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101 | good king rode forth from his castle, saw the suffering workers and healed them.'" Of his father Ginsberg said "My father would go around the house either reciting Emily Dickinson and Longfellow under his breath or attacking T. S. Eliot for ruining poetry with his 'obscurantism.' I grew suspicious of both sides." Naomi... | "Allen Ginsberg" | [
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102 | little pet", as Bill Morgan says in his biography of Ginsberg, titled, "I Celebrate Myself: The Somewhat Private Life of Allen Ginsberg". She also tried to kill herself by slitting her wrists and was soon taken to Greystone, a mental hospital; she would spend much of Ginsberg's youth in mental hospitals. His experience... | "Allen Ginsberg" | [
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103 | Ginsberg – he mentioned it and other moments from his childhood in "Kaddish". His experiences with his mother's mental illness and her institutionalization are also frequently referred to in "Howl". For example, "Pilgrim State, Rockland, and Grey Stone's foetid halls" is a reference to institutions frequented by his mo... | "Allen Ginsberg" | [
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104 | Solomon in section three, "I'm with you in Rockland where you imitate the shade of my mother," once again showing the association between Solomon and his mother. Ginsberg received a letter from his mother after her death responding to a copy of "Howl" he had sent her. It admonished Ginsberg to be good and stay away fro... | "Allen Ginsberg" | [
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105 | In a letter she wrote to Ginsberg's brother Eugene, she said, "God's informers come to my bed, and God himself I saw in the sky. The sunshine showed too, a key on the side of the window for me to get out. The yellow of the sunshine, also showed the key on the side of the window." These letters and the absence of a faci... | "Allen Ginsberg" | [
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106 | key is in the window". In Ginsberg's freshman year at Columbia he met fellow undergraduate Lucien Carr, who introduced him to a number of future Beat writers, including Jack Kerouac, William S. Burroughs, and John Clellon Holmes. They bonded, because they saw in one another an excitement about the potential of American... | "Allen Ginsberg" | [
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107 | long infatuation. In the first chapter of his 1957 novel "On the Road" Kerouac described the meeting between Ginsberg and Cassady. Kerouac saw them as the dark (Ginsberg) and light (Cassady) side of their "New Vision", a perception stemming partly from Ginsberg's association with communism, of which Kerouac had become ... | "Allen Ginsberg" | [
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108 | by the Pony Stable patrons and was writing poetry there the night of their meeting. Ginsberg claims he was immediately attracted to Corso, who was straight, but understanding of homosexuality after three years in prison. Ginsberg was even more struck by reading Corso's poems, realizing Corso was "spiritually gifted." G... | "Allen Ginsberg" | [
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109 | with during one of his forays into heterosexuality. Ginsberg took Corso over to their apartment. There the woman proposed sex with Corso, who was still very young and fled in fear. Ginsberg introduced Corso to Kerouac and Burroughs and they began to travel together. Ginsberg and Corso remained lifelong friends and coll... | "Allen Ginsberg" | [
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110 | extensively read the poetry of Ezra Pound and T. S. Eliot, when she met Joyce Johnson and Leo Skir, among other Beat players. As Cowen had felt a strong attraction to darker poetry most of the time, Beat poetry seemed to provide an allure to what suggests a shadowy side of her persona. While at Barnard, Cowen earned th... | "Allen Ginsberg" | [
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111 | books, including "Minor Characters" and "Come and Join the Dance", which expressed the two women's experiences in the Barnard and Columbia Beat community. Through his association with Elise Cowen, Ginsberg discovered that they shared a mutual friend, Carl Solomon, to whom he later dedicated his most famous poem "Howl".... | "Allen Ginsberg" | [
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112 | (later referred to as his "Blake vision"). At first, Ginsberg claimed to have heard the voice of God, but later interpreted the voice as that of Blake himself reading "Ah! Sun-flower", "The Sick Rose", and "Little Girl Lost", also described by Ginsberg as "voice of the ancient of days". The experience lasted several da... | "Allen Ginsberg" | [
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113 | was the hand that crafted itself. He explained that this hallucination was not inspired by drug use, but said he sought to recapture that feeling later with various drugs. Ginsberg stated: "living blue hand itself. Or that God was in front of my eyes - existence itself was God" and "And it was a sudden awakening into a... | "Allen Ginsberg" | [
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114 | met Peter Orlovsky (1933–2010), with whom he fell in love and who remained his lifelong partner. Selections from their correspondence have been published. Also in San Francisco, Ginsberg met members of the San Francisco Renaissance (James Broughton, Robert Duncan, Madeline Gleason and Kenneth Rexroth) and other poets w... | "Allen Ginsberg" | [
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115 | at Reed College: Gary Snyder, Philip Whalen, and Lew Welch. In 1959, along with poets John Kelly, Bob Kaufman, A. D. Winans, and William Margolis, Ginsberg was one of the founders of the "Beatitude" poetry magazine. Wally Hedrick — a painter and co-founder of the Six Gallery — approached Ginsberg in mid-1955 and asked ... | "Allen Ginsberg" | [
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116 | the most important events in Beat mythos, known simply as "The Six Gallery reading" took place on October 7, 1955. The event, in essence, brought together the East and West Coast factions of the Beat Generation. Of more personal significance to Ginsberg, the reading that night included the first public presentation of ... | "Allen Ginsberg" | [
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117 | passionately, drunken, with arms outstretched. Ginsberg's principal work, "Howl", is well known for its opening line: "I saw the best minds of my generation destroyed by madness, starving hysterical naked ..." "Howl" was considered scandalous at the time of its publication, because of the rawness of its language. Short... | "Allen Ginsberg" | [
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118 | manager who was jailed for selling "Howl," became lifelong friends. Ginsberg claimed at one point that all of his work was an extended biography (like Kerouac's "Duluoz Legend"). "Howl" is not only a biography of Ginsberg's experiences before 1955, but also a history of the Beat Generation. Ginsberg also later claimed ... | "Allen Ginsberg" | [
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119 | saw the best minds of my generation destroyed by madness", which sets the stage for Ginsberg to describe Cassady and Solomon, immortalizing them into American literature. This madness was the "angry fix" that society needed to function — madness was its disease. In the poem, Ginsberg focused on "Carl Solomon! I'm with ... | "Allen Ginsberg" | [
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120 | is still perhaps the best place to start. In 1957, Ginsberg surprised the literary world by abandoning San Francisco. After a spell in Morocco, he and Peter Orlovsky joined Gregory Corso in Paris. Corso introduced them to a shabby lodging house above a bar at 9 rue Gît-le-Coeur that was to become known as the Beat Hote... | "Allen Ginsberg" | [
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121 | previous writings. This period was documented by the photographer Harold Chapman, who moved in at about the same time, and took pictures constantly of the residents of the "hotel" until it closed in 1963. During 1962–1963, Ginsberg and Orlovsky travelled extensively across India, living half a year at a time in Calcutt... | "Allen Ginsberg" | [
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122 | when the authorities were eager to expel him. In May 1965, Ginsberg arrived in London, and offered to read anywhere for free. Shortly after his arrival, he gave a reading at Better Books, which was described by Jeff Nuttall as "the first healing wind on a very parched collective mind". Tom McGrath wrote: "This could we... | "Allen Ginsberg" | [
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123 | Hall in London on June 11, 1965. The event attracted an audience of 7,000, who heard readings and live and tape performances by a wide variety of figures, including Ginsberg, Adrian Mitchell, Alexander Trocchi, Harry Fainlight, Anselm Hollo, Christopher Logue, George MacBeth, Gregory Corso, Lawrence Ferlinghetti, Micha... | "Allen Ginsberg" | [
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124 | title by Lorrimer in the UK and Grove Press in US. Though the term "Beat" is most accurately applied to Ginsberg and his closest friends (Corso, Orlovsky, Kerouac, Burroughs, etc.), the term "Beat Generation" has become associated with many of the other poets Ginsberg met and became friends with in the late 1950s and e... | "Allen Ginsberg" | [
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125 | mistaken identification of Ginsberg as the leader. Ginsberg never claimed to be the leader of a movement. He claimed that many of the writers with whom he had become friends in this period shared many of the same intentions and themes. Some of these friends include: David Amram, Bob Kaufman; Diane di Prima; Jim Cohn; p... | "Allen Ginsberg" | [
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126 | a sheet of toilet paper. Through a party organized by Amiri Baraka, Ginsberg was introduced to Langston Hughes while Ornette Coleman played saxophone. Later in his life, Ginsberg formed a bridge between the beat movement of the 1950s and the hippies of the 1960s, befriending, among others, Timothy Leary, Ken Kesey, and... | "Allen Ginsberg" | [
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127 | studying Buddhism and shared what he learned from Dwight Goddard's "Buddhist Bible" with Ginsberg. Ginsberg first heard about the Four Noble Truths and such sutras as the Diamond Sutra at this time. Ginsberg's spiritual journey began early on with his spontaneous visions, and continued with an early trip to India with ... | "Allen Ginsberg" | [
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128 | Rumtek Monastery. Continuing on his journey, Ginsberg met Dudjom Rinpoche in Kalimpong who taught him "If you see something horrible, don't cling to it, and if you see something beautiful, don't cling to it." After returning to the United States, a chance encounter on a New York City street with Chögyam Trungpa Rinpoch... | "Allen Ginsberg" | [
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129 | Colorado. Ginsberg was also involved with Krishnaism. He had started incorporating chanting the Hare Krishna mantra into his religious practice in the mid-1960s. After learning that A. C. Bhaktivedanta Swami Prabhupada, the founder of the Hare Krishna movement in the Western world had rented a store front in New York, ... | "Allen Ginsberg" | [
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130 | him to promote his cause. Despite disagreeing with many of Bhaktivedanta Swami's required prohibitions, Ginsberg often sang the Hare Krishna mantra publicly as part of his philosophy and declared that it brought a state of ecstasy. He was glad that Bhaktivedanta Swami, an authentic swami from India, was now trying to s... | "Allen Ginsberg" | [
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131 | hippie community. On January 17, 1967, Ginsberg helped plan and organize a reception for Bhaktivedanta Swami at San Francisco International Airport, where fifty to a hundred hippies greeted the Swami, chanting Hare Krishna in the airport lounge with flowers in hands. To further support and promote Bhaktivendata Swami's... | "Allen Ginsberg" | [
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132 | Moby Grape, who performed there along with the Hare Krishna founder Bhaktivedanta Swami and donated proceeds to the Krishna temple. Ginsberg introduced Bhaktivedanta Swami to some three thousand hippies in the audience and led the chanting of the Hare Krishna mantra. Music and chanting were both important parts of Gins... | "Allen Ginsberg" | [
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133 | relatives, including his cousin Savitri Banerjee. When Ginsberg asked if he could sing a song in praise of Lord Krishna on William F. Buckley, Jr.'s TV show "Firing Line" on September 3, 1968, Buckley acceded and the poet chanted slowly as he played dolefully on a harmonium. According to Richard Brookhiser, an associat... | "Allen Ginsberg" | [
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134 | a sound system for hours on end. Ginsberg further brought mantras into the world of rock and roll when he recited the Heart Sutra in the song "Ghetto Defendant". The song appears on the 1982 album Combat Rock by British first wave punk band The Clash. Ginsberg came in touch with the Hungryalist poets of Bengal, especia... | "Allen Ginsberg" | [
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135 | that he, like Whitman, adhered to an "American brand of mysticism" that was "rooted in humanism and in a romantic and visionary ideal of harmony among men." In 1960, he was treated for a tropical disease, and it is speculated that he contracted hepatitis from an unsterilized needle administered by a doctor, which playe... | "Allen Ginsberg" | [
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136 | strokes which were first diagnosed as Bell's palsy, which gave him significant paralysis and stroke-like drooping of the muscles in one side of his face. Later in life, he also suffered constant minor ailments such as high blood pressure. Many of these symptoms were related to stress, but he never slowed down his sched... | "Allen Ginsberg" | [
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137 | Struga, he met with the other Golden Wreath winners Bulat Okudzhava and Andrei Voznesensky. In 1993, the French Minister of Culture made him a Chevalier des Arts et des Lettres. Ginsberg continued to help his friends as much as he could, going so far as to give money to Herbert Huncke out of his own pocket, and housing... | "Allen Ginsberg" | [
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138 | 16, 1996. After returning home from the hospital for the last time, where he had been unsuccessfully treated for congestive heart failure, Ginsberg continued making phone calls to say goodbye to nearly everyone in his addressbook. Some of the phone calls, including one with Johnny Depp, were sad and interrupted by cryi... | "Allen Ginsberg" | [
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139 | complications of hepatitis. He was 70 years old. Gregory Corso, Roy Lichtenstein, Patti Smith and others came by to pay their respects. One third of Ginsberg's ashes were buried in his family plot in Gomel Chesed Cemetery in Newark, NJ. He was survived by Orlovsky. When Orlovsky died, as per Ginsberg's wishes, another ... | "Allen Ginsberg" | [
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140 | and the beatniks. Ginsberg's willingness to talk about taboo subjects made him a controversial figure during the conservative 1950s, and a significant figure in the 1960s. In the mid-1950s, no reputable publishing company would even consider publishing "Howl". At the time, such "sex talk" employed in "Howl" was conside... | "Allen Ginsberg" | [
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141 | including "Lady Chatterley's Lover". The sex that Ginsberg described did not portray the sex between heterosexual married couples, or even longtime lovers. Instead, Ginsberg portrayed casual sex. For example, in "Howl", Ginsberg praises the man "who sweetened the snatches of a million girls". Ginsberg used gritty descr... | "Allen Ginsberg" | [
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142 | up on charges for publishing pornography, and the outcome led to a judge going on record dismissing charges, because the poem carried "redeeming social importance", thus setting an important legal precedent. Ginsberg continued to broach controversial subjects throughout the 1970s, 1980s, and 1990s. From 1970–1996, Gins... | "Allen Ginsberg" | [
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143 | heroin was a taboo subject and Huncke was left with nowhere to go for help. Ginsberg was a signer of the anti-war manifesto "A Call to Resist Illegitimate Authority," circulated among draft resistors in 1967 by members of the radical intellectual collective RESIST. Other signers and RESIST members included Mitchell Goo... | "Allen Ginsberg" | [
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144 | riot in 1988 and provided an eyewitness account to "The New York Times". Allen Ginsberg called attention to the suffering of victims during the Bangladesh Liberation War in 1971. He wrote his legendary 152-line poem, "September on Jessore Road", after visiting refugee camps and witnessing the plight of millions fleeing... | "Allen Ginsberg" | [
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145 | shade. Where is America's Air Force of Light? Bombing North Laos all day and all night? Out of the poem, he made a song that was performed by Bob Dylan, other musicians and Ginsberg himself. The last few lines of the poem read: Ginsberg talked openly about his connections with communism and his admiration for past comm... | "Allen Ginsberg" | [
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146 | I was a kid I'm not sorry". Biographer Jonah Raskin has claimed that, despite his often stark opposition to communist orthodoxy, Ginsberg held "his own idiosyncratic version of communism". On the other hand, when Donald Manes, a New York City politician, publicly accused Ginsberg of being a member of the Communist Part... | "Allen Ginsberg" | [
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147 | Communist and Capitalist that I have observed". Ginsberg travelled to several communist countries to promote free speech. He claimed that communist countries, such as China, welcomed him, because they thought he was an enemy of capitalism, but often turned against him when they saw him as a troublemaker. For example, i... | "Allen Ginsberg" | [
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148 | for alleged drug use and public drunkenness, and the security agency StB confiscated several of his writings, which they considered to be lewd and morally dangerous. Ginsberg was then deported from Czechoslovakia on May 7, 1965 by order of the StB. Václav Havel points to Ginsberg as an important inspiration. One contri... | "Allen Ginsberg" | [
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149 | note for gay marriage by listing Peter Orlovsky, his lifelong companion, as his spouse in his Who's Who entry. Subsequent gay writers saw his frank talk about homosexuality as an opening to speak more openly and honestly about something often before only hinted at or spoken of in metaphor. In writing about sexuality in... | "Allen Ginsberg" | [
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150 | member of North American Man/Boy Love Association (NAMBLA), a pedophilia and pederasty advocacy organization in the United States that works to abolish age of consent laws and legalize sexual relations between adults and children, saying that "Attacks on NAMBLA stink of politics, witchhunting for profit, humorlessness,... | "Allen Ginsberg" | [
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151 | discussion society not a sex club. I joined NAMBLA in defense of free speech." In 1994, Ginsberg appeared in a documentary on NAMBLA called "" (playing on the gay male slang term "Chickenhawk"), in which he read a "graphic ode to youth". Ginsberg talked often about drug use. He organized the New York City chapter of Le... | "Allen Ginsberg" | [
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152 | against the hazards of tobacco in his "Put Down Your Cigarette Rag (Don't Smoke):" "Don't Smoke Don't Smoke Nicotine Nicotine No / No don't smoke the official Dope Smoke Dope Dope." on the latter's book "The Politics of Heroin in Southeast Asia", which claimed that the CIA was knowingly involved in the production of he... | "Allen Ginsberg" | [
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153 | Allen wrote many essays and articles, researching and compiling evidence of the CIA's alleged involvement in drug trafficking, but it would take 10 years, and the publication of McCoy's book in 1972, before anyone took him seriously. In 1978 Ginsberg received a note from the chief editor of "The New York Times", apolog... | "Allen Ginsberg" | [
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154 | proof." Subsequent investigations by the Inspector General of the CIA, United States House Committee on Foreign Affairs, and United States Senate Select Committee to Study Governmental Operations with Respect to Intelligence Activities, a.k.a. the Church Committee, also found the charges to be unsubstantiated. Most of ... | "Allen Ginsberg" | [
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155 | entire life to poetry. Soon after, he wrote "Howl", the poem that brought him and his Beat Generation contemporaries to national attention and allowed him to live as a professional poet for the rest of his life. Later in life, Ginsberg entered academia, teaching poetry as Distinguished Professor of English at Brooklyn ... | "Allen Ginsberg" | [
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156 | draft of "Howl" he disliked the fact that Ginsberg had made editorial changes in pencil (transposing "negro" and "angry" in the first line, for example). Kerouac only wrote out his concepts of Spontaneous Prose at Ginsberg's insistence because Ginsberg wanted to learn how to apply the technique to his poetry. The inspi... | "Allen Ginsberg" | [
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157 | to go to a mental institution and demand a lobotomy. The institution refused, giving him many forms of therapy, including electroshock therapy. Much of the final section of the first part of "Howl" is a description of this. Ginsberg used Solomon as an example of all those ground down by the machine of "Moloch". Moloch,... | "Allen Ginsberg" | [
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158 | mentioned a few times in the Torah and references to Ginsberg's Jewish background are frequent in his work. Ginsberg said the image of Moloch was inspired by peyote visions he had of the Francis Drake Hotel in San Francisco which appeared to him as a skull; he took it as a symbol of the city (not specifically San Franc... | "Allen Ginsberg" | [
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159 | subsequently been interpreted as any system of control, including the conformist society of post-World War II America, focused on material gain, which Ginsberg frequently blamed for the destruction of all those outside of societal norms. He also made sure to emphasize that Moloch is a part of humanity in multiple aspec... | "Allen Ginsberg" | [
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160 | The personal aspects of "Howl" are perhaps as important as the political aspects. Carl Solomon, the prime example of a "best mind" destroyed by defying society, is associated with Ginsberg's schizophrenic mother: the line "with mother finally fucked" comes after a long section about Carl Solomon, and in Part III, Ginsb... | "Allen Ginsberg" | [
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161 | 1959's "Kaddish", which had its first public reading at a Catholic Worker Friday Night meeting, possibly due to its associations with Thomas Merton. Ginsberg's poetry was strongly influenced by Modernism (most importantly the American style of Modernism pioneered by William Carlos Williams), Romanticism (specifically W... | "Allen Ginsberg" | [
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162 | poet Federico García Lorca. The power of Ginsberg's verse, its searching, probing focus, its long and lilting lines, as well as its New World exuberance, all echo the continuity of inspiration that he claimed. He corresponded with William Carlos Williams, who was then in the middle of writing his epic poem "Paterson" a... | "Allen Ginsberg" | [
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163 | Ginsberg, "In this mode perfection is basic, and these poems are not perfect." Though he disliked these early poems, Williams loved the exuberance in Ginsberg's letter. He included the letter in a later part of "Paterson". He encouraged Ginsberg not to emulate the old masters, but to speak with his own voice and the vo... | "Allen Ginsberg" | [
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164 | Early breakthrough poems include "Bricklayer's Lunch Hour" and "Dream Record". Carl Solomon introduced Ginsberg to the work of Antonin Artaud ("To Have Done with the Judgement of God" and "Van Gogh: The Man Suicided by Society"), and Jean Genet ("Our Lady of the Flowers"). Philip Lamantia introduced him to other Surrea... | "Allen Ginsberg" | [
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165 | as: Franz Kafka, Herman Melville, Fyodor Dostoevsky, Edgar Allan Poe, and Emily Dickinson. Ginsberg also made an intense study of haiku and the paintings of Paul Cézanne, from which he adapted a concept important to his work, which he called the "Eyeball Kick". He noticed in viewing Cézanne's paintings that when the ey... | "Allen Ginsberg" | [
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166 | strong, an artifact of high culture with an artifact of low culture, something holy with something unholy. The example Ginsberg most often used was "hydrogen jukebox" (which later became the title of a song cycle composed by Philip Glass with lyrics drawn from Ginsberg's poems). Another example is Ginsberg's observatio... | "Allen Ginsberg" | [
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167 | Ginsberg also found inspiration in music. He frequently included music in his poetry, invariably composing his tunes on an old Indian harmonium, which he often played during his readings. He wrote and recorded music to accompany William Blake's "Songs of Innocence" and "Songs of Experience". He also recorded a handful ... | "Allen Ginsberg" | [
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168 | maintained a friendship with him over many years. In 1996, he also recorded a song cowritten with Paul McCartney and Philip Glass, "The Ballad of the Skeletons", which reached number 8 on the Triple J Hottest 100 for that year. From the study of his idols and mentors and the inspiration of his friends — not to mention ... | "Allen Ginsberg" | [
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169 | poetry sexualized aspects of the male form. Many of Ginsberg's early long line experiments contain some sort of anaphora, repetition of a "fixed base" (for example "who" in "Howl", "America" in "America") and this has become a recognizable feature of Ginsberg's style. He said later this was a crutch because he lacked c... | "Allen Ginsberg" | [
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170 | his style in later poems. In the original draft of "Howl", each line is in a "stepped triadic" format reminiscent of William Carlos Williams. However, he abandoned the "stepped triadic" when he developed his long line although the stepped lines showed up later, most significantly in the travelogues of "The Fall of Amer... | "Allen Ginsberg" | [
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171 | the epic, free verse style of the 19th-century American poet Walt Whitman. Both wrote passionately about the promise (and betrayal) of American democracy, the central importance of erotic experience, and the spiritual quest for the truth of everyday existence. J. D. McClatchy, editor of the "Yale Review", called Ginsbe... | "Allen Ginsberg" | [
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172 | its contradictory urges." McClatchy's barbed eulogies define the essential difference between Ginsberg ("a beat poet whose writing was ... journalism raised by combining the recycling genius with a generous mimic-empathy, to strike audience-accessible chords; always lyrical and sometimes truly poetic") and Kerouac ("a ... | "Allen Ginsberg" | [
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173 | the evolving quicksilver mind of America's only literary virtuoso ..."): Allen Ginsberg Irwin Allen Ginsberg (; June 3, 1926 – April 5, 1997) was an American poet, philosopher and writer. He is considered to be one of the leading figures of both the Beat Generation during the 1950s and the counterculture that soon foll... | "Allen Ginsberg" | [
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174 | Algebraically closed field In abstract algebra, an algebraically closed field "F" contains a root for every non-constant polynomial in "F"["x"], the ring of polynomials in the variable "x" with coefficients in "F". As an example, the field of real numbers is not algebraically closed, because the polynomial equation "x"... | "Algebraically closed field" | [
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175 | algebraically closed, because if "a", "a", …, "a" are the elements of "F", then the polynomial ("x" − "a")("x" − "a") ··· ("x" − "a") + 1 has no zero in "F". By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed. Another example of an algebraically closed field... | "Algebraically closed field" | [
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176 | degree one. The assertion "the polynomials of degree one are irreducible" is trivially true for any field. If "F" is algebraically closed and "p"("x") is an irreducible polynomial of "F"["x"], then it has some root "a" and therefore "p"("x") is a multiple of "x" − "a". Since "p"("x") is irreducible, this means that "p"... | "Algebraically closed field" | [
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177 | roots in "F", "q"("x") also has no roots in "F". Therefore, "q"("x") has degree greater than one, since every first degree polynomial has one root in "F". The field "F" is algebraically closed if and only if every polynomial "p"("x") of degree "n" ≥ 1, with coefficients in "F", splits into linear factors. In other word... | "Algebraically closed field" | [
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178 | words, "F" is algebraically closed. On the other hand, that the property stated here holds for "F" if "F" is algebraically closed follows from the previous property together with the fact that, for any field "K", any polynomial in "K"["x"] can be written as a product of irreducible polynomials. J. Shipman showed in 200... | "Algebraically closed field" | [
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179 | has no proper algebraic extension, let "p"("x") be some irreducible polynomial in "F"["x"]. Then the quotient of "F"["x"] modulo the ideal generated by "p"("x") is an algebraic extension of "F" whose degree is equal to the degree of "p"("x"). Since it is not a proper extension, its degree is 1 and therefore the degree ... | "Algebraically closed field" | [
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180 | it has no finite algebraic extension because if, within the previous proof, the word "algebraic" is replaced by the word "finite", then the proof is still valid. The field "F" is algebraically closed if and only if, for each natural number "n", every linear map from "F" into itself has some eigenvector. An endomorphism... | "Algebraically closed field" | [
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181 | "F"["x"]. Dividing by its leading coefficient, we get another polynomial "q"("x") which has roots if and only if "p"("x") has roots. But if "q"("x") = "x" + "a""x"+ ··· + "a", then "q"("x") is the characteristic polynomial of the "n×n" companion matrix The field "F" is algebraically closed if and only if every rational... | "Algebraically closed field" | [
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182 | closed then, since the irreducible polynomials in "F"["x"] are all of degree 1, the property stated above holds by the theorem on partial fraction decomposition. On the other hand, suppose that the property stated above holds for the field "F". Let "p"("x") be an irreducible element in "F"["x"]. Then the rational funct... | "Algebraically closed field" | [
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183 | irreducible, it must divide this product and, therefore, it must also be a first degree polynomial. For any field "F", if two polynomials "p"("x"),"q"("x") ∈ "F"["x"] are relatively prime then they do not have a common root, for if "a" ∈ "F" was a common root, then "p"("x") and "q"("x") would both be multiples of "x" −... | "Algebraically closed field" | [
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184 | field "F" is algebraically closed, let "p"("x") and "q"("x") be two polynomials which are not relatively prime and let "r"("x") be their greatest common divisor. Then, since "r"("x") is not constant, it will have some root "a", which will be then a common root of "p"("x") and "q"("x"). If "F" is not algebraically close... | "Algebraically closed field" | [
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185 | then "F" contains all "n"th roots of unity, because these are (by definition) the "n" (not necessarily distinct) zeroes of the polynomial "x" − 1. A field extension that is contained in an extension generated by the roots of unity is a "cyclotomic extension", and the extension of a field generated by all roots of unity... | "Algebraically closed field" | [
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186 | If a proposition which can be expressed in the language of first-order logic is true for an algebraically closed field, then it is true for every algebraically closed field with the same characteristic. Furthermore, if such a proposition is valid for an algebraically closed field with characteristic 0, then not only is... | "Algebraically closed field" | [
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187 | extension is called an algebraically closed extension. Among all such extensions there is one and only one (up to isomorphism, but not unique isomorphism) which is an algebraic extension of "F"; it is called the algebraic closure of "F". The theory of algebraically closed fields has quantifier elimination. Algebraicall... | "Algebraically closed field" | [
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188 | Anatoly Karpov Anatoly Yevgenyevich Karpov (; born May 23, 1951) is a Russian chess grandmaster and former World Champion. He was the official world champion from 1975 to 1985 when he was defeated by Garry Kasparov. He played three matches against Kasparov for the title from 1986 to 1990, before becoming FIDE World Cha... | "Anatoly Karpov" | [
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189 | the greatest players in history. His tournament successes include over 160 first-place finishes. He had a peak Elo rating of 2780, and his 102 total months at world number one is the second longest of all-time, behind only Garry Kasparov, since the inception of the FIDE ranking list in 1970. Karpov was born on May 23, ... | "Anatoly Karpov" | [
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190 | Mikhail Botvinnik's prestigious chess school, though Botvinnik made the following remark about the young Karpov: "The boy does not have a clue about chess, and there's no future at all for him in this profession." Karpov acknowledged that his understanding of chess theory was very confused at that time, and wrote later... | "Anatoly Karpov" | [
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191 | Spassky in 1952. Karpov finished first in his first international tournament in Třinec several months later, ahead of Viktor Kupreichik. In 1967, he won the annual European Junior Championship at Groningen. Karpov won a gold medal for academic excellence in high school, and entered Moscow State University in 1968 to st... | "Anatoly Karpov" | [
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192 | player. In 1969, Karpov became the first Soviet player since Spassky (1955) to win the World Junior Chess Championship, scoring an undefeated 10/11 in the finals at Stockholm. In 1970, he tied for fourth place at an international tournament in Caracas, Venezuela, and was awarded the grandmaster title. He won the 1971 A... | "Anatoly Karpov" | [
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193 | the Leningrad Interzonal Tournament. The latter success qualified him for the 1974 Candidates Matches, which would determine the challenger to the reigning world champion, Bobby Fischer. Karpov defeated Lev Polugaevsky by the score of +3=5 in the first Candidates' match, earning the right to face former champion Boris ... | "Anatoly Karpov" | [
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194 | from Karpov secured him overall victory by +4−1=6. The Candidates' final was played in Moscow with Korchnoi. Karpov took an early lead, winning the second game against the Sicilian Dragon, then scoring another victory in the sixth game. Following ten consecutive draws, Korchnoi threw away a winning position in the seve... | "Anatoly Karpov" | [
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195 | prevailed +3−2=19, moving on to challenge Fischer for the world title. Though a world championship match between Karpov and Fischer was highly anticipated, those hopes were never realised. Fischer not only insisted that the match be the first to ten wins (draws not counting), but also that the champion would retain the... | "Anatoly Karpov" | [
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196 | negotiations fell through. This thrust the young Karpov into the role of World Champion without having faced the reigning champion. Garry Kasparov argued that Karpov would have had good chances, because he had beaten Spassky convincingly and was a new breed of tough professional, and indeed had higher quality games, wh... | "Anatoly Karpov" | [
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197 | the very strong Milan tournament in 1975, and captured his first of three Soviet titles in 1976. He created a phenomenal streak of tournament wins against the strongest players in the world. Karpov held the record for most consecutive tournament victories (9) until it was shattered by Garry Kasparov (14). As a result, ... | "Anatoly Karpov" | [
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198 | Karpov took an early lead, winning the eighth game after seven draws to open the match. When the score was +5−2=20 in Karpov's favour, Korchnoi staged a comeback, and won three of the next four games to draw level with Karpov. However, Karpov won the very next game to retain the title (+6−5=21). Three years later Korch... | "Anatoly Karpov" | [
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199 | a peak at the Montreal "Tournament of Stars" tournament in 1979, where he finished joint first (+7−1=10) with Mikhail Tal, ahead of a field of strong grandmasters completed by Jan Timman, Ljubomir Ljubojević, Boris Spassky, Vlastimil Hort, Lajos Portisch, Robert Hübner, Bent Larsen and Lubomir Kavalek. He dominated Las... | "Anatoly Karpov" | [
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200 | Union at six Chess Olympiads, in all of which the USSR won the team gold medal. He played first reserve at Skopje 1972, winning the board prize with 13/15. At Nice 1974, he advanced to board one and again won the board prize with 12/14. At La Valletta 1980, he was again board one and scored 9/12. At Lucerne 1982, he sc... | "Anatoly Karpov" | [
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