In mathematics, the Fibonacci numbers denoted Fn, form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is, F 0 = 0, F 1 = 1, F n = F n − 1 + F n − 2, for n > 1. The beginning of the sequence is thus: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … In some older books, the value F 0 = 0 is omitted, so that the sequence starts with F 1 = F 2 = 1, the recurrence F n = F n − 1 + F n − 2 is valid for n > 2. Fibonacci numbers are related to the golden ratio: Binet's formula expresses the nth Fibonacci number in terms of n and the golden ratio, implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases. Fibonacci numbers are named after Italian mathematician Leonardo of Pisa known as Fibonacci. In his 1202 book Liber Abaci, Fibonacci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics, as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths.

Fibonacci numbers appear unexpectedly in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, graphs called Fibonacci cubes used for interconnecting parallel and distributed systems, they appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, an uncurling fern, the arrangement of a pine cone's bracts. Fibonacci numbers are closely related to Lucas numbers L n, in that the Fibonacci and Lucas numbers form a complementary pair of Lucas sequences: U n = F n and V n = L n; the Fibonacci sequence appears in Indian mathematics in connection with Sanskrit prosody, as pointed out by Parmanand Singh in 1985. In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long syllables of 2 units duration, juxtaposed with short syllables of 1 unit duration.

Counting the different patterns of successive L and S with a given total duration results in the Fibonacci numbers: the number of patterns of duration m units is Fm + 1. Knowledge of the Fibonacci sequence was expressed as early as Pingala. Singh cites Pingala's cryptic formula misrau cha and scholars who interpret it in context as saying that the number of patterns for m beats is obtained by adding one to the Fm cases and one to the Fm−1 cases. Bharata Muni expresses knowledge of the sequence in the Natya Shastra. However, the clearest exposition of the sequence arises in the work of Virahanka, whose own work is lost, but is available in a quotation by Gopala: Variations of two earlier meters... For example, for four, variations of meters of two three being mixed, five happens.... In this way, the process should be followed in all mātrā-vṛttas. Hemachandra is credited with knowledge of the sequence as well, writing that "the sum of the last and the one before the last is the number... of the next mātrā-vṛtta."

Outside India, the Fibonacci sequence first appears in the book Liber Abaci by Fibonacci. Using it to calculate the growth of rabbit populations. Fibonacci considers the growth of an idealized rabbit population, assuming that: a newly born breeding pair of rabbits are put in a field. Fibonacci posed the puzzle: how many pairs will there be in one year? At the end of the first month, they mate. At the end of the second month they produce a new pair, so there are 2 pairs in the field. At the end of the third month, the original pair produce a second pair, but the second pair only mate without breeding, so there are 3 pairs in all. At the end of the fourth month, the original pair has produced yet another new pair, the pair born two months ago produces their first pair, making 5 pairs. At the end of the nth month, the number of pairs of rabbits is equal to t

Sheila Colleen Bair was the 19th Chair of the U. S. Federal Deposit Insurance Corporation, during which time she assumed a prominent role in the government's response to the 2008 financial crisis, she was appointed to the post for a five-year term on June 2006 by George W. Bush. On August 1, 2015, she became the 28th president of Washington College in Chestertown, MD, she left Washington College on June 30, 2017 shortly after taking a position as a non-executive director for the state-owned Industrial and Commercial Bank of China. Bair served as a member of the FDIC Board of Directors through July 8, 2011. Bair is a native of Kansas, her father, was a surgeon. Her mother, was a nurse and housewife, she received her bachelor's degree in philosophy from the University of Kansas in 1975, worked as a bank teller for a brief period, before receiving a J. D. from the University of Kansas School of Law in 1978. In 1981, she was recruited by Senator Bob Dole, a Republican from her state, to serve as counsel on his staff in Washington.

Prior to her appointment at the FDIC, Bair was the Dean's Professor of Financial Regulatory Policy for the Isenberg School of Management at the University of Massachusetts Amherst, a post she had held since 2002. She served as Assistant Secretary for Financial Institutions at the U. S. Department of the Treasury, Senior Vice President for Government Relations of the New York Stock Exchange, a Commissioner and Acting Chair of the Commodity Futures Trading Commission, Research Director, Deputy Counsel and Counsel to Kansas Republican Senate Majority Leader Bob Dole. While an academic, Bair served on the FDIC's Advisory Committee on Banking Policy. Bair pursued a seat in the U. S. Congress. Bair began her career in the General Counsel's office of the former U. S. Department of Health and Welfare. Bair left the FDIC on July 2011, when her five-year term expired, she became a senior advisor to The Pew Charitable Trusts in August 2011. She is chair of the Systemic Risk Council, a volunteer effort formed by the CFA Institute and the Pew Charitable Trusts to monitor and comment on regulation.

Bull by the Horns: Fighting to Save Main Street from Wall Street and Wall Street from Itself was published September 25, 2012. Bair has written two books for children that encourage savings: Rock and the Savings Shock and Isabel's Car Wash. Bair joined the board of Banco Santander in January 2014 though she has been a critic of revolving door. In March 2017, she joined the state-owned Industrial and Commercial Bank of China as a non-executive director. Bair is married to Scott P. Cooper and has two children and Colleen. Bair pressed the Basel Committee on Banking Supervision to adopt strong capital and leverage standards. In a fictional TV movie about the crises, Patricia Randell played Bair in the 2011 HBO movie Too Big to Fail, based on the popular book of the same name by New York Times journalist Andrew Ross Sorkin. In 2009, Bair was named one of Time magazine's "Time 100" most influential people. In 2008, Bair topped The Wall Street Journal's annual 50 "Women to Watch List." In 2008 and 2009, Forbes ranked her as the second most powerful woman in the world behind German chancellor Angela Merkel.

Forbes described her FDIC office as "the last stop for capital-starved banks before going under." Bair received the John F. Kennedy Profile in Courage Award and Hubert H. Humphrey Civil Rights Award. In 2009, Bair was presented the Consumer Federation of America's Philip Hart Public Service Award. On March 29, 2012 Bair was honored by the Romney Institute of Public Management as the Administrator of the Year. Bair, Sheila. Bull by the horns: fighting to save Main Street from Wall Street and Wall Street from itself. New York: Free Press. ISBN 9781451672480. LCCN 2012039342. Bair, Sheila. Isabel's car wash. Morton Grove, Illinois: Albert Whitman & Co. ISBN 9780807536520. LCCN 2007030956. Bair, Sheila. Rock and the savings shock. Morton Grove, Illinois: Albert Whitman & Co. ISBN 9780807570944. LCCN 2005026974. Appearances on C-SPAN Sheila Bair on Charlie Rose Sheila Bair on IMDb "Sheila Bair collected news and commentary". Bloomberg News. "Sheila Bair collected news and commentary". The New York Times

Barelvi is a movement following the Sunni Hanafi school of jurisprudence, with over 200 million followers in South Asia. The name derives from the north Indian town of Bareilly, the hometown of its founder and main leader Ahmed Raza Khan. Although Barelvi is the used term, the followers of the movement prefer to be known by the title of Ahle Sunnat wa Jama'at, or as Sunnis, a reference to their perception as forming an international majority movement; the movement emphasizes personal devotion to Allah and the Muslim prophet Muhammad and a synthesis of Sharia with Sufi practices such as veneration of saints. Because of this, they are called Sufi. Ahmad Raza Khan and his supporters never used the term'Barelvi' to identify themselves or their movement, as they saw themselves as Sunni Muslims defending traditional Sunni beliefs from deviations. Only was the term'Barelvi' used; the Barelvi movement is named after the town of Bareilly, from where this movement was originated. To its followers, the Barelvi movement is the Ahle Sunnat wal Jama'at, or "People of the traditions and the community," and they refer to themselves as Sunnis.

This terminology is used to lay exclusive claim to be the only legitimate form of Sunni Islam in South Asia, in opposition to the Deobandi, Ahl-i Hadith and Darul Uloom Nadwatul Ulama followers. The Barelvi movement became known as Barelvi due to their leader Ahmad Raza Khan who established Islamic schools in 1904 with the Manzar-e-Islam; the Barelvi movement formed as a defense of the traditional mystic practices of South Asia, which it sought to prove and support. Although the Darul Uloom Nadwatul Ulama was founded in 1893 to reconcile South Asia's Muslim sectarian differences, the Barelvis withdrew their support from the council and criticized its efforts as heretical and counter to the Islamic values. In contrast with the Deobandi movement, the Barelvis showed unequivocal support for the Movement for Pakistan. In the aftermath of the 1948 Partition, they formed an association to represent the movement in Pakistan, called Jamiyyat-u Ulam-i Pakistan. Like ulema of the Deobandi and Ahl-i Hadith movements, Barelvi ulema have advocated application of sharia law across the country.

As a reaction to the anti-Islam film Innocence of Muslims, a conglomerate of forty Barelvi parties called for a boycott of Western goods, while at the same time condemning violence which had taken place in protest against the film. India Today estimates that the vast majority of Muslims in India adhere to the Barelvi movement, The Heritage Foundation and The Washington Post give similar assessments for the vast majority of Muslims in Pakistan. Political scientist Rohan Bedi estimates; the majority of people in the United Kingdom of Pakistani and Kashmir origin are descended from immigrants from Barelvi-majority areas. The Barelvi movement in Pakistan has received funding from Barelvis in the UK, in part as a reaction to rival movements in Pakistan receiving funding from abroad. According to an editorial in the English-language Pakistani newspaper The Daily Times, many of these mosques have been however usurped by Saudi-funded radical organizations. Like other Sunni Muslims, Barelvis base their beliefs on the Quran and Sunnah and believe in monotheism and the prophethood of Muhammad.

Although Barelvis may follow any one of the Ashari and Maturidi schools of Islamic theology and one of the Hanafi, Shafi'i and Hanbali madhhabs of fiqh in addition to optionally choosing from one of the Sunni Sufi orders like the Qadiri, Chishti or the Suhrawardi tariqas. Most Barelvis in South Asia follow the Maturidi school of Islamic theology and the Hanafi madhhab of fiqh. A central doctrine of the Barelvi movement is that Muhammad is both light. According to the doctrine, Muhammad's physical birth was preceded by his existence as light which pre-dates creation. According to this doctrine the primordial reality of Muhammad existed before creation and that God created creation for the sake of Muhammad. Proponents of this doctrine believe. Sahl al-Tustari the famous 9th century Sufi commentator of the Quran, describes the creation of the primordial light of Muhammad in his tafsir. Al-Tustari's student, Mansur Al-Hallaj, affirms this doctrine in his book ‘’Ta Sin Al-Siraj’’. According to Stūdīyā Islāmīkā, all Sufi orders are united in the belief of the light of Muhammad and generate practices with this concept as a foundational belief.

Another central doctrine of the Barelvi movement is that Muhammad can witness and be present in multiple places as the same time. The doctrine is present in various Sufi works prior to the Barelvi movement, such as Sayyid Uthman Bukhari's Jawahir al-Quliya, where he instructs how Sufis may have manifested to them the presence of Muhammad. Proponents of this doctrine assert that the term Shahid in Quran 33:45 4:41 refers to this ability of Muhammad and provide various hadiths as sources to support this belief. A fundamental belief of the Barelvi movement is; this relates to the concept of Ummi as mentioned in the Quran 7:157. Barelvis do not see this word as referring to unlettered or illiterate, but rather see it as referring to one, not taught by man; the consequence of this belief is that Muhammad therefore learns directly from God and his knowledge is universal in nature and encompasses the seen and unseen realms. This belief predates the Barelvi movement and can be found in Sufi books such as Rumi's Fihi Ma Fihi in which he states: Sunni Muslims of th