Since N and N' are both nilpotent and they commute, the operator Af is nilpotent ; ;

Since a ruled surface of order N with N concurrent generators is necessarily a cone, it follows finally that every line through a point, P, of **zg meets its image at P, as asserted.

The number of neutrons, N, is known as the neutron number of the atom ; thus, A = Z + N. Since protons and neutrons have approximately the same mass ( and the mass of the electrons is negligible for many purposes ), and the mass defect is usually very small compared to the mass, the atomic mass of an atom is roughly equal to A.

Since the remainders decrease with every step but can never be negative, a remainder r < sub > N </ sub > must eventually equal zero, at which point the algorithm stops.

In the first step, the final nonzero remainder r < sub > N − 1 </ sub > is shown to divide both a and b. Since it is a common divisor, it must be less than or equal to the greatest common divisor g. In the second step, it is shown that any common divisor of a and b, including g, must divide r < sub > N − 1 </ sub >; therefore, g must be less than or equal to r < sub > N − 1 </ sub >.

Since r < sub > N − 1 </ sub > is a common divisor of a and b, r < sub > N − 1 </ sub > ≤ g.

Since the first part of the argument showed the reverse ( r < sub > N − 1 </ sub > ≤ g ), it follows that g = r < sub > N − 1 </ sub >.

Since the inverse DFT is the same as the DFT, but with the opposite sign in the exponent and a 1 / N factor, any FFT algorithm can easily be adapted for it.

* The atomic mass unit ( u ) is defined so that a single carbon-12 atom has a mass of 12 u ; 1 u is approximately .< ref group =" note "> Since the Avogadro constant N < sub > A </ sub > is defined as the number of atoms in 12 g of carbon-12, it follows that 1 u is exactly 1 /( 10 < sup > 3 </ sup > N < sub > A </ sub >) kg .</ ref > The atomic mass unit is convenient for expressing the masses of atoms and molecules.

Since N. G. Herreshoff, in 1877, many have been convinced about the superiority of multihulls.

Since there are N atoms in total,

Since E < sub > 2 </ sub >-E < sub > 1 </ sub > ≫ kT, it follows that the argument of the exponential in the equation above is a large negative number, and as such N < sub > 2 </ sub >/ N < sub > 1 </ sub > is vanishingly small ; i. e., there are almost no atoms in the excited state.

Since 1972, the U. N. Decolonization Committee has called for Puerto Rico's decolonization and for the U. S. to recognize the island's right to self-determination and independence.

" Since World War II, every major military action has been technically a U. S. military operation or a U. N. " police action ", which are deemed legally legitimate by Congress, and various United Nations Resolutions because of decisions such as the Gulf of Tonkin Resolution or the Authorization for Use of Force.

Since the amplitude at these reflections grows linearly with the number N of scatterers, the observed intensity of these spots should grow quadratically, like N < sup > 2 </ sup >.

Since the definition of the gram is not () mathematically tied to that of the dalton, the number N < sub > A </ sub > of molecules in a mole ( Avogadro's number ) must be determined experimentally.

Since the 1998 – 99 Ashes series, a Waterford Crystal representation of the Ashes urn has been presented to the winners of an Ashes series as the official trophy of that series.

Since 1995 the UK government has advised that regular consumption of 3 – 4 units a day for men, or 2 – 3 units a day for women, would not pose significant health risks, but that consistently drinking four or more units a day ( men ), or three or more units a day ( women ), is not advisable.

Since 1977, New Zealand has celebrated Arbor Day on June 5, which is also World Environment Day, prior to then Arbor Day, in New Zealand, was celebrated on August 4 – which is rather late in the year for tree planting in New Zealand hence the date change.

Since re-capitalization in 2003 – 04 season, Roma had a short-lived financial self-sustainability.

During a 1935 – 1942 colonization attempt, the island was most likely on Hawaii time, which was then 10. 5 hours behind UTC .< ref > Since it is uninhabited the island's time zone is unspecified, but it lies within a nautical time zone 12 hours behind UTC.

Since there is no heir to inherit Elimelech's land, levirate custom required a close relative ( usually the dead man's brother ) to marry the widow of the deceased in order to continue his family line ( Deuteronomy 25: 5 – 10 ).

Since the donation of the Abbey Moutier-Grandvalto and all its possessions to Bishop Adalbero II in 999 till the Reformation, Basel was ruled by prince-bishops ( see Bishop of Basel, whose memory is preserved in the crosier shown on the Basel coat-of-arms – see above ).

Since the Roman consular fasti make it somewhat easy to confuse 87 – 57 BC with 84 – 54 BC, many scholars accept the dates 84 BC – 54 BC, supposing that his latest poems and the publication of his libellus coincided with the year of his death.

Since there is no evidence that late Maastrichtian nonavian dinosaurs could burrow, swim or dive, they were unable to shelter themselves from the worst parts of any environmental stress that occurred at the K – T boundary.

Since craters are caused by explosions, they are nearly always circular – only very low-angle impacts cause significantly elliptical craters.

Since the Universe could, under different circumstances, conceivably not exist ( contingency ), its existence must have a cause – not merely another contingent thing, but something that exists by necessity ( something that must exist in order for anything else to exist ).

Since then they have had four separate periods in the top flight of English football: between 1936 – 1957, 1986 – 1990, 1998 – 1999 and 2000 – 2007.

* 1824 – United States presidential election, 1824: Since no candidate received a majority of the total electoral college votes in the election, the United States House of Representatives is given the task of deciding the winner in accordance with the Twelfth Amendment to the United States Constitution.

Since the beginning of Norman rule in the 12th century, the city has functioned as the capital in varying geopolitical entities: Lordship of Ireland ( 1171 – 1541 ), Kingdom of Ireland ( 1541 – 1800 ), island as part of the United Kingdom of Great Britain and Ireland ( 1801 – 1922 ), and the Irish Republic ( 1919 – 1922 ).

* Since 1972 – Josh Freese's solo album

Since 1993, the Eritrean People ’ s Liberation Front army has been involved in tree planting ; the annual average rate of deforestation during 1990 – 2000 was 0. 3 percent.

Since accurate base maps are necessary for any planning program, the first step taken by the planning division to implement the long-range state plan has been to prepare two series of base maps -- one at a scale of 1 inch to a mile, and the second a series of 26 sheets at a scale of 1 inch to 2000 feet, covering the entire state.

Since the provision of skywave service requires adequate freedom from interference, only class 1, stations are capable of rendering skywave service.

Since Parts 1, and 2, were prepared independently, the abbreviation schemes and the chemical symbols used differ in the two parts.

Since we are considering here relatively small diameter ( 1 - 1.5 inches ) fiber plates, their average thickness can be kept below 1/4 inch and their internal losses may be assumed as 15 percent ( per plate ).

Since 1949, the St. Louis club has been below on May 1 just four times.

Since then the corporate headquarters have moved to New York City at 1 Park Avenue in Manhattan.

Since 1 April 1956, Aalen has the status of Große Kreisstadt (“ major district town ”).

This subextension is called a separable closure of K. Since a separable extension of a separable extension is again separable, there are no finite separable extensions of K < sup > sep </ sup >, of degree > 1.

Since the bigger 1 / 4 " tapioca pearls are separately sold as " black pearls " ( 黑珍珠 ) in markets, this name was the logical first choice and is more popular with the consumers of the drink.

Since B < sub > n </ sub >= 0 for all odd n > 1, and many formulas only involve even-index Bernoulli numbers, some authors write B < sub > n </ sub > instead of B < sub > 2n </ sub >.

Since 2003, a 1, 070 km pipeline has been used to export crude oil from the oilfields around Doba to offshore oil-loading facilities on Cameroon's Atlantic coast at Kribi.

Since the U. S .- Chile Free Trade Agreement went into effect on January 1, 2004, U. S .- Chilean trade has increased by 154 %.

Since these stock additions retain the short pistol barrel ( as short as 100 mm ( 4 inches )) they are highly restricted under the NFA unless the shoulder stocks are of original manufacture for the gun and the gun has been " delisted " as outlined in links 1 & 2 in the paragraph above, as is the case with so-equipped " Broomhandle " Mausers and Lugers.

Since then, she has donated over US $ 1 million to the program.

Since a comic book shop in his area traded 2 old comics for 1 new, he only had 2 Duck comics left from his sister's collection by the 70s, one of them being The Golden Helmet.

Since all probabilities p < sub > i </ sub > add up to one: p < sub > 1 </ sub > + p < sub > 2 </ sub > + ... + p < sub > k </ sub > = 1, the expected value can be viewed as the weighted average, with p < sub > i </ sub >’ s being the weights:

Since its unification with Pakistan, the East Pakistan Army had consisted of only one infantry brigade, which was made up of two battalions, the 1st East Bengal Regiment and the 1 / 14 or 3 / 8 Punjab Regiment in 1948.

Since the algorithm ensures that the remainders decrease steadily with every step, r < sub > k − 1 </ sub > is less than its predecessor r < sub > k − 2 </ sub >.