Though there is no closed form for F ( x ), there are a number of algorithms that estimate it numerically.

Though the Rohirrim had inflicted enormous damage on their enemies, Sauron's forces were still numerically superior, and Gothmog, the lieutenant of Minas Morgul, who had assumed field command on the death of the Witch-king, summoned reserves from nearby Osgiliath.

Though numerically speaking, the church continues to grow, in recent times numbers of priests, nuns and brothers have declined, and the involvement of laypeople has increased.

Though the last to be built, it was not the last 9F numerically as Crewe had already completed engines with higher numbers.

Though he was at least in his seventies, he led an early-morning counterattack on numerically superior British forces.

Though somewhat numerically inferior the Germans were vastly superior in terms of training and firepower, possessing numerous modern submachine guns, light machine guns and hand grenades.

Though only a minority of the population, the movement rebelled on Sunday 7 December, the day after the news of the coup arrived.

Though a minority of WASG members opposed the merger of the two parties scheduled for June 2007, the new party – The Left – was on Germany's political stage before the federal elections.

Though Democrats reinstated the disappearing quorum when they took control of the House the following year, Reed as minority leader proved so adroit at using the tactic against them that Democrats reinstated Reed Rules in 1894.

Though coalition and minority governments have been an occasional feature of parliamentary politics, the first-past-the-post electoral system used for general elections tends to maintain the dominance of these two parties, though each has in the past century relied upon a third party to deliver a working majority in Parliament.

Though a minority of scholars follow the Augustinian hypothesis, most scholars ( including Christians and non-Christians ) date the composition of the Christian gospels to between 70 and 100 CE, a time after Christianity had separated from Judaism ( and after Pharisaism emerged as the dominant form of Judaism ).

Though proclaimed king when only three years of age, he was regarded as merely nominal by the unruly nobles to whom a minority was convenient.

Though not facing a majority, the inability of the Liberal parties ( themselves unable to form a single faction until 1934 ), the Conservatives and Agrarians to form a majority government pressed for a minority government led by Hansson, expecting support from the Farmers ' League through an agriculture policy favoring the interests of the League ( kohandeln ), although stopping short of inviting it into the cabinet.

Though only a minority investor, Sony closed MGM's distribution system and folded most of their staff into their own studio, and the movies UA had completed and planned for release ( Capote, Art School Confidential, The Woods, and Romance and Cigarettes ) were reassigned to Sony Pictures Classics.

Though still trained in Osteopathic Manipulative Medicine ( OMM ), the modern derivative of Still's techniques, a minority of osteopathic physicians use it in actual practice.

Though the minority of the split, this body remained in fellowship with the European churches.

Though progress has been made, increasing minority enrollment at CSU has been a challenge for school administrators, one made yet more difficult by high dropout rates in many Colorado high schools with concentrated minority populations.

Though Sunni Muslim, Adygs are seen as a loyal minority within Israel, who serve in the armed forces.

Though they are minority populations, they are treated as equals with the elephants, and are given the same rights, liberties, and respects that the elephants have, and are fully incorporated into the elephant society.

Though not all the inhabitants are Christian, the wall will target principally Christian lands, and further the hardships for an dwindling minority in the Holy Land.

Though most of these groups are recent immigrants, since the Middle Ages, Jews have formed the oldest minority to retain its own identity.

Though technically a vote on whether the bill can proceed, the bill's supporters usually claimed that the vote was a vote on the bill itself, providing opportunities to the minority party to, at the least, force the majority party to be put on record as opposing a popular bill.

Though a small minority, Canadian Jews have had an open presence in the country since the arrival of the first Jewish immigrants after the British took possession of nearly all of New France after the 1763 Treaty of Paris ending the Seven Years ' War.

Though attended only by a tradition-minded minority, Vienna folk dance events are carefully organized events often taking place at extraordinary locations, such as, the Alpenverein, the Autumn Dance at baroque Belvedere ( palace ), the Ländler dances ( the traditional music of which bears the rhythm from which in the 19th century the Vienna waltz music was developed ), and the summer dances which take place twice each week during summer holiday season ( July & August ).

Though the Conservatives were reduced to a minority government in 1975 and 1977, they stayed in power as they moved to the left of the rural-based Liberals.

Though achievement gaps persist between white and minority students, the academic performance gap is far narrower for DoDEA students than for students throughout the United States.

Though a minority of its yearly meetings ( New York, New England, Baltimore, Southeastern and Canada ) were also affiliated with Friends General Conference and over the decades became more theologically liberal and predominantly unprogrammed in worship style, the theological position of the majority of its constituent yearly meetings continues to be often similar in flavor to the Protestant Christian mainstream in Indiana and North Carolina.

Though a significant minority are also speakers of a dialect of Min Nan known as Zhenan Min.

Though this line of thinking has been criticized for requiring the establishment of a utopian socialist society, the notion that the disproportionality observed in minority representation in crime rate statistics could be understood as the result of systematic economic disadvantage found its way into many of the theories developed in subsequent generations.

Though Americans usually lived in groups segregated by national origin or religious belief, they liked to work and shop in the noise and vitality of downtown.

Though Nancy was a controversial First Lady, 56 percent of Americans had a favorable opinion of her when her husband left office on January 20, 1989, with 18 percent having an unfavorable opinion and the balance not giving an opinion.

Though less than 3 % of Native Americans owned slaves, divisions grew among the Native Americans over slavery.

Though less than 3 % of Native Americans owned slaves, bondage created destructive cleavages among those who were slaveholders.

Though their schools were segregated and underfunded, as were services, due to a lack of political representation, African Americans were able to create their own communities.

Though produced in Denmark, these Horace stories were written by Americans: Stefan Petrucha, Sarah Kinney, and Don Markstein.

Though Native Americans comprise less than 1 % of the city ’ s population, they doubled between the 2000 and 2010 Census, and combined with West New York ’ s Native Americans, comprise 38 % of the county ’ s Native American population.

Though Native Americans comprise less than 1 % of the city ’ s population, they doubled in the 2000s, and combined with Union City ’ s Native American ’ s comprise 38 % of the county ’ s Native American population.

Though Levit did not consider himself to be a racist, he was bowing to the prevailing attitudes of many European Americans during the 1950s, considering housing and racial relations entirely separate matters.

Though tactically defeated, the Americans all but destroyed Cornwallis ' army as an effective fighting force and gained a permanent strategic advantage in the South.

Though both Jackson and Commodore Patterson reported that the retreating forces had spiked their cannon, leaving no guns to turn on the Americans ' main defense line, this is contradicted by Major Mitchell's diary which makes it clear this was not so, as he states he had " Commenced cleaning enemy's guns to form a battery to enfilade their lines on the left bank ".

Though it had never been a full critical success relative to other Rodgers and Hammerstein productions such as South Pacific and The King and I, it inspired another generation of Asian Americans to re-imagine the musical.

Though the bill granted Native Americans citizenship it did not grant them the right to vote and Zitkala-Ša continued to work for civil rights and better access to health care and education for Native Americans up until her death in 1938.

Though he managed to complete all the courses easily, a lack of qualified teachers meant that Baker was first sent to a school in England for more training and then back to France to teach Americans what he had learned in England, in an attempt to create pilots as quickly as possible.

Though the American loss was 673 ; the British loss was 575, but along with the Army's success under Brigadier General Horatio Gates at Saratoga on October 17 when John Burgoyne surrendered, the battle led to the official recognition of the Americans by France, which formed an alliance with the Americans afterward.

Though the Americans won a clear victory, it did not have decisive strategic results as York was a less important objective in military terms than Kingston, where the British armed vessels on Lake Ontario were based.

According to GRU defector Kalanbe, " Though most Americans do not realize it, America is penetrated by Russian military intelligence to the extent that arms caches lie in wait for use by Russian special forces ".

Though Faubus later lost general popularity as a result of his stand against desegregation, at the time he was included among the " Ten Men in the World Most Admired by Americans ", according to Gallup's most admired man and woman poll for 1958.

Though a Spanish counterattack on San Juan was quickly broken up, the Americans on Kettle Hill faced a more serious attack from some 600 Spanish regulars.

Though the Americans demanded that the Italian authorities extradite Abu Abbas of the PLO, Craxi stood firm on the grounds that the crime had been perpetrated on Italian territory, on which the Italian Republic had sole jurisdiction.

Though these Mormon women were poor in material goods, they felt the need of Native Americans exceeded their own.