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Miller–Rabin primality test

The Miller–Rabin primality test or Rabin–Miller primality test is a primality test: an algorithm which determines whether a given number is prime, similar to the Fermat primality test and the Solovay–Strassen primality test. It was first discovered by Russian mathematician M. M. Artjuhov in 1967. Gary L. Miller rediscovered it in 1976. Michael O. Rabin modified it to obtain an unconditional probabilistic algorithm in 1980. Just like the Fermat and Solovay–Strassen tests, the Miller–Rabin test relies on an equality or set of equalities that hold true for prime values checks whether or not they hold for a number that we want to test for primality. First, a lemma about square roots of unity in the finite field Z/pZ, where p is prime and p > 2. 1 and −1 always yield 1 when squared modulo p. There are no nontrivial square roots of 1 modulo p. To show this, suppose that x is a square root of 1 modulo p. Then: x 2 ≡ 1 ≡ 0. In other words, prime p divides the product. By Euclid's lemma it divides one of the factors x − 1 or x + 1, implying that x is congruent to either 1 or −1 modulo p. Now, let n be prime, n > 2.

It follows that n − 1 is and we can write it as 2s·d, where s and d are positive integers and d is odd. For each a in *, either a d ≡ 1 or a 2 r ⋅ d ≡ − 1 for some 0 ≤ r ≤ s − 1. To show that one of these must be true, recall Fermat's little theorem, that for a prime number n: a n − 1 ≡ 1. By the lemma above, if we keep taking square roots of an−1, we will get either 1 or −1. If we get −1 the second equality holds and it is done. If we never get −1 when we have taken out every power of 2, we are left with the first equality; the Miller–Rabin primality test is based on the contrapositive of the above claim. That is, if we can find an a such that a d ≢ 1 and a 2 r d ≢ − 1 for all 0 ≤ r ≤ s − 1 n is not prime. We call a a witness for the compositeness of n. Otherwise a is called a strong liar, n is a strong probable prime to base a; the term "strong liar" refers to the case where n is composite but the equations hold as they would for a prime. Note that Miller–Rabin pseudoprimes are called strong pseudoprimes.

Every odd composite n has many witnesses a. However, no simple way of generating such an a is known; the solution is to make the test probabilistic: we choose a non-zero a in Z/nZ randomly, check whether or not it is a witness for the compositeness of n. If n is composite, most of the choices for a will be witnesses, the test will detect n as composite with high probability. There is a small chance that we are unlucky and hit an a, a strong liar for n. We may reduce the probability of such error by repeating the test for several independently chosen a. However, there are diminishing returns in doing tests to many bases, because if n is a pseudoprime to base a it seems more to be a pseudoprime to another base b. For testing large numbers, it is common to choose random bases a, as, a priori, we don't know the distribution of witnesses and liars among the numbers 1, 2... N − 1. In particular, Arnault gave a 397-digit composite number for which all bases a less than 307 are strong liars; as expected this number was reported to be prime by the Maple isprime function, which implemented the Miller–Rabin test by checking the specific bases 2,3,5,7, 11.

However, selection of a few specific small bases can guarantee identification of composites for n less than some maximum determined by said bases. This maximum is quite large compared to the bases; as random bases lack such determinism for small n, specific bases are better in some circumstances. Suppose we wish to determine if n = 221 is prime. We write n − 1 = 220 as 22·55, so that we have s = 2 and d = 55. We randomly select a number a such that 1 < a < n - 1, say a = 174. We proceed to compute: a20·d mod n = 17455 mod 221 = 47 ≠ 1, n − 1 a21·d mod n = 174110 mod 221 = 220 = n − 1. Since 220 ≡ −1 mod n, either 221 is prime, or 174 is a strong liar for 221. We try another random a, this time choosing a = 137: a20·d mod n = 13755 mod 221 = 188 ≠ 1, n − 1 a21·d mod n = 137110 mod 221 = 205 ≠ n − 1. Hence 137 is a witness for the compositeness of 221, 174 was in fact a strong liar. Note that this tells us nothing about the factors of 221. However, the example with 341

H. S. Shivaprakash

H. S. Shivaprakash is a leading playwright writing in Kannada, he is professor at the School of Jawaharlal Nehru University, New Delhi. He heads the Cultural Centre at Berlin, known as the Tagore Centre, as Director run by Indian Council for Cultural Relations, he has seven anthologies of poems, twelve plays, several other books to his credit. His works have been translated into English, Italian, German, Hindi, Marathi and Telugu, his plays have been performed in Kannada, Meitei, Assamese, Bodo and Malayalam. Shivaprakash is a well-known authority on vachana literature, Bhakti movements of India, Sufi and other mystic traditions. Shivaprakash was born in Bangalore in June 1954, his father Shivamurthy Shastri was an eminent Lingayat scholar and served under the erstwhile Maharaja of Mysore. After obtaining his MA in English literature from Bangalore University, Shivaprakash joined the Karnataka government service as an English lecturer and taught for over two decades at various colleges in Bangalore and Tumkur.

In 1996, he was appointed the editor of Indian Literature, the bimonthly journal of Sahitya Akademi in New Delhi. Shivaprakash joined the School of Arts and Aesthetics in Jawaharlal Nehru University as Associate professor in 2001, where he is professor of Aesthetics and Performance studies. In 2000, he was selected for the International Writing Program of the School of Letters, University of Iowa, is honorary fellow of the school. Professor H. S. Shiva Prakash has been appointed Director of Tagore Centre in Berlin; the appointment is for three years Shivaprakash published his first anthology of poems Milarepa in 1977, when he was still 23. It was recognized as a fresh voice in Kannada poetry, but Shivaprakash gained popularity and acclaim only with his second anthology, Malebidda Neladalli in 1983. The poem "Samagara Bhimavva" became an instant hit, which brought him to the centre-stage of post-Bandaya Kannada poetry. Since Shivaprakash has published four collections of poetry, Anukshana Charite, Maleye Mantapa and Matte Matte and two anthologies of poems in translation and Nanna Mainagara, edited the translation of contemporary Gujarati poetry, Samakaleena Gujarati Kavitegalu and Malayalam poetry Manasantara.

Shivaprakash's poems make use of mystic symbolism, dream-images and motifs from everyday life to portray the nature of power and the contradictions of modern life. Shivaprakash published his first play Mahachaitra in 1986; the stage-adaptation of the play by C. G. Krishnaswamy for the troop Samudaya became a major hit; the play was based on the life and times of the 12th century Lingayat saint Basavanna and narrated the struggles of the artisan saints of the city of Kalyana through a Marxist analytic. The play was acknowledged as a landmark in Kannada literature. Mahachaitra is recognized as one among the three greatest plays out of the 25-odd plays on Basavanna written in Kannada, the other two being P. Lankesh's Sankranti and Girish Karnad's Taledanda. Shivaprakash won the Karnataka Sahitya Akademi award for this play, his other plays include Sultan Tipu, Shakespeare Swapnanauke, Manteswamy Kathaprasanga, Madari Madayya, Madhavi, Makarachandra, Sati and Maduvehennu. He has translated Shakespeare's King Lear and adapted Federico García Lorca's The Shoemaker's Prodigious Wife into Kannada under the title Mallammana Mane Hotlu and Shakespeare's Macbeth under the name Maranayakana Drishtanta.

Manteswamy Kathaprasanga, a play about a 16th-century Dalit saint was made into a successful stage adaptation by director Suresh Anagalli and produced over 300 shows. It kindled interest in this obscure saint and the life of Manteswamy has since been a major area of research in Kannada academia. Most of Shivaprakash's plays are inspired by Marxism and Shaiva mysticism Lingayatism and Kashmir Shavism; the plays employ motifs from Sufism and other forms of mysticism like Mahayana and Zen Buddhism. Structurally, the plays are inspired by Brecht's epic theatre. Mahachaitra was prescribed as a textbook for undergraduate courses in three universities of Karnataka. In 1995, nearly a decade after its publication, when it was prescribed as a textbook in Gulbarga University, it caused a heated controversy. A section of Lingayats under the leadership of the nun Shri Shri Jagadguru Mate Mahadevi accused the work of portraying Basavanna in poor light and urged the Government of Karnataka to ban the play, it led to a legal battle and the play was withdrawn from the university syllabus.

The Mahachaitra controversy seems to have inspired Githa Hariharan's English novel In Times of Siege, which narrates the story of a professor in an open university in Delhi, who finds himself in the midst of a controversy over a chapter on Basavanna which he wrote for an undergraduate textbook. Milarepa Malebidda Neladalli Anukshana Charite Suryajala Maleye Mantapa Matte Matte Mabbina Haage Kanive Haasi Maruroopagalu Nanna Mainagara Navilu Nagara Maatu Mantravaaguvavrege Maretuhoda Dombaraake Kavite Indinavarege Autumn Ways Maagiparva I Keep Vigil of Rudra Mahachaitra Sultan Tipu Shakespeare Swapnanauke Manteswamy Kathaprasanga Madari Madayya Madurekanda Madhavi Matrika Makarachandra Sati Cassandra Maduve Hennu King Lear (Kannada translation of Shakespea

William K. Thierfelder

William K. Thierfelder is an American businessman, academic administrator, former athlete serving as the 20th President of Belmont Abbey College in Belmont, North Carolina. William K. Thierfelder was born in Manhattan and raised in the Bronx where his father, William P. Thierfelder, was Vice President of the New York Yankees, he holds a Bachelor of Arts in psychology from the University of Maryland and both a master's degree and a doctoral degree in sports psychology and human movement from Boston University. He is a licensed psychologist, a two-time NCAA Division I All-American, a former NCAA Division I coach, a former member of the United States Olympic Committee's Sports Psychology Registry, he did not compete due to an injury. Prior to his selection as the 20th President of Belmont Abbey College, Thierfelder served as President of the York Barbell Company in York, Pennsylvania, he has served as the Executive Director of the Player Management Group, LLC. Thierfielder is a former certified member of the National Strength and Conditioning Association as well as the American College of Sports Medicine.

He is a member of the American Psychological Association and has served as an adjunct Professor of Surgery for Penn State University's Milton S. Hershey College of Medicine. In 2007, Dr. Thierfelder was named Boston University School of Education's Alumnus of the Year, he is a Knight of the Sovereign Military Order of Malta and is the author of Less Than A Minute To Go: The Secret To World-Class Performance in Sport and Everyday Life. He and his wife and their ten children, live outside Charlotte, North Carolina. Dr. Thierfelder's leadership of Belmont Abbey College has been marked by controversy related to issues of implementation regarding the Affordable Care Act. In 2011, the college under his leadership filed a lawsuit against the Obama administration, arguing that its mandated coverage of contraceptives for employees participating in employer-sponsored healthcare violated the college's right to religious liberty; the lawsuit was a source of division for the campus, with many students and faculty members both in favor of the measure and opposed.

Those opposed argued that Belmont Abbey College had never been a religious institution, since many of its faculty members and administrators came from non-Catholic backgrounds. Those in favor of the lawsuit argued that the college's roots as a monastery, with a Catholic basilica on the main campus, were indicators that its faith tradition and daily operations were rooted in Catholicism; as filed, the original lawsuit was dismissed by a federal judge in 2012 due to the Obama administration's failure, at that point in time, to add and clarify religious exemptions to specific parts of the Affordable Care Act and its coverage mandates. Further, U. S. District Judge James Boasberg wrote in his decision at the time that "Belmont’s injury is too speculative to confer standing and that the case is not ripe for decision.” Belmont Abbey College refiled the lawsuit in late 2013, after the Obama administration granted religious exemptions to another religious institution. In 2016, the college, along with other religious colleges and universities throughout the United States came under increasing criticism from LGBT advocates for refusing to implement policies advocating lesbian, gay and transgender behaviors.

Belmont Abbey College argued that its status as a Catholic institution was in conflict with these policies. In a statement, the college claimed that such recognition and approval of LGBT behavior would "abdicate the responsibility of the college community as a whole to act in accord with its fundamental identity as a community which publicly identifies itself as in communion with the Catholic Church."

Reverse chronology

Reverse chronology is a method of story-telling whereby the plot is revealed in reverse order. In a story employing this technique, the first scene shown is the conclusion to the plot. Once that scene ends, the penultimate scene is shown, so on, so that the final scene the viewer sees is the first chronologically. Many stories employ flashback, showing prior events, but whereas the scene order of most conventional films is A-B-C-etc. A film in reverse chronology goes Z-Y-X-etc. In interactive fiction, reverse chronology is a well-known technique.. For instance, the story Beanstalk the and Jack, tells the fairy tale Jack and the Beanstalk using reverse chronology: the opening scene depicts Jack chopping the beanstalk down and killing the giant; the next scene features Jack being discovered by the giant and climbing down the beanstalk in fear of his life. We see Jack running into the man with the infamous magic beans at the end of the story, being sent off by his mother to sell the cow; the unusual nature of this method means.

For example, Memento features a man with anterograde amnesia, meaning he is unable to form new memories. The film parallels the protagonist's perspective by unfolding in reverse chronological order, leaving the audience as ignorant of the events that occurred prior to each scene as the protagonist is. In the film Irréversible, an act of homicidal violence takes place at the start of the movie. During the remainder of the film we learn not only that the violence is an act of vengeance, but what is being avenged; the film was controversial for its graphic nature. However, as it is, told in reverse, the audience is made to consider the exact consequences of each action, there is "more than meets the eye." The epic poem Aeneid, written by Virgil in the 1st century BC, uses reverse chronology within scenes. In "The Three Apples", a murder mystery in the One Thousand and One Nights, the middle part of the story shows a flashback of events leading up to the discovery of a dead body at the beginning of the story.

The action of W. R. Burnett's novel, Goodbye to the Past, moves continually from 1929 to 1873; the Long View by Elizabeth Jane Howard describes a marriage in reverse chronology from 1950s London back to its beginning in 1926. Edward Lewis Wallant uses flashbacks in reverse chronology in The Human Season; the novel Christopher Homm, by C. H. Sisson, is told in reverse chronology. Philip K. Dick, in his 1967 novel Counter-Clock World, describes a future in which time has started to move in reverse, resulting in the dead reviving in their own graves, living their lives in reverse ending in returning to the womb, splitting into an egg and a sperm during copulation between a recipient woman and a man; the novel was expanded from Dick's short story "Your Appointment Will Be Yesterday", first published in the August 1966 edition of Amazing Stories. Martin Amis's 1991 novel Time's Arrow tells the story of a man who, it seems, brings dead people to life, it is revealed that the story is being seen backwards, he was a doctor at Auschwitz who brought death to live people.

He escaped to the United States, the novel starts with his death and ends with his birth. Amis writes in the Afterword that he had a "certain paragraph" from Kurt Vonnegut's Slaughterhouse Five in mind; as he waits to be taken by aliens to the planet Tralfamadore, the protagonist, Billy Pilgrim, watches a war movie backwards. American planes full of holes fly backwards as German planes suck bullets from them; the American fliers became high school kids again, Billy guesses, Hitler returns to babyhood. Iain Banks's novel Use of Weapons interweaves two parallel stories, one told in standard chronology and one in reverse, both concluding at a critical moment in the main character's life. Julia Alvarez's novel How the García Girls Lost Their Accents opens in 1989 with one of the characters returning to her native Dominican Republic; the story of why the family left and their attempts to succeed in New York are told in reverse chronological order, with the last events happening in 1956. A number of plays have employed this technique.

George S. Kaufman and Moss Hart's 1934 play, Merrily We Roll Along, is told in reverse order, as is the Harold Pinter play Betrayal. Kaufman and Hart's play was adapted as a musical comedy by Stephen Sondheim in 1981, Pinter's play was made into a film in 1983. In 1927, Jean Epstein's La glace à trois faces features a sequence where the events happen in reverse, beginning with the protagonist's exit from a room until the viewer sees the entrance; the Czech comedy Happy End is a farce which starts with a guillotined man finding his head popped back on his shoulders and ends with him as a new-born being pushed back into his mother's womb. Atom Egoyan, influenced by Pinter's plays, tells the story of The Sweet Hereafter in reverse chronology, with the first scene of the film set in 1977 and the last in 1968; the technique was employed in Peppermint Candy, by South Korean director Lee Chang-dong. In Irréversible, the technique is used so that the end credits are not only shown at t

Sahaptin language

Sahaptin or Shahaptin is one of the two-language Sahaptian branch of the Plateau Penutian family spoken in a section of the northwestern plateau along the Columbia River and its tributaries in southern Washington, northern Oregon, southwestern Idaho, in the United States.. Many of the tribes that surrounded the land were skilled with horses and trading with one another; the word Sahaptin/Shahaptin is not the one used by the tribes that speak it, but from the Columbia Salish name, Sħáptənəxw / S-háptinoxw, which means "stranger in the land". This is the name Kawaxchinláma, traditionally call the Nez Perce people. Early white explorers mistakenly applied the name to all the various Sahaptin speaking people, as well as to the Nez Perce. Sahaptin is spoken by various tribes of the Washington Reservations; the Yakama tribal cultural resources program has been promoting the use of the traditional name of the language, Ichishkíin Sɨ́nwit, instead of the Salish term Sahaptin. Sahaptin has three mutually intelligible dialects in a dialect cluster: Northern Sahaptin group Northwest Sahaptin dialect cluster: Klickitat, Upper Nisqually, Kittitas Northeast Sahaptin dialect cluster: Wanapum, Lower Snake, Walla Walla Southern Sahaptin group: Umatilla, Skin-pah, Tenino There are published grammars, a recent dictionary, a corpus of published texts.

Sahaptin has a split ergative syntax, with direct-inverse voicing and several applicative constructions. The ergative case inflects third-person nominals only when the direct object is first- or second-person: 1) i-q̓ínu-šana yáka paanáy 3nom-see-asp bear ‘the bear saw him’2) i-q̓ínu-šana=aš yáka-nɨm 3nom-see-asp=1sg bear-erg ‘the bear saw me’The direct-inverse contrast can be elicited with examples such as the following. In the inverse, the transitive direct object is coreferential with the subject in the preceding clause. Direct: 3) wínš i-q̓ínu-šana wapaanłá-an ku i-ʔíƛ̓iyawi-ya paanáy man 3nom-see-asp grizzly-acc and:3nom-kill-pst ‘the man saw the grizzly and he killed it’Inverse: 4) wínš i-q̓ínu-šana wapaanłá-an ku pá-ʔiƛ̓iyawi-ya man 3nom-see-asp grizzly-acc and inv-kill-pst ‘the man saw the grizzly and it killed him’The inverse retains its transitive status, a patient nominal is case marked accusative. 5) ku pá-ʔiƛ̓iyawi-ya wínš-na and inv-kill-pst man-acc ‘and it killed the man’ A semantic inverse is marked by the same verbal prefix pá-.

Direct: 6) q̓ínu-šana=maš see-asp=1sg/2sg ‘I saw you’Inverse: 7) pá-q̓inu-šana=nam inv-see-asp =2sg ‘you saw me’In Speech Act Participant and third-person transitive involvement, direction marking is as follows: Direct: 8) á-q̓inu-šana=aš paanáy obv-see-asp=1sg 3sg.acc ‘I saw him/her/it’Inverse: 9) i-q̓ínu-šana=aš pɨ́nɨm 3nom-see-asp=1sg 3erg ‘he/she/it saw me’ The charts of consonants and vowels below are used in the Yakima Sahaptin language: Vowels can be accented. Sahaptian languages Sahaptin people Cayuse Palus Umatilla Walla Walla Yakama Beavert and Sharon Hargus. Ichishkiin Sɨ́nwit Yakama/Yakima Sahaptin Dictionary. Toppenish and Seattle: Heritage University and University of Washington Press. Hargus and Virginia Beavert.. Yakima Sahaptin clusters and epenthetic. Anthropological Linguistics, 44.1-47. Jacobs, Melville. Northwest Sahaptin Texts, 1. University of Washington Publications in Anthropology 2:6:175-244. Seattle: University of Washington Press. Jacobs, Melville. A Sketch of Northern Sahaptin Grammar.

University of Washington Publications in Anthropology 4:2:85-292. Seattle: University of Washington Press. Jacobs, Melville. Northwest Sahaptin Texts. English language only. Columbia University Contributions to Anthropology 19, Part 1. New York: Columbia University Press. Jacobs, Melville. Northwest Sahaptin Texts. Sahaptin language only. Columbia University Contributions to Anthropology 19, Part 2. New York: Columbia University Press. Mithun, Marianne.. The languages of Native North America. Cambridge: Cambridge University Press. ISBN 0-521-23228-7. Rigsby and Noel Rude.. Sketch of Sahaptin, a Sahaptian Language. In Languages, ed. by Ives Goddard, pp. 666–692. Handbook of North American Indians, Volume 17. Washington, D. C.: Smithsonian Institution. Rude, Noel.. Pronominal prefixes in Klikitat Sahaptin. In Papers from the 1988 Hokan-Pe

Table tennis at the 2012 Summer Olympics – Qualification

An NOC may enter up to six athletes, two male and two female athletes in singles events and up to one men's and one women's team in team events. A total of 69 athletes qualified for each singles event as follows:†: Athlete qualified for team event only. * Highest ranked South Asian athlete. * Guo Yan is replaced by Ding Ning. ** Highest ranked South Asian athlete. Sixteen teams, including one team from each of six continents and the host NOC, qualified the team events; the qualification of an NOC was based on the number of the qualified players and the ranking of the 2012 World Team Championships. The 2012 World Team Championships were held from March 25 to April 2012 in Dortmund, Germany. A total of fourteen quota places per gender were reserved for completing qualified teams representing NOCs with less than three qualified individual athletes. Eight men's quotas and four women's quotas remained unused, were transferred to the singles competition of the same gender and added to the Final Qualification Tournament.

International Table Tennis Federation