F}_0^\diamondsuit$, and $\gamma_\alpha$ is the identity of $\mathcal{O}_{\mathbb{P}^1}^{\oplus 2}$.
Write $\Phi_0$ for the morphism ``$\Phi_\alpha$" in this case.
On the open subscheme $U := \mathrm{Spec}(k[u])$ of $\mathbb{P}^1$,
where $u = x/y$,
we have $\mathbb{A}(\mathcal{O}_{\mathbb{P}^1}(-1))^\times |_{U} = \mathrm{Spec}(k[u, y, y^{-1}])$, $\mathbb{A}^{2 \times}|_U = \mathrm{Spec}(k[x, y, y^{-1}])$ and $\Phi_0 |_U$ may be given by
$x \mapsto u \cdot y$ and $y \mapsto y$.
Hence,
\begin{align} \label{E8999}
\Phi_0^* (\omega^\mathrm{Weyl})|_U = \Phi^*_0 (dy \wedge dx)= dy \wedge d(u \cdot y) = y \cdot dy \wedge du.
\end{align}
On the other hand, it follows from the definition of $\psi_0$ that
$\psi_0^\mathbb{A} : \mbA (\mathcal{O}_{\mathbb{P}^1}(-1))^\times \rightarrow \mbA (\Omega_{\mathbb{P}^1})^\times$ is given by assigning $f \mapsto (y \cdot f)^2\cdot du$ for each $f \in \Gamma (U, \mathcal{O}_{\mathbb{P}^1}(-1)^\times)$.
If $u^\vee$ denote the dual coordinate of $u$ in $\mbA (\Omega_{\mathbb{P}^1})^\times$,
then
\begin{align} \label{E8989}
\check{\omega}_{\mathbb{L}_0}|_{U} = (\psi^\mathbb{A}_{0})^*(\check{\omega}^\mathrm{can}|_{U}) = (\psi^\mathbb{A}_{0})^*\left(\frac{1}{2} \cdot d u^\vee \wedge du\right) = \frac{1}{2} \cdot dy^2 \wedge du = y \cdot dy \wedge du.
\end{align}
By (\ref{E8999}) and (\ref{E8989}), we obtain the desired equality
\begin{align} \label{Ertgbn}
\Phi_0^* (\omega^\mathrm{Weyl}) = \check{\omega}_{\mathbb{L}_0}
\end{align}
of this case.
(This result will be used in (\ref{Fgghyn}).)
Now, let us go back to our situation.
Denote by
\begin{align}
\gamma^\mathbb{P}_{\alpha} : \mathcal{E} |_{U_\alpha} \stackrel{\sim}{\migi} U_\alpha \times \mathbb{P}^1
\end{align}
the isomorphism induced from $\gamma^\mathbb{A}_{\alpha}$ via projectivization.
Denote by $\phi_\alpha$ the composite
\begin{align}
\phi_\alpha : U_\alpha \xrightarrow{\sigma_\mathcal{E} |_{U_\alpha}} \mathcal{E} |_{U_\alpha} \xrightarrow{\gamma_{\alpha}^\mathbb{P}} U_\alpha \times \mathbb{P}^1 \xrightarrow{\mathrm{pr}_2} \mathbb{P}^1,
\end{align}
which is verified to be \'{e}tale since the Kodaira-Spencer map
associated to $\sigma_\mathcal{E}$ is an isomorphism.
Under the natural identifications
$U_\alpha \times \mathbb{A}^{2 \times} \stackrel{\sim}{\migi} \phi_\alpha^*(\mbA (\mathcal{O}_{\mathbb{P}^1}^{\oplus 2})^\times)$
and
$U_\alpha \times \mathbb{P}^1 \stackrel{\sim}{\migi} \phi_\alpha^*(\mathcal{E}_0)$ (where $\phi_\alpha^*(-)$ denotes base-change by $\phi_\alpha$), we obtain a commutative diagram
\begin{align}
\vcenter{\xymatrix{ & & & \mbA (\mathcal{F} |_{U_\alpha})^\times \ar[dd]^{\pi_{\mathcal{F}|_{U_\alpha}}} \ar[dl]^\sim_{\gamma^\mathbb{A}_{\alpha}} \\
& & \phi_\alpha^*(\mbA (\mathcal{O}_{\mathbb{P}^1}^{\oplus 2})^\times) \ar[dd]^(.35){\phi_\alpha^*(\pi_{\mathcal{O}_{\mathbb{P}^1}^{\oplus 2}})}& \\
& U_\alpha \ar[dl]^\sim_{\mathrm{id}_{U_\alpha}} \ar[rr]^{\hspace{-25mm}\sigma_\mathcal{E} |_{U_\alpha}}|(.45)\hole& & \mathcal{E} |_{U_\alpha} \ar[ld]_\sim^{\gamma_{\alpha}^\mathbb{P}} \\
U_\alpha \ar[rr]_{\phi_\alpha^*(\sigma_0)}& & \phi_\alpha^*(\mathcal{E}_0).&
}}
\end{align}
Since (\ref{Ddd01}) is cartesian, the above diagram induces an isomorphism (of $\mathbb{G}_m$-torsors)
\begin{align} \label{E3321}
\underline{\gamma}_{\alpha}^\mathbb{A} : \mbA (\mathcal{L} |_{U_\alpha})^\times \stackrel{\sim}{\migi} \phi_\alpha^*(\mathbb{A} (\mathcal{O}_{\mathbb{P}^1}(-1))^\times)
\end{align}
over $U_\alpha$ such that the following diagram is commutative:
\begin{align} \label{Ertyui}
\vcenter{\xymatrix{ & \mbA (\mathcal{L} |_{U_\alpha})^\times \ar[rr]^{\eta^\mathbb{A}_\mathcal{F} |_{U_\alpha}} \ar[dl]_{\underline{\gamma}_{\alpha}^\mathbb{A}}^\sim \ar[dd]|(.50)\hole& & \mbA (\mathcal{F} |_{U_\alpha})^\times \ar[dd]^{\pi_{\mathcal{F}|_{U_\alpha}}} \ar[dl]_\sim^{\gamma_{\alpha}^\mathbb{A}} \\
\phi^*_\alpha (\mbA (\mathcal{O}_{\mathbb{P}^1}(-1))^\times) \ar[rr]^{\hspace{20mm}\phi^*_\alpha (\eta^\mathbb{A}_{0})} \ar[dd] & & \phi_\alpha^*(\mbA (\mathcal{O}_{\mathbb{P}^1}^{\oplus 2})^\times) \ar[dd]^(.35){\phi_\alpha^*(\pi_{\mathcal{O}_{\mathbb{P}^1}^{\oplus 2}})}& \\
& U_\alpha \ar[dl]^-\sim_-{\mathrm{id}_{U_\alpha}} \ar[rr]^{\hspace{-20mm}\sigma_\mathcal{E} |_{U_\alpha}}|(.50)\hole& & \mathcal{E} |_{U_\alpha} \ar[ld]_\sim^{\gamma_{\alpha}^\mathbb{P}} \\
U_\alpha \ar[rr]_{\phi_\alpha^*(\sigma_0)}& & \phi_\alpha^*(\mathcal{E}_0).&
}}
\end{align}
Moreover, the morphism (\ref{E3321}) induces an $\mathcal{O}_{U_\alpha}$-linear isomorphism
\begin{align} \label{E4291}
\underline{\gamma}_\alpha : \mathcal{L}|_{U_\alpha} \stackrel{\sim}{\migi} \phi_\alpha^*(\mathcal{O}_{\mathbb{P}^1}(-1))
\end{align}
fitting into the following isomorphism of short exact sequences:
\begin{align}
\vcenter{\xymatrix{
0 \ar[r]& \mathcal{L} |_{U_\alpha} \ar[r]^{\eta_\mathcal{F}} \ar[d]_\wr^{\underline{\gamma}_\alpha}& \mathcal{F} |_{U_\alpha} \ar[d]_\wr^{\gamma_\alpha} \ar[r] & \mathcal{L}^\vee |_{U_\alpha} \ar[r] \ar[d]_\wr^{(\underline{\gamma}_\alpha^{\vee})^{-1}}& 0 \\
0 \ar[r] & \phi^*_\alpha (\mathcal{O}_{\mathbb{P}^1}(-1)) \ar[r] & \phi_\alpha (\mathcal{O}_{\mathbb{P}^1}^{\oplus 2}) \ar[r] & \phi^*_\alpha (\mathcal{O}_{\mathbb{P}^1}(-1)) \ar[r]& 0,
}}
\end{align}
where the upper right-hand horizontal arrow $\mathcal{F} |_{U_\alpha} \rightarrow \mathcal{L}^\vee |_{U_\alpha}$ arises from $\delta_{\mathcal{F}^\diamondsuit} : \mathrm{det}(\mathcal{F}) \stackrel{\sim}{\migi} \mathcal{O}_X$ (cf. (\ref{Effgh3})).
Since $\gamma_\alpha$ is, moreover, compatible with the connections $\nabla_\mathcal{F}$ and $d^{\oplus 2}$, the respective Kodaira-Spencer maps
give rise to
a commutative square diagram of $\mathcal{O}_{U_\alpha}$-modules
\begin{align}
\vcenter{\xymatrix@C=40pt{
\mathcal{L}^{\otimes 2} |_{U_\alpha} \ar[r]_\sim^{\psi_{\mathcal{L}}|_{U_\alpha}} \ar[d]^\wr_{\underline{\gamma}_\alpha^{\otimes 2}} & \Omega_{U_\alpha} \ar[d]_\wr^{\phi_\alpha} \\
\phi_\alpha^*(\mathcal{O}_{\mathbb{P}^1}(-1)^{\otimes 2}) \ar[r]^-\sim_-{\phi_\alpha^*(\psi_{0})} &\phi_\alpha^*(\Omega_{\mathbb{P}^1}),
}}
\end{align}
as well as a commutative diagram of $U_\alpha$-schemes
\begin{align} \label{E687ff}
\vcenter{\xymatrix@C=44pt{
\mbA (\mathcal{L} |_{U_\alpha})^\times \ar[d]^{\wr}_{\underline{\gamma}_\alpha^\mathbb{A}} \ar[r]^{\psi_\mathcal{L}^\mathbb{A} |_{U_\alpha}}& \mbA (\Omega_{U_\alpha})^\times \ar[d]_\wr^{\phi_\alpha^{\mathbb{A}}} \\
\phi_\alpha^*(\mbA (\mathcal{O}_{\mathbb{P}^1}(-1))^\times) \ar[r]_-{\phi_\alpha^*(\psi_{\mathbb{L}_0}^\mathbb{A})}& \phi_\alpha^*(\mathbb{A} (\Omega_{\mathbb{P}^1})).
}}
\end{align}
Hence, the following sequence of equalities holds:
\begin{align} \label{Fgghyn}
\Phi_\alpha^*(\omega^\mathrm{Weyl}) &\stackrel{(\ref{Ert52})}{=} (\gamma_{\alpha}^\mathbb{A} \circ \eta_\mathcal{F}^\mathbb{A} |_{U_\alpha})^*(\mathrm{pr}_2^*(\omega^{\mathrm{Weyl}})) \\
&\stackrel{(\ref{Ertyui})}{=} (\phi_\alpha^*(\eta_0^\mathbb{A}) \circ \underline{\gamma}_{\alpha}^\mathbb{A})^*(\mathrm{pr}_2^*(\omega^\mathrm{Weyl})) \notag \\
&= \underline{\gamma}_{\alpha}^{\mathbb{A}*}(\Phi_0^*(\omega^\mathrm{Weyl})) \notag \\
&\stackrel{(\ref{Ertgbn})}{=} \underline{\gamma}_{\alpha}^{\mathbb{A}*} (\check{\omega}_{\mathbb{L}_0}) \notag \\
&\stackrel{(\ref{Erww23})}{=} \underline{\gamma}_{\alpha}^{\mathbb{A}*} \left(\frac{1}{2} \cdot (\psi_0^\mathbb{A})^*(\omega^\mathrm{can}_{\mathbb{P}^1})\right) \notag \\
&\stackrel{(\ref{E687ff})}{=} \frac{1}{2} \cdot (\phi_\alpha^{\mathbb{A}*}\circ\psi_\mathcal{L}^\mathbb{A})^*(\omega_{\mathbb{P}^1}^\mathrm{can}) \notag \\
& = \frac{1}{2} \cdot \psi_{\mathcal{L}}^{\mathbb{A}*}(\omega_{U_\alpha}^\mathrm{can}) \notag \\
& = \check{\omega}_\mathbb{L} |_{U_\alpha}. \notag
\end{align}
This completes the proof of the latter assertion.
The former assertion, i.e., the \'{e}taleness of $\Phi_\alpha$ follows from the
\'{e}taleness of $\phi_\alpha$ and the fact that the square diagram
\begin{align}
\vcenter{\xymatrix{
\mathbb{A}(\mathcal{L} |_{U_\alpha})^\times \ar[r]^-{\Phi_\alpha} \ar[d] & \mathbb{A}^{2 \times} \ar[d]\\
U_\alpha \ar[r]_{\phi_\alpha} & \mathbb{P}^1
}}
\end{align}
is cartesian, where the vertical arrows are the natural projections.
This completes the proof of the lemma.
\end{proof}
\vspace{3mm}
By the above lemma and the discussion in \S\,\ref{S10} applied to $\Phi_\alpha$,
the pull-back of the $\mu_2$-FC quantization $\mathcal{W}^2_k$ (cf. (\ref{Efgklo})) on $(\mathbb{A}^{2\times}, \omega^\mathrm{Weyl})$ specifies
a $\mu_2$-FC quantization
\begin{align} \label{W201}
\Phi^*_\alpha(\mathcal{W}^2_k)
\end{align}
on the symplectic variety $(\mbA (\mathcal{L} |_{U_\alpha})^\times, \check{\omega}_\mathbb{L} |_{U_\alpha})$.
The (isomorphism class of the) FC quantization on $(\mbA (\Omega_{U_\alpha}), \check{\omega}^\mathrm{can})$
corresponding to
$\Phi^*_\alpha(\mathcal{W}^2_k)$ via (\ref{W1005}) does not depend on the choice of $\mathbb{L}$.
\vspace{5mm}
\subsection{Step II: Global construction} \label{S11}
\leavevmode\\ \vspace{-4mm}
In this second step, we glue together the locally defined quantizations constructed above to obtain an FC quantization on the entire space $X$, as follows.
After possibly replacing $\{U_\alpha \}_\alpha$ with its refinement, we can assume that each $U_\alpha$ is affine.
Let us take a pair $(\alpha, \beta) \in I \times I$ with $U_{\alpha\beta} := U_\alpha \cap U_\beta \neq \emptyset$.
Since ($X$ is separated, which implies that) $U_{\alpha \beta}$ is affine, we can write $U_{\alpha \beta} = \mathrm{Spec}(R_{\alpha \beta})$ for some $k$-algebra $R_{\alpha \beta}$.
In what follows, we shall use the notation $(-)^{(1)}$ to denote the base-change of objects via $F_k$.
In particular, we obtain the $k$-algebra $R^{(1)}_{\alpha \beta}$ equipped with a $k$-algebra (injective) homomorphism
$R^{(1)}_{\alpha \beta} \rightarrow R_{\alpha \beta}$.
Also, write $R_{\mathcal{L}, \alpha \beta} := \Gamma (\mbA (\mathcal{L} |_{U_{\alpha \beta}})^\times, \mathcal{O}_{\mbA (\mathcal{L} |_{U_{\alpha \beta}})^\times})$.
If $\mathcal{I}_\alpha$ denotes the ideal of $R_{\alpha \beta}^{}[x, y]$ determined by the closed immersion
\begin{align}
(\gamma_{\alpha}^\mathbb{A}\circ \eta_\mathcal{F}^\mathbb{A}|_{U_\alpha}) |_{U_{\alpha \beta}}:
\mbA (\mathcal{L} |_{U_{\alpha \beta}})^\times \ \left(= \mathrm{Spec}(R_{\mathcal{L}, \alpha \beta}) \right) \hookrightarrow \mathbb{A}^{2 \times} \times U_{\alpha \beta} \ \left(= \mathrm{Spec} (R_{\alpha \beta}[x, y]) \right).
\end{align}
Hence, $R_{\alpha \beta}[x, y]/\mathcal{I}_\alpha \cong R_{\mathcal{L}, \alpha \beta}$ and
we have a natural isomorphism of $R^{(1)}_{\alpha \beta}[x^p, y^p]/\mathcal{I}_\alpha^{(1)}$ ($= \Gamma ((\mbA (\mathcal{L} |_{U_{\alpha \beta}})^\times)^{(1)}, \mathcal{O}_{(\mbA (\mathcal{L} |_{U_{\alpha \beta}})^\times)^{(1)}})$)-algebras
\begin{align} \label{W210}
\Gamma (\mbA (\mathcal{L} |_{U_{\alpha \beta}})^\times, \Phi_\alpha^*(\mathcal{W}_k^2)) \cong W^2_{R_{\alpha \beta}} \otimes_{R_{\alpha \beta}^{(1)}[x^p, y^p]} (R^{(1)}_{\alpha \beta}[x^p, y^p]/\mathcal{I}_\alpha^{(1)}).
\end{align}
Next,
let $\gamma_{\alpha \beta}$ be the automorphism of $R_{\alpha \beta}[x, y]$
corresponding to
$(\gamma_\alpha |_{U_{\alpha \beta}}) \circ (\gamma_\beta |_{U_{\alpha \beta}})^{-1} \in \mathrm{SL}_2 (R_{\alpha \beta}) \ \left(= \mathrm{Sp}(R_{\alpha \beta}) \right)$.
This automorphism restricts to an automorphism of $R_{\alpha \beta}[x^p, y^p]$, and hence, we have the following diagram
\begin{align} \label{E02jj5}
\xymatrix@R=6pt{
&& R_{\mathcal{L}, \alpha \beta} &&
\\
&R_{\alpha \beta}[x, y] \ar[ru]&& R_{\alpha \beta}[x, y] \ar[ll]^{\hspace{20mm}\gamma_{\alpha \beta}}_{\hspace{19mm}\sim} \ar[lu]&
\\
&& R_{\mathcal{L}, \alpha \beta}^{(1)} \ar[uu]|(.51)\hole &&
\\
&R_{\alpha \beta}^{(1)}[x^p, y^p] \ar[uu] \ar[ru] && R_{\alpha \beta}^{(1)}[x^p, y^p]. \ar[ll]^{\gamma_{\alpha \beta}}_\sim \ar[uu]\ar[lu]&
}
\end{align}
The bottom triangle in (\ref{E02jj5}) turns out to be commutative since the other various small diagrams are commutative.
This implies that
$\gamma_{\alpha \beta} (\mathcal{I}_\alpha^{(1)}) = \mathcal{I}_\beta^{(1)}$.
Hence, the automorphism
of the $R_{\alpha \beta}^{(1)}$-algebras $W^2_{R_{\alpha \beta}}$ given by
$\gamma_{\alpha \beta}$ via (\ref{E4522})
induces an isomorphism
\begin{align}
W^2_{R_{\alpha \beta}} \otimes_{R_{\alpha \beta}^{(1)}[x^p, y^p]} (R_{\alpha \beta}^{(1)}[x^p, y^p]/\mathcal{I}_\beta^{(1)}) \stackrel{\sim}{\migi} W^2_{R_{\alpha \beta}} \otimes_{R_{\alpha \beta}^{(1)}[x^p, y^p]} (R_{\alpha \beta}^{(1)}[x^p, y^p]/\mathcal{I}_\alpha^{(1)})
\end{align}
By passing to (\ref{W210}), we obtain an isomorphism
\begin{align}
\Phi_{\alpha \beta} : \Phi_\beta^*(\mathcal{W}_k^2) |_{\mbA (\mathcal{L} |_{U_{\alpha \beta}})^\times} \stackrel{\sim}{\migi} \Phi_\alpha^*(\mathcal{W}_k^2)|_{\mbA (\mathcal{L} |_{U_{\alpha \beta}})^\times}
\end{align}
of sheaves of $k[[\hslash]]$-algebras on $\mbA (\mathcal{L} |_{U_{\alpha \beta}})^\times$.
This isomorphism is verified to be compatible with the $\mu_2$-actions, and the induced isomorphism between the respective quotient FC quantizations does not depend on the choice of $\mathbb{L}$.
By means of the isomorphisms $\Phi_{\alpha \beta}$ (for various $\alpha$, $\beta$),
the sheaves $\Phi_\alpha^*(\mathcal{W}_k^2)$
may be glued together to obtain a sheaf
\begin{align}
\mathcal{W}_{\mathcal{S}^\heartsuit},
\end{align}
which forms a $\mu_2$-FC quantization on $(\mbA (\mathcal{L})^\times, \check{\omega}_\mathbb{L})$.
The isomorphism class of this quantization does not depend on the choice of $\{ (U_\alpha, \gamma_\alpha) \}_{\alpha \in I}$, and moreover, the isomorphism class of its quotient by the $\mu_2$-action does not depend on the choice of $\mathbb{L}$.
Thus, we have obtained a well-defined map
\begin{align}
\bigstar_{X, \mathbb{L}} : \mathfrak{P} \mathfrak{S}^F_X \rightarrow \mathfrak{Q}^{\mu_2 \text{-}\mathrm{FC}}_{(\mbA (\mathcal{L})^\times, \check{\omega}_\mathbb{L})}
\end{align}
such that the composite
\begin{align}
\bigstar_X : \mathfrak{P} \mathfrak{S}^F_X \rightarrow \mathfrak{Q}^{\mathrm{FC}}_{(\mbA (\Omega_X)^\times, \check{\omega}^\mathrm{can})}
\end{align}
of this map and the bijection $\mathfrak{Q}_{(\mbA (\mathcal{L})^\times, \check{\omega}_\mathbb{L})}^{\mu_2 \text{-}\mathrm{FC}}\stackrel{\sim}{\migi} \mathfrak{Q}^\mathrm{FC}_{(\mbA (\Omega_X)^\times, \check{\omega}^\mathrm{can})}$ does not depend on the choice of $\mathbb{L}$.
\vspace{5mm}
\subsection{Step III: Injectivity} \label{S62}
\leavevmode\\ \vspace{-4mm}
The remaining portion of the proof is the injectivity of $\bigstar_{X, \mathbb{L}}$.
Here, let $(S, \omega_S)$ be a symplectic variety and $\mathcal{O}_S^\hslash$ an FC quantization on $(S, \omega_S)$.
Given two local sections $a, b \in \mathcal{O}_S$, we express $a * b$ as
\begin{align}
a * b = \delta_b^0(a) + \delta_b^1 (a) \cdot \hslash + \delta_b^2 (a) \cdot \hslash^2 + \cdots \in \mathcal{O}_{S} [[\hslash ]]
\end{align}
for some local sections $\delta_b^i (a) \in \mathcal{O}_S$ ($i=0,1,2, \cdots$).
For each $i$, the assignment $b \mapsto \delta_b^i (a)$ defines a (locally defined)
$k$-linear endomorphism $\delta_b^i \in \mathcal{E} nd_k (\mathcal{O}_{S})$.
The assignment $b \mapsto \delta_b^i$ gives an $\mathcal{O}_{S}$-linear morphism
\begin{align}
\delta^i : \mathcal{O}_{S} \rightarrow \mathcal{E} nd_k (\mathcal{O}_{S}),
\end{align}
where we equip $\mathcal{E} nd_k (\mathcal{O}_{S})$ with a structure of $\mathcal{O}_S$-module given by multiplication on the left.
\vspace{3mm}
\begin{lemma} \label{L200} \leavevmode\\
\ \ \
If the triple $(S, \omega_S, \mathcal{O}_S^\hslash)$ is taken to be $(\mbA (\mathcal{L})^\times, \check{\omega}_\mathbb{L}, \mathcal{W}_{\mathcal{S}^\heartsuit})$ (as discussed in the previous subsection),
then
for each $i < p$ the image of $\delta^i$ is contained in $\mathcal{D}^{\leq i}_{\mbA (\mathcal{L})^\times}$.
\end{lemma}
\begin{proof}
By the local nature of the assertion, it suffices to prove this lemma with $\mbA (\mathcal{L})^\times$ replaced with each open subscheme $\mbA (\mathcal{L} |_{U_\alpha})^\times$.
Moreover, since
$\Phi_\alpha^*({^l \mathcal{D}}^{\leq i}_{\mbA^{2\times}})\cong {^l \mathcal{D}}^{\leq i}_{\mbA (\mathcal{L}|_{U_\alpha})^\times}$,
it suffices to consider the case where $(S, \omega_S, \mathcal{O}_S^\hslash)$ is taken to be $(\mathbb{A}^{2 \times}, \omega^\mathrm{Weyl}, \mathcal{W}_k^2)$.
Then, it follows from the definition of the multiplication in $W_k^2$ that for each $a, b \in k[x, y]$, we have $\delta^i_b(a) = \frac{1}{i!} \cdot \frac{\partial^i a}{\partial y^i}\cdot \frac{\partial^i b}{\partial x^i}$.
This completes the proof of the lemma.
\end{proof}
\vspace{3mm}
We shall finish the proof of the main theorem.
Since $\pi_*( \mathcal{O}_{\mbA (\mathcal{L})^\times})$ is naturally identified with $\bigoplus_{j \in \mathbb{Z}}\mathcal{L}^{\otimes j}$, we have the inclusion
\begin{align} \label{Errt}
\mathcal{L}^{\otimes (-i+1)} \hookrightarrow
\pi_*( \mathcal{O}_{\mbA (\mathcal{L})^\times})
\end{align}
into the $(-i+1)$-st factor.
Next, if $\pi$ denotes the natural projection $\mathbb{A} (\mathcal{L})^\times \rightarrow X$, then
the kernel of the surjection $\mathcal{T}_{\mbA (\mathcal{L})^\times} \twoheadrightarrow\pi^*(\mathcal{T}_{X})$
obtained by differentiating $\pi$ is isomorphic to $\pi^*(\mathcal{L})$.
The resulting injection $\pi^*(\mathcal{L}) \hookrightarrow \mathcal{T}_{\mbA (\mathcal{L})^\times}$ induces an injection
\begin{align} \label{Ertygfd}
\pi^*(\mathcal{L}^{\otimes i}) \hookrightarrow \mathcal{T}_{\mbA (\mathcal{L})^\times}^{\otimes i}.
\end{align}
The projection formula gives the composite isomorphism
\begin{align} \label{F9090}
\pi_*(\pi^*(\mathcal{L}^{\otimes i})) \stackrel{\sim}{\migi} \pi_*(\mathcal{O}_{\mbA (\mathcal{L})^\times})\otimes \mathcal{L}^{\otimes i} \stackrel{\sim}{\migi} \left(\bigoplus_{j \in \mathbb{Z}} \mathcal{L}^{\otimes j}\right) \otimes \mathcal{L}^{\otimes i} = \bigoplus_{j \in \mathbb{Z}} \mathcal{L}^{\otimes (j+i)}.
\end{align}
Hence, we have an injection
\begin{align} \label{Eghj}
\mathcal{L}^{\otimes (i+1)} \hookrightarrow \bigoplus_{j \in \mathbb{Z}} \mathcal{L}^{\otimes (j+i)} \xrightarrow{(\ref{F9090})} \pi_*(\pi^*(\mathcal{L}^{\otimes i})) \xrightarrow{(\ref{Ertygfd})}\pi_*(\mathcal{T}_{\mbA (\mathcal{L})^\times}^{\otimes i}),
\end{align}
where the first arrow is the inclusion into the first factor.
\vspace{3mm}
\begin{prop} \label{L203} \leavevmode\\
\ \ \
Let $i$ be an integer with $2 \leq i < p$ and let $\delta^i : \mathcal{O}_{\mathbb{A} (\mathcal{L})^\times} \rightarrow \mathcal{D}_{\mathbb{A} (\mathcal{L})^\times}^{\leq i}$ be the morphism resulting from Lemma \ref{L200}.
Denote by $D^\clubsuit$ the projective connection for $\mathbb{L}$ corresponding to $\mathcal{S}^\heartsuit$ via the upper horizontal bijections in the diagram (\ref{E46090}).
Then, the following diagram is commutative:
\begin{align} \label{Efgmaob}
\vcenter{\xymatrix@C=46pt{
\mathcal{L}^{\otimes (-i +1)} \ar[rr]^{\frac{1}{i!} \cdot \xi^{2 \rightarrow i} (D^\clubsuit)} \ar[d]_{(\ref{Errt})} & & \mathcal{L}^{\otimes (i+1)} \ar[d]^{(\ref{Eghj})}
\\
\pi_*(\mathcal{O}_{\mbA (\mathcal{L})^\times}) \ar[r]_{\pi_*( \delta^i)} & \pi_* (\mathcal{D}^{\leq i}_{\mathbb{A} (\mathcal{L})^\times}) \ar[r]_{\pi_*(\Sigma)} &\pi_*(\mathcal{T}^{\otimes i}_{\mbA (\mathcal{L})^\times}).
}}
\end{align}
In particular, (by considering the case of $i=2$), the map $\bigstar_X$ is injective.
\end{prop}
\begin{proof}
First, we shall consider the former assertion.
Just as in the proof of Lemma \ref{L200},
it suffices to consider the case where the triple $(X, \mathbb{L}, \mathcal{S}^\heartsuit)$ is taken to be $(\mathbb{P}^1, \mathbb{L}_0, \mathcal{S}_0^\heartsuit)$.
Let us identify $\mathbb{A} (\mathcal{O}_{\mathbb{P}^1}(-1))^\times$ with $\mathbb{A}^{2 \times}$ via $\Phi_0$ (cf. the proof in Lemma \ref{Lff1} for the definition of $\Phi_0$).
Over the open subsheme $U:= \mathrm{Spec}(k[u])$ (where $u:= x/y$) of $\mathbb{P}^1$,
the composite $\Sigma \circ \delta^i$ sends each element $f (u) \cdot y^{i-1} \in \Gamma (U, \mathcal{O}_{\mathbb{P}^1}(i-1))$ to
\begin{align} \label{Efghhy}
\frac{1}{i!} \cdot \frac{\partial^i}{\partial x^i}(f(u) \cdot y^{i-1}) \cdot \left(\frac{\partial}{\partial y} \right)^{\otimes i} = \frac{1}{y \cdot i!} \cdot \frac{\partial^if}{\partial u^i}(u) \cdot \left(\frac{\partial}{\partial y} \right)^{\otimes i} \in \Gamma (\pi^{-1}(U), \mathcal{T}_{\mathbb{A}^{2\times}}^{\otimes i}).
\end{align}
On the other hand, since the injection $\mathcal{O}_{\mathbb{P}^1}(-i-1) \rightarrow \pi_*(\mathcal{T}^{\otimes i}_{\mathbb{A} (\mathcal{O}_{\mathbb{P}^1}(-1))^\times})$ of (\ref{Eghj}) is given by $\frac{1}{y^{i+1}} \mapsto \frac{1}{y} \cdot \big(\frac{\partial}{\partial y}\big)^{\otimes i}$, the image of
$\frac{1}{i!} \cdot \xi^{2\rightarrow u}(D_0^\clubsuit)(f(u) \cdot y^{i-1})$ via this injection coincides with $\frac{1}{y \cdot i!}\cdot \frac{\partial^i f}{\partial u^i} (u) \cdot \big(\frac{\partial}{\partial y}\big)^{\otimes i}$ (cf. Remark \ref{R001}), which is identical to (\ref{Efghhy}).
This implies the commutativity of (\ref{Efgmaob}), as desired.
The latter assertion follows from the former assertion.
\end{proof}
\vspace{3mm}
According to the above results, we complete the proof of the main theorem.
\vspace{10mm}
\section{A higher-dimensional variant of the main theorem} \label{S010}\vspace{3mm}
In this final section, we shall prove (cf. Theorem \ref{T009h1gh} described later) a higher dimensional variant of our main theorem, which may be thought of as a positive characteristic analogue of ~\cite{Bi}, Proposition 4.3.
\vspace{5mm}
\subsection{Frobenius-$\mathrm{Sp}$ structures} \label{Sp62}
\leavevmode\\ \vspace{-4mm}
Let $n$ be an integer with $n >1$ and
$S$ a smooth variety of dimension $2n-1$.
\vspace{3mm}
\begin{defi}
\label{D0799} \leavevmode\\
\ \ \
A {\bf Frobenius-$\mathrm{Sp}$ structure (of level $1$)} on $S$ is a triple
\begin{align}
\mathcal{S}^{\heartsuit \diamondsuit} := (\mathcal{S}^\heartsuit, \mathcal{S}^\diamondsuit, \kappa)
\end{align}
consisting of a Frobenius-projective structure (of level $1$) $\mathcal{S}^\heartsuit$ on $S$, an $(\mathrm{Sp}_{2n})_S^\mathrm{F}$-torsor $\mathcal{S}^\diamondsuit$ on $S$, and an isomorphism $\kappa : \mathcal{S}^\diamondsuit \times^{\mathrm{Sp}_{2n}} \mathrm{PGL}_{2n} \stackrel{\sim}{\migi} \mathcal{S}^\heartsuit$, where
$\mathcal{S}^\diamondsuit \times^{\mathrm{Sp}_{2n}} \mathrm{PGL}_{2n}$ denotes
the $(\mathrm{PGL}_{2n})_S^\mathrm{F}$-torsor induced from $\mathcal{S}^\diamondsuit$ via change of structure group by the composite $\mathrm{Sp}_{2n} \hookrightarrow \mathrm{GL}_{2n} \twoheadrightarrow \mathrm{PGL}_{2n}$.
\end{defi}\vspace{1mm}
\vspace{3mm}
In what follows, we shall observe that each Frobenius-$\mathrm{Sp}$ structure gives rise to a theta characteristic and an FC quantization on a certain symplectic variety.
First, let us consider a procedure for constructing a theta characteristic by means of a Frobenius-$\mathrm{Sp}$ structure.
Let $\mathcal{S}^{\heartsuit \diamondsuit} := (\mathcal{S}^\heartsuit, \mathcal{S}^\diamondsuit, \kappa)$ be a Frobenius-$\mathrm{Sp}$ structure on $S$.
The $(\mathrm{SL}_{2n})^\mathrm{F}_S$-torsor corresponding to $\mathcal{S}^\diamondsuit$ via change of structure group by the natural inclusion $\mathrm{Sp}_{2n} \hookrightarrow \mathrm{SL}_{2n}$ determines a vector bundle on $S^{(1)}$ with trivial determinant.
If $\mathcal{F}_\mathcal{S}$ denotes its pull-back via $F_{S/k}$, then $\mathcal{F}_\mathcal{S}$ admits a canonical connection $\nabla_\mathcal{S}$ with vanishing $p$-curvature (cf. ~\cite{Kat}, \S\,5, Theorem 5.1).
In particular, we have $(\mathrm{det}(\mathcal{F}_\mathcal{S}), \nabla_{\mathrm{det}(\mathcal{F}_\mathcal{S})}) \cong (\mathcal{O}_S, d)$.
Denote by $\mathcal{E}_\mathcal{S}$ the $\mathbb{P}^{2n-1}$-bundle on $S$ defined to be the pull-back via $F_{S/k}$ of the $\mathbb{P}^{2n-1}$-bundle on $S^{(1)}$ corresponding to $\mathcal{S}^\heartsuit$.
The local sections $U \rightarrow U \times \mathbb{P}^n$ (for various open subschemes $U$ of $S$) classified by sections in $\mathcal{S}^\heartsuit$ may be glued together to obtain a well-defined global section
$\sigma_\mathcal{S} : S \rightarrow \mathcal{E}_\mathcal{S}$.
Then, there exists a unique line subbundle $\mathcal{L}_\mathcal{S}$ of $\mathcal{F}_\mathcal{S}$
such that the square diagram
\begin{align}
\vcenter{\xymatrix{
\mbA (\mathcal{L}_\mathcal{S})^\times \ar[r] \ar[d] & \mbA (\mathcal{F}_\mathcal{S})^\times\ar[d]
\\
S \ar[r]_{\sigma_\mathcal{S}} & \mathcal{E}_\mathcal{S}
}}
\end{align}
is commutative and cartesian, where the upper horizontal arrow arises from the inclusion $\mathcal{L}_\mathcal{F} \hookrightarrow \mathcal{F}_\mathcal{S}$ and the vertical arrows denote the natural projections.
By the definitions of $\sigma_\mathcal{S}$ and $\mathcal{L}_\mathcal{S}$,
the $\mathcal{O}_{S}$-linear morphism
\begin{align}
\Omega_S^\vee \otimes \mathcal{L}_\mathcal{S} \stackrel{\sim}{\migi} \mathcal{F}_\mathcal{S} / \mathcal{L}_\mathcal{S}
\end{align}
induced naturally by the composite
\begin{align}
\mathcal{L}_\mathcal{S} \hookrightarrow \mathcal{F}_\mathcal{S} \xrightarrow{\nabla_\mathcal{S}} \Omega_S \otimes \mathcal{F}_\mathcal{S} \twoheadrightarrow \Omega_S \otimes (\mathcal{F}_\mathcal{S}/ \mathcal{L}_\mathcal{S})
\end{align}
turns out to be an isomorphism.
This isomorphism yields, via taking determinants, an isomorphism
\begin{align}
\omega_S^\vee \otimes \mathcal{L}_\mathcal{S}^{\otimes (2n-1)} \ \left(=\mathrm{det}(\Omega_S^\vee \otimes \mathcal{L}_\mathcal{S}) \right) \stackrel{\sim}{\migi} \left(\mathrm{det}(\mathcal{F}_\mathcal{S}/\mathcal{L}_\mathcal{S}) = \right) \mathcal{L}_\mathcal{S}^\vee,
\end{align}
or equivalently, an isomorphism
\begin{align}
\psi_\mathcal{S} : \mathcal{L}_\mathcal{S}^{\otimes 2n} \stackrel{\sim}{\migi} \omega_S.
\end{align}
Thus, we obtain a theta characteristic
\begin{align}
\mathbb{L} := (\mathcal{L}_\mathcal{S}, \psi_\mathcal{S})
\end{align}
on $S$.
\vspace{3mm}
\begin{rema} \label{R156}
\leavevmode\\
\ \ \
Let us mention the case where
$n=1$, i.e., $S$ is a smooth curve.
As discussed above, each Frobenius-$\mathrm{Sp}$ structure
yields a theta characteristic.
Conversely, suppose that we are given a theta characteristic $\mathbb{L}$ and a projective structure $\mathcal{S}^\heartsuit$ on $S$.
Denote by $\mathcal{F}^\diamondsuit := (\mathcal{F}, \nabla_\mathcal{F}, \mathcal{N}_\mathcal{F}, \eta_\mathcal{F})$ the $(\mathrm{SL}_2, \mathbb{L})$-oper corresponding to $\mathcal{S}^\heartsuit$.
Since it has vanishing $p$-curvature, there exists a unique $(\mathrm{SL}_2)_S^\mathrm{F}$-torsor $\mathcal{S}^\diamondsuit$ such that the induced $\mathrm{SL}_2$-torsor on $S$ equipped with a canonical connection
is isomorphic to $(\mathcal{F}, \nabla_\mathcal{F})$.
Then,
(since $\mathrm{SL}_2 = \mathrm{Sp}_{2}$) the triple consisting of $\mathcal{S}^\heartsuit$, $\mathcal{S}^\diamondsuit$, and the natural isomorphism $\kappa : \mathcal{S}^\diamondsuit \times^{\mathrm{SL}_2} \mathrm{PGL}_2 \stackrel{\sim}{\migi} \mathcal{S}^\heartsuit$, specifies a Frobenius-$\mathrm{Sp}$ structure on $S$.
According to this construction,
giving a Frobenius-$\mathrm{Sp}$ structure on a smooth curve $S$ is equivalent to giving a pair of a theta characteristic $\mathbb{L}$ and a Frobenius-projective structure on $S$.
\end{rema}
\vspace{5mm}
\subsection{An FC quantizations arising from a Frobenius-$\mathrm{Sp}$ structure} \label{Sj162}
\leavevmode\\ \vspace{-4mm}
Next, let us construct a certain symplectic variety and an FC quantization on it.
We shall keep the above notation.
By the definition of a Frobenius-$\mathrm{Sp}$ structure,
there exists a collection
\begin{align}
\{ (U_\alpha, \gamma_\alpha) \}_{\alpha \in I},
\end{align}
of pairs $(U_\alpha, \gamma_\alpha)$ (indexed by a set $I$),
where $\{ U_\alpha \}_{\alpha \in I}$ is an open covering of $S$ and $\gamma_\alpha$ (for each $\alpha \in I$) denotes an $\mathcal{O}_{U_\alpha}$-linear isomorphism $\mathcal{F}_\mathcal{S} |_{U_\alpha} \stackrel{\sim}{\migi} \mathcal{O}_{U_\alpha}^{\oplus 2n}$ inducing, via taking determinants, the fixed isomorphism $\mathrm{det}(\mathcal{F}_\mathcal{S}) \stackrel{\sim}{\migi} \mathcal{O}_S$ (restricted to $U_\alpha$).
Moreover, we can assume that for any pair $(\alpha, \beta) \in I \times I$ with $U_{\alpha \beta} := U_\alpha \cap U_\beta \neq \emptyset$,
the automorphism $\gamma_{\alpha \beta} := (\gamma_\alpha |_{U_{\alpha \beta}})\circ (\gamma_\beta |_{U_{\alpha\beta}})^{-1}$ of $\mathcal{O}_{U_{\alpha \beta}}^{\oplus 2n}$ corresponds to a $U_{\alpha \beta}$-rational point of $\mathrm{Sp}_{2n} \ \left(\subseteq \mathrm{SL}_{2n} \right)$.
Let
\begin{align}
\gamma_\alpha^\mbA : \mbA (\mathcal{F}_\mathcal{S} |_{U_\alpha})^\times
\stackrel{\sim}{\migi} \left( \mbA (\mathcal{O}_{U_\alpha}^{\otimes 2n}) \stackrel{\sim}{\migi} \right) U_\alpha \times \mathbb{A}^{2n \times}
\end{align}
be the isomorphism induced by $\gamma_\alpha$, and let
$\gamma_\alpha^\mathbb{P}$ be the isomorphism $\mathcal{E}_\mathcal{S} |_{U_\alpha} \stackrel{\sim}{\migi} U_\alpha \times \mathbb{P}^{2n-1}$ obtained from $\gamma_\alpha^\mathbb{A}$ via projectivization.
Then, we have
two composites
\begin{align}
\Phi_\alpha &: \mathbb{A} (\mathcal{L}_\mathcal{S} |_{U_\alpha})^\times \xrightarrow{\mathrm{incl.}} \mathbb{A} (\mathcal{F}_\mathcal{S} |_{U_\alpha})^\times \xrightarrow{\gamma_\alpha^\mathbb{A}} U_\alpha \times \mathbb{A}^{2n \times} \xrightarrow{\mathrm{pr}_2} \mathbb{A}^{2n \times}, \\
\phi_\alpha &: U_\alpha \xrightarrow{\sigma_\mathcal{S} |_{U_\alpha}} \mathcal{E}_\mathcal{S} |_{U_\alpha} \xrightarrow{\gamma_\alpha^\mathbb{P}} U_\alpha \times \mathbb{P}^{2n -1} \xrightarrow{\mathrm{pr}_2} \mathbb{P}^{2n-1}. \notag
\end{align}
Since $\phi_\alpha$ is \'{e}tale and the square diagram
\begin{align}
\vcenter{\xymatrix{
\mathbb{A} (\mathcal{L}_\mathcal{S} |_{U_\alpha})^\times \ar[r]^-{\Phi_\alpha}\ar[d] & \mathbb{A}^{2n \times} \ar[d] \\
U_\alpha \ar[r]_-{\phi_\alpha}& \mathbb{P}^{2n-1}
}}
\end{align}
(where the vertical arrows denote the natural projections) is commutative and cartesian,
$\Phi_\alpha$ turns out to be \'{e}tale.
The pull-back $\Phi_\alpha^* (\omega^\mathrm{Weyl})$ of $\omega^\mathrm{Weyl}$ via $\Phi_\alpha$ speficies
a symplectic structure on $\mathbb{A} (\mathcal{L}_\mathcal{S} |_{U_\alpha})^\times$.
For each pair $(\alpha, \beta) \in I \times I$ with $U_{\alpha \beta} \neq \emptyset$, we shall denote by $\gamma_{\alpha \beta}^\mathbb{A}$ the automorphism of $U_{\alpha\beta} \times \mathbb{A}^{2n \times}$ corresponding to $\gamma_{\alpha \beta}$.
Since $\gamma_{\alpha \beta} \in \mathrm{Sp}_2(U_{\alpha \beta})$, the equality $\gamma_{\alpha \beta}^{\mathbb{A} *}(\mathrm{pr}_2^*(\omega^\mathrm{Weyl}))= \mathrm{pr}_2^*(\omega^\mathrm{Weyl})$ holds,
which implies that $\Phi_\alpha^*(\omega^\mathrm{Weyl})|_{U_{\alpha \beta}} = \Phi_\beta^*(\omega^\mathrm{Weyl})|_{U_{\alpha \beta}}$.
Thus, the $\Phi_\alpha^*(\omega^\mathrm{Weyl})$'s may be glued together to obtain a symplectic structure
$\omega_\mathcal{S}$
on $\mathbb{A} (\mathcal{L}_\mathcal{S})^\times$.
In particular, we have a symplectic variety
\begin{align}
(\mathbb{A} (\mathcal{L}_\mathcal{S})^\times, \omega_\mathcal{S}).
\end{align}
Moreover, for each $\alpha \in I$,
the pull-back $\Phi_\alpha^*(\mathcal{W}_k^{2n})$ of $\mathcal{W}_k^{2n}$ via $\Phi_\alpha$ specifies
an FC quantization
on $(\mathbb{A} (\mathcal{L}_\mathcal{S} |_{U_\alpha})^\times, \Phi_\alpha^*(\omega^\mathrm{Weyl}))$.
It follows from an argument similar to the argument in \S\,\ref{S11} (together with the homomorphism (\ref{E4522})) that
$\Phi_\alpha^*(\mathcal{W}_k^{2n})$ may be glued together to obtain an FC quantization
\begin{align}
\mathcal{W}_\mathcal{S}
\end{align}
on $(\mathbb{A} (\mathcal{L}_\mathcal{S})^\times, \omega_\mathcal{S})$.
Consequently, we have obtained the following assertion.
(In the case $n=1$, one verifies that the symplectic structure $\omega_\mathcal{S}$ coincides with $\check{\omega}_\mathbb{L}$ and the asserted construction of FC quantizations is consistent with $\bigstar_{X, \mathbb{L}}$ mentioned in our main theorem.)
\vspace{3mm}
\begin{thm} \label{T009h1gh}\leavevmode\\
\ \ \
Let $S$ be a smooth variety of dimension $2n$ (where $n$ is a positive integer).
Then, by means of a Frobenius-$\mathrm{Sp}$ structure $\mathcal{S}^{\heartsuit \diamondsuit}$ on $S$, we can construct canonically a theta characteristic $\mathbb{L} := (\mathcal{L}_\mathcal{S}, \psi_\mathcal{S})$ on $S$, a symplectic structure $\omega_\mathcal{S}$ on $\mathbb{A} (\mathcal{L}_\mathcal{S})^\times$, and an FC quantization $\mathcal{W}_\mathcal{S}$ on the resulting symplectic variety $(\mathbb{A} (\mathcal{L}_\mathcal{S})^\times, \omega_\mathcal{S})$.
\end{thm}
\vspace{3mm}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,247 |
using System;
using System.Collections.Generic;
using System.Linq;
using System.Net;
using System.Threading.Tasks;
using Moq;
using NUnit.Framework;
using SFA.DAS.NLog.Logger;
using SFA.DAS.ReferenceData.Application.Services.OrganisationSearch;
using SFA.DAS.ReferenceData.Domain.Interfaces.Services;
using SFA.DAS.ReferenceData.Domain.Models.Company;
using SFA.DAS.ReferenceData.Types.DTO;
using Address = SFA.DAS.ReferenceData.Types.DTO.Address;
namespace SFA.DAS.ReferenceData.Application.UnitTests.Services.CompanySearchServiceTests
{
public class WhenISearchForACompany
{
private Mock<ILog> _logger;
private Mock<ICompaniesHouseEmployerVerificationService> _verificationService;
private CompanySearchService _searchService;
[SetUp]
public void Arrange()
{
_logger = new Mock<ILog>();
_verificationService = new Mock<ICompaniesHouseEmployerVerificationService>();
_searchService = new CompanySearchService(_verificationService.Object, _logger.Object);
}
[Test]
public async Task ShouldSearchForCompaniesUsingCompanyHouseService()
{
//Arrange
var searchTerm = "test";
var results = new CompanySearchResults();
_verificationService.Setup(x => x.FindCompany(It.IsAny<string>(), 10)).ReturnsAsync(results);
//Act
await _searchService.Search(searchTerm, 10);
//Assert
_verificationService.Verify(x => x.FindCompany(searchTerm, 10), Times.Once);
}
[Test]
public async Task ShouldReturnFoundCompany()
{
//Arrange
const string searchTerm = "test";
var resultItem = new CompanySearchResultsItem
{
CompanyName = "Test Company",
Address = new Domain.Models.Company.Address
{
Premises = "12",
CompaniesHouseLine1 = "Test Street",
CompaniesHouseLine2 = "Test Park",
TownOrCity = "Test Town",
County = "Testshire",
PostCode = "TE51 3TS"
},
DateOfIncorporation = DateTime.Now,
CompanyNumber = "12345678"
};
_verificationService.Setup(x => x.FindCompany(It.IsAny<string>(), 10)).ReturnsAsync(new CompanySearchResults
{
Companies = new List<CompanySearchResultsItem> { resultItem }
});
//Act
var results = await _searchService.Search(searchTerm, 10);
//Assert
Assert.IsNotNull(results.FirstOrDefault());
}
[Test]
[TestCase("", OrganisationStatus.None)]
[TestCase(null, OrganisationStatus.None)]
[TestCase("active", OrganisationStatus.Active)]
[TestCase("dissolved", OrganisationStatus.Dissolved)]
[TestCase("liquidation", OrganisationStatus.Liquidation)]
[TestCase("receivership", OrganisationStatus.Receivership)]
[TestCase("administration", OrganisationStatus.Administration)]
[TestCase("voluntary-arrangement", OrganisationStatus.VoluntaryArrangement)]
[TestCase("converted-closed", OrganisationStatus.ConvertedClosed)]
[TestCase("insolvency-proceedings", OrganisationStatus.InsolvencyProceedings)]
public async Task ShouldSetOrganisationStatus(string companiesHouseStatus, OrganisationStatus expectedMappedStatus)
{
//Arrange
const string searchTerm = "test";
var resultItem = new CompanySearchResultsItem
{
CompanyName = "Test Company",
Address = new Domain.Models.Company.Address
{
Premises = "12",
CompaniesHouseLine1 = "Test Street",
CompaniesHouseLine2 = "Test Park",
TownOrCity = "Test Town",
County = "Testshire",
PostCode = "TE51 3TS"
},
DateOfIncorporation = DateTime.Now,
CompanyNumber = "12345678",
CompanyStatus = companiesHouseStatus
};
_verificationService.Setup(x => x.FindCompany(It.IsAny<string>(), 10)).ReturnsAsync(new CompanySearchResults
{
Companies = new List<CompanySearchResultsItem> { resultItem }
});
//Act
var results = await _searchService.Search(searchTerm, 10);
//Assert
Assert.AreEqual(expectedMappedStatus, results.FirstOrDefault().OrganisationStatus);
}
[TestCase("12", "Test Street", "TestPark", "Test Town", "Testshire", "TE51 3TS")]
[TestCase(null, "Test Street", "TestPark", "Test Town", "Testshire", "TE51 3TS")]
[TestCase("12", "Test Street", null, "Test Town", "Testshire", "TE51 3TS")]
[TestCase(null, "Test Street", null, "Test Town", "Testshire", "TE51 3TS")]
public async Task ShouldFormatAddressCorrectlyForFoundCompanies(string premises, string companiesHouseLine1, string companiesHouseLine2,
string townOrCity, string county, string postcode)
{
//Arrange
const string searchTerm = "Test";
var resultItem = new CompanySearchResultsItem
{
CompanyName = "Test Company",
Address = new Domain.Models.Company.Address
{
Premises = premises,
CompaniesHouseLine1 = companiesHouseLine1,
CompaniesHouseLine2 = companiesHouseLine2,
TownOrCity = townOrCity,
County = county,
PostCode = postcode
},
DateOfIncorporation = DateTime.Now,
CompanyNumber = "12345678"
};
_verificationService.Setup(x => x.FindCompany(It.IsAny<string>(), 10)).ReturnsAsync(new CompanySearchResults
{
Companies = new List<CompanySearchResultsItem> { resultItem }
});
//Act
var results = await _searchService.Search(searchTerm, 10);
var organisation = results.FirstOrDefault();
//Assert
Assert.IsNotNull(organisation);
Assert.AreEqual(resultItem.CompanyName, organisation.Name);
Assert.AreEqual(!string.IsNullOrEmpty(resultItem.Address.Premises) ? resultItem.Address.Premises : resultItem.Address.CompaniesHouseLine1, organisation.Address.Line1);
Assert.AreEqual(!string.IsNullOrEmpty(resultItem.Address.Premises) ? resultItem.Address.CompaniesHouseLine1 : resultItem.Address.CompaniesHouseLine2, organisation.Address.Line2);
Assert.AreEqual(!string.IsNullOrEmpty(resultItem.Address.Premises) ? resultItem.Address.CompaniesHouseLine2 : null, organisation.Address.Line3);
Assert.AreEqual(resultItem.Address.TownOrCity, organisation.Address.Line4);
Assert.AreEqual(resultItem.Address.County, organisation.Address.Line5);
Assert.AreEqual(resultItem.Address.PostCode, organisation.Address.Postcode);
Assert.AreEqual(resultItem.CompanyNumber, organisation.Code);
Assert.AreEqual(resultItem.DateOfIncorporation, organisation.RegistrationDate);
Assert.AreEqual(OrganisationType.Company, organisation.Type);
Assert.AreEqual(OrganisationSubType.None, organisation.SubType);
}
[Test]
public async Task ThenAnEmptyAddressIsReturnedWhenNullIsReturnedFromTheApi()
{
//Arrange
var resultItem = new CompanySearchResultsItem
{
CompanyName = "Test Corp",
Address = null,
DateOfIncorporation = DateTime.Now,
CompanyNumber = "12345678"
};
_verificationService.Setup(x => x.FindCompany(It.IsAny<string>(), 10)).ReturnsAsync(new CompanySearchResults
{
Companies = new List<CompanySearchResultsItem> { resultItem }
});
//Act
var actual = await _searchService.Search("test", 10);
//Assert
Assert.IsNotNull(actual);
Assert.IsNotNull(actual.FirstOrDefault());
Assert.IsAssignableFrom<Address>(actual.First().Address);
}
[Test]
public async Task ShouldReturnNullIfExceptionIsThrowAndLogTheError()
{
//Arrange
var exception = new WebException();
_verificationService.Setup(x => x.FindCompany(It.IsAny<string>(), 10))
.Throws(exception);
//Act
var result = await _searchService.Search("test", 10);
//Assert
_logger.Verify(x => x.Error(exception, It.IsAny<string>()));
Assert.IsNull(result);
}
[Test]
public async Task ShouldReturnNullIfNoCompaniesFound()
{
//Arrange
_verificationService.Setup(x => x.FindCompany(It.IsAny<string>(), 10)).ReturnsAsync(new CompanySearchResults{Companies = new CompanySearchResultsItem[0]});
//Act
var result = await _searchService.Search("test", 10);
//Assert
Assert.IsNotNull(result);
Assert.IsEmpty(result);
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,119 |
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</div><hr /><div id="function.fdf-get-status" class="refentry">
<div class="refnamediv">
<h1 class="refname">fdf_get_status</h1>
<p class="verinfo">(PHP 4, PHP 5)</p><p class="refpurpose"><span class="refname">fdf_get_status</span> — <span class="dc-title">Get the value of the /STATUS key</span></p>
</div>
<div class="refsect1 description" id="refsect1-function.fdf-get-status-description">
<h3 class="title">Description</h3>
<div class="methodsynopsis dc-description">
<span class="type">string</span> <span class="methodname"><strong>fdf_get_status</strong></span>
( <span class="methodparam"><span class="type">resource</span> <code class="parameter">$fdf_document</code></span>
)</div>
<p class="para rdfs-comment">
Gets the value of the <em>/STATUS</em> key.
</p>
</div>
<div class="refsect1 parameters" id="refsect1-function.fdf-get-status-parameters">
<h3 class="title">Parameters</h3>
<p class="para">
<dl>
<dt>
<code class="parameter">fdf_document</code></dt>
<dd>
<p class="para">
The FDF document handle, returned by <span class="function"><a href="function.fdf-create.html" class="function">fdf_create()</a></span>,
<span class="function"><a href="function.fdf-open.html" class="function">fdf_open()</a></span> or <span class="function"><a href="function.fdf-open-string.html" class="function">fdf_open_string()</a></span>.
</p>
</dd>
</dl>
</p>
</div>
<div class="refsect1 returnvalues" id="refsect1-function.fdf-get-status-returnvalues">
<h3 class="title">Return Values</h3>
<p class="para">
Returns the key value, as a string.
</p>
</div>
<div class="refsect1 seealso" id="refsect1-function.fdf-get-status-seealso">
<h3 class="title">See Also</h3>
<p class="para">
<ul class="simplelist">
<li class="member"><span class="function"><a href="function.fdf-set-status.html" class="function" rel="rdfs-seeAlso">fdf_set_status()</a> - Set the value of the /STATUS key</span></li>
</ul>
</p>
</div>
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,527 |
Q: .htaccess multi subfolder redirection based on URL I want to redirect the request to the corresponding file using httaccess for multiple subfolders.
For example (GOAL):
1) www.mysite.com -> /var/www/mysite/website/index.php
2) www.mysite.com/app -> /var/www/mysite/app/web/index.php
3) www.mysite.com/app2 -> /var/www/mysite/app2/web/index.php
With the following piece of code i can achieve step 2 for example:
RewriteCond %{REQUEST_URI} ^/app/
RewriteCond %{HTTP_HOST} ^(www\.)?mysite\.
RewriteRule ^(.*)$ ../app/web/$1 [L]
but if i add for example..
RewriteCond %{REQUEST_URI} !^/website/
RewriteCond %{HTTP_HOST} ^(www\.)?mysite\.
RewriteRule ^(.*)$ /website/$1 [L]
1) works great, but 2) stop working... obviously the second piece of code override the first.. but i don't know how to deal with this.
If there is an answer with easy understanding and / or with easy implementation about this goal in particular i don't found it.
can anyone share an example of how can achieve step 1 and step 2 together? or even if it is inpossible and the only way is using the "category" (app, app2, website) in all cases in the url.
any help is thankful
A: You can use this code in your DOCUMENT_ROOT/.htaccess file:
DirectoryIndex index.php
RewriteEngine On
RewriteBase /
RewriteCond %{ENV:REDIRECT_STATUS} 200
RewriteRule ^ - [L]
RewriteRule ^/?$ website/ [L]
RewriteCond %{REQUEST_FILENAME} !-f
RewriteRule ^([^/]+)(/.*)?$ /$1/web$2 [L]
A: You don't need to check for HTTP_HOST, because all requests will match. You also don't need to check for REQUEST_URI, because you can do that in the pattern part of the RewriteRule. For the rest, you can just use easy, specific rules
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule ^app2 /app2/web/index.php [L]
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule ^app /app/web/index.php [L]
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule ^ /website/index.php [L]
You must pay attention to the order of the rules. If app were first, it would also match app2. These rules assume, that the document root is in /var/www/mysite. If it is in /var/www, you must prefix the substitution with /mysite, e.g. /mysite/app2/web/index.php, etc.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,862 |
/* jshint undef: true, unused: true */
/* global module */
var exports = module.exports = {};
function User() {
this.id = "";
this.name = "";
}
exports.user = User;
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,794 |
Russo, Karl, Widmaier & Cordano PLLC is dedicated to guiding both businesses and individuals through civil litigation whenever it becomes necessary to get good results.
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Civil litigation is a detail-oriented branch of the law. We examine cases from all angles and make sure that our clients understand all their options before proceeding. No matter what course of action you choose, we make certain that your rights are protected and your goals are in focus.
We litigate in both the state and federal courts. We also handle representations during arbitration proceedings, cases before the New York State Division of Human Rights and cases involving the Equal Employment Opportunity Commission (EEOC).
We take a tough, no-nonsense approach to every case. We also offer free initial consultations so that our clients have nothing to lose by scheduling an appointment and discussing their case.
Please contact Russo, Karl, Widmaier & Cordano PLLC, either online or at our Long Island office by calling 631-265-7200 or 718-737-7181. Our attorneys look forward to speaking with you. | {
"redpajama_set_name": "RedPajamaC4"
} | 251 |
\section{Introduction}
Outflows in the form of jets and winds are observed from many disc accreting
objects ranging from young stars to systems with white
dwarfs, neutron stars, and black holes.
A large body of observations exists for outflows
from young stars at different stages of their
evolution, ranging from protostars, where powerful
collimated outflows - jets - are observed, to classical T Tauri stars (CTTSs) where the outflows are weaker and often less collimated (see review by Ray et al. 2007).
Correlation between the disc's radiated power and the jet power has been found in many CTTSs (Cabrit et al. 1990;
Hartigan, Edwards \& Gandhour 1995).
A significant number of CTTSs show signs
of outflows in spectral lines, in particular in
He I where two distinct components of outflows
had been found (Edwards et al. 2003, 2006; Kwan, Edwards, \& Fischer 2007).
Outflows are also observed from accreting compact stars such as accreting white dwarfs in symbiotic binaries (Sokoloski \& Kenyon 2003),
or from the vicinity of neutron stars, such as from Circinus X-1 (Heinz et al. 2007).
Different theoretical models have been proposed
to explain the outflows from protostars and CTTSs
(see review by Ferreira, Dougados, \& Cabrit 2006).
The models include those where the outflow
originates from a radially distributed disc
wind (Blandford \& Payne 1982; K\"onigl \& Pudritz 2000;
Casse \& Keppens 2004; Ferreira et al. 2006) or
from the innermost region of the accretion disc (Lovelace, Berk \& Contopoulos 1991). The latter model is related to the
X-wind model (Shu et al. 1994; 2007; Najita \& Shu 1994; Cai et al. 2008) where
the outflow originates from the vicinity of the disc-magnetosphere boundary.
Progress in understanding the theoretical models
has come from MHD simulations of accretion discs around rotating
magnetized stars as discussed below. Laboratory experiments are
also providing insights into jet formation processes (Hsu \& Bellan 2002;
Lebedev et al. 2005) but these are not discussed here.
Outflows or jets from the disc-magnetosphere boundary were found in early axisymmetric MHD simulations by
Hayashi, Shibata \& Matsumoto (1996) and Miller \& Stone (1997).
A one-time episode
of outflows from the inner disc and inflation of the
innermost field lines connecting the star and the disc were observed
for a few dynamical time-scales.
Somewhat longer simulation runs were performed
by Goodson et al. (1997, 1999), Hirose et al. (1997),
Matt et al. (2002) and K\"uker, Henning \& R\"udiger (2003)
where several episodes of field inflation and outflows were observed.
These simulations hinted at a possible long-term nature for the outflows.
However, the simulations were not sufficiently long to establish
the behavior of the outflows.
MHD simulations showing long-lasting (thousand of
orbits of the inner disc) outflows from the disc-magnetosphere have been obtained by our group
(Romanova et al. 2009; Lii et al. 2012).
We obtained these outflows/jets
in two main cases: (1) where the star rotates slowly but the field lines are bunched up into an X-type configuration, and (2) where
the star rotates rapidly, in the ``propeller regime''
(Illarionov \& Sunyaev 1975;
Alpar \& Shaham 1985; Lovelace, Romanova \& Bisnovatyi-Kogan 1999) .
Figure 1 shows the equatorial angular rotation rate $\Omega(r,z=0)$
of the plasma in the two cases. Here, $r_*$ is the radius
of the star; $r_{\rm m}$ is the magnetospheric radius where the
kinetic energy density of the disc matter is about equal to the energy
density of the magnetic field; and
$r_{\rm cr}= (GM/\Omega_*^2)^{1/3}$
is the co-rotation radius where the angular rotation rate of the
star $\Omega_*$ equals that of the Keplerian disc $\Omega_K=
(GM/r^3)^{1/2}$. For a slowly rotating star $r_{\rm m} < r_{\rm cr}$
whereas for a rapidly rotating star in the propeller regime
$r_{\rm m} >r_{\rm cr}$.
\begin{figure}
\begin{center}
\includegraphics[width=5in]{Fig1.eps}
\caption{Schematic profiles of the midplane angular velocity
of the plasma for the case of a slowly rotating star (left-hand panel)
and a rapidly rotating star (right-hand panel) which is in the
propeller regime. Here, $\Omega_*$ is the angular
rotation rate of the star, $\Omega_K$ is the Keplerian rotation
rate of the disc, $r_*$ is the star's radius, $r_{\rm m}$ is the
radius of the magnetosphere, and $r_{\rm cr}$ is the co-rotation
radius.}
\end{center}
\end{figure}
Figure 2 shows examples of the outflows in the
two cases.
In both cases, two-component outflows are observed:
One component originates at the inner edge
of the disc near $r_{\rm m}$ and has a narrow-shell conical shape close to the disc and therefore is termed a ``conical wind". It is matter dominated
but can become collimated at large distances due to its toroidal magnetic
field.
The other component is a magnetically dominated high-velocity
``axial jet'' which flows along the open stellar magnetic field lines.
The axial jet may be very strong in the propeller regime.
A detailed discussion of the simulations and analysis can
be found in Romanova et al. (2009) and Lii et al. (2012).
Sec. 2 describes the simulations. Sec. 3 discusses the conical winds and axial jets, the driving and collimation forces, and the variability of
the winds and jets. Sec. 4 discusses simulation
results on one-sided and lop-sided jets. Sec. 5 gives the conclusions.
\begin{figure}
\begin{center}
\includegraphics[width=5.in]{Fig2.eps}
\caption{Two-component outflows observed in slowly (left) and rapidly (right) rotating magnetized stars adapted from Romanova et al. (2009).
The background shows
the poloidal matter flux $F_m=\rho v_p$, the arrows are the poloidal
velocity vectors, and the lines are sample magnetic field lines.
The labels point to the main outflow components.}
\end{center}
\end{figure}
\section{MHD Simulations}
We simulate the outflows resulting from disc-magnetosphere interaction
using the equations of axisymmetric MHD.
Outside of the disc the flow is described by the
equations of ideal MHD.
Inside the disc the flow is described by
the equations of viscous, resistive MHD.
In an inertial reference frame the equations are:
\begin{equation}\label{eq1}
\displaystyle{ \frac{\partial \rho}{\partial t} + {\bf \nabla}\cdot
\left( \rho
{\bf v} \right)} = 0~,
\end{equation}
\begin{equation}\label{eq2}
{\frac{\partial (\rho {\bf v})}{\partial t} + {\bf
\nabla}\cdot
{\cal T} } = \rho ~{\bf g}~,
\end{equation}
\begin{equation}\label{eq3}
{\frac{\partial {\bf B}}{\partial t} - {\bf
\nabla}\times ({\bf v} \times {\bf B}) + {\bf \nabla} \times\left(
\eta_t {\bf \nabla}\times {\bf B} \right)} = 0~,
\end{equation}
\begin{equation}\label{eq4}
{\frac{\partial (\rho S)}{\partial t} + {\bf
\nabla}\cdot ( \rho S {\bf v} )} = Q~.\
\end{equation}
Here, $\rho$ is the density, $S$ is the specific entropy, $\bf
v$ is the flow velocity, $\bf B$ is the magnetic field, $\eta_t$
is the magnetic diffusivity, $\cal{T}$ is the
momentum flux-density tensor, $Q$ is
the rate of change of entropy per unit volume, and ${\bf g} = - (GM /r^{2})\hat{{\bf r}}$
is the gravitational acceleration due to the star, which has mass $M$.
The total mass of the disc is assumed to
be negligible compared to $M$.
Here, ${\cal T}$ is the sum of the ideal plasma terms {\it and}
the $\alpha-$viscosity terms discussed in the next paragraph.
The plasma is considered to be an
ideal gas with adiabatic index $\gamma =5/3$, and $S=\ln(p/
\rho^{\gamma})$. We use spherical coordinates $(r, \theta, \phi)$ with $\theta$ measured from the symmetry axis.
The equations in spherical coordinates are given in
Ustyugova et al. (2006).
Both the viscosity
and the magnetic diffusivity of the disc plasma are considered
to be due to turbulent fluctuations of the velocity and the magnetic field.
Both effects are non-zero only inside the disc as determined by
a density threshold.
The microscopic transport coefficients are replaced by
turbulent coefficients.
The values of these coefficients are assumed to be given by
the $\alpha$-model of Shakura and Sunyaev (1973), where the
coefficient of the turbulent kinematic viscosity is $\nu_t =
\alpha_\nu c_s^{2}/\Omega_K$, where $c_s$ is the isothermal sound
speed and $\Omega_K(r)$ is the Keplerian angular velocity.
We take into account the viscous stress terms ${\cal T}_{r\phi}^{\rm vis}$
and ${\cal T}_{\theta \phi}^{\rm vis}$ (Lii et al. 2012).
Similarly, the coefficient of the turbulent magnetic
diffusivity $\eta_t=\alpha_\eta c_s^{2}/\Omega_K$. Here,
$\alpha_\nu$ and $\alpha_\eta$ are dimensionless coefficients which
are treated as parameters of the model.
The MHD equations are solved in
dimensionless form so that the results can be readily applied
to different accreting stars.
\begin{table*}
\centering
\caption{Reference values for different types of stars. We choose the mass $M$, radius $R_*$,
equatorial magnetic field $B_*$ and the period $P_*$ of the star and derive the other reference values.
The reference
mass $M_0$ is taken to be the mass $M$ of the star.
The reference radius is taken to be twice the radius
of the star, $R_0=2 R_*$.
The surface
magnetic field $B_*$ is
different for different types of stars.
The reference velocity is $v_0=(GM/R_0)^{1/2}$.
The reference time-scale $t_0=R_0/v_0$, and the reference angular velocity $\Omega_0=1/t_0$.
We measure time
in units of $P_0=2\pi t_0$ (which is the Keplerian rotation period at $r=R_0$).
In the plots we use the dimensionless time $T=t/P_0$.
The reference magnetic field is $B_0=B_*(R_*/R_0)^{3}/{\tilde\mu}$,
where $\tilde\mu$ is the dimensionless magnetic moment which has
a numerical value of $10$ in the simulations discussed here.
The reference density is taken to
be $\rho_0 = B_0^{2}/v_0^{2}$.
The reference pressure is $p_0=B_0^{2}$.
The reference temperature is
$T_0=p_0/{\cal R} \rho_0 = v_0^{2}/{\cal R}$, where ${\cal R}$ is the gas constant.
The reference accretion rate is $\dot M_0
= \rho_0 v_0 R_0^{2}$.
The reference energy flux is $\dot
E_0=\dot M_0 v_0^{2}$.
The reference angular momentum flux is $\dot{L}_0=\dot M_0 v_0
R_0$. The poloidal magnetic field
of the star (in the absence of external plasma) is an aligned dipole field.
}
\begin{tabular}{l@{\extracolsep{0.2em}}l@{}lllll}
\hline
& & Protostars & CTTSs & Brown dwarfs & White dwarfs & Neutron stars \\
\hline
\multicolumn{2}{l}{$M(M_\odot)$} & 0.8 & 0.8 & 0.056 & 1 & 1.4 \\
\multicolumn{2}{l}{$R_*$} & $2R_\odot$ & $2R_\odot$ & $0.1R_\odot$ & 5000 km & 10 km \\
\multicolumn{2}{l}{$R_0$ (cm)} & $2.8\cdot10^{11}$ & $2.8\cdot10^{11}$ & $1.4\cdot10^{10}$ & $10^9$ & $2\cdot10^6$ \\
\multicolumn{2}{l}{$v_0$ (cm s$^{-1}$)} & $1.95\cdot10^7$ & $1.95\cdot10^7$ & $1.6\cdot10^7$ & $3.6\cdot10^8$ & $9.7\cdot10^{9}$ \\
\multicolumn{2}{l}{$P_*$} & $1.04$ days & $5.6$ days & $0.13$ days & $89$ s & $6.7$ ms \\
\multicolumn{2}{l}{$P_0$} & $1.04$ days & $1.04$ days & $0.05$ days & $17.2$ s & $1.3$ ms \\
\multicolumn{2}{l}{$B_*$ (G)} & $3.0\cdot10^3$ & $10^{3}$ & $2\cdot10^{3}$ & $10^{6}$ & $10^{9}$ \\
\multicolumn{2}{l}{$B_0$ (G)} & 37.5 & 12.5 & 25.0 & $1.2\cdot10^4$ & $1.2\cdot10^{7}$ \\
\multicolumn{2}{l}{$\rho_0$ (g cm$^{-3}$)} & $3.7\cdot10^{-12}$ & $4.1\cdot10^{-13}$ & $1.4\cdot10^{-12}$ & $1.2\cdot10^{-9}$ & $1.7\cdot10^{-6}$ \\
\multicolumn{2}{l}{$n_0$ (cm$^{-3}$)} & $2.2\cdot10^{12}$ & $2.4\cdot10^{11}$ & $8.5\cdot10^{11}$ & $7\cdot10^{14}$ & $10^{18}$ \\
\multicolumn{2}{l}{$\dot M_0$($M_\odot$yr$^{-1}$)} & $1.8\cdot10^{-7}$ & $2\cdot10^{-8}$ & $1.8\cdot10^{-10}$ & $1.3\cdot10^{-8}$ & $2\cdot10^{-9}$ \\
\multicolumn{2}{l}{$\dot E_0$ (erg s$^{-1}$)} & $2.1\cdot10^{33}$ & $2.4\cdot10^{32}$ & $2.5\cdot10^{30}$ & $5.7\cdot10^{34}$ & $6\cdot10^{36}$ \\
\multicolumn{2}{l}{$\dot L_0$ (erg s$^{-1}$)} & $3.1\cdot10^{37}$ & $3.4\cdot10^{36}$ & $1.7\cdot10^{33}$ & $1.6\cdot10^{35}$ & $1.2\cdot10^{33}$ \\
\multicolumn{2}{l}{$T_d$ (K)} & $2290$ & $4590$ & $5270$ & $1.6\cdot10^{6}$ & $1.1\cdot10^{9}$ \\
\multicolumn{2}{l}{$T_c$ (K)} & $2.3\cdot10^{6}$ & $4.6\cdot 10^{6}$ & $5.3\cdot10^{6}$ & $8\cdot10^{8}$ & $5.6\cdot10^{11}$ \\
\hline
\end{tabular}
\end{table*}
The system of MHD equations (1-4) have been
integrated numerically in spherical $(r,\theta,\phi)$ coordinates
using a Godunov-type numerical scheme.
The calculations were done
in the region $R_{\rm in} \leq r \leq R_{\rm out}$, $0\leq \theta
\leq \pi/2$. The grid is uniform in the $\theta$-direction with $N_\theta$ cells.
The $N_r$ cells in the radial direction have $ dr_{j + 1} = (1 + 0.0523)dr_j$ ($j=1..N_r$) so
that the poloidal-plane cells are curvilinear rectangles with approximately
equal sides. This choice results in high spatial resolution near the star
where the disc-magnetosphere interaction takes place while
also permitting a large simulation region.
We have used a range of resolutions going from
$N_r\times N_\theta = 51\times 31$ to $121\times 51$.
\begin{figure*}
\centering
\includegraphics[width=5in]{Fig3.eps}
\caption{Matter flux $\rho v_p$ (background), sample field lines, and poloidal velocity vectors in a propeller-driven outflow at time $t=1400$ (rotations at $r=1$) adapted from Romanova et al. (2009).
Sample numerical values are given for the poloidal $v_p$ and total $v_t$ velocity, and for the density $\rho$ for different parts of the simulation region.
One can see from the Table 1 that for CTTSs $v_p=1$ corresponds to $v_p=195$ km/s in dimensional units.
Unit density corresponds to $\rho_0=4.1\times 10^{-13}$ g cm$^{-3}$. Dimensionless data shown on the plot can be converted to dimensional units for other types of stars using the reference values from the Table 1.}
\end{figure*}
\section{Conical Winds and Axial Jets}
A large number of simulations were done in order
to understand the origin and nature of conical winds.
All of the key parameters were varied in order to ensure that
there is no special dependence on any parameter.
We observed that the formation of conical winds
is a common phenomenon for a wide range of parameters.
They are most persistent and strong in cases where
the viscosity and diffusivity coefficients are not very
small, $\alpha_\nu\gtrsim 0.03$, $\alpha_\eta \gtrsim 0.03$.
Another important condition is that $\alpha_\nu \gtrsim\alpha_\eta$;
that is, the magnetic Prandtl number
of the turbulence, ${\cal P}_m=\alpha_\nu/\alpha_\eta \gtrsim 1$.
This condition favors the bunching of the stellar magnetic
field by the accretion flow.
Figure 3 shows a snapshot from our simulations at time $t=1400$ for the propeller regime.
The figure shows the dimensionless density and velocity at sample points.
One can see that the velocities in the conical wind component are similar to those in conical winds around slowly rotating stars.
Matter launched from the disc initially has
an approximately Keplerian azimuthal velocity, $v_K=\sqrt{GM_*/r}$.
It is gradually accelerated to poloidal velocities $v_p\sim (0.3-0.5) v_K$
and the azimuthal velocity decreases.
The flow has a high density and carries most of the disc mass into the outflows.
The situation is the opposite in the axial jet component where the density is $10^2-10^3$ times lower, while the poloidal and total velocities are
significantly higher.
Thus we find a {\it two-component outflow}: a matter
dominated conical wind and a magnetically dominated axial jet.
\begin{figure*}
\centering
\includegraphics[width=5in]{Fig4.eps}
\caption{The conical wind/jet from a slowly rotating star
at time $t= 860$ adapted from Lii et al. (2012).
The background shows the poloidal matter flux density $\rho{\bf v}_p$ and the lines show
the poloidal projections of the magnetic magnetic field.
The red vectors show the poloidal matter velocity ${\bf v}_p$.
Dimensional values can be obtained from Table 1.
For example for a CTTS, $t_0 = 0.366$ days, $R_0= 2\ensuremath{{\rm R}_{\odot}}$, $ t=860$ corresponds
to $315$ days, and the simulation region is $0.39$ AU in radius.
The horizontal axis shows the distance from the star in units of the reference radii $R_0$. For this case $\alpha_\nu=0.3$ and $\alpha_\eta=0.1$. }
\end{figure*}
We observe conical winds in both slowly and rapidly rotating stars.
In both cases, matter in the conical winds passes through the Alfv\'en surface
(and shortly thereafter through the fast magnetosonic point),
beyond which the flow is matter-dominated in the sense that the energy flow is carried mainly by the matter.
The situation is different for the axial jet component where the flow
is sub-Alfv\'enic within the simulation region.
For this component the energy flow is carried by the
Poynting flux and the angular moment flow is carried by the
magnetic field.
\noindent{\bf Collimation and Driving of the Outflows:}
Figure 4 shows the long-distance development of a conical wind from a slowly rotating star. At large distances
the conical wind becomes collimated.
To understand the collimation we analyzed
total force (per unit mass)
perpendicular to a poloidal magnetic field line (Lii et al. 2012).
For distances beyond the Alfv\'en surface of the flow this
force is approxiamtely
\begin{equation}
f_{\rm tot, \perp} = - v_{\rm p}^2 \frac{\partial \Theta}{\partial s}
- \frac{1}{8\pi \rho}\frac{\partial{\bf B}_p^2}{\partial n}
-\frac{1}{8\pi\rho (r\sin\theta)^2}\frac{\partial(r\sin\theta B_\phi)^2}{\partial n} + \frac{v_\phi^2}{r}\frac{\cos\Theta}{\sin\theta}.
\end{equation}
(Ustyugova et al. 1999). Here, $\Theta$ is the angle between the poloidal magnetic field and the symmetry axis, $s$ is the arc length
along the poloidal field line, $n$ is a coordinate normal to
the poloidal field, and the $p-$subscripts indicate
the poloidal component of a vector.
Once the jet begins to collimate, the curvature term $-v_{\rm p}^2\partial \Theta/\partial s$ also becomes negligible.
The magnetic force may act to either collimate or decollimate the jet, depending on the relative magnitudes of the toroidal $(r\sin\theta B_\phi)^2$ gradient (which collimates the outflow) and poloidal ${\mathbf B}_p^2$ gradient (which ``decollimates'').
In our simulations, the collimation of the matter implies that the magnetic hoop stress is larger than the poloidal field gradient. Thus the main perpendicular forces acting in the jet are the collimating effect of the toroidal magnetic field and the decollimating effect of the centrifugal force
and the gradient of ${\bf B}_p^2$. The collimated effect of $B_\phi$ dominates. Note that in MKS units $2\pi r\sin\theta B_\phi/\mu_0$ is the poloidal current flowing through a surface of radius $r$ from colatitude zero to $\theta$. For the jets from young stars this current is of the order of
$2 \times 10^{13}$ A.
\begin{figure*}[b]
\centering
\includegraphics[width=5.in]{Fig5.eps}
\caption{Forces along a field line in the jet adapted from Lii
et al. (2012).
{\bf Panel (a)} shows the poloidal matter flux density \ensuremath{\rho \left| {\mathbf{v}_p} \right|}\ as a background overplotted with poloidal magnetic field lines. The vectors show the {\it total} force ${\mathbf f}_{\rm tot}$ along a representative field line originating from the disk at $r = 1.24$.
{\bf Panel (b)} plots the angular velocity $\Omega$ as the background. The vectors show the sum of the {\it gravitational + centrifugal} forces ${\mathbf f}_{\rm G+C}$ along the representative field line.
{\bf Panel (c)} shows the poloidal current $I_p$ as the background. The vectors show the total {\it magnetic} force ${\bf f}_{\rm M}$ along the representative field line. }
\end{figure*}
The driving force for the outflow
is simply the force parallel to the poloidal magnetic field of the flow $f_{{\rm tot},\parallel}$.
This is obtained by taking the dot product of the Euler equation with the $\hat{\mathbf b}$ unit vector which is parallel to the poloidal magnetic field line ${\mathbf B}_p$. The derivation by Ustyugova et al. (1999) gives
\begin{equation}
f_{\rm tot, \parallel}
= - \frac{1}{\rho}\frac{\partial P}{\partial s} - \frac{\partial \Phi}{\partial s} + \frac{v_\phi^2}{r\sin\theta} \sin\Theta + \frac{1}{4 \pi \rho} \hat{\mathbf b} \cdot [(\nabla \times \bvec) \times \bvec].
\end{equation}
Here, the terms on the right-hand side correspond to the
pressure, gravitational, centrifugal and magnetic forces, respectively
denoted ${\bf f}_{\rm P, G, C, M}$.
The pressure gradient force, ${\bf f}_{\rm P}$, dominates within the disk. The matter in the disk is approximately in Keplerian rotation such that the sum of the gravitational and centrifugal forces roughly cancel (${\bf f}_{\rm G+C} \approx 0$). Near the slowly rotating star, however, the matter is strongly coupled to the stellar magnetic field and the disk orbits at sub-Keplerian speeds, giving ${\bf f}_{\rm G+C} \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}} 0$.
The magnetic
driving force (the last term of Eq. 6) can be expanded as
\begin{equation}
f_{\rm M, \parallel}
= -\frac{1}{8 \pi \rho (r\sin\theta)^2}{\partial \over \partial s}
(r\sin\theta B_\phi)^2~,
\end{equation}
(Lovelace et al. 1991).
Figure 5 shows the variation of the total force ${\bf f}_{\rm tot}$,
the gravitational plus centrifugal force, and the magnetic force
along a representative field line.
This analysis establishes that the predominant driving force for the outflow is the magnetic force (Eq. 7) and not the centrifugal force.
This in agreement with the analysis of Lovelace et al. (1991).
{\bf Variability:} For both rapidly and slowly
rotating stars the magnetic field lines
connecting the disc and the star have the tendency to
inflate and open (Lovelace, Romanova \& Bisnovatyi-Kogan 1995).
Quasi-periodic reconstruction of the magnetosphere due to
inflation and reconnection has been discussed theoretically (Aly
\& Kuijpers 1990) and has been
observed in a number of axisymmetrtic simulations (Hirose et al.
1997; Goodson et al. 1997, 1999; Matt et al. 2002; Romanova et al.
2002). Goodson \& Winglee (1999) discuss the physics of inflation cycles.
They have shown that each cycle of inflation
consists of a period of matter accumulation near
the magnetosphere, diffusion of this matter through the magnetospheric field,
inflation of the corresponding field lines,
accretion of some matter onto the star, and outflow of some matter as winds,
with subsequent expansion of the magnetosphere. There simulations show $5-6$
cycles of inflation and reconnection.
Our simulations often show $30-50$ cycles of inflation and reconnection.
Figure 6 shows the time evolution of the accretion rates
for a of a slowly rotating star (Romanova et al. 2009).
\begin{figure*}[h]
\centering
\includegraphics[width=3.in]{Fig6.eps}
\caption{Matter flux onto the star $\dot M_s$, into the
conical wind $\dot M_w$, and through the disk $\dot M_d$. }
\end{figure*}
\begin{figure*}[h]
\centering
\includegraphics[width=4.in]{Fig7.eps}
\caption{The initial poloidal magnetic field lines
and constant magnetic pressure lines for
the case of an aligned dipole and quadrupole field adapted from
Lovelace et al. (2010).
The funnel flow (ff) and the wind in this figure are suggested.
The dashed lines are constant values of ${\bf B}^2$.}
\end{figure*}
Kurosawa and Romanova (2012) have calculated spectra from modeled conical winds using the radiative transfer code TORUS
and have shown that conical winds may explain different features in the hydrogen spectral lines, in the He I line and
also a relatively narrow, low-velocity blue-shifted absorption
components in the He I $\lambda 10830$ which is often seen in observations (Kurosawa et al. 2011).
\section{One-Sided and Lop-Sided Jets}
There is clear evidence, mainly from
Hubble Space Telescope (HST) observations, of the asymmetry
between the approaching and receding jets
from a number of young stars.
The objects include the jets in HH 30 (Bacciotti et al. 1999),
RW Aur (Woitas et al. 2002), TH 28 (Coffey et al. 2004),
and LkH$\alpha$ 233 (Perrin \& Graham 2007).
Specifically, the radial speed of the approaching
jet may differ by a factor of two from that of
the receding jet.
For example, for RW Aur the radial redshifted
speed is $\sim 100$ km/s whereas the blueshifted
radial speed is $\sim175$ km/s.
The mass and momentum fluxes are also
significantly different
for the approaching and receding jets in a number of cases.
Of course, it is possible that the observed asymmetry
of the jets could be due to say differences in the gas densities
on the two sides of the source. Here, we discuss
the case of intrinsic asymmetry where the asymmetry of outflows is
connected with asymmetry of the star's magnetic field.
Substantial observational evidence points to the fact that
young stars often have {\it complex} magnetic fields consisting
of dipole, quadrupole, and higher order poles
misaligned with respect to each other and the rotation
axis (Jardine et al. 2002; Donati et al. 2008).
Analysis of the plasma flow around stars with realistic fields
have shown that a fraction
of the star's magnetic field lines are open
and may carry outflows (e.g., Gregory et al. 2006).
It is evident that the complex magnetic field
of a star will destroy the commonly assumed symmetry of the magnetic field and the plasma about the equatorial plane.
Figure 7 shows an illustrative complex magnetic field consisting of
the combination of a dipole and a quadrupole field both of which are axisymmetric.
The figure includes the suggested locations of the funnel
flow to the star (Romanova et al. 2002) and the conical wind outflows.
The MHD simulations fully support the qualitative picture
suggested in Figure 7 (Lovelace et al. 2010). The time-scale during
which the jet comes from the upper hemisphere is set
by the evolution time-scale for the stellar magnetic field.
This is determined by the dynamo processes responsible for
the generation of the field.
Remarkably, once the assumption of symmetry about the
equatorial plane is dropped, we find that the conical winds
alternately come from one hemisphere and then the other
{\it even} when the stellar magnetic field is a centered
axisymmetric dipole (Lovelace et al. 2010).
An illustrative case of this spontaneous symmetry breaking is shown
in Figure 8. The time-scale for the `flipping' is the accretion time-scale
of the inner part of the disc which is expected to be much less
than the evolution time of the star's magnetic field.
\begin{figure*}[t]
\centering
\includegraphics[width=5.in]{Fig8.eps}
\caption{"Flip-flop"of outflows in the case
where the stellar magnetic field is a
centered axisymmetric dipole adapted from Lovelace
et al. (2010).
The color background shows the matter flux
distribution, and the lines are the poloidal magnetic field lines.}
\end{figure*}
\section{Conclusions}
Detailed magnetohydrodynamic simulations have
established that long-lasting outflows of cold disc matter into a hot low-density corona from the disc-magnetosphere boundary in cases of slowly and rapidly rotating stars. The main results are the following:
For slowly rotating stars
a new type of outflow --- a conical wind --- has been discovered.
Matter flows out forming a {\it conical wind} which has the shape of a thin conical shell with a half-opening angle $\theta \sim 30^\circ$.
The outflows appear in cases where the magnetic flux of the star is bunched up by the inward accretion flow of the disc.
We find that this occurs when the turbulent magnetic Prandtl number
(the ratio of viscosity to diffusivity) ${\cal P}_m > 1$, and when the viscosity is sufficiently high, $\alpha_\nu\gtrsim 0.03$.
Winds from the disc-magnetosphere boundary
have been proposed earlier by Shu and collaborators and referred to as
X-winds (Shu et al. 1994).
In this model, the wind originates from a small region near the corotation radius $r_{\rm cr}$,
while the disc truncation radius $r_t$ (or, the magnetospheric radius $r_{\rm m}$) is only slightly smaller than
$r_{\rm cr}$ ($r_{\rm m}\approx 0.7 r_{\rm cr}$, Shu et al. 1994).
It is suggested that excess angular
momentum flows from the star to the disc and from there into the X-winds.
The model aims to explain the slow rotation of the star and the formation of jets.
In the simulations discussed here we have obtained outflows from both
slowly and rapidly rotating stars.
Both have conical wind components which are reminiscent of X-winds.
In some respects the conical winds are similar to
X-winds: They both require {\it bunching} of the poloidal field lines and show outflows from the inner disc; and they both have high rotation and show gradual poloidal acceleration (e.g., Najita \& Shu 1994).
The main differences are the following:
{(1)} The conical/propeller outflows have {\it two components}: a slow high-density conical wind (which can be considered as an analogue of the X-wind), and a fast low-density jet.
No jet component is discussed in the X-wind model.
{(2)} Conical winds form around stars with {\it any rotation rate} including very slowly rotating stars. They do not require fine tuning of the corotation and truncation radii.
For example, bunching of field lines is often expected during periods of enhanced or unstable accretion when the disc comes closer to the surface of
the star and $r_{\rm m}<<r_{\rm cr}$. Under this condition conical winds will form. In contrast, X-winds require $r_{\rm m}\approx r_{\rm cr}$.~ { (3)}
The base of the conical wind component in both slowly and rapidly rotating stars is associated with the region where the field lines are bunched up, and not with the corotation radius.
{(4)} X-winds are driven by the {\it centrifugal force}, and as a result matter flows over a wide range of directions below the ``dead zone" (Shu et al. 1994; Ostriker \& Shu 1995).
In conical winds the matter is driven by the {\it magnetic force} (Lovelace et al. 1991) which acts such that
the matter flows into a {\it thin shell} with a cone half-angle $\theta\sim 30^\circ$. The same force tends to collimate the flow.
For rapidly rotating stars in the propeller regime where
$r_{\rm m} > r_{\rm cr}$ and where the condition for bunching, ${\cal P}_m>1$, is satisfied
we find two distinct outflow components (1) a relatively low-velocity conical wind and (2) a high-velocity axial jet.
A significant part of the disc matter and angular momentum flows into the conical winds.
At the same time a significant part of the rotational energy of the star flows
into the magnetically-dominated axial jet.
This regime is particularly relevant to protostars, where the star rotates rapidly and has a high accretion rate.
The star spins down rapidly due to the angular momentum flow into the axial jet along the field lines connecting the star and the corona. For typical parameters a protostar spins down in $3\times10^5$ years. The axial jet is powered by the spin-down of the star rather than by disc accretion.
The matter fluxes into both components (wind and jet) strongly oscillate due to events of inflation and reconnection. Most powerful outbursts occur every $1-2$ months. The interval between outbursts is expected to be longer for smaller diffusivities in the disc.
Outbursts are accompanied by higher outflow velocities and stronger self-collimation of both components. Such outbursts may
explain the ejection of knots in some CTTSs every few months.
When the artificial requirement of symmetry about the
equatorial plane is dropped, MHD simulations reveal
that the conical winds may
alternately come from one side of the disc and then the other
even for the case where the stellar magnetic field is a
centered axisymmetric dipole (Lovelace et al. 2010).
In recent work we have studied the disc accretion
to rotating magnetized stars in the propeller regime
using a new code with very high resolution in the region
of the disc (Lii et al. 2013).
In this code {\it no} turbulent viscosity or diffusivity is incorporated,
but instead strong turbulence occurs due to the magneto-rotational
instability. This turbulence drives the accretion and it leads
to episodic outflows.
The authors thank G.~V. Ustyugova and A.~V. Koldoba for the development of the codes used in the
reported simulations. This research was supported in part by NSF grants AST-1008636 and AST-1211318 and by a NASA ATP grant NNX10AF63G; we thank NASA for use of the NASA High Performance Computing Facilities.
\section*{References}
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\end{harvard}
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 576 |
Chamber membership is vital to the success and growth of your business and our community. Membership with the Regional Chamber provides exclusive member perks like cost-saving discount programs and services that help you maximize your bottom line. Your membership can pay for itself!
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Consider us your Business Resource Center as we assist with all of your business needs.
Hover and click on the photos below to access information about Membership levels, services and much more. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,310 |
La elección presidencial de Israel de 2014 fue llevada a cabo de forma indirecta el 13 de junio de 2007. Los 120 miembros de la Knesset votaron por mayoría absoluta del total de los miembros al nuevo presidente de Israel por un mandato único (sin posibilidad de reelección) de 7 años.
La Knesset eligió a Shimon Peres, ex primer ministro y miembro del partido Kadima. Sus oponentes eran Reuven Rivlin, ex presidente de la Knesset, del Likud, y Colette Avital, de HaAvodá. Después de que la primera ronda de votación puso a Peres a la cabeza, pero apenas por debajo de la mayoría absoluta requerida para la elección, Rivlin y Avital se retiraron y Peres fue elegido fácilmente en la segunda ronda.
El presidente de Israel tiene fundamentalmente tareas formales y protocolares. Ostenta el título de Jefe del Sanedrín, el ente judicial y legislativo supremo del pueblo judío en la Tierra de Israel. Es el jefe de Estado del país, representa la unidad del país, del Estado y del pueblo por arriba de las diferencias partidarias. Por esto, en general las personas que han sido elegidas tuvieron gran recorrido en la política del país o del pueblo de Israel.
Resultado
Referencias
Israel
2007
Israel
Israel en 2007
Shimon Peres | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,204 |
module GraphQL
module StaticValidation
module FragmentsAreOnCompositeTypes
def on_fragment_definition(node, parent)
validate_type_is_composite(node) && super
end
def on_inline_fragment(node, parent)
validate_type_is_composite(node) && super
end
private
def validate_type_is_composite(node)
node_type = node.type
if node_type.nil?
# Inline fragment on the same type
true
else
type_name = node_type.to_query_string
type_def = context.warden.get_type(type_name)
if type_def.nil? || !type_def.kind.composite?
add_error(GraphQL::StaticValidation::FragmentsAreOnCompositeTypesError.new(
"Invalid fragment on type #{type_name} (must be Union, Interface or Object)",
nodes: node,
type: type_name
))
false
else
true
end
end
end
end
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,922 |
Q: Listing profiles and show related user email field in template Need to list my profilemodel-records. Would like to include the user.email field in the template.(The users email related to the profile...)
To get the users email is just
{{ user.email }}
I was sure I had done it like this before, but now nothing.....
{{ profile.user.email }}
This is my view:
def profileall(request):
profilelist = Profile.objects.all()
return render(request, 'docapp/profilelist.html',{'profilelist':profilelist})
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,851 |
The Doebner reaction is the chemical reaction of an aniline with an aldehyde and pyruvic acid to form quinoline-4-carboxylic acids.
The reaction serves as an alternative to the Pfitzinger reaction.
Reaction mechanism
The reaction mechanism is not exactly known; two proposals are presented here. One possibility is at first an aldol condensation, starting from the enol form of the pyruvic acid (1) and the aldehyde, forming an β,γ-unsaturated α-ketocarboxylic acid (2). This is followed by a Michael addition with aniline to form an aniline derivative (3). After a cyclization at the benzene ring and two proton shifts, the quinoline-4-carboxylic acid (4) is formed by water elimination:
An alternative mechanism is based on the aniline and the aldehyde forming at first the Schiff base upon water elimination. The subsequent reaction with the enol form of pyruvic acid (1) leads to the formation of the above-mentioned aniline derivative (3) followed by the above-described reaction mechanism:
Side reactions
It is reported in the literature that the Doebner reaction fails in case of 2-chloro-5-aminopyridine. In this case the cyclization would take place at the amino group instead of the benzene ring and lead to a pyrrolidine derivative.
Alternative reactions
Alternative syntheses of quinoline derivatives are for example:
Pfitzinger reaction
Conrad-Limpach reaction
Doebner-Miller reaction
Combes quinoline synthesis
References
Carbon-carbon bond forming reactions
Condensation reactions
Quinoline forming reactions
Multiple component reactions
Name reactions | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,708 |
import { deepStrictEqual } from 'assert';
import { deserializeEnvironmentVariableCollection, serializeEnvironmentVariableCollection } from 'vs/workbench/contrib/terminal/common/environmentVariableShared';
import { EnvironmentVariableMutatorType, IEnvironmentVariableMutator } from 'vs/workbench/contrib/terminal/common/environmentVariable';
suite('EnvironmentVariable - deserializeEnvironmentVariableCollection', () => {
test('should construct correctly with 3 arguments', () => {
const c = deserializeEnvironmentVariableCollection([
['A', { value: 'a', type: EnvironmentVariableMutatorType.Replace }],
['B', { value: 'b', type: EnvironmentVariableMutatorType.Append }],
['C', { value: 'c', type: EnvironmentVariableMutatorType.Prepend }]
]);
const keys = [...c.keys()];
deepStrictEqual(keys, ['A', 'B', 'C']);
deepStrictEqual(c.get('A'), { value: 'a', type: EnvironmentVariableMutatorType.Replace });
deepStrictEqual(c.get('B'), { value: 'b', type: EnvironmentVariableMutatorType.Append });
deepStrictEqual(c.get('C'), { value: 'c', type: EnvironmentVariableMutatorType.Prepend });
});
});
suite('EnvironmentVariable - serializeEnvironmentVariableCollection', () => {
test('should correctly serialize the object', () => {
const collection = new Map<string, IEnvironmentVariableMutator>();
deepStrictEqual(serializeEnvironmentVariableCollection(collection), []);
collection.set('A', { value: 'a', type: EnvironmentVariableMutatorType.Replace });
collection.set('B', { value: 'b', type: EnvironmentVariableMutatorType.Append });
collection.set('C', { value: 'c', type: EnvironmentVariableMutatorType.Prepend });
deepStrictEqual(serializeEnvironmentVariableCollection(collection), [
['A', { value: 'a', type: EnvironmentVariableMutatorType.Replace }],
['B', { value: 'b', type: EnvironmentVariableMutatorType.Append }],
['C', { value: 'c', type: EnvironmentVariableMutatorType.Prepend }]
]);
});
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,847 |
\section{Introduction}
The study of nonlinear waves in discrete media has been a prolific subject of great interest for more than thirty years. Among such waves, one can highlight solitons, which are localized waves emerging in nonlinear dispersive media \cite{daux}. A relevant example concerns optical waveguide arrays (see, e.g., Ref.~\cite{kiva}), where the inherent periodicity produces discreteness, which in turn gives rise to dispersion. The latter, may be counterbalanced by nonlinear effects, as e.g., the Kerr effect (the dependence of the refractive index on the intensity of the light pulse). This leads to the emergence of discrete solitons or discrete breathers, depending on the context they emerge or the equation by which they are described \cite{daux,kiva,reviews_breathers}.
In the case of discrete optical settings, such as the one of waveguide arrays mentioned above, the electric field of light beams can be described, in the framework of the paraxial approximation, by the Discrete Nonlinear Schr\"odinger (DNLS) equation \cite{kiva,solitons_optics}. The properties of solitons in the DNLS equation depend strongly on the type of nonlinearity (which, at the same time, depends on the particular type of dielectric the waveguides consists of). The most ubiquitous nonlinearity is the cubic one, which appears not only in optical settings caused by the above mentioned Kerr effect (occurring, e.g., in AlGaAs), but also in Bose-Einstein condensates (BECs) trapped in deep optical lattices, granular crystals or biomolecules \cite{book_panos}. Other nonlinearities of interest are the cubic-quintic, power-law, or saturable (in the case of photorefractive materials, such as LiNbO$_3$ or SBN) ones \cite{efremidis,morandotti,luther}. In this work, we focus on the last type of nonlinearity, and dub the DNLS equation with such nonlinearity as saturable DNLS or SDNLS equation, making comparisons with the analogously called cubic DNLS equation.
Contrary to the cubic DNLS equation, where the allowed frequencies for solitons are bounded only from below (with the lower bound corresponding to the linear limit), the frequencies available for SDNLS solitons have two bounds, with the upper one corresponding to the linear limit. For both equations, solitons become sharper when going beyond the linear limit. However, contrary to cubic solitons whose width monotonically decreases with the frequency, the width of SDNLS solitons has a minimum when decreasing their frequency, so that their width increases diverging when the frequency reaches the lower boundary \cite{maluckov}. Consequently, except for frequencies in the vicinity of the linear modes band, SDNLS solitons are in general wider than cubic ones. This is responsible, among other phenomena, for the possibility of radiationless motion of solitons in the SDNLS equation \cite{melvin_prl}, or the mobility of solitons in a two-dimensional saturable NLS lattice \cite{vicencio}, contrary to two-dimensional cubic NLS lattices where the generation of mobile solitons is impossible.
Having introduced one of the topics of this work, namely solitons in the SDNLS equation, we proceed by presenting the second one, i.e., the interaction of solitons with point impurities. This problem was already studied for the continuous NLS equation in the 1990s, with the works of Refs.~\cite{kosevich,Cao} (whose results were analytically proved and enlarged in Ref.~\cite{Goodman}) showing that, for attractive delta-like impurities, solitons can be trapped if their velocities are small and the impurity strength large enough, so that a nonlinear resonance with a defect mode occurs. In addition, this trapping can be supplemented with the splitting of the incoming soliton. The interaction with repulsive delta-like impurities shows a similar scenario but excluding the possibility of trapping \cite{Holmer}. In other words, the soliton behaves as a quantum particle which tunnels a potential barrier. This fact has been exploited in the generation of recombining solitons in harmonically trapped BECs \cite{Hulet}; such a generation process, has recently been demonstrated in experiments \cite{Wales}, and was shown to find applications in interferometry \cite{Polo,Sakaguchi}.
Notice, however, that this scenario becomes more complex with extended defects \cite{Ernst}. By virtue of Ehrenfest's theorem, the soliton can behave as a quantum particle if its length scale is larger than that of the defect \cite{Brand}. Consequently, narrow solitons (compared to the defect size), behave as classical particles and cannot split when interacting with defects \cite{Hansen}.
Once the main ingredients are presented, it is time to develop the main topic of the paper: the interaction of solitons with point impurities in the discrete NLS equation with saturable nonlinearity. Here it is important to point out that, despite the large number of studies devoted to the scattering of solitons from defects in continuous settings, there exist only a few works concerning discrete settings. Indeed, from the seminal work of Ref.~\cite{Bishop} where the dynamics of solitons on extended defects and point impurities was considered, very few examples can be found. For instance, in Ref.~\cite{Morales}, the trapping of low amplitude DNLS solitons by point impurities was considered, in Ref.~\cite{ricardo}
the scattering of highly localized discrete solitons from attractive and repulsive point impurities was analyzed,
while in Ref.~\cite{Yaounde}, the interaction of multipeaked solitons with point impurities was studied. Surprisingly, all of these studies have been circumscribed to the cubic DNLS equation. To the best of our knowledge,
relevant studies in the framework of the SDNLS model have not been performed so far (even in continuum settings). The aim of the present work is to fill this gap.
Coming back to Ref.~\cite{ricardo}, it is relevant to briefly recopilate the observed phenomenology described therein. In particular, it was found that, when the impurity is repulsive, the soliton behaves as a classical particle interacting with a potential barrier, so that the soliton can ``cross'' the impurity if its speed is high enough (decelerating where the crossing occurs) and being reflected otherwise. For the case of an attractive impurity, a trapping of the soliton at the impurity site is mainly observed, creating a stationary state. If the impurity is strong enough, apart from the trapped breathing soliton, a reflected one emerges and, for very strong impurities, the trapped soliton cannot be generated. Finally, if the attractive impurity is weak, the soliton can cross it, while its velocity increases. In other words, the splitting of solitons has not been observed yet in DNLS settings.
However, for the case of the SDNLS equation that we study herein, we find several differences. The most remarkable one is the possibility of soliton splitting for both attractive and repulsive impurities. In the former case, a soliton trapped at the impurity site can coexist simultaneously with the reflected and refracted solitons. We also observe that it is possible to find values of the velocity and the impurity strength for which the splinters possess the same power, although they can have different amplitude or width.
Our presentation is
organized as follows. In Section~\ref{sec:model} we introduce the model and some properties of stationary and moving discrete solitons of the saturable DNLS. The numerical results concerning the interaction between solitons and impurities are presented in Section~\ref{sec:numerics}. Finally, in Section~\ref{sec:conclusions}, we summarize the results of the study and present some ideas for further work.
\section{The model and its soliton solutions}
\label{sec:model}
\subsection{Model setup}
We consider
an array of coupled optical waveguides, filled with photorefractive dielectrics, which can be described by a
SDNLS equation of the form:
\begin{equation}\label{eq:dyn}
i\frac{d{\psi}_n}{dt}+C(\psi_{n+1}+\psi_{n-1}-2\psi_n)-\frac{\gamma}{1+|\psi_n|^2}\psi_n+\alpha_n\psi_n=0\,,\, n=-N/2,\ldots,N/2.
\end{equation}
Here, the wavefunction $\psi_n$ represents
the electric field envelope, $t$ denotes the propagation direction, $\gamma$ is the nonlinearity coefficient, $C$ the coupling constant, and $\alpha_n$ accounts for a linear inhomogeneity term. Experimentally, this kind of term can be implemented, e.g., by varying the separation between
a single pair of adjacent waveguides in an homogeneous array. In what follows, we consider this term to be of the form $\alpha_n=\alpha\delta_{n,0}$, i.e., it describes a single point defect in the lattice that can be either attractive ($\alpha>0$) or repulsive ($\alpha<0$). We consider here the case of a focusing nonlinearity, with $\gamma>0$ (and a positive coupling constant $C$); upon suitable parameter renormalization, we will fix the value of $\gamma$, without lack of generality, to $\gamma=1$.
The SDNLS equation possesses two conserved quantities, namely the power (squared $\ell^2$ norm):
\begin{equation}\label{eq:norm}
P=\sum_n|\psi_n|^2,
\end{equation}
and the Hamiltonian:
\begin{equation}\label{eq:hamiltonian}
H=\sum_n\left[C|\psi_{n+1}-\psi_n|^2+\gamma\log(1+|\psi|^2)-\alpha_n|\psi_n|^2\right].
\end{equation}
\subsection{Stationary and moving solitons}
Stationary solutions of equation (\ref{eq:dyn}) can be sought in the form:
\begin{equation}
\psi_n(t)=\phi_n\exp(-i\omega t)\,,
\end{equation}
with $\omega$ being the soliton frequency. With this assumption, (\ref{eq:dyn}) transforms into an algebraic set of equations:
\begin{equation}\label{eq:stat}
\omega\phi_n+C(\phi_{n+1}+\phi_{n-1}-2\phi_n)-\frac{\gamma}{1+|\phi_n|^2}\phi_n+\alpha_n\phi_n=0.
\end{equation}
Two important quantities to account for are the soliton's center of mass ($X$) and width ($W$) defined as follows:
\begin{equation}\label{eq:XW}
X=\frac{1}{P}\sum_n n|\psi_n|^2, \qquad W=\sqrt{\frac{1}{P}\sum_n n^2|\psi_n|^2-X^2}.
\end{equation}
We will be interested in generating moving solitons far from the defect site that are launched towards the latter. To this aim, first we need to calculate a stationary soliton centered at $n=n_0$, which is solution of Eq.~(\ref{eq:stat}) with $\alpha_n=0$; this equation is solved by means of a fixed-point algorithm (e.g., Newton-Raphson) using a quasi-continuum seed, $\phi_n=\sech(n-n_0)$ for $\omega=0.9$ and $C=1$. Once the stationary soliton for such values of $\omega$ and $C$ is found,
it can be continued to the desired values of those parameters. Figure~\ref{fig:stationary} shows the frequency dependence of the power, energy (Hamiltonian) and width of solitons for $\alpha_n=0$ and $C=1$; note that this value of the coupling constant will be fixed for the rest of the paper for reasons explained below. It can be observed that $0<\omega<\gamma$ and, since $\gamma=1$ (as stated in the previous subsection), $\omega\in(0,1)$, with the energy and power tending to zero when $\omega\rightarrow1$ and diverging when $\omega\rightarrow0$. As pointed out in the Introduction, the frequency is bounded, contrary to the cubic DNLS equation where the frequency takes values in a semi-infinite interval. Regarding the width, we can observe that it diverges in the limits $\omega\rightarrow0$ and $\omega\rightarrow1$, having a minimum at $\omega=0.43$; this behaviour is in stark contrast with the one observed in the cubic DNLS, where the width decreases when the frequency departs from the linear modes band. For the sake of comparison, Fig.~\ref{fig:stationary} shows a dashed (red) line indicating the width of the soliton considered in \cite{ricardo}, which is smaller than that of any SDNLS soliton.
\begin{figure}
\begin{tabular}{cc}
\includegraphics[width=6cm]{norm.pdf} &
\includegraphics[width=6cm]{energy.pdf} \\
\includegraphics[width=6cm]{width.pdf} &
\includegraphics[width=6cm]{profile.pdf}
\end{tabular}%
\caption{Frequency dependence
of the power (top left panel), energy (top right panel) and width (bottom left panel) of solitons in a homogeneous lattice ($\alpha_n=0$) with $C=1$. In the
bottom left panel, the dashed (red) line
corresponds to the width of the cubic DNLS soliton considered in the work of \cite{ricardo}. Bottom right panel depicts the profile of the soliton density, $|\psi_n|^2$, for $\omega=0.9$ (the moving soliton profile is, obviously, the same).}
\label{fig:stationary}
\end{figure}
Once a stationary soliton is attained (see Fig.~\ref{fig:stationary} for an example of its profile), it is kicked with a thrust $q$ in the following form:
\begin{equation}
\psi_n(0)=\phi_n\mathrm{e}^{iq(n-n_0)}.
\end{equation}
When this initial condition is used for simulating Eq.~(\ref{eq:dyn}), a moving soliton can be generated, as long as the Peierls-Nabarro barrier is small enough. This can only be attained when $C$ is large enough, although, as explained in \cite{melvin_prl}, this does not warrant the mobility. Nevertheless, we have checked that the solitons can be set into motion emitting a low amount of radiation for $C=1$ and $\omega=0.9$ whenever $q\in[0.05,0.5]$. Because of this, we implement such a restriction in $q$ in what follows. As the free soliton can be considered as a classical quasi-particle of effective mass equal to $P$, and $\phi_n\in\mathbb{R}^n$, the velocity $v=dX/dt$ is related to the thrust $q$ via
\begin{equation}\label{eq:velocity}
v=\frac{2\sin(q)}{P}\sum_n\phi_{n+1}\phi_n.
\end{equation}
For our particular case, $v=1.9687\sin(q)$. From this relation, it is easy to find the kinetic energy of the quasi-particle as $K=5.9584\sin^2(q/2)$.
\section{Interaction of solitons with impurities} \label{sec:numerics}
In what follows we consider solitons with frequency $\omega=0.9$ and coupling constant $C=1$, which are launched from $n_0=-50$ (unless stated otherwise) in a lattice with $N$ nodes and periodic boundary conditions, and a delta-like impurity of strength $\alpha$ is located at $n=0$; the number $N$ has been taken large enough so that the boundaries do not affect the outcome. In order to monitor the outcome, we define the reflection ($R$), transmission ($T$), and trapping ($L$) coefficients as follows:
\begin{equation}\label{eq:coefs}
R=\lim_{t\rightarrow\infty}\frac{1}{P}\sum_{n=-N/2}^{-(\delta+1)}|\psi_n(t)|^2\,,\quad
T=\lim_{t\rightarrow\infty}\frac{1}{P}\sum_{n=\delta+1}^{N/2}|\psi_n(t)|^2\,, \quad
L=\lim_{t\rightarrow\infty}\frac{1}{P}\sum_{n=-\delta}^{\delta}|\psi_n(t)|^2\,,
\end{equation}
where $P$ is soliton power [see Eq.~(\ref{eq:norm})] and $\delta=20$ (i.e. we are assuming a typical trapped soliton width of $\lesssim 41$ nodes, cf. Fig.~\ref{fig:simsplit_attractive} below). The limit
$t\rightarrow\infty$ in the above definitions has the following meaning: the quantities $R$, $T$ and $L$ are determined at values of $t$
large enough
after the collision event but, at the same time, small enough for neglecting the interaction between solitons via the periodic boundaries.
First, we consider the case of repulsive impurity, i.e., $\alpha<0$. In this case, the outcome is quite simple: the soliton breaks into a reflected and a transmitted soliton, similarly to the scattering of a quantum particle from a potential barrier. As is expected, $T$ decreases with $|\alpha|$ and, at the same time, $R$ grows with $|\alpha|$. Figure~\ref{fig:TR_repulsive} shows the dependence of $T$ on $\alpha$ and $q$ ($R$ is simply equal to $1-T$). Consequently, there is a critical value of $\alpha$, namely $\alpha_c$, depending on $q$, for which equal splitting takes place, i.e. $T=R=0.5$. This equal splitting
results in the generation of two solitons with the same power, although it does not mean that the solitons have the same width, amplitude or velocity. Specifically, as can be seen in the example of Fig.~\ref{fig:simsplit_repulsive} depicting the evolution of a soliton with $\alpha=-0.7$ and $q=0.41$ (corresponding to a velocity $v=0.785$), the splinters move with velocities $0.737$ and $-0.690$ in spite that $R=T=0.5$.
We can compare this outcome with relevant phenomenology reported previously in the literature. In particular, the work of \cite{Holmer} considers the scattering of a soliton in the NLS from a Dirac delta potential obtaining that, if the soliton velocity $V$ is large, the transmission coefficient is the same as that of a quantum particle in the linear Schr\"odinger equation $i\partial_t\psi=[-(1/2)\partial_{x}^2+\alpha\delta(x)]\psi$, namely:
\begin{equation}\label{eq:T_linear}
T(V)=\frac{V^2}{V^2+\alpha^2},
\end{equation}
so a perfect splitting would take place at $|\alpha|=|\alpha_c|$ with $|\alpha_c|=V$. In our case, the DNLS equation (\ref{eq:dyn}) can be considered as a finite differences discretization of the NLS equation, with the 2nd-order spatial derivative being
$(1/2)\partial_{x}^2\psi\approx(1/2h^2)(\psi_{n+1}+\psi_{n-1}-2\psi_n)$. Upon comparing this expression with the discrete Laplacian term of (\ref{eq:dyn}), we get that $C=1/2h^2$. As $x=nh$ the discrete soliton velocity $v$ is related to that of the continuum soliton by $v=V/h$, or in terms of the coupling constant, $v=V\sqrt{2C}$.
With this in mind, one can see that in the discrete case, $|\alpha_c|$ should tend to $v\sqrt{2C}$ for fast enough solitons if the scattering behaves for the DNLS in a similar fashion to the NLS case. We can check that this is the case in Fig.~\ref{fig:split_repulsive} which, apart from $\alpha_c(q)$ and $\alpha_c(v)$, shows the velocities of the splinters as a function of $q(\alpha_c)$ in the case $R=T=0.5$. From the latter figure, it can be seen that the velocity of the transmitted splinter is always higher than that of the reflected one, so that the scattering process seems to have a preference to ``remember'' the direction of the incoming soliton. In addition, one can see that the velocities of the splinters approach to the velocity of the incoming soliton when the latter increases.
\begin{figure}
\begin{tabular}{cc}
\includegraphics[width=6cm]{T_repul.pdf} &
\includegraphics[width=6cm]{RT_repul.pdf} \\
\end{tabular}%
\caption{Repulsive impurity. Left panel: dependence of $T$ on $\alpha$ and $q$. Right panel: dependence of $R$ and $T$ on $\alpha$ for $q=0.2$. In all the simulations, $N=1000$
}
\label{fig:TR_repulsive}
\end{figure}
\begin{figure}
\begin{tabular}{cc}
\includegraphics[width=6cm]{evol_repul_1.pdf} &
\includegraphics[width=6cm]{evol_repul_2N.pdf} \\
\end{tabular}%
\caption{Evolution of a soliton with $q=0.41$ interacting with a repulsive impurity with $\alpha=-0.70$, which results in equal splitting. Left panel shows the space-time diagram of the density $|\psi_n|^2$, and right panel shows the profiles of the splinters a long time ($t=2500$).}
\label{fig:simsplit_repulsive}
\end{figure}
\begin{figure}
\begin{tabular}{cc}
\includegraphics[width=6cm]{split_alphac_repul.pdf} &
\includegraphics[width=6cm]{split_vel_repul.pdf} \\
\end{tabular}%
\caption{Analysis of equal splitting for repulsive impurities. Left panel shows the value of $|\alpha_c|$ versus $q$ and $v$; the (red) dashed line in the bottom plot corresponds to a line with slope $\sqrt{2C}$, corresponding to the theoretical prediction of the NLS equation. Right panel shows the velocities of the transmitted $v_t$ and reflected $v_r$ splinters together with the velocity of the incoming soliton.}
\label{fig:split_repulsive}
\end{figure}
Next, we proceed with the case of an attractive impurity, i.e., $\alpha>0$. In this case, the outcome is similar to
that occurring for repulsive impurities, except for the excitation of trapped solitons at the impurity site in some regions of the $(\alpha,q)$ plane. Consequently, the equal splitting scenario is found for $R=T<0.5$, as part of the energy remains trapped. The dependence of $T$, $R$ and $L$ with $\alpha$ and $q$ is depicted in Figs.~\ref{fig:TRL_attractive1} and Figs.~\ref{fig:TRL_attractive2}, where it is clear
to see that $L$ is nonzero in a localized region of small $q$ and $\alpha$; this is an indication of a nonlinear resonance occurring between the incoming and trapped solitons. We can also observe that for low velocities and small enough $\alpha$, the behaviour of the coefficients is not smooth.
Figure~\ref{fig:simsplit_attractive} shows the evolution of a trapping and splitting process occurring for $\alpha=0.40$ and $q=0.2340$ (corresponding to a velocity $v=0.456$). In this case, we find equal splitting with $R=T=0.4$, and trapping with $L=0.2$; the velocities of the splinters are $0.440$ and $-0.354$. Notice that the trapped soliton has a peaked profile $\sim\exp(a|n|)$, contrary to the incoming and outgoing solitons which preserve the typical of SDNLS solitons sech / Gaussian shape; this is caused by the peaked shape of nonlinear impurity modes, i.e., solitons centered at the impurity site. On the other hand, the trapped soliton oscillates with a frequency $0.915$; the nonlinear impurity mode with this frequency has been depicted also in Fig.~\ref{fig:simsplit_attractive} with a perfect matching with the trapped soliton, demonstrating that the soliton trapping is actually caused by a nonlinear resonance.
Here, it is worth mentioning that, as predicted in Ref.~\cite{Holmer}, the equal splitting takes place for high speed at $\alpha=\alpha_c=v\sqrt{2C}$, similarly to the repulsive impurity case ---see Fig.~\ref{fig:split_attractive}. When comparing to the repulsive case, the velocity of the transmitted soliton is very much closer to that of the incoming soliton, being even higher than the latter for small values of velocities, as can be seen in the middle panel of Fig.~\ref{fig:split_attractive}. This figure also shows the trapping coefficient $L$, for the equal splitting case, featuring a decaying behavior with $q$.
\begin{figure}
\begin{tabular}{ccc}
\includegraphics[width=6cm]{T_attr.pdf} &
\includegraphics[width=6cm]{R_attr.pdf} &
\includegraphics[width=6cm]{L_attr.pdf} \\
\end{tabular}%
\caption{Attractive impurity. Dependence of $T$ (left panel), $R$ (middle panel) and $L$ (right panel) on $\alpha$ and $q$. In all simulations, $N=1000$.
}
\label{fig:TRL_attractive1}
\end{figure}
\begin{figure}
\begin{tabular}{cc}
\includegraphics[width=6cm]{LRT_attr1.pdf} &
\includegraphics[width=6cm]{LRT_attr2.pdf} \\
\includegraphics[width=6cm]{LRT_attr3.pdf} &
\includegraphics[width=6cm]{LRT_attr4.pdf} \\
\end{tabular}%
\caption{Dependence of $T$, $R$ and $L$ on $\alpha$, for fixed $q=0.1$ (top left panel) and $q=0.2$ (top right panel), and on $q$, for fixed $\alpha=0.2$ (bottom left panel) and $\alpha=0.5$ (bottom right panel).}
\label{fig:TRL_attractive2}
\end{figure}
\begin{figure}
\begin{tabular}{cc}
\includegraphics[width=6cm]{evol_attr_1.pdf} &
\includegraphics[width=6cm]{evol_attr_2N.pdf} \\
\end{tabular}%
\caption{Evolution of a soliton with $q=0.2340$ interacting with an attractive impurity $\alpha=0.40$ which results in equal splitting and trapping. Left panel shows the space-time diagram of the density $|\psi_n|^2$ and right panel shows the profiles of the splinters and the trapped soliton for a long time ($t=2500$). Notice the red dots overlapping the trapped soliton; it corresponds to a defect mode with the same frequency $\omega=0.915$ as the trapped soliton.}
\label{fig:simsplit_attractive}
\end{figure}
\begin{figure}
\begin{tabular}{ccc}
\includegraphics[width=6cm]{split_alphac_attr.pdf} &
\includegraphics[width=6cm]{split_vel_attr.pdf} &
\includegraphics[width=6cm]{split_coef_attr.pdf}
\end{tabular}%
\caption{Analysis of equal splitting for attractive impurity. Left panel shows the value of $\alpha_c$ versus $q$ and $v$; the red dashed line in the bottom plot corresponds to a line with slope $\sqrt{2C}$, corresponding to the theoretical prediction; middle panel depicts the velocities of the transmitted $v_t$ and reflected $v_r$ splinters together with the velocity of the incoming soliton; right panel displays the trapping coefficient $L$ versus $q$ when $\alpha=\alpha_c$.}
\label{fig:split_attractive}
\end{figure}
Some of the results presented here are typical for the quantum scattering of nonlinear waves from a localized potential. They are, however, different from those reported in \cite{ricardo} for the cubic DNLS, which are closer to a classical scattering. This is mainly due to the fact that the width of the SDNLS soliton is larger than the effective width of the defect potential, contrary to the case of the soliton of \cite{ricardo}, which is very much narrower. Nevertheless, it should be pointed out that this fact does not preclude the
occurrence of quantum scattering in the cubic DNLS; in fact, if a soliton with $\omega=2.1$ were chosen in the model of \cite{ricardo}, a qualitatively similar behaviour to the reported herein could be observed.
\section{Conclusions} \label{sec:conclusions}
In this work, we have analyzed the interaction of a discrete soliton with an impurity (considered to be either attractive or repulsive) in the saturable nonlinear Schr\"odinger equation. We have found that the scattering process is characterized by typical features of quantum scattering of nonlinear localized waves from narrow inhomogeneities, namely the soliton splits into two splinters and, in the case of attractive impurities, a nonlinear defect mode can be excited resulting in the emergence of trapped solitons. This behaviour is in stark contrast with the observations for narrow solitons interacting with an impurity in the cubic DNLS (see, e.g., the work of Ref.~\cite{ricardo}), where the soliton does not split. Particular attention has been paid to the case of equal splitting and results were successfully compared with the outcome observed for high-speed continuum NLS solitons \cite{Holmer}.
The results presented herein
pave the way for quite interesting future studies. In particular,
the possibility of storing a large amount of energy of SNDLS solitons (compared to cubic ones), specially for frequencies far from the linear modes band, suggests a systematic investigation of
the outcome of the scattering from the impurity, which is expected to be very rich when
weak attractive impurities are considered. In addition, a similar study in the two-dimensional settings, which is impossible in the case of the usual DNLS with the
the cubic nonlinearity but can be realized in the SDNLS equation \cite{vicencio}, may open
avenues for novel unexplored phenomena. These studies are in progress and will be published elsewhere.
\begin{acknowledgments}
We acknowledge support from EU (FEDER program 2014-2020) through both MCIN/AEI/10.13039/501100011033 (under the projects PID2019-108508GB-I00 (FP), PID2019-110430GB-C21 (JCM) and PID2020-112620GB-I00 (JCM)), and Consejer\'{\i}a de Econom\'{\i}a, Conocimiento, Empresas y Universidad de la Junta de Andaluc\'{\i}a (under the projects P18-RT-3480 and US-1380977) (JCM). JFT and AP acknowledge the funding of this study via the project HPC-EUROPA3 (INFRAIA-2016-1-730897), with the support of the EC Research Innovation Action under the H2020 Programme. JFT
gratefully acknowledges computer resources and technical support provided by the Greek Research and Technology Network (GRNET) at HPC-ARIS supercomputer center.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 668 |
\section{Introduction}
It is well known that classical Newtonian dynamics fails on galactic
scales. There is astronomical and cosmological evidence for a
discrepancy between the dynamically measured mass-to-light ratio of
any system and the minimum mass-to-light ratios that are compatible
with our understanding of stars, of galaxies, of groups and clusters
of galaxies, and of superclusters. It turns out that on large scales
most astronomical systems have much larger mass-to-light ratios than
the central parts. Observations on the rotation curves have turn out
that galaxies are not rotating in the same manner as the Solar
System. If the orbits of the stars are governed solely by
gravitational force, it was expected that stars at the outer edge of
the disc would have a much lower orbital velocity than those near
the middle. In fact, by the Virial theorem the total kinetic energy
should be half the total gravitational binding energy of the
galaxies. Experimentally, however, the total kinetic energy is found
to be much greater than predicted by the Virial theorem. Galactic
rotation curves, which illustrate the velocity of rotation versus
the distance from the galactic center, cannot be explained by only
the visible matter. This suggests that either a large portion of the
mass of galaxies was contained in the relatively dark galactic halo
or Newtonian dynamics does not apply universally.
The dark matter proposal is mostly referred to Zwicky (1957) who
gave the first empirical evidence for the existence of the unknown
type of matter that takes part in the galactic scale only by its
gravitational action. He found that the motion of the galaxies of
the clusters induced by the gravitational field of the cluster can
only be explained by the assumption of dark matter in addition to
the matter of the sum of the observed galaxies. Later, It was
demonstrated that dark matter is not only an exotic property of
clusters but can also be found in single galaxies to explain their
flat rotation curves.
The second proposal results in the modified Newtonian dynamics
(MOND), proposed by Milgrom, based on a modification of Newton's
second law of motion (Milgrom, 1983). This well known law states
that an object of mass $m$ subject to a force $F$ undergoes an
acceleration $a$ by the simple equation $F=ma$. However, it has
never been verified for extremely small accelerations which are
happening at the scale of galaxies. The modification proposed by
Milgrom was the following
$$
F=m\mu(\frac{a}{a_0})a,
$$
$$
\mu(x)=\left \{ \begin{array}{ll} 1 \:\: \mbox{if}\:\: x\gg 1
\\
x \:\: \mbox{if}\:\: \|x|\ll 1,
\end{array}\right.
$$
where $a_0=1.2\times10^{-10} ms^{-2}$ is a proposed new constant.
The acceleration $a$ is usually much greater than $a_0$ for all
physical effects in everyday life, therefore $\mu(a/a_0)$=1 and
$F=ma$ as usual. However, at the galactic scale where $a \sim a_0$
we have the modified dynamics $F=m(\frac{a^2}{a_0})$ leading to a
constant velocity of stars on a circular orbit far from the center
of galaxies.
Another interesting model in this direction has been recently
proposed by Sanders. In this model, it is assumed that gravitational
attraction force becomes more like $1/r$ beyond some galactic scale
(Sanders, 2003). A test particle at a distance $r$ from a large mass
$M$ is subject to the acceleration
$$
a = \frac{GM}{r^2} g(r/r_0),
$$
where $G$ is the Newtonian constant, $r_0$ is of the order of the
sizes of galaxies and $g(r/r_0)$ is a function with the asymptotic
behavior
$$
g(r/r_0)=\left \{ \begin{array}{ll} 1 \:\: \mbox{if}\:\: r\gg r_0
\\
r/r_0 \:\: \mbox{if}\:\: r\ll r_0.
\end{array}\right.
$$
Dark matter as the manifestation of Mach principle has also been
considered as one of the solutions for the dark matter problem.
According to Mach principle the distant mass distribution of the
universe has been considered as being responsible for generating the
local inertial properties of the close material bodies.
Borzeszkowski and Treder have shown that the dark matter problem may
be solved by a theory of Einstein-Mayer type (Borzeszkowski and
Treder, 1998). The field equations of this gravitational theory
contain hidden matter terms, where the existence of hidden matter is
inferred solely from its gravitational effects. In the
nonrelativistic mechanical approximation, the field equations
provide an inertia-free mechanics where the inertial mass of a body
is induced by the gravitational action of the cosmic masses. From
the Newtonian point of view, this mechanics shows that the effective
gravitational mass of astrophysical objects depends on $r$ such that
one expects the existence of new type of matter, the so called dark
matter.
\section{The model}
We introduce a new interpretation of Mach principle by which a
particle with the mass $m$ and at the radial position $r$ interacts
gravitationally with the matter $M(r)$ encompassed by the region of
radius $r$ as follows
\begin{equation}
V= -\frac{G m M(r)}{R},\label{1}
\end{equation}
where $R$ is the radius of the galactic disc. This is a non-local
interaction of the particle with the mass distribution inside the
galactic disc.
The motivation for taking this type of gravitational potential at
galactic scale is the main observation that in the rotation curve of
galaxies the linear curve turns into a flat one in a rather sudden
way. So, this behavior may be explained if we interpret it as a
result of a rather sudden change in the mass distribution of that
structure. In fact, this is really the case because the turning
region of the curve from linear to flat case corresponds to the
region in which the central massive disc of the structure with the
typical radius $R$ turns into the outer void region without mass.
A particle of gravitational mass $m$ located at a distance $r<R$
from the center of the galaxy will acquire the total energy
\begin{equation}
E=\frac{1}{2}mv^2 -\frac{G m M(r)}{R},\label{2}
\end{equation}
where $v$ is the circular velocity around the center of galaxy. If
we roughly assume the mass $M$ of the galaxy is uniformly
distributed over a disc of radius $R$ then $M(r)=\sigma \pi r^2$,
where $\sigma$ is the constant surface mass density. Newton's law is
then written as
\begin{equation}
\frac{mv^2}{r}=\frac{G m}{R}2\sigma \pi r,\label{3}
\end{equation}
which results in a linear behavior
\begin{equation}
v=r\sqrt{\frac{G}{R}2\sigma \pi}, \label{4}
\end{equation}
and zero total energy, $E=0$.\\ At distances $r>R$ where $M=Const$,
the potential energy as well as total energy become constant
\begin{equation}
E=\frac{1}{2}mv^2 -\frac{G m M}{R},\label{5}
\end{equation}
which results in a constant velocity
\begin{equation}
v=\sqrt{\frac{2}{m}(E+\frac{GmM}{R})}.\label{6}
\end{equation}
\newpage
\section{Concluding remarks}
In conclusion, we obtained a linear rotation curve for $r<R$ which
turns into a flat curve for $r>R$, $R$ being the radius of the
galaxy disc. It is interesting to note that since there is no sharp
demarcation between the massive disk and the outer void space, there
is no sharp turning point in the rotation curve extending from $r<R$
to $r>R$. In fact, the curve turns in a rather gentle way since the
surface mass density $\sigma$ does not change suddenly, rather it
decays smoothly over the turning region in passing from massive part
toward the outer void region. Therefore, one may define an effective
characteristic radius of the disc, namely $R_{eff}$, for each galaxy
so that its substitution in Eqs. (\ref{4}), (\ref{6}) would lead to
rotation curves in good agreement with observations.
\section*{Acknowledgment}
This work has been supported financially by Research Institute for
Astronomy and Astrophysics of Maragha (RIAAM).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,476 |
Rasbora einthovenii és una espècie de peix de la família dels ciprínids i de l'ordre dels cipriniformes.
Morfologia
Els mascles poden assolir els 9 cm de longitud total.
Distribució geogràfica
Es troba a Malàisia i Indonèsia.
Referències
Bibliografia
Bleeker, P. 1851. Vijfde bijdrage tot de kennis der ichthyologische fauna van Borneo, met beschrijving van eenige nieuwe soorten van zoetwatervisschen. Natuurkd. Tijdschr. Neder. Indië v. 2: 415-442.
.
Roberts, T.R., 1989. The freshwater fishes of Western Borneo (Kalimantan Barat, Indonesia). Mem. Calif. Acad. Sci. 14:210 p.
einthovenii | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,577 |
Anthomyia vittiventris är en tvåvingeart som beskrevs av Ackland 1987. Anthomyia vittiventris ingår i släktet Anthomyia och familjen blomsterflugor. Inga underarter finns listade i Catalogue of Life.
Källor
Blomsterflugor
vittiventris | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,471 |
I couldn't think of a better day to share this quick little story, then on Valentine's Day. I had been contemplating whether or not to share as this is mostly a doll review blog, but then I figured...it's just for fun anyway so why not!?
About 2 and a half years ago, a week after I had my daughter "Little Button", my father passed away unexpectedly from cardiac complications. It completely shocked and rattled my world, but I was so incredibly fortunate enough to have a brand new, precious, sweet little newborn to focus on and love that my family and I dealt with his passing extremely well and in time, came to easily feel joy again whenever we think and share stories of him.
He was very young, though, only 52 years old.
One of his most striking characteristics was that he had a really amazing sense of humor. That is pretty much how everyone remembers him when they share their own personal memories of what their time with him was like.
Ever since then, I talk to him daily and feel his presence near me quite often. He was a really funny, witty guy and extremely caring. He was an amazing person and now that he has passed "to the other side" I almost feel like my relationship with him is even better. Happier and clearer and without worry.
Shortly after we moved here to the Farm, I went through a few weeks of uncertainty regarding an issue I was having in my personal life. On this particular day a few months ago, I was really missing him and asked for a sign from him that all was well, that he really was there watching out for me and that I was on the "right path,"etc.
Later that day, I got an urge to suddenly look at the ceiling in my office while I was working and there was a bright, heart-shaped sun "spot". I had never seen anything like this! Now I know it was just the way the sun happened to be shining through the curtains, but it was such a coincidence that I happened to catch it, at that moment because not long after, it had morphed into a different pattern.
Since this day, I have never looked up at the ceiling again to see this exact shape or anything even resembling heart!
Throughout the next few days, I continued to see heart shaped objects during my walks outside, so I started to tote my camera around as I knew I eventually wanted to do a blog post about it.
Aw, that was touching. Thanks for sharing your story (and the photos of those random hearts out in the world). Have a great weekend!
What a sweet story. I do believe that those were signs from your father! I lost my mother when I was 15, my sister when I was 16 and my father when I was 24. They all died at very young ages. My mom was 48, my sister was 24 and my father was 51. I took my mother's and sister's deaths the hardest and even to this day I "talk" to them. For example I have lost an item and will ask them for help in finding it. Much to my surprise, I will turn around as see the item in a spot that I had just frantically searched through! I get a feeling of comfort when this happens, knowing that they are somehow out there still giving me guidance. I hop you have a wonderful valentine's day with your husband and your daughters. Thanks for sharing all the little hearts!
Awww that was a lovely story :) I'm glad to hear that you get comfort from the heart shapes you've seen in your world.
The beloved ones always leave us too early and too unexpectedly. It's a blessing that they stay in our memory though. Maybe we wouldn't love them so much if they lived forever? Maybe we wouldn't care at all if we knew that they were supposed to be with us all the time?
So sweet! I have had similar experiences so I know what you are feeling. I hope you had the best Valentines day ever!
I am so sorry for your loss, but that is amazing, I am sure he is with you always.
Farrah, what a sweet story you shared! You were given such a special gift of comfort and love! I was close to my Grandpa that passed away a year ago last week and I love the little moments I come across that remind me of him.
I missed this lovely post...for some reason it didn't come up on my feed. What a sad but also lovely story and I am so glad you still feel your Dad near. He really was far too young.
Feel Good Friday-I Love You! | {
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} | 88 |
Starcraft II: Wings of Liberty testing begins
Tags: Starcraft II (2), Vivendi Universal Interactive (NYSE:VIV), PC, Strategy
Quick Link: HEXUS.net/qav7k
The closed beta phase for the highly anticipated real-time strategy game, StarCraft II: Wings of Liberty has begun.
Thousands of gamers around the world have started to receive invitations from the company to participate in the first phase of the beta test. In addition to providing feedback on the multiplayer balance of StarCraft II, testers are also getting a first look at Blizzard Entertainment's revamped Battle.net service, which will be the online platform for StarCraft II, World of Warcraft, and future Blizzard Entertainment games.
StarCraft II: Wings of Liberty is the sequel to Blizzard Entertainment's 1998 hit StarCraft, which has been hailed by players and critics worldwide as one of the top real-time strategy games of all time. Sporting a new 3D-graphics engine, StarCraft II will once again center on the clash between the protoss, terrans, and zerg, with each side deploying legions of veteran, upgraded, and brand-new unit types. StarCraft II: Wings of Liberty is currently slated to ship in the first half of 2010.
A new version of Battle.net will be launched alongside StarCraft II with several enhancements and new features.
For more information on StarCraft II and the beta test, visit the official StarCraft II website at http://eu.starcraft2.com . To set up a Battle.net account and sign up for a chance to participate in the StarCraft II beta test, please visit the official Battle.net website at http://eu.battle.net/.
Doom Eternal official trailer 2 released Developer: our free game torrent resulted in 4x Steam sales Elite: Dangerous pays tribute to Terry Pratchett Blitzkrieg 3 microtransactions cut, champions one-off payment idea Sony accidentally outs Xperia Z4 Tablet ahead of its official release | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,611 |
\section{Introduction}
\label{sec:1}
The detection of 3 very high energy (VHE) astrophysical neutrinos with the IceCube detector has recently opened up a whole new window on the energetic Universe ~\cite{aar13}. The present work proposes a viable physical model of the interactions taking place in source regions for the prediction of astrophysical neutrinos.
The possible origins of these PeV neutrino events have been discussed by many in recent times ~\cite{lah13}, but still remains a topic of much speculation. A fundamental question is therefore being raised: what classes of astronomical objects could accelerate hadrons to very high energies, and in which types of interactions are neutrinos then produced?
In this situation, acceleration of protons in the vicinity of the surface of relatively young local neutron stars with super strong magnetic fields ($B \sim 10^{14-17}$ G), widely known as magnetars, is in fact supposed to be a possibility. The subsequent production of photomesons by interactions with radiative background proceeds, and that has already been studied by many authors. Here, we have proposed an additional contribution to the target photon fields for photomeson production from the photon splitting mechanism. The photon splitting process is expected to modify the radiative background, and enhance the neutrino flux at PeV energies from the object.
The phase diagram $P - B$, corresponds a phase in which the star could pass a neutrino-loud regime ~\cite{zha03}. If the spin-down power of a local magnetar for a favorable set of $P$ and $B$ at a particular evolutionary phase is consumed to accelerate protons, then the object might emit PeV muon neutrinos ($\nu_{\mu}$) through photomeson interactions. The radiative background of the star is believed to be filled mainly with soft ultraviolet (UV)-A or B photons that are in turn produced from the effect of photon splitting mechanism on magnetar's unmodified radiative background.
\section{Photon splitting process in the magnetosphere}
\label{sec:2}
The photon splitting, a QED process that splits a high-energy photon into pair of low energy photons in presence of a pure magnetic field and/or magnetized plasma. No detailed calculation on the probabilities of these processes in the current scenario ($B > B_{\rm{cr}} = 4.41 \times 10^{13}$ G with plasma) is available except a numerical calculation used in ~\cite{chi12}. The environment affects the rate by changing photo dispersion properties of the region.
Photon splitting may occur via various possible channels which are determined by the electric field vector ($\bf{X}$), momentum vector ($\bf{q}$) associated with a photon, and the magnetic field vector ($\bf{B}$). The state {\lq 1\rq} refers to a configuration where $\bf{X}$ is perpendicular to the plane containing $\bf{q}$ and $\bf{B}$ while {\lq 2\rq} corresponds the $\bf{X}$ being parallel to their plane. For low energy background target UV-C photons ($T_{\rm{kin.}} \cong 0.1 - 0.2$ keV $<< m_{\rm e}c^{2}$) the only physical mode, $\gamma_{1} \rightarrow \gamma_{2} + \gamma_{2}$ is responsible for splitting and, thereby soften background photon spectra in the magnetized plasma. But in the absence of plasma, the physical mode, $\gamma_{1} \rightarrow \gamma_{1} + \gamma_{2}$ is the significant one for photon splitting over the channel, $\gamma_{1} \rightarrow \gamma_{2} + \gamma_{2}$.
\section{PeV neutrinos from magnetars: Physical model}
\label{sec:3}
In particle astrophysics, it has been presumed that the VHE protons and/or ions are injected, and accelerated in surrounding regions of cosmic accelerators. These accelerated ions then interact with the radiative background and subsequently VHE neutrinos and gamma-rays are generated via dominant photo-meson interactions,
\begin{equation}
p + \gamma \rightarrow \Delta^{+} \rightarrow \left\{\begin{array}{ll}
p + \pi^{\rm o} \rightarrow p + 2\gamma \\
n\pi^{+} \rightarrow n + e^{+} + \nu_{\rm e} + \nu_{\mu} + \bar{\nu_{\mu}}
\end{array}
\right.
\end{equation}
The final products of all neutrino flavours turn into a ratio of ${\nu}_{\rm{e}} : {\nu}_{\mu} : {\nu}_{\tau} = 1:1:1$ at earth.
In magnetar's early life, the spin-down power is consumed to accelerate protons/ions, and the magnetic field driven power supplies ambient photon targets. The kinematic threshold for photo-meson interaction process in eq.(1) is determined by the accessible photon energies in the radiative field that are mostly UV type photons.
The neutron star (NS) remnant or merger that appears from massive binary NSs ($M \sim 2{\rm M}_{\odot}$) coalescence, may form a millisecond magnetar with thin ejecta walls across polar caps in its very early phase. As the star receives huge angular momentum from the binary it possesses a rapid rotation at the moment of its birth. These magnetars also have super-strong magnetic fields ~\cite{zra13}. Here, we suggest that in the evolutionary phase of a magnetar after NSs merger, the star may transit a state when spin-down power is comparable with its magnetic power, and spin period falls in the range $200 - 500$ ms.
An extension of previous calculations for young pulsars to magnetars reveals that protons or heavier ions undergo acceleration in the magnetar's polar caps attaining energies close to $10^{16} - 10^{17}$ eV, provided the magnetar's magnetic moment vector $\bf{\mu}$ and $\bf{B}$ parameter satisfy the strong condition; $\bf{\mu}.\bf{B} <0$. These VHE protons will interact with soft UV-A and UV-B photons close to the magnetar's polar caps, the $\Delta$ resonance state may form satisfying the kinematic threshold condition for the process in eq.(1). The energy of the modified target photons is $2.8{kT}_{\infty}(1+z_{\rm g}) \sim 0.01$ keV, where $z_{\rm g}$ $\sim 0.4$ being the gravitational red shift. Thus the proton threshold energy $\epsilon_{\rm{p},{\rm Th}}$ for the $\Delta^{+}$ resonance state ranges $\geq 3\times10^{16}$ eV. The VHE proton flux emitted from the polar cap region would therefore be
\begin{equation}
\Phi_{\rm{PC}}\simeq c f_{\rm d}(1-f_{\rm d}) n_{\rm o} A_{\rm{PC}},
\end{equation}
where $A_{\rm{PC}}$ denotes polar cap area, and it is $\eta_{\rm A} (4 \pi R^{2})$ with $\eta_{\rm A}$ accounts the ratio of polar cap area to the magnetar surface area. Earlier calculations in ~\cite{lin05} for estimating proton/ion flux in pulsar's polar caps took the parameter $\eta_{\rm A}$ as unity. The characteristic polar cap radius can be given by, $r_{\rm{PC}}= R (\Omega {R/c)}^{1/2}$, and hence $\eta_{\rm A}$ takes the form $\Omega R/(4c)$.
It is seen from the process in eq.(1) that the charge-changing reaction goes on just $\frac{1}{3}$-rd of the reaction time, about three high-energy neutrinos (or a pair of $\nu_{\mu},\bar{\nu_{\mu}}$) will accompany with four high-energy gamma-rays on the average when a significant number of such reactions proceed successfully. The total flux of neutrinos that is originated from the disintegration of $\Delta^{+}$ resonance state will be
\begin{equation}
\Phi_{\nu}(r \simeq 1.2R) = 2 c f_{\xi} A_{\rm{pc}} f_{\rm d} (1-f_{\rm d}) n_{\rm o} P_{\rm c},
\end{equation}
with $f_{\xi}$ is $2/3$. If now the duty cycle factor $f_{\rm{dc}}$ of the muon neutrino is taken into account, the phase averaged $\nu_{\mu}$ flux on the Earth from a magnetar at a distance $D$ is given by
\begin{equation}
\Phi_{{\nu}_{\mu},\bar{\nu_{\mu}}} \simeq 2 c f_{\xi} f_{\zeta} \eta_{\rm A} f_{\rm{dc}} f_{\rm s} f_{\rm d} (1-f_{\rm d}) n_{\rm o}\left(\frac{R}{D}\right)^{2}P_{\rm c}
\end{equation}
The effect of neutrino oscillations is represented by the parameter $f_{\zeta}$ (here, it is 1/2). The factor $f_{\rm s}$ is set equal to 1 for $\nu_{\mu}$.
We now calculate numerical values for $\nu_{\mu}$ flux using the formula in the eq.(4) for a typical galactic magnetar with $D \sim 2$ kpc, $P \sim 350$ ms, $B_{15} \sim 1.5$, $T_{0.1 \; \rm{keV}} \sim 0.0255$, and $f_{\rm{dc}} \leq 0.10$ in both the cases when $\eta_{\rm A}$ is equal to (i) 1 and (ii) $\Omega R/(4c)$. We have taken star radius equal to $R = 10$ km for the present calculation. For the purpose, we choose $Z = 1$ and $f_{\rm d} = 1/2$ here.
The corresponding $\nu_{\mu}$ flux ($E^{2}\phi_{\nu_{\mu}}$) calculated out are $6.03 \times 10^{-10}$ in $\rm{GeV cm}^{-2}s^{-1}$ for $\eta_{\rm A} = 1$. These values are $0.0009 \times 10^{-10}$ according to the case (ii) in $\rm{GeV cm}^{-2}s^{-1}$. If we compare with IceCube estimated integral PeV neutrino flux, that is, $\sim 2.4\times 10^{-9}$ $\rm{GeV cm}^{-2}s^{-1}$, these predicted values look quite low, particularly in (ii). IceCube has measured neutrino flux as $E^{2}\phi_{\nu_{\mu}+{\tilde{\nu}_{\mu}}} \sim 3\times 10^{-8}$ GeV cm$^{-2}$ s$^{-1}$ sr$^{-1}$ corresponding to neutrino energy range $0.2 - 2$ PeV.
\section{Conclusions}
If protons reach $10 - 100$ PeV energy scale in a magnetar then their interactions with modified UV-A/B photon targets may generate PeV neutrino events with energies between $1 - 10$ PeV as observed by the IceCube experiment. The model suggests no possible indication of any statistically significant excess from the direction of any local magnetar to be observed by IceCube in near future.
\begin{acknowledgement}
This work was supported by SERB, DST, Govt. of India through its grant no. EMR/2015/001390.
\end{acknowledgement}
\input{reference}
\end{document}
| {
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} | 7,348 |
What is 310S Stainless Steel Round Bar?
Why use Stainless Steel 310S Bar?
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Grade 310S stainless steel can be hot worked after heating at 1177°C (2150°F). It should not be forged below 982°C (1800°F). It is rapidly cooled to increase the corrosion resistance.
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Grade 310S stainless steel is annealed at 1038-1121°C (1900-2050°F) followed by quenching in water.
Grade 310S stainless steel does not react to heat treatment. Strength and hardness of this alloy can be increased by cold working. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,226 |
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"redpajama_set_name": "RedPajamaC4"
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\section{Introduction}
In game theory the mathematical models for decision-making create challenging and important problems binding computer science, economy, mathematics and psychology. In one of such models there is a set $D$ of all possible decisions and a finite number $n$ of players (decision makers) with their individual nonnegative weights (efforts) and decisions. Obviously, the sum of all weights cannot be zero and thus (as a vector) it belongs to the set
\Eq{*}{
W_n:=[0,+\infty)^n \setminus \{(0,\dots,0)\}.
}
The issue is to aggregate all the individual decisions with the corresponding weights to one (common) decision. For this purpose, we need the notion of an \emph{aggregation function} on $D$, which is defined to be a map
\Eq{*}{
\M\colon \WD \to D,
\qquad\mbox{where}\qquad\WD:=\bigcup_{n=1}^\infty D^n \times W_n.
}
For instance, when the set $D$ of decisions is a convex subset of a linear space $X$, then the weighted arithmetic mean $\A$, which is defined as
\Eq{*}{
\A(x,\lambda)
:=\frac{\lambda_1x_1+\dots+\lambda_nx_n}{\lambda_1+\dots+\lambda_n}
\qquad(x=(x_1,\dots,x_n)\in D^n,\,
\lambda=(\lambda_1,\dots,\lambda_n)\in W_n),
}
is a well-known aggregation function. Further examples for an aggregation function are as follows (see for example \cite{Hen98}):
\begin{enumerate}
\item The \emph{Primacy Effect} $\D_{PE} \colon \WD \to D$ is defined by
\Eq{*}{
\D_{PE}(x,\lambda):=\big\{x_i\:\big|\ \lambda_i \ne 0 \text{ and }\lambda_j =0\text{ for all } j\in\{1,\dots,i-1\} \big\}.
}
\item The \emph{Recency Effect} $\D_{RE} \colon \WD \to D$ is defined by
\Eq{*}{
\D_{RE}(x,\lambda):=\big\{x_i\:\big|\ \lambda_i \ne 0 \text{ and }\lambda_j =0\text{ for all }j\in\{i+1,\dots,n\}\big\}.
}
\item The \emph{First Dominating Decision} $\D_{FDD} \colon \WD \to D$ is given by
\Eq{*}{
\D_{FDD}(x,\lambda):=\big\{x_i\:\big|\ \lambda_i =\max(\lambda) \text{ and }\lambda_j <\lambda_i \text{ for all }j\in\{1,\dots,i-1\}\big\}.
}
\item The \emph{First Dominant} $\D_{FD} \colon \WD \to D$ is given by
\Eq{*}{
\D_{FD}(x,\lambda):=\D_{FDD}(x,\lambda^*),
\qquad\mbox{where }\lambda_i^*:=\sum_{j\colon x_j=x_i} \lambda_j.
}
\end{enumerate}
All functions listed in (1)--(4) are reflexive, eliminative, nullhomogeneous in the weights but not symmetric (see the relevant definitions below). Furthermore, they are all \emph{conservative} (or \emph{selective}), which means that the aggregated decision is always one of the individual ones. For a detailed study of (nonweighted) conservative aggregation functions, we refer the reader to the recent study by Couceiro--Devillet--Marichal \cite{CouDevMar18} and Devillet--Kiss--Marichal \cite{DevKisMar19}.
In many settings, $D$ is an infinite set which often refers to the position of the players in a space before the game. An aggregation function unites the positions of all the players into one. An individual nonnegative weight measures the impact of the decision of the corresponding players to the final outcome. In order to introduce plausible and natural properties for aggregation functions, we introduce the concept of decision-making functions on an arbitrary set $D$. For this aim, we adopt the notion of weighted means (which were defined on an interval) from the paper \cite{PalPas18b} to our more general setting. An aggregation function $\M \colon \WD \to D$ is called a \emph{decision-making function (on $D$)} if it satisfies the following five conditions:
\begin{enumerate}[(i)]
\item $\M$ is \emph{reflexive}: For all $x \in D$ and $\lambda \in \R_+$,
we have $\M(x,\lambda)=x$.
\item $\M$ is \emph{nullhomogeneous in the weights}: For all $n \in \N$, $(x_1,\dots,x_n)\in D^n$, $(\lambda_1,\dots,\lambda_n)\in W_n$, and $t \in \R_+$, we have
\Eq{*}{\qquad
\M\big((x_1,\dots,x_n),(t\lambda_1,\dots,t\lambda_n)\big)
=\M\big((x_1,\dots,x_n),(\lambda_1,\dots,\lambda_n)\big).
}
\item $\M$ is \emph{symmetric}: For all $n \in \N$, $(x_1,\dots,x_n)\in D^n$, $(\lambda_1,\dots,\lambda_n)\in W_n$ and for all permutations $\sigma$ of $\{1,\dots,n\}$, we have
\Eq{*}{\qquad
\M\big((x_{\sigma(1)},\dots,x_{\sigma(n)}),
(\lambda_{\sigma(1)},\dots,\lambda_{\sigma(n)})\big)
=\M\big((x_1,\dots,x_n),(\lambda_1,\dots,\lambda_n)\big).
}
\item $\M$ is \emph{eliminative or neglective}: For all $n\geq 2$, $(x_1,\dots,x_n)\in D^n$ and $(\lambda_1,\dots,\lambda_n)\in W_n$ with $\lambda_1=0$, we have
\Eq{*}{\qquad
\M\big((x_1,\dots,x_n),(\lambda_1,\dots,\lambda_n)\big)
=\M\big((x_2,\dots,x_n),(\lambda_2,\dots,\lambda_n)\big).
}
\item $\M$ is \emph{reductive}: For all $n\geq 2$, $(x_1,\dots,x_n)\in D^n$ with $x_1=x_2$ and $(\lambda_1,\dots,\lambda_n)\in W_n$, we have
\Eq{*}{\qquad
\M\big((x_1,\dots,x_n),(\lambda_1,\dots,\lambda_n)\big)
=\M\big((x_2,x_3,\dots,x_n),(\lambda_1+\lambda_2,\lambda_3,\dots,\lambda_n)\big).
}
\end{enumerate}
We also introduce the concept of the effort function, which is aiming to aggregate the individual weights (efforts) into one positive number: A function $\E \colon \WD \to \R_+$ is called an \emph{effort function (on $D$)} if it satisfies the following five conditions:
\begin{enumerate}[(i)]
\item $\E$ is \emph{reflexive in the weights}: For all $x \in D$ and $\lambda \in \R_+$, we have $\E(x,\lambda)=\lambda$.
\item $\E$ is \emph{homogeneous in the weights}: For all $n \in \N$, $(x_1,\dots,x_n)\in D^n$, $(\lambda_1,\dots,\lambda_n)\in W_n$, and $t \in \R_+$, we have
\Eq{*}{\qquad
\E\big((x_1,\dots,x_n),(t\lambda_1,\dots,t\lambda_n)\big)
=t\E\big((x_1,\dots,x_n),(\lambda_1,\dots,\lambda_n)\big).
}
\item $\E$ is \emph{symmetric}.
\item $\E$ is \emph{eliminative or neglective}.
\item $\E$ is \emph{reductive}.
\end{enumerate}
One can easily see that the map $\alpha \colon \WD \to \R_+$ given by
\Eq{*}{
\alpha(x,\lambda)
:=\lambda_1+\dots+\lambda_n
\qquad(x\in D^n,\,
\lambda=(\lambda_1,\dots,\lambda_n)\in W_n),
}
is an effort function, which we call the \emph{arithmetic effort function}.
The symmetry property of decision-making and effort functions means that there is no distinction between players and also their order is irrelevant for the decision. This property has a far-reaching consequences especially for conservative functions, as it determines the anty-symmetric preference relation on $D$ by $x\succ y :\!\!\iff x=\M((x,y),(1,1))$ (cf.\ \cite{Dev19} for details).
It was proved experimentally that this relation cannot be generalized to multivariable choice; this phenomena is known as a decoy effect (see for example Huber--Payne--Puto \cite{HubPayPut82}).
The nullhomogeneity of $\M$ and the homogeneity of $\E$ in the weights states that if the weights are scaled by the same factor, then the decision remains unchanged and the effort is scaled by the same factor. The meaning of the elimination principle is that players with zero weight do not affect the decision and the effort. One can easily check that the arithmetic mean is a decision-making function over any convex subset of a linear space.
We introduce now an aggregation-type property which will play a significant role in the sequel. We say that a decision-making function $\M$ on $D$ is \emph{delegative} (admits the delegation principle or partial aggregation principle) if, for all $(y,\mu)\in \WD$, there exists a pair $(y_0,\mu_0) \in D \times \R_+$ such that
\Eq{ts}{
\M((x,y),(\lambda,\mu))=\M((x,y_0),(\lambda,\mu_0))
\qquad((x,\lambda)\in \WD).
}
Analogously, we can speak about the delgativity of an effort function $\E$ on $D$ which means that, for all $(y,\mu)\in \WD$, there exists a pair $(y_0,\mu_0) \in D \times \R_+$ such that
\Eq{ts+}{
\E((x,y),(\lambda,\mu))=\E((x,y_0),(\lambda,\mu_0))
\qquad((x,\lambda)\in \WD).
}
\begin{lem} Let $(y,\mu)\in \WD$ be fixed. If $\M\colon\WD \to D$ is a delegative decision-making function, then \eq{ts} holds if and only if $y_0=\M(y,\mu)$. Analogously, if $\E\colon\WD \to \R_+$ is a delegative effort function, then \eq{ts+} holds if and only if $\mu_0=\E(y,\mu)$.
\end{lem}
\begin{proof} Using the properties of decision making functions and applying the delegativity of $\M$ for $(x,\lambda)=(y,\mu)$ twice, we get
\Eq{*}{
\M(y,\mu)=\M((y,y),(\mu,\mu))
=\M((y,y_0),(\mu,\mu_0))=\M((y_0,y_0),(\mu_0,\mu_0))
=y_0.
}
Similarly, the properties of effort functions and applying the delegativity
of $\E$ yield
\Eq{*}{
\E(y,\mu)=\frac12\E((y,y),(\mu,\mu))
=\frac12\E((y,y_0),(\mu,\mu_0))=\frac12\E((y_0,y_0),(\mu_0,\mu_0))
=\mu_0,
}
which completes the proof of the lemma.
\end{proof}
Motivated by the above statement, a delegative decision-making function $\M$ and a delegative effort function $\E$ are called \emph{associated} if, for all $(y,\mu)\in\WD$, the equalities \eq{ts} and \eq{ts+} hold with $(y_0,\mu_0)=(\M(y,\mu),\E(y,\mu))$.
The aim of this paper is to present a construction of a broad class of decision-making and effort functions which arise from studying so-called weighted Bajraktarević means. Despite of the analytical background of this paper, we are convinced that our construction could provide useful models for game theoretical and for decision-making problems.
Our motivation is to present a mean-type approach to studying the Farm Structure Optimization Problem -- cf. for example
Abd El-Wahed--Abo-Sinna \cite{AbdAbo01}, Czy\.{z}ak \cite{Czy90}, S\l{}owi\'{n}ski--Teghem \cite{SloTeg90}, Tzeng--Huang \cite{HuaTze14}, Xu--Zhou \cite{XuZho11}.
\section{Observability and conical convexity}
A subset $S$ of a linear space $X$ over $\R$ is called a \emph{ray} if $S=R_x:=\R_+x:=\{\lambda x\mid\lambda\in\R_+\}$ holds for some nonzero element $x\in X$. If $y\in R_x$, then the unique positive numbers $\lambda$ for which $y=\lambda x$ holds will be denoted by $[y:x]$.
A set $S$ is called a \emph{cone} if it is the union of rays of $X$. One can see that $S$ is a cone precisely if it is closed under multiplication by positive scalars. The cone generated by the set $S$ -- denoted by $\cone(S)$ -- is the smallest cone containing $S$. It is clear that $\cone(S)=\R_+S=\bigcup_{x\in S}R_x$.
A subset $S$ of $X$ over $\R$ is called \emph{observable (from the origin)} if the rays generated by two distinct elements of $S$ are disjoint. Note that for an observable subset $S$, every ray contained in $\cone(S)$ intersects $S$ at exactly one point. Moreover, the family $\{R_x \mid x \in S\}$ is a partition of $\cone(S)$. Due to this fact, for every observable set $S$ one can define the projection along rays (briefly the \emph{ray projection}) $\pi_S \colon \cone(S) \to S$ as follows: For every $x\in S$ and $y\in R_x$, we have $\pi_S(y)=x$. As a matter of fact, observability of $S$ is not only sufficient but also necessary to define such map.
We say that a function $f\colon D\to X$ is \emph{observable} if it is injective and has an observable image. Analogously to the previous setup, we define $\pi_f:=\pi_{f(D)}$. The extended inverse of $f$, denoted by $f^{(-1)}\colon \cone (f(D))\to D$ is defined by
\Eq{*}{
f^{(-1)}:=f^{-1}\circ \pi_f.
}
Clearly, $f^{(-1)}(x)=f^{-1}(x)$ if $x\in f(D)$.
We say that a subset $S$ of $X$ is \emph{conically convex} if $\cone(S)$ is a convex set and convex hull of $S$ does not contain the origin.
Note that conical convexity is a weaker property than convexity: every convex set which does not contain zero is conically convex, the converse implication is not true.
Hereafter an observable function $f \colon D \to X$ such that $f(D)$ is conically convex is called \emph{admissible}.
In the next result we give a characterization of admissibility for functions with 2-dimensional range.
\begin{thm}\label{thm:det}
Let $f=(f_1,f_2):I\to\R^2$ be continuous. Then $f$ is admissible if and only if, for all distinct elements $x,y\in I$, $f(x)$ and $f(y)$ are linearly independent, that is,
\Eq{det}{
\left|\begin{matrix}
f_1(x) & f_1(y)\\f_2(x) & f_2(y)
\end{matrix}\right|
\neq 0.
}
\end{thm}
\begin{proof}
Assume first that $f$ is admissible but \eq{det} is not valid for some $x,y\in I$ with $x\neq y$. Then the two vectors $f(x)$ and $f(y)$ are linearly dependent. Thus, there exist $(\alpha,\beta)\neq(0,0)$ such that $\alpha f(x)+\beta f(y)=0$.
Because $(0,0)$ is not contained in $f(I)$, we have that $f(x)\neq (0,0)\neq f(y)$, hence $\alpha\beta\neq0$. In the case when $\alpha\beta<0$, we get that the rays generated by $f(x)$ and $f(y)$ are the same, which contradicts the observability. If $\alpha\beta>0$, then $(0,0)$ is in the segment connecting $[f(x),f(y)]$, which contradicts the property that the convex hull of $f(I)$ does not contain the origin.
Assume now that \eq{det} holds for all $(x,y)\in \Delta(I):=\{(x,y)\mid x,y\in I,\,x<y\}$. This immediately shows that $f(I)$ is observable.
Since $\Delta(I)$ is a convex set and $f$ is continuous, therefore the determinant is either positive on $\Delta(I)$ or negative on $\Delta(I)$. We may assume that it is positive everywhere on $\Delta(I)$.
Then the pair $(f_1,f_2)$ is a so-called Chebyshev system on $I$. According to \cite[Theorem 2]{BesPal03}, there exist two real constants $\alpha$ and $\beta$ such that $\alpha f_1+\beta f_2$ is positive over the interior of $I$ and thus it is nonnegative over $I$. This implies that the curve $f(I)$ is contained in the closed half plain $P=\{(u,v)\in\R^2\mid \alpha u+\beta v\geq0\}$. Consider now the unit circle and project the curve $f(I)$ into it. Then, by the continuity of the projection mapping, the projection of the curve is an arc of the unit circle which is contained in $P$. It is obvious that the conical hull of the curve $f(I)$ and the conical hull of the arc are identical. On the other hand, the property that the arc is contained in the half plain $P$, implies that its conical hull is convex. This shows the conical hull of $f(I)$ is also convex.
The origin cannot be contained in the conical hull of $f(I)$ because then there were two distinct points of $f(I)$ belonging to the boundary line $\{(u,v)\in\R^2\mid \alpha u+\beta v=0\}$ such that the origin were in the segment connecting them. This contradicts the property that every two distinct points
of $f(I)$ are linearly independent.
\end{proof}
\section{Generalized Bajraktarevi\'c means and their properties}
Given an admissible function $f\colon D\to X$, we can define the \emph{(weighted) generalized Bajraktarevi\'c mean} $\B_f\colon\WD\to D$ by
\Eq{*}{
\B_f(x,\lambda)=f^{(-1)} \Big( \sum_{k=1}^n \lambda_k f(x_k) \Big),
}
where $x =(x_1,\dots,x_n)\in D^n$ are the entries and $\lambda=(\lambda_1,\dots,\lambda_n) \in W_n$ are the corresponding weights. We also define $\beta_f \colon \WD\to \R_+$ by
\Eq{*}{
\beta_f(x,\lambda):=\bigg[\sum_{k=1}^n \lambda_k f(x_k):f(\B_f(x,\lambda))\bigg],
}
which we call the \emph{effort function associated to $\B_f$}.
Intuitively $D$ is the space of all possible decisions, $x$ is a vector of all considered decisions, and $\lambda$ is the effort of players invested in the option. In this way $\B_f(x,\lambda)$ is the decision derived from the possible decisions and their weights.
\begin{lem}\label{lem:afy}
Let $f\colon D \to X$ be an admissible function. Then, for all $n \in \N$ and $(x,\lambda) \in D^n \times W_n$, there exists exactly one pair $(u,\eta) \in D \times \R_+$ satisfying
\Eq{E:afy}{
\eta f(u)= \sum_{k=1}^n \lambda_k f(x_k).
}
Furthermore, $u=\B_f(x,\lambda)$ and $\eta=\beta_f(x,\lambda)$.
\end{lem}
\begin{proof}
As $f(D)$ is conically convex, we know that
\Eq{*}{
\frac{\sum_{k=1}^n \lambda_k f(x_k)}{\sum_{k=1}^n \lambda_k} \in \cone f(D).
}
Thus there exists $\eta \in \R_+$ such that
\Eq{*}{
\eta^{-1} \sum_{k=1}^n \lambda_k f(x_k) \in f(D).
}
Moreover, as $f(D)$ is observable, the constant $\eta$ is uniquely determined by $x$ and $\lambda$. Now, as $f$ is injective, there exists exactly one $u\in D$ such that
\Eq{*}{
\eta^{-1} \sum_{k=1}^n \lambda_k f(x_k)=f(u),
}
which is trivially equivalent to \eq{E:afy}. The last assertion simply follows from the definition of $\B_f(x,\lambda)$ and $\beta_f(x,\lambda)$.
\end{proof}
Now $\B_f$ is the equivalent (or aggregated) decision, $\beta_f$ is the equivalent (or aggregated) effort. In fact $\beta_f$, can be considered as the amount of goods we need to invest into a single decision to be irrelevant between the diversed an the aggregated situation. This property is much more transparent in view of Theorem~\ref{thm:subs}.
\begin{thm}
Let $f\colon D\to X$ be an admissible function. Then $\B_f$ is a decision-making function and $\beta_f$ is an effort function on $D$.
\end{thm}
\begin{proof}
Fix $n \in \N$ and $(x,\lambda) \in D^n \times W_n$. In view of Lemma~\ref{lem:afy} both $\B_f(x,\lambda)$ and $\beta_f(x,\lambda)$ depends on its arguments implicitly via the sum $\sum_{k=1}^n \lambda_k f(x_k)$. This immediately implies that $\B_f$ and $\beta_f$ are both symmetric, reductive and eliminative. The nullhomogeneity of $\B_f$ is obvious in view of the definition of $f^{(-1)}=f^{-1} \circ \pi_f$. The reflexivity properties of $\B_f$ and $\beta_f$ are immediate consequences of the equalities
\Eq{*}{
\B_f(y,\lambda)&=f^{(-1)}(\lambda f(y))=f^{-1}\circ f(y)=y,\\
\beta_f(y,\lambda)
&=[\lambda f(y)\colon f(\B_f(y,\lambda))]
=[\lambda f(y)\colon f(y)]=\lambda\qquad (y \in D,\lambda\in\R_+).
}
Now we only need to verify the homogeneity of $\beta_f$ in the weights. To this end, take additionally $t \in \R_+$.
By the definition of $\beta_f$ and the nullhomogeneity of $\B_f$, we have
\Eq{*}{
\beta_f(x,t\lambda)&=\bigg[\sum_{k=1}^n t\lambda_k f(x_k):f(\B_f(x,t\lambda))\bigg]\\
&=t\bigg[\sum_{k=1}^n \lambda_k f(x_k):f(\B_f(x,\lambda))\bigg]=t\beta_f(x,\lambda),
}
which completes the proof.
\end{proof}
\begin{thm}\label{thm:f1f2}
Let $f_1,f_2\colon I \to \R$ be continuous functions such that $f_2$ is nowhere zero and $f_1/f_2$ is strictly monotone. Then $f=(f_1,f_2)$ is admissible and we have the equalities
\begin{align}
\B_f(x,\lambda)&=\Big(\frac{f_1}{f_2}\Big)^{-1} \bigg(\frac{\lambda_1 f_1(x_1)+\dots+\lambda_nf_1(x_n)}{\lambda_1 f_2(x_1)+\dots+\lambda_nf_2(x_n)} \bigg) &\qquad ((x,\lambda)\in I^n \times W_n), \label{Bf}\\
\beta_f(x,\lambda)
&=\frac{\lambda_1 f_2(x_1)+\dots+\lambda_nf_2(x_n)}
{f_2(\B_f(x,\lambda))} &\qquad ((x,\lambda)\in I^n \times W_n)\label{bf}.
\end{align}
\end{thm}
\begin{proof} First we show that $f=(f_1,f_2)$ is admissible. In view of Theorem~\ref{thm:det}, it is sufficient to show that \eq{det} holds for all distinct elements $x,y$ of $I$. Indeed,
\Eq{*}{
\left|\begin{matrix}
f_1(x) & f_1(y)\\f_2(x) & f_2(y)
\end{matrix}\right|
=f_1(x)f_2(y)-f_1(y)f_2(x)
=f_2(x)f_2(y)\bigg(\frac{f_1(x)}{f_2(x)}-\frac{f_1(y)}{f_2(y)}\bigg),
}
which is nonzero by our assumptions.
To prove the identity \eq{Bf},
let $n \in \N$ and a pair $(x,\lambda)\in I^n \times W_n$ be fixed. As $f(\B_f(x,\lambda))$ and $\sum_{i=1}^n \lambda_i f(x_i)$ are on the same ray, we have
\Eq{*}{
\frac{f_1(\B_f(x,\lambda))}{f_2(\B_f(x,\lambda))}= \frac{\lambda_1 f_1(x_1)+\dots+\lambda_nf_1(x_n)}{\lambda_1 f_2(x_1)+\dots+\lambda_nf_2(x_n)}.
}
Using that $f_1/f_2$ is invertible and applying its inverse side-by-side, we obtain the desired equality \eq{Bf}. The equality
\Eq{*}{
\beta_f(x,\lambda)
=\bigg[\sum_{k=1}^n \lambda_k f(x_k):f(\B_f(x,\lambda))\bigg]
=\bigg[\sum_{k=1}^n \lambda_k f_2(x_k):f_2(\B_f(x,\lambda))\bigg]
}
implies \eq{bf} immediately.
\end{proof}
A classical particular case of the above theorem is when $f_1$ and $f_2$ are power functions on $I=\R_+$: $f_1(x)=x^p$ and $f_2(x)=x^q$, where $p,q\in\R$ with $p\neq q$. Then
\Eq{E:Ginipq}{
\B_f(x,\lambda)&=G_{p,q}(x,\lambda)
:=\bigg(\frac{\lambda_1 x_1^p+\dots+\lambda_n x_n^p}
{\lambda_1 x_1^q+\dots+\lambda_n x_n^q}\bigg)^{\frac{1}{p-q}},\\
\beta_f(x,\lambda)&=\gamma_{p,q}(x,\lambda)
:=\frac{(\lambda_1 x_1^q+\dots+\lambda_n x_n^q)^{\frac{p}{p-q}}}
{(\lambda_1 x_1^p+\dots+\lambda_n x_n^p)^{\frac{q}{p-q}}},
}
which are called the \emph{Gini mean} and the \emph{Gini effort function} of parameter $(p,q)$ where $p\neq q$ (cf.\ \cite{Gin38}). For the case $p=q$, let $f_1(x):=x^p\ln(x)$ and $f_2(x):=x^p$. Then we have
\Eq{*}{
\B_f(x,\lambda)&=G_{p,p}(x,\lambda)
:=\exp\bigg(\frac{\lambda_1 x_1^p\ln x_1+\dots+\lambda_n x_n^p\ln x_p}
{\lambda_1 x_1^p+\dots+\lambda_n x_n^p}\bigg),\\
\beta_f(x,\lambda)&=\gamma_{p,p}(x,\lambda)
:=(\lambda_1 x_1^p+\dots+\lambda_n x_n^p) \exp\bigg(( -p) \cdot \frac{\lambda_1 x_1^p\ln x_1+\dots+\lambda_n x_n^p\ln x_p}
{\lambda_1 x_1^p+\dots+\lambda_n x_n^p}\bigg),
}
\section{Aggregation-type properties}
\subsection{Delegativity}
In this subsection we establish the delegativity of the generalized Bajraktarević means and the corresponding effort functions, moreover, we show that these two maps are associated to each other.
\begin{thm}\label{thm:subs}
Let $f \colon D \to X$ be an admissible function.
Then both $\B_f$ and $\beta_f$ are delegative and associated.
\end{thm}
\begin{proof} Fix $m\in\N$ and $(y,\mu)\in D^m\times W_m$. For an arbitrary $(x,\lambda)\in D^n\times W_n$, according to the Lemma~\ref{lem:afy}, we have
\Eq{*}{
\beta_f((x,y),(\lambda,\mu)) f(\B_f((x,y),(\lambda,\mu)))
&=\sum_{k=1}^n \lambda_k f(x_k)+\sum_{j=1}^m \mu_jf(y_j),\\
\qquad \beta_f(y,\mu) f(\B_f(y,\mu))&=\sum_{j=1}^m \mu_jf(y_j).
}
Then, these equalites and Lemma~\ref{lem:afy} imply
\Eq{*}{
\beta_f((x,y),(\lambda,\mu)) f(\B_f((x,y),(\lambda,\mu)))
&=\sum_{k=1}^n \lambda_k f(x_k)+\beta_f(y,\mu) f(\B_f(y,\mu))\\
&=\beta_f((x,y_0),(\lambda,\mu_0))f(\B_f((x,y_0),(\lambda,\mu_0))),
}
which then yields \eq{ts}.
\end{proof}
The following result could be derived from the above theorem, but we shall provide a direct and short proof for it.
\begin{cor}
Let $f \colon D \to X$ be an admissible function. Let $n,m\in\N$, $x\in D^n$ and $\lambda^{(1)},\dots,\lambda^{(m)}\in W_n$. Denote $y_j:=\B_f(x,\lambda^{(j)})$ and $\mu_j:=\beta_f(x,\lambda^{(j)})$ for $j\in\{1,\dots,m\}$. Then, for all $(t_1,\dots,t_m)\in W_m$,
\Eq{tt}{
\B_f(x,t_1\lambda^{(1)}+\dots+t_m\lambda^{(m)})
&=\B_f(y,(t_1\mu_1,\dots,t_m\mu_m)),\\
\beta_f(x,t_1\lambda^{(1)}+\dots+t_m\lambda^{(m)})
&=\beta_f(y,(t_1\mu_1,\dots,t_m\mu_m)).
}
\end{cor}
\begin{proof} By the definitions of $y_1,\dots,y_m$ and $\mu_1,\dots,\mu_m$, according to Lemma~\ref{lem:afy}, we have
\Eq{*}{
\mu_jf(y_j)
=\beta_f(x,\lambda^{(j)})f(\B_f(x,\lambda^{(j)}))
=\sum_{i=1}^n \lambda^{(j)}_i f(x_i) \qquad (j\in\{1,\dots,m\}).
}
Multiplying this equality by $t_j$ side by side, and then summing up the equalities so obtained for $j\in\{1,\dots,m\}$, we get
\Eq{*}{
\beta_f&(y,(t_1\mu_1,\dots,t_m\mu_m))
f(\B_f(y,(t_1\mu_1,\dots,t_m\mu_m)))\\
&=\sum_{j=1}^m t_j\mu_jf(y_j)
=\sum_{j=1}^m t_j\bigg(\sum_{i=1}^n \lambda^{(j)}_if(x_i)\bigg) =\sum_{i=1}^n \bigg(\sum_{j=1}^m t_j\lambda^{(j)}_i\bigg) f(x_i)\\
&=\beta_f(x,t_1\lambda^{(1)}+\dots+t_m\lambda^{(m)})
f(\B_f(x,t_1\lambda^{(1)}+\dots+t_m\lambda^{(m)})).
}
The above equality, by the observability of $f$, yields \eq{tt}.
\end{proof}
The latter corollary can be also rewritten in the matrix form.
\begin{cor}\label{cor:subs}
Let $f \colon D \to X$ be an admissible function. Let $x\in D^n$, and
$\Lambda=(\lambda^{(1)},\dots,\lambda^{(m)})\in W_n^m$. Define a vector $y:=\big(\B_f(x,\lambda^{(i)})\big)_{i=1}^m \in D^m$ and $\mu:=\big(\beta_f(x,\lambda^{(i)})\big)_{i=1}^m \in \R_+^m$. Then,
for all $t\in W_m$,
\Eq{E:Lt}{
\B_f(x,\Lambda t)=B_f(y,\mu\!\cdot\! t)\quad\text{ and }\quad \beta_f(x,\Lambda t)=\beta_f(y,\mu\!\cdot\! t),
}
where ``$\cdot$'' stands for the coordinate-wise multiplication of the elements of $W_m$.
\end{cor}
\subsection{Casuativity}
We say that a decision making function $\M$ on $D$ is \emph{casuative} if for all $(x,\lambda)\in\WD$ and for all pair $(y,\mu) \in D \times (0,+\infty)$, we have
\Eq{E:casu}{
\M(x,\lambda)=\M\big((x,y),(\lambda,\mu)\big) \iff y=\M(x,\lambda).
}
Casuativity is somehow opposite to conservativity. Namely there holds the following easy-to-see lemma.
\begin{lem}
Let $\M$ be a symmetric and casuative decision-making function on $D$. Then for all distinct $x_1,x_2 \in D$ and $\lambda_1,\lambda_2 \in (0,+\infty)$ we have $\M((x_1,x_2),(\lambda_1,\lambda_2)) \in D \setminus\{x_1,x_2\}$.
\end{lem}
It is also reasonable to define \emph{weak casuativity} in the case when the $(\Leftarrow)$ implication of \eq{E:casu} holds. Observe that there are number of weakly casuative decision making functions which are not casuative, for example decision making functions which are induced by a preference relation.
\begin{prop}
Generalized Bajraktarević mean $\B_f$ is casuative for every admissible function $f \colon D \to X$.
\end{prop}
\begin{proof}
As $\B_f$ is delegative it is sufficient to show that \eq{E:casu} holds for all $(x,\lambda)\in D \times \R_+$. The $(\Leftarrow)$ part is then trivial. To prove the converse implication let $x,y \in D$ and $\lambda, \mu \in (0,+\infty)$ such that
\Eq{*}{
\B_f\big((x,y),(\lambda,\mu)\big)=\B_f(x,\lambda)=x.
}
Thus, by Lemma~\ref{lem:afy}, there exists $\eta \in \R_+$ such that $\lambda f(x)+\mu f(y)=\eta f(x)$.
Consequently, as $\mu \ne 0$ we get $f(y)=\tfrac{\eta-\lambda}\mu f(x)$. Now admissibility of $f$ implies $0 \notin \conv f(X)$ and therefore $\tfrac{\eta-\lambda}\mu>0$. Then observability of $f$ yields $y=x$.
\end{proof}
This property plays an important role in the definition of effort function.
\begin{cor}\label{cor:Bbeta}
Let $f\colon D \to X$ and $g \colon D \to Y$ be two admissible function such that $\B_f=\B_g$. Then $\beta_f=\beta_g$.
\end{cor}
\begin{proof}
Let $(x,\lambda)\in\WD$, set $m:=\B_f(x,\lambda)=\B_g(x,\lambda)$ and assume that $d:=\beta_g(x,\lambda)-\beta_f(x,\lambda)\neq 0$. Without loss of generality, we may assume that $d>0$.
Take $y_0 \in D \setminus \{m\}$ arbitrarily. By Theorem~\ref{thm:subs} and the equality $\B_f=\B_g$ we have
\Eq{*}{
\B_f((m,y_0),(\beta_f(x,\lambda),1))
&=\B_f((x,y_0),(\lambda,1))
=\B_g((x,y_0),(\lambda,1))\\
&=\B_g((m,y_0),(\beta_g(x,\lambda),1))
=\B_f((m,y_0),(\beta_g(x,\lambda),1))\\
&=\B_f((m,y_0,m),(\beta_f(x,\lambda),1,d)).
}
Now by casuativity of $\B_f$ and $d>0$, we obtain
\Eq{*}{
\B_f((m,y_0),(\beta_f(x,\lambda),1))=m,
}
which implies $y_0=m$, a contradiction. Therefore, $d=0$, i.e., $\beta_g(x,\lambda)=\beta_f(x,\lambda)$.
\end{proof}
\section{Convexity induced by admissible functions}
For an admissible function $f \colon D \to X$, a subset $S\subseteq D$ is called \emph{$f$-convex} if, for all $n\in\N$, $(x,\lambda)\in S^n\times W_n$, we have that $\B_f(x,\lambda)\in S$.
Observe that if $I \subset \R$ is an interval and $f=(f_1,f_2) \colon I \to \R^2$ satisfies conditions of Theorem~\ref{thm:f1f2} then the mean-value property for a Bajraktarević mean can be written as $\B_{f}(x,\lambda)\in J$ for every subinterval $J \subseteq I$, $n \in \N$ and $(x,\lambda)\in J^n \times W_n$. This yields that every subinterval of $I$ is $f$-convex.
Furthermore, in view of continuity of $f$, the converse implication is also valid.
The first assertion characterizes $f$-convexity in terms of standard convexity.
\begin{prop} \label{prop:fconv}
Let $f \colon D \to X$ be an admissible function. Then $S\subseteq D$ is $f$-convex if and only if $\cone(f(S))$ is convex.
\end{prop}
\begin{proof}
Assume first that $S$ is $f$-convex.
It suffices to show that for every $n \in \N$ and a pair $(x,\lambda) \in S^n \times W_n$ we have $\xi:=\sum_{i=1}^n \lambda_if(x_i) \in \cone (f(S))$.
But $\pi_f(\xi)=f(\B_f(x,\lambda))$, thus $\xi \in R_{f(\B_f(x,\lambda))}\subseteq \cone (f(S))$.
Conversely, if $\cone(f(S))$ is convex then, as it is positively homogeneous we obtain $r:=\sum_{i=1}^n \lambda_i f(x_i) \in \cone(f(S))$ for every $n \in \N$ and $(x,\lambda)\in S^n \times W_n$. By the definition of projection we obtain $\pi_f(r) \in f(S)$ and therefore $\B_f(x,\lambda)=f^{(-1)}(r)=f^{-1}\circ \pi_f(r) \in S$, which shows that $S$ is $f$-convex.
\end{proof}
Now we show that $f$-convex sets admit two very important properties of convex sets.
\begin{lem}
Let $f \colon D \to X$ be an admissible function. Then the class of $f$-convex subsets of $D$ is closed with respect to intersection and chain union.
\end{lem}
\begin{proof}
Let $\mathcal{S}$ be an arbitrary family of $f$-convex subsets of $D$. Fix $n \in \N$, and $(x,\lambda) \in \big(\bigcap \mathcal{S}\big)^n \times W_n$.
Then for every $S \in \mathcal{S}$ we have $x \in S^n$ and, in view \mbox{$f$-convexity} of $S$, we also have $\B_f(x,\lambda)\in S$.
Therefore $\B_f(x,\lambda) \in \bigcap \mathcal{S}$ which shows that $\bigcap \mathcal{S}$ is $f$-convex.
Now take a chain $\mathcal{Q}$ of $f$-convex sets.
Take $n \in \N$ and $(x,\lambda)\in (\bigcup \mathcal{Q})^n \times W_n$ arbitrarily. Then, by the chain property of $\mathcal{Q}$, there exists $Q \in \mathcal{Q}$ such that $x \in Q^n$. As $Q$ is $f$-convex, we obtain $\B_f(x,\lambda)\in Q\subseteq \bigcup \mathcal{Q}$. Whence $\bigcup \mathcal{Q}$ is $f$-convex subset of $D$, too.
\end{proof}
Applying this lemma, for every admissible function $f \colon D \to X$ and $S \subset D$ we define
\emph{$f$-convex hull} of $S$ as the smallest $f$-convex subset of $D$ containing $S$ and denote it by $\conv_f(S)$.
In the next lemma we show that, similarly to the ordinary convex hull, this definition can be also expressed as a set of all possible combinations of elements in $S$.
\begin{lem}
Let $f \colon D \to X$ be an admissible function and $S \subseteq D$. Then
\Eq{*}{
\conv_f(S)=\big\{\B_f(x,\lambda)\mid (x,\lambda)\in \WS\big\}.
}
\end{lem}
\begin{proof}
Denote the set on the right-hand-side of the latter equality by $T$. The inclusion $T \subseteq \conv_f(S)$ is the obvious implication of $f$-convexity of the hull.
To prove the converse inclusion, we need to show that $T$ is $f$-convex.
Let $m\in \N$, $y=(y_1,\dots,y_m)\in T^m$ and $\nu \in W_m$ be arbitrary. By the definition of $T$, in view of reduction priciple and Corollary~\ref{cor:subs} there exists $n \in \N$, a vector $x \in S^n$ and $\Lambda=(\lambda^{(1)},\dots,\lambda^{(n)}) \in W_n^m$ such that $y_i=\B_f(x,\lambda^{(i)})$ for all $i \in \{1,\dots,m\}$.
Now, Corollary~\ref{cor:subs} implies that there exists $\mu\in \R_+^m$ such that \eq{E:Lt} holds.
In particular for $t:=\big(\tfrac{\nu_i}{\mu_i}\big)_{i=1}^m \in W_m$, we get
\Eq{*}{
\B_f(y,\nu)=\B_f(y,(t_1\mu_1,\dots,t_m\mu_m))=\B_f(x,\Lambda t)\in T,
}
which shows that $T$ is $f$-convex, indeed.
\end{proof}
We now establish the fundamental relationship between the $f$-convex hull in $D$ and the standard convex hull in $X$.
\begin{prop}
Let $f \colon D \to X$ be an admissible function and $S \subseteq D$. Then
\Eq{convf}{
\conv_f(S)=f^{(-1)} \big( \conv(f(S)) \big).
}
\end{prop}
\begin{proof}
Observe that for every $y \in \conv_f(S)$ there exists $n \in \N$ and $(x,\lambda)\in S^n \times W_n$ such that $y=\B_f(x,\lambda)$. Then
\Eq{*}{
y=f^{(-1)} \bigg( \sum_{k=1}^n \lambda_kf(x_k) \bigg) \in f^{(-1)} \big( \R_+\conv(f(S)) \big)=
f^{(-1)} \big( \conv(f(S)) \big).
}
Conversely, for every $y \in f^{(-1)} \big( \conv(f(S)) \big)$ there exists $n \in \N$ and $(x,\lambda)\in S^n \times W_n$ with $\sum_{k=1}^n \lambda_k=1$ such that $y=f^{(-1)} \big(\sum_{k=1}^n\lambda_kf(x_k)\big)$.
Then, by the definition of $\B_f$, we obtain $y=\B_f(x,\lambda)\in\conv_f(S)$ which completes the proof.
\end{proof}
\section{Equality of generalized Bajraktarevi\'c means}
Let us now state our main result which characterizes the equality of two generalized Bajraktarevi\'c means. In the one-dimensional case this is a classical result due to Acz\'el--Dar\'oczy \cite{AczDar63c} and Dar\'oczy--P\'ales \cite{DarPal82}.
\begin{thm}\label{lem:sufequ}
Let $X$ and $Y$ be linear spaces, $D$ be an arbitrary set and $f \colon D \to X$, $g \colon D \to Y$ be admissible functions. Then $\B_f=\B_g$ if and only if $g=A\circ f$ for some linear map $A\colon X\to Y$.
\end{thm}
\begin{proof}[Unverified alternative proof]
Take $n \in \N$, a pair $(x,\lambda)\in D^n\times W_n$, and denote briefly $y:=\B_f(x,\lambda)\in D$. By Lemma~\ref{lem:afy}, applying $A$ side-by-side in \eq{E:afy} and using the equality $g=A\circ f$ twice, we get
\Eq{*}{
\beta_f(x,\lambda) g(y)=A \big( \beta_f(x,\lambda) f(y)\big)=A \bigg( \sum_{k=1}^n \lambda_k f(x_k)\bigg)=
\sum_{k=1}^n \lambda_k Af(x_k)=\sum_{k=1}^n \lambda_k g(x_k).
}
Now the converse implication in Lemma~\ref{lem:afy} implies $y=\B_g(x,\lambda)$. As $x$ and $\lambda$ are arbitrary, we obtain the equality $\B_f=\B_g$.
To prove the converse implication assume that $\B_f=\B_g$. Then by Corollary~\ref{cor:Bbeta} we have $\beta_f=\beta_g$.
We shall use to following claim several times in the proof:
\begin{claim}
Let $n \in \N$ and $(x,\lambda)\in D^n\times \R^n$ .
Then
\Eq{*}{
\sum_{i=1}^n \lambda_i f(x_i)=0
\qquad \text{
if and only if} \qquad
\sum_{i=1}^n \lambda_i g(x_i)=0.
}
\end{claim}
\begin{proof}
For $\lambda\equiv 0$ the statement is trivial. From now on assume that $\lambda$ is a nonzero vector and define
\Eq{la+-}{
\lambda^+:=(\max(0,\lambda_i))_{i=1}^d,\qquad
\lambda^-:=(\max(0,-\lambda_i))_{i=1}^d.
}
Then $\lambda^+$ and $\lambda^-$ are disjointly supported with nonnegative entries, and $\lambda=\lambda^+-\lambda^-$.
By the first equality we have
\Eq{*}{
\sum_{i=1}^n \lambda_i^+ f(x_i)= \sum_{i=1}^n \lambda_i^- f(x_i).
}
Therefore as $\lambda$ is nonzero and $0\notin \conv f(D)$, we obtain that both $\lambda^-,\lambda^+ \in W_n$. Furthermore by the definition
$m:=\B_f(x,\lambda^-)=\B_f(x,\lambda^+)$ and $\mu:=\beta_f(x,\lambda^-)=\beta_f(x,\lambda^+)$. Therefore as $\B_f=\B_g$ and $\beta_f=\beta_g$ we obtain
\Eq{*}{
\sum_{i=1}^n \lambda_i g(x_i)&=\sum_{i=1}^n \lambda_i^+ g(x_i)-\sum_{i=1}^n \lambda_i^- g(x_i)\\
&=\beta_g(x,\lambda^+) g(\B_g(x,\lambda^+))-\beta_g(x,\lambda^-) g(\B_g(x,\lambda^-))
=\mu g(m)-\mu g(m)=0.
}
The second implication is analogous.
\end{proof}
Denote the linear span of $f(D)$ and $g(D)$ by $X_0$ and $Y_0$, respectively. Let $H_f\subseteq f(D)$ be a Hamel base for $X_0$. Then one can choose a system of elements $\{x_\gamma\mid \gamma\in\Gamma\}\subseteq D$ such that $H_f=\{f(x_\gamma)\mid \gamma\in\Gamma\}$.
We are now going to show that $H_g:=\{g(x_\gamma)\mid \gamma\in\Gamma\}$ is a Hamel base for $Y_0$. Indeed, for every collection of pairwise-distinct elements $\gamma_1,\dots,\gamma_d \in \Gamma$, the system $\{f(x_{\gamma_1}),\dots,f(x_{\gamma_d})\}$ is linearly independent and thus, by our Claim, so is $\{g(x_{\gamma_1}),\dots,g(x_{\gamma_d})\}$.
To show that it is a Hamel base for $Y_0$, we have to prove that $H_g$ is also a generating system. If not, then there exists an element $x^*\in D$ such that $H_g\cup\{g(x^*)\}$ is linearly independent. Repeating the same argument (by interchanging the roles of $f$ and $g$) it follows that $H_f\cup\{f(x^*)\}$ is linearly independent, which contradicts that $H_f$ is a generating system.
As $H_f$ and $H_g$ are Hamel bases for $X_0$ and $Y_0$, respectively, there exists a unique linear mapping $A \colon X_0\to Y_0$ such that
\Eq{defA}{
g(x)=Af(x)\qquad \text{for all }x \in D_\Gamma:=\{x_{\gamma}\mid \gamma\in\Gamma\}.
}
Our aim is to extend the latter equality to all $x \in D$. To this end, take $x \in D\setminus D_\Gamma$ arbitrarily.
Using that $H_f$ is a Hamel base for $X_0$, we can find elements $\gamma_1,\dots,\gamma_d\in \Gamma$ and nonzero real numbers $\lambda_1,\dots,\lambda_d$ such that
\Eq{*}{
f(x)=\sum_{i=1}^d \lambda_i f(x_{\gamma_i}).
}
Then $\sum_{i=1}^d \lambda_i f(x_{\gamma_i})-f(x)=0$, and by our claim $\sum_{i=1}^d \lambda_i g(x_{\gamma_i})-g(x)=0$.
Finally we obtain
\Eq{*}{
g(x)=\sum_{i=1}^d \lambda_i g(x_{\gamma_i})=\sum_{i=1}^d \lambda_i Af(x_{\gamma_i})=A\bigg(\sum_{i=1}^d \lambda_i f(x_{\gamma_i})\bigg)=Af(x).
}
Therefore \eq{defA} holds for all $x \in D$ which completes the proof.
\end{proof}
\section{Synergy}
Before we introduce the notion of synergy let us present some interpretation of the aggregated effort. The initial issue of coalitions in decision making theory (and, more general, theory of cooperation in games) is the problem how to measure the coalition quality. Intiuitively, synergy is the difference between the aggregated effort and the sum of the individual efforts, i.e., the arithmetic effort. For the detailed study of synergy, we refer the reader to \cite{ShuMoj12} and references therein.
\begin{exa}\label{ex:toy}
In a toy model we have three parties in a parliament with a total number of $100$ votes and three parties: Party~A ($\lambda_1$ votes), Party~B ($\lambda_2$ votes), Party~C ($\lambda_3$ votes). Assume that $\lambda_1\ge \lambda_2\ge\lambda_3$.
The are three possible coalitions AB, AC and BC. From the point of view of the dominant decision system (for example $\D_{FDD}$) each coalition above $50$ votes, is equivalent to the same number, the smallest majority which is $51$. Thus we have for all $x \in D^2$ and $i,j \in \{1,2,3\}$ with $i \ne j$,
\Eq{*}{
\alpha(\lambda)=\begin{cases}
100 & \sum_{i=1}^n \lambda_i\ge 51;\\
\sum_{i=1}^n \lambda_i & \sum_{i=1}^n \lambda_i\le 50.
\end{cases}
}
Now we could compare the sum of weights with the equivalent weight in two cases:
\noindent {\it Situation I:} $(\lambda_1,\lambda_2,\lambda_3)=(45,35,20)$:
\Eq{*}{
s_{AB}&=\alpha(\lambda_1,\lambda_2)-(\alpha(\lambda_1)+\alpha(\lambda_2))=\lambda_3=20;\\
s_{AC}&=\alpha(\lambda_1,\lambda_3)-(\alpha(\lambda_1)+\alpha(\lambda_3))=\lambda_2=35;\\
s_{BC}&=\alpha(\lambda_2,\lambda_3)-(\alpha(\lambda_2)+\alpha(\lambda_3))=\lambda_1=45;\\
s_{ABC}&=\alpha(\lambda_1,\lambda_2,\lambda_3)-(\alpha(\lambda_1)+\alpha(\lambda_2)+\alpha(\lambda_3))&=0.
}
Obviously, each party wants to be in a coalition. However $A$ prefers $C$ than $B$ (as $s_{AC}\ge s_{AB}$) but both $B$ and $C$ prefer to make a coalition with each other (as $s_{BC}\ge s_{AB}$ and $s_{BC}\ge s_{AC}$). Consequently the coalition $BC$ is the unique Nash equilibrium.
\noindent {\it Situation II:} $(\lambda_1,\lambda_2,\lambda_3)=(55,30,15)$:
\Eq{*}{
s_{AB}&=\alpha(\lambda_1,\lambda_2)-(\alpha(\lambda_1)+\alpha(\lambda_2))=-\lambda_2=-30;\\
s_{AC}&=\alpha(\lambda_1,\lambda_3)-(\alpha(\lambda_1)+\alpha(\lambda_3))=-\lambda_1=-15;\\
s_{BC}&=\alpha(\lambda_2,\lambda_3)-(\alpha(\lambda_2)+\alpha(\lambda_3))=0;\\
s_{ABC}&=\alpha(\lambda_1,\lambda_2,\lambda_3)-(\alpha(\lambda_1)+\alpha(\lambda_2)+\alpha(\lambda_3))=0.
}
Then $A$ does not want to make a coalition with either $B$ or $C$ (as the synergy is negative). Similarly neither $B$ nor $C$ wants to make a coalition with $A$ (this essentially follows from the real situation). The coalition $BC$ is irrelevant (which refers to the zero synergy).
\end{exa}
\bigskip
Obviously, as it was announced, the examples above are instrumental to dominant decision systems only. For more complicated decision making systems, we need to define the synergy in a different way.
In general, the synergy depends on the players' decisions. There are essentially two important assertions:
\begin{enumerate}
\item zero synergy refers to the situation when aggregation is irrelevant to the rest of the system;
\item positive synergy should be profitable from the point of view of a decision making system.
\end{enumerate}
In our model, for an effort function $\E\colon \WD \to \R_+$, the $\E$-\emph{synergy} is a function $\sigma_\E\colon \WD \to \R$ defined as follows
\Eq{*}{
\sigma_\E(x,\lambda):=\E(x,\lambda)-(\lambda_1+\cdots+\lambda_n),\qquad n \in \N\text{ and }(x,\lambda) \in D^n \times W_n.
}
In other words, $\sigma_\E$ measures the difference between the given effort and the arithmetic effort, which is the sum of individual efforts.
If $f\colon D\to X$ is an admissible function and $\E=\beta_f$, then $\sigma_\E$ will simply be denoted as $\sigma_f$. Furthermore, in view of Corollary~\ref{cor:Bbeta}, we can see that the synergy depends only on the mean $\B_f$, that is, the equality $\B_f=\B_g$ implies $\sigma_f=\sigma_g$. Therefore, we can define $\sigma_{\B_f}:=\sigma_f$.
This property has an important interpretation in the theory of coalitional games. The case when the synergy is negative corresponds to the situation when there appear some distractions in the cooperation (see Example~\ref{ex:toy}.II). The case of positive synergy refers to the situation when the aggregated effort of the group is greater then sum of efforts of the individuals (see Example~\ref{ex:toy}.I).
\begin{exa}\label{ex:hyperboloid}
Given a manifold $S:=\{(x,y,z)\mid x^2+y^2-z^2=-1 \wedge z\ge 0\}\subseteq\R^3$ with a parametrization $f \colon \R^2 \to S$ given by $f(x,y)=(x,y,\sqrt{1+x^2+y^2})$. Then $S$ is observable, $f$ is admissible and the Bajraktarevi\'c-type mean $\B_f \colon \mathscr{W}(\R^2)\to \R^2$ is of the following form (here and below $n \in \N$ is fixed, $x,y\in\R^n$, and $\lambda \in W_n$) :
\Eq{*}{
\B_f((x,y),\lambda)
&=f^{(-1)} \Big( \sum_{i=1}^n \lambda_i f(x_i,y_i) \Big)\\
&=f^{(-1)} \Big( \sum_{i=1}^n \lambda_i x_i,\sum_{i=1}^n \lambda_i y_i,\sum_{i=1}^n \lambda_i \sqrt{1+x_i^2+y_i^2} \Big).
}
Now define
\Eq{*}{
\Delta((x,y),\lambda)
:=\Big(\sum_{i=1}^n \lambda_i \sqrt{1+x_i^2+y_i^2}\Big)^2-\Big(\sum_{i=1}^n \lambda_i x_i\Big)^2-\Big(\sum_{i=1}^n \lambda_i y_i\Big)^2.
}
We can easily verify that
\Eq{*}{
\frac1{\sqrt{\Delta((x,y),\lambda)}} \Big( \sum_{i=1}^n \lambda_i x_i,\sum_{i=1}^n \lambda_i y_i,\sum_{i=1}^n \lambda_i \sqrt{1+x_i^2+y_i^2} \Big) \in S,
}
which yields
\Eq{*}{
\B_f((x,y),\lambda)
&=\Big( \frac1{\sqrt{\Delta((x,y),\lambda)}}\sum_{i=1}^n \lambda_i x_i,\frac1{\sqrt{\Delta((x,y),\lambda)}}\sum_{i=1}^n \lambda_i y_i \Big);\\
\beta_f((x,y),\lambda)&=\sqrt{\Delta((x,y),\lambda)}.
}
Furthermore the inverse triangle (Minkowski's) inequality (for $\ell^{1/2}$) applied to the vectors $(\lambda_i^2)$, $(\lambda_i^2x_i^2)$, $(\lambda_i^2y_i^2)$ implies $\beta_f((x,y),\lambda) \ge \lambda_1+\dots+\lambda_n$ or, equivalently, $\sigma_f((x,y),\lambda) \ge 0$.
Observe that the above mean is not the standard convex combination of its arguments. Indeed, for the entries $(x,y)=((1,0),(0,1))$ and weights $\lambda=(1,1)$, we get $\B_f((x,y),\lambda)=(\frac{\sqrt6}6,\frac{\sqrt6}6)$, which obviously does not belong to the segment $\conv((0,1),(1,0))$.
\end{exa}
\subsection{Generalized quasi-arithmetic means}
In the next lemma we characterize the subfamily of zero-synergy generalized Bajraktarevi\'c means. Its single variable counterpart was proved in \cite{Pas1910}.
\begin{thm}\label{lem:degen}
Let $f \colon D \to X$ be an admissible function. Then the following conditions are equivalent:
\begin{enumerate}[(i)]
\item $f(D)$ is a convex set and
\Eq{E:flat}{
\B_f(x,\lambda)=f^{-1} \bigg(\frac{\sum_{i=1}^n \lambda_i f(x_i)} {\sum_{i=1}^n\lambda_i}\bigg)\qquad \text{for all }n \in \N \text{ and }(x,\lambda)\in D^n \times W_n,
}
in particular the right hand side is well-defined for all such pairs;
\item $\sigma_f\equiv 0$;
\item \label{BfAss} $\B_f$ is associative, that is,
\Eq{iii}{
\B_f\big((x,y),(\lambda,\mu)\big)
=\B_f\big(\big(\B_f(x,\lambda),y\big),(\alpha(x,\lambda),\mu) \big)
}
for all pairs $(x,\lambda),(y,\mu)\in \WD$ (where $\alpha\colon\WD\to\R_+$ stands for the arithmetic effort function);
\item Equality \eq{iii} holds for all $(x,\lambda)\in\WD$ and $(y,\mu)\in D\times\R_+$.
\end{enumerate}
\end{thm}
\begin{proof} If $D$ is a singleton, then all of the above conditions are satisfied. Therefore, we may assume that $D$ has at least two distinct elements.
The implications $(i) \Rightarrow (ii)$, $(ii) \Rightarrow (iii)$, and $(iii) \Rightarrow (iv)$ are easy to check. To prove the implication $(iv) \Rightarrow (i)$, assume that $\B_f$ satisfies \eq{iii} for all $(x,\lambda)\in\WD$ and $(y,\mu)\in D\times\R_+$.
Fix $n \in \N$ and a pair $(x,\lambda) \in D^n \times W_n$. We denote briefly $\bar x:=\B_f(x,\lambda)$, $\bar\lambda:=\beta_f(x,\lambda)$ and $\bar\alpha:=\alpha(x,\lambda)$. Then we have that $\sum_{i=1}^n \lambda_i f(x_i)=\bar\lambda f(\bar x)$.
Now fix $y \in D \setminus\{\bar x\}$ and $\mu>0$. Applying the delegativity of $\B_f$ and condition (iv), we have
\Eq{*}{
\B_f\big((\bar x,y),(\bar\lambda,\mu)\big)
=\B_f\big((x,y),(\lambda,\mu)\big)
=\B_f\big((\bar x,y),(\bar\alpha,\mu)\big).
}
Consequently $\bar\lambda f(\bar x)+\mu f(y)$ and $\bar\alpha f(\bar x)+\mu f(y)$ are on the same ray, i.e., there exists a constant $C>0$ such that
\Eq{*}{
C \cdot \big(\bar\lambda f(\bar x)+\mu f(y)\big)
=\bar\alpha f(\bar x)+\mu f(y),
}
which reduces to
\Eq{*}{
0=(C\bar\lambda-\bar\alpha)f(\bar x)+\mu (C-1)f(y).
}
As $y \ne \bar x$, the admissibility implies that $f(y)$ and $f(\bar x)$ are linearly independent. Consequently, the above equality implies $C=1$ and $\bar\lambda=\bar\alpha$. Thus,
\Eq{*}{
\frac{\sum_{i=1}^n \lambda_i f(x_i)}{\sum_{i=1}^n \lambda_i}
=\frac{\sum_{i=1}^n \lambda_i f(x_i)}{\bar\alpha}
=\frac{\sum_{i=1}^n \lambda_i f(x_i)}{\bar\lambda}
=f(\bar x)\in f(D),
}
which implies that $f(D)$ is a convex set. Finally, applying $f^{-1}$ side-by-side we get that \eq{E:flat} holds.
\end{proof}
\subsection{Gini means}
We are now going to calculate the sign of the synergy for Gini means. Before we go into the details, we recall a few properties of this family. First, it is easy to observe that $\G_{p,q}=\G_{q,p}$ for all $p,q \in \R$. Furthermore, in a case $q=0$, the Gini mean $\G_{p,0}$ equals the $p$-th Power mean (in particular it is associative). These means are monotone with respect to their parameters, more precisely, for all $p,q,r,s \in \R$, we have that $\G_{p,q}\le \G_{r,s}$ if and only if $\min(p,q)\le \min(r,s)$ and $\max(p,q)\le \max(r,s)$ (cf.\ \cite{DarLos70}). Finally, a Gini mean $\G_{p,q}$ is monotone as a mean (in each of its argument) if and only if $pq \le 0$ (see \cite{Los71a,Los71b}). We show that the sign of $pq$ is also important in characterizing the sign of their synergy.
\begin{prop}
Sign of the synergy of the Gini mean $\G_{p,q}$ coincides with that of $-pq$.
More precisely, for all $p,q \in \R$, $n\in \N$, nonconstant vector $x\in \R_+^n$ and $\lambda \in \R_+^n$, we have
$\sign\big(\sigma_{\G_{p,q}}(x,\lambda)\big)=-\sign(pq)$.
\end{prop}
\begin{proof}
If $pq=0$, then $\G_{p,q}$ is associative and thus Lemma~\ref{lem:degen} implies $\sigma_{\G_{p,q}}\equiv 0$. From now on assume that $pq\ne 0$.
Fix $n \in \N$, $\lambda \in \R_+^n$ and nonconstant vector $x\in \R_+^n$. Let
\Eq{*}{
\varphi_p:=\lambda_1x_1^p+\cdots+\lambda_nx_n^p\quad\text{ and }\quad \psi_p:=\lambda_1x_1^p\ln(x_1)+\cdots+\lambda_nx_n^p\ln(x_n)\quad (p \in \R).
}
Assume first that $p \ne q$. As $\gamma_{p,q}=\gamma_{q,p}$, without loss of generality, we can assume that $p>q$. Then by \eq{E:Ginipq}, we have
\Eq{*}{
\gamma_{p,q}(x,\lambda)=\frac{(\lambda_1 x_1^q+\dots+\lambda_n x_n^q)^{\frac{p}{p-q}}}
{(\lambda_1 x_1^p+\dots+\lambda_n x_n^p)^{\frac{q}{p-q}}}=\bigg(\frac{\varphi_q^p}{\varphi_p^q}\bigg)^{\frac1{p-q}}.
}
Whence by the definition
\Eq{*}{
\sigma_{\G_{p,q}}(x,\lambda)=\gamma_{p,q}(x,\lambda)-(\lambda_1+\cdots+\lambda_n)=\bigg(\frac{\varphi_q^p}{\varphi_p^q}\bigg)^{\frac1{p-q}}-\varphi_0.
}
In view of the inequality $p >q$, we obtain
\Eq{*}{
\sigma_{\G_{p,q}}(x,\lambda) > 0 \iff \bigg(\frac{\varphi_q^p}{\varphi_p^q}\bigg)^{\frac1{p-q}}>\varphi_0 \iff \varphi_q^p > \varphi_0^{p-q} \varphi_p^q \iff \bigg(\frac{\varphi_q}{\varphi_0}\bigg)^p > \bigg(\frac{\varphi_p}{\varphi_0}\bigg)^q.
}
For $pq<0$ we obtain that $\sigma_{\G_{p,q}}(x,\lambda) > 0$ is equivalent to $\big(\tfrac{\varphi_q}{\varphi_0}\big)^{1/q}< \big(\tfrac{\varphi_p}{\varphi_0}\big)^{1/p}$. But the last inequality is just the equality between power means. Thus we have $\sigma_{\G_{p,q}}(x,\lambda) > 0$ whenever $pq<0$.
If $pq>0$ then $\sigma_{\G_{p,q}}(x,\lambda) > 0$ is equivalent to $\big(\tfrac{\varphi_q}{\varphi_0}\big)^{1/q} > \big(\tfrac{\varphi_p}{\varphi_0}\big)^{1/p}$. But, as $p>q$ we know that the converse inequality holds. So in this case we obtain $\sigma_{\G_{p,q}}(x,\lambda) < 0$.
In the last case, when $p=q\ne0$, we have
\Eq{*}{
\sigma_{\G_{p,p}}(x,\lambda)=\gamma_{p,p}(x,\lambda)-(\lambda_1+\dots+\lambda_n)=\varphi_p \exp\Big(\frac{-p\psi_p}{\varphi_p}\Big)-\varphi_0.
}
Therefore
\Eq{*}{
\sigma_{\G_{p,p}}(x,\lambda) < 0 \iff \varphi_p \exp\Big(\frac{-p\psi_p}{\varphi_p}\Big)-\varphi_0<0 \iff \exp\Big(\frac{-p\psi_p}{\varphi_p}\Big) < \frac{\varphi_0}{\varphi_p}.
}
We can apply the strictly decreasing mapping $\R_+ \ni \xi \mapsto \sign(p)\xi^{-1/p}$ side-by-side to obtain
\Eq{*}{
\sigma_{\G_{p,p}}(x,\lambda) < 0 &\iff \sign(p) \exp\Big(\frac{\psi_p}{\varphi_p}\Big) > \sign(p) \Big(\frac{\varphi_p}{\varphi_0}\Big)^{1/p}\\
&\iff \sign(p) \G_{p,p}(x,\lambda) >\sign(p) \G_{p,0}(x,\lambda).
}
But the inequality on the right-hand-side holds for all $p\in \R \setminus\{0\}$, which completes the proof.
\end{proof}
\begin{xrem}
Observe that Gini mean has a positive synergy if the graph of $\gamma_{p,q}$ is hyperbolic and negative for parabolic graphs. In the case of hyperboloid in Example~\ref{ex:hyperboloid} the synergy was also positive.
\end{xrem}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,336 |
\section{Introduction}
\subsection{Summary}
\label{subsec:summary}
The main result of this paper is that for a $(m,n)$-matrix $A$ over the complex group ring of
${\mathbb Z}^d$ the Novikov-Shubin invariant of the bounded ${\mathbb Z}^d$-equivariant operator
$r_A^{(2)} \colon L^2({\mathbb Z}^d)^m \to L^2({\mathbb Z}^d)^n$ given by right multiplication with $A$ is larger
than zero. Actually rather explicit lower bounds in terms of elementary invariants of the
minors of the matrix $A$ will be given.
This is a direct consequence of a polynomial bound of the
spectral density function of $r_A^{(2)} $ which is interesting in its own right.
It will play a role in the forthcoming paper~\cite{Friedl-Lueck(2015twisting)}, where
we will twist $L^2$-torsion with finite dimensional representations and it
will be crucial that we allow complex coefficients and not only integral coefficients.
Novikov-Shubin invariants were originally defined analytically in
\cite{Novikov-Shubin(1986b),Novikov-Shubin(1986a)}. More information about them can be found
for instance in~\cite[Chapter~2]{Lueck(2002)}.
Before we state the main result, we need the following notions.
\subsection{The width and the leading coefficient}
\label{subsec:The_Width_and_the_leading_coefficient}
Consider a non-zero element
$p = p(z_1^{ \pm 1}, \ldots, z_{d}^{\pm 1}) $ in
${\mathbb C}[{\mathbb Z}^d] = {\mathbb C}[z_1^{ \pm 1}, \ldots, z_{d}^{\pm 1}]$ for some integer $d \ge 1$.
There are integers $n_d^-$ and $n_d^+$ and elements $q_n(z_1^{\pm 1},\ldots ,z_{d-1}^{ \pm 1})$ in
${\mathbb C}[{\mathbb Z}^{d-1}] = {\mathbb C}[z_1^{ \pm 1}, \ldots, z_{d-1}^{\pm 1}]$ uniquely determined by the properties
that
\begin{eqnarray*}
n_d^-
& \le &
n_d^+;
\\
q_{n_d^-}(z_1^{\pm 1},\ldots ,z_{d-1}^{ \pm 1})
& \not= &
0;
\\
q_{n_d^+}(z_1^{\pm 1},\ldots ,z_{d-1}^{ \pm 1})
& \not= &
0;
\\
p(z_1^{ \pm 1}, \ldots, z_{d}^{\pm 1})
& = &
\sum_{n = n_d^-}^{n_d^+}
q_n(z_1^{\pm 1},\ldots ,z_{d-1}^{ \pm 1}) \cdot z_d^n.
\end{eqnarray*}
In the sequel denote
\begin{eqnarray*}
w(p) & = & n_d^+ -n_d^-;
\\
q^+(p) & = & q_{n_d^+}(z_1^{\pm 1},\ldots ,z_{d-1}^{ \pm 1}).
\end{eqnarray*}
Define inductively elements $p_i(z_1^{ \pm 1}, \ldots, z_{d-i}^{\pm 1}) $ in
${\mathbb C}[{\mathbb Z}^{d-i}] = {\mathbb C}[z_1^{ \pm 1}, \ldots, z_{d-i}^{\pm 1}]$ and integers
$w_i(p) \ge 0$ for $i = 0,1,2, \ldots, d$ by
\begin{eqnarray*}
p_0(z_1^{ \pm 1}, \ldots, z_{d}^{\pm 1})
& := &
p(z_1^{ \pm 1}, \ldots, z_{d}^{\pm 1});
\\
p_1(z_1^{\pm 1},\ldots ,z_{d-1}^{ \pm 1})
& := &
q^+(p)
\\
p_{i}
& := &
q^+(p_{i-1}) \quad \text{for } i = 1,2 \ldots, d;
\\
w_0(p)
& := & w(p)
\\
w_i(p)
& := &
w(p_i) \quad \text{for } i = 1,2 \ldots, (d-1).
\end{eqnarray*}
Define the \emph{width} of $p = p(z_1^{ \pm 1}, \ldots, z_{d}^{\pm 1})$ to be
\begin{eqnarray}
\operatorname{wd}(p) = \max\{w_0(p), w_1(p), \ldots, w_{d-1}(p)\},
\label{width(p)}
\end{eqnarray}
and the leading \emph{coefficient of $p$} to be
\begin{eqnarray}
\operatorname{lead}(p) & = & p_d.
\label{lead(p)}
\end{eqnarray}
Obviously we have
\begin{eqnarray*}
& \operatorname{wd}(p) \ge \operatorname{wd}(p_1) \ge \operatorname{wd}(p_2) \ge \cdots \ge \operatorname{wd}(p_d) = 0; &
\\
& \operatorname{lead}(p) = \operatorname{lead} (p_1) = \ldots = \operatorname{lead}(p_0) \not= 0.
\end{eqnarray*}
Notice that $p_i$, $\operatorname{wd}(p)$ and $\operatorname{lead}(p)$ do depend on the ordering of the variables $z_1, \ldots , z_d$.
\begin{remark}[Leading coefficient]\label{rem:leading_coefficients}
The name ``leading coefficient'' comes from the following alternative definition.
Equip ${\mathbb Z}^d$ with the lexicographical order, i.e., we put $(m_1, \ldots, m_d) < (n_1,
\ldots, n_d)$, if $m_d < n_d$, or if $m_d = n_d$ and $m_{d-1} < n_{d-1}$, or if $m_d = n_d$, $m_{d-1}
= n_{d-1}$ and $m_{d-2} < n_{d-2}$, or if $\ldots$, or if $m_i = n_i$ for $i = d, (d-1), \ldots, 2$ and
$m_1 < n_1$. We can write $p$ as a finite sum with complex coefficients $a_{n_1, \ldots, n_d}$
\[p(z_1^{\pm}, \ldots, z_d^{\pm})
= \sum_{(n_1, \ldots, n_d) \in {\mathbb Z}^d} a_{n_1, \ldots, n_d} \cdot z_1^{n_1} \cdot z_2^{n_2} \cdot \cdots \cdot z_d^{n_d}.
\]
Let $(m_1, \ldots m_d) \in {\mathbb Z}^d$ be maximal with respect to the lexicographical order among those elements
$(n_1, \ldots, n_d) \in {\mathbb Z}^d$ for which $a_{n_1, \ldots, n_d} \not= 0$. Then the leading coefficient
of $p$ is $a_{m_1, \ldots, m_d}$.
\end{remark}
\subsection{The $L^1$-norm of a matrix}
\label{subsec:The_L1-norm_of_a_matrix}
For an element $p = \sum_{g \in {\mathbb Z}^d} \lambda_g \cdot g \in {\mathbb C}[{\mathbb Z}^d]$ define
$||p||_1 := \sum_{g \in G} |\lambda_g|$. For a matrix $A \in M_{m,n}({\mathbb C}[{\mathbb Z}^d])$ define
\begin{eqnarray}
||A||_1 & = &
\max\{||a_{i,j}||_1 \mid 1 \le i \le m, 1 \le j \le n\}.
\label{L1-norm_of_a_matrix}
\end{eqnarray}
The main purpose of this notion is that it gives an a priori upper bound on the norm $r_A^{(2)} \colon L^2({\mathbb Z}^d) \to L^2({\mathbb Z}^d)$,
namely, we get from~\cite[Lemma~13.33 on page~466]{Lueck(2002)}
\begin{eqnarray}
||r_A^{(2)}|| & \le & m \cdot n \cdot ||A||_1.
\label{norm_estimate_by_L1-norm}
\end{eqnarray}
\subsection{The spectral density function}
\label{subsec:The_spectral_density_function}
Given $A \in M_{m,n}({\mathbb C}[{\mathbb Z}^d])$, multiplication with $A$ induces a bounded
${\mathbb Z}^d$-equivariant operator $r_A^{(2)} \colon L^2({\mathbb Z}^d)^m \to L^2({\mathbb Z}^d)^n$. We will denote by
\begin{eqnarray}
F\bigl(r_A^{(2)}\bigr) \colon [0,\infty) & \to & [0,\infty)
\label{spectral_density_function}
\end{eqnarray}
its \emph{spectral density function} in the sense
of~\cite[Definition~2.1 on page~73]{Lueck(2002)}, namely, the von Neumann dimension of
the image of the operator obtained by applying the functional calculus to the characteristic function
of $[0,\lambda^2]$ to the operator $(r^{(2)}_A)^*r_A^{(2)}$. In the special case $m = n = 1$,
where $A$ is given by an element $p \in {\mathbb C}[{\mathbb Z}^d]$,
it can be computed in terms of the Haar measure $\mu_{T^d}$ of the $d$-torus $T^d$
see~\cite[Example~2.6 on page~75]{Lueck(2002)}
\begin{eqnarray}
F\bigl(r_A^{(2)}\bigr)(\lambda) & = & \mu_{T^d}\bigl(\{(z_1, \ldots, z_d) \in T^d \mid \; |p(z_1, \ldots, z_d)| \le \lambda\}\bigr).
\label{F(P)_in_terms_of_volume}
\end{eqnarray}
\subsection{The main result}
\label{subsec:The_main_result}
Our main result is:
\begin{theorem}[Main Theorem]
\label{the:Main_Theorem}
Consider any natural numbers $d,m,n$ and a non-zero matrix $A \in M_{m,n}({\mathbb C}[{\mathbb Z}^d])$.
Let $B$ be a quadratic submatrix of $A$ of maximal size $k$ such that the corresponding minor
$p = {\det}_{{\mathbb C}[{\mathbb Z}^d]}(B)$ is non-trivial. Then:
\begin{enumerate}
\item \label{the:Main_Theorem:spectral_density_estimate}
If $\operatorname{wd}(p) \ge 1$, the spectral density function
of $r_A^{(2)} \colon L^2({\mathbb Z}^d)^m \to L^2({\mathbb Z}^d)^n$ satisfies for all $\lambda \ge 0$
\begin{multline*}
\lefteqn{F\bigl(r_A^{(2)}\bigr)(\lambda) - F\bigl(r_A^{(2)}\bigr)(0)}
\\ \le
\frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot k \cdot d \cdot \operatorname{wd}(p)
\cdot \left(\frac{k^{2k -2} \cdot (||B||_1)^{k-1} \cdot \lambda}{|\operatorname{lead}(p)|}\right)^{\frac{1}{d \cdot \operatorname{wd}(p)}}.
\end{multline*}
If $\operatorname{wd}(p) = 0$, then $F\bigl(r_A^{(2)}\bigr)(\lambda) = 0$ for all $\lambda < |\operatorname{lead}(p)|$
and $F\bigl(r_A^{(2)}\bigr)(\lambda) = 1$ for all $\lambda \ge |\operatorname{lead}(p)|$;
\item \label{the:Main_Theorem:Novikov-Shubin}
The Novikov-Shubin invariant of $r_A^{(2)}$ is $\infty$ or $\infty^+$ or a real number satisfying
\[
\alpha\bigl(r_A^{(2)}\bigr) \ge \frac{1}{d \cdot \operatorname{wd}(p)},
\]
and is in particular larger than zero.
\end{enumerate}
\end{theorem}
It is known that the Novikov-Shubin invariants of $r_A^{(2)}$ for a matrix $A$ over the integral group ring of ${\mathbb Z}^d$
is a rational numbers larger than zero unless its value is $\infty$ or $\infty^+$.
This follows from Lott~\cite[Proposition~39]{Lott(1992a)}.
(The author of~\cite{Lott(1992a)}
informed us that his proof of this statement is correct
when $d = 1$ but has a gap when $d > 1$.
The nature of the gap is
described in~\cite[page 16]{Lott(1999b)}.
The proof in this case can be completed
by the same basic method used in~\cite{Lott(1992a)}.) This confirms a
conjecture of Lott-L\"uck~\cite[Conjecture~7.2]{Lott-Lueck(1995)} for $G = {\mathbb Z}^d$.
The case of a finitely generated free group $G$ is taken care of by Sauer~\cite{Sauer(2003)}.
Virtually finitely generated free abelian groups and virtually finitely generated free groups
are the only cases of finitely generated groups, where the positivity of the Novikov-Shubin invariants for
all matrices over the complex group ring is now known.
In this context we mention the preprints~\cite{Grabowski(2014),Grabowski-Virag(2013)},
where examples of groups $G$ and matrices $A \in M_{m,n}({\mathbb Z} G)$ are constructed for which
the Novikov-Shubin invariant of $r_A^{(2)}$ is zero, disproving a
conjecture of Lott-L\"uck~\cite[Conjecture~7.2]{Lott-Lueck(1995)}.
\subsection{Example}
\label{sec:Example}
Consider the case $d = 2$, $m = 3$ and $n = 2$ and the $(3,2)$-matrix over ${\mathbb C}[{\mathbb Z}^2]$
\[
A =
\begin{pmatrix}
z_1^3 & -1 & 1
\\
2 \cdot z_1 \cdot z_2^2 -16 & z_2 & z_1z_2
\end{pmatrix}
\]
Let $B$ be the $(2,2)$-submatrix obtained by deleting third column.
Then $k = 2$,
\[
B =
\begin{pmatrix}
z_1^3 & -1
\\
2 \cdot z_1 \cdot z_2^2 -16 & z_2
\end{pmatrix}
\]
and we get
\[
p := {\det}_{{\mathbb C}[{\mathbb Z}^2]}(B) = z_1^3 \cdot z_2 + 2 \cdot z_1 \cdot z_2^2 -16.
\]
Using the notation of Section~\ref{subsec:The_Width_and_the_leading_coefficient}
one easily checks $p_1(z_1) = 2 \cdot z_1$, $\operatorname{wd}(p) = 2$, and $\operatorname{lead}(p) = 2$.
Obviously $||A||_1 = \max\{|1|,|-1|,|2| + |16|,|1|\} = 18$. Hence
Theorem~\ref{the:Main_Theorem} implies for all $\lambda \ge 0$
\begin{eqnarray*}
F\bigl(r_A^{(2)}\bigr)(\lambda) - F\bigl(r_A^{(2)}\bigr)(0)
& \le &
\frac{192\cdot \sqrt{2}}{\sqrt{47}}
\cdot \lambda^{\frac{1}{4}}.
\\
\alpha\bigl(r_A^{(2)}\bigr)
& \ge &
\frac{1}{4}.
\end{eqnarray*}
\subsection{Acknowledgments}
\label{sec:Acknowledgments.} This paper is financially supported
by the Leibniz-Preis of the author granted by the Deutsche Forschungsgemeinschaft {DFG}.
The author wants to thank the referee for his useful comments.
\typeout{------------------------ Section 2: The case $m = n = 1$ -------------------------------}
\section{The case $m = n = 1$}
\label{subsec:The_case_m_is_n_is_1}
The main result of this section is the following
\begin{proposition}
\label{pro:case_m_is_n_is_1}
Consider an non-zero element
$p$ in ${\mathbb C}[{\mathbb Z}^d] = {\mathbb C}[z_1^{ \pm 1}, \ldots, z_{d}^{\pm 1}]$.
If $\operatorname{wd}(p) = 0$, then $F\bigl(r_A^{(2)}\bigr)(\lambda) = 0$ for all $\lambda < |\operatorname{lead}(p)|$
and $F\bigl(r_A^{(2)}\bigr)(\lambda) = 1$ for all $\lambda \ge |\operatorname{lead}(p)|$. If $\operatorname{wd}(p) \ge 1$,
we get for the spectral density function of $r_p^{(2)}$ for all $\lambda \ge 0$
\[
F\bigl(r_p^{(2)}\bigr)(\lambda)
\le
\frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot d \cdot \operatorname{wd}(p)
\cdot \left(\frac{\lambda}{|\operatorname{lead}(p)|}\right)^{\frac{1}{d \cdot \operatorname{wd}(p)}}.
\]
\end{proposition}
For the case $d = 1$ and $p$ a monic polynomial, a similar estimate of the shape
$F\bigl(r_p^{(2)}\bigr)(\lambda) \le C_k \cdot \lambda^{\frac{1}{k-1}}$ can be found
in~\cite[Theorem~1]{Lawton(1983)}, where the $k \ge 2$ is the number of non-zero coefficients,
and the sequence of real numbers $(C_k)_{k \ge 2}$ is recursively defined and satisfies $C_k \ge k-1$.
\subsection{Degree one}
\label{subsec:Proof_of_Proposition_ref(pro:case_m_is_n_is_1)_in_degree_one}
In this subsection we deal with Proposition~\ref{pro:case_m_is_n_is_1} in the case $d = 1$.
We get from the Taylor expansion of $\cos(x)$ around $0$ with the Lagrangian remainder term
that for any $x \in {\mathbb R}$ there exists
$\theta(x) \in [0,1]$ such that
\[
\cos(x) = 1 - \frac{x^2}{2} + \frac{\cos(\theta(x) \cdot x)}{4!} \cdot x^4.
\]
This implies for $x \not= 0$ and $|x| \le 1/2$
\[
\left|\frac{2 - 2 \cos(x)}{x^2} - 1\right|
=
\left|\frac{2 \cdot \cos(\theta(x) \cdot x)}{4!} \cdot x^2\right|
\le
\left|\frac{2 \cdot \cos(\theta(x) \cdot x)}{4!} \right| \cdot |x|^2
\le \frac{1}{12} \cdot \frac{1}{4} = \frac{1}{48}.
\]
Hence we get for $x \in [-1/2,1/2]$
\begin{eqnarray}
\frac{47}{48} \cdot x^2 \le 2 - 2\cos(x).
\label{cos(x)_versus_x2}
\end{eqnarray}
\begin{lemma}\label{lem:polynomial_F(lambda)-estimate_z-a}
For any complex number $a \in {\mathbb Z}$ we get for the spectral density function of $(z-a) \in {\mathbb C}[{\mathbb Z}] = {\mathbb C}[z,z^{-1}]$
\[
F\bigl(r_{z-a}^{(2)}\bigr)(\lambda) \le \frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot \lambda \quad \text{for}\; \lambda \in [0,\infty).
\]
\end{lemma}
\begin{proof}
We compute using~\eqref{F(P)_in_terms_of_volume}, where $r := |a|$,
\begin{eqnarray*}
F\bigl(r_{z-a}^{(2)}\bigr)(\lambda)
& = &
\mu_{S^1}\{z \in S^1 \mid |z-a| \le \lambda\}
\\
& = &
\mu_{S^1}\{z \in S^1 \mid |z-r| \le \lambda\}
\\
& = &
\mu_{S^1}\{\phi \in [-1/2,1/2] \mid |\cos(\phi) + i \sin(\phi) -r| \le \lambda\}
\\
& = &
\mu_{S^1}\{\phi \in [-1/2,1/2] \mid |\cos(\phi) + i \sin(\phi) -r|^2 \le \lambda^2\}
\\
& = &
\mu_{S^1}\{\phi \in [-1/2,1/2] \mid (\cos(\phi) -r)^2 + \sin(\phi)^2 \le \lambda^2\}
\\
& = &
\mu_{S^1}\{\phi \in [-1/2,1/2] \mid r \cdot (2 - 2\cos(\phi) + (r-1)^2 \le \lambda^2\}.
\end{eqnarray*}
We estimate using~\eqref{cos(x)_versus_x2} for $\phi \in [-1/2,1/2]$
\[
r \cdot (2 - 2\cos(\phi)) + (r-1)^2 \ge r \cdot (2 - 2\cos(\phi)) \ge \frac{47}{48} \cdot \phi^2.
\]
This implies for $\lambda \ge 0$
\begin{eqnarray*}
F\bigl(r_{z-a}^{(2)}\bigr)(\lambda)
& = &
\mu_{S^1}\{\phi \in [-1/2,1/2] \mid r \cdot (2 - 2\cos(\phi) + (r-1)^2 \le \lambda^2\}
\\
& \le &
\mu_{S^1}\{\phi \in [-1/2,1/2] \mid \frac{47}{48} \cdot \phi^2 \le \lambda^2\}
\\
& = &
\mu_{S^1}\left\{\phi \in [-1/2,1/2] \;\left|\; |\phi| \le \sqrt{\frac{48}{47}} \cdot \lambda \right.\right\}
\\
& \le &
2 \cdot \sqrt{\frac{48}{47}} \cdot \lambda
\\
& = &
\frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot \lambda.
\end{eqnarray*}
\end{proof}
\begin{lemma}\label{pro:case_m_is_n_is_1_in_degree_one}
Let $p(z) \in {\mathbb C}[{\mathbb Z}] = {\mathbb C}[z,z^{-1}]$ be a non-zero element.
If $\operatorname{wd}(p) = 0$, then $F\bigl(r_p^{(2)}\bigr)(\lambda) = 0$ for all $\lambda < |\operatorname{lead}(p)|$
and $F\bigl(r_p^{(2)}\bigr)(\lambda) = 1$ for all $\lambda \ge |\operatorname{lead}(p)|$. If $\operatorname{wd}(p) \ge 1$,
we get
\[
F\bigl(r_p^{(2)}\bigr)(\lambda)
\le
\frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot \operatorname{wd}(p) \cdot \left(\frac{\lambda}{|\operatorname{lead}(p)|}\right)^{\frac{1}{\operatorname{wd}(p)}}
\quad \text{for}\; \lambda \in [0,\infty).
\]
\end{lemma}
\begin{proof}
If $\operatorname{wd}(p) = 0$, then $p$ is of the shape $C \cdot z^n$, and the claim follows
directly from~\eqref{F(P)_in_terms_of_volume}.
Hence we can assume without loss of generality that $\operatorname{wd}(p) \ge 1$. We can write $p(z)$ as a product
\[
p(z) = \operatorname{lead}(p) \cdot z^k \cdot \prod_{i=1}^r (z - a_i)
\]
for an integer $r \ge 0$, non-zero complex numbers $a_1, \ldots, a_r$ and an integer $k$.
Since for any polynomial $p$ and complex number $c \not= 0$ we have for all $\lambda \in [0,\infty)$
\[
F\bigl(r_{c \cdot p}^{(2)}\bigr)(\lambda) = F\bigl(r_p^{(2)}\bigr)\left(\frac{\lambda}{|c|}\right),
\]
we can assume without loss of generality $\operatorname{lead}(p) = 1$.
If $r = 0$, then $p(z) = z^k$ for some $k \not = 0$ and the claim follows by a direct inspection.
Hence we can assume without loss of generality $r \ge 1$.
Since the width, the leading coefficient
and the spectral density functions of $p(z)$ and
$z^{-k} \cdot p(z)$ agree, we can assume without loss of generality $k = 0$, or equivalently,
that $p(z)$ has the form for some $r \ge 1$
\[
p(z) = \prod_{i=1}^r (z - a_i).
\]
We proceed by induction over $r$.
The case $r = 1$ is taken care of by
Lemma~\ref{lem:polynomial_F(lambda)-estimate_z-a}.
The induction step from $r-1\ge 1$ to $r$ is done as follows.
Put $q(z) = \prod_{i=1}^{r-1} (z - a_i)$. Then $p(z) = q(z) \cdot (z-a_r)$.
The following inequality for elements $q_1,q_2 \in {\mathbb C}[z,z^{-1}]$ and $s \in (0,1)$
is a special case of~\cite[Lemma~2.13~(3) on page~78]{Lueck(2002)}
\begin{eqnarray}
F\bigl(r_{q_1 \cdot q_2}^{(2)}\bigr)(\lambda)
& \le &
F\bigl(r_{q_1}^{(2)}\bigr)(\lambda^{1-s}) + F\bigl(r_{q_2}^{(2)}\bigr)(\lambda^{s}).
\label{F(pq)_estimated_by_F(p)_and_F(q)}
\end{eqnarray}
We conclude from~\eqref{F(pq)_estimated_by_F(p)_and_F(q)} applied to
$p(z) = q(z) \cdot (z-a_r)$ in the special case $s = 1/r$
\begin{eqnarray*}
F\bigl(r_p^{(2)}\bigr)(\lambda)
& \le &
F\bigl(r_q^{(2)}\bigr)(\lambda^{\frac{r-1}{r}}) + F\bigl(r_{z-a_r}^{(2)}\bigr)(\lambda^{1/r}).
\end{eqnarray*}
We conclude from the induction hypothesis for $\lambda \in [0,\infty)$
\begin{eqnarray*}
F\bigl(r_q^{(2)}\bigr)(\lambda)
& \le &
\frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot (r-1) \cdot \lambda^{\frac{1}{r-1}};
\\
F\bigl(r_{z-a_r}^{(2)}\bigr)(\lambda)
& \le &
\frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot \lambda.
\end{eqnarray*}
This implies for $\lambda \in [0,\infty)$
\begin{eqnarray*}
F\bigl(r_p^{(2)}\bigr)(\lambda)
& \le &
F\bigl(r_q^{(2)}\bigr)(\lambda^{\frac{r-1}{r}}) + F\bigl(r_{z-a_r}^{(2)}\bigr)(\lambda^{1/r})
\\
&\le &
\frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot (r-1) \cdot \left(\lambda^{\frac{r-1}{r}}\right)^{\frac{1}{r-1}}
+ \frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot \lambda^{\frac{1}{r}}
\\
&\le &
\frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot (r-1) \cdot \lambda^{\frac{1}{r}}
+ \frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot \lambda^{\frac{1}{r}}
\\
& = &
\frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot r \cdot \lambda^{\frac{1}{r}}.
\end{eqnarray*}
\end{proof}
\subsection{The induction step}
\label{subsec:The_induction_step_in_the_proof_of_Proposition_ref(pro:case_m_is_n_is_1)}
Now we finish the proof of Proposition~\ref{pro:case_m_is_n_is_1}
by induction over $d$.
If $\operatorname{wd}(p) = 0$, then $p$ is of the shape $C \cdot z_1^{n_1} \cdot z_2^{n_2} \cdot \cdots \cdot z_d^{n_d}$,
and the claim follows directly from~\eqref{F(P)_in_terms_of_volume}.
Hence we can assume without loss of generality that $\operatorname{wd}(p) \ge 1$.
The induction beginning $d =1$ has been taken care of
by Lemma~\ref{pro:case_m_is_n_is_1_in_degree_one}, the induction step from
$d-1$ to $d \ge 2$ is done as follows.
Since $F\bigl(r_p^{(2)}\bigr)(\lambda) \le 1$, the claim is obviously true for $\frac{\lambda}{|\operatorname{lead}(p)|} \ge 1$.
Hence we can assume in the sequel $\frac{\lambda}{|\operatorname{lead}(p)|} \le 1$.
We conclude from~\eqref{F(P)_in_terms_of_volume} and Fubini's Theorem applied to $T^d = T^{d-1} \times S^1$,
where $\chi_A$ denotes the characteristic function of a subset $A$ and $p_1(z_1^{\pm}, \ldots ,z_{d-1}^{\pm 1})$
has been defined in Subsection~\ref{subsec:The_Width_and_the_leading_coefficient}
\begin{eqnarray}
\label{F(p)(lambda_estimate:1} & &
\\
\lefteqn{F\bigl(r_p^{(2)}\bigr)(\lambda)}
& &
\nonumber
\\
& = &
\mu_{T^d}\bigl(\{(z_1, \ldots , z_d) \in T^d \mid\; |p(z_1,\ldots, z_d)| \le \lambda\}\bigr)
\nonumber
\\
& = &
\int_{T^d} \chi_{\{(z_1, \ldots , z_d) \in T^d \mid \; |p(z_1,\ldots, z_d)| \le \lambda\}} \; d\mu_{T^n}
\nonumber
\\
& = &
\int_{T^{d-1} } \left(\int_{S^1} \chi_{\{(z_1, \ldots , z_d) \in T^d \mid \; |p(z_1,\ldots, z_d)| \le \lambda\}} \; d\mu_{S^1} \right) \;d\mu_{T^{d-1}}
\nonumber
\\
& = &
\int_{T^{d-1} } \chi_{\{(z_1, \ldots, z_{d-1}) \in T^{d-1}\mid\; |p_1(z_1, \ldots, z_{d-1}) \le |\operatorname{lead}(p)|^{1/d} \cdot \lambda^{(d-1)1/d}\}}
\nonumber
\\
& & \hspace{5mm}
\cdot \left(\int_{S^1} \chi_{\{(z_1, \ldots , z_d) \in T^d \mid \; |p(z_1,\ldots, z_d)| \le \lambda\}} \; d\mu_{S^1} \right) \;d\mu_{T^{d-1}}
\nonumber
\\
& & \hspace{10mm}
+
\int_{T^{d-1} } \chi_{\{(z_1, \ldots, z_{d-1}) \in T^{d-1} \mid\; |p_1(z_1, \ldots, z_{d-1}) > |\operatorname{lead}(p)|^{1/d} \cdot\lambda^{(d-1))/d}\}}
\nonumber
\\
& & \hspace{15mm}
\cdot \left(\int_{S^1} \chi_{\{(z_1, \ldots , z_d) \in T^d \mid \; |p(z_1,\ldots, z_d)| \le \lambda\}} \; d\mu_{S^1} \right) \;d\mu_{T^{d-1}}
\nonumber
\\
& \le &
\int_{T^{d-1} } \chi_{(z_1, \ldots, z_{d-1}) \mid\; |p_1(z_1, \ldots, z_{d-1})| \le |\operatorname{lead}(p)|^{1/d} \cdot \lambda^{(d-1)1/d}\}}
+
\nonumber
\\
& & \hspace{5mm}
\max\left. \biggl\{\int_{S^1} \chi_{\{(z_1, \ldots , z_d) \in T^d \mid \; |p(z_1,\ldots, z_d)| \le \lambda\}} \; d\mu_{S^1} \right| (z_1, \ldots, z_{d-1} ) \in T^{d-1}
\nonumber
\\
& & \hspace{30mm}
\; \text{with} \; |p_1(z_1, \ldots, z_{d-1})| > |\operatorname{lead}(p)|^{1/d} \cdot\lambda^{(d-1)/d}\biggr\}.
\nonumber
\end{eqnarray}
We get from the induction hypothesis applied to $p_1(z_1, \ldots, z_{d-1})$
and~\eqref{F(P)_in_terms_of_volume}
since $\frac{\lambda}{|\operatorname{lead}(p)|} \le 1$, $\operatorname{wd}(p_1) \le \operatorname{wd}(p)$ and $\operatorname{lead}(p) = \operatorname{lead}(p_1)$
\begin{eqnarray}
& &
\label{F(p)(lambda_estimate:2}
\\
\lefteqn{\int_{T^{d-1} } \chi_{(z_1, \ldots, z_{d-1}) \mid\; |p_1(z_1, \ldots, z_{d-1})| \le |\operatorname{lead}(p)|^{1/d}\cdot \lambda^{(d-1)1/d}\}}}
& &
\nonumber
\\
& = &
\int_{T^{d-1} } \chi_{(z_1, \ldots, z_{d-1}) \mid\; |p_1(z_1, \ldots, z_{d-1})| \le |\operatorname{lead}(p_1)|^{1/d}\cdot \lambda^{(d-1)1/d}\}}
\nonumber
\\
& = &
F\bigl(r_{p_1}^{(2)}\bigr)\bigl(|\operatorname{lead}(p_1)|^{1/d}| \cdot \lambda^{(d-1)/d}\bigr)
\nonumber \\
& \le &
\frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot (d-1) \cdot \operatorname{wd}(p_1)
\cdot \left(\frac{|\operatorname{lead}(p_1)|^{1/d} \cdot \lambda^{(d-1)/d}}{|\operatorname{lead}(p_1)|}\right)^{\frac{1}{(d-1) \cdot \operatorname{wd}(p_1)}}
\nonumber
\\
& = &
\frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot (d-1) \cdot \operatorname{wd}(p_1)
\cdot \left(\frac{\lambda}{|\operatorname{lead}(p_1)|}\right)^{\frac{1}{d \cdot \operatorname{wd}(p_1)}}
\nonumber
\\
& = &
\frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot (d-1) \cdot \operatorname{wd}(p_1)
\cdot \left(\frac{\lambda}{|\operatorname{lead}(p)|}\right)^{\frac{1}{d \cdot \operatorname{wd}(p_1)}}
\nonumber
\\
& \le &
\frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot (d-1) \cdot \operatorname{wd}(p)
\cdot \left(\frac{\lambda}{|\operatorname{lead}(p)|}\right)^{\frac{1}{d \cdot \operatorname{wd}(p_1)}}
\nonumber
\\
& \le &
\frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot (d-1) \cdot \operatorname{wd}(p)
\cdot \left(\frac{\lambda}{|\operatorname{lead}(p)|}\right)^{\frac{1}{d \cdot \operatorname{wd}(p)}}.
\nonumber
\end{eqnarray}
Fix $(z_1, \ldots, z_{d-1} ) \in T^{d-1}$ with $|p_1(z_1, \ldots, z_{d-1})| > \operatorname{lead}(p)^{1/d} \cdot \lambda^{(d-1)/d}$.
Consider the element $f(z_d^{\pm 1}) := p(z_1, \ldots z_{d-1},z_d^{\pm}) \in {\mathbb C}[z_d^{\pm}]$. It has the shape
\[
f(z_d^{\pm})
= \sum_{n = n^-}^{n^+} q_n(z_1, \ldots, z_{d-1}) \cdot z_d^n.
\]
The leading coefficient of $f(z_d^{\pm 1})$ is $p_1(z_1, \ldots z_{d-1}) = q_{n_+}(z_1, \ldots, z_{d-1}) $.
Hence we get from Lemma~\ref{pro:case_m_is_n_is_1_in_degree_one}
applied to $f(z_d^{\pm 1})$ and~\eqref{F(P)_in_terms_of_volume}
since $\frac{\lambda}{|\operatorname{lead}(p)|} \le 1$, $\operatorname{wd}(f) \le \operatorname{wd}(p)$ and
$|\operatorname{lead}(f)| = |p_1(z_1, \ldots z_{d-1}))| > |\operatorname{lead}(p)|^{1/d} \cdot \lambda^{(d-1)/d}$
\begin{eqnarray}
& &
\label{F(p)(lambda_estimate:3}
\\
\lefteqn{\int_{S^1} \chi_{\{(z_1, \ldots , z_d) \in T^d \mid \; |p(z_1,\ldots, z_d)| \le \lambda\}} \; d\mu_{S^1}}
& &
\nonumber
\\
& = &
\int_{S^1} \chi_{\{z_d \in S^1 \mid \; |f(z_d)| \le \lambda\}} \; d\mu_{S^1}
\nonumber
\\
& = &
\frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot \operatorname{wd}(f)
\cdot \left(\frac{\lambda}{\operatorname{lead}(f)}\right)^{\frac{1}{\operatorname{wd}(f)}}
\nonumber
\\
& \le &
\frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot \operatorname{wd}(f)
\cdot \left(\frac{\lambda}{\operatorname{lead}(p)^{1/d} \cdot \lambda^{(d-1)/d}}\right)^{\frac{1}{\operatorname{wd}(f)}}
\nonumber
\\
& = &
\frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot \operatorname{wd}(f)
\cdot \left(\frac{\lambda}{\operatorname{lead}(p)}\right)^{\frac{1}{d \cdot \operatorname{wd}(f)}}
\nonumber
\\
& \le &
\frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot \operatorname{wd}(p)
\cdot \left(\frac{\lambda}{\operatorname{lead}(p)}\right)^{\frac{1}{d \cdot \operatorname{wd}(p)}}.
\nonumber
\end{eqnarray}
Combining~\eqref{F(p)(lambda_estimate:1},~\eqref{F(p)(lambda_estimate:2} and~\eqref{F(p)(lambda_estimate:3}
yields for $\lambda$ with $\frac{\lambda}{|\operatorname{lead}(p)|} \le 1$
\begin{eqnarray*}
F\bigl(r_p^{(2)}\bigr)(\lambda)
& \le &
\frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot (d-1) \cdot \operatorname{wd}(p)
\cdot \left(\frac{\lambda}{|\operatorname{lead}(p)|}\right)^{\frac{1}{d \cdot \operatorname{wd}(p)}}
\\
& & \hspace{40mm}+
\frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot \operatorname{wd}(p)
\cdot \left(\frac{\lambda}{|\operatorname{lead}(p)|}\right)^{\frac{1}{d \cdot \operatorname{wd}(p)}}
\\
& = &
\frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot d \cdot \operatorname{wd}(p)
\cdot \left(\frac{\lambda}{|\operatorname{lead}(p)|}\right)^{\frac{1}{d \cdot \operatorname{wd}(p)}}.
\end{eqnarray*}
This finishes the proof of Proposition~\ref{pro:case_m_is_n_is_1}.
\typeout{--------------- Section 2: Proof of the main Theorem ---------------------}
\section{Proof of the main Theorem~\ref{the:Main_Theorem}}
\label{sec:Proof_of_Main_Theorem}
Now we can complete the proof of our Main Theorem~\ref{the:Main_Theorem}.
We need the following preliminary result
\begin{lemma}\label{lem:preparation}
Consider $B \in M_{k,k}({\mathbb C}[{\mathbb Z}^d])$ such that $p:= \det_{{\mathbb C}[{\mathbb Z}^d]}(B)$ is non-trivial.
Then we get for all $\lambda \ge 0$
\[
F\bigl(r_B^{(2)}\bigr)(\lambda) \le k \cdot F\bigl(r_p^{(2)}\bigr)\bigl(||r_B^{(2)}||^{k-1} \cdot \lambda\bigr).
\]
\end{lemma}
\begin{proof}
In the sequel we will identify $L^2({\mathbb Z}^d)$ and $L^2(T^d)$ by the Fourier transformation.
We can choose a unitary ${\mathbb Z}^d$-equivariant operator $U \colon L^2({\mathbb Z}^d)^k \to L^2({\mathbb Z}^d)^k$
and functions $f_1, f_2, \ldots, f_k \colon T^d\to {\mathbb R}$
such that $0 \le f_1(z) \le f_2(z) \le \ldots \le f_k(z) $ holds for all $z \in T^d$ and
we have the following equality of bounded ${\mathbb Z}^d$-equivariant operators
$L^2({\mathbb Z}^d)^k = L^2(T^d)^k \to L^2({\mathbb Z}^d)^k= L^2(T^d)^k$, see~\cite[Lemma~2.2]{Lueck-Roerdam(1993)}
\begin{eqnarray}
(r_B^{(2)})^* \circ r_B^{(2)} = U \circ
\begin{pmatrix}
r_{f_1}^{(2)} & 0 & 0 & \cdots & 0 & 0
\\
0 & r_{f_2}^{(2)} & 0 & \cdots & 0 & 0
\\
0 & 0 & r_{f_3}^{(2)} & \cdots & 0 & 0
\\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots
\\
0 & 0 & 0 & \cdots & r_{f_{k-1}}^{(2)} & 0
\\
0 & 0 & 0 & \cdots & 0 & r_{f_k}^{(2)}
\end{pmatrix} \circ U^*.
\label{diagonalization}
\end{eqnarray}
Since $p \not= 0$ holds by assumption and hence the rank of $B$ over ${\mathbb C}[{\mathbb Z}^d]^{(0)}$ is maximal,
we conclude from~\cite[Lemma~1.34 on page~35]{Lueck(2002)}
that $r_B^{(2)}$ and hence $r_{f_i}^{(2)}$ for each $i = 1,2, \ldots, k$
are weak isomorphisms, i.e., they are injective and have dense images. We conclude
from~\cite[Lemma~2.11~(11) on page~77 and Lemma~2.13 on page~78]{Lueck(2002)}
\[
F(r_B^{(2)})(\lambda)
=
F\left((r_B^{(2)})^* \circ r_B^{(2)} \right)(\lambda^2)
=
\sum_{i = 1}^k F(r_{f_i}^{(2)})(\lambda^2).
\]
For $i = 1,2, \ldots, k$ we have $f_1(z) \le f_i(z)$ for all $z \in T^d$ and hence
$F\bigl(r_{f_i}^{(2)}\bigr)(\lambda) \le F\bigl(r_{f_1}^{(2)}\bigr)(\lambda)$ for all $\lambda \ge 0$. This implies
\begin{eqnarray}
F\bigl(r_B^{(2)}\bigr)(\lambda) \le k \cdot F\bigl(r_{f_1}^{(2)})(\lambda^2).
\label{estimate_for_spectral_density_after_diagonalization}
\end{eqnarray}
Let $B^* \in M_{k,k}({\mathbb C}[{\mathbb Z}^d)$ be the matrix obtain from $B$ by transposition and applying to each entry the involution
${\mathbb C}[{\mathbb Z}^d] \to {\mathbb C}[{\mathbb Z}^d|$ sending $\sum_{g \in G} \lambda_g \cdot g$ to $\sum_{g \in G}\overline{\lambda_g} \cdot g^{-1}$.
Then $\bigl(r_B^{(2)}\bigr)^* = r_{B^*}^{(2)}$.
Since $(r_B^{(2)})^* \circ r_B^{(2)} = r_{BB^*}^{(2)}$ and $\det_{{\mathbb C}[{\mathbb Z}^d]}(BB^*) = \det_{{\mathbb C}[{\mathbb Z}^d]}(B) \cdot \det_{{\mathbb C}[{\mathbb Z}^d]}(B^*)
= p \cdot p^*$ holds, we conclude from~\eqref{diagonalization} the equality of functions $T^d \to [0,\infty]$
\[
pp^* = \prod_{i = 1}^k f_i.
\]
Since $\sup\{|f_i(z)| \mid z \in T^d\}$ agrees with the operatornorm $||r_{f_i}^{(2}||$
and we have $||r_B^{(2)}||^2 = ||(r_B^{(2)})^* r_B^{(2)}|| = \max\bigl\{||r_{f_i}^{(2)}|| \; \bigl| \; i = 1,2, \ldots, k\} = ||r_{f_k}^{(2)}||$,
we obtain the inequality of functions $T^d \to [0,\infty]$
\[
pp^* \le \left(\prod_{i = 2}^k ||r_{f_i}^{(2)}|| \right) \cdot f_1 \le \bigl(||r_B^{(2)}||^2\bigr)^{k-1} \cdot f_1.
\]
Hence we get for all $\lambda \ge 0$
\begin{eqnarray*}
F\bigl(r_{pp^*}^{(2)}\bigr)\left(\bigl(||r_B^{(2)}||^{k-1} \lambda\bigr)^2\right)
& = &
F\bigl(r_{pp^*}^{(2)}\bigr)\left(||r_B^{(2)}||^2\bigr)^{k-1} \lambda^2\right)
\\
& \ge &
F\left((||r_B^{(2)}||^2\bigr)^{k-1} \cdot r_{f_1}^{(2)}\right)\left(||r_B^{(2)}||^2\bigr)^{k-1} \cdot \lambda^2\right)
\\
& = &
F\bigl(r_{f_1}^{(2)})(\lambda^2).
\end{eqnarray*}
This together with~\eqref{estimate_for_spectral_density_after_diagonalization}
and~\cite[Lemma~2.11~(11) on page~77]{Lueck(2002)} implies
\begin{eqnarray*}
F\bigl(r_B^{(2)}\bigr)(\lambda)
&\le &
k \cdot F\bigl(r_{f_1}^{(2)})(\lambda^2)
\\
& \le &
k \cdot F\bigl(r_{pp^*}^{(2)}\bigr)\left(\bigl(||r_B^{(2)}||^{k-1} \lambda\bigr)^2\right)
\\
& \le &
k \cdot F\bigl(r_{p}^{(2)}\bigr)\bigl(||r_B^{(2)}||^{k-1} \lambda\bigr).
\end{eqnarray*}
\end{proof}
\begin{proof}[Proof of the Main Theorem~\ref{the:Main_Theorem}]
\eqref{the:Main_Theorem:spectral_density_estimate}
In the sequel we denote by $\dim_{{\mathcal N}(G)}$ the von Neumann dimension, see for
instance~\cite[Subsection~1.1.3]{Lueck(2002)}.
The rank of the matrices $A$ and $B$ over the quotient field ${\mathbb C}[{\mathbb Z}^d]^{(0)}$ is $k$.
The operator $r_{B}^{(2)} \colon L^2({\mathbb Z}^d)^k \to L^2({\mathbb Z}^d)^k$ is a weak isomorphism,
and $\dim_{{\mathcal N}({\mathbb Z}^d)}(\overline{\operatorname{im}(r_A^{(2)}))} = k$
because of~\cite[Lemma~1.34~(1) on page~35]{Lueck(2002)}.
In particular we have $F\bigl(r_{B}^{(2)}\bigr)(0) = 0$.
Let $i^{(2)} \colon L^2({\mathbb Z}^d)^k \to L^2({\mathbb Z}^d)^m $ be the inclusion
corresponding to $I \subseteq \{1,2, \ldots, m\}$ and let
$\operatorname{pr}^{(2)} \colon L^2({\mathbb Z}^d)^n \to L^2({\mathbb Z}^d)^k$ be
the projection corresponding to $J \subseteq \{1,2, \ldots, n\}$,
where $I$ and $J$ are the subsets specifying the submatrix $B$.
Then $r_{B}^{(2)} \colon L^2({\mathbb Z}^d)^k \to L^2({\mathbb Z}^d)^k$
agrees with the composite
\[
r_{B}^{(2)} \colon L^2({\mathbb Z}^d)^k
\xrightarrow{i^{(2)}} L^2({\mathbb Z}^d)^m
\xrightarrow{r_A^{(2)}} L^2({\mathbb Z}^d)^n
\xrightarrow{\operatorname{pr}^{(2)}}L^2({\mathbb Z}^d)^k.
\]
Let $p^{(2)} \colon L^2({\mathbb Z}^d)^m \to \ker(r_A^{(2)}))^{\perp}$ be the orthogonal projection
onto the orthogonal complement $\ker(r_A^{(2)})^{\perp} \subseteq L^2(G)^m$
of the kernel of $r_A^{(2)}$.
Let $j^{(2)} \colon \overline{\operatorname{im}(r_A^{(2)})} \to L^2(G)^n$
be the inclusion of the closure of the image of $r_A^{(2)}$.
Let $(r_A^{(2)})^{\perp} \colon \ker(r_A^{(2)})^{\perp} \to \overline{\operatorname{im}(r_A^{(2)})}$ be the
${\mathbb Z}^d$-equivariant bounded operator uniquely determined by
\begin{eqnarray*}
r_A^{(2)}
& = &
j^{(2)} \circ (r_A^{(2)})^{\perp} \circ p^{(2)}.
\end{eqnarray*}
The operator $(r_A^{(2)})^{\perp} $
is a weak isomorphism by construction. We have the decomposition of the weak isomorphism
\begin{eqnarray}
& & r_{B}^{(2)} = \operatorname{pr}^{(2)} \circ \; r_A^{(2)} \circ i^{(2)}
=
\operatorname{pr}^{(2)} \circ j^{(2)} \circ (r_A^{(2)})^{\perp} \circ p^{(2)} \circ i^{(2)}.
\label{lem:Estimate_in_terms_of_minors_decomposition}
\end{eqnarray}
This implies that the morphism
$p^{(2)} \circ i^{(2)} \colon L^2({\mathbb Z}^d)^k)\to \ker(r_A^{(2)})^{\perp}$
is injective and the morphism
$\operatorname{pr}^{(2)} \circ j^{(2)} \colon \overline{\operatorname{im}(r_A^{(2)})} \to L^2({\mathbb Z}^d)^k$
has dense image. Since we already know
$\dim_{{\mathcal N}(G)}\bigl(\overline{\operatorname{im}(r_A^{(2)})}\bigr) = k = \dim_{{\mathcal N}(G)}\bigl(L^2({\mathbb Z}^d)^k\bigr)$,
the operators $p^{(2)} \circ i^{(2)} \colon L^2({\mathbb Z}^d)^k\to \ker(r_A^{(2)})^{\perp}$
and $\operatorname{pr}^{(2)} \circ j^{(2)} \colon \overline{\operatorname{im}(r_A^{(2)})} \to L^2({\mathbb Z}^d)$
are weak isomorphisms. Since the operatornorm of $ \operatorname{pr}^{(2)} \circ j^{(2)}$
and of $p^{(2)} \circ i^{(2)}$ is less or equal to $1$,
we conclude from~\cite[Lemma~2.13 on page~78]{Lueck(2002)}
and~\eqref{lem:Estimate_in_terms_of_minors_decomposition}
\begin{eqnarray*}
\lefteqn{F\bigl(r_A^{(2)}\bigr)(\lambda) - F\bigl(r_A^{(2)}\bigr)(0)}
& &
\\
& = &
F\bigl((r_A^{(2)})^{\perp}\bigr)(\lambda)
\\
& \le &
F\bigl(\operatorname{pr}^{(2)} \circ j^{(2)} \circ (r_A^{(2)})^{\perp} \circ p^{(2)} \circ i^{(2)}\bigr)
\bigl(||\operatorname{pr}^{(2)} \circ j^{(2)}|| \cdot ||p^{(2)} \circ i^{(2)}|| \cdot \lambda\bigr)
\\
& = &
F\bigl(r_{B}^{(2)}\bigr)\bigl(||\operatorname{pr}^{(2)} \circ j^{(2)}|| \cdot ||p^{(2)} \circ i^{(2)}|| \cdot \lambda\bigr)
\\
& \le &
F\bigl(r_{B}^{(2)}\bigr)(\lambda).
\end{eqnarray*}
Put $p = \det_{{\mathbb C}[{\mathbb Z}^d]}(B)$.
If $\operatorname{wd}(p) = 0$, the claim follows directly from
Proposition~\ref{pro:case_m_is_n_is_1}.
It remains to treat the case $\operatorname{wd}(p) \ge 1$.
The last inequality together with~\eqref{norm_estimate_by_L1-norm} applied to $B$,
Proposition~\ref{pro:case_m_is_n_is_1} applied to $p$ and Lemma~\ref{lem:preparation} applied to $B$
yields for $\lambda \ge 0$
\begin{eqnarray*}
\lefteqn{F\bigl(r_A^{(2)}\bigr)(\lambda) - F\bigl(r_A^{(2)}\bigr)(0)}
& &
\\
& \le &
F\bigl(r_{B}^{(2)}\bigr)(\lambda)
\\
&\le &
k \cdot F\bigl(r_p^{(2)}\bigr)\bigl(||r_{B}^{(2)}||^{k-1} \cdot \lambda)
\\
&\le &
k \cdot F\bigl(r_p^{(2)}\bigr)\bigl((k^2 \cdot ||B||_1)^{k-1} \cdot \lambda)
\\
& \le &
\frac{8 \cdot \sqrt{3}}{\sqrt{47}} \cdot k \cdot d \cdot \operatorname{wd}(p)
\cdot \left(\frac{k^{2k -2} \cdot (||B||_1)^{k-1} \cdot \lambda}{|\operatorname{lead}(p)|}\right)^{\frac{1}{d \cdot \operatorname{wd}(p)}}.
\end{eqnarray*}
This finishes the proof of assertion~\eqref{the:Main_Theorem:spectral_density_estimate}.
Assertion~\eqref{the:Main_Theorem:Novikov-Shubin} is a direct consequence of
assertion~\eqref{the:Main_Theorem:spectral_density_estimate} and the definition of the Novikov-Shubin invariant.
This finishes the proof of Theorem~\ref{the:Main_Theorem}.
\end{proof}
\typeout{--------------- Section 3: Example ---------------------}
\typeout{-------------------------------------- References ---------------------------------------}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,089 |
Home » Information » What Are The Japanese Yokai Spirits?
What Are The Japanese Yokai Spirits?
Yordan Zhelyazkov
What are Yokai?
Symbolism of the Yokai
Importance of Yokai in Modern Culture
FAQs About Yokai
Together with kami (gods), yokai is one of the words most often used when talking about Japanese Shinto mythology. The yokai are spirits or ghosts and a type of supernatural monsters in Japanese mythology. However, they're more complex than the general view of ghosts and spirits.
The Tengu Is A Type Of Yokai
Yokai in Japanese Shintoism are most types of supernatural animalistic spirits or phenomena. Yokai can be supernatural animals, demons, animated objects, ghosts, mutated or cursed people, and even some minor kami and demi-gods.
The word yokai comes from:
yō, meaning attractive, bewitching, calamity
kai meaning mystery, wonder
Put together, the idea is of an attractive yet calamitious mystery. And that's the best way to describe yokai in English as there's no better direct translation.
Simply calling yokai spirits doesn't really cut it as many yokai are not spirits. Calling them "supernatural beings" is probably closer to the truth but there are supernatural beings in Shintoism that aren't yokai.
So, it's best to just use the word yokai in English as it is, as there isn't a direct translation.
Yokai vs. Spirits
Not all yokai are spirits, however, and not all spirits are yokai either. In traditional Japanese mythology, all things are believed to be possessed by spirits – people, animals, plants, and even inanimate objects. Even natural phenomena such as rain and earthquakes are said to be possessed by spirits.
Spirits: Spirits have both personalities and emotions but they are usually peaceful and don't interact with the physical world, outside of simply possessing whatever it is they are possessing. Usually called nigi-mitama (when they are "good") or ara-mitama (when they are "bad"), these spirits are not yokai. Instead, they are their own thing.
Yokai: Yokai spirits are separate beings that can operate in the real world without possessing anyone or anything. Many of them can forcefully possess people or animals but that's not their "natural" form. Yokai, whether spirits, demons, goblins, or ghosts, are self-contained beings that can roam around on their own.
Are Yokai Good or Evil?
Portrait of a Japanese Oni Demon – A Type of Yokai
Yokai can be either good or evil depending on the specific yokai in question, but they are usually morally ambiguous.
Some yokai are almost always good and are even dedicated to the service of a particular kami. Such is the case with the famous zenko kitsune (fox-like) yokai – the famous nine-tailed foxes that serve the kami Inari.
Other yokai, however, like the flying demons, called Tengu, are usually evil and are only rarely viewed as morally gray.
Types of Yokai
There are many different ways to categorize yokai and none is "more true" than the others. Because yokai are such an unclear type of beings, many scholars can't even agree on what exactly yokai are and which beings are yokai or which aren't.
For example, some people call minor kami gods yokai while others insist that they are strictly kami. Another point of contention are the henge beings – changed people/things or mutants. Some view those as a type of yokai while others believe they are their own category.
Most people nowadays seem to view yokai as broadly as possible – with all supernatural Shinto creatures and spirits (aside from the kami gods) seen as different types of yokai.
Generally, however, most yokai are divided into the following categories:
Supernatural animal spirits
Supernatural plant spirits
Cursed or mutated people, animals, or objects
Reincarnated or afterworld spirits
Demon-like or goblin-like evil spirits
Yokai symbolize everything supernatural in Japanese mythology. Most come from Shintoism but many are also influenced by Japanese Buddhism, by Chinese Taoism, or even by Hinduism.
Depending on the yokai in question, these beings can symbolize anything from cataclysmic natural events to simple, everyday animal quirks. Together, however, yokai symbolize the magical mystery of the world around us – everything we can't yet explain and everything our imaginations can conjure.
It's fair to say that yokai are living a "second life" in Japanese culture right now.
For countless centuries yokai used to be viewed as invisible and unseen spirits. When paintings and illustrative arts became popular in Japan, however, yokai started gaining visual representations. They truly rose to prominence during the Edo period (1603-1868) when portraying them in art became much easier and more captivating. Unfortunately, began to be ignored during the Meiji period of modernization, when they began to be viewed as outdated and silly superstitions.
After World War II, however, Japan rediscovered its roots, and yokai, kami, and other beautiful myths started captivating people's imaginations again. Manga artist Shigeru Mizuki was instrumental for that switch thanks to his now-legendary comic series GeGeGe no Kitaro.
Today, yokai can be found in every other Japanese manga, anime, or video game. Hayao Miyazaki's movies are especially famous for their gorgeous and imaginative yokai spirits but so are other works of art such as Mushishi, Kamisama Kiss, Hakkenden – Eight Dogs of the East, Zakuro, The Morose Mononokean, and many more.
Are yokai demons?
Yokai are supernatural monsters and creatures with diverse habits, behaviors and appearances.
Are yokai evil?
Depending on the type of yokai, it can be evil and dangerous while others can be benevolent and bringers of good fortune.
What are some famous yokai?
Some popular yokai include tengu, oni, kitsune, obake and kappa.
The yokai are among the most distinctly Japanese creations, populating the myths and bringing life to them. While there's no generally accepted consensus on what exactly the yokai are, the overview is that they're seen as supernatural, mystical entities that are either mischievous or benevolent, with each type of yokai having its own characteristics.
Tags: Japanese mythology
Yordan Zhelyazkov is a published fantasy author and an experienced copywriter. While he has degrees in both Creative Writing and Marketing, much of his research and work are focused on history and mythology. He's been working in the field for years and has amassed a great deal of knowledge on Norse, Greek, Egyptian, Mesoamerican, Japanese mythology, and others. | {
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But what exactly is it that makes a business want to hire commercial photographers? Here's everything you need to know about what businesses want when they hire photographers.
Portfolios are the best way to showcase freelance work. A commercial photographer should put together nothing but the best of their previous work into a portfolio to show potential employing businesses how they've been successful. Without proof of talent, skill and ability to fulfill the project at hand, a business won't even consider hiring that photographer. There must be a mutually beneficial relationship between both parties. The business needs the photographer, and vice versa.
A business will seek nothing but top talent when it comes to the photographer they will hire. The photographer is primarily responsible for creating, shooting and delivering material that essentially will represent the business. The photographer needs to be able to adapt to requirements of the job.
Maybe the business has rules for what can be photographed. Maybe the business has a strict timeline and budget. A talented photographer will work as a professional to ensure the very best for the business.
In order to create material that will represent a business, the business looks for freelance photographers that can meet their criteria. This can take dedication on the photographer's part. When it comes down to it, a business has a goal they need to meet, and they are hiring someone that can help them get to that goal.
As a freelance photographer, you should answer honestly to yourself whether or not you can help a business achieve their goal. What are you skilled at? What types of businesses have you successfully worked with before? Because the business has a specific goal in mind, you need to make sure you're the right fit. If they're hiring someone to take pictures of their building, and you photograph portraits, then you might not be the right fit.
Meanwhile, if a business wants someone to take pictures of their new cupcake display and you specialize in photographing food, consider yourself the one for the job.
The same goes for businesses when they're in the process of looking for a photographer to hire. If a company is hiring someone to photograph personalized business cards for its employees, it's going to need someone who specializes in head shots. There are essential questions the hiring manager of a business should ask in order to find the right photographer.
Choosing the right photographer can potentially make or break a business, especially if it's a startup. Branding is what makes a business just that — a business. Branding is a way businesses establish themselves, their products and their media.
There are specific ways a business can ensure it's not making any mistakes when it comes to its brand. Brand guidelines are something that can't be overstepped when a commercial photographer comes in to do work. A visual style guide might be in place, and the photographer absolutely must be able to meet those guidelines for the business.
The quality of images businesses look for are ones that will make an impact and reach their target audience. Then, a business's target audience becomes the photographer's target audience. Being able to capture images that reach the audience is exactly what the business needs from the photographer they are going to hire. A business wants to know that they are hiring someone who will bring in more customers, sales, awareness or whatever else it is that will improve their bottom line.
The quality of an image has been proven to affect sales. When businesses sell their products online, the image can make consumers want to buy the product. Online shoppers say that a picture really does say a thousand words, as 67% of consumers note the importance of image quality in showing them the product they're purchasing.
A business wants a photographer that can give them images that provide results. Whether the image increases sales, boosts subscribers or does something else entirely, the images photographers create for businesses have to be of the highest quality.
Being able to afford a commercial photographer is a huge consideration for a business. Professional photographers may already have pricing set for the freelance work they do. On the other hand, businesses can sometimes have a budget that sometimes doesn't line up with what the photographer would charge.
A business might have an offer for the photographer, which is up to the photographer to accept. Either way, it's vital for the business to stay under budget.
The outcome of a photo shoot proves whether or not the professional relationship between a business and photographer was beneficial. If it was, it means that photographer produced images for the business that it will be able to use with successful results.
A professional collaboration between a commercial photographer and a business to create images and media that consumers want to see isn't always a fast and easy thing. Businesses and commercial photographers must be able to work together for a successful and mutually beneficial outcome. | {
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Skara Brae é um assentamento neolítico, situado na Baía de Skaill, a maior das ilhas Órcades. É composto por dez casas agrupadas que foram habitadas aproximadamente entre 3180 a 2500 AEC. Skara Brae é a aldeia neolítica mais completa e bem preservada do norte da Europa, tendo sido eleita como património mundial pela UNESCO, como um dos locais que constituem o "Coração Neolítico das Órcades". Mais antiga do que Stonehenge ou a Grande Pirâmide de Giza, este povoado tem sido denominado de "Scottish Pompeii" (Pompeia escocesa), devido ao seu excelente estado de preservação.
Localização Geográfica
Skara Brae situa-se na ilha principal de Orkney na ponta norte da Escócia. Situadas no Mar do Norte o clima destas ilhas é muito severo tendo por isso uma paisagem desolada com poucas árvores.
História
A povoação de Skara Brae foi construída na ilha principal de Orkney, a população neolítica utilizou uma alternativa à rara madeira da área para construir a povoação: Placas de pedra. Isto permitiu um vislumbre sobre o que seria uma casa neolítica.
No centro da ilha temos uma série de depressões criadas por um monte de lixo, que foram aproveitadas para a construção de oito casas, todas feitas de pedra, com um comprimento entre os 4 e os 6 metros numa área rectangular com um forno central.
Tendo em conta a falta de madeira julga-se que os telhados fossem feitos com as costelas de baleias, cobertas com peles de animais e depois colocada vegetação em cima para uma melhor protecção contra o clima severo.
Nas casas as lajes de pedra serviam de mobília tanto para criar prateleiras, arcas e camas que depois eram cheias com vegetação e pele de animais.
Esta povoação foi abandonada por volta de 2500 a.c. a mudança de clima tornando-se mais frio e úmido terá sido a razão do abandono.
Foi em 1850 exposta por uma tempestade, sendo posteriormente escavada por Vere Gordon Childe. Até hoje continua a ser alvo de estudos por parte de arqueólogos e historiadores.Atingiu em 1999 juntamente com outros locais neolíticos de Orkney o estatuto de Património da Humanidade da UNESCO
Sociedade
A sociedade de Skara Brae estava ligada a diversas actividades, pesca, agricultura e caça numa pequena escala. Os artefactos encontrados nas escavações efectuadas atestam estas actividades com ferramentas em osso e marfim, potes com entalhe, contas feitas de conchas e pedras e esculturas feitas em pedra com motivos geométricos.
A povoação de Skara Brae foi construída na ilha principal de Orkney, a população neolítica utilizou uma alternativa á rara madeira da área para construir a povoação: Placas de pedra. Isto permitiu um vislumbre sobre o que seria uma casa neolítica. No centro da ilha temos uma série de depressões criadas por um monte de lixo, que foram aproveitadas para a construção de oito casas, todas feitas de pedra, com um comprimento entre os 4 e os 6 metros numa área rectangular com um forno central. Tendo em conta a falta de madeira julga-se que os telhados fossem feitos com as costelas de baleias, cobertas com peles de animais e depois colocada vegetação em cima para uma melhor protecção contra o clima severo. Nas casas as lajes de pedra serviam de mobília tanto para criar prateleiras, arcas e camas que depois eram cheias com vegetação e pele de animais. Esta povoação foi abandonada por volta de 2500 a.c. a mudança de clima tornando-se mais frio e úmido terá sido a razão do abandono. Foi em 1850 exposta por uma tempestade, sendo posteriormente escavada por Vere Gordon Childe. Até hoje continua a ser alvo de estudos por parte de arqueólogos e historiadores.Atingiu em 1999 juntamente com outros locais neolíticos de Orkney o estatuto de Património da Humanidade da UNESCO.
Ver também
Círculo de Brodgar
Bibliografia
História da Escócia
Arqueologia da Escócia
Património Mundial da UNESCO na Escócia | {
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The Lal Bahadur Shastri National Award is an annual prestigious award instituted from Lal Bahadur Shastri Institute of Management, Delhi. This consists of a cash award of 5,00,000 rupees plus a citation and a plaque.
Background
The award was started in 1999 and is provided to a business leader, management practitioner, public administrator, educator or institution builder for his/her sustained individual contributions for achievements of high professional order and excellence. This award is given by the President of India.
Awardees
Award gallery
See also
Lal Bahadur Shastri Institute of Management
References
External links
LBS National Award: Excellence in Public Administration, Academics, and Management
Civil awards and decorations of India
Memorials to Lal Bahadur Shastri | {
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Bozoum Airport är en flygplats i Centralafrikanska republiken. Den ligger i prefekturen Préfecture de l'Ouham-Pendé, i den västra delen av landet, km nordväst om huvudstaden Bangui. Bozoum Airport ligger meter över havet.
Terrängen runt Bozoum Airport är platt. Den högsta punkten i närheten är Gougoun, meter över havet, km väster om Bozoum Airport. Närmaste större samhälle är Bozoum, km öster om Bozoum Airport.
I omgivningarna runt Bozoum Airport växer huvudsakligen savannskog. Runt Bozoum Airport är det mycket glesbefolkat, med invånare per kvadratkilometer. Savannklimat råder i trakten. Årsmedeltemperaturen i trakten är °C. Den varmaste månaden är mars, då medeltemperaturen är °C, och den kallaste är juli, med °C. Genomsnittlig årsnederbörd är millimeter. Den regnigaste månaden är september, med i genomsnitt mm nederbörd, och den torraste är december, med mm nederbörd.
Kommentarer
Källor
Flygplatser i Préfecture de l'Ouham-Pendé | {
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\section{Introduction}
\label{sec:1}
It is fitting, in this conference in memory of N. Voglis, to recall that
I became interested in the investigation of regular and chaotic motions
in elliptical galaxies thanks to a paper of his \cite{VKS02}. By that time,
I had been working on N--body problems for two decades, and on regular
and chaotic motion for seven or eight years, but I had never been involved
in research on elliptical galaxies. The paper by Voglis and his coworkers
showed me that, with the computers and the numerical tools I had at my
disposal, I might be able to contribute significantly to a very interesting
subject and, in fact, I have been devoted to that subject ever since.
Having worked in this field for a few years only, it would be presumtuous
from my part to attempt to present here a complete review of the subject.
Alternatively, to be a relative newcomer to the field has the advantage of
bringing to it views and opinions different from the prevailing ones: they
may be wrong, but they stimulate progress.
Therefore, I will limit the scope of this review to a few items that have
been of particular interest to me and which I have strived to clarify with
my research: 1) Can we build stable triaxial models of stellar systems
that contain high fractions of chaotic orbits?; 2) Is the distinction
between partially and fully chaotic orbits of any use?; 3) Is figure
rotation significant in triaxial stellar systems?; 4) Which are the
usefulness and limitations of frequency analysis for the classification of
large numbers of regular orbits in model stellar systems?
Since galactic dynamics is not the only subject of this conference, which
includes other fields like celestial mechanics, it may be useful to recall
that the time scales pertinent to galaxies are completely different from
those that rule the Solar System. While the age of the latter is of the
order of $10^8$ orbital periods, galactic ages are of the order of $10^2$
orbital periods only. Thus, the chaotic orbits we will refer here are much
more strongly chaotic (i.e., their Lyapunov times measured in orbital periods
are much shorter) than those of the Solar System. Technical tools, such as
frequency analysis, should also be considered with this fact in mind.
\section{Highly chaotic triaxial stellar systems}
\label{sec:2}
\subsection{Building self--consistent triaxial stellar systems}
\label{sec:3}
A popular method to build a self--consistent triaxial stellar system is the one
due to Schwarzschild \cite{Sch79}. One chooses a density distribution and
obtains the potential that it creates; a library of thousands of orbits is
then computed in that potential and weights are assigned according to the
time that a body on that orbit spends in different regions of space;
finally, those weights are used to compute the relative numbers of those orbits
that are needed to obtain the original density distribution.
Another way to proceed is to use an N--body code to build a triaxial stellar
system (e.g., through the collapse of an N--body system initially out of
equilibrium), then to smooth and to freeze the potential fitting it with adequate
formulae, to use these formulae, together with the positions and velocities of the
bodies as initial conditions, to compute a representative sample of orbits in that
potential and, finally, to classify those orbits to get the orbital structure of
the system
\cite{VKS02}.
Those two methods should be regarded as complementary. Schwarzschild's one
allows a very precise definition of the density distribution of the system
one wants to study; alternatively, some properties of the models dictated by
mathematical simplicity (e.g., constant axial ratios over the whole system)
might bias its results, while the N--body method is free of that problem.
\subsection{The problem of chaotic orbits in Schwarzschild's method}
\label{sec:4}
Schwarzschild \cite{Sch93} found it necessary to include chaotic orbits in his
models but, then, these were not fully stable. He built several models using
orbits computed over a Hubble time and, subsequently, followed those orbits for
two additional Hubble times. When he computed the axial ratios obtained using
the data for the third Hubble time, he found significant differences with
respect to the ratios computed over the first Hubble time, from a low of about
4\% for his second and fourth models, to a high of about 17\% for his fifth model.
The cause of that evolution is that chaotic orbits change their behaviour with
time, resembling that of regular orbits at certain intervals, behaving more
chaotically at other intervals and exploring different regions of space in the
meantime. Moreover, that weaker or stronger chaotic behaviour can be traced with
Lyapunov exponents computed over finite intervals which decrease and increase
their values accordingly \cite{KM94}, \cite{Muz07}. Merritt and his coworkers
tried to solve this problem using what they called "fully mixed solutions"
\cite{MF96} and, more recently, integrating orbits over five Hubble times
\cite{Cap07}. In the former work, they found solutions for the weak cusp model,
but not for their strong cusp model; the subsequent evolution of these models to
test their stability was not investigated, however. In the latter work
they indicate that there is "a slight evolution toward a more prolate shape",
but they provide no quantitative estimates other than indicating that differences
in velocity dispersions are "almost always below 10\%". Clearly, it is very
difficult to incorporate chaotic orbits in Schwarzschild's method: as some
chaotic orbits begin to behave more chaotically, one needs to have other chaotic
orbits that behave more regularly as compensation; such a delicate equilibrium
cannot be attained simply obtaining the weights of chaotic orbits over longer
integration times and, moreover, the relatively low number of orbits used
(typically a few thousands) makes even more difficult that task. Finally, the
usual imposition of constant axial ratios over the whole system in
Schwarzschild's method prevents the existence of a rounder halo of chaotic
orbits that seems to be a necessary condition to have highly chaotic triaxial
stellar systems \cite{VKS02}, \cite{MCW05}, \cite{AMNZ07}.
\subsection{The stability of highly chaotic triaxial stellar systems}
\label{sec:5}
The models of the N--body method are built self--consistently from the
start and typically contain hundreds of thousands, or even millions, of bodies so
that they should be free of the difficulties that plage the construction of highly
chaotic triaxial stellar systems with Schwarzschild's method. In fact, stable
models with high fractions of chaotic orbits were obtained with the N-body method,
using about $10^5$ particles \cite{VKS02}, \cite{MCW05}; moreover, the fractions
of the different types of orbits were not significantly altered when the potential
was fitted to the N--body distribution at different times. A stable cuspy model
that was mildly triaxial and made up of 512,000 particles, plus several others with
128,000 particles, were also built \cite{HB01}; later on, it was shown that the
introduction of a black hole, although affecting the inner regions of the model,
did not alter the triaxiality at larger radii and the authors concluded that the
triaxiality of elliptical galaxies is not inconsistent with the presence of
supermassive black holes at their centers \cite{HB02}.
Highly stable models of $10^6$ particles were built by us with the N--body
method \cite{AMNZ07},
\cite{MNZ08}: all of them have decreasing flattening from center to border,
which arised naturally from the N--body evolution during the generation of the
systems; they have different degrees of flattening and triaxiality, two
of them are moderately cuspy ($\gamma \approx 1.0$), and all have high
fractions (between 36\% and 71\%) of chaotic orbits. When integrated with the
N--body code, our models suffer changes in their central density and minor
semiaxis values which do not exceed, respectively, about 4\% and 2\% over a
Hubble time. Nevertheless, these changes are most likely due to collisional
effects of the N--body code \cite{HB90} because, when the number of bodies is
reduced by a factor of 10 (and their masses are increased by the same factor),
those changes increase by factors between 3 and 10. Alternatively, integrating
the motion of the bodies in the fixed smooth potential, which suppresses the
collisional effects (and which, by the way, is what Schwarzschild did) reduces
those changes to 0.1\% only (i.e., between one and two orders of magnitude smaller
than those found by Schwarzchild \cite{Sch93}).
Thus, we may conclude that highly stable triaxial models with large fractions
of chaotic orbits can be built with the N--body method. The difficulties to
build such models with Schwarzschild's method should thus be attributed to the
method itself and not to physical reasons.
\section{Partially and fully chaotic orbits}
\label{sec:6}
Since we are dealing with stationary systems, the orbits of the particles that
make them up obey the
energy integral, but they need two additional isolating integrals to be regular
orbits. Thus, we distinguish between partially chaotic orbits (one additional
integral besides energy) and fully chaotic orbits (energy is the only integral
they obey). A practical way to make the distinction is to compute the six
Lyapunov exponents: they come in three pairs of equal value and opposite signs,
due to the conservation of phase space volume, and each isolating integral
makes zero one pair. Thus, in our case, two Lyapunov exponents are always
zero (due to energy conservation); of the remaining four, if two are positive
the orbit is fully chaotic, if only one is positive the orbit is partially
chaotic and, finally, if all are zero the orbit is regular.
It was noted in \cite{PV84} that orbits obeying two isolating integrals have
smaller fractal dimension than orbits obeying only one, but earlier hints of
the differences between them can also be found in \cite{GS81} (whose
semi-stochastic orbits are probably what we now call partially chaotic orbits)
and in \cite{CGG78} (whose orbits in their big and small seas can be identified,
respectively, with the fully and partially chaotic orbits).
The reason why distinguishing partially from fully chaotic orbits in galactic
dynamics is important is that, since they obey different numbers of isolating
integrals, they have different spatial distributions as shown in \cite{Muz03},
\cite{MM04}, \cite{MCW05}, \cite{AMNZ07} and \cite{MNZ08}. In triaxial systems,
partially chaotic orbits usually exhibit a distribution intermediate between
those of regular and of fully chaotic orbits, and a possible explanation is that
some of the partially chaotic orbits lie in the stochastic layer surrounding
the resonances and thus behave similarly to regular orbits \cite{NPM07}.
Nevertheless, that is not the whole story as some partially chaotic orbits
seem to obey a global integral, rather than local ones \cite{AMNZ07}.
Partially chaotic orbits should not be confused with fully chaotic orbits with
low Lyapunov exponents, which also tend to have distributions more similar to
those of regular orbits than those of fully chaotic orbits with high Lyapunov
exponents \cite{MM04}, \cite{MCW05}. It is worth recalling that, no matter how
small their Lyapunov exponents are, fully chaotic orbits obey only one
isolating integral while partially chaotic orbits obey two so that, from a
theoretical point of view, they are indeed different kinds of orbits. From a
practical point of view, it is also easy to see that they have different
distributions: Table 1 gives the axial ratios of the distributions of
different kinds of orbits for models E4, E5 and E6 from \cite{AMNZ07} and E4c
and E6c from \cite{MNZ08}; the x, y and z axes are parallel, respectively, to
the major, intermediate and minor axes of the models. The third column gives
the axial ratios for the distributions of partially chaotic orbits, and the
fourth and fifth columns give the same ratios for weakly fully chaotic orbits
for two choices of the limiting value of the Lyapunov exponents used to
define "weakly", 0.050 and 0.100. Although for some models (e.g. E4 and E4c)
the possible differences are masked by the rather large statistical errors,
it is clear from the Table that the distributions of partially chaotic orbits
are significantly different from those of weakly fully chaotic orbits (at the
$3 \sigma$ level) for the other models.
\begin{table}
\centering
\caption{Axial ratios of the different classes of orbits in our models.}
\label{tab:1}
\begin{tabular}{lllll}
\hline\noalign{\smallskip}
Ratio & ~~System & ~~Partially Ch. & ~~~W.F.Ch. (0.050) & ~~W.F.Ch. (0.100)\\
\noalign{\smallskip}\hline\noalign{\smallskip}
y/x & ~~E4 & ~~$0.896 \pm 0.064$ & ~~~$0.692 \pm 0.027$ & ~~$0.745 \pm 0.019$ \\
& ~~E5 & ~~$0.808 \pm 0.036$ & ~~~$0.764 \pm 0.024$ & ~~$0.797 \pm 0.017$ \\
& ~~E6 & ~~$0.658 \pm 0.035$ & ~~~$0.789 \pm 0.027$ & ~~$0.845 \pm 0.019$ \\
& ~~E4c & ~~$0.748 \pm 0.027$ & ~~~$0.733 \pm 0.016$ & ~~$0.730 \pm 0.013$ \\
& ~~E6c & ~~$0.528 \pm 0.020$ & ~~~$0.693 \pm 0.013$ & ~~$0.700 \pm 0.010$ \\
\noalign{\smallskip}\hline\noalign{\smallskip}
z/x & ~~E4 & ~~$0.790 \pm 0.054$ & ~~~$0.802 \pm 0.035$ & ~~$0.826 \pm 0.024$ \\
& ~~E5 & ~~$0.477 \pm 0.018$ & ~~~$0.684 \pm 0.021$ & ~~$0.708 \pm 0.014$ \\
& ~~E6 & ~~$0.286 \pm 0.013$ & ~~~$0.644 \pm 0.022$ & ~~$0.673 \pm 0.015$ \\
& ~~E4c & ~~$0.692 \pm 0.024$ & ~~~$0.762 \pm 0.017$ & ~~$0.757 \pm 0.013$ \\
& ~~E6c & ~~$0.334 \pm 0.010$ & ~~~$0.466 \pm 0.007$ & ~~$0.490 \pm 0.006$ \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
At any rate, it is clear that the distributions of partially and fully chaotic
orbits differ significantly and that they should not be bunched together as a
single group of chaotic orbits. The problem is that the computation of the
Lyapunov exponents demands long computation times and there are not yet faster
methods that allow to distinguish partially from fully chaotic orbits. The fact
that many chaotic orbits can be frequency analyzed and are found to lie in
regions of the frequency map corresponding to regular orbits \cite{KV05} might,
perhaps, lead to a faster method of separation in the future. Nevertheless, many
fully chaotic orbits can be frequency analyzed, while many partially chaotic
orbits cannot \cite{AMNZ07}, so that much remains to be done before a workable
method based on frequency analysis can be designed.
\section{Figure rotation in triaxial systems}
\label{sec:7}
Although the system investigated in \cite{MCW05} had been regarded as stationary,
integrations much longer than those used in that work revealed that, in fact, it
was very slowly rotating around its minor axis \cite{Muz06}. The total angular
momentum of the system was zero, so that this was an unequivocal case of figure
rotation.
Figure rotation was also found in most of the models studied in \cite{AMNZ07} and
\cite{MNZ08} and it is clear that the rotational velocity increases with the
flattening of the system; only model E4 from \cite{AMNZ07}, which is almost axially
symmetric, prolate and with axial ratio close to 0.6 has no significant rotation. It
should be stressed, however, that even the highest rotational velocities found thus
far are extremely low: the systems can complete only a fraction of a revolution in a
Hubble time or, put in a different way, the radii of the Lindblad and corotation
resonances are at least an order of magnitude larger than the systems themselves.
It had been suggested that figure rotation might produce important changes in the
degree of chaoticity \cite{MF96} and it turned out that, in spite of the extremely
low rotational velocity, a significant difference in the fraction of chaotic orbits
was found between the models of \cite{MCW05} and \cite{Muz06} which only differ in
that the former is stationary and the latter is rotating. Alternatively, no
significant difference was found for the different kinds of regular orbits in those
two models. The most likely explanation is that, although the rotational velocity
is too low to produce a measurable effect on the regular orbits, the break of
symmetry caused by the presence of rotation suffices to increase chaos significantly.
\section{Musings on orbital classification through frequency analysis}
\label{sec:8}
\subsection{Classification methods}
\label{sec:9}
The spectral properties of galactic orbits were investigated by Binney and Spergel
\cite{BS82} and, more recently, Papaphilippou and Laskar \cite{PL96} and
\cite{PL98} applied to stellar systems the frequency analysis techniques developed
by the latter for celestial mechanics. Following the ideas of Binney and Spergel,
Carpintero and Aguilar \cite{CA98} developed an automatic orbit classification code.
Kalapotharakos and Voglis \cite{KV05} developed a classification system based on
the frequency map of Laskar and, later on, I \cite{Muz06} improved it somewhat.
Having used extensively both the Carpintero and Aguilar \cite{CMW99},
\cite{MCW00}, \cite{CVM01}, \cite{CMVW03} and \cite{MCW05}, and the Kalapotharakos
and Voglis methods \cite{Muz06}, \cite{AMNZ07} and \cite{MNZ08}, I strongly prefer
the latter. The main advantage of the Kalapotharakos and Voglis method is that one
can see what is happening throughout the process. It is very easy to detect problems
from the anomalous positions that the corresponding frequency ratios yield on the
frequency map and, thus, to improve the method. This is an aspect that deserves to
be emphasized: the need to use frequencies different from those corresponding to the
maximum amplitudes had not been noted in \cite{KV05}, but it was in \cite{Muz06},
probably because a somewhat cuspier potential was investigated in the latter
work; similarly, that distinction was unnecessary for the long axis tubes (LATs
hereafter) of \cite{Muz06}, but had to be made for those of the almost axially
symmetric E4 system of \cite{AMNZ07}. In other words, as one explores different
stellar system models (cuspier, closer to axisymmetry, and so on)
the orbital classification system may need to be improved and that need is quite
evident with the Kalapotharakos and Voglis method. Thanks to these improvements,
virtually all the regular orbits can be classified with the frequency map,
while usually between 10\% and 15\% of them remain unclassified with the other
method \cite{MCW05} and \cite{J05}.
Besides, separation of chaotic from regular
orbits with the method of Carpintero and Aguilar is erratic, at least in rotating
systems \cite{CMVW03}. Since the problem seems to arise from the presence of nearby
lines in the spectra, which is worse in rotating systems but not exclusive of them,
I strongly suspect that orbit classification in non--rotating systems may also be
affected. That is why in our last work with that method \cite{MCW05} we used
it only to classify regular orbits, previously selected using Lyapunov exponents.
\subsection{Which frequency to choose?}
\label{sec:10}
Frequency analysis is usually performed on complex variables formed taking one
coordinate as the real part and the corresponding velocity as the imaginary part.
One thus gets the frequencies Fx, Fy and Fz corresponding, respectively, to motion
along the (x, y, z) axes which, in turn, are parallel to the main axes of the
stellar system. The frequencies usually selected for the frequency map are those
corresponding to the maximum amplitudes in each coordinate \cite{WFM98}, \cite{KV05},
but it has been known since 1982 \cite{BS82} that, due to a libration effect, one
should not always take those. Besides, another effect linked to very highly elongated
orbits also demands to adopt frequencies which are not the ones corresponding to the
maximum amplitudes \cite{Muz06}, \cite{AMNZ07}. Nevertheless, it is just fair to note
that these exceptions are not too common: out of 17,103 orbits investigated in
\cite{AMNZ07} and \cite{MNZ08} only 265 (1.5\%) needed the former correction and 153
(0.9\%) the latter one. These fractions vary considerably from one model to another,
however, and as the affected orbits tend to concentrate at low absolute values of
energy and/or are extremely elongated, not taking these effects into account
might bias the sample of classified orbits.
\subsection{The usefulness of the energy vs. frequency plane}
\label{sec:11}
Regular orbits obey two additional isolating integrals, besides energy, and the
values of the orbital frequencies are related to these integrals. For a given
energy, different frequencies imply different values of the other integrals and,
thus, different types of orbits. Inner and outer LATs,
short axis tubes, boxes and even different resonant orbits can be separated on the
energy vs. frequency (or frequency ratio) plane, but that does not mean that it is
practical to use it, because those separations are more easily done on the
frequency map.
Nevertheless, some insight can be gained from the use of the energy vs.
frequency plane. Figure 1 of \cite{Muz06} offers a good example, because that
plane was used there to show that one should not always use the frequency
corresponding to the maximum amplitude as the principal frequency. Besides,
while in \cite{KV05} it was correctly stated that outer LATs had larger
Fx/Fz values than inner LATs, no indication of which was the separating
value was provided there. Actually, as shown in Figure 2 of \cite{AMNZ07}, one has
to use the energy vs. frequency ratio plane to separate inner from outer LATs,
because the separating value varies with the energy of the orbit.
\begin{figure}
\vskip 5.5mm
\centering
\includegraphics[width=5.5truecm]{muzzio_fig1ul.eps}~\hfill
\includegraphics[width=5.5truecm]{muzzio_fig1ur.eps}
\vskip 6.5mm
\centering
\includegraphics[width=5.5truecm]{muzzio_fig1ml.eps}~\hfill
\includegraphics[width=5.5truecm]{muzzio_fig1mr.eps}
\vskip 6.5mm
\centering
\includegraphics[width=5.5truecm]{muzzio_fig1ll.eps}~\hfill
\includegraphics[width=5.5truecm]{muzzio_fig1lr.eps}
\caption{Projection on the (x, y) plane of several examples of LATs; see text for details}
\label{fig:1}
\end{figure}
Figure 1 presents the (x, y) projections of several LATs from model
E4 of \cite{AMNZ07}. The orbits on the left column have similar energy values,
close to the minimum energy of -5.96 and, although their Fx/Fz values range from
0.6516 to 0.8118, they are all inner LATs, as evidenced by their concave
upper and lower limits. We also notice that their extension along the x axis is
reduced as their Fx/Fz values increase and, in fact, the regions of space occupied
by orbits 0872 and 0009 resemble more those occupied by outer LATs than
those occupied by inner LATs. We found a similar effect on the x extension
of the orbits at other energy values although, when the separation shown in Figure 2
of \cite{AMNZ07} is crossed, there is also of course a change from inner to outer
LATs. The upper and middle parts of the right column of Figure 1 correspond
to orbits 3141 and 1513 that are virtually face to face at each side of the separation
on the energy vs. frequency ratio plot: they have similar Fx/Fz values but, due to their
energy difference, the former is an outer, and the latter an inner, LAT. Notice also that
the Fx/Fz value of (outer LAT) orbit 3141 is lower than that of (inner LAT) orbit 0872.
Finally, the lower right section of Figure 1 corresponds to (outer LAT) orbit 3307, whose
Fx/Fz value is lower than those of (inner LAT) orbits 0009 and 0872.
\begin{figure}
\vskip 5.5mm
\centering
\includegraphics[width=5.5truecm]{muzzio_fig2ul.eps}~\hfill
\includegraphics[width=5.5truecm]{muzzio_fig2ur.eps}
\vskip 6.5mm
\centering
\includegraphics[width=5.5truecm]{muzzio_fig2ll.eps}~\hfill
\includegraphics[width=5.5truecm]{muzzio_fig2lr.eps}
\caption{Projection on the (x, y) plane of two examples of boxes; see text for details}
\label{fig:2}
\end{figure}
Interestingly, the shortening of the x axis as the Fx/Fz ratio increases, shown above for
the LATs, affects the boxes as well. Figure 2 presents the (x, y) and (x, z) projections of
orbits 0104 and 0097 from model E4 of \cite{AMNZ07}, which have both essentially the same
energy. Nevertheless, while the former, with Fx/Fz = 0.6332, lies straight on the line
occupied by the boxes on the energy vs. frequency ratio plane, the latter, with Fx/Fz =
0.8076, lies well above that line. We see on the left part of Figure 2 that 0104 is indeed
a typical box, but the right part shows that 0097, although still a box, is strongly
compressed along the x axis.
Due to their elongation along the major axis, inner LATs and boxes are usually considered
as the main building blocks of highly elongated triaxial systems, but we now see that
there are inner LATs and boxes that are, in fact, strongly compressed along that axis. To
put things in the proper perspective we should emphasize, however, that these orbits were
found in the almost rotationally symmetric model E4 of \cite{AMNZ07} and that they are not
very abundant.
\section{Discussion}
\label{sec:12}
We have reviewed several papers on triaxial stellar systems built with the N--body method
that show that it is perfectly possible to have strongly chaotic triaxial stellar systems
that are also highly stable over periods of the order of a Hubble time. The difficulties
to build such systems with Schwarzschild's method should thus be attributed to the
method itself and not to physical reasons.
It is clear, both from a theoretical and from a practical point of view, that partially
and fully chaotic orbits populate different regions of space and should not be bunched
together under the single banner of chaotic orbits. The main problem here is that the
single method thus far available to separate them, that of Lyapunov exponents, is very
slow and faster methods are wanted. We also showed that the distribution of partially
chaotic orbits is different from that of weakly fully chaotic orbits, in accordance
with the fact that the former obey two isolating integrals of motion and the latter only
one.
Very slow figure rotation seems to be an ordinary trait of strongly elongated triaxial
stellar models formed through the collapse of cold N--body systems. The rotational velocity
diminishes, and even disappears entirely, as one goes to less elongated and less triaxial
models.
Frequency analysis offers a very useful tool for the classification of large numbers of
regular orbits. I strongly favor the use of the method of Ka\-la\-po\-tha\-ra\-kos and Voglis
\cite{KV05}, with the improvements we introduced in \cite{Muz06} and \cite{AMNZ07}. Since
the need for those improvements became apparent when models with different characteristics
(cuspiness, approximate rotational symmetry) were considered, it would not be surprising
that further refinements will be necessary as the method is applied to other systems.
Nevertheless, a nice feature of this method is that, when there is such need, it becomes
plainly evident. Besides, plots of known integrals, such as energy, and the orbital frequencies
(or frequency ratios), that are related to the values of the integrals, are very useful to
reveal peculiarities of the orbits as one explores different models; a good example of this
is provided by the compression along the major axis of some LATs and boxes from an almost
axisymmetric system, shown in our Figures 1 and 2.
\section{Acknowledgements}
\label{sec:13}
I am very grateful to H\'ector R. Viturro and to Ruben E. Mart\'{\i}nez for their
technical assistance, and to Lilia P. Bassino and to an anonimous referee for carefully
reading the first version of this paper and suggesting some language improvements.
This work was supported with grants from the Consejo Nacional de
Investigaciones Cient\'{\i}ficas y T\'ecnicas de la Rep\'ublica Argentina, the
Agencia Nacional de Promoci\'on Cient\'{\i}fica y Tecnol\'ogica and the Universidad
Nacional de La Plata.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 568 |
Yellowxander
"The stuff what I do, innit!"
Bands Over Breakfast
M2tM Notts
Nordic Noize
Nottingham Rocks
The Djentleman's Lunchbox
The Launch Pad
Dead By April, Skindred
Skindred (Live)
Skindred – 01/10/09
The Rescue Rooms, NottinghamUnfortunately due to such an early start and the DxA interview, I only caught the last minute of Aussies Karnivool, but they sounded pretty kick ass, a less intense Tool perhaps? The bass at this venue is pretty heavy and almost makes implode, and that's just from the music playing in-between acts, so I had a feeling tonights sound wouldn't be great, and it wasn't, but that didn't effect the bands performance.
Dead By April were so much more brutal and high energy than I was expecting. It did confuse me occasionally when a backing track filled in for some of the layers of vocals which I would have though they would have done them selves. This almost made me question were they actually playing or not, as they were performance was very tight and well rehearsed. But, as they said in the interview, for them coming over here, it's like starting over all again from the bottom, so they do have something to prove. Jamie's screaming stunned me only to then astound me at how good his singing voice is also, this added to Pontus' almost angelic tones makes for a great vocal set up. ★★★✩✩
Skindred had to contend playing to a room smaller than they're used to, and I think they enjoyed the challenge. You know that at any Skindred you're gonna get jumping, well they even got the floor bouncing at The rescue Rooms, literally, I was actually perturbed by this! Added with perhaps no air con, massive bass and an un-padded metal pole with only me between it and the rest of the audience, the night could have been quite poor to be honest, thankfully, there was a lineup great enough to not dwell on these negative points. I was literally wet all over from my sweat and other people sweat, it really was that hot, it looked like we were all at a foam party. We know how funny Benji can be, but tonight he concentrated on getting personal with everyone and getting some dancing action. That said, the crowd didn't' stop for one minute, jumping, head banging, moshing, fisting the air and I think the only thing that didn't happen was any crowd surfing, makes a change! This was actually the 6th time I've seen the band and I'm really glad to have seen them at such an intimate venue, certainly at this stage in their career.
From → Reviews
« Papa Roach (Live)
Playlist 28/09/09 »
Playlist 18/09/20 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,580 |
The SSL master keys can be logged by mitmproxy so that external programs can decrypt TLS connections both from and to the proxy.
Key logging is enabled by setting the environment variable <samp>SSLKEYLOGFILE</samp> so that it points to a writable
text file. Recent versions of WireShark can use these log files to decrypt packets.
You can specify the key file path in WireShark via<br>
<samp>Edit → Preferences → Protocols → SSL → (Pre)-Master-Secret log filename</samp>.
Note that <samp>SSLKEYLOGFILE</samp> is respected by other programs as well, e.g. Firefox and Chrome.
If this creates any issues, you can set <samp>MITMPROXY_SSLKEYLOGFILE</samp> alternatively. | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,977 |
Q: Bootstrap mobile menu button not visible I'm not able to view the menu button of bootstrap menu when it is displayed in mobile view.In the same time I need to remove all the ui styling offered by the Bootstrap and have my own styling for navbar.
<nav class="navbar-fixed-top" role="navigation">
<div class="container-fluid">
<div class="navbar-header">
<button type="button" class="navbar-toggle" data-toggle="collapse" data-target="#myNavbar">
<span class="icon-bar"></span>
<span class="icon-bar"></span>
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</button>
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A: You should add navbar navbar-default or navbar navbar-inverse so you navigation links, button etc have color/states applied to them.
<nav class="navbar navbar-default navbar-fixed-top" role="navigation">
If you want a transparent or white background uss CSS.
Docs
With navbar-default
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js"></script>
<script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.6/js/bootstrap.min.js"></script>
<link href="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.6/css/bootstrap.min.css" rel="stylesheet" />
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NS_CC_EXT_BEGIN
#pragma mark - CCHierarchiesAnimate
CCHierarchiesAnimate* CCHierarchiesAnimate::create (const char* animationName,
const char* spriteAnimationName)
{
CCHierarchiesAnimate* ret = new CCHierarchiesAnimate();
if (ret->initWithName(animationName, spriteAnimationName))
{
ret->autorelease();
return ret;
}
CC_SAFE_DELETE(ret);
return NULL;
}
CCHierarchiesAnimate* CCHierarchiesAnimate::createWholeAnimation (const char* spriteAnimationName)
{
CCHierarchiesAnimate* ret = new CCHierarchiesAnimate();
if (ret->initWholeAnimation(spriteAnimationName))
{
ret->autorelease();
return ret;
}
CC_SAFE_DELETE(ret);
return NULL;
}
CCHierarchiesAnimate::CCHierarchiesAnimate ()
: _spriteAnimation(NULL), _curFrameIndex(0)
{
}
CCHierarchiesAnimate::~CCHierarchiesAnimate ()
{
CCHierarchiesSpriteAnimationCache::sharedHierarchiesSpriteAnimationCache()->removeAnimation(_spriteAnimationName.c_str());
}
bool CCHierarchiesAnimate::initWithName (const char* animationName,
const char* spriteAnimationName)
{
CCAssert(animationName != NULL && spriteAnimationName != NULL, "CCHierarchiesAnimate: arguments must be non-nil");
_spriteAnimation = CCHierarchiesSpriteAnimationCache::sharedHierarchiesSpriteAnimationCache()->addAnimation(spriteAnimationName);
if (!_spriteAnimation->getAnimationByName(animationName, _currentAnimation)) {
CCLOG("no animation (%s) in (%s) while create CCHierarchiesAnimate", animationName, spriteAnimationName);
CCHierarchiesSpriteAnimationCache::sharedHierarchiesSpriteAnimationCache()->removeAnimation(spriteAnimationName);
CC_SAFE_RELEASE(this);
return false;
}
float duration = (float)(_currentAnimation.endFrameIndex - _currentAnimation.startFrameIndex + 1) / _spriteAnimation->getFrameRate();
if (CCActionInterval::initWithDuration(duration)) {
_animationName = animationName;
_spriteAnimationName = spriteAnimationName;
return true;
}
else {
CC_SAFE_RELEASE(this);
return false;
}
}
bool CCHierarchiesAnimate::initWholeAnimation (const char* spriteAnimationName)
{
CCAssert(spriteAnimationName != NULL, "CCHierarchiesAnimate: arguments must be non-nil");
_spriteAnimation = CCHierarchiesSpriteAnimationCache::sharedHierarchiesSpriteAnimationCache()->addAnimation(spriteAnimationName);
_currentAnimation.name = HIERARCHIES_WHOLE_ANIMATION_NAME;
_currentAnimation.startFrameIndex = 0;
_currentAnimation.endFrameIndex = _spriteAnimation->getFrameCount() - 1;
float duration = (float)(_currentAnimation.endFrameIndex - _currentAnimation.startFrameIndex + 1) / _spriteAnimation->getFrameRate();
if (CCActionInterval::initWithDuration(duration)) {
_animationName = HIERARCHIES_WHOLE_ANIMATION_NAME;
_spriteAnimationName = spriteAnimationName;
return true;
}
else {
CC_SAFE_RELEASE(this);
return false;
}
}
const char* CCHierarchiesAnimate::getAnimationName ()
{
return _animationName.c_str();
}
const char* CCHierarchiesAnimate::getSpriteAnimationName ()
{
return _spriteAnimationName.c_str();
}
void CCHierarchiesAnimate::startWithTarget (CCNode* pTarget)
{
CCHierarchiesSpriteBase* target = dynamic_cast<CCHierarchiesSpriteBase*>(pTarget);
CCAssert(target != NULL, "CCHierarchiesAnimate can only valid on CCHierarchiesSpriteBase class");
CCAssert(_spriteAnimationName.compare(target->getAnimationName()) == 0, "CCHierarchiesAnimate effect on a invalid CCHierarchiesSpriteBase instance");
CCActionInterval::startWithTarget(pTarget);
_curFrameIndex = _currentAnimation.startFrameIndex;
target->displayFrameAtIndex(_curFrameIndex);
}
CCActionInterval* CCHierarchiesAnimate::reverse(void)
{
CCHierarchiesAnimate* animate = NULL;
if (_animationName == HIERARCHIES_WHOLE_ANIMATION_NAME) {
animate = CCHierarchiesAnimate::createWholeAnimation(_spriteAnimationName.c_str());
}
else {
animate = CCHierarchiesAnimate::create(_animationName.c_str(), _spriteAnimationName.c_str());
}
unsigned int tmp = animate->_currentAnimation.endFrameIndex;
animate->_currentAnimation.endFrameIndex = animate->_currentAnimation.startFrameIndex;
animate->_currentAnimation.startFrameIndex = tmp;
return animate;
}
CCObject* CCHierarchiesAnimate::copyWithZone (CCZone* pZone)
{
CCZone* pNewZone = NULL;
CCHierarchiesAnimate* pCopy = NULL;
if(pZone && pZone->m_pCopyObject)
{
//in case of being called at sub class
pCopy = dynamic_cast<CCHierarchiesAnimate*>(pZone->m_pCopyObject);
}
else
{
pCopy = new CCHierarchiesAnimate();
pZone = pNewZone = new CCZone(pCopy);
}
CCActionInterval::copyWithZone(pZone); // call super class init once
pCopy->initWithName(_animationName.c_str(), _spriteAnimationName.c_str()); // call super class init twice ??
CC_SAFE_DELETE(pNewZone);
return pCopy;
// if (pCopy->initWithName(_animationName.c_str(), _spriteAnimationName.c_str())) {
// CC_SAFE_DELETE(pNewZone);
// return pCopy;
// }
// else {
// CC_SAFE_DELETE(pNewZone);
// return NULL;
// }
}
void CCHierarchiesAnimate::update (float time)
{
CCHierarchiesSpriteBase* target = dynamic_cast<CCHierarchiesSpriteBase*>(m_pTarget);
if (_currentAnimation.endFrameIndex >= _currentAnimation.startFrameIndex) {
unsigned int curFrameIndex = _currentAnimation.startFrameIndex + time * (_currentAnimation.endFrameIndex - _currentAnimation.startFrameIndex);
while (_curFrameIndex != curFrameIndex) {
_curFrameIndex++;
target->displayFrameAtIndex(_curFrameIndex);
}
}
else { // animation reverse play
unsigned int curFrameIndex = _currentAnimation.endFrameIndex + (1 - time) * (_currentAnimation.startFrameIndex - _currentAnimation.endFrameIndex);
while (_curFrameIndex != curFrameIndex) {
_curFrameIndex--;
target->displayFrameAtIndex(_curFrameIndex);
}
}
}
#pragma mark - CCHierarchiesFlipX
CCHierarchiesFlipX* CCHierarchiesFlipX::create (bool x)
{
CCHierarchiesFlipX* ret = new CCHierarchiesFlipX();
if (ret && ret->initWithFlipX(x)) {
ret->autorelease();
return ret;
}
CC_SAFE_DELETE(ret);
return NULL;
}
bool CCHierarchiesFlipX::initWithFlipX (bool x)
{
_flipX = x;
return true;
}
void CCHierarchiesFlipX::update(float time)
{
CC_UNUSED_PARAM(time);
CCHierarchiesSpriteBase* target = dynamic_cast<CCHierarchiesSpriteBase*>(m_pTarget);
CCAssert(target, "CCHierarchiesFlipX only valid on CCHierarchiesSpriteBase class");
target->setFlipX(_flipX);
}
CCFiniteTimeAction* CCHierarchiesFlipX::reverse()
{
return CCHierarchiesFlipX::create(!_flipX);
}
CCObject * CCHierarchiesFlipX::copyWithZone(CCZone* zone)
{
CCZone* newZone = NULL;
CCHierarchiesFlipX* ret = NULL;
if (zone && zone->m_pCopyObject) {
ret = (CCHierarchiesFlipX*) (zone->m_pCopyObject);
} else {
ret = new CCHierarchiesFlipX();
zone = newZone = new CCZone(ret);
}
CCActionInstant::copyWithZone(zone);
ret->initWithFlipX(_flipX);
CC_SAFE_DELETE(newZone);
return ret;
}
#pragma mark - CCHierarchiesFlipY
CCHierarchiesFlipY* CCHierarchiesFlipY::create (bool y)
{
CCHierarchiesFlipY* ret = new CCHierarchiesFlipY();
if (ret && ret->initWithFlipY(y)) {
ret->autorelease();
return ret;
}
CC_SAFE_DELETE(ret);
return NULL;
}
bool CCHierarchiesFlipY::initWithFlipY (bool y)
{
_flipY = y;
return true;
}
void CCHierarchiesFlipY::update(float time)
{
CC_UNUSED_PARAM(time);
CCHierarchiesSpriteBase* target = dynamic_cast<CCHierarchiesSpriteBase*>(m_pTarget);
CCAssert(target, "CCHierarchiesFlipY only valid on CCHierarchiesSpriteBase class");
target->setFlipY(_flipY);
}
CCFiniteTimeAction* CCHierarchiesFlipY::reverse()
{
return CCHierarchiesFlipY::create(!_flipY);
}
CCObject * CCHierarchiesFlipY::copyWithZone(CCZone* zone)
{
CCZone* newZone = NULL;
CCHierarchiesFlipY* ret = NULL;
if (zone && zone->m_pCopyObject) {
ret = (CCHierarchiesFlipY*) (zone->m_pCopyObject);
} else {
ret = new CCHierarchiesFlipY();
zone = newZone = new CCZone(ret);
}
CCActionInstant::copyWithZone(zone);
ret->initWithFlipY(_flipY);
CC_SAFE_DELETE(newZone);
return ret;
}
#pragma mark - CCHierarchiesAvatarMapInsert
CCHierarchiesAvatarMapInsert* CCHierarchiesAvatarMapInsert::create (const char* avatarMapFrom, const char* avatarMapTo)
{
CCHierarchiesAvatarMapInsert* ret = new CCHierarchiesAvatarMapInsert();
if (ret && ret->init(avatarMapFrom, avatarMapTo)) {
ret->autorelease();
return ret;
}
CC_SAFE_DELETE(ret);
return NULL;
}
bool CCHierarchiesAvatarMapInsert::init (const char* avatarMapFrom, const char* avatarMapTo)
{
_avatarMapFrom = avatarMapFrom;
_avatarMapTo = avatarMapTo;
return true;
}
void CCHierarchiesAvatarMapInsert::update (float time)
{
CC_UNUSED_PARAM(time);
CCHierarchiesSpriteDynamic* target = dynamic_cast<CCHierarchiesSpriteDynamic*>(m_pTarget);
CCAssert(target, "CCHierarchiesAvatarMapInsert only valid on CCHierarchiesSpriteDynamic class");
target->setAvatarMap(_avatarMapFrom.c_str(), _avatarMapTo.c_str());
}
CCObject* CCHierarchiesAvatarMapInsert::copyWithZone (CCZone *zone)
{
CCZone* newZone = NULL;
CCHierarchiesAvatarMapInsert* ret = NULL;
if (zone && zone->m_pCopyObject) {
ret = (CCHierarchiesAvatarMapInsert*) (zone->m_pCopyObject);
} else {
ret = new CCHierarchiesAvatarMapInsert();
zone = newZone = new CCZone(ret);
}
CCActionInstant::copyWithZone(zone);
ret->init(_avatarMapFrom.c_str(), _avatarMapTo.c_str());
CC_SAFE_DELETE(newZone);
return ret;
}
#pragma mark - CCHierarchiesAvatarMapReset
CCHierarchiesAvatarMapReset* CCHierarchiesAvatarMapReset::create ()
{
CCHierarchiesAvatarMapReset* ret = new CCHierarchiesAvatarMapReset();
if (ret && ret->init()) {
ret->autorelease();
return ret;
}
CC_SAFE_DELETE(ret);
return NULL;
}
bool CCHierarchiesAvatarMapReset::init ()
{
return true;
}
void CCHierarchiesAvatarMapReset::update (float time)
{
CC_UNUSED_PARAM(time);
CCHierarchiesSpriteDynamic* target = dynamic_cast<CCHierarchiesSpriteDynamic*>(m_pTarget);
CCAssert(target, "CCHierarchiesAvatarMapReset only valid on CCHierarchiesSpriteDynamic class");
target->resetAvatarMap();
}
CCObject* CCHierarchiesAvatarMapReset::copyWithZone (CCZone *zone)
{
CCZone* newZone = NULL;
CCHierarchiesAvatarMapReset* ret = NULL;
if (zone && zone->m_pCopyObject) {
ret = (CCHierarchiesAvatarMapReset*) (zone->m_pCopyObject);
} else {
ret = new CCHierarchiesAvatarMapReset();
zone = newZone = new CCZone(ret);
}
CCActionInstant::copyWithZone(zone);
ret->init();
CC_SAFE_DELETE(newZone);
return ret;
}
#pragma mark - CCHierarchiesAvatarMapSet
CCHierarchiesAvatarMapSet* CCHierarchiesAvatarMapSet::create (const AvatarMapType& avatarMap)
{
CCHierarchiesAvatarMapSet* ret = new CCHierarchiesAvatarMapSet();
if (ret && ret->init(avatarMap)) {
ret->autorelease();
return ret;
}
CC_SAFE_DELETE(ret);
return NULL;
}
bool CCHierarchiesAvatarMapSet::init (const AvatarMapType& avatarMap)
{
_avatarMap = avatarMap;
return true;
}
void CCHierarchiesAvatarMapSet::update (float time)
{
CC_UNUSED_PARAM(time);
CCHierarchiesSpriteDynamic* target = dynamic_cast<CCHierarchiesSpriteDynamic*>(m_pTarget);
CCAssert(target, "CCHierarchiesAvatarMapSet only valid on CCHierarchiesSpriteDynamic class");
target->resetAvatarMap();
AvatarMapType::const_iterator avatarItemIter;
for (avatarItemIter = _avatarMap.cbegin(); avatarItemIter != _avatarMap.cend(); avatarItemIter++) {
target->setAvatarMap(avatarItemIter->first.c_str(), avatarItemIter->second.c_str());
}
}
CCObject* CCHierarchiesAvatarMapSet::copyWithZone (CCZone *zone)
{
CCZone* newZone = NULL;
CCHierarchiesAvatarMapSet* ret = NULL;
if (zone && zone->m_pCopyObject) {
ret = (CCHierarchiesAvatarMapSet*) (zone->m_pCopyObject);
} else {
ret = new CCHierarchiesAvatarMapSet();
zone = newZone = new CCZone(ret);
}
CCActionInstant::copyWithZone(zone);
ret->init(_avatarMap);
CC_SAFE_DELETE(newZone);
return ret;
}
NS_CC_EXT_END
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,291 |
Kit Harington on life after Game of Thrones John Snow
Posted by WHISNews21 ⋅ November 6, 2017 ⋅ Leave a comment
Kit Harington Life After Game Of Thrones
'I'm Not Always Going To Play Swords And Horses'
Kit Harington is reminiscing about his drama school days. "You think you know the world, and you're so overly serious about what you're doing, and how important it is, sitting on the steps, with coffee and a script, in black…"
He smiles, but his arms are folded defensively across his chest, and he occasionally brings a hand up to play with his moustache. He's wearing take-me-seriously wire-framed specs and is dressed entirely in black. He looks down and has a moment of recognition, as if he's just walked into a trap. "I am actually completely in black. And I'm talking about important things," he smiles. "I haven't changed at all, have I?"
He might not think so, but at 30 Harington is living a very different life from the one he led at 19. Shortly after leaving drama school, he signed up to shoot the pilot for a promising new television fantasy series, Game Of Thrones, playing a moody young northerner called Jon Snow. It was his first screen role. "We didn't know if it was going to go, and we didn't know if it was going to be any good," he explains. "But it was HBO, and it was American TV, so it felt like a huge deal."
Game of Thrones' first incarnation wasn't promising, however, and the pilot turned out to be a dud. "They made a lot of mistakes. It didn't look right, didn't feel right, had nothing different about it." In it, the nascent Jon Snow was wearing a wig, was clean-shaven and made use of the baby face Harington hides under his beard. He claims that hardly anyone has seen that pilot, not even him, but that the show's creators, DB Weiss and David Benioff, have a copy they use to keep him in line. "They say, if I ever piss them off too much, they'll release it on YouTube. Every now and then, they send me a screengrab, just as a threat."
Weiss and Benioff went back to the drawing board. Harington, then 23, grew his hair long, slapped some mud on his face and surprised himself by sprouting a beard for the first time. The whole show became dirtier, grittier and more grown up. By the end of the first season, millions of viewers who might have balked at the idea of swordfights and spells suddenly found that, when sex and violence were added to the mix, they could get behind a fantasy series, after all. As Game Of Thrones grew into the behemoth it is today, Snow was manoeuvred into the very heart of the story.
Harington is now about to go to work on the show's eighth and final season. How is Snow going to feel when he finds out that the woman he's in love with, Daenerys Targaryen, is also his aunt? "I really hope that he just nods slowly and goes, 'Damned right'," he says, doing a mock-leer. "Something really horribly inappropriate, and you find out Jon's had a really sick mind the whole time. That's the way I'd love to play it. I'll try it for one take, anyway."
He is close to Emilia Clarke, who plays Daenerys, though owing to the series' vast geography, the two did not film together until season seven. "But we were very good friends by the time we got to the sex scene," Harington laughs and squirms. "Which is really weird. Usually, you'd turn up and you'd know the actor, but you're not best mates. The main thing was trying not to laugh. It was like, OK, if we laugh, we'll never get this scene done, so we've got to do minimal takes, then we can crack up about it afterwards."
500 fans camped outside the hotel every day, and you have to get through them. It feels like being Bieber or something
For reasons both personal and professional, Harington thinks the show is ending at the perfect time. He's one of the few cast members who's made it this far, though he did have to negotiate a gruesome death and resurrection to get here. "I wouldn't have wanted to go on for another year, but if it had finished last year, it wouldn't have felt long enough. Maybe the most special year was the first. We weren't being recognised in the street, we didn't know what we were doing, we were having a great time."
Thronesmania is, he acknowledges, "bizarre and weird". "Like, being in Spain and there being a crowd of 500, maybe 600 fans camped outside the hotel every day, and you have to get through them. It feels like being Bieber or something." When Snow became the brooding heart-throb of the Thrones world, Harington became a kind of boyband pin-up, too. "Yeah," he sighs. "I don't particularly enjoy that." He catches himself. He speaks of his "privilege" often and is acutely conscious of anything that sounds like he's whining. "I don't know. Do I? I'm glad I've experienced it, but that's what I mean about it being eight years, then it's done. You couldn't go on for much longer. It's a bit incessant."
Fame of that kind makes him moody, he explains. He's sceptical of all it brings, though he was thrilled he got to take his brother to the Italian Grand Prix recently. When he says the word "celebrity", he makes air quotes. "It makes me snappy and it makes me uncomfortable, and I turn into a grumpy person." It's unfair on his friends, he says, who bear the brunt of it, but the main problem is selfies or, as he puts it, "the photo thing". Now, he'll set himself days where he won't take pictures with people. He performs his rejection spiel, which is polite and apologetic, but can piss people off. "But you just have to, otherwise you start feeling like a mannequin. Especially me and Rose, we never do a photo together. Because then it makes our relationship feel like… puppets." He feels around for the right word. "Like we're a walking show."
Rose is Rose Leslie, his one-time Game Of Thrones co-star, who played his love interest, the wildling Ygritte, from 2012 to 2014. The pair fell for each other in real life in the freezing Icelandic countryside, but were reluctant even to acknowledge that they were together for a long time. Recently, they've edged out on to the red carpet, holding hands. "We're living together and we're very much a couple," he says, reading from the script. He's nice enough to stumble over why he doesn't want to talk about her, though he's clearly smitten. "I just don't – and I won't say much about it now – but I don't believe in talking about one's relationship in the press because it's… I strongly believe it's her relationship and mine, and anything I say here, she may not want me to say. So I just don't say it. I genuinely think it's our private life." Yet there are Buzzfeed lists dedicated to how cute their relationship is, whole YouTube compilations that show them being nice about each other in interviews. He grins a wide grin. "Are there?" It's adorbs, I tease. "She'd approve of adorbs," he smiles, but that's all he'll say. Two days after we speak, their engagement announcement is announced.
Harington is here to talk about Gunpowder, a grim new historical drama in which he plays Robert Catesby, the mastermind behind the plot to blow up the Houses of Parliament in 1605. Harington's middle name is Catesby; it's his mother's maiden name and Robert was an ancestor. "It's always been a piece of family curiosity, really. 'Do you know, if you go all the way back, your ancestor was the leader of the gunpowder plot, even though everyone thinks it was Guy Fawkes?'" He was so taken with the tale that it became the first project of the production company Thriker Films, which Harington set up with his best mate from drama school. It will appear in the blockbuster Saturday night slot on BBC1.
In 2014, Harington told his agent that, when he wasn't playing Snow, he'd like to branch out: "No more swords, no more horses – and maybe I can cut my hair." Yet in Gunpowder he's a sword-wielding, horse-riding, long-haired hothead. "Yeah. Hahaha. I always say things and end up backing down on them. It was weird, though, because I walked right into it. I pitched a TV show where I have long hair and a beard, and it got made. It doesn't mean I'm always going to play swords and horses."
During his time off from Game Of Thrones, he squeezed in other roles. He was the lead in the feature-length Spooks spin-off MI-5, and the historical action romp Pompeii, though it's the forthcoming Xavier Dolan movie The Death And Life Of John F Donovan that looks most promising. While Harington has got world-weariness down to a fine art in Game Of Thrones, he's also very good at playing the clown: in HBO's silly tennis spoof 7 Days In Hell, he pulled off the dizzy posh Brit player with gusto. "I love playing a thicko," he says, then corrects himself, in case anyone is offended. "That's probably an incredibly terrible term, thicko. But, you know, someone who is wonderfully well-meaning, but isn't… I've always been the kind of person who's well-meaning but slightly vacant at times."
This vagueness has its drawbacks. "Because you don't want to be defined like that. There's an element that I can get, 'Ahhh, sweet Kit, little Kit.'" He's 5ft 6in, he says, and there's that baby face, under the hair. "And I've worked very hard against that. I don't want to be patronised." Do people patronise him? "No, I don't think so. If they do, you shut it down pretty quick." He grows so quiet that it's hard to hear him. "You have to stand up for yourself. But I won't go into that." He takes a deep breath, and shuts himself up. "Mmm."
Harington is worried he doesn't come across well in interviews. He frets that, in print, his dry sense of humour can sound "really fucking arrogant". As a result, he's a concertina of openness and caution. He comes across as more serious than I suspect he might be under normal circumstances. He'll give an answer, and then say, "I don't know" as if warning you not to take what he's just said as gospel.
Has being objectified made me feel uncomfortable? Yes. Do I think my position is the same as a woman's in society? No
Last year, an interview he gave was picked apart over comments he made about being objectified, and how the film industry could be sexist towards men as well as women. "I was wrong there, though," he shrugs. "Sexism against men is not something I should have really said. I think what I meant was, being objectified. At that time, I did feel objectified, and now I've learned how to control that." How? "Just shutting it down. Look, I do think men can get objectified. I do feel I have been objectified in the past, sexually as well, in pieces that have been written about me." I've seen a couple, I say, thinking of one article that highlighted the bulge under his loincloth with the aid of several arrows. "Has that made me feel uncomfortable in the past? Yes. Do I think my position is the same as a woman's in society? No. They're very different things, and I should have separated them. I was wrong."
Harington was raised in Acton, west London, and has an older brother, who works in IT. His mother Deborah is a playwright; his dad, Sir David Harington, a businessman. They moved to Worcestershire when Harington was 11. "There was a very good comprehensive school there, and that's why. It's something I'd like to emulate with my kids – when I have kids."
He's already thought about how he wants to raise his children. "Um, they get brought up in London, hopefully, and see a very multicultural society, and hopefully go to a state primary school, and have the first 11 years of seeing the city I love. Then get the beauty of going to the country and being given space and air, and have the beautiful halcyon memories that I have. It's the sense of space, the big open sky, that in those years can be good for thinking and emotions."
The word "halcyon" makes me wonder. Did he, as a teenager, write bad poetry? "Oh yeah," he says proudly, unfolding his arms. "Yeah, I did that. I was a horrible kind of romantic creative, and I had all sorts of ideas about what I wanted to do. Acting, journalism…" It was acting that stuck, in the end, and sent him back to London, to drama school. "Do you know what," he suddenly announces, grinning. "I'm going to stick my neck out and say a couple of [the poems] were quite good. But I haven't written poetry in ages."
If any fans weren't sufficiently convinced of Harington's glossy-haired, woke-Byronic-boyfriend appeal, then the fact that he writes poetry may tip them over the edge. "I still read a lot of poetry. There's one I read recently by Jack Underwood, called Happiness. I'd urge anyone to go and read it." What's it about? "Just the happiness he finds in his domestic life, and it's beautifully written, and it's profound in its exploration of happiness." He looks as if he thinks he's revealed too much. "I don't know why I went into that."
Harington's dad is a baronet, and he's descended not only from Robert Catesby, but also from Charles II, and John Harington, who invented the first flushing toilet for Elizabeth I: I'm surprised he went to a state school, I say. "Mum and Dad didn't have the money to send us to private schools, first and foremost, but second, they wouldn't have wanted to. They believe in the state system, they believe in the NHS, they believe in state education, and they've instilled that in me." He says his upbringing was privileged – "I was very middle class: not loads of money, not no money" – but that he's not quite what you might think he is. "One's family history is one thing, and I'm very proud of my family history, but it doesn't directly speak of who I am," he says, which, really, is a polite response to being told by a stranger that you're not as posh as they thought.
Harington is keenly aware that there's an ongoing debate about working-class access to the arts, and to acting, but thinks we're kicking the wrong target by criticising Etonian actors rather than drama cuts. "There has to be more effort put in at an educational level, to give people those opportunities. Let's face it, Eddie Redmayne, Benedict Cumberbatch, they are very good actors who deserve to be where they are, and they got there because their educational systems recognised their talent. That needs to happen [in state schools]."
Last year, he read a poem alongside Cate Blanchett and Chiwetel Ejiofor, in support of the UN refugee council. A few months earlier, while playing Doctor Faustus in the West End, he stood up for young theatregoers when one critic slammed them for taking pictures ("If the theatre is going to die of anything," he responded, "it will be from exactly this type of stereotyping and prejudice").
What makes him stick up for the underdog? "I don't know. What's the point of sticking up for the guy at the top? I'm not an underdog. I've been given every opportunity and I've really ended up in a place of great privilege. I do believe it's your duty to try to share that privilege."
When Snow died at the end of season five, Harington spent months fibbing his locks off about how that was definitely it for him on Game Of Thrones; he was done with it, and that, no, he'd never return. At the start of season six, he was conjured back into existence, to live another day, and to have incestuous boat sex with his aunt. Since he lied so well then, how can I believe anything he's just told me? "You can't." He gets up for a quick cigarette and grins. "I might change my mind about anything I've just said. But that's my privilege, I guess."
Jim Rose Remembers Radio Monday Nov 06, 2017
November 06, 2017 [Monday] Issue #1666
HAPPY 97TH BIRTHDAY KDKA-1020 On November 2, 1920, the Westinghouse Electric Company of East Pittsburgh, PA launches the world's first broadcast of the 1920 election results to listeners on KDKA-1020. Happy 97th Birthday to The Pioneer Broadcasting Station of The World Newsradio KDKA-1020 The Voice of Pittsburgh. Frank Conrad Assistant Chief Engineer of Westinghouse Electric builds a transmitter on the second floor of his Wilkinsburg garage with license 8-X-K which becomes KDKA-1020.
CUMULUS FAILS TO MAKE $24M INTEREST PAYMENT As Cumulus Media's restructuring talks intensify, Cumulus board of directors forgo scheduled interest payment of $23.6 million due November 1, 2017. In its 8-K filing with the SEC, Cumulus acknowledges its private discussions with lenders and note holders. If Cumulus fails to make the interest payment by December 1, it matures into an Event of Default. Cumulus hopes that progress on its restructuring can be made in this month of November.
ENTERCOM Q3 NET REVENUE IS UP 1% Inside Radio reports before closing its merger with CBS Radio, for the third quarter of 2017, Entercom's net revenue is $122.3 million, which is an increase of 1% over 2016's $121.6 million. Entercom operating income is $13.5 million, after $8.8 million in merger and acquisition costs, compares to $25.7 million in the third quarter of 2016. On Wednesday, November 1, 2017 the pending Entercom-CBS Radio merger takes a giant leap forward with Justice Department approval.
CBS RADIO BECOMES ENTERCOM During my second time as a DJ on KILT FM Houston, TX from 1991-96, its owner Westinghouse buys CBS Radio. Westinghouse scoops up all of its radio stations under their new CBS Radio umbrella. Lots of other drastic changes occur. The Telecommunications Act of 1996 drains radio of its life. This causes 1996 to be my final year in radio. With its purchase of CBS Radio, Entercom creates a bigger footprint that contains a nationwide spoor of 235 radio stations, that includes radio stations in 22 of the top 25 markets.
TRIVIA QUESTION: What is the first full time radio Sports Talk station? The answer appears below.
•MARV NYREN Chicago native former VP of corporate sponsorships of Chicago Public Media WBEZ FM (91.5) is new Chicago VP/Market Manager of Cumulus Media News/Talk WLS-890, Classic Hits WLS FM (94.7), Alternative WKQX FM (101.1) and Classic Rock WLUP FM (97.9). •JOANNA CAMPBELL moves on up from weekends on Times-Shamrock Classic Rock WZBA FM (100.7 The Bay) Baltimore, MD to middays fulltime. •WXNR FM Curtis Media Group CHR (99X) Greenville-New Bern-Jacksonville, NC flips to Rhythmic CHR as HOT 99.5.
•1952 MICHAEL CUNNINGHAM writer born on November 6th. •1960 JOHNNY HORTON (Battle of New Orleans #1 – 6 weeks-1959) dies in car crash on November 5th near Milano, TX. Former Elkins Radio classmate Jimmy Tomlinson's brother guitarist Tommy suffers broken leg. •1979 AYATOLLA KHOMEINI takes over Iran on November 6th. •1985 ALABAMA wins two American Music Awards on January 28th. •1991 YELTSIN outlaws Communist Party on November 6th. •2004 PHIL VASSAR's In A Real Love is #1 in Billboard Country on November 6th.
WINK MARTINDALE [Host of over 20 TV game shows-ex DJ on WHBQ-560/KHJ-930/KMPC-1540] (Los Angeles/Calabasas, CA) Jim: I thoroughly enjoy your columns. I keep saying, "Wink, you must write a word of thanks to Jim!" Just as our dear friend and colleague Jack Parnell does from time to time. Finally, here I am with a good reason for writing. Radio is, and always will be, my first love. It's so sad to see radio in a somewhat downward spiral. In far too many markets (large and small), radio has become a background activity rather than foreground. Now this: I'm sure you're aware of the FCC saying goodbye, so long to the MAIN STUDIO RULE. Another death knell for local radio, in my view. "A politically divided FCC" voted to eliminate the 80-year-old main studio rule that required radio and TV broadcasters maintain a main studio in or near their community of license. FCC Chairman Ajit Pai said the requirement had become OUTDATED in the digital age, which allows the community to engage with stations via social media or email without having a physical studio nearby. He also said maintaining a physical address is an expense better put to other uses, like adding more local programming. (That I like). Stations are still required to have a local, toll-free telephone number, and to maintain any portion of their public files that is not online at a publicly accessible location within their community of license. Democrat Mignon Clyburn remarked, "Today is a solemn one in the history of radio and television broadcasting. By eliminating the main studio rule in its entirety for all broadcast stations – regardless of size or location – the FCC signals that it no longer believes, those awarded a license to use the public airwaves, should have a local presence in their community." Clyburn also said a toll-free number just didn't cut it. I totally agree. The thinking of an inept, impotent Federal Communications Commission and Dept. of Justice allowed companies to expand far above their ability to operate profitably. An owner with offices in New York cannot run a radio station in Jackson, Tennessee! Bring back ownership caps and give preference to local control. Let iHeart, CBS, Cumulus and EMF die a deserved death and let local owners bring listeners back to radio with localized, compelling programming that makes people WANT to listen. That's my opinion and I'm stickin' to it. Best blessings, Wink Martindale Calabasas, CA
BILL CHERRY [Realtor, In late 1950s Music 'til Dawn host on WWL-870] (Dallas, TX) Subject: Jim Rose Remembers Radio (#1665) November 03, 2017 [Friday] Jim, loved the mention of Gene Nobles (sponsored by Randy's Record Shop). Bill Cherry
NEWS BREAKS! JRRR NOW IN LYNN WOOLLEY's WBDaily (www.wbdaily.com/) in the Radio and Television Categories section. JRRR IS IN CLAUDE HALL's Vox Jox. www.VoxJox.org/ CASH BOX MAGAZINE FEATURES JRRR EVERYDAY ALL WEEK LONG http://cashboxmagazine.com/ Click the Jim Rose Remembers Radio Banner on the Home Page. MEDIA CONFIDENTIAL 'Liked' Links http://mediaconfidential.blogspot.com/JRRR IS IN RadioDX. Click http://radionx.com/radio-blog-news/. JRRR is in Bob Dearborn's THE OLDE DISC JOCKEY's ALMANAC http://oldediscjockeysalmanac.blogspot.com/. Send Your Radio News And Comments To Rosekkkj@earthlink.net.
CHUCK DUNAWAY [ex WABC-770, KILT-610, KLIF-1190, KBOX-1480, WKY-930 & WIXY-1260] (Houston-Katy, TX) Yes, Dave (Jarrott) worked for me at (KHFI FM) K-98…he did mornings. I did quite (well) in Austin (a)t K-98 and later at KLBJ (590) am and fm never got involved in the local politics at all…the best thing about Austin is that I met Kendall while there…the best thing in my life…Take care pal…chuck
TRIVIA ANSWER: In 1987, WFAN-770 New York City debuts as the first full-time Sports Talk radio station.
CHARLES GEORGE [KC5RAI, ex KCHU FM] (Dallas, TX) My right shoulder was hurting now it has shifted to my left shoulder. So I limit my Guitar playing and play my Soprano Recorder and Record Player more. I just got a new LP from The Sound Of Vinyl of Big Bill Broonzy, I Love My Whiskey. I think it interesting how many Copy a Cover when the original is available. My father said he often had people ask about getting a book from the Library but they could only tell him about the Front Cover and maybe the Subject matter. My father and mother often had views about Book Value. Like a song I learned as a child in church, In Times Like These. 73, Charles, KC5RAI
BIG BILL BROONZY (June 26, 1893 – August 14, 1958) St. Peter don't you call me 'cause I can't go. I owe my soul to the company store.
Today's Inspiration Station With Rhonnie Scheuerman
Author Ruth Bell Graham: The Fools
"THE FOOL says in his heart "There is no God.""This designation goes for those who doubt God's sovereignty as well as those that deny him. Either he is sovereign, or He is not God. Therefore when we become so preoccupied with and dismayed by circumstances and certain people that we doubt God's ability to Handel things in His own way, and in His own time, then we, too are fools.
Ruth Bell Graham | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,816 |
\section{Introduction}
Surface diffusion coefficient is one of the factors that decide about the crystal growth process. In order to understand formation of nanostructures it is necessary to understand the underlying microscopic processes, such as adsorption, desorption and diffusion. A lot of studies focus on dynamics of a single adatom \cite{gomer,liu,tringides}. However,
diffusion of small clusters, e.g., dimers and trimers, which
occurs with lower barriers \cite{ehrlich,Michely1,Evans1,kellog}, can be also significant and can affect island formation.In particular on metallic surfaces even diffusion of larger clusters can be very effective\cite{Evans1,kellog}.
Diffusive motion of atoms and clusters at the surface is studied experimentally by scanning tunneling microscopy (STM) and field ion microscopy (FIM), both used for direct imaging of individual adatoms \cite{Morgenstern1,Morgenstern2,Repp1,bogicevic}. With the help of other techniques, such as field emission microscopy (FEM), photoemission electron microscopy diffusion coefficient can be measured.
As for theoretical approaches, the first step is always calculation of the adiabatic potential, that means all the adsorption sites and energy barriers between them \cite{Marinica1,karim,wu,Wang1,Hayat1}. Then these data are used to analyze the diffusive motion. It can be done by means of simulation methods, such as kinetic Monte Carlo (kMC) or molecular dynamics \cite{Wang1,Hayat1,Michailov1,Michailov2,Michailov3,Flores1}. The latter solves the equations of motion for adatoms in a given potential. Their results are, however, limited to shorter time scales than those obtained in kMC simulations. Both these methods must be applied for specified potential or jump rates and at specified temperature. By changing any of these, one has to perform the simulation again. Below we derive analytic expressions for diffusion coefficients for adatoms and dimers based on the existing picture of the potential energy landscape.
Otherwise than MC or molecular dynamic calculations, analytic formulas express diffusion coefficient as a function of temperature and other model parameters \cite{ala,penev,penev2,haus}. It gives much more insight into the real physics of the system. Moreover, if new, more accurate data about the energy landscape are obtained, it is possible to improve the diffusion coefficient values on using the same formulas. Among theoretical approaches there are methods based on Langevin equation, Fokker-Planck equation or path integral formalism \cite{ala}. Our approach, on the other hand, is built on the master equation. Similarly to the method developed by Titulaer and Deutch \cite{Titulaer1}, which was later applied in \cite{Sanchez1,Salo1} for metal surfaces, we find the diffusive eigenvalue of the rate matrix. However, we do that by means of a variational method.
In this work we calculate the diffusion coefficient for Cu monomer and Cu dimer on Cu(111) and Ag(111) surfaces. In such a way dynamics of homo- and heteroepitaxial system is compared for the same adatom. Such systems have been investigated both experimentally \cite{Morgenstern1,Morgenstern2,Repp1} and theoretically \cite{Marinica1,karim,wu,Wang1,Hayat1,Flores1}. In addition to diffusion of a dimer, there have been also studies concerning adatom clusters of larger size \cite{karim,wu,Wang1,Flores1,signor}. Hetero-diffusion appears to be easier with lower activation barrier and homo-diffusion is dominated by relatively high energy barriers between neighboring cells, whereas rotation inside each cell can run freely.
We calculate diffusion coefficient using adsorption energies and energetic barriers for jumps between sites obtained in Refs.\cite{karim,Hayat1} by means of ab-initio calculations. We get analytical expressions for diffusion coefficients from which general relations between different diffusion modes at various temperatures are derived. Effective diffusion coefficients are represented by prefactor and activation energy parameters. In general both these parameters show temperature dependence which is in contrast with the global Arrhenius behaviour of the diffusion known for simple systems.
Such temperature dependence of diffusion could explain some discrepancies in results obtained from different experiments \cite{chvoj,chvoj2,bietti}.
Analysis of diffusion coefficients also shows that whereas dimer diffusion on Ag surface happens via translation together with rotation, on Cu surface both moves are separated, and fast rotation is limited only to cells, whereas slow intercell jumps decide about rate of translational diffusion of dimers.
\section{Calculation of diffusion coefficients}
Variational approach to the calculation of the diffusion coefficient from a set of master equations was proposed in Ref. \cite{Gortel1}. Since then it has been applied to many different systems of adatoms jumping on a crystal surface. So far mostly strictly theoretical systems have been considered. However, recently we have succesfully applied the variational approach to the diffusion of a single Ga adatom on the GaAs(001) surface c(4x4) $\alpha$ and $\beta$ reconstructions \cite{M;Z-K1} using the energy landscapes from Refs. \cite{Roehl1} and \cite{Roehl2}. It is also possible to solve a problem of correlated particles using our approach. Till now single adatom has always been treated as a simple particle, that is a particle whose state could be completely described by giving its position on the lattice. It could jump to one of the neighbouring sites on the lattice provided it wasn't already occupied. Such an adatom didn't possess any additional degrees of freedom like for example orientation or shape.
In this paper we describe dimer diffusion, namely Cu dimer on Cu(111) and Ag(111) surfaces. A dimer could be thought of as a compound particle or a cluster, that is an ensemble consisting of a certain number (two in the case of dimer) of simple particles that are bound together. Therefore, the move of the dimer should always be considered as the move of the whole, not as that of the individual constituents. The state of the dimer can be described fully by giving the position of its centre of mass, its length (the distance between the constituents) and its orientation along the underlying lattice. Therefore, the approach can be generalised to compound particles without any modifications.
Diffusion coefficient of a single particle jumping on a crystal lattice is defined as the mean square displacement over time \cite{gomer,tringides,ala,haus}
\begin{equation}
D_{nm}=\lim_{t\rightarrow\infty}\frac{1}{4t}<\Delta r_{n}(t)\Delta r_{m}(t)>.\label{diff_def}
\end{equation}
Indices n and m denote one of the space coordinates (x or y) on the lattice. $\Delta r_{n(m)}(t)$ is the displacement of the particle's centre of mass in the n(m) direction after time t. Since we follow the centre of mass, it doesn't matter if by a particle we mean a simple point particle or a cluster. Position of its centre of mass is always well defined. The above definition is used for extracting the diffusion coefficient from the data obtained from scanning tunneling microscopy or molecular dynamics methods.
In order to analyse diffusive motion on a crystal surface we can divide arbitrarily the lattice into unit cells which repeat themselves periodically over infinite surface. Origin of each cell has position $\vec{r}_{j}$ in the space and within given cell there are $m$ adsorption sites, whose locations relative to the cell's origin are given by $\vec{a}_{\alpha}$, where $\alpha=1,\ldots m$. Therefore, each adsorption site on the lattice is specified unambiguously by parameters (j,$\alpha$) and its position in the space is equal to $\vec{r}_{j}^{\alpha}=\vec{r}_{j}+\vec{a}_{\alpha}$. Physically, position of a given adsorption site corresponds to the position of the centre of mass of the particle (simple or compound) occupying that site. A particle occupying the site (j,$\alpha$) has the adsorption energy $E(\alpha)$. In general, it is possible for two or more different adsorption sites to have the same position. They will be distinguished by the index $\alpha$ though. For the coverage $\rho$ (ratio of adparticles at the surface to the total number of adsorption sites) the probability of finding a particle in the site of the energy $E_{\alpha}$ is equal to $P_{eq}(\alpha)=\rho\exp[-\beta E(\alpha)]/\sum_{\gamma}\exp[-\beta E(\gamma)]$. The summation is done over all the sites in the cell and $\beta=1/k_{B}T$ is the inverse temperature factor.
Diffusive motion on a surface consists of a series of thermally activated jumps between adsorption sites. According to the transition state theory, the jump rate (number of jumps per unit of time) is a function of the energy barrier along the path of the jump \cite{TST}. For a single jump from the (j,$\alpha$) site to the (l,$\gamma$) site it is equal to
\begin{equation}
W(j,\alpha;l,\gamma)=W_{\alpha,\gamma}=\nu\exp\{-\beta[\tilde{E}(j,\alpha;l,\gamma)-E(\alpha)]\},\label{jump_rate}
\end{equation}
$E(\alpha)$ is the adsorption energy of the initial site of the jump and $\tilde{E}(j,\alpha;l,\gamma)$ is the energy at the saddle point between the two sites involved in the jump. These are the energies that are found by means of ab-initio calculations \cite{Hayat1}. Variable $\nu$ is the diffusion prefactor, whose exact value is a subject for a separate study, however, its value is very often assumed to be of order of $10^{13}/s$. It is sometimes also called the attempt frequency and its physical meaning is the number of attempts of the particle to jump out of its site per unit of time. The exponential part in (\ref{jump_rate}), on the other hand, is the probability of a successful attempt.
Time evolution of a system is described by the Master equation
\begin{equation}
\frac{d}{dt}P(j,\alpha;t)=\sum_{l,\gamma}[W_{\gamma,\alpha}P(l,\gamma;t)-W_{\alpha,\gamma}P(j,\alpha;t)].\label{master_eq}
\end{equation}
$W_{\gamma,\alpha}$ are jump rates defined by (\ref{jump_rate}) and P(j,$\alpha$;t) is the probability that the site (j,$\alpha$) is occupied. We are looking for the diffusive eigenvector of Eqs (\ref{master_eq}).
We will calculate this eigenvector by the variational method. Details of this method are described in Appendix A.
\section{Diffusion of $Cu$ monomer and dimer on $Cu(111)$ and $Ag(111)$ surfaces}
Both copper and silver crystallize in the face centered cubic structure and their (111) lattices consist of two types of sites called fcc and hcp, and arranged in the way shown in Fig. \ref{fcc111}. The lattice constant for silver is equal to $4.09\AA$ compared to $3.61\AA$ for copper. This difference in lattice constant is due to the interaction strength between substrate atoms and it also determines the shape of the energetic surface that is seen by adsorbed Cu atoms.
The energy landscape for Cu monomer and Cu dimer on Cu(111) was calculated in Ref. \cite{Marinica1} and on Ag(111) in Ref. \cite{Hayat1} using ab-initio methods. On the base of these calculations we can see that Ag surface observed by Cu adatom in hetero-diffusion process is more smooth. There is no evident difference between jump rates inside and outside surface cell. At the same time in the homo-diffusion process Cu/Cu(111) jumps from one cell to the neighboring one are the slowest ones and determine the global diffusion rate.
We use here the potential energy surfaces from Refs. \cite{Marinica1,Hayat1} to calculate the diffusion coefficient for both Cu monomer and dimer. For the dimer we calculate also rotational diffusion coefficient and compare it with the translational one.
\begin{figure}
\begin{center}
\includegraphics[width=6cm]{Ag111.png}
\end{center}
\caption{Fcc(111) lattice with two possible energy minima for a Cu monomer.}
\label{fcc111}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{konfiguracje.png}
\end{center}
\caption{Possible configurations of Cu dimer on the fcc(111) surface. The sfh configurations appear only on the Ag(111).}
\label{conf}
\end{figure}
A single adatom can occupy either fcc or hcp site. These sites differ from each other by the position of the atoms lying directly below. An adatom occupying an fcc site will therefore form bonds of different lengths and directions with underlying surface atoms than the one that occupies an hcp site. It means that the potential energy of the adatom will depend on whether it resides at an fcc or an hcp site. However, according to both the experiment \cite{Morgenstern1,Morgenstern2,Repp1} and the calculations \cite{Marinica1,Hayat1}, the energy difference of these two sites is very small, of order of single meV at both Cu(111) and Ag(111) surfaces.
At the same time dimers formed at these surfaces see completely different landscape of potential energy. It is observed that dimers become quite stable and long-lived structures \cite{Morgenstern1,Morgenstern2} and it is possible to study their diffusional movement. Dissociation of the Cu dimer on Cu(111) has been never observed experimentally or in simulations. According to \cite{Marinica1} the energy barrier for such a process is 450 meV, which is much higher than for most of the possible jumps of the dimer. On the other hand, dissociation of the Cu dimer on Ag(111) has occured at 700 K in molecular dynamics simulations \cite{Hayat1}, which suggests that the corresponding energy barrier is reduced on that surface compared to Cu(111). However, the dissociation was only temporary and after a few ps the constituents recombined forming the dimer again. Therefore, the dimers can be treated as stable particles in our temperature range.
According to Ref. \cite{Hayat1} Cu dimer on Ag(111) can be found in one of the four possible energetic minima (see Fig. \ref{conf}). They differ by the type of site occupied by each single particle in the dimer (fcc or hcp) and by the distance between those particles. Different dimer configurations are called ff (two fcc sites), hh (two hcp sites), sfh (short fh) and lfh (long fh). The ff and hh states differ only by the type of occupied sites while the distance between the particles is the same in both cases. On the other hand, in sfh and lfh states, particles occupy the same pairs of sites but the distance between them can be either short (sfh) or long (lfh). At Cu(111) surface Cu dimer can be found in ff, hh or lfh configurations only. The sfh state is not built on this lattice \cite{Marinica1}. Each of the dimer's energy minima on this surface can appear in three variants which differ by their relative angular orientation. All the possible configurations of the Cu dimer on the fcc(111) lattice are shown in Fig. \ref{conf}.
It has been argued \cite{Hayat1} whether the sfh and lfh states are actually potential minima on the Ag(111) lattice or just metastable states. They lie only a few meV below the calculated transition states to the ff and hh states and the potential energy landscape around sfh and lfh states is almost flat. However, authors treat these configurations as separate states and in our calculations below we shall also consider the sfh and lfh sites as energy minima. In other situation all barriers for jumps between states should be calculated from the beginning and then they can be used in our formula giving similar results as the one shown below.
\begin{figure}
\begin{center}
\includegraphics[angle=-90,width=7cm]{Cu_Cu_dyfuzja}
\includegraphics[angle=-90,width=7cm]{Cu_Ag_dyfuzja.eps}
\end{center}
\caption{Diffusion coefficient for Cu monomer and Cu dimer on the Cu(111) (top) and Ag(111) (bottom) surfaces as the function of the temperature. $\nu=10^{13}/s$.}
\label{diff_coeff}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[angle=-90,width=7cm]{Cu_Cu_en_akt.eps}
\includegraphics[angle=-90,width=7cm]{Cu_Ag_en_akt.eps}
\end{center}
\caption{Effective activation energy for the diffusive motion of Cu monomer and dimer, and for the rotational diffusive motion of dimer on the Cu(111) (top) and Ag(111) (bottom) surfaces as the function of the temperature.}
\label{eff_act_en}
\end{figure}
\subsection{Monomer}
Structure of the energy landscape as seen by Cu monomer is the same for both Cu(111) and Ag(111) surfaces. Both the energetic minima and the barriers are found at the same positions. The only difference is the exact values of the energies. Therefore, we can derive a general formula for the monomer and then substitute specific values for both Cu(111) and the Ag(111) surfaces in order to calculate the diffusion coefficients.
As it is apparent from the first-principles calculations \cite{Marinica1,Hayat1} and from the experiments \cite{Morgenstern1,Morgenstern2,Repp1}, Cu monomers are slightly more stable at fcc sites than at hcp sites. For Cu(111) surface the difference between energies at those two sites is equal to 5 meV \cite{Marinica1} and for the Ag(111) lattice it is equal to 6 meV \cite{Hayat1}. The energetic barrier for the jump from an fcc to an hcp site on Cu is equal to 0.041 eV and to 0.075 eV on Ag. The barriers for inverse processes are 0.036 eV and 0.069 eV, respectively. We use these data in order to calculate the diffusion coefficient for a single Cu adatom on the Cu(111) and the Ag(111) surfaces. The variational parameters in (\ref{var_form}) are equal to $\vec{0}$ because of the system symmetry. Therefore, we just have a sum of two terms, each related to one type of jump, which can be also expressed as
\begin{equation}
D=\frac{a^2}{2}\frac{W_{fcc\rightarrow hcp}W_{hcp\rightarrow fcc}}{W_{fcc\rightarrow hcp}+W_{hcp\rightarrow fcc}}\label{diff_mon}
\end{equation}
The parameter $a$ in the above formula (and in all equations in this article) is the length of the lattice constant of the underlying surface, which is also equal to the bond length of Cu or Ag. It is related to the bulk lattice constant $A$ by $a=\frac{A\sqrt{2}}{2}$ what gives $a_{Cu}=2.55 \AA$ for Cu and $a_{Ag}=2.89 \AA$ for Ag surface.
According to (\ref{diff_mon}), the diffusion of Cu monomer on Cu(111) and Ag(111) is given by the same formula as this for the one-dimensional diffusion of a single particle in the potential with two alternating minima. The temperature dependence of the monomer's diffusion coefficients is shown in Fig.~\ref{diff_coeff} for Cu surface in top panel and for Ag surface in bottom panel, in both cases plotted as top lines. It can be seen that monomer diffusion on Ag surface is slightly slower than this on Cu lattice. Assuming a local Arrhenius behavior of the diffusion coefficient $D=D_{0}\exp(-\beta E_{a})$ it is possible to calculate the effective activation energy $E_a=-\frac{\partial\ln{D}}{\partial\beta}$. For the monomer it is equal to
\begin{equation}
E_{a}=\frac{W_{fcc\rightarrow hcp}\Delta E_{hcp\rightarrow fcc}+W_{hcp\rightarrow fcc}\Delta E_{fcc\rightarrow hcp}}{W_{fcc\rightarrow hcp}+W_{hcp\rightarrow fcc}}
\end{equation}
It is obvious that a quantity defined in such a way depends on the temperature. $E_a$ for monomer on Cu lattice is shown in top panel of Fig.~\ref{eff_act_en}, drawn with the middle line and $E_a$ for monomer on Ag lattice it is shown with the top line in the bottom panel of Fig.~\ref{eff_act_en}. As seen in both cases $E_a$ does not change much, it starts at lower temperatures from the value equal to the height of the larger of the two barriers and at higher temperatures it goes down approaching the value which is the average of both barriers. Therefore, for Cu(111) surface the effective activation energy goes from 41 meV to 39 meV. We can compare then with low temperature experimental value 37$\pm$5 \cite{Repp1}. For Ag(111) surface activation energy starts at 75 meV at low temperatures compared to 65$\pm$9 meV in the experiment \cite{Morgenstern1}, then it goes down reaching 72 meV at high temperatures.
Knowing the effective activation energy, it is also possible to calculate the effective prefactor $D_{0}=D\exp(\beta E_{a})$. Its behaviour is shown in Fig. \ref{eff_pref} with the bottom line in the top panel and with the top line in bottom panel. In both cases at low temperatures they decrease rapidly up to about 100 K and then remain constant at higher temperatures.
\begin{figure}
\begin{center}
\includegraphics[angle=-90,width=7cm]{Cu_Cu_pref.eps}
\includegraphics[angle=-90,width=7cm]{Cu_Ag_pref.eps}
\end{center}
\caption{Effective prefactor for the diffusive motion of Cu monomer and dimer on the Cu(111) (top) and Ag(111) (bottom) surfaces as the function of the temperature. $\nu=10^{13}/s$}
\label{eff_pref}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=9cm]{Cu_Cu111+path.png}
\includegraphics[width=6cm]{Cu_Cu111_potential.png}
\caption{Scheme of the transitions on the Cu(111) (top) surface and energy profile (bottom) along the marked path for a single Cu dimer. Names of the sites are assigned according to Fig.~\ref{conf}.}
\label{jumps}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{path_intercell.png}
\caption{Realizations of intercell jumps via hh$\rightarrow$ff jumps at Cu(111) surface (top) and via sfh state at Ag(111) surface (bottom). \label{intercell}}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=14cm]{path_dimer.png}
\caption{Different diffusion modes on Cu(111) (top) and Ag(111) (bottom) surfaces. \label{path_dimer}}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[angle=-90,width=8cm]{Cu_dyfuzja_obr.eps}
\end{center}
\caption{Rotational diffusion coefficient for Cu dimer on the Cu(111) (top) and Ag(111) (bottom) surfaces as the function of the temperature. $\nu=10^{13}/s$}
\label{diff_rot_coeff}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[angle=-90,width=8cm]{pref_obr.eps}
\end{center}
\caption{Effective prefactor for the rotational diffusive motion of dimer on the Cu(111) (top) and Ag(111) (bottom) lattices as the function of the temperature. $\nu=10^{13}/s$.}
\label{eff_pref_rot}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=9cm]{Cu_Ag111+path.png}
\includegraphics[width=6cm]{Cu_Ag111_potential.png}
\end{center}
\caption{Scheme of the transitions on the Ag(111) (top) surface and energy profile (bottom) along the marked path for a single Cu dimer. Names of the sites are assigned according to Fig.~\ref{conf}.}
\label{jumps2}
\end{figure}
\subsection{Dimer diffusion}
For Cu dimer on the fcc(111) surfaces the situation is more complex than for the monomer. As discussed above we have the following dimer configurations: ff, hh, fh that differ by the positions of two atoms of dimer. As we shall see below the distance between the atoms also matters and in general we will have sfh (short fh) and lfh (long fh) positions. Moreover each configuration occurs in three different orientations. All the possible configurations found on the fcc(111) lattices are shown in Fig. \ref{conf}. The map of the possible transitions between all those sites is quite complex and differs significantly between Cu and Ag surfaces, therefore, below we shall discuss both cases separately. The energy landscape and values of energy for both the sites and the barriers have been taken from Ref. \cite{Marinica1} (for Cu(111)) and from Ref. \cite{Hayat1} (for Ag(111)) and used below to calculate the diffusion coefficients of Cu dimers on those lattices.
\subsubsection{Cu dimer on Cu(111) lattice}
Only three types of equilibrium states from Fig. \ref{conf} are found for dimer on Cu(111) lattice \cite{Repp1,Marinica1}. These are ff, hh and lfh. Configuration sfh has too high energy at this lattice hence it cannot be used by dimer to realize intercell jumps. Energies of dimer states and barriers for transitions between those states has been worked out in Ref. \cite{Marinica1}. We plotted the map of all the possible jumps in our system in Fig. \ref{jumps}. Yellow circles represent adsorption sites for the dimer (its centre of mass on the lattice) and the lines represent transitions. Three lfh sites with different subscripts have actually the same position but have been drawn as separate ones for clarity. The black lines represent intracell jumps, which are limited to a finite area on the lattice. The dashed black line between ff and hh sites means that this jump is directly between these sites and does not pass through the lfh site. Blue, red and green lines represent intercell motion in different directions, that is the one which goes beyond given cell. Energy landscape along the path marked in red is plotted in the lower part of Fig. \ref{jumps}. Plotted path leads dimer from one cell to the next one. Each cell is visible in top panel of Fig \ref{jumps} in the form of star. Intercell jumps have much higher energy barrier than intracell ones. In our calculations we shall take into account all the intracell jumps and the lowest in energy intercell motion, namely the concerted sliding between ff and hh sites, which has an energetic barrier of 120 meV. It is shown in Fig. \ref{intercell} how this particular intercell jump is realized and how it differs from the intracell jump of the same type. Barriers for all the remaining intercell jumps (including the dimer's dissociation) are at least three times larger, therefore these jumps are highly unlikely and will be omitted in further considerations.
We denote rate of the intercell jump by $V_{ff\rightarrow hh}$, while the intracell jumps will have rates $W_{ff\rightarrow hh}$, $W_{ff\rightarrow lfh}$, $W_{hh\rightarrow lfh}$ and $W_{lfh\rightarrow lfh}$. Symbols with tilde over them will be equal to corresponding rates multiplied by the equilibrium probability of the initial site's occupancy, for example $\tilde{V}_{ff\rightarrow hh}=P_{eq}^{ff}V_{ff\rightarrow hh}$. We shall use this notation in the whole article.
It can be seen in Fig. \ref{jumps} that the system possesses three axes of symmetry (one along the a sites, one along the b sites and one along the c sites). The lattice sites labeled by different subscripts (a, b or c) are energetically equivalent, that is they have exactly the same potential energy and the same neighboring sites with the same jump rates to them. In order to apply our variational method, we need to introduce some parameters that depend on the state of the system. The state of the system is given completely by specifying in which of the sites shown in Fig. \ref{conf} the dimer currently resides. Each of those sites should have its own geometric vector phase, which can be written for convenience in the angular form $\vec{\delta}=\delta_{r}\left(\cos{\delta_{\phi}},\sin{\delta_{\phi}}\right)$. The form of these phases should reflect the symmetry of the potential felt by the dimer at given site. For site $a$ it can be seen that there is symmetry with respect to the y axix, therefore we should have $\delta_{\phi}^{a}=\frac{\pi}{2}$, which gives $\vec{\delta^{a}}=\left(0,\delta_{r}^{a}\right)$. The angular parts of the corresponding $b$ and $c$ phases should then be equal to $\frac{7\pi}{6}$ and $\frac{11\pi}{3}$, respectively. Therefore, the only variational parameters will be the radial parts, that is $\delta_{r}^{ff}$, $\delta_{r}^{hh}$ and $\delta_{r}^{lfh}$. From now on we will omit the subscript r.
We insert such defined phases into the variational formula (\ref{var_form}) and, using simple differentiation, minimise the expression with respect to the parameters. Resulting expression for phases are given in Appendix B. Finally the diffusion coefficient for Cu dimer at Cu(111) surface is
\begin{align}
D=&\frac{3}{4}a^{2}\tilde{V}_{ff\rightarrow hh} \left\{ 1+3\left[\tilde{W}_{ff\rightarrow hh}(\tilde{W}_{ff\rightarrow lfh}+\tilde{W}_{hh\rightarrow lfh}) \right. \right. \nonumber \\
& \left. + 2\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{hh\rightarrow lfh}] \right] \left [(\tilde{W}_{ff\rightarrow hh}+2\tilde{V}_{ff\rightarrow hh}) \right.\nonumber \\
& \left.\left.(\tilde{W}_{ff\rightarrow lfh}+\tilde{W}_{hh\rightarrow lfh})+2\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{hh\rightarrow lfh}\right] ^{-1} \right \}.
\label{11}
\end{align}
As expected, the intercell jump rate $\tilde{V}_{ff\rightarrow hh}$ is fundamental for the diffusive motion of the dimer. The first term in the above sum describes the diffusion controlled exclusively by the $ff\rightarrow hh$ intercell jumps, which are quite slow. However, the second term has a large contribution to the total value of the diffusion coefficient. It describes the motion which is partially realised by the much faster intracell jumps. Therefore, we see that even though those jumps are unable to trigger the diffusion by themselves, they can enhance it significantly by bypassing some of the slow intercell jumps. These two modes of diffusion: one omitting intracell jumps given by first term of sum (\ref{11}) and the second through the inter and intracell jumps are illustrated in top panel of Fig.~\ref{path_dimer}. Fig. \ref{diff_coeff} shows diffusion coefficients for both modes. It is seen that the one plotted with the bottom line is much slower than the second one what means that appropriate diffusion path is less probable.
It should be also noted that jump rate $\tilde{W}_{lfh\rightarrow lfh}$ is not included in the above expression. It is understandable, since those jumps only rotate dimer between its different angular orientations. They do not move the dimer centre of mass at all. However, it is also possible to calculate the rotational diffusion coefficient. By analogy to (\ref{diff_def}) it is defined as
\begin{equation}
D_{r}=\lim_{t\rightarrow\infty}\frac{1}{2t}<\theta^2>.
\end{equation}
The variational formula (\ref{var_form}) can be used in the same form, if instead of the translational degree of freedom $\vec{r}$ we use the angular one $\theta$. The variational phases coupled to the angle $\theta$ go to zero, and it is easy to understand as all the sites are symmetric with respect to change of the angle. Thus the rotational diffusion coefficient for the Cu dimer on the Cu(111) is equal to
\begin{equation}
D_{r}=\frac{\pi^{2}}{18}(\tilde{W}_{ff\rightarrow lfh}+\tilde{W}_{hh\rightarrow lfh}+2\tilde{W}_{lfh\rightarrow lfh}).
\end{equation}
It implies that the rotational diffusion is realised in three independent ways. The contribution of the $lfh\rightarrow lfh$ jump is twice as large as these of $ff\rightarrow lfh$ and $hh\rightarrow lfh$ because there is only one $lfh\rightarrow lfh$ jump needed in order to rotate the dimer by 120 degrees. Both $ff\rightarrow lfh$ and $hh\rightarrow lfh$ need to be performed twice for the same effect.
It can be seen in Fig. \ref{diff_coeff} that the dimer diffusion coefficient is much smaller than that for the monomer. It is more difficult to compare translational and rotational motion of dimer because one is measured in $nm^2/s$ and the second in $rad^2/s$. In order to compare them both we multiplied $ D_r$ by $d^2/4$, square of half of the dimer length. Such value can be attributed to the diffusion of one of the atoms of dimer in the rotational movement. In Fig. \ref{diff_rot_coeff} the upper line shows Cu dimer rotation in this case and it can be seen that it is as fast as monomer diffusion over the surface and at the same time much faster than dimer translational diffusion. It means that dimer rotates at given cell and this rotation does not couple to the translational diffusion. We also calculated the activation energies for these motions just as we have already done for the monomer. The formula for the translational dimer's motion is too complex to write it down, therefore we just plot the result in Fig. \ref{eff_act_en}. The expression for activation energy for the dimer's rotation is
\begin{align}
E_{a}&=\left[ (E_{ff}+\Delta E_{ff\rightarrow lfh})\tilde{W}_{ff\rightarrow lfh}+(E_{hh}+\Delta E_{hh\rightarrow lfh}) \right. \nonumber \\ & \left. \tilde{W}_{hh\rightarrow lfh}+2(E_{lfh}+\Delta E_{lfh\rightarrow lfh})\tilde{W}_{lfh\rightarrow lfh} \right] / \nonumber \\ &( \tilde{W}_{ff\rightarrow lfh}+\tilde{W}_{hh\rightarrow lfh}+2\tilde{W}_{lfh\rightarrow lfh})-<E>\label{act_en}
\end{align}
where
\begin{align}
&<E>=\nonumber \\
&\frac{E_{ff}\exp(-\beta E_{ff})+E_{hh}\exp(-\beta E_{hh})+E_{lfh}\exp(-\beta E_{lfh})}{\exp(-\beta E_{ff})+\exp(-\beta E_{hh})+\exp(-\beta E_{lfh})}
\end{align}
is mean value of dimer energy in the equilibrium conditions. The first term in \ref{act_en} is the weighted arithmetic mean of the energies at the saddle points over which the dimer jumps while changing its orientation. The second term is the average energy that the dimer has at a given temperature. We show all activation energies for Cu(111) surface in the top panel of Fig. \ref{eff_act_en}. We see that the activation energy for the dimer translational motion is the largest one and this for dimer rotation is lower even than this of the monomer diffusion. Therefore dimer motion consists mostly of rotation, while the probability for the move of the center of mass is much lower. Rotation is realized via intracell jumps and we can compare the calculated value $E_a$=20 meV with measured at low temperatures $E_a$ intracell diffusion as 18$\pm$3 meV \cite{Repp1}. Monomer diffusion is somewhere in between. We can also see how prefactors for translational dimer motion (Fig. \ref{eff_pref}) and rotational motion (Fig. \ref{eff_pref_rot}) depend on temperature. It can be seen that prefactor for dimer diffusion is higher than that for monomer, but decreases with temperature in the same way, while prefactor for rotational diffusion increases with temperature reaching constant value from below. Below we will compare this behavior with this on Ag surface.
\begin{figure}
\begin{center}
\includegraphics[angle=-90,width=6.2cm]{dyf_dim-1.eps}
\includegraphics[angle=-90,width=6.2cm]{dyf_mon+dim_temp-1.eps}
\end{center}
\caption{Logarithmic scale plot of diffusion coefficient as a function of temperature (left) and as a function of inverse temperature (right) for Cu monomers and dimers on the Cu(111) and Ag(111) surfaces at low temperatures. Curves are described from top down. Two lines in the middle show monomer and dimer at Ag surface $\nu=10^{13}/s$.}
\label{diff_coeff_log}
\end{figure}
\subsubsection{Cu dimer on Ag(111) lattice}
For Cu dimer on Ag(111) there is one additional lattice position, namely the $sfh$ site (short $fh$), which wasn't present in the homoepitaxial case. According to Ref. \cite{Hayat1} there are four types of transitions between Cu/Ag(111) sites. The geometry of jumps on this lattice is significantly different than that on Cu/Cu(111) and it requires a separate consideration. The map of the energy minima and the possible jumps in the system is shown in Fig.~\ref{jumps2}. The path along which we plot energy potential is shown in top panel of Fig.~\ref{jumps2}. It can be seen that it leads from one cell to the next one. As before three neighboring cells can be seen in Fig~\ref{jumps2}. Three $lfh$ sites in each cell have the same position of the center of mass despite being drawn separately and the dashed line between $ff$ and $hh$ sites represents the direct transition between these sites.
We should point out some differences between the homoepitaxial and the heteroepitaxial case. First of all, as we have already mentioned, $sfh$ position is added. It means that the direct intercell $ff\rightarrow hh$ jump is replaced by jumps through the $sfh$ site (Fig.~\ref{intercell}). Because of this there exist more sequences of jumps that are possible, since the dimer from the $sfh$ site can jump either to one of the two $ff$ sites or to one of the $hh$ sites. Therefore, it is difficult to state unambiguously which jumps are intracell and which are intercell. Another difference is lack of $hh\rightarrow lfh$ and $lfh\rightarrow lfh$ connections.
We apply the variational method to the hetoroepitaxial system in the same way as we did for the homoepitaxial one. First we assign geometrical phases to each of the sites taking into account the symmetry, which is the same as in the previous case. Then we insert phases into Eq.~(\ref{var_form}) and minimize this expression with respect to them. Resulting formulas for phases can be found in Appendix B. And final expression for diffusion coefficient is
\begin{align}
D&=3a^{2}\{\tilde{W}_{ff\rightarrow hh}\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{hh\rightarrow sfh}(\tilde{W}_{ff\rightarrow sfh}+\tilde{W}_{hh\rightarrow sfh})\nonumber\\
&+\frac{1}{2}\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{hh\rightarrow sfh}(\tilde{W}_{ff\rightarrow hh}\tilde{W}_{ff\rightarrow lfh}\nonumber\\
&+\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{ff\rightarrow sfh}+\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{hh\rightarrow sfh} \nonumber \\&+\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{hh\rightarrow sfh})+\frac{1}{4}\tilde{W}_{ff\rightarrow hh}\tilde{W}_{ff\rightarrow lfh}(\tilde{W}_{ff\rightarrow sfh}^{2}\nonumber\\
&+\tilde{W}_{hh\rightarrow sfh}^{2})\} /\{2\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{hh\rightarrow sfh}(\tilde{W}_{ff\rightarrow hh}+ \tilde{W}_{ff\rightarrow lfh}\nonumber \\ &+\tilde{W}_{ff\rightarrow sfh}+\tilde{W}_{hh\rightarrow sfh})+\tilde{W}_{ff\rightarrow hh}(\tilde{W}_{ff\rightarrow sfh}^{2}\nonumber\\
&+\tilde{W}_{hh\rightarrow sfh}^{2}+\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{ff\rightarrow sfh}+\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{hh\rightarrow sfh}) \nonumber \\ &+\frac{3}{2}\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{hh\rightarrow sfh}^2\}.
\label{15}
\end{align}
The above formula is quite complex, however, there are still a few things that can be deduced from it. First of all, we can see that unlike the Cu(111) surface, there is no single type of jump on Ag(111) that is crucial for the diffusion. On the other hand, the diffusion needs at least two different types of jumps in order to occur. One single type of jump is unable to trigger diffusion by itself. However, it seems that the $ff\rightarrow sfh$ jump is the most important one since it appears in almost all the diffusion paths. As we can see, the distinction between intracell and intercell jumps is not so obvious here as it was on the Cu(111) surface.
When we ignore the two slowest jumps i.e. $W_{ff\rightarrow hh}=W_{hh\rightarrow sfh}=0$ the diffusion coefficient formula simplifies to
\begin{equation}
D=\frac{3}{4}a^2P_{ff}^{eq}\frac{W_{ff\rightarrow sfh}W_{ff\rightarrow lfh}}{W_{ff\rightarrow sfh}+W_{ff\rightarrow lfh}}.
\label{16}
\end{equation}
It corresponds to the zigzag motion observed by STM in Ref. \cite{Morgenstern1}.
For a more complex case, where the least probable transition is still $W_{ff\rightarrow hh}=0$ but $W_{hh\rightarrow sfh}\neq 0$,
the diffusion coefficient is
\begin{align}
&D=3a^2 \left[ \tilde{W}_{ff\rightarrow sfh}\left(\tilde{W}_{ff\rightarrow lfh}
\tilde{W}_{hh\rightarrow sfh}+\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{ff\rightarrow lfh} \right. \right. \nonumber \\
& \left.\left. +\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{hh\rightarrow sfh} \right) \right] / \left[ 3\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{hh \rightarrow sfh} \right. \nonumber \\
& \left.+4\tilde{W}_{ff\rightarrow sfh} \left( \tilde{W}_{ff\rightarrow sfh}+\tilde{W}_{ff\rightarrow lfh}+\tilde{W}_{hh\rightarrow sfh}\right) \right]
\label{17}
\end{align}
Here, in addition to the zigzag motion, we have also two other ones. It is also seen that jumps $W_{ff\rightarrow lfh}$ and $W_{hh\rightarrow sfh}$ by themselves are not able to make the dimer diffuse along the surface. It is understandable because these jumps are between two completely different pairs of states. In the bottom panel of Fig. \ref{diff_coeff} we plotted both approximations Eqs. (\ref{16},\ref{17}). In Fig. \ref{path_dimer} the successive dimer paths are shown. It can be seen that it is not zigzag motion (plotted in the lowest dashed line) alone that is responsible for the dimer diffusion. With two more modes added to this simplest one we can reproduce diffusion curve more precisely. The most probable dimer movement can be quite complicated as shown in Fig.\ref{path_dimer}.
The effective activation energy in the most general case is expressed by a complicated formula, therefore we do not write the equation, but just plot the results in Fig. \ref{eff_act_en}. Its behavior is a bit more complex than for the monomer's case, what is not surprising since the dimer's map of transitions is also more complex as is the diffusion coefficient. At low temperatures the activation energy for the dimer is 75 meV compared to 73 meV in the experiment \cite{Morgenstern1} and the molecular dynamics calculations \cite{Hayat1}. Between 0 and 100 K activation energy for dimer diffusion is higher than this for monomers and for higher temperatures $E_a$ for dimers decreases below value of $E_a$ for monomers. Effective prefactor for the dimer plotted in Fig.~\ref{eff_pref} is lower than this for monomers what compensates the difference in activation energy. At lower temperatures where $E_a$ for dimers is higher than that for monomers both diffusion coefficients have the same value as seen in Fig. \ref{diff_coeff_log}. This is different than at Cu surface where even at low temperatures dimer diffusion coefficient is much lower than this for monomers. It can be also seen that at low temperatures dimers at Ag surface move much faster than at Cu surface. The same diffusion coefficient is shown in the right panel of Fig.~\ref{diff_coeff_log}. In this scale all lines are straight. Each curve has different slope, given by activation energy. However, it is impossible to see any difference in activation energies as a function of temperature in such plot.
The rotational diffusion coefficient for the Cu dimer on Ag(111) lattice is equal to
\begin{equation}
D_r=\frac{\pi^2}{18}(\tilde{W}_{ff\rightarrow lfh}+\tilde{W}_{ff\rightarrow sfh}+\tilde{W}_{hh\rightarrow sfh})
\end{equation}
It implies that the rotational diffusive motion in our case is a sum of three parallel motions, each of them related to a different type of jump. Transition $W_{ff\rightarrow hh}$ is not present in the expression, which is understandable because this transition does not contribute to the change of the dimer's orientation as do the other three transitions. In contrast to the translational diffusion, the rotational one requires just one type of jump in order to occur. It is with agreement with a simple analysis of the dimer's angular motion on the lattice. Each of the jumps whose rates appear in the expression can by itself rotate the dimer by the full angle. As was in the case of Cu/Cu(111) it is not possible to compare diffusion coefficients for different types of motion (translational and rotational) directly, however, as in previous case in Fig. \ref{diff_rot_coeff} we plotted value $D_r$ multiplied by square of the half of dimer length. It can be seen that rotation on Ag surface is much slower than rotation on Cu and it is very close to the translational diffusion of dimers on Ag. It can be understood that particles rotate while they move forward.
As was in the case of the translational diffusion, we can also calculate the effective activation energy for the rotational diffusion
\begin{align}
&E_{a}=[(E_{ff}+\Delta E_{ff\rightarrow lfh})\tilde{W}_{ff\rightarrow lfh}+(E_{ff}+\Delta E_{ff\rightarrow sfh})\nonumber \\ &\tilde{W}_{ff\rightarrow sfh}
+(E_{hh}+\Delta E_{hh\rightarrow sfh})\tilde{W}_{hh\rightarrow sfh}]
\nonumber \\ &/[\tilde{W}_{ff\rightarrow lfh}+\tilde{W}_{ff\rightarrow sfh}+\tilde{W}_{hh\rightarrow sfh}] -< E>
\end{align}
where $<E>$ is mean energy of the dimer in the equilibrium state averaged over all four possible states. Note that even if not all transitions on the lattice are engaged in the rotational diffusion, the energy average is taken over all possible energies in the system.
Because there are three parallel paths of rotational diffusion, the dimer can choose any of them independently. At low temperatures the effective activation energy for the rotational diffusion is 72 meV, which is also the energy barrier for the $ff\rightarrow sfh$ transition. Even though the barrier for the $hh\rightarrow sfh$ transition is even lower, this transition is less favourable because the hh site lies 13 meV higher than the $ff$ site and consequently, the saddle point for the $hh\rightarrow sfh$ jump lies higher than for the $ff\rightarrow sfh$ jump. Therefore, the $ff\rightarrow sfh$ jumps make the easiest diffusion path. The effective activation energy then slightly rises reaching 73 meV at 50 K, possibly due to the other jumps becoming more important. Then above that temperature, as the dimer gets more and more energy, the activation energy starts to fall down very quickly reaching
\begin{align}
E_{a}^{\infty}&=\frac{2E_{ff}+\Delta E_{ff\rightarrow lfh}+\Delta E_{ff\rightarrow sfh}+E_{hh}+\Delta E_{hh\rightarrow sfh}}{3}\nonumber\\
&-\frac{E_{ff}+E_{hh}+E_{sfh}+E_{lfh}}{4}\approx 36 meV
\end{align}
in the limit of infinite temperature.
In Fig. \ref{eff_pref_rot} we show an effective prefactor for the rotational diffusion. Its behaviour is completely different than that of the translational prefactor. First it rises very quickly and about 50 K reaches a maximum slightly above $6^.10^{12}/s$, which is almost twice as high as the value at 0 K. After that maximum the prefactor goes down, just as in the translational cases.
The different behaviour of both the activation energy and the prefactor for the rotational motion compared to the translational one can be explained due to the fact that the former is more parallel while the latter (especially for the monomer) is more serial.
\section{Conclusions}
Systems of diffusing Cu adatoms and dimers were compared at (111) surfaces for homo and heteroepitaxial systems. Cu and Ag crystals have different lattice constants and as a result potential energy surface seen by adatoms varies. However, for both surfaces diffusion of single adatoms is simple and has the same character and values given by appropriate activation energies for these processes. For dimers the situation is completely different. At Ag surface even if the lattice of dimer jumps is quite complicated, involving all types of possible moves, all jumps are of the same rate and as an effect at low temperatures diffusion is quite fast. At higher temperatures it slows down. For homogeneous diffusion of Cu dimers the energy surface is more complex. Jumps inside hexagonal cells are much faster than these between cells. Diffusion either happens only via intercell jumps - this is slower path, or it partly goes via intracell jumps, which is slightly faster. Both diffusion channels together give diffusion coefficient that is lower than this for Ag surface, but at higher temperatures, around 600 K the relation turns back. It is interesting to compare temperature dependence of rotational movement in both cases. They both have rather simple form being a sum of independent diffusional modes, but their effective activation energies are very different. Whereas activation energy for rotational motion at Cu surface is very low and almost independent of temperature, $E_a$ at Ag is higher and decreases with temperature. As an effect rotation at Cu surface is as fast as single particle movement at this surface and it is localized inside cells, whereas rotation at Ag lattice is as fast as dimer movement and it happens simultaneously with translational movement of dimers. At low temperatures it is the same as monomer diffusion at Ag surface. It can be seen that even if both surfaces have similar geometry and not very distinct lattice constants, the character of resulting monomer and dimer diffusion is different.
The general consequence of diffusion over complex energy surface is temperature dependence of activation energy and prefactor of diffusion coefficient. For such an effect to appear there is no need for special interactions or other correlations in the system. Interestingly prefactors in our case drastically change on coming from low to high temperatures as shown in Fig 5. Such temperature dependence of prefactors for monomer or dimer diffusion could explain some discrepancies in results obtained from different experiments \cite{chvoj,chvoj2,bietti}. Especially change of prefactors ratio for diffusion at step and on terraces for Ag/Ag(111) system \cite{chvoj,chvoj2} can be considered as an effect of multistate particle diffusion. We have shown above that depending on the temperature the character of diffusion on smooth (111) metallic surfaces changes due to small differences of the energy barriers for adatom or dimer jumps. We have also shown that at these surfaces dimer diffusion is more complicated, so it can be regarded as a good candidate for the interpretation of the experimental data.
\section{Acknowledgment}
Research supported by the National Science Centre(NCN) of Poland
(Grant NCN No. 2013/11/D/ST3/02700)
\section{Appendix A}
Because we deal with an infinite periodic lattice it is convenient to take the Fourier transform of the equation (\ref{master_eq})
\begin{equation}
\frac{d}{dt}P_{\alpha}(\vec{k};t)=\sum_{\gamma\neq\alpha}M(\gamma;\alpha)P_{\gamma}(\vec{k};t)+M(\alpha;\alpha)P_{\alpha}(\vec{k};t),\label{eq_matrix}
\end{equation}
where $P_{\alpha}(\vec{k};t)=\sum_{j}\exp(i\vec{k}\vec{r}_{j}^{\alpha})P(j,\alpha;t)$. Elements $M(\gamma;\alpha)$ are expressed in terms of the jump rates in the following way:
\begin{eqnarray}
M(\alpha;\alpha)=-\sum_{\gamma\neq\alpha}W_{\alpha,\gamma},\nonumber\\
M(\gamma;\alpha)=W_{\gamma,\alpha}e^{i\vec{k}(\vec{r}_{l\rq{}}^{\gamma}-\vec{r}_{l}^{\alpha})}.
\end{eqnarray}
Sum in the first expression is over all possible transitions from the state $\alpha$ to the neighboring states. The second expression is written for each transition from $\gamma$ to $\alpha$ and $l\rq{}=l$ as long as jump is within chosen cell, whereas $l\rq{}=l\pm1$ when jump of adparticle transforms it to the neighboring cell.
The matrix $\hat{M}$ contains all the information about system dynamics. Its eigenvalues correspond to dynamic modes that describe relaxation of the system towards the equilibrium. It can be shown that of all the eigenvalues the diffusive one has the smallest absolute value \cite{haus,Titulaer1,Gortel1}. Therefore, we can use the variational method in order to obtain the diffusion coefficient
\begin{equation}
\lim_{|\vec{k}|\rightarrow 0}\lambda_{D}=\lim_{|\vec{k}|\rightarrow 0}\frac{\vec{w}\hat{M}\vec{v}}{\vec{w}\vec{v}}=-\vec{k}\hat{D}\vec{k},
\end{equation}
where $\vec{w}$ and $\vec{v}$ are left and right trial eigenvectors, respectively. Their elements are related to each other by $w_{\alpha}^{*}P_{eq}(\alpha)=v_{\alpha}$. Assuming $w_{\alpha}=e^{-i\vec{k}\vec{\phi}_{\alpha}}$, where $\vec{\phi}_{\alpha}$ is the geometric phase of the adsorption site $\alpha$, we get the variational formula for the diffusion coefficient in the form
\begin{align}
\vec{k}\hat{D}_{var}\vec{k}&=\lim_{|\vec{k}|\rightarrow 0}\sum_{\alpha>\gamma}W_{\alpha;\gamma}P_{eq}(\alpha)|e^{i\vec{k}(\vec{r}_{l}^{\gamma}+\vec{\phi}_{\gamma})}-e^{i\vec{k}(\vec{r}_{j}^{\alpha}+\vec{\phi}_{\alpha})}|^2\nonumber\\
&=\sum_{\alpha>\gamma}W_{\alpha;\gamma}P_{eq}(\alpha)[\vec{k}(\vec{r}_{l}^{\gamma}+\vec{\phi}_{\gamma}-\vec{r}_{j}^{\alpha}-\vec{\phi}_{\alpha})]^2\label{var_form}.
\end{align}
The above choice of variational vector has been so far valid in the cases of single adatom diffusion. Each of the vector phases corresponds to exactly one adsorption site within a single cell and each of its vector components is coupled either to x or y direction on the surface. The same approach can be used to describe the diffusion of the dimer center of mass. Moreover, the procedure can be also applied for rotational degrees of freedom instead of the translational one in order to calculate rotational diffusion coefficient.
The total number of variational parameters, whose values should be chosen in such a way that the above expression is minimized, is twice the number of different types of adsorption sites. However, since only the phase differences contribute to the expression, we can always shift all the phases simultaneously by the same value. In such a way we can set one of the vector phases to $\vec{0}$. Moreover, the remaining phases can be related to each other due to the system's symmetry, which leads to further reduction of independent parameters. Ultimately we have a set of several linear equation, one for each independent parameter. We insert solution of these equations to Eq. (\ref{var_form}) and receive final expression for the diffusion coefficient.
\section{Appendix B}
First step in the calculations of diffusion coefficient described above is to find variational parameters from the equation (\ref{var_form}). Phases are associated with sites of the lattice that represents possible jumps in the phase space. Dimer configurations are defined by positions of both particles. In the case of Cu dimer diffusion at Cu(111) surface sites are marked by $ff, lfh$ and $hh$. Each site has attributed phase vector of orientation consistent with the lattice symmetry. Final expression for diffusion coefficient given by (\ref{11}) is obtained for $\delta_{lfh}$ phase which turns out to be zero and the other two
\begin{align}
& \delta_{ff}=-\{\tilde{W}_{ff\rightarrow lfh}^{2}(\frac{\tilde{W}_{ff\rightarrow hh}}{2}+\tilde{V}_{ff\rightarrow hh}) \nonumber \\ &+\tilde{W}_{hh\rightarrow lfh}^{2}(\frac{\tilde{W}_{ff\rightarrow hh}}{2}-2\tilde{V}_{ff\rightarrow hh})\nonumber\\
&+\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{hh\rightarrow lfh}(\tilde{W}_{ff\rightarrow hh}-\tilde{V}_{ff\rightarrow hh} \nonumber \\
& +\tilde{W}_{ff\rightarrow lfh}+\tilde{W}_{hh\rightarrow lfh}+2\tilde{W}_{lfh\rightarrow lfh})\nonumber\\
&+\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{lfh\rightarrow lfh}(\tilde{W}_{ff\rightarrow hh}+2\tilde{V}_{ff\rightarrow hh}) \nonumber \\
&+\tilde{W}_{hh\rightarrow lfh}\tilde{W}_{lfh\rightarrow lfh}(\tilde{W}_{ff\rightarrow hh}-4\tilde{V}_{ff\rightarrow hh})\}\nonumber\\
&/\{[(\tilde{W}_{ff\rightarrow hh}+2\tilde{V}_{ff\rightarrow hh})(\tilde{W}_{ff\rightarrow lfh}+\tilde{W}_{hh\rightarrow lfh}) \nonumber \\
&+2\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{hh\rightarrow lfh}][\tilde{W}_{ff\rightarrow lfh}+\tilde{W}_{hh\rightarrow lfh}\nonumber\\
&+2\tilde{W}_{lfh\rightarrow lfh}]\}
\end{align}
\begin{align}
&\delta_{hh}=\{\tilde{W}_{ff\rightarrow lfh}^{2}(\frac{\tilde{W}_{ff\rightarrow hh}}{2}-2\tilde{V}_{ff\rightarrow hh})+\tilde{W}_{hh\rightarrow lfh}^{2}(\frac{\tilde{W}_{ff\rightarrow hh}}{2}\nonumber\\
&+\tilde{V}_{ff\rightarrow hh})+\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{hh\rightarrow lfh}(\tilde{W}_{ff\rightarrow hh}-\tilde{V}_{ff\rightarrow hh} \nonumber \\ &+\tilde{W}_{ff\rightarrow lfh}+\tilde{W}_{hh\rightarrow lfh}+2\tilde{W}_{lfh\rightarrow lfh})\nonumber\\
&+\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{lfh\rightarrow lfh}(\tilde{W}_{ff\rightarrow hh}-4\tilde{V}_{ff\rightarrow hh})\nonumber \\ &+\tilde{W}_{hh\rightarrow lfh}\tilde{W}_{lfh\rightarrow lfh}(\tilde{W}_{ff\rightarrow hh}+2\tilde{V}_{ff\rightarrow hh})\}\nonumber\\
&/\{[(\tilde{W}_{ff\rightarrow hh}+2\tilde{V}_{ff\rightarrow hh})(\tilde{W}_{ff\rightarrow lfh}+\tilde{W}_{hh\rightarrow lfh}) \nonumber \\ &+2\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{hh\rightarrow lfh}][\tilde{W}_{ff\rightarrow lfh}+\tilde{W}_{hh\rightarrow lfh}+2\tilde{W}_{lfh\rightarrow lfh}]\}
\end{align}
When Cu dimer diffuses on Ag(111) surface it has four different positions $ff, sfh, lfh$ or $hh$.
In the most general case, with all the jumps taken into account diffusion is given by Eq. (\ref{15}) where the following variational parameters were used
\begin{align}
&\delta_{ff}=-\{\frac{3}{4}\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{hh\rightarrow sfh}^{2}+\frac{3}{2}\tilde{W}_{ff\rightarrow hh}\tilde{W}_{hh\rightarrow sfh}^{2} \nonumber \\
&-\tilde{W}_{ff\rightarrow sfh}^{2}\tilde{W}_{hh\rightarrow sfh}+\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{hh\rightarrow sfh}\nonumber\\
&-\frac{1}{2}\tilde{W}_{ff\rightarrow sfh}^{2}\tilde{W}_{ff\rightarrow hh}+\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{ff\rightarrow hh}\tilde{W}_{hh\rightarrow sfh}\nonumber\\
&+\frac{1}{2}\tilde{W}_{ff\rightarrow lfh}(\tilde{W}_{ff\rightarrow hh}\tilde{W}_{hh\rightarrow sfh}+\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{ff\rightarrow hh})\}\nonumber\\
&/\{2\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{hh\rightarrow sfh}^{2}+\frac{3}{2}\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{hh\rightarrow sfh}^{2} \nonumber \\
&+\tilde{W}_{ff\rightarrow hh}\tilde{W}_{hh\rightarrow sfh}^{2}+2\tilde{W}_{ff\rightarrow sfh}^{2}\tilde{W}_{hh\rightarrow sfh}\nonumber\\
&+\tilde{W}_{ff\rightarrow sfh}^{2}\tilde{W}_{ff\rightarrow hh}+2\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{hh\rightarrow sfh} \nonumber \\ &
+2\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{ff\rightarrow hh}\tilde{W}_{hh\rightarrow sfh}+\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{ff\rightarrow hh}\tilde{W}_{hh\rightarrow sfh}\nonumber\\
&+\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{ff\rightarrow hh}\}
\end{align}
\begin{align}
&\delta_{hh}=-\{\frac{3}{4}\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{hh\rightarrow sfh}^{2}+\frac{1}{2}\tilde{W}_{ff\rightarrow hh}\tilde{W}_{hh\rightarrow sfh}^{2}\nonumber\\
&
+\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{hh\rightarrow sfh}^{2}+\frac{1}{2}\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{hh\rightarrow sfh} \nonumber \\ &-\frac{3}{2}\tilde{W}_{ff\rightarrow sfh}^{2}\tilde{W}_{ff\rightarrow hh}
-\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{ff\rightarrow hh}\tilde{W}_{hh\rightarrow sfh}\nonumber\\
&-\frac{1}{2}\tilde{W}_{ff\rightarrow lfh}(\tilde{W}_{ff\rightarrow hh}\tilde{W}_{hh\rightarrow sfh}-\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{ff\rightarrow hh})\}\nonumber\\
&/\{2\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{hh\rightarrow sfh}^{2}+\frac{3}{2}\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{hh\rightarrow sfh}^{2} \nonumber \\
&+\tilde{W}_{ff\rightarrow hh}\tilde{W}_{hh\rightarrow sfh}^{2}+2\tilde{W}_{ff\rightarrow sfh}^{2}\tilde{W}_{hh\rightarrow sfh}\nonumber\\
&+\tilde{W}_{ff\rightarrow sfh}^{2}\tilde{W}_{ff\rightarrow hh}+2\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{hh\rightarrow sfh} \nonumber \\ &
+2\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{ff\rightarrow hh}\tilde{W}_{hh\rightarrow sfh}+\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{ff\rightarrow hh}\tilde{W}_{hh\rightarrow sfh}\nonumber\\
&+\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{ff\rightarrow hh}\}
\end{align}
\begin{align}
\delta_{sfh}=&\frac{\tilde{W}_{ff\rightarrow hh}}{\tilde{W}_{hh\rightarrow sfh}}\left(1+\delta_{ff}-\delta_{hh}\right)-2\delta_{hh}-\frac{1}{2}\nonumber\\
\delta_{lfh}=&-\frac{1}{4}-\frac{\delta_{ff}}{2}
\end{align}
For the case $W_{ff\rightarrow hh}=W_{hh\rightarrow sfh}=0$ the phases simplify to
\begin{align}
\delta_{ff}&=\frac{\tilde{W}_{ff\rightarrow sfh}-\tilde{W}_{ff\rightarrow lfh}}{2(\tilde{W}_{ff\rightarrow sfh}+\tilde{W}_{ff\rightarrow lfh})}\nonumber\\
\delta_{hh}&=0\nonumber\\
\delta_{sfh}&=\delta_{lfh}=-\frac{\tilde{W}_{ff\rightarrow sfh}}{2(\tilde{W}_{ff\rightarrow sfh}+\tilde{W}_{ff\rightarrow lfh})}
\end{align}
and we obtain Eq. (\ref{16}).
For a more complex case, where the least probable transition is still $W_{ff\rightarrow hh}=0$ but $W_{hh\rightarrow sfh}\neq 0$, we have the following values of phases
\begin{align}
&\delta_{ff}=\left[-\frac{3}{4}\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{hh\rightarrow sfh}+\tilde{W}_{ff\rightarrow sfh}^{2}\right.\nonumber\\&\left.-\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{ff\rightarrow lfh}\right]
/ \left[\frac{3}{4}\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{hh\rightarrow sfh}\right.\nonumber \\
&\left.+2\tilde{W}_{ff\rightarrow sfh}(\tilde{W}_{hh\rightarrow sfh}+\tilde{W}_{ff\rightarrow sfh}+\tilde{W}_{ff\rightarrow lfh}) \right]
\end{align}
\begin{align}
&\delta_{hh}=-\left[\frac{3}{4}\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{hh\rightarrow sfh}+\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{hh\rightarrow sfh}\right.\nonumber \\
&\left.+\frac{1}{2}\tilde{W}_{ff\rightarrow sfh}\tilde{W}_{ff\rightarrow lfh}\right] / \left[\frac{3}{2}\tilde{W}_{ff\rightarrow lfh}\tilde{W}_{hh\rightarrow sfh}\right.\nonumber\\&\left.+2\tilde{W}_{ff\rightarrow sfh}(\tilde{W}_{hh\rightarrow sfh}+\tilde{W}_{ff\rightarrow sfh}+\tilde{W}_{ff\rightarrow lfh}) \right]
\end{align}
\begin{align}
\delta_{sfh}&=-\frac{1}{2}-2\delta_{hh}\nonumber\\
\delta_{lfh}&=-\frac{1}{4}-\frac{\delta_{ff}}{2}
\end{align}
and these phases lead to Eq.(\ref{17})
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,226 |
/*****************************************************************************
* timing.h
*
* Making a coarse estimation for programm's running time. The time unit
* is second.
*
* Zhang Ming, 2010-01, Xi'an Jiaotong University.
*****************************************************************************/
#ifndef TIMING_H
#define TIMING_H
#include <ctime>
inline static double seconds()
{
const double secs_per_tick = 1.0 / CLOCKS_PER_SEC;
return ( double(clock()) ) * secs_per_tick;
}
namespace splab
{
class Timing
{
public:
Timing();
~Timing();
void start();
void stop();
double stopAndRead();
double read();
double currentTime();
private:
bool running;
double startTime;
double total;
};
// class Timing
#include <timing-impl.h>
}
// namespace splab
#endif
// TIMING_H
| {
"redpajama_set_name": "RedPajamaGithub"
} | 474 |
Home • Art for Foodies • Hot Dog and I
Hot Dog and I
by Fernanda Cohen
11"x8.5" Sold Out
22"x17" Only 5 left!
30x40 - Black - Framed to Image (Our Pick)
30x40 - White - Framed to Image
We all communicate through words, as words translate into visuals. My visuals are my words. I believe in drawing not as mere media or style, but rather as the ultimate way to deliver ideas; a combination of self-expression with an essentially bold interaction with the viewer. As a visual artist as well as an idealist, I comprehend almost everything through subtlety and humor. I play with universal ideas as I watch them mock my existence. Ultimately, I simply continue an infinite dialogue on a blank piece of paper.
Museo PR
11"x8.5" | edition of 200
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Fernanda Cohen
Fernanda grew up in Buenos Aires, and moved to New York City in 2000 to study illustration at the School of Visual Arts. Fernanda has illustrated the cover of the New York Times Magazine, and her work has been featured by The New Yorker, Target, Travel and Leisure, Continental Airlines, SONY, MTV, The Guardian, Soho House Hotels, Cosmopolitan, Paper, W Hotels and Harvard Business Review, among others. Her work has received over 50 awards worldwide, including gold and silver medals from the Society of Illustrators of New York and Los Angeles, first prize by Creative Futures (U.K.), Communication Arts, HOW,... Read More
Graphis, American Illustration, Lurzer's Archive (Austria), Applied Arts (Canada), and Curvy (Australia), among others. Fernanda has also had three solo exhibitions of her personal work: A Taste of Art Gallery in TriBeCa, NYC (2004), Galeria Sonoridad Amarilla in Buenos Aires (2005), and the Consulate of Argentina in New York (2006). Her work has been exhibited in group shows around the world, including the U.S., Germany, Argentina, Singapore, Japan, Italy, Australia and England. Other projects include Fernanda's own line of porcelain, Reina Renée, in Argentina; her collection of post-it notes, "Les Filles," in France; a children's book for the U.S. publishing house Scholastic; Christmas window illustration in Buenos Aires and Manhattan, and her independent line of handbags, Lilah Bags. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,898 |
\section{Introduction}
Phenomenology at the LHC often involves high multiplicity final
states. For example, backgrounds to Higgs searches involve processes
such as $PP\rightarrow W^+W^- + 2~\mbox{jets}$ and $PP\rightarrow
t\bar{t}\ +\ b\bar{b}$. Both these examples involve $2\rightarrow 4$
scatterings. At leading order (LO) such high multiplicity final state
amplitudes can be evaluated using either numerical recursive
techniques~\cite{Berends:1987me,Mangano:2002ea,Draggiotis:2002hm} or
other numerical and/or algebraic
techniques~\cite{Ishikawa:1993qr,Stelzer:1994ta,Krauss:2001iv,Maltoni:2002qb,Boos:2004kh}.
However, ${\cal O}\left(\alpha_S\right)$, next-to-leading order (NLO)
corrections to the scattering amplitudes are desirable. Not only do
NLO corrections give a first reliable prediction of total rates, they
also give a good error estimate on the shapes of distributions. At
NLO the current state of the art for hadron colliders are $2
\rightarrow 3$ processes.
Thus NLO predictions for $PP\rightarrow 3\
\mbox{jets}$~\cite{Kilgore:1996sq,Nagy:2003tz} (based on virtual
corrections of ref.~\cite{Bern:1993mq,Bern:1994fz,Kunszt:1994tq}) and
$PP\rightarrow V\ +\ 2\ \mbox{jets}$~\cite{Campbell:2002tg} (based on
virtual corrections of
ref.~\cite{Glover:1996eh,Campbell:1997tv,Bern:1997sc}) are known, and
codes for $PP\rightarrow t\bar{t}\ +\
\mbox{jet}$~\cite{Brandenburg:2004fw,Uwer:2005tq} and $PP\rightarrow
H\ +\ 2\ \mbox{jets}$ via gluon fusion~\cite{Ellis:2005qe} are under
construction. Other processes such as $PP\rightarrow V_1V_2\ +\
\mbox{jet}$ and $PP\rightarrow V_1V_2V_3$ are now feasible.
By contrast the consideration of $2\rightarrow 4$ processes is still
in its infancy. In electroweak physics the full one-loop electroweak
corrections to $e^+e^-\rightarrow\ 4\ \mbox{fermions}$ were calculated
in Ref.~\cite{Denner:2005fg,Denner:2005nd}. However the calculation of
NLO $2\rightarrow 4$ QCD scattering cross sections is currently
unexplored. Such a calculation involves both the evaluation of the
one-loop six-point virtual corrections and the inclusion of the
$2\rightarrow 5$ scattering bremsstrahlung contributions through Monte
Carlo integration.
In this paper we consider the virtual corrections to six-gluon
scattering which is relevant for a calculation of $PP\rightarrow
4$~jets. By considering the one-loop corrections to $gg\rightarrow
gggg$ we select the most complicated QCD six-point processes. If the
amplitude is calculated in terms of Feynman diagrams, the number of
diagrams is very large and the gauge cancellations between these
diagrams is the most severe. These cancellations could be a concern
in a semi-numerical procedure; the six-gluon amplitude therefore
provides a stringent test of the method. In this paper we consider
neither the bremsstrahlung contributions, nor the one-loop processes
involving external quarks, which are needed to obtain results for a
physical cross section.
The technique for the analytic calculation of the one-loop corrections
to multi-gluon amplitudes which is relevant for this paper
is the decomposition of the calculation into simpler
pieces with internal loops of ${\cal{N}}=4$ and ${\cal{N}}=1$
multiplets of super-symmetric Yang-Mills particles and a residue
involving only scalar particles in the
loops~\cite{Bern:1993mq,Bern:1994zx,Bern:1994cg}.
After recent
advances~\cite{Bidder:2004tx,Bern:2005ji,Bern:2005cq,Britto:2005ha},
all supersymmetric contributions have been computed analytically,
however not all of the scalar contributions for six-gluon amplitudes
(or higher) are known yet. We present here numerical results for
six-gluon contributions. For supersymmetric pieces we provide
completely independent cross-checks of analytical results.
Although all one-loop $2\rightarrow 2$ and almost all of the currently
known $2\rightarrow 3$ amplitudes were calculated using analytic
techniques, we believe that semi-numerical or hybrid
numerical/analytic techniques offer promise for more rapid progress.
This technique was demonstrated recently for the case of the one-loop
$\mbox{H}\ +\ 4\ \mbox{partons}$ amplitude~\cite{Ellis:2005qe}.
Many methods have been proposed to calculate NLO amplitudes, both
semi-numerical~\cite{Fleischer:1999hq,Passarino:2001jd,Binoth:2003ak,Duplancic:2003tv,
Nagy:2003qn,Belanger:2003sd,delAguila:2004nf,Denner:2002ii,Giele:2004iy,vanHameren:2005ed,Denner:2005nn}
or numerical~\cite{Soper:2001hu,Anastasiou:2005cb}. Of these methods
only a few have actually been used to evaluate one-loop amplitudes.
Only by using the methods in explicit calculations one can be sure
that all numerical issues have been addressed properly.
In section II we discuss the colour algebra involved with the
evaluation of a six-gluon amplitude. The numerical techniques used in
this paper are discussed in section III, while in section IV the
comparison is made with numerous super-symmetric and the few scalar
results, which exist in the literature. Finally, our conclusions in
section V summarize the paper.
\section{Six-gluon amplitude at one-loop}
At tree-level, amplitudes with $n$ external gluons can be decomposed
into colour-ordered sub-amplitudes, multiplied by a trace of $n$
colour matrices, $T^a$. The traceless, hermitian, $N_c\times N_c$
matrices, $T^a$, are the generators of the $SU(N_c)$ algebra.
Following the usual conventions for this branch of the QCD literature,
they are normalized so that $\mathop{\rm Tr}\nolimits( T^a T^b) = \delta^{ab}$. Summing
over all non-cyclic permutations the full amplitude ${\cal A}^{\rm
\scriptsize \mbox{\rm tree}}_ n$ is reconstructed from the sub-amplitudes
$A_n^{\rm \scriptsize
\mbox{\rm tree}}(\sigma)$~\cite{Berends:1987me,Mangano:1987xk},
\begin{equation}
{\cal A}_{n}^{\rm tree}(\{p_i,\lambda_i,a_i\}) =
g^{n-2} \sum_{\sigma \in S_n/Z_n} \mathop{\rm Tr}\nolimits( T^{a_{\sigma(1)}}
\cdots T^{a_{\sigma(n)}} )
\ A^{\rm tree}_n (p_{\sigma(1)}^{\lambda_{\sigma(1)}},\ldots,
p_{\sigma(n)}^{\lambda_{\sigma(n)}})\ .
\end{equation}
The momentum, helicity ($\pm$), and colour index of the $i$-th
external gluon are denoted by $p_i$, $\lambda_i$, and $a_i$
respectively. $g$ is the coupling constant, and $S_n/Z_n$ is the set
of $(n-1)!$ non-cyclic permutations of $\{1,\ldots, n\}$.
The expansion in colour sub-amplitudes is slightly more complicated at
one-loop level. Let us consider the case of massless internal
particles of spin $J=0,1/2,1$ corresponding to a complex scalar, a
Weyl fermion or a gluon. If all internal particles belong to the
adjoint representation of SU$(N_c)$, the colour decomposition for
one-loop $n$-gluon amplitudes is given by~\cite{Bern:1990ux},
\begin{equation}
{\cal A}_n^{[J]} ( \{p_i,h_i,a_i\} ) = g^n
\sum_{c=1}^{\lfloor{n/2}\rfloor+1}
\sum_{\sigma \in S_n/S_{n;c}}
{\rm Gr}_{n;c}( \sigma ) \,A_{n;c}^{[J]}(\sigma) \,,
\label{Oneloopform}
\end{equation}
where ${\lfloor{x}\rfloor}$ denotes the largest integer less than or
equal to $x$ and $S_{n;c}$ is the subset of $S_n$ which leaves the
double trace structure in ${\rm Gr}_{n;c}(1)$ invariant.
The leading-colour structure is simply given by,
\begin{equation}
{\rm Gr}_{n;1}(1) = N_c\ \mathop{\rm Tr}\nolimits (T^{a_1}\cdots T^{a_n} ) \,.
\end{equation}
The subleading-colour structures are given by products of colour traces
\begin{equation}
{\rm Gr}_{n;c}(1) = \mathop{\rm Tr}\nolimits( T^{a_1}\cdots T^{a_{c-1}} )\,
\mathop{\rm Tr}\nolimits ( T^{a_c}\cdots T^{a_n}) \,.
\end{equation}
The subleading sub-amplitudes $A_{n;c>1}$ are determined by the
leading ones $A^{[1]}_{n;1}$ through the merging
relation~\cite{Kleiss:1988ne,Bern:1990ux,Bern:1994zx,DelDuca:1999rs}
\begin{equation}
A^{[1]}_{n;c>1}(1,2,\ldots,c-1;c,c+1,\ldots,n)\ =\
(-1)^{c-1} \sum_{\sigma \in {\rm OP}\{\alpha\}\{\beta\}}
A^{[1]}_{n;1}(\sigma_1,\ldots,\sigma_n) \, ,
\label{Kleiss-Kuijf}
\end{equation}
where $\alpha_i \in \{\alpha\} \equiv \{c-1,c-2,\ldots,2,1\}$,
$\beta_i \in \{\beta\} \equiv \{c,c+1,\ldots,n-1,n\}$, and
${\rm OP}\{\alpha\}\{\beta\}$ is the set of ordered permutations of
$\{1,2,\ldots,n\}$ but with the last element $n$ fixed. The ordered
permutations are defined as a set of all mergings of $\alpha_i$ with
respect to the $\beta_i$, such that the cyclic ordering of the
$\alpha_i$ within the set $\{\alpha\}$ and of the $\beta_i$ within the
set $\{\beta\}$ is unchanged. In practice, since $n$ is fixed, no
further cycling of the set $\{\beta\}$ is required. Thus a complete
description can be given in terms of the leading colour sub-amplitudes
$A_{n;1}$ alone.
The contribution of a single flavour of Dirac fermion
in the fundamental representation, (relevant for quarks in QCD) is
\begin{equation}
{\cal A}_{n}^{\rm Dirac}(\{p_i,\lambda_i,a_i\}) =
g^n \sum_{\sigma \in S_n/Z_n} \mathop{\rm Tr}\nolimits( T^{a_{\sigma(1)}}
\cdots T^{a_{\sigma(n)}} )
\ A^{[1/2]}_{n;1} (p_{\sigma(1)}^{\lambda_{\sigma(1)}},\ldots,
p_{\sigma(n)}^{\lambda_{\sigma(n)}})\ .
\end{equation}
Simple colour arguments~\cite{Bern:1990ux} allow one to demonstrate
that this colour sub-amplitude is the same as the leading colour
sub-amplitude for a single Weyl fermion in the adjoint representation
defined in Eq.~(2.2).
Since the subleading colour amplitudes are not independent, we shall
henceforth drop them from our discussion. To simplify the notation we
shall also drop the subscripts $n$ and $c$. The amplitude denoted by
$A$ will thus refer to leading colour amplitude with six external
gluons.
\section{Method of calculation}
The method we use is purposely kept as simple as possible. Especially
in numerical methods this is desirable for both keeping track of
numerical accuracy and code transparency.
To generate all the required Feynman diagrams we use
Qgraf~\cite{Nogueira:1991ex}. The Qgraf output is easily manipulated
using Form~\cite{Vermaseren:2000nd} to write the amplitude in the form
\begin{equation}
A(1,2,3,4,5,6)=\sum_{N=2}^6\sum_{M=0}^N
K_{\mu_1\cdots\mu_M}(p_1,\epsilon_1;\ldots;p_6,\epsilon_6)
I_N^{\mu_1\cdots\mu_M}(p_1,\ldots,p_6) \, ,
\end{equation}
where the kinematic tensor $K$ depends on the purely four-dimensional
external vectors and contains all the particle and process
information. The $N$-point tensor integrals of rank $M$ are defined in
$D$ dimensions as
\begin{equation}
I_N^{\mu_1\cdots\mu_M}(p_1,\ldots,p_6)=
\int \frac{d^Dl}{i \pi^{D/2}} \frac{l^{\mu_1}\ldots l^{\mu_M}}{d_1d_2 \ldots d_N},
\;\;\; d_i \equiv (l+q_i)^2,\;\;\; q_i \equiv \sum_{j=1}^i p_j\,,
\end{equation}
and can be evaluated semi-numerically.
For $N\leq 4$ we use the method of
\cite{Giele:2004ub,Giele:2004iy,Ellis:2005zh} which we already
developed, tested and used in the calculation of $\mbox{H} + 4\
\mbox{partons}$ at one-loop~\cite{Ellis:2005qe}. In general, the
basis integrals will contain divergences in $\epsilon=(4-D)/2$ from
soft, collinear and ultraviolet divergences and the answer returned by
the semi-numerical procedure will be a Laurent series in inverse
powers of $\epsilon$.
For the five~(six)-point tensor integrals the method we use relies on
the completeness (over-completeness) of the basis of external momenta
for a generic phase space point. We therefore use a technique for
tensor reduction which generalizes the methods of
ref.~\cite{vanNeerven:1983vr,vanOldenborgh:1989wn}. This technique is
valid as long as the basis of external momenta is
complete\footnote{For exceptional momentum configurations (such as
threshold regions or planar event configurations) this is not the
case. Exceptional configurations can be treated using a
generalization of the expanded relations proposed in
refs.~\cite{Giele:2004ub,Ellis:2005zh}. This is beyond the scope of
this paper.}. Assuming we have a complete basis of external momenta
we can select a set of 4 momenta $\{p_{k_1},p_{k_2},p_{k_3},p_{k_4}\}$
which form the basis of the four-dimensional space. We can then
decompose the loop momentum
\begin{equation}
l^\mu=\sum_{i=1}^4 l\cdot p_{k_i} v_{k_i}^\mu
=V^\mu+\frac{1}{2}\sum_{i=1}^4 \left(d_{k_i}-d_{k_i-1}\right) v_{k_i}^\mu\,,
\end{equation}
where the $v_{k_i}$ are defined as linear combinations of the basis
vectors
\begin{equation}\label{axial}
v^{\mu}_{k_i} = \sum_{j=1}^4 [G^{-1}]_{ij} p^\mu_{k_j}, \;\;\; G_{ij} =p_{k_i} \cdot p_{k_j}\,,
\end{equation}
where $G$ is the Gram matrix and
\begin{equation}
V^\mu=-\frac{1}{2}\sum_{i=1}^4 (r_{k_i}-r_{k_i-1})v^\mu_{k_i},\;\;\;
r_k=q_k^2\,.
\end{equation}
With this relation it is now easy to reduce an $N$-point function of
rank $M$ to a lower rank $N$-point function and a set of lower rank
$(N-1)$-point functions
\begin{equation}
I_N^{\mu_1\cdots\mu_M}=I_N^{\mu_1\cdots\mu_{M-1}}V^{\mu_M}
+\frac{1}{2}\sum_{i=1}^4\left(I_{N,k_i}^{\mu_1\cdots\mu_{M-1}}-I_{N,k_i-1}^{\mu_1\cdots\mu_{M-1}}\right)v_{k_i}^{\mu_M}\,,
\end{equation}
where $I_{N,j}$ is a $(N-1)$-point integral originating from $I_N$
with propagator $d_j$ removed. More explicitly, choosing without loss
of generality the base set $\{p_1,p_2,p_3,p_4\}$, we get
\begin{eqnarray}
\lefteqn{I_N^{\mu_1\cdots\mu_M}(p_1,p_2,p_3,p_4,p_5,\ldots,p_N)=
I_N^{\mu_1\cdots\mu_{M-1}}(p_1,p_2,p_3,p_4,p_5,\ldots,p_N) V^{\mu_M}(p_1,p_2,p_3,p_4)}
\nonumber\\&+&\frac{1}{2}
\left(I_{N-1}^{\mu_1\cdots\mu_{M-1}}(p_1+p_2,p_3,p_4,p_5,\ldots,p_N)
-I_{N-1}^{\mu_1\cdots\mu_{M-1}}(p_2,p_3,p_4,p_5,\ldots,p_N)\right)
\nonumber\\&&\times
v_1^{\mu_M}(p_1,p_2,p_3,p_4)
\nonumber\\&+&\frac{1}{2}
\left(I_{N-1}^{\mu_1\cdots\mu_{M-1}}(p_1,p_2+p_3,p_4,p_5,\ldots,p_N)
-I_{N-1}^{\mu_1\cdots\mu_{M-1}}(p_1+p_2,p_3,p_4,p_5,\ldots,p_N)\right)
\nonumber\\&&\times
v_2^{\mu_M}(p_1,p_2,p_3,p_4)
\nonumber\\&+&\frac{1}{2}
\left(I_{N-1}^{\mu_1\cdots\mu_{M-1}}(p_1,p_2,p_3+p_4,p_5,\ldots,p_N)
-I_{N-1}^{\mu_1\cdots\mu_{M-1}}(p_1,p_2+p_3,p_4,p_5,\ldots,p_N)\right)
\nonumber\\&&\times
v_3^{\mu_M}(p_1,p_2,p_3,p_4)
\nonumber\\&+&\frac{1}{2}
\left(I_{N-1}^{\mu_1\cdots\mu_{M-1}}(p_1,p_2,p_3,p_4+p_5,\ldots,p_N)
-I_{N-1}^{\mu_1\cdots\mu_{M-1}}(p_1,p_2,p_3+p_4,p_5,\ldots,p_N)\right)
\nonumber\\&&\times
v_4^{\mu_M}(p_1,p_2,p_3,p_4)\,.
\nonumber\\
\end{eqnarray}
For example, applying this relation repeatedly to the tensor six-point
integrals we will be left with the scalar six-point integral and
five-point tensor integrals. The five-point tensor integrals can be
reduced using the same technique. Subsequently we can use the method
of~\cite{Giele:2004ub,Giele:2004iy,Ellis:2005zh} to further
numerically reduce all remaining integrals to the basis of scalar 2-,
3- and 4-point integrals. This procedure turns out to be efficient and
straightforward to implement numerically.
\section{Comparison with the literature}
Since we have directly calculated the loop amplitudes with internal
gluons and fermions we can easily obtain the result for QCD with an
arbitrary number $n_f$ of flavours of quarks,
\begin{equation}
{A}^{\rm QCD} = A^{[1]} + \frac{n_f}{N} A^{[1/2]}\, .
\end{equation}
However since the analytic calculations in the literature are
presented in terms of supersymmetric theories we need to re-organize
our results to compare with other authors.
\subsection{Supersymmetry}
Since we have calculated the amplitudes with massless spin $1$, spin
$1/2$ and spin $0$ particles in the internal loop we can combine our
results as follows
\begin{eqnarray}
{A}^{{\cal{N}}=4}&=&A^{[1]}+4 A^{[1/2]}+3 A^{[0]}\, , \\
{A}^{{\cal{N}}=1}&=& A^{[1/2]}+A^{[0]}.
\end{eqnarray}
${A}^{{\cal{N}}=4}$, so constructed, describes an amplitude where the full
supersymmetric ${\cal{N}}=4$ multiplet runs in the loop, and ${A}^{{\cal{N}}=1}$
denotes the contribution from an ${\cal{N}}=1$ super-multiplet running in
the loop.
In analytic calculations the intention is to proceed in the opposite
direction. Amplitudes with multiplets of supersymmetric Yang-Mills in
internal loops have much improved ultra-violet behavior and are
four-dimensional cut-constructible. For this reason, all of these
supersymmetric amplitudes have been calculated and most have been
presented in a form suitable for numerical evaluation. As far as
six-gluon amplitudes with scalars in the loop, ${A}^{[0]}$, are
concerned three of the needed eight independent helicity amplitudes
have been published so far. Only in the helicity combinations where
all contributions are known can one reconstruct the ingredients needed
for QCD amplitudes
\begin{eqnarray}
{A}^{[1]}&=&{A}^{{{\cal{N}}}=4}-4{A}^{{\cal{N}}=1}+{A}^{[0]} \, ,\\
{A}^{[1/2]}&=& {A}^{{\cal{N}}=1}-{A}^{[0]}\, .
\end{eqnarray}
\subsection{Numerical results}
As a preparatory exercise we performed a check of the four- and
five-point gluon one-loop amplitudes. We found agreement with the
literature~\cite{Ellis:1985er,Kunszt:1993sd,Bern:1993mq}.
We now turn to the amplitude for six-gluons which is the main result
of this paper. Our numerical program allows the evaluation of the
one-loop amplitude at an arbitrary phase space point and for arbitrary
helicities. For a general phase space point it is useful to re-scale
all momenta so that the momenta of the gluons, (and the elements of
the Gram matrix), are of $O(1)$ before performing the tensor
reduction. Without loss of generality we can assume that this has been
done.
To present our numerical results we choose a particular phase space
point with the six momenta $p_i$ chosen as follows, $(E,p_x,p_y,p_z)$,
\begin{eqnarray}
\label{specificpoint}
p_1 & = & \frac{\mu}{2} (-1, +\sin\theta, +\cos\theta \sin\phi, +\cos\theta \cos\phi ), \nonumber \\
p_2 & = & \frac{\mu}{2} (-1, -\sin\theta, -\cos\theta \sin\phi, -\cos\theta \cos\phi ), \nonumber \\
p_3 & = & \frac{\mu}{3} (1,1,0,0), \nonumber \\
p_4 & = & \frac{\mu}{7} (1,\cos\beta,\sin\beta,0), \nonumber \\
p_5 & = & \frac{\mu}{6} (1,\cos\alpha \cos\beta, \cos\alpha \sin\beta,\sin\alpha), \nonumber \\
p_6 & = & -p_1-p_2-p_3-p_4-p_5\, ,
\end{eqnarray}
where $\theta= \pi/4,\phi= \pi/6,\alpha= \pi/3,\cos \beta= -7/19$.
Note that the energies of $p_1$ and $p_2$ are negative and $p_i^2=0$.
In order to have energies of $O(1)$ we make the choice for the scale
$\mu=n=6$~[GeV]. As usual $\mu$ also denotes the scale which is used
to carry the dimensionality of the $D$-dimensional integrals. The
results presented contain no ultraviolet renormalization.
Analytic results require the specification of eight helicity
combinations: all other amplitudes can be obtained by the parity
operation or cyclic permutations. We choose these eight combinations
to be the two finite amplitudes ($++++++,-+++++$), the maximal
helicity violating amplitudes ($--++++,-+-+++,-++-++$), and the
next-to-maximal helicity violating amplitudes
($---+++,--+-++,-+-+-+$). These eight amplitudes would not be
sufficient for a numerical evaluation, but the numerical approach
allows the evaluation of any helicity configuration at will.
In Table~\ref{tableneq4} we give results for a particular colour
sub-amplitude ${A}^{{\cal{N}}=4}(1,2,3,4,5,6)$ for the above eight choices
of the helicity. An overall factor of $i c_\Gamma$ has been removed from
all the results in the Tables~\ref{tableneq4}, \ref{tableneq1}, and
\ref{tablescalar}
\begin{equation}
c_\Gamma = {(4 \pi)^\epsilon \over 16 \pi^2 }
{\Gamma(1+\epsilon)\Gamma^2(1-\epsilon)\over\Gamma(1-2\epsilon)}\ .
\label{cgdef}
\end{equation}
The results for the ${\cal{N}}=4$ amplitudes depend on the number of
helicities of gluons circulating in internal loops.
For a recent description
of regularization schemes see, for example, ref.~\cite{Bern:2002zk}.
Our results are
presented in the 't Hooft-Veltman scheme.
The translation to the four-dimensional helicity scheme is immediate
\begin{equation}
{A}^{{\cal{N}}=4}_{\rm FDH} = {A}^{{\cal{N}}=4}_{\rm t-HV} + \frac{c_\Gamma}{3}
{A}_{\rm tree}\,.
\label{THVtoFDH}
\end{equation}
Note that analytic results from the literature are quoted in the
four-dimensional helicity scheme, which respects supersymmetry. These
results have been translated to the 't Hooft-Veltman scheme using
Eq.~(\ref{THVtoFDH}) before insertion in our tables.
\begin{scriptsize}
\TABLE{
\begin{tabular}{|c|c|c|c|c|}
\hline
Helicity & $1/\epsilon^2$ & $1/\epsilon$ & 1 &[Ref]/(Eq.\#) \\
\hline
$++++++$ & $0$ & $0$ & $0$ & \\
$++++++$ & $(-1.034+i~2.790 ) 10^{-8}$&$ (-9.615+i~3.708 ) 10^{-8}$&$ -(0.826+i~2.514) 10^{-7}$ & [SN-A] \\
\hline
$-+++++$ & $0$ & $0$ & $0$ & \\
$-+++++$ & $(1.568+ i~2.438) 10^{-8}$ & $ (-0.511 +i~1.129) 10^{-7}$&$ -(3.073+i~0.1223) 10^{-7}$ & [SN-A] \\
\hline
\hline
$--++++$ & $-161.917+i~54.826 $ & $ -489.024-i~212.415 $ & $ -435.281-i~1162.971 $ & \cite{Bern:1994zx}/(4.19)\\
$--++++$ & $(-0.933 +i~1.513) 10^{-8} $ & $ -(7.655+i~0.440)10^{-8} $ & $ -(-0.221+i~1.834)10^{-7} $ & [SN-A] \\
\hline
$-+-+++$ & $ -33.024 + i~44.423 $ & $ -169.358 + i~33.499 $ & $ -330.119 -i~229.549 $ & \cite{Bern:1994zx}/(4.19) \\
$-+-+++$ & $(-7.542+i~0.939) 10^{-8} $ & $ -(1.157 +i~0.363)10^{-8} $ & $ -(3.474 +i~2.856)10^{-8} $ & [SN-A] \\
\hline
$-++-++$ & $ -0.5720 - i~3.939 $ & $ 6.929 - i~10.302 $ & $ 28.469 -i~5.058 $ & \cite{Bern:1994zx}/(4.19) \\
$-++-++$ & $(-2.279 +i~1.803)10^{-8} $ & $ -(1.176 +i~0.399)10^{-7} $ & $ (0.054-i~3.307)10^{-7} $ & [SN-A] \\
\hline
\hline
$---+++$ & $ -6.478 -i~10.407 $ & $ 6.825 -i~37.620 $ & $ 75.857 - i~47.081 $ & \cite{Bern:1994cg}/(6.19) \\
$---+++$ & $ (2.686-i~1.668)10^{-8} $ & $ (1.232+i~0.554)10^{-7} $ & $ (0.020+i~3.334 )10^{-7} $ & [SN-A] \\
\hline
$--+-++$ & $ 14.074-i~22.908 $ & $ 80.503- i~23.464 $ & $ 169.047 + i~93.601 $ & \cite{Bern:1994cg}/(6.24) \\
$--+-++$ & $ -(1.619+i~0.943)10^{-8} $ & $ -(1.030+i~8.234)10^{-8} $ & $ (1.560 -i~0.801)10^{-8} $ & [SN-A] \\
\hline
$-+-+-+$ & $ 13.454+i~13.177 $ & $ 3.495+i~58.632 $ & $ -88.32+i~103.340 $ & \cite{Bern:1994cg}/(6.26) \\
$-+-+-+$ & $ (1.045-i~0.113)10^{-9} $ & $ (-0.772+i~1.652))10^{-8} $ & $ (-7.795+i~7.881))10^{-8} $ & [SN-A] \\
\hline
\hline
\end{tabular}
\caption{$\cal{N}$=4 color ordered sub-amplitudes evaluated at the specific point, Eq.~(\ref{specificpoint}).
The results are given in the 'tHooft-Veltman regularization scheme.
[SN-A] means the difference between the semi-numerical result and
the analytical one.}
\label{tableneq4}
}
\end{scriptsize}
\begin{scriptsize}
\TABLE{
\begin{tabular}{|c|c|c|c|c|}
\hline
Helicity & $1/\epsilon^2$ & $1/\epsilon$ & 1 &[Ref]/(Eq.\#) \\
\hline
$++++++$ & 0 & 0 & 0 & \\
$++++++$ & $(-3.470+i~9.320) 10^{-9}$&$(-3.226+i~1.253) 10^{-8}$&$ -(3.899+i~8.969) 10^{-8}$ & [SN-A] \\
\hline
$-+++++$ & 0 & 0 & 0 & \\
$-+++++$ & $(5.228+i~8.127) 10^{-9}$&$(-1.678+i~3.775) 10^{-8} $&$ -(1.013+i~0.2066) 10^{-7} $ & [SN-A] \\
\hline
\hline
$--++++$ & 0 & $26.986-i~9.1376$ & $101.825-i~52.222$ & \cite{Bern:1994cg}/(5.9)\\
$--++++$ &$(-3.297+i~5.194) 10^{-9}$ & $ -(-2.104+i~0.344) 10^{-8} $ & $(0.949 -i~4.895) 10^{-8} $ & [SN-A] \\
\hline
$-+-+++$ & $0$ & $ 5.504-i~7.404 $ & $ 21.811-i~29.051 $ & \cite{Bern:1994cg}/(5.12)\\
$-+-+++$ & $(-1.847 + i~0.8566) 10^{-10} $ & $ -(6.141+i~4.633 ) 10^{-10} $ & $ (3.095+i~2.138) 10^{-7} $ & [SN-A] \\
\hline
$-++-++$ & $0$ & $0.09533+i~0.6565$ & $ -2.183+i~3.260 $ & \cite{Bern:1994cg}/(5.12)\\
$-++-++$ & $(-7.599+i~6.018) 10^{-9}$ & $ -(3.929+i~1.304)10^{-8} $ & $(0.008-i~1.100)10^{-7} $ & [SN-A] \\
\hline
\hline
$---+++$ & $0$ & $1.080 +i~1.735$ & $ 0.722+i~5.285$ & \cite{Bidder:2004tx}/(9) \\
$---+++$ & $(8.965-i~5.555) 10^{-9}$ & $(4.107 +i~1.858)10^{-8} $ & $ (0.002+i~1.114)10^{-7} $ & [SN-A] \\
\hline
$--+-++$ & $0$ & $-2.346+i~3.819$ & & \cite{Britto:2005ha}/(5.4,2.3)\\
$--+-++$ & $(-5.351-i~2.825) 10^{-9}$ & $-2.346+i~3.819$ & $-2.238+i~17.687$ & [SN] \\
\hline
$-+-+-+$ & $0$ & $-2.242-i~2.196$ & & \cite{Britto:2005ha}/(5.13,2.3)\\
$-+-+-+$ & $(1.124-i~0.2060) 10^{-10}$ & $-2.242-i~2.196$ & $-1.721-i~7.433$ & [SN] \\
\hline
\hline
\end{tabular}
\caption{$\cal{N}$=1 color ordered sub-amplitudes evaluated at the specific point, Eq.~(\ref{specificpoint}).
[SN] means that the result is obtained using our semi-numerical
code, while [SN-A] denotes the difference between the semi-numerical
result and the analytical one.}
\label{tableneq1}
}
\end{scriptsize}
In Table~\ref{tableneq1} we give results for the colour sub-amplitudes
${A}^{{\cal{N}}=1}(1,2,3,4,5,6)$ for the same eight helicity choices and
where possible compare with analytical results.~\footnote{In
Eq.~(5.16) of ref.~\cite{Bern:1994cg} for the degenerate case
m=j-1=2 one has $\hat{{\cal C}}_m = \{j+1, \ldots, n-1 \} $, as can
be seen from Fig.~8 of this same paper. This point has also been
made in ref.~\cite{Cachazo:2004zb}. }
Note that because of the relation
\begin{equation}
{A}^{{\cal{N}}=1}|_{\rm singular} = \frac{c_\Gamma}{\epsilon} A^{\rm tree}\, ,
\end{equation}
the column giving the single pole can as well be considered as a
listing of the results for the colour-ordered sub-amplitudes at tree
graph level (stripped only of the overall factor of $i$).
We note that for two of the helicity amplitudes $--+-++$ and $-+-+-+$
we were unable to evaluate the analytic results numerically. This was
due to the fact that calculating the residue of certain poles as
required by the formula in ref.~\cite{Britto:2005ha}, resulted in zero
value denominators of sub-expressions\footnote {We thank the authors
of ref.~\cite{Britto:2005ha} for confirming that there are problems
with the numerical evaluation of the formula for these amplitudes in
their paper.}.
\begin{scriptsize}
\TABLE{
\begin{tabular}{|c|c|c|c|c|}
\hline
Helicity & $1/\epsilon^2$ & $1/\epsilon$ & 1 & [Ref]/(Eq.\#) \\
\hline
$++++++$ & $0$ & $0$ & $ (4.867 + i~2.092) 10^{-1}$&\cite{Bern:2005ji}/(4.3)\\
$++++++$ & $(3.672 +i~9.749) 10^{-9} $ & $(-3.404 + i~1.238) 10^{-8}$& $ -(3.016+ i~9.169) 10^{-8} $& [SN-A] \\
\hline
$-+++++$ & 0 & 0 & $-3.194 + i~0.6503 $ & \cite{Bern:2005ji}/(4.10)\\
$-+++++$ & $(5.921 +i~8.411) 10^{-9}$ & $(-1.606 +i~4.051) 10^{-8} $ & $ -(1.086 +i~0.038) 10^{-7} $ & [SN-A] \\
\hline
\hline
$--++++$ & $0$ & $8.995-i~3.046 $& {$43.089-i~20.288 $} &\cite{Bern:2005cq}/(4.27,4.28) \\
$--++++$ & $(1.280 + i~0.002) 10^{-8}$ & $(2.768+i~4.232) 10^{-8} $ & $ (-1.004+i~0.955)10^{-7} $ & [SN-A] \\
\hline
$-+-+++$ & $(1.045-i~0.580) 10^{-8}$ & $1.835-i~2.468 $ & $9.752-i~11.791$ & [SN] \\
\hline
$-++-++$ & $(-7.791+i~6.717) 10^{-9}$ & $3.178\cdot 10^{-2}+i~0.2188 $ & $-1.447+i~0.1955$ & [SN] \\
\hline
\hline
$---+++$ & $(8.934-i~5.359) 10^{-9}$ & $0.3599+ i~0.5782$ & $ 0.5617+i~5.8166$ & [SN] \\
\hline
$--+-++$ & $(0.1016 +i~1.276) 10^{-8}$ & $ -0.7819 +i~1.273 $ & $ -0.6249+i~6.552$ & [SN] \\
\hline
$-+-+-+$ & $(1.065- i~0.5417) 10^{-8}$ & $ -0.7475-i~0.7321 $ & $ -1.298 - i~3.255$ & [SN] \\
\hline
\hline
\end{tabular}
\caption{One loop six gluon colour ordered sub-amplitudes with a scalar loop
evaluated the specific point Eq.~(\ref{specificpoint}). [SN] means
that the result is obtained using our semi-numerical code, while
[SN-A] denotes the difference between the semi-numerical result and
the analytical one.}
\label{tablescalar}
}
\end{scriptsize}
Lastly in Table~\ref{tablescalar} we give results for the colour
sub-amplitudes $A^{[0]}(1,2,3,4,5,6)$ for scalar gluons, for the same
eight helicity choices.\footnote{In ref.~\cite{Bern:2005cq} [v1-v3]
the definition of $F_f$ has an overall sign missing, a typographical
error not present in the original calculation of the $\cal{N}$ = 1 term
in ref.~\cite{Bern:1994cg}.}
For all amplitudes for which no analytic result exists, we checked the
gauge invariance of the amplitudes by changing the gluon polarization.
The gauge invariance was obeyed with a numerical accuracy of ${\cal
O}\left(10^{-8}\right)$. To evaluate a single colour-ordered
sub-amplitude for a complex scalar took 9 seconds on a 2.8GHz Pentium
processor. To evaluate the complete set of 64 possible helicities will
be less than 64 times longer, because the scalar integrals are stored
during the calculation of the first amplitude are applicable to all
other configurations with the same external momenta.
\section{Conclusions}
In this paper we have presented numerical results which demonstrate
that the complete one-loop amplitude for six-gluon scattering is now
known numerically.
By forming multiplets of SUSY Yang Mills in the internal loops,
we were able compare with most of the known analytic results.
In addition, we have presented numerical results for amplitudes which
are currently completely unknown. Note that the analytic and
semi-numerical results are complementary. The hardest piece to
calculate analytically is the scalar contribution $A^{[0]}$, which is
the easiest for the semi-numerical approach. Thus it is possible that
a numerical code involving both semi-numerical and analytic results
will be the most efficient and expedient. Our results demonstrate the
power of the semi-numerical method, which can supplant the analytic
method where it is too arduous and provide a completely independent
check where analytic results already exist.
After inclusion of the one-loop corrections to the other parton
subprocesses involving quarks it would be possible to proceed to a NLO
evaluation of the rate for four jet production. We intend to use
these methods to calculate NLO corrections to other processes which we
consider to be of more pressing phenomenological interest.
\section*{Acknowledgements}
We would like to thank Zvi Bern, Lance Dixon and David Kosower for
providing helpful comments on the draft of this manuscript. We also
acknowledge useful discussions with John Campbell, Vittorio Del Duca
and Fabio Maltoni.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,008 |
Q: Redux , state.concat is not a function at rootReducer. And being forced to reRender an element for it to see the state change So I have this sidebar component where I load my store and my dispatcher
//select
const mapStateToProps = state => {
return { renderedEl: state.renderedEl }
}
function mapDispatchToProps(dispatch) {
return{
renderLayoutElement: element => dispatch(renderLayoutElement(element))
}
}
Then inside the same component this Is how I trigger the dispatcher
renderEl = (el) => {
var elementName = el.target.getAttribute('id');
var renderedElements = this.props.renderedEl; //this is data from the store
for (let key in renderedElements) {
if (key == elementName) {
renderedElements[key] = true
}
}
this.props.renderLayoutElement({renderedElements});
}
Then as I understand it gets sent to the reducer
import {RENDER_LAYOUT_ELEMENT} from "../constants/action-types"
const initialState = {
renderedEl: {
heimdall: false,
skadi: false,
mercator: false
}
}
function rootReducer(state = initialState, action){
if(action.type === RENDER_LAYOUT_ELEMENT){
return Object.assign({},state,{
renderedEl: state.renderedEl.concat(action.payload)
})
}
return state
}
export default rootReducer;
This is its action
import {RENDER_LAYOUT_ELEMENT} from "../constants/action-types"
export function renderLayoutElement(payload) {
return { type: RENDER_LAYOUT_ELEMENT, payload }
};
Now the thing is. Im receiving a
state.renderedEl.concat is not a function at rootreducer / at dispatch
I dont understand why does that happen.
Becuase, actually the store gets updated as I can see, but the console returns that error. And I have to reload the render that uses the props of that store (with an onhover) in order to be able to see the changes. It doesnt happen automatically as it would happen with a state
A: if(action.type === RENDER_LAYOUT_ELEMENT){
return { ...state, renderedEl: { ...state.renderedEl, ...action.payload } };
}
Duplicate from comments maybe it can be helpful to someone else :)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,906 |
{"url":"https:\/\/www.rdocumentation.org\/packages\/dbscan\/versions\/1.1-1\/topics\/sNN","text":"# sNN\n\n0th\n\nPercentile\n\n##### Shared Nearest Neighbors\n\nCalculates the number of shared nearest neighbors.\n\nKeywords\nmodel\n##### Usage\nsNN(x, k, kt = NULL, sort = TRUE, search = \"kdtree\", bucketSize = 10,\nsplitRule = \"suggest\", approx = 0)\n##### Arguments\nx\n\na data matrix, a dist object or a kNN object.\n\nk\n\nnumber of neighbors to consider to calculate the shared nearest neighbors.\n\nkt\n\nthreshold on the number of shared nearest neighbors graph. Edges are only preserved if kt or more neighbors are shared.\n\nsearch\n\nnearest neighbor search strategy (one of \"kdtree\", \"linear\" or \"dist\").\n\nsort\n\nsort the neighbors by distance? Note that this is expensive and sort = FALSE is much faster. kNN objects can be sorted using sort().\n\nbucketSize\n\nmax size of the kd-tree leafs.\n\nsplitRule\n\nrule to split the kd-tree. One of \"STD\", \"MIDPT\", \"FAIR\", \"SL_MIDPT\", \"SL_FAIR\" or \"SUGGEST\" (SL stands for sliding). \"SUGGEST\" uses ANNs best guess.\n\napprox\n\nuse approximate nearest neighbors. All NN up to a distance of a factor of 1+approx eps may be used. Some actual NN may be omitted leading to spurious clusters and noise points. However, the algorithm will enjoy a significant speedup.\n\n##### Details\n\nThe number of shared nearest neighbors is the intersection of the kNN neighborhood of two points. Note: that each point is considered to be part of its own kNN neighborhood. The range for the shared nearest neighbors is [0,k].\n\n##### Value\n\nAn object of class sNN containing a list with the following components:\n\nid\n\na matrix with ids.\n\ndist\n\na matrix with the distances.\n\nshared\n\na matrix with the number of shared nearest neighbors.\n\nk\n\nnumber of k used.\n\n%% ...\n\nNN and kNN for k nearest neighbors.\n\n\u2022 sNN\n\u2022 snn\n\u2022 sort.sNN\n##### Examples\n# NOT RUN {\ndata(iris)\nx <- iris[, -5]\n\n# finding kNN and add the number of shared nearest neighbors.\nk <- 5\nnn <- sNN(x, k = k)\nnn\n\n# shared nearest neighbor distribution\ntable(as.vector(nn$shared)) # explore neighborhood of point 10 i <- 10 nn$shared[i,]\n\nplot(nn, x)\n\n# apply a threshold to create a sNN graph with edges\n# if more than 3 neighbors are shared.\nplot(sNN(nn, kt = 3), x)\n# }\nDocumentation reproduced from package dbscan, version 1.1-1, License: GPL (>= 2)\n\n### Community examples\n\nLooks like there are no examples yet.","date":"2018-07-16 04:43:55","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.21643564105033875, \"perplexity\": 9334.228648315651}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-30\/segments\/1531676589179.32\/warc\/CC-MAIN-20180716041348-20180716061348-00471.warc.gz\"}"} | null | null |
Q: Unable to POST my data using retrofit android using kotlin I am new to android studio and kotlin. I have trying to post my data(object) onto the server but I'm getting a 404 response code.
My retrofit:
object RetrofitClient {
private var OurInstance : Retrofit?=null
val client = OkHttpClient.Builder()
val instance:Retrofit
get() {
if (OurInstance==null)
{
OurInstance =Retrofit.Builder()
.baseUrl("http://coreapi.imagin8ors.org:8080/")
.addConverterFactory(GsonConverterFactory.create())
.addCallAdapterFactory(RxJava2CallAdapterFactory.create())
.build()
}
return OurInstance!!
}
}
My interface:
interface IMyAPI {
@GET("v1/child/{dynamic}/learningpods_new")
fun getDynamic(@Path("dynamic")dynamic:String):Observable<ArrayList<Data>>
@POST("v1/authenticate/create")
fun createParent(@Body parentDetails: ParentDetails):Call<ParentDetails>
}
Code snippet in my activity:
fun sendNetworkRequest(parentDetails: ParentDetails){
val retrofit=Retrofit.Builder().baseUrl("http://coreapi.imagin8ors.org:8080/")
.addConverterFactory(GsonConverterFactory.create())
.build()
val client=retrofit.create(IMyAPI::class.java)
val call = client.createParent(parentDetails)
call.enqueue(object :Callback<ParentDetails>{
override fun onFailure(call: Call<ParentDetails>?, t: Throwable?) {
Toast.makeText(this@ParentmainActivity,"something went wrong",Toast.LENGTH_LONG).show()
}
override fun onResponse(call: Call<ParentDetails>?, response: Response<ParentDetails>?) {
Toast.makeText(this@ParentmainActivity,"successful :"+response?.code(),Toast.LENGTH_LONG).show()
}
})
}
Function call: sendNetworkRequest(new_parent)
Now, the response code returned is 404. This issue has been raised previously but those solutions didn't work for me. That's why I have posted as another question.
The links I have referred to: link1, link2, link3 and many more..
Initially I was using the retrofitclient but then it didn't work so I tried with the sendNetworkResquest() function. However, this too doesn't work.
A: I think what you need is a PUT request.
When I try your API with POST I actually get a 405 - Method Not Allowed. But PUT works.. except that I am sending no data, so I get a 400. But definitely not a 404
curl -X POST http://.../create -> Method not allowed
curl -X PUT http://.../create -> Bad Request
Also double check that your body actually contains valid data in the required format, as specified by your backend endpoint, otherwise you'll get further errors
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,710 |
{"url":"https:\/\/mirjamglessmer.com\/2017\/08\/15\/reflecting-the-sky\/","text":"# Reflecting the sky\n\nEven though I mostly look at the water to see waves, sometimes it is also really nice to just watch the reflections of the sky\u2026","date":"2022-07-07 14:17:07","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.803217351436615, \"perplexity\": 1299.6746662324867}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656104692018.96\/warc\/CC-MAIN-20220707124050-20220707154050-00597.warc.gz\"}"} | null | null |
Comic-Con is just getting underway in San Diego—with Bill Murray and Hellboy kicking off the wave of nerdery earlier today—and we doubt there's any better way to continue these festivities than with everybody's favorite 900-something-year-old alien and his plucky, young companion. That's right, today was Doctor Who day at Comic-Con, and though the BBC didn't have much to say about the future of what is probably its most popular series that isn't a dry comedy about humorous misunderstandings, it did show off a new trailer and reveal that the new season (or "series," to our English friends) will premiere on September 19.
The trailer itself features the usual sort of things you'd expect from a Doctor Who trailer, including mysterious voiceovers, Daleks, and some weird monsters, but that's not to say that there aren't some surprises. For one, Michelle Gomez's Missy shows up and she seems surprisingly friendly with Jenna Coleman's Clara. Also, the Doctor plays an electric guitar for a second, which is pretty cool. The big hook, though, is a quick appearance from Game Of Thrones' Maisie Williams that is obviously designed to get Doctor Who fans talking about who she could be playing. All executive producer Steven Moffat would say is that "it's going to be surprising, what she gets up to." We'll have to wait until (at least) September to find out. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,944 |
{"url":"http:\/\/gate-exam.in\/ME\/PaperSolution\/GATE2014","text":"# Mechanical Engineering - GATE 2014 Paper Solution\n\nQuestion No.\u00a01\n\n#### SET - 1\n\nChoose the most appropriate phrase from the options given below to complete the following sentence.\n\nThe aircraft_________ take off as soon as its flight plan was filed.\n\nQuestion No.\u00a02\n\nAll women are entrepreneurs.\n\nSome women are doctors.\n\nWhich of the following conclusions can be logically inferred from the above statements?\n\nQuestion No.\u00a03\n\nChoose the most appropriate word from the options given below to complete the following sentence.\n\nMany ancient cultures attributed disease to supernatural causes. However, modern science has largely helped _________ such notions.\n\nQuestion No.\u00a04\n\nThe statistics of runs scored in a series by four batsmen are provided in the following table. Who is the most consistent batsman of these four?\n\n Batsman Average Standard deviation K 31.2 5.21 L 46.0 6.35 M 54.4 6.22 N 17.9 5.90\n\nQuestion No.\u00a05\n\nWhat is the next number in the series?\n\n12 35 81 173 357 ____\n\nQuestion No.\u00a06\n\nFind the odd one from the following group:\n\nW,E,K,O\u00a0\u00a0 I,Q,W,A\u00a0\u00a0\u00a0 F,N,T,X\u00a0\u00a0\u00a0 N,V,B,D\n\nQuestion No.\u00a07\n\nFor submitting tax returns, all resident males with annual income below Rs 10 lakh should fill up Form P and all resident females with income below Rs 8 lakh should fill up Form Q. All people with incomes above Rs 10 lakh should fill up Form R, except non residents with income above Rs 15 lakhs, who should fill up Form S. All others should fill Form T. An example of a person who should fill Form T is\n\nQuestion No.\u00a08\n\nA train that is 280 metres long, travelling at a uniform speed, crosses a platform in 60 seconds and passes a man standing on the platform in 20 seconds. What is the length of the platform in metres?\n\nQuestion No.\u00a09\n\nThe exports and imports (in crores of Rs.) of a country from 2000 to 2007 are given in the following bar chart. If the trade deficit is defined as excess of imports over exports, in which year is the trade deficit 1\/5th of the exports?\n\nQuestion No.\u00a010\n\nYou are given three coins: one has heads on both faces, the second has tails on both faces, and the third has a head on one face and a tail on the other. You choose a coin at random and toss it, and it comes up heads. The probability that the other face is tails is\n\nQuestion No.\u00a011\n\nGiven that the determinant of the matrix $\\left[\\begin{array}{ccc}1& 3& 0\\\\ 2& 6& 4\\\\ -1& 0& 2\\end{array}\\right]$ is -12, the determinant of the matrix $\\left[\\begin{array}{ccc}2& 6& 0\\\\ 4& 12& 8\\\\ -2& 0& 4\\end{array}\\right]$ is\n\nQuestion No.\u00a012\n\n$\\underset{x\\to 0}{Lt}\\frac{x-\\mathrm{sin}x}{1-\\mathrm{cos}x}$ is\n\nQuestion No.\u00a013\n\nThe argument of the complex number $\\frac{1+i}{1-i}$ , where $i=\\sqrt{-1},$,is\n\nQuestion No.\u00a014\n\nThe matrix form of the linear system $\\frac{\\mathrm{dx}}{\\mathrm{dt}}=3x-5y$ and $\\frac{\\mathrm{dy}}{\\mathrm{dt}}=4x+8y$ is\n\nQuestion No.\u00a015\n\nWhich one of the following describes the relationship among the three vectors,$\\widehat i+\\widehat j+\\widehat k,2\\widehat i+3\\widehat j+\\widehat k$ and $5\\widehat i+6\\widehat j+4\\widehat k$ ?\n\nQuestion No.\u00a016\n\nA circular rod of length \u2018L\u2019 and area of cross-section \u2018A\u2019 has a modulus of elasticity \u2018E\u2019 and coefficient of thermal expansion \u2018\u03b1\u2019. One end of the rod is fixed and other end is free. If the temperature of the rod is increased by \u0394T, then\n\nQuestion No.\u00a017\n\nA metallic\u00a0 rod of 500 mm\u00a0 length and 50mm diameter when subjected to a tensile force of 100KN at the ends,experinces an increse an its length by 0.5mm and a reduction in its diameter by 0.015mm.The poission's ratio of the road material is__________\n\nQuestion No.\u00a018\n\nCritical damping is the\n\nQuestion No.\u00a019\n\nA\u00a0 circular object of radius r rolls without slipping on a horizontal level floor with center having velocity V. The velocity at the point of contact between the object and floor is","date":"2018-01-21 16:47:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 17, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.606534481048584, \"perplexity\": 1333.0217623604542}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-05\/segments\/1516084890795.64\/warc\/CC-MAIN-20180121155718-20180121175718-00609.warc.gz\"}"} | null | null |
\section{Introduction}
Thermal states of non relativistic particles interacting by the sole
Coulomb potential are known to provide an adequate description of many
states of matter. The introduction of magnetic interactions between the
particles poses a novel problem since they are mediated by the coupling to
the transverse part of the electromagnetic field. This immediately leads to
consider the full system of matter in equilibrium with radiation : the
relevant theory becomes then the thermal quantum electrodynamics (thermal
QED).
In order to go beyond pure electrostatics without facing the full QED, a
number of studies rely on the Darwin approximation. Darwin has shown
\cite{Darwin}, \cite{Landau} that one can eliminate the transverse degrees
of freedom of the field within the Lagrangian formalism up to order
$c^{-2}$ ($c$ is the speed of light). A nice review of the derivation of
the Darwin Lagrangian and a lucid discussion of its consequences can be
found in \cite{Essen}. The resulting Darwin Hamiltonian can be used to
investigate the equilibrium properties of the so called weakly relativistic
plasmas; see the recent works of Appel and Alastuey
\cite{Alastuey-Appel-Physica-238}, \cite{Appel-Alastuey-Physica-252},
\cite{Appel-Alastuey-Phys.Rev.} and earlier references therein. These
authors have done a careful analysis of the domain of validity of the
Darwin approximation and shown in particular that the predictions of the
Darwin Hamiltonian on the tail of particle correlations in thermal states
cannot be correct. Indeed the well-known Bohr--van Leeuwen theorem
\cite{Alastuey-Appel-Physica-276} asserts that classical (non-quantum)
matter completely decouples from the radiation field. Thus the Darwin
Hamiltonian, which treats the particles classically, should not predict any
effect of the transverse field when used for thermal equilibrium
computations. The Darwin approximation is, however, not deprived of any
meaning in statistical physics. Indeed, the authors show in
\cite{Appel-Alastuey-Phys.Rev.} that Darwin predictions about current
correlations coincide with those of thermal QED in the restricted window of
distances $\lambda_\text{part}\ll r \ll \lambda_\text{ph}$, where
$\lambda_\text{part}=\hbar\sqrt{\beta/m}$ is the de Broglie thermal
wavelength of the particles and $\lambda_\text{ph}=\beta\hbar c$ the
thermal wavelength of the photons.
But to determine the tail $r\gg \lambda_\text{ph}$ of the correlations in
the presence of the radiation field, matter has to be treated quantum
mechanically to avoid the conclusion of the Bohr--van Leeuwen theorem. The
situation is similar to orbital diamagnetism in equilibrium, which is of
quantum-mechanical origin.
In this work, we consider equilibrium states of non-relativistic spinless
quantum charges coupled with the radiation field in the standard way
(section 2). We shall, however, treat the field classically on the ground
that the large distances $r\gg \lambda_\text{ph}$ are controlled by the
small wave numbers $k\sim \tfrac{1}{r}\ll\tfrac{1}{\lambda_\text{ph}}$,
implying $\beta\hbar \omega_{\mathbf{k}}\sim \frac{\lambda_\text{ph}}{r}\ll
1$. Hence only long-wavelength photons will contribute to the asymptotics
which is expected to be adequately described by classical fields. The full
QED model with quantized electromagnetic field will be studied in a
subsequent work (see also comments in the concluding remarks, section 8).
Our main tool will be the Feynman-Kac-It\^o path integral representation of
the degrees of freedom of the charges.
The Feynman-Kac integral representation has been widely used to derive
various properties of quantum Coulomb systems, in particular to determine
the exact large-distance behaviour of the correlations; see \cite{Cornu1},
\cite{Cornu3}, and \cite{Alastuey}, \cite{Brydges-Martin} for reviews. In
this representation quantum charges become fluctuating charged loops
(closed Brownian paths), formally analogous to classical fluctuating wires
carrying multipoles of all orders. These fluctuations are responsible for
the lack of exponential screening in the quantum plasma and for an
algebraic tail $\sim r^{-6}$ of the particle correlations
\cite{Alastuey-Martin}.
Adding an external magnetic field produces a phase factor in the
Feynman-Kac-It\^o formula, whose argument is the flux of the magnetic field
across the random loop. Correlations in the case of an homogeneous external
magnetic field have been studied in \cite{Cornu2}. When the particles are
thermalized with the field, the latter becomes itself random and
distributed according to the thermal weight of the free radiation. The
system can be viewed as a classical-like system of random loops immersed in
a random electromagnetic field. At this point, the field degrees of
freedom can be exactly integrated out by means of a simple Gaussian
integral since the Hamiltonian of free radiation is quadratic in the field
amplitudes. One is then left with an effective pairwise current-current
interaction between the loops which has a form similar to the magnetostatic
energy between a pair of classical currents. For the sake of illustrating
the basic mechanisms in a simple setting, this program is carried out in
section 3
with particles obeying Maxwell-Boltzmann statistics.
Appropriate modifications needed to take into account the particle
statistics (Bose or Fermi) are given in section 7.
In section 4 we apply the formalism to the determination of the asymptotic
form of the correlation between two quantum particles embedded in a
classical plasma. This simple model already illustrates the main features
occurring in the general system. The effective magnetic interaction
contributes to the $r^{-6}$ tail, but its ratio to the Coulombic
contribution is of the order of the square of the relativistic parameter
$(\beta mc^{2})^{-1}=(\lambda_\text{part}/\lambda_\text{ph})^2$.
In section 5 we consider the generalization of the results obtained for two
particles to the full system of quantum charges. The analysis relies on the
technique of quantum Mayer graphs previously developed for Coulomb systems,
and we merely point out the few changes that are needed to include the
effective magnetic interactions.
Field fluctuations in plasmas have been studied for a long time at
macroscopic scales, much larger than interparticle distances; see
\cite{Landau}, \cite{Felderhof} and references cited therein. In section 6,
we reexamine this question from a microscopic viewpoint and show that
electromagnetic field correlations are always long ranged due to the
quantum nature of the particles. This is in disagreement with the
prediction of macroscopic theories. We come back to this point in the
concluding remarks (section 8). However, in the classical limit, we recover
the fact already observed in \cite{Felderhof} that the long-range behaviour
of the longitudinal and transverse parts of the electric field correlations
compensate exactly.
In section 7, we generalise the formalism developed in section 3 to include
Bose and Fermi particle statistics. This is done as usual by decomposing
the permutation group into cycles and grouping particles belonging to a
cycle into an extended Brownian loop. When this is combined with the
Feynman-Kac-It\^o path integral representation of the particles, the system
takes again a classical-like form: a collection of Brownian loops immersed
in a classical random electromagnetic field. At this point the physical
quantities can again be analyzed in terms of Mayer graphs comprising
pairwise Coulomb and effective magnetic interactions, as in section 5.
The methods presented in this paper have been applied to the study of the
semi-classical Casimir effect \cite{Buenzli-Martin},
\cite{Buenzli-Martin2}.
\section{The model}
We first consider the QED model for non-relativistic quantum charges
(electrons, nuclei, ions) with masses $m_{\gamma}$ and charges $e_{\gamma}$
contained in a box $\Lambda\in\mathbb{R}^3$ of linear size $L$ and
appropriate statistics. The index $\gamma$ labels the ${\cal S}$ different
species and runs from $1$ to ${\cal S}$. The particles are in equilibrium
with the radiation field at temperature $T$. The field is itself enclosed
into a large box $K$ with sides of length $R,\,R\gg L$. The Hamiltonian of
the total finite volume system reads, in Gaussian units,
\begin{align}
H_{L,R}=\sum_{i=1}^{n}\frac{\({\bf
p}_{i}-\frac{e_{\gamma_{i}}}{c}\mathbf{A}(\r_{i})\)^{2}}{2m_{\gamma_{i}}}+
\sum_{i<j}^{n}
\frac{e_{\gamma_{i}}e_{\gamma_{j}}}{|\r_{i}-\r_{j}|}+\sum_{i=1}^{n} V_{{\rm
walls}}(\gamma_{i},\r_{i})+H_{0}^{{\rm rad}}. \label{B.1}
\end{align}
The sums run on all particles with position $\r_{i}$, momentum
$\mathbf{p}_i$, and species index $\gamma_{i}$; $V_{{\rm
walls}}(\gamma_{i},\r_{i})$ is a steep external potential that
confines a particle in $\Lambda$. It can eventually be taken infinitely
steep at the wall's position, implying Dirichlet boundary conditions---{\it i.e.},
vanishing of the particle wave functions at the boundaries of $\Lambda$.
The electromagnetic field is written in the Coulomb (or transverse) gauge
so that the vector potential $\mathbf{A}(\r)$ is divergence free and $H_{0}^{{\rm
rad}}$ is the Hamiltonian of the free radiation field. The Coulomb
gauge is usually preferred for simplicity in situations where the particles
are non-relativistic and high-energy processes are neglected
\cite{Cohen}. It has the advantage to clearly disentangle electrostatic and
magnetic couplings in the Hamiltonian.
We impose periodic boundary conditions on the faces of the large box $K$
\footnote{Periodic conditions are convenient here. We could as well choose
metallic boundary conditions. Since the field region $K$ will be
extended over all space right away, the choice of conditions on the
boundaries of $K$ are expected to make no differences for the particles
confined in $\Lambda$.}. Hence expanding $\mathbf{A}(\r)$ and the free photon
energy $H_{0}^{{\rm rad}}$ in the plane-wave modes $\mathbf{k}= (\frac{2\pi
n_{x}}{R},\frac{2\pi n_{y}}{R},\frac{2\pi n_{z}}{R})$ gives
\begin{align}
\mathbf{A}(\r)&=\(\frac{4\pi \hbar c^{2}}{R^{3}}\)^{1/2}\sum_{\mathbf{k}\lambda}g(\mathbf{k})
\frac{ {\bf e}_{\mathbf{k}\lambda}}{\sqrt{2\omega_{\mathbf{k}}}}
(a_{\mathbf{k}\lambda}^{*}\mathrm{e}^{-i\mathbf{k}\cdot\r}+a_{\mathbf{k}\lambda}\mathrm{e}^{i\mathbf{k}\cdot\r})\label{B.2}\\
H_{0}^{{\rm rad}}&={\sum_{\mathbf{k}\lambda}}\hbar
\omega_\mathbf{k}\,a_{\mathbf{k}\lambda}^{*}a_{\mathbf{k}\lambda} \label{B.2a}
\end{align}
where $a_{\mathbf{k}\lambda}^{*}$ and $\,a_{\mathbf{k}\lambda}$ are the creation and
annihilation operators for photons of modes $({\mathbf{k}\lambda})$, ${\bf
e}_{\mathbf{k}\lambda}$ ($\lambda=1,2$) are two unit polarization vectors
orthogonal to ${\mathbf{k}}$, and $\omega_\mathbf{k}=ck,\; k=|\mathbf{k}|$. In (\ref{B.2}),
$g(\mathbf{k}), \; g(0)=1$, is a real spherically symmetric smooth form factor
needed to take care of the ultraviolet divergencies. It is supposed to
decay rapidly beyond the characteristic wave number $k_c=m c/\hbar$ (see
\cite{Cohen}, chap. 3). Since we are interested in the large-distance
$r\to\infty$ asymptotics, related to the small-$\mathbf{k}$ behaviour $k\to 0$,
the final result will be independent of this cut-off function.
The total partition function
\begin{align}
Z_{L,R}= \mathrm{Tr}\ \mathrm{e}^{-\beta H_{L,R}}
\label{B.3}
\end{align}
is obtained by carrying the trace $\mathrm{Tr}=\mathrm{Tr}_\text{mat} \mathrm{Tr}_\text{rad}$ of
the total Gibbs weight over particles' and the field's degrees of freedom~:
namely, on the particle wave functions with appropriate quantum statistics
and on the Fock states of the photons. The average values of observables
$\langle O_\text{mat}\rangle=Z_{L,R}^{-1}\mathrm{Tr}\( \mathrm{e}^{-\beta
H_{L,R}}O_\text{mat}\)$ concerning only the particle degrees of freedom
can be computed from the reduced thermal weight
\begin{align}
\rho_{L,R}=\frac{\mathrm{Tr}_\text{rad}\ \mathrm{e}^{-\beta H_{L,R}}}{Z^\text{rad}_{0, R}},
\label{B.4}
\end{align}
where $Z_{0, R}^\text{rad}=\mathrm{Tr}_\text{rad}\exp{(-\beta H_{0}^{{\rm rad}})}$
is the partition function of the free radiation field, as follows from the
obvious identity
\begin{align}
\langle O_\text{mat} \rangle= \frac{\mathrm{Tr}_\text{mat}
\left(O_\text{mat}\,\rho_{L,R}\right)}{\mathrm{Tr}_\text{mat}\, \rho_{L,R}} \label{B.4a}.
\end{align}
We shall perform the thermodynamic limit in two stages by first letting
$R\to\infty$. Then $\rho_{L}=\lim_{R\to\infty}\rho_{L,R}$ defines the
effective statistical weight of the particles in $\Lambda$ immersed in an
infinitely extended thermalized radiation field.
As discussed in the Introduction, in this paper we treat the
electromagnetic field classically. This amounts to replacing the photon
creation and annihilation operators in (\ref{B.2}) and (\ref{B.2a}) by
complex amplitudes $\alpha_{\mathbf{k}\lambda}^{*}$ and $\alpha_{\mathbf{k}\lambda}$. In
this case, the free field distribution factorizes out as $\exp\({-\beta
H_{R,L}}\)=\exp\({-\beta H_{0}^{{\rm rad}}}\)\exp\({-\beta
H_\mathbf{A}}\)$, where
\begin{align}
H_{R,L}=H_\mathbf{A}+H_{0}^{{\rm rad}},\quad H_\mathbf{A}= \sum_{i=1}^n \frac{\({\bf
p}_{i}-\frac{e_{\gamma_{i}}}{c}\mathbf{A}(\r_{i})\)^{2}}{2m_{\gamma_{i}}}
+U_\text{pot}(\r_{1},\gamma_{1},\ldots,\r_{n},\gamma_{n}), \label{B.4b}
\end{align}
and $U_\text{pot}$ is the total potential energy. Since the free radiation
weight $\exp\({-\beta H_{0}^{{\rm rad}}}\)$ is Gaussian,
$\mathbf{A}(\r)=\mathbf{A}(\r,\{\alpha_{\mathbf{k}\lambda}\})$ can be viewed as a realization of a
Gaussian random field, and the term
$H_\mathbf{A}=H_{\mathbf{A}}(\{\alpha_{\mathbf{k}\lambda}\})$ becomes the energy of the
particles in a given realization of the vector potential having Fourier
amplitudes $\{\alpha_{\mathbf{k}\lambda}\}$.
The partial trace (\ref{B.4}) becomes, explicitly,
\begin{align}
\rho_{L,R}=\left\langle \mathrm{e}^{-\beta H_\mathbf{A}} \right\rangle_{{\rm rad}},
\label{B.5}
\end{align}
where for a general function $F(\{\alpha_{\mathbf{k}\lambda}\})$ of the mode
amplitudes $\left\langle F \right\rangle_{{\rm rad}}$ denotes the
normalized Gaussian average over all modes \footnote{The classical field is
expanded as in (\ref{B.2}) and (\ref{B.2a}) with dimensionless
amplitudes $\alpha_{\mathbf{k} \lambda}$. In fact there will be no $\hbar$
dependence arising from the field, as seen by changing everywhere
$\alpha_{\mathbf{k}\lambda} \mapsto \alpha_{\mathbf{k}\lambda}/\sqrt{\hbar}$.}
\begin{align}
\left\langle F \right\rangle_{{\rm rad}}=\prod_{\mathbf{k}\lambda}\int
\frac{\d^{2}\alpha_{\mathbf{k}\lambda}}{\pi}\left[\beta\hbar \omega_{\mathbf{k}}
\mathrm{e}^{-\beta\hbar \omega_{\mathbf{k}}|\alpha_{\mathbf{k} \lambda}|^{2}}\right]
F(\{\alpha_{\mathbf{k}\lambda}\}). \label{B.5a}
\end{align}
Note that the stability of Coulombic matter and the existence of
thermodynamics for extended systems are assured if at least one of the
species obeys Fermi statistics \cite{Lieb}. In the next section, merely as
a matter of simplifying the presentation, we compute the effective particle
interactions defined by $\rho_{L}$ ignoring quantum statistics. In this
case, Maxwell-Boltzmann statistics requires the presence of an additional
short-range repulsive potential
$V_\text{sr}(\gamma_i,\gamma_j,|\r_i-\r_j|)$ in the Hamiltonian (\ref{B.1})
to prevent the collapse of opposite charges and guarantee thermodynamical
stability. The generalization to Fermi and Bose statistics will be given in
section 7.
\section{The gas of charged loops and the effective magnetic interaction}
We now introduce the Feynman-Kac-It\^o path integral representation of the
configurational matrix element $\langle \r_1,...,\r_n | \mathrm{e}^{-\beta H_{\mathbf{A}}}
| \r_1,..., \r_n\rangle $ for the particles interacting with a fixed
realization of the field. For a single particle of mass $m$ and charge $e$
in a scalar potential $V^\text{ext}(\r)$ and vector potential $\mathbf{A}(\r)$, we
first recall that this matrix element reads \cite{Feynman-Hibbs},
\cite{Roepstorff}, \cite{Simon}
\begin{align}
&\langle \r |
\exp\left(-\beta\left[\frac{\(\mathbf{p}-\frac{e}{c}\mathbf{A}(\r)\)^{2}}{2m}+V^\text{ext}(\r)\right]\right)|\r\rangle=
\(\frac{1}{2\pi\lambda^2}\)^{3/2} \!\int\!\! \mathrm{D}(\b\xi) \nonumber\\
&\times
\exp\left(-\beta\left[\int_{0}^{1}\!\!\! \d s\ V^\text{ext}\big(\r +\lambda \b\xi
(s)\big)-i\frac{e}{\sqrt{\beta m c^{2}}}\int_{0}^{1} \!\!\! \d\b\xi(s) \cdot
\mathbf{A}\big(\r+\lambda \b\xi (s)\big)\right] \right).
\label{3.1}
\end{align}
Here $\b\xi (s),\;0\leq s \leq1,\; \b\xi (0)=\b\xi (1)=\mathbf{0}$, is a
closed dimensionless Brownian path and $\mathrm{D}(\b\xi)$ is the corresponding
conditional Wiener measure normalized to $1$. It is Gaussian, formally
written as
\\
$\exp\Big(-\frac{1}{2}\int_0^1 \d s \left|\frac{\d\b\xi (s)}{\d
s}\right|^2\Big)\d[\b\xi (\cdot)]$, with zero mean and covariance
\begin{align}
\int\!\! \mathrm{D}(\b\xi)\,\xi^{\mu}(s_1)\xi^{\nu}(s_2)=\delta^{\mu\nu}(\min(s_1,\:s_2)-s_1 s_2)
\label{3.2}
\end{align}
where $\xi^{\mu}(s)$ are the Cartesian coordinates of $\b\xi(s)$. In this
representation a quantum point charge looks like a classical charged closed
loop denoted by $\mathcal{F}=(\r,\b\xi)$, located at $\r$ and with a random shape
$\b\xi(s)$ having an extension given by the de Broglie length
$\lambda=\hbar\sqrt{\beta/m}$ (the quantum fluctuation). The magnetic
phase in (\ref{3.1}) is a stochastic line integral: it is the flux of the
magnetic field across the closed loop. The correct interpretation of this
stochastic integral is given by the rule of the middle point; namely, the
integral on a small element of line $\mathbf{x}-\mathbf{x}'$ is defined by
\begin{align}
\int_{\mathbf{x}}^{\mathbf{x}'}\!\!\!\d\b\xi\cdot {\bf f}(\b\xi)=(\mathbf{x}-\mathbf{x}')\cdot{\bf f}
\(\frac{\mathbf{x}+\mathbf{x}'}{2}\),\quad \mathbf{x}-\mathbf{x}'\to 0 \label{3.3}
\end{align}
We shall stick to this rule when performing explicit
calculations.\footnote{Other prescriptions are possible for the path
integral to correctly represent the quantum mechanical Gibbs weight in
presence of a magnetic field. The It\^o rule may be used when ${\bf f}$
is divergence free \cite{Roepstorff}.} Note the dimensionless
relativistic factor $(\beta m c^{2})^{-1/2}$ in front of the vector
potential term.
This is readily generalized to a system of $n$ interacting particles~: The
weight in the space of $n$ loops $\mathcal{F}_{1}=(\r_{1},\gamma_{1},
\b\xi_{1}),\ldots,\mathcal{F}_{n}=(\r_{n},\gamma_{n},\b\xi_{n})$ coming from the
path integral representation of \\ \mbox{$\langle \r_1,...,\r_n |
\mathrm{e}^{-\beta H_{\mathbf{A}}} | \r_1,..., \r_n\rangle$} is $\exp(-\beta
U(\mathcal{F}_1,...,\mathcal{F}_n,\mathbf{A}))$ where
\begin{align}
U(\mathcal{F}_1,...,\mathcal{F}_n, \mathbf{A}) =&\sum_{i<j}^{n}e_{\gamma_{i}}e_{\gamma_{j}}V_\text{c}
(\mathcal{F}_{i},\mathcal{F}_{j}) \nonumber
\\&-i\sum_{j=1}^{n}\frac{e_{\gamma_{j}}} {\sqrt{\beta
m_{\gamma_{j}}c^{2}}} \int_{0}^{1}\!\!\! \d\b\xi_j(s) \cdot
\mathbf{A}(\r_j+\lambda_{\gamma_j} \b\xi_j (s)) \label{3.4}
\end{align}
The matrix element $\langle \r_1,...,\r_n | \mathrm{e}^{-\beta H_{\mathbf{A}}} | \r_1,...,
\r_n\rangle $ is obtained by integrating $\exp(-\beta U(\mathcal{F}_1,...,\mathcal{F}_n,\mathbf{A}))$
over the random shapes $\pmb{\xi}_1,...,\pmb{\xi}_n$ of the loops, as in (\ref{3.1}).
In (\ref{3.4}),
\begin{align}
V_\text{c}(\mathcal{F}_{i},\mathcal{F}_{j})=\int_{0}^{1}\!\!\d s\ \frac{1}
{|\r_{i}+\lambda_{\gamma_{i}}\b\xi _{i}(s)-\r_{j}-\lambda_{\gamma_{j}}\b\xi
_{j}(s)|} \label{B.6}
\end{align}
is the Coulomb potential between two loops, and for the sake of brevity, we
have omitted the non electromagnetic terms
\begin{align}
\sum_{i<j}^n V_\text{sr}(\mathcal{F}_i,\mathcal{F}_j) + \sum_{i=1}^n V_\text{walls}(\mathcal{F}_i) \label{non-em-terms}
\end{align}
corresponding to the short-range regularization and to the confinement
potential. The vector potential term can be written as \\ \mbox{$-i\int\!\!
\d\mathbf{x}\ \mathbf{A}(\mathbf{x})\cdot \pmb{\mathcal{J}}(\mathbf{x})$} in terms of current densities associated
with the Brownian loops~:
\begin{align}
\pmb{\mathcal{J}}(\mathbf{x})=\sum_{i=1}^{n}\mathbf{j}(\mathcal{F}_i,\mathbf{x}),\quad
\mathbf{j}(\mathcal{F}_i,\mathbf{x})=\frac{e_{\gamma_{i}}}{\sqrt{\beta m_{\gamma_i} c^{2}}}\int_{0}^{1}
\!\!\!\d\b\xi_{i}(s)\ \delta(\mathbf{x}-\r_{i}-\lambda_{\gamma_{i}}\b\xi_{i}(s)).
\label{B.8}
\end{align}
If one interprets the (ill-defined) derivative
$\lambda_{\gamma_{i}}\d\b\xi_{i}(s)/\d s={\bf v}_{i}(s)$ as the
``velocity'' of a particle of charge $e_{\gamma_{i}}$ moving along the loop
$\b\xi_{i}(s)$, the quantity $e_{\gamma_{i}}{\bf
v}_{i}(s)\delta(\mathbf{x}-\r_{i}-\lambda_{\gamma_{i}}\b\xi_{i}(s)) $
corresponds to a classical current density. This is just a formal
analogy. In subsequent calculations of stochastic integrals arising from
(\ref{B.8}), we will always use the mathematically well-defined rule of the
middle point (\ref{3.3}). Moreover, such ``imaginary time'' currents
appearing in the Feynman-Kac-It\^o representation are not the physical
``real-time'' current observables. Our definition (\ref{B.8}) also includes
the relativistic factor $(\beta m_{\gamma_i} c^{2})^{-1/2}$.
A remarkable fact is that the transverse part of the field enters in
\mbox{$\exp(-\beta U(\mathcal{F}_1,...,\mathcal{F}_n,\mathbf{A}))$} as a phase factor linear in
$\mathbf{A}$ and its Fourier amplitudes (contrary to the Hamiltonian (\ref{B.1})
written in operatorial form). Since the statistical weight $\mathrm{e}^{-\beta
H_{0}^{{\rm rad}}}$ (\ref{B.2a}) is a Gaussian function of these
Fourier amplitudes, it makes it possible to perform explicitly the partial
trace over the field degrees of freedom in (\ref{B.5}) according to the
following steps~:
\begin{align}
&\left\langle\exp\left[i\beta\int\!\! \d\mathbf{x}\ \mathbf{A}(\mathbf{x})\cdot \pmb{\mathcal{J}}(\mathbf{x})\right]
\right\rangle_{{\rm rad}}= \left\langle\prod_{\mathbf{k}\lambda}\exp
\left[i(u_{\mathbf{k}\lambda}^{*}\alpha_{\mathbf{k}\lambda}+u_{\mathbf{k}\lambda}\alpha_{\mathbf{k}\lambda}^{*})\right]
\right\rangle_{{\rm rad}}=\nonumber\\
&\exp\left[-\frac{\beta}{2R^{3}}\sum_{\mathbf{k}\lambda}\frac{4\pi
g^{2}(\mathbf{k})}{k^{2}}\left|\pmb{\mathcal{J}}(\mathbf{k})\cdot{\bf
e}_{\mathbf{k}\lambda}\right|^{2}\right] =
\exp\left[-\frac{\beta}{2}\int\!\!\frac{\d\mathbf{k}}{(2\pi)^{3}}
(\mathcal{J}^\mu(\mathbf{k}))^{*}G^{\mu\nu}(\mathbf{k})\mathcal{J}^\nu(\mathbf{k})\right].
\label{B.9}
\end{align}
The first equality is obtained by introducing the mode expansion
(\ref{B.2}), yielding
\begin{align}
u_{\mathbf{k}\lambda}=\beta\(\frac{4\pi\hbar c^{2}}{R^{3}}\)^{1/2}
\frac{g(\mathbf{k})}{\sqrt{2\omega_{\mathbf{k}}}}\ \pmb{\mathcal{J}}(\mathbf{k})\cdot{\bf e}_{\mathbf{k}\lambda}, \quad
\pmb{\mathcal{J}}(\mathbf{k})=\int\!\! \d\mathbf{x}\ \mathrm{e}^{-i\mathbf{k}\cdot\mathbf{x}}\pmb{\mathcal{J}}(\mathbf{x}) \label{B.10}
\end{align}
The second equality results from (\ref{B.5}), (\ref{B.5a}) and the Gaussian
integral
\mbox{$\int\!\frac{d^{2}\alpha}{\pi}\,\mathrm{e}^{-b|\alpha|^{2}+i(u^{*}\alpha+u\alpha^{*})}
= b^{-1}\mathrm{e}^{-b^{-1}|u|^{2}}$}, $b>0$, whereas the infinite volume limit
$R\to\infty$ and the polarization sum have been performed in the last
equality. We have denoted by $G^{\mu\nu}(\k)$ the covariance of the free
transverse field~:
\begin{align}
G^{\mu\nu}(\mathbf{k})=\frac{4\pi g^{2}(\mathbf{k})}{k^{2}}\delta_{{\rm
tr}}^{\mu\nu}(\mathbf{k}), \quad \delta_{{\rm
tr}}^{\mu\nu}(\mathbf{k})=\delta^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{k^{2}}, \quad k^\mu G^{\mu\nu}(\k) \equiv 0
\label{B.11}
\end{align}
($\delta_{{\rm tr}}^{\mu\nu}(\mathbf{k})$ is the transverse Kronecker symbol). In
(\ref{B.9}) and throughout the paper, summation on repeated vector
components $\mu,\nu=1,2,3$ is understood. In the configuration space, the
asymptotic behaviour of $G^{\mu\nu}(\mathbf{x})$ is obtained by approximating
$g^{2}(\mathbf{k})\sim 1$ in the inverse Fourier transform of $G^{\mu\nu}(\mathbf{k})$:
\begin{align}
G^{\mu\nu}(\mathbf{x})\sim\int\!\! \frac{\d\mathbf{k}
}{(2\pi)^{3}}\,\mathrm{e}^{i\mathbf{k}\cdot\mathbf{x}}\,\frac{4\pi}{k^{2}}
\(\delta^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{k^{2}}\)
=\frac{1}{2r}\(\delta_{\mu\nu}+\frac{x^\mu
x^{\nu}}{r^{2}}\),\;r=|\mathbf{x}|\to\infty \label{B.12}.
\end{align}
Decomposing the total current (\ref{B.8}) into the individual loop currents
we see that the effective weight (\ref{B.9}) takes the form
\begin{align}
\left\langle\exp\left[i\beta\int\!\! \d\mathbf{x}\ \mathbf{A}(\mathbf{x})\cdot \pmb{\mathcal{J}}(\mathbf{x})\right]
\right\rangle_{{\rm rad}}=&\prod_{i=1}^{n}\exp\(-\frac{\beta
e_{\gamma_{i}}^{2}}{2}{W_\text{m}}(i,i)\)\nonumber\\
&\times\exp\(-\beta\sum_{i<j}^{n}e_{\gamma_{i}}e_{\gamma_{j}}{W_\text{m}}(i,j)\),
\label{B.13}
\end{align}
where for two loops $i=\mathcal{F}_{i}$ and $j=\mathcal{F}_{j}$ we have introduced the
loop-loop effective magnetic potential
\begin{align}
&e_{\gamma_{i}}e_{\gamma_{j}}{W_\text{m}}(i,j)=\int\!\! \d\mathbf{x} \!\!\int
\!\!\d\mathbf{y}\ (j^{\mu}(\mathcal{F}_i,\mathbf{x}))^{*}G^{\mu\nu}(\mathbf{x}-\mathbf{y})j^{\nu}(\mathcal{F}_j,\mathbf{y})=
\label{B.14}\\
&=\frac{e_{\gamma_{i}}e_{\gamma_{j}}}{\beta
\sqrt{m_{\gamma_i}m_{\gamma_j}}c^{2}}\int\!\! \frac{\d\mathbf{k} }{(2\pi)^{3}}\,
\mathrm{e}^{i\mathbf{k}\cdot(\r_i-\r_j)}\!\!\int_{0}^{1}\!\!\d\xi_{i}^\mu(s_1)\,
\mathrm{e}^{i\mathbf{k}\cdot\lambda_{\gamma_{i}}\b\xi_{i}(s_1)}\!\! \int_{0}^{1}\!\!
\d\xi_{j}^\nu(s_2)\, \mathrm{e}^{-i\mathbf{k}\cdot\lambda_{\gamma_{j}}\b\xi_{j}(s_2)}
G^{\mu\nu}(\mathbf{k}). \nonumber
\end{align}
As a consequence of Gaussian integration, one recovers pairwise
interactions (\ref{B.14}) between loops. The product in (\ref{B.13})
contains the magnetic self-energies of the loops.
It is pleasing and convenient that after averaging over the field modes,
the energy of the system of loops becomes an exact and explicit sum of pair
potentials (and self-energies)\footnote{We omit again in (\ref{loop-Gibbs})
the non-electromagnetic terms (\ref{non-em-terms}).}~:
\begin{align}
\left\langle \mathrm{e}^{-\beta U(\mathcal{F}_1,...,\mathcal{F}_n,\mathbf{A})} \right\rangle_\text{rad} =
\Big[\prod_{i=1}^{n}\mathrm{e}^{-\frac{\beta
e_{\gamma_{i}}^{2}}{2}{W_\text{m}}(i,i)}\Big]\ \mathrm{e}^{-\beta \sum_{i<j}
e_{\gamma_i}e_{\gamma_j}\big( V_\text{c}(i,j) + {W_\text{m}}(i,j)
\big)}. \label{loop-Gibbs}
\end{align}
It is interesting to ask for the status of the partial density matrix
(\ref{B.4}) compared to that generated by the Darwin Hamiltonian
$\rho_\text{Darwin} \propto \mathrm{e}^{-\beta H_\text{Darwin}}$ or, more
generally, if $\rho_{L,R}$ can be cast in the form $\rho_{L,R}\propto
\mathrm{e}^{-\beta H_\text{eff}}$ for some tractable Hamiltonian
$H_\text{eff}(\{\mathbf{p}_{i},\r_{i}\})$ depending on the canonical variables of
the particles. The answer to this last question is very presumably
negative. Indeed the magnetic interaction (\ref{B.14}) is a two times
functional of the Brownian loops; namely, it lacks the equal-time constraint
occurring in the Coulomb potential (\ref{B.6}) (see the discussion before
(\ref{B.17}) below) necessary to come back to a simple operator form by
using the Feynman-Kac-It\^o formula backwards. This is a well-known common
feature of interactions resulting from integrating out external degrees of
freedom \cite{Feynman-Hibbs}.
The long-distance asymptotics of ${W_\text{m}}(i,j)$ as $|\r_{i}-\r_{j}|\to\infty$ is
determined by the small $\mathbf{k}$ behaviour in the integrand of (\ref{B.14}).
Noting that $\int_{0}^{1}\d\b\xi(s)=0$ for a closed loop (It\^o's lemma), one
has
\begin{align}
\int_{0}^{1}\!\!\d\xi_{i}^\mu(s)\,
\mathrm{e}^{i\mathbf{k}\cdot\lambda_{\gamma_{i}}\b\xi_{i}(s)} \sim i
\lambda_{\gamma_i}
\int_{0}^{1}\!\!\d\xi_{i}^\mu(s)\ \mathbf{k}\cdot\b\xi_{i}(s),\quad \mathbf{k}\to 0,
\end{align}
and thus
\begin{align}
&{W_\text{m}}(i,j)\sim
\label{B.16}\\
&\sim\frac{\lambda_{\gamma_{i}} \lambda_{\gamma_{j}}}{\beta
\sqrt{m_{\gamma_{i}} m_{\gamma_{j}}}c^{2}} \!\int\!\!\! \frac{\d\mathbf{k}
}{(2\pi)^{3}}\mathrm{e}^{i\mathbf{k}\cdot(\r_{i}-\r_{j})} \!\!\int_{0}^{1}\!\!\! \d\xi^\mu_{i}(s_{1})
(\mathbf{k}\cdot\b\xi_{i}(s_{1})) \!\!\int_{0}^{1}\!\!\! \d\xi^\nu_{j}(s_{2})
(\mathbf{k}\cdot\b\xi_{j}(s_{2}))G^{\mu\nu}(\mathbf{k})\nonumber\\
&=\frac{\lambda_{\gamma_{i}} \lambda_{\gamma_{j}}}{\beta
\sqrt{m_{\gamma_{i}} m_{\gamma_{j}}}c^{2}} \!\int_{0}^{1}\!\!\!
\d\xi_{i}^{\mu}(s_{1}) (\b\xi_{i}(s_{1})\cdot {\bf \nabla}_{\r_{i}})
\!\!\int_{0}^{1}\!\!\! \d\xi_{j}^{\nu}(s_{2}) (\b\xi_{j}(s_{2})\cdot{\bf
\nabla}_{\r_{j}})\,G^{\mu\nu}(\r_{i}-\r_{j}),\nonumber
\end{align}
as $ |\r_{i}-\r_{j}|\to \infty$. Upon using the asymptotic form
(\ref{B.12}) of $G^{\mu\nu}(\r_{i}-\r_{j})$, it is clear that for fixed
loop shapes $\b\xi_{i}$ and $\b\xi_{j}$ the decay of ${W_\text{m}}(i,j)$ is $\sim
|\r_{i}-\r_{j}|^{-3}$. It is of dipolar type modified by the constraint
imposed by the transversality.
The Coulombic part (\ref{B.6}) of the loop-loop interaction still decays as
$r^{-1}$ and deserves the following remark. From the Feynman-Kac formula
the potential (\ref{B.6}) inherits the quantum-mechanical equal-time
constraint; {\it i.e.}, every element of charge
$e_{\gamma_{i}}\lambda_{\gamma_{i}}\d\b\xi_i(s_{1})$ of the first loop does
not interact with every other element
$e_{\gamma_{j}}\lambda_{\gamma_{j}}\d\b\xi_j(s_{2})$ as would be the case
in classical physics, but the interaction takes place only if $s_1
=s_2$. It is therefore of interest to split
\begin{align}
V_\text{c}(i,j)={V_\text{elec}}(i,j)+{W_\text{c}}(i,j), \label{B.17}
\end{align}
where
\begin{align}
{V_\text{elec}}(i,j)= \int_{0}^{1}\!\!\! \d s_{1} \!\int_{0}^{1}\!\!\! \d
s_{2}\ \frac{1} {|\r_{i}+\lambda_{\gamma_{i}}\b\xi
_{i}(s_{1})-\r_{j}-\lambda_{\gamma_{j}}\b\xi _{j}(s_{2})|} \label{B.18}
\end{align}
is a genuine classical electrostatic potential between two charged loops
and
\begin{align}
{W_\text{c}}(i,j)=\int_{0}^{1}\!\!\! \d s_{1} \!\int_{0}^{1}\!\!\! \d
s_{2}\,(\delta(s_{1}\!-\!s_{2})\!-\!1)\frac{1}
{|\r_{i}+\lambda_{\gamma_{i}}\b\xi
_{i}(s_{1})-\r_{j}-\lambda_{\gamma_{j}}\b\xi _{j}(s_{2})|} \label{B.19}
\end{align}
is the part of $V_\text{c}(i,j)$ due to intrinsic quantum fluctuations (${W_\text{c}}(i,j)$
vanishes if $\hbar$ is set equal to zero). Because of the identities
\begin{align}
\int_{0}^{1}\!\!\!\d s_{1}\,(\delta(s_{1}-s_{2})-1) = \int_{0}^{1}\!\!\! \d
s_{2}\,(\delta(s_{1}-s_{2})-1)=0, \label{B.20}
\end{align}
the large-distance behaviour of ${W_\text{c}}$ originates again from the term
bilinear in $\b\xi_{i}$ and $\b\xi_{j}$ in the multipolar expansion of the
Coulomb potential in (\ref{B.19})
\begin{align}
&{W_\text{c}}(i,j)\sim \label{B.21}
\int_{0}^{1}\!\!\! \d s_{1}\!\! \int_{0}^{1}\!\!\! \d s_{2}\,
(\delta(s_{1}\!-\!s_{2})\!-\!1)\(\lambda_{\gamma_{i}}\b\xi_{i}(s_{1})\cdot
\nabla_{\r_{i}} \)\(
\lambda_{\gamma_{j}}\b\xi_{j}(s_{2})\cdot\nabla_{\r_{j}}\)
\frac{1}{|\r_{i}-\r_{j}|}.
\end{align}
It is dipolar and formally similar to that of two electrical dipoles of
sizes $e_{\gamma_{i}}\lambda_{\gamma_{i}}\b\xi_{i}$ and
$e_{\gamma_{j}}\lambda_{\gamma_{j}}\b\xi_{j}$.
\section{Two quantum charges in a classical plasma}
In order to exhibit the effect of the magnetic potential on the particle
correlations, we consider the simple model of two quantum charges $e_a$ and
$e_b$ with corresponding loops $\mathcal{F}_a=(\r_{a},\b\xi_{a})$ and
$\mathcal{F}_b=(\r_{b},\b\xi_{b})$ immersed in a configuration $\omega$ of classical
charges, following section VII of \cite{Alastuey-Martin} or section IV.C of
\cite{Brydges-Martin}. According to (\ref{B.17}) one can decompose the
total energy as $\mathcal{U}(\mathcal{F}_{a},\mathcal{F}_{b},\omega)=e_a e_b
W(\mathcal{F}_{a},\mathcal{F}_{b})+U_\text{cl}(\mathcal{F}_{a},\mathcal{F}_{b},\omega) $ where
$W(\mathcal{F}_a,\mathcal{F}_b)={W_\text{c}}(\mathcal{F}_a,\mathcal{F}_b)+{W_\text{m}}(\mathcal{F}_a,\mathcal{F}_b)$ is the sum of the electric and
magnetic quantum dipolar interactions and $U_\text{cl}(\mathcal{F}_a,\mathcal{F}_b,\omega)$
is the purely classical Coulomb energy (\ref{B.18}) of the two loops $\mathcal{F}_a$
and $\mathcal{F}_b$ together with that of the particles in the configuration
$\omega$. The correlation $\rho(\mathcal{F}_a,\mathcal{F}_b)$ between the loops is obtained
by integrating out the coordinates $\omega$ of the classical charges~:
\begin{align}
\rho(\mathcal{F}_{a},\mathcal{F}_{b})=\frac{1}{\Xi_\text{cl}}\int_{\Lambda}\!\!\!
\d\omega\
\mathrm{e}^{-\beta\mathcal{U}(\mathcal{F}_{a},\mathcal{F}_{b},\omega)}=\mathrm{e}^{-\beta
e_a e_b
W(\mathcal{F}_{a},\mathcal{F}_{b})}\rho_\text{cl}(\mathcal{F}_{a},\mathcal{F}_{b}),
\label{B.22}
\end{align}
where $\Xi_\text{cl}$ is the partition function of the classical plasma and
$\rho_\text{cl}(\mathcal{F}_{a},\mathcal{F}_{b})$ is the correlation of the
two loops embedded in the plasma interacting with genuine classical Coulomb
forces. In the latter quantity, the classical theory of screening applies
so that effective interaction between the loops decay exponentially fast
\footnote{The usual Debye theory of screening has been rigorously shown to be
valid at least at sufficiently high temperature
\cite{Brydges-Federbush}.}. Thus one can approximate
$\rho_\text{cl}(\mathcal{F}_{a},\mathcal{F}_{b})$ in (\ref{B.22}) by
$\rho(\mathcal{F}_{a})\rho(\mathcal{F}_{b})$ up to a term exponentially decaying
as $|\r_{a}-\r_{b}|\to\infty$. Furthermore, integrating
$\rho(\mathcal{F}_{a},\mathcal{F}_{b})$ on the loop shapes leads to the
following expression for the positional correlation of the quantum charges
\begin{align}
& \rho(\r_{a},\r_{b})=
\int\!\! \mathrm{D}(\b\xi_{a}) \!\!\int\!\! \mathrm{D}(\b\xi_{b})\ \mathrm{e}^{-\beta e_a e_b
W(\mathcal{F}_{a},\mathcal{F}_{b})}\rho(\mathcal{F}_{a})
\rho(\mathcal{F}_{b}) + \mathcal{O}(\mathrm{e}^{-C|\r_{a}-\r_{b}|})= \nonumber \\
& = \rho_{a}\rho_{b}-\beta e_a e_b
\int\!\! \mathrm{D}(\b\xi_{a}) \!\!\int\!\! \mathrm{D}(\b\xi_{b})\
W(\mathcal{F}_{a},\mathcal{F}_{b}) \rho(\b\xi_{a})\rho(\b\xi_{b}) +
\nonumber \\
& \phantom{=} + \frac{1}{2}\beta^{2}e_a^2 e_b^2 \int\!\! \mathrm{D}(\b\xi_{a})
\!\!\int\!\! \mathrm{D}(\b\xi_{b})\ W^{2}(\mathcal{F}_{a},\mathcal{F}_{b})
\rho(\b\xi_{a})\rho(\b\xi_{b}) +...+
\mathcal{O}(\mathrm{e}^{-C|\r_{a}-\r_{b}|})
\label{B.23}
\end{align}
Since $W(\mathcal{F}_{a},\mathcal{F}_{b})\sim |\r_{a}-\r_{b}|^{-3}$ (see
(\ref{B.16}), (\ref{B.21})), the above expansion in powers of $W$ generates
algebraically decaying terms at large separation. It is known that in a
homogeneous and isotropic phase, the electric dipole part ${W_\text{c}}$ does not
contribute at linear order \cite{Alastuey-Martin},
\cite{Brydges-Martin}. The same is true for the magnetic part. To see
this, it is convenient to write the linear ${W_\text{m}}$ term of (\ref{B.23}) as
\begin{align}
&-\beta e_a e_b \int\!\! \mathrm{D}(\b\xi_{a}) \!\!\int\!\! \mathrm{D}(\b\xi_{b})\
{W_\text{m}}(\mathcal{F}_{a},\mathcal{F}_{b}) \rho(\b\xi_{a}) \rho(\b\xi_{b})
\nonumber\\
&=-\frac{\beta e_a e_b}{\sqrt{\beta m_{a}c^{2}} \sqrt{\beta m_{b}c^{2}}}
\int\!\!\! \frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\,
\mathrm{e}^{i\mathbf{k}\cdot(\r_{a}-\r_{b})}\ {t_{a}^{\mu}}^\ast(\mathbf{k}) t_{b}^{\nu}(\mathbf{k})
G^{\mu\nu}(\mathbf{k}) \label{B.24}.
\end{align}
The stochastic $\b\xi_{a}$-line-integral is now included in the definition
of the tensor
\begin{align}
t_{a}^{\mu}(\mathbf{k})= \int\!\! \mathrm{D}(\b\xi_{a})\rho(\b\xi_{a})
\!\int_{0}^{1}\!\!\!\d\xi_{a}^{\mu}(s)\ \mathrm{e}^{-i\lambda_{a}\mathbf{k}\cdot \b\xi_{a}(s)}
\label{B.25}
\end{align}
and likewise for $t_{b}^{\nu}(\mathbf{k})$. Since both the measure $\mathrm{D}(\b\xi_{a})$
and $\rho(\b\xi_{a})$ are invariant under a rotation of $\b\xi_{a}$ in an
isotropic system, $t_{a}^{\mu}(\mathbf{k})$ transforms in a covariant manner under
rotations of $\mathbf{k}$. Thus it is necessarily of the form
$t_{a}^{\mu}(\mathbf{k})=k^{\mu}f_{a}(|\mathbf{k}|)$, implying the vanishing of
(\ref{B.24}) because of the transversality of $G^{\mu\nu}(\mathbf{k})$. One
concludes that the slowest non-vanishing contribution comes from the
$W^{2}$ term in (\ref{B.23})
\begin{align}
\rho(\r_{a},\r_{b})- \rho_{a} \rho_{b}=
\frac{A(\beta)}{|\r_{a}-\r_{b}|^{6}}+\mathrm{O}\(\frac{1}{|\r_{a}-\r_{b}|^{8}}\) \label{B.26aaa}.
\end{align}
The temperature-dependent amplitude
$A(\beta)=A_\text{cc}(\beta)+A_\text{mm}(\beta)+A_\text{cm}(\beta)$
involves in principle electric and magnetic contributions from ${W_\text{c}}^{2}$
and ${W_\text{m}}^{2}$, as well as a cross contribution from $2{W_\text{c}}{W_\text{m}}$. These
contributions can be calculated explicitly at lowest order in $\hbar$ (or
equivalently in the high-temperature limit $\beta\to 0$). The electric
contribution in this limit is known to be \cite{Alastuey-Martin},
\cite{Brydges-Martin}
\begin{align}
A_\text{cc}(\beta)\sim
\hbar^{4}\frac{\beta^{4}}{240}\frac{e_{a}^{2}e_{b}^{2}}{m_{a}m_{b}}\rho_{a}\rho_{b}.
\label{B.27}
\end{align}
To compute the magnetic contribution in the same limit, we write
the quadratic term
\begin{align}
& \frac{\beta^{2} e_a^2 e_b^2}{2}\int\!\! \mathrm{D}(\b\xi_{a})\rho(\b\xi_{a})\!\!
\int\!\!\mathrm{D}(\b\xi_{b})
\rho(\b\xi_{b})\ {W_\text{m}}^{2}(\mathcal{F}_{a},\mathcal{F}_{b})
=\frac{e_{a}^{2}e_{b}^{2}}{2 m_{a}c^{2}m_{b}c^{2}}\nonumber\\
& \times \int\!\!\! \frac{d^{3}\mathbf{k}_{1}}{(2\pi)^{3}} \!\int\!\!\!
\frac{d^{3}\mathbf{k}_{2}}{(2\pi)^{3}}\, \mathrm{e}^{i(\mathbf{k}_{1}+\mathbf{k}_{2})\cdot(\r_{a}-\r_{b})}
\,\left(T_{a}^{\mu\nu}(\mathbf{k}_{1},\mathbf{k}_{2})\right)^\ast {T_{b}^{\sigma\tau}}(\mathbf{k}_{1},\mathbf{k}_{2})
G^{\mu\sigma}(\mathbf{k}_{1}) G^{\nu\tau}(\mathbf{k}_{2})
\label{B.28}
\end{align}
in terms of the tensors
\begin{align}
T_{a}^{\mu\nu}(\mathbf{k}_{1},\mathbf{k}_{2})=\int\!\!\mathrm{D}(\b\xi_{a}) \rho(\b\xi_{a})
\!\!\int_{0}^{1}\!\!\! \d\xi_{a}^{\mu}(s_{1}) \!\!\int_{0}^{1}\!\!\!
\d\xi_{a}^{\nu}(s_{2})\ \mathrm{e}^{-i\lambda_{a}\mathbf{k}_{1}\cdot\b\xi_{a}(s_{1})}
\mathrm{e}^{-i\lambda_{a}\mathbf{k}_{2} \cdot \b\xi_{a}(s_{2})} \label{B.29}
\end{align}
and $T_{b}^{\sigma\tau}(\mathbf{k}_{1},\mathbf{k}_{2})$, defined likewise. As usual the
behaviour at large distances is controlled by that of the integrand of
(\ref{B.28}) at small wave numbers. Expanding (\ref{B.29}) at lowest order
in $\mathbf{k}_{1}$ and $\mathbf{k}_{2}$ gives
\begin{align}
T_{a}^{\mu\nu}(\mathbf{k}_{1},\mathbf{k}_{2}) & \sim \int\!\! \mathrm{D}(\b\xi_{a})
\rho(\b\xi_{a}) \!\!\int_{0}^{1}\!\!\! \d\xi_{a}^{\mu}(s_{1})
\!\!\int_{0}^{1}\!\!\! \d\xi_{a}^{\nu}(s_{2})
\left(-i\lambda_{a}\mathbf{k}_{1}\cdot\b\xi_{a}(s_{1})\right)
\left(-i\lambda_{a}\mathbf{k}_{2}\cdot\b\xi_{a}(s_{2})\right) \nonumber \\
& = -\lambda_{a}^{2}k_{1}^{\epsilon}k_{2}^{\eta}
\int\!\! \mathrm{D}(\b\xi_{a})\rho(\b\xi_{a}) \!\!\int_{0}^{1}\!\!\! \d\xi_{a}^{\mu}(s_{1})
\!\!\int_{0}^{1}\!\!\! \d\xi_{a}^{\nu}(s_{2})\ \xi_{a}^{\epsilon}(s_{1})
\xi_{a}^{\eta}(s_{2})
\label{B.30}
\end{align}
and likewise for $T_{b}^{\sigma\tau}(\mathbf{k}_{1},\mathbf{k}_{2})$. One sees that
because of the factor $\lambda_{a}^{2}\lambda_{b}^{2}$, the overall
contribution in (\ref{B.28}) will have a $\hbar^{4}$ factor so that at this
order we can neglect the quantum fluctuation in the density setting
$\rho(\b\xi_{a})\sim \rho_{a}$ independent of $\b\xi_{a}$. Thus the
stochastic integral to be calculated becomes (appendix A)
\begin{align}
\int\!\!\mathrm{D}(\b\xi) \!\!\int_{0}^{1}\!\!\! \d\xi^{\mu}(s)
\!\!\int_{0}^{1}\!\!\! \d\xi^{\nu}(t)\ \xi^{\epsilon}(s) \xi^{\eta}(t) =
\frac{1}{12}(\delta^{\mu\nu}\delta^{\eta\epsilon} -
\delta^{\mu\eta}\delta^{\nu\epsilon}),
\label{B.31}
\end{align}
leading to
\begin{align}
T_{a}^{\mu\nu}(\mathbf{k}_{1},\mathbf{k}_{2}) &\sim -\frac{\lambda_{a}^{2}
\rho_{a}}{12}(\delta^{\mu\nu}\mathbf{k}_{1} \cdot \mathbf{k}_{2}-k_{2}^{\mu}
k_{1}^{\nu}),\nonumber\\
T_{b}^{\sigma\tau}(\mathbf{k}_{1},\mathbf{k}_{2}) &\sim
-\frac{\lambda_{b}^{2}\rho_{b}}{12}(\delta^{\sigma\tau}\mathbf{k}_{1} \cdot
\mathbf{k}_{2}-k_{2}^{\sigma} k_{1}^{\tau}).
\label{B.32}
\end{align}
When this is inserted into (\ref{B.28}) and summation on vectorial indices
are performed, one finds the expression
\begin{align}
A\int\!\!\! \frac{\d\mathbf{k}_{1}}{(2\pi)^{3}} \!\!\int\!\!\!
\frac{\d\mathbf{k}_{2}}{(2\pi)^{3}}\, \mathrm{e}^{i(\mathbf{k}_{1}+\mathbf{k}_{2})\cdot(\r_{a}-\r_{b})}
\,(4\pi)^{2} |g(\mathbf{k}_{1})|^{2} |g(\mathbf{k}_{2})|^{2}
\left[1+\frac{(\mathbf{k}_{1}\cdot\mathbf{k}_{2})^{2}}{k_{1}^{2}k_{2}^{2}} \right],
\label{B.33}
\end{align}
with $A=\frac{\lambda_{a}^{2} \lambda_{b}^{2} e_{a}^{2} e_{b}^{2} \rho_{a}
\rho_{b}}{288m_{a}m_{b}c^{4}}$. The first term in the large brackets
gives a rapidly decaying contribution since it involves the Fourier
transform of the form factor $g^{2}(\mathbf{k})$. The algebraic large-distance
contribution comes from the second term which reads, after Fourier
transformation (approximating $g(\mathbf{k})\sim 1$, $\mathbf{k}\to 0$),
\begin{align}
A\(\partial_{\mu}\partial_{\nu}\frac{1}{|\r_{a}-\r_{b}|}\)
\(\partial_{\mu}\partial_{\nu}\frac{1}{|\r_{a}-\r_{b}|}\)
=A\;\frac{6}{|\r_{a}-\r_{b}|^{6}}. \label{B.34}
\end{align}
Finally one checks that there is no cross Coulomb-magnetic contribution
$A_\text{cm}(\beta)$ at the dominant order $\r^{-6}$ as a consequence of
transversality (appendix B). So adding (\ref{B.27}) and (\ref{B.34}) gives
the final result
\begin{align}
\rho(\r_{a},\r_{b})- \rho_{a}\rho_{b} \sim \hbar^{4}\beta^{4}\frac{\rho_{a}
\rho_{b}e_{a}^{2}e_{b}^{2}}{240 \,m_{a}m_{b}}\left[ 1+\frac{5}{(\beta
m_{a}c^{2})(\beta m_{b}c^{2})}\right]\frac{1}{|\r_{a}-\r_{b}|^{6}}
\label{B.26}
\end{align}
as $|\r_{a}-\r_{b}|\to\infty$ and at lowest order in $\hbar$. One sees from
(\ref{B.6}) and (\ref{B.14}) that the order of magnitude of the ratio ${W_\text{m}}/V_\text{c}$ is
$(\beta m c^2)^{-1}$. In an electrolyte at room temperature $T=300 K$, this
ratio is found to be $\approx 10^{-11}$. The magnetic correction to the correlation
decay (\ref{B.26}) is negligible in this case.
\section{Particle correlations in the many-body system}
We apply the formalism developed in section 3 to the determination of the
large-distance decay of the particle density correlations in the more
general case where all particles are quantum-mechanical, but still obeying
Maxwell-Boltzmann statistics.
We show hereafter that the algebraic $r^{-6}$ decay of the (truncated) particle
density correlations
\begin{align}
\rho_\text{T}(\gamma_a,\r_a,\gamma_b,\r_b) \sim
\frac{A_{ab}(\beta,\{\rho_\gamma\})}{|\r_a-\r_b|^{6}}, \qquad |\r_a-\r_b|\to\infty
\label{particle-correl-decay}
\end{align}
found in the absence of the radiation field \cite{Cornu3},
\cite{Brydges-Martin} is not altered, but that the coefficient
$A_{ab}(\beta,\{\rho_\gamma\})$ contains in addition small magnetic terms
of the order $(\beta m c^2)^{-2}$, as in (\ref{B.26}). As an illustration,
we give the lowest order contribution of this coefficient with respect to
Planck's constant $\hbar$.
By the Feynman-Kac-It\^o representation, the full system composed of
quantum point charges coupled to the radiation field has reduced to a
classical-like system of extended charged loops $\mathcal{F}=(\r,\gamma,\pmb{\xi})$ for
which all the methods of classical statistical mechanics apply. The only
novelty comes from the additional magnetic potential ${W_\text{m}}$. In the
following, we merely summarize the arguments since they are essentially the
same as those found in \cite{Cornu3}, \cite{Brydges-Martin} when no
radiation field is present.
As usual, we express the truncated two-loop correlation
$\rho_\text{T}(\mathcal{F}_a,\mathcal{F}_b)$ $=$ $\rho(\mathcal{F}_a)\rho(\mathcal{F}_b)h(\mathcal{F}_a,\mathcal{F}_b)$ in terms
of the loop Ursell function $h(\mathcal{F}_a,\mathcal{F}_b)$. The latter function can be
expanded in a formal diagrammatic Mayer series of powers of the loop
densities $\rho(\mathcal{F})$. One needs to resum the long-range part of the Coulomb
potential $V_\text{c}$, which is responsible for the non-integrability of the
Mayer bonds \\$f(\mathcal{F}_i,\mathcal{F}_j)$ $=$ \mbox{$\exp(-\beta
e_{\gamma_i}e_{\gamma_j} [V_\text{c}(\mathcal{F}_i,\mathcal{F}_j)+{W_\text{m}}(\mathcal{F}_i,\mathcal{F}_j)])-1$} at
infinity. Using the decomposition (\ref{B.17}) we resum the convolution
chains built with the purely electrostatic long-range part ${V_\text{elec}}(\mathcal{F},\mathcal{F}')$
into a Debye-H\"uckel-type screened potential ${\Phi_\text{elec}}(\mathcal{F},\mathcal{F}')$. Then
reorganizing the diagrams leads to a representation of the loop Ursell
function by terms of so-called prototype diagrams, built with the two kinds
of bonds
\begin{align}
&F(\mathcal{F},\mathcal{F}') = -\beta e_\gamma e_{\gamma'} {\Phi_\text{elec}}(\mathcal{F},\mathcal{F}'),
\\&{F^\text{R}}(\mathcal{F},\mathcal{F}') = \mathrm{e}^{-\beta e_\gamma e_{\gamma'} [{\Phi_\text{elec}}(\mathcal{F},\mathcal{F}') +
W(\mathcal{F},\mathcal{F}')]} - 1 + \beta e_\gamma e_{\gamma'} {\Phi_\text{elec}}(\mathcal{F},\mathcal{F}'),
\label{FR}
\end{align}
where we have defined $W = {W_\text{c}} + {W_\text{m}}$ as in section 4. \footnote{Strictly
speaking, the short-range repulsive potential needed in the framework
of Maxwell-Boltzmann statistics would arise here in the exponent of
(\ref{FR}). It has no implication in this discussion about long-range
behaviours, and we simply omit it.}
The potential ${\Phi_\text{elec}}(\mathcal{F},\mathcal{F}')$ has been studied in
\cite{Ballenegger-etal}. It corresponds to the term $n=0$ of the full
quantum analog of the Debye-H\"uckel potential given by formula (89) of
\cite{Ballenegger-etal}. This contribution $n=0$ is shown to be decaying at
infinity faster than any inverse power of $|\r-\r'|$ (see formula (58) of
\cite{Ballenegger-etal}, and the comment following it).
The asymptotic decay of the two-particle correlation
$\rho_\text{T}(\gamma_a,\r_a,\gamma_b,\r_b)$ is inferred from that of the
loop correlation $\rho_\text{T}(\mathcal{F}_a, \mathcal{F}_b)$ by integrating it over the
Brownian shapes $\pmb{\xi}_a$ and $\pmb{\xi}_b$. The bond $F$ is rapidly decreasing,
and the asymptotic decay of ${F^\text{R}}$ is dominated by the dipolar decays of
${W_\text{c}}$ and ${W_\text{m}}$~: ${F^\text{R}}(\mathcal{F},\mathcal{F}') \sim -\beta e_\gamma e_{\gamma'} W(\mathcal{F},\mathcal{F}')$
as $|\r-\r'|\to\infty$. We further extract this dipolar part from ${F^\text{R}}$ and
define the bond
\begin{align}
\widetilde{{F^\text{R}}}(\mathcal{F},\mathcal{F}') &= {F^\text{R}}(\mathcal{F},\mathcal{F}') + \beta e_\gamma e_{\gamma'}
W(\mathcal{F},\mathcal{F}') \nonumber
\\&\sim \tfrac{1}{2}[\beta e_\gamma e_{\gamma'} W(\mathcal{F},\mathcal{F}')]^2
=\mathrm{O}({|\r-\r'|^{-6}})
\label{FRtilde}
\end{align}
and work now with the three bonds $F$, $\widetilde{{F^\text{R}}}$, and
$W$. \footnote{In \cite{Brydges-Martin}, the bond $F$ is further decomposed
into a multipole expansion. Our bonds ${F^\text{R}}$ and $\widetilde{{F^\text{R}}}$
differ formally from their bonds $F_\text{l}$ and $\tilde{F}_\text{l}$
only by the inclusion of the magnetic contribution ${W_\text{m}}$ into $W$.}
To find out the slowest-decaying diagrams, we write the truncated two-loop
correlation $\rho_\text{T}(\mathcal{F}_a,\mathcal{F}_b)$ in an exact Dyson series of
convolution chains involving $W$ and $H$~:
\begin{align}
\rho_\text{T}(\mathcal{F}_a,\mathcal{F}_b) = &\rho(\mathcal{F}_a)\rho(\mathcal{F}_b)H(\mathcal{F}_a,\mathcal{F}_b) -\beta (K \star W
\star K)(\mathcal{F}_a,\mathcal{F}_b) \nonumber
\\&+ \beta^2 (K \star W \star K \star W \star K)(\mathcal{F}_a,\mathcal{F}_b)+...
\label{KWseries}
\end{align}
where $H$ denotes the sum of the diagrams that remain connected under
removal of one $W$-bond and
$K(\mathcal{F}_1,\mathcal{F}_2)=\rho(\mathcal{F}_1)\rho(\mathcal{F}_2)H(\mathcal{F}_1,\mathcal{F}_2)+\delta(\mathcal{F}_1,\mathcal{F}_2)\rho(\mathcal{F}_1)$. This
topological constraint ensures that $H$ decays at least as $r^{-6}$. The
series (\ref{KWseries}) is conveniently analysed in Fourier representation
with respect to $\r_a-\r_b$. After expanding $W$ into the sum ${W_\text{c}}+{W_\text{m}}$, we
have three types of chains~: pure ${W_\text{c}}$ or ${W_\text{m}}$ chains and mixed ${W_\text{c}},
{W_\text{m}}$ chains. It is shown in \cite{Cornu3}, \cite{Brydges-Martin} that the
contribution of pure ${W_\text{c}}$ chains to the particle correlation
$\rho_\text{T}(\gamma_a,\r_a,\gamma_b,\r_b)=\int \!\mathrm{D}(\pmb{\xi}_a)
\int\!\mathrm{D}(\pmb{\xi}_b)\, \rho_\text{T}(\mathcal{F}_a,\mathcal{F}_b)$ decays strictly faster than
$\mathrm{o}(|\r_a-\r_b|^{-6})$. \footnote{In this proof, only the invariance of
$H$ under rotations is used, which also holds when the magnetic
potential is included.} We show below that all other chains containing
${W_\text{m}}$ bonds vanish identically as the consequence of transversality. This
implies that the longest-range part of the correlations originates from the
function $H$ in the first term of the right-hand side of (\ref{KWseries}),
hence the result (\ref{particle-correl-decay}).
A chain mixing ${W_\text{c}}$ and ${W_\text{m}}$ bonds must have at least one element ${W_\text{c}}
\star K \star {W_\text{m}}$ or ${W_\text{m}} \star K \star {W_\text{c}}$. In Fourier space, one can
write, from (\ref{B.19}) and (\ref{B.14}),
\begin{align}
({W_\text{c}} \star K \star {W_\text{m}})&(\gamma_a, \pmb{\xi}_a, \gamma_b,\pmb{\xi}_b,\k) =
\int_0^1\!\!\!\!\d s_a \!\! \int_0^1\!\!\!\!\d s_1\, (\delta(s_a-s_1)-1)
\frac{4\pi}{k^2} \mathrm{e}^{i \k\cdot\lambda_{\gamma_a} \pmb{\xi}_a(s_a)}\notag
\\&\times \big[T^{\nu_2} (\k,s_1) G^{\nu_2,\nu_b}(\k)\big] \int_0^1\!\!\!
d\xi_b^{\nu_b}(s_b)
\mathrm{e}^{-i\k\cdot\lambda_{\gamma_b}\pmb{\xi}_b(s_b)}, \label{WcKWm}
\end{align}
where
\begin{align}
T^{\nu_2}(\k,s_1) =
&\sum_{\gamma_1}\!\!\int\!\!\mathrm{D}(\pmb{\xi}_1)\!\sum_{\gamma_2}\!\!\int\!\!\mathrm{D}(\pmb{\xi}_2)
\,\mathrm{e}^{-i\k\cdot\lambda_{\gamma_1}\pmb{\xi}_1(s_1)} K(\gamma_1, \pmb{\xi}_1,
\gamma_2,\pmb{\xi}_2,\k) \nonumber
\\& \times\int_0^1
\!\!\!\d\xi_2^{\nu_2}(s_2) \mathrm{e}^{i\k\cdot\lambda_{\gamma_2}\pmb{\xi}_2(s_2)}
\end{align}
and $K(\gamma_1, \pmb{\xi}_1, \gamma_2,\pmb{\xi}_2,\k)$ is the Fourier transform of
$K(\mathcal{F}_1,\mathcal{F}_2)$ with respect to $\r_1-\r_2$. As the measures $\mathrm{D}(\pmb{\xi}_1)$
and $\mathrm{D}(\pmb{\xi}_2)$ and the function $K(\gamma_1, \pmb{\xi}_1, \gamma_2,\pmb{\xi}_2,\k)$
are invariant under spatial rotations, $T^{\nu_2}(\k,s_1)$ transforms as a
tensor, implying that it is necessarily of the form $T^{\nu_2}(\k,s_1) =
k^{\nu_2}\, a(k,s_1)$ for some rotationally invariant function $a$ of
$\k$. Using $k^\mu G^{\mu\nu} (\k) \equiv 0$ one deduces immediately that
(\ref{WcKWm}) vanishes. The case of ${W_\text{m}}\star K\star {W_\text{c}}$ is similar. To
see that there are no chains containing only ${W_\text{m}}$ bonds in
$\rho_\text{T}(\gamma_a,\r_a,\gamma_b,\r_b)$, it is sufficient to notice
that the integrated root element $\int\! \mathrm{D}(\pmb{\xi}_a)\, K \star {W_\text{m}}$ also
involves a factor $[T^{\nu_2}(\k) G^{\nu_2,\nu_b}(\k)]$ (for another
function $T^{\nu_2}(\k)$ transforming in a covariant manner), and thereby
vanishes for the same reason.
The graphs that do contribute to the coefficient
$A_{ab}(\beta,\{\rho_\gamma\})$ of (\ref{particle-correl-decay}) are those
of $H$ that contain bonds with algebraic decay~: namely, $\widetilde{{F^\text{R}}}$
and $W$. To select the lowest contribution in $\hbar$, one notes first that
$W$ is at least of order $\hbar^2$, as seen in (\ref{B.16}), (\ref{B.21})
which correspond to the lowest-order terms in the multipolar expansions of
${W_\text{c}}$ and ${W_\text{m}}$. (Higher-order multipoles generate higher powers of the de
Broglie wavelengths.) Since ${\Phi_\text{elec}}$ is rapidly decreasing, the algebraic
part of $\widetilde{{F^\text{R}}}$ is of order $\hbar^4$ and is given by
$\tfrac{1}{2}[\beta e_\gamma e_{\gamma'} W(\mathcal{F},\mathcal{F}')]^2$, as in
(\ref{FRtilde}). Thus, up to order $\hbar^4$, graphs with an algebraic
decay can contain only one bond $W$, two bonds $W$, or one bond
$\widetilde{{F^\text{R}}}$ belonging to paths connecting the two root points. If
there is a single such link $W$, by the topological structure of $H$ there
exists another path connecting the root points made of the more rapidly
decreasing bonds $F$ and $\widetilde{{F^\text{R}}}$. Hence the whole graph has a
decay faster than $r^{-6}$. If there are two $W$ bonds in between the root
points, as each of them is of order $\hbar^2$ all the other bonds and
vertices can be evaluated in the classical limit $\hbar\to
0$. Consequently, at least one of the extremities of either bond $W$ is
attached to a purely classical part of the graph, which is independent of
the Brownian shapes. We call such a point a classical end of $W$. At such
points, integration over the Brownian shape of the loop ``kills'' the
$r^{-3}$ decay of $W$ (see Appendix C), leading to an overall decay faster
than $r^{-6}$. Finally, at order $\hbar^4$, the only graphs that contribute
to (\ref{particle-correl-decay}) are constituted by a single
$\widetilde{{F^\text{R}}}$ bond linked to the root points by purely classical
subgraphs. The sum of such graphs contributes to the particle correlation
in the large-distance limit as
\begin{align}
\rho_\text{T}(\gamma_a,\r_a,\gamma_b,\r_b) \sim &\sum_{\gamma_1,\gamma_2} \left[
\int\!\! \d \r\ n_\text{T}^\text{cl}(\gamma_a,\gamma_1,\r)\right] \left[ \int\!\! \d \r
\ n_\text{T}^\text{cl}(\gamma_2,\gamma_b,\r)\right] \label{particle-correl-asympt-graph}
\\&\times \int\!\! \mathrm{D}(\pmb{\xi}_1) \!\!\int\!\! \mathrm{D}(\pmb{\xi}_2)\ \tfrac{1}{2} \left[
\beta e_{\gamma_1} e_{\gamma_2} W^\text{dip}(\gamma_1,\pmb{\xi}_1,\gamma_2,
\pmb{\xi}_2, \r_a-\r_b)\right]^2, \nonumber
\end{align}
where $W^\text{dip}={W_\text{c}}^\text{dip} + {W_\text{m}}^\text{dip}$ is the sum of the
dipolar parts (\ref{B.21}) and (\ref{B.16}) of ${W_\text{c}}$ and ${W_\text{m}}$, and
$n_\text{T}^\text{cl}(\gamma_a,\gamma_1,\r)$ is the classical truncated
two-point density correlation (including coincident points). The functional
integrals in (\ref{particle-correl-asympt-graph}) have been calculated in
section 4, see (\ref{B.28})-(\ref{B.26}), yielding the final result
\begin{align}
\rho_\text{T}(\gamma_a,\r_a,\gamma_b,\r_b) \sim &\frac{\hbar^4
\beta^4}{240}\sum_{\gamma_1,\gamma_2} \left[ \int\!\! \d
\r\ n_\text{T}^\text{cl}(\gamma_a,\gamma_1,\r)\right] \left[ \int\!\! \d \r
\ n_\text{T}^\text{cl}(\gamma_2,\gamma_b,\r)\right]\nonumber\\
&\times \frac{e_{\gamma_1}^2 e_{\gamma_2}^2}{m_{\gamma_1} m_{\gamma_2}}
\left[ 1 + \frac{5}{\beta m_{\gamma_1} c^2 \beta m_{\gamma_2} c^2}\right]
\frac{1}{|\r_a-\r_b|^{6}}
\end{align}
as $|\r_a-\r_b|\to\infty$ and at lowest order in $\hbar$. To this order, the
only difference with (\ref{B.26}) is the occurrence of the classical correlation
functions $n_\text{T}^\text{cl}$, a manifestation of the fact that in the quantum many-body
problem, every pair of particles contribute to the tail of the
correlation function. This generalizes the result of
\cite{Alastuey-Martin}, formula (5.12), to the inclusion of the magnetic
interactions.
As a final comment, we observe that the inclusion of the transverse degrees of
freedom of the field does not modify the charge sum rule in the system of loops
and hence it also holds for the charge correlations in the particle system. This
sum rule reads
\begin{align}
\int\!\!\d\r \!\int\!\! \mathrm{D}(\pmb{\xi}) \sum_{\gamma} \frac{e_{\gamma} \rho_\text{T}(\mathcal{F},
\mathcal{F}_1)}{\rho(\mathcal{F}_1)} = -e_{\gamma_1}.
\end{align}
It states that the charge of the cloud of loops induced around a fixed loop
$\mathcal{F}_1$ exactly compensates that of $\mathcal{F}_1$. The proof can be carried out word by
word as in \cite{Ballenegger-etal}, section 6.1.2. It relies exclusively on the
long-range part $r^{-1}$ of the Coulomb potential $V_\text{c}$ and is not altered by
the presence of the magnetic potential ${W_\text{m}}$.
\section{Transverse field correlations}
A characteristic feature of charged systems is that longitudinal field
correlations always remain long ranged in spite of the screening mechanisms
that reduce the range of the particle correlations. It has been established
on a microscopic basis that the correlations of the longitudinal electric
field $\mathbf{E}_\text{l}$ behave as \cite{Lebowitz-Martin},
\cite{martin-sumrules}
\begin{align}
\langle E_\text{l}^\mu(\mathbf{x})E_\text{l}^\nu(\mathbf{y})\rangle_\text{T} \sim -\partial_\mu
\partial_\nu \frac{1}{|\mathbf{x}-\mathbf{y}|} \left[-\tfrac{2\pi}{3}\int\!
\d\r\ |\r|^{2} S(\r)\right] , \quad |\mathbf{x}-\mathbf{y}|\to \infty, \label{F.1}
\end{align}
where $S(\r)$ is the (classical or quantum-mechanical) charge-charge
correlation function.
In order to obtain the correlations of the transverse fields we first
consider correlations $\langle
A^{\mu}(\mathbf{x})A^{\nu}(\mathbf{y})\rangle_\text{T}$ of the vector
potential at free points $\mathbf{x}$ and $\mathbf{y}$ in space. These correlations are
easily obtained by functional differentiation, adding to the original
Hamiltonian (\ref{B.1}) a coupling to an external current
$\pmb{\mathcal{J}}_\text{ext}(\mathbf{x})$
\begin{align}
H_{L,R}(\pmb{\mathcal{J}}_\text{ext})=H_{L,R}-i\! \int\!\!
\d\mathbf{x}\ \pmb{\mathcal{J}}_\text{ext}(\mathbf{x})\cdot\mathbf{A}(\mathbf{x}), \label{F.2}
\end{align}
so that
\begin{align}
\langle A^{\mu}(\mathbf{x})A^{\nu}(\mathbf{y})\rangle_\text{T}=\left.
-\frac{1}{\beta^{2}} \frac {\delta^{2}}
{\delta\mathcal{J}_\text{ext}^{\mu}(\mathbf{x})
\delta\mathcal{J}_\text{ext}^{\nu}(\mathbf{y})}\ln \mathrm{Tr}\ \mathrm{e}^{-\beta
H_{L,R}(\pmb{\mathcal{J}}_\text{ext})} \right|_{\pmb{\mathcal{J}}_\text{ext}=0}. \label{F.3}
\end{align}
Decomposing $H_{L,R}$ as in (\ref{B.4b}) one can write
\begin{align}
&\langle A^{\mu}(\mathbf{x})A^{\nu}(\mathbf{y})\rangle_\text{T} = \nonumber\\
&=\left. -\frac{1}{\beta^{2}} \frac {\delta^{2}}
{\delta\mathcal{J}_\text{ext}^{\mu}(\mathbf{x})\delta
\mathcal{J}_\text{ext}^{\nu}(\mathbf{y})}\ln \mathrm{Tr}_\text{mat} \left\langle
\mathrm{e}^{-\beta H_\mathbf{A}}\mathrm{e}^{i\beta \int\! \d\mathbf{x}\ \pmb{\mathcal{J}}_\text{ext}(\mathbf{x})\cdot
\mathbf{A}(\mathbf{x})}
\right\rangle_\text{rad}\right|_{\pmb{\mathcal{J}}_\text{ext}=0}.
\label{F.4}
\end{align}
Using the Feynman-Kac formula as in section 3 one sees that the only
modification in (\ref{B.9}) is the replacement of the loop current
$\pmb{\mathcal{J}}(\mathbf{x})$ by the total current\footnote{As a consequence of the imaginary
coupling constant in the Hamiltonian (\ref{F.2}), the total current is
real, so that we can still apply the Gaussian integration formula used
in (\ref{B.9}).}
\begin{align}
\pmb{\mathcal{J}}_\text{tot}(\mathbf{x})=\pmb{\mathcal{J}}(\mathbf{x})+\pmb{\mathcal{J}}_\text{ext}(\mathbf{x}). \label{F.5}
\end{align}
The Gaussian integration on the field variables replaces (\ref{B.9})
by
\begin{align}
&\exp\Big\{-\frac{\beta}{2} \int\!\!\! \frac{\d\mathbf{k}}{(2\pi)^{3}}
\left(\mathcal{J}_\text{tot}^{\mu}(\mathbf{k})\right)^{*}
G^{\mu\nu}(\mathbf{k})\mathcal{J}_\text{tot}^{\nu}(\mathbf{k}) \Big\}=
\exp\Big\{-\frac{\beta}{2} \int\!\!\! \frac{\d\mathbf{k}}{(2\pi)^{3}}
G^{\mu\nu}(\mathbf{k})
\nonumber\\
&\times\left[\left(\mathcal{J}^{\mu}\right)^{*}
\mathcal{J}^{\nu} +
\left(\mathcal{J}_\text{ext}^{\mu}\right)^{*}
\mathcal{J}^{\nu} +
\left(\mathcal{J}^{\mu}\right)^{*}
\mathcal{J}_\text{ext}^{\nu} +
\left(\mathcal{J}_\text{ext}^{\mu}\right)^{*}
\mathcal{J}_\text{ext}^{\nu} \right](\k)\Big\}.
\label{F.6}
\end{align}
Therefore, from (\ref{F.6}), functional differentiation with
respect to $\pmb{\mathcal{J}}_\text{ext}$ according to (\ref{F.4}) produces two terms
\begin{align}
\langle A^{\mu}(\mathbf{x})A^{\nu}(\mathbf{y})\rangle_\text{T} =
\langle A^{\mu}(\mathbf{x})A^{\nu}(\mathbf{y})\rangle^{0}_\text{T} + \langle A^{\mu}(\mathbf{x})
A^{\nu}(\mathbf{y})\rangle^\text{mat}_\text{T}.
\label{F.7}
\end{align}
The first contribution (written in Fourier form)
\begin{align}
\langle A^{\mu}(\mathbf{x})A^{\nu}(\mathbf{y})\rangle^{0}_\text{T} &=
\frac{1}{\beta}\int\!\!\! \frac{\d\mathbf{k}}{(2\pi)^{3}}
\mathrm{e}^{i\mathbf{k}\cdot (\mathbf{x}-\mathbf{y})} G^{\mu\nu}(\mathbf{k})\nonumber\\
&\sim \frac{1}{2\beta r}\(\delta^{\mu\nu} + \frac{r^{\mu}r^{\nu}}{r^{2}}\),
\quad r\to\infty, \;\;\r=\mathbf{x}-\mathbf{y},
\label{F.8}
\end{align}
arises from the part quadratic in $\pmb{\mathcal{J}}_\text{ext}$ in (\ref{F.6}). It
describes the thermal fluctuations of the free field, and in view of
(\ref{B.12}), decays as $r^{-1}$. The second term, coming from the part
linear in $\pmb{\mathcal{J}}_\text{ext}$,
\begin{align}
&\langle A^{\mu}(\mathbf{x})A^{\nu}(\mathbf{y})\rangle^\text{mat}_\text{T}=\nonumber\\
&=-\int\!\!\! \frac{\d\mathbf{k}}{(2\pi)^{3}}\, \mathrm{e}^{i\mathbf{k\cdot x}}
\!\!\int\!\!\! \frac{\d\mathbf{k}'}{(2\pi)^{3}}\, \mathrm{e}^{i\mathbf{k}'\cdot\mathbf{y}}
G^{\mu\sigma}(\mathbf{k}) G^{\nu\tau}(\mathbf{k}')
\ \langle\mathcal{J}^{\sigma}(\mathbf{k})
\mathcal{J}^{\tau}(\mathbf{k}')\rangle_\text{T}, \label{F.9}
\end{align}
represents the modification to the free-field fluctuations caused by the
presence of matter. It involves the loop current correlation function
$\langle\mathcal{J}^{\sigma}(\mathbf{k}) \mathcal{J}^{\tau}(\mathbf{k}')
\rangle_\text{T}$ where the average is taken with respect to the thermal
weight (\ref{loop-Gibbs}) for the statistical-mechanical system of
loops. Expressing the currents $\pmb{\mathcal{J}}(\k) = \int \d
\mathcal{F}\ \mathbf{j}(\mathcal{F},\k) \hat\rho(\mathcal{F})$ in terms of the density of loops
$\hat\rho(\mathcal{F})=\sum_i \delta(\mathcal{F},\mathcal{F}_i)$ (see (\ref{B.8})), one can write this
current correlation in terms of the loop density correlation function
$n_\text{T}(\gamma_{1},\pmb{\xi}_{1},\gamma_{2},\pmb{\xi}_{2},\mathbf{k})$ (including
the contribution of coincident points)~:
\begin{align}
&\langle\mathcal{J}^{\sigma}(\mathbf{k})\mathcal{J}^{\tau}(\mathbf{k}')\rangle_\text{T} =
(2\pi)^{3}\delta(\mathbf{k} + \mathbf{k}')\nonumber\\
&\times\sum_{\gamma_{1}, \gamma_{2}} \!\int\!\! \mathrm{D}(\pmb{\xi}_{1})
\!\!\int\!\! \mathrm{D}(\pmb{\xi}_{2})\ \mathcal{T}^{\sigma}(\gamma_{1},
\pmb{\xi}_{1}, \mathbf{k}) \big(\mathcal{T}^{\tau}(\gamma_{2},
\pmb{\xi}_{2}, \mathbf{k})\big)^\ast n_\text{T}(\gamma_{1}, \pmb{\xi}_{1}, \gamma_{2},
\pmb{\xi}_{2}, \mathbf{k}).
\label{F.10}
\end{align}
The $\delta(\mathbf{k}+\mathbf{k}')$ is the manifestation of the translational
invariance of the state, and we have set
\begin{align}
\mathcal{T}^{\sigma}(\gamma_{i}, \pmb{\xi}_{i}, \mathbf{k}) =
\frac{e_{\gamma_{i}}}{\sqrt{\beta m_{\gamma_{i}}c^{2}}} \int_{0}^{1}\!\!\!
\d\xi_{i}^{\sigma}(s_{i})\ \mathrm{e}^{i\lambda_{\gamma_{i}} \mathbf{k} \cdot\pmb{\xi}_{i}(s_{i})}.
\label{F.11}
\end{align}
When (\ref{F.10}) is introduced into (\ref{F.9}), one obtains the final form
\begin{align}
\langle A^{\mu}(\mathbf{x})A^{\nu}(\mathbf{y})\rangle_\text{T}^\text{mat}=
-\int\!\!\!\frac{\d\mathbf{k}}{(2\pi)^{3}}\, \mathrm{e}^{i\mathbf{k}\cdot (\mathbf{x}-\mathbf{y})}
G^{\mu\sigma}(\mathbf{k})G^{\nu\tau}(\mathbf{k})\mathcal{Q}^{\sigma\tau}(\mathbf{k}),
\label{F.12}
\end{align}
where $\mathcal{Q}^{\sigma\tau}(\mathbf{k})$ is the tensor
\begin{align}
\mathcal{Q}^{\sigma\tau}(\mathbf{k}) = \sum_{\gamma_{1}, \gamma_{2}}
\!\int\!\! \mathrm{D}(\pmb{\xi}_{1}) \!\!\int\!\! \mathrm{D}(\pmb{\xi}_{2})\,
\mathcal{T}^{\sigma}(\gamma_{1}, \pmb{\xi}_{1}, \mathbf{k})
\big(\mathcal{T}^{\tau}(\gamma_{2}, \pmb{\xi}_{2}, \mathbf{k})\big)^\ast
n_\text{T}(\gamma_{1}, \pmb{\xi}_{1}, \gamma_{2}, \pmb{\xi}_{2}, \mathbf{k}).
\label{F.13}
\end{align}
To obtain the long-distance behaviour of this correlation we examine the
integrand in (\ref{F.13}) at small $\mathbf{k}$. Because of isotropy, the tensor
$\mathcal{Q}^{\sigma\tau}(\mathbf{k})$ transforms covariantly under the
rotations, and thus is of the form
\begin{align}
\mathcal{Q}^{\sigma\tau}(\mathbf{k})=a(k)\delta^{\sigma\tau}+b(k)
k^{\sigma}k^{\tau}, \quad k=|\mathbf{k}|
\label{F.14}
\end{align}
The term $k^{\sigma}k^{\tau}$ does not contribute to (\ref{F.13}) since
$G^{\mu\sigma}(\mathbf{k})$ is transversal. Because of It\^o's lemma,
$\mathcal{T}^{\sigma}(\gamma_{i},\pmb{\xi}_{i},\mathbf{k})$ is linear in $\mathbf{k}$
as $\mathbf{k}\to 0$, implying $a(k)=a\;k^{2}[1 + o(k)]$. Hence, using
$\delta^{\mu\sigma}_{tr}(\mathbf{k})\delta^{\nu\sigma}_{tr}(\mathbf{k})=\delta^{\mu\nu}_{tr}(\mathbf{k})$
one finds
\begin{align}
G^{\mu\sigma}(\mathbf{k})G^{\nu\tau}(\mathbf{k})\mathcal{Q}^{\sigma\tau}(\mathbf{k})=4\pi
\,a\, \frac{4\pi}{k^{2}}\,\delta^{\mu\nu}_{tr}(\mathbf{k}) [1+o(k)]=4\pi \,a\,
G^{\mu\nu}(\mathbf{k})[1+o(k)] \label{F.15}
\end{align}
as $\ k\to 0$. This shows that $\langle
A^{\mu}(\mathbf{x})A^{\nu}(\mathbf{y})\rangle_\text{T}^\text{mat}$ has the
same type of decay as the free field part (\ref{F.8}) with a modified
amplitude. Summing up the two contributions (\ref{F.7}) gives
\begin{align}
\langle A^{\mu}(\mathbf{x})A^{\nu}(\mathbf{y})\rangle_\text{T}\sim
\frac{1}{2r}\left(\delta^{\mu\nu}+\frac{r^{\mu}r^{\nu}}{r^{2}}\right)
\left(\frac{1}{\beta}-4\pi a\right), \quad r\to\infty. \label{F.16}
\end{align}
For $\mathbf{B}(\mathbf{x})=\nabla\times\mathbf{A}(\mathbf{x})$, one finds, from (\ref{F.16}),
\begin{align}
\langle B^{\mu}(\mathbf{x})B^{\nu}(\mathbf{y})\rangle_\text{T} \sim
\left(\partial_\mu \partial_\nu \frac{1}{r}\right)
\left(\frac{1}{\beta}-4\pi a\right), \quad r\to\infty. \label{correl-B}
\end{align}
The constant $a=a(\hbar,\beta,\rho)$ embodies the effects of matter on the
transverse field fluctuations. It has a relativistic factor $(m c^2)^{-1}$
and vanishes in the classical limit $\hbar\to 0$ (in accordance to the
Bohr--van Leeuwen decoupling) as $\mathrm{O}(\hbar^4)$ (see Appendix D).
In order to find the correlations of the transverse electric field
\begin{align}
\mathbf{E}_\text{t}(\mathbf{x})&=-\frac{1}{c}\left.\frac{\partial \mathbf{A}(\mathbf{x},t)}{\partial t}\right\vert_{t=0} \nonumber
\\&= -\(\frac{4\pi \hbar c^2}{R^{3}}\)^{1/2}\sum_{\mathbf{k}\lambda}g(\mathbf{k})
\frac{ {\bf e}_{\mathbf{k}\lambda}}{\sqrt{2\omega_{\mathbf{k}}}} \Big(\frac{i
\omega_\k}{c} \alpha_{\mathbf{k}\lambda}^{*}\mathrm{e}^{-i\mathbf{k}\cdot\mathbf{x}}- \frac{i
\omega_\k}{c} \alpha_{\mathbf{k}\lambda}\mathrm{e}^{i\mathbf{k}\cdot\mathbf{x}}\Big),
\end{align}
we couple the latter to an external polarisation $\pmb{\mathcal{P}}_\text{ext}(\mathbf{x})$,
\begin{align}
H_{L,R}(\pmb{\mathcal{P}}_\text{ext})=H_{L,R}-i\! \int\!\!
\d\mathbf{x}\ \pmb{\mathcal{P}}_\text{ext}(\mathbf{x})\cdot\mathbf{E}_\text{t}(\mathbf{x}), \label{F.2a}
\end{align}
and proceed as after (\ref{F.2}). This amounts to replacing everywhere
$\pmb{\mathcal{J}}_\text{ext}(\k)$ by $i k \pmb{\mathcal{P}}_\text{ext}(\k)$ so that the right-hand
side of equation (\ref{F.6}) is changed into
\begin{align}
&\exp\Big\{-\frac{\beta}{2} \int\!\!\! \frac{\d\mathbf{k}}{(2\pi)^{3}}
G^{\mu\nu}(\mathbf{k}) \label{cov-Pext}
\\
&\times\left[\left(\mathcal{J}^{\mu}\right)^{*}
\mathcal{J}^{\nu} - i k
\left(\mathcal{P}_\text{ext}^{\mu}\right)^{*}
\mathcal{J}^{\nu} + i k
\left(\mathcal{J}^{\mu}\right)^{*}
\mathcal{P}_\text{ext}^{\nu} + k^2
\left(\mathcal{P}_\text{ext}^{\mu}\right)^{*}
\mathcal{P}_\text{ext}^{\nu} \right](\k)\Big\}. \nonumber
\end{align}
As $\pmb{\mathcal{P}}_\text{ext}(\r)$ and $\pmb{\mathcal{J}}(\r)$ are real,
$\pmb{\mathcal{P}}_\text{ext}^\ast(\k) = \pmb{\mathcal{P}}_\text{ext}(-\k)$
and likewise for $\pmb{\mathcal{J}}$. From the change of variable
$\k\mapsto -\k$, one sees that the second term in the integrand in
(\ref{cov-Pext}) is exactly compensated by the third term. Only the term
quadratic in $\pmb{\mathcal{P}}_\text{ext}$ remains, which is responsible
upon functional differentiation for the thermal fluctuations of the free
field, as in (\ref{F.8}). Hence, the correlations of the transverse part of
the electric field are decoupled from matter and one finds
\begin{align}
\langle E_\text{t}^{\mu}(\mathbf{x})E_\text{t}^{\nu}(\mathbf{y})\rangle_\text{T} =
\langleE_\text{t}^{\mu}(\mathbf{x})E_\text{t}^{\nu}(\mathbf{y})\rangle_\text{T}^0
\sim \left(\partial_\mu \partial_\mu \frac{1}{r}\right)
\frac{1}{\beta}, \quad r\to\infty. \label{correl-E-trans}
\end{align}
The asymptotic correlation of the complete electric field $\mathbf{E}(\mathbf{x})
= \mathbf{E}_\text{l}(\mathbf{x}) +\mathbf{E}_\text{t}(\mathbf{x})$ follows from
(\ref{F.1}) and (\ref{correl-E-trans}) (one can check by similar
calculations that the cross correlation $\langle
E_\text{l}^{\mu}(\mathbf{x})E_\text{t}^{\nu}(\mathbf{y})\rangle_\text{T}$ vanishes
identically)~:
\begin{align}
\langle E^{\mu}(\mathbf{x})E^{\nu}(\mathbf{y})\rangle_\text{T} &=
\langle E_\text{l}^{\mu}(\mathbf{x})E_\text{l}^{\nu}(\mathbf{y})\rangle_\text{T}
+\langle E_\text{t}^{\mu}(\mathbf{x})E_\text{t}^{\nu}(\mathbf{y})\rangle_\text{T}
\nonumber
\\&=\left(\partial_\mu \partial_\mu \frac{1}{r}\right)
\left(\tfrac{2\pi}{3}\int\! \d\r\ |\r|^{2}
S(\r)+\frac{1}{\beta}\right), \quad r\to\infty. \label{correl-E}
\end{align}
In the classical limit, $S(\r)$ satisfies the second-moment
Stillinger--Lovett sum rule \cite{martin-sumrules} $-\tfrac{2\pi}{3}\int\!
\d\r\ |\r|^{2} S(\r) = 1/\beta$. Hence, the asymptotic longitudinal
electric field correlations in the matter are exactly compensated by those
of the free radiation part, as noted in \cite{Felderhof}. However, this no
longer holds for quantum plasmas. As an illustration, for the quantum
one-component plasma (jellium), one has \cite{pines-nozieres}
\begin{align}
-\tfrac{2\pi}{3}\int\! \d\r\ |\r|^{2} S(\r) = \tfrac{\hbar \omega_p}{2}
\text{coth}\left(\frac{\hbar \omega_p \beta}{2}\right) = \frac{1}{\beta} +
\frac{\beta}{3} \left(\frac{\hbar \omega_p}{2}\right)^2
+\mathrm{O}(\hbar^4), \label{ocp-sumrule}
\end{align}
where $\omega_p$ is the plasmon frequency. The long range of the electric
field correlations is thus a purely quantum-mechanical effect. These
findings are further discussed in the concluding remarks (section $8$).
\section{Bose and Fermi statistics}
In this final section we introduce the needed modifications when the
Fermionic or Bosonic particle statistics are taken into account.
The Bose or Fermi statistics of the particles can be incorporated in the
formalism following the same procedure as described in \cite{Cornu1},
\cite{Brydges-Martin} (section V). The matrix elements of (\ref{B.5}),
which is still an operator depending on the particle variables, are to be
evaluated with properly symmetrized (antisymmetrized) states. When
combining the decomposition of the permutation into cycles with the
Feynman-Kac-It\^o path integral representation this leads to generalize the
previous loops $\mathcal{F}=(\r, \gamma, \b\xi)$ to Brownian loops
$\mathcal{L}=(q,\mathbf{R},\gamma,\mathbf{X})$ that carry $q$ particles (a set of particles
labeled by indices belonging to a permutation cycle of length $q$). The
loop is located at $\mathbf{R}$ and has a random shape which is a Brownian bridge
$\mathbf{X}(s)$, $0\leq s\leq q$, $\mathbf{X}(0)=\mathbf{X}(q)=\mathbf{0}$ with zero mean and
covariance
\begin{align}
\int\!\! \mathrm{D}({\bf X })\, X_{\mu} (s_{1})
X_{\nu}(s_{2})=\delta_{\mu\nu}\,q\left[\min\left(\frac{s_{1}}{q},\frac{s_{2}}{q}\right)
-\frac{s_{1}}{q}\frac{s_{2}}{q}\right].
\label{7.14}
\end{align}
We merely give the final formulae since all steps are essentially identical
as those presented in the above mentioned works.
The grand canonical partition function of the particle system, with the
field degrees of freedom integrated out, takes the following classical-like
form in the space of loops
\begin{align}
\Xi_\Lambda=\sum_{n=0}^\infty\frac{1}{n!}\int\!\prod_{i=1}^n\d\mathcal{L}_i\ z(\mathcal{L}_i)
\exp\big(-\beta \mathcal{U}(\mathcal{L}_1,\ldots,\mathcal{L}_n)\big).
\label{7.15}
\end{align}
Integration on phase space means integration over space and summation over
all internal degrees of freedom of the loops~:
\begin{align}
\int\!\! \d\mathcal{L}\cdots=\int\!\! \d\mathbf{R} \sum_{\gamma}\sum_{q=1}^\infty \int\!\!
\mathrm{D}(\mathbf{X})\cdots.
\label{7.16}
\end{align}
$\mathcal{U}(\mathcal{L}_1,\ldots,\mathcal{L}_n)$ is the sum of the pair interactions
between two different loops
$e_{\gamma_i}e_{\gamma_j}[\mathcal{V}_\text{c}(\mathcal{L}_i,\mathcal{L}_j)+\mathcal{W}_\text{m}(\mathcal{L}_i,\mathcal{L}_j)]$
with
\begin{align}
\mathcal{V}_\text{c}(\mathcal{L}_i,\mathcal{L}_j) =\int_0^{q_{i}}\!\!\! \d s_{1}
\!\!\int_0^{q_{j}}\!\!\! \d s_{2}\ \delta( \tilde{s}_1-\tilde{s}_2)
\frac{1}{\big|\mathbf{R}_i+\lambda_{\gamma_i}\mathbf{X}_{i}(s_1) -
\mathbf{R}_j-\lambda_{\gamma_j}\mathbf{X}_{j}(s_2)\big|} \label{7.17}
\end{align}
the Coulomb potential, and
\begin{align}
\mathcal{W}_\text{m}(\mathcal{L}_i,\mathcal{L}_j)&=\frac{1}{\beta
\sqrt{m_{\gamma_i}m_{\gamma_j}}c^{2}}\int\!\!\! \frac{\d\mathbf{k} }{(2\pi)^{3}}\,
\mathrm{e}^{i\mathbf{k}\cdot(\r_i-\r_j)}\label{7.18}\\
&\times\int_{0}^{q_{i}}\!\!\! \d X_{i}^\mu(s_1)\,
\mathrm{e}^{i\mathbf{k}\cdot\lambda_{\gamma_{i}} \mathbf{X}_{i}(s_1)}\! \int_{0}^{q_{j}}\!\!\! \d
X_{j}^\nu(s_2)\,
\mathrm{e}^{-i\mathbf{k}\cdot\lambda_{\gamma_{j}}\mathbf{X}_{j}(s_2)}\
G^{\mu\nu}(\mathbf{k}) \nonumber
\end{align}
the effective magnetic potential obtained after integrating out the field
variables. Here ${\tilde s}=s\ \text{mod}\ 1$ and
$
\delta({\tilde s})=\sum_{n=-\infty}^{\infty}\mathrm{e}^{2i\pi ns}
$
is the periodic Dirac function of period $1$ that takes into account the
equal time constraint imposed by the Feynman-Kac formula. Finally, the
activity $z(\mathcal{L}_i)$ of a loop
\begin{align}
z(\mathcal{L}_{i})=\frac{(\eta_{\gamma_i})^{q_{i}-1}}{q_{i}}\;
\frac{z_{\gamma_{i}}^{q_i}}{(2\pi q_i \lambda_{\gamma_{i}}^2)^{3/2}}
\;\exp(-\beta [\mathcal{U}_\text{self}(\mathcal{L}_{i})+\mathcal{V}_\text{walls}(\mathcal{L}_i)]), \;\;\;z_{\gamma_{i}}=\mathrm{e}^{\beta
\mu_{\gamma_{i}}}
\label{7.20}
\end{align}
incorporates the chemical potential $\mu_{\gamma_{i}}$ of the particle, the
effects of quantum statistics ($\eta_{\gamma_{i}}=1$ for bosons and
$\eta_{\gamma_{i}}=-1$ for fermions), and the internal interaction
$\mathcal{U}_\text{self}(\mathcal{L}_{i})=-\frac{\beta
e_i^2}{2}(\mathcal{V}_\text{c}+\mathcal{W}_\text{m})(\mathcal{L}_i,\mathcal{L}_i)$ of
the particles belonging to the same loop (omitting the infinite Coulomb
self-energies of the point particles). The addition of the magnetic
potential $\mathcal{W}_\text{m}$ is the only modification compared to the
formalism previously developed for pure Coulombic
interactions. Maxwell-Boltzmann statistics and the potentials (\ref{B.14})
and (\ref{B.6}) of section 3 are recovered when only single-particle loops
($q=1$) are retained.
At this point, due to the classical-like structure of the partition
function (\ref{7.15}), methods of classical statistical mechanics can be
used in the auxiliary system of loops, in particular the technique of Mayer
graphs, as in section 5. The statistics of the particles affects the
coefficients of the tails of the density and field correlations, but not
their general forms (\ref{particle-correl-decay}), (\ref{correl-B}) and
(\ref{correl-E}).
\section{Concluding remarks}
The Feynman--Kac--It\^o path integral representation of the Gibbs weight
enables one to integrate out the (classical) field variables. This yields
an exact pairwise magnetic potential in the space of loops, which is
asymptotically dipolar. It generates small ($\mathrm{O}((\beta m c^2)^{-2})$)
corrections to the tail of the particle correlation due to the pure
Coulombic interaction.
A word is necessary about spin interactions that have not been included in
the above discussion. Spin-$1/2$ electrons give rise to the additional term
$-\nu\sum_{i=1}^{n}\pmb{\sigma}_{i}\cdot \mathbf{B}(\r_{i})$ in the Hamiltonian, with
$\mathbf{B}(\r)=\nabla \wedge \mathbf{A}(\r)$, $\nu=\tfrac{g_{s}e\hbar}{4mc}$, $g_{s}$ the
gyromagnetic factor, and $\pmb{\sigma}$ the Pauli matrices. Using spin coherent
states \cite{klauder}, a functional integral representation of the Gibbs
weight can be constructed including the coupling of the spins to the
field. Since this coupling is linear with respect to the vector potential,
Gaussian integration on the field variables leads again to an effective
spin-spin interaction $W_\text{s}(i,j)$ and effective cross interactions
$W_\text{sm}(i,j)$ and $W_\text{ms}(i,j)$ between spin and orbital
magnetism; details can be found in \cite{Diplome}. One finds that these
interactions are of dipolar type $\sim r^{-3}, r\to\infty$ and they have to
be added to the magnetic potential ${W_\text{m}}(i,j)$. In a homogeneous and
isotropic phase, the spin interaction terms contribute to the $r^{-6}$ tail
of the particle correlations with the same amplitude
$\frac{\lambda_{a}^{2}\lambda_{b}^{2}e_{a}^{2}e_{b}^{2}\rho_{a}\rho_{b}}{m_{a}m_{b}c^{4}}$,
up to numerical factors, as that found in section 4 in the case of the
magnetic potential ${W_\text{m}}$.
Regarding the electric field correlations in the plasma, we also find that
they have an algebraic decay of dipolar type. This is in disagreement with
the macroscopic calculation presented by Landau and Lifshitz \cite{Landau},
\S $88$, based on linear response theory and the fluctuation-dissipation
theorem. Indeed, the electric field fluctuations obtained in this theory
are short ranged (exponentially fast decaying)~: unlike in our calculation,
the algebraic parts of the longitudinal and transverse correlations
compensate exactly in the Landau and Lifshitz formulae
\cite{jancovici-private-comm}. Understanding the relation between our
strictly microscopic approach and the macroscopic theory of field
fluctuations is an open problem.
Let us, however, briefly point out some differences between the two
approaches. The microscopic approach involves all length scales, whereas
Landau and Lifshitz assume that the correlations of the polarisation are
local ($\delta$ correlated in space) and thus deal with a
wave-number-independent dielectric function $\epsilon(\omega)$. Taking into
account the wave-number dependence, it is likely that $\epsilon(\k, \omega)$
has terms non-analytic in $\k$, reflecting the fact that Coulombic matter
has algebraically decaying correlations. In fact, for the jellium model,
the static dielectric function $\epsilon(\k,\omega=0)$ has a singular term
$\sim |\k|$ in its small-$\k$ expansion \cite{Cornu-Martin}. It is possible
that in a non-local generalization of the Landau--Lifshitz theory such
singular terms also generate power-law decays of the field
correlations. Furthermore, the magnetic permeability is usually set equal
to that of the vacuum, thus ignoring the magnetization fluctuations,
whereas in our calculation the latter are properly included.
We stress again that the results of this paper hold when the
electromagnetic field is classical, which is supposed to correctly depict
the small-wave-number regime, as said in the Introduction. Hence, the
occurrence of the Planck constant originates exclusively from the
quantum-mechanical nature of matter. If the field is quantized, we can,
however, not exclude an interplay between $\hbar_\text{matter}$ and
$\hbar_\text{field}$, which could lead to a modification of the asymptotic
formulae presented in the paper.
\subsection*{Acknowledgements}
We thank A. Alastuey and B. Jancovici for useful discussions. P.R.B. is
supported by the Swiss National Foundation for Scientific Research.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,593 |
This is a list of convention centers named after people. It details the name of the convention center, its location and eponym.
List
Aiwan-e-Iqbal, Lahore, Pakistan, named for Muhammad Iqbal
Anthony Wayne Ballroom at the Grand Wayne Convention Center, Fort Wayne, Indiana, United States, named for Anthony Wayne (American Revolutionary War general)
Auditoria Benito Juarez, Los Mochis, Mexico, named for Benito Juárez (former president of Mexico)
Bandaranaike Memorial International Conference Hall, Colombo, Sri Lanka, named for Solomon Ridgeway Dias Bandaranaike
Bartle Hall Convention Center, Kansas City, Missouri, United States, named for Harold Roe Bartle
Calvin L. Rampton (Salt Palace) Convention Center, Salt Lake City, Utah, United States, named for Cal Rampton (former Utah governor)
Cobo Hall, Detroit, Michigan, United States, named for Albert Cobo (former Detroit mayor)
David Eccles Conference Center, Ogden, Utah, United States, named for David Eccles (businessman)
David L. Lawrence Convention Center, Pittsburgh, Pennsylvania, United States, named for David L. Lawrence
DeVos Place Convention Center, Grand Rapids, Michigan, United States, named for Richard DeVos
Donald E. Stephens Convention Center, Rosemont, Illinois, United States, named for Donald E. Stephens (former Rosemont mayor)
Ernest N. Morial Convention Center, New Orleans, Louisiana, United States, named for Ernest N. Morial
George R. Brown Convention Center, Houston, Texas, United States, named for George R. Brown (entrepreneur)
Hynes Convention Center, Boston, Massachusetts, United States, named for John Hynes
Jacob K. Javits Convention Center, New York City, United States, named for Jacob K. Javits
James H. Rainwater Convention Center, Valdosta, Georgia, United States, named for James H. Rainwater (former Valdosta mayor)
Jinnah Convention Centre, Islamabad, Pakistan, named for Quaid-e-Azam Muhammad Ali Jinnah
Julius Nyerere International Convention Centre, Dar es Salaam, Tanzania. Named after Julius Nyerere
Kenyatta International Conference Centre, Nairobi, Kenya. Named after Jomo Kenyatta.
King Hussein Bin Talal Convention Center, east coast of the Dead Sea in Jordan. Named for Hussein I, former King of Jordan
Lloyd Erskine Sandiford Conference and Cultural Centre, Two Mile Hill, Saint Michael, Barbados, named after Lloyd Erskine Sandiford (former prime minister of Barbados)
Mahatma Mandir, Gandhinagar, Gujarat, India, named after Mahatma Gandhi
McCormick Place, Chicago, Illinois, United States, named for Robert R. McCormick
Michael Fowler Centre, Wellington, New Zealand, named for Sir Michael Fowler (former Wellington mayor)
Moscone Center, San Francisco, California, United States, named for George Moscone
Obi Wali International Conference Center, Port Harcourt, Rivers State, named for senator Obi Wali
Prime F. Osborn III Convention Center, Jacksonville, Florida, United States, named for Prime F. Osborn III (former CSX chairman)
Putra World Trade Centre, Kuala Lumpur, Malaysia, named for Tunku Abdul Rahman (first Prime Minister of Malaysia)
Queen Sirikit National Convention Center, Bangkok, Thailand, named for Sirikit
Sime Darby Convention Centre, Bukit Kiara, Kuala Lumpur, named for William Sime and Henry Darby (founders of Sime Darby
Sultan Ahmad Shah International Convention Centre, Kuantan, Malaysia, named for Ahmad Shah of Pahang (Sultan of Pahang)
Walter E. Washington Convention Center, Washington, D.C., named for former mayor Walter E. Washington
William A. Egan Civic & Convention Center, Anchorage, Alaska, United States, named for William Allen Egan
See also
List of convention and exhibition centers
List of eponyms
List of places named after people
References
Convention Centers Named After People
Convention Centers
Casinos | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,115 |
Q: MongoDb conditional relationship Suppose I have following 4 collections:
1- posts
2- companies
3- groups
4- users
Bellow is my current structure in post:
and their relation is:
*
*A company has an owner and many other members (user collection).
*A group has many members (users).
*A user has many posts.
*A group has many posts that published by one of its members.
*A company has many posts that published by its owner or members.
Now i have a problem on storing relation of users, company, and group with posts collection.
Bellow is my current structure:
I have decided to have a field postable inside my post document, and has a type field that will be 'user', or 'group', or 'company', and two other fields name, and id that will be company/group id and company/group name in cases that post is belonged to company or group but not user means type="group" || type="company".
Now how i can handle this to map id as FK of group and company collection (one field FK of two collection) ?
Is it the right structure ?
A: What you have here is a polymorphic association. In relational databases, it is commonly implemented with two fields, postable_id and postable_type. The type column defines which table to query and id column determines the record.
You can do the same in mongodb (in fact, that is what you came up with, minus the naming convention). But mongodb has a special field type precisely for this type of situations: DBRef. Basically, it's an upgraded id field. It carries not only the id, but also collection name (and database name).
how i can handle this to map id as FK of group and company collection (one field FK of two collection)?
Considering that mongodb doesn't have joins and you have to load all references manually, I don't see how this is any different from a regular FK field. Just the collection name is stored in the type field now, instead of being hardcoded.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,871 |
package org.mustbe.consulo.unity3d.shaderlab.lang.psi;
import org.jetbrains.annotations.NotNull;
import org.jetbrains.annotations.Nullable;
import com.intellij.lang.ASTNode;
/**
* @author VISTALL
* @since 08.05.2015
*/
public class ShaderPropertyAttribute extends ShaderLabElement
{
public ShaderPropertyAttribute(@NotNull ASTNode node)
{
super(node);
}
@Nullable
public ShaderReference getReferenceExpression()
{
return findChildByClass(ShaderReference.class);
}
@Override
public void accept(SharpLabElementVisitor visitor)
{
visitor.visitPropertyAttribute(this);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,873 |
{"url":"http:\/\/math.stackexchange.com\/questions\/494719\/interpolating-g1-sum-a-1-infty-frac1aa-g2-sum-a-1-i","text":"# Interpolating $G(1)=\\sum_{a=1}^{\\infty} \\frac{1}{a^{a}}$, $G(2) = \\sum_{a=1}^{\\infty} \\sum_{b=1}^{\\infty} \\frac{1}{(ab)^{ab}}$ on $\\mathbb{C}$\n\nGiven that: $$G(1) =\\sum_{a=1}^{\\infty} \\frac{1}{a^{a}}$$\n\n(this is just the Sophomore's dream series, but the rest are not)\n\n$$G(2) = \\sum_{a=1}^{\\infty} \\sum_{b=1}^{\\infty} \\frac{1}{(ab)^{ab}}$$\n\n$$G(3) = \\sum_{a=1}^{\\infty} \\sum_{b=1}^{\\infty}\\sum_{c=1}^{\\infty} \\frac{1}{(abc)^{abc}}$$\n\nI'd like to interpolate $G(z)$ for $z\\in\\mathbb{C}$ given the above sequence. I can probably compute (with difficulty) $G(n)$ for many $n$.\n\n\u2022 Does the sequence above specify a unique $G(z)$?\n\u2022 Are there analytic tricks which would make finding such an interpolation easy? (the less I have to compute here, the better) *\n-","date":"2015-10-04 09:33:11","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9933207035064697, \"perplexity\": 1189.0653019367596}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-40\/segments\/1443736673081.9\/warc\/CC-MAIN-20151001215753-00085-ip-10-137-6-227.ec2.internal.warc.gz\"}"} | null | null |
\section{Introduction}
The need to analyze the combinatorial complexity of a $k$-fold maximum of function classes (see (\ref{eq:Fmax}) for the formal definition) arises in a number of diverse settings. Two natural instances of this problem include learning polytopes with margin
\citep{GKKN-nips18} and robust learning~\citep{DBLP:conf/alt/AttiasKM19};
see also \citet{DBLP:journals/jmlr/RavivHO18}.
\paragraph{Notation.}
We write $\mathbb{N}=\set{0,1,\ldots}$ to denote the natural numbers.
For $n\in\mathbb{N}$, we write
$[n]:=\set{1,2,\ldots,n}$.
All of our logarithms are base $e$, unless explicitly denoted otherwise.
We use $\max\set{u,v}$
and $u\vee v$ interchangeably,
and write
$\operatorname{Log}(x):=\log(e\vee x)$.
For any function class $F$
over a set $\Omega$ and $E\subset\Omega$,
$
F(E)=\evalat{F}{E}
$
denotes the projection
on (restriction to) $E$.
In line with the common convention
in functional analysis,
absolute numerical constants
will be denoted by letters such as
$C,c$,
whose value
may change from line to line.
Any transformation $\phi:\mathbb{R}\to\mathbb{R}$
may be applied to a function $f\in\mathbb{R}^\Omega$
via $\phi(f):=\phi\circ f$,
as well as to $F\subset\mathbb{R}^\Omega$
via $\phi(F):=\set{\phi(f);f\in F}$.
The sign function thresholds at $0$:
$\operatorname{sign}(t)=\pred{t\ge0}$.
\paragraph{Fat-shattering dimension.}
This ``parametrized variant of the $P$-dimension'' \citep{alon97scalesensitive}
was first proposed by \citet{183126};
its key role in learning theory
lies in characterizing the PAC learnability
of real-valued function classes
\citep{alon97scalesensitive,DBLP:journals/jcss/BartlettL98}.
Let $\Omega$ be a set and $F\subset\mathbb{R}^\Omega$. For $\gamma>0$,
a set $S=\set{x_1,\ldots,x_m}\subset\Omega$
is said to be $\gamma$-shattered by
$F$
if
\begin{eqnarray}
\label{eq:gamma-shatter}
\sup_{r\in\mathbb{R}^m}
\;
\min_{y\in\set{-1,1}^m}
\;
\sup_{f\in F}
\;
\min_{i\in[m]}
\;
y_i(f(x_i)-r_i)\ge \gamma.
\end{eqnarray}
The $\gamma$-fat-shattering dimension, denoted by $\operatorname{fat}_\gamma(F)$,
is the size of the largest $\gamma$-shattered set (possibly $\infty$).
As in
\citet{DBLP:journals/tit/GottliebKK14+colt},
we also define the notion
of $\gamma$-shattering at $0$,
where the ``shift'' $r$
in (\ref{eq:gamma-shatter})
is constrained to be $0$. Formally, the shattering condition
is
$
\min_{y\in\set{-1,1}^m}
\sup_{f\in F}
\min_{i\in[m]}
y_if(x_i)\ge \gamma
,
$
and the
we denote
corresponding
dimension
by
$\operatorname{\textup{f\aa t}}_\gamma(F)$.
\citet[Lemma 13]{DBLP:conf/alt/AttiasKM19}
shows that for all $F\subset\mathbb{R}^\Omega$,
\begin{eqnarray}
\label{eq:fatfat0}
\operatorname{fat}_\gamma(F)
=\max_{r\in\mathbb{R}^\Omega}\operatorname{\textup{f\aa t}}_\gamma(F-r),
\qquad
\gamma>0,
\end{eqnarray}
where $F-r=\set{f-r; f\in F}$
is the $r$-shifted class (the maximum is
always achieved).
Lemma~\ref{lem:fat-faat}
presents another, apparently novel, connection between $\operatorname{fat}$
and $\operatorname{\textup{f\aa t}}$.
For a set $\Omega$ and $k$ function classes $F_1,\ldots,F_k\subseteq \mathbb{R}^\Omega$, define the
$k$-fold
maximum
operator
\begin{eqnarray}
\label{eq:Fmax}
F_{\max}
\equiv
\max
\paren{F_{i\in[k]}}
:=
\set{
x\mapsto\max_{i\in[k]}f_i(x); f_i\in F_i};
\end{eqnarray}
in words,
each element of $F_{\max}$
is
obtained by choosing
a function $f_i$
from each of the $k$ classes $F_i$,
and computing
their pointwise maximum.
\paragraph{Main results.}
Our main results involve upper-bounding
the fat-shattering dimension
of a $k$-fold maximum in terms
of the dimensions of the component classes.
We begin with the simplest (to present):
\begin{theorem}
\label{thm:max-fat}
For
$F_1,\ldots,F_k\subseteq\mathbb{R}^\Omega$
and
$F_{\max}$ as defined in (\ref{eq:Fmax}),
we have
\begin{eqnarray}
\label{eq:max-fat}
\operatorname{fat}_\gamma(F_{\max})
&\le&
25
D_\gamma \log^2(
90
D_\gamma),
\qquad
\gamma>0,
\end{eqnarray}
where
$D_\gamma:=\sum_{i=1}^k\operatorname{fat}_\gamma(F_i)>0$.
In the degenerate case
where $D_\gamma=0$,
$\operatorname{fat}_\gamma(F_{\max})=0$.
\end{theorem}
\paragraph{Remark.}
We made no attempt to optimize the constants;
these are only provided to give a rough
order-of-magnitude sense. In the sequel,
we forgo numerical estimates
and state the results in terms of
unspecified universal constants.
The next result
provides an alternative bound
based on an entirely
different technique:
\begin{theorem}
\label{thm:max-fat-RV}
For
$0<\varepsilon<\log 2$,
$F_1,\ldots,F_k\subseteq
[-R,R]
^\Omega$,
and
$F_{\max}$ as defined in (\ref{eq:Fmax}),
we have
\begin{eqnarray}
\label{eq:max-fat-RV}
\operatorname{fat}_\gamma(F_{\max})
&\le&
CD\operatorname{Log}^{1+\varepsilon}\frac{Rk}{\gamma}
,
\qquad
0<\gamma<R,
\end{eqnarray}
where
\begin{eqnarray*}
D &=&
\sum_{i=1}^k\operatorname{fat}_{c\varepsilon \gamma}(F_i)
\end{eqnarray*}
and $C,c>0$ are universal constants.
\end{theorem}
\paragraph{Remark.}
The bounds in Theorems~\ref{thm:max-fat}
and \ref{thm:max-fat-RV} are, in general, incomparable --- and not just because of the unspecified constants in the latter. One notable difference is that
Theorem~\ref{thm:max-fat} only depends on the
shattering scale $\gamma$,
while
Theorem~\ref{thm:max-fat-RV} additionally
features a (weak) explicit dependence on the aspect
ratio $R/\gamma$.
In particular,
Theorem~\ref{thm:max-fat} is applicable to semi-bounded affine classes
(see Section~\ref{sec:affine}),
while Theorem~\ref{thm:max-fat-RV} is not.
Still, for fixed $R,\gamma$
and large $k$,
the latter
presents a significant asymptotic
improvement over the former.
For the special case of affine function classes,
the technique of Theorem~\ref{thm:max-fat-RV}
yields a
considerably sharper estimate:
\begin{theorem}
\label{thm:max-fat-lin-R}
Let $B\subset\mathbb{R}^d$ be the $d$-dimensional
Euclidean unit ball
and
\begin{eqnarray}
\label{eq:Fi-max-fat-lin-R}
F_i=\set{x\mapsto w\cdot x+b;
\nrm{w}\vee |b|\le R_i},
\qquad
i\in[k].
\end{eqnarray}
be $k$ collections of
$R_i$-bounded affine functions
on $\Omega=B$
and
$F_{\max}$ as defined in (\ref{eq:Fmax}).
Then
\begin{eqnarray}
\label{eq:Fi-max-fat-lin-R-bd}
\operatorname{fat}_{\gamma}(F_{\max})
&\le&
\frac{
c\operatorname{Log}(k)
}{
\gamma^2
}
\sum_{i=1}^k
{R_i^2}
,
\qquad
0<\gamma<\min_{i\in[k]}
R_i
,
\end{eqnarray}
where $c>0$
is a universal constant.
\end{theorem}
\paragraph{Remark.}
A bound
of this form
was claimed in
\citet{kontorovich2018rademacher},
however the argument there
was flawed,
see Section~\ref{sec:discuss}.
The result of Theorem~\ref{thm:max-fat-lin-R}
can easily be generalized to hinge-loss
affine classes. Let $F$ be as in (\ref{eq:Fi-max-fat-lin-R}),
define $F'$
as the function class
on $B\times\set{-1,1}$
given by $F'=\set{(x,y)\mapsto yf(x); f\in F}$,
and
the {\em hinge-loss affine class}
$F''$
as the function class
on $B\times\set{-1,1}$
given by $F''=\set{(x,y)\mapsto
\max\set{0,1-f'(x,y)}
; f'\in F'}$.
One first observes that
the restriction of $F'$
to any $\set{(x_1,y_1),\ldots,(x_n,y_n)}$,
as a body in $\mathbb{R}^n$,
is identical to the
the restriction of $F$
to
$\set{x_1,\ldots,x_n}$.
Interpreting $F''$ as a $2$-fold maximum
over the singleton class $G=\set{g\equiv0}$
and a bounded affine class lets us invoke
Theorem~\ref{thm:max-cov}
to argue that $F$
and $F''$ have the same $L_\infty$ covering numbers.
An inspection of the proof
of Theorem~\ref{thm:max-fat-lin-R}
shows
that
$\operatorname{fat}_\gamma$
of
the $k$-fold maximum of
bounded hinge-loss classes
also behaves as (\ref{eq:Fi-max-fat-lin-R-bd}).
\citet[Theorem 7]{DBLP:journals/jmlr/RavivHO18}
upper-bounded the Rademacher complexity
this class by $k/\sqrt n$,
and the analysis above, combined with
the calculation in
\citet{kontorovich2018rademacher}
yields a bound of
$O\paren{
\sqrt{
\frac{k\log k\cdot \log^3 n}{n}
}
}$.
\begin{conjecture}
\label{conj:fat-lin-R-tight}
Theorem~\ref{thm:max-fat-lin-R} is tight --- i.e.,
has a matching lower bound.
\end{conjecture}
The estimate in Theorem~\ref{thm:max-fat-lin-R} is {\em dimension-free} in the sense of being independent of $d$. In applications where
a dependence on $d$ is admissible,
an
optimal bound can be obtained:
\begin{theorem}
\label{thm:max-fat-lin-d}
Let
$\Omega=\mathbb{R}^d$
and
$F_i\subset\mathbb{R}^\Omega$ be $k$ (identical)
function classes consisting of all
real-valued affine functions:
\begin{eqnarray*}
F_i=\set{x\mapsto w\cdot x+b
},
\qquad
i\in[k]
\end{eqnarray*}
and let $F_{\max}$ be their $k$-fold maximum.
Then
\begin{eqnarray*}
\operatorname{fat}_\gamma(
F_{\max}
)
&\le&
cdk\operatorname{Log} k
,
\qquad
\gamma>0,
\end{eqnarray*}
where $c>0$ is a universal constant.
\end{theorem}
The
optimality of the
upper bound in
Theorem~\ref{thm:max-fat-lin-d}
is witnessed by the
matching lower bound:
\begin{theorem}
\label{thm:logk-lb}
For $k\ge1$ and $d\ge4$,
let
$F_1=F_2=\ldots=F_k$ be the collection of all
affine functions over $
\Omega=
\mathbb{R}^d$
and let $F_{\max}$
be their $k$-fold maximum.
Then
\begin{eqnarray*}
\operatorname{fat}_\gamma(
F_{\max}
)
&\ge&
c\log(k)
\sum_{i=1}^k
\operatorname{fat}_\gamma(F_i)
=
cdk\log k
,
\qquad
\gamma>0,
\end{eqnarray*}
where $c>0$ is a universal constant.
\end{theorem}
The scaling argument employed in the
proof of Theorem~\ref{thm:logk-lb}
can be invoked to show that
the claim continues to hold
for $\Omega=B$.
Together, Theorems~\ref{thm:max-fat-lin-d}
and \ref{thm:logk-lb} show that
the logarithmic
dependence on $k$
is optimal.
More ambitiously, we pose the
\begin{conjecture}
\label{conj:holy-grail}
For all $F_i\subseteq\mathbb{R}^\Omega$, $i\in[k]$,
\begin{eqnarray}
\label{eq:AKM}
\operatorname{fat}_\gamma(
F_{\max}
)
&\le&
c\operatorname{Log}(k)\sum_{j=1}^k
\operatorname{fat}_{\gamma}(F_i)
,
\qquad
\gamma>0
\end{eqnarray}
for some universal $c>0$.
\end{conjecture}
In light of Theorem~\ref{thm:logk-lb},
this is the best one could
hope for in general.
\section{Related work}
It was claimed in
\citet[Theorem 12]{DBLP:conf/alt/AttiasKM19}
that
$
\operatorname{fat}_\gamma(
F_{\max}
)
\le
2\log(3k)\sum_{j=1}^k
\operatorname{fat}_{\gamma}(F_i),
$
but the proof had a mistake (see Section~\ref{sec:discuss});
our Conjecture~\ref{conj:holy-grail}
posits
that the general form of the bound
does hold.
Using the recent
disambiguation result
of
\citet
{DBLP:journals/corr/abs-2107-08444}
presented in
Lemma~\ref{lem:disambig} here,
\citet[Theorem 6.4]{attias2021improved}
obtained the
bound
$
\operatorname{fat}_\gamma(
F_{\max})
\le
O\paren{\log(k)\log^2(|\Omega|)\sum_{j=1}^k\operatorname{fat}_\gamma(F_i)}$.
The latter is, in general, incomparable
to Theorem~\ref{thm:max-fat}
--- but is clearly
a considerable improvement for large
or infinite $\Omega$.
Using
the covering number
results of
\citet{MR1965359,talagrand2003vapnik}
(see Section~\ref{sec:cov-num}),
\citet[Theorem 6.2]{duan11} obtained a general result, which, when specialized to
$k$-fold maxima, yields
\begin{eqnarray}
\label{eq:duan}
\operatorname{fat}_{\gamma}(F_{\max})
&\le&
C
{\log\frac{
k}{\gamma}}
\cdot
\sum_{i=1}^k
\operatorname{fat}_{c\gamma/\sqrt k}(F_i)
\end{eqnarray}
for universal constants
$C,c>0$;
(\ref{eq:duan})
is an immediate consequence of
Theorem~\ref{thm:max-cov} (with $p=2$),
Lemma~\ref{lem:talag},
and
Lemma~\ref{lem:MV}.
Our results improve
over (\ref{eq:duan}) by
removing the dependence on $k$
in the scale of the fat-shattering
dimensions; however, \citeauthor{duan11}'s
general method is applicable to
a much wider class of $k$-fold aggregations
than the maximum.
Results on the VC-dimension of
aggregations of Boolean function
classes were obtained by
\citet{78414,
MR1072253,
DBLP:journals/ipl/EisenstatA07,
DBLP:journals/ipl/Eisenstat09},
including the lower bound of
\citet{DBLP:journals/jmlr/CsikosMK19}.
\citet[Lemma A.2]{DBLP:conf/nips/SrebroST10}
bounded the fat-shattering dimension
in terms of the Rademacher complexity.
\citet{DBLP:journals/corr/abs-1911-06468}
bounded the Rademacher complexity
of a smooth $k$-fold aggregate, see
also references therein.
Closure results for other dimensions
were obtained by \citet{ghazi2021near}.
\section{Proofs}
\subsection{Proof of Theorem~\ref{thm:max-fat}}
\label{sec:max-fat}
\paragraph{Partial concept classes and disambiguation.}
We say that $F^\star\subseteq\set{0,1,\star}^\Omega$
is a
{\em partial}
concept class over
$\Omega$;
this usage is consistent with
\citet{DBLP:journals/corr/abs-2107-08444},
while \citet{DBLP:conf/alt/AttiasKM19,attias2021improved}
used the descriptor
{\em ambiguous}.
For $f^\star\in
F^\star
$,
define its {\em disambiguation set}
$\mathscr{D}(f^\star)\subseteq\set{0,1}^\Omega$
as
\begin{eqnarray*}
\mathscr{D}(f^\star)=
\set{
g\in\set{0,1}^\Omega:
\forall x\in\Omega,~f^\star(x)\neq\star\implies f^\star(x)=g(x)
};
\end{eqnarray*}
in words, $\mathscr{D}(f^\star)$ consists of the
{\em total} concepts $g:\Omega\to\set{0,1}$
that agree pointwise with $f^\star$, whenever the latter takes a value in $\set{0,1}$.
We say that $\bar F\subseteq\set{0,1}^\Omega$
disambiguates $F^\star$
if for all $
f^\star\in
F^\star$,
we have $\bar F\cap\mathscr{D}(f^\star)\neq\emptyset$;
in words, every
$f^\star\in
F^\star$
must have a disambiguated representative in
$\bar F$.\footnote{\citet{attias2021improved}
additionally required that $\bar F
\subseteq
\bigcup_{f^\star\in F^\star}\mathscr{D}(f^\star)$,
but this is an unnecessary restriction,
and does not affect any of the results.}
As in \citet{DBLP:journals/corr/abs-2107-08444,attias2021improved}, we say\footnote{
\citet{DBLP:conf/alt/AttiasKM19}
had
incorrectly
given
$F^\star(S)=\set{0,1}^S$
as the shattering condition.
}
that
$S\subset\Omega$ is VC-shattered by $F^\star$
if $F^\star(S)\supseteq\set{0,1}^S$. We write $\operatorname{vc}(F^\star)$ to denote
the size of the largest VC-shattered set
(possibly, $\infty$).
The obvious relation
$
\operatorname{vc}(F^\star)
\le
\operatorname{vc}(\bar F)
$
always holds
between a partial concept class
and any of its disambiguations.
\citet[Theorem 13]{DBLP:journals/corr/abs-2107-08444},
proved the following variant of the
Sauer-Shelah-Perles Lemma for partial
concept classes:
\begin{lemma}[\citet{DBLP:journals/corr/abs-2107-08444}]
\label{lem:disambig}
For every $F^\star\subseteq\set{0,1,\star}^\Omega$
with $d=\operatorname{vc}(F^\star)<\infty$,
there is an $\bar F$
disambiguating $F^\star$
such that
\begin{eqnarray}
\label{eq:disamb}
|\bar F(\Omega)|
&\le&
(|\Omega|+1)^{(d+1)\log_2|\Omega|+2}.
\end{eqnarray}
For $d>0$ and $|\Omega|>1$, this
implies the somewhat more wieldy
estimate\footnote{
The estimate (\ref{eq:disamb-c})
does not appear in
\citet{DBLP:journals/corr/abs-2107-08444},
but is an elementary consequence of
(\ref{eq:disamb}).} to
\begin{eqnarray}
\label{eq:disamb-c}
|\bar F(\Omega)|
&\le&
|\Omega|^{5d\log_2|\Omega|}.
\end{eqnarray}
\end{lemma}
We will make use of the
elementary fact
\begin{eqnarray}
\label{eq:log2x}
x\le A\log_2 x
&\implies&
x\le 3 A\log(3 A)
,
\qquad x,A\ge 1
\end{eqnarray}
and its corollary
\begin{eqnarray}
\label{eq:log2y}
y\le A(\log_2y)^2
&\implies&
y\le 5 A\log^2(18 A)
,
\qquad y,A\ge 1.
\end{eqnarray}
\begin{proof}[of Theorem~\ref{thm:max-fat}]
We follow the basic techniques of discretization and $r$-shifting, employed in
\citet{DBLP:conf/alt/AttiasKM19,attias2021improved}. Fix $\gamma>0$
and define the operator
$\discr{\cdot}:\mathbb{R}\to\set{0,1,\star}$
as
\begin{eqnarray*}
\discr{t}
&=&
\begin{cases}
0, & t\le\gamma \\
1, & t\ge\gamma \\
\star, & \text{else}.
\end{cases}
\end{eqnarray*}
Observe that for all
$F\subseteq\mathbb{R}^\Omega$
and
$\discr{F}:=\set{\discr{f};f\in F}$,
we have
$
\operatorname{\textup{f\aa t}}_\gamma(F)=\operatorname{vc}(\discr{F}).
$
Suppose that some
$S=\set{x_1,\ldots,x_\ell}\subset\Omega$
is $\gamma$-shattered by $F_{\max}$.
Proving the claim amounts to
upper-bounding $\ell$ appropriately.
By (\ref{eq:fatfat0}),
there is an $r\in\mathbb{R}^\Omega$
such that $\operatorname{fat}_\gamma(F_{\max})
=\operatorname{\textup{f\aa t}}_\gamma(F_{\max}-r)
=\operatorname{vc}(\discr{F_{\max}-r})
$.
Put $F'_i:=F_i-r$
and
\begin{eqnarray}
\label{eq:Fmax'}
F_{\max}':=F_{\max}-r
=\max(F'_{i\in[k]})
\end{eqnarray}
($k$-fold max commutes with $r$-shift).
Hence, $S$ is VC-shattered
by $\discr{F_{\max}'}$
and
\begin{eqnarray}
\label{eq:di}
v_i:=
\operatorname{vc}(\discr{F_i'})
=
\operatorname{\textup{f\aa t}}_\gamma(F_i')
\le
\operatorname{fat}_\gamma(F_i')
=
\operatorname{fat}_\gamma(F_i)
,
\qquad
i\in[k].
\end{eqnarray}
Let us assume for now that
each $v_i>0$;
in this case,
there is no loss of
generality in assuming $\ell>1$.
Let
$\bar F_i$
be a ``good'' disambiguation
of $\discr{F_i'}$
on $S$, as furnished by
Lemma~\ref{lem:disambig}:
\begin{eqnarray}
|\bar F_i(S)|
&\le&
\ell^{5
v_i
\log_2\ell}.
\end{eqnarray}
Observe that $
{\bar F}_{\max}
:=\max(\bar F_{i\in[n]})$ is a valid disambiguation of $\discr{F_{\max}'}$.
It follows that
\begin{eqnarray}
\label{eq:2^ell}
2^\ell
\;=\;
|{\bar F}_{\max}
(S)|
\;\le\;
\prod_{i=1}^k |\bar F_i(S)|
\;\le\;
\ell^{
5\log_2\ell
\sum_{i=1}^k
v_i
}
.
\end{eqnarray}
Thus,
(\ref{eq:log2y})
implies
that
$
\ell\le
25
(\sum v_i)\log^2(
90
\sum v_i)
$,
and the latter is an
upper bound on
$\operatorname{vc}(
{\bar F}_{\max}
)
$
---
and hence, also on
$\operatorname{vc}(
\discr{F_{\max}'}
)
=\operatorname{fat}_\gamma(F_{\max})
$.
The claim now follows from~(\ref{eq:di}).
If any one given $v_i=0$, we claim that (\ref{eq:2^ell}) is unaffected.
This is because
any $G^\star\in\set{0,1,\star}^\Omega$
with $\operatorname{vc}(G^\star)=0$ has a
singleton
disambiguation
$\bar G=\set{g}$.
Indeed, any
given
$x\in\Omega$ can
receive at most one of
$\set{0,1}$
as a label
from the members of
$G$ (otherwise, it would be shattered, forcing
$\operatorname{vc}(G^\star)\ge1$).
If {\em any} $g^\star\in G^\star$ labels $x$
with $0$, then {\em all} members of $G^\star$
are disambiguated to label $x$ with $0$
(and, {\em mutatis mutandis}, $1$).
Any $x$ labeled with $\star$ by {\em every} $g^\star\in G^\star_i$ can be disambiguated arbitrarily (say, to $0$).
Disambiguating the degenerate
$\discr{F_i'}$
to the singleton
$
\bar F_i(S)
$
has no effect on the product in
(\ref{eq:2^ell}).
The foregoing argument continues to hold
if more than one $v_i=0$. In particular,
in the degenerate case where
$d_1=d_2=\ldots=d_k=0$,
we have $\prod|\bar F_i(S)|=1$,
which forces $\ell=0$.
\end{proof}
\subsection{Proof of Theorem~\ref{thm:max-fat-RV}}
We use the notation and results
from the Appendix,
and in particular, from
Section~\ref{sec:cov-num}.
\begin{proof}[of Theorem~\ref{thm:max-fat-RV}]
Suppose that some
$\Omega_\ell=\set{x_1,\ldots,x_\ell}\subset\Omega=B$
is $\gamma$-shattered by $F_{\max}$
and
let $F_i(\Omega_\ell)=\evalat{F_i}{\Omega_\ell}$.
By Lemma~\ref{lem:RV} (with $n=\ell$ and $p=
\infty$),
\begin{eqnarray*}
\log\mathcal{N}(F_i(\Omega_\ell),L_\infty(\mu_\ell),\gamma)
&\le&
Cv_i\log(R\ell/v_i\gamma)\log^\varepsilon(\ell/v_i),
\qquad
0<\gamma<R,
\end{eqnarray*}
where
$v_i=\operatorname{fat}_{c\varepsilon \gamma}(F_i)$.
Then Theorem~\ref{thm:max-cov} implies that
\begin{eqnarray*}
\log\mathcal{N}(F_{\max}(\Omega_\ell),L_\infty(\mu_\ell),
\gamma/2
)
&\le&
\sum_{i=1}^k
\log\mathcal{N}(F_i(\Omega_\ell),L_\infty(\mu_\ell),
\gamma/2)
\\
&\le&
C\sum_{i=1}^k
v_i\log(R\ell/v_i\gamma)\log^\varepsilon(\ell/v_i)
\\
&\maavar{a}{\le}&
C\sum_{i=1}^k
v_i\log^{1+\varepsilon}(R\ell/v_i\gamma)
\\
&\maavar{b}{\le}&
CD\log^{1+\varepsilon}\frac{R\ell k}{D\gamma},
\end{eqnarray*}
where $D:=\sum_{i=1}^k v_i$,
(a) follows because $R/\gamma>1$,
and
(b) follows from Lemma~\ref{lem:log^eps}
(we can assume $\ell\ge 2$ without loss of generality).
Combining
(\ref{eq:covn-pq}) and
(\ref{eq:talag}) yields
\begin{eqnarray*}
\log\mathcal{N}(F_{\max}(\Omega_\ell),L_\infty(\mu_\ell),\gamma/2)
&\ge&
C\operatorname{fat}_{\gamma}(F_{\max})
=C\ell,
\end{eqnarray*}
whence
\begin{eqnarray*}
\ell
&\le&
CD\log^{1+\varepsilon}\frac{R\ell k}{D\gamma}.
\end{eqnarray*}
Using the elementary fact
\begin{eqnarray*}
x\le A\operatorname{Log}^{1+\varepsilon} x
&\implies&
x\le c A\operatorname{Log}^{1+\varepsilon} A
\qquad
x,A>0
\end{eqnarray*}
(with
$x=R\ell k/D\gamma$
and
$A=cRk/\gamma$),
we get
\begin{eqnarray*}
\ell&\le& C D\operatorname{Log}^{1+\varepsilon}\frac{Rk}{\gamma},
\end{eqnarray*}
which implies the claim.
\end{proof}
\subsection{Proof of Theorem~\ref{thm:max-fat-lin-R}}
We use the notation and results
from the Appendix,
and in particular, from
Section~\ref{sec:cov-num-lin}.
\begin{proof}[of Theorem~\ref{thm:max-fat-lin-R}]
Suppose that some
$\Omega_\ell=\set{x_1,\ldots,x_\ell}\subset\Omega=B$
is $\gamma$-shattered by $F_{\max}$,
let $F_i(\Omega_\ell)=\evalat{F_i}{\Omega_\ell}$
(where $F_i$
is defined in (\ref{eq:Fi-max-fat-lin-R}))
and $\mu_\ell$ be the uniform distribution
on $\Omega_\ell$.
By Lemma~\ref{lem:ramon} (with $m=\ell$),
\begin{eqnarray*}
\log\mathcal{N}(F_i(\Omega_\ell),L_\infty(\mu_\ell),\gamma)
&\le&
C\frac{R_i^2}{\gamma^2}\operatorname{Log}\frac{\ell \gamma}{R_i}
,
\qquad
0<\gamma<R_i\\
&=:&
C v_i \operatorname{Log}\frac{\ell}{v_i}
\qquad\quad
\text{(i.e., $v_i:=R_i^2/\gamma^2)$}
.
\end{eqnarray*}
Then Theorem~\ref{thm:max-cov} implies that
\begin{eqnarray*}
\log\mathcal{N}(F_{\max}(\Omega_\ell),L_\infty(\mu_\ell),
\gamma/2)
&\le&
\sum_{i=1}^k
\log\mathcal{N}(F_i(\Omega_\ell),L_\infty(\mu_\ell),
\gamma/2)
\\
&\le&
C\sum_{i=1}^k
v_i \operatorname{Log}\frac{\ell}{v_i}
\\
&\maavar{a}{\le}&
C D \operatorname{Log}\frac{k\ell}{D},
\end{eqnarray*}
where $D:=\sum_{i=1}^k v_i$
and
(a) follows from
Corollary~\ref{cor:xlogn/x}.
Combining
(\ref{eq:covn-pq}) and
(\ref{eq:talag}) yields
\begin{eqnarray*}
\log\mathcal{N}(F_{\max}(\Omega_\ell),L_\infty(\mu_\ell),\gamma/2)
&\ge&
C\operatorname{fat}_{\gamma}(F_{\max})
=C\ell
,
\end{eqnarray*}
whence
\begin{eqnarray*}
\ell
&\le&
C D \operatorname{Log}\frac{k\ell}{D}.
\end{eqnarray*}
Using the elementary fact
\begin{eqnarray*}
x\le A\operatorname{Log} x
&\implies&
x\le cA\operatorname{Log} A,
\qquad
x,A>0
\end{eqnarray*}
(with $x=k\ell/D$
and
$A=ck$)
we get
$\ell\le cD\operatorname{Log} k$,
which implies the claim.
\end{proof}
\subsection{Proof of Theorem~\ref{thm:max-fat-lin-d}}
\begin{proof}[of Theorem~\ref{thm:max-fat-lin-d}]
By (\ref{eq:fatfat0}),
there is an $r\in\mathbb{R}^\Omega$
such that $\operatorname{fat}_\gamma(F_{\max})
=\operatorname{\textup{f\aa t}}_\gamma(F_{\max}-r)
$.
As in (\ref{eq:Fmax'}),
put $F'_i:=F_i-r$
and
$
F_{\max}':=F_{\max}-r
=\max(F'_{i\in[k]})
$.
Define $\bar F_{\max}=\operatorname{sign}(F_{\max}')$
and
$\bar F_i=\operatorname{sign}(F_i')$.
Since $\operatorname{sign}$
and $\max$ commute, we have
$\bar F_{\max}
=\max(\bar F_{i\in[k]})
$.
We claim that
\begin{eqnarray}
\label{eq:faatmax}
\operatorname{\textup{f\aa t}}_\gamma(F_{\max}')
&\le&
\operatorname{vc}(\bar F_{\max}).
\end{eqnarray}
Indeed, any $S\subset\Omega$
that is $\gamma$-shattered
with shift $r=0$ by any $G\subset\mathbb{R}^\Omega$
is also VC-shattered by $\operatorname{sign}(G)$.
(See Section~\ref{sec:max-fat},
and
notice that the converse implication
--- and the reverse inequality --- do not hold.)
Lemmas
\ref{lem:fat-faat}
and
\ref{lem:fat-hyp}
imply
that
\begin{eqnarray*}
\operatorname{\textup{f\aa t}}_\gamma(F_i)
=
\operatorname{fat}_\gamma(F_i)
=
\operatorname{fat}_\gamma(F_i')
=
\operatorname{vc}(\bar F_i)
=d+1
.
\end{eqnarray*}
Now the argument of
\citet[Lemma 3.2.3]{MR1072253}
applies:
\begin{eqnarray}
\label{eq:blumer}
\operatorname{vc}(\bar F_{\max})
&\le&
2(d+1)k\log(3k)
\end{eqnarray}
(this holds for any $k$-fold aggregation function,
not just the maximum).
Combining (\ref{eq:faatmax})
with (\ref{eq:blumer}) proves the claim.
\end{proof}
\subsection{Proof of Theorem~\ref{thm:logk-lb}}
\begin{proof}[of Theorem~\ref{thm:logk-lb}]
It follows from
\citet[Example 3.2]{mohri-book2012}
that $\operatorname{vc}(\operatorname{sign}(F_i))=d+1$.
Since $F_i$ is closed under
scalar multiplication,
a scaling argument
shows that
any $S\subset\mathbb{R}^d$
that is VC-shattered by
$\operatorname{sign}(F_i)$
is also
$\gamma$-shattered
by $F_i$
with shift $r=0$,
whence
$\operatorname{\textup{f\aa t}}_\gamma(F_i)
=d+1$
for all $\gamma>0$;
invoking
Lemma~\ref{lem:fat-faat}
extends this to $\operatorname{fat}_\gamma(F_i)$
as well.
Now
\citet[Theorem 1]{DBLP:journals/jmlr/CsikosMK19}
shows that the $k$-fold unions
of
half-spaces
necessarily shatter some set $S\subset\mathbb{R}^d$
of size at least
$cdk\log k$.
Since union is a special case of
the max operator,
and the latter commutes with $\operatorname{sign}$,
the scaling argument shows that
this
$S$ is $\gamma$-shattered
by
$F_{\max}$
with shift $r=0$.
Hence,
$\operatorname{fat}_\gamma(F_{\max})
\ge
\operatorname{\textup{f\aa t}}_\gamma(F_{\max})
\ge
|S|
$,
which proves the claim.
\end{proof}
\section{Discussion}
\label{sec:discuss}
Throughout the paper,
we mentioned several mistaken claims in the literature.
In this section, we briefly discuss the nature
of these mistakes --- which are, in a sense,
variations on the same kind of error.
We begin with
\citet[Lemma 14]{DBLP:conf/alt/AttiasKM19},
which incorrectly claimed that
any partial function class $F^\star$
has a disambiguation $\bar F$
such that $\operatorname{vc}(\bar F)\le\operatorname{vc}(F^\star)$
(see Section~\ref{sec:max-fat}
for the definitions).
The mistake was pointed out to us by
Yann Guermeur,
and later,
\citet[Theorem 11]{DBLP:journals/corr/abs-2107-08444}
showed
that there exist partial classes
$F^\star$ with $\operatorname{vc}(F^\star)=1$
for which every disambiguation
$\bar F$ has $\operatorname{vc}(\bar F)=\infty$.
\citet{kontorovich2018rademacher}
attempted to prove the bound
stated in our Theorem~\ref{thm:max-fat-lin-R}
(up to constants, and only for linear classes).
The argument proceeded via a reduction to
the Boolean case,
as in our proof of Theorem~\ref{thm:max-fat-lin-d}.
It was correctly observed that
if, say,
some finite $S\subset\Omega$
is $1$-shattered
by
$F_i$ with shift $r=0$,
then it is also VC-shattered
by $\operatorname{sign}(F_i)$. Neglected was the fact
that $\operatorname{sign}(F_i)$ might shatter additional points in $\Omega\setminus S$ --- and,
in sufficiently high dimension, it necessarily will.
The crux of the matter is that
(\ref{eq:faatmax}) holds
in the dimension-dependent but
not the dimension-free setting;
again, this may be seen as a variant
of the disambiguation mistake.
Finally, the proof of
\citet[Lemma 6]{DBLP:journals/tcs/HannekeK19}
claims,
in the first display,
that the shattered set can be classified
with large margin, which is incorrect
--- yet another variant of mistaken
disambiguation.
\acks{We thank
Steve Hanneke
and
Ramon van Handel for very helpful
discussions;
the latter, in particular,
patiently explained to us how to prove
Lemma~\ref{lem:ramon}.
Roman Vershynin kindly gave us permission
to share his example in Remark~\ref{rem:vers}.
This research was partially supported by
the Israel Science Foundation
(grant No. 1602/19) an Amazon Research Award.
}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 742 |
{"url":"https:\/\/im.kendallhunt.com\/HS\/students\/1\/6\/16\/index.html","text":"# Lesson 16\n\nGraphing from the Vertex Form\n\n\u2022 Let\u2019s graph equations in vertex form.\n\n### 16.1: Which Form to Use?\n\nExpressions in different forms can be used to define the same function. Here are three ways to define a function $$f$$.\n\n$$f(x)=x^2-4x+3$$\n\n(standard form)\n\n$$f(x)=(x-3)(x-1)$$\n\n(factored form)\n\n$$f(x)=(x-2)^2-1$$\n\n(vertex form)\n\nWhich form would you use if you want to find the following features of the graph of $$f$$? Be prepared to explain your reasoning.\n\n1. the $$x$$-intercepts\n2. the vertex\n3. the $$y$$-intercept\n\n### 16.2: Sharing a Vertex\n\nHere are two equations that define\u00a0quadratic functions.\n\n$$\\displaystyle p(x) =\\text-(x-4)^2 + 10\\\\ q(x) = \\frac12(x-4)^2 + 10$$\n\n1. The graph of $$p$$ passes through $$(0,\\text-6)$$ and $$(4,10)$$, as shown\u00a0on the coordinate plane.\n\nFind the coordinates of another point on the graph of $$p$$. Explain or show your reasoning. Then, use the points to sketch and label the graph.\n\n2. On the same coordinate plane, identify the vertex and two other points that are on the graph of $$q$$. Explain or show your reasoning. Sketch and label the graph of $$q$$.\n3. Priya says, \"Once I know the vertex is $$(4,10)$$, I can find out, without graphing, whether the vertex is the maximum or the minimum of function $$p$$. I would just compare the coordinates of the vertex with the coordinates of a point on either side of it.\"\n\nComplete the\u00a0table and then explain how Priya might have reasoned about whether the vertex is the minimum or maximum.\n\n $$x$$ $$p(x)$$ 3 4 5 10\n\n1. Write a the equation for a quadratic function whose graph has the vertex at $$(2,3)$$ and contains the point $$(0,\\text-5)$$.\n2. Sketch a graph of your function.\n\n### 16.3: Card Sort: Matching Equations with Graphs\n\nYour teacher will give you a set of cards. Each card contains an equation or a graph that represents a quadratic function. Take turns matching each equation to a graph that represents the same function.\n\n\u2022 For each pair of cards that you match, explain to your partner how you know they belong together.\n\u2022 For each pair of cards that your partner matches, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.\n\u2022 Once all the cards are matched, record the equation, the label and a sketch of the corresponding graph, and write a brief note or explanation about how you knew they were a match.\n\nEquation:\n\nExplanation:\n\nEquation:\n\nExplanation:\n\nEquation:\n\nExplanation:\n\nEquation:\n\nExplanation:\n\nEquation:\n\nExplanation:\n\nEquation:\n\nExplanation:\n\n### Summary\n\nWe saw that vertex form is especially helpful for finding the vertex of a graph of a quadratic function. For example, we can tell that the function $$p$$ given by $$p(x) = (x-3)^2 + 1$$ has a vertex at $$(3,1)$$.\n\nWe also noticed that, when the squared expression $$(x-3)^2$$ has a positive coefficient, the graph opens upward. This means that the vertex $$(3,1)$$ represents the minimum function value.\n\nBut why does the function $$p$$ take on its minimum value when $$x$$ is 3?\n\nHere is one way to explain it: When $$x=3$$,\u00a0the squared term $$(x-3)^2$$ equals 0, as $$(3-3)^2 =0^2=0$$. When $$x$$ is any other value besides 3, the squared term\u00a0$$(x-3)^2$$ is a positive number\u00a0greater than 0. (Squaring any number results in a positive number.)\u00a0This means that the output when $$x \\neq 3$$ will always be greater than the output when $$x=3$$, so the function\u00a0$$p$$\u00a0has a minimum value at $$x=3$$.\n\nThis table shows some values of the function for some values of $$x$$. Notice that the output is the least when $$x=3$$ and it increases both as $$x$$ increases and as it decreases.\n\n $$x$$ $$(x-3)^2+1$$ 0 1 2 3 4 5 6 10 5 2 1 2 5 10\n\nThe squared term sometimes has a negative coefficient, for instance: $$h(x)= \\text-2(x+4)^2$$. The $$x$$ value that makes $$(x+4)^2$$ equal 0 is -4, because $$(\\text-4+4)^2=0^2=0$$. Any other $$x$$ value makes $$(x+4)^2$$ greater than 0. But when $$(x+4)^2$$ is multiplied by a negative number (-2), the resulting expression,\u00a0$$\\text-2(x+4)^2$$, ends up being negative. This means that the output when $$x \\neq \\text-4$$ will always be less than the output when $$x=\\text-4$$, so the function\u00a0$$h$$\u00a0has\u00a0its maximum value when $$x =\\text-4$$.\n\nRemember that we can find the $$y$$-intercept of the graph representing any function we have seen. The $$y$$-coordinate of the\u00a0$$y$$-intercept is the value of the function when $$x = 0$$. If $$g$$ is defined by $$g(x)=(x+1)^2-5$$, then the $$y$$-intercept is $$(0,\\text-4)$$ because\u00a0$$g(0)=(0+1)^2 -5=\\text-4$$. Its vertex is at $$(\\text-1,\\text-5)$$.\u00a0 Another point on the graph with the same $$y$$-coordinate is located the same horizontal distance from the vertex but on the other side.\n\n### Glossary Entries\n\n\u2022 vertex form (of a quadratic expression)\n\nThe vertex form of a quadratic expression in $$x$$ is\u00a0$$a(x-h)^2 + k$$, where $$a$$, $$h$$, and $$k$$ are constants, and $$a$$ is not 0.","date":"2023-02-08 11:10:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7777470350265503, \"perplexity\": 277.60081822928515}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764500758.20\/warc\/CC-MAIN-20230208092053-20230208122053-00485.warc.gz\"}"} | null | null |
\section{Background}
\subsection{JOREK solver}
The \verb|JOREK| \cite{Huysmans_2007} solver always involves LU factorisations, either to solve the entire matrix directly (generally used when the simulation is run axisymmetrically) or as a preconditioner before using the iterative solver GMRes. We focus on the latter case.
In the preconditioner, the blocks of toroidal harmonics are assumed to be decoupled from one another (i.e.\ off-diagonal blocks are not considered in the preconditioner), which greatly reduces the memory requirements and the runtime. This approach comes to its limits when the system is very nonlinear, as the preconditioner will be inaccurate and many GMRes iterations may be needed to converge to the solution.
\subsection{LU factorisation of sparse matrix}
The blocks of harmonics are sparse matrices (i.e.\ a small percentage of entries is non-zero) and naive algorithms used for dense matrices should thus not be applied here. Indeed, these would lead to a substantial fill-in: the combined storage cost for the $L$ and $U$ factors would greatly exceed that of the original matrix. The \verb|JOREK| code makes use of either \verb|PaStiX| \cite{henon:inria-00346017} or \verb|MUMPS| \cite{doi:10.1137/S0895479899358194}, both of which are able to efficiently handle sparse matrices.
In the following, we will briefly present the main steps of these solvers, without going into the details. More information can be found e.g. in \cite{mary:tel-01929478}.
In the \textbf{analysis} phase, a graph ordering tool is called, generally \verb|SCOTCH| or \verb|METIS|. These represent the matrix as a graph and an elimination tree is constructed to reduce fill-in. This tree is traversed as each node is eliminated and its contribution to nodes further up the tree are computed (corresponding to fill-in). This corresponds to an update of nodes up the tree which can be done either immediately after elimination (fan-out or right-looking), as late as possible before the elimination of the node further up the tree (fan-in or left-looking) or it can be carried up the tree, accumulating further updates from other nodes in so called fronts (fan-out or right-looking)\footnote{
PaStiX is using a fan-in strategy in the communication. In the sense that $A\cdot B$ are accumulated locally on the origin node, and then sent to the remote node when all contributions are applied, such that the remote node can perform the operation $C=C+sum(A\cdot B)$. Except for this specific case in distributed, all the algorithm is fan-out/right-looking oriented to get more parallelism, even for the accumulation of the remote contribution. MUMPS, in contrast, based on the multi-frontal approach, can be considered as fan-in/left-looking oriented algorithm.}.
In the \textbf{factorisation} phase, the factors of the $L$ and $U$ matrices are computed, following the elimination tree obtained in the analysis phase. This is typically the most CPU intensive step, although the factorisation does not need to be repeated for every time step in a typical \verb|JOREK| simulation, since the preconditioning matrix is reused as long as reasonable convergence is obtained in GMRes.
In the \textbf{solve} phase, the solution is computed using standard forward and back-substitution methods. A solve also has to be repeated for every iteration in GMRes.
\subsection{BLR compression}
Even with the use of graph ordering tools, the fill-in can be substantial and the required memory storage for the factors of $L$ and $U$ can be very high. As a possible solution to this problem, the factors could be compressed using Block-Low-Rank (BLR) compression.
Let $A$ be a matrix of size $m\times n$. Let $k_\epsilon$ be the approximated numerical rank of $A$ at accuracy $\epsilon$. $A$ is a low-rank matrix if there exist three matrices $U$ of size $m\times k_\epsilon$ , $V$ of size $n\times k_\epsilon$ and $E$ of size $m\times n$ such that :
\begin{equation}
A= U \cdot V^T +E,
\end{equation}
where $||E||_2 \leq \epsilon$ and $k_\epsilon \cdot (m + n) < mn$. The last condition implies lower dimensionality and thus also lower storage costs for combined U and V than for original A, provided the matrix is sufficiently dense. The basic idea is to represent the matrix as the product of two skinny matrices. Note that lossless compression is possible as the accuracy $\epsilon$ can be set to $0$.
Besides possible memory gains, using BLR compression can in principle also reduce the runtime, as basic matrix operations are accelerated. Indeed, for $m=n$, matrix-matrix products go from $2n^3$ (for dense matrices) to $2kn(n+2k)$ operations, while a triangular solve goes from $n^3$ to $3kn^2$ operations. The complexity is thus reduced for $k<n/2$ and $k<n/3$ respectively.
The $L$ and $U$ matrices are generally not low-rank, but individual sub-matrices may be efficiently compressed. There is no general admissibility criterion to determine if a block should be stored in low-rank. This depends on the solver used, see \cite{amestoy:hal-01505070} (\verb|MUMPS|) and \cite{pichon:hal-01824275} (\verb|PaStiX|) for more details on the implementation of BLR compression in these solvers.
\subsubsection{Runtime performance in JOREK}
\label{sec_runtime_perf}
The runtime of the \verb|JOREK| solver will be impacted by the use of BLR compression. Before testing this in detail in the rest of this report, we layout how the performance is affected by BLR compression.
The \textbf{analysis} step takes a longer time due to the additional search for suitable matrices to compress. However, the impact on the total runtime is negligible, as the analysis only has to be done once at the simulation (re-)start.
Two opposing effects will play a role in the \textbf{factorisation} step: the compression of the LU blocks leads to an overhead but the compressed matrices lead to faster basic matrix operations (see above). In practice, the time for factorisation seems to be generally increased with BLR in our tests, although higher values of $\epsilon$ can reduce or even revert this effect.
The same conclusions hold for the \textbf{solve} step, as the (de-)compression induces an overhead, which the faster matrix operations can cancel\footnote{This overhead can be avoided and will be avoided in both PaStiX and MUMPS in the future.}
The runtime performance of the \textbf{GMRes} iterative solver is set by the number of iterations needed to reach convergence and the time needed for a basic solve (see step above), as it is repeated for every GMRes iteration. As for the factorisation and solve steps above, this means that the use of BLR causes an overhead but larger values of $\epsilon$ will not always be beneficial here. Indeed, they may reduce the number of matrix operations and thereby reduce runtime, but a too large value of $\epsilon$ deteriorates the accuracy of the preconditioner and leads to worse convergence in GMRes, increasing the number of iterations necessary to reach convergence. Taken to the extreme, this can stop the simulation, as the GMRes iterative solver is given a maximum number of steps to reach convergence in \verb|JOREK|.
\section{BLR compression in MUMPS}
\subsection{Resolution scan}
\label{sec_MUMPS_resol}
\subsubsection{Basic setup}
All simulations in this report were carried out on a Linux cluster, where each compute node is equipped with 2x Intel Xeon Gold 6130 CPUs with 16 cores and 2.1 GHz base clock speed, AVX 512, ``Skylake'' architecture, 22 MB L3 cache, and fast Omnipath interconnect.
We first perform basic resistive ballooning mode simulations (a test case called \verb|inxflow| in \verb|JOREK|) with 2 toroidal harmonics ($n=0,6$) and 4 toroidal planes. The simulations are restarted during the linear growth phase for 10 time steps, with the factorisation enforced to be carried out in every time step. The iterative solver GMRes is given a tolerance of $10^{-7}$ and $50$ maximal iterations to reach convergence. Each simulation is run on one compute node with 2 MPI tasks, with 2 OpenMP threads each.
The resolution was varied from \verb|(n_flux, n_tht) = (16, 20)| up to \\\verb|(n_flux, n_tht) = (128, 160)|, where \verb|n_flux| denotes the radial and \verb|n_tht| the poloidal number of grid points, in 6 steps of approximate factors $(\sqrt{2}, \sqrt{2})$. For each resolution, the runtime performance and memory consumption of the solver MUMPS are investigated, with and without BLR compression, and with different values of the BLR tolerance: $\epsilon = 0, 10^{-16}, 10^{-12}, 10^{-8}, 10^{-4}$.
\subsubsection{Memory consumption}
The total memory consumption in MB of the MUMPS solver depending on resolution is shown in Tab~\ref{tab_memory} and Fig.~\ref{fig_resol_memory}. Note that the blocks corresponding to nonzero toroidal harmonics have twice the dimensionality of the $n=0$ block, such that $1/5$ of the total memory indicated is used in the $n=0$ block and $4/5$ in the $n=6$ block.
Significant memory gains can be made using BLR compression. These gains are exacerbated for large values of $\epsilon$ (as the matrices can be compressed more effectively) and for large problem sizes (as there are more opportunities for compression).
\begin{table}[!ht]
\begin{tabular}{c|c c c c c c}%
\verb|n_flux,n_tht| & No BLR & $\epsilon=0$ & $\epsilon=10^{-16}$ & $\epsilon=10^{-12}$ & $\epsilon=10^{-8}$ & $\epsilon=10^{-4}$ \\[1pt]\hlin
$16,20$ & $ 799$ & $ 822$ & $ 815$ & $ 813$ & $ 806$ & $ 733$\\
$22,28$ & $ 1762$ & $ 1751$ & $ 1742$ & $ 1740$ & $ 1690$ & $ 1466$\\
$32,40$ & $ 4365$ & $ 4023$ & $ 4001$ & $ 3956$ & $ 3743$ & $ 3118$\\
$44,56$ & $ 9047$ & $ 8151$ & $ 8086$ & $ 7886$ & $ 7286$ & $ 5928$\\
$64,80$ & $ 20891$ & $ 18603$ & $ 18193$ & $ 17375$ & $ 15787$ & $ 12644$\\
$88,112$ & $ 45628$ & $ 40353$ & $ 38148$ & $ 36015$ & $ 32494$ & $ 26119$\\
$128,160$ & $ 101035$ & $ 88741$ & $ 81299$ & $ 75508$ & $ 67516$ & $ 53717$
\end{tabular}
\caption{Memory consumption in the MUMPS resolution scan in Megabytes}
\label{tab_memory}
\end{table}
\begin{figure}[!ht]
\includegraphics[scale=1]{memory_relative.eps}
\caption{Memory gains from BLR compression}
\label{fig_resol_memory}
\end{figure}
Note, however, that large values of $\epsilon$ can significantly deteriorate the quality of the preconditioning, leading to an increased number of iterations for GMRes (see Sec.~\ref{sec_runtime_perf}). Depending on the required tolerance for GMRes and the number of iterations, this can even prevent convergence, which was the case in the highest resolution simulations with $\epsilon = 10^{-4}$.
\subsubsection{Runtime}
The runtimes listed below are averages over the simulations' 10 time steps. The activation of the Block-Low-Rank feature leads to an increase in the analysis phase's runtime by a factor of approximately $2$. It was however left out in the following, as the analysis only has to be performed once and its impact on the performance is thus negligible.
\begin{table}[!ht]
\begin{tabular}{c|c c c c c c}%
\verb|n_flux,n_tht| & No BLR & $\epsilon=0$ & $\epsilon=10^{-16}$ & $\epsilon=10^{-12}$ & $\epsilon=10^{-8}$ & $\epsilon=10^{-4}$ \\[1pt]\hlin
$16,20$ & $ 1.1$ & $ 1.5$ & $ 1.5$ & $ 1.4$ & $ 1.4$ & $ 1.2$\\
$22,28$ & $ 2.6$ & $ 3.7$ & $ 3.7$ & $ 3.6$ & $ 3.5$ & $ 2.7$\\
$32,40$ & $ 7.2$ & $ 10.3$ & $ 10.0$ & $ 10.1$ & $ 9.0$ & $ 7.0$\\
$44,56$ & $ 16.6$ & $ 23.5$ & $ 23.7$ & $ 22.5$ & $ 19.3$ & $ 14.2$\\
$64,80$ & $ 44.1$ & $ 64.6$ & $ 62.5$ & $ 56.5$ & $ 46.1$ & $ 32.9$\\
$88,112$ & $ 119.5$ & $ 168.6$ & $ 144.6$ & $ 126.6$ & $ 101.9$ & $ 74.1$\\
$128,160$ & $ 330.1$ & $ 496.0$ & $ 388.6$ & $ 319.6$ & $ 253.4$ & $ 164.9$
\end{tabular}
\caption{Runtime for factorisation step (in s)}
\label{tab_resol_facto}
\end{table}
\begin{table}[!ht]
\begin{tabular}{c|c c c c c c}%
\verb|n_flux,n_tht| & No BLR & $\epsilon=0$ & $\epsilon=10^{-16}$ & $\epsilon=10^{-12}$ & $\epsilon=10^{-8}$ & $\epsilon=10^{-4}$ \\[1pt]\hlin
$16,20$ & $ 2.0$ & $ 2.7$ & $ 2.7$ & $ 2.6$ & $ 2.6$ & $ 2.5$\\
$22,28$ & $ 4.5$ & $ 6.1$ & $ 6.1$ & $ 6.1$ & $ 5.8$ & $ 5.3$\\
$32,40$ & $ 11.1$ & $ 15.7$ & $ 15.2$ & $ 15.5$ & $ 14.2$ & $ 13.2$\\
$44,56$ & $ 24.4$ & $ 34.0$ & $ 34.3$ & $ 33.3$ & $ 29.6$ & $ 25.8$\\
$64,80$ & $ 59.9$ & $ 88.2$ & $ 86.1$ & $ 80.1$ & $ 84.0$ & $ \infty$\\
$88,112$ & $ 151.7$ & $ 217.1$ & $ 191.6$ & $ 175.0$ & $ 151.0$ & $ \infty$\\
$128,160$ & $ 410.8$ & $ 626.2$ & $ 499.2$ & $ 442.7$ & $ 428.8$ & $ \infty$
\end{tabular}
\caption{Runtime for entire time step (in s)}
\label{tab_resol_iter}
\end{table}
The runtime for the factorisation step of the MUMPS solver is shown in Tab.~\ref{tab_resol_facto} and Fig.~\ref{fig_resol_runtime_facto}. Here, the use of BLR compression can be detrimental (overhead caused by the compression of LU factors) or beneficial (speedup of matrix operations). Which of these two effects dominates depends on the size of the problem and the choice of $\epsilon$.
The runtime for the solution step of the MUMPS solver is shown in Fig.~\ref{fig_resol_runtime_solve}. The solution step generally incurs an overhead from compression, which can however be mitigated at high resolutions, where the decrease in the number of operations can compensate for the overhead.
The runtime for the GMRes step of the JOREK solver is shown in Fig.~\ref{fig_resol_runtime_gmres}, where the strong spike for $\epsilon = 10^{-4}$ reflects the loss of convergence. A slowdown is observed at all resolutions as the same overhead mentioned in the solution step comes into play here, as well as larger number of iterations in the GMRes solver when $\epsilon$ is too high and the preconditioner is too inaccurate.
The reduction of the time for the solution phase in Fig.~\ref{fig_resol_runtime_solve} at the highest resolution leaves open the possibility that this phase is actually accelerated at even larger resolutions. This could then also possibly lead to a reduction in the time for GMRes, if it was possible to compensate for both the (de-)compression overhead and the higher number of iterations caused by the preconditioner's inaccuracy.
\begin{figure}[!ht]
\includegraphics[scale=1]{time_facto_relative.eps}
\caption{Runtime for factorisation step}
\label{fig_resol_runtime_facto}
\end{figure}
\begin{figure}[!ht]
\includegraphics[scale=1]{time_solve_relative.eps}
\caption{Runtime for solution step}
\label{fig_resol_runtime_solve}
\end{figure}
\begin{figure}[!ht]
\includegraphics[scale=1]{time_gmres_relative.eps}
\caption{Runtime for GMRes}
\label{fig_resol_runtime_gmres}
\end{figure}
\begin{figure}[!ht]
\includegraphics[scale=1]{time_iter_relative.eps}
\caption{Average runtime per time step}
\label{fig_resol_runtime_iter}
\end{figure}
The average total runtime per time step shown in Tab.~\ref{tab_resol_iter} confirms the generally increased runtime when using BLR compression. For large values of $\epsilon$, this increase is quite modest. In our simulations at highest resolution, the optimal choice is $\epsilon = 10^{-8}$, where the increase in runtime is minimal (no BLR: $411$ s, $\epsilon = 10^{-8}$: $429$ s) while the memory requirements are substantially lower (no BLR: $101$ GB, $\epsilon = 10^{-8}$: $68$ GB). The high performance observed here is due to the reduced runtime for factorisation, largely offsetting the increase in runtime from GMRes. The peculiar spike in the runtime for $\epsilon = 10^{-8}$ and a resolution of \verb|(n_flux, n_tht) = (64, 80)| is due to an increase in the number of iterations of GMRes for $3$ out of the $10$ time steps, suggesting that fluctuations in the preconditioner's accuracy can come into play for larger values of $\epsilon$.
However, the \verb|JOREK| solver generally keeps the preconditioning LU factors for several timesteps, until it does not satisfactorily precondition the matrix anymore, i.e. the number of GMRes iterations in the previous time step has passed a certain threshold. As a consequence, for many time steps only the solve step and GMRes have to be performed. For too large $\epsilon$, this could increase the runtime substantially, as GMRes is repeated several times for each LU factorisation. Additionally, the LU factorisation might need to be repeated after a smaller number of time steps. However, the runtimes for the solution phase and GMRes phases might be reduced at very high resolutions due to the reduced number of operations, as Figs.~\ref{fig_resol_runtime_solve} and \ref{fig_resol_runtime_gmres} seem to suggest.
The accuracy $\epsilon$ should be chosen as large as possible to allow for significant memory gains and keep the runtime as low as possible, without being too high, such that GMRes convergence is not too greatly impaired. This choice of $\epsilon$ will depend on the exact problem at hand as well as the setup of the GMRes solver.
\subsection{More realistic simulation}
\label{sec_spi}
The goal of this section is to ascertain the benefits of BLR compression during the nonlinear phase, where the preconditioning matrix is not as effective and the convergence of GMRes is thus even more critical, as well as its general usefulness in a typical \verb|JOREK| simulation.
We thus turn to simulations of shattered pellet injections (SPI), running for a larger number of time steps ($7330$) through linear and nonlinear phases, with 6 toroidal harmonics ($n=0...5$) and 32 toroidal planes. The resolution of all simulations is \verb|(n_flux,n_tht)| = (56, 138). The maximal number of iterations of the GMRes iterative solver was set to $100$ while its tolerance was set to $10^{-6}$ and $10^{-7}$, to investigate its effects on the convergence properties when using compression. Indeed, we force the recalculation of the preconditioner (i.e. a factorisation) every time the number of steps in GMRes exceeds $20$, such that a lower GMRes tolerance will lead to more factorisations.
We again compare simulations with different MUMPS setups: without BLR and with BLR, using different values of $\epsilon$. For the higher GMRes tolerance value of $10^{-6}$, the simulations with $\epsilon = 10^{-8}, 10^{-4}$ did not yield satisfactory convergence in GMRes, the former in the linear and the latter in the nonlinear phase. We thus examine only the values $\epsilon = 0, 10^{-16}, 10^{-12}, 10^{-10}$.
The total memory consumption of the MUMPS solver in GB amounts to
\begin{itemize}
\item No BLR: \hspace{30pt} $166.0$ ($100\%$)
\item BLR, $\epsilon = 0$: \qquad $147.1$ ($88.6\%$)
\item BLR, $\epsilon = 10^{-16}$: $139.8$ ($84.2\%$)
\item BLR, $\epsilon = 10^{-12}$: $132.6$ ($79.9\%$)
\item BLR, $\epsilon = 10^{-10}$: $126.8$ ($76.4\%$)
\end{itemize}
The runtime performance data of all simulations is given in Tab.~\ref{tab_data_spi_runs}. Note that the runtime for the factorisation and solve steps is independent of the GMRes tolerance value.
\begin{table}[h!]
\centering
\small
\begin{tabular}{l l | c c c c c}
GMRes tol. & & No BLR & $\epsilon=0$ & $\epsilon=10^{-16}$ & $\epsilon=10^{-12}$ & $\epsilon=10^{-10}$\\[1pt]\hline
& Nbr. of factorisations & $786$ & $716$ & $950$ & $805$ & $816$ \\
$10^{-6}$ & Avg. time per iter. (s) & $47.0$ & $65.7$ & $68.2$ & $63.3$ & $61.9$ \\
& Avg. nbr. of GMRes iter. & $14.5$ & $14.5$ & $15.6$ & $14.7$ & $14.7$ \\
& Avg. time per GMRes (s) & $13.5$ & $28.8$ & $28.9$ & $26.4$ & $25.0$ \\[1pt]\hline
& Nbr. of factorisations & $3145$ & $3033$ & $3044$ & $3026$ & \\
$10^{-7}$ & Avg. time per iter. (s) & $76.6$ & $109.4$ & $104.9$ & $99.7$ & \\
& Avg. nbr. of GMRes iter. & $21.2$ & $21.0$ & $21.3$ & $20.8$ & \\
& Avg. time per GMRes (s) & $18.8$ & $40.9$ & $38.8$ & $35.8$ & \\[1pt]\hline
& Avg. time per facto. (s) & $50.7$ & $67.9$ & $63.1$ & $58.3$ & $55.2$ \\
& Avg. time per solve (s) & $0.46$ & $1.78$ & $1.63$ & $1.58$ & $1.45$
\end{tabular}
\caption{Performance of BLR compression in SPI simulations}
\label{tab_data_spi_runs}
\end{table}
\subsubsection{Runtime performance for a GMRes tolerance of $10^{-6}$}
In the first set of simulations with a GMRes tolerance of $10^{-6}$, the number of factorisations varied somewhat unexpectedly, as can be seen in Tab.~ \ref{tab_data_spi_runs}. Indeed, the number of factorisations rose substantially for $\epsilon = 10^{-16}$, largely exceeding the number of factorisations for $\epsilon = 10^{-12}$ and $\epsilon = 10^{-10}$. Indeed, the frequency (number of occurences) for each occuring number of GMRes iterations shown in Fig.~\ref{fig_spi_histo_gmres} reveals an increased number of GMRes iterations between $21$ and $26$ for $\epsilon = 10^{-16}$. This in turns leads to a higher average time per iteration (Tab.~\ref{tab_data_spi_runs}).
\begin{figure}[ht!]
\includegraphics[scale=1]{histo_nb_iter_gmres_report.eps}
\caption{Histogram of GMRes iteration numbers for a tolerance of $10^{-6}$}
\label{fig_spi_histo_gmres}
\end{figure}
This discrepancy can be explained by the fact that a GMRes tolerance of $10^{-6}$ is too high, possibly leading to different physical results. Note that in this simulation, most factorisations take place during the physically violent thermal quench of the plasma core. Even small variations in the duration or intensity of this process can lead to the observed differences. Indeed, a longer thermal quench is observed for $\epsilon = 10^{-16}$, as shown in Fig.~\ref{fig_spi_core_temp} (at $t \sim 1300-1500$). Note that this does not directly explain why the number of factorisations increases for $\epsilon = 10^{-16}$, but it does make this fact less surprising. The reduction of the relative differences in the number of factorisations at a higher GMRes tolerance value of $10^{-7}$ (see next section) seems to confirm that the original tolerance of $10^{-6}$ is too high.
\begin{figure}[ht!]
\includegraphics[scale=1]{core_temps.eps}
\caption{Core temperature evolution for GMRes tolerances of $10^{-6}$ and $10^{-7}$.}
\label{fig_spi_core_temp}
\end{figure}
At all observed converging values of $\epsilon$, the runtime is substantially increased, as shown by the average runtime per iteration in Tab.~\ref{tab_data_spi_runs}. This is only partly due to longer and more numerous factorisations, as the biggest contributing factor to the increased runtime is the increased runtime in GMRes, which originates from an increased runtime per solve, not from an increased number of iterations, as can be seen from Tab.~\ref{tab_data_spi_runs}. At the highest converging value of $\epsilon = 10^{-10}$, this leads to a $\sim32\%$ runtime increase for a memory gain of $\sim24\%$.
\subsubsection{Runtime performance for a GMRes tolerance of $10^{-7}$}
In the case of a GMRes tolerance of $10^{-7}$, the number of factorisations seems to be roughly constant for all values of $\epsilon$, and it is surprisingly slightly reduced compared to the simulations without BLR compression. The latter is probably again due to small differences in the physical results, as the former indicates that the accuracy of the preconditioner is determined here by the linearity of the system. Indeed, the average number of GMRes iterations and the number of factorisations does not increase for large values of $\epsilon$ up to $10^{-12}$. For $\epsilon = 10^{-8}$, this does not hold anymore, as the GMRes solver did not reach convergence in this case.
The constancy of the number of factorisations in this case means that the time per iteration is now set by the time for factorisation, solve and GMRes, all of which incur increased runtimes from (de-)compression. However, these are partially offset by the use of larger values of $\epsilon$. Even at the high number of factorisations for this GMRes tolerance value, the main contributor to the increased runtime is the runtime in GMRes, although the increased runtime for factorisation now plays a larger role compared to the higher GMRes tolerance value. At the highest observed converging value of $\epsilon = 10^{-12}$, a memory gain of $\sim20\%$ for a $\sim30\%$ runtime increase.
The above analysis suggests that the optimal $\epsilon$ value should be found somewhere between $10^{-12}$ and $10^{-8}$, which could unfortunately not be investigated due to time constraints for this study. Logically, the optimal $\epsilon$ value should be such that the inaccuracy in the preconditioner induced by the compression accuracy is similar to that inherent in our assumption of decoupled harmonics in the preconditioner. This should ensure that the time for factorisation, solve and GMRes is decreased without incurring too great an increase in the preconditioner's inaccuracy.
Moreover, it might even be worth going beyond this limit by increasing the maximal number of GMRes iterations when using BLR compression, allowing for still larger values of $\epsilon$ which further reduce memory consumption and possibly offset the runtime increase caused by the larger number of GMRes iterations.
\newpage
\section{PaStiX version 6.x}
The new version 6.0 of the \verb|PaStiX| sparse matrix solver is still in development, but seems to be the way forward for \verb|JOREK|, as it brings new features such as Block-Low-Rank compression~\footnote{For the tests shown here, a development version equivalent to release 6.0.2 with some additional corrections was used. The hash key of the respective commit in the git repository is 17f18ce6e87de504580c27d52e5672311c413d21.}. It is however not yet MPI-parallelised, which will be implemented in version 6.1, so its current use in \verb|JOREK| is restricted to cases with one MPI task per \verb|PaStiX| instance (i.e. one task per toroidal harmonic). Apart from this, the solver is already fully functional in \verb|JOREK|, as an updated interface has been implemented in \verb|mod_poiss.f90| (equilibrium), \verb|solve_pastix_all.f90| (direct LU factorisation of entire matrix, generally used in axisymmetric runs), \verb|solve_mat_n.f90| and \verb|gmres_precondition.f90| (LU factorisation as preconditioner followed by GMRes, generally used for multiple harmonics).
The implementation can handle multiple degrees of freedom to make analysis and factorisation more efficient (switched on through the flag \verb|USE_BLOCK| in \verb|JOREK|), although the underlying matrix structure and the analysis results currently have to be expanded during the analysis phase, as the thereafter invoked calls do not yet support multiple degrees of freedom. As this expansion is as of now only implemented in the analysis phase, the analysis is currently being repeated for every time step when using \verb|PaStiX 6| with multiple degrees of freedom in \verb|JOREK|\footnote{This overhead can be avoided in the future.}. Once the \verb|PaStiX| developers have remedied to these problems, the expansion and repeated analysis should be removed in the \verb|JOREK| implementation, giving a further boost to the runtime performance.
\subsection{Benchmarking}
In the following, we present small benchmark tests of \verb|PaStiX 6| with the previously generally used \verb|PaStiX| 5.2.1 ("Release 4492"). We again use a peeling-ballooning scenario (\verb|inxflow|): where the simulation is run with 2 toroidal harmonics ($n=0,6$) in the phase of linear growth (Sec.~\ref{sec_pastix6_ntor3}).
The time evolution is computed with and without the \verb|USE_BLOCK| feature, which also the evaluation of its usefulness in different versions of \verb|PaStiX|. Furthermore, in Sec.~\ref{sec_pastix6_openmp_scan}, the scaling in the number of OpenMP threads is checked for the different \verb|PaStiX| versions by running 10 time steps in the linear growth phase with different numbers of OpenMP threads. Finally, 10 time steps are again rerun in the linear growth phase for various spatial resolutions in Sec.~\ref{sec_pastix6_resolution_scan} to check how the new \verb|PaStiX| version scales with the problem size. The basic setup has a spatial resolution of \verb|(n_flux,n_tht) = (32,40)| and the number of OpenMP threads is set to $2$.
\subsubsection{Basic simulation}
\label{sec_pastix6_ntor3}
The simulation is run for 90 time steps in the linear growth phase to obtain a first idea of the typical runtime performance but also to later assess if the OpenMP and resolution scans which are run over only 10 time steps yield the correct results. The runtimes measured in this first benchmark are given in Tab.~\ref{tab_benchmark_basic}. They are all averaged except the analysis for single-dof which is performed only once. Both single and multiple degrees of freedom (dof) are investigated (\verb|USE_BLOCK| feature).
\begin{table}[h!]
\centering
\begin{tabular}{l | c c c c}
\multirow{2}{*}{Runtime (in s)} & \multicolumn{2}{c}{single dof} & \multicolumn{2}{c}{multiple dofs} \\
& PaStiX 5.2.1 & PaStiX 6.0.2 & Release 5.2.1 & PaStiX 6.0.2 \\[1pt]\hline
Analysis & $26.1$ & $4.9$ & $0.14$ & $0.47$\\
Factorisation & $10.9$ & $9.5$ & $7.81$ & $7.77$ \\
Solve & $0.20$ & $0.15$ & $0.17$ & $0.15$ \\
GMRes & $1.96$ & $1.56$ & $1.79$ & $1.55$ \\
\end{tabular}
\caption{Basic benchmark of PaStiX 6}
\label{tab_benchmark_basic}
\end{table}
Considering a single degree of freedom, the time for analysis is subtantially reduced in \verb|PaStiX 6|, by a factor of $\sim 5$. However, the analysis phase only has to be repeated once per simulation (restart), such that this gain is appreciated but should not heavily influence the total runtime.
The reason why the use of multiple degrees of freedom seems to lead to a greater speed-up in the older \verb|PaStiX| version is that the time listed under Analysis also includes the conversion of the matrix to an input usable by \verb|PaStiX|. This conversion takes slightly longer for \verb|PaStiX 6| because the latter necessitates an additional new sparse matrix structure, whereas the matrix was directly passed to \verb|PaStiX| beforehand. This additional overhead could be reduced in the future by directly using the new sparse matrix structure in \verb|JOREK|'s \verb|distribute_harmonics| routine.
The time for factorisation is also reduced in this simple case for the new \verb|PaStiX| version. The difference shrinks when multiple degrees of freedom are used, such that the times for factorisation are very similar here. Indeed, multiple degrees of freedom are not yet implemented in the factorisation part of \verb|PaStiX 6|, such that the speed-up is reduced to the contribution from a more performant analysis.
Finally, the runtimes for solution and for GMRes are reduced in the new \verb|PaStiX| version, by approximately $25\%$ when a single degree of freedom is assumed. Here, the use of multiple degrees of freedom does not lead to a speed-up for \verb|PaStiX 6|, as it is not yet implemented in the solve part.
\subsubsection{OpenMP thread scan}
\label{sec_pastix6_openmp_scan}
In this scan, 10 time steps were performed during the linear growth phase and the factorisation was forced to take place at every time step. The number of OpenMP threads was varied in factors of $2$ from $2$ to $32$. The resulting runtimes for the factorisation and solve steps, as well as the GMRes solver, are shown in Fig.~\ref{fig_benchmark_openmp}. The analysis step was left out as it did not vary depending on the number of OpenMP threads, so the result from the previous basic benchmark stands (old version: $\sim 25$ s, new version: $\sim 5$ s).
\begin{figure}[ht!]
\includegraphics[scale=1]{times_benchmark_openmp.eps}
\caption{Runtimes in OpenMP scan for the single-dof case}
\label{fig_benchmark_openmp}
\end{figure}
Although the scaling in the number of OpenMP threads seems to be worse for the factorisation step in the new \verb|PaStiX| version, the difference is minimal even at $32$ threads, where the runtimes for factorisation are basically the same between the two \verb|PaStiX| versions. The difference also becomes smaller for the solve and GMRes steps, but the new version of \verb|PaStiX| was still always faster for these compared to the old version. This can also be seen in Fig.~\ref{fig_benchmark_openmp_speedup}, where the speed-up in \verb|PaStiX 6| has been computed by dividing the runtime of the older \verb|PaStiX| version by that of \verb|PaStiX 6|. The analysis phase speed-up is $\sim 5$ for all OpenMP configurations.
\begin{figure}[ht!]
\includegraphics[scale=1]{times_benchmark_openmp_relative.eps}
\caption{Speedup from PaStiX 6 in the OpenMP scan}
\label{fig_benchmark_openmp_speedup}
\end{figure}
\subsubsection{Resolution scan}
\label{sec_pastix6_resolution_scan}
The same 10 time steps were now performed at resolutions between \verb|(n_flux, n_tht)| = (16, 20) and (128, 160) in 6 steps of approximate size $(\sqrt{2}, \sqrt{2})$. Only $2$ OpenMP threads were used, and the factorisation was again forced to be repeated at every time step.
The speed-up in this scan can be seen in Fig.~\ref{fig_benchmark_resol_speedup}, showing that \verb|PaStiX 6| leads to a speed-up for all phases (Factorisation, Solution, GMRes) and all resolutions, with the notable exception of the solution and GMRes phases at the very highest resolutions. The analysis phase was not included in Fig.~\ref{fig_benchmark_resol_speedup}, as it stays constant around $\sim 500\%$ for all resolutions.
\begin{figure}[ht!]
\includegraphics[scale=1]{times_benchmark_resol_relative.eps}
\caption{Speedup from PaStiX 6 in the resolution scan}
\label{fig_benchmark_resol_speedup}
\end{figure}
The speed-up in the factorisation is most encouraging as it does not plummet at larger resolutions, which is positive considering how significant this part of the solver is (e.g. runtime for factorisation at the highest resolution employed here, \verb|PaStiX| Release 4492: $631$ s, \verb|PaStiX 6|: $470$ s). The solve and GMRes phases show a more mixed picture, as the speed-up seems to be lost when going to very high resolutions. Whether this trend continues at even higher resolutions remains to be investigated, and this trend could of course change during the further development of \verb|PaStiX 6|. Moreover, these phases are generally less critical for the total runtime (e.g. runtime for solution phase at the highest resolution employed here, \verb|PaStiX| Release 4492: $6.93$ s, \verb|PaStiX 6|: $11.9$ s). This prompts the need for a future benchmarking of \verb|PaStiX 6| on a realistic \verb|JOREK| simulation, where the effect on the total runtime could be meaningfully investigated.
\subsection{Preliminary results from BLR compression with PaStiX 6}
The new BLR compression feature in \verb|PaStiX 6| could not yet be thoroughly tested, as some convergence problems occured when using the memory-optimal settings. The following results are derived from a resolution scan with the same setup as in Sec.~\ref{sec_MUMPS_resol} but using the \verb|PaStiX 6| solver with the "just-in-time" setting, which optimises runtime gains instead of memory gains \cite{pichon:hal-01824275}.
The memory consumption, obtained from a \verb|PaStiX 6| diagnostic, is listed in Tab.~\ref{tab_memory_pastix6}. Note that the numbers may not all be accurate to the last digit, and can not necessarily be directly compared to those given in the MUMPS resolution scan. Nevertheless, they indicate that very good compression can be attained, even for small (albeit non-zero) values of $\epsilon$. This also seems to indicate that the accuracy of the BLR solver here cannot be directly compared to that from the \verb|MUMPS| solver, possibly due to an additional internal scaling in \verb|MUMPS|.
\begin{table}[!ht]
\begin{tabular}{c|c c c c c c}%
\verb|n_flux,n_tht| & No BLR & $\epsilon=0$ & $\epsilon=10^{-16}$ & $\epsilon=10^{-12}$ & $\epsilon=10^{-8}$ & $\epsilon=10^{-4}$ \\[1pt]\hlin
$16,20$ & $ 477$ & $ 476 $ & $ 473$ & $ 460$ & $ 385$ & $ 249$\\
$22,28$ & $ 1070$ & $ 1060$ & $ 1045$ & $ 968$ & $ 770$ & $ 470$\\
$32,40$ & $ 2579$ & $ 2549$ & $ 2393$ & $ 2112$ & $ 1681$ & $ 982$\\
$44,56$ & $ 5570$ & $ 5520$ & $ 4910$ & $ 4273$ & $ 3398$ & $ 1907$\\
$64,80$ & $ 13320$ & $ 13120$ & $ 10970$ & $ 9410$ & $ 7410$ & $ 4020$\\
$88,112$ & $ 27770$ & $ 27460$ & $ 21620$ & $ 18470$ & $ 14600$ & $ 8020$\\
$128,160$ & $ 65100$ & $ 64300$ & $ 47600$ & $ 40680$ & $ 31950$ & $ 17350$
\end{tabular}
\caption{Memory consumption in the PaStiX 6 resolution scan in Megabytes}
\label{tab_memory_pastix6}
\end{table}
The speed-up in the average time per iteration in Fig.~\ref{fig_resol_runtime_iter_pastix6} confirms that the $\epsilon$ values cannot be directly compared to those of the \verb|MUMPS| solver, as many more simulations could not reach convergence here (all runs with $\epsilon = 10^{-4}$ and almost all with $\epsilon = 10^{-8}$)\footnote{Note, that version 6.0.2 of PaStiX also had a bug, which will be resolved in 6.0.3, which might partially explain this behaviour}. However, it seems easier to obtain a speed-up here, as many simulations with $\epsilon = 10^{-12}, 10^{-16}$ demonstrate, especially at high resolutions.
\begin{figure}
\includegraphics[scale=1]{time_iter_relative_pastix6.eps}
\caption{Average runtime per time step}
\label{fig_resol_runtime_iter_pastix6}
\end{figure}
This speed-up can mostly be traced back to the speed-up in the factorisation step, shown in Fig.~\ref{fig_resol_runtime_facto_pastix6}. Note that the factorisation is forced to be repeated every time step in this scan, such that the factorisation will be the main contribution to the total time per iteration. This is not necessarily the case in a typical JOREK simulation, as Sec.~\ref{sec_spi} demonstrated.
\begin{figure}
\includegraphics[scale=1]{time_facto_relative_pastix6.eps}
\caption{Runtime for factorisation step}
\label{fig_resol_runtime_facto_pastix6}
\end{figure}
It is thus instructive to investigate the speed-up in the solve and GMRes steps, shown in Figs.~\ref{fig_resol_runtime_solve_pastix6} and \ref{fig_resol_runtime_gmres_pastix6}. The speed-up or slow-down of the solve phase depends on the value of $\epsilon$ employed, a higher resolution only mildly increases the speed-up for a given $\epsilon$. In comparison, the GMRes runtimes seem to always be longer, as a result of the worse preconditioner. At high resolutions however, the fact that the time for solve is reduced with BLR suggests that the runtime for GMRes could also be reduced in a realistic \verb|JOREK| simulation. Indeed, the number of GMRes iterations in Tab.~\ref{tab_data_spi_runs} showed that for small enough values of $\epsilon$ the accuracy of the preconditioner is primarily determined by the nonlinearity of the system, not by the BLR accuracy. In other words, the increased runtime for GMRes observed in Fig.~\ref{fig_resol_runtime_gmres_pastix6} merely reflects the highly increased number of iterations in GMRes, which were not observed in the realistic simulation (Sec.~\ref{sec_spi}) when using small enough values of $\epsilon$.
The results in this section are preliminary, as many aspects of BLR compression in \verb|PaStiX 6| remain to be investigated. However, they are already very encouraging as the memory consumption is greatly reduced and the factorisation step seems to enjoy a subtantial speed-up. Whether this speed-up can rival the slow-down in the GMRes phase in a realistic \verb|JOREK| simulation remains to be investigated.
\begin{figure}
\includegraphics[scale=1]{time_solve_relative_pastix6.eps}
\caption{Runtime for solution step}
\label{fig_resol_runtime_solve_pastix6}
\end{figure}
\begin{figure}
\includegraphics[scale=1]{time_gmres_relative_pastix6.eps}
\caption{Runtime for GMRes}
\label{fig_resol_runtime_gmres_pastix6}
\end{figure}
\section{Conclusions}
Interfaces in the JOREK MHD code have been updated for MUMPS and PaStiX in order to test block low rank compression offered by these solver libraries. First tests show promising trends. Further tests are necessary to fully evaluate the benefit in production simulations.
\section*{Acknowledgements}
First and foremost, the authors would like to thank Mathieu Faverge and the rest of the \verb|PaStiX 6| development team for their quick and helpful responses to the diverse problems we encountered during the implementation of the new \verb|PaStiX 6| interface in \verb|JOREK|.
The authors would also like to thank Guido Huijsmans for his instructive comments on the use of Block-Low-Rank compression in \verb|JOREK|.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,304 |
Michal Martyniuk Trio + Jakub Skowronski
March 30, 2018 / JazzLocal32.com
When Michal Martyniuk left Auckland for Poland last year, it was hot on the heels of a successful appearance at Java Jazz; the biggest Jazz festival in the world. It was always on the cards that Martyniuk's Auckland trio would fare well, as they are the epitome of an inventive, high energy unit and all of that is wrapped up in a very European sound.
While it was obvious to Kiwis and to the enthusiastic Java Jazz festival goers, I wondered how Martyniuk would be received in Europe. I have travelled there often and there are thousands of good Jazz musicians and many fine trios vying for attention. Jazz is valued there, especially in the northeast, and audiences are inclined to be very discriminating. I got my answer shortly after Martyniuk's arrival, as notifications of media events, club gigs, radio and TV interviews started appearing. He had broken through the clamour and received acclaim in his birthplace. His co-released warm as toast Jazz-soul-funk album 'After 'Ours' and his Jazz gigs, equally acclaimed.
The journey back to the country of his birth had been important for Martyniuk and he has returned with heightened confidence, exuding a sense that anything is possible. This was evidenced by the trio's live performance at the Lewis Eady showroom. Many New Zealand improvising bands have a laid back organic feel as that is generally our thing. In contrast, this band is tightly focussed, but without that in any way detracting from its appeal. The tunes by Martyniuk are melodic and often rhythmically complex. This is counterbalanced nicely by Samsom and McArthur who create contrast and interwoven texture. The first set was a mix of old and new tunes. His older tunes like The Awakening and New Beginning, familiar in the same way standards are – always pleasing, always yielding up something fresh. His more recent compositions a mix of burners and ballads.
The Lewis Eady gig was augmented by the addition of visiting Polish saxophonist Jakub Skowronski. Skowronski has a beautiful even tone on tenor and like Samsom and McArthur, he's the perfect foil for Martyniuk. While he made it all look effortless, his solos took us deep inside the music. These guys were made to play together and I hope they remain a unit. They have a lot more to tell us yet and with any luck, we will get to enjoy the continuing story as it unfolds. Those who wish to be part of this journey can contribute via a recently set up 'Kickstarter' campaign following this link. There was some really exciting new material recorded in Poland over the last year and the Kickstarter campaign is about getting that released into the world. No one ever regretted supporting great music like this.
Michal Martyniuk (piano, compositions, leader), Cameron McArthur (upright bass), Ron Samsom (drums, percussion) + Jakub Skowronski (tenor). You can follow this band and order albums from Empire Agency Co. Bands / Michal Martyniuk Trio
Concerts - visiting Musicians, Lewis Eadys, New Zealand Jazz Gigs, Piano Jazz
Auckland Jazz, Cameron McArthur, Jakub Skowronski, Lewis Eadys, Michal Martyniuk, Piano Jazz, Ron Samsom
← Brian Smith Quintet
Emerging Artists – Orr / Fritsch → | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,441 |
\section{Introduction}
\label{sec:intro}
Moment-angle complexes have attracted a great deal of interest recently
because they are a nexus for important problems arising in algebraic topology,
algebraic geometry, combinatorics, complex geometry and commutative
algebra. They are best described as a special case of the polyhedral
product functor, popularized in~\cite{BBCG} as a generalization of
moment-angle complexes and $K$-powers~\cite{BP2}, which were in
turn generalizations of moment-angle manifolds~\cite{DJ}.
Let $K$ be a simplicial complex on $m$ vertices. For $1\leq i\leq m$,
let $(X_{i},A_{i})$ be a pair of pointed $CW$-complexes, where $A_{i}$ is
a pointed subspace of $X_{i}$. Let $\uxa=\{(X_{i},A_{i})\}_{i=1}^{m}$ be
the sequence of $CW$-pairs. For each simplex (face) $\sigma\in K$, let
$\uxa^{\sigma}$ be the subspace of $\prod_{i=1}^{m} X_{i}$ defined by
\[\uxa^{\sigma}=\prod_{i=1}^{m} Y_{i}\qquad
\mbox{where}\qquad Y_{i}=\left\{\begin{array}{ll}
X_{i} & \mbox{if $i\in\sigma$} \\
A_{i} & \mbox{if $i\notin\sigma$}.
\end{array}\right.\]
The \emph{polyhedral product} determined by \uxa\ and $K$ is
\[\uxa^{K}=\bigcup_{\sigma\in K}\uxa^{\sigma}\subseteq\prod_{i=1}^{m} X_{i}.\]
For example, suppose each $A_{i}$ is a point. If $K$ is a disjoint union
of $n$ points then $(\underline{X},\underline{\ast})^{K}$ is the wedge
$X_{1}\vee\cdots\vee X_{n}$, and if $K$ is the standard $(n-1)$-simplex
then $(\underline{X},\underline{\ast})^{K}$ is the product
$X_{1}\times\cdots\times X_{n}$.
In the case when each pair of spaces $(X_{i},A_{i})$ equals $(D^{2},S^{1})$,
the polyhedral product $\uxa^{K}$ is called a \emph{moment-angle complex},
and is written in more traditional notation as $\zk$. Two important properties
of~$\zk$ are: the cohomology ring $\cohlgy{\zk;\mathbb{Z}}$ is the Tor-algebra
$\mbox{Tor}_{\mathbb{Z}[v_{1},\ldots,v_{m}]}(\mathbb{Z}[K],\mathbb{Z})$
where $\mathbb{Z}[K]$ is the Stanley-Reisner face ring of~$K$ and
$\vert v_{i}\vert=2$ for $1\leq i\leq m$; and $\zk$ is homotopy equivalent
to the complement of the coordinate subspace arrangement in~$\mathbb{C}^{m}$
determined by $K$. Stanley-Reisner face rings are a subject of intense interest
in commutative algebra (even having its own MSC number), and complements
of coordinate subspace arrangements are an area of major importance
in combinatorics. The connection to moment-angle complexes allows
for topological methods to be used to inform upon problems posed in
commutative algebra and combinatorics.
To date, a great deal of work has been done to determine when $\zk$
is homotopy equivalent to a wedge of spheres, or to produce analogous
statements in the case of certain polyhedral
products~\cite{GT2,GT3,GPTW,GW,IK1,IK2}. When this is the case,
the complement of the corresponding coordinate subspace arrangement
is homotopy equivalent to a wedge of spheres, and $\cohlgy{\zk;\mathbb{Z}}$
is \emph{Golod}, meaning that all cup products and higher Massey products
are zero. In all cases thus far, the arguments start by using combinatorics
to identify a good class of simplicial complexes to consider, then homotopy
theory is used to prove that $\zk$ is homotopy equivalent to a wedge of
spheres for this class of simplicial complexes, and finally it is deduced
that $\cohlgy{\zk;\mathbb{Z}}$ is Golod.
Moment-angle complexes arise in complex geometry and algebraic geometry
in a different way. Let $P$ be a simple polytope, let $P^{\ast}$ be its dual, and
let $\partial P^{\ast}$ be the boundary complex of $P^{\ast}$. Then
$K=\partial P^{\ast}$ is a simplicial complex, and we let
$\mathcal{Z}(P)=\mathcal{Z}_{\partial P^{\ast}}$.
In this case, $\mathcal{Z}(P)$ has the richer structure of a manifold, and is called
a \emph{moment-angle manifold}. These manifolds can be interpreted
as intersections of complex quadrics, each fibring over a projective toric
variety. The topology and geometry of these manifolds have been studied
in considerable depth~\cite{BM,BP2,DJ,GL}. In particular, in~\cite{BM,GL}
a large class of simple polytopes~$P$ was identified for which $\mathcal{Z}(P)$
is diffeomorphic to a connected sum of products of two spheres.
Panov~\cite{P} observed that the two directions of work produce very similar
results in the following way. If $K$ is a simplicial complex consisting of $\ell$
disjoint points, then by~\cite{GT1} there is a homotopy equivalence
\[\zk\simeq\bigvee_{k=3}^{\ell+1} (S^{k})^{\wedge (k-2)\binom{\ell}{k-1}}\]
where $(S^{k})^{\wedge n}$ is the $n$-fold smash product of $S^{k}$ with itself.
On the other hand, if $P$ is a simple polytope that has been obtained
from the $d$-simplex by iteratively cutting off a vertex $\ell -1$ times
(the cuts occuring in any order), then by~\cite{BM} there is a diffeomorphism
\[\mathcal{Z}(P)\cong
\#_{k=3}^{\ell+1} (S^{k}\times S^{\ell+2d-k})^{\#(k-2)\binom{\ell}{k-1}}\]
where $(S^{k}\times S^{\ell +2d-k})^{\# n}$ is the $n$-fold connected sum
of $S^{k}\times S^{\ell +2d-k}$ with itself. The coefficients and the sphere
dimensions in both decompositions coincide. This led Panov to pose the following.
\medskip
\noindent
\textbf{Problem}: Describe the nature of this correspondence.
\medskip
The purpose of the paper is to answer this problem, thereby establishing
a bridge between two very different approaches to moment-angle manifolds.
Let $P$ be a simple
polytope obtained from a $d$-simplex by $\ell -1$ vertex cuts. To study
polyhedral products, we consider the dual simplicial complex~$P^{\ast}$,
which is a \emph{stacked polytope} (defined explicitly in Section~\ref{sec:vertex}).
We show that the homotopy type of $\mathcal{Z}_{\partial P^{\ast}}$
is independent of the stacking order for $P^{\ast}$
(dual to the result in~\cite{BM,GL} that the diffeomorphism type
of $\mathcal{Z}(P)$ is independent of the order in which the vertex
cuts occur for $P$). This lets us choose a stacking order, yielding a
stacked polytope $\LL$ on the vertex set $[m]$ for $m=d+\ell$,
which is more convenient to analyze (see Section~\ref{sec:deletion}
for details). We prove the following. Let $\partial\LL-\{1\}$ be the
full subcomplex of $\partial\LL$ obtained by deleting the vertex $\{1\}$.
\begin{theorem}
\label{Panovsoln}
The stacked polytope $\LL$ has the following properties:
\begin{letterlist}
\item there is a homotopy equivalence
$\mathcal{Z}_{\partial\LL-\{1\}}\simeq\mathcal{Z}_{P_{\ell}}$
where $P_{\ell}$ is $\ell$ disjoint points;
\item the inclusion
\(\namedright{\partial\LL-\{1\}}{}{\partial\LL}\)
induces a map
\(\namedright{\mathcal{Z}_{\partial\LL-\{1\}}}{}{\mathcal{Z}_{\partial\LL}}\),
which up to homotopy equivalences, is a map
\[f\colon\namedright
{\bigvee_{k=3}^{\ell+1} (S^{k})^{\wedge (k-2)\binom{\ell}{k-1}}}
{}{\#_{k=3}^{\ell+1} (S^{k}\times S^{\ell+2d-k})^{\#(k-2)\binom{\ell}{k-1}}};\]
\item $f$ has a left homotopy inverse $g$;
\item when restricted to a factor $\cohlgy{S^{k}\times S^{\ell+2d-k}}$ in
the cohomology of the connected sum, $f^{\ast}$ is zero on precisely
one of the ring generators.
\end{letterlist}
\end{theorem}
It is helpful to point out one consequence of Theorem~\ref{Panovsoln}. Since $f$ has
a left homotopy inverse, $f^{\ast}$ is an epimorphism. By part~(d), $f^{\ast}$
is nonzero on precisely one ring generator when restricted to any factor
$\cohlgy{S^{k}\times S^{\ell+2d-k}}$. Let $A$ be the collection of such
generators, one from each factor in the connected sum. The matching
coefficients in the wedge decomposition of $\mathbb{Z}_{\partial\LL-\{1\}}$
and the connected sum decomposition of $\mathbb{Z}_{\partial\LL}$ then
implies that $f^{\ast}$ maps $A$ isomorphically onto
$\cohlgy{\bigvee_{k=3}^{\ell+1} (S^{k})^{\wedge (k-2)\binom{\ell}{k-1}}}$.
Along the way, we phrase as many of the intermediate results as possible
in terms of polyhedral products $\cxx^{K}$, where $CX$ is the cone on a
space $X$, or in terms of $\ensuremath{(CX,X)}^{K}$, where all the coordinate spaces $X_{i}$
equal a common space $X$. This is of interest because, when
$K=\partial P^{\ast}$ for~$P$ a simple polytope obtained from a
$d$-simplex by $\ell -1$ vertex cuts, $\ensuremath{(CX,X)}^{K}$ is analogous to the connected
sum of products of two spheres. This analogue is \emph{not} a connected
sum, in general, nor is it even a manifold. So understanding its homotopy
theory helps distinguish how much of the homotopy theory of a connected
sum depends on the actual geometry.
The author would like to thank the referee for many helpful comments.
\section{Preliminary homotopy theory}
\label{sec:prelim}
In this section we give preliminary results regarding the homotopy
theory of polyhedral products that will be used later on. In particular,
in Proposition~\ref{htype} we identify a family of simplicial complexes whose
polyhedral products have the same homotopy type as the polyhedral
product corresponding to a disjoint union of points.
For spaces $A$ and $B$, the \emph{right half-smash} $A\rtimes B$
is the space $(A\times B)/\sim$ where $(\ast,b)\sim\ast$. The
\emph{join} $A\ast B$ is the space $(A\times I\times B)/\sim$ where
$(a,1,b)\sim (\ast,1,b)$ and $(a,0,b)\sim (a,0,\ast)$; it is well known
that there is a homotopy equivalence $A\ast B\simeq\Sigma A\wedge B$.
The following lemma was proved in~\cite{GT2}.
\begin{lemma}
\label{polemma}
Suppose that there is a homotopy pushout
\[\diagram
A\times B\rto^-{\pi_{1}}\dto^{\ast\times 1} & A\dto \\
C\times B\rto & Q
\enddiagram\]
where $\pi_{1}$ is the projection onto the first factor. Then there
is a homotopy equivalence $Q\simeq (A\ast B)\vee (C\rtimes B)$.~$\hfill\Box$
\end{lemma}
Suppose that $K$ is a simplicial complex on the vertex set $\{1,\ldots,m\}$.
If $L$ is a sub-complex of~$K$ on vertices $\{i_{1},\ldots,i_{k}\}$ then when
applying the polyhedral product to $K$ and $L$ simultaneously, we must
regard $L$ as a simplicial complex $\overline{L}$ on the vertices
$\{1,\ldots,m\}$. By definition of the polyhedral product, we therefore
obtain
\[\cxx^{\overline{L}}=\cxx^{L}\times\prod_{t=1}^{m-k} X_{j_{t}}\]
where $\{j_{1},\ldots,j_{m-k}\}$ is the complement of $\{i_{1},\ldots,i_{k}\}$
in $\{ 1,\ldots,m\}$.
The following lemma describes the homotopy type of $\cxx^{K}$
when $K=K_{1}\cup\Delta^{k}$, where $K_{1}$ and $\Delta^{k}$
have been glued along a common face $\Delta^{k-1}$. A similar
gluing lemma was proved in~\cite{GT2} that was stated more generally in
terms of two simplicial complexes joined along a common face, although
it was stated only in the more restrictive case of
$(\underline{C\Omega X},\underline{\Omega X})$. For our purposes,
it is helpful to be more explicit about the vertices in $\Delta^{k-1}$,
which affects the homotopy type of $\cxx^{K}$, so a proof is included.
\begin{lemma}
\label{glue}
Let $K$ be a simplicial complex on the vertex set $\{1,\ldots,m\}$.
Suppose that $K=K_{1}\cup\Delta^{k}$ where: (i) $K_{1}$ is a simplicial
complex on the vertex set $\{1,\ldots,m-1\}$ and $\{i\}\in K_{1}$ for
$1\leq i\leq m-1$; (ii)~$\Delta^{k}$ is on the vertex set $\{m-k,\ldots,m\}$,
and (iii)~$K_{1}\cap\Delta^{k}$ is a $(k-1)$-simplex on the vertex set
$\{m-k,\ldots,m-1\}$. Then there is a homotopy equivalence
\[\cxx^{K}\simeq\bigg(\big(\prod_{i=1}^{m-k-1} X_{i}\big)\ast X_{m}\bigg)\vee
\bigg(\cxx^{K_{1}}\rtimes X_{m}\bigg).\]
\end{lemma}
\begin{proof}
The simplicial complex $K$ can be written as a pushout
\[\diagram
\Delta^{k-1}\rto\dto & \Delta^{k}\dto \\
K_{1}\rto & K.
\enddiagram\]
Regarding $K_{1}$, $\Delta^{k}$ and $\Delta^{k-1}$ as simplicial complexes on
the vertex set $\{1,\ldots,m\}$ and applying the polyhedral product functor,
we obtain a pushout
\begin{equation}
\label{cxxpo}
\diagram
\cxx^{\overline{\Delta^{k-1}}}\rto\dto & \cxx^{\overline{\Delta^{k}}}\dto \\
\cxx^{\overline{K_{1}}}\rto & \cxx^{K}.
\enddiagram
\end{equation}
We now identify the spaces and maps in~(\ref{cxxpo}).
By hypothesis, $K_{1}$ is a simplicial complex on the vertex set
$\{1,\ldots,m-1\}$, $\Delta^{k}$ is on the vertex set
$\{m-k,\ldots,m\}$ and $\Delta^{k-1}$ is on the vertex set
$\{m-k,\ldots,m-1\}$. So by definition of the polyhedral product we have
\begin{align*}
\cxx^{\overline{\Delta^{k-1}}} & =\prod_{i=1}^{m-k-1} X_{i}\times
\prod_{i=m-k}^{m-1} CX_{i}\times X_{m} \\
\cxx^{\overline{\Delta^{k}}} & =\prod_{i=1}^{m-k-1} X_{i}\times
\prod_{i=m-k}^{m} CX_{i} \\
\cxx^{\overline{K_{1}}} & =\cxx^{K_{1}}\times X_{m}.
\end{align*}
Further, under these identifications, the map
\(\namedright{\cxx^{\overline{\Delta^{k-1}}}}{}{\cxx^{\overline{\Delta^{k}}}}\)
is the identity on each factor indexed by $1\leq i\leq m-1$ and is the
inclusion
\(\namedright{X_{m}}{}{CX_{m}}\)
on the $m^{th}$ factor, and the map
\(\namedright{\cxx^{\overline{\Delta^{k-1}}}}{}{\cxx^{\overline{K_{1}}}}\)
is the identity on the $m^{th}$ factor. Therefore, as the cone $CX_{i}$ is
contractible, up to homotopy equivalences (\ref{cxxpo}) is the same as
the homotopy pushout
\begin{equation}
\label{cxxpo2}
\diagram
(\prod_{i=1}^{m-k-1} X_{i})\times X_{m}\rto^-{\pi_{1}}\dto^{f\times 1}
& \prod_{i=1}^{m-k-1} X_{i}\dto \\
\cxx^{K_{1}}\times X_{m}\rto & \cxx^{K}
\enddiagram
\end{equation}
where $\pi_{1}$ is the projection and $f$ is some map.
By~\cite{GT3}, any simplicial complex $L$ on vertices $\{1,\ldots,\ell\}$
for which $\{i\}\in L$ for $1\leq i\leq\ell$ has the property that the inclusion
\(\namedright{\prod_{i=1}^{\ell} X_{i}}{}{\cxx^{L}}\)
is null homotopic. In our case, by hypothesis, $\{i\}\in K_{1}$ for
$1\leq i\leq m-1$, so the inclusion
\(\namedright{\prod_{i=1}^{m-1} X_{i}}{}{\cxx^{K_{1}}}\)
is null homotopic. Since the map $f$ in~(\ref{cxxpo2}) factors
through this inclusion, it too is null homotopic. Therefore
Lemma~\ref{polemma} applies to the homotopy pushout~(\ref{cxxpo2}),
giving a homotopy equivalence
\[\cxx^{K}\simeq\bigg(\big(\prod_{i=1}^{m-k-1} X_{i}\big)\ast X_{m}\bigg)\vee
\bigg(\cxx^{K_{1}}\rtimes X_{m}\bigg).\]
\end{proof}
For example, let $P_{m}$ be $m$ disjoint points. Then
$P_{m}=P_{m-1}\cup\Delta^{0}$ where $\Delta^{0}$ is a single point,
and the union is taken over the emptyset. Applying Lemma~\ref{glue}
then immediately gives the following.
\begin{corollary}
\label{disjointpts}
There is a homotopy equivalence
\[\cxx^{P_{m}}\simeq\bigg(\big(\prod_{i=1}^{m-1} X_{i}\big)\ast X_{m}\bigg)
\vee\bigg(\cxx^{P_{m-1}}\rtimes X_{m}\bigg).\]
$\hfill\Box$
\end{corollary}
In Proposition~\ref{htype} we will consider the polyhedral product
$\cxx^{K}$ where all the coordinate spaces $X_{i}$ are equal to a
common space $X$. In this case, we write $\ensuremath{(CX,X)}^{K}$. In particular,
in the case of $m$ disjoint points, Corollary~\ref{disjointpts} implies
that there is a homotopy equivalence
\begin{equation}
\label{disjptscxix}
\ensuremath{(CX,X)}^{P_{m}}\simeq\bigg(\big(\prod_{i=1}^{m-1} X\big)\ast X\bigg)
\vee\bigg(\ensuremath{(CX,X)}^{P_{m-1}}\rtimes X\bigg).
\end{equation}
\begin{proposition}
\label{htype}
Let $k\geq 1$ and suppose that there is a sequence of simplicial complexes
\[K_{1}=\Delta^{k}\subseteq K_{2}\subseteq\cdots\subseteq K_{\ell}\]
such that, for $i>1$, $K_{i}=K_{i-1}\cup_{\sigma_{i}}\Delta^{k}$ where
$\sigma_{i}=\Delta^{k-1}$. That is, $K_{i}$ is obtained from $K_{i-1}$
by gluing on a $\Delta^{k}$ along the common face $\sigma_{i}$. Let
$K=K_{\ell}$ and observe that $K$ is a simplicial complex on $k+\ell$
vertices. Then there is a homotopy equivalence
\[\ensuremath{(CX,X)}^{K}\simeq\ensuremath{(CX,X)}^{P_{\ell}}.\]
\end{proposition}
\begin{remark} It may be useful to note that Proposition~\ref{htype}
also makes sense for $k=0$, in which case $\Delta^{0}$ is a point and
each $\sigma_{i}$ is the emptyset, in which case $K=K_{\ell}$ is $\ell$
disjoint points, and the conclusion is a tautology. In the case when $k=1$,
notice that $K=K_{\ell}$ is formed by iteratively taking an interval at
stage $i$ and gluing one of its endpoints to a vertex of the preceeding
simplicial complex at stage $i-1$. One example of this is the boundary of
the $(\ell+2)$-gon with one vertex removed, another is all $\ell$
intervals joined at a common vertex.
\end{remark}
\begin{proof}
Fix $k\geq 1$. The proof is by induction on $\ell$.
When $\ell=1$, we have $K=K_{1}=\Delta^{k}$. By definition of
the polyhedral product,
$\cxx^{K}=\prod_{i=1}^{k+1} CX_{i}$, so
$\ensuremath{(CX,X)}^{K}=\prod_{i=1}^{k+1} CX$. On the other hand, as $P_{1}$ is a single
point, $\cxx^{P_{1}}=\ensuremath{(CX,X)}^{P_{1}}=CX$. Thus $\ensuremath{(CX,X)}^{K}\simeq\ensuremath{(CX,X)}^{P_{1}}$ as
both spaces are contractible.
Suppose that the proposition holds for all integers $t$ satisfying $t<\ell$.
Consider $K_{\ell}=K_{\ell-1}\cup_{\sigma_{\ell}}\Delta^{k}$ where
$\sigma_{\ell}=\Delta^{k-1}$. Reordering the vertices if necessary,
we may assume that $K_{\ell-1}$ is a simplicial complex on the vertex
set $\{1,\ldots,k+\ell -1\}$, $\Delta^{k}$ is on the vertex set
$\{\ell,\ldots,k+\ell\}$, and $\sigma_{\ell}=\Delta^{k-1}$ is on the vertex
set $\{\ell,\ldots,k+\ell -1\}$. By Lemma~\ref{glue}, there is a
homotopy equivalence
\[\cxx^{K_{\ell}}\simeq\bigg(\big(\prod_{i=1}^{\ell-1} X_{i}\big)\ast X_{k+\ell}\bigg)
\vee\bigg(\cxx^{K_{\ell-1}}\rtimes X_{k+\ell}\bigg).\]
Therefore, there is a homotopy equivalence
\[\ensuremath{(CX,X)}^{K_{\ell}}\simeq\bigg(\big(\prod_{i=1}^{\ell-1} X\big)\ast X\bigg)
\vee\bigg(\ensuremath{(CX,X)}^{K_{\ell-1}}\rtimes X\bigg).\]
This formula is exactly the same as that in~(\ref{disjptscxix}) for
$\ensuremath{(CX,X)}^{P_{\ell}}$. By inductive hypothesis,
$\ensuremath{(CX,X)}^{K_{\ell -1}}\simeq\ensuremath{(CX,X)}^{P_{\ell -1}}$,
so we obtain $\ensuremath{(CX,X)}^{K_{\ell}}\simeq\ensuremath{(CX,X)}^{P_{\ell}}$. The proposition
therefore holds by induction.
\end{proof}
\section{Vertex cuts and stacked polytopes}
\label{sec:vertex}
In this section we discuss some constructions obtained from
simple polytopes, and discuss some of their properties in the
context of polyhedral products. We begin with some definitions
(see, for example,~\cite[Chapter 1]{BP2}).
A \emph{(convex) polytope} is the convex hull of a finite set of points
in $\mathbb{R}^{n}$. Its dimension is the dimension of its affine hull.
Let $P$ be a $d$-dimensional polytope. A \emph{facet} of $P$ is
a $(d-1)$-dimensional face. The polytope $P$ is \emph{simple} if
each vertex lies in exactly $d$ facets of $P$. A partial ordering
may be defined on the faces of $P$ by inclusion. This determines
a poset called the \emph{face poset} of $P$. The opposite poset,
given by reversing the order, determines another polytope $P^{\ast}$
called the \emph{dual} of $P$. If $P$ is simple then $P^{\ast}$ is a
simplicial complex. Dualizing has the property that $P^{\ast\ast}=P$.
Let $\partial P^{\ast}$ be the boundary of $P^{\ast}$.
Suppose that $P$ is a simple polytope. Following~\cite{BP2,DJ}, a
moment-angle complex $\mathcal{Z}(P)$ can be associated to $P$ by
defining $\mathcal{Z}(P)=\mathcal{Z}_{\partial P^{\ast}}$. Generalizing to
polyhedral products in the case where each coordinate space equals a
common space $X$, define $\ensuremath{(CX,X)}(P)$ as $\ensuremath{(CX,X)}^{\partial P^{\ast}}$.
The moment-angle complex $\mathcal{Z}(P)$ is in fact a manifold,
but this property does not extend in general to $\ensuremath{(CX,X)}(P)$.
An operation that produces new simple polytopes from existing
ones is by doing vertex cuts.
\begin{definition}
Let $P$ be a simple polytope of dimension $d$ and let
$V(P)$ be its vertex set. A hyperplane $H$ in $\mathbb{R}^{d}$
cuts a vertex $x$ of $P$ if $x$ and $V(P)/\{x\}$ lie
in different open half-spaces of $H$. Let $Q$ be the intersection of
$P$ with the closed half-space of $H$ containing $V(P)/\{x\}$.
We say that~$Q$ is obtained from $P$ be a \emph{vertex cut operation}.
\end{definition}
Diagrammatically, this is pictured as follows:
\[\begin{tikzpicture}
\draw (0,0)--(1.2,2)--(2,-1)--(0,0);
\draw (1.2,2)--(2.3,0.5)--(2,-1);
\draw [dashed, ultra thin] (0,0)--(2.3,0.5);
\node at (1.25,-1.7) {$P$};
\end{tikzpicture}\qquad\qquad
\begin{tikzpicture}
\draw (0,0)--(1.2,2)--(1.75,-0.3)--(1.3,-0.7)--(0,0);
\draw (1.2,2)--(2.3,0.5)--(2.1,-0.7)--(1.75,-0.3);
\draw (1.3,-0.7)--(2.1,-0.7);
\draw [dashed, ultra thin] (0,0)--(2.3,0.5);
\node at (1.25,-1.7) {$Q$};
\end{tikzpicture}\]
The dual of a vertex cut operation is a stacking operation.
\begin{definition}
Let $K$ be a simplicial complex of dimension $d$ and let $\sigma$
be a facet of $K$. Define $L$ as $K\cup_{\sigma}\Delta^{d}$, that is,
$L$ is obtained from $K$ by gluing a $d$-simplex onto $K$ along the
facet $\sigma$. We say that $L$ is obtained from $K$ by a
\emph{stacking operation}.
\end{definition}
Diagrammatically, this is pictured as follows:
\[\begin{tikzpicture}
\draw (0,0)--(1.2,2)--(2,-1)--(0,0);
\draw (1.2,2)--(2.3,0.5)--(2,-1);
\draw [dashed, ultra thin] (0,0)--(2.3,0.5);
\node at (1.25,-1.7) {$K$};
\end{tikzpicture}\qquad\qquad
\begin{tikzpicture}
\draw (0,0)--(1.2,2)--(2,-1)--(0,0);
\draw (1.2,2)--(2.3,0.5)--(2,-1);
\draw (0,0)--(0.9,0.8)--(1.2,2);
\draw (0.9,0.8)--(2,-1);
\draw [dashed, ultra thin] (0,0)--(2.3,0.5);
\node at (1.25,-1.7) {$L$};
\end{tikzpicture}\]
Notice that it is immediate from the definitions that the vertex cut
and stacking operations preserve dimension.
The objects we wish to study are the moment-angle manifold
$\mathcal{Z}(P)$ and the polyhedral product $\ensuremath{(CX,X)}(P)$ where
$P$ is a simple polytope obtained from $\Delta^{d}$ by iterated
vertex cut operations. Equivalently, we study the polyhedral products
$\mathcal{Z}_{\partial P^{\ast}}$ and $\ensuremath{(CX,X)}^{\partial P^{\ast}}$
where $\partial P^{\ast}$ is the boundary of a simple polytope
obtained from $\Delta^{d}$ by iterated stacking operations.
An important property of the vertex cut operation is that the diffeomorphism
type of $\mathcal{Z}(P)$ is independent of the order in which the vertices
were cut~\cite[Theorem 2.1]{GL}. Dually, the diffeomorphism type of
$\mathcal{Z}(P)$ is independent of the stacking order for $P^{\ast}$.
Weakening to homotopy type, we generalize this property to polyhedral
products.
\begin{proposition}
\label{cutindep}
Let $P$ be a simple polytope and let $Q$ be a simple polytope
obtained from $P$ by a vertex cut operation. Then the homotopy
type of $\ensuremath{(CX,X)}(Q)$ is independent of which vertex was~cut.
\end{proposition}
\begin{remark}
It is easy to see that Proposition~\ref{cutindep} does not hold when
$\ensuremath{(CX,X)}$ is replaced by $\cxx$, that is, when the coordinate spaces $X_{i}$
may be different. For example, let $P=\Delta^{2}$ with vertex set $\{1,2,3\}$.
Cut vertex $1$ to obtain a new polytope $Q_{1}$ on the vertex set $\{2,3,4,5\}$
or cut vertex $2$ to obtain a new polytope $Q_{2}$ on the vertex set
$\{1,3,4,5\}$. Both $Q_{1}$ and $Q_{2}$ equal the square~$I^{2}$. Notice
that $Q_{1}$ and $Q_{2}$ are self-dual, so
$\partial Q_{1}^{\ast}=\partial Q_{2}^{\ast}=\partial I^{2}$.
Next, observe that $\partial I^{2}=A\ast B$ where $A$ and $B$ are $2$
points and $\ast$ is the join operation, defined in general by
$K_{1}\ast K_{2}=\{\sigma_{1}\cup\sigma_{2}\mid \sigma_{i}\in K_{i}\}$.
A straightforward property of the polyhedral product~\cite{BBCG} is
that $\cxx^{K_{1}\ast K_{2}}\simeq\cxx^{K_{1}}\times\cxx^{K_{2}}$.
In our case, this gives $\cxx^{\partial I^{2}}=\cxx^{A}\times\cxx^{B}$.
Therefore, taking coordinate spaces $X_{i}$ for $1\leq i\leq 5$, we obtain
$\cxx^{\partial Q_{1}^{\ast}}\simeq (X_{2}\ast X_{3})\times(X_{4}\ast X_{5})$ while
$\cxx^{\partial Q_{2}^{\ast}}\simeq (X_{1}\ast X_{3})\times(X_{4}\ast X_{5})$. These
have distinct homotopy types, but if each $X_{i}$ equals a common
space $X$ then
$\ensuremath{(CX,X)}(Q_{1})=\ensuremath{(CX,X)}^{\partial Q_{1}^{\ast}}\simeq
\ensuremath{(CX,X)}^{\partial Q_{2}^{\ast}}=\ensuremath{(CX,X)}(Q_{2})$.
\end{remark}
We will prove the equivalent, dual statement to Proposition~\ref{cutindep}.
\begin{proposition}
\label{stackindep}
Let $K$ be a simplicial complex of dimension $d$ which is dual
to a simple polytope~$P$. Let $L$ be a simplicial complex obtained
from $K$ by stacking along a facet of $K$. Then the homotopy type
of $\ensuremath{(CX,X)}^{\partial L}$ is independent of which facet of $K$ was stacked.
\end{proposition}
\begin{proof}
Let $\sigma_{1}$ and $\sigma_{2}$ be two facets of $K$. For $t=1,2$,
let $\Delta^{d}_{t}$ be a $d$-simplex stacked onto $\sigma_{t}$.
Then there are pushouts
\[\diagram
\sigma_{t}\rto\dto & \Delta^{d}_{t}\dto \\
K\rto & L_{t}
\enddiagram\]
which define the simplicial complexes $L_{1}$ and $L_{2}$. Since
$\sigma_{1}$ and $\sigma_{2}$ are faces of $\partial K$, the
stacking operation also induces pushouts
\[\diagram
\sigma_{t}\rto\dto & \partial\Delta^{d}_{t}\dto \\
\partial K\rto & \partial L_{t}.
\enddiagram\]
We will prove the proposition by showing that
$\ensuremath{(CX,X)}^{\partial L_{1}}\simeq\ensuremath{(CX,X)}^{\partial L_{2}}$.
It is useful to first consider $\cxx^{\partial L_{1}}$ and
$\cxx^{\partial L_{2}}$ where we have to more explicitly keep track of
coordinates. Suppose that the vertex set of $K$ is $\{1,\ldots,m\}$.
Stacking introduces one additional vertex in $L_{t}$ which we label
in both cases as $m+1$. Suppose that $\sigma_{t}$ is on the vertex set
$\{i_{t,1},\ldots,i_{t,d}\}$. Let $\{j_{t,1},\ldots,j_{t,m-d}\}$ be the
complement of $\{i_{t,1},\ldots,i_{t,d}\}$ in $\{1,\ldots,m\}$. Observe
that $\Delta^{d}_{t}$ is on the vertex set $\{i_{t,1},\ldots,i_{t,d},m+1\}$.
Regarding each of $\sigma_{t}$, $\Delta^{d}_{t}$, $K$
and $L_{t}$ as being on the vertex set $\{1,\ldots,m+1\}$, we can take
polyhedral products to obtain pushouts
\begin{equation}
\label{stackdgrm1}
\diagram
\cxx^{\overline{\sigma_{t}}}\rto\dto & \cxx^{\overline{\partial\Delta^{d}_{t}}}\dto \\
\cxx^{\overline{\partial K}}\rto & \cxx^{\partial L_{t}}.
\enddiagram
\end{equation}
In general, if $\tau$ is a $d$-simplex on the vertex set \mbox{$\{1,\ldots,d+1\}$},
then by definition of the polyhedral product we have
$\cxx^{\tau}=\prod_{r=1}^{d+1} CX_{r}$,
and
$\cxx^{\partial\tau}=\bigcup_{r=1}^{d+1}(CX_{1}\times\cdots\times X_{r}\times
\cdots\times CX_{d+1})$,
where, in each term of the union, all the factors are cones except for one.
Applying this to our case, we obtain
\begin{align*}
\cxx^{\overline{\sigma_{t}}} & =\cxx^{\sigma_{t}}\times\prod_{s=d+1}^{m} X_{j_{t,s}}
\times X_{m+1}=
\prod_{s=1}^{d} CX_{i_{t,s}}\times\prod_{s=d+1}^{m} X_{j_{t,s}}\times X_{m+1} \\
\cxx^{\overline{\partial\Delta^{d}_{t}}} &
= \cxx^{\partial\Delta^{d}_{t}}\times\prod_{s=d+1}^{m} X_{j_{t,s}}=
\bigcup_{s=1}^{d+1}(CX_{i_{t,1}}\times\cdots\times X_{i_{t,s}}\times\cdots
\times CX_{i_{t,d+1}})\times\prod_{s=d+1}^{m} X_{j_{t,s}} \\
\cxx^{\overline{K}} & =\cxx^{K}\times X_{m+1}
\end{align*}
where, in the second line, to compress notation we have used $i_{t,d+1}$ to
refer to the vertex $m+1$ for both $t=1,2$.
Further, under these identifications, the map
\(\namedright{\cxx^{\overline{\sigma_{t}}}}{}{\cxx^{\overline{\partial K}}}\)
is the product of the identity map on $X_{m+1}$ and a map
\(f_{t}\colon\namedright{\cxx^{\sigma_{t}}\times\prod_{s=d+1}^{m} X_{j_{t,s}}}{}
{\cxx^{\partial K}}\)
induced by the inclusion of the face
\(\namedright{\sigma_{t}}{}{\partial K}\),
and the map
\(\namedright{\cxx^{\overline{\sigma_{t}}}}{}{\cxx^{\overline{\partial\Delta^{d}}}}\)
is a coordinate-wise inclusion which we label as $i$.
Thus~(\ref{stackdgrm1}) can be identified with the pushouts
\begin{equation}
\label{stackdgrm2}
\diagram
\bigg(\prod_{s=1}^{d} CX_{i_{t,s}}\times\prod_{s=d+1}^{m} X_{j_{t,s}}\bigg)
\times X_{m+1}\rto^-{i}\dto^{f_{t}\times 1}
& \bigcup_{s=1}^{d+1}(CX_{i_{t,1}}\times\cdots\times X_{i_{t,s}}\times
\cdots\times CX_{i_{t,d+1}})\dto \\
\cxx^{\partial K}\times X_{m+1}\rto & \cxx^{\partial L_{t}}.
\enddiagram
\end{equation}
Now simplifying to the case of $\ensuremath{(CX,X)}$ when each coordinate space $X_{i}$
equals a common space $X$, we obtain pushouts
\begin{equation}
\label{stackdgrm3}
\diagram
\bigg(\prod_{s=1}^{d} CX\times\prod_{s=d+1}^{m} X\bigg)
\times X\rto^-{i}\dto^{f_{t}\times 1}
& \bigcup_{s=1}^{d+1}(CX\times\cdots\times X\times\cdots\times CX)\dto \\
\ensuremath{(CX,X)}^{\partial K}\times X\rto & \ensuremath{(CX,X)}^{\partial L_{t}}.
\enddiagram
\end{equation}
Observe that the only difference in the pushouts for $\ensuremath{(CX,X)}^{\partial L_{1}}$ and
$\ensuremath{(CX,X)}^{\partial L_{2}}$ in~(\ref{stackdgrm3}) are the maps $f_{1}$ and $f_{2}$.
We will show that there is a self-homotopy equivalence $e$ of $\ensuremath{(CX,X)}^{\partial K}$
which satisfies a homotopy commutative square
\begin{equation}
\label{stackdgrm4}
\diagram
\prod_{s=1}^{d} CX\times\prod_{s=d+1}^{m} X\rto^-{p}\dto^{f_{1}}
& \prod_{s=1}^{d} CX\times\prod_{s=d+1}^{m} X\dto^{f_{2}} \\
\ensuremath{(CX,X)}^{\partial K}\rto^-{e} & \ensuremath{(CX,X)}^{\partial K}
\enddiagram
\end{equation}
where $p$ permutes coordinates. Granting this, observe that we obtain
a map from the $t=1$ pushout in~(\ref{stackdgrm3}) to the $t=2$ pushout
by using $p\times 1$ on the upper left corner, $e\times 1$ on the lower
right corner, and noting that $i$ is a coordinate-wise inclusion, we can
also use $p\times 1$ on the upper right corner. This induces a map
of pushouts
\(h\colon\namedright{\ensuremath{(CX,X)}^{\partial L_{1}}}{}{\ensuremath{(CX,X)}^{\partial L_{2}}}\).
As~$p$ and $e$ are homotopy equivalences, so is $h$, and this completes
the proof.
It remains to construct the self-homotopy equivalence $e$
of $\ensuremath{(CX,X)}^{\partial K}$. First consider the simple polytope $P$ that is dual to $K$.
Let $v_{1}$ and $v_{2}$ be vertices of $P$. Consider the permutation that
interchanges $v_{1}$ and $v_{2}$ while leaving the other vertices fixed.
Since the polytope $P$ is simple, this permutation induces a self-map
of the face poset of $P$ which permutes the $k$-dimensional faces
for each $0\leq k\leq d$. Dually, the face poset for $K$ is obtained by
reversing the arrows on the face poset for $P$, so we obtain a self-map
of the face poset of $K$ which permutes the $k$-dimensional faces
for each $0\leq k\leq d$ . Consequently, if we let $v_{1}$ and $v_{2}$
be the vertices of $P$ that are dual to the facets $\sigma_{1}$ and
$\sigma_{2}$ of $K$, we obtain a map
\(g^{\prime}\colon\namedright{K}{}{K}\)
of simplicial complexes which permutes the facets $\sigma_{1}$
and~$\sigma_{2}$. This induces a map
\(g\colon\namedright{\partial K}{}{\partial K}\)
of simplicial complexes which permutes the faces $\sigma_{1}$
and $\sigma_{2}$. Now apply the polyhedral product $\ensuremath{(CX,X)}$ to the
face poset of $K$. Any face $\tau$ of $K$ has $\ensuremath{(CX,X)}^{\tau}$ equal
to a product of copies of $CX$ or $X$, depending on whether a
vertex is in or not in $\tau$. So the self-map of the face poset of $K$
induces a self-map of $\cxx^{\tau}$ for each face $\tau$ of $K$
which permutes the $CX$ factors and permutes the $X$ factors.
Any such permutation is a homotopy equivalence. The morphism
of face posets ensures that these permuations are compatible
under face-wise inclusions, so there are induced maps
\(e^{\prime}\colon\namedright{\ensuremath{(CX,X)}^{K}}{}{\ensuremath{(CX,X)}^{K}}\)
and
\(e\colon\namedright{\ensuremath{(CX,X)}^{\partial K}}{}{\ensuremath{(CX,X)}^{\partial K}}\)
which are homotopy equivalences, and $e$ satisfies~(\ref{stackdgrm4}).
\end{proof}
Starting with a simplicial complex $K$ of dimension $d$, there are many
ways of iteratively stacking to produce a new simplicial complex $L$.
A particular sequence of stacks is called a \emph{stack history} of~$L$.
\begin{corollary}
\label{stackcor}
Let $K$ be a simplicial complex of dimension $d$ which is dual
to a simple polytope~$P$ and let $L$ be a simplicial complex obtained
from $K$ by iterated stacking operations. Then the homotopy type
of $\ensuremath{(CX,X)}^{\partial L}$ is independent of the stack history of $L$.~$\hfill\Box$
\end{corollary}
\section{Deleting a vertex from the boundary of a stacked polytope}
\label{sec:deletion}
In this section we consider a special case of iterated stacking operations.
Let $P=\Delta^{d}$ be the $d$-simplex. Then $P$ is a simple polytope,
and the dual $K=P^{\ast}$ of $P$ is again $\Delta^{d}$. In this case,
if $L$ is obtained from $K$ by a sequence of stacking operations, then
$L$ is also a simple polytope of dimension~$d$, as well as a simplicial
complex. The simple polytope $L$ is called a \emph{stacked polytope}.
Each copy of $\Delta^{d}$ in $L$ is called a \emph{stack}, so if $L$ is
formed by $\ell -1$ stacking operations, then it has $\ell$ stacks.
Suppose that $L$ is a stacked polytope with $\ell$ stacks. So there is
a sequence of stacked polytopes
\[L_{1}=\Delta^{d}\subseteq L_{2}\subseteq\cdots\subseteq L_{\ell}=L\]
where, for $2\leq i\leq\ell$, $L_{i}$ has been formed by gluing a $\Delta^{d}$
to $L_{i-1}$ along a common facet. By Corollary~\ref{stackcor}, the homotopy
type of $\ensuremath{(CX,X)}^{\partial L}$ is independent of the stack history of $L$.
Thus we can choose a stacking order which is more convenient for
analyzing $\ensuremath{(CX,X)}^{\partial L}$.
The prescribed stacking order we choose is as follows. Let
$\LL_{1}=\Delta^{d}$. Label the vertices of $\Delta^{d}$ as
$\{1,\ldots,d+1\}$. Form $\LL_{2}$ by stacking a copy of $\Delta^{d}$
to $\LL_{1}$ on the facet $(1,\ldots,d)$. Label the one extra vertex
of $\LL_{2}$ as $d+2$, and notice that if $d=1$ then the vertex $\{3\}$
is a facet of $\LL_{2}$ and if $d>1$ then $(1,\ldots,d-1,d+2)$ is a
facet of $\LL_{2}$. Now stack onto this facet and iterate the procedure.
We obtain, for $2< k\leq\ell$, a stacked polytope $\LL_{k-1}$ on the
vertex set $\{1,\ldots,d+k-1\}$ where if $d=1$ then the vertex
$\{k=d+k-1\}$ is a fact and if $d>1$ then $(1,\ldots,d-1,d+k-1)$ is
a facet. Form $\LL_{k}$ by stacking a copy of $\Delta^{d}$ on this facet.
Label the one extra vertex of $\LL_{k}$ as $d+k$, and observe that
if $d=1$ then the vertex $\{d+k\}$ is a facet of $\LL_{k}$ and if $d>1$ then
$(1,\ldots,d-1,d+k)$ is a facet of~$\LL_{k}$. Finally, let $\LL=\LL_{\ell}$.
Now we identify the simplicial complex obtained by deleting the
vertex $\{1\}$ from $\partial\LL$.
\begin{lemma}
\label{delete}
The simplicial complex $\LL=\LL_{\ell}$ has the following properties:
\begin{letterlist}
\item $\LL$ has $\ell d+1$ facets;
\item there are $\ell$ facets in part~(a) which do not contain the vertex $\{1\}$:
these are $(2,3,\ldots,d,d+1)$, $(2,3,\ldots,d,d+2)$, and for
$2< k\leq\ell$, $(2,3,\ldots,d-1,d+k-1,d+k)$;
\item the simplicial complex $\partial\LL-\{1\}$ filters as a sequence
\[M_{1}=\Delta^{d-1}\subseteq M_{2}\subseteq\cdots\subseteq
M_{\ell}=\partial L-\{1\}\]
where, for $2\leq k\leq\ell$, $M_{k}=M_{k-1}\cup_{\sigma_{k}}\Delta^{d-1}$,
where $\sigma_{k}=\Delta^{d-2}$.
\end{letterlist}
\end{lemma}
\begin{proof}
For part~(a), observe that $\LL_{1}=\Delta^{d}$ has $d+1$ facets. As $\LL_{\ell}$ is formed
by gluing on $(\ell -1)$ more $\Delta^{d}$'s, the total of $\ell$ copies of $\Delta^{d}$
have $\ell(d+1)$ facets. But each gluing occurs along a common facet,
so at each of the $(\ell -1)$ gluings $1$ facet is removed. Thus $\LL_{\ell}$
has $\ell(d+1)-(\ell-1)=\ell d+1$ facets.
For part~(b), observe that in $\LL_{1}=\Delta^{d}$ there are $d+1$ facets but
only one of them, $(2,3,\ldots,d,d+1)$, does not contain the vertex $\{1\}$.
In forming $\LL_{2}$, we stack on the facet $(1,2,\ldots,d)$ of $\LL_{1}$, and label
the extra vertex $d+2$. This operation removes $(1,2,\ldots,d)$ as a facet
of $\LL_{1}$ and introduces $d$ new facets: all $d+1$ facets of $\Delta^{d}$ on
the vertex set $\{1,2,\ldots,d,d+2\}$ except for $(1,2,\ldots,d)$. Of the new facets,
only one of them, $(2,3,\ldots,d,d+2)$, does not contain the vertex $\{1\}$.
Iterating, for $2<k\leq\ell$, in forming $\LL_{k}$, we stack on the facet
$(1,2,\ldots,d-1,d+k-1)$ of $\LL_{k-1}$, and label the extra vertex $d+k$.
This operation removes $(1,2,\ldots,d-1,d+k-1)$ as a facet of $\LL_{k}$
and introduces $d$ new facets: all $d+1$ facets of $\Delta^{d}$ on the vertex
set $\{1,2,\ldots,d-1,d+k-1,d+k\}$ except for $(1,2,\ldots,d-1,d+k-1)$.
Of the new facets, only one of them, $(2,3,\ldots,d-1,d+k-1,d+k)$,
does not contain the vertex $\{1\}$. Thus, precisely $\ell$ of the
$\ell d+1$ total facets of $\LL_{\ell}$ do not contain the
vertex $\{1\}$, and these are: $(2,3,\ldots,d,d+1)$, $(2,3,\ldots,d,d+2)$,
and for $2< k\leq\ell$, $(2,3,\ldots,d-1,d+k-1,d+k)$.
For part~(c), since $\LL$ is a simple polytope which is also a
simplicial complex, the geometric realization of $\partial\LL$
can be obtained by gluing together the facets of $\LL$. The
geometric realization of the simplicial complex $\partial\LL-\{1\}$
is therefore obtained by gluing together those facets of $\LL$
which do not contain the vertex $\{1\}$. We perform this gluing
procedure one simplex at a time. Let $M_{1}=\Delta^{d-1}$
be $(2,3,\ldots,d,d+1)$. Form $M_{2}$ by gluing the $(d-1)$-simplex
$(2,3,\ldots,d,d+2)$ to $M_{1}$ along the common $(d-2)$-simplex
$(2,3,\ldots,d)$. For $2\leq k\leq\ell$, form $M_{k}$ by gluing the
$(d-1)$-simplex $(2,3,\ldots,d-1,d+k-1,d+k)$ to $M_{k-1}$ along
the common $(d-2)$-simplex $(2,3,\ldots,d-1,d+k-1)$. Then
$M_{\ell}=\partial\LL-\{1\}$.
\end{proof}
Applying Proposition~\ref{htype} to Lemma~\ref{delete}~(c), we immediately
obtain the following.
\begin{proposition}
\label{deletiontype}
There is a homotopy equivalence
\[\ensuremath{(CX,X)}^{\partial\LL-\{1\}}\simeq\ensuremath{(CX,X)}^{P_{\ell}}\]
where $P_{\ell}$ is $\ell$ disjoint points.~$\hfill\Box$
\end{proposition}
Now specialize to polyhedral products on the pairs $(D^{2},S^{1})$
and write $\zk$ for $\ensuremath{(CX,X)}^{K}$. In~\cite{GT1} the homotopy type of
$\mathcal{Z}_{P_{\ell}}$ was identified, giving the following.
\begin{corollary}
\label{deletioncor}
There is a homotopy equivalence
\[\mathcal{Z}_{\partial\LL-\{1\}}\simeq\mathcal{Z}_{P_{\ell}}\simeq
\bigvee_{k=3}^{\ell+1} (S^{k})^{\wedge (k-2)\binom{\ell}{k-1}}.\]
$\hfill\Box$
\end{corollary}
\section{Cup products in $\cohlgy{\mathbb{Z}_{\partial\LL}}$}
\label{sec:cup}
On the one hand, since $\LL$ is a stacked polytope of dimension $d$
with $\ell$ stacks, it is dual to a simple polytope obtained from $\Delta^{d}$
by $\ell -1$ vertex cuts. So by~\cite{BM,GL} there is a diffeomorphism
$\mathcal{Z}_{\partial\LL}\cong
\#_{k=3}^{\ell+1} (S^{k}\times S^{\ell+2d-k})^{\#(k-2)\binom{\ell}{k-1}}$.
The cup products in $\cohlgy{\mathcal{Z}_{\partial\LL}}$ are then clear
from the description of the space as a connected sum of products of spheres.
On the other hand, there is a combinatorial description of the cup product
structure in $\mathcal{Z}_{K}$ for any simplicial complex $K$, proved
in~\cite{BBP,BP1,F}. Take homology with integer coefficients. The \emph{join}
of two simplicial complexes $K_{1}$ and $K_{2}$ is
$K_{1}\ast K_{2}=\{\sigma_{1}\cup\sigma_{2}\mid \sigma_{i}\in K_{i}\}$.
\begin{theorem}
\label{zkcohlgy}
There is an isomorphism of graded commutative algebras
\[\cohlgy{\zk}\cong\bigoplus_{I\subset[m]}\rcohlgy{K_{I}}.\]
Here, $\rcohlgy{K_{I}}$ denotes the reduced simplicial cohomology
of the full subcomplex $K_{I}\subset K$ (the restriction of $K$
to $I\subset[m]$). The isomorphism is the sum of isomorphisms
\[H^p(\zk)\cong\sum_{I\subset[m]}\widetilde{H}^{p-|I|-1}(K_I)\]
and the ring structure (the Hochster ring) is given by the maps
\[\namedright{H^{p-|I|-1}(K_{I})\otimes H^{q-|J|-1}(K_{J})}{}
{H^{p+q-|I|-|J|-1}(K_{I\cup J})}\]
which are induced by the canonical simplicial maps
\(\namedright{K_{I\cup J}}{}{K_{I}*K_{J}}\)
for $I\cap J=\emptyset$ and zero otherwise.~$\hfill\Box$
\end{theorem}
Theorem~\ref{zkcohlgy} implies that the Hochster ring structure on
$\mathcal{Z}_{\partial\LL}$ matches the ring product structure arising
from the geometry of the connected sum, at least up to an isomorphism.
We need information from both, so we are led to geometrically realize
the isomorphism, via a homotopy equivalence.
In general, if $M$ is an $n$-dimensional manifold, let $M-\ast$ be
$M$ with a point in the interior of the $n$-disc removed. As a
$CW$-complex, $M-\ast$ is homotopy equivalent to the $(n-1)$-skeleton
of $M$. By definition of the connected sum, if $M$ and $N$ are two
$n$-dimensional manifolds then $(M\#N)-\ast\simeq (M-\ast)\vee (N-\ast)$.
In our case, as
$\mathcal{Z}_{\partial\LL}=
\#_{k=3}^{\ell+1} (S^{k}\times S^{\ell+2d-k})^{\#(k-2)\binom{\ell}{k-1}}$,
the $(\ell+2d-1)$-skeleton of $\mathcal{Z}_{\partial\LL}$ is the wedge
\[W=\bigvee_{k=3}^{\ell+1} (\bigvee_{t=1}^{(k-1)\binom{\ell}{k-1}}
(S^{k}\vee S^{\ell+2d-k})).\]
Therefore, there is one ring generator in $\cohlgy{\mathcal{Z}_{\partial\LL}}$
for each sphere in the wedge $W$.
Applying Theorem~\ref{zkcohlgy} to $\mathcal{Z}_{\partial\LL}$ we obtain
an abstract isomorphism of algebras
\(h\colon\namedright{\cohlgy{\mathcal{Z}_{\partial\LL}}}{}
{\cohlgy{\mathcal{Z}_{\partial\LL}}}\),
where on the left the generating set is given by the Hochster ring structure,
on the right the generating set is given by the $CW$-structure of
$\mathcal{Z}_{\partial\LL}$, and $h$ maps generators to
generators. Restricting to degrees less than $\ell+2d$, we obtain an
abstract isomorphism of modules
\(h'\colon\namedright{\cohlgy{W}}{}{\cohlgy{W}}\).
Dualizing, we obtain an abstract isomorphism of modules
\(h''\colon\namedright{\hlgy{W}}{}{\hlgy{W}}\).
Since~$W$ is a wedge of spheres, the abstract map $h''$ may be
realized geometrically, as follows. Let $n$ be the number of spheres
in the wedge $W$ and label the spheres from $1,\ldots, n$. For
$1\leq i\leq n$, let
\(j_{i}\colon\namedright{S_{n}}{}{W}\)
be the inclusion into the wedge, and let $x_{i}\in\hlgy{W}$ be the Hurewicz
image of $j_{i}$. Suppose that
$h''(x_{i})=t_{i,1}x_{1}+\cdots + t_{i,n}x_{n}$ for some integers $t_{i,1},\ldots,t_{i,n}$.
Define
\(g_{i}\colon\namedright{S_{i}}{}{W}\)
by $g_{i}=t_{i,1}j_{1}+\cdots t_{i,n}j_{n}$. Let
\(g\colon\namedright{W}{}{W}\)
be the wedge sum of the maps $g_{i}$ for $1\leq i\leq n$. Then
$g_{\ast}=h''$. Dualizing, $g^{\ast}=h'$. As $h'$ is an isomorphism,
by Whitehead's Theorem $g$ is a homotopy equivalence.
Next, the map attaching the top cell to $W$ to form $\mathcal{Z}_{\partial\LL}$ is
a sum of Whitehead products, one Whitehead product for each
$S^{k}\times S^{\ell+2d-k}$. This Whitehead product is detected in
cohomology by a nonzero cup product. Since Theorem~\ref{zkcohlgy}
gives a ring isomorphism between the cup product structures on
$\cohlgy{\mathcal{Z}_{\partial\LL}}$ from the connected sum and the Hochster ring,
$g$ can be extended to a map
\[\Gamma\colon\namedright{\mathcal{Z}_{\partial\LL}}{}{\mathcal{Z}_{\partial\LL}}\]
which induces an isomorphism in cohomology and so is a homotopy
equivalence. Thus we have the following.
\begin{lemma}
\label{cupprods}
Altering
$\mathcal{Z}_{\partial\LL}\cong
\#_{k=3}^{\ell+1} (S^{k}\times S^{\ell+2d-k})^{\#(k-2)\binom{\ell}{k-1}}$
by a self-homotopy equivalence if necessary, we may assume that each
Hochster ring generator in $\cohlgy{\mathcal{Z}_{\partial\LL}}$ is represented
by a map
\(\namedright{S^{t}}{}{\#_{k=3}^{\ell+1} (S^{k}\times S^{\ell+2d-k})^{\#(k-2)\binom{\ell}{k-1}}}\)
which is the inclusion of one of the spheres in the $(\ell+2d-1)$-skeleton of
the connected sum.~$\hfill\Box$
\end{lemma}
Lemma~\ref{cupprods} lets us use combinatorial information from
the Hochster ring to deduce cup product information for the cohomology
of the connected sum. We apply this to deduce some cup product
information in $\cohlgy{\mathcal{Z}_{\partial\LL}}$.
Let $\mathcal{I}$ be an index set which runs over
all the products of two spheres in the connected sum
$\#_{k=3}^{\ell+1} (S^{k}\times S^{\ell+2d-k})^{\#(k-2)\binom{\ell}{k-1}}$.
There are $\Sigma_{k=3}^{\ell+1} (k-1)\binom{\ell}{k-1}$
elements in $\mathcal{I}$. Each $\alpha\in\mathcal{I}$ corresponds
to a product of spheres $S^{k}\times S^{\ell +2d-k}$ which
determines a nontrivial cup product in $\cohlgy{\mathcal{Z}_{\partial\LL}}$:
if $x_{\alpha},y_{\alpha}\in\cohlgy{\mathcal{Z}_{\partial\LL}}$ are generators
corresponding to the inclusions of $S^{k}$ and $S^{\ell+2d-k}$ into
the $(\ell+2d-1)$-skeleton of the connected sum, then
$x_{\alpha}\cup y_{\alpha}\neq 0$. By Lemma~\ref{cupprods}, we
may assume that $x_{\alpha}$ and $y_{\alpha}$ are Hochster ring
generators. Thus $x_{\alpha}\in\rcohlgy{\partial\LL_{I_{\alpha}}}$
and $y_{\alpha}\in\rcohlgy{\partial\LL_{J_{\alpha}}}$ for some
index sets $I_{\alpha}$ and $J_{\alpha}$ of $[m]$, where $m=\ell+d$
is the number of vertices of $\partial\LL$.
The inclusion
\(\namedright{\partial\LL-\{1\}}{}{\partial\LL}\)
induces a map
\[f\colon\namedright{\mathcal{Z}_{\partial\LL-\{1\}}}{}{\mathcal{Z}_{\partial\LL}}.\]
\begin{lemma}
\label{cupseparate}
$\mbox{Ker}\, f^{\ast}$ contains one and only one of $x_{\alpha}$ or $y_{\alpha}$.
\end{lemma}
\begin{proof}
First, in the Hochster ring for $\cohlgy{\mathcal{Z}_{\partial\LL}}$ we have
$x_{\alpha}\cup y_{\alpha}\neq 0$. So by Theorem~\ref{zkcohlgy},
$I_{\alpha}\cap J_{\alpha}=\emptyset$.
We claim that $1\in I_{\alpha}\cup J_{\alpha}$. For if not,
then $I_{\alpha}\cup J_{\alpha}$ is contained
in the vertex set for $\partial\LL -\{1\}$. Therefore, by
Theorem~\ref{zkcohlgy} all three of
$\rcohlgy{\partial\LL_{I_{\alpha}}},\rcohlgy{\partial\LL_{J_{\alpha}}},
\rcohlgy{\partial\LL_{I_{\alpha}\cup J_{\alpha}}}$
are contained in $\cohlgy{\mathcal{Z}_{\partial\LL-\{1\}}}$.
That is, $x_{\alpha},y_{\alpha}\in\cohlgy{\mathcal{Z}_{\partial\LL-\{1\}}}$
and so $x_{\alpha}\cup y_{\alpha}\in\cohlgy{\mathcal{Z}_{\partial\LL-\{1\}}}$.
But by Corollary~\ref{deletioncor}, $\mathcal{Z}_{\partial\LL-\{1\}}$
is homotopy equivalent to a wedge of spheres, implying that all
the cup products in its cohomology are zero, a contradiction.
Now, since $1\in I_{\alpha}\cup J_{\alpha}$ and
$I_{\alpha}\cap J_{\alpha}=\emptyset$, either $1\in I_{\alpha}$
or $1\in J_{\alpha}$. If $1\in I_{\alpha}$ then $1\notin J_{\alpha}$,
implying that
$x_{\alpha}\in\rcohlgy{\partial\LL_{I}}$ is not an elment of
$\cohlgy{\mathcal{Z}_{\partial\LL-\{1\}}}$ while
$y_{\alpha}\in\rcohlgy{\partial\LL_{J}}$ is. That is, $f^{\ast}(x_{\alpha})=0$
while $f^{\ast}(y_{\alpha})\neq 0$. Similarly, if $1\in J_{\alpha}$
then $f^{\ast}(x_{\alpha})\neq 0$ and $f^{\ast}(y_{\alpha})=0$.
\end{proof}
\section{Panov's problem}
\label{sec:panov}
Recall from the Introduction that if $K$ is $\ell$ disjoint points
then there is a homotopy equivalence
\begin{equation}
\label{ptequiv}
\zk\simeq\bigvee_{k=3}^{\ell+1} (S^{k})^{\wedge (k-2)\binom{\ell}{k-1}}
\end{equation}
and if $P$ is a simple polytope of dimension $d$ obtained from $\Delta^{d}$
by $\ell -1$ vertex cut operations (in any order) then there is a diffeomorphism
\begin{equation}
\label{connequiv}
\mathcal{Z}(P)\cong
\#_{k=3}^{\ell+1} (S^{k}\times S^{\ell+2d-k})^{\#(k-2)\binom{\ell}{k-1}}.
\end{equation}
Panov posed the problem of identifying the nature of the correspondence
between the decompositions in~(\ref{ptequiv}) and~(\ref{connequiv}).
In this section we give an answer to the problem.
Let $P$ be a simple polytope of dimension $d$ which has been
obtained from $\Delta^{d}$ by $\ell -1$ vertex cuts. Dualizing,
$P^{\ast}$ is a stacked polytope of dimension $d$ with $\ell$
stacks. By Proposition~\ref{stackindep}, the homotopy type of
$\mathcal{Z}(P)=\mathcal{Z}_{\partial P^{\ast}}$ is independent
of the stacking order of $P^{\ast}$. We may therefore analyze the
homotopy type of $\mathcal{Z}(P)$ by analyzing the homotopy
type of $\mathcal{Z}_{\partial\LL}$.
\begin{proof}[Proof of Theorem~\ref{Panovsoln}]
Consider the inclusion
\[\namedright{\partial\LL-\{1\}}{}{\partial\LL}.\]
The moment-angle complex, regarded as a polyhedral product, is natural
for maps of simplicial complexes, so we obtain an induced map of
moment-angle complexes
\[f\colon\namedright{\mathcal{Z}_{\partial\LL-\{1\}}}{}{\mathcal{Z}_{\partial\LL}}.\]
By Corollary~\ref{deletioncor} and~(\ref{connequiv}), up to homotopy
equivalences $f$ can be regarded as a map
\[f\colon\namedright{\bigvee_{k=3}^{\ell+1} (S^{k})^{\wedge (k-2)\binom{\ell}{k-1}}}
{}{\#_{k=3}^{\ell+1} (S^{k}\times S^{\ell+2d-k})^{\#(k-2)\binom{\ell}{k-1}}}.\]
In general, whenever $K^{\prime}$ is a full subcomplex of $K$, by~\cite{BBCG}
there is a retract of $\cxx^{K^{\prime}}$ off of $\cxx^{K}$. In our case,
since $\partial\LL-\{1\}$ is a full subcomplex of $\partial\LL$, the map $f$
has a left homotopy inverse
\[g\colon\namedright{\#_{k=3}^{\ell+1}
(S^{k}\times S^{\ell+2d-k})^{\#(k-2)\binom{\ell}{k-1}}}{}
{\bigvee_{k=3}^{\ell+1} (S^{k})^{\wedge (k-2)\binom{\ell}{k-1}}}.\]
This proves parts~(a), (b) and (c) of the theorem. Part~(d) is Lemma~\ref{cupseparate}.
\end{proof}
More is true than stated in Theorem~\ref{Panovsoln}, and it may be useful
to elaborate on it. As in Section~\ref{sec:cup}, let $\mathcal{I}$ be an index
set which runs over all the products of two spheres in the connected
sum~(\ref{connequiv}). Each $\alpha\in\mathcal{I}$ corresponds
to a product of spheres $S^{k}\times S^{\ell +2d-k}$ which determines
a nontrivial cup product in $\cohlgy{\mathcal{Z}_{\partial\LL}}$:
if $x_{\alpha},y_{\alpha}\in\cohlgy{\mathcal{Z}_{\partial\LL}}$ are generators
corresponding to the spheres
$S^{k}\vee S^{\ell +2d-k}\subset S^{k}\times S^{\ell +2d-k}$
then $x_{\alpha}\cup y_{\alpha}\neq 0$. By
Proposition~\ref{cupseparate}, $f^{\ast}$ is nonzero for one and only
one of $x_{\alpha}$ or $y_{\alpha}$. It is not immediately clear which
of $x_{\alpha}$ or $y_{\alpha}$ is sent nontrivially by~$f_{\ast}$ to
$\cohlgy{\mathcal{Z}_{\partial\LL-\{1\}}}$, so write $z_{\alpha}$
for the generator which has nontrivial image. By Lemma~\ref{cupprods},
$z_{\alpha}$ is the dual of the Hurewicz image of the composite of inclusions
\[i_{\alpha}\colon S^{t_{\alpha}}\hookrightarrow S^{k}\vee S^{\ell+2d-k}\hookrightarrow
{\#_{k=3}^{\ell+1} (S^{k}\times S^{\ell+2d-k})^{\#(k-2)\binom{\ell}{k-1}}}\]
where $t_{\alpha}$ is $k$ or $\ell+2d-k$ depending on whether
$z_{\alpha}$ is $x_{\alpha}$ or $y_{\alpha}$. Taking the wedge sum of
all the maps $i_{\alpha}$ for every $\alpha\in\mathcal{I}$ we obtain a map
\[i\colon\namedright{\bigvee_{\alpha\in\mathcal{I}} S^{t_{\alpha}}}{}
{\#_{k=3}^{\ell+1} (S^{k}\times S^{\ell+2d-k})^{\#(k-2)\binom{\ell}{k-1}}}\]
with the property that $i^{\ast}$ factors through $f^{\ast}$. That is,
there is a commutative diagram
\begin{equation}
\label{cohlgyfactor}
\diagram
& \cohlgy{\bigvee_{\alpha\in\mathcal{I}} S^{t_{\alpha}}} \\
\cohlgy{\bigvee_{k=3}^{\ell+1} (S^{k})^{\wedge (k-2)\binom{\ell}{k-1}}}
\urto^{\phi}
& \cohlgy{\#_{k=3}^{\ell+1} (S^{k}\times S^{\ell+2d-k})^{\#(k-2)\binom{\ell}{k-1}}}
\uto^{i^{\ast}}\lto^-{f^{\ast}}
\enddiagram
\end{equation}
for some ring map $\phi$. Note at this point that $\phi$
need not be induced by a map of spaces, it exists only on the level of cohomology.
By construction, $i$ is the inclusion of one factor in each product of
spheres in the connected sum. So $i^{\ast}$ is an epimorphism taking
ring generators to ring generators.
The commutativity of~(\ref{cohlgyfactor}) therefore implies that $\phi$ is also
an epimorphism, and must take ring generators to ring generators. Now observe
that both $\bigvee_{\alpha\in\mathcal{I}} S^{t_{\alpha}}$ and
$\bigvee_{k=3}^{\ell+1} (S^{k})^{\wedge (k-2)\binom{\ell}{k-1}}$
are wedges of precisely the same number of spheres. So the
domain and range of $\phi$ have the same number of ring
generators. Hence $\phi$ must be an isomorphism.
Finally, we geometrically realize $\phi$. Consider
the composite
\[\nameddright{\bigvee_{\alpha\in\mathcal{I}} S^{t_{\alpha}}}
{i}{\#_{k=3}^{\ell+1} (S^{k}\times S^{\ell+2d-k})^{\#(k-2)\binom{\ell}{k-1}}}
{g}{\bigvee_{k=3}^{\ell+1} (S^{k})^{\wedge (k-2)\binom{\ell}{k-1}}}.\]
Taking cohomology, by~(\ref{cohlgyfactor}) we obtain
$i^{\ast}\circ g^{\ast}=\phi\circ f^{\ast}\circ g_{\ast}$. Since
$g$ is a left homotopy inverse of~$f$, we therefore have
$i^{\ast}\circ g^{\ast}=\phi$. Since $\phi$ is an isomorphism,
so is $i^{\ast}\circ g^{\ast}$, implying by Whitehead's Theorem
that $g\circ i$ is a homotopy equivalence. Thus $\phi$ is
the map induced in cohomology by the homotopy equivalence
$g\circ i$. Note, however, that it may not be the case that
there is a homotopy $i\simeq f\circ(g\circ i)$, that is, it may
not be the case that~(\ref{cohlgyfactor}) can be improved to a homotopy
commutative diagram on the level of spaces.
\bibliographystyle{amsalpha}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,231 |
{"url":"https:\/\/micro-manager.org\/Micro-Manager_File_Formats","text":"# Micro-Manager File Formats\n\nMicro-Manager can save files in two formats, which referred to as \u201cseparate image files\u201d and \u201cImage file stack\u201d.\n\n# Separate image files\n\nAcquired images are saved to disk as separate TIFF files, each containing a single grayscale image. The file naming convention is \u201cimg\u201d prefix followed by frame number, channel name and slice number (img_00000000t_channel_00z.tif). In addition, the folder will contain a file named \u201cmetadata.txt\u201d that contains the metadata in JSON format.\n\n# Image file stack (in brief)\n\nA TIFF file or group of TIFF files that contain multiple acquired images per a single file. These files conform to the OME-TIFF specification, allowing them to be easily imported into a variety of analysis applications including anything that utilizes the Bio-Formats library.\n\nImage file stacks are designed to be easily imported into ImageJ without the need for a special reader plugin. A stack file can be dragged onto the ImageJ toolbar and will automatically open as a hyperstack with the same contrast settings used in Micro-Manager. Any acquisition comments typed into the Multi-Dimensional Acquisition window or the comments tab of the main Micro-Manager GUI can be viewed by pressing \u201ci\u201d with one of these files open in ImageJ.\n\nBy default, one file is created for each XY stage position (up to a maximum of 4 GB per file). In the tools-options menu, this can be changed to save all XY positions in a single file. This is especially useful for acquisitions using a large number XY positions. Since OME-TIFFs require that an identical String of XML metadata be embedded in each file in an acquisition, acquisitions that have a large number of XY positions with a small amount of data at each one waste space on disk and time by writing the same String of metadata in each file at the acquisition\u2019s conclusion.\n\nWriting to these files results in faster performance than writing to Seperate Image Files, in part because it minizes the number of system calls to create new files. This can be advantageous in situations where disk write speed is a limiting factor (i.e. writing to a server or collecting data at a high rate).\n\n# Programming using Image file stacks\n\nExisting Micro-Manager libraries can be used to easily read these files. in order to do so a java project must use MMCoreJ.jar and MMJ_.jar as libraries. These JARs can be found in the Micro-Manager-1.4\/plugins\/Micro-Manager\/ directory. Both file formats sit behind a common interface for reading and writing, org\/micromanager\/api\/TaggedImage.java. Image file stacks are implement by org\/micromanager\/acquisition\/TaggedImageStorageMultipageTiff.java. To create an instance of this class, capable of reading an existing Image file stack data set, use:\n\nTaggedImageStorageMultipageTiff stackReader = new\nTaggedImageStorageMultipageTiff(\"C:\\\\Data\\\\Directory where data set is\",\nfalse, null, false, false);\n\n\nImportant methods for utilizing this class are:\n\npublic TaggedImage getImage(int channelIndex, int\nsliceIndex, int frameIndex, int positionIndex)\npublic Set<String> imageKeys()\npublic void close()\n\n\nimageKeys() returns a [java.util.Set](http:\/\/docs.oracle.com\/javase\/6\/docs\/api\/java\/util\/Set.html) containing the image labels for each image present in the data set. An image label is simply the image's channel, slice, frame, and position indices separated by underscores. For example, the label for the image of channel 1, slice 2, frame 3, position 4 would be \"1\\_2\\_3\\_4\".\n\nclose() should be called to release the connection the files when they are no longer needed.\n\nA TaggedImage simply consists of two public fields. TaggedImage.tags is a reference to the image metadata, stored in a JSONObject. TaggedImage.pix is a pointer to the image pixels, stored in a byte[], short[], or int[] depending on the image type.\n\n## Writing Images\n\nIn addition to the MMCoreJ.jar and MMJ_.jar libraries that are required for reading Image File Stacks, writing to these files requires 4 more libraries, which can also be found in Micro-Manager-1.4\/plugins\/Micro-Manager\/: loci-common.jar, ome-xml.jar, scifio.jar, and slf4j-api-1.7.1.jar.\n\nWriting images into the Image File Stack format uses the same class as reading one of these datasets, but requires a different parameters be passed to the constructor. When writing an Image File Stack, you will need to pass a JSONObject containing a minimal amount of summary metadata to the constructor. Code for creating this summary metadata with the minimal amount of tags needed for saving is listed below. In this example, a data set of 512x512 16 bit monochrome images is created with 10 time points, 8 z slices, 2 channels, and 3 positions.\n\nsummary = new JSONObject();\nsummary.put(\"Slices\", 18;\nsummary.put(\"Positions\", 1);\nsummary.put(\"Channels\", 2);\nsummary.put(\"Frames\", 10);\nsummary.put(\"Positions\",3);\nsummary.put(\"SlicesFirst\",true);\nsummary.put(\"TimeFirst\",false);\nsummary.put(\"PixelType\", \"GRAY16\");\nsummary.put(\"Width\",512);\nsummary.put(\"Height\",512);\nsummary.put(\"Prefix\",\"Put the desired base filename here\");\n\/\/these are used to create display settings\nsummary.put(\"ChColors\", new org.json.JSONArray(\"[1,1]\"));\nsummary.put(\"ChNames\", new org.json.JSONArray(\"[\"DAPI\",\"FITC\"]\"));\nsummary.put(\"ChMins\", new org.json.JSONArray(\"[0,0]\"));\nsummary.put(\"ChMaxes\", new org.json.JSONArray(\"[65535,65535]\"));\n\n\nSlicesFirst and TimeFirst tell the image storage what order to expect images to arrive in. If the full complement of expected images does not arrive by the time the image closes, but all images up to that point have come in the expected order, the storage will automatically complete the current frame with blank images (this behavior is useful for correctly opening aborted acquisitions in ImageJ). SlicesFirst being true means a whole set of z slice images arrive before moving on to another channel. TimeFirst being false means all positions are collected at a given time point before moving on to the next time point, rather than running successive time lapses at each position.\n\nNext, create the MultipageTiffWriter. The fourth argument is a boolean specifying whether separate metadata.txt files should also be written (does not affect functionality, merely an easy extra way to view metadata). The fifth argument is a boolean flag for whether XY positions should be placed in separate files or combined into a single one. In this case we don\u2019t create metadata.txt, and we create seperate files for XY positions:\n\nTaggedImageStorageMultipageTiff storage = new\nTaggedImageStorageMultipageTiff(\"C:\/Data\/Directory where you want to\nsave\",true,summary,false,true);\n\n\nImportant methods for writing images are:\n\npublic void putImage(TaggedImage taggedImage)\npublic void finished()\npublic void close()\n\n\nfinished() should be called after no more image are going to be added. The storage becomes read only after this call\n\nclose() should be called after images are done being both read and written.\n\n# Image file stack specification\n\nMicro-Manager Image file stacks conform to both the TIFF Specification and OME TIFF Specification, contain data allowing them to be easily imported into ImageJ, store acquisition comments and display settings, and store an index map of the byte offsets of images within a file to allow for optimal reading performance.\n\n Bytes 0-7 (0x0-0x7) Standard TIFF Header 8-11 (0x8-0xb) Index map offset header (54773648 = 0x0343C790) 12-15 (0xc-0xf) Index map offset 16-19 (0x10-0x13) Display settings offset header (483765892 = 0x1CD5AE84) 20-23 (0x14-0x17) Display settings offset 24-27 (0x18-0x1b) Comments offset header (99384722 = 0x05EC7D92) 28-31 (0x1c-0x1f) Comments offset 32-35 (0x20-0x23) Summary metadata header (2355492 = 0x0023F124) 36-39 (0x24-0x27) Summary metadata length 40- (0x28-) summary metadata (UTF-8 JSON)\n\n## Image File Directories\n\nThe first IFD starts immediately after the summary metadata. Each IFD will contain the same set of TIFF tags, except for the first one in each file, which contains two ImageJ metadata tags, and two copies of the ImageDescription tag. One of these contains a string needed by ImageJ to recognize these files, and the other contains OME metadata. Although these tags appear in the first IFD, their values will not be written until the end of the file, when it is closed. The tags are written in the following order (non-standard TIFF tags have the values listed after them), following the TIFF specification requirement that they be sorted numerically:\n\nImageWidth (256 = 0x0100)\n\nImageHeight (257 = 0x0101)\n\nBitsPerSample (258 = 0x0102)\n\nCompression (259 = 0x0103)\n\nPhotometricInterpretation (262 = 0x0106)\n\nImageDescription (270 = 0x010e) (first IFD only)\u2013contains OME XML metadata\n\na 2nd ImageDescription (270 = 0x010e) (first IFD only)-\u2013contains ImageJ file opening information\n\nStripOffsets (273 = 0x0111)\n\nSamplesPerPixel (277 = 0x0115)\n\nRowsPerStrip (278 = 0x0116)\n\nStripByteCounts (279 = 0x0117)\n\nXResolution (282 = 0x011a)\n\nYResolution (283 = 0x011b)\n\nResolutionUnit (296 = 0x0128)\n\nIJMetadataByteCounts (first IFD only) (50838 = 0xc696)\n\nIJMetadata (first IFD only) (50839 = 0xc697)\n\nImmediately after these tags are written:\n\n-4 bytes containg the offset of the next IFD (per the TIFF specification)\n\n-The pixel data\n\n-In RGB files only, 6 bytes containing the values of the BitsPerSample tag Pixel values\n\n-16 bytes containing the values of the XResolution and YResolution tags\n\n## End of file\n\nAfter the last IFD, the following constructs are written:\n\n### Index map\n\nA listing of all the images contained in the file and their byte offsets. This allows a specific image to be quickly accessed without having to parse the entire file and read in image metadata. It consists of the following:\n\n-A 4 byte header (3453623 = 0x0034b2b7)\n\n-4 bytes containing the number of entries in the index map\n\n-20 bytes for each entry, with 4 bytes each allocated to the image\u2019s channel index, slice index, frame index, position index, and byte offset of the image\u2019s IFD within the file\n\nIf for some reason a file fails to write out its index map (i.e. the application crashes during file writing), opening this file will present a dialog asking if you would like to \u201cfix\u201d the data set. This fixing process consists of reading through all the IFDs present in the file to reconstruct the index map and then writing it to the end of the file.\n\nA subset of the metadata used by the ImageJ TIFF writer (ij.io.TiffEncoder.java), which allows contrast settings and acquisition comments to propagate into ImageJ. The position and size of this metadata is specified by the IJMetadataCounts and IJMetadata tags in the first IFD.\n\nA string containing the OME XML metadata for this data set. This String is referenced by the first of the two ImageDescription tags in the first IFD of the file, in accordance with the OME-TIFF specification. Since this String must be identical for all files in a data set, it is not written for any file until the entire data set is closed at the conclusion of an acquisition.\n\n### ImageJ Image Description String\n\nThe ImageJ image description String that allows these files to opened correctly as hyperstacks in ImageJ. This String is referenced by the second of the two ImageDescription tags in the first IFD of the file.\n\n### Image display settings\n\nImage display settings (channel contrast and colors), which are automatically rewritten whenever these are changed in an open data set. The first 4 bytes of this block contain the Display Settings Header (347834724 = 0x14BB8964), and the next 4 contain the number of subsequent bytes reserved for display settings. A UTF-8 JSON string containing display settings is written.","date":"2022-01-26 11:31:39","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.19016127288341522, \"perplexity\": 4342.777126432531}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-05\/segments\/1642320304947.93\/warc\/CC-MAIN-20220126101419-20220126131419-00009.warc.gz\"}"} | null | null |
The 1999 Indian general election in Jammu and Kashmir to the 13th Lok Sabha were held for 6 seats. Jammu and Kashmir National Conference won 4 seats and Bharatiya Janata Party won 2 seats.
Constituency Details
Results
Party-wise Results
List of Elected MPs
See also
Results of the 2004 Indian general election by state
Elections in Jammu and Kashmir
References
Jammu
1999
1999 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,898 |
Johnson O. Akinleye is NCCU's permanent chancellor.
NCCU coach Levelle Moton did New Edition's whole damn "If It Isn't Love" routine during the BET Awards.
Are organizers for HBCU Pay-For-Pay League "talking out the side of their neck" or making boss moves?
Hampton shakes up football coaching staff as it gears up for make-or-break season. | {
"redpajama_set_name": "RedPajamaC4"
} | 80 |
{"url":"https:\/\/puzzling.stackexchange.com\/questions\/3118\/t-knights-l-knaves-and-r-jokers-with-one-question-type-only","text":"# T Knights, L Knaves and R Jokers with one question type only\n\nI would like to try more strict conditions for puzzle, which frodoskywalker solved so simply. Basically all the same, but only one type of questions is allowed.\n\n1. We have T Knights (truth-tellers), L Knaves (liars) and R Jokers (random-tellers). $T > 0$. We know all 3 numbers, but we do not know who is who.\n2. We can chose any two of them and ask the first about the second one \"Is he a Knight?\". We can repeat this questioning procedure any number of times.\n3. We need to find one Knight.\n\nHow must T, L and R be related to make the task possible to complete?\n\nMy analysis:\n1. If $T+L \\le R$ it is impossible. $T$ jokers can simulate Knights, $L$ Knaves and we would never be able to distinguish between them and find a real Knight.\n2. If $T>R+L$ the task is possible. It is possible with $L = 0, T > R$ and we can treat Knaves as Jokers.\n3. If $R=0, L = T$ then this is impossible.\nIndeed, if you get an answer \"Yes\", then there is two possibilities: \"Kni Kni\" or \"Kna Kna\" (both Knights or both Knaves). If we get \"No\", then it is either \"Kni Kna\" or \"Kna Kni\". From this we can see that if we replace all Knights by Knaves and vice versa we will get exactly the same answers. So even if we are able to divide them all into two groups according to their type we will never know who is who.\n4. If $R=0, L > T$ then it can be solved in exactly the same way as for $L=0, T > R$ case:\nWe need to exclude all pairs who answer \"No\" and create a chain of \"Yes\" longer then half of the not excluded people. Then the last in the chain will be a Knave. Just ask him about all the rest until you hear \"No\".\n\nSo the question is what happens when $R \\neq 0$, $L>R-T$ and $L \\ge T - R$. For example, when $L = 1, T = R = 10$.\n\nWe must have $L + T > R$ and $T \\neq L + R$.\n\nThe distinguishing feature of knights is that they form a group of $T$ people who identify one another as knights and everyone else as not knights. We only have trouble if we have two groups of people who do that.\n\nIf $T = L + R$, and the jokers always lie, then the set of non-knights will be indistinguishable from the set of knights.\n\nFrom here down, consider only cases where $T \\neq L + R$. In this case, the group of knights identify $T$ people as knights and the knaves identify $L+R \\neq T$ people as knights, so we can separate those groups.\n\nThe only way to have confusion is if we have a group of $T$ jokers who identify one another as knights and everyone else as not knights. This can only happen if $T \\leq R$.\n\nIn this case, we have two groups of potential knights and everyone else, who is potentially a knave. The potential knaves have $L$ knaves and $R-T$ jokers. If $L > R-T$, then we know most of the potential knaves will reliably lie about the groups of potential knights, so we can identify which group are the actual knights.\n\nIf $L \\leq R-T$, then we can have a set of $L$ jokers who tell the truth about the people in the two sets of potential knights, which is the opposite of what the $L$ knaves say. Thus, we cannot identify which group of potential knights are the actual knights.\n\nReiterate: We have ambiguity if:\n\nA. We have $T = L+R$ and all the jokers lie.\n\nB. We have one group of $T$ jokers who all claim everyone in that group is a knight and everyone else is not and we also have another group of $L$ jokers who tell the truth about the knights and the first set of jokers, but lie about everyone else.\n\nIn any other case, we can distinguish which group are the real knights. Thus, the problem is solvable if $L + T > R$ and $T \\neq L + R$.","date":"2022-01-25 18:41:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6534201502799988, \"perplexity\": 554.9306913023663}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-05\/segments\/1642320304859.70\/warc\/CC-MAIN-20220125160159-20220125190159-00380.warc.gz\"}"} | null | null |
\section{Introduction}
In this paper we introduce the notion of stable conic bundle. This notion
appears as the stability
condition in the GIT construction of the moduli space of these
objects.
Let $X$ be a smooth complex curve of genus $g$.
Let $r>0$ and $d$ be two integer numbers. Let ${\curly L}$ be a line bundle
over $X$. These data will be fixed throughout the paper.
\begin{definition}
A conic bundle\ on $X$ of type $(r,d,{\curly L})$ is a pair $({\curly E},Q)$ where ${\curly E}$ is a
vector bundle on $X$ of rank $r$ and degree $d$, and $Q$ is a morphism
$$
Q: Sym^2 {\curly E} \to {\curly L}.
$$
A morphism between conic bundles\ $\varphi:({\curly E},Q)\to ({\curly E}',Q')$ is a
morphism
$f:{\curly E} \to {\curly E}'$ such that there is a commutative diagramme
$$
\CD
\operatorname{Sym}^2 {\curly E} @>\operatorname{Sym}^2 f>> \operatorname{Sym}^2 {\curly E}' \\
@V{Q}VV @V{Q'}VV \\
{\curly L} @>g>> {\curly L}
\endCD
$$
where $g$ is a scalar multiple of identity.
\end{definition}
Then two conic bundles\ $({\curly E},Q)$ and $({\curly E}',Q')$ will be isomorphic when
there is an isomorphism ${\curly E}\cong {\curly E}'$
that takes $Q$ into a scalar multiple of $Q'$. The name conic bundle\ comes
from the case $r=3$. We will be mostly interested in this case, and in
fact we will only define stability for $r \leq 3$.
If ${\curly L}={\mathcal{O}}_X$ and $Q$ gives a nondegenerate quadratic form on each
fiber, then the conic bundle is equivalent to an orthogonal bundle
(see \cite{R}). In this case there is already a definition of stability,
and we check in section \ref{orthogonalbundles} that it is a particular
case of our definition.
\begin{definition}
Consider a conic bundle\ $({\curly E},Q)$ and a subbundle ${\curly E}'$ of ${\curly E}$ of rank
$r'$. Let $x$ be a general point in $X$. If ${\curly F}_1$ and ${\curly F}_2$ are
subbundles of ${\curly E}$, we denote by ${\curly F}_1 {\curly F}_2$ the subbundle of
$\operatorname{Sym}^2 {\curly E}$
generated by elements of the form $f_1 f_2$ where $f_1$ and $f_2$
are local sections of ${\curly F}_1$ and ${\curly F}_2$. We define a function
$c_Q({\curly E}')$ as follows:
$$
c_Q({\curly E}')=\left\{
\begin{array}{lcl}
2, & \text{if} & Q|_{{\curly E}'{\curly E}'}\neq 0\\
1, & \text{if} & Q|_{{\curly E}'{\curly E}}\neq 0 = Q|_{{\curly E}'{\curly E}'} \\
0, & \text{if} & Q|_{{\curly E}'{\curly E}}=0.
\end{array}
\right.
$$
\end{definition}
Sometimes it will be convenient to write this type of conditions on
$Q$ in matrix form. Choosing a basis compatible with the filtration
${\curly E}' \subset {\curly E}$ these three cases can be expressed as follows
$$
\left(
\begin{array}{cc}
\times & \cdot \\
\cdot & \cdot
\end{array}
\right)
,\qquad
\left(
\begin{array}{cc}
0 & \times \\
\times & \cdot
\end{array}
\right)
,\qquad
\left(
\begin{array}{cc}
0 & 0 \\
0 & \times
\end{array}
\right)
,\qquad
$$
where $\times$ means that that block is nonzero, $0$ means that it is
zero and $\cdot$ means that it can be anything.
\begin{definition}
\label{gcritical}
Let $({\curly E},Q)$ be a conic bundle. We say that two subbundles ${\curly E}_1 \subset {\curly E}_2
\subset {\curly E}$ give a critical filtration of $({\curly E},Q)$, if $\operatorname{rk}({\curly E}_1)=1$,
$\operatorname{rk}({\curly E}_2)=2$, $\operatorname{rk}({\curly E})=3$, $Q|_{{\curly E}_1 {\curly E}_2}=0$,
and $Q|_{{\curly E}_1 {\curly E}}\neq 0
\neq Q|_{{\curly E}_2 {\curly E}_2}$.
\end{definition}
The fact that ${\curly E}_1 \subset {\curly E}_2 \subset {\curly E}$ is a critical filtration of
$({\curly E},Q)$ means that for a generic point $x\in X$, the conic $Q_x$
defined by $Q$ on the fibre of $\mathbb{P}({\curly E})$ over $x$ is smooth, the point
defined by ${\curly E}_1$ is in the conic and the line defined by ${\curly E}_2$ is
tangent to the conic. In matrix form with a basis adapted to the
filtration ${\curly E}_1\subset {\curly E}_2 \subset {\curly E}$ this can be expressed
as
$$
Q=\left(
\begin{array}{ccc}
0 & 0 & \times \\
0 & \times & \cdot \\
\times & \cdot & \cdot
\end{array}
\right)
$$
Later on (definition \ref{gcritfiltvectspac})
we will introduce a similar definition for filtrations of vector
spaces.
Now we are ready to define the notion of stability. We will only
define it for $r\leq 3$. As it is usual when one is
working with vector bundles with extra structure, this notion will
depend on a positive rational number $\tau$. We could as well take
$\tau$ to be a real number, but this wouldn't give anything new
because when we vary $\tau$ the stability of a conic bundle\ can only
change at rational values of $\tau$.
We follow the notation of \cite{H-L}: Whenever the word '(semi)stable'
appears in a statement with the symbol '$(\leq)$', two statements should
be read. The first with the word 'stable' and strict inequality, and the
second with the word 'semistable' and the relation '$\leq$'.
\begin{definition}
Let $\tau$ be a positive rational number. Let $({\curly E},Q)$ be a conic bundle\
with $r\leq 3$. We say that $({\curly E},Q)$ is (semi)stable with respect to
$\tau$ if the following conditions hold
(ss.1) If ${\curly E}'$ is a proper subbundle of ${\curly E}$, then
$$
\frac{\deg({\curly E}')-c_Q({\curly E}')\tau}{\operatorname{rk}({\curly E}')} (\leq) \frac{\deg({\curly E}) -
2\tau }{r}.
$$
(ss.2) If ${\curly E}_1\subset {\curly E}_2 \subset {\curly E}$ is a critical filtration, then
$$
\deg({\curly E}_1)+\deg({\curly E}_2) (\leq) \deg({\curly E}).
$$
\end{definition}
Note that condition (ss.1) is reminiscent of the stability conditions
for vector bundles with extra structure in the literature, but
condition (ss.2) is new. It is due to the fact that in a conic bundle, $Q$ is
a nonlinear object. So far all objects that have been considered were
linear, and this is why this kind of conditions didn't appear. This
nonlinearity is responsible for the fact that the proof is more
involved, and we have to consider only conic bundles\ with $r\leq 3$.
For higher $r$ we expect to have more conditions of the form (ss.2).
\begin{lemma}
\label{gautom}
Let $({\curly E}_1,Q_1)$ and $({\curly E}_2,Q_2)$ be stable conic bundles\ of the same type
$(r,d,{\curly L})$. Then any nontrivial morphism $\phi:({\curly E}_1,Q_1) \to
({\curly E}_2,Q_2)$ is an isomorphism, and furthermore it is a scalar
multiple of identity.
\end{lemma}
\begin{proof}
Assume that $\phi$ is nontrivial. Let $f:{\curly E}_1 \to {\curly E}_2$ be the
corresponding morphism of sheaves. Consider the subsheaves ${\curly E}'=\ker f$
of ${\curly E}_1$ and ${\curly E}''=\operatorname{im} f$ of ${\curly E}_2$. Assume ${\curly E}' \neq 0$. By
commutativity of the diagramme
$$
\CD
\operatorname{Sym}^2 {\curly E}' @>>> \operatorname{Sym}^2 {\curly E}_1 @>\operatorname{Sym}^2 f>> \operatorname{Sym}^2 {\curly E}_2 \\
@V{Q_1}VV @V{Q_1}VV @V{Q_2}VV \\
{\curly L} @= {\curly L} @>g>> {\curly L}
\endCD
$$
we have that $c_{Q_1}({\curly E}')=0$, and then by stability
$$
\frac{\deg({\curly E}')}{\operatorname{rk}({\curly E}')} < \frac{d-2 \tau}{r} <\frac{\deg({\curly E}'')-2\tau}
{\operatorname{rk}({\curly E}'')} \leq \frac{\deg({\curly E}'')-c_{Q''}({\curly E}'')\tau}{\operatorname{rk}({\curly E}'')}
<\frac{d-2\tau}{r}
$$
which is a contradiction. Then ${\curly E}'=0$ and $f$ is an isomorphism.
Now let $x\in X$ be a point, and let $\lambda$ be an eigenvalue of $f$
at the fibre over $x$. Then $h=f-\lambda \operatorname{id}_{{\curly E}_1}$ is not surjective
at $x$, hence $h$ cannot be an isomorphism and then $h=0$.
\end{proof}
A flat family of (semi)stable conic bundles\ of type $(r,d,{\curly L})$ parametrized by a
scheme $T$ is a triple $({\curly E}_T,Q_T,{\curly N})$ where ${\curly E}$ is a
vector bundle on $X\times T$, flat over $T$, that restricts to a
vector bundle of rank
$r$ and degree $d$ on each fibre $X\times t$, and $Q_T$ is a
morphims $Q_T:\operatorname{Sym}^2 {\curly E}_T \to p^*_X {\curly L} \otimes p^*_T {\curly N}$
where ${\curly N}$ is a line
bundle on $T$, and this morphim restricts to (semi)stable conic bundles\ on
each fibre. Two families $({\curly E}_T,Q_T,{\curly N})$ and
$({\curly E}'_T,Q'_T,{\curly N}')$ will be considered equivalent if there is a line
bundle ${\curly M}$ on $T$, an isomorphism $f:{\curly E}_T\otimes p^*_T {\curly M} \to
{\curly E}'_T$ and a commutative diagramme
$$
\CD
\operatorname{Sym}^2 {\curly E}_T\otimes p^*_T {\curly M}^2 @>\operatorname{Sym}^2 f>> \operatorname{Sym}^2 {\curly E}'_T \\
@V{Q}VV @V{Q'}VV \\
p^*_X{\curly L}\otimes p^*_T{\curly N} \otimes p^*_T{\curly M}^2 @>{\cong}>>
p^*_X{\curly L}\otimes p^*_T{\curly N}'
\endCD
$$
Let $\mathfrak{M}_\tau (r,d,{\curly L})^\natural$ (resp. $\overline \mathfrak{M}_\tau (r,d,{\curly L})
^\natural$) be
the functor that sends a scheme $T$ to the set of flat families of
stable (resp. semistable) conic bundles\
of type $(r,d,{\curly L})$ parametrized by $T$. The moduli space for this
functor will be denoted by $\mathfrak{M}_\tau (r,d,{\curly L})$ (resp. $\overline
\mathfrak{M}_\tau (r,d,{\curly L})$).
\smallskip
\noindent
\textbf{Theorem I.}
\label{gmaintheorem}
\textit{Let $X$ be a Riemann surface. Let $\tau>0$ be a rational number. There
exist a projective coarse moduli space $\overline \mathfrak{M}_\tau (r,d,{\curly L})$ of
semistable conic bundles\ with respect to $\tau$ of fixed type $(r,d,{\curly L})$. The
closed points of $\overline \mathfrak{M}_\tau (r,d,{\curly L})$ correspond to
S-equivalence classes of conic bundles . There is an open set $\mathfrak{M}_\tau
(r,d,{\curly L})$ corresponding to stable conic bundles. This open set is a fine moduli
space of stable conic bundles. Points in this open set
correspond to isomorphism classes of conic bundles.}
For a definition of S-equivalence, see subsection \ref{gsequiv}
At the same time we wrote this article, I. Mundet i Riera found the
conditions for existence of solutions to a generalization of the
vortex equation associated to Kaehler fibrations.
As expected, the condition he finds is,
in the case of conic bundles, the same as the condition we have found for
stability. This is explained in the appendix.
\section{GIT Construction}
\label{ggit}
In this section we will construct the moduli space of semistable
conic bundles. This construction is based on the ideas of Simpson
for the construction of the moduli space of semistable sheaves
(\cite{S}). We will follow closely the paper \cite{K-N} of King
and Newstead and the paper \cite{H-L} of Huybrechts
and Lehn. In \ref{gboundnesstheorems} we prove some boundness
theorems that are needed later, and in \ref{gconstruction} we give
the construction of the moduli space and prove the semistability
condition. The base field $k$ can be any algebraically closed field of
characteristic zero, but we are mainly interested in $\mathbb{C}$.
\subsection{Boundness theorems}
\label{gboundnesstheorems}
\begin{proposition}
\label{gboundnessb}
Let $X$ be a genus $g$ curve. Let ${\mathcal{S}}$ be a set of vector bundles
on $X$ with degree $d$ and rank $r$. Assume that there is a constant
$b$ such that if ${\curly E} \in {\mathcal{S}}$ and ${\curly E}'$ is a nonzero subsheaf of ${\curly E}$,
then
$$
\mu({\curly E}')=\frac{\deg ({\curly E}')}{\operatorname{rk} ({\curly E}')} \leq b.
$$
Then there is a constant $m_0$ such that if $m\geq m_0$, for all ${\curly E}
\in {\mathcal{S}}$, we have $h^1({\curly E}(m))=0$ and ${\curly E}(m)$ is generated by global
sections. Hence ${\mathcal{S}}$ is bounded.
\end{proposition}
\begin{proof}
Let $x$ be a point of the curve $X$ and ${\curly E}\in {\mathcal{S}}$. The exact sequence
$$
0 \to {\curly E}(m)\otimes {\mathcal{O}}_X(-x) \to {\curly E}(m) \to {\curly E}(m)|_x \to 0
$$
gives that if $h^1({\curly E}(m)\otimes {\mathcal{O}}_X(-x))=0$ for all $x\in X$, then
${\curly E}(m)$ is
generated by global sections and $h^1({\curly E}(m))=0$.
Assume that $h^1({\curly E}(m)\otimes {\mathcal{O}}_X(-x)) \neq 0$. Then by Serre
duality there is a
nonzero morphism ${\curly E}(m)\otimes {\mathcal{O}}_X(-x) \to {\curly K}_X$, where ${\curly K}_X$
is the canonical divisor. This gives an
effective divisor
$D$ on $X$ and an exact sequence
$$
0 \to {\curly E}'(m) \to {\curly E}(m) \to {\curly K}_X(x-D) \to 0.
$$
Let $d'=\deg({\curly E}')$. We have $\operatorname{rk} ({\curly E}')=r-1$. Then
$$
d'=(1-r)m+d+rm-(2g-1-\deg (D))\geq d-2g+1+m
$$
On the other hand, by hypothesis $d'\leq (r-1)b$, and combining both
inequalities we get
$$
m\leq (r-1)b-d+2g-1.
$$
Then if we take $m_0>(r-1)b-d+2g-1$, for any $m\geq m_0$ and $x\in X$
we will have $h^1({\curly E}(m)\otimes {\mathcal{O}}_X(-x))=0$, thus ${\curly E}(m)$ is
generated by global
sections and $h^1({\curly E}(m))=0$. By standard methods using the Quot
scheme, this implies that ${\mathcal{S}}$ is bounded.
\end{proof}
\begin{corollary}
\label{gboundcor1}
The same conclusion is true for the set of vector bundles
${\curly E}$ occurring in semistable conic bundles\ $({\curly E},Q)$ of fixed type.
The constant $m_0$ depends on $X$, $\tau$, $r$ and $d$, but not on
${\curly L}$.
\end{corollary}
\begin{proof}
By condition (ss.1) we have that for every subsheaf ${\curly E}'$ of ${\curly E}$
$$
\frac{\deg ({\curly E}')}{\operatorname{rk} ({\curly E}')} \leq \frac{d-2\tau}{r}+\frac{c_Q
({\curly E}') \tau}{\operatorname{rk} ({\curly E}')} \leq \frac{d-2\tau}{r}+2\tau.
$$
Take $b=\frac{d-2\tau}{r}+2\tau$ and apply the proposition.
\end{proof}
\begin{corollary}
\label{gboundcor2}
Let ${\mathcal{S}}$ be the set of semistabilizing sheaves, i.e. sheaves ${\curly E}'$,
${\curly E}_1$, ${\curly E}_2$ that give equality in condition (ss.1) or (ss.2). Then
the conclusions of proposition \ref{gboundnessb} are also true for
${\mathcal{S}}$.
\end{corollary}
\begin{proof}
By semistability, the slope of a subsheaf of a sheaf in ${\mathcal{S}}$ is
bounded. On the other hand there are only a finite number of
possibilities for rhe rank and degree of a sheaf in ${\mathcal{S}}$, then we
can apply proposition \ref{gboundnessb}.
\end{proof}
Now we will state two lemmas of King and Newstead
(\cite[lemma 2.2]{K-N} and \cite[corollary 2.6.2]{K-N}).
\begin{lemma}
Let ${\curly E}$ be a torsion free sheaf such that for all subsheaf ${\curly F}$ of ${\curly E}$,
$\mu({\curly F})\leq b$. If $b<0$, then $h^0({\curly E})=0$. If $b\geq 0$ then
$h^0({\curly E})\leq \operatorname{rk}({\curly E}) (b+1)$.
\end{lemma}
\hfill $\Box$
\begin{lemma}
\label{gbound2}
Fix $R$, $b$, $k$. Then there exists an $n_0$ such that if ${\mathcal{S}}$
is a set of torsion free sheaves with
(i) $\operatorname{rk}({\curly E}) \leq R$
(ii) $\mu({\curly F})\leq b$ for all nonzero subsheaves ${\curly F}$ of ${\curly E}$
(iii) For some $n\geq n_0$
$$
h^0({\curly E}(n))\geq \operatorname{rk}({\curly E}) (\chi({\mathcal{O}}_X (n))+k)
$$
Then the set ${\mathcal{S}}$ is bounded.
\end{lemma}
\hfill $\Box$
\subsection{Construction and proof of main theorem}
\label{gconstruction}
\hfil
\medskip
Now we will give the GIT construction of the moduli space. We will
assign a point in a projective scheme $Z$ to a
conic bundle\ $({\curly E},Q)$ of fixed type $(r,d,{\curly L})$. Let $P$ be the Hilbert
polynomial of ${\curly E}$, i.e. $P(m)=rm+d+r(1-g)$. We will assume that $m$ is
large enough so that corollaries \ref{gboundcor1} and \ref{gboundcor2} are
satisfied. Let $V$ be a vector space of dimension $p=P(m)$. Let ${\mathcal{H}}$
be the Hilbert scheme $\operatorname{Hilb}(V\otimes {\mathcal{O}}_X(-m),P)$ parametrizing
quotients of $V\otimes {\mathcal{O}}_X(-m)$ with Hilbert polynomial $P$.
Let $l>m$ be an integer, $W=H^0({\mathcal{O}}_X (l-m))$, and $G$ be the
Grassmannian $\operatorname{Grass} (V\otimes W,p)$ of quotients of $V\otimes W$ of
dimension $p$. For $l$ large enough we have embeddings
$$
{\mathcal{H}} \to G \to \mathbb{P}(\Lambda ^{P(l)}(V\otimes W))
$$
Let $B=H^0({\curly L})$ and ${\mathcal{P}}=\mathbb{P}(\operatorname{Sym} ^2(V^\vee \otimes B))$. Given $({\curly E},Q)$
and an isomorphism $V\cong H^0({\curly E}(m))$ we get a point $(\tq,\tQ)$ in
${\mathcal{H}} \times {\mathcal{P}}$ as follows:
The vector bundle ${\curly E}(m)$ is generated by global sections (corollary
\ref{gboundcor1}), then we have a quotient
$$
q:V\otimes {\mathcal{O}}_X(-m) \cong H^0({\curly E}(m))\otimes {\mathcal{O}}_X(-m) \twoheadrightarrow {\curly E}.
$$
Denote by $\tilde q$ the point in ${\mathcal{H}}$ corresponding to this
quotient. On the other hand, we get a point $\tilde Q$ in ${\mathcal{P}}$ by the
composition
$$
\operatorname{Sym} ^2 V \cong \operatorname{Sym}^2 H^0({\curly E}(m)) \to H^0(\operatorname{Sym} ^2{\curly E}(m)) \to H^0({\curly L}(2m))=B
$$
Let $\tilde Q$ be a point in ${\mathcal{P}}$. We will denote by $Q'$ a
representative of $\tilde Q$, i.e. $Q':\operatorname{Sym}^2 V \to B$. This gives
(up to multiplication by a scalar) an evaluation
$$
\operatorname{ev}:\operatorname{Sym}^2 V \otimes {\mathcal{O}}_X(-2m) \to B\otimes {\mathcal{O}}_X(-2m)
\to {\curly L}.
$$
Let $Z$ be the closed subset of ${\mathcal{H}} \times {\mathcal{P}}$ of points $(\tq,\tQ)$ such
that (some multiple) of this evaluation map factors
through $\operatorname{Sym}^2 {\curly E}$
$$
\operatorname{Sym}^2 V \otimes {\mathcal{O}}_X(-2m) \to \operatorname{Sym}^2 {\curly E} \to {\curly L}.
$$
The group $\operatorname{SL}(V)$ acts in a natural way on ${\mathcal{H}} \times {\mathcal{P}}$.
A point in $Z$ will be called ``good'' if the quotient
$$
q:V\otimes {\mathcal{O}}_X(-m) \twoheadrightarrow {\curly E}
$$
induces an isomorphism $V \stackrel{\cong}\to H^0({\curly E}(m))$, and ${\curly E}$ is
torsion free. Note that a conic bundle\ $({\curly E},Q)$ gives a ``good'' point in $Z$
and conversely we can recover the conic bundle\ from the point, and two
``good'' points correspond to the same conic bundle\ iff they are in the same
orbit of the action of $\operatorname{SL}(V)$. This action on ${\mathcal{H}} \times {\mathcal{P}}$
preserves the subscheme $Z$ and the subset of ``good'' points.
Let ${\curly M}$ be the line bundle on ${\mathcal{H}}$ given by the embedding ${\mathcal{H}} \to
\mathbb{P}(\Lambda ^{P(l)}(V\otimes W))$. Embedd $Z$ in projective space with
${\mathcal{O}}_Z(n_1,n_2)=p_{{\mathcal{H}}}^*{\curly M} ^{\otimes n_1} \otimes p_{{\mathcal{P}}}^*
{\mathcal{O}}_{{\mathcal{P}}} (n_2)$
$$
Z \hookrightarrow \mathbb{P}(\operatorname{Sym} ^{n_1}[\Lambda ^{P(l)}(V\otimes W)] \otimes
\operatorname{Sym} ^{n_2}[\operatorname{Sym}^2 V^\vee \otimes B])
$$
The group $\operatorname{SL}(V)$ acts naturally on $\operatorname{Sym} ^{n_1}[\Lambda ^{P(l)}
(V\otimes W)] \otimes \operatorname{Sym} ^{n_2}[\operatorname{Sym}^2 V^\vee \otimes B]$, and
this gives a linearization for the action of $\operatorname{SL}(V)$ on $Z$.
Now we will characterize the (semi)stable points of $Z$ under the
action of $\operatorname{SL}(V)$ with the linearization induced by
${\mathcal{O}}_X(n_1,n_2)$. We will take
$$
\frac{n_2}{n_1}=\frac{P(l)-P(m)}{P(m)-2\tau} \tau.
$$
\textbf{Notation.} Given a point $(\tq,\tQ)\in Z$ and a subspace $V' \subset
V$ we denote by ${\curly E}_{V'}$ the image of $V'\otimes {\mathcal{O}}_X(-m)$ under the
quotient $q:V\otimes {\mathcal{O}}_X(-m) \to {\curly E}$. Note that $V' \subset
H^0({\curly E}_{V'}(m))$, but in general they are not equal. If ${\curly E}' \subset {\curly E}$
is a subsheaf of ${\curly E}$ we have
${\curly E}_{H^0({\curly E}'(m))} \subset {\curly E}'$, with equality if ${\curly E}'(m)$ is generated by
global sections. Given a sheaf ${\curly F}$, we will denote
by $P_{\curly F}$ its Hilbert polynomial.
The following definition is analogous to definition \ref{gcritical}.
\begin{definition}
\label{gcritfiltvectspac}
Let $(\tq,\tQ)$ be a point in $Z$. Let
$V_1\subset V_2 \subset V_3=V$ be a filtration of $V$. Let $Q'_{ab}$
be the restriction of $Q':\operatorname{Sym} ^2 V \to B$ to
$V_a\otimes V_b$. We say that $V_1$, $V_2$ give a critical filtration of
$(\tq,\tQ)$, if $\operatorname{rk}({\curly E}_{V_1})=1$, $\operatorname{rk}({\curly E}_{V_2})=2$,
$Q'_{12}=0$, and $Q'_{13}\neq 0 \neq
Q'_{22}$.
\end{definition}
\begin{proposition}
\label{gmainprop}
For $l$ large enough the point $(\tq,\tQ) \in Z$ is
(semi)stable by the action of $\operatorname{SL}(V)$ with respect to the linearization
by ${\mathcal{O}}_Z(n_1,n_2)$ iff:
(*.1) If $V' \varsubsetneq V$ is a subspace of $V$, then
$$
\dim V' (n_1 P(l)+2n_2) (\leq) \dim V (n_1 P_{{\curly E}_{V'}}(l) +c_Q
({\curly E}_{V'})n_2).
$$
(*.2) If $V_1\subset V_2 \subset V$ is a critical
filtration, then
$$
(\dim V_1 + \dim V_2)(n_1 P(l)+2 n_2) (\leq) \dim V
(n_1(P_{{\curly E}_{V_1}}(l)+ P_{{\curly E}_{V_2}}(l)) +2 n_2).
$$
\end{proposition}
\begin{proof}
We will apply the Hilbert-Mumford criterion: a point $(\tilde q,
\tilde Q)$ is (semi)stable iff for all one-parameter subgroup (1-PS)
$\lambda$ of $\operatorname{SL}(V)$ we
have $\mu((\tq,\tQ),\lambda) (\leq) 0$, where
$\mu((\tq,\tQ),\lambda)$ is the minimum weight of the action
of $\lambda$ on $(\tq,\tQ)$.
Let $p=P(m)$. A 1-PS $\lambda$ of $\operatorname{SL}(V)$ is equivalent to a basis
$\{v_1,\dots,v_p\}$ of $V$ and a weight vector
$(\gamma_1,\dots,\gamma_p)$ with $\gamma_i \in \mathbb{Z}$, $\gamma_1\leq
\dots \leq \gamma_p$, and $\sum \gamma_i=0$. The set
${\mathcal{C}}$ of all weight vectors is a cone in
$\mathbb{Z}^p$. If a basis of $V$ has been chosen, then by a slight abuse of
notation we will denote $\mu((\tq,\tQ),\lambda)$ by $\mu((\tq,\tQ),\gamma)$, where
$\gamma \in {\mathcal{C}}$.
We will choose a set of one-parameter subgroups, calculate
$\mu((\tq,\tQ),\lambda)$, and then imposing $\mu((\tq,\tQ),\lambda)(\leq)0$ we will
obtain necessary conditions for $(\tq,\tQ)$ to be (semi)stable.
Then we will show that the chosen set of one-parameter subgroups is
sufficient, in the sense that if we check that
$\mu((\tq,\tQ),\lambda)(\leq)0$ for all one-parameter subgroups in this set,
then the same will hold for any arbitrary one-parameter subgroup
in ${\mathcal{C}}$.
We have $\mu((\tq,\tQ),\lambda)=n_1 \mu(\tilde q,\lambda)+n_2 \mu(\tilde Q,\lambda)$,
where $\mu(\tilde q,\lambda)$ (resp. $\mu(\tilde Q,\lambda)$) is the minimum
weight of the action of $\lambda$ on $\tilde q\in {\mathcal{H}}$ (resp. $\tilde Q\in {\mathcal{P}}$).
Fix a basis $\{v_1,\dots,v_p\}$ of $V$. Define $\varphi(i)=\dim q'(
\langle v_1,\dots,v_i \rangle \otimes W)$, where $q':V\otimes W
\twoheadrightarrow k^{P(l)}$ is
the quotient corresponding to the point $\tilde q \in {\mathcal{H}}$. We have (see
\cite{S})
$$
\mu(\tilde q,\gamma)=\sum_{i=1}^p \gamma_i (\varphi(i)-\varphi(i-1)).
$$
On the other hand
$$
\mu(\tilde Q,\gamma)=\min _{i,j\in \{1,\dots,p\}} \{\gamma_i+\gamma_j :
Q'(v_i,v_j)\neq 0 \}.
$$
Note that $\mu(\tilde q,\gamma)$ is linear on $\gamma \in {\mathcal{C}}$, but
$\mu(\tilde Q,\gamma)$ is not.
\bigskip
\noindent\textit{\textbf{GIT (semi)stable implies conditions (*)}}
\bigskip
Let $(\tq,\tQ)$ be a (semi)stable point in $Z$. Let $\{v_1,\dots,v_p\}$ be a
basis of $V$. Define
$$
i_k=\min \{i: \operatorname{rk}({\curly E}_{\langle v_1,\dots,v_i\rangle}) \geq k\}.
$$
Note that if $(\tq,\tQ)$ is ``good'', then the map $V\to H^0({\curly E}(m))$ is an
isomorphism (in particular injective), and then $i_1=1$. Later on we
will see that for sufficiently large $m$, a semistable point is
``good'', but now we won't assume that $(\tq,\tQ)$ is ``good''. Define a
filtration of $V$
$$
V_1=\langle v_1,\dots,v_{i_1}\rangle \subset V_2=\langle
v_1,\dots,v_{i_2} \rangle
\subset V_3=V.
$$
Let $Q'_{ab}$ be the restriction of $Q':\operatorname{Sym} ^2 V \to B$ to
$V_a\otimes V_b$. To calculate $\mu(\tilde Q,\lambda)$ we distinguish seven
cases.
$$
\begin{array}{rll}
1)& Q'_{11}\neq 0 & \mu(\tilde Q,\lambda)=2\gamma_{i_1} \\
2)& Q'_{11}=0,\; Q'_{12}\neq 0 &
\mu(\tilde Q,\lambda)=\gamma_{i_1}+\gamma_{i_2} \\
3)& Q'_{12}=0,\; Q'_{13}\neq 0\neq Q'_{22} & \mu(\tilde Q,\lambda)=
\min(2\gamma_{i_2},\gamma_{i_1}+\gamma_{i_3}) \\
4)& Q'_{13}=0,\; Q'_{22}\neq 0 & \mu(\tilde Q,\lambda)= 2\gamma_{i_2} \\
5)& Q'_{22}=0,\; Q'_{13}\neq 0 & \mu(\tilde Q,\lambda)=
\gamma_{i_1}+\gamma_{i_3} \\
6)& Q'_{13}=Q'_{22}=0,\; Q'_{23}\neq 0 & \mu(\tilde Q,\lambda)=
\gamma_{i_2}+\gamma_{i_3} \\
7)& Q'_{23}=0,\; Q'_{33}\neq 0 & \mu(\tilde Q,\lambda)= 2\gamma_{i_3} \\
\end{array}
$$
Note that in all cases, except case 3, $\mu(\tilde Q,\lambda)$ is a linear
function of $\gamma \in {\mathcal{C}}$.
First we will consider weight vectors of the form
\begin{eqnarray}
\gamma^{(i)}=(\overbrace{i-p,\dots,i-p}^i,\overbrace{i,\dots,i}^{p-i})
\; \; \;
(1\leq i < p).
\label{ggenerator1}
\end{eqnarray}
and define $V'=\langle v_1,\dots,v_i\rangle $ (it is clear that any
subspace of $V$ can
be written in this form, after choosing an appropriate bases for $V$).
We have $\mu(\tilde q,\gamma^{(i)})=-p \varphi(i)+i \varphi(p)$. To obtain
a formula for $\mu(\tilde Q,\gamma^{(i)})$ we have to analyze each of the
seven cases. We will only work out the details for cases 2 and 3, the
remaining cases being similar to case 2.
In case 2 we have
$\mu(\tilde Q,\gamma^{(i)})=\gamma^{(i)}_{i_1}+\gamma^{(i)}_{i_2}$. Then,
according to the value of $i$ we have
$$
\mu(\tilde Q,\gamma^{(i)})=
\left\{
\begin{array}{lll}
2i &,i<i_1 &(\operatorname{rk}({\curly E}_{V'})=0)\\
2i-p &,i_1\leq i < i_2&(\operatorname{rk}({\curly E}_{V'})=1)\\
2i-2p &,i_2 \leq i&(\operatorname{rk}({\curly E}_{V'})\geq 2)\\
\end{array}
\right .
$$
In case 3 we have $\mu(\tilde Q,\gamma^{(i)})= \min (2\gamma^{(i)},
\gamma^{(i)}_{i_1}+\gamma^{(i)}_{i_3})$, hence
$$
\mu(\tilde Q,\gamma^{(i)})=
\left\{
\begin{array}{lll}
2i &,i<i_1 &(\operatorname{rk}({\curly E}_{V'})=0)\\
2i-p &,i_1\leq i < i_2&(\operatorname{rk}({\curly E}_{V'})=1)\\
2i-2p &,i_2 \leq i&(\operatorname{rk}({\curly E}_{V'})\geq 2)\\
\end{array}
\right .
$$
Doing the calculation for the seven cases we check that in every case
we have
$$
\mu(\tilde Q,\gamma^{(i)})=2i-c_Q({\curly E}_{V_i})p
$$
Then $\mu((\tq,\tQ),\gamma^{(i)})(\leq)0$ gives
$$
n_1(-p \varphi(i) +i \varphi(p))+ n_2(2i-c_Q({\curly E}_{V'}))(\leq) 0
$$
If we vary $V'$ (allowing $V'=V$), the submodules ${\curly E}_{V'}$ are
bounded, so we can take $l$ large enough such that
$\varphi(i)=P_{{\curly E}_{V'}}(l)$. We have $i=\dim V'$, and $\varphi(p)=P(l)$,
and then we obtain condition (*.1).
To obtain condition (*.2), assume that we have subspaces $V_1 \subset
V_2 \subset V$ giving a critical filtration. Let $i=\dim V_1$ and
$j=\dim V_2 - \dim V_1$. Take a bases $\{v_1,\dots,v_p\}$ of $V$ adapted
to this filtration, i.e. such that $V_1=\langle v_1,\dots,v_i\rangle $ and
$V_2=\langle v_1,\dots v_{i+j}\rangle $. Consider the weight vector
\begin{eqnarray}
\lefteqn{\gamma^{(i)}+\gamma^{(i+j)}=} \nonumber \\
& &(\overbrace{2i+j-2p,\dots,2i+j-2p}^{i},
\overbrace{2i+j-p,\dots,2i+j-p}^{j},
\overbrace{2i+j,\dots,2i+j}^{p-i-j}).\label{ggenerator2}
\end{eqnarray}
An easy computation then shows
$\mu(\tilde Q,\gamma^{(i)}+\gamma^{(i+j)})= 2(\dim V_1+\dim V_2 -\dim V)$.
On the other hand $\mu(\tilde q,
\gamma^{(i)}+\gamma^{(i+j)})= (\dim V_1+\dim V_2)P(l) -\dim
V(P_{V_1}(l) + P_{V_2}(l))$, and then
$\mu((\tq,\tQ),\gamma^{(i)}+\gamma^{(i+j)}) (\leq) 0$ gives condition (*.2).
\bigskip
\noindent\textit{\textbf{ Conditions (*) imply GIT (semi)stable}}
\bigskip
Now we have to show that the one-parameter subgroups that we have used
are sufficient. As we did before, we will fix an arbitrary base $V$,
and we consider the seven different cases. In all cases except 3,
$\mu(\tilde Q,\gamma)$, and hence $\mu((\tq,\tQ),\gamma)$, is a linear function of
$\gamma \in {\mathcal{C}}$, and then to prove that $\mu((\tq,\tQ),\gamma)(\leq)0$ for
all $\gamma$ it is enough to check it on the generators $\gamma^{(i)}$
defined above (\ref{ggenerator1}).
In case 3 we have $\mu(\tilde Q,\gamma)= \min(2\gamma_{i_2} , \gamma_{i_1} +
\gamma_{i_3})$, hence it is no longer linear on $\gamma$, and it is
not enough to check the condition on the generators
$\gamma^{(i)}$. But it is a piecewise linear function. The cone ${\mathcal{C}}$
of weights is divided in two cones
\begin{eqnarray*}
{\mathcal{C}}^>=\{(\gamma_1,\dots,\gamma_p)\in {\mathcal{C}}: 2\gamma_{i_2} \geq \gamma_{i_1} +
\gamma_{i_3}\} \\
{\mathcal{C}}^<=\{(\gamma_1,\dots,\gamma_p)\in {\mathcal{C}}: 2\gamma_{i_2} \leq \gamma_{i_1} +
\gamma_{i_3}\}
\end{eqnarray*}
Observe that $\mu(\tilde Q,\gamma)$ is linear on each of these cones. We
will use the following lemma.
\begin{lemma}
\label{gconelemma}
Let ${\mathcal{C}}$ be a cone in $\mathbb{Z}^p$, let $\gamma^{(i)}$ be a set of
generators of ${\mathcal{C}}$, i.e. ${\mathcal{C}}=(\oplus_i \mathbb{Q}^+ \gamma^{(i)}) \cap
\mathbb{Z}^p$. Let $A:\mathbb{Z}^p \to \mathbb{Q}$ be a linear function such that
$A(\gamma^{(i)}) \in \{1,0,-1\}$. Let ${\mathcal{C}}^>$ be the subcone $\{v\in
{\mathcal{C}}: A(v)\geq 0\}$. Then the set of vectors
$$
v_{i,j}=
\left\{
\begin{array}{ll}
\gamma^{(i)} &,\; A(e_i)\geq 0 \\
\gamma^{(i)}+\gamma^{(i+j)} &,\; A(e_i)=-1,\; A(e_{i+j})=1 \\
0 &,\; \text{otherwise.}\\
\end{array}
\right .
$$
generate ${\mathcal{C}}^>$.
\end{lemma}
\hfill $\Box$
We apply this lemma with $A(\gamma)=(2\gamma_{i_2}- \gamma_{i_1}-
\gamma_{i_3})/p$ (and then with the negative of this, for ${\mathcal{C}}^<$),
and we obtain a set of generators for ${\mathcal{C}}^>$ and ${\mathcal{C}}^<$. But all
these vectors are either of the form $\gamma^{(i)}$ with $1\leq i < p$,
or of the form $\gamma^{(i)}+
\gamma^{(i+j)}$ with $\operatorname{rk}({\curly E}_{\langle v_1,\dots,v_i\rangle })=1$ and
$\operatorname{rk}({\curly E}_{\langle v_1,\dots,v_{i+j}\rangle })=2$, and we have already
considered
them.
\end{proof}
\begin{remark}
\textup{In the following propositions we will prove that conditions
(*) are equivalent to the stability conditions (s). Recall that $\dim
V=p=P(m)$, $\varphi(i)=P_{E_{V'}}(l)$, $\varphi(p)=P(l)$ and $\dim
V'=i$. The idea is to show that for $l\gg m \gg 0$, we can replace
$P(l)$ by $\operatorname{rk}(E)l$, $P_{E_{V'}}(l)$ by $\operatorname{rk}(E')l$, $P(m)$ by
$\deg(E)+ rm$ and $\dim V'$ by $P_{E_{V'}}(m)$, and this by $\deg(E')+
rm$.}
\end{remark}
\begin{proposition}
\label{gtrans}
For $m$ and $l$ large enough we have that conditions (*) are equivalent
to:
(**.1) If $V'\varsubsetneq V$ is a subspace of $V$, then
$$
r(\dim V' - c_Q({\curly E}_{V'})) \leq \operatorname{rk}({\curly E}_{V'})(\dim V -
2 \tau),
$$
and in case of equality we also require $\dim
V'(\leq) P_{{\curly E}_{V'}}(m)$.
(**.2) If $V_1\subset V_2 \subset V$ is a critical filtration, then
$$
\dim V_1 + \dim V_2 \leq \dim V
$$
and in case of equality we also require $\dim V_1 + \dim V_2 (\leq)
P_{{\curly E}_{V_1}}(m)+P_{{\curly E}_{V_2}}(m)$.
\end{proposition}
\begin{proof}
We rewrite (*.1) using
$$
\frac{n_2}{n_1}=\frac{P(l)-P(m)}{P(m)-2\tau} \tau.
$$
We obtain
\begin{eqnarray*}
[(\dim V_1 - c_Q({\curly E}_{V'})\tau)r - \operatorname{rk}({\curly E}_{V'})(\dim V -
2\tau )](l-m) + \\
+(\dim V - 2 \tau) \dim V (\dim V' - P_{{\curly E}_{V'}})(m)
(\leq) 0.
\end{eqnarray*}
We have $(l-m)\gg 0$ and $m\gg 0$, hence $\dim V >2\tau$ and the
result follows. Now we rewrite (*.2), using
$r=3$, $\operatorname{rk}({\curly E}_{V_1})=1$ and $\operatorname{rk}({\curly E}_{V_2})=2$.
\begin{eqnarray*}
3(\dim V_1 + \dim V_2 - \dim V)(l-m) + \\
+(\dim V -2\tau)\dim V (\dim
V_1 + \dim V_2 - P_{{\curly E}_{V_1}}(m)-P_{{\curly E}_{V_2}}(m))(\leq)0,
\end{eqnarray*}
and the result follows.
\end{proof}
\begin{proposition}
\label{giden}
For $m$ and $l$ large enough, we have
(i) If $({\curly E},Q)$ is a (semi)stable conic bundle,
then the corresponding point $(\tq,\tQ)$ in $Z$ is $GIT$ (semi)stable under
the action of $\operatorname{SL}(V)$.
(ii) If $(\tq,\tQ)\in Z$ is a $GIT$ semistable point, then $\tilde q$ is
``good'' and $h^1({\curly E}(m))=0$.
(iii) If $(\tq,\tQ)\in Z$ is a $GIT$ (semi)stable point, then the
corresponding conic bundle\ $({\curly E},Q)$ is (semi)stable.
Note that thanks to (ii), in (iii) we know that ${\curly E}$ is torsion free.
\end{proposition}
\begin{proof}
We will proof the three items in three steps
\bigskip
\noindent\textit{\textbf{Step 1. (Semi)stable conic bundle\ $\Rightarrow$
GIT (semi)stable $(\tq,\tQ)$}}
\bigskip
We will use proposition \ref{gtrans}. We will start checking (**.1).
Let ${\mathcal{S}}$ be the set of vector bundles ${\curly E}'$ that are subsheaves of
bundles ${\curly E}$ occurring in semistable conic bundles. It satisfies hypothesis
(i) and (ii) of lemma \ref{gbound2} with $R=3$ and $b=\frac{d-2 \tau}
{r} +2 \tau$. Let $k=\frac{d-2\tau}{r}$, $n$ large enough, so that
propositions \ref{gbound2}, \ref{gboundcor1} and \ref{gboundcor2} hold,
and let ${\mathcal{S}}_n$ be the subset of ${\mathcal{S}}$ consisting of bundles ${\curly E}'$
that satisfy hypothesis (iii) of lemma \ref{gbound2}. Then the
set ${\mathcal{S}}_n$ is bounded. Taking $m>n$ large enough we then have
$h^1({\curly E}'(m))=0$ for ${\curly E}' \in {\mathcal{S}}_n$. In other words,
\begin{eqnarray}
h^0({\curly E}'(m))=\operatorname{rk}({\curly E}')(\chi(\SO_X(m)) + \frac{\det ({\curly E}')}{\operatorname{rk}({\curly E}')}),\;\;\;\;\;
\text{for}\;\; {\curly E}'\in {\mathcal{S}}_n
\end{eqnarray}
On the other hand, we still have
\begin{eqnarray}
\label{gineq1}
h^0({\curly E}'(m))<\operatorname{rk}({\curly E}')(\chi(\SO_X(m)) + \frac{d-2\tau}{r}),\;\;\;\;\;
\text{for}\;\; {\curly E}'\in {\mathcal{S}} \setminus {\mathcal{S}}_n
\end{eqnarray}
Let $V'$ be a subspace of $V$, and ${\curly E}_{V'}$ the corresponding
sheaf. If ${\curly E}_{V'}$ belongs to ${\mathcal{S}}_n$, we get that condition (ss.1)
implies (**.1), because
\begin{eqnarray*}
\frac{\dim V' - c_Q({\curly E}_{V'})\tau}{\operatorname{rk}({\curly E}_{V'})} \leq
\frac{h^0({\curly E}_{V'}(m))-c_Q({\curly E}_{V'})\tau}{\operatorname{rk}({\curly E}_{V'})} = & \\
=\frac{\deg ({\curly E}_{V'})-c_Q({\curly E}_{V'})\tau}{\operatorname{rk}({\curly E}_{V'})}+\chi(\SO_X(m))
(\leq) & \\
(\leq) \frac{d-2\tau}{r}+\chi(\SO_X(m)) = \frac{\dim V - 2\tau}{r}
\end{eqnarray*}
and $\dim V' \leq h^0({\curly E}_{V'}(m))=P_{{\curly E}_{V'}}(m)$, because
$h^1({\curly E}'(m))=0$.
On the other hand, if ${\curly E}_{V'}$ belongs to ${\mathcal{S}}\setminus{\mathcal{S}}_n$,
inequality (\ref{gineq1}) implies (**.1)
$$
\frac{\dim V' - c_Q({\curly E}_{V'})}{\operatorname{rk}({\curly E}_{V'})} \leq
\frac{h^0({\curly E}_{V'}(m))}{\operatorname{rk}({\curly E}_{V'})} <
\frac{d-2\tau}{r}+\chi(\SO_X(m))=\frac{\dim V -2\tau}{r}
$$
In both cases,
if inequality (ss.1) is strict, then inequality (**.1) is
also strict. But assume that there is a semistabilizing subsheaf
${\curly E}'$ of ${\curly E}$ (i.e. giving equality in (ss.1)). By corollary
\ref{gboundcor2}, ${\curly E}'(m)$ is generated by global sections. Let
$V'=H^0({\curly E}'(m)) \subset H^0({\curly E}(m))=V$. Then ${\curly E}'={\curly E}_{V'}$,
and we have
\begin{eqnarray*}
\lefteqn{\frac{\dim V' -c_Q({\curly E}_{V'})\tau}{\operatorname{rk}({\curly E}_{V'})} =
\frac{\deg({\curly E}_{V'})-c_Q({\curly E}_{V'})}{\operatorname{rk}({\curly E}_{V'})}+\chi(\SO_X(m))=} \\
& & \frac{d-2\tau}{r}+\chi(\SO_X(m))=\frac{\dim V -2\tau}{r}
\end{eqnarray*}
and $\dim V'=P_{{\curly E}_{V'}}(m)$.
Now we will check condition (**.2). Let ${\mathcal{T}}$ be the set of vector
bundles of the form ${\curly E}_1\oplus {\curly E}_2$ such that ${\curly E}_1 \subset {\curly E}_2 \subset
{\curly E}$ gives a critical filtration of a (semi)stable conic bundle\
$({\curly E},Q)$. Hypothesis (i) and (ii) of lemma
\ref{gbound2} are satisfied with $R=3$ and
$b=\frac{d-2\tau}{r}+2\tau$. Let $k=d/3$, and $n$ large
enough. Let ${\mathcal{T}}_n$ be the subset of ${\mathcal{T}}$ consisting of vector
bundles ${\curly E}_1\oplus {\curly E}_2$ satisfying hypothesis (iii). Then ${\mathcal{T}}_n$ is
bounded, and taking $m$ large enough we have $0=h^1(({\curly E}_1\oplus
{\curly E}_2)(m))=h^1({\curly E}_1(m))+h^1({\curly E}_2(m))$ for ${\curly E}_1\oplus {\curly E}_2 \in {\mathcal{T}}_n$.
Hence for ${\curly E}_1\oplus {\curly E}_2\in {\mathcal{T}}_n$,
\begin{eqnarray}
h^0({\curly E}_1(m))+h^0({\curly E}_2(m))=3\chi(\SO_X(m)) + \deg ({\curly E}_1)+\deg({\curly E}_2)
\end{eqnarray}
On the other hand, for ${\curly E}_1\oplus {\curly E}_2\in {\mathcal{T}}\setminus {\mathcal{T}}_n$ we
still have
\begin{eqnarray}
\label{gineq2}
h^0({\curly E}_1(m))+h^0({\curly E}_2(m))<3\chi(\SO_X(m)) + d
\end{eqnarray}
Let $V_1 \subset V_2 \subset V$ be a critical filtration of $V$. If
${\curly E}_{V_1}\oplus {\curly E}_{V_2} \in {\mathcal{T}}_n$, we get that (ss.2) implies (**.2),
because
\begin{eqnarray*}
\dim V_1 + \dim V_2 \leq
h^0({\curly E}_{V_1}(m))+h^0({\curly E}_{V_2}(m))= \\
=3\chi(\SO_X(m)) + \deg ({\curly E}_{V_1})+ \deg({\curly E}_{V_2}) (\leq)
3\chi(\SO_X(m)) + \deg({\curly E})=\dim V
\end{eqnarray*}
and also $\dim V_1 + \dim V_2 \leq h^0({\curly E}_{V_1}(m)) + h^0({\curly E}_{V_2}(m)) =
P_{{\curly E}_{V_1}}(m) + P_{{\curly E}_{V_2}}(m)$.
On the other hand, if ${\curly E}_{V_1}\oplus {\curly E}_{V_2} \in {\mathcal{T}}\setminus {\mathcal{T}}_n$,
inequality (\ref{gineq2}) implies (**.2)
$$
\dim V_1 + \dim V_2 \leq h^0({\curly E}_{V_1}(m)) + h^0({\curly E}_{V_2}(m)) < 3\chi(\SO_X(m)) +d
=\dim V
$$
In both cases, if inequality (ss.2) is strict, also (**.2) is
strict. But assume that we have subsheaves ${\curly E}_1 \subset {\curly E}_2 \subset {\curly E}$
giving a critical filtration of a semistable conic bundle\ $({\curly E},Q)$. By lemma
\ref{gboundcor2} ${\curly E}_1(m)$ and ${\curly E}_2(m)$ are generated by global sections
and $h^1({\curly E}_1(m))=h^1({\curly E}_2(m))=0$. Taking $V_1=H^0({\curly E}_1(m))$ and $V_2=
H^0({\curly E}_2(m))$ we have ${\curly E}_{V_1}={\curly E}_1$ and ${\curly E}_{V_2}={\curly E}_2$, hence
\begin{eqnarray*}
\dim V_1 +\dim V_2 = \deg({\curly E}_1)+\deg({\curly E}_2) +3\chi(\SO_X(m)) =\\
= \deg({\curly E})+3\chi(\SO_X(m)) =\dim V
\end{eqnarray*}
and also $\dim V_1 + \dim V_2 = P_{{\curly E}_{V_1}}(m)+P_{{\curly E}_{V_2}}(m)$
\bigskip
\noindent\textit{\textbf{Step 2. $(\tq,\tQ)$ GIT (semi)stable $\Rightarrow$
$h^1({\curly E}(m))=0$ and $\tilde q$ good}}
\bigskip
If $h^1({\curly E}(m))\neq 0$, then by Serre duality $\operatorname{Hom} ({\curly E},{\curly K}_X)\neq 0$. Take
$\psi \in \operatorname{Hom} ({\curly E},{\curly K}_X)$. The composition $V\otimes {\mathcal{O}}_X \to {\curly E}(m) \to
{\curly K}_X$ gives a linear map
$$
f: V \to H^0({\curly K}_X).
$$
Let $U$ be the kernel of $f$. We have $\dim U \geq \dim V - \dim
H^0({\curly K}_X) = p-g$. Then by (semi)stability of $(\tq,\tQ)$ we have
$$
r(p-g-c_Q({\curly E}_U)\tau) \leq r(\dim U- c_Q({\curly E}_U)\tau)
(\leq) \operatorname{rk} (E_U) ( p- 2\tau),
$$
$$
(r-\operatorname{rk}({\curly E}_U))p \leq r(g+c_Q({\curly E}_U)\tau) - \operatorname{rk}({\curly E}_U) 2 \tau
$$
We have $\operatorname{rk}({\curly E}_U)\leq r$. Then if $m$ is large enough the inequality
forces $r=\operatorname{rk}({\curly E}_U)$. By definition of $U$ we have ${\curly E}_U(m)\subset
\ker \psi$, then $\operatorname{rk}(\ker \psi)=r$, $\operatorname{rk}(\operatorname{im} \psi)=0$, and then
$\psi=0$ because ${\curly K}_X$ is torsion free. We conclude that (for $m$
large enough) $h^1({\curly E}(m))=0$.
Then $\dim V=p=h^0({\curly E}(m))$, and to show that $\tilde q$ is ``good'' it is
enough to show that the induced linear map
$$
V \to H^0({\curly E}(m))
$$
is injective. Let $V'$ be the kernel. Then we have
$\operatorname{rk}({\curly E}_{V'})=0$. By semistability we have (**.1)
$$
r(\dim V' - c_Q({\curly E}_{V'})) \leq 0,
$$
but $c_Q({\curly E}_{V'})=0$, and then $\dim V'$ must be zero.
To show that ${\curly E}$ is torsion free, let ${\curly T}\subset {\curly E}$ be the torsion
subsheaf. We have $V \cong H^0({\curly E}(m))$, and then $U=H^0({\curly T}(m))$ is a
subspace of $V$. The associated sheaf ${\curly E}_U$ has rank equal to zero,
and arguing as above we get $U=0$.
\bigskip
\noindent\textit{\textbf{Step 3. GIT (Semi)stable $(\tq,\tQ)$ $\Rightarrow$
(semi)stable conic bundle}}
\bigskip
By the previous step we know that we can choose $m$ large enough so
that $\tilde q$ is ``good''. We will check first (ss.1). Let ${\curly E}'$ be a
subsheaf of ${\curly E}$. Define $V'=H^0({\curly E}'(m))$. We have ${\curly E}_{V'}\subset {\curly E}'$,
$\operatorname{rk} ({\curly E}_{V'})\leq \operatorname{rk}({\curly E}')$, $\dim V'\geq P_{{\curly E}'}(m)$, and
$c_Q({\curly E}') \geq c_Q({\curly E}_{V'})$. Then
\begin{eqnarray*}
r(P_{{\curly E}'}(m)-c_Q({\curly E}')\tau ) \leq
r(\dim V' - c_Q({\curly E}_{V'}) \tau ) (\leq)\\
(\leq) \operatorname{rk}({\curly E}_{V'})(\dim V -2 \tau)
\leq \operatorname{rk}({\curly E}')(\dim V -2 \tau).
\end{eqnarray*}
Note that if (**.1) is strict, then also (ss.1) is strict. But assume
that there is a subspace $V'\subset V$ that is semistabilizing,
i.e. both conditions in (**.1) are equalities. Then
\begin{eqnarray*}
\frac{\deg ({\curly E}_{V'})-c_Q({\curly E}_{V'}) \tau}{\operatorname{rk}({\curly E}_{V'})} =
\frac{\dim V' -c_Q ({\curly E}_{V'}) \tau}{\operatorname{rk}({\curly E}_{V'})} -\chi(\SO_X(m)) =\\
=\frac{\dim V - 2 \tau}{r} -\chi(\SO_X(m)) =
\frac{\deg({\curly E}) - 2 \tau}{r},
\end{eqnarray*}
and we get that (ss.1) for ${\curly E}_{V'}$ also gives equality.
Now we are going to check (ss.2). As in step 1, consider the set ${\mathcal{T}}$
of vector bundles of the form ${\curly E}_1 \oplus {\curly E}_2$ such that ${\curly E}_1 \subset
{\curly E}_2 \subset {\curly E}$ gives a critical filtration. We have already proved
condition (ss.1), thus hypothesis (i) and (ii) of lemma \ref{gbound2}
are again satisfied. Then, as in step 1, we define the subset ${\mathcal{T}}_n
\subset {\mathcal{T}}$, and taking $m$ large enough we can assume that the vector
bundles ${\curly E}_1$ and ${\curly E}_2$ are generated by global sections if
${\curly E}_1\oplus {\curly E}_2 \in {\mathcal{T}}$.
Let ${\curly E}_1 \subset {\curly E}_2 \subset {\curly E}$ be a
critical filtration of $({\curly E},Q)$. Let $V_1=H^0({\curly E}_1(m))$ and
$V_2=H^0({\curly E}_2(m))$.
If ${\curly E}_{V_1}\oplus {\curly E}_{V_2} \in {\mathcal{T}}_n$, then ${\curly E}_{V_1}$ and ${\curly E}_{V_2}$ are
generated by global sections and then ${\curly E}_{V_1}={\curly E}_1$, ${\curly E}_{V_2}={\curly E}_2$,
and $V_1\subset V_2 \subset V$ is a critical
filtration of $V$ and (**.2) holds
\begin{eqnarray*}
\deg ({\curly E}_1) + \deg({\curly E}_2) =
\dim V_1 + \dim V_2 -3\chi(\SO_X(m)) (\leq)\\
(\leq)
\dim V - 3\chi(\SO_X(m)) = \deg({\curly E}).
\end{eqnarray*}
On the other hand, if ${\curly E}_{V_1}\oplus {\curly E}_{V_2} \in {\mathcal{T}} \setminus {\mathcal{T}}_n$,
inequality (\ref{gineq2}) implies (**.2)
$$
\deg ({\curly E}_1) + \deg({\curly E}_2) \leq
h^0({\curly E}_1(m)) + h^0({\curly E}_2(m)) -3\chi(\SO_X(m)) <\deg({\curly E}).
$$
Note that if (**.2) is strict then (ss.2) is also
strict.
But assume that there is a semistabilizing critical sequence $V_1
\subset V_2 \subset V$, i.e. a critical sequence giving equality in
both conditions of (**.2). Then
\begin{eqnarray*}
\deg({\curly E}_{V_1})+\deg({\curly E}_{V_2}) = \dim V_1 + \dim V_2 -3\chi(\SO_X(m)) =\\
=\dim V -3 \chi(\SO_X(m)) = \deg {\curly E},
\end{eqnarray*}
and we also get an equality in (ss.2).
\end{proof}
Once we have established proposition \ref{giden} and lemma
\ref{gautom} we can prove theorem I using standard techniques. We
follow closely \cite{H-L}.
\begin{proof2}\textit{ of theorem I.}
Let ${\overline \mathfrak{M}_\tau (r,d,{\curly L})}$ (resp. $\mathfrak{M}_\tau (r,d,{\curly L})$) be
the GIT quotient of $Z$ (resp. $Z^s$) by $\operatorname{SL}(V)$.
First we construct a universal family on $Z^{ss}$ using the universal
families of the Quot scheme ${\mathcal{Q}}$ and on ${\mathcal{P}}=\mathbb{P}(\operatorname{Sym}^2V^\vee \otimes
B)$ (we think of ${\mathcal{P}}$ as the Grassmannian of one dimensional
subspaces of $\operatorname{Sym}^2 V^\vee \otimes B$, and hence the universal subbundle of
subspaces is ${\mathcal{O}}_{{\mathcal{P}}}(-1)$).
Recall that $Z^{ss}$ is in ${\mathcal{H}} \times {\mathcal{P}}$. The universal quotient
${\curly E}_{\mathcal{H}}$ on
${\mathcal{H}} \times X$ pulls back to a vector bundle ${\curly E}_{Z^{ss}}=
p^*_{{\mathcal{H}} \times {\mathcal{Q}}} {\curly E}_{\mathcal{H}}$ on $Z^{ss}$. On the other hand
the universal subbundle on
${\mathcal{P}} \times X$ gives a morphism $\operatorname{Sym}^2 V \otimes p^*_X {\mathcal{O}}_X(-2m)
\to p^*_X{\curly L} \otimes {\mathcal{O}}_{{\mathcal{P}}}(1)$ of sheaves over $Z^{ss} \times
X$. By the definition of $Z$, there
is a line bundle ${\curly N}$ on $Z^{ss}$ such that this last morphism
factors and gives $Q_{Z^{ss}}:\operatorname{Sym}^2{\curly E}_{Z^{ss}} \to p^*_X {\curly L}
\otimes p^*_{Z^{ss}}{\curly N}$.
Note that the line bundle ${\curly N}$ is needed because the factorization
on $Z$ is only up to scalar multiplication.
The triple
$({\curly E}_{Z^{ss}},Q_{Z^{ss}},{\curly N})$ is a universal conic bundle.
Given a family
$({\curly E}_T,Q_T)$ of conic bundles\ parametrized by $T$, and using the universal
family on $Z^{ss}$, we obtain a morphism $T
\to \overline \mathfrak{M}_\tau (r,d,{\curly L})$. This is done in the following way:
Let $m$ be large enough so that proposition \ref{giden} holds. Given a
family $({\curly E}_T,Q_T,{\curly N})$ of conic bundles\ parametrized by $T$, consider the
locally free sheaf ${\mathcal{V}}={p_T}_*({\curly E}_T\otimes p^*_X{\mathcal{O}}_X(m))$, and note
that $p^*_T{\mathcal{V}} \otimes p^*_X{\mathcal{O}}_X(-m) \to {\curly E}_T$ is a surjection. Cover
$T$ with open sets $U_i$ such that there are isomorphisms $\phi_i:
V\otimes {\mathcal{O}}_{U_i} \to {\mathcal{V}}$. Then we have quotients $q_i:V\otimes
p^*_X {\mathcal{O}}_X(-m) \twoheadrightarrow {\curly E}_{U_i}$ and families of subspaces ${\mathcal{O}}_{U_i} \hookrightarrow
\operatorname{Sym}^2V^\vee \otimes B \otimes {\mathcal{O}}_{U_i}$, and these give maps $U_i \to
Z^{ss}$. On the intersections $U_i \cap U_j$ this maps in general will
differ by the action of $\operatorname{SL}(V)$, then they combine to give a well
defined morphism $T \to \overline \mathfrak{M}_\tau (r,d,{\curly L})$.
It is straightforward to check the universal property for
$\overline \mathfrak{M}_\tau (r,d,{\curly L})$, and then $\overline \mathfrak{M}_\tau
(r,d,{\curly L})$ is a coarse moduli space.
Now we will show that the universal family restricted to $Z^s$
descends to $\mathfrak{M}_\tau (r,d,{\curly L})$, making it a fine moduli space.
Applying Luna's
\'etale slice theorem \cite{L}, we can find an \'etale cover $U'$ of
$\mathfrak{M}_\tau (r,d,{\curly L})$ over
which there is a universal family $({\curly E}'_{U'},Q'_{U'})$. Consider $U''=
U' \times _{\mathfrak{M}_\tau (r,d,{\curly L})} U'$ and take an isomorphism $\Phi:
p^*_1({\curly E}'_{U'},Q'_{U'}) \to p^*_2({\curly E}'_{U'},Q'_{U'})$ with the condition
$p^*_1 Q'_{U'} = p^*_2 Q'_{U'} \circ \operatorname{Sym}^2 \Phi$. This isomorphism
exists and is unique by lemma \ref{gautom}, and then it satisfies the
cocycle condition of descend theory \cite[Chap. VII]{Mr}, and hence the
family $({\curly E}'_{U'},Q'_{U'})$ descends to $\mathfrak{M}_\tau (r,d,{\curly L})$.
\end{proof2}
\subsection{S-equivalence}
\label{gsequiv}
\hfil
\medskip
Let $({\curly E},Q)$ and $({\curly E}',Q')$ be two nonisomorphic conic bundles. If they
are strictly semistable, it
could still happen that the corresponding points in the moduli space
$\overline\FM_\tau (r,d,\CrL)$ coincide. In this
case we say that they are S-equivalent (note that this is not the usual
definition. Usually one defines two bundles as S-equivalent if the
graded objects of their
Jordan-H\"older filtrations coincide, and then proves that S-equivalence
classes corresponds to points of the moduli space). In this section, given a
strictly semistable conic bundle\ $({\curly E},Q)$, we will show
how to obtain a canonical representative $({\curly E}^S,Q^S)$ of its
S-equivalent class. In other words, given two semistable conic bundles\
$({\curly E},Q)$ and $({\curly E}',Q')$, they will be S-equivalent iff $({\curly E}^S,
Q^S)$ is isomorphic to $({{\curly E}'}^S,{Q'}^S)$.
Let $({\curly E},Q)$ be a strictly semistable conic bundle. Then there exists at least
one ``semistabilizing object'', i.e. there exists either a subbundle
${\curly E}' \subset {\curly E}$ that gives equality on (ss.1)(and then we say
that ${\curly E}'$ is a semistabilizing object of type I), or there is a critical
filtration ${\curly E}_1 \subset {\curly E}_2 \subset {\curly E}$ giving equality on
(ss.2)(and then we say that the filtration is a semistabilizing object
of type II). Choose one semistabilizing object. We define a new conic bundle\
$({\curly E}_0,Q_0)$ as follows (it will depend on which semistabilizing object
we choose):
In the first case (corresponding to (ss.1)), the vector bundle is
defined to be ${\curly E}_0={\curly E}' \oplus {\curly E}/{\curly E}'$ (note that if ${\curly E}$
is semistable and
${\curly E}'$ gives equality on (ss.1), then ${\curly E}/{\curly E}'$ is torsion free). To define
$Q_0$, let $v$ and $w$ be local sections of ${\curly E}_0$ on an open set $U$. We
distinguish three cases:
$$
\text{If}\;\; c_Q({\curly E}')=
\left\{
\begin{array}{ll}
2 \text{, then}\;
Q_0(v,w)=&
\left\{
\begin{array}{ll}
Q(v,w)&v,w \in {\curly E}'(U)\\
0&\text{otherwise}\\
\end{array}
\right .\\
1 \text{, then}\;
Q_0(v,w)=&
\left\{
\begin{array}{ll}
Q(v,w)&v\in {\curly E}'(U) \; \text{or}\; w\in {\curly E}'(U)\\
0&\text{otherwise}\\
\end{array}
\right .\\
0 \text{, then}\;
Q_0(v,w)=&
Q(v,w)\\
\end{array}
\right .
$$
In matrix form this can be written as follows
$$
\text{If} \; Q=
\left(
\begin{array}{cc}
\times & \cdot \\
\cdot & \cdot
\end{array}
\right)
,\quad\text{then}\; Q_0=
\left(
\begin{array}{cc}
\times & 0 \\
0 & 0
\end{array}
\right)
$$
$$
\text{If} \; Q=
\left(
\begin{array}{cc}
0 & \times \\
\times & \cdot
\end{array}
\right)
,\quad\text{then}\; Q_0=
\left(
\begin{array}{cc}
0 & \times \\
\times & 0
\end{array}
\right)
$$
$$
\text{If} \; Q=
\left(
\begin{array}{cc}
0 & 0 \\
0 & \times
\end{array}
\right)
,\quad\text{then}\; Q_0=
\left(
\begin{array}{cc}
0 & 0 \\
0 & \times
\end{array}
\right)
$$
It is easy to see that this is well defined.
In the second case
(corresponding to (ss.2)) we define the vector bundle to be ${\curly E}_0={\curly E}_1
\oplus {\curly E}_2/{\curly E}_1 \oplus {\curly E}/{\curly E}_2$. Again let $v$ and $w$ be
local sections
of ${\curly E}_0$ on an open set $U$. Then we set
$$
Q_0(v,w)=
\left\{
\begin{array}{ll}
Q(v,w)&v \; \text{and}\; w \in {\curly E}_2(U)\\
Q(v,w)& v \; \text{or}\; w \in {\curly E}_1(U)\\
0&\text{otherwise,}\\
\end{array}
\right .
$$
and in matrix form
$$
Q=
\left(
\begin{array}{ccc}
0 & 0 & \times \\
0 & \times & \cdot \\
\times & \cdot & \cdot
\end{array}
\right)
\quad Q_0=
\left(
\begin{array}{ccc}
0 & 0 & \times \\
0 & \times & 0 \\
\times & 0 & 0
\end{array}
\right)
.
$$
Again it is easy to see that this is well defined.
\begin{proposition}
\label{sequivcrit}
The conic bundle\ $({\curly E}_0,Q_0)$ is also semistable. Furthermore, if we repeat
this process, eventually we will get a conic bundle\ that we will call
$({\curly E}^S,Q^S)$ with the following properties
(i) $({\curly E}^S,Q^S)$ is semistable, and if we apply this process to it with any
object we obtain an isomorphic conic bundle\ (i.e. this process stops).
(ii) $({\curly E}^S,Q^S)$ only depends on the isomorphism class of $({\curly E},Q)$.
(iii) Two conic bundles\ $({\curly E},Q)$ and $({\curly E}',Q')$ are S-equivalent if and only if
$({\curly E}^S,Q^S)$ is isomorphic to $({{\curly E}'}^S,{Q'}^S)$.
\end{proposition}
\begin{remark}
\textup{
The conic bundle\ $({\curly E}^S,Q^S)$ is the analogue of the graded object $\operatorname{gr}({\curly E})$
of the Jordan-H\"older filtration of a semistable torsion-free sheaf.
Note that $\operatorname{gr}({\curly E})$ can also be obtained by a process similar to this.}
\end{remark}
\begin{proof}
We start with a general observation about GIT quotients. Let $Z$ be a
projective variety with a linearized action by a group $G$.
Two points in the open subset $Z^{ss}$ of semistable points are
S-equivalent (they are mapped to the same point in the moduli space)
if there is a common closed orbit in the closures (in $Z^{ss}$) of
their orbits. Let $z\in Z^{ss}$. Let $B$ be the unique closed orbit in
the closure $\overline{G\cdot z}$ in $Z^{ss}$ of its orbit $G \cdot
z$. Assume that $z$
is not in $B$. Then there exists a one-parameter
subgroup $\lambda$ such that the limit $z_0=\lim_{t\to 0}
\lambda (t) \cdot z$ is in $\overline{G
\cdot z}\setminus G \cdot z$. Note that we must have
$\mu(z,\lambda)=0$ (otherwise $z_0$ would be unstable).
Note that $G \cdot z_0 \subset \overline{G
\cdot z}\setminus G \cdot z$, and then $\dim G \cdot z_0 < \dim G
\cdot z$. Repeating this process with $z_0$ we then get a sequence
of points that eventually stops and gives $\tilde z \in B$. Two points
$z_1$ and $z_2$ will then be S-equivalent if and only if after
applying this procedure to both of them the orbits of
$\tilde z_1$ and $\tilde z_2$ are the same.
We will use the notation introduced in subsection
\ref{gconstruction}.
We will prove the proposition using the previous observation. The fact
that choosing a ``semistabilizing object'' of $({\curly E},Q)$ induces a one
parameter subgroup
with $\mu(z,\lambda)=0$ (where $z$ is the corresponding point on
$Z^{ss}$) follows from proposition \ref{giden} and the
proof of proposition \ref{gmainprop}.
The fact that the limit point $z_0$
corresponds to $({\curly E}_0,Q_0)$ is an easy calculation (see
\cite[lemma 1.26]{S}). The conic bundle\ $({\curly E}_0,Q_0)$ is semistable by
proposition \ref{conj}.
It is easy to check that $\tilde z$ corresponds to $({\curly E}^S,Q^S)$,
and then items (ii) and (iii) follow from the fact that $\tilde
z$ is in $B$.
\end{proof}
\begin{proposition}
\label{conj}
Let $\lambda$ be a 1-PS of $\operatorname{SL}(V)$.
Let $\operatorname{SL}(V)$ act on $Z$. Asume this action is linearized with respect to
an ample line bundle ${\curly H}$.
Let $z\in Z^{ss}$. Let $z_0=\lim_{t\to 0}
\lambda (t) \cdot z$. If $\mu(z,\lambda)=0$, then $z_0\in Z^{ss}$.
\end{proposition}
\begin{proof}
This proof was given to us by A. King. We can assume, without loss of
generality, that the polarization ${\curly H}$ of $Z$ is very ample, and then $Z$
embedds in $\mathbb{P} (H^0({\curly H})^\vee)$ and $\operatorname{SL}(V)$ acts on $H^0({\curly H})^\vee$.
A point $x\in Z$ is (semi)stable iff its image in $\mathbb{P}
(H^0({\curly H})^\vee)$ is (semi)stable, and then we can assume
$(Z,{\curly H})=(\mathbb{P}(\mathbb{C}^n), {\mathcal{O}} (1))$, with $\operatorname{SL}(V)$ acting on $\mathbb{C}^n$.
Let $\pi:\mathbb{C}^n\setminus \{0\} \to \mathbb{P}(\mathbb{C}^n)$ be the projection. Let
$z\in \mathbb{P}(\mathbb{C}^n)$ be a semistable point and $\lambda$ a 1-PS with
$\mu(z,\lambda) =0$. Let $\tilde z \in \mathbb{C}^n$ be a point in the fibre
$\pi^{-1}(z)$, and let
$$
\tilde z_0=\lim_{t\to 0} \lambda (t)\cdot \tilde z.
$$
This limit exists and it is not the origin because $\mu(z,\lambda)
=0$. We have $z_0:=\lim_{t\to 0} \lambda (t)\cdot z= \pi(\tilde z_0)$
(by continuity of $\pi$). Assume that the point $z_0$ is unstable.
Then the closure of the orbit of $\tilde z_0$ contains the origin, but this
closure is included in the closure of the orbit of $\tilde z$, and this
doesn't contain the origin because $z$ is semistable. Then $z_0$ is
semistable. Furthermore, $z_0$ cannot be stable because
$\mu(z_0,\lambda) =\mu(z,\lambda)=0$, then $z_0$ is strictly
semistable.
\end{proof}
\section{Properties of conic bundles}
\label{properties}
\subsection{Irreducibility of moduli space}
\hfil
\medskip
First we will show that the semistability and stability of conic bundles\ are
open conditions.
\begin{proposition}
Let $({\curly E}_T,Q_T,{\curly N})$ be a flat family of conic bundles\ parametrized by
$T$. The subset $T^{s}$ (resp. $T^{ss}$) corresponding to stable
(resp. semistable) conic bundles\ is open.
\end{proposition}
\begin{proof}
Let $m$ and $l$ be large enough so that ${\curly V}={p^{}_T}_*
({\curly E}_T \otimes p^*_X{\mathcal{O}}_X(m))$ is locally free,
$p^*_T {\curly V} \otimes p^*_X {\mathcal{O}}_X(-m) \to {\curly E}_T$
is a surjection and proposition \ref{giden} holds.
Note that the universal family that was constructed on $Z^{ss}$ in the
proof of theorem I can be extended to the set $Z^{good}$ of ``good'' points.
Arguing as in the proof of theorem I, there is a finite
open cover $\{U_i\}_
{i\in I}$ of $T$ and morphisms $f_i:U_i \to Z^{good} \subset Z$. These
morphisms depend on the choices made (the choice of local
trivializations of ${\curly V}$), but the $\operatorname{SL}(V)$ orbit of $f_i(t)$ are
independent of the choices. In particular, the property of $f_i(t)$
belonging to $Z^{ss}$ only depends on $t$. By proposition
\ref{giden}, $f_i(t)$ lies in $Z^{s}$ (resp. $Z^{ss}$) iff
the conic bundle\ $({\curly E}_t,Q_t)$ is stable (resp. semistable). Then
$$
T^{s}=\bigcup_{i\in I} f^{-1}_i(Z^{s})
$$
and the openness of $Z^{s}$ in $Z$ proves that $T^{s}$ is open
(the same argument works for $Z^{ss}$).
\end{proof}
\begin{theorem}
Let $X$ be a Riemann surface. Fix $r$, $d$ and $\tau$. Then
there exists an integer $l_0$ such that if $\deg {\curly L} > l_0$, then $\overline
\mathfrak{M}_\tau(r,d,{\curly L})$ and $\mathfrak{M}_\tau (r,d,{\curly L})$ are
irreducible or empty.
\end{theorem}
\begin{proof}
We will construct a flat family of conic bundles\ parametrized by an irreducible
scheme $\widetilde Y$ with the property that every semistable conic bundle\ of type
$(r,d,{\curly L})$ belongs to the family. Then there is a surjective
morphism $\widetilde Y^{ss} \to \overline \mathfrak{M}_\tau(r,d,{\curly L})$, where $\widetilde
Y^{ss}$ is the open subset representing semistable points, and this
proves
that $\overline \mathfrak{M}_\tau(r,d,{\curly L})$ is irreducible. Repeating this
with the open subset $\widetilde Y^{s}$ corresponding to stable points, we prove
that $\mathfrak{M}_\tau(r,d,{\curly L})$ is also irreducible.
Let $m$ be large enough so that for any semistable conic bundle\ $({\curly E},Q)$
in $\overline \mathfrak{M}_\tau(r,d,{\curly L})$, the
vector bundle ${\curly E}(m)$ is generated by global sections (corollary
\ref{gboundcor1}), and such that
\begin{equation}
\label{gext1const}
2g-2-d-rm<0.
\end{equation}
Note that $m$ only depends on $X$, $r$, $d$ and $\tau$, but not on
${\curly L}$. If we choose $r-1$ generic sections of ${\curly E}(m)$, we have an
exact sequence
$$
0 \to {\mathcal{O}}_X^{\oplus r-1}(-m) \to {\curly E} \to {\curly M}(-m) \to 0
$$
where ${\curly M}$ is a line bundle of degree $d+rm$.
By standard methods we
can construct a universal family ${\curly F}_Y$ of extensions of line
bundles of degree $d+(r-1)m$ by ${\mathcal{O}}_X^{\oplus r-1}(-m)$. This will be
parametrized by a scheme $Y$ that has a morphism to $\operatorname{Pic}^{d+rm}(X)$,
and the fibre over a line bundle ${\curly M}$ is naturally isomorphic to
$\operatorname{Ext}^1 ({\curly M}(-m),{\mathcal{O}}_X^{\oplus r-1}(-m))$. Note that to construct this
family we need that the dimension of this $\operatorname{Ext}^1$ group is constant
when we vary ${\curly M}$, but this is true thanks to (\ref{gext1const}).
Each point $y\in Y$ corresponds to an extension of the form
$$
0 \to {\mathcal{O}}_X^{\oplus r-1}(-m) \to {\curly F}_y \to {\curly M}(-m) \to 0.
$$
It follows from the argument in the previous paragraph that all vector
bundles in semistable conic bundles\ do occur in this family.
Note that, if $({\curly E},Q)$ is a conic bundle, $Q$ can be thought of as an
element of $H^0(\operatorname{Sym}^2 {\curly F}_y^\vee \otimes {\curly L})$.
Now choose $l_0$ large enough so that for any line bundle ${\curly L}$ of
degree $\deg({\curly L})>l_0$ the following holds
$$
H^1(\operatorname{Sym}^2 {\curly F}_y^\vee \otimes {\curly L})=0
$$
for any $y \in Y$. Then $H^0(\operatorname{Sym}^2 {\curly F}_y^\vee \otimes {\curly L})$
is constant when we vary $y$, and we can construct a (flat) family of
conic bundles\ parametrized by $\widetilde Y=\mathbb{V}(\operatorname{Sym}^2 {\curly F}^\vee \otimes p^*_X
{\curly L})$, and every semistable conic bundle\ of type $(r,d,{\curly L})$ belongs to
this family.
\end{proof}
\subsection{Orthogonal bundles}
\label{orthogonalbundles}
\hfil
\medskip
An orthogonal bundle is a vector bundle associated to a principal
bundle with (complex) orthogonal structure group.
Equivalently, it is a conic bundle\ $({\curly E},Q)$
with ${\curly L}={\mathcal{O}}_X$, such that the bilinear form $Q:\operatorname{Sym} ^2
{\curly E} \to {\mathcal{O}}_X$ induces an isomorphism $Q:{\curly E} \to {\curly E}^\vee$.
We will call such a conic bundle\ a smooth conic bundle.
In this case the conic bundle\ gives a smooth conic ${\curly C}_x$ for each point
$x\in X$. Note that the isomorphism $Q:{\curly E} \to {\curly E}^\vee$ induces an
isomorphism $\det Q: \det {\curly E} \to \det {\curly E}^\vee$, and then
$\deg({\curly E})=0$ (in fact $(\det({\curly E}))^{\otimes 2}={\mathcal{O}}_X$).
There is a notion of stability for orthogonal bundles (see \cite{R}):
a bundle ${\curly E}$ is orthogonal (semi)stable iff for every proper
isotropic subbundle ${\curly F}$, $\deg({\curly F})(\leq)0$. The notion of stability
that we have defined for conic bundles\ depends in principle on a parameter $\tau$,
but we will show
that in the case of a smooth conic bundle, the notion of stability
doesn't depend on the particular value of the parameter. In fact we
will prove that
a smooth conic bundle\ is $\tau$-(semi)stable iff it is (semi)stable as an
orthogonal bundle.
\begin{lemma}
\label{lemmaorthogonal}
Let $({\curly E},Q)$ be a smooth conic bundle, and let ${\curly F}$ be a proper vector
subbundle of ${\curly E}$. Then
(i) There is an exact sequence
$$
0 \to {\curly F}^\perp \to {\curly E} \to {\curly F}^\vee \to 0,
$$
and $\deg({\curly F})=\deg({\curly F}^\perp)$.
(ii) If ${\curly F}$ is isotropic ($c_Q({\curly F})\leq 1$), then $\operatorname{rk}({\curly F})=1$.
(iii) If $\operatorname{rk}({\curly F})=1$, then $c_Q({\curly F})\geq 1$
\end{lemma}
\begin{proof}
(i) Follows from the exact sequence
$$
0 \to {\curly F}^\perp \to {\curly E}\cong{\curly E}^\vee \to {\curly F}^\vee \to 0,
$$
and the fact that $\deg({\curly E})=0$.
(ii) Assume that $\operatorname{rk}({\curly F})=2$. Then, in a basis adapted to ${\curly F}
\subset {\curly E}$
$$
Q=\left(
\begin{array}{ccc}
0 & 0 & \cdot \\
0 & 0 & \cdot \\
\cdot & \cdot & \cdot
\end{array}
\right)
$$
and then $\det Q = 0$, contradicting the fact that the conic bundle\ is
smooth.
(iii) If $c_Q({\curly F})=0$, then
$$
Q=\left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & \cdot & \cdot \\
0 & \cdot & \cdot
\end{array}
\right)
$$
and then $\det Q = 0$, again contradicting the fact that the conic bundle\ is
smooth.
\end{proof}
\begin{proposition}
A smooth conic bundle\ $({\curly E},Q)$ is $\tau$-semistable iff the vector
bundle ${\curly E}$ is semistable as an
orthogonal bundle. Furthermore, it is $\tau$-stable iff it is stable
as an orthogonal bundle.
\end{proposition}
\begin{proof}
Let $({\curly E},Q)$ be a smooth $\tau$-semistable conic bundle. Let ${\curly F}$ be
an isotropic vector subbundle. By lemma \ref{lemmaorthogonal} (ii),
$\operatorname{rk}({\curly F})=1$. We have ${\curly F} \subset {\curly F}^\perp$,
$\operatorname{rk}({\curly F}^\perp)=2$ (by
lemma \ref{lemmaorthogonal} (i)), and we check that ${\curly F} \subset {\curly F}^\perp
\subset {\curly E}$ is a critical filtration. Then $\deg({\curly F}) +
\deg({\curly F}^\perp) \leq 0$, but
$\deg({\curly F}) = \deg ({\curly F}^\perp)$ (lemma \ref{lemmaorthogonal} (i)),
and then $\deg({\curly F})\leq 0$, which proves that ${\curly E}$ is semistable as
an orthogonal bundle. Furthermore, if $({\curly E},Q)$ is
$\tau$-stable, then $\deg({\curly F}) + \deg({\curly F}^\perp) < 0$,
$\deg({\curly F})< 0$ and ${\curly E}$ is stable as an orthogonal bundle.
Conversely, let ${\curly E}$ be an orthogonal semistable bundle. Let ${\curly F}$
be any vector subbundle. Following \cite{R} let ${\curly N}={\curly F} \cap
{\curly F}^\perp$, and let ${\curly N}'$ be the vector subbundle generated by
${\curly N}$. We have an exact sequence
\begin{equation}
\label{orthoshort}
0 \to {\curly N}' \to {\curly F} \oplus {\curly F}^\perp \to {\curly M}' \to 0
\end{equation}
where ${\curly M}'$ is the subbundle of ${\curly E}$ generated by ${\curly F} +
{\curly F}^\perp$. We have ${\curly M}'=({\curly N}')^\perp$.
If ${\curly N}'=0$, then ${\curly E}={\curly F}\oplus {\curly F}^\perp$, $c_Q({\curly F})=2$, and
$\deg({\curly F}) = 0$ (lemma \ref{lemmaorthogonal} (i)). Then
$$
\frac{\deg({\curly F}) - c_Q({\curly F})\tau}{\operatorname{rk}({\curly F})}=\frac{-2\tau}{\operatorname{rk}({\curly F})}<
\frac{-2\tau}{3}=\frac{\deg({\curly E}) - 2\tau}{3}.
$$
If ${\curly N}'\neq 0$, then $\deg({\curly F})=\deg({\curly N}')$ (by lemma
\ref{lemmaorthogonal} (i) and the exact sequence (\ref{orthoshort})),
and then $\deg({\curly F})\leq 0$ (because ${\curly E}$ is orthogonal semistable
and ${\curly N}'$ is isotropic). If $\operatorname{rk}({\curly F})=2$, then $c_Q({\curly F})=2$ (by
lemma \ref{lemmaorthogonal} (ii)), and if $\operatorname{rk}({\curly F})=1$, then
$c_Q({\curly F})\geq 1$ (by lemma \ref{lemmaorthogonal} (iii)). In any case
$$
\frac {\deg({\curly F}) - c_Q({\curly F})\tau}{\operatorname{rk}({\curly F})} \leq \frac{-
c_Q({\curly F})\tau}{\operatorname{rk}({\curly F})} < \frac{-2\tau}{3}=\frac{\deg({\curly E}) - 2\tau}{3}
$$
Now let ${\curly E}_1 \subset {\curly E}_2 \subset {\curly E}$ be a critical
filtration. Then ${\curly E}_1$ is isotropic, ${\curly E}_2 = {\curly E}_1 ^\perp$, and
then
$$
\deg({\curly E}_1)+\deg({\curly E}_2)=2\deg({\curly E}_1) \leq 0,
$$
because ${\curly E}_1$ is isotropic and ${\curly E}$ is orthogonal semistable.
This finishes the proof that $({\curly E},Q)$ is $\tau$-semistable. Furthermore, if
${\curly E}$ is orthogonal stable, the last inequality is strict, and
we obtain that $({\curly E},Q)$ is $\tau$-stable.
\end{proof}
\section*{Appendix: Hitchin-Kobayashi correspondence for conic
bundles}
\centerline{\textbf{(By I. Mundet i Riera)}}
\bigskip
In this appendix I use the result in \cite{Mu} to
relate the notion of stability for conic bundles
to the existence of solutions to a certain PDE. This is similar
to the well known relation between stability of vector bundles and
existence of Hermite--Einstein metrics, or between stability of
holomorphic pairs and solutions to the vortex equations. As usual
in the literature, I call such a relation a Hitchin--Kobayashi
correspondence (see \cite{Mu} and the references therein).
Take a non-degenerate conic bundle $Q:\operatorname{Sym}^2{\curly E}\to{\curly L}$ on
a Riemann surface $X$. Let $E$ be the smooth bundle underlying ${\curly E}$.
We denote $\overline{\partial}_{{\curly E}}$ the $\overline{\partial}$
operator on $E$ given by ${\curly E}$.
Fix a metric (In this appendix {\it metric} will always mean
{\it Hermitian metric}). on ${\curly L}$ and consider the following
equation on a metric $h$ on $E$:
\begin{equation}
i\Lambda F_{\overline{\partial}_{{\curly E}},h}+
\frac{\tau}{2}\frac{Q\otimes Q^{*_h}}{\|Q\|^2_h}=c\mbox{Id},
\label{equ0}
\end{equation}
where $F_{\overline{\partial}_{{\curly E}},h}$ is the curvature of the
Chern connection of $\overline{\partial}_{{\curly E}}$ with respect to $h$,
$\Lambda:\Omega^2(X)\to\Omega^0(X)$ is the adjoint of wedging with
the Kaehler form of $X$ and the subscript $h$ in $*$ and $\|\cdot\|$
is to recall that both depend on $h$. Finally, $\tau>0$ and $c$ are
real numbers.
We will take a (rather standard) point of view putting equation
(\ref{equ0}) inside the setting considered in \cite{Mu}.
Then we will study the existence criterion to solutions of the
equation given in \cite{Mu} applied to this particular case, thus
arriving again at the notion of stability for conic bundles.
Fix a metric $h_0$ in $E$. Let ${\mathcal{G}}^c$ be the complex gauge group of
$E$, i.e., the group of its smooth automorphisms covering the identity
on $X$. The group ${\mathcal{G}}^c$ acts on the space of
$\overline{\partial}$ operators on $E$ by pullback. So $g\in{\mathcal{G}}^c$
sends $\overline{\partial}_{{\curly E}}$ to $g^*\overline{\partial}_{{\curly E}}=
g\circ \overline{\partial}_{{\curly E}}\circ g^{-1}$.
Any metric $h$ on ${\curly E}$ is the pullback $h=g^*h_0$
by some $g\in{\mathcal{G}}^c$. Furthermore, for any metric $h$ and gauge
transformation $g$
$$g\left(F_{(g^{-1})^*\overline{\partial}_{{\curly E}},h}\right) g^{-1}
=F_{\overline{\partial}_{{\curly E}},g^*h}
\qquad\mbox{and}\qquad
g\left(\frac{g^{-1}Q\otimes (g^{-1}Q)^{*_h}}{\|g^{-1}Q\|^2_h}
\right) g^{-1}=\frac{Q\otimes Q^{*_{(g^*h)}}}{\|Q\|^2_{g^*h}}.$$
So if $h=g^*h_0$ solves (\ref{equ0}) then, conjugating by $g$, we get
\begin{equation}
i\Lambda F_{(g^{-1})^*\overline{\partial}_{{\curly E}},h_0}+
\frac{\tau}{2}
\frac{g^{-1}Q\otimes g^{-1}Q^{*_{h_0}}}{\|g^{-1}Q\|^2_{h_0}}=c\mbox{Id}.
\label{equ1}
\end{equation}
We will now see that equation (\ref{equ1}) on $g\in{\mathcal{G}}^c$ is a
particular case of the equations considered in \cite{Mu}.
Let $F=\mathbb{P}(\operatorname{Sym}^2(\mathbb{C}^3)^*)$.
Take on $F$ the symplectic structure $\tau\omega$, where $\omega$
is the symplectic structure on $F$ obtained from the canonical metric
on $\operatorname{Sym}^2(\mathbb{C}^3)^*$.
(For that we view $F$ as the symplectic quotient $F=\mu_0^{-1}(-i)/S^1$,
where $\mu_0(z)=-i|z|^2$ is the moment map of the action of $S^1$ on
$\operatorname{Sym}^2(\mathbb{C}^3)^*$.)
Consider on $F$ the action of $U(3,\mathbb{C})$ induced by the canonical
action on $\mathbb{C}^3$. This action is Hamiltonian, and the moment map
evaluated at $x\in F$ is
\begin{equation}
\mu(x)=-i\frac{\tau}{2}\left(\frac{\hat{x}\otimes\hat{x}^*}{\|\hat{x}\|^2}
\right),
\label{mom}
\end{equation}
where $\hat{x}\in\operatorname{Sym}^2(\mathbb{C}^3)^*$ is any lift of $x$.
Let $P$ be the $U(3,\mathbb{C})$ principal bundle of $h_0$-unitary
frames of $E$, and let ${\mathcal{F}}=P\times_{U(3,\mathbb{C})}F$. The conic bundle
$Q$ gives a section $\Phi\in\Gamma({\mathcal{F}})$, and by formula (\ref{mom})
the term in (\ref{equ1}) involving $Q$ is $i\mu(\Phi)$.
Let ${\mathcal{A}}$ be the set of
connections on $P$. Let $A=A_{\overline{\partial}_{{\curly E}},h_0}$ be the
Chern connection. The action of ${\mathcal{G}}^c$ on ${\mathcal{A}}$ considered in
\cite{Mu} is as follows: $g\in{\mathcal{G}}^c$ sends $A$ to
$g(A)=A_{g^*\overline{\partial}_{{\curly E}},h_0}$.
Finally, since the conic bundle $Q:\operatorname{Sym}^2{\curly E}\to{\curly L}$ is
non-degenerate, the pair $(A,\Phi)$ is simple.
So by the theorem in \cite{Mu} there is a solution $g\in{\mathcal{G}}^c$ to equation
(\ref{equ1}) if and only if $(A,\Phi)$ is $c$-stable.
Furthermore, the metric $g^*h_0$ is unique.
The previous discussion applies also to bundles of quadrics
on projective bundles of arbitrary dimension. In the next section
we will study the $c$-stability condition on any rank and in the next
one we will give a more precise description of $c$-stability
for conic bundles.
\subsection*{Stability for bundles of quadrics}
\hfil
\medskip
We will suppose from now on
that $\mbox{Vol}(X)=1$. The stability condition stated in \cite{Mu}
refers to reductions of the structure group of our bundle to parabolic
subgroups plus antidominant characters of those parabolic subgroups.
In our case the structure group is $GL(n,\mathbb{C})$, so a parabolic reduction
is equivalent to a filtration by subbundles:
$$0\subset E_1\subset\dots\subset E_r=E,$$
where the ranks strictly increase.
The action on $E$ of any antidominant character for this reduction
is given by a matrix of this form (written using any splitting
$E=E_1\oplus E_2/E_1\oplus\dots\oplus E_r/E_{r-1}$)
\begin{equation}
\chi=\left(
\begin{array}{cccc}
z+m_1+\dots+m_{r-1} & 0 & \dots & 0 \\
0 & z+m_2+\dots+m_{r-1} & \dots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \dots & z
\end{array}
\right)-\sum_{k=1}^{r-1} m_k\frac{\operatorname{rk}(E_k)}{\operatorname{rk}(E)}\mbox{Id},
\label{caracter}
\end{equation}
where $z$ is any real number and $m_j\leq 0$ are negative real numbers
(strictly speaking this is the action of $i$ times an antidominant
character; however, we will ignore this in the sequel. Using the
notation of \cite{Mu} this matrix is $i g_{\sigma,\chi}$, where
$\chi$ is an antidominant character).
\textbf{Stability of a quadric.}
To write the stability notion for $(A,\Phi)$ we need to compute
the maximal weight of the action of $\chi$ on the section $\Phi$.
So fix a point $x\in X$ and write $$0\subset W_1\subset\dots\subset W_r=W$$
the induced filtration in the fibre $W=E_x$ over $x$.
Take a basis $e_1,\dots,e_n$ of $W$ such that for any $1\leq k\leq r$,
$\{e_1,\dots,e_{\operatorname{rk}(W_k)}\}$ is also a basis of $W_k$.
Write $e_1^*,\dots,e_n^*$ the dual basis, so that $Q$ gives
on $W$ the quadratic form
$$Q=Q(x)=\sum_{i\leq j} \alpha_{ij}(e_i^* e_j^*).$$
The action of $\chi$ on $\operatorname{Sym}^2W^*$ diagonalizes in the basis
$\{e_i^* e_j^*\}_{i\leq j}$, and one has
\begin{eqnarray*}
\lefteqn{\chi(e_i^* e_j^*)=}\\
& & \left(-(2z+2m_I+\dots+2m_{J-1}+m_J+\dots+m_{r-1})
+2\sum_{k=1}^{r-1} m_k\frac{\dim(W_k)}{\dim(W)}\right) (e_i^* e_j^*).
\end{eqnarray*}
Here and in the sequel we follow this convention: the index $I$
(resp. $J$) is the minimum one such that $e_i$ belongs to $W_I$ (resp. $e_j$
belongs to $W_J$). From this one deduces that
$$\mu(Q(x);\chi)=\max_{\alpha_{ij}\neq 0}
\{-(2z+2m_I+\dots+2m_{J-1}+m_J+\dots+m_{r-1})\}
+2\sum_{k=1}^{r-1} m_k\frac{\dim(W_k)}{\dim(W)}.$$
Define $M_I=-(m_I+\dots+m_{r-1})$. Given two subspaces $W',W''\subset W$,
we will write $Q(W',W'')=0$ if for any $w'\in W'$ and
$w''\in W''$, $Q(w',w'')=0$. Otherwise we will write
$Q(W',W'')\neq0$. Then,
\begin{equation}
\mu(Q(x);\chi)=\max_{Q(W_I,W_J)\neq 0}\{M_I+M_J-2z\}
+2\sum_{k=1}^{r-1} m_k\frac{\dim(W_k)}{\dim(W)}\mbox{Id}.
\end{equation}
\textbf{Stability for the bundle of quadrics.}
The pair $(A,\Phi)$ is by definition
$c$-stable if for any filtration $\sigma$ of $E$ by subbundles
$$0\subset E_1\subset\dots\subset E_r=E$$
and any antidominant character $\chi$ as in (\ref{caracter}) one has
\begin{equation}
\deg(\sigma,\chi)+\tau\int_{x\in X}\mu(Q(x);\chi)-\langle\chi,c\mbox{Id}\rangle>0.
\label{estable}
\end{equation}
Here the degree of the pair $(\sigma,\chi)$ is
$$\deg(\sigma,\chi)=z\deg(E)+\sum_{j=1}^{r-1}m_j
\left(\deg(E_j)-\frac{\operatorname{rk}(E_j)}{\operatorname{rk}(E)}\deg(E)\right),$$
and, on the other hand, $\langle\chi,c\mbox{Id}\rangle=zc\operatorname{rk}(E)=zcn$.
The map $Q$ is holomorphic, and the function $\mu(Q(x);\chi)$ is lower
semicontinuous and takes a finite number of values as $x$ moves on $X$.
Hence, $\mu(Q(x);\chi)$ takes its maximal value in a Zariski open
dense subset of $X$, and so
$$\int_{x\in X}\mu(Q(x);\chi)=
\mbox{Vol}(X)\max_{x\in X}\mu(Q(x);\chi)=\max_{x\in X}\mu(Q(x);\chi).$$
For any pair of subbundles $E',E''\subset E$,
define $Q(E',E'')=\max_{x\in X} Q(E'_x,E''_x).$ Then
\begin{equation}
\max_{x\in X}\mu(Q(x);\chi)=\max_{Q(E_i,E_j)\neq 0}\{M_i+M_j-2z\}+
\sum_{k=1}^{r-1}2m_k\frac{\operatorname{rk}(E_k)}{\operatorname{rk}(E)}.
\label{maxim2}
\end{equation}
Putting everything together (\ref{estable}) becomes
\begin{align}
0 &< z\deg(E)-2\tau z-zc\operatorname{rk}(E)
+\sum_{k=1}^{r-1}m_k
\left(\deg(E_k)-\frac{\operatorname{rk}(E_k)}{\operatorname{rk}(E)}\deg(E)
+2\tau \frac{\operatorname{rk}(E_k)}{\operatorname{rk}(E)}
\right) \notag \\
& +\tau\max_{Q(E_i,E_j)\neq 0}\{M_i+M_j\}.\notag
\end{align}
This must be true for any real number $z$, so the pair $(A,\Phi)$
can only be stable if
$$c=\frac{\deg(E)-2\tau }{\operatorname{rk}(E)}.$$
Define now
$$d_k=\deg(E_k)-\frac{\operatorname{rk}(E_k)}{\operatorname{rk}(E)}\deg(E)
+2\tau \frac{\operatorname{rk}(E_k)}{\operatorname{rk}(E)}.$$
Then the stability condition reduces to
\begin{equation}
\sum_{k=1}^{r-1} m_kd_k+\tau\max_{Q(E_i,E_j)\neq 0}\{M_i+M_j\}>0.
\label{defi}
\end{equation}
And this must hold for any choice of (not all zero) negative numbers
$m_1,\dots,m_{r-1}$.
\section{The case $\operatorname{rk}(E)=3$}
In the sequel we will use the following notation. If $E'$ is a
vector bundle and $\alpha$ is any real number,
$$\mu_{\alpha}(E')=\frac{\deg(E')-\alpha}{\operatorname{rk}(E')}.$$
In this section we assume that $\operatorname{rk}(E)=3$. Hence, $Q$ describes a
bundle of conics in a bundle of projective planes $\mathbb{P}(E)$ on $X$.
Recall that we assume that $Q$ is (generically) non-degenerate.
We have seen above that the pair $(A,\Phi)$ cannot be $c$-stable unless
$$c=\mu_{2\tau}(E).$$
Suppose this holds. Now, according to formula (\ref{defi}),
$(A,\Phi)$ is stable if and only if for any filtration
$0\subset E_1\subset E_2\subset E$ and
for any pair of (not all zero) real numbers $m_1,m_2\leq 0$,
\begin{equation}
m_1d_1+m_2d_2+\tau\max_{Q(E_i,E_j)\neq 0}\{M_i+M_j\}>0,
\end{equation}
where, as before,
$d_k=\deg(E_k)-\frac{\operatorname{rk}(E_k)}{\operatorname{rk}(E)}\deg(E)+2\tau \frac{\operatorname{rk}(E_k)}{\operatorname{rk}(E)}$.
There are three cases to consider:
\begin{itemize}
\item $Q(E_1,E_1)=Q(E_1,E_2)=0$, $Q(E_2,E_2)\neq 0$.
Geometrically, $E_1$ gives fibrewise a point on the conic
and $E_2$ a tangent line to the conic at the point given by $E_1$.
In this case,
$$\max_{Q(E_i,E_j)\neq 0}\{M_i+M_j\}=\max\{-2m_2,-m_1,-m_2\}.$$
Hence,
\begin{align*}
0 &> d_1-\tau=\deg(E_1)-\frac{\operatorname{rk}(E_1)}{\operatorname{rk}(E)}\deg(E)+
2\tau\frac{\operatorname{rk}(E_1)}{\operatorname{rk}(E)}-\tau, \\
0 &> d_2-2\tau=\deg(E_2)-\frac{\operatorname{rk}(E_2)}{\operatorname{rk}(E)}\deg(E)+
2\tau\frac{\operatorname{rk}(E_2)}{\operatorname{rk}(E)}-2\tau, \\
0 &> d_1+d_2-2\tau.
\end{align*}
Simplifying, we obtain the following conditions:
$$\mu_{\tau}(E_1)<\mu_{2\tau}(E)
\mbox{, }\qquad\mu_{2\tau}(E_2)<\mu_{2\tau}(E)
\mbox{, }\qquad\deg(E_1)+\deg(E_2)<\deg(E).$$
\item
$Q(E_1,E_1)=0$, $Q(E_1,E_2)\neq 0$ ($\Rightarrow\ Q(E_2,E_2)\neq 0$).
Geometrically, $E_1$ is a point on the conic and $E_2$ a line
passing through $E_1$ but generically not tangent to the conic.
In this case,
$$\max_{Q(E_i,E_j)\neq 0}\{M_i+M_j\}=-m_1-2m_2.$$
Hence,
\begin{align}
0 &> d_1-\tau=\deg(E_1)-\frac{\operatorname{rk}(E_1)}{\operatorname{rk}(E)}\deg(E)+
2\tau\frac{\operatorname{rk}(E_1)}{\operatorname{rk}(E)}-\tau, \notag \\
0 &> d_2-2\tau=\deg(E_2)-\frac{\operatorname{rk}(E_2)}{\operatorname{rk}(E)}\deg(E)+
2\tau\frac{\operatorname{rk}(E_2)}{\operatorname{rk}(E)}-2\tau.\notag
\end{align}
Simplifying, we obtain the following two conditions:
$$\mu_{\tau}(E_1)<\mu_{2\tau}(E)
\mbox{, }\qquad\mu_{2\tau}(E_2)<\mu_{2\tau}(E).$$
\item
$Q(E_1,E_1)\neq 0$ ($\Rightarrow\ Q(E_1,E_2)\neq 0
\mbox{ and }Q(E_2,E_2)\neq 0$).
Geometrically, $E_1$ gives a point generically not on the conic
and $E_2$ any line through $E_1$.
In this case,
$$\max_{Q(E_i,E_j)\neq 0}\{M_i+M_j\}=-2m_1-2m_2.$$
Hence,
\begin{align}
0 &> d_1-2\tau=\deg(E_1)-\frac{\operatorname{rk}(E_1)}{\operatorname{rk}(E)}\deg(E)+
2\tau\frac{\operatorname{rk}(E_1)}{\operatorname{rk}(E)}-2\tau, \notag \\
0 &> d_2-2\tau=\deg(E_2)-\frac{\operatorname{rk}(E_2)}{\operatorname{rk}(E)}\deg(E)+
2\tau\frac{\operatorname{rk}(E_2)}{\operatorname{rk}(E)}-2\tau. \notag
\end{align}
Simplifying, we obtain the following two conditions:
$$\mu_{2\tau}(E_1)<\mu_{2\tau}(E)
\mbox{, }\qquad\mu_{2\tau}(E_2)<\mu_{2\tau}(E).$$
\end{itemize}
\vskip 1cm
In conclusion, and as claimed at the beginning, the condition of
$c$-stability obtained from studying equation (\ref{equ0}) coincides
with that of stability obtained from the GIT construction of the
moduli space of conic bundles.
\bigskip
\textbf{Acknowledgements.}
We would like to thank R. Hern\'andez, A. King, S. Ramanan and
C.S. Seshadri for discussions and comments.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,339 |
With the determination that p-cresol sulfate was the dominant component of urinary MBPLM, studies were then undertaken to measure p-cresol sulfate in urine by MRM and to correlate it with the level of MBPLM quantitated in urine by RIA. p-Cresol sulfate and urinary MBPLM values were highly correlated (r = 0.780), and in a small series values for all MS patients, especially those with SP and PP MS, were higher than those for normal controls.42 With the RIA for urinary MBPLM, an assay on serum could not be validated due to the interference by serum proteins. Using MRM, p-cresol sulfate has been detected in serum at a level of about 2-5% that of urine. Serum p-cresol sulfate correlates well (r = 0.822) with urine p-cresol sulfate values, adjusted to reflect renal function and urine dilution, for creatinine content. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,870 |
On a lighter note my copy of Super Mario All Stars finally shipped. Looking forward to the goodies in the package as well as replaying some classics yet again ala Nintendo. I will be sure to post a short review and some pictures as soon as it arrives. Makes me all nostalgic, which makes me want to go to Midwest Gaming Classic, which I'm not going to because I have no money from buying a house. Thanks for making me sad classic Mario, Thanks a lot.
I wish I was talking about Punchline but I'm not. My father was recently hospitalized with a heart attack and received a triple bypass the next day. The technology we have now is both amazing and creepy, that we can clip clip connect you to a heart lung machine, reroute some pipes, and sew you back up in a couple hours leaves me floored. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,853 |
\section{Introduction}
A folded-symplectic manifold is a $2m$-dimensional manifold $M$ equipped with a closed $2$-form $\sigma$ so that $\sigma^m$ vanishes transversally and $\sigma$ restricted to its degenerate hypersurface $Z$ is maximally non-degenerate. We call the hypersurface, $Z$, the \emph{folding hypersurface}. Folded-symplectic forms arise naturally: the connected sum of two symplectic manifolds possesses a folded-symplectic structure \cite{CGW}, every four manifold possessses a folded-symplectic structure \cite{CdS}, and the pullback of a symplectic form by a map with fold singularities is a folded-symplectic form. A toric, folded-symplectic manifold is a (connected) folded-symplectic manifold $(M^{2m},\sigma)$ equipped with an effective, Hamiltonian action of a torus of dimension $m$. These completely integrable systems are generalizations of toric, symplectic manifolds, where we have simply inserted hypersurfaces on which the $2$-form governing the dynamics is allowed to degenerate.
The study of toric folded-symplectic manifolds is a union between two seemingly opposed mathematical viewpoints: toric symplectic geometry, where degeneracies are few and far between, and singularity theory, where one allows for smooth functions to degenerate in a controlled manner. From the latter point of view, one could say folded-symplectic geometry began in 1969 with Martinet's study of generic singularities of differential forms \cite{M}. Indeed, on a general four-manifold, the author shows that a two form has a singularity of type $\Sigma_{2,0}$ if it can be written in coordinates as $xdx\wedge dy + dz\wedge dt$, which is the canonical example of a folded-symplectic form. Incidentally, one observes that $xdx\wedge dy + dz\wedge dt$ is the pullback of the standard symplectic structure $\omega_{\R^4}$ on $\R^4$ by the fold map $\psi(x,y,z,t)=(\frac{x^2}{2},y,z,t)$, which validates the name \emph{folded-symplectic}.
From the symplectic point of view, Cannas da Silva, Guillemin, and Woodward began a series of studies in folded-symplectic geometry in 1999-2000 focused on relating the existence of \emph{spin}-c structures to the existence of folded-symplectic forms (on connected sums of symplectic manifolds). In \cite{CGW}, they describe an unfolding operation in which one may decompose a folded-symplectic manifold into disjoint symplectic pieces, provided the intersection $\ker(\sigma)\cap TZ$ defines a $1$-dimensional foliation of the folding hypersurface by circles. This assumption that the null foliation induces a circle fibration permeates much of the literature on folded-symplectic manifolds. In \cite{CGP}, Cannas da Silva, Guillemin, and Pires study \emph{toric origami manifolds}, which are compact toric folded-symplectic manifolds where the null foliation on the fold is generated by a locally free circle action, so that the folding hypersurface $Z$ fibrates over a compact base $B$. They show that there is a one-to-one correspondence between toric origami manifolds and \emph{origami templates}, which are collections of unimodular polytopes in the dual of the Lie algebra of the torus, $\fg^*$. This is a generalization of Delzant's classical theorem \cite{De} which states that there is a one-to-one correspondence between compact toric symplectic manifolds and unimodular polytopes in $\fg^*$.
The primary goal of this thesis is to extend these classification results to non-compact, toric, folded-symplectic manifolds where there is no assumption on the null-foliation on the fold. Some work has already been done in this area: in \cite{LeeC} Lee gives a sufficient condition for the existence of isomorphisms between toric, folded-symplectic four manifolds and in \cite{KL} Karshon-Lerman classify all \emph{symplectic} toric manifolds. In our classification, we make a mild assumption that the folding hypersurface is co-orientable. As we show in chapter $4$, every folding hypersurface $Z$ with an Hamiltonian action of a Lie group $G$ may be equivariantly embedded into a folded-symplectic, Hamiltonian $G$-manifold $(M,\sigma)$ as a co-orientable folding hypersurface (q.v. corollary \ref{cor:preserves}), so this assumption isn't unfounded. Furthermore, one can show that if the folding hypersurface is \emph{not} co-orientable, then the ambient folded-symplectic manifold $M$ cannot be orientable, in which case one need only pass to the orientable double cover $\tilde{M}$ of $M$ to be back in a situation where the folding hypersurface is co-orientable. The primary reason we make this assumption about co-orientability is to avoid cases where the action of the torus $G$ on the folding hypersurface is \emph{not} effective. We follow the approach of \cite{KL} in our classification of non-compact toric symplectic manifolds. We first prove:
\begin{theorem}
Let $(M,\sigma,\mu:M\to \fg^*)$ be a toric, folded-symplectic manifold with moment map $\mu:M \to \fg^*$, where $\fg$ is the Lie algebra of the torus $G$ acting on $M$. Assume the fold $Z\subseteq M$ is co-orientable. Then,
\begin{enumerate}
\item $M/G$ is naturally a manifold with corners and
\item the moment map $\mu$ descends to a smooth map $\bar{\mu}:M/G \to \fg^*$, which is a unimodular map with folds (q.v. definition \ref{def:umf}).
\end{enumerate}
\end{theorem}
We then fix a manifold with corners $W$ and a unimodular map with folds $\psi:W \to \fg^*$ and we define a category $\mathcal{M}_{\psi}(W)$ (q.v. definition \ref{def:empsi} or below).
\begin{definition}
Let $W$ be a manifold with corners and let $\psi:W\to \fg^*$ be a unimodular map with folds, where $\fg$ is the Lie algebra of a torus $G$. We define $\mathcal{M}_{\psi}(W)$ to be the category whose objects are triples:
\begin{displaymath}
(M,\sigma, \pi:M \to W)
\end{displaymath}
where $\pi$ is a quotient map and $(M,\sigma, \psi \circ \pi)$ is a toric, folded-symplectic manifold with co-orientable folding hypersurface, where the torus is $G$, with moment map $\psi \circ \pi$. We refer to an object as a \emph{toric, folded-symplectic manifold over $\psi$}. A morphism between two objects $(M_i,\sigma_i,\pi_i:M \to W)$, $i=1,2$, is an equivariant diffeomorphism $\phi:M_1 \to M_2$ that induces a commutative diagram:
\begin{displaymath}
\xymatrix{
M_1 \ar[rr]^{\phi} \ar[dr]^{\pi_1} & & M_2 \ar[dl]_{\pi_2} \\
&W \ar[r]^{\psi}& \fg^*
}
\end{displaymath}
and satisfies $\phi^*\sigma_2=\sigma_1$. That is, $\phi$ is an equivariant folded-symplectomorphism that preserves moment maps. By definition, every morphism is invertible, hence $\mathcal{M}_{\psi}(W)$ is a groupoid.
\end{definition}
We seek to classify objects in $\mathcal{M}_{\psi}(W)$ up to isomorphism and we prove the following classifcation result.
\begin{theorem}
Isomorphism classes of objects in $\mathcal{M}_{\psi}(W)$ are in bijection with $H^2(W; \mathbb{Z}_G \times \R)$, where $\mathbb{Z}_G= \ker(\exp:\fg \to G)$ is the integral lattice of the torus $G$ that acts on objects of $\mathcal{M}_{\psi}(W)$.
\end{theorem}
The proof of these two results constitute the bulk of the material presented in chapters $5-8$. The secondary goal of this thesis, comprising the remaining $3$ chapters, is to provide a foundational framework for the study of folded-symplectic forms. Given the fact that folded-symplectic forms have recently found their way into fields such as four manifolds \cite{CdS} and Higgs bundles \cite{H}, it is the author's belief that a rigorous study of some of the structures associated to a fold form will be useful.
The organization of the thesis is as follows. In chapter $2$ we begin with an introduction to jet bundles on manifolds with corners. Jet bundles are typically only studied on manifolds (without corners). Michor defines jet bundles for manifolds with corners in \cite{Mi2}, though they are defined using restrictions of jet bundles over $\R^n$ to quadrants. We offer an equivalent approach to the construction of the first jet bundle which avoids restrictions and excessive choices of coordinates. We then discuss the various structures associated with the first jet bundle, such as jet fields and connections, and move on to define fold maps for manifolds with corners. There are two reasons we rigorously develop the notion of a fold singularity for maps between manifolds with corners. First, the definition of a fold singularity of a map between manifolds with corners doesn't appear in the literature. It is tempting to take the definition of a fold singularity (e.g. in \cite{Ho} or \cite{GG}) and simply replace \emph{manifold} by \emph{manifold with corners}. However, it is not difficult to construct examples of maps with only fold singularities under this definition whose locus of degeneracies is neither a manifold or a manifold with corners. The issue is that a fold singularity is defined via a transversal intersection (q.v. definition \ref{def:folds}) and if one uses the traditional notion of transversality in the category of manifolds with corners, then inverse images of submanifolds with corners may fail to be submanifolds with corners (q.v. \cite{Mi2}). Thus, the second reason we take time to develop the theory is because we must adopt an adequate notion of transversality and show that this notion leads to a theory of fold singularities in the category of manifolds with corners that is consistent with the theory in the category of manifolds. We conclude the chapter with an exercise in the theory of first jet bundles where we give a generalization of Morse functions to Morse sections of $1$-dimensional fiber bundles.
We begin our study of folded-symplectic manifolds in chapter $3$, giving the definition along with some examples and then constructing a folded version of the cotangent bundle, which may be seen as an attempt to lengthen the list of naturally occurring sources of folded-symplectic forms. This construction dualizes the construction of the $b$-cotangent bundle found in \cite{GMP} (example $9$). We then discuss Moser's argument for folded-symplectic manifolds in detail, ultimately arriving at a characterization of folded-symplectic forms in terms of their induced maps on a distinguished pair of sheaves. We conclude by discussing a non-equivariant normal form for a neighborhood of a co-orientable folding hypersurface, which was proven in the case where the manifold is compact and oriented in \cite{CGW} (the equivariant version may be found in chapter $4$). To this end, we show that the fold inherits a canonical orientation from the fold form $\sigma$. This fact is briefly discussed in \cite{M} using a choice of orientation near each point in the fold. We give an alternate approach to show that the orientation arises from the intrinsic derivative of the contraction mapping $C_{\sigma}:TM \to T^*M$, $C_{\sigma}(X)=i_X\sigma$, from tangent bundle to cotangent bundle.
In chapters $4$ and $5$ we discuss Hamiltonian actions of Lie groups on folded-symplectic manifolds. Most of the material in chapter $4$ is standard material in symplectic geometry that we will need to prove that the orbit space of a toric, folded-symplectic manifold (with co-orientable folding hypersurface) is a manifold with corners. The two new results we prove are the following. First, there is a well-defined symplectic normal bundle to orbit-type strata in an Hamiltonian folded-symplectic manifold.
\begin{prop}
Let $G$ be a compact, connected Lie group and let $(M,\sigma)$ be a folded-symplectic Hamiltonian $G$-manifold with moment map $\mu:M\to \fg^*$, where $\fg$ is the Lie algebra of $G$. Suppose the folding hypersurface $Z$ is co-orientable. Let $H\le G$ be a subgroup and suppose $M_H$ is nonempty. Then there exists a vector bundle $\widetilde{(TM_H)}^{\sigma}\to M_H$ with the following properties:
\begin{enumerate}
\item $\widetilde{(TM_H)}^{\sigma}$ is a subbundle of $TM\big\vert_{M_H}$.
\item The restriction $\widetilde{(TM_H)}^{\sigma}\big\vert_{M\setminus Z}$ to the symplectic portion of $M$ is the vector bundle $T(M_H \setminus Z)^{\sigma}\to (M_H\setminus Z)$.
\item $TM\big\vert_{M_H}$ splits $H$-equivariantly as $TM\big\vert_{M_H} = TM_H \oplus \widetilde{(TM_H)}^{\sigma}$.
\item $\widetilde{(TM_H)}^{\sigma}$ equipped with the restriction of $\sigma$ is a symplectic vector bundle over $M_H$.
\item $\widetilde{(TM_H)}^{\sigma}\big\vert_{Z_H}$ is a subbundle of $TZ_H$.
\end{enumerate}
In other words, the symplectic normal bundle to $M_H\setminus Z$ extends across the fold $Z$ to give us a symplectic normal bundle to $M_H$ and, at points of the intersection $Z_H=M_H\cap Z$, it is tangent to the fold.
\end{prop}
\noindent Second, we have the penultimate structure theorem for toric, folded-symplectic manifolds.
\begin{theorem}\label{thm:intro}
Let $(M,\sigma,\mu:M\to \fg^*)$ be a toric, folded-symplectic manifold with co-orientable folding hypersurface. Then,
\begin{enumerate}
\item The orbit type strata $M_H$ are transverse to the folding hypersurface and each $(M_H,i_{M_H}^*\sigma, \mu\vert_{M_H}:M_H \to \frak{h}^o)$ is a toric, folded-symplectic manifold, hence $M$ is stratified by toric, folded-symplectic manifolds.
\item The orbit space $M/G$ is a manifold with corners and the boundary strata of $M/G$ are given by the images of the orbit-type strata $M_H/G$.
\item The moment map descends to $\psi:M/G\to \fg^*$, a unimodular map with folds. Furthermore, since each $(M_H,i_{M_H}^*\sigma)$ is a toric, folded-symplectic manifold, the restriction of $\psi$ to $M_H/G$ is a map with fold singularities if we view it as a map into $\frak{h}^o$. Hence $\psi:M/G \to \fg^*$ is a unimodular map with folds that restricts to maps with fold singularities on the boundary strata.
\item The null-foliation on $Z$ may be recovered from $\psi$, along with its orientation induced by $\sigma$ using the intrinsic derivative of $\psi$ and the map $(\cdot)_Z: \pi^*\operatorname{Im}(d\psi)^o \to \ker(\sigma)\cap TZ$ (q.v. lemma \ref{lem:nullfoliation}).
\item The remainder of the bundle $\ker(\sigma)$ can be constructed by choosing lifts of elements of $\ker(d\psi)$.
\item The representation of $H$ on the fibers of $(\widetilde{TM_H})^{\sigma}$ at $Z$ may be read from the orbital moment map.
\item The local structure of the folding hypersurface is determined by the image of $\psi(Z/G)$ (q.v. corollary \ref{cor:foldnorm}).
\end{enumerate}
Thus, the fold, the null foliation, the orientation, the kernel bundle, and the symplectic slice representation may all be recovered from the orbital moment map. And, one may recover all symplectic invariants away from the fold just by reading the weights of the symplectic slice representation from the orbital moment map.
\end{theorem}
\noindent In the end of chapter $5$, we introduce the two categories $\mathcal{M}_{\psi}(W)$ and $\mathcal{B}_{\psi}(W)$ of toric folded-symplectic manifolds and bundles, respectively. We have already seen $\mathcal{M}_{\psi}(W)$ in this introduction. The category of toric, folded-symplectic bundles consists of principal $G$ bundles $\pi:P \to W$ equipped with a folded-symplectic structure so that the action of $G$ is Hamiltonian with moment map $\psi\circ \pi$. We then argue that all objects of $\mathcal{M}_{\psi}(W)$ are locally isomorphic (q.v. lemma \ref{lem:locunique}).
In chapter $6$, we develop our last tool before we begin the process of classifying toric, folded-symplectic manifolds: folded-symplectic reduction. We precisely describe the conditions under which the reduced space is symplectic and the conditions under which the reduced space is folded-symplectic with nonempty fold. In particular, we show that one need only understand how the zero level set of the moment map intersects the folding hypersurface in order to accurately predict when the reduced space is symplectic or not. We conclude the chapter by adapting the minimal coupling procedure of Sternberg \cite{St} to symplectic vector bundles over folded-symplectic manifolds and describing how this relates to the orbit-type strata of toric, folded-symplectic manifolds. In particular, the structure theorem (q.v theorem \ref{thm:intro}) implies that the orbit type strata are themselves folded-symplectic with well-defined symplectic normal bundles, hence we may apply the minimal coupling procedure to equip a neighborhood of the zero section of the normal bundle with a folded-symplectic structure.
In chapters $7$ and $8$ we classify toric, folded-symplectic manifolds over a fixed unimodular map with folds up to isomorphism. We begin by classifying objects in the category $\mathcal{B}_{\psi}(W)$ of toric, folded-symplectic bundles (with corners) over $\psi:W \to \fg^*$ (q.v. definition \ref{def:bpsi}), following the approach of \cite{KL}. We prove:
\pagebreak
\begin{theorem}
Let $\psi:W \to \fg^*$ be a unimodular map with folds, where $\fg$ is the Lie algebra of a torus $G$. Let $\mathcal{B}_{\psi}(W)$ be the category of toric, folded-symplectic bundles over $\psi$. Then there is a bijection $b:\pi_0(\mathcal{B}_{\psi}) \to H^2(W; \mathbb{Z}_G \times \R)$. That is, isomorphism classes of toric, folded-symplectic bundles over $\psi$ are parameterized by cohomology classes in $H^2(W,\mathbb{Z}_G\times \R)$.
\end{theorem}
\noindent We then develop a functor $c:\mathcal{B}{\psi}(W) \to \mathcal{M}_{\psi}(W)$ by cutting away the corners to introduce stabilizers. This comprises the first three sections of chapter $8$. To finish the classification, we show that $c$ is an equivalence of categories, hence it induces a bijection on isomorphism classes of objects.
As a final note to the reader, we will often use transversality for manifolds with corners (q.v. definition \ref{def:A1:trans}), which we denote as $\pitchfork_s$ for \emph{strong transversality}. As we have already stated, the traditional notion of transversality does not suffice in the category of manifolds with corners. If $f:W \to N$ is a map between manifolds with corners, $S\subseteq N$ a submanifold, and $f\pitchfork S$ in the sense of manifolds, then $f^{-1}(S)$ may simply be a topological space or a differential space (q.v. \cite{Mi2}). To give it the structure of a submanifold with corners, we need to require that the restriction of $f$ to each boundary stratum of $W$ is transverse to $S$ (q.v. \cite{CD}). This condition is clearly quite limiting, which is why we choose to call this type of transversality \emph{strong transversality}.
\pagebreak
\section{First Order Jet Bundles over Manifolds with Corners and Fold Maps}
The purpose of this section is two-fold. First, we will need jets on manifolds with corners, so we would like to introduce the basic theory of the first-order jet bundle $J^1(E)$ of a fiber bundle $\pi:E \to M$ over a manifold with corners. This task is somewhat cumbersome if we use the usual approach to jet bundles, which involves plastering the total space $J^1(E)$ with derivative coordinate charts as in \cite{GG, Mi2, Sa} and then either realizing $J^1(E)$ as a restriction $J^1(\tilde{E})\big\vert_E$, where $\tilde{E}$ is a manifold without corners containing $E$ or defining it locally as such a restriction and gluing the pieces together (as in \cite{Mi2}). We present an alternate approach which does not rely on excessive coordinate computations and avoids the problem of embedding $E$ into a manifold $\tilde{E}$ without corners. The approach relies on the understanding of $J^1(M\times F)$, the first jet bundle of the trivial fiber bundle $M\times F \to F$. In short, if one knows what $J^1(M\times F)$ should be as a manifold with corners, then $J^1(E)$ is many copies of $J^1(U\times F)$, $U\subseteq M$ open, glued together which gives $J^1(E)$ the structure of a smooth manifold with corners. This is proposition \ref{prop:jettopology}. The advantage of our approach is that it is very clean while the obvious disadvantage is that it does not generalize to higher order jet bundles. We prove that every first order jet bundle $J^1(E)$ is isomorphic to $\hom(\pi^*TM,V)$, where $V\to E$ is the vertical bundle, and such isomorphisms may be specified by a choice of connection $\chi$ on $E$. Thus, a first order jet bundle has a very recognizable form as a $\hom$ bundle, albeit perhaps non-canonically. This is the content of proposition \ref{prop:C2}. We then use our understanding of the first-order jet bundle to generalize Morse functions to fiber bundles with fiber-dimension $1$, which represents original work.
The main purpose of this section is to introduce the notion of an equidimensional map with fold singularities on manifolds with corners, or simply a map with fold singularities between manifolds (with corners) of the same dimension. Maps with fold singularities have found their place in much of mathematical literature, but there does not appear to be a definition of a fold singularity for maps between manifolds with corners. The definition of a map with fold singularities using the intrinsic derivative, given in the appendix of \cite{Ho}, would seem to be adequate for generalizing fold maps to manifolds with corners, but then pathological examples arise where fold maps $f:M \to N$ become homeomorphisms or the folding hypersurface has connected components of varying dimensions. Indeed, consider the following set of examples.
\begin{example}
Let $f:\R^2 \to \R^2$ be the map given by $f(x,y)=(x,y^2)$, which has fold singularities along the $x$-axis. We consider three scenarios:
\begin{itemize}
\item Let $W=\{(x,y)\vert \mbox{ } y\ge 0\}$. Then, using the definition of a fold singularity found in \cite{Ho}, $f\big\vert_W:W \to \R^2$ has fold singularities along $y=0$, but no folding is accomplished by $f\big\vert_W$. In fact, it is a homeomorphism of $W$ onto itself. This example may not raise too many objections, so consider the next example.
\item Let $W=\{(x,y) \vert \mbox{} y\ge \vert x \vert\}$. Then $f\big\vert_W: W \to \R^2$ has a fold singularity at $(0,0)$ (in the sense of \cite{Ho}). If we are to mimic the behaviour of fold maps in the category of manifolds, the locus of degeneracies should be a codimension $1$ submanifold with corners, but it is a codimension $2$ submanifold with corners here. Again, this may not be unreasonable, so let us consider another example.
\item Let $W\subset \R^2$ be any manifold with corners that lies inside the upper half plane ($y\ge 0$) whose intersection with the $x$-axis has a component of dimension $0$ and a component of dimension $1$. Then $f\big\vert_W:W \to \R^2$ has fold singularities and its locus of degeneracies is a disjoint union of manifolds of varying dimensions. This is highly undesirable.
\end{itemize}
\end{example}
\noindent As we have stated in the introduction, the primary issue is transversality: if $f:M\to N$ is a map of manifolds with corners and $S$ is a submanifold with corners, then $f\pitchfork S$ in the traditional sense of manifolds is not enough to guarantee that $f^{-1}(S)$ is a submanifold with corners. One needs to require that $f$ restricted to each stratum of $M$ is transverse to $S$.
Our approach to defining fold maps is to use the theory of first order jet bundles on manifolds with corners that we develop in section 2.1 and adapt Guillemin and Golubitsky's definition of a submersion with fold singularities (definition 4.1 in \cite{GG}) to our needs. In particular, we generalize the definition of a submersion with folds to arbitrary fiber bundles, we present a definition of an equidimensional map with fold singularities between manifolds with corners, we develop an extremely useful computational tool for understanding equidimensional fold maps, and we provide a normal form for fold maps, all of which is original work since it hasn't been done in the case of manifolds with corners. The most important results that the reader ought to carry on to the subsequent chapters are the computational corollary \label{cor:folds4-0}:
\begin{cor}
Let $f:M \to N$ be a smooth map between two $m$-dimensional manifolds with corners. Then $f$ is a map with fold singularities if and only if
\begin{enumerate}
\item The induced map $(df)^m: M \to \hom(\Lambda^m(TM), \Lambda^m(TN))$ is transverse (in the sense of manifolds with corners) to the zero section $\mathcal{O}$ of $\hom(\Lambda^m(TM),\Lambda^m(TN)) \to M \times N$ (i.e. locally the determinant of $df$ vanishes transversally), and
\item $\ker(df) \pitchfork ((df)^m)^{-1}(\mathcal{O})$.
\end{enumerate}
\end{cor}
\noindent which we will use without reserve to show that certain maps are fold maps, and the following factorization corollary to proposition \ref{prop:folds5},
\begin{cor}\label{cor:folds5}
Let $f:M^n \to N^n$ be an equidimensional map with fold singularities, suppose $f$ is strata-preserving, and suppose for all $p\in Z$, the folding hypersurface, $\ker(df_p)$ is tangent to the stratum of $M$ containing $p$ (i.e. $\ker(df) \to Z$ is stratified). Then, for each $p\in Z$ there exist a neighborhood $U\subset M$, a strata-preserving fold map $\psi:U\to U$, and a strata-preserving open embedding $\phi:U \to N$ so that $f\vert_U=\phi\circ \psi$. Hence, locally, every map $f$ satisfying the conditions of the proposition factors as a diffeomorphism composed with a strata-preserving fold map.
\end{cor}
\noindent which we will use when we construct a cutting procedure for toric, folded-symplectic bundles (q.v. chapters $7$ and $8$).
\subsection{First Order Jet Bundles}
Our constructions are motivated by remark 2 on page 41 of \cite{GG} and part 7 of theorem 21.5 in \cite{Mi1}, which state that $J^1(M\times N)$ is canonically isomorphic to a $\hom$ bundle, $\hom(TM,TN)$, which we define below (q.v. \ref{def:hombdle}). In particular, since we know what the first jet bundle of a product \emph{should} be, we essentially define the first jet bundle of a product to be $\hom(TM,TN)$ and argue that the first jet bundle of a general fiber bundle is a collection of these bundles glued together. While the presentation of the first jet bundle found here is somewhat nonstandard, most of these results are \emph{not} new. We will alert the reader to known results along the way.
\subsubsection{First Order Jets}
Let $\pi:E \to M$ be a smooth fiber bundle with typical fiber $F$ over a smooth $m$-dimensional manifold with corners, $M$, where $F$ may have corners. We wish to define the first jet bundle $J^1(E)$ and give it a smooth atlas.
\begin{definition}\label{def:locsec}
A \emph{local section} of $E$ near $p\in M$ is a neighborhood $U$ of $p$ and a section $\phi:U \to E\vert_U \subset E$. A local section near $p$ will be denoted as a pair $(U, \phi)$. The space of local sections near $p$ will be denoted $S_p=\{(U,\phi)\vert \mbox{ } p\in U\}$.
\end{definition}
\begin{definition}\label{def:1equiv}
Two local sections $(U,\phi)$, $(V,\psi)$ are $1$-equivalent at $p\in M$ if:
\begin{enumerate}
\item $p\in U\cap V$,
\item $\phi(p)=\psi(p)$, and
\item $d\phi_p = d\psi_p$.
\end{enumerate}
\end{definition}
\noindent In coordinates, $(U,\phi)$ and $(V,\psi)$ are $1$-equivalent at $p$ if they agree at $p$ and their first partials are equal at $p$. The following lemma is an immediate consequence of the definition.
\begin{lemma}\label{lem:equivrelation}
$1$-equivalence at $p\in M$ defines an equivalence relation $\sim$ on the set of local sections near $p$, $S_p=\{(U,\phi)\vert \mbox { } p\in U\}$.
\end{lemma}
\begin{definition}\label{def:1jetfiber}
We define the \emph{space of $1$-jets at $p\in M$} to be $(J^1_pE) = S_p/\sim$. If $(U,\phi)$ is a local section near $p$ then we denote its equivalence class in $S_p/\sim$ by $j^1_p\phi$.
\end{definition}
\begin{definition}\label{def:firstjet}
We define \emph{the first order jet space} $J^1(E)$ (as a set) to be the set $J^1(E) = \sqcup_{p\in M}J^1_pE$. It is equipped with a projection map to $M$, $\pi_1:J^1(E) \to M$, given by $\pi_1(j^1_p\phi)=p$ and a projection map to $E$, $\pi_{1,0}:J^1(E) \to E$, given by $\pi_{1,0}(j^1_p\phi)=\phi(p)$. If $U\subset M$ is an open set, we define $J^1(E)\vert_U = \sqcup_{p\in U}J^1_pE = \pi_1^{-1}(U)$.
\end{definition}
Note that $\pi_{1,0}$ is well-defined since all elements of the equivalence class $j^1\phi$ have the same value $\phi(p)$ at $p$. We can assign a topology and smooth structure to $J^1(E)$ as follows. First, we need the following definition:
\begin{definition}\label{def:hombdle}
Let $M,F$ be two manifolds with corners. Let $pr_1:M\times F \to M$ and $pr_2:M\times F \to F$ be the projections. We define:
\begin{displaymath}
\hom(TM,TF):=\hom(pr_1^*TM, pr_2^*TF)
\end{displaymath}
which is a bundle over $M\times F$ with fiber $\hom(T_mM,T_fF)$.
\end{definition}
Now, let $U\subset M$ be an open set so that $E\vert_U$ is trivializable. That is, there exists an isomorphism $\Phi_U$ of fiber bundles:
\begin{equation}\label{eq:triv}
\xymatrixcolsep{5pc}\xymatrix{
E\vert_U \ar[r]^{\Phi_U} \ar[dr]^{\pi} & U\times F \ar[d]^{pr_1}\\
& U
}
\end{equation}
The map $\Phi_U$ of equation \ref{eq:triv} defines a map of sets
\begin{displaymath}
\begin{array}{l l}
\tilde{\Phi}_U & : J^1(E)\vert_U \to \hom(TU,TF) \\
& \mbox{\hspace{6mm}} j^1_p\phi \mbox{\hspace{2mm}}\mapsto (p,\Phi_U(\phi(p)),dpr_2(d\Phi_U(d\phi_p))) \\
\end{array}
\end{displaymath}
giving us a commutative diagram:
\begin{displaymath}
\xymatrixcolsep{5pc}\xymatrix{
J^1(E)\vert_U \ar[r]^{\tilde{\Phi}_U} \ar[d]^{\pi_{1,0}} & \hom(TU,TF) \ar[d]^{pr_U\times pr_F} \\
E\vert_U \ar[r]^{\Phi_U} \ar[dr]^{\pi} & U\times F \ar[d]^{pr_1} \\
& U \\
}
\end{displaymath}
\begin{lemma}\label{lem:jettopology}
The map $\tilde{\Phi}_U$ is a well-defined bijection with an inverse given by:
\begin{displaymath}
\tilde{\Phi}_U^{-1}(p,f,A) = j^1_p\phi
\end{displaymath}
where $j^1_p\phi$ is the equivalence class of local sections $(U,\phi)$ at $p$ whose value at $p$ is $\Phi_U^{-1}(p,f)$ and differential is $d\phi_p= (d\Phi_U)_{(p,f)}^{-1}(id_{T_pU}\oplus A)$.
\end{lemma}
\begin{remark}
Before we begin, let us explain the notation $id_{T_pU \oplus A}$. That is, let us review how we may recover the differential of a section $\phi$ of $U\times F$ given its differential $dpr_2(d\phi):TU \to TF$ in the vertical direction. If we are given a section $\phi:U \to U\times F$ of the trivial fiber bundle $U\times F$, then it may be written as $\phi(p)=(p,g(p))$ for some smooth map $g:U\to F$. Then $d\phi_p = id_{T_pU} \oplus dg_p$, using the canonical splitting $T(U\times F) = pr_1^*TU \oplus pr_2^*TF$, where $pr_i$ is projection onto the $i^{th}$ factor of $U\times F$. Note that $dg_p\in \hom(T_pU, T_{g(p)}F)$. Thus, the differential of any section of $U\times F$ has the form $id_{T_pU} \oplus A$ for some $A\in \hom(T_pU, T_fF)$.
\vspace{2mm}
Conversely, if we are given an element $A\in \hom(T_pU,T_fF)$, then there is a map $g:V \to F$ defined on a neighborhood $V$ of $p$ satisfying $g(p)=f$ and $dg_p= A$. Then $id_{T_pU} \oplus dg_p: T_pU \to T_pU \oplus T_fF$ is the differential of the local section $\phi(u)=(u,g(u))$. Hence, we may append the identity map $id_{T_pU}$ to any element $A\in \hom(T_pU,T_fF)$ to transform it into the differential of a section.
\end{remark}
\begin{proof}[Proof of lemma \ref{lem:jettopology}] \mbox{ } \newline
\begin{enumerate}
\item $\tilde{\Phi}_U$ is well-defined since $(U_1,\phi_1)\sim (U_2,\phi_2)$ at $p$ if and only if $\phi_1(p)=\phi_2(p)$ and $(d\phi_1)_p = (d\phi_2)_p$, hence $dpr_2(d\Phi_U((d\phi_1)_p))=dpr_2(d\Phi_U((d\phi_2)_p))$.
\item We now show that the so-called inverse map, $\tilde{\Phi}_U^{-1}$ is a well-defined map whose image is in $J^1(E)$. Namely, we must show that if we are given $(p,f,A)$, where $A\in \hom(T_pU, T_fF)$, then there exists a local section $(V,\phi)$ near $p$ whose value is $\Phi_U^{-1}(p,f)$ and whose differential is $d\phi_p=(d\Phi_U)_{(p,f)}^{-1}(id_{TU}\oplus A)$.
\vspace{5mm}
There exist neighborhoods $V\subset U$ of $p$ and $W\subset F$ of $f$ and a map $\psi:V\to W$ so that:
\begin{enumerate}
\item $\psi(p)=f$ and
\item $d\psi_p = A$.
\end{enumerate}
Then $\psi$ defines a local section of $U\times F$ near $p\in U$: $\phi(x) = (x, \psi(x))$ and $(V, \Phi_U^{-1}\circ \phi)$ is a local section of $E$ near $p$. The value of $\Phi_U^{-1}\circ \phi$ at $p$ is $\Phi_U^{-1}(p,f)$. The differential of $\Phi_U^{-1}\circ \psi$ at $p$ is:
\begin{displaymath}
d(\Phi_U^{-1}\circ \phi)_p = d(\Phi_U)_{(p,f)}^{-1}(id_{T_pU}\oplus A)
\end{displaymath}
hence our definition of $\tilde{\Phi}\vert_U^{-1}(p,f,A)$ gives us a well-defined element of $J^1_pE$.
\item To see that the two maps are inverses, we compute:
\begin{displaymath}
\tilde{\Phi}_U \circ \tilde{\Phi}_U^{-1}(p,f,A) = \tilde{\Phi}_U(j^1_p\phi) = (\Phi_U(\phi(p)),dpr_2(d\Phi_U(d\phi_p)))=(p,f,dpr_2(id_{T_pU}\oplus A)) = (p,f,A)
\end{displaymath}
where we have used that $\tilde{\Phi}_U^{-1}(p,f,A)=j^1_p\phi$, where $d\phi_p = d\Phi_U^{-1}(id_{T_pU}\oplus A)$. We also have,
\begin{displaymath}
\tilde{\Phi}\vert_U^{-1} \circ \tilde{\Phi}_U(j^1_p\phi) = \tilde{\Phi}\vert_U^{-1}(\Phi_U(\phi(p)),dpr_2(d\Phi_U(d\phi_p)))=j^1\eta
\end{displaymath}
where $j^1\eta$ is the equivalence class of local sections $(U',\eta)$ near $p$ whose value at $p$ is $\Phi_U^{-1}(\Phi_U(\phi(p)))=\phi(p)$ and whose differential at $p$ is:
\begin{displaymath}
(d\Phi\vert_U)_{(p,f)}^{-1}(id_{T_pU}\oplus dpr_2(d\Phi_U(d\phi_p))) = d\phi_p
\end{displaymath}
Therefore, $j^1\eta = j^1\phi$.
\end{enumerate}
\end{proof}
\noindent Since $\hom(TU,TF)$ is a topological space and $\tilde{\Phi}_U$ is a bijection, we can pull back the topology on $\hom(TU,TF)$ to $J^1(E)\vert_U$ so that $\tilde{\Phi}_U$ is a homeomorphism. This gives us a global topology on $J^1(E)$ making the maps $\pi_{1,0}$ and $\pi_1$ continuous.
\begin{definition}\label{def:jettopology}
Let $\pi:E\to M$ be a fiber bundle and $J^1(E)$ the first order jet space. The topology $\mathcal{T}$ on $J^1(E)$ is the finest topology so that for each open set $U\subset M$ where $E$ is trivializable, the induced map $\tilde{\Phi}_U$ of lemma \ref{lem:jettopology} is a homeomorphism. We refer to $J^1(E)$ with the topology $\mathcal{T}$ as the \emph{first order topological jet space}.
\end{definition}
\begin{definition}\label{def:chart}
Let $\pi:E \to M$ be a fiber bundle over a manifold with corners $M$ with typical fiber $F$. Let $U\subset M$ be an open subset so that $E\vert_U$ is trivializable and let $\tilde{\Phi}_U:J^1(E)\vert_U \to \hom(TU,TF)$ be the induced map of lemma \ref{lem:jettopology}. We will say that the pair $(U,\tilde{\Phi}_U)$ is a \emph{chart} on $J^1(E)$ even though $U\subset M$. This is because the domain of $\tilde{\Phi}_U$ is $\pi_1^{-1}(U)$, hence specifying $U$ uniquely specifies the domain of $\tilde{\Phi}_U$. Therefore, to save on notation, we will use $(U,\tilde{\Phi}_U)$ instead of $(\pi_1^{-1}(U), \tilde{\Phi}_U)$ when referring to charts on $J^1(E)$. We say a collection of charts $\{(U,\tilde{\Phi}_U)\}$ is a $C^{\infty}$-\emph{atlas} if the transition maps $\tilde{\Phi}_V \circ \tilde{\Phi}_U^{-1}$ are diffeomorphisms of manifolds with corners.
\end{definition}
\noindent Now, we have a collection of charts $\{(U,\tilde{\Phi}_U)\vert \mbox{ } E\vert_U \simeq U \times F\}$ on $J^1(E)$. Our next task is to show that this collection forms a smooth atlas.
\begin{lemma}\label{lem:jettopology1}
Let $\pi:E \to M$ be a fiber bundle and let $J^1(E)$ be the first order topological jet space. Let $(U,\tilde{\Phi}_U)$, $(V,\tilde{\Phi}_V)$ be two charts on $J^1(E)$ so that $U\cap V \ne \emptyset$. Then $\tilde{\Phi}_V\circ \tilde{\Phi}_U^{-1}: \hom(T(U\cap V), TF) \to \hom(T(U\cap V), TF)$ is a diffeomorphism of manifolds with corners.
\end{lemma}
\begin{proof} \mbox{ } \newline
We compute:
\begin{displaymath}
\tilde{\Phi}_V (\tilde{\Phi}_U(p,f,A) = (\Phi_V (\Phi_U^{-1}(p,f)), dpr_2(d\Phi_V\circ d\Phi_U^{-1}(id_{TU}\oplus A))) = ((\Phi_V\circ \Phi_U^{-1})(p,f),d(pr_2 \circ \Phi_V \circ \Phi_U^{-1})(id_{TU}\oplus A))
\end{displaymath}
Since $\Phi_V\circ \Phi_U^{-1}$ is a diffeomorphism of manifolds with corners, the result follows.
\end{proof}
\begin{remark}
Recall that if we have a fiber bundle $\pi:E \to M$, then we can define the vertical subbundle $V$ of $TE$ to be the bundle whose fiber at $e$ is $V_e=\ker(d\pi_e)$.
\end{remark}
\begin{prop}\label{prop:jettopology}
Let $\pi:E \to M$ be a smooth fiber bundle and let $J^1(E)$ be the first order topological jet space with charts $\mathcal{A}=\{(U,\tilde{\Phi}_U) \vert \mbox{ } E\vert_U \simeq U \times F\}$. Then $\mathcal{A}$ is a $C^{\infty}$ atlas and $J^1(E)$ is a smooth $C^{\infty}$ manifold with corners. Furthermore,
\begin{enumerate}
\item The projection map $\pi_{1,0}:J^1(E) \to E$ gives $J^1(E)$ the structure of a smooth fiber bundle with fiber at $e$ isomorphic to $\hom(T_{\pi(e)}U, V_e)$, where $V_e$ is the fiber of the vertical bundle $V \to E$ of $E$.
\item The projection map $\pi_1:J^1(E) \to M$ gives $J^1(E)$ the structure of a smooth fiber bundle with fiber at $p$ isomorphic to $\hom(T_pU, TF)$.
\item The transition maps $\tilde{\Phi}_V \circ \tilde{\Phi}_U^{-1}: \hom(T(U\cap V), TF) \to \hom(T(U\cap V), TF)$ are \emph{affine} maps of vector bundles, hence $J^1(E)$ is generally not a vector bundle over $E$ or $M$.
\end{enumerate}
\end{prop}
\begin{remark}
Proposition 4.1.7, lemma 4.1.9, and corollary 4.1.10 of \cite{Sa} imply the first two results.
\end{remark}
\begin{remark}
The purpose of explicitly stating part $3$ of proposition \ref{prop:jettopology} is to make it clear that $\pi_1$ and $\pi_{1,0}$ are not necessarily vector bundles, despite their role as a generalization of the tangent and cotangent bundles. However, they are \emph{close} to being vector bundles in the sense that the transition maps are affine maps when restricted to the fibers. This result is encapsulated in proposition 4.6.3 in \cite{Sa}, which states that $J^1(E)$ is an affine bundle modeled on $\pi^*TM\otimes V$ with $V\to E$ being the vertical bundle.
\end{remark}
\begin{proof}[Proof of proposition \ref{prop:jettopology}] \mbox{ }
\begin{enumerate}
\item The charts $(U, \tilde{\Phi}_U)$ identify $J^1(E)\vert_U$ with $\hom(TU,TF)$ and $\pi_{1,0}$ with \newline $pr_U\times pr_F:\hom(TU,TF)\to U\times F$ and the transition maps are smooth maps of \emph{fiber} bundles. That is, for any two charts $(U,\tilde{\Phi}_U)$, $(V,\tilde{\Phi}_V)$ the diagram:
\begin{displaymath}
\xymatrixcolsep{5pc}\xymatrix{
\hom(T(U\cap V),TF) \ar[dr]^{pr_U\times pr_F} \ar[r]^{\tilde{\Phi}_V \circ \tilde{\Phi}_U^{-1}} & \hom(T(U\cap V),TF) \ar[d]^{pr_U \times pr_F} \\
& U \times F
}
\end{displaymath}
commutes. Therefore, $\pi_{1,0} : J^1(E) \to E$ is a smooth fiber bundle. We can identify the typical fiber by identifying the fiber of $pr_U\times pr_F: \hom(TU,TF) \to U\times F$, which is $\hom(T_uU, T_fF) \simeq \hom(T_uU,V_e)$, where $\pi(e)=u$.
\item The charts $(U,\tilde{\Phi}_U)$ identify $\pi_1$ with the projection $pr_U: \hom(TU,TF) \to U$, which is a fiber bundle with typical fiber $\hom(T_uU, TF)$, hence $\pi_1\vert_U:J^1(E)\vert_U \to U$ has the structure of a fiber bundle with fiber $\hom(T_uU,TF)$. The transition maps $\tilde{\Phi}_V \circ \tilde{\Phi}_U^{-1}$ make the diagram:
\begin{displaymath}
\xymatrixcolsep{5pc}\xymatrix{
\hom(T(U\cap V), TF) \ar[dr]^{pr_U} \ar[r]^{\tilde{\Phi}_V \circ \tilde{\Phi}_U^{-1}} & \hom(T(U\cap V),TF) \ar[d]^{pr_U} \\
& U\cap V
}
\end{displaymath}
commute, hence they are isomorphisms of fiber bundles. Thus, $\pi_1:J^1(E) \to M$ is a smooth fiber bundle over $M$.
\item We check that $\tilde{\Phi}_V \circ \tilde{\Phi}_U^{-1}: \hom(T_u(U\cap V), T_fF) \to \hom(T_u(U\cap V), T_{f'}F)$ is affine (linear up to the addition of a constant term), where $f'=pr_2(\Phi_V \circ \Phi_U^{-1})(u,f)$. Let $A\in \hom(T_u(U\cap V),T_fF)$. Then $\tilde{\Phi}_V \circ \tilde{\Phi}_U^{-1}(A)$ is the composition:
\begin{displaymath}
A \to id_{T_uU} \oplus A \to dpr_2(d(\Phi_V \circ \Phi_U^{-1})(id_{T_uU} \oplus A))
\end{displaymath}
where the last arrow is a linear map and the first arrow is an affine map. Therefore, the composition is an affine map.
\end{enumerate}
\end{proof}
\noindent Maps between jet bundles are simply maps of fiber bundles. However, there is a distinguished class of maps that are induced by maps of fiber bundles. They are studied in section 4.2 of \cite{Sa}.
\begin{definition}\label{def:jetmap}
Let $\pi:E \to M$, $\pi':E' \to M$ be two fiber bundles over a manifold with corners $M$. Let $\gamma:E \to E'$ be a map of fiber bundles. Then the induced map $\tilde{\gamma}:J^1(E) \to J^1(E')$ is defined as:
\begin{displaymath}
\tilde{\gamma}(j^1_p\phi) = j^1_p(\gamma(\phi))
\end{displaymath}
where $j^1_p\phi$ is the $1$-equivalence class of sections at $p$ with representative $\phi$.
\end{definition}
\begin{lemma}\label{lem:jetmap}
Let $\pi:E \to M$, $\pi':E'\to M$ be two fiber bundles over a manifold with corners $M$. Let $\gamma:E\to E'$ be a map of fiber bundles and let $\tilde{\gamma}:J^1(E)\to J^1(E')$ be the induced map. Then,
\begin{enumerate}
\item $\tilde{\gamma}$ is a smooth map of fiber bundles over $M$.
\item There is a commutative diagram:
\begin{displaymath}
\xymatrix{
J^1(E) \ar[r]^{\tilde{\gamma}} \ar[d]^{\pi_{1,0}} & J^1(E') \ar[d]^{\pi'_{1,0}} \\
E \ar[r]^{\gamma} & E '
}
\end{displaymath}
\item $\tilde{\gamma}$ is a diffeomorphism if and only if $\gamma$ is a diffeomorphism.
\end{enumerate}
\end{lemma}
\begin{proof} \mbox{} \newline
\begin{enumerate}
\item We use a chart $(U,\tilde{\Phi}_U)$ on $J^1(E)\vert_U\simeq \hom(TU,TF)$ and a chart $(U,\tilde{\Phi}'_U)$ on $J^1(E')\vert_U \simeq \hom(TU,TF') $ to see that the map $\tilde{\gamma}$ in coordinates is:
\begin{displaymath}
\tilde{\Phi}'_U \circ \tilde{\gamma} \circ \tilde{\Phi}_U^{-1} (p,f,A) = (p,pr_2(\Phi'_U(\gamma(\Phi_U^{-1}(p,f)))), dpr_2(d(\Phi_U' \circ \gamma \circ \Phi_U^{-1})(id_{T_pU} \oplus A)))
\end{displaymath}
which is smooth since $\Phi_U'$, $\Phi_U^{-1}$, $\gamma$, and $pr_2:U\times F' \to F'$ are smooth. It is a map of fiber bundles because:
\begin{displaymath}
\pi'(\tilde{\gamma}(j^1_p\phi))= \pi'(j^1_p(\gamma(\phi))) = p = \pi(j^1_p\phi)
\end{displaymath}
hence $\pi'\circ \tilde{\gamma} = \pi$.
\item We need to verify that $\pi'_{1,0}\circ \tilde{\gamma} = \gamma\circ \pi_{1,0}$. We have that for any $s\in J^1(E)$, $s=j^1_p\phi$ and:
\begin{displaymath}
\pi'_{1,0}(\tilde{\gamma}(j^1_p\phi)) = \pi'_{1,0}(j^1_p(\gamma(\phi))) = \gamma(\phi(p)) = \gamma(\pi_{1,0}(j^1_p\phi))
\end{displaymath}
which proves that $\pi'_{1,0}(\tilde{\gamma}(s))=\gamma(\pi_{1,0}(s)).$
\item If $\gamma$ is a diffeomorphism, then it has a smooth inverse $\gamma^{-1}$, which is also a map of fiber bundles. Hence $\gamma^{-1}$ induces a map of jet bundles $\tilde{\gamma^{-1}}:J^1(E') \to J^(E)$ and it is straightforward to see that $\tilde{\gamma}^{-1} = \tilde{\gamma^{-1}}$. On the other hand, if $\tilde{\gamma}$ is a diffeomorphism then we can construct the inverse, $\gamma^{-1}$, of $\gamma$ as follows:
\begin{enumerate}
\item For each $e' \in E'$, let $(U_{e'},\phi_{e'})$ be a local section near $p=\pi'(e')$ so that $\phi_{e'}(u)=e'$ for all $u\in U_{e'}$. That is, we choose a local section that is constant. Similarly, for $e\in E$ we can define a local section $\phi_e$ near $\pi(e)$ satisfying $\phi(u)=e$ for all $u$ in a neighborhood of $p=\pi(e)$. Then we may define:
\begin{displaymath}
\gamma^{-1}(e') = \pi_{1,0}(\tilde{\gamma}^{-1}(j^1_p\phi_{e'})
\end{displaymath}
and, using the definition of $\phi_e$, we have
\begin{displaymath}
\begin{array}{l c l}
\gamma^{-1}(\gamma(e)) &= & \pi_{1,0}(\tilde{\gamma}^{-1}(j^1_p\phi_{\gamma(e)})) \\
&= & \pi_{1,0}(\tilde{\gamma}^{-1}(\tilde{\gamma}(j^1_p\phi_e))) \\
& = & \pi_{1,0}(j^1_p\phi_e) \\
& = & e
\end{array}
\end{displaymath}
Where the second line follows since $\phi_{\gamma(e)}$ and $\gamma(\phi_e)$ are both local, constant sections near $p$ with value $\gamma(e)$ and $\tilde{\gamma}$ is one-to-one, hence $j^1_p\phi$ is the unique $1$-jet mapping to $j^1_p\phi_{\gamma(e)}$. From part $2$, we also have that $\gamma \circ \pi_{1,0} \circ \tilde{\gamma}^{-1} = \pi_{1,0}'$, hence:
\begin{displaymath}
\begin{array}{lcl}
\gamma(\gamma^{-1}(e')) & = & \gamma(\pi_{1,0}(\tilde{\gamma}^{-1}(j^1_p\phi_{e'}))) \\
& = & \pi_{1,0}'(j^1_p\phi_{e'}) \\
& = & e'
\end{array}
\end{displaymath}
Therefore, $\gamma^{-1}$ is a set-theoretic inverse of $\gamma$.
\item To see that $\gamma^{-1}$ is smooth, use charts $(U,\tilde{\Phi}_U)$, $(U,\tilde{\Phi}_U')$ on $J^1(E)$ and $J^1(E')$ respectively. We have the diagram:
\begin{displaymath}
\xymatrixcolsep{5pc}\xymatrix{
\hom(TU,TF) \ar[d]_{pr_U\times pr_F} \ar[r]^{\tilde{\Phi}_U'\circ \tilde{\gamma}\circ\tilde{\Phi}_U^{-1}} & \hom(TU,TF')\ar[d]^{pr_U\times pr_F} \\
U \times F \ar[r]^{\Phi_U' \circ \gamma \circ \Phi_U^{-1}} & U \times F' \ar@/^/[u]^{\mathcal{O}}
}
\end{displaymath}
where the map $\mathcal{O}$ is the embedding of $U\times F'$ as the zero section. Then $\gamma^{-1}$ in these coordinates is the composition:
\begin{displaymath}
\Phi_U \circ \gamma^{-1} \circ (\Phi_U')^{-1} = (pr_U \times pr_F) \circ (\tilde{\Phi}_U \circ \tilde{\gamma}^{-1} \circ (\tilde{\Phi}_U')^{-1}) \circ \mathcal{O}
\end{displaymath}
This is true because we may define the map $e \to j^1_{\pi(e)}\phi_e$, where $\phi_e$ is a locally constant section with value $e$ at $\pi(e)$, on $E\vert_U\simeq U\times F$. In coordinates near $\pi(e)$, this map is exactly the zero section $\mathcal{O}$: $\phi_e$ is constant, hence all of its derivatives near $e$ vanish.
\end{enumerate}
\end{enumerate}
\end{proof}
\begin{remark}
At the end of the proof of lemma \ref{lem:jetmap}, we defined a map $e \to j^1_{\pi(e)}\phi_e$ which sent $e$ to the equivalence class of a locally constant section defined near $\pi(e)$ with value $e$. The reader may wonder why this map is only locally defined. The concise answer is that the transition maps of $E$ may not send locally constant sections to locally constant sections. To be more specific, let us suppose that it extends to a well defined section $\chi:E \to J^1(E)$. Using a chart $(U,\tilde{\Phi}_U)$, one may readily see that
\begin{displaymath}
\tilde{\Phi}_U\circ \chi \circ \Phi_U^{-1}:U\times F \to \hom(TU,TF)
\end{displaymath}
\emph{must} be the zero section $\mathcal{O}$ since $\chi(e)$ is the equivalence class of a locally constant section whose derivatives vanish. However, if we choose another chart $(V,\tilde{\Phi}_V)$ and consider:
\begin{displaymath}
\tau=(\tilde{\Phi}_V\circ \tilde{\Phi}_U^{-1}) \circ \mathcal{O} \circ (\Phi_V \circ \Phi_U^{-1})
\end{displaymath}
then there is no guarantee that this is the zero section of $\hom(T(U\cap V),TF)$ since the maps $\tilde{\Phi}_V\circ \tilde{\Phi}_U^{-1}$ are affine maps when restricted to the fibers (q.v. proposition \ref{prop:jettopology}, statement 3). In diagrammatic form:
\begin{displaymath}
\xymatrixcolsep{5pc}\xymatrix{
\hom(T(U\cap V),TF) \ar[d]^{pr_U \times pr_F} & \ar[l]_{\tilde{\Phi}_V^\circ \tilde{\Phi}_U^{-1}}^{\text{affine}} \hom(T(U\cap V), TF) \ar[d]^{pr_U \times pr_F} \\
(U\cap V) \times F \ar@/^/[u]^{\tau} & \ar[l]^{\Phi_V\circ \Phi_U^{-1}} (U\cap V)\times F \ar@/^/[u]^{\mathcal{O}}
}
\end{displaymath}
where $\tau$ is the pushforward of the zero section $\mathcal{O}$ using the top and bottom arrows. Since the top arrow is an affine transformation, $\tau$ may not be the zero section.
\end{remark}
\begin{example}\label{ex:products}
Trivial fiber bundles are the focus of \cite{GG}, hence all of the facts (in the case of manifolds) we are about to present may be found there. Let $M$ and $N$ be smooth manifolds with corners and let $M\times N \to M$ be the trivial fiber bundle over $M$ with fiber $N$. We have essentially \emph{defined} $J^1(M\times N)$ to be $\hom(TM,TN)\to M\times N$ using the canonical map $\Phi:J^1(M\times N) \to \hom(TM,TN)$ given by:
\begin{displaymath}
\Phi(j_p^1\phi) = (p,\phi(p), dpr_2(d\phi_p))
\end{displaymath}
where $pr_2:M\times N \to N$ is the projection. By lemma \ref{lem:jettopology}, this map is a bijection and therefore induces a smooth structure on $J^1(E)$ by pulling back the smooth structure on $\hom(TM,TN)$, which identifies $J^1(E)$ with $\hom(TM,TN)$.
\vspace{3mm}
Two interesting cases of jets of trivial bundles arise when the base space is $\R$ or the fiber is $\R$.
\begin{enumerate}
\item If $M=\R$, then the fiber bundle is $\R \times N \to \R$. Let $pr_2:\R \times N \to N$ be the projection onto $N$. Then any section $\phi(t)=(t,\gamma(t))$ gives rise to a curve $\gamma(t) = pr_2(\phi(t))$ and any curve gives rise to a section. The first jet bundle $J^1(\R \times N)$ keeps track of the derivatives of these curves, hence it is no surprise that we obtain:
\begin{displaymath}
J^1(\R \times N) = \hom(T\R, TN) \simeq TN \times \R
\end{displaymath}
The extra factor of $\R$ arises because we are allowed to differentiate our curves $\gamma(t)$ at any $t\in \R$, hence the arbitrary convention of differentiating at $0$ and taking $\gamma'(0)$ as a tangent vector is removed.
\item If the fiber $N$ satisfies $N=\R$, then the bundle is $M\times \R$ and we have:
\begin{displaymath}
J^1(M\times \R) = \hom(TM,T\R) \simeq \hom(TM,\R)\times \R = T^*M \times \R
\end{displaymath}
The extra factor of $\R$ arises due to the fact that we must keep track of the image of the functions $f:M\to \R$ that we differentiate at $p \in M$ to obtain covectors $df_p\in T_pM$. That is, the convention of forming the cotangent space $T_p^*M$ by differentiating functions $f:M\to \R$ at $p$ with value $f(p)=0$ is removed.
\end{enumerate}
\end{example}
\begin{example}\label{ex:vectorbundles}
This example is proposition 4.1.12 in \cite{Sa}. Let $\pi:E \to M$ be a vector bundle. Then there exists a canonical vector bundle structure on the bundle $\pi_1:J^1(E) \to M$. We define:
\begin{itemize}
\item $j^1_p\phi + j^1_p\psi := j^1_p(\phi + \psi)$ and
\item $cj^1_p\phi := j^1_p(c\phi)$.
\end{itemize}
These operations are smooth since we are simply differentiating the addition and scalar multiplication operations on $E$. We will see that vector bundle structures on $\pi_{1,0}:J^1(E) \to M$ are defined choices of connection on $E$, hence the vector bundle structures on $\pi_{1,0}$ are not canonically defined unless there is a canonical choice of connection.
\end{example}
\begin{example}\label{ex:mobius}
This is example 4.1.18 in \cite{Sa}, but we add a few more details. Let $\R\times[0,1]/(t,0)\sim(-t,1)$ be the M\"{o}bius bundle over $S^1$. It is a vector bundle, hence example \ref{ex:vectorbundles} reveals that $\pi_1:J^1(E) \to S^1$ has the structure of a vector bundle. We claim that $J^1(E)\simeq E \oplus E$ as vector bundles over $S^1$. We prove this fact by showing the two bundles have the same transition maps over the same cover of $S^1$.
\vspace{3mm}
For simplicity's sake, embed $S^1$ in $\R^2$ as the unit circle and cover it with the open sets $U_0=S^1\setminus \{(1,0)\}$ and $U_1=S^1\setminus \{(-1,0)\}$. Let $U_{10}=U_1\cap U_0$ be the intersection. The transition map for $E$ is:
\begin{displaymath}
\xymatrixrowsep{.5pc}\xymatrix{
\Phi_{10}:U_{10}\times \R \ar[r] & U_{10}\times \R \\
(p,t) \ar[r] & (p,-t)
}
\end{displaymath}
meaning the induced transition map on $J^1(E)$ is given by:
\begin{displaymath}
\xymatrixrowsep{.5pc}\xymatrix{
\tilde{\Phi}_{10}:\hom(TU_{10},T\R) \ar[r] & \hom(TU_{10},T\R) \\
\tilde{\Phi}_{10}(p,t,A) \ar[r] & (p,-t,-A)
}
\end{displaymath}
Since $\hom(TU_{10},T\R)$ is a trivial $2$-plane bundle over $U_{10}$, we see that $J^1(E)$ and $E\oplus E$ have the same defining cover and transition map, meaning they are isomorphic. Here, we use the vector field $\frac{\partial}{\partial \theta}$ on $S^1$ to trivialize $TS^1$.
\end{example}
\begin{example}\label{ex:principal}
Assume $G$ is a compact Lie group and $\pi:P\to M$ is a principal $G$-bundle. Then $J^1(P)$ is equipped with an action of $G$ defined by:
\begin{displaymath}
g\cdot (j^1_p\phi) := j^1_p(g\cdot \phi)
\end{displaymath}
where $g\in G$. This action is free:
\begin{displaymath}
g\cdot j_p^1\phi = j_p^1\phi \mbox{ } \to \mbox{ } g\cdot\phi(p)=\phi(p) \mbox{ } \to \mbox{ } g=e
\end{displaymath}
since the action of $G$ on $P$ is free. The action is proper since $G$ is compact, hence $J^1(P)/G$ is a manifold with corners. The map $\pi_1:J^1(P) \to M$ is $G$-invariant and descends to a smooth map on the orbit space:
\begin{displaymath}
\bar{\pi}_1: J^1(P)/G \to M
\end{displaymath}
We will see that sections of $\bar{\pi}_1$ correspond to principal connections on $P$.
\end{example}
\subsubsection{Sections of $J^1(E)$ and Prolongations}
\noindent Given a local section $(U,\phi)$ of a fiber bundle $\pi:E \to M$, we would like to describe its prolongation $(U,j^1\phi)$ to a local section of $J^1(E)$ as in \cite{Sa}. The prolongation will give us information about the derivative of the section $\phi$, hence first order singularities of $\phi$ will be encoded in the $1$-jet prolongation $j^1\phi$.
\begin{remark}
Here, a singularity is not a point where $\phi$ ceases to be smooth: a singular point will be a point where the differential of $\phi$ drops rank or corank in some way, which we cover in section 2.1.4.
\end{remark}
\begin{definition}\label{def:prolongation}
Let $(U,\phi)$ be a local section of a fiber bundle $\pi:E \to M$. We define its \emph{prolongation} or $1$-\emph{jet}, $(U,j^1\phi)$, to a local section of $\pi_1:J^1(E)\to M$ to be the map:
\begin{displaymath}
(j^1\phi)(p) = j^1_p\phi
\end{displaymath}
where $j^1_p\phi$ is the equivalence class of sections at $p$ whose value is $\phi(p)$ and derivative is $d\phi_p$.
\end{definition}
\begin{lemma}\label{lem:prolongation}
Let $(U,\phi)$ be a local section of $\pi:E\to M$. Then $(U,j^1\phi)$ is a smooth local section of $\pi_1:J^1(E)\to M$.
\end{lemma}
\begin{proof}\mbox{} \newline
Using a trivializiation $(V,\Phi_V)$ of $E$, we get a chart $(V,\tilde{\Phi}_V)$ and obtain the diagram:
\begin{displaymath}
\xymatrixcolsep{5pc}\xymatrix{
J^1(E)\vert_{U\cap V} \ar[d]^{\pi_{1,0}} \ar[r]^{\tilde{\Phi}_V} & \hom(T(U\cap V),TF) \ar[d]_{pr_U\times pr_F} \\
E \ar[d]^\pi \ar[r]^{\Phi_V} & (U\cap V)\times F \ar[d]^{pr_1} \\
U\cap V \ar@<-.5ex>[r]^{id} \ar@/^/[u]^{\phi} \ar@/^2pc/[uu]^{j^1\phi} & U \cap V \ar@/^/[u]^{\Phi_V(\phi)} \ar@/_3pc/[uu]_{d(\Phi_V(\phi))}
}
\end{displaymath}
and we see that $j^1\phi$ is identified with the local section of $\hom(T(U\cap V),TF)$ given by $d(\Phi_V\circ \phi)$. Since $\phi$ and $\Phi_V$ are smooth, $j^1\phi$ is smooth.
\end{proof}
\noindent We may also define smooth sections of $\pi_{1,0}:J^1(E) \to E$.
\begin{definition}\label{def:jetfields}
Let $\pi:E \to M$ be a fiber bundle over a manifold with corners and let $J^1(E)$ be its first jet bundle with projection map $\pi_{1,0}:J^1(E) \to E$. A \emph{jet field} $\chi$ is a (global) section $\chi:E \to J^1(E)$ of $\pi_{1,0}$.
\end{definition}
\subsubsection{Connections, Jet Fields, and Vector Bundle Structures on $J^1(E)$}
Let $\pi:E\to M$ be a fiber bundle over a manifold with corners and let $J^1(E)$ be its first jet bundle. We would like to study the relationships between Ehresmann connections on $E$, jet fields, and vector bundle structures on $J^1(E)$.
\begin{remark}\label{rem:connections}
Recall, if $\pi:E \to M$ is a fiber bundle over a manifold with corners $M$, then a (Ehresmann) connection $\chi$ is defined to be a smooth splitting of the tangent bundle $TE$ into $TE \simeq H \oplus V$, where $V=\ker(d\pi)$ is the \emph{vertical bundle} of $E$ and $H$ is a choice of \emph{horizontal bundle} transverse to $V$. This choice gives us two projections:
\begin{itemize}
\item $p_H:TE \to H$ and
\item $p_V:TE \to V$.
\end{itemize}
where $p_H\big\vert_H = id_H$, $p_V\big\vert_V=id_V$, $p_H\big\vert_V=0$, and $p_V\big\vert_H=0$. Conversely, given a projection operator $p_V:TE \to V$ satisfying $p_V\big\vert_V=id_V$, we may define $H=\ker(p_V)$ and $p_H:= id_{TE}-p_V$. Similarly, if we are given an operator $p_H:TE \to TE$ satisfying $p_H^2=p_H$, $\ker(p_H)=V$, and $p_H\big\vert_{p_H(TE)} = id$, then we may define $H:= p_H(TE)$ as the horizontal subbundle and we obtain a vertical projection $p_V:= id_{TE}-p_V$. Thus, a choice of an Ehresmann connection is equivalent to a choice of projection operator $p_V$ onto the vertical bundle or a choice of a projection operator $p_H$ onto a horizontal bundle.
\end{remark}
\begin{remark}
The following proposition is proposition 4.6.3 in \cite{Sa}. Our proof is modeled closely on the one found in \cite{Sa}, but we avoid coordinates.
\end{remark}
\begin{prop}\label{prop:C1}
Let $\pi:E \to M$ be a fiber bundle over a manifold with corners and let $J^1(E)$ be its first jet bundle. Let $\Gamma(J^1(E))$ be the space of jet fields, which are sections of $\pi_{1,0}:J^1(E) \to E$. Let $\operatorname{Con}(E)$ denote the space of Ehresmann connections on $E$, which we will view as the space of projection operators $p_H:TE \to TE$ satisfying:
\begin{enumerate}
\item $p_H^2 = p_H$,
\item $\ker(p_H)=V$, and
\item $p_H\big\vert_{p_H(TE)} = id$.
\end{enumerate}
Then there is a bijection:
\begin{displaymath}
\Sigma:\Gamma(J^1(E)) \to \operatorname{Con(E)}
\end{displaymath}
hence a jet field on $E$ uniquely specifies a connection on $E$.
\end{prop}
\begin{proof} \mbox{ } \newline
Using the notation of proposition \ref{prop:C1}, we let $\Gamma(J^1(E))$ be the space of jet fields (sections of $\pi_{1,0}:J^1(E) \to E$) and let $\operatorname{Con}(E)$ denote the space of Ehresmann connections on $E$. We will construct the map
\begin{displaymath}
\Sigma: \Gamma(J^1(E) \to \operatorname{Con}(E)
\end{displaymath}
show it is injective, and then argue that injectivity implies surjectivity using a gluing argument.
\begin{enumerate}
\item The map $\Sigma$ is relatively easy to define. Let $\chi$ be a jet field on $E$. Then for each $e\in E$, $\chi(e)=j^1_p\phi$ for some local section $\phi$ with $\phi(p)=e$. This defines a map $(p_H)_e:T_eE \to TE$ given by $(p_H)_e:=d\phi_p \circ d\pi_e$ with $(p_H)_e^2=(p_H)_e$ and $\ker(p_H)_e=V_e$, hence $H_e=\operatorname{Image}(p_H)_e$ is a horizontal subspace of $TE$. Now, $(p_H)_e$ is well-defined since it only depends on the derivative $d\phi_p$, which is the same for all elements of $j^1_p\phi$. Performing the construction for each $e\in E$ gives us a projection operator $p_H:TE \to TE$. It is smooth since $\chi$ is smooth (the derivatives $d\phi_p$ vary smoothly) and its image is a horizontal subbundle $H$ of $TE$. Thus, $p_H$ defines an Ehresmann connection on $E$ which we'll denote $\Sigma(\chi)$.
\item Suppose $\Sigma(\chi_1)=\Sigma(\chi_2)$. Then the horizontal subbundles $H_1,H_2$ agree, i.e. $H_1=H_2$, and the projection operators $p_H^i:TE \to H$, $i=1,2$ agree. In order to show that $\chi_1(e)=\chi_2(e)$, we need only check that the derivatives of the representative sections agree. For each point $e\in E$ we have $d\phi^1_p \circ d\pi_e = d\phi^2_p\circ d\pi_e$, where $\phi^i_p$ is a representative of $\chi_i(e)$. Since $\pi$ is a submersion, this implies that $d\phi^1_p=d\phi^2_p$, hence $\chi_1(e)=\chi_2(e)$.
\item Now, let $p_V:TE \to V$ be a connection on $E$ with $p_H:TE \to H$ given by $p_H=id_{TE} - p_V$. We will construct a jet field $\chi$ on $E$ so that $\Sigma(\chi)$ is $p_H:TE \to H$. For each $p\in M$, let $U_p$ be a neighborhood so that $E\vert_{U_p}$ is trivializable, hence we may assume $E=U\times F$ with projection $p_V:TU\times TF \to pr_2^*TF$, where $pr_2:U\times F \to F$ is the projection onto the second factor. We will construct the requisite jet field on each $U_p$ and show that these jet fields agree on overlaps $U_{p_1}\cap U_{p_2}$, $p_1,p_2\in M$, hence they glue together to give a global jet field. Let $pr_1:U\times F \to U$ be the projection on the first factor. Since $dpr_1(p_V)=0$, we have:
\begin{displaymath}
dpr_1(p_H)=dpr_1(id_{TU\times TF} - p_V) = dpr_1
\end{displaymath}
That is, we have:
\begin{displaymath}
p_H = dpr_1(p_H) \oplus dpr_2(p_H) = dpr_1 \oplus dpr_2(p_H)
\end{displaymath}
The term $dpr_2(p_H)$ is a $TF$-valued $1$-form on $U\times F$ that vanishes on vertical vectors, meaning it vanishes on $pr_2^*TF$. We have an injective map $\circ dpr_1$:
\begin{displaymath}
\xymatrixcolsep{4pc}\xymatrix{
\hom(TU,TF) \ar[r]^{\circ dpr_1} \ar[d] & \hom(TU\times TF, TF) \ar[d] \\
U\times F \ar[r]^{id} & U \times F
}
\end{displaymath}
given by pre-composition with $dpr_1$. The image consists of all maps for which the kernel is the vertical bundle $pr_2^*TF \to U\times F$. Because $dpr_2(p_H)$ vanishes on vertical vectors, it is in the image of $\circ dpr_1$ and we have that $dpr_2(p_H)=\beta\circ dpr_1$ where $\beta$ is a unique (by injectivity of $\circ dpr_1$) smooth section of $\hom(TU,TF) \to U\times F$. Note that $\hom(TU,TF)\simeq J^1(U\times F)$ so that $\beta$ corresponds to a jet field on $U\times F$. At a point $(u,f)\in U\times F$, the jet corresponding to $\beta_{(u,f)}$ is the equivalence class:
\begin{displaymath}
j^1_u\phi = [(\phi(u)=(u,f),d\phi_u=(id_{T_uU} \oplus \beta))]
\end{displaymath}
Then the projection operator we obtain at $(u,f)$, using the recipe of part $1$ of the proof, is:
\begin{displaymath}
d\phi_u\circ (dpr_1)_{(u,f)} = id_{T_uU}\circ dpr_1 \oplus \beta \circ dpr_1 = dpr_1 \oplus dpr_2(p_H) = p_H
\end{displaymath}
which means $\beta$ gives us the correct connection on $U\times F$.
\vspace{5mm}
Now, by part $2$ of the proof, $\beta$ is the unique jet field on $E\vert_U$ corresponding to the connection $p_V:T(E\vert_U) \to V$. Thus, we may cover $E$ by subsets of the form $E\vert_U$, where $E\vert_U$ is trivializable, and construct a local jet field $\beta_U$ on each such set corresponding to $p_V$ restricted to $E\vert_U$. On overlaps $E\vert_{U_1} \cap E\vert_{U_2}=E\vert_{U_1\cap U_2}$, $\beta_{U_1}$ and $\beta_{U_2}$ both correspond to the unique jet field that map to the connection $p_V$, hence they agree. Thus, the $\beta_{U_i}'s$ define a global jet field $\beta$ so that $\Sigma(\beta)=\chi$. We therefore have that $\Sigma$ is a surjection.
\item $\Sigma$ is injective by part $2$ and surjective by part $3$, so we have proven that $\Sigma$ is a bijection.
\end{enumerate}
\end{proof}
\begin{prop}\label{prop:C2}
Let $\pi:E \to M$ be a fiber bundle over a manifold with corners, let $\tau_E:TE\to E$ be the projection, let $\tau_E:V\to E$ be the vertical bundle of $E$, and let $J^1(E)$ be the first jet bundle of $E$. Then,
\begin{enumerate}
\item A choice of connection $\chi$ on $E$ specifies an isomorphism of fiber bundles, $F_{\chi}$, over $E$ given by:
\begin{displaymath}
\xymatrixcolsep{4pc}\xymatrixrowsep{1pc}\xymatrix{
F_{\chi}: J^1(E) \ar[r] & \hom(\pi^*TM, V) \\
j^1_p\phi \ar[r] & p_V(d\phi_p)
}
\end{displaymath}
where $p_V:TE \to V$ is the vertical projection afforded by $\chi$.
\item The isomorphism $F_{\chi}$ induces a vector bundle structure on $J^1(E)$ so that $F_{\chi}$ is an isomorphism of vector bundles. Furthermore, all such structures are isomorphic.
\item If $U_1,U_2$ are two neighborhoods with trivializations $\Phi_i:E\vert_{U_i} \to U_i\times F$ and induced charts \newline $\tilde{\Phi}_{U_i}:J^1(E)\vert_{U_i} \to \hom(TU_i, TF)$, then the transition maps
\begin{displaymath}
\tilde{\Phi}_{12}=\tilde{\Phi}_{U_2}\circ \tilde{\Phi}_{U_1}^{-1}
\end{displaymath}
are linear with respect to the vector bundle structure induced by $F_{\chi}$.
\end{enumerate}
\end{prop}
\begin{remark}
The third claim about linearity of the transition maps is proven in \cite{Sa} by first showing $J^1(E)$ is an affine bundle modeled on $\pi^*TM\otimes V$ (theorem 4.1.11) and then proving lemma 2.4.8, which states that a section of an affine bundle modeled on a vector bundle induces a vector bundle structure on the affine bundle. Since connections \emph{are} sections of $J^1(E)$, this is one way to see that connections induce vector bundle structures on $J^1(E)$. In proposition \ref{prop:C2}, the vector bundle structure is made explicit when we construct the isomorphism with $\hom(\pi^*TM,V)$.
\end{remark}
\begin{proof} \mbox{ } \newline
\begin{enumerate}
\item Let $\chi$ be a connection on $\pi:E\to M$ and let $p_V:TE \to V$ be the corresponding projection onto the vertical bundle. We define the map $F_{\chi}$ to be:
\begin{displaymath}
F_{\chi}(j^1_p\phi) = p_V(d\phi_p)
\end{displaymath}
which is well defined since it only depends on $\phi(p)$ and $d\phi_p$. It is a map of fiber bundles since $\pi_{1,0}(j^1_p\phi)=\phi(p) = pr_E(p_V(d\phi_p))$, where $pr_E:\hom(\pi^*E, V) \to E$ is the projection. We can see smoothness as follows. A choice of trivialization $\Phi_U:E\vert_U \to U\times F$ gives us an identification $J^1(E)\simeq \hom(TU,TF)$, $\pi^*TM \simeq pr_1^*TU$, $V=pr_2^*TF$, and $\hom(\pi^*TM,V) = \hom(TU,TF)$, where $pr_1:U\times F \to U$, $pr_2:U\times F \to F$ are projection onto the first and second factors, respectively. The map $F_{\chi}$ is then the composition:
\begin{displaymath}
\xymatrixrowsep{.5pc}\xymatrixcolsep{5pc}\xymatrix{
\hom(TU,TF) \ar[r] & \hom(TU,TU\times TF) \ar[r] & \hom(TU,TF) \\
(u,f,A) \ar[r] & (u,f,id_{T_uU} \oplus A) \ar[r] & (u,f, p_V(id_{T_uU} \oplus A))
}
\end{displaymath}
where $(u,f) \in U\times F$. Each operation is smooth, hence $F_{\chi}$ is locally a composition of smooth operations. Furthermore, we can see that $F_{\chi}$ is a fibrewise isomorphism using this local picture. The map is injective since, for any $A_1,A_2 \in \hom(T_uU,T_fF)$:
\begin{displaymath}
p_V(id_{T_uU}\oplus A_1) = p_v(id_{T_uU}\oplus A_2) \iff p_V(0\oplus (A_1-A_2)) = 0 \iff A_1-A_2 = 0 \iff A_1=A_2
\end{displaymath}
since $p_V\vert_V = id_V$ and $(0\oplus A_1-A_2)$ maps into the vertical bundle of $U\times F$. It is an affine map of vector bundles since $A \to id_{T_uU} \oplus A$ is affine and projection onto $V$ is linear. Since it is an injective, affine map of vector of the same rank, it is an isomorphism of $\hom(TU,TF)$ with itself.
This shows that $F_{\chi}$ is locally an isomorphism of fiber bundles. Since it is globally a map of fiber bundles, it is therefore a global isomorphism of fiber bundles.
\item It is a general fact that if $\phi:E_1 \to E_2$ is an isomorphism of fiber bundles and $E_2$ is a vector bundle, then one can use $\phi$ to pull back the vector bundle structure on $E_2$ to $E_1$. In our case, it will be useful to give explicit formulas for addition and scalar multiplication. To do so, we'll need a formula for the inverse of $F_{\chi}$. Let $\tau_E:H\to E$ be the horizontal bundle and consider the pullback diagram:
\begin{displaymath}
\xymatrixcolsep{4pc}\xymatrix{
H \ar@/_2pc/[ddr]^{\tau_E} \ar@/^2pc/[drr]^{d\pi} \ar@{-->}[dr]^{\tilde{d\pi}} & & \\
& \pi^*TM \ar[d]^{\pi^*\tau_M} \ar[r]^{\bar{\pi}} & TM \ar[d]^{\tau_M} \\
& E \ar[r]^{\pi} & M
}
\end{displaymath}
The universal property of pullbacks guarantees the existence of a smooth map $\tilde{d\pi}:H \to \pi^*TM$. Since $d\pi:H \to TM$ is a fibrewise isomorphism, we must have that $\tilde{d\pi}$ is an isomorphism of vector bundles over $E$. We can now write:
\begin{displaymath}
F_{\chi}^{-1}(e, A)= [(\pi(e),e, \tilde{d\pi}_e^{-1} + A)]
\end{displaymath}
where $[(\pi(e),e, \tilde{d\pi}_e^{-1} + A)]$ is the equivalence class of local sections at $\pi(e)$ with value $e$ and derivative $\tilde{d\pi}_e^{-1} + A$. Note that all we have done is added the horizontal portion of the derivative of a section, $\tilde{d\pi}^{-1}$, to the vertical portion, $A$, to reconstruct the whole derivative of the local section. To see this explicitly, note that if $p_H:TE \to H$ is the horizontal projection then for any local section $\phi$ of $E$ at $p\in M$ we have:
\begin{displaymath}
\tilde{d\pi}_{\phi(p)} \circ p_H \circ d\phi_p = id_{T_pM} \iff p_H\circ d\phi_p = \tilde{d\pi}_{\phi(p)}^{-1}
\end{displaymath}
hence the horizontal portion $p_H\circ d\phi$ is indeed $\tilde{d\pi}^{-1}$. We may use this fact to write out the vector bundle operations on $J^1(E)$.
\vspace{2mm}
We will use $+_J$ and $(c)_J$ to denote addition and scalar multiplication on the jet bundle $J^1(E)$.
\begin{displaymath}
\begin{array}{lcl}
j^1_p\phi_1 +_J j^1_p\phi_2 & := & F_{\chi}^{-1}(F_{\chi}(j^1_p\phi_1) + F_{\chi}(j^1_p\phi_2)) \\
& = & F_{\chi}^{-1}(p_V((d\phi_1)_p + (d\phi_2)_p)) = [(p,\phi_1(p), \tilde{d\pi}^{-1} + p_V((d\phi_1)_p + (d\phi_2)_p)] \\
(c)_J(j^1_p\phi) & := & F_{\chi}^{-1}(cF_{\chi}(j^1_p\phi)) = [(p,\phi(p), \tilde{d\pi}^{-1} + cp_V(d\phi_p))]
\end{array}
\end{displaymath}
Note that we are defining addition in the fiber of $J^1(E)$ over $e$, hence $\phi_1(p)=\phi_2(p)$ in the definition of $+_J$. Now, it is an easy consequence of the definitions of $F_{\chi}$ and the vector space structure that any two choices of connection, $\chi_1,\chi_2$ give isomorphic vector bundle structures on $J^1(E)$. Note that we have:
\begin{displaymath}
\xymatrixcolsep{5pc}\xymatrix{
J^1(E) \ar[r]^{F_{\chi_1}} & \hom(\pi^*TM, V) & \ar[l]_{F_{\chi_2}} J^1(E)
}
\end{displaymath}
where each map is an isomorphism of vector bundles with respect to the induced vector bundle structures. Hence the composite $F_{\chi_2}^{-1} \circ F_{\chi_1}:J^1(E) \to J^1(E)$ is an isomorphism of vector bundles.
\item We have two trivializations of $E$ which give us a commutative diagram:
\begin{displaymath}
\xymatrixcolsep{4pc}\xymatrix{
\hom(T(U_1 \cap U_2), TF) \ar[r]^-{\tilde{\Phi}_{U_1}^{-1}} \ar[d]& J^1(E) \ar[d]^{\pi_{1,0}} \ar[r]^-{\tilde{\Phi}_{U_2}} & \hom(T(U_1 \cap U_2), TF) \ar[d] \\
(U_1\cap U_2) \times F \ar[r]^{\Phi_{U_1}^{-1}} \ar[dr]^{pr_1} & E \ar[r]^{\Phi_{U_2}} & (U_1 \cap U_2)\times F \ar[dl]^{pr_1} \\
& U_1\cap U_2 &
}
\end{displaymath}
and we wish to check that the composite $\tilde{\Phi}_{12}:= \tilde{\Phi}_{U_2} \circ \tilde{\Phi}_{U_1}^{-1}$ is linear with respect to the induced vector bundle structures on $\hom(T(U_1\cap U_2), TF)$. Let us give formulas for these structures. We have a collection of maps:
\begin{displaymath}
\begin{array}{ll}
pr_1:(U_1\cap U_2)\times F \to (U_1 \cap U_2) & \text{Projection onto the first factor.} \\
pr_2:(U_1\cap U_2) \times F \to F & \text{Projection onto the second factor.} \\
p_V^i:T(U_1\cap U_2) \times TF \to pr_2^*TF & \text{Projection onto the vertical bundle in $U_i$.} \\
id_{TU} \oplus 0: TU \rightarrow TU\times TF & \text{The section of $\hom(TU,TU\times TF)$ given by}\\
& (id_{TU}\oplus 0)_{(u,f)}(v)= (v,0)\in T_uU\times T_fF
\end{array}
\end{displaymath}
We have the relations:
\begin{enumerate}
\item $p_V^i(0\oplus dpr_2)=dpr_2$, which follows since $p_V^i$ restricted to the vertical bundle is the identity.
\item $dpr_2(0 \oplus p_V^i) = dpr_2 + p_V^i(dpr_1 \oplus 0)$, which follows from the first identity since $p_V^i=p_V^i(dpr_1\oplus dpr_2)$.
\item Letting $\Phi_{12}=\Phi_{U_2}\circ \Phi_{U_1}^{-1}$, we have that $d\Phi_{12}(0 \oplus p_V^1) = (0\oplus p_V^2)(d\Phi_{12})$. That is, the transition maps are connection preserving. This is because the connections on $U_i\times F$ are induced by the global connection on $E$.
\item We also have $(dpr_1\oplus 0)\circ d\Phi_{12} = dpr_1 \oplus 0$ since $\Phi_{12}$ is a map of fiber bundles, hence
\begin{displaymath}
(dpr_1\oplus 0)(d\Phi_{12}(id_{T_uU} \oplus A)) = (dpr_1\oplus 0)(id_{T_uU}\oplus A) = id_{T_uU} \oplus 0
\end{displaymath}
\end{enumerate}
We also have nice formulas for addition, $+_J$, and scalar multiplication, $(c)_J$. For $A_1,A_2\in \hom(T_uU,T_fF)$ we compute:
\begin{displaymath}
\begin{array}{lcl}
A_1 +_J A_2 & = & dpr_2(\tilde{d\pi}^{-1} + p_V^i((id_{T_uU} \oplus A_1) + (id_{T_uU} \oplus A_2)))\\
& = & dpr_2(\tilde{d\pi}^{-1} + p_V^i(id_{T_uU} \oplus 0) + p_V^i(id\oplus (A_1 + A_2))) \\
& = & dpr_2(id_{T_uU} \oplus p_V^i(id_{T_uU}\oplus(A_1 + A_2))) \\
& = & p_V^i(id_{T_uU}\oplus(A_1 + A_2)) \\
& & \text{where the last line follows since $dpr_2(p_V)=p_V$ at $(u,f)$.} \\
& \mbox{ } & \\
(c)_J(A) & = & dpr_2(\tilde{d\pi}^{-1} + cp_V^i(id_{T_uU} \oplus A)) \\
& = & dpr_2(id + p_V^i(c(id_{T_uU}\oplus A) - id_{T_uU}\oplus 0))\\
& = & p_V^i(c(id_{T_uU}\oplus A) - id_{T_uU}\oplus 0)
\end{array}
\end{displaymath}
Recall that $\tilde{\Phi}_{12}(A) = dpr_2(d\Phi_{12}(id_{T_uU}\oplus A))$. We now show the transition maps are linear.
\begin{displaymath}
\begin{array}{lcl}
\tilde{\Phi}_{12}(A_1+_J A_2) &=& dpr_2(d\Phi_{12}(id_{T_uU}\oplus(A_1+_J A_2))) \\
&=& dpr_2(d\Phi_{12}(id_{T_uU}\oplus(p_V^i(id_{T_uU}\oplus(A_1+A_2))))) \\
&=& dpr_2(d\Phi_{12}(id_{T_uU}\oplus 0) + dpr_2(d\Phi_{12}(0\oplus p_V^i(id_{T_uU}\oplus(A_1+A_2)))) \\
&=& dpr_2(d\Phi_{12}(id_{T_uU}\oplus 0) + dpr_2((0\oplus p_V^2)d\Phi_{12}(id_{T_uU}\oplus(A_1+A_2))) \\
&=& dpr_2(d\Phi_{12}(id_{T_uU}\oplus 0) + dpr_2(d\Phi_{12}(id_{T_uU}\oplus(A_1+A_2))) +\\
& & p_V^2(dpr_1\oplus 0)d\Phi(id_{T_uU}\oplus A_1 + A_2) \\
&=& dpr_2(d\Phi_{12}(id_{T_uU}\oplus 0) + dpr_2(d\Phi_{12}(id_{T_uU}\oplus(A_1+A_2))) + p_V^2(id_{T_uU}\oplus 0)\\
& & \text{where we have appealed to two of the previously listed identities.} \\
&\mbox{} & \\
\tilde{\Phi}_{12}(A_1)+_J \tilde{\Phi}_{12}(A_2) &=& p_V^2(id_{T_uU}\oplus(\tilde{\Phi}_{12}(A_1)+ \tilde{\Phi}_{12}(A_2))) \\
&=& p_V^2(id_{T_uU}\oplus 0) + p_V^2(0\oplus dpr_2(d\Phi_{12}(id_{T_uU}\oplus A_1 + id_{T_uU}\oplus A_2)))\\
&=& p_V^2(id_{T_uU}\oplus 0) + dpr_2(d\Phi_{12}(id\oplus 0)) + dpr_2(d\Phi_{12}(id\oplus(A_1+A_2)))
\end{array}
\end{displaymath}
Evidently, $\tilde{\Phi}_{12}(A_1+_J A_2)=\tilde{\Phi}_{12}(A_1)+_J \tilde{\Phi}_{12}(A_2)$. For scalar multiplication, we have:
\begin{displaymath}
\begin{array}{lcl}
\tilde{\Phi}_{12}((c)_J A) & = & dpr_2(d\Phi_{12}(id_{T_uU} \oplus p_V^1(c(id_{T_uU}\oplus A) -id_{T_uU}\oplus 0))) \\
& = & dpr_2(d\Phi_{12}(id_{T_uU}\oplus 0))+dpr_2(d\Phi_{12}(0\oplus p_V^1)(c(id_{T_uU}\oplus A) - id_{T_uU}\oplus 0)) \\
& = & dpr_2(d\Phi_{12}(id_{T_uU}\oplus 0)) +dpr_2(0\oplus p_V^2)(d\Phi_{12}(c(id_{T_uU}\oplus A) -id_{T_uU}\oplus 0)) \\
& = & dpr_2(d\Phi_{12}(id_{T_uU}\oplus 0)) +dpr_2(d\Phi_{12}(c(id_{T_uU}\oplus A) -id_{T_uU}\oplus 0)) + \\
& & p_V^2(dpr_1 \oplus 0)(c(id_{T_uU}\oplus A) -id_{T_uU}\oplus 0) \\
& = & cdpr_2(d\Phi_{12}(id \oplus A)) + (c-1)p_V^2(id_{T_uU}\oplus 0) \\
& \mbox {} & \\
(c)_J\tilde{\Phi}_{12}(A) & = & p_V^2(c(id_{T_uU}\oplus \tilde{\Phi}_{12}(A)) - id_{T_uU} \oplus 0) \\
& = & (c-1)p_V^2(id_{T_uU}\oplus 0) + cp_V^2(0 \oplus \tilde{\Phi}_{12}(A)) \\
& = & (c-1)p_V^2(id_{T_uU}\oplus 0) + cp_V^2(0 \oplus dpr_2(d\Phi_{12}(id\oplus A)))\\
& = & (c-1)p_V^2(id_{T_uU}\oplus 0) + cdpr_2(d\Phi_{12}(id\oplus A))
\end{array}
\end{displaymath}
hence $\tilde{\Phi}_{12}((c)_J A)=(c)_J\tilde{\Phi}_{12}(A)$.
\end{enumerate}
\end{proof}
\begin{prop}\label{prop:C3}
Let $\pi:E \to M$ be a fiber bundle over a manifold with corners, $M$, equipped with a connection $\chi$. Let $F_{\chi}:J^1(E) \to \hom(\pi^*TM,V)$ be the induced isomorphism of fiber bundles, which equips $J^1(E)$ with the structure of a vector bundle. Let $\Sigma : \Gamma(\pi_{1,0}:J^1(E) \to E) \to \operatorname{Con}(E)$ be the bijection between jet fields and connections from proposition \ref{prop:C1}. Let $\mathcal{O}\in \Gamma(\hom(\pi^*TM, V))$ be the zero section. Then:
\begin{displaymath}
\Sigma^{-1}(\chi)=F_{\chi}^{-1}(\mathcal{O})
\end{displaymath}
Thus, the zero section of $J^1(E)$ corresponds to the connection $\chi$.
\end{prop}
\begin{proof} \mbox{} \newline
We need only check that $\Sigma(F_{\chi}^{-1}(\mathcal{O}))$ corresponds to the horizontal projection operator $p_H:TE \to H$ induced by $\chi$. We have:
\begin{displaymath}
j^1_p\phi \in F_{\chi}^{-1}(\mathcal{O}) \mbox{ } \iff p_V(d\phi_p)=0 \mbox{ } \iff (id_{TE})_{\phi(p)} = d\phi_p\circ d\pi_{\phi(p)} + (p_V)_{\phi(p)} \end{displaymath}
hence $(p_H)_{\phi_(p)} = (id_{TE})_{\phi_(p)} - (p_V)_{\phi_(p)} = d\phi_p\circ d\pi_{\phi(p)}$ and we see that the horizontal projections associated with $\chi$ and $\Sigma(F_{\chi}^{-1}(\mathcal{O}))$ are the same, hence they are the same connection and, since $\Sigma$ is invertible, we have:
\begin{displaymath}
\Sigma(F_{\chi}^{-1}(\mathcal{O}))=\chi \iff \Sigma^{-1}(\chi)=F_{\chi}^{-1}(\mathcal{O})
\end{displaymath}
To clarify the last statement of the proposition, note that with the vector bundle structure on $J^1(E)$ induced by $F_{\chi}$, the zero section is $F_{\chi}^{-1}(\mathcal{O})$.
\end{proof}
\begin{example}\label{ex:mobius2}
Let $\pi:E \to S^1$ be the M\"{o}bius bundle, which we view as the space $E=\R\times [0,1]/(t,0)\sim(-t,1)$. We have seen that $\pi_1:J^1(E)\to S^1$ is endowed with a canonical vector bundle structure arising from addition and scalar multiplication on $E$. By propositions \ref{prop:C1} and \ref{prop:C2}, we may specify a vector bundle structure on $\pi_{1,0}:J^1(E) \to E$ by specifying a jet field (connection) $\chi:E \to J^1(E)$.
\vspace{3mm}
Let $\chi:E \to J^1(E)$ be the jet field defined by $\chi(e)=j^1_p\phi$, where $\phi(p)=e$ and $\phi$ is constant in a neighborhood of $p$. The transition map for $E$ corresponds to multiplication by $\pm 1$, hence if $\phi$ is locally constant in one trivialization, the transition map sends it to a locally constant section in another trivialization. Therefore, $\chi$ defines a jet field on $E$. Note that the collection of tangent spaces defined by $\chi$, i.e. the connection defined by $\chi$, have integral submanifolds:
\begin{itemize}
\item The zero section, which we view as an embedded copy of $S^1$ in $E$, and
\item the connected double covers of $S^1$ given by the orbits of the lifted action on $S^1$.
\end{itemize}
In the model $\R \times [0,1]/\sim$, we may equivalently describe these submanifolds as:
\begin{itemize}
\item The zero section $\{[(0,s)]\}$ and
\item the strips $\{[(t,s)]\} \cup \{[(-t,s)]\}$, where $t\in \R$ is fixed and nonzero.
\end{itemize}
\end{example}
\begin{example}\label{ex:vectorbundle2}
We have seen that if $\pi:E \to M$ is a vector bundle, then $\pi_1:J^1(E) \to M$ is endowed with a canonical vector bundle structure induced by addition and scalar multiplication on $E$. A linear connection on $E$ is defined to be a choice of horizontal subspace $H_e \subset T_eE$ at each $e\in E$ that is invariant under (the derivatives of) the addition and scalar multiplication operations of $E$. We can encode this in the language of jet fields by defining a linear connection on $E$ to be a jet field that induces a map of vector bundles:
\begin{displaymath}
\xymatrix{
\chi: E \ar[r] \ar[d]^\pi & J^1(E) \ar[d]^{\pi_1} \\
M \ar[r] & M}
\end{displaymath}
\end{example}
\begin{example}\label{ex:principal2}
As in example \ref{ex:principal}, let $G$ be a compact Lie group. Let $\pi:P\to M$ be a principal $G$ bundle and recall from example \ref{ex:principal} that $J^1(P)$ is equipped with a free and proper action of $G$. A principal connection on $P$ is a choice of $G$-invariant splitting $H \oplus V$ of $TP$. We can encode this in the language of jet fields by defining a principal connection to be a $G$-equivariant section $\chi:P \to J^1(P)$.
\vspace{3mm}
To see that this is the correct definition, let $p\in P$ and let $H_p$ be the horizontal distribution at $p$. We may represent $H_p$ as the image of the differential of some section $\phi$ at $\pi(p)$. That is, $H_p = \operatorname{Im}(d\phi_m)$, where $m=\pi(p)$. Let $\tau_g:P \to P$ be the action of $g\in G$. $G$-invariance then implies that:
\begin{displaymath}
H_{g\cdot p} = d\tau_g(\operatorname{Im}(d\phi_m)) = \operatorname{Im}(d(g\cdot \phi)_m) = d\tau_g(H_p).
\end{displaymath}
Thus $\chi(g\cdot p) = j^1_m(g\cdot \phi) = g\cdot (j^1_m\phi)$. Now, $G$-equivariance of $\chi$ implies that $\chi$ descends to a section:
\begin{displaymath}
\bar{\chi}:P/G \to J^1(P)/G
\end{displaymath}
of $\bar{\pi}_1:J^1(P)/G \to M$. Conversely, any section $\bar{\chi}:M \to J^1(P)/G$ lifts to a $G$-equivariant section $\chi:P \to J^1(P)/G$ as follows:
\begin{itemize}
\item $\bar{\chi}(m) = [j^1_m\phi]$ is an equivalence class of jets where $j^1_m\phi$ and $j^1_m\psi$ are in this equivalence class if and only if $j^1_m\phi = j^1_m(g\cdot \psi)$ for some $g\in G$. Consequently, if $\psi(m)=\phi(m)$, then $g=e$ since the action of $G$ is free on $P$ and $d\phi_m = d\psi_m$.
\item The transitivity of the action of $G$ on $P$ implies that for each point $p\in \pi^{-1}(m)$ in the fiber over $m$, there exists an element $j^1_m\psi \in \bar{\chi}(m)$ so that $\psi(m)=p$. By the preceding bullet, this element is unique.
\item Thus, we may define the lift $\chi(p)$ to be the unique element $j^1_m\phi \in \bar{\chi}(\pi(p))$ satisfying $\phi(m)=p$. To see that $\chi$ is a \emph{smooth} lift, it's enough to check the claim locally, hence we may assume $P=M\times G$. We leave this computation to the reader.
\end{itemize}
\end{example}
\subsubsection{1-Jets of Corank r}
In this section, we closely follow the constructions of \cite{GG}, but we adapt everything to arbitrary fiber bundles. Given a local section $(U,\phi)$ of a fiber bundle $\pi:E \to M$, we have seen that $j^1\phi:U \to J^1(E)$, the $1$-jet of $\phi$, is a smooth local section of $\pi_1:J^1(E)\to M$. We would like to garner as much information as possible about $\phi$ from the properties of the $1$-jet, $j^1\phi$, of $\phi$. One way to do this is to study the rank or corank of the $1$-jet, which will give us information about the subset of points where the derivative of $\phi$ fails to be injective or surjective, depending whether we study rank or corank, respectively.
\begin{definition}\label{def:rank}
Let $\pi:E \to M$ be a fiber bundle over a manifold with corners $M$ equipped with a connection $\chi$. That is, we have a projection $p_V:TE \to V$ to the vertical bundle of $E$ induced by the connection $\chi$. We define $\operatorname{rank}:J^1(E) \to \Z$ and $\operatorname{corank}:J^1(E) \to \Z$ to be:
\begin{displaymath}
\begin{array}{l c l}
\operatorname{rank}(j^1_p\phi) &=& \operatorname{rank}(p_V(d\phi_p))\\
\operatorname{corank}(j^1_p\phi) &=& \operatorname{corank}(p_V(d\phi_p))
\end{array}
\end{displaymath}
\end{definition}
It is immediate from the definition that rank and corank are nonnegative. In light of proposition \ref{prop:C2}, we have defined the rank of a $1$-jet $j^1_p\phi$, given $\chi$, to be the rank of the corresponding element $F_{\chi}(j^1_p\phi) \in \hom(\pi^*TM,V)_{\phi(p)}$. We are interested in a particular subset of $J^1(E)$ defined as follows:
\begin{definition}\label{def:Sr}
Let $\pi:E \to M$ be a fiber bundle over a manifold with corners equipped with a connection $\chi$. We define $S_r \subset J^1(E)$ to be the set:
\begin{displaymath}
S_r := \{j^1_p\phi \vert \mbox{ } \operatorname{corank}(j^1_p\phi)=r\}
\end{displaymath}
\end{definition}
Our next goal is to prove the following proposition:
\begin{prop}\label{prop:Sr}
Let $\pi:E \to M$ be a fiber bundle over a manifold with corners equipped with a connection $\chi$. Then $S_r \subset J^1(E)$ is a sub-fiber bundle of $\pi_{1,0}:J^1(E) \to E$ with typical fiber given by $L^r_{m,n}$, where $m=\dim(M)$, $n=\dim(E)-\dim(M)$, and:
\begin{displaymath}
L^r_{m,n}:= \{ A \in \hom(\R^m, \R^n) \vert \mbox{ } \operatorname{corank}(A)=r\}
\end{displaymath}
Furthermore, $S_r$ has codimension $(m-q+r)(n-q+r)$, where $q=\min\{m,n\}$.
\end{prop}
The proof of the proposition is straightforward once we have the following lemma, which is lemma 5.2 and proposition 5.3 of \cite{GG} combined.
\begin{lemma}\label{lem:Lrmn}
Let $L^r_{m,n} = \{A \in \hom(\R^m, \R^n) \vert \mbox{ } \operatorname{corank}(A)=r\}$. Then $L^r_{m,n}$ is a smooth submanifold of $\hom(\R^m,\R^n)$ of codimension $(m-q+r)(n-q+r)$, where $q=\min\{m,n\}$.
\end{lemma}
\begin{proof} \mbox{ } \newline
Let $S$ be an $m\times n$ matrix where $S=\begin{pmatrix}A & B \\ C & D \\ \end{pmatrix}$, where $A$ is a $q-r \times q-r$ invertible matrix. Define
\begin{displaymath}
T=\begin{pmatrix}
I_{q-r} & 0 \\
-CA^{-1} & I_r \\
\end{pmatrix}
\end{displaymath}
and note that $T$ is invertible. Hence $\operatorname{rank}(S)=\operatorname{rank}(TS) = \operatorname{rank}(\begin{pmatrix} A & B \\ 0 & D-CA^{-1}B\\ \end{pmatrix})$. The latter matrix has rank $q-r$ if and only if $D-CA^{-1}B=0$. We can use this fact to see $L^r_{m,n}$ is a submanifold as follows.
Let $S \in L^r_{m,n}$ and choose bases of $\R^n$, $\R^m$ so that $S=\begin{pmatrix}A & B \\ C & D \\ \end{pmatrix}$ where $A$ is a $q-r\times q-r$ invertible matrix. Let $U$ be the open neighborhood of $S$ consisting of all matrices of the form $S'=\begin{pmatrix}A' & B' \\ C' & D' \\ \end{pmatrix}$, with $A'$ a $q-r\times q-r$ invertible matrix.
\vspace{2mm}
Define $f:U\to \hom(\R^{n-q+r}, \R^{m-q+r})$ by $f(S')=D'-C'(A')^{-1}B'$. We can see that $f$ is a submersion as follows. Fix $A'$, $B'$, $C'$, and $D'$, let $D\in T_{f(S')}\hom(\R^{n-q+r}, \R^{m-q+r})$, and define $\gamma(t)= \begin{pmatrix}A' & B' \\ C' & D' + tD \\ \end{pmatrix}$. Then
\begin{displaymath}
\frac{\partial}{\partial t} f(\gamma(t)) = D
\end{displaymath}
which shows that $df_{S'}$ is surjective. Consequently, $f^{-1}(0)$ is a smooth submanifold of $U$. Since $\operatorname{rank}(S')=q-r \iff D'-C'(A')^{-1}B=0$, we have that $f^{-1}(0) = L^r_{m,n}\cap U$, which shows $L^r_{m,n}$ is a smooth submanifold of $\hom(\R^n, \R^m)$. Since $\{0\}$ has codimension $(n-q+r)(m-q+r)$ in $\hom(\R^{n-q+r}, \R^{m-q+r})$, $L^r_{m,n}$ has codimension $(n-q+r)(m-q+r)$.
\end{proof}
\begin{proof}[proof of proposition \ref{prop:Sr}] \mbox{ } \newline
We have a fiber bundle $\pi:E \to M$ and a connection $\chi$. Let $\tau_E:V\to E$ denote the vertical bundle of $E$ with projection $p_V:TE \to V$. By proposition \ref{prop:C2}, $\chi$ defines an isomorphism of fiber bundles:
\begin{displaymath}
\xymatrixrowsep{.5pc}\xymatrix{
F_{\chi}:J^1(E) \ar[r] & \hom(\pi^*TM, V)\\
j^1_p\phi \ar[r] & p_V(d\phi_p)
}
\end{displaymath}
Note that the typical fiber of $\hom(\pi^*TM,V)$ is $\hom(\R^m,\R^n)$, where $m=\dim(M)$ and $n=\dim(E)-\dim(M)$ is the dimension of the fibers of $E$. By lemma \ref{lem:Lrmn}, $L^r_{m,n}$ is a submanifold of $\hom(\R^m,\R^n)$. Since it is invariant under the structure group $GL(\R,n) \times GL(\R,m)$ of $\hom(\pi^*TM, V)$, it defines a smooth sub-fiber bundle $L_r\subset \hom(\pi^*TM, V)$ with typical fiber isomorphic to $L^r_{m,n}$. The fibers of $L_r$ are exactly the elements with corank $r$ and
\begin{displaymath}
\operatorname{corank}(j^1_p\phi) = r \mbox{ } \iff \mbox{ } \operatorname{corank}(p_V(d\phi)p=r \mbox{ } \iff \mbox{ } \operatorname{corank}F_{\chi}(j^1_p\phi) = r
\end{displaymath}
by the definitions of corank and $F_{\chi}$, hence $F_{\chi}\vert_{S_r}:S_r \to L_r$ is a bijection. Therefore,
\begin{displaymath}
S_r = F_{\chi}^{-1}(L_r)
\end{displaymath}
Since $L_r$ is a sub-fiber bundle and $F_{\chi}^{-1}$ is an isomorphism of fiber bundles, $S_r$ is a sub-fiber bundle of $J^1(E)$. Since the codimension of $L_r$ is $(m-q+r)(n-q+r)$, where $q=\min\{m,n\}$, we have that the codimension of $S_r$ is $(m-q+r)(n-q+r)$.
\end{proof}
Since rank and corank depend on the connection, they do not behave well with general maps of fiber bundles. However, these properties are preserved by connection-preserving diffeomorphisms of fiber bundles.
\begin{lemma}\label{lem:rank}
Let $\pi:E\to M$ and $\pi':E\to M$ be two fiber bundles over a manifold with corners $M$ with connections $\chi$, $\chi'$, respectively. That is, we have projection maps $p_V:TE \to V$ and $p_{V'}:TE' \to V'$ to the vertical bundles of $E$ and $E'$, respectively. Let $\tau:E\to E'$ be a diffeomorphism of fiber bundles so that $\tau^*\chi' = \chi$. Then $\operatorname{rank}\circ \tilde{\tau} = \operatorname{rank}:J^1(E) \to \Z$ and $\operatorname{corank}\circ \tilde{\tau}=\operatorname{corank}:J^1(E)\to \Z$. That is, the notion of rank and corank is preserved by connection-preserving diffeomorphisms of fiber bundles.
\end{lemma}
\begin{proof} \mbox{} \newline
$\tau^*\chi' = \chi$ is equivalent to the projection operators satisfying:
\begin{displaymath}
p_V = d\tau^{-1} \circ p_{V'} \circ d\tau \iff d\tau \circ p_V = p_{V'} \circ d\tau
\end{displaymath}
Hence, for any element $j^1_p\phi \in J^1(E)$ we have:
\begin{displaymath}
\begin{array}{lcl}
\operatorname{rank}(\tilde{\tau}(j^1_p\phi)) & = & \operatorname{rank}(j^1_p(\tau \circ \phi)) \\
& = & \operatorname{rank}(p_{V'}(d\tau(d\phi_p))) \\
& = & \operatorname{rank}(d\tau(p_V(d\phi_p))) \\
& = & \operatorname{rank}(p_V(d\phi_p)) \\
& = & \operatorname{rank}(j^1_p\phi)
\end{array}
\end{displaymath}
where the second-to-last line follows since $d\tau\vert_V:V \to V'$ is an isomorphism, hence it doesn't change the rank. Replacing rank by corank and performing the same computations, one sees that:
\begin{displaymath}
\operatorname{corank}(\tilde{\tau}(j^1_p\phi)) = \operatorname{corank}(j^1_p\phi)
\end{displaymath}
\end{proof}
\subsubsection{The Intrinsic Derivative}
We now discuss the intrinsic derivative of maps $f:M \to N$ between manifolds with corners and of maps $\rho:E \to F$ between vector bundles $E$, $F$ over a fixed manifold with corners, $M$. We will need this notion when we study orientations on folding hypersurfaces in folded-symplectic manifolds. The construction presented here is the same as the construction for manifolds (without corners) presented in \cite{GG}, p. 150. The notion of an intrinsic derivative is due to Porteous \cite{P} and an interpretation in coordinates can be found in appendix C of \cite{Ho}.
Before we describe the intrinsic derivative of a map between manifolds, we first consider maps between vector bundles over the same base space. Suppose we are given two vector bundles $E$ and $F$ over a manifold with corners $M$ and suppose $\operatorname{ranke}(E)\ge \operatorname{rank}(F)$. If we have a smooth map $\rho: E\to F$ of vector bundles, then we may view $\rho$ as a smooth section:
\begin{displaymath}
\rho: M \hookrightarrow \hom(E,F)
\end{displaymath}
where we are abusing notation by referring to both the map and the section as $\rho$. From now on, let us think of $\rho$ as a section of $\hom(E,F)\to M$. Fix a point $p\in M$, let $K_p$ be the kernel $\ker(\rho_p)$, and let $L_p$ be the cokernel $\operatorname{coker}(\rho_p) := F_p/\rho_p(E_p)$. Similar to the bundle $S_r$, we define a subfiber bundle of $\hom(E,F)$ whose fiber consists of the elements of corank $r$.
\begin{definition}\label{def:LrEF}
Let $E$ and $F$ be two vector bundles over a manifold with corners $M$. We define the subset of $\hom(E,F)$ of elements of corank $r$:
\begin{displaymath}
L^r(E,F):= \{ A \in \hom(E_p,F_p) \mbox{ } \vert \mbox { } \operatorname{corank}(A)=r, \mbox{ } p\in M\}
\end{displaymath}
which is a subfiber bundle of $\hom(E,F)$ with typical fiber $L^r_{mn}$ by lemma \ref{lem:Lrmn}.
\end{definition}
If $r=\dim(L_p)$ is the corank of $\rho$ at the point $p\in M$, then we have that the section $\rho \in \Gamma(\hom(E,F))$ intersects $L^r(E,F)$ at $p$. If we differentiate the section, we obtain a sequence of arrows:
\begin{displaymath}
\xymatrix{
T_pM \ar[r]^-{d\rho_p} & T_{\rho_p}(\hom(E_p,F_p)) \ar[r]^-q & \nu(L^r(E,F))_{\rho_p}:= T_{\rho_p}(\hom(E_p,F_p))/T_{\rho_p}L^r(E,F)
}
\end{displaymath}
where the map $q:T_{\rho_p}(\hom(E_p,F_p)) \to \nu(L^r(E_p,F_p))_{\rho_p}$ is just the projection. Since $T_{\rho_p}\hom(E_p,F_p)$ is canonically identified with $\hom(E_p,F_p)$, there is a canonical map:
\begin{equation}\label{eq:canmap}
RP:T_{\rho_p}\hom(E_p,F_p) \to \hom(K_p,L_p)
\end{equation}
given by restricting a linear map in $T_{\rho_p}\hom(E_p,F_p)$ to the kernel of $\rho_p$, $K_p$, and projecting onto the cokernel of $\rho_p$, $L_p$. As shown in lemma 3.2 in \cite{GG} (p.150), the kernel of this map is exactly $T_{\rho_p}L^r(E,F)_p$, hence it descends to a canonical isomorphism:
\begin{equation}\label{eq:canmap1}
RP: \nu(L^r(E,F))_{\rho_p} \to \hom(K_p,L_p)
\end{equation}
hence we may add another arrow to the above sequence to obtain a map:
\begin{displaymath}
\xymatrix{
T_pM \ar[r]^-{d\rho_p} & T_{\rho_p}(\hom(E_p,F_p)) \ar[r]^-q & \nu(L^r(E,F))_{\rho_p}\ar[r]^-{RP} & \hom(K_p,L_p).
}
\end{displaymath}
We define the intrinsic derivative $(D\rho)_p$ of $\rho$ at $p$ to be the composition of the above arrows, hence it is a map:
\begin{displaymath}
(D\rho)_p:T_pM \to \hom(K_p,L_p)
\end{displaymath}
Now, let us return to smooth maps between manifolds with corners $f:M \to N$ and let us assume that the dimensions of source and target spaces satisfy $\dim(M) \ge \dim(N)$. Our discussion will be similar to the case of maps between vector bundles, but we'll need to be more careful since we won't have vector bundles over the same base space: $TM \to M$ and $TN \to N$ are vector bundles over different spaces, in general.
Differentiating $f$ gives us a map $df:TM \to TN$, which is a section of the fiber bundle $\hom(TM,TN) \to M$ and this is \emph{not} a vector bundle unless $N$ is a vector space: its fiber is $T_pM \times TN$. Let $p\in M$ and let $r=\operatorname{corank}(df_p)$ be the corank of the differential at $p$. Let $K_p = \ker(df_p)\subseteq T_pM$ be the kernel and let $L_p = T_{f(p)}N / df_p(T_pM)$ be the cokernel. Note that, while the kernel is a subspace of the domain of $df_p$, the cokernel is not canonically identified with a subspace of $T_{f(p)}N$. Using the above discussion of vector bundles with $E$ as $TM\to M\times N$, $F$ as $TN \to M \times N$, $\hom(E,F)$ as $\hom(TM,TN) \to M\times N$, and $L^r(E,F)$ as $L^r(TM,TN) \to M\times N$, we obtain a sequence of maps:
\begin{displaymath}
\xymatrix{
T_pM \ar[r]^-{(d(df))_p} & T_{df_p}(\hom(TM, TN)) \ar[r]^-{q} & \nu(L^r(TM,TN))_{\rho_p}\ar[r]^-{RP} & \hom(K_p,L_p)
}
\end{displaymath}
The composition give us a linear map:
\begin{displaymath}
F_p:T_pM \to \hom(\ker(df_p),\operatorname{coker}(df_p))
\end{displaymath}
and restriction to $\ker(df_p)$ gives us a quadratic map, which we call the intrinsic second derivative of $f$ at $p$, or just \emph{the intrinsic derivative of f at $p$}:
\begin{displaymath}
(Df)_p:\ker(df_p)\otimes \ker(df_p) \to \operatorname{coker}(df_p)
\end{displaymath}
given by $(Df)_p(\eta \otimes v) = F_p(\eta)(v)$. The following examples are more of a guide for computing the intrinsic derivative in the case of maps between vector bundles and the case of maps between manifolds.
\begin{example}\label{ex:intrVB}
Let $E$ and $F$ be two vector bundles over a fixed manifold with corners $M$. Let $\rho\in \Gamma(\hom(E,F))$ be a section of $\hom(E,F)$ and let $p\in M$. Let's see how we can compute the intrinsic derivative at $p$. Choose a basis $\{e_1,\dots,e_n\}$ for $E_p$ so that $e_i\in \ker(\rho_p)$ for $1\le i \le j$, where $j\le n$. Choose a local frame of $F$ near $p$, $\{f_1,\dots,f_k\}$.
\begin{itemize}
\item Let $e_i \in \ker(\rho_p)$ and let $\tilde{e_i}$ be an extension to a local section of $E$ near $p$.
\item Then $\rho(\tilde{e}_i)= \sum_{l=1}^k a_lf_l$, where $a_l \in C^{\infty}(M)$ is smooth for each $l$.
\item Then, for each $X\in T_pM$, we can consider $\sum_{l=1}^k d(a_l)_p(X)f_l \in F_p$.
\item We can then send $\sum_{l=1}^kd(a_l)_p(X)f_l$ to its image in $F_p/\rho_p(E_p)$ and we claim that this gives us $D\rho_p(X)(e_i)$.
\item Indeed, if $\tilde{e}_i'$ is any other extension, then the difference $\tilde{e}_i-\tilde{e}_i'$ vanishes at $p$. If $\{v_1,\dots,v_n\}$ is a local frame for $E$ near $p$, then we can write:
\begin{displaymath}
\tilde{e}_i-\tilde{e}_i'= \sum_{l=1}^n g_lv_l
\end{displaymath}
where the $g_l's$ vanish at $p$. Then,
\begin{displaymath}
\rho_p(\tilde{e}_i-\tilde{e}_i')=\sum_{l=1}^n g_l\rho_p(v_l) = \sum_{l=1}^{n}\sum_{r=1}^k g_la_{lr}f_r
\end{displaymath}
where the $a_{lr}'s$ are smooth. Then $d(g_la_{lr})_p = a_{lr}(p)d(g_l)_p$, since $g_l(p)=0$. Thus, for any $X\in T_pM$,
\begin{displaymath}
\sum_{l=1}^{n}\sum_{r=1}^k d(g_l)_p(X)a_{lr}(p)f_r(p) = \sum_{l=1}^n d(g_l)_p(X)\rho_p(v_l)=\rho_p(\sum_{l=1}^n d(g_l)_p(X))
\end{displaymath}
hence if we apply the above recipe to any two extensions of $e_i$, the results differ by an element in the image of $\rho_p$, meaning we get the same element of the cokernel when we project.
\item If we choose a different local frame of $F$ near $p$, then the two frames are related by a local automorphism of $F$, which will induce an automorphism of $\operatorname{coker}(\rho_p)$ sending one coordinate representation of $D\rho_p(e_i)$ to the other, hence the map we get from $\ker(\rho_p)$ to $\operatorname{coker}(\rho_p)$ is coordinate independent.
\item Lastly, the reason that this is the intrinsic derivative is that we may extend the $e_i$'s to a local frame of $E$ and then $\rho$ becomes a matrix: all we are doing is differentiating the coefficients in this matrix, restricting to the kernel of $\rho_p$, and projecting to the cokernel of $\rho_p$, which is exactly the recipe of the intrinsic derivative.
\end{itemize}
\end{example}
\begin{example}\label{ex:intrFunc}
Given a smooth map $f:M \to N$ between manifolds with corners, we will show how one may compute the intrinsic derivative in coordinates. In \cite{Ho}, H\"{o}rmander uses Taylor expansions in order to show the existence of the quadratic map defined above. We offer an alternate approach in coordinates, but the reader is invited to peruse either.
Fix a point $p\in M$ and choose coordinates around $p$, $f(p)$ so that we may assume $f$ is a smooth map $f:\R^m \to \R^n$. Technically, it is a smooth map defined on a quadrant since $M$ and $N$ have corners, but smoothness implies it extends to an open subset of $\R^m$ and $\R^n$ so we just extend it for the sake of simplicity. In general, almost everything we are about to say only makes sense in coordinates, but the intrinsic derivative is a local construction so this is acceptable. Now,
\begin{itemize}
\item choose a vector $v\in \ker(df_p)$,
\item extend it to a local vector field $\tilde{v}$ near $p$,
\item define the map $g:\R^m \to \R^n$ given by $g(x)=df_x(\tilde{v}(x))$, which has values in the fiber $\R^n$ of $T(\R^n)$,
\item and differentiate $g$ using $v$: $dg_p(v) = v(df(\tilde{v}))$.
\end{itemize}
Because $g$ takes values in the fiber, $\R^n$, of $T(\R^n)$, we may view it as a tangent vector to $\R^n$ at $f(p)$. Any two extensions $\tilde{v}_1$ and $\tilde{v}_2$ of $v$ agree at $p$, hence the difference $\tilde{v}_1 - \tilde{v}_2$ vanishes at $p$. Thus, if one differentiates $(df(\tilde{v}_1-\tilde{v}_2))$ in any direction at $p$, the result may be interpreted (in coordinates) as an element in the image of $df_p$. Indeed, if $p$ is the origin in $\R^n$ and $\displaystyle X=\sum_{i=1}^n a_i\frac{\partial}{\partial x_i}$ is a vector field vanishing at the origin, then,
\begin{displaymath}
\frac{\partial}{\partial x_j}\big\vert_0 df(\sum_{i=1}^n a_i \frac{\partial}{\partial x_i}) = \sum_{i=1}^n\frac{\partial a_i}{\partial x_j} df_0(\frac{\partial}{\partial x_i})
\end{displaymath}
since the $a_i's$ vanish at $0$. Thus, the difference $vdf(\tilde{v}_1) -vdf(\tilde{v}_2)$ is an element of $df_p(T_pM)$, hence the vector $v(df(\tilde{v}))$ is well defined as an element of $T_p\R^n/df_p(T_p\R^m)$. That is, we have a well-defined element of $T_pN/df_p(T_M)$ and the above construction gives us a map from $T_pM$ into $\hom(\ker(df_p),\operatorname{coker}(df_p)$. As in the vector bundle example, this is the intrinsic derivative since we have simply differentiated how the derivative of $f$ acts on elements of $T_pM$ and then restricted to $\ker(df_p)$ followed by projection to $\operatorname{coker(df_p)}$.
\end{example}
\subsection{Sections With Fold Singularities}
We now generalize the definition of a submersion with folds, definitions 4.1 and 4.2, found in \cite{GG} to arbitrary fiber bundles.
\subsubsection{Definition of a Section with Fold Singularities}
\begin{definition}\label{def:folds}
Let $\pi:E \to M$ be a fiber bundle over a manifold with corners $M$. Let $\chi$ be a connection on $E$, let $p_V:TE \to V$ be the induced projection to the vertical bundle of $E$, and let $S_r$ be the submanifold of $J^1(E)$ of jets of corank $r$. We say a local section $(U,\phi)$ has a fold singularity at $p\in U$ if:
\begin{enumerate}
\item $j^1\phi_p \in S_1$ (the derivative drops rank by $1$, where rank is determined by the connection),
\item $j^1\phi \pitchfork_s S_1$ at $p$, meaning $Z= j^1\phi^{-1}(S_1)$ is a submanifold with corners near $p$, and
\item $\ker(p_V(d\phi_p))\pitchfork T_p(Z)$.
\end{enumerate}
If for each $p\in U$ we have $j^1\phi_p\in S_0$ or $j^1\phi_p\in S_1$ and the above three conditions are satisfied at such points, then we say that $(U,\phi)$ is a section with fold singularities. We call $Z=(j^1\phi)^{-1}(S_1)$ the \emph{fold locus} of $\phi$. It has codimension $(\dim(M)-q + 1)(\dim(F) - q + 1)$, where $q=\min(\dim(M),\dim(F))$, by proposition \ref{prop:Sr} and lemma \ref{lem:Lrmn}.
\end{definition}
\begin{remark}
Let $F$ bet the typical fiber of a fiber bundle $\pi:E \to M$ and let $(U,\phi)$ be a section with fold singularities. Then, we necessarily have that $\dim(M) \ge \dim(F)$. To see this, recall that $\chi$ gives us an isomorphism of fiber bundles:
\begin{displaymath}
\xymatrixcolsep{4pc}\xymatrixrowsep{.5pc}\xymatrix{
F_{\chi}: J^1(E) \ar[r] & \hom(\pi^*TM, V) \\
(j^1_p\phi) \ar[r] & (p,\phi(p),p_V\circ d\phi_p)
}
\end{displaymath}
If $\dim(F) > \dim(M)$, then $\operatorname{rank}(V) > \operatorname{rank}(\pi^*TM)$, hence any element of the fiber of $\hom(\pi^*TM, V)$ would necessarily have corank $\ge 1$. This means that if $(U,\phi)$ is a local section then $\operatorname{corank}(j^1_p\phi)\ge 1$ for all $p\in U$, meaning $(U,\phi)$ either does not satisfy condition $1$ or it satisfies condition $1$ and violates condition $2$ since the image of $j^1\phi$ would be contained in $S_1$.
We therefore have that the fold locus $(j^1\phi)^{-1}(S_1)$ has codimension $(\dim(M)-\dim(F)+1)$ by proposition \ref{prop:Sr}.
\end{remark}
\begin{definition}\label{def:folds1}
Let $f:M \to N$ be a smooth map of manifolds with corners. We will say that $f:M \to N$ is a submersion with folds if the section $\phi(m)=(m,f(m))$ of $M\times N$ is a section with fold singularities, where we equip $M\times N$ with the standard connection. The fold locus, $Z=(j^1\phi)^{-1}(S_1)$ is a smooth submanifold with corners of $M$ of codimension $(\dim(M)-\dim(N) + 1)$.
\end{definition}
\begin{lemma}\label{lem:folds0}
Let $\pi:E \to M$ be a fiber bundle over a manifold with corners with connection $\chi$. Let $(U,\phi)$ be a section with fold singularities. Suppose $\pi':E' \to M$ is another fiber bundle with connection $\chi'$ and $\tau:E\to E'$ is a connection-preserving isomorphism of fiber bundles. Then $(U,\tau\circ \phi)$ is a section with fold singularities.
\end{lemma}
\begin{proof} \mbox{ } \newline
Let $V$ and $V'$ be the vertical bundles of $E$ and $E'$ respectively. The map $\tau$ induces an isomorphism of fiber bundles (over $M$):
\begin{displaymath}
\tilde{\tau}: \hom(\pi^*TM, V) \to \hom(\pi'^*TM, V')
\end{displaymath}
and by proposition \ref{prop:C2} we have a series of isomorphisms of fiber bundles (over $M$) :
\begin{displaymath}
\xymatrixcolsep{4pc}\xymatrix{
J^1(E) \ar[r]^{F_{\chi}} & \hom(\pi^*TM,V) \ar[r]^{\tilde{\tau}} & \hom(\pi'^*TM, V') \ar[r]^{F_{\chi'}^{-1}} & J^1(E')
}
\end{displaymath}
which sends $S_r\subset J^1(E)$ to $S_r'\subset J^1(E')$ by lemma \ref{lem:rank}. Therefore, $\phi$ is a section with fold singularities if and only if $\tau(\phi)$ is a section with fold singularities.
\end{proof}
\begin{cor}\label{cor:folds-diffeo}
Let $M$ and $N$ be smooth manifolds with corners and let $f:M\to N$ be a submersion with folds. Let $\tau:N \to P$ be a diffeomorphism of manifolds with corners. Then $\tau \circ f$ is a submersion with folds.
\end{cor}
\begin{proof} \mbox{ } \newline
Apply lemma \ref{lem:folds0} with $E=M\times N$, $E'=M\times N_1$, and with $\chi, \chi'$ as the standard connections. $f$ is a submersion with folds if and only if the section $\phi(m)=(m,f(m))$ is a section with folds if and only if $\tau(\phi)(m)=(m,\tau(f(m)))$ is a section with folds if and only if $\tau(f(m))$ is a submersion with folds.
\end{proof}
\subsubsection{Equidimensional Fold Maps and Computations}
\begin{definition}\label{def:folds2}
Let $\pi:E \to M$ be a fiber bundle over a manifold with corners with connection $\chi$ and typical fiber $F$ satisfying $\dim(F)=\dim(M)$. We say a local section $(U,\phi)$ is an \emph{equidimensional section with fold singularities} if $(U,\phi)$ is a section with fold singularities (q.v. definition \ref{def:folds1}). When the dimensions of $F$ and $M$ are understood to be equal, we simply refer to $(U,\phi)$ as a section with fold singularities. We call $Z=(j^1\phi)^{-1}(S_1)$ the \emph{folding hypersurface} of $\phi$ since it has codimension $1$.
\end{definition}
\begin{definition}\label{def:folds3}
Let $M$ and $N$ be two $m$-dimensional manifolds with corners and let $E=M\times N$ be the trivial fiber bundle over $M$ with standard flat connection, $H=TM\oplus 0$. We say $f:M \to N$ is an \emph{equidimensional map with fold singularities} if the section $\phi(m)=(m,f(m))$ is a section with fold singularities. When the dimensions of $M$ and $N$ are understood to be equal, we simply refer to $f$ as a \emph{map with fold singularities}. We call $Z=(j^1\phi)^{-1}(S_1)$ the \emph{folding hypersurface} of $f$ since it is a codimension $1$ submanifold with corners of $M$.
\end{definition}
\begin{remark}
Recall that if $A:V_1 \to V_2$ is a linear map of vector spaces then $A^n$ denotes the induced map between the $n^{th}$ exterior powers: $A^n:\Lambda^n(V_1) \to \Lambda^n(V_2)$. Similarly, if $A:V_1 \to V_2$ is a map of vector bundles then we may write $A^n$ for the induced map $A:\Lambda^n(V_1) \to \Lambda^n(V_2)$ between maps of vector bundles.
\end{remark}
We have an important computational tool allowing us to detect when sections have fold singularities.
\begin{prop}\label{prop:folds4}
Let $\pi:E \to M$ be a fiber bundle over a manifolds with corners $M$ with connection $\chi$, induced projection $p_V:TE \to V$ onto the vertical bundle, and typical fiber $F$ satisfying $\dim(F)=\dim(M)=n$. Let $\mathcal{O}$ be the zero section of $\hom(\Lambda^n(\pi^*TM), \Lambda^n(V))$ (a vector bundle over $E$). Then $(U,\phi)$ satisfies
\begin{enumerate}
\item $\operatorname{corank}(j^1_\phi)>0$ $\iff$ $j^1_p\phi \in S_1$ and
\item $j^1\phi \pitchfork_s S_1$
\end{enumerate}
if and only if $(p_V(d\phi))^n:M \to \hom(\Lambda^n(\pi^*TM),\Lambda^n(V))$ satisfies $(p_V(d\phi))^n\pitchfork_s \mathcal{O}$.
\end{prop}
The proposition is a direct consequence of the following lemma.
\begin{lemma}\label{lem:folds1}
Let $M$ be a manifold with corners and let $\psi:M\to \hom(\R^n,\R^n)$ be a smooth map. Let $L^r:=L^r_{nn} \subset \hom(\R^n,\R^n)$ be the subset of matrices of corank $r$ (rank $n-r$). Then $\det(\psi):M\to \R$ is (strongly) transverse to $0$ if and only if the two conditions:
\begin{enumerate}
\item $\operatorname{corank}(\psi(p))>0$ if and only if $\psi(p)\in L^1$ and
\item $\psi \pitchfork_s L^1$.
\end{enumerate}
are satisfied
\end{lemma}
\begin{proof} \mbox{ } \newline
\begin{enumerate}
\item Assume that $\operatorname{corank}(\psi(p))>0$ if and only if $\psi(p)\in L^1$ and $\psi \pitchfork_s L^1$. We show that $\det(\psi)\pitchfork_s 0$.
\begin{itemize}
\item First, note that at any point $A$ of $L^1$ $\det: \hom(\R^n, \R^n) \to \R$ is a submersion. To see this, write $A=\begin{bmatrix} c_1 & \dots & c_n \end{bmatrix}$ and, without loss of generality, assume $c_n$ is in the span of the first $n-1$ columns. Pick any vector $v$ not in the span of the first $n-1$ rows and consider $\gamma(t)=\begin{bmatrix} c_1& \dots &c_{n-1} & c_n+tv\end{bmatrix}$. Then:
\begin{displaymath}
\frac{d}{dt} \det(\gamma(t)) = \frac{d}{dt} \det(c_1,\dots,c_{n-1},c_n+tv) = \frac{d}{dt}(\det(A) + t\det(c_1,\dots, c_{n-1},v)) = \det(c_1,\dots,c_{n-1},v) \ne 0
\end{displaymath}
which shows $\det$ is a submersion at $A$.
\item Note that the restriction $\det \vert_{L^1}$ is identically zero since all elements in this subset have rank $n-1$, hence its derivative restricted to vectors tangent to $L^1$ vanishes. We also note that $L^1$ is a hypersurface in $\hom(\R^n,\R^n)$ by lemma \ref{lem:Lrmn}. Therefore, if $A \in L^1$ and $v\in T_A\hom(\R^n,\R^n)$ is transverse to $T_AL^1$, we must have that $d(\det)_A(v) \ne 0$ since it is surjective at $A$ by the first bullet. Otherwise, the differential would vanish at $A\in L^1$ since it would vanish along directions tangent to a hypersurface and a direction transverse to that hypersurface.
\item Let $p\in M$ be a point for which $\psi(p)\in L^1$. $\psi \pitchfork_s L^1$ by assumption, so there exists a vector $v_0 \in T_pM$, tangent to the stratum containing $p$, so that $d\psi_p(v_0) \pitchfork T_{\psi(p)}L^1$. Then $d(\det\circ \psi)p(v_0)\ne 0$ by the previous bullet.
\item Since this is true for any $p\in M$ with $\psi(p)\in L^1$ and $\psi(p)\in L^r \iff \mbox{ }r=0,1$, we have $\det(\psi)\pitchfork_s 0$.
\end{itemize}
\item Conversely, assume that $\det(\psi)\pitchfork_s 0$.
\begin{itemize}
\item We first show that $\psi$ cannot intersect the subset of matrices of corank $r$, $L^r$, unless $r=1$ or $r=0$. Assume that $\psi(p)\in L^r$ and $r>1$. Since $\det(\psi)\pitchfork_s 0$, $\det(\psi)^{-1}(0)$ is a smooth submanifold with corners of $M$ transverse to boundary strata and there are coordinates $(x_1,\dots,x_{m-1},t)$ around $p$ with $\det(\psi)^{-1}(0)$ identified with the zero set $t=0$ and $p$ identified with the origin.
\item We write $\psi(\vec{x},t)$ in column form as $\psi(\vec{x},t) = \begin{bmatrix} c_1&\dots& c_{n-2}&c_{n-1}&c_n \end{bmatrix}$ for some smooth maps $c_i:M \to \R^n$.
\item We are assuming $\psi(0)\in L^r$ with $r>1$ so, without loss of generality, $c_{n-1}(0)$ and $c_n(0)$ are in the span of the first $n-2$ columns at $0$. That is, there are constants $a_i$ so that:
\begin{displaymath}
c_n(0)=\sum_{i=1}^{n-2} a_i c_i(0)
\end{displaymath}
\item Consider the curve $\gamma(t) = (0,t)$ passing through $p$. Since $c_n(\gamma(t))-\sum_{i=1}^{n-2} a_i c_i(\gamma(t))$ vanishes at $t=0$, we have that $c_n(\gamma(t))= tG(\vec{x},t) + \sum_{i=1}^{n-2} a_i c_i(\gamma(t))$ for some smooth map $G(\vec{x},t)$.
\item By assumption, $\det(\psi) \pitchfork_s 0$. Since $Z=\det(\psi)^{-1}(0)$ is a hypersurface, this implies that for any direction $v$ transverse to $Z$ at $p$, $v(\det(\psi)) \ne 0$. Therefore, $\frac{d}{dt}\det(\psi(\gamma(t)))$ should be nonzero at $t=0$.
\item However, we compute:
\begin{displaymath}
\begin{array}{lcl}
\det(\psi(\gamma(t)) &=& \det( c_1 , \dots , c_{n-2} , c_{n-1} , tG(\vec{x},t) + \sum_{i=1}^{n-2} a_i c_i(\gamma(t))) \\
&=& \det(c_1, \dots, c_{n-2}, c_{n-1}, tG(\vec{x},t)) \\
&=& t\det(c_1, \dots, c_{n-2}, c_{n-1}, G(\vec{x},t)) \\
\end{array}
\end{displaymath}
hence $\frac{d}{dt} \det(\psi(\gamma(t)))\big\vert_{t=0} = \det(c_1(0),\dots, c_{n-2}(0), c_{n-1}(0),G(0)) =0$ since $c_{n-1}(0)$ is in the span of the first $n-2$ columns by assumption. This contradicts the assumption that $\det(\psi)\pitchfork_s 0$, hence we must have $r\le 1$ and $\psi$ may only intersect $L^1$ if $r>0$. That is $\psi(p) \in L^r \mbox{ } \iff \mbox{ } r=1,0$.
\item Now, since $L^1$ is a hypersurface (q.v. lemma \ref{lem:Lrmn}), we can deduce that $\psi \pitchfork_s L^1$ fairly easily. If $p\in \det(\psi)^{-1}(0)$ and $d\psi_p \not\pitchfork T_{\psi(p)}L^1$, then we must have that $d\psi_p(T_pM)\subset T_{\psi(p)}L^1$, since $L^1$ is a hypersurface by lemma \ref{lem:Lrmn}. But then $d(\det\circ \psi)_p =0$ since the derivative of the determinant vanishes along directions tangent to $L^1$, meaning $\det(\psi) \not\pitchfork_s 0$ at $p$, contradicting our assumption.
\item Therefore, $\psi$ intersects $L^r$ if and only if $r\le 1$ and $\psi \pitchfork_s L^1$.
\end{itemize}
\end{enumerate}
\end{proof}
\begin{proof}[proof of proposition \ref{prop:folds4}]
Lemma \ref{lem:folds1} gives us a straightforward proof of proposition \ref{prop:folds4}. Recall that in the setting of proposition \ref{prop:folds4} we have a fiber bundle $\pi:E \to M$ with connection $\chi$, which gives us a projection $p_V:TE \to V$ onto the vertical bundle, and we have $S_1\subset J^1(E)$, the subset of jets of corank $1$. Our goal is to show that $j^1\phi \pitchfork_s S_1$ (and only intersects $S_1, S_0$) if and only if $(p_V(d\phi))^n\pitchfork_s \mathcal{O}$. By proposition \ref{prop:C2}, the connection $\chi$ induces an isomorphism of fiber bundles:
\begin{displaymath}
F_{\chi}: J^1(E) \to \hom(\pi^*TM, V)
\end{displaymath}
Let $L_r$ denote the submanifold of $\hom(p^*TM, V)$ of maps of corank $r$. As a note in what follows, $L_r$ is always the subfiber bundle of $\hom(p^*TM,V)$ of elements of corank $r$ while $L^r$ is the subset of $\hom(\R^n, \R^n)$ of elements of corank $r$. The relationship is that $L^r$ is the typical fiber of $L_r$. We have:
\begin{displaymath}
\begin{array}{lc}
\text{$j^1\phi$ intersects $S_r$ only when $r=1,0$ and $j^1\phi \pitchfork_s S_1$} & \iff \\
\text{$F_{\chi}(j^1\phi)=p_V(d\phi)$ intersects $L_r$ only when $r=1,0$ and $p_V(d\phi)\pitchfork_s L_1$} & \iff \\
\text{The previous line is true in any trivialization of $\hom(\pi^*TM, V)$} &\iff \\
\text{it is true for $p_V(d\phi):U \to E\vert_U \times \hom(\R^n \times \R^n)$, $p_V(d\phi)(u)=(\phi(u),A(u))$ } &\iff \\
\text{$A:U \to \hom(\R^n, \R^n)$ only intersects $L^1$ and $A\pitchfork_s L^1$ } &\iff \\
\text{$\det(A):U \to \R$ satisfies $\det(A)\pitchfork_s 0$ by lemma \ref{lem:folds1}} &\iff \\
\text{$A^n:\Lambda^n(\R^n) \to \Lambda^n(\R^n)$ is transverse to the zero section $\mathcal{O}\in \Gamma(\hom(\Lambda^n(\R^n),\Lambda^n(\R^n)))$} & \iff \\
p_V(d\phi)^n \pitchfork_s \mathcal{O} \in \Gamma(\hom(\Lambda^n(\pi^*TM),\Lambda^n(V)))
\end{array}
\end{displaymath}
where the first $5$ statements are a local restatement of what it means to only intersect $S_1$ and $S_0$ and to intersect $S_1$ transversally. The last line is a global restatement of the previous line.
\end{proof}
\begin{cor}\label{cor:folds4-0}
Let $f:M \to N$ be a smooth map between two $m$-dimensional manifolds with corners. Then $f$ is a map with fold singularities if and only if
\begin{enumerate}
\item The induced map $(df)^m: M \to \hom(\Lambda^m(TM), \Lambda^m(TN))$ is transverse (in the sense of manifolds with corners) to the zero section $\mathcal{O}$ of $\hom(\Lambda^m(TM),\Lambda^m(TN)) \to M \times N$, and
\item $\ker(df) \pitchfork ((df)^m)^{-1}(\mathcal{O})$.
\end{enumerate}
\end{cor}
\begin{proof} \mbox{ } \newline
This is just a restatement of the proposition in the case that $E$ is the trivial fiber bundle and $\chi$ is the standard connection.
\end{proof}
\begin{remark}
While corollary \ref{cor:folds4-0} follows easily from proposition \ref{prop:folds4}, its utility should not be overlooked. What do we need to check to determine if a map of manifolds with corners $f:M^n \to N^n$ has fold singularities?
\begin{enumerate}
\item Compute the determinant $\det(df)$ in local coordinates and check that it vanishes transversally. Note that we are dealing with manifolds with corners, so this means that we must check the condition $\det(df) \pitchfork_s 0$ on each stratum of $M$.
\item In local coordinates, check that $\ker(df_p)\pitchfork \det(df)^{-1}(0)$ for all $p\in \det(df)^{-1}(0)$.
\end{enumerate}
This makes the problem of understanding fold maps $f:M^n \to N^n$ very simple from a computational point of view.
\end{remark}
\begin{cor}\label{cor:folds4-2}
Let $f:M\to N$ be a smooth map with fold singularities between two $m$-dimensional manifolds with corners with folding hypersurface $Z\subset M$. Suppose $\phi:M_1 \to M$ is a diffeomorphism of manifolds with corners. Then $f\circ\phi:M_1 \to N$ is a (equidimensional) map with fold singularities with folding hypersurface $\phi^{-1}(Z)$.
\end{cor}
\begin{proof} \mbox{ } \newline
This statement follows from the definitions, but we list it here as a corollary to proposition \ref{prop:folds4} since the criteria for detecting fold singularities are now easier to describe.
\vspace{2mm}
We have that $(df)^m: M \to \hom(\Lambda^m(TM), \Lambda^m(TN))$ satisfies $(df)^m\pitchfork_s \mathcal{O}$, where $\mathcal{O}$ is the zero section, if and only if $(df\circ d\phi)^m : M_1 \to \hom(\Lambda^m(TM_1), \Lambda^m(TN))$ is transverse to the zero section since $\phi$ is a diffeomorphism, hence $(df\circ d\phi)^m$ is transverse the zero section and the degenerate hypersurface of $df\circ d\phi$ is $\phi^{-1}(Z)$. $\ker(df\circ d\phi)=d\phi^{-1}(\ker(df))$ is transverse to $\phi^{-1}(Z)$ since $\ker(df)\pitchfork Z$ and $\phi$ is a diffeomorphism of manifolds with corners. Thus, $f\circ \phi$ is a map with fold singularities.
\end{proof}
\begin{cor}\label{cor:folds4-1}
Let $f:M \to N$ be an equidimensional map with fold singularities. Let $S\subset M$ be an $s$-dimensional submanifold with corners of $M$ and suppose $f(S) \subset R \subset N$, where $R$ is an $s$-dimensional submanifold with corners of $N$. Then $f\vert_{S}:S \to R$ is an equidimensional map with fold singularities.
\vspace{2mm}
Furthermore, at points $p\in S$ where $\ker(df_p)\subset T_pS$, $df_p$ induces an isomorphism on the fibers of the normal bundles, $\nu(S)$ and $\nu(R)$, or $S$ and $R$. That is:
\begin{displaymath}
df_p:\nu(S)_p \to \nu(R)_{f(p)}
\end{displaymath}
is an isomorphism.
\end{cor}
\begin{proof}\mbox{ } \newline
In light of proposition \ref{prop:folds4}, we need only check that the determinant of $d(f\vert_S)$ vanishes transversally in coordinates and the kernel of $d(f\vert_S)$ is transverse to the degenerate hypersurface. By the definition of a submanifold with corners, we may choose coordinates $(x_1,\dots, x_m)$ near $S$ so that $S=\{x_1=0, \dots,x_{m-s}=0\}$, where some of these coordinates may be defined on half spaces $\R^+$, but it will be of no consequence. Since $\dim(R)$ is also $s$, we may choose coordinates $(y_1,\dots,y_m)$ near $R$ so that $R=\{y_1=0,\dots,y_{m-s}=0\}$. We now have three cases to consider:
\begin{enumerate}
\item If $Z\cap S =\emptyset$, then $d(f\vert_S)$ never drops rank and so it is trivially a map with fold singularities.
\item If $p\in Z\cap S$ but the one-dimensional subspace $\ker(df_p)$ is not contained in $T_pS$, then $d(f\vert_{S_1})$ has maximal rank at $p$.
\item If $p\in Z\cap S$ and $\ker(df_p)\subset T_pS$, then we may choose a curve $\gamma(t)$ in $S$ so that $\gamma(0)=p$ and $\gamma'(0)\ne 0$ is in $\ker(d\psi_p)$. Along this curve, the differential $df$ has the form:
\begin{equation}\label{eq:restriction}
df_{\gamma(t)} =
\begin{bmatrix}
A(\gamma(t)) & 0 \\
B(\gamma(t)) & d(f\vert_S)_{\gamma(t)}
\end{bmatrix}
\end{equation}
Since $\gamma'(0)$ is transverse to the folding hypersurface $Z$ and $f$ has fold singularities, we must have that $\frac{d}{dt}\big\vert_0(\det(df_{\gamma(t)}))\ne 0$ by proposition \ref{prop:folds4}. Using the form of $df_{\gamma(t)}$ in equation \ref{eq:restriction}, we find that:
\begin{equation}\label{eq:restriction2}
\begin{array}{lcl}
\frac{d}{dt}\big\vert_0 \det(df_{\gamma(t)}) & = & \frac{d}{dt}\big\vert_0 \det(A(\gamma(t)))\det(d(f\vert_S)_{\gamma(t)}) \\
& = & \det(d(f\vert_S)_{\gamma(0)})\frac{d}{dt}\big\vert_0 \det(A(\gamma(t))) + \det(A(\gamma(0)))\frac{d}{dt}\big\vert_0\det(d(f\vert_S)_{\gamma(t)}) \\
& = & \det(A(p))\frac{d}{dt}\big\vert_0\det(d(f\vert_S)_{\gamma(t)}) \ne 0
\end{array}
\end{equation}
Thus, neither $\det(A(p))$ or $\frac{d}{dt}\big\vert_0\det(d(f\vert_S)_{\gamma(t)})$ may be $0$. In particular, $\det(d(f\vert_S))$ vanishes transversally. The computation also reveals that the degenerate hypersurface of $f\vert_S$ is exactly $S\cap Z$ near $p$. Since $\ker(df)\pitchfork TZ$, we have that $\ker(d(f\vert_S)_p)\pitchfork (T_p(Z\cap S))$, hence $f\vert_S$ is an equidimensional map with fold singularities.
\vspace{2mm}
To prove that $df_p$ induces an isomorphism on the fibers of the normal bundles, we appeal to equation \ref{eq:restriction} and our computation in \ref{eq:restriction2}. In our chosen coordinates, we use the standard metric on $\R^n$ to see that $A(p)$ maps $(T_pS)^{\perp}$ to $(T_{f(p)}R)^{\perp}$. Since $\det(A(p))\ne 0$ by \ref{eq:restriction2}, we have that $A(p)$ has maximal rank. Since $\dim(S)=\dim(R)$ and $\dim(M)=\dim(N)$, the normal bundle $\nu(S)$ and $\nu(R)$ have the same rank. Thus, $A(p)$ is an isomorphism.
\end{enumerate}
\end{proof}
\begin{example}
Here is an example of corollaries \ref{cor:folds4-0} and \ref{cor:folds4-1} in action. Consider the map $f:\R^3 \to \R^3$ given by $f(x,y,z)=(x,y,z^2)$. $\det(df)= 2z$ vanishes transversally at $z=0$ and $\ker(df_{z=0}) = \frac{\partial}{\partial z}$ is transverse to $z=0$, hence $f$ has fold singularities. Let $S$ be the paraboloid $y-z^2-x^2=0$ and let $R$ be the parabolic sheet $y-z-x^2=0$. $f(S) \subset R$ and so $f\vert_S:S \to R$ is a map with fold singularities, which is straightforward to compute directly:
\vspace{3mm}
\noindent Both surfaces are the graphs of functions $y(x,z)$, so we may identify them with the $xz$-plane by projecting. Under this identification, $f\vert_S(x,z)=(x,z^2)$ is a map with fold singularities with folding hypersurface given by $z=0$. Thus, $f\vert_S$ has fold singularities along the parabola $y=x^2$, $z=0$ with kernel spanned by $\frac{\partial}{\partial z}$.
\end{example}
\begin{prop}\label{prop:folds5}
Let $f:M^n \to N^n$ be an equidimensional map with fold singularities, suppose $f$ is strata-preserving, and suppose for all $p\in Z$, the folding hypersurface, $\ker(df_p)$ is tangent to the stratum of $M$ containing $p$. Then, for each $p\in Z$ there exist coordinates near $p$ and $f(p)$ so that:
\begin{displaymath}
f(x_1,\dots,x_{n-1}, t) = (x_1, \dots, x_{n-1}, 0 ) + t^2F(x_1,\dots,x_{n-1})
\end{displaymath}
where $t\in \mathbb{R}$ and $F(\vec{x})$ is a smooth map such that $\phi(\vec{x},t) := (\vec{x},0) + tF(\vec{x})$ is a strata-preserving diffeomorphism in a neighborhood of $0$. Hence, locally, every map $f$ satisfying the conditions of the proposition factors as a diffeomorphism composed with a strata-preserving fold map.
\end{prop}
\begin{proof} \mbox{ } \newline
First, we show that one may assume $M=N=Z\times \R$ and $f:Z\times \R \to Z\times \R$ is a map that folds along $Z\times \{0\}$ with kernel $\frac{\partial}{\partial t}$ and $f(z,0)=(z,0)$ for all $z\in Z$.
\begin{itemize}
\item Let $p\in Z$. Because $\ker(df_x)$ is tangent to strata for all $x\in Z$, we may choose a local section of the bundle $\ker(df) \to Z$ near $p$ and extend it to a stratified vector field in a neighborhood of $p$. Its flow then induces coordinates near $p$ of the form $(z,t)$, where $t\in \R$ and $z\in Z$, so we may assume that $M=Z \times \R$ where the folding hypersurface is $Z\times \{0\}$ and $\ker(df_{(z,0)})=\span\{\frac{\partial}{\partial t}\}$. Furthermore, we may choose coordinates on $N$ near $f(p)$ and so we may assume $N=\R^n$.
\item To summarize, we have a map $f:Z\times \R \to \R^n$, $\dim(Z)=n-1$, with fold singularities at $Z\times \{0\}$ and $\ker(df)$ is spanned by $\frac{\partial}{\partial t}$.
\item Since $f(z,t)-f(z,0)$ vanishes at $t=0$ we can write $f(z,t)= f(z,0)+ tG(z,t)$ for some smooth map $G(z,t)$. Since $df_{(z,0)}(\frac{\partial}{\partial t})=0$, we have that $G(z,0)=0$ for all $z\in Z$, hence $G(z,t)=tF(z,t)$ for some smooth map $F(z,t)$. This gives us the general formula $f(z,t)=f(z,0) + t^2F(z,t)$. Note that since $f$ is strata-preserving, $f\big\vert_{t=0}$ is strata-preserving. Since the $t$ coordinate is defined on $\R$ and not a half-space $\R^+$, this implies that if $k$ half-space coordinates of $f(z,t)$ vanish, the same $k$ coordinates of $f(z,0)$ must vanish, which means that the same $k$ coordinates of $t^2F(z,t)$ must vanish, meaning that $F(z,t)$ is a vector tangent to the stratum containing $f(z,t)$, at $f(z,t)$, for all $t\ne 0$. Smoothness of $F(z,t)$ then implies that we may also interpret $F(z,0)$ as a vector tangent to the stratum containing $f(z,0)$, hence $F(z,t)$ may be interpreted as a vector tangent to the stratum containing $f(z,t)$ at the point $f(z,t)$.
\item Choose a local frame for $Z$ near $p$, $\{e_1,\dots, e_{n-1}\}$ and extend it to a local frame on $Z\times \R$ near $(p,0)$, $\{e_1, \dots, e_{n-1}, \frac{\partial}{\partial t}\}$. Then $df$, near $(p,0)$ may be written:
\begin{displaymath}
df=
\begin{bmatrix}
df(e_1) & \dots & df(e_{n-1}) & 2tF + t^2dF(\frac{\partial}{\partial t}) \\
\end{bmatrix}
\end{displaymath}
Since $\det(df)\pitchfork_s 0$ by corollary \ref{cor:folds4-0}, we have:
\begin{displaymath}
\frac{\partial}{\partial t}\big\vert_{t=0}(\det(df)) = \det(df(e_1),\dots, df(e_{n-1}), F) \ne 0
\end{displaymath}
Which means that at points of $Z\times \{0\}$, the differential of the mapping $\phi(z,t)= f(z,0) + tF(z,0)$ has maximum rank. We claim that $\phi$ is a stratified map in a neighborhood of any point $p=(z_0,0)$. By corollary \ref{cor:folds4-1}, $df_p$ maps directions transverse to the stratum containing $p$ isomorphically onto the normal bundle of the stratum containing $f(p)$. Since $f\big\vert_{t=0}$ is an immersion, this implies that $f\big\vert_{t=0}$ locally embeds $Z\times \{0\}$ as a codimension $1$ submanifold with corners of $N$ transverse to strata. The vector $F(z,0)$ is tangent to the stratum containing $f(z,0)$ and is transverse to the embedded image of $Z\times \{0\}$. Thus, there is a neighborhood of $p$ on which $\phi$ is stratified since is a map that embeds a hypersurface transverse to strata and sends a path $(z,t)$ to a path through $f(z,0)$ along the stratum containing $f(z,0)$, transverse to the embedded hypersurface. Since $\phi$ is stratified and $d\phi\big\vert_{t=0}$ has maximal rank, there is a neighborhood of $p=(z_0,0)$ on which $\phi$ is a diffeomorphism of manifolds with corners. Notice that $\phi(z,0)=f(z,0)$ for all $z\in Z$, hence $\phi^{-1}(f(z,0))=(z,0)$ and, in particular, $\phi^{-1}(f(z_0,0))=(z_0,0)$.
\item Thus, if we postcompose with $\phi^{-1}$ then we have a map $\phi^{-1} \circ f : V_1\subset \to V_2$, where $V_i\subset Z\times \R$ is an open neighborhood of $(z_0,0)$ for each $i$. Consequently, we may assume that we have a map $f:Z\times \R \to Z \times \R$ that folds along $Z\times \{0\}$ with kernel $\frac{\partial}{\partial t}$ and $f(z,0)=(z,0)$ for all $z\in Z$.
\end{itemize}
To finish the proof, we may choose coordinates $\vec{x}=(x_1,\dots x_{n-1})$ near $p$ that identify $p$ with the origin. We then have a map $f:\R^{n-1}\times \R \to \R^{n-1}\times \R$ satisfying $f(\vec{x},0)= (\vec{x},0)$. Using the same tricks in the first part of the proof, we may write $f(\vec{x},t)= (\vec{x},0) + t^2F(\vec{x},t)$ where $\phi(\vec{x},t)= (\vec{x},0) + tF(\vec{x},0)$ is a diffeomorphism near $0$. Let $\gamma(\vec{x},t)=(\vec{x},t^2)$ and note that it folds along the set $\{t=0\}$. Furthermore, the map $\psi(\vec{x},t) = (\vec{x},0) + t^2F(\vec{x},0)$ is the composition $\psi= \phi \circ \gamma$. The maps:
\begin{displaymath}
\begin{array}{ll}
\psi^+ = & \phi \circ \gamma\big\vert_{t\ge 0} \\
\psi^- = & \phi \circ \gamma\big\vert_{t\le 0}
\end{array}
\end{displaymath}
are homeomorphisms near $0$ since $\phi$ is a homeomorphism near $0$ and $\gamma(\vec{x},t)=(\vec{x},t^2)$ restricted to $t\ge 0$ or $t\le 0$ is a homeomorphism. We now define a continuous map:
\begin{displaymath}
\Phi(\vec{x},t) =
\begin{cases}
((\psi^+)^{-1}\circ f)(\vec{x},t) & \mbox{if } t\ge 0 \\
((\psi^-)^{-1}\circ f)(\vec{x},t) & \mbox{if } t\le 0
\end{cases}
\end{displaymath}
Note that $(\psi^+)^{-1}(f(\vec{x},0)) = (\psi^+)^{-1}(\vec{x},0) = (\vec{x},0) = (\psi^-)^{-1}(\vec{x},0) = (\psi^-)^{-1}(f(\vec{x},0))$ so that $\Phi$ is well-defined and continuous at $t=0$. It is strata-preserving since it is a composition of strata-preserving maps.
\vspace{5mm}
Our first claim is that $\Phi$ is a diffeomorphism of manifolds with corners near $0$. Let $e_1, \dots, e_{n-1}$ be the standard basis vectors for $\R^{n-1}$ and observe that $\{e_1,\dots, e_{n-1},F(\vec{x},0)\}$ is a linearly independent set for $\vec{x}$ near $0$, which is equivalent to $\phi$ being a diffeomorphism near $0$ since the image of $d\phi_0$ using the standard basis is precisely this set. Therefore, there exist smooth functions $a_1, \dots, a_n$ so that:
\begin{displaymath}
F(\vec{x},t) = (a_1, \dots, a_{n-1}, a_nF(\vec{x},0))
\end{displaymath}
near $0$, where $a_n \ne 0$ near $0$ since $F(\vec{x},0)$ and $F(\vec{x},t)$ agree at $t=0$. Then, we may write:
\begin{displaymath}
f(\vec{x},t) = (\vec{x},0) + t^2F(\vec{x},t)= (x_1 +t^2a_1, \dots, x_{n-1}+t^2a_{n-1}, a_nt^2F(\vec{x},0))
\end{displaymath}
and if we apply $(\gamma\big\vert_{t\ge0})^{-1}$ or $(\gamma\big\vert_{t\le 0})^{-1})$ we obtain the formula:
\begin{displaymath}
\Phi(\vec{x},t)=
\begin{cases}
(\gamma\big\vert_{t\ge0})^{-1}(\phi^{-1}(f(\vec{x},t))) = (x_1 +t^2a_1, \dots, x_{n-1}+t^2a_{n-1}, \sqrt{a_n}t) & t\ge 0 \\
(\gamma\big\vert_{t\le0})^{-1}(\phi^{-1}(f(\vec{x},t))) = (x_1 +t^2a_1, \dots, x_{n-1}+t^2a_{n-1}, \sqrt{a_n}t) & t\le 0
\end{cases}
\end{displaymath}
where $\sqrt{a_n}$ is smooth near $0$ since $a_n$ is nonzero in a neighborhood of $p$. Hence,
\begin{displaymath}
\Phi(\vec{x},t)= (x_1 +t^2a_1, \dots, x_{n-1}+t^2a_{n-1}, a_nt^2, \sqrt{a_n}t)
\end{displaymath}
for some smooth functions $a_i$ with $a_n\ne 0$ near $0$, meaning $\Phi$ is a smooth map near $0$. It is a diffeomorphism near $0$ since $\Phi$ restricted to the fold, $\{t=0\}$, is the identity map and $d\Phi_0(\frac{\partial}{\partial t}) = \sqrt{a_n(0)}\ne 0$. Thus, $d\Phi_0$ is an isomorphism and there exists a neighborhood of $0$ on which it is a diffeomorphism since it is a strata-preserving map.
Our second claim is that $\Phi$ leads to a normal form and factorization of $f$ as a diffeomorphism composed with a fold map. By definition of $\Phi$, if we apply the map $\gamma(z,t)=(z,t^2)$ on the left, we get:
\begin{displaymath}
\gamma \circ \Phi = \phi^{-1} \circ f
\end{displaymath}
where $\phi(\vec{x},t)=(\vec{x},0) + tF(\vec{x},0)$ is a diffeomorphism. Since $\gamma$ has fold singularities and $\Phi$ is a diffeomorphism, $\gamma \circ \Phi$ is a map with fold singularities, hence $f=\phi \circ (\gamma \circ \Phi)$ is a factorization of $f$ into a diffeomorphism composed with a fold map, which is folded by corollary \ref{cor:folds-diffeo}. If we precompose with $\Phi^{-1}$, we get:
\begin{displaymath}
(f\circ \Phi^{-1})(\vec{x},t) = (\phi\circ \gamma)(\vec{x},t) = (\vec{x},0) + t^2F(\vec{x},0)
\end{displaymath}
which proves the proposition.
\end{proof}
\begin{example}\label{ex:stdfoldmap}
The map $f:\R^n\times \R \to \R^n \times \R$ given by $f(\vec{x},t)=(\vec{x},t^2)$ is a (equidimensional) map with fold singularities. This example of a fold map is, in some sense, the \emph{only} example. By proposition \ref{prop:folds5}, one can write every fold map as $\psi(\vec{x},t)=(\vec{x},0) + t^2F(\vec{x})$, where $F$ is transverse to the image of $d\psi$ at $\{t=0\}$. Furthermore, if the map $\psi$ is strata-preserving then, as we saw in the proof of \ref{prop:folds5}, $F(\vec{x})$ may be viewed as a vector tangent to the stratum containing $\psi(\vec{x},t)$. Thus, we may define a new coordinate on the target space near $\psi(\vec{x},t)$ using $F(\vec{x})$. Then the map $\psi$ has the form $\psi(\vec{x},t)=(\vec{x},t^2)$.
\end{example}
\begin{example}\label{ex:foldspheres}
Let $S^2$ be the $2$-sphere embedded into $\R^3$ as the level set $x^2+y^2+z^2=1$. Then the projection map $p:S^2 \to \R^2$ given by $p(x,y,z)=(x,y)$ has fold singularities along the equator $S^1\subset S^2$. For example, near the point $(1,0,0)$, the map looks like the composition:
\begin{displaymath}
(y,z) \to (\sqrt{1-y^2-z^2},y,z) \to (\sqrt{1-y^2-z^2},y)
\end{displaymath}
The determinant of the differential of this map is $\displaystyle \frac{-z}{\sqrt{1-y^2-z^2}}$, which vanishes transversally at $z=0$. The kernel of the differential of the map is given by $\displaystyle \frac{\partial}{\partial z}$, which is transverse to the equator. By corollary \ref{cor:folds4-0}, $p$ is a map with fold singularities along $S^1$. One may perform the same construction for each sphere $S^n$ so that every sphere admits a fold map into $\R^n$ with folding hypersurface given by the equator $S^{n-1}$. Note that one may also perform the same construction for closed surfaces of genus $g$.
\end{example}
\begin{example}
Let $W\subset \R^2$ be the half space $W=\{(x,y)\vert y\ge 0\}$ and consider the map $f:W \to \R^2$ given by $f(x,y)=(x,y^2)$. As we discussed in the beginning of the chapter, this map has fold singularities in the traditional sense. However, it does not have fold singularities according to definition \ref{def:folds3}. According to corollary \ref{cor:folds4-0}, we can compute the determinant of $df$ to determine if $f$ has fold singularities. The determinant is $2y$, which vanishes along the boundary $y=0$. Thus, $\det(df)$ is not transverse (in the sense of manifolds with corners) to $0$ and so it does not have fold singularities.
\end{example}
\subsection{Example: Generalizing Morse Functions to Fiber Bundles}
\subsubsection{What is a Morse Function?}
We give a definition of a Morse function and relate it to $1$-jets for the purpose of generalizing Morse functions to sections of fiber bundles with $1$-dimensional fibers. We do not give the standard definition where one defines the Hessian and requires it to be non-degenerate at critical points. Instead, we give an equivalent, geometric formulation which will facilitate our discussion.
\begin{remark}
Let us assume throughout this section that all manifolds and fiber bundles are without boundary, hence they are simply $C^{\infty}$ manifolds. We will use the term \emph{manifold} to mean a manifold without boundary.
\end{remark}
\begin{definition}\label{def:Morse}
Let $M$ be a manifold (without corners) and let $\mathcal{O}\in \Gamma(T^*M)$ be the zero section. We say a smooth map $f:M\to \R$ is a \emph{Morse function} if $df \pitchfork \mathcal{O}$ as a map $df:M \to T^*M$.
\end{definition}
We now discuss how this definition relates to our study of jet bundles. Let $E=M\times \R$ and let \newline $\pi:E\to M$ be the trivial fiber bundle over $M$ with standard flat connection. That is, the horizontal bundle is $TM \oplus 0 \subset TM \times T\R$ and the vertical projection $p_V:TM \times T\R \to T\R$ is $p_V(X,Y) = Y$. The first jet bundle $J^1(E)$ is $\hom(TM,T\R)$ (q.v. example \ref{ex:products}), which is canonically isomorphic to $T^*M \times \R$ as a fiber bundle over $M\times \R$. A section $\phi(m)=(m,f(m))$ is the graph of a function $f:M\to \R$ and its $1$-jet $j^1\phi$ at $m$ is $j^1\phi(m)=(m,df_m,f(m))$ in $T^*M \times \R$.
The submanifold $S_r\subset J^1(E)$ of jets of corank $1$ is empty for $r\ge 1$ since $\operatorname{corank}(df)=0,1$. The submanifold $S_1\subset J^1(E)$ is exactly the zero section of $T^*M \times \R \to M\times \R$:
\begin{displaymath}
\operatorname{corank}(j^1_m\phi) = 1 \iff \mbox{ } \operatorname{corank}(p_V(d\phi_m)) = 1 \iff \mbox{ } \operatorname{corank}(df_m)=1 \iff df_p=0
\end{displaymath}
Recall, we may view $1$-jet fields on $E$ as connections on $E$ by proposition \ref{prop:C1}. We claim that the zero section $\chi:M \times \R \to T^*M \times \R$ of the first jet bundle is exactly the standard flat connection on $M\times \R$:
\begin{displaymath}
\begin{array}{lcl}
(X,Y)\in T_mM \times T_t\R \text{ is horizontal} & \iff & \text{for $\phi(m)=(m,f(m))$ representing $\chi(m,t)$}\\
& & \text{we have $d\phi_m\circ d\pi_{(m,t)}(X,Y)=(X,Y)$} \\
& \iff & d\phi_m(X)=(X,Y)\\
& \iff & (X,df_m(X))=(X,Y) \\
& \iff & Y = df_m(X) = 0 \text{ (since $\chi$ is the zero section, we have $df_m=0$) }\\
& \iff & (X,Y)=(X,0) \\
& \iff & \text{The horizontal bundle is $\pi^*TM\oplus 0$.}
\end{array}
\end{displaymath}
By our definition of a Morse function, we have:
\begin{displaymath}
\begin{array}{lcl}
\text{$f$ is Morse} & \iff & df \pitchfork \mathcal{O}, \text{ where $\mathcal{O}$ is the zero section of $T^*M$,} \\
& \iff & j^1\phi \pitchfork \mathcal{O}\times \R, \text{ where $\phi(m)=(m,f(m))$} \\
& \iff & j^1\phi \text{ is transverse to the zero section, $\chi$, of $J^1(M\times \R)$} \\
& \iff & j^1\phi \pitchfork \chi \text{ (just a restatement of the previous line)}
\end{array}
\end{displaymath}
Thus, a Morse function $f:M\to \R$ may be viewed as a section (graph) $\phi:M \to M\times \R$ whose $1$-jet is transverse to the standard flat connection on $M\times \R \to M$, viewed as either a section of $\pi_{1,0}:J^1(M\times \R) \to M\times \R$ or a submanifold of $J^1(E)$. This leads us to a possible definition of what a Morse section of a fiber bundle should be, where the fiber has dimension $1$.
\subsubsection{$\chi$-Morse Functions and Sections}
\begin{remark}
Again, we are assuming all manifolds appearing in this section are manifolds without boundary.
\end{remark}
\begin{definition}\label{def:Morse1}
Let $\pi:E \to M$ be a fiber bundle with typical fiber $F$, $\dim(F)=1$, and a connection $\chi$, which we view as a section $\chi:E \to J^1(E)$ by proposition \ref{prop:C1}. A local section $(U,\phi)$ is $\chi$-Morse if $j^1\phi \pitchfork_s \chi$. That is, the $1$-jet of $\phi$ is transverse to the connection $\chi$.
\end{definition}
\begin{definition}\label{def:Morse2}
Let $\pi:M\times \R \to M$ be the trivial fiber bundle with a connection $\chi$, which we view as a section $\chi:M\times \R \to J^1(M\times \R)$. Then a $\chi$-\emph{Morse function} $f:M\to \R$ is a function such that $\phi_f(m)=(m,f(m))$ is a $\chi$-Morse section of $M\times \R$.
\end{definition}
We have a nice geometric interpretation of what it means to be $\chi$-Morse.
\begin{lemma}\label{lem:Morse1}
Let $\pi:E \to M$ be a fiber bundle with typical fiber $F$, $\dim(F)=1$, and a connection $\chi:E \to J^1(E)$. Let $p_V:TE \to V$ be the projection onto the vertical bundle. Let $(U,\phi)$ be a local section of $E$. Then,
\begin{displaymath}
\phi \pitchfork \chi \mbox{ } \iff \mbox { } p_V(d\phi) \pitchfork \mathcal{O}
\end{displaymath}
where $\mathcal{O}\in \Gamma(\hom(\pi^*TM,V))$ is the zero section.
\end{lemma}
\begin{proof}
Proposition \ref{prop:C2} shows that the connection $\chi$ induces an isomorphism of fiber bundles:
\begin{displaymath}
\xymatrixcolsep{4pc}\xymatrixrowsep{3pc}\xymatrix{
J^1(E) \ar[r]^{F_{\chi}} \ar[d]^{\pi_{1,0}} & \hom(\pi^*TM, V) \ar[d] \\
E \ar[d]^{\pi} \ar@/^/[u]^{\chi} \ar[r]^{id}& E \ar[d]^{\pi} \\
M \ar@/^2pc/[uu]^{j^1\phi} \ar[r]^{id} & M \ar@/_2pc/[uu]_{p_V(d\phi)}
}
\end{displaymath}
where $F_{\chi}(j^1_m\phi)=p_V(d\phi_m)$. Furthermore, $F_{\chi}\circ \chi$ is the zero section by proposition \ref{prop:C3}. Thus, $j^1\phi \pitchfork \chi$ if and only if $F_{\chi}(j^1\phi) \pitchfork \mathcal{O}$ if and only if $p_V(d\phi) \pitchfork \mathcal{O}$.
\end{proof}
We may now make sense of a critical point.
\begin{definition}\label{def:Morse3}
Let $\pi:E \to M$ be a fiber bundle with typical fiber $F$, $\dim(F)=1$, and a connection $\chi: E \to J^1(E)$. Let $p_V:TE \to V$ be the projection onto the vertical bundle. Let $(U,\phi)$ be a local section. We say $m\in M$ is a \emph{$\chi$-critical point} of $M$ if $p_V(d\phi_m)=0$. Equivalently, $\operatorname{corank}(j^1_m\phi)=1$, hence $j^1_m\phi$ intersects $S_1$.
\end{definition}
\begin{example}
Let $M=\R$ and let $E=\R \times \R$ with coordinates $(s,t)$ and let $\chi$ be the connection defined by the vertical projection:
\begin{displaymath}
\begin{array}{c}
p_V:T\R \times T\R \to T\R \\
\displaystyle p_V(a\frac{\partial}{\partial s}, b\frac{\partial}{\partial t}) = (b-a)\frac{\partial}{\partial t}
\end{array}
\end{displaymath}
with corresponding horizontal projection $\displaystyle p_H(a\frac{\partial}{\partial s}, b\frac{\partial}{\partial t}) = a\frac{\partial}{\partial s} + a\frac{\partial}{\partial t}$. A section $\phi(m)=(m,f(m))$ is $\chi$-Morse if $p_V(d\phi)=(df-ds)$ is transverse to the zero section $\iff$ $(\frac{\partial f}{\partial s} -1) \pitchfork_s 0$ $\iff$ the slope of the graph of $f$ passes through $1$ transversally, which is true if and only if $\frac{\partial^2 f}{\partial^2 s} \ne 0$ at $\chi$-critical points.
Note that the horizontal distribution is involutive, hence the Frobenius theorem allows us to to integrate this distribution and obtain smooth submanifolds parameterized by the fiber $\R$, $S_c\subset \R \times \R$, where $c\in \R$. These submanifolds are simply the graphs of $f(s)=s+c$, $c\in \R$. At a $\chi$-critical point $p$ of a Morse section $\phi$, the section becomes tangent to $S_c$ for some $c$. Since $\frac{\partial^2 f}{\partial^2 s}(p) \ne 0$, the section $\phi$ is concave up or concave down, meaning it locally looks like the parabola $x^2$ intersecting $f(x)=x$ at $x=1$ or the parabola $-x^2$ intersecting $f(x)=x$ at $x=-1$.
\end{example}
\begin{example}
Building on the previous example, we may consider a flat connection $\chi$ on fiber bundle $\pi:E \to M$ with fiber $F$, $\dim(F)=1$. Then the horizontal bundle $H$ is involutive and we may integrate it to obtain a foliation of $E$ by submanifolds $S_p$, parameterized by $p\in F$, of dimension $\dim(M)$ transverse to the fibers of $\pi$ whose tangent spaces are exactly the horizontal subspaces.
A local section $(U,\phi)$ of $E$ is $\chi$-Morse if whenever $\phi(U)$ is tangent to $S_p$, its vertical differential $p_V(d\phi)$ vanishes transversely in all directions. Hence, in any direction through a $\chi$-critical point, $\phi$ will look like a the graph of a parabola intersecting a submanifold above or below it tangentially.
\end{example}
\subsubsection{$\chi$-Morse Functions are generic}
There are several questions one can, and probably should, ask about $\chi$-Morse functions $f:M\to \R$, where $\chi$ is a connection on $M\times \R$:
\begin{enumerate}
\item First and foremost, how useful are they? For a choice of connection $\chi$, do we get a homology theory using the $\chi$-critical points of $f$ as we do in the standard case of a Morse function?
\item To this end, how do we define the index of a $\chi$-critical point $p\in M$? Presumably, the connection $\chi$ will allow us to produce a non-degenerate quadratic form on $T_pM$ whose index will be the index of the $\chi$-critical point. But, how will this work and what will it tell us?
\item And, if one is to produce a homology theory from a $\chi$-Morse function, then one ought to be able to show $\chi$-Morse functions \emph{exist}. Do they exist? And, are they generic?
\end{enumerate}
We address the last question. To this end, we will need the Thom Transversality Theorem, which is Theorem 4.9 in \cite{GG}. The general statement involves $k$-jets: we will only provide the statement in the case where $k=1$, but the reader should note that Thom proved a more powerful theorem than the one we give.
\begin{theorem}\label{thm:Thom}
Let $M$ and $N$ be smooth manifolds (without corners), let $M\times N \to M$ be the trivial fiber bundle, and for each $f\in C^{\infty}(M,N)$ let $\phi_f(m)=(m,f(m))$ denote the induced section of $M\times N$. Let $W \subset J^1(M\times N)$ be a smooth submanifold of the first jet bundle and let:
\begin{displaymath}
T_W = \{f\in C^{\infty}(M,N) \vert \mbox{ } j^1\phi_f \pitchfork W\}
\end{displaymath}
Then $T_W$ is a residual subset of $C^{\infty}(M,N)$ in the $C^{\infty}$ topology and it is open and dense if $W$ is a closed subset.
\end{theorem}
Almost for free, we obtain:
\begin{prop}\label{prop:Morse}
Let $\pi:M \times \R \to M$ be the trivial fiber bundle over $M$ and let $\chi: M\times \R \to J^1(M\times \R)$ be a connection on $M\times \R$. Then the set:
\begin{displaymath}
C^{\infty}_{\chi}(M) = \{f\in C^{\infty}(M) \vert \mbox{ } \text{f is $\chi$-Morse} \}
\end{displaymath}
is an open, dense subset of $C^{\infty}(M)$.
\end{prop}
\begin{proof} \mbox{ } \newline
Let $W$ be the submanifold $\chi(M\times \R)$ of $J^1(M\times \R)$. $W$ is then a closed submanifold and $f$ is $\chi$-Morse if and only if the induced section $\phi_f$ satisfies $j^1\phi_f \pitchfork \chi$ $\iff$ $j^1\phi_f \pitchfork W$. Therefore, the set $T_W$ described in the hypotheses of the Thom transversality theorem is exactly the set $C^{\infty}_{\chi}(M)$. Since $W$ is closed, $T_W$ is open and dense in $C^{\infty}(M)$. We therefore have that $C^{\infty}_{\chi}(M)\subset C^{\infty}(M)$ is an open dense subset.
\end{proof}
As a last exercise, we show that if $f\in C^{\infty}(M)$, then it is $\chi$-Morse for some connection $\chi$ on $M\times \R$.
\begin{lemma}\label{lem:Morse2}
Let $M$ be a smooth manifold (without corners) and let $f\in C^{\infty}(M)$ be a smooth function. Then there exists a connection $\chi$ on the bundle $M\times \R \to M$ so that $f$ is $\chi$-Morse.
\end{lemma}
\begin{proof}
Given a smooth function $f:M \to \R$, its differential $df$ defines a section of $T^*M$ which we may use to define a connection $\chi_f$ on $M\times \R$. Recall that $J^1(M\times \R) = T^*M \times \R$ as a bundle over $M\times \R$. The connection $\chi_f:M\times \R \to T^*M \times \R$ is then given as:
\begin{displaymath}
\chi_f(m,t) = (m,df_m, t)
\end{displaymath}
By proposition \ref{prop:Morse} there exists a $\chi_f$-Morse function $g:M \to \R$, meaning, if $\phi_g$ is the graph of $g$, $j^1\phi_g \pitchfork \chi_f$, but this is true if and only if $dg \pitchfork df$ as sections of $T^*M$. Using this fact, we define a connection:
\begin{displaymath}
\chi_g(m,t)=(m,dg_m,t)
\end{displaymath}
and note that $df \pitchfork dg$ if and only if $j^1\phi_f \pitchfork_s \chi_g$ if and only if $f$ is $\chi_g$-Morse.
\end{proof}
Thus, with our new definition of $\chi$-Morse, every function may be realized as a $\chi$-Morse function for some $\chi$ and therefore has a home inside a generic subset of $C^{\infty}(M)$.
\section{Folded Symplectic Manifolds}
We introduce the notion of a folded-symplectic manifold and show how they arise in a very fundamental way by dualizing a standard construction in $b$-symplectic geometry. Some constructions will involve manifolds with corners while others will use manifolds without boundary: we will always explicitly state which type of manifold we are using in the definitions, lemmas, and propositions. Our main goals in this section, listed in order of desirability, are:
\begin{enumerate}
\item to develop a normal form for the folding hypersurface inside a folded-symplectic manifold,
\item to show that being folded-symplectic is equivalent to inducing an isomorphism of sheaves $\sigma^\#: \Gamma(TM) \to S$, where $S$ is a distinguished sheaf of $1$-forms on $M$, discussed below, and finally
\item to develop a Moser-type argument for deformations of folded-symplectic structures.
\end{enumerate}
Of course, we will see that these goals are all related: we will need a Moser argument to develop the normal form for the fold and, to this end, we will need to understand when one can solve Moser's equation, which is intimately related to the second goal.
\subsection{Definition and Examples}
\begin{definition}\label{def:fsform}
Let $M$ be a $2m$-dimensional manifold with corners. We say $\sigma\in\Omega^2(M)$ is \emph{folded-symplectic} if
\begin{enumerate}
\item $d\sigma = 0$
\item $\sigma^m \pitchfork_s \mathcal{O}$, where $\mathcal{O} \subset \Lambda^{2m}(T^*M)$ is the zero section, hence $Z=(\sigma^m)^{-1}(\mathcal{O})$ is a codimension $1$ submanifold with corners intersecting the strata of $M$ transversally.
\item If $i_Z:Z\hookrightarrow M$ is the inclusion, $i_Z^*\sigma$ has maximal rank, $2m-2$.
\end{enumerate}
We say $(M,\sigma)$ is a \emph{folded-symplectic manifold} with corners and we call $Z\subset M$ the \emph{fold} or the \emph{folding hypersurface}.
\end{definition}
\begin{definition}\label{def:fsmaps}
Let $(M,\sigma_1)$, $(N,\sigma_2)$ be two folded-symplectic manifolds with corners. We say a smooth map $\phi:M\to N$ is \emph{folded-symplectic} if $\phi^*\sigma_2 = \sigma_1$. If $\phi$ is a diffeomorphism, we say it is a \emph{folded-symplectomorphism}.
\end{definition}
\begin{definition}\label{def:nullbundles}
Let $(M,\sigma)$ be a folded-symplectic manifold with corners and let $Z\subset M$ be the folding hypersurface of $\sigma$ with inclusion $i_Z:Z\hookrightarrow M$. Assume that $Z$ is nonempty. We define two vector subbundles of $i_Z^*TM$:
\begin{enumerate}
\item $\ker(\sigma) \to Z$ is $2$-plane bundle over $Z$ whose fiber at a point $z\in Z$ is:
\begin{displaymath}
\ker(\sigma_z) = \{v \in T_zZ \vert \mbox{ } i_v\sigma_z=0\}
\end{displaymath}
\item $\ker(i_Z^*\sigma) \to Z$ is the rank $1$ vector bundle over $Z$ defined to be the intersection $\ker(i_Z^*\sigma):= \ker(\sigma) \cap TZ$. It may be viewed as a vector subbundle of $i_Z^*TM$ or $TZ$.
\end{enumerate}
\end{definition}
\begin{remark}
Note that since $\ker(i_Z^*\sigma)$ is a rank $1$ vector subbundle of $TZ$, it is trivially involutive. Hence, by the Frobenius theorem, it defines an integrable distribution on $Z$ and we obtain a foliation of $Z$ by $1$-dimensional leaves.
\end{remark}
\begin{definition}\label{def:nullfibration}
Let $(M,\sigma)$ be a folded-symplectic manifold with corners and let $Z\subset M$ be the folding hypersurface of $\sigma$. Assume that $Z$ is nonempty and let $\ker(i_Z^*\sigma)$ be the rank $1$ vector subbundle of $TZ$ of definition \ref{def:nullbundles}. Let $\mathcal{F}$ be the foliation of $Z$ induced by this bundle. We refer to this foliation as the \emph{null-foliation}.
\end{definition}
\begin{example}\label{ex:fs1}
Let $(M,\omega)$ be a symplectic manifold (with or without corners). It is trivially folded-symplectic with folding hypersurface $Z=\emptyset$. Since $Z$ is empty, the bundles $\ker(\sigma)$ and $\ker(i_Z^*\sigma)$ are not defined, though one could view them as just empty sets. Consequently, there is no null-foliation to consider.
\end{example}
\begin{example}\label{ex:fs2}
Let $(M^{2n},\omega)$ be a symplectic manifold with corners and let $\psi:N^{2n} \to M^{2n}$ be a map with folds. Then $\sigma=\psi^*\omega$ is a folded-symplectic form on $M$. To see why this is true, note that in any choice of coordinates near $p\in M$ $\det(d\psi) \pitchfork_s 0$ by corollary \ref{cor:folds4-0}, hence in any choice of coordinates coordinates $(\psi^*\omega)^n = \det(d\psi) \omega^n \pitchfork_s \mathcal{O}$. This computation also shows that the folding hypersurface $Z$ of $\psi$ is the folding hypersurface of $\psi^*\omega$. Furthermore:
\begin{enumerate}
\item The bundle $\ker(\sigma)$ has fiber: $\ker(\sigma_z) = (d\psi_z)^{-1}(d\psi_z(T_zM)^{\omega})$.
\item In particular, it contains $\ker(d\psi_z)$. Since $\ker(d\psi_z) \pitchfork T_zZ$ and $Z$ is a hypersurface, we may write:
\begin{displaymath}
\ker(\sigma_z) = (d\psi_z)^{-1}(d\psi_z(T_zZ)^{\omega})
\end{displaymath}
Since $Z$ is a hypersurface and $d\psi_z\big\vert_{TZ}$ is injective, $d\psi_z(T_zZ)$ is a codimension $1$ subspace of $T_{\psi(z)}M$, meaning it is co-isotropic. Therefore, the $1$-dimensional subspace $d\psi_z(T_zZ)^{\omega}$ is contained in $d\psi_z(T_zZ)$ and has a unique $1$-dimensional preimage in $T_zZ$, which shows that $i_Z^*(\psi^*\omega)$ has a $1$-dimensional kernel at any point $z\in Z$, meaning it has maximal rank. This verifies the third condition of definition \ref{def:fsform}.
\item Since $\omega$ is closed, $\psi^*\omega$ is closed, which completes the verification of all three conditions in definition \ref{def:fsform}.
\end{enumerate}
\end{example}
\begin{example}\label{ex:fs3}
Let $\sigma \in \Omega^2(\R^{2n})$ be defined by
\begin{displaymath}
\sigma = x_1dx_1\wedge dx_2 + dx_3\wedge dx_4 + \dots + dx_{2n-1}\wedge dx_{2n}
\end{displaymath}
Then $\sigma$ is folded-symplectic with fold $Z$ defined by $x_1=0$.
\begin{enumerate}
\item The bundle $\ker(\sigma)\to Z$ is framed by the vector fields $\displaystyle \{\frac{\partial}{\partial x_1}, \frac{\partial}{\partial x_2}\}$ along $Z$.
\item The bundle $\ker(i_Z^*\sigma)$ is framed by the vector field $\displaystyle \frac{\partial}{\partial x_2}$ along $Z$.
\end{enumerate}
\end{example}
\begin{example}\label{ex:fs4}
Let $\pi: S^{2n} \to \R^{2n}$ be the projection $\pi(\vec{x},z) =\vec{x}$ and let $\omega_{\R^{2n}}$ be the standard symplectic form on $\R^{2n}$. $\pi$ is a map which folds along the equator $S^{2n-1}$, hence $\sigma=\pi^*\omega_{\R^{2n}}$ is a folded-symplectic form on $S^{2n}$ by example \ref{ex:fs2}. \begin{enumerate}
\item The bundle $\ker(\sigma) \to S^{2n-1}$ is spanned by $\displaystyle \frac{\partial}{\partial z}$ (the kernel of $d\pi$ at the equator) and the vector field(s) induced by the diagonal action of $S^1$ on $S^{2n-1}$, $\lambda\cdot (z_1, \dots, z_n) = (\lambda z_1, \dots , \lambda z_n)$, where $z_i$ are complex numbers. This is because the kernel of the restriction of $\omega_{\R^{2n}}$ to $S^{2n-1}$ is spanned by this vector field.
\item Consequently, the null foliation on the fold $Z=S^{2n-1}$ is given by the orbits of the diagonal $S^1$ action on $S^{2n-1}$.
\end{enumerate}
\end{example}
\begin{example}\label{ex:fs5}
For $m>0$, let $G=\mathbb{Z}/2\mathbb{Z}$ act on $\R^{m}$ by reflection: $\vec{x} \to -\vec{x}$. Then the projection $\pi:S^{2n} \to \R^{2n}$ of example \ref{ex:fs4} is $G$-equivariant. Since $\omega_{\R^{2n}}$ is $G$-invariant, $\sigma=\pi^*\omega_{\R^{2n}}$ is $G$-invariant. Let $p:S^{2n} \to \R P^{2n} (=S^{2n}/G)$ be the orbit map. Since $\sigma$ is $G$-invariant, we have $\sigma=p^*\bar{\sigma}$ for some $\bar{\sigma}\in \Omega^2(\R P^{2n})$. Since $p$ is a covering map, it is a local diffeomorphism and $\bar{\sigma}$ has fold singularities along $\R P ^{2n-1}$.
\begin{enumerate}
\item The null-foliation on $\R P^{2n-1}$ is given by the image of the orbits of the diagonal action of $S^1$ on $S^{2n-1}$.
\item The null direction transverse to $\R P^{2n-1}$ at a point $[(\vec{x},0)]$ is given by $dp_{(\vec{x},0)}(\frac{\partial}{\partial z})$.
\end{enumerate}
\end{example}
The following lemma demonstrates how one may construct a folded-symplectic manifold from an odd-dimensional manifold equipped with a closed $2$-form of maximal rank. We will also see how this lemma allows us to decide when some $2$-forms are folded-symplectic or not.
\begin{lemma}\label{lem:symplectize}
Let $Z^{2n-1}$ be an odd-dimensional manifold with corners. Let $\omega \in \Omega^2(Z)$ be a closed $2$-form of maximal rank $2n-2$ and let $\ker(\omega)\to Z$ be the bundle whose fiber is $\ker(\omega)_z:= \{v\in T_zZ \vert \mbox{ } i_v\omega=0\}$. Let $\mu \in \Omega(Z\times \R)$ and let $p:Z\times \R \to Z$ be the projection. Then
\begin{displaymath}
\sigma :=p^*\omega + t\mu
\end{displaymath}
is folded-symplectic in a neighborhood of $Z\times \{0\}$ with fold $Z\times \{0\}$ if and only if for all $z\in Z$ we have $\mu\big\vert_{\ker{p^*\omega}}$ is nondegenerate in a neighborhood of $Z\times \{0\}$.
\end{lemma}
\begin{proof}
The top power of $\sigma$ is:
\begin{displaymath}
\sigma^n = t(p^*\omega)^{n-1} \wedge \mu + t^2 \beta
\end{displaymath}
where $\beta \in \Omega^{2n}(Z\times \R)$ is some $2n$-form, which shows that $\sigma^n$ vanishes at $Z\times \{0\}$. We have $\sigma^n \pitchfork_s \mathcal{O}$, where $\mathcal{O}$ is the zero section, if and only if $\mu$ doesn't vanish on the directions on which $p^*\omega$ vanishes. That is, $\mu$ doesn't vanish on $\ker(p^*\omega)$ at points of $Z\times \{0\}$, hence it doesn't vanish on $\ker(p^*\omega)$ in a neighborhood of $Z\times \{0\}$. In particular, ince $p^*\omega$ vanishes on $\frac{\partial}{\partial t}$ and $p^*\ker(\omega)$, we have:
\begin{displaymath}
\sigma^n \pitchfork_s \mathcal{O} \mbox{} \iff \text{ $\mu_{(z,0)}(\frac{\partial}{\partial t}, v)\ne 0$ for all $v\in \ker(\omega_z)$}
\end{displaymath}
By definition of $\sigma$, its restriction to the hypersurface $Z\times \{0\}$ has maximal rank, since it is essentially $\omega$ on $Z$, which shows that it is folded-symplectic if and only if the hypotheses of the lemma are satisfied.
\end{proof}
\begin{cor}\label{cor:symplectize}
Let $Z^{2n-1}$ be a manifold with corners and suppose $\sigma \in \Omega^2(Z\times \R)$ is folded-symplectic with fold $Z\times \{0\}$. Furthermore, suppose $\ker(\sigma)$ contains the subbundle framed by $\frac{\partial}{\partial t}$ along $Z\times \{0\}$. Let $i:Z \to Z\times \R$ be the inclusion as the zero section and let $p:Z\times \R \to Z$ be the projection. Then:
\begin{enumerate}
\item $\sigma = p^*i^*\sigma + t\mu$ for some $2$-form $\mu\in \Omega^2(Z\times \R)$.
\item $\mu\vert_{\ker(\sigma)}$ is non-degenerate.
\end{enumerate}
\end{cor}
\begin{proof}
\begin{enumerate}
\item Consider the difference $\sigma -p^*i^*\sigma$. Since both $p^*i^*\sigma$ and $\sigma$ vanish on $\frac{\partial}{\partial t}$ along $Z\times \{0\}$, this difference vanishes at $Z\times \{0\}$. Thus, $\sigma - p^*i^*\sigma = t\mu$ for some $2$-form $\mu\in \Omega^2(Z\times \R)$ by lemma \ref{lem:had}, which shows $\sigma = p^*i^*\sigma + t\mu$.
\item $(Z,i^*\sigma)$ is an odd-dimensional manifold with a closed $2$-form of maximal rank. By lemma \ref{lem:symplectize}, $\sigma$ is folded-symplectic with fold $Z\times \{0\}$ if and only if $\mu$ doesn't vanish on $\ker(p^*i^*\sigma)$ at $Z\times \{0\}$. Since $\ker(p^*i^*\sigma)$ is spanned by $\frac{\partial}{\partial t}$ and elements of $\ker(i^*\sigma)$ along $Z\times \{0\}$, we have that $\ker(p^*i^*\sigma)\big\vert_{Z\times \{0\}} = \ker(\sigma)$. Since $\sigma$ is folded, we must have that $\mu\big\vert_{\ker{\sigma}}$ is non-degenerate.
\end{enumerate}
\end{proof}
\begin{example}\label{ex:symplectize1}
Consider $S^1$ equipped with the $2$-form $0\in \Omega^2(S^1)$, which is a closed $2$-form of maximal rank. Then the cylinder $S^1\times \R$ may be given a fold structure using $\sigma = 0 + t(2dt\wedge d\theta) = d(t^2d\theta)$.
\end{example}
\begin{example}\label{ex:symplectize2}
Take any contact manifold $(Z,\alpha)$, where $\alpha$ is a contact $1$-form on $Z$. Let $p:Z\times \R \to Z$ be the projection. Then:
\begin{displaymath}
\sigma := p^*(d\alpha) + t(2dt\wedge p^*\alpha) + t^2p^*d\alpha = d(1+t^2)(p^*\alpha)
\end{displaymath}
is folded-symplectic in a neighborhood of $Z\times \{0\}$ with folding hypersurface $Z\times \{0\}$.
\end{example}
\begin{example}\label{ex:symplectize3}
To generalize example \ref{ex:symplectize2}, consider any oriented, odd-dimensional manifold $Z$ with a closed $2$-form $\omega$ of maximal rank. Since $Z$ is oriented, the bundle $\ker(\omega)$ is oriented, meaning we may choose a non-vanishing $1$-form $\alpha \in \Omega^1(Z)$ that doesn't vanish on $\ker(\omega)\to Z$. Then $\sigma = p^*\omega + d(t^2p^*\alpha)$ is folded-symplectic in a neighborhood of $Z\times \{0\}$ since $dt\wedge p^*\alpha$ is non-degenerate on $\ker(p^*\omega)$ along $Z\times \{0\}$.
\end{example}
\begin{remark}
We will show that if $(M,\sigma)$ is a folded-symplectic manifold with folding hypersurface $Z\subset M$, then $Z$ is canonically oriented by $\sigma$, hence $(Z,i_Z^*\sigma)$ is an odd-dimensional, oriented manifold with a closed $2$-form of maximal rank. Thus, \emph{every} folding hypersurface ever conceived is the type of hypersurface discussed in example \ref{ex:symplectize3}. Furthermore, each such hypersurface admits a symplectization and, by example \ref{ex:symplectize3}, a folded-symplectization. We will see that neighborhoods of co-orientable folding hypersurfaces look like the symplectization of the folding hypersurface pulled back by a folding map.
\end{remark}
We will now give some more utility to the constructions of examples \ref{ex:symplectize1}, \ref{ex:symplectize2}, and \ref{ex:symplectize3} by providing a partial normal form for a neighborhood of the fold $Z$ of a folded-symplectic manifold $(M,\sigma)$ in the case that $Z$ is co-orientable. We will need this normal form when we discuss vector bundle and sheaf-theoretic characterizations of folded-symplectic forms.
\begin{lemma}\label{lem:fsnormal}
Let $(M,\sigma)$ be a folded-symplectic manifold with corners and suppose for all $p\in Z$ $\ker(\sigma_p)$ is tangent to the stratum of $M$ containing $p$. Furthermore, suppose $Z$ is co-orientable. Then there exists a a neighborhood $U$ of $Z$, a neighborhood $V$ of the zero section of $Z\times \R$, and an isomorphism $\phi:V \to U$ so that:
\begin{displaymath}
\phi^*\sigma = p^*i^*\sigma + t\mu
\end{displaymath}
where $i:Z \hookrightarrow Z\times \R$ is the inclusion as the zero section, $p:Z\times \R \to Z$ is the projection, and $\mu\in \Omega^2(U)$ is some $2$-form satisfying $\mu\big\vert_{\ker(\phi^*\sigma)}$ is non-degenerate.
\end{lemma}
\begin{remark}\label{rem:whatismu}
We will use \ref{lem:fsnormal} to construct the induced orientation on the fold, first discovered by Martinet in \cite{M}, and we will also use it when we develop a sheaf-theoretic characterization of the notion of being folded-symplectic. The most important detail we would like to emphasize is that the $2$-form $\mu$ appearing in lemma \ref{lem:fsnormal} functions as the intrinsic derivative of the contraction map $C_{\sigma}:TM \to T^*M$, $C_{\sigma}(p,X)=i_X\sigma_p$, at points of the folding hypersurface. We'll discuss this more in the future.
\end{remark}
\begin{proof}
We will construct $\phi$ by considering the flow of a stratified vector field on $M$ whose values at $Z$ lie inside $\ker(\sigma)$ and whose image under $\phi^{-1}$ will be $\frac{\partial}{\partial t}$. We have a short exact sequence of vector bundles:
\begin{displaymath}
0 \to \ker(i_Z^*\sigma) \to \ker(\sigma) \to \ker(\sigma)/\ker(i_Z^*\sigma) \to 0
\end{displaymath}
The bundle $\ker(\sigma)/\ker(i_Z^*\sigma)$ is canonically isomorphic to the normal bundle $\nu(Z)= i_Z^*TM/ TZ$ of $Z$ via the inclusion $j:\ker(\sigma) \hookrightarrow i_Z^*TM$. The kernel of the inclusion followed by projection to $\nu(Z)$ is exactly $\ker(i_Z^*\sigma)$, hence $j$ descends to an isomorphism $\ker(\sigma)/\ker(i_Z^*\sigma) \simeq \nu(Z)$.
Since $Z$ is co-orientable, $\nu(Z)$ is trivializable. Thus, there exists a non-vanishing section $\bar{w} \in \Gamma(\ker(\sigma)/\ker(i_Z^*\sigma))$ which lifts to a non-vanishing section $w \in \Gamma(\ker(\sigma))$ so that $w(z)\notin T_zZ$ for all $z\in Z$. Since $\ker(\sigma)$ is tangent to the strata of $M$, $w$ is stratified, hence we may extend $w$ to a stratified vector field $\tilde{w}$ on $M$. Since stratified vector fields may be integrated, we may consider the flow $\tilde{\phi}$ of $\tilde{w}$, which is defined on an open subset of $M\times \R$ containing $M\times \{0\}$. In particular, it is defined on an open neighborhood $\tilde{V}$ of the zero section of $Z\times \R$. We may then define:
\begin{displaymath}
\xymatrixcolsep{3pc}\xymatrixrowsep{.5pc}\xymatrix{
\phi: V\subset Z\times \R \ar[r] & M \\
(z,t) \in V \ar[r] & \tilde{\phi}(z,t)}
\end{displaymath}
It is a bijection in a neighborhood of $Z\times \{0\}$ since $\tilde{w}$ is non-vanishing in a neighborhood of $Z\times \{0\}$ and integral curves do not intersect. It is a strata-preserving map since $\tilde{w}$ is a stratified vector field. Hence, it is locally a diffeomorphism of manifolds with corners in a neighborhood of $Z\times \{0\}$ since $d\phi$ has maximal rank at $Z\times \{0\}$:
\begin{enumerate}
\item $\phi\big\vert_{Z\times \{0\}}(z,0)= z$ and
\item $d\phi_{(z,0}(\frac{\partial}{\partial t}) = \tilde{w}(z)$
\end{enumerate}
This means that there is a neighborhood $V\subset \tilde{V}$ of $Z\times \{0\}$ and a neighborhood $U=\phi(V)$ of $Z$ so that $\phi:V \to U$ is a diffeomorphism of manifolds with corners.
\vspace{5mm}
Since $d\phi_{(z,0)}(\frac{\partial}{\partial t})= \tilde{w}(z) \in \ker(\sigma_z)$, $\frac{\partial}{\partial t} \in \ker(\phi^*\sigma)_{(z,0)}$ for all $z\in Z$. By corollary \ref{cor:symplectize}, $\phi^*\sigma = p^*i^*\sigma + t\mu$ for some $2$-form $\mu$ and $\mu\big\vert_{\ker(\phi^*\sigma)}$ is non-degenerate.
\end{proof}
\begin{cor}\label{cor:fsnormal}
Let $(M^{2n},\sigma)$ be a folded-symplectic manifold with corners with folding hypersurface $Z\subset M$. Suppose that for all $z\in Z$ $\ker(\sigma_z)$ is tangent to the stratum of $M$ containing $z$. Then if $Z$ is co-orientable, it is orientable.
\end{cor}
\begin{proof}
Let $p:Z\times \R \to Z$ be the projection and let $i:Z \to Z\times \R$ be the inclusion as the zero section. By lemma \ref{lem:fsnormal}, we may assume a neighborhood of $Z$ is a neighborhood of the zero section $V\subset Z\times \R$ with fold form $p^*i^*\sigma + t\mu$, where $\mu\big\vert_{\ker(p^*i^*\sigma)}$ is non-degenerate in a neighborhood of $Z\times \{0\}$. Consequently, $i^*(i_{\frac{\partial}{\partial t}}\mu)$ is a $1$-form on $Z$ that does not vanish on $\ker(i_Z^*\sigma)$. We may then define an orientation on $Z$ using the form:
\begin{displaymath}
\Omega= (i_Z^*\sigma)^{n-1}\wedge \alpha
\end{displaymath}
\end{proof}
\begin{remark}
We will see that the orientation defined in the proof of corollary \ref{cor:fsnormal} may be canonically defined using the the folded symplectic form $\sigma$ and its kernel bundle $\ker(\sigma) \to Z$. In fact, we will see that if $(M,\sigma)$ is any folded-symplectic manifold, the folding hypersurface $Z\subset M$ is endowed with an orientation induced by $\sigma$, regardless of co-orientability.
\end{remark}
\subsection{Cotangent Bundles and Folded Structures}
While we now have the basic facts and examples of folded-symplectic manifolds established, we are left wondering where one might find a naturally occurring folded-symplectic structure. To answer this question, let us consider for a moment the case of $b$-symplectic structures on manifolds without boundary. A $b$-symplectic manifold is a $2n$-dimensional manifold (without boundary) equipped with a Poisson structure $\Pi \in \Gamma(\Lambda^2(TM))$ such that $\Pi^n \pitchfork \mathcal{O}$, where $\mathcal{O}$ is the zero section of $\Gamma(\Lambda^{2n}(TM))$. Thus, folded-symplectic manifolds are, in a sense, dual to the notion of a $b$-symplectic manifold where we consider $T^*M$ instead of $TM$.
Now, $b$-symplectic structures occur quite naturally (see \cite{GMP,GMPS} for details). Given a manifold without boundary $M$ and a hypersurface $Z\subset M$, one may form the $Z$-tangent bundle $T_ZM$, specified up to isomorphism, whose space of global sections is isomorphic to the space of vector fields $\Gamma_Z(TM)$ that are tangent to $Z$ at points of $Z$. Dualizing, one obtains the $Z$-cotangent bundle $T_Z^*M := (T_ZM)^*$. There is a map $i:T_ZM \to TM$ induced by the inclusion $\Gamma_Z(TM) \hookrightarrow \Gamma(TM)$ into the space of vector fields on $M$. Its dual $i^*$ gives us a map $i^*:T^*M \to T^*_ZM$ and the pushforward of the canonical Poisson structure on $T^*M$ gives us a $b$-symplectic structure on $T^*_ZM$. In what follows, we dualize this construction to produce vector bundles that \emph{always} have a folded-symplectic structure, albeit perhaps non-canonically.
\begin{remark}
We work with manifolds without boundary for the sake of convenience. However, in what follows, one need only assume all vector bundles are stratified subbundles of $TM$ in order to generalize to the case of manifolds with corners. There is one instance where we use a tubular neighborhood theorem for manifolds without boundary (q.v. proof of lemma \ref{lem:Vcotangent3}): it is also true in the case of manifolds with corners but the proof appears to be outside the scope of this document (q.v. \cite{MN} p. 4).
\end{remark}
\subsubsection{Constructing the bundles $T^*_V M$}
\begin{definition}\label{def:Vcotangent}
Let $M$ be a manifold without boundary and let $Z\subset M$ be a closed hypersurface, i.e. a codimension $1$ submanifold of $M$. Let $V$ be a rank $1$ vector subbundle of $i_Z^*TM$ so that for all $z\in Z$ the fiber $V_z$ over $z$ is transverse to $T_zZ$. For each open set $U\subset M$ we define:
\begin{displaymath}
\Omega_V^1(U):= \{\alpha \in \Omega^1(U) \vert \mbox{ } \alpha\vert_V = 0\}
\end{displaymath}
to be the space of all $1$-forms on $U$ vanishing on $V$. If $U\cap Z = \emptyset$, then this is just $\Omega^1(U)$. The restriction maps are defined using the pullbacks $i^*$ by the inclusion maps $i:U \hookrightarrow V$.
\end{definition}
\begin{lemma}\label{lem:Vcotangent1}
Let $M$ be a manifold without boundary, let $Z\subset M$ be a closed hypersurface, and let $V$ be a rank $1$ vector subbundle of $i_Z^*TM$ transverse to $TZ$. Then $\Omega_V^1(\cdot)$ is a sheaf of $C^{\infty}(M)$ modules on $M$.
\end{lemma}
\begin{proof} \mbox{ } \newline
$\Omega_V^1(\cdot)$ is a sub-presheaf of $\Omega^1(M)$, hence we need only check the conditions that ensure it's a sheaf.
\begin{itemize}
\item If $U\subset M$ is open, $\{U_i\}$ is a cover of $U$, and we have $\omega \in \Omega_V^1(U)$ such that $\omega\big\vert_{U_i}=0$ for all $i$, then $\omega=0$ since $\omega$ evaluated at any point $p\in U$ must be zero.
\item Using the same cover, if we are given $\omega_i \in \Omega_V^1(U_i)$ for each $i$ so that $\omega_i\big\vert_{U_i\cap U_j} = \omega_j\big\vert_{U_i\cap U_j}$, then we may choose a partition of unity $\{\psi_i\}$ subordinate to $\{U_i\}$, refining the cover to guarantee local finiteness if necessary, and define $\omega= \sum_i \psi_i \omega_i$. If $p\in Z\cap U$, then for each $v\in V_p$ we have:
\begin{displaymath}
\omega_p(v) = \sum_i (\psi_i\omega_i)(v) = 0
\end{displaymath}
since either $\psi_i(p)=0$ or $(\omega_i)_p(v)=0$. Therefore, the $\omega_i's$ glue together to give a unique section $\omega \in \Omega_V^1(U)$.
\end{itemize}
\end{proof}
\begin{lemma}\label{lem:Vcotangent2}
Let $M$ be a manifold without boundary, $Z\subset M$ a closed hypersurface, and $V$ a rank $1$ vector subbundle of $i_Z^*TM$ transverse to $TZ$. Let $\Omega_V^1(\cdot)$ be the corresponding sheaf of $1$-forms vanishing on $V$ along $Z$. Then $\Omega_V^1(\cdot)$ is a locally free sheaf of $C^{\infty}(M)$ modules. Furthermore, for each $z\in Z$ there exists a neighborhood $U$ of $z$ and coordinates $(x_1,\dots, x_{n-1},t)$ on $U$ so that $\Omega_V^1(U)$ is generated by $\{dx_1,\dots, dx_{n-1}, tdt\}$.
\end{lemma}
\begin{proof}
Let $p\in M$ be a point. If $p\in M\setminus Z$ then there exists a neighborhood $U$ of $p$ such that $U\cap Z = \emptyset$ and an isomorphism $\phi:U\to V$ onto a neighborhood $V$ of the origin in $\R^n$, where $n=\dim(M)$. $\Omega_V^1(U)$ is then generated by the pullbacks of $dx_1,\dots,dx_n$ under $\phi$.
If $p\in Z$, then we may choose a local, nonvanishing section $w$ of $V$ and extend it to a nonvanishing vector field $\tilde{w}$ in a neighborhood $U$ of $p$. Furthermore, we may shrink $U$ so that the intersection $U\cap Z$ has coordinates defined on it. If we shrink $U$ sufficiently and require that its closure is compact, then the flow $\tilde{\phi}$ of $\tilde{w}$ gives us an isomorphism:
\begin{displaymath}
\phi: U\cap Z \times (-\epsilon, \epsilon) \to U
\end{displaymath}
for some $\epsilon > 0$, defined by $\phi(z,t)= \tilde{\phi}(z,t)$. Here, we are using the fact that $V$ is transverse to $TZ$ and is thus identifiable with the normal bundle $\nu(Z)$ to $Z$. Furthermore, $d\phi_{(z,0)}(\frac{\partial}{\partial t})$ is a nonzero element of $V_z$, hence the fibers of $V$ are identified with the span of $\frac{\partial}{\partial t}$ along $(U\cap Z)\times \{0\}$. Choosing coordinates $x_1,\dots, x_{n-1}$ on $U\cap Z$, we have that $\Omega^1_V(U)$ is spanned by the pullbacks of $\{dx_1,\dots dx_{n-1},tdt\}$.
\end{proof}
\begin{cor}\label{cor:Vcotangent}
Let $M$ be a manifold without boundary, $Z\subset M$ a closed hypersurface, and $V\to Z$ a rank $1$ subbundle of $i_Z^*TM$ transverse to $TZ$. Then there exists a vector bundle $T_V^*M \to M$ whose rank is $\dim(M)$ and whose space of global sections $\Gamma(T_V^*M)$ is isomorphic to $\Omega^1_V(M)$. Furthermore, the inclusion $i:\Omega^1_V(M) \to \Omega^1(M)$ induces a map $i:T_V^*M \to T^*M$.
\end{cor}
\begin{proof} \mbox{ }\newline
The category of locally free sheaves of $C^{\infty}(M)$ modules on $M$ and the category of vector bundles over $M$ are equivalent by the non-compact version of the Serre-Swan theorem. For a discussion of this equivalence, see \cite{R}.
\end{proof}
\begin{remark}
The bundle $T_V^*M$ is defined up to isomorphism, hence once we have fixed a representative $T_V^*M$ the map $i:T_V^*M \to T^*M$ is unique up to a precomposition $i\circ \Phi$ with an automorphism $\Phi:T_V^*M \to T_V^*M$. In coordinates around $p\in Z$, we may frame $T_V^*M$ by $\{dx_1,\dots,dx_{n-1},tdt\}$ (q.v. lemma \ref{lem:Vcotangent2}) and the map $i:T_V^*M\to T^*M$ is:
\begin{displaymath}
i(x_1,\dots,x_{n-1},t,p_1,\dots,p_{n-1},p_n) = (x_1,\dots,x_{n-1},t,p_1,\dots,p_{n-1},tp_n)
\end{displaymath}
Note that this map is \emph{not} a map with fold singularities: while its determinant vanishes transversally, its kernel $\displaystyle \frac{\partial}{\partial p_n}$ is always tangent to the degenerate hypersurface, $\{t=0\}$.
\end{remark}
\subsubsection{Uniqueness up to Isomorphism}
The isomorphism class of $T_V^*M$ is independent of $V$. Indeed, we have the following lemma:
\begin{lemma}\label{lem:Vcotangent3}
Let $M$ be a manifold without boundary and let $Z\subset M$ be a closed hypersurface. Suppose we have chosen two vector subbundles $V_1$, $V_2$ of $i_Z^*TM$ satisfying $V_i \pitchfork TZ$ for $i=1,2$. There exists an isomorphism $\Phi:TM \to TM$ inducing an isomorphism of locally free sheaves $\Phi^*:\Omega_{V_2}^1(\cdot) \to \Omega_{V_1}^1(\cdot)$.
\end{lemma}
\begin{proof}
We produce an isomorphism $\Phi_1:TU \to TU$ in a neighborhood $U$ of $Z$ that induces an isomorphism $\Phi\big\vert_Z:V_1 \to V_2$ and then glue it to the identity map $id:TM \to TM$ using a partition of unity. For the first step, we'll need to extend the bundles $V_1$ and $V_2$ to neighborhoods of $Z$.
\begin{enumerate}
\item For each $i\in \{1,2\}$, $V_i$ is transverse to $TZ$, hence the tubular neighborhood theorem gives us a neighborhood $U$ of $Z$, neighborhoods $W_i\subseteq V_i$ of the zero sections, and isomorphisms $\phi_i:W_i \to U$ satisfying:
\begin{itemize}
\item $\phi_i(z,0)=z$ and
\item $d\phi_{(z,0)}(e_i) \in (V_i)_z$ for all $e_i$ tangent to the fiber of $V_i$ at $(z,0)$.
\end{itemize}
This isomorphism allows us to extend the bundles $V_i$ to the neighborhood $U$ as follows. If $\pi_i:V_i \to Z$ is the projection, then we define the extension of $V_i$ to be $d\phi_i(\ker(d\pi_i))$, where we restrict $\ker(d\pi_i)$ to $W_i$.
\item Thus, we have a neighborhood $U$ of $Z$ and extensions $\tilde{V}_i$ of $V_i$ to vector subbundles of $TU$. Using the same technique as in step $1$, we may extend $TZ \to Z$ to a vector subbundle of $TU$ (this requires a choice of connection). Let us call this extension $E$. Now, $E$ is complementary to $\tilde{V}_i$ at points of $Z$ for each $i$, hence we may shrink $U$ so that $E$ is complementary to both $\tilde{V}_i's$ on $U$, giving us two splittings:
\begin{itemize}
\item $TU \simeq E \oplus \tilde{V}_1$ with projection $p_1:TU \to \tilde{V}_1$ and inclusion $i_1:\tilde{V}_1 \to E\oplus \tilde{V}_1$,
\item $TU \simeq E \oplus \tilde{V}_2$ with projection $p_2:TU \to \tilde{V}_2$ and inclusion $i_2:\tilde{V}_2 \to E \oplus \tilde{V}_2$.
\end{itemize}
We may use these maps to define a an isomorphism $\phi_1:\tilde{V}_1 \to \tilde{V}_2$ given by $p_2 \circ i_1$. To see this, note that at each point $p\in U$, $p_2(i_1(v))= 0$ if and only if $i_1(v)\in E_p$ if and only if $v=0$, hence it is an injective map of line bundles and is therefore an isomorphism.
\item Now, define $\Phi_1:TU \to TU$ by $\Phi_1 = Id_E \oplus \phi_1$.
\item We finish the construction by choosing a partition of unity $\psi_1, \psi_2$ subordinate to the cover $\{U, M\setminus Z\}$ and defining $\Phi:TM \to TM$ by $\Phi= \psi_1(\Phi_1) + \psi_2 Id$. $\Phi$ is an isomorphism of vector bundles since it is an isomorphism over $M\setminus U$ and on $U$ we have:
\begin{itemize}
\item $\Phi\vert_E = Id_E$ and
\item $\Phi\vert_{V_1} = \phi_1$
\end{itemize}
hence $\Phi\vert_U$ is an isomorphism of $TU$.
\end{enumerate}
Now, note that along $Z$, $\Phi:V_1 \to V_2$ is an isomorphism. Therefore, $\Phi^*:T^*M \to T^*M$ induces a map of sections $\Phi^*:\Omega^1(M) \to \Omega^1(M)$ which restricts to a map of $C^{\infty}(M)$-modules $\Phi^*: \Omega_{V_1}^1(M) \to \Omega_{V_2}^1(M)$. Since $\Phi$ is invertible, $\Phi^*$ is invertible and is therefore an isomorphism of modules. Restricting $\Phi^*$ to open subsets of $M$ gives the requisite isomorphism of sheaves.
\end{proof}
\begin{cor}\label{cor:Vcotangent1}
Let $M$ be a manifold without boundary, let $Z\subset M$ be a closed hypersurface, and let $V_1$, $V_2$ be two rank $1$ vector subbundles of $i_Z^*TM$ transverse to $TZ$. Then there exists an isomorphism $\Phi: T_{V_2}^*M \to T_{V_1}^*M$.
\end{cor}
\begin{proof} \mbox{ } \newline
By lemma \ref{lem:Vcotangent3}, there exists an isomorphism of locally free sheaves of $C^{\infty}(M)$ modules $\Phi^*:\Omega_{V_2}^1(\cdot) \to \Omega_{V_1}^1(\cdot)$. Since the category of locally free sheaves of $C^{\infty}(M)$ modules and the category of vector bundles over $M$ are equivalent (q.v. \cite{R}), this map induces an isomorphism between any choice of representatives $T_{V_1}^*M$ and $T_{V_2}^*M$ of the vector bundles isomorphism classes assocatiated to $\Omega_{V_1}^1(M)$ and $\Omega_{V_2}^1(M)$, respectively.
\end{proof}
\subsubsection{Folded Structures on $T_V^*M$}
\begin{lemma}\label{lem:Vcotangent4}
Let $M$ be a manifold without boundary, let $Z\subset M$ be a closed hypersurface, and let $V$ be a rank $1$ subbundle of $i_Z^*TM$ transvserse to $TZ$. Let $\pi_V:T_V^*M\to M$ be a representative of the isomorphism class of vector bundles associated to $\Omega_V^1(M)$ and let $i:T_V^*M \to T^*M$ be the map induced by the inclusion $\Omega_V^1(M) \to \Omega^1(M)$. Let $\omega_{T^*M}$ be the canonical symplectic structure on $T^*M$. Then $i^*\omega_{T^*M}$ is a folded-symplectic form with folding hypersurface $\pi_V^{-1}(Z)$.
\end{lemma}
\begin{proof} \mbox{ } \newline
Away from $Z\subset M$, $i$ is an isomorphism of vector bundles, hence it is a local diffeomorphism, and $i^*\omega_{T^*M}$ is nondegenerate. At a point $p\in Z$, we have seen that we may choose coordinates $x_1,\dots,x_{n-1}, t$ near $p$ so that $V$ is framed by $\frac{\partial}{\partial t}$ along $Z=\{t=0\}$ and $T_V^*(M)$ is framed by $\{dx_1,\dots, dx_{n-1},tdt\}$. That is, $T_V^*M$ is isomorphic to a trivial vector bundle spanned by these sections. Furthermore, in these coordinates, the map $i$ is given by:
\begin{displaymath}
i(x_1,\dots,x_{n-1},t,p_1,\dots, p_{n-1},p_n) = (x_1,\dots,x_{n-1},t,p_1,\dots,p_{n-1},tp_n)
\end{displaymath}
In these coordinates, the canonical symplectic form $\omega_{T^*M}$ is $\omega_{T^*M}= (\sum_{i=1}^{n-1}dq_i\wedge dx_i) + dq_n\wedge dt$ and we have:
\begin{displaymath}
i^*\omega_{T^*M}= (\sum_{i=1}^{n-1}dp_i\wedge dx_i) + tdp_n \wedge dt
\end{displaymath}
which is the folded-symplectic form of example \ref{ex:fs3} whose folding hypersurface (in these coordinates) is $\{t=0\}=\pi_V^{-1}(Z)$. Thus, $i^*\omega_{T^*M}$ is a folded-symplectic form whose folding hypersurface is $\pi_V^{-1}(Z)$. As a final note, $i^*\omega_{T^*M}$ vanishes on $\frac{\partial}{\partial t}$ when $t=0$, illustrating that, locally, null directions transverse to the folding hypersurface are given by the fibers of $V$.
\end{proof}
\begin{remark}
As a closing remark to the construction, consider the map $i:T_V^*M \to T^*M$. We know that $T_V^*M$ is unique up to isomorphism, hence the map $i$ is canonical up to a precomposition with an automorphism of $T_V^*M$. Precomposing with any automorphism $\Phi:T_V^*M\to T_V^*M$ gives us a new map $i\circ \Phi: T_V^*M \to T^*M$ which again induces a folded-symplectic structure $(i\circ \Phi)^*\omega_{T^*M}$. However, $(\Phi^{-1})^*$ is a folded-symplectomorphism between $(i\circ \Phi)^*\omega_{T^*M}$ and $i^*\omega_{T^*M}$, hence the folded-symplectic structure we have induced on $T_V^*M$ is canonical up to folded-symplectomorphism.
\end{remark}
\subsection{Isotopies Between Fold Forms}
Now that we know where one can find a plethora of examples of folded-symplectic manifolds, let us determine how to construct isomorphisms between folded-symplectic structures. Our method of choice will be isotopies. Suppose we are given a folded-symplectic manifold with corners $(M,\sigma)$ and a $1$-form $\beta$. When can we solve the equation
\begin{equation}\label{eq:Moser1}
i_X\sigma = \beta
\end{equation}
for some smooth vector field $X$ on $M$? Certainly, if $\sigma$ has a nonempty folding hypersurface $Z$, then one cannot always solve equation \ref{eq:Moser1} for $X$. Indeed, let $M=\R^2$, $\sigma=x_1dx_1\wedge dx_2$, and $\beta= dx_1$. Then the solution to equation \ref{eq:Moser1} is $X=\displaystyle \frac{1}{x_1}\frac{\partial}{\partial x_2}$, which is undefined at the fold $Z=\{x_1=0\}$. The reasons for understanding equation \ref{eq:Moser1} are two-fold:
\begin{enumerate}
\item First, we would like to understand when a closed $2$-form is folded-symplectic in terms of maps of vector bundles, sheaves, or modules. For example, a closed $2$-form $\omega\in \Omega^2(M)$ is symplectic if and only if the map $\omega^\#(X)=i_X\omega$ from $TM$ to $T^*M$ is an isomorphism of vector bundles. Equivalently, $\omega$ induces an isomorphism of sheaves $\omega^\#:\operatorname{Vec}(\cdot) \to \Omega^1(\cdot)$ or $C^{\infty}(M)$ modules, $\omega^\#:\operatorname{Vec}(M) \to \Omega^1(M)$. What is the corresponding condition for folded-symplectic forms?
\item Secondly, and perhaps most importantly, we would like to develop a Moser-type argument to produce folded-symplectomorpisms from families of fold forms, thereby providing a convenient way to construct folded-symplectomorphisms. We'll use the discussion of $1$ to determine when solutions to Moser's equation (eq. \ref{eq:Moser1}) exist.
\end{enumerate}
\subsubsection{Isotopies and Moser's argument}
We begin by reviewing Moser's argument in the context of symplectic geometry (q.v. \cite{MS} for details). Given a manifold with corners $M$ and a smooth path of symplectic forms $\omega_t\in \Omega^2(M)$, we would like to produce a family of diffeomorphisms $\phi_t:M \to M$ satisfying:
\begin{equation}\label{eq:Moser2}
\psi_t^*\omega_t=\omega_0
\end{equation}
Generally, this cannot be done. But, if we assume the derivative is exact:
\begin{equation}\label{eq:Moser3}
\frac{d}{dt}\omega_t=d\beta_t
\end{equation}
where $\beta_t\in \Omega^1(M)$ is a smooth path of $1$-forms, then we have a better chance of producing $\psi_t$ by representing it as a flow of a time-dependent vector field $X_t$ on $M$. If we assume $\displaystyle \frac{d}{dt}\psi_t = X_t\circ \psi_t$ and $\psi_0=Id_M$, then differentiating equation \ref{eq:Moser3} gives us:
\begin{equation}\label{eq:Moser4}
0=\psi_t^*(\frac{d}{dt}\omega_t + \pounds_{X_t}\omega_t) = \psi_t^*(\dot{\omega}_t + d(i_{X_t}\omega_t))
\end{equation}
Since $\psi_t$ is assumed to be a diffeomorphism, this will be true if and only if:
\begin{equation}\label{eq:Moser5}
\dot{\omega}_t + d(i_{X_t}\omega_t) =0 \mbox{ } \iff \mbox{ } d(i_{X_t}\omega_t) = -d\beta_t
\end{equation}
where we are using the assumption $\dot{\omega}_t = d\beta_t$. Equation \ref{eq:Moser5} may be solved by solving:
\begin{equation}\label{eq:Moser6}
i_{X_t}\omega_t = -\beta_t
\end{equation}
Since we are assuming $\omega_t$ is symplectic for all $t$, we have a smooth path of vector bundle isomorphisms $\omega_t^\#:TM \to T^*M$ with smooth inverse $\omega_t^{\flat}:T^*M \to TM$. Equation \ref{eq:Moser6} is then solved by setting $X_t = -\omega_t^{\flat}(\beta_t)$. If $X_t$ is stratified and $M$ is compact, we may integrate $X_t$ to obtain a path of diffeomorphisms $\psi_t:M\to M$ satisfying $\psi_t^*\omega_t=\omega_0$.
In general, $M$ is not compact and so $X_t$ needn't have a globally defined flow $\psi_t$. To circumvent this problem, we use the following lemma:
\begin{lemma}\label{lem:Moser1}
Let $M$ be a manifold with corners and suppose $X_t$ is a stratified, time-dependent vector field on $M$ (i.e. it is tangent to the boundary strata at points of $\partial(M)$). Let $N\subset M$ be a submanifold with corners so that $X_t(p)=0$ for all $p\in N$ and for all $t$. Then there exists a neighborhood $U$ of $N$ on which the flow $\psi_t$ is defined for all $t\in [0,1]$. Thus, we have a smooth path of open embeddings $\psi_t:U \to M$ satisfying $\psi_t(n)=n$ for all $n\in N$.
\end{lemma}
\begin{remark}
We impose the condition that $X_t$ is stratified so that its flow defines diffeomorphisms of manifolds with corners.
\end{remark}
\begin{proof}
The trick is to pass to the product $M\times \R$ and consider the vector field $Y=(X_t, \frac{\partial}{\partial t})$. We claim that its flow will give us the time-dependent flow of $X_t$. Its flow $F_s$ has the form
\begin{displaymath}
F_s(m,t)=(G_s(m,t), t+s)
\end{displaymath}
and we have:
\begin{displaymath}
F_{-s}(F_s(m,t))=(G_{-s}(G_s(m,t), t+s),t)=(m,t)
\end{displaymath}
Hence, for fixed $s$ and $t$ the map $m \to G_s(m,t)$ is a diffeomorphism with smooth inverse given by $y \to G_{-s}(y, s+t)$. Furthermore, we have:
\begin{displaymath}
\frac{d}{ds}(G_s(m,0),s)=\frac{d}{ds}F_s(m,0) = (Y \circ F_s)(m,0) = (X_s(G_s(m,0)), \frac{\partial}{\partial t})
\end{displaymath}
hence $\displaystyle \frac{d}{ds}G_s(m,0) = X_s(G_s(m,0))$ and so $m \to G_s(m,0)$ is the flow of $X_s$. Now, if the flow $F_s$ exists at a point $(m,0)$ for all $s\in [0,1]$, then there exists a neighborhood $V_m$ of $(m,0)$ so that $F_s$ is defined on $U$ for all $t\in[0,1]$ since the flow is defined on an open domain and $(m,0)\times [0,1]$ is compact. If we define $U_m=V_m\cap (M\times \{0\})$, then $U_m$ defines a neighborhood of $m\in M$ on which the flow $G_s(p,0)$ exists for all time $s\in [0,1]$. We form such neighborhoods for each $m\in N$ and take the union $U=\bigcup_{m\in N} U_m$. Then the flow $\psi_s(p):=G_s(p,0)$ is defined for all time $s\in [0,1]$ on $U$. For each $s$, $\psi_s:U \to M$ is an open embedding into $M$. Furthermore, since the vector field $X_s$ satisfies $X_s(m)= 0$ for all $m\in N$, $\psi_s(m)=G_s(m,0)=m$ for all $s$, hence $\psi_s\vert_N = Id_N$.
\end{proof}
Now, returning to Moser's argument, let us consider the folded-symplectic setting. Suppose we have a $2m$-dimensional manifold with corners $M$ and a path $\sigma_t\in \Omega^2(M)$ of folded-symplectic forms satisfying:
\begin{displaymath}
Z=((\sigma_t)^m)^{-1}(\mathcal{O}) \text{ is independent of $t$.}
\end{displaymath}
That is, we assume the folding hypersurface is fixed in time. Furthermore, let us assume that the kernel bundles $\ker(\sigma_t)\to Z$ have fibers tangent to the faces of $M$ for all $t$. Following the setup for Moser's argument in the symplectic setting, we assume:
\begin{displaymath}
\frac{d}{dt}\sigma_t = d\beta_t \text{ for a family of smooth $1$-form $\beta_t\in \Omega^1(M)$.}
\end{displaymath}
If we begin with the equation $\psi_t^*\sigma_t = \sigma_0$ and differentiate, then we arrive at the equation:
\begin{equation}\label{eq:Moser7}
-d\beta_t = di_{X_t}\sigma_t
\end{equation}
where $X_t$ is the time-dependent vector field generating $\psi_t$. It is therefore sufficient to solve:
\begin{equation}\label{eq:Moser8}
-\beta_t=i_{X_t}\sigma_t \text{ (just as in the symplectic case)}
\end{equation}
Thus, we have arrived at the question we originally posed:
\begin{itemize}
\item Given a folded-symplectic form $\sigma$ and a $1$-form $\beta$, can we find a smooth vector field $X$ such that $i_X\sigma=\beta$?
\end{itemize}
This is the subject of the next section.
\subsubsection{Solving $i_X\sigma= \beta$ to Obtain Isotopies}
Suppose $(M,\sigma)$ is a folded-symplectic manifold with corners with folding hypersurface $Z\subset M$. Let $\ker(\sigma)\to Z$ be the kernel bundle over $Z$ and suppose it is stratified. That is, suppose its fibers are tangent to the boundary strata of $M$. Similar to the construction of the folded cotangent bundle $T_V^*M$, we will construct a module of $1$-forms vanishing on $\ker(\sigma)$ at points of $Z$, which will give us a vector bundle $T_{\ker(\sigma)}^*M$. We will use this vector bundle to characterize folded-symplectic forms on $M$ whose fold is $Z$ and whose kernel bundle is $\ker(\sigma)$.
\begin{remark}
In much of what follows, it is not necessary to assume that $\ker(\sigma)$ is stratified. It will only be used when we show that $\sigma^\#:TM \to T_{\ker(\sigma)}^*M$ is an isomorphism if and only if $\sigma$ is folded-symplectic. Furthermore, we needn't begin any constructions assuming we have a folded-symplectic form. The main ingredients in what follows are:
\begin{itemize}
\item A hypersurface $Z\subset M$ inside a manifold with corners, transverse to strata.
\item A $2$-plane bundle $E \to Z$ so that $E\cap TZ$ is a rank $1$ vector bundle.
\end{itemize}
which is devoid of any folded-symplectic geometry: we just have a hypersurface and a vector bundle. One may then study all forms whose degenerate set is $Z$ and whose kernel distribution on $Z$ contains the bundle $E\to Z$. What we will effectively show is that if $\sigma$ is such a form, then it is folded-symplectic with folding hypersurface $Z$ and kernel bundle $E\to Z$ if and only if it induces an isomorphism of sheaves $\sigma^\#:\operatorname{Vec}(\cdot) \to \Omega^1_E(\cdot)$, where $\operatorname{Vec}(\cdot)$ is the sheaf of vector fields and $\Omega^1_E(\cdot)$ is the sheaf of $1$-forms vanishing on $E$ at $Z$. However, on a first run through the constructions, it is perhaps easier to begin with a folded-symplectic form and a fixed kernel bundle $\ker(\sigma)$ so that one has a grounding in the folded-symplectic world.
\end{remark}
\begin{definition}\label{def:Ecotangent1}
Let $(M,\sigma)$ be a folded-symplectic manifold with corners with folding hypersurface $Z\subset M$. Let $\ker(\sigma)\to Z$ be the kernel bundle over the fold $Z$ and suppose it is stratified. That is, for all $p\in Z$, $\ker(\sigma)_p$ is tangent to the stratum of $M$ containing $p$. We define a presheaf $\Omega_{\ker(\sigma)}^1(\cdot)$ of $C^{\infty}(M)$ modules on $M$ as follows. Given an open subset $U\subset M$,
\begin{displaymath}
\Omega_{\ker(\sigma)}^1(U) = \{\beta \in \Omega^1(U) \vert \mbox{ } \beta\big\vert_{\ker(\sigma)} = 0 \text{ at $Z\cap U$}\}
\end{displaymath}
to be the space of $1$-forms on $U$ that vanish on $\ker(\sigma)$ at points of $Z\cap U$. The restriction maps are given by pullbacks $i^*$ induced by inclusions $i:V \hookrightarrow U$.
\end{definition}
\begin{remark}
We take a moment to make sense of everything stated in the definition. Note that vanishing on $\ker(\sigma)$ is a linear condition, hence $\Omega_{\ker(\sigma)}^1(U)$ is a submodule of $\Omega^1(U)$ for all $U\subset M$. Since the restriction maps of $\Omega_{\ker(\sigma)}^1(\cdot)$ are induced by those of $\Omega^1(\cdot)$, we have that $\Omega_{\ker(\sigma)}^1(\cdot)$ is a sub-presheaf of $\Omega^1(\cdot)$, hence it is a presheaf.
\end{remark}
\begin{lemma}\label{lem:Ecotangent1}
Let $(M,\sigma)$ be a folded-symplectic manifold with corners with folding hypersurface $Z\subset M$. Suppose that the fibers of $\ker(\sigma)\to Z$ are tangent to the strata of $M$. Then the presheaf $\Omega^1_{\ker(\sigma)}(\cdot)$ of forms vanishing on $\ker(\sigma)$ is a sheaf of $C^{\infty}(M)$ modules on $M$.
\end{lemma}
\begin{proof}
Let $U\subseteq M$ be open and let $\{U_i\}_{i\in I}$ be an open cover of $U$.
\begin{enumerate}
\item Suppose we have $\tau \in \Omega_{\ker(\sigma)}^1(U)$ such that $\tau\big\vert_{U_i} =0 $ for each $U_i$. Then $\tau=0$ since it is $0$ at every point of $U$.
\item Suppose we have a collection of forms $\{\tau_i\in \Omega_{\ker(\sigma)}^1(U_i)\}_{i\in I}$ satisfying:
\begin{displaymath}
\tau_i\big\vert_{U_i\cap U_j} = \tau_j\big\vert_{U_i\cap U_j}
\end{displaymath}
for all $i,j\in I$. Then, refining the cover if necessary, we may choose a partition of unity $\{\psi_i\}$ subordinate to the cover $\{U_i\}$ and define:
\begin{displaymath}
\tau:= \sum_{i\in I} \psi_i \tau_i
\end{displaymath}
Since vanishing on $\ker(\sigma)$ is a linear condition, $\tau$ vanishes on $\ker(\sigma)$ at points of $U\cap Z$.
\end{enumerate}
\end{proof}
\begin{lemma}\label{lem:Ecotangent2}
Let $(M,\sigma)$ be a manifold with corners with folding hypersurface $Z\subset M$ and let $\ker(\sigma) \to Z$ be the kernel bundle of $\sigma$. Suppose that the fibers of $\ker(\sigma)$ are tangent to the strata of $M$. Let $\Omega^1_{\ker(\sigma)}(\cdot)$ be the sheaf of $1$-forms vanishing on $\ker(\sigma)$ along $Z$. Then $\Omega^1_{\ker(\sigma)}(\cdot)$ is a locally free sheaf of $C^{\infty}(M)$ modules.
\end{lemma}
\begin{proof}
Let $2m=\dim(M)$. Suppose $p\in M\setminus Z$. Then there exists a neighborhood $U$ of $p$ diffeomorphic to an open subset of the quadrant $(\R^+)^{2m}$ satisfying $U\cap Z=\emptyset$. Then $\Omega_{\ker(\sigma)}^1(U)$ is spanned by the pullbacks of $dx_1,\dots,dx_{2m}$.
\vspace{5mm}
Now, suppose that $p\in Z$. Since $Z$ is transverse to the boundary strata $\partial^k(M)$ for all $k\ge 0$, we may choose a stratified vector field transverse to $Z$ whose flow gives us a diffeomorphism of a neighborhood $U$ of $p$ with $U\cap Z \times (-\epsilon, \epsilon)$, where $U\cap Z$ is identified with the zero section $U\cap Z \times \{0\}$. This gives us coordinates $(z,t)$ around $p$, where $z$ lies in the folding hypersurface. Choose a local frame $\{f_1,f_2\}$ of $\ker(\sigma)$ and extend it to a local frame $\{e_1,\dots, e_{2m-2}, f_1,f_2\}$ of $i_Z^*TM$ on $U\cap Z$. Extend this frame to a frame on $U\cap Z \times (-\epsilon, \epsilon)$ by lifting the vector fields on $U\cap Z$ back to the product. We redefine the $e_i$'s and $f_i$'s to be these vector fields. Then the set:
\begin{displaymath}
\{e_1^*, \dots, e_{2m-2}^*, tf_1^*, tf_2^*\}
\end{displaymath}
is a basis for $\Omega_{\ker(\sigma)}^1(U)$, where $t$ is the coordinate on $(-\epsilon, \epsilon)$. This is because $\{e_1^*,\dots,e_{2m-2}^*, f_1^*, f_2^*\}$ is a coframe for $T^*U$ and any form vanishing on $\ker(\sigma)$ at $t=0$ must have vanishing $f_1^*$ and $f_2^*$ terms at $t=0$.
\end{proof}
\begin{cor}\label{cor:Ecotangent1}
Let $(M,\sigma)$ be a folded-symplectic manifold with corners, let $Z\subset M$ be its folding hypersurface, let $\ker(\sigma)\to Z$ be the kernel bundle of $\sigma$ and suppose it is stratified (its fibers are tangent to the strata of $M$), and let $\Omega_{\ker(\sigma)}^1(\cdot)$ be the locally free sheaf of $1$-forms vanishing on $\ker(\sigma)$ along $Z$. Then there exists a vector bundle $T_{\ker(\sigma)}^*M$ whose space of global sections is isomorphic to $\Omega_{\ker(\sigma)}^1(\cdot)$. Furthermore, $\sigma$ induces a map of locally free sheaves:
\begin{displaymath}
\sigma^\#: \operatorname{Vec}(\cdot) \to \Omega_{\ker(\sigma)}^1(\cdot)
\end{displaymath}
given by $X \to i_X\sigma$. Hence $\sigma$ induces a map $\sigma^\#:TM \to T_{\ker(\sigma)}^*M$.
\end{cor}
\begin{proof} \mbox{ } \newline
Since $\Omega_{\ker(\sigma)}^1(\cdot)$ is a locally free sheaf of $C^{\infty}(M)$ modules, there exists a vector bundle $T_{\ker(\sigma)}^*M$ whose space of sections is isomorphic to $\Omega_{\ker(\sigma)}^1(M)$. Now, if $X\in \operatorname{Vec}(U)$ is a vector field on an open subset $U\subset M$, then for all $z\in U\cap Z$ and for all $v\in \ker(\sigma)_z$, we have:
\begin{displaymath}
(i_X\sigma)(v)=\sigma(X,v)=0
\end{displaymath}
hence $i_X\sigma \in \Omega_{\ker(\sigma)}^1(U)$, thus $\sigma^\#(X) = i_X\sigma$ defines a map of sheaves since it commutes with restrictions. It is a map of locally free sheaves since $\sigma^\#$ is $C^{\infty}(U)$ linear for each open subset $U\subset M$. It therefore induces a map of vector bundles, which we also denote $\sigma^\#$:
\begin{displaymath}
\sigma^\#: TM \to T_{\ker(\sigma)}^*M
\end{displaymath}
\end{proof}
\begin{prop}\label{prop:Ecotangent1}
Let $(M,\sigma)$ be a folded-symplectic manifold with corners, $Z\subset M$ its folding hypersurface, $\ker(\sigma)\to Z$ its kernel bundle, where we assume $\ker(\sigma)$ is stratified (its fibers are tangent to the strata of $M$), and let $\Omega^1_{\ker(\sigma)}(M)$ be the space of $1$-forms vanishing on $\ker(\sigma)$ along $Z$. Then,
\begin{enumerate}
\item The map $\sigma^\#:\operatorname{Vec}(\cdot) \to \Omega_{\ker(\sigma)}^1(\cdot)$ is an isomorphism of locally free sheaves and therefore induces an isomorphism of vector bundles $\sigma^\#:TM \to T_{\ker(\sigma)}^*M$. In particular, $\sigma^\#:\Gamma(TM) \to \Omega_{\ker(\sigma)}^1(M)$ is an isomorphism of $C^{\infty}(M)$ modules.
\item If $\sigma_0$ is another closed $2$-form on $M$ so that the degenerate set of $\sigma_0$ is $Z$ and $\sigma_0$ vanishes on $\ker(\sigma)$, then $\sigma_0$ is folded-symplectic if and only if the induced map $\sigma_0^\#$ is an isomorphism.
\end{enumerate}
\end{prop}
\begin{remark}
We are showing that $\sigma_0^n \pitchfork_s 0$ and $i_Z^*\sigma_0$ has maximal rank if and only if $\sigma_0^\# :\Gamma(TM) \to \Omega^1_{\ker(\sigma)}$ is an isomorphism of locally free sheaves of $C^{\infty}(M)$ modules.
\end{remark}
\begin{proof}
\begin{enumerate}
\item Let $U\subset M$ be open. If $U\cap Z=\emptyset$, then there is nothing to prove since $\sigma\vert_U$ is non-degenerate, hence $\sigma^\#\vert_U$ is invertible. Now, assume $U\cap Z\ne \emptyset$. We show that $\sigma^\#: \operatorname{Vec}(U) \to \Omega^1_{\ker(\sigma)}(U)$ is injective, hence local solutions to the equation $i_X\sigma = \beta$ are unique. We use this uniqueness to show surjectivity by gluing local solutions together.
\begin{enumerate}
\item Suppose $X,Y \in \operatorname{Vec}(U)$ and $\sigma^\#(X) = \sigma^\#(Y)$. Since $\sigma^\#$ is a map of sheaves, this identity holds on $U\setminus Z$, which is open and dense in $U$. We have:
\begin{displaymath}
\begin{array}{lcl}
\sigma^\#(X)=\sigma^\#(Y) & \iff & i_X\sigma = i_Y\sigma \\
& \iff & i_{X-Y}\sigma = 0 \\
& \iff & i_{X-Y}\sigma = 0 \text{ on $U\setminus Z$} \\
& \iff & X-Y = 0 \text{ on $U\setminus Z$ since $\sigma\vert_{U\setminus Z}$ is symplectic}\\
& \iff & X=Y \text{ on $U\setminus Z$} \\
& \iff & X=Y \text{ on $U$ since $U\setminus Z$ is dense.}
\end{array}
\end{displaymath}
Thus $\sigma^\#$ is injective on $U$.
\item Now, let $\beta \in \Omega_{\ker(\sigma)}^1(U)$. If $V\subset U$ is an open subset, then the injectivity of $\sigma^\#$ implies that local solutions $i_X\sigma\vert_V = \beta\vert_V$ are unique. Hence, if $\{V_i\}$ is a cover of $U$ with a set $\{X_i\}$ of solutions to $i_X\sigma\vert_{V_i} = \beta\vert_{V_i}$ on each $V_i$, then we have:
\begin{displaymath}
X_i\big\vert_{V_i\cap V_j} = X_j \big\vert_{V_i\cap V_j}
\end{displaymath}
hence these local solutions glue to give a global solution $X\in \operatorname{Vec}(U)$ to the equation $i_X\sigma = \beta$. We show that such a cover of $U$ exists.
\begin{itemize}
\item On $U\setminus Z$, $\sigma$ is symplectic and so the equation $i_X\sigma\big\vert_{U\setminus Z} = \beta\big\vert_{U\setminus Z}$ has a solution.
\item For each $z\in U\cap Z$, $\ker(\sigma)$ being stratified implies that we may choose a neighborhood $V_z$ of $z$ and a diffeomorphism
\begin{displaymath}
\phi: V_z\cap Z \times (-\epsilon, \epsilon) \to V_z
\end{displaymath}
so that $\phi^*\sigma=p^*i^*\sigma +t\mu$ for some $2$-form $\mu$, where $p:V_z\cap Z \times (-\epsilon, \epsilon) \to V_z\cap Z$ is the projection and $i:V_z\cap Z \to V_z \cap Z \times (-\epsilon, \epsilon)$ is the inclusion as the zero section.
\vspace{3mm}
Choose a local frame $f$ of $\ker(i_Z^*\sigma)$ near $p$ and extend it to a local frame $\{e_1,\dots,e_{2m-2},f\}$ of $Z$ near $z$. We may lift these vector fields to vector fields on the product $V_z\cap Z \times (-\epsilon, \epsilon)$ (using the standard connection), where we use the same notation $e_i$ and $f$, and extend to a local frame:
\begin{displaymath}
\{e_1, \dots, e_{2m-2},f,\frac{\partial}{\partial t}\}
\end{displaymath}
where $f$ and $\frac{\partial}{\partial t}$ satisfy:
\begin{displaymath}
\begin{array}{l}
i_f\sigma = i_f(p^*i^*\sigma + t\mu) = 0 + ti_f\mu = t(i_f\mu) \\
i_{\frac{\partial}{\partial t}}\sigma = i_{\frac{\partial}{\partial t}}(p^*i^*\sigma + t\mu) = t(i_{\frac{\partial}{\partial t}}\mu)
\end{array}
\end{displaymath}
Now, since $\sigma$ is folded-symplectic, lemma \ref{lem:symplectize} implies that $\mu$ is non-degenerate on $\ker(\phi^*\sigma)$, hence:
\begin{displaymath}
\mu(f,\frac{\partial}{\partial t}) \ne 0
\end{displaymath}
in a neighborhood of $(V_z\cap Z) \times \{0\}$. Hence, the set:
\begin{displaymath}
\{i_{e_1}\sigma, \dots, i_{e_{2m-2}}\sigma, i_f\mu, i_{\frac{\partial}{\partial t}}\mu\}
\end{displaymath}
is a local frame of the cotangent bundle in a neighborhood of $(z,0)$. Hence, we may write:
\begin{displaymath}
\beta = \sum_{i=1}^{2m-2}a_i(i_{e_i}\sigma) + b(i_f\mu) + c(i_{\frac{\partial}{\partial t}}\mu)
\end{displaymath}
for some smooth functions $a_i, b,$ and $c$. Since $\beta$ vanishes on $f$ and $\frac{\partial}{\partial t}$ at $t=0$, we must have that $b=tb_0$ and $c=tc_0$ for some smooth functions $b_0$ and $c_0$. Define:
\begin{displaymath}
X=\sum_{i=1}^{2m-2} a_ie_i + b_0f + c_0\frac{\partial}{\partial t}
\end{displaymath}
Then,
\begin{displaymath}
i_X\sigma = \sum_{i=1}^{2m-2} a_i(i_{e_i}\sigma) + tb_0(i_f\mu) + tc_0(i_{\frac{\partial}{\partial t}}\mu) = \beta
\end{displaymath}
Thus, for each $z\in U\cap Z$ there exists a neighborhood $U_z$ and a solution $X_z$ of the equation $i_X\sigma =\beta$ on $U_z$. Since $\{U_z\}_{z\in U\cap Z} \cup U\setminus Z$ covers $U$ and we have solutions on each open subset, we may glue them together by uniqueness to obtain a global solution $X \in \operatorname{Vec}(U)$ to $i_X\sigma = \beta$. Thus, $\sigma^\#$ is surjective.
\end{itemize}
\item Since $\sigma^\#$ is a map of locally free sheaves of $C^{\infty}(M)$-modules and it is an isomorphism on each open subset of $U$ it is an isomorphism of locally free sheaves of $C^{\infty}(M)$-modules. Therefore, it induces an isomorphism of vector bundles:
\begin{displaymath}
\sigma^\#: TM \to T_{\ker(\sigma)}^*(M)
\end{displaymath}
\end{enumerate}
\item Now, suppose $\sigma_0$ is another closed $2$-form on $M$ that degenerates at all points of $Z$ (and only at $Z$) and vanishes on $\ker(\sigma)$. Part $1$ of the proposition implies that $\sigma_0^\#:\operatorname{Vec}(\cdot) \to \Omega^1_{\ker(\sigma)}(\cdot)$ is an isomorphism if $\sigma_0$ is folded-symplectic. We must show the \emph{only if} portion, so we assume that $\sigma_0^\#$ is an isomorphism. We first show that $i_Z^*\sigma_0$ has maximal rank and then use this to argue it is folded-symplectic.
\begin{enumerate}
\item Consider a point $p\in Z$. Since $\ker(\sigma)$ is stratified and transverse to $Z$, we may choose a neighborhood $U$ of $p$ and assume that $M = U\cap Z \times (-\epsilon,\epsilon)$ where $U\cap Z$ is identified with the zero section and $\frac{\partial}{\partial t}$ is in $\ker(\sigma)$ (q.v. lemma \ref{lem:fsnormal}). Hence, we may assume $M=Z\times (-\epsilon, \epsilon)$ and drop the intersection notation $U\cap Z$ from our discussion. Since this is a local calculation, we may also assume that $Z$ is contractible so that any vector bundle over $M$ is trivializable. Lastly, since $\sigma_0$ vanishes on $\ker(\sigma)$, it vanishes on $\frac{\partial}{\partial t}$ so that:
\begin{displaymath}
\sigma_0 = p^*i^*\sigma_0 + t\mu
\end{displaymath}
\vspace{3mm}
Choose a local frame $f$ of $\ker(i_Z^*\sigma)$, extend it to a local frame $\{e_1,\dots, e_{2m-2},f\}$ of $Z$, and then extend this to a local frame of $M$ by lifting the vector fields to the product $Z\times \R$ using the standard connection. We are first going to show that $i^*\sigma_0$, the restriction of $\sigma_0$ to $Z$, has maximal rank.
Since $e_i^*$ vanishes on $\ker(\sigma)$ at $Z\times \{0\}$, it is an element of $\Omega^1_{\ker(\sigma)}(M)$. Since $\sigma_0^\#$ is an isomorphism (by assumption), we know there exists a vector field $X_i$ satisfying:
\begin{displaymath}
i_{X_i}\sigma_0=e_i^*
\end{displaymath}
If we write $\displaystyle X_i=\sum_{j=1}^{2m-2} a_{ij}e_j + b_if + c_i\frac{\partial}{\partial t}$ and use the fact that $\displaystyle i_{\frac{\partial}{\partial t}}\sigma_0\big\vert_Z = i_f \sigma_0\big\vert_Z=0$, we have:
\begin{equation}\label{eq:surjects}
i_{X_i}\sigma_0 = e_i^* \to \mbox{ } i^*(i_{X_i}\sigma_0) = e_i^* \to \mbox{ } i_{\sum a_{ij}e_j}(i^*\sigma_0) = e_i^*\big\vert_Z \in \Gamma(TZ)
\end{equation}
where $i:Z\hookrightarrow Z\times \R$ is the inclusion as the zero section. Since $\sum a_{ij}e_j$ a section of $TZ$ at points of $Z\times \{0\}$, equation \ref{eq:surjects} shows that $e_i^*$ is in the image of the contraction mapping $(i^*\sigma_0)^\#:TZ \to T^*Z$. Since we can choose any $e_i$ and equation \ref{eq:surjects} holds, we have that the map $(i^*\sigma_0)^\#:TZ \to T^*Z$ sends each fiber of $TZ$ onto a $2m-2$-dimensional subspace of the corresponding fiber of $T^*Z$, meaning $i^*\sigma_0$, the restriction of $\sigma_0$ to the fold, has maximal rank.
\item Now, recall that we have written $\sigma_0$ as $\sigma_0=p^*i^*\sigma_0 + t\mu$ for some $2$-form $\mu$. Since $i^*\sigma_0$ has maximal rank, lemma \ref{lem:symplectize} implies that $\sigma_0$ is folded if and only if $\mu\big\vert_{\ker(\sigma)}$ is non-degenerate, hence we would like to show $\mu$ is non-degenerate. We are again assuming that we have a frame $\{e_1,\dots,e_{2m-2},f,\frac{\partial}{\partial t}\}$, where $f\in \ker(\sigma)$ at points of $Z\times \{0\}$. Assume towards a contradiction that there is a point $(p,0)\in Z\times \{0\}$ where $\mu(f,\frac{\partial}{\partial t})=0$. Then we claim there is no vector field $X$ satisfying $i_X\sigma_0 = tdt$. Indeed, $X$ would need to satisfy:
\begin{itemize}
\item $(i_X\sigma_0)(e_i)=0$ for all $i$ and
\item $(i_X\sigma_0)(\frac{\partial}{\partial t}) = t\mu(X,\frac{\partial}{\partial t})=t$
\end{itemize}
The first condition implies that at points $(z,0)$ we have $X\in \ker(\sigma)$, hence $\displaystyle X=af + b\frac{\partial}{\partial t}$ for some constants $a$ and $b$ at $(z,0)$. The second condition implies that $\mu(X,\frac{\partial}{\partial t})=1$, hence $a\ne 0$. At $(z,0)$, we then have that
\begin{displaymath}
1=\mu(X,\frac{\partial}{\partial t})=a\mu(f,\frac{\partial}{\partial t})=0
\end{displaymath}
since we are assuming $\mu$ is degenerate on the $\ker(\sigma)$ at $(z,0)$. Thus, $\mu(X,\frac{\partial}{\partial t})=0\ne 1$, which means $tdt$ cannot be in the image of $\sigma_0^\#$, which contradicts the assumption that $\sigma_0^\#: \operatorname{Vec}(\cdot)\to \Omega^1_{\ker(\sigma)}(\cdot)$ is an isomorphism of sheaves since it cannot be surjective near $(z,0)$.
\vspace{5mm}
This means that $\mu\big\vert_{\ker(\sigma)}$ is non-degenerate and lemma \ref{lem:symplectize} implies that $\sigma_0=p^*i^*\sigma_0 + t\mu$ is folded-symplectic in a neighborhood of $Z\times \{0\}$. Since this is true at each point $z\in Z$ and, by assumption, $\sigma_0$ only degenerates at $Z$, we have that $\sigma_0$ is folded-symplectic on $M$.
\end{enumerate}
\end{enumerate}
\end{proof}
\begin{cor}\label{cor:Ecotangent2}
Let $(M,\sigma)$ be a folded-symplectic manifold with corners with folding hypersurface $Z\subset M$ and stratified kernel bundle $\ker(\sigma)\to Z$ (i.e. the fibers of $\ker(\sigma)$ are tangent to the boundary strata of $M$). Let $\beta \in \Omega^1(M)$ be a $1$-form. Then there is a smooth vector field $X\in \Gamma(TM)$ satisfying $i_X\sigma = \beta$ if and only if $\beta$ vanishes on $\ker(\sigma)$ at points of $Z$. That is, if and only if $\beta \in \Omega^1_{\ker(\sigma)}(M)$.
\end{cor}
\begin{proof} \mbox{ } \newline
By proposition \ref{prop:Ecotangent1}, the map $\sigma^\#:\operatorname{Vec}(\cdot) \to \Omega^1_{\ker(\sigma)}(\cdot)$ is an isomorphism of sheaves, hence $\sigma^\#:\Gamma(TM) \to \Omega^1_{\ker(\sigma)}(M)$ is an isomorphism of global sections. Thus, $i_X\sigma = \beta$ if and only if $\sigma^\#(X)=\beta$ if and only if $\beta \in \Omega^1_{\ker(\sigma)}(M)$.
\end{proof}
\begin{prop}\label{prop:Moser}
Let $(M,\sigma)$ be a $2m$-dimensional folded-symplectic manifold with corners and suppose $\sigma_t$ is a smooth path of folded-symplectic forms satisfying:
\begin{enumerate}
\item $\sigma_0 = \sigma$,
\item The fold $Z=(\sigma_t^m)^{-1}(\mathcal{O})$ is independent of $t$, and
\item The bundles $\ker(\sigma_t)$ are stratified (their fibers are tangent to the boundary strata of $M$).
\end{enumerate}
Then, if $\dot{\sigma}_t=d\beta_t$ for some path of $1$-forms $\beta_t$ so that $\beta_t \in \Omega^1_{\ker(\sigma_t)}(M)$ for all $t$, there exists a smooth time-dependent vector field $X_t$ satisfying $i_{X_t}\sigma_t = -\beta_t$. If $X_t$ is stratified and vanishes at $Z$ for all $t$, then we may integrate $X_t$ in a neighborhood $U$ of $Z$ to obtain a path of open embeddings:
\begin{displaymath}
\phi_t:U \to M
\end{displaymath}
satisfying $\phi_t(z)=z$ and $\phi_t^*\sigma_t = \sigma_0 =\sigma$ for all $t\in [0,1]$.
\end{prop}
\begin{proof} \mbox{ } \newline
For each $t\in \R$, $\sigma_t$ defines an isomorphism of $C^{\infty}(M)$ modules:
\begin{displaymath}
\sigma_t^\#: \Gamma(TM) \to \Omega_{\ker(\sigma_t)}^1(M)
\end{displaymath}
by proposition \ref{prop:Ecotangent1}. Since $\beta_t\in \Omega_{\ker(\sigma_t)}^1(M)$ for all $t\in \R$, we have that $X_t:=(\sigma_t^\#)^{-1}(-\beta_t)$ defines a smooth vector field on $M$ for each $t\in \R$. Even though the target space of $\sigma_t^\#$ varies in $t$, we claim that $X_t$ is also smooth in $t$. However, we put this issue on hold momentarily and address the remaining claim of the proposition: assume $X_t$ is smooth, stratified, and vanishes on $Z$. Then lemma \ref{lem:Moser1} implies that there exists a neighborhood $U$ of $Z$ on which the flow $\phi_t$ of $X_t$ exists for all $t\in [0,1]$. $X_t$ satisfies:
\begin{displaymath}
i_{X_t}\sigma=-\beta_t
\end{displaymath}
which is equation \ref{eq:Moser8}, hence its flow satisfies $\phi_t^*\sigma_t = \sigma_0$.
\vspace{4mm}
We now address the smoothness of $X_t$. At first glance, this would appear to be obvious since $\sigma_t^\#$ and $\beta_t$ are both smooth in $t$. However, the target space $\Omega_{\ker(\sigma_t)}^1(M)$ varies in $t$, so it isn't immediately clear that the composite $(\sigma_t^\#)^{-1}(\beta_t)$ is smooth in $t$ unless we say something like ``$\Omega_{\ker(\sigma_t)}^1(M)$ varies smoothly in $t$.'' But, its not clear what we mean by such a statement unless we have built some machinery. If the reader is convinced that $X_t$ is smooth because $\Omega_{\ker(\sigma_t)}^1(M)$ is a smoothly varying submodule of $\Omega^1(M)$, then he or she may pass on to the next section.
\vspace{4mm}
We show smoothness in $t$ by adapting our constructions on $M$ to the product $M\times \R$. Also, we only check smoothness of $X_t$ in $t$ near points of the fold $Z\subset M$ since each $\sigma_t$ is symplectic away from $Z$, hence $(\sigma_t^\#)^{-1}(-\beta_t)$ is smooth away from $Z$. The family of $1$-forms $\sigma_t$ defines a smooth map $\Phi$ of vector bundles over $M\times \R$:
\begin{displaymath}
\xymatrix{
TM \times \R \ar[r]^\Phi \ar[d]^{\tau_M\times Id} & T^*M\times \R \ar[d]^{\tau^*_M \times Id} \\
M\times \R \ar[r] & M\times \R
}
\end{displaymath}
where $\Phi$ is defined at a point $(m,t)$ as $\Phi(m,X,t) = i_X(\sigma_t)_m$, the contraction of $X$ with $\sigma_t$. Let $\tilde{\sigma}$ be the $2$-form $\tilde{\sigma}\in \Omega^2(M\times \R)$ defined at a point $(m,t)$ by:
\begin{displaymath}
\tilde{\sigma}_{(m,t)}(X,\frac{\partial}{\partial t}) = (\sigma_t)_m(X)
\end{displaymath}
where $X\in T_mM$ is a tangent vector. Let $\tilde{\beta}\in \Omega^1(M\times \R)$ be the $1$-form defined similarly as:
\begin{displaymath}
\tilde{\beta}_{(m,t)}(X,\frac{\partial}{\partial t}) = (\beta_t)_m(X)
\end{displaymath}
Now, $TM\times \R$ is a vector subbundle of $TM\times T\R$, hence $E:=\ker(\Phi\vert_{Z\times \R})$ defines a vector subbundle of $i_Z^*TM \times T\R \to Z\times \R$. Note that $E\to Z\times \R$ has fiber $E_{(z,t)} = \ker(\sigma_t)_z$, hence $E$ is a stratified vector bundle over $Z\times \R$ and gathers the bundles $\ker(\sigma_t)\to Z$ into a smooth vector bundle over $Z\times \R$. Note that $\beta$ is a $1$-form on $Z\times \R$ that vanishes on $\frac{\partial}{\partial t}$ everywhere on $M\times \R$ and vanishes on $E$ at $Z\times \R$.
\vspace{3mm}
Since $\ker(\sigma_t)\pitchfork TZ$ and $\ker(\sigma_t) = E\vert_{Z\times \{t\}}$, we have that $E \pitchfork TZ \times T\R$. Therefore, near a point $(z,t) \in Z\times \R$, we may choose a nonvanishing section $w\in \Gamma(E)$ transverse to $Z\times \R$. The section $w$ is stratified and tangent to the leaves $Z\times \{t\}$ for all $t\in \R$. This is because $E$ is stratified and tangent to the leaves $Z\times \{t\}$ for all $t\in \R$. Consequently, we may extend it to a non-vanishing stratified vector field $\tilde{w}$ in a neighborhood $U$ of $(z,t)$ that is tangent to $M\times \{t\}$ for all $t\in \R$. Its flow then defines a diffeomorphism (on a perhaps smaller neighborhood of $(z,t)$ in $U$):
\begin{displaymath}
\phi:U\cap(Z\times \R)\times (-\epsilon,\epsilon) \to U
\end{displaymath}
where the coordinates are $(z,t,s)$ and $\frac{\partial}{\partial s}$ is identified with $\tilde{w}$ under $d\phi$. Consequently, we may assume that our manifold is $Z\times \R \times (-\epsilon, \epsilon)$ with coordinates $(z,t,s)$, $M\times \{t\}$ is identified with $Z\times \{t\}\times (-\epsilon, \epsilon)$, $Z\times \R$ is identified with $Z\times \R \times \{0\}$, and $E \to Z\times \R \times \{0\}$ has fiber containing $\frac{\partial}{\partial s}$ at all points.
\vspace{3mm}
Let $p:Z\times \R \times (-\epsilon,\epsilon) \to Z\times \R$ be the projection and $i:Z\times \R \to Z \times \R \times (-\epsilon, \epsilon)$ the inclusion as the zero section. $i_{\frac{\partial}{\partial s}}\tilde{\sigma}$ vanishes at $s=0$, we may write:
\begin{displaymath}
\tilde{\sigma} = p^*i^*\tilde{\sigma} + s\mu
\end{displaymath}
for some $2$-form $\mu$ where:
\begin{itemize}
\item $\tilde{\sigma}_t = p^*i^*\tilde{\sigma}_t + s\mu_t = \sigma_t + p^*i^*\sigma_t +s\mu_t$ is folded on $Z\times \{t\} \times (-\epsilon, \epsilon)$ for all $t\in \R$,
\item $\mu\big\vert_E$ is non-degenerate on $E$ since $\mu_t \big\vert_{\ker(\sigma)_t}$ is non-degenerate by lemma \ref{lem:symplectize}, and
\item $i_{\frac{\partial}{\partial t}}\mu = 0$ and $i_{\frac{\partial}{\partial t}}\sigma =0$.
\end{itemize}
Now, choose a local frame $\{e_1,\dots, e_{2m-2}, f, \frac{\partial}{\partial t}\}$ of $Z\times \R$ near $(z,t)$, where the $e_i's$ are tangent to the leaves $Z\times \{t\}$ and $f \in \Gamma(E)$ is a section of the kernel bundle tangent to the leaves $Z\times \{t\}$. Extend this frame to the product:
\begin{displaymath}
\{e_1,\dots, e_{2m-2},f,\frac{\partial}{\partial t}, \frac{\partial}{\partial s}\}
\end{displaymath}
by lifting each $e_i$ and $\frac{\partial}{\partial t}$ to the product and appending $\frac{\partial}{\partial s}$.
\vspace{3mm}
Since $f$ and $\frac{\partial}{\partial s}$ span $E$ near $(z,t)$ and $\mu\big\vert_E$ is non-degenerate, we have $\mu(f,\frac{\partial}{\partial s})\ne 0$. We then obtain a coframe:
\begin{displaymath}
\{i_{e_1}\sigma, \dots, i_{e_{2m-2}}\sigma, i_f\mu, i_{\frac{\partial}{\partial s}}\mu, dt\}
\end{displaymath}
near $(z,t)$. Thus, $\tilde{\beta}$ may be written:
\begin{displaymath}
\tilde{\beta} = \sum_{i=1}^{2m-2} a_i(i_{e_i}\sigma) + c_1(i_f\mu) + c_2(i_{\frac{\partial}{\partial s}}\mu) + c_3dt
\end{displaymath}
for some choice of smooth function $a_i$ and $c_i$ near $(z,t)$ on $Z\times \R \times (-\epsilon, \epsilon)$.
\begin{enumerate}
\item Since $\tilde{\beta}(\frac{\partial}{\partial t})=0$, $c_3=0$.
\item Since $\tilde{\beta}(\frac{\partial}{\partial s})$ and $\tilde{\beta}(f)$ vanish at $s=0$, the condition $\mu(\frac{\partial}{\partial s}, f)\ne 0$ at $Z\times \R \times \{0\}$ implies that $c_2$ and $c_1$ vanish at $s=0$. Thus, $c_2=s\tilde{c}_2$ and $c_1=s\tilde{c}_2$ for some smooth function $\tilde{c}_1$ and $\tilde{c}_2$.
\item The time dependent vector field is then given by:
\begin{displaymath}
X = \sum_{i=1}^{2m-2}a_ie_i + \tilde{c}_1f + \tilde{c}_2\frac{\partial}{\partial s}
\end{displaymath}
restricted to the leaves $Z\times \{t\} \times (-\epsilon, \epsilon)$. This is because each of the vectors appearing in the defining equation of $X$ are tangent to the leaves $Z\times \{t\} \times (-\epsilon, \epsilon)$ by construction.
\item Since $a_i$, $\tilde{c}_1$, and $\tilde{c}_2$ are smooth function of $z$, $t$, and $s$, we have that $X$ defines a smooth, time-dependent vector field on $Z\times (-\epsilon,\epsilon)$, which shows that $X_t$ is smooth at points of the fold.
\end{enumerate}
\end{proof}
\subsection{Local Structure Near the Fold}
We now seek to strengthen corollary \ref{cor:fsnormal} to include non-orientable folded-symplectic manifolds and refine lemma \ref{lem:fsnormal} to a statement about folded-symplectic manifolds without boundary. We will prove the following two propositions:
\begin{prop}\label{prop:orientation}
Let $(M,\sigma)$ be a folded-symplectic manifold with corners. Let $Z\subset M$ be the folding hypersurface. Then $Z$ possesses a canonical orientation induced by $\sigma$. Equivalently, the null bundle $\ker(\sigma)\cap TZ$ possesses a canonical orientation induced by $\sigma$.
\end{prop}
\begin{remark}
This strengthens corollary \ref{cor:fsnormal} by showing that there are no choices in defining the orientation on $Z$ \emph{and} the orientation exists even in the case where $Z$ is not co-orientable.
\end{remark}
\begin{definition}\label{def:orientation}
Let $(M,\sigma)$ be a folded-symplectic manifold with corners. Let $Z\subset M$ be the folding hypersurface. We call the orientation on $Z$ afforded by proposition \ref{prop:orientation} the $\sigma$-induced orientation or the \emph{orientation induced by} $\sigma$. We also refer to the orientation on $\ker(\sigma)\cap TZ$ as the \emph{orientation induced by} $\sigma$.
\end{definition}
\begin{prop}\label{prop:fsnormal}
Let $(M,\sigma)$ be a folded-symplectic manifold without boundary. Let $Z\subset M$ be the folding hypersurface with kernel bundle $\ker(\sigma)\to Z$ and suppose $Z$ is co-orientable. Then there exists a neighborhood $U$ of $Z$, a neighborhood $V\subset Z\times \R$ of the zero section, and a diffeomorphism $\phi: V \to U$ of manifolds (without boundary) satsifying:
\begin{enumerate}
\item $\phi(z,0)= z$ for all $z\in Z$ and
\item $\phi^*\sigma = p^*i^*\sigma + d(t^2p^*\alpha)$, where $p:Z\times \R \to Z$ is the projection, $i:Z \to Z\times \R$ is the inclusion as the zero section, and $\alpha\in \Omega^1(Z)$ is a $1$-form that doesn't vanish on $\ker(i_Z^*\sigma)$ (and orients it in the canonical way, necessarily).
\end{enumerate}
\end{prop}
\begin{remark}
Proposition \ref{prop:fsnormal} is a slight generalization of theorem $1$ in \cite{CGW}. In particular, we do not require $M$ to be compact and orientable. The equivariant version of this proposition is proposition \ref{prop:eqfsnormal}.
\end{remark}
The advantage of strengthening lemma \ref{lem:fsnormal} to the form of proposition \ref{prop:fsnormal} may be found in its corollary:
\begin{cor}\label{cor:fsnormal1}
Let $(M,\sigma)$ be a folded-symplectic manifold without boundary. Let $Z\subset M$ be the folding hypersurface with kernel bundle $\ker(\sigma) \to Z$ and suppose $Z$ is co-orientable. Let $\psi:Z\times \R \to Z\times \R$ be the fold map $\psi(z,t)=(z,t^2)$. Then there exists a neighborhood $U$ of $Z$, a neighborhood $V$ of the zero section of $Z\times \R$, and a diffeomorphism of manifolds (without boundary) $\phi:V \to U$ so that:
\begin{displaymath}
\phi^*\sigma = \psi^*\omega
\end{displaymath}
for some symplectic form $\omega\in \Omega^2(V)$.
\end{cor}
Hence, if $Z$ is co-orientable then there is a neighborhood $U$ of $Z$, a symplectic form $\omega\in \Omega^2(U)$ and a fold map $f:U \to U$ satisfying $\sigma = f^*\omega$. This means that every folded-symplectic form with co-orientable folding hypersurface, $Z$, looks like the pullback of a symplectic structure by a fold map in the neighborhood of $Z$.
\subsubsection{Canonical Orientation on the Fold}
Before we begin, let us recall a construction of the Hessian in Morse Theory. Given a manifold without boundary, a smooth function $f:M \to \R$, and a critical point $p\in M$, one may define the Hessian at $p$ $Hf_p:T_pM \times T_pM \to \R$ as follows:
\begin{displaymath}
Hf_p(X,Y) = X(df(\tilde{Y}))
\end{displaymath}
where $\tilde{Y}$ is any extension of the vector $Y\in T_pM$ to a local vector field. It turns out that the degeneracy of $df$ at $p$ is enough to guarantee that the Hessian is independent of the choice of extension $Y$. Note that, using our discussion of the intrinsic derivative, we may also interpret the Hessian as the intrinsic derivative of $f$ at $p$.
\vspace{5mm}
Given a folded-symplectic manifold with corners $(M,\sigma)$, we will perform a similar construction using the degeneracies of $\sigma$ in order to ensure that we have a well-defined result. The spirit of the proof is very similar to that of the proof that $Hf_p$ is well-defined, though it is slightly more technical since we now have a $2$-form instead of a $1$-form $df$.
\begin{remark}\label{rem:intrinsic}
In what follows, we are computing an intrinsic derivative and using it to define the orientation on $Z$. However, since the reader may not be familiar with the intrinsic derivative (q.v. section 2.1.5), we simply compute everything directly. We outline the intrinsic derivative approach in this remark.
Consider the map of vector bundles given by contraction with $\sigma$ (we avoid calling it $\sigma^\#$ since we have reserved that notation for a different map):
\begin{displaymath}
\xymatrixrowsep{.5pc}\xymatrixcolsep{3pc}\xymatrix{
C_{\sigma}: TM \ar[r] & T^*M \\
C_{\sigma}(p,X) \ar[r] & i_X\sigma_p
}
\end{displaymath}
This is a map of vector bundles over $M$, hence we may view it as a section of $\hom(TM,T^*M)$ and compute its intrinsic derivative at points $z\in Z$. Its kernel at $z\in Z$ is just $\ker(\sigma_z)$ and its cokernel may be identified with any $2$-dimensional subspace $V$ of $T_z^*Z$ transverse to $\ker(\sigma_z)^o$. The intrinsic derivative at $z$ gives us a map:
\begin{displaymath}
\xymatrixcolsep{3pc}\xymatrix{
(DC_{\sigma})_z:T_zM \ar[r] & \hom(\ker(\sigma_z), V)
}
\end{displaymath}
Because $\operatorname{corank}(C_{\sigma}) = 2$ along $Z$, we have that $d(C_{\sigma})_z(T_zZ)$ is tangent to $L^2(TM, T^*M)_{C_{\sigma_z}}$, hence projecting to the normal bundle of $L^2(TM,T^*M)$ gives us the zero map. Thus, $(Df)_z:T_zM \to \hom(\ker(\sigma_z),V)$ descends to a map:
\begin{displaymath}
\xymatrixcolsep{3pc}\xymatrix{
(DC_{\sigma})_z:\nu(Z)_z \ar[r] & \hom(\ker(\sigma_z), V)
}
\end{displaymath}
where $\nu(Z)=TM\big\vert_Z / TZ$ is the normal bundle. We can define an orientation at $z$ as follows: pick a direction $w\in T_zM$ transverse to $z$, which gives us a nonzero element of $\nu(Z)_z$. We then define a nonzero element $v$ of $\ker(\sigma_z)\cap T_zZ$ to be \emph{positively oriented} if:
\begin{displaymath}
((DC_{\sigma})_z([w])(w))(v) > 0
\end{displaymath}
that is, if the element $(Df)_z([w])(w) \in V\subset T_z^*Z$ evaluated on $v$ gives a positive number. There is something to prove here. Namely, one must show that $((DC_{\sigma})_z([w])(w))(v)$ is nonzero, but this follows from the fact that $\sigma$ induces a symplectic structure on $\ker(\sigma)$ by differentiating in a direction normal to $Z$ (q.v. lemma \ref{lem:fsnormal}). If the reader is convinced, he or she may skip the proof of proposition \ref{prop:orientation}. On the other hand, if the reader isn't convinced, then let us offer a more explicit proof below.
\end{remark}
\newpage
\begin{proof}[Proof of proposition \ref{prop:orientation}]
\mbox{ }
Let $\dim(Z)=2m-1$. We define the orientation on the fold $Z$ as follows. Let $p\in Z$ be any point in the fold and choose a local non-vanishing section $w\in \Gamma(\ker(\sigma))$ near $p$ transverse to $Z$. Fix an extension $\tilde{w}$ of $w$ to a local vector field on $M$ near $p$. We then define a $1$-form $\alpha\in \Omega^1(Z)$ on $Z$ near $p$ by the equation:
\begin{equation}\label{eq:alpha}
\alpha_x(v) = w_x(\sigma(\tilde{w},\tilde{v}))
\end{equation}
where $x\in Z$ is a point near $p$ and $\tilde{v}$ is any extension of $v\in T_xZ$ to a local vector field on $M$ near $p$. This is just differentiating the local function $\sigma(\tilde{w},\tilde{v})$ in the direction $w_x$. We claim that the $2m-1$ form:
\begin{equation}\label{eq:Omega}
\Omega := (i_Z^*\sigma)^{m-1}\wedge \alpha
\end{equation}
defines an orientation near $p$ which is independent of the choices we have made along the way. So, given $p\in Z$, we have several goals:
\begin{enumerate}
\item Show that the $1$-form $\alpha$ defined near $p$ by equation \ref{eq:alpha} does not depend on the choice of extension $\tilde{v}$.
\item Show that the induced orientation in a neighborhood of $p$ defined by $\Omega$ in equation \ref{eq:Omega} is independent of the choice of $w$ and its extension $\tilde{w}$.
\item Show that the preceding two facts allow us to glue the orientations together on $Z$ to give a globally defined orientation.
\end{enumerate}
We proceed in the above order:
\begin{enumerate}
\item To show that $\alpha$ is well-defined, we use the local model of lemma \ref{lem:fsnormal}. That is, we assume $M$ is a neighborhood of the zero section of $Z\times \R$ and $\sigma=p^*i^*\sigma + t\mu$, where $p:Z\times \R \to Z$ is the projection, $i:Z \to Z\times \R$ is the inclusion as the zero section, and $\mu$ is some $2$-form that is non-degenerate on $\ker(\sigma)$. In this local model, we have that $\frac{\partial}{\partial t} \in \Gamma(\ker(\sigma))$ at points of $Z\times \{0\}$. Therefore, any section $w\in \Gamma(\ker(\sigma))$ transverse to $Z\times \{0\}$ has the form:
\begin{displaymath}
w=g\frac{\partial}{\partial t} + X, \text{ where $X\in \ker(i_Z^*\sigma)$ and $g$ is non-vanishing}.
\end{displaymath}
Its extension $\tilde{w}$ then has the form:
\begin{displaymath}
\tilde{w} = \tilde{g}\frac{\partial}{\partial t} + \tilde{X}
\end{displaymath}
where $\tilde{X}$ is an extension of $X$ to a local vector field in a neighborhood of $Z\times \{0\}$ and $\tilde{g}$ is a local extension of $g$ to a smooth function in a neighborhood of $Z\times \{0\}$. Note that, by definition of extension, $\tilde{X}(z,0)=X(z)\in \ker(i_Z^*\sigma)$ and $\tilde{g}(z,0)=g(z) \ne 0$. We compute:
\begin{displaymath}
\begin{array}{l}
w(\sigma(\tilde{w},\tilde{v})) = \\
\displaystyle (g\frac{\partial}{\partial t}\big\vert_{t=0} + X(z))(p^*i^*\sigma(\tilde{w},\tilde{v}) + t\mu(\tilde{w},\tilde{v})) = \\
\displaystyle g\frac{\partial}{\partial t}\big\vert_{t=0}(\tilde{g}t\mu(\frac{\partial}{\partial t},\tilde{v}) + p^*i^*\sigma(\tilde{X},\tilde{v}) + t\mu(\tilde{X},\tilde{v}) = \\
\displaystyle (g^2\mu(\frac{\partial}{\partial t}, v) + g\mu(X,v))\big\vert_{t=0} + g\frac{\partial}{\partial t}\big\vert_{t=0} p^*i^*\sigma(\tilde{X},\tilde{v})
\end{array}
\end{displaymath}
where the third line follows since $X(z)$ is tangent to $Z$ and the quantity $\sigma(\tilde{w},\tilde{v})_{(z,0)} = \sigma(w,\tilde{v})_{(z,0)}=0$ vanishes along $Z\times \{0\}$. Now, $\tilde{X}\big\vert_{t=0} = X \in \ker(i_Z^*\sigma)$, hence $\tilde{X}=p^*X + tY$, where $p^*X=(X,0)$ is the pullback of the vector field $X\in \Gamma(TZ)$ to the product $Z\times \R$ and $Y$ is some vector field. We then have:
\begin{displaymath}
g\frac{\partial}{\partial t}\big\vert_{t=0}p^*i^*\sigma(\tilde{X},\tilde{v}) = gp^*i^*\sigma(Y,v)\big\vert_{t=0}
\end{displaymath}
which depends only on the extension $\tilde{w}$ of $w$. But, we fixed this extension, hence each of the terms of:
\begin{equation}\label{eq:alpha1}
\alpha(v)=w(\sigma(\tilde{w},\tilde{v})) = (g^2\mu(\frac{\partial}{\partial t}, v) + g\mu(X,v) + gp^*i^*\sigma (Y,v))\big\vert_{t=0}
\end{equation}
do not depend on the extension of $v$ to $\tilde{v}$ and we obtain a well-defined $1$-form $\alpha$ on $Z$ in a neighborhood of $p$. Furthermore, if $v\in \Gamma(\ker(i_Z^*\sigma))$ lies in the kernel of $\sigma$, then equation \ref{eq:alpha1} yields:
\begin{displaymath}
\alpha(v) = g^2\mu(\frac{\partial}{\partial t},v)\big\vert_{t=0}
\end{displaymath}
which is nonzero since $\mu$ is non-degenerate on the kernel bundle $\ker(\sigma) \to Z$. Thus, $\alpha\big\vert_{\ker(i_Z^*\sigma)}$ is nonvanishing, which allows us to define the orientation:
\begin{equation}\label{eq:Omega1}
\Omega:= (i_Z^*\sigma)^{m-1}\wedge \alpha = g^2(i_Z^*\sigma)^{m-1}\wedge \mu(\frac{\partial}{\partial t}, \cdot)
\end{equation}
To see the rightmost equality, simply choose a local frame $v\in \Gamma(\ker(i_Z^*\sigma))$, extend to a local frame of $Z$ near $p$, and evaluate $\Omega$ on this frame. Many of the terms will vanish since $p^*i^*\sigma(v,\cdot) = 0$. Since $g$ is nowhere vanishing on $Z$, $g^2$ is nowhere vanishing and $\Omega$ defines an orientation form on $Z$. Note that we are using the fact that $i_Z^*\sigma$ has maximal rank $2m-2$.
\item Now, if $w_0$ is any other section of $\ker(\sigma)$ transverse to $Z\times \{0\}$, then we may write:
\begin{displaymath}
w_0 = f\frac{\partial}{\partial t} + X_0
\end{displaymath}
for some $X_0\in \ker(i_Z^*\sigma)$ and some smooth function $f\in C^{\infty}(Z)$. Fix an extension $\tilde{w}_0$ of $w_0$ and let $\alpha_0$ be the $1$-form generated by our construction. If we let $v\in \Gamma(\ker(i_Z^*\sigma))$ be a section of the kernel and use equation \ref{eq:alpha1}, we obtain:
\begin{displaymath}
\alpha_0(v) = f^2\mu(\frac{\partial}{\partial t}, v)
\end{displaymath}
hence the induced orientation form $\Omega_0$ is:
\begin{displaymath}
\Omega_0 = (i_Z^*\sigma)^{m-1} \wedge \alpha_0 = f^2(i_Z^*\sigma)^{m-1}\wedge \mu(\frac{\partial}{\partial t} ,\cdot)
\end{displaymath}
That is, all orientation forms constructed in this manner are positive multiples of each other since they are positive multiples of:
\begin{displaymath}
(i_Z^*\sigma)^{m-1} \wedge \mu(\frac{\partial}{\partial t},\cdot)
\end{displaymath}
Thus, given $p\in Z$, we have defined an orientation on a neighborhood $U_p\subset Z$ of $p$ that is independent of our choices.
\item Now, cover $Z$ by the neighborhoods $\{U_p\}_{p\in Z}$, where each $U_p$ is equipped with our orientation. On an overlap $U_{p_0}\cap U_{p_1}$, these orientations must agree since any section $w_i$ of $\ker(\sigma)$ on $U_i$ restricts to a section of $\ker(\sigma)$ on the intersection $U_{p_0}\cap U_{p_1}$. Hence, if we perform the construction of the orientation on $U_{p_0}\cap U_{p_1}$ using $w_0$ and $w_1$, we find that the induced orientations agree since they are independent of the choices of sections of $\ker(\sigma)$. That is, the orientation of $U_{p_0}$ agrees with that of $U_{p_1}$ on the intersection. Thus, our construction produces a global orientation on $Z$.
\end{enumerate}
\end{proof}
\subsubsection{Normal Form for the Fold}
We will need the following weak version of the Poincar\'{e} lemma to prove proposition \ref{prop:fsnormal}.
\begin{lemma}\label{lem:poincare}
Let $Z$ be a manifold with corners and let $\sigma\in \Omega^k(Z\times \R)$ be a $k$-form such that $i_Z^*\sigma = 0$, where $i_Z:Z\times \{0\} \to Z\times \R$ is the inclusion as the zero section. Let $r_s(z,t)=(z,st)$ be the deformation retract of $Z\times \R$ onto $Z\times \{0\}$. Then:
\begin{displaymath}
\sigma = d\int_0^1 \frac{1}{s}r^*_s(i_{t\frac{\partial}{\partial t}}\sigma)ds = d(t\int_0^1 r_s^*(i_{\frac{\partial}{\partial t}}\sigma)ds
\end{displaymath}
\end{lemma}
\begin{proof} \mbox{ } \newline
This is lemma 31.16 in \cite{Mi1}.
\end{proof}
\begin{proof}[Proof of proposition \ref{prop:fsnormal}] \mbox{ }
By assumption, we have a folded-symplectic maniold $(M,\sigma)$ without boundary and a co-orientable folding hypersurface $Z\subset M$. By lemma \ref{lem:fsnormal}, we may assume that a neighborhood of $Z$ is a neighborhood $W$ of the zero section of $Z\times \R$ with folded-symplectic form:
\begin{displaymath}
\sigma = p^*i^*\sigma + t\mu
\end{displaymath}
where $p:Z\times \R \to Z$ is the projection, $i:Z \to Z\times \R$ is the inclusion as the zero section, and $\mu$ is a $2$-form on $W$. Let $\alpha \in \Omega^1(Z)$ be the $1$-form defined by:
\begin{displaymath}
\alpha := i^*(i_{\frac{\partial}{\partial t}}\mu)
\end{displaymath}
By lemma \ref{lem:symplectize}, $\mu\big\vert_{\ker(\sigma)}$ is non-degenerate, hence $\alpha \big\vert_{\ker(i_Z^*\sigma)}$ is non-vanishing. In fact, it defines the canonical orientation of $Z$ in proposition \ref{prop:orientation}. Define:
\begin{displaymath}
\sigma_0:= p^*i^*\sigma + d(\frac{t^2}{2}p^*\alpha)
\end{displaymath}
and
\begin{displaymath}
\sigma_1:= p^*i^*\sigma + t\mu
\end{displaymath}
and let $\sigma_s$ be the path:
\begin{equation}\label{eq:foldpath}
\sigma_s:= (1-s)\sigma_0 + s\sigma_1= p^*i^*\sigma + t((1-s)(dt\wedge p^*\alpha + \frac{t}{2}p^*d\alpha) + s\mu)
\end{equation}
Note that $\ker(\sigma_s)=\ker(\sigma)$ is independent of $s$. Define $\mu_s:= (1-s)(dt\wedge p^*\alpha + \frac{t}{2}p^*d\alpha) + s\mu$ and note that for any oriented section $f \in \Gamma(\ker(i_Z^*\sigma))$, we have:
\begin{displaymath}
\mu_s\big\vert_{t=0}(\frac{\partial}{\partial t}, f) = \mu(\frac{\partial}{\partial t},f) \ne 0
\end{displaymath}
hence $\mu_s\big\vert_{\ker(\sigma)} = \mu_s\big\vert_{\ker(\sigma_s)}$ is non-degenerate. Lemma \ref{lem:symplectize} implies that $\sigma_s$ is folded for each $s$. Thus, there exists a neighborhood $\tilde{V}$ of $Z\times \{0\}$ on which $\sigma_s$ is folded for all $s\in [0,1]$. Now, $\sigma_1-\sigma_0$ is closed and vanishes on $Z\times \{0\}$, hence lemma \ref{lem:poincare} implies:
\begin{displaymath}
\dot{\sigma}_s = \sigma_1-\sigma_0 = d(t\int_0^1r_s^*(i_{\frac{\partial}{\partial t}}(\sigma_1-\sigma_0))ds)
\end{displaymath}
where $r_s(z,t)=(z,st)$ is the deformation retract onto $Z\times \{0\}$. Since $i_{\frac{\partial}{\partial t}}(\sigma_0-\sigma_1)$ vanishes at $Z\times \{0\}$, we may write it as $t\eta$ for some $1$-form $\eta$, hence:
\begin{displaymath}
\dot{\sigma}_s = d(t\int_0^1r_s^*(t\eta)ds) = d(t^2\int_0^1 sr_s^*(\eta)ds)
\end{displaymath}
Let $\beta_s = t\int_0^1 sr_s^*(\eta)ds$ and note that it vanishes at $Z\times \{0\}$, hence it vanishes on $\ker(\sigma)$ and therefore gives us an element $\beta_s \in \Omega^1_{\ker(\sigma_s)}(M) = \Omega^1_{\ker(\sigma)}(M)$ for all $s$. By proposition \ref{prop:Moser}, there exists a smooth time-dependent vector field $X_s$ on $\tilde{V}$ satisfying:
\begin{displaymath}
i_{X_s}\sigma_s = -\beta_s
\end{displaymath}
defined as $X_s:=(\sigma_s^\#)^{-1}(-\beta_s)$. Now, $\dot{\sigma}_s= t\beta_s$, hence $tX_s$ is the unique vector field satisfying:
\begin{displaymath}
i_{tX_s}\sigma_s = -t\beta_s = -\dot{\sigma}_s
\end{displaymath}
Hence, the flow $\phi_s$ of $tX_s$ satisfies $\phi_s^*\sigma_s =\sigma_0$ by proposition \ref{prop:Moser}. Since $tX_s$ vanishes at $Z\times \{0\}$, we may again use proposition \ref{prop:Moser} to obtain a neighborhood $V\subset{\tilde{V}}$ of $Z\times \{0\}$ on which the flow exists for all time $s\in [0,1]$. We then have an open embedding:
\begin{displaymath}
\phi_1: V \to Z\times \R
\end{displaymath}
satisfying $\phi_1(z,0)=(z,0)$ and $\displaystyle\phi_1^*\sigma_1 = \phi_1^*\sigma = \sigma_0 = p^*i^*\sigma + d(\frac{t^2}{2}p^*\alpha)$.
\end{proof}
\begin{remark}
There is a significant difficulty in generalizing proposition \ref{prop:fsnormal} to manifolds with corners. Namely, the time-dependent vector field $X_s$ constructed in the proof may $not$ be stratified, meaning its flow does $not$ generate diffeomorphisms of manifolds with corners. Consider the following example:
\vspace{3mm}
\begin{example}
This example discusses the difficulties inherent in producing stratified, time-dependent vector fields. Let $M= (\R^+)^2 \times \R^2$ with coordinates $(x_1,x_2, t, y)$ and let
\begin{displaymath}
\begin{array}{lcl}
\sigma_0 & =& dx_1\wedge dx_2 + tdt\wedge dy +dx_1 \wedge dy \\
\sigma_1 & =& dx_1\wedge dx_2 + tdt\wedge dy +dx_1 \wedge dy + 3t^2dt\wedge dx_1 \\
\end{array}
\end{displaymath}
and let $\sigma_s$ be the linear path $\sigma_s = (1-s)\sigma_0 + s\sigma_1$, which we may write as:
\begin{displaymath}
\sigma_s = \sigma_0 + s3t^2dt\wedge dx_1 = dx_1\wedge dx_2 + tdt\wedge dy +dx_1 \wedge dy + 3st^2dt\wedge dx_1
\end{displaymath}
Then $\sigma_s$ is folded for all $s$ since:
\begin{itemize}
\item $\sigma_s^2 = tdx_1\wedge dx_2 \wedge dt\wedge dy$, hence $Z=\{t=0\}$ and
\item $i_Z^*\sigma_s= dx_1\wedge dx_2 + dx_1\wedge dy$ has rank $2$ on $(\R^+) \times \{0\}\times \R$.
\end{itemize}
The derivative $\dot{\sigma}_s$ is:
\begin{displaymath}
\dot{\sigma}_s = 3t^2dt\wedge dx_1
\end{displaymath}
There are infinitely many candidates for primitives $\beta_s$ of $\dot{\sigma}_s$, where \emph{primitive} means $d\beta_s = \dot{\sigma}_s$. We consider two of them and show that the vector fields produced from the equation $i_{X_s}\sigma_s = -\beta_s$ are not stratified.
\begin{enumerate}
\item In the proof of proposition \ref{prop:fsnormal}, the primitive of $\dot{\sigma}_s$ would be $\beta_s= t^3dx_1$. The vector field satisfying the equation $i_{X_s}\sigma_s = -\beta_s$ is then $ct^3\frac{\partial}{\partial x_2}$, which is not a stratified vector field on $M$. Indeed, it is transverse to the stratum $\{x_2=0\}$ when $t\ne 0$, meaning its flow is not strata-preserving and therefore does not induce diffeomorphisms of manifolds with corners.
\item Let us try using the primitive $\beta_s= -3x_1t^2dt$. Then the vector field $X_s$ satisfying $i_{X_s}\sigma_s = -\beta_s$ is:
\begin{displaymath}
X_s = 3x_1t\frac{\partial}{\partial y} + 3x_1 t \frac{\partial}{\partial x_2}
\end{displaymath}
which is not stratified since it is transverse to the set $\{x_2=0\}$ when $x_1\ne 0$ and $t\ne 0$.
\end{enumerate}
The point is that one must find a family of primitives $\beta_s$ for $\dot{\sigma}_s$ so that for each $s\in \R$ $\beta_s$ lies in the image of the submodule of stratified vector fields under the map:
\begin{displaymath}
\sigma_s^\#:\Gamma(TM) \to \Omega_{\ker(\sigma_s)}^1(M)
\end{displaymath}
and it is not clear that this is always possible.
\end{example}
\end{remark}
\pagebreak
\section{Hamiltonian Group Actions on Folded-Symplectic Manifolds}
The purpose of this chapter is two-fold. We first review group actions and symplectic group actions for the benefit of the reader, standardizing our notation and recalling some important results about both general group actions and torus actions. This material comprises the bulk of the first two sections and could rightfully be placed in an appendix. However, because Hamiltonian group actions are an integral component of our study, we wish to place the background alongside our development of new theory for folded-symplectic manifolds. In particular, we review symplectic representations of tori, Hamiltonian actions on symplectic manifolds, and the normal form of Guillemin and Sternberg \cite{GS2} for isotropic orbits in Hamiltonian $G$-manifolds. We will need all of these tools in order to discuss the local (equivariant) invariants of a toric, folded-symplectic manifold. Along the way, we will show that every folding hypersurface admits an equivariant symplectization and an equivariant folded-symplectization where the fold is co-orientable. Coupling the normal form of Guillemin-Sternberg with the symplectization of the folding hypersurface produces a local uniqueness result for folding hypersurfaces in toric, folded-symplectic manifolds (q.v. lemma \ref{lem:locunique}), which we will extend to a local uniqueness statement for toric, folded-symplectic manifolds. After we have reviewed the requisite group action theory, we will develop an equivariant analog of the normal form proposition \ref{prop:fsnormal} for the folding hypersurface, which will facilitate our proofs of structural results about toric, folded-symplectic manifolds.
\subsection{Group Actions on Manifolds}
We give a definition of a group action on manifolds \emph{with corners}. However, most subsequent lemmas, definitions, theorems, and propositions will consider only group actions on manifolds \emph{without corners}. We will distinguish between manifolds with corners and those without when necessary.
\begin{definition}\label{def:action}
Let $G$ be a Lie group and let $M$ be a manifold with corners. An action of $G$ on $M$ is a homomorphism:
\begin{displaymath}
\tau:G \to \operatorname{Diff}(M)
\end{displaymath}
so that the action map $\mathcal{A}:G\times M \to M \times M$ given by $\mathcal{A}(g,m)=(\tau(g)(m),m)$ is smooth. For the sake of notation, for each $g\in G$ we write$\tau_g:=\tau(g)$ and refer to it as \emph{the action of g on M}. We often write $g\cdot p$ for the action of an element $g\in G$ on $p\in M$. We reserve the right to switch between $g\cdot p$ and $\tau_g(p)$ freely.
\end{definition}
\begin{definition}\label{def:orbit}
Let $G$ be a Lie group that acts on a manifold without corners $M$. Let $p\in M$ be a point. We define the \emph{orbit of p} to be the set:
\begin{displaymath}
G\cdot p:= \{m\in M\vert \mbox{ } \exists g\in G \text{ such that } m=g\cdot p\}
\end{displaymath}
We define the \emph{stabilizer of p} to be the subgroup:
\begin{displaymath}
G_p:=\{ g\in G \vert \mbox{ }g\cdot p=p\}
\end{displaymath}
\end{definition}
\begin{definition}\label{def:proper}
Let $G$ be a Lie group and let $M$ be a manifold with corners. Suppose $\tau:G \to \operatorname{Diff}(M)$ is an action of $G$ on $M$. We say the action is \emph{proper} if the action map $\mathcal{A}:G\times M \to M \times M$ is proper. That is, if $K\subset M\times M$ is compact, then $\mathcal{A}^{-1}(K)$ is also compact.
\end{definition}
The following lemma is standard in the study of group actions on manifolds. Its proof may be found in \cite{Ka, Mi1}.
\begin{lemma}\label{lem:orbits}
Let $G$ be a Lie group acting properly on a manifold without corners $M$. For each $p\in M$, the stabilizer $G_p$ is a compact subgroup of $G$, hence it is a Lie subgroup of $G$, $G/G_p$ is a smooth manifold, and the action map $\mathcal{A}:G\times M \to M \times M$ induces a map $\mathcal{A}': G/G_p \to M$ given by $\mathcal{A}'([g])=g\cdot p$. The map $\mathcal{A}'$ is a smooth embedding whose image is the orbit $G\cdot p$, hence the orbits of a proper Lie group action are embedded submanifolds.
\end{lemma}
We will need the analog of lemma \ref{lem:orbits} in the case where $M$ is a manifold with corners.
\begin{cor}\label{cor:orbits}
Let $G$ be a Lie group acting properly on a manifold with corners. Then for each $p\in M$, the stabilizer, $G_p$ is a closed subgroup of $G$, $G/G_p$ is a smooth manifold, and the orbit $G\cdot p$ is an embedded submanifold of the stratum containing $p$. In particular, they are embedded submanifolds with corners of $M$ whose boundary is empty.
\end{cor}
\begin{proof} \mbox{ } \newline
Let $\partial^k(M)$ be the stratum of $M$ containing $p$. By definition of a group action, $G$ acts by diffeomorphisms of manifolds with corners, hence it preserves the $k$-boundary $\partial^k(M)$ for each $k\ge 0$. This means that we may consider the restricted action map $\mathcal{A}:G\times \partial^k(M) \to \partial^k(M) \times \partial^k(M)$, which is also proper since any compact subset $K\subset \partial^k(M)\times \partial^k(M)$ is also a compact subset of $M\times M$, hence $\mathcal{A}^{-1}(K)\subset G\times \partial^k(M)$ is compact. It follows from lemma \ref{lem:orbits} that $G\cdot p$ is an embedded submanifold of $\partial^k(M)$, hence it is an embedded submanifold with corners of $M$ whose boundary is empty.
\end{proof}
\begin{definition}\label{def:induced}
Let $G$ be a Lie group acting properly on a manifold $M$ with corners. Let $\fg=\operatorname{Lie}(G)$ and let $\exp:\fg \to G$ be the exponential map. For each $X\in \fg$ we define the induced vector field $X_M\in \Gamma(TM)$ pointwise by the equation:
\begin{displaymath}
X_M(p):=\frac{d}{dt}\big\vert_0 \exp(tX)\cdot p
\end{displaymath}
If we write $\mathcal{A}_p:G \to M$ for the map given by $\mathcal{A}_p(g)=g\cdot p$ then $X_M(p)= d(\mathcal{A}_p)_e(X)$.
\end{definition}
The following statement is a corollary to lemma \ref{lem:orbits}.
\begin{cor}\label{cor:orbits1}
Let $G$ be a Lie group acting properly on a manifold $M$ with corners. Let $\fg=\operatorname{Lie}(G)$ be the Lie algebra of $G$. Then for each point $p\in M$, the tangent space to the orbit is:
\begin{displaymath}
T_p(G\cdot p) := \{X_M(p) \vert \mbox{ } X\in \fg\}
\end{displaymath}
That is, the tangent space to the orbit at $p$ is generated by the induced vector fields at $p$.
\end{cor}
\begin{proof}\mbox{ } \newline
Let $p\in M$ and let $G_p$ be its stabilizer. The map $\mathcal{A}_p:G \to G\cdot p$ given by $\mathcal{A}_p(g)=g \cdot p$ factors through the projection $\pi:G \to G/G_p$:
\begin{displaymath}
\xymatrix{
G \ar[r]^{\mathcal{A}_p} \ar[d]^{\pi} & G\cdot p \\
G/G_p \ar[ur]^{\mathcal{A}'}
}
\end{displaymath}
where $\mathcal{A}'$ is the embedding of $G/G_p$ into $M$ given by lemma $\mathcal{A}'([g])=g\cdot p$ (q.v. lemma \ref{lem:orbits}). Since $\mathcal{A}'$ is an embedding and $\pi$ is a submersion, the composite $\mathcal{A}'\circ \pi$ is a submersion, hence $\mathcal{A}_p$ is a submersion. The image of $d\mathcal{A}_p$ is therefore a surjection, meaning the tangent space $T_p(G\cdot p)$ is generated by the image of $d(\mathcal{A}_p)_e$, which is the space of induced vector fields at $p$.
\end{proof}
\begin{definition}\label{def:diffslice}
Let $G$ be a Lie group acting properly on a manifold without corners $M$. Let $p\in M$ and let $T_p(G\cdot p)$ be the tangent space to the orbit at $p$, which is well-defined by lemma \ref{lem:orbits}. We define the \emph{differential slice} $W$ at $p$ to be the quotient space
\begin{displaymath}
W:=T_pM/(T_p(G\cdot p).
\end{displaymath}
We have a linear action of $G_p$ on $T_pM$ given by
\begin{displaymath}
g \cdot v = d\tau_g(v)
\end{displaymath}
which turns $T_pM$ into a representation of $G_p$. That is, we have a homomorphism $\rho:G_p \to GL(T_pM)$. Since $T_p(G\cdot p)$ is an invariant subspace of this representation, the representation $\rho:G_p \to GL(T_pM)$ descends to a representation $\bar{\rho}:G_p \to W$, hence we refer to $W$ as the \emph{differential slice representation}.
\end{definition}
\begin{remark}
Let $G$ be a Lie group acting acting properly on a manifold without corners $M$. Let $p\in M$, $H=G_p$, and let $W$ be the differential slice representation at $p$. We may form the vector bundle $G\times_H W$, which is the quotient of $G\times W$ by the action $h\cdot(g,v)=(gh^{-1},h\cdot v)$. It inherits an action of $G$ given by $g_0\cdot [(g,v)]=[(g_0g,v)]$. We will see that this vector bundle equipped with this action of $G$ is a local model for the action of $G$ on $M$ near $p$.
\end{remark}
The following theorem, due to Palais, is critical in the study of group actions on manifolds and reduces the study of neighborhoods of orbits to an exercise in representation theory. A proof can be found in \cite{ACL} (in the case where $G$ is compact), \cite{Ka}, \cite{Me}, or \cite{Mi1}.
\begin{theorem}[The Differential Slice Theorem]\label{thm:slice}
Let $G$ be a Lie group acting properly on a manifold without corners $M$ and let $p\in M$ be a point with stabilizer $G_p$ and differential slice representation $W$. There exists an invariant neighborhood $U_1$ of $p$, an invariant neighborhood $U_2$ of the zero section of $G \times _{G_p} W$, and an equivariant diffeomorphism $\phi:U_1 \to U_2$ satisfying $\phi(p)=[(e,0)]$, where $e\in G$ is the identity.
\end{theorem}
\begin{definition}\label{def:orbittype}
Let $G$ be a Lie group acting on a manifold without corners $M$. Let $H\le G$ be a Lie subgroup and let $(H)$ denote the conjugacy class of $H$. We define:
\begin{displaymath}
\begin{array}{l}
M_H = \{p\in M \vert \mbox{ } G_p=H\} \\
M_{(H)} =\{p \in M \vert \mbox{ } G_p\in (H)\}
\end{array}
\end{displaymath}
We refer to $M_{(H)}$ as the \emph{orbit-type stratum of type} $H$.
\end{definition}
\begin{remark}\label{rem:abelian}
Suppose $G$ is an abelian Lie group acting on a manifold with corners. For any Lie subgroup $H\le G$ we have $(H)=\{H\}$, hence $M_H=M_{(H)}$. Thus, when we begin to consider toric actions, we will refer to $M_H$ as an \emph{orbit-type stratum}. Note that, in general, the relationship between $M_H$ and $M_{(H)}$ is that $G\cdot M_H = \{g\cdot p \vert \mbox{ } p\in M_H, \mbox{ } g\in G\}= M_{(H)}$ since the stabilizers of points in the same orbit are related by conjugation.
\end{remark}
The following is a standard corollary to the slice theorem (q.v. \cite{Ka}).
\begin{cor}\label{cor:slice1}
Let $G$ be a Lie group acting properly on a manifold without corners $M$. Let $H\le G$ be a Lie subgroup. Then $M_H$ is a smooth submanifold of $M$ and $M_{(H)}$ is a smooth submanifold of $M$, hence $\bigsqcup_{H\le G} M_{(H)}$ is a decomposition of $M$ into disjoint, smooth submanifolds.
\end{cor}
\begin{proof} \mbox{ } \newline
Let $p$ be a point in $M$ with stabilizer $H=G_p$ and let $W$ be the differential slice representation. Let $W^H$ be the invariant subspace of vectors fixed by the action of $H$. We will show that, around $p$, $M_H$ is isomorphic to $\nu(H) \times_H W^H$ and $M_{(H)}=G\times_H W^H$.
\begin{enumerate}
\item Let $p\in M_H$. By theorem \ref{thm:slice} there is an invariant neighborhood of $p$ equivariantly diffeomorphic to a neigbhorhood of the zero section in $G\times_H W$, where $W=T_pM/(T_p(G\cdot p))$ is the differential slice at $p$. Let $W^H$ be the subspace of $W$ fixed by $H$ and let $\nu(H)$ be the normalizer of $H$ in $G$. We claim that in these coordinates $M_H$ corresponds to $\nu(H)\times_H W^H$. We first show that $M_H \subset \nu(H)\times_H W^H$. Indeed, suppose $[(g,v)]\in G\times_H W$ is a point with stabilizer $H$ and $h\in H$ is an arbitrary element. Then, we have:
\begin{equation}\label{eq:subman}
h\cdot[(g,v)] = [(hg,v)]=[(g,v)] \iff \text{There exists $g_0\in H$ such that $(hgg_0^{-1},g_0\cdot v)=(g,v)$.}
\end{equation}
Thus, $g^{-1}hg=g_0\in H$. Since $h$ was arbitrary, we have $g^{-1}hg\in H$ for all $h\in H$, hence $g$ is in the normalizer of $H$: $g\in \nu(H)$. We still need to show that $v$ is fixed by $H$. We have
\begin{equation}\label{eq:subman1}
h\cdot([g,v]) = ([hg,v])=[(g(g^{-1}hg),v)]=[(g,gh^{-1}g^{-1}v)]=[(g,v)]
\end{equation}
which is true if and only if there exists $g_1\in H$ such that $gg_1^{-1}=g$ and $g_1gh^{-1}g^{-1}v=v$. The first equation implies $g_1=e$, hence $gh^{-1}g^{-1}v=v$. Since $h$ was arbitrary and $g\in \nu(H)$, we have that $H$ fixes $v$. Thus, $[(g,h)]\in \nu(H)\times_H W^H$ and $M_H \subseteq \nu(H)\times_H W^H$.
Let us show the reverse inclusion. If we begin with a point $[(g,v)]\in \nu(H)\times_H W^H$ then for all $h\in H$, $g\in \nu(H)$ and the calculation of equation \ref{eq:subman1} modulo the last equality demonstrates that $h\cdot [(g,v)] = [(g,v)]$, hence $H$ is contained in the stabilizer of $[(g,v)]$. If $\eta\in G$ is some element such that $\eta \cdot[(g,v)]=[(g,v)]$, then the calculation of equation \ref{eq:subman} demonstrates that $\eta g= gh^{-1}$ for some $h\in H$, hence $\eta = gh^{-1}g^{-1}$, which is an element of $H$ since $g$ lies in the normalizer of $H$. Thus, near $p$, $M_H$ is equivariantly diffeomorphic to a neighborhood of the zero section of $\nu(H) \times_H W^H$.
\item By remark \ref{rem:abelian}, the set $M_{(H)}$ is simply the set $G\cdot M_H$. Thus, in the model $G\times_H W$ we have that $M_{(H)}$ corresponds to $G\cdot(\nu(H) \times_H W^H)$. Since the action of $G$ on itself is transitive, this set is $G\times _H W^H$, which gives $M_{(H)}$ the structure of a smooth submanifold.
\end{enumerate}
\end{proof}
\begin{cor}\label{cor:slice2}
Let $M$ be a manifold without corners and let $G$ be a Lie group acting properly on $M$. Let $H\le G$ be a subgroup and consider $M_H$. For each $p\in M_H$, we have:
\begin{displaymath}
T_pM_H= (T_pM)^H
\end{displaymath}
where $(T_pM)^H$ is the subspace of vectors fixed by the action of $H$.
\end{cor}
\begin{remark}
This is proposition 3.3 in \cite{GS1}.
\end{remark}
\begin{proof}\mbox{ } \newline
Since $H$ acts trivially on $M_H$, its induced action on $T_pM_H$ is trivial, so we certainly have $T_pM_H\subseteq (T_pM)^H$. Thus, we focus on showing the reverse inclusion. Using an $H$-invariant metric on $T_pM$, we may write $T_pM = T_p(G\cdot p ) \oplus E$, where $E$ is some invariant complimentary subspace isomorphic to the differential slice representation. Then $(T_pM)^H = T_p(G\cdot p)^H \oplus E^H$. The vectors in $T_p(G\cdot p)$ fixed by $H$ are given by $T_p(\nu(H)\cdot p)$, where $\nu(H)$ is the normalizer of $H$.
By the slice theorem, a neighborhood of the orbit in $M_H$ is isomorphic to a neighborhood of the zero section of $\nu(H)\times_H W^H = \nu(H)/H \times W^H$, where $W\simeq E$ is the differential slice. The point $p$ corresponds to $[(e,0)]$ and the tangent space at $[(e,0)]$ is exactly $T_{[e]}\nu(H)/H \oplus W^H$, which demonstrates that the dimension of the tangent space of $T_pM_H$ is exactly the dimension of $(T_pM)^H = T_p(G\cdot p)^H \oplus E^H$, hence the two spaces are the same.
\end{proof}
A highly nontrivial consequence of the slice theorem is the existence of what is known as a \emph{principal orbit type} in $M$.
\begin{prop}\label{prop:princ}
Let $G$ be a compact Lie group acting on a manifold $M$ without corners. If the orbit space $M/G$ is connected, then there exists a unique conjugacy class $(H)$ of subgroups of $G$ such that the orbit type stratum $M_{(H)}$ is open and dense in $M$.
\end{prop}
\begin{proof}[Summary of the proof] \mbox{ } \newline
This is theorem 4.27 in \cite{Ka}. The proof is several pages, so we opt to summarize it for the purpose of explaining how it is derived from the slice theorem. A condensed version of the proof may be found in Eckhard Meinrenken's notes, \cite{Me}. The strategy is as follows: one first fixes a point $p\in M$ with stabilizer $G_p$ and shows that such an orbit-type stratum exists in the model $G\times_{G_p} W$, where $W$ is the differential slice. This is done using induction on the dimension of $M$ and constructing an equivariant retraction of $G\times_{G_p}(W\setminus \{0\})$ onto the unit sphere bundle $G\times_{G_p} S(V)$, where $S(V)$ is the unit sphere in $V$ for some choice of invariant metric. One then covers $M$ by invariant neighborhoods isomorphic to the local models $G\times_H W$, where $H$ and $W$ depend on the choice of point $p\in M$. A unique, open, dense orbit type stratum exists on each neighborhood. Uniqueness guarantees that the orbit-type strata agree on overlaps, hence there is a unique orbit-type stratum $M_{(H)}\subset M$ that is open and dense.
\end{proof}
We'll need proposition \ref{prop:princ} in our discussion of effective, abelian group actions.
\begin{definition}\label{def:effective}
Let $G$ be a Lie group acting on a manifold with corners $M$. We say that the action of $G$ is \emph{effective} if the homomorphism $\tau:G \rightarrow \operatorname{Diff}(M)$ is injective. Equivalently, for each $g\in G$ with $g\ne e$ there exists a point $p\in M$ such that $\tau_g(p)\ne p$.
\end{definition}
We have several structural lemmas pertaining to effective, abelian group actions.
\begin{lemma}\label{lem:eff1}
Let $M$ be a manifold without corners and suppose $G$ is a compact abelian group acting effectively on $M$. Then the set $M_{e}$ is open and dense in $M$. That is, the set where the action of $G$ is free is open and dense in $M$.
\end{lemma}
\begin{proof} \mbox{ } \newline
By proposition \ref{prop:princ}, there exists an orbit-type stratum $M_{(H)}$ that is open and dense in $M$. Since $G$ is abelian, $M_{(H)}=M_H$, hence there is a subgroup $H\le G$ such that $M_H$ is open and dense in $M$. If $H$ is not $\{e\}$, then there exists $h\in H$, $h\ne e$ and $\tau_h$ acts trivially on $M_H$. Since $M_H$ is open and dense and the action is smooth, $\tau_h$ fixes all of $M$. This means that the action is not effective, contradicting our assumptions.
\end{proof}
\begin{lemma}\label{lem:eff2}
Let $M$ be a manifold without corners and suppose $G$ is a compact abelian group acting effectively on $M$. Then $G$ acts effectively on every invariant, open subset of $M$.
\end{lemma}
\begin{proof}\mbox{ } \newline
Assume that there is an invariant open subset $U\subset M$ so that the action of $G$ is not effective. Then there exists $g\in G$, $g\ne e$, such that $\tau_g\vert_U = id_U$, which means the stabilizer of every point in $U$ is nontrivial. However, lemma \ref{lem:eff1} implies that $G$ acts freely on an open dense subset of $M$, hence it acts freely on an open dense subset of $U$. Thus, we have arrived at a contradiction and we must have that $G$ acts effectively on \emph{all} invariant, open subsets of $M$.
\end{proof}
\begin{lemma}\label{lem:eff3}
Let $M$ be a manifold without corners and suppose $G$ is a compact abelian group acting effectively on $M$. Let $p\in M$, let $H=G_p$ be the stabilizer, and let $W=T_pM/T_p(G\cdot p)$ be the differential slice representation of $H$ at $p$. Then $H$ acts effectively on $W$. Equivalently, the representation $\rho:H \to GL(W)$ is faithful.
\end{lemma}
\begin{proof}\mbox{ } \newline
Suppose there is an element $h\in H$ such that $h\ne e$ and $h$ fixes all elements of $W$. By the slice theorem, an invariant neighborhood of $p$ is isomorphic to an invariant neighborhood of the zero section of $G\times_H W$. We then have that for any element $[(g,v)]\in G\times_H W$:
\begin{displaymath}
h\cdot [(g,v)] = [(hg,v)]= [(gh,v)] = [(g,h^{-1}\cdot v)]= [(g,v)]
\end{displaymath}
which means that $h$ fixes an invariant neighborhood of $p$. This contradicts the conclusion of lemma \ref{lem:eff3}, hence we must have that the action of $h$ on $W$ is effective and the representation $\rho: H \to GL(W)$ is faithful.
\end{proof}
\subsection{Hamiltonian Actions in the Symplectic Case}
We introduce Hamiltonian actions for symplectic manifolds, study symplectic representations on vector spaces, and introduce a local normal form for proper, Hamiltonian actions on symplectic manifolds. The two key components of this sections are the study of symplectic weights for representations of tori and the existence of a local normal form. Everything in this section is review.
First, recall that a symplectic manifold is a manifold $M$ equipped with a closed, non-degenerate two form $\omega$. Equivalently, it is a folded-symplectic manifold $(M,\omega)$ where the folding hypersurface $Z\subset M$ is empty. A symplectic vector space is a vector space $V$ equipped with a non-degenerate element $\omega \in \Lambda^2(V^*)$, hence $(V,\omega)$ is a symplectic manifold since the form is constant, hence closed. The group $\operatorname{Symp}(V,w)$ is the group of linear automorphisms of $V$ preserving $\omega$. Let us recall some basic constructions:
\begin{enumerate}
\item If $V_1\subseteq V$ is a subspace, then $V_1^{\omega} = \{v\in V \mbox{ } \vert \mbox { } \omega(v,v_1)=0 \text{ for all $v_1\in V_1$}\}$
\item A subspace $V_1\subseteq V$ is isotropic if $V_1\subseteq V_1^{\omega}$.
\item A subspace $V_1 \subseteq V$ is coisotropic if $V_1^{\omega} \subseteq V_1$.
\item A subspace is Lagrangian if $V_1=V_1^{\omega}$. Lagrangian subspaces are maximally isotropic: there are no larger isotropic subspaces in which they are contained.
\item Finally, if $(M,\omega)$ is a symplectic manifold, then a submanifold $N$ is isotropic/coisotropic/Lagrangian if the fibers of $TN$ are isotropic/coisotropic/Lagrangian in $TM\big\vert_N$
\end{enumerate}
\begin{definition}\label{def:Ham}
Let $(M,\omega)$ be a symplectic manifold and let $G$ be a Lie group which acts properly on $M$. Let $\fg$ be the Lie algebra of $G$ and let $\fg^*$ be its dual. We say the action is \emph{Hamiltonian} if:
\begin{enumerate}
\item for all $g\in G$, $\tau_g^*\omega=\omega$, where $\tau_g:M \to M$ is the action of $g$ on $M$, and
\item there exists an equivariant map $\mu:M\to \fg^*$ satisfying:
\begin{displaymath}
i_{X_M}\omega = -d\langle \mu, X\rangle, \text{ for all $X\in \fg$}
\end{displaymath}
\end{enumerate}
That is, the action preserves the symplectic form $\omega$ and the induced vector fields $X_M$ are Hamiltonian vector fields for the functions $\langle \mu, X \rangle$.
\end{definition}
\subsubsection{Symplectic Representations of Tori}
\begin{definition}\label{def:symrep}
Let $G$ be a Lie group. A \emph{symplectic representation} of $G$ is a homomorphism $\rho:G \to \operatorname{Symp}(V,\omega)$, where $(V,\omega)$ is some symplectic vector space.
\end{definition}
\begin{lemma}\label{lem:symrep}
Let $G$ be a Lie group and let $\rho:G \to \operatorname{Symp}(V,\omega)$ be a symplectic representation. Then the action of $G$ on $V$ is Hamiltonian with moment map defined by:
\begin{displaymath}
\langle \mu(V), X \rangle = -\frac{1}{2}\omega(d\rho_e(X)v,v)
\end{displaymath}
where $X\in \fg$ is a Lie algebra element.
\end{lemma}
\begin{proof}\mbox{ } \newline
We first note that the vector field induced by $X\in \fg$ is $X_V(v)= \frac{d}{dt}\big\vert_0 \rho(\exp(tX))v=d\rho_e(X)v$. Fix a vector $v\in V$ and let $\eta \in T_vV$. We compute:
\begin{displaymath}
\begin{array}{lcl}
\langle d\mu_v(\eta),X \rangle & = & \frac{d}{dt}\big\vert_0 (-\frac{1}{2}\omega(d\rho_e(X)(v+t\eta), v+t\eta)) \\
& = & -\frac{1}{2}(\omega(d\rho_e(X)v, \eta) + \omega(d\rho_e(X)\eta,v)) \\
& = & -\frac{1}{2}(\omega(d\rho_e(X)v,\eta) - \omega(\eta, d\rho_e(X)v) \\
& = & -\frac{1}{2}(\omega(d\rho_e(X)v,\eta)+\omega(d\rho_e(X)v,\eta) \\
& = & -\omega(d\rho_e(X)v,\eta) \\
& = & -(i_{X_V(v)}\omega)(\eta)
\end{array}
\end{displaymath}
Thus, $\mu$ is a moment map for the action of $G$.
\end{proof}
We now discuss symplectic representations of tori and their weights in a modicum of detail. These representations may be viewed as complex representations of tori and the theory of these representations has been well studied (q.v. \cite{Ad,Th}), hence the material in this section is review. We'll need the material in this section when we study orbit spaces of toric, folded-symplectic manifolds.
\begin{definition}
Let $G$ be a torus with Lie algebra $\fg$. Then the exponential map $\exp:\fg \to G$ is a group epimorphism, i.e. it is surjective. We define its kernel $\mathbb{Z}_G:=\ker(\exp)$ to be the \emph{integral lattice of} $G$. The dual lattice $\mathbb{Z}_G^*$ is the set of all elements $\beta\in \fg^*$ for which $\beta(X)\in \mathbb{Z}$ for every $X\in \mathbb{Z}_G$.
\end{definition}
\begin{remark}
Given a torus $G$ with exponential map $\exp:\fg \to G$, we can define an action of $G$ on $\C^n$ if we are given a (multi)set of weights $\{\beta_1, \dots,\beta_n\}$. We have:
\begin{displaymath}
\exp(X) \cdot (z_1, \dots, z_n) := (e^{2\pi i \beta_1(X)}z_1, \dots, e^{2\pi i \beta_n(X)} z_n)
\end{displaymath}
which gives us a well-defined action of $G$ since the integral lattice $\mathbb{Z}_G$ is mapped to the identity via this recipe, hence $\ker(\exp)$ is sent to the identity transformation and so the above recipe gives a well-defined action of $\fg/\ker(\exp) = G$ on $\C^n$.
\end{remark}
\begin{definition}\label{def:weights}
Let $G$ be a torus. A \emph{character} is a homomorphism $\chi:G \to U(1)$ and a \emph{weight} is a differential of a character at the identity, $\beta=d\chi_e$. Note that there is a one-to-one correspondence between weights and characters which can be seen from the commutative diagram:
\begin{displaymath}
\xymatrix{
\fg \ar[r]^{\beta} \ar[d]^{\exp_G} & \operatorname{Lie}(U(1)) \ar[d]^{\exp_{U(1)}} \\
G \ar[r]^{\xi} & U(1)
}
\end{displaymath}
If we identify $\operatorname{Lie}(U(1))$ with $\R$, then a weight $\beta$ is an element of $\fg^*$. The character associated to $\beta$ is a homomorphism of tori, hence $\beta$ maps $\mathbb{Z}_G$ to $\mathbb{Z}\subset \R$ and so $\beta \in \mathbb{Z}_G^*$. Thus, we refer to the set $\mathbb{Z}_G^*$ as the set of weights for $G$.
\end{definition}
\begin{example}\label{ex:complexrep}
Let $\rho:G \to GL(\C^n)$ be a complex representation of a torus $G$. Then the standard theory of toric representations tells us that there is a multiset of weights associated to $\rho$, $\{\beta_1,\dots, \beta_n\}$, which specifies the representation up to isomorphism. These weights exist because every complex representation of a torus splits as a direct sum of $1$-dimensional complex representations by Schur's lemma (q.v. proposition 3.7 in \cite{Ad}) and the action of $G$ preserves an invariant metric on each summand, hence $\rho$ is really a map $\rho:G \to \oplus_{i=1}^n U(1)$. Projection onto the $i^{th}$ factor gives us a character, which gives us a corresponding weight $\beta_i$. The action of $G$ on a summand $\C$ is given explicitly by $\exp(X)\cdot z = e^{2\pi\beta_i(X)}z$, where $X\in \fg$ is a Lie algebra element. Thus, the weight specifies the representation up to isomorphism.
If we include the symplectic structure $\omega_{\C}=\displaystyle \frac{i}{2\pi}dz \wedge d\bar{z}$ on $\C$ into our calculations, then the action $\exp(X)\cdot z = e^{2\pi\beta_i(X)}z$ is an Hamiltonian action and the moment map is $\mu(z)=\vert z \vert^2\beta_i$. Thus, if we consider the action of $G$ on $\C^n$ with weights $\{\beta_1, \dots, \beta_n\}$, the action is Hamiltonian with moment map:
\begin{displaymath}
\mu(z_1,\dots,z_n)=\sum_{i=1}^n \vert z_i \vert^2 \beta_i
\end{displaymath}
\end{example}
\begin{lemma}\label{lem:sympweights}
Let $G$ be a torus. There exists a one-to-one correspondence between multisets of weights $\{\beta_1, \dots, \beta_n\}$, $\beta_i\in \mathbb{Z}_G^*$, and isomorphism classes of symplectic representations $\rho:G \to \operatorname{Symp}(V,\omega)$, where $\dim(V)=2n$.
\end{lemma}
\begin{proof}\mbox{ } \newline
Choose an invariant almost complex structure $J:V \to V$ compatible with $\omega$, so that $\omega(\cdot, J\cdot)$ is an invariant metric. The choice of $J$ identifies $V$ with $\C^n$, hence we have a complex representation of a torus. Using the metric, for example, we may split $\C^n$ into a direct sum of $1$-dimensional complex representations: $V\simeq \oplus_{i=1}^n V_i$. This is the combination of the facts that all representations split into a direct sum of unique irreducibles (unique up to permutation of summands) and that $G$ is a torus (q.v. lemma 3.25 and theorem 3.24 in \cite{Ad}). These $V_i's$ will also be irreducible \emph{real} representations (q.v. \cite{B,Th}). Now, the $V_i's$ are mutually orthogonal with respect to the metric, hence they are symplectically orthogonal as well since the metric is $\omega(\cdot, J\cdot)$. $J$ restricts to an almost complex structure on each $V_i$, hence we have a splitting $(V,\omega) = \oplus_{i=1}^n (V_i,\omega_i)$ where each factor is linearly, symplectically isomorphic to $(\C, \frac{i}{2\pi}dz\wedge d\bar{z})$ via the choice of $J$. We note that this splitting is independent of the choice of $J$ since the splitting of a representation into isotypicals is unique up to a reordering of the factors (q.v. \cite{Ad,B,Th}). Since the action preserves the metric, each representation $\rho_i: G \to \operatorname{Symp}(V_i,\omega_i)$ is really a character $\chi_i:G \to U(1)$. Thus, a choice of a $J$ gives us a multi-set of weights $\{\beta_1, \dots, \beta_n\}$.
\vspace{5mm}
The weights do not depend on the choice of $J$. Indeed, the space of all invariant almost complex structures is contractible since it is diffeomorphic to the space of invariant metrics, which is an affine space. Thus, for two different choices of invariant almost complex structures, $J_1$ and $J_2$, there is a continuous path $\gamma(t)$ connecting them. For each $(V_i,\omega_i)$, the path of almost complex structures, $\gamma(t)$, gives us a continuous path of characters $\chi_t: G\to U(1)$, hence a continuous path of weights $(\beta_i)_t$. Since the set of weights is discrete, we have that $(\beta_i)_t$ does not depend on $t$, hence the weights do not depend on the choice of invariant almost complex structure.
\vspace{5mm}
Now, because the splitting of $(V,\omega)$ into isotypicals $\oplus_{i=1}^n(V_i,\omega_i)$ is unique up to reordering of the factors, the multiset of weights corresponding to $(V,\omega)$ is unique. That is, $G$ cannot act on a summand $(V_i,\omega_i)$ with two different weights, hence the correspondence $\text{Representations} \to \text{Weights}$ is injective. Furthermore, if we are given a multiset of weights $\{\beta_1,\dots,\beta_n\}$ then we may construct a symplectic representation of $G$ using the recipe $\exp(X)\cdot(z_1,\dots, z_n) = (e^{2\pi i\beta_i(X)}z_1,\dots,e^{2\pi i\beta_n(X)}z_n)$, where $X\in \fg$ is a Lie algebra element. By our discussion, every symplectic representation of $G$ is isomorphic to a complex representation of this form and the multiset of weights corresponding to this representation is unique. Thus, the correspondence between symplectic representations of tori and multisets of weights is both injective and surjective.
\end{proof}
\begin{cor}\label{cor:sympweights}
Let $\rho: G \to \operatorname{Symp}(V,\omega)$ be a symplectic representation of a torus $G$. Then there exist weights $\beta_1,\dots, \beta_n$ associated to $\rho$ and an isomorphism of symplectic representations $\phi:(V,\omega) \to (\C^n,\omega_{\C^n})$, where $G$ acts on the $i^{th}$ factor of $\C^n$ with weight $\beta_i$. Furthermore, the moment map for the action of $G$ on $\C^n$ is:
\begin{displaymath}
\mu(z_1,\dots,z_n)=\sum_{i=1}^n \vert z_i\vert^2\beta_i
\end{displaymath}
\end{cor}
\begin{proof} \mbox{ } \newline
This result follows from lemma \ref{lem:sympweights} and example \ref{ex:complexrep}.
\end{proof}
\begin{lemma}\label{lem:sympweights2}
Suppose $\rho:G \to \operatorname{Symp}(V,\omega)$ is a faithful symplectic representation of a compact abelian group, $G$. Then
\begin{enumerate}
\item $\dim(G)\le \frac{1}{2}\dim(V)$ and
\item if $\dim(G)=\frac{1}{2}\dim(V)$, then $G$ is connected, hence it is a torus. The weights corresponding to this representation form a $\mathbb{Z}$-basis for the set of weights $\mathbb{Z}_G^*$.
\end{enumerate}
\end{lemma}
\begin{proof} \mbox{ } \newline
\begin{enumerate}
\item Let $\dim(V)=2n$. A choice of invariant almost complex structure $J$ on $V$ compatible with $\omega$ allows us to view $\rho$ as a homomorphism $\rho:G \to U(n)$, where $U(n)$ is the unitary group, since $G$ preserves the metric $\omega(\cdot, J\cdot)$. Let $G^0$ be the connected component of the identity in $G$, which is a torus since $G$ is compact and abelian. The image $\rho(G^0)$ of $G^0$ is then a subtorus of $U(n)$, which lies inside a maximal torus of $U(n)$. All such tori are conjugate (q.v. corollary 4.23 in \cite{Ad}), so we may assume it is the standard $n$-torus. Thus,
\begin{displaymath}
\dim(G)=\dim(G_0) = \dim(\rho(G^0)) \le n
\end{displaymath}
where we have used that $\rho$ is injective. Since $n=\frac{1}{2}\dim(V)$, we have $\dim(G) \le \frac{1}{2}\dim(V)$.
\item If $\dim(G)=\frac{1}{2}\dim(V)=n$, then $\rho(G^0)$ is a torus of dimension $n$ in $U(n)$, where $G^0$ is the connected component of the identity. Since all maximal tori in $U(n)$ are of dimension $n$ and $\rho(G^0)$ has dimension $n$, we conclude that $\rho(G^0)$ is itself a maximal torus, hence $\rho\vert_{G^0}$ surjects onto an $n$-dimensional torus in $U(n)$. Since $\rho(G)$ is contained in this torus and $\rho$ is injective, we must have $G^0=G$, else there are two points sent to the same element in $U(n)$.
\item Finally, if we decompose $(V,\omega)$ into its $2$-dimensional isotypical subspaces $(V,\omega)=\oplus_{i=1}^n (V_i,\omega_i)$, then $\rho$ is a map $\rho:G \to U(1)^n$, the characters $\chi_i:G \to U(1)$ are given by projection onto the $i^{th}$ factor of $U(1)^n$, and the weights are $\beta_i=(d\chi_i)_e$. By part $2$, if we assume that $\dim(G)=n$, then $\rho$ is an isomorphism of tori, meaning $d\rho_e$ maps $\ker(\exp_G)$ isomorphically onto $\ker(\exp_{U(1)^n})=\mathbb{Z}^n$. Since $d\rho_e= (\beta_1,\dots,\beta_n)$, this can only happen if the $\mathbb{Z}$ span of $\{\beta_1,\dots,\beta_n\}$ is all of $\mathbb{Z}_G^*$.
\end{enumerate}
\end{proof}
\subsubsection{Symplectic Normal Form}
We would now like to state a few important results in the theory of Hamiltonian actions on symplectic manifolds, which we will use to study the local uniqueness of toric, folding hypersurfaces and, consequently, toric folded-symplectic manifolds.. The following lemma is due to Guillemin and Sternberg and is a staple of the study of Hamiltonian group actions. It is theorem 3.5 in $\cite{GS1}$.
\begin{lemma}\label{lem:symporbit}
Let $(M,\omega)$ be a symplectic manifold with a proper, Hamiltonian action of a Lie group $G$ and corresponding moment map $\mu:M\to \fg^*$. Let $H\le G$ be a subgroup. Then the set $M_H$ is a symplectic submanifold of $M$.
\end{lemma}
\begin{proof} \mbox{ } \newline
We need only show that the tangent space to $M_H$ is symplectic. By corollary \ref{cor:slice2}, for $p\in M_H$ the tangent space $T_pM_H$ is $(T_pM)^H$, the subspace of vectors fixed by the action of $H$. We claim this subspace is symplectic. We may assume $H$ is compact since the action is proper, hence we may choose an invariant almost complex structure $J$ compatible with $\omega$. Then, for $v\in (T_pM)^H$ with $v\ne 0$, $(d\tau_h)_p(v)=v$ implies $(d\tau_h)_p(Jv)=J(d\tau_h)_p(v)=Jv$, hence $Jv$ is also fixed by the action of $H$. Since $\omega(v,Jv)>0$, we see that $\omega$ is nondegenerate on $(T_pM)^H$, hence it is nondegenerate on $T_pM_H$. Thus, $(M_H, i^*\omega)$ is a symplectic submanifold of $M$.
\end{proof}
Since $M_H$ is a symplectic submanifold, the bundle $TM_H^{\omega}$ is a symplectic vector bundle over $TM_H$. Since it is complementary to $TM_H$ in $TM\big\vert_{M_H}$, it is canonically isomorphic to the normal bundle $\nu(M_H)$ and we see that there is a canonical symplectic structure on $\nu(M_H)$. Instead of studying the symplectic normal bundle to $M_H$, we could study the symplectic normal bundle to an orbit $G\cdot p$, the fibers of which are called the \emph{symplectic slice representation}.
\begin{definition}\label{def:sympslice}
Let $(M,\omega)$ be a symplectic manifold with a proper, Hamiltonian action of a Lie group $G$ and let $p\in M$ be a point with stabilizer $H:=G_p$. We define the \emph{symplectic slice} to be:
\begin{displaymath}
V:= \frac{T_p(G\cdot p)^{\omega}}{T_p(G\cdot p) \cap T_p(G\cdot p)^{\omega}}
\end{displaymath}
Since the form $\omega$ is $G$ invariant, it is $H$-invariant and $T_p(G\cdot p)^{\omega}$ is an invariant subspace of $T_pM$. $T_p(G\cdot p)$ is also an invariant subspace since the orbits are invariant under the action of $G$, hence their intersection $T_p(G\cdot p)^{\omega}\cap T_p(G\cdot p)$ is invariant and $V$ inherits the structure of a representation of $H$. Since $H$ preserves $\omega$, the representation is symplectic, hence we often call $V$ the \emph{symplectic slice representation}.
\end{definition}
The significance of the symplectic slice representation is that its data along with the pullback of the symplectic form $\omega$ to the orbit $G\cdot p$ determines an invariant neighborhood of $G\cdot p$ up to equivariant symplectomorphism. This is the content of the equivariant constant rank embedding theorem, which we do not discuss here (q.v. \cite{LS, Me}). We will be concerned with group actions for which the orbits are isotropic. That is, the pullback to the orbit is $0$: $i^*\omega =0$. In this case, the equivariant constant rank embedding theorem is the equivariant version of Weinstein's isotropic embedding theorem. There is a convenient normal form for neighborhoods of such orbits discovered by Guillemin and Sternberg (\cite{GS1}).
\begin{theorem}\label{thm:sympnorm}
Let $(M,\omega)$ be a symplectic manifold with a proper, Hamiltonian action of a Lie group $G$ with moment map $\mu$ and let $p\in M$ be a point with stabilizer $G_p=H$. Suppose the orbit $G\cdot p$ is an isotropic submanifold of $M$ ($i^*\omega =0$) and let $V$ be the symplectic slice representation at $p$. Let $\fg$ be the Lie algebra of $G$ and choose an $H$-equivariant splitting $\fg^*=\frak{h}^*\oplus \frak{h}^o$, where $\frak{h}$ is the Lie algebra of $h$ and $\frak{h}^o$ is its annihilator. Then there exists a symplectic structure on the total space of
\begin{displaymath}
E=G\times_H \frak{h}^o \oplus V,
\end{displaymath}
an invariant neighborhood $U_1$ of the zero section of $E$, an invariant neighborhood $U_2$ of $G\cdot p$, and an equivariant symplectomorphism $\phi:U_1\to U_2$ satisfying $\phi([e,0,0])=p$ so that the moment map $\mu \circ \phi$ is given by:
\begin{equation}
\mu\circ \phi(g,\eta,v) = Ad^*(g)(\eta + \Phi_V(v)) + \mu(p)
\end{equation}
where the action of $G$ on $E$ is given by $g_0\cdot [(g,\eta,v)] = [(g_0g,\eta,v)]$ and $\Phi_V$ is the canonically defined moment map for the action of $H$ on $V$ (q.v. lemma \ref{lem:symrep}).
\end{theorem}
\begin{proof}[Sketch of a Proof] \mbox{ } \newline
We provide a short sketch of how one might prove theorem \ref{thm:sympnorm}. Suppose $(M_i,\omega_i)$, $i=1,2$ are two symplectic manifolds, each with a proper Hamiltonian action of $G$ and moment maps $\mu_i$. Let $p_i\in M_i$ be a point in $M_i$ for $i=1,2$ and suppose the stabilizers $G_{p_1}$, $G_{p_2}$ agree, say $G_{p_i}=H$. Furthermore, suppose that the orbits $G_\cdot p_1$ and $G\cdot p_2$ are isotropic and that $\mu_1(p_1)=\mu_2(p_2)$. The equivariant constant rank embedding theorem states that there is an equivariant symplectomorphism between neighborhoods $U_i$ of $G\cdot p_i$ if and only if the symplectic slice representations are linearly, symplectically isomorphic. Thus, to prove the theorem it suffices to prove that
\begin{itemize}
\item there exists a symplectic structure on $E$,
\item the orbit $G\cdot ([e,0,0])$ is isotropic, and
\item the symplectic slice representations for $G\cdot p$ and $G\cdot([e,0,0])$ are isomorphic.
\end{itemize}
The symplectic structure comes from the fact that $E$ is the reduced space $(T^*G \times V)//_0H$. We will discuss symplectic reduction in chapter 6, so the reader may need to take this as a black box for now. The orbit of $([e,0,0])$ is then the image of the $0$ section of $T^*G \times V$ in the reduced space. Since the zero section in $T^*G \times V$ is isotropic, its image in the reduced space is isotropic. By construction, the symplectic slice representation at $([e,0,0])$ is precisely $V$. Thus, the equivariant constant rank embedding theorem gives us an isomorphism between a neighborhood of $G\cdot p$ in $M$ and a neighborhood of the zero section of $G\times_H \frak{h}^o \oplus V$. The equation for the moment map comes from computing the moment map for the reduced space.
\end{proof}
We will come back to this normal form theorem when we discuss toric actions on folded-symplectic manifolds.
\subsection{Hamiltonian Actions on Folded-Symplectic Manifolds}
We define Hamiltonian actions for folded-symplectic manifolds and construct an equivariant analog of proposition \ref{prop:fsnormal} for folded-symplectic manifolds with co-orientable folding hypersurfaces, which gives us a useful normal form for a neighborhood of the folding hypersurface. Our normal form generalizes theorem $1$ in \cite{CGW} since we do not require compactness or orientability of the folded-symplectic manifold. We then use this normal form to study the structure of folded-symplectic manifolds equipped with Hamiltonian actions of Lie groups. Along the way, we prove a new result that reveals that every folding hypersurface in a folded-symplectic Hamiltonian $G$-manifold may be realized as:
\begin{enumerate}
\item a hypersurface in a \emph{symplectic}, Hamiltonian $G$-manifold or
\item a co-orientable folding hypersurface in a folded-symplectic, Hamiltonian $G$-manifold.
\end{enumerate}
That is, regardless of whether or not a folding hypersurface is co-orientable in the original ambient manifold, we can always extract it and equivariantly embed it into a folded-symplectic manifold as a co-orientable folding hypersurface.
We begin with the definition of an Hamiltonian action.
\begin{definition}\label{def:fsham}
Let $(M,\sigma)$ be a folded-symplectic manifold without corners and let $G$ be a Lie group. We say an action of $G$ on $M$ is Hamiltonian if:
\begin{enumerate}
\item $\tau_g^*\sigma = \sigma$ for all $g\in G$ and
\item There exists an equivariant map $\mu:M\to \fg^*$ satisfying
\begin{displaymath}
i_{X_M}\sigma = -d\langle \mu,X\rangle, \text{ for all lie algebra elements $X\in \fg$.}
\end{displaymath}
\end{enumerate}
We call $\mu$ the moment map and refer to $(M,\sigma)$ with the action of $G$ as a \emph{folded-symplectic Hamiltonian $G$-manifold}.
\end{definition}
\begin{lemma}\label{lem:preserves}
Let $G$ be a Lie group and let $(M,\sigma)$ be a folded-symplectic Hamiltonian $G$-manifold with folding hypersurface $Z$. Then the action of $G$ preserves the fold, $Z$. That is, $Z$ is an invariant submanifold of $M$. Furthermore, the action of $G$ preserves $\ker(\sigma)\to Z$ and if $Z$ is equipped with its orientation induced by $\sigma$, then the action of $G$ on $Z$ is orientation-preserving.
\end{lemma}
\begin{proof}\mbox{ } \newline
We first show that the action preserves $Z$ and then show it is orientation preserving.
\begin{itemize}
\item For all $g\in G$, we have $\tau_g^*\sigma = \sigma$ by the definition of an Hamiltonian action, where $\tau_g$ is the action of $g$ on $M$. Let $2m=\dim(M)$ be the dimension of $M$. Then $\tau_g^*\sigma = \sigma$ implies $\tau_g^*(\sigma^m)=\sigma^m$. Since $\tau_g$ is a diffeomorphism, we must have that $\tau_g(Z) \subset Z$. Since the same logic holds for $g^{-1}$ and $\tau_g^{-1}=\tau_{g^{-1}}$, we have that $\tau_{g}^{-1}(Z) \subset Z$ and so $\tau_g(Z)=Z$.
\item To see that the action of $G$ preserves $\ker(\sigma)$, we pick a point $p\in Z$, a vector $v\in \ker(\sigma_p)$, an element $g\in G$, and we compute:
\begin{displaymath}
i_{d\tau_g(v)}\sigma = \sigma_{g\cdot p}(d\tau_g(v), \cdot ) = \sigma_p(v, d\tau_{g^{-1}}(\cdot)) = 0
\end{displaymath}
hence $\ker(\sigma)$ is an invariant subbundle of $TM\big\vert_Z$. Since $Z$ is invariant, we also have that $\ker(\sigma)\cap TZ$ is an invariant subbundle of $TZ$.
\item We now want to show the action is orientation preserving. It is enough to show the action on $\ker(\sigma)\cap TZ$ is orientation preserving since the orientation on $Z$ is equivalent to a choice of orientation on $\ker(\sigma)\cap TZ$. Let $p\in Z$ and choose a co-orientable neigbourhood $U\subseteq Z$ of $p$. We may choose any non-vanishing section $w$ of $\ker(\sigma)$ on $U$ that is transverse to $Z$ and extend it to a local vector field $\tilde{w}$ in a neighborhood $\tilde{U}$ of $p$. An element $v$ of $\ker(\sigma_p)\cap T_pZ$ is then positively oriented if, for any extension of $v$ to a vector field $\tilde{v}$, we have:
\begin{equation}\label{eq:recipe}
w_p(\sigma(\tilde{w},\tilde{v}))= d(\sigma(\tilde{w},\tilde{v}))_p(w_p) >0
\end{equation}
Fix an element $g\in G$. We define a function:
\begin{displaymath}
\xymatrixrowsep{.5pc}\xymatrixcolsep{3pc}\xymatrix{
f:\tilde{U} \ar[r] & \R \\
f(x) \ar[r] & \sigma_x(\tilde{w}(x),\tilde{v}(x))
}
\end{displaymath}
and a function $h:\tau_g(\tilde{U})\to \R$ given by:
\begin{displaymath}
\xymatrixrowsep{.5pc}\xymatrixcolsep{3pc}\xymatrix{
h:\tau_g(\tilde{U}) \ar[r] & \R \\
h(\tau_g(x)) \ar[r] & \sigma_{\tau_g(x)}(d\tau_g(\tilde{w}(x)),d\tau_g(\tilde{v}(x)))
}.
\end{displaymath}
Then $h\circ \tau_g = f$ by construction. We then have:
\begin{displaymath}
d(\sigma(\tilde{w},\tilde{v}))_p(w_p) = df(w_p) = dh(d\tau_g(w_p))= d(\sigma(d\tau_g(\tilde{w}),d\tau_g(\tilde{v})))_{g\cdot p}(d\tau_g(w_p))
\end{displaymath}
Since $w_p$ is an element of $\ker(\sigma_p)$, $d\tau_g(w_p)$ is an element of $\ker(\sigma_p)$ by part $2$. Since $w_p$ is transverse to $Z$ and $\tau_g$ is a diffeomorphism that preserves $Z$, $d\tau_g(w_p)$ is transverse to $Z$. Since $\ker(\sigma)\cap TZ$ is an invariant subbundle of $TZ$, $v \in \ker(\sigma_p)\cap T_pZ$ if and only if $d\tau_g(v) \in \ker(\sigma_{g\cdot p})\cap T_{g\cdot p}Z$. Lastly, we note that $d\tau_g(\tilde{w})$ and $d\tau_g(\tilde{v})$ are extensions of $d\tau_g(w_p)$ and $d\tau_g(v)$, respectively. Thus, $d\tau_g(v)$ is positively oriented if and only if:
\begin{displaymath}
d(\sigma(d\tau_g(\tilde{w}),d\tau_g(\tilde{v})))_{g\cdot p}(d\tau_g(w_p)) >0
\end{displaymath}
by equation \ref{eq:recipe}. But $d(\sigma(d\tau_g(\tilde{w}),d\tau_g(\tilde{v})))_{g\cdot p}(d\tau_g(w_p))>0$ if and only if $d(\sigma(\tilde{w},\tilde{v}))_p(w_p) = df(w_p)>0$, as we have shown. Hence $v$ is positively oriented if and only if $d\tau_g(v)$ is positively oriented, which shows that the action is orientation-preserving.
\end{itemize}
\end{proof}
\begin{cor}\label{cor:preserves}
Let $G$ be a compact Lie group and let $(M,\sigma)$ be a folded-symplectic Hamiltonian $G$-manifold with folding hypersurface $Z$ and moment map $\mu:M \to \fg^*$. Then there exists an equivariant co-isotropic embedding of $Z$ into a symplectic, Hamiltonian $G$-manifold $(M_0,\omega_0)$. There exists an equivariant fold map $\psi:M_0 \to M_0$ which folds along the image of $Z$, hence $Z$ also embeds into the folded-symplectic Hamiltonian $G$-manifold $(M_0,\psi^*\omega_0)$ as a co-orientable folding hypersurface.
\end{cor}
\begin{proof}\mbox{ } \newline
Since $\ker(\sigma)\cap TZ$ is oriented, we may choose a non-vanishing section $v$ and choose a corresponding $1$-form $\alpha \in \Omega^1(TZ)$ so that $\alpha(v)=1$. Since $G$ is compact, we may average $\alpha$ via the equation:
\begin{displaymath}
\frac{1}{\vert G\vert} \int_G \tau_g^*\alpha dg
\end{displaymath}
so that it is $G$ invariant. Since the action of $G$ is orientation preserving, $\alpha(v)\ne 0$. We then form the $G$-space:
\begin{displaymath}
(Z\times \R, \omega_0=p^*(i_Z^*\sigma) + d(tp^*\alpha)), \text{ $g\cdot (z,t)=(g\cdot z, t)$}.
\end{displaymath}
where $p:Z\times \R \to Z$ is the projection. Since $\sigma$ is closed, $\omega_0$ is closed. Since $\sigma$ is $G$-invariant and $\alpha$ is $G$- invariant, $\omega_0$ is $G$-invariant. We have $\displaystyle dt\wedge p^*\alpha(\frac{\partial}{\partial t},v) >0$, hence $\omega_0$ is non-degenerate at $Z\times \{0\}$ and is non-degenerate in an invariant neighborhood $M_0$ of the zero section $Z\times \{0\}$. A moment map for the action of $G$ on $M_0$ is given by $\mu_s(z,t)= \mu\big\vert_Z(z) + tF(z)$, where $F(z)$ is defined by the equation $\langle F(z), X \rangle = \alpha(X_Z)$, hence the action of $G$ on $M_0$ is Hamiltonian. We therefore have an equivariant co-isotropic embedding into a symplectic, Hamiltonian $G$-manifold. To obtain the fold map, simply restrict $M_0$ to the set of points where $\vert t \vert \le 1$ and set $\psi(z,t)=(z,t^2)$.
\end{proof}
\begin{lemma}\label{lem:preserves1}
Suppose $G$ is a connected Lie group and $(M,\sigma)$ is a folded-symplectic Hamiltonian $G$-manifold, where the action of $G$ is proper. Then for each $p\in Z$, $\ker(\sigma_p)$ is a representation of $G_p$. If the fold is co-orientable, this representation is trivial.
\end{lemma}
\begin{proof} \mbox{ } \newline
For all $g\in G$, we have $\tau_g^*\sigma = \sigma$ by definition of an Hamiltonian action. Thus, for $p\in Z$, $v\in T_pM$, and $h\in G_p$, we have $i_v\sigma_p = 0$ if and only if $i_{(d\tau_h)_pv}\sigma_p =0$, hence $\ker(\sigma_p)$ is an invariant subspace of $T_pM$ for the action of $G_p$. Thus, $\ker(\sigma_p)$ is a representation of $G_p$.
\vspace{3mm}
By lemma \ref{lem:preserves}, the action of $G$ preserves $\ker(\sigma)\cap TZ$, hence $\ker(\sigma_p)\cap T_pZ$ is a representation of $G_p$. Lemma \ref{lem:preserves} also implies that $G$ acts on $\ker(\sigma)\cap TZ$ by orientation-preserving isomorphisms, hence the action of $G_p$ on $\ker(\sigma_p)\cap T_pZ$ is orientation preserving. Since the action of $G$ is proper, $G_p$ is compact by lemma \ref{lem:orbits}. $\ker(\sigma_p)\cap T_pZ$ is $1$-dimensional, hence we may assume that we have a representation $\rho: G_p \to GL(\R)=\R\setminus \{0\}$ of a compact Lie group on $\R$ that is also orientation-preserving. For each $h\in G_p$, $\rho(h)=\pm 1$. Otherwise, the set $\{\rho(h)^n\vert n\in \mathbb{Z}\}$ is unbounded if $\vert\rho(h)\vert>1$, hence the image of $\rho$ is not compact, or its closure contains $0$ if $\vert\rho(h)\vert<1$ and the image is similarly non-compact. In either case, we have a contradiction since $G_p$ itself is compact. Since the action is orientation preserving, $\rho(G_p)=\{1\}$, hence the representation of $G_p$ on $\ker(\sigma_p)\cap T_pZ$ is trivial.
Similarly, if $Z$ is co-orientable then $G$ preserves a choice of co-orientation on $Z$. Co-orientability of $Z$ is equivalent to the orientability of the bundle $\ker(\sigma)/\ker(i_Z^*\sigma)$, since this bundle is isomorphic to the normal bundle of $Z$. Thus, if we fix a co-orientation on $Z$ and choose an invariant complement $V$ to $\ker(\sigma_p)\cap T_pZ$ in $\ker(\sigma_p)$, then $V$ is an oriented $1$-dimensional real representation of $G_p$ and the action is orientation preserving. Thus, the representation is trivial on $V$. Since $\ker(\sigma_p)\simeq V \oplus (\ker(\sigma_p))\cap T_pZ$, we have that the representation $\ker(\sigma_p)$ of $G_p$ is trivial.
\end{proof}
The following is an equivariant analog of proposition \ref{prop:fsnormal}. As with proposition \ref{prop:fsnormal}, it is a generalization of theorem 1 in \cite{CGW}: we do not require that the folded-symplectic manifold be orientable and we do not require compactness.
\begin{prop}\label{prop:eqfsnormal}
Let $G$ be a compact, connected Lie group and suppose $(M,\sigma)$ is a folded-symplectic Hamiltonian $G$-manifold with moment map $\mu:M\to \fg^*$, where $\fg$ is the Lie algebra of $G$. If the folding hypersurface $Z$ is co-orientable, then there exists an invariant neighborhood $U_1$ of the zero section of $Z\times \R$, an invariant neighborhood $U_2$ of $Z$ in $M$, and an equivariant diffeomorphism:
\begin{displaymath}
\phi:U_1 \to U_2, \text{ where the action on $U_1$ is $g\cdot (z,t)=(g\cdot z, t)$}
\end{displaymath}
such that $\phi^*\sigma = p^*i^*\sigma + d(t^2p^*\alpha)$ and $\phi(z,0)=z$ for all $z\in Z$. Here, $p:Z\times \R \to Z$ is the projection, $i:Z\to Z\times \R$ is the inclusion as the zero section, and $\alpha\in \Omega^1(Z)^G$ is an invariant $1$-form that does not vanish on $\ker(i_Z^*\sigma)$ and orients it in the canonical way.
\end{prop}
\begin{proof} \mbox{ } \newline
The proof is essentially the same as the proof of proposition \ref{prop:fsnormal}, except that we must ensure all constructions are equivariant. We therefore walk the reader through the relevant steps and assume all claims unrelated to invariance or equivariance have been proven (as they have been in proposition \ref{prop:fsnormal}).
\begin{itemize}
\item We first note that $G$ preserves the orientation on $\ker(\sigma)\cap TZ$ induced by $\sigma$ (q.v. lemma \ref{lem:preserves}). Furthermore, since $G$ is connected, it preserves any choice of co-orientation on the fold.
\item Since $\ker(\sigma)\cap TZ$ is oriented, we may choose a $1$-form $\alpha \in \Omega^1(Z)$ so that $\alpha\big\vert_{\ker(\sigma)\cap TZ}$ is nonvanishing and positive for any oriented section. Since $G$ is compact, we may average $\alpha$ using the formula:
\begin{displaymath}
\frac{1}{\vert G\vert} \int_G \tau_g^*\alpha dg
\end{displaymath}
hence we may assume $\alpha$ is $G$-invariant. Since $G$ preserves the orientation on $\ker(\sigma)\cap TZ$, we have that this invariant form is still non-vanishing on $\ker(\sigma)\cap TZ$.
\item Now, as in proposition \ref{prop:fsnormal}, we choose a vector $w$ on $M$ so that for each point $z\in Z$ we have $w(z) \in \ker(\sigma_z)$ and $w(z)$ is transverse to $T_zZ$. We may then average $w$ via the formula:
\begin{displaymath}
\frac{1}{\vert G \vert} \int_G d\tau_g(w_{(\tau_g^{-1}(z))})dg
\end{displaymath}
hence we may assume $w$ is $G$-invariant. Since $G$ preserves the co-orientation of $Z$ induced by $w$, the averaged vector field is still transverse to $Z$ at points of $Z$. Since the action of $G$ preserves the subbundle $\ker(\sigma)$, the averaged vector field still has values in $\ker(\sigma)$ at points of $Z$.
\item Since $w$ is $G$-invariant, its flow $\Phi(z,t)$ is $G$-equivariant.
\item We use the flow $\Phi$ of $w$ to define $\tilde{\phi}:Z\times \R \to M$ by
\begin{equation}\label{eq:flowdiff}
\phi(z,t)=\Phi(z,t)
\end{equation}
which satisfies $\phi(z,0)=z$. Furthermore, it is a diffeomorphism in a neighborhood $U$ of $Z\times \{0\}$. By the slice theorem, every neighborhood of a point in $Z\times \R$ contains an invariant neighborhood of the same point, hence we may shrink the neighborhood $U$ to assume it's invariant.
\item $\tilde{\phi}^*\sigma$ and $p^*i^*\sigma + d(t^2p^*\alpha)$ are $G$-invariant, agree at $Z\times \{0\}$, and induce the same orientation on $Z$, hence the linear path
\begin{displaymath}
\sigma_s := s\tilde{\phi}^*\sigma + (1-s)(p^*i^*\sigma + d(t^2p^*\alpha)=p^*i^*\sigma + (1-s)(d(t^2p^*\alpha) + s\mu= p^*i^*\sigma +\mu_s
\end{displaymath}
is folded-symplectic in a neighborhood of $Z\times \{0\}$ by lemma \ref{lem:symplectize} ($\mu_s$ is non-degenerate on the kernel bundle $\ker(\sigma_s)=\ker(\sigma_0)$) and invariant under the action of $G$.
\item As discussed in proposition \ref{prop:fsnormal}, the derivative $\dot{\sigma}_s$ is exact: $\dot{\sigma}_s = -d\beta_s$. Furthermore, we may choose $\beta_s$ so that it vanishes to second order at $Z\times \{0\}$. Since the group $G$ is compact, we may average $\beta_s$:
\begin{displaymath}
\tilde{\beta}_s=\frac{1}{\vert G \vert} \int_G \tau_g^*\beta_s dg
\end{displaymath}
We have
\begin{displaymath}
-d\tilde{\beta}_s = -\frac{1}{\vert G \vert} \int_G \tau_g^*d\beta_s dg = \frac{1}{\vert G \vert} \int_G \sigma_s dg = \dot{\sigma}_s
\end{displaymath}
hence we may assume that the primitive for $\dot{\sigma}_s$ is $G$-invariant. Since the action of $G$ preserves the fold and $\beta_s$ vanishes to second order at $Z\times \{0\}$, we have that $\tilde{\beta}_s$ also vanishes to second order at $Z\times \{0\}$.
\item Since $\tilde{\beta}_s$ vanishes on $\ker(\sigma_s)=\ker(\sigma_0)$ at $Z\times \{0\}$, proposition \ref{prop:Moser} gives us a unique time-dependent vector field $X_s$ so that $i_{X_s}\sigma_s = -\tilde{\beta}$. We claim that $X_s$ is $G$-invariant:
\begin{displaymath}
\sigma_s(d\tau_g(X_s),\cdot )_{g\cdot p} = \sigma_s(X_s,d\tau_{g^{-1}} \cdot)_p = (i_{X_s}\sigma_s)(d\tau_{g^{-1}}\cdot)_p = -(\tau_{g^{-1}}^*\tilde{\beta_s})_p = -(\tilde{\beta_s})_p
\end{displaymath}
Since $X_s$ is unique, we must have $d\tau_g(X_s)=X_s$.
\item The flow of $X_s$ is therefore equivariant and gives us an equivariant isotopy $\phi_s$ so that $\phi_s^*\sigma_s = \sigma_0$. Taking $\phi=\tilde{\phi}\circ \phi_1$, where $\tilde{\phi}$ is defined using a flow in equation \ref{eq:flowdiff}, gives us our requisite diffeomorphism from a neighborhood of the zero section of $Z\times \R$ onto a neighborhood of $Z$ in $M$.
\end{itemize}
\end{proof}
\begin{cor}\label{cor:eqfsnormal1}
Let $G$ be a compact, connected Lie group and let $(M,\sigma)$ be a folded-symplectic Hamiltonian $G$-manifold with moment map $\mu:M\to \fg^*$, where $\fg$ is the Lie algebra of $G$. If the folding hypersurface $Z$ is co-orientable then there exists an invariant neighborhood $U$ of $Z$, a symplectic form $\omega\in \Omega^2(U)$ for which the action of $G$ is Hamiltonian, an equivariant map with fold singularities $\psi:U \to U$ which folds along $Z$ so that $\psi^*\omega= \sigma$, and a symplectic moment map $\mu_s$ so that $\psi^*\mu_s = \mu$ on $U$.
\end{cor}
\begin{proof}\mbox{ } \newline
According to the equivariant normal form proposition \ref{prop:eqfsnormal}, a local model for an invariant neighborhood of $Z$ is given by a neighborhood $V$ of the zero section of $Z\times \R$ equipped with the fold form
\begin{displaymath}
\sigma_0=p^*i^*\sigma + d(t^2p^*\alpha),
\end{displaymath}
where $\alpha\in \Omega^1(Z)$ positively orients the bundle $\ker(\sigma)\cap TZ$ and $\ker(\sigma_0) = (\ker(\sigma)\cap TZ) \oplus (\span(\frac{\partial}{\partial t}))$. As a reminder, $p:Z\times \R \to Z$ is the projection and $i:Z \to Z\times \R$ is the inclusion as the zero section. The form:
\begin{displaymath}
\omega = p^*i^*\sigma + d(tp^*\alpha)
\end{displaymath}
is non-degenerate in a neighborhood of $Z\times \{0\}$ since $dt \wedge p^*\alpha$ is symplectic on $\ker(\sigma_0)$. It is closed since both of the summands are closed $2$-forms. Thus, $\omega$ is symplectic in a neighborhood $V_1$ of $Z\times \{0\}$.
Take the intersection of $V\cap V_1$ with the set of points where $\vert t \vert <1$ and redefine $V$ to be this set. Then the map $\psi(z,t)=(z,t^2)$ maps $V$ to $V$ and folds along $Z\times \{0\}$, $\omega$ is symplectic on $V$, and $\psi^*\omega = \sigma$. A symplectic moment map for the action of $G$ is given by:
\begin{displaymath}
\mu_s(z,t) = \mu_Z(z) + tF
\end{displaymath}
where $\mu_Z = \mu\vert_Z$ and $F$ is defined by $\langle F, X \rangle = \alpha(X_M)$. The folded-symplectic moment map is given by
\begin{displaymath}
\mu(z,t)=\mu_Z(z) + t^2F
\end{displaymath}
hence $\psi^*\mu_s = \mu$. To summarize, we have a commutative diagram:
\begin{displaymath}
\xymatrix{
(U,\sigma) \ar[r]^\psi \ar[dr]^\mu &(U,\omega) \ar[d]^{\mu_s} \\
& \fg^*
}
\end{displaymath}
\end{proof}
The following isn't entirely a corollary of proposition \ref{prop:eqfsnormal}: one can show the orbit-type strata are transverse to $Z$ without constructing the normal form. However, the normal form makes this fact obvious, so we list it as a corollary.
\begin{cor}\label{cor:eqfsnormal2}
Let $G$ be a compact, connected Lie group and let $(M,\sigma)$ be a folded-symplectic Hamiltonian $G$-manifold (without corners) with moment map $\mu:M \to \fg^*$, where $\fg$ is the Lie algebra of $G$. If the folding hypersurface $Z$ is co-orientable, then the orbit-type strata $M_{(H)}$ intersect $Z$ transversely. Furthermore, if $H\le G$ is a subgroup, then $M_H \pitchfork Z$ and $(M_H, i^*\sigma)$ is a folded-symplectic manifold.
\end{cor}
\begin{proof}\mbox{ } \newline
In the local model of proposition \ref{prop:eqfsnormal}, the orbit type strata are given by $M_{(H)}=Z_{(H)} \times \R$, hence they intersect the fold $Z\times\{0\}$ transversely. To prove the second claim, we begin by noting that $M_H = Z_H\times \R$ in the local model, hence $M_H\pitchfork Z$. By lemma \ref{lem:symporbit}, $M_H\setminus Z$ is a symplectic submanifold of $M\setminus Z$, hence we need only check that $(M_H,i^*\sigma)$ is folded-symplectic near $Z$. By corollary \ref{cor:eqfsnormal1} we have a commutative diagram:
\begin{displaymath}
\xymatrix{
(U,\sigma) \ar[r]^\psi \ar[dr]^\mu &(U,\omega) \ar[d]^{\mu_s} \\
& \fg^*
}
\end{displaymath}
where $U$ is an invariant neighborhood of the fold, $\omega$ is symplectic, and $\mu_s$ is a symplectic moment map for the action of $G$. The map $\psi$ is an equivariant fold map. Equivariance implies that $\psi$ restricts to a map $\psi:M_H\cap U \to M_H\cap U$. By corollary \ref{cor:folds4-1}, $\psi$ is a map with fold singularities. Note that $\psi$ is guaranteed to have singularities along $M_H\cap Z$ since $\omega$ is non-degenerate, $\sigma = \psi^*\omega$, and $\sigma$ has singularities along $M_H\cap Z$. Now, lemma \ref{lem:symporbit} implies that $(M_H\cap U,i^*\omega)$ is a symplectic submanifold, hence $(M_H\cap U, \psi^*i^*\omega)$ is folded-symplectic. However, $\psi \circ i = i \circ \psi$, where $i:M_H\cap U \to M$ is the inclusion, hence $\psi^*i^*\omega = i^*\psi^*\omega = i^*\sigma$ and $(M_H\cap U, i^*\sigma)$ is folded-symplectic. Thus, $(M_H,i^*\sigma)$ is a folded-symplectic submanifold of $M$.
\end{proof}
The following proposition allows us to study the normal bundle of $M_H$ and argue that it is naturally a symplectic vector bundle. This proposition will also give us a canonical, invariant complement to the bundle $TM_H$ inside $TM\vert_{M_H}$, which we will use when we study orbit spaces.
\begin{prop}\label{prop:normbundle}
Let $G$ be a compact, connected Lie group and let $(M,\sigma)$ be a folded-symplectic Hamiltonian $G$-manifold with moment map $\mu:M\to \fg^*$, where $\fg$ is the Lie algebra of $G$. Suppose the folding hypersurface $Z$ is co-orientable. Let $H\le G$ be a subgroup and suppose $M_H$ is nonempty. Then there exists a vector bundle $\widetilde{(TM_H)}^{\sigma}\to M_H$ with the following properties:
\begin{enumerate}
\item $\widetilde{(TM_H)}^{\sigma}$ is a subbundle of $TM\big\vert_{M_H}$.
\item The restriction $\widetilde{(TM_H)}^{\sigma}\big\vert_{M\setminus Z}$ to the symplectic portion of $M$ is the vector bundle $T(M_H \setminus Z)^{\sigma}\to (M_H\setminus Z)$.
\item $TM\big\vert_{M_H}$ splits $H$-equivariantly as $TM\big\vert_{M_H} = TM_H \oplus \widetilde{(TM_H)}^{\sigma}$.
\item $\widetilde{(TM_H)}^{\sigma}$ equipped with the restriction of $\sigma$ is a symplectic vector bundle over $M_H$.
\item $\widetilde{(TM_H)}^{\sigma}\big\vert_{Z_H}$ is a subbundle of $TZ_H$.
\end{enumerate}
In other words, the symplectic normal bundle to $M_H\setminus Z$ extends across the fold $Z$ to give us a symplectic normal bundle to $M_H$ and, at points of the intersection $Z_H=M_H\cap Z$, it is tangent to the fold.
\end{prop}
\begin{remark}
There's a straightforward way to see why this bundle should exist. By corollary \ref{cor:eqfsnormal1}, we have a local model for a neighborhood $U$ of the folding hypersurface:
\begin{displaymath}
\xymatrix{
(U,\sigma) \ar[r]^{\psi} & (U,\omega)
}
\end{displaymath}
where $\psi$ is an equivariant fold map that folds along $Z$, $\omega$ is symplectic, and $\psi^*\omega=\sigma$. The bundle $TM_H^{\omega}$ is well-defined and complementary to $TM_H$ in $TM \big\vert_H$. By corollary \ref{cor:folds4-1}, $\psi:M_H \to M_H$ induces an isomorphism on each fiber of the normal bundle $d\psi:\nu(M_H)_p \to \nu(M_H)_{\psi(p)}$. Since $\nu(M_H) \simeq TM_H^{\omega}$, we are led to believe that $\psi$ will allow us to simply pull back the bundle $TM_H^{\omega}$. We do so using the recipe:
\begin{displaymath}
(\widetilde{M_H})^{\sigma}_p :=
\begin{cases}
d\psi_{\psi(p)}^{-1})(TM_H^\omega) & \text{if } p\notin Z_H \\
(d\psi\big\vert_Z)_{\psi(p)}^{-1}(TM_H^{\omega}) & \text{if } p\in Z_H
\end{cases}
\end{displaymath}
Of course, it's not immediately obvious that this works, which is why we need to prove something. Furthermore, we don't want to use a model to define a bundle which appears to arise from the data intrinsic to a folded-symplectic Hamiltonian $G$-manifold, so we define this symplectic normal bundle without appealing to fold maps and symplectic forms, which require choices.
\end{remark}
\begin{proof}\mbox{ } \newline
Let $p\in M_H$. We define the fiber of $\widetilde{(TM_H)}^{\sigma}$ at $p$ to be the elements of $T_pM_H$ that extend to local sections of the restricted tangent bundle $TM\big\vert_{M_H}$ with values in the distribution $(TM_H)^{\sigma}$. It is straightforward to check that this is a vector subspace of $(T_pM_H)^{\sigma}$. We argue that the dimension of $E_p$ is constant and, since all elements of $E_p$ extend to local sections with values in $(TM_H)^{\sigma}$, this fact implies $\widetilde{(TM_H)}^{\sigma}$ is a vector subbundle of $TM\big\vert_{M_H}$.
If $p\in M_H \setminus Z$, then by definition of $\widetilde{(TM_H)}^{\sigma}$, the fiber $\widetilde{(TM_H)}^{\sigma}_p$ is just $(T_pM_H)^{\sigma}$, which proves the second claim of the proposition. Thus, we need only consider the case where we are at the fold: $p\in M_H\cap Z$. To this end, we may use proposition \ref{prop:eqfsnormal} to assume that our manifold is $Z\times \R$ with folded-symplectic form $\sigma=p^*i^*\sigma + d(t^2p^*\alpha)$ for some $1$-form $\alpha \in \Omega^1(Z)$ orienting $\ker(\sigma)\cap TZ$ positively. Here, $p:Z\times \R \to Z$ is the projection and not the point $p$. Let us fix some notation. The kernel of $\sigma$ at $Z\times \{0\}$ is:
\begin{displaymath}
\ker(\sigma) = \span(\frac{\partial}{\partial t}) \oplus (\ker(\sigma)\cap TZ)
\end{displaymath}
and the orientability of $\ker(\sigma)\cap TZ$ means we may choose an invariant, nonvanishing section $v\in\Gamma(\ker(\sigma)\cap TZ)$ so that:
\begin{equation}\label{eq:kersigma}
\ker(\sigma) = \span(\frac{\partial}{\partial t}) \oplus \span(v)
\end{equation}
where $v$ extends to a globally defined vector field on $M_H=Z_H\times \R$ via the recipe $\tilde{v}(z,t)=v(z)$. We'll refer to the extension as $v$ from now on and note that $v$ takes values in $\ker(\sigma)\cap TZ$ at $Z\times \{0\}$. We may also assume that $p^*\alpha(v)=1$ since $p^*\alpha(v)$ is nonvanishing.
As a reminder, the connectedness of the group $G$ implies that the representation of $H$ on $\ker(\sigma_p)$ is trivial by lemma \ref{lem:preserves1}. Thus, $H$ fixes $\ker(\sigma)\big\vert_{M_H\cap Z}$ and corollary \ref{cor:slice2} implies that $\ker(\sigma)\big\vert_{M_H\cap Z}$ is a subbundle of $T(M_H\cap Z)$. That is, $\ker(\sigma)$ is tangent to $M_H$ at points of $M_H\cap Z$. Now, we wish to prove the following claim which gives a precise description of when a vector in $T_p(M_H\cap Z)$ admits an extension to a local section of $TM\big\vert_{M_H}$ with values in $TM_H^{\sigma}$:
\vspace{3mm}
\noindent\textbf{Claim:}Let $p\in Z\times \{0\}$. A vector $X\in \widetilde{(TM_H)}^{\sigma}_p$ extends to a local section of $TM\big\vert_{M_H}$ with values in $(TM_H)^{\sigma}$ if and only if the covector $i_X(dt\wedge p^*\alpha_p)$ vanishes on $\ker(\sigma_p)$. In particular, $\ker(\sigma_p)\cap \widetilde{(TM_H)}^{\sigma}_p= \{0\}$.
\vspace{3mm}
To prove the \emph{only if} portion, we show that if the covector $i_X(dt\wedge p^*\alpha_p)$ doesn't vanish on $\ker(\sigma_p)$, then any extension of $X$ to a local vector field will have values not in $(TM_H)^{\sigma}$ in arbitrarily small neighborhoods of $p$. If the covector $i_X(dt\wedge p^*\alpha_p)$ doesn't vanish on $\ker(\sigma_p)$, then there is an element $Y\in \ker(\sigma_p)$ such that $(dt\wedge p^*\alpha)(X,Y) \ne 0$. As discussed above, $Y$ is a linear combination of $v(p)\in \ker(\sigma_p)\cap T_pZ$ and $\frac{\partial}{\partial t}$, which are both tangent to $M_H$ at $p$, hence $Y$ admits an extension $\tilde{Y}$ to a vector field on $M_H$ which is a linear combination of $v$ and $\frac{\partial}{\partial t}$. In particular, the extension of $Y$ satisfies:
\begin{equation}
i_{\tilde{Y}}p^*i^*\sigma = i_{gv + f\frac{\partial}{\partial t}}p^*i^*\sigma = 0
\end{equation}
hence its contraction with $\sigma$ is:
\begin{equation}\label{eq:contraction}
i_{\tilde{Y}}\sigma = i_{\tilde{Y}}d(t^2p^*\alpha)
\end{equation}
Any extension $\tilde{X}$ of $X$ satisfies $(dt\wedge p^*\alpha)(\tilde{X},\tilde{Y})\ne 0$ in some neighborhood of $p\in M_H$, which depends on the choice of extension. Thus, by equation \ref{eq:contraction} we have:
\begin{equation}
\sigma(\tilde{X},\tilde{Y}) = 2t(dt\wedge p^*\alpha)(\tilde{X},\tilde{Y}) + t^2p^*d\alpha(\tilde{X},\tilde{Y})
\end{equation}
which vanishes transversally at $t=0$ since $(dt\wedge p^*\alpha(\tilde{X},\tilde{Y})\big\vert_{t=0}=(dt\wedge p^*\alpha)(X,Y)\ne 0$, hence $\sigma(\tilde{X},\tilde{Y})$ is nonzero for arbitrarily small, nonzero values of $t$. Thus, $\tilde{X}$ does not take values in $(TM_H)^{\sigma}$ in a neighborhood of $p$.
\vspace{3mm}
For the \emph{if} part, we'll need to construct a vector bundle on $M_H\cap Z$ first. We begin by noting that the local model for a neighborhood of the fold gives us a diagram:
\begin{displaymath}
\xymatrixcolsep{5pc}\xymatrix{
(Z\times \R,\sigma=p^*i^*\sigma + d(t^2p^*\alpha) \ar[r]^{\psi(z,t)=(z,t^2)} & (Z\times \R, \omega=p^*i^*+d(tp^*\alpha))
}
\end{displaymath}
where $\psi^*\omega=\sigma$. At a point $(z,0)\in Z_H\times \{0\}=M_H\cap Z)$ we must have that $(T_{(z,0)}M_H)^{\omega}\subset T_{(z,0)}(Z\times \{0\})$. This is because both null directions $\frac{\partial}{\partial t}$ and $v$ are tangent to $M_H$ and $(i_v\omega)_{(z,0)}=-dt$, hence any vector $X$ transverse to $Z$ at $(z,0)$ pairs with $v$ to give $\omega(v,X)\ne 0$.
Since $(T_{(z,0)}M_H)^{\omega}\subset T_{(z,0)}(Z\times \{0\})$ for any $z\in Z_H$ and $\psi\big\vert_{Z\times \{0\}}$ is the identity, we can form the bundle $d\psi\big\vert_Z^{-1}(TM_H^{\omega})\vert_Z=TM_H^\omega\big\vert_Z$ along $M_H\cap Z$. Since $\psi^*\omega=\sigma$, we have that the fibers of $TM_H^{\omega}\big\vert_{M_H\cap Z}$ are subspaces of the fibers of $TM_H^{\sigma} \to M_H\cap Z$.
Now, to finish the proof of our claim, we note that if $X\in (T_pM_H)^{\sigma}$ satisfies $i_X(dt\wedge p^*\alpha)\big\vert_{\ker(\sigma_p)}=0$, then $X$ must be an element of the fiber of $d\psi^{-1}(TM_H^{\omega})\vert_Z$. Certainly, it must lie in $T_pZ$ since it cannot be transverse to $Z$ else $i_X(dt\wedge p^*\alpha)$ would not vanish on $v$. We then have:
\begin{displaymath}
i_{d\psi(X)}\omega = i_X\omega = (i_Xp^*i^*\sigma + ti_Xp^*d\alpha)\big\vert_{t=0} = i_Xp^*i^*\sigma = i_X\sigma
\end{displaymath}
hence $i_X\omega$ vanishes on $T_pM_H$ because $i_X\sigma$ vanishes on $T_pM_H$ by assumption. Thus, we may extend $X$ to a local section of the bundle $TM_H^{\omega}\big\vert_{M_H\cap Z}$ and then extend it to a section of $TM\big\vert_{M_H}$ via $\tilde{X}(z,t)=X(z)$. This extension does not necessarily have values in $(TM_H)^{\sigma}$, so we correct it. The contraction is:
\begin{equation}
i_{\tilde{X}}\sigma = i_{\tilde{X}}p^*i^*\sigma -2tdtp^*\alpha(\tilde{X}) + t^2i_{\tilde{X}}p^*d\alpha
\end{equation}
We then compute:
\begin{equation}
i_{\tilde{X} - p^*\alpha(\tilde{X})v} \sigma = i_{\tilde{X}}p^*i^*\sigma +t^2i_{\tilde{X} - p^*\alpha(\tilde{X})v}dp^*\alpha
\end{equation}
Note that the $1$-form $t^2i_{\tilde{X} - p^*\alpha(\tilde{X})v}dp^*\alpha$ vanishes at the fold, hence proposition \ref{prop:Ecotangent1} guarantees the existence of a vector field $\Gamma$ such that $i_{\Gamma}=-t^2i_{\tilde{X} - p^*\alpha(\tilde{X})v}dp^*\alpha$. Then we have:
\begin{equation}
i_{\tilde{X}-p^*\alpha(\tilde{X})v + \Gamma}\sigma = i_{\tilde{X}}p^*i^*\sigma
\end{equation}
which vanishes on $M_H$ if and only if it vanishes on $TM_H$ at points of $Z\times \{0\}$ since
\begin{itemize}
\item $(i_{\tilde{X}}p^*i^*\sigma)_{(z,t)} = (i_Xp^*i^*\sigma)_{(z,0)}$ by definition of $\tilde{X}$ and
\item $M_H=Z_H\times \R$, hence the tangent bundle is $TZ_H \times T\R$, hence a $1$-form that is independent of $t$ and vanishes on $TZ_H \times T\R$ at points of $Z\times \{0\}$ will vanish on $TZ_H\times T\R$.
\end{itemize}
Thus, we have shown that
\begin{itemize}
\item if $i_X(dt\wedge p^*\alpha)\big\vert_{\ker(\sigma)}=0$ at a point $(z,0)$, then it lies in $TM_H^{\omega}$ at $(z,0)$ and
\item if $X$ is in the fiber of $TM_H^{\omega}$ at $(z,0)$, then it extends to a local section of $TM\big\vert_{M_H}$ with values in $TM_H^{\sigma}$,
\end{itemize}
which finishes the proof of the claim. Now, from our discussion, we have show that the dimension of the subspace of vectors $X\in (T_pM_H)^{\sigma}$ satisfying $i_X(dt\wedge p^*\alpha)\big\vert_{\ker(\sigma)}=0$ is exactly the rank of $(TM_H)^{\omega}$, which is fixed, hence the dimension of $\widetilde{(TM_H)}^{\sigma}_p$ is independent of the choice of $p\in M_H$ and $\widetilde{(TM_H)}^{\sigma}$ is a vector bundle over $M_H$.
\vspace{3mm}
To prove the remaining claims of the proposition, note that there is nothing to prove away from $Z$ since the bundle away from $Z$ is $TM_H^{\sigma}$, which is a symplectic vector bundle and complementary to $M_H$. Thus, we consider a point $p\in Z$. Using the local model, we have shown that the fiber at $p$ is $T_pM_H^{\omega}$, where $\omega$ is some symplectic form for which the action is Hamiltonian. Since $T_pM_H^{\omega}$ is complementary to $T_pM_H$, we have that $\widetilde{(TM_H)}^{\sigma}$ is complementary to $TM_H$. Secondly, we have shown that $\sigma$ restricted to $T_pM_H^{\omega}$ is $\omega$ restricted to $T_pM_H^{\omega}$. Since $\omega$ is symplectic, $\sigma$ must be symplectic. Thus, $\widetilde{(TM_H)}^{\sigma}$ is a symplectic vector bundle.
\end{proof}
\begin{cor}\label{cor:normbundle}
Let $G$ be a compact, connected Lie group and let $(M,\sigma)$ be a folded-symplectic Hamiltonian $G$-manifold with moment map $\mu:M\to \fg^*$, where $\fg$ is the Lie algebra of $G$. Suppose the folding hypersurface $Z$ is co-orientable. Let $H\le G$ be a subgroup and suppose $M_H$ is nonempty. Then the normal bundle $\nu(M_H) = (TM\vert_{M_H})/TM_H$ is canonically a symplectic vector bundle.
\end{cor}
\begin{proof}\mbox{ } \newline
According to proposition \ref{prop:normbundle}, the bundle $\widetilde{(TM_H)}^\sigma$ is a symplectic vector bundle complementary to $TM_H$ inside the restricted tangent bundle $TM\big\vert_{M_H}$. The projection:
\begin{displaymath}
p: TM\big\vert_{M_H} \to TM\big\vert_{M_H}/(TM_H)
\end{displaymath}
therefore restricts to an isomorphism
\begin{displaymath}
p: \widetilde{(TM_H)}^\sigma \to \nu(M_H).
\end{displaymath}
We define the symplectic structure on $\nu(M_H)$ to be $\omega$ such that $p^*\omega = \sigma \big\vert_{\widetilde{(TM_H)}^\sigma}$, which must be non-degenerate since $p$ is an isomorphism.
\end{proof}
\begin{cor}\label{cor:normbundle1}
Let $G$ be a torus and let $(M,\sigma)$ be a folded-symplectic Hamiltonian $G$-manifold with moment map $\mu:M\to \fg^*$, where $\fg$ is the Lie algebra of $G$. Suppose the folding hypersurface $Z$ is co-orientable. Let $H\le G$ be a subgroup and suppose $M_H$ is nonempty. If the action of $G$ is effective, then the action of $H$ on the fibers of $\widetilde{(TM_H)}^{\sigma}$ is effective, hence the representations $\rho:H\to GL(\widetilde{(TM_H)}^{\sigma}_p)$ are faithful, symplectic representations of $H$.
\end{cor}
\begin{proof}\mbox{ } \newline
Let $p\in M_H$. The action of $H$ on the differential slice $T_pM/T_p(G\cdot p)$ is effective by lemma \ref{lem:eff3}, hence the action of $H$ on $T_pM$ is effective. Since the tangent space $T_pM$ splits as $\tilde{(TM_H)}^{\sigma}_p \oplus T_pM_H$ and the action of $H$ on $T_pM_H$ is trivial, we must have that the action of $H$ on $\widetilde{(TM_H)}^{\sigma}_p$ is effective. Equivalently, the representation $\rho:H \to GL(\widetilde{(TM_H)}^{\sigma}_p)$ is faithful.
\end{proof}
\begin{cor}\label{cor:normbundle2}
Let $(M,\sigma)$ be a folded-symplectic manifold with an Hamiltonian action of a Lie group $G$ and moment map $\mu:M\to \fg*$, where $\fg=\operatorname{Lie}(G)$. Consider the fold $Z$. For each subgroup $H\le G$ there exists a vector subbundle of $TZ\big\vert_{Z_H}$, $(\widetilde{TZ_H})^{\sigma}$ which is a symplectic vector bundle and such that $TZ\big\vert_{Z_H}= TZ_H \oplus (\widetilde{Z_H})^{\sigma}$, $H$-equivariantly.
\end{cor}
\begin{proof}
By corollary \ref{cor:preserves}, we can equivariantly embed $Z$ into a folded-symplectic Hamiltonian $G$ manifold $(M,\sigma)$ as a co-orientable folding hypersurface, where the action of $G$ preserves the co-orientation. The result then follows by taking $(\widetilde{TM_H})^{\sigma}\big\vert_{Z_H}$ since the fibers along $Z_H$ are tangent to $Z$. Note that we aren't assuming $G$ is connected, but we only did this in the statement of proposition \ref{prop:normbundle} to ensure that the action of $G$ preserved the co-orientation of $Z$. This is automatically built into the equivariant embedding of $Z$ into $(M,\sigma)$, hence we can freely drop the connectedness assumption.
\end{proof}
\begin{remark}
Let $(M,\sigma)$ be a folded-symplectic manifold with an Hamiltonian action of a Lie group compact, connected Lie group $G$. Corollary \ref{cor:normbundle2} essentially states that the data encoded in the fibers of the generalized symplectic normal bundle $(\widetilde{TM_H})^{\sigma}$ along $Z_H = M_H\cap Z$ are intrinsic to the folding hypersurface. Again, as in corollary \ref{cor:preserves}, it appears as if the folding hypersurface does not seem to mind how it is embedded into a folded-symplectic manifold.
\end{remark}
\pagebreak
\section{Toric Folded-Symplectic Manifolds}
We now turn our attention to toric folded-symplectic manifolds. These manifolds have a maximal number of commuting Hamiltonian functions whose differentials become linearly dependent at the folding hypersurface. Thus, toric folded-symplectic manifolds may be viewed as somewhat tractable examples of degenerate, completely integrable systems. We define these manifolds in the case that they do not have corners. We'll need a separate definition for corners when we introduce the category $\mathcal{B}_{\psi}$ and we choose to delay it for now to avoid confusion between the cases with corners and the cases without corners.
The main results of this section are as follows. We show that the orbit space of a toric, folded-symplectic manifold with a co-orientable folding hypersurface is a manifold with corners. To prove this result, we first show that the stabilizers in a toric folded-symplectic manifold are tori and then argue that these manifolds are locally standard. We then show that the moment map descends to what we call a unimodular map with folds. We spend a fair amount of time studying what information one can read from the orbital moment map, which turns out to be a great deal. In particular, one can reproduce the null foliation on the folding hypersurface \emph{and} its induced orientation directly from the orbital moment map. These results all represent original work which is motivated by results seen in the studies of origami manifolds (q.v. \cite{CGP, HP}).
\subsection{Definitions and Basic Properties}
\begin{definition}\label{def:TFSmanifold}
A \emph{toric folded-symplectic manifold} (without corners) is a \emph{connected} folded-symplectic manifold $(M,\sigma)$ with an effective, Hamiltonian action of a torus $G$, where $\dim(G)=\frac{1}{2}\dim(M)$. We denote a toric, folded-symplectic manifold as a triple $(M,\sigma,\mu:M \to \fg^*)$, where $\mu$ is a moment map for the action of $G$. We often to omit $G$ from the notation as it is usually implied that we have fixed a torus, $G$.
\end{definition}
\begin{remark}
We are going to classify toric, folded-symplectic manifolds with co-orientable folding hypersurfaces. Hence, throughout much of this section, the reader will see the phrase \emph{with co-orientable folding hypersurface} appear in the hypotheses of the lemmas and propositions.
\end{remark}
\begin{lemma}\label{lem:isoorbits}
Let $(M,\sigma,\mu:M\to \fg^*)$ be a toric, folded-symplectic manifold. Let $p\in M$ be a point and let $G\cdot p$ be the orbit through $p$. Then $G\cdot p$ is istotropic: $i_{G\cdot p}^*\sigma =0$.
\end{lemma}
\begin{proof}
By corollary \ref{cor:orbits1}, the tangent space $T_p(G\cdot p)$ is generated by the induced vector fields: $T_p(G\cdot p)= \{X_M(p) \vert \mbox{ } X\in \fg\}$. Thus, we need only show that $\sigma_p(X_M(p), Y_M(p))$ for every pair of induced vector fields. Let $X,Y \in\fg$ be Lie algebra elements. We compute:
\begin{equation}\label{eq:isotropic}
\sigma(X_M(p),Y_M(p)) = (i_{X_M}\sigma)(Y_M(p)) = -d(\langle \mu, X \rangle)_p(Y_M(p))
\end{equation}
The action of $G$ on $\fg^*$ is trivial since $G$ is abelian, hence the equivariance of $\mu$ implies $G$-invariance of $\mu$: $\mu(g\cdot p ) =Ad^*(g)\mu(p)=\mu(p)$. $G$-invariance of $\mu$ implies the $G$-invariance of $\langle \mu, X \rangle$, hence $\langle \mu, X\rangle$ is constant along orbits and its derivative vanishes along directions tangent to orbits. Thus, the right-hand side of equation \ref{eq:isotropic} is zero and we have:
\begin{displaymath}
\sigma(X_M(p),Y_M(p))=0
\end{displaymath}
for any choice of $X,Y\in \fg$, which means $\sigma$ restricted to orbits is $0$.
\end{proof}
\begin{lemma}\label{lem:stabtori}
Let $(M,\sigma,\mu:M\to \fg^*)$ be a toric, folded-symplectic manifold. Suppose the folding hypersurface $Z\subset M$ is co-orientable. Then the stabilizer of a point $p\in M$ is a subtorus $H=G_p$ of $G$ and $\dim(M_H)=2(\dim(G)-\dim(H))$. Consequently, $(M_H,i_{M_H}^*\sigma, \mu\big\vert_{M_H}:M_H \to \frak{h}^o)$ is a toric, folded-symplectic $G/H$ manifold, where $\frak{h}^o$ is the annihilator of $\frak{h}=Lie(H)$ in $\frak{g}^*$. The kernel bundle $i^*_{M_H}\sigma \to Z_H$ is given by $\ker(\sigma)\big\vert_{Z_H} \to Z_H$.
\end{lemma}
\begin{proof}
Suppose $H\le G$ and $M_H$ is nonempty. We argue that $\dim(M_H)=2(\dim(G)-\dim(H))$, which means that the symplectic normal bundle $(\widetilde{TM_H})^{\sigma}$ has rank $\dim(H)$ by proposition \ref{prop:normbundle}. The symplectic representation of $H$ on the fibers of $(\widetilde{TM_H})^{\sigma}$ is faithful by corollary \ref{cor:normbundle1}, hence lemma \ref{lem:sympweights2} will imply that $H$ is a torus. We now compute the dimension of $M_H$.
By corollary \ref{cor:eqfsnormal2}, $M_H \pitchfork_s Z$, hence we may choose a point $p\in M_H\setminus Z$. The torus $G$ is abelian by definition, hence $H\le G$ fixes $G\cdot p$ and so it fixes $T_p(G\cdot p)$. By corollary \ref{cor:slice2}, we have $T_p(G\cdot p)\subseteq T_pM_H$. Since $M\setminus Z$ is symplectic and $(M\setminus Z)_H=M_H\setminus Z$, lemma \ref{lem:symporbit} implies $M_H\setminus Z$ is symplectic. Since $T_p(G\cdot p)$ is isotropic in $T_p(G\cdot p)$ (q.v. lemma \ref{lem:isoorbits}), we must have $\dim(M_H)\ge 2\dim(G\cdot p) = 2(\dim(G)-\dim(H))$.
By proposition \ref{prop:normbundle}, the symplectic normal bundle $(\widetilde{TM_H})^\sigma$ is complementary to $TM_H$ in $TM\big\vert_{M_H}$, hence its rank $r$ is $r=\dim(M)-\dim(M_H) \le 2\dim(G) - 2(\dim(G)-\dim(H)) = 2\dim(H)$. Thus, $r\le 2\dim(H)$. By corollary \ref{cor:normbundle1}, the representation of $H$ on a fiber of $(\widetilde{TM_H})^{\sigma}$ is symplectic and faithful, hence $2\dim(H) \le r$ by lemma \ref{lem:sympweights2}. We therefore have:
\begin{displaymath}
2\dim(H) \le \operatorname{rank}((\widetilde{TM_H})^{\sigma}) \le 2\dim(H)
\end{displaymath}
hence $\operatorname{rank}((\widetilde{TM_H})^{\sigma})=2\dim(H)$, $H$ is a torus by lemma \ref{lem:sympweights2}, and $\dim(M_H) = \dim(H) - \operatorname{rank}((\widetilde{TM_H})^{\sigma}) = 2\dim(G) - 2\dim(H) = 2(\dim(G)-\dim(H))$.
\vspace{3mm}
To see that $(M_H,i_{M_H}^*\sigma, \mu\big\vert_{M_H})$ is a toric, folded-symplectic manifold we first invoke corollary \ref{cor:eqfsnormal1}, which states that $(M_H,i_{M_H}^*\sigma)$ is folded-symplectic, where the kernel bundle is simply $\ker(i_{M_H}^*\sigma) = \ker(\sigma)\big\vert_{Z_H}$ by inspection of the normal form in proposition \ref{prop:eqfsnormal}. Now, $G$ is a torus and $H$ is a torus, hence $G/H$ is a compact, connected abelian group. That is, it is also a torus. Since the action of $G$ preserves $\sigma$, the action of $G/H$ on $M_H$ preserves $i_{M_H}^*\sigma$. Since $H$ fixes $M_H$, we have that for each element $X \in \frak{h}$ of the Lie algebra of $H$
\begin{displaymath}
-\langle d(\mu\big\vert_{M_H}), X \rangle = i_{M_H}^*(-i_{X_M}\sigma) = 0
\end{displaymath}
since $X_M$ is zero along $M_H$. Thus, $d\mu$ maps into $\frak{h}^o$, hence each connected component of $M_H$ must map into an affine subspace $\eta +\frak{h}^o$, which is isomorphic to $(\frak{g}/\frak{h})^* = Lie(G/H)^*$ via projection, hence $\mu\big\vert_{M_H}$ gives a moment map for the action of $G/H$ on $M_H$.
\end{proof}
\begin{cor}\label{cor:stabtori}
Let $(M,\sigma,\mu:M\to \fg^*)$ be a toric folded-symplectic manifold with co-orientable folding hypersurface $Z$. Let $M_{(H)}=M_H$ be an orbit-type stratum. Then,
\begin{enumerate}
\item There exist well-defined weights $\beta_i$, $1\le i \le h=\dim(H)$, for the symplectic representations of $H$ on the fibers of $(\widetilde{TM_H})^{\sigma}$ and these weights do not change along the connected components of $M_H$.
\item The weights $(\beta_1,\dots,\beta_h)$ form a $\mathbb{Z}$-basis for the weight lattice of $H$.
\end{enumerate}
\end{cor}
\begin{proof}
By lemma \ref{lem:stabtori}, $H$ is a subtorus of $G$, where $G$ is the torus acting on $M$. By corollay \ref{cor:normbundle}, the representations of $H$ on the fibers of $(\widetilde{TM_H})^{\sigma}$ are faithful and symplectic. By lemma \ref{lem:sympweights}, symplectic representations of tori have well-defined weights which specify the representation up to isomorphism, which gives us a (multi)set of weights $\{\beta_1,\dots,\beta_h\}$. By lemma \ref{lem:stabtori}, $\operatorname{rank}((\widetilde{TM_H})^{\sigma}) = 2\dim(H)$. By lemma \ref{lem:sympweights2}, we have that $h=\dim(H)$, the weights $\{\beta_1,\dots,\beta_h\}$ are distinct, and they form a $\mathbb{Z}$-basis for the weight lattice $\mathbb{Z}_H^*$ of $H$.
\end{proof}
\begin{remark}
The following fixed-point corollary of lemma \ref{lem:stabtori} can be proven independently of lemma \ref{lem:stabtori} by studying the folding hypersurface directly. However, lemma \ref{lem:stabtori} helps us to make the claim obvious, so we list it as a corollary. The reader should also be aware of the fact that one needn't require co-orientability of the fold, but for our proof to work it is necessary.
\end{remark}
\begin{cor}\label{cor:stabtori}
Let $(M,\sigma,\mu:M\to \fg^*)$ be a toric, folded-symplectic manifold with co-orientable folding hypersurface $Z\subset M$. Then there are no points in $Z$ fixed by the torus $G$.
\end{cor}
\begin{proof}
Let $M_G$ be the fixed-point stratum of $M$. By lemma \ref{lem:stabtori}, $\dim(M_G)=2(\dim(G)-\dim(G))=0$, hence it has codimension $\dim(M)$ in $M$. By corollary \ref{cor:eqfsnormal1}, $M_G\pitchfork Z$, hence the intersection has codimension $\dim(M)$. But, $Z$ is a hypersurface in $M$, hence it has dimension $\dim(M)-1$ and the maximum codimension would be $\dim(M)-1$. Thus, $M_G\cap Z$ must be empty.
\end{proof}
There is another more interesting way to see corollary \ref{cor:stabtori}. We will show that the null foliation on the folding hypersurface $Z$ is generated by the group action. Towards the end of this chapter, we will show that the generators of the null foliation fit together to give us a vector bundle over $Z$, which will be an invariant of a toric-folded symplectic manifold, albeit a superfluous invariant. As in corollary \ref{cor:stabtori}, one needn't require co-orientability of the fold $Z$, but we are focusing on such manifolds so we require it.
\begin{lemma}\label{lem:generator}
Let $(M,\sigma,\mu:M\to \fg^*)$ be a toric, folded-symplectic manifold with co-orientable folding hypersurface $Z$. Let $p\in Z$ be a point in the fold and let $\ker(\sigma)\cap TZ$ be the line bundle on which $i_Z^*\sigma$ vanishes. Then the fiber $\ker(\sigma_p)\cap T_pZ$ is tangent to the orbit through $p$: $\ker(\sigma_p)\cap T_pZ \subseteq T_p(G\cdot p)$, hence the torus action generates the null foliation.
\end{lemma}
\begin{proof}
Let $p\in Z$, let $H=G_p$ be its stabilizer, and suppose the claim is false at $p$. Since $\ker(\sigma)\cap TZ$ is a line bundle, this means we are assuming $\ker(\sigma_p)\cap T_pZ \cap T_p(G\cdot p)=\{0\}$. We will show that this leads to the dimension of $Z_H=M_H\cap Z$ being too large. By lemma \ref{lem:stabtori}, $\dim(M_H) = 2(\dim(G)-\dim(H))$ and by corollary \ref{cor:eqfsnormal1} $M_H \pitchfork Z$, hence $\dim(Z_H) = \dim(M_H\cap Z) = 2(\dim(G)-\dim(H))-1$.
By corollary \ref{cor:eqfsnormal1}, $M_H$ is folded-symplectic with folding hypersurface $Z_H=M_H\cap Z$. The kernel bundle of $i_{Z_H}^*\sigma$ is just $\ker(i_Z^*\sigma)\big\vert_{Z_H}$ since $\ker(\sigma)$ is tangent to $M_H$ at points of $M_H$, which is true because $H$ acts trivially on the fibers of $\ker(\sigma)$ by lemma \ref{lem:preserves}. Notice that since the orbit is contained in $Z$ and $H$ fixes all elements in $T_p(G\cdot p)$, which is true since $G$ is abelian, we have $T_p(G\cdot p)\subset T_pZ_H$. We therefore have that $\ker(\sigma_p)\cap T_pZ$ and $T_p(G\cdot p)$ are two non-intersecting subspaces of $T_pZ_H$.
Now, $i_{Z_H}^*\sigma$ has maximal rank and the kernel is \emph{not} contained in the isotropic subspace $T_p(G\cdot p)$ by assumption, hence there must be a subspace $V_p\subseteq T_pZ_H$ complementary to $T_p(G\cdot p)+ (\ker(\sigma_p)\cap T_pZ)$ so that $i_{Z_H}^*\sigma$ is nondegenerate on $V_p + T_p(G\cdot p)$. Since $T_p(G\cdot p)$ is isotropic, $\dim(V_p)$ is at least $\dim(G\cdot p)=\dim(G)-\dim(H)$. But, this gives us:
\begin{displaymath}
\dim(Z_H) = \dim(V_p) + \dim(T_p(G\cdot p) + \dim(\ker(\sigma_p)\cap T_pZ) \ge 2(\dim(G)-\dim(H)) + 1 > \dim(Z_H)
\end{displaymath}
which is a contradiction, so we must have that $\ker(\sigma_p)\cap T_pZ$ is contained in $T_p(G\cdot p)$.
\end{proof}
Corollary \ref{cor:stabtori} can now be seen as follows: if $G$ fixes a point $p\in Z$, then we have $\ker(\sigma_p)\cap T_pZ \subseteq T_p(G\cdot p) = \{0\}$ by lemma \ref{lem:generator}. But, $\ker(\sigma_p)\cap T_pZ$ is $1$-dimensional by definition of folded-symplectic, hence $G$ cannot fix any point in $Z$. The following corollary is a computational restatement of lemma \ref{lem:generator}. We will need it to see how one may recover the null foliation on $Z$ using the moment map.
\begin{cor}\label{cor:generator}
Let $(M,\sigma,\mu:M\to \fg^*)$ be a toric folded-symplectic manifold and suppose the folding hypersurface $Z\subset M$ is co-orientable. Then for all $p\in Z$ there exists a lie algebra element $X\in \fg$ such that $X_M(p)\ne 0$ and $X_M(p)\in \ker(\sigma_p)\cap T_pZ$. That is, the induced vector field $X_M$ generates the tangent space to the leaf of the null foliation on $Z$ at $p$.
\end{cor}
\subsection{Toric Symplectic Local Normal Form and the Fold}
We now seek to describe the local structure of a toric symplectic manifold, which will allow us to describe the local structure of the folding hypersurface inside a toric folded-symplectic manifold quite easily. Let us make a few remarks before stating the toric symplectic normal form proposition.
\begin{remark}\label{rem:complement}
Let $G$ be a torus and suppose $H\le G$ is a subtorus with $\dim(H)<\dim(G)$ so that $H\ne G$. Then one can find a complementary subtorus $K\le G$ so that $G\simeq H\times K$ as Lie groups. This works as follows:
\begin{itemize}
\item Since $H\le G$ is a subgroup, $Lie(H)=\frak{k}$ is a Lie subalgebra of $\fg$, hence the integral lattice $\mathbb{Z}_H$ is a sublattice.
\item Recall that an element $\eta$ of $\mathbb{Z}_G$ is \emph{primitive} if there is no element $\eta_0\in \mathbb{Z}_G$ and no positive integer $n$ such that $\eta=n\eta_0$. Since $H$ is a subtorus, the integral lattice $\mathbb{Z}_H$ must contain $h=\dim(H)$ elements, $\eta_1,\dots,\eta_h$, that are primitive in both $\mathbb{Z}_H$ and $\mathbb{Z}_G$.
\item Thus, there exist $k=\dim(G)-\dim(H)$ other primitive elements, $v_1,\dots,v_k$, in $\mathbb{Z}_G$ so that $\{\eta_1,\dots,\eta_h,v_1,\dots,v_k\}$ is a $\mathbb{Z}$-basis for $\mathbb{Z}_G$. Consequently, $K=\exp(\span\{v_1,\dots,v_k\})$ is a subtorus of $G$.
\end{itemize}
\end{remark}
\begin{remark}\label{rem:slicerep}
Let $(M,\sigma,\mu:M\to \fg^*)$ be a toric, folded-symplectic manifold. Choose a point $p\in M\setminus Z$ and let $H=G_p$ be its stabilizer, which must be a subtorus of $G$ by lemma \ref{lem:stabtori} . By lemma \ref{lem:isoorbits}, the orbit $G\cdot p$ is isotropic, hence the symplectic slice at $p$ is:
\begin{displaymath}
V_p = \frac{T_p(G\cdot p)^{\sigma}}{T_p(G\cdot p)}
\end{displaymath}
We claim that this is canonically isomorphic, as a representation of $H$, to the fiber of $(\widetilde{TM_H})^\sigma$. Indeed, we have:
\begin{itemize}
\item $T_p(G\cdot p)^{\sigma}\cap T_pM_H = T_p(G\cdot p)$ since $T_p(G\cdot p)$ is Lagrangian in $T_pM_H$ by lemmas \ref{lem:isoorbits} and \ref{lem:stabtori}.
\item $T_p(G\cdot p)^{\sigma} \cap (\widetilde{TM_H})^{\sigma}_p = (\widetilde{TM_H})^{\sigma}_p$ since, by definition, $(\widetilde{TM_H})^{\sigma}_p$ is the $\sigma$ perpendicular of $T_pM_H$ at points of $M\setminus Z$, hence it must vanish on $T_p(G\cdot p)\subset T_pM_H$.
\end{itemize}
By a dimension count, facilitated by lemma \ref{lem:stabtori}, we have the decomposition into invariant subspaces:
\begin{displaymath}
T_p(G\cdot p)^{\sigma} = (\widetilde{TM_H})^{\sigma}_p \oplus T_p(G\cdot p)
\end{displaymath}
hence $T_p(G\cdot p)^{\sigma}/T_p(G\cdot p) = (\widetilde{TM_H})^{\sigma}_p$ as representations of $H$. Thus, for toric, folded-symplectic manifolds, the existence of the bundle $(\widetilde{TM_H})^{\sigma}$ implies that the notion of a symplectic slice extends across the fold: at a point $z\in Z$ in the fold, we could define the fiber $(\widetilde{TM_H})^{\sigma}_z$ to be the symplectic slice.
\end{remark}
\begin{remark}\label{rem:slicerep2}
Let $(M,\sigma,\mu:M\to \fg^*)$ be a toric, folded-symplectic manifold. Let $p\in M$, let $H=G_p$ be the stabilizer of $p$, and consider the representation of $H$ on $(\widetilde{TM_H})^{\sigma}$. By lemma \ref{lem:stabtori}, $H$ is a torus and corollary \ref{cor:stabtori} tells us that there exist $h=\dim(H)$ weights, $\{\beta_1,\dots,\beta_h\}$, associated to the representation $(\widetilde{TM_H})^{\sigma}_p$ that form a basis for the integral lattice $\mathbb{Z}_H^*$ of $H$. By lemma \ref{lem:sympweights}, there exists an isomorphism of symplectic representations between $(\widetilde{TM_H})^{\sigma}_p$ and the representation of $H$ on $\C^h$ given by:
\begin{displaymath}
\exp(X)\cdot(z_1,\dots,z_h) = (e^{2\pi i\beta_1(X)}z_1, \dots, e^{2\pi i\beta_h(X)},z_h)
\end{displaymath}
\end{remark}
The following normal form proposition is an application of theorem \ref{thm:sympnorm} and remarks \ref{rem:complement} and \ref{rem:slicerep2}. It is lemma B.5 in \cite{KL}.
\begin{prop}\label{prop:torsympnorm}
Let $(M,\omega,\mu:M\to \fg^*)$ be a toric \emph{symplectic} manifold. That is, assume it is folded-symplectic with empty folding hypersurface. Let $p$ be a point in $M$ and let $H=G_p$ be its stabilizer.
\begin{enumerate}
\item Let $\mathbb{T}^h=\R^h/\mathbb{Z}^h$ be the standard torus. There exists an isomorphism $\tau_H: H \to \mathbb{T}^h$ of Lie groups such that the symplectic slice representation at $p$ is isomorphic to the action of $H$ on $\C^h$ obtained from the composition of $\tau_H$ with the standard action of $\mathbb{T}^h$ on $\C^h$, which is:
\begin{equation}\label{eq:action}
[t_1,\dots,t_h]\cdot (z_1,\dots, z_h)=(e^{2\pi i t_1}z_1, \dots, e^{2\pi i t_h}z_h)
\end{equation}
and this isomorphism can be constructed using the weights $\{\beta_1,\dots,\beta_h\}$ of the symplectic slice representation of $H$.
\item Choose a complementary subtorus $K\le G$ and let $\tau:G \to K \times H$ be an isomorphism of Lie groups such that $\tau(a)=(a,e)$ for all $a\in K$. Then there exists a $G$-invariant open neighbourhood $U$ of $p$ in $M$ and a $\tau$-equivariant open symplectic embedding
\begin{displaymath}
j:U \hookrightarrow T^*K \times \C^h
\end{displaymath}
with $j(G\cdot p)=K\times \{0\}$. Here, $K$ acts on $T^*K$ by the lift of the left multiplication and $H$ acts on $\mathbb{C}^h$ by the recipe of equation \ref{eq:action} composed with $\tau_H$. The moment map is:
\begin{equation}\label{eq:momentmap}
\mu\vert_U = \mu(p) + \tau^* \circ \phi \circ j
\end{equation}
where $\phi:T^*K \times \C^h \to \frak{k}^* \oplus \frak{h}^*$ is given by
\begin{equation}\label{eq:stdmomentmap}
\phi((\lambda,\eta),(z_1,\dots,z_h))=(\eta,\sum_{i=1}^h \vert z_j \vert^2\beta_j).
\end{equation}
where the $\beta_j's$ are the weights for the representation of $H$ on $\C^h$ and $\tau^*:\frak{k}^* \times \frak{h}^* \to \fg^*$ is the isomorphism on the duals of the Lie algebras induced by $\tau$.
\end{enumerate}
\end{prop}
\begin{proof}[Sketch of Proof]
By theorem \ref{thm:sympnorm}, a neighborhood of the orbit $G\cdot p$ in $M$ is isomorphic to a neighborhood of the zero section of:
\begin{displaymath}
G\times_H (\frak{h}^o\oplus V) = (T^*G \times V)//_0 H
\end{displaymath}
with its natural symplectic structure. The slice representation has weights $\{\beta_1,\dots,\beta_h\}$ associated to it and these weights form a basis for the weight lattice $\mathbb{Z}_H^*$ of $H$. Consequently, they define an isomorphism $\tau_H$ between $H$ and the standard torus $\mathbb{T}^h$:
\begin{displaymath}
\xymatrixcolsep{5pc}\xymatrix{
\frak{h} \ar[r]^{(\beta_1,\dots,\beta_h)} \ar[d]^{\exp_H} & \mathbb{R}^h \ar[d]^{\exp_{\mathbb{T}^h}} \\
H \ar[r]^{\tau_H} & \mathbb{T}^h
}
\end{displaymath}
By remark \ref{rem:slicerep2}, $V$ is isomorphic, as a symplectic representation, to $\C^h$ where $H$ acts via:
\begin{displaymath}
\exp(X)\cdot(z_1,\dots,z_h) = (e^{2\pi i\beta_1(X)}z_1, \dots, e^{2\pi i\beta_h(X)},z_h)
\end{displaymath}
This is exactly the action generated by the standard torus action composed with the map $\tau_H$. If we choose a complementary subtorus $K$ so that $G\simeq K \times L$, then the reduced space $(T^*G \times \C^h)//_0 H$ becomes:
\begin{displaymath}
(T^*K \times T^*H \times \C^h)//_0 H = T^*K \times (T^*H \times \C^h)//_0 H = T^*K \times C^h
\end{displaymath}
and the moment map for the $K\times H$ action is
\begin{equation}\label{eq:momentmap1}
\phi(\lambda,\eta, z_1,\dots,z_h)= \eta + \sum_{i=1}^h \vert z_i \vert^2 \beta_i + \mu(p)
\end{equation}
Hence, a neighborhood of $G\cdot p$ is isomorphic to a neighborhood of the zero section of $T^*K \times C^h$ with its standard symplectic structure and moment map given by equation \ref{eq:momentmap1}. The isomorphism $\tau:G \to K \times H$ induces a map $\tau^*:\frak{h}^*\times \frak{k}^* \to \frak{g}^*$ and it is straightforward to check that if $\phi$ is the moment map in equation \ref{eq:stdmomentmap}, then $\mu\big\vert_U=\tau^*\circ \phi + \mu(p)$.
\end{proof}
The following corollary states that we can realize $Z$ as a unique embedded hypersurface in the standard symplectic model of proposition \ref{prop:torsympnorm}. This may be taken as a consequence of corollary \ref{cor:preserves}, which shows we can always form an equivariant symplectization of the folding hypersurface. It may also be taken as a consequence of corollary \ref{cor:normbundle2}, which states that the symplectic slice data is intrinsic to the fold.
\begin{cor}\label{cor:foldnorm}
Let $(M,\sigma,\mu:M\to \fg^*)$ be a toric, folded-symplectic manifold with co-orientable folding hypersurface $Z\subset M$. Let $p\in Z$ and let $H=G_p$ be its stabilizer, which is a subtorus by lemma \ref{lem:stabtori}, with $h:=\dim(H)$. Let $K\le G$ be a subtorus of $G$ complementary to $H$ in $G$. Then there exists a neighborhood $\mathcal{U}\subseteq Z$ of $p$ and a $K\times H$ equivariant, co-isotropic embedding:
\begin{displaymath}
j_Z:\mathcal{U} \hookrightarrow T^*K \times \C^h
\end{displaymath}
so that $\mu\vert_{\mathcal{U}}(z)= (\phi \circ j_Z)(z) + \mu(p)$ where
\begin{equation}\label{eq:mmtmap}
\phi(\lambda, \eta, z_1,\dots, z_h)= \eta + \sum_{i=1}^{h}\vert z_i \vert^2 \beta_i
\end{equation}
and $\{\beta_1,\dots,\beta_h\}$ is the set of weights for the action of $H$ on $(\widetilde{TM_H})^{\sigma}_p$. The image of $\mathcal{U}$ in $T^*K \times \C^h$ is uniquely specified by the image of the moment map $\mu$:
\begin{displaymath}
j_Z(Z)=\phi^{-1}(\mu(\mathcal{U})-\mu(p)).
\end{displaymath}
Here, $\mu(\mathcal{U})-\mu(p)$ is the set of points in the image of $\mu$ shifted by the value $\mu(p)$.
\end{cor}
\begin{proof}
We begin by showing that $Z$ admits an equivariant, co-isotropic embedding into a toric, symplectic manifold and that the symplectic slice data at the fold is intrinsic to the fold: the symplectic slice obtained in the image of the embedding is independent of the embedding. By proposition \ref{prop:eqfsnormal}, there exists an invariant neighborhood $U$ of $Z$ and a commutative diagram:
\begin{equation}\label{eq:model}
\xymatrix{
(U,\sigma) \ar[r]^{\psi} \ar[dr]^{\mu} & (U,\omega) \ar[d]^{\mu_s} \\
& \fg^*
}
\end{equation}
where $\omega$ is a symplectic structure for which the action of $G$ is Hamiltonian with moment map $\mu_s$, $\psi$ is an equivariant map with fold singularities that folds along $Z$ with $\psi\big\vert_Z=id_Z$, and $\psi^*\omega=\sigma$. Note that $(U,\omega)$ is a toric symplectic manifold. We want to study the symplectic slice representation $T_z(G\cdot z)^{\omega}/T_z(G\cdot z)$ at points $z\in Z$ and show that it is isomorphic to $\C^h$ and independent of our choice of symplectization. By proposition \ref{prop:normbundle}, the fiber of the symplectic normal bundle at $z$, $(\widetilde{TM_H})^{\sigma}_z$, is contained in $T_zZ$. Since the restriction of the fold map $\psi\big\vert_Z$ is the identity on $Z$ and $\psi^*\omega=\sigma$, we have
\begin{displaymath}
d\psi_z((\widetilde{TM_H})^{\sigma}_z) \subset (TM_H)^{\omega}_z.
\end{displaymath}
Since they have the same dimension and $d\psi_z\big\vert_{T_zZ}=id_{T_zZ}$ is injective, we have that
\begin{displaymath}
d\psi_z((\widetilde{TM_H})^{\sigma}_z) = (TM_H)^{\omega}_z
\end{displaymath}
By remark \ref{rem:slicerep}, the fiber $(TM_H)^{\omega}_z$ is isomorphic to the symplectic slice representation at $z$. We therefore have an equivariant coisotropic embedding:
\begin{displaymath}
j\circ \psi\vert_Z:(Z,i_Z^*\sigma) \hookrightarrow (U,\omega)
\end{displaymath}
into a toric symplectic manifold $(U,\omega)$ and the symplectic slice representation $V$ of the stabilizer $H$ at the point $z\in U$ depends only on the representation $(\widetilde{TM_H})^{\sigma}_z$, hence the weights $\{\beta_1, \dots, \beta_h\}$ associated to $V$ are independent of our choice of model in equation \ref{eq:model}.
\vspace{2mm}
Now, apply proposition \ref{prop:torsympnorm} to a neighborhood $U_1$ of $z\in U$. We obtain an equivariant, open symplectic embedding:
\begin{displaymath}
j:U_1 \hookrightarrow T^*K \times \C^h
\end{displaymath}
where $K\le G$ is a subtorus complementary to $H$ in $G$. Precomposing with the restriction of the fold map $\psi\big\vert_Z$ and defining $\mathcal{U}:=U_1\cap Z$, we obtain the requisite embedding:
\begin{displaymath}
j_Z:\mathcal{U} \hookrightarrow T^*K \times \C^h
\end{displaymath}
Lastly, we show the hypersurface is uniquely specified by the image of $\mu$. Let $\phi(\lambda, \eta, z_1,\dots, z_h) = \eta + \sum_{i=1}^h \vert z_i \vert^2 \beta_i$ be the moment map for the action of $K\times H$ on $T^*K \times \C^h$. Let $t_1,\dots, t_h \in \R^+$ be nonnegative real numbers. Then,
\begin{displaymath}
\phi^{-1}(\eta,t_1^2\beta_1, \dots, t_h^2\beta_h) = (K\times H) \cdot (\eta,t_1,\dots,t_h)
\end{displaymath}
hence the inverse images of points are orbits. Thus, $(\phi \circ j_Z)(z) + \mu(p) = (\eta,t_1^2\beta_1, \dots, t_h^2\beta_h)$ if and only if $j_Z(z)$ is in the orbit $\phi^{-1}((\eta,t_1^2\beta_1, \dots, t_h^2\beta_h)-\mu(p))$. Since the fold $Z$ is $G$-invariant (q.v. lemma \ref{lem:preserves}), the image $j_Z(Z)$ contains the entire orbit, hence
\begin{displaymath}
\phi^{-1}(\mu(\mathcal{U})-\mu(p)) \subseteq j_Z(z).
\end{displaymath}
The reverse inclusion follows from the fact that the moment maps satisfy $\phi\circ j_Z + \mu(p) = \mu$:
\begin{displaymath}
\phi \circ j_Z = \mu-\mu(p) \mbox{ }\rightarrow \mbox{ } j_Z(\mathcal{U}) \subseteq \phi^{-1}(\mu(\mathcal{U})-\mu(p))
\end{displaymath}
hence $\phi^{-1}(\mu(\mathcal{U})-\mu(p))=j_Z(\mathcal{U})$ and the hypersurface is uniquely determined by the moment map image.
\end{proof}
\subsubsection{Unimodular Maps with Folds}
We are almost ready to describe the invariants of a toric, folded-symplectic manifold $(M,\sigma, \mu:M \to \fg^*)$ for the purposes of classifying them up to isomorphism. However, before we can describe these invariants, we'll need a few definitions so we can give them a name. The following definitions and facts about unimodular local embeddings are taken, nearly verbatim, from \cite{KL}. The results and discussions about unimodular maps with folds are generalizations of those related to unimodular local embeddings found in \cite{KL}.
\begin{definition}\label{def:unicone}
Let $G$ be a torus and let $\fg$ be its Lie algebra. A \emph{unimodular cone} in $\fg^*$ is a subset $C$ of the form:
\begin{displaymath}
C= \{\eta \in \fg^* \vert \mbox{ } \langle \eta - \epsilon, v_i \rangle \ge 0 \text{ for all $1\le i \le k$}\}
\end{displaymath}
where $\epsilon$ is a point in $\fg^*$, $k$ is an integer greater than $0$, and $\{v_1,\dots,v_k\}$ is a $\mathbb{Z}$-basis of the integral lattice of a subtorus of $G$. We write:
\begin{displaymath}
C=C_{(v_1,\dots,v_k),\epsilon}
\end{displaymath}
when we wish to make the dependence on the $v_i$ and $\epsilon$ explicit. The \emph{closed facets} of $C$ are the sets:
\begin{displaymath}
F_i = \{\eta \in C \mbox{ } \vert \mbox{ } \langle \eta - \epsilon, v_i \rangle =0\}, \mbox{ } 1\le i\le k
\end{displaymath}
which are subsets of affine hyperplanes and we call $v_i$ the \emph{inward pointing primitive normal} to $F_i$.
\end{definition}
\begin{remark}
A unimodular cone $C$ in $\fg^*$ is a manifold with corners. The $k$-boundary $\partial^k(C)$ is given by $k$-fold intersections of the closed facets $F_i$ of $C$. That is, the boundary of $C$ is determined by the intersections of affine hyperplanes in $\fg^*$.
\end{remark}
\begin{lemma}\label{lem:face}
Let $G$ be a torus and $\fg$ its Lie algebra. Let $C$ be a unimodular cone in $\fg^*$. Then the primitive inward pointing normal $v_i$ to a facet $F_i$ of $C$ is uniquely determined by any open neighborhood of a point $x\in F_i$ in $C$.
\end{lemma}
\begin{proof}
A facet of $C$ is the closure of an open set inside an affine hyperplane inside $\fg^*$, hence we may assume that $F=\epsilon + H$, where $\epsilon \in \fg^*$, $H$ is a codimension $1$ subspace, and $\epsilon + H$ is the set of elements of the form $\epsilon + \eta$, where $\eta\in H$. The annihilator $H^o\subset \fg$ is a unique $1$-dimensional subspace of $\fg$, which is the Lie algebra $\frak{k}$ of a subtorus $K$ of $G$ by the definition of a unimodular cone. Thus, we may choose a primitive normal $v_i$ in the integral lattice $\mathbb{Z}_K\subset \mathbb{Z}_G$ of $K$. This normal is determined up to sign and the sign is determined by the convention that $v_i$ points into the cone.
\end{proof}
\begin{definition}\label{def:ule}
Let $W$ be a manifold with corners and let $G$ be a torus with Lie algebra $\fg$. A smooth map $\bar{\mu}:W\to \fg^*$ is a \emph{unimodular local embedding} if for each $w\in W$ there exists an open neighbourhood $\mathcal{T}\subset W$ of $w$ and a unimodular cone $C\subset \fg^*$ such that $\bar{\mu}(\mathcal{T})\subset C$ and $\bar{\mu}\big\vert_{\mathcal{T}}:\mathcal{T} \hookrightarrow C$ is an open embedding. That is, $\bar{\mu}\big\vert_{\mathcal{T}}:\mathcal{T} \to \bar{\mu}(\mathcal{T})$ is a diffeomorphism of manifolds with corners.
\end{definition}
\begin{lemma}\label{lem:attach}
Let $G$ be a torus with Lie algebra $\fg$ and let $\bar{\mu}:W\to \fg^*$ be a unimodular local embedding. Then $\bar{\mu}$ attaches to every point $w\in W$ a subtorus $K_w$ of $G$, a basis $\{v_1^{w}, \dots, v_k^w\}$ of its integral lattice, and a faithful symplectic representation of $K_w$ on $\C^k$ with weights $\{(v_1^w)^*, \dots, (v_k^w)^*\}$. This assignment is constant on the stratum of $W$ containing $w$. Moreover, for each $w\in W$ there exists a neighborhood $U_w$ such that $\bar{\mu}$ restricts to an open embedding:
\begin{displaymath}
\bar{\mu}\big\vert_{U_w}:U_w \hookrightarrow C_{(v_1^w,\dots,v_k^w),\bar{\mu}(w)}
\end{displaymath}
\end{lemma}
\begin{proof}
By definition of a unimodular local embedding, for each point $w\in W$ there exists a neighborhood $\mathcal{T}\subseteq W$ of $w$ and a unimodular cone $C$ so that $\bar{\mu}(\mathcal{T})\subseteq C$ and $\bar{\mu}\big\vert_{\mathcal{T}}:\mathcal{T} \to \bar{\mu}(\mathcal{T})$ is a diffeomorphism of manifolds with corners. By shrinking $\mathcal{T}$, we may assume that $w$ lies in the intersection of the closures of exactly $k=\operatorname{depth}_W(w)$ distinct components of $\partial^1(W)$. Call these components $B_1,\dots, B_k$.
Since $\mathcal{T}$ is open and $\bar{\mu}\big\vert_{\mathcal{T}}$ is a diffeomorphism, $\bar{\mu}(\mathcal{T}\cap B_i)$ is an open subset of a face $F_i$ of $C$. Furthermore, the sets $\bar{\mu}(\mathcal{T}\cap B_i)$ are distinct for distinct values of $i$ since the components $B_1,\dots, B_k$ are all distinct and $\bar{\mu}$ is a diffeomorphism, hence the faces $F_i$ are all distinct.
By lemma \ref{lem:face}, $\bar{\mu}(\mathcal{T}\cap B_i)$ determines the inward pointing normal $v_i$ of $F_i$ uniquely, hence we obtain a linearly independent set $\{v_1^w,\dots, v_k^w\}$ from the normals of $C$. By the definition of a unimodular cone, this set is a $\mathbb{Z}$-basis for the integral lattice of a subtorus $K_w\subset G$, where
\begin{displaymath}
K_w := \exp(\span\{v_1^w,\dots, v_k^w\})
\end{displaymath}
This basis does not depend on the cone $C$. Indeed, the normals only depend on the images of the strata $\bar{\mu}(\mathcal{T}\cap B_i)$, which are fixed by $\bar{\mu}$, hence the assignment is independent of our choices. The faithful, symplectic representation of $K_w$ on $\C^k$ is given by the representation with weights $\{(v_1^w)^*,\dots, (v_k^w)^*\}$. Explicitly, $K_w$ acts via
\begin{displaymath}
\exp(X)\cdot (z_1, \dots, z_k) = (e^{2\pi i (v_1^w)^*(X)}z_1,\dots,e^{2\pi i (v_k^w)^*(X)}z_k)
\end{displaymath}
Now, if $p$ is another point in $\mathcal{T}$ in the same stratum as $w$, then $p\in \mathcal{T} \bigcap \cap_{i=1}^k(\bar{B}_i)$. That is, the stratum of $p$ is determined by the same codimension $1$ strata $B_i$ that determine the stratum containing $w$. Hence, the subtorus and lattice basis assigned to $p$ is determined by the images of the codimension $1$ strata $\bar{\mu}(\mathcal{T}\cap B_i)$. We therefore obtain the same lattice basis and same subtorus constructed for $w$ since we only used the normals to the affine hyperplanes containing $\bar{\mu}(\mathcal{T}\cap B_i)$. Thus, the assignments are locally constant along strata, hence they are constant on each connected component of the stratification:
\begin{displaymath}
W = \bigsqcup_{j=1}^{\dim(W)} \partial^j(W)
\end{displaymath}
\end{proof}
\begin{remark}
Suppose we choose another point $w'\in U_w$, where $U_w$ is the neighborhood of $w\in W$ constructed in lemma \ref{lem:attach}. The subtorus $K_{w'}$, lattice basis $\{v_1^{w'}, \dots, v_{k'}^{w'}\}$, and representation are determined by the boundary structure of $U_w$ along with the set $\{v_1^w, \dots, v_k^w\}$. Explicitly, we have:
\begin{displaymath}
\{v_1^{w'},\dots, v_{k'}^{w'}\} = \{v_i^w \mbox{ } \vert \mbox{ } \langle \bar{\mu}(w')-\bar{\mu}(w), v_i^w \rangle =0\}
\end{displaymath}
and
\begin{displaymath}
K_{w'}=\exp(\span_{\R}(\{v_i^{w} \mbox{ } \vert \mbox{ } \langle \bar{\mu}(w')-\bar{\mu}(w), v_i^w \rangle =0\}))
\end{displaymath}
The representation is determined by taking the duals to the lattice vectors $v_i^{w'}$.
\end{remark}
We now generalize the unimodular local embeddings of \cite{KL} by introducing fold singularities. The reason is that one of the invariants of a toric, folded-symplectic manifold will be what is known as its orbital moment map, which will be a unimodular map with folds.
\begin{definition}\label{def:umf}
Let $G$ be a torus with Lie algebra $\fg$ and let $W$ be a manifold with corners. A smooth map $\psi:W \to \fg^*$ is a \emph{unimodular map with folds} if:
\begin{enumerate}
\item It is a map with fold singularities, hence the folding hypersurface $\hat{Z}$ is a codimension $1$ submanifold with corners transverse to the strata of $W$. We require that $\hat{Z}$ be co-orientable.
\item The fibers of the bundle $\ker(d\psi) \to \hat{Z}$ are tangent to the strata of $W$. That is, if $z\in \partial^k\cap \hat{Z}=\partial^k(\hat{Z})$, then $\ker(d\psi_z)\subset T_z\partial^k(W)$.
\item The map away from the fold, $\psi\big\vert_{W\setminus \hat{Z}}$, is a unimodular local embedding.
\end{enumerate}
\end{definition}
Since $\psi$ only has fold singularities, we have an analog of lemma \ref{lem:attach} for unimodular maps with folds, which we will use to produce local factorizations of $\psi$ into unimodular local embeddings composed with fold maps. This will be useful for when we perform folded-symplectic reduction on manifolds with corners.
\begin{lemma}\label{lem:attach1}
Let $G$ be a torus with Lie algebra $\fg$, let $W$ be a manifold with corners, and let $\psi:W \to \fg^*$ be a unimodular map with folds with folding hypersurface $\hat{Z}\subset W$. $\psi$ attaches to every point $w\in W$ a subtorus $K_w$ of $G$, a $\mathbb{Z}$-basis $\{v_1^w,\dots,v_k^w\}$ of its integral lattice, and a faithful symplectic representation of $K_w$ on $\C^k$ with weights defined by the duals of each $v_1^w$.
\end{lemma}
\begin{proof}
Since $\psi\big\vert_{W\setminus \hat{Z}}$ is a unimodular local embedding, we know that such an assignment exists on $W\setminus \hat{Z}$ by lemma \ref{lem:attach}, hence lemma \ref{lem:attach1} will be proven if we can show this assignment extends across the fold $\hat{Z}$. To this end, we will show that the connected components of the codimension $1$ stratum $\partial^1(W)$ are mapped to affine hyperplanes under $\psi$. Then, at a point $z\in \hat{Z}$ which is in the intersection of the closures of $k$ distinct components $B_i$ of $\partial^1(W)$, i.e. $z\in \cap_{i=1}^k\bar{B}_i$, we'll have that the normals corresponding to the affine hypersurfaces $\psi(B_i)$ comprise a $\mathbb{Z}$-basis for a the integral lattice of a subtorus $K_w$ of $G$ since they form such a basis away from $\hat{Z}$, which will prove the theorem.
We let $B$ be a connected component of the codimension $1$ boundary $\partial^1(W)$ and consider $z\in \hat{Z}\cap B$. Since $\hat{Z}\pitchfork_s B$, we may choose a neighborhood $U$ of $z$ in $B$ such that $U\setminus \hat{Z}$ has two connected components, $U^+$ and $U^-$. Choose a curve $\gamma(t)$ in $U$ satisfying:
\begin{enumerate}
\item $\gamma(0)=z$, i.e. $\gamma$ passes through $z$,
\item $\gamma'(0)\in \ker(d\psi_z)$ with $\gamma'(0)\ne 0$, and
\item $\gamma(t) \in U^+$ for $t>0$ and $\gamma(t) \in U^-$ for $t<0$.
\end{enumerate}
Since $\psi\big\vert_{W\setminus \hat{Z}}$ is a unimodular local embedding, lemma \ref{lem:attach} reveals that $\psi(U^+)$ is an open subset of an affine hyperplane $\psi(w) + H^+$, where $H^+\subset \fg^*$ is a codimension $1$ subspace, and $\psi(U^-)$ is an open subset of an affine subspace $\psi(w) + H^-$, where $H^-\subset \fg^*$ is a codimension $1$ subspace. We let $v^+$ and $v^-$ denote the primitive normal vectors to these hypersurfaces. Our goal is to show they are the same, hence $H^+ = H^-$ and $\psi(w) + H^+ =\psi(w)+H^-$. Now, if we identify the tangent space of a vector space at a point with the vector space itself, we have the following series of calculations:
\begin{itemize}
\item $d\psi_z(T_zZ)$ is contained in both $H^-$ and $H^+$ as a codimension $1$ subspace. Indeed, $d\psi_{\gamma(t)}(T_zZ)$ is a smoothly varying subspace of $H^-$ for $t<0$ and is a smoothly varying subspace of $H^+$, hence in the limit as $t$ goes to $0$ it lies in both the subspace $H^-$ and $H^+$. If we can show that $H^-$ and $H^+$ both contain a common nonzero vector transverse to $d\psi_z(T_zZ)$, then we will have that $H^+=H^-$.
\item We have $d\psi(\gamma'(t)) \in H^+$ for $t>0$ and $d\psi(\gamma'(t)) \in H^-$ for $t<0$. Hence $d\psi(\gamma'(t))$ is a path in $H^+$ for $t>0$ and a path in $H^-$ for $t<0$, meaning its derivative satisfies:
\item $\frac{d}{dt}(d\psi(\gamma'(t)))$ is in $H^+$ for $t>0$ and $H^-$ for $t<0$.
\item Now, $\frac{d}{dt}\big\vert_{t=0}(d\psi(\gamma'(t)))$ is transverse to the image of $d\psi_z$ since $\gamma'(0)\in \ker(d\psi_z)$ and $\psi$ has fold singularities, hence $H^+$ and $H^-$ both contain $\frac{d}{dt}\big\vert_{t=0}(d\psi(\gamma'(t)))$, which is transverse to $d\psi_z(T_zZ)$ in $H^+$ and $d\psi_z(T_zZ)$ in $H^-$.
\item Thus, $H^+=H^-$ and there is a unique affine hypersurface $\psi(w) + H$ into which $B$ is mapped and there is a unique primitive element $v \in \mathbb{Z}_G$ corresponding to $B$.
\end{itemize}
As we have discussed, if $z$ is contained in the intersection of the closure of exactly $k$ codimension $1$ strata, $z\in \cap_{i=1}^k \bar{B}_i$ and $\operatorname{depth}_W(z)=k$, then there are $k$ primitive normal vectors $\{v_1^z, \dots ,v_k^z\}$ obtained from reading the normals to the images of the $B_i$ away from $\hat{Z}$. Away from $\hat{Z}$, these normals form a $\mathbb{Z}$-basis for the integral lattice of a subtorus of $G$ since $\psi$ is a unimodular local embedding away from $\hat{Z}$. As we have shown, these normals do not change as we cross the fold, $\hat{Z}$, hence they form a $\mathbb{Z}$-basis for the integral lattice of a subtorus $K_z$ of $G$.
\end{proof}
\begin{cor}\label{cor:attach}
Let $G$ be a torus with Lie algebra $\fg$, let $W$ be a manifold with corners, and let $\psi:W\to \fg^*$ be a unimodular map with folds, where the folding hypersurface is denoted $\hat{Z}$. Then for each $z\in \hat{Z}$, there exists a neighborhood $U_z$, a map with fold singularities $\gamma:U_z \to U_z$ with folding hypersurface $\hat{Z}\cap U_z$, and a unimodular local embedding $\bar{\mu}:U_Z \to \fg^*$ so that
\begin{displaymath}
\psi\big\vert_{U_z} = \bar{\mu}\circ \gamma.
\end{displaymath}
Thus, the image of a unimodular map with folds is locally described by a unimodular cone folded across a hypersurface transverse to its faces.
\end{cor}
\begin{proof}
Let $z\in \hat{Z}$ be a point in the fold. Corollary \ref{cor:folds5} implies that there exists a neighborhood $U_z$ and a factorization $\psi\big\vert_{U_z} =\bar{\mu}\circ \gamma$, where $\gamma:U_z \to U_z$ is a fold map which folds across $U_z \cap \hat{Z}$ and $\bar{\mu}$ is an open embedding. By lemma \ref{lem:attach1}, the codimension $1$ strata $B\in \partial^1(W)$ are mapped into affine hyperplanes whose normals are primitive elements of $\mathbb{Z}_G$. Furthermore, at the intersections of their closures, $\cap_{i=1}^k\bar{B}_i$, these normals fit together to give a basis for the integral lattice of a subtorus of $G$. We may shrink $U_z$, if necessary, to assume that it is a manifold with faces. That is, it is diffeomorphic to an open subset of $\R^m \times (\R^+)^n$ for some $m,n\in \mathbb{N}$. Thus, $\bar{\mu}$ maps the faces of $U_z$ to affine hyperplanes whose normals form a basis for the integral lattice of a subtorus, hence $\bar{\mu}$ is a unimodular local embedding.
\end{proof}
\subsection{The Orbit Space of a Toric, Folded-Symplectic Manifold and Invariants}
We are now ready to produce the two primary invariants of a toric, folded-symplectic manifold $(M,\sigma,\mu:M\to \fg^*)$. The first is the orbit map $\pi:M \to M/G$, where $M/G$ has the structure of a manifold with corners.
\begin{remark}\label{rem:invntfxns}
It will be useful to know about some of the invariant structures on the standard representation $\C$ of the circle $S^1$. If a function $f:\C \to \R$ is invariant under the action of $S^1$, then its values are determined by its restriction to the real line $f\big\vert_{\R}$. The restriction is invariant under reflections across the origin since this is the action of $\{1,-1\}\subset S^1$ restricted to the real line. Thus, every $S^1$-invariant smooth function $f(z)$ on $\mathbb{C}$ may be written as $f(z) = f\big\vert_{\R}(\vert z \vert) = g(\vert z \vert ^2)$ for some smooth function $g: \R^+ \to \R$. Conversely, any function that can be written as $g(\vert z \vert ^2)$ is $S^1$-invariant, hence the map
\begin{displaymath}
\xymatrixrowsep{.5pc}\xymatrix{
\C^{\infty}(R^+) \ar[r] & C^{\infty}(\C)^{S^1} \\
g \ar[r] & g(\vert z \vert^2)
}
\end{displaymath}
is surjective. It is also injective and is therefore an isomorphism. This is the content of theorem $1$ of \cite{Sch} in the special case of $S^1$ acting on $\C$. Consequently, $\R^+$ may be identified with the orbit space $\C/S^1$ and the quotient map $q:\C \to \R^+$ is given by $q(z) = \vert z \vert ^2$.
Of course, all of this applies to the product version of this problem. If $\mathbb{T}^h$ acts on $\C^h$ in the standard way via rotations, then the space of invariant smooth functions $C^{\infty}(\C^h)^{\mathbb{T}^h}$ is isomorphic to $C^{\infty}((R^+)^h)$, where the map sends a function on $(\R^+)^h$ $g(t_1,\dots, t_h)$ to $g(\vert z_1 \vert ^2, \dots, \vert z_h \vert ^2)$.
\end{remark}
\begin{prop}\label{prop:corners}
Let $(M,\sigma,\mu:M\to \fg^*)$ be a toric, folded-symplectic manifold with co-orientable folding hypersurface $Z$. Then $M/G$ is naturally a manifold with corners and $Z/G$ is a co-orientable submanifold with corners transverse to the strata of $M/G$. Furthermore, $\pi:M \to M/G$ is a quotient map and $\mu:M \to \fg^*$ descends to a smooth map $\psi:M/G \to \fg^*$.
\end{prop}
\begin{proof} \mbox{ } \newline
\begin{enumerate}
\item We first show $M/G$ is a manifold with corners. We will show that it is locally a manifold with corners by constructing charts using the slice theorem. We will then show that the transition maps are smooth by applying Theorem 1 of \cite{Sch}. The key is that $M$ is locally standard: the differential slice representation decomposes as a trivial representation and a faithful, symplectic representation of a torus $H$ on $\C^h$, where $h=\dim(H)$.
More precisely, let $p\in M$ and let $H=G_p$ be the stabilizer. By proposition \ref{prop:normbundle}, the tangent space $T_pM$ splits as a representation of $H$ into $T_pM= T_pM_H \oplus (\widetilde{TM_H})^{\sigma}_p$, where $(\widetilde{TM_H})^\sigma$ is the symplectic normal bundle to $M_H$ constructed in proposition \ref{prop:normbundle}. Thus, the differential slice, as a representation, is:
\begin{displaymath}
T_pM/T_p(G\cdot p) = (T_pM_H/T_p(G\cdot p)) \oplus (\widetilde{TM_H})^{\sigma}_p
\end{displaymath}
$H$ acts trivially on $T_pM_H/T_p(G\cdot p)$ since $H$ fixes $T_pM_H$ and the dimension of this space is $\dim(M_H)-\dim(G\cdot p) = 2\dim(\dim(G)-\dim(H)) -\dim(G) + \dim(H) = \dim(G)-\dim(H)$. Thus, $T_pM_H/T_p(G\cdot p)$ is isomorphic, as a representation, to the annihilator of $\frak{h}=\operatorname{Lie}(H)$, $\frak{h}^o\subset \fg^*$.
By corollary \ref{cor:stabtori}, there exist linearly independent weights $\{\beta_1,\dots,\beta_h\}$ for the faithful, symplectic representation of $H$ on $(\widetilde{TM_H})^{\sigma}_p$, hence $(\widetilde{TM_H})^{\sigma}_p$ is isomorphic to $\C^h$ where $H$ acts by rotations. Explicitly,
\begin{displaymath}
\exp(X)\cdot (z_1,\dots,z_h) = (e^{2\pi i\beta_1(X)}z_1, \dots, e^{2\pi i\beta_n(X)}z_n)
\end{displaymath}
If we choose a complementary subtorus $K\le G$ and specify an isomorphism $\tau: G \to K\times H$ of Lie groups, then a neighborhood of $G\cdot p$ is $G$-equivariantly isomorphic to a neighborhood of the zero section of:
\begin{displaymath}
G\times_H \frak{h}^o \oplus \C^h = K\times \frak{h}^o \oplus \C^h = T^*K \times \C^k
\end{displaymath}
The orbit space of this neighborhood is then $(T^*K\times \C^k)/G = \frak{h}^o \times C^h/H$. Now, by remark \ref{rem:invntfxns} any $S^1$-invariant function $f(z)$ on $\C$ may be written as $g(\vert z\vert^2)$, where $g$ is a smooth function on $\R$. This extends to the product $\C^h$: a $\mathbb{T}^h$-invariant function $f(z_1,\dots,z_h)$ may be written as $g(\vert z_1 \vert^2, \dots, \vert z_h \vert^2)$ for some smooth map $g:\R^h \to \R$. That is, the functions $p_i(\vec{z})=\vert z_i \vert^2$ generate the ring of $\mathbb{T}^h$ invariant functions on $\C^h$. By theorem 1 of \cite{Sch}, the space $C^h/\mathbb{T}^h$ is $(\R^+)^h$ and the quotient map $q$ is given by:
\begin{displaymath}
q(z_1,\dots,z_h) = (\vert z_1 \vert^1, \dots, \vert z_h \vert^2)
\end{displaymath}
Consequently, $\frak{h}^o \times C^h/H \simeq \R^k \times (\R^+)^h$ and the orbit space $M/G$ is a manifold with corners. We have thus shown that for each point $[p]\in M/G$ there is a neighborhood $U$ of $[p]$ and a homeomorphism $\phi:U \to V$ onto an open subset $V$ of $\R^k \times (\R^+)^h$, where $h$ and $k$ depend on $p$. $\phi$ satisfies the following key property:
\begin{displaymath}
\phi^*C^{\infty}(V) = C^{\infty}(U)
\end{displaymath}
where $C^{\infty}(U)$ is the ring of functions on $U\subset M/G$ that lift to smooth, invariant functions on $\pi^{-1}(U)$, hence $C^{\infty}(U)$ is $C^{\infty}(\pi^{-1}(U))^G$, the ring of invariant smooth functions on $\pi^{-1}(U)$. Thus, for any two neighborhoods $U_1$, $U_2$ and homeomorphisms $\phi_1$, $\phi_2$, the transition maps $\phi_1\circ \phi_2^{-1}$ satisfy:
\begin{displaymath}
(\phi_1\circ\phi_2^{-1})^*C^{\infty}(V_1)=C^{\infty}(V_2)
\end{displaymath}
hence they send smooth functions to smooth functions, meaning they are smooth maps. Thus, the transition maps are diffeomorphisms of manifolds with corners and $M/G$ inherits the structure of a manifold with corners.
\item The proof that $Z/G$ is a submanifold with corners of $M/G$ is very similar, but we first equivariantly identify a neighborhood of $Z$ with a neighborhood of the zero section of $Z\times \R$, so that we may assume $M=Z\times \R$ where $G$ doesn't act on the second factor. Consider the differential slice representation $T_Z/T_z(G\cdot z)$. By proposition \ref{prop:normbundle}, we have $T_zZ = T_zZ_H \oplus (\widetilde{TM_H})^{\sigma}_z$. Hence the slice representation is:
\begin{displaymath}
T_zZ/T_z(G\cdot z) = (T_zZ_H/T_z(G\cdot z)) \oplus (\widetilde{TM_H})^{\sigma}_z
\end{displaymath}
where the first summand is a trivial representation. The same arguments used to show $M/G$ is a manifold with corners now also show that $Z/G$ is a manifold with corners. Thus, the quotient space $(Z\times \R)/G = Z/G \times \R$ is a manifold with corners and $Z/G$ intersects the strata transversally:
\begin{displaymath}
\partial^k(Z/G\times \R) = \partial^k(Z/G) \times \R
\end{displaymath}
Since a neighborhood of $Z$ in $M$ is isomorphic to a neighborhood of the zero section of $Z\times \R$, we have that a neighborhood $\bar{U}$ of $Z/G$ in $M/G$ is isomorphic to a neighborhood of the zero section of $Z/G \times \R$, hence $Z/G$ intersects the strata of $M/G$ transversally.
\item Finally, the fact that $\pi:M \to M/G$ is a quotient map follows from the fact that the map
\begin{displaymath}
q:\C^h \to (\R^+)^h, \text{ $q(z_1,\dots,z_h)=(\vert z_1 \vert^2, \dots, \vert z_h \vert^2)$},
\end{displaymath}
is a quotient map for the standard $\mathbb{T}^h$ action on $\C^h$. Since $M$ is locally isomorphic to $T^*K \times \C^h$, where $\C^h$ is a representation of a subtorus $H$ of $G$ and $K$ is a complementary subtorus, we have that $\pi:M \to M/G$ is locally isomorphic to $p \times q: T^*K \times \C^h \to \frak{h}^o \times (\R^+)^h$, where $p:T^*K \to \frak{h}^o$ is the projection. Since $\mu$ is $G$-invariant and $\pi:M \to M/G$ is a quotient map, there exists a smooth map $\psi:M/G \to \fg^*$ such that $\mu =\psi \circ \pi$.
\end{enumerate}
\end{proof}
\begin{definition}\label{def:orbitalmomentmap}
Let $(M,\sigma,\mu:M\to \fg^*)$ be a toric folded-symplectic manifold. By proposition \ref{prop:corners}, $M/G$ is a manifold with corners and $\mu:M \to \fg^*$ descends to $\psi:M/G \to \fg^*$. We call $\psi$ the \emph{orbital moment map}.
\end{definition}
\begin{prop}\label{prop:umf}
Let $(M,\sigma,\mu:M\to \fg^*)$ be a toric, folded-symplectic manifold with co-orientable folding hypersurface $Z\subset M$. Let $\psi:M/G \to \fg^*$ be the orbital moment map. Then $\psi$ is a unimodular map with folds whose folding hypersurface is $Z/G$. Furthermore, at a point $[z]\in Z/G$, $\ker(d\psi_{[z]})$ is the image of $\ker(\sigma_z)$ under the differential $d\pi_z$ of the quotient map $\pi:M\to M/G$ at $z\in Z$.
\end{prop}
\begin{proof}
Let us first consider the case where $(M,\sigma,\mu:M\to \fg^*)$ is a toric \emph{symplectic} manifold. Let $p\in M$, let $H=G_p$ be its stabilizer with $h=\dim(H)$, let $\{\beta_1,\dots,\beta_h\}$ be the weights of the symplectic slice representation of $H$, and let $K$ be a complementary subtorus in $G$. By lemma \ref{prop:torsympnorm}, we have an invariant neighborhood $U$ of $G\cdot p$ and a commutative diagram:
\begin{displaymath}
\xymatrix{
(U,\sigma) \ar[r]^j \ar[dr]^{\mu} & T^*K \times \C^h \ar[d]^{\varphi} \\
& \fg^* = \frak{k}^* \oplus \frak{h}^* }
\end{displaymath}
where $j$ is an open, equivariant symplectic embedding and $\varphi(\lambda, \eta, z_1,\dots, z_h) = \eta + \sum_{i=1}^h \vert z_i \vert^2 \beta_i$. The image of $\mu\big\vert_U$ is therefore an open subset of a unimodular cone $C$ in $\fg^*$ whose normals are given by $\beta_1^*, \dots, \beta_h^*$, where the dual makes sense as an element of $\fg^*$ since we have chosen a splitting $\fg^* = \frak{k}^* \oplus \frak{h}^*$. The ring of $\mathbb{T}^h$ invariant functions on $\C^h$ is generated by the functions $p_i(\vec{z}) = \vert z_i \vert^2$, $1\le i\le h$, hence the ring of $H$ invariant functions on $\C^h$ are also generated by the $p_i$'s since $H$ acts via:
\begin{displaymath}
\exp(X)\cdot (z_1,\dots,z_h)=(e^{2\pi i \beta_1(X)} z_1, \dots, e^{2\pi i \beta_h(X)} z_h)
\end{displaymath}
Thus, the map $\varphi:T^*K \times \C^h \to C$ is a quotient map by theorem 1 in \cite{Sch}. Consequently, the commutativity of the diagram implies $\mu: U \to C$ is the quotient map and so it descends to the identity map $id_C:C \to C$, hence $\mu$ descends to a unimodular local embedding.
Now, if $(M,\sigma,\mu:M\to \fg^*)$ is toric folded-symplectic, then $(M\setminus Z, \sigma,\mu\big\vert_{M\setminus Z})$ is a toric \emph{symplectic} manifold, hence the orbital moment map $\psi$ restricts to a unimodular local embedding on $(M\setminus Z)/G = M/G \setminus Z/G$. At the fold, $Z$, proposition \ref{prop:eqfsnormal} gives us an invariant neighborhood $U$ of $Z$ and a commutative diagram:
\begin{equation}\label{eq:bigdiagram}
\xymatrixcolsep{4pc}\xymatrix{
(U,\sigma) \ar[r]^-{\phi} \ar[d]^{\pi} & (Z\times \R, p^*i^*\sigma + d(t^2p^*\alpha)) \ar[r]^{\tilde{\gamma}(z,t)=(z,t^2)}\ar[d] & (Z\times \R, p^*i^*\sigma +d(tp^*\alpha)) \ar[dr]^{\mu_s} \ar[d] \\
U/G \ar[r]^{\bar{\phi}} & Z/G \times \R \ar[r]^{\gamma([z],t)=([z],t^2)} & Z/G\times \R \ar[r]^{\bar{\mu}_s} & \fg^*
}
\end{equation}
where
\begin{itemize}
\item $\phi$ is an equivariant open embedding satisfying $\phi(z)=(z,0)$, $\bar{\phi}$ is an open embedding,
\item $\phi^*(p^*i^*\sigma+d(t^2p^*\alpha))=\sigma$,
\item $p^*i^*\sigma + d(tp^*\alpha)$ is nondegenerate in a neighborhood of $Z\times\{0\}$,
\item $\mu_s$ is a symplectic moment map in a neighborhood of $Z\times \{0\}$, and
\item $\bar{\mu}_s$ is the induced map on the quotient space.
\end{itemize}
From our study of the toric \emph{symplectic} case, $\bar{\mu}_s$ is a unimodular local embedding. The composition of the arrows along the top row with $\mu_s$ gives the moment map $\mu\vert_U$, hence the composition of the arrows in the bottom row gives us $\psi\big\vert_{U/G}$. Let $\gamma:Z/G \times \R \to Z/G \times \R$ be the map $\gamma([z],t)=([z],t^2)$. Since $Z/G$ is a manifold with corners by proposition \ref{prop:corners}, $\gamma$ is a smooth map with fold singularities along $Z/G \times \{0\}$. We therefore have:
\begin{displaymath}
\psi = \bar{\mu}_s \circ \gamma \circ \phi
\end{displaymath}
where $\bar{\mu}_s$ is a unimodular local embedding, $\gamma$ is a fold map, and $\phi$ is an open embedding of manifolds with corners. Since $\phi$ is an open embedding, the map $\gamma \circ \phi$ is a map with fold singularities that folds along $Z/G$ by corollary \ref{cor:folds4-2}. Since $\bar{\mu}_s$ is a local embedding, the composition $\bar{\mu}_s \circ \gamma \circ \phi$ is a map with fold singularities that folds along $Z/G$ by corollary \ref{cor:folds-diffeo}. Thus, $\psi$ has fold singularities at points of $Z/G$.
We can read the kernel of $d\psi$ at $Z/G$ from the diagram. The kernel of $p^*i^*\sigma + d(t^2p^*\alpha)$ contains $\frac{\partial}{\partial t}$ at points of $Z\times \{0\}$. The kernel of $d\gamma$ at $Z/G \times \{0\}$ is given by $\frac{\partial}{\partial t}$, which is the image of the kernel of the fold form under the projection map, hence $d\pi_z(\ker(\sigma_z))=\ker(d\psi_z)$.
\end{proof}
\begin{remark}
Let $(M,\sigma,\mu:M\to \fg^*)$ be a toric, folded-symplectic manifold with co-orientable folding hypersurface $Z$. By proposition \ref{prop:umf}, the kernel of the differential of the orbital moment map, $d\psi$, at points of $Z/G$ is given by the image of the fibers of the bundle $\ker(\sigma) \to Z$ under the projection $d\pi:TM \to T(M/G)$. The bundle $\ker(\sigma)\to Z$ has rank $2$, while the bundle $\ker(d\psi) \to Z/G$ has rank $1$. The reason that the dimensions are not preserved under $d\pi$ is because the fibers of the bundle $\ker(\sigma)\cap TZ$ are generated by the group action by lemma \ref{lem:generator}. Thus, a $1$-dimensional subspace of each fiber of $\ker(\sigma)$ is eliminated by the differential of the orbit map $d\pi$. On the other hand, we are about to show that one may recover $\ker(\sigma)\cap TZ$ with its orientation induced by $\sigma$ from the orbital moment map.
\end{remark}
\begin{definition}\label{def:annihilator}
Let $(M,\sigma,\mu:M\to \fg^*)$ be a toric, folded-symplectic manifold with co-orientable folding hypersurface $Z$. Then $M/G$ is a manifold with corners by proposition \ref{prop:corners} and $\mu$ descends to $\psi:M/G \to \fg^*$, a unimodular map with folds. We may define two vector bundles over $Z/G$. We have the rank $1$ \emph{kernel bundle} of $\psi$:
\begin{displaymath}
\ker(d\psi) \to Z/G, \mbox{ } \ker(d\psi)_{[z]}:= \{v \in T_{[z]}M/G \mbox{ } \vert \mbox{ } d\psi_{[z]}(v)=0\}
\end{displaymath}
and the rank $1$ \emph{annihilator bundle} of $\psi$, which functions as the cokernel of the differential:
\begin{displaymath}
\operatorname{Im}(d\psi)^o \to Z/G, \mbox{ } \operatorname{Im}(d\psi)^o_{[z]}:= \{\eta \in \fg \mbox{ } \vert \mbox{ } \langle d\psi_{[z]}, \eta \rangle =0 \}
\end{displaymath}
where we have identified $T\fg^* = \fg^* \times \fg^*$.
\end{definition}
\begin{remark}
There is a fast way to see that $\operatorname{Im}(d\psi)^o \to Z/G$ is in fact a rank $1$ vector bundle over $Z/G$. Consider the trivial bundle $Z/G \times \fg$ over $Z/G$ and the cotangent bundle $T^*(Z/G)$. The differential $d\psi$ coupled with the canonical pairing $\langle \cdot, \cdot \rangle$ between $\fg^*$ and $\fg$ gives us a map of vector bundles over $Z/G$:
\begin{displaymath}
\xymatrixcolsep{4pc}\xymatrix{
Z/G \times \fg \ar[r]^{\langle d\psi, \cdot \rangle} & T^*(Z/G)
}
\end{displaymath}
where the map is given pointwise by $([z],X) \to \langle d\psi_{[z]},X \rangle$, which is a covector in $T^*(Z/G)$. The annihilator bundle $\operatorname{Im}(d\psi)^o \to Z/G$ is then the kernel of this map.
\end{remark}
We focus our attention on $\operatorname{Im}(d\psi)^o \to Z/G$ for now and show there is an orientation on $\operatorname{Im}(d\psi)^o \to Z/G$ induced by $\psi$. We will then show that there is a smooth, orientation-preserving isomorphism of vector bundles $(\cdot)_Z:\pi^*\operatorname{Im}(d\psi)^o \to (\ker(\sigma)\cap TZ)$, where $\pi:M \to M/G$ is the quotient map.
\begin{lemma}\label{lem:orientation1}
Let $(M,\sigma,\mu:M\to \fg^*)$ be a toric, folded-symplectic manifold and let $\psi:M/G \to \fg^*$ be the orbital moment map. Let $\operatorname{Im}(d\psi)^o \to Z/G$ be the annihilator bundle. There exists an orientation on $\operatorname{Im}(d\psi)^o \to Z/G$ induced by $\psi$.
\end{lemma}
\begin{proof}
We mimic the construction of the orientation on $\ker(\sigma)\cap TZ$. As in the case of the orientation on the fold, we will be computing the intrinsic derivative $D\psi_{[z]}$ of $\psi$ at a point $[z]\in Z/G$, which will give us a quadratic map $D\psi_{[z]}: \ker(d\psi_{[z]})\otimes \ker(d\psi_{[z]}) \to \operatorname{coker}(d\psi_{[z]})$. An element $v\in \operatorname{Im}(d\psi)^o_{[z]}$ is then positively oriented if for any nonzero element $w\in \ker(d\psi_{[z]})$, we have $\langle D\psi_{[z]}(w\otimes w), v \rangle >0$, where we identify $\operatorname{coker}(d\psi_{[z]})$ with $\operatorname{Im}(d\psi_{[z]})^o\subset \frak{g}$.
Of course, we have an alternative approach for those who are less familiar with the intrinsic derivative. The orientation is defined at a point $[z]\in Z/G$ as follows:
\begin{enumerate}
\item Choose a nonzero element $Y\in \ker(d\psi_{[z]})$ and extend it to a local vector field $\tilde{Y}$ near $[z]$.
\item Then, for any $X \in \operatorname{Im}(d\psi)^o_{[z]}$, we obtain a function $f(p)=\langle d\psi_p(\tilde{Y}), X \rangle$ defined in a neighborhood of $[z]$.
\item An element $X \in \operatorname{Im}(d\psi)^o_{[z]}$ will be positively oriented if $df_{[z]}(Y)>0$.
\end{enumerate}
We first show that $df_{[z]}(Y)$ must be nonzero. In coordinates $(p,t)\in Z/G\times \R$ near $[z]$ with the fold identified with $\{t=0\}$, we may write $\psi$ as:
\begin{displaymath}
\psi(p,t)= \psi\big\vert_{Z/G}(p) + t^2F(p)
\end{displaymath}
where $F([z])$ is some nonvanishing smooth map $F:Z/G \to \fg^*$ and
\begin{displaymath}
\phi(p,t)=\psi\big\vert_{Z/G}(p) + tF(p)
\end{displaymath}
is a local embedding. We have that $F(p)$ is transverse to the image of $d\psi_p$, hence any nonzero $X\in \operatorname{Im}(d\psi)^o_p$ has a nonzero pairing $\langle F(p), X \rangle$ ($F$ plays the role of the intrinsic derivative). If $\langle F(p), X \rangle=0$, then $X$ annihilates $d\psi_p(T_p(Z/G)) + \R(F(p)) =\fg^*$, hence $X$ annihilates $\fg^*$ and must be $0$. Now, at $([z],0)$, our choice of $Y$ must be $c\frac{\partial}{\partial t}$ for some $c\ne 0$. If we extend it to a local vector field and compute, we obtain:
\begin{displaymath}
c\frac{\partial}{\partial t}\big\vert_{t=0}\langle d\psi_{([z],0)}(\tilde{Y}), X \rangle = c^2\langle F([z]), X \rangle \ne 0
\end{displaymath}
Now, the choice of $Y$ doesn't affect the sign of the answer: changing the sign of $c$ doesn't change the sign of $c^2$. Thus, imposing the condition that $X$ is positively oriented if and only if $c^2\langle F([z]), X \rangle >0$ is independent of the choice of $Y$. The orientation induced on $\operatorname{Im}(d\psi)^o$ is therefore intrinsic to the map $\psi$.
\end{proof}
\begin{lemma}\label{lem:nullfoliation}
Let $(M,\sigma,\mu:M \to \fg^*)$ be a toric, folded-symplectic manifold with co-orientable folding hypersurface. Let $\ker(\sigma)\cap TZ$ be the null bundle over $Z$ with its orientation induced by $\sigma$. Let $\pi:M \to M/G$ be the quotient map and let $\operatorname{Im}(d\psi)^o$ be the annihilator bundle of the orbital moment map $\psi:M/G \to \fg^*$ with its orientation induced by $\psi$. Then the map:
\begin{displaymath}
(\cdot)_Z: \pi^*\operatorname{Im}(d\psi)^o \to (\ker(\sigma)\cap TZ)
\end{displaymath}
given pointwise by $(\cdot)_Z(z,X)=(X)_Z(z)$, is an orientation-preserving isomorphism of vector bundles over $Z$.
\end{lemma}
\begin{proof}
We first need to check that the image of the map $(\cdot)_Z$ actually lands inside $\ker(\sigma)\cap TZ$. We will then show that it is an orientation-preserving map and, since the two bundles are line bundles, this will show that the map is an isomorphism.
Consider an element $X$ in the fiber of the annihilator bundle $\pi^*\operatorname{Im}(d\psi)^o_z \subset \fg$. By definition of the annihilator bundle,
\begin{equation}\label{eq:image}
\begin{array}{ll}
\langle d\psi_{\pi(z)}, X \rangle =0 & \implies \\
\langle d(\psi\circ \pi)_z, X \rangle = 0 & \iff \\
\langle d\mu_z,X \rangle =0 & \iff \\
i_{X_Z(z)}\sigma_z =0 & \text{By definition of the moment map.}
\end{array}
\end{equation}
Here, we have used the fact that $X_Z(z)=X_M(z)$ since $G$ preserves the fold $Z$. We need to show that the last line of \ref{eq:image} implies $X_Z\in \ker(\sigma)$. If we can show that this implication is true on an open dense subset of $Z$, then the smoothness of $(\cdot)_Z$ implies that its image is inside $\ker(\sigma)\cap TZ$ over all of $Z$. Let us find the requisite open dense subset.
The action of $G$ on $M$ is effective, hence the action is free on an open dense subset $M_{\{e\}}$ of $M$. By corollary \ref{cor:eqfsnormal1}, $M_{\{e\}}$ is transverse to the fold $Z$, hence $Z_{\{e\}} = M_{\{e\}} \cap Z$ is nonempty. Otherwise, every point in $Z$ would have a nontrivial stabilizer, hence every point in some invariant neighborhood of $Z$ would have a nontrivial stabilizer since the orbit-type strata are transverse to $Z$. Since $M_{\{e\}}$ is open and dense, $Z_{\{e\}}$ is open and dense in $Z$. Otherwise, there is a point $z\in Z$ and a neighborhood $U\subseteq$ of $z$ in $Z$ so that all points $p\in U$ have nontrivial stabilizer, meaning there is an invariant neighbourhood of $U$ in $M$ where every point $p$ has a nontrivial stabilizer. Thus, $M_{\{e\}}$ is not dense, which is a contradiction.
Thus, on an open dense subset $Z_{\{e\}}$ of $Z$, the induced vector fields $X_Z$, $X\in \fg$, vanish if and only if $X=0$. Let us restrict our attention to this subset. If $z\in Z_{\{e\}}$, then $i_{X_Z(z)}\omega_z = 0$ if and only if $X_Z(z) \in \ker(\omega_z)$ and $X_Z(z)\ne 0$. Thus, the last line of equation \ref{eq:image} shows that the $X_Z(z)$ is a nonzero element of $\ker(\sigma_z)$. Since $X_Z(z)=(\cdot)_Z(z,X)$, we have shown that the image of $(\cdot)_Z$ is in $\ker(\sigma)\cap TZ$ on an open dense subset, hence it lies in $\ker(\sigma)\cap TZ$ everywhere.
The fact that the map $(\cdot)_Z$ is orientation preserving is most easily seen using the local model of proposition \ref{prop:eqfsnormal}. We have a diagram:
\begin{displaymath}
\xymatrixcolsep{4pc}\xymatrix{
(Z \times \R, p^*i^*\sigma + d(t^2p^*\alpha)) \ar[d] \ar[dr]^-{\mu\vert_Z + t^2F} & \\
Z/G \times \R \ar[r]^-{\bar{\mu}\vert_{Z/G} + t^2\bar{F}} & \fg^*
}
\end{displaymath}
where $\alpha\in \Omega^1(Z)$ orients the null foliation and $F:Z \to \fg^*$ is defined by the pairing $\langle F(z), X \rangle = \alpha_z(X_M(z))$. The orientation of $\operatorname{Im}(d\psi)^o$ is such that $X \in \operatorname{Im}(d\psi)^o_{[z]}$ is positively oriented if and only if $\langle \bar{F}([z]), X \rangle > 0$, but
\begin{displaymath}
\begin{array}{ll}
\langle \bar{F}([z]),X \rangle > 0 & \implies \\
\langle F(z),X \rangle > 0 & \implies \\
\alpha_z(X_M(z)) > 0 &
\end{array}
\end{displaymath}
which means that $X_M(z) = (\cdot)_Z(z,X)$ is positively oriented.
\end{proof}
Thus, the orbital moment map $\psi:M/G \to \fg^*$ allows us to completely reconstruct $\ker(\sigma)\cap TZ$ and, hence, the null foliation on the folding hypersurface $Z\subset M$. On the other hand, there is an easy way to reconstruct the rest of the kernel bundle $\ker(\sigma)\to Z$ using a lifting recipe.
\begin{lemma}\label{lem:lifts}
Let $(M,\sigma,\mu:M\to \fg^*)$ be a toric folded-symplectic manifold with co-orientable folding hypersurface and orbital moment map $\psi:M/G \to \fg^*$. If $[z]\in Z/G$ is a point in the folding hypersurface of $\psi$, then we may lift $\ker(d\psi_{[z]})$ stratum-by-stratum as follows. If $v\in \ker(d\psi_{[z]})$ is a nonzero kernel element, then we may choose a lift of $v$ to a vector $\tilde{v}\in T_zM$ such that:
\begin{enumerate}
\item $d\pi_z(\tilde{v}) = v$,
\item $i_{\tilde{v}}\sigma=0$
\end{enumerate}
\end{lemma}
\begin{remark}
We are not defining a smooth lift in lemma \ref{lem:lifts}; we are simply saying that one may make a choice of a lift at each point in $Z/G$. The two conditions guarantee that $\tilde{v}$ is transverse to $T_zZ$ and lies inside the kernel $\ker(\sigma_z)$. Thus, if we couple the span of each lift with the image of the map $(\cdot)_Z: \pi^*\operatorname{Im}(d\psi)^o \to (\ker(\sigma)\cap TZ)$ from lemma \ref{lem:nullfoliation}, we recover the bundle $\ker(\sigma)$ in its entirety.
\end{remark}
\begin{proof}
Recall from the proof of proposition \ref{prop:umf} that we have a commutative diagram in equation \ref{eq:bigdiagram}, giving us a local factorization of the orbital moment map:
\begin{equation}
\xymatrixcolsep{4pc}\xymatrix{
(U,\sigma) \ar[r]^-{\phi} \ar[d]^{\pi} & (Z\times \R, p^*i^*\sigma + d(t^2p^*\alpha)) \ar[r]^{\tilde{\gamma}(z,t)=(z,t^2)}\ar[d] & (Z\times \R, p^*i^*\sigma +d(tp^*\alpha)) \ar[dr]^{\mu_s} \ar[d] \\
U/G \ar[r]^{\bar{\phi}} & Z/G \times \R \ar[r]^{\gamma([z],t)=([z],t^2)} & Z/G\times \R \ar[r]^{\bar{\mu}_s} & \fg^*
}
\end{equation}
At a point $[z]\in Z/G$, the kernel is spanned by $\displaystyle\frac{\partial}{\partial t}$ and we may certainly lift it to $\displaystyle\frac{\partial}{\partial t}$ on $Z\times \R$, which satisfies the requisite conditions of the lemma.
\end{proof}
We have therefore (almost) proved the penultimate structure theorem regarding toric, folded-symplectic manifolds with co-orientable folding hypersurfaces.
\begin{theorem}\label{thm:structure}
Let $(M,\sigma,\mu:M\to \fg^*)$ be a toric, folded-symplectic manifold with co-orientable folding hypersurface. Then,
\begin{enumerate}
\item The orbit type strata $M_H$ are transverse to the folding hypersurface and each $(M_H,i_{M_H}^*\sigma, \mu\vert_{M_H})$ is a toric, folded-symplectic manifold, hence $M$ is stratified by toric, folded-symplectic manifolds.
\item The orbit space $M/G$ is a manifold with corners and the boundary strata of $M/G$ are given by the images of the orbit-type strata $M_H/G$.
\item The moment map descends to $\psi:M/G\to \fg^*$, a unimodular map with folds. Furthermore, since each $(M_H,i_{M_H}^*\sigma)$ is a toric, folded-symplectic manifold, the restriction of $\psi$ to $M_H/G$ is a map with fold singularities if we view it as a map into $\frak{h}^o$. Hence $\psi:M/G \to \fg^*$ is a unimodular map with folds that restricts to maps with fold singularities on the boundary strata.
\item The null-foliation on $Z$ may be recovered from $\psi$, along with its orientation induced by $\sigma$ using the intrinsic derivative of $\psi$ and the map $(\cdot)_Z: \pi^*\operatorname{Im}(d\psi)^o \to \ker(\sigma)\cap TZ$ (q.v. lemma \ref{lem:nullfoliation}).
\item The remainder of the bundle $\ker(\sigma)$ can be constructed by choosing lifts of elements of $\ker(d\psi)$.
\item The representation of $H$ on the fibers of $(\widetilde{TM_H})^{\sigma}$ at $Z$ may be read from the orbital moment map.
\item The local structure of the folding hypersurface is determined by the image of $\psi(Z/G)$ (q.v. corollary \ref{cor:foldnorm}.
\end{enumerate}
Thus, the fold, the null foliation, the orientation, the kernel bundle, and the symplectic slice representation may all be recovered from the orbital moment map. And, by proposition \ref{prop:torsympnorm}, one may recover all symplectic invariants away from the fold just by reading the weights of the symplectic slice representation from the orbital moment map.
\end{theorem}
\begin{proof}\mbox{ } \newline
The only detail we haven't proven is that, if $p\in M/G$, the representation of the stabilizer of a point in $\pi^{-1}(p)$ on the symplectic normal bundle to the orbit-type stratum containing $p$ is encoded in the moment map. Let's summarize the proofs of all of the claims of theorem \ref{thm:structure} and prove this last detail along the way.
\begin{enumerate}
\item Lemma \ref{lem:stabtori} implies that $(M_H,i^*_{M_H}\sigma, \mu\big\vert_{M_H})$ is a toric, folded-symplectic manifold.
\item Proposition \ref{prop:corners} implies that $M/G$ is a manifold with corners. Locally, the boundary strata of $M/G$ are determined by the images of the orbit type strata $M_H$, hence globally the boundary strata of $M/G$ are determined by $M_H/G$.
\item Proposition \ref{prop:umf} implies that $\psi:M/G \to \fg^*$ is a unimodular map with folds. By part $1$, $M_H$ is a toric, folded-symplectic manifold, hence proposition \ref{prop:umf} implies that the restriction $\psi:M_H/G \to \frak{h}^o$ is a unimodular map with folds. However, $M_H/G$ is a manifold without corners, hence unimodularity is a vacuous condition and we may remove it: $\psi:M_H/G \to \frak{h}^o$ is a map with fold singularities.
\item Lemmas \ref{lem:lifts} and \ref{lem:nullfoliation} imply parts $4$ and $5$ of the theorem.
\item The fact that the representation of $H$ can be read from $\psi$ follows from proposition \ref{prop:torsympnorm} and remark \ref{rem:slicerep}. In particular, at a point $p$ away from $Z/G$, there is a neighborhood $U_p$ and a unimodular cone $C$ such that $\psi\big\vert_U : U \hookrightarrow C$ is an open embedding. By proposition \ref{prop:torsympnorm}, the normals to the facets of $C$ span the integral lattice of a subtorus $H$ of $G$, hence their duals in $\frak{h}^*$ define the weights of the symplectic slice representation. By remark \ref{rem:slicerep}, the representation on the symplectic slice at points away from $Z\subset M$ is canonically isomorphic to the representation on the fibers of $(\widetilde{TM_H})^{\sigma}$ and this representation doesn't change along connected components of $M_H$. By lemma \ref{lem:attach1}, the assignment of a basis of the integral lattice of $H$ to points extends across the fold. Thus, we may read the representation from $\psi$.
\item By corollary \ref{cor:foldnorm}, the folding hypersurface is equivariantly isomorphic to a hypersurface $\Sigma \subset T^*K \times \C^h$ for some subtori $K\le G$ and $H\le G$ of $G$, where $H$ acts on $\C^h$ via rotations. The corollary states that this hypersurface is uniquely determined by the moment map image, hence $\psi$ locally determines $\Sigma \subset T^*K \times \C^h$ up to isomorphism.
\end{enumerate}
\end{proof}
We need one last lifting lemma before we leave the basic theory of toric, folded-symplectic manifolds behind. To this end, it may be easiest to lead with an example.
\begin{example}\label{ex:invntvfs}
Consider the action of $S^1$ on $\C$ by rotations. The orbit map is $q(z) = \vert z \vert^2 \in \R^+$. We would like to know when one may lift a vector field on $\R^+$ to an invariant vector field on $\C$ via the quotient map $q$. We claim that if a vector field $\displaystyle X=f\frac{\partial}{\partial t}$ is stratified, then it lifts to an invariant vector field on $\C$, namely the radial vector field. Indeed, if $X$ is stratified on $\R^+$ then it must vanish at the origin. Consequently, we may write it as $\displaystyle X=tg\frac{\partial}{\partial t}$ for some smooth function $g$. We then define the lift of $X$ to be:
\begin{displaymath}
\tilde{X}(z)= \frac{1}{2}g\circ q R
\end{displaymath}
where $R(z)$ is the radial vector field. In cartesian coordinates, we may write this lift as $\displaystyle\tilde{X}(x,y)=\frac{1}{2}(g\circ q)x\frac{\partial}{\partial x} + y\frac{\partial}{\partial y}$. Since the quotient map is $q(x,y)=x^2+y^2$, we have:
\begin{displaymath}
dq_{(x,y)}(\tilde{X}) = \frac{1}{2}(g\circ q)(2x^2 + 2y^2)\frac{\partial}{\partial t}= (g\circ q)(x^2+y^2)\frac{\partial}{\partial t} = (X \circ q)(x,y)
\end{displaymath}
hence $dq(\tilde{X})=X\circ q$. Now, this procedure applies more generally to the $\mathbb{T}^h$ action on $\C^h$ by rotations. The orbit space is $(\R^+)^h$ with coordinates $(x_1,\dots,x_h)$. A stratified vector field is a linear combination of the vector fields $\displaystyle \frac{\partial}{\partial x_i}$ and each of these lift to the radial vector fields on each factor of $\C$, hence any stratified vector field has a lift to $\C^h$.
\end{example}
\begin{lemma}\label{lem:liftbro}
Let $(M,\sigma,\mu:M\to \fg^*)$ be a toric, folded-symplectic manifold with orbit map $\pi:M\to M/G$ and orbital moment map $\psi:M/G \to \fg^*$. Let $[z]\in Z/G$ be a point in the fold of $\psi$. Suppose $X$ is a stratified vector field on $M/G$ so that $X_p \in \ker(d\psi_p)$ and $X_p\ne 0$ for all $p\in Z/G$. Then there exists a neighborhood $U$ of $[z]$ and lift of $X$ to an invariant vector field $\tilde{X}$ on $\pi^{-1}(U)$ so that $\tilde{X}_z \in \ker(\sigma_z)$ for all $z\in \pi^{-1}(U)\cap Z$. That is, we may lift stratified vector fields passing through the kernel of $d\psi$ to invariant vector fields passing through $\ker(\sigma)$.
\end{lemma}
\begin{proof}
Let $[z]\in Z/G$ and let $z\in \pi^{-1}([z])$ and let $H$ be the stabilizer of $p$. Since the claim is local, we may assume that $M=Z\times \R$ where the kernel of $\sigma$ contains $\displaystyle \frac{\partial}{\partial t}$ and $Z=K\times\R^{g-h-1} \times \C^h$, where $g=\dim(G)$, $h=\dim(H)$, and $K$ is a subtorus of $G$ complementary to $H$. Here, we are using the fact that the differential slice $T_zZ/T_z(G\cdot z)$ is isomorphic to $TZ_H/(T_z(G\cdot z)) \oplus (\widetilde{TZ_H})^{\sigma}$, where the second summand is a faithful, symplectic representation of $H$ with dimension $2\dim(H)$. The orbit map is then:
\begin{displaymath}
q(k,x_1,\dots,x_{g-h-1},z_1,\dots,z_h,t) = (x_1,\dots,x_{g-h-1}, \vert z_1 \vert^2, \dots, \vert z_h \vert^2, t)
\end{displaymath}
If $t_1, \dots, t_h$ are the coordinates on $(\R)^+$, then any stratified vector field may be written as a linear combination of:
\begin{itemize}
\item the vector field $\displaystyle \frac{\partial }{\partial t}$,
\item the vector fields of the form $\displaystyle \frac{\partial}{\partial x_i}$, and
\item the vector fields of the form $\displaystyle t_i\frac{\partial}{\partial t_i}$.
\end{itemize}
Since each of these has a lift, there is no problem with producing a lift of any linear combination of them. We want a specific lift, though: one which passes through the fold tangent to the kernel $\ker(\sigma)$. A vector field that is tangent to $\ker(d\psi)$ at $t=0$ must have coefficients that vanish at $t=0$ for all terms except the $\displaystyle \frac{\partial}{\partial t}$ term. When we lift, we pull back these coefficients via the quotient map, so the lifted vector field has the property that all terms except for the $\displaystyle \frac{\partial}{\partial t}$ term vanish at $t=0$, hence the lifted vector field takes values in $\ker(\sigma)$ at the fold.
\end{proof}
\subsection{The Categories $\mathcal{M}_{\psi}$ and $\mathcal{B}_{\psi}$}
We have made a strong case for the fact that the only two invariants of a toric, folded-symplectic manifold are the orbit space $M/G$, which is a manifold with corners, and the orbital moment map $\psi:M/G \to \fg^*$, which is a unimodular map with folds. We therefore fix a manifold with corners, $W$, and a unimodular map with folds $\psi:W \to \fg^*$, and we ask: is it possible to classify all toric folded-symplectic manifolds whose orbit space is $W$ and whose orbital moment map is $\psi:W\to \fg^*$? The answer will be \emph{yes}, but we will need a bit of machinery to prove it. We first begin by collecting the data into a category.
\begin{definition}\label{def:empsi}
Let $W$ be a manifold with corners and let $\psi:W\to \fg^*$ be a unimodular map with folds, where $\fg$ is the Lie algebra of a torus $G$. We define the category $\mathcal{M}_{\psi}(W)$ to be the category whose objects are triples:
\begin{displaymath}
(M,\sigma, \pi:M \to W)
\end{displaymath}
where $\pi$ is a quotient map and $(M,\sigma, \psi \circ \pi)$ is a toric, folded-symplectic manifold with co-orientable folding hypersurface, where the torus is $G$, with moment map $\psi \circ \pi$. We refer to an object as a \emph{toric, folded-symplectic manifold over $\psi$}. A morphism between two objects $(M_i,\sigma_i,\pi_i:M \to W)$, $i=1,2$, is an equivariant diffeomorphism $\phi:M_1 \to M_2$ that induces a commutative diagram:
\begin{displaymath}
\xymatrix{
M_1 \ar[rr]^{\phi} \ar[dr]^{\pi_1} & & M_2 \ar[dl]^{\pi_2} \\
&W \ar[r]^{\psi}& \fg^*
}
\end{displaymath}
and satisfies $\phi^*\sigma_2=\sigma_1$, hence $\phi$ is an equivariant folded-symplectomorphism that preserves moment maps. By definition, every morphism is invertible, hence $\mathcal{M}_{\psi}(W)$ is a groupoid.
\end{definition}
\begin{remark}\label{rem:empsi}
It turns out that $\mathcal{M}_{\psi}(W)$ is more than just a groupoid. If $U\subset W$ is an open subset, then for any object $(M,\sigma, \pi:M \to W)$ we have the restricted object $(\pi^{-1}(U),\sigma, \pi\big\vert_{\pi^{-1}(U)}:\pi^{-1}(U) \to U)$, which we denote as $(M\big\vert_U, \sigma, \pi\big\vert_U)$, despite the fact that $U$ is an open subset of $W$. Since $\psi\big\vert_U$ is a unimodular map with folds, we have that $(M\big\vert_U, \sigma \pi\big\vert_U)$ is an object of $\mathcal{M}_{\psi}(U)$. Thus, for each pair of open sets $U, V\subseteq W$ of $M$, we have a restriction functor:
\begin{displaymath}
\big\vert_V^U: \mathcal{M}_{\psi}(U) \to \mathcal{M}_{\psi}(V)
\end{displaymath}
For any three open subsets $U,V,T$ with $T \subset V \subset U$, the restriction maps satisfy $\big\vert_T^V \circ \big\vert_V^U=\big\vert_T^U$, hence $\mathcal{M}_{\psi}: \operatorname{Open}(W)^{op} \to \mathcal{M}_{\psi}(\cdot)$ is a presheaf. Each category $\mathcal{M}_{\psi}(U)$ is a groupoid by definition, hence $\mathcal{M}_{\psi}$ is a presheaf of groupoids.
\vspace{5mm}
If $(M_1,\sigma_1,\pi_1)$ and $(M_2,\sigma_2,\pi_2)$ are two toric, folded-symplectic manifolds over $\psi:W \to \fg^*$ and their restrictions to open subsets agree, then they must agree as toric-folded-symplectic manifolds. If we have an open subset $U\subseteq W$ and an open cover of $\{U_i\}_{i\in I}$ with objects $(M_i, \sigma_i, \pi_i)$ in $\mathcal{M}_{\psi}(U_i)$ for each $i$ that satisfy:
\begin{displaymath}
(M_i\big\vert_{U_i\cap U_j}, \sigma_i ,\pi_i \big\vert_{U_i \cap U_j}) = (M_j\big\vert_{U_i\cap U_j}, \sigma_j, \pi_j\big\vert_{U_i\cap U_j})
\end{displaymath}
then we may form the space $(M,\sigma,\pi) = (\sqcup_i (M_i,\sigma_i \pi_i))/\sim$, where $\sim$ is the equivalence relation that states that two points are equivalent $p_i \sim p_i$ if they are both in $U_i\cap U_j$ and the form $\sigma$ at an equivalence class $[p_i]$, $\sigma_{[p_i]}$, is defined to be $\sigma_{p_i}$. Similarly, the quotient map $\pi$ at $[p_i]$ is $\pi(p_i)$, which doesn't depend on $i$. This space is a toric, folded-symplectic manifold. Thus, $\mathcal{M}_{\psi}$ is a \emph{sheaf} of groupoids.
\end{remark}
It is not particularly efficient to study $\mathcal{M}_{\psi}$ directly. For example, consider the question of whether or not $\mathcal{M}_{\psi}(W)$ is nonempty. At this stage, we cannot answer such a question since we don't know how one might construct objects in $\mathcal{M}_{\psi}$. On the other hand, we \emph{could} answer the question if we were able to assume that the action is free, which we discuss below in remark \ref{rem:nonempty}. Thus, we construct a category where we assume all actions are principal. Since we are keeping the orbit space $W$ fixed, this will mean that the total space of a principal bundle over $W$ will be a manifold with corners. Since we are interested in Hamiltonian torus actions, we will need a definition of an Hamiltonian action on a manifold with corners. To obtain such a definition, simply replace the words \emph{folded-symplectic manifold without corners} in definition \ref{def:fsham} with the words \emph{folded-symplectic manifold with corners}. We now form the category of principal, toric, folded-symplectic bundles over a fixed unimodular map with folds, $\psi:W \to \fg^*$.
\begin{definition}\label{def:bpsi}
Let $\psi: W \to \fg^*$ be a fixed unimodular map with folds, where $\fg$ is the Lie algebra of a torus $G$. We define $\mathcal{B}_{\psi}(W)$ to be the category whose objects are principal $G$-bundles $\pi:P \to W$ equipped with an invariant folded-symplectic form $\sigma$ with co-orientable folding hypersurface, denoted as a pair
\begin{displaymath}
(\pi:P \to W, \sigma)
\end{displaymath}
so that $\psi\circ \pi$ is a moment map for the principal action of the torus $G$ on $P$. A morphism $\phi$ between two objects $(\pi_1:P_1\to W, \sigma_1)$ and $(\pi_2:P_2 \to W, \sigma_2)$ is a map of principal $G$ bundles:
\begin{displaymath}
\xymatrix{
P_1 \ar[rr]^{\phi} \ar[dr]^{\pi_1}& & P_2 \ar[dl]^{\pi_2} \\
&W\ar[r]^{\psi} & \fg^*
}
\end{displaymath}
so that $\phi^*\sigma_2 = \sigma_1$, hence $\phi^*(\psi\circ \pi_2) =\psi \circ \pi_1$. That is, $\phi$ preserves moment maps.
\end{definition}
\begin{remark}\label{rem:bpsi}
As in the case of $\mathcal{M}_{\psi}$, it is straightforward to show that $\mathcal{B}_{\psi}$ is a sheaf of groupoids on $W$.
\end{remark}
\begin{remark}\label{rem:nonempty}
Unlike the case of $\mathcal{M}_{\psi}(W)$, it is easy to show that $\mathcal{B}_{\psi}(W)$ is nonempty. Consider the cotangent bundle $T^*G=G\times \fg^*$, where $G$ is a torus, with its canonical symplectic structure $\omega_{T^*G}$ and canonical moment map $\mu(\lambda, \eta)= -\eta$ given by projection. We have a pullback diagram:
\begin{displaymath}
\xymatrixcolsep{3pc}\xymatrix{
\psi^*(T^*G) \ar[d]^{\pi} \ar[r]^{id_G \times \psi} & G\times \frak{g}^* \ar[d]^{\mu} \\
W \ar[r]^{\psi} & \fg^*
}
\end{displaymath}
Since $\psi$ is a map with fold singularities, $\psi\times id_G$ is a map with fold singularities: its determinant vanishes transversally in any coordinate chart and $\ker(d\psi)=\ker(d\psi \times d(id_G))$, hence the kernel is transverse to the folding hypersurface $\pi^{-1}(Z)$ (q.v. corollary \ref{cor:folds4-1}). Thus, $\sigma=\psi^*\omega_{T^*G}$ is a folded-symplectic form and $\psi \circ \pi$ is a moment map for the action of $G$. Since $\mu:T^*G \to \fg^*$ is a principal $G$-bundle, $\psi^*(T^*G)$ is a principal $G$ bundle. Thus $(\pi:\psi^*(T^*G) \to W, \psi^*(\omega_{T^*G}))$ is an object in $\mathcal{B}_{\psi}(W)$.
\end{remark}
What is the purpose of defining $\mathcal{B}_{\psi}(W)$? In a way, we will see that studying $\mathcal{B}_{\psi}(W)$ allows us to bypass the intricate global structures of objects in $\mathcal{M}_{\psi}(W)$. After all, any object of $\mathcal{M}_{\psi}(W)$ is \emph{almost} a toric, folded-symplectic bundle over $\psi:W \to \fg^*$ since the action is free on an open dense subset. So, we could simply replace points with stabilizers by corners, which is what $\mathcal{B}_{\psi}(W)$ does, and then remove the corners using a local cutting procedure. This can be condensed into the following strategy:
\begin{enumerate}
\item Fix a unimodular map with folds, $\psi:W \to \fg^*$.
\item Classify objects in $\mathcal{B}_{\psi}(W)$ up isomorphism. We call the isomorphism classes of objects in $\mathcal{B}_{\psi}(W)$ $\pi_0(\mathcal{B}_{\psi}(W))$.
\item Construct a functor $c:\mathcal{B}_{\psi}(W) \to \mathcal{M}_{\psi}(W)$.
\item Show the functor $c$ is an equivalence of categories, which amounts to showing it is an isomorphism of sheaves of groupoids.
\item Reap the rewards by noticing any equivalence of categories will induce a bijection on isomorphism classes of objects, hence $\pi_0(\mathcal{B}_{\psi}(W))= \pi_0(\mathcal{M}_{\psi}(W))$.
\end{enumerate}
Before we leave the basic theory of toric, folded-symplectic manifolds behind and endeavour to classify objects of $\mathcal{M}_{\psi}(W)$, let us record a local uniqueness statement for objects of $\mathcal{M}_{\psi}(W)$, which we will use to show that the functor $c$, which we have yet to construct or define, is an isomorphism of sheaves of groupoids.
\begin{lemma}\label{lem:locunique}
Let $\psi:W \to \fg^*$ be a unimodular map with folds with folding hypersurface $\hat{Z}$, where $\fg$ is the Lie algebra of a torus $G$. Suppose we have two objects $(M_i,\sigma_i,\pi_i:M \to W)$, $i=1,2$, in $\mathcal{M}_{\psi}(W)$. Then for any point $w\in W$, there exists a neighborhood $U$ of $w$ and an isomorphism
\begin{displaymath}
\phi:(M_1\big\vert_U ,\sigma_1, \pi_1\big\vert_U) \rightarrow (M_2\big\vert_U, \sigma_2, \pi_2\big\vert_U)
\end{displaymath}
of toric, folded-symplectic manifolds. That is, there exists a morphism between $(M_1\big\vert_U ,\sigma_1, \pi_1\big\vert_U)$ and $(M_2\big\vert_U, \sigma_2, \pi_2\big\vert_U)$ in the category $\mathcal{M}_{\psi}(U)$.
\end{lemma}
\begin{proof} \mbox{ }\newline
We first show that one can obtain a local isomorphism of folding hypersurfaces $\gamma:U_1 \to U_2$, where $U_i\subseteq Z_i$ is an open subset, so that $\gamma^*(i_{Z_2}^*\sigma_2) = i_{Z_1}^*\sigma_1$ and $\gamma$ induces a commutative diagram:
\begin{displaymath}
\xymatrix{
(U_1,i_{Z_1}^*\sigma_1, \pi_1) \ar[dr]^{\pi_1} \ar[rr]^{\gamma} & & (U_2,i_{Z_2}^*\sigma_2,\pi_2) \ar[dl]^{\pi_1} \\
&U_0\subseteq \hat{Z} \ar[r]^{\psi}& \fg^*
}
\end{displaymath}
We then show that one may extend this isomorphism to an isomorphism $\phi$ of neighborhoods $\tilde{U}_1\subseteq M_1$ and $\tilde{U}_2\subseteq M_2$ of $U_1$ and $U_2$, which makes the diagram commute:
\begin{displaymath}
\xymatrix{
(\tilde{U}_1,\sigma_1,\pi_1) \ar[dr]^{\pi_1} \ar[rr]^{\phi} & & (\tilde{U}_2,\sigma_2, \pi_2 \ar[dl]^{\pi_2}) \\
&U\subset W \ar[r]^{\psi} & \fg^*
}
\end{displaymath}
The local isomorphism of the folds is constructed as follows. Pick a point $z\in \hat{Z}$ in the folding hypersurface of $\psi$. By theorem \ref{thm:structure}, we may read the stabilizer and symplectic slice representation at points $p_i\in \pi_i^{-1}(\hat{z})$, $i=1,2$, from $\psi$ at $z$. Thus, the stabilizer of $p_1$ is the same as the stabilizer of $p_2$ and we denote it as $H\le G$. Let $\beta_1,\dots, \beta_h$ be the weights associated to $\hat{z}$ via $\psi$ (q.v. lemma \ref{lem:attach1}). By theorem \ref{thm:structure}, these are the weights of the symplectic slice representations of $H$ at $p_1$ and $p_2$. Since they have the same weights, they are isomorphic as representations. In particular, they are both isomorphic to $\C^h$. Choose a subtorus $K$ complementary to $H$. Corollary \ref{cor:foldnorm} then gives us invariant neighborhoods $U_i$ of $\pi_i^{-1}(\hat{z})$ in $Z_i$ and a commutative diagram:
\begin{displaymath}
\xymatrixcolsep{3pc}\xymatrix{
U_1 \ar[d]^{\pi_1} \ar[r]^-{j_{Z_1}} & T^*K \times \C^h \ar[d]^{\phi} & \ar[l]_-{j_{Z_2}} U_2 \ar[d]^{\pi_2}\\
U_0 \ar[r]^{\psi} & \fg^* & \ar[l]^{\psi} U_0
}
\end{displaymath}
where the arrows on the top row are $K\times H$ equivariant open embeddings, or $G$-equivariant depending on ones perspective since we can precompose with an isomorphism. The arrow $\phi$ is the moment map for the action of $G=K\times H$ on $T^*K \times \C^h$. By corollary \ref{cor:foldnorm}, the image of $j_{Z_i}(U_i)$ is uniquely determined by the moment map image $\psi(U_0)$, hence $j_{Z_2}^{-1}\circ j_{Z_1}$ is an equivariant isomorphism of the folding hypersurfaces. To obtain the requisite commuting diagram, we shrink $U_0$ so that $\psi$ is an embedding on $U_0$ and is thus invertible as a map onto its image. The map $\gamma:=j_{Z_2}^{-1}\circ j_{Z_1}$ then covers $\psi\big\vert_{U_0}^{-1}\circ \psi\big\vert_{U_0}= id_{U_0}$.
\vspace{5mm}
To extend the isomorphism, we choose a stratified vector field $w$ in a neighborhood $U$ of $U_0$ so that $w$ takes nonzero values in $\ker(d\psi)$ at points of $U_0\subseteq \hat{Z}$. By lemma \ref{lem:liftbro}, this vector field lifts to an invariant local vector field $w_1$ and $w_2$ defined in a neighborhood of $U_1$ and $U_2$ respectively. Furthermore, at points of $U_i$, we may assume that $w_i$ takes values in the kernel of $\sigma_i$ by lemma \ref{lem:liftbro}. The integral curves of $w_i$ are mapped to the integral curves of $w$ under the map $\pi_i$ by definition of a lift. Let $\Phi_i$ be the local flow of $w_i$. We define a diffeomorphism from a neighborhood $\tilde{U}_1$ onto a neighborhood $\tilde{U}_2$ as follows. For each point $z\in U_1$, define
\begin{displaymath}
\phi(\Phi_1(z,t))=\Phi_2(\gamma(z),t)
\end{displaymath}
which is simply the map that takes the flow line of $w_1$ through $z$ to the flow line of $w_2$ through $\gamma(z)$. By construction, this map covers the identity $id_U:U\to U$, where $U\subseteq W$ is the image of $\tilde{U}_i$ under $\pi_i$. This is because $\gamma$ covers the identity on $U_0$ and the flowlines $\Phi_1(z,t)$, $\Phi_2(\gamma(z),t)$ project to the flow line of $w$ through $\pi_1(z)=\pi_2(\gamma(z))$. Note that the map $\phi$ may not be a diffeomorphism, but it has maximal rank at points of $Z_1$ so there is a neighborhood on which it is a diffeomorphism. It is equivariant:
\begin{displaymath}
\phi(g\cdot \Phi_1(z,t)) = \phi(\Phi_1(g\cdot z, t)) = \Phi_2(\gamma(g\cdot z),t)= \Phi_2(g\cdot \gamma(z),t) = g\cdot \Phi_2(\gamma(z),t) = g\cdot \phi(\Phi_1(z,t)).
\end{displaymath}
It also restricts to $\gamma$ on $Z_1$ since $\phi(\Phi_1(z,0))=\Phi_2(\gamma(z),0)= \gamma(z)$. Lastly, $\phi$ doesn't necessarily satisfy $\phi^*\sigma_2= \sigma_1$. However, $\phi^*\sigma_2$ and $\sigma_1$ agree at the folding hypersurface and they induce the same orientation on $U_1\subseteq Z_1$ since this orientation can be read from $\psi$, which is fixed (q.v. lemma \ref{lem:nullfoliation}). These two conditions are enough to guarantee that the linear path $\sigma_s=(1-s)\sigma_1 + s\phi^*\sigma_2$ is folded-symplectic in a neighborhood of $U_1\subseteq Z_1$. As we have seen (q.v. proof of proposition \ref{prop:eqfsnormal}), this path will generate an invariant time-dependent vector field $X_s$ and an isotopy $\phi_s$ such that $\phi_s^*\sigma_s=\sigma_0$.
This time dependent vector field must be tangent to orbits for all $s$. Indeed, the map $\phi_s$ preserves the moment map $\pi_1\circ \psi$, hence $d\mu(X_s)=0$ for all $s$. In particular, $X_s(p)\in T_p(G\cdot p)^{\sigma}$ for each $p$ near $U_1$. Since the orbits are Lagrangian on an open dense subset, this implies that $X_s(p)$ is inside $T_p(G\cdot p)$ on an open dense subset, meaning it is everywhere tangent to orbits.
Now, since the orbits are compact, we may integrate $X_s$ to obtain a time-dependent flow for all $s$ that preserves orbits. That is, $\phi_s$ isn't just a family of open embeddings: it's actually a family of equivariant diffeomorphisms. Thus, $\phi \circ \phi_1$ is a local isomorphism of toric, folded-symplectic manifolds covering the identity map.
\end{proof}
\pagebreak
\section{Folded-Symplectic Reduction}
The goal of this section is to generalize symplectic reduction to folded-symplectic manifolds and everything here, save the symplectic reduction theorem, is original work. The following theorem gives the recipe for constructing a reduced space, or symplectic quotient.
\begin{theorem}\label{thm:sred}
Let $(M,\omega,\mu:M\to \fg^*)$ be a symplectic manifold with a proper, Hamiltonian action of a Lie group $G$ and corresponding moment map $\mu:M\to \fg^*$. Suppose $G$ acts on $\mu^{-1}(0)$ freely. Then $\mu^{-1}(0)$ is a smooth manifold of codimension $\dim(G)$, $\pi:\mu^{-1}(0)\to \mu^{-1}(0)/G$ is a principal $G$ bundle, there exists $\omega_0 \in \Omega^2(\mu^{-1}(0)/G)$ such that $\pi^*\omega_0=\omega\big\vert_{\mu^{-1}(0)}$, and $\omega_0$ is symplectic.
\end{theorem}
\begin{proof}[Sketch of the proof] \mbox{ } \newline
\begin{itemize}
\item Let $p\in \mu^{-1}(0)$ and let $X\in \fg$ be a nonzero element. If we identify $\fg^*$ with $T_0\fg^*$, then the differential $d\mu_p:T_pM \to \fg^*$ is a map into $\fg^*$. We can show it is surjective by proving the annihilator of its image, $\operatorname{Im}(d\mu_p)^o$, is $\{0\}\subset \fg$. We compute:
\begin{displaymath}
0=\langle d\mu_p,X \rangle \iff (d\langle \mu, X \rangle)_p=0 \iff -i_{X_M}\omega_p =0 \iff X_M(p)=0
\end{displaymath}
The action is free at $p$ by assumption, hence $X_M(p) \iff X=0$. Thus, $X$ annihilates the image of $d\mu_p$ if and only if it is $0$. We therefore have that $p$ is a regular point of $\mu$, hence $0$ is a regular value since $p$ was arbitrary. We then have that $\mu^{-1}(0)$ is a smooth manifold of codimension $\dim(G)$.
\item The action of $G$ on $\mu^{-1}(0)$ is smooth, free, and proper by assumption, hence $\pi:\mu^{-1}(0)\to \mu^{-1}(0)/G$ inherits the structure of a principal $G$ bundle. One could demonstrate this explicitly using the slice theorem: a neighborhood of an orbit in $\mu^{-1}(0)$ looks like a neighborhood of the zero section of $G\times V$, where $V$ is the differential slice.
\item The form $\omega\big\vert_{\mu^{-1}(0)}$ is basic since it is invariant under the action of $G$ and, for any $X\in \fg$ and $Y\in T_p\mu^{-1}(0)$, we have $(i_{X_M}\omega)(Y)= -d\langle \mu,X\rangle(Y)=\langle -d\mu(Y),X\rangle =0$ since $Y$ is tangent to a the $0$ level set. Thus, there exists $\omega_0$ such that $\pi^*\omega_0= \omega\big\vert_{\mu^{-1}(0)}$.
\item $\pi$ is a submersion, so $\pi^*$ is injective. Since $0=d(\omega\big\vert_{\mu^{-1}(0)})=d\pi^*\omega_0 = \pi^*d\omega_0$ and $\pi^*$ is injective, we must have $d\omega_0=0$. Proving that $\omega_0$ is non-degenerate requires one to show that, for each $p\in \mu^{-1}(0)$, $(T_p\mu^{-1}(0))^{\omega}=T_p(G\cdot p)$.
\end{itemize}
\end{proof}
\subsection{The Technique}
We will prove the following analog of theorem \ref{thm:sred}.
\begin{theorem}\label{thm:fsred}
Let $(M,\sigma,\mu:M\to \fg^*)$ be a folded-symplectic manifold without boundary with an Hamiltonian action of a compact, connected Lie group $G$ and moment map $\mu$. If
\begin{enumerate}
\item $\mu^{-1}(0)$ is a manifold of codimension $\dim(G)$ and
\item $G$ acts on $\mu^{-1}(0)$ freely,
\end{enumerate}
then $\pi:\mu^{-1}(0) \to \mu^{-1}(0)/G$ is a principal $G$ bundle, there exists $\sigma_0 \in \Omega^2(\mu^{-1}(0)/G)$ such that $\pi^*\sigma_0=\sigma\big\vert_{\mu^{-1}(0)}$, and $\sigma_0$ is folded-symplectic.
\end{theorem}
\begin{definition}\label{def:fsred}
Let $(M,\sigma,\mu:M\to \fg^*)$ be a folded-symplectic manifold without boundary with an Hamiltonian action of a compact, connected Lie group $G$ and moment map $\mu$. If
\begin{enumerate}
\item $\mu^{-1}(0)$ is a manifold of codimension $\dim(G)$ and
\item $G$ acts on $\mu^{-1}(0)$ freely,
\end{enumerate}
then theorem \ref{thm:fsred} implies that $\mu^{-1}(0)/G$ is a folded-symplectic manifold with fold form $\sigma_0$. We define $M_{red}:=\mu^{-1}(0)/G$ to be the \emph{reduced space at 0}. We define $\sigma_{red}:=\sigma_0$ to be the \emph{reduced form} on $M_{red}$.
\end{definition}
\begin{remark}
We will show via examples that one \emph{must} assume $\mu^{-1}(0)$ is a manifold, $\mu^{-1}(0)$ has codimension $\dim(G)$, and $G$ acts on $\mu^{-1}(0)$ freely in order to guarantee that $M_{red}$ is a folded-symplectic manifold. Removing any one of these assumptions on $\mu^{-1}(0)$ allows one to construct examples where the reduced space either fails to be a manifold or fails to be a folded-symplectic space.
\end{remark}
Note that theorem \ref{thm:fsred} does not imply that $(Z\cap\mu^{-1}(0))/G$ is a folding hypersurface for $\sigma_0$. It's possible that $\sigma_0$ could be symplectic, in which case the folding hypersurface is empty. To rectify this deficit, we have a structural theorem that allows one to definitively state whether or not $\sigma_0$ has singularities by studying the intersection of $\mu^{-1}(0)$ with $Z$.
\begin{theorem}\label{thm:fsred1}
Let $(M,\sigma,\mu:M\to \fg^*)$ be a folded-symplectic manifold without boundary with an Hamiltonian action of a compact, connected Lie group $G$ and moment map $\mu$. Let $Z$ be the folding hypersurface. Suppose
\begin{enumerate}
\item $\mu^{-1}(0)$ is a manifold of codimension $\dim(G)$
\item $G$ acts freely on $\mu^{-1}(0)$.
\end{enumerate}
Then we may form the reduced space $(M_{red},\sigma_{red})$ by theorem \ref{thm:fsred}, where $M_{red}=\mu^{-1}(0)/G$ and $\sigma_{red}$ is folded-symplectic. Let $\chi$ be a connected component of $\mu^{-1}(0)$. Then either $\chi \subset Z$ or $\chi \pitchfork Z$.
\begin{enumerate}
\item If $\chi\subset Z$ then $(\chi/G, \sigma_{red})$ is a symplectic manifold.
\item If $\chi\pitchfork Z$ then $(\chi/G,\sigma_{red})$ is folded-symplectic with folding hypersurface $(\chi\cap Z)/G$.
\end{enumerate}
\end{theorem}
Our proof of theorem \ref{thm:fsred} requires a lemma.
\begin{lemma}\label{lem:pplfold}
Let $G$ be a Lie group. Suppose $M_1$ and $M_2$ are manifolds with corners satisfying $\dim(M_1)=\dim(M_2)$. Let $\pi_1:P_1\to M_1$ and $\pi_2:P_2\to M_2$ be two principal $G$ bundles and suppose $\psi:P_1 \to P_2$ is an equivariant map with fold singularities. Then $\psi$ descends to a smooth map $\bar{\psi}:M_1 \to M_2$ with fold singularities.
\end{lemma}
\begin{proof}[Proof of theorem \ref{thm:fsred}]
First, let us note that the content of the theorem is that $\sigma_0$ is folded-symplectic: the proof that $\pi:\mu^{-1}(0)\to \mu^{-1}(0)/G$ is a principal $G$ bundle and that $\sigma\big\vert_{\mu^{-1}(0)}=\pi^*\sigma_0$ is the same as the proof given for theorem \ref{thm:sred}. Thus, throughout our proof of theorem \ref{thm:sred}, we assume that $\sigma\big\vert_{\mu^{-1}(0)}=\pi^*\sigma_0$ and devote our attention to showing that $\sigma_0$ is folded-symplectic.
We first prove the theorem in the case where the fold, $Z\subset M$, is co-orientable and then use this result to study the non-coorientable case. The normal form proposition \ref{prop:eqfsnormal} implies that for each point $p\in \mu^{-1}(0)$ there exists an invariant neighborhood $U$, an invariant symplectic form $\omega\in \Omega^2(U)$, and an equivariant fold map $\psi:U \to U$ so that $\psi^*\omega= \sigma$. Furthermore, the action of $G$ is Hamiltonian for $\omega$ with symplectic moment map $\mu_s:U\to \fg^*$ and $\mu=\mu_s\circ\psi$. Thus, $\mu^{-1}(0)\cap U = \psi^{-1}(\mu_s^{-1}(0))$ and we have a map:
\begin{displaymath}
\psi:\mu^{-1}(0) \to \mu_s^{-1}(0)
\end{displaymath}
By assumption, $G$ acts on $\mu^{-1}(0)$ freely, hence the stabilizer of $p$ is trivial and we may assume that $G$ acts on $U$ freely using the slice theorem to construct such a neighborhood. Thus, $G$ acts on $\mu_s^{-1}(0)$ freely, meaning $\mu_s^{-1}(0)$ is a smooth manifold of codimension $\dim(G)$ since $\mu_s$ is a symplectic moment map. We now have two principal bundles $\pi:\mu^{-1}(0) \to \mu^{-1}(0)/G$ and $\pi_1:\mu_s^{-1}(0) \to \mu_s^{-1}(0)/G$ and an equivariant map $\psi:\mu^{-1}(0) \to \mu_s^{-1}(0)$, giving us a commutative diagram:
\begin{displaymath}
\xymatrixcolsep{5pc}\xymatrix{
(\mu^{-1}(0),\sigma) \ar[r]^{\psi} \ar[d]^{\pi} & (\mu_s^{-1}(0),\omega) \ar[d]^{\pi_1} \\
(\mu^{-1}(0)/G,\sigma_0) \ar[r]^{\bar{\psi}} \ar[r]^{\bar{\psi}} & (\mu_s^{-1}(0)/G,\omega_0)
}
\end{displaymath}
where $\pi_1^*\omega_0=\omega\big\vert_{\mu^{-1}(0)}$. By the symplectic reduction theorem (q.v. theorem \ref{thm:sred}), $\omega_0$ is symplectic. Since the dimensions of the level sets agree, $\dim(\mu^{-1}(0))=\dim(U)-\dim(G)=\dim(\mu_s^{-1}(0))$, and $\psi:U\to U$ is a map with fold singularities, corollary \ref{cor:folds4-1} implies that $\psi\big\vert_{\mu^{-1}(0)}$ is a map with fold singularities and lemma implies $\bar{\psi}$ is a map with fold singularities. Thus, $\bar{\psi}^*\omega_0$ is a folded-symplectic form on $\mu^{-1}(0)/G$. It remains to show that $\bar{\psi}^*\omega_0 = \sigma_0$. The commutativity of the diagram implies:
\begin{displaymath}
\sigma=\psi^*\omega = \psi^*\pi_1^*\omega_0 = \pi^*(\bar{\psi}^*\omega_0
\end{displaymath}
Since $\pi^*\omega_0=\omega$ and $\pi^*$ is injective, we have that $\pi^*(\bar{\psi}^*\omega_0)=\pi^*\sigma_0$ if and only if $\bar{\psi}^*\omega_0=\sigma_0$. Thus, $\sigma_0$ is folded-symplectic.
Now, if $(M,\sigma)$ is not orientable, we may consider its orientable double cover $\tilde{M}$ given by its orientation covering. Recall that this space is the set of pairs $(m,\eta)$, where $m\in M$ and $\eta$ is an orientation at $m$. We have a covering map $p:\tilde{M} \to M$ whose fiber is $\mathbb{Z}/(2\mathbb{Z})$. Since it is a local diffeomorphism, $p^*\sigma$ is a folded-symplectic form on $\tilde{M}$ with folding hypersurface $\tilde{Z}=p^{-1}(Z)$. Since all folding hypersurfaces are orientable by proposition \ref{prop:orientation}, we have that $\tilde{Z}$ is an orientable hypersurface in an oriented manifold, hence it is co-orientable.
If we have an Hamiltonian action of $G$ on $(M,\sigma)$ with moment map $\mu:M \to \fg^*$, then there is a canonical lift of the action of $G$ to $\tilde{M}$: if $\tau_g:M \to M$ is the action of an element $g\in G$, $\tilde{\tau}_g:\tilde{M} \to \tilde{M}$ is given pointwise by $\tilde{\tau}_g(m,\eta)=(\tau_g(m),d\tau_g(\eta))$. This action makes the projection map $p:\tilde{M}\to M$ equivariant: $\tau_g\circ p = p \circ \tilde{\tau}_g$. Thus, $\tilde{\tau}_g^*p^*\sigma = p^*\tau_g^*\sigma=p^*\sigma$ and the action preserves the fold-form. If $X_{\tilde{M}}$ is the vector field on $\tilde{M}$ induced by $X\in \fg$, then $i_{X_{\tilde{M}}}(p^*\sigma)=-d\langle p^*\mu,X\rangle$, hence the action of $G$ on $\tilde{M}$ is Hamiltonian with moment map $\tilde{\mu}:=\mu\circ p:\tilde{M}\to \fg^*$ and $\tilde{\mu}^{-1}(0)=p^{-1}(\mu^{-1}(0))$. We therefore have a commutative diagram:
\begin{equation}\label{diagram:fsred1}
\xymatrixcolsep{5pc}\xymatrix{
(\tilde{\mu}^{-1}(0)),p^*\sigma) \ar[r]^{\tilde{\pi}} \ar[d]^p & (\tilde{\mu}^{-1}(0)/G,\tilde{\sigma}_0) \ar[d]^{\bar{p}} \\
(\mu^{-1}(0),\sigma) \ar[r]^{\pi} & (\mu^{-1}(0)/G,\sigma_0)
}
\end{equation}
where $\tilde{\pi}^*\tilde{\sigma}_0=p^*\sigma$. The action of $G$ on $\mu^{-1}(0)$ is smooth, free, and proper by assumption, hence the lifted action on $\tilde{\mu}^{-1}$ is smooth, free, and proper since $p:\tilde{\mu}^{-1}(0)\to \mu^{-1}(0)$ is a double cover. The top row is therefore a principal $G$ bundle and the quotient $\tilde{\mu}^{-1}(0)/G$ is a smooth manifold. By our study of the co-orientable case, $\tilde{\sigma}_0$ is folded-symplectic. Commutativity of the diagram means that $\pi\circ p = \bar{p}\circ \tilde{\pi}$, meaning
\begin{displaymath}
\tilde{\pi}^*\bar{p}^*\sigma_0=p^*\pi^*\sigma_0 = p^*\sigma= \tilde{\pi}^*\tilde{\sigma}_0.
\end{displaymath}
Since $\tilde{\pi}$ is a submersion, $\tilde{\pi}^*$ is injective and we have $\tilde{\sigma}_0 = \bar{p}^*\sigma_0$. Since $\pi\circ p= \bar{p}\circ\tilde{\pi}$ is a subermsion, $\bar{p}$ must be a submersion. Since $\tilde{\mu}^{-1}(0)/G$ and $\mu^{-1}(0)$ have the same dimensions, $\bar{p}$ is a local diffeomorphism, meaning $\sigma_0$ is folded-symplectic if and only if $\bar{p}^*\sigma_0=\tilde{\sigma}_0$ is folded-symplectic. Since $\tilde{\sigma}_0$ is folded-symplectic, $\sigma_0$ must be folded-symplectic.
\end{proof}
\begin{proof}[Proof of theorem \ref{thm:fsred1}]
Again, we are assuming that the compact, connected group $G$ acts on $\mu^{-1}(0)$ freely and that $\mu^{-1}(0)$ is a smooth manifold of codimension $\dim(G)$. We will also assume that the folding hypersurface $Z\subset M$ is co-orientable and study the non-coorientable case at the end. We will first prove that if $\chi$ is a connected component of $\mu^{-1}(0)$, then $\chi$ is transverse to the fold, $Z$, or $\chi$ is contained in $Z$. We will then study each case separately and argue that if $\chi \subset Z$, then $\chi/G$ is symplectic and if $\chi\pitchfork_s Z$, then $\chi/G$ is folded-symplectic with folding hypersurface $Z\cap \chi)/G$.
\begin{itemize}
\item Let us define two sets:
\begin{displaymath}
\begin{array}{ll}
\mathcal{A}_1:= & \{p\in \mu^{-1}(0) \vert \mbox{ } \text{$p$ is not in $Z$ or } T_p\mu^{-1}(0)\pitchfork T_pZ\} \\
\mathcal{A}_2:= & \{p\in \mu^{-1}(0) \cap Z \vert \mbox{ } T_p\mu^{-1}(0) \subset T_p Z\}
\end{array}
\end{displaymath}
where $\mathcal{A}_1$ is the set of all points where $\mu^{-1}(0)$ is transverse to the fold and $\mathcal{A}_2$ is the set of all points where $\mu^{-1}(0)$ intersects the fold tangentially. Note that these are the only types of intersections $\mu^{-1}(0)$ can have with $Z$ since $Z$ is a hypersurface, hence $\mathcal{A}_1 \cup \mathcal{A}_2 = \mu^{-1}(0)$. We also have $\mathcal{A}_1\cap \mathcal{A}_2=\emptyset$ by definition. Now, $\mathcal{A}_1$ is an open set in $\mu^{-1}(0)$ since transversal intersections are preserved under perturbations. Thus, if we can show $\mathcal{A}_2$ is also open, then $\mathcal{A}_1 \cup \mathcal{A}_2$ is an open cover of $\mu^{-1}(0)$ by disjoint open sets. Hence, any connected component $\chi$ of $\mu^{-1}(0)$ must lie in $\mathcal{A}_1$ or $\mathcal{A}_2$. If $\chi\subset \mathcal{A}_1$ then it intersects $Z$ transversally by definition. If $\chi\subset \mathcal{A}_2$ then it is a subset of $Z$ by definition.
We begin by noting that if the fiber of the null bundle $\ker(\sigma)\cap TZ$ at $z\in \mu^{-1}(0)$ is not tangent to the orbit $G\cdot z$ at $z$, then $z$ is a regular point. Indeed, if $X \in \fg$ and we assume $\langle d\mu_z, X\rangle=0$, then the induced vector field $X_M$ satisfies $(i_{X_M}\sigma)_z=0$, meaning $X_M \in (\ker(\sigma_z)\cap T_zZ) \cap T_z(G\cdot z) = \{0\}$. Since the action is free at $z$, $X_M = 0 \iff X=0$. Thus, the annihilator of the image of $d\mu_z$, $\operatorname{Im}(d\mu_z)^o$, is $\{0\}\subset \fg$ and $z$ is a regular point. Thus, $T_z\mu^{-1}(0) =\ker(d\mu_z)$, meaning $\ker(\sigma_z)\subset \ker(d\mu_z)$ is tangent to $\mu^{-1}(0)$ at $z$. Since $\ker(\sigma_z) \pitchfork T_zZ$, we have that $\mu^{-1}(0)$ intersects $Z$ transversally at $z$. Thus,
\begin{property}\label{property}
If the fiber of $\ker(\sigma)\cap TZ$ at $z\in\mu^{-1}(0)$ is not tangent to the orbit $G\cdot z$ at $z$, then $\mu^{-1}(0)$ intersects $Z$ transversally at $z$, hence $z\in \mathcal{A}_1$.
\end{property}
Thus, if $z\in \mathcal{A}_2$ then $T_z\mu^{-1}(0)\subset T_zZ$ and the contrapositive of property \ref{property} shows that $\ker(\sigma_z)\cap T_zZ$ is a subspace of $T_z(G\cdot z)$. Let us assume we have a point $z\in \mathcal{A}_2\subset \mu^{-1}(0)$. By the equivariant normal form proposition (q.v. proposition \ref{prop:eqfsnormal}), we may assume that there is an invariant neighborhood $U\subset M$ of $z$, an equivariant fold map $\psi:U \to U$ with fold $Z\cap U$ and $\psi\vert_Z=id_Z$, and a symplectic structure $\omega\in \Omega^2(U)$ for which the action of $G$ is Hamiltonian satisfying $\psi^*\omega=\sigma$. We let $\mu_s$ denote the symplectic moment map and note that $\mu_s\circ \psi = \mu$. We may assume that $G$ acts on $U$ freely since the action of $G$ on $\mu^{-1}(0)$ is free by assumption. We then have that $0$ is a regular value of the symplectic moment map, $\mu_s$, and $\mu_s^{-1}(0)$ is a submanifold of codimension $\dim(G)$ inside $U$. Its tangent space is defined by $\ker(d\mu_s)$. This construction gives us a commutative diagram:
\begin{equation}\label{diagram:fsred2}
\xymatrixcolsep{5pc}\xymatrix{
(\mu^{-1}(0),\sigma) \ar[r]^{\psi} \ar[d]^{\pi} & (\mu_s^{-1}(0),\omega) \ar[d]^{\pi_1} \\
(\mu^{-1}(0)/G,\sigma_0) \ar[r]^{\bar{\psi}} \ar[r]^{\bar{\psi}} & (\mu_s^{-1}(0)/G,\omega_0)
}
\end{equation}
where $\bar{\psi}$ is a map with fold singularities, $\omega_0$ is symplectic (by theorem \ref{thm:sred}), and $\sigma_0=\bar{\psi}^*\omega_0$ is folded-symplectic. Note that proposition \ref{prop:eqfsnormal} also gives us an involution $i:U\to U$ satisfying $i\vert_Z=id_Z$ and $\psi\circ i = i$. Since $\mu^{-1}(0)=\psi^{-1}(\mu_s^{-1}(0))= i^{-1}(\psi^{-1}(\mu_s^{-1}(0)))$, we see that the involution $i$ restricts to an involution:
\begin{equation}\label{eq:involution}
i:\mu^{-1}(0) \to \mu^{-1}(0), \mbox{ } \psi\circ i = \psi
\end{equation}
which we will use below.
Now, since $\ker(\sigma_z)\cap T_zZ\subset T_z(G\cdot z)$, there exists $X\in \fg$ such that $X_M(z)\in \ker(\sigma_z)$. Since $z$ is a regular point of $\mu_s$, the annihilator of the image of $(d\mu_s)_z$ is $\{0\}$ and we must have $\langle (d\mu_s)_z, X \rangle \ne 0$. The restriction of the fold map $\psi$ to $Z$ is the identity, hence $\mu_s\vert_Z=(\mu_s\circ \psi \vert_Z)=\mu\vert_Z$. We then have:
\begin{equation}\label{eq:fsred1}
\langle (d\mu_s)_z, X \rangle \vert_{TZ} = \langle d\mu_z, X \rangle \vert_{TZ} =0
\end{equation}
where the rightmost equality follows since $X_M(z)\in \ker(\sigma)$, hence $0=(i_{X_M}\sigma)_z = -\langle d\mu_z, X \rangle$. Since $\langle d\mu_s, X \rangle \ne 0$ and $\langle d\mu_s, X \rangle \big\vert_{TZ}=0$, we must have that $\langle (d\mu_s)_z(Y),X \rangle \ne 0$ for \emph{any} nonzero $Y\in T_zM$ not in $T_zZ$. That is, $(d\mu_s)_z$ does not vanish on any direction transverse to $T_zZ$. Thus, $\ker(d\mu_s)_z=T_z\mu_s^{-1}(0)$ must be contained in $T_zZ$ and so $\mu_s^{-1}(0)$ intersects $Z$ tangentially at $z$.
By assumption, $T_z\mu^{-1}(0)\subset T_zZ$ and we have just shown $T_z\mu_s^{-1}(0)\subset T_zZ$. The map $\psi:U \to U$ restricts to the identity on the fold $Z$, hence $d\psi_z=id_{T_zZ}:T_z\mu^{-1}(0) \to T_z\mu_s^{-1}(0)$ is injective. Since $\mu^{-1}(0)$ and $\mu_s^{-1}(0)$ have the same codimension in $U$, they have the same dimension and the differential $d\psi_z$ is an isomorphism. Thus, there is a connected neighborhood $V_1\subset \mu^{-1}(0)$ of $z$ and a connected neighborhood $V_2\subset \mu_s^{-1}(0)$ of $\psi(z)=z$ so that $\psi:V_1 \to V_2$ is a diffeomorphism. We claim that this is only possible if $V_1\subset Z$. We use the involution $i:\mu^{-1}(0) \to \mu^{-1}(0)$ of equation \ref{eq:involution}, whose fixed point set is $\mu^{-1}(0)\cap Z$. For each $p\in V_1\cap (Z\cap \mu^{-1}(0))$, there exists an involutive neighborhood $U_p\subset \mu^{-1}(0)$ of $p$. That is, $i(U_p)=U_p$. If there exists $x\in U_p\setminus Z$, then $i(x)\ne x$ since $x$ is not in the fixed-point set of of $i$. By definition of $i$, we have $\psi\circ i =\psi$, meaning $\psi(i(x))=\psi(x)$. Since $i(x)$ and $x$ are distinct points in $U_p\subset V_1$ and $\psi\vert_{V_1}$ is injective by definition of $V_1$, we have found a contradiction. Thus, $U_p\subset (\mu^{-1}(0)\cap Z)$ and each $p\in V_1\cap (\mu^{-1}(0)\cap Z)$ has a neighborhood $U_p\subset V_1$ contained entirely in $\mu^{-1}(0)\cap Z$. This means that the set of points in $V_1$ fixed by $i$ is closed \emph{and} open. Since $V_1$ is connected, we have that all of $V_1$ is fixed and so $V_1\subset \mu^{-1}(0)\cap Z$. Note that at each point $p\in V_1$ we have $T_p\mu^{-1}(0)=T_pV_1 \subset T_pZ$, hence $p\in \mathcal{A}_2$. We have therefore shown that $V_1$ is a neighborhood of $z$ contained entirely within the set $\mathcal{A}_2$ which means $\mathcal{A}_2$ is open.
Since $\mathcal{A}_1$ and $\mathcal{A}_2$ are two disjoint open sets covering $\mu^{-1}(0)$, any connected component $\chi$ of $\mu^{-1}(0)$ must be contained in either $\mathcal{A}_1$ or $\mathcal{A}_2$. If $\chi\subset \mathcal{A}_1$, then $\chi\pitchfork_s Z$ by definition of $\mathcal{A}_1$. If $\chi\subset\mathcal{A}_2$, then $\chi\subset Z$, which proves the first claim of theorem \ref{thm:fsred1}.
\item Now, assume that $\chi$ is a connected component of $\mu^{-1}(0)$ and $\chi\subset Z$. Diagram \ref{diagram:fsred2}
\begin{displaymath}
\xymatrixcolsep{5pc}\xymatrix{
(\mu^{-1}(0),\sigma) \ar[r]^{\psi} \ar[d]^{\pi} & (\mu_s^{-1}(0),\omega) \ar[d]^{\pi_1} \\
(\mu^{-1}(0)/G,\sigma_0) \ar[r]^{\bar{\psi}} & (\mu_s^{-1}(0)/G,\omega_0)
}
\end{displaymath}
becomes
\begin{displaymath}
\xymatrixcolsep{5pc}\xymatrix{
(\chi,\sigma) \ar[r]^{id_Z} \ar[d]^{\pi} & (\chi,\omega) \ar[d]^{\pi_1} \\
(\chi/G,\sigma_0) \ar[r]^{id} & (\chi/G,\omega_0)
}
\end{displaymath}
and $\sigma_0=\omega_0$ is symplectic.
\item Now suppose $\chi\subset \mu^{-1}(0)$ is a connected component transverse to $Z$. We will show that at each point $z\in \chi\cap Z$, $\ker(\sigma_z)\subset T_z\chi$. Assume for a moment that this is true. If $\pi:\mu^{-1}(0) \to \mu^{-1}(0)/G$ is the quotient map, then $\ker(\sigma_z)\not\subset \ker(d\pi_z)$ since the kernel of $d\pi_z$ is the tangent space $T_z(G\cdot z)\subset T_zZ$ while $\ker(\sigma_z)$ contains a subspace transverse to $T_zZ$. Therefore, there is nonzero element $v\in \ker(\sigma_z)\subset T_z\chi$ such that $d\pi_z(v)\ne 0$. We then have $\sigma_0(d\pi_z(v),\cdot)= 0$ since $d\pi_z$ is surjective and $\sigma_0(d\pi_z(v),d\pi(c\dot))=i_v(\pi^*\sigma_0)_z=i_v\sigma_z=0$, which means $\sigma_0$ has a degeneracy at $\pi(z)$. By theorem \ref{thm:fsred}, this degeneracy is a fold singularity.
It remains to show that if $z\in\mu^{-1}(0)\cap Z$, then $\ker(\sigma_z)\subset T_z\mu^{-1}(0)$. To this end, we use the setting of diagram \ref{diagram:fsred2} where we have a an invariant neighborhood $U$ of $z$ on which $G$ acts freely, a symplectic structure $\omega\in \Omega^2(U)$ for which the action of $G$ is Hamiltonian with moment map $\mu_s$, and an equivariant fold map $\psi:U \to U$ so that $\psi^*\omega=\sigma$, hence $\mu_s\circ \psi = \mu$.
Recall from our derivation of property \ref{property} that if $\ker(\sigma_z)\cap T_zZ$ is \emph{not} a subspace of $T_z(G\cdot z)$, then $z$ is a regular point of $\mu$ and $T_z\mu^{-1}(0)=\ker(d\mu_z)$. Since $\ker(\sigma_z)\subset \ker(d\mu_z)$, we have $\ker(\sigma_z)\subset T_z\mu^{-1}(0)$ as required. We may therefore assume that the null space $\ker(\sigma_z)\cap T_zZ \subset T_z(G\cdot z)$ is tangent to the orbit through $z$. As we have seen, this tangency implies that $T_z\mu_s^{-1}(0)\subset T_zZ$. We also have that the map:
\begin{equation}\label{eq:fsred3}
d\psi_z: T_z\mu^{-1}(0)\cap T_zZ \rightarrow T_z\mu_s^{-1}(0)
\end{equation}
is injective. We are assuming $\mu^{-1}(0)$ intersects $Z$ transversally, hence $\dim(\mu^{-1}(0)\cap Z)=\dim(U)-\dim(G)-1$. Since $\dim(\mu_s^{-1}(0))=\dim(U)-\dim(G)$, we see that the image of $d\psi_z$ in equation \ref{eq:fsred3} has codimension $1$ by injectivity. Choose a nonzero vector $v\in T_z\mu^{-1}(0)$ not in $T_zZ$, which is possible since we are assuming a transversal intersection. If $d\psi(v)=0$ then $v\in \ker(d\psi_z)$ and we are done since, by diagram \ref{diagram:fsred2}, $\ker(\sigma_z) = \ker(d\psi_z) +(\ker(\sigma_z)\cap T_zZ)$, hence $\ker(\sigma_z)\subset T_z\mu^{-1}(0)$. If $d\psi_z(v)\ne0$, then we must have that $d\psi(v) \in d\psi_z(T_z\mu^{-1}(0)\cap T_zZ)$. Otherwise, $d\psi_z$ in equation \ref{eq:fsred3} would be injective and surjective, hence it would map $T_z\mu^{-1}(0)$ isomorphically onto $T_z\mu_s^{-1}(0)$. This means $\psi:\mu^{-1}(0)\to \mu_s^{-1}(0)$ would be a diffeomorphism in a neighborhood of $z$. We have seen that this implies a neighborhood of $z$ in $\mu^{-1}(0)$ is contained in $Z$, contradicting our assumption that $\mu^{-1}(0)$ intersects $Z$ transversally at $z$. Thus, there is a nonzero vector $w\in T_z\mu^{-1}(0) \cap T_zZ$ such that $d\psi_z(v)=d\psi_z(w)$, which means $d\psi_z(v-w)=0$. We then have that $v-w\in \ker(d\psi_z)\cap T_z\mu^{-1}(0)$. It cannot be zero since $w\subset T_zZ$ and $v$ is not in $T_zZ$, implying that $\ker(\sigma_z)\subset \mu^{-1}(0)$. As initially discussed, this fact is enough to guarantee that the reduced form $\sigma_0$ has a fold singularity.
\item Finally, if $Z$ is not co-orientable then $M$ cannot be orientable. As in the proof of theorem \ref{thm:fsred}, we lift the action of $G$ to the orientation covering $p:\tilde{M} \to M$ and pull back the fold form $\sigma$ to the fold form $p^*\sigma$, which makes $\tilde{M}$ into an Hamiltonian $G$-manifold $(\tilde{M},p^*\sigma, \tilde{\mu}=\mu\circ p)$. The folding hypersurface of $p^*\sigma$ is the inverse image of the folding hypersurface in $M$, $p^{-1}(Z)$. Let $\chi \subset \mu^{-1}(0)$ be a connected component of $\mu^{-1}(0)$. Since $p:\tilde{M} \to M$ is a covering map, it is a local diffeomorphism and:
\begin{enumerate}
\item $\chi \pitchfork Z$ if and only if $p^{-1}(\chi) \pitchfork p^{-1}(Z)$
\item $\chi \subset Z$ if and only if $p^{-1}(\chi)\subset p^{-1}(Z)$.
\end{enumerate}
Since $p^{-1}(\chi)$ is a collection of connected components of $\tilde{\mu}^{-1}(0)$, our previous discussion implies that these connected components are either transverse to $p^{-1}(Z)$ or contained in $p^{-1}(Z)$. Thus, $\chi\pitchfork_s Z$ or $\chi\subset Z$. To finish, recall that in the proof of theorem \ref{thm:fsred} we constructed a commutative diagram:
\begin{displaymath}
\xymatrixcolsep{5pc}\xymatrix{
(\tilde{\mu}^{-1}(0),p^*\sigma) \ar[r]^{\tilde{\pi}} \ar[d]^p & (\tilde{\mu}^{-1}(0)/G,\tilde{\sigma}_0) \ar[d]^{\bar{p}} \\
(\mu^{-1}(0),\sigma) \ar[r]^{\pi} & (\mu^{-1}(0)/G,\sigma_0)
}
\end{displaymath}
where $(\tilde{\mu}^{-1}(0)/G,\tilde{\sigma}_0)$ is a folded-symplectic manifold, $\bar{p}$ is a local diffeomorphism, and $\bar{p}^*\sigma_0=\tilde{\sigma}_0$. If $\chi\subset \mu^{-1}(0)$ is a connected component and $\chi\subset Z$, then $p^{-1}(\chi)\subset p^{-1}(Z)$ and $(p^{-1}(\chi)/G,\tilde{\sigma}_0)$ is symplectic. Since $\bar{p}$ is a local diffeomorphism, $\sigma_0$ must be symplectic. Now suppose $\chi\pitchfork Z$. Then $p^{-1}(\chi)\pitchfork Z$ and $(p^{-1}(\chi)/G,\tilde{\sigma}_0)$ is folded-symplectic with folding hypersurface $p^{-1}(\chi\cap Z)/G$, which means $\sigma_0$ is folded-symplectic with folding hypersurface $\bar{p}(p^{-1}(\chi\cap Z)/G) = (\chi\cap Z)/G$.
\end{itemize}
\end{proof}
\subsection{Removing Assumptions on $\mu^{-1}(0)$}
We take a moment to discuss the differences between the symplectic and folded-symplectic reduction hypotheses. In the symplectic setting, a free action of $G$ on $\mu^{-1}(0)$ is enough to guarantee that $0$ is a regular value, hence $\mu^{-1}(0)$ is a manifold of codimension $\dim(G)$. In the folded-symplectic setting, a free action alone is not enough to guarantee that $\mu^{-1}(0)$ is a smooth manifold.
\begin{example}
Consider $M=(\R^2\setminus\{0\}) \times \R^2$ with coordinates $(r,\theta, x,y)$ and fold form $\sigma = (r-1)dy\wedge d\theta + ydr\wedge d\theta + dx\wedge dy + dx\wedge dr$. To see that $\sigma$ is folded, we consider the top power $\omega^2 = (r-1-y)dy\wedge d\theta \wedge dx \wedge dr$, which is transverse to $0$. The folding hypersurface is defined by $r-1-y=0$ and the kernel of $\sigma$ at the hypersurface $r-1=y$ contains $\frac{\partial}{\partial r}-\frac{\partial}{\partial y}$, which is not in $\ker(d(r-1-y))$, hence it is transverse to $\{r-1-y=0\}$.
Now, $S^1$ acts freely on $M$ by rotations in $\R^2\setminus \{0\}$. A corresponding moment map is
\begin{displaymath}
\mu(r,\theta, x,y) = y(r-1)
\end{displaymath}
and $\mu^{-1}(0)$ is the union of the hyperplane $y=0$ with the product of the cylinder $r=1$ with $\R^2$. These two submanifolds intersect transversally at the folding hypersurface when $r=y+1$, hence their union is not a submanifold. Alternatively, $\mu^{-1}(0)$ is the product of $S^1$ with the union of the hyperplanes $r=1$ and $y=0$ in the $(x,y,r)$ coordinate system. Since the these hyperplanes intersect transversally, we again see that $\mu^{-1}(0)$ is not a submanifold.
\end{example}
Thus, if we drop the assumption that $\mu^{-1}(0)$ is a manifold and simply require that the action of $G$ is free, $\mu^{-1}(0)$ may only be a topological space. It therefore makes sense to require that $\mu^{-1}(0)$ is a manifold and that the action of $G$ is free. However, in this scenario, the reduced space may only be a presymplectic manifold.
\begin{example}
Consider $M=\mathbb{T}^2 \times S^1 \times \R$ with coordinates $(\theta_1,\theta_2,\theta, h)$ and fold form
\begin{displaymath}
\sigma=\sin(\theta_1)d\theta_1 \wedge d\theta_2 + e^hdh\wedge d\theta_2 - dh\wedge d\theta
\end{displaymath}
where the folding hypersurface is $\{\theta_1=0\}$. $S^1$ acts on $M$ via $\lambda \cdot (\lambda_1, \lambda_2, \lambda_3, h) = (\lambda_1, \lambda \lambda_2, \lambda \lambda_3, h)$. A corresponding moment map is
\begin{displaymath}
\mu(e^{2\pi i \theta_1}, e^{2\pi i \theta_2}, \lambda_3, h) = \cos(\theta_1) + e^h - h.
\end{displaymath}
Then $\mu^{-1}(0) = \{h=0, \lambda_1 = -1\} \simeq \mathbb{T}^2$. The quotient by $S^1$ is isomorphic to $\mathbb{T}$, which cannot be folded-symplectic since it is odd dimensional.
\end{example}
We therefore include the assumption that $\mu^{-1}(0)$ has codimension $\dim(G)$. Lastly, if we assume that $\mu^{-1}(0)$ is a manifold of codimension $\dim(G)$ and drop the assumption that $G$ acts on $\mu^{-1}(0)$ freely, then the reduced space may not be a manifold.
\subsection{Application: Minimal Coupling}
We give a generalization of a construction due to Sternberg \cite{St} which will allow us to combine folded-symplectic structure on a manifold $M$ with a symplectic structure on a symplectic vector bundle $\pi:E \to M$ to produce a folded-symplectic structure in a neighborhood of the zero section of $E$. We follow the approach found in \cite{LS}, hence we begin with the case of a principal bundle.
\begin{definition}\label{def:mincoupform}
Let $(M,\sigma)$ be a folded-symplectic manifold (without corners), let $G$ be a Lie group. For the purposes of the discussion, we'll assume $G$ is compact and connected. Let $\pi:P \to M$ be a principal $G$-bundle, let $\fg$ be the Lie algebra of $G$, and let $A\in \Omega^1(P,\fg)^G$ be a connection $1$-form on $P$. Let $\langle \cdot, \cdot \rangle$ be the canonical pairing between $\fg$ and $\fg^*$.
Consider the space $P\times \fg^*$ with projection maps $pr_i$ onto the $i^{th}$ factor. Let $G$ act diagonally via $g\cdot (p,\eta) = (pg^{-1},Ad^*(g)\eta)$. We define:
\begin{displaymath}
\Omega_A:= \pi^*\sigma - d\langle pr_2, A\rangle
\end{displaymath}
to be the \emph{minimal coupling form} associated to the connection $A$.
\end{definition}
\begin{lemma}\label{lem:mincoupform}
The minimal coupling form $\Omega_A$ is nondegenerate in a neighborhood of the zero section $P\times \{0\}$ of $P\times \fg^*$. Furthermore, the action is Hamiltonian with moment map given by $-pr_2:P\times \fg^* \to \fg^*$.
\end{lemma}
\begin{proof}\mbox{ } \newline
\begin{itemize}
\item Let $H\to P$ be the horizontal bundle of $TP$ afforded by the connection $A$, let $V\to P$ be the vertical bundle, and let $\dim(M)=2m$.
\item Since $\sigma$ is folded-symplectic, we have $\sigma^m \pitchfork 0$ by definition. We then have that $((\pi^*\sigma)\big\vert_H)^m \pitchfork 0$, where we view $0$ as the zero section of $\Lambda^{2m}(H^*)$.
\item The $2$-form $d\langle pr_2,A\rangle$ is $\langle dpr_2, A\rangle$ at points of the zero section of $P\times \fg^*$, which is non-degenerate when restricted to $V\oplus T\fg^*$.
\item Thus, at points of the zero section, the top power of $\Omega_A$:
\begin{equation}\label{eq:mincoupform}
(\Omega_A)^{m + \dim(G)} = ((\pi^*\sigma)\big\vert_H)^m \wedge (\langle dpr_2, A \rangle\big\vert_{V\oplus T\fg^*})^{\dim(G)}
\end{equation}
vanishes transversally along the zero section. Thus, it vanishes transversally in a neighborhood of the zero section.
\item From equation \ref{eq:mincoupform}, the intersection of the degenerate hypersurface of $\Omega_A$ with $P\times \{0\}$ is the degenerate hypersurface of $(\pi^*\sigma)\big\vert_H)^m$ intersected with $P\times \{0\}$, which is $\pi^{-1}(Z)\cap P\times \{0\}$. The kernel of $\Omega_A$ at $\pi^{-1}(Z)\cap P \times \{0\}$ is the horizontal lift of the kernel of $\sigma$ at $Z$. Since $\ker(\sigma)\pitchfork Z$, we must have that $\ker(\Omega_A)\pitchfork \pi^{-1}(Z)$ at points of $P\times \{0\}$. Thus, $\ker(\Omega_A)$ is transverse to the degenerate hypersurface of $\Omega_A$ in a neighborhood of $P\times \{0\}$ and $\Omega_A$ is folded-symplectic in a neighborhood of $P\times \{0\}$.
\item The $G$-invariance of $\langle pr_2, A\rangle$ implies:
\begin{displaymath}
i_{X_P}\Omega_A = -i_{X_p}d\langle pr,A \rangle = di_{X_P}\langle pr_2,A \rangle = d\langle pr_2, X \rangle
\end{displaymath}
hence $\mu = -pr_2$ by the definition of a moment map.
\end{itemize}
\end{proof}
\begin{lemma}\label{lem:mincoupform1}
Let $G$ be a compact, connected Lie group and suppose $(F,\omega)$ is a symplectic manifold with an Hamiltonian action of $G$ and moment map $\mu:F \to \fg^*$. Let $(M,\sigma)$ be a folded-symplectic manifold and let $\pi:P \to M$ be a principal $G$-bundle over a folded-symplectic manifold. Then, for each choice of connection $1$-form $A$ on $P$, the associated bundle $P\times_G F$ carries a natural folded-symplectic structure in a neighborhood of $P\times_G \mu^{-1}(0)$.
\end{lemma}
\begin{proof}\mbox{ } \newline
\begin{itemize}
\item We exhibit $P\times_G F$ as a folded-symplectic reduced space, which will endow it with a natural folded-symplectic structure by theorem \ref{thm:fsred}.
\item Consider the product $(P\times \fg^*)\times F$ with the action of $G$ given by $g\cdot (p,\eta,f) = (pg^{-1},Ad^*(g)\eta, g\cdot f)$. The folded-symplectic structure is given by $\Omega_A \oplus \omega$ where we assume, for the time being, that this structure is folded-symplectic on the whole space and not simply on a neighborhood of $P\times \{0\} \times F$. Then the action is Hamiltonian with moment map:
\begin{displaymath}
J(p,\eta,f)= \mu(f) - \eta
\end{displaymath}
\item $J$ is a submersion, hence $0$ is a regular value and $J^{-1}(0)$ is a codimension $\dim(G)$ submanifold. It has a free action of $G$ since the action of $G$ on $P$ is free. Thus, $(P\times \fg^* \times F)//_0G :=J^{-1}(0)/G$ is a folded-symplectic manifold. Since $0$ is a regular value, $J^{-1}(0)$ intersects the folding hypersurface transversally, hence the reduced form on $J^{-1}(0)/G$ has a nonempty folding hypersurface.
\item The map $q: J^{-1}(0) \to P\times_G F$ given by $q(p,\eta,f) = [p,f]$ is a surjective submersion and the fibers are the $G$-orbits inside $J^{-1}(0)$. Thus, $q$ descends to a diffeomorphism $\bar{q}:J^{-1}(0)/G \to P\times_G F$. Since there is a natural folded-symplectic structure on $J^{-1}(0)/G$, $P\times_G F$ inherits a folded-symplectic structure.
\item Now, lemma \ref{lem:mincoupform} tells us that $\Omega_A$ is only symplectic in a neighborhood $U$ of $P\times \{0\}$. Thus, the map $q$ restricted to $J^{-1}(0)\cap (U\times F)$ descends to an open embedding of $(J^{-1}(0) \cap (U\times F))/G$ into $P\times_G F$. The image of this embedding contains a neighborhood of $P\times_G \mu^{-1}(0)$ since $J^{-1}(0)\cap (U\times F)$ contains $P\times \{0\} \times \mu^{-1}(0)$. This neighborhood inherits a folded-symplectic structure from the reduced space $(J^{-1}(0)\cap (U\times F))/G$.
\end{itemize}
\end{proof}
Now, we apply lemma \ref{lem:mincoupform1} to the case where we have a symplectic vector bundle $E$ over a folded-symplectic base $(M,\sigma)$.
\begin{lemma}\label{lem:mincoupform2}
Let $(M,\sigma)$ be a folded-symplectic manifold and let $\pi:E \to M$ be a symplectic vector bundle over $M$. Then for each choice of connection $1$-form $A$ on the frame bundle $Fr(E)$ of $E$, there exists a folded-symplectic form defined in a neighborhood $U$ of the zero section of $E$.
\end{lemma}
\begin{remark}
Note that we are not claiming anything about uniqueness of this folded-symplectic structure. It is not clear that one may deform one minimal coupling form into another, hence it is not clear that we may deform the resultant folded-symplectic structures on $E$ into one another.
\end{remark}
\begin{proof} \mbox{ } \newline
\begin{itemize}
\item If we choose an almost complex structure $J$ on $E$, then we may identify the structure group of $E$ with $U(n)$, which is compact and connected. The frame bundle $Fr(E)$ of $E$ is then a principal $U(n)$ bundle.
\item The typical fiber of $V$ is a symplectic vector space with a symplectic action of the structure group $U(n)$. As we have discussed, such actions are Hamiltonian. The moment map $\mu:V \to \operatorname{Lie(U(n))}^*$ satisfies $0\subset \mu^{-1}(0)$.
\item Now, $E$ is isomorphic to the associated bundle $Fr(E)\times_{U(n)} V$. A choice of connection $A$ on $Fr(E)$ induces a folded-symplectic structure on a neighborhood of $Fr(E)\times_{U(n)} \mu^{-1}(0)$ by lemma \ref{lem:mincoupform1}, which is a neighborhood of $Fr(E)\times_{U(n)} \{0\}$. Since this is the zero section of $E$, we have that a choice of connection $A$ on $Fr(E)$ endows a neighborhood of the zero section of $E$ with a folded-symplectic structure.
\end{itemize}
\end{proof}
\begin{remark}
The above construction is somewhat relevant in the case of toric, folded-symplectic manifolds. Let $(M,\sigma,\mu:\to \fg^*)$ be a toric, folded-symplectic manifold. We have shown that the orbit-type strata $M_H$ are folded-symplectic submanifolds. We have also shown that there is a well-defined symplectic normal bundle $(\widetilde{TM_H})^{\sigma}$. Lemma \ref{lem:mincoupform} provides us with a method of constructing a folded-symplectic structure in a neighborhood of the zero section of $(\widetilde{TM_H})^{\sigma}$. If one could prove a uniqueness statement, then this construction would give a local model for a neighborhood of an orbit-type stratum.
\end{remark}
\pagebreak
\section{Classifying Toric, Folded-Symplectic Bundles: $\pi_0(\mathcal{B}_{\psi}(W))$}
Classifying toric folded-symplectic bundles up to isomorphism is a straightforward task and we will proceed as follows. Let $G$ be a torus and let $\psi:W\to \fg^*$ be a unimodular map with folds, where $W$ is a manifold with corners. We first observe that isomorphism classes of principal $G$ bundles over $W$ are parameterized by $H^2(W,\mathbb{Z}_G)$. That is, for each principal $G$ bundle over $W$ there exists an element $c_1(P)\in H^2(W,\mathbb{Z}_G)$ called the \emph{first Chern class of} $P$, which specifies $P$ up to isomorphism (of principal $G$ bundles). Next, we fix an element $c_1(P)\in H^2(W,\mathbb{Z}_G)$ and choose a representative bundle $\pi:P\to W$ from this diffeomorphism class. In other words, we fix the structure of $P$ as a principal $G$ bundle. We ask the question: how can we parameterize the folded-symplectic structures on $\pi:P\to W$ for which the action of $G$ is Hamiltonian with moment map $\psi\circ \pi$. It turns out that the answer lies in the basic cohomology of $P$, $H^2(W,\R)$. We will show there exists a characteristic class $c_{hor}(P) \in H^2(W,\R)$ that specifies the Hamiltonian, folded-symplectic structure of $P$ up to isomorphism. Combining the two characteristic classes, we obtain a map:
\begin{displaymath}
c_1(\cdot) \times c_{hor}(\cdot):\pi_0(\mathcal{B}_{\psi}(W)) \to H^2(W,\mathbb{Z}_G\times \R)
\end{displaymath}
where $\mathcal{B}_{\psi}(W)$ is the category of toric, folded-symplectic bundles over $\psi:W\to \fg^*$ of definition \ref{def:bpsi} and $\pi_0(\mathcal{B}_{\psi}(W))$ is the set of isomorphism classes of objects. We will argue that this map is a bijection, completing the classification.
\subsection{Classifying Principal Torus Bundles}
We begin by discussing the classification of principal $G$ bundles over $W$, where $G$ is a torus. This task is perhaps not even deserving of its own section, but we would like to discuss it. Our arguments are very similar to those found in \cite{Mi1} for the classification of vector bundles. In particular, the example on p. 103 may be useful to the reader. All we are doing is replacing the structure group with the torus $G$. Given a torus, we have the following short exact sequence:
\begin{displaymath}
\xymatrix{
0 \ar[r] & \mathbb{Z}_G \ar[r]^{i} & \fg \ar[r]^{\exp} & G \ar[r] & 0.
}
\end{displaymath}
This induces a short exact sequence of locally constant sheaves on $W$,
\begin{displaymath}
\xymatrix{
0 \ar[r] & \underline{\mathbb{Z}_G} \ar[r]^{i} & \underline{\fg} \ar[r]^{\exp} & \underline{G} \ar[r] & 0.
}
\end{displaymath}
We then have a long exact sequence in \v{C}ech cohomology with coefficients in sheaves of abelian groups:
\begin{displaymath}
\xymatrix{
\ldots \ar[r] & \check{\mathrm{H}}^k(W,\underline{\mathbb{Z}_G}) \ar[r] & \check{\mathrm{H}}^k(W,\underline{\fg}) \ar[r] & \check{\mathrm{H}}^k(W,\underline{G}) \ar[r] & \ldots.
}
\end{displaymath}
Now, principal $G$-bundles are parameterized by $\check{\mathrm{H}}^1(W,\underline{G})$ since a principal $G$ bundle $\pi:P \to W$ may be specified by a cover $\{U_i\}$ of $W$ and trivializations $\phi_i:P\big\vert_{U_i} \to U_i \times G$. The trivializations give us transition maps $\phi_i\circ\phi_j^{-1}$, which are equivalent to maps $\phi_{ij}:(U_i\cap U_j) \to G$. The $\phi_{ij}'s$ satisfy the cocycle condition, hence they give us a $1$-cocycle in \v{C}ech cohomology which specifies the bundle up to isomorphism. The sheaf $\underline{\fg}$ is a fine sheaf since it admits partitions of unity, hence the \v{C}ech cohomology groups are $0$ in dimensions greater than $0$. In particular, we have a short exact sequence of cohomology groups:
\begin{displaymath}
\xymatrix{
0 \ar[r] & \check{\mathrm{H}}^1(W,\underline{G}) \ar[r] & \check{\mathrm{H}}^2(W,\underline{\mathbb{Z}_G}) \ar[r] & 0
}
\end{displaymath}
meaning $\check{\mathrm{H}}^1(W,\underline{G}) \simeq \check{\mathrm{H}}^2(W,\underline{\mathbb{Z}_G})$. On the other hand $\check{\mathrm{H}}^2(W,\underline{\mathbb{Z}_G})$ is isomorphic to the singular cohomology group $H^2(W,\mathbb{Z}_G)$, which finishes the classification of the structure as a principal $G$-bundle.
\begin{definition}\label{def:firstchern}
Let $P$ be a principal $G$ bundle over a manifold with corners $W$. Let $c_1(P)$ be the unique element in $H^2(W,\mathbb{Z}_G)$ corresponding to the isomorphism class of $P$. We call $c_1(P)$ the \emph{first Chern class of} $P$.
\end{definition}
\subsection{Classifying the Toric, Folded-Symplectic Structure}
Recall that we are assuming we have a unimodular map with folds $\psi:W \to \fg^*$. Fix an isomorphism class $c_1(P)\in H^2(W,\mathbb{Z}_G)$ of principal $G$ bundles and fix a representative member $\pi:P \to W$. Suppose we have two folded-symplectic structures $\sigma_1$,$\sigma_2$ on $P$ for which the action of $G$ is Hamiltonian with moment map $\psi\circ \pi$. We are going to give a necessary and sufficient condition for them to be isomorphic. In particular, we will show that their difference $\sigma_1-\sigma_2$ must be an exact, basic $1$-form. First, we give a recipe for how one can construct such folded-symplectic structures.
\begin{lemma}\label{lem:construct}
Let $\psi:W\to \fg^*$ be a unimodular map with folds, where $\fg$ is the Lie algebra of a torus $G$. Let $\pi:P \to W$ be a principal $G$-bundle and let $A\in \Omega^1(P,\fg)^G$ be a principal connection $1$-form on $P$. Then,
\begin{enumerate}
\item The two form $\sigma =d\langle \psi\circ \pi, A \rangle$ is a folded-symplectic form on $P$, where $\langle \cdot , \cdot \rangle:\fg^*\otimes \fg \to \R$ is the standard pairing.
\item Furthermore, the action of $G$ on $(P,\sigma)$ is Hamiltonian with moment map $\mu=\psi \circ \pi$.
\end{enumerate}
\end{lemma}
\begin{proof} \mbox{ } \newline
Let $n=\dim(G)=\dim(W)$, where the dimensions must be the same since $\psi$ is a unimodular map with folds.
\begin{enumerate}
\item $\sigma$ is a closed $2$-form by definition, hence we need only show that $\sigma^n \pitchfork_s 0$ and $\sigma$ restricted to its degenerate hypersurface has maximal rank. It may be written as:
\begin{displaymath}
\sigma = \langle (d\psi \circ d\pi) \wedge A \rangle + \langle \psi \circ \pi, dA \rangle
\end{displaymath}
Since $P$ is a principal torus bundle, the curvature form $dA$ is a basic $\fg$-valued $2$-form. Thus, the top power satisfies:
\begin{displaymath}
\sigma^n = (\langle (d\psi \circ d\pi \wedge A \rangle ) ^n
\end{displaymath}
If we choose a basis $\{e_1,\dots,e_n\}$ of $\fg$, then we may write:
\begin{displaymath}
\begin{array}{l}
A=\sum_{i=1}^n A_i\otimes e_i, \text{ where $A_i \in \Omega^1(P)^G$.} \\
\psi=\sum_{i=1}^n \psi_i \otimes e_i^*, \text{ where $\psi_i\in C^{\infty}(W)$ is smooth.}
\end{array}
\end{displaymath}
hence,
\begin{displaymath}
\sigma^n = C\pi^*(d\psi_1 \wedge\dots \wedge d\psi_n) \wedge(A_1\wedge \dots \wedge A_n)
\end{displaymath}
where $C\in \R$ and $C\ne 0$. The form $A_1\wedge \dots \wedge A_n$ is non-degenerate on the fibers of $P$ and the form $(d\psi_1 \wedge \dots \wedge d\psi_n)$ vanishes transversally on $W$ by corollary \ref{cor:folds4-1}, hence $\sigma^n \pitchfork_s 0$, where we are using transversality in the sense of manifolds with corners. From this calculation, we also have that the degenerate hypersurface of $\sigma$ is given by:
\begin{displaymath}
Z= \pi^{-1}(\hat{Z})
\end{displaymath}
where $\hat{Z}$ is the folding hypersurface of $\psi$. We still need to show that the restriction of $\sigma$ to the degenerate hypersurface, $i_Z^*\sigma$, has maximal rank. To this end, we will explicitly describe the kernel of $\sigma$ and see that it is $2$-dimensional and transverse to $Z$, hence its intersection with $TZ$ is $1$-dimensional.
\begin{itemize}
\item The first piece of the kernel will be constructed using the annihilator bundle of $\psi$, $\operatorname{Im}(d\psi)^o$ (q.v. definition \ref{def:annihilator}. If $p\in Z$ is a point in the degenerate hypersurface of $\sigma$, then we take any nonzero element $V\in \operatorname{Im}(d\psi)^o_{\pi(p)}$ in the fiber of the annihilator bundle. We then have that the induced vector field $V_P$ at $p$ is nonzero and lies in the kernel of $\sigma_p$:
\begin{displaymath}
(i_{V_P}\sigma)_p = \langle d\psi\circ d\pi_p \wedge A_p(V_P) \rangle = \langle d\psi \circ d\pi_p, V \rangle = 0
\end{displaymath}
\item The second portion of the kernel is slightly trickier to construct. We first pick a nonzero element $X\in \ker(d\psi_{\pi(p)})$ and choose a horizontal lift $\tilde{X}$ to an element of $T_pP$. Note that a horizontal lift satisfies $A(\tilde{X})=0$. We then have that the contraction with $\sigma$ satisfies:
\begin{displaymath}
(i_{\tilde{X}}\sigma)_p = \langle d\psi\circ d\pi(\tilde{X}), A\rangle_p + \langle \psi \circ \pi, i_{\tilde{X}}dA\rangle_p = \langle \psi \circ \pi, i_{\tilde{X}}dA \rangle_p
\end{displaymath}
hence the contraction isn't necessarily $0$. We would like to show that there is a vertical vector that we can add to $\tilde{X}$ to kill this extra term. $dA$ is basic, hence $dA=\pi^*\beta$ for some $\beta\in \Omega^2(W)$ and the contraction at $p$ is $\beta_{\pi(p)}(X, d\pi(\cdot))= \pi^*(i_X\beta)$. Consider the map:
\begin{displaymath}
\xymatrixrowsep{.5pc}\xymatrix{
F:\fg \ar[r] & T_{\pi(p)}^*W \\
\eta \ar[r] & \langle d\psi_{\pi(p)}, \eta\rangle
}
\end{displaymath}
The source and target space have the same dimension and the annihilator of the image of $d\psi$ is one dimensional, hence $F$ maps onto an $n-1$ dimensional subspace. The image of $F$ is contained in the space of covectors which vanish on $\ker(d\psi_{\pi(p)})$. Since this space also has dimension $n-1$, we have that $F$ surjects onto the space of covectors which vanish on $\ker(d\psi_{\pi(p)})$.
Since $X\in \ker(d\psi_{\pi(p)})$, we have that $(i_X\beta)_{\pi(p)}$ is a covector vanishing on $\ker(d\psi_{\pi(p)})$. Thus, there is an element $\eta \in \fg$ such that $\langle d\psi_{\pi(p)}, \eta \rangle = i_X\beta$. We now add $\eta_P$ to our horizontal lift $\tilde{X}$ and compute the contraction:
\begin{displaymath}
(i_{\tilde{X} +\eta_P}\sigma)_p = \pi^*(i_X\beta)_p - \langle d\psi \circ d\pi ,A(\eta_P) \rangle _p = \pi^*(i_X\beta)_p -\pi^*(i_X\beta) =0
\end{displaymath}
hence $\tilde{X} + \eta_P$ is in the kernel of $\sigma$ which is a vector transverse to the hypersurface $Z$.
\item Thus, the kernel of $\sigma$ at $p$ contains $\span\{\tilde{X} + \eta_P, V_P\}$. We check that the dimension of $\ker(\sigma_p)$ is no larger than $2$. The projection of $\ker(\sigma_p)$ via $d\pi_p$ is a surjection onto $\ker(d\psi_p)$ by proposition \ref{prop:umf}, hence any two vectors in $X_1, X_2$ in $\ker(\sigma_z)$ projecting onto $X\in \ker(d\psi_{\pi(p)})$ differ by a vector $V'_P$ tangent to the fibers of $P$. We then have that:
\begin{displaymath}
(i_{V'_P}\sigma)_p = -\langle d\psi \circ d\pi ,A(V'_P)\rangle = \langle d\psi \circ d\pi, V'\rangle
\end{displaymath}
hence $V'$ annihilates the image of $d\psi$. But the annihilator bundle is a line bundle, hence $V'_P$ is parallel to $V_P$ and so the kernel is spanned by $\tilde{X} + \eta_P$ and $V_P$, hence it is $2$-dimensional.
\end{itemize}
Finally, the action of $G$ is Hamiltonian with moment map $\psi\circ \pi$ since for all $X\in \fg$ we have:
\begin{displaymath}
i_{X_P}\sigma = i_{X_P}d\langle \psi \circ \pi, A\rangle = -d(i_{X_P}\langle \psi \circ \pi, A \rangle) = -d\langle \psi \circ \pi, X \rangle
\end{displaymath}
where the second equality follow from the $G$-invariance of the form $\langle \psi\circ \pi, A\rangle$.
\end{enumerate}
\end{proof}
\begin{remark}\label{rem:kernelconstruction}
Given a unimodular map with folds, $\psi:W\to \fg^*$, lemma \ref{lem:construct} gives us a recipe for constructing a folded-symplectic structure $\sigma$ on a principal $G$-bundle $\pi:M \to W$ so that the $G$-action is Hamiltonian, where the moment map is $\psi\circ \pi$. Incidentally, we also found a recipe for constructing the kernel of $\sigma$ at each point $p$ in the folding hypersurface $Z=\pi^{-1}(\hat{Z})$, where $\hat{Z}$ is the folding hypesurface of $\psi$. The steps are as follows:
\begin{itemize}
\item Fix a connection $A$ on $P$. Indeed, lemma \ref{lem:construct} requires us to fix a connection on $P$.
\item At each $p\in Z$ in the folding hypersurface, we consider any nonzero element $V$ of the fiber of the annihilator bundle $\operatorname{Im}(d\psi)^{o}_{\pi(p)}$. Then the induced vector field $V_P$ is in the kernel of $\sigma$ at $p$: $V_P(p)\in \ker(\sigma_p)$.
\item Choose a horizontal lift $\tilde{X}$ of any nonzero element of $\ker(d\psi_{\pi(p)})$. Let $\eta \in \fg$ be a Lie algebra element satisfying $i_{\eta_P}\sigma = -\langle \psi\circ \pi, i_{\tilde{X}}dA\rangle$, which can be done since $i_{\tilde{X}}dA$ is the pullback of a covector vanishing on $\ker(d\psi_{\pi(p)})$. Then $\tilde{X} + \eta_P(p)$ is in the kernel of $\sigma$ at $p$. It is transverse to $Z$ since $X$ is transverse to the folding hypersurface $\hat{Z}$ of $\psi$ and $Z=\pi^{-1}(\hat{Z})$.
\item As we showed in the proof of \ref{lem:construct}, these two vectors span the kernel of $\sigma$ at $p$.
\end{itemize}
\end{remark}
We now begin the process of constructing a bijection between $H^2(W,\R)$ and isomorphic folded-symplectic structures on $P$ for which the action of $G$ is Hamiltonian with moment map $\psi\circ \pi$. The first step is to produce a map from the space of closed two forms $\Omega^2(W)^c$ to the space of folded-symplectic structures on the principal bundle $P$.
\begin{lemma}\label{lem:map1}
Let $\psi:W \to \fg^*$ be a unimodular map with folds, where $\fg$ is the Lie algebra of a torus $G$. Suppose $\pi:P \to W$ is a principal $G$ bundle and suppose $\sigma$ is a closed, invariant $2$-form for which the action of $G$ is Hamiltonian. Let $A\in \Omega^1(P,\fg)^G$ be a connection $1$-form on $P$ and let $\Omega^2(W)^c$ be the space of closed $2$-forms. Then
\begin{enumerate}
\item $\sigma = d(\langle \psi\circ \pi, A\rangle) + \pi^*\beta$ for some closed $2$-form $\beta\in\Omega^2(W)^c$, hence $\sigma$ is folded-symplectic.
\item Let $\mathcal{S}$ denote the space of folded-symplectic structures on $P$ for which the action of $G$ is Hamiltonian with moment map $\psi\circ \pi$. Then the map $F_A:\Omega^2(W)^c \to \mathcal{S}$ given by $F_A(\beta) = d\langle \psi\circ \pi \rangle + \pi^*\beta$ is a bijection.
\end{enumerate}
\end{lemma}
\begin{proof}\mbox{ } \newline
\begin{enumerate}
\item We consider the difference $\Omega=\sigma -d\langle\psi\circ \pi, A \rangle$. Since the action of $G$ is Hamiltonian for both forms with the same moment maps $\psi\circ \pi$, we have that for each $X\in \fg$ the contraction of $X_P$ with $\Omega$ satisfies:
\begin{displaymath}
i_{X_P}\Omega = i_{X_P}\sigma -i_{X_P}d\langle \psi\circ \pi, A \rangle = -d\langle \psi\circ \pi, X \rangle + d\langle \psi\circ\pi, X\rangle = 0
\end{displaymath}
Since the action of $G$ preserves both $\sigma$ and $d\langle \psi\circ \pi, A\rangle$, the action of $G$ also preserves $\Omega$. Thus, $\Omega$ is a closed, invariant $2$-form that vanishes on vertical vectors, so it must be the pullback of a closed, basic $2$-form $\beta$. That is, $\Omega=\pi^*\beta$ for some $\beta \in \Omega^2(W)^c$. We therefore have:
\begin{displaymath}
\sigma = d\langle \psi\circ \pi, A \rangle + \pi^*\beta
\end{displaymath}
To see that $\sigma$ is folded-symplectic we compute its top power and we compute its kernel at degenerate points.
\begin{itemize}
\item We have $\sigma^n = (d\langle \psi\circ \pi, A \rangle + \pi^*\beta)^n = (d\langle \psi\circ \pi, A \rangle)^n$ since $\pi^*\beta$ is basic. Thus, $\sigma^n \pitchfork_s 0$ by our considerations in lemma \ref{lem:construct} and $\sigma$ degenerates on the hypersurface $Z=\pi^{-1}(\hat{Z})$.
\item The kernel of $\sigma$ may be computed in the same way we computed the kernel of $d\langle \psi\circ \pi, A \rangle$ in remark \ref{rem:kernelconstruction}. At each point $p\in Z$ in the degenerate hypersurface, we may choose a nonzero element $V\in \operatorname{Im}(d\psi)^o_{\pi(p)}$ and then $(i_{V_P}\sigma)_p=0$. To obtain the piece of the kernel transverse to $Z$, we begin with $X\in \ker(d\psi_{\pi(p)})$ and take a horizontal lift to $\tilde{X} \in T_pP$. Then,
\begin{displaymath}
(i_{\tilde{X}}\sigma)_p =\langle \psi\circ \pi, i_{\tilde{X}}dA \rangle_p + \pi^*(i_X\beta)_p
\end{displaymath}
which is the pullback of a covector that vanishes on $\ker(d\psi)_{\pi(p)}$. Thus, there is an element $\eta \in \fg$ such that
\begin{displaymath}
\langle d\psi \circ d\pi_p, \eta\rangle = \langle \psi\circ \pi, i_{\tilde{X}}dA \rangle_p + \pi^*(i_X\beta)_p.
\end{displaymath}
The contraction of $\tilde{X} + \eta_P(p)$ with $\sigma$ at $p$ is then zero.
\item The kernel cannot have a larger dimension since $\ker(d\psi_{\pi(p)})$ is $1$-dimensional and $\operatorname{Im}(d\psi)^o_{\pi(p)}$ is $1$-dimensional. A larger kernel would imply that at least one of these spaces would have a larger dimension.
\end{itemize}
\item We have therefore shown that the map $F_A(\beta) = d\langle \psi \circ \pi, A\rangle +\pi^*\beta$ has image in the space of folded-symplectic forms for which the action of $G$ is Hamiltonian with moment map $\psi\circ \pi$. We have also shown that it is surjective. It is injective because the map $\pi^*$ is injective, hence $F_A$ is a bijection.
\end{enumerate}
\end{proof}
\begin{definition}\label{def:connectionmap}
Let $\psi:W \to \fg^*$ be a unimodular map with folds, where $\fg$ is the Lie algebra of a torus, $G$. Let $\pi:P \to W$ be a principal $G$-bundle with connection $A\in \Omega^1(P,\fg)^G$, let $\mathcal{S}$ be the space of folded-symplectic forms for which the action of $G$ is Hamiltonian with moment map $\psi\circ \pi$, let $\Omega^2(W)^c$ be the space of closed $2$ forms on $W$, and let $F_A:\Omega^2(W) \to \mathcal{S}$ be the bijection of lemma \ref{lem:map1}. We define $f_A: \mathcal{S} \to \Omega^2(W)$ to be the inverse of $F_A$. We call $f_A$ the \emph{horizontal map} induced by the connection $A$.
\end{definition}
\begin{lemma}\label{lem:map2}
Let $\psi:W \to \fg^*$ be a unimodular map with folds, let $\pi:P \to W$ be a principal $G$ bundle, and let $\mathcal{S}$ be the space of folded-symplectic forms for which the action of $G$ is Hamiltonian with moment map $\psi\circ \pi$. Then for any choice of connection $A\in \Omega^1(P,\fg)^G$, we obtain the horizontal map $f_A:\mathcal{S} \to \Omega^2(W)^c$. If $p:\Omega^2(W)^c \to H^2(W,\R)$ is the projection map to homology, then the composite:
\begin{displaymath}
p\circ f_A: \mathcal{S} \to H^2(W,\R)
\end{displaymath}
is independent of the choice of connection $A$.
\end{lemma}
\begin{proof}\mbox{ } \newline
Suppose we choose a second connection $A'$ and perform the same constructions as in lemma \ref{lem:map1} to obtain $F_{A'}$ and, consequently, its inverse $f_{A'}$. We let $\sigma \in \mathcal{S}$ and we let $\beta_A = f_A(\sigma)$, $\beta_{A'}=f_{A'}(\sigma)$ be the two different images in $\Omega^2(W)^c$. We compute:
\begin{displaymath}
\sigma = d\langle \psi\circ \pi, A \rangle + \pi^*\beta_A = d\langle \psi\circ \pi, A'\rangle + \pi^*\beta_{A'}
\end{displaymath}
by definition of $f_{A}$ and $f_{A'}$. Consequently,
\begin{displaymath}
\pi^*(\beta_{A'}-\beta_A) = d(\langle \psi\circ \pi, A-A'\rangle )
\end{displaymath}
since $\langle\psi \circ \pi, A-A'\rangle$ is basic, i.e. it is $\pi^*\alpha$ for some $\alpha \in \Omega^1(W)$, we have:
\begin{displaymath}
\pi^*(\beta_A-\beta_{A'}) = d\pi^*\alpha = \pi^*d\alpha
\end{displaymath}
which means $\beta_A - \beta_{A'} = d\alpha$ since $\pi^*$ is injective. Thus, $p\circ f_A(\sigma)=p\circ f_{A'}(\sigma)$ and so the map is independent of the choice of connection.
\end{proof}
\begin{definition}
Let $\psi:W \to \fg^*$ be a unimodular map with folds, where $\fg$ is the Lie algebra of a torus $G$. Let $\pi:P \to W$ be a principal $G$ bundle
and let $\mathcal{S}$ be the space of folded-symplectic forms on $P$ for which the action of $G$ is Hamiltonian with moment map $\psi\circ \pi$. For any connection $A\in \Omega^1(P,\fg)^G$, we obtain the horizontal map $f_A:\mathcal{S} \to \Omega^2(W)^c$. By lemma \ref{lem:map2}, the composite $p\circ f_A: \mathcal{S} \to H^2(W,\R)$ is independent of the choice of connection $A$. We define this map to be the \emph{horizontal chern class} $c_{hor}: \mathcal{S} \to H^2(W,\R)$.
\end{definition}
The horizontal chern class of a folded-symplectic structure will be a characteristic class: it behaves well under isomorphisms of principal $G$ bundles.
\begin{cor}\label{cor:map2}
Let $(\pi_i:P_i\to W, \sigma_i)$, $i=1,2$, be two toric, folded-symplectic bundles over $\psi:W\to \fg^*$ Suppose $\phi:P_1\to P_2$ is an isomorphism of principal $G$ bundles over $W$. Then $c_{hor}(\phi^*\sigma_2)=c_{hor}(\sigma_2)$.
\end{cor}
\begin{proof}\mbox{ } \newline
Note that we actually don't need the folded structure $\sigma_1$, but we have inserted it since we are studying objects inside $\mathcal{B}_{\psi}(W)$ and all such objects come equipped with a folded-symplectic structure. In any case, we fix a connection $A_2$ on $P_2$ and write:
\begin{displaymath}
\sigma_2 = d\langle \psi\circ \pi_2, A_2 \rangle \pi_2^*\beta_2
\end{displaymath}
and
\begin{displaymath}
\phi^*\sigma_2 = d\langle \psi\circ \pi_1, \phi^*A_2 \rangle + \pi_1^*\beta_2,
\end{displaymath}
where $\phi^*A_2$ is the pullback connection. We then have that $c_{hor}(\phi^*\sigma_2)= [\beta_2] = c_{hor}(\sigma_2)$.
\end{proof}
We now show that if $\pi:P\to W$ is a fixed principal $G$ bundle over $\psi:W \to \fg^*$ and $\sigma_0$, $\sigma_1$ are two folded-symplectic structures on $P$ for which the action of $G$ is Hamiltonian with moment map $\psi\circ \pi$, then $(P,\sigma_0)$ and $(P,\sigma_1)$ are isomorphic if and only if their horizontal chern classes agree.
\begin{prop}\label{prop:map3}
Let $\psi:W \to \fg^*$ be a unimodular map with folds and let $\pi:P \to W$ be a principal $G$ bundle. Let $\sigma_0$, $\sigma_1$ be two folded-symplectic forms on $P$ for which the action of $G$ is Hamiltonian with moment map $\psi\circ \pi$. Then there exists a gauge transformation $\phi:P \to P$ satisfying $\phi^*\sigma_1=\sigma_0$ if and only if $c_{hor}(\sigma_0)=c_{hor}(\sigma_1)$. That is, the cohomology class of the basic $2$-forms associated to $\sigma_0$ and $\sigma_1$ must be the same.
\end{prop}
\begin{proof}\mbox{ } \newline
\begin{enumerate}
\item We show the \emph{only if} portion first. Suppose $\phi:P\to P$ is a gauge transformation satisfying $\phi^*\sigma_1=\sigma_0$. Fix a connection $A$ on $P$. We have
\begin{equation}\label{eq:sigmai}
\sigma_i = d\langle \psi\circ \pi, A\rangle + \pi^*\beta_i
\end{equation}
for $i=0,1$ by lemma \ref{lem:map1}. Consequently,
\begin{displaymath}
\begin{array}{lcl}
\phi^*\sigma_1 & = & d\langle \psi\circ \pi, \phi^*A \rangle + \pi^*\beta_1\\
& = & d\langle \psi\circ \pi, A \rangle + \pi^*\beta_0
\end{array}
\end{displaymath}
since, by assumption, $\phi^*\sigma_1=\sigma_0$. We then have:
\begin{displaymath}
\pi^*(\beta_0-\beta_1)=d(\psi\circ \pi,\phi^*A - A \rangle.
\end{displaymath}
Since $\langle \psi \circ \pi, \phi^*A -A \rangle$ is a basic $1$-form, it is $\pi^*\gamma$ for some $\gamma\in\Omega^1(W)$. Thus,
\begin{displaymath}
\pi^*(\beta_0-\beta_1)= d\pi^*\gamma = \pi^*d\gamma
\end{displaymath}
and since $\pi^*$ is injective, we have $\beta_0-\beta_1 =d\gamma$, hence $c_{hor}(\sigma_1)=[\beta_1]=[\beta_0]=c_{hor}(\sigma_0)$.
\item Now, suppose that $c_{hor}(\sigma_0)=c_{hor}(\sigma_1)$. That is, by equation \ref{eq:sigmai}, $[\beta_1]=[\beta_0]$. Then there is a one form $\gamma\in \Omega^1(W)$ such that $\beta_1=\beta_0 + d\gamma$. We define the linear path of fold forms:
\begin{equation}\label{eq:foldpath1}
\sigma_s:=\sigma_0 + s\pi^*(d\gamma)
\end{equation}
which is folded for all $s$ since:
\begin{itemize}
\item $(\sigma_s)^n=\sigma_0^n$, hence $\sigma_s\pitchfork_s 0$ and $Z=(\sigma_s)^{-1}(0) = (\sigma_0)^{-1}(0)$ is an embedded submanifold with corners of $P$.
\item Using the ideas of remark \ref{rem:kernelconstruction}, we have that
\begin{equation}\label{eq:kernel}
\ker(\sigma_s)_p=\span\{\tilde{X}+v_s, V\},
\end{equation}
where $\tilde{X}$ is some lift of $X\in \ker(d\psi)_{\pi(p)}$, $v_s$ is some vertical vector, and $V$ is a vertical vector generated by an element in the fiber of $\operatorname{Im}(d\psi)_{\pi(p)}^o$. Because the kernel element $X\in \ker(d\psi)$ is transverse to the folding hypersurface $\hat{Z}$ of $\psi$ and $Z= \pi^{-1}(\hat{Z})$, we have that $\tilde{X} +v_s$ is transverse to $T_pZ$, hence $i_Z^*\sigma_s$ has maximal rank.
\end{itemize}
Using our usual Moser-type argument, we seek an isotopy satisfying $\phi_s^*\sigma_s=\sigma_0$, which amounts to solving:
\begin{equation}\label{eq:pplmoser}
di_{X_s}\sigma_s = -d\pi^*\gamma
\end{equation}
for a time-dependent vector field $X_s$, whose flow is $\phi_s$. It is then sufficient to solve
\begin{equation}\label{eq:pplmoser1}
i_{X_s}\sigma_s = -\pi^*\gamma
\end{equation}
which has a smooth solution if and only if $\pi^*\gamma$ vanishes on $\ker(\sigma_s)$ at the points of $Z$ for all $s$ (q.v. proposition \ref{prop:Moser}). By our description of the kernel of $\sigma_s$ in equation \ref{eq:kernel}, we will have a solution if and only if $\gamma \in \Omega^1(W)$ vanishes on $\ker(d\psi)$ at points of the folding hypersurface $\hat{Z}$ of $\psi$. It is not clear that $\gamma$ satisfies this condition so we modify the right hand side of equation \ref{eq:pplmoser1} by adding the pullback of a \emph{closed} basic $1$ form $\pi^*\gamma_0$ so that
\begin{equation}\label{eq:pplmoser2}
i_{X_s}\sigma_s = -(\pi^*\gamma + \pi^*\gamma_0)=-\pi^*(\gamma - \gamma_0)
\end{equation}
has a smooth solution. Note that this solution $X_s$ will satisfy:
\begin{displaymath}
di_{X_s}\sigma_s = -\pi^*d\gamma -\pi^*d\gamma_0 = -\pi^*d\gamma
\end{displaymath}
since $\gamma_0$ is closed, hence $X_s$ will solve equation \ref{eq:pplmoser} as required. Since $\hat{Z}$ is co-orientable in $W$ and $\ker(d\psi)$ is stratified (by definition \ref{def:umf}), we may make the following choices:
\begin{itemize}
\item We choose an nonvanishing section $X \in \Gamma(\ker(d\psi))$ which we extend to a vector field $\tilde{X}$ on $W$.
\item We choose a smooth function $g:W \to \R$ such that $g\big\vert_{\hat{Z}}=0$ and $dg(X)\big\vert_{\hat{Z}}=1$.
\end{itemize}
Set let $f$ be the product $f=(g)(\gamma(\tilde{X}))\in C^{\infty}(W)$. We define:
\begin{displaymath}
\gamma_0 = -df =-)\gamma(\tilde{X})dg - (g)d(\gamma(\tilde{X})).
\end{displaymath}
For all $z\in \hat{Z}$, we have:
\begin{displaymath}
\begin{array}{lcl}
i_{X(z)}(\gamma_z + (\gamma_0)_z) & = & \gamma(X) + i_X(\gamma_0) \\
& = & \gamma(X) -\gamma(\tilde{X})dg(X) - (g)(d(\gamma(\tilde{X}))(X)) \\
& = & \gamma(X) - \gamma(X)(1) - (0)(d(\gamma(\tilde{X}))(X)) \\
& = & 0
\end{array}
\end{displaymath}
where we have suppressed the subscript $z$ after the first step to avoid notational clutter. Consequently equation \ref{eq:pplmoser2} has a smooth, $G$-invariant solution $X_s$. The $G$-invariance of $X_s$ follows from the fact that $\sigma_s$ and $\pi^*(\gamma + \gamma_0)$ are both $G$-invariant.
Now, we claim that $X_s$ is tangent to orbits for all $s$. Indeed, for all $X\in \fg$ we have $(i_{X_s}\sigma_s)(X_P) = \pi^*(\gamma + \gamma_0)(X_P) =0$ since the right hand side is basic. Thus, for all $p\in P$ we have that $X_s(p)\in T_p(G\cdot p)^{\sigma_s}$. Since $\sigma_s$ is symplectic on an open dense subset and the orbits are Lagrangian, this implies that $X_s(p)\in T_p(G\cdot p)$ for $p$ in an open dense subset of $P$. Thus, smoothness of $X_s$ implies $X_s$ is tangent to orbits everywhere. In other words, $X_s$ is tangent to the fibers of $\pi:P \to W$. Since the fibers are compact, we may integrate $X_s$ to obtain $\phi_s$, which maps fibers to fibers and is thus a gauge transformation: $\pi \circ \phi_s = \pi$. We take $\phi_1$ as our requisite isomorphism.
\end{enumerate}
\end{proof}
We are now ready to classify objects in $\mathcal{B}_{\psi}(W)$.
\begin{theorem}\label{thm:bundleclassification}
Let $\psi:W \to \fg^*$ be a unimodular map with folds, where $\fg$ is the Lie algebra of a torus $G$. Let $\mathcal{B}_{\psi}(W)$ be the category of toric, folded-symplectic bundles over $\psi$. Then there is a bijection $b:\pi_0(B_{\psi}) \to H^2(W; \mathbb{Z}_G \times \R)$ given by:
\begin{displaymath}
b([(\pi:P \to W, \sigma)]) = c_1(P)\times c_{hor}(\sigma).
\end{displaymath}
That is, isomorphism classes of toric, folded-symplectic bundles over $\psi$ are parameterized by cohomology classes in $H^2(W,\mathbb{Z}_G\times \R)$.
\end{theorem}
\begin{proof}\mbox { } \newline
Let $(\pi_1:P \to W, \sigma_1)$ and $(\pi_2:P\to W, \sigma_2)$ be two toric, folded-symplectic bundles over $\psi:W \to \fg*$. First, let us show that the map $b$ is defined. Recall that isomorphism classes of principal $G$-bundles over $W$ are parameterized by their first Chern class $c_1 \in H^2(W,\mathbb{Z}_G)$. Thus, there exists an isomorphism of principal $G$-bundles $\phi:P_1 \to P_2$ if and only if $c_1(P_1)=c_1(P_2)$.
Suppose $\phi:P_1\to P_2$ is an isomorphism of toric folded-symplectic bundles over $\psi:W \to \fg^*$. That is $\phi$ is a map of principal bundles and $\phi^*\sigma_2= \sigma_1$. Then corollary \ref{cor:map2} implies that $c_{hor}(\sigma_1)=c_{hor}(\phi^*\sigma_2) = c_{hor}(\sigma_2)$. Since $\phi$ is an isomorphism of principle bundles, we have $c_1(P_1)=c_1(P_2)$ by the above remarks. Thus, the map is well-defined.
Now, suppose $c_1(P_1)=c_1(P_2)$ and $c_{hor}(\sigma_1)=c_{hor}(\sigma_2)$. Then there exists an isomorphism of principal bundles $\phi:P_1\to P_2$. By corollary \ref{cor:map2}, $c_{hor}(\phi^*\sigma_2) = c_{hor}(\sigma_2)=c_{hor}(\sigma_1)$. By proposition \ref{prop:map3} there exists a gauge transformation $\phi_1:P_1\to P_1$ such that $\phi_1^*(\phi^*\sigma_2)=\sigma_1$, hence $\phi\circ \phi_1:P_1 \to P_2$ is an isomorphism of toric, folded-symplectic bundles over $\psi:W \to \fg^*$.
\end{proof}
\pagebreak
\section{Construction of the Functor $c:\mathcal{B}_{\psi}(W) \to \mathcal{M}_{\psi}(W)$.}
The construction of the functor $c:\mathcal{B}_{\psi}(W) \to \mathcal{M}_{\psi}(W)$ is local in nature and is accomplished in three steps. In the first step, we take an object $(\pi:P\to W, \sigma)$ of $\mathcal{B}_{\psi}(W)$ and we collapse its boundary strata in a natural way to obtain a topological space, which we call $c_{top}(P)$. We then argue that there is a natural smooth structure on $c_{top}(P)$ which is constructed using a local cutting procedure which we explain below. Furthermore, the charts on $c_{top}(P)$ are such that the transition maps are folded-symplectic maps, hence there is a global folded-symplectic structure on the space that we call $c(P)$, which is just $c_{top}(P)$ with its smooth structure constructed below. Finally, we show that the assignment $P \to c(P)$ is functorial, hence $c$ is a well-defined functor. This is the strategy used in \cite{KL} and we follow it very closely, the reason being that the ingredients for the construction are essentially the same: we have the notion of a symplectic slice representation (q.v. proposition \ref{prop:normbundle}), we have the ability to read the slice representation from the orbital moment map (q.v. theorem \ref{thm:structure}), and we have folded-symplectic reduction. With these tools in hand, one need only add in a few extra remarks to show that the constructions in \cite{KL} extend across the folding hypersurface of $W$. Before we begin, we recall the definitions of the presheaves (q.v. remark \ref{rem:empsi}) $\mathcal{M}_{\psi}$ and $\mathcal{B}_{\psi}$:
\begin{definition}
Let $W$ be a manifold with corners and let $\psi:W\to \fg^*$ be a unimodular map with folds, where $\fg$ is the Lie algebra of a torus $G$. We define the category $\mathcal{M}_{\psi}(W)$ to be the category whose objects are triples:
\begin{displaymath}
(M,\sigma, \pi:M \to W)
\end{displaymath}
where $\pi$ is a quotient map and $(M,\sigma, \psi \circ \pi)$ is a toric, folded-symplectic manifold with co-orientable folding hypersurface, where the torus is $G$, with moment map $\psi \circ \pi$. We refer to an object as a \emph{toric, folded-symplectic manifold over $\psi$}. A morphism between two objects $(M_i,\sigma_i,\pi_i:M \to W)$, $i=1,2$, is an equivariant diffeomorphism $\phi:M_1 \to M_2$ that induces a commutative diagram:
\begin{displaymath}
\xymatrix{
M_1 \ar[rr]^{\phi} \ar[dr]^{\pi_1} & & M_2 \ar[dl]^{\pi_2} \\
&W \ar[r]^{\psi}& \fg^*
}
\end{displaymath}
and satisfies $\phi^*\sigma_2=\sigma_1$, hence $\phi$ is an equivariant folded-symplectomorphism that preserves moment maps. By definition, every morphism is invertible, hence $\mathcal{M}_{\psi}(W)$ is a groupoid.
\end{definition}
\begin{definition}
Let $\psi: W \to \fg^*$ be a fixed unimodular map with folds, where $\fg$ is the Lie algebra of a torus $G$. We define $\mathcal{B}_{\psi}(W)$ to be the category whose objects are principal $G$-bundles $\pi:P \to W$ equipped with an invariant folded-symplectic form $\sigma$ with co-orientable folding hypersurface, denoted as a pair
\begin{displaymath}
(\pi:P \to W, \sigma)
\end{displaymath}
so that $\psi\circ \pi$ is a moment map for the principal action of the torus $G$ on $P$. A morphism $\phi$ between two objects $(\pi_1:P_1\to W, \sigma_1)$ and $(\pi_2:P_2 \to W, \sigma_2)$ is a map of principal $G$ bundles:
\begin{displaymath}
\xymatrix{
P_1 \ar[rr]^{\phi} \ar[dr]^{\pi_1}& & P_2 \ar[dl]^{\pi_2} \\
&W\ar[r]^{\psi} & \fg^*
}
\end{displaymath}
so that $\phi^*\sigma_2 = \sigma_1$, hence $\phi^*(\psi\circ \pi_2) =\psi \circ \pi_1$. That is, $\phi$ preserves moment maps.
\end{definition}
\begin{remark}
If $\psi:W\to \fg^*$ is a unimodular map with folds and $W$ has no boundary, i.e. $W$ is a manifold without corners, then
\begin{displaymath}
\mathcal{B}_{\psi}(W) = \mathcal{M}_{\psi}(W).
\end{displaymath}
In general, if $W$ is a manifold with corners then its interior $\mathring{W}$ is a manifold, hence
\begin{displaymath}
\mathcal{B}_{\psi}(\mathring{W})=\mathcal{M}_{\psi}(\mathring{W})
\end{displaymath}
\end{remark}
Per our custom, we are assuming folding hypersurfaces are \emph{co-orientable}.
\subsection{Step 1 - Define $c_{top}(P)$}
Fix an object $(\pi:P\to W, \sigma)$ of $\mathcal{B}_{\psi}(W)$. Recall that for each point $w\in W$, $\psi$ attaches to $w$ an integral basis $\{v_1,\dots, v_k\}$ of the integral lattice of a subtorus $K_w$ of $G$ (q.v. lemma \ref{lem:attach1}). We let $\sim$ be the smallest equivalence relation on $P$ such that for all $p_1, p_2\in P$, $p_1 \sim p_2$ if and only if:
\begin{enumerate}
\item $\pi(p_1)=\pi(p_2)=:w$
\item There exists an element $k\in K_w$ satisfying $k\cdot p_1=p_2$.
\end{enumerate}
We let $c_{top}(P)$ be the topological space $P/\sim$. Since the action of $G$ commutes with the action of each $K_w$, the action of $G$ on $P$ descends to a continuous action of $G$ on $c_{top}(P)$. Furthermore, given a morphism $\phi\in Hom((P_1,\sigma_1),(P_2,\sigma_2))$ between two objects, the equivariance of $\phi$ implies that $\phi$ descends to an equivariant homeomorphism $c_{top}(\phi):c_{top}(A)\to c_{top}(B)$ and that $c_{top}:\mathcal{B}_{\psi} \to Top_G$ is a functor.
\begin{remark}\label{rem:restriction}
More specifically, $Top_G$ is the category whose objects are topological $G$-spaces and for any $X,Y \in \operatorname{Ob}(Top_G)$, $\hom(X,Y) = \{\phi:X \to Y \vert \space \text{ $\phi$ is an equivariant homeomorphism}\}$. One can show that $Top_G$ is a presheaf of groupoids and that:
\begin{displaymath}
c_{top}:\mathcal{B}_{\psi} \to Top_G
\end{displaymath}
is a map of presheaves of groupoids. Furthermore, note that the two spaces $c_{top}(P\vert_U)$ and $c_{top}(P)\vert_U$ are the same set and have the same topology, hence:
\begin{displaymath}
c_{top}(P\vert_U) = c_{top}(P)\vert_U
\end{displaymath}
Thus, for any morphism $\phi:P_1 \to P_2$ between objects in $\mathcal{B}_{\psi}$ and any open subset $U\subset W$, we will write $c_{top}(\phi)\vert_U$ to mean $c_{top}(\phi\vert_U)$.
\end{remark}
\subsection{Step 2 - Construction of Smooth and Folded-Symplectic Structures}
For each $w\in W$ lemma \ref{lem:attach1} attaches an integral basis $\{v_1,\dots, v_k\}$ of the integral lattice of a subtorus $K_w$ of $G$ and an effective symplectic representation $V$ of $K_w$ with weights $\{v_1^*,\dots, v_k^*\}$. Given such a representation, we can form the global reduced space $cut(P):=(P \times V)//_0 K_w$. Of course, this may not be a manifold, but we have the following existence lemma to remedy this problem locally.
\begin{lemma}\label{lem:cut}
Let $(\pi:P\to W, \sigma)$ be a toric folded-symplectic bundle over $\psi:W\to \fg^*$. For each $w\in W$, $\exists$ a neighborhood $U_w$ of $w$ so that $cut(P\vert_{U_w}):= (P\vert_{U_w} \times V)//_0 K_w$ is a toric folded symplectic manifold over $\psi\vert_{U_w}: U_w \to \fg^*$.
\end{lemma}
\begin{proof}
Let $K_w$ be the subtorus of $G$ asociated to $w$ and $\frak{k}_w$ its Lie algebra. Since faithful representations of tori are classified by their weights, we may assume $V= \C^k$. Let $j:\frak{k}_w \to \fg$ be the inclusion of the Lie algebra of $K_w$, $j^*:\fg^*\to \frak{k}_w^*$ the corresponding projection, and $\xi_0 = j^*\psi(w)$.
\begin{itemize}
\item A moment map for the action of $K_w$ on $P\vert_{U_w} \times V$ is given by
\begin{equation}\label{eq:cutmoment}
\Phi(p,z)=(j^*\psi)\circ \pi(p) - \sum_{i=1}^k \vert z_i \vert^2 v_i^* -\xi_0.
\end{equation}
\item We note that $j^*\psi$ is a submersion at $w$. This is because $\frak{k}_w$ is normal to the image of the stratum containing $w$. The fold map induces fiberwise isomorphisms on normal bundles (q.v. corollary \ref{cor:folds4-1}), hence it sends directions normal to the stratum containing $w$ isomorphically onto directions transverse to the affine hyperplane corresponding to $w$ in the image of $\psi$. Hence, there is a neighborhood $U$ of $w$ on which $j^*\psi$ is a submersion.
\item We may choose $U$ to be contractible so that $P\vert_U \simeq U\times G$. More precisely, we have a commuting diagram:
\begin{displaymath}
\xymatrix{U\times G \ar[r]^{\simeq} \ar[dr]^{pr_1} & P\vert_U \ar[d]^{\pi} \ar[dr]^{\mu} & \\
& U \ar[r]^{\psi} &\fg^*}
\end{displaymath}
\item By corollary \ref{cor:attach}, we may choose $U$ so that $\psi\vert_U=\bar{\mu}\circ\Psi$, where $\bar{\mu}:U \to \fg^*$ is a unimodular local embedding and $\Psi:U\to U$ is a map with fold singularities. If $U$ is isomorphic to a subset of $R^n$, then both $\bar{\mu}$ and $\Psi$ extend to $\tilde{\mu}:\tilde{U}\to \fg^*$ and $\tilde{\Psi}:\tilde{U}\to \tilde{U}$, where $\tilde{U}$ is a manifold without corners containing $U$, $\tilde{\mu}$ is a local embedding, and $\tilde{\Psi}$ is a map with fold singularities. This is because these maps are smooth, hence they extend by definition, and being an embedding or a fold map is an open property.
\item Because $\mu$ is a local embedding, we may assume that $\tilde{\mu}\vert_{\tilde{U}}$ is an embedding, $\mu(\tilde{U})$ is an open ball in $\fg^*$, and $\tilde{\mu}(\tilde{U}\setminus U)$ lies outside of the unimodular cone into which $U$ is mapped by $\psi$. Recall that this cone is:
\begin{displaymath}
\{ \xi \in \fg^* \vert \langle \xi-\psi(w), v_i \rangle \ge 0 \}
\end{displaymath}
where the $v_i's$ are the primitive normals assigned to $w$ via lemma \ref{lem:attach1}. Equivalently, we have:
\begin{displaymath}
\{\xi \in \fg^* \vert \langle j^*\xi - \xi_0, v_i \rangle \ge 0 \}
\end{displaymath}
\item Consequently, if we consider $\tilde{\Phi}(u,g,z) = j^*\tilde{\mu}\circ\tilde{\Psi} \circ pr_1(u,g) - \sum_{i=1}^k \vert z_i \vert^2 v_i^* -\xi_0$, then $\tilde{\Phi}^{-1}(0) = \Phi^{-1}(0)$.
\item $0$ is a regular value of $\Phi$ since $j^*\psi$ is a submersion (shrink $U$ if necessary). Thus, $\Phi^{-1}(0)$ is a manifold of codimension $\dim(K_w)$ on which $K_w$ acts freely. We set $U_w=\tilde{U}\cap W$. By theorem \ref{thm:fsred}, we have that $\Phi^{-1}(0)/K_w = (P\vert_{U_w} \times \C^k)//_0 K_w = cut(P\vert_{U_w})$ is a folded-symplectic manifold.
\item The action of $G$ on ($P\vert_{U_w} \times \C^k)$ commutes with the action of $K_w$ and the moment map $\mu$ and the projection $\pi$ descend to a moment map $\bar{\mu}: cut(P\vert_{U_w}) \to \fg^*$ and a quotient map $\bar{\pi}:cut(P\vert_{U_w})\to U_w$ satisfying $\bar{\mu} =\psi \circ \bar{\pi}$.
\item To see that the action of $G$ is toric, consider $\eta \in int(U_w)$. This implies:
\begin{equation}\label{eq:toricaction}
\langle j^*\psi(\eta) -\xi_0, v_i \rangle > 0 \: \forall i, \: 0\le i \le k
\end{equation}
Consider $g_0\in G$ and $[\eta, g, z] \in cut(P\vert_{U_w})$. We have:
\begin{displaymath}
g_0 \cdot [\eta,g,z] = [\eta, g_0g, z] = [\eta, g, z] \iff (\eta,g_0g,z) = (\eta, kg, kz)
\end{displaymath}
for some $k\in K_w$. This implies that $k=g_0\in K_w$ and $g_0 z = z$. But, $\psi(\eta) = \sum_{i=1}^n \vert z_i \vert^2v_i^* + \xi_0$ and no $z_i$ can be $0$ by \ref{eq:toricaction}, hence $z$ has no nontrivial stabilizer and $g_0=e$. Thus, the action is effective.
\end{itemize}
\end{proof}
\begin{remark}
Suppose $(P_1,\sigma_1)$ and $(P_2,\sigma_2)$ are two objects of $\mathcal{B}_{\psi}$ and $\phi:A \to B$ is a morphism. $\phi$ is $G$-equivariant, hence $\phi\times id: P_1\vert_{U_w} \times V \to P_2\vert_{U_w} \times V$ is $G\times K_w$-equivariant and induces an isomorphism of toric, folded-symplectic manifolds:
\begin{displaymath}
cut(\phi):cut(P_1\vert_{U_w}) \to cut(P_2\vert{U_w}), \text{ $cut(\phi)[p,z]=[\phi(p),z]$}.
\end{displaymath}
Thus, for each $w\in W$ we have a functor $cut: B_{\psi\vert_{U_w}} \to M_{\psi\vert_{U_w}}$
\end{remark}
We will now construct natural $G$-equivariant homeomorphisms $\alpha^P_w: c_{top}(P\vert_{U_w}) \to cut(P\vert_{U_w})$. As before, let $\{v_1, \dots, v_k\}$ be the integral basis attached to $w$ and let $K_w$ be the corresponding subtorus of $G$. Let $\xi_0=j^*\psi(w)$, fix the representation of $K_w$ to be $\C^k$, and let $\frak{k}_w$ be $Lie(K_w)$. Let $\mu_w:\C^k \to \frak{k}_w$ be the moment map for the action of $K_w$. The construction of $\alpha^P_w$ depends on the fact that $(\C^k, \omega_w, \mu_w)$ is a toric symplectic $K_w$-manifold over the cone $\mu_w(\C^k) = \{\eta\in \frak{k}_w^* \vert \: \langle \eta, v_i \rangle \le 0 \text{ for } 1\le i \le k \}$. Moreover,
\begin{enumerate}
\item the map $\mu_w:\C^k \to \mu_w(\C^k)$ has a continuous section
\begin{displaymath}
s:\mu_w(\C^k)\to \C^k, s(\eta) = (\sqrt{\langle -\eta, v_1 \rangle}, \dots , \sqrt{\langle -\eta, v_k \rangle})
\end{displaymath}
which is smooth over the interior of the cone $\mu_w(\C^k)$.
\item The stabilizer $K_z$ of $z\in \C^k$ deponds only on the face of the cone $\mu_w(\C^k)$ containing $\mu_w(z)$ in its interior:
\begin{displaymath}
K_z = \exp(\span_{\R} \{v_i \in \{v_1,\dots,v_k\} \vert \: \langle \mu_w(z),v_i \rangle = 0\})
\end{displaymath}
\end{enumerate}
Let $U=U_w$ and $\nu=j^*\circ \mu: P\vert_U \to \frak{k}_w^*$, the $K_w$-moment map. Then for any point $p\in P\vert_U$
\begin{displaymath}
\xi_0-\nu(p) \in \mu_w(\C^k)
\end{displaymath}
and
\begin{displaymath}
\begin{array}{lcl}
s(\xi_0 - \nu(p)) & = & (\sqrt{\langle \nu(p)-\xi_0, v_1 \rangle}, \dots \sqrt{\langle \nu(p)-\xi_0, v_k \rangle}) \\
& = & (\sqrt{\langle \mu(p)-\psi(w), v_1 \rangle}, \dots \sqrt{\langle \mu(p)-\psi(w), v_k \rangle})
\end{array}
\end{displaymath}
where $\mu=\psi\circ \pi:P\to \fg^*$ is the moment map for the action of $G$ on $P$. This gives us a continuous map
\begin{displaymath}
\phi: P\vert_U \to \Phi^{-1}(0) \subset P\vert_U\times \C^k, \: \phi(p)=(p,s(\xi_0 - \nu(p)))
\end{displaymath}
where $\xi_0 = j^*\psi(w)$ and $\Phi$ is the moment map of equation \ref{eq:cutmoment}. The image of $\phi$ intersects every $K_w$ orbit in $\Phi^{-1}(0)$. Hence the composite
\begin{displaymath}
f=\tau\circ \phi: P\vert_U \to \Phi^{-1}(0)/K_w,
\end{displaymath}
where $\tau:\Phi^{-1}(0)\to \Phi^{-1}(0)/K_w$ is the orbit map, is surjective. Next we show that the fibers of $f$ are precisely the equivalence classes of the relation $\sim$ defined in Step 1. By definition, two points $p_1, p_2 \in P\vert_U$ are equivalent with respect to $\sim$ if and only if $\pi(p_1)=\pi(p_2)$ and there is an $a\in K_{\pi(p_1)}$ with $a\cdot p_2 = p_1$. On the other hand $f(p_1)=f(p_2)$ if and only if there is an $a\in K_w$ with:
\begin{displaymath}
(p_1, s(\xi_0-\nu(p_1))) = (a\cdot p_2, a\cdot s(\xi_0 - \nu(p_2)))
\end{displaymath}
For any point $\eta \in \mu_w(\C^k)$
\begin{displaymath}
a\cdot s(\eta) = s(\eta) \iff \text{ $a$ lies in the stabilizer $K_{s(\eta)}$ of $s(\eta)$}.
\end{displaymath}
For $\eta= \xi_0 - \nu(p_2) = \xi_0 -\nu(p_1) = j^*(w - \psi(\pi(p_1)))$,
\begin{displaymath}
\begin{array}{lll}
K_{s(\eta)} & = & \exp(\span_{\R}(\{v_i \in \{v_1,\dots,v_k\} \vert \: \langle \xi_0-\nu(p_1), v_i \rangle =0\})) \\
& = & \exp(\span_{\R}(\{v_i \in \{v_1,\dots,v_k\} \vert \: \langle j^*\psi(w)-j^*\psi(\pi(p_1)),v_i \rangle =0\}))\\
& = & \exp(\span_{\R}(\{v_i \in \{v_1,\dots,v_k\} \vert \: \langle \psi(w)-\psi(\pi(p_1),v_i\rangle =0\})) \\
& = & K_{\pi(p_1)}.
\end{array}
\end{displaymath}
Therefore, $f(p_1)=f(p_2) \iff \: \pi(p_1)=\pi(p_2) \text{ and } \exists a\in K_{\pi(p)} \text{ satisfying } a\cdot p_1=p_2$. We conclude that the fibers of $f$ are precisely the equivalence classes of the relation $\sim$. Therefore $f$ descends to a continuous bijection:
\begin{displaymath}
\alpha_w^P: c_{top}(P\vert_U) = (P\vert_U)/\sim \to \Phi^{-1}(0)/K = cut(P\vert_U), \: \alpha^P_w([p]) = [p,s(\xi_0-\nu(p))].
\end{displaymath}
The properness of $f$ implies that $f$ is closed, hence $\alpha^P_w$ is closed. Thus, $\alpha^P_w$ is a homeomorphism.
\begin{remark}\label{rem:commute}
Notice that for any morphism $\phi:P_1\vert_U \to P_2\vert_U$ we have:
\begin{displaymath}
\begin{array}{lcl}
cut(\phi)(\alpha_w^{P_1}[p]) & = & cut(\phi)([p,s(\xi_0 - \nu(p))]) \space = \space [\phi(p),s(\xi_0 - \nu(p))] \\
& = & \alpha^{P_2}_w([\phi(p))] \space = \space \alpha^{P_2}_w(c_{top}([p]).
\end{array}
\end{displaymath}
\end{remark}
To finish the construction of the functor $c$, we must show that the transition maps $v= (\alpha_{w_2}^P) \circ (\alpha_{w_1}^P)^{-1}$ are isomorphisms of toric, folded-symplectic manifolds over $\psi:U_{w_1}\cap U_{w_2} \to \fg^*$. We will need a lemma:
\begin{lemma}\label{lem:fsmaps}
Let $(M_1,\sigma_1)$ and $(M_2,\sigma_2)$ be two toric folded-symplectic manifolds of dimension $2n$. Suppose $\phi:M_1\to M_2$ is a smooth bijection satisfying:
\begin{enumerate}
\item $\phi(Z_1)=Z_2$, where $Z_i\subset M_i$ is the fold for each $i$ and
\item $\phi^*\sigma_2=\sigma_1$.
\end{enumerate}
Then $\phi$ is a diffeomorphism.
\end{lemma}
\begin{proof} \mbox{ } \newline
We simply check that $d\phi$ has maximal rank everywhere. Condition $1$ implies that $(\phi^*\sigma_2)_m=(\sigma_1)_m$ drops rank at $m$ if and only if $m\in Z_1 \: \iff \: \phi(m) \in Z_2$. Since $\sigma_1$ is symplectic on $M_1\setminus Z_1$, we have that $m\in M_1\setminus Z_1$ implies $\operatorname{rank}(d\phi_m)=2n$.
At a point $z\in Z_1$, it suffices to perform a local computation to check that $d\phi_z$ has maximal rank. Thus, we may assume that $M_1=M_2=\R^{2n}$ so that
\begin{displaymath}
(\sigma_1)^n=\phi^*(\sigma_2)^n = \det(d\phi)(\sigma_2\circ \phi)^n = \det(d\phi)f(dx_1\wedge \dots \wedge dx_{2n})
\end{displaymath}
where $f:\R^{2n} \to \R$ is a function vanishing on $Z_1$. Since $\sigma_1^n \pitchfork_s 0$ at $z$, we must have that $\det(d\phi)f \pitchfork_s 0$ at $z$. This implies $\det(d\phi)_z \ne 0$ and so $\operatorname{rank}(d\phi_z)=2n$ and $\phi$ is a diffeomorphism.
\end{proof}
We first show that $(\alpha_{w_2}^P)\circ (\alpha_{w_1}^P)^{-1}$ satisfies condition $1$ of lemma \ref{lem:fsmaps} and then discuss why condition $2$ is satisfied.
\begin{lemma}\label{lem:foldtofold}
Let $(M_i,\sigma_i,\pi_i:M_i\to W)$, $i\in \{1,2\}$, be two toric folded-symplectic manifolds over a unimodular map with folds $\psi:W\to \fg^*$. Suppose $\phi:M_1\to M_2$ is an equivariant map satisfying $\pi_2\circ \phi = \pi_1$. Then $\phi(Z_1)=Z_2$.
\end{lemma}
\begin{proof}
Let $\hat{Z}$ be the folding hypersurface of $\psi$. We have:
\begin{displaymath}
Z_1 = \pi_1^{-1}(\hat{Z}) = \phi^{-1}(\pi_2^{-1}(\hat{Z})) = \phi^{-1}(Z_2)
\end{displaymath}
Thus, $\phi(Z_1)\subset Z_2$. To show equality, we show that $\phi\vert_{Z_1}$ surjects onto $Z_2$. Fix $z_2\in Z_2$ and let $z=\pi_2(z_2)$. There exists an element $z_1\in \pi_1^{-1}(z)$ and $\pi_2\circ\phi=\pi_1$ implies $\phi(z_1) \in \pi_2^{-1}(z)$, hence $h\cdot\phi(z_1)=z_2$ for some $h\in G$. Thus, $\phi(h\cdot z_1) = h\cdot\phi(z_1) = z_2$ and $\phi(Z_1)=Z_2.$
\end{proof}
By construction of $cut(P\vert_U)$, we have a commuting diagram:
\begin{displaymath}
\xymatrix{
cut(P\vert_{U_{w_1}}) \ar[dr]_{\pi_1} & c_{top}(P\vert_{(U_{w_1}\cap U_{w_2})})\ar[l]_-{\alpha^P_{w_1}} \ar[r]^-{\alpha^P_{w_2}}& cut(P\vert_{U_{w_2}}) \ar[dl]^{\pi_2} \\
& U_{w_1}\cap U_{w_2} \ar[r]^{\psi} & \fg^*}
\end{displaymath}
where $\pi_1, \pi_2$ are the quotient maps. Technically, we are considering $cut(P\vert_{U_{w_i}})\vert_{(U_{w_1}\cap U_{w_2})}$, but we omit this extra notation. Since the maps on the top line are equivariant and the diagram commutes, we see that $(\alpha_{w_2}^P)\circ (\alpha_{w_1}^P)^{-1}$ satisfies the conditions of lemma \ref{lem:foldtofold} and so $(\alpha_{w_2}^P)\circ (\alpha_{w_1}^P)^{-1}(Z_1)=Z_2$, where $Z_i \subset cut(P\vert_{U_{w_i}})$ is the folding hypersurface.
It remains to show that $v=(\alpha_{w_2}^P)\circ (\alpha_{w_1}^P)^{-1}$ is a folded-symplectic map. It suffices to produce a smooth folded-symplectic map $\mathfrak{v}$ satisfying
\begin{equation}
\mathfrak{v} \circ \alpha_{w_1}^P = \alpha^P_{w_2}
\end{equation}
We then have that $\mathfrak{v} = v$ on the domain of $(\alpha_{w_1}^P)^{-1}$, meaning $v$ is a folded-symplectic map. By our discussion and lemma \ref{lem:fsmaps}, $v$ will be an isomorphism of toric folded-symplectic manifolds. It suffices to consider the case when $U_{w_1} \subset U_{w_2}$ since one may then compose with inclusions and restrictions to get a commutative diagram:
\begin{displaymath}
\xymatrix{
cut(P\vert_{U_{w_1}})\vert_{(U_{w_1}\cap U_{w_2})} \ar[d] \ar[r]^{\simeq} & cut(P\vert_{(U_{w_1}\cap U_{w_2})}) \ar[d]& cut(P\vert_{U_{w_2}})\vert_{(U_{w_1}\cap U_{w_2})} \ar[l]_{\simeq} \ar[d]\\
(U_{w_1}\cap U_{w_2}) \ar[r]^{id} & (U_{w_1}\cap U_{w_2}) & (U_{w_1}\cap U_{w_2}) \ar[l]_{id}}
\end{displaymath}
The composition of the top row, which is $v$ or $v^{-1}$ depending on the order of composition, will then be a diffeomorphism.
\begin{enumerate}
\item First, consider the special case when $K_{w_1}=K_{w_2}$. Then the collections of the corresponding weights $\{v_j^{w_1}\}_{j=1}^k$, $\{(v_j^{w_2})\}_{j=1}^k$ are the same set. Hence, there exists a linear, symplectic isomorphism $\tilde{\mathfrak{v}}:\C^k \to \C^k$ permuting the coordinates and intertwining the two representations and corresponding moment maps, which we denote $\mu_{w_1}^1$ and $\mu_{w_1}^2$. Consequently, $id\times \tilde{\mathfrak{v}}: P\vert_{U_{w_1}} \times \C^k \to P\vert_{U_{w_1}} \times \C^k$ induces a folded-symplectomorphism of reduced spaces:
\begin{displaymath}
\mathfrak{v}: (P\vert_{U_{w_1}} \times \C^k)//_0 K_w \to (P\vert_{U_{w_1}} \times \C^k)//_0K_w, \: [p,z] \to [p,\tilde{\mathfrak{v}}(z)].
\end{displaymath}
Note that $\mu_{w_1}^1(\C^k)= \mu_{w_1}^2(\C^k)$ and denote the corresponding sections as $s_1:\mu_{w_1}^1(\C^k)\to \C^k$ and $s_2:\mu_{w_1}^1(\C^k) \to \C^k$. By definition, $\tilde{\mathfrak{v}}(s_1)=s_2$. We have:
\begin{displaymath}
\begin{array}{lcl}
\mathfrak{v}(\alpha_{w_1}^P([p])) & = & \mathfrak{v}([p,s_1(\xi_0 - \nu(p))]) \\
& = & [p, \tilde{\mathfrak{v}}(s_1(\xi_0-\nu(p)))] \\
& = & [p, s_2(\xi_0 - \nu(p))] \\
& = & \alpha_{w_2}^P([p])
\end{array}
\end{displaymath}
hence $\bar{\mathfrak{v}}\circ \alpha_{w_1}^P = \alpha_{w_2}^P$ and $(\alpha_{w_2}^P) \circ (\alpha_{w_1}^P)^{-1}$ is a folded-symplectomorphism.
\item In general, we have a strict inclusion $\{v_j^{w_1}\}_{j=1}^{k_1} \subset \{v_j^{w_2}\}_{j=1}^{k_2}$ since $w_2$ lies in a boundary component of possibly larger codimension in the quadrant $U_{w_1}\cap U_{w_2}$. By our study of case $1$, we may assume that $v_j^{w_1}=v_j^{w_2}$ for all $1\le j \le k_1$. We may then reduce the clutter in the notation by dropping the superscripts $^{(w_1)}$ and $^{(w_2)}$ and setting $K_i= K_{w_i}$, $i=1,2$.
By construction of the neighbourhoods $U_{w_i}$ (q.v. lemma \ref{lem:cut}), we have:
\begin{itemize}
\item $\langle \psi(w_1) - \psi(w_2), v_i \rangle =0$ for $i=1,\dots,k_1$ and
\item for all $w\in U_{w_1}$,
\begin{displaymath}
\langle \psi(w) - \psi(w_2), v_i \rangle > 0 \text{ for } i=k_1+1,\dots, k_2.
\end{displaymath}
\end{itemize}
Consequently, for any point $p\in P\vert_{U_{w_1}}$, the functions
\begin{displaymath}
p \to \sqrt{\langle \mu(p) - \psi(w_2), v_i\rangle }
\end{displaymath}
are smooth for $i=k_1 + 1, \dots, k_2$. Also, for $p\in P\vert_{U_{w_1}}$
\begin{displaymath}
\langle \mu(p) - \psi(w_2), v_i \rangle = \langle \mu(p) - \psi(w_1), v_i \rangle
\end{displaymath}
for $i=1,\dots, k_1$. Now consider the map:
\begin{displaymath}
\tilde{\mathfrak{v}}:P\vert_{U_{w_1}} \times \C^{k_1} \to P\vert_{U_{w_1}} \times \C^{k_2}
\end{displaymath}
given by
\begin{equation}
\tilde{\mathfrak{v}}(p,z_1,\dots,z_{k_1}) = (p,z_1,\dots, z_{k_1}, \sqrt{\langle \mu(p)-\psi(w_2), v_{k_1+1} \rangle}, \dots, \sqrt{\langle \mu(p)-\psi(w_2), v_{k_2} \rangle}).
\end{equation}
The map $\tilde{\mathfrak{v}}$ is smooth and $K_1$-equivariant. Since $\tilde{\mathfrak{v}}^*(dz_j\wedge d\bar{z}_j)$=0 for $j>k_1$, it is folded-symplectic. We have that:
\begin{displaymath}
\tilde{\mathfrak{v}}^{-1}(\Phi_2^{-1}(0)) \subset \Phi_1^{-1}(0)
\end{displaymath}
where the $\Phi_j: P\vert_{U_{w_1}} \times \C^{k_j} \to \frak{k}^*_j$, $j=1,2$ are the corresponding $K_j$ moment maps (q.v. \ref{eq:cutmoment}). This is because:
\begin{displaymath}
(p,z)\in \Phi_j^{-1}(0) \iff \langle \psi(\pi(p)) - \psi(w_j), v_i \rangle = \vert z_i \vert^2 \text{ for all } i=1,\dots, k_j
\end{displaymath}
Consequently, $\tilde{\frak{v}}$ descends to a well-defined smooth folded-symplectic map
\begin{equation}
\frak{v}: \Phi_1^{-1}(0)/K_1 \to \Phi_2^{1}(0)/K_2
\end{equation}
given by
\begin{displaymath}
\frak{v}([p,z_1,\dots,z_{k_1}]) = [p,z_1,\dots, z_{k_1}, \sqrt{\langle \mu(p) - \psi(w_2), v_{k_1+1} \rangle}, \dots, \sqrt{\langle \mu(p) - \psi(w_2), v_{k_2} \rangle}].
\end{displaymath}
We have that $\frak{v}(\alpha_{w_1}^P([p]))= \alpha_{w_2}^P([p])$, which means the transition maps are isomorphisms of toric folded-symplectic manifolds over $U_{w_1}$.
\end{enumerate}
We define $c(P)$ to be $c_{top}(P)$ equipped with the structure of a toric folded-symplectic manifold endowed by the charts $\{c_{top}(P)\vert_{U_w}, \alpha_{w}^P\}_{w\in W}$.
\subsection{Step 3- Show $c$ is Functorial}
We finish by showing that any isomorphism $\phi:P_1\to P_2$ of toric folded-symplectic bundles induces an isomorphism $c(\phi)$. Recall that $\phi:P_1\to P_2$ induces a continuous map $c_{top}(\phi):c_{top}(P_1) \to c_{top}(P_2)$ given by $c_{top}(\phi)([p])=[\phi(p)]$.
By remark \ref{rem:commute}, we have that $cut(\phi)\circ \alpha_{w}^{P_1} = \alpha^{P_2}_w\circ c_{top}(\phi)$, hence $c_{top}(\phi)\vert_{U_w}= (\alpha^{P_2}_w)^{-1}\circ cut(\phi)\circ \alpha_w^{P_1}$ is smooth, equivariant, and folded-symplectic. Because $\phi:P_1\to P_2$ covers $id:W\to W$, it follows that $c_{top}(\phi)$ covers $id:W\to W$, hence $c_{top}(\phi)$ is an isomorphism of toric folded-symplectic manifolds, which we denote $c(\phi)$.
Finally, because $c_{top}$ is a functor, it follows that $c$ satisfies the requirements to be a functor. Hence, we have constructed a functor $c:\mathcal{B}_{\psi} \to \mathcal{M}_{\psi}$ as required.
\begin{remark}\label{rem:restriction1}
We use remark \ref{rem:restriction} and restrict the structure maps $\alpha^P_w$ to see that for all $P\in \operatorname{Ob}(\mathcal{B}_{\psi})$ and for all open subsets $U\subset W$
\begin{displaymath}
c(P)\vert_U = c(P\vert_U)
\end{displaymath}
and for any morphism $\phi:P_1\to P_2$,
\begin{displaymath}
c(\phi)\vert_U = c(\phi\vert_U).
\end{displaymath}
Hence, we write $c(\phi)\vert_U$ when referring to $c(\phi\vert_U)$.
\end{remark}
We end the section with a lemma that will be used to prove that $c:\mathcal{B}_{\psi}(W) \to \mathcal{M}_{\psi}(W)$ is an equivalence of categories.
\begin{lemma}\label{lem:idfunctor}
The functor $c:\mathcal{B}_{\psi}\to \mathcal{M}_{\psi}$ is a map of presheaves of groupoids (q.v. remark \ref{rem:empsi}). Moreover, if $\mathring{W}$ denotes the interior of $W$, then $c_{\mathring{W}}:\mathcal{B}_{\psi}(\mathring{W})\to \mathcal{M}_{\psi}(\mathring{W})$ is isomorphic to the identity functor.
\end{lemma}
\begin{proof}
The fact that $c$ is a map of presheaves of groupoids follows from the fact that $c_{top}$ is a map of presheaves of groupoids. Over the interior $\mathring{W}$, the functor $c_{top}$ is isomorphic to the identity functor since the subtorus associated to any point $w\in \mathring{W}$ is $\{e\}$. Because this subtorus is trivial we have
\begin{displaymath}
cut(P\vert_{U_w}) = (P\vert_{U_w} \times \{0\})//_0\{e\} \simeq P\vert_{U_w}
\end{displaymath}
as toric folded-symplectic manifolds over $U_w$.
\end{proof}
\subsection{$c:\mathcal{B}_{\psi}(W) \to \mathcal{M}_{\psi}(W)$ is an Equivalence of Categories}
We now prove the following theorem which states that $c$ is an equivalence of categories. At the very end of the section, we will use this equivalence to provide the classification result for toric, folded-symplectic manifolds with co-orientable folding hypersurface.
\begin{theorem}\label{thm:equi-cats}
Let $\psi:W\to \fg^*$ be a unimodular map with folds. The functor:
\begin{displaymath}
c:\mathcal{B}_{\psi}(W) \to \mathcal{M}_{\psi}(W)
\end{displaymath}
is an equivalence of categories.
\end{theorem}
\begin{remark}
The strategy for proving theorem \ref{thm:equi-cats} is borrowed from \cite{KL} and our proof is virtually identical: the main ingredients are the functor $c$, the classification of objects in $\mathcal{B}_{\psi}(W)$, and local equivalence (q.v. lemma \ref{lem:locunique}) of objects in $\mathcal{M}_{\psi}(W)$. We list it here for the sake of completeness. It works as follows:
\begin{enumerate}
\item We first recall that $c:\mathcal{B}_{\psi}\to \mathcal{M}_{\psi}$ is a map of presheaves of groupoids (q.v. lemma \ref{lem:idfunctor}).
\item Show that $c:\mathcal{B}_{\psi}(U)\to \mathcal{M}_{\psi}(U)$ is an equivalence of categories for every open subset $U\subseteq W$. Hence, it is an isomorphism of presheaves of groupoids.
\end{enumerate}
\end{remark}
\begin{remark}
As a brief reminder, recall our notation for restrictions of objects and maps. Suppose $U\subseteq W$ is open.
\begin{itemize}
\item If $\pi:M\to W$ is an object of $\mathcal{M}_{\psi}$ or $\pi:P\to W$ is an object of $\mathcal{B}_{\psi}$, we will use the notation $M\vert_U$, $P\vert_U$ to stand for the objects $\pi: \pi^{-1}(U)\to W$ in $\mathcal{M}_{\psi}(U), \mathcal{B}_{\psi}(U)$, respectively.
\item Given $\phi\in \hom(P_1,P_2)$, we write $\phi\vert_U$ to mean $\phi\vert_{P\vert_U}$. Similarly, for $\varphi\in \hom(c(P_1),c(P_2))$ we write $\varphi\vert_U$ to mean $\varphi\vert_{c(P_1)\vert_{U}}$.
\end{itemize}
\end{remark}
The proof will be given in a series of lemmas. We begin by showing that $c$ is fully faithful, which we will use to prove it is essentially surjective.
\begin{lemma}\label{lem:faithful}
For any open subset $U\subseteq W$ and for any two objects $P_1,P_2 \in \operatorname{Ob}(\mathcal{B}_{\psi}(U))$, the map:
\begin{displaymath}
c_U:\hom(P_1,P_2) \to \hom(c(P_1),c(P_2))
\end{displaymath}
is injective.
\end{lemma}
\begin{proof}
Let $\mathring{W}$ denote the interior of $W$ and let $\mathring{U}=U\cap \mathring{W}$. By lemma \ref{lem:idfunctor}, $c_{\mathring{U}}:\hom(P_1\vert_{\mathring{U}}, P_2\vert_{\mathring{U}}) \to \hom (c(P_1)\vert_{\mathring{U}}, c(P_2)\vert_{\mathring{U}})$ is isomorphic to the identity. There are isomorphisms $\delta_{P_1}, \delta_{P_2}$ so that for all $\phi\in \hom(P_1\vert_{\mathring{U}},P_2\vert_{\mathring{U}})$ the diagram
\begin{displaymath}
\xymatrix{
P_1\vert_{\mathring{U}} \ar[d]^{\phi} \ar[r]^{\delta_{P_1}} & c(P_1)\vert_{\mathring{U}} \ar[d]^{c(\phi)} \\
P_2\vert_{\mathring{U}} \ar[r]^{\delta_{P_2}} & c(P_2)\vert_{\mathring{U}}}
\end{displaymath}
commutes. Consequently, if $\phi_1,\phi_2 \in \hom(P_1,P_2)$ and $c(\phi_1)=c(\phi_2)$ then their restrictions $\mathring{\phi}_i:=\phi_i\vert_{\mathring{U}}$ satisfy:
\begin{displaymath}
\mathring{\phi}_i = (\delta_{P_2})^{-1}\circ c(\phi_i) \circ \delta_{P_1}
\end{displaymath}
Since $c(\phi_1)=c(\phi_2)$ by assumption, we have that $\mathring{\phi}_1=\mathring{\phi}_2$. Since $\mathring{U}$ is dense in $U$, this implies that $\phi_1=\phi_2$.
\end{proof}
We will use the following theorem, which is theorem 3.1 in \cite{HaS}, to show that $c$ is also surjective as a map on $\hom$-sets.
\begin{theorem}\label{thm:has} Let $M$ be a manifold with an action of a torus $G$ and $h:M\to M$ a $G$-eequivariant diffeomorphism with $h(x)\in G\cdot x$ for all points $x\in M$. Let $\pi:M\to M/G$ be the orbit map. Then there exists a map $f:M/G \to G$ such that
\begin{displaymath}
h(x)= f(\pi(x))\cdot x
\end{displaymath}
for all $x\in M$ and such that $f\circ\pi$ is smooth.
\end{theorem}
\begin{lemma}\label{lem:surjective1}
For any open subset $U$ of $W$ and for any $P\in \operatorname{Ob}(\mathcal{B}_{\psi}(U))$ the map:
\begin{displaymath}
c:\hom(P,P) \to \hom(c(P),c(P))
\end{displaymath}
is onto.
\end{lemma}
\begin{proof}
By theorem \ref{thm:has}, given $\varphi\in \hom(c(P),c(P))$ there is a function $f:U\to G$ so that
\begin{displaymath}
\varphi(x)=f(\pi(x))\cdot x
\end{displaymath}
where $\pi:c(P)\to U$ is the quotient map and $f\circ \pi$ is smooth. As in the proof of lemma \ref{lem:faithful}, we have
\begin{displaymath}
\varphi\vert_{\mathring{U}} = c(\mathring{\phi})
\end{displaymath}
where $\mathring{\phi}$ is given by:
\begin{displaymath}
\mathring{\phi}=(\delta_P)^{-1}\circ \varphi\vert_{\mathring{U}} \circ \delta_P.
\end{displaymath}
Hence for $p\in P\vert_{\mathring{U}}$,
\begin{displaymath}
\mathring{\phi}(p) = (\delta_P)^{-1}(f(\pi(\delta_P(p)))\cdot \delta_P(p))=(\delta_P)^{-1}(f(\pi(p))\cdot \delta_P(p)) = f(\pi(p))\cdot (\delta_P)^{-1}(\delta_P(p)) = f(\pi(p))\cdot p.
\end{displaymath}
Define the map $\phi:P \to P$ by
\begin{displaymath}
\phi(p) := f(\pi(p))\cdot p \text{ for all $p\in P$.}
\end{displaymath}
This map is $G$-equivariant and commutes with $\pi:P\to U$. Since $f\circ \pi$ is smooth, the map $\phi$ is a diffeomorphism with inverse $\phi^{-1}(p)=f(\pi(p))^{-1}\cdot p$. Moreover, since the restriction of $\phi$ to $P\vert_{\mathring{U}}$ is $\mathring{\phi}$, the map $\phi$ is folded-symplectic on $P\vert_{\mathring{U}}$. Since this set is dense in $P$, we conclude that $\phi$ is folded-symplectic on all of $P$. Since it maps the folding hypersurface to itself, lemma \ref{lem:foldtofold} implies that $\phi\in \hom(P,P)$. It remains to check that $c(\phi)=\varphi$. The functor $c$ commutes with restrictions to $P\vert_{\mathring{U}}$, hence
\begin{displaymath}
c(\phi)\vert_{c(P)\vert_{\mathring{U}}} = c(\mathring{\phi}) = \varphi\vert_{c(P)\vert_{\mathring{U}}}
\end{displaymath}
by construction. Hence $c(\phi)$ and $\varphi$ are the same on an open dense subset. Smoothness implies that they are the same on all of $c(P)$.
\end{proof}
\begin{lemma}\label{lem:fullyfaithful}
Suppose $U\subseteq W$ is an open subset with $H^2(U,\mathbb{Z})=0$. Then for any $P_1,P_2 \in \operatorname{Ob}(\mathcal{B}_{\psi}(U))$ the map
\begin{displaymath}
c = c_U: \hom(P_1,P_2) \to \hom(c(P_1),c(P_2)),
\end{displaymath}
is a bijection.
\end{lemma}
\begin{proof}
Lemma \ref{lem:faithful} shows that $c_U$ is injective, hence we only need to show that it is surjective. Let $\varphi\in \hom(c(P_1),c(P_2))$. By theorem \ref{thm:bundleclassification}, there exists $\phi\in \hom(P_1,P_2)$. Then $c(\phi)\in \hom(c(P_1),(P_2))$ and $c(\phi)^{-1}\circ\varphi \in \hom(c(P_1),c(P_1))$. By lemma \ref{lem:surjective1}, there exists $\nu \in \hom(P_1,P_1)$ such that $c(\nu)=c(\phi)^{-1}\circ \varphi$. Consequently,
\begin{displaymath}
\varphi = c(\phi) \circ c(\nu) = c(\phi\circ \nu)
\end{displaymath}
Since $\phi \circ \nu \in \hom(P_1,P_2)$, we are finished.
\end{proof}
To finish the proof that $c$ is fully faithful, it is enough to note that the functors $\underline{\hom}(P_1,P_2)$ and $\underline{\hom}(c(P_1),c(P_2))$ given by
\begin{displaymath}
\underline{\hom}(P_1,P_2)(U):= \hom(P_1\vert_U, P_2\vert_U)
\end{displaymath}
and
\begin{displaymath}
\underline{\hom}(c(P_1),c(P_2))(U):= \hom(c(P_1)\vert_U, c(P_2)\vert_U)
\end{displaymath}
are sheaves. We have shown that $c$ commutes with restrictions, hence
\begin{displaymath}
c=c_u: \underline{\hom}(P_1,P_2)(U) \to \underline{\hom}(c(P_1),c(P)2))(U)
\end{displaymath}
is a map of sheaves. By lemma \ref{lem:fullyfaithful} the map $c_U$ is a bijection for any contractible open set $U$. It follows that $c:\underline{\hom}(P_1,P_2) \to \underline{\hom}(c(P_1),c(P_2))$ is an isomorphism of sheaves. Consequently, it is a bijection on global sections. That is,
\begin{displaymath}
c:\hom(P_1,P_2)\to \hom(c(P_1),c(P_2))
\end{displaymath}
is a bijection. It remains to show that $c$ is essentially surjective.
\begin{lemma}\label{lem:descent}
Let $\{U_i\}_{i\in I}$ be an open cover of $W$, $U_{ij}:= U_i\cap U_j$, and $U_{ijk}:= U_i\cap U_j\cap U_k$ for all $i,j,k \in I$. Suppose we have a collection of objects $P_i\in \mathcal{B}_{\psi}(U_i)$ and isomorphisms $\Phi_{ij}:P_j\vert_{U_{ij}} \to P_i\vert_{U_{ij}}$ defining a cocycle: $\Phi_{ii}=id, \Phi_{ji}=\Phi_{ij}^{-1}$, and
\begin{displaymath}
\Phi_{ij}\vert_{U_{ijk}} \circ \Phi_{jk}\vert_{U_{ijk}} \circ \Phi_{ki}\vert_{U_{ijk}} = id
\end{displaymath}
for all triples $i,j,k\in I$. Then there exists an object $P\in \mathcal{B}_{\psi}(W)$ and isomorphisms $\gamma_i:P\vert_{U_i} \to P_i$ so that
\begin{equation}\label{eq:descent}
\xymatrix{
P_j\vert_{U_{ij}} \ar[d]^{\Phi_{ij}} & \ar[l]^{\gamma_j} P\vert_{U_{ij}} \ar[d]^{\simeq} \\
P_i\vert_{U_{ij}} & \ar[l]^{\gamma_i} P\vert_{U_{ij}}}
\end{equation}
commutes.
\end{lemma}
\begin{proof}
We may take $P=(\sqcup_{i\in I} P_i)/{\sim}$ where $\sim$ is the equivalence relation defined by the $\Phi_{ij}'s$. Then $P$ is a principal $G$-bundle over $W$ and the symplectic $G$-invariant folded-symplectic forms on the $P_i's$ define a $G$-invariant folded-symplectic form on $P$ (because the $\Phi_{ij}'s$ are folded-symplectic maps). The maps $\gamma_i^{-1}:P_i \to P\vert_{U_i}$ are induced by the inclusions $P_i\hookrightarrow \sqcup_{j\in I}P_j$.
\end{proof}
\begin{lemma}\label{lem:essentially-surjective}
For any open subset $U\subseteq W$ the functor $c:\mathcal{B}_{\psi}(U) \to \mathcal{M}_{\psi}(U)$ is essentially surjective.
\end{lemma}
\begin{proof}
Given $M \in \mathcal{M}_{\psi}(U)$, we want to show that it is isomorphic to $c(P)$ for some $P\in \mathcal{B}_{\psi}(U)$. Since $\mathcal{B}_{\psi}(U)$ is nonempty, we may choose an object $P'\in \mathcal{B}_{\psi}(U)$. By lemma \ref{lem:locunique} $c(P')$ and $M$ are locally isomorphic. Therefore there is a cover $\{U_i\}_{i\in I}$ of $U$ and a family of isomorphisms $\{\phi_i:c(P')\vert_{U_i} \to M\vert_{U_i}\}.$ Set
\begin{displaymath}
P_i:= P'\vert_{U_i}.
\end{displaymath}
Consider the collection of isomorphisms
\begin{displaymath}
\phi_{ij}:= (\phi_i\vert_{U_{ij}})^{-1} \circ \phi_j\vert_{U_{ij}} : c(P_j)\vert_{U_{ij}} \to c(P_i)\vert_{U_{ij}}, \space i,j \in I
\end{displaymath}
Since $c$ is fully faithful, there are unique isomorphisms
\begin{displaymath}
\Phi_{ij}:P_j\vert_{U_{ij}} \to P_i\vert_{U_{ij}}
\end{displaymath}
with $c(\Phi_{ij}) = \phi_{ij}$. Since $c$ commutes with restrictions to open subsets and since $\{\phi_{ij}\}_{i,j\in I}$ form a cocycle and the $\Phi_{ij}$ are unique, $\{\Phi_{ij}\}_{i,j\in I}$ form a cocycle as well. By lemma \ref{lem:descent} there is $P\in \operatorname{Ob}(B_{\psi}(U))$ and a family of isomorphisms $\{\gamma_i:P\vert_{U_i} \to P_i\}$ so that \ref{eq:descent} commutes. Then
\begin{displaymath}
\xymatrix{
M\vert_{U_{ij}} \ar[d] & c(P_j)\vert_{U_{ij}} \ar[l]_{\phi_j} \ar[d]^{c(\Phi_{ij})} & c(P)\vert_{U_{ij}} \ar[l]_{c(\gamma_j)} \ar[d]^{Id} \\
M\vert_{U_{ij}} & \ar[l]^{\phi_i} c(P_i)\vert_{U_{ij}} & c(P)\vert_{U_{ij}} \ar[l]^{c(\gamma_i)}}
\end{displaymath}
commutes as well. Consequently
\begin{displaymath}
\phi_i\circ c(\gamma_i)\vert_{U_{ij}} = \phi_j \circ c(\gamma_j)\vert_{U_{ij}}.
\end{displaymath}
Since $\hom(c(P),M)$ is a sheaf on $U$, the family $\{\phi_i\circ c(\gamma_i) : c(P)\vert_{U_i} \to M_{U_i}\}$ gives rise to a well defined isomorphism $\varphi:c(P)\to M$.
\end{proof}
Since $U$ was arbitrary, we can take $U=W$ and lemma \ref{lem:essentially-surjective} shows that $c:\mathcal{B}_{\psi}(W)\to \mathcal{M}_{\psi}(W)$ is essentially surjective. This completes the proof of theorem \ref{thm:equi-cats} and gives us the following classification theorem for toric, folded-symplectic manifolds with co-orientable folding hypersurface.
\begin{theorem}\label{thm:Classification}
Let $\psi:W\to \fg^*$ be a unimodular map with folds, where $\fg$ is the Lie aglebra of a torus $G$. Let $\mathcal{M}_{\psi}(W)$ be the category of toric, folded-symplectic manifolds over $\psi$ (necessarily with co-orientable folding hypersurface). Then
\begin{displaymath}
\pi_0(\mathcal{M}_{\psi}(W)) = H^2(W,\mathbb{Z}_G \times \R)
\end{displaymath}
That is, isomorphism classes of toric, folded-symplectic manifolds are in bijection with cohomology classes in $H^2(W,\mathbb{Z}_G \times \R)$. In particular, to every toric, folded-symplectic manifold $(M,\sigma,\pi:M\to W)$ over $\psi:W \to \fg^*$, we may associate a first Chern class $c_1(M)\in H^2(W,\mathbb{Z}_G)$ and a horizontal Chern class $c_{hor}(\sigma)$.
\end{theorem}
\begin{proof}\mbox{ } \newline
An equivalence of categories induces a bijection on isomorphism classes of objects. Since $\pi_0(\mathcal{B}_{\psi}(W))= H^2(W,\mathbb{Z}_G \times \R)$ by theorem \ref{thm:bundleclassification}, theorem \ref{thm:equi-cats} implies that $H^2(W,\mathbb{Z}_G\times \R) = \pi_0(\mathcal{B}_{\psi}(W) = \pi_0(\mathcal{M}_{\psi}(W))$.
\end{proof}
\pagebreak
\begin{appendix}
\section{Manifolds with Corners}
\subsection{Definitions and Conventions}
We give basic definitions of manifolds with corners and do not go into much depth regarding how manifolds with corners should be treated. The goal of the appendix is to define what it means for a map $f:M \to N$ of manifolds with corners to be transverse to a submanifold with corners $S\subset N$. We then show that $f^{-1}(S)$ is a submanifold with corners of $M$ if $f$ is transverse to $S$.
\begin{definition}
A manifold with corners $W$ is a Hausdorff, second countable topological space with a collection of charts $(U_i,\phi_i)$, where $\phi_i:U_i \to \R^k \times (\R^+)^h$ is a homeomorphism onto an open subset of the quadrant $\R^k \times (\R^+)^h$. The transition maps $\phi_i\circ \phi_j^{-1}$ are required to be diffeomorphisms in the sense that they are restrictions of diffeomorphisms defined on open subsets of $\R^n$ to the quadrants. A \emph{submanifold with corners} $S\subset W$ is a topological subspace $S$ so that for each point $p\in S$ there is a chart $(U,\phi)$, $\phi:U \hookrightarrow \R^k \times (\R^+)^h$, where $\phi(S\cap U)$ is the zero set of a subset of the coordinates.
\end{definition}
\begin{definition}
Let $W$ be a manifold with corners. In each coordinate chart $(U_i,\phi_i)$ one has the notion of the depth of a point, which is given by how many half-space coordinates are $0$. Let $x_i$ be the coordinates on $\R^k$ and let $y_j$ be the coordinates on $(\R^+)^h$. Then,
\begin{displaymath}
\operatorname{depth}_W(x_1,\dots,x_k,y_1,\dots, y_j) = \vert \{y_l \vert \mbox{ } y_l=0\} \vert
\end{displaymath}
That is, the depth is the number of $y_j's$ that are $0$. Since the transition maps are diffeomorphisms of manifolds with corners, they preserve the depth function, hence $\operatorname{depth}_W$ is a well-defined map from $W$ to the integers. The $k$-boundary $\partial^k(W)$ of $W$ consists of all points of depth $k$. It is a smooth manifold, hence we have a decomposition of $W$ into smooth manifolds:
\begin{displaymath}
\sqcup_{i=1}^n \partial^k(W)
\end{displaymath}
We often simply refer to the $\partial^k(W)'s$ as the strata of $W$.
\end{definition}
\subsection{Transversality and Submanifolds with Corners}
Throughout this section, we will assume the following two statements from the differential geometry of manifolds (without corners) are true:
\begin{prop}\label{prop:A1:man1}
Let $M$,$N$ be two smooth manifolds (without corners) and let $S\subset N$ be a codimension $s$ submanifold (without corners). If $f:M \to N$ is a smooth map satisfying $f\pitchfork S$, then $f^{-1}(S)$ is a smooth codimension $s$ submanifold (without corners) of $M$.
\end{prop}
\begin{prop}\label{prop:A1:man2}
Suppose $f:M \to N$ is a smooth map of $m$-dimensional manifolds (without corners). If $p\in M$ and $\operatorname{rank}(df_p)=m$, then there exists a neighborhood $U\subset M$ of $p$ such that $f\vert_U$ is a diffeomorphism.
\end{prop}
\begin{cor}\label{cor:A1:man2}
Suppose $f:M \to N$ is a smooth map between manifolds with corners and suppose it is strata-preserving. That is $f(\partial^k(M))\subseteq \partial^k(N)$ for all $k\ge 0$. Suppose $p\in M$ is a point where $df_p$ is an isomorphism. Then there exists a neighborhood $U$ of $p$ such that $f\big\vert_U$ is an open embedding of manifolds with corners. In particular, it is a diffeomorphism onto its image.
\end{cor}
\begin{proof}
This is more of an observation than anything. If $p\in M$ is a regular point, we may choose a neighborhood $U$ around $p$ that is isomorphic to a quadrant in $\R^m$. The map $f$ extends to a smooth map $\tilde{f}$ in a neighborhood of $p$ in $\R^n$. Since $df_p$ is an isomorphism at $p$, $d\tilde{f}_p$ is an isomorphism and there is a neighborhood $V$ of $p$ on which it is an open embedding. The restriction of $f$ to $V\cap U$ is then an open embedding of manifolds with corners since, by assumption, $f$ is strata-preserving.
\end{proof}
The following definition is definition 4 of \cite{CD}. Proposition \ref{prop:A1:trans} is theorem 6 of \cite{CD}. Our proof of proposition \ref{prop:A1:trans} is similar, but we make a few modifications.
\begin{definition}\label{def:A1:trans}
Let $M$ be an $m$-dimensional manifold with corners, $N$ an $n$-dimensional manifold with corners, $S\subset N$ an $s$-dimensional submanifold with corners of $N$, and suppose $f:M\to N$ is smooth. We say $f\pitchfork_s S$ if for all $k>0$ we have:
\begin{center}
\begin{displaymath}
f\vert_{\partial^k(M)} \pitchfork S,
\end{displaymath}
\end{center}
meaning $df_p(T_p\partial^k(M)) + T_{f(p)}S = T_{f(p)}N$ whenever $p\in f^{-1}(S)$. In other words, we say $f\pitchfork_s S$ if its restriction to each stratum of $M$ is transverse to $S$ in the traditional sense of manifolds (without corners).
\end{definition}
\begin{prop}\label{prop:A1:trans}
Let $M$ be an $m$-dimensional manifold with corners, $N$ an $n$-dimensional manifold with corners, $S\subset N$ an codimension $s$ submanifold with corners of $N$, and suppose $f:M\to N$ is a smooth map such that $f\pitchfork_s S$ in the sense of definition \ref{def:A1:trans}. Then $f^{-1}(S)$ is a smooth submanifold with corners of $M$ with codimension $s$.
\end{prop}
\begin{proof} \mbox{} \newline
Let $p\in f^{-1}(S)$. We will prove the proposition in three stages. First, we'll show that it is sufficient to study the case when $N=\R^l$ for some $l$ and $S=\{0\}$. Next, we'll produce a preliminary change of coordinates near $p$ that will exhibit $S$ as a zero set on the stratum containing $p$, but may not do so away from the stratum. We'll finish by applying a secondary diffeomorphism to fix the problem.
\begin{enumerate}
\item By definition of submanifold with corners, there exists a projection $P$ defined in a neighborhood $V$ of $f(p)$, $P:V \to \R^s$, so that $P^{-1}(0) = V\cap S$. Then $F= P \circ f : f^{-1}(U) \to R^s$ is a smooth map satisfying
\begin{enumerate}
\item $F^{-1}(0)=S\cap f^{-1}(U)$ and
\item $F$ restricted to a stratum is transverse to $0$, hence
\item $0$ is a regular value of $F$,
\end{enumerate}
Thus, proving $f^{-1}(S)$ is a submanifold with corners near $p$ is equivalent to showing $F^{-1}(0)$ is a submanifold with corners near $p$
\item We may assume that a neighborhood of $p$ in $f^{-1}(U)$ is the product $W=[0,\epsilon)^k \times (-\epsilon, \epsilon)^{m-k}$ for some $\epsilon >0$ with coordinates $(x_1,\dots, x_k, y_1,\dots, y_{m-k})$ and $p=0$ is the origin. Then we have a smooth map $F:W \to \R^s$, which extends to a smooth map $\tilde{F}:\tilde{W} \to \R^s$, where $\tilde{W}$ is a neighborhood of the origin. This is simply the definition of what it means to be smooth on a subset of $\R^m$.
\begin{itemize}
\item Since $0$ is a regular value of $F$, we can assume $0$ is a regular value of $\tilde{F}$, shrinking $\tilde{W}$ if necessary. We may also assume that $\tilde{W} = (-\epsilon, \epsilon)^m$ by shrinking $\epsilon$.
\item The transversality assumption, $f\vert_{\partial^i(M)} \pitchfork_s S$, implies that $\tilde{F}\vert_{\{0\}\times (-\epsilon,\epsilon)^{m-k}} \pitchfork_s S$, hence $\tilde{F}^{-1}(0) \cap \{0\}\times (-\epsilon, \epsilon)^{m-k}$ is a codimension $s$ submanifold of $\{0\} \times (-\epsilon, \epsilon)^k$.
\item Consequently, there is a diffeomorphism $\phi: \{0\} \times (-\epsilon, \epsilon)^{m-k}\to \{0\} \times (-\epsilon, \epsilon)$ of the stratum containing the $p=0$ so that:
\begin{displaymath}
\phi(\tilde{F}^{-1}(0)\cap \{0\}\times (-\epsilon,\epsilon)^{m-k})= \{(\vec{0}, y_1, \dots, y_{m-k})\vert \mbox{ } y_i=0 \text{ for } i>(m-k-s)\}
\end{displaymath}
\item Extend $\phi$ to the product neighborhood $\tilde{W}= (-\epsilon,\epsilon)^k \times (-\epsilon, \epsilon)^{m-k}$ using $\tilde{\phi}=id \times \phi$, where $id:(-\epsilon,\epsilon)^k \to (-\epsilon,\epsilon)^k$ is the identity.
\end{itemize}
\item Now, note that our discussion in $2$ shows that $\tilde{F}^{-1}(0)\pitchfork_s \{0\} \times (-\epsilon,\epsilon)^{m-k}$. That is, $\tilde{F}^{-1}(0)$ is transverse to the stratum containing $p=0$. This is because $d\tilde{F}$ vanishes on $m-k-s$ directions in the stratum and \emph{doesn't} vanish on the other $s$ directions, hence it must vanish on some $k$ directions transverse to the stratum. The same is then true for $\tilde{\phi}(\tilde{F}^{-1}(0))$ since $\tilde{\phi}$ is a diffeomorphism that preserves $\{0\} \times (-\epsilon,\epsilon)^{m-k}$. Let $S'=\tilde{\phi}(\tilde{F}^{-1}(0))$.
\begin{itemize}
\item Consequently, the projection map $\gamma(x_1,\dots,x_k, y_1, \dots ,y_{m-k}) = (x_1,\dots,x_k,y_1,\dots, y_{m-k-s},0,\dots, 0)$ restricted to $S'$ is a diffeomorphism in a neighborhood of $0$.
\item Let $pr:(-\epsilon, \epsilon)^{k} \times (-\epsilon,\epsilon)^{m-k} \to (-\epsilon, \epsilon)^{k}$ be the projection onto the first $k$ factors. We define the map:
\begin{displaymath}
\Gamma(x_1,\dots,x_k,y_1,\dots,y_{m-k-s},\vec{z}) = (x_1,\dots,x_k,y_1,\dots,y_{m-k-s},\vec{z}-pr(\gamma\vert_{S'}^{-1}(\vec{x},\vec{y},0)))
\end{displaymath}
where $\vec{z}=(y_{m-k-s+1},\dots,y_{m-k})$, $\vec{x}=(x_1,\dots x_k)$, and $\vec{y}=(y_1,\dots,y_{m-k-s})$.
\item Since $\gamma^{-1}$ doesn't depend on the coordinates $y_{m-k-s+1},\dots, y_{m-k}$ it is straightforward to show that $\Gamma$ is a submersion, hence it is a diffeomorphism in a neighborhood of $0$.
\item Note that if $(x_1,\dots,x_k, y_1,\dots,y_{m-k-s},\vec{z})\in S'$, then $\vec{z}=pr((\gamma\vert_{S'})^{-1}(\vec{x},\vec{y},0))$, hence
\begin{displaymath}
\Gamma(x_1,\dots,x_k, y_1,\dots,y_{m-k-s},\vec{z}) = (x_1,\dots,x_k,y_1,\dots,y_{m-k-s},\vec{0})
\end{displaymath}
Conversely, $\Gamma(p_0)$ has vanishing $y_{m-k-s+1},\dots,y_{m-k}$ coordinates if and only if it has a preimage in $S'$. Thus, $\Gamma$ maps $S'$ diffeomorphically onto the set $\{(x_1,\dots,x_k,y_1,\dots,y_{m-k-s},\vec{0})\vert \mbox{} x_i,y_i\in \R\}$.
\end{itemize}
To finish the proof, we simply compose the two diffeomorphisms $\Gamma$ and $id\times \phi$ to get a diffeomorphism exhibiting $\tilde{F}^{-1}(0)$ as the set $\{(x_1,\dots,x_k,y_1,\dots,y_{m-k-s},\vec{0})\vert \mbox{} x_i,y_i \in \R\}$.
\end{enumerate}
\end{proof}
\begin{remark}
In particular, the folding hypersurface is transverse to the faces of $M$.
\end{remark}
We have a version of Hadamard's lemma for vector bundles over manifolds with corners.
\begin{lemma}\label{lem:had}
Let $Z$ be a smooth manifold with corners and let $\pi:E\to Z\times \R$ be a rank $k$ vector bundle over $Z\times \R$. Denote the coordinate on $\R$ by $t$. Suppose $\beta:Z\times \R \hookrightarrow E$ is a smooth section satisfying $\beta(z,0)=0$ for all $z\in Z$. Then there exists a unique smooth section $\mu:Z\times \R \hookrightarrow E$ satisfying $\beta= t\mu$.
\end{lemma}
\begin{proof}
We will show that if $\mu$ exists, then it is unique. We will then show that one may always solve $\beta=t\mu$ for $\mu$ locally, after which we invoke uniqueness to patch together the local solutions.
\begin{enumerate}
\item We first address uniqueness. If $\mu_1, \mu_2$ are two sections satisfying $t\mu_1=\beta=t\mu_2$, then $\mu_1=\frac{\beta}{t}=\mu_2$ on the open, dense subset $\{(z,t)\vert t\ne0\}$. Since $\mu_1,\mu_2$ are smooth, $\mu_1=\mu_2$ everywhere.
\item We now show that $\mu$ exists. Choose a trivialization $(U\times \R,\Phi)$ of $\pi$, where $U\subset Z$, so we have that $\Phi:E\vert_{U\times \R} \to U\times \R \times \R^k$ is an isomorphism of vector bundles. Let $p:U\times \R\times \R^k\to \R^k$ be the standard projection. Then $f:=p(\Phi(\beta\vert_U)):U\times \R \to \R^k$ is a vector-valued function on $U\times \R$ that satisfies $f(z,0)=0$ for each $z\in U$. If $g:U\times \R \to \R$ is a function satisfying $g(z,0)=0$ for each $z\in U$, then we have:
\begin{equation}
g(z,t)=\int_0^1\frac{\partial}{\partial s} g(z,st)ds = \int_0^1 dg_{(z,st)}(t\pt)ds = t\int_0^1 dg_{(z,st)}(\pt)ds
\end{equation}
hence $g=th$ for some smooth function $h$. The same reasoning then applies to vector-valued functions and so we have $f=tF$ for some smooth map $F:U\times \R \to \R^k$. Therefore, $\Phi(\beta\vert_U)(z,t)=(z,t, tF(z,t))$ and if we define $\mu_U(z,t) = \Phi^{-1}(z, t, F(z,t))$ we obtain a local section $\mu_U$ of $\pi$ satisfying $t\mu_U = \beta\vert_U$. Here, we are using that $\Phi$ is a linear map on the fibers of $E\vert_U$ and $U\times \R \times \R^k$.
Cover $Z\times \R$ by a collection of neighborhoods $C=\{U\times \R\}$ so that $E\vert_{U\times \R}$ is trivializable for each $U\times \R \in C$. Then we obtain a collection of sections $\{\mu_U\}_{U\times \R \in C}$ which glue together by uniqueness to give us $\mu$ satisfying $t\mu = \beta$.
\end{enumerate}
\end{proof}
\end{appendix}
\pagebreak
\addcontentsline{toc}{section}{References}
| {
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UPDATE: Our engineers have resolved the issue that was causing difficulty accessing New Digication ePortfolios. Users should be able to access New Digication ePortfolios now.
We are aware that users are experiencing issues accessing New Digication ePortfolios. Our engineers are aware of the issue and are working to resolve it as quickly as possible.
We are very sorry for the frustration and difficulty that these issues have caused our community and will update this post with more information as it becomes available! | {
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A pottery mask of a ferocious monster, the rounded face moulded in high relief with fierce features. The monster has large bulging eyes and pronounced eyebrows over a wide mouth, which he opens to reveal sharp teeth. The monster's large nose is upturned and two antler-shaped horns protrude from the top of its head. The big ears and the mane are spread around the face. The mask has several holes for attachment. There are some traces of red and black pigments on the surface of the pottery, which further serve to emphasise the ferocious features.
This device, known as pushou in Chinese, is rare in the Tang period and appears to be revived in earthenware as a deliberate archaism. In a tradition dating from at least the Han dynasty (206 BC – 220 AD), the doors of the inner burial chambers of large tombs bore the head of a monster, possibly a taotie in origin, combined with a ring in the form of a door knocker, together known as 'pushou'. Taotie masks are the faces found on Chinese archaic bronzes and are generally referred to as 'monster masks'. Pushou were normally made of metal, such as a gilded bronze example excavated from Chengdu city, Sichuan province in 1942, which is dated to the Great Shu kingdom (907 – 925) of the Five Dynasties and is now in the collection of the Yongling Museum in Chengdu city. A closely comparable Tang dynasty pottery horned monster mask of similar size and also with bared teeth from the collection of Anthony M. Solomon was exhibited at Harvard in 2002 (fig. 1). A smaller green-glazed pottery horned monster biting a ring, similar in the ferocious manner and dated to the first half of the 8th century, is in the collection of the Fitzwilliam Museum, Cambridge. An example of a 7th century monster mask of square shape, which was possibly used as decoration of a Buddhist shrine, is in the collection of the British Museum. | {
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Q: How to remove hanging indentation from bibliographic entries I would like to use a bibliography style that aligns the text of the second and later lines of each reference entry with the text of the first. I have been using \bibliographystyle{apalike} but, as can be seen from in image below, the text of each line is not aligned. Which styles will achieve this? Thanks...
A: To remove the hanging indentation, you needn't change bibliography styles. Just set the \bibhang length parameter to zero.
Assuming the Alziary et al entry is in a file called adk.bib, running latex, bibtex, and latex twice more on the MWE below results in the following output:
\documentclass{article}
\usepackage{natbib}
\bibliographystyle{apalike}
\setlength{\bibhang}{0pt}
\begin{document}
\nocite{*}
\bibliography{adk}
\end{document}
Remark: You should encase the words "P.D.E.", "Asian", and "Analytical" in the title field in curly braces to prevent them from being converted to lowercase.
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Q: Coplanarity of 4 tangent points to a sphere. Suppose that segments AB, BC, CD and DA are tangents to a sphere. I need to prove that the four tangent points are coplanar.
I really appreciate any approach or solution.
A: Let $P$, $Q$, $R$ and $S$ be touching points of segments $AB$, $BC$, $CD$ and $AD$ to the sphere respectively.
Thus, $$AP=AS,$$ $$BP=BQ,$$ $$CR=CQ$$ and $$DR=DS.$$
Now, let the plane $(PQR)\cap AD=\{S'\}$ and $AA'$, $BB',$ $CC'$ and $DD'$ be perpendiculars to $(PQR).$
Thus, by similarity we obtain:
$$\frac{DS'}{S'A}=\frac{DD_1}{AA'},$$
$$\frac{AP}{PB}=\frac{AA'}{BB'},$$
$$\frac{BQ}{QC}=\frac{BB'}{CC'}$$ and
$$\frac{CR}{RD}=\frac{CC'}{DD'},$$ which says
$$\frac{DS'}{S'A}\cdot\frac{AP}{PB}\cdot\frac{BQ}{QC}\cdot\frac{CR}{RD}=1.$$
Can you end it now?
A: COMMENT: Consider an arbitrary axis passing the center of sphere S. Also n planes $P_i$ parallel with this axis and tangent to the sphere. These planes intersect at lines $L_i$ with their adjacent planes. The radii $R_i$ of sphere connecting the center and the touching points of tangent planes are perpendicular to planes $P_i$, also to arbitrary axis as the result. Hence these radii are in a plane it's normal is the arbitrary axis, let's denote it as $P_a$ . This plane intersect the planes $P_i$ and constructs lines which have following specifications:
-Their ends $V_t$ is the cross section of plane $P_a$ and lines $L_i$.
-they are tangent to the sphere S.
-They make a closed n-gonal polygon which is located in $P_a$ and has vertices $V_t$.
-They are located in one plane $P_a$ perpendicular to arbitrary axis i.e the axis is it's normal, because they all have one common normal which is the arbitrary axis.
Now the four lines you mentioned make a closed qua-dragon it's sides tangent to sphere; this provides the fourth specification, i.e their location in one plane,.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,043 |
{{ partial "header.html" . }}
<body>
{{ if .Params.totop }}
<div id="nav-to-top">
<span class="decorative-marker">//</span><a href="#top">To Top</a>
</div>
{{ end }}
<div id="layout" class="pure-g">
{{ partial "sidebar.html" . }}
{{ if .Params.banner }}
<div class="content-banner-image pure-u-1 pure-u-md-3-4" style="{{ if .Params.bannerheight }}height: {{ .Params.bannerheight }}px;{{ end }}">
<img src="{{ .Params.banner }}" class="pure-img" style="{{ if .Params.bannerfill }}width: 100%;{{ end }}" />
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{{ end }}
<div class="content pure-u-1 pure-u-md-3-4">
<a name="top"></a>
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<span class="post-author-single">By <a class="post-author" {{ if .Params.authorlink }}href="{{ .Params.authorlink }}"{{ end }} target="{{ .Params.authorlinktarget }}">{{ .Params.author }}</a></span>
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{{ $Site := .Site }}
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<div class="post-categories">
{{ range .Params.categories }}
<a class="post-category post-category-{{ . | urlize }}" href="{{ $Site.BaseUrl}}/categories/{{ . | urlize }}">{{ . }}</a>
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<img src="{{ .Params.bannerinline }}" class="pure-img content-banner-image-inline" />
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<div class="tags-list">
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"redpajama_set_name": "RedPajamaGithub"
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import React from "react";
import PropTypes from "prop-types";
import prefixer from "../utils/prefixer";
import cx from "classnames";
import CSSModules from "react-css-modules";
import styleSheet from "./ProgressBar.css";
//TODO Change this to a self contained component from the current version that requires too much logic in the parent component
export class ProgressBar extends React.PureComponent {
calculateRatio(value) {
if (value < this.props.min) {
return 0;
}
if (value > this.props.max) {
return 1;
}
return (value - this.props.min) / (this.props.max - this.props.min);
}
circularStyle() {
if (this.props.mode !== "indeterminate") {
return {
strokeDasharray: `${2 *
Math.PI *
25 *
this.calculateRatio(this.props.value)}, 400`
};
} else {
return {};
}
}
linearStyle() {
if (this.props.mode !== "indeterminate") {
return {
buffer: prefixer({
transform: `scaleX(${this.calculateRatio(this.props.buffer)})`,
transitionDuration: this.props.transitionDuration
}),
value: prefixer({
transform: `scaleX(${this.calculateRatio(this.props.value)})`,
transitionDuration: this.props.transitionDuration
})
};
} else {
return {};
}
}
renderCircular() {
const strokeDasharray = this.circularStyle();
let color = this.props.color;
const styles = {
stroke: color
};
const style = Object.assign(strokeDasharray, styles);
return (
<svg styleName={"circle"} viewBox="0 0 60 60">
<circle styleName={"path"} style={style} cx="30" cy="30" r="25" />
</svg>
);
}
renderLinear() {
const { buffer, value } = this.linearStyle();
let color = this.props.color;
const styles = prefixer(
{
backgroundColor: color
},
value
);
return (
<div>
<span data-ref="buffer" styleName={"buffer"} style={buffer} />
<span data-ref="value" styleName={"value"} style={styles} />
</div>
);
}
renderRange() {
let rangeStyle = prefixer({
transform: `translateX(${this.calculateRatio(this.props.value.from) *
100}%)
scaleX(${this.calculateRatio(
this.props.value.to - this.props.value.from
)})`
});
return <span data-ref="value" styleName={"value"} style={rangeStyle} />;
}
renderInner() {
if (this.props.type === "circular") {
return this.renderCircular();
}
if (isNaN(this.props.value)) {
return this.renderRange();
}
return this.renderLinear();
}
render() {
const { type, mode, value, min, max, className, style } = this.props;
const classes = [type, mode, className];
return (
<div
style={style}
className={cx(className)}
aria-valuenow={value}
aria-valuemin={min}
aria-valuemax={max}
styleName={cx(classes)}
>
{this.renderInner()}
</div>
);
}
}
ProgressBar.propTypes = {
/**
* The value of a second progress bar
* @examples ''
*/
buffer: PropTypes.number,
/** An object, array, or string of CSS classes to apply to ProgressBar.*/
className: PropTypes.oneOfType([
PropTypes.string,
PropTypes.object,
PropTypes.array
]),
/**
* Determines what color the ProgressBar will be.
* @examples '<ProgressBar color="red"/>'
*/
color: PropTypes.string,
/**
* The max value of the ProgressBar
*/
max: PropTypes.number,
/**
* The min value of the ProgressBar
*/
min: PropTypes.number,
/**
* Mode can be one of: 'determinate', 'indeterminate'
* indeterminate will show a cycling ProgressBar; determinate will show progress based on value.
*/
mode: PropTypes.string,
/**
* Pass inline styling here.
*/
style: PropTypes.object,
/**
* Length of time in seconds for the transition (can use decimals)
* @examples '35'
*/
transitionDuration: PropTypes.string,
/**
* Type of ProgressBar; 'circular' or 'linear'
*/
type: PropTypes.oneOf(["linear", "circular"]),
/**
* The default value(s) of the progress bar. Can be a number or an object containing keys of "from" and "to"
* @examples '{"from": 10, "to": 80" }'
*/
value: PropTypes.oneOfType([
PropTypes.number,
PropTypes.shape({
from: PropTypes.number,
to: PropTypes.number
})
])
};
ProgressBar.defaultProps = {
buffer: 0,
className: "",
max: 100,
min: 0,
transitionDuration: ".35s",
mode: "indeterminate",
type: "linear",
value: 0
};
export default CSSModules(ProgressBar, styleSheet, { allowMultiple: true });
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,447 |
DRAGONBREATH
NO SUCH THING
AS GHOSTS
DRAGONBREATH
NO SUCH THING AS GHOSTS
BY
URSULA VERNON
DIAL BOOKS
an imprint of Penguin Group (USA) Inc.
_For Mom. I still think the javelina costume was cool._
DIAL BOOKS
An imprint of Penguin Group (USA) Inc.
Published by The Penguin Group • Penguin Group (USA) Inc., 375 Hudson Street, New York, NY 10014, U.S.A. • Penguin Group (Canada), 90 Eglinton Avenue East, Suite 700, Toronto, Ontario, Canada M4P 2Y3 (a division of Pearson Penguin Canada Inc.) • Penguin Books Ltd, 80 Strand, London WC2R 0RL, England • Penguin Ireland, 25 St. Stephen's Green, Dublin 2, Ireland (a division of Penguin Books Ltd) • Penguin Group (Australia), 250 Camberwell Road, Camberwell, Victoria 3124, Australia (a division of Pearson Australia Group Pty Ltd) Penguin Books India Pvt Ltd, 11 Community Centre, Panchsheel Park, New Delhi - 110 017, India • Penguin Group (NZ), 67 Apollo Drive, Rosedale, Auckland 0632, New Zealand (a division of Pearson New Zealand Ltd) • Penguin Books (South Africa) (Pty) Ltd, 24 Sturdee Avenue, Rosebank, Johannesburg 2196, South Africa • Penguin Books Ltd, Registered Offices: 80 Strand, London WC2R 0RL, England
Copyright © 2011 by Ursula Vernon
All rights reserved
The publisher does not have any control over and does not assume any responsibility for author or third-party websites or their content.
Designed by Jennifer Kelly
Library of Congress Cataloging-in-Publication Data
Vernon, Ursula.
No such thing as ghosts / by Ursula Vernon. p. cm. — (Dragonbreath ; 5)
Summary: Not only must Danny and Wendell trick-or-treat with skeptical classmate Christiana, school bully Big Eddy dares them to enter a haunted house on Halloween night, where they may have to sacrifice their candy to a ghost.
ISBN: 978-1-101-52937-9
[1. Halloween—Fiction. 2. Haunted houses—Fiction. 3. Ghosts—Fiction. 4. Bullies—Fiction. 5. Dragons—Fiction. 6. Iguanas—Fiction. 7. Humorous stories.] I. Title.
PZ7.V5985No 2011
[Fic]—dc22 2011001164
# Table of Contents
Pity Candy
Tricking and Treating
The Skeptic
Panic Time
The Haunted Bathroom
Something's Hungry
Thuds and Footsteps
Rat Leader
Danny Gets Floaty
Operation Mongoose!
A Call for Mommy
When Clowns Attack
What's Behind Door #1?
The Great Sacrifice
Little White Lies
"Okay," said Danny Dragonbreath, "I give up. What are you supposed to be?"
His best friend, Wendell, sighed. The iguana had a pie plate taped to his chest and was carrying another one.
"I'm a hydrogen atom," said Wendell glumly. He waved the pie plate. "This is my electron."
Danny had only the vaguest idea what an atom or an electron was, but he knew one thing for sure. "You let your mom make your costume, didn't you?"
Wendell sighed again.
Danny shook his head. Wendell's mother believed in education the way other parents believed in sports or health food. She also couldn't sew. Danny's mother couldn't sew either, but she understood the importance of Halloween in a young dragon's life and was willing to take him shopping for batwing capes and fake vampire teeth.
"Bummer."
Wendell shrugged.
Danny nodded. Pity candy was just as good as any other candy, and there was usually a lot more of it.
"And you're a vampire," said Wendell. "Not bad."
"I wanted to go as a giant false vampire bat, but Mom couldn't find a costume."
Wendell shuddered. There had been an incident with a giant false vampire bat monster over the summer, and while the iguana was more sympathetic to bats in general as a result—was even occasionally glad to see them fly overhead—he wasn't going to get over the giant slavering one any time soon. At least, not until Danny dragged him into something even more horrible, and giant bats started to seem friendly and non-threatening.
That was the nice thing about being friends with Danny—your traumas never had time to settle in.
Danny's father came down the stairs. "Okay, guys, ready to go trick-or-treating? Got your pillowcases?"
"Great! We'll go up and down our street, then drive over to..."
He stopped. He gazed at Wendell.
"Wendell, are you a...pie salesman?"
"Hydrogen atom," said Wendell wearily.
Danny's father looked briefly at the ceiling and said something under his breath that might have been Wendell's mother's name. "Your mom came up with your costume, didn't she?"
Wendell nodded.
"It's okay," said Wendell. "I'm hoping for the pity candy."
"Oh, well, that's okay then," said Mr. Dragonbreath. He picked up the car keys and called, "Going trick-or-treating, honey!" up the stairs.
"Try not to lose them!" Mrs. Dragonbreath yelled back down.
"Right. Onward! Candy awaits!" He held open the door, and the boys tromped out into the Halloween night.
The Dragonbreaths lived on a quiet street,* and only about half the houses had their lights on, advertising the availability of Halloween candy. One or two had cardboard ghosts on the door or jack-o'-lanterns on the step, but generally Danny's neighbors didn't get into Halloween. The tradition for Danny and Wendell was to trick-or-treat down the block and pick up a few pieces of starter candy.
Then Danny's dad would drive them over to the neighborhood in the rich suburb, with the really good candy. Some of the people over there gave out whole candy bars, not the little "fun size" ones. It was not to be missed.
Danny's dad was a good sport about trick-or-treating too, staying well back from Danny and Wendell so that nobody could see that a dad was taking them around. And when he claimed the grown-up candy tax, he usually took things like Milk Duds that no one would want to eat anyway.
The half-dozen houses on Danny's block worked out well for Wendell, because nobody could figure out his costume.
Then they'd drop an extra piece of candy in his bag. Danny, watching the third miniature candy bar land in Wendell's pillowcase, started to regret how good his vampire costume looked.
When they finished the street, they piled into the car, ready for the big haul.
"Where are we stopping?" asked Danny.
"I promised Christiana Vanderpool's mom we'd come over and take her out trick-or-treating," said Mr. Dragonbreath.
"Daaaaaad!"
"What?" Danny's father glanced in the rearview mirror. "Don't you like Christiana? I thought she brought a sheep brain to school last year."
"Well, yeah," Danny admitted, "that was pretty cool. But...she's _weird_."
"I hate to break it to you, but you're kinda weird yourself, son," said his dad.
Wendell snickered.
"Yeah, but..." Danny sighed.
What he couldn't say was that Christiana was a Junior Skeptic and didn't even believe he was a dragon. She didn't believe in anything, even stuff like Santa Claus that you weren't really supposed to believe in anymore, but you sort of pretended to because everybody else did. When Ms. Brown had them read "'Twas the Night Before Christmas" in class, Christiana had said it was a Victorian romanticization of an outmoded pagan belief system, and c'mon, that wasn't normal. She sometimes used words even Wendell had to look up.
On the other hand, the sheep brain in the jar _had_ been awfully cool.
Danny slouched down in the seat.
"She's probably really nice if you get to know her," his dad said from the front seat.
Which just went to show that grown-ups did not understand a lot of things, because A) Danny had been in school with Christiana for years, and knew her just fine, and B) whatever words you were going to use to describe her, "nice" was not among them.
"Suki liked her," offered Wendell.
"Huh," said Danny. That was unexpected. Suki the salamander had been awesome (even if she was a girl), and if Suki liked Christiana...well, anyway, it was only for one night of trick-or-treating. How bad could it be?
Christiana Vanderpool got into the car wearing a large purple suit and a sarcastic expression. She was a stocky crested lizard shorter than Wendell, although the antenna on her suit made her look taller.
She looked at Danny. "Vampire. Typical." She looked at Wendell. "Mmm. Hydrogen atom?"
Wendell nodded glumly.
"Your electron and your proton aren't in scale," she said, settling herself into the seat.
"I know," said Wendell, who did.
Danny would have said something in defense of his friend—although the pie plates were pretty hard to defend—but his dad got back in the driver's seat, which cut down on the conversation.
"I thought salmon were fish. With...like...fins and things," said Danny.
"She's the germ that gives you food poisoning," Wendell explained. He looked down his snout at her. " _You're_ not to scale either."
Unexpectedly, Christiana grinned. "Yeah, I know. Cool suit though, huh?" She wiggled her cilia at them.
It was a short drive to the really good neighborhood, and bands of kids were already roaming the streets. Danny scrambled out of the car, followed by the others. The nearest house had a strobe light and a smoking cauldron. The sound of recorded hysterical laughter echoed down the sidewalk. It was perfect.
The trio settled into a pattern. Christiana rang the doorbell. The door opened, and Danny pushed Wendell forward. Wendell held out his pillowcase and looked hopeful, whereupon the homeowner gazed at the iguana's pie plates, was seized by pity, and dumped candy with a generous hand.
By the time they'd walked up one side of the street and two cul-de-sacs, Wendell's pillowcase was dragging on the ground. Christiana and Danny weren't quite so fortunate, but they still had very respectable hauls. Danny was beginning to wish he'd brought a second pillowcase.
They'd also encountered boiling cauldrons, shrieking hinges, animatronic skeletons, overstuffed scarecrows, and three more strobe lights. The flashing lights caused Wendell to mutter darkly about seizures and made Danny's father do a horribly embarrassing dance that he said was called "The Robot."
Still, it had been a good night. At least until—
Danny and Christiana followed Wendell's gaze and made identical "ungh" noises.
Big Eddy the Komodo dragon and his cronies were walking up the sidewalk toward them.
"Hold on to your candy," muttered Christiana.
Danny looked over his shoulder, to where his dad was talking to some other parents, probably about doing The Robot. The presence of nearby adults would keep Big Eddy from beating them up and taking their candy...probably...but it was dark enough out that you didn't want to take any chances.
"C'mon," said Wendell, pointing at another knot of kids coming up the sidewalk behind them. "Safety in numbers."
Danny was willing to give it a try. They slowed their footsteps up to the next house, and pretended to be very interested in a couple of jack-o'-lanterns lining the next porch. By the time they reached the street again, the other kids were only a few feet behind them.
"Oh look," said Big Eddy, "it's dorkbreath and his sidekick." He loomed over Danny, who backed up, not wanting to get shoved. "Got tired of pretending to be a dragon and decided to be a vampire, huh?"
"I _am_ a dragon," said Danny, but he said it under his breath. He had a fond memory of a rat riding Big Eddy down a hallway, and preferred not to tarnish it with a memory of Big Eddy's fist hitting him in the snout.
Christiana gave him a skeptical look. Danny felt his scales get hot.
"Nothing," said Wendell wretchedly as Big Eddy yanked away his electron pie plate.
The other kids had arrived. Danny recognized a few of them from school, although they were mostly in other grades. None of them seemed to want to press past Big Eddy, but the sidewalk was getting awfully crowded.
Somebody pushed Danny in the back, not hard, but enough to move him forward a little. Big Eddy's eyes snapped back to Danny. Danny winced.
"I'm surprised you weren't too scared to come out, dorkbreath. Aren't you afraid a ghost's gonna get you?"
"I'm not scared of any stupid ghosts," said Danny, stepping off the curb and circling around Big Eddy. Maybe if they started walking away, the bully would forget about them. Big Eddy didn't have a very long attention span.
No such luck. The big Komodo dragon turned around and started following him. The other kids followed at a cautious distance.
"You're totally scared," said Big Eddy. "You're a big chicken."
"So prove it," said Big Eddy. "I bet you're too scared to go up to the haunted house."
Danny rolled his eyes. "I've _been_ to haunted houses. They're not scary.* It's just people in masks."
"Not _that_ kind of haunted house, dorkbreath. A real one. That one." Big Eddy spun around and pointed.
"How do you know the house is haunted?" asked Christiana. "It just looks abandoned to me."
" _Everybody_ knows it's haunted," said Big Eddy.
"Then _everybody_ ought to have proof," said Christiana.
Big Eddy looked confused, which was generally a prelude to Big Eddy getting mad. Danny jumped in. "Anyway, even if it is haunted, we can't go in. It's trespassing."
The Komodo dragon sneered at him. "You're just chicken."
"I'm not chicken," said Danny, "but my dad is back there"—he jerked a thumb in the direction of his father and the other grown-ups—"and I can't just break into somebody's house in front of him. He'd ground me for a _month_."
There was a murmur of agreement from the kids gathered around them. You couldn't do that sort of thing in front of grown-ups. They really didn't understand.
"So go trick-or-treat it," said Big Eddy, shoving Danny in the shoulder. "Prove you're not chicken."
Danny gulped.
It occurred to him that having all the other kids around might mean that there were more witnesses if Big Eddy decided to beat him up...but it also meant that there were more witnesses to Big Eddy calling him a chicken.
He looked up at the house. The driveway was long and dark and overhung with trees. There were no lights on and the windows were boarded up.
He looked back toward his dad, but apparently the grown-ups were having a really interesting conversation about life insurance or vegetables or something else of interest to grown-ups. They were definitely not paying attention except for an occasional glance to make sure the kids weren't being hit by cars.
"Bet you won't," said Big Eddy. "Bet you're scared."
"I am NOT," said Danny, which was mostly true. He wasn't scared of ghosts, exactly—they couldn't be that much worse than a giant squid—but he wasn't sure how you dealt with them. Ghosts could go all invisible and stuff, and they probably weren't bothered by having fire breathed on them.
It occurred to Danny that he had possibly backed himself into a bit of a corner.
"Um..."
"I'll come with you," said Christiana firmly. "I don't believe in ghosts."
The crowd of kids all took a step back. Saying you didn't believe in ghosts on Halloween was kind of like standing in a thunderstorm and saying you didn't believe in lightning.
Danny was stuck. If Christiana was going, and Big Eddy was watching...well, there was nothing else to do. He slung his pillowcase of candy over his shoulder, cast a last longing look back at his dad—no help there—and started walking.
Wendell found himself with a dilemma.
He could go with Danny and Christiana, who were probably about to be mauled by ghosts, or he could stay here. With Big Eddy.
You might believe in ghosts or not—but the school bully was definitely real. And Wendell had stuck by Danny in worse situations than just haunted houses...there had been the squid, and the sewers, and the giant bat, and the thing with the ninjas...
The iguana sighed and scurried after Danny and Christiana, up the long driveway to the (possibly) haunted house.
The house looked even worse close up. The porch was sagging, the windows were boarded, and there were real cobwebs, even thicker and denser than the fake Halloween kind. Danny's parents had always said that you weren't supposed to go to a house that didn't have the lights on, and this place didn't even have bulbs in the porch light.
"It sure looks haunted," said Danny.
Wendell gulped.
"Nobody's ever proved that ghosts exist," said Christiana, striding determinedly toward the door.
"I have a hard time breathing fire under pressure," said Danny. (Actually, under pressure seemed to be the only time he _could_ breathe fire—if by "pressure" you meant "stark raving terror"—but this was a difficult distinction to explain to somebody like Christiana.)
She smirked. "Yeah, that's what they all say."
" _I've_ seen him breathe fire," said Wendell staunchly.
Any gratitude Danny might have been feeling to Christiana for coming with them was evaporating rapidly. He gritted his teeth and stomped up the steps onto the porch.
"No doorbell," said Wendell. "Nobody's here. Can we go back now?"
"I guess we could knock," said Danny. "I mean...nobody's gonna come, but as long as we knock, that ought to be enough..."
He made a fist and rapped on the peeling paint of the door. The sound seemed to boom through the inside of the house, much more loudly than it should have. Wendell cringed.
"Well," said Christiana, "I guess that—"
With a long wooden moan, the door swung open.
Danny knew he should turn around right now and walk away, at least until he was off the porch, and then he should run like his tail was on fire.
It was Halloween that made him do it, he decided later. On any other day of the year, he would have run away, but on Halloween, you went up to scary haunted houses and the doors creaked open and that was _normal_.
He took a step forward, onto the threshold.
"Danny, what are you doing?" squeaked Wendell.
Christiana, however, stepped up beside him and poked her head around the edge of the door frame.
"Must've been unlocked," she said. "What a dump!"
As if in a dream, Danny took another step forward, into the dark room. There was moonlight coming around the edges of the boarded windows, casting long, pale stripes across the floor.
The door slammed shut. It knocked Wendell into Christiana, and both of them into the room, pinching Wendell's tail cruelly in the door frame. He yelped. Danny jumped.
"What happened?" yelled Wendell, scrambling to his feet. "Why'd it close!?"
"It was just the wind," said Christiana, grabbing for the doorknob. "Don't freak out."
She turned the doorknob and pushed.
Nothing happened.
She shoved at it harder, twisting the knob back and forth, but the door didn't budge.
"Um," she said.
Danny pushed her aside and grabbed the doorknob himself.
"It's stuck!" he said.
"We're going to die..." Wendell moaned. "We're trapped..."
Danny slammed his shoulder against the door, which did more damage to his shoulder than the door. It didn't budge.
"We're not stuck," said Danny. "We just need to find another way out." He peered around the room.
There was a dark fireplace and a sofa covered by an old sheet, neither of which was any help at all. Most of the windows were still intact, but at least one was broken out, leaving a frame edged with daggers of glass that glittered in the moonlight. If they tried to go out through the window, they'd have to find a way to break the boards out, then crawl through without disemboweling themselves on the broken glass.
This did not seem promising.
"There's got to be a back door," said Christiana.
A dark doorway led deeper into the house. While moonlight lit their room unevenly, the hallway beyond the door was pitch-black.
"I'm not going in there!" said Wendell.
Danny couldn't blame him. The doorway looked like an open mouth. He rubbed the back of his neck nervously. "Do the lights work?"
Christiana found a light switch and flicked it a few times. Nothing happened.
"I was afraid of that," said Danny. "Um. I don't have a flashlight..."
"Me neither."
Wendell coughed. "Err...wait. Mom made me bring one...." He dug around in his pillowcase of candy and eventually pulled out a flashlight. "She was worried that if it got dark I'd get hit by a car."
"I take back everything I've ever said about your mother," said Danny, grabbing the flashlight.
The flashlight didn't work. Danny unscrewed the cap to check the batteries. Behind him, he heard Christiana say, "Periodic table bandages, huh?"
"Yeah."
"My dad gets the countries-of-the-world ones. I got a splinter in my hand last month and spent three days staring at the gross national product of Belgium."
Wendell laughed. It was a nervous, strangled sort of laugh, but it was still a laugh. Danny, whose knowledge of Belgium began and ended with waffles, poured the batteries into his hand, blew on them, and shoved them back in the flashlight. He smacked the flashlight into his palm a few times, and it lit up grudgingly.
The little circle of light seemed very small against the darkness. Danny flicked the light down the hallway, over the dusty floorboards, revealing another open doorway.
"I guess we have to go down there," he said.
He took a step forward.
A shriek echoed through the house, a horrible keening noise that sank to a dull moan and finally died away. It sounded like somebody being tortured. A shriek from Wendell followed, although it wasn't quite as loud. Danny jumped.
"What was _that_?!" he said.
"It's the ghost," said Wendell, arms over his head. "It knows we're coming, it's gonna get us, we're gonna dieeeee...."
"Ha!" said Christiana suddenly. She strode forward into the hallway and stomped on the floor.
Another shriek rang out, shorter this time.
"It's not a ghost, it's the floor," she said. She bounced up and down on a particularly creaky board, producing a series of short groaning noises, like a donkey with hiccups. "I bet nobody's walked on it in _years_."
Danny rolled his eyes, feeling embarrassed. He hadn't been scared. Not exactly. Startled, maybe. It had been loud, that was all. He stepped onto the floorboard, which yelped again.
"What if the ghost heard it?" asked Wendell.
"There are no ghosts," said Christiana. "Nobody's ever proved ghosts exist, anyway."
Wendell was generally pretty scientifically minded, but he'd seen too many weird things while hanging around with Danny. Also, he'd checked out a book of ghost stories from the library a week ago, and it had some pretty alarming stuff in it. There had been one about a hitchhiker who turned out to be a ghost who always came back on the anniversary of her death. It made his scales crawl.
Danny took another few steps down the hallway and reached an open doorway on the left wall. He shone the light into it. Christiana and Wendell came up behind him, the iguana taking big steps to avoid the creaky floorboard.
The room was a bathroom, and it was _nasty_.
The toilet was missing a lid, and the water was rust-brown and slimy. The paint was peeling and had formed big blisters, and some of the blisters had burst. Mold crept up the walls.
There was a picture of a clown over the toilet tank. The clown was crying, which was probably meant to be tragic, but was mostly just creepy.
"If there's a ghost here, they're a real slob," said Danny.
"How long did you leave that sandwich in your locker that one time?" asked Wendell. "Purely as a matter of curiosity..."
"Yeah, but it was a sandwich. Not a whole _bathroom_." (The sandwich in question had turned a variety of interesting colors and then grown fur. He'd thrown it away eventually, although Wendell claimed that it was alive and trying to communicate.)
They retreated from the bathroom. Wendell said under his breath, "I hear clown attacks are up this year..."
"I'm pretty sure I can kick your tail, Wendell," said Christiana through gritted teeth. Danny snickered.
The young crested lizard shut the bathroom door behind them. When the boys looked at her, she said, "What?"
"Afraid the clown will sneak up on us?" asked Danny.
"Look," said Christiana, sounding annoyed, "I just don't—"
The bathroom door swung slowly open. The three of them stared at it. "Maybe it didn't latch," said Danny, because somebody had to say something.
"The door frame's probably warped," said Christiana nervously. She wiped her hands on the sides of her bacteria suit.
"Yeah," said Danny. "Still..." He took a deep breath and shone the flashlight back into the bathroom.
Nothing. The paint was still peeling, the toilet was still disgusting, the painting of flowers over the toilet was still—
"Didn't that used to be a clown?" asked Danny.
Christiana actually took the flashlight away from him and shone it on the painting, which was of a vase full of flowers.
"Well," she said, after a minute, and then stopped. She looked up and down the hall, as if expecting there to be another bathroom, possibly with a clown painting in it.
"I don't know." She frowned. "I suppose we might not have seen it clearly the first time, but I sure _thought_ it was a clown..."
Danny didn't know what to think. It had definitely been a clown, and it was definitely _not_ a clown now. At the same time, it seemed like a weird thing for a ghost to do. Ghosts were supposed to rattle chains and moan, not switch around the artwork.
"Anyway," said Christiana, sounding a little more confident, "just because we can't explain it doesn't mean there isn't an explanation. It just means that we don't know what it is right now."
"Uh-huh," said Wendell. "This sounds like a great philosophical discussion. Maybe we could have it sometime. You know, maybe when we're not _standing in the hallway of a haunted house_?!"
Christiana looked at Danny. Danny looked at Christiana.
"Right!" said Danny. "Back door. Let's find it."
The kitchen lay at the end of the hallway. Danny felt for a light switch and flicked it hopefully, even though he was pretty sure it wouldn't work.
It didn't. Wendell sighed.
In the beam of the flashlight, they saw stairs leading up to the second story. All three of them moved away from the stairs. The rest of the kitchen looked ordinary, if dusty—stove, sink, refrigerator, all dark and dim and silent. The linoleum was peeling up in the corners and the window over the sink was boarded up.
"We don't know that anybody died here," said Christiana. "They might have just moved out."
"They could have moved out because it was haunted!" Wendell waved his pie plate in the air. "I read a book about this one haunted house where they had poltergeists and they tried everything to get them out and the poltergeists kept making weird noises and turning on the faucets and the people had to move."
Danny had an urge to borrow this book from Wendell, which did not happen often. Christiana was less impressed.
"You're being irrational," she said, folding her arms. "None of that stuff is real."
Wendell opened his mouth to say that after several years of running around with Danny, he'd seen giant squid and ninja frogs, were-hot-dogs and ancient bat gods, and his notion of what was possible had expanded quite a bit as a result. But then he closed it again, without saying anything. Christiana would demand that he prove it, and he couldn't prove anything standing in a dark kitchen in an abandoned and probably haunted house.
He wished he could. A few years ago, he would have agreed with her. Now she just thought he was stupid, and Wendell really hated it when people thought he was stupid.
"I think there's a door through here," said Danny, pointing the flashlight through another doorway.
They walked as quietly as they could, practically sneaking. Even Christiana was doing it. There was something about the empty house that made you not want to make loud noises.
On the far wall of the new room was a closed door with a window in it. The window was boarded over, but it looked like it might lead to the outside.
Wendell rushed forward and grabbed the handle. All three kids held their breath.
The handle turned. The deadbolt did not. Wendell threw himself back, clinging to the handle, and only succeeded in making the door rattle in its frame.
Danny said a word that his mother said occasionally when somebody cut her off in traffic.
Wendell slumped against the door frame. "Do you think your dad will find us? It's been _hours_."
"It's been about twenty minutes," said Christiana. "And he'd have to figure out what house we went to. Unless Big Eddy tells him, he won't know."
"Maybe we could break a window," said Danny. "Like...with a chair or something." He hated the thought of breaking a window—mostly because he _knew_ it was going to come out of his allowance—but if there wasn't any other way out...
"It occurs to me—" began Christiana.
Wendell said, "...Eep."
Danny followed the iguana's gaze and felt his stomach do an unpleasant sort of flop.
An enormous white shape was looming up behind Christiana, nearly twice as tall as she was. Pale lumps, like arms or wings, flared out to either side.
"Oh," said Wendell, "oh, oh—"
"Christiana," hissed Danny, "behind you!" He lifted the flashlight with nerveless fingers. The vast shape seemed to rise even higher.
Christiana spun around, blinked, and then made an exasperated noise.
"Come _on,_ you guys..."
She reached out, caught a corner of the shape, and yanked.
The sheet came off the ancient wingback chair in a cloud of dust.
"It's a _chair,_ " she said. "It's not scary."
"I find that upholstery rather alarming," said Wendell, eyeing the garish floral pattern.
"I can't believe you guys. Every little noise and it's 'Oh, help, it's a ghost, it's a poltergeist!'"
"It's dark," said Danny indignantly, "and the white sheet, it really _did_ look like a ghost."
"I keep telling you, there's no such thing as—"
She was cut off by the sound of footsteps. Loud, heavy footsteps, coming from upstairs. All three kids froze, listening as someone—or something—walked overhead.
A door opened and shut. Then silence...
...or not quite silence.
_"Hungry..."_ whispered a soft, hissing voice. It didn't sound very close, but it wasn't exactly far away, either. _"I'm hungry..."_
"I don't know if that was a ghost," whispered Danny, "but there's definitely somebody up there."
Danny looked at Wendell. Wendell looked at Danny. Christiana looked at both of them.
Wendell opened his mouth to say something—probably "We're all going to die," or something equally upbeat—but never got the words out.
_Something_ began hammering violently on the back door. It sounded like thunder, like drums, like a herd of horses running up the side of the building.
Most of all, it sounded like something wanted in. Bad.
Wendell shrieked, dropped his candy, and bolted. Danny took off after him—to make sure he didn't trip in the dark. Yeah. Absolutely. Not because he was terrified. Definitely not.
They skidded into the front room. Danny thought for a second that Wendell might run right _through_ the front door, leaving a cut-out opening behind him, like a cartoon. They hoped the footsteps behind them were Christiana's. (If they weren't Christiana's, Danny didn't want to know.)
Another flurry of pounding shook the house, this time on the _front_ door. Wendell let out another ear-splitting shriek and stopped so fast that Danny ran into his back. More hammering hit the boarded windows, making the glass rattle in the frames.
"We're surrounded," moaned Wendell, backing toward the hallway. "There's no way out..."
Danny didn't know what was going on, but he was pretty sure he didn't like it.
He inhaled, feeling smoke roil at the bottom of his lungs. He didn't know if there was anything there to breathe fire on, or whether he'd just burn the house down, but he was going to be prepared.
"It's the ghost!" said Wendell.
"Maybe it's vampires!" said Danny.
"Maybe it's your dad trying to find us," said Christiana.
Hammer-blows struck both the front and back doors. It sounded like the doors would fall down at any second.
_"Hungry..."_ hissed the spectral voice, practically in their ears.
It was kind of ironic, Danny thought—just a minute ago, he'd wanted the doors to open, and now he wanted them to stay closed.
"Here!" whispered Christiana beside him. "Get the other end!" She grabbed one side of the couch and tried to drag it toward the door.
Danny tossed the flashlight to Wendell, who promptly dropped it. (Wendell always did catch like a nerd.) The light spun crazily over the ceiling as Danny threw himself at the other end of the couch.
Between the two of them, they managed to drag the couch so that the end was in front of the door.
"I don't know how long that'll hold," Danny panted, watching the couch vibrate with every blow.
"We've got to hide," gasped Wendell, clutching his pie plate to his chest.
"We should—um—retreat and gather more data," said Christiana.
They fled down the hallway.
Danny didn't know who "they" were, and didn't feel like finding out. He grabbed for the door to his right and yanked it open.
Stairs led down into the dark. Danny hesitated and shot another look at the front door.
The pounding started up again, from both the front and back. Christiana spread her hands and shook her head, clearly out of ideas.
" _Do_ something..." moaned Wendell.
Gulping, Danny pulled Wendell down onto the first step and waited for Christiana to follow. He reached past the crested lizard and closed the door behind them.
It closed with a sinister _thunk!_ shutting the three of them inside the cellar.
The sounds were much more muffled inside the cellar. Danny turned and shined the flashlight down the stairs.
The floor was concrete, not very clean. Cobwebs hung thickly from the ceiling. An old washer and dryer stood in one corner, surrounded by rusty water stains. The other side of the basement was full of old boxes.
"Guess we should go down," Danny whispered.
They went down, wedged so closely together that it was a wonder they didn't trip each other, fall down the stairs, break their necks and save the ghosts the trouble.
If it _was_ ghosts.
"I don't think it was ghosts," said Christiana.
Wendell jumped at her voice, and Danny had to grab for the banister.
"If it wasn't ghosts, then why did you run?" he snapped.
"Because it was freaky!" she shot back. "I happen to believe in serial killers and cannibals, thank you very much!"
"Don't forget clowns," muttered Wendell.
Christiana gave him a look that would have burned through his pie plate, if he hadn't dropped it in the headlong flight to the cellar. He prudently decided to stand on the other side of Danny.
"You think it was cannibals?" asked Danny. "Cool! I always wanted to meet a real cannibal..." He ran the flashlight over the boxes. Most of them had things written on the side like "Summer clothes" and "Kitchen stuff," but then again, if you were a cannibal, you probably didn't put labels like "Yummy dead bodies" and "Fresh corpses" on your boxes, did you?
After all, if you used a moving company, they were bound to get suspicious—
"No, I don't think it was cannibals," said Christiana. "Let's look at this logically."
She paced back and forth at the foot of the stairs. Danny backed up a step.
"Who knows we're in this house?" she asked.
"The trick-or-treaters behind us," said Danny.
"Big Eddy," said Wendell.
"Bingo." She pointed at Wendell. "And who would think it was hysterical if we were locked up in this house?"
"He probably snuck back here to scare us," she said.
Danny nodded. "And then he and his buddies—you know, the two losers he hangs around with—started pounding on the doors."
Wendell frowned. "Then how do you explain the footsteps upstairs? Or that voice?"
"One of them could have snuck in," said Christiana. "Or even waited for us upstairs. He might even be the one who locked us in. And maybe he just hasn't had dinner yet."
"It makes sense," said Danny. He felt like an idiot. And he was slightly disappointed he wouldn't get to meet a cannibal.
"No," said Wendell, after a minute, "the _point_ is that if someone got in up there, there must be a way out too, right?"
Christiana frowned. "Not necessarily. Isn't one of his buddies a chameleon? He could have hidden downstairs..."
"He could have, but I bet there's an open window or something upstairs." Wendell thought about it. "You'd think Big Eddy would have gotten in too, though..."
"He's the size of a moose," said Christiana disdainfully.
Danny slapped his forehead. "You're right! He might not have been able to fit in a window—but I bet we could!"
Christiana nodded. "It's worth checkin—EEEP!"
She jumped about six inches sideways and into a box marked "Kitchen." Danny spun around, prepared to see anything from Big Eddy to a cannibalistic clown from Mars, and saw—
"You brought a sheep brain to school, but you're scared of _rats_?" asked Wendell.
"I'm not scared of rats," muttered Christiana, extracting herself from the pile of boxes she'd fallen into. A large copper bowl rolled in a circle, teetering, and stopped with a faint _booonnng_.
"Rats are highly intelligent and serve a valuable place in the ecosystem. I was just startled," she said. "I was looking over there, and it moved, and...look, it's been a rough night, okay?"
Danny put his hands on his knees and addressed the rat. "Hey there! Do you know our friend the potato salad?"
The rat hopped into the cardboard boxes and vanished for a moment. It returned, looked Danny up and down, and shook its whiskers. "Squeak."
"Nothing we can fit through," Danny guessed. "Crud."
"You're talking to a rat," said Christiana.
"Hey, you _said_ they were highly intelligent," said Danny. "We've had good luck with rats."
Christiana rubbed her forehead, looked like she was about to say something scathing about rats, changed her mind, and said instead, "If we're staying down here, I'm taking off my head."
This was the sort of statement that would normally have required a great deal of analysis and caused concern, but then she reached up and pulled the head off her costume. "Phew. That thing is _hot_."
"Squeak?"
Danny looked over. The rat had returned, and was perching on the edge of a box, at eye level.
"Hey again," he said to the rat.
"Squeak!" it said. It made a beckoning gesture, then ran down the box to the floor.
The rat hopped from box to box, and finally darted through a hole in the back wall. Danny played the flashlight over the hole and saw the frame of a low, square door.
"It's a hidden passage!" he said. "Cool!" Danny approved of hidden passages. You could hide treasure in them, or smugglers or skeletons. Or maybe even all three.
"It's just a crawlspace," said Christiana. "Probably so people can work on the plumbing."
"I am _not_ going in there," said Wendell.
The rat poked its head out again and squeaked.
"Let's see if we can get it open," said Danny. He knelt and tugged on the door. It stuck a bit, then yawned open.
The crawlspace was dark and cobwebby and smelled like rotted wood and his aunt Shirley's casserole. The rat ran a few feet down and squeaked again.
"Maybe it's a way out," said Danny. He stuck his head through the doorway. The rat squeaked excitedly. Wood creaked under Danny's hands.
Wendell fidgeted. On the one hand, Danny was going into the hideous unknown darkness. On the other hand, he was taking the flashlight with him. "Danny..."
"I'm fine," Danny called. "Stinks in here, though. Smells all moldy."
"Some mold spores can be really toxic," Christiana said helpfully.
"Neat!" Danny said over his shoulder. "Can we throw them at the ghost?"
Wendell wrung his hands.
"Are you okay!?" Wendell rushed forward. He could just make out Danny lying on a pile of boxes down below.
"I'm...fine...I think..." Danny sat up, rubbing his head groggily. "Guess the junk broke my fall..."
The rat, squeaking worriedly, flipped over the broken edges of the boards and dropped into the opening below. A minute later, the rat and Danny were eye to eye.
Christiana shouldered Wendell aside and peered over the edge of the hole. "Looks like you're in some kind of sub-basement," she said. "Must have been another storage room. I don't think we can pull you up, but maybe we can find the stairs and meet you down there."
"Okay." Danny sat up. The pile of boxes made some interesting sproinging and twanging noises. "Man, they sure had a lot of junk..." He found the flashlight and swept it around the room. "Gotta be a door somewhere..."
The rat caught his attention again. It was waving him over to one side of the room.
"Is that the door?"
Moving through the pile of debris was harder than Danny expected. He couldn't walk. Instead he made a series of lunges, crushing boxes and old magazines underfoot. It was like swimming in a sea of moldy cardboard.
When Danny finally reached the rat, it was perched on the lip of a box next to the wall. It squeaked at him, and pointed.
"What's down there?" Danny trained the light down, through a gap in the boxes, and saw...something. It looked faded and fuzzy and rather grungy.
The rat squeaked again, very seriously, and pointed up at the ceiling, then back down at the object.
Danny had no idea what that meant, but if the rat thought it was important...He dropped to his knees and reached for whatever it was.
It was a stuffed animal.
It was impossible to tell what it had been—a bear, maybe, or a sheep, or something else. It had a generic animal shape, and one dusty button eye.
The rat squeaked again.
"This?" Danny turned it over, puzzled. "What's so important about—"
And then the world went away.
Danny was floating in a dark place.
This did not bother him as much as it could have. He didn't seem to have a body, but hey, these things happened. Maybe he was dreaming, or he'd hit his head harder than he thought and now he was having an out-of-body experience. He'd seen a TV show about those. There had been weird floaty music on the show, and he was a bit disappointed that there wasn't any music now.
Still, you couldn't have everything.
A door opened, and light spilled into the room. At first Danny thought Wendell and Christiana had found him, but he seemed to be looking down on the door, and the light coming through the doorway was shining on a bedroom, not a pile of junk.
"How are you feeling, honey?" asked an unfamiliar voice.
Danny wasn't sure if she was talking to him, and furthermore wasn't sure how to talk in his bodiless state, but fortunately somebody else answered, so it didn't matter.
"I'm _fine,_ Mom," said a little kid's voice. It sounded crabby and tired and sort of thick. Then it started coughing.
"Oh, honey..." said the mom, and she walked from the doorway, across Danny's vision to the bed. "I'm making you breakfast. It's eggs."
There was a kid in the bed, Danny could see. It was only about the size of Danny's seven-year-old cousin. Danny couldn't tell if it was a boy or a girl, but he could definitely tell that it was grumpy.
"Don't want eggs. I wanna go trick-or-treating," it said, folding its arms.
"Honey, you're too sick to go out," said the mom. "I know it's disappointing..."
"I wanna GO!"
Come to think of it, it kinda _sounded_ like his seven-year-old cousin too.
"Honey—"
"I WANNA!"
Fortunately a full-blown tantrum was broken off by a coughing fit. It was a bad cough too, the kind that had bubbly snotty stuff in it.
Danny tried to see if he could move around without a body. He thought he managed to wobble back and forth a bit, but it was hard to tell.
He wondered what the people below him would see if they looked up—a ghostly dragon hanging around the ceiling? Nothing at all?
"I _like_ trick-or-treating," said the little kid. "I hate eggs. You're _mean._ "
"I know it seems mean," said the mom, "but you're sick. Next year you can go out. Won't that be fun?"
She bustled around the bed, tucking something in alongside the kid. Danny caught the gleam of familiar button eyes.
"Don't wanna go next year..." said the little kid, hugging the stuffed animal. "Wanna go _now_..."
The scene was fading. Danny had a sense that there was something important here, something he was supposed to remember, and he tried to grab for it, but it was like having a dream and waking up and trying to remember—
—and then he was back in the basement, staring into the dusty eye of the stuffed animal.
"Whoa!" he said. He patted himself down—yup, that was his body, all right, and he seemed to be back in it. "That was _awesome!_ Did I just have an out-of-body experience?"
The rat spread its paws and shrugged.
"I gotta tell Wendell!"
Wendell would have liked nothing better than to be having a long talk about out-of-body experiences with Danny. Instead, he was trying to find his way to the basement stairs in the pitch-black.
The next ten minutes were not among the best of Wendell's life.
He and Christiana ran into each other. They ran into the boxes. Christiana yelled at him. He yelled back. They flung junk aside and hit each other with it. Christiana upended an entire box of plastic forks on Wendell's head. (She claimed it was an accident. The iguana had his suspicions.)
Finally, they saw a rectangle of light.
And once they got close to the door, they could hear Danny on the other side.
Eventually, with Danny pushing from his side and Wendell and Christiana pulling on their side, they got the door open.
"I missed you! I missed you so much!" Wendell cried.
"Uh—" said Danny.
Wendell snatched the flashlight away from him and hugged it. "Never leave me again," he told the light.
"...right, then," said Danny. "Guys, you're not gonna believe this. I found this stuffed animal, and I had this vision—and I think I know what we're up against."
"You're right, I don't believe it," said Christiana.
"I'm not saying you didn't see something," said the iguana. "It could have been a vision. But you did fall and hit your head, and there _were_ all those mold spores..."
Danny sighed. He should have known that Christiana wouldn't believe anything like a vision, but he had to admit that Wendell's defection stung a little. Danny glared at the stuffed whatever-it-was. "The rat thought it was important," he said stubbornly.
"Look," said Christiana, "this should be easy. We'll do what scientists do. We'll get proof."
"We can't prove something like that," said Christiana, "but it should be easy to prove whether it's Big Eddy making the noise and not a ghost." She started up the stairs, and Danny and Wendell followed her into the hallway.
They didn't dare use the flashlight, for fear of giving themselves away. Danny stepped carefully over the creaky board, and Christiana and Wendell stopped next to it.
Danny knelt down by one of the windows and listened carefully.
After a minute or two, he heard somebody say, "I'm bored." It sounded like Jason the salamander, one of Big Eddy's cronies.
"Shut up," hissed a deeper voice. Danny jerked back from the window, startled—from the sound of it, Big Eddy was standing directly on the other side of the wall.
"They're probably freaking out in there," said Big Eddy. "When they're good and scared, we'll throw open the door and they'll go running. Then we can grab their candy and get out of here."
"Couldn't you just have taken their candy?" asked Jason.
Danny rolled his eyes.
"They haven't made a noise for _ages,_ " said Jason. "Maybe they left."
"Shut up, or I'll take _your_ candy."
Danny sighed. It looked like Christiana was right—but then what was the meaning of his vision in the basement? Why had the rat led him there? It didn't make _sense_.
He looked over his shoulder. Christiana and Wendell were waiting in the hallway. Clearly they'd both heard Big Eddy talking too. He tiptoed back to join them,.
"So," said Wendell. "What do we do now? Wait until Big Eddy goes away?"
"I've got an idea," said Danny. "There's got to be stuff in all of those cardboard boxes downstairs that we can use..."
Christiana set her costume head down. "An idea for what?"
"Big Eddy was trying to scare us, right? So we're gonna turn around and scare _him_."
He paused dramatically. "It's time for Operation Mongoose!"
"I think we're getting away from the point here," Danny said.
"Can we call it Operation Dark Thunder instead?" asked Wendell.
"No," said Danny, somewhat annoyed, because Operation Dark Thunder was a way cooler name.
"You know, Operation Dark Thunder is a way cooler name," said Christiana.
"We're calling it Operation Mongoose! Now shut up and grab some stuff to scare bullies with!"
Wendell was nervous. It had been a bad evening, and they were about to play a joke on Big Eddy that might work, but might just make him really mad.
When Big Eddy was mad, someone usually got stuffed head-first in a toilet.
Danny crept back across the room, and listened at the window. Apparently Big Eddy was still out there, because Danny quickly turned back and waved a hand at Wendell. The iguana waved to Christiana.
Operation Mongoose was under way.
Christiana stomped a foot down on the squeaky board, which let out a scream like a dying cow.
"What was _that_?" Jason yelped. His voice seemed to suddenly come from farther away, as if he'd jumped back. Danny grinned and waved to Wendell.
Wendell stood in the middle of the room, holding a device cobbled together from an old jump rope and a rusty tin can they'd found in the cellar. He grabbed the rope and began whipping the can around over his head in a circle.
An eerie keening began as the can picked up speed, rising in pitch as Wendell got into the rhythm.
Danny could hardly keep from crowing with delight. The iguana said he'd read about it in a magazine—it was called a "bull-roarer" or something like that—and he thought he'd be able to make one, but he hadn't been sure it would work. You were supposed to use wooden planks, not tin cans and old jump ropes.
Nevertheless, it was working beautifully.
"That's not normal!" Jason yelled. "It's a ghost!"
"Shut up!" yelled Big Eddy—whether at Jason or at the noises coming from the house, it was hard to tell.
Christiana joined in with another series of shrieks from the floorboards.
"I want to get out of here!" said Jason. He was definitely farther away now. It sounded like he'd run clear off the porch.
Danny decided it was time for him to contribute. Holding the big copper bowl in both hands, he slammed it hard against the wall.
Danny pressed his ear to the crack in the boards again. The sounds of footsteps, and of Big Eddy yelling "Shut up, shut up, _shut up!_ " were fading into the distance. He felt the warm glow of a job well done.
It actually took longer to extricate Wendell from the bull-roarer than it had to scare the bully off, but nobody minded.
Unfortunately, whatever Big Eddy had done to jam the doorknob, it was still jammed. Danny stood on the couch and twisted the knob as hard as he could, but nothing happened.
"Well, crud."
"We should still try upstairs," said Christiana.
"What if Big Eddy's chameleon buddy is still up there?" asked Wendell.
Danny grinned. "Then he's probably scared out of his mind."
They picked up their bags of candy and Christiana's costume head. Danny didn't even begrudge the fact that the rat had gone through his bag and made off with most of the Tootsie Rolls.
The only brief scare came when Wendell bent over to get his candy and stood up into a huge cobweb.
"It's just a cobweb," said Danny wearily, pulling the strands off his friend. "It's not like it's the ghost."
"And now we know it wasn't a ghost at all," said Christiana, "it was Big Eddy."
"I'm still not entirely sure of that," said Wendell, scraping off the last of the cobwebs. "I mean, Big Eddy couldn't have done anything about that painting."
"We probably just weren't seeing it clearly because of the dark," said Christiana. "Maybe it was one of those optical illusions that look different when you look at them from different angles. Anyway, haven't you heard of Occam's Razor?"
"Can you kill ghosts with it?" asked Danny.
"Well...I suppose _metaphorically_...Well, anyway, Occam's Razor is this principle that the fewer assumptions you have to make, the more likely you are to be right."
Christiana rubbed her snout. "To assume that it's ghosts, we have to assume that ghosts exist—despite centuries of failed attempts to prove their existence—that there's one here, that it's playing tricks on us for some unknown reason..."
Danny and Wendell waited.
"But if it's Big Eddy, all we have to assume is that Big Eddy was here and he's a jerk."
"I believe _that's_ been proven conclusively," muttered Wendell.
"So there." Christiana dusted her hands. "No ghosts needed."
Danny rolled his eyes. "What is your problem with people believing in ghosts, anyway? I mean, wouldn't the world be _cooler_ with ghosts?"
"The world would be cooler if there were no bullies and it rained candy, but that doesn't mean it's gonna happen."
"Sheesh," said Danny. "Lighten up already!"
Christiana sighed. "Look, it's just...sloppy thinking, okay? Like a hundred years ago, there were people called Spiritualists who claimed they could talk to ghosts, right? So they'd take money from these poor people who'd lost their kids or their husbands or whatever and claim they were talking to the ghosts of their dead kids. And the Spiritualists were all frauds, but they'd just bleed those poor people dry, and leave them with no money and their family still dead, and it was all just sad and stupid."
It seemed to Danny that the fact that the Spirit-whatsit people had been jerks who lied about talking to ghosts didn't necessarily mean ghosts didn't exist. He'd once tried to convince his father that the tracks left in the lawn by some careless handling of a bowling ball were the work of a rogue elephant, and the fact that it hadn't been a real elephant did not mean that no elephants anywhere were real.*
Christiana did not seem entirely reasonable on the issue however, and anyway, standing around arguing about long-dead con artists was not getting them out of the house any faster.
"It'd be awfully hard. And sticky," said Wendell suddenly.
They both looked at him.
"If it rained candy," said Wendell. "It'd be worse than hail, because it wouldn't melt. You'd get all those dents in your car, and then there'd be candy stuck to everything, and you couldn't even eat it because it would have been all over the ground."
"And have you ever gotten hit with hail? It hurts! It's like a little chip of glass! Imagine if you got whacked with a jawbreaker or something!"
Christiana looked at her bag of candy as if she was thinking of whacking the iguana with a jawbreaker _right now_.
"We'd get reinforced umbrellas," said Danny. "And instead of rain gutters, we'd have candy gutters."
Danny was mulling over this mental image when they reached the kitchen and Christiana let out a hiss.
Wendell kept going for a couple of seconds—something about pollution causing acid candy rain—then noticed what the other two were looking at. His voice choked off with a squeak.
The hallway was still empty and clown-free, but the wallpaper was suddenly oozing something that looked like blood.
Christiana stood there with her hands on her hips, glaring at it.
Wendell didn't really know what to say. It hadn't been an attractive wallpaper to begin with, but the ooze didn't help. On the other hand, it wasn't quite as scary as it could have been. It was so over-the-top, midnight-movie-on-the-oldies-station cheesy that it seemed more like a Halloween decoration than an actual horror.
He was careful not to touch it, though. Might be a biohazard. The house couldn't possibly have had its shots.
Danny, being Danny, put out a finger and poked it.
Christiana wiggled her cilia at him.
"Eggs or blood...I don't really see how Big Eddy could have done that," said Wendell.
"Let's just keep moving," Christiana said, who clearly needed a minute to think of a reasonable explanation.
Bags of candy in hand, the three of them crept toward the kitchen, but things didn't look any better in there. An eerie green light oozed from one of the corners of the ceiling. It made a thin bright line, ran down the wall to the kitchen counter, then skittered over it and down onto the floor. The light traced the edge of a floorboard, zigzagging across the floor until it reached the stairs.
"What...is... _that_?" squeaked Wendell.
"Maybe it's somebody with a laser pointer?" whispered Christiana.
It didn't look like any laser pointer Danny had ever seen. The light wasn't at all jittery or flickery. Instead it oozed like glowing green honey up the stairs, flowing over each step, and vanished finally into the second floor.
" _That_ wasn't normal," said Danny. "Cool, but not normal."
"I knew there were ghosts!" hissed Wendell. "I knew it, I knew it!"
"It wasn't a ghost!" snarled Christiana. "It was Big Eddy! We saw him!"
The last word wasn't quite out of her mouth before the now-familiar voice whispered, _"I'm hungry..."_ It seemed to be coming from behind the wallpaper. Wendell let out a yelp and tried to hide behind his pillowcase.
"You think _this_ is Big Eddy too?" Danny didn't bother to hide his disbelief. Hammering on walls was one thing. Big Eddy was good at hammering on things—walls, nails, smaller kids who wouldn't hand over their lunch money...
Setting up eggs and sophisticated light shows was something else entirely. Danny wasn't sure that Big Eddy knew how to use a light switch, never mind whatever that green light had been.
She stomped toward the stairs.
She was halfway across the kitchen, and Danny and Wendell were giving each other should-we-follow-her/what-are-you-crazy looks, when a sound came from down the hallway behind them.
It was a very ordinary sound. Under normal circumstances, it might even have been a funny sound.
It was the sound of a toilet flushing.
Christiana stopped as if she'd run into a brick wall.
All three of them turned.
Footsteps creaked down the hallway and halted in the entryway to the kitchen.
Something stood in the doorway.
Something glowing.
Something _grinning_.
It was the clown from the painting.
The flashlight wasn't on it, but it didn't matter. The clown shone in the dark as if it were made of fireflies.
Wendell grabbed Danny's shoulder so hard it hurt. Danny didn't blame him. His stomach felt like someone had wrapped an invisible hand around it and squeezed.
The clown looked at each of them in turn, its painted eyes settling finally on Christiana. Its mouth yawned open. It had a great many teeth, and they were very long and sharp.
Then it giggled.
It was a high, humorless giggle, and it stayed in the air a lot longer than it should have, like a crow cawing.
Christiana tried. Danny gave her credit. She really, really tried. She actually stood her ground, even through that awful giggling, even when the clown took a step forward. Danny wasn't sure if he'd be able to stand there through that, and Wendell would have been in the next county and picking up speed.
"Wanna see a trick?" said the clown, and giggled again.
Christiana didn't speak. Wendell made a noise to indicate that he very much did not want to see a trick.
The clown reached up, popped its red nose off the end of its snout, and held it in the air. "Nothing up my sleeve..." it said.
Then it casually popped out both of its eyeballs—Wendell yelped—and began juggling them.
It was too much for Christiana. She let out a yell of disgust and bolted around the kitchen table— _no, don't run,_ Danny thought, _they can't chase you unless you run,_ which made no sense yet nevertheless struck him as absolutely true—and dove behind Danny and Wendell.
The tightness in Danny's chest made it hard to breathe, but he tried to suck in air anyway. If the clown got any closer, he was going to breathe fire. He might burn the house down, and the clown might be a ghost, and ghosts probably didn't burn, but he had to do _something_.
Meanwhile, a tiny little voice in the back of his head was going _How is he doing that? If you're juggling your own eyeballs, how can you see what you're doing?_
He gulped air. His sinuses felt smoky, but he couldn't seem to get a deep breath.
The clown, still juggling, walked to the edge of the table, across from them. "No?" it said. It popped its eyes and nose back in, none the worse for wear.
"Bet you can't do that," said the clown. It leaned forward.
As fire-breathing went, it wasn't worth much. There was a lot of smoke, and the flame was better suited to a birthday candle. The clown snickered.
But there _was_ a clown.
That meant there was a ghost.
That meant that his vision had been real.
Danny shoved a hand into his pillowcase, found the battered edge of the stuffed animal, and yanked it out.
You used crosses on vampires and silver on were-wolves...maybe you used stuffed animals on clowns.
The clown recoiled. "That's _mine,_ " it said in a thin, childlike voice. But instead of coming toward them, it backed away, up the stairs. The firefly light coming off it cast flickering bars of shadow across the kitchen, and then it too was gone.
Slowly, painfully, Danny's stomach unclenched.
He turned. Wendell was wide-eyed, but seemed okay. Maybe if you believed that there were ghosts, it came as less of a shock to find out you were right.
"You saw it too?" asked the iguana.
Danny couldn't think of anything much to say to that—they hadn't heard running water after the toilet flush, so Wendell was probably right. He looked at Christiana.
The crested lizard was in a bad way. She was on her knees, breathing in short, panicky gasps, and she had gripped the costume head so tightly that she'd bent some of the wiggly bits in half.
"Christiana? Christiana? It's gone!"
Danny grabbed her shoulder and shook it. "Christiana?"
She wrinkled her snout at him. "What's that _smell_?"
"Danny breathed fire," said Wendell.
It was the best possible thing that anyone could have said. Christiana's eyes focused on him with sudden intensity, and she said, "Nuh- _uh!_ "
"Uh- _huh!_ " said Wendell. "He totally did!"
She sat up. "Do it again," she said.
"It doesn't work like that," said Danny. "I only did it because that clown was coming at us. I can't just do it on command."
"Sure," said Christiana, in a tone that indicated she didn't believe a word of it.
"He did!" said Wendell.
"I've been thinking," said Danny. "I'm not sure it's murderous."
"That clown," said Wendell firmly, "was up to no good."
"Clowns are _never_ up to anything good," muttered Christiana.
Danny shrugged. "Look, I know it was scary, but it didn't hurt us. And it definitely recognized the stuffed animal."
The trio considered.
"Well...not all ghosts are bad," said Wendell slowly. "A lot of them—from what I've read—just want to be acknowledged, or laid to rest, or something like that. Maybe it just wants our attention."
"It's got _my_ attention," said Danny. "And everything we've seen—the footsteps, the clown, the light—they've all led upstairs." He pointed upward.
The kids stood in the dusty kitchen, flashlight trained on the stairs. Nothing moved. Very distantly, through the boarded windows, they could hear the sound of crickets.
"Are you suggesting we go up there?" asked Wendell, gulping.
Danny nodded.
"Okay," said Christiana, "okay, let's say for the sake of argument—not that I believe it—that this _is_ a ghost. Why would it want us upstairs?"
"But anyway, maybe its remains are up there, and we're supposed to lay them to rest."
"Do you know how to lay bones to rest?" asked Christiana skeptically.
Wendell frowned. "I've read a couple of things. I could probably wing it."
Danny slapped the iguana on the back. "My buddy the exorcist!"
"A skeleton is one thing," said Christiana. "It's bound to be less squishy than the sheep brain, anyway. But what if the...fine, all right, the _ghost_...is hostile, and it's luring us into a trap?"
Danny rubbed the back of his neck. "Well, I don't know. But I keep thinking that if it could hurt us, it probably would have already. I mean, the clown was _right there,_ and none of us ended up murdered."
"Yeah, but it wasn't nice at all," said Wendell. "It might not be trying to hurt us, but it sure is trying to scare us."
"Well, what kind of ghost is it?" asked Christiana. "Hypothetically."
Wendell threw his hands in the air. "What am I, the expert? I don't know!"
"C'mon, Wendell, you read like every ghost book in the library last week. You've got to know all about ghosts." Danny folded his arms and leaned against the table. "So what is it?"
Wendell took his glasses off and cleaned them on the hem of his shirt. "Well...I can tell you what it's _not_."
"That's a start," said Christiana.
The iguana began ticking off ghosts on his fingers. "It's probably not a poltergeist. I think they just throw things. There's a Babylonian ghost called an enkimmu that shows up if it isn't buried, but they live underground, and there was nothing wrong in the basement. There's a ghost from Thailand that looks like a skull that flies around with all its guts flapping around behind it—"
_"Cool!"_ said Danny.
"—but we're not in Thailand and anyway, the clown didn't have its guts flapping around or anything."
Christiana, sensing a long list coming, dug into her bag of candy, pulled out a roll of Smarties, and began munching.
Wendell stopped in mid-recitation and stared at her.
"No, but the ghost might," said Wendell slowly. "In a whole bunch of cultures, you give ghosts offerings of food. They have days where you leave out meals for them, or even candy. In fact, on All Hallow's Eve, back in the old days, you had to leave gifts of food for the wandering dead."
"All Hallow's Eve?" asked Danny.
"What they used to call Halloween," said Christiana, gazing thoughtfully at the candy.
"Are you saying we have to cook the ghost _dinner_?"
Wendell shrugged helplessly. "Maybe. It might be hungry. It might want to be buried. It might want to kill us all and wear our livers as little hats! I don't _know!_ "
Danny exhaled. There was still a little bit of smoke on his breath, but Christiana was contemplating her candy and didn't seem to notice.
"Well," he said. "I don't know what kind of ghost it is, or what it wants, but I do know where to find it."
The trip up the stairs took longer than it should have, because nobody was willing to be the last one up the stairs—where any monsters waiting below could snatch them from behind—and nobody quite wanted to be first up the stairs, where they would be the first to meet the terrifying truth, which may or may not have been wearing somebody's liver as a hat.
Eventually, with much shoving and wiggling, they went up the stairs three abreast, although Danny noticed he was always the first one setting his foot on the next step.
They made it to the landing and nothing spectacularly horrible happened. Well, the wallpaper started oozing again, but by that point, nobody was really that worried by it.
Danny had managed to wrestle the flashlight back from Wendell, but had given him the stuffed animal instead. He had a feeling that it was going to be important.
The second flight of stairs was much shorter. The hallway at the top was moonlit, with dark doorways leading onto it.
At the very end of the hallway was a closed door.
Danny opened his mouth to say something—"Well, here we go," or "Look out, ghosts!" or maybe just "I hope this works!"—when the strange green light from before slid down the wall, wrapped once around the banister, and slipped down the hallway to the closed door. A thin light of green fire outlined the doorway, and then it faded.
"I guess that's where we're supposed to go," he said.
Wendell gulped again. Christiana looked grim.
Danny set his foot on the first step and began to climb.
He was only a single step from the top, with Wendell and Christiana behind him, when the closed door opened, slowly swinging with that thin creak of hinges found in horror movies the world over.
Wendell stopped. Danny took another step, and so did Christiana, which left Wendell standing on the lower step alone. He squeaked and crowded so close against Danny's back that the dragon nearly dropped the flashlight.
There was a brief moment of jostling on the stairs, and then Danny gritted his teeth and stepped up into the hallway.
A shivering wind seemed to swirl around him, and then every door except the one at the end of the hall slammed shut, one after the other— _WHAM! WHAM! WHAM! WHAM!_ Wendell let out a shriek.
With the echoes of the slamming doors ringing in his ears, it took a moment for Danny to recognize that Christiana had come up beside him. A second later, Wendell was on his other side. The iguana looked terrified, and he was holding his bag of candy up like a shield, but he was there.
The hallway was about twenty-five feet long. The door at the end was cracked open just enough to see darkness through it.
It should have taken under five seconds to walk down the hallway and push the door open the rest of the way.
Danny couldn't swear to it, but he was pretty sure it took more like five years.
The first few steps weren't so bad. They made it as far as the first set of closed doors before the noises began.
_"Hungry...hungry...hungry..."_ chanted the voice.
Something was scrabbling at the bottom of the door on their left, pawing at it the way a cat paws when it wants to get out. Danny didn't look at it. He kept his eyes locked forward and kept walking.
Wendell, being Wendell, did look. "It's got red claws..." he moaned.
"Keep walking," said Danny. "Don't look at it."
There was a soft _snick_ as the door on their right opened behind them.
Danny didn't look at it. He did glance over at Christiana, and saw her leaning forward like a lizard in a strong wind. "I don't believe in ghosts," she hissed under her breath. "I don't believe in ghosts."
This did not strike Danny as a terribly productive statement, but if it got her down the hallway, he wasn't going to argue.
The wallpaper was now oozing so furiously that they should have been ankle-deep in egg yolk, but somehow they weren't.
They were nearly at the second pair of doors when one popped open, and the clown stuck its head out.
The clown grinned. It was inches away. It could reach out and grab him right now if it wanted to.
Just behind his left shoulder, he could hear Christiana saying, "It won't get me twice. It isn't real, it isn't real..."
He took another step forward. For some reason, all he could think of was Wendell saying: "It didn't wash its hands!"
"Come...closer..." the clown whispered. And then it disappeared back through the door.
Danny looked at Christiana. Christiana looked at Danny. Together they reached out and pushed open the door.
The ghost sat on the bed inside.
It was a very small ghost. It looked younger than any of them. It was hard to tell what species it had been—some kind of lizard, but it was mostly transparent and had only the hint of scales. An unboarded window behind it shone with moonlight, which fell through the ghost and across the dusty bed without casting a shadow on the floor.
"Trick or treat!" it cried, bouncing on the bed. "Did I scare you? Did I?"
Wendell wiped a hand over his face and made a noise that was either relief or disbelief or something in between.
"Uh...yeah," said Danny. "Definitely." He glanced at the other two. Christiana looked murderous. He elbowed Wendell instead.
"I _haunt_ here," said the ghost, sounding somewhat snooty (almost _exactly_ like Danny's seven-year-old cousin, now that he thought about it). "This is where I died."
"Um," said Danny. What did you say to something like that? Stinks to be you? "Uh...sorry for your loss?"
"I missed Halloween," said the ghost. "I was sick and I wanted to go out trick-or-treating, and Mom said I could go next year. But instead I got sicker and I _died_ and didn't get to go at _all_."
_Just like my vision!_ Danny thought.
The ghost sounded peeved. It occurred to Danny that while trick-or-treating would be something he'd miss if he was dead, there were a lot of other things he'd miss more. His mom, say. His dad. Hanging around with Wendell. Comic books. Bacon. You know, the really _important_ stuff.
On the other hand, he'd never been dead, so maybe things changed when you were a ghost. Still, Danny was getting a feeling that he wouldn't have liked the ghost very much when it was alive. (Then again, he didn't like his cousin much either.)
"So I come back here for Halloween every year," said the ghost. "Sometimes people come in, and I get to scare them, but usually they just run away." It bounced on the bed.
The threesome looked at one another.
"I think this is yours," said Danny, taking the stuffed animal from Wendell. "I found it in the basement, and—"
_"Stuffy!"_ shrieked the ghost, and pounced.
It felt like cobwebs, maybe, or a puff of cool air when it touched him. Then the ghost had the stuffed animal, which had assumed the same odd transparency, and Danny was holding empty air.
"Stuffy?" asked Wendell, of no one in particular.
The ghost hugged the mangy stuffed animal tightly. "Stuffy! I _missed_ you!"
Danny exhaled. "So..." he said. "You've got your stuffed animal back, so I suppose you can rest now, and we can get going..."
The ghost looked up from Stuffy, eyes narrowing. "No. Not yet. I said trick or treat!"
There was an awkward pause. The wallpaper split open and oozed a whole barnyard worth of egg yolk down over the baseboards.
"Why eggs, anyway?" asked Wendell.
"I hate eggs," said the ghost. " _And_ mashed potatoes."
"We didn't see any mashed potatoes..."
"You didn't go in the closet."
Danny was getting frustrated. He'd been sure that the stuffed animal was the key, and if they gave it back, the ghost would be at rest, or at least let them go. Now he wasn't sure what to do.
"'Scuse us a minute," said Wendell brightly, and pulled Danny backward into a brief huddle. "I think I've got it. Remember what I said about food offerings?" asked the iguana. "And that voice—it kept saying it was hungry, right?"
"We need to cook it a meatloaf?" asked Danny.
"No, dummy! It wants to go trick-or-treating," said Wendell. "Well, what's the point of trick-or-treating?"
"Candy," said Danny immediately—and then bit his lip. "Oh, no! You mean it's hungry for our _candy_?"
Wendell nodded grimly.
Danny winced. Ghosts were ghosts...but candy was candy!
"I think...between the stuffed animal and the offering...we might be able to buy it off." Wendell shoved his glasses up his nose. "I'm not sure if that'll lay it to rest, but maybe it'll at least let us go."
Danny grimaced. Figuring out the secret of a haunted house had been cool—a little spooky, sure, but a good Halloween kind of spooky. But giving up your Halloween candy...that wasn't cool at all.
Christiana, who had been silent through this whole exchange, stepped forward. "I've got a couple of questions..." she said.
The ghost frowned at her, but she plunged ahead anyway. "So you're a ghost. What happened, exactly, after you died?"
"I was _dead,_ " said the ghost. "Duh."
"Right, right. But your existence postulates the existence of some form of afterlife, so what does that entail? Clearly you can manifest visually and to a limited extent physically, but is your range constrained? Do you have a sense of the passage of time? Are there other ghosts with which you can communicate?"
Wendell was nodding, so he apparently understood this display of vocabulary, but Danny had started to flounder somewhere around "postulates the existence."
He wasn't the only one.
The ghost narrowed its eyes. A door slammed somewhere in the hallway.
Danny leaned over to Christiana and hissed, "Maybe this isn't the best idea..."
"But if it's _really_ a ghost—the loss to science—!"
Another door slammed. Somewhere, the clown giggled.
Danny decided that he really, really wouldn't have liked the ghost when it was alive. It was probably one of those snotty little kids that threw tantrums on the floor of the grocery store because their mom wouldn't buy them a gumball.
He figured the clown was probably some kind of illusion, like a puppet the ghost was controlling. Maybe the ghost couldn't hurt them...but maybe it could.
"If we give it the candy, maybe it'll unlock the door!" hissed Wendell.
Danny had to think about this for a minute. It wasn't like they'd tried to break the windows...sure, they'd probably slice their arms off on the way out, but they'd have lots of candy for the hospital...
Christiana sighed. Her shoulders slumped. She opened up her pillowcase and dumped about half of it out on the floor.
"Offerings of food, huh?" she said to Wendell.
It was a lot of candy. Christiana gazed at it with longing and no skepticism whatsoever.
"You can keep the Milk Duds," said the ghost. "I hate Milk Duds."
"Everybody hates Milk Duds," said Christiana gloomily, but she swept a half-dozen little yellow boxes back into her bag anyway. Her pillowcase seemed pitifully light. It was the sort of haul you got when you were a little tiny kid going around with your mom to ring the doorbell for you.
"So that's plenty, right?" said Danny. "That's a lot of candy." He tried to hide his bag behind his back to make it look smaller.
The ghost glared at him. "I'm _really_ hungry," it said. The clown giggled again, sounding as if it was right behind the door.
Christiana punched him in the arm.
Faced with this example, Danny really couldn't refuse. Christiana had extremely pointy knuckles. He upended half his pillowcase, feeling a wrench. So much sugar, lost. So much hard-earned chocolate. There were a couple full-sized candy bars in there too, the really good ones.
"Oh well," he said, "I guess it's better than Big Eddy getting it."
Wendell looked at the resulting mound of candy. He looked at his pillowcase, then up at the expectant ghost.
"This is the only really good candy I get all year," he said sadly. "My mom buys sugar-free stuff the rest of the time. And carob. I don't care what she says, it doesn't taste like chocolate at all."
Danny and Christiana both put their hands on his shoulders.
"Good-bye, chocolate," said the iguana, drawing a candy bar slowly from the bag. "Good-bye, licorice. Good-bye, lollipops. Good-bye, thing that tastes sort of vaguely like chocolate but with that weird waxy shell—"
The ghost squinted at Wendell's pie plate. "What are you supposed to be, anyway?"
"A hydrogen...you know, never mind. A pie salesman."
"That's a stupid costume," said the ghost.
"Yes," said the iguana, "yes, it is." He gazed into his pillowcase. "I guess it's only _pity_ candy." He poured a generous quantity onto the mound.
"Hey," said Danny as the ghost burrowed into the candy, "can you open the door and let us out of here now?"
"Huh? Oh, sure..." It waved a hand. There was a distant creaking noise from downstairs.
Wendell did not exactly bolt, but he was out the door and headed down the hall calling "Thanks-nicetomeetyougottagobye!" with a speed better suited to a whip snake than an iguana. Christiana wasn't far behind him.
Danny paused in the doorway and turned back to say something to the ghost—he hadn't liked it, but he felt a little bad that it had died and not gotten a chance to go trick-or-treating—
The moonlight from the unboarded window fell through the place that it had been sitting, and neither ghost nor candy was anywhere to be seen.
"Well," said Danny as they walked down the driveway. The world seemed very bright in the glow of the streetlights. "That was an adventure."
"I'm never going anywhere near that house again," said Wendell, sadly examining the remains of his candy.
"Wuss," muttered Christiana.
"Heck, yeah!" Christiana turned around and walked backward, gazing at the house. "I'm gonna get my dad's video camera and come back. If that really is a ghost, it needs to be documented! We need proof of what we saw!"
"It might not be there," said Danny. "We gave it candy AND its stuffed animal. Wouldn't that lay it to rest?"
Wendell shrugged. "It depends on the ghost. In some cultures, they come back for food offerings every year. There's a big festival in Mexico every year called the Day of the Dead just to feed the wandering ghosts."
Danny frowned. "If it'll be back next year, maybe we should warn people."
"Or feed Big Eddy to it."
They all snickered.
They were halfway to the sidewalk when Wendell tripped over something and fell down with an "OOF!"
"You okay?" Danny helped him up.
Surprisingly, Wendell was grinning. "I'm great! Look what I found!"
"Big Eddy must have dropped his candy when he was running off," said Christiana. "Awesome!"
They quickly split the loot three ways. It didn't quite make up for the candy lost to the ghost, but it certainly helped.
Wendell's mood improved dramatically. "So...Christiana...if you believe that's a ghost, I guess that means you believe Danny's a dragon too?"
Danny sighed.
"Don't be ridiculous," said Christiana. "Just because ghosts might be real, it doesn't mean _everything_ is real. Ghosts have nothing to do with UFOs or those weirdos who think the Mayans made spaceships...or fire-breathing dragons. Sorry, Danny."
Danny, who hadn't expected anything else and kind of wished Wendell would drop the subject completely, stared at the sidewalk. There weren't any other kids out trick-or-treating. He wasn't sure how long they'd been in the house, but he figured it was at least an hour.
His dad was gonna kill him. Then ground him. Then kill him again.
As if in answer to his fears, the family car pulled up to the curb with a screech of brakes. His father leaped out of the car, looking relieved, terrified, and furious in equal measure.
"Thank god you're all right! Young man, you are in _so much trouble..._ "
Danny hung his head and waited. Ghosts were one thing, but his dad on a rampage was _scary_.
Christiana looked at him, looked at his father, made a faintly exasperated noise under her breath, and stepped in front of Danny and Wendell.
Danny blinked.
"We were following these other kids," said Christiana, managing just the faintest trace of a sob, "and we thought you were back with their parents, but then they went inside and it turned out it wasn't you after all, and by then we were a whole bunch of streets away and we didn't know how to get back—"
Wendell gazed at her in awe.
Danny tried to look quietly heroic. Wendell had that expression that meant he was fighting a snicker.
"Where were you?" asked Mr. Dragonbreath plaintively. "I drove up and down for half an hour looking..."
"Uh—" Christiana glanced at the others.
"There were a lot of cul-de-sacs," volunteered Wendell. "We kept thinking we'd found the street, and then it would end in another cul-de-sac. We probably weren't that far away, but it feels like we've been walking for _hours_."
"Well..." said Danny's father. He exhaled. "I suppose...no harm done. If you're more careful next time, we don't need to say anything more about it."
"So, kids..." Mr. Dragonbreath put an arm across the back of the seat. "I've been hearing radio reports of an escaped mental patient, with hooks for hands—"
"Oh, come ON, Dad!" Danny rolled his eyes. "We tell that one at summer camp!"
"Oh." His father considered. "So there was this ghostly hitchhiker—"
"Daaaaad!"
* Well, quiet except for Danny. The regular parade of ambulances, fire trucks, and emergency plumbers livened up the street substantially, and Danny could never figure out why the neighbors weren't more grateful.
* Actually, this was not entirely true. There was a place way out in the country that put on a haunted house and hayride that had guys in ski masks carrying chain saws, and even though you _knew_ they were actors, the sound of the chain saw starting up in the dark was pretty terrifying. Danny had made his parents go through it three times.
* His father had grounded Danny immediately—not, he said, for the bowling ball marks, but for insulting his intelligence with the elephant bit.
| {
"redpajama_set_name": "RedPajamaBook"
} | 7,354 |
\section*{Package summary}
\begin{description}
\item[Title of the Package] HPL
\item[Version] 1.0
\item[Package obtained from] {\verb+http://www-theorie.physik.unizh.ch/~maitreda/HPL/+}
\item[E-Mail] maitreda@physik.unizh.ch
\item[License] none
\item[Computers] Computers running Mathematica under Windows or Linux
\item[Operating system] Linux or Windows
\item[Program language] Mathematica
\item[Memory required to execute] depending on the libraries loaded
\item[Other Package needed] none
\item[External file required] all needed files are provided
\item[Keywords] Harmonic polylogarithm, multiple zeta values
\end{description}
\clearpage
\section{Introduction}
The harmonic polylogarithms (HPL's)\cite{Remiddi} are a generalization of the usual polylogarithms \cite{Lewin} and of the Nielsen polylogarithms \cite{Nielsen}. The HPL's appear in many calculations in high energy physics. They are found in three-loop deep inelastic splitting and coefficient functions \cite{Vermaseren:2005qc,Moch:2004xu,Vogt:2004mw,Moch:2004pa}, in two-loop massive vertex form factors \cite{Bonciani:2003te,Bonciani:2003ai,Bonciani:2003hc,Bernreuther:2004ih,Bernreuther:2004th,Bernreuther:2005rw,Mastrolia:2003yz,Degrassi:2005mc}, in two-loop Bhabha scattering \cite{Bonciani:2003cj,Bonciani:2004gi,Bonciani:2004qt,Czakon:2004wm,Bonciani:2005im,Penin:2005kf} and in multi-loop three-point and four-point functions \cite{Bern:2005iz,Birthwright:2004kk,Aglietti:2004ki,Heinrich:2004iq,Aglietti:2004nj,Aglietti:2004tq,Smirnov:2003vi,Aglietti:2003yc,Gehrmann:2001ck,Gehrmann:2000zt,Gehrmann:2002zr}. The HPL's also show up the expansion of hypergeometric functions around their parameters \cite{Weinzierl:2002hv,weinzierl1,Weinzierl:2004bn,Huber:2005yg} and in more formal developments \cite{Blumlein:2005jg}.
The HPL's have already been implemented for the algebraic manipulation language FORM \cite{FORM}. This implementation is described in \cite{Vermaseren_harmpol} and can be found at \cite{harmpol_homepage}. A FORTRAN code for the numerical evaluation of the HPL's is also available \cite{Gehrmann:2001pz}. A numerical implementation is also available for GiNaC \cite{Bauer:2000cp}, described in \cite{Vollinga:2004sn}. The aim of this work is to provide an implementation of the HPL's, including numerical evaluation. A {\tt Mathematica} implementation of the HPL's will in particular allow the usage of these functions in connection with many of the multi-purpose features of {\tt Mathematica}. Concerning speed, it is not expected that our {\tt Mathematica} implementation outperforms the FORM implementation\cite{Vermaseren_harmpol}.
The paper is articulated as follows. In section \ref{definition} we review the definition of the HPL's. Their analytic properties are described in section \ref{derivative} (derivative), \ref{algebra} (product algebra) and \ref{divergences} (singular behavior). In section \ref{basis} we treat the different identities relating different HPL's. Section \ref{series} presents the series expansion of the HPL's. The HPL's of related arguments are treated in section \ref{related}. Section \ref{analytic} treats the analytic continuation of the HPL's. Values of HPL of argument 1 are related to the multiple zeta values (MZV), which we discuss in section \ref{MZV}. In section \ref{implementation} we describe our implementation and provide examples of its usage. Finally we conclude in section \ref{conclusion}. Some tables and calculational details have been relegated to the appendix. In appendix \ref{MZVtable} we list some identities for the MZV. In appendix \ref{zetaproof} we prove the relation between the HPL's of unity argument and the MZV's. Finally in appendix \ref{identities} we list the representation of some HPL's in terms of more common functions.
\section{Definition}\label{definition}
The harmonic polylogarithms (HPL) $H\left(a_1,\dots,a_k;x\right)$ are functions of one variable $x$ labeled by a vector $a=(a_1,\dots,a_k)$. The dimension $k$ of the vector $a$ is called the weight of the HPL. We define the functions
\begin{eqnarray}
f_1(x)&=&\frac{1}{1-x}\nonumber\\
f_0(x)&=&\frac{1}{x}\nonumber\\
f_{-1}(x)&=&\frac{1}{1+x}
\end{eqnarray}
The HPL's are defined recursively through integration of these functions. For weight one we have
\begin{eqnarray}
H(1;x)&=&\int\limits_0^xf_1(t)\d t=\int\limits_0^x\frac{1}{1-t}\d t=-\log(1-x)\nonumber\\
H(0;x)&=&\log(x)\nonumber\\
H(-1;x)&=&\int\limits_0^xf_{-1}(t)\d t=\int\limits_0^x\frac{1}{1+t}\d t=\log(1+x),
\end{eqnarray}
and for higher weights
\begin{eqnarray}
H(^n0;x)&=&\frac{1}{n!}\log^n x\nonumber\\
H(a,a_{1,\dots,k};x)&=&\int\limits_0^x f_a(t)H(a_{1,\dots,k};t)\d t\, ,
\end{eqnarray}
where we used the notations
\[^n i=\underbrace{i,\dots,i}_n,\quad \textnormal{and }\quad a_{1,\dots,k}=a_1,\dots,a_k .\]
A useful notation introduced in\cite{Remiddi} for harmonic polylogs with non-zero right-most index is given by dropping the zeros in the vector $a$, adding 1 to the absolute value of the next right non-zero index for each dropped 0. This gives for example $(3,-2)$ for $(0,0,1,0,-1)$. We can extend this notation to all index vectors by allowing zeros to take place in the extreme right of the new index vector. This gives for example $(3,-2,0,0)$ for $(0,0,1,0,-1,0,0)$. We will enclose index vectors in this notation in curly brackets and refer to it as the "m"-notation, as opposed to the "a"-notation. Some formulae or transformations are easier expressed in the one or the other notation, therefore we keep both notations in parallel.
\section{Derivatives}\label{derivative}
The formula for the derivative of the HPL's follows directly from their definition
\begin{equation}
\frac{\d}{\d x}H(a,a_{1,\dots,k};x)=f_a(x)H(a_{1,\dots,k};x).
\end{equation}
In the "m"-notation, the derivation reads for $n$ positive
\begin{eqnarray}
\frac{\d}{\d x}H\left(\{n,m_{1,\dots,k}\};x\right)&=&\frac{1}{x}H\left(\{n-1,m_{1,\dots,k}\};x\right),\qquad n>1\nonumber\\
\frac{\d}{\d x}H\left(\{-n,m_{1,\dots,k}\};x\right)&=&\frac{1}{x}H\left(\{-(n-1),m_{1,\dots,k}\};x\right),\qquad n>1\nonumber\\
\frac{\d}{\d x}H\left(\{1,m_{1,\dots,k}\};x\right)&=&\frac{1}{1-x}H\left(\{m_{1,\dots,k}\};x\right)\nonumber\\
\frac{\d}{\d x}H\left(\{-1,m_{1,\dots,k}\};x\right)&=&\frac{1}{1+x}H\left(\{m_{1,\dots,k}\};x\right)\nonumber\\
\frac{\d}{\d x}H\left(\{^n0\};x\right)&=&\frac{1}{x}H\left(\{^{n-1}0\};x\right)
\end{eqnarray}
\section{Product algebra}\label{algebra}
The product of two HPL of weights $w_1$ and $w_2$ can be expressed as a linear combination of HPL's of weight $w=w_1+w_2$. The formula in the "a"-notation for two HPL's with index vectors $\mathbf{p}$ and $\mathbf{q}$ is given by
\[H(p_1,\dots,p_{w_1};x)H(q_1,\dots,q_{w_2};x)=H(\mathbf{p};x)H(\mathbf{q};x)=\sum\limits_{r\in \mathbf{p}\uplus\mathbf{q}}H(\mathbf{r};x)\]
where $\mathbf{p}\uplus\mathbf{q}$ is the set of all arrangements of the elements of $\mathbf{p}$ and $\mathbf{q}$ such that the internal order of the elements of $\mathbf{p}$ and $\mathbf{q}$ is kept. For example one has for $\mathbf{p}=(a,b)$ and $\mathbf{q}=(y,z)$
\begin{eqnarray}
H(a,b;x)H(y,z;x)&=&H(a,b,y,z;x)+H(a,y,b,z;x)\nonumber\\
&&+H(a,y,z,b;x)+H(y,a,b,z;x)\nonumber\\
&&+H(y,a,z,b;x)+H(y,z,a,b;x)
\end{eqnarray}
\section{Extraction of the singular behavior}\label{divergences}
The HPL's can have divergences in $x=0$ and $x=1$. The divergent part can be extracted with the help of the above product rules. Divergences in 0 appear as logarithmic divergence with behavior $\simeq \log^n(x)$ if there are $n$ 0's at the right end of the index vector. In this case one can make the divergent behavior explicit by writing
\begin{eqnarray}
\lefteqn{H(a_{1,\dots,k};x)H(0;x)=}&&\nonumber\\
&&H(a_{1,\dots,k},0;x)+H(a_{1,\dots,k-1},0,a_k;x)+\dots+H(0,a_{1,\dots,k};x).
\end{eqnarray}
which one can solve for $H(a_{1,\dots,k},0;x)$. Recalling that $H(0;x)=\log(x)$, the divergent $\log$ stands now explicitely. If there are more zeros in the right end of the index vector, one has to use this method recursively, until all end zeros have been exchanged against $\log$'s.
Similarly, when there are $n$ 1's at the very left of the index vector there appears a logarithmic divergence $\simeq\log^n(1-x)$. This can be extracted in the same way as above. One uses the product rule
\begin{eqnarray}
\lefteqn{H(1;x)H(a_{1,\dots,k};x)=}&&\nonumber\\
&&H(1,a_{1,\dots,k};x)+H(a_1,1,a_{2,\dots,k};x)+\dots+H(a_{1,\dots,k},1;x).
\end{eqnarray}
Here we solve for $H(1,a_{1,\dots,k};x)$. Again we can apply the method recursively to make the singular behavior by $x=1$ explicit.
This method allows to express all HPL with left $1$'s and/or right $0$'s in terms of products of $H(1;x)$, $H(0;x)$ and HPL's without left $1$'s and right $0$'s.
Using the same method, it is also possible to extract simultaneously the $H(-1;x)$ and $H(0;x)$ coming from the $-1$ on the left and $0$ on the right of the index vector. The factors $H(-1;x)$ will correspond to divergences in $-1$ which are relevant for the analytical continuation to the interval $(-\infty,-1)$, see section \ref{analytic}. One cannot however extract simultaneously the $H(-1;x)$ and $H(1;x)$ by
\[H(1,-1;x)=H(-1;x)H(1;x)-H(-1,1;x).\]
We will make extensive use of these relations in the next sections.
\section{Minimal set}\label{basis}
The procedure for extracting the divergent parts of an HPL described in section \ref{divergences} allows to express many HPL's in terms of HPL's without divergences at 0 and 1 (so called "irreducible") and products of HPL's of smaller weight. The product rules described in section \ref{algebra} also provide relations between HPL's of a given weight and HPL's of lower weight. One can combine all these relations to get a minimal set of HPL's for a given weight from which one can construct all other HPL's of this given weight, up to products of HPL's of lower weight. Table \ref{basistable} shows the number of HPL's, irreducible HPL's and the dimension of the minimal set as a function of the weight.
\begin{table}[h]
\begin{center}
\begin{tabular}[h]{|c|ccc|}
\hline
\rule[-1.5ex]{0cm}{4ex}Weight&Full basis&Irreducible set&Minimal set\\\hline
1&3&3&3\\
2&9&4&3\\
3&27&12&8\\
4&81&36&18\\
5&243&108&48\\
6&729&324&116\\
7&2187&972&312\\
8&6561&2916&810\\\hline
\end{tabular}
\end{center}
\caption{Dimension of the different basis}
\label{basistable}
\end{table}\\
Only the number of elements in the minimal set is fixed, there is a freedom left for the choice of which elements are to be taken as independent. Our choice was first to exclude all HPL's whose divergent behavior can be extracted along the lines of section \ref{divergences} from the minimal set. For the remaining ones, we ordered the index vectors with the following procedure. One adds one to all indices of the index vector (in the "a"-notation), the result is to be interpreted as the expansion in basis 3 of a number. This number describs the index vector on a unique way. We used this numbering to sort the irreducible HPL's and choose to express the last as a function of the first.
\section{Series expansion}\label{series}
The series expansion of the HPL's can be defined recursively. Let us call $Z_i(a_{1,\dots,k})$ the coefficients of the expansion of $H\left(a_{1,\dots,k};x\right)$.
\begin{equation}
H(a_{1,\dots,k};x)=\sum\limits_{i=0}^\infty x^i Z_i(a_{1,\dots,k})
\end{equation}
We assume that the index vector has no trailing 0, as these lead to $\log(x)$ divergences. These divergences have to be made explicit by the procedure of section \ref{divergences}. Now we use the definition of the HPL's
\[H(a,a_{1,\dots,k};x)=\int_0^x \d x'f_a(\prim{x})H(a_{1,\dots,k};\prim{x})=\sum\limits_{i=0}^\infty Z_i(a_{1,\dots,k})\int_0^x \d x'f_a(\prim{x})\prim{x}^i.\]
For the three different possibilities $1,0,-1$ for $a$ we get
\begin{eqnarray}
\int_0^x \frac{\d \prim{x}}{1-\prim{x}}\prim{x}^i&=&\int_0^x\d \prim{x}\sum\limits_{j=i}^{\infty}x^j=\sum\limits_{j=i+1}^\infty\frac{x^j}{j}\nonumber\\
\int_0^x \frac{\d \prim{x}}{\prim{x}}\prim{x}^i&=&\frac{x^i}{i}\nonumber\\
\int_0^x \frac{\d \prim{x}}{1+\prim{x}}\prim{x}^i&=&\!\!\int_0^x\d \prim{x}(x)^{i}\left(\sum\limits_{j=1}^{\infty}(-x)^j\right)=(-1)^{i+1}\sum\limits_{j=i+1}^\infty\frac{(-x)^j}{j}.
\end{eqnarray}
The recursion relation for the $Z_i$'s is found by interchanging the order of the summations over $i$ and $j$
\begin{eqnarray}
Z_j(1,a_{1,\dots,k})&=&\frac{1}{j}\sum\limits_{i=2}^j Z_{i-1}(a_{1,\dots,k})\nonumber\\
Z_j(0,a_{1,\dots,k})&=&\frac{1}{j} Z_{j}(a_{1,\dots,k})\nonumber\\
Z_j(-1,a_{1,\dots,k})&=&\frac{(-1)^{j}}{j}\sum\limits_{i=2}^j(-1)^{i+1}Z_{i-1}(a_{1,\dots,k})
\end{eqnarray}
For the "m"-notation, we can use the same notation for the coefficient of the series expansion, as no confusion is possible.
\[H\left(\{m_{1,\dots,k}\};x\right)=\sum\limits_{j=0}^\infty x^i Z_j(m_{1,\dots,k})\]
Here again we assume that all $\log(x)$ divergences have been made explicit by the procedure of section \ref{divergences}, so that the vector $m_{1,\dots,k}$ has no trailing 0. The recursion relations for $n$ positive reads
\begin{eqnarray}\label{coeffseries}
Z_j(n,m_{1,\dots,k})&=&\frac{1}{j^n}\sum\limits_{i=2}^j Z_{i-1}(m_{1,\dots,k})\nonumber\\
Z_j(-n,m_{1,\dots,k})&=&\frac{(-1)^j}{j^n}\sum\limits_{i=2}^j(-1)^{i+1}Z_{i-1}(m_{1,\dots,k}).
\end{eqnarray}
The recursion is settled by the expansions of $H(\{n\};x)$ and $H(\{-n\};x)$
\begin{eqnarray}
H(\{n\};x)&=&Li_n(x)=\sum\limits_{i=1}^\infty\frac{x^i}{i^n}\nonumber\\
H(\{-n\};x)&=&-Li_{n}(-x)=\sum\limits_{i=1}^\infty\frac{-(-x)^i}{i^n},
\end{eqnarray}
which give
\[Z_i(\{n\})=\left\{\begin{array}{cc}\displaystyle\frac{\mathrm{sgn}(n)^{i+1}}{i^{|n|}},\qquad& i>0\\0,&i\le 0\; .\end{array}\right.\]
\section{HPL's of related arguments}\label{related}
In this section, we present the identities between HPL's of different (but related) arguments. All the identities are obtained through change of variable in the definition of the HPL's. The easiest case is the change of variables
\[x=-t.\]
This transformation is only allowed if the right-most index is not zero, as in this case one would get negative argument in a logarithm\footnote{See section \ref{analytic} for the analytical continuation in this case}.
\begin{equation}\label{minustrans}
H\left(\{m_1,\dots,m_k\},-x\right)=(-1)^kH\left(\{-m_1,\dots,-m_k\};x\right)
\end{equation}
This identity holds in the "m"-notation. For index vectors in the "a"-notation, the exponent $k$ is not the length of the vector, but the number of $1$ and $-1$ in the index vector.
We now consider the change of variable
\[x\rightarrow x^2.\]
Since we cannot express $1+x^2$ as sum of our basis functions $f_1,f_0$ and $f_{-1}$, we will exclude index vectors with negative indices for our considerations. The identities for the weight $1$ are
\begin{eqnarray}
H(0;x^2)&=&\log(x^2)=2H(0;x),\nonumber\\
H(1;x^2)&=&\log(1-x^2)=H(1;x)-H(-1;x).
\end{eqnarray}
There we chose $x$ to be the positive root of $x^2$. Furthermore we see in the second equation that the identity only holds for $x$ smaller than 1. For higher weights we use recursively the relations
\begin{eqnarray}
H(0,m_{2\dots,k};x^2)&=&\int_0^{x^2}\frac{\d \prim{x}}{\prim{x}}H(m_{2,\dots,k};\prim{x})\nonumber\\
&=&2\int_0^{x}\frac{\d \prim{t}}{\prim{t}}H(m_{2,\dots,k};\prim{t}^2),\nonumber\\
H(1,m_{2\dots,k};x^2)&=&\int_0^{x^2}\frac{\d \prim{x}}{1-\prim{x}}H(m_{2,\dots,k};\prim{x})\nonumber\\
&=&\int_0^{x}\d \prim{t}\left(\frac{1}{1-\prim{t}}-\frac{1}{1+\prim{t}}\right)H(m_{2,\dots,k};\prim{t}^2).
\end{eqnarray}
where $H(m_{2,\dots,k};\prim{t}^2)$ is expressed as HPL's of argument $\prim{t}$, which is known in a recursive approach.
The next transformation we consider is
\[x\rightarrow 1-x.\]
Since $1/(2-x)$ (the transform of $1/(1+x)=f_{-1}$) can not be expressed as linear combinations of the $f_i$'s, we will only consider index vectors without negative index. We first have the identities for weight 1
\begin{eqnarray}
H(0;1-x)&=&\log(1-x)=-H(1;x),\nonumber\\
H(1;1-x)&=&-\log(x)=-H(0;x).
\end{eqnarray}
These identities only hold for $x$ between 0 and 1. Here again, we process recursively in the depth. We use the fact that one can express HPL's with 1's on the left of the index vector as product of $H(1;x)$ (whose transformation we know from above) and HPL's without left 1's and treat only the case of a 0 as the left index. For this we use the formula
\begin{eqnarray}
H(0,m_{2\dots,k};1-x)&=&\int_0^{1-x}\frac{\d \prim{x}}{\prim{x}}H(m_{2,\dots,k};\prim{x})\nonumber\\
&=&\int_0^{1}\frac{\d \prim{t}}{\prim{t}}H(m_{2,\dots,k};\prim{x})-\int_{1-x}^{1}\frac{\d \prim{x}}{\prim{x}}H(m_{2,\dots,k};\prim{x})\nonumber\\
&=&H(0,m_{1,\dots,k};1)-\int_0^{x}\frac{\d \prim{t}}{1-\prim{t}}H(m_{2,\dots,k};1-\prim{t}),\nonumber\\
\end{eqnarray}
where one has to insert for $H(m_{2,\dots,k};1-\prim{t})$ the expansion in terms of HPL's of argument $\prim{t}$ known from the recursion. The HPL's (with positive $m$'s) evaluated for argument 1 are related to the multiple zeta values (MZV). This aspect is treated in section \ref{MZV}.
We consider now the transformation\footnote{The convention for the sign of the imaginary part of $x$ is different from \cite{Remiddi}.}
\[x=\frac{1}{t} +i\epsilon\,, \qquad\qquad t=\frac{1}{x}-i\epsilon\,,\qquad\qquad x>1.\]
The identities for weight 1 read
\begin{eqnarray}
H(0;x)&=&-H(0;t),\nonumber\\
H(1;x)&=&H(1;t)+H(0;t)+ i\pi,\nonumber\\
H(-1;x)&=&H(-1;t)-H(0;t).
\end{eqnarray}
For higher weight we proceed by induction. We will assume that the leftmost element in the vector index is $0$ or $-1$, since we can extract the one's with the procedure described in section \ref{divergences}.
\begin{eqnarray}
H(a,a_{2,\dots,k};x)&=&\int\limits_0^x\d x'f_a(x')H(a_{2,\dots,k};t)\nonumber\\
&=&\left(\int\limits_0^1\d x'-\int\limits_x^1\d x'\right)f_a(x')H(a_{2,\dots,k};x')\nonumber\\
&=&H(a,a_{2,\dots,k};1)+\int\limits_t^1\frac{\d t'}{t'^2}f_a\left(\frac{1}{t'}\right)H\left(a_{2,\dots,k};\frac{1}{t'}\right)
\end{eqnarray}
We now distinguish the two cases $a=0$ and $a=-1$,
\begin{eqnarray}
\int\limits_t^1\frac{\d t'}{t'^2}f_0\left(\frac{1}{t'}\right)&=&\int\d t'\frac{1}{t'}\nonumber\\
\int\limits_t^1\frac{\d t'}{t'^2}f_{-1}\left(\frac{1}{t'}\right)&=&\int\d t'\left(\frac{1}{t'}-\frac{1}{1+t'}\right) .
\end{eqnarray}
Inserting for
\[H\left(a_{2,\dots,k};\frac{1}{t'}\right)\]
its representation in terms of HPL's of argument $t'$ (which is known in a recursive approach) one gets terms of the kind
\begin{equation}
\int\limits_t^1 \d t' f_{b}(t')H(b_{2,\dots,k};t')=H(b,b_{2,\dots,k};1)-H(b,b_{2,\dots,k};t).
\end{equation}
The term $H(b,b_{2,\dots,k};1)$ is a finite constant, as $b$ is either $-1$ or $0$ and not one.
The next transformation is
\[x=t/(t-1).\]
Since $\log((2t-1)/(t-1))$ (the transform of $\log(1+x)$) and $\log(t/(t-1))$ (the transform of $\log(x)$) can not be expressed in terms of $f_{1,0,-1}(t)$, we restrict the following consideration to HPL without trailing $0$'s and without negative index. We have only one identity of weight 1,
\[H(1;x)=-H(1;t)\; ,\]
which only holds for $x,t<1$. For the higher weights we use the recursion relations
\begin{eqnarray}
H(0,m_{2\dots,k};x)&=&\int_0^{x}\frac{\d \prim{x}}{\prim{x}}H(m_{2,\dots,k};\prim{x})\nonumber\\
&=&\int_0^{t}\d \prim{t}\left(-\frac{1}{1-\prim{t}}\right)H\left(m_{2,\dots,k};\frac{\prim{t}}{\prim{t}-1}\right),\nonumber\\
H(1,m_{2\dots,k};x)&=&\int_0^{x}\frac{\d \prim{x}}{\prim{x}}H(m_{2,\dots,k};\prim{x})\nonumber\\
&=&\int_0^{t}\d \prim{t}\left(\frac{1}{1-\prim{t}}+\frac{1}{\prim{t}}\right)H\left(m_{2,\dots,k};\frac{\prim{t}}{\prim{t}-1}\right) \,,
\end{eqnarray}
where the HPL in the integrand is replaced by its expansion in terms of HPL's of $t$, as known from the recursion.
The last transformation we consider is
\[x=\frac{1-t}{1+t}\,.\]
Here we can also consider negative index in the index vector. The identities for weight 1 are
\begin{eqnarray}
H(0;x)&=&-H(1;t)-H(-1;t),\nonumber\\
H(1;x)&=&-H(0;t)-H(-1;1)+H(-1;t),\nonumber\\
H(-1;x)&=&-H(-1;t)+H(-1;1).
\end{eqnarray}
For the identities of higher weight, one uses again the fact that HPL's with 1's on the left of the vector index can be reduced. The recursion relations for the two other cases are
\begin{eqnarray}
H(0,m_{2\dots,k};x)&=&\int_0^{x}\frac{\d \prim{x}}{\prim{x}}H(m_{2,\dots,k};\prim{x})\nonumber\\
&=&\int_0^{1}\frac{\d \prim{x}}{\prim{x}}H(m_{2,\dots,k};\prim{x})-\int_{x}^{1}\frac{\d \prim{x}}{\prim{x}}H(m_{2,\dots,k};\prim{x})\nonumber\\
&=&H(0,m_{2,\dots,k};1)\nonumber\\
&&-\int_0^{t}\d \prim{t}\left(\frac{1}{1-\prim{t}}+\frac{1}{1+\prim{t}}\right)H\left(m_{2,\dots,k};\frac{1-\prim{t}}{1+\prim{t}}\right),\nonumber\\ \\
H(-1,m_{2\dots,k};x)&=&\int_0^{x}\frac{\d \prim{x}}{\prim{x}}H(m_{2,\dots,k};\prim{x})\nonumber\\
&=&H(0,m_{2,\dots,k};1)-\int_0^{t}\frac{\d \prim{t}}{1+\prim{t}}H\left(m_{2,\dots,k};\frac{1-\prim{t}}{1+\prim{t}}\right),\nonumber\\
\end{eqnarray}
where we insert for \[H\left(m_{2,\dots,k};\displaystyle\frac{1-\prim{t}}{1+\prim{t}}\right)\]
the identity of lower weight known from the recursion.
\section{Analytical continuation}\label{analytic}
The transformations of the preceding section were valid for restricted intervals. In this section we consider the analytical continuation to the remaining real axis. We first consider the continuation to the interval $(-1,0)$. We define
\[x=-t+\delta \epsilon i\;,\]
where $0<t<1$ and $\epsilon$ is infinitesimally small and positive. $\delta$ is either $+1$ or $-1$. The continuation for the HPL of weight 1 are
\begin{eqnarray}\label{acm10}
H(1;x)&=&H(-1;t)\,,\nonumber\\
H(0;x)&=&H(0;t)+\delta i\pi\,,\nonumber\\
H(-1;x)&=&H(1;t)\,.
\end{eqnarray}
For higher weight one extracts the factors $H(0;x)$ with the method of section \ref{divergences} and substitutes the above result for it. For the remainder, one can use the formula (\ref{minustrans}) of the preceding section.
The next interval to consider is the interval $(1,\infty)$. There we use the transformation
\[x\rightarrow t=\frac{1}{x}\,,\]
which transforms the real axis greater 1 into the interval $(0,1)$. We consider
\[x=\frac{1}{t} +i\delta \epsilon\,, \qquad\qquad t=\frac{1}{x}-i\delta\epsilon\,, \qquad\qquad x>1.\]
This transformation has already been treated in the preceding section for $\delta =1$. We can use the same procedure to keep track of the sign of the infinitesimal imaginary part by modifying the transformation of $H(1;x)$ to
\[H(1;x)=H(1;t)+H(0;t)+ i\delta\pi\,.\]
The last interval is $(-\infty,-1)$. There one also uses the transformation $x\rightarrow -x$. Again we consider
\begin{equation}
x=-t+\delta \epsilon i,\qquad t=-x-\delta \epsilon i.
\end{equation} The identities of weight 1 are
\begin{eqnarray}\label{acminfm1}
H(1;x)&=&-H(-1;t)=-H\left(-1;\frac{1}{t}\right)+H\left(0;\frac{1}{t}\right)\nonumber\\
H(0;x)&=&H(0;t)+\delta i\pi=-H\left(0;\frac{1}{t}\right)+\delta i\pi\nonumber\\
H(-1;x+\delta \epsilon i)&=&-H(1;t-\delta \epsilon i)=-H\left(1;\frac{1}{t}\right)-H\left(0;\frac{1}{t}\right)+\delta i\pi
\end{eqnarray}
For higher weight one extracts both the $H(0;x)$ and the $H(-1;x)$ with the method of section \ref{divergences} and substitutes the above expressions. The remainder is transformed with (\ref{minustrans}). Since $t$ is larger than 1, one should use the analytical continuation described above to bring the argument of the HPL into the interval $(0,1)$.
\section{Values at unity and Multiple Zeta Values}\label{MZV}
Some transformations of the preceding section, there appear HPL's of argument $1$. These are related for positive $m$'s to the Multiple Zeta Values (MZV) and for general $m$'s to colored MZV. The relation can be found through induction\footnote{see Appendix \ref{zetaproof}.} and reads
\begin{eqnarray}\label{HPLMZVlink}
H(\{m_{1,\dots,k}\};1)&=&N(m_{1,\dots,k})\zeta(\tilde m_{1,\dots,k}),\quad k>1\\
H(\{m\};1)&=&\zeta(m)\,,\qquad\qquad \qquad \qquad m>0\\
H(\{-m\};1)&=&(1-2^{1-m})\zeta(m)\qquad\qquad m>0\,.
\end{eqnarray}
The MZV $\zeta$ are defined as
\begin{equation}\label{MZVdef}
\zeta(m_1,\dots,m_k)=\sum\limits_{i_1=1}^\infty\sum\limits_1^{i_1-1}\dots\sum\limits_1^{i_{k-1}-1}\prod\limits_{j=1}^k\frac{\mathrm{sgn}(m_j)^{i_j}}{i_j^{|m_j|}}\,.
\end{equation}
and are described in the literature, for example in \cite{math.CA/9910045,Borwein:1996yq}. The vector $\tilde m$ is obtained from the vector $m$ with
\begin{equation}
\tilde m=(m_1,\mathrm{sgn}(m_1)m_2,\dots, \mathrm{sgn}(m_{i-1}) m_i,\dots ,\mathrm{sgn}(m_{k-1}) m_k)
\end{equation}
The factor $N(m_{1,\dots,k})$ is given by
\begin{equation}
N(m_{1,\dots,k})=(-1)^{\#(m_i<0)}
\end{equation}
The MZV's also form an algebra. Due to this fact, they can be expressed in terms of a few mathematical constants like powers of $\pi$, $\zeta$-functions and polylogs at specified values. We list some of the identities of \cite{math.CA/9910045,Borwein:1996yq} in Appendix \ref{MZVtable}.
For the implementation of the HPL at unity, we translated the tables of the FORM package {\tt harmpol.h} \cite{Remiddi} and their expansions for weight 7 and 8 {\tt htable7.prc} and {\tt htable8.prc} for Mathematica.
In these tables, there appear some constants that are not expressible through known constants like $\pi$, $\zeta(n)$, $\log(2)$, or $Li_n(1/2)$. Using the different relations between the different MZV, one can reduce the number of independent constants. Which constants are kept is a matter of choice. In the tables we translated, the choice of the independent constants is
\begin{eqnarray}
s_6 &=& S(\{-5,-1\},\infty) =\sum\limits_{i_1=1}^{\infty}\frac{(-1)^{i_1}}{i_1^5} \sum\limits_{i_2=1}^{i_1}\frac{(-1)^{i_2}}{i_2}\nonumber\\
&=&\zeta(-5,-1)+\zeta(6)\simeq 0.98744142640329971377\\
s_{7a} &=& S(\{-5,1,1\},\infty) =\sum\limits_{i_1=1}^{\infty}\frac{(-1)^{i_1}}{i_1^5} \sum\limits_{i_2=1}^{i_1}\frac{1}{i_2}\sum\limits_{i_3=1}^{i_2}\frac{1}{i_3}\nonumber\\
&=&\zeta(-5,1,1)+\zeta(-6,1)+\zeta(-5,2)+\zeta(-7)\\
&\simeq& -0.95296007575629860341\\
s_{7b} &=& S(\{5,-1,-1\},\infty) =\sum\limits_{i_1=1}^{\infty}\frac{1}{i_1^5} \sum\limits_{i_2=1}^{i_1}\frac{(-1)^{i_2}}{i_2}\sum\limits_{i_3=1}^{i_2}\frac{(-1)^{i_3}}{i_3}\nonumber\\
&=&\zeta(7)+\zeta(5,2)+\zeta(-6,-1)+\zeta(5,-1,-1)\\
&\simeq& 1.02912126296432453422\nonumber\\
s_{8a} &=& S(\{5,3\},\infty) =\sum\limits_{i_1=1}^{\infty}\frac{1}{i_1^5} \sum\limits_{i_2=1}^{i_1}\frac{1}{i_2^3}\nonumber\\
&=&\zeta(8)+\zeta(5,3)\simeq 1.0417850291827918834\nonumber\\
s_{8b} &=& S(\{-7,-1\},\infty) =\sum\limits_{i_1=1}^{\infty}\frac{(-1)^{i_1}}{i_1^7} \sum\limits_{i_2=1}^{i_1}\frac{(-1)^{i_2}}{i_2}\nonumber\\
&=&\zeta(8)+\zeta(-7,-1)\simeq 0.99644774839783766600\\
s_{8c} &=& S(\{-5,-1,-1,-1\},\infty) \nonumber\\&=&\sum\limits_{i_1=1}^{\infty}\frac{(-1)^{i_1}}{i_1^5} \sum\limits_{i_2=1}^{i_1}\frac{(-1)^{i_2}}{i_2}\sum\limits_{i_3=1}^{i_2}\frac{(-1)^{i_3}}{i_3}\sum\limits_{i_3=1}^{i_3}\frac{(-1)^{i_4}}{i_4}\nonumber\\
&=&\zeta(8)+\zeta(-7,-1)+\zeta(-5,-3)+\zeta(6,2)+\zeta(-5,-1,2)\nonumber\\
&&+\zeta(-5,2,-1)+\zeta(6,-1,-1)+\zeta(-5,-1,-1,-1)\\
&\simeq& 0.98396667382173367094\nonumber\\
s_{8d} &=& S(\{-5,-1,1,1\},\infty) =\sum\limits_{i_1=1}^{\infty}\frac{(-1)^{i_1}}{i_1^5} \sum\limits_{i_2=1}^{i_1}\frac{(-1)^{i_2}}{i_2}\sum\limits_{i_3=1}^{i_2}\frac{1}{i_3}\sum\limits_{i_3=1}^{i_3}\frac{i}{i_4}\nonumber\\
&=&\zeta(8)+\zeta(-5,-3)+\zeta(6,2)+\zeta(7,1)+\zeta(-5,-2,1)+\zeta(-5,-1,2)\nonumber\\
&&+\zeta(6,1,1)+\zeta(-5,-1,1,1)\simeq 0.99996261346268344768
\end{eqnarray}
For index vectors with left 1's, there appear divergences of the form
\[\sum\limits_{i=1}^{\infty}\frac{1}{i}=S(1,\infty)=H(1;1).\]
These divergences are well defined\footnote{see \cite{Remiddi}.} and can cancel during a calculation. Right 0's can be extracted with the method of section \ref{divergences}. All factors $H(0;1)=\log(1)=0$ vanish, even multiplied with $H(1;1)=-\log(0)$, since
\[ \log(x)\log^n(1-x)\rightarrow 0,\qquad x \rightarrow 0,\quad n>0.\]
\section{Implementation}\label{implementation}
In this section we present our implementation of the HPL's. The package can be found at\cite{HPLhomepage}. Installation instructions can be found there. After instalation, the package can be called with
{\tt <<HPL`.}
\noindent This call should be done at the beginning of the {\tt Mathematica} session.
\subsection{New functions}
In the package {\tt HPL} we defined the following new functions
\begin{itemize}
\item {\tt HPL[m,x]} is the harmonic polylogarithm $H(m;x)$. {\tt m} is a list representing the index vector. We chose the "m"-notation as the standard notation. It is possible to give as argument a vector in the "a"-notation, or even a mix between the two notations, the result will be automatically converted to the "m"-notation.
\vspace{0.2cm}\\
\fbox{\parbox{0.75\textwidth}{\includegraphics[scale=0.5]{HPLExample1.eps}}
}\vspace{0.1cm}\\
\item {\tt HPLMtoA[m\_List]},{\tt HPLAtoM[a\_List]} convert vectors of the "m"- to the "a"- and from "a"- to "m"-notation respectively. Both can convert vectors which mix the two notations.
\vspace{0.2cm}\\
\fbox{\parbox{0.75\textwidth}{\includegraphics[scale=0.5]{HPLExample2.eps}}
}\vspace{0.1cm}\\
\item {\tt HPLLogExtract} extracts the logarithmic divergences of the HPL in its argument at 0 and 1. The result is displayed as function of $\log(x),\log(1-x)$ or $H(1;x),H(0;x)$ depending on the option settings (see section \ref{options}).
\vspace{0.2cm}\\
\fbox{\parbox{0.75\textwidth}{\includegraphics[scale=0.5]{HPLExample3.eps}}
}\vspace{0.1cm}\\
\item {\tt HPLConvertToKnownFunctions} returns its argument with HPL's replaced by their representation in terms of more common functions, whenever possible. This is only needed if the option {\tt \$HPLAutoConvertToKnownFunctions} is set {\tt False} (see section \ref{options}).
\vspace{0.2cm}\\
\fbox{\parbox{0.75\textwidth}{\includegraphics[scale=0.5]{HPLExample4.eps}}
}\vspace{0.1cm}\\
\item {\tt HPLProductExpand} returns its argument where the products of HPL's of weight $w_1,\dots,w_k$ are replaced by their representation as a linear combination of HPL's of weight $w_1+\dots+w_k$. In order to expand all products, {\tt HPLProductExpand} expands the argument (using {\tt Expand}), so that terms of the form
\[H(\dots;x)\big(H(\dots;x)+H(\dots;x)+\dots\big).\]
are also replaced.
\vspace{0.2cm}\\
\fbox{\parbox{0.75\textwidth}{\includegraphics[scale=0.5]{HPLExample5.eps}}
}\vspace{0.1cm}
\item {\tt HPLConvertToSimplerArgument} returns its argument with HPL's of arguments $-x$, $x^2$, $1-x$, $x/(x-1)$ and $(1-x)/(1+x)$ replaced by their expansion as a sum of HPL's of argument $x$, as described in section \ref{related}.
\vspace{0.2cm}\\
\fbox{\parbox{0.75\textwidth}{\includegraphics[scale=0.5]{HPLExample6.eps}}
}\vspace{0.1cm}
\item {\tt HPLReduceToMinimalSet} returns its argument with the HPL's projected to the minimal set, as described in section \ref{basis}.
\vspace{0.2cm}\\
\fbox{\parbox{0.75\textwidth}{\includegraphics[scale=0.5]{HPLExample18.eps}}
}\vspace{0.1cm}
\item {\tt HPLAnalyticContinuation} returns its argument with HPL's replaced by their analytic continuation. The arguments of the HPL's are taken to belong to the interval specified by the option {\tt HPLAnalyticContinuationRegion} which can be either
\begin{description}
\item[{\tt minftom1}] the interval $-\infty$ to $-1$
\item[{\tt m1to0}] the interval $-1$ to $0$
\item[{\tt onetoinf}] the interval $1$ to $\infty$.
\end{description}
The HPL's are replaced by their representation in terms of HPL's of argument in the interval $0,1$. The choice of the side of the branch cut from which the argument is approached is set by the option
{\tt HPLAnalyticContinuationSign}
which can take values $-1$, $1$ or any symbol.
\vspace{0.2cm}\\
\fbox{\parbox{0.75\textwidth}{\includegraphics[scale=0.5]{HPLExample11.eps}}
}\vspace{0.1cm}\\
If the option {\tt HPLAnalyticContinuationRegion} is omitted, and if the argument is numerical, {\tt HPLAnalyticContinuation} will use automatically the appropriate setting. If the option {\tt HPLAnalyticContinuationSign} is omitted, {\tt HPLAnalyticContinuation} will use the value stored in the variable {\tt \$HPLAnalyticContinuationSign} which is set by default to $1$\footnote{This is the same convention as \cite{Gehrmann:2001pz}, but opposite to that of \cite{Vollinga:2004sn}}.
\vspace{0.2cm}\\
\fbox{\parbox{0.75\textwidth}{\includegraphics[scale=0.5]{HPLExample12.eps}}
}\vspace{0.1cm}\\
It is to be noted that the {\tt Mathematica} conventions for the analytic continuation are not always the same as that of the {\tt HPL} package. This is illustrated by the following example
\vspace{0.2cm}\\
\fbox{\parbox{0.75\textwidth}{\includegraphics[scale=0.5]{HPLExample17.eps}}
}\vspace{0.1cm}\\
Since the substitution of the HPL's through more common functions has precedence over the analytic continuation, the option
{\tt \$HPLAutoConvertToKnownFunctions}
can interfere with the analytic continuation, as shown in the following example.
\vspace{0.2cm}\\
\fbox{\parbox{0.75\textwidth}{\includegraphics[scale=0.5]{HPLExample13.eps}}
}\vspace{0.1cm}\\
This example shows that with the option
{\tt \$HPLAutoConvertToKnownFunctions}
set {\tt True} we loose the control over the sign of the imaginary part (as in the first case it will now depend on the {\tt Mathematica} conventions).
\item {\tt MZV[m]} is the Multiple Zeta Value (see section \ref{MZV}) corresponding to the index vector $m$. Their value in terms of mathematical constants are tabulated\footnote{these tables are those of the FORM package {\tt harmpol}\cite{Vermaseren_harmpol}} up to weight 8 and systematically replaced. For higher weights, the cases covered by Appendix \ref{MZVtable} are also replaced.
\vspace{0.2cm}\\
\fbox{\parbox{0.75\textwidth}{\includegraphics[scale=0.5]{HPLExample7.eps}}
}\vspace{0.1cm}
\item The function {\tt \$HPLOptions} gives a list of the options of the package and their current values.
\vspace{0.2cm}\\
\fbox{\parbox{0.75\textwidth}{\includegraphics[scale=0.5]{HPLExample19.eps}}
}
\end{itemize}
\subsection{Functions modified}
\begin{itemize}
\item We defined the derivatives of the HPL's as described in section \ref{derivative}. The integration showing up in the recursive definition of the HPL's is also implemented.
\vspace{0.2cm}\\
\fbox{\parbox{0.75\textwidth}{\includegraphics[scale=0.5]{HPLExample8.eps}}
}\vspace{0.1cm}
\item The function {\tt Series} is able to expand HPL's around $x=0$ and $x=1$.
\vspace{0.2cm}\\
\fbox{\parbox{0.75\textwidth}{\includegraphics[scale=0.5]{HPLExample9.eps}}
}\end{itemize}
\subsection{Working with options}\label{options}
The package {\tt HPL} has some options to control its behavior. They set the preferred form in which expressions are displayed. The option can be locally overridden by the functions described above. The effects of the options are described in the following.
\begin{description}
\item[{\tt \$HPLAutoLogExtract}:] If {\tt True} the logarithmic divergences $\log(1-x)$ and $\log(x)$ are automatically extracted from the HPL's following the procedure described in section \ref{divergences}. These divergences are displayed as $\log$ or as $H(0/1;x)$, depending on the value of the option {\tt \$HPLAutoConvertToKnownFunctions}. The default setting is {\tt False}.
\item[{\tt \$HPLAutoProductExpand}:] If {\tt True} the products of HPL's are automatically converted into a sum of HPL of weight equal to the sum of the weights of the two factors, as described in section \ref{algebra}. Default is {\tt False}.
\item[{\tt \$HPLAutoConvertToKnownFunctions}:] If set {\tt True}, the HPL's will be converted to more common functions (logs, polylogarithms, polynoms, Nielsen polylogs) if possible, using the identities of Appendix \ref{identities}. This might be counterproductive when the properties of the HPL's are more explicit in the HPL form than in their equivalent representation. Furthermore, if this options is set {\tt True} while using the analytic continuation described above, the result may be wrong, as {\tt Mathematica} does not have different conventions for the analytic continuation. Default is {\tt False}.
\item[{\tt \$HPLAutoReduceToMinimalSet}:] If set {\tt True}, the HPL's will be automatically reduced to a minimal basis (up to weight 8). This only makes sense if one does not expand the obtained products again, or if the factors of smaller weight can be replaced by their expression in terms of known functions. Therefore, for the reduction to be performed, one has to have the option
{\tt \$HPLAutoProductExpand} equal {\tt False} or
{\tt \$HPLAutoConvertToKnownFunctions} equal {\tt True}.
If this is not fulfilled, the option will have no effect. Default is {\tt False}.
\item[{\tt \$HPLAutoConvertToSimplerArgument}:] If set {\tt True}, the HPL's of arguments $-x$, $x^2$, $1-x$, $1/x$, $x/(x-1)$ and $(1-x)/(1+x)$ will be automatically substituted by their representation in term of HPL's of argument $x$ along the lines of section \ref{related}. Default is {\tt False}.
\item[{\tt \$HPLAnalyticContinuationSign}:] If set to 1 the analytic continuation of the HPL's is taken for arguments with an positive infinitesimal imaginary part, if set to -1 with an negative one. This is only the default setting and can be overridden by specifying the option {\tt AnalyticContinuationSign} in the function {\tt HPLAnalyticContinuation}, as described above. Default setting is $+1$.
\end{description}
\subsection{Numerical evaluation}
If the argument of the HPL is numerical, the package evaluates its value and gives the result in the same precision as the precision of $x$.
In \cite{Remiddi}, the authors propose to use the argument transformation $x\rightarrow (1-t)/(1+t)$ for the evaluation of HPL's with argument near one or negative with a large absolute value. This is only of advantage if one wants to evaluate the whole set of HPL's for this given $x$. Since for low weight, we have a representation in terms of usual functions (which {\tt Mathematica} can evaluate fast and precisely), this change of variable is not useful in this case. For high weights, the expression of the HPL's in term of HPL's of argument $\frac{1-x}{1+x}$ is quite large, so that the time gain in the convergence is compensated by the number of different HPL's to evaluate. Therefore, we used the series expansion systematically for the evaluation of the numerical value of the HPL. The convergence is not very good for values near 1, where not all digits displayed are accurate. There is a FORTRAN package for the precise numerical evaluation of HPL's \cite{Gehrmann:2001pz}.
\vspace{0.2cm}\\
\fbox{\parbox{0.75\textwidth}{\includegraphics[scale=0.5]{HPLExample10.eps}}
}\vspace{0.1cm}\\
For argument outside the interval $0,1$, one can give the sign of the infinitesimal imaginary part to be taken in account for the analytic continuation, as for the function {\tt HPLAnalyticContinuation}
\vspace{0.2cm}\\
\fbox{\parbox{0.75\textwidth}{\includegraphics[scale=0.5]{HPLExample15.eps}}
}\vspace{0.1cm}\\
For the numerical values of the MZV, we implemented the procedure described in \cite{math.CA/9910045,Vollinga:2004sn}. The numerical values of the constants $s_6,s_{7a},s_{7b},s_{8a},s_{8b},s_{8c}$ and $s_{8d}$ correspond to the results given by the {\tt EZface} application\cite{EZface}.
\vspace{0.2cm}\\
\fbox{\parbox{0.75\textwidth}{\includegraphics[scale=0.5]{HPLExample16.eps}}}\vspace{0.2cm}\\
Due to the possibility of evaluating the HPL numerically, {\tt Plot} is able to represent HPL's
\vspace{0.2cm}\\
\fbox{\parbox{0.75\textwidth}{\includegraphics[scale=0.5]{HPLExample14.eps}}
}\vspace{0.1cm}\\
We checked numerical agreement with \cite{Gehrmann:2001pz} at double precision accuracy.
\section{Conclusion}\label{conclusion}
In this paper, we presented an implementation of the harmonic polylogarithm of Remiddi and Vermaseren \cite{Remiddi} for Mathematica. It contains an implementation of the product algebra, the derivative properties, series expansion and numerical evaluation. The analytic continuation has been treated carefully, allowing the user the full control over the definition of the sign of the imaginary parts. Many options enable the user to adapt the behavior of the package to his specific problem. This package is already used by the package {\tt HypExp}\cite{Huber:2005yg} to treat the HPL's and MZV's appearing in the expansion of hypergeometric functions around integer-valued parameters.
\section*{Acknowledgement}
We would like to thank Thomas Gehrmann for many useful discussions and Jos Vermaseren, Ettore Remiddi, Thomas Gehrmann for carefully reading our manuscript. We also wish to thank the Swiss National Science Foundation (SNF) which supported this work under contract 200021-101874.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,522 |
\section{Introduction}
Graph-based data analysis identifies one of the most effective tools for modern data analysis, as it allows the investigation of complex datasets by means of a compact representation of mathematical manifolds, groups and varieties across different domains \cite{GEOMDL,fluid_graphsignalproc}.
In particular, the samples in the considered dataset are typically represented as nodes in the graph structure, whilst the edges are meant to model and quantify the similarity between each pair of nodes, according to a given informativity criterion and/or distance metric \cite{fluid_graph_learn_smooth,fluid_graphlearn_DF,fluid_graphlearn_SP,fluid_graphsignalproc,CommunityDetection_Fortunato,GEOMDL}.
In this way, it is possible to obtain flexible and efficient investigation of complex data acquired by multiple acquisition strategies (or \textit{modalities}), i.e., \textit{multimodal} datasets.
These properties are particularly interesting when semisupervised and unsupervised characterization of the records is required, e.g., clustering, outlier detection, ranking, label propagation \cite{COUILLET20,fluid_graphsignalproc}.
In technical literature, several strategies have been proposed to adequately model the graph structure and topology, from geometrical to statistical analysis, to probabilistic description of the similarities among samples (see for instance\cite{COUILLET18,COUILLET20,CommunityDetection_SC1,CommunityDetection_SC2,CommunityDetection_SC3,CommunityDetection_SC4}).
In fact, these frameworks assume that the information would propagate across the graph according to the heat diffusion mechanism \cite{fluid_diffusion1,fluid_diffusion3}.
Specifically, assuming that the data distribution would be uniform across the samples and smooth in the feature space, it is possible to consider the connections as transition probabilities in larger Markov models, of which the nodes represent the states.
In this analogy, the weight that each edge of the graph can be associated with
is then computed in terms of affinity between the nodes according to a dynamical system considered at equilibrium, which results in the ability to use Euclidean distances to quantify the similarities among samples \cite{fluid_diffusion1,fluid_diffusion3}.
This idea represents the bedrock on which the most advanced data analytics methods - from nonlinear optimization to eigenanalysis, from inference to regression, from geometrical deep learning to motif networks - rely on \cite{fluid_graphsignalproc,GEOMDL,fluid_graphDL2,COUILLET20,COUILLET18,CD1}.
Indeed, this assumption has been proven to be successful in a number of applications, and numerous artificial intelligence schemes take advantage of this to design the main characterization of decision making procedures.
Nonetheless, it is also true that this approach might fail in detailing the observations and records acquired for the most modern challenges in data analysis \cite{multimod1,multimod2}.
In fact, when investigating
large scale datasets that are collected by heterogeneous modalities, the records can be affected by multiple and diverse sources of noise, leading to incomplete, non-uniform, and/or unbalanced data to be analyzed \cite{multimod9,multimod1}.
Moreover, the data would show different degrees of reliability, so that information ends up being extracted by datasets showing
corrupted and/or missing records.
Finally, the efficiency and the explainability of the data analysis must be addressed, so that near real time applications and thorough interpretation of the data can be carried out.
This situation is typically not addressed by the graph signal processing methods in technical literature, as these architectures might not be able to adequately extract information from multimodal datasets, especially in an unsupervised fashion \cite{multimod1,capacity,multimod2,multimod9}.
To give an intuitive understanding of these points, let us consider a practical example.
Let us say that we are interested in studying genes within genome-wide association studies in order to determine specific markers that lead to a disease. In some cases, this is simple, with one DNA marker resulting in the expression of the disease. Other diseases may have multiple genes that together result in the expression of a disease. When studying large volumes of genes from sources taken globally (e.g., from nations as globally diverse as Finland, Kenya, and Japan) the common mistake made by learning methods is to classify at a level that may group the results into three classes, corresponding to the three nations. In reality, there are other markers (less 'obvious' to the machine) that could correspond to the same disease expressed in genes that are common across all three classes. This population stratification leads to differences in gene frequencies as a result of systematic differences in ancestry rather than association of genes with disease \cite{limitsGWAS}.
To address these points, in this work we propose the consideration of a new model for information propagation across graphs.
Specifically, we introduce a \textit{fluid diffusion model} to shape the graph design, with special focus to the topology and connectivity of the data structure \cite{fluid_FokkerPlanck1,fluid_FokkerPlanck2}.
In this way, the global and local interactions among samples and records are taken into account in terms of tensor representation, which can be expressed as permeability, diffusivity and flow velocity across the graph.
This representation allows one to take into account a large variety of data characteristics, so to ensure versatility and applicability of this approach in several operational scenarios.
By taking advantage of this novel data representation, we provide an efficient method for community detection that can be easily implemented in terms of spectral clustering approach.
Then, we demonstrate that this approach enables an accurate understanding of the data structures, so that a strong enhancement with respect to the state-of-the-art methods can be observed.
The main contributions of this paper can be then summarized as follows:
\begin{itemize}
\item a new paradigm to model information propagation - based on fluid diffusion - on graph structures which is able to grasp global and local scale interactions and patterns induced by multimodal datasets
\item the analysis of the proposed fluid diffusion system in terms of eigenvectors of the flow velocity matrix that can be employed to characterize the dependency among samples and the relevance of the features the considered dataset consists of. In this way, an effective understanding of the graph can be carried out in terms of eigenanalysis, enabling valid characterization of the data structures and topologies;
\item an efficient method for non-overlapping community detection, taking advantage of the eigenanalysis of the flow velocity matrix used to describe the graph connections.
\end{itemize}
The paper is organized as follows.
Section \ref{sec_fluidgraph} introduces the theoretical aspects of classic graph representation of datasets based on the heat diffusion model. It continues with the motivation of the proposed novel graph representation based on fluid diffusion, and its main properties.
Section \ref{sec_fluidCD} reports a thorough overview of the main works introduced in technical literature for the application task used as test case in this paper - community detection - as well as the proposed method for community detection based on the novel fluid graph representation of multimodal datasets.
Section \ref{secresult} reports the performance results obtained over three multimodal datasets, as well as heuristic confirmation of the motivation and validity of the proposed fluid graph representation.
Finally, Section \ref{secconcl} delivers our final remarks and some ideas on future research.
\section{Fluid graph representation}
\label{sec_fluidgraph}
In this Section, we introduce the main motivations for the novel graph representation for multimodal data analysis we proposed in this work.
First, the connection between classic graph representation and heat diffusion is summarized.
Then, the main issues for classic graph representation are presented, leading to the motivation and the description of the graph representation based on fluid diffusion that we introduce in this paper.
\subsection{Classic graph representation and heat diffusion}
\label{sec_meth_heat}
Graph representation of data manifolds is a valuable tool in extracting information from records, understanding their interactions,
and providing a thorough interpretation of their semantics.
Indeed, graph-based signal processing has enabled exploiting data
structure and relational priors, improving data and computational
efficiency, and enhancing model interpretability in various domains \cite{fluid_graphsignalproc,GEOMDL}.
The structure and meaning of the edges and nodes, as defined within graph representation, affects the accuracy and reliability of any information then derived from it \cite{fluid_graphsignalproc,specclust3,specclust2,specclust1}.
In fact, graph representation identifies a favorable trade-off between simplicity and explainability of the relationships between the samples in the dataset.
The similarity and interactions among samples are represented by means of the weights of the edges of the graphs.
The edge weight is then typically computed as function of the proximity of the corresponding
data points in the feature space induced by the records collected in the considered dataset.
Hence, a connection between two samples in the dataset could be considered stronger as the proximity of their corresponding feature vectors increases \cite{fluid_diffusion3}.
Characterizing the complex geometry of the data in the feature space is therefore crucial to obtain an accurate graph representation and therefore a reliable understanding of the interactions among samples.
To this aim, combining the main properties of random walks and spectral analysis is a proven approach in finding relevant structures in complex manifolds, enabling the detection and classification of thematic clusters within the data \cite{fluid_diffusion2,fluid_diffusion3,COUILLET18}.
Indeed,
using the eigenfunctions of a Markov matrix defining a random walk on the data can help in achieving a new description of data sets, as well as in providing a thorough interpretation of the similarity modeled by the edge weights \cite{fluid_diffusion2,fluid_diffusion3}.
To embed these samples in a Euclidean space, these quantities can be associated with transition mechanisms described in terms of diffusion processes.
Further, processing the higher order moments of the Markov matrix this strategy aims to connect the spectral properties of the diffusion process to the geometrical characteristics of the dataset.
Specifically, let $\textbf{X} = \{ \textbf{x}_i\}_{i=1, \ldots, N}$, $\textbf{x}_i \in \mathbb{R}^n$ be the considered dataset consisting of $N$ samples characterized by $n$ features.
In general, it is possible to translate $\textbf{X}$ into a graph structure consisting of nodes and edges connecting them.
Specifically, the $i$-th node identifies the sample $\textbf{x}_i$ in the dataset $\textbf{X}$.
On the other hand, the weight of the edge connecting node $i$ to node $j$ is computed according to a function (or kernel) $\eta(\textbf{x}_i, \textbf{x}_j)$ of the features associated with the considered samples.
In the classic derivation of graph structure,
the goal of the metric $\eta$ is to capture the characteristics of the local geometry of the given dataset.
It is then possible to construct a Markov matrix associated with $\textbf{X}$ that can describe the local geometry of the dataset by summarizing the node-to-node similarities.
In other terms, the $(i,j)$ element of the Markov matrix is defined as probability
of transition in one time step from node $i$ to node $j$ in the graph.
As such, the $(i,j)$ element of the Markov matrix is also proportional to the edge weight $\eta(\textbf{x}_i, \textbf{x}_j)$.
Moreover, it is possible to retrieve the transition probability in $t$ steps by elevating the Markov matrix to the power $t$
\cite{fluid_diffusion2,fluid_diffusion3,fluid_diffusion1}.
These properties of the Markov matrix are particularly interesting for the characterization of the graph structure and connections.
Analyzing the behavior of the Markov matrix for long transitions, i.e., large power of the Markov matrix, can help in detecting and understanding the actual relationships among the samples in the dataset \cite{fluid_diffusion1,fluid_diffusion3}.
To this aim, spectral theory plays a crucial role.
In particular, the eigenanalysis of the aforesaid Markov matrix can help unveil hidden patterns among the samples, leading to a precise understanding of the interactions among samples.
Moreover, a compact description of the random walk processes based on the eigenvectors and eigenvalues of the Markov matrix can be used to identify the information propagation mechanisms that can be drawn within the dataset according to the geometrical properties of the samples in the feature space \cite{fluid_diffusion2}.
The metric $\eta$ is expected to provide a characterization of the local geometry of the dataset \cite{fluid_diffusion1}.
On the other hand, the Markov matrix defines the direction of propagation according to the transition probabilities, which can lead to an exhaustive understanding of the overall properties of the dataset when long random walks induced by the Markov matrix are considered \cite{fluid_diffusion2}.
This scenario can be investigated in terms of a stochastic dynamical system where the transitions summarized in the Markov matrix can be described as results of a differential equation.
This can lead to a global characterization of the system when integrated on a long term scale \cite{fluid_diffusion1,fluid_diffusion3}.
Hence, the graph is considered as a realization of a dynamical process at equilibrium \cite{fluid_diffusion1}.
This analogy is particularly interesting, since it enables a robust description of the data interactions with respect to noise perturbation, as well as a multiscale analysis of the considered dataset \cite{fluid_diffusion3}.
This investigation relies once again on the transition probability proportional to the weight of the edge
connecting the two nodes.
In particular, the inference mechanism is based on the transition probability density $p(\textbf{x}(t+\epsilon) = \textbf{x}_j|\textbf{x}(t)=\textbf{x}_i)$ of finding the system at location $\textbf{x}_j$ at time
$t+\epsilon$, given an initial location $\textbf{x}_i$ at time $t$, where $\textbf{x}_i$ identifies the point in the $n$-dimensional feature space corresponding to the $i$-th sample, using the notation previously introduced in this Section \cite{fluid_diffusion1,fluid_diffusion2,fluid_randomwalk}.
In this way, the analysis of the relationships and interactions among samples can be less affected by the density of the data and the local geometry of the dataset \cite{fluid_diffusion3}.
Nevertheless, it is also true that the characterization of the dataset in terms of dynamic system analysis requires that the Markov matrix and the metric used to quantify the edge weight in the graph representation would fulfill a number of conditions.
Specifically:
\begin{itemize}
\item the transition probabilities must only depend on the current state and
not on the past ones (first-order Markov chain).
In this way, since the graph is connected, the
Markov chain is irreducible, that is, every state can be reached from any other state \cite{fluid_diffusion2,fluid_diffusion1};
\item the Markov matrix must be characterized by a unique stationary distribution to allow the existence of the eigenvalues of the Markov matrix \cite{fluid_diffusion2,fluid_diffusion3};
\item the Markov chain must be ergodic since the state space of the Markov chain associated with the matrix of the node transitions is finite and the corresponding random walk is aperiodic
\cite{fluid_diffusion3};
\item the kernel $\eta(\textbf{x}_i, \textbf{x}_j)$ used to quantify the edge weight must capture the relationships between pair of samples in $\textbf{X}$, so it is not surprising that it must be
non-negative. Moreover, the function $\eta$ must be a rotation invariant kernel \cite{rotinvkernel}, so that is possible to retrieve the manifold structure regardless of the distribution of the samples \cite{fluid_diffusion3}.
\end{itemize}
When these conditions are satisfied, it is possible to prove that the solution of the aforementioned problem satisfies the forward Fokker-Planck equation associated with the \textit{heat diffusion process} which can be written for the density $p(\textbf{x}(t+\epsilon) = \textbf{x}_j|\textbf{x}(t)=\textbf{x}_i)$ as follows \cite{fluid_diffusion1}:
\begin{equation}
\frac{\partial p}{\partial t} = \nabla \cdot (\nabla p + p\nabla U(\textbf{x})),
\label{eq_FokkerPlanck_heat}
\end{equation}
\noindent where $\nabla = [\frac{\partial}{\partial x_i}]_{i=1, \ldots, n}$, and the state $\textbf{x}(t) \in \mathbb{R}^n$ (i.e., each sample in the dataset) is a realization of the dynamical system that can be written as follows:
\begin{equation}
\dot{\textbf{x}} = - \nabla U(\textbf{x}) + \sqrt{2}\dot{\textbf{w}},
\label{eq_dynsyst_heat}
\end{equation}
\noindent where $\dot{\textbf{x}}$ and $\dot{\textbf{w}}$ identify the derivatives with respect to $t$ of $\textbf{x}$ and $\textbf{w}$, respectively.
Moreover, $U$ is the free energy at
$\textbf{x}$ (which can be also called the potential at $\textbf{x}$), and $\textbf{w}$(t) is
an $n$-dimensional Brownian motion process.
From a practical point of view, in this scenario the considered dataset $\textbf{X} = \{ \textbf{x}_i\}_{i=1, \ldots, N}$, $\textbf{x}_i \in \mathbb{R}^n$ is assumed to be sampled from the aforesaid dynamical system in
equilibrium \cite{fluid_diffusion1,fluid_diffusion3}.
In general, the solution of (\ref{eq_FokkerPlanck_heat})
can be written in terms of an eigenfunction expansion of the Fokker-Planck operator \cite{fluid_diffusion1,fluid_diffusion3}.
In low dimensions, it is possible to calculate approximations to
these eigenfunctions via numerical solutions of the relevant partial differential equations. In
high dimensions, however, this approach is in general infeasible and one typically resorts to
simulations of trajectories of (\ref{eq_dynsyst_heat}). In this case, there is a need to
employ statistical methods to analyze the simulated trajectories, identify the slow variables,
the meta-stable states, the reaction pathways connecting them and the mean transition times
between them \cite{fluid_diffusion1,fluid_diffusion2,fluid_diffusion3}.
In particular, for the analysis of (\ref{eq_dynsyst_heat}), a key role is played by the normalized graph Laplacian matrix, i.e., the matrix which $(i,j)$ element is set to $\sum_{k=\{1,\ldots,N\}\setminus i} \eta(\textbf{x}_i,\textbf{x}_k)$ if $i=j$, and $- \eta(\textbf{x}_i,\textbf{x}_j)$ otherwise \cite{fluid_diffusion1,fluid_diffusion3,fluid_diffusion2,fluid_randomwalk}.
In fact,
it is possible to prove that the eigenvalues and eigenfunctions
of the normalized graph Laplacian operator asymptotically correspond to the Fokker-Planck equation with
a potential $2U(\textbf{x})$ \cite{fluid_diffusion1}.
The crucial role of the Laplacian operator is further highlighted by considering that under special conditions the Fokker-Planck equation and consequently its solution can be strongly simplified.
Specifically, assuming that the considered domain is regularly sampled,
the solution $p(\textbf{x}(t+\epsilon) = \textbf{x}_j|\textbf{x}(t)=\textbf{x}_i)$ reduces to the value of the $(i,j)$ element of the
Laplacian matrix associated with the Markov matrix
\cite{fluid_diffusion1,fluid_diffusion3,fluid_diffusion2,fluid_randomwalk}.
This approach has then enabled the main contributions to graph-based data analysis, from normalized cut ratio derivation, to the development of spectral clustering techniques, to the most recent strategies for graph-based deep learning \cite{fluid_laplacian,specclust3,GEOMDL,fluid_graphsignalproc}.
At this point, it is worth noting that this approach allows us to obtain a thorough description of the data interactions
since affine data would result in high values of correlation among samples
\cite{fluid_diffusion3}.
It is also true that the correlation is quantified by taking into account the characterization of the local geometry of the dataset,
which in turn relies on the choice of the kernel metric, $\eta$.
Nevertheless, a given kernel will grasp specific properties and characteristics of the data set, so that the definition of the $\eta$ function should be based on the application scenarios and analysis tasks under exam, such as those in \cite{Luxburg,COUILLET20,CommunityDetection_SC1,CommunityDetection_SC2,CommunityDetection_SC3,CommunityDetection_SC4}.
Therefore, the kernel choice is critical in achieving an accurate and reliable characterization of the data interactions \cite{fluid_diffusion1,fluid_diffusion3,COUILLET16}.
However, some kernel functions could be preferred for their versatility (e.g., Gaussian kernel).
It is therefore difficult to understand (and eventually compare) \textit{a priori} the actual effectiveness and validity of the metrics used to quantify the edge weights in the graph representation derived according to the aforesaid approach \cite{fluid_diffusion1,COUILLET16}.
The distribution of the higher order moments of the feature statistics might affect the validity of the different $\eta$ functions, thus making the search for an optimal kernel to derive the graph representation of the data very difficult to target \cite{COUILLET16,fluid_diffusion3}.
On the other hand, it is possible to analyze the Laplacian matrix associated with the Markov matrix in terms of its eigenvalues to assess the ability of the considered graph representation to capture information about the data relationships \cite{COUILLET16}.
Specifically, when the graph Laplacian matrix is able to retrieve information on the underlying phenomena and processes captured by the considered observations, its eigenvalues tend to isolate from each other, leading to a spiked model of the graph Laplacian \cite{Luxburg,COUILLET16,CommunityDetection_Fortunato}.
Hence, understanding how the data covariance distribution could affect the eigenvalue spectrum of the Laplacian matrix would provide key information for the assessment of the kernel metric effectiveness.
To this aim, describing the Laplacian matrix characteristics in terms of random matrix theory can help \cite{COUILLET16}.
In particular, it is possible to prove that a Laplacian matrix with isolated eigenvalues corresponding to thematic clusters (e.g., classes, communities) within the data generated under the heat diffusion model can be written as a random matrix generated according to a spiked model with the same eigenvalues \cite{COUILLET16,randommatrix1, randommatrix2, randommatrix3}.
This allows the analytic investigation of the characteristics of the graph Laplacian, with special focus to its covariance \cite{COUILLET16}.
In fact, it is possible to identify several conditions that the mean and covariance matrices associated with each thematic cluster in the dataset must show
so that the Laplacian matrix can have isolated eigenvalues \cite{COUILLET16,randommatrix3,fluid_diffusion2}.
These conditions can be quantified by taking into account a few metrics derived from the heat diffusion model set-up and the inter-class covariance matrices \cite{COUILLET16}.
Specifically, let us assume that the samples in the dataset $\textbf{X}$ can be associated with $k$ classes.
It is therefore possible to compute for each class a length-$n$ vector $\textbf{m}_l = [m_{l_i}]_{i=1, \ldots, n}$ ($l \in \{1, \ldots, k\}$), where $m_{l_i} \in \mathbb{R}$ identifies the average value that the $i$-th record assumes across the samples belonging to class $l$.
Each class can be analogously
characterized by a non-negative definite covariance matrix $\textbf{C}_l \in \mathbb{R}^{n \times n}$ computed across the samples associated with the $l$-th class.
Moreover, let $N_l$ be the population of the $l$-th class within the dataset, i.e., the amount of samples belonging to the $l$-th class: hence, $\sum_{l=1}^{k} N_l = N$.
It is thus possible to write $\textbf{m}_l'= \textbf{m}_l - \bar{\textbf{m}} = \textbf{m}_l - \sum_{j=1}^{k} \frac{N_j}{N} \textbf{m}_j$.
Analogously,
$\textbf{C}_l'= \textbf{C}_l - \bar{\textbf{C}} = \textbf{C}_l - \sum_{j=1}^{k} \frac{N_j}{N} \textbf{C}_j$. Finally, we can define $\textbf{T} = \{T_{ij}\}_{(i,j) \in \{1, \ldots, k\}^2} \in \mathbb{R}^{k \times k}$, where $T_{ij}= \frac{1}{n} \text{Tr}[\textbf{C}_i'\textbf{C}_j']$, and $\textbf{t} = [t_l = \frac{1}{\sqrt{n}} \text{Tr}[\textbf{C}_l']]_{l=1, \ldots, k}$, $t_l \in \mathbb{R}$.
With this in mind, it is possible to prove \cite{COUILLET16} that the Laplacian matrix would show isolated eigenvalues if
the inter-class mean and covariance matrices must show as much energy (modeled by the $\textbf{m}$, $\textbf{T}$ and $\textbf{t}$ factors) as possible when the number of samples and/or features to be considered in the dataset increase, assuming that the first derivatives of the kernel function $\eta$ would not tend to 0 when $N \rightarrow +\infty$ and/or $n \rightarrow +\infty$ \cite{COUILLET16,randommatrix4,randommatrix1}.
In other terms, one or more of the following conditions must hold:
\begin{eqnarray}
||\textbf{m}_l'|| & \xrightarrow{N \rightarrow +\infty, n \rightarrow +\infty}& +\infty \nonumber \\
t_l & \xrightarrow{N \rightarrow +\infty, n \rightarrow +\infty}& +\infty \label{eq_isoleigen_cond}\\
T_{lj} & \xrightarrow{N \rightarrow +\infty, n \rightarrow +\infty}& +\infty,\nonumber
\end{eqnarray}
\noindent for some $l,j \in \{1, \ldots, k\}$ \cite{COUILLET16}.
These conditions are sufficient to guarantee that the graph Laplacian matrix derived under the heat diffusion model can show isolated eigenvalues. In other terms, it
is able to summarize the main properties of the dataset, i.e., to lead to a thorough characterization of the interactions and relationships among the samples \cite{COUILLET16}.
Nonetheless, when the data are sparse, these conditions might not be matched \cite{COUILLET20}.
In this case, the graph Laplacian matrix should be regularized to ensure that the energy of the higher order statistics could be concentrated, thus avoiding the aforementioned vanishing phenomenon that could jeopardize the presence of isolated eigenvalues \cite{COUILLET18,COUILLET20,COUILLET16}.
On the other hand, it is possible to show that this process can be valid only when the number of thematic clusters the considered records are meant to describe is very low (e.g., two) \cite{COUILLET20}.
\subsection{From heat to fluid: a new graph representation}
\label{sec_methmot}
\subsubsection{Motivations of a new graph representation}
\label{sec_meth_motnew}
As previously mentioned, investigating the graph structures induced by the datasets by exploiting the heat propagation analogy in terms of information inference has been proven to be effective and efficient for a wide range of applications and methodological research instances.
Nonetheless, these architectures might fail in addressing several data analysis issues that can occur when dealing with multimodal records, especially in operational scenarios \cite{multimod1,capacity}.
Specifically, we can summarize the major limitations of the classic heat diffusion model for graph investigation in the following points \cite{multimod1,GEOMDL,fluid_graphsignalproc}:
\begin{itemize}
\item \textit{adaptivity}: The learning system would have to deal with records showing multiple resolutions (either in time, space, metrical units). Moreover, noise (i.e., any undesired effect) might affect attributes/features/classes in different ways across the whole dataset, as well as in intra- and inter-class relationships. Hence, a single data model (in terms of propagation mechanisms, label assignment, similarity computation) might not be adequate for obtaining accurate and solid characterization of the records;
\item \textit{sparsity/missing data}: Not all the attributes of each sample might be relevant (by corruption, or by linear correlation). Using all the records to compute the similarity among samples might lead to dramatic degradation and/or bias of the analysis.
Further, the complexity of the data to be investigated might make classic impainting/interpolation techniques inadequate, thus jeopardizing the validity of the outcomes;
\item \textit{data mismatch/unbalance}:
the distribution of the thematic clusters in the dataset might be strongly unbalanced, and/or the training set might not contain samples associated with all the classes actually present in the dataset.
Thus, relying on a uniform statistical distribution as the source of the samples to be investigated might be an assumption too hard to match;
\item \textit{prediction/inference}: The dynamics of the phenomena captured by the data may be too complex to follow on a large scale, so classic graph-based approaches may lead to a strong informative outlier removal. Furthermore, the amount of samples to be used for training the learning/assimilation/inference models is typically very scarce, either in quantity and quality.
Not all estimates can therefore be drawn with sufficient confidence.
\end{itemize}
These issues would result in strong limitations of the data analysis schemes used to characterize the records.
They would in fact limit the full exploitation of the available training set,
either in terms of information extraction or context-aware inference.
Moreover, the aforementioned points would reverberate in terms of degradation of confidence and precision of the analysis, as well as restriction of the ability to fully explain and interpret the records under exam \cite{capacity,multimod2,multimod1,multimod7}.
With this in mind, the graph representation based on the heat diffusion model might sound intuitively inadequate to deal with all these limitations induced by modern data analysis.
Nevertheless, several approximate solutions have been proposed in technical literature, in order to mitigate the effect of these conditions whilst maintaining the data analysis steps compliant to the main assumptions presented in Section \ref{sec_meth_heat} \cite{GEOMDL,fluid_graphDL1,fluid_graphDL2,fluid_graphsignalproc,fluid_structeqmodel1}.
Thus, it is useful to provide a practical example to show how the properties in the previous Section that motivate the use of heat diffusion model are not matched when multimodal datasets are considered.
In particular, we can focus on the conditions in (\ref{eq_isoleigen_cond}), as they must be fulfilled for the classic graph representation to be adequate for information extraction from the considered datasets.
To this aim, we report in
Appendix \ref{app_motiv}
an analysis we conducted on a multimodal dataset that is considered as a benchmark in the remote sensing community \cite{Trentodata}.
Investigating this dataset from a theoretical, methodological, and experimental perspective supports the need for a novel graph representation so that the major limitations of the classic heat diffusion model could be addressed.
In particular, we have shown how the necessary conditions for the heat diffusion model to reliably characterize the data interactions (i.e., the properties in (\ref{eq_isoleigen_cond})) might not hold.
For these reasons, we propose using a fluid diffusion model to derive a new graph representation, that could be more flexible and versatile to address the modern data analysis needs and limitations.
Our findings are reported in the following subsection.
\subsubsection{Proposed approach}
\label{sec_fluid_repr}
We need to define the graph topology and the diffusion model to be applied in order to take into account the data analysis needs mentioned in the previous subsection.
To this aim, the definition of the process underlying the diffusion mechanisms across the graph should reflect a higher flexibility of the system, so to address the relevance of the features and the modeling of a confidence score for the propagation structure \cite{fluid_FokkerPlanck5,fluid_FokkerPlanck6}.
Hence, the system in (\ref{eq_dynsyst_heat}) should be replaced by a more complex stochastic differential model, such as follows:
\begin{equation}
\dot{\textbf{x}} = \textbf{a}(\textbf{x}) + \textbf{B}(\textbf{x})\dot{\textbf{w}},
\label{eq_dynsyst_fluid}
\end{equation}
\noindent where $\textbf{w}(t)$ is a $N$-dimensional Wiener process,
$\textbf{a}(\textbf{x})$ is a length-$n$ vector, whilst
$\textbf{B}(\textbf{x})$ identifies a $n \times N$ matrix \cite{fluid_FokkerPlanck3,fluid_FokkerPlanck5,fluid_FokkerPlanck6,fluid_FokkerPlanck4}.
In a fluid diffusion system model, \textbf{a} regulates the \textit{flow rate}, i.e., the velocity by which the diffusion can take place from one node to another in the system \cite{fluid_FokkerPlanck5,fluid_FokkerPlanck2}.
In general, it depends on the characteristics of the $\textbf{x}$ state, as well as on the local \textit{conductivity} properties of the fluid diffusion at local scale, and on the \textit{diffusivity} properties of the model at global scale \cite{fluid_FokkerPlanck2,fluid_FokkerPlanck3}.
In particular, the $\textbf{B}(\textbf{x})$ matrix summarizes the rate by which the diffusion can take place across the features of the considered system \cite{fluid_FokkerPlanck5,fluid_FokkerPlanck2}.
In more detail, the \textit{conductivity} properties (typically summarized by a $N \times N \times n$ tensor $\cal K$) model the ease with which the fluid diffusion can take place from one node to another in the system \cite{fluid_FokkerPlanck2,fluid_FokkerPlanck1,fluid_FokkerPlanck3}.
In our analogy, $\cal K$ would model the relevance of each single feature in computing the weight of each edge in the graph, hence quantifying the information diffusion at \textit{local} scale in the dataset.
On the other hand, the aforementioned \textit{diffusivity} is expected to model the ability of each feature to permit diffusion across the nodes in the diffusion system.
As such, it represents a \textit{global} quantity, that is summarized by the matrix $\tilde{\textbf{B}}(\textbf{x}) = \frac{1}{2} \textbf{B}(\textbf{x})\textbf{B}^T(\textbf{x})$.
In our analogy, $\tilde{\textbf{B}}(\textbf{x})$ quantifies a contextual weight,
aiming to model the ability of each edge in the graph
to convey the information, and is hence linked with a notion of
confidence that can be associated with local portions of
the whole manifold \cite{fluid_FokkerPlanck4,fluid_FokkerPlanck5,fluid_FokkerPlanck2}.
With this in mind, the exchange of information across nodes (modeled by $\textbf{a}$) would depend on the convection process ruled by the conductivity tensor ${\cal K}$ and the diffusion rate, computed as a derivative over the features of the diffusivity matrix $\tilde{\textbf{B}}$ \cite{fluid_FokkerPlanck8}.
Thus, the \textbf{a} vector is typically written as follows:
\begin{equation}
\textbf{a}(\textbf{x}) = \textbf{v}(\textbf{x}, {\cal K}) + \nabla \tilde{\textbf{B}}(\textbf{x}),
\label{eq_flowrate}
\end{equation}
\noindent where $\nabla = [\frac{\partial}{\partial x_i}]_{i=1, \ldots, n}$ as in (\ref{eq_dynsyst_heat}), and $\textbf{v}$ is the \textit{fluid transport velocity}, i.e., a function of the conductivity and the state \textbf{x}, ultimately modelling a dynamic weight based on ${\cal K}$ for the different components of \textbf{x} \cite{fluid_FokkerPlanck1,fluid_FokkerPlanck2}.
Analogously to the case reported in Section \ref{sec_meth_heat}, we are interested in deriving the expression of the transition probability density for the system in (\ref{eq_dynsyst_fluid}).
To achieve this goal, we can investigate the properties of this stochastic differential equation by taking advantage of the results provided by the It\^{o}'s lemma \cite{fluid_FokkerPlanck7} to the diffusion process in the form of (\ref{eq_dynsyst_fluid}).
This approach first introduces an arbitrary twice-differentiable scalar function $g(\textbf{x})$ where $\textbf{x}$ is defined as in (\ref{eq_dynsyst_fluid}).
Then, it considers the expansion in Taylor series of $g(\textbf{x})$.
Specifically, considering that $\textbf{w}$ is by definition a Wiener process, the Taylor expansion of $g(\textbf{x})$ can be truncated at the second order.
As such, it is possible to write as follows \cite{fluid_FokkerPlanck7}:
\begin{eqnarray}
dg(\textbf{x}) = & \{ \textbf{a}(\textbf{x}) \cdot \nabla g(\textbf{x}) \\
+ & \frac{1}{2} \sum_{i,j=1}^{n} \tilde{B}_{ij} \partial x_i \partial x_j g(\textbf{x}) \} dt \nonumber \\
+ & \nabla g(\textbf{x}) \cdot \textbf{B}(\textbf{x}) \textbf{w}, \nonumber
\label{eq_Ito1_fluid}
\end{eqnarray}
\noindent where $\tilde{B}_{ij} = \frac{1}{2}[\textbf{B}(\textbf{x}) \textbf{B}^T(\textbf{x})]_{ij}$.
At this point, recalling that expected value of an It\^{o} integral is zero \cite{fluid_FokkerPlanck7} and that $\textbf{w}$ identifies a Wiener process with $\frac{d}{dt}\mathbb{E}[\textbf{w}] = 0$, it is possible to write as follows:
\begin{equation}
\frac{d}{dt}\mathbb{E}[g(\textbf{x})] = \mathbb{E}\left[ \textbf{a}(\textbf{x}) \cdot \nabla g(\textbf{x}) + \sum_{i,j=1}^{n} \tilde{B}_{ij} \partial x_i \partial x_j g(\textbf{x}) \right].
\label{eq_Ito2_fluid}
\end{equation}
\noindent
The term $\frac{d}{dt}\mathbb{E}[g(\textbf{x})]$ can be computed taking into account the conditional probability density of a particle starting at $(\textbf{x}_0,t_0)$, i.e., $p(\textbf{x},t) = p(\textbf{x},t| \textbf{x}_0,t_0)$ \cite{fluid_FokkerPlanck7,fluid_FokkerPlanck5}.
Specifically, we can write (\ref{eq_Ito2_fluid}) as follows:
\begin{eqnarray}
\frac{d}{dt}\mathbb{E}[g(\textbf{x})] = & \frac{d}{dt} \int g(\textbf{x}) p(\textbf{x},t) d\textbf{x} = \int g(\textbf{x}) \frac{\partial p(\textbf{x},t)}{\partial t} d\textbf{x} \nonumber \\
= & \int \textbf{a}(\textbf{x}) \cdot \nabla g(\textbf{x}) p(\textbf{x},t) d\textbf{x} \\
+ & \sum_{i,j=1}^{n} \int \tilde{B}_{ij} \partial x_i \partial x_j g(\textbf{x})p(\textbf{x},t) d\textbf{x}. \nonumber
\label{eq_Ito3_fluid}
\end{eqnarray}
\noindent
Integrating by parts, this equation can be rewritten as follows \cite{fluid_FokkerPlanck7}:
\begin{equation}
\int \left[ \frac{\partial p(\textbf{x},t)}{\partial t} + \nabla \cdot \left[\textbf{a}(\textbf{x})p(\textbf{x},t) \right] - \beta'(\textbf{x},t) \right] g(\textbf{x}) d\textbf{x} = 0,
\label{eq_Ito4_fluid}
\end{equation}
\noindent where $\beta'(\textbf{x},t) = \sum_{i,j=1}^{n} \partial x_i \partial x_j (\tilde{B}_{ij} p(\textbf{x},t))$.
Then, since the $g$ function is arbitrary by construction, the aforesaid equation is satisfied when the term inside the square brackets is null.
With this in mind and expanding the $\beta'$ term, it is possible to write as follows (the details of the algebraic steps are detailed in
Appendix \ref{app_deriv1}):
\begin{eqnarray}
\frac{\partial p(\textbf{x},t)}{\partial t} + & \nabla \cdot \left[ \textbf{a}(\textbf{x})p(\textbf{x},t) \right] - \left[ \nabla \cdot (\nabla \tilde{\textbf{B}}(\textbf{x})) \right] p(\textbf{x},t) \nonumber \\
= & \left[ \nabla \tilde{\textbf{B}}(\textbf{x}) \right] \cdot \nabla p(\textbf{x},t) - \nabla \cdot \left[ \tilde{\textbf{B}}(\textbf{x}) \nabla p(\textbf{x},t)\right]
\label{eq_Ito5_fluid}
\end{eqnarray}
At this point,
using the representation in (\ref{eq_flowrate}), and considering the linearity of the divergence operator and its product rule \cite{fluid_FokkerPlanck7}, it is now possible to write the diffusion equation for this system as follows:
\begin{eqnarray}
\frac{\partial p(\textbf{x},t)}{\partial t} = & -& \nabla \cdot \left \{ \left[\textbf{a}(\textbf{x}) - \nabla \tilde{\textbf{B}}(\textbf{x})\right]p(\textbf{x},t) \right \} \nonumber \\
&+& \nabla \cdot \tilde{\textbf{B}}(\textbf{x})\nabla p(\textbf{x},t).
\label{eq_FokkerPlanck_fluid}
\end{eqnarray}
It is possible to recognize in this expression the Fokker-Planck equation for \textit{fluid diffusion in porous media}
\cite{fluid_FokkerPlanck1,fluid_FokkerPlanck2}.
Analogously to the heat diffusion analysis, the solution of (\ref{eq_FokkerPlanck_fluid}) can be retrieved by eigenanalysis of the Fokker-Planck operator.
Furthermore, the asymptotic analysis of the trajectories of the system in (\ref{eq_dynsyst_fluid}) can help in obtaining a thorough characterization of its solution \cite{fluid_FokkerPlanck1}.
It is indeed possible to analyze the geometry of the dataset by investigating the data by means of an approach based on Markov chain scheme \cite{fluid_diffusion1}.
In fact, it is possible to characterize the transition probability density
$p(\textbf{x}(t+\epsilon) = \textbf{x}_j|\textbf{x}(t)=\textbf{x}_i)$ by using a time domain random walk approach \cite{Fluid_TDRW1,Fluid_TDRW2}.
This scheme achieves the characterization of the whole system by exploiting the adjacency of the nodes.
Specifically, it aims to solve a Green function problem derived from (\ref{eq_dynsyst_fluid}) by imposing initial conditions and absorbing boundary conditions to the diffusion system centered on the node $i$ \cite{Fluid_TDRW1}.
This strategy aims to determine the transition probability density $p(\textbf{x}(t+\epsilon) = \textbf{x}_j|\textbf{x}(t)=\textbf{x}_i)$ by means of the first arrival time density $\phi_{ij}$ at the boundary between nodes $i$ and $j$, that would denote the joint probability of the transition to occur from node $i$ to node $j$ \cite{Fluid_TDRW1}.
Specifically, it is possible to write this transition probability density as follows:
\begin{equation}
p(\textbf{x}(t+\epsilon) = \textbf{x}_j|\textbf{x}(t)=\textbf{x}_i) = p_{ij} = \int_0^{+\infty} \phi_{ij} dt
\label{eq_fluid_trans_prob_ij}
\end{equation}
The details of the time domain random walk strategy are detailed in Appendix \ref{app_deriv2}.
By projecting the solution of the Green function problem in a Laplace space, we can write $p_{ij}$ as follows \cite{Fluid_TDRW1,Fluid_TDRW2}:
\begin{equation}
p_{ij} = \frac{|v_{+}^\dagger| \exp\left[v_{+}^\dagger \right] \csch \left[|v_{+}^\dagger|\right]}{\sum_{u \in \{+,-\}} |v_{u}^\dagger| \exp\left[\tilde{u}\cdot v_{u}^\dagger \right] \csch \left[|v_{u}^\dagger|\right]},
\label{eq_fluid_trans_prob_pij}
\end{equation}
\noindent where $\csch[z] = 1/\sinh[z] = 2/(\exp[z] - \exp[-z])$. Moreover,
$\tilde{u}$ is set to 1 when $u=+$, whilst $\tilde{u}=-1$ if $u$ is -. Finally, $v_{\pm}^\dagger = v_{\pm}/2\tilde{B}_\pm$, being:
\begin{eqnarray}
v_+ & = & v_{ij}, \\
v_- & = & \sum_{m \in {\cal N}(i) \setminus j} \frac{v_{im}}{|{\cal N}(i)|- 1}, \nonumber \\
\tilde{B}_+ & = & \tilde{B}_{ij}, \nonumber \\
\tilde{B}_- & = & \sum_{m \in {\cal N}(i) \setminus j} \frac{\tilde{B}_{im}}{|{\cal N}(i)|- 1}, \nonumber
\label{eq_v_B_pm}
\end{eqnarray}
\noindent where ${\cal N}(i)$ identifies the neighborhood of node $i$, i.e., the set of nodes adjacent to node $i$.
Finally, $\tilde{B}_{ij} \in [0,1]$ is the $(i,j)$-th element of the matrix $\tilde{\textbf{B}}$, whereas $v_{ij}$ is the transport velocity between node $i$ and node $j$ as in (\ref{eq_flowrate}).
This quantity is typically computed as $v_{ij} = - || {\cal K}_{ij:}^T \odot (\textbf{x}_i - \textbf{x}_j) ||_2 \in [0,1]$, where ${\cal K}_{ij:}$ is the length-$n$ row vector collecting the third dimension elements of the conductivity tensor $\cal K$ on the $(i,j)$ coordinates and $\odot$ is the Hadamard product \cite{fluid_FokkerPlanck1,fluid_FokkerPlanck2,fluid_FokkerPlanck3,fluid_FokkerPlanck4, fluid_FokkerPlanck5, fluid_FokkerPlanck6, Fluid_TDRW1, Fluid_TDRW2}.
Hence, it is possible to define a Markov matrix $\textbf{Q} = \{Q_{ij}\}_{(i,j) \in \{1, \ldots, N\}^2}$ that summarizes the edge weights of the new graph representation according to the transition probabilities in (\ref{eq_fluid_trans_prob_pij}), i.e., $Q_{ij}= p_{ij}/\sum_{i} p_{ij}$ \cite{fluid_diffusion1,fluid_FokkerPlanck6,fluid_diffusion3}.
The matrix $\textbf{Q}$ can be explored and used to address several tasks in multimodal data analysis and to improve the information extraction from the considered datasets.
In particular, the eigenanalysis of the
Laplacian matrix associated with $\textbf{Q}$ (whose $(i,j)$ element can be defined as $\sum_{k=\{1,\ldots,N\}\setminus i}Q_{ik}$ if $i=j$, and as $- Q_{ij}$ otherwise) can be directly connected to the solution of the system in (\ref{eq_FokkerPlanck_fluid}).
In fact, in general the solution of the fluid diffusion equation can be written in terms of the eigenfunction expansion, i.e., $p(\textbf{x},t)$ can be expressed as follows:
\begin{equation}
p(\textbf{x},t) = \sum_{i=0}^{+ \infty} \omega_i \exp[-\lambda_i t] \varphi_i(\textbf{x}),
\label{eq_FokkerPlanck_solution}
\end{equation}
\noindent where $\lambda_i$ are the sorted eigenvalues of the fluid Fokker-Planck operator (with $\lambda_0 =0$),
$\varphi_i$ are
their corresponding eigenfunctions, and the coefficients $\omega_i$ depend on the initial conditions \cite{fluid_diffusion1,fluid_FokkerPlanck6,fluid_diffusion3}.
It is worth noting that numerical approximations of these eigenfunctions can be computed
when the considered data are characterized by a low amount of records (e.g., three).
On the other hand, when high dimensional data are considered - such as the multimodal data we are considering in this work - it is not possible to use numerical solutions to solve this equation.
The only valid approach would be to simulate the trajectories of the stochastic differential model in (\ref{eq_dynsyst_fluid}), which implies the use of statistical methods to analyze the simulated results and explore the validity of low and high frequency trends, as well as the mean transition times among them \cite{fluid_FokkerPlanck5,fluid_FokkerPlanck6,fluid_diffusion1}.
It has been proven that the solution of (\ref{eq_FokkerPlanck_solution}) can be described by reduced set of $\kappa$ eigenfunctions, which can carry significant information on the density and geometry of the data under exam \cite{fluid_FokkerPlanck1,fluid_diffusion1,fluid_FokkerPlanck6}.
To obtain a reliable characterization of the relevant eigenvectors and eigenvalues,
it is useful to explore the asymptotic behavior of the diffusion process in the probability space.
This analysis shows that the eigenvalues of the matrix $\textbf{Q}$ asymptotically
correspond to the relevant $\kappa$ eigenfunctions that can be used to achieve a solid understanding of the solution in (\ref{eq_FokkerPlanck_solution}), as previously mentioned \cite{fluid_FokkerPlanck2,fluid_FokkerPlanck1,fluid_FokkerPlanck5,fluid_FokkerPlanck6,fluid_diffusion1,fluid_diffusion3}.
This result is extremely interesting, and it summarizes the key-role that the matrix $\textbf{Q}$ can have in providing a thorough and reliable understanding of the properties underlying the graph topology induced by the considered datasets, as well as the information propagation mechanisms.
In Appendix \ref{app_Qmat},
we provide an example to visualize how the proposed definition of the \textbf{Q} matrix could improve the characterization of data interactions with respect to the classic graph representation based on heat diffusion mechanism.
In this work, we focus our attention on the investigation of $\textbf{Q}$ in order to learn the structure of the data under exam, and to enable an effective functional analysis of the records, with special focus to multimodal data analysis.
The next Section summarizes the main steps of the proposed method for fluid community detection.
\section{Fluid community detection}
\label{sec_fluidCD}
\subsection{Background and related works}
\label{sec_methback}
Several
criteria at global and local scale can be used to identify communities in graphs.
Since in this work we focus our attention towards the detection of communities that are separated, we report in this Section an overview of non-overlapping community detection methods.
Moreover, we summarize the main categories community detection algorithms can be grouped in, according to the strategy they employ \cite{CommunityDetection_Fortunato}.
In particular, it is possible to categorize these methods in seven main groups: 1) graph partitioning; 2) hierarchical clustering; 3) partitional clustering; 4) spectral clustering; 5) dynamic community detection; 6) statistical inference-based community detection; and 7) hybrid methods.
We report in \ref{app_review}
a brief overview of these algorithms.
The methods
are typically designed to address problems that could be described in a monovariate data analysis system \cite{CommunityDetection_Fortunato,Fluid_multimodML, Luxburg}.
In particular, these architectures are developed at theoretical level to address community detection problems when a single source of information is used to generate the data to be analyzed.
Nevertheless, multimodal community detection is typically addressed by extending these approaches to records acquired by multiple modalities \cite{multimod1,multimod7,multimod2,Fluid_multimodML}.
In particular, the similarity between samples (either in terms of edge betweenness, modularity, Euclidean distance, geodesic metric) is computed along all the features available \cite{GEOMDL,graphclust_schaub,fluid_graphsignalproc}.
In other terms, the aforesaid methods are applied to datasets where the records acquired by the diverse modalities are stacked and vectorized, so that each sample could be considered as a point in an extended multidimensional feature space.
This approach is very popular within the scientific community, because of its high degree of implementability.
However, this does not always reflect in good performance in terms of community detection, especially in operational scenarios \cite{multimod1,fluid_graphDL2,fluid_graphsignalproc,capacity,Fluid_multimodML}.
Hence, directly applying these architectures to multimodal data analysis might lead to strong limitations of the multimodal community detection performance.
In other terms,
successful community detection is achieved only when datasets characterized by low diversity, low sparsity, high reliability, and low variability can be found across the considered records (see for instance \cite{CommunityDetection_Bertozzi1, CommunityDetection_Bertozzi2, CommunityDetection_Bertozzi3, CommunityDetection_Bertozzi4}).
As these characteristics are hard to be found in multimodal datasets, especially when addressing operational scenarios \cite{multimod1,Fluid_multimodML}.
Hence, this makes this research avenue an open field for a successful development of multimodal data analysis methods \cite{Fluid_multimodML,CommunityDetection_DL}.
Nevertheless, it is also true in recent years methods designed to address multimodal data analysis have been proposed.
One first approach relies on the design of $n$-partite graphs, where different kind of nodes are used to represent the diverse modalities under exam \cite{CommunityDetection_multi1, CommunityDetection_multi2,CommunityDetection_multi3}.
On these graph structures, partitional clustering techniques are applied to retrieve the community structures hidden within the data.
These methods are typically showing pretty high computational complexity, such that it is proven that they might achieve successful performance when the trade-off between diversity and thematic clusters to be identified is good (i.e., when few modalities with numerous communities are considered, or when multiple modalities and a small amount of communities are taken into account).
This is a major factor that must be considered when employing these techniques \cite{Fluid_multimodML,CommunityDetection_Fortunato,CommunityDetection_DL}.
Similar results can be registered when the records acquired by diverse modalities are separately processed, to be then fused at a later stage \cite{CommunityDetection_align1,CommunityDetection_align2,CommunityDetection_align3}.
In this case, methods retrieved from graph partitioning, partitional clustering, and/or hybrid approaches are used to analyze the different sources of information.
Then, the obtained information is integrated according to optimization criteria designed to maximize the alignment between modalities and therefore identify communities are are more homogeneous across the diverse sources of information \cite{CommunityDetection_align2}.
Although this approach can be performed with pretty low latency (especially when high performance computing platforms are available), it can hardly be generalized for multimodal datasets where diverse records, characterized by various statistical distributions, high variability and sparsity ae considered, limiting the range of applications that could actually benefit of this strategy.
Recently, methods relying on the multiview data analysis approach have been introduced for community detection \cite{CommunityDetection_view1,CommunityDetection_view2}.
In this case, the different $n$ modalities are assumed to generate $n$ graphs that are then investigated by means of spectral clustering techniques.
Then, an optimization process based on normalized cut approach is performed, in order to identify the most informative clusters across the whole dataset.
This approach typically shows low computational complexity.
However, it is also true that it requires the modalities to be as less as diverse as possible, so that the joint cut across the graphs can be accurately performed \cite{CommunityDetection_view1}.
Moreover, the multiple graphs are expected to show homogeneous characteristics imposed by the communities underneath. This is a pretty strong requirement, since in multimodal data analysis not all the features are typically reliable, significant and/or informative at the same level \cite{CommunityDetection_view2}.
As such, using this scheme to integrate heterogeneous sources of information at large scale might be cumbersome.
\subsection{Proposed approach}
\label{secmeth}
Taking into account the definition of the $\textbf{Q}$ matrix as a result of the fluid diffusion model (as introduced in Section \ref{sec_fluid_repr}), there are several properties that can be particularly interesting to address the community detection task in an accurate and efficient way.
In this work, we take advantage of the aforementioned characteristics of the eigenanalysis of the flow velocity matrix to identify the most informative clusters within the considered multimodal dataset.
As such, the approach we propose in this paper could fall within the spectral clustering category of community detection mentioned in Section \ref{sec_methback}.
This choice helps us in achieving accurate and reliable understanding of the data interactions in closed form and with rigorous convergence, while guaranteeing a simple implementation and high efficiency of the unsupervised community detection approach.
It is also worth noting that the focus of this paper is on unsupervised analysis of the functional relationships among samples, i.e., no contextual information (either in shape of side information, or \textit{a priori} knowledge, nor semantic knowledge) could be taken into account to achieve a fully data drive investigation.
As such, the diffusivity term $\tilde{\textbf{B}}$ in (\ref{eq_flowrate}) can be then set to the identity matrix throughout the following Sections.
\subsubsection{Definition of the \textbf{Q} matrix}
\label{sec_meth_ADR}
In order to provide a thorough investigation of the complex relationships hidden within the records in multimodal datasets according to the fluid graph representation previously introduced, we first need to define the permeability tensor ${\cal K}$ in (\ref{eq_flowrate}).
To this aim, it would be instrumental to investigate the significance of the features associated with each sample in the considered dataset.
In this work, we propose to address this task by exploring the relevance of the features at global (i.e., across all modalities) and local (i.e., across samples for each feature) scale.
Following the successful approach proposed in \cite{ensembleADR}, we quantify the multiscale significance of the features by using information theory-based metrics.
Specifically, we consider to measure the degree of redundancy and intercorrelation between features across the whole dataset by employing mutual information \cite{mackaybook,MUTINF6}.
This choice allows us to assess the redundancy and dependence among features we could record across the dataset.
In fact, mutual information quantifies the shared information between two random variables \cite{MUTINF6}.
This is especially relevant when complex datasets, which lead to fully connected graph representations, are taken into account \cite{ELdata_pruning5,mackaybook}.
In other terms, let us consider a dataset $\textbf{X}$ that consists of $N$ samples and $n$ features, i.e., $\textbf{X} = \{\textbf{x}_i^T\}_{i=1, \ldots, N}$, $\textbf{x}_i = [x_{ij}]_{j=1, \ldots, n}$, that induces a graph ${\cal G} =({\cal V}, {\cal E})$ where ${\cal V}$ and ${\cal E}$ identify the node and edge sets, respectively.
Moreover, the $i$-th node $v_i$ is associated with the $i$-th sample $\textbf{x}_i$.
Then, we can write the mutual information between two features $n_1$ and $n_2$ (where $n_j \in \{1, \ldots, n\}$) as follows:
\begin{eqnarray}
w_{n_1 n_2}^{\text{MI}} =& \sum_{i = 1}^N\sum_{j = 1}^N p(x_{in_1},x_{jn_2})\log \frac{p(x_{in_1},x_{jn_2})}{p(x_{in_1})p(x_{jn_2})}, \label{eq_GKMI2}
\end{eqnarray}
\noindent where $p(y,z)$ is the joint density function of $y$ and $z$, and $p(z)$ is marginal of $z$.
It is worth to note that, according to this definition, large values of $w_{n_1 n_2}^{\text{MI}}$
imply redundancy in information. On the other hand, low values of $w_{n_1 n_2}^{\text{MI}}$
imply synergy (novelty) \cite{mackaybook}.
On the other hand, it is important to evaluate
the significance of the local properties of the features for each sample.
In this way, we can take into account the local characteristics of each feature, so to address
the variability of the statistical properties of the data across the complete dataset.
In other terms, we should identify a metric for which, if the features $n_1$ and $n_2$ are very similar across the $m$-th sample, the value will be large.
In this case, it would be possible to assume that using just one of these features would be enough to obtain a robust and precise understanding of the given sample.
Conversely, a small value of this metric would mean that the two features are independent from each other such that they should be both taken into account to characterize the sample \cite{Luxburg,specclust1}.
The distance metric based on Gaussian kernel would shows all these properties.
Thus, we propose to quantify the difference between two features $n_1$ and $n_2$ for the $m$-th sample as follows:
\begin{eqnarray}
w_{m_{n_1 n_2}}^{\text{GK}} & =& \exp \left[-\frac{|| \underline{x}_{m n_1}- \underline{x}_{m n_2} ||^2}{2\sigma^2} \right], \label{eq_GKMI1}
\end{eqnarray}
\noindent where the value of $\sigma$ controls the width of the Gaussian kernel \cite{Luxburg,COUILLET16}.
At this point, we can build for each sample $m$ a graph $\mathbb{G}_m=({\cal V},{\cal E}_m^{\text{GK}}, {\cal E}^{\text{MI}})$.
The $n$-th vertex $v_n$ in ${\cal V}$ identifies the $n$-th feature of the $m$-th sample.
Two vertices $v_{n_1}$ and $v_{n_2}$ are connected by two kinds of edges, ${\cal E}_{m_{n_1 n_2}}^{\text{GK}}$ and ${\cal E}_{{n_1 n_2}}^{\text{MI}}$, whose weights are computed according to (\ref{eq_GKMI1}) and (\ref{eq_GKMI2}), respectively.
This structure can be used as a platform to perform adaptive feature selection across the dataset according to the guidelines of spectral clustering approach \cite{Luxburg,specclust1,specclust2,specclust3,COUILLET18,COUILLET20}.
In particular, the aforesaid weights can be arranged in matrix form, so to generate two adjacency matrices associated with $\mathbb{G}_m$, i.e., $\textbf{W}_m^{\text{GK}} = \{w_{m_{ij}}^{\text{GK}}\}_{(i,j)\in \{1, \ldots, n\}^2}$ and $\textbf{W}^{\text{MI}} = \{w_{ij}^{\text{MI}}\}_{(i,j)\in \{1, \ldots, n\}^2}$.
For the first matrix, it is possible to define a degree matrix $\textbf{D}_m^{\text{GK}} =
\text{diag}(\textbf{d}_m^{\text{GK}})$, $\textbf{d}_m^{\text{GK}} = [d_{m_j}^{\text{GK}} = \sum_{l=1}^{n} w_{m_{jl}}^{\text{GK}}]_{j=1, \ldots, n}$.
Hence, it is possible to define a normalized Laplacian matrix associated with $\textbf{W}_m^{\text{GK}}$ as $\bar{\textbf{L}}_m^{\text{GK}} = \textbf{I} - \textbf{D}_m^{\text{GK}^{-1/2}}
\textbf{W}_m^{\text{GK}}
\textbf{D}_m^{\text{GK}^{-1/2}}$.
Analogously, we can define $\textbf{D}^{\text{MI}} = \text{diag}(\textbf{d}^{\text{MI}})$, $\textbf{d}^{\text{MI}} = [d_j^{\text{MI}} = \sum_{l=1}^{n} w_{jl}^{\text{MI}}]_{j=1, \ldots, n}$, as the degree matrix associated with $\textbf{W}^{\text{MI}}$, and
$\bar{\textbf{L}}^{\text{MI}} = \textbf{I} - \textbf{D}^{\text{MI}^{-1/2}}
\textbf{W}^{\text{MI}}
\textbf{D}^{\text{MI}^{-1/2}}$ as its associated normalized Laplacian matrix.
With this in mind, identifying the relevant features for each sample in the dataset can be described as partitioning the graph $\mathbb{G}_m$
such that the vertices of the same subgraph have strong connections via both links, while the vertices from different subgraphs have one or two weak connections.
In spectral clustering, this problem can be written as follows:
\begin{eqnarray}
\begin{cases}
\min_{\textbf{H}} & \text{Tr}( \textbf{H}^T \bar{\textbf{L}}_m^{\text{GK}} \textbf{H})
\\
\min_{\textbf{H}} & \text{Tr}( \textbf{H}^T \bar{\textbf{L}}^{\text{MI}} \textbf{H})
\end{cases}
\label{eq_GKMI3}
\end{eqnarray}
\noindent where $\textbf{H}$ represents the matrix of the indicator vectors, and $\textbf{H}\textbf{H}^T = \textbf{I}$ \cite{Luxburg,specclust1,specclust2}.
The solution of (\ref{eq_GKMI3}) is given by the common
eigenspace of the two normalized Laplacian matrices.
Hence, this problem translates in identifying the set of joint eigenvectors $\textbf{V}_m = [\textbf{v}_{m1}, \dots, \textbf{v}_{mn}]$ that solves the following \cite{Ablin2018,Pham}:
\begin{eqnarray}
\min_{\textbf{V}_m}
\log \frac{\left|\text{diag}(\textbf{V}_{m}^T\bar{\textbf{L}}_m^{\text{GK}} \textbf{V}_{m})\right|}{\left|\textbf{V}_{m}^T\bar{\textbf{L}}_m^{\text{GK}} \textbf{V}_{m}\right|}
+ \log \frac{\left|\text{diag}(\textbf{V}_{m}^T\bar{\textbf{L}}^{\text{MI}} \textbf{V}_{m})\right|}{\left|\textbf{V}_{m}^T\bar{\textbf{L}}^{\text{MI}} \textbf{V}_{m}\right|},
\label{eqn:criterion}
\end{eqnarray}
At this point, $\textbf{H}$ in (\ref{eq_GKMI3}) contains the eigenvectors corresponding to the $K_m$ lowest and non-null eigenvalues.
It is worth recalling that the cardinality $K_m$ identifies the number of relevant features in the $m$-th sample according to the spectral clustering guidelines \cite{Luxburg,COUILLET18,COUILLET20,specclust1,specclust2,specclust3}.
In fact, the number of relevant features equals the number of informative eigenvalues that
can be defined as the local minima of the eigenvalues' difference curve \cite{Luxburg,COUILLET20,specclust1}.
To this aim, the kneedle algorithm can be employed to select the optimal $K_m$ \cite{kneedle}.
It is therefore crucial that the difference among the eigenvalues is well pronounced, so that the separation between eigenvalues associated with relevant and non-informative features can be easily carried out.
Once the $K_m$ eigenvalues have been identified, it is possible to select the set of relevant features for the $m$-th sample as the centroids of the associated clusters.
This information will finally be used to define the elements of the ${\cal K}$ tensor in (\ref{eq_flowrate}).
Specifically, if the $l$-th feature has been selected as relevant for both sample $m_1$ and $m_2$, then ${\cal K}_{m_1 m_2 l} =1$; otherwise, ${\cal K}_{m_1 m_2 l} =0$.
It is worth noting that this simple set-up could be made more sophisticated by allowing the values of ${\cal K}$ to live in $\mathbb{R}$.
Future works will be dedicated to investigate the impact of this choice in the effective use of the proposed fluid diffusion model in multimodal data analysis.
Analogously, the definition of the distance operator to be used to determine the $v$ values in (\ref{eq_v_B_pm}) can be subject for deep investigation in the near future.
Nevertheless, in this work we can assume without losing generality (and considering the observations on the continuity of the information propagation drawn in Section \ref{sec_methmot}) that the each $v_{ij}$ in (\ref{eq_v_B_pm}) will be based on the norm-2 between nodes \cite{Fluid_TDRW1,Fluid_TDRW2}.
At this point, we can compute the \textbf{Q} matrix that has been introduced in Section \ref{sec_fluid_repr}.
The next steps of the proposed community detection strategy consist in the eigenanalysis of the \textbf{Q} matrix. The next paragraphs summarize the main steps of the approach we present in this work.
\subsubsection{Community detection based on fluid Laplacian matrix}
\label{sec_meth_fluidLapl}
As previously mentioned, the \textbf{Q} matrix is the core of the community detection algorithm based on fluid diffusion that we introduce in this work.
We can indeed build a new matrix $\textbf{F}= \textbf{D}-\textbf{Q}$, where \textbf{D} is a $N \times N$ diagonal matrix such that $D_{ii} = \sum_{j=\{1,\ldots,N\}\setminus i} Q_{ij}$.
As such, \textbf{F} is a matrix where all the
diagonal elements are positive, and the other elements
are negative.
Therefore, \textbf{F} is invertible.
Let us further analyze the properties of \textbf{F}.
Specifically, let us consider a generic vector \textbf{z}, and let us derive the analytical solution of the $\textbf{z}^T \textbf{F} \textbf{z}$ function.
It is possible to prove that the following holds \cite{CommunityDetection_SC2,COUILLET20}:
\begin{eqnarray}
\textbf{z}^T\textbf{F}\textbf{z} & = & \textbf{z}^T \textbf{D} \textbf{z} - \textbf{z}^T \textbf{Q} \textbf{z} \nonumber \\
& = & \sum_{i=1}^{N} D_{ii} z_i^2 - \sum_{i,j=1}^{N} Q_{ij}z_i z_j \\
& = & \frac{1}{2} \left( \sum_{i=1}^{N} D_{ii} z_i^2 - 2\sum_{i,j=1}^{N} Q_{ij}z_i z_j + \sum_{j=1}^{N} D_{jj} z_j^2 \right) \nonumber \\
& = & \frac{1}{2} \sum_{i,j=1}^{N} Q_{ij} (z_i - z_j)^2. \nonumber
\label{eq_fluidLapl_deriv}
\end{eqnarray}
\noindent Therefore, the matrix \textbf{F} can be considered as a Laplacian matrix, and will take the name of \textit{fluid Laplacian matrix}.
Moreover, this system can be used to construct a graph that could be partitioned in communities.
In particular, in order to find a partition of the graph such
that the edges between different communities have lower
weight and the edges within the same community have a
higher weight, we can apply the Normalized Cut algorithm \cite{normalizedCut}.
In other terms, the graph induced by \textbf{Q} can be partitioned in $K_F$ connected components $C_\kappa$, $\kappa=1, \ldots, K_F$ (where $\bigcup_{\kappa=1}^{K_F} C_\kappa = {\cal V}$, and $C_{\kappa_1} \cap C_{\kappa_2} = \emptyset$ $\forall \kappa_1, \kappa_2 \in \{1, \ldots,K_F\}^2$, $\kappa_1 \neq \kappa_2$) by minimizing over $\{C_\kappa\}_{\kappa = 1, \ldots, K_F}$ the NormalizedCut function $NC_{K_F}$, which can be written as follows:
\begin{eqnarray}
NC_{K_F} &=& \frac{1}{2} \sum_{\kappa=1}^{K_F} \frac{\zeta(C_\kappa,\bar{C}_\kappa)}{\tilde{C}_\kappa},
\label{eq_specclust2}
\end{eqnarray}
\noindent where $\bar{C}_\kappa$ represents the complement of the $\kappa$-th partition over the vertex set ${\cal V}$, $\tilde{C}_\kappa$ is a measure of the width of the $\kappa$-th partition (typically expressed in volume $\mathsf{vol}(C_\kappa) = \sum_{v_i \in C_\kappa} \sum_{j=1}^{|{\cal V}|} Q_{ij}$), and $\zeta(C_\kappa,\bar{C}_\kappa) = \sum_{v_i \in C_\kappa, v_j \in \bar{C}_\kappa} Q_{ij}$. \cite{normalizedCut}.
The minimization of $NC_{K_F}$ leads to have large weights for the edges connecting the nodes within $C_\kappa$, while the edges connecting nodes within $C_\kappa$ with the nodes in its complement $\bar{C}_\kappa$ will show small weights.
Furthermore, this operation can be described in terms of the eigenvectors of the normalized fluid Laplacian matrix $\bar{\textbf{F}} = \textbf{D}^{-1/2} \textbf{F} \textbf{D}^{-1/2}$.
In other terms, the $NC_{K_F}$ optimization can be written as follows:
\begin{eqnarray}
\min_{\textbf{J}} \text{Tr}( \textbf{J}^T \bar{\textbf{F}} \textbf{J}) & \text{subject to} & \textbf{J}^T\textbf{J}=\textbf{I},
\label{eq_specclust4}
\end{eqnarray}
\noindent where \textbf{J} is the matrix of the first smallest $K_F$ eigenvectors of $\bar{\textbf{F}}$.
Hence, in order to solve this problem, it is possible to employ the kneedle algorithm \cite{kneedle} to select the best value of $K_F$, and then run a traditional $K_F$-means algorithm over \textbf{J} (considering the rows of \textbf{J} as nodes) in order to identify the $K_F$ communities \cite{COUILLET18,Luxburg,CommunityDetection_SC2}.
In this way, we can guarantee a high degree of implementability and efficiency of the system, whilst achieving a thorough unsupervised characterization of the multimodal data under exam.
The following Section provides tests to validate this set-up with respect to state-of-the-art methods.
\section{Experimental results and discussion}
\label{secresult}
Several multimodal datasets representative of different research fields have been used to validate the novel graph representation based on fluid diffusion model and to test the performance of the proposed community detection method.
In this Section, we first summarize the main characteristics of the datasets we have taken into account.
Then,
we report the performance of the proposed community detection framework based on fluid diffusion model.
\subsection{Datasets}
\label{sec_exp_res}
We tested the proposed approach on three very diverse datasets, focusing on three different research fields: remote sensing, brain-computer interface, and photovoltaic energy.
\subsubsection{Multimodal remote sensing (RS)}
\label{sec_exp_RS}
First,
we considered a multimodal
dataset consisting of LiDAR and hyperspectral records acquired over the University of Houston campus and the neighboring urban area, and was distributed for the 2013 IEEE GRSS Data Fusion Contest~\cite{data_houston}.
Specifically:
\begin{itemize}
\item the size of the dataset is 1905$\times$349 pixels, with spatial resolution equal to 2.5m;
\item the final dataset consists of $N$=151 features. In fact, the hyperspectral dataset includes 144 spectral bands ranging from 0.38 to 1.05 $\mu{m}$, whilst the LiDAR records includes one band and 6 textural features;
\item the available ground truth labels consists of $K_C=15$ classes.
\end{itemize}
\subsubsection{Multimodal brain-computer interface (BCI)}
\label{sec_exp_BCI}
The second dataset we considered was collected by means of brain-computer interface \cite{BrainComputer1}.
Specifically, the records were collected by means of
60-channel electroencephalography
(EEG), 7-channel electromyography
(EMG) and 4-channel electro-oculography (EOG) on $K_C=11$ intuitive upper extremity
movements from 25 participants.
A 3-sessions experiment was carried out, and 3300 trials per participant were collected.
According to the notation we have used in Section \ref{secmeth}, the final dataset consists of $N=71 (=60+4+7)$ multimodal features for a total sum of 82500 samples across all the participants.
\subsubsection{Multimodal photovoltaic energy (PV)}
\label{sec_exp_PV}
The final dataset was acquired in order to monitor the photovoltaic energy produced between July 21 and Aug. 17, 2018 at the University of Queensland, Australia \cite{data_PhotoVoltaic}.
The records that have been collected by weather ground stations can be listed as follows:
\begin{itemize}
\item instantaneous and average wind speed [km/h] and direction [deg];
\item temperature [deg];
\item relative humidity [\%];
\item mean surface level pressure [hPa];
\item accumulated rain [mm];
\item rain intensity [mm/h];
\item accumulated hail [hits/cm2];
\item hail intensity [hits/cm2hr];
\item solar mean [W/m2].
\end{itemize}
This summed to 1440 samples acquired for each day, summing up to a dataset of 1440 $\times$ 28 records.
For each sample, the photovoltaic energy [W/h] is recorded: $K_C=10$ classes uniformly drawn based on this parameter are considered.
The considered data analysis task consists of assigning a class to all the samples by investigating the $N=12$ heterogeneous features.
\subsection{Results}
\label{sec_eval}
In order to provide a thorough investigation of the actual impact of the proposed approach, we conducted several experiments at different levels.
Specifically, we tested the proposed architecture in terms of design choices, validity of the proposed model, parameter sensitivity, and community detection performance.
First, we investigated whether the proposed adaptive dimensionality reduction method for the definition of the ${\cal K}$ tensor in (\ref{eq_fluid_trans_prob_pij}) that is used then to determine the fluid Laplacian matrix in Section \ref{sec_meth_fluidLapl}.
In particular, we tested the ability of the method in Section \ref{sec_meth_ADR} to reliably identify relevant features across complex datasets, so that the construction of the ${\cal K}$ tensor could be carried out.
To this aim, we considered the three aforementioned datasets, and added records generated by noise-like process characterized by a Gaussian distribution and a signal-to-noise ratio (SNR) set to 20dB to each sample.
Hence, we obtained for each dataset an additional subset of attributes (approximately 30$\%$ of the original amount, i.e., we added 50, 23, and 4 features for the datasets in Section \ref{sec_exp_RS}-\ref{sec_exp_PV}, respectively) that were clearly irrelevant for the characterization of the phenomena under exam and therefore simulating a set of corrupted, noisy, and non-informative records.
Then, we applied the adaptive dimensionality reduction approach for the determination of ${\cal K}$, as well as other four state-of-the-art feature selection methods on the new total datasets: the algorithms we considered were based on diverse strategies, i.e., genetic algorithm (GA) \cite{CommunityDetection_FS4}, structure preserving feature selection (SPFS) \cite{CommunityDetection_FS1}, regularization-based feature selection (RegFS) \cite{CommunityDetection_FS2}, and filter-based feature selection (FilterFS) \cite{CommunityDetection_FS3}.
Our goal is to check whether these methods are able to discriminate the real records from the added (irrelevant) ones in the obtained extended datasets.
We carried out 100 runs, and summarized the results in Fig. \ref{fig_res_FS_RS}-\ref{fig_res_FS_PV}.
\begin{figure}[htb]
\centering
\includegraphics[width=1\columnwidth]{hist_FluidCD_FS_RS_210804-eps-converted-to.pdf}
\caption{Fraction of non-corrupted features selected for the definition of the ${\cal K}$ in (\ref{eq_fluid_trans_prob_pij}) when analyzing the extended dataset in Section \ref{sec_exp_RS} (multimodal remote sensing) achieved by using the method described in Section \ref{secmeth} (ADR), genetic algorithm (GA) \cite{CommunityDetection_FS4}, structure preserving feature selection (SPFS) \cite{CommunityDetection_FS1}, regularization-based feature selection (regFS) \cite{CommunityDetection_FS2}, and filter-based feature selection (FilterFS) \cite{CommunityDetection_FS3}. The ADR method is the approach we propose to establish the fluid graph representation in this paper.}
\label{fig_res_FS_RS}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=1\columnwidth]{hist_FluidCD_FS_BCI_210804-eps-converted-to.pdf}
\caption{Fraction of non-corrupted features selected for the definition of the ${\cal K}$ in (\ref{eq_fluid_trans_prob_pij}) when analyzing the extended dataset in Section \ref{sec_exp_BCI} (multimodal brain-computer interface). The same notation as in Fig. \ref{fig_res_FS_RS} applies here.}
\label{fig_res_FS_BCI}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=1\columnwidth]{hist_FluidCD_FS_PV_210804-eps-converted-to.pdf}
\caption{Fraction of non-corrupted features selected for the definition of the ${\cal K}$ in (\ref{eq_fluid_trans_prob_pij}) when analyzing the extended dataset in Section \ref{sec_exp_PV} (multimodal photovoltaic energy). The same notation as in Fig. \ref{fig_res_FS_RS} applies here.}
\label{fig_res_FS_PV}
\end{figure}
Taking a look to these histograms, it is possible to appreciate how the approach we proposed in this work is actually able to outperform the state-of-the-art methods by selecting almost exclusively the original (relevant) features for dimensionality reduction.
Moreover, the high kurtosis of the histograms achieved by means of the adaptive dimensionality reduction method emphasizes the robustness of this algorithm to select relevant features across all the tests we have performed.
On the other hand, the variability of the performance of the other methods appear very high, so that their outcomes do not appear solid.
Thes observations are valid for all the datasets we presented in Section \ref{sec_exp_res}, hence supporting our choice of using the adaptive dimensionality reduction method presented in Section \ref{sec_meth_ADR} for the construction of the fluid Laplacian matrix.
Let us now focus our attention on the actual procedure for community detection based on the fluid diffusion model that we propose in this work.
In particular, it is worth to recall that the main steps for the community detection technique reported in Section \ref{sec_meth_fluidLapl} are fundamentally based on the eigenanalysis of the fluid Laplacian matrix \textbf{F}.
Thus, the method in Section \ref{sec_meth_fluidLapl} could be considered as an instance of the spectral clustering approach for community detection, according to the characteristics summarized in Appendix \ref{sec_back_spec}.
As such, the ability to discriminate the lowest eigenvalues from the overall eigenvalues set, so that the identification of the communities in the dataset can be accurately carried out \cite{Luxburg}.
Therefore, it is important to analyze the eigenspectrum of the computed eigenvalues, so to retrieve a solid understanding of the actual characterization ability the considered spectral clustering-based architecture might have.
Hence, in order to obtain a first assessment of the actual impact of the proposed community detection method, we assessed the improvement provided by the use of the fluid Laplacian matrix for spectral clustering.
Specifically, we computed the eigenspectrum resulting from the analysis of the datasets in Section \ref{sec_exp_res} by means of the scheme proposed in Section \ref{sec_meth_fluidLapl} and several spectral clustering methods introduced in technical literature, i.e., using \textit{unnormalized} and \textit{normalized} Laplacian matrix \cite{Luxburg}, \textit{graph distance}-based spectral clustering \cite{CommunityDetection_SC2}, \textit{covariate}-assisted spectral clustering \cite{CommunityDetection_SC1}, spectral clustering using \textit{probability} matrix \cite{CommunityDetection_SC3}, \textit{self tuning} spectral clustering \cite{CommunityDetection_SC4}, and \textit{regularized} Laplacian matrix \cite{COUILLET20}.
It is worth noting that all these methods are relying on the graph representation based on heat diffusion.
In particular, the parameter $\zeta_p$ for the regularized Laplacian matrix was set to $\sqrt{c\Phi}$, according to the guidelines in \cite{COUILLET20}.
We reported the retrieved eigenspectra associated with these approaches applied to the three datasets.
In other terms, we displayed in these figures the histogram showing on the y-axis the percentage of occurrence of all the values each eigenvalue can assume, that are reported on the x-axis.
By visual inspection, it is possible to appreciate that the proposed approach is able to provide a better identification of the isolated eigenvalues in the dataset by increasing the spread among them.
On the other hand, the eigenvalues obtained by using the Laplacian matrices defined in the aforesaid methods appear to be all compressed close to 0.
As such, it is possible to assume that the subsequent eigenanalysis for the detection of communities in the data might be cumbersome and difficult to perform.
In this respect, when considering a dataset composed by $K_C$ communities, the gap between the $K_C$-th and the $K_C+1$-th eigenvalues plays a crucial role to the achievement of an effective clustering of the datasets, according to the theoretical aspects of spectral clustering (briefly reported in Appendix \ref{sec_back_spec}).
In particular, it is possible to achieve a more accurate community detection for large eigengaps.
Thus, to quantify the difference between the approaches considered in Fig. \ref{fig_res_eigen_RS}-\ref{fig_res_eigen_PV}, we computed the difference $|\lambda_{K_C+1} - \lambda_{K_C}|$ (where $\lambda_i$ identifies the $i$-th eigenvalue) for all the methods in these figures. We reported the eigengaps we obtained for the considered datasets in Table \ref{tab1}, where $K_C$ is set to 15, 11, and 10 for the datasets in Section \ref{sec_exp_RS}, \ref{sec_exp_BCI}, and \ref{sec_exp_PV}, respectively.
At this point, the improvement provided by using the fluid Laplacian matrix as in Section \ref{sec_meth_fluidLapl} with respect to state-of-the-art methods appears dramatic.
Indeed, the results in Table \ref{tab1} emphasize the ability of the fluid diffusion model in addressing the complex interactions among samples that can occur at global and local scale in multimodal datasets.
This impact of this result is further highlighted by Table \ref{tab2}, where the gap between the $K_C+1$-th and $K_C+2$-th eigenvalue is displayed.
In fact, it is possible to appreciate how this difference is sensibly smaller than their corresponding values in Table \ref{tab1}.
\begin{figure}[htb]
\centering
\includegraphics[width=1\columnwidth]{hist_eigen_RS_210421_2-eps-converted-to.pdf}
\caption{Spectrum of the eigenvalues obtained when analyzing the dataset in Section \ref{sec_exp_RS} (multimodal remote sensing) by means of spectral clustering methods based on different set-ups of the Laplacian matrix: \textit{fluid} Laplacian matrix as in Section \ref{sec_meth_fluidLapl}, \textit{unnormalized} and \textit{normalized} Laplacian matrix \cite{Luxburg}, \textit{Graph Distance}-based Laplacian matrix \cite{CommunityDetection_SC2}, \textit{covariate-assisted} Laplacian matrix \cite{CommunityDetection_SC1}, \textit{probability}-based Laplacian matrix \cite{CommunityDetection_SC3}, \textit{self tuning} Laplacian matrix \cite{CommunityDetection_SC4}, \textit{regularized} Laplacian matrix \cite{COUILLET20}.}
\label{fig_res_eigen_RS}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=1\columnwidth]{hist_eigen_BCI_210421-eps-converted-to.pdf}
\caption{Spectrum of the eigenvalues obtained when analyzing the dataset in Section \ref{sec_exp_BCI} (multimodal brain-computer interface) by means of spectral clustering methods based on different set-ups of the Laplacian matrix: the same notation as in Fig. \ref{fig_res_eigen_RS} applies here.}
\label{fig_res_eigen_BCI}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=1\columnwidth]{hist_eigen_PV_210421-eps-converted-to.pdf}
\caption{Spectrum of the eigenvalues obtained when analyzing the dataset in Section \ref{sec_exp_PV} (multimodal photovoltaic energy) by means of spectral clustering methods based on different set-ups of the Laplacian matrix: the same notation as in Fig. \ref{fig_res_eigen_RS} applies here.}
\label{fig_res_eigen_PV}
\end{figure}
\begin{table}[!th]
\renewcommand{\arraystretch}{1.3}
\caption{Eigengap $|\lambda_{K_C+1} - \lambda_{K_C}|$ for the methods in Fig. \ref{fig_res_eigen_RS} to \ref{fig_res_eigen_PV}}
\label{tab1}
\centering
\begin{tabular}{|c|c|c|c|}
\hline
\bfseries Method & \bfseries RS & \bfseries BCI & \bfseries PV \\
& (multimodal & (multimodal & (multimodal \\
& remote sensing) & brain-computer & photovoltaic \\
& & interface) & energy) \\
\hline
\textbf{Fluid} & 45 & 48 & 54 \\
\hline
\textbf{Unnormalized} & $2 \times 10^{-14}$ & $1.4 \times 10^{-14}$ & $3 \times 10^{-15}$ \\
\hline
\textbf{Normalized} & $6 \times 10^{-15}$ & $3 \times 10^{-15}$ & $ 2\times 10^{-15}$ \\
\hline
\textbf{Graph Distance} & $6 \times 10^{-14}$ & $3 \times 10^{-14}$ & $4 \times 10^{-14}$ \\
\hline
\textbf{Covariate} & $7 \times 10^{-14}$ & $3 \times 10^{-14}$ & $2.9 \times 10^{-14}$ \\
\hline
\textbf{Probability} & $8 \times 10^{-15}$ & $2 \times 10^{-15}$ & $3 \times 10^{-14}$ \\
\hline
\textbf{Self tuning} & $4 \times 10^{-15}$ & $1.4 \times 10^{-15}$ & $2.3 \times 10^{-14}$ \\
\hline
\textbf{Regularized} & $3 \times 10^{-15}$ & $1 \times 10^{-15}$ & $1 \times 10^{-14}$ \\
\hline
\end{tabular}
\end{table}
\begin{table}[!th]
\renewcommand{\arraystretch}{1.3}
\caption{Eigengap $|\lambda_{K_C+2} - \lambda_{K_C+1}|$ for the methods in Fig. \ref{fig_res_eigen_RS} to \ref{fig_res_eigen_PV}}
\label{tab2}
\centering
\begin{tabular}{|c|c|c|c|}
\hline
\bfseries Method & \bfseries RS & \bfseries BCI & \bfseries PV \\
& (multimodal & (multimodal & (multimodal \\
& remote sensing) & brain-computer & photovoltaic \\
& & interface) & energy) \\
\hline
\textbf{Fluid} & 0.2 & 1.2 & 0.3 \\
\hline
\textbf{Unnormalized} & $3.2 \times 10^{-14}$ & $1.8 \times 10^{-14}$ & $2.2 \times 10^{-15}$ \\
\hline
\textbf{Normalized} & $6.4 \times 10^{-15}$ & $3.9 \times 10^{-15}$ & $ 2.3\times 10^{-15}$ \\
\hline
\textbf{Graph Distance} & $5.5 \times 10^{-14}$ & $2.8 \times 10^{-14}$ & $3.8 \times 10^{-14}$ \\
\hline
\textbf{Covariate} & $6.8 \times 10^{-14}$ & $2.9 \times 10^{-14}$ & $2.77 \times 10^{-14}$ \\
\hline
\textbf{Probability} & $7.7 \times 10^{-15}$ & $2.1 \times 10^{-15}$ & $2.8 \times 10^{-14}$ \\
\hline
\textbf{Self tuning} & $3.8 \times 10^{-15}$ & $1.5 \times 10^{-15}$ & $2.4 \times 10^{-14}$ \\
\hline
\textbf{Regularized} & $2.8 \times 10^{-15}$ & $1.26 \times 10^{-15}$ & $1.03 \times 10^{-14}$ \\
\hline
\end{tabular}
\end{table}
With this in mind, we can further explore the capacity of the proposed method by assessing its actual ability of detecting communities within the considered datasets.
In order to obtain a quantitative assessment in this sense, we compute cluster assignments on the graph and evaluate the partitions that are delivered.
We evaluate the performance of the methods by computing three indices: modified purity ($mP$), modified adjusted Rand index ($mARI$), and modified normalized mutual information ($mNMI$) \cite{fluid_metrics,fluid_metrics2}.
These metrics are able to quantify the ability of the methods to understand the graph properties, and
to provide a proper set-up to extract information at semantic and functional level from the considered dataset.
Moreover, in order to obtain a more reliable evaluation of the graph learning performance, these metrics take into account graph topology whilst their "traditional" counterparts do not \cite{fluid_metrics,fluid_metrics2}.
In order to give a compact definition of the aforesaid metrics, let $\Omega =\{\omega_k\}_{k=1, \ldots, C_\Omega}$ be the set of detected clusters, whilst
$\Psi =\{\psi_l\}_{l=1, \ldots, C_\Psi}$ is the set of ground-truth classes.
Then, let us consider the modified purity index.
To define $mP$, it is important to take into account the purity of a node for a partition $\Omega$
relatively to
another partition $\Psi$, i.e., a function that identifies if the class of $\Psi$ containing node $u$ is majority in that of $\Omega$
also containing $u$,
and otherwise. This function can be written as:
\begin{equation}
P(u,\Omega, \Psi)=\delta(\psi_j \: s.t. \: |\omega_i \cap \psi_j| is \; maximum),
\label{eq_purity_1}
\end{equation}
\noindent where $u \in \omega_i$, $u \in \psi_j$, and $\delta$ is the Kronecker delta function.
At this point, assuming that $w_u$ is the weight of node $u$ the modified purity ($mP$) can be defined as follows:
\begin{equation}
mP(\Omega,\Psi) = \sum_{i} \sum_{u \in \omega_i} \frac{w_u}{\sum_j w_j} P(u, \Omega, \Psi )
\label{eq_mP}
\end{equation}
\noindent It is worth noting that $mP$ measures whether each detected community is assigned to the ground-truth label which is most frequent in the community without bias (that could affect the classic purity metric \cite{fluid_metrics,fluid_metrics2}). Moreover, the upper bound of this metric is 1, which corresponds to a perfect match between the partitions $\Omega$ and $\Psi$. On the other hand, its
lower bound is 0 and corresponds to a complete mismatch between partitions $\Omega$ and $\Psi$.
With the notion of weight of a node in mind, it is possible to modify the classic definition of the adjusted Rand index in order to take into account the topological configuration of the graph.
In this respect, it is necessary to define for any subset of nodes $\Phi$ the quantity $\kappa(\Phi) = \sum_{t,u \in \Phi} w_t w_u$.
Hence, the modified adjusted Rand index ($mARI$) can be written as follows \cite{fluid_metrics,fluid_metrics2}:
\begin{gather}
mARI(\Omega, \Psi) = \nonumber \\
\frac{\sum_{ij} \kappa(\omega_i \cap \psi_j) - \frac{1}{\kappa({\cal V})}\sum_i \kappa(\omega_i) \sum_{j} \kappa(\psi_j)}{\frac{1}{2} (\sum_i \kappa(\omega_i) + \sum_{j} \kappa(\psi_j)) - \frac{1}{\kappa({\cal V})}\sum_i \kappa(\omega_i) \sum_{j} \kappa(\psi_j)}
\label{eq_mARI}
\end{gather}
\noindent This metric interprets the evaluation of community detection performance in terms of the decisions that have been taken for the nodes in the graph.
In particular, $mARI$ is used to assess the community detection performance against the case for which the detected clusters $\Omega$ would be randomly generated, taking into account the individual effect of each node in the graph and in a chance-corrected manner \cite{fluid_metrics,fluid_metrics2}.
Specifically, $mARI$ is upper bounded to 1.
I.e., $mARI$ assumes the value 1 when $\Omega$ and $\Psi$ perfectly match.
On the other hand, $mARI$ is equal to or less than 0 when the similarity between $\Omega$ and $\Psi$ is equal or less than what is expected from
two random partitions.
Finally, to define the modified normalized mutual information index ($mNMI$), we should consider a modified joint probability of cluster $\omega_i$ and class $\psi_j$, which can be written as $\tilde{p}_{ij} = \sum_{u \in \omega_i \cap \psi_j} \frac{w_u}{\sum_l w_l}$.
Accordingly, we can define $mNMI = mNMI(\Omega, \Psi)$ as follows:
\begin{equation}
mNMI= \frac{-2 \sum_{ij} \tilde{p}_{ij} \log( \frac{\tilde{p}_{ij}}{\sum_i \tilde{p}_{ij} \sum_j \tilde{p}_{ij} } )}{\sum_i \sum_j \tilde{p}_{ij} \log (\sum_j \tilde{p}_{ij}) + \sum_j \sum_i \tilde{p}_{ij} \log(\sum_i \tilde{p}_{ij}) }
\label{eq_mNMI}
\end{equation}
It is possible to notice that the metrics in (\ref{eq_mP}), (\ref{eq_mARI}), and (\ref{eq_mNMI}) all rely on the definition of weight of a node, that is supposed to help in emphasize the role of the graph topology in computing how well the proposed graph representation can help in extracting functional information from the given dataset.
To this aim, several definitions of $w_u$ can be drawn out \cite{fluid_metrics,fluid_metrics2}.
In particular, three configurations can be considered in order to cover the main characteristics to be taken into account when assessing the relevance of a node in graph learning frameworks:
\begin{enumerate}
\item $w_u = d_u / \max_t d_t$, where $d_u$ identifies the degree of node $u$ - in this case, we can talk about a \textit{degree measure};
\item $w_u = d_u^{\mathtt{INT}} /d_u$, where $d_u^{\mathtt{INT}}$ is the internal degree of node, i.e. the number of connections it has
in the cluster it belongs to - in this case, we can talk about an \textit{embeddedness measure};
\item $w_u = d_u^{\mathtt{INT}} / \max_t d_t$ - in this case, we can talk about a \textit{weighted embeddedness measure}.
\end{enumerate}
Thus, when assessing the functional information retrieval performance of the different graph learning frameworks, we considered the metrics as defined according to these set-ups, i.e., three values for each metric, for a total of nine metrics for performance comparison.
In particular, we used these metrics to compare the strategy we introduced in this work with the following state-of-the-art methods:
\begin{itemize}
\item clustering via hypergraph modularity (\textit{CNM}) \cite{CommunityDetection_GP7};
\item hierarchical community detection (\textit{HCD}) \cite{CommunityDetection_HC5};
\item community detection based on distance dynamics (\textit{Attractor}) \cite{CommunityDetection_HC6};
\item joint criterion for community detection (\textit{JCDC}) \cite{CommunityDetection_HC7};
\item a standard \textit{k}-means algorithm, where $k=K_C$;
\item variational Bayes community detection (\textit{VB}) \cite{CommunityDetection_SI5};
\item node importance-based label propagation (\textit{NI-LPA}) \cite{CommunityDetection_LP3};
\item fluid label propagation (\textit{FLP}) \cite{CommunityDetection_LP4}
\item weighted stochastic block model (\textit{WSBM}) \cite{CommunityDetection_Hyb4};
\item multiview spectral clustering (\textit{Multiview}) \cite{CommunityDetection_view1};
\item covariate-assisted spectral clustering (\textit{CASC}) \cite{CommunityDetection_SC1};
\item regularized spectral clustering (\textit{RegularizedSC}) \cite{COUILLET20};
\item deep multimodal clustering (\textit{MMClustering}) \cite{CommunityDetection_multi3}.
\end{itemize}
\noindent All these methods rely on graph representation based on heat diffusion. It is worth to recall that the method introduced in \cite{CommunityDetection_LP4} does not show any overlap whatsoever with the strategy for community detection we introduce in this work. The authors in \cite{CommunityDetection_LP4} do not discuss fluid diffusion indeed, nor introduce any novel graph representation of datasets.
Figures \ref{fig_res_heat_RS} to \ref{fig_res_heat_PV} report the results we achieved by assessing the ability of extracting functional information by applying clustering methods to the outcomes of graph learning frameworks.
For each column we report the value of $mP$, $mARI$, and $mNMI$ computed according to the setting mentioned in the previous bullet point list.
Independently by the configuration we used to assess the results, the proposed method is able to outperform the other schemes in all datasets and set-ups.
Indeed, the approach we introduce appears to be more suitable to characterize multimodal data.
In particular, it is worth noting that the proposed diffusion model provides a solid platform that can be used by several graph analysis approaches to effectively explore the data properties (stronger fluctuations of the proposed metrics are registered for heat diffusion-based schemes across different clustering algorithms).
This is crucial to extract functional characteristics of the records taken into account, so that an analysis at semantic level can be accurately performed.
Hence, the flexibility produced by the fluid diffusion model to analyze the graph representation provides a great advantage with respect to the heat diffusion-based approaches.
\begin{figure}[htb]
\centering
\includegraphics[width=1\columnwidth]{heatmap_FluidCD_RS_210830-eps-converted-to.pdf}
\caption{Heatmap of the community detection results achieved when analyzing the dataset in Section \ref{sec_exp_RS} (multimodal remote sensing), for each method considered. The results in terms of modified purity index (mP), modified adjusted Rand index (mARI), and modified normalized mutual information (mNMI) for different configurations discussed in Section \ref{sec_exp_res} (degree measure - 1; embeddedness measure - 2; weighted embeddedness measure - 3) are reported.}
\label{fig_res_heat_RS}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=1\columnwidth]{heatmap_FluidCD_BCI_210830-eps-converted-to.pdf}
\caption{Heatmap of the community detection results achieved when analyzing the dataset in Section \ref{sec_exp_BCI} (multimodal brain-computer interface), for each method considered. The same notation as in Fig. \ref{fig_res_heat_RS} is used here.}
\label{fig_res_heat_BCI}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=1\columnwidth]{heatmap_FluidCD_PV_210830-eps-converted-to.pdf}
\caption{Heatmap of the community detection results achieved when analyzing the dataset in Section \ref{sec_exp_PV} (multimodal photovoltaic energy), for each method considered. The same notation as in Fig. \ref{fig_res_heat_RS} is used here.}
\label{fig_res_heat_PV}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=1\columnwidth]{PAMI_CD_CompCompl_211205_2-eps-converted-to.pdf}
\caption{Execution time (in log(seconds)) required by the community detection algorithms to achieve the results in Fig. \ref{fig_res_heat_RS}-\ref{fig_res_heat_PV} for the multimodal remote sensing, brain computer interface, and photovoltaic energy datasets (blue, red, and orange bars, respectively).}
\label{fig_compcompl}
\end{figure}
Finally, Fig. \ref{fig_compcompl} displays the execution time (in $\log$[sec]) for the aforesaid algorithms to achieve community detection on the multimodal remote sensing, brain-computer interface, and photovoltaic energy datasets: for each scheme, these results are shown in blue, red, and orange bars, respectively.
It is possible to appreciate how the size of the datasets is typically the driving force behind these outcomes.
The proposed community detection algorithm based on fluid graph representation delivers a performance that is comparable with the other methods in this respect.
Nevertheless, these results show how taking advantage of the modularity property of deep learning-based approach (such as that in \cite{CommunityDetection_multi3}) could reduce the computational load of the architecture.
Hence, to improve the scalability of the approach we presented in this work, a deep learning analysis relying on the proposed fluid graph representation will be considered in future works.
\section{Conclusion}
\label{secconcl}
In this paper, we introduce a novel approach for graph representation with special focus of multimodal data analysis.
The proposed scheme is based on the use of a fluid diffusion model to characterize the interactions among samples, and hence the mechanism for information propagation in graphs.
This approach is meant to address several issues in modern multimodal data analysis, when large scale datasets collected by heterogeneous sources of information are investigated.
In particular, the proposed framework aims to provide an accurate and versatile automatic characterization of the relationships among samples, so that a robust community detection can be derived for complex datasets where multiple statistical, geometrical, and semantic distributions are collected.
In this respect, the main contributions of this work are:
\begin{itemize}
\item the introduction of a novel model for graph information propagation based on fluid diffusion;
\item the development of a compact description of the interactions among data that takes advantage of the eigenanalysis of flow velocity matrix, so to guarantee a data driven set-up for multimodal data characterization;
\item the development of an architecture for community detection based on fluid dynamics, which allows to obtain a solid characterization of the connections among samples in complex datasets (e.g., where samples show different levels of reliability and where the relevance of the feature might vary across the data).
\end{itemize}
We tested our approach on three diverse real multimodal datasets in terms of functional information retrieval.
The experimental results we achieved show the solidity of our approach, as the proposed framework is able to outperform the state-of-the-art methods in community detection, which are all based on heat diffusion model.
Thus, it is possible to state that the fluid diffusion model could be a valid option to improve the characterization of multimodal data analysis and to enhance the understanding of the functions and phenomena underlying the multimodal records, so to fully exploit the potential provided by the diversity of modalities collected in the datasets under exam.
This approach can thus represent the platform on which multimodal data analysis could be based to address the main issues in modern multimodal data analysis.
Future works will be then devoted to explore the development of fluid diffusion-based data analysis schemes for specific tasks, from semisupervised learning to explainable data analysis, to prediction and inference in complex operational scenarios.
Moreover, the results we achieved might signal the need to investigate the geometry of multimodal spaces by means of new computational strategies, which theoretical and methodological basis will be studied in the next steps of this work.
\section{Acknowledgements}
This work is funded in part by Centre for Integrated Remote
Sensing and Forecasting for Arctic Operations (CIRFA) and
the Research Council of Norway (RCN Grant no. 237906), the Visual Intelligence Centre for
Research-based Innovation funded by the Research Council of
Norway (RCN Grant no. 309439),
the Automatic Multisensor remote sensing for Sea Ice
Characterization (AMUSIC) Framsenteret "Polhavet" flagship project 2020, and the IMPETUS project funded by the European Union Horizon 2020 research and innovation program under grant agreement nr. 101037084.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,592 |
Tag: Arshamomaque Pond Preserve
Make that the Paul Stoutenburgh Preserve
by Cyndi Zaweski
CYNDI MURRAY PHOTO | Town Supervisor Scott Russell presented Barbara Stoutenburgh with a plaque during the ceremony.
In honor of a lifelong commitment to environmental conservation, the 120-acre Arshamomaque Pond Preserve in Southold was officially re-named the Paul Stoutenburgh Preserve Thursday.
In room full of friends and elected officials, Mr. Stoutenburgh's family received a plaque immortalizing Mr. Stoutenburgh's dedication to the environment. Mr. Stoutenburgh, 91, was unable to attend.
"He taught me to appreciate nature and to give back to a community," Mr. Stoutenburgh's grandson, also named Paul Stoutenburgh, said. "This preserve means a lot to all of us."
Mr. Stoutenburgh, a Cutchogue resident, wrote the popular weekly column "Focus on Nature" for The Suffolk Times for 50 years. A dedicated public servant, he served three terms as a Southold Town Trustee, including one year as board president, and four years as a town councilman championing environmental causes.
"He helped changed the culture of the town," County Legislator Al Krupski (D-Cutchogue) said. "He really had a vision of the town going into the future."
Mr. Stoutenburgh and his wife, Barbara, are largely responsible for getting Southold Town to adopt its first wetlands code and preventing development of the type that has taken over much of Long Island. In 2011, the couple was named The Suffolk Times People of the Year for their efforts.
During the ceremony Barbara Stoutenburgh spoke about the man behind the causes and read aloud one of her husband's poems entitled "Looking Back."
"I know where the largest rock on the island lies and how the warm sandy beach was formed," he wrote. "I have seen the wonders of a boy and a girl and then another boy… and so with it that a sense of place was found and I a part of it."
Al Krupski, Arshamomaque Pond Preserve, conservation, environment, paul stoutenburgh, paul stoutenburgh preserve
Southold preserve to be named for Stoutenburgh
BARBARAELLEN KOCH FILE PHOTO | Paul Stoutenburgh, left, with his wife Barbara, being interviewed in their Cutchogue home in 2011.
In recognition of a lifelong commitment to environmental causes, the 120-acre Arshamomaque Pond Preserve in Southold will officially become the Paul Stoutenburgh Preserve on Thursday, June 13.
Mr. Stoutenburgh dedicated much of his life fighting to prevent development of the type that has taken over much of Long Island.
"People here can enjoy the natural world around us because of Mr. Stoutenburgh," Supervisor Scott Russell said. "This recognition is long overdue."
Mr. Stoutenburgh and wife, Barbara, were instrumental in getting Southold to adopt its first wetlands code. The Suffolk Times named the couple its "People of the Year" in 2011.
The renaming ceremony will take place at 1 p.m. at the preserve, located on the north side of Route 25 in Southold, just east of Port of Egypt Marina. The preserve entrance is just before the bridge. Parking is available on the south side of Route 25 along Old Main Road.
In case of rain the ceremony will be held at Southold Town Hall.
Arshamomaque Pond Preserve, Cutchogue, paul stoutenburgh, Southold
'Climate smart' could lower insurance rates in Southold Town
by Beth Young
GIANNA VOLPE FILE PHOTO | An aerial view of Rabbit Lane in East Marion following Hurricane Sandy.
Southold Town is considering signing a pledge to become a "Climate Smart Community," a program sponsored by the state Department of Environmental Conservation that helps towns reduce carbon emissions, protect shorelines and provide flood insurance rate reductions for residents.
David Bert of Cameron Engineering was hired by the State Energy Research and Development Authority to help towns implement the program.
He told the Town Board at Tuesday's work session that the state is making $30 million available to municipalities in each of the next three years to help combat climate change.
"There's certainly more emphasis coming out of Albany on that these days," Mr. Bert said, referring to Gov. Andrew Cuomo's strong stance on fighting climate change.
Mr. Bert said the competitive grant program will likely begin this summer and communities that sign on to the pledge will receive more points toward their grant applications.
In the pledge, the town would agree to prepare a climate action plan, which Cameron Engineering could help the town develop at NYSERDA's expense. The firm also provides mapping assistance.
Program organizers are planning a workshop on alternative fuels for municipal vehicle fleets on the East End on April 25, which will include a discussion of potential charging and fueling stations.
One of the ancillary benefits of signing the pledge, Mr. Bert said, is that it enables the town to qualify for the National Flood Insurance Program's Community Rating Program, which can help property owners get discounts ranging from 5 to 45 percent.
"There are a few municipalities on the island that do participate," he said, adding that some of Southold's public awareness programs and new building codes already in place would make residents eligible for discounts.
"It's quite a process, but it's useful," he said.
PRESERVE NAMED FOR STOUTENBURGH
Also on Tuesday, board members approved renaming Arshamomaque Pond Preserve on Route 25 in Southold as the Paul Stoutenburgh Preserve, after the longtime North Fork environmentalist.
After learning of the honor, Mr. Stoutenburgh and his wife, Barbara, suggested the following wording for a plaque to be placed on a rock at the entrance to the park:
"With knowledge and the will to do what's right, our world will blossom and keep on returning to us the delicate fragrance of May pinks in the woods, a spring run of flounder for dinner and ospreys to delight our heart and spirit."
A public ribbon-cutting will be scheduled for this spring.
Arshamomaque Pond Preserve, Cameron Engineering, Climate Smart Community, paul stoutenburgh, Southold Town | {
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} | 8,048 |
Campus Reform | Old 'cheap' Republicans 'raped' America: Video captures award-winning professor's anti-Republican rant
Old 'cheap' Republicans 'raped' America: Video captures award-winning professor's anti-Republican rant
Prof pushes students at virtual 'die-in' to use 'radical activist tactics' for stricte…
By Alexa Schwerha
Nikole Hannah-Jones brags about tricking audience into thinking MLK quotes were her words
By Robert Schmad '23
Josiah Ryan | Editor-in-Chief Emeritus
Tuesday, September 3, 2013 4:12 PM
A professor at Michigan State University (MSU) opened the first day of his creative writing class on Thursday by bashing Mitt and Ann Romney and ranting against "old Republicans" who he says "raped" the country, according to a student who made a secret recording of the class.
The eight-minute secretly recorded video also reveals Prof. William S. Penn bullying a student who apparently disagreed with his Democratic politics and arguing that Republicans want to prevent "black people" from voting.
"If you go to the Republican convention in Florida, you see all of the old Republicans with the dead skin cells washing off them," said Penn. "They are cheap. They don't want to pay taxes because they have already raped this country and gotten everything out of it they possibly could."
"They don't want to pay for your tuition because who are you? Well, to me you are somebody," he continued.
WATCH: Secret video of MSU professor bashing Republicans, Romney family
Penn then appeared to harass a student who had apparently displayed displeasure with his politics.
"You can frown if you want," said Penn, gesticulating towards the student sitting near the front. "You look like you are frowning. Are you frowning?"
Penn, who was a Distinguished Faculty Award recipient at MSU in 2003, and a two time winner of the prestigious Stephen Crane Prize for Fiction, then went on to accuse Republicans of crafting a systematic plan for "getting black people not to vote."
"This country still is full of closet racists," he said. "What do you think is going on in South Carolina and North Carolina. Voter suppression. Its about getting black people not to vote. Why? Because black people tend to vote Democratic."
"Why would would Republicans want to do it?" he asked. "Because Republicans are not a majority in this country anymore. They are a bunch of dead white people. Or dying white people."
In a brief conversation with Campus Reform on Tuesday, Penn would not confirm or deny it was him in the recording, which was given to Campus Reform by a current in his student who asked to remain anonymous.
"I'm not interested," Penn told Campus Reform.
Tom Oswald, a spokesman for MSU told Campus Reform in an email Tuesday, that the university had no further comment on the situation.
"Thank you for reaching out to us," wrote Oswald. "If you've already reached out to the professor in this situation, I don't think we in the university's central PR office would have anything more to say on this matter."
Update 8:29 p.m., EDT, Sept 3, 2013: A MSU spokesman said the school will "looking into" the matter.
"MSU is thankful we've been made aware of the situation; we will be looking into it. At MSU it is important the classroom environment is conducive to a free exchange of ideas and is respectful of the opinions of others," wrote Kent Cassella, an MSU spokesman in am email to Campus Reform on Tuesday night. Read more...
Follow the author of this article on Twitter: @JosiahRyan and @CalebBonham
Josiah Ryan
Editor-in-Chief Emeritus | {
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Q: searching with choice text in django admin interface I am extending user model with AbstractUser and added CharField with choices to it.
code looks like this
Location_choices = (
('IN', 'India'),
('USA', 'America'),
('O', 'Other Country')
)
class User(AbstractUser):
location = models.CharField(choices=Location_choices, max_length=3)
def __str__(self):
return self.username
And registered in admin.py like below
class AdminUser(admin.ModelAdmin):
search_fields = ('username', 'email', 'id', 'location')
# Register your models here.
admin.site.register(User, AdminUser)
The problem is I am able to search with location choices like (IN, USA, O)
But I want to search with choice text india or america
How do I achieve It. Thanks in advance
A: How about Django's list_filter? I think it is very helpful widget to have in admin site.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,369 |
New York, April 14, 2016 /3BL Media/ – Pronaca, a major producer of poultry and pork products, and the leading buyer of hard yellow corn in Ecuador, has joined the Business Call to Action (BCtA) with a commitment to engage 500 smallholder farmers in the company's corn value chain as well as increase their production 5 percent by the end of 2016.
Historically from indigenous populations, generally poor and often with only a rudimentary education, Ecuador's farmers have been locked in a cycle of poverty. Intermediaries traditionally lend them money at extremely high interest rates and are lax in accurately weighing farm produce. In addition, the farmers receive little technical assistance or information about advanced farming and sustainable agriculture practices. As a result, yields are low, farmers do not receive a fair price for their harvests and they pay prohibitive rates to repay their loans.
To change the dynamic in Ecuador, Pronaca started by cutting out the middlemen and began buying directly from farmers. The company provides financial assistance to enable the purchase of certified seeds, fertilizers and other agriculture supplies at affordable rates, and delivers technical assistance to improve crop productivity, quality and sustainability. Pronaca also guarantees the purchase of 100 percent of farmers' harvests at prices fixed by the government.
With this new process, Pronaca is looking to increase farmers' corn productivity by 5 percent – from 5.6 mt per hectare to 5.88 mt per hectare – by the end of 2016. It will engage 500 smallholder farmers in its corn value chain, training them on sustainable agricultural practices and equipment management, and providing access to financing. And it will expand their production options beyond corn by transferring the technical know-how needed to grow additional vegetables – improving nutrition while increasing incomes.
About Business Call to Action (BCtA): Launched at the United Nations in 2008, the Business Call to Action (BCtA) aims to accelerate progress towards the Sustainable Development Goals (SDGs) by challenging companies to develop inclusive business models that offer the potential for both commercial success and development impact. BCtA is a unique multilateral alliance between key donor governments including the Dutch Ministry of Foreign Affairs, Swedish International Development Cooperation Agency (Sida), UK Department for International Development, US Agency for International Development, and the Ministry of Foreign Affairs of the Government of Finland, and the United Nations Development Programme — which hosts the secretariat. For more information, please visit www.businesscalltoaction.org or on Twitter at @BCtAInitiative.
About Pronaca: Established in 1957, Pronaca is one of the largest companies in Ecuador, with more than US$800 million in annual sales. Its main activity is producing and marketing poultry and pork products. In addition, the company is the leading buyer in Ecuador of hard yellow corn, which is a main component of animal feed – an essential input for Pronaca's livestock activities. The company generates approximately 8,000 direct jobs. For more information, please visit www.pronaca.com. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,167 |
\section{Introduction}
\section{Invariant subspaces}
Let $ \lambda $ be an eigenvalue of a complex $n \times n$ matrix
$A$ and let
\[
E _{ \lambda }(A) = \mathop{\mathrm{Ker}}\nolimits ( A - \lambda I) ^n
\]
be the corresponding generalized eigenspace.
Suppose \,
$ \dim E _{ \lambda }(A) = k $. If
\[
( s - \lambda ) ^{t_1}, \, \dots \, , ( s - \lambda ) ^{t_k}, \,\, \,
t_1 \leq \cdots \leq t_k ,
\]
are the corresponding elementary divisors
then \,$E _{ \lambda }(A) $
is a direct sum of $t_i$-dimensional
cyclic subspaces,
i.e.
\[
E _{ \lambda }(A) = K_1 \oplus \cdots \oplus K_k
\]
with
\begin{equation} \label{eq.jch}
K_i
=
{\rm{span}} \bigl\{ u_i, \, ( A - \lambda I)u_i,
\, \dots , ( A - \lambda I) ^{ t_i -1 } u_i \bigr\} ,
\end{equation}
and \,$ ( A - \lambda I) ^{ t_i} u_i = 0$, \,
$ i= 1,\dots , k$ .
We call \begin{equation} \label{eq.uu}
U = ( u_1, \dots , u_k )
\end{equation}
a tuple of \emph{generators} of
$ E _{ \lambda }(A) $.
From a given $U$ one can construct
$A$-invariant subspaces
in the following way.
Let $r = (r_1, \dots , r_k )$ be such that
\begin{equation} \label{eq.co1}
0 \, \leq \, r_i \, < \, t_i , \, \, i = 1, \dots ,k .
\end{equation}
We set
\begin{equation} \label{eq.ri}
W_{r_i} (U) = {\rm{span}} \bigl\{
( A - \lambda I)^{r_i}u_i, \, ( A - \lambda I) ^{r_i +1}u_i, \,
\dots , ( A - \lambda I) ^{t_i -1} u_i \bigr\}
\end{equation}
and
\begin{equation} \label{eq.dis}
W(r,U) \, = \,
W_{r_1}(U) \, \oplus \, \cdots \, \oplus \, W_{r_k}(U) .
\end{equation}
The construction of invariant subspaces
of the from $W(r,U)$
is a standard procedure in linear algebra and systems theory
(see e.g. \cite{Ku}, \cite[p.61]{GLR0}, \cite{Fr}, \cite{RR},
\cite[p.28]{BEGO}).
If $U$ and $\tilde{U}$ are two different tuples of generators of
$ E _{ \lambda }(A) $ then the restrictions of $A$ to $W(r,U)$
and $ W(r, \tilde{U})$ have the same elementary divisors,
namely
$ (s- \lambda) ^{t_i- r_i}$, $ i = 1, \dots ,k$.
However,
in general, the subspaces
$ W(r,U) $ and $ W(r, \tilde{U})$ will be different.
Consider the following example with
$k= 2$, $t_1= 2$, $t_2 = 3$, and
\begin{equation} \label{eq.mtr}
A = \mathop{\mathrm{diag}}\nolimits (N_2, N_3),
\,
N_2 = \begin{pmatrix} 0 & 1 \\ 0 & 0
\end{pmatrix}\!,
\,
N_3 = \begin{pmatrix} 0 & 1 & 0
\\
0 & 0 & 1 \\
0 & 0 & 0
\end{pmatrix}\!.
\end{equation}
Let $e_i$
be a unit vector of ${\mathbb{C}}^5$.
Then
$U = \{ e_2 , e_5 \}$ and
$ \tilde{U} = \{ e_2, e_5 + e_2 \} $ are tuples of generators of
$E_0(A) = \mathop{\mathrm{Ker}}\nolimits A^5 = {\mathbb{C}}^5$.
If we choose \,$ r = (1,0) $, then
$ W(r,U)
= {\rm{span}} \{ e_1, e_3, e_4, e_5 \} $
and $ W(r, \tilde{U})
=
{\rm{span}} \{e_1, e_3 , e_4 + e_1, e_5 + e_2 \}$.
Thus
\begin{equation} \label{eq.uuti}
W(r,U) \neq W(r, \tilde{U}).
\end{equation}
On the other hand, if we choose \,$ r = (1,2) $, then
\begin{equation} \label{eq.sm}
W(r,U) = W(r, \tilde{U}).
\end{equation}
It is the purpose of our note to determine those
tuples $r = ( r_1, \dots , r_k)$
which have the property that the space
$ W(r,U) $ given by \eqref{eq.ri} and \eqref{eq.dis}
is independent of the generator tuple $U$.
The motivation for our study comes from
Kucera's survey article \cite{Ku}, which
deals with independence of generator tuples in
the case of Hamiltonian matrices.
In Section \ref{sec.app}
we make the connection with \cite[p.60]{Ku}
applying a corollary of our main theorem
to Hamiltonian matrices and algebraic Riccati
equations.
In the sequel we
assume that $\lambda = 0 $
is an eigenvalue of $A$ and we
focus on $E_0(A) = \mathop{\mathrm{Ker}}\nolimits A ^n $.
With each nonzero vector $v \in E _{0}(A)$
we associate a {\emph{height}} $ \mathop{ \mathrm{h} {} }\nolimits (v)$ and an {\emph{exponent}} $ \mathop{ \mathrm{e} {} }\nolimits (v)$
as follows.
Suppose
\[
v \in \mathop{\mathrm{Im}}\nolimits A^q, \, v \notin
\mathop{\mathrm{Im}}\nolimits A ^{q +1}, \,\,
v \in \mathop{\mathrm{Ker}}\nolimits A ^p, \,
v \notin \mathop{\mathrm{Ker}}\nolimits A ^{p-1}.
\]
Then we set
$ \mathop{ \mathrm{h} {} }\nolimits (v) = q$ and $ \mathop{ \mathrm{e} {} }\nolimits (v) = p$.
Thus, if $ \lambda = 0 $ in \eqref{eq.jch}
then the elements of $U$ in \eqref{eq.uu}
satisfy
$ \mathop{ \mathrm{e} {} }\nolimits (u_1) = t_1 \leq \cdots \leq \mathop{ \mathrm{e} {} }\nolimits (u_k) = t_k$\,
and $ \mathop{ \mathrm{h} {} }\nolimits (u_i) = 0$.
We define
\[
\langle v \rangle = {\rm{span}} \{ A ^{ \nu } v, \nu \geq 0 \}.
\]
Then $ \langle v \rangle $
is a cyclic subspace generated by
$v$, and \,$\dim \langle v \rangle = \mathop{ \mathrm{e} {} }\nolimits (v) $.
\section{The main result}
\begin{theorem} \label{thm.mn}
Let
$A \in {\mathbb{C}}^{n \times n} $
and let
\begin{equation} \label{eq.elth}
s^{ t_1 }, \dots , s^{ t_k }, \,
t_1 \leq \cdots \leq t_k ,
\end{equation}
be the
elementary divisors corresponding to the eigenvalue $\lambda = 0$.
Let
\[
U = (u_1, \dots , u_k )
\]
be a tuple of generators of
$E_0(A) = \mathop{\mathrm{Ker}}\nolimits A^n$
such that
\, $
\mathop{ \mathrm{e} {} }\nolimits (u_i) = t_i$, $i = 1, \dots , k$,
and
\[
E_0(A) = \langle u_1 \rangle \, \oplus \, \cdots \,
\oplus \,
\langle u_k\rangle .
\]
Let \,$ r = ( r_1, \dots , r_k) $ be a $k$-tuple of integers
with \,$ 0 \leq r_i < t_i$, $i = 1, \dots , k$.
Define
\begin{equation} \label{eq.agn}
W(r,U)
=
\langle A ^{r_1} u_1 \rangle \, \oplus \, \cdots \,
\oplus \,
\langle A ^{r_k} u_k \rangle
\end{equation}
and
\begin{equation} \label{eq.wr}
W(r) = \bigl(\mathop{\mathrm{Im}}\nolimits A ^{r_1} \cap \mathop{\mathrm{Ker}}\nolimits A^{t_1 - r_1} \bigr)
+
\cdots + \bigl(\mathop{\mathrm{Im}}\nolimits A ^{r_k} \cap \mathop{\mathrm{Ker}}\nolimits A^{t_k - r_k} \bigr).
\end{equation}
Then the following statements are equivalent:
\\
{\rm{(i)}}
The $k$-tuple $r = ( r_1, \cdots , r_k) $
satisfies
\begin{equation} \label{eq.rmo}
r_1 \leq \cdots \leq r_k,
\end{equation}
and
\begin{equation} \label{eq.df}
t_1 - r_1 \leq \cdots \leq t _k - r_k.
\end{equation}
{\rm{(ii)}}
The space
$ W(r,U) $
is independent of \,$U$.\\
Moreover, if \eqref{eq.rmo} and \eqref{eq.df} hold then
\,$W(r,U) = W(r)$.
\end{theorem}
\medskip
\noindent Proof.
(i) $\Rightarrow$ (ii).
We show that
\eqref{eq.rmo} and \eqref{eq.df}
imply
$ W(r,U) = W(r)$.
Define
\,$
W_{r_s}(U) = \langle A ^ { r_s } u_s \rangle $
such that \eqref{eq.dis} holds.
From
\[
W_{r_s} (U) \subseteq
\mathop{\mathrm{Im}}\nolimits A ^{r_s}\, \cap \, \mathop{\mathrm{Ker}}\nolimits A^{t_s - r_s}
\]
we immediately obtain $ W(r,U) \subseteq W(r) $.
Now let
$ x $ be in $\mathop{\mathrm{Im}}\nolimits A ^{r_s} \cap \mathop{\mathrm{Ker}}\nolimits A^{t_s - r_s} $.
Then
\, $ x = A ^{r_s} y$ \, for some \, $ y \in E_0(A)$,
and
\begin{equation} \label{eq.zro}
A^{t_s - r_s} x = A^ {t_s} y = 0.
\end{equation}
With respect to the basis
\begin{equation} \label{eq.mclb}
\mathcal{B} _U
=
\{A^{\nu_i}u_i; \, 0 \leq \nu _i \leq t_i -1 , \,
i = i, \dots , k \}
\end{equation}
we have
\[ y
=
\sum _{i = 1}^ k \, \sum _{\nu_i = 0} ^{t_i -1}
\,
\alpha _{i \nu _i} A^{ \nu_i } u_i .
\]
Let $\ell$ be the
largest integer sucht that \,$t_{ \ell } \leq t_s$.
Then $ A^{ t_s} u_i = 0$ for \, $i = 1, \dots,
\mbox{$\ell$}$.
Moreover
$ A^{ t_s + \nu _i } u_i = 0 $ if $ t_s + \nu _i >
t_i $.
Therefore
\[
A^{ t_s} y =
\sum _{i > \ell}
\, \sum _{\nu _i = 0} ^{ t_i - t_s -1}
\,
\alpha _{i \nu _i } A^{ t_s + \nu _i } u_i = 0.
\]
Since the vectors of $\mathcal{B}_U$
are linearly independent we obtain
\, $\alpha _{i \nu _i} = 0$ for $i > \ell$
and
\, $ \nu _i = 0, \dots , t_i - t_s -1 $. Hence
\[
y
=
\sum _{i = 1}^ {\ell } \, \sum _{\nu _i = 0}
^{ t_i -1}
\alpha _{i \nu _i }
A^{ \nu _i } u_i \,
+ \,
\sum _{i > \ell}
\sum _{\nu _i = t_i - t_s } ^{t_i -1}
\alpha _{i \nu _i }
A^{ \nu _i } u_i
\]
and
\[
x
=
\sum _{i = 1}^ {\ell} \, \sum _{\nu _i = 0}
^{ t_i -1}
\alpha _{i \nu _i }
A^{ r_s + \nu _i } u_i \,
+ \,
\sum _{i > \ell}
\sum _{\nu _i = t_i - t_s } ^{t_i -1}
\alpha _{i \nu _i }
A^{r_s + \nu _i } u_i.
\]
Note that
$ t_s = \cdots = t_{\ell} $
implies $ r_s = \cdots = r_{\ell}.$
Hence,
if \,$ 1 \leq i \leq \ell$ then
$ r_i \leq r_s $, and therefore
\begin{equation} \label{eq.dp}
A^{ r_s + \nu _i } u_i \in W_{r_i} (U).
\end{equation}
On the other hand, if
\,
$i > \ell $ %
then
$t_s - r_s \leq t_i - r_i$.
In that case
$ \nu _i \in
\{ t_i - t_s, \dots , t_i -1 \} $
implies
\[
r_s + \nu _i \geq r_s + (t_i - t_s) \geq r_i.
\]
Thus, we again have \eqref{eq.dp}.
Hence $x \in W(r,U)$ and therefore $ W(r) \subseteq W(r,U)$.
\begin{comment}
(ii) $\Rightarrow$ (i).
Let
\begin{equation} \label{eq.mclb}
\mathcal{B} _U
=
\{A^{\nu_i}u_i; \, 0 \leq \nu _i \leq t_i -1 , \,
i = i, \dots , k \}
\end{equation}
be the Jordan basis of $E_0(A)$ corresponding to $U$.
Suppose
$ V = (
v_{1}, \dots , v_{k } )$
is another tuple of generators of $ E_{0} (A)$ satisfying
$ \mathop{ \mathrm{e} {} }\nolimits (v_i) = t_i$.
It suffices to show that
the generators \, $ A ^{r_1} v_1 , \dots , A ^{r_k} v_k $
of $ W(r,V) $ are
contained in $ W(r,U) $.
Consider the vector
\begin{equation} \label{eq.licb}
A ^{r_s} v_s
=
\sum _{i = 1} ^k \sum_{ \nu_{i} = 0 }^{t_i -1 }
\alpha _{ i \nu_{i} }
A ^{ \nu_{i} } u_i
\end{equation}
with
$ \mathop{ \mathrm{h} {} }\nolimits ( A ^{r_s} v_s ) = r_s$.
The fact that
is $\mathcal{B}_U $ is a Jordan basis implies
(see e.g.~\cite{Bru})
\[
{\mathrm{h}} \Bigl( \sum _{i = 1} ^k
\sum_{ \nu_{i} = 0 }^{ t_i -1}
\alpha _{ i \nu_{i} }
A ^{ \nu_{i} } u_i \Bigr)
=
\min \{
\nu_{i}
; \, 0 \leq \nu_{i} \leq t_i -1 , \,
1 \leq i \leq k, \,
\alpha _{ i \nu_{i} } \neq 0 \}.
\]
Thus we have
\begin{equation} \label{eq.if}
\nu_i \geq r_s \quad {\rm{if}}
\quad \alpha _{ i \nu_{i} } \neq 0.
\end{equation}
Let
$\ell$ be the greatest integer such that
$ r_{\ell} \leq r_s $.
Set
\[
w =
\sum _{i = 1} ^{\ell}
\sum_{ \nu_{i} = r_s }^{ t_i -1}
\alpha _{ i \nu_{i} }
A ^{ \nu_{i} } u_i,
\]
and
\begin{equation} \label{eq.zwy}
y = \sum _{i = \ell + 1} ^{k}
\sum_{ \nu_{i} = r_s }^{ t_i -1}
\alpha _{ i \nu_{i} }
A ^{ \nu_{i} } u_i \,\,\, {\rm{if}} \,\,\, \ell < k,
\end{equation}
and \,$ y = 0 $\, if \,$ \ell = k $.
Because of \eqref{eq.if} we can write \eqref{eq.licb}
as \,$ A^{r_s} v_s = w + y$.
We shall see that both $w$ and $y$ are in $W(r,U)$.
If \, $\alpha _{ i \nu_{i} } \neq 0$ then it follows
from \eqref{eq.rmo} and \eqref{eq.if}
that
\begin{equation} \label{eq.llu}
\nu_i \, \geq \, r_{\ell} \, = \, \cdots \, =
\, r_s \, \geq \, r_{s-1} \,
\geq \, \cdots \, \geq \, r_1,
\end{equation}
and we see that
$ w \in W(r,U) $.
Now suppose $ \ell < k $.
From
\, $
A^{t_s - r_s} A^{r_s} v_s = 0 $ \,
follows
\begin{equation} \label{eq.awy}
A^{t_s - r_s} w + A^{t_s - r_s} y = 0 .
\end{equation}
Because of \eqref{eq.if}
we have
\[
t_s - r_s + \nu _i \geq t_s \quad {\rm{if}} \quad
\alpha _{ i \nu_{i} } \neq 0 .
\]
Note that \eqref{eq.llu} yields \,$ A^{t_s} u_i = 0$,
for $i = 1, \dots , \ell $. Therefore
\,$ A^{t_s - r_s + \nu _i} u_i = 0$.
Hence
\, $
A^{t_s - r_s} w = 0 $.
Thus \eqref{eq.awy} yields
\[
A^{ t_s - r_s } y
=
\sum _{i = \ell +1} ^{k} \,
\sum_{ \nu_{i} = r_s }^{ t_i -1}
\alpha _{ i \nu_{i} }
A ^{ t_s - r_s + \nu_{i} } u_i
= 0.
\]
The vectors of $\mathcal{B}_U$
are linearly independent. Hence, if
$ \alpha _{ i \nu_{i} } \neq 0 $,
then we have
$ A ^{ t_s - r_s + \nu_{i} } u_i
= 0$, or equivalently,
\,$ t_s - r_s + \nu_{i} \geq \mathop{ \mathrm{e} {} }\nolimits (u_i) = t_i $.
Thus
\begin{equation} \label{eq.ung}
( t_s - r_s) \, + \, ( \nu_{i} - r_i ) \, \geq \, t_i - r_i .
\end{equation}
From \eqref{eq.df}
follows
\,$
t_i - r_i \geq t_s - r_s
$\,
for $i = \ell +1, \dots , k$.
Hence \eqref{eq.ung} yields $ \nu_{i} \geq r_i $
in \eqref{eq.zwy}, which implies $y \in W(r,U)$.
\end{comment}
\medskip
\noindent
(ii) $\Rightarrow$ (i).
We assume that $ W(r,U)$ is independent of $U$.
Let us show first that
\begin{equation} \label{eq.co2}
r_i = r_j \,\,\, {\rm{if}} \,\,\, t_i = t_j.
\end{equation}
Suppose \,$r = (r_1, \dots , r_k) $ is such that
\,$ t_s = t _{s+1}$ and
\,$r_{s} \neq r _{s+1}$,
e.g.
\begin{equation} \label{eq.sptt}
r_{s+1} < r _{s} \,\,\, {\rm{for \,\, some}} \,\,\,
s \in \{1, \dots , k-1 \}.
\ee
Let
\,$ V = (v_1, \dots , v_k) $
be such that
\,$(v_ {s}, v _{s+1 } ) = (u _{s+1 }, u_{s})$,
and \,$v_i = u_i$ if $i \notin \{s, s +1 \}$.
Then
$A ^{ r_{s+1 } } u_{s+1} \in W(r,U)$
but
$ A ^{ r_{s+1 } } u_{s+1} = A ^{ r_{s+1 } }v_{s} \notin W(r, V)$.
Therefore the tuples
$U $ and $V$ contain the same elements, but
$ W(r,U) \neq W(r, V )$.
Now suppose that \eqref{eq.rmo} is not satisfied.
Then
we have \eqref{eq.sptt}, and
\begin{equation} \label{eq.wzv}
A ^ { r_{s+1} } u _{s} \notin W(r,U) .
\end{equation}
Let $V = (v_1, \dots , v_k )$
be given by
\, $ v_{s+1} = u _{s+1} + u _{s} $, and $ v_i = u_i $, if
$i \neq s+1$.
Thus $V$ is
a tuple of generators of $ E_{0}(A) $ with
$ \mathop{ \mathrm{e} {} }\nolimits (v_i) = \mathop{ \mathrm{e} {} }\nolimits (u _i)$.
Consider
\[
A ^ { r_{s+1} } u _{s+1}
+
A ^ { r_{s+1} } u _{s } \, =
\, A ^ { r_{s+1} } v_{s+1} \, \in \, W(r,V).
\]
Then $ A ^ { r_{s+1} } v_{s+1} \notin W(r,U) $.
Otherwise $A ^ { r_{s+1} } u_{s+1} \in W(r,U) $
would imply \,
$ A ^ { r_{s+1} } u _{s} \in W(r,U) $, which is a
contradiction
to \eqref{eq.wzv}.
\medskip
Suppose $r = (r_1, \dots , r_k)$
does not satisfy \eqref{eq.df}.
Then
\, $ t_s - r _s > t_{s+1} - r _{s +1} $ \,
for some $s \in \{1, \dots , k -1 \} $.
Because of \eqref{eq.co2} we
have $t_{s+1} \neq t_s$. Hence
\, $
r _ {s+1} - r _s > t_{s+1} - t_s > 0 $,
and \, $ r _s < r _ {s+1} $, and
\begin{equation} \label{eq.dec}
r _s + ( t_{s+1} - t_s ) < r _ {s+1}.
\end{equation}
Because \eqref{eq.agn} it is obvious
that \eqref{eq.dec} implies
\begin{equation} \label{eq.spt}
A ^ { r _{s} + ( t_{s+1} - t_{s} ) } u_{s+1} \notin W (r, U) .
\end{equation}
Define
\,$
v_s = u _s + A ^{ t_{s+1} - t_s } u _ {s+1}$.
Then \,$ \mathop{ \mathrm{e} {} }\nolimits (v_s) = \mathop{ \mathrm{e} {} }\nolimits (u_s) = t_s$.
Therefore
\begin{equation} \label{eq.vv}
V = \{ u_1, \dots, u_{s-1}, v_s, u_{s+1}, \dots, u_k \}
\end{equation}
is another tuple of generators of $E_0(A)$.
Let us show that $ W (r, V) \neq W (r, U) $. Clearly, the vector
$ A ^{r_s} v_s $ belongs to $ W (r, V) $. Suppose \,
\[
A ^{r_s} u_s + A ^{ r_s + ( t_{s+1} - t_s ) } u _ {s+1}
\,
=
\,
A ^{r_s} v_s \, \in \, W (r, U) .
\]
Because of $ A ^{r_s} u_s \in W (r, U) $
that would imply
\[
A ^ {r_s + (t_{s+1} - t_s) } u _ {s+1} \in W (r, U) ,
\]
which is a contradiction to \eqref{eq.spt}.
\hfill $\square$
\medskip
\bigskip
Let us consider again Example \eqref{eq.mtr}.
We have $(t_1,t_2) = (2,3)$.
In the case of $r = (1,0)$ condition \eqref{eq.rmo} is violated,
which accounts for
\eqref{eq.uuti}.
In the case of $r = (1,2)$
both \eqref{eq.rmo} and \eqref{eq.df} hold, which ensures
\eqref{eq.sm}.
In accordance with a definition in
\cite[p.\,83]{GLR} and \cite{Bru} the space $ W(r,U) $ is a
\emph{marked} $A$-invariant subspace of $E_0(A)$.
That means
$ W(r,U) $
has a Jordan basis, in our case
\[
\{A^{r_i+ \mu_i}u_i; \, 0 \leq \mu _i \leq t_i -r_i -1, \,
i = i, \dots , k \},
\]
which can be extended to a Jordan basis of $E_0(A)$,
namely to
$\mathcal{B}_U$ in \eqref{eq.mclb}.
Let
\,$ \mathcal{M}_r $ be the set of marked subspaces $M$ of $E_0(A)$
such that
the elementary divisors of
the restriction $ A _{\mid M} $ are
$ s^{t_1 - r_1}, \dots , s^{t_k - r_k} $.
We have noted before that for each tuple of generators $U$
the corresponding space
$ W(r,U) $ is in $ \mathcal{M}_r $.
Suppose \eqref{eq.rmo} and \eqref{eq.df}
hold. Then all the spaces $ W(r,U) $ coincide with
$W(r)$ and one might ask whether $W(r)$
is the only subspaces in $ \mathcal{M}_r $.
In the following we have an example
where $ \mathcal{M}_r \supsetneqq \{ W(r) \}$.
Let $n = 10$, $k = 2$,
and $ t = (t_1,t_2 ) = (4,6)$, and $ r = ( 2, 3) $.
Then $ t - r = ( 2, 3)$. Hence
the conditions \eqref{eq.rmo} and \eqref{eq.df}
are satisfied.
Let
$U = ( u_1, u_2) $ be a tuple of generators
such that
$ \mathop{ \mathrm{e} {} }\nolimits ( u_1 ) = 4$ and $ \mathop{ \mathrm{e} {} }\nolimits ( u_2 ) = 6$.
The subspaces
\[
M = W(r,U) = W(r) = \langle A^2 u_1 \rangle
\oplus \langle A^3 u_2 \rangle
\]
and
\,$
\tilde{M } =
\langle A u_1 \rangle
\oplus \langle A^4 u_2 \rangle
$\,
are marked,
the elementary divisors of $A_{\mid M } $ and
$A_{\mid \tilde{M } } $ are $s^2, s^3$.
Hence $ \tilde{M } \in \mathcal{M}_r $, but
$ \tilde{M } \neq W(r)$.
\medskip
Let $[m]$ denote the greatest integer of $m$.
If we assume $(t_1, \dots, t_k) $ as in \eqref{eq.elth} and take
$r = ( [\tfrac{1}{2}t_1 ], \dots , [\tfrac{1}{2}t_ k ] )$
then the conditions \eqref{eq.rmo} and \eqref{eq.df}
are satisfied and we note the
following corollary of Theorem \ref{thm.mn}.
\begin{corollary} \label{eq.spc}
Let
$A \in {\mathbb{C}}^{n \times n}$ and $ 0 \in \sigma(A) $.
Let \,$
s^{2m_1}, \dots , s^{2m_k}$,
be the elementary
divisors of $A$
corresponding to $\lambda = 0$.
If
$ U = ( u_1, \dots , u_k ) $ is a tuple of generators of \,
$\mathop{\mathrm{Ker}}\nolimits A^n$
such that $ \mathop{ \mathrm{e} {} }\nolimits (u_i) = 2m_i$,
$i = 1, \dots , k$,
then $ \mathop{ \mathrm{e} {} }\nolimits ( A^{m_i}u_i) = m_i$ \,
for all $i$, and
\begin{multline*}
\langle A ^{m_1} u_1 \rangle \oplus
\cdots \oplus
\langle A ^{m_k} u_k \rangle
=
\bigl(\mathop{\mathrm{Im}}\nolimits A ^{m_1} \cap \mathop{\mathrm{Ker}}\nolimits A ^{m_1}\bigr)
+
\cdots +
\bigl( \mathop{\mathrm{Im}}\nolimits A ^{m_k} \cap \mathop{\mathrm{Ker}}\nolimits A ^{m_k}\bigr).
\end{multline*}
\end{corollary}
\section{An application} \label{sec.app}
\begin{comment}
The use of Jordan bases for the construction of invariant subspaces
of the from $W(r,U)$
is a standard procedure in linear algebra and systems theory
(see e.g. \cite{Ku}, \cite[p.61]{GLR0}, \cite{Fr}, \cite[p.28]{BEGO},
\cite{RR}).
\end{comment}
In this section
we apply
Corollary~\ref{eq.spc} to the algebraic Riccati equation
\begin{equation} \label{eq.are}
Q + F^* X + XF - XDX = 0
\end{equation}
and its
associated
Hamiltonian matrix
\begin{equation} \label{eq.ham}
H = \begin{pmatrix}
F & - D \\ -Q & - F^*
\end{pmatrix}.
\end{equation}
Here $F, D , Q $ are complex $m \times m$ matrices,
$D$ and $Q$ are hermitian, $D \geq 0$, and the pair
$(F, D) $ is assumed to be controllable.
Then (see \cite[p.59]{Ku}
all elementary divisors corresponding to eigenvalues
$ i \alpha \in i {\mathbb{R}} $ have even degree.
To fix ideas we assume
\, $ \sigma (H) = \{ 0\} $.
The subsequent result complements Lemma 3.2.3 of
\cite[p.60]{Ku}.
\begin{prop}
Let
\,$ s ^{2m_1 } , \dots s ^{ 2m_k }$
be the elementary divisors of $H$.
Set
\begin{equation} \label{eq.een}
W = \bigl(\mathop{\mathrm{Im}}\nolimits H ^{m_1} \cap \mathop{\mathrm{Ker}}\nolimits H^{m_1} \bigr)
+
\cdots + \bigl(\mathop{\mathrm{Im}}\nolimits H ^{m_k} \cap \mathop{\mathrm{Ker}}\nolimits H^{m_k} \bigr).
\end{equation}
Then $W$ is an $H$-invariant subspace of
$ {\mathbb{C}}^{2m} $ and
$ \dim W = m$.
Let $ Y , Z \in {\mathbb{C}}^{m \times m}$ be such that
the columns of \,$ \left( \begin{smallmatrix} Y \\ Z
\end{smallmatrix} \right) $
are a basis of $W$.
Then
$ Y $ is nonsingular and
$X = Z Y ^{-1} $ is the unique hermitian solution of
\eqref{eq.are}.
\end{prop}
Proof. Set $ t = (2 m _1, \dots , 2 m_k) $.
Let $ U = (u_1, \dots , u_k) $ be a tuple of
generators of $E_0(H) = {\mathbb{C}} ^{2m}$.
According to \cite{Ku} we have
\[
W ( \tfrac{1}{2} t , U) = {\rm{span}} \left( \begin{matrix} I_m \\ X
\end{matrix} \right),
\]
where
$X \in {\mathbb{C}}^{m \times m}$ is the unique hermitian
solution of \eqref{eq.are}.
From Corollary~\ref{eq.spc} we know that
$ W ( \tfrac{1}{2} t, U) $ is independent of the choice of $U$.
Moreover,
$ W ( \tfrac{1}{2} t, U) = W $ where
$W $ is given by \eqref{eq.een}.
Hence, if \,$ W = {\rm{span}} \left( \begin{smallmatrix}
Y \\ Z
\end{smallmatrix} \right) $
then $ Y $ is nonsingular, and
\[
{\rm{span}}
\left( \begin{matrix}
Y \\ Z
\end{matrix} \right)
=
{\rm{span}} \left( \begin{matrix}
I \\ Z Y ^{-1}
\end{matrix} \right)
\]
implies that $ X = Z Y ^{-1} $ is the solution of
\eqref{eq.are}.
\hfill $\square$
\section{Conclusions}
Results of this note can
be considered in a module theoretic framework.
In a subsequent paper we shall make the connection
of Theorem~\ref{thm.mn}
with marked subspaces in \cite{FPP} and with
torsion modules over discrete valuation domains
in~\cite{AW}.
\bigskip
\textbf{Acknowledgement.} We would like to thank Dr. G. Dirr for a
valuable comment.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,099 |
Unfortunately, we do not send any keys ourselves, but the devs send it themselves.
If you have questions on keys you have been asking for I would recommend getting into contact with them.
Obviously, not everyone can have every key they are asking for as devs usually have between 100-1000 keys they send out on average with multiple thousands of people asking for a key.
i Requested a Atlas Key, it said i owned the game but i didnt what happened can help me with that??
Hello, how do I get a replacement key for hitman 2? I was contacted by one of your staff on twitter saying they was looking into the matter and I have not heard anything back since. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,465 |
Iona Jones is a Welsh television executive, the former head of Welsh language channel S4C.
Born in Haverfordwest, Pembrokeshire, she was brought up in Lampeter, Ceredigion, before moving to Cardiff with her family when she was six. Jones graduated in economic and social history from Exeter University.
On graduation in 1986, she joined BBC Cymru Wales as a trainee journalist. She later became editor of the current affairs programme Taro Naw and daily news programme Newyddion, produced by the BBC for S4C. In 1995, Jones joined S4C as the director of corporate affairs. In 2000 she joined ITV Wales, where she represented Carlton Television's regional companies throughout the UK on the Communications Act 2003.
Jones was appointed Director of Programmes at S4C in 2003, and became Chief Executive in 2005, the fourth S4C CEO and first woman to hold the position. Jones left S4C in July 2010, after discussion over the future funding of S4C.
She has three children and lives in Cardiff.
References
Year of birth missing (living people)
Living people
People from Haverfordwest
Alumni of the University of Exeter
Welsh television executives
Women television executives
Journalists from Cardiff
Women chief executives
Welsh women in business | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,414 |
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